SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS

JEAN-MICHEL BISMUT AND FRANÇOIS LABOURIE

Abstract. The purpose of this paper is to give a proof of the Verlinde formulas by applying the Riemann-Roch-Kawasaki theorem to the moduli space of flat G-bundles on a Riemann surface Σ with marked points, when G is a connected simply connected compact Lie group G. Conditions are given for the moduli space to be an orbifold, and the strata are described as moduli spaces for semisimple centralizers in G. The contribution of the strata are evaluated using the formulas of Witten for the symplectic volume, methods of symplectic geometry, including formulas of Witten-Jeffrey-Kirwan, and residue formulas. Our paper extends prior work by Szenes and Jeffrey-Kirwan for SU(n) to general groups G.

Contents

Introduction 4 1. Simple Lie groups and their centralizers 8 1.1. Roots and coroots 8 1.2. The basic scalar product on the Lie algebra 9 1.3. The volume of T 10 1.4. The quadratic form attached to a representation 10 1.5. The dual Coxeter number 11 1.6. An embedding of the center in the Weyl group 13 1.7. The element ρ/c 16 1.8. Some properties of the characters of G 18 1.9. The elements whose centralizer is semisimple 21 1.10. Some properties of semisimple centralizers 23 1.11. The first homotopy group in a semisimple centralizer 24 1.12. Centralizers in a connected and non simply connected semisimple Lie

group 25

1.13. The intersection of an adjoint orbit with a centralizer 27 1.14. The stabilizer of an element of the Lie algebra, and coadjoint orbits 28 2. Fourier analysis on the centralizers of semisimple Lie groups 31 2.1. Some linear algebra 31 2.2. The linear algebra of the basis of a root system 34 2.3. Fourier series and integration along the fibre 36 2.4. Iterated residues and the series Q_{u}(t, x) 39 2.5. The Fourier transform on quotient of lattices 44 2.6. Fourier series on T 44 2.7. Iterated residues and the series R_{u}(t, x) 46 2.8. Fourier series for the universal cover of a semisimple centralizer 47 2.9. Bernouilli polynomials and the Fourier series P_{n}(t) 48 2.10. The Fourier series P̃_{u,n}(t) 51 2.11. The function P_{u,n,q}(t) 53 2.12. The action of a differential operator on P_{u,n,q}(t) 57 3. Symplectic manifolds and moment maps 58 3.1. Orbifolds 58 3.2. Symplectic manifolds and moment maps 60 3.3. Symplectic reduction 62 3.4. Symplectic reduction with respect to a fixed stabilizer. 63 3.5. The image of the symplectic volume by the moment map 64 3.6. The symplectic volume as a polynomial near the origin. 66 3.7. A formula of Witten-Jeffrey-Kirwan 68 3.8. The volume of symplectic coadjoint orbits 71 4. The affine space of connections 75 4.1. The central extension of the loop group LG 75 4.2. The coadjoint orbits of LG 76 4.3. A central extension of (LG)^{s} 81 4.4. The holonomy of L̃G 82 4.5. The symplectic space of connections on Σ 83 4.6. The canonical line bundle on A 86 4.7. The non simply connected case. 90 4.8. The case of a connected subgroup of a simply connected group. 92 4.9. The action of stabilizers on the line bundle L. 93 4.10. The action of stabilizers on the line bundle λ_{p}. 96 5. The moduli space of flat bundles on a Riemann surface 99 5.1. Combinatorial description of the Riemann surface Σ 99 5.2. The combinatorial complexes on Σ 100 5.3. The map ϕ 105 5.4. The set of regular values of the map ϕ. 108 5.5. The stabilizers Z′(x). 112 5.6. The tangent bundle to the moduli space and its symplectic form 115 5.7. A metric on ( T M/G) 119 5.8. The Witten formula for the symplectic volume distribution 122 5.9. Logarithms 129 5.10. A symplectic structure on an open set in X. 130 5.11. The integral of certain characteristic classes on the strata of M/G 137 5.12. An evaluation of certain Euler characteristics 144 5.13. Evaluation of c_{1}(T M/G) 148 6. The Riemann-Roch-Kawasaki formula on the moduli space of flat bundles152 6.1. Almost complex orbifolds 152 6.2. The Theorem of Riemann-Roch-Kawasaki 154 6.3. The line bundle λ^{p} 155 6.4. The Theorem of Riemann-Roch-Kawasaki on the moduli space of flat

Date: October 20, 1998.

Supported by Institut Universitaire de France (I.U.F.).

Supported by Institut Universitaire de France (I.U.F.).

bundles 156

6.5. Evaluation of the Atiyah-Bott-Lefschetz Todd class on a stratum of

the moduli space 158

6.6. The dimensions of the splitting of the normal bundle to the strata 160 6.7. The contribution of a stratum of the moduli space 163 6.8. The case where ∑_{j=1}^{s} pt_{j} ∉ \overline{R} 172 6.9. A residue formula for the index 172 6.10. A formula for the index for large p 173 6.11. A perturbation of the index problem 175 7. Residues and the Verlinde formula 181 7.1. A residue formula for L_{q}(t, x) 181 7.2. A residue formula for M_{q}(t, x) 187 7.3. The Verlinde sums 190 7.4. A residue formula for the Verlinde sums 193 7.5. The generic case with g ≥ 2 196 7.6. The non generic case for g ≥ 2 200 7.7. The general case 204 8. The Verlinde formulas 206 8.1. The generic case 206 8.2. The perturbed case at \frac{pt_{j}}{p+c} 209 8.3. The perturbed case at \frac{ρ+pt_{j}}{p+c} 210 References 211

Introduction

The Verlinde formula [58], [3] computes the dimension of spaces of holomorphic sections of canonical line bundles over the moduli space M of semistable G_{C}- bundles on a Riemann surface Σ with marked points, with G a connected and simply connected compact Lie group. For a given “level” p, the Verlinde formula is a sum over the finite collection of weights parametrizing the representations of the central extension of the loop group LG at level p. This formula, discovered by Verlinde in the context of quantum field theory, has received a number of rigorous proofs first for G = SU(2) by Thaddeus [53], Bertram and Szenes [7] and Szenes [50] (see also Donaldson [16] and Jeffrey-Weitsman [29] for related questions), and for more general groups by Tsuchiya-Ueno-Yamada [55], Beauville-Laszlo [4] (for G = SU(n)), Faltings [19], Kumar, Narasimhan, Ramanathan [36] for general groups G. A common feature of many of these proofs is that establishing the fusion rules for the Verlinde numbers is an essential step in the proof, a second step being the description of the fusion algebra.

The purpose of this paper is to give a proof of the Verlinde formula for connected simply connected compact Lie groups, by methods of symplectic geometry. This program has been already carried out by Szenes [51] for SU(3), and Jeffrey-Kirwan [28] in the case of SU(n). The main point of the paper is to extend their approach a to general groups. More precisely, we will obtain the Verlinde formula by an application of Riemann-Roch.

The theorem of Narasimhan-Sheshadri [44] asserts that M can be identified with the set of representations of π_{1}(Σ) with values in G, with given conjugacy classes of holonomies at the marked points. For generic choices of holonomies, this last space is a symplectic orbifold (M/G, ω), which carries an orbifold Hermitian line bundle with connection (λ^{p}, ∇^{λ^{p}}), such that c_{1}(λ^{p}, ∇^{λ^{p}}) = pω. In particular the orbifold M/G is complex. This orbifold carries a canonical Dirac operator D_{p,+}, unique up to homotopy. Its index Ind(D_{p,+}) is the Euler characteristic of λ^{p}. We will compute the index Ind(D_{p,+}) using the formula of Riemann-Roch-Kawasaki [30, 31]. We show that, for p large, it is given by the Verlinde formula. The fact that p has to be large may be related to the fact that a priori, higher cohomology may well not vanish for small p. In fact, the Verlinde is a formula for dim H^{0}(M, λ^{p}), while Ind(D_{p,+}) is the corresponding Euler characteristic. For non generic holonomies, we also show that small perturbations of the holonomies still produce an orbifold moduli space. For suitable perturbations, we show that for any p, the index of the corresponding Dirac operator D_{p,+} is still given by the Verlinde formula. The typical case where such a perturbation is needed is when Σ does not have marked points.

Our proof contains various interrelated steps.

• A first step is the description of the strata of the orbifold moduli space M/G. These strata are in fact moduli spaces for the semisimple centralizers in G. Up to conjugacy, there is only a finite family of such centralizers. In general, they are non simply connected. The strata split into a union of substrata indexed by the fundamental group of the centralizers. The description of the geometry of M/G involves results contained in Sections 1, 4, 5, 6. Observe that if G = SU(n), there are no non trivial semisimple centralizers, which explains the smoothness of the moduli space M/G (this case which was already considered by Jeffrey-Kirwan [28]).

• A second step consists in reproving Witten’s formula [59] for the symplectic volume of the moduli spaces.

• Another step is the detailed construction of the orbifold line bundle λ^{p}. Also we have to compute the action of the finite stabilizers of elements of M on λ^{p}. This is done in Sections 4 and 5.

• In Section 5, we show that the formalism of the moment map can be applied to each stratum of the moduli space. In Section 6, we use a formula of Witten [60] and Jeffrey-Kirwan [26] (see Vergne [57]) to express the contribution of each stratum as the action of a differential operator on a locally polynomial function on a maximal torus T. This locally polynomial function is just the symplectic volume of a deformation of the moduli space M/G.

• Witten’s formula [59] for the deformed symplectic volume of each stratum is a Fourier series on T. In order to calculate the contribution of each stratum explicitly, it is of critical importance to express Witten’s formula using residues techniques, which will make obvious the fact that the given Fourier series is indeed a local polynomial on T. These residue techniques are developed in Section 2.

• In Section 7, we give a residue formula for the index of the Dirac operator on M/G, by putting together the contribution of all strata.

• Another step is to express the Verlinde sums as residues. This step, which is carried out of Section 7, has many formal similarities with what is done in Section 2 for the Fourier series on T.

• In Section 8, a comparison of the results of Sections 6 and 7 leads us to our main result.

We now review our techniques in more detail.

1. The residue techniques Residue techniques play an important role in the whole paper, in order to con-

vert the Witten Fourier series [59] for the symplectic volumes into expressions which make them local polynomials in an “obvious” way, so that differential operators can be applied to these polynomials. Similar residue techniques are also applied to the Verlinde sums.

Szenes [51, 52] initiated the use of residue techniques to treat Verlinde formulas. In [51], Szenes applied such residue techniques to the case of SU(3) and obtained the corresponding Verlinde formulas. In [52], Szenes developed a cohomological approach in terms of arrangement of hyperplanes to treat the Witten sums for any group G. In [28], Jeffrey and Kirwan gave a related treatment of the Verlinde formulas for SU(n).

In the present paper, multidimensional residues are used in a rather “naive” way. The Witten sums are expressed as sums over a lattice identified to Z^{r}. We compute the given sums by summing in succession in the variables k_{1}, …, k_{r} ∈ Z, and by applying standard residue techniques to these one dimensional sums. Handling the recursion requires the development of a trivial, but heavy linear algebra. We believe that Szenes techniques [52] can put put to fruitful use to give a more conceptual approach to this part of our work.

2. A combinatorial description of the moduli spaces Let O_{1}, …, O_{s} be s adjoint orbits in G. Put X = G^{2g}×∏_{j=1}^{s} O_{j}. Let ϕ : X → G be given by

ϕ(u_{1}, v_{1}, …, u_{g}, v_{g}, w_{1}, …, w_{s}) = ∏_{i=1}^{g}[u_{i}, v_{i}] ∏_{j=1}^{s} w_{j}. (0.1)

Put M = ϕ^{−1}(1). Under a genericity assumption on the O_{j}, 1 ≤ j ≤ s , the condition (A) of Definition 5.17, which requires that s ≥ 1, in Theorem 5.18, we show that M is a smooth manifold on which G acts locally freely. The moduli space is the orbifold M/G.

To prove Witten’s formula [59], we show in Theorem 5.44 that the image by ϕ of the Haar measure on X has a density with respect to the Haar measure on G, which is essentially given by the symplectic volume of the quotient fibres of ϕ, , which are themselves moduli spaces with an extra marked point . This approach was initiated by Liu [37, 38], who showed in particular that the differential of ϕ can be expressed in terms of the combinatorial complexes which compute the absolute and relative cohomology of the flat adjoint bundle E. Inspired by Witten [59, 60], Liu gave a special role to the heat kernel on the group G to establish Witten’s formula, while in our approach, we do not use any heat kernel. Needless to say, the heat kernel on G remains crucial in understanding connections with 2-dimensional Yang-Mills theory, and also with Witten’s non Abelian localization [60].

3.Moment maps and the quantization conjecture

In Section 5.10, under genericity assumptions, we show that a G-invariant neighborhood X̂ of M in X can be equipped with a symplectic form, that G acts on X̂ with a moment map, and that the standard symplectic structure on the quotients on the fibres of ϕ included in X̂ come from the symplectic structure on X̂. We can then use directly the formulas of Witten [60] and Jeffrey-Kirwan [26] to express the integrals of certain characteristic classes in terms of differential operators acting on the symplectic volume of the fibres.

When the genericity assumptions are not verified, we replace the given moduli space by a generic perturbation, which still carries a Dirac operator to which the Riemann-Roch-Kawasaki theorem [30, 31] can be applied. We then show that the above index results still hold.

We will now put our results in perspective from the point of view of geometric quantization, especially in connection with the Guillemin-Sternberg conjecture [22]. By Atiyah-Bott [2, Section 9], we know that when there is one marked point with central holonomy, M/G is a symplectic reduction of the affine space A of Gconnections with respect to the action of the gauge group ΣG, which acts on A with a moment map μ, which is the curvature, so that M/G = μ^{−1}(0)/ΣG. Similarly the line bundle λ^{p} is itself the reduction of a universal line bundle L^{p} on A. This theory can be extended to the case with marked points (this we do in part in Section 4 and 5). If A was instead a compact manifold, and ΣG a compact connected Lie group, the Guillemin-Sternberg conjecture [22] asserts that the Riemann-Roch number of (M/G, λ^{p}) is equal to the multiplicity of the trivial representation in the action of ΣG on the cohomology of L^{p}. The Guillemin-Sternberg conjecture has been proved in various stages, the most general result being given by Meinrenken [39],[40] (for a more analytic proof, see Tian and Zhang [54]). In Meinrenken’s formalism, when the considered group does not act locally freely on μ^{−1}(0) , one replaces 0 by any regular value of μ close to 0 , and one still gets a corresponding version of Guillemin-Sternberg’s conjecture.

A new twist has been introduced to the story of the proof of the Verlinde formula in work by Meinrenken and Woodward [41], [42], Alekseev, Malkin, Meinrenken [1], and later work by these authors. In [41, 42], Meinrenken and Woodward gave a symplectic proof of the fusion rules for the Verlinde numbers. In [1], the authors develop a theory of group actions with moment maps taking their values in the given Lie group G. This theory is in fact a theory of the standard moment map for an action of a central extension of the loop group LG. The space M is the prototype of such a manifold, the moment map being just ϕ. These authors then develop a localization formula in equivariant cohomology, which is an analogue of the formula of Duistermaat-Heckman [18], Berline-Vergne [5]. By using the theory of symplectic cuts and the previous results by Meinrenken [39, 40] on the Guillemin-Sternberg conjecture, they announce a proof of the Verlinde formulas.

In some way, our paper represents a direct attempt to prove Verlinde formulas directly, by a method which resembles the proof given by Jeffrey-Kirwan [27] of the Guillemin-Sternberg conjecture. The proof of [27] consists in extracting the Riemann-Roch number of μ^{−1}(0)/ΣG from the Lefschetz formulas.

Our paper is organized as follows. In Section 1, we establish basic simple facts on compact simply connected simple Lie groups and their semisimple centralizers. In Section 2, we develop our basic residue techniques in several variables. In particular, we express certain Fourier series on T, which are local polynomials, as residues. In Section 3, we reestablish well-known results on symplectic actions with moment maps, and we give a proof of the formula of Jeffrey-Kirwan. In Section 4, we construct the canonical line bundle L on the moduli space of G-connections on the Riemann surface Σ with fixed holonomy at the given marked points. We show that a suitable central extension of the gauge group ΣG acts on L, and we compute the action of certain stabilizers on L and on a related line bundle λ_{p}. In Section 5, we describe the moduli space M/G. We show that the formalism of the moment map can be applied to the action of G on M. We apply the formula of Witten [60] and Jeffrey-Kirwan [26] to the moduli spaces associated to semisimple centralizers. Also we give a formula for c_{1}(T M/G). In Section 6, we apply the theorem of Riemann-Roch-Kawasaki to the orbifold M/G, and we give a residue formula for the index Ind(D_{p,+}). In Section 7, we give a residue formula for the Verlinde sums. Finally in Section 8, we compare Ind(D_{p,+}) and the Verlinde formula, and give a number of conditions under which they coincide.

The results contained in this paper were announced in [12].

1. Simple Lie groups and their centralizers

Let G be a connected and simply connected compact simple Lie group. The purpose of this Section is to give the basic facts which will be needed in the description of the strata of the moduli space of flat G-bundles on a Riemann surface Σ. This involves in particular a complete description of the semisimple centralizers in G.

This Section is organized as follows. In Section 1.1, we recall elementary properties of roots and coroots. In Section 1.2, we construct the basic scalar product on the Lie algebra 𝔤 of G. In Section 1.3, we compute the volume of a maximal torus T. In Section 1.4, we relate the quadratic form attached to a representation to the basic quadratic form. In Section 1.5, we introduce the dual Coxeter number. In Section 1.6, we construct an embedding of the center Z(G) in the Weyl group W. In Section 1.7, we give simple properties of the element ρ/c ∈ T, in particular in its relations with the center Z(G). In Section 1.8, we review elementary properties of the characters of G. In Sections 1.9-1.12, we describe the semisimple centralizers, and give some of their properties. In Section 1.13, we consider the intersection of an adjoint orbit with such a centralizer. Finally in Section 1.14, we recall various properties of the stabilizers of elements of 𝔤, and we construct the symplectic structure on the coadjoint orbits, and the corresponding line bundles.

1.1. Roots and coroots. Let G be a connected simply connected simple compact Lie group of rank r. Let 𝔤 be its Lie algebra, let 𝔤^{∗} be its dual. Let T be a maximal torus in G, let 𝔱 be its Lie algebra, let 𝔱^{∗} be its dual. Let W be the corresponding Weyl group.

We will denote the composition law multiplicatively in G, but often additively in T.

Let Γ ⊂ 𝔱 be the lattice of integral elements in 𝔱, i.e.

Γ = {t ∈ 𝔱 , exp(t) = 1 in T } . (1.1)

Let Λ = Γ^{∗} ⊂ 𝔱^{∗} be the lattice of weights in 𝔱^{∗}, so that if h ∈ Γ, λ ∈ Λ,

⟨λ, h⟩ ∈ Z . (1.2)

Let R = {α} ⊂ Λ be the root system of G. Then R is a finite family of elements of Λ, which span 𝔱^{∗}. Let \overline{R} ⊂ Λ be the lattice generated by R, let \overline{R}^{∗} ⊃ Γ be the lattice dual to \overline{R}.

Let CR = \{h_{α}\} ⊂ 𝔱 be the family of coroots attached to R. Let \overline{CR} ⊂ 𝔱 be the lattice generated by CR, let \overline{CR}^{∗} ⊂ 𝔱^{∗} be the corresponding dual lattice. Since G is simply connected, by [15, Theorem V.7.1],

(1.3)  Γ = \overline{CR,}
Λ = \overline{CR}^{∗} .

Let Z(G) be the center of G. By [15, Proposition V.7.16],

\overline{R}^{∗}/\overline{CR} = Z(G) . (1.4)

Let R_{s}, R_{ℓ} and CR_{s}, CR_{ℓ} be the short, long roots and coroots. Note that R_{s} corresponds to CR_{ℓ} and R_{ℓ} to CR_{s}. Recall that G is said to be simply laced if all the roots (or coroots) have the same length. In this case, all the roots will be considered as long, and the coroots as short, so that

(1.5)  R_{s} = ∅ ,
CR_{ℓ} = ∅ .

In the sequel, when G is simply laced, any statement concerning R_{s} or CR_{ℓ} should be disregarded.

Let \overline{R}_{ℓ}, \overline{R}_{s}, \overline{CR_{ℓ}}, \overline{CR_{s}} be the lattices generated by R_{ℓ}, R_{s}, CR_{ℓ}, CR_{s}. It follows from the classification of Lie groups that

(1.6)  \overline{CR_{s}} = \overline{CR,}
\overline{R}_{s} = \overline{R} .

Note that when G is simply laced, the second equality in (1.6) is empty.

1.2. The basic scalar product on the Lie algebra.

Definition 1.1. Let ⟨ , ⟩ be the G-invariant scalar product on 𝔤 such that if ∥.∥ is the corresponding norm, if h_{α} ∈ CR_{s},

∥h_{α}∥^{2} = 2 . (1.7)

If G is not simply laced, there is one m ∈ N (equal to 2 or 3) such that if h_{α} ∈ CR_{ℓ},

∥h_{α}∥^{2} = 2m . (1.8)

By [15], if f, f′ ∈ \overline{CR},

⟨f, f′⟩ ∈ Z . (1.9)

Using ⟨ , ⟩, we may and we will identify 𝔱 and 𝔱^{∗}. By [15, V, eq. (2.14)], under this identification, if α ∈ R, if h_{α} ∈ CR corresponds to α, then

α = \frac{2}{∥h_{α}∥^{2}} h_{α} . (1.10)

By (1.10) , we find that

(1.11)  \overline{CR} ⊂ \overline{R} ∩ \overline{R}^{∗} .

Also, by (1.6), (1.8), (1.10),

(1.12)  \overline{CR} = \overline{R}_{ℓ},
\overline{CR}_{ℓ} = m\overline{R} .

By (1.12),

m\overline{R} ⊂ \overline{R}_{ℓ} ⊂ \overline{R}, (1.13)

m\overline{CR} ⊂ \overline{CR_{ℓ}} ⊂ \overline{CR} .

From (1.12),(1.13),

(1.14)  m\overline{CR} = m\overline{R}_{ℓ} ⊂ \overline{CR_{ℓ}} = m\overline{R} ⊂ \overline{CR}

= \overline{R}_{ℓ} ⊂ \frac{\overline{CR_{ℓ}}}{m} = \overline{R} ⊂ \frac{\overline{CR}}{m} = \frac{\overline{R}_{ℓ}}{m} .

Also

Γ = \overline{CR} ⊂ \overline{R} ∩ \overline{R}^{∗} ⊂ \overline{CR}^{∗} = Λ . (1.15)

10 JEAN-MICHEL BISMUT AND FRANÇOIS LABOURIE

Observe that the Weyl group W preserves all the objects which have been constructed above. Also note that T = 𝔱/\overline{CR}. (1.16) Moreover T′ = 𝔱/\overline{R}^{∗} is a maximal torus in the adjoint group G′ = G/Z(G). 1.3. The volume of T. Let Vol(T) be the volume of T for the metric ⟨, ⟩. Proposition 1.2. The following identity holds Vol(T)^{2} = |\overline{CR}^{∗}/\overline{CR}|. (1.17) Proof. Consider the exact sequence of lattices 0 → \overline{CR} → \overline{CR}^{∗} → \overline{CR}^{∗}/\overline{CR} → 0, (1.18) which induces the exact sequence 0 → \overline{CR}^{∗}/\overline{CR} → 𝔱/\overline{CR} → 𝔱/\overline{CR}^{∗} → 0 . (1.19) From (1.19), we obtain Vol(𝔱/\overline{CR}) = |\overline{CR}^{∗}/\overline{CR}|Vol(𝔱/\overline{CR}^{∗}) . (1.20)

Now 𝔱/\overline{CR} and 𝔱^{∗}/\overline{CR}^{∗} ≃ 𝔱/\overline{CR}^{∗} are dual tori. Therefore Vol(𝔱/\overline{CR})Vol(𝔱/\overline{CR}^{∗}) = 1 . (1.21) By (1.20), (1.21), we obtain (1.17). The proof of our Proposition is completed. Proposition 1.3. The following identity holds Vol(T)^{2} = |Z(G)| |\overline{R}/\overline{R}_{ℓ}| . (1.22) In particular, if G is simply laced, Vol(T)^{2} = |Z(G)| . (1.23) Proof. Clearly |\overline{CR}^{∗}/\overline{CR}| = |\overline{CR}^{∗}/\overline{R}| |\overline{R}/\overline{CR}| . (1.24) Also by (1.4), (\overline{CR}^{∗}/\overline{R})^{∗} = \overline{R}^{∗}/\overline{CR} = Z(G) . (1.25) From (1.12), (1.17 ), (1.24), (1.25), we get (1.22) . The identity (1.23) follows. 1.4. The quadratic form attached to a representation. Assume temporarily that G is a compact connected semisimple Lie group, which is not necessarily simply connected. Otherwise, we use the notation of Sections 1.1-1.3.

Let σ : G → Aut(V) be a finite dimensional representation of G. Definition 1.4. If A, B ∈ 𝔤, put

⟨A, B⟩^{σ} = \frac{1}{4π^{2}} Tr^{V} [σ(A)σ(B)] . (1.26)

Then ⟨ , ⟩ is an ad-invariant symmetric bilinear form on 𝔤.

Let x ∈ R ↦ [x] ∈ [0, 1[ be the periodic function of period 1 such that for x ∈ [0, 1[, [x] = x.

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS 11

Proposition 1.5. If u ∈ \overline{R}^{∗}, h ∈ Γ, then ⟨u, h⟩^{σ} ∈ Z . (1.27) Proof. Let M ⊂ Λ be the set of T -weights in the representation σ. Then

⟨u, h⟩^{σ} = − ∑_{λ∈M}⟨λ, u⟩⟨λ, h⟩ . (1.28)

Also for λ ∈ M, ⟨λ, h⟩ ∈ Z. Therefore, mod(Z),

(1.29)  ⟨u, h⟩^{σ} = − ∑_{λ∈M}[⟨λ, u⟩]⟨λ, h⟩

= − ∑_{s∈[0,1[}⟨ ∑_{{λ∈M \atop [⟨λ,u⟩]=s}} λ, h⟩ .

Also the image of u in T = 𝔱/Γ lies in Z(G).Therefore the representation σ splits into a sum of representations on which u acts like e^{2iπs}, 0 ≤ s < 1. The corresponding T-weights are given by {λ ∈ M, [⟨λ, u⟩] = s}. Since G is semisimple

∑_{{λ∈M \atop [⟨λ,u⟩]=s}} λ = 0 (1.30)

From (1.29), (1.30), we get (1.27) . The proof of our Proposition is completed. 1.5. The dual Coxeter number. Again we assume that G is a connected and simply connected simple compact Lie group. Also we use the assumptions and notation of Sections 1.1-1.3.

Let K ⊂ 𝔱 be a Weyl chamber. Let R_{+} be the corresponding system of positive roots, so that R = R_{+} ∪ R_{−}. (1.31) Then K = {t ∈ 𝔱, for any α ∈ R_{+}, ⟨α, t⟩ > 0}. (1.32) Let P ⊂ 𝔱 be the alcove in K whose closure contains 0. Then P = {t ∈ 𝔱, for any α ∈ R_{+}, 0 < ⟨α, t⟩ < 1}. (1.33)

Since G is simple, the adjoint representation of G′ on 𝔤 is irreducible. The corresponding T -weights are given by {0} ∪ R. Let α_{0} ∈ R_{+} ∩ R_{ℓ} be the corresponding highest root. Then P = {t ∈ K, ⟨α_{0}, t⟩ < 1}. (1.34) Definition 1.6. Let ρ ∈ 𝔱^{∗} be given by

ρ = \frac{1}{2} ∑_{α∈R_{+}} α. (1.35)

By [15, Proposition V.4.12], if α ∈ R_{+} is a simple root, ⟨ρ, h_{α}⟩ = 1. In particular ρ ∈ \overline{CR}^{∗}, and ρ ∈ K. Definition 1.7. Let c ∈ N be the dual Coxeter number, given by c = ⟨ρ, h_{α_{0}}⟩ + 1. (1.36)

12 JEAN-MICHEL BISMUT AND FRANÇOIS LABOURIE

Let τ : G′ → Aut(𝔤) be the adjoint representation. Then by [45, p 285], [21, eq. (1.6.45)], ⟨, ⟩^{τ} = −2c⟨, ⟩. (1.37) Proposition 1.8. If t ∈ 𝔱,

ct = ∑_{α∈R_{+}}⟨α, t⟩α. (1.38)

In particular c\overline{R}^{∗} ⊂ \overline{R}. (1.39) Proof. Since {0} ∪ R are the T -weights of τ, if t, t′ ∈ 𝔱,

⟨t, t′⟩^{τ} = −2 ∑_{α∈R_{+}}⟨α, t⟩⟨α, t′⟩. (1.40)

Using (1.37), (1.40), we get (1.38). From (1.38), (1.39) follows.

Recall that we have identified 𝔱 and 𝔱^{∗} by ⟨, ⟩. Then α_{0} = h_{α_{0}}. In particular, for any α ∈ R_{+}, 0 < ⟨α, ρ/c⟩ < 1, (1.41) i.e. ρ/c ∈ P. Definition 1.9. For t ∈ 𝔱, put

σ(t) = ∏_{α∈R_{+}}(e^{iπ⟨α,t⟩} − e^{−iπ⟨α,t⟩}). (1.42)

Equivalently,

σ(t) = e^{2iπ⟨ρ,t⟩} ∏_{α∈R_{+}}(1 − e^{−2iπ⟨α,t⟩}). (1.43)

By (1.43), we find that σ(t) descends to a function defined on T = 𝔱/\overline{CR}.

Recall that W is the Weyl group of G . If w ∈ W, set ϵ_{w} = det(w|_{𝔱}). (1.44) Then by [15, Theorem VI.1.7], if w ∈ W, t ∈ 𝔱, σ(wt) = ϵ_{w}σ(t). (1.45)

Put ℓ = |R_{+}| . (1.46) Now we recall a result stated in [3, Lemma 9.17]. Proposition 1.10. The following identity holds (i^{ℓ}σ(e^{ρ/c}))^{2} = |\overline{CR}^{∗}/c\overline{CR}|. (1.47)

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS 13

1.6. An embedding of the center in the Weyl group. By [15, Theorem V.4.1], wK ∩ K ≠ ∅ if and only if w = 1.

Take q ∈ Z^{∗}. Then ⋃_{w∈W} w(qP ∩ \overline{CR}^{∗}) is a disjoint union of finite sets. Proposition 1.11. The set ⋃_{w∈W} w(qP ∩ \overline{CR}^{∗}) embeds naturally as a subset of \overline{CR}^{∗}/q\overline{CR}. More precisely,

⋃_{w∈W} w(qP ∩ \overline{CR}^{∗}) = \{λ ∈ \overline{CR}^{∗}/q\overline{CR}, σ^{2}(λ/q) ≠ 0\}. (1.48)

Also cP ∩ \overline{CR}^{∗} = \{ρ\} . (1.49) Proof. Let W_{aff} = W ⋉ \overline{CR} be the affine Weyl group. By [15, Proposition V.7.10], W_{aff} acts freely and effectively on the set of alcoves in 𝔱. Thus we get (1.48). If λ ∈ cP ∩ \overline{CR}^{∗}, then λ ∈ \overline{CR}^{∗} ∩ K. By [15, Note V.4.14], λ − ρ ∈ \overline{CR}_{+}^{∗} = \overline{CR}^{∗} ∩ \overline{K} . (1.50)

Also, by (1.36), ⟨ρ, h_{α_{0}}⟩ = c − 1 . (1.51)

Since α_{0} = h_{α_{0}}, by (1.34), since λ ∈ cP, ⟨λ, h_{α_{0}}⟩ < c . (1.52) By (1.50)-(1.52), we obtain 0 ≤ ⟨λ − ρ, h_{α_{0}}⟩ < 1 . (1.53) Since ⟨λ − ρ, h_{α_{0}}⟩ ∈ N, from (1.53), we obtain λ = ρ . (1.54) The proof of our Proposition is completed. Proposition 1.12. Let f be a W-invariant function on \overline{CR}^{∗}/q\overline{CR}. Then

∑_{{λ∈\overline{CR}^{∗}/q\overline{CR} \atop σ^{2}(λ/q)≠0}} f(λ) = |W | ∑_{λ∈qP ∩\overline{CR}^{∗}} f(λ) (1.55)

Proof. This is a trivial consequence of Proposition 1.11. Proposition 1.13. The set qP ∩\overline{CR}^{∗} embeds naturally into \{λ ∈ \overline{CR}^{∗}/q\overline{R}^{∗}, σ^{2}(λ/q) ≠ 0}. Proof. By (1.33), (1.42), if λ ∈ qP, then σ^{2}(λ/q) ≠ 0. Let λ, λ′ ∈ qP ∩ \overline{CR}^{∗}, and assume there is μ ∈ \overline{R}^{∗} such that λ − λ′ = qμ. (1.56) By (1.33), for α ∈ R_{+}, −q < ⟨α, λ − λ′⟩ < q, (1.57) and so −1 < ⟨α, μ⟩ < 1. (1.58)

Since μ ∈ \overline{R}^{∗}, for α ∈ R_{+}, ⟨α, μ⟩ ∈ Z. By (1.58), we get

μ = 0. (1.59)

The proof of our Proposition is completed.

Proposition 1.14. If λ ∈ \overline{CR}^{∗}, σ^{2}(λ/q) ≠ 0, there is w ∈ W, λ′ ∈ qP ∩\overline{CR}^{∗} such that wλ − λ′ ∈ q\overline{R}^{∗}.

Proof. Take λ ∈ \overline{CR}^{∗}. Then by [15, Proposition V.7.10], there exists w ∈ W, h ∈ \overline{P}, f ∈ \overline{CR}, such that

\frac{λ}{q} = wh + f . (1.60)

Clearly qf ∈ q\overline{CR}. Also by (1.45),

σ^{2}(λ/q) = σ^{2}(h) . (1.61)

So if σ^{2}(λ/q) ≠ 0, then σ^{2}(h) ≠ 0, so that h ∈ P. The proof of our Proposition is completed.

Recall that Z(G) = \overline{R}^{∗}/\overline{CR}. Also W acts trivially on Z(G) ⊂ G. Equivalently if f ∈ \overline{R}^{∗}, w ∈ W

wf − f ∈ \overline{CR} . (1.62)

If g ∈ G, let Z(g) ⊂ G be the centralizer of g, let 𝔷(g) be its Lie algebra. By [14, Corollaire 5.3.1] , since G is simply connected, Z(g) is a connected Lie subgroup of G.

An element t ∈ T is said to be regular (resp. very regular) if 𝔷(t) = 𝔱 (resp. Z(t) = T). By the above, t ∈ T is regular if and only if t is very regular. Let T_{reg} ⊂ T be the set of regular elements in T. By [15, Proposition V.7.10] , P embeds into T_{reg}. More precisely,

T_{reg} = ⋃_{w∈W} wP (1.63)

and the union in (1.63) is disjoint.

Let u ∈ Z(G). Then u + P ⊂ T is another alcove in T_{reg}. Therefore there is a well-defined w_{u} ∈ W such that

u + P = w_{u}P . (1.64)

Proposition 1.15. The map u ∈ Z(G) ↦ w_{u} ∈ W embeds Z(G) as a commutative subgroup of W. In particular |Z(G)| divides |W |.

Proof. If u ∈ Z(G), w_{u} = 1, then u + P = P in T. Therefore there is v ∈ \overline{R}^{∗} mapping into u ∈ \overline{R}^{∗}/\overline{CR} such that v + P = P in T. By proceeding as in (1.57), (1.58), we find that v = 0, i.e. u = 1.

If u, u′ ∈ Z(G), then

u + u′ + P = u′ + w_{u}P = w_{u}(u′ + P) = w_{u}w_{u′}P in T . (1.65)

From (1.65) , we get

w_{u+u′} = w_{u}w_{u′} . (1.66)

The proof of our Theorem is completed.

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS 15

By Proposition 1.13 , we can view qP ∩ \overline{CR}^{∗} as a subset of \overline{CR}^{∗}/q\overline{R}^{∗}, which itself is stable by W. In particular if λ ∈ qP ∩ \overline{CR}^{∗}, w ∈ W, wλ will be viewed as an element of \overline{CR}^{∗}/q\overline{R}^{∗}. So the equality wλ = λ says here that wλ − λ ∈ q\overline{R}^{∗}. Proposition 1.16. If λ ∈ qP ∩ \overline{CR}^{∗} ⊂ \overline{CR}^{∗}/q\overline{R}^{∗}, w ∈ W, then wλ ∈ qP ∩ \overline{CR}^{∗} ⊂ \overline{CR}^{∗}/q\overline{R}^{∗} if and only if there is u ∈ Z(G) such that w = w_{u}. Proof. Take λ ∈ qP ∩ \overline{CR}^{∗}, u ∈ \overline{R}^{∗} representing an element of Z(G) = \overline{R}^{∗}/\overline{CR}. Since λ/q ∈ P, by (1.64) , there is μ ∈ P such that w_{u}λ/q − μ − u ∈ \overline{CR} . (1.67) Therefore w_{u}λ − qμ ∈ q\overline{R}^{∗} . (1.68) Then λ′ = qμ ∈ qP ∩ \overline{CR}^{∗}, and w_{u}λ − λ′ ∈ q\overline{R}^{∗}.

Conversely, if λ, λ′ ∈ qP ∩ \overline{CR}^{∗}, u ∈ \overline{R}^{∗} are such that wλ − λ′ = qu , (1.69) then wλ/q = λ′/q + u . (1.70) From (1.70) , since λ/q, λ′/q ∈ P, we get w = w_{u}. The proof of our Proposition is completed.

Put

h = \frac{|W |}{|Z(G)|} . (1.71)

Let w^{1}, …, w^{s} ∈ W be distinct representatives in W of the classes of W/Z(G).

Theorem 1.17. The set ⋃_{1}^{n} w^{i}(qP ∩ \overline{CR}^{∗}) is a disjoint union, which embeds into

\overline{CR}^{∗}/q\overline{R}^{∗}. More precisely

⋃_{1}^{n} w_{i}(qP ∩ \overline{CR}^{∗}) = \{λ ∈ \overline{CR}^{∗}/q\overline{R}^{∗} , σ^{2}(λ/q) ≠ 0\} . (1.72)

Proof. Our Theorem follows from Propositions 1.14 and 1.16. Theorem 1.18. Let f a W -invariant function on \overline{CR}^{∗}/q\overline{R}^{∗}. Then

∑_{{λ∈\overline{CR}^{∗}/q\overline{R}^{∗} \atop σ^{2}(λ/q)≠0}} f(λ) = \frac{|W |}{|Z(G)|} ∑_{λ∈qP ∩\overline{CR}^{∗}} f(λ) . (1.73)

Proof. Our identity follows from Theorem 1.17. Remark 1.19. Our Theorem is also a consequence of Proposition 1.12. In fact, if f is a W -invariant function on \overline{CR}^{∗}/q\overline{R}^{∗},

∑_{{λ∈\overline{CR}^{∗}/q\overline{CR} \atop σ^{2}(λ/q)≠0}} f(λ) = |\frac{\overline{R}^{∗}}{\overline{CR}}| ∑_{{λ∈\overline{CR}^{∗}/q\overline{R}^{∗} \atop σ^{2}(λ/q)≠0}} f(λ) . (1.74)

Now \overline{R}^{∗}/\overline{CR} ≃ Z(G), and so using Proposition 1.12 , we finally obtain (1.73).

1.7. The element ρ/c.

Proposition 1.20. If s ∈ 𝔱, t ∈ P, then

∑_{α∈R}⟨α, s⟩[⟨α, t⟩] = 2⟨ct − ρ, s⟩ . (1.75)

Proof. By (1.35),(1.38),

(1.76)  ∑_{α∈R}⟨α, s⟩[⟨α, t⟩] = ∑_{α∈R_{+}} (⟨α, s⟩⟨α, t⟩ − ⟨α, s⟩(1 − ⟨α, t⟩))
= ∑_{α∈R}⟨α, s⟩⟨α, t⟩ − ⟨2ρ, s⟩
= 2⟨ct − ρ, s⟩ .

The proof of our Proposition is completed.

Let 𝔱 ∈ T_{reg}. Then t determines a Weyl chamber K and an alcove P of the above type. In the sequel, we consider t as an element of P ⊂ T. The element ρ is still given by (1.35 ). Of course ρ depends implicitly on t ∈ T_{reg}.

Theorem 1.21. The following identity holds

ct − ρ = \frac{1}{2} ∑_{α∈R}[⟨α, t⟩]α . (1.77)

In particular the map t ∈ T_{reg} ↦ ct−ρ ∈ 𝔱 descends to a map T′_{reg} = T_{reg}/Z(G) → 𝔱.

Proof. By Proposition 1.20 , we get (1.77) . Also if u ∈ Z(G), if t ∈ T_{reg} is replaced by t + u ∈ T_{reg}, [⟨α, t⟩] is unchanged. By (1.77) , we find that t ∈ T_{reg} ↦ ct − ρ ∈ 𝔱 descends to a map from T′_{reg} into 𝔱. The proof of our Theorem is completed.

Observe that since ρ ∈ \overline{CR}^{∗} and 2ρ ∈ \overline{R}, then u ∈ Z(G) = \overline{R}^{∗}/\overline{CR} ↦ exp(2iπ⟨ρ, u⟩) = ±1 is a character of Z(G).

Recall that K is a Weyl chamber, and that Z(G) has been embedded in W, by an embedding which depends explicitly on K.

Theorem 1.22. If w ∈ W, then wρ/c − ρ/c ∈ \overline{CR} if and only if w = 1. Also if w ∈ W , then wρ/c − ρ/c = u ∈ \overline{R}^{∗} if and only if w = w_{u}, so that w ∈ Z(G). The map w ∈ Z(G) ↦ wρ/c − ρ/c ∈ \overline{R}^{∗}/\overline{CR} = Z(G) is a group isomorphism. Also if u ∈ Z(G),

exp(2iπ⟨ρ, u⟩) = ϵ_{w_{u}}. (1.78)

Proof. By (1.41), ρ/c ∈ P. Using [15, Proposition V.7.10] , we find that if wρ/c − ρ/c ∈ \overline{CR}, then w = 1. By Theorem 1.21 , if u ∈ Z(G) = \overline{R}^{∗}/\overline{CR},

ρ/c + u = w_{u}(ρ/c) in 𝔱/\overline{CR} . (1.79)

By (1.79) , we find that w_{u}(ρ/c) − ρ/c ∈ \overline{R}^{∗}. Conversely let w ∈ W be such that

u = w(ρ/c) − ρ/c ∈ \overline{R}^{∗} . (1.80)

By (1.79), (1.80), we get

w_{u}(ρ/c) = w(ρ/c) in T = 𝔱/\overline{CR}, (1.81)

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS 17

so that w = w_{u}. Clearly if u ∈ Z(G), by (1.45), σ(w_{u}e^{ρ/c}) = ε_{w_{u}}σ(e^{ρ/c}) . (1.82) Also by (1.79) and by the above, σ(w_{u}e^{ρ/c}) = σ(e^{ρ/c+u}) = e^{2iπ⟨ρ,u⟩}σ(e^{ρ/c}) . (1.83) Since σ(e^{ρ/c}) ≠ 0, from (1.82), (1.83), we get (1.78) . The proof of our Theorem is completed.

Let \overline{P} be the closure of the alcove P. By (1.33), \overline{P} = \{t ∈ 𝔱, for any α ∈ R_{+}, 0 ≤ ⟨α, t⟩ ≤ 1\}. (1.84) Proposition 1.23. For h ∈ \overline{CR}, then \overline{P} ∩ (\overline{P} + h) ≠ ∅ if and only if h = 0. Proof. Recall that W_{aff} = W ⋉ \overline{CR}. By [15, Lemma V.4.3], if f ∈ \overline{P} , v ∈ W_{aff}, then vf ∈ \overline{P} if and only if vf = f. In particular, if f ∈ \overline{P} , h ∈ \overline{CR}, then f + h ∈ \overline{P} if and only if h = 0. The proof of our Proposition is completed. Proposition 1.24. The center Z(G) = \overline{R}^{∗}/\overline{CR} embeds as a finite subset of \overline{P}. Proof. If u ∈ Z(G) = \overline{R}^{∗}/\overline{CR}, there is an alcove Q containing 0 such that u is represented in 𝔱 by an element v ∈ \overline{Q}. By [15, Theorem V.4.1], there is w ∈ W such that wQ = P. Also since u ∈ \overline{R}^{∗}/\overline{CR}, then wv − v ∈ \overline{CR}. Therefore wv ∈ \overline{P} still represents u. So any element u ∈ Z(G) has a representative in \overline{P}. By Proposition 1.23, this representative is unique. The proof of our Proposition is completed. Remark 1.25. If u ∈ Z(G), we still denote by u the corresponding representative in \overline{P}. Then if w ∈ W, wu ∈ w\overline{P} is the unique representative of u in w\overline{P}. Of course, if wu ∈ \overline{P}, the above implies that wu = u. However this also follows from [15, Theorem V.4.1].

Let w_{o} ∈ W be the unique element of the Weyl group such that wK = −K. Theorem 1.26. Let u ∈ \overline{P} be the unique representative in \overline{P} of an element of Z(G). Then if t ∈ \overline{P}, if [t + u] ∈ w_{u}\overline{P} represents t + u ∈ T, then [t + u] = t + w_{u}u. (1.85) Also w_{u}u = w_{o}u. (1.86) In particular w_{u}\overline{P} = w_{u}u + \overline{P} . (1.87)

Moreover u = ρ/c − w_{u}ρ/c in 𝔱. (1.88) In particular, if t ∈ \overline{P}, w_{u}ρ − c[t + u] = ρ − ct in 𝔱. (1.89)

Proof. Clearly w_{o}(−u) lies in \overline{P}, and so it is the unique representative in \overline{P} of u^{−1} ∈ T. If t ∈ P, by (1.84), it is clear that t + w_{o}u lies in some alcove containing 0. Also t + w_{o}u represents t + u ∈ T. From the above, we get

[t + u] = t + w_{o}u. (1.90)

The alcove which contains t + w_{o}u is necessarily equal to w_{u}P. Therefore w_{u}u and w_{o}u both lie in w_{u}\overline{P}, and represent u ∈ Z(G). By Proposition 1.24, we get (1.86). By (1.86) and (1.90), we obtain (1.85) and (1.87).

Using Theorem 1.21 and (1.85), if t ∈ P,

c(t + w_{u}u) − w_{u}ρ = ct − ρ in 𝔱. (1.91)

From (1.91), we get (1.88) and (1.89). The proof of our Theorem is completed.

1.8. Some properties of the characters of G. Put

\overline{CR}_{+}^{∗} = \overline{CR}^{∗} ∩ \overline{K} . (1.92)

If λ ∈ \overline{CR}_{+}^{∗}, let χ_{λ} be the character of G which is the character of the irreducible representation of G with highest weight λ. By the Weyl character formula, if t ∈ T_{reg},

χ_{λ}(t) = ∑_{w∈W} \frac{e^{2iπ⟨wλ,t⟩}}{∏_{α∈R_{+}}(1 − e^{2iπ⟨wα,t⟩})} . (1.93)

Equivalently

χ_{λ}(t) = \frac{1}{σ(t)} ∑_{w∈W} ε_{w}e^{2iπ⟨w(ρ+λ),t⟩} . (1.94)

Recall that by [15, LemmaVI.1.2] , if μ ∈ \overline{CR}^{∗} is not included in a Weyl chamber,

∑_{w∈W} ε_{w}e^{2iπ⟨wμ,t⟩} = 0 . (1.95)

If μ ∈ \overline{CR}^{∗} lies in K, then by [15, Note V.4.14] , there is λ ∈ \overline{CR}_{+}^{∗} such that μ = ρ + λ .

If λ ∈ \overline{CR}^{∗}, there exists a Weyl chamber K such that λ ∈ \overline{K}. In general K is not unique.

Proposition 1.27. The character χ_{λ} does not depend on K.

Proof. Assume that λ ∈ \overline{K}, λ ∈ \overline{K}′. Then by [15, Lemma V.4.2] , there is w′ ∈ W such that w′K = K′. Let R_{+}, R′_{+} be the system of positive roots attached to K, K′. Clearly

R′_{+} = w′R_{+} . (1.96)

Also since λ ∈ \overline{K} ∩ \overline{K}′, and w′\overline{K} = \overline{K}′, then λ ∈ \overline{K}, w′^{−1}λ ∈ \overline{K}. By [15, Theorem V.4.1], it follows that

w′^{−1}λ = λ . (1.97)

Let χ_{λ}^{K}(t), χ_{λ}^{K′}(t) be the characters attached to K, K′, of highest weight λ. Using (1.93), (1.97) , we get

χ_{λ}^{K}(t) = χ_{λ}^{K′}(t) . (1.98)

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS 19

Our Proposition follows.

For λ ∈ \overline{CR}^{∗}, we can then write χ_{λ}(t) without explicitly mentioning a Weyl chamber K. Proposition 1.28. If λ ∈ \overline{CR}^{∗}, w ∈ W, χ_{wλ} = χ_{λ} . (1.99) Proof. We may and we will assume that λ ∈ \overline{CR}_{+}^{∗} = \overline{CR}^{∗} ∩ \overline{K}. Then wλ ∈ \overline{CR}^{∗} ∩ wK. Equation (1.99) now follows from (1.93).

Recall that if λ ∈ \overline{CR}^{∗}, w ∈ W, then wλ − λ ∈ \overline{R}. It follows that if θ_{1}, …, θ_{s} ∈

\overline{CR}^{∗}, then ∑_{j=1}^{s} θ_{j} ∈ \overline{R} if and only if, given (w^{1}, …, w^{s}) ∈ W^{s}, then ∑_{j=1}^{s} w^{j}θ_{j} ∈ \overline{R}.

Proposition 1.29. If ∑_{j=1}^{s} θ_{j} ∈ \overline{R}, then ∏_{j=1}^{s} χ_{θ_{j}} descends to a function on the adjoint group G′ = G/Z′(G). Proof. By Proposition 1.27 , we may and will assume that the θ_{j}’s lie in \overline{CR}_{+}^{∗} = \overline{CR}^{∗} ∩ K. By (1.93) , for t ∈ T_{reg},

∏_{j=1}^{s} χ_{θ_{j}}(t) = ∑_{(w^{1},... ,w^{s})∈W^{s}} \frac{e2iπ⟨∑_{k=1}^{s} w^{k}θ_{k}, t⟩}{∏_{j=1}^{s} ∏_{α∈R_{+}}(1 − e^{−2iπ⟨w^{j}α,t⟩})} . (1.100)

If ∑_{j=1}^{s} θ_{j} ∈ \overline{R}, then ∑_{j=1}^{s} w^{j}θ_{j} ∈ \overline{R}, and so by (1.100), ∏_{j=1}^{s} χ_{θ_{j}}(t) descends to a function on the adjoint torus T′ = 𝔱/\overline{R}^{∗}. The proof of our Proposition is completed.

Proposition 1.30. If ∑_{j=1}^{s} θ_{j} ∉ \overline{R}, then

∑_{μ∈\overline{R}^{∗}/\overline{CR}} ∏_{j=1}^{s} χ_{θ_{j}}(e^{t+μ}) = 0 . (1.101)

Proof. This follows from (1.100). Proposition 1.31. If λ ∈ \overline{CR}^{∗}, λ ∉ \overline{R}, then χ_{λ}(e^{ρ/c}) = 0 . (1.102) Proof. We may and we will assume that λ ∈ \overline{CR}_{+}^{∗}. If w ∈ W, then χ_{λ}(e^{ρ/c}) = χ_{λ}(e^{wρ/c}) . (1.103) Take u ∈ \overline{R}^{∗}. Then by Theorem 1.22, w_{u}ρ/c − ρ/c = u in T. (1.104)

By (1.103), (1.104),

χ_{λ}(e^{ρ/c}) = χ_{λ}(e^{ρ/c+u}) . (1.105)

Now since the representation associated to λ is irreducible, the central element e^{u} acts in the λ representation like e^{2iπ⟨λ,u⟩}. From (1.105) , we get

χ_{λ}(e^{ρ/c}) = e^{2iπ⟨λ,u⟩}χ_{λ}(e^{ρ/c}) . (1.106)

If λ ∉ \overline{R}, it is then clear that (1.102) holds. The proof of our Proposition is completed.

Now we have the result of Kostant [34].

Theorem 1.32. If λ ∈ \overline{R}, then

χ_{λ}(e^{ρ/c}) = 0, +1 or − 1 . (1.107)

Take now p ∈ N^{∗} and λ ∈ p\overline{P}, so that λ ∈ \overline{CR}_{+}^{∗}. Take u ∈ Z(G) = \overline{R}^{∗}/\overline{CR}. Then λ/p + u ∈ T is represented uniquely by an element [λ/p + u] ∈ w_{u}\overline{P}. Observe that by ((1.85),

[λ/p + u] = λ/p + w_{u}u in 𝔱. (1.108)

Also

p[λ/p + u] − (λ + pu) ∈ p\overline{CR} . (1.109)

Theorem 1.33. If μ ∈ \overline{CR}^{∗}/(p + c)\overline{CR}, σ(μ/(p + c)) ≠ 0, then

χ_{p[λ/p+u]}(e^{μ/(p+c)}) = ε_{w_{u}}e^{2iπ⟨u,μ⟩}χ_{λ}(e^{μ/(p+c)}) . (1.110)

In particular

χ_{pu}(e^{μ/(p+c)}) = ϵ_{w_{u}}e^{2iπ⟨u,μ⟩}. (1.111)

Proof. By (1.94),

(1.112)  χ_{λ}(e^{μ/(p+c)}) = \frac{1}{σ(e^{μ/(p+c)})} ∑_{w∈W} ε_{w}e^{2iπ⟨w(λ+ρ), \frac{μ}{p+c} ⟩}

= \frac{1}{σ(e^{μ/(p+c)})} ∑_{w∈W} ε_{w}e^{2iπ⟨wλ/p,μ⟩}
e^{2iπ⟨w(ρ−cλ/p), \frac{μ}{p+c} ⟩} .

Also p[λ/p + u] ∈ w_{u}\overline{P} ∩ \overline{CR}^{∗}. When replacing \overline{P} by w_{u}\overline{P}, ρ is replaced by w_{u}ρ. When replacing λ by p[λ/p + u], using (1.109), we find that e^{2iπ⟨wλ/p,μ⟩} is replaced by e^{2iπ⟨wλ/p,μ⟩}e^{2iπ⟨wu,μ⟩}. Also by equation (1.89) in Theorem 1.26, we find that when replacing λ by p[λ/p + u], ρ − cλ/p ∈ 𝔱 is unchanged. Finally σ(e^{μ/(p+c)}) is changed into ε_{w_{u}}σ(e^{μ/(p+c)}). Equation (1.110) follows from the above arguments. Equation (1.111) is a consequence of (1.110).

Remark 1.34. If we use (1.110) with p = 0, and μ = ρ, we get

1 = ϵ_{w_{u}}e^{2iπ⟨ρ,u⟩}, (1.113)

which is precisely equation (1.78).

Theorem 1.33 will be used in Remark 8.4 as a consistency check on our index theoretic computations.

1.9. The elements whose centralizer is semisimple. We still assume that G is a connected and simply connected simple compact Lie group.

Definition 1.35. Let C ⊂ 𝔱 be the set of u ∈ 𝔱 such that {α ∈ R; ⟨α, u⟩ ∈ Z} spans 𝔱^{∗}.

Clearly

\overline{R}^{∗} ⊂ C . (1.114)

Since by (1.6), (1.12), \overline{CR} = \overline{CR}_{s} = \overline{R}_{ℓ}, then

\overline{R}^{∗} ⊂ \overline{CR}^{∗} ⊂ C . (1.115)

In particular \overline{R}^{∗} ⊂ C acts by translations on C. Also the Weyl group W acts on C.

Proposition 1.36. The set C/\overline{R}^{∗} is a finite subset of the adjoint torus T′ = 𝔱/\overline{R}^{∗}, which contains \overline{CR}^{∗}/\overline{R}^{∗}. Also W acts on C/\overline{R}^{∗}.

Proof. Let R′ ⊂ R be such that the elements of R′ span 𝔱^{∗}, and let \overline{R}′ be the associated lattice. Clearly \overline{R}/\overline{R}′ is a finite set. Then

C/\overline{R}^{∗} = ⋃_{R′} \overline{R}′^{∗}/\overline{R}^{∗} . (1.116)

From (1.116) , its follows that C/\overline{R}^{∗} is finite. The proof of our Proposition is completed.

Let now K be a Weyl chamber in 𝔱. Let R_{+} be the corresponding system of positive roots so that

R = R_{+} ∪ (−R_{+}) . (1.117)

Definition 1.37. If u ∈ G′ = G/Z(G), let Z(u) ⊂ G be the centralizer of u.

Recall that by [14, Corollaire 5.3.1], since G is connected and simply connected, Z(u) is a connected Lie subgroup of G. Clearly if u ∈ T′ = 𝔱/\overline{R}^{∗}, T is a maximal torus in Z(u).

Also W acts like the identity on Z(G) = \overline{R}^{∗}/\overline{CR}. Therefore, if u ∈ T′ = 𝔱/\overline{R}^{∗}, w ∈ W, then wu − u is well-defined in T = 𝔱/\overline{CR}. Put

W_{u} = \{w ∈ W ; wu − u = 0 in T = 𝔱/\overline{CR}^{∗}\} . (1.118)

Theorem 1.38. If u ∈ T′ = 𝔱/\overline{R}^{∗}, the root system R_{u} of Z(u) is given by

R_{u} = {α ∈ R, ⟨α, u⟩ ∈ Z} . (1.119)

Also R_{u,+} = R_{u} ∩ R_{+} is a system of positive roots for Z(u). If Z(Z(u)) ⊂ Z(u) is the center of Z(u), its Lie algebra 𝔷(Z(u)) is given by

𝔷(Z(u)) = {f ∈ 𝔱 , for any α ∈ R_{u} , ⟨α, f⟩ = 0} . (1.120)

Also the Weyl group W_{Z(u)} of Z(u) is given by

W_{Z(u)} = W_{u}. (1.121)

Finally

(1.122) C/\overline{R}^{∗} = \{u ∈ T′ , Z(u) is semisimple\} .

22 JEAN-MICHEL BISMUT AND FRANÇOIS LABOURIE

Proof. Let 𝔷(u) ⊂ 𝔤 be the Lie algebra of Z(u). Then 𝔷(u) = {f ∈ 𝔤, u · f = f} . (1.123) As a T -space, 𝔤 ⊗_{R} C splits as

𝔤 ⊗_{R} C = (𝔱 ⊗_{R} C) ⊕ (⨁_{α∈R} 𝔤_{α}) . (1.124)

From (1.124), we get

𝔷(u) ⊗_{R} C = (𝔱 ⊗_{R} C) ⊕ ( ⨁_{{α∈R \atop ⟨α,u⟩∈Z}} 𝔤_{α}) . (1.125)

From (1.125), it is clear that (1.119) holds.

Since the forms α in R do not vanish on K, the same is true for elements in R_{u}, i.e. K is included in a Z(u) Weyl chamber K_{u}. If R_{u,+} is the corresponding system of positive roots, then R_{u,+} = R_{u} ∩ R_{+} . (1.126)

The identity (1.120) is trivial. Let N(T) ⊂ G be the normalizer of T. Then W = N(T)/T . (1.127) Similarly let N_{u}(T) be the normalizer of T in Z(u). Then W_{Z(u)} = N_{u}(T)/T . (1.128) Clearly N_{u}(T) = N(T) ∩ Z(u). (1.129) Therefore W_{Z(u)} is a subgroup of W. Since u ∈ Z(Z(u))/Z(G), if w ∈ W_{Z(u)}, wu − u = 0 in T . (1.130) Conversely if w ∈ W, let w′ ∈ N(T) represent w. If wu − u = 0 in T, then w′uw′^{−1} = u in T, (1.131) i.e. w′ ∈ Z(u). Therefore w′ ∈ N_{u}(T), and w lies in W_{Z(u)}.

By definition, Z(u) is semisimple if and only if 𝔷(Z(u)) = 0. From (1.120), we get (1.122).

The proof of our Theorem is completed. Remark 1.39. Clearly, if we identify u to a corresponding element in 𝔱, then W_{u} = \{w ∈ W, wu − u ∈ \overline{CR}\}. (1.132) By Theorem 1.38, W_{u} = W_{Z(u)}. Let \overline{CR}_{u} be the lattice generated by the coroots of Z(u). Since u ∈ Z(Z(u)), if w ∈ W_{Z(u)}, wu − u ∈ \overline{CR}_{u}. Therefore W_{u} = \{w ∈ W, wu − u ∈ \overline{CR}_{u}\}. (1.133) Proposition 1.40. If n ≥ 2 and G = SU(n), then C = \overline{R}^{∗}. (1.134) Proof. By [15, Proposition V.6.3], it is clear that if α_{1}, …, α_{r} is a basis of 𝔱^{∗} over R, then α_{1}, …, α_{r} spans \overline{R}. Equation (1.134) follows.

1.10. Some properties of semisimple centralizers. We still assume that G is a connected simply connected simple compact Lie group of rank r. Also we use the notation of Sections 1.1-1.3 and 1.9.

Take u ∈ C/\overline{R}^{∗}. To the Lie group Z(u), we can associate the objects we considered in Section 1.1 for G, with the reservation that since the action of W_{u} on 𝔱 may be reducible, Z(u) is semisimple and in general not simple.

However, we equip 𝔷(u) with the scalar product induced from the scalar product of 𝔤, ⟨ , ⟩. Therefore 𝔱 is equipped with the scalar product ⟨ , ⟩ , and the identification 𝔱 ≃ 𝔱^{∗} will still be the one we used for G.

The objects we considered before which are attached to Z(u) will be denoted with the index u. The lattices Γ_{u} ⊂ 𝔱, Λ_{u} = Γ_{u}^{∗} ⊂ 𝔱^{∗} are given by

Γ_{u} = \overline{CR}, Λ_{u} = \overline{CR}^{∗}. (1.135)

The roots R_{u} have already been described in Theorem 1.38 . Clearly

CR_{u} = {h_{α}, α ∈ R_{u}} . (1.136)

Note that in general

π_{1}(Z(u)) = \overline{CR}/\overline{CR_{u}}, (1.137)

and so Z(u) is in general not simply connected.

If u ∈ C/\overline{R}^{∗}, put as in (1.35),

ρ_{u} = \frac{1}{2} ∑_{α∈R_{u,+}} α . (1.138)

Then ρ_{u} ∈ \overline{CR}_{u}^{∗} .

Theorem 1.41. For any u ∈ C,

2cu ∈ \overline{R}, (1.139)

2ρ_{u} ∈ \overline{R} .

If h ∈ \overline{R}^{∗}/\overline{CR}_{u}, then 2c⟨u, h⟩ ∈ Z, 2⟨ρ − ρ_{u}, h⟩ ∈ Z, and moreover

c⟨u, h⟩ = ⟨ρ − ρ_{u}, h⟩ mod(Z). (1.140)

In particular, if h ∈ π_{1}(Z(u)) = \overline{CR}/\overline{CR}_{u} , then 2c⟨u, h⟩ ∈ Z, 2⟨ρ_{u}, h⟩ ∈ Z, and

c⟨u, h⟩ = ⟨ρ_{u}, h⟩ mod(Z) . (1.141)

Proof. Let τ : G′ → End(𝔤) be the adjoint representation. Then by (1.37), if A, B ∈ 𝔤

⟨A, B⟩^{τ} = −2c⟨A, B⟩ . (1.142)

Also τ induces a representation Z(u)/Z(G) → End(𝔤). Then \overline{R}^{∗} ⊂ 𝔱 is exactly the lattice of integral elements in 𝔱 with respect to Z(u)/Z(G). By Proposition 1.5 and by (1.142), we then find that 2cu ∈ \overline{R}. Also 2ρ_{u} ∈ \overline{R}_{u} ⊂ \overline{R}.

If h ∈ \overline{R}^{∗}, it follows that 2c⟨u, h⟩ ∈ Z, 2⟨ρ_{u}, h⟩ ∈ Z. Also since \overline{CR}_{u} ⊂ \overline{R}_{u}, and u ∈ \overline{R}_{u}^{∗}, if h ∈ \overline{CR}_{u}, ⟨u, h⟩ ∈ Z, and, since ρ_{u} ∈ \overline{CR}_{u}^{∗}, then ⟨ρ_{u}, h⟩ ∈ Z. Therefore mod (Z), if h ∈ \overline{R}^{∗}, c⟨u, h⟩ and ⟨ρ_{u}, h⟩ only depend on the class of h in \overline{R}^{∗}/\overline{CR}_{u}.

Now we establish (1.141). By [15, Proposition V.7.10], we may suppose that u ∈ C is such that for α ∈ R

|⟨α, u⟩| ≤ 1 . (1.143)

Take h ∈ \overline{R}^{∗}. Since Z(u) is semisimple, and since the α ∈ R are the weights of the restriction of τ to Z(u), then by proceeding as in (1.29),

∑_{α∈R}[⟨α, u⟩]⟨α, h⟩ = ∑_{s∈[0,1[} s⟨ ∑_{{α∈R \atop [⟨α,u⟩]=s}} α , h⟩ = 0 . (1.144)

Also,

(1.145)  ∑_{α∈R}[⟨α, u⟩]⟨α, h⟩ = ∑_{α∈R\R_{u}}[⟨α, u⟩]⟨α, h⟩

= ∑_{α∈R_{+}\R_{u,+}} (⟨α, u⟩⟨α, h⟩ − (1 − ⟨α, u⟩)⟨α, h⟩)

= ∑_{α∈R\R_{u}}⟨α, u⟩⟨α, h⟩ − ⟨ ∑_{α∈R_{+}\R_{u,+}} α, h⟩ .

If α ∈ R_{u}, ⟨α , u⟩ ∈ Z, and ⟨α, h⟩ ∈ Z. Since roots in R_{u} come by pairs,by (1.40),(1.142), (1.144), (1.145),

2c⟨u, h⟩ = ⟨ ∑_{α∈R_{+}\R_{u,+}} α, h⟩ mod (2Z) , (1.146)

which is equivalent to (1.140).

If h ∈ \overline{CR}, then ⟨ρ, h⟩ ∈ Z. From (1.140), we get (1.141). The proof of our Theorem is completed.

Remark 1.42. Needless to say, the class of ρ_{u} in 𝔱/\overline{R} does not depend on the choice of R_{+}. This fact fits with (1.141).

1.11. The first homotopy group in a semisimple centralizer. Take u ∈ C/\overline{R}^{∗}. Then Z(u) is a connected semisimple subgroup of G. Also by (1.135), (1.137) and by [15, Theorem V.7.1],

(1.147)  π_{1}(Z(u)) = \overline{CR}/\overline{CR}_{u},
Z(Z(u)) = \overline{R}_{u}^{∗}/\overline{CR} .

By (1.147),

\frac{Z(Z(u))}{Z(G)} = \overline{R}_{u}^{∗}/\overline{R}^{∗} . (1.148)

Let π_{u} : Z̃(u) → Z(u) be the universal cover of Z(u). Then by [15, Proposition V.7.16],

Z(Z̃(u)) = \overline{R}_{u}^{∗}/\overline{CR}_{u}, (1.149)

and π_{1}(Z(u)) is a subgroup of Z(Z̃(u)).

Clearly

\frac{Z(Z(u))}{Z(G)} = \overline{R}_{u}^{∗}/\overline{R}^{∗} ⊂ C/\overline{R}^{∗} . (1.150)

Take v ∈ Z(Z(u))/Z(G). Then Z(v) ⊃ Z(u), CR_{v} ⊃ CR_{u}. Therefore π_{1}(Z(u)) = \overline{CR}/\overline{CR}_{u} surjects on π_{1}(Z(v)) = \overline{CR}/\overline{CR}_{v}. Let τ_{u,v} : \overline{CR}/\overline{CR}_{u} → \overline{CR}/\overline{CR}_{v} be this surjection. Clearly \overline{R}_{u}^{∗}/\overline{R}^{∗} maps into \overline{CR}_{u}^{∗}/\overline{R}^{∗}.

Let h ∈ π_{1}(Z(u)) = \overline{CR}/\overline{CR}_{u}. Then h defines a character δ_{h} of Z(Z(u))/Z(G) given by

δ_{h} : v ∈ Z(Z(u))/Z(G) ↦ exp(2iπ⟨h, v⟩) . (1.151)

Recall that by Proposition 1.36, \overline{CR}^{∗}/\overline{R}^{∗} ⊂ C/\overline{R}^{∗}.

Proposition 1.43. The element u ∈ C/\overline{R}^{∗} is such that for any h ∈ \overline{CR}/\overline{CR}_{u}, δ_{h}(u) = 1 if and only if u ∈ \overline{CR}^{∗}/\overline{R}^{∗}.

Proof. This is trivial.

Proposition 1.44. For any u ∈ C/\overline{R}^{∗}, v ∈ \frac{Z(Z(u))}{Z(G)} = \overline{R}_{u}^{∗}/\overline{R}^{∗}, h ∈ π_{1}(Z(u)) =

\overline{CR}/\overline{CR_{u}},

exp(2iπ⟨h, v⟩) = exp(2iπ⟨τ_{u,v}h, v⟩) . (1.152)

Proof. We have the exact sequence

0 → \overline{CR}_{v}/\overline{CR}_{u} → \overline{CR}/\overline{CR}_{u} →^{τ_{u,v}} \overline{CR}/\overline{CR}_{v} → 0 . (1.153)

Also if h′ ∈ \overline{CR}_{v}/\overline{CR}_{u}, exp(2iπ⟨h′, v⟩) = 1. Our Proposition follows.

Remark 1.45. Proposition 1.44 will be used in Remark 4.40.

1.12. Centralizers in a connected and non simply connected semisimple Lie group. Let G be a compact connected semisimple Lie group. We use otherwise the same notation as in Sections 1.1-1.3 and 1.11.

Let π_{1}(G) be the first homotopy group of G. Let G̃ be the universal cover of G. Then G̃ is a compact connected and simply connected semisimple Lie group, π_{1}(G) is a subgroup of of Z(G̃) and

G = G̃/π_{1}(G) (1.154)

Let T be a maximal torus in G. Let u ∈ T′ = 𝔱/\overline{R}^{∗}. Let Z(u) ⊂ G be the centralizer of u.By [14, Corollaire 5.3.1], if G is simply connected, then Z(u) is connected. Let Z(u)_{o} be the connected component of the identity in Z(u). Then Z(u)_{o} is a normal subgroup of Z(u), and so Z(u)_{o}\Z(u) is a finite group. Moreover T ⊂ Z(u)_{o}, so that u ∈ Z(u)_{o}.

If g ∈ Z(u), let g̃ ∈ G̃ be a lift of g, let ũ ∈ G̃ be a lift of u. Then [g̃, ũ] ∈ π_{1}(G) does not depend on g̃, ũ.

Proposition 1.46. The map g ∈ Z(u) ↦ [g̃, ũ] ∈ π_{1}(G) induces an embedding of Z(u)_{o}\Z(u) into π_{1}(G). In particular Z(u)_{o}\Z(u) is commutative.

Proof. The Lie group Z(ũ) ⊂ G̃ is connected. Therefore Z(ũ)/π_{1}(G) is a connected subgoup of Z(u).Since Z(ũ)/π_{1}(G) and Z(u)_{o} have the same Lie algebra, it follows that

Z(u)_{o} = Z(ũ)/π_{1}(G). (1.155)

Let g, g′ ∈ Z(u), let g̃, g̃′ ∈ G̃ lift g, g′. Let h, h′ ∈ π_{1}(G) be such that

g̃ũg̃^{−1} = ũh̃,

g̃′ũg̃′^{−1} = ũh̃′. (1.156)

26 JEAN-MICHEL BISMUT AND FRANÇOIS LABOURIE

Then g̃′g̃ũg̃^{−1}g̃′^{−1} = g̃′ũhg̃^{−1} = g̃′ũg̃′^{−1}h = ũh′h. (1.157) If g ∈ Z(u)_{o}, if g̃ ∈ Z(ũ) lifts g, then [g̃, ũ] = 1. (1.158) Conversely if ( 1.158) holds, then g̃ ∈ Z(ũ) and g ∈ Z(u)_{o}. The proof of our Proposition is completed.

Let N(T) ⊂ G be the normalizer of T in G. Then W = N(T)/T is the Weyl group. Put W_{u} = {w ∈ W, wu − u = 0 in T = 𝔱/Γ}. (1.159) Proposition 1.47. The following identity holds W_{u} = (N(T) ∩ Z(u))/T. (1.160) Proof. The proof of our Proposition is the same as in (1.128)-(1.131).

Let W_{Z(u)_{o}} be the Weyl group of Z(u)_{o}. Then W_{Z(u)_{o}} = W_{Z(ũ)}. Also W_{Z(u)_{o}} is a normal subgroup of W_{u}. Then W_{Z(u)_{o}}\W_{u} is a finite group.

Let \overline{CR} ⊂ 𝔱 be the lattice in 𝔱 spanned by the coroots of G̃. Let Γ ⊂ 𝔱 be the lattice of integrals elements in 𝔱, i.e. whose exponential in T ⊂ G is equal to 1. Then \overline{CR} ⊂ Γ, and Γ/\overline{CR} = π_{1}(G) (1.161)

We define \overline{R}, \overline{R}^{∗} as in Section 1.1 . Let \overline{u} ∈ 𝔱 represent u ∈ T′ = 𝔱/\overline{R}^{∗}.Then w ∈ W ↦ w\overline{u} − \overline{u} ∈ Γ/\overline{CR} = π_{1}(G) is a well-defined map.

Recall that Z(G̃) = \overline{R}^{∗}/\overline{CR}, and that π_{1}(G) = Γ/\overline{CR} is a subgroup of Z(G̃). In particular, by Proposition 1.15, π_{1}(G) embeds as a commutative subgroup of W. Theorem 1.48. The map w ∈ W_{u} ↦ w\overline{u} − \overline{u} ∈ Γ/\overline{CR} = π_{1}(G) induces an embedding of W_{Z(u)_{o}}\W_{u} into π_{1}(G). In particular W_{Z(u)_{o}}\W_{u} is commutative. If w ∈ W_{Z(u)_{o}}\W_{u} is identified to the corresponding element w′ ∈ π_{1}(G) ⊂ Z(G̃), the image of w′ in W lies in W_{u} and represents w in W_{Z(u)_{o}}\W_{u}. Proof. Let w, w′ ∈ W_{u}. Let h, h′ ∈ Γ be such that w\overline{u} − \overline{u} = h, w′\overline{u} − \overline{u} = h′. (1.162) Then w′w\overline{u} − \overline{u} = w′h + h′. (1.163) Now Γ ⊂ \overline{R}^{∗}, and so w′h − h ∈ \overline{CR}. (1.164) Therefore by (1.163), (1.164) w′w\overline{u} − \overline{u} = h + h′ in Γ/\overline{CR}. (1.165)

If w ∈ W_{Z(u)_{o}}, since u ∈ Z(u)_{o} is in the center of Z(u)_{o}, then w\overline{u} − \overline{u} = 0 in Γ/\overline{CR}. (1.166)

Conversely let w ∈ W be such that

w\overline{u} − \overline{u} ∈ \overline{CR}. (1.167)

Let g̃ ∈ G̃ represent w.By (1.167),

[g̃, ũ] = 1, (1.168)

so that g̃ maps into an element of Z(u)_{o}. In particular w ∈ W_{Z(u)_{o}}. The proof of our Proposition is completed.

Observe that

W_{Z(u)_{o}}\W_{u} = (N(T) ∩ Z(u)_{o})\(N(T) ∩ Z(u)). (1.169)

Therefore W_{Z(u)_{o}}\W_{u} embeds naturally into Z(u)_{o}\Z(u).

Theorem 1.49. We have the identity

W_{Z(u)_{o}}\W_{u} ≃ Z(u)_{o}\Z(u). (1.170)

This identity is compatible with the embeddings of both groups into π_{1}(G) = Γ/\overline{CR}.

Proof. Let g ∈ Z(u). Then gT g^{−1} is a maximal torus in Z(u)_{o}.Therefore there is h ∈ Z(u)_{o} such that

(hg)T (hg)^{−1} = T, (1.171)

so that hg ∈ N(T) ∩ Z(u). Therefore the embedding W_{Z(u)_{o}}\W_{u} → Z(u)_{o}\Z(u) is in fact one to one. It is trivial to verify that the above identification is compatible with the given embeddings into π_{1}(G). The proof of our Theorem is completed.

1.13. The intersection of an adjoint orbit with a centralizer. We make the same assumptions as in Section 1.12.

Let t ∈ T. Let O_{t} be the adjoint orbit of t in G. By [15, Lemma IV.2.5],

O_{t} ∩ T = {wt}_{w∈W} (1.172)

More generally, if H is a Lie subgroup of G, and t ∈ H, let O_{H}(t) be the adjoint orbit of t in H. In particular O_{t} = O_{G}(t).

Theorem 1.50. If G is simply connected, if u ∈ T′, then

O_{t} ∩ Z(u) = ⋃_{w∈W_{u}\W} O_{Z(u)}(wt). (1.173)

If t is regular, the above union is disjoint.

If G is not necessarily simply connected, if t ∈ T is very regular,

(1.174)  O_{t} ∩ Z(u) = ⋃_{w∈W_{Z(u)o} \W} O_{Z(u)_{o}}(wt),

O_{t} ∩ Z(u) = ⋃_{w∈W_{u}\W} O_{Z(u)}(wt),

and the above unions are disjoint.

Proof. If G is simply connected, then Z(u) is connected, and T is a maximal torus in Z(u). Let g ∈ O_{t} ∩ Z(u). There is g′ ∈ Z(u) such that g′gg′^{−1} ∈ O_{t} ∩ T. By [15, Lemma IV.2.5], there is w ∈ W such that

g′gg′^{−1} = wt, (1.175)

so that

g ∈ O_{Z(u)}(wt). (1.176)

Therefore (1.173) holds. If t ∈ T is regular, since G is simply connected, t is very regular. Therefore the {wt} are distinct in T. Moreover two elements in T lie in the same Z(u)-orbit if and only if they lie in the same W_{u}-orbit. Using Theorem 1.38, it follows that when t ∈ T is regular, the union in (1.173) is disjoint.

If G is non necessarily simply connected, if g ∈ O_{t} ∩ Z(u), then u ∈ Z(g). If t is very regular, Z(g) is a maximal torus, which is included in Z(u)_{o}. Therefore g ∈ Z(u)_{o}. The above argument shows that there is g′ ∈ Z(u)_{o}, and w ∈ W such that

g′gg′^{−1} = wt, (1.177)

which is equivalent to

g ∈ O_{Z(u)_{o}}(wt). (1.178)

So we have proved the first identity in (1.174). Since t is very regular in G, it is very regular in Z(u)_{o}. So the union in the first identity of (1.174) is disjoint.

Clearly

⋃_{w∈W_{u}\W} O_{Z(u)}(wt) ⊂ O_{t} ∩ Z(u), (1.179)

and also

⋃_{w∈W_{Z(u)o} \W} O_{Z(u)_{o}}(wt) ⊂ ⋃_{w∈W_{u}\W} O_{Z(u)}(wt). (1.180)

Therefore the second identity in (1.174) holds.

If g ∈ Z(u), w ∈ W are such that gtg^{−1} = wt, if g′ ∈ N(T) represents w, then g′^{−1}gt(g′^{−1}g)^{−1} = t. Since t is very regular, g′^{−1}g ∈ T, so that g ∈ Z(u) ∩ N(T).It follows that w ∈ W_{u}. Therefore the second union in (1.174) is also disjoint.

The proof of our Theorem is completed.

1.14. The stabilizer of an element of the Lie algebra, and coadjoint orbits. Let G be a compact connected semisimple Lie group. We use otherwise the notation of Section 1.1. Recall that τ : G → Aut(𝔤) is the adjoint representation.

Definition 1.51. If p ∈ 𝔱, put

Z(p) = {g ∈ G ; τ(g) · p = p} . (1.181)

By [15, Theorem IV.2.3], Z(p) is a connected Lie subgroup of G. Then T is a maximal torus in Z(p). Let 𝔷(p) be the Lie algebra of Z(p).

Theorem 1.52. If p ∈ 𝔱, then

𝔷(p) ⊗_{R} C = 𝔱 ⊕ ⨁_{{α∈R \atop ⟨α,p⟩=0}} 𝔤_{α} . (1.182)

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS 29

Also the root system R_{p} of Z(p) is given by R_{p} = {α ∈ R ; ⟨α, p⟩ = 0} . (1.183) If Z(Z(p)) ⊂ Z(p) is the center of Z(p), its Lie algebra 𝔷(Z(p)) is given by 𝔷(Z(p)) = {f ∈ 𝔱, for any α ∈ R_{p}, ⟨α, f⟩ = 0} . (1.184) Proof. The proof of these results is left to the reader. It is essentially the same as the proof of Theorem 1.38. Definition 1.53. Let π : 𝔱 ⊗_{R} C → C be the monomial

π(t) = ∏_{α∈R_{+}}⟨2iπα, t⟩ . (1.185)

By [15, Corollary V.4.6 and Lemma V.4.10], if w ∈ W π(wt) = ε_{w}π(t) . (1.186) Also if t ∈ 𝔱, one has the obvious det Ad(t)_{|𝔤/𝔱} = π^{2}(t/i) . (1.187) By (1.187), we find that π^{2}(t/i) does not depend on K, and lifts to a G-invariant function on 𝔤. Definition 1.54. Set

(1.188)  𝔤_{reg} = {p ∈ 𝔤 , Z(p) is a maximal torus} ,
𝔱_{reg} = {t ∈ 𝔱 ; Z(t) = T } .

Clearly 𝔱_{reg} = 𝔤_{reg} ∩ 𝔱 . (1.189) Proposition 1.55. The following identity holds

(1.190)  𝔤_{reg} = {p ∈ 𝔤 , π^{2}(p/i) ≠ 0} ,
𝔱_{reg} = {t ∈ 𝔱 ; π^{2}(t/i) ≠ 0} .

Proof. Take t ∈ 𝔱. Since Z(t) is connected, t ∈ 𝔱_{reg} if and only if 𝔷(t) = 𝔱 . (1.191) Using Theorem 1.52, we get the second identity in (1.190). Also by [15, Theorem IV.1.6], any G-orbit in 𝔤 intersects 𝔱. Our Proposition follows.

By (1.190), t ∈ 𝔱_{reg} if and only if t lies in a Weyl chamber. The G-orbit of t in 𝔤 intersects 𝔱 at |W | distinct elements, which form the W -orbit of t.

If p ∈ 𝔤, let O_{p} be the G-orbit of p. Clearly, π_{p} : g ∈ G/Z(p) ↦ gpg^{−1} ∈ O_{p} (1.192) is a one to one map. Also by [6, Lemma 7.22], O_{p} is equipped with a canonical symplectic form σ_{O_{p}}. In fact G acts on the left on O_{p}. If X ∈ 𝔤, let X^{O_{p}} be the corresponding vector field on O_{p}. Then if X, Y ∈ 𝔤, q ∈ O_{p}, σ_{O_{p},q}(X^{O_{p}}, Y^{O_{p}}) = ⟨q, [X, Y]⟩. (1.193)

Let f_{p} be the left invariant 1-form on G f_{p} = ⟨p, g^{−1}dg⟩ . (1.194) Let π_{p} be the projection G → G/Z(p) ≃ O_{p}.

30 JEAN-MICHEL BISMUT AND FRANÇOIS LABOURIE

Proposition 1.56. The following identity holds df_{p} = −π_{p}^{∗}σ_{p} . (1.195) In particular the restriction of f_{p} to Z(p) is a closed 1-form. Proof. Clearly

df_{p} = ⟨p, −\frac{1}{2}[g^{−1}dg, g^{−1}dg]⟩ (1.196)

from which (1.195) follows. Since π_{p} maps Z(p) to a constant in G/Z(p), our Proposition follows. Proposition 1.57. If p ∈ \overline{CR}^{∗}, f_{p} is an integral closed 1-form on Z(p). Proof. Clearly T is a maximal torus in Z(p). Then by [15, Proposition V.7.6], Z(p)/T is simply connected. Therefore π_{1}(T) surjects on π_{1}(Z(p)). To verify that f_{p} is an integral 1-form, we only need to check that if s ∈ S_{1} ↦ g_{s} ∈ T is smooth, the integral of f_{p} on this loop lies in Z. Since p ∈ \overline{CR}^{∗}, this is obvious. The proof of our Proposition is completed.

Now we assume that p ∈ \overline{CR}^{∗}. Take g ∈ Z(p). Let s ∈ [0, 1] ↦ g_{s} ∈ Z(p) be a smooth path such that g_{0} = 1, g_{1} = g.

Proposition 1.58. The map g ↦ exp (2iπ ∫_{0}^{1} g_{s}^{∗}f_{p}ds) ∈ S_{1} defines a representation ρ_{p} of Z(p). Proof. This follows from Proposition 1.57. Definition 1.59. Let L_{p} be the Hermitian line bundle on O_{p} L_{p} = G ×_{Z(p)} C . (1.197)

Clearly the connection d + 2iπf_{p} (1.198) descends to a connection ∇^{L_{p}} on L_{p}. Proposition 1.60. The following identity holds c_{1}(L_{p}, ∇^{L_{p}}) = σ_{p} . (1.199) Proof. This is obvious by (1.195).

2. Fourier analysis on the centralizers of semisimple Lie groups

The purpose of this Section is to express certain Fourier series on T (which will later turn out to be the symplectic volumes of the stratas of the moduli space of flat G-bundles on the Riemann surface Σ, first computed by Witten [59, 60] ) as residues of certain holomorphic functions in several complex variables. The main point is that it is then possible to compute explicitly the action of certain differential operators on these Fourier series. The results of this Section will be used in Sections 5, 6 and 7.

This Section is organized as follows. In Section 2.1, we make elementary constructions in linear algebra. In Section 2.2, we apply these constructions to the root system of G. In Section 2.3, given u ∈ C/\overline{R}^{∗} and the corresponding semisimple centralizer Z(u), we consider an associated Fourier series Q_{u}(t, x), which we express as a simple integral along the fibre of a torus fibration. In Section 2.4, we express Q_{u}(t, x) in terms of iterated residues. In Sections 2.5-2.8, we introduce other related Fourier series, which are related in particular to the universal cover of semisimple centralizers. In Sections 2.9-2.11, we introduce our symplectic volume Fourier series, which are local polynomials on T. We express these Fourier series as residues. Finally, in Section 2.12, we compute the action of any power series of differential operators on these local polynomials.

As explained in the Introduction, residue techniques have been developed by Szenes [51], [52] to handle the Witten Fourier series [59, 60]. The methods of Szenes are more conceptual than ours, which only usesimple linear algebra. It is probable that the results of this Section can be rephrased using Szenes’s formalism.

2.1. Some linear algebra. Let V be a real vector space of dimension r. Let e_{1}, …, e_{r} be a basis of V, let e^{1}, …, e^{r} be the dual basis of V^{∗}.

For 1 ≤ i ≤ r, if u_{1}, …, u_{i} ∈ V, put

⟨u_{1}, …, u_{i}⟩ = ⟨u_{1} ∧ … ∧ u_{i}, e^{1} ∧ … ∧ e^{i}⟩ . (2.1)

Equivalently

⟨u_{1}, …, u_{i}⟩ = \frac{u_{1} ∧ … u_{i} ∧ e_{i+1} ∧ … ∧ e_{r}}{e_{1} ∧ … ∧ e_{r}} . (2.2)

Let f_{1}, …, f_{r} be another basis of V, let f^{1}, …, f^{r} be the corresponding dual basis of V^{∗}.

Let E, E′, F, F′ be the flags in V,

(2.3)  E : 0 ⊂ {e_{1}} ⊂ {e_{1}, e_{2}} … ⊂ {e_{1}, …, e_{r}} = V,
E′ : V = {e_{1}, …, e_{r}} ⊃ {e_{2}, …, e_{r}} … ⊃ {e_{r}} ⊃ 0,
F : 0 ⊂ {f_{1}} ⊂ {f_{1}, f_{2}} … ⊂ {f_{1}, …, f_{r}} = V,
F′ : {f_{1}, …, f_{r}} ⊃ {f_{2}, …, f_{r}} ⊃ … {f_{r}} ⊃ 0 .

We assume that F lies in the orbit of E. Equivalently for i, 1 ≤ i ≤ r, f_{1}, …, f_{i}, e_{i+1}, …, e_{r} is a basis of E, i.e.

⟨f_{1}, …, f_{i}⟩ ≠ 0 . (2.4)

Let A be the (r, r) matrix expressing f_{1}, …, f_{r} on the basis e_{1}, …, e_{r}. Then (2.4) says that the principal minors of A do not vanish.

On V^{∗}, we can define the flags E^{∗}, E′^{∗}, F^{∗}, F′^{∗} associated to the basis e^{1}, …, e^{r} and f^{1}, …, f^{r}. Now E^{∗} lies in the orbit of F^{∗}, i.e. for any i, e^{1}, …, e^{i}, f^{i+1}, …, f^{r} is a basis of V^{∗}.

For 0 ≤ i ≤ r − 1, let p_{i} be the projection from V on {e_{i+1}, …, e_{r}} with kernel {f_{1}, …, f_{i}}. Clearly

p_{i+1} = p_{i+1}p_{i} . (2.5)

Also for 0 ≤ i ≤ r − 1, p_{i}f_{i+1}, …, p_{i}f_{r} is a basis of {e_{i+1}, …, e_{r}}. More precisely for 0 ≤ j ≤ r − i, p_{i}f_{i+1}, …, p_{i}f_{i+j}, e_{i+j+1}, …, e_{r} is a basis of {e_{i+1}, …, e_{r}}.

Clearly, if t ∈ V,

p_{1}t = t − \frac{⟨t, e^{1}⟩}{⟨f_{1}, e^{1}⟩}f_{1} . (2.6)

By the above, there are similar formulas for p_{2}, …, p_{r}. In particular, by (2.5), (2.6), for 1 ≤ i ≤ r, and t ∈ V,

p_{i}t = p_{i−1}t − \frac{⟨p_{i−1}t, e^{i}⟩}{⟨p_{i−1}f_{i}, e^{i}⟩}p_{i−1}f_{i}. (2.7)

From (2.7), we get

p_{i}t = t − ∑_{j=1}^{i} \frac{⟨p_{j−1}t, e^{j}⟩}{⟨p_{j−1}f_{j}, e^{j}⟩}p_{j−1}f_{j}. (2.8)

Using (2.8) with i = r, we obtain

t = ∑_{j=1}^{r} \frac{⟨p_{j−1}t, e^{j}⟩}{⟨p_{j−1}f_{j}, e^{j}⟩}p_{j−1}f_{j}. (2.9)

For 0 ≤ i ≤ r − 1, let q_{i} be the projection from V^{∗} on {f^{i+1}, …, f^{r}} with kernel {e^{1}, …, e^{i}}. Then q_{i} is the transpose of p_{i}, i.e.

q_{i} = p̃_{i}. (2.10)

Also, as in (2.6), if x ∈ V^{∗},

q_{1}x = x − \frac{⟨x, f_{1}⟩}{⟨e^{1}, f_{1}⟩}e^{1} . (2.11)

Moreover the results which hold for the p_{i}’s also hold for the q_{i}’s.

Theorem 2.1. For 1 ≤ i ≤ r,

∑_{j=1}^{i} \frac{⟨p_{j−1}t, e^{j}⟩⟨f_{j}, q_{j−1}x⟩}{⟨p_{j−1}f_{j}, q_{j−1}e^{j}⟩} = ⟨t, x⟩ − ⟨p_{i}t, q_{i}x⟩ . (2.12)

In particular,

∑_{i=1}^{r} \frac{⟨p_{i−1}t, e_{i}⟩⟨q_{i−1}x, f_{i}⟩}{⟨p_{i−1}f_{i}, q_{i−1}e^{i}⟩} = ⟨t, x⟩ . (2.13)

Proof. Equation (2.12) follows from (2.8), and equation (2.13) from (2.9).

Proposition 2.2. If t ∈ V, x ∈ V^{∗}, for 0 ≤ i ≤ r,

(1 − p_{i})t = ∑_{j=1}^{i} \frac{⟨f_{1} ∧ … ∧ f_{j−1} ∧ t ∧ f_{j+1} ∧ … ∧ f_{i}, e^{1} ∧ … ∧ e^{i}⟩}{⟨f_{1} ∧ … ∧ f_{i}, e^{1} ∧ … ∧ e^{i}⟩}f_{j},

(2.14) p_{i}t = ∑_{j=i+1}^{r} \frac{⟨f_{1} ∧ … ∧ f_{i} ∧ t, e^{1} ∧ … ∧ e^{i} ∧ e^{j}⟩}{⟨f_{1} ∧ … ∧ f_{i}, e^{1} ∧ … ∧ e^{i}⟩}e_{j}

= ∑_{j=i+1}^{r}(⟨t, e^{j}⟩ − ∑_{k=1}^{i}

\frac{⟨f_{1} ∧ … ∧ f_{i}, e^{1} ∧ … ∧ e^{k−1} ∧ e^{j} ∧ e^{k+1} ∧ … ∧ e^{i}⟩⟨t, e^{k}⟩}{⟨f_{1} ∧ … ∧ f_{i}, e^{1} ∧ … ∧ e^{i}⟩})e_{j},

(1 − q_{i})x = ∑_{j=1}^{i} \frac{⟨f_{1} ∧ … ∧ f_{i}, e^{1} ∧ … ∧ e^{j−1} ∧ x ∧ e^{j+1} ∧ … ∧ e^{i}⟩}{⟨f_{1} ∧ … ∧ f_{i}, e^{1} ∧ … ∧ e^{i}⟩}e^{j},

q_{i}x = ∑_{j=i+1}^{r} \frac{⟨f_{1} ∧ … ∧ f_{i} ∧ f_{j}, e^{1} ∧ … ∧ e^{i} ∧ x⟩}{⟨f_{1} ∧ … ∧ f_{i}, e^{1} ∧ … ∧ e^{i}⟩}f^{j}

= ∑_{j=i+1}^{r}(⟨f_{j}, x⟩ − ∑_{k=1}^{i}

\frac{⟨f_{1} ∧ … ∧ f_{k−1} ∧ f_{j} ∧ f_{k+1} ∧ … ∧ f_{i}, e^{1} ∧ … ∧ e^{i}⟩⟨f_{k}, x⟩}{⟨f_{1} ∧ … ∧ f_{i}, e^{1} ∧ … ∧ e^{i}⟩})f^{j} .

Proof. Clearly we only need to prove the first two series of identities in (2.14). The first identity and the first part of the second identity are standard linear algebra. Also for j ≥ i + 1,

⟨f_{1} ∧ … ∧ f_{i} ∧ t, e^{1} ∧ … ∧ e^{i} ∧ e^{j}⟩ = ⟨f_{1} ∧ … ∧ f_{i}, e^{1} ∧ … ∧ e^{i}⟩⟨t, e^{j}⟩

− ∑_{k=1}^{i}(−1)^{k−i}⟨f_{1} ∧ … ∧ f_{i}, e^{1} ∧ … ∧ e^{k−1} ∧ e^{k+1} ∧ … ∧ e^{i} ∧ e^{j}⟩⟨t, e^{k}⟩

= ⟨f_{1} ∧ … ∧ f_{i}, e^{1} ∧ … ∧ e^{i}⟩⟨t, e^{j}⟩ (2.15)

− ∑_{k=1}^{i}⟨f_{1} ∧ … ∧ f_{i}, e^{1} ∧ … ∧ e^{k−1} ∧ e^{j} ∧ e^{k+1} ∧ … ∧ e^{i}⟩⟨t, e^{k}⟩ .

The proof of our Proposition is completed.

Proposition 2.3. For any j, 1 ≤ j ≤ r

⟨f_{j}, q_{j−1}x⟩ = (2.16)

⟨f_{j}, x⟩ − ∑_{k=1}^{j−1} \frac{⟨f_{1}∧...∧f_{k−1}∧f_{j}∧f_{k+1}∧...∧f_{j−1},e^{1}∧...∧e^{j−1}⟩}{⟨f_{1}∧...∧f_{j−1},e^{1}∧...∧e^{j−1}⟩}⟨f_{k}, x⟩ .

Proof. This is a consequence of the last equality in (2.14).

Remark 2.4. From (2.16), we find that ⟨f_{j}, q_{j−1}x⟩ = ⟨p_{j−1}f_{j}, x⟩ depends only on the ⟨f_{k}, x⟩, 1 ≤ k ≤ j.

2.2. The linear algebra of the basis of a root system. Let G be a compact connected and simply connected simple Lie group of rank r. Let T be a maximal torus in G. Let 𝔱 be the Lie algebra of T. Let K be a Weyl chamber in 𝔱. Otherwise, we use the notation of Section 1.1.

We will also use the notation of Section 2.1, with V = 𝔱^{∗}. We identify 𝔱 and 𝔱^{∗} by the scalar product ⟨ , ⟩ of Section 1.2. Then

r = dim 𝔱 . (2.17)

Let e_{1}, …, e_{r} ⊂ R_{+} be the simple basis of 𝔱^{∗} associated to K [15, Proposition V.4.5]. Any α ∈ R_{+} is a linear combination with non negative integral coefficients of e_{1}, …, e_{r}. Then e_{1}, …, e_{r} generate \overline{R}. Let e^{1}, …, e^{r} be the corresponding dual basis of \overline{R}^{∗} ⊂ 𝔱.

Let {α_{1}, …, α_{ℓ}} be an ordering of R_{+}, such that

α_{i} = e_{i} for 1 ≤ i ≤ r . (2.18)

Recall that we use the notation of Section 2.1 . Clearly all the ⟨α_{i_{1}}, …, α_{i_{j}}⟩ lie in Z.

If G is not simply laced, m was defined in (1.8) and is equal to 2 or to 3. By convention, if G is simply laced, we take m = 1. By (1.12),

m\overline{R} ⊂ \overline{CR}. (2.19)

Definition 2.5. Let d ∈ N^{∗} be a common multiple of the m|⟨α_{i_{1}} ∧ … ∧ α_{i_{r}}⟩|.

Observe that since e_{1}, …, e_{r} ∈ R_{+}, for any j, 1 ≤ j ≤ r, d is a multiple of m⟨α_{i_{1}}, …, α_{i_{j}}⟩ ∈ Z. Also by (2.19),

d\overline{R} ⊂ \overline{CR}. (2.20)

Recall that C ⊂ 𝔱 was defined in Definition 1.35.

Proposition 2.6. The following identity holds

dC ⊂ \overline{R}^{∗} . (2.21)

Proof. Let u ∈ C, let α_{i_{1}}, …, α_{i_{r}} ∈ R_{+} be a basis of 𝔱^{∗} such that for 1 ≤ j ≤ r, ⟨α_{i_{j}}, u⟩ ∈ Z. Since e_{1}, …, e_{r} is a basis of \overline{R}, we find that if H is the lattice generated by α_{i_{1}}, …, α_{i_{r}},

d\overline{R} ⊂ H , (2.22)

which is equivalent to

dH^{∗} ⊂ \overline{R}^{∗} . (2.23)

Since u ∈ H^{∗}, from (2.23) , we get (2.21). The proof of our Proposition is completed.

Remark 2.7. Take n ≥ 2. Since SU(n) is simply laced, m = 1. Also by [15, Proposition V.6.3], all the |⟨α_{i_{1}}, …, α_{i_{j}}⟩| are equal to 1. Therefore, for G = SU(n), we can take d = 1. Using (1.114) and (2.21), we recover Proposition 1.40.

Definition 2.8. A family I = (i_{1}, …, i_{r}) of distinct indices in {1, …, ℓ} is said to be generic if

⟨α_{i_{1}}, …, α_{i_{j}}⟩ ≠ 0 for 1 ≤ j ≤ r . (2.24)

Given a generic family I = (i_{1}, …, i_{r}), we now use the notation in Section 2.1 associated to the given basis e_{1}, …, e_{r} and α_{i_{1}}, …, α_{i_{r}} of 𝔱^{∗} ≃ 𝔱. In particular the operators which appear in Section 2.1 will be denoted with the superscript I, to mark their dependence on I.

Let α^{i_{1}}, …, α^{i_{r}} be the basis of 𝔱^{∗} ≃ 𝔱 which is dual to α_{i_{1}}, …, α_{i_{r}}.

Recall that (2.18) holds.

Definition 2.9. A family I = (i_{1}, …, i_{j−1}) of j − 1 distinct elements of {1, …, ℓ} is said to be generic if I_{j−1} = {i_{1}, …, i_{j−1}, j, j + 1, …, r} is generic.

If I = (i_{1}, …, i_{r}) is generic, I_{j−1} = (i_{1}, …, i_{j−1}) is also generic, and by construction (or by (2.14)),

(2.25)  p_{j−1}^{I} = p_{j−1}^{I_{j−1}} ,
q_{j−1}^{I} = q_{j−1}^{I_{j−1}} .

Definition 2.10. If x ∈ C^{ℓ}, and if I = (i_{1}, …, i_{r}) is generic, put

x^{I} = ∑_{1}^{r} x_{i_{j}}α^{i_{j}} ∈ 𝔱 ≃ 𝔱^{∗} . (2.26)

By (2.25),

q_{j−1}^{I}x^{I} = q_{j−1}^{I_{j−1}}x^{I} . (2.27)

Also by Proposition 2.3,

⟨α_{i_{j}}, q_{j−1}^{I}x^{I}⟩ = ⟨p_{j−1}^{I}α_{i_{j}}, x^{I}⟩ = (2.28)

x_{i_{j}} − ∑_{k=1}^{j−1} \frac{⟨α_{i}_{1}, …, α_{i_{k−1}}, α_{i_{j}}, α_{i_{k+1}}, …, α_{i_{j−1}}⟩}{⟨α_{i_{1}}, …, α_{i_{j−1}}⟩}x_{i_{k}} .

Note that the right hand side of (2.28) only depends on x_{i_{1}}, …, x_{i_{j}}.

Assume that I = (i_{1}, …, i_{r}) is generic. Then by Proposition 2.2, if t ∈ 𝔱, y ∈ 𝔱^{∗}

(2.29)  ⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩ = \frac{⟨α_{i_{1}}, …, α_{i_{j}}⟩}{⟨α_{i_{1}}, …, α_{i_{j−1}}⟩},
⟨p_{j−1}^{I}α_{i_{j}}, t⟩ = \frac{⟨α_{i_{1}} ∧ … ∧ α_{i_{j}}, e^{1} ∧ … ∧ e^{j−1} ∧ t⟩}{⟨α_{i_{1}}, …, α_{i_{j−1}}⟩} ,
⟨p_{j−1}^{I}y, e^{j}⟩ = \frac{⟨α_{i_{1}}, …, α_{i_{j−1}}, y⟩}{⟨α_{i_{1}}, …, α_{i_{j−1}}⟩} .

In particular, by (2.29), we find that

\frac{d}{⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩} ∈ Z . (2.30)

Also by (2.28), (2.29), if x ∈ Z^{ℓ},

d\frac{⟨p_{j−1}^{I}α_{i_{j}}, x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩} ∈ Z . (2.31)

Definition 2.11. For u ∈ C/\overline{R}^{∗}, let I_{u} ⊂ \{1, …, ℓ\} be given by

I_{u} = {i, 1 ≤ i ≤ ℓ, α_{i} ∈ R_{u,+}} . (2.32)

36 JEAN-MICHEL BISMUT AND FRANÇOIS LABOURIE

Put ℓ_{u} = |I_{u}|. (2.33) 2.3. Fourier series and integration along the fibre. Definition 2.12. For u ∈ C/\overline{R}^{∗}, let R^{I_{u}} ⊂ R^{ℓ} be given by R^{I_{u}} = {s = (s^{1}, …, s^{ℓ}) ∈ R^{ℓ} , s^{i} = 0 for i ∉ I_{u}} . (2.34)

Let a_{u} : R^{I_{u}} → 𝔱^{∗} ≃ 𝔱 be given by

a_{u}(s) = ∑_{i∈I_{u}} s^{i}α_{i} . (2.35)

The transpose ã_{u} : 𝔱 ≃ 𝔱^{∗} → R^{I_{u}} is given by ã_{u}(t) = (⟨α_{i}, t⟩)_{i∈I_{u}} . (2.36)

Set V_{u} = ker a_{u}. (2.37) Then we have the exact sequence

\xymatrix{0 → V_{u} → R^{I_{u}} \ar[r]_{a_{u}} & 𝔱 → 0} . (2.38)

We define Z^{I_{u}} ⊂ Z^{ℓ}, (R/Z)^{I_{u}} ⊂ (R/Z)^{ℓ} as before. Then a_{u}(Z^{I_{u}}) = \overline{R}_{u} . (2.39) Also a_{u} induces a surjection (R/Z)^{I_{u}} → 𝔱/\overline{R} . Set K_{u} = ker a_{u} ⊂ (R/Z)^{I_{u}}. (2.40) We have the exact sequence

\xymatrix{0 → K_{u} → (R/Z)^{I_{u}} \ar[r]_{a_{u}} & 𝔱/\overline{R} → 0} . (2.41)

Set γ_{u} = Z^{I_{u}} ∩ V_{u} . (2.42) Then K_{u} is a union of |\overline{R}/\overline{R}_{u}| tori V_{u}/γ_{U} . Definition 2.13. For u ∈ C/\overline{R}^{∗}, let H_{u} ⊂ 𝔱/\overline{R} be given by

(2.43) H_{u} = \{t ∈ 𝔱/\overline{R}, t = ∑_{j∈J ⊂I_{u}} t^{j}α_{j}, and \{α_{j}, j ∈ J \} does not span 𝔱^{∗} ≃ 𝔱\} .

Then H_{u} is a finite union of hypertori in T. Put H = H_{0} . (2.44) Clearly,

H = \{t ∈ 𝔱/\overline{R}, t = ∑_{j∈J ⊂\{1,... ,ℓ\}} t^{j}α_{j}, and \{α_{j}, j ∈ J \} does not span 𝔱^{∗} ≃ 𝔱\} . (2.45)

For any u ∈ C/\overline{R}^{∗}, H_{u} ⊂ H . (2.46)

Proposition 2.14. Let I = (i_{1}, …, i_{r}) ⊂ I_{u} be generic. Then if t ∈ (𝔱/\overline{R}) \ H is represented by \overline{t} ∈ 𝔱,

⟨p_{j−1}^{I}\overline{t}, e^{j}⟩ ∉ Z. (2.47)

Proof. By (2.29),

⟨p_{j−1}^{I}\overline{t}, e^{j}⟩ = \frac{⟨α_{i_{1}}, …, α_{i_{j−1}}, \overline{t}⟩}{⟨α_{i_{1}}, …, α_{i_{j−1}}⟩} . (2.48)

Now for \overline{t} = e_{j}, the expression (2.48) is equal to 1. Also ⟨α_{i_{1}}, …, α_{i_{j−1}}, \overline{t}⟩ vanishes if and only if \overline{t} is a linear combination of α_{i_{1}}, …, α_{i_{j−1}}, e_{j+1}, …, e_{r}. Therefore the condition

⟨p_{j−1}^{I}\overline{t}, e^{j}⟩ ∈ Z (2.49)

is equivalent to

\overline{t} = ∑_{k=1}^{j−1} a^{k}α_{i_{k}} + be_{j} + ∑_{k=j+1}^{r} c^{k}e_{k} , a^{k}, c^{k} ∈ R , b ∈ Z . (2.50)

Then,

\overline{t} = ∑_{k=1}^{j−1} a^{k}α_{i_{k}} + ∑_{k=j+1}^{r} c^{k}e_{k} in 𝔱/\overline{R}, (2.51)

so that t ∈ H.

The proof of our Proposition is completed.

Let dt be the Lebesgue measure on 𝔱 associated with ⟨ , ⟩. We still denote by dt the Lebesgue measure on 𝔱/\overline{R}. We map L^{1}(𝔱/\overline{R}) into D′(𝔱/\overline{R}) by the map

f ↦ \frac{fdt}{Vol(𝔱/\overline{R})}. We use the same convention for other tori.

Clearly a_{u} induces a map a_{u∗} from D′((R/Z)^{I_{u}}) into D′(t/\overline{R}).

Proposition 2.15. If g ∈ L^{1}((R/Z)^{I_{u}}), then a_{u∗}g ∈ L^{1}(𝔱/\overline{R}). More precisely,

a_{u}_{∗}g(t) = ∫_{K_{u}} g(t + s)\frac{ds}{Vol(K_{u})} . (2.52)

Also if k ∈ Z^{I_{u}}, then

(2.53) a_{u∗}[e^{2iπ⟨k,s⟩}] = e^{2iπ⟨λ,t⟩} if there is λ ∈ \overline{R}^{∗} such that k_{i} = ⟨λ, α_{i}⟩ , i ∈ I_{u} ,
= 0  otherwise .

Proof. The proof of this result is trivial.

Definition 2.16. For u ∈ C/\overline{R}^{∗} , x ∈ (C \ 2iπZ)^{ℓ}, t ∈ 𝔱/\overline{R}, set

Q_{u}(t, x) = ∑_{λ∈\overline{R}^{∗}} \frac{exp(2iπ⟨λ, t⟩)}{∏_{i∈I_{u}}(2iπ⟨α_{i}, λ⟩ − x_{i})} . (2.54)

Clearly as a function of t, Q_{u}(t, x) is a well-defined distribution on 𝔱/\overline{R}. For M ∈ N, we also consider the partial sums

Q_{u}^{M}(t, x) = ∑_{{λ∈\overline{R}^{∗} \atop |λ|≤M}} \frac{exp(2iπ⟨λ, t⟩)}{∏_{i∈I_{u}}(2iπ⟨α_{i}, λ⟩ − x_{i})} . (2.55)

Note that Q_{u}(t, x) depends only on the projection of x on C^{I_{u}}.

Theorem 2.17. For any n ∈ N^{∗},u ∈ C/\overline{R}^{∗}, v ∈ \frac{\overline{R}^{∗}}{n}, t ∈ 𝔱/\overline{R}, the following identity holds

n^{ℓ_{u}−r} ∑_{h∈\overline{R}/n\overline{R}} exp(2iπ⟨v, t + h⟩)Q_{u}(\frac{t + h}{n}, nx) = Q_{u}(t, x + 2iπã_{u}v) . (2.56)

Proof. Clearly, if λ ∈ \overline{R}^{∗}, v ∈ \overline{R}^{∗}/n,

(2.57) \frac{1}{n^{r}} ∑_{h∈\overline{R}/n\overline{R}} exp(2iπ⟨\frac{λ}{n} + v, t + h⟩) = exp(2iπ⟨\frac{λ}{n} + v, t⟩) if \frac{λ}{n} + v ∈ \overline{R}^{∗} ,
= 0 otherwise .

From (2.54),(2.57) , we get

n^{ℓ_{u}−r} ∑_{h∈\overline{R}/n\overline{R}} exp(2iπ⟨v, t + h⟩)Q_{u}(\frac{t + h}{n}, nx) (2.58)

= ∑_{λ∈\overline{R}^{∗}} \frac{exp(2iπ⟨λ, t⟩)}{∏_{i∈I_{u}}(2iπ⟨α_{i}, λ⟩ − (x_{i} + 2iπ⟨α_{i}, v⟩))}

= Q_{u}(t, x + 2iπã_{u}v) .

The proof of our Theorem is completed.

Theorem 2.18. The partial sums Q_{u}^{M}(t, x) converge uniformly together with their derivatives to Q_{u}(t, x) on compact subsets of (𝔱/\overline{R})\H_{u}×(C\2iπZ)^{ℓ}. The following identity of distributions holds

(2.59) Q_{u}(t, x) = (−1)^{ℓ_{u}}a_{u∗} [\frac{1}{∏_{i∈I_{u}} ( exp(x_{i}) − 1)} exp(⟨x, s⟩)] in D′(𝔱/\overline{R}) .

In particular for x ∈ (C\2iπZ)^{ℓ}, Q_{u}(t, x) is a distribution in L^{∞}(𝔱/\overline{R}). Also (2.59) is an identity of smooth functions on (𝔱/\overline{R}) \ H_{u} × (C \ 2iπZ)^{ℓ}.

Proof. For x ∈ C \ 2iπZ, the Fourier series for e^{xs} (s ∈ [0, 1]) is given by

e^{xs} = (e^{x} − 1) ∑_{k∈Z} \frac{e^{2iπks}}{−2iπk + x} . (2.60)

Since e^{xs} is smooth on [0, 1], the partial sums in (2.60) converge uniformly together with their derivatives to e^{xs} on compact subsets of R/Z \ {0} × C \ 2iπZ.

From Proposition 2.15 and from (2.60) , we get

a_{u∗}[e^{⟨x,s⟩}] = ∏_{i∈I_{u}}(e^{x_{i}} − 1) ∑_{λ∈\overline{R}^{∗}} \frac{e^{2iπ⟨λ,t⟩}}{∏_{i∈I_{u}}(−2iπ⟨α_{i}, λ⟩ + x_{i})} , (2.61)

which coincides with (2.59).

Also for x ≠ 0, the wave front set of the distribution e^{xs} on S^{1} is just {0} × R^{∗}. By [24, Theorem 8.2.13], we see that (t, p) ∈ 𝔱/\overline{R} × R^{I_{u}} \ \{0\} lies in the wave front set of Q_{u}(t, x) only if t = ∑_{i∈I_{u}} t^{i}α_{i}, and ⟨p, α_{i}⟩ = 0 when t^{i} ∉ Z. Therefore Q_{u}(t, x)

is smooth on (𝔱/\overline{R}) \ H_{u}.

The proof of our Theorem if completed.

2.4. Iterated residues and the series Q_{u}(t, x). Recall that by Theorem 2.18, for generic values of x, Q_{u}(t, x) is smooth on (𝔱/\overline{R}) \ H_{u}. Therefore by (2.46), Q_{u}(t, x) is smooth on (𝔱/\overline{R}) \ H.

Let x ∈ R ↦ [x] ∈ [0, 1[ be the periodic function of period 1, such that [x] = x on [0, 1[.

In the sequel, if t ∈ 𝔱/\overline{R}, we represent t by a given element in 𝔱, which we also denote by t.

Also the map (f^{1}, …, f^{r}) ∈ \{0, 1, …, d − 1\}^{r} ↦ f = ∑_{1}^{r} f^{i}e_{i} ∈ \overline{R} defines a

one to one map into \overline{R}/d\overline{R}. In the sequel, we will always identify f ∈ \overline{R}/d\overline{R} to the corresponding element in {0, 1, …, d − 1}^{r}.

Theorem 2.19. For any u ∈ C/\overline{R}^{∗}, for generic values of x ∈ (C \ 2iπZ)^{ℓ}, for t ∈ (𝔱/\overline{R}) \ H, if we still denote by t a representative in 𝔱,

Q_{u}(t, x) = (−1)^{r} ∑_{{I=(i_{1},... ,ir)⊂Iu \atop I generic, f∈ \frac{\overline{R}}{d\overline{R}}}} \frac{1}{⟨α_{i_{1}}, …, α_{i_{r}}⟩}

exp \{d ∑_{j=1}^{r} \frac{⟨p_{j−1}^{I}α_{i_{j}}, x^{I}⟩}{⟨p_{j−1}^{I}, α_{i_{j}}, e^{j}⟩} [\frac{1}{d}⟨p_{j−1}^{I}(t + f), e^{j}⟩]\} (2.62)

\frac{1}{∏_{i∈I_{u}\I}(⟨α_{i}, x^{I}⟩ − x_{i})} ∏_{j=1}^{r} \frac{1}{exp (\frac{d⟨p_{j−1}^{I}α_{i_{j}} ,x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}} ,e^{j}⟩}) − 1} .

Proof. Take 1 ≤ j ≤ r, I = (i_{1}, …, i_{j−1}) ⊂ I_{u} such that ⟨α_{i_{1}}⟩ ≠ 0, …, ⟨α_{i_{1}}, …, α_{i_{j−1}}⟩ ≠

0. For k̂ = (k_{j+1}, …, k_{r}) ∈ Z^{r−j}, we identify k̂ with ∑_{i=j+1}^{r} k_{i}e^{i} ∈ 𝔱 ≃ 𝔱^{∗}. In the

sequel, p_{j−1}^{I} denotes the projection 𝔱 → {e_{j}, …, e_{r}} with kernel {α_{i_{1}}, …, α_{i_{j−1}}}. In view of (2.28), for i ∈ I_{u}, i ∉ I, we will use the abusive notation

⟨p_{j−1}^{I}α_{i}, x^{I}⟩ = (2.63)

x_{i} − ∑_{k=1}^{j−1} \frac{⟨α_{i_{1}}, …, α_{i_{k−1}}, α_{i}, α_{i_{k+1}}, …, α_{i_{j−1}}⟩}{⟨α_{i_{1}}, …, α_{i_{j−1}}⟩}x_{i_{k}} .

Equation (2.63) will in fact be a definition for the left-hand side. For s ∈ R, if 1 ≤ j ≤ r, set

Q_{I} = ∑_{k∈Z} \frac{e^{2iπks}}{∏_{i∈I_{u}\I}(⟨p_{j−1}^{I}α_{i}, 2iπ(ke^{j} + k̂) − x^{I}⟩)} . (2.64)

We claim that for at least one i ∈ I_{u} \ I,

⟨p_{j−1}^{I}α_{i}, e^{j}⟩ ≠ 0 . (2.65)

Figure 2.1

In fact by (2.29),

⟨p_{j−1}^{I}α_{i}, e^{j}⟩ = \frac{⟨α_{i_{1}}, …, α_{i_{j−1}}, α_{i}⟩}{⟨α_{i_{1}}, …, α_{i_{j−1}}⟩} . (2.66)

Since the elements of R_{u,+} span 𝔱^{∗} ≃ 𝔱, for at least one i ∈ I_{u} \ I, (2.66) does not vanish.

Clearly

Q_{I} = \frac{1}{d} ∑_{{k∈Z \atop 0≤f<d}} \frac{exp(2iπ \frac{k}{d} (s + f))}{∏_{i∈I_{u}\I} (2iπ⟨p_{j−1}^{I}α_{i}, e^{j}⟩\frac{k}{d} + ⟨p_{j−1}^{I}α_{i}, 2iπk̂ − x^{I}⟩)} . (2.67)

Then for generic x ∈ C^{ℓ}, we can write (2.67) in the form

(2.68) Q_{I} = \frac{1}{d} ∑_{0≤f<d} ∑_{k∈Z}

Res_{a=2iπk}\{\frac{e^{a[ \frac{s+f}{d} ]}}{( ∏_{i∈I_{u}\I} (\frac{⟨p_{j−1}^{I}α_{i}, e^{j}⟩a}{d} + ⟨p_{j−1}^{I}α_{i}, 2iπk̂ − x^{I}⟩))(e^{a} − 1)}\} .

Assume that s ∉ Z. Then for f ∈ N, 0 ≤ f < d,

0 < [\frac{s + f}{d} ] < 1 . (2.69)

Also by (2.29) , if ⟨α_{i_{1}}, …, α_{i_{j−1}}, α_{i}⟩ ≠ 0, then

\frac{d}{⟨p_{j−1}^{I}α_{i}, e^{j}⟩} ∈ Z , (2.70)

\frac{d⟨p_{j−1}^{I}α_{i}, k̂⟩}{⟨p_{j−1}^{I}α_{i}, e^{j}⟩} ∈ Z .

For n ∈ N, consider the contour Γ_{n} = Γ_{n,+} ∪ Γ_{n,−} given in Figure 2.1, and its interior Δ_{n}. For f ∈ N, 0 ≤ f < d, put

g_{f}(a) = \frac{e^{a[ \frac{s+f}{d} ]}}{( ∏_{i∈I_{u}\I} (\frac{⟨p_{j−1}^{I}α_{i}, e^{j}⟩a}{d} + ⟨p_{j−1}^{I}α_{i}, 2iπk̂ − x^{I}⟩))(e^{a} − 1)} . (2.71)

By (2.69), as a ∈ Δ_{n}, |a| → +∞,

g_{f}(a) → 0 . (2.72)

Also the integral ∫_{Γ_{n}} g_{f}(a)da converges. We can then use the residue theorem to

evaluate ∫_{Γ_{n}} g_{f}(a)da. Finally by (2.65), (2.71), as n → +∞, ∫_{Γ_{n}} g_{f}(a)da → 0.

Then we find that for generic x ∈ C^{ℓ},

∑_{a∈C} Res_{a}g_{f}(a) = 0 . (2.73)

Now for generic x ∈ C^{ℓ}, the poles of g_{f}(a) other than {2iπk}_{k∈Z} are simple (this follows in particular from (2.63)), and given by

a = −d\frac{⟨p_{j−1}^{I}α_{i_{j}}, 2iπk̂ − x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩} , i_{j} ∈ I_{u} \ I, ⟨α_{i_{1}}, …, α_{i_{j}}⟩ ≠ 0 . (2.74)

In the sequel, we use the notation

(I, i_{j}) = (i_{1}, …, i_{j}) . (2.75)

Observe that if (i_{1}, …, i_{j}) is generic, if y ∈ 𝔱^{∗} for 1 ≤ k ≤ j − 1,

\frac{⟨α_{i_{1}}, …, α_{i_{k−1}}, y, α_{i_{k+1}}, …, α_{i_{j}}⟩}{⟨α_{i_{1}}, …, α_{i_{j}}⟩} = (2.76)

\frac{1}{⟨α_{i_{1}}, …, α_{i_{j−1}}⟩}(⟨α_{i_{1}}, …, α_{i_{k−1}}, y, α_{i_{k+1}}, …, α_{i_{j−1}}⟩

−\frac{⟨α_{i_{1}}, …, α_{i_{k−1}}, α_{i_{j}}, α_{i_{k+1}}, …, α_{i_{j−1}}⟩}{⟨α_{i_{1}}, …, α_{i_{j}}⟩}⟨α_{i_{1}}, …, α_{i_{j−1}}, y⟩) .

In fact if y lies the vector space spanned by α_{i_{1}}, …, α_{i_{k−1}}, α_{i_{k+1}}, …, α_{i_{j−1}}, α_{i_{j}}, e_{j+1}, …, e_{r} both sides of (2.76) vanish, and if y = α_{i_{k}} both sides are equal to 1. So by (2.6), (2.63), (2.66), (2.68), (2.70), (2.73) and (2.76) with y = α_{i}, i ∉ (I, i_{j}), we get

Q_{I} = − ∑_{{i_{j} ∈Iu\I \atop ⟨α_{i_{1}} ,... ,α_{i_{j}} ⟩≠0 ,0≤f<d}} \frac{⟨α_{i_{1}}, …, α_{i_{j−1}}⟩}{⟨α_{i_{1}}, …, α_{i_{j}}⟩} (2.77)

exp (−2iπ \frac{⟨p_{j−1}^{I}α_{i_{j}}, k̂⟩}{⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩}(s + f) + \frac{d⟨p_{j−1}^{I}α_{i_{j}}, x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩} [\frac{s + f}{d} ])

\frac{1}{( ∏_{i∈I_{u}\(I,i_{j})} (⟨p_{j}^{(I,i_{j})}α_{i}, 2iπk̂ − x^{I}⟩)) (exp (\frac{d⟨p_{j−1}^{I}α_{i_{j}} ,x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}} ,e^{j}⟩}) − 1)} .

42 JEAN-MICHEL BISMUT AND FRANÇOIS LABOURIE

Now by (2.7), for j′ ≥ j + 1,

⟨p_{j}^{(I,i_{j})}e_{j}, e^{j′}⟩ = −\frac{⟨p_{j−1}^{I}α_{i_{j}}, e^{j′}⟩}{⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩} . (2.78)

Using (2.77), (2.78), we obtain

Q_{I} = − ∑_{{i_{j} ∈Iu\I \atop ⟨α_{i_{1}} ,... ,α_{i_{j}} ⟩≠0 ,0≤f<d}} \frac{⟨α_{i_{1}}, …, α_{i_{j−1}}⟩}{⟨α_{i_{1}}, …, α_{i_{j}}⟩} (2.79)

exp (2iπ⟨p_{j}^{(I,i_{j})}(s + f)e_{j}, k̂⟩ + \frac{d⟨p_{j−1}^{I}α_{i_{j}}, x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩} [\frac{s + f}{d} ])

\frac{1}{( ∏_{i∈I_{u}\(I,i_{j})} (⟨p_{j}^{(I,i_{j})}α_{i}, 2iπk̂ − x^{I}⟩)) (exp (\frac{d⟨p_{j−1}^{I}α_{i_{j}} ,x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}} ,e^{j}⟩}) − 1)} .

Clearly

Q_{u}(t, x) = ∑_{k=(k_{1},... ,k_{r})∈Z^{r}} \frac{exp(2iπ ∑_{i=1}^{r} k_{i}⟨t, e^{i}⟩)}{∏_{i∈I_{u}}(2iπ⟨α_{i}, ∑_{i=1}^{r} k_{i}e^{i}⟩ − x_{i})} . (2.80)

Also with the notation in (2.63), ⟨p_{0}^{∅}α_{i}, x^{I}⟩ = x_{i} . (2.81) So using (2.79) with I = ∅, s = ⟨t, e^{1}⟩, we find that for ⟨t, e^{1}⟩ ∉ Z,

Q_{u}(t, x) = − ∑_{i_{1}∈I_{u}_{⟨α_{i_{1}} ⟩≠0} ; 0≤f^{1}<d} \frac{1}{⟨α_{i_{1}}⟩} ∑_{k=(k^{2},... ,k^{r})∈Z^{r−1}}

exp(2iπ⟨p_{1}^{(i_{1})}(t + f^{1}e_{1}), k⟩ + d\frac{x_{i_{1}}}{⟨α_{i_{1}}⟩} [\frac{⟨t + f^{1}e_{1}, e^{1}⟩}{d}]) (2.82)

\frac{1}{( ∏_{i∈I_{u}\\{i_{1}\}} (⟨p_{1}^{(i_{1})}α_{i}, 2iπk − x^{I}⟩))( exp(\frac{dx_{i_{1}}}{⟨α_{i_{1}}⟩}) − 1)} .

Clearly, if t ∈ (t/\overline{R}) \ H, then ⟨t, e^{1}⟩ ∉ Z, and so (2.82) holds. More generally, if t ∈ (𝔱/\overline{R}) \ H, f ∈ \overline{R}, by Proposition 2.14, ⟨p_{j−1}^{I}(t + f), e_{j}⟩ ∉ Z. Therefore using (2.79), (2.82), we can iterate the procedure. Finally observe that by (2.63), if I = (i_{1}, …, i_{r}) is generic, for i ∉ I,

(2.83) ⟨p_{r}^{I}α_{i}, x^{I}⟩ = x_{i} − ∑_{k=1}^{r} \frac{⟨α_{i_{1}}, …, α_{i_{k−1}}, α_{i}, α_{i_{k+1}}, …, α_{i_{r}}⟩}{⟨α_{i_{1}}, …, α_{i_{r}}⟩}x_{i_{k}}
= x_{i} − ⟨α_{i}, x^{I}⟩ .

Using (2.83), we get (2.62).

The proof of our Theorem is completed.

Remark 2.20. If we had assumed instead that e_{1}, …, e_{r} is a simple basis of R_{u,+}, we would have obtained a better equality in (2.62) , with t ∈ (𝔱/\overline{R}) \ H_{u}. However, it is essential here that we have use the same simple basis e_{1}, …, e_{r} of R_{+} for all the u ∈ C/\overline{R}^{∗} simultaneously. Finally observe that Theorem 2.19 can possibly be reformulated using the formalism of Szenes [52].

Remark 2.21. Take n ∈ N^{∗}. Then we have the exact sequence

\xymatrix{0 → \overline{R}/d\overline{R} \ar[r]_{n} & \overline{R}/dn\overline{R} \ar[r]_{p} & \overline{R}/n\overline{R} → 0} . (2.84)

In (2.84), the map n is just multiplication by n, and p is the obvious projection. Now in (2.62), we may replace d by nd. We get

Q_{u}(t, x) = n^{ℓ_{u}−r}(−1)^{r} ∑_{{I=(i_{1},... ,ir)⊂Iu,I generic \atop f∈\overline{R}/d\overline{R},h∈\overline{R}/n\overline{R}}} (2.85)

\frac{1}{⟨α_{i_{1}}, …, α_{i_{r}}⟩} exp\{d ∑_{j=1}^{r} \frac{⟨p_{j−1}^{I}α_{i_{j}}, nx^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩}

[\frac{1}{d}⟨p_{j−1}^{I}(\frac{(t + h)}{n} + f), e^{j}⟩]\}\frac{1}{∏_{i∈I_{u}\I}(⟨α_{i}, nx^{I}⟩ − nx_{i})}

\frac{1}{∏_{j=1}^{r} (exp (\frac{d⟨p_{j−1}^{I}α_{i_{j}} ,nx^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}} ,e^{j}⟩}) − 1)} .

Comparing (2.62) and (2.85) gives the identity (2.56), with v = 0.

By definition, Q_{u}(t, x) is well-defined on 𝔱/\overline{R}. However it is not entirely clear that the right-hand side of (2.62) is indeed well-defined on 𝔱/\overline{R}. We will now check this fact directly.

Theorem 2.22. As a function of t ∈ 𝔱, the right-hand side of (2.62) descends to a function on 𝔱/\overline{R}.

Proof. By Proposition 2.2, ⟨α_{i_{1}}, …, α_{i_{k}}⟩p_{k}^{I}e_{k} is an integral linear combination of the (e_{ρ})_{ρ≥k+1}. Therefore for k ≤ j − 1, dp_{j−1}^{I}e_{k} is an integral linear combination of the (p_{j−1}^{I}e_{ρ})_{ρ≥k+1}. So for k ≤ j − 1, d⟨p_{j−1}^{I}e_{k}, e^{j}⟩ is an integral linear combination of the ⟨p_{j−1}^{I}e_{k′}, e^{j}⟩(k + 1 ≤ k′ ≤ j).

To prove that the right-hand side of (2.62) is well-defined on (𝔱/\overline{R}) \ H, we only need to show that if we add to t an element of d\overline{R}, the total expression does not change.

Take f ∈ \overline{R}/d\overline{R}. Then f is uniquely represented by an element we also note f,

f = ∑_{1}^{r} f^{k}e_{k} , 0 ≤ f^{k} < d . (2.86)

Clearly

⟨p_{j−1}^{I}(t + f), e^{j}⟩ = ⟨p_{j−1}^{I}(t + ∑_{1}^{j} f^{k}e_{k}), e^{j}⟩ . (2.87)

Put

A_{j} = [\frac{1}{d}⟨p_{j−1}^{I}(t + f), e^{j}⟩] . (2.88)

Since p_{j−1}^{I}e_{j} = e_{j}, ⟨p_{j−1}^{I}e_{j}, e^{j}⟩ = 1. Therefore if we add to f^{j} an integral multiple of d, A_{j} is unchanged.

By the above, if we add to f^{r} an integral multiple of d, the right-hand side of (2.62) is unchanged. If we add to f^{r−1} an integral multiple of d, then only the term A_{r} is possibly affected. However as we saw before, d⟨p_{r−1}^{I}e_{r−1}, e^{r}⟩ is an integral multiple of ⟨p_{r−1}^{I}e_{r}, e^{r}⟩ = 1, i.e.

d⟨p_{r−1}^{I}e_{r−1}, e^{r}⟩ = q⟨p_{r−1}^{I}e_{r}, e^{r}⟩ , q ∈ Z . (2.89)

Therefore adding to f^{r−1} an integral multiple of d is equivalent to adding to f^{r} an integer. This show the right-hand side of (2.62) is invariant under this change.

A trivial downward recursion procedure shows that Q_{u}(t, x) is indeed well-defined on 𝔱/\overline{R}.

The proof of our Theorem is completed.

2.5. The Fourier transform on quotient of lattices. Let Δ, Δ′ be lattices in 𝔱, with Δ ⊂ Δ′. Then there is a projection 𝔱/Δ → 𝔱/Δ′. Let f ∈ D′(t/Δ). For μ ∈ Δ^{∗}/Δ′^{∗}, put

f̂_{μ}(t) = \frac{1}{|\frac{Δ′}{Δ} |} ∑_{k∈Δ′/Δ} e^{−2iπ⟨μ,k⟩}f(t + k) . (2.90)

Then f̂_{μ} ∈ D′(t/Δ). Moreover if k ∈ Δ′/Δ,

f̂_{μ}(t + k) = exp(2iπ⟨μ, k⟩)f̂_{μ}(t) . (2.91)

Also

f(t) = ∑_{μ∈Δ^{∗}/Δ′^{∗}} f̂_{μ}(t) . (2.92)

By (2.90)

f(t + k) = ∑_{μ∈Δ^{∗}/Δ′^{∗}} exp(2iπ⟨μ, k⟩)f̂_{μ}(t) . (2.93)

Equation (2.93) is just an aspect of Fourier transform. Note here that if A ⊂ 𝔱/Δ is such that f is C^{∞} on (𝔱/Δ) \ A, then f̂_{μ} is C^{∞} on (𝔱/Δ) \ ⋃_{k∈Δ′/Δ}(A + k). Then

formula (2.93) only expresses f as a function which is smooth on (𝔱/Δ)\ ⋃_{k∈Δ′/Δ}(A+

k), i.e. there is a loss of regularity in (2.93).

2.6. Fourier series on T. In the constructions of Section 2.3 , we may as will replace R by CR, \overline{R} by \overline{CR}, \overline{R}^{∗} by \overline{CR}^{∗}. For u ∈ C/\overline{R}^{∗}, we still define I_{u}, R^{I_{u}} as in (2.32), (2.34). However in (2.36), a_{u} : R^{I_{u}} → 𝔱^{∗} ≃ 𝔱 is replaced by b_{u} : R^{I_{u}} → 𝔱 ≃ 𝔱^{∗} given by

b_{u}(t) = ∑_{i∈I_{u}} t^{i}h_{α_{i}} . (2.94)

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS 45

Clearly b_{u}(Z^{I_{u}}) = \overline{CR_{u}} ⊂ \overline{CR} . (2.95) Therefore there is a a surjection b_{u} : (R/Z)^{I_{u}} → T = 𝔱/\overline{CR}. Then we have an exact sequence

\xymatrix{0 → L_{u} → (R/Z)^{I_{u}} \ar[r]_{b_{u}} & T → 0} . (2.96)

Definition 2.23. For u ∈ C/\overline{R}^{∗}, let S_{u} ⊂ T = 𝔱/\overline{CR} be given by

(2.97) S_{u} = \{t ∈ 𝔱/\overline{CR}, t = ∑_{j∈J ⊂I_{u}} t^{j}h_{α_{j}}, and \{h_{α_{j}}, j ∈ J \} do not span 𝔱\} .

Put S = S_{0} . (2.98) Then

S = \{t ∈ T = t/\overline{CR}, t = ∑_{j∈J ⊂\{1,... ,ℓ\}} t^{j}h_{α_{j}}, and \{h_{α_{j}}, j ∈ J \} do not span 𝔱\} . (2.99)

As in (2.46) , for any u ∈ C/\overline{R}^{∗}, S_{u} ⊂ S . (2.100)

By (1.12) , it is clear that if λ ∈ \overline{CR}^{∗}, α ∈ R_{ℓ}, β ∈ R, ⟨λ, α⟩ ∈ Z , (2.101)

m⟨λ, β⟩ ∈ Z .

In the sequel, when G is simply laced, we will make m = 1. Definition 2.24. For u ∈ C/\overline{R}^{∗}, x ∈ (C \ (2iπ \frac{Z}{m})^{ℓ}, t ∈ T = 𝔱/\overline{CR}, put

R_{u}(t, x) = ∑_{λ∈\overline{CR}^{∗}} \frac{exp(2iπ⟨λ, t⟩)}{∏_{i∈I_{u}}(2iπ⟨α_{i}, λ⟩ − x_{i})} . (2.102)

Clearly, by (1.8), (1.10),

R_{u}(t, x) = m^{|R_{u,+}∩R_{s}|} ∑_{λ∈\overline{CR}^{∗}} \frac{exp(2iπ⟨λ, t⟩)}{∏_{i∈I_{u}}(2iπ⟨h_{α_{i}}, λ⟩ − \frac{∥h_{α_{i}} ∥^{2}}{2} x_{i})} . (2.103)

Of course in (2.103) , \frac{∥h_{α_{i}}∥^{2}}{2} = 1 or m. From (2.103) , it should now be clear that Theorem 2.18 can be applied to R_{u}(t, x). In particular on compact subsets of T \ S_{u} × (C \ \frac{2iπZ}{m})^{ℓ}, R_{u}(t, x) is a smooth function of (t, x).

If G is simply laced, the objects we just constructed are the ones we already obtained in Section 2.3.

Now we use the notation in (2.35). Put a = a_{0} . (2.104) Then if s ∈ R^{ℓ},

as = ∑_{i=1}^{ℓ} s^{i}α_{i} . (2.105)

Let ã : 𝔱 ≃ 𝔱^{∗} → R^{ℓ} be the transpose of a. Then

ãt = (⟨α_{1}, t⟩, …, ⟨α_{ℓ}, t⟩) , (2.106)

and ã maps \overline{R}^{∗} into Z^{ℓ}.

Now we will use (2.90)-(2.93), with Δ = \overline{CR}, Δ′ = \overline{R}. In the sequel, we view R_{u}(t, x) as an element of D′(𝔱/\overline{CR}). If μ ∈ \overline{CR}^{∗}/\overline{R}^{∗}, we define (R̂_{u})_{μ}(t, x) as in (2.90).

Proposition 2.25. If μ ∈ \overline{CR}^{∗}/\overline{R}^{∗}, if λ_{μ} ∈ \overline{CR}^{∗} represents μ, then

(R̂_{u})_{μ}(t, x) = exp(2iπ⟨λ_{μ}, t⟩)Q_{u}(t, x − 2iπãλ_{μ}) . (2.107)

Proof. If λ ∈ \overline{CR}^{∗}, then

(2.108) { \atop exp(2iπ⟨λ, t⟩)_{μ}} = exp(2iπ⟨λ, t⟩) if λ ∈ \overline{CR}^{∗} maps to μ ∈ \overline{CR}^{∗}/\overline{R}^{∗} ,

= 0 otherwise .

Then

(R̂_{u})_{μ}(t, x) = ∑_{λ∈\overline{R}^{∗}} \frac{exp(2iπ⟨λ + λ_{μ}, t⟩)}{∏_{i∈I_{u}}(2iπ⟨α_{i}, λ⟩ − (x_{i} − 2iπ⟨α_{i}, λ_{μ}⟩)} . (2.109)

From (2.109) , we get (2.107). The proof of our Proposition is completed.

Remark 2.26. By Proposition 2.25 , we get the otherwise obvious fact that the right-hand side of (2.107) only depends on μ and not on λ_{μ}.

For any μ ∈ \overline{CR}^{∗}/\overline{R}^{∗}, we choose λ_{μ} ∈ \overline{CR}^{∗} representing μ.

Theorem 2.27. The following identity holds

R_{u}(t, x) = ∑_{μ∈\overline{CR}^{∗}/\overline{R}^{∗}} exp(2iπ⟨λ_{μ}, t⟩)Q_{u}(t, x − 2iπãλ_{μ}) . (2.110)

Proof. This follows from (2.92) and (2.107).

2.7. Iterated residues and the series R_{u}(t, x). Let τ : 𝔱/\overline{CR} → t/\overline{R} be the obvious projection. For u ∈ C/\overline{R}^{∗}, put

(2.111)  \overline{S}_{u} = τ^{−1}(H_{u})
\overline{S} = τ^{−1}(H) .

Clearly τ maps S_{u} into H_{u}, S into H. Therefore

(2.112)  S_{u} ⊂ \overline{S}_{u} ,
S ⊂ \overline{S} .

If G is simply laced

(2.113)  S_{u} = \overline{S}_{u} ,
S = \overline{S} .

Recall that by (2.20), d\overline{R} ⊂ \overline{CR}.

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS 47

Theorem 2.28. For any u ∈ C/\overline{R}^{∗}, t ∈ T \ \overline{S}, then

R_{u}(t, x) = |\frac{\overline{R}}{\overline{CR}}| (−1)^{r} ∑_{{I=(i_{1},... ,i_{r})⊂I_{u} \atop I generic} ; f∈\overline{CR}/d\overline{R}} \frac{1}{⟨α_{i_{1}}, …, α_{i_{r}}⟩} (2.114)

\frac{1}{∏_{i∈I_{u}\I}(⟨α_{i}, x^{I}⟩ − x_{i})} exp \{d ∑_{j=1}^{r} \frac{⟨p_{j−1}^{I}α_{i_{j}}, x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩}[\frac{1}{d}⟨p_{j−1}^{I}(t + f), e^{j}⟩]\}

∏_{j=1}^{r} \frac{1}{exp(\frac{d⟨p_{j−1}^{I}α_{i_{j}} ,x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}} ,e^{j}⟩}) − 1}.

Proof. We use Theorems 2.19 and 2.27 . Also, if h ∈ 𝔱, (ãh)^{I} = h . (2.115) So we deduce from (2.115) that if i ∈ I_{u} \ I, ⟨α_{i}, (ãλ_{μ})^{I}⟩ − ⟨α_{i}, λ_{μ}⟩ = 0 . (2.116) By (2.19), (2.28), if λ ∈ \overline{CR}^{∗},

\frac{d⟨p_{j−1}^{I}α_{i_{j}}, ãλ⟩}{⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩} ∈ Z. (2.117)

From (2.13), (2.62), (2.110), (2.116), (2.117), we get (2.114). The proof of our Theorem is completed. 2.8. Fourier series for the universal cover of a semisimple centralizer. Take u ∈ C/\overline{R}^{∗}. Recall that by π_{1}(Z(u)) = \overline{CR}/\overline{CR_{u}}. (2.118) Also \overline{CR} ⊂ \overline{R}_{u}^{∗} . (2.119)

Let Z̃(u) be the universal cover of Z(u). Then by [15, Theorem V.7.1], Z(Z̃(u)) = \overline{R}_{u}^{∗}/\overline{CR_{u}} . (2.120) Therefore π_{1}(Z(u)) = \overline{CR}/\overline{CR_{u}} is a subgroup of Z(Z̃(u)). Also \overline{CR}_{u}^{∗} is the lattice of weights of Z̃(u). Definition 2.29. For u ∈ C/\overline{R}^{∗}, t ∈ 𝔱/\overline{CR}_{u}, x ∈ (C \ \frac{2iπZ}{m})^{ℓ}, put

R̃_{u}(t, x) = ∑_{λ∈\overline{CR}_{u}^{∗}} \frac{exp(2iπ⟨λ, t⟩)}{∏_{i∈I_{u}}(2iπ⟨λ, α_{i}⟩ − x_{i})} . (2.121)

Clearly π_{1}(Z(u))^{∗} = \overline{CR}_{u}^{∗}/\overline{CR}^{∗} . (2.122) Also \overline{R}_{u}^{∗}/\overline{R}^{∗} maps into \overline{CR}_{u}^{∗}/\overline{CR}^{∗}, with kernel \frac{\overline{R}_{u}^{∗} ∩ \overline{CR}^{∗}}{\overline{R}^{∗}}. In particular u ∈ C/\overline{R}^{∗} maps to an element of π_{1}(Z(u))^{∗}.

48 JEAN-MICHEL BISMUT AND FRANÇOIS LABOURIE

Definition 2.30. For u ∈ C/\overline{R}^{∗}, q ∈ Z, t ∈ T = 𝔱/\overline{CR}, put

R_{u,q}(t, x) = \frac{1}{|\frac{\overline{CR}}{\overline{CR}_{u}} |} ∑_{h∈ \frac{\overline{CR}}{\overline{CR}u}} exp(2iπ⟨qu, h⟩)R̃_{u}(t + h, x) . (2.123)

Proposition 2.31. If u ∈ C, the following identity holds R_{u,q}(t, x) = R_{u}(t, x + 2iπqãu) exp(−2iπ⟨qu, t⟩) . (2.124) Proof. Since u ∈ \overline{CR}_{u}^{∗}, our identity follows from (2.123). Theorem 2.32. For any u ∈ C/\overline{R}^{∗}, q ∈ Z, t ∈ T \ \overline{S}, then

(2.125) R_{u,q}(t, x) = |\frac{\overline{R}}{\overline{CR}}| (−1)^{r} ∑_{{I=(i_{1},... ,i_{r})⊂I_{u} \atop I generic} ; f∈\overline{CR}/d\overline{R}} \frac{1}{⟨α_{i_{1}}, …, α_{i_{r}}⟩} exp(2iπ(⟨qu, f⟩))

\frac{1}{∏_{i∈I_{u}\I}(⟨α_{i}, x^{I}⟩ − x_{i})} exp \{d ∑_{j=1}^{r} \frac{⟨p_{j−1}^{I}α_{i_{j}}, x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩}[\frac{1}{d}⟨p_{j−1}^{I}(t + f), e^{j}⟩]\}

∏_{j=1}^{r} \frac{1}{exp(\frac{d⟨p_{j−1}^{I}α_{i_{j}} ,x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}} ,e^{j}⟩}) − 1}.

Proof. We use Theorem 2.28 and Proposition 2.31. Since for i ∈ I_{u}, ⟨u, α_{i}⟩ ∈ Z, by (2.13), (2.29), (2.116), we get (2.125). The proof of our Theorem is completed. 2.9. Bernouilli polynomials and the Fourier series P_{n}(t). Definition 2.33. For n ∈ N, t ∈ T = 𝔱/\overline{CR}, put

P_{n}(t) = − ∑_{{λ∈\overline{CR}^{∗} \atop π(λ)≠0}} \frac{exp(2iπ⟨λ, t⟩)}{[π(λ)]^{n}} . (2.126)

For M ∈ N^{∗}, we will consider the partial sums

P_{n}^{M}(t) = − ∑_{{λ∈\overline{CR}^{∗} \atop π(λ)≠0} ; |λ|≤M} \frac{exp(2iπ⟨λ, t⟩)}{[π(λ)]^{n}} . (2.127)

Clearly P_{n}(t) is a well-defined distribution on T = 𝔱/\overline{CR}. Recall that K = {t ∈ 𝔱, for α in R_{+}, ⟨α, t⟩ > 0} . (2.128) Then by [15, Note V.4.14] and (1.186),

(2.129)  P_{n}(t) = − ∑_{{λ∈\overline{CR}^{∗}∩\overline{K} \atop w∈W}} \frac{ε_{w} exp(2iπ⟨w(λ + ρ), t⟩)}{[π(λ + ρ)]^{n}} for n odd ,

= − ∑_{{λ∈\overline{CR}^{∗}∩\overline{K} \atop w∈W}} \frac{exp(2iπ⟨w(λ + ρ), t⟩)}{[π(λ + ρ)]^{n}} n even .

By (1.186), (2.126), for w ∈ W,

P_{n}(wt) = ε_{w}^{n}P (t) . (2.130)

Since π(t) is a polynomial, π (\frac{∂/∂t}{2iπ}) is a differential operator. Then we have the

identity of distributions on T = 𝔱/\overline{CR},

π (\frac{∂/∂t}{2iπ}) P_{n}(t) = P_{n−1}(t) for n ≥ 1 . (2.131)

Definition 2.34. For t ∈ [0, 1], n ≥ 0, put

p_{n}(t) = − ∑_{k≠0} \frac{e^{2iπkt}}{(2iπk)^{n}} . (2.132)

If G = SU(2), then T ≃ S_{1} ≃ R/Z. One verifies easily that the p_{n}(t)’s are exactly the P_{n}’s associated to G = SU(2). Then (2.131) is the equation of distribution on S_{1}

p′_{n}(t) = p_{n−1}(t) , n ≥ 1 . (2.133)

Also

p_{0}(t) = 1 − δ_{{0}} , (2.134)

So by (2.133), (2.134), we get

(\frac{d}{dt})^{n} p_{n}(t) = 1 − δ_{\{0\}} . (2.135)

By (2.135) , it is clear that the p_{n}(t)’s are polynomials on S_{1} \ {0}. Also for n ≥ 2, the series in (2.132) is absolutely convergent on [0, 1]. For n ≥ 1, the series in (2.132) converges uniformly together with its derivatives on compact sets of S_{1}, not containing 0.

By [43, Appendix B] , the p_{n}’s are exactly the Bernouilli polynomials. In the sequel, we will consider the p_{n}’s as polynomials on R, whose restriction to ]0, 1[ is given by (2.132).

Recall that

Td(x) = \frac{x}{1 − e^{−x}} . (2.136)

Then

Td(x) − Td(−x) = 1 . (2.137)

Finally by [49, p147] , if the B_{k}, k ≥ 1, are the Bernoulli numbers,

Td(x) = 1 + \frac{x}{2} + ∑_{k=1}^{+∞}(−1)^{k+1}B_{k} \frac{x^{2k}}{(2k)!} . (2.138)

Proposition 2.35. For n ≥ 0, t ∈ R,

p_{n}(t) = Res_{a=0} [\frac{e^{ta}}{a^{n}} \frac{1}{e^{a} − 1}] . (2.139)

Proof. For 0 ≤ t ≤ 1, put

f_{t,n}(a) = \frac{e^{ta}}{a^{n}} \frac{1}{e^{a} − 1} . (2.140)

Clearly f_{t,n}(a) has simple poles at a = 2iπk, k ∈ Z^{∗}, and a pole of order n + 1 at a = 0. Then, for k ∈ Z^{∗}

Res_{a=2iπk}f_{t,n}(a) = \frac{e^{2iπkt}}{(2iπk)^{n}} . (2.141)

Now we use the Cauchy residue theorem inside a circle of centre 0 and radius 2π(M + 1/2), as M → +∞. For n ≥ 2, or for n = 1, t ∈]0, 1[, the integral of f_{t,n}(a) on the circle tends to 0 as M → +∞. So we find that

∑_{k∈Z} Res_{a=2iπk}f_{t,n}(a) = 0 . (2.142)

From (2.142) , we get (2.139) for n ≥ 2, or n = 1, 0 < t < 1. Then since p_{n}(t) is a polynomial, (2.139) holds for any t ∈ R. For n = 0, (2.139) is trivial. The proof of our Proposition is completed.

Proposition 2.36. For n ≥ 0, t ∈ R,

p_{n}(t) = Td(−∂/∂t)\frac{t^{n}}{n!} , (2.143)

p_{n+1}(t + 1) − p_{n+1}(t) = \frac{t^{n}}{n!} .

Proof. Clearly,

\frac{∂}{∂t}e^{at} = ae^{at} . (2.144)

By (2.144) , we find that for |a| < 2π,

Td(−∂/∂t)e^{at} = Td(−a)e^{at} . (2.145)

Using (2.139), (2.145), we get

(2.146)  p_{n}(t) = Res_{a=0} [\frac{e^{ta}}{a^{n+1}} Td(−a)] = Td(−∂/∂t)Res_{a=0} [\frac{e^{ta}}{a^{n+1}} ]

= Td(−∂/∂t)\frac{t^{n}}{n!} .

By (2.139)

p_{n+1}(t + 1) − p_{n+1}(t) = Res [\frac{e^{ta}}{a^{n+1}} ] = \frac{t^{n}}{n!} . (2.147)

The proof of our Proposition is completed.

Recall that R_{s,+} denote the set of short positive roots.

Theorem 2.37. For n ≥ 1, as M → +∞, the partial sums P_{n}^{M}(t) converge to P_{n}(t) uniformly on the compact subsets of T \ S. The following identity of distributions holds on T = 𝔱/\overline{CR},

P_{n}(t) = (−1)^{ℓ+1}m^{n|R_{s,+}|}b_{∗} [∏_{i=1}^{ℓ} p_{n}(t^{i})] . (2.148)

In particular P_{n}(t) ∈ L_{∞}(T). Also (2.148) is an identity of smooth functions on T \ S. Finally P_{n}(t) is a polynomial on T \ S.

Proof. By proceeding as in the proof of Theorem 2.18 , we get (2.148). By the same argument as in Theorem 2.18 , we find that P_{n}(t) is smooth on T \ S. Also the uniform convergence results for the p_{n}^{M}(t)’s and (2.148) imply the corresponding uniform convergence for P_{n}^{M}(t).

Now, we will show that P_{n}(t) is a polynomial on T \ S. In fact we will prove that for 1 ≤ i ≤ ℓ, for p large enough,

h_{α_{i}}^{p}P_{n}(t) = 0 on T \ S . (2.149)

Clearly b_{∗}∂/∂t_{i} = h_{α_{i}}. Then using (2.135) , we obtain

(2.150)  h_{α_{1}}^{n}b_{∗}[p_{n}(t^{1}) … p_{n}(t^{ℓ})] = b_{∗}[p_{n}(t^{2}) … p_{n}(t^{ℓ})]
− b_{∗}[δ_{t_{1}=0}p_{n}(t^{2}) … p_{n}(t^{ℓ})] .

Clearly

h_{α_{1}}b_{∗}[p_{n}(t^{2}) … p_{n}(t^{ℓ})] = 0 . (2.151)

Let V_{1} be the vector space in 𝔱 spanned by h_{α_{2}}, …, h_{α_{ℓ}}. If V_{1} is not equal to 𝔱, the support of b_{∗}[δ_{t_{1}=0}p_{n}(t^{2}) … p_{n}(t^{ℓ})] is included in S. From (2.150) , we then get

h_{α_{1}}^{n+1}b_{∗}[p_{n}(t^{1}) … p_{n}(t^{ℓ})] = 0 on T \ S . (2.152)

If V_{1} = 𝔱, we can express h_{α_{1}} in the form

h_{α_{1}} = ∑_{j=2}^{ℓ} a^{j}h_{α_{j}} . (2.153)

By (2.135),

(2.154)  h_{α_{2}}^{n}b_{∗}[δ_{t_{1}=0}p_{n}(t^{2}) … p_{n}(t^{ℓ})] = b_{∗}[δ_{t_{1}=0}p_{n}(t^{3}) … p_{n}(t^{ℓ})]
− b_{∗}[δ_{t_{1}=0,t_{2}=0}p_{n}(t^{3}) … p_{n}(t^{ℓ})] .

Let V_{2} be the vector space spanned by h_{α_{2}}, …, h_{α_{ℓ}}. If V_{2} ≠ 𝔱, then by the analogue of (2.152),

h_{α_{2}}^{n+1}q_{∗}[δ_{t_{1}=0}p_{n}(t^{2}) … p_{n}(t^{ℓ})] = 0 on T \ S. (2.155)

Now using (2.155) , and iterating the above argument, it should be clear that for p ∈ N large enough,

h_{α_{1}}^{p}P_{n}(t) = 0 on T \ S . (2.156)

In (2.156), we may as well replace the index 1 by the index i, 1 ≤ i ≤ ℓ. Therefore we have established (2.149).

The proof of our Theorem is completed.

2.10. The Fourier series P̃_{u,n}(t). Take u ∈ C/\overline{R}^{∗}. Recall that Z̃(u) is the universal cover of Z(u). Then Z̃(u) is connected and simply connected. Therefore to Z̃(u), we can associate the objects we just constructed for G. Observe that 𝔱/\overline{CR}_{u} is a maximal torus in Z̃(u), and 𝔱 is its Lie algebra.

Definition 2.38. Let π_{u}(t) be the function on 𝔱

π_{u}(t) = ∏_{α∈R_{u,+}}⟨2iπα, t⟩ . (2.157)

Then π_{u}(t) is the analogue of π(t).

Definition 2.39. For u ∈ C/\overline{R}^{∗}, n ∈ N, t ∈ 𝔱/\overline{CR_{u}}, put

P̃_{u,n}(t) = − ∑_{{λ∈\overline{CR}_{u}^{∗} \atop π_{u}(λ)≠0}} \frac{exp(2iπ⟨λ, t⟩)}{[π_{u}(λ)]^{n}} . (2.158)

Then P̃_{u,n}(t) is the analogue of P_{n}(t) for Z̃(u). Take w ∈ W. We identify w with a representative in N(T)/T. Then if u ∈ C/\overline{R}^{∗},

(2.159)  Z(wu) = w Z(u)w^{−1} ,
\overline{CR}_{wu} = w\overline{CR}_{u},
R_{Z(wu)} = wR_{u},
R_{Z(wu),+} = wR_{u} ∩ R_{+} .

By (2.159), we get

π_{wu}(t) = (−1)^{|R_{+}∩w(−R_{u,+})|}π_{u}(w^{−1}t) . (2.160)

Proposition 2.40. If u ∈ C/\overline{R}^{∗}, n ∈ N, w ∈ W, then

P̃_{u,n}(t) = (−1)^{n|R_{+}∩w(−R_{u,+})|}P̃_{wu,n}(wt) . (2.161)

Proof. Clearly by (2.159), (2.160),

(2.162)  P̃_{u,n}(t) = − ∑_{{λ∈\overline{CR}{∗ \atop wu} \atop πu(w^{−1}λ)≠0}} \frac{e^{2iπ⟨λ,wt⟩}}{[π_{u}(w^{−1}λ)]^{n}}

= −(−1)^{n|R_{+}∩w(−R_{u,+})|} ∑_{{λ∈\overline{CR}{∗ \atop wu} \atop πwu(λ)≠0}} \frac{e^{2iπ⟨λ,wt⟩}}{[π_{wu}(λ)]^{n}}

= (−1)^{n|R_{+}∩w(−R_{u,+})|}P̃_{wu,n}(wt),

which coincides with (2.161).

Let S̃_{u} ⊂ 𝔱/\overline{CR_{u}} be the analogue of S ⊂ 𝔱/\overline{CR}. Of course S̃_{u} projects into S_{u}. Then by Theorem 2.37 , P̃_{u,n}(t) is polynomial on (𝔱/\overline{CR_{u}}) \ S̃_{u}.

Let i_{1}, …, i_{ℓ_{u}} be the elements of I_{u} = {i ∈ {1, …, ℓ}, α_{i} ∈ R_{u,+}} arranged in increasing order, i.e. i_{1} < i_{2} … < i_{ℓ_{u}}.

If f(x) = f(x_{i_{1}}, …, x_{i_{ℓu}}) is a meromorphic function of x ∈ R^{I_{u}}, we denote by Res_{x=0}^{I_{u}}f the expression

(2.163) Res_{x=0}^{I_{u}}f = Res_{x_{i_{ℓu}} =0} … Res_{x_{i_{1}} =0}f(x_{i_{1}}, …, x_{i_{ℓu}}) .

It will be of fundamental importance that the order in (2.163) is fixed once and for all.

Theorem 2.41. For u ∈ C/\overline{R}^{∗}, n ≥ 1,

P̃_{u,n}(t) = −Res_{x=0}^{I_{u}} \frac{1}{( ∏_{i∈I_{u}} x_{i})^{n}} R̃_{u}(t, x) . (2.164)

Proof. Clearly

(2.165)  Res_{x_{i}=0} \frac{1}{x_{i}^{n}} \frac{1}{2iπ⟨α_{i}, λ⟩ − x_{i}} = \frac{1}{(2iπ⟨α_{i}, λ⟩)^{n}} if ⟨α_{i}, λ⟩ ≠ 0,
= 0 if ⟨α_{i}, λ⟩ = 0 .

Using (2.165) , we get (2.164). The proof of our Theorem is completed.

Remark 2.42. The identity (2.164) can be viewed as an identity of distributions on 𝔱/\overline{CR_{u}}, of smooth functions on (𝔱/\overline{CR_{u}}) \ S̃_{u}, of elements of L^{∞}(𝔱/\overline{CR_{u}}). Also for the moment, the ordering in (2.164) is still irrelevant.

2.11. The function P_{u,n,q}(t). If t ∈ 𝔱, we still denote by t the corresponding element in T = 𝔱/\overline{CR}.

Definition 2.43. For u ∈ C/\overline{R}^{∗}, q ∈ Z, n ∈ N, t ∈ 𝔱/\overline{CR}, put

P_{u,n,q}(t) = \frac{1}{|\frac{\overline{CR}}{\overline{CR_{u}}} |} ∑_{h∈ \frac{\overline{CR}}{\overline{CRu}}} exp(2iπ⟨qu, h⟩)P̃_{u,n}(t + h) . (2.166)

Observe that the function exp(2iπ⟨qu, t⟩)P_{u,n,q}(t) descends to a well defined function on T = 𝔱/\overline{CR}. Equivalently, we may consider P_{u,n,q}(t) as a section of the flat line bundle L_{u} on T, associated to u ∈ 𝔱^{∗}/\overline{CR}^{∗}.

Proposition 2.44. If u ∈ C/\overline{R}^{∗}, q ∈ Z, n ∈ N, w ∈ W, then

P_{u,n,q}(t) = (−1)^{n|R_{+}∩w(−R_{u,+})|}P_{wu,q,n}(wt) . (2.167)

Proof. Clearly w : \frac{\overline{CR}}{\overline{CR_{u}}} → \frac{\overline{CR}}{\overline{CR}_{wu}} is one to one. Then (2.167) follows from Propo-

sition 2.40.

Theorem 2.45. The section P_{u,n,q}(t) is a polynomial on T \ S_{u}.

Proof. Since P̃_{u,n}(t) is polynomial on (𝔱/\overline{CR_{u}}) \ S̃_{u}, it is clear from (2.166) that P_{u,n,q} is polynomial on T \ S_{u}.

Theorem 2.46. For u ∈ C/\overline{R}^{∗}, n ∈ N, q ∈ Z, t ∈ 𝔱/\overline{CR}, the following identity holds

P_{u,n,q}(t) = −Res_{x=0}^{I_{u}} \frac{1}{( ∏_{i∈I_{u}} x_{i})^{n}} R_{u,q}(t, x) . (2.168)

Proof. This follows from (2.123), (2.164), (2.166) .

Definition 2.47. For u ∈ C/\overline{R}^{∗}, let I_{u} be the set of generic I = (i_{1}, …, i_{r}) ⊂ I_{u} such that if σ^{I} ∈ S_{r} is defined by i_{σ^{I}(1)} < i_{σ^{I}(2)} < … < i_{σ^{I}(r)}, if j ∈ I_{u} \ I, either j < σ^{I}(1), or if p_{j} is defined by the condition

i_{σ^{I}(1)} < … < i_{σ^{I}(p_{j})} < j < i_{σ^{I}(p_{j}+1)} < …, (2.169)

then

α_{i_{σ^{I} (1)}} ∧ … ∧ α_{i_{σ^{I} (p_{j})}} ∧ α_{j} ≠ 0 . (2.170)

54 JEAN-MICHEL BISMUT AND FRANÇOIS LABOURIE

Condition (2.170) is equivalent to α_{j} ∉ {α_{i_{σ^{I} (1)}}, …, α_{i_{σ^{I} (p_{j})}}} . (2.171)

If I ∈ I_{u}, we define Res_{x=0}^{I} by the formula (2.172) Res_{x=0}^{I} = Res_{x_{i_{σ(r)}} =0} … Res_{x_{i_{σ(1)}} =0} . Theorem 2.48. For u ∈ C/\overline{R}^{∗}, n ≥ 1, q ∈ N, if t ∈ 𝔱 represents an element of T \ \overline{S}, then

P_{u,n,q}(t) = |\frac{\overline{R}}{\overline{CR}}| (−1)^{r+1} ∑_{{I=(i_{1},... ,ir)∈Iu \atop f∈\overline{CR}/d\overline{R}}} \frac{1}{⟨α_{i_{1}}, …, α_{i_{r}}⟩} exp(2iπ(⟨qu, f⟩)) (2.173)

Res_{x=0}^{I} (\frac{1}{∏_{α∈R_{u,+}}⟨α, x^{I}⟩})^{n} exp\{d ∑_{j=1}^{r} \frac{⟨p_{j−1}^{I}α_{i_{j}}, x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩}

[\frac{1}{d}⟨p_{j−1}^{I}(t + f), e^{j}⟩]\} ∏_{j=1}^{r} \frac{1}{exp(\frac{d⟨p_{j−1}^{I}α_{i_{j}} ,x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}} ,e^{j}⟩}) − 1}.

Proof. By Theorems 2.32 and 2.46, for t ∈ T \ \overline{S}, we get

P_{u,n,q}(t) = |\frac{\overline{R}}{\overline{CR}} | (−1)^{r+1} ∑_{I=(i_{1},... ,ir)⊂Iu ; {Igeneric \atop f∈\overline{CR}/d\overline{R}}} \frac{1}{⟨α_{i_{1}}, …, α_{i_{r}}⟩} exp(2iπ(⟨qu, f⟩))

Res_{x=0}^{I_{u}} \frac{1}{( ∏_{i∈I_{u}} x_{i})^{n}} \frac{1}{∏_{i∈I_{u}\I}(⟨α_{i}, x^{I}⟩ − x_{i})} exp\{d ∑_{j=1}^{r} \frac{⟨p_{j−1}^{I}α_{i_{j}} ,x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}} ,e^{j}⟩} (2.174)

[\frac{1}{d}⟨p_{j−1}^{I}(t + f), e^{j}⟩]\} ∏_{j=1}^{r} \frac{1}{exp \frac{d⟨p_{j−1}^{I}α_{i_{j}} ,x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}} ,e^{j}⟩}) − 1}.

Now x^{I} depends only on x_{i_{1}}, …, x_{i_{r}}. Therefore in (2.174), the dependence on the

(x_{i})_{i∈I_{u}\I} is only via the term \frac{1}{∏_{i∈I_{u}\I} x_{i}^{n}(⟨α_{i}, x^{I}⟩ − x_{i})}.

Take ϵ ≥ 0. Let v ∈ C, |v| ≪ ϵ, w ∈ C, |w| > ϵ. Then by the theorem of residues, for n ≥ 1,

\frac{1}{2πi} ∫_{{x_{j} ∈C \atop |x_{j} |=ϵ}} \frac{dx_{j}}{x_{j}^{n}(v + w − x_{j})} = \frac{1}{(v + w)^{n}} , (2.175)

\frac{1}{2πi} ∫_{{x_{j} ∈C \atop |x_{j} |=1}} \frac{dx_{j}}{x_{j}^{n}(v − x_{j})} = 0 .

Take j ∈ I_{u} \ I. If

α_{j} = ∑_{k=1}^{r} a^{k}α_{i_{σ^{I} (k)}} , (2.176)

then

⟨α_{j}, x^{I}⟩ = ∑_{k=1}^{r} a^{k}x_{i_{σ^{I} (k)}} . (2.177)

Put

v = ∑_{k=1}^{p_{j}} a^{k}x_{i_{σ^{I} (k)}} , w = ∑_{k=p_{j}+1}^{r} a^{k}x_{i_{σ^{I} (k)}} . (2.178)

Observe that w is identically 0 if and only if a^{k} = 0 for k ≥ p_{j} + 1, i.e. if α_{j} ∈ {α_{i_{σ^{I} (1)}}, …, α_{i_{σ^{I} (p_{j})}}}.

To evaluate (2.174), we will use (2.175). Namely take a sequence ϵ_{1}, …, ϵ_{ℓ_{u}} in R_{+}, with 0 < ϵ_{i_{1}} < … < ϵ_{ℓ_{u}}. Then if f(x_{i_{1}}, …, x_{i_{ℓu}}) is a meromorphic function, by definition,

Res_{x=0}^{I_{u}}f = ∫_{{|x_{i_{j}} |=ϵ_{i_{j}} \atop 1≤j≤ℓu}} f(x_{i_{1}}, …, x_{i_{ℓu}})\frac{dx_{i_{1}}}{2iπ} … \frac{dx_{i_{ℓu}}}{2iπ} . (2.179)

If the sequenceϵ_{1}, …, ϵ_{ℓ_{u}} is enough decreasing, when taking the x_{i_{j}} as in (2.179), if α_{j} ∉ {α_{i_{σ^{I} (1)}}, …, α_{i_{σ^{I} (r)}}}, in (2.178), then |w| > ϵ_{j}.

Using (2.175), (2.179), we find easily that for j ∈ I_{u} \ I,

(2.180) Res_{x_{j}=0} \frac{1}{x_{j}^{n}(⟨α_{j}, x^{I}⟩ − x_{j})} = 0  if α_{j} ∈ \{α_{i_{σ^{I} (1)}}, …, α_{i_{σ^{I} (p_{j})}}\} ,
=\frac{1}{⟨α_{j}, x^{I}⟩^{n}} if α_{j} ∉ \{α_{i_{σ^{I} (1)}}, …, α_{i_{σ^{I} (p_{j})}}\} .

A related argument is as follows. Define v, w by (2.178). If w = 0, then since the x_{i_{σ^{I} (k)}}, k ≤ p_{j} have been made nearly equal to 0 before x_{j},

\frac{1}{v + w − x_{j}} = − ∑_{k=0}^{+∞} \frac{v^{k}}{x_{j}^{k+1}} . (2.181)

From (2.181) , we get

Res_{x_{j}=0} [\frac{1}{x_{j}^{n}} \frac{1}{v + w − x_{j}} ] = 0 . (2.182)

If w ≠ 0, then

\frac{1}{v + w − x_{j}} = ∑_{k=0}^{+∞} \frac{(x_{j} − v)^{k}}{w^{k+1}} . (2.183)

By (2.183) , we get

Res_{x_{j}=0} \frac{1}{x_{j}^{n}} \frac{1}{v + w − x_{j}} = ∑_{k=n−1}^{+∞} \frac{C_{k}^{n−1}(−v)^{k−n+1}}{w^{k+1}} = \frac{1}{(v + w)^{n}} , (2.184)

which gives another proof of (2.180). Finally observe that if j ∈ I,

⟨α_{j}, x^{I}⟩ = x_{j} . (2.185)

From (2.174), (2.180), (2.185), we get (2.173). The proof of our Theorem is completed.

By Theorem 2.22, we already know that, as a section of a line bundle, P_{u,n,q}(t) is polynomial on T \\overline{S}. By Theorem 2.22, the right-hand side of (2.173) descends to a section of the same line bunlde . We will give a direct proof that the right-hand side of (2.173) is polynomial on T \ \overline{S}.

Theorem 2.49. The right-hand side of (2.173) is a polynomial on T \ \overline{S}.

Proof. In the right-hand side of (2.173), up to a locally constant factor on T \ \overline{S},

we may and will take off the brackets in [\frac{1}{d}⟨p_{j−1}^{I}(t + f), e^{j}⟩]. By (2.13),

∑_{j=1}^{r} \frac{⟨p_{j−1}^{I}α_{i_{j}}, x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩}⟨p_{j−1}^{I}(t + f), e^{j}⟩ = ⟨t + f, x^{I}⟩ . (2.186)

Then

⟨t + f, x^{I}⟩ = ∑_{1}^{r} F^{j}(t)x_{σ^{I}(j)} , (2.187)

and the F^{j}(t) are affine functions of t ∈ 𝔱. Therefore

exp(⟨t + f, x^{I}⟩) = ∏_{j=1}^{r} (∑_{k=0}^{+∞} \frac{(F^{j}(t)x_{σ^{I}(j)})^{k}}{k!} ) . (2.188)

Moreover

\frac{1}{exp (\frac{d⟨p_{j−1}^{I}α_{i_{j}} ,x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}} ,e^{j}⟩}) − 1} = \frac{Td (\frac{−d⟨p_{j−1}^{I}α_{i_{j}} ,x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}} ,e^{j}⟩})}{\frac{d⟨p_{j−1}^{I}α_{i_{j}} ,x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}} ,e^{j}⟩}} . (2.189)

Also by (2.28), (2.29),

\frac{⟨p_{j−1}^{I}α_{i_{j}}, x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩} = \frac{1}{⟨α_{i_{1}}, …, α_{i_{j}}⟩} (2.190)

(⟨α_{i_{1}}, …, α_{i_{j−1}}⟩x_{i_{j}} − ∑_{k=1}^{j−1}⟨α_{i_{1}}, …, α_{i_{k−1}}, α_{i_{j}}, α_{i_{k+1}}, …, α_{i_{j−1}}⟩x_{i_{k}}) .

Finally the ⟨α_{j}, x^{I}⟩ are linear combinations of x_{i_{1}}, …, x_{i_{r}}.

Freeze now x_{i_{σ^{I} (2)}}, …, x_{i_{σ^{I} (r)}}, and consider one of the terms in (2.173) as a function of x_{i_{σ^{I} (1)}}. Then it is clear that this term is a meromorphic function of x_{i_{σ^{I} (1)}} with a pole of finite order at x_{i_{σ^{I} (1)}=0}. When taking the residue at 0 in the variable x_{i_{σ^{I} (1)}}, by the above, it is clear this will introduce a finite power F^{1}(t)^{k}, i.e. a polynomial function of t.

It is now clear that the procedure can be iterated. The proof of our Theorem is completed.

2.12. The action of a differential operator on P_{u,n,q}(t). Let s ∈ 𝔱^{∗} ↦ F (s) ∈ C[s] be a power series, which converges on a neighborhood of 0. Then F (∂/∂t) is a formal power series of differential operators. Since P_{u,n}(t) is a polynomial on T \ S_{u}, F (∂/∂t)P_{u,n}(t) is a well-defined polynomial on T \ S_{u}.

Recall that we have identified 𝔱 and 𝔱^{∗}. Then if I = (i_{1}, …, i_{r}) is generic, if x ∈ C^{ℓ}, then x^{I} ∈ 𝔱 ≃ 𝔱^{∗}.

Theorem 2.50. For u ∈ C/\overline{R}^{∗}, n ∈ N^{∗}, q ∈ N, if t ∈ 𝔱 represents an element of T \ \overline{S}, then

F (∂/∂t)P_{u,n,q}(t) = |\frac{\overline{R}}{\overline{CR}}| (−1)^{r+1} ∑_{{I=(i_{1},... ,i_{r})∈I_{u} \atop f∈\overline{CR}/d\overline{R}}} \frac{1}{⟨α_{i_{1}}, …, α_{i_{r}}⟩} (2.191)

exp (2iπ(⟨qu, f⟩)) Res_{x=0}^{I}(F (x^{I})\frac{1}{( ∏_{j∈I_{u}}⟨α_{j}, x^{I}⟩)^{n}}

exp\{d\frac{⟨p_{j−1}^{I}α_{i_{j}}, x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩} [\frac{1}{d}⟨p_{j−1}^{I}(t + f), e^{j}⟩]\}

\frac{1}{∏_{j=1}^{r} (exp (\frac{⟨p_{j−1}^{I}α_{i_{j}}, x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩}) − 1)}.

Proof. For y ≠ 0, [y]−y is locally constant. Using (2.186), if s ∈ 𝔱 is identified with the corresponding vector field, then

s\{exp(d ∑_{j=1}^{r} \frac{⟨p_{j−1}^{I}α_{i_{j}}, x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩}[\frac{1}{d}⟨p_{j−1}^{I}(t + f), e^{j}⟩])\} (2.192)

= ⟨s, x^{I}⟩ exp(d ∑_{j=1}^{r} \frac{⟨p_{j−1}^{I}α_{i_{j}}, x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩}[\frac{1}{d}⟨p_{j−1}^{I}(t + f), e^{j}⟩]).

From (2.173), (2.192), we get (2.191).

The proof of our Theorem is completed.

3. Symplectic manifolds and moment maps

In this Section, we recall known results on symplectic manifolds and moment maps. In particular we prove the formula of Witten [60] and Jeffrey-Kirwan [26], which expresses the integral of certain characteristic classes on symplectic reductions in terms of the action of differential operators on the symplectic volume. The results of this Section will be used in Section 5, where we will give a formula for the integral of certain characteristic classes on the strata of the moduli space.

This Section is organized as follows. In Section 3.1, we recall elementary facts on orbifolds. In Section 3.2-3.4, we give elementary properties of moment maps. In Section 3.5, we give a direct simple proof that the image of the symplectic volume measure by the moment map can be evaluated in terms of the symplectic volume of the symplectic reductions. In Section 3.6, when 0 is a regular value of the moment map μ, we recall Jeffrey-Kirwan’s expression of the volume of the neighbouring fibres in terms of integrals of characteristic classes on the symplectic reduction of μ^{−1}(0). In Section 3.7, we prove the formula of Witten-Jeffrey-Kirwan. Finally in Section 3.8, we apply the above to the symplectic coadjoint orbits of G.

3.1. Orbifolds. Let X be a smooth compact manifold. Let G be a compact connected Lie group, and let 𝔤 be its Lie algebra. We assume that G acts on X on the right. If Y ∈ 𝔤, let Y^{X} ∈ Vect(X) be the corresponding vector field.

We assume that G acts locally freely on X, i.e. for any non zero Y ∈ 𝔤, Y^{X} is a non vanishing vector field on X.

Let 𝔤^{X} be the subvector bundle of T X which is the image of 𝔤 by Y → Y^{X}. Then we have an exact sequence of G vector bundles

0 → 𝔤^{X} → T X → T X/𝔤^{X} → 0 . (3.1)

The above data define an orbifold X/G, and the G-bundle T X/G^{X} is also called the tangent bundle T X/G to X/G. If the G-bundle T X/𝔤^{X} is orientable (or equivalently if T X is orientable), we will say that the orbifold X/G is orientable.

If G acts freely on X, then X/G is just the standard quotient.

If y ∈ X, let Z(y) = {g ∈ G; yg = y} be the stabilizer of y. Then Z(y) is a finite subgroup of G. By [?, Proposition 27.4] , there are finitely many conjugacy classes of finite subgroups of G, which occur as stabilizers.

Inclusion induces a partial ordering on the set of conjugacy classes of finite subgroup of G. On each connected component of X, there is a unique minimal conjugacy class of stabilizers S, called the generic conjugacy class of stabilizers. This minimal conjugacy class then acts as the identity on the considered connected component. The order |S| of a generic stabilizer is locally constant on X.

Let X_{reg} be the set of y ∈ X such that Z(y) lies in the minimal conjugacy class. Then X_{reg} is open in X, and X_{reg}/G is a smooth manifold included in the orbifold X/G.

Definition 3.1. A 1-form θ : T X → 𝔤 is said to be a connection form on π : X → X/G if

• For Y ∈ 𝔤,

θ(Y^{X}) = Y. (3.2)

• For g ∈ G, Y ∈ 𝔤,

g^{∗}θ = θ.g. (3.3)

One verifies trivially that G-connections exist. Then the curvature Θ of θ is defined by

dθ = −\frac{1}{2}[θ, θ] + Θ. (3.4)

Also if Y ∈ 𝔤, g ∈ G,

i_{Y^{X}} Θ = 0, g^{∗}Θ = Θ.g. (3.5)

Put

T^{H}X = {U ∈ T X, θ(U) = 0}. (3.6)

Then T^{H}X is a G-invariant subbundle of T X such that

T X = T^{H}X ⊕ 𝔤^{X}. (3.7)

A G-invariant form α on X is said to be basic if for Y ∈ 𝔤, i_{Y^{X}} α = 0. From now on, we suppose that X/G is an oriented orbifold. Then X_{reg}/G is an oriented manifold. By definition

∫_{X/G} α = ∫_{X_{reg}/G} α . (3.8)

Let E_{1}, …, E_{n} be a basis of 𝔤, let E^{1}, …, E^{n} be the corresponding dual basis of 𝔤^{∗}. We write the connection θ in the form

θ = Σ_{1}^{n}θ^{i}E_{i} . (3.9)

Then E^{1} ∧ … ∧ E^{n} defines a volume form on G. Let Vol(G) be the corresponding volume of G.

We equip 𝔤^{X} with the orientation induced by the orientation of 𝔤. Then T X ≃ T X/𝔤^{X} ⊕ 𝔤^{X} is naturally oriented. If α is a G-invariant basic form on X,

∫_{X/G} α = \frac{|S|}{Vol(G)} ∫_{X} α ∧ θ^{1} ∧ … ∧ θ^{n} . (3.10)

If α is a G-invariant basic form on X, dα is also G-invariant and basic. Then since (X/G) \ (X_{reg}/G) is a union of submanifolds of codimension ≥ 2,

∫_{X/G} dα = 0 . (3.11)

In fact note that

∫_{X} dα ∧ θ^{1} ∧ … ∧ θ^{n} = (−1)^{deg(α+1)} ∫_{X} α ∧ d(θ^{1} ∧ … ∧ θ^{n}) . (3.12)

By (3.4),

∫_{X} α ∧ d(θ^{1} ∧ … ∧ θ^{n}) = ∫_{X} α ∧ ∑_{i=1}^{n}(−1)^{i−1}θ^{1} ∧ … ∧ θ^{i−1} ∧ Θ^{i} ∧ … ∧ θ^{n} = 0, (3.13)

which provides another proof of (3.11).

Let E → X be a complex G-vector bundle on E. Let ∇^{E} be a G-invariant horizontal connection on X. Namely suppose first that G acts freely on X. Then ∇^{E} is just a connection on the vector bundle X ×_{G} E over X/G. More generally, if G only acts locally freely on X, the above construction still makes sense. Let F^{E} = ∇^{E,2} be the curvature of ∇^{E}. Then F^{E} is a G-invariant basic 2 form on X with values in 𝔤. Let P be an ad-invariant function on 𝔤. Then P (E, ∇^{E}) = P (\frac{−F^{E}}{2iπ}) is a G-invariant basic closed form on X. If X/G is an oriented orbifold, the integral ∫_{X/G} P (F, ∇^{E}) is well-defined and does not depend on ∇^{E}. 3.2. Symplectic manifolds and moment maps. Let (X, σ) be a symplectic manifold, so that σ is a nondegenerate closed 2-form. Let H : X → R be a smooth function. The corresponding hamiltonian vector field Y_{H} is defined by the equation dH = i_{Y_{H}} σ, (3.14) which we rewrite in the form (d − i_{Y_{H}})(H + σ) = 0. (3.15) From (3.15), we get L_{Y_{H}} σ = 0. (3.16) If H, H′ are two smooth functions, put {H, H′} = −Y_{H′}.H = Y_{H}.H′ = −σ(Y_{H}, Y_{H′}). (3.17) Then Y_{{H,H′}} = [Y_{H}, Y_{H′}]. (3.18)

Let G be a compact connected Lie group acting on the right on M, and preserving σ. Let 𝔤 be the Lie algebra of G. Let η : G → Aut(𝔤^{∗}) be the coadjoint representation. If Y ∈ 𝔤, p ∈ 𝔤^{∗}, let {Y, p} ∈ 𝔤^{∗} be the infinitesimal (left) action of Y on p.

We say that μ : X → 𝔤^{∗} is a moment map if

• For x ∈ X, g ∈ G, μ(xg) = μ(x).g. (3.19)

• If Y ∈ 𝔤, if Y^{X} is the corresponding vector field on X, then ⟨μ, Y ⟩ is a

Hamiltonian for Y^{X}. In particular, by (3.17),(3.19), ⟨μ, [Y, Y′]⟩ = −σ(Y^{X}, Y′^{X}). (3.20) Also by (3.19), Y^{X}μ = −{Y, μ}. (3.21)

Assume now that G acts locally freely on X. Of course, this never occurs if X is compact. Let θ be a G-connection form on π : X → X/G. Proposition 3.2. There is a unique closed 2-form η on X/G such that σ = π^{∗}η − d⟨μ, θ⟩. (3.22) Proof. By (3.3) , ( 3.19) , the 1-form ⟨μ, θ⟩ is G-invariant, so that if Y ∈ 𝔤, L_{Y^{X}} ⟨μ, θ⟩ = 0. (3.23) Also by (3.16), L_{Y^{X}} σ = 0. (3.24)

Then by (3.14), (3.23),

i_{Y^{X}} (σ + d⟨μ, θ⟩) = i_{Y^{X}} σ + L_{Y^{X}} ⟨μ, θ⟩ − di_{Y^{X}} ⟨μ, θ⟩ = 0 (3.25)

By (3.25), we find that the G-invariant 2-form σ + d⟨μ, θ⟩ is basic. Therefore it is of the form π^{∗}η. The proof of our Proposition is completed.

Let σ^{H} be the restriction of σ to T^{H}X. Since σ^{H} is G-invariant, it descends to a 2-form on X/G. The same is true for the 2-form ⟨μ, Θ⟩.

Theorem 3.3. The following identity of 2-forms holds on X/G,

σ^{H} = π^{∗}η − ⟨μ, Θ⟩. (3.26)

Proof. By (3.4),

d⟨μ, θ⟩ = ⟨dμ, θ⟩ + ⟨μ, −\frac{1}{2}[θ, θ] + Θ⟩. (3.27)

From (3.27), we get

[d⟨μ, θ⟩]^{H} = ⟨μ, Θ⟩. (3.28)

From (3.22) , (3.28), we get (3.26).

Remark 3.4. If X ∈ 𝔤, by (3.15),

(d − i_{Y^{X}})(σ + ⟨μ, Y ⟩) = 0. (3.29)

Classically [57], (3.29) shows that σ^{H} + ⟨μ, Θ⟩ descends to a closed 2-form on X/G. Of course this also follows from Proposition 3.2 and Theorem 3.3.

Remark 3.5. Let ⟨, ⟩ be a G- invariant scalar product on 𝔤. Then 𝔤 and 𝔤^{∗} can be identified. Let H ⊂ G be a Lie subgroup of G, and let 𝔥 ⊂ 𝔤 be the corresponding Lie algebra. Let 𝔥^{⊥} be the orthogonal space to 𝔥 in 𝔤. Then 𝔤 = 𝔥 ⊕ 𝔥^{⊥} is a H-invariant splitting. Let P^{𝔥} : 𝔤 → 𝔥 be the corresponding projection. Put

θ^{𝔥} = P^{𝔥}θ, θ^{𝔥^{⊥}} = P^{𝔥^{⊥}}θ. (3.30)

Then θ^{𝔥} is a connection on X → X/H, whose curvature Θ^{𝔥} is given by

Θ^{𝔥} = P^{𝔥}Θ − \frac{1}{2}P^{𝔥}[θ^{𝔥^{⊥}}, θ^{𝔥^{⊥}}]. (3.31)

Take μ ∈ 𝔤^{∗}, and suppose that Z(μ) = H. Then

⟨μ, \frac{1}{2}[θ, θ]⟩ = ⟨μ, \frac{1}{2}[θ^{𝔥^{⊥}}, θ^{𝔥^{⊥}}]⟩ = ⟨μ, \frac{1}{2}P^{𝔥}[θ^{𝔥^{⊥}}, θ^{𝔥^{⊥}}]⟩. (3.32)

Therefore by (3.31), (3.32),

⟨μ, −\frac{1}{2}[θ, θ] + Θ⟩ = ⟨μ, Θ^{𝔥}⟩. (3.33)

So by (3.4), (3.22),(3.27), (3.33), if Z(μ) = H,

σ = π^{∗}η − ⟨dμ, θ⟩ − ⟨μ, Θ^{𝔥}⟩. (3.34)

Equation (3.34) is of special interest when μ ∈ 𝔱^{∗} ≃ 𝔱, and H = T.

Put

E = X ×_{G} 𝔤^{∗}. (3.35)

Then E is an orbifold vector bundle over X/G. Moreover (3.19) says that μ descends to a section of E over X/G.

3.3. Symplectic reduction. We make the same assumptions as in Section 3.2. We do no longer assume that G acts locally freely on X. If p ∈ 𝔤^{∗}, put

X_{p} = μ^{−1}(p). (3.36)

Let Z(p) = {g ∈ G, g.p = p} be the stabilizer of p in G, and let 𝔷(p) = {X ∈ 𝔤, {X, p} = 0} be its Lie algebra. By [15, Theorem IV.2.3], Z(p) is a connected Lie subgroup of G. Clearly Z(p) acts on X_{p}.

Proposition 3.6. The element p ∈ 𝔤^{∗} is a regular value of μ if and only if for x ∈ X_{p}, Y ∈ 𝔤 ↦ Y^{X}(x) ∈ T_{x}X is injective, or equivalently if and only if Z(p) acts locally freely on X_{p}.

Proof. By (3.29) if x ∈ X_{p}, Y ∈ 𝔤 lies in coker(dμ(x)) if and only if Y^{X}(x) = 0. Also by (3.21), if μ(x) = p, Y^{X}(x) = 0, then {Y, p} = 0, i.e. Y ∈ 𝔷(p). The proof of our Proposition is completed.

Assume now that p is a regular value of μ, and that X_{p} ≠ ∅. Then X_{p} is a smooth submanifold of X, on which Z(p) acts locally freely, so that X_{p}/Z(p) is a Z(p)-orbifold.

Let i_{p} be the embedding X_{p} → X. Then by (3.21), (3.29), for Y ∈ 𝔷(p),

i_{Y^{X}} i_{p}^{∗}σ = 0. (3.37)

By (3.37) , we find that the Z(p) -invariant closed 2-form i_{p}^{∗}σ descends to a closed 2-form σ_{p} on X_{p}/Z(p). Then (X_{p}/Z(p), σ_{p}) is a symplectic orbifold.

If p ∈ 𝔤^{∗}, g ∈ G maps X_{p} into X_{p.g}. Also p ∈ 𝔤^{∗} is regular if and only if p.g is regular, and there is an obvious symplectomorphism (X_{p}/Z(p), σ_{p}) → (X_{p.g}/Z(p.g), σ_{p.g}).

Let O ⊂ 𝔤^{∗} be a coadjoint orbit. If p ∈ O, Y, Z ∈ 𝔤, put

σ_{O}(Y, Z) = ⟨p, [Y, Z]⟩. (3.38)

By (1.193), σ_{O} is a G-invariant symplectic form on O. Moreover μ_{O} : p ∈ O ↦ −p ∈ 𝔤^{∗} is a moment map for the right action of G on O.

Put

X_{O} = μ^{−1}(O). (3.39)

Then G acts on the right on X_{O} ⊂ X.

Let p_{1}, p_{2} be the projections X ×O → X, X ×O → O. Then (X ×O, p_{1}^{∗}σ+p_{2}^{∗}σ_{O}) is a symplectic manifold on which G acts symplectically on the right, with moment map p_{1}^{∗}μ + p_{2}^{∗}μ_{O}.Then

(X × O)_{0} = {(x, p) ∈ X × O, μ(x) − p = 0}. (3.40)

So, if p ∈ O,

(X × O_{p})_{0} ≃ X_{O_{p}}, (3.41)

(X × O_{p})_{0}/G ≃ X_{p}/Z(p).

Moreover p ∈ 𝔤^{∗} is a regular value of μ if and only if 0 is a regular value of p_{1}^{∗}μ + p_{2}^{∗}μ_{O}. Then one finds easily that (3.41) identifies the symplectic forms on the corresponding orbifolds.

3.4. Symplectic reduction with respect to a fixed stabilizer. Let G be a compact connected semisimple Lie group. We use the same notation as in Section 1.14.

Let ⟨, ⟩ be a G-invariant scalar product on 𝔤. So we can identify 𝔤 and 𝔤^{∗}, 𝔱 and 𝔱^{∗}. Let η : G → Aut(𝔤^{∗}) be the coadjoint representation. Of course we can now identify the representations τ and η.

Take p_{0} ∈ 𝔤^{∗}. Put

(3.42)  Z = Z(p_{0}),
𝔷 = 𝔷(p_{0}).

Definition 3.7. Put

ℜ = {p ∈ 𝔱^{∗}, Z(p) = Z}. (3.43)

Since the Z(p) are connected,

ℜ = {p ∈ 𝔱^{∗}, 𝔷(p) = 𝔷}. (3.44)

By (1.182), (3.44), ℜ is the complement of a finite union of hyperplanes in {p ∈ 𝔱^{∗}, if ⟨p_{0}, h_{α}⟩ = 0, ⟨p, h_{α}⟩ = 0}.

Now we make the same assumptions as in Section 3.2. Suppose that G acts locally freely on X. Then by Proposition 3.6 , any p ∈ 𝔤^{∗} is a regular value of μ.

Definition 3.8. Put

𝔖 = μ^{−1}(ℜ). (3.45)

Then 𝔖 is a submanifold of X, on which Z acts locally freely. Let σ_{𝔖} be the restriction of σ to S. Then σ_{𝔖} is a closed 2-form on 𝔖, which in general is not symplectic. Let j : 𝔤^{∗} → 𝔷^{∗} be the obvious projection. Then jμ : 𝔖 → 𝔷^{∗} is a moment map for the action of Z on 𝔖 with respect to σ_{𝔖}. Finally 𝔖/Z embeds into X/G.

Let P be the orthogonal projection 𝔤 → 𝔷. Let θ be a connection form on X → X/G. Put

θ̃ = P θ. (3.46)

Then θ̃ defines a Z-connection on X → X/Z. Let Θ̃ be the curvature of θ. Clearly, over 𝔖, since μ ∈ 𝔷,

⟨μ, θ⟩ = ⟨μ, θ̃⟩. (3.47)

By (3.34), over 𝔖,

σ = π^{∗}η − (⟨dμ, θ̃⟩ + ⟨μ, Θ̃⟩). (3.48)

In particular, if p ∈ ℜ,

i_{p}^{∗}σ = i_{p}^{∗}(π^{∗}η − ⟨μ, Θ̃⟩). (3.49)

Recall that the vector bundle E was defined in (3.35). Let ˜∇^{E} be the connection on E_{|𝔖/Z} induced by θ̃. Then over X_{p}/Z,

˜∇^{E}μ = 0. (3.50)

Equation (3.50) explains why over X_{p}, σ given by (3.49) is closed. In fact it is the pull-back of the symplectic form σ_{p} on X_{p}/Z(p).

Proposition 3.9. Locally, over 𝔖, the cohomology class of σ_{p} depends linearly on p ∈ ℜ.

Proof. This is obvious by (3.49).

3.5. The image of the symplectic volume by the moment map. Let G be a compact connected semisimple Lie group. We use otherwise the notation of Section 1. Let (X, σ) be a compact symplectic manifold, on which G acts symplectically, with moment map μ : X → 𝔤^{∗}.

We make the assumption that

Z(x) = 1 for a.e. x ∈ X. (3.51)

Then a.e., G acts locally freely on X. By Proposition 3.6, a.e., dμ(x) surjects on 𝔤^{∗}. By Sard’s theorem, a.e. p ∈ 𝔤^{∗} is a regular value of μ.

Let dp, dt be the Lebesgue measures on 𝔤, 𝔱 with respect to ⟨, ⟩. Let dg be the Lebesgue measure on G. If p ∈ 𝔤, let dg_{p} be the Lebesgue measure on Z(p).

Clearly

(3.52) \{e^{σ}\}^{max} = \frac{σ^{dim X/2}}{(dim X/2)!}.

Then \frac{σ^{dim X/2}}{(dim X/2)!} is a dim X form on X, which does not vanish, and so defines an orientation of X. If f : X → R is a bounded measurable function, put

∫_{X} f(x)|e^{σ}| = ∫_{X} f(x)e^{σ}, (3.53)

The notation (3.53) emphasizes the fact that the left-hand side is an integral with respect to a nonnegative measure. We will use a similar notation over X_{p}/Z(p).

Now we will compute the disintegration of the symplectic volume on X with respect to the moment map μ. The point of this proof is that it is parallel to a corresponding for moduli spaces given in Theorem 5.44.

Recall that the monomial π : 𝔱 → C was defined in Definition 1.53.

Theorem 3.10. Let f : X → R be a bounded measurable function. Then

(3.54)  ∫_{X} f(x)|e^{σ}| = ∫_{𝔱/W} |π(t)|dt ∫_{X_{t}/T} |e^{σ_{t}}| ∫_{G} f(x.g)dg,
∫_{X} f(x)|e^{σ}| = ∫_{𝔤} \frac{dp}{|π(p)|} ∫_{X_{p}/Z(p)} |e^{σ_{p}}| ∫_{Z(p)} f(x.g)dg_{p}.

Proof. Since dμ(x) is a.e. surjective, it is clear that μ_{∗} \frac{σ^{dim X/2}}{(dim X/2)!} is absolutely continuous with respect to dp. Also for a.e. x, dμ(x) is surjective and μ(x) ∈ 𝔤_{reg}.

Let K ⊂ 𝔱 be a Weyl chamber. Let q be the projection 𝔤_{reg} → 𝔱_{reg}/W ≃ K. If x ∈ X is such that μ(x) ∈ 𝔤_{reg}, we have a G-equivariant complex

(C′_{x}, ∂) : 0 → 𝔤 → T_{x}X →^{dqμ} 𝔱 → 0, (3.55)

where the map 𝔤 → T_{x}X is just Y → Y^{X}(x). If dμ(x) is surjective, by Proposition 3.6, the cohomology of the complex (C′_{x}, ∂) is concentrated in degree 1. More precisely, one finds easily that in this case,

H^{1}(C′_{x}, ∂) ≃ T_{π(x)}X_{qμ(x)}/Z(qμ(x)). (3.56)

In particular, by [33] and by (3.55), (3.56), we have a canonical isomorphism of real lines

(det C′_{x})^{−1} ≃ det(T_{π(x)}X_{qμ(x)}/Z(qμ(x))). (3.57)

Now det(T_{x}X) is equipped with the volume form associated to σ_{x}.Also 𝔤 and 𝔱 are equipped with the volume forms dp and dt. Therefore det(C′_{x}) is equipped with a natural metric. Let dv_{X_{qμ(x)}/Z(qμ(x))} be the corresponding volume form on T_{π(x)}X_{qμ(x)}/Z(qμ(x)) via the isomorphism (3.57).

By the formula of change of variables,we get

∫_{X} f(x)|e^{σ}| = ∫_{𝔱_{reg}/W} dt ∫_{X_{t}/T} dv_{X_{t}/T} (x) ∫_{G} f(x.g)dg. (3.58)

Take t ∈ 𝔱_{reg}, x ∈ X such that μ(x) = t. Consider the double complex

(3.59)  \xymatrix{ & 0 & 0 & 0 \\ 0 \ar[r] & 𝔤 \ar[u] \ar[r] & T_{x}X \ar[u] \ar[r]^{d(qμ)} & 𝔱 \ar[u] \ar[r] & 0 \\ 0 \ar[r] & 𝔤 \ar[u] \ar[r] & T_{x}X ⊕ Im Ad(t) \ar[u] \ar[r] & 𝔤 \ar[u] \ar[r] & 0 \\ 0 \ar[r] & 0 \ar[u] \ar[r] & Im Ad(t) \ar[u] \ar[r] & Im Ad(t) \ar[u] \ar[r] & 0 \\  & 0 \ar[u] & 0 \ar[u] & 0 \ar[u]}

In (3.59) , the map 𝔤 → ImAd(t) is just f ↦ −Ad(t)f = −[t, f], the map T_{x}X → 𝔤 is dμ(x). Also the columns and the lowest row are acyclic. In particular the cohomology groups of the two upper rows are isomorphic. Therefore they are concentrated in degree 1 and equal to H^{1}(C′_{x}, ∂).

We equip ImAd(t) ≃ T_{t}O_{t} with the 2-form σ_{O_{t}} given in (3.38). By (3.22), (3.27), the volume induced by the second row on its determinant is just \frac{σ^{dim(X_{t}/T)/2}}{(dim(X_{t}/T)/2)!} on det H^{1}(C′_{x}, ∂). On the other hand, let R be the lowest row in (3.59). Since it is acyclic, R ≃ (det R)^{−1} has a canonical section η. Clearly

|η| = |π(t)|. (3.60)

Therefore

dv_{X_{t}/T} = |π(t)|\frac{|σ_{t}^{dim(X_{t}/T)/2}|}{(dim(X_{t}/T)/2)!}. (3.61)

By (3.58), (3.61), we get the first identity in (3.54) .

By [14, Proposition 6.3.4], if h : 𝔤 → R is a bounded measurable function,

∫_{𝔤} h(p)dp = \frac{1}{Vol(T)} ∫_{𝔱/W} |π(t)|^{2}dt ∫_{G} h(t.g)dg. (3.62)

From the first identity in (3.54) and from (3.62), we obtain

∫_{X} f(x)|e^{σ}| = ∫_{𝔱/W} |π(t)|dt ∫_{X_{t}/T} |e^{σ_{t}}| ∫_{T \G} d g^{.} ∫_{T} f(x.t′ g^{.})dt′ = (3.63)

∫_{𝔤} \frac{dp}{|π(p)|} ∫_{X_{p}/Z(p)} |e^{σ_{p}}| ∫_{Z(p)} f(x.g)dg_{p},

which is the second identity in (3.54). The proof of our Theorem is completed.

Definition 3.11. If t ∈ 𝔱 is a regular value of μ, let |V (t)| be the absolute value of the symplectic volume of X_{t}/Z(t) with respect to σ_{t}.

Theorem 3.12. Let f : 𝔤^{∗} → R be a bounded measurable function. Then

(3.64)  ∫_{X} f(μ(x))|e^{σ}| = ∫_{𝔱/W} [∫_{G} f(t.g)dg] |π(t)||V (t)|dt,
∫_{X} f(μ(x))|e^{σ}| = Vol(T) ∫_{𝔤^{∗}} f(p)\frac{|V (p)|}{|π(p)|}dp.

Proof. Our Theorem follows from Theorem 3.10.

Clearly, if 0 is a regular value of μ, μ_{∗}|e^{σ}| has a smooth density with respect to dp near p = 0. In particular

lim_{{p∈𝔤reg \atop p→0}} \frac{|V (p)|}{|π(p)|} (3.65)

exists and is the value of the above density at p = 0. A more precise statement is as follows.

Proposition 3.13. If 0 ∈ 𝔤^{∗} is a regular value of μ, and if

Z(x) = 1 a.e. on X_{0}, (3.66)

then

(3.67)  lim_{{p∈𝔤reg \atop p→0}} \frac{|V (p)|}{|π(p)|} = \frac{Vol(G)}{Vol(T)}|V (0)|.

Proof. By 3.12, one gets (3.67) easily.

3.6. The symplectic volume as a polynomial near the origin. In the sequel, we will assume that 0 is a regular value of μ, and also that

Z(x) = 1 a.e. on X_{0}. (3.68)

Let U be a W -invariant open neighborhood of 0 in 𝔱^{∗} consisting of regular values of μ. Let 𝔤_{U}^{∗} ⊂ 𝔤^{∗} be the corresponding union of coadjoint orbits. Then 𝔤_{U}^{∗} is an open neighborhood of 0 in 𝔤.

Classically, U is small enough, there is a G-invariant open neighborhood V of X_{0} in X such that we have the identification of G-spaces

V ≃ X_{0} × 𝔤_{U}^{∗}, (3.69)

so that in the right hand-side of (3.69), μ is just the projection M × 𝔤_{U}^{∗} → 𝔤_{U}^{∗}. By (3.69), if p ∈ U,

X_{O_{p}} ≃ X_{0} × O_{p}, (3.70) so that

X_{O_{p}}/G ≃ X_{0} ×_{G} O_{p} ≃ X_{0}/Z(p) (3.71)

is a O_{p}-fibre bundle over the orbifold X_{0}/G.

Now we construct an orientation over X_{0}/T. In fact if t ∈ T_{reg}, then O_{t} ≃ G/T, and so

X_{0}/T = X_{0} ×_{G} G/T. (3.72)

Let K ⊂ 𝔱 be a Weyl chamber, and let t ∈ K. Then the symplectic form σ_{O_{t}} orients G/T ≃ O_{t}, and the corresponding orientation on G/T does not depend on t ∈ K. Also X_{0}/G carries the symplectic form σ_{0}. Therefore once K is fixed, X_{0}/T carries a canonical orientation.

Take t ∈ U. Let i_{t} : X_{t} → X be the obvious embedding. Then by (3.37), i_{t}^{∗}σ descends to a closed 2-form on X_{t}/T, which we denote σ̄_{t}. Note that if t ∈ 𝔱_{reg}, σ_{t} = σ̄_{t}. If t ∉ 𝔱_{reg}, T ⊂ Z(t), T ≠ Z(t), and σ̄_{t} is in general not symplectic.

By (3.69),

X_{t} ≃ X_{0} × {t}. (3.73)

We equip X_{t}/T ≃ X_{0}/T with the given fixed orientation.

Definition 3.14. Put

P (t) = ∫_{X_{t}/T} eσ̄_{t}. (3.74)

Then P (t) is a smooth function on U.

Theorem 3.15. One has the identity

(3.75)  |P (t)| = |V (t)| if t ∈ 𝔱_{reg},
= 0 if t ∉ 𝔱_{reg}.

If w ∈ W, t ∈ U,

P (wt) = ϵ_{w}P (t). (3.76)

Also, near 0, P (t) is a polynomial. More precisely, if θ is a connection form on π : X_{0} → X_{0}/G, and if Θ is its curvature,

P (t) = ∫_{X_{0}/T} exp (π^{∗}σ_{0} − ⟨t, Θ⟩ + ⟨t, \frac{1}{2}[θ, θ]⟩) . (3.77)

Near 0, P (t) and π(t/i) either vanish together or are nonzero, and then they have the same sign. In particular

π(t/i)P (t) ≥ 0 near 0. (3.78)

Proof. By the considerations we made after (3.72), (3.75) holds. If w ∈ W, let g ∈ N(T) represent w ∈ W = N(T)/T. Since G is connected, it acts on X_{0}, 𝔤, 𝔤^{∗} by orientation preserving maps. Clearly

T X_{0} = π^{∗}(T X_{0}/G) ⊕ 𝔤^{X}. (3.79)

Also

𝔤 ⊗_{R} C = 𝔱 ⊗_{R} C ⊕ (⨁_{α∈R} 𝔤_{α}) , (3.80)

so that

(𝔤/𝔱) ⊗_{R} C = ⨁_{α∈R} 𝔤_{α}. (3.81)

Then (𝔤/𝔱) ⊗_{R} C descends to the complexified tangent space to the fibre G/T over X_{0}/G. Since g ∈ N(T), g preserves 𝔱 and its orthogonal 𝔱^{⊥} in 𝔤. Since g changes the orientation of 𝔱 by the factor ϵ_{w}, it changes the orientation of 𝔤/𝔱 by the same factor.

So g changes the orientation of X_{0}/T by the factor ϵ_{w}. Since σ is G-invariant, and g acts on X_{0}/T,

σ̄_{gt} = σ̄_{t}. (3.82)

From the above, we get (3.76).

By (3.22),(3.27),

σ = π^{∗}η − ⟨dμ, θ⟩ − ⟨μ, Θ − \frac{1}{2}[θ, θ]⟩. (3.83)

Clearly i_{t}^{∗}π^{∗}η is cohomologous to i_{0}^{∗}π^{∗}η = π^{∗}σ_{0}. So by (3.83), we get (3.77). From (3.77), it is cldear that P (t) is a polynomial near 0.

If t ∈ U\𝔱_{reg}, by the considerations after (3.72), σ̄_{t} is an everywhere degenerate 2-form on X_{t}/T. Therefore

P (t) = 0 on U\𝔱_{reg}. (3.84)

For t ∈ U ∩ 𝔱_{reg}, σ̄_{t} is a symplectic 2-form on X_{t}/T, so that

P (t) ≠ 0 on U ∩ 𝔱_{reg}. (3.85)

By (1.190), (3.85), P (t) and π(t/i) have the same zeroes on U.

By (3.38), on X_{0}/T, the form ⟨t, \frac{1}{2}[θ, θ]⟩ is just the symplectic 2-form along the fibre G/T ≃ O_{t}. If t ∈ K ∩ U is close to 0, by (3.72), (3.77), P (t) > 0. Therefore, on K ∩ U, P (t) and π(t/i) have the same sign. Using (1.186) and (3.76), we get the end of our Theorem, the proof of which is now completed.

Remark 3.16. Recall that we have a fibration X_{0}/T →^{G/T} X_{0}/G. Then we can rewrite (3.77) in the form

P (t) = ∫_{X_{0}/G} e^{σ_{0}} ∫_{G/T} e^{−⟨t,Θ⟩+⟨t, \frac{1}{2} [θ,θ]⟩}. (3.86)

In (3.86), ∫_{G/T} … is an integral along the fibre. In Remark 3.26, we will reinterpret identity (3.86).

3.7. A formula of Witten-Jeffrey-Kirwan. We make the same assumptions as in Section 3.6. Also we suppose the set U to be bounded. Let ⟨, ⟩ be a Ginvariant scalar product on 𝔤. If f : 𝔤^{∗} → R is a bounded measurable function, let Mf : 𝔤^{∗} → R be given by

Mf(p) = \frac{1}{Vol(G)} ∫_{G} f(p.g)dg. (3.87)

Recall that 𝔤 and 𝔤^{∗} have been identified by ⟨, ⟩, so that (3.87) also makes sense when f : 𝔤 → R is bounded and measurable.

Definition 3.17. If f : 𝔤^{∗} → R is a bounded measurable function with support in 𝔤_{U}^{∗}, for u ∈ 𝔤, put

f̂_{𝔤}(u) = ∫_{𝔤^{∗}} f(p)e^{⟨u,p⟩}dp. (3.88)

Clearly

Mf̂_{𝔤} = M̂f_{𝔤}. (3.89)

Also f̂_{𝔤}(u) is an analytic function of u.

If f : 𝔱^{∗} → R is a bounded measurable function with support in U, for t ∈ 𝔱, set

f̂_{𝔱}(t) = ∫_{𝔱^{∗}} f(p)e^{⟨t,p⟩}dp. (3.90)

If f is W -invariant, then f̂_{𝔱} is also W -invariant.

The maps f ↦ f_{|𝔤} or f ↦ f_{|𝔱} are injective.

Also by Rossmann’s formula [48], [56], if f : 𝔤^{∗} → R is taken as before, then,

π̂f_{𝔱} = π(\frac{.}{2π})f̂_{𝔤}. (3.91)

In the sequel, we orient X by the symplectic form σ. Now we prove a formula due to Jeffrey-Kirwan [26] in a work where they prove a formula by Witten [60]. Our approach is closely related to Vergne [57].

Theorem 3.18. Let f : 𝔤^{∗} → R be a bounded measurable function with support in 𝔤_{U}^{∗}. Let θ be a connection form on X_{0} → X_{0}/G, and let Θ be its curvature. Then

∫_{X} f(μ)e^{σ} = Vol(G) ∫_{X_{0}/G} M̂f_{𝔤}(−Θ)e^{σ_{0}}. (3.92)

Proof. By Theorem 3.12

∫_{X} f(μ)e^{σ} = Vol(G) ∫_{𝔱/W} Mf(t)|π(t)||V (t)|dt. (3.93)

By Theorem 3.15,

|π(t)||V (t)| = π(t/i)P (t). (3.94)

So (3.93) can be rewritten in the form

∫_{X} f(μ)e^{σ} = Vol(G) ∫_{𝔱/W} (Mf)(t)π(t/i)P (t)dt. (3.95)

By (3.77),

P (t) = ∫_{X_{0}/T} e^{π^{∗}σ_{0}−⟨t,− \frac{1}{2} [θ,θ]+Θ⟩}. (3.96)

It follows from the above that we may as well assume that over X_{0} × 𝔤^{∗}, σ is given by

σ = π^{∗}σ_{0} − d⟨p, θ⟩. (3.97)

Equivalently,

σ = π^{∗}σ_{0} − ⟨dp, θ⟩ − ⟨p, −\frac{1}{2}[θ, θ] + Θ⟩. (3.98)

Then

∫_{X} f(μ)e^{σ} = ∫_{X_{0}×𝔤^{∗}} f(p)e^{π^{∗}σ_{0}−⟨p,Θ⟩−⟨dp,θ⟩+ \frac{1}{2} ⟨p,[θ,θ]⟩}. (3.99)

Since only the term of top degree in Λ(𝔤^{∗}) contributes to the integral in the right hand-side of (3.99), we can rewrite (3.99) in the form

∫_{X} f(μ)e^{σ} = ∫_{X_{0}×𝔤^{∗}} f(p)e^{π^{∗}σ_{0}−⟨p,Θ⟩−⟨dp,θ⟩}. (3.100)

Now recall that X is oriented by σ, and X_{0}/G by σ_{0}. In particular, near X_{0} ⊂ X_{0} × 𝔤^{∗}, X_{0} × 𝔤^{∗} is oriented by σ. Also ⟨p, Θ⟩ is G-invariant. From (3.100), we get

∫_{X} f(μ)e^{σ} = ∫_{X_{0}/G} e^{σ_{0}} ∫_{𝔤^{∗}} e^{−⟨p,Θ⟩} [∫_{G} f(p.g)dg] dp, (3.101)

which coincides with (3.92). The proof of our Theorem is completed.

Remark 3.19. In [26], Jeffrey-Kirwan use instead the coisotropic embedding theorem [Theorem 39.2] [23], which asserts there is a G-equivariant identification V ≃ X_{0} × 𝔤_{U}^{∗}, so that σ is exactly given by

σ = p_{1}^{∗}σ_{0} − d⟨p, θ⟩. (3.102)

Recall that by Theorem 3.15, P (t) is a polynomial near t = 0. Therefore P (\frac{∂}{∂t}) is a differential operator.

Theorem 3.20. If f : 𝔱 → R is a bounded measurable W -invariant function with support in U, then

∫_{X} f(μ)e^{σ} = \frac{Vol(G)}{|W |} [P (\frac{∂}{∂t}){ \atop π(t/i)f}_{|𝔱}(t)]_{|t=0} . (3.103)

Proof. By (3.95),

∫_{X} f(μ)e^{σ} = \frac{Vol(G)}{|W |} ∫_{𝔱} f(t)π(t/i)P (t)dt. (3.104)

From (3.90), (3.104), we get (3.103). The proof of our Theorem is completed.

Let θ be a G-connection form on X_{0} → X_{0}/G, and let Θ be its curvature. Let Q be a G- invariant C^{∞} function defined on 𝔤 with values in R. We define Q(−Θ) by its Taylor expansion, which only contains a finite number of terms. Then Q(−Θ) is a closed form on X_{0}/G, and ∫_{X_{0}/G} Q(−Θ)e^{σ_{0}} does not depend on θ.

Similarly, we define the differential operator Q(\frac{∂}{∂t}) as the formal power series associated to the Taylor expansion of Q. When applying the power series Q(\frac{∂}{∂t})π(\frac{∂/∂t}{2iπ}) to the polynomial P (t), only a finite number of terms in the Taylor expansion contribute, so that Q(\frac{∂}{∂t})π(\frac{∂/∂t}{2iπ})P (t) is a well-defined polynomial.

Theorem 3.21. The following identity holds,

∫_{X_{0}/G} Q(−Θ)e^{σ_{0}} = \frac{1}{|W |} [Q(∂/∂t)π(\frac{∂/∂t}{2iπ})P (t)]_{|t=0} . (3.105)

Proof. Let f : 𝔤^{∗} → R be a G- invariant bounded measurable function with support in 𝔤_{U}^{∗}. By Theorems 3.18 and 3.20,

∫_{X_{0}/G} f̂_{𝔤}(−Θ)e^{σ_{0}} = \frac{1}{|W |} [P (∂/∂t){ \atop π(./i)f}_{|𝔱}] (0). (3.106)

By Rossmann’s formula (3.91) and by (3.106), we obtain

∫_{X_{0}/G} f̂_{𝔤}(−Θ)e^{σ_{0}} = \frac{1}{|W |} [P (∂/∂t)π(\frac{t}{2iπ})f̂_{𝔤}(t)] (0). (3.107)

Equivalently

∫_{X_{0}/G} f̂_{𝔤}(−Θ)e^{σ_{0}} = \frac{1}{|W |} [f̂_{𝔤}(∂/∂t)π(\frac{∂/∂t}{2iπ})P (t)] (0). (3.108)

Then (3.108) is exactly (3.105) when Q = f̂_{𝔤}.

Clearly, it is enough to verify (3.105) when Q is a polynomial. Then Q is the Fourier transform of a distribution whose support is {0}. Using (3.108) for f with support in 𝔤_{U}^{∗}, and a simple limit procedure, we get (3.105) when Q is a polynomial. The proof of our Theorem is completed.

3.8. The volume of symplectic coadjoint orbits. Let t ∈ 𝔱, and let O_{t} ⊂ 𝔤 = 𝔤^{∗} be the G- orbit of t. Recall that σ_{O_{t}} is the canonical symplectic form on O_{t} given in 1.193.

Let p_{t} : G/T → O_{t} be given by

p_{t}g = g.t. (3.109)

Then p_{t} is one to one if and only if t ∈ 𝔱_{reg}. If t ∉ 𝔱_{reg},

dim(G/T) > dim O_{t}. (3.110)

Definition 3.22. If t ∈ 𝔱, X ∈ 𝔤 , put

H(t, X) = ∫_{G/T} e^{⟨p_{t}x,X⟩+p_{t}^{∗}σ_{O_{t}}} . (3.111)

Then since G acts on the right on O_{t} and preserves σ_{O_{t}}, H(t, X) is a G- invariant function of X ∈ 𝔤^{∗}.

Let K ⊂ 𝔱 be a Weyl chamber, and let π(t) be the corresponding function on 𝔱 defined in (1.185).

Proposition 3.23. If t ∉ 𝔱_{reg},

H(t, X) = 0. (3.112)

If t ∈ \overline{K}, X ∈ 𝔱_{reg}, then

H(t, X) = \frac{1}{π(X/2iπ)} ∑_{w∈W} ϵ_{w}e^{⟨wt,X⟩}. (3.113)

Proof. Using (3.110), we get (3.112).Recall that G acts on the right on O_{t}. Also if X ∈ 𝔤,

d⟨p, X⟩ + i_{X^{O_{t}}} σ_{O_{t}} = 0 (3.114)

Then when t ∈ K, we get (3.113) from the localization formula in equivariant cohomology of Duistermaat-Heckman[18], Berline-Vergne [5], [6, Theorem 7.11]. This equality obviously extends to the case where t ∈ K̄. The proof of our Proposition is completed.

Remark 3.24. From (3.112), (3.113), we recover the well-known fact [15, Lemma VI.1.2] that if t ∉ 𝔱_{reg},

∑_{w∈W} ϵ_{w}e^{⟨wt,.⟩} = 0. (3.115)

Recall that H(t, X) is a G- invariant function of X ∈ 𝔤^{∗}. Also the function P (t) was defined in Definition 3.14.

Proposition 3.25. For t_{0} ∈ U ∩ K̄,

P (t_{0}) = \frac{1}{|W |} [H(t_{0}, ∂/∂t)π (\frac{∂/∂t}{2iπ}) P (t)]_{|t=0} . (3.116)

Proof. By (3.113), if X ∈ 𝔱_{reg},

H(t_{0}, X)π(\frac{X}{2iπ}) = ∑_{w∈W} ϵ_{w}e^{⟨wt_{0},X⟩}. (3.117)

Of course the identity extends to arbitrary X ∈ 𝔱. In particular, by (3.117), we have the identity of formal power series of differential operators,

H(t_{0}, ∂/∂t)π(\frac{∂/∂t}{2iπ}) = ∑_{w∈W} ϵ_{w}e^{⟨wt_{0},∂/∂t⟩}. (3.118)

Now in view of (3.76),

∑_{w∈W} ϵ_{w}e^{⟨wt_{0},∂/∂t⟩}P (t)_{|t=0} = |W |P (t_{0}). (3.119)

From (3.118), (3.119), we get (3.116). The proof of our Proposition is completed.

Remark 3.26. In view of (3.105), (3.116), we get

P (t_{0}) = ∫_{X_{0}/G} H(t_{0}, −Θ)e^{σ_{0}} (3.120)

One then verifies that (3.120) is just a reformulation of (3.77).

Recall that we have identified 𝔤 and 𝔤^{∗}, 𝔱 and 𝔱^{∗} by the scalar product ⟨, ⟩. For t ∈ 𝔱 ≃ 𝔱^{∗}, the orbit O_{t} is equipped with the symplectic form σ_{O_{t}}.

Theorem 3.27. The following identity holds

|∫_{G/T} e^{p_{t}^{∗}σ_{O_{t}}} | = \frac{Vol(G)}{Vol(T)}|π(t)|, (3.121)

\frac{Vol(G)}{Vol(T)} \frac{1}{|W |}π(\frac{∂/∂t}{2iπ})π(t/i) = 1.

Proof. If t ∈ 𝔱_{reg}, then O_{t} ≃ G/T. So the first identity in (3.121) is trivial.

Let T^{∗}G be the cotangent bundle of G. We identify X = T^{∗}G to G × 𝔤^{∗} via the left action of G on T^{∗}G. If θ is the canonical left-invariant 1-form on G with values in 𝔤, if p ∈ 𝔤^{∗}, ⟨p, θ⟩ is the canonical real 1-form on T^{∗}G ≃ G × 𝔤^{∗}, and −d⟨p, θ⟩ is the canonical symplectic form σ on T^{∗}G. Also G acts on the right on T^{∗}G and preserves σ. Then (g, p) ∈ G × 𝔤^{∗} ↦ p ∈ 𝔤^{∗} is a moment map for this action.

Now we use the notation of Section 3.7. Take t ∈ 𝔱. Then

X_{t} = G × {t}, (3.122)

and X_{t}/Z(t) ≃ G/Z(t) can be identified with O_{t} by g ∈ G/Z(t) ↦ g.(t) ∈ O_{t}. Then one verifies easily that the symplectic form σ_{t} on X_{t}/Z(t) is just σ_{O_{t}}. By the first identity in (3.121), we get

∫_{X_{t}/T} eσ̄_{t} = \frac{Vol(G)}{Vol(T)}π(t/i), (3.123)

so that

P (t) = \frac{Vol(G)}{Vol(T)}π(t/i). (3.124)

Finally observe that the moment map μ : (g, p) ∈ G × 𝔤^{∗} ↦ p ∈ 𝔤^{∗} is regular, that G acts freely on X_{0} ≃ G, and X_{0}/G = 1. We apply Theorem 3.21 with Q = 1, use (3.124), and we get

\frac{Vol(G)}{Vol(T)} \frac{1}{|W |} [π(\frac{∂/∂t}{2iπ})π(t/i)]_{|t=0} = 1. (3.125)

Also π(\frac{∂/∂t}{2iπ})π(t/i) is constant. It is now clear that the second equation in 3.121) follows from (3.125). The proof of our Theorem is completed.

Remark 3.28. It is of some interest to verify that as should be the case, the righthand side in the second equation in (3.121) is positive. In fact

π(\frac{∂/∂t}{2iπ})π(t/i) = (2π)^{ℓ} ∑_{σ∈S_{ℓ}}⟨α_{1}, α_{σ(1)}⟩ … ⟨α_{ℓ}, α_{σ(ℓ)}⟩. (3.126)

Put

(α_{1} ⊗ … ⊗ α_{ℓ})^{σ} = \frac{1}{l!} ∑_{σ∈S_{ℓ}} α_{σ(1)} ⊗ … α_{σ(ℓ)} ∈ S^{ℓ}𝔱^{∗}. (3.127)

Then (3.126) can be written as

π(\frac{∂/∂t}{2iπ})π(t/i) = (2π)^{ℓ}ℓ!∥(α_{1} ⊗ … ⊗ α_{ℓ})^{σ}∥_{S^{ℓ}𝔱^{∗}}^{2}, (3.128)

which is indeed positive.

Recall that ρ ∈ K, so that π(ρ/i) > 0. We now recover a well-known formula for Vol(G/T) [6, Corollary 7.27].

Theorem 3.29. The following identity holds

\frac{Vol(G)}{Vol(T)} = \frac{1}{π(ρ/i)}. (3.129)

Proof. We proceed as in [6]. Let λ ∈ Λ ∩ \overline{K}. Let χ_{λ} be the character of highest weight λ. Then by Kirillov’s formula [32], [5], [6, Theorem 8.4], for t ∈ 𝔱, |t| small enough,

χ_{λ}(e^{t}) = \frac{π(t)}{σ(t)} ∫_{O_{ρ+λ}} e^{2iπ⟨μ,t⟩+σ_{O_{ρ+λ}}} . (3.130)

74 JEAN-MICHEL BISMUT AND FRANÇOIS LABOURIE

In particular χ_{λ}(1) = ∫_{O_{ρ+λ}} e^{σ_{O_{ρ+λ}}} . (3.131) By Theorem 3.27,

∫_{O_{ρ+λ}} e^{σ_{O_{ρ+λ}}} = \frac{Vol(G)}{Vol(T)}π((ρ + λ)/i). (3.132)

Also by Weyl’s dimension formula [Theorem VI.1.7][15],

χ_{λ}(1) = \frac{π(ρ + λ)}{π(ρ)}. (3.133)

From (3.131)-(3.133), we get (3.129). The proof of our Theorem is completed.

4. The affine space of connections

In this Section, we construct a canonical line bundle L on the affine space of connections on the trivial G-bundle on a Riemann surface Σ with marked points, and an action of a central extension of the gauge group ΣG on L. We compute the action of the stabilizers of certain connections on the line bundle L. We also construct a line bundle λ_{p}, which, as we shall see in Section 6.3, will descend to the moduli space of flat G-bundles. The main purpose of this Section is to evaluate the angles of the action of the stabilizers on the line bundle λ_{p}, in order to apply the theorem of Riemann-Roch-Kawasaki [30, 31] to this moduli space, which we will do in Section 6.

Our Section is organized as follows. In Section 4.1, we briefly recall the construction of the central extension L̃G of the loop space LG. In Section 4.2, we consider the coadjoint orbits of L̃G, their symplectic form, and the corresponding line bundles. In Section 4.3, we construct a central extension of (LG)^{s}. In Section 4.4, we give a formula for the holonomy of the canonical connection on the

S^{1}-bundle L̃G →^{S^{1}} LG. In Section 4.5, we describe the symplectic affine manifold A of G-connections on Σ, and a symplectic action of a central extension Σ̃G of the gauge group ΣG. In Section 4.6, we construct the line bundle L on A, and an action of Σ̃G on L. In Section 4.7, when G is not simply connected, we classify the G-bundles on Σ. In Section 4.8, we specialize the results of Section 4.7 when H is a connected subgroup of a simply connected group G. In Section 4.9, we compute the action of certain elements of Σ̃G on L. Finally, in Section 4.10, we define the line bundle λ_{p} , on which ΣG acts, and we compute the action of certain elements of ΣG on λ_{p}.

4.1. The central extension of the loop group LG. Let G be a compact connected and simply connected simple Lie group. We will use the notation of Section 1. In particular ⟨, ⟩ denotes the basic scalar product on 𝔤.

Let T be a maximal torus in G, let 𝔱 ⊂ 𝔤 be its Lie algebra. Let W be the corresponding Weyl group. Then W acts on \overline{CR}. Let W_{aff} be the affine Weyl group

W_{aff} = W ⋉ \overline{CR}. (4.1)

Let S^{1} ≃ R/Z, and let t ∈ [0, 1[ be the canonical coordinate on S^{1}. Then ∂/∂t trivalizes T S^{1} and dt trivializes T^{∗}S^{1}.

Let LG be the loop group of G, i.e. the group of smooth maps s ∈ S^{1} ↦ g_{s} ∈ G. Let L𝔤 be the Lie algebra of LG, i.e. the set of smooth maps s ∈ S^{1} ↦ f_{s} ∈ 𝔤.

If α, b ∈ L𝔤, put

η(α, β) = ∫_{S^{1}}⟨α, dβ⟩ . (4.2)

Then by [45, Section 4.2], η is a cocycle on L𝔤. By [45, Theorem 4.4.1], there is a

unique central extension \xymatrix{ρ : L̃G \ar[r]^{S^{1}} & LG} associated to η. The Lie algebra of L̃G

is L̃𝔤 = L𝔤 ⊕ R. If (α, a), (α′, a′) ∈ L̃𝔤 = L𝔤 ⊕ R,

[(α, a), (α′, a′)] = ([α, α′], ∫_{S^{1}}⟨α, dα′⟩) . (4.3)

Observe that the Lie group G × S^{1} embeds as a Lie subgroup of L̃G, and that the Lie algebra of G × S^{1}, 𝔤 ⊕ R (equipped with its standard Lie algebra structure) embeds correspondingly as a Lie subalgebra of L𝔤 ⊕ R.

Clearly s ∈ S^{1} acts on LG by g ∈ LG ↦ k_{s}g = g_{.+s} ∈ LG. We can then form the semidirect product L̂G = S^{1} ⋉ LG. The Lie algebra L̂𝔤 of L̂G is given by L̂𝔤 = R ⊕ L𝔤, and can be identified to the Lie algebra of differential operators a\frac{d}{dt} + α, a ∈ R, α ∈ L𝔤, so that

[a\frac{d}{dt} + α, a′ \frac{d}{dt} + α′] = [α, α′] + a\frac{d}{dt}α′ − a′ \frac{d}{dt}α . (4.4)

By [45, Theorem 4.4.1], the action of S^{1} on LG lifts to L̃G, so that we can form

the semidirect product ˆ L̃G = S^{1} ⋉ L̃G. Its Lie algebra ˆ L̃𝔤 is given by

(4.5)  ˆ  L̃𝔤 = R ⊕ L𝔤 ⊕ R
= R ⊕ L̃𝔤
= L̂𝔤 ⊕ R .

The Lie bracket in ˆ L̃𝔤 is given by

[(a, α, b), (a′, α′, b′)] = (0, [α, α′] + a\frac{dα′}{dt} − a′ \frac{dα}{dt} , ∫_{S^{1}}⟨α, dα′⟩) . (4.6)

Also S^{1} × G × S^{1} embed as a Lie subgroup of ˆ L̃𝔤, and R ⊕ 𝔤 ⊕ R (with its standard structure of Lie algebra) embeds correspondingly as a Lie subalgebra of R⊕L𝔤⊕R.

Observe that we can reverse the orientation of S^{1}. Namely let ψ : LG → LG

be the morphism g_{s} ↦ g_{−s}. Then ψ lifts to a morphism ˆ L̃G → ˆ L̃G, so that if (x, y) ∈ S^{1} × S^{1},

ψ(x, y) = (x^{−1}, y^{−1}) . (4.7)

By [45, Section 4.9], on ˆ L̃𝔤, there is a ˆ L̃G-invariant symmetric bilinear form

⟨(a, α, b), (a′, α′, b′)⟩ = ∫_{S^{1}}⟨α, α′⟩dt − ab′ − ba′ . (4.8)

By (4.8), we have the embedding

R ⊕ L𝔤 ⊂ (L𝔤 ⊕ R)^{∗} . (4.9)

Equivalently

L̂𝔤 ⊂ (L̃𝔤)^{∗} . (4.10)

4.2. The coadjoint orbits of LG. The group LG has a coadjoint action on the right on (L̃𝔤)^{∗}. If we restrict this action to L̂𝔤 ⊂ (L̃𝔤)^{∗}, for g ∈ LG, we get

(a\frac{d}{dt} + α) · g = a\frac{d}{dt} + ag^{−1} \frac{dg}{dt} + g^{−1}αg . (4.11)

Let P = S^{1} × G be the trivial G-bundle on S^{1}. Let A^{S^{1}} be the affine space of G-connections on S^{1}. A connection in A^{S^{1}} can be written as

d + A , A ∈ Ω^{1}(S^{1}, 𝔤) . (4.12)

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS 77

In the sequel, we identify d+A ∈ A^{S^{1}} with the differential operator \frac{d}{dt} +A(∂/∂t) ∈ L̂𝔤. So A^{S^{1}} is an affine subspace of L̂𝔤.

Clearly LG acts on the right on A^{S^{1}}, so that if g ∈ LG, d + A · g = g^{−1}(d + A)g (4.13) i.e. A · g = g^{−1}dg + g^{−1}Ag . (4.14) By (4.11), (4.14), A^{S^{1}} embeds as an affine suspace of (L̃𝔤)^{∗} = L̂𝔤 and the action of LG on (L̃𝔤)^{∗} induces the corresponding action of LG on A^{S^{1}}.

Now we briefly develop the theory of coadjoint orbits for L̃G. Definition 4.1. If A ∈ L𝔤, let w ∈ G be the holonomy of the operator \frac{d}{dt} + A on S^{1}, i.e. if g_{t} is the solution of

\frac{dg_{t}}{dt} g_{t}^{−1} + A_{t} = 0 , (4.15)

then w = g_{1} . (4.16)

Then Floquet’s theory of differential equations (Frenkel [20, Section 3.2], Segal [45, Proposition 4.3.6]) shows that the orbits of \frac{d}{dt} + A in L̂𝔤 can be expressed in terms of the adjoint orbits in G. Namely \frac{d}{dt} + A and \frac{d}{dt} + A′ lie in the same LG-orbit if and only if the corresponding holonomies w and w′ lie in the same Gorbit. In particular the LG orbits of the differential operator \frac{d}{dt} + A always contain representatives of the form \frac{d}{dt} + λ, λ ∈ 𝔱, and two \frac{d}{dt} + λ lie in the same LG-orbit if and only if λ and λ′ lie in the same W_{aff}-orbit.

If λ ∈ 𝔱, let Z(d/dt + λ) ⊂ LG be the stabilizer of d/dt + λ. Proposition 4.2. If λ ∈ 𝔱, Z(d/dt + λ) ≃ Z(e^{−λ}) . (4.17) Namely if g ∈ Z(e^{−λ}), the corresponding element of Z(d/dt + λ) is t ∈ S^{1} ↦ e^{−tλ}ge^{tλ} ∈ G. If e^{−λ} ∈ T_{reg}, then Z(d/dt + λ) = T ⊂ LG . (4.18) Proof. The proof of this simple result is left to the reader.

Let O_{d/dt+λ} ⊂ L̂𝔤 be the LG orbit of d/dt + λ. Clearly the map g ∈ LG ↦ g(d/dt + λ)g^{−1} ∈ L̂𝔤 (4.19) induces the identification LG/Z(d/dt + λ) ≃ O_{d/dt+λ} . (4.20) In particular if λ ∈ P, LG/T ≃ O_{d/dt+λ} . (4.21)

Also the theory of coadjoint orbits developed in Section 1.14 tells us that O_{d/dt+λ} is equipped with a symplectic form σ_{O_{d/dt+λ}}. Namely let d/dt+A = D^{A} ∈ O_{d/dt+λ}.

Recall that LG acts on the right on O_{d/dt+λ}. If α ∈ L𝔤, let α^{O_{d/dt+λ}} be the corresponding vector field on O_{d/dt+λ}. Clearly

α_{A}^{O_{d/dt+λ}} = D^{A}α . (4.22)

Then formula (1.193) for the symplectic form σ_{O_{d/dt+λ}} is

σ(D^{A}α, D^{A}β) = ⟨d/dt + A, [α, β]⟩ . (4.23)

By (4.3), (4.8), (4.23), we get

σ(D^{A}α, D^{A}β) = ∫_{S^{1}}⟨D^{A}α, β⟩ . (4.24)

Finally L̃G acts symplectically on the left and on the right O_{d/dt+λ}, and d/dt+A ↦ (d/dt + A) ∈ (L̃𝔤)^{∗} is a moment map for the left action of L̃G on O_{d/dt+λ}.

If p ∈ R^{∗}, λ ∈ 𝔱, if pd/dt + A ∈ O_{pd/dt+A}, then d/dt + A/p = D^{A/p} is a connection, and

σ_{O_{pd/dt+λ}}(D^{A}α, D^{A}β) = p ∫_{S^{1}}⟨D^{A/p}α, β⟩ . (4.25)

Recall that T × S^{1} ⊂ L̃G.

Definition 4.3. If (λ, p) ∈ \overline{CR}^{∗} × Z^{∗}, if ρ_{(λ,−p)} is the one dimensional representation of T × S^{1} of weight (λ, −p), let H_{(λ,p)} be the line bundle on L̃G/(T × S^{1}) = LG/T,

H_{(λ,p)} = L̃G ×_{ρ_{(λ,−p)}} C . (4.26)

In the same way as the Weyl group W acts on the right on G/T, the affine Weyl group W_{aff} ≃ W ⋊ \overline{CR} acts on the right LG/T. In fact if w ∈ W, w acts on LG/T by g ↦ gw and μ ∈ \overline{CR} acts on LG/T by g ↦ ge^{μt}. Then W_{aff} acts on the left on \overline{CR}^{∗} × Z^{∗} by the formula

(4.27)  w(λ, p) = (wλ, p)
μ(λ, p) = (λ + pμ, p) .

We can then restrict (λ, p) to vary in a fundamental domain of the action of W_{aff}. This just says that

p ∈ Z^{∗} , λ ∈ |p|\overline{P} . (4.28)

Equivalently if p ∈ Z^{∗},

for α ∈ R_{+} , 0 ≤ ⟨α, λ⟩ ≤ ⟨α_{0}, λ⟩ ≤ p . (4.29)

Let now p ∈ Z^{∗}, λ ∈ |p|P ∩ \overline{CR}^{∗}. Then by (4.18), (4.20),

O_{pd/dt+λ} ≃ LG/T . (4.30)

Therefore, the line bundle H_{(λ,p)} is well defined on O_{pd/dt+λ}. Also by proceeding as in Section 1.14, the Hermitian line bundle H_{(λ,p)} is equipped with a unitary connection ∇^{H_{(λ,p)}} and

c_{1}(H_{(λ,p)}, ∇^{H_{(λ,p)}}) = σ_{O_{pd/dt+λ}} . (4.31)

If p ∈ Z^{∗}, λ ∈ |p|\overline{P} ∩ \overline{CR}^{∗}, but λ ∉ pP, the theory is slightly more involved.

Now, we will just assume that λ ∈ \overline{CR}^{∗} and explain the construction of the line bundle H_{(λ,p)} on O_{pd/dt+λ} in full generality.

By (4.17),

Z(pd/dt + λ) ≃ Z(e^{−λ/p}) . (4.32)

Definition 4.4. Let Z̃(pd/dt + λ) be the stabilizer of pd/dt + λ in L̃G.

Then Z̃(pd/dt+λ) is a central extension of Z(pd/dt+λ). Let 𝔷(e^{−λ/p}), 𝔷̃(pd/dt+ λ) be the Lie algebras of Z(e^{−λ/p}), Z̃(pd/dt + λ). Then

𝔷̃(pd/dt + λ) ≃ 𝔷(e^{−λ/p}) ⊕ R . (4.33)

Proposition 4.5. If (X, a), (Y, b) ∈ 𝔷̃(pd/dt + λ) ≃ 𝔷(e^{−λ/p}) ⊕ R, then

[(X, a), (Y, b)] = ([X, Y], \frac{1}{p}⟨λ, [X, Y]⟩) . (4.34)

Proof. Clearly, if X ∈ 𝔷(e^{−λ/p}), the corresponding element in the Lie algebra 𝔷(pd/dt + λ) of Z(pd/dt + λ) is just e^{−tλ/p}Xe^{tλ/p}. In view of (4.3),

[(e^{−tλ/p}Xe^{tλ/p}, 0) , (e^{−tλ/p}Y e^{tλ/p}, 0)] (4.35)

(e^{−tλ/p}[X, Y]e^{tλ/p} , ⟨X, −[λ/p, Y]⟩) .

Also

⟨X, −[λ, Y]⟩ = ⟨λ, [X, Y]⟩ . (4.36)

The proof of our Proposition is completed.

Proposition 4.6. For any λ ∈ \overline{CR}^{∗},

(X, a) ∈ 𝔷(e^{−λ/p}) ⊕ R ↦ (X, ⟨\frac{λ}{p}, X⟩ + a) ∈ 𝔷̃(pd/dt + λ) (4.37)

is an isomorphism of Lie algebras.

Proof. This is clear by (4.34).

Remark 4.7. If λ ∈ pP ∩\overline{CR}^{∗}, so that e^{−λ/p} ∈ T_{reg}, then 𝔷(e^{−λ/p}) = 𝔱. In this case, in (4.34), ⟨λ, [X, Y]⟩ = 0. Therefore (λ, a) ∈ 𝔷(e^{−λ/p}) ⊕ R ↦ (X, a) ∈ 𝔷̃(pd/dt + λ) is also an isomorphism of Lie algebras.

Observe that T is a maximal torus in Z(e^{−λ/p}) . By [14, Corollaire 5.3.1], Z(e^{−λ/p})/T is simply connected.

Definition 4.8. Put

R_{e^{−λ/p}} = {α ∈ R ; ⟨α, λ/p⟩ ∈ Z} . (4.38)

We define CR_{e^{−λ/p}} as in (1.136). Let \overline{CR}_{e^{−λ/p}} ⊂ 𝔱 be the lattice spanned by CR_{e^{−λ/p}}.

By [15, Theorem V.7.1],

π_{1}(Z(e^{−λ/p})) = \frac{\overline{CR}}{\overline{CR}_{e^{−λ/p}}} . (4.39)

Put

\overline{R}_{e^{−λ/p}}^{∗} = \{t ∈ 𝔱 ; if α ∈ R_{e^{−λ/p}} , ⟨α, t⟩ ∈ Z\} . (4.40)

By [15, Proposition V.7.16],

Z(Z(e^{−λ/p})) = \frac{\overline{R}_{e^{−λ/p}}^{∗}}{\overline{CR}} . (4.41)

In particular

−λ/p ∈ \frac{\overline{R}_{e^{−λ/p}}^{∗}}{\overline{CR}} . (4.42)

Now \frac{\overline{R}_{e^{−λ/p}}^{∗}}{\overline{CR}} maps into the group π_{1}(Z(e^{−λ/p}))^{∗}. In particular the map

ψ(pd/dt + λ) : h ∈ π_{1}(Z(e^{−λ/p})) ↦ e^{2iπ⟨λ/p,h⟩} ∈ C (4.43)

is well-defined. Since λ ∈ \overline{CR}^{∗}, e^{2iπ⟨λ/p,h⟩} is a p^{th} root of unity.

Recall that pd/dt is identified to the 1-form a ∈ R ↦ −pa ∈ R. Also λ ∈ \overline{CR}^{∗} is a left invariant 1-form on L̃G.

By Proposition 1.56, pd/dt + λ is a closed 1-form on Z̃(pd/dt + λ).

Let Z̃(e^{−λ/p}) be the universal cover of Z(e^{−λ/p}). Then pd/dt is a closed form on Z̃(e^{−λ/p}) ×_{ψ(pd/dt+λ)} S^{1}.

Theorem 4.9. We have the identity of Lie groups Z̃(pd/dt + λ) ≃ Z̃(e^{−λ/p}) ×_{ψ_{(pd/dt+λ)}} S^{1} . (4.44)

In particular Z̃(pd/dt+λ) contains a p^{th} cover of Z(e^{−λ/p}). Under the identification (4.44), pd/dt + λ ≃ pd/dt, (4.45) and the forms in (4.45) are closed and integral.

Proof. Clearly by (4.37), Z̃(pd/dt + λ) is a locally trivial central extension of Z(e^{−λ/p}). Let s ∈ S^{1} ↦ t_{s} ∈ T be a smooth loop. The corresponding loop with values in Z(pd/dt + λ) is given by s ∈ S^{1} ↦ (e^{−tλ/p}t_{s}e^{tλ/p}) ∈ LG, i.e. by s ∈ S^{1} ↦ t_{s} ∈ G ⊂ LG . (4.46)

Now recall that G ∈ L̃G, so that we have a loop s ∈ S^{1} ↦ t_{s} ∈ G ∩ Z̃(pd/dt + λ) . (4.47)

By (4.37), the horizontal lift of s ∈ S^{1} ↦ t_{s} ∈ G ⊂ LG in Z̃(pd/dt+λ) with respect to the flat connection on Z̃(pd/dt + λ) → Z(e^{−λ/p}) is given by

s ∈ S^{1} ↦ t_{s} exp (2iπ ∫_{S^{1}}⟨λ/p, t_{s}^{−1}dt_{s}⟩) . (4.48)

From (4.48), we get (4.44).

If S_{p}^{1} denotes the group of p^{th} roots of unity in C, Z̃(e^{−λ/p}) ×_{ψ_{(pd/dt+λ)}} S_{p}^{1} is a subgroup of Z̃(pd/dt + λ), which is a p-covering of Z(e^{−λ/p}).

Observe that

⟨pd/dt + λ, (X, ⟨\frac{λ}{p}, X⟩ + a)⟩ = −pa = ⟨pd/dt, a⟩ . (4.49)

From (4.49) we get (4.45).

By Proposition 1.56, the 1-form pd/dt + λ is closed on Z̃(pd/dt + λ). It is even easier to see that pd/dt is closed on Z̃(e^{−λ/p}) ×_{ψ_{(pd/dt+λ)}} S^{1}. Finally it is trivial to

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS 81

verify that pd/dt is integral on Z̃(e^{−λ/p}) ×_{ψ(pd/dt+λ)} S^{1}. The proof of our Theorem is completed.

By Theorem 4.9, as in Proposition 1.58, the integral 1-form pd/dt + λ defines a representation ρ_{1} : Z̃(p\frac{d}{dt} + λ) → S^{1}. Also ρ_{2} : (g, x) ∈ Z̃(e^{−λ/p}) ×_{ψ_{(pd/dt+λ)}} S^{1} ↦ x^{−p} ∈ S^{1} is also a representation. Proposition 4.10. Under the isomorphism (4.44), ρ_{1} ≃ ρ_{2} . (4.50) Proof. By (4.45), we get (4.50). Remark 4.11. Assume that λ ∈ pP ∩ \overline{CR}^{∗}, so that Z(e^{−λ/p}) ≃ T. Then Z̃(e^{−λ/p}) ≃ 𝔱 . (4.51) Also the map (f, a) ∈ 𝔱 × S^{1} ↦ (f, e^{2iπ⟨λ/p,f⟩}a) ∈ 𝔱 × S^{1} (4.52) descends to an isomorphism T × S^{1} ≃ 𝔱 ×_{ψ_{(pd/dt+λ)}} S^{1} . (4.53) Then the 1-form pd/dt + λ on T × S^{1} corresponds to pd/dt on 𝔱 ×_{ψ_{(pd/dt+λ)}} S^{1}.

In fact in this case, Z̃(pd/dt + λ) = T × S^{1}, (4.54) and (4.53) is a special case of (4.44).

Clearly

\frac{L̃G}{Z̃(pd/dt + λ)} = \frac{LG}{Z(pd/dt + λ)} ≃ O_{pd/dt+λ} . (4.55)

Definition 4.12. Let H_{(λ,p)} be the line bundle on O_{pd/dt+λ}, H_{(λ,p)} = L̃G ×_{ρ_{1}} S^{1} . (4.56)

Then H_{(λ,p)} is a Hermitian line bundle. As in (1.198), we can equip H_{(λ,p)} with a unitary connection ∇^{H_{(λ,p)}}. By proceeding as in (1.199), we get c_{1}(H_{(λ,p)}, ∇^{H_{(λ,p)}}) = σ_{O_{pd/dt+λ}} . (4.57) 4.3. A central extension of (LG)^{s}. Definition 4.13. Put

P_{s} = {(a_{1}, …, a_{s}) ∈ (S^{1})^{s} , ∏_{j=1}^{s} a_{j} = 1} . (4.58)

Then P_{s} is a Lie subgroup of (S^{1})^{s}, and its Lie algebra 𝔭_{s} is given by

𝔭_{s} = {(b_{1}, …, b_{s}) ∈ R^{s} , ∑_{j=1}^{s} b_{j} = 0} . (4.59)

Clearly P_{s} ⊂ (S^{1})^{s} is a Lie subgroup of (L̃G)^{s}.

Definition 4.14. Put

(4.60)  (L̃G)^{s} = (L̃G)^{s}/P_{s},
(L̃𝔤)^{s} = (L̃𝔤)^{s}/𝔭_{s} .

Then (L̃G)^{s} is a central extension of (LG)^{s} and (L̃𝔤)^{s} is its Lie algebra. Clearly

(L̃𝔤)^{s} = (L𝔤)^{s} ⊕ R . (4.61)

Also if (α_{1}, …, α_{s}, a), (α′_{1}, …, α′_{s}, a′) ∈ (L̃𝔤)^{s}, then

[(α_{1}, …, α_{s}, a), (α′_{1}, …, α′_{s}, a′)] = ([α_{1}, α′_{1}], …, [α_{s}, α′_{s}] , ∑_{j=1}^{s} ∫_{S^{1}}⟨α_{j}, dα′_{j}⟩) . (4.62)

Also G^{s} × S^{1} embeds as a subgroup of (L̃G)^{s}, and 𝔤^{s} ⊕ R embeds as a Lie subalgebra of (L̃𝔤)^{s}. Finally recall that in Section 4.1, we defined an action of S^{1} on LG. Therefore S^{1} acts on (LG)^{s}.

Definition 4.15. Put

(4.63)  (L̂G)^{s} = S^{1} ⋉ (LG)^{s},
(L̂𝔤)^{s} = R ⊕ (L𝔤)^{s} .

Then (L̂𝔤)^{s} is the Lie algebra of (L̂G)^{s}. If we embed S^{1}, R into (S^{1})^{s}, R^{s} by the diagonal embeddings, then (L̂G)^{s} ⊂ (L̂G)^{s}, and (L̂𝔤)^{s} is a Lie subalgebra of (L̂𝔤)^{s}. Also by (4.10),

(L̂𝔤)^{s} ⊂ (L̃𝔤)^{s∗}. (4.64)

4.4. The holonomy of L̃G. Let κ be the closed left invariant 3-form on G,

κ(X, Y, Z) = \frac{1}{2}⟨X, [Y, Z]⟩ . (4.65)

Then by [45, Proposition 4.4.5],

κ ∈ H^{3}(G, Z) . (4.66)

Recall that η is the left-invariant 2-form on LG,

η = \frac{1}{2} ∫_{S^{1}}⟨g^{−1}dg, \frac{∂}{∂t}g^{−1}dg⟩dt . (4.67)

Put

Δ = {z ∈ C , |z| ≤ 1} . (4.68)

Let g(s, t) : S^{1} × S^{1} → G be a smooth map. We identify g(s, t) with the smooth loop in LG s ∈ S^{1} ↦ g_{s} = g(s, ·) ∈ LG.

Clearly

∂Δ = S^{1} . (4.69)

Since π_{i}(G) = 0, 0 ≤ i ≤ 2, g(s, t) extends to a smooth map g(z, t) : Δ × S^{1} → G. Therefore we get a map \overline{g} : z ∈ Δ ↦ g(z, .) ∈ LG.

Theorem 4.16. The following identity holds

\frac{1}{2} ∫_{S^{1}×S^{1}}⟨g^{−1} \frac{∂g}{∂t} , g^{−1} \frac{∂g}{∂s} ⟩ dtds (4.70)

− ∫_{Δ×S^{1}} g^{∗}κ + ∫_{Δ} \overline{g}^{∗}η = 0 .

Proof. Let h be the left-invariant 1-form on LG,

h = \frac{1}{2} ∫_{S^{1}}⟨g^{−1} \frac{∂g}{∂t} , g^{−1}dg⟩dt . (4.71)

A straightforward computation shows that

dh = −η + \frac{1}{2} ∫_{S^{1}}⟨g^{−1} \frac{∂g}{∂t} , \frac{1}{2} [g^{−1}dg, g^{−1}dg]⟩dt . (4.72)

Using (4.72), we get (4.70).

Clearly

L̃𝔤 = L𝔤 ⊕ R . (4.73)

The splitting (4.73) defines a connection on the S^{1} bundle \xymatrix{L̃G \ar[r]^{S^{1}} & LG} . Let

s ∈ S^{1} ↦ g̃_{s} ∈ L̃G be a horizontal lift of s ∈ S^{1} ↦ g_{s} ∈ LG. Since g_{0} = g_{1}, then

g̃_{1}g_{0}^{−1} ∈ S^{1} . (4.74)

Theorem 4.17. The following identity holds

(4.75) g̃_{1}g_{0}^{−1} = exp (2iπ ∫_{Δ} \overline{g}^{∗}η)
= exp (2iπ (−\frac{1}{2} ∫_{S^{1}}⟨g^{−1} \frac{∂g}{∂t} , g^{−1} \frac{∂g}{∂s} ⟩dsdt + ∫_{Δ×S^{1}} g^{∗}κ)) .

Proof. The first identity is a straightforward consequence of (4.2), (4.67). The second identity follows from Theorem 4.16.

4.5. The symplectic space of connections on Σ. Let X be a compact Riemann surface of genus g. Let x_{1}, …, x_{s} be s distinct elements of X. Let Δ_{1}, …, Δ_{s} be small non intersecting small open disks of center x_{1}, …, x_{s}.

Put

Σ = X\ ⋃_{j=1}^{s} Δ_{j} . (4.76)

Then Σ is a compact Riemann surface with boundary ∂Σ. Also Σ being oriented, ∂Σ is also naturally oriented.

Let G be a compact connected and simply connected compact simple Lie group. We use otherwise the same notation as in Section 1. In particular 𝔤 denotes the Lie algebra of G, and ⟨, ⟩ denotes the basic scalar product on 𝔤 defined in Section 1.2.

Let P = Σ × G be the trivial G-bundle on Σ. Observe that since G is simply connected, a G-bundle on Σ is necessarily trivial.

Definition 4.18. Let A be the affine space of G-connections on P.

84 JEAN-MICHEL BISMUT AND FRANÇOIS LABOURIE

Any element in A can be written in the form d+A, with A ∈ Ω^{1}(Σ, 𝔤). Therefore A is an affine space with underlying vector space Ω^{1}(Σ, 𝔤). Also T A = A × Ω^{1}(Σ, 𝔤) . (4.77)

Let ΣG, Σ𝔤 be the sets of smooth maps Σ → G, Σ → 𝔤. The Σ𝔤 is a Lie algebra, which will be considered as the Lie algebra of ΣG. Definition 4.19. If U, V ∈ Ω^{1}(Σ, 𝔤), put σ(U, V) = ∫_{Σ} −⟨U ∧ V ⟩ . (4.78)

Then σ is a symplectic form on A.

Observe that ΣG acts on the right on A, so that if g ∈ ΣG, d + A · g = g^{−1}(d + A)g , (4.79) i.e. A · g = g^{−1}Ag + g^{−1}dg . (4.80) Clearly ΣG preserves the symplectic form σ. Definition 4.20. If α ∈ Σ𝔤, let α^{A} be the corresponding vector field on A.

Let ∇^{A} be the covariant derivative associated to the connection A ∈ A. Then α_{A}^{A} = ∇_{.}^{A}α . (4.81)

For simplicity, we now fix an oriented parametrization of the s circles in ∂Σ, so that ∂Σ = (S^{1})^{s} . (4.82) Definition 4.21. Let r be the restriction map

(4.83)  g ∈ ΣG ↦ g_{|∂Σ} ∈ (LG)^{s} ,
α ∈ Σ𝔤 ↦ α_{|∂Σ} ∈ (L𝔤)^{s} ,
d + A ∈ A ↦ (d + A)_{|∂Σ} ∈ (A^{S^{1}})^{s} .

All these maps are equivariant. By (4.83), we get a map

d + A ↦ (\frac{d}{dt} + A(\frac{d}{dt}))_{|∂Σ} ∈ (L̂𝔤)^{s} . (4.84)

We still denote by ρ the projection (L̃G)^{s} → (LG)^{s}. Definition 4.22. Put

(4.85) Σ̃G = {(g, g′) ∈ ΣG × (L̃G)^{s} ; g_{|∂Σ} = ρg′ in (LG)^{s}} ,
Σ̃𝔤 = {(α, α′) ∈ Σ𝔤 × (L̃𝔤)^{s} ; α_{|∂Σ} = ρα′ in (L𝔤)^{s}} .

Then Σ̃𝔤 is the Lie algebra of the Lie group Σ̃G. More precisely Σ̃𝔤 = Σ𝔤 ⊕ R . (4.86) Let τ be the projection Σ̃𝔤 = Σ𝔤 ⊕ R → R. If (α, a), (α′, a′) ∈ Σ̃𝔤, then

[(α, a), (α′, a′)] = ([α, α′], ∫_{∂Σ}⟨α, dα′⟩) . (4.87)

Also G × S^{1} embeds as a subgroup of Σ̃G by the map (g, t) ↦ (g, (g, …, g), t) ⊂ ΣG × (L̃G)^{s}.

Clearly the restriction map r extend to a map

(4.88)  Σ̃G → (L̃G)^{s} ,
Σ̃𝔤 → (L̃𝔤)^{s} .

In particular, there is a dual map

(L̃𝔤)^{s∗} → (Σ̃𝔤)^{∗} . (4.89)

If A ∈ A, then by (4.64),

\frac{d}{dt} + A(∂/∂t) ∈ (L̃𝔤)^{s∗} . (4.90)

So by (4.89), (4.90), we may view (d + A)_{|∂Σ} as an element of (Σ̃𝔤)^{∗}.

As we just saw, ΣG acts on A and preserves the symplectic form σ. Therefore Σ̃G also acts symplectically on A. Of course S_{1} ⊂ Σ̃G acts trivially on A. If α̃ ∈ Σ̃𝔤, let α^{A} be the associated vector field on A. If α = ρα̃,

α^{A} = α^{A} = ∇_{.}^{A}α . (4.91)

Clearly

Σ𝔤 = Ω^{0}(Σ, 𝔤) . (4.92)

Also

Ω^{2}(Σ, 𝔤) ⊂ Ω^{0}(Σ, 𝔤)^{∗} = (Σ𝔤)^{∗} . (4.93)

Definition 4.23. If A ∈ A, let F^{A} = dA + \frac{1}{2}[A, A] ∈ Ω^{2}(Σ, 𝔤) be the curvature of A.

By (4.93), if A ∈ A, F^{A} ⊂ (Σ𝔤)^{∗} ⊂ (Σ̃𝔤)^{∗}. Also by (4.89), (4.90) if A ∈ A, (d + A)_{|∂Σ} ∈ (Σ̃𝔤)^{∗}.

Recall that in Section 3.2, moment maps were defined with respect to symplectic actions of compact Lie groups on symplectic manifolds. Here we will use the same definition with respect to an action of Σ̃G on A. We now state an extension of a result of Atiyah-Bott [2, Section 9].

Theorem 4.24. The map

A ∈ A ↦ μ(A) = F^{A} − (d + A)_{|∂Σ} ∈ (Σ̃𝔤)^{∗} (4.94)

is a moment map for the action of Σ̃G on A.

Proof. First we will prove that if α̃ ∈ Σ̃𝔤,

d⟨F^{A} − (d + A)_{|∂Σ}, α̃⟩ = i_{α^{A}}σ . (4.95)

Clearly if α̃ = (α, a_{1}, …, a_{s}) ∈ Σ̃𝔤⊕R^{s} is identified with the corresponding element in Σ̃𝔤,

⟨F^{A} − (d + A), α̃⟩ = ∫_{Σ}⟨F^{A}, α⟩ − ∫_{∂Σ}⟨A, α⟩ + ∑_{j=1}^{s} a_{j} . (4.96)

If B ∈ T A ≃ Ω^{1}(Σ, 𝔤), then

(4.97)  ⟨B, d⟨F^{A} − (d + A)_{|∂Σ}, α̃⟩⟩ = ∫_{Σ}⟨D^{A}B, α⟩ − ∫_{∂Σ}⟨B, α⟩
= − ∫_{Σ}⟨∇^{A}α, B⟩
= i_{α^{A}}σ(B) .

By (4.97), ⟨F^{A} − (d + A)_{|∂Σ}, α̃⟩ is a Hamiltonian for the vector field α^{A}. Also by definition, if g ∈ Σ̃G, F^{A·g} − (d + A · g)_{|∂Σ} = (F^{A} − (d + A)_{|∂Σ}) · g . (4.98) We have completed the proof of our Theorem. 4.6. The canonical line bundle on A. Now we describe the construction of the canonical line bundle L on A. In the case where there are no marked points, our construction follows closely earlier work by Ramadas, Singer and Weitsman [46]. Definition 4.25. Let (L, ∥ ∥_{L}) be the trivial Hermitian line bundle on A. Let ∇^{L} be the unitary connection (L, ∥ ∥_{L}) :

∇^{L} = d − iπ ∫_{Σ}⟨., A⟩ . (4.99)

Proposition 4.26. The following identity holds c_{1}(L, ∇^{L}) = σ . (4.100) Proof. This follows from (4.99). Definition 4.27. If α̃ ∈ Σ̃𝔤, put Lα̃ = ∇_{α^{A}}^{L} − 2iπ⟨μ(A), α̃⟩ . (4.101)

Clearly for α̃ ∈ Σ̃𝔤, Lα̃ acts on C^{∞}(A, L). Proposition 4.28. If α̃ ∈ Σ̃𝔤, then [Lα̃, ∇^{L}] = 0 . (4.102) Also if α̃, β̃ ∈ Σ̃𝔤, [Lα̃, Lβ̃] = L_{[α̃,β̃]} . (4.103) Proof. The identities (4.102) and (4.103) are trivial consequences of (3.120) and of Theorem 4.24. Proposition 4.29. If α̃ = (α, a) ∈ Σ̃𝔤 = Σ𝔤 ⊕ R, then

Lα̃ = ∇_{α^{A}} − iπ ∫_{Σ}⟨A, dα⟩ − 2iπa . (4.104)

Proof. By (4.99), (4.101), we get

(4.105)  Lα̃ = ∇_{α^{A}} − iπ ∫_{Σ}⟨∇^{A}α, A⟩ − 2iπ ∫_{Σ}⟨F^{A}, α⟩
+2iπ ∫_{∂Σ}⟨A, α⟩ − 2iπa .

Also

(4.106)  ∫_{Σ}⟨∇^{A}α, A⟩ = − ∫_{Σ}⟨A, dα⟩ − ∫_{Σ}⟨[A, A], α⟩ ,
∫_{Σ}⟨F^{A}, α⟩ = ∫_{Σ}⟨dA + \frac{1}{2}[A, A], α⟩
= ∫_{∂Σ}⟨A, α⟩ + ∫_{Σ}(⟨A, dα⟩ + \frac{1}{2}⟨[A, A], α⟩) .

From (4.100), (4.106), we get (4.104). The proof of our Theorem is completed.

Remark 4.30. By (4.104), if α, β ∈ Σ𝔤,

[L_{α}, L_{β}] = L_{[α,β]} − 2iπ ∫_{∂Σ}⟨α, dβ⟩ . (4.107)

In view of (4.87), (4.104), (4.107) fits with (4.103).

Now we will show that the action of Σ̃𝔤 on L lifts to an action of Σ̃G. This result has been proved by Ramadas, Singer and Weitsman [46] in the case where there are no marked points, so that Σ̃G is just ΣG.

Theorem 4.31. The action of Σ̃𝔤 on L defined in (4.101) lifts uniquely to a unitary action of Σ̃G on L which preserves ∇^{L}. In particular if (g, t) ∈ G × S^{1} ⊂ Σ̃G, (g, t) acts on L by

f ∈ L_{A} ↦ f · (g, t) = e^{2iπt}f ∈ L_{A·g} . (4.108)

Proof. Let g ∈ ΣG. Since π_{i}(G) = 0, 0 ≤ i ≤ 2, there is a smooth path s ∈ [0, 1] ↦ g_{s} ∈ ΣG such that

(4.109)  g_{0} = 1 ,
g_{1} = g .

Recall that

Σ̃𝔤 = Σ𝔤 ⊕ R . (4.110)

The splitting (4.110) defines a left-invariant connection on the S^{1}-bundle \xymatrix{Σ̃G \ar[r]^{S^{1}} & ΣG} . Let s ∈ [0, 1] ↦ g̃_{s} be the horizontal lift of s ∈ [0, 1] ↦ g_{s} ∈ ΣG with respect to this connection, with g̃_{0} = 1. Put

(4.111)  α_{s} = g_{s}^{−1} \frac{dg_{s}}{ds} ∈ Σ𝔤 ,
g̃ = g̃_{1} .

Then

g_{s}^{−1} \frac{dg̃}{ds} = α_{s} ∈ Σ̃𝔤 . (4.112)

Now we will show that if A ∈ A, f ∈ L_{A},

f · g̃ = exp (iπ ∫_{0}^{1} ds [∫_{Σ}⟨A · g_{s}, dα_{s}⟩]) f ∈ L_{A·g} (4.113)

is an unambigous formula for the action of Σ̃G on L. Clearly we may assume that g_{1} = g_{0} = 1, so that s ∈ S_{1} = R/Z ↦ g_{s} ∈ ΣG is a loop in ΣG.

88 JEAN-MICHEL BISMUT AND FRANÇOIS LABOURIE

Recall that by(4.61), (L̃𝔤)^{s} = (L𝔤)^{s} ⊕ R . (4.114) The splitting (4.114) defines a connection on the S^{1} bundle (L̃G)^{s} → (LG)^{s}. Then by definition s ∈ [0, 1] ↦ g̃_{s|∂Σ} ∈ (L̃G)^{s} is the horizontal lift of s ∈ S^{1} ↦ g_{s} ∈ (LG)^{s}. In particular g̃_{1|∂Σ} ∈ S^{1}.

By (4.104), we only need to verify that in this case, exp (iπ ∫_{0}^{1} ds [∫_{Σ}⟨A · g_{s}, dα_{s}⟩]) = g̃_{1} . (4.115) Observe that A · g_{s} = g_{s}^{−1}Ag_{s} + g_{s}^{−1}dg_{s} . (4.116) Let θ^{g}, θ^{d} be the 𝔤-valued left and right invariant forms on G, i.e. θ^{g} = g^{−1}dg, θ^{d} = dgg^{−1} . (4.117) Then

dθ^{g} = −\frac{1}{2}[θ^{g}, θ^{g}] dθ^{d} = \frac{1}{2}[θ^{d}, θ^{d}] . (4.118)

From (4.111), (4.118), we get

(4.119)  dα_{s} = \frac{∂}{∂s}g_{s}^{∗}θ^{g} + [α_{s}, g_{s}^{∗}θ^{g}]
= g_{s}^{−1} \frac{∂}{∂s}(g_{s}^{∗}θ^{d})g_{s} .

Therefore

⟨g_{s}^{−1}Ag_{s}, dα_{s}⟩ = \frac{∂}{∂s}⟨A, g_{s}^{∗}θ^{d}⟩ . (4.120)

Since g_{0} = g_{1} = 1, by (4.120), we obtain ∫_{0}^{1} ds ∫_{Σ}⟨g_{s}^{−1}Ag_{s}, dα_{s}⟩ = 0 . (4.121) By (4.121), ∫_{0}^{1} ds ∫_{Σ}⟨A · g_{s}, dα_{s}⟩ = ∫_{S^{1}} ds ∫_{Σ}⟨g_{s}^{−1}dg_{s}, dα_{s}⟩ (4.122) and so (4.122) does not depend on A. Observe that this last fact follows from general principles, and that the right-hand side of (4.122) is just the left-hand side evaluated at A = 0.

Also

∫_{Σ}⟨g_{s}^{−1}dg_{s}, dα_{s}⟩ = − ∫_{∂Σ}⟨g_{s}^{∗}θ^{g}, α_{s}⟩ − \frac{1}{2} ∫_{Σ}⟨g_{s}^{∗}[θ^{g}, θ^{g}], α_{s}⟩ . (4.123)

In view of (4.65), (4.123), we get

\frac{1}{2} ∫_{0}^{1} ds ∫_{Σ}⟨g_{s}^{−1}dg_{s}, dα_{s}⟩ = −\frac{1}{2} ∫_{S^{1}} ds ∫_{∂Σ}⟨g_{s}^{∗}θ^{g}, α_{s}⟩ − ∫_{Σ×S^{1}} g^{∗}κ . (4.124)

Recall that Σ is obtained from the Riemann surface X by deleting s disks Δ_{1}, …, Δ_{s} centered at x_{1}, …, x_{s}. Since π_{i}(G) = 0, 0 ≤ i ≤ 2, the smooth map g : S^{1} × Σ ↦ G extends to a smooth map g : S^{1} × X → G such that

g = 1 near S^{1} × {x_{j}} , 1 ≤ j ≤ s . (4.125)

From (4.124), we get

\frac{1}{2} ∫_{0}^{1} ds ∫_{Σ}⟨g_{s}^{−1}dg_{s}, dα_{s}⟩ = −\frac{1}{2} ∫_{S^{1}} ds ∫_{∂Σ}⟨g_{s}^{∗}θ^{g}, α_{s}⟩ + ∑_{j=1}^{s} ∫_{S^{1}×Δ_{j}} g^{∗}κ − ∫_{S^{1}×Δ_{j}} g^{∗}κ. (4.126)

Since κ ∈ H^{3}(G, Z),

∫_{S^{1}×X} g^{∗}κ ∈ Z. (4.127)

For 0 ≤ r ≤ 1, t ∈ S^{1}, then rt ∈ Δ. If (z, t) ∈ Δ × S_{j}^{1}, put

h(z, t) = g(\frac{z}{|z|}, |z|t). (4.128)

By (4.125), h is a smooth map from Δ × S_{j}^{1} into G. Let \overline{h} : Δ → (LG)^{s} be the smooth map defined the way we did after (4.69).

If we orient S_{j}^{1} as a component of ∂Σ, we get

−\frac{1}{2} ∫_{S^{1}} ds ∫_{∂Σ}⟨g_{s}^{∗}θ^{g}, α_{s}⟩ = −\frac{1}{2} ∑_{j=1}^{s} ∫_{S^{1}×S_{j}^{1}}⟨h^{−1} \frac{∂h}{∂t}, h^{−1} \frac{∂h}{∂s}⟩dtds. (4.129)

Also for 1 ≤ j ≤ s, by orienting S_{j}^{1} as before,

∫_{S^{1}×Δ_{j}} g^{∗}κ = ∫_{Δ×S_{j}^{1}} h^{∗}κ. (4.130)

Using Theorem 4.16 and (4.129)-(4.130), we get

(4.131)  −\frac{1}{2} ∫_{S^{1}} ds ∫_{∂Σ} ⟨g_{s}^{∗}θ^{g}, α_{s}⟩ + ∑_{j=1}^{s} ∫_{S^{1}×Δ_{j}} g^{∗}κ
= ∫_{Δ} \overline{h}^{∗}η .

Using now Theorem 4.17 and (4.122), (4.126), (4.127), (4.131), we get (4.115). So formula (4.113) for f · g̃ is unambiguous. Also by (4.102), Σ̃G preserves ∇^{L}.

Assume now that g ∈ G. Let α ∈ 𝔤 be such that g = exp(α). Put

g_{s} = exp(sα) . (4.132)

Recall that G ⊂ Σ̃G. Then for s ∈ [0, 1], g_{s} ∈ Σ̃G. By (4.113), if f ∈ L_{A},

f · g = f ∈ L_{A·g} . (4.133)

Finally, by (4.104), if 𝔱 ∈ S^{1},

f · t = e^{2iπt}f . (4.134)

The proof of our Theorem is completed.

Remark 4.32. By (4.120), in (4.124), we get

\frac{1}{2} ∫_{0}^{1} ds ∫_{Σ}⟨A · g_{s}, dα_{s}⟩ = \frac{1}{2} ∫_{Σ}⟨A, dg_{1}g_{1}^{−1}⟩ (4.135)

−\frac{1}{2} ∫_{0}^{1} ds ∫_{∂Σ}⟨g_{s}^{−1}dg, g_{s}^{−1} \frac{∂g}{∂s} ⟩

− ∫_{Σ×[0,1]} g^{∗}κ .

So (4.135) makes more explicit the dependence of (4.113) on A. Also if f ∈ C^{∞}(A, L), g̃ ∈ Σ̃G, let g̃f ∈ C^{∞}(A, L) be given by

g̃f(A) = g^{−1}f(A · g̃) . (4.136)

Then (4.136) defines a representation of Σ̃G on C^{∞}(A, L). In this representation t ∈ S^{1} acts like e^{−2iπt}. Also if α̃ ∈ Σ̃G,

\frac{d}{dt}e^{tα̃} · f = L_{α^{A}}f . (4.137)

4.7. The non simply connected case. Assume now that G is a connected semisimple compact Lie group, which is not necessarily simply connected. Let G̃ be the universal cover of G. Then π_{1}(G) ⊂ Z(G̃) and G = G̃/π_{1}(G).

Let P_{g,s} be a 4g + 3s polygon covering Σ. The edges of P_{g,s} are denoted a_{1}, b_{1}, a_{1}^{−1}, b_{1}^{−1}, …, a_{g}^{−1}, b_{g}^{−1}, c_{1}, d_{1}, c_{1}^{−1}, …, c_{s}, d_{s}, d_{s}^{−1}. For 1 ≤ j ≤ s, set w_{j} = c_{j}d_{j}c_{j}^{−1}. Let q be an element of Σ which is a common point to the a_{i}, b_{i}, w_{j}. Then π_{1}(q, Σ) is generated by the circles a_{1}, b_{1}, …, a_{g}, b_{g}, w_{1}, …, w_{s}. The case g = 1, s = 1 is represented in Figure 5.1.

Let \xymatrix{P \ar[r]^{G} & Σ} be a G-bundle on Σ. Take x ∈ Σ \ (∂Σ ⋃ ∪_{i=1}^{g}(a_{i} ∪ b_{i})). Then Σ\{x} retracts on (∪_{i=1}^{g}a_{i} ∪ b_{i}) ⋃(∪_{j=1}^{s}w_{j}). Since G is connected, the restriction of P to the 1-skeleton (∪_{i=1}^{g}a_{i} ∪ b_{i}) ⋃(∪_{j=1}^{s}w_{j}) is trivial. Therefore P_{Σ\{x}} is trivial, i.e.

P_{|Σ\{x}} ≃ Σ \ {x} × G . (4.138)

Let Δ be a small disk in Σ centered at x. We orient ∂Δ as part of ∂(Σ \ Δ). Then

P_{|Δ} ≃ Δ × G . (4.139)

Let σ : Δ \ {x} → G be the transition map describing the G-bundle P on Σ, i.e.

(y, g) ∈ P_{|Δ} ≃ (y, σg) ∈ P_{|Σ\{x}} , y ∈ Δ \ {x} . (4.140)

Then the homotopy classes of G-bundles are classified by the homotopy classes of maps : Δ \ {x} → G. Since Δ \ {x} retracts on ∂Δ ≃ S^{1}, the homotopy classes of G bundles are specified by the element [P] of π_{1}(G) associated to σ_{|∂Δ} : S^{1} → G.

Clearly [σ] does not depend on the trivialization of π on Δ.

The map π → [P] ∈ π_{1}(G) depends explicitly on the trivialization of P on Σ\{x}. More precisely the homotopy classes of trivializations of P on the 1-skeleton are given by π_{1}(G)^{2g+s}. This just says that if a trivialization of P on Σ\{x} is given, all the other trivializations are given by the action of π_{1}(G)^{2g+s} on this trivialization.

Let p be the map :

p : (a_{1}, b_{1}, …, a_{g}, b_{g}, c_{1}, …, c_{s}) ∈ π_{1}(G)^{2g+s} ↦ ∑_{j=1}^{s} c_{j} ∈ π_{1}(G) . (4.141)

Then if given c ∈ π_{1}(G)^{2g+s}, we replace the given trivialization of P on Σ \ {x} by the action of c, [P] is changed into [P] − p(c).

Finally, with the above provisos, the map P → [P] ∈ π_{1}(G) does not depend on the choice of x.

Clearly, if \xymatrix{P \ar[r]^{G} & Σ} is trivial, P lifts to a G̃-bundle. Conversely, if \xymatrix{P \ar[r]^{G} & Σ}

lifts to a G̃-bundle \xymatrix{Q \ar[r]^{G̃} & Σ} , since Q is trivial, and P = Q/π_{1}(G), P is also trivial. Therefore [P] ∈ π_{1}(G) describes the obstruction to a lifting of the G-bundle P to a G̃-bundle Q.

As before we fix a trivialization of P on Σ \ {x}.

Let ∇^{P} be a connection on the G-bundle \xymatrix{P \ar[r]^{G} & Σ} . Since

P_{|Σ\{x}} ≃ Σ \ {x} × G , (4.142)

P_{|Σ\{x}} lifts to the trivial G̃-bundle.

Q = Σ \ {x} × G̃ . (4.143)

The connection ∇^{P} lifts to a connection ∇^{Q} on Q. Clearly, this construction of Q depends on the choice of the trivialization of P on Σ \ {x}.

We will write that a family of circles Δ of center x tends to x if their radius tends to 0. Clearly as Δ → x, the holonomy of ∇^{P} around x tends to 1.

Proposition 4.33. As Δ → x, the holonomy of ∇^{Q} around ∂Δ ⊂ ∂(Σ \ Δ) tends to [P] ∈ π_{1}(G) ⊂ Z(G̃). In particular if ∇^{P} if flat, for Δ small enough, the holonomy of ∇^{Q} around ∂Σ \ Δ is equal to [P].

Proof. We fix an origin in S^{1} ≃ ∂Δ. Let t ∈ [0, 1] ↦ τ_{t} ∈ G, τ_{0} = 1 be a horizontal lift of S^{1} ⊂ ∂(Σ \ Δ) in P_{|∂Δ} with respect to ∇^{P}, in the trivialization P_{|Δ} ≃ Δ × G. Then t ∈ [0, 1] ↦ σ_{t}τ_{t}σ_{0}^{−1} ∈ G denotes the corresponding horizontal lift of S^{1} in the trivialization P_{|Σ\{x}} ≃ Σ \ {x} × G.

Let t ∈ [0, 1[↦ σ̃_{t} ∈ G̃, t ∈ [0, 1] ↦ τ̃_{t} ∈ G̃ be lifts of t ∈ [0, 1[↦ σ_{t} ∈ G, t ∈ [0, 1[↦ τ_{t} ∈ G, with τ̃_{0} = 1. Then t ∈ [0, 1] ↦ σ̃_{t}τ̃_{t}σ_{0}^{−1} ∈ G̃ is a horizontal lift of S^{1} ≃ ∂Δ in Q_{|∂Δ} in the trivialization Q_{|Σ\{x}} ≃ Σ \ {x} × G̃, with respect to ∇^{Q}. In particular, parallel transport along S^{1} ≃ ∂Δ is given by σ̃_{1}τ̃_{1}σ_{0}^{−1}. Now recall that by definition

σ̃_{1} = [P]σ̃_{0} , [P] ∈ π_{1}(G) ⊂ Z(G̃) . (4.144)

Therefore the above parallel transport is just [P]σ̃_{0}τ̃_{1}σ_{0}^{−1}.

Now as Δ → x, τ_{1} → 1. Therefore τ̃_{1} → 1, and the parallel transport tends to [P].

The proof of our Proposition is completed.

4.8. The case of a connected subgroup of a simply connected group. Let now G be a connected and simply connected compact simple Lie group. Let H ⊂ G be a connected compact semisimple Lie subgroup of G.

Let \xymatrix{P \ar[r]^{G} & Σ} be a G-bundle on Σ. Then

P ≃ Σ × G . (4.145)

Reducing the G-bundle P to a H-bundle Q is equivalent to finding a section of P ×_{G} G/H ≃ Σ × G/H. Let ˙1 ∈ G/H be the image of 1 ∈ G. Then if η ∈ Σ(G/H), the corresponding H-bundle Q is given by

Q = {(y, g) ∈ Σ × G , g^{−1}η = ˙1} . (4.146)

Therefore homotopy classes of H-reductions of P are just homotopy classes in Σ(G/H).

Since π_{i}(G) = 0, 0 ≤ i ≤ 2, we get

(4.147)  π_{2}(G/H) ≃ π_{1}(H) ,
π_{i}(G/H) = 0 , 0 ≤ i ≤ 1 .

Take x ∈ Σ \ (∂Σ ⋃ ∪_{i=1}^{g}(a_{i} ∪ b_{i})) as in Section 4.7. Recall that Σ \ {x} is homotopy equivalent to ∪_{i=1}^{g}(a_{i} ∪ b_{i}) ⋃ ∪_{j=1}^{s}w_{j}. Since π_{i}(G/H) = 0, 0 ≤ i ≤ 1, there is only one homotopy class of sections Σ \ {x} → G/H.

If η is a section of Σ × G/H, we may and we will assume that η = ˙1 on Σ \ Δ. In this case

Q_{|Σ\Δ} = Σ × H, (4.148)

and Q has a canonical section over Σ \ Δ.

Therefore homotopy classes of H-reductions of P are classified by homotopy classes of maps η : Δ → G/H such that η_{|∂Δ} = ˙1, i.e. by π_{2}(G/H) ≃ π_{1}(H).

Take z ∈ π_{2}(G/H) ≃ π_{1}(H). Let Q_{z} be the H-reduction of P associated to z. By the above, Q_{|Σ\{x}} has been canonically trivialized on Σ \ {x}. Since Q is a

H-bundle, the class [Q_{z}] ∈ π_{1}(H) is well-defined.

Proposition 4.34. The following identity holds

[Q_{z}] = z in π_{1}(H) . (4.149)

Proof. Let y ∈ Δ ↦ h(y) ∈ G/H be a smooth map such that h_{|∂Δ} = ˙1, representing z in π_{1}(H) ≃ π_{2}(G/H). Then

Q_{z|Δ} = {(y, g) ∈ Σ × G ; g^{−1}h(y) = ˙1} . (4.150)

To trivialize Q_{z} on Δ, we fix a connection ∇^{Q_{z}} on Q_{z}. If (x, g) ∈ Q_{z,x}, we parallel transport g along radial lines in Δ. This way, we obtain g_{1} : ∂Δ → H. By definition

[g_{1}] = z in π_{1}(H) . (4.151)

Also σ = g_{1} exactly defines the H-bundle Q_{z} on Σ in the sense of (4.140). From (4.151), we get (4.149). The proof of our Proposition is completed.

4.9. The action of stabilizers on the line bundle L. Now we use the notation of Section 1, and more especially of Section 1.9.

Let u ∈ C/\overline{CR}. We will specialize the results of Section 4.8 to the case H = Z(u). Clearly the map

g ∈ G/Z(u) ↦ gug^{−1} ∈ O_{u} ⊂ G (4.152)

is one to one. It maps ˙1 into u.

If η is a section of Σ × O_{u} over Σ, the corresponding Z(u)-bundle Q is given by

Q = {(x, g) ∈ Σ × G , g^{−1}ηg = u} . (4.153)

Since Q_{|Σ\{x}} ≃ Σ \ {x} × Z(u), Q has a section over Σ \ {x}. Equivalently, there is g ∈ Σ \ {x} × G such that

η = gug^{−1} on Σ \ {x} . (4.154)

Homotopy classes of trivializations of Q on Σ \ {x} are just homotopy classes of g ∈ Σ \ {x} such that (4.154) holds. We have already classified these homotopy classes by π_{1}(Z(u))^{2g+s}. By (4.147), we know that η_{|Σ\{x}} is homotopy equivalent to the constant η = ˙1 ≃ u.

In the sequel, we will assume that η = u on Σ \ Δ which corresponds to η = ˙1 on Σ \ Δ .

Similarly

Q_{|Δ} ≃ Δ × Z(u) (4.155)

i.e. Q_{|Δ} has a section. Therefore there exists g ∈ ΔG such that

η = gug^{−1} on Δ . (4.156)

In particular by (4.156), g_{|∂Δ} takes its values in Z(u), so that g_{|∂Δ} ∈ LZ(u). Also there is only one homotopy class of g ∈ ΔG such that (4.156) holds.

Then σ = g_{|∂Δ} ∈ LZ(u) is exactly the σ in (4.140) defining the Z(u)-bundle Q on Σ.

Recall that [σ] ∈ π_{1}(Z(u)) ≃ \overline{CR}/\overline{CR}_{u}. We identify [σ] to a given element [σ] ∈ \overline{CR}. Let s ∈ S^{1} ↦ b_{s} ∈ T be a loop which represents [σ] in π_{1}(Z(u)). Put

g^{0} = (bg^{−1})_{|∂Δ} ∈ LZ(u) . (4.157)

Then g_{|∂Δ}^{0} is homotopic to 0. Equivalently g_{|∂Δ}^{0} defines an element of LZ̃(u). Since

π_{i}(Z̃(u)) = 0, 0 ≤ i ≤ 2, g^{0} extends to g^{0} ∈ ΣZ(u).

By conjugation of η by g^{0} ∈ ΣZ(u), we may and we will assume in the sequel that

η = u on Σ \ Δ , (4.158)

η = gug^{−1} on Δ ,
g ∈ ΔG, g_{|∂Δ} ∈ LT .

Clearly ∫_{∂Σ\Δ} g^{−1}dg ∈ \overline{CR} is the homotopy class of g ∈ LT.

Let now η ∈ ΣG. Put

A^{η} = {A ∈ A ; A · η = A} . (4.159)

Assume that

A^{η} ≠ ∅ . (4.160)

94 JEAN-MICHEL BISMUT AND FRANÇOIS LABOURIE

Put u = η(x) . (4.161) In the sequel we assume that u ∈ C/\overline{CR} . (4.162)

Then by (4.159), (4.160), it is clear that η is a section of the orbit O = O_{u}. In the sequel we assume that the conditions in (4.158) hold.

Let B ∈ 𝔱 be such that u = exp(B) . (4.163) For s ∈ [0, 1], x ∈ Σ \ Δ, put \overline{η}(x, s) = exp(sB) . (4.164) Then since π_{i}(G) = 0, 0 ≤ i ≤ 2, the map \overline{η} : Σ \ Δ × [0, 1] → G extends to a map \overline{η} : Σ × [0, 1] → G such that

• \overline{η}_{|Σ\Δ×[0,1]} = exp(sB) , (4.165)

• \overline{η}_{|Σ×\{1\}} = η .

For s ∈ [0, 1], put η_{s}(x) = \overline{η}(x, s) . (4.166) Then s ∈ [0, 1] ↦ η_{s} ∈ ΣG is a smooth path. Let s ∈ [0, 1] ∈ η̃_{s} ∈ Σ̃G be the corresponding horizontal lift in Σ̃G. Theorem 4.35. The following identity holds

η̃_{1}|_{L|A^{η}} = exp (−2iπ⟨∫_{∂Σ\Δ} g^{−1}dg, B⟩) . (4.167)

Proof. We consider the splitting Σ = Σ \ Δ ∪ Δ . (4.168) Now both Σ \ Δ and Δ are objects like Σ. They carry the trivial G-bundles P_{|Σ\Δ} and P_{Δ}. Therefore to Σ \ Δ and Δ, we associate the spaces of G-connections A^{Σ\Δ} and A^{Δ}, the line bundles L^{Σ/Δ} and L^{Δ}, equipped with the actions of { \atop (Σ/Δ)G} and Δ̃G.

Clearly A embeds into A^{Σ\Δ} × A^{Δ}, and ΣG embeds into (Σ \ Δ)G × ΔG. First we claim that L = (L^{Σ\Δ} ⊗ L^{Δ})_{|A} (4.169) and the isomorphism (4.169) identifies the metrics and the connections. In fact this is clear by (4.99).

Put ε = {(x, y) ∈ S^{1} × S^{1} , xy = 1} . (4.170) Then ({ \atop (Σ \ Δ)G} × Δ̃G)/ε acts naturally on L^{Σ\Δ} ⊗ L^{Δ}.

We claim that Σ̃G embeds as a subgroup of ({ \atop (Σ \ Δ)G} × Δ̃G)/ε. In fact recall that we orient ∂Σ \ Δ as the boundary of Σ \ Δ. Also in Section 4.1, we defined the morphism f ∈ L̃G ↦ ψf ∈ L̃G. So if f ∈ Σ̃G, let g̃ ∈ L̃G be such that

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS 95

ρg̃ = f_{|∂Σ\Δ}. Then to f ∈ Σ̃G, we associate ((f_{|Σ\ΔG}, g̃), (f_{ΔG}, ψg̃)) ∈ ({ \atop (Σ \ Δ)G}× Δ̃G)/ε. Since as we saw in (4.7), if s ∈ S^{1}, ψs = s^{−1}, we find that this element of ({ \atop (Σ \ Δ)G} × Δ̃G)/ε does not depend on the choice of g̃.

Now Σ̃G acts on L. Similarly, by the above embedding, Σ̃G also acts on (L^{Σ\Δ} ⊗ L^{Δ})_{|A}. We claim that both actions coincide. In fact this is obvious by the explicit formula in (4.113).

Let η_{s}^{Σ\Δ} and η_{s}^{Δ} be the restrictions of η_{s} to Σ \ Δ and Δ. Let s ∈ [0, 1] ↦ η_{s}^{Σ\Δ} ∈ { \atop Σ \ ΔG}, s ∈ [0, 1] ↦ η_{s}^{Δ} ∈ Δ̃G be the corresponding horizontal lifts. By the above, we get the equality in S^{1}, η̃_{1|L} = η_{1|L^{Σ\Δ}}^{Σ\Δ}η_{1|L^{Δ}}^{Δ} . (4.171) Now we will compute both terms in (4.171). By (4.165), η_{s}^{Σ/Δ} = exp(sB) . (4.172) Using (4.113), we obtain η_{1|L^{Σ\Δ}}^{Σ\Δ} = 1 . (4.173)

Consider the path s ∈ [0, 1] ↦ θ_{s} ∈ ΔG, with θ_{s} = g exp(sB)g^{−1} . (4.174) Clearly by (4.158), (4.165), (4.174), η_{1}^{Δ} = θ_{1} . (4.175) Moreover since g_{|∂Δ} takes its values in T, η_{s|∂Δ}^{Δ} = θ_{s|∂Δ} . (4.176) Let s ∈ [0, 1] ↦ θ̃_{s} ∈ Δ̃G be the horizontal lift of s ∈ [0, 1] ↦ θ_{s} ∈ ΔG. By (4.175), (4.176), we get η_{1}^{Δ} = θ̃_{1} . (4.177)

Set κ_{s} = exp(sB) ∈ ΔG . (4.178) Let s ∈ [0, 1] ↦ κ̃_{s} ∈ Δ̃G be the horizontal lift of s ∈ [0, 1] ↦ κ_{s} ∈ ΔG.

Recall that L̃𝔤 = L𝔤 ⊕ R . (4.179) By [45, Proposition 4.3.2], we get g(B, 0)g^{−1} = (gBg^{−1}, − ∫_{∂Δ}⟨g^{−1}dg, B⟩) . (4.180)

Using (4.180), we obtain

gκ̃_{1}g^{−1} = θ̃_{1} exp (−2iπ ∫_{∂Δ}⟨g^{−1}dg, B⟩) . (4.181)

Clearly κ_{1} = u , (4.182)

96 JEAN-MICHEL BISMUT AND FRANÇOIS LABOURIE

and so θ_{1} = gκ_{1}g^{−1} . (4.183) If A ∈ A^{Δ,θ_{1}}, then A · g ∈ A^{Δ,κ_{1}}. By (4.181), we get

κ_{1|L^{Δ}} = θ̃_{1|L^{Δ}} exp (−2iπ ∫_{∂Δ}⟨g^{−1}dg, B⟩) . (4.184)

Finally using (4.113) again, we find κ_{1|L^{Δ}} = 1 . (4.185) From (4.177), (4.184), (4.185), we obtain

(4.186)  η_{1|L^{Δ}}^{Δ} = exp (2iπ ∫_{∂Δ}⟨g^{−1}dg, B⟩)
= exp (−2iπ ∫_{∂Σ\Δ}⟨g^{−1}dg, B⟩) .

By (4.171), (4.173), (4.186), we get (4.167). The proof of our Theorem is completed.

4.10. The action of stabilizers on the line bundle λ_{p}. Take now θ_{1}, …, θ_{s} ∈ \overline{CR}^{∗}. Take p ∈ Z^{∗}. Let L_{1}, …, L_{s} be the canonical line bundles constructed in Sections 1.14 and 4.2, which are associated to the L̃G orbits of (−\frac{pd}{dt} + θ_{1}, …, −\frac{pd}{dt} + θ_{s}). Definition 4.36. Put

(4.187)  A_{p}(θ_{1}, …, θ_{s}) = \{A ∈ A, −p(\frac{d}{dt} + A)_{|S_{1}^{1}} ∈ O_{−p \frac{d}{dt} +θ_{1}}, …,

−p(\frac{d}{dt} + A)_{|S_{s}^{1}} ∈ O_{−p \frac{d}{dt} +θ_{s}} .

Let ν_{p} : A_{p}(θ_{1}, …, θ_{s}) ↦ ∏_{j=1}^{s} O_{−p \frac{d}{dt} +θ_{j}} be given by

A ↦ ν_{p}(A) = ( − p(\frac{d}{dt} + A_{|S_{1}^{1}}), …, −p(\frac{d}{dt} + A_{|S_{s}^{1}})) . (4.188)

Equivalently, A_{p}(θ_{1}, …, θ_{s}) is the set of A ∈ A such that the holonomy of A on S_{1}^{1} lies in the G-orbit of e^{θ_{1}/p}, … the holonomy of A on S_{s}^{1} lies in the G-orbit of e^{θ_{s}/p}. Definition 4.37. Let λ_{p} be the line bundle on A_{p}(θ_{1}, …, θ_{s}), λ_{p} = L^{p} ⊗ ν_{p}^{∗}( ⊗_{j=1}^{s} L_{j}) . (4.189)

Clearly ΣG acts on the right on A_{p}(θ_{1}, …, θ_{s}). By Theorem 4.31, this action lifts to an action on the right of Σ̃G on λ_{p}. Proposition 4.38. The action of Σ̃G on λ_{p} descends to an action of ΣG on λ_{p}. Proof. We only must show that S^{1} ⊂ Σ̃G acts trivially on λ_{p}. However by (4.108), if t ∈ S^{1}, t acts on the right on L^{p} like e^{2iπpt}. Also t acts on the right on ⊗_{j=1}^{s}L_{j} like e^{−2iπpt}. Our Proposition follows.

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS 97

Let η ∈ ΣG. Assume that A^{η} ∩ A_{p}(θ_{1}, …, θ_{s}) ≠ ∅ . (4.190) Put u = η(x) . (4.191) By conjugating η by g ∈ G, we may and we will assume that u ∈ T. We identify u to a corresponding element in 𝔱, which we still denote by u. In the sequel, we assume that u ∈ C/\overline{CR}, (4.192) so that Z(u) is semisimple.

Now if A ∈ A^{η}, the G-bundle P on Σ reduces to a Z(u)-bundle Q on Σ.

Recall that by (4.138), Q_{|Σ\{x}} ≃ Σ \ {x} × Z(u) . (4.193) Put Q̃_{|Σ\{x}} = Σ \ {x} × Z̃(u) . (4.194) Then the connection A lifts to a Z̃(u) connection on Q̃_{|Σ\{x}}. Still two different trivializations of Q_{|Σ\{x}} produce two distinct Z̃(u) bundles Q̃_{|Σ\{x}}.

Recall that by (1.173),

O_{θ_{j}/p} ∩ Z(u) = ⋃_{w∈W_{u}\W} O_{Z(u)}(wθ_{j}/p) . (4.195)

To normalize Q_{|Σ\{x}}, we impose that the holonomy of the connection A along S_{j}^{1} be conjugate in Z̃(u) to e^{w^{j}θ_{j}/p} ∈ Z̃(u), with w^{j} ∈ W_{u}\W.

Let then [Q] ∈ π_{1}(Z(u)) ≃ \frac{\overline{CR}}{\overline{CR}_{u}} be defined in Sections 4.7-4.9. Theorem 4.39. The following identity holds

η_{|λ_{p}}_{|A^{η}∩A_{p}(θ_{1},... ,θ_{s})} = exp (−2iπ⟨∑_{j=1}^{s} w^{j}θ_{j} + p[Q], u⟩) . (4.196)

Proof. As we saw in (4.158), by conjugating η by an element of ΣG, we may and we will assume that

(4.197)  η = u on Σ \ Δ ,
= gug^{−1} on Δ with g ∈ ΔG , g_{|∂Δ} ∈ T .

Then Q_{|Σ\Δ} = Σ \ Δ × Z(u) . (4.198) Let ∇^{P} be a connection on P which preserves η. Then ∇^{P} induces a connection ∇^{Q}, which, in the trivialization associated to (4.198), is given by ∇^{Q} = d + A , A ∈ Ω^{1}(Σ \ Δ, 𝔷(u)) . (4.199) The connection ∇^{Q} lifts to a connection ∇Q̃ given by ∇Q̃ = d + A . (4.200)

By our choice of Q̃, we may and we will assume that the holonomy of ∇Q̃ on S_{j}^{1} lies in the Z̃(u) conjugacy class of e^{w^{j}θ_{j}/p}.

By [20, Section 3.2], [45, Proposition 4.3.6] for 1 ≤ j ≤ s, there is h̃_{j} ∈ LZ̃(u) such that

h̃_{j}^{−1} (\frac{d}{dt} + A_{|S_{j}^{1}}) h̃_{j} = \frac{d}{dt} − w_{j} \frac{θ_{j}}{p} . (4.201)

Since Z̃(u) is simply connected, there is h ∈ ΣZ̃(u) such that

h̃_{|S_{j}^{1}} = h̃_{j} , 1 ≤ j ≤ s . (4.202)

Let h ∈ ΣZ(u) be the image of h̃. Then h_{|∂Δ} ∈ LZ(u) is homotopic to the constant loop 1. By proceeding as in (4.157), we may and we will assume that h_{|∂Δ} ∈ LT is homotopic to the constant loop 1 in Z(u).

Now we may as well replace η by h^{−1}ηh and A ∈ A^{η} by A · h ∈ A^{h^{−1}ηh}. By (4.197),

(4.203)  h^{−1}ηh_{|Δ} = h^{−1}gug^{−1}h_{|Δ} ,
h^{−1}g_{|∂Δ} ∈ T.

Finally the homotopy class of h^{−1}g_{|S_{j}^{1}} ∈ LZ(u) is the same as the homotopy class of g_{|S_{j}^{1}} ∈ LZ(u), 1 ≤ j ≤ s.

So basically, we may and we will assume that η verifies the assumptions in (4.197), but we also have the extra assumptions

(\frac{d}{dt} + A)_{|S_{j}^{1}} = \frac{d}{dt} − w^{j} \frac{θ_{j}}{p} , 1 ≤ j ≤ s . (4.204)

Now by definition, the right action of u ∈ L̃G on L_{j,−pd/dt+w^{j}θ_{j}} is given by exp(−2iπ⟨u, w^{j}θ_{j}⟩) ∈ S^{1}. Using (4.167), we see that

η_{|λ_{p}} = exp (2iπ⟨− ∑_{j=1}^{s} w^{j}θ_{j} − p ∫_{∂Σ\Δ} g^{−1}dg, u⟩) . (4.205)

By Proposition 4.34,

∫_{∂Σ\Δ} g^{−1}dg = [Q] ∈ π_{1}(Z(u)) . (4.206)

From (4.205), (4.206), we get (4.196).

The proof of our Theorem is completed.

Remark 4.40. It should be pointed out that the expression (4.196) is natural. In fact it only depends on the w^{j} ∈ W_{u}\W. Also as we saw after (4.141), we know how [Q] changes if we change the trivialization of Q_{|Σ\Δ}. One verifies easily that (4.196) is compatible with this formula.

Also by Proposition 1.44, one verifies easily that (4.196) is compatible with the fact that ΣG acts as a group on λ_{p}.

5. The moduli space of flat bundles on a Riemann surface

The purpose of this Section is to construct the moduli space M/G of flat Gbundles over a Riemann surface Σ with marked points. We establish the Witten formula [59, 60] for the symplectic volume of M/G. Also we show that the formula of Witten [60] and Jeffrey-Kirwan [26] can be applied to M/G. In particular, we express the integrals of certain characteristic classes over M/G in terms of the action of certain differential operators on local polynomials over the maximal torus T. Also we give a formula for c_{1}(T M/G) . All these results will be needed in Section 6, when we apply the Theorem of Riemann-Roch-Kawasaki [30, 31] to M/G .

As explained in the introduction , our derivation of the Witten formula is closely related to earlier work by Liu [37, 38].

This Section is organized as follows. In Section 5.1, we give the standard combinatorial description of Σ. In Section 5.2, we introduce the corresponding combinatorial complexes, which compute the absolute and relative cohomology of flat vector bundles over Σ. In Section 5.3, we introduce the G-equivariant map ϕ : X = G^{2g} × ∏_{1}^{s} O_{j} → G, such that M = {x ∈ X, ϕ(x) = 1}. Also we relate the differential of ϕ to the combinatorial complexes on Σ which compute the absolute and relative cohomology of the flat adjoint vector bundle E. In Section 5.4, we give natural conditions on the orbits O_{j} under which 1 is a regular value of ϕ, so that M/G is an orbifold. In Section 5.5, we give conditions under which the set of elements in the fibres of ϕ or in X with non trivial stabilisers are of codimension ≥ 2. In Section 5.6, we describe T M/G and the symplectic form ω. In Section 5.7, we show that the symplectic volume form on M/G can be evaluated in terms of the corresponding combinatorial complexes. In Section 5.8, using the results of the previous subsections, we prove Witten’s formula [59, 60] for the symplectic volume of the reductions of M/G.

In Section 5.9, we define logarithms from certain subsets of G into 𝔤. In Section 5.10, we show that the invariant open sets of X where G acts locally freely are naturally equipped with a symplectic form, and that the action of G on these sets has a moment map. In Section 5.11, we compute the integrals of certain characteristic classes on the moduli spaces associated to the centralizers Z(u), u ∈ C/\overline{R}^{∗}. As we will see in Section 6.4, these moduli spaces correspond to the strata of M/G. In Section 5.12, we compute certain Euler characteristics. Finally, in Section 5.13, we give a formula for c_{1}(T M/G).

5.1. Combinatorial description of the Riemann surface Σ. If a, b lie in a group Γ, put

[a, b] = aba^{−1}b^{−1} . (5.1)

Let g ∈ N, s ∈ N, g + s > 0. Let Γ be the discrete group generated by 1, u_{1}, v_{1}, …, u_{g}, v_{g}, w_{1}, …, w_{s}, and the relation

∏_{i=1}^{g}[u_{i}, v_{i}] ∏_{j=1}^{s} w_{j} = 1 . (5.2)

Let X be an oriented connected compact surface of genus g. Let x_{1}, …, x_{s} be s distinct elements of X. Put

X′ = X \ {x_{1}, …, x_{s}} . (5.3)

Figure 5.1

Let Δ_{1}, …, Δ_{s} be small non intersecting open disks in X, centered at p_{1}, …, p_{s}. Let Σ be the surface with boundary

Σ = X\ ⋃_{j=1}^{s} Δ_{j} . (5.4)

Let q ∈ Σ. Then Γ can be canonically identified with π_{1}(q, Σ) = π_{1}(q, X′). Let P_{g,s} be a 4g+3s polygon covering Σ. The edges of P_{g,s} are denoted a_{1}, b_{1}, a_{1}^{−1}, b_{1}^{−1}, …, a_{g}^{−1}, b_{g}^{−1}, c_{1}, d_{1}, c_{1}^{−1}, …, c_{s}, d_{s}, c_{s}^{−1}. Then P_{g,s} induces a cell decomposition of Σ with one two-cell, 2g + 2s one-cells, 1 + s zero-cells. The two-cell is the interior

P^{∘}_{g,s} of P_{g,s}. The 2g + 2s one-cells are the circles a_{1}, b_{1}, …, a_{g}, b_{g}, d_{1}, …, d_{s}, and

the segments c_{1}, …, c_{s}. The 1 + s zero-cells consist of q which lies in the a_{i}, b_{i}, and is a boundary point for the c_{i}, and also r_{1}, …, r_{s} which lie respectively in c_{1}, …, c_{s} and in d_{1}, …, d_{s}. The case g = 1, s = 1 is represented in Figure 5.1, where q^{1}, …, q^{5} represent q, and r_{1}^{1}, r_{2}^{2} represent r_{1}. In the description given above, the group π_{1}(q, Σ) is generated by the circles u_{1} = a_{1}, v_{1} = b_{1}, …, u_{g} = a_{g}, v_{g} = b_{g}, w_{1} = c_{1}d_{1}c_{1}^{−1}, …, w_{s} = c_{s}d_{s}c_{s}^{−1}.

Also the above decomposition induces a cell decomposition of ∂Σ, with s 1-cells d_{1}, …, d_{s}, and s 0-cells y_{1}, …, y_{s}.

To the above cell-decompositions of Σ, we can associate the corresponding complexes over Z, (C_{.}^{Σ}, ∂), (C_{.}^{∂Σ}, ∂) which calculate the homology groups H_{.}(Σ, Z), H_{.}(∂Σ, Z). Let (C_{.}^{Σ,r}, ∂) be the quotient complex defined by the exact sequence

0 → (C_{.}^{∂Σ}, ∂) → (C_{.}^{Σ}, ∂) → (C_{.}^{Σ,r}, ∂) → 0 . (5.5)

Then the homology of (C_{.}^{Σ,r}, ∂) is the relative homology H((Σ, ∂Σ), Z).

Note in particular that C_{2}^{Σ,r} is generated by P^{∘}, C_{1}^{Σ,r} by the 2g + s one-cells a_{1}, b_{1}, …, a_{g}, b_{g}, c_{1}, …, c_{s}, C_{0}^{Σ,r} by the zero-cell q.

5.2. The combinatorial complexes on Σ. Let V be a finite dimensional complex vector space. Let τ : Γ → Aut(V) be a represention of Γ. Equivalently let u_{i}, v_{i} (1 ≤ i ≤ g), w_{j} (1 ≤ j ≤ s) in Aut(V) such that

∏_{i=1}^{i}[u_{i}, v_{i}] ∏_{j=1}^{s} w_{j} = 1 (5.6)

Let π : Σ̃ → Σ be the universal cover of Σ. Put

F = Σ̂ ×_{Γ} V . (5.7)

Then F is a flat complex vector bundle on Σ. Let ∇^{F} be the flat connection on F.

To the above cell decompositions of Σ, ∂Σ, we associate the corresponding combinatorial complexes (C_{.}^{Σ}(F), ∂), (C_{.}^{∂Σ}(F), ∂), (C_{.}^{Σ,r}(F), ∂), whose homologies are respectively H_{.}(Σ, F), H_{.}(∂Σ, F), H_{.}((Σ, ∂Σ), F), and which fit in the exact sequence of complexes

0 → (C_{.}^{∂Σ}(F), ∂) → (C_{.}^{Σ}(F), ∂) → (C_{.}^{Σ,r}(F), ∂) → 0 . (5.8)

For s > 0, we have the associated long exact sequence

0 → H_{2}((Σ, ∂Σ), F) → H_{1}(∂Σ, F) → H_{1}(Σ, F) → H_{1}((Σ, ∂Σ), F) (5.9)

→ H_{0}(∂Σ, F) → H_{0}(Σ, F) → 0

(in (5.9), we used the fact that if s > 0, H_{2}(Σ, F) = 0, H_{0}((Σ, ∂Σ), F) = 0).

Let p be the barycenter of the 2-cell P^{∘}, let q^{1}, …, q^{4g+1}, r_{1}^{1}, r_{1}^{2}, …, r_{s}^{1}, r_{s}^{2} be the vertices of P (see Figure 5.1 for the case g = 1, s = 1).

Then C_{2}^{Σ}(F) is the space of flat sections of π^{∗}F on P^{∘}, and can be identified to F_{p}. Similarly C_{1}^{Σ}(F) is the space of flat sections of F in the “interior” of the corresponding one-cells. We identify (π^{∗}F)_{q_{1}} to π^{∗}F_{p} by parallel transport with respect to π^{∗}∇^{F} along a radial line connecting p to q^{1}. Along ∂P , we identify π^{∗}F to (π^{∗}F)_{q_{1}} by clockwise parallel transport with respect to π^{∗}∇^{F}.

Then we identify C_{1}^{Σ}(F)

• over a_{1}, with (π^{∗}F)_{q^{1}} .

• over b_{1}, with (π^{∗}F)_{q^{2}} .

• over a_{2}, with (π^{∗}F)_{q^{5}} .

• over b_{2}, with (π^{∗}F)_{q^{6}} .

⋮

• over c_{1}, with (π^{∗}F)_{q^{4g+1}} .

• over d_{1}, with (π^{∗}F)_{r_{1}^{1}}

⋮

Of course ultimately, (π^{∗}F)_{q^{1}}, (π^{∗}F)_{q^{2}}, (π^{∗}F)_{q^{5}}, … are identified to π^{∗}F_{p} as indicated before.

Finally C_{0}^{Σ}(F) is identified to (π^{∗}F)_{q^{1}} ⊕ (⨁_{j=1}^{s} F_{r_{j}^{1}}), which itself is identified to a sum of 1 + s copies of (π^{∗}F)_{p}.

So C_{2}^{Σ}(F) is just (π^{∗}F)_{p}, C_{1}^{Σ}(F) is a sum of 2g+3s copies of (π^{∗}F)_{p}, and C_{0}^{Σ}(F) a sum of 1 + s copies of (π^{∗}F)_{p}.

102 JEAN-MICHEL BISMUT AND FRANÇOIS LABOURIE

Then one verifies easily that if f ∈ C_{2}^{Σ}(F) ≃ (π^{∗}F)_{p}, ∂f = (1 − u_{1}v_{1}^{−1}u_{1}^{−1})f_{|a_{1}} + (1 − v_{1}u_{1}v_{1}^{−1})u_{1}^{−1}f_{|b_{1}} +

(1 − u_{2}v_{2}^{−1}u_{2}^{−1})[v_{1}, u_{1}]f_{|a_{2}} + (1 − v_{2}u_{2}v_{2}^{−1})u_{2}^{−1}[v_{1}, u_{1}]f_{|b_{2}} + … +

(1 − w_{1}^{−1})[v_{g}, u_{g}] … [v_{1}, u_{1}]f_{|c_{1}} + [v_{g}, u_{g}] … [v_{1}, u_{1}]f_{|d_{1}} (5.10)

+(1 − w_{2}^{−1})w_{1}^{−1}[v_{g}, u_{g}] … [v_{1}, u_{1}]f_{|c_{2}} + w_{1}^{−1}[v_{g}, u_{g}] … [v_{1}, u_{1}]f_{|d_{2}} + … Similarly ∂ : C_{1}^{Σ}(F) → C_{0}^{Σ}(F) is such that if f ∈ (π^{∗}F)_{p},

(5.11)  ∂(f_{|a_{i}}) = (1 − u_{i}^{−1})f_{|q^{1}} ,
∂(f_{|b_{i}}) = (1 − v_{i}^{−1})f_{|q^{1}} ,
∂(f_{|c_{j}}) = f_{|q^{1}} − f_{r_{j}^{1}},
∂(f_{|d_{j}}) = (1 − w_{j}^{−1})f_{|r_{j}^{1}}.

In view of (5.6), (5.10), (5.11) one verifies that, as should be the case, ∂^{2} = 0.

By restriction to ∂Σ, we obtain the chain map ∂ : C_{1}^{∂Σ}(F) → C_{0}^{∂Σ}(F) given by ∂f_{|d_{j}} = (1 − w_{j}^{−1})f_{|r_{j}^{1}} . (5.12)

The complex (C^{Σ,r}(F), ∂) is obtained by making formally f_{|r_{j}^{1}} = 0, f_{|d_{j}} = 0 in (5.10), (5.11).

Note that for s > 0, we recover the fact that H_{2}(Σ, F) = 0 , H_{0}((Σ, ∂Σ), F) = 0 . (5.13) Proposition 5.1. The following identity holds H_{2}((Σ, ∂Σ), F) ≃ {f ∈ (π^{∗}F)_{p}, (u_{1} − 1)f = 0, (v_{1} − 1)f = 0, …, (5.14)

(u_{g} − 1)f = 0, (v_{g} − 1)f = 0 ,
(w_{1} − 1)f = 0, …, (w_{s} − 1)f = 0} .

Proof. By (5.10), if f ∈ C_{2}^{Σ,r}(F), the condition ∂f = 0, can be written as u_{1}v_{1}^{−1}u_{1}^{−1}f = f , (5.15)

[v_{1}, u_{1}]f = u_{1}^{−1}f ,
u_{2}v_{2}^{−1}u_{2}^{−1}[v_{1}, u_{1}]f = [v_{1}, u_{1}]f ,
[v_{2}, u_{2}][v_{1}, u_{1}]f = u_{2}^{−1}[v_{1}, u_{1}]f,
⋮
(1 − w_{1}^{−1})[v_{g}, u_{g}] … [v_{1}, u_{1}]f = 0 ,
(1 − w_{2}^{−1})w_{1}^{−1}[v_{g}, u_{g}] … [v_{1}, u_{1}]f = 0 .
⋮

From the first two equalities in (5.15), [v_{1}, u_{1}]f = v_{1}f = u_{1}^{−1}f , (5.16) so that v_{1}^{−1}u_{1}^{−1}f = f . (5.17)

From (5.16), (5.17), we find that the first two equalities in (5.15) are equivalent to

u_{1}f = v_{1}f = f . (5.18)

By proceeding as before, we get from (5.15),

u_{i}f = f , v_{i}f = f , 1 ≤ i ≤ g , (5.19)

so that

w_{j}f = f , 1 ∈ j ≤ s . (5.20)

The proof of our Proposition is completed.

Let F^{∗} be the flat vector bundle on Σ, which is dual to F. Then if V^{∗} is dual to V, F^{∗} is the flat bundle on Σ associated to the dual representation τ^{∗} : Γ → Aut(V^{∗}).

Let (C^{Σ.}(F), ∂), (C^{∂Σ.}(F), ∂), (C^{Σ,r}(F), ∂) be the complexes dual to (C_{.}^{Σ}(F^{∗}), ∂), (C_{.}^{∂Σ}(F^{∗}), ∂), (C_{.}^{Σ,r}(F^{∗}), ∂). Then we have the exact sequence of complexes

0 → (C^{Σ,r.}(F), ∂) → (C^{Σ.}(F), ∂) → (C^{∂Σ.}(F), ∂) → 0 . (5.21)

The associated cohomology groups are respectively H^{.}((Σ, ∂Σ), F), (H^{.}(Σ, F), H(∂Σ, F). For s > 0, the corresponding long exact sequence

0 → H^{0}(Σ, F) → H^{0}(∂Σ, F) → H^{1}((Σ, ∂Σ), F) → H^{1}(Σ, F) → (5.22)

H^{1}(∂Σ, F) → H^{2}((Σ, ∂Σ), F) → 0

is dual to the exact sequence (5.9) for F^{∗}.

Now we describe the chain map ∂ in the complexes (5.21). We trivialize the flat vector bundle F as indicated above. In particular the complexes in (5.21) are now direct sums of copies of (π^{∗}F)_{p}. By (5.10), (5.11), we find that if f ∈ π^{∗}F_{p}, ∂ : C^{Σ,0}(F) → C^{Σ,1}(F) is such that

∂(f|q^{1}) = ∑_{i=1}^{g}((1 − u_{i})f|a_{i} + (1 − v_{i})f|b_{i}) + ∑_{j=1}^{s} f|c_{j} , (5.23)

∂(f|r_{j}^{1}) = −f|c_{j} + (1 − w_{j})f|d_{j},

and ∂ : C^{Σ,1}(F) → C^{Σ,2}(F) by

(5.24)  ∂(f|a_{i}) = [u_{1}, v_{1}] … [u_{i−1}, v_{i−1}](1 − u_{i}v_{i}u_{i}^{−1})f ,
∂(f|b_{i}) = [u_{1}, v_{1}] … [u_{i−1}, v_{i−1}]u_{i}(1 − v_{i}u_{i}^{−1}v_{i}^{−1})f ,
∂(f|c_{j}) = [u_{1}, v_{1}] … [u_{g}, v_{g}]w_{1} … w_{j−1}(1 − w_{j})f ,
∂(f|d_{j}) = [u_{1}, v_{1}] … [u_{g}, v_{g}]w_{1} … w_{j−1}f .

The complex (C^{Σ,r,.}(F), ∂) is obtained formally from (5.23), (5.24) by making the components indexed by d_{j} or r_{j}^{1} equal to 0. Finally ∂ : C^{0,∂Σ}(F) → C^{1,∂Σ}(F) is given by

∂(f|r_{j}^{1}) = (1 − w_{j})f|d_{j} . (5.25)

Observe that by (5.23), (5.24), it is clear that for s > 0,

H^{2}(Σ, F) = 0 , H^{0}((Σ, ∂Σ), F) = 0 . (5.26)

Proposition 5.2. The following identity holds,

(5.27) H^{0}(Σ, F) = {f|q^{1} + ∑_{j=1}^{s} f|r_{j}^{1} , with f ∈ π^{∗}F_{p} such that

(1 − u_{i})f = 0, (1 − v_{i})f = 0 (1 ≤ i ≤ g) , (1 − w_{j})f = 0 (1 ≤ j ≤ s)} .

Proof. Our identity follows from (5.23).

Remark 5.3. By Poincaré duality

H_{2}((Σ, ∂Σ), F) ≃ H^{0}(Σ, F) . (5.28)

Using Propositions 5.1 and 5.2, the identification (5.28) has been made explicit. Another interpretation is to view the complex (C^{Σ.}(F), ∂) as the relative homology complex associated to the dual cell decomposition of Σ.

Suppose that s ≥ 1. Let (K^{.}, ∂) be the trivial complex concentrated in degree 1,

K^{1} = ⨁_{j=1}^{s} ker(1 − w_{j})|c_{j} ≃ H^{0}(∂Σ, F) . (5.29)

Then by (5.24), (K^{.}, ∂) is a subcomplex of (C^{Σ,r.}(F), ∂).

We have the obvious exact sequences

\xymatrix{0 \ar[r] & ker(1 − w_{j}) \ar[r] & π^{∗}F_{p} \ar[r]_{T_{j}=1−w_{j}} & Im(1 − w_{j}) \ar[r] & 0} . (5.30)

The exact sequences (5.30) induce an exact sequence of complexes

\xymatrix{0 \ar[r] & (K^{.}, ∂) \ar[r] & (C^{Σ,r.}(F), ∂) \ar[r]_{T} & (Ĉ^{Σ.}(F), ∂) \ar[r] & 0} . (5.31)

By (5.23), (5.24),

(5.32)  Ĉ^{Σ,k}(F) = C^{Σ,r,k}(F) , k = 0 or 2 ,
= ⨁_{i=1}^{g} π^{∗}F_{p_{|a_{i}}} ⊕ π^{∗}F_{p_{|b_{i}}} ⊕ ⨁_{j=1}^{s} Im(1 − w_{j})_{|c_{j}} , k = 1 .

By (5.23), (5.24), if f ∈ π^{∗}F_{p}, ∂ : Ĉ^{Σ,0}(F) → Ĉ^{Σ,1}(F) is given by

∂f = ∑_{1}^{g} ((1 − u_{i})f_{|a_{i}} + (1 − v_{i})f_{|b_{i}}) + ∑_{j=1}^{s}(1 − w_{j})f_{|c_{j}} , (5.33)

and ∂ : Ĉ^{Σ,1}(F) → Ĉ^{Σ,2}(F) by

(5.34)  ∂(f|a_{i}) = [u_{1}, v_{1}] … [u_{i−1}, v_{i−1}](1 − u_{i}v_{i}u_{i}^{−1})f ,
∂(f|b_{i}) = [u_{1}, v_{1}] … [u_{i−1}, v_{i−1}]u_{i}(1 − v_{i}u_{i}^{−1}v_{i}^{−1})f ,
∂(f|c_{j}) = [u_{1}, v_{1}] … [u_{g}, v_{g}]w_{1} … w_{j−1}f .

Let Ĥ^{.}(Σ, F) be the cohomology of (Ĉ^{Σ.}(F), ∂). By (5.31), for s > 0, we have the long exact sequence

(5.35)

[figure: 0; Ĥ^{0}(Σ, F); K^{1} = H^{0}(∂Σ, F); H^{1}((Σ, ∂Σ), F); T; Ĥ^{1}(Σ, F); 0; H^{2}((Σ, ∂Σ), F); Ĥ^{2}(Σ, F); 0 .]

Theorem 5.4. For s > 0, the following identities hold,

Ĥ^{0}(Σ, F) = H^{0}(Σ, F) , (5.36)

Ĥ^{2}(Σ, F) = H^{2}((Σ, ∂Σ), F) .

Also Ĥ^{1}(Σ, F) is the image of H^{1}((Σ, ∂Σ), F) in H^{1}(Σ, F). In (5.35), the map Ĥ^{0}(Σ, F) → K^{1} is just the map H^{0}(Σ, F) → H^{0}(∂Σ, F), the map K^{1} → H^{1}((Σ, ∂Σ), F) is minus the map H^{0}(∂Σ, F) → H^{1}((Σ, ∂Σ), F), and T : H^{1}((Σ, ∂Σ), F) → Ĥ^{1}(Σ, F) is induced by the map H^{1}((Σ, ∂Σ), F) → H^{1}(Σ, F).

Proof. By (5.33),

Ĥ^{0}(Σ, F) = {f ∈ π^{∗}F_{p} , (1 − u_{i})f = 0 , (1 ≤ i ≤ g) (5.37)

(1 − v_{i})f = 0 , (1 − w_{j})f = 0 , 1 ≤ j ≤ s} .

Usin Proposition 5.2, and (5.37), we get the first equality in (5.36). The second equality in (5.36) follows from (5.24), (5.34).

Inspection of (5.23), (5.24), (5.37) shows that the map Ĥ^{0}(Σ, F) → K^{1} is the canonical H^{0}(Σ, F) → H^{0}(∂Σ, F). Take f ∈ ker(1 − w_{j}). By (5.23), if f|r_{j}^{1} is viewed as lying in C^{0}(Σ, F), then

∂f|r_{j}^{1} = −f|c_{j} . (5.38)

Therefore the map K^{1} → H^{1}((Σ, ∂Σ), F) is minus the map H^{0}(∂Σ, F) → H^{1}((Σ, ∂Σ), F). By (5.35), we have the canonical isomorphism

Ĥ^{1}(Σ, F) ≃ \frac{H^{1}((Σ, ∂Σ), F)}{Im H^{0}(∂Σ, F)} . (5.39)

By (5.22), (5.39), we find that Ĥ^{1}(Σ, F) is exactly the image of H^{1}((Σ, ∂Σ), F) in H^{1}(Σ, F).

The proof of our Theorem is completed.

Remark 5.5. Assume that F and F^{∗} are naturally isomorphic as flat bundles. This is the case when F is real and carries a flat scalar product. Then H^{0}(Σ, F) and H^{2}((Σ, ∂Σ), F) are Poincaré dual, and the image of H^{1}((Σ, ∂Σ), F) in H^{1}(Σ, F) is equipped with a non degenerate 2-form, the intersection product. The conclusion is that in this case, the cohomology of (Ĉ^{Σ.}(F), ∂) exhibits Poincaré duality.

5.3. The map ϕ. Let G be a compact connected semisimple Lie group. Let Z(G) be the center of G. Then Z(G) is a finite subgroup of G. Let G′ = G/Z(G) be the adjoint group. Then G′ acts naturally on G by conjugation. If u ∈ G′, v ∈ G, we will write

u · v = uvu^{−1} . (5.40)

106 JEAN-MICHEL BISMUT AND FRANÇOIS LABOURIE

Similarly G′ acts on G^{2g+s} by conjugation. We will use the same notation for the action of G on G or on G^{2g+s} by conjugation.

Take g, s ∈ N, with g + s > 0. Definition 5.6. Let ϕ : G^{2g+s} → G be the map

ϕ(u_{1}, v_{1}, …, u_{g}, v_{g}, w_{1}, …, w_{s}) = ∏_{i=1}^{g}[u_{i}, v_{i}] ∏_{j=1}^{s} w_{j} . (5.41)

Then ϕ is G′-equivariant, i.e. if g ∈ G′, x ∈ G^{2g+s}, ϕ(g · x) = g · ϕ(x) . (5.42)

Let 𝔤 be the Lie algebra of G. If g ∈ G, we identify T_{g}G to 𝔤 = T_{e}G via the right multiplication operator R_{g∗}. Then if z ∈ G^{2g+s}, T_{z}G^{2g+s} = 𝔤^{2g+s} . (5.43) Definition 5.7. Take x ∈ G^{2g+s}. Let δ : 𝔤 → 𝔤^{2g+s} be the derivative at g = 1 of the map g ∈ G ↦ g · x ∈ G^{2g+s}. Similarly let δ : 𝔤^{2g+s} → 𝔤 be the derivative at x ∈ G^{2g+s} of x′ = G^{2g+s} ↦ ϕ(x′) ∈ G.

Clearly G acts on 𝔤 by the adjoint action. Proposition 5.8. Take x = (u_{1}, v_{1}, …, u_{g}, v_{g}, w_{1}, …, w_{s}) ∈ G^{2g+s}. Then if f ∈ 𝔤, δf = ((1 − u_{1})f, (1 − v_{1})f, …, (1 − u_{g})f, (1 − v_{g})f, (5.44)

(1 − w_{1})f, …, (1 − w_{s})f) .

Similarly, if (f_{1}, …, f_{2g+s}) ∈ 𝔤^{2g+s},

δ(f_{1}, …, f_{2g+s}) = ∑_{i=1}^{g}[u_{1}, v_{1}] … [u_{i−1}, v_{i−1}] (5.45)

((1 − u_{i}v_{i}u_{i}^{−1})f_{2i−1} + u_{i}(1 − v_{i}u_{i}^{−1}v_{i}^{−1})f_{2i})

+ ∑_{j=1}^{s}[u_{1}, v_{1}] … [u_{g}, v_{g}]w_{1} … w_{j−1}f_{2g+j} .

Proof. This is a trivial computation, which is left to the reader.

Let O ⊂ G be an orbit in G. More precisely, take g ∈ G, and put O = {g′ · g, g′ ∈ G} . (5.46) Of course g ∈ O. If Z(g) ⊂ G is the centralizer of g, O ≃ G/Z(g) . (5.47)

Clearly the tangent space T_{g}O ⊂ T_{g}G ≃ 𝔤 is given by T_{g}O_{g} = Im(1 − g) . (5.48) Also T_{e}Z(g) ⊂ 𝔤 is given by T_{e}Z(g) = ker(1 − g) . (5.49) Then the exact sequence 0 → T_{e}Z(g) → T_{e}G → T_{g}O_{g} → 0 (5.50)

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS 107

corresponds to the obvious

\xymatrix{0 \ar[r] & ker(1 − g) \ar[r] & 𝔤 \ar[r]_{1−g} & Im(1 − g) \ar[r] & 0} . (5.51)

Let O_{1}, …, O_{s} be s adjoint orbits of G in G. Put

X = G^{2g} × ∏_{j=1}^{s} O_{j} . (5.52)

If x = (u_{1}, v_{1}, …, u_{g}, v_{g}, w_{1}, …, w_{s}) ∈ X, δ : 𝔤 → 𝔤^{2g+s} is in effect a linear map : 𝔤 → 𝔤^{2g} ⊕ ⨁_{j=1}^{s} Im(1 − w_{j}). By restriction we obtain the linear maps

\xymatrix{(D, δ) : 0 \ar[r] & 𝔤 \ar[r]_{δ} & 𝔤^{2g} ⊕ ⨁_{j=1}^{s} Im(1 − w_{j}) \ar[r]_{δ} & 𝔤 \ar[r] & 0} . (5.53)

Note that in general δ^{2} ≠ 0 . (5.54) Also observe that if g ∈ G′, g maps (D, ∂)_{x} into (D, ∂)_{gx}. Definition 5.9. If x ∈ X, let Z(x) ⊂ G, Z′(x) ⊂ G′ be the stabilizers of x, and let 𝔷(x) be the Lie algebra of Z(x) and Z′(x).

Of course Z′(x) = Z(x)/Z(G). Proposition 5.10. For x ∈ X, then {f ∈ 𝔤, δf = 0} = 𝔷(x) . (5.55) Proof. This is obvious by 5.44.

Let ⟨ , ⟩ be a G-invariant scalar product on 𝔤. If 𝔥 is a vector subspace of 𝔤, let 𝔥^{⊥} be the orthogonal space to 𝔥 in 𝔤. Proposition 5.11. For x ∈ X, then

[δ(𝔤^{2g} ⊕ ⨁_{j=1}^{s} Im(1 − w_{j}))]^{⊥} = 𝔷(x) . (5.56)

Proof. Let δ^{∗} : 𝔤 → 𝔤^{2g} ⊕ ⨁_{j=1}^{s} Im(1 − w_{j}) be the adjoint of δ : 𝔤^{2g} ⊕ ⨁_{j=1}^{s} Im(1 − w_{j}) → 𝔤. Clearly

δ(𝔤^{2g} ⊕ ⨁_{j=1}^{s} Im(1 − w_{j}))^{⊥} = ker δ^{∗} . (5.57)

Let P^{Im(1−w_{j})} be the orthogonal projection operator 𝔤 → Im(1 − w_{j}). Then (1 − w_{j}^{−1})P^{Im(1−w_{j})} = (1 − w_{j}^{−1}), (5.58) and 1 − w_{j}^{−1} is one to one from Im(1 − w_{j}) into itself. Using 5.45, we obtain a formula for δ^{∗} involving the projection P^{Im(1−w_{j})}. By (5.58), to calculate ker δ^{∗}, we may replace P^{Im(1−w_{j})} by (1 − w_{j}). By proceeding as in Proposition 5.1, we get (5.56).

Recall that ϕ : X → G is said to be regular at x ∈ X if dϕ(x) : T_{x}X → T_{ϕ(x)}G is surjective. Theorem 5.12. The map ϕ : X → G is regular at x ∈ X if and only if 𝔷(x) = 0.

Proof. Our Theorem is a trivial consequence of Proposition 5.11.

Definition 5.13. Put

M = {x ∈ X, ϕ(x) = 1} . (5.59)

If x = (u_{1}, …, w_{s}) ∈ M, then x defines a morphism Γ → G. Also G acts on 𝔤 via the adjoint representation. Let E be the flat real vector bundle on Σ constructed as in (5.7), i.e.

E = Σ̂ ×_{Γ} 𝔤 . (5.60)

The following result was first obtained by Liu [37, 38]

Theorem 5.14. If x ∈ M, then in (5.53), δ^{2} = 0, i.e. (D, ∂)_{x} is a complex. The complex (D, δ)_{x} can then be canonically identified to the complex (Ĉ^{Σ.}(E), ∂)_{x}.

Proof. By (5.42), if x ∈ M,

ϕ(gx) = 1 . (5.61)

By (5.61), we find that δ^{2} = 0. Comparing (5.33), (5.34) and (5.44), (5.45), our Theorem follows.

Remark 5.15. By Theorem 5.4, if x ∈ M,

(5.62)  Ĥ^{0}(Σ, E) = H^{0}(Σ, E)
Ĥ^{2}(Σ, E) = H^{2}((Σ, ∂Σ), E) .

In view of Propositions 5.1 and 5.10, we find that if x ∈ M,

(5.63)  Ĥ^{0}(Σ, E) = 𝔷(x) ,
Ĥ^{2}(Σ, E) = 𝔷(x)^{∗} .

So when x ∈ M, Proposition 5.11 follows from (5.63).

5.4. The set of regular values of the map ϕ. Recall that G and G′ act by the adjoint action on 𝔤. Then one verifies easily that if x ∈ M, 𝔤 ∈ G,

\xymatrix{(Ĉ_{x}^{Σ.}(E), ∂) \ar[r]_{g} & (Ĉ_{g.x}^{Σ.}(E), ∂)} (5.64)

is an isomorphism of complexes. In particular the induced map

\xymatrix{Ĥ_{x}^{Σ.}(E) \ar[r]_{g} & Ĥ_{g.x}^{Σ.}(E)} (5.65)

is an isomorphism.

Let T be a maximal torus in G, let 𝔱 be the Lie algebra of T. Let W be the Weyl group of G with respect to T. Let R = {α} ⊂ 𝔱^{∗} be the root system of G, let CR = {h_{α}} ⊂ 𝔱 be the corresponding coroots.

Now we recall the definition of S ⊂ T given in Definition 2.23.

Definition 5.16. Let S ⊂ T be given by

(5.66) S = {t ∈ T; 𝔱 = ∑_{α∈R} t^{α}h_{α}; the h_{α} for which t^{α} ≠ 0 do not span 𝔱} .

Let t_{1}, …, t_{s} ∈ T. Let O_{1}, …, O_{s} be the orbits of t_{1}, …, t_{s} in G. By [15, Lemma IV.2.5],

O_{j} ∩ T = W t_{j} , 1 ≤ j ≤ s . (5.67)

Definition 5.17. We will say that (t_{1}, …, t_{s}) verify (A) if for any (w^{1}, …, w^{s}) ∈ W^{s},

∑_{1}^{s} w^{j}t_{j} ∉ S . (5.68)

Theorem 5.18. For g ≥ 1, the map ϕ : X → G is surjective. For g ≥ 1, t ∈ T is a regular value of ϕ if and only if (t_{1}, t_{2}, …, t_{s}, −t) ∈ T^{s+1} verify (A).

For g = 0, t ∈ T is a regular value of ϕ in the following two cases :

• t ∉ ϕ(X). (5.69)

• t ∈ ϕ(X) , and (t_{1}, …, t_{s}, −t) verify (A) .

Proof. By [14, Corollaire 4.5], since G is semisimple, the map (u, v) ∈ G^{2} ↦ [u, v] ∈ G is surjective. Therefore, for g ≥ 1, the map ϕ is surjective.

First we will prove the remainder of our Theorem for t = 1. By Theorem 5.12, 1 ∈ G is a non regular value of ϕ if and only if there exists x ∈ M such that 𝔷(x) ≠ 0. Equivalently there exists p ∈ 𝔤, p ≠ 0 and x = (u_{1}, …, w_{s}) ∈ M such that x ∈ Z(p)^{2g+s}. By [15, Theorem IV.2.3], Z(p) is a connected Lie subgroup of G, which is not semisimple, since Rp is contained in the Lie algebra of its center. Let T_{0} = Z_{0}(Z(p)) be the connected component of the identity in Z(Z(p)). Then we have the exact sequence

1 → T_{0} → Z(p) → Z(p)/T_{0} → 1, (5.70)

and Z(p)/T_{0} is semisimple. Put

γ = π_{1}(Z(p)/T_{0}) , (5.71)

and let { \atop Z(p)/T_{0}} be the universal cover of Z(p)/T_{0}. Then (5.70) fits in the diagram

(5.72)  \xymatrix{ & 1 \ar[d] & 1 \ar[d] & 1 \ar[d] \\ 1 \ar[r] & 1 \ar[r] \ar[d] & γ \ar[r] \ar[d] & γ \ar[r] \ar[d] & 1 \\ 1 \ar[r] & T_{0} \ar[r] \ar[d] & Ẑ(p) \ar[r] \ar[d] & { \atop Z(p)}/T_{0} \ar[r] \ar[d] & 1 \\ 1 \ar[r] & T_{0} \ar[r] \ar[d] & Z(p) \ar[r] \ar[d] & Z(p)/T_{0} \ar[r] \ar[d] & 1 \\  & 1 & 1 & 1}

In (5.72), Ẑ(p) → Z(p) is the obvious γ covering of Z(p).

Since { \atop Z(p)/T_{0}} is semisimple and simply connected, any central extension of { \atop Z(p)/T_{0}} by a torus is trivial, so that

Ẑ(p) = T_{0} × { \atop Z(p)/T_{0}} . (5.73)

Therefore γ ⊂ { \atop Z(p)/T_{0}} embeds as a finite subgroup of T_{0} × { \atop Z(p)/T_{0}}.

Let T̃(p) be a maximal torus in { \atop Z(p)/T_{0}}. Then T_{0} × T̃(p) is a maximal torus in Ẑ(p). From (5.72), (5.73), we get the complex

(5.74)  \xymatrix{ & 1 \ar[d] & 1 \ar[d] & 1 \ar[d] \\ 1 \ar[r] & 1 \ar[r] \ar[d] & γ \ar[r] \ar[d] & γ \ar[r] \ar[d] & 1 \\ 1 \ar[r] & T_{0} \ar[r] \ar[d] & T_{0} × T̃(p) \ar[r] \ar[d] & T̃(p) \ar[r] \ar[d] & 1 \\ 1 \ar[r] & T_{0} \ar[r] \ar[d] & T \ar[r] \ar[d] & T/T_{0} \ar[r] \ar[d] & 1 \\  & 1 & 1 & 1}

Now we use the notation of Theorem 1.52. Let 𝔱_{0} = 𝔷(Z(p)) be the Lie algebra of T_{0}. Let 𝔱_{1} ⊂ 𝔱 be the vector subspace of 𝔱 spanned by the {h_{α}}_{α∈R_{p}}. Clearly

𝔱 = 𝔱_{0} ⊕ 𝔱_{1} . (5.75)

Also 𝔱_{1} is the Lie algebra of T/T_{0} or of T̃(p).

Let [Ẑ(p), Ẑ(p)] be the commutator subgroup Ẑ(p), i.e. the group spanned by commutators in Ẑ(p). By (5.73), since { \atop Z(p)/T_{0}} is semisimple,

[Ẑ(p), Ẑ(p)] = { \atop Z(p)/T_{0}} . (5.76)

Clearly [Ẑ(p), Ẑ(p)] maps onto [Z(p), Z(p)]. Therefore [Z(p), Z(p)] is a compact connected subgroup of Z(p), and so it is a connected Lie subgroup of Z(p).

Let now t ∈ T ∩ [Z(p), Z(p)]. By the above, there is a ∈ { \atop Z(p)/T_{0}}, b = (b_{0}, b_{1}) ∈ T_{0} × T̃(p) which map to t. Therefore, if c = ab^{−1}, then c ∈ γ. Clearly

c = (b_{0}^{−1}, ab_{1}^{−1}) . (5.77)

By (5.74), (5.77),

ab_{1}^{−1} ∈ γ ⊂ T̃(p) . (5.78)

From (5.77), we get

a ∈ T̃(p) . (5.79)

We thus find that if t ∈ T ∩ [Z(p), Z(p)], t is the projection of an element of T̃(p). Equivalently t can be represented by \overline{t} ∈ 𝔱_{1}.

Recall that

∏_{i=1}^{g}[u_{i}, v_{i}] ∏_{j=1}^{s} w_{j} = 1 , (5.80)

and u_{i}, v_{i} ∈ Z(p), 1 ≤ i ≤ g, w_{j} ∈ Z(p), 1 ≤ j ≤ s. Clearly T is a maximal torus in Z(p). So there is g ∈ Z(p) such that gw_{j}g^{−1} ∈ T. Since gw_{j}g^{−1} ∈ T ∩ O_{t_{j}}, by (5.67), there is w^{j} ∈ W such that

gw_{j}g^{−1} = w^{j}t_{j} . (5.81)

Therefore

w_{j} = w^{j}t_{j} in Z(p)/[Z(p), Z(p)] . (5.82)

So by (5.80), (5.82), we get

∑_{j=1}^{s} w^{j}t_{j} ∈ T ∩ [Z(p), Z(p)] . (5.83)

By the above, we find that

∑_{j=1}^{s} w^{j}t_{j} = ∑_{α∈R_{p}} s^{α}h_{α} in T . (5.84)

Now since p ≠ 0, {h_{α}, α ∈ R_{p}} does not span 𝔱, i.e. (t_{1}, …, t_{s}) does not verify (A).

Conversely, suppose that g ≥ 1, and (t_{1}, …, t_{s}) does not verify (A). Then there is (w^{1}, …, w^{s}) ∈ W^{s}, such that ∑_{j=1}^{s} w^{j}t_{j} can be expressed as a linear combination of {h_{α}} which do not span 𝔱. Let p ∈ 𝔱, p ≠ 0 be orthogonal to the corresponding {α}. By the above,

∑_{j=1}^{s} w^{j}t_{j} ∈ [Z(p), Z(p)] . (5.85)

Using the fact that in a compact semisimple Lie group, any element is a commutator [14, Corollaire 4.5], and also the considerations after (5.76), by (5.85), there is u_{g}, v_{g} ∈ Z(p) such that

[u_{g}, v_{g}] ∏_{j=1}^{s} w^{j}t_{j} = 1 . (5.86)

If x = (1, …, 1, u_{g}, v_{g}, w^{1}t_{1}, …, w^{s}t_{s}), then x ∈ M, and p ∈ 𝔷(x), so that 𝔷(x) ≠ {0}. By Theorem 5.12, we find that 1 is not a regular value of ϕ.

Let t = t_{s+1}^{−1} ∈ T. Let O_{s+1} ⊂ G be the G-orbit of t_{s+1}. Put X_{+1} = G^{2g} × ∏_{j=1}^{s+1} O_{j}. More generally we denote with the index +1 the objects we constructed above, which are associated to X_{+1}. Clearly

X_{+1} = X × O_{s+1} . (5.87)

Also if (x, w_{s+1}) ∈ X_{+1},

ϕ_{+1}(x, w_{s+1}) = ϕ(x)w_{s+1} . (5.88)

Clearly ϕ(x) = t if and only if (x, t_{s+1}) ∈ M_{+1}. By (5.88), if ϕ is regular at x, ϕ_{+1} is regular at (x, t_{s+1}). Conversely, suppose that ϕ_{+1} is regular at (x, t_{s+1}). By (5.88),

Im(dϕ_{+1})(x, t_{s+1}) = Im(dϕϕ^{−1})_{x} + (t_{s+1} − 1)(𝔤) . (5.89)

By (5.42), we find that

(t_{s+1} − 1)(𝔤) ⊂ Im(dϕϕ^{−1})_{x} . (5.90)

From (5.89), (5.90), we find that ϕ is regular at t if and only if ϕ_{+1} is regular at (x, t_{s+1}). Now we use our Theorem for t = 1 and we obtain the stated result in full generality.

Remark 5.19. Instead of studying the case t = 1 first, we could as well have used Theorem 5.12 and proceeded directly.

5.5. The stabilizers Z′(x).

Theorem 5.20. If G is simply connected, under one of the following three conditions, if t ∈ T is a regular value of ϕ, {x ∈ ϕ^{−1}{t}, Z′(x) ≠ 1} is included in a union of submanifolds of {x ∈ ϕ^{−1}(t)} of codimension ≥ 2 :

• For g ≥ 2.

• For g = 1, if one of the t_{j}’s or t is regular.

• For g = 0, s = 1, in which case ϕ^{−1}{t} is empty, or if at least 3 of the elements {t_{1}, …, t_{s}, t} are regular.

If G is only connected, if t_{1}, …, t_{s} are very regular, if t ∈ T is a regular value of ϕ, {x ∈ ϕ^{−1}(t), Z′(x) ≠ 1} is included in a union of submanifolds of {x ∈ ϕ^{−1}(t)} of codimension ≥ 2 :

• For g ≥ 2.

• For g = 1, if s ≥ 1 or if t is very regular.

• For g = 0, s = 1, in which case ϕ^{−1}(t) is empty, if s = 2 and t is very regular, or if s ≥ 3.

Proof. We will prove our Theorem in various stages.

• The case where G is simply connected and t = 1.

If 1 is a regular value of ϕ, either M = ϕ^{−1}{1} is empty, or it is a smooth submanifold of X.

By Theorem 5.12, when M is non empty, the group G acts locally freely on M. Therefore M/G is an orbifold. If dim(M/G) is the dimension of the maximal statum of M/G, then

(5.91) dim M/G = (2g − 2) dim(𝔤) + ∑_{j=1}^{s} dim(O_{j}) .

Of course the same observation applies to the action of G′ on M. Clearly M/G ≃ M/G′. In the sequel we will use the notation M/G or M/G′ indifferently. However in Section 6.3, we will construct an orbifold G-line bundle on M/G, which may well not be an orbifold G′-line bundle.

Take now x ∈ M such that Z′(x) ≠ 1. Let u ∈ Z′(x), u ≠ 1. If x = (u_{1}, …, w_{s}), then u_{1}, …, w_{s} ∈ Z(u). By conjugation, we may as well assume that u ∈ T′ = T/Z(G). Since G acts locally freely on M, the group Z(u) is semisimple. By Theorem 1.38, u ∈ C/\overline{R}^{∗}. Of course here u ≠ 0.

Since G is simply connected, Z(u) is connected. By Theorem 1.50, for any t ∈ T,

O(t) ∩ Z(u) = ⋃_{w∈W_{u}\W} O_{Z(u)}(wt) . (5.92)

For 1 ≤ j ≤ s, let w^{j} ∈ W such that

w_{j} ∈ O_{Z(u)}(w^{j}t_{j}), 1 ≤ j ≤ s . (5.93)

Put

X^{u} = Z(u)^{2g−2} × ∏_{j=1}^{s} O_{Z(u)}(w^{j}t_{j}) . (5.94)

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS 113

We define ϕ_{u} : X^{u} → Z(u) as in (5.41). Let M^{u} = ϕ_{u}^{−1}{1}. Then x ∈ M^{u}. Moreover Z(u) acts locally freely on M^{u}. Therefore, by Theorem 5.12, 1 is a regular value of ϕ_{u}, i.e. M^{u} is a smooth submanifold of M. Note that this also follows from the fact that M is a smooth manifold, G acts on M, and M^{u} is a component of the fixed point set of u in M. Then

(5.95) dim M^{u}/Z(u) = (2g − 2) dim 𝔷(u) + ∑_{j=1}^{s} dim O_{Z(u)}(w^{j}t_{j}) .

Now by Theorem 1.52, dim 𝔷(u) ≤ dim(𝔤) − 2 . (5.96) Also dim O_{Z(u)}(w^{j}t_{j}) ≤ dim O_{j} . (5.97)

Now we consider 3 cases :

• If g ≥ 2, by (5.91), (5.95)-(5.97), dim M^{u}/Z(u) ≤ dim M/G − 2 . (5.98) By (5.98), M^{u}/Z(u) maps to a submanifold of codimension ≥ 2 in M/G. Therefore the G-orbit of M^{u} in M is a submanifold of codimension ≥ 2 in M.

• If g = 1, assume that t_{1} ∈ T is regular. Then (5.99) dim O_{t_{1}} = dim(𝔤) − dim(𝔱) ,

dim O_{Z(u)}(w^{1}t_{1}) = dim 𝔷(u) − dim(𝔱) .

By (5.91), (5.95), (5.96), (5.97), (5.99), we find again that dim M^{u}/Z(u) ≤ dim M/G − 2 . (5.100)

• If g = 0, if s = 2, then (5.101) dim O_{t_{j}} ≤ dim(𝔤) − dim(𝔱) ; s = 1, 2 , so that using (5.91), dim M/G < 0, (5.102) i.e. M = ∅.

If s ≥ 3, assume that t_{1}, t_{2}, t_{3} are regular. Then

(5.103) dim M/G = dim(𝔤) − 3 dim(𝔱) + ∑_{j=4}^{s} dim(O_{j}) ,

dim M^{u}/Z(u) = dim 𝔷(u) − 3 dim(𝔱) + ∑_{j=4}^{s} dim O_{Z(u)}(w^{j}t_{j}) .

By (5.96), (5.97), (5.103) dim M^{u}/Z(u) ≤ dim M/G − 2 . (5.104) So we have proved our Theorem in this case.

• The case where G is non simply connected and t = 1.

First we proceed as above. In this case for u ∈ C/\overline{R}^{∗}, u ≠ 1, Z(u) is in general non connected. However by (1.174), since t_{1}, …, t_{s} are very regular,

O_{j} ∩ Z(u) = ⋃_{w^{j}∈W_{Z(u)o} \W} O_{Z(u)_{o}}(w^{j}t_{j}) . (5.105)

For 1 ≤ j ≤ s, let w^{j} ∈ W be such that

w_{j} ∈ O_{Z(u)_{o}}(w^{j}t_{j}) . (5.106)

Again we define X^{u} by (5.94), and we define ϕ_{u} as before. Put M^{u} = ϕ_{u}^{−1}{1}. By the same argument as before, 1 is a regular value of ϕ_{u}. Then (5.95) still holds. If g ≥ 2, (5.104) is still true. If g = 1, since t_{1} is very regular in G, then t_{1} lies in Z(u)_{0} and is very regular in Z(u), so that (5.100) still holds.

If g = 0, under the stated assumptions, the argument in the proof above can still be reproduced.

The proof of our Theorem is completed in this case.

• The case where t ∈ T is an arbitrary regular value of ϕ.

Let t = t_{s+1}^{−1} ∈ T. We use the notation in the proof of Theorem 5.18. Then t is a regular value of ϕ if and only if 1 is a regular value of ϕ_{+1}. By the above, we obtain our Theorem in full generality.

The proof of our Theorem is completed.

Theorem 5.21. If G is simply connected, under one of the following conditions, {x ∈ X; Z′(x) ≠ 1} is included in a union of submanifolds of codimension ≥ 2:

• g ≥ 1.

• g = 0, s ≥ 2, and at least 2 of the t_{j}’s is regular.

If G is non necessarily simply connected, if all the t_{j}’s are very regular, then {x ∈ X, Z′(x) ≠ 1} is included in a union of submanifolds of codimension ≥ 2 if

• g ≥ 1.

• g = 0, s ≥ 2, and at least 2 of the t_{j}’s are very regular.

Proof. By [], the set of conjugacy classes of stabilizers Z′(x) is finite. Take x ∈ X and assume that Z′(x) ≠ e. Let G′(x) be the orbit of G′ through x. Then

G′(x) ≃ G′/Z′(x) . (5.107)

Let N_{x} be the normal bundle to G′(x) at x. Then Z′(x) acts on N_{x}. By [?, Proposition 27.2], there is a G′-invariant open neighborhood of G′(x) in X which can be identified as a G-space to a neighborhood of the zero section in N = G′×_{Z′(x)}N_{x}.

Let N_{x}^{f} be the invariant part of N_{x} under Z′(x). Then N_{x}^{f} extends to a vector subbundle of the vector bundle N on G′/Z′(x), of the form G′/Z′(x) × N_{x}^{f}.

If y ∈ N_{x}, then Z′(y) ⊂ Z′(x) and Z′(y) = Z′(x) if and only if y ∈ N_{x}^{f}. In particular Z′(y) is conjugate to Z′(x) if and only if y ∈ N_{x}^{f}, in which case Z′(y) = Z′(x). It follows that near G′(x), the elements of X whose stabilizer is conjugate to Z′(x) form a neighborhood of the zero section of N^{f}.

Let u ∈ Z′(x), u ≠ 1. We may and we will assume that u ∈ T. Let (G′/Z′(x))^{u} be the fixed point set of u in G′/Z′(x). Let codim(G′/Z′(x))^{u}, G′/Z′(x)) be the codimension of (G′/Z′(x))^{u} in G′/Z′(x). Clearly

(5.108) codim(G′/Z′(x))^{u}, G′/Z′(x)) ≤ dim(𝔤) − dim(𝔷(u)) .

Let N_{x}^{u} be the vector subspace of N_{x} fixed under u. Then N_{x}^{f} ⊂ N_{x}^{u}. If N^{u} is the fixed point set of N under u, then N^{u} is a vector bundle over (G′/Z′(x))^{u}, with fibre modelled on N_{x}^{u}.

Let dim(N^{f}) = dim(G′/Z′(x)) + dim N_{x}^{f} be the dimension of the total space of N^{f}. Similarly let dim(N^{u}) be the dimension of the total space of N_{u}. By (5.108), we get

(5.109) dim(N^{f}) ≤ dim(N^{u}) + dim(𝔤) − dim 𝔷(u) .

Clearly

(5.110) dim X = 2g dim(𝔤) + ∑_{j=1}^{s} dim(O_{j}) .

Let X^{u} be the fixed point set of X under u. Then

X^{u} = Z(u)^{2g} × ∏_{j=1}^{s}(O_{j} ∩ Z(u)), (5.111)

so that

(5.112) dim(X^{u}) = 2g dim 𝔷(u) + ∑_{j=1}^{s} dim(O_{j} ∩ Z(u)) .

By (5.109), (5.112),

(5.113) dim(N^{f}) ≤ (2g − 1) dim 𝔷(u) + dim 𝔤 + ∑_{j=1}^{s} dim(O_{j} ∩ Z(u)) .

If g ≥ 1, using (5.96), (5.110), (5.113),

dim(N^{f}) ≤ dim(X) − 2 . (5.114)

Assume now that g = 0, that s ≥ 2, that G is simply connected and t_{1}, t_{2} are regular, or more generally that t_{1}, t_{2} are very regular. By Theorem 1.50,

(5.115)  dim O_{j} = dim(𝔤) − dim(𝔱) ,
dim O_{j} ∩ Z(u) = dim 𝔷(u) − dim 𝔱 .

So if g = 0, using (5.96), (5.110), (5.113), (5.115), we get

(5.116) dim(N^{f}) ≤ dim 𝔷(u) + dim 𝔤 − 2 dim(𝔱) + ∑_{j=3}^{s} dim(O_{j} ∩ Z(u))
≤ dim(X) − 2 .

From (5.114), (5.116), our Theorem follows.

5.6. The tangent bundle to the moduli space and its symplectic form. We make the same assumptions as in Sections 5.1-5.4.

If x ∈ X, let G′(x) be the orbit of G′ at x, and let T_{x}G′(x) be the tangent space to G′(x) at x. Recall that T_{x}X ⊂ 𝔤^{2g+s}, so that T_{x}G′(x) ⊂ 𝔤^{2g+s}.

Proposition 5.22. If x ∈ M,

T_{x}G′(x) = ∂(Ĉ_{x}^{Σ,0}(E)) ⊂ Ĉ_{x}^{Σ,1}(E). (5.117)

If ϕ is regular at x ∈ M, then M is a submanifold of X near x, and

T_{x}M = ker ∂_{|Ĉ^{Σ,1}(E)→Ĉ^{Σ,2}(E)}. (5.118)

Proof. By Theorem 5.14, the first part of our Proposition is clear. If ϕ is regular at x ∈ M,

T_{x}M = ker dϕ(x). (5.119)

Using Theorem 5.14 and (5.119), we get (5.118).

Remark 5.23. Of course, if ϕ is regular at x, then T_{x}G′(x) ⊂ T_{x}M. This fits with (5.117), (5.118), because ∂^{2} = 0. By Proposition 5.10 and Theorem 5.12, T_{x}G′(x) has dimension dim 𝔤.

Let π : M → M/G be the obvious projection. If ϕ is regular at x ∈ M, Z′(x) is a finite group.

Theorem 5.24. If ϕ is regular at x ∈ M, M is smooth near x, and M/G is an orbifold near π(x). More precisely, near π(x) ∈ M/G,

T M/G = M ×_{G} Ĥ^{Σ,1}(E). (5.120)

Also near π(x),

M/G ≃ Ĥ_{x}^{1}(Σ, E)/Z′(x). (5.121)

Proof. These are classical facts from the theory of orbifolds (see [?, Proposition 27.7].

Now we assume temporarily that G is simply connected. We use the notation of Section 4.

Definition 5.25. Put

A^{flat}(t_{1}, …, t_{s}) = \{A ∈ A, F^{A} = 0, −(\frac{d}{dt} + A)_{S_{j}^{1}} ∈ O_{− \frac{d}{dt} +t_{j}}\}, 1 ≤ j ≤ s. (5.122)

Equivalently, A^{flat}(t_{1}, …, t_{s}) is the set of flat connections A such that for 1 ≤ j ≤ s, the holonomy w_{j} of A along S_{j}^{1} lies in the orbit O_{j}.

Clearly ΣG acts on the right on A^{flat}(t_{1}, …, t_{s}).

Definition 5.26. Let Σ_{q}G be the group of smooth maps g : Σ → G such that g_{q} = 1.

It is easy to see that Σ_{q}G acts freely on A^{flat}(t_{1}, …, t_{s}) . Also there is an obvious continuous map f : A^{flat}(t_{1}, …, t_{s}) ↦ M which, to A, associates (u_{1}, v_{1}, … w_{1}, …, w_{s}), the holonomies of A along the circles a_{1}, b_{1}, …, c_{1}d_{1}c_{1}^{−1}, …, c_{s}d_{s}c_{s}^{−1}.

We equip A^{flat}(t_{1}, …, t_{s})/Σ_{q}G with the quotient topology.

Proposition 5.27. The map f induces an identification of compact spaces

A^{flat}(t_{1}, …, t_{s})/Σ_{q}G ≃ M . (5.123)

Proof. Clearly f descends to a one to one map A^{flat}(t_{1}, …, t_{s})/Σ_{q}G ≃ M. This map is continuous. By [17, Proposition 2.2.3], f is a pointwise identification. The fact that the topologies coincide follow for instance from the techniques of [17, Section 4.2], in a much simpler context.

Assume now that (t_{1}, …, t_{s}) verify (A). Then by Theorem 5.18, M is a smooth manifold, possibly empty if g = 0. Using the techniques of [17, Section 4.2], we find that A^{flat}(t_{1}, …, t_{s}) is a smooth manifold. Note that here, we use explicitly the fact that for any x ∈ M, Ĥ^{2}(Σ, E) = 0. By [17, Proposition 4.2.23], A^{flat}(t_{1}, … t_{s})/ΣG is a smooth manifold. Also M → M/G is a G-orbifold. By the above, ΣG acts locally freely on A^{flat}(t_{1}, … t_{s}).

Proposition 5.28. We have the identification of orbifolds,

A^{flat}(t_{1}, … t_{s})/ΣG ≃ M/G. (5.124)

Proof. By proceeding as in [17, Proposition 4.2.23], we find easily that the identification f in Proposition 5.27 is an identification of smooth manifolds. Our Proposition follows.

We do no longer assume G to be simply connected.

Let ⟨, ⟩ be a G-invariant bilinear symmetric form on 𝔤. Then E is equipped with the corresponding flat bilinear symmetric form. Recall that by Theorem 5.4, Ĥ^{1}(Σ, E) is the image of H^{1}((Σ, ∂Σ), E) into H^{1}(Σ, E).

Definition 5.29. If x ∈ M, α, α′ ∈ Ĥ_{x}^{1}(Σ, E), put

ω_{x}(α, α′) = ∫_{Σ} −⟨α, α′⟩. (5.125)

Then ω_{x} is an intersection form, so that it is non degenerate.

Theorem 5.30. The 2-form ω is G-invariant. It descends to a symplectic 2-form on the orbifold M/G.

Proof. Suppose first that G is simply connected. By Proposition 5.28, the space A^{flat}(t_{1}, … t_{s})/ΣG is an orbifold. Then the results of Section 4.1 and Theorem 4.24 show that A^{flat}(t_{1}, … t_{s})/ΣG is a symplectic reduction of the symplectic manifold A. Since it is a symplectic reduction of the symplectic affine space A, it carries a symplectic form. One verifies easily that ω is just this form. Therefore ω is closed.

If G is not simply connected, the homotopy types of G-bundles have been classified in Section 4.7. Let G̃ be the universal cover of G. By introducing an extra holonomy h ∈ Z(G̃), we can then replace G by G̃. The fact that ω is closed is now a consequence of the corresponding result for G̃.

The proof of our Theorem is completed.

Recall that G acts on the right and on the left on X. If x ∈ M, a ∈ Γ, then x(a) ∈ G. We define a right action of Γ on X by the formula

x.a = x(a).x, x ∈ M, a ∈ Γ (5.126)

Then if a ∈ Γ, g ∈ G, x ∈ M,

(x.g).a = (x.g)(a).(x.g) = (x.a).g, (5.127)

i.e. the actions of Γ and G on M commute.

Also Γ × G acts on the right on Σ̂ (of course, the factor G acts trivially on Σ̂). Therefore Γ × G acts on the right on M × Σ̂. Also Γ × G acts locally freely on M, and Γ acts freely on Σ̂.

Let V be a complex vector space. Let ρ : G → Aut(V) be a representation of G. Then ρ induces a representation Γ × G → Aut(V). Put

F = (M × Σ̂) ×_{Γ×G} V. (5.128)

Then F is an orbifold vector bundle on M/G×Σ. It is obtained via the identification

(x, σ, f) ≃ (x.g, σ.a, g^{−1}x(a)^{−1}f) , (x, σ, f) ∈ M × Σ̂ × V , (a, g) ∈ Γ × G . (5.129)

For a given x ∈ M, the restriction of F to the fibre Σ is exactly the flat vector bundle considered in Section 5.2.

If σ ∈ Σ, then σ^{∗}F is an orbifold vector bundle on M/G. If σ, σ′ ∈ Σ, if t ∈ [0, 1] ↦ σ_{t} ∈ Σ is a smooth path, with σ_{0} = σ, σ_{1} = σ′, parallel transport with respect to the flat connection identifies σ^{∗}F and σ′^{∗}F.

Over Σ, there are two distinguished points p and q. Recall that G acts on 𝔤 by conjugation. When V = 𝔤, let E be the corresponding real vector bundle on M/G × Σ.

Definition 5.31. Let E be the vector bundle on M/G,

E = p^{∗}E. (5.130)

Also G acts by conjugation on G and on the O_{j}’s. Put

Ĝ = M ×_{G} G, Ô_{j} = M ×_{G} O_{j}. (5.131)

Then Ĝ → M/G is a G-bundle, and Ô_{j} → M/G is a O_{j}-bundle. Also u_{1}, …, v_{g} are sections of Ĝ and w_{1}, …, w_{s} are sections of Ô_{1}, …, Ô_{s}.

Elements of Ĝ act naturally on E. As explained in Section 5.2, u_{1} can be considered as the parallel transport operator along the closed curve a_{1} …

For 1 ≤ j ≤ s, let T_{w_{j}} Ô_{j}/(M/G) be the relative tangent bundle to the fibre Ô_{j} . As we saw in (5.48),

T_{w_{j}} Ô_{j}/M = Im(1 − w_{j}) ⊂ E. (5.132)

As explained in Theorem 5.14, on M/G, we have the bundle of complexes

0 → Ĉ^{Σ,0}(E) → Ĉ^{Σ,1}(E) → Ĉ^{Σ,2}(E) → 0, (5.133)

which, by (5.53), (5.132) can also be written as

0 → E → E^{2g} ⊕ (⨁_{j=1}^{s} T_{w_{j}}(Ô_{j}/(M/G))) → E → 0. (5.134)

Theorem 5.32. On M/G, we have the identity

T M/G = E^{2g} ⊕ (⨁_{j=1}^{s} T_{w_{j}} Ô_{j}/(M/G)) ⊖ E^{2} in K(M/G). (5.135)

Proof. By (5.120),

T M/G = Ĥ^{1}(Σ, E). (5.136)

Also over M, by (5.63),

Ĥ^{j}(Σ, E) = 0, j = 0 , 2. (5.137)

Hence, over M/G, by (5.134), (5.136), (5.137),

Ĥ^{1}(Σ, E) = E^{2g} ⊕ (⨁_{j=1}^{s} T_{w_{j}} Ô_{j}/(M/G)) ⊖ E^{2} in K(M/G). (5.138)

From (5.137), (5.138), we get (5.135).

Theorem 5.33. If the orbits O_{1}, …, O_{s} are very regular (resp. regular), then

(5.139) T M/G = E^{2(g−2)+s} ⊖ R^{s dim 𝔱} in K(M/G) (resp. in K(M/G) ⊗_{Q} Z.)

Proof. Let t ∈ T be very regular, let O be the orbit of t in G. Then Z(t) = T. If g ∈ G,

Z(g.t) = g.T, 𝔷(g.t) = g.𝔱. (5.140)

If w ∈ O, we have the splitting

𝔤 = Im(1 − w) ⊕ ker(1 − w), (5.141)

which corresponds to the splitting

E = T_{w}O ⊕ N_{O/E}. (5.142)

By (5.140), the vector bundle 𝔷(w) = ker(1 − w) is trivial on O. The action of G on the orbit O lifts to an action on the vector bundle 𝔷(w). From (5.141), (5.142), we find that the normal bundle N_{O/E} is equivariantly trivial. From (5.135), (5.142), we get (5.139).

If t ∈ T is only regular, then 𝔷(t) = 𝔱. Also O = G/Z(t). Then if w = g.t, g ∈ G ↦ g.𝔷(t) = 𝔷(w) induces a G-invariant flat connection on 𝔷(w) ≃ N_{O/𝔤}. Consider the splitting of vector bundles

E = T_{w_{j}}(Ô_{j}/(M/G)) ⊕ ker(1 − w_{j}). (5.143)

If the orbit O_{j} is very regular, ker(1 − w_{j}) is a trivial vector bundle on M/G. Therefore

T_{w_{j}} Ô_{j}/(M/G) = E ⊖ R^{dim 𝔱} in K(M/G). (5.144)

If the orbit O_{j} is only regular, then one finds easily by the above that ker(1−w_{j}) is equipped over M/G with a flat connection. We thus get (5.139) in this case. The proof of our Theorem is completed.

5.7. A metric on det(T M/G). If λ is a line, let λ^{−1} be the dual line. If E is a vector space, set

det(E) = Λ^{max}(E) . (5.145)

If E = ⨁_{i=0}^{m} E_{i} is a Z graded vector space, set

det(E) = ⨂_{i=0}^{m}(det E_{i})^{(−1)^{i}} . (5.146)

Let G be a compact connected semisimple Lie group.

Definition 5.34. If x ∈ M, let λ_{x} be the real line

λ_{x} = (det Ĥ_{x}(Σ, E))^{−1}. (5.147)

Let ⟨, ⟩ be a G-invariant scalar product on 𝔤. Then E is equipped with a fibrewise flat scalar product. By Theorem 5.4 and Remark 5.5,

Ĥ^{2}(Σ, E) ≃ (Ĥ^{0}(Σ, E))^{∗}, Ĥ^{1}(Σ, E) ≃ (Ĥ^{1}(Σ, E))^{∗}. (5.148)

Observe that in (5.148), the identifications depend explicitly on ⟨, ⟩. By (5.148), for x ∈ M,

λ_{x}^{2} ≃ R. (5.149)

Now R carries a canonical metric ∥ ∥_{R} such that ∥1∥_{R} = 1.

Definition 5.35. Let ∥ ∥_{λ_{x}} be the metric on λ_{x} such that the identification (5.149) is an isometry.

120 JEAN-MICHEL BISMUT AND FRANÇOIS LABOURIE

In the sequel, we assume that (t_{1}, …, t_{s}) verify (A). Proposition 5.36. If x ∈ M, the metric ∥ ∥_{λ_{x}} defines a volume element on Λ^{max}T_{x}(M/G), which is exactly the volume associated to the symplectic form ω_{x}. Proof. If x ∈ M, then Ĥ_{x}^{0}(Σ, E) = 0, Ĥ_{x}^{1}(Σ, E) = 0, so that λ_{x} = det Ĥ_{x}^{1}(Σ, E) . (5.150) By (5.120), T_{x}M/G = Ĥ_{x}^{1}(Σ, E) . (5.151) Finally in (5.148), the identification Ĥ_{x}^{1}(Σ, E) ≃ (Ĥ_{x}^{1}(Σ, E))^{∗} is done via the symplectic form ω_{x}. Our Proposition follows.

Now recall that Ĥ_{x}(Σ, E) is the cohomology of the complex (Ĉ_{x}^{Σ}(E), ∂). Put λ̂_{x} = (det(Ĉ_{x}^{Σ}(E)))^{−1}. (5.152) Then by [33], there is a canonical isomorphism λ_{x} ≃ λ̂_{x}. (5.153) Also, since E is equipped with a flat scalar product, λ̂_{x} is also naturally equipped with a metric ∥ ∥λ̂_{x}, which also depends on ⟨, ⟩. Let ∥ ∥λ̂_{x} be the corresponding metric on λ_{x} via the canonical isomorphism (5.153). . Definition 5.37. For 1 ≤ j ≤ s, put |σ_{O_{j}}| = | det(1 − w_{j})_{|Im(1−w_{j})}|^{1/2}, w_{j} ∈ O_{j}. (5.154)

As the notation indicates, |σ_{O_{j}}| is a constant on O_{j}. We make the convention that if Im(1 − w_{j}) ⊂ 𝔤 is reduced to 0, |σ_{O_{j}}| = 1 (5.155) Remark 5.38. Recall that the function σ : G → C was defined in (1.42). If O_{j} is regular, if w_{j} ∈ O_{j}, |σ_{O_{j}}| = |σ(w_{j})|. (5.156)

The following result has been proved by Witten [59, Section 4], who exhibited the role of the Reidemeister torsion [47] in this context. Theorem 5.39. For any x ∈ M,

∥.∥λ̂_{x} = ∏_{j=1}^{s} |σ_{O_{j}}|∥.∥_{λ_{x}}. (5.157)

Proof. By (5.31), (5.158) det(Ĉ^{Σ}(E)) ≃ det(C^{Σ,r}(E)) ⊗ (det(K))^{−1}, which can also be written in the form, (5.159) det(Ĉ^{Σ}(E)) ≃ det(C^{Σ,r}(E)) ⊗ det(H^{0}(∂Σ, E)). By (5.29),

H^{0}(∂Σ, E) = ⨁_{j=1}^{s} ker(1 − w_{j}). (5.160)

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS 121

Let ∥.∥_{det H^{0}(∂Σ,E)} be the metric on det H^{0}(∂Σ, E) induced by the scalar product of E. By (5.31) , we find that under the isomorphism (5.159), (5.161) ∥ ∥_{det Ĉ^{Σ}(E)} = ∥ ∥_{det C^{Σ,r}(E)}∥ ∥_{det H^{0}(∂Σ,E)}

[∏_{j=1}^{s} det(1 − w_{j})_{|Im(1−w_{j})}]^{−1} .

Also by (5.21), there is a canonical isomorphism (5.162) det C^{Σ}(E) ≃ det C^{Σ,r}(E) ⊗ det C^{∂Σ}(E) . If ∥ ∥_{det C^{Σ}(E)} denote the obvious metric on det(C^{Σ}(E)), by (5.21), (5.162), we get (5.163) ∥ ∥_{det C^{Σ}(E)} = ∥ ∥_{det C^{Σ,r}(E)}∥ ∥_{det C^{∂Σ}(E)} . Also, we have a canonical isomorphism (5.164) det(C^{∂Σ}(E)) ≃ det(H(∂Σ, E)) . By Poincaré duality, H^{1}(∂Σ, E) = (H^{0}(∂Σ, E))^{∗}, (5.165) so that by (5.164), (5.165), (5.166) det(C^{∂Σ}(E)) ≃ det(H^{0}(∂Σ, E))^{2} . By (5.25), (5.166),

(5.167) ∥ ∥_{det(C^{∂Σ}(E))} = ∥ ∥_{det H^{0}(∂Σ,E)}^{2} [∏_{j=1}^{s} det(1 − w_{j})_{|Im(1−w_{j})}]^{−1} .

Using (5.159)-(5.166),we get (5.168) (det Ĉ^{Σ}(E))^{2} ≃ det C^{Σ}(E) ⊗ det C^{Σ,r}(E). Also under the isomorphism (5.168), (5.169) ∥ ∥_{det Ĉ^{Σ}(E)}^{2} = ∥ ∥_{det C^{Σ}(E)}∥ ∥_{det C^{Σ,r}(E)}

[∏_{j=1}^{s} det(1 − w_{j})_{|Im(1−w_{j})}]^{−1} .

Now recall that by [33], there are canonical isomorphisms

(5.170)  det(C^{Σ}(E)) ≃ det(H(Σ, E)) ,
det(C^{Σ,r}(E)) ≃ det(H((Σ, ∂Σ), E) .

By Poincaré duality, (5.171) det H(Σ, E) ⊗ det H((Σ, ∂Σ), E) ≃ R . So by (5.170), (5.171), (5.172) det C^{Σ}(E) ⊗ det C^{Σ,r}(E) ≃ R .

Let ∥ ∥_{R} be the trivial metric on R, such that ∥1∥_{R} = 1. We claim that under the Poincaré duality isomorphism (5.172), (5.173) ∥ ∥_{det C^{Σ}(E)}∥ ∥_{det C^{Σ,r}(E)} = ∥ ∥_{R} .

In fact (5.173) is a simple consequence of the existence of the Reidemeister torsion [47], or more precisely of the Reidemeister metric [13, Section 1] on det H(Σ, E)) and det H((Σ, ∂Σ), E). In fact given any cell decomposition of the manifold with boundary Σ, one can construct Reidemeister metrics on det H(Σ, E) and det H((Σ, ∂Σ), E) by the procedure indicated in [13]. The basic fact is that these metrics do not depend on the choice of the cell decomposition. By applying this argument to a cell decomposition and the corresponding dual decomposition, one deduces immediately that the Reidemeister metrics on det H(Σ, E)det and det H((Σ, ∂Σ), E) correspond by Poincaré duality. As a consequence, (5.173) follows.

From (5.169), (5.173), we get (5.157). The proof of our Theorem is completed.

From (5.156), (5.157), it follows that if O_{1}, …, O_{s} are regular, if x ∈ B,

∥ ∥λ̂_{x} = ∏_{j=1}^{s} |σ(w_{j})∥ ∥_{λ_{x}} . (5.174)

5.8. The Witten formula for the symplectic volume distribution. Let G be a compact connected semisimple Lie group. Let ⟨ , ⟩ be a G-invariant scalar product on 𝔤. Let dg be the Haar measure on G associated to ⟨ , ⟩. We us otherwise the same notation as in Section 5.7. In particular T ⊂ G is a maximal torus in G.

Let O_{1}, …, O_{s} be s adjoint orbits of G in G. Take w_{j} ∈ O_{j}. Let Z(w_{j}) be the centralizer of w_{j}. Then the map

a ∈ G/Z(w_{j}) ↦ a · w_{j} ∈ O_{j} (5.175)

is an identification of smooth manifolds.

The scalar product ⟨ , ⟩ induces a volume form dv_{O_{j}} on O_{j} ⊂ G.

Definition 5.40. Let dv_{X} be the volume form on X = G^{2g} × ∏_{j=1}^{s} O_{j},

dv_{X}(u_{1}, v_{1}, …, u_{g}, v_{g}, w_{1}, …, w_{s}) = ∏_{i=1}^{2g} du_{i}dv_{i} ∏_{j=1}^{s} dv_{O_{j}}(w_{j}) . (5.176)

Let dt be the Lebesgue measure on 𝔱 induced by ⟨, ⟩.

Definition 5.41. If g ∈ G, let X_{O_{g}} ⊂ G^{2g} × ∏_{j=1}^{s} O_{j} × O_{g} be the set M in (5.59) associated to the orbits O_{1}, …, O_{s}, O_{g}.

More precisely

X_{O_{g}} = {(u_{1}, v_{1}, …, w_{1}, …, w_{s}, w) ∈ G^{2g} × ∏_{j=1}^{s} O_{j} × O_{g} , (5.177)

∏_{i=1}^{g}[u_{i}, v_{i}] ∏_{j=1}^{s} w_{j}w = 1} . (5.178)

Clearly, X_{O_{g}} can be identified with {x ∈ X, h(x) ∈ O_{g^{−1}}}. Set

X_{g} = {x ∈ X; h(x) = g^{−1}} . (5.179)

Now G acts naturally on X_{O_{g}}, and Z(g) acts on X_{g}. Then one has the obvious

X_{O_{g}}/G ≃ X_{g}/Z(g) . (5.180)

Clearly, if x ∈ X, ∫_{G} f(x · g)dg depends only on πx ∈ X/G.

Let G_{vreg} (resp. G_{reg} ) be the set of very regular (resp. regular) elements in G. If g ∈ G_{vreg}, let dt_{g} be the Haar measure on the maximal torus Z(g), which is associated to ⟨ , ⟩.

By Sard’s theorem, a.e. every g ∈ G is a regular value of ϕ. For such a g ∈ G, X_{O_{g}} is smooth, and G acts locally freely on X_{O_{g}}. Also by Theorem 5.30, the orbifold X_{O_{g}}/G is equipped with a symplectic 2 form ω_{g}. Let dv_{X_{Og} /G} be the corresponding volume element on X_{O_{g}}/G. Since X_{O_{g}}/G = X_{g}/Z(g), let dv_{X_{g}/Z(g)} be the associated volume on X_{g}/Z(g). Observe here that dv_{X_{g}/G} is unsensitive to orientation. In particular the integral of a nonnegative function with respect to dv_{X_{Og} /G} is nonnegative.

In the sequel, we make the following assumptions:

• G is simply connected, and one of the following assumptions is verified :

• g ≥ 2 .

• g ≥ 1, s ≥ 1 and at least one of the t_{j}’s is regular.

• g = 0, s ≥ 3, and at least 3 of the t_{j}’s are regular.

or

• G is connected, the t_{j}’s are very regular , and either

• g ≥ 2 .

• g ≥ 1, s ≥ 1 .

• g = 0, s ≥ 3 .

Now we will get an analogue of Theorem 3.10.

Theorem 5.42. Let f : X → R be a bounded measurable function. Then

(5.181)  ∫_{X} f(x)dv_{X}(x) = \frac{∏_{j=1}^{s} |σ_{O_{j}}|}{|Z(G)|} ∫_{T/W} |σ(t)|dt ∫_{X_{t}/T} dv_{X_{t}/T} (x)
∫_{G} f(x · g)dg ,
∫_{M} f(x)dv_{X}(x) = \frac{∏_{j=1}^{s} |σ_{O_{j}}|}{|Z(G)|} ∫_{G} \frac{dg}{|σ(g)|} ∫_{X_{g}/Z(g)} dv_{X_{g}/Z(g)}(x)
∫_{Z(g)} f(x · t)dt_{g} .

Proof. By Theorem 5.21, we know that a.e., Z′(x) = 1. By Theorem 5.12, for a.e. x ∈ X, dϕ(x) is surjective. Therefore the image ϕ_{∗}dv_{X} of dv_{X} by ϕ is absolutely continuous with respect to dg. Also on the set {x ∈ X, ϕ is regular at x, Z′(x) = 1}, which has full measure and is stable by G, we can use the implicit function theorem, and also integrate along the fibres of the action of G , which are diffeomorphic to G.

Let p : G_{vreg} → T_{vreg}/W be the obvious projection. Since ϕ_{∗}dv_{X} is absolutely continuous with respect to dg, ϕ^{−1}(G_{vreg}) is an open set in X, whose complement is dv_{X} negligible. Then x ∈ ϕ^{−1}(G_{vreg}) ↦ pϕ(x) ∈ T_{vreg}/W is a smooth map. Also if x ∈ ϕ^{−1}(G_{vreg}), g ∈ G,

pϕ(x · g) = pϕ(x) . (5.182)

By (5.182), if x ∈ ϕ^{−1}(G_{vreg}), the obvious analogue of (D, δ) in (5.53) is the complex

\xymatrix{(C′_{x}^{.}, ∂) : 0 \ar[r] & 𝔤 \ar[r]_{δ} & 𝔤^{2g} ⊕ ⨁_{j=1}^{s} Im(1 − w_{j}) \ar[r]_{d(pϕ)_{x}} & 𝔱 \ar[r] & 0 .} (5.183)

By Proposition 5.10 and Theorem 5.12, if x ∈ G is such that Z′(x) = 1, the cohomology of the complex (5.183) is concentrated in degree 1, and the first cohomology group of (5.183) is isomorphic to T_{π(x)}X_{pϕ(x)}/T. In particular,

(det C′_{x})^{−1} ≃ det(T_{π(x)}X_{pϕ(x)}/T) . (5.184)

Now the complex C′_{x} is equipped with a scalar product. Let ∥ ∥_{det(C′_{x})} be the obvious metric on det(C′_{x}), and let ∥ ∥_{det(T_{x}X_{(x)}/T)} be the corresponding metric on det(T_{π(x)}X_{hp(x)/T}). Then ∥ ∥_{det(T_{x}X_{(x)}/T)} defines a smooth volume form on (X_{π(x)}/T)_{reg}, which will be denoted d̃v_{X_{pϕ(x)}/T}. Then the formula of change of variable asserts that

∫_{X} f(x)dv_{X}(x) = \frac{1}{|Z(G)|} ∫_{T/W} dt ∫_{X_{t}/T} d̃v_{X_{t}/T} (x) ∫_{G} f(x · g)dg . (5.185)

If x ∈ ϕ^{−1}(T_{vreg}), put t = [ϕ(x)]^{−1} ∈ T_{vreg}. Then we have the double complex

\xymatrix{ & 0 & 0 & 0 \\ 0 \ar[r] & 𝔤 \ar[u] \ar[r] & 𝔤^{2g} ⊕ ⨁_{j=1}^{s} Im(1 − w_{j}) \ar[u] \ar[r]_{d(pϕ)} & 𝔱 \ar[u] \ar[r] & 0 \\ 0 \ar[r] & 𝔤 \ar[u] \ar[r] & 𝔤^{2g} ⊕ ⨁_{j=1}^{s} Im(1 − w_{j}) ⊕ Im(1 − t) \ar[u] \ar[r] & 𝔤 \ar[u] \ar[r] & 0 \\ 0 \ar[r] & 0 \ar[u] \ar[r] & Im(1 − t) \ar[r]_{t^{−1}} \ar[u] & Im(1 − t) \ar[r] \ar[u] & 0 \\  & 0 \ar[u] & 0 \ar[u] & 0 \ar[u]} (5.186)

Let x ∈ Σ\∂Σ, and let Δ be a small disk of center x. Put Σ_{+1} = Σ\Δ. In (5.186), starting from below, the first row is trivial. The second row is just the complex (Ĉ_{(x,t)}^{Σ_{+1}}(E), ∂) we constructed in (5.31) (with s replaced by s + 1). The third row is the complex (C′_{x}, ∂). Also the columns are acyclic.

Let us now explain why the diagram commutes. By (5.34), if x = (u_{1}, v_{1}, …, w_{s}),

∂(f_{1}, …, f_{2g+s+1}) = ∑_{i=1}^{g}([u_{1}, v_{1}] … [u_{i−1}, v_{i−1}](1 − u_{i}v_{i}u_{i}^{−1})f_{2i−1} (5.187)

+u_{i}(1 − v_{i}u_{i}v_{i}^{−1})f_{2i}) + ∑_{j=1}^{s}[u_{1}, v_{1}] … [u_{g}, v_{g}]w_{1} … w_{j−1}f_{2g+j} +

∏_{i=1}^{g}[u_{i}, v_{i}] ∏_{j=1}^{s} w_{j}f_{2g+s+1} .

By construction,

∏_{i=1}^{g}[u_{i}, v_{i}] ∏_{j=1}^{s} w_{j} = t^{−1} . (5.188)

So in (5.187),

∂(f_{2g+s+1}) = t^{−1}f_{2g+s+1} . (5.189)

So from (5.189), we find the diagram (5.186) commutes.

Now the determinants of the columns of (5.186) are canonically trivial, and the norm of the associated canonical sections is equal to 1. Also since the first row is acyclic, the cohomology groups of the second and third rows are canonically identified, using the obvious long exact sequence. Finally the first row is acyclic, and the norm of the canonical section of the determinant of the first row is equal to 1.

By an abvious extension of [11, Theorem 1.10], it follows that if x ∈ ϕ^{−1}(T_{vreg}) is regular, then the metric on det(Ĥ^{1}(Σ_{+1}, E)) (where Ĥ^{1}(Σ_{+1}, E) is the first cohomology group of the complexes (C′_{x}, ∂) or (Ĉ_{(x,t)}^{Σ_{+1}}(E), ∂)) induced by the complexes

(C′_{x}, ∂) or (Ĉ_{(x,t)}^{Σ_{+1}}(E), ∂) are identical. Using now (5.156) and Theorem 5.39, , we get

d̃v_{X_{t}/T} = ∏_{j=1}^{s} |σ_{O_{j}}||σ(t)|dv_{X_{t}/T} . (5.190)

By (5.185), (5.190), we obtain the first equality in (5.181).

The scalar product ⟨ , ⟩ induces a scalar product on 𝔤/𝔱 = 𝔱^{⊥}. Let dġ be the corresponding volume element on T/G. Tautologically if k : G → R is bounded and measurable

∫_{G} k(g)dg = ∫_{T \G} dġ ∫_{T} k(tg)dt . (5.191)

Also Weyl’s integration formula [15, Theorem IV.1.11] asserts that

∫_{G} k(g)dg = \frac{1}{Vol(T)} ∫_{T/W} |σ|^{2}(t)dt ∫_{G} f(t · g)dg . (5.192)

Using the first identity in (5.181) and (5.191), (5.192), we obtain

∫_{X} f(x)dv_{X}(x) = \frac{∏_{j=1}^{s} |σ_{O_{j}}|}{|Z(G)|} ∫_{T/W} |σ(t)|dt (5.193)

∫_{X_{t}/T} dv_{X_{t}/T} (ẋ) ∫_{T \G} dġ ∫_{T} f(x · t′g)dt′

= \frac{∏_{j=1}^{s} |σ_{O_{j}}|}{|Z(G)|} ∫_{G} \frac{dg}{|σ(g)|} ∫_{X_{g}/Z(g)} dv_{X_{g}/Z(g)}(x) ∫_{Z(g)} f(xt)dt_{g} .

The proof of our Theorem is completed.

Definition 5.43. If t ∈ T, let |V (t)| be the absolute value of volume of X_{t}/Z(t) with respect to ω_{t}.

Then V (t) is a W -invariant function on T, which extends to a central function on G.

Now we prove a result which was already essentially proved by Liu [37, 38].

Theorem 5.44. Let f : G → R be a bounded measurable function. Then

∫_{X} f(ϕ^{−1}(x))dv_{X}(x) = \frac{∏_{j=1}^{s} |σ_{O_{j}}|}{|Z(G)|} ∫_{T/W} [ ∫_{G} f(g · t)dg]|σ(t)||V (t)|dt , (5.194)

∫_{X} f(ϕ^{−1}(x))dv_{X}(x) = \frac{Vol(T) ∏_{j=1}^{s} |σ_{O_{j}}|}{|Z(G)|} ∫_{G} \frac{f(g)|V (g)|}{|σ(g)|}dg .

Proof. Our formula follows from Theorem 5.42.

Remark 5.45. In [37, 38], Liu uses an essentially similar argument in his proof of the Witten formula [59, 60]. In fact let p_{t}(g) be the convolution heat kernel on G. Liu considers the quantity ∫_{X} p_{t}(ϕ^{−1}(x))dv_{X}, and following Witten [59], he studies its limit as t → 0. The arguments he uses to evaluate the limit i as being (up to a normalization constant) the absolute value of the symplectic volume of M/G are essentially the ones which are used in the proofs of Theorems 5.42 and 5.44.

Let A be the set of regular values of ϕ^{−1} in G. Then A is an open dense set, such that ^{c}A is negligible. Also on A, ϕ_{∗}dv_{X} has a smooth density with respect to dg. By Theorem 5.44, it follows that \frac{|V (g)|}{|σ(g)|} is smooth on A ∩ G_{vreg}.

Assume temporarily that (t_{1,} …, t_{s}) verify (A). By Theorem 5.18, 1 is a regular value of ϕ. By Theorem 5.20, Z′(x) = 1 a.e. on M = X_{1}. Using Theorem 5.44, and proceeding as in the proof of Theorem 3.13, we get

(5.195)  lim_{{g∈Gvreg \atop g→1}} \frac{|V (g)|}{|σ(g)|} = \frac{Vol(G)}{Vol(T)}V (1) .

Using (5.195) we find that

lim_{t→0} ∫_{X} p_{t}(ϕ^{−1}(x))dv_{X} = \frac{Vol(G) ∏_{j=1}^{s} |σ_{O_{j}}|}{|Z(G)|}V (1), (5.196)

which is a formula obtained by Liu [37, 38] by arguments essentially similar to the ones we gave in our proof of Theorems 5.42 and 5.44.

Let K be a positive Weyl chamber in 𝔱. Let Λ_{+} = Λ∩\overline{K} be the set of nonnegative weights. Then the irreducible representations of G are indexed by Λ_{+}. If λ ∈ Λ_{+}, let χ_{λ} be the character of the corresponding representation of G.

By Theorem 5.44, \frac{|V (g)|}{|σ(g)|} defines a L_{1} G-invariant function on G. Therefore, it defines an invariant distribution on G, which can be expanded as a linear combination of the characters χ_{λ} of G.

Let w_{j} ∈ O_{j}. Then O_{j} ≃ G/Z(w_{j}). Also Vol(Z(w_{j})) does not depend on the choice of w_{j}. Finally χ_{λ} takes the complex value χ_{λ}(t_{j}) on the orbit O_{j}.

Now we prove a result of Witten [59, 60].

Theorem 5.46. The following identity of G-invariant distributions holds on G,

(5.197) \frac{|V (g)|}{|σ(g)| ∏_{j=1}^{s} |σ_{O_{j}}|} = |Z(G)|\frac{Vol(G)^{2g+s−1}}{Vol(T) ∏_{j=1}^{s} Vol(Z(t_{j}))}

∑_{λ∈Λ_{+}} \frac{∏_{j=1}^{s} χ_{λ}(t_{j})χ_{λ}(g)}{χ_{λ}(1)^{2g+s−1}} .

Proof. By Theorem5.44, we get

\frac{1}{Vol(G)} ∫_{G} \overline{χ}_{λ}(g)\frac{|V (g)|}{|σ(g)|}dg = (5.198)

\frac{|Z(G)|}{Vol(G)Vol(T)} \frac{1}{∏_{j=1}^{s} |σ_{O_{j}}|} ∫_{X} χ_{λ}(ϕ(x))dv_{X}(x) .

To evaluate the integral in the right-hand side of (5.198), we follow Witten [59]. By [15, Theorem II.4.5], if a, b ∈ G,

∫_{G} χ_{λ}(agbg^{−1})dg = \frac{Vol(G)}{χ_{λ}(1)}χ_{λ}(a)χ_{λ}(b) , (5.199)

∫_{G} χ_{λ}(ag)χ_{λ′}(g^{−1}b)dg = Vol(G)δ_{λ,λ′} \frac{χ_{λ}(ab)}{χ_{λ}(1)} .

Finally, if h : O_{j} → R is a bounded measurable function, we have the easy formula

∫_{O_{j}} h(g)dv_{O_{j}}(g) = \frac{|σ_{O_{i}}|^{2}}{Vol(Z(t_{j}))} ∫_{G} h(gt_{j}g^{−1})dg . (5.200)

By (5.199), (5.200), we obtain

∫_{X} χ_{λ}(ϕ(x))dv_{X}(x) = \frac{Vol(G)^{2g+s} ∏_{j=1}^{s} |σ_{O_{j}}|^{2}}{∏_{j=1}^{s} Vol(Z(t_{j}))} \frac{χ_{λ}(t_{1}) … χ_{λ}(t_{s})}{χ_{λ}^{2g+s−1}(1)} . (5.201)

So by (5.198), (5.201), we get

\frac{1}{Vol(G)} ∫_{G} \overline{χ}_{λ}(g)\frac{|V (g)|}{|σ(g)|}dg (5.202)

= |Z(G)|\frac{Vol(G)^{2g+s−1}}{Vol(T)} ∏_{j=1}^{s} \frac{|σ_{O_{j}}|}{Vol(Z(t_{j}))}χ_{λ}(t_{j})\frac{1}{χ_{λ}(1)^{2g+s−1}} .

Now (5.202) is exactly the λ-Fourier coefficient of the invariant distribution \frac{|V (g)|}{|σ(g)|}. The proof of our Theorem is completed.

Clearly the distribution \frac{|V (g)|}{|σ(g)|} in C^{∞} at the regular values of the function ϕ. Now we will make a crude analysis of the Sobolev regularity of |V (g)|.

Proposition 5.47. If p < 2g − 1 − dim(𝔱)/2, the invariant distribution in the right-hand side of (5.197) lies in H^{p}(G). If 2g − 2 − dim(𝔱) > 0, this distribution is continuous.

Proof. Let ρ be the half-sum of the positive roots. By Weyl’s dimension formula [15, Theorem VI.1.7], if R_{+} is the set of positive roots,

χ_{λ}(1) = ∏_{α∈R_{+}} \frac{⟨λ + ρ, α⟩}{⟨ρ, α⟩} . (5.203)

By [15, Proposition V.4.12], if α ∈ R_{+}, ⟨ρ, α⟩ > 0. Therefore there is c > 0 such that if α ∈ R_{+}, λ ∈ Λ_{+},

⟨λ + ρ, α⟩ ≥ sup(c, ⟨λ, α⟩) . (5.204)

By (5.203), (5.204), we find that since the α ∈ R_{+} form a basis of 𝔱^{∗}, there is C > 0 such that

χ_{λ}(1) ≥ C∥λ∥ . (5.205)

Also

|χ_{λ}| ≤ χ_{λ}(1) . (5.206)

Let Δ_{G} be the Casimir operator on G [45, Section 9.4]. By [45, Proposition 9.4.2],

Δ_{G}χ_{λ} = \frac{1}{2}(∥λ + ρ∥^{2} − ∥ρ∥^{2})χ_{L} . (5.207)

Using (5.207), we get

(Δ^{G})^{p/2} ∑_{λ∈Λ_{+}} \frac{∏_{j=1}^{s} χ_{λ}(t_{j})χ_{λ}(g)}{(χ_{λ}(1))^{2g+s−1}} (5.208)

= \frac{1}{2^{p/2}} ∑_{λ∈Λ_{+}} \frac{(∥λ + ρ∥^{2} − ∥ρ∥^{2})^{p/2}}{χ_{λ}(1)^{2g+s−1}} ∏_{j=1}^{s} χ_{λ}(t_{j})χ_{λ}(g) .

By (5.205), (5.206),

\frac{|(∥λ + ρ∥^{2} − ∥ρ∥^{2})^{p/2} ∏_{1}^{s} χ_{λ}(t_{j})|}{χ_{λ}(1)^{2g+s−1}} ≤ C(1 + |λ|)^{−(2g−1−p)} . (5.209)

Also

∫_{𝔱} \frac{dλ}{(1 + |λ|)^{2(2g−1−p)}} < +∞ (5.210)

if and only if p < 2g − 1 − \frac{dim(𝔱)}{2}. From (5.209), (5.210), we get the first part of our Proposition. Also,

∫_{𝔱} \frac{dλ}{(1 + |λ|)^{2g−2}} < +∞ (5.211)

if and only if 2g − 2 − dim(𝔱) > 0. By (5.210), we obtain the second part of our Proposition.

5.9. Logarithms. Let U be a nonempty open set in G, stable by the adjoint action of G. We assume that there is a well-defined logarithm log : U → 𝔤, i.e. a smooth function U → 𝔤 such that if g ∈ U, g′ ∈ G,

g = exp(log(g)) , (5.212)

log(g′gg′^{−1}) = g′ · log(g) .

In particular by (5.212), log(g) is Z(g)-invariant.

Example 5.48. If U is a small ad-invariant open neighborhood of a central element in G, a logarithm is well-defined on U.

Example 5.49. Let O ⊂ G be a very regular orbit. Then O ∩ T consists of |W | distinct elements. Let t ∈ O ∩ T, and let h ∈ 𝔱 such that exp(h) = t. Then t ↦ h extends into a well defined logarithm O → G.

Example 5.50. Suppose that G is simply connected. Let K be a Weyl Chamber in 𝔱, let P be the alcove in K whose closure contains 0. Then by [15, Proposition V.7.10]

W \T_{reg} = P . (5.213)

Also by [15, Proposition V.7.11], the map

(g, t) ∈ G/T × P ↦ g exp(t)g^{−1} ∈ G_{reg} (5.214)

is one to one. Let log : G_{reg} → 𝔤 be the G-invariant function on G_{reg} such that if t ∈ P,

log(exp(t)) = t ∈ P . (5.215)

Then log : G_{reg} → 𝔤 is a logarithm.

We still assume that G is simply connected and simple. Let u ∈ C/\overline{R}^{∗}. Let Z(u) ⊂ G be the centralizer of u. Let Z(u)_{vreg} be the set of very regular elements in Z(u). Clearly

Z(u) ∩ G_{reg} ⊂ Z(u)_{vreg} . (5.216)

Also the function log maps Z(u)∩G_{reg} into 𝔷(u), and gives a logarithm on Z(u)_{vreg}.

Let B_{U} be the set of connections on the trivial G bundle over S_{1}, whose holonomy w lies in U. Needless to say, in a given trivialization, any element of B_{u} can be written in the form

\frac{D}{Dt} = \frac{d}{dt} + a_{t} , a_{t} ∈ L𝔤_{t} . (5.217)

Let τ_{t}^{0} be the parallel transport operator along s ∈ [0, t], so that w = τ_{1}^{0}. Set

w_{t} = τ_{t}^{0}τ_{1}^{0}(τ_{t}^{0})^{−1} . (5.218)

Then w_{t} is just w under translation of the origin in S_{1} by t.

Clearly log(w_{t}) is well-defined, and

log(w_{t}) = τ_{t}^{0} log(w)(τ_{t}^{0})^{−1} . (5.219)

Also

\frac{D}{Dt} log(w_{t}) = 0 . (5.220)

By (5.219),

e^{t log(w_{t})} = τ_{t}^{0}e^{t log(w)}(τ_{t}^{0})^{−1}. (5.221)

Put

\frac{^{0}D}{Dt} = e^{−t log(w_{t})} \frac{D}{Dt}e^{t log(w_{t})} . (5.222)

By (5.220), (5.222),

\frac{^{0}D}{Dt} = \frac{D}{Dt} + log(w_{t}) . (5.223)

From (5.220), (5.223), we get

\frac{^{0}D}{Dt} log(w_{t}) = 0 . (5.224)

The parallel transport operator ^{0}τ_{t}^{0} for \frac{^{0}D}{Dt} is given by

0τ_{t}^{0} = e^{−t log(w_{t})}τ_{t}^{0} , (5.225)

so that

0τ_{1}^{0} = 1 . (5.226)

By (5.226), the parallel transport trivialization with respect to \frac{^{0}D}{Dt} is globally defined on S^{1}, and in this trivialization,

\frac{^{0}D}{Dt} = \frac{d}{dt} . (5.227)

5.10. A symplectic structure on an open set in X. In the sequel, we assume that the assumptions before Theorem 5.42 are in force, and also that (t_{1}, …, t_{s}) verify (A).

Let U_{1}, …, U_{s} be ad-invariant open subsets of G on which a logarithm is well defined. Then G acts on the right on G^{2g} × ∏_{j=1}^{s} U_{j}.

Recall that x ∈ G^{2g+s} ↦ ϕ(x) ∈ G was defined in (5.41). Let x ∈ G^{2g} ×∏_{j=1}^{s} U_{j} be such that ϕ(x) = 1, and that G acts locally freely at x. Let U be an open neighbhorhood of 1 in G, such that a logarithm is still well-defined on U.

Let V be an open neighbhorhood of x in G^{2g}×∏_{j=1}^{s} U_{j} such that h(V) ⊂ U. Then we can find an open neighborhood A ⊂ G^{2g} × ∏_{j=1}^{s} U_{j} of x, which is G-invariant, such that h(A) ⊂ U, and on which G acts locally freely.

Let ϕ_{+1} : G^{2g+s+1} → G be given as in (5.41), with s replaced by s + 1. Set

A′ = {x′ ∈ G^{2g+s+1}; ϕ_{+1}(x′) = 1} . (5.228)

Clearly x ∈ G^{2g+s} ↦ (x, ϕ^{−1}(x)) ∈ A′ is a one to one G-equivariant map. Let Ã ⊂ A′ be the image of A by this map.

Let Σ_{+1} be the Riemann surface Σ with s + 1 small disks deleted. Equivalently Σ_{+1} is obtained from Σ by deleting an extra small disk. Let Σ̂_{+1} be the universal cover of Σ_{+1}. Put Γ_{+1} = π_{1}(Σ_{+1}). Then Γ_{+1} is generated by u_{1}, v_{1}, …, u_{g}, v_{g}, w_{1}, …, w_{s+1}, and the relation

∏_{i=1}^{g}[u_{i}, v_{i}] ∏_{j=1}^{s+1} w_{j} = 1 . (5.229)

As in (5.126), we find that Γ_{+1} and G both act on the right on Ã, and these actions commute. Also Γ_{+1} × G acts on the right on Σ̂_{+1} (and the action of G on Σ̂_{+1} is trivial). Then Γ_{+1} × G acts on the right on Ã× Σ̂_{+1}. Since Γ_{+1} acts freely on Σ̂_{+1}, Γ_{+1} × G acts locally freely on Ã × Σ̂_{+1}.

Set

C = (Ã × Σ̂_{+1})/Γ_{+1} . (5.230)

Then C is a fibre bundle over Σ_{+1} with fibre Ã. Also G acts locally freely on Ã and this action descends to a locally free action on C. We can then form the orbifolds Ã/G and C/G = Ã/G × Σ_{+1}.

Let V be a complex vector space. Let ρ : G → Aut(V) be a representation of G. We still denote by ρ the corresponding representation of Γ_{+1} × G in Aut(V).

Put

F = Ã × Σ̂_{+1} ×_{Γ_{+1}×G} V . (5.231)

Then F is a vector bundle on Ã/G × Σ_{+1}. Also,verifies easily that

F = C ×_{G} V . (5.232)

Moreover F is obained by the identification in (5.129), i.e.

(5.233)  (x, σ, f) ≃ (xg, σa, g^{−1}x(a^{−1})f) , x ∈ Ã,
σ ∈ Σ̃_{+1} , f ∈ V , g ∈ G , a ∈ Γ_{+1} .

Recall that r_{1}, …, r_{s+1} are the origin in S_{1}^{1}, …, S_{s+1}^{1}. For 1 ≤ j ≤ s + 1, put

\frac{^{0}D_{j}}{Dt} = \frac{D}{Dt} + log(w_{j,t}) . (5.234)

Then \frac{^{0}D_{j}}{Dt} is a connection on the G-bundle P on S_{j}, with holonomy 1.

For 1 ≤ j ≤ s + 1, let ∇_{j} be a connection on the G-bundle Ã → Ã/G. Consider

the G-bundle \xymatrix{Ã × Σ̂_{+1} \ar[r]^{G} & Ã/G × Σ̂_{+1}} . Along the fibres Σ̂_{+1}, we can equip this G-bundle with the trivial connection. This connection is Γ_{+1} invariant. Therefore it

descends to a G-connection on \xymatrix{(Ã × Σ̂_{+1})/Γ_{+1} \ar[r]^{G} & Ã/G × Σ_{+1}} along the fibres Σ_{+1}. This connection along Σ_{+1} is exactly the flat G-connection associated to the given element of Ã.

Along S_{j}^{1}, 1 ≤ j ≤ s+1, we trivialize the G-bundle \xymatrix{(Ã × Σ̂_{+1})/Γ_{+1} \ar[r]^{G} & Ã/G × Σ_{+1}}

with respect to \frac{^{0}D_{j}}{Dt}. Then over Ã/G × S_{j}^{1}, 1 ≤ j ≤ s + 1, the connection ∇_{j}

induces a G-connection on \xymatrix{(Ã × Σ̂_{+1})/Γ_{+1} \ar[r]^{G} & Ã/G × Σ_{+1}} along Â/G. Since

the Ã × S_{j}^{1} are disjoint, we can extend this connection to a G-connection on

\xymatrix{(Ã × Σ̂_{+1})/Γ_{+1} \ar[r]^{G} & Ã/G × Σ̂_{+1}} along Ã/G.

Ultimately the G-bundle \xymatrix{(Ã × Σ̂_{+1})/Γ_{+1} \ar[r]^{G} & Ã/G × Σ_{+1}} is equipped with a G-connection ∇. Let F be the curvature of this connection. Then F is a Gequivariant basic 2-form on (Ã × Σ̂_{+1})/Γ_{+1} with values in 𝔤.

Let E be the orbifold vector bundle on Ã/G × Σ_{+1} associated to the adjoint representation G → Aut(𝔤). Let ∇^{E} be the induced connection on E. Then F descends to a 2-form on Ã/G × Σ_{+1} with values in E. Set

E_{j} = r_{j}^{∗}E ; 1 ≤ j ≤ s + 1. (5.235)

Then E_{j} is a vector bundle on Ã/G. Let ∇^{E_{j}} be the connection induced by ∇^{E} (or ∇_{j}) on E_{j}. Observe that for 1 ≤ j ≤ s + 1, log(w_{j}) is a section of E_{j}.

Recall that C/G = Ã/G × Σ_{+1}. Then

Λ(T^{∗}C/G) = Λ(T^{∗}Ã/G)ˆ⊗Λ(T^{∗}Σ_{+1}) . (5.236)

If ω ∈ Λ(T^{∗}C), we can write ω in the form

ω = Σω^{(p,q)} , ω^{(p,q)} ∈ Λ^{p}(T^{∗}Ã/G)ˆ⊗Λ^{q}(T^{∗}Σ_{+1}) . (5.237)

Let k be the embedding Ã/G × ∂Σ_{+1} → C/G. Let Θ_{j} be the curvature of ∇_{j}.

Theorem 5.51. The following identities hold

(5.238) F^{(0,2)} = 0 ,
k^{∗}F = Θ_{j} − ∇^{E_{j}} log(w_{j})dt on Ã/G × S_{j}^{1} , 1 ≤ j ≤ s + 1 .

Proof. By construction the first identity holds in (5.238). Also on Ã/G × S_{j}^{1},

k^{∗}F = k^{∗} (∇^{E_{j}} + dt(\frac{^{0}D}{Dt} − log(w_{j})))^{2} ^{(1,1)} = Θ_{j} − ∇^{E_{j}} log(w_{j})dt . (5.239)

The proof of our Theorem is completed.

Let ⟨ , ⟩ be a G-invariant bilinear symmetric form on 𝔤. Clearly F^{2} ∈ Λ^{4}(T^{∗}C)⊗ 𝔤 ⊗ 𝔤. Then ⟨F^{2}⟩ ∈ Λ^{4}(T^{∗}C).

Let π, π′ be the projections Ã/G × Σ_{+1} → Ã/G, Ã/G × ∂Σ_{+1} → Ã/G. Let ρ be the projection Ã → Ã/G.

For 1 ≤ j ≤ s + 1, let θ_{j} be the connection 1-form on the G-bundle Ã → Ã/G associated to the connection ∇_{j}. Then θ_{j} is a 1 form with values in 𝔤^{∗}.

Definition 5.52. Let α be the 2-form on Ã,

α = ρ^{∗} [π_{∗} [\frac{⟨F^{2}⟩}{2}] + ∑_{j=1}^{s+1}⟨log(w_{j}), Θ_{j}⟩] − ∑_{j=1}^{s+1} d⟨log(w_{j}), θ_{j}⟩ . (5.240)

Theorem 5.53. The 2-form α is G-invariant and closed. It does not depend on the choices made in its construction.

Proof. By Chern-Weil theory, the form \frac{⟨F^{2}⟩}{2} is closed on Ã/G × Σ_{+1}. Using Stokes formula, we get

dπ_{∗} [\frac{⟨F^{2}⟩}{2}] = π′_{∗} [\frac{k^{∗}⟨F^{2}⟩}{2}] . (5.241)

By (5.238), (5.241), we get

dπ_{∗} [\frac{⟨F^{2}⟩}{2}] = − ∑_{j=1}^{s+1}⟨∇^{E_{j}} log(w_{j}), Θ_{j}⟩ . (5.242)

Also by Bianchi’s identity,

(5.243) d⟨log(w_{j}), Θ_{j}⟩) = ⟨∇^{E_{j}} log(w_{j}), Θ_{j}⟩ .

By (5.242), (5.243), we find that the form α is closed.

Now we replace Σ̂_{+1} by Σ̂_{+1} × R. We still denote by π the projection Ã × Σ̂_{+1} × R → Ã/G × R with fibre Σ̂_{+1}. The G- bundle Ã → Ã/G is replaced by Ã × R → Ã/G × R. We consider a smooth family of data, which are used to construct the form α_{ℓ}. In particular the connections ∇_{j} depends on ℓ. We extend these fibrewise connections: ∇_{j} Ã → Ã/G to a connection dℓ\frac{∂}{∂ℓ} + ∇_{j} on Ã × R → Ã/G × R. We will denote with a the analogue of the above objects over Ã × Σ̂_{+1} × R. This way, we obtain a form α̃ on Ã × R such that

α̃ = α_{ℓ} + dℓ ∧ β_{ℓ} , (5.244)

where β_{ℓ} is a 1-form on Ã/G. By the above arguments, α̃ is closed.

Now we will show that

β_{ℓ} = 0 . (5.245)

This will imply that

\frac{∂α_{ℓ}}{∂ℓ} = 0 . (5.246)

So we will have established our claim that α_{ℓ} does not depend on ℓ.

Recall that along the fibres Σ̂_{+1}, the connection on the considered G-bundle does not depend on ℓ ∈ R, and is flat. It then follows easily that π_{∗} [\frac{⟨F̃^{2}⟩}{2}] does not contain dℓ.

Also the connection form θ̃_{j} does not contain dℓ, i.e. it is of the form

θ̃_{j} = θ_{j,ℓ} . (5.247)

Then

Θ̃_{j} = Θ_{j,ℓ} + dℓ\frac{∂}{∂ℓ}θ_{j,ℓ} . (5.248)

Therefore

(5.249)  ⟨log(w_{j}), Θ̃_{j}⟩ = ⟨log(w_{j}), Θ̃_{j,ℓ} + dℓ\frac{∂}{∂ℓ}θ_{j,ℓ}⟩,
d̃⟨log(w_{j}), θ̃_{j}⟩ = ⟨d log(w_{j}), θ_{j,ℓ}⟩ + ⟨log(w_{j}), dθ_{j,ℓ}⟩,

+⟨log(w_{j}), dℓ\frac{∂}{∂ℓ}θ_{j,ℓ}⟩ .

From (5.142), we find the sum of the last terms in (5.240) does not contain dℓ either.

The proof of our Theorem is completed.

Now we fix t_{1} ∈ T ∩ U_{1}, …, t_{s} ∈ T ∩ U_{s}. For 1 ≤ j ≤ s, let O_{j} ⊂ G be the orbit of t_{j}.

Put

X̂ = {x ∈ Ãw_{j} ∈ O_{j} , 1 ≤ j ≤ s} . (5.250)

Then X̂ ⊂ X is stable under G. Let m be one of the embeddings X̂ → Ã, X̂/G → Ã/G.

For 1 ≤ j ≤ s, let Õ_{j} ⊂ 𝔤 be the G-orbit of log(t_{j}). Since there is a well-defined logarithm on U_{j}, O_{j} and Õ_{j} are in one to one correspondance.

For 1 ≤ j ≤ s, let σ_{j} be the canonical symplectic form on the orbit Õ_{j}. If Y ∈ 𝔤, let Y Õ_{j} be the corresponding vector field associated to the right action of G on Õ_{j}.

134 JEAN-MICHEL BISMUT AND FRANÇOIS LABOURIE

Then if Y, Y′ ∈ 𝔤, p ∈ Ô_{j}, as in (1.193), σ_{j,p}(Y, Y′) = ⟨p, [Y, Y′]⟩ . (5.251) Definition 5.54. Put

σ = m^{∗}α + ∑_{j=1}^{s} log(w_{j})^{∗}σ_{j} . (5.252)

Clearly σ is a G-invariant closed 2-form on X̂. We will calculate σ.

For 1 ≤ j ≤ s, the G-bundle X̂ → X̂/G can be reduced to the Z(t_{j})-bundle {x ∈ X̂, w_{j} = t_{j}} ↦ {x ∈ X̂, w_{j} = t_{j}}/Z(t_{j}). Let ∇_{j} be a Z(t_{j})-connection on this last bundle. This connection lifts to a G-connection on X̂ → X̂/G. We will make this choice of ∇_{j} in our construction of σ. Then for 1 ≤ j ≤ s, ∇_{j}w_{j} = 0 , (5.253)

∇_{j} log(w_{j}) = 0 .

Theorem 5.55. The following identity holds

σ = m^{∗}ρ^{∗}(π_{∗} \frac{⟨F^{2}⟩}{2}) + ⟨log(w_{s+1}), Θ_{s+1}⟩) (5.254)

−d⟨log(w_{s+1}), θ_{s+1}⟩ .

Proof. The second identity in (5.253) can be written in the form d log(w_{j}) + [θ_{j}, log(w_{j})] = 0 . (5.255) By (5.251), we get

log(w_{j})^{∗}σ_{j} = ⟨log(w_{j}), \frac{1}{2}[θ_{j}, θ_{j}]⟩ . (5.256)

Also

dθ_{j} = −\frac{1}{2}[θ_{j}, θ_{j}] + Θ_{j} . (5.257)

From (5.255)-(5.257), we obtain for 1 ≤ j ≤ s, (5.258) ρ^{∗}⟨log(w_{j}), Θ_{j}⟩ − d⟨log(w_{j}), θ_{j}⟩ + log(w_{j})^{∗}σ_{j}

= −⟨d log(w_{j}), θ_{j}⟩ + ⟨log w_{j}, [θ_{j}, θ_{j}]⟩ = 0 .

The proof of our Theorem is completed.

Set M̂ = {x ∈ X̂, w_{s+1} = 1} . (5.259) Our notation in (5.259) is compatible with (5.59). By Propositions 5.10 and 5.11, we know that M̂ is a submanifold of X̂, and that G acts locally freely on M̂. Let i : M̂ → X̂, M̂/G → X̂/G be the obvious embeddings.

By Theorem 5.30, M̂/G carries a symplectic form ω. Recall that 𝔤 is equipped with the scalar product ⟨, ⟩. We identify 𝔤 and 𝔤^{∗} by this scalar product. Theorem 5.56. If U is small enough, σ is a symplectic form on X̂. Also x = (u_{1}, v_{1}, …, u_{g}, v_{g}, w_{1}, …, w_{s+1}) ∈ X̂ ↦ log(w_{s+1}) ∈ 𝔤 is a moment map for the action of G on X with respect to the 2-form σ. The associated symplectic form on the symplectic reduction M̂/G coincides with i^{∗}π_{∗}(\frac{⟨F^{2}⟩}{2}) and with ω.

Proof. By (5.254),

i^{∗}σ = i^{∗}ρ^{∗}π_{∗} (\frac{⟨F^{2}⟩}{2}) . (5.260)

Let ∇^{H}, ∇^{V} be the components of ∇ along Ã/G, Σ_{+1}. Since ∇ is flat along Σ_{+1},

F = ∇^{H,2} + [∇^{H}, ∇^{V}]. (5.261)

From (5.261) we get

π_{∗}(\frac{⟨F^{2}⟩}{2}) = π_{∗}(\frac{[∇^{H}, ∇^{V}]}{2}). (5.262)

Moreover by (5.238), (5.253),

k^{∗}F^{(1,1)} = 0. (5.263)

From (5.261)-(5.263), we get easily

i^{∗}π_{∗}(\frac{⟨F^{2}⟩}{2}) = ρ_{∗}ω. (5.264)

From (5.260), (5.264), we obtain

i^{∗}σ = ρ^{∗}ω. (5.265)

Now by Theorem 5.30, ω is a symplectic form. Therefore by (5.254), (5.265), if U is small enough, σ is also a symplectic form.

If Y ∈ 𝔤, let Y X̂ be the corresponding vector field on X̂. Then by (5.254),

(5.266)  i_{Y^{X}} σ = −i_{Y X̂}d⟨log(w_{s+1}), θ_{s+1}⟩
= di_{Y X̂}⟨log(w_{s+1}), θ_{s+1}⟩
= d⟨log(w_{s+1}), Y ⟩ .

From (5.266), we find that x ∈ X ↦ log(w_{s+1}) ∈ 𝔤 ≃ 𝔤^{∗} is a moment map for the action of G on X̂.

The proof of our Theorem is completed.

By Theorem 5.12, since G acts locally freely on X̂, the derivative of x ∈ X̂ ↦ w_{s+1}^{−1} ∈ G is surjective.

Take t ∈ U close enough to 0. Put

X_{t} = {x ∈ X̂, w_{s+1} = t} . (5.267)

Then Z(t) acts locally freely on X_{t} . So X_{t}/Z(t) can be equipped with the symplectic form σ_{t}, the reduction of the symplectic form σ. Also recall that by Theorem 5.30, M_{t}/Z(t) is equipped with a symplectic form ω_{t}.

Theorem 5.57. For t ∈ U,

σ_{t} = ω_{t} . (5.268)

Proof. Clearly the G-bundle Ã → Ã/G reduces to X_{t} → X_{t}/Z(t). Let ∇_{s+1} be a Z(t) connection on this last bundle. Then ∇_{s+1} lifts to a G-connection on Ã → Ã/G. We will use ∇_{s+1} to calculate the restriction of σ to X_{t}. On X_{t},

∇^{E_{s+1}} log(w_{s+1}) = 0 , (5.269)

d log(w_{s+1}) + [θ_{s+1}, log(w_{s+1})] = 0 .

Over log(w_{s+1}) = − log(t), we get

[θ_{s+1}, log(w_{s+1})] = 0 . (5.270)

Then over log(w_{s+1}) = − log(t), using (5.257), (5.269), (5.270),

d⟨log(w_{s+1}), θ_{s+1}⟩ = ⟨log(w_{s+1}), Θ_{s+1}⟩ − ⟨log(w_{s+1}), \frac{1}{2}[θ_{s+1}, θ_{s+1}]⟩ (5.271)
= ⟨log(w_{s+1}), Θ_{s+1}⟩ .

From (5.254), (5.271), we find that over X_{t},

σ = ρ^{∗}π_{∗} (\frac{⟨F^{2}⟩}{2}) . (5.272)

Using (5.272) and proceeding as in the proof of Theorem 5.56, our Theorem follows.

Take j, 1 ≤ j ≤ s. Put

X̂_{j} = {x ∈ Ã; w_{j′} ∈ O_{j′}, j′ ≠ j, w_{j} ∈ U_{j}, w_{s+1} = 1} . (5.273)

Let n_{j} be the embedding X̂_{j} → Ã.

Definition 5.58. Put

κ_{j} = n_{j}^{∗}α + ∑_{j′=1 , j′≠j}^{s} log(w_{j})^{∗}σ_{j} . (5.274)

We will choose the connection ∇_{j′}, 1 ≤ j ≤ s, j′ ≠ j as in (5.253). Then by proceeding as in the proof of Theorem 5.55, we get

(5.275)  κ_{j} = n_{j}^{∗}ρ^{∗} (π_{∗} (\frac{⟨F^{2}⟩}{2}) + ⟨log(w_{j}), Θ_{j}⟩)
−d⟨log(w_{j}), θ_{j}⟩ .

Theorem 5.59. If U_{j} is small enough, κ_{j} is a symplectic form on X̂_{j}. Also x = (u_{1}, v_{1}, …, u_{g}, v_{g}, w_{1}, …, w_{s}) ∈ X̂_{j} ↦ log(w_{j}) ∈ 𝔤 is a moment map for the action of G on X̂_{j} with respect to κ_{j}. Finally for t_{j} ∈ U_{j}, the symplectic form on the symplectic reduction {x ∈ X̂_{j}, w_{j} = t_{j}}/Z(t_{j}) is the symplectic form defined in Theorem 5.30.

Proof. The proof of our Theorem is the same as the proof of Theorems 5.56 and 5.57.

Remark 5.60. If t_{1}, …, t_{s} are restricted to be very regular, we may and will assume that θ_{1}, …, θ_{s} are T -connections. By Theorems 5.57 and 5.59, we find that π_{∗}(\frac{⟨F^{2}⟩}{2}) restricts to the symplectic form of Theorem 5.30. Also by (5.275), the cohomology class of

ω + ∑_{j=1}^{s}⟨log(w_{j}), Θ_{j}⟩ (5.276)

is locally constant, which is a consequence of the theory of the moment map for torus actions obtained by Duistermaat-HeckmanDH.

5.11. The integral of certain characteristic classes on the strata of M/G. Let G be a compact connected and simply connected simple Lie group. We use the notation of Section 1. In particular, ⟨, ⟩ denotes the basic scalar product on 𝔤.

Let t_{1}, …, t_{s} be regular elements in T. Then t_{1}, …, t_{s} are very regular.

We assume that s ≥ 1, and that

• (t_{1}, …, t_{s}) verify (A).

• If g = 0, then s ≥ 3. In particular

2g − 2 + s ≥ 1 (5.277)

Let u ∈ C/\overline{R}^{∗}. Recall that π_{u} : Z̃(u) → Z(u) is the universal cover of Z(u), with fibre π_{1}(Z(u)) ≃ \overline{CR}/\overline{CR}_{u} ⊂ Z(Z̃(u)). Also T̃_{u} = 𝔱/\overline{CR}_{u} is a maximal torus in Z̃(u).

Remember that T_{reg} is the set of regular elements in T with respect to G. Let t ∈ T_{reg}. Then Z(t) = T, t is very regular in Z(u), and O_{Z(u)}(t) ≃ Z(u)/T.

Let t̃ ∈ T̃_{u} be a lift of t in Z̃(u). Then t̃ is still regular in Z̃(u). Since Z(t) = T, the centralizer Z(t̃) of t̃ in Z̃(u) is just Z(t̃) = 𝔱/\overline{CR}_{u}. Then O_{Z̃(u)}(t̃) ≃

Z̃(u)/(𝔱/\overline{CR}_{u}) ≃ Z(u)/T. Equivalently the projection O_{Z̃(u)}(t̃) → O_{Z(u)}(t) is one to one.

As before, we identify t_{1}, …, t_{s} with corresponding elements of G-alcoves in 𝔱 whose closure contains 0. This way, we get elements of T̃_{u} = 𝔱/\overline{CR}_{u}, which lift t_{1}, …, t_{s} unambiguously in Z̃(u). We still denote these elements by t_{1}, …, t_{s}.

Clearly

Z̃′(u) = Z′(u). (5.278)

Set

(5.279)  X_{u} = Z(u)^{2g} × ∏_{j=1}^{s} O_{Z(u)}(t_{j}),

X̃_{u} = Z̃(u)^{2g} × ∏_{j=1}^{s} O_{Z̃(u)}(t_{j}).

Then Z(u) acts on X_{u} and on X̃_{u}. Moreover (\overline{CR}/\overline{CR}_{u})^{2g} acts freely on X̃_{u}. Namely if f = (α_{1}, β_{1}, … α_{g}, β_{g}) ∈ (\overline{CR}/\overline{CR}_{u})^{2g}, if x = (ũ_{1}, ṽ_{1}, … ũ_{g}, ṽ_{g}, w̃_{1}, … w̃_{s}) ∈ X̃_{u},, put

f.x = (ũ_{1}α_{1}, ṽ_{1}β_{1}, …, ũ_{g}α_{g}, ṽ_{g}β_{g}, w̃_{1}, … w̃_{s}). (5.280)

The actions of Z(u) and of (\overline{CR}/\overline{CR}_{u})^{2g} commute. Also the map π_{u} : Z̃(u) → Z(u) extends to a map X̃_{u} → X_{u}. Clearly π_{u} is Z(u)-equivariant. Also if f ∈ (\overline{CR}/\overline{CR}_{u})^{2g},

π_{u}f = π_{u}. (5.281)

More precisely π_{u} is a (\overline{CR}/\overline{CR}_{u})^{2g} cover.

Definition 5.61. Let ϕ̃_{u} : X̃_{u} → Z̃(u) and ϕ_{u} : X_{u} → Z(u) be the analogues of ϕ defined in (5.41).

Clearly

ϕ_{u}π_{u} = π_{u}ϕ̃_{u}. (5.282)

Also ϕ̃_{u} and ϕ_{u} are Z(u)- equivariant .

Clearly (t_{1}, …, t_{s}) verify (A) with respect to Z(u) or Z̃(u). Therefore by Theorem 5.18, 1 is a regular value of ϕ_{u}. Equivalently, by Theorem 5.18, 1 is a regular value of ϕ. By Theorem 5.12, G acts locally freely on M = ϕ^{−1}(1), and so by Theorem 5.12, 1 is a regular value of ϕ_{u}.

Proposition 5.62. Any h ∈ \overline{CR}/\overline{CR}_{u} is a regular value of ϕ̃_{u}.

Proof. Since 1 is a regular value of ϕ_{u}, any h ∈ π_{u}^{−1}(1) is a regular value of ϕ̃_{u}.

Recall that the Lie algebra 𝔷(u) is equipped with the scalar product induced by the scalar product ⟨, ⟩ on 𝔤.

Let U be a G- invariant open neighborhood of 1 in G, such that a logarithm log : U → 𝔤 is defined, with

log(1) = 0. (5.283)

Put

U_{u} = U ∩ Z(u), Ũ_{u} = π_{u}^{−1}(U_{u}). (5.284)

Clearly log maps U_{u} into 𝔷(u). Then Ũ_{u} is an open neighborhood of \overline{CR}/\overline{CR}_{u} in Z̃(u), which only consists of regular values of ϕ̃_{u}.

In the sequel, we will view exp(log(ϕ_{u}(x)) as an element of Z̃(u). Observe that if x̃ ∈ X̃_{u} is such that x̃ ∈ ϕ̃_{u}^{−1}(Ũ_{u}),

π_{u} [ϕ̃_{u}(x̃) exp (− log(ϕ_{u}π_{u}(x̃)))] = 1. (5.285)

Therefore

ϕ̃_{u}(x̃) exp (− log(ϕ_{u}π_{u})(x̃))) ∈ \overline{CR}/\overline{CR}_{u}. (5.286)

Finally note that if x ∈ ϕ_{u}^{−1}(U), if x̃ is such that π_{u}(x̃) = x, then ϕ̃_{u}(x̃) does not depend on x̃. Therefore ϕ̃_{u}(x̃) exp (− log(ϕ_{u}π_{u}(x̃))) ∈ \overline{CR}/\overline{CR}_{u} does not depend on x̃.

Definition 5.63. If h ∈ \overline{CR}/\overline{CR}_{u}, let ϕ_{u}^{−1}(U_{u})_{h} ⊂ ϕ_{u}^{−1}(U) be given by

ϕ_{u}^{−1}(U_{u})_{h} = {x ∈ ϕ_{u}^{−1}(U_{u}), if π_{u}(x̃) = x, ϕ̃_{u}(x̃) exp (− log(ϕ_{u}π_{u}(x̃))) h = 1}. (5.287)

Similarly, let ϕ̃_{u}^{−1}(Ũ_{u})_{h} ⊂ ϕ̃_{u}^{−1}(Ũ_{u}) be given by

ϕ̃_{u}^{−1}(Ũ_{u})_{h} = {x̃ ∈ ϕ̃_{u}^{−1}(Ũ_{u}), ϕ̃_{u}(x̃) exp (− log(ϕ_{u}π_{u}(x̃))) h = 1}. (5.288)

Then we have the disjoint union

ϕ_{u}^{−1}(U_{u}) = ⋃_{h∈\overline{CR}/\overline{CR}_{u}} ϕ_{u}^{−1}(U_{u})_{h}, (5.289)

ϕ̃_{u}^{−1}(Ũ_{u}) = ⋃_{h∈\overline{CR}/\overline{CR}_{u}} ϕ̃_{u}^{−1}(Ũ_{u})_{h},

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS 139

and for any h ∈ \overline{CR}/\overline{CR}_{u}, π_{u} : ϕ̃_{u}^{−1}(Ũ_{u})_{h} → ϕ_{u}^{−1}(U_{u})_{h} is a (\overline{CR}/\overline{CR}_{u})^{2g}-cover. Moreover Z(u) preserves ϕ̃_{u}^{−1}(Ũ_{u})_{h} and ϕ_{u}^{−1}(U_{u})_{h}, and (\overline{CR}/\overline{CR}_{u})^{2g} acts freely on ϕ̃_{u}^{−1}(Ũ_{u})_{h}.

If U is small enough, there is an open neignborhood Ṽ_{u} of 1 in Z̃(u) such that π_{u} : (Ṽ_{u}) → U_{u} is one to one. Also if t ∈ U, and t̃ ∈ Ṽ_{u} is the lift of t, then π_{u} is one to one from O_{Z̃(u)}(t̃) into O_{Z(u)}(t). Definition 5.64. If h ∈ \overline{CR}/\overline{CR}_{u}, t ∈ U_{u}, t̃∈ Ṽ_{u}, with π_{u}(t̃) = t, put M_{t}(Z(u), O_{Z(u)}(t_{1}), …, O_{Z(u)}(t_{s}), h) = {x ∈ ϕ_{u}^{−1}(U_{u})_{h}, ϕ_{u}(x) = t^{−1}}, (5.290)

M_{t}(Z̃(u), O_{Z̃(u)}(t_{1}), …, O_{Z̃(u)}(t_{s}), h) = {x̃ ∈ ϕ̃_{u}^{−1}(Ũ_{u})_{h}, ϕ̃_{u}(x̃) = t̃^{−1}h^{−1}}.

Then π_{u} : M_{t}(Z̃(u), O_{Z̃(u)}(t_{1}), …, O_{Z̃(u)}(t_{s}), h) → M_{t}(Z(u), O_{Z(u)}(t_{1}), …, O_{Z(u)}(t_{s}) , h) is a |\overline{CR}/\overline{CR}_{u}|^{2g} cover. If Z_{u}(t) is the centralizer of t in Z(u), π_{u} is a Z_{u}(t) equivariant map.

By Theorem 5.12 and Proposition 5.62, Z_{u}(t) acts locally freely on M_{t}(Z̃(u), O_{Z̃(u)}(t_{1}), …, O_{Z̃(u)}(t_{s}), h) and on M_{t}(Z(u), O_{Z(u)}(t_{1}), …, O_{Z(u)}(t_{s}), h). Proposition 5.65. The map π_{u} : M_{t}(Z̃(u), O_{Z̃(u)}(t_{1}), …, O_{Z̃(u)}(t_{s}), h)/Z_{u}(t) → M_{t}(Z(u), O_{Z(u)}(t_{1}), …, O_{Z(u)}(t_{s}), h)/Z_{u}(t) is a |\overline{CR}/\overline{CR}_{u}|^{2g} cover on the regular part of the orbifold M_{t}(Z̃(u), O_{Z̃(u)}(t_{1}), …, O_{Z̃(u)}(t_{s}), h)/Z_{u}(t).

Proof. Let x̃ ∈ M_{t}(Z̃(u), O_{Z̃(u)}(t_{1}), …, O_{Z̃(u)}(t_{s}), h), and let f ∈ (\overline{CR}/\overline{CR}_{u})^{2g} , g ∈ Z_{u}(t) be such that f.x̃ = x̃.g. (5.291) Then by (5.281), (5.291), π_{u}(x̃) = π_{u}(x̃).g. (5.292) If x = π_{u}(x) is in the regular part of M_{t}(Z(u), O_{Z(u)}(t_{1}), …, O_{Z(u)}(t_{s}), h), from (5.292), we get g = 1. (5.293) So by (5.291), (5.293), f = 1. (5.294) The proof of our Proposition is completed.

For t ∈ U_{u}, let σ_{t}^{u} be the symplectic form on M_{t}(Z(u), O_{Z(u)}(t_{1}), …, O_{Z(u)}(t_{s}) , h)/Z_{u}(t), and let σ̃_{t}^{u} be the symplectic form on M_{t}(Z̃(u), O_{Z̃(u)}(t_{1}), …, O_{Z̃(u)}(t_{s}), h)/Z_{u}(t). Definition 5.66. For u ∈ C/\overline{R}^{∗}, t ∈ U_{u}, put

V_{u}(t_{1}, … t_{s}, t, h) = |∫_{M_{t}(Z(u),O_{Z(u)}(t_{1}),... ,O_{Z(u)}(t_{s}),h)/Z′_{u}(t)} e^{σ̃_{t}^{u}} | , (5.295)

Ṽ_{u}(t_{1}, … t_{s}, t, h) = |∫_{M_{t}(Z̃(u),O_{Z̃(u)}(t_{1}),... ,O_{Z̃(u)}(t_{s}),h)/Z′_{u}(t)} e^{σ̃_{t}^{u}} | .

Proposition 5.67. The following identity holds

σ̃_{t}^{u} = π_{u}^{∗}σ_{t}^{u}. (5.296)

Moreover

|V_{u}|(t_{1}, … t_{s}, t, h) = \frac{1}{|\overline{CR}/\overline{CR}_{u}|^{2g}} |Ṽ_{u}|(t_{1} … t_{s}, t, h). (5.297)

Proof. Equation (5.296) is trivial. Using (5.296), we get (5.297).

Let K ⊂ 𝔱 be a Weyl chamber for G which is fixed once and for all. Then if u ∈ C/\overline{R}^{∗}, K is included in a unique Weyl chamber K_{u} for Z(u) or Z̃(u). Put

\overline{CR}_{u,+}^{∗} = \overline{CR}_{u}^{∗} ∩ \overline{K}_{u}. (5.298)

Then the irreducible representations of Z̃(u) are parametrized by \overline{CR}_{u,+}^{∗}. If λ ∈

\overline{CR}_{u,+}^{∗}, let χ_{λ}^{Z̃(u)} be the representation of Z̃(u) with highest weight λ. For t ∈ T̃_{u} = 𝔱/\overline{CR}_{u}, put

σ_{Z(u)}(T̃) = ∏_{α∈R_{u,+}} (e^{iπ⟨α,t⟩} − e^{−iπ⟨α,t⟩}) . (5.299)

Then |σ_{Z(u)}(t)| is well defined on 𝔱/\overline{R}^{∗}, and so is well defined on T = 𝔱/\overline{CR}.

Theorem 5.68. For any u ∈ C/\overline{R}^{∗}, h ∈ \overline{CR}/\overline{CR}_{u}, the following identity of Z(u)- invariant distributions in the variable t ∈ U holds,

\frac{|Ṽ_{u}|(t_{1}, … t_{s}, h, t)}{|σ_{Z(u)}(t + h) ∏_{j=1}^{s} |σ_{Z(u)}(t_{j})|} = |Z(Z̃(u))|Vol(𝔱/\overline{CR}_{u})|^{2g−2} (5.300)

|\frac{Vol(Z̃(u)}{Vol(T̃_{u})}|^{2g+s−1} ∑_{λ∈\overline{CR}_{u,+}^{∗}} \frac{∏_{j=1}^{s} χ_{λ}^{Z̃(u)}(t_{j})χ_{λ}^{Z̃(u)}(t + h)}{χ_{λ}^{Z̃(u)}(1)^{2g+s−1}}.

Proof. Recall that t ∈ U is identified to the corresponding element in Ṽ_{u} ⊂ Z̃(u). Since t_{j} ∈ Z̃(u) is very regular, the centralizer of t_{j} in Z̃(u) is equal to T̃_{u}. Then by (5.156) and by Theorem 5.46, we get (5.300) . The proof of our Theorem is completed.

Remark 5.69. Note that \overline{CR}^{∗} ∩ \overline{K}_{u} is exactly the set of nonnegative weights for Z(u). Also since h ∈ \overline{CR}/\overline{CR}_{u} lies in Z(Z̃(u)),

χ_{λ}^{Z̃(u)}(t + h) = e^{2iπ⟨λ,h⟩}χ_{λ}^{Z̃(u)}(t). (5.301)

From (5.300), (5.301), we get easily,

\frac{1}{|\overline{CR}/\overline{CR}_{u}|^{2g}} \frac{∑_{h∈\overline{CR}/\overline{CR}_{u}} |Ṽ_{u}|(t_{1}, …, t_{s}, h, t)}{|σ_{Z(u)}(t)|| ∏_{j=1}^{s} |σ_{Z(u)}(t_{j})|} = (5.302)

|Z(Z̃(u))||Vol(𝔱/\overline{CR}_{u})|^{2g−2} |\overline{CR}/\overline{CR}_{u}|^{1−2g} |\frac{Vol(Z̃_{u})}{Vol(T̃_{u})}|^{2g+s−1}

∑_{λ∈\overline{CR}^{∗}∩\overline{K}_{u}} \frac{∏_{j=1}^{s} χ_{λ}^{Z(u)}(t_{j})χ_{λ}^{Z(u)}(t)}{χ_{λ}^{Z(u)}(1)^{2g+s−1}}.

Clearly

Z(Z(u)) = Z(Z̃(u))/(\overline{CR}/\overline{CR}_{u}), (5.303)

Vol(𝔱/\overline{CR}_{u}) = Vol(T)|\overline{CR}/\overline{CR}_{u}|,

\frac{Vol(Z̃(u))}{Vol(T̃_{u})} = \frac{Vol(Z(u))}{Vol(T)}.

From (5.302), (5.303), we get

\frac{1}{|\overline{CR}/\overline{CR}_{u}|^{2g}} \frac{∑_{h∈\overline{CR}/\overline{CR}_{u}} Ṽ_{u}(t_{1}, …, t_{s}, h, t)}{|σ_{Z(u)}(t)|| ∏_{j=1}^{s} |σ_{Z(u)}(t_{j})|} = (5.304)

|Z(Z(u))|Vol(𝔱/\overline{CR})^{2g−2} |\frac{Vol(Z(u)}{Vol(T)}|^{2g+s−1}

∑_{λ∈\overline{CR}^{∗}∩\overline{K}_{u}} \frac{∏_{j=1}^{s} χ_{λ}^{Z(u)}(t_{j})χ_{λ}^{Z(u)}(t)}{χ_{λ}^{Z(u)}(1)^{2g+s−1}}.

Identity (5.304) fits with (5.197) and with (5.297).

Since t_{1}, …, t_{s} are very regular in Z̃(u), there are Z(u)-invariant open neighborhoods U_{1}^{u}, …, U_{s}^{u} of t_{1}, …, t_{s} in Z̃(u) on which a logarithm is well defined. Since Z(u) acts locally freely on M_{0}(Z̃(u), O_{Z̃(u)}(t_{1}), …, O_{Z̃(u)}(t_{s}), h), all the results of Section 5.10 can be used in this situation.

In particular, by Theorem 5.56, there is a Z(u)-invariant open neighborhood X̂_{u,h} of M_{0}(Z̃(u), O_{Z̃(u)}(t_{1}), …, O_{Z̃(u)}(t_{s}), h) in X̃_{u}, equipped with a symplectic form σ̃^{u}, such that the σ̃_{t}^{u} are the symplectic reductions of σ̃^{u}. So we may use the results of Section 3 in this situation.

Let T′_{u} = 𝔱/\overline{R}_{u}^{∗} be the obvious maximal torus in Z′(u) = Z(u)/Z(Z(u)). Now we will use the notation in Section 3.6. As in Section 3.6, the choice of a Weyl chamberK and of the corresponding Weyl chamber K_{u} for Z(u) defines an orientation on the M_{t}(Z̃(u), O_{Z̃(u)}(t_{1}), …, O_{Z̃(u)}(t_{s}), h)/T′_{u}, for t ∈ T ∩ U_{u}.

Definition 5.70. For t ∈ T ∩ U_{u}, put

H_{(u,g,s)}(t_{1}, … t_{s}, t) = ∫_{M_{t}(Z̃(u),O_{Z̃(u)}(t_{1}),... ,O_{Z̃(u)}(t_{s}),h)/T′_{u}} e^{σ̃_{t}^{u}}. (5.305)

Using Theorems 3.15, 5.56 and 5.59 , we know that H_{(u,g,s)}(t_{1}, … t_{s}, t) is locally a polynomial in t_{1}, …, t_{s}, t. Also recall that P̃_{u,2g+s−1}(t), t ∈ T̃_{u} was defined in Definition 2.39. By Theorem 2.37, P̃_{u,2g+s−1}(t) is a polynomial on T̃_{u}\S̃_{u}. Finally remember that in (2.33) , we set ℓ_{u} = |R_{u,+}|.

Theorem 5.71. The following identity of local polynomials in (t_{1}, …, t_{s}, t) holds

H_{(u,g,s)}(t_{1}, …, t_{s}, h, t) = (−1)^{ℓ_{u}(g−1)+1}|Z(Z̃(u))| (5.306)

|Vol(𝔱/\overline{CR}_{u})|^{2g−2} ∏_{j=1}^{s} sgn((−i)^{ℓ_{u}}σ_{Z(u)}(t_{j}))e^{−2iπ⟨ρ_{u},h⟩}

∑_{(w′^{1},... ,w′^{s})∈W_{u}^{s}} ∏_{j=1}^{s} ϵ_{w′^{j}} P̃_{u,2g+s−1}(t + h + ∑_{j=1}^{s} w′^{j}t_{j}).

Proof. Clearly, if t ∈ U_{u} ∩ T_{reg}, then Z′(t) = T′, so that Z_{u}(t) = T′_{u}. By (5.295), (5.305),

|H_{(u,g,s)}(t_{1}, …, t_{s}, h, t)| = Ṽ_{u}(t_{1}, … t_{s}, h, t) on U_{u} ∩ T_{reg}. (5.307)

Also

|σ_{Z(u)}(t)| = sgn((−i)^{ℓ_{u}}σ_{Z(u)}(t))(−i)^{ℓ_{u}}σ_{Z(u)}(t). (5.308)

Moreover, by (1.43), if h ∈ \overline{CR}/\overline{CR}_{u},

σ_{Z(u)}(t + h) = e^{2iπ⟨ρ_{u},h⟩}σ_{Z(u)}(t), (5.309)

By Theorem 1.41,

e^{2iπ⟨ρ_{u},h⟩} = ±1. (5.310)

Moreover for t close enough to 0, (−i)^{ℓ_{u}}σ_{Z(u)}(t) and π_{u}(t/i) have the same sign. Also by (3.129), (3.133),

\frac{χ_{λ}^{Z̃(u)}(1)}{Vol(Z̃(u))/Vol(T̃_{u})} = π_{u}(\frac{ρ_{u} + λ}{i}). (5.311)

Finally by Theorem 3.15, for t close enough to 0, H_{(u,g,s)}(t) and π_{u}(t/i) either vanish together, or they are nonzero, and then they have the same sign. Using (5.300), (5.310), (5.311) , we get the identity of distributions

H_{(u,g,s)}(t_{1}, …, t_{s}, h, t) = (−1)^{ℓ_{u}(g−1)}|Z(Z̃(u))||Vol(𝔱/\overline{CR}_{u})|^{2g−2} (5.312)

∏_{j=1}^{s} sgn((−i)^{ℓ_{u}}σ_{Z(u)}(t_{j}))e^{−2iπ⟨ρ_{u},h⟩}

∑_{λ∈\overline{CR}_{u,+}^{∗}} \frac{∏_{j=1}^{s} (σ_{Z(u)}(t_{j})χ_{λ}^{Z̃(u)}(t_{j}))σ_{Z(u)}(t + h)χ_{λ}^{Z̃(u)}(t + h)}{(π_{u}(ρ_{u} + λ))^{2g+s−1}}.

By (1.94), Theorem 1.38, and by (5.312), we obtain

H_{(u,g,s)}(t_{1}, …, t_{s}, h, t) = (−1)^{ℓ_{u}(g−1)}|Z(Z̃(u))|Vol(𝔱/\overline{CR}_{u})|^{2g−2}

∏_{j=1}^{s} sgn((−i)^{ℓ_{u}}σ_{Z(u)}(t_{j}))e^{−2iπ⟨ρ_{u},h⟩} (5.313)

∑_{λ∈\overline{CR}_{u,+}^{∗}} ∑_{(w′^{1},... ,w′^{s},w)∈W_{u}^{s+1}}  \frac{∏_{j=1}^{s} ϵ_{w′^{j}} ϵ_{w}^{s−1} exp(2iπ⟨w(λ+ρ),t+h+∑_{j=1}^{s} w′^{j}t_{j}⟩)}{(π_{u}(ρ_{u}+λ))^{2g+s−1}}.

Now by (1.186),if w ∈ W_{u},

π_{u}(w(ρ_{u} + λ)) = ϵ_{w}π_{u}(ρ_{u} + λ). (5.314)

Also by using in particular [15, Note V.4.14],

\{w(ρ_{u} + λ)\}_{{w∈Wu, \atop λ∈\overline{CR}{∗ \atop u,+}}} = \{λ ∈ \overline{CR}_{u}^{∗}, π_{u}(λ) ≠ 0\}. (5.315)

Using (2.158), ((5.313)-(5.315), we get the identity of distributions on U_{u},

H_{(u,g,s)}(t_{1}, …, t_{s}, h, t) = (−1)^{ℓ_{u}(g−1)+1}|Z(Z̃(u))||Vol(𝔱/\overline{CR}_{u})|^{2g−2}

∏_{j=1}^{s} sgn((−i)^{ℓ_{u}}σ_{Z(u)}(t_{j}))e^{−2iπ⟨ρ_{u},h⟩}

∑_{(w′^{1},... ,w′^{s})∈W_{u}^{s}} ∏_{j=1}^{s} ϵ_{w′^{j}} P̃_{u,2g+s−1}(t + h + ∑_{j=1}^{s} w′^{j}t_{j}). (5.316)

Now by Theorems 2.37, 3.15, 5.56, 5.59, we know that both sides of (5.316) are local polynomials of (t_{1}, …, t_{s}, t). Therefore (5.316) extends to an identity of local polynomials. The proof of our Theorem is completed.

Put

M(Z(u), O_{Z(u)}(t_{1}), …, O_{Z(u)}(t_{s}), h) = M_{0}(Z(u), O_{Z(u)}(t_{1}), …, O_{Z(u)}(t_{s}), h). (5.317)

Then M(Z(u), O_{Z(u)}(t_{1}), …, O_{Z(u)}(t_{s}), h) ⊂ M, and Z(u) acts locally freely on M(Z(u), O_{Z(u)}(t_{1}), …, O_{Z(u)}(t_{s}), h).

Let θ be a Z(u) connection form on the Z(u)-bundle M(Z(u), O_{Z(u)}(t_{1}), …, O_{Z(u)}(t_{s}), h) → M(Z(u), O_{Z(u)}(t_{1}), …, O_{Z(u)}(t_{s}), h)/Z(u), and let Θ be its curvature. Let θ_{1}, …, θ_{s} be connection forms taken as in Section 5.10, and such that (5.253) holds, and let Θ_{1}, …, Θ_{s} be their curvatures. Then Θ_{1}, …, Θ_{s} take their values in 𝔱.

Let Q be a Z(u)-invariant C^{∞} function on 𝔷(u), let Q_{1}, …, Q_{s} be C^{∞} functions on 𝔱. Recall that ω is the canonical symplectic form on M/G which is associated to the basic scalar product ⟨, ⟩ on 𝔤.

Proposition 5.72. If p ∈ R^{∗}, the following identity holds

∫M(Z(u),O_{Z(u)}(t_{1}),... ,O_{Z(u)}(t_{s}),h)/Z(u) Q(-Θ) ∏_{j=1}^{s} Q_{j}(−Θ_{j})e^{pω} = (5.318)

\frac{1}{|\overline{CR}/\overline{CR}_{u}|^{2g}} \frac{p^{(g−1) dim 𝔷(u)+ \frac{s}{2} dim(𝔷(u)/𝔱)}}{|W_{u}|}

Q(\frac{∂/∂t}{p}) ∏_{j=1}^{s} Q_{j}(\frac{∂/∂t_{j}}{p})π_{u}(\frac{∂/∂t}{2iπ})H_{(u,g,s)}(t_{1}, …, t_{s}, h, t).

Proof. Clearly

∫_{M(Z(u),O_{Z(u)}(t_{1}),... ,O_{Z(u)}(t_{s}),h)/Z(u)} Q(−Θ)e^{pω} ∏_{1}^{s} Q_{j}(−Θ_{j}) = (5.319)

p^{(dim M(Z(u),O_{Z(u)}(t_{1}),... ,O_{Z(u)}(t_{s}),h)/Z(u))/2}

∫_{M(Z(u),O_{Z(u)}(t_{1}),... ,O_{Z(u)}(t_{s}),h)/Z(u)} Q(−Θ/p) ∏_{1}^{s} Q_{j}(−Θ_{j}/p)e^{ω}.

Also byTheorems 5.56 and Proposition 5.65,

∫_{M(Z(u),O_{Z(u)}(t_{1}),... ,O_{Z(u)}(t_{s}),h)/Z(u)} Q(−Θ/p)e^{ω} = (5.320)

\frac{1}{|\overline{CR}/\overline{CR}_{u}|^{2g}} ∫_{M(Z̃(u),O_{Z̃(u)}(t_{1}),... ,O_{Z̃(u)}(t_{s}),h)/Z(u)Z(u)} Q(−Θ/p)e^{σ̃_{0}^{u}}.

Finally our assumptions on the t_{j}’s, Theorems 5.20 and 5.21 guarantee that Z′_{u}(x) = 1 a.e. on M(Z̃(u), O_{Z̃(u)}(t_{1}), …, O_{Z̃(u)}(t_{s}), h). We can then apply Theorem 3.21 to (5.320) and get (5.318). The proof of our Theorem is completed.

5.12. An evaluation of certain Euler characteristics. Recall that x ∈ R ↦ [x] ∈ [0, 1[ is the periodic function of period 1 such that for x ∈ [0, 1[, [x] = x.

Proposition 5.73. Let m ∈ N. Put z = e^{\frac{2iπ}{m}}. Then for ℓ ∈ Z,

\frac{1}{m} ∑_{k=1}^{m−1} \frac{z^{kℓ}}{1 − z^{−k}} = \frac{1}{2} − [\frac{ℓ}{m}] − \frac{1}{2m} . (5.321)

Proof. Take ρ ∈] − 1, +1[, and ℓ with 0 ≤ ℓ < m. Then

∑_{k=1}^{m−1} \frac{z^{−kℓ}}{1 − ρz^{k}} = ∑_{n=0}^{+∞} ∑_{k=1}^{m−1} ρ^{n}z^{k(n−ℓ)} = ∑_{n=0}^{+∞} ρ^{n} ( ∑_{k=0}^{m−1} z^{k(n−ℓ)} − 1) (5.322)

= ∑_{n=0}^{+∞} ρ^{n}m1_{m|(n−ℓ)} − \frac{1}{1 − ρ}

= \frac{mρ^{ℓ}}{1 − ρ^{m}} − \frac{1}{1 − ρ} .

By making ρ → 1 in (5.322), we get

∑_{k=1}^{m−1} \frac{z^{−kℓ}}{1 − z^{k}} = \frac{m − 1}{2} − ℓ , (5.323)

which is equivalent to

\frac{1}{m} ∑_{k=1}^{m−1} \frac{z^{kℓ}}{1 − z^{−k}} = \frac{1}{2} − [\frac{ℓ}{m}] − \frac{1}{2m} . (5.324)

So we have established (5.321) for 0 ≤ ℓ < m. Similarly if −m ≤ ℓ < 0, using (5.324), we obtain

\frac{1}{m} ∑_{k=1}^{m−1} \frac{z^{kℓ}}{1 − z^{−k}} = \frac{1}{m} ∑_{k=1}^{m−1} \frac{z^{k(m+ℓ)}}{1 − z^{−k}} = \frac{1}{2} − [1 + \frac{ℓ}{m}] − \frac{1}{2m} (5.325)

\frac{1}{2} − [\frac{ℓ}{m}] − \frac{1}{2m} .

The proof of our Proposition is completed.

Let Σ_{g} be a Riemann surface of genus g. Here, we have fixed a complex structure on Σ_{g}. Let x_{1}, …, x_{s} be s distinct points in Σ_{g}. Let D be the divisor

D = ∑_{j=1}^{s} x_{j} . (5.326)

Let [D] be the corresponding holomorphic line bundle on Σ_{g}. Let σ_{D} be the canonical section of [D]. Clearly

∫_{Σ_{g}} c_{1}([D]) = s . (5.327)

Let m ∈ N such that m divides s. Let λ be a holomorphic line bundle on Σ_{g} such that

λ^{m} = [D] . (5.328)

Put

Σ_{g}^{b} = {t ∈ λ , σ_{D} = t^{m}} . (5.329)

Then p = Σ_{g}^{b} → Σ_{g} is a branched covering of order m, with branching points x_{1}, …, x_{s}. Also by Hurwitz’s formula, the genus g′ of Σ_{g}^{b} is given by

g′ = mg + \frac{1}{2}(m − 1)(s − 2) . (5.330)

If ℓ ∈ Z/mZ, t ∈ Σ_{g}^{b}, put

ℓ(t) = e^{\frac{2iπℓ}{m}} t . (5.331)

Then (5.331) defines an action of Z/mZ over Σ_{g}^{b} such that pℓ = p. Also if ℓ ∈ Z/mZ, l ≠ 0, then x_{1}, …, x_{s} ∈ Σ_{g}^{b} are the only fixed points of ℓ.

Let Σ be the Riemann surface with boundary, which is obtained from Σ_{g} by deleting s small disks centered at x_{1}, …, x_{s}. Set

Σ^{b} = p^{−1}Σ . (5.332)

Then Σ^{b} is also a Riemann surface with boundary, obtained from Σ_{g}^{b} by deleting s small disks Δ_{1}, …, Δ_{s} centered at x_{1}, …, x_{s}.

Let G be a compact connected and simply connected simple Lie group.

Definition 5.74. We will say that g ∈ G is of order m if g^{m} = 1. More generally if O ⊂ G is an adjoint orbit, O will be said to be of order m if for one (or any) element g∈ O, g^{m} = 1.

In the sequel, we assume that O_{1}, … O_{s} are of order m.

Let x ∈ M. Then the trivial G-bundle P over Σ is equipped with the corresponding flat G-connection. Therefore the G-bundle p^{∗}P → Σ^{b} is equipped with the corresponding flat connection. Moreover Z/mZ acts naturally on this G-bundle and preserves the flat connection.

Observe that for 1 ≤ j ≤ s, the holonomy of the flat connection over the circle p^{−1}(S_{j}^{1}) —which is a m cover of S_{j}— is w_{j}^{m} = 1, i.e. π^{∗}P has trivial holonomy around π^{−1}(S_{j}). Therefore the flat connection on Σ^{b} extends to a flat connection on the trivial G-bundle p^{∗}P → Σ_{g}^{b}.

We claim the action of Z/mZ extends to the bundle p^{∗}P → Σ_{g}^{b}. To define the flat bundle p^{∗}P near x_{j}, we use the identification by parallel transport along the circle S_{j}^{1}. In fact recall that for 1 ≤ j ≤ s, w_{j} is the holonomy of the flat connection along S_{j}^{1} considered as lying in ∂Σ. The holonomy along S_{j} considered as the boundary of the disk Δ_{j} is w_{j}^{−1}. If ℓ ∈ Z/mZ, t ∈ Σ, f ∈ P, then

ℓ(t, f) = (e^{\frac{2iπℓ}{m}} t, f) . (5.333)

However using the trivialization of parallel transport along p^{∗}S_{j}^{1}, in this trivialization

ℓ(t, f) = (e^{\frac{2iπℓ}{m}} t, w_{j}^{ℓ}f) . (5.334)

In particular the action of ℓ on p^{∗}P_{x_{j}} is given by f ∈ P ↦ w_{j}^{ℓ}f ∈ P.

Let V be a complex vector space. Let ρ : G → Aut(V) be a representation of G. Let F be the flat vector bundle on Σ

F = P ×_{G} V . (5.335)

Then by the above construction, p^{∗}F extends to a flat vector bundle on Σ_{g}^{b}, on which Z/mZ acts. In particular Z/mZ acts on H^{.}(Σ_{g}^{b}, p^{∗}F). Let [H(Σ_{g}^{b}, p^{∗}F)]^{Z/mZ} be the invariant part of H(Σ_{g}^{b}, p^{∗}F) under the action of Z/mZ.

Let ⟨ , ⟩ be a G-invariant bilinear symmetric form on V. Let ⟨ , ⟩_{Ĥ^{1}(Σ,F)} and

⟨ , ⟩_{H^{1}(Σ_{g}^{b},p^{∗}F)} be the coresponding intersection forms on Ĥ^{1}(Σ, F) and H^{1}(Σ_{g}^{b}, p^{∗}F).

Proposition 5.75. The following identity holds

Ĥ^{.}(Σ, F) = [H^{.}(Σ_{g}^{b}, p^{∗}F)]^{Z/mZ}. (5.336)

Under the identification (5.336), if α, α′ ∈ Ĥ^{1}(Σ, F),

⟨α, α′⟩_{Ĥ^{1}(Σ,F)} = \frac{1}{m}⟨α, α′⟩_{H^{1}(Σ_{g}^{b},p^{∗}F)} . (5.337)

Proof. Clearly, [H^{0}(Σ_{g}^{b}, p^{∗}F)]^{Z/mZ} consists of flat Z/mZ-invariant sections of p^{∗}F on Σ_{g}^{b}. In particular by for 1 ≤ j ≤ s, w_{j}f_{|x_{j}} = f_{|x_{j}}. Therefore these sections descend to flat sections of F on Σ. Using (5.27) and (5.36), (5.336) holds in degree 0. By Theorem 5.4, using Poincaré duality, (5.336) holds in degree 2.

Let α be a Z/mZ-invariant closed form in Ω^{1}(Σ_{g}^{b}, p^{∗}F) representing [α] ∈ [H^{1}(Σ_{g}^{b}, p^{∗}F)]^{Z/mZ}. We may and will assume that α vanish near x_{1}, …, x_{s}. Then α descends to a smooth closed 1 form on Σ, which vanishes on ∂Σ. Also α is defined up to the coboundary of a Z/mZ-invariant form in Ω^{0}(Σ_{g}^{b}, p^{∗}F). Using Theorem 5.4, we find there is a well-defined map [H^{1}(Σ_{g}^{b}, p^{∗}F)]^{Z/mZ} → Ĥ^{1}(Σ, F). Conservely, if β is closed in Ω^{1}(Σ, F) and vanishes near ∂Σ, then p^{∗}β is a smooth Z/mZinvariant closed 1 form in Ω^{1}(Σ_{g}^{b}, p^{∗}F) . So we have defined a map Ĥ^{1}(Σ, F) → [H^{1}(Σ_{g}^{b}, p^{∗}F)]^{Z/mZ}. It is now easy to verify these two maps are inverse to each other. The identity (5.337) follows trivially.

The proof of our Theorem is completed.

Let χ̂(F) be the Euler characteristic of the complex (Ĉ^{Σ}(F), ∂). Then

(5.338)  χ̂(F) = ∑_{i=0}^{2}(−1)^{i} dim(Ĥ^{i}(Σ, F))

= ∑_{i=0}^{2}(−1)^{i} dim(Ĉ^{Σ,i}(F)) .

So by (5.32), (5.338),

χ̂(F) = (2 − 2g) dim F − ∑_{j=1}^{s} dim(1 − w_{j})(F)) . (5.339)

Let χ^{Z/mZ}(p^{∗}F) be the invariant Euler characteric of p^{∗}F on Σ_{g}^{b}, i.e.

(5.340) χ^{Z/mZ}(p^{∗}F) = ∑_{i=0}^{2}(−1)^{i} dim[H^{i}(Σ_{g}^{b}, p^{∗}F)]^{Z/mZ} .

By Proposition 5.75,

χ̂(F) = χ^{Z/mZ}(p^{∗}F) . (5.341)

We will prove (5.341) again using the Theorem of Riemann-Roch-Kawasaki [30, 31], stated in Theorem 6.8. We get

(5.342) χ^{Z/mZ}(p^{∗}F) = \frac{1}{m} [dim(F) ∫_{Σ_{g}^{b}} e(T Σ_{g}^{b}) + ∑_{j=1}^{s} Tr^{F} [ ∑_{k=1}^{m−1} w_{j}^{k}]] .

Now using (5.330), we get

∫_{Σ_{g}^{b}} e(T Σ_{g}^{b}) = 2 − 2g′ = m(2 − 2g) − (m − 1)s . (5.343)

Moreover

\frac{1}{m} ∑_{k=1}^{m−1} w_{j}^{k} = \frac{1}{m} ∑_{k=0}^{m−1} w_{j}^{k} − \frac{1}{m} . (5.344)

Now \frac{1}{m} ∑_{1}^{m−1} w_{j}^{k} is a projection on \{f ∈ F, w_{j}f = f\}. So by (5.344),

(5.345) \frac{1}{m}Tr^{F} [ ∑_{1}^{m−1} w_{j}^{k}] = dim ker((w_{j} − 1)_{|F}) − \frac{dim F}{m} .

By (5.342)-(5.345),

(5.346) χ^{Z/mZ}(p^{∗}F) = (2 − 2g) dim F − ∑_{j=1}^{s} dim((1 − w_{j})(F)) ,

which fits with (5.339), (5.341).

Now we assume that ρ : G → Aut(V) is a real representation of G. We will now give another proof of (5.346). Let χ^{h,Z/mZ}(p^{∗}F) be the holomorphic invariant Euler characteristic

(5.347) χ^{h,Z/mZ}(p^{∗}F) = ∑_{i=0}^{1} dim [H^{0,i}(Σ_{g}^{b}, p^{∗}F)]^{Z/mZ} .

By Hodge theory,

χ^{Z/mZ}(p^{∗}F) = 2χ^{h,Z/mZ}(p^{∗}F) . (5.348)

Also by the Riemann-Roch-Kawasaki theorem [30, 31],

χ^{h,Z/mZ}(p^{∗}F) = \frac{1}{m}[ dim(F) ∫_{Σ_{g}^{b}} \frac{1}{2}c_{1}(T Σ_{g}^{b}) + (5.349)

∑_{j=1}^{s} ∑_{k=1}^{m−1} \frac{1}{1 − e^{−2iπk/m}} Tr^{F} [w_{j}^{k}] .

By (5.343),

\frac{1}{m} ∫_{Σ_{g}^{b}} \frac{1}{2}c_{1}(T Σ_{g}^{b}) = 1 − g − \frac{1}{2}(1 − \frac{1}{m})s . (5.350)

The eigenvalues of w_{j|V} have absolute value 1. Since V is real, they are either ±1, or they come as complex conjugate pairs. By Proposition 5.73, we get

(5.351) \frac{1}{m} ∑_{k=1}^{m−1} \frac{1}{1 − e^{−2iπk/m}} Tr^{V} [w_{j}^{k}] = −\frac{1}{2m} dim F + \frac{1}{2} dim ker(w_{j} − 1)_{|F} .

By (5.349)-(5.351), we obtain

(5.352) χ^{h,Z/mZ}(p^{∗}F) = (1 − g) dim F − \frac{1}{2} ∑_{1}^{s} dim((1 − w_{j})(F)),

which fits with (5.346).

5.13. Evaluation of c_{1}(T M/G). In the sequel, we assume that G is a connected simply connected compact simple Lie group. Otherwise we use the notation in Section 1. In particular ⟨ , ⟩ denotes the basic scalar product on 𝔤 defined in Section 1.2.

As in Section 5.10, we construct a connection on the G-bundle

[figure: \frac{M×Σ̂}{Γ}; G; M/G × Σ] such that the assumptions after (5.252) hold. In particular, for 1 ≤ j ≤ s,

(5.353)  ∇_{j}_{.}w_{j} = 0 ,
∇_{j}_{.} log(w_{j}) = 0 .

Also since t_{j} lies in an alcove P, it determines ρ_{j} ∈ \overline{CR}^{∗} ∩ P by formula (1.35). Over M/G, t_{j}, ρ_{j} descend to sections of E_{j}. By (5.353), for 1 ≤ j ≤ s,

∇_{j}ρ_{j} = 0 . (5.354)

Let θ_{j} be the connection form on associated to ∇_{j}, let Θ_{j} be its curvature. Then θ_{j} can be considered as a 𝔱-connection, and Θ_{j} is a 𝔱-valued 2 form on M/G.

Recall that T M/G is a symplectic vector bundle. Let J^{T M/G} be any almost complex structure polarizing σ, i.e. σ is J^{T M/G} invariant, and U, V ∈ T M/G ↦ σ(J^{T M/G}U, V) is a scalar product. Such J^{T M/G} exist and are homotopic. Therefore c_{1}(T M/G) is a well-defined element of H^{2}(M/G, Q).

Recall that c is the dual Coxeter number defined in Definition 1.7.

Theorem 5.76. The following identity hold,

c_{1}(T M/G) = 2(cω + ∑_{j=1}^{s}⟨ct_{j} − ρ_{j}, Θ_{j}⟩). (5.355)

Proof. First we assume that m∈ N, that t_{1}, …, t_{s} are of order m, and s|m. We

claim that the G-bundle \xymatrix{(M × Σ̂)/Γ \ar[r]^{G} & M/G × Σ} lifts to a G-bundle \xymatrix{Q \ar[r]^{G} & M/G × Σ_{g}^{b}} on which Z/mZ acts naturally. This consists only in doing the lifting constructions of Section 5.12 fibrewise.

Recall that a complex structure has been fixed on Σ_{g} and Σ_{g}^{b}, and that Z/mZ acts holomorphically on Σ_{g}^{b}.

By (5.120), we have the identity

T_{R}M/G = Ĥ^{1}(Σ, E) . (5.356)

Using Proposition 5.75 and (5.356), we get

T_{R}M/G = [H^{1}(Σ_{g}^{b}, p^{∗}E)]^{Z/mZ} . (5.357)

Now

H^{1}(Σ_{g}^{b}, p^{∗}E) ⊗_{R} C = H^{(1,0)}(Σ_{g}^{b}, p^{∗}E) ⊕ H^{(0,1)}(Σ_{g}^{b}, p^{∗}E) , (5.358)

and the splitting (5.358) is Z/mZ invariant. By (5.356)-(5.359), we get

(5.359) Ĥ^{1}(Σ, E) ⊗_{R} C = [H^{(1,0)}(Σ_{g}^{b}, p^{∗}E)]^{Z/mZ} ⊕ [H^{(0,1)}(Σ_{g}^{b}, p^{∗}E)]^{Z/mZ} .

Let J be the complex structure on H^{1}(Σ_{g}^{b}, p^{∗}E) which is i on H^{(1,0)}(Σ_{g}^{b}, p^{∗}E), −i on H^{(0,1)}(Σ_{g}^{b}, p^{∗}E). We claim that J polarizes the symplectic form ω on Ĥ^{1}(Σ, E) = T_{R}M/G. In fact by if α, α′ ∈ Ĥ^{1}(Σ, E) are represented by the forms η, η′ ∈ Ω^{1}(Σ_{j}^{b}, p^{∗}E) which are closed and Z/mZ invariant, then by Proposition 5.75,

ω(α, α′) = −\frac{1}{m} ∫_{Σ_{g}^{b}}⟨η, η′⟩ . (5.360)

It is now trivial to verify that J polarizes ω.

So by (5.356), (5.359),

c_{1}(T M/G) = c_{1} ([H^{(1,0)}(Σ_{g}^{b}, p^{∗}E)]^{Z/mZ}) , (5.361)

which is equivalent to

c_{1}(T M/G) = −c_{1} ([H^{(0,1)}(Σ_{g}^{b}, p^{∗}E)]^{Z/mZ}) . (5.362)

Let θ be a Z/mZ invariant connection on the G bundle over \xymatrix{Q \ar[r]^{G} & Σ_{g}^{b} × M/G} . Observe that ({x_{j}}×M/G)_{1≤j≤s} are exactly the fixed point of the action of Z/mZ over Σ_{g}^{b} ×M/G. Since the connection θ is Z/mZ-invariant, and since, for 1 ≤ j ≤ s, 1 ∈ Z/mZ acts on E_{x_{j}} like w_{j}, we find that over M/G

∇w_{j_{|x_{j}}} = 0 . (5.363)

By (5.63),

H^{0}(Σ, E) = 0. (5.364)

By (5.336), (5.364),

[H^{0}(Σ_{g}^{b}, p^{∗}E)]^{Z/mZ} = 0. (5.365)

Now we will use an equivariant version of the curvature theorem of Bismut and Freed [10]. Namely we equip T Σ_{g}^{b} with a Z/mZ-invariant metric. Recall that p^{∗}E is also equipped with a Z/mZ-invariant metric. Since Z/mZ is a finite group, the construction sof [9, Definition 2.2] provide us with a like metric on the line bundle det([H^{(0,.)}(Σ_{g}^{b}, p^{∗}E)]^{Z/mZ}). By proceeding as in [10], we also obtain a unitary connection on this line bundle. Incidently, observe that since all our data are holomorphic, we could instead use the holomorphic constructions of [11] in an equivariant context. The curvature of the connection on det([H^{(0,.)}(Σ_{g}^{b}, p^{∗}E)]^{Z/mZ}) is obtained by applying the techniques of [10] or [11]. An important technical point is to prove an equivariant version of the local families index theorem of [8]. A large part of the steps which are needed in extending the results of [8] is already done in [9]. By mixing the techniques used in Lefschetz fixed point theory and in the prooof of the local families index theorem as in [8], [6], [9], one finds easily that the curvature is given by a differential form version of the theorem of Riemann-Roch-Kawasaki [30, 31].

Using (5.365) and the above considerations, we obtain

(5.366) c_{1} ([H^{(0,1)}(Σ_{g}^{b}, E)]^{Z/mZ}) = −\frac{1}{m}( ∫_{Σ_{g}^{b}} Td(T Σ_{g}^{b})ch(p^{∗}E, ∇^{p^{∗}E})

+ ∑_{j=1}^{s} ∑_{k=1}^{m−1} \frac{1}{1 − e^{−2iπk/m}} Tr^{E} [w_{j}^{k} exp (−\frac{Θ_{j}}{2iπ} )] )^{(2)} .

By (1.37) and by Theorems 5.55 and 5.56,

(\frac{1}{m} ∫_{Σ_{g}^{b}} Td(T Σ_{g}^{b})ch(p^{∗}E, ∇^{p^{∗}E}))^{(2)} = 2cω . (5.367)

Recall that

𝔤 ⊗_{R} C = 𝔱 ⊕ (⨁_{α∈R} g_{α}) . (5.368)

Let S_{j} be the operator acting on 𝔤 ⊗_{R} C,

S_{j} = \frac{1}{m} ∑_{k=1}^{m−1} \frac{1}{1 − e^{−2iπk/m}} τ(t_{j}^{k}) . (5.369)

By Proposition 5.73, we find that

(5.370)  S_{j|𝔱} = \frac{1}{2} − \frac{1}{2m} ,
S_{j|𝔤_{α}} = \frac{1}{2} − [⟨α, t_{j}⟩] − \frac{1}{2m}, α ∈ R .

By conjugation by an element G, we may and we will assume that w_{j} = t_{j}. Then

(5.371) ( ∑_{k=1}^{m−1} \frac{1}{1 − e^{−2iπk/m}} Tr^{E} [w_{j}^{k} exp (−\frac{Θ_{j}}{2iπ} )])^{(2)} = −Tr^{𝔤} [S_{j} \frac{Θ_{j}}{2iπ} ] .

Also because w_{j} = t_{j}, ∇_{x_{j}}^{E,2} is a 2 form on M/G with values in 𝔱. From (5.370), we get easily

(5.372)  −Tr^{𝔤} [S_{j} \frac{Θ_{j}}{2iπ} ] = − ∑_{α∈R} (\frac{1}{2} − [⟨α, t_{j}⟩]) ⟨α, Θ_{j}⟩
= ∑_{α∈R}[⟨α, t_{j}⟩]⟨α, Θ_{j}⟩ .

Now by Proposition 1.20, and by (5.372), we get

−Tr^{𝔤} [S_{j} \frac{∇_{x_{j}}^{E,2}}{2iπ} ] = 2⟨ct_{j} − ρ_{j}, Θ_{j}⟩ . (5.373)

From (5.362), (5.366), (5.367), (5.371)-(5.373), we get

c_{1}(T M/G) = 2(cω + ∑_{j=1}^{s}⟨ct_{j} − ρ_{j}, Θ_{j}⟩) . (5.374)

We claim now that in the special case when the orbits O_{1}, …, O_{m} are of order m, (5.374) is exactly (5.355). In spite of the formal simultanities, the objects introduced in both equation are not exactly of the same kind. However we leave to the reader the verification that they indeed coincide .

Now we establish (5.355) in full generality. Clearly if (t_{1}, …, t_{s}) are regular and verify (A), they can be approximated by a sequence of regular elements (t_{1}^{m}, …, t_{s}^{m}) which verify (A) and are of order m. Let s_{m} ≥ s be such that m|s_{m}. Then we consider the above situation, with s replaced by s_{m}. At x_{1}, …, x_{s}, we assume that the holonomies are w_{1}, …, w_{s} and at x_{s+1}, …, x_{s_{m}}, they are 1. Needless to say, since 1 is far from being regular, we do not impose any restriction on the connection ∇_{j}, j ≥ s + 1. Since for j ≥ s + 1, Tr^{𝔤}[Θ_{j}] = 0, it is clear that in the final formula, the j ≥ s+1 do not contribute, so that (5.374) still hold. Then by the above (5.355) holds. A trivial limit procedure shows that (5.355) still holds in full generality.

Remark 5.77. By Remark 5.66, we find that the cohomology class of ω+∑_{j=1}^{s}⟨log(w_{j}), Θ_{j}⟩ is locally constant. Observe that in (5.355), ⟨ρ_{j}, Θ_{j}⟩ is a closed form on M/G whose cohomology class does not depend locally on t_{1}, …, t_{s}. So Theorem 5.76 fits with the above considerations.

Assume temporarily that some t_{j} lie in Z(G) = \overline{R}^{∗}/\overline{CR}. By (5.372), we find that such t_{j} do not contribute to formula (5.355) for c_{1}(T M/G). In other words if the t_{j} are either regular or central, formula (5.355) still holds, where in the right-hand side, the summation is limited to a sum over the regular orbits.

6. The Riemann-Roch-Kawasaki formula on the moduli space of flat bundles

The purpose of this Section is to give formula for the index of a Dirac operator on the moduli space M/G of flat vector G bundles, by using the theorem of Riemann-Roch-Kawasaki [30, 31]. To do this, we describe the strata of M/G and we express the contribution of each stratum as a residue in several variables, using the results of Sections 2 and 5. The results of this Section were already obtained by Szenes [51] for G = SU(3) and Jeffrey-Kirwan [28] in the case G = SU(n), s = 1, with a central holonomy at the marked point for which M/G is smooth.

This Section is organized as follows. In Section 6.1, we describe the strata of a general orbifold, and we introduce various associated characteristic classes. In Section 6.2, we state the theorem of Riemann-Roch-Kawasaki for almost complex orbifolds. In Section 6.3, we construct the orbifold line bundle λ^{p} on the orbifold M/G. In Section 6.4, we describe the strata of the moduli space M/G as moduli spaces associated to semisimple centralizers in G. In Section 6.5, we compute the Atiyah-Bott-Lefschetz-Todd class of a given stratum. In Section 6.6, we compute the dimension of certain vector spaces which appear naturally in the evaluation of the Atiyah-Bott-Lefschetz class.

Then we make a number of genericity assumptions on the t_{j}’s. In Section 6.7, we compute the contribution of a stratum to the Riemann-Roch-Kawasaki formula in terms of differential operators acting on symplectic volumes. In Section 6.8, we briefly show that under an obvious condition on the holonomies t_{j}, 1 ≤ j ≤ s, the index of the considered Dirac operator vanishes identically. In Section 6.9, we give a residue formula for the index. In Section 6.10, we give another related asymptotic formula for |p| large.

Then we drop the genericity assumptions. In Section 6.11, we compute the index of a Dirac operator on a perturbed moduli space, for which the genericity assumptions hold. The point is that, as we shall see in Section 7, the index of the Dirac operator for the perturbed moduli space is exactly given by the Verlinde formula. Under genericity assumptions, this is only true asymptotically for the given moduli space M/G.

6.1. Almost complex orbifolds. Let M be a smooth compact manifold. Let G be a compact connected Lie group, and let 𝔤 be its Lie algebra. We assume that G acts on M on the left. If X ∈ 𝔤, let X^{M} ∈ Vect(M) be the corresponding vector field.

We assume that G acts locally freely on M, i.e. for any non zero X ∈ 𝔤, X^{M} is a non vanishing vector field on M.

We will use here the notation of Section 3.1, with X replaced by M. Then M/G is an orbifold.

Definition 6.1. If g ∈ G, put

M^{g} = {x ∈ M , gx = x} . (6.1)

Set

H = {g ∈ G; M^{g} ≠ ϕ} . (6.2)

Then H is a finite union of conjugacy classes in G. Let (H) be the corresponding finite set of conjugacy classes.

If g ∈ G, then Z(g) acts locally freely on M^{g}. We can then apply the above constructions to M^{g}. Let H_{g} ⊂ Z(g) be the generic stabilizer of M^{g}.

Observe that if g′ ∈ G is conjugate to g, the above constructions correspond by conjugation.

Take g ∈ H. Let N_{M^{g}/M} ≃ T M/T M^{g} be the normal bundle to M^{g} in M. Then we have the complex of Z(g) vector bundles over M^{g},

(6.3)  \xymatrix{ & 0 & 0 & 0 \\ 0 \ar[r] & 𝔤^{M}/𝔷^{M}(g) \ar[r] \ar[u] & N_{M^{g}/M} \ar[r] \ar[u] & N_{M^{g}/M}/(𝔤^{M}/𝔷^{M}(g)) \ar[r] \ar[u] & 0 \\ 0 \ar[r] & 𝔤^{M} \ar[r] \ar[u] & T M \ar[r] \ar[u] & T M/𝔤^{M} \ar[r] \ar[u] & 0 \\ 0 \ar[r] & 𝔷^{M}(g) \ar[r] \ar[u] & T M^{g} \ar[r] \ar[u] & T M^{g}/𝔷^{M}(g) \ar[r] \ar[u] & 0 \\  & 0 \ar[u] & 0 \ar[u] & 0 \ar[u]}

and the rows and columns in (6.3) are acyclic.

Clearly if g′ ∈ G, g′ maps M^{g} into M^{g′gg′^{−1}}. Also g′ acts on the complex (6.3). Put

N^{g} = N_{M^{g}/M}/(𝔤^{M}/𝔷^{M}(g)). (6.4)

Then the third column in (6.3) is the exact sequence of Z(g) vector bundles on M^{g},

0 → T M^{g}/Z(g) → T M/G → N^{g} → 0 . (6.5)

Equivalently N^{g} is the “normal bundle” to T M^{g}/Z(g) into M/G.

Clearly g acts on each of the vector bundles in (6.3). Then g acts like 1 on 𝔷^{M}(g), T M^{g} and T M^{g}/Z(g). In particular , there are locally constants θ, 0 < θ < π on M^{g} such that

N_{M^{g}/M} ⊗_{R} C = ⨁_{0<θ<π}(N_{M^{g}/M}^{θ} ⊕ N_{M^{g}/M}^{−θ}) ⊕ N^{π} . (6.6)

In (6.6), the θ’s are distinct, g acts on the left on N_{M^{g}/M}^{θ}, N_{M^{g}/M}^{−θ} by multiplication by e^{iθ}, e^{−iθ}, and on N^{π} by multiplication by −1. Therefore N^{g} ⊗_{R} C splits as

N^{g} ⊗_{R} C = ⨁_{0<θ<π}(N^{g,θ} ⊕ N^{g,−θ}) ⊕ N^{g,π} . (6.7)

Definition 6.2. The orbifold M/G will be said to be almost complex if T M/G = T M/𝔤^{M} is equipped with a G-invariant almost complex structure J^{T M/G}.

Since in (6.5), T M^{g}/Z(g) is the +1 eigenspace of g on T M/G, we find that for any g ∈ G, M^{g}/Z(g) is an almost complex orbifold. Therefore N^{g} is also equipped with an almost complex structure.

Now we denote with the superscript (1, 0) the +i eigenspace of the given complex structure. In particular T^{(1,0)}M^{g}/Z(g) is well defined, and N^{g,(1,0)} splits as

N^{g,(1,0)} = ⨁_{θ∈]−π,π]\{0}} N^{g,(1,0),θ} . (6.8)

In the sequel, we will assume that the orbifold M/G is almost complex.

6.2. The Theorem of Riemann-Roch-Kawasaki. Let ∇^{T^{(1,0)}M_{g}/Z(g)}, …, ∇^{N^{g,(1,0),θ}} be Z(g) invariant horizontal connections on T^{(1,0)}M_{g}/Z(g), N^{g,(1,0),θ}.

Let E be a complex G-vector bundle on M, equipped with a G-invariant horizontal connection ∇^{E}. Let F^{E} be the curvature of ∇^{E}. Then (E, ∇^{E})_{|M^{g}} is a Z(g)-vector bundle equipped with a Z(g) invariant connection.

Definition 6.3. Let ch_{g}(E, ∇^{E}) be the closed form on M^{g}/Z(g)

ch_{g}(E, ∇^{E}) = Tr [g exp (\frac{−F^{E}}{2iπ} )] . (6.9)

In (6.9), g denotes the left action of the given element of G on on E. Let ch_{g}(E) be the cohomology class associated to the form ch_{g}(E, ∇^{E}).

Definition 6.4. If B is a square matrix, put

(6.10)  Td(B) = det (\frac{B}{1 − e^{−B}} ) ,
T̂d(B) = det (\frac{1}{1 − e^{−B}} ) ,
Â(B) = det (\frac{B}{2 sinh(B/2)}) .

Observe that

Td(B) = Â(B)e^{\frac{1}{2} Tr[B]} (6.11)

Definition 6.5. Put

Td(T^{(1,0)}M_{g}/Z_{g}, ∇^{T^{(1,0)}M_{g}/Z_{g})}) = Td (−\frac{F^{T^{(1,0)}M_{g}/Z_{g}}}{2iπ} ) , (6.12)

T̂d(N^{g(1,0)}, ∇^{N^{g(1,0)}}) = ∏_{θ∈]−π,π]\\{0\}} T̂d (iθ − \frac{F^{N^{g,(1,0),θ}}}{2iπ} ) .

Then the forms in (6.12) are closed on M_{g}/Z(g). Let Td(T^{(1,0)}M_{g}/Z_{g}), T̂d(N^{g,(1,0)}) be the corresponding cohomology classes.

Definition 6.6. Put

L(g, E) = Td(T^{(1,0)}M_{g}/Z_{g})T̂d(N^{g,(1,0)})ch_{g}(E) . (6.13)

Then ∫_{M^{g}/Z(g)} L(g, E) depends only on the conjugacy class of g in G.

Let h^{T^{(1,0)}M}, h^{E} be G-invariant metrics on T^{(1,0)}M, E. Let ∇^{T^{(1,0)}M/G}, ∇^{E} be G-invariant unitary horizontal connections on T^{(1,0)}M, E. Then ΛT^{∗(1,0)}M/G⊗E is naturally equipped with a G-invariant unitary horizontal connection ∇^{ΛT^{∗(1,0)}M/G⊗E}.

Recall that by [6, p 135] , ΛT^{∗(1,0)}M/G ⊗ E is a T M/G Clifford module. If X ∈ T M(G), let c(X) be the corresponding Clifford multiplication operator.

Let dv be the volume element on M_{reg}/G associated to h^{T^{(1,0)}M/G}.

Let K = K_{+}⊕K_{−} be the vector space of G-invariant C^{∞} sections of Λ(T^{∗(0,1)}M/G)⊗ E = (Λ^{even}(T^{∗(0,1)}M/G) ⊗ E) ⊕ (Λ^{odd}(T^{∗(0,1)}M/G) ⊗ E). We equip H with the Hermitian product

s, s′ ∈ H ↦ ⟨s, s′⟩ = ∫_{M_{reg}/G}⟨s, s′⟩_{Λ(T^{∗(0,1)}M/G)⊗E}dv . (6.14)

Let e_{1}, …, e_{n} be an orthonormal basis of T M/G.

Definition 6.7. Let D^{M} be the Dirac operator acting on K

D^{M} = ∑_{1}^{n} c(e_{i})∇_{e_{i}}^{Λ(T^{∗(0,1)}M/G)⊗E} . (6.15)

Then D^{M} is a formally left-adjoint operator which exchanges K_{+} and K_{−}. Let D_{+}^{M} be the restriction of D^{M} to K_{+}. Then we write D^{M} in matrix form as

D^{M} = [\begin{matrix}0 & D_{−}^{M} \\ D_{+}^{M} & 0 \end{matrix}] (6.16)

Then by [30, 31], D_{+}^{M} is a Fredholm operator. Its index Ind(D_{+}^{M}) is given by

(6.17) Ind(D_{+}^{M}) = dim ker D_{+}^{M} − dim ker D_{−}^{M} .

If γ ∈ (H), let g_{γ} ∈ γ be any representative in G of the conjugacy class γ. Now we state the theorem of Riemann-Roch-Kawasaki [30, 31].

Theorem 6.8. The following identity holds

Ind(D_{+}^{M}) = ∑_{γ∈(H)} ∫_{M^{gγ} /Z(g_{γ})} \frac{1}{|H_{g_{γ}} |}L(g_{γ}, E) . (6.18)

6.3. The line bundle λ^{p}. From now on, we suppose that all the assumptions of Section 5.10 are in force. Also we fix once and for all a positive Weyl chamber K and the corresponding alcove P ⊂ K whose closure contains 0. Finally we may and we will assume that

t_{j} ∈ P, 1 ≤ j ≤ s. (6.19)

Definition 6.9. Let M ∈ Z be given by

M = \{p ∈ Z, pt_{1}, …, pt_{s} ∈ \overline{CR}^{∗}\}. (6.20)

In the sequel we assume that M is not reduced to 0. Then there is p_{0} ∈ N^{∗} such that

M = p_{0}Z. (6.21)

Let p ∈ M. Put

θ_{j} = pt_{j}, 1 ≤ j ≤ s. (6.22)

Recall that the set of connections A_{p}(θ_{1}, …, θ_{s}) was defined in Definition 4.36. Also the Hermitian line bundle with unitary connection (λ_{p}, ∇^{λ_{p}}) on A_{p}(θ_{1}, …, θ_{s}) was defined in Definition 4.37. By Proposition 4.38, ΣG acts on the right on λ_{p} and preserves ∇^{λ_{p}}. Finally A^{flat}(t_{1}, …, t_{s}) was defined in Definition 5.25.

Clearly if p ∈ M, if θ_{1}, … θ_{s} are given by (6.22), then

A^{flat}(t_{1}, …, t_{s}) ⊂ A_{p}(θ_{1}, …, θ_{s}). (6.23)

Therefore the line bundle λ_{p} restricts to A^{flat}(t_{1}, …, t_{s}). Also ΣG acts on A^{flat}(t_{1}, …, t_{s}). Finally by Proposition 5.27,

A^{flat}(t_{1}, …, t_{s})/Σ_{q}G ≃ M. (6.24)

Definition 6.10. Let λ^{p}, ∇^{λ^{p}} be the Hermitian line bundle with unitary connection over M of the Σ_{q}G-invariant sections of λ_{p} on A^{flat}(t_{1}, …, t_{s}).

The notation for λ^{p} is justified by the fact that if p ∈ M, p′ ∈ Z, then

λ^{pp′} = (λ^{p})^{⊗p′}. (6.25)

It is then clear that the action of G on M lifts to λ^{p}. Recall that ω is the canonical symplectic form on M/G which was defined in Definition 5.29, which is associated to the basic scalar product ⟨, ⟩ on 𝔤.

Proposition 6.11. The following identity of closed 2-forms holds on M,

c_{1}(λ^{p}, ∇^{λ^{p}}) = pω. (6.26)

Proof. Let A ∈ A^{flat}(t_{1}, …, t_{s}). Let α, α′ be 2 closed forms in Ω^{1}(Σ, E), which are exact on ∂Σ, i.e.. there are β, β′ ∈ Ω^{0}(∂Σ, E) such that α_{|∂Σ} = ∇^{A}β, α′_{|∂Σ} = ∇^{A}β′. By (4.24),(4.78), (4.100), (4.189),

c_{1}(λ^{p}, ∇^{λ^{p}})(α, α′) = p ∫_{Σ} −⟨α, α′⟩ + p ∫_{∂Σ}⟨β, ∇^{A}β′⟩. (6.27)

If [α], [α′] are the classes of α, α′ in Ĥ^{1}(Σ, E), from (6.27), we get

c_{1}(λ^{p}, ∇^{λ^{p}})(α, α′) = pω([α], [α′]). (6.28)

The proof of our Theorem is completed.

Since ω is a symplectic form, there is an almost complex structure J on T M/G which polarizes ω, i.e. ω(JX, Y) is a Riemannian metric on T M/G. Also J is unique up to homotopy. In the sequel, we will always equip T M/G with such a complex structure.

Then we will apply the theorem of Riemann-Roch-Kawasaki [30, 31] the orbifold M/G and the orbifold line bundle λ^{p}. Up to now, we have made G act on M or on λ^{p} on the right. However to fit with the formalism of Sections 6.1 and 6.2 , we will now make G act on the left by setting gx = xg^{−1}.

6.4. The Theorem of Riemann-Roch-Kawasaki on the moduli space of flat bundles. Recall that

M = {x ∈ G^{2g} × ∏_{j=1}^{s} O_{j} , h(x) = 1} . (6.29)

We will use the notation

M = M(G, O_{1}, …, O_{s}) . (6.30)

Let τ be projection T = 𝔱/\overline{CR} → T′ = 𝔱/\overline{R}^{∗}.

Theorem 6.12. The following identity holds

(H) = W \C/\overline{CR} . (6.31)

Also if v ∈ C/\overline{CR}, if u = τv ∈ C/\overline{R}^{∗}, M^{v} = M^{u}. Moreover

M^{u} = ⋃_{(w^{1},... ,w^{s})∈(W_{u}\W)^{s}} M(Z(u), O_{Z(u)}(w^{1}t_{1}), …, O_{Z(u)}(w^{s}t_{s})), (6.32)

and the union in (6.32) is disjoint.Finally, if v ∈ C/\overline{CR},

H_{v} = Z(Z(v)) . (6.33)

Proof. Clearly, if u ∈ G′, then ugu^{−1} = g if and only if g ∈ Z(u). It is then clear that if u ∈ G′,

M^{u} = {x ∈ Z(u)^{2g} × ∏_{j=1}^{s}(O_{j} ∩ Z(u)) , h(x) = 1} . (6.34)

Since G acts locally freely on M, if M^{u} ≠, Z(u) is semisimple. By Theorem 1.38, we get (6.31). If u ∈ C/\overline{CR}, using Theorem 1.50 and (6.34), we get (6.32). Finally by Theorem 5.20, we obtain (6.33). The proof of our Theorem is completed.

Remark 6.13. By Proposition 1.40, if G = SU(n), n ≥ 2, if u ∈ C, then u ∈ \overline{R}^{∗}. From Theorem 6.12, it follows that G acts freely on M, so that M/G is a smooth manifold.

Clearly Z(G) = \overline{R}^{∗}/\overline{CR} ⊂ T is fixed by W. Therefore if v ∈ T, W_{v} depends only on u = τv ∈ 𝔱/\overline{R}^{∗}. We will then write W_{u} instead of W_{v}.

We use the notation of Section 1. In particular π_{u} : Z̃(u) → Z(u) is the universal cover of Z(u).

Let u ∈ C/\overline{R}^{∗}, let t ∈ T = 𝔱/\overline{CR} be regular. Then Z(t) = T, and O_{Z(u)}(t) ≃ Z(u)/T.

Let t̃ ∈ 𝔱/\overline{CR}_{u} be a lift of t in Z̃(u). Then t̃ is still regular in Z̃(u). Since Z(t) = T, the centralizer Z(t̃) in Z̃(u) is just Z(t̃) = 𝔱/\overline{CR}_{u}. Then O_{Z̃(u)}(t̃) ≃

Z̃(u)/(𝔱/\overline{CR}_{u}) ≃ Z(u)/T. Equivalently the projection O_{Z̃(u)}(t̃) → O_{Z(u)}(t) is one to one.

Take u ∈ C/\overline{R}^{∗}, (w^{1}, …, w^{s}) ∈ W^{s}. Recall that here t_{1}, …, t_{s} are also considered as elements of 𝔱, so that w^{1}t_{1}, …, w^{s}t_{s} ∈ 𝔱 . Ultimately, we may consider w^{1}t_{1}, …, w^{s}t_{s} as element of Z̃(u). Then by the above, O_{Z̃(u)}(w^{1}t_{1}), …, O_{Z̃(u)}(w^{s}t_{s}) lift O_{Z(u)}(w^{1}t_{1}), …, O_{Z(u)}(w^{s}t_{s}), and the projection π_{u} identifies the corresponding orbits.

Let x = (u_{1}, v_{1}, …, u_{g}, v_{g}, w_{1}, …, w_{s}) ∈ M(Z(u), O_{Z(u)}(w^{1}t_{1}), …, O_{Z(u)}(w^{s}t_{s})). Let ũ_{1}, ṽ_{1}, …, ũ_{g}, ṽ_{g} ∈ Z̃(u) be lifts of u_{1}, v_{1}, …, u_{g}, v_{g} ∈ Z(u). Also w_{1}, …, w_{s} ∈ O_{Z(u)}(w^{1}t_{1}), …, O_{Z(u)}(w^{s}t_{s}) lift uniquely to w̃_{1}, …, w̃_{s} ∈ O_{Z̃(u)}(w^{1}t_{1}), …, O_{Z̃(u)}(w^{s}t_{s}).

Proposition 6.14. The element ∏_{i=1}^{g}[ũ_{i}, ṽ_{i}] ∏_{j=1}^{s} w̃_{j} ∈ Z̃(u) does not depend on ũ_{1}, …, ṽ_{g}.

It lies in π_{1}(Z(u)) = \overline{CR}/\overline{CR}_{u}.

Proof. Since π_{1}(Z(u)) ⊂ Z(Z̃(u)), the first part of the Proposition is trivial. Also,

π_{u} [∏_{i=1}^{g}[ũ_{i}, ṽ_{i}] ∏_{j=1}^{s} w̃_{j}] = ∏_{i=1}^{g}[u_{i}, v_{i}] ∏_{j=1}^{s} w_{j} = 1 . (6.35)

From (6.35), we get the second part of the Proposition.

Using Proposition 6.14 , we can now define:

Definition 6.15. If u ∈ C/\overline{CR}, (w^{1}, …, w^{s}) ∈ W^{s}, h ∈ π_{1}(Z(u)) = \overline{CR}/\overline{CR_{u}} put

M(Z(u), O_{Z(u)}(w^{1}t_{1}), …, O_{Z(u)}(w^{s}t_{s}), h) = (6.36)

{x = (u_{1}, …, u_{g}, v_{g}, w_{1}, …, w_{s}) ∈ M(Z(u), O_{Z(u)}(w^{1}t_{1}), …, O_{Z(u)}(w^{s}t_{s})),

∏_{i=1}^{g}[ũ_{i}, ṽ_{i}] ∏_{j=1}^{s} w̃_{j}h = 1} .

Clearly, we have the disjoint union

M(Z(u), O_{Z(u)}(w^{1}t_{1}), …, O_{Z(u)}(w^{s}t_{s})) = (6.37)

⋃_{h∈\overline{CR}/\overline{CR}_{u}} M(Z(u), O_{Z(u)}(w^{1}t_{1}), …, O_{Z(u)}(w^{s}t_{s}), h) .

Also since \overline{CR}/\overline{CR}_{u} ⊂ Z(Z̃(u)), Z′(u) preserves each M(Z(u), O_{Z(u)}(w^{1}t_{1}), …, O_{Z(u)}(w^{s}t_{s}), h).

By Definition 6.10, if p ∈ M, there is a well-defined G-orbifold line bundle λ^{p} on M/G. Let D_{p} be the corresponding Dirac operator acting on smooth sections of Λ(T^{∗(0,1)}M/G) ⊗ λ^{p} over M/G.

Theorem 6.16. For p ∈ M, the following identity holds

Ind(D_{p,+}) = ∑_{u∈C/\overline{CR}} \frac{|W_{u}|}{|W |} \frac{1}{|Z(Z(u))|} ∫_{M^{u}/Z(u)} L(u, λ^{p}) . (6.38)

Proof. By Theorems 6.8 and 6.12,

Ind(D_{p,+}) = ∑_{u∈W \C/\overline{CR}} \frac{1}{|Z(Z(u))|} ∫_{M^{u}/Z(u)} L(u, λ^{p}) (6.39)

from which (6.38) follows. The proof of our Theorem is completed.

6.5. Evaluation of the Atiyah-Bott-Lefschetz Todd class on a stratum of the moduli space. Now we take u ∈ C/\overline{R}^{∗}, x ∈ M^{u}. Then u descends to a flat section of the G bundle Ĝ, which acts naturally on the left on the vector bundle E as a flat section of Aut(E). In particular, on M^{u}, the vector bundle E ⊗_{R} C splits as a direct sum of vector bundles

E ⊗_{R} C = ⨁_{0<θ<π}(E^{θ} ⊕ E^{−θ}) ⊕ E^{π}. (6.40)

In (6.40), for −π < θ ≤ π, u acts on E^{θ} like e^{iθ}.

For 0 < θ < π, E^{θ} is a complex vector bundle on M^{u}/Z′(u). Let ∏(e^{\frac{x+iθ}{2}} −

e^{− \frac{x+iθ}{2}} )(E^{θ} be the corresponding characteristic class. Also E^{π} is a real vector bun-

dle on M^{u}/Z′(u). Moreover 2i cosh(\frac{x}{2}) is an even function of x. Let ∏(e^{\frac{x+iπ}{2}} − e^{− \frac{x+iπ}{2}} )(E^{π}) = ∏ 2i cosh(\frac{x}{2})(E^{π}) be the corresponding Pontryagin class of E^{π}.

Needless to say, if E^{π} has a complex structure , then this class is exactly ∏ 2i cosh(\frac{x}{2})(E^{π,(1,0)}).

Theorem 6.17. For u ∈ C/\overline{R}^{∗}, the following identity of characteristic classes holds on M^{u}/Z′(u),

Td(T^{(1,0)}M^{u}/Z′(u)) = Â^{2g−2+s}(E^{0})e^{\frac{1}{2} c_{1}(T^{(1,0)}M^{u}/Z′(u))} , (6.41)

T̂d(N^{u(1,0)}) = [\frac{1}{∏_{0<θ≤π} Π (e^{\frac{x+iθ}{2}} − e^{\frac{−(x+iθ)}{2}} ) (E^{θ})}]^{2g−2+s}

e^{\frac{1}{2} c_{1}(N^{u(1,0)})}e^{\frac{i}{2} ∑_{θ∈]−π,π]\\{0\}} θdim(N^{u(1,0),θ})}(−1)^{∑_{−π<θ<0} dim(N^{u(1,0),θ})} .

Proof. Using Theorem 5.33 and (6.11), we get the first identity in (6.41). Clearly

T̂d(N^{u(1,0)}) = [∏_{θ∈]−π,π]\\{0\}} (e^{\frac{1}{2} (x+iθ)} − e^{− \frac{1}{2} (x+iθ)}) (N^{u(1,0),θ})]^{−1} (6.42)

c^{\frac{1}{2} c_{1}(N^{u(1,0)})+ \frac{i}{2} ∑_{θ∈]−π,π]\\{0\}} θdim(N^{u(1,0),θ})} .

Also, for −π < θ < 0,

N^{u(1,0),θ} = \overline{N^{u(0,1),−θ}} (6.43)

From (6.43), we get

∏_{θ∈]−π,π]\\{0\}} (e^{\frac{1}{2} (x+iθ)} − e^{− \frac{1}{2} (x+iθ)}) (N^{u(1,0),θ}) (6.44)

= ∏_{0<θ≤π} (e^{\frac{1}{2} (x+iθ)} − e^{− \frac{1}{2} (x+iθ)}) (N^{u(1,0),θ})

∏_{0<θ<π} (e^{− \frac{1}{2} (x+iθ)} − e^{\frac{1}{2} (x+iθ)}) (N^{u(0,1),θ})

= ∏_{0<θ <π} (e^{\frac{1}{2} (x+iθ)} − e^{− \frac{1}{2} (x+iθ)}) (N^{u,θ}) ∏(ie^{x/2} + ie^{−x/2})(N^{u(1,0),π})

(−1)^{∑_{−π<θ<0} dim(N^{u(1,0),θ})} .

Also one finds easily that over M^{u}/Z(u), equality (5.139) of Theorem 5.33 can be split according to the values of θ. So we find that for 0 < θ < π,

Π (e^{\frac{1}{2} (x+iθ)} − e^{− \frac{1}{2} (x+iθ)}) (N^{u,θ}) (6.45)

= [Π (e^{\frac{1}{2} (x+iθ)} − e^{− \frac{1}{2} (x+iθ)}) (E^{θ})]^{2g−2+s} .

Also 2i cosh(x/2) is an even function of x. Then

∏(ie^{x/2} + ie^{−x/2})(N^{u(1,0),π}) = ∏(2i cosh(x/2))(N^{u,π}) (6.46)

= [∏(2i cosh(\frac{x}{2})]^{2g−2+s} (E^{π}) = ∏ (e^{\frac{x+iπ}{2}} − e^{− \frac{(x+iπ)}{2}} )^{2g−2+s} (E^{π}) .

By (6.42)-(6.46), we get the second identity in (6.41). The proof of our Theorem is completed.

6.6. The dimensions of the splitting of the normal bundle to the strata. Let J be a G-invariant almost complex structure on T M/G which polarizes the symplectic form ω. We know that J is unique up to homotopy.

Take u ∈ C/\overline{R}^{∗}. Let x ∈ M^{u}. Then by (5.120),

T M/G = Ĥ^{1}(Σ, E) . (6.47)

Recall that u defines a flat section of the bundle \xymatrix{Ĝ \ar[r]^{G} & Σ} . Clearly

𝔤 ⊗_{R} C = (𝔷(u) ⊗_{R} C) ⊕ ( ⨁_{α∈R\R_{u}} 𝔤_{α}) . (6.48)

Let x ∈ R ↦ [x]′ ∈] − 1/2, +1/2] be the function periodic of period 1, such that

[x]′ = x for x ∈] − 1/2, +1/2] . (6.49)

Clearly u acts on the left on 𝔷(u) like the identity and on 𝔤_{α} like e^{2iπ⟨α,u⟩}. For −π < θ ≤ π, put

𝔤^{θ} = ⨁_{{α∈R\Ru \atop [⟨α,u⟩]′= \frac{θ}{2π}}} 𝔤^{α} , (6.50)

with the convention that if θ = 0, 𝔤 = 𝔷(u).

Then (6.48) can be written as

𝔤 ⊗_{R} C = ⨁_{−π<θ≤π} 𝔤^{θ} . (6.51)

Also (6.51) is a Z(u)-invariant splitting. It induces a corresponding flat splitting of the flat bundle E of the form

E = ⨁_{−π<θ≤π} E^{θ} . (6.52)

In (6.52), E^{0} is just the analogue of E when replacing G by Z(u).

By (6.52), we get

Ĥ^{1}(Σ, E) ⊗_{R} C = ⨁_{−π<θ≤π} Ĥ^{1}(Σ, E^{θ}) . (6.53)

Also T M/G = Ĥ^{1}(Σ, E), and J acts on Ĥ^{1}(Σ, E). Since J is G-invariant, J commutes with u. In particular J acts on each Ĥ^{1}(Σ, E^{θ}). Let Ĥ^{(1,0)}(Σ, E^{θ}), Ĥ^{(0,1)}(Σ, E^{θ}) be the +i, −i eigenspaces of J, so that

Ĥ^{1}(Σ, E^{θ}) = Ĥ^{(1,0)}(Σ, E^{θ}) ⊕ Ĥ^{(0,1)}(Σ, E^{θ}) . (6.54)

Observe that since Ĥ^{1}(Σ, E^{θ}) is equipped with the u-invariant symplectic form ω, then

[Ĥ^{(1,0)}(Σ, E^{θ})]^{∗} = Ĥ^{(0,1)}(Σ, E^{θ}) , if θ = 0, π, (6.55)

[Ĥ^{(1,0)}(Σ, E^{θ})]^{∗} = Ĥ^{(0,1)}(Σ, E^{−θ}) , θ ∈] − π, π[ .

Since J is unique up to homology, the dimensions of the vector spaces which appear in (6.55) do not depend on J.

Since Z(u) is semisimple, for −π < θ ≤ π,

∑_{{α∈R \atop [⟨α,u⟩]′= \frac{θ}{2π}}} α = 0 . (6.56)

Therefore for any t ∈ 𝔱,

∑_{[⟨α,u⟩]′= \frac{θ}{2π}} [⟨α, t⟩] ∈ Z . (6.57)

Theorem 6.18. For −π < θ ≤ π, over M^{u},

dim Ĥ^{(0,1)}(Σ, E^{θ}) = (g − 1) dim 𝔤^{θ} + ∑_{j=1}^{s} ∑_{{α∈R \atop [⟨α,u⟩]′= \frac{θ}{2π}}} [⟨α, t_{j}⟩] . (6.58)

Proof. First assume that m ∈ N, that t_{1}, …, t_{s} are of order m, and m|s. Then we make the construction in Section 5.12. In particular u lifts to a Z/mZ-invariant parallel section over Σ_{g}^{b}. Then we have the Z/mZ invariant splitting

H^{1}(Σ_{g}^{b}, E) ⊗_{R} C = H^{(1,0)}(Σ_{g}^{b}, E) ⊕ H^{(0,1)}(Σ_{g}^{b}, E) . (6.59)

By arguing as in the proof of Theorem 5.76, when taking the Z/mZ invariant part of (6.59), we get a complex structure on Ĥ^{1}(Σ, E) which polarizes σ. It is then feasible to take

Ĥ^{(0,1)}(Σ, E^{θ}) = [H^{(0,1)}(Σ, E^{θ})]^{Z/mZ} . (6.60)

By construction,

[Ĥ^{0}(Σ, E^{θ})] = 0 . (6.61)

Therefore

(6.62) dim[H^{0,1}(Σ, E^{θ})]^{Z/mZ} = −χ^{h,Z/mZ}(Σ_{g}^{b}, p^{∗}E^{θ}) .

Using (5.321) and the theorem of Riemann-Roch-Kawasaki [30, 31] as in (5.349), (5.350), and also (6.62), we get (6.58).

Let us now consider the general case. As in the proof of Theorem 5.76, we approximate (t_{1}, …, t_{s}) by regular (t_{1}^{m}, …, t_{s}^{m}) which are of order m and such that (A) holds. Recall that over Σ_{g}^{b}, we still have a Z(u) bundle, so that we may and we will assume that at x_{j} (j ≥ s + 1), we have in fact a Z(u) connection. Since Z(u) is semisimple, for j ≥ s + 1,

Tr^{E_{θ}}[Θ_{j}] = 0 . (6.63)

Again the extra points x_{s+1}, …, x_{s_{m}} do not contribute to the computation, so that (6.58) still holds. Since for 1 ≤ j ≤ s, [⟨α, t_{j}^{m}⟩] → [⟨α, t_{j}⟩], we get (6.58) in full generality.

The proof of our Theorem is completed.

Theorem 6.19. For θ ∈] − π, π],

dim Ĥ^{(1,0)}(Σ, E^{θ}) = (g − 1) dim 𝔤^{θ} + ∑_{j=1}^{s} ∑_{{α∈R \atop [⟨α,u⟩]′= \frac{θ}{2π}}} (1 − [⟨α, t_{j}⟩]) . (6.64)

162 JEAN-MICHEL BISMUT AND FRANÇOIS LABOURIE

Proof. By (6.55), for θ ∈] − π, π[, (6.65) dim H^{1,0}(Σ, E^{θ}) = dim H^{0,1}(Σ, E^{−θ}) . So by Theorem 6.18, we get

dim Ĥ^{1,0}(Σ, E^{θ}) = (g − 1) dim 𝔤^{−θ} + ∑_{j=1}^{s} ∑_{{α∈R \atop [⟨α,u⟩]′= \frac{−θ}{2π}}} [⟨α, t_{j}⟩] . (6.66)

Now for θ ∈] − π, π[, dim 𝔤^{θ} = dim 𝔤^{−θ} . (6.67) Also since for 1 ≤ j ≤ s, α ∈ R, then ⟨α, t_{j}⟩ ∉ Z,

∑_{{α∈R \atop [⟨α,u⟩]′= \frac{−θ}{2π}}} [⟨α, t_{j}⟩] = ∑_{{α∈R \atop [⟨α,u⟩]′= \frac{θ}{2π}}} [⟨−α, t_{j}⟩] = (6.68)

∑_{{α∈R \atop [⟨α,u⟩]′= \frac{θ}{2π}}} (1 − [⟨α, t_{j}⟩]) .

From (6.66), (6.68), we get (6.64) when θ ∈] − π, π[.

Also E^{π} is a real vector bundle. Then (6.69) dim Ĥ^{(1,0)}(Σ, E^{π}) = dim Ĥ^{(0,1)}(Σ, E^{π}) . By Theorem 6.18, we get

dim H^{(1,0)}(Σ, E^{π}) = (g − 1) dim 𝔤^{π} + ∑_{j=1}^{s} ∑_{{α∈R \atop [⟨α,u⟩]′= \frac{1}{2}}} [⟨α, t_{j}⟩] . (6.70)

Observe that if α ∈ R, then [⟨α, u⟩]′ = \frac{1}{2} if and only if [−⟨α, u⟩]′ = \frac{1}{2}. Also when changing α into −α, [⟨α, t_{j}⟩] is changed into 1 − [⟨α, t_{j}⟩]. So for θ = π, we find that (6.70) is still equivalent to (6.64).

The proof of our Theorem is completed. Theorem 6.20. For u ∈ C/\overline{R}^{∗}, θ, θ ∈] − π, π]\\{0\},

dim N^{u(1,0),θ} = (g − 1) dim 𝔤^{θ} + ∑_{j=1}^{s} ∑_{{α∈R \atop [⟨α,u⟩]′= \frac{θ}{2π}}} (1 − [⟨α, t_{j}⟩]) . (6.71)

Proof. Recall that T M/G = Ĥ^{1}(Σ, E) . (6.72) Also (6.72) is an identification of u-spaces. Then our Theorem is just another formulation of Theorem 6.19.

6.7. The contribution of a stratum of the moduli space. Recall that by Theorem 1.41,

h ∈ \overline{CR}/\overline{CR}_{u} ↦ exp(2iπ⟨ρ_{u}, h⟩) ∈ S_{1} (6.73)

is a character with values in ±1.

By Theorem 5.18 and by Proposition 5.62, for g ≥ 1, M(Z(u), O_{Z(u)}(w^{1}t_{1}), …, O_{Z(u)}(w^{s}t_{s}), h) is a non empty smooth manifold, and for g = 0, M(Z(u), O_{Z(u)} (w^{1}t_{1}), …, O_{Z(u)}(w^{s}t_{s}), h) is either non empty and smooth, or it is empty.

In the sequel, it will be understood that any geometric statement about the empty set is empty.

Recall that by [32], [6, Lemma 7.22] and (1.193), the orbits O(w^{j}t_{j}) carry a natural complex structure, which polarizes the canonical symplectic form σ_{O(w^{j}t_{j})} on O(w^{j}t_{j}). Let N_{O_{Z(u)}(w^{j}t_{j})/O(w^{j}t_{j})}^{(1,0)} be the holomorphic normal bundle to O_{Z(u)}(w^{j}t_{j}) in O(w^{j}t_{j}).

As we saw after (5.305), H_{u,g,s}(t_{1}, …, t_{s}, t) is locally a polynomial of (t_{1}, …, t_{s}, t).

Theorem 6.21. For u ∈ C/\overline{CR}, p ∈ M, p ≠ −c, (w^{1}, …, w^{s}) ∈ W^{s}, h ∈ \frac{\overline{CR}}{\overline{CR}_{u}},

∫_{M(Z(u),O_{Z(u)}(w^{1}t_{1}),... ,O_{Z(u)}(w^{s}t^{s}),h)/Z(u)} L(u, λ^{p}) = (p + c)^{(g−1) dim 𝔷(u)+ \frac{s}{2} dim 𝔷(u)/𝔱}

\frac{(−1)}{|W_{u}|}^{\frac{(g−1)}{2} (dim(𝔤)−dim(𝔷(u))} ∏_{j=1}^{s} e^{iπ ∑_{α∈R_{+}} [⟨w^{j}α,u⟩]+2iπ⟨w^{j}pt_{j}+ph,u⟩}

[ ∏_{α∈R_{u,+}} Â(⟨α, \frac{∂/∂t}{p + c}⟩) (6.74)

∏_{α∈R_{+}\R_{u,+}}  \frac{1}{2 sinh (\frac{1}{2}(⟨α, \frac{∂/∂t}{p+c}⟩ + 2iπ[⟨α, u⟩]))}]^{2g−2+s}

∏_{j=1}^{s} e^{\frac{1}{p+c} ⟨(ρ−ct_{j}),∂/∂t_{j}⟩}π_{u} (\frac{∂/∂t}{2πi}) \frac{1}{|\frac{\overline{CR}}{\overline{CR}_{u}} |^{2g}} H_{u,g,s}(w^{1}t_{1}, …, w^{s}t_{s}, h, t) .

Also

{T̂d(N^{u(1,0)})_{|M(Z(u),O_{Z(u)}(w^{1}t_{1}),... ,O_{Z(u)}(w^{s}t_{s}))/Z′(u)}}^{(0)} = (6.75)

∏_{α∈R_{+}\R_{u,+}} (\frac{1}{4 sin^{2}(π⟨α,u⟩)})^{g−1} ∏_{j=1}^{s} _{\frac{1}{det(1−u^{−1})_{|N{(1,0) \atop O_{Z(u)}(w^{j} t_{j})/O(w^{j} t_{j})_{|w^{j} t_{j}}}}}} .

Proof. First we consider the case where g = 0, and M(Z(u), O_{Z(u)}(w^{1}t_{1}), …, O_{Z(u)}(w^{s}t_{s}), h) = ∅. By definition, the left-hand side of (6.74) is 0. Also for t ∈ T close enough to 1, M_{t}(Z(u), O_{Z(u)}(w^{1}t_{1}, …, O_{Z(u)}(w^{s}t_{s}), h) is also empty. By (5.305), it follows that H_{u,g,s}(t_{1}, …, t_{s}, t) vanishes identically near t = 1. Then (6.74) is just the identity 0 = 0.

Now we assume that M(Z(u), O_{Z(u)}(w^{1}t_{1}), …, O_{Z(u)}(w^{s}t_{s}), h) is non empty. If w = (w^{1}, …, w^{s}) ∈ W^{s}, put

M_{w,h}^{u} = M(Z(u), O_{Z(u)}(w^{1}t_{1}), …, O_{Z(u)}(w^{s}t_{s}), h) . (6.76)

We use the notation of Section 4.7. If x ∈ M_{w,h}^{u}, the corresponding flat connection reduces the G-bundle P to a Z(u)-bundle we still denote by P. This Z(u)-bundle lifts to a Z̃(u)-bundle Q. By Proposition 4.33, and by (6.36),

[P] = e^{h}. (6.77)

By Theorem 4.39, the right action of u on λ^{p} over M_{w,h}^{u} is given by e^{−2iπp⟨∑_{j=}^{s} w^{j}t_{j}+h,u⟩}.

The corresponding left action is then given by e^{2iπp⟨∑_{j=}^{s} w^{j}t_{j}+h,u⟩}. By (6.41) , and by the above result,we get

∫_{M_{w,h}^{u}/Z(u)} L(u, λ^{p}) = ∫_{M_{w,h}^{u}/Z(u)} [Â(E^{0})\frac{1}{∏_{0<θ≤π} Π (e^{\frac{x+iθ}{2}} − e^{− \frac{(x+iθ)}{2}} ) (E^{θ})}]^{2g−2+s}

e^{\frac{1}{2} c_{1}(T^{1,0}M/G)+pωe^{\frac{i}{2} ∑_{θ∈]−π,π]\\{0\}} θdim(N^{u(1,0),θ})}} (6.78)

(−1)^{∑_{−π<θ<0} dim(N^{u(1,0),θ})} ∏_{j=1}^{s} e^{2iπ⟨w^{j}pt_{j},u⟩}e^{2iπ⟨pu,h⟩} .

Now we use Theorem 5.76, which gives us a formula for c_{1}(T M/G). By Proposition 5.72 and by (6.78),

∫_{M_{w,h}^{u}/Z(u)} L(u, λ^{p}) = \frac{1}{|W_{u}|}e^{\frac{i}{2} ∑_{θ∈]−π,π]\\{0\}} θ dim(N^{u(1,0),θ})} (6.79)

(−1)^{∑_{−π<θ<0} dim(N^{u(1,0),θ})}(p + c)^{(g−1) dim 𝔷(u)+ \frac{s}{2} dim(𝔷(u)/𝔱)}[ ∏_{α∈R_{u,+}} Â(⟨α, \frac{∂/∂t}{p + c}⟩)

∏_{{α∈R \atop 0<[⟨α,u⟩]< \frac{1}{2}}} \frac{1}{2 sinh (\frac{1}{2} (⟨α, \frac{∂/∂t}{p+c}⟩ + 2iπ[⟨α, u⟩]))} ∏_{{α∈R_{+} \atop [⟨α,u⟩]= \frac{1}{2}}} \frac{1}{2i cosh (\frac{1}{2}⟨α, \frac{∂/∂t}{(p+c)}⟩)}]^{2g−2+s}

∏_{j=1}^{s} e^{\frac{1}{p+c} ⟨ρ−ct_{j},∂/∂t_{j}⟩}π_{u}(\frac{∂/∂t}{2πi})\frac{1}{|\frac{\overline{CR}}{\overline{CR}_{u}} |^{2g}} H_{u,g,s}(w^{1}t_{1}, …, w^{s}t_{s}, h, t) ∏_{j=1}^{s} e^{2iπ⟨w^{j}pt_{j},u⟩}e^{2iπ⟨ph,u⟩} .

Observe that if α ∈ R\R_{u}, when changing α into −α, 2 sinh (\frac{1}{2}(⟨α, \frac{∂/∂t}{p+c}⟩+

2iπ[⟨α, u⟩])) is unchanged. Therefore

∏_{{α∈R \atop 0<[⟨α,u⟩]< \frac{1}{2}}} \frac{1}{2 sinh (\frac{1}{2} (⟨α, \frac{∂/∂t}{p+c}⟩ + 2iπ[⟨α, u⟩]))} ∏_{{α∈R_{+} \atop [⟨α,u⟩]= \frac{1}{2}}} (6.80)

\frac{1}{2i cosh \frac{1}{2} (⟨α, \frac{∂/∂t}{p+c}⟩)} = ∏_{α∈R_{+}\R_{u,+}} \frac{1}{2 sinh (\frac{1}{2} (⟨α, \frac{∂/∂t}{p+c}⟩ + 2iπ[⟨α, u⟩]))} .

By Theorem 6.20,

(6.81)  exp(i/2 ∑_{θ∈]−π,π]\{0}} θ dim(N^{u(1,0),θ}))  =

exp (i/2( ∑_{θ∈]−π,π]} θ(dim 𝔤^{θ}(g − 1) + ∑_{j=1}^{s} ∑_{{α∈R \atop [⟨α,u⟩]′=θ/2π}} (1 − [⟨α, w^{j}t_{j}⟩])))) .

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS 165

Now the θ ∈] − π, π[\{0} come by opposite pairs. Therefore

∑_{θ∈]−π,π]\{0}} θ dim 𝔤^{θ} = π dim 𝔤^{π} , (6.82)

and so e^{\frac{i}{2} Σθ dim 𝔤^{θ}(g−1)} = (−1)^{\frac{dim 𝔤^{π}}{2} (g−1)} . (6.83)

For 1 ≤ j ≤ s, put

(6.84)  t′_{j} = w^{j}t_{j}
R_{j,+} = w^{j}R_{+}

Then R_{j,+} is the positive root system associated to t′_{j}. For 1 ≤ j ≤ s,

∑_{s∈]−1/2,1/2]} s ∑_{{α∈R \atop [⟨α,u⟩]′=s}} (1 − [⟨α, t′_{j}⟩])

= ∑_{α∈R}[⟨α, u⟩]′(1 − [⟨α, t′_{j}⟩]) (6.85)

= ∑_{α∈R_{j,+}} ([⟨α, u⟩]′(1 − ⟨α, t′_{j}⟩) + [−⟨α, u⟩]′⟨α, t′_{j}⟩)
= − ∑_{α∈R}[⟨α, u⟩]′⟨α, t′_{j}⟩ + ∑_{α∈R_{j,+}}[⟨α, u⟩]′ .

Clearly,

∑_{α∈R}[⟨α, u⟩]′⟨α, t′_{j}⟩ = ∑_{s} s⟨ ∑_{{α∈R \atop [⟨α,u⟩]′=s}} α, t′_{j}⟩ . (6.86)

Now for a given s, the {α ∈ R, [⟨α, u⟩]′ = s} are exactly the weights of the representation of Z(u) on {f ∈ 𝔤 ⊗_{R} C, uf = e^{2iπs}f}. Since Z(u) is semisimple

∑_{{α∈R \atop [⟨α,u⟩]′=s}} α = 0 . (6.87)

From (6.81)-(6.87) , we get

exp(\frac{i}{2} ∑_{θ∈]−π,π]\\{0\}} θdim(N^{u(1,0),θ})) = (−1)^{\frac{dim 𝔤^{π}}{2} (g−1)} ∏_{j=1}^{s} exp(iπ ∑_{α∈R_{+}}[⟨w^{j}α, u⟩]′) . (6.88)

Also by Theorem 6.20, using (6.87) , we get

∑_{−π<θ<0} dimN^{u(1,0),θ)} (6.89)

= (g − 1) ∑_{−π<θ<0} dim𝔤^{θ} + ∑_{j=1}^{s} ∑_{{α∈R \atop −1/2<[⟨α,u⟩]′<0}} (1 − [⟨α, t′_{j}⟩])
= (g − 1)|\{α ∈ R, −\frac{1}{2} < [⟨α, u⟩]′ < 0\}|

+ ∑_{j=1}^{s} (− ∑_{−1/2<s<0}⟨∑_{{α∈R \atop [⟨α,u⟩]′=s}} α, t′_{j}⟩ + |\{α ∈ R, −\frac{1}{2} < [⟨α, u⟩]′ < 0, ⟨α, t′_{j}⟩ > 0\}|)

= (g − 1)|\{α ∈ R, −\frac{1}{2} < [⟨α, u⟩]′ < 0\}|

+ ∑_{j=1}^{s} |\{α ∈ R, −\frac{1}{2} < [⟨α, u⟩]′ < 0, ⟨α, t′_{j}⟩ > 0\}|

= (g − 1)|\{α ∈ R, −\frac{1}{2} < [⟨α, u⟩]′ < 0\}| + ∑_{j=1}^{s} |\{α ∈ R_{+} , −\frac{1}{2} < [⟨w^{j}α, u⟩]′ < 0\} .

So by (6.88),(6.89),

(6.90) exp(\frac{i}{2} ∑_{θ∈]−π,π]\\{0\}} θ dim(N^{u(1,0),θ}))(−1)^{∑_{−π<θ<0} dim(N^{u,(1,0),θ}))} =

(−1)^{(g−1)(|\{α∈R,0<[⟨α,u⟩]′< \frac{1}{2} \}|+ \frac{1}{2} |\{α∈R,0<[⟨α,u⟩]′= \frac{1}{2} \}|)}

∏_{j=1}^{s} exp(iπ ∑_{α∈R_{+}}[⟨w^{j}α, u⟩]) .

Moreover

|\{α ∈ R, 0 < [⟨α, u⟩]′ < \frac{1}{2}\}| + \frac{1}{2}|\{α ∈ R, [⟨α, u⟩]′ = \frac{1}{2}\}| = \frac{1}{2}(dim(𝔤) − dim𝔷(u)) . (6.91)

From (6.79), (6.80), (6.90), (6.91) , we get (6.74).

Using (6.74) , or by proceeding directly, we get

{T̂d(N^{u(1,0)})_{|M_{w,h}^{u}/Z′(u)}}^{(0)} = (−1)^{(g−1)(dim𝔤−dim𝔷(u))/2} ∏_{j=1}^{s} exp(iπ ∑_{α∈R_{+}}[⟨w^{j}α, u⟩])

(∏_{α∈R_{+}\R_{u,+}} \frac{1}{2 sinh(iπ[⟨α,u⟩]})^{2g−2+s} . (6.92)

Using the invariance of sinh(iπ[⟨α, u⟩]) when α ∈ R\R_{u} is changed into −α, from (6.92), we get

\{T̂d(N^{u(1,0)})_{|M_{w,h}^{u}/Z′(u)}\}^{0} = (∏_{α∈R_{+}\R_{u,+}} \frac{1}{4 sin^{2}(π⟨α,u⟩)})^{g−1} (6.93)

∏_{j=1}^{s} ∏_{{α∈R_{+} \atop α∉w{−1 \atop j}Ru}} \frac{e^{iπ[⟨α,(w^{j})^{−1}u⟩]}}{2 sinh(iπ[⟨α,(w^{j})^{−1}u⟩])} =

= ∏_{α∈R_{+}\R_{u,+}} [\frac{1}{4 sin^{2}(π⟨α,u⟩)}]^{g−1} ∏_{j=1}^{s} ∏_{{α∈w_{j} R_{+}, \atop α∉Ru}} \frac{1}{1−e^{−2iπ⟨α,u⟩}} =
= ∏_{α∈R_{+}\R_{u,+}} [\frac{1}{4 sin^{2}(π⟨α,u⟩)}]^{g−1} ∏_{j=1}^{s} _{\frac{1}{det(1−u^{−1})_{|N{(1,0) \atop O_{Z(u)}(w^{j} t_{j})/O(w^{j} t_{j})_{|w^{j} t_{j}}}}}} ,

which is just (6.75). The proof of our Theorem is completed.

Recall that

(6.94)  ℓ = |R_{+}|
ℓ_{u} = |R_{u,+}| ,
r = dim 𝔱 .

Our assumptions on g, s guarantee that 2g − 2 + s ≥ 1. Also by (2.100) and by Theorem 2.45, the function P_{u,p+c,2g−2+s}(t) is a polynomial on T \ S.

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS 167

Theorem 6.22. For u ∈ C/\overline{CR}, p ∈ M, p ≠ −c, then

∫_{M^{u}/Z(u)} L(u, λ^{p}) = Vol(T)^{2g−2} \frac{|Z(Z(u))|}{|W_{u}|}(p + c)^{(g−1)dim(𝔷(u))+ \frac{s}{2} dim(𝔷(u)/𝔱)} (−1)^{ℓ(g−1)+1} ∏_{α∈R_{u,+}} (6.95)

[Â(⟨α, \frac{∂/∂t⟩}{p+c}) ∏_{α∈R_{+}\R_{u,+}} _{\frac{1}{2 sinh (\frac{1}{2}(⟨α, \frac{∂/∂t⟩}{p+c} ⟩+2iπ[⟨α,u⟩]))}}]^{2g−2+s}
∑_{(w^{1},... ,w^{s})∈W^{s}} ∏_{j=1}^{s}[sgn((−i)^{ℓu}σ_{Z(u)}(w^{j}t_{j}))e^{⟨w^{j}(ρ−ct_{j}), \frac{∂/∂t}{p+c} ⟩_{+}}
+iπ ∑_{α∈R_{+}}[⟨w^{j}α, u⟩] + 2iπ⟨w^{j}pt_{j}, u⟩]P_{u,2g−2+s,p+c}(t)_{|t=∑_{1}^{s} w^{j}t_{j}}

Proof. By (6.32), (6.37), ∫_{M^{u}/Z(u)} L(u, λ^{p}) = (6.96)

∑_{{(w^{1},... ,w^{s})∈(Wu\W)^{s} \atop h∈\overline{CR}/\overline{CRu}}} ∫_{M(Z(u),O_{Z(u)}(w^{1}t_{1}),... ,O_{Z(u)}(w^{s}t_{s}),h)/Z(u)} L(u, λ^{p}) .

We use Theorem 6.21 to calculate the term in the right-hand side of (6.96) . Also by (2.131) and by Theorem 5.71, π_{u} (\frac{∂/∂t}{2iπ}) H_{u,g,s}(w^{1}t_{1}, …, w^{s}t_{s}, h, t) = (−1)^{ℓ_{u}(g−1)+1} (6.97)

|Z(Z̃(u))|Vol(𝔱/\overline{CR_{u}})|^{2g−2} ∑_{(w′^{1},... ,w′^{s})∈W_{u}^{s}}

∏_{j=1}^{s} ε_{w′^{j}}sgn((−i)^{ℓ_{u}}σ_{Z(u)}(w^{j}t_{j}))e^{−2iπ⟨ρ_{u},h⟩}P̃_{u,2g−2+s}(t + h + ∑_{j=1}^{s} w′^{j}w^{j}t_{j}) . Clearly

Z(Z(u)) = \frac{Z(Z̃(u))}{\overline{CR}/\overline{CR_{u}}} , (6.98)

and so

|Z(Z̃(u))| = |Z(Z(u)| |\frac{\overline{CR}}{\overline{CR_{u}}} | . (6.99)

Moreover

(6.100) Vol(𝔱/\overline{CR_{u}}) = Vol(𝔱/\overline{CR})|\frac{\overline{CR}}{\overline{CR_{u}}} | = Vol(T)|\frac{\overline{CR}}{\overline{CR_{u}}} | .

By (6.97)-(6.100) , we get

π_{u} (\frac{∂/∂t}{2iπ}) \frac{H_{u,g,s}}{|\frac{\overline{CR}}{\overline{CRu}} |^{2g}} (w^{1}t_{1}, …, w^{s}t_{s}, h, t) = (−1)^{ℓ_{u}(g−1)+1} (6.101)

|Z(Z(u))|Vol(T)^{2g−2} ∑_{(w′^{1},... ,w′^{s})∈W^{s}(u)} ∏_{j=1}^{s} ε_{w′^{j}}((−i)^{ℓu}σ_{Z(u)})(w^{j}t_{j})

exp(−2iπ⟨ρ_{u}, h⟩)\frac{P̃_{u,2g−2+s}}{|\frac{\overline{CR}}{\overline{CRu}} |} (t + h + ∑_{j=1}^{s} w′^{j}w^{j}t_{j}) .

Also by Theorem 1.41 , for h ∈ \overline{CR}/\overline{CR_{u}}, exp(−2iπ⟨ρ_{u}, h⟩) = exp(2iπ⟨cu, h⟩) . (6.102)

168 JEAN-MICHEL BISMUT AND FRANÇOIS LABOURIE

Now observe that if w′^{j} ∈ W_{u}, by (1.45),

ε_{w′^{j}} = \frac{σ_{Z(u)}(w′^{j}w^{j}t_{j})}{σ_{Z(u)}(w^{j}t_{j})} , 1 ≤ j ≤ s . (6.103)

Therefore sgn ((−i)^{ℓ_{u}}σ_{Z(u)}(w^{j}t_{j})) ε_{w′^{j}} = sgn ((−i)^{ℓ_{u}}σ_{Z(u)}(w′^{j}w^{j}t_{j})) . (6.104) Also ℓ_{u} + (dim(𝔤) − dim(𝔷(u))/2 = ℓ . (6.105) From (2.166), (6.74), (6.96), (6.101)-(6.105), we get (6.95). The proof of our Theorem is completed. Remark 6.23. As shown in Section 2.11, the function exp(2iπ⟨qu, t⟩)P_{u,2g−2+s,p+c}(t) descends to a function which is well-defined on T = 𝔱/\overline{CR}. Ultimately, this explains why equation (6.95) is unambiguous.

Recall that the function σ(t) = ∏_{i=1}^{ℓ} (e^{iπ⟨α,t⟩} − e^{−iπ⟨α,t⟩}) is well-defined on T =

𝔱/\overline{CR}, and the function \frac{e^{2iπ⟨ρ,t⟩}}{σ(t)} is well-defined on T′ = 𝔱/\overline{R}^{∗}.

Proposition 6.24. For any u ∈ C/\overline{R}^{∗}, for any s ∈ 𝔱_{reg}, x ∈ P, w ∈ W,

sgn((−i)σ_{Z(u)}(wx)) ∏_{α∈R_{+}} \frac{1}{2 sinh(\frac{1}{2}(⟨α, s⟩ + 2iπ[⟨α, u⟩]))} (6.106)

exp(iπ ∑_{α∈R_{+}}[⟨wα, u⟩]) = ε_{w} ∏_{α∈R_{+}} \frac{1}{2 sinh(\frac{1}{2}⟨α, s + 2iπu⟩)} e^{2iπ⟨wρ,u⟩} .

Proof. As we just saw, both sides are unchanged when replacing u by u+γ, γ ∈ \overline{CR}. By [15, Proposition V.7.10], we may as well assume that for any α ∈ R, |⟨α, u⟩| ≤ 1 . (6.107) If α ∈ R\R_{u}, sinh(\frac{1}{2}(⟨α, s⟩ + 2iπ[⟨α, u⟩])) is unchanged when α is replaced by −α. Therefore using (1.45), we get ∏_{α∈R_{+}} \frac{1}{2 sinh( \frac{1}{2} (⟨α,s⟩+2iπ[⟨α,u⟩]))} = (−1)^{|R_{+}∩w^{−1}(−R_{u,+})|} (6.108)

∏_{α∈wR_{+}} \frac{1}{2 sinh( \frac{1}{2} (⟨α,s⟩+2iπ[⟨α,u⟩]))} =
(−1)^{|R_{+}∩w^{−1}(−R_{u,+})|+|{α∈R_{+},−1<⟨wα,u⟩<0}|+|{α∈R_{+},⟨wα,u⟩=±1}|}
∏_{α∈R_{+}} \frac{1}{2 sinh( \frac{1}{2} ⟨α,w^{−1}(s+2iπu)⟩)} =
ε_{w}(−1)^{|R_{+}∩w^{−1}(−R_{u,+})|+|{α∈R_{+},−1<⟨wα,u⟩<0}|+|{α∈R_{+},⟨wα,u⟩=±1}|}
∏_{α∈R_{+}} \frac{1}{2 sinh( \frac{1}{2} ⟨α,s+2iπu⟩)} .

Moreover, for x ∈ P, sgn((−i)^{ℓ_{u}}σ_{Z(u)}(wx)) = (−1)^{|R_{+}∩w^{−1}(−R_{u,+})|} . (6.109)

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS 169

Also by (6.107),

exp(iπ ∑_{α∈R_{+}}[⟨wα, u⟩]) = (6.110)

exp(iπ⟨w ∑_{α∈R_{+}} α, u⟩)(−1)^{|{α∈R_{+},−1<⟨wα,u⟩<0}|+|{α∈R_{+},⟨wα,u⟩=±1}|} =

e^{2iπ⟨wρ,u⟩}(−1)^{|{α∈R_{+},−1<⟨wα,u⟩<0}|+|{α∈R_{+},⟨wα,u⟩=±1}|} .

From (6.108)-(6.110) , we get (6.106). The proof of our Proposition is completed.

Remark 6.25. It is clear that the right-hand side of (6.106) only depends on the class of u in C/\overline{R}^{∗}.

Observe that the function

s ∈ 𝔱 ↦ \frac{∏_{α∈R_{u,+}}⟨α, s⟩e^{2iπ⟨wρ,u⟩}}{∏_{α∈R_{+}} 2 sinh(\frac{1}{2}⟨α, s + 2iπu⟩)} ∈ C (6.111)

is a well-defined holomorphic function near s = 0. Theorem 6.26. For any u ∈ C/\overline{CR}, p ∈ M, p ≠ −c, the following identity holds,

∫_{M^{u}/Z(u)} L(u, λ^{p}) = Vol(T)^{2g−2} \frac{|Z(Z(u))|}{|W_{u}|}(p + c)^{(g−1)dim(𝔷(u))+ \frac{s}{2} dim(𝔷(u)/𝔱)}

(−1)^{ℓ(g−1)+1} [\frac{∏_{α∈R_{u,+}}⟨α, \frac{∂/∂t}{p + c}⟩}{∏_{α∈R_{+}} 2 sinh (\frac{1}{2}⟨α, \frac{∂/∂t}{p + c} + 2iπu⟩)}]^{2g−2+s} (6.112)

∑_{(w^{1},... ,w^{s})∈W^{s}} ∏_{j=1}^{s} ε_{w^{j}} exp(⟨w^{j}(ρ − ct_{j}), \frac{∂/∂t}{p+c}⟩
+2iπ⟨w^{j}(ρ + pt_{j}), u⟩)P_{u,2g−2+s,p+c}(t)_{|t=∑_{j=1}^{s} w^{j}t_{j}} .

Proof. Clearly, since 2g − 2 is even,

[ ∏_{α∈R_{u,+}} Â(⟨α, \frac{∂/∂t}{p + c}⟩) (6.113)

∏_{α∈R_{+}\R_{u,+}} \frac{1}{2 sinh(\frac{1}{2}(⟨α, \frac{∂/∂t}{p + c}⟩ + 2iπ[⟨α, u⟩]))}]^{2g−2}

= [\frac{∏_{α∈R_{u,+}}⟨α, \frac{∂/∂t}{p + c}⟩}{∏_{α∈R_{+}} 2 sinh(\frac{1}{2}⟨α, \frac{∂/∂t}{p + c} + 2iπu⟩)}]^{2g−2} .

Also, since t_{j} ∈ P, by Proposition 6.24 , we get

∏_{α∈R_{u,+}} Â(⟨α, \frac{∂/∂t}{p+c}⟩) ∏_{α∈R_{+}\R_{u,+}} \frac{1}{2 sinh(\frac{1}{2}(⟨α, \frac{∂/∂t}{p + c}⟩ + 2iπ[⟨α, u⟩]))}

sgn((−i)^{ℓ_{u}}σ_{Z(u)}(w^{j}t_{j}))e^{iπ ∑_{α∈R_{+}} [⟨w^{j}α,u⟩]} = (6.114)

ε_{w} \frac{∏_{α∈R_{u,+}}⟨α, \frac{∂/∂t}{p + c}⟩}{∏_{α∈R_{+}} 2 sinh(\frac{1}{2}⟨α, \frac{∂/∂t}{p + c} + 2iπu⟩)}e^{2iπ⟨w^{j}ρ,u⟩} .

By Theorem 6.22 and by (6.113),(6.114) , we get (6.112). The proof of our Theorem is completed.

Remark 6.27. Suppose temporarily that t_{1}, …, t_{s−1} verify assumption (A), and that ϵ > 0 is small enough so that if |t_{s}| < ϵ, (t_{1}, …, t_{s}) still verify (A). Then (6.112) can be written in the form

∫_{M^{u}/Z(u)} L(u, λ^{p}) = (6.115)

Vol(T)^{2g−2} \frac{|Z(Z(u))|}{|W_{u}|} (p + c)^{(g−1)dim(𝔷(u))+ \frac{s}{2} dim(𝔷(u)/𝔱)}

(−1)^{ℓ(g−1)+1} [\frac{∏_{α∈R_{u,+}}⟨α, \frac{∂/∂t}{p + c}⟩}{∏_{α∈R_{+}} 2 sinh (\frac{1}{2}⟨α, \frac{∂/∂t}{p + c} + 2iπu⟩)}]^{2g−2+s}

∑_{w∈W} ε_{w}e^{⟨w(ρ+pt_{s}), \frac{∂/∂t}{p+c} +2iπu⟩} ∑_{(w^{1},... ,w^{s−1})∈W^{s−1}}

∏_{j=1}^{s−1} ε_{w^{j}}e^{⟨w^{j}(ρ−ct_{j}), \frac{∂/∂t}{p+c} ⟩_{+}2iπ⟨w^{j}(ρ+pt_{j}),u⟩}P_{u,2g−2+s,p+c}(t)_{|t=∑_{j=1}^{s−1} w^{j}t_{j}} .

Using (1.94), (2.131), since pt_{s} = θ_{s} ∈ \overline{CR}_{+}^{∗}, we may rewrite (6.115) in the form

∫_{M^{u}/Z(u)} L(u, λ^{p}) = (6.116)

Vol(T)^{2g−2} \frac{|Z(Z(u))|}{|W_{u}|}(p + c)^{(g−1)dim𝔷(u)+ \frac{(s−1)}{2} dim(𝔷(u)/𝔱)}

(−1)^{ℓ(g−1)+1} [\frac{∏_{α∈R_{u,+}}⟨α, \frac{∂/∂t}{p + c}⟩}{∏_{α∈R_{+}} 2 sinh (\frac{1}{2}⟨α, \frac{∂/∂t}{p + c} + 2iπu⟩)}]^{2g−2+s−1} χ_{pt_{s}}(e^{\frac{∂/∂t}{2iπ(p+c)} +u})

∑_{(w^{1},... ,w^{s−1})∈W^{s−1}} ∏_{j=1}^{s−1} ε_{w^{j}}e^{⟨w^{j}(ρ−ct_{j}), \frac{∂/∂t}{p+c} ⟩+2iπ⟨w^{j}(ρ+pt_{j}),u⟩}

P_{u,2g−2+s−1,p+c}(t)_{|t=∑_{j=1}^{s−1} w^{j}t_{j}} .

With respect to (6.112) , s has been replaced by s − 1, and we have the extra differential operator χ_{pt_{s}}(e^{\frac{∂/∂t}{2iπ(p+c)} +u}).

Let M′ be the manifold attached to G, O_{1}, …, O_{s−1}. Let F be the vector bundle over M′/G associated to the representation of highest weight pt_{s} ⊂ \overline{CR}^{∗} of G. Then if we still denote by L(u, λ^{p}) the corresponding class (6.13) over M′^{u}/Z(u), in view of (5.318), we can rewrite (6.115),(6.116) in the form

∫_{M^{u}/Z(u)} L(u, λ^{p}) = ∫_{M′^{u}/Z(u)} L(u, λ^{p})ch_{u}(F) . (6.117)

Also by Theorem 5.56, for |t_{s}| small enough, the orbifold M/G fibres over M′/G with fibre G/T. Let ρ be the projection M/G → M′/G. Then one verifies easily that

ρ_{∗}L(u, λ^{p}) = L(u, λ^{p})ch_{u}(E), (6.118)

which makes (6.117) tautological.

Remark 6.28. If u ∈ C/\overline{CR}, w ∈ W, one should have the equality

∫_{M^{u}/Z(u)} L(u, λ^{p}) = ∫_{M^{wu}/Z(wu)} L(wu, λ^{p}) . (6.119)

We will briefly explain why the right-hand side of (6.112) is unchanged when replacing u by wu. Put

(6.120) C = Vol(T)^{2g−2} \frac{|Z(Z(u))|}{|W_{u}|}(p + c)^{(g−1)dim 𝔷(u))+s/2 dim(𝔷(u)/𝔱)}(−1)^{ℓ(g−1)+1} .

Then using (1.45), (2.167), (6.112), we get

∫_{M^{wu}/Z(wu)} L(wu, λ^{p}) = C [\frac{∏_{α∈R_{u,+}} \frac{⟨wα,∂/∂t⟩}{p+c}}{∏_{α∈R_{+}} 2 sinh (\frac{1}{2} ⟨wα, \frac{∂/∂t}{p+c} +2iπwu⟩)}]^{2g−2+s}

(ε_{w}(−1)^{|R_{+}∩w(−R_{u,+})|})^{s} ∑_{(w^{1},... ,w^{s})∈W^{s}} ∏_{j=1}^{s} ε_{w^{j}}e^{⟨w^{j}(ρ−ct_{j}), \frac{∂/∂t}{p+c} ⟩_{+}+2iπ⟨w^{j}(ρ+pt_{j}),wu⟩}

P_{wu,2g−2+s,p+c}(t)_{|t=∑_{1}^{s} w^{j}t_{j}} = C [\frac{∏_{α∈R_{u,+}} \frac{⟨α,∂/∂t⟩}{p+c}}{∏_{α∈R_{+}} 2 sinh (\frac{1}{2} ⟨α, \frac{∂/∂t}{p+c} +2iπu⟩)}]^{2g−2+s}

∑_{(w^{1},... ,w^{s})∈W^{s}} ∏_{j=1}^{s} ε_{w^{j}}e^{⟨w^{j}(ρ−ct_{j}), \frac{∂/∂t}{p+c} ⟩_{+}+2iπ⟨w^{j}(ρ+pt_{j}),u⟩}
P_{u,2g−2+s,p+c}(t)_{|t=∑_{1}^{s} w^{j}t_{j}} = ∫_{M^{u}/Z(u)} L(u, λ^{p}) . (6.121)

So we find that (6.112) is compatible with (6.119).

6.8. The case where ∑_{j=1}^{s} pt_{j} ∉ \overline{R}. Recall that in Section 1.8, we saw that if ∑_{j=1}^{s} pt_{j} ∈ \overline{R} , then for any (w^{1}, …, w^{s}) ∈ W^{s}, then ∑_{1}^{s} w^{j}pt_{j} ∈ \overline{R}.

Theorem 6.29. If ∑_{1}^{s} pt_{j} ∉ \overline{R}, if v ∈ C/\overline{R}^{∗}, then

∑_{{u∈C/\overline{CR} \atop τu=v}} ∫_{M^{u}/Z(u)} L(u, λ^{p}) = 0. (6.122)

In particular, if p ∈ M, p ≠ −c,

Ind(D_{p,+}) = 0 (6.123)

Proof. To establish (6.122), we use Theorem 6.26. In fact, in the right hand-side of formula (6.112), we observe that the various terms depend on the image v = τu, with the exception of ∏_{j=1}^{s} exp(2iπ⟨w^{j}pt_{j}, u⟩). Also

(6.124) ∑_{v∈\overline{R}^{∗}/\overline{CR}} exp (2iπ⟨∑_{j=1}^{s} w^{j}pt_{j}, v⟩) = 0  if ∑_{j=1}^{s} w^{j}pt_{j} ∉ \overline{R},

= |\frac{\overline{R}^{∗}}{\overline{CR}}|  if ∑_{j=1}^{s} w^{j}pt_{j} ∈ \overline{R} .

(6.125)

It is now clear that (6.122) holds. Equation (6.123) follows from Theorem 6.16 and from (6.122).

6.9. A residue formula for the index. In view of Theorem 6.29, we may and we will assume that

∑_{j=1}^{s} pt_{j} ∈ \overline{R}. (6.126)

We now use the notation of Section 2.

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS 173

Theorem 6.30. For any u ∈ C/\overline{R}^{∗}, p ∈ M, p ≠ −c, the following identity holds

∫_{M^{u}/Z(u)} L(u, λ^{p}) = |\frac{\overline{R}}{\overline{CR}} | Vol(T)^{2g−2} \frac{|Z(Z(u))|}{|W_{u}|}(p + c)^{(g−1)r}(−1)^{ℓ(g−1)+r}

∑_{{I=(i_{1},... ,ir)∈Iu \atop f∈\overline{CR}/d\overline{R}}} \frac{1}{⟨α_{i_{1}}, …, α_{i_{r}}⟩} Res_{x=0}^{I} [ ∏_{α∈R_{+}} \frac{1}{2 sinh (\frac{1}{2}⟨α, \frac{x^{I}}{p+c} + 2iπu⟩)}]^{2g−2+s}

∑_{(w^{1},... ,w^{s})∈W^{s}} ε_{w^{1}} … ε_{w^{s}} exp (⟨∑_{j=1}^{s} w^{j}(ρ − ct_{j}), \frac{x^{I}}{p+c}⟩) (6.127)

+2iπ⟨∑_{j=1}^{s} w^{j}(ρ + pt_{j}) + (p + c)f, u⟩) exp (d ∑_{j=1}^{r} \frac{⟨p_{j−1}^{I}α_{i_{j}} ,x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}} ,e^{j}⟩}
[\frac{1}{d}⟨p_{j−1}^{I}(∑_{k=1}^{s} w^{k}t_{k} + f), e^{j}⟩])_{\frac{1}{∏_{j=1}^{r} exp (d \frac{⟨p{I \atop j−1}α_{i_{j}} ,x^{I} ⟩}{⟨p{I \atop j−1}α_{i_{j}} ,e^{j} ⟩})−1}}.

Proof. If (w^{1}, …, w^{s}) ∈ W^{s}, ∑_{j=1}^{s} w^{j}t_{j} ∉ \overline{S}. Then we use Theorems 2.50 and 6.26 and we get (6.127). Theorem 6.31. If p ∈ M, p ≠ −c, then

Ind(D_{p,+}) = |\frac{\overline{R}}{\overline{CR}}| Vol(T)^{2g−2} \frac{|Z(G)|}{|W |}(p + c)^{(g−1)r}(−1)^{ℓ(g−1)+r} ∑_{u∈C/\overline{R}^{∗}} (6.128)

∑_{{I=(i_{1},... ,ir)∈Iu \atop f∈\overline{CR}/d\overline{R}}} \frac{1}{⟨α_{i_{1}}, …, α_{i_{r}}⟩} Res_{x=0}^{I} [ ∏_{α∈R_{+}} \frac{1}{2 sinh (\frac{1}{2}⟨α, \frac{x^{I}}{p+c} + 2iπu⟩)}]^{2g−2+s}

∑_{(w^{1},... ,w^{s})∈W^{s}} ε_{w^{1}} … ε_{w^{s}} exp (⟨∑_{j=1}^{s} w^{j}(ρ − ct_{j}), \frac{x^{I}}{p + c}⟩

+2iπ ∑_{j=1}^{s} w^{j}(ρ + pt_{j}) + (p + c)f, u⟩) exp (d ∑_{j=1}^{r} \frac{⟨p_{j−1}^{I}α_{i_{j}}, x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩}

[\frac{1}{d}⟨p_{j−1}^{I}(∑_{k=1}^{s} w^{k}t_{k} + f), e^{j}⟩])\frac{1}{∏_{j=1}^{r} exp (d\frac{⟨p_{j−1}^{I}α_{i_{j}} ,x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}} ,e^{j}⟩}) − 1}.

Proof. Recall that Z(G) = \overline{R}^{∗}/\overline{CR}. Using Theorems 6.16, (6.124) and 6.30, we get (6.128). The proof of our Theorem is completed. Remark 6.32. As we saw in Proposition 1.40 and Remark 2.7, for n ≥ 2 and G = SU(n), then C/\overline{R}^{∗} = 0, and we can choose d = 1. Then formula (6.128) has an especially simple form. 6.10. A formula for the index for large p.

Theorem 6.33. For p ∈ M, and |p| large enough, the following identity holds

Ind(D_{p,+}) = |\frac{\overline{R}}{\overline{CR}}| Vol(T)^{2g−2} \frac{|Z(G)|}{|W |}(p + c)^{(g−1)r}(−1)^{ℓ(g−1)+r} ∑_{u∈C/\overline{R}^{∗}}

∑_{{I=(i_{1},... ,ir)∈Iu \atop f∈\overline{CR}/d\overline{R}}} \frac{1}{⟨α_{i_{1}}, …, α_{i_{r}}⟩} Res_{x=0}^{I} [ ∏_{α∈R_{+}} \frac{1}{2 sinh (\frac{1}{2}⟨α, \frac{x^{I}}{p+c} + 2iπu⟩)}]^{2g−2+s}

∑_{(w^{1},... ,w^{s})∈W} ε_{w^{1}} … ε_{w^{s}} exp (⟨∑_{j=1}^{s} w^{j} \frac{ρ}{p + c}, x^{I}⟩ (6.129)

+2iπ⟨∑_{j=1}^{s} w^{j}(ρ + pt_{j} + (p + c)f, u⟩)

exp (d ∑_{j=1}^{r} \frac{⟨p_{j−1}^{I}α_{i_{j}}, x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩}[\frac{1}{d}⟨p_{j−1}^{I}(∑_{k=1}^{s} \frac{w^{k}pt_{k}}{p + c} + f), e^{j}⟩])

\frac{1}{∏_{j=1}^{r} ( exp (d\frac{⟨p_{j−1}^{I}α_{i_{j}} ,x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}} ,e^{j}⟩}) − 1)}.

Proof. Clearly, for p → +∞, ∑_{1}^{s} \frac{w^{j}ct_{j}}{p + c} → 0. Also by Proposition 2.14, since

∑_{j=1}^{s} w^{j}t_{j} ∉ \overline{S}, if f ∈ \overline{R},

⟨p_{j−1}^{I}( ∑_{k=1}^{s} w^{k}t_{k} + f), e^{j}⟩ ∉ Z . (6.130)

Therefore by (2.13) , for |p| large enough,

d ∑_{j=1}^{r} \frac{⟨p_{j−1}^{I}α_{i_{j}}, x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩} [\frac{1}{d}⟨p_{j−1}^{I}( ∑_{k=1}^{s} \frac{w^{k}pt_{k}}{p + c} + f), e^{j}⟩] = (6.131)

d ∑_{j=1}^{r} \frac{⟨p_{j−1}^{I}α_{i_{j}}, x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩} [\frac{1}{d}⟨p_{j−1}^{I}( ∑_{k=1}^{s} w^{k}t_{k} + f), e^{j}⟩] − ⟨∑_{j=1}^{s} \frac{cw^{j}t_{j}}{p + c} , x^{I}⟩ .

By (2.115), (6.131) , we find that for |p| large enough,

exp(∑_{j=1}^{r} \frac{⟨p_{j−1}^{I}α_{i_{j}} ,x^{I}⟩}{⟨p_{j−1}^{I}α_{i},e^{j}⟩} [\frac{1}{d}⟨p_{j−1}^{I}( ∑_{k=1}^{s} \frac{w^{k}pt_{k}}{p+c} + f), e^{j}⟩])

= exp (⟨− ∑_{j=1}^{s} \frac{w^{j}ct_{j}}{p+c} , x^{I}⟩) (6.132)

exp(d ∑_{j=1}^{r} \frac{⟨p_{j−1}^{I}α_{i_{j}}, x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩} [\frac{1}{d}⟨p_{j−1}^{I}( ∑_{k=1}^{s} w^{k}t_{k} + f), e^{j}⟩]).

From (6.128), (6.132) , we get (6.129). The proof of our Theorem is completed.

Remark 6.34. The formulas in Theorems 6.31 and 6.33 are essentially identical.The point is that, as we will see in Theorems 7.23 and 8.1, the right-hand side of (6.129) is just Verlinde’s formula.

6.11. A perturbation of the index problem. We still assume that s ≥ 1 and that if g = 0, then s ≥ 3, so that 2g − 2 + s ≥ 1. Note that these assumptions are almost irrelevant, since by adding as many marked points as one wishes with holonomy equal to 1, one can make s arbitrarily large.

Recall that, by Proposition 1.23, \overline{P} embeds into T. Also the orbit under the Weyl group W of any element in T always intersects \overline{P}.

Let t_{1}, …, t_{s} be s elements in T. We may and we will assume that they all lie in \overline{P}. In the sequel we will consider t_{1}, …, t_{s} as elements of 𝔱. We still define M ⊂ Z as in (6.20), and we assume that M is not reduced to 0. For p ∈ M, set

θ_{j} = pθ_{j}, 1 ≤ j ≤ s. (6.133)

Then θ_{j} ∈ \overline{CR}^{∗}, 1 ≤ j ≤ s.

Let δ = (δ_{1}, …, δ_{s}) ∈ 𝔱^{s} and assume that t_{1} + δ_{1}, …, t_{s} + δ_{s} lie in P and verify (A). Put

X^{δ} = G^{2g} × ∏_{j=1}^{s} O_{t_{j}+δ_{j}}, (6.134)

and let M^{δ} correspond to X^{δ} as in (5.59). Then M^{δ} is a smooth submanifold of X^{δ}.

We will briefly show how to equip M^{δ} with an orbifold line bundle λ^{p}. We use the notation of Section 4.10. For 1 ≤ j ≤ s, consider the orbit O_{−pd/dt+p(t_{j}+δ_{j})} ⊂ L̂𝔤. By (4.30),

O_{−pd/dt+p(t_{j}+δ_{j})} ≃ LG/T. (6.135)

Also as in Definition 4.3, L̃G/(T × S^{1}) ≃ LG/T can be equipped with the line bundle H_{(θ_{j},−p)} of weight (θ_{j}, p). Therefore O_{−pd/dt+p(t_{j}+δ_{j})} can be equipped with the corresponding line bundle L_{j}. Recall that pt_{j} = θ_{j}. Then we define the line bundle λ_{p} as in (4.189), and the line bundle λ^{p} on M^{δ} as in Definition 6.10.

We can then define a Dirac operator D_{p}^{(t_{j}+δ_{j})} in the same way as in Sections 6.3 and 6.4.

Theorem 6.35. For p ∈ M, p ≠ −c,

Ind(D_{p,+}^{(t_{j}+δ_{j})}) = |\frac{\overline{R}}{\overline{CR}}| Vol(T)^{2g−2} \frac{1}{|W |}(p + c)^{(g−1)r} (6.136)

(−1)^{ℓ(g−1)+r} ∑_{u∈C/\overline{CR}} ∑_{{I=(i_{1},... ,ir)∈Iu \atop f∈\overline{R}/\overline{dR}}} \frac{1}{⟨α_{i_{1}}, …, α_{i_{r}}⟩}

Res_{x=0}^{I} [ ∏_{α∈R_{+}} \frac{1}{2 sinh (\frac{1}{2}⟨α, \frac{x^{I}}{p+c} + 2iπu⟩)}]^{2g−2+s}

∑_{(w^{1},... ,w^{s})∈W^{s}} ε_{w^{1}} … ε_{w^{s}} exp (⟨∑_{j=1}^{s} w^{j}(\frac{ρ − ct_{j}}{p + c} − δ_{j}), x^{I}⟩

+2iπ⟨∑_{j=1}^{s} w^{k}(ρ + pt_{j}) + (p + c)f, u⟩)

exp (d ∑_{j=1}^{r} \frac{⟨p_{j−1}^{I}α_{i_{j}}, x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩} [\frac{1}{d}⟨p_{j−1}^{I} (∑_{k=1}^{s} w^{k}(t_{k} + δ_{k}) + f) , e^{j}⟩])

\frac{1}{∏_{j=1}^{r} ( exp (\frac{d⟨p_{j−1}^{I}α_{i_{j}}, x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩}) − 1)} .

Proof. The proof of our Theorem follows the same steps as the proof of Theorem 6.31, where the case δ = 0 was considered. We briefly describe the main steps where the proof of our more general result differs.

• Instead of (6.26) in Proposition 5.11, we have

c_{1}(λ^{p}, ∇^{λ^{p}}) = p(ω + ∑_{j=1}^{s}⟨δ_{j}, Θ_{j}⟩). (6.137)

This follows from an easy computation which is left to the reader.

• In formula (6.74) in Theorem 6.21, the main point is in fact that ∏_{j=1}^{s} e^{2iπ⟨w^{j}pt_{j}+ph,u⟩} is unchanged. The argument is in fact the same as in the proof of Theorem 6.21. Also using (6.137) instead of Theorem 5.76, one finds that in formula (6.74) in Theorem 6.21, ∏_{j=1}^{s} e^{\frac{1}{p+c} ⟨(ρ−ct_{j}),∂/∂t_{j}⟩} should be replaced

by ∏_{j=1}^{s} e^{⟨ \frac{ρ−ct_{j}}{p+c} −δ_{j},∂/∂t_{j}⟩}.

By proceeding otherwise as before, we get (6.136). The proof of our Theorem is completed.

Remark 6.36. Using (6.136) and proceeding as in the proof of Theorem 6.29, we find that (6.123) is still true. So in the sequel, we may and we will assume that (6.126) holds.

Now we will slightly modify the statement of Theorem 6.35. Recall that t_{1}, …, t_{s} all lie in \overline{P}. Therefore for p ∈ M, p ≥ 0, for 1 ≤ j ≤ s, pt_{j}/(p + c) still lies in \overline{P}.

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS 177

Theorem 6.37. For p ∈ M, p ≥ 0, if ε_{1}, …, , ε_{s} ∈ 𝔱 are such that \frac{pt_{1}}{p+c}+ε_{1}, … \frac{pt_{s}}{p+c}+ ε_{s} ∈ P, and that, for any (w^{1}, …, w^{s}) ∈ W^{s}, ∑_{j=1}^{s} w^{j} (\frac{pt_{j}}{p+c} + ε_{j}) ∉ \overline{S}, then

Ind(D_{p,+}^{( \frac{pt_{j}}{p+c} +ε_{j})}) = |\frac{\overline{R}}{\overline{CR}}| Vol(T)^{2g−2} \frac{|Z(G)|}{|W |}(p + c)^{(g−1)r}

(−1)^{ℓ(g−1)+r} ∑_{u∈C/\overline{R}^{∗}} ∑_{{I=(i_{1},... ,ir)∈Iu \atop f∈\overline{R}/\overline{dR}}} \frac{1}{⟨α_{i}, …, α_{i_{r}}⟩}

Res_{x=0}^{I} [ ∏_{α∈R_{+}} \frac{1}{2 sinh (\frac{1}{2}⟨α, \frac{x^{I}}{p+c} + 2iπu⟩)}]^{2g−2+s} (6.138)

∑_{(w^{1},... ,w^{s})∈W^{s}} ε_{w^{1}} … ε_{w^{s}} exp (⟨∑_{j=1}^{s} w^{j}(\frac{ρ}{p + c} − ε_{j}), x^{I}⟩

+2iπ⟨∑_{j=1}^{s} w^{j}(ρ + pt_{j}) + (p + c)f, u⟩)

exp (d ∑_{j=1}^{r} \frac{⟨p_{j−1}^{I}α_{i_{j}}, x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩} [\frac{1}{d}⟨p_{j−1}^{I} (∑_{k=1}^{s} w^{k}(\frac{pt_{k}}{p + c} + ε_{k}) + f) , e^{j}⟩])

\frac{1}{∏_{j=1}^{r} ( exp (\frac{d⟨p_{j−1}^{I}α_{i_{j}}, x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩}) − 1)} .

Proof. In Theorem 6.35, we take

δ_{j} = \frac{−ct_{j}}{p + c} + ε_{j} , 1 ≤ j ≤ s . (6.139)

Then we get (6.138). Proposition 6.38. For p ∈ M, p ≥ 0, there is ε_{1}, …, ε_{s} ∈ 𝔱 such that for ℓ ∈]0, 1], 1 ≤ j ≤ s, \frac{pt_{j}}{p+c} + ℓε_{j} ∈ T is regular, and for any (w^{1}, …, w^{s}) ∈ W^{s}, ℓ ∈]0, 1], ∑_{j=1}^{s} w^{j} (\frac{pt_{j}}{p+c} + ℓε_{j}) ∉ \overline{S}. Proof. Recall that \overline{S} is a union of hypertori in T. Therefore \overline{S} ∩\overline{P} is the intersection of \overline{P} ⊂ 𝔱 with a union of hyperplanes in 𝔱. Our Proposition now follows easily.

Let [x]_{+}, [x]_{−} be the functions defined on R with values in [0, 1], which are periodic of period 1 and such that [x]_{+} = x for x ∈ [0, 1[, (6.140)

[x]_{−} = x for x ∈]0, 1] .

Observe that by 2.14 , for ℓ ∈]0, 1], f ∈ \overline{R}/d\overline{R}, 1 ≤ j ≤ r,

⟨p_{j−1}^{I} (∑_{k=1}^{s} w^{k} (\frac{pt_{k}}{p + c} + ℓε_{k}) + f) , e_{j}⟩ ∉ Z . (6.141)

From (6.141) , it follows that either

⟨p_{j−1}^{I} (∑_{j=1}^{s} w^{j} (\frac{pt_{j}}{p + c}) + f) , e^{j}⟩ ∉ Z , (6.142)

or

⟨p_{j−1}^{I} (∑_{j=1}^{s} w^{j} (\frac{pt_{j}}{p + c}) + f) , e^{j}⟩ ∈ Z (6.143)

⟨p_{j−1}^{I} (∑_{j=1}^{s} w^{j}ε^{j}) , e^{j}⟩ ≠ 0 .

Definition 6.39. Given (w^{1}, …, w^{s}) ∈ W^{s}, for f ∈ \overline{R}/d\overline{R}, I = (i_{1}, …, i_{r}), put

(6.144) η_{j}^{(w^{1},... ,w^{s})}(f, I) = + if ⟨p_{j−1}^{I} (∑_{k=1}^{s} w^{k} (\frac{pt_{j}}{p + c}) + f) , e^{j}⟩ ∉ Z ,

or if ⟨p_{j−1}^{I} (∑_{k=1}^{s} w^{k} (\frac{pt_{k}}{p + c}) + f) , e^{j}⟩ ∈ Z ,

⟨p_{j−1}^{I} (∑_{k=1}^{s} w^{k}ε_{k}) , e^{j}⟩ > 0

= − if ⟨p_{j−1}^{I} (∑_{k=1}^{s} w^{k} (\frac{pt_{k}}{p + c}) + f) , e^{j}⟩ ∈ Z ,

⟨p_{j−1}^{I} (∑_{k=1}^{s} w^{k}ε_{k}) , e^{j}⟩ < 0 .

Observe that if f = ∑_{k=1}^{r} f^{j}e_{j}

⟨p_{j−1}^{I}(∑_{k=1}^{s} w^{k}(\frac{pt_{k}}{p + c} + f), e^{j}⟩ = ⟨p_{j−1}^{I}(∑_{k=1}^{s} w^{k} \frac{pt_{k}}{p + c} + ∑_{k=1}^{j} f^{k}e_{k}), e^{j}⟩ . (6.145)

By (6.145), it is clear that η_{j}^{(w^{1},... ,w^{s})}(f, I) depend only on f^{1}, …, f^{j}, i_{1}, …, i_{j−1}.

Theorem 6.40. Given p ∈ M, p ≥ 0, the integer Ind D_{p,+}^{( \frac{pt_{j}}{p+c} +ℓε_{j})} does not depend on ℓ ∈]0, 1]. More precisely

Ind D_{p,+}^{( \frac{pt_{i}}{p+c} +ℓε_{i})} = |\frac{\overline{R}}{\overline{CR}}| Vol(T)^{2g−2} \frac{|Z(G)|}{|W |}(p + c)^{(g−1)r} (6.146)

(−1)^{ℓ(g−1)+r} ∑_{u∈C/\overline{R}^{∗}} ∑_{{I=(i_{1},... ,ir)∈Iu \atop f∈\overline{CR}/d\overline{R}}} \frac{1}{⟨α_{i}, …, α_{i_{r}}⟩}

Res_{x=0}^{I} [ ∏_{α∈R_{+}} \frac{1}{2 sinh (\frac{1}{2}⟨α, \frac{x^{I}}{p + c} + 2iπu⟩)}]^{2g−2+s}

∑_{(w^{1},... ,w^{s})∈W^{s}} ε_{w^{1}} … ε_{w^{s}} exp (⟨∑_{j=1}^{s} w^{j}(\frac{ρ}{p + c}), x^{I}⟩

+2iπ⟨∑_{1}^{s} w^{j}(ρ + pt_{j}) + (p + c)f, u⟩)

exp (d ∑_{j=1}^{s} \frac{⟨p_{j−1}^{I}α_{i_{j}}, x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩}

[\frac{1}{d}⟨p_{j−1}^{I} (∑_{k=1}^{s} w^{k}(\frac{pt_{k}}{p + c}) + f) , e_{j}⟩]_{η_{j}^{(w^{1},... ,w^{s})}(f,I)} )

\frac{1}{∏_{j=1}^{s} ( exp (\frac{d⟨p_{j−1}^{I}α_{i_{j}}, x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩}) − 1) .}

Proof. Clearly, for ℓ ∈]0, 1],

[\frac{1}{d}⟨p_{j−1}^{I} ∑_{k=1}^{s} w^{k} (\frac{pt_{k}}{p + c} + ℓε_{k}) + f, e_{j}⟩] (6.147)

depends continuously on ℓ ∈]0, 1], and moreover as ℓ → 0, it converges to

[\frac{1}{d}⟨p_{j−1}^{I} (∑_{1}^{s} \frac{w^{j}t_{j}}{p + c} + f) , e_{j}⟩]_{η_{j}^{(w^{1},... ,w^{s})}(f,I)} . (6.148)

By (6.138) and the above, Ind D_{p,+}^{( \frac{pt_{j}}{p+c} +ℓε_{j})} depends continuously on ℓ ∈]0, 1], and so remains constant. This last fact should also be clear by the construction of

D_{p,+}^{( \frac{pt_{j}}{p+c} +ℓε_{j})}.

Using (6.138), (6.148) , we get (6.146). The proof of our Theorem is completed.

Remark 6.41. Observe that in our direct index theoretic computations, we have avoided introducing central holonomies, because they are non generic. We dealt with this case by a perturbation argument. However we may assume that one of the t_{j}, say t_{s}, is equal to h_{o} ∈ Z(G), while the other holonomies are generic. By Proposition 1.24, we can identify h_{o} to a unique element in \overline{P}, which we still denote h_{o}, so that h_{o} ∈ \overline{P} ∩ \overline{R}^{∗}. When the moduli space M/G is an orbifold (which is the case when condition (\overline{A}) holds), all the computations we did in Sections 6.1-6.9 can be done directly. By proceeding as in Remark 6.27, it is obvious that in both cases, we compute the same index. This is less obvious at the level of explicit computations. Using in particular equations (1.78) and (1.140), we find that equation (6.112) is now replaced by

∫_{M^{u}/Z(u)} L(u, λ^{p}) = ϵ_{w_{ho}} Vol(T)^{2g−2} \frac{|Z(Z(u))|}{|W_{u}|}
(p + c)^{(g−1)dim(𝔷(u))+ \frac{(s−1)}{2} dim(𝔷(u)/𝔱)}

(−1)^{ℓ(g−1)+1} [\frac{∏_{α∈R_{u,+}}⟨α, \frac{∂/∂t}{p + c}⟩}{∏_{α∈R_{+}} 2 sinh (\frac{1}{2}⟨α, \frac{∂/∂t}{p + c} + 2iπu⟩)}]^{2g−2+s−1} (6.149)

∑_{(w^{1},... ,w^{s−1})∈W^{s−1}} ∏_{j=1}^{s−1} ε_{w^{j}} exp(⟨w^{j}(ρ − ct_{j}), \frac{∂/∂t}{p+c}⟩
+2iπ⟨w^{j}(ρ + pt_{j}), u⟩) exp(2iπ⟨(p + c)h_{o}, u⟩)
P_{u,2g−2+s−1,p+c}(t)_{|t=∑_{j=1}^{s−1} w^{j}t_{j}+h_{o}} .

The fact that ultimately, the two explicit index formulas coincide will be verified via the Verlinde formulas in Remark 8.4, by using in particular Theorem 1.33.

7. Residues and the Verlinde formula

In this Section, we apply residues techniques to the finite Verlinde sum, we express it as a residue in several complex variables, and we prove that for p large enough, the Riemann-Roch number of M/G is given by the Verlinde formula. Without any condition on p, the Verlinde formula is the Riemann-Roch number of some perturbation of the moduli space of M/G.

As indicated in the introduction, higher cohomology groups on M/G may well not vanish. This would account for the discrepancy between the index and the Verlinde formula for small p.

This Section is organized as follows. In Section 7.1, we give a residue formula for a Fourier series over \overline{R}^{∗}/q\overline{R}^{∗}, L_{q}(t, x). In Section 7.2, we consider a related series M_{q}(t, x), which we also evaluate as a sum of residues, the sum being indexed by the semisimple centralizers Z(u). In Section 7.3, we introduce the Verlinde sums V_{g,q}(θ_{1}, …, θ_{s}). In Section 7.4, we express the Verlinde sums as residues.

Finally in Sections 7.5-7.7, we “localize” the Verlinde formulas to put them a form which is very similar to the form we obtained in Section 6 for Ind(D_{p,+}) . This is done using the techniques we already developed in Section 2.

7.1. A residue formula for L_{q}(t, x).

Definition 7.1. For q ∈ Z^{∗}, t ∈ \overline{R}/q\overline{R} , x ∈ (C\2iπZ)^{ℓ}, put

L_{q}(t, x) = ∑_{λ∈\overline{R}^{∗}/q\overline{R}^{∗}} \frac{exp(2iπ⟨λ, t/q⟩)}{∏_{i=1}^{ℓ} 2q tanh (\frac{2iπ⟨α_{i},λ⟩−x_{i}}{2q} )} . (7.1)

If t ∈ \overline{R}/q\overline{R}, then t/q ∈ (\overline{R}/q)/\overline{R} ⊂ 𝔱/\overline{R}. Recall that the subset H of t/\overline{R} was defined in (2.44), (2.45). Then by (2.45),

\{t ∈ \overline{R}/q\overline{R}, t/q ∈ H\} = \{t ∈ \overline{R}/q\overline{R}, t = ∑_{j∈J ⊂\{1,... ,r\}} t^{j}α_{j} in 𝔱/q\overline{R} , (7.2)

and the {α_{j}, j ∈ J } do not span 𝔱^{∗} ≃ 𝔱} .

As in Section 2.4 , we identify \overline{R}/d\overline{R} to \{0, 1, …, d − 1\}^{r}, i.e. f ∈ \overline{R}/d\overline{R} is

identified to ∑_{1}^{r} f^{i}e_{i}, f^{i} ∈ {0, …, d − 1}.

By (2.14), if I = (i_{1}, …, i_{r}) is generic, if x ∈ Z^{ℓ}, then d⟨e_{j}, x^{I}⟩ ∈ Z, 1 ≤ j ≤ r, and so for 1 ≤ i ≤ ℓ, d⟨α_{i}, x^{I}⟩ ∈ Z.Therefore the function x ∈ C^{ℓ} ↦ tanh (\frac{⟨α_{i},x^{I}⟩−x_{i}}{2q} ) is periodic of period (2iπqd, …, 2iπqd).

Similarly if t ∈ \overline{R}, we claim that the function

x ∈ C^{ℓ} ↦ exp (d ∑_{j=1}^{r} \frac{⟨p_{j−1}^{I}α_{i_{j}}, x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩}[\frac{1}{d}⟨p_{j−1}^{I}t, \frac{e^{j}}{q} ⟩]) (7.3)

is periodic of period (2iπqd, …, 2iπqd). In fact by (2.13), (2.28), (2.29), if x ∈ (2iπZ)^{ℓ},

exp (d ∑_{j=1}^{r} \frac{⟨p_{j−1}^{I}α_{i_{j}}, x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩}[\frac{1}{d}⟨p_{j−1}^{I}t, \frac{e^{j}}{q} ⟩]) = exp (⟨\frac{t}{q} , x^{I}⟩) . (7.4)

Also, as we just saw, if t ∈ \overline{R}, z ∈ Z^{ℓ}, d⟨t, x^{I}⟩ ∈ Z. From (7.4), it follows that if x ∈ (2iπqdZ)^{ℓ}, t ∈ \overline{R}, (4.4) is equal to 1, which proves the periodicity of (7.3).

If g = (g_{1}, …, g_{r}) ∈ Z^{r}, if I = (i_{1}, …, i_{r}) ⊂ {1, …, ℓ}, let g_{I} ⊂ C^{ℓ} be such that

(7.5)  (g_{I})_{i_{j}} = g_{j}, 1 ≤ j ≤ r ,
(g_{I})_{i} = 0, i ∉ I .

Theorem 7.2. For generic values of x ∈ (C\2iπZ)^{ℓ}, for t ∈ \overline{R}/q\overline{R}, t/q ∉ H, if we still denote by t a representative of t in \overline{R}, then

(7.6) L_{q}(t, x) = \frac{(−1)^{r}}{d^{r}} ∑_{I=(i_{1},... ,ir)⊂\{1,... ,ℓ\},I generic f∈\overline{R}/d\overline{R},g∈(Z/dZ)^{r}} sgn (⟨α_{i_{1}}, …, α_{i_{r}}⟩)

[exp(d ∑_{j=1}^{r} \frac{⟨p_{j−1}^{I}α_{i_{j}}, x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩}[\frac{1}{d}⟨p_{j−1}^{I}(t + qf), \frac{e^{j}}{q} ⟩])

\frac{1}{∏_{i∈^{c}I} 2q tanh (\frac{⟨α_{i},x^{I}⟩−x_{i}}{2q} )}](x + 2iπqg_{I}) ∏_{j=1}^{r} \frac{1}{exp (\frac{d⟨p_{j−1}^{I}α_{i_{j}} ,x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}} ,e^{j}⟩}) − 1} .

Proof. Take 1 ≤ j ≤ r, I = (i_{1}, …, i_{j−1}) ⊂ {1, …, ℓ} such that ⟨α_{i_{1}}⟩ ≠ 0, ⟨α_{i_{1}}, …, α_{i_{j−1}}⟩ ≠ 0. We use otherwise the notation and the conventions in the proof of Theorem 2.19, in particular in equation (2.63).

For k̂ = (k_{j+1}, …, k_{r}) ⊂ Z^{r−j}, we identify k̂ to ∑_{i=j+1}^{r} k^{i}e_{i} ∈ \overline{R}^{∗}. For a ∈ C,

x ∈ C^{ℓ}, t ∈ \overline{R}, put

S_{I}(a, x) = (7.7)

∑_{{0≤gp<d \atop 1≤p≤j−1}}\{exp(a⟨p_{j−1}^{I}t, \frac{e^{j}}{q} ⟩ + d ∑_{n=1}^{j−1} \frac{⟨p_{n−1}^{I}α_{i_{n}}, x^{I}⟩}{⟨p_{n−1}^{I}α_{i_{n}}, e^{n}⟩}[\frac{1}{d}⟨p_{n−1}^{I}t, \frac{e^{n}}{q} ⟩])

\frac{1}{∏_{i∈^{c}I} 2q tanh (\frac{⟨p_{j−1}^{I}α_{i},ae^{j}+2iπk̂−x^{I}⟩}{2q})}\}(x + 2iπq(g_{1}, …, g_{j−1})_{I}) .

We claim that as a function of x ∈ C^{ℓ}, each of the expressions { }, with (g_{1}, …, g_{j−1}) = 0, is periodic of period 2iπqd, …, 2iπqd. In fact by (2.63), if x ∈ Z^{ℓ},

d⟨p_{j−1}^{I}, α_{i}, x^{I}⟩ ∈ Z . (7.8)

Also by (2.28), (2.63), for 1 ≤ n ≤ j − 1, x ∈ Z^{ℓ}

\frac{d⟨p_{n−1}^{I}α_{i_{n}}, x^{I}⟩}{⟨p_{n−1}^{I}α_{i_{n}}, e^{n}⟩} ∈ Z , (7.9)

so that if x ∈ Z^{ℓ}, using (2.12),

exp (2iπd ∑_{n=1}^{j−1} \frac{⟨p_{n−1}^{I}α_{i_{n}}, x^{I}⟩}{⟨p_{n−1}^{I}α_{i_{n}}, e^{n}⟩} [\frac{1}{d}⟨p_{n−1}^{I}t, \frac{e^{n}}{q} ⟩]) (7.10)

= exp (\frac{2iπ}{q} ∑_{n=1}^{j−1} \frac{⟨p_{n−1}^{I}α_{i_{n}}, x^{I}⟩}{⟨p_{n−1}^{I}α_{i_{n}}, e^{n}⟩}⟨p_{n−1}^{I}t, e^{n}⟩)

exp (\frac{2iπ}{q} (⟨t, x^{I}⟩ − ⟨p_{j−1}^{I}t, x^{I}⟩)) .

Also by (2.14), (2.63), if t ∈ \overline{R}, x ∈ Z^{ℓ},

d⟨t, x^{I}⟩ ∈ Z , (7.11)

d⟨p_{j−1}^{I}t, x^{I}⟩ ∈ Z .

By (7.11), it follows that if x ∈ qdZ^{ℓ}, (7.10) is equal to 1.

Ultimately, we find that indeed, each of the expression { } in (7.7) is periodic in x ∈ C^{ℓ} of period 2iπqd, …, 2iπqd. Therefore, we will write (7.7) in the form

S_{I}(a, x) = ∑_{g∈(Z/dZ)^{j−1}}\{exp(a⟨p_{j−1}^{I}t, \frac{e^{j}}{q} ⟩ + d (7.12)

∑_{n=1}^{j−1} \frac{⟨p_{n−1}^{I}α_{i_{n}}, x^{I}⟩}{⟨p_{n−1}^{I}α_{i_{n}}, e^{n}⟩}[\frac{1}{d}⟨p_{n−1}^{I}t, \frac{e^{n}}{q} ⟩])

\frac{1}{∏_{i∈^{c}I} 2q tanh(\frac{⟨p_{j−1}^{I}α_{i},ae^{j}+2iπk̂−x^{I}⟩}{2q})}\}(x + 2iπqg_{I}) .

In particular S_{I}(a, x) is periodic in x ∈ C^{ℓ} of period 2iπq, …, 2iπq.

Now we claim that S_{I}(a, x) is periodic in a ∈ C, with period 2iπq. In fact, since e^{j} = (ãe^{j})^{I},

⟨p_{j−1}^{I}α_{i}, e^{j}⟩ = ⟨p_{j−1}^{I}α_{i}, (ãe^{j})^{I}⟩ . (7.13)

Also by (2.12),

⟨p_{j−1}^{I}t, e^{j}⟩ = ⟨t, e^{j}⟩ − ∑_{n=1}^{j−1} \frac{⟨p_{n−1}^{I}α_{i_{n}}, (ãe^{j})^{I}⟩}{⟨p_{n−1}^{I}α_{i_{n}}, e^{n}⟩}⟨p_{n−1}^{I}t, e^{n}⟩ . (7.14)

Since t ∈ \overline{R}, then ⟨t, e^{j}⟩ ∈ Z and so, by (7.14), we get

⟨p_{j−1}^{I}t, e^{j}⟩ = − ∑_{n−1}^{j−1} \frac{⟨p_{n−1}^{I}α_{i_{n}}, (ãe^{j})^{I}⟩}{⟨p_{n−1}^{I}α_{i_{n}}, e^{n}⟩}⟨p_{n−1}^{I}t, e^{n}⟩ mod (Z) . (7.15)

From (7.12), (7.13), (7.15), we find that

(7.16)  S_{I}(a + 2iπq, x) = S_{I}(a, x − 2iπqãe^{j})
= S_{I}(a, x) ,

i.e. S_{I}(a, x) is periodic in a with period 2iπq.

Put

L_{I} = ∑_{k∈Z/qZ} S_{I}(2iπk, x) . (7.17)

184 JEAN-MICHEL BISMUT AND FRANÇOIS LABOURIE

Clearly

L_{I} = \frac{1}{d} ∑_{k∈Z/dqZ} S_{I}(2iπk, x) . (7.18)

From (7.18), we obtain

L_{I} = \frac{1}{d^{2}} ∑_{{0≤f<d \atop k∈Z/d^{2}qZ}} exp (\frac{2iπfk}{d} ) S_{I} (\frac{2iπk}{d} , x) . (7.19)

By (7.12), for generic x, we rewrite (7.19) in the form

(7.20) L_{I} = \frac{1}{d^{2}} ∑_{k∈Z/d^{2}qZ}\{ ∑_{{0≤f<d \atop g∈(Z/dZ)^{j−1}}} Res_{a=2iπk}[exp(a[\frac{⟨p_{j−1}^{I}(t + qfe_{j}), e^{j}/q⟩}{d}]

+d ∑_{n=1}^{j−1} \frac{⟨p_{n−1}^{I}α_{i_{n}}, x^{I}⟩}{⟨p_{n−1}^{I}α_{i_{n}}, e^{n}⟩}[\frac{1}{d}⟨p_{n−1}^{I}t, e^{n}/q⟩])

\frac{1}{∏_{i∈^{c}I} 2q tanh (\frac{⟨p_{j−1}^{I}α_{i},ae^{j}/d+2iπk̂−x^{I}⟩}{2q})} \frac{1}{e^{a} − 1}]\}(x + 2iπqg_{I}) .

Now we just saw that the function of a, ∑_{{0≤f<d \atop g∈(Z/dZ)^{j−1}}} …, which appears in the right-hand side of(7.20), is periodic of 2iπqd.

We claim that for f, 0 ≤ f < d, g ∈ (Z/dZ)^{j−1}, the function of a,

h_{f,g}(a) = \frac{exp(a[\frac{p_{j−1}^{I}(t+qfe_{j}),e^{j}/q⟩}{d}])}{∏_{i∈^{c}I} 2q tanh(\frac{⟨p_{j−1}^{I}α_{i},ae^{j}/d+2iπk̂−x^{I}⟩}{2q})} \frac{1}{e^{a} − 1} (7.21)

is periodic of period 2iπd^{2}q. This follows from the fact that since t ∈ \overline{R}, d⟨p_{j−1}^{I}(t + qfe_{j}, e^{j}⟩ ∈ Z , (7.22)

d⟨p_{j−1}^{I}α_{i}, e^{j}⟩ ∈ Z .

By Proposition 2.14, if t/q ∉ H,

0 < [\frac{⟨p_{j−1}^{I}(t + qfe^{j}), e^{j}/q⟩}{d}] < 1 . (7.23)

Then for given f, 0 ≤ f < d, g ∈ (Z/dZ)^{j−1}, we will apply the theorem of residues to the function h_{f,g}(a) on the domain given in Figure 7.1. By (2.65), for at least one i_{j} ∉ I, ⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩ ≠ 0. For generic x, mod (2iπd^{2}q), the poles of the function h_{f,g}(a) other than the 2iπk are simple and given by

a = \frac{d}{⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩}(⟨p_{j−1}^{I}α_{i_{j}}, x^{I} − 2iπk̂⟩ + 2iπg^{j}q) , (7.24)

0 ≤ g^{j} < d|⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩| , i_{j} ∉ I , ⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩ ≠ 0 .

Figure 7.1

Needless to say, in (7.24), d⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩ ∈ Z. In view of (2.28) , we rewrite (7.24) in the form

a = \frac{d}{⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩}⟨p_{j−1}^{I}α_{i_{j}}, (x + 2iπq(g^{j})_{I,i_{j}})^{I} − 2iπk̂⟩ , (7.25)

0 ≤ g^{j} < d|⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩| .

By periodicity, the integrals of h_{g,j}(a) over δ_{+} and δ_{−} cancel each other.

Using the theorem of residues and (2.70), (2.76), (2.78), (7.20), (7.23), (7.25), we get

L_{I} = −\frac{1}{d} ∑_{{0≤f<d ,g∈(Z/dZ)^{j−1} \atop i_{j}:⟨α_{i_{1}} ,... ,α_{i_{j}} ⟩≠0 ,0≤g^{j} <d|⟨p{I \atop j−1}α_{i_{j}} ,e^{j} ⟩|}} (7.26)

\{\frac{1}{⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩} exp(2iπ⟨p_{j}^{(I,i_{j})}e_{j}, k̂⟩⟨p_{j−1}^{I}(t + qfe_{j}), e^{j}/q⟩

+d ∑_{n=1}^{j} \frac{⟨p_{n−1}^{I}α_{i_{n}}, x^{I}⟩}{⟨p_{n−1}^{I}α_{i_{n}}, e^{n}⟩}[\frac{1}{d}⟨p_{n−1}^{I}(t + qfe_{j}), e^{n}/q⟩])

\frac{1}{∏_{i∉(I,i_{j})} 2q tanh (^{\frac{⟨p_{j}^{(I,i_{j})}α_{i},2iπk̂−x^{I}⟩}{2q}})}\} (x + 2iπq(g, g_{j})_{I,i_{j}})

\frac{1}{exp (\frac{d⟨p_{j−1}^{I}α_{i_{j}} ,x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}} ,e^{j}⟩}) − 1} .

We claim that as functions of x_{i_{j}}, the terms in the right-hand side of (7.26) are periodic of period 2iπqd. In fact this follows from what we saw after (7.7), by replacing j − 1 by j. We also claim these functions, are periodic as functions of x_{i_{j}}, with period 2iπqd⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩. In fact, using (2.29),

\frac{d⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩}{⟨α_{i_{1}}, …, α_{i_{j}}⟩} = \frac{d}{⟨α_{i_{1}}, …, α_{j−1}⟩} ∈ Z . (7.27)

In view of (2.63), (7.27) takes care of the terms tanh (\frac{⟨p_{j−1}^{(I,i_{j})}α_{i}, 2iπk̂ − x^{I}⟩}{2q}). Also

among the term ⟨p_{n−1}^{I}α_{i_{n}}, x^{I}⟩, n ≤ j, only ⟨p_{j−1}^{I}α_{i_{j}}, x^{I}⟩ depends on x_{i_{j}}. More precisely by (2.63),

\frac{∂}{∂x_{i_{j}}} ⟨p_{j−1}^{I}α_{i_{j}}, x^{I}⟩ = 1 . (7.28)

Clearly,

qd\frac{⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩}{⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩}d [\frac{1}{d}⟨p_{j−1}^{I}(t + qfe_{j}), e^{j}/q⟩] = (7.29)

d⟨p_{j−1}^{I}(t + qfe_{j}), e^{j}⟩ mod (Z) .

Using (2.29), we find that (7.29) is 0 mod (Z), i.e. the left-hand side of (7.29) lies in Z. Ultimately, this guarantees the periodicity of the functions appearing in (7.26) in x_{i_{j}}, with period 2iπqd⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩.

Now by (2.29),

d⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩ = \frac{d⟨α_{i_{1}}, …, α_{i_{j}}⟩}{⟨α_{i_{1}}, …, α_{i_{j−1}}⟩} . (7.30)

From (7.30), we find that

d⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩ | d^{2} . (7.31)

Since the functions in (7.26) are periodic in x_{i_{j}} of period 2iπqd⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩, using (7.31), in (7.26), we may and will replace the condition 0 ≤ g^{j} < d|⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩| by

the condition 0 ≤ g^{j} < d^{2}, and this introduces a correcting factor \frac{|⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩|}{d} in the right-hand side of (7.26). Using the periodicity in x_{i_{j}} of period 2iπqd, we may finally replace the condition 0 ≤ g^{j} < d^{2} by 0 ≤ g^{j} < d, with a new correcting factor d. Ultimately in (7.26), we replace the condition 0 ≤ g^{j} < d|⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩| by 0 ≤ g^{j} < d, with a correcting factor |⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩|. Also by (7.30),

(7.32) sgn⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩ = sgn⟨α_{i_{1}}, …, α_{i_{j}}⟩sgn⟨α_{i_{1}}, …, α_{j−1}⟩ .

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS 187

So, from (7.26), (7.32), we get

L_{I} = \frac{−1}{d} ∑_{{0≤f<d,g∈(Z/dZ)^{j} \atop i_{j}:⟨α_{i_{1}} ,... ,α_{i_{j}} ⟩≠0}} sgn(⟨α_{i_{1}}, …, α_{i_{j−1}}⟩⟨α_{i_{1}}, …, α_{i_{j}}⟩) (7.33)

\{exp (2iπ⟨p_{j}^{(I,i_{j})}e_{j}, k̂⟩⟨p_{j−1}^{I}(t + qfe_{j}), e^{j}⟩ + d ∑_{n=1}^{j} \frac{⟨p_{n−1}^{I}α_{i_{n}}, x^{I}⟩}{⟨p_{n−1}^{I}α_{i_{n}}, e^{n}⟩}

[\frac{1}{d}⟨p_{n−1}^{I}(t + qfe_{j}), e^{n}/q⟩] )

\frac{1}{∏_{i∉(I,i_{j})} 2q tanh (^{\frac{⟨p_{j}^{(I,i_{j})}α_{i},2iπk̂−x^{I}⟩}{2q}})}\}(x + 2iπqg_{(I,i_{j})})

\frac{1}{exp (\frac{d⟨p_{j−1}^{I}α_{i_{j}} ,x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}} ,e^{j}⟩}) − 1} .

Clearly

(7.34)  L_{q}(t, x) = ∑_{k=(k_{1},... ,k_{r})∈(Z/qZ)^{r}} \frac{exp(2iπ ∑_{j=1}^{r} k_{j}⟨t, e^{j}/q⟩)}{∏_{i=1}^{ℓ} 2q tanh (^{\frac{2iπ⟨α_{i},∑_{j=1}^{r} k_{j}e^{j}⟩−x_{i}}{2q}} )} .

Also with the notation in (2.63), ⟨p_{0}^{ϕ}α_{i}, x^{I}⟩ = x_{i} . (7.35) Then we use (7.33), so that if t ∈ \overline{R}/q\overline{R}, t/q ∉ H,

L_{q}(t, x) = \frac{−1}{d} ∑_{{0≤f^{1}<d,g∈Z/dZ \atop ⟨α_{i_{1}} ⟩≠0,k=(k^{2},... ,k^{r})∈(Z/qZ)^{r−1}}} (7.36)

sgn⟨α_{i_{1}}⟩ exp (2iπ⟨p_{1}^{(i_{1})}(t + qf^{1}e_{1}), k⟩ + d\frac{x_{i_{1}}}{⟨α_{i_{1}}⟩} [\frac{⟨t + qf′e_{1}, e^{1}/q⟩}{d}] )

\frac{1}{∏_{i≠i_{1}} 2q tanh (\frac{⟨p_{1}^{(i_{1})}α_{i},2iπk−x^{I}⟩}{2q})}(x + 2iπqg_{i_{1}})\frac{1}{exp (\frac{dx_{i_{1}}}{⟨α_{i_{1}} ⟩}) − 1} .

Using (7.33), (7.36) and an obvious iteration, in view of (2.83), we obtain (7.6). The proof of our Theorem is completed. 7.2. A residue formula for M_{q}(t, x). Definition 7.3. For q ∈ N^{∗}, t ∈ \overline{R}/q\overline{CR}, x ∈ (C\2iπZ/m)^{ℓ}, put

M_{q}(t, x) = ∑_{λ∈\overline{CR}^{∗}/q\overline{R}^{∗}} \frac{exp(2iπ⟨λ, t/q⟩)}{∏_{i=1}^{ℓ} 2q tanh (\frac{2iπ⟨α_{i},λ⟩−x_{i}}{2q} )} . (7.37)

Now we will adapt the formalism of Sections 2.5 and 2.6, with 𝔱/\overline{CR} replaced by \overline{R}/q\overline{CR}. If f(t) is a function on \overline{R}/q\overline{CR}, if μ ∈ \overline{CR}^{∗}/\overline{R}^{∗}, put

f̂_{μ}(t) = \frac{1}{|\overline{R}/\overline{CR}|} ∑_{v∈\overline{R}/\overline{CR}} e^{−2iπ⟨μ,v⟩}f(t + qv) . (7.38)

188 JEAN-MICHEL BISMUT AND FRANÇOIS LABOURIE

Then by (2.92),

f(t) = ∑_{μ∈\overline{CR}^{∗}/\overline{R}^{∗}} f̂_{μ}(t) . (7.39)

Proposition 7.4. If μ ∈ \overline{CR}^{∗}/\overline{R}^{∗}, if λ_{μ} ∈ \overline{CR}^{∗} represents μ, for t ∈ \overline{R}/q\overline{CR}, (M̂_{q})_{μ}(t, x) = exp(2iπ⟨λ_{μ}, t/q⟩)L_{q}(t, x − 2iπãλ_{μ}) . (7.40) Proof. If λ ∈ \overline{CR}^{∗}/q\overline{R}^{∗}, then

{ \atop exp(2iπ⟨λ, t/q⟩)_{μ}} = exp(2iπ⟨λ, t/q⟩) if λ maps to μ

in \overline{CR}^{∗}/\overline{R}^{∗} (7.41)

0 otherwise .

Then

(M̂_{q})_{μ}(t, x) = ∑_{λ∈\overline{R}^{∗}/q\overline{R}^{∗}} \frac{exp(2iπ⟨λ + λ_{μ}, t/q⟩}{∏_{i=1}^{ℓ} 2q tanh (\frac{2iπ⟨α_{i},λ⟩−(x_{i}−2iπ⟨α_{i},λ_{μ}⟩)}{2q})} , (7.42)

which is equivalent to (7.40). Proposition 7.5. For q ∈ N^{∗}, t ∈ \overline{R}/q\overline{CR}, x ∈ (C\2iπZ/m)^{ℓ},

M_{q}(t, x) = ∑_{μ∈\overline{CR}^{∗}/\overline{R}^{∗}} exp(2iπ⟨λ_{μ}, t/q⟩)L_{q}(t, x − 2iπãλ_{μ}) . (7.43)

Proof. This follows from (7.39), (7.40).

Recall that τ is the projection 𝔱/\overline{CR} → 𝔱/\overline{R}. Then τ induces the projection \overline{R}/q\overline{CR} → \overline{R}/q\overline{R}. Also recall that the set \overline{S} ⊂ 𝔱/\overline{CR} was defined in (2.111). By (2.45), (2.111),

(7.44)  \{t ∈ \overline{R}/q\overline{CR}, t/q ∈ \overline{S}\} = \{t ∈ \overline{R}/q\overline{CR}, τt = ∑_{j∈J} t^{j}α_{j} ,
{α_{j}, j ∈ J } does not span 𝔱^{∗}} .

Theorem 7.6. For generic values of x ∈ (C\2iπZ/m)^{ℓ}, t ∈ \overline{R}/q\overline{CR}, t/q ∉ \overline{S}, if we still denote by t a representative of t in \overline{R}, then M_{q}(t, x) = (7.45)

\frac{(−1)^{r}}{d^{r}} ∑_{I=(i_{1},... ,ir)⊂\{1,... ,ℓ\},I generic f∈\overline{R}/d\overline{R},g∈(Z/dZ)^{r} ,μ∈\overline{CR}^{∗}/\overline{R}^{∗}} sgn(⟨α_{i_{1}}, …, α_{i_{r}}⟩) exp(2iπ⟨λ_{μ}, t/q⟩)

[exp(d ∑_{j=1}^{r} \frac{⟨p_{j−1}^{I}α_{i_{j}}, x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩} [\frac{1}{d}⟨p_{j−1}^{I}(t + qf), e^{j}/q⟩])

\frac{1}{∏_{i∈^{c}I} 2q tanh (\frac{⟨α_{i},x^{I}⟩−x_{i}}{2q} )}](x + 2iπqg_{I} − 2iπãλ_{μ})

\frac{1}{∏_{j=1}^{r} (exp (\frac{d⟨p_{j−1}^{I}α_{i_{j}} ,x^{I}⟩}{⟨p_{j−1}^{I_{j−1}} α_{i_{j}} ,e^{j}⟩}) − 1)}(x − 2iπãλ_{μ}) .

Proof. This follows from Theorem 7.2 and Proposition 7.5.

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS 189

Definition 7.7. If I = (i_{1}, …, i_{r}) is generic, let H^{I} be the lattice in 𝔱 generated by α^{i_{1}}, …, α^{i_{r}}.

Recall that C ⊂ \overline{R}^{∗} was defined in Definition 1.35. Proposition 7.8. The following identity holds

C = ⋃_{I generic} H^{I} . (7.46)

In particular for I generic, dH^{I} ⊂ \overline{R}^{∗} . (7.47) Proof. The identity (7.46) follows from the definition of C. Then (7.47) follows from (2.21), (7.46). The proof of our Proposition is completed.

By Proposition 7.8, we have a natural projection h^{I} : H^{I}/dH^{I} → H^{I}/\overline{R}^{∗} , (7.48) whose kernel is just ker h^{I} = \overline{R}^{∗}/dH^{I} . (7.49) Proposition 7.9. For any generic I,

|\overline{R}^{∗}/dH^{I}| = \frac{d^{r}}{|⟨α_{i_{1}}, …, α_{i_{r}}⟩|} . (7.50)

Proof. Recall that e_{1}, …, e_{r} span \overline{R}, and e^{1}, …, e^{r} span \overline{R}^{∗}. Therefore

(7.51)  |\overline{R}^{∗}/dH^{I}| = d^{r}|⟨α^{i_{1}} ∧ … ∧ α^{i_{r}}, e_{1} ∧ … ∧ e_{r}⟩|
= \frac{d^{r}}{|⟨α_{i_{1}} ∧ … ∧ α_{i_{r}}⟩|} .

The proof of our Proposition is completed. Theorem 7.10. For generic values of x ∈ (C\2iπZ/m)^{ℓ}, for t ∈ \overline{R}/q\overline{CR}, t/q ∉ \overline{S}, if we still denote by t a representative of t in \overline{R}, then

M_{q}(t, x) = |\frac{\overline{R}}{\overline{CR}}| (−1)^{r} ∑_{u∈C/\overline{R}^{∗}} ∑_{{I=(i_{1},... ,ir)⊂Iu,I generic \atop f∈\overline{CR}/d\overline{R}}} (7.52)

\frac{1}{⟨α_{i_{1}}, …, α_{i_{r}}⟩} \frac{exp (2iπ (⟨u, t + qf⟩))}{∏_{i∈^{c}I} 2q tanh (\frac{⟨α_{i},x^{I}+2iπqu⟩−x_{i}}{2q} )}

exp (d ∑_{j=1}^{r} \frac{⟨p_{j−1}^{I}α_{i_{j}}, x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩} [\frac{1}{d}⟨p_{j−1}^{I}(t + qf), e^{j}/q⟩] )

\frac{1}{∏_{j=1}^{r} (exp (\frac{d⟨p_{j−1}^{I}α_{i_{j}} ,x^{I}⟩}{⟨p_{j−1}^{I_{j−1}} α_{i_{j}} ,e^{j}⟩}) − 1)}.

Proof. Clearly the map g ∈ (Z/dZ)^{ℓ} ↦ g_{I}^{I} ∈ H^{I}/dH^{I} is one to one. Also if t ∈ \overline{R}, f ∈ \overline{R}, by (2.13), (2.28), (2.29),

exp (d ∑_{j=1}^{r} \frac{⟨p_{j−1}^{I}α_{i_{j}}, 2iπqg_{I}^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩} [\frac{1}{d}⟨p_{j−1}^{I}(t + qf), e^{j}/q⟩]) (7.53)

= exp(2iπ⟨t + qf, g_{I}^{I}⟩) = exp(2iπ⟨t + qf, h^{I}g_{I}^{I}⟩) .

Moreover by (2.115),

⟨α_{i}, (ãλ_{μ})^{I}⟩ − (ãλ_{μ})_{i} = 0 . (7.54)

Also if i ∉ I, using (7.47),

⟨α_{i}, g_{I}^{I}⟩ − (g_{I})_{i} = ⟨α_{i}, g_{I}^{I}⟩ , (7.55)

q⟨α_{i}, g_{I}^{I}⟩ = q⟨α_{i}, h^{I}g_{I}^{I}⟩ in Z/qZ .

Finally by Proposition 7.8, the map h^{I} : H^{I}/dH^{I} → C/\overline{R}^{∗} surjects on \{u ∈ C/\overline{R}^{∗}, I ⊂ I_{u}\}, and by (7.49), (7.50), the fibre has \frac{d^{r}}{|⟨α_{i_{1}} ,... ,α_{ir} ⟩|} elements.

From (2.13), (2.119A), (7.45), (7.53)-(7.55), and proceeding as in the proof of Theorem 2.28, we get (7.52). The proof of our Theorem is completed.

7.3. The Verlinde sums. Recall that by Proposition 1.27, if λ ∈ \overline{CR}^{∗}, the character χ_{λ} of G does not depend on the choice of a Weyl chamber K such that λ ∈ \overline{K}. Also by Proposition 1.28, if w ∈ W

χ_{wλ} = χ_{λ} . (7.56)

Moreover the function (i^{ℓ}σ)^{2}(t) is well defined on T′ = 𝔱/\overline{R}^{∗}.

Now we introduce the Verlinde sums [58], [3].

Definition 7.11. For g ∈ N, q ∈ Z^{∗}, θ_{1}, …, θ_{s} ∈ \overline{CR}^{∗}, put

V_{g,q}(θ_{1}, …, θ_{s}) = |\frac{\overline{CR}^{∗}}{q\overline{CR}}|^{g−1} \frac{1}{|W |} ∑_{{λ∈\overline{CR}^{∗}/q\overline{CR} \atop σ^{2}(λ/q)≠0}} (7.57)

\frac{1}{(i^{ℓ}σ(λ/q))^{2g−2}} ∏_{j=1}^{s} χ_{θ_{j}}(e^{λ/q}) .

By Proposition 1.12, we get

V_{g,q}(θ_{1}, …, θ_{s}) = |\frac{\overline{CR}^{∗}}{q\overline{CR}}|^{g−1} ∑_{λ∈qP ∩\overline{CR}^{∗}} \frac{1}{(i^{ℓ}σ(λ/q))^{2g−2}} ∏_{j=1}^{s} χ_{θ_{j}}(e^{λ/q}) . (7.58)

Recall that in Section 1.8, it was shown that ∑_{j=1}^{s} θ_{j} ∈ \overline{R}, if and only if for (w^{1}, …, w^{s}) ∈ W^{s}, ∑_{j=1}^{s} w^{j}θ_{j} ∈ \overline{R}.

By Proposition 1.29, observe that if ∑_{j=1}^{s} θ_{j} ∈ \overline{R}, then ∏_{j=1}^{s} χ_{θ_{j}}(e^{λ/q}) is a well-defined function of λ ∈ \overline{CR}^{∗}/q\overline{R}^{∗}.

Theorem 7.12. If ∑_{j=1}^{s} θ_{j} ∉ \overline{R}, then

V_{g,q}(θ_{1}, …, θ_{s}) = 0 . (7.59)

If ∑_{j=1}^{s} θ_{j} ∈ \overline{R}, then

V_{g,q}(θ_{1}, …, θ_{s}) = |\frac{\overline{CR}^{∗}}{q\overline{CR}}|^{g−1} \frac{|Z(G)|}{|W |} ∑_{λ∈\overline{CR}^{∗}/q\overline{R}^{∗}} (7.60)

\frac{1}{(i^{ℓ}σ(λ/q))^{2g−2}} ∏_{j=1}^{s} χ_{θ_{j}}(e^{λ/q}) .

Proof. Clearly

V_{g,q}(θ_{1}, …, θ_{s}) = |\frac{\overline{CR}^{∗}}{q\overline{CR}}|^{g−1} \frac{1}{|W |} ∑_{{λ∈\overline{CR}/q\overline{R}^{∗} \atop σ^{2}(λ/q)≠0}} \frac{1}{(i^{ℓ}σ(λ/q))^{2g−2}} (7.61)

∑_{μ∈\overline{R}^{∗}/\overline{CR}} ∏_{j=1}^{s} χ_{θ_{j}}(e^{λ/q+μ}) .

Using Proposition 1.29 and 1.30, (7.61), and the fact that Z(G) = \overline{R}^{∗}/\overline{CR}, we get our Theorem. Theorem 7.13. The following identity holds

V_{g,c}(θ_{1}, …, θ_{s}) = ∏_{j=1}^{s} χ_{θ_{j}}(e^{ρ/c}) . (7.62)

If one of the θ_{j}’s does not lie in \overline{R}, V_{g,c}(θ_{1}, …, θ_{s}) = 0 . (7.63) More generally V_{g,c}(θ_{1}, …, θ_{s}) = 0 , +1 or − 1 . (7.64) Proof. By Proposition 1.10 and 1.11, we get (7.62). By Proposition 1.31 or by Theorem 7.12 when s = 1 and by (7.62), we get (7.63). By (7.62) and by Theorem 1.32, (7.64) follows. The proof of our Theorem is completed.

Let K be a Weyl chamber. Definition 7.14. If θ ∈ \overline{CR}^{∗}, put

\overline{χ}_{θ}^{K}(t) = \frac{1}{σ(t)} ∑_{w∈W} ε_{w}e^{2iπ⟨w(ρ+θ),t⟩} . (7.65)

By (3.115) , if ρ + θ does not lie in a Weyl chamber, \overline{χ}_{θ}^{K}(t) = 0 . (7.66) If ρ + θ lies in the Weyl chamber w_{0}K, then \overline{χ}_{θ}^{K}(t) = χ_{ρ+θ−w_{0}ρ}(t) . (7.67) Also if w ∈ W, \overline{χ}_{wθ}^{wK}(t) = \overline{χ}_{θ}^{K}(t) . (7.68)

192 JEAN-MICHEL BISMUT AND FRANÇOIS LABOURIE

Definition 7.15. For g ∈ N, q ∈ Z^{∗}, θ_{1}, …, θ_{s}, θ_{s+1} ∈ \overline{CR}^{∗}, put

\overline{V} _{g,q}^{K}(θ_{1}, …, θ_{s}, θ_{s+1}) = |\frac{\overline{CR}^{∗}}{q\overline{CR}}|^{g−1} \frac{1}{|W |} ∑_{{λ∈\overline{CR}^{∗}/q\overline{CR} \atop σ^{2}(λ/q)≠0}} (7.69)

\frac{1}{(i^{ℓ}σ(λ/q))^{g−1}} ∏_{j=1}^{s} χ_{θ_{j}}(e^{λ/q})\overline{χ}_{θ_{s+1}}^{K}(e^{λ/q}) .

Theorem 7.16. For g ∈ N, q ∈ Z^{∗}, the following identity holds

V_{g,q}(θ_{1}, …, θ_{s}) = \frac{(−1)^{ℓ}}{|\frac{\overline{CR}^{∗}}{q\overline{CR}} |} ∑_{(v^{1},v^{2})∈W^{2}} ε_{v^{1}}ε_{v^{2}} (7.70)

\overline{V} _{g+1,q}^{K}(θ_{1}, …, θ_{s}, v^{1}ρ + v^{2}ρ) .

Proof. Recall that χ_{0} = 1 . (7.71) By (7.57), (7.71),

V_{g,q}(θ_{1}, …, θ_{s}) = |\frac{\overline{CR}^{∗}}{q\overline{CR}}|^{g−1} \frac{1}{|W |} ∑_{{λ∈\overline{CR}^{∗}/q\overline{CR} \atop σ^{2}(λ/q)≠0}} (7.72)

\frac{1}{(i^{ℓ}σ(λ/q))^{2g−2}} ∏_{j=1}^{s} χ_{θ_{j}}(e^{λ/q})χ_{0}^{3}(e^{λ/q}) .

By (7.56), we may and we will assume that θ_{1}, …, θ_{s} ∈ \overline{CR}_{+}^{∗} = \overline{CR}^{∗} ∩ \overline{K}.

Using (1.94), (7.72) we obtain

V_{g,q}(θ_{1}, …, θ_{s}) = \frac{|\frac{\overline{CR}^{∗}}{q\overline{CR}} |^{g−1}}{|W |} ∑_{{λ∈\overline{CR}^{∗}/q\overline{CR} \atop σ^{2}(λ/q)≠0}} (7.73)

\frac{1}{(i^{ℓ}σ(λ/q))^{2g−2}} \frac{1}{(σ(λ/q))^{s+3}} ∑_{(w^{1},... ,w^{s})∈W^{s+3}} ∏_{j=1}^{s+3} ε_{w^{j}}

exp (2iπ⟨∑_{j=1}^{s} w^{j}(ρ + θ_{j}) + ∑_{j=s+1}^{s+3} w^{j}ρ, λ/q⟩)

= (−1)^{ℓ} \frac{|\overline{CR}^{∗}/q\overline{CR}|^{g−1}}{|W |} ∑_{{λ∈\overline{CR}^{∗}/q\overline{CR} \atop σ^{2}(λ/q)≠0}} \frac{1}{(i^{ℓ}σ(λ/q))^{2g}} \frac{1}{(σ(λ/q))^{s+1}}

∑_{(v^{1},v^{2})∈W^{2}} ε_{v^{1}}ε_{v^{2}} ∑_{(w^{1},... ,w^{s+1})∈W^{s+1}} ∏_{j=1}^{s+1} ε_{w^{j}}

exp (2iπ⟨∑_{j=1}^{s} w^{j}(ρ + θ_{j}) + w^{s+1}(ρ + v^{1}ρ + v^{2}ρ), λ/q⟩) ,

which is just (7.70).

The proof of our Theorem is completed.

Remark 7.17. If w ∈ W, ρ + wρ ∈ \overline{R}. Therefore (7.59) and (7.70) are compatible. Also observe that the fusion rules [3], [58] express V_{g,q}(.) in terms of V_{g−1,q}(.). Here equation (7.70) goes the opposite way.

7.4. A residue formula for the Verlinde sums. In the sequel, we fix a Weyl chamber K ⊂ 𝔱, and we use the notation in Section 2.

Also by (7.56) and (7.59), we may and we will assume that

θ_{j} ∈ \overline{CR}_{+}^{∗} = \overline{CR}^{∗} ∩ \overline{K} , 1 ≤ j ≤ s, (7.74)

∑_{j=1}^{s} θ_{j} ∈ \overline{R} .

Definition 7.18. If w ∈ W, let σ_{w} ∈ S_{ℓ}, let ε_{w}(1), …, ε_{w}(ℓ) ∈ {−1, +1} be such that for 1 ≤ i ≤ ℓ,

w^{−1}α_{i} = ε_{w}(i)α_{σ_{w}(i)} . (7.75)

By [15, Corollary V.4.6 and Lemma V.4.10] , we know that

ε_{w} = ∏_{i=1}^{ℓ} ε_{w}(i) . (7.76)

Definition 7.19. For x ∈ C^{ℓ}, w ∈ W, put

φ_{w}(x) = ∏_{i=1}^{ℓ} \frac{1}{1 − e^{−ε_{w}(i)x_{i}}} , . (7.77)

Equivalently, by (7.76),

φ_{w}(x) = ε_{w} ∏_{i=1}^{ℓ} \frac{1}{2 sinh(\frac{x_{i}}{2} )}e^{\frac{1}{2} ∑_{1}^{ℓ} ε_{w}(i)x_{i}} . (7.78)

Theorem 7.20. For g ≥ 2, q ∈ Z^{∗}, the following identity holds

V_{g,q}(θ_{1}, …, θ_{s}) = Vol(T)^{2g−2}q^{(g−1)r} \frac{|Z(G)|}{|W |} (7.79)

(−1)^{ℓ(g−1)} Res_{x=0}^{(1,... ,ℓ)} \frac{1}{[∏_{i=1}^{ℓ} 2 sinh(\frac{x_{i}}{2q})]^{2g−2}} ∑_{(w^{1},... ,w^{s})∈W^{s}} ∏_{j=1}^{s} φ_{w^{j}}(x/q)

M_{q} (∑_{j=1}^{s} w^{j}θ_{j}, x) .

For g ≥ 0, for s even, 2g − 2 + s ≥ 1 and q ∈ Z^{∗}, the following identity holds

V_{g,q}(θ_{1}, …, θ_{s}) = Vol(T)^{2g−2}q^{(g−1)r} \frac{|Z(G)|}{|W |} (7.80)

(−1)^{ℓ(g−1)}Res_{x=0}^{(1,... ,ℓ)} \frac{1}{[∏_{i=1}^{ℓ} 2 sinh(\frac{x_{i}}{2q})]^{2g−2+s}} ∑_{(w^{1},... ,w^{s})∈W^{s}} ∏_{j=1}^{s} ϵ_{w^{j}}

M_{q}(∑_{j=1}^{s} w^{j}(ρ + θ_{j}), x).

Figure 7.2

Proof. First we assume that g ≥ 2. For λ ∈ \overline{CR}^{∗}/q\overline{R}^{∗}, w = (w^{1}, …, w^{s}) ∈ W^{s}, x ∈ (C\2iπZ/m)^{ℓ}, put

U_{λ,w}(x) = \frac{1}{[∏_{i=1}^{ℓ} 2 sinh(\frac{x_{i}}{2q})]^{2g−2}} ∏_{j=1}^{s} φ_{w^{j}}(x/q) (7.81)

∏_{i=1}^{ℓ} \frac{1}{2q tanh (\frac{2iπ⟨λ,α_{i}⟩−x_{i}}{2q} )} .

Then U_{λ,w}(x) is a meromorphic function of x_{1}, …, x_{ℓ}, which is periodic of period 2iπq, …, 2iπq.

For ε > 0, let Δ_{ε} be the domain in C given in Figure 7.2. For ε > 0 small enough, as a function of x_{i} ∈ Δ_{ε}, the function U_{λ,w}(x) has poles at 0, and also at 2iπq [\frac{⟨λ,α_{i}⟩}{q}]. Here q [\frac{⟨λ,α_{i}⟩}{q}] is the real number in [0, q[ which represents ⟨λ, α_{i}⟩ mod q. For ε > 0 small enough, the poles do not meet the boundary ∂Δ_{ε} = δ_{−}∪δ_{+}.

Assume first that [\frac{⟨λ,α_{i}⟩}{q}] ≠ 0. Then 2iπq [\frac{⟨λ,α_{i}⟩}{q}] is a simple pole of U_{λ,w}(x). We now use the theorem of residues on Δ_{ε}. Since g ≥ 2, as x_{i} → ±∞ inside Δ_{ε}, the function U_{λ,w}(x) tends to 0. Also since U_{λ,w}(x) is periodic of period 2iπq in the variable x_{i}, the boundary integrals cancel each other. Therefore

Res_{x_{i}=0}U_{λ,w}(x) + Res_{x_{i}=2iπq[ \frac{⟨λ,α_{i}⟩}{q} ]}U_{λ,w}(x) = 0 . (7.82)

Clearly

Res_{x_{i}=2iπq[ \frac{⟨λ,α_{i}⟩}{q} ]}U_{λ,w}(x) = (7.83)

− [\frac{1}{2i sin (\frac{⟨λ,α_{i}⟩π}{q} )} ∏_{j≠i} \frac{1}{2 sinh (\frac{x_{j}}{2q} )}]^{2g−2}

∏_{j=1}^{s} φ_{w^{j}} (\frac{x_{1}}{q} … \frac{x_{i−1}}{q} , \frac{2iπ⟨λ, α_{i}⟩}{q}, …, \frac{x_{ℓ}}{q} ) ∏_{j≠i} \frac{1}{2q tanh (\frac{2iπ⟨λ,α_{i}⟩−x_{i}}{2q} )} .

Assume now that [\frac{⟨λ,α_{i}⟩}{q}] = 0. Then as a function of x_{i}, the function U_{λ,w}(x) has a single pole at x_{i} = 0. By the theorem of residues used as before, we get

Res_{x_{i}=0}U_{λ,w}(x) = 0 . (7.84)

By (7.83), (7.84), we obtain

(−1)^{ℓ(g−1)}Res_{x=0}^{1,... ,ℓ} \frac{1}{[∏_{1}^{ℓ} 2 sinh (\frac{x_{i}}{2q} )]^{2g−2}} ∑_{(w^{1},... ,w^{s})∈W^{s}} φ_{w^{j}}(\frac{x}{q}) (7.85)

M_{q} (∑_{j=1}^{s} w^{j}θ_{j}, x) = ∑_{{λ∈\overline{CR}^{∗}/q\overline{R}^{∗},σ^{2}(λ/q)≠0 \atop (w^{1},... ,w^{s})∈W^{s}}} \frac{1}{(i^{ℓ}σ(λ/q))^{2g−2}}

∏_{j=1}^{s} φ_{w^{j}} (2iπ \frac{⟨λ, α_{1}⟩}{q}, …, 2iπ \frac{⟨λ, α_{ℓ}⟩}{q}) exp (2iπ⟨w^{j}θ_{j}, λ/q⟩) .

Now by (1.94), (7.75), (7.78), for 1 ≤ j ≤ s,

∑_{w^{j}∈W} φ_{w^{j}} (2iπ \frac{⟨λ, α_{i}⟩}{q}, …, 2iπ \frac{⟨λ, α_{1}⟩}{q}) exp (2iπ⟨w^{j}θ_{j}, λ/q⟩) (7.86)

= χ_{θ_{j}}(e^{λ/q}) .

From (1.17), (7.85), (7.86), we get (7.79).

Now we consider the case where g ≥ 0, s is even and 2g − 2 + s ≥ 1. Put

(7.87) H_{λ,w}(x) = \frac{1}{[∏_{i=1}^{ℓ} 2 sinh(\frac{x_{i}}{2q})]^{2g−2+s}} ∏_{i=1}^{ℓ} \frac{1}{2q tanh (\frac{2iπ⟨λ,α_{i}⟩−x_{i}}{2q} )} .

Then H_{λ,w}(x) is a meromorphic function of x_{1}, …, x_{ℓ}. Since s is even, it is periodic of period 2iπq, …, 2iπq in x_{1}, …, x_{ℓ}.

Assume that [\frac{⟨λ,α_{i}⟩}{q}] ≠ 0. Since 2g − 2 + s ≥ 1, as x_{i} tends to ±∞ inside Δ_{ϵ}, H_{λ,w}(x) tends to 0. We thus find that the analogue of (7.82) holds, with U_{λ,w} replaced by H_{λ,w}. The analogue of (7.83) is now

Res_{x_{i}=2iπq[ \frac{⟨λ,α_{i}⟩}{q} ]}H_{λ,w}(x) = − [_{\frac{1}{2i sin( \frac{⟨λ,α_{i}⟩π}{q} )}} ∏_{j≠i} _{\frac{1}{2 sinh(\frac{x_{j}}{2q})}}]^{2g−2+s}

∏_{j≠i} _{\frac{1}{2q tanh( \frac{2iπ⟨λ,α_{i}⟩−x_{i}}{2q} )}} . (7.88)

Also the analogue of (7.84) still holds. So we get the analogue of (7.85),

(−1)^{ℓ(g−1)}Res_{x=0}^{1,... ,ℓ} \frac{1}{[∏_{1}^{ℓ} 2 sinh (\frac{x_{i}}{2q} )]^{2g−2+s}} ∑_{(w^{1},... ,w^{s})∈W^{s}} ϵ_{w^{1}} … ϵ_{w^{s}} (7.89)

M_{q} (∑_{j=1}^{s} w^{j}(ρ + θ_{j}), x) = ∑_{{λ∈\overline{CR}^{∗}/q\overline{R}^{∗},σ^{2}(λ/q)≠0 \atop (w^{1},... ,w^{s})∈W^{s}}} \frac{1}{(i^{ℓ}σ(λ/q))^{2g−2}} \frac{1}{(σ(λ/q))^{s}}

∏_{j=1}^{s} ϵ_{w^{j}} exp (2iπ⟨w^{j}(ρ + θ_{j}), λ/q⟩) .

Now by (1.94), for 1 ≤ j ≤ s,

\frac{1}{σ(λ/q)} ∑_{w^{j}∈W} ϵ_{w^{j}} exp(2iπ⟨w^{j}(ρ + θ_{j}), λ/q⟩ = χ_{θ_{j}}(e^{λ/q}). (7.90)

From (??), (7.89), (7.90), we get (7.80).

The proof of our Theorem is completed.

Remark 7.21. It is crucial that in (7.80), s is even. In fact under the given conditions on the θ_{j}’s, ∑_{j=1}^{s} w^{j}θ_{j} ∈ \overline{R}, but we know that ∑_{j=1}^{s} w^{j}ρ ∈ \overline{R} only if s is even.

7.5. The generic case with g ≥ 2.

Definition 7.22. We will say that (θ_{1}, …, θ_{s}) verify assumption (\overline{A}_{q}) if for any (w^{1}, …, w^{s}) ∈ W^{s}, \frac{1}{q} ∑_{j=1}^{s} w^{j}θ_{j} ∉ \overline{S}.

In the sequel, if u ∈ C/\overline{R}^{∗}, we choose once and for all a representative of u in C, which we still denote by u.

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS 197

Theorem 7.23. If q ∈ Z^{∗}, g ≥ 2, if (θ_{1}, …, θ_{s}) verify (\overline{A}_{q}), then

V_{g,q}(θ_{1}, …, θ_{s}) = |\frac{\overline{R}}{\overline{CR}}| Vol(T)^{2g−2} \frac{|Z(G)|}{|W |}q^{(g−1)r}(−1)^{ℓ(g−1)+r}

∑_{u∈C/\overline{R}^{∗}} ∑_{{I=(i_{1},... ,ir)∈Iu \atop f∈\overline{CR}/d\overline{R}}} \frac{1}{⟨α_{i_{1}}, …, α_{i_{r}}⟩}
Res_{x=0}^{I} [ ∏_{α∈R_{+}} \frac{1}{2 sinh (\frac{1}{2}⟨α, x^{I}/q + 2iπu⟩)}]^{2g−2+s}

∑_{(w^{1},... ,w^{s})∈W^{s}} ε_{w^{1}} … ε_{w^{s}} exp (⟨∑_{j=1}^{s} w^{j}ρ, x^{I}/q⟩ (7.91)

+2iπ⟨∑_{j=1}^{s} w^{j}(ρ + θ^{j}) + qf, u⟩)

exp (d ∑_{j=1}^{r} \frac{⟨p_{j−1}^{I}α_{i_{j}}, x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩}[\frac{1}{d}⟨p_{j−1}^{I}( ∑_{k=1}^{s} w^{k}θ_{k}/q + f), e^{j}⟩])

\frac{1}{∏_{j=1}^{r} ( exp (\frac{d⟨p_{j−1}^{I}α_{i_{j}} ,x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}} ,e^{j}⟩}) − 1)}.

Proof. Take t ∈ \overline{R}. By Theorem 7.10, if t ∈ \overline{R}, t/q ∉ \overline{S}, then

Res_{x=0}^{(1,... ,ℓ)} \frac{1}{[ ∏_{i=1}^{ℓ} 2 sinh (\frac{x_{i}}{2q} )]^{2g−2}} ∏_{j=1}^{s} φ_{w^{j}}(\frac{x}{q})M_{q}(t, x)

= |\frac{\overline{R}}{\overline{CR}}| (−1)^{r} ∑_{u∈C/\overline{R}^{∗}} ∑_{{I=(i_{1},... ,ir)⊂Iu,I generic \atop f∈\overline{CR}/d\overline{R}}}

Res_{x=0}^{(1,... ,ℓ)}\{\frac{1}{[ ∏_{i=1}^{ℓ} 2 sinh (\frac{x_{i}}{2q} )]^{2g−2}} ∏_{j=1}^{s} φ_{w^{j}}(x/q) (7.92)

\frac{1}{∏_{i∈^{c}I} 2q tanh (\frac{⟨α_{i},x^{I}+2iπqu⟩−x_{i}}{2q} )}

\frac{1}{⟨α_{i_{1}}, …, α_{i_{r}}⟩} exp(2iπ⟨u, t + qf⟩)

exp (d ∑_{j=1}^{r} \frac{⟨p_{j−1}^{I}α_{i_{j}}, x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩}[\frac{1}{d}⟨p_{j−1}^{I}(t + qf), \frac{e^{j}}{q} ⟩])

\frac{1}{∏_{j=1}^{r} ( exp (\frac{d⟨p_{j−1}^{I}α_{i_{j}} ,x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}} ,e^{j}⟩}) − 1)}.

Observe that in the right-hand side of (7.92), the expressions starting after exp(2iπ⟨u, t+ qf⟩) does not depend on the x_{i}, i ∉ I.

Now we explain how to evaluate the residue in (7.92). We use the notation in Definition 2.47.

1. The case i ∉ I, i ∉ I_{u} or i ∉ I, i ∈ I_{u}, α_{i} ∉ {α_{σ^{I}(1)}, …, α_{σ^{I}(p_{i})}}.

Note that the above condition just says that i ∉ I, and ⟨α_{i}, u⟩ ∈ Z, α_{i} ∉ {α_{σ^{I}(1)}, …, α_{σ^{I}(p_{i})}}, or ⟨α_{i}, u⟩ ∉ Z. If ⟨α_{i}, u⟩ ∈ Z, α_{i} ∉ {α_{σ^{I}(1)}, …, α_{σ^{I}(p_{i})}}, for at least one k ∈ I, k > i, then ⟨α_{i}, α^{k}⟩ ≠ 0. So since the {x_{k}}_{k>i} have not yet been made “small”, for generic x, ⟨α_{i}, x^{I}⟩ is “large”. So we find that under the assumptions we made in 1., for generic x, mod (2iπqZ), ⟨α_{i}, x^{I} + 2iπqu⟩ should be considered as different of 0. Then by proceeding as in (7.82) and using the fact that g ≥ 2, we get

Res_{x_{i}=0}[\frac{1}{[2 sinh (\frac{x_{i}}{2q} )]^{2g−2+s}} e^{\frac{1}{2q} ∑_{j=1}^{s} ε_{w^{j}} (i)x_{i}} (7.93)

\frac{1}{2q tanh (\frac{⟨α_{i},x^{I}+2iπqu⟩−x_{i}}{2q} )}] =

[\frac{1}{2 sinh (\frac{⟨α_{i},x^{I}+2iπqu⟩}{2q})}]^{2g−2+s}e^{\frac{1}{2q} ⟨∑_{j=1}^{s} ε_{w^{j}} (i)α_{i},x^{I}+2iπqu⟩} .

2. The case where i ∉ I, i ∈ I_{u} and α_{i} ∈ {α_{σ^{I}(1)}, …, α_{σ^{I}(p_{i})}}.

In this case ⟨α_{i}, u⟩ ∈ Z, and so

(7.94) \frac{1}{2q tanh (\frac{⟨α_{i},x^{I}+2iπqu⟩−x_{i}}{2q} )} = \frac{1}{2q tanh (\frac{⟨α_{i},x^{I}⟩−x_{i}}{2q} )}

= −\frac{1}{2q} \frac{1 − tanh (\frac{x_{i}}{2q} ) tanh (\frac{⟨α_{i},x^{I}⟩}{2q})}{tanh (\frac{x_{i}}{2q} )(1 − \begin{matrix}tanh & ( & \frac{⟨α_{i},x^{I}⟩}{2q} & ) \\ tanh & ( & \frac{x_{i}}{2q} & ) \end{matrix})} .

Now since α_{i} ∈ {α_{σ^{I}(1)}, …, α_{σ^{I}(p_{i})}}, ⟨α_{i}, x^{I}⟩ is a linear combination of the x_{σ^{I}(k)}, k ≤ p_{i}, which have been made “small”. Therefore we get the expression

(7.95) \frac{1}{1 − \frac{tanh (\frac{⟨α_{i},x^{I}⟩}{2q})}{tanh(x_{i}/2q)}} = 1 + \frac{tanh (\frac{⟨α_{i},x^{I}⟩}{2q})}{tanh(x_{i}/2q)} + \frac{tanh^{2} (\frac{⟨α_{i},x^{I}⟩}{2q})}{tanh^{2}(x_{i}/2q)} + ….

In view of (7.92), (7.94), (7.95), for k ≥ 1, we should evaluate

(7.96) Res_{x_{i=0}}[\frac{1}{[2 sinh(x_{i}/2q)]^{2g−2+s}} e^{\frac{1}{2q} ∑_{j=1}^{s} ε_{w^{j}} (i)x_{i}} \frac{1}{tanh^{k}(x_{i}/2q)}] .

The function of x_{i} which appears in (7.96) is periodic of period 2iπq. Recall that the contour Δ_{ε} was defined in the proof of Theorem 7.20. For ε > 0, 0 is the only pole inside Δ_{ε}. By using the theorem of residues as in (7.84), and the fact that g ≥ 2, we find that (7.96) vanishes.

From (7.94)-(7.96), we see that if i ∉ I, i ∈ I_{u}, α_{i} ∈ {α_{σ^{I}(1)}, …, α_{σ^{I}(p_{i})}}, then

Res_{x_{i=0}}[\frac{1}{[2 sinh(x_{i}/2q)]^{2g−2+s}} e^{\frac{1}{2q} ∑_{j=1}^{s} ε_{w^{j}} (i)x_{i}} (7.97)

\frac{1}{2q tanh (\frac{⟨α_{i},x^{I}+2iπqu⟩−x_{i}}{2q} )}] = 0 .

Using (7.79) in Theorem 7.20 and (7.92), (7.93), (7.97), we get

V_{g,q}(θ_{1}, …, θ_{s}) = |\frac{\overline{R}}{\overline{CR}}| Vol(T)^{2g−2} \frac{|Z(G)|}{|W |}q^{(g−1)r}(−1)^{ℓ(g−1)+r} (7.98)

∑_{u∈C/\overline{R}^{∗}} ∑_{{I=(i_{1},... ,ir)∈Iu \atop f∈\overline{CR}/d\overline{R}}} \frac{1}{⟨α_{i_{1}}, …, α_{i_{r}}⟩}

Res_{x=0}^{I}\{[∏_{i∈I} \frac{1}{2 sinh(\frac{x_{i}}{2q})} ∏_{i∉I} \frac{1}{2 sinh (\frac{⟨α_{i},x^{I}+2iπqu⟩}{2q})}]^{2g−2+s}

∑_{(w^{1},... ,w^{s})∈W^{s}} ε_{w^{1}} … ε_{w^{s}} exp(\frac{1}{2} ∑_{j=1}^{s}(⟨∑_{i∉I} ε_{w^{j}}(i)α_{i}, \frac{x^{I}}{q} ⟩ + \frac{∑_{i∈I} ε_{w^{j}}(i)x_{i}}{q} ))

exp (2iπ⟨u, ∑_{j=1}^{s}(\frac{∑_{i∉I} ε_{w^{j}}(i)α_{i}}{2} + w^{j}θ_{j}) + qf⟩)

exp(d ∑_{j=1}^{r} \frac{⟨p_{j−1}^{I}α_{i_{j}}, x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩}[\frac{1}{d}⟨p_{j−1}^{I}( ∑_{k=1}^{s} w^{k}θ_{k} + qf), e^{j}/q⟩])

\frac{1}{∏_{j=1}^{r} (exp (\frac{d⟨p_{j−1}^{I}α_{i_{j}} ,x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}} ,e^{j}⟩}) − 1)}\}.

Now observe that if i ∈ I, ⟨α_{i}, x^{I}⟩ = x_{i} . (7.99) Therefore

∑_{i∉I}⟨ε_{w^{j}}(i)α_{i}, x^{I}⟩ + ∑_{i∈I} ε_{w^{j}}(i)x_{i} = (7.100)

⟨∑_{i=1}^{ℓ} ε_{w^{j}}(i)α_{i}, x^{I}⟩ = ⟨w^{j} ∑_{i=1}^{ℓ} α_{i}, x^{I}⟩ = 2⟨w^{j}ρ, x^{I}⟩ .

Also if i ∈ I ⊂ I_{u}, then ⟨α_{i}, u⟩ ∈ Z, and so

[\frac{1}{2 sinh (\frac{⟨α_{i},x^{I}+2iπqu⟩}{2q})}]^{2g−2} = [\frac{1}{2 sin(\frac{x_{i}}{2q})}]^{2g−2} , (7.101)

\frac{1}{2 sinh (\frac{⟨α_{i},x^{I}+2iπqu⟩}{2q} + iπ)} exp (\frac{2iπ⟨u, ε_{w^{j}}(i)α_{i}⟩}{2}) = \frac{1}{2 sinh (\frac{x_{i}}{2q} )} .

From (7.98)-(7.101), we get (7.91). The proof of our Theorem is completed.

Remark 7.24. Clearly,

[\frac{1}{2 sin (\frac{⟨α_{i},x^{I}/q+2iπu⟩}{2})}]^{2g−2} (7.102)

does not depend on the representative u in C. Also

∏_{α∈R_{+}} \frac{1}{2 sinh (⟨α_{i}, x^{I}/q + 2iπu⟩)}ε_{w^{j}} exp (⟨w^{j}ρ, x^{I}/q + 2iπu⟩) = (7.103)

∏_{α∈R_{+}} \frac{1}{1 − exp ( − ⟨w^{j}α, x^{I}/q + 2iπu⟩)}

does not depend either on the representative u ∈ C. This explains why the terms in the right-hand side of (7.91) do not depend on the choice of the representatives of u.

7.6. The non generic case for g ≥ 2. Recall that the functions [x]_{+}, [x]_{−} were defined in (6.140).

Let η_{1}, …, η_{r} be r functions from \overline{R}/d\overline{R} × \{I, Igeneric\} into \{+, −\} such that if

f = ∑_{1}^{r} f^{j}e_{j} ∈ \overline{R}/d\overline{R}, I = (i_{1}, …, i_{r}), then η_{j}(f, I) depends only on f^{1}, …, f^{j}, i_{1}, …, i_{j−1}.

First, we will extend Theorem 7.2 to the case where t ∈ \overline{R}/q\overline{R} is arbitrary.

Theorem 7.25. For any t ∈ \overline{R}/q\overline{R}, there are meromorphic functions φ_{1}(t, x_{2}, …, x_{ℓ}), …, φ_{i}(t, x_{1}, …, x_{i−1}, x_{i+1}, …, x_{ℓ}) … which vanish identically when t/q ∉ H, such that

(7.104)  L_{q}(t, x) = \frac{(−1)^{r}}{d^{r}} ∑_{I=(i_{1},... ,ir)⊂\{I,... ,ℓ\},I generic f∈\overline{R}/d\overline{R},g∈(Z/dZ)^{r}} sgn (⟨α_{i_{1}}, …, α_{i_{r}}⟩)

[exp(d ∑_{j=1}^{r} \frac{⟨p_{j−1}^{I}α_{i_{j}}, x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩}[\frac{1}{d}⟨p_{j−1}^{I}(t + qf), \frac{e^{j}}{q} ⟩]_{η_{j}(f,I)})

\frac{1}{∏_{i∈^{c}I} 2q tanh (\frac{⟨α_{i},x^{I}⟩−x_{i}}{2q} )}](x + 2iπqg_{I}) ∏_{j=1}^{r} \frac{1}{exp (\frac{d⟨p_{j−1}^{I}α_{i_{j}} ,x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}} ,e^{j}⟩}) − 1}

+ ∑_{i=1}^{ℓ} φ_{i}(t, x_{1}, …, x_{i−1}, x_{i+1}, …, x_{ℓ}) .

Proof. We use the notation in the proof of Theorem 7.2. Let η_{1}, …, η_{j−1} ∈ \{+, −\}. We define S_{I}(a, x) as in (7.7), except that [\frac{1}{d}⟨p_{i_{n−1}}^{I}t, \frac{e^{n}}{q} ⟩] is replaced

by [\frac{1}{d}⟨p_{i_{n−1}}^{I}t, \frac{e^{n}}{q} ⟩]_{η_{n}}, 1 ≤ n ≤ j − 1.

Figure 7.3

Take η_{j}(f) : {0, …, d − 1} ↦ {+, −}. Then we rewrite (7.20) in the form

L_{I} = \frac{1}{d^{2}} ∑_{k∈Z/d^{2}qZ}\{ ∑_{{0≤f<d \atop g∈(Z/dZ)^{j−1}}} (7.105)

Res_{a=2iπk}[exp(a[\frac{⟨p_{j−1}^{I}(t + qfe_{j}), e^{j}/q⟩}{d}]_{η_{j}(f)}

+d ∑_{n=1}^{j−1} \frac{p_{n−1}^{I}α_{i_{n}}, x^{I}⟩}{⟨p_{n−1}^{I}α_{i_{n}}, e^{n}⟩}[\frac{1}{d}⟨p_{n−1}^{I}t, e^{n}/q⟩]_{η_{n}})

\frac{1}{∏_{i∈^{c}I} 2q tanh (\frac{⟨p_{j−1}^{I}α_{i},ae^{j}/d+2iπk̂−x^{I}⟩}{2q})} \frac{1}{e^{a} − 1}]\}(x + 2iπqg_{I}) .

Instead of (7.21), we now set

h_{f,g}(a) = \frac{exp (a [^{\frac{⟨p_{j−1}^{I}(t+qfe_{j}), \frac{e^{j}}{q} ⟩}{d}}]_{η_{j}(f)})}{∏_{i∈^{c}I} 2q tanh (^{\frac{⟨p_{j−1}^{I}α_{i}, \frac{ae^{j}}{d} +2iπk̂−x^{I}⟩}{2q}})} \frac{1}{e^{a} − 1} . (7.106)

The key point is that we do no longer assume that (7.23) holds. Equivalently

\frac{⟨p_{j−1}^{I}(t+qfe_{j}),e^{j}/q⟩}{d} may now well be an integer.

For n ∈ N, we now replace the contour in Figure 7.1 by the contour of Figure 7.3. We apply the theorem of residues to h_{f,g}(a) on this contour, and let n → +∞.

Clearly, if \frac{⟨p_{j−1}^{I}(t+q/e_{j}),e^{j}/q⟩}{d} is an integer, if η_{j}(f) = ±, then

lim_{a→±∞} h_{f,g}(a) = 0 , (7.107)

lim_{a→∓∞} h_{f,g}(a) = ∓ \frac{1}{∏_{{i∈^{c}I \atop ⟨p{I \atop j−1}α_{i},e^{j} ⟩=0}} tanh (\frac{⟨p_{j−1}^{I}α_{i}, 2iπk̂ − x^{I}⟩}{2q})}

\frac{1}{(2q)^{|^{c}I|}} (−1)^{|\{i∈^{c}I,⟨p_{j−1}^{I}α_{i},e^{j}⟩≠0, sgn⟨p_{j−1}^{I}α_{i},e^{j}⟩=±1\}|} .

Observe here that by (2.65), {i ∈ ^{c}I; ⟨p_{j−1}^{I}α_{i}, e^{j}⟩ ≠ 0} is non empty. Then by (2.63), the right-hand side of the second equation in (7.107) does not depend on {x_{i}}_{i∈^{c}I ,⟨p_{j−1}^{I}α_{i},e^{j}⟩≠0}.

So by proceeding as in (7.21)-(7.25), we find that

• if \frac{⟨p_{j−1}^{I}(t+qfe_{j}),e^{j}/q⟩}{d} is not an integer, (7.26) still holds, with [ ] replaced by [ ]_{η_{1}}, …, [ ]_{η_{j−1}}, [ ]_{η_{j}(f)} .

• if \frac{⟨p_{j−1}^{I}(t+qfe_{j}),e^{j}/q⟩}{d} is an integer, writing η_{j} instead of η_{j}(f), then (7.26) is replaced by

L_{I} = −\frac{1}{d} ∑_{{0≤f<d ,g∈(Z/dZ)^{j−1} \atop i_{j}:⟨α_{i_{1}} ,... ,α_{i_{j}} ⟩≠0 ,0≤g^{j} <d|⟨p{I \atop j−1}α_{i_{j}} ,e^{j} ⟩|}} (7.108)

\{\frac{1}{⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩} exp(2iπ⟨p_{j}^{(I,i_{j})}e_{j}, k̂⟩⟨p_{j−1}^{I}(t + qfe_{j}), e^{j}/q⟩

+d ∑_{n=1}^{j} \frac{⟨p_{n−1}^{I}α_{i_{n}}, x^{I}⟩}{⟨p_{n−1}^{I}α_{i_{n}}, e^{n}⟩}[\frac{1}{d}⟨p_{n−1}^{I}(t + qfe_{j}), e^{n}/q⟩]_{η_{n}})

\frac{1}{∏_{i∉(I,i_{j})} 2q tanh (^{\frac{⟨p_{j}^{(I,i_{j})}α_{i},2iπk̂−x^{I}⟩}{2q}})}\} (x + 2iπq(g, g_{j})_{I,i_{j}})

\frac{1}{exp (\frac{d⟨p_{j−1}^{I}α_{i_{j}} ,x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}} ,e^{j}⟩}) − 1} + q ∑_{{0≤f<d \atop g∈(Z/dZ)^{g−1}}}

[exp (d ∑_{n=1}^{j−1} \frac{⟨p_{n−1}^{I}α_{i_{n}}, x^{I}⟩}{⟨p_{n−1}^{I}α_{i}, e^{n}⟩}[\frac{1}{d}⟨p_{n−1}^{I}t, e^{n}/q⟩]_{η_{n}})

\frac{1}{∏_{{i∈^{c}I \atop ⟨p{I \atop j−1}α_{i},e^{j} ⟩=0}} tanh (\frac{⟨p_{j−1}^{I}α_{i}, 2iπk̂ − x^{I}⟩}{2q})}](x + 2iπqg_{I})\frac{1}{(2q)^{|^{c}I|}}

(−1)^{|{i∈^{c}I,⟨p_{j−1}^{I}α_{i},e^{j}⟩≠0, sgn⟨p_{j−1}^{I}α_{i},e^{j}⟩=η_{j}(f)}|} ,

the key point being that the last sum in (7.108) is a sum of functions which do not depend on the x_{i}, i ∈ ^{c}I , ⟨p_{j−1}^{I}α_{i}, e_{j}⟩ ≠ 0).

Using (7.108) and proceeding by recursion as in the proof of Theorem 7.2, we get (7.104). The proof of our Theorem is completed.

Remark 7.26. The last term is the right-hand side of (7.108) depends in a non trivial way on the choice of η_{j}. More precisely, it is a sum of terms depending on η_{j} up to sign. As a corollary, we find that the function ∑_{1}^{ℓ} φ_{i}(t, x_{1}, …, x_{i−1}, x_{i+1}, …, x_{ℓ}) also depends on the choice of η_{1}, …, η_{r} in a non trivial way.

Now we extend Theorem 7.6 to the general case. We still take η_{1}, …, η_{r} as in Theorem (7.25.

Theorem 7.27. For any t ∈ \overline{R}/q\overline{R}, there are meromorphic functions ψ_{1}(t, x_{2}, …, x_{ℓ}), …, ψ_{i}(t, x_{1}, …, x_{i−1}, x_{i+1}, …, x_{ℓ}) which vanish identically when t/q ∉ \overline{S}, such that

M_{q}(t, x) = (7.109)

\frac{(−1)^{r}}{d^{r}} ∑_{I=(i_{1},... ,ir)⊂\{1,... ,ℓ\},I generic f∈\overline{R}/d\overline{R},g∈(Z/dZ)^{r} ,μ∈\overline{CR}^{∗}/\overline{R}^{∗}} sgn(⟨α_{i_{1}}, …, α_{i_{r}}⟩) exp(2iπ⟨λ_{μ}, t/q⟩)

[exp(d ∑_{j=1}^{r} \frac{⟨p_{j−1}^{I}α_{i_{j}}, x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩} [\frac{1}{d}⟨p_{j−1}^{I}(t + qf), e^{j}/q⟩]_{η_{j}(f,I)}]

\frac{1}{∏_{i∈^{c}I} 2q tanh (\frac{⟨α_{i},x^{I}⟩−x_{i}}{2q} )}(x + 2iπqg_{I} − 2iπãλ_{μ})

∏_{j=1}^{r} \frac{1}{exp (\frac{d⟨p_{j−1}^{I}α_{i_{j}} ,x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}} ,e^{j}⟩}) − 1}(x − 2iπqãλ_{μ}) +

∑_{i=1}^{ℓ} ψ_{i}(t, x_{1}, …, x_{i−1}, x_{i+1}, …, x_{ℓ}) .

Proof. Using Proposition 7.5 and Theorem 7.25, we get (7.109).

Given (w^{1}, …, w^{s}) ∈ W^{s}, let η_{1}^{(w^{1},... ,w^{s})}, …, η_{s}^{w^{1},... ,w^{s})} be s functions from \overline{R}/d\overline{R}×\{I, I generic\} into \{+, −\} having the properties listed before Theorem 7.25.

Theorem 7.28. If q ∈ Z^{∗}, g ≥ 2, then

V_{g,q}(θ_{1}, …, θ_{s}) = |\frac{\overline{R}}{\overline{CR}}| Vol(T)^{2g−2} \frac{|Z(G)|}{|W |}q^{(g−1)r}(−1)^{ℓ(g+1)+r} (7.110)

∑_{u∈C/\overline{R}^{∗}} ∑_{{I=(i_{1},... ,ir)∈Iu \atop f∈\overline{CR}/d\overline{R}}} \frac{1}{⟨α_{i_{1}}, …, α_{i_{r}}⟩}

Res_{x=0}^{I} [ ∏_{α∈R_{+}} \frac{1}{2 sinh (\frac{1}{2}⟨α_{i}, x^{I}/q + 2iπu⟩)}]^{2g−2+s}

∑_{(w^{1},... ,w^{s})∈W^{s}} ε_{w^{1}} … ε_{w^{s}} exp (⟨∑_{j=1}^{s} w^{j}ρ, x^{I}/q⟩

+2iπ⟨∑_{j=1}^{s} w^{j}(ρ + θ^{j}) + qf, u⟩)

exp (d ∑_{j=1}^{r} \frac{⟨p_{j−1}^{I}α_{i_{j}}, x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩}[\frac{1}{d}⟨p_{j−1}^{I}( ∑_{k=1}^{s} \frac{w^{k}θ_{k}}{q} + f), e^{j}⟩]_{η_{j}^{(w^{1},... ,w^{s})}(f,I)})

\frac{1}{∏_{j=1}^{r} ( exp (\frac{d⟨p_{j−1}^{I}α_{i_{j}} ,x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}} ,e^{j}⟩}) − 1)}.

Proof. We use(7.79) in Theorem 7.20 and 7.27, and also the arguments in the proof of Theorem 7.23. We will show that for 1 ≤ i ≤ ℓ,

Res_{x=0}^{1,... ,ℓ} \frac{1}{[ ∏_{1}^{ℓ} 2 sinh (\frac{x_{i}}{2q} )]^{2g−2}} ∏_{j=1}^{s} φ_{w^{j}}(x/q)ψ_{i}(x_{1}, …, x_{i−1}, x_{i+1}, …, x_{ℓ}) = 0 (7.111)

from which (7.110) will follows. To establish (7.111), we only need to show that

Res_{x_{i}=0} \frac{1}{[2 sinh (\frac{x_{i}}{2q} )]^{2g−2}} ∏_{j=1}^{s} φ_{w^{j}}(x/q) = 0 . (7.112)

The proof of (7.112) is the same as the proof of (7.84).

We have thus established our Theorem.

7.7. The general case. We will now establish another form of Theorem 7.28.

Theorem 7.29. If g ≥ 0, if s is even and if 2g − 2 + s ≥ 1, the following identity holds

V_{g,q}(θ_{1}, …, θ_{s}) = |\frac{\overline{R}}{\overline{CR}}| Vol(T)^{2g−2} \frac{|Z(G)|}{|W |}q^{(g−1)r}(−1)^{ℓ(g+1)+r} (7.113)

∑_{u∈C/\overline{R}^{∗}} ∑_{{I=(i_{1},... ,ir)∈Iu \atop f∈\overline{CR}/d\overline{R}}} \frac{1}{⟨α_{i_{1}}, …, α_{i_{r}}⟩}

Res_{x=0}^{I} [ ∏_{α∈R_{+}} \frac{1}{2 sinh (\frac{1}{2}⟨α_{i}, x^{I}/q + 2iπu⟩)}]^{2g−2+s}

∑_{(w^{1},... ,w^{s})∈W^{s}} ε_{w^{1}} … ε_{w^{s}} exp(2iπ⟨∑_{j=1}^{s} w^{j}(ρ + θ^{j}) + qf, u⟩)

exp (d\frac{∑_{j=1}^{r}⟨p_{j−1}^{I}α_{i_{j}}, x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩}[\frac{1}{d}⟨p_{j−1}^{I}( ∑_{k=1}^{s} \frac{w^{k}(ρ + θ_{k})}{q} + f), e^{j}⟩]_{η_{j}^{(w^{1},... ,w^{s})}(f,I)})

\frac{1}{∏_{j=1}^{r} ( exp (\frac{d⟨p_{j−1}^{I}α_{i_{j}} ,x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}} ,e^{j}⟩}) − 1)}.

Proof. We proceed as in the proof of Theorems 7.23 and 7.28. Instead of (7.79) in Theorem 7.20, we use (7.80) and we still use Theorem 7.27. In particular the obvious analogues of (7.93) and (7.97) still hold because s is even and 2g−2+s ≥ 1. For the same reason, the analogue of (7.111), (7.112), with ϕ replaced by 1 still holds.

The proof of our Theorem is completed.

8. The Verlinde formulas

In this Section, we prove the Verlinde formulas, in the restricted sense which was described in the Introduction. Namely, we show essentially that for p large enough, the Riemann-Roch number of M/G is given by the Verlinde formulas. Also we show that the Riemann-Roch numbers of suitable perturbations of M/G are given by the Verlinde formulas.

This Section is organized as follows. In Section 8.1, we establish our results under a suitable genericity assumption on the holonomies. In Sections 8.2 and 8.3, we consider suitable perturbations of the moduli space.

8.1. The generic case. Here we will assume that s ≥ 1, 2g − 2 + s ≥ 1, that t_{1}, …, t_{s} are regular and lie in the alcove P, and that (t_{1}, …, t_{s}) verify assumption (\overline{A}) of Section 6.8. Namely we assume that for any (w^{1}, …, w^{s}) ∈ W^{s}, ∑_{j=1}^{s} w^{j}t_{j} ∉ \overline{S}.

Recall that M ⊂ Z was defined in (6.20). Here we assume that M is not reduced to 0.

Theorem 8.1. For p ∈ M, p ≥ 0 large enough,

Ind(D_{p,+}) = V_{g,p+c}(pt_{1}, …, pt_{s}). (8.1)

Proof. By Theorem 6.29 and by Theorem 7.12, we know that if ∑_{j=1}^{s} pt_{j} ∉ \overline{R}, then both sides of (8.1) are equal to 0. So we may as well assume that ∑_{j=1}^{s} pt_{j} ∈ \overline{R}.

First we consider the case where g ≥ 2. Observe that for p ∈ M large enough, (pt_{1}, …, pt_{s}) verify (\overline{A}_{p+c}). By comparing formula (6.129) in Theorem 6.33 and formula (7.91) in Theorem 7.23, we get (8.1).

Now we consider the case g = 1. By Theorem 7.16, we get

V_{1,p+c}(pt_{1}, …, pt_{s}) = \frac{(−1)^{ℓ}}{|\frac{\overline{CR}^{∗}}{(p+c)\overline{CR}} |} ∑_{(v^{1},v^{2})∈W^{2}} ϵ_{v^{1}}ϵ_{v^{2}} \overline{V} _{2,p+c}^{K}(pt_{1}, …, pt_{s}, v^{1}ρ + v^{2}ρ). (8.2)

SYMPLECTIC GEOMETRY AND THE VERLINDE FORMULAS 207

Observe that for p large enough, (pt_{1}, …, pt_{s}, v^{1}ρ + v^{2}ρ) verify (\overline{A}_{p+c}). Using (1.17), Theorem 7.23 with g = 2 and (8.2), we obtain

V_{1,p+c}(pt_{1}, …, pt_{s}) = \frac{|Z(G)|}{|W |}(−1)^{r} (8.3)

∑_{u∈C/\overline{R}^{∗}} ∑_{{I=(i_{1},... ,ir)∈Iu \atop f∈\overline{R}/d\overline{R},μ∈\overline{CR}^{∗}/\overline{R}^{∗}}} \frac{1}{⟨α_{i_{1}}, …, α_{i_{r}}⟩}

Res_{x=0}^{I} [ ∏_{α∈R_{+}} \frac{1}{2 sinh (\frac{1}{2}⟨α, x^{I}/q + 2iπu⟩)}]^{s+3} ∑_{(w^{1},... ,w^{s+3})∈W^{s+3}} ε_{w^{1}} … ε_{w^{s+3}}

exp (⟨∑_{j=1}^{s+1} w^{j}ρ, x^{I}/(p + c)⟩ + 2iπ⟨∑_{j=1}^{s} w^{j}pt_{j}/(p + c) + ∑_{j=s+2}^{s+3} w^{j}ρ/(p + c), λ_{μ}⟩

+2iπ⟨∑_{j=1}^{s} w^{j}(ρ + pt^{j}) + ∑_{j=s+1}^{s+3} w^{j}ρ + (p + c)f, u⟩)

\{exp (d ∑_{j=1}^{r} \frac{⟨p_{j−1}^{I}α_{i_{j}}, x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩}[\frac{1}{d}⟨p_{j−1}^{I}( ∑_{k=1}^{s} w^{k}pt_{k}/(p + c) +

∑_{k=s+2}^{s+3} w^{k}ρ/(p + c) + f), e^{j}⟩])

\frac{1}{∏_{j=1}^{r} ( exp (\frac{d⟨p_{j−1}^{I}α_{i_{j}} ,x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}} ,e^{j}⟩}) − 1)}\}(x − 2iπãλ_{μ}) .

Since (t_{1}, …, t_{s}) verify (\overline{A}), by Proposition 2.14,

⟨p_{j−1}^{I} ∑_{k=1}^{s} w^{k}t_{k} + f, e^{j}⟩ ∉ Z. (8.4)

By proceeding as in (6.131) and using (8.4), we get for p ∈ M large enough,

[\frac{1}{d}⟨p_{j−1}^{I}(∑_{k=1}^{s} w^{k}pt_{k}/(p + c) + ∑_{k=s+2}^{s+3} w^{k}ρ/(p + c) + f), e^{j}⟩] = (8.5)

[\frac{1}{d}⟨p_{j−1}^{I}(∑_{k=1}^{s} w^{k}pt_{k}/(p + c) + f), e^{j}⟩] + \frac{1}{d}⟨p_{j−1}^{I} ∑_{k=s+2}^{s+3} w^{k}ρ/(p + c), e^{j}⟩.

Also by (2.13),

∑_{j=1}^{s} \frac{⟨p_{j−1}^{I}α_{i_{j}}, x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}}, ej⟩}⟨p_{j−1}^{I} ∑_{k=s+2}^{s+3} w^{k}ρ/(p + c), e^{j}⟩ = ⟨ ∑_{k=s+2}^{s+3} w^{k}ρ/(p + c), x^{I}⟩. (8.6)

So by (8.3), (8.5), (8.6), we get

V_{1,p+c}(pt_{1}, …, pt_{s}) = \frac{|Z(G)|}{|W |}(−1)^{r} (8.7)

∑_{u∈C/\overline{R}^{∗}} ∑_{{I=(i_{1},... ,ir)∈Iu \atop f∈\overline{R}/d\overline{R},μ∈\overline{CR}^{∗}/\overline{R}^{∗}}} \frac{1}{⟨α_{i_{1}}, …, α_{i_{r}}⟩}

Res_{x=0}^{I} [ ∏_{α∈R_{+}} \frac{1}{2 sinh (\frac{1}{2}⟨α, x^{I}/(p + c) + 2iπu⟩)}]^{s+3} ∑_{(w^{1},... ,w^{s+3})∈W^{s+3}} ε_{w^{1}} … ε_{w^{s+3}}

exp (⟨∑_{j=1}^{s} w^{j}ρ, x^{I}/(p + c)⟩ + ⟨ ∑_{j=s+1}^{s+3} w^{j}ρ, \frac{x^{I}}{p + c} + 2iπu⟩ +

2iπ⟨∑_{j=1}^{s} w^{j}pt_{j}/(p + c), λ_{μ}⟩ + 2iπ⟨∑_{j=1}^{s} w^{j}(ρ + pt^{j}) + (p + c)f, u⟩)

\{exp (d ∑_{j=1}^{r} \frac{⟨p_{j−1}^{I}α_{i_{j}}, x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩}[\frac{1}{d}⟨p_{j−1}^{I}( ∑_{k=1}^{s} w^{k}pt_{k}/(p + c) + f), e^{j}⟩])

\frac{1}{∏_{j=1}^{r} ( exp (\frac{d⟨p_{j−1}^{I}α_{i_{j}} ,x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}} ,e^{j}⟩}) − 1)}\}(x − 2iπãλ_{μ}) .

Now by (1.94), (7.71),

\frac{1}{∏_{α∈R_{+}} sinh (\frac{1}{2}⟨α, \frac{x^{I}}{p+c} + 2iπu⟩)} ∑_{w∈W} ϵ_{w} exp(⟨wρ, \frac{x^{I}}{p + c} + 2iπu⟩) = 1. (8.8)

By (8.7), (8.8), we obtain

V_{1,p+c}(pt_{1}, …, pt_{s}) = \frac{|Z(G)|}{|W |}(−1)^{r} (8.9)

∑_{u∈C/\overline{R}^{∗}} ∑_{{I=(i_{1},... ,ir)∈Iu \atop f∈\overline{R}/d\overline{R},μ∈\overline{CR}^{∗}/\overline{R}^{∗}}} \frac{1}{⟨α_{i_{1}}, …, α_{i_{r}}⟩}

Res_{x=0}^{I} [ ∏_{α∈R_{+}} \frac{1}{2 sinh (\frac{1}{2}⟨α, x^{I}/(p + c) + 2iπu⟩)}]^{s} ∑_{(w^{1},... ,w^{s})∈W^{s}} ε_{w^{1}} … ε_{w^{s}}

exp (⟨∑_{j=1}^{s} w^{j}ρ, x^{I}/(p + c)⟩ + 2iπ⟨∑_{j=1}^{s} w^{j}pt_{j}/(p + c), λ_{μ}⟩

+2iπ⟨∑_{j=1}^{s} w^{j}(ρ + pt^{j}) + (p + c)f, u⟩)

\{exp (d ∑_{j=1}^{r} \frac{⟨p_{j−1}^{I}α_{i_{j}}, x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}}, e^{j}⟩}[\frac{1}{d}⟨p_{j−1}^{I}( ∑_{k=1}^{s} w^{k} \frac{pt_{k}}{p + c} + f), e^{j}⟩])

\frac{1}{∏_{j=1}^{r} ( exp (\frac{d⟨p_{j−1}^{I}α_{i_{j}} ,x^{I}⟩}{⟨p_{j−1}^{I}α_{i_{j}} ,e^{j}⟩}) − 1)}\}(x − 2iπãλ_{μ}) .

By comparing (6.129) and (8.9), we get (8.1) for g = 1.

A similar proof can be given for the case g = 0. The proof of our Theorem is completed.

8.2. The perturbed case at \frac{pt_{j}}{p+c}. Now we make the same assumptions as in Section 6.11. Namely take p ∈ M, p ≥ 0. We assume that ϵ_{1}, …, ϵ_{s} ∈ 𝔱 are such that form ℓ ∈]0, 1], ≤ j ≤ s, \frac{pt_{j}}{p+c} + ℓϵ_{j} ∈ T is regular, and for any (w^{1}, …, w^{s}) ∈ W^{s}, ℓ ∈]0, 1], ∑_{j=1}^{s} w^{j} \frac{pt_{j}}{p+c} + ℓϵ_{j} ∉ \overline{S}.

Theorem 8.2. For g ≥ 2,

Ind(D_{p,+}^{( \frac{pt_{j}}{p+c} +ℓϵ_{j})}) = V_{g,p+c}(pt_{1}, …, pt_{s}). (8.10)

For g = 0 or g = 1, if (t_{1}, …, t_{s}) verify (\overline{A}), for p ∈ M and p ≥ 0 large enough, equation (8.10) still holds.

Proof. By Remark 6.36, we find that if ∑_{j=1}^{s} pt_{j} ∉ \overline{R}, then the left-hand side of (8.10) vanishes. By (7.59) in Theorem 7.12, the right hand-side also vanishes. As in the proof of Theorem 8.1, we may and we will assume that ∑_{j=1}^{s} pt_{j} ∈ \overline{R}.

First we consider the case g ≥ 2. Then (8.10) follows by comparing (6.146) in Theorem 6.40, and (7.110) in Theorem 7.28.

In the case 0 ≤ g ≤ 1, we proceed as in the proof of Theorem 8.1, and use (8.10) in the case g ≥ 2.

The proof of our Theorem is completed.

8.3. The perturbed case at \frac{ρ+pt_{j}}{p+c}. Observe that if t ∈ \overline{P}, for p ∈ N, \frac{ρ+pt}{p+c} ∈ P, i.e. \frac{ρ+pt}{p+c} is regular. This follows follows from (1.33), (1.34), (1.36).

The same argument as in the proof of Proposition 6.38 shows that if s ≥ 1, given p ≥ 0, p ∈ M, there exist ϵ_{1}, …, ϵ_{s} ∈ 𝔱 such that for ℓ ∈]0, 1], 1 ≤ j ≤ s, \frac{pt_{j}+ρ}{p+c} +ℓϵ_{j} ∈ T is regular, and for any (w^{1}, …, w^{s}) ∈ W^{s}, ℓ ∈]0, 1], ∑_{j=1}^{s} w^{j}(\frac{pt_{j}}{p+c} + ℓϵ_{j}) ∉ \overline{S}.

In Theorem 6.35 we now take

δ_{j} = \frac{ρ − ct_{j}}{p + c} + ℓϵ_{j}. (8.11)

By the construction of Section 6.11, we get a Dirac operator Dp,+^{\frac{ρ+pt_{j}}{p+c}}.

Theorem 8.3. For g ≥ 0, and 2g − 2 + s ≥ 1, for ℓ ∈]0, 1],

Ind(D_{p,+}^{( \frac{ρ+pt_{j}}{p+c} +ℓϵ_{j})}) = V_{g,p+c}(pt_{1}, …, pt_{s}). (8.12)

Proof. As in the proof of Theorems 8.1 and 8.2, we may restrict ourselves to the case where ∑_{j=1}^{s} pt_{j} ∈ \overline{R}. First assume that s is even. Then (8.12) follows from (6.136) in Theorem 6.35, with δ_{j} given by (8.11), and from (7.113) in Theorem 7.29. When s is odd, first we perturb the s holonomies as indicated before. Then we add an extra marked point x_{s+1}, with holonomy equal to 1. We perturb the holonomy 1 to a generic holonomy, and we use the result we just prove with s + 1 marked points. We can then gently make our holonomy t_{s+1} tend to 1. This is in fact possible because the perturbation of the first s holomnomies are generic and verify condition (\overline{A}. The proof of our Theorem is completed.

Remark 8.4. Suppose temporarily that the assumptions of Remark 6.41 are satisfied. In particular we assume the holonomy t_{s} to be central and represented by h_{o} ∈ \overline{P}. A direct treatment would show that an analogue of (8.12) would still hold, where, in the Verlinde sum (7.57), the term χ_{ph_{o}}(e^{λ/(p+c)} should apparently be replaced by ϵ_{w_{ho}} exp(2iπ⟨h_{o}, λ⟩). However Theorem 1.33 tells us that indeed, this is just χ_{ph_{o}}(e^{λ/(p+c)}, so that we recover indeed the standard Verlinde formula.

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Département de Mathématique, Université Paris-Sud, Bâtiment 425, 91405 Orsay, France

E-mail address: bismut@topo.math.u-psud.fr

Département de Mathématique, Université Paris-Sud, Bâtiment 425, 91405 Orsay, France

E-mail address: labourie@topo.math.u-psud.fr