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Multiple Bernoulli series and volumes of moduli spaces of flat bundles over surfaces

Using Szenes formula for multiple Bernoulli series we explain how to compute Witten series associated to classical Lie algebras. Particular instances of these series compute volumes of moduli spaces of flat bundles over surfaces, and also certain multiple zeta values.


Introduction
Let V be a finite dimensional real vector space, and Λ a lattice in V . We denote the dual of Λ by Γ.
In this paper we compute is piecewise polynomial (see [13]).
Szenes multidimensional residue formula (see Theorem 1.33) is the generalization of this formula to higher dimension, and it is the tool that we use for computing B(Φ, Λ)(v) as a piecewise polynomial function.
A particular but crucial instance of multiple Bernoulli series is when Λ is the coroot lattice of a compact connected simple Lie group G, and Φ is comprised of positive coroots of G. The series B(Φ 2g−2 , Λ), where the argument Φ 2g−2 refers to taking elements of Φ with multiplicity 2g − 2, appeared in the work of E. Witten ([17], §3), where Witten shows that its value at v = 0 is (up to a scalar depending on G and g) the symplectic volume of the moduli space of flat G-connections on a Riemann surface of genus g. Similarly, for a regular element v of the Cartan Lie algebra of G, Witten shows that the value of B(Φ 2g−1 , Λ)(v) is (up to a scalar depending on G and g) the symplectic volume of the moduli space of flat G-connections on a Riemann surface of genus g with one boundary component, around which the holonomy is determined by v.
More generally, for the above choice of Λ and Φ, when v = {v 1 , . . . , v s } is a collection of s regular elements of the Cartan Lie algebra, certain linear combinations of B(Φ 2g−2+s , Λ) at some particular values (depending on v) is the symplectic volume of the moduli space of flat G-connections on a Riemann surface of genus g with s boundary components, around which the holonomy is determined by v. Then, its dependance on v is piecewise polynomial.
Multiple Bernoulli series have also been studied by P.E. Gunnells and R. Sczech ( [6]) in view of applications to zeta functions of real number fields. Explicit computations of volumes of moduli spaces of flat bundles on Riemann surfaces are also obtained in [6]. The techniques they used is a generalization of the continued fraction algorithm and is different from ours. Y. Komori, K. Matsumoto and H. Tsumura ( [7], [8], [9]) studied the restriction of the series (0.0.1), by summing it over the cone of dominant regular weights of a semi-simple Lie group G, and defined functions ζ(s, v, G) (cf. Section 5.2). They also obtained relations between these functions over Q. When Λ is the coroot lattice of a compact connected simple Lie group G and the sequence Φ is the set of its positive coroots with equal even multiplicity for long roots and (possibly different) equal even multiplicity for short roots, due to the Weyl group symmetry, the summation B(Φ, Λ)(0) over the full (regular) weight lattice is just (up to multiplication by an appropriate power of (2π)) Komori-Matsumoto-Tsumura zeta function ζ(s, 0, G). Thus, the value of ζ(s, 0, G) (up to a certain power of (2π)) is a rational number which can be computed explicitly, and we give examples of such computations.
As it is observed in [7], some instances of the series ζ(s, v, G) also compute certain multiple zeta values. In the last part of the article we give various such computations of multiple zeta values using B(Φ, Λ).
Here is the outline of individual sections.
In Section 1, we recall a formula due to A. Szenes, which allows an efficient computation of B(Φ, Λ).
In Section 2, we give an outline of an algorithm that efficiently computes the needed ingredients of this formula for classical root systems. We also give several simple examples.
In Section 3, we show how this applies to the symplectic volume of the moduli space of flat G-connections on a Riemann surface of genus g with s boundary components. We obtain an expression for the symplectic volume by taking the limit of the Verlinde formula. We then show that our formula thus obtained coincides with that of Witten (including the constants) given in terms of the Riemannian volumes of G and T . We also give examples of these functions.
In Sections 4 and 5, we give several examples and tables of Witten volumes, which include some examples from [7], [8] and [9]. We give an idea of computational limitation of our algorithm (written as a simple Maple program) in terms of the rank of the group G and the number of elements in Φ. Following Y. Komori, K. Matsumoto, H. Tsumura, we also give some examples of rational multiple zeta values.
To compute more examples, our Maple program is available on the webpage of the last author.
Finally, in the appendix, for completeness, we include a proof of Szenes formula.
Research Fund 5076. The third author wishes to thank Bogazici University for support of research visits.
We thank Shrawan Kumar for pointing out a minor mistake in the earlier version of this manuscript in the formula of Proposition 3.1 for the case of Lie group of type G 2 .
We also thank the referees for their suggestions on the long abstract version of this manuscript.
the pairing between U and V . Γ a lattice in U; γ ∈ Γ. Λ := Γ * dual lattice in V ; Γ, Λ ⊂ Z, λ ∈ Λ. Φ a sequence of vectors in V ; φ ∈ Φ. In this subsection, U is an r-dimensional complex vector space, and we recall briefly some structure theorems for the ring of rational functions that are regular on the complement of a union of hyperplanes [4].
Let V be the dual vector space to U. If φ ∈ V , we denote by H φ = {u ∈ U; φ, u = 0}.
Let H = {H 1 , . . . , H N } be a set of hyperplanes in U. Then, we may choose φ k ∈ V such that H k = H φ k ; the element φ k will be called an equation of H k . Clearly, an equation φ k is not unique, it is determined up to a nonzero scalar multiple.
We consider an open subset of U. An element of U H will be called regular.
Definition 1.1. We denote by S(V ) the symmetric algebra of V and identify it with the ring of polynomial functions on U.
We denote by R H the ring of regular rational functions on U that are regular on U H . That is, the ring generated by S(V ) together with inverses of the linear forms φ k defining H.
The ring D(U) of differential operators on U with polynomial coefficients acts on R H . In particular, U operates on R H by differentiation. We denote by ∂(U)R H the space of functions in R H obtained by differentiation.
If V is one dimensional, then the ring R H is the ring of Laurent polynomials C[z, z −1 ], and the function z i , for i = −1, is obtained as a derivative d If f = n a n z n is an element of C[z, z −1 ], we denote by Res z=0 f the coefficient a −1 of z −1 in the expression of f . This linear form is characterized by the fact that it vanishes on d dz R H . By analogy to the one dimensional case, a linear functional on R H vanishing on ∂(U)R H will be called a 'residue'.
Let us thus analyze the space R H modulo ∂(U)R H .
Let us consider a set Φ eq := {φ 1 , φ 2 , . . . , φ N } of equations for H. A subset σ of Φ eq will be called a basis if the elements φ k in σ form a basis of V . We denote by B(Φ eq ) the set of such subsets σ. A subset ν of Φ eq will be called generating if the elements φ k in ν generate the vector space V . .
We denote by S H the subspace of R H generated by the elements f σ , σ ∈ B(Φ eq ).
• We denote by G H the subspace of R H generated by the elements θ(ν, n) where ν is generating.
As the notation suggests, the spaces R H , S H and G H depend only on H. The term simple fraction comes from the fact that if σ = {φ 1 , φ 2 , . . . , φ r } is a basis, then we can choose coordinates z i on U so that φ i (z) = z i , so that for this system of coordinates f σ (z) = 1 r i=1 z i . We recall the following 'partial fraction' decomposition theorem. Lemma 1.3. Let ν be a subset of Φ eq generating a t dimensional subspace of V . Then θ(ν, n) may be written as a linear combination of elements θ(σ, m) = Finally, the following theorem is proved in Brion-Vergne [4].
The projector, denoted by R, from R H to S H will be called the total residue.
In view of this theorem, a residue is just a linear form on S H . When H is the set of hyperplanes with equations the positive coroots of a simple compact Lie group G, the dimension of S H is given by the product of exponents of G ( [11]). In Section 2, we will give an explicit basis for S H for simple Lie algebras of type A, B and C (which defines the same set of hyperplanes as B) with dual basis consisting of iterated residues.
1.2. Szenes polynomial. In this section and for the rest of the article, V will denote a real vector space of dimension r.
Let U be the dual vector space of V . Let Λ be a lattice in V with dual lattice Γ in U.
Let H := {H 1 , H 2 , . . . , H N } be a real arrangement of hyperplanes in U. We say that Λ and H are compatible if the hyperplanes in H are rational with respect to Λ, that is, they can be defined by equations φ k ∈ Λ. If Λ ′ is another lattice commensurable with Λ, then Λ ′ and H are also compatible.
Thus we now consider a lattice Λ and a real arrangement of hyperplanes H = {H 1 , H 2 , . . . , H N } in U rational with respect to Λ.
We choose Φ eq = {φ 1 , φ 2 , . . . , φ N }, a set of defining equations for H, with each φ i in Λ. We sometimes refer to H only via its set of equations Φ eq and write H = ∪{φ k = 0}.
We denote the complex arrangement defined by ∪{φ k = 0} in U C by the same letter H. We denote by An admissible hyperplane W in V (for the system H) is an hyperplane generated by (r−1) linearly independent elements φ k of Φ eq . Such an hyperplane will also be called an (admissible) wall. An admissible affine wall is a translate of a wall by an element of Λ.
An element v ∈ V is called regular for (H, Λ) if v is not on any affine wall (we will just say that v is regular). The meaning of the word regular is thus different for elements v ∈ V (v is not on any affine wall) and u ∈ U C (u is such that k φ k , u = 0). However, it will be clear what regular means in the context. A tope τ is a connected component of the complement of all affine hyperplanes. Thus a tope τ is a connected open subset of V consisting of regular elements. We denote the set of topes by T (H, Λ). As the notation indicates, T (H, Λ) does not depend on the choice of equations for H. Example 1.6. Let V = Re 1 ⊕ Re 2 and Λ = Ze 1 ⊕ Ze 2 . Let U be its dual with basis {e 1 , e 2 }. We express z ∈ U C as z = z 1 e 1 + z 2 e 2 , and consider the set of hyperplanes with the set of equations Φ eq = {e 1 , e 2 , e 1 + e 2 }. Figure 1 depicts topes associated to this pair. Example 1.7. Let V = Re 1 ⊕ Re 2 and Λ = Ze 1 ⊕ Ze 2 . Let U be its dual with basis {e 1 , e 2 }. We express z ∈ U C as z = z 1 e 1 + z 2 e 2 , and consider the set of hyperplanes with the set of equations Φ eq = {e 1 , e 2 , e 1 +e 2 , e 1 −e 2 }. Figure 2 depicts topes associated to this pair.
We denote by V reg (H, Λ) (or simply V reg ) the set of (H, Λ) regular elements of V . It is an open subset of V which is the disjoint union of all topes.
A locally constant function on V reg is a function on V reg which is constant on each tope. A piecewise polynomial function on V reg is a function on V reg which is given by a polynomial expression on each tope. If t ∈ R, we denote by [t] the integral part of t, and by {t} = t − [t] the fractional part of t. If γ ∈ Γ vanishes on an admissible hyperplane W , and c is a constant, then the function v → { γ, v + c} is piecewise polynomial (piecewise linear) and is periodic with respect to Λ. Szenes residue formula provides an algorithm to describe Bernoulli series in terms of these basic functions.
Definition 1.8. Let M H be the space of functions h/Q where Q is a product of linear forms belonging to Φ eq , and h a holomorphic function defined in a neighborhood of 0 in U C .
We define the spaceR H as the space of functionsĥ/Q whereĥ = ∞ k=0 P k is a formal power series and Q is a product of linear forms belonging to Φ eq as before.
where B(k, t) is the k th Bernoulli polynomial in t as before, is a polynomial in t.
Let f ∈ R H , z ∈ U H and γ ∈ Γ. Then if z is small, 2iπγ − z is still a regular element of U. Consider the series When f decreases sufficiently quickly at infinity, the series S(f, z, v) is absolutely convergent and defines a continuous function of v. In general, as f is of at most polynomial growth, the series is the Fourier series of a generalized function on V /Λ.
Multiplying S(f, z, v) by the exponential e − z,v we introduce the following definition. Definition 1.9. Let f ∈ R H , z ∈ U H and small. We define the generalized function of v by The meaning of A Λ (f ) is clear : average the function z → f (−z)e − v,z over 2iπΓ in order to obtain a function on the complex torus U C /2iπΓ.
We consider A Λ (f )(z, v) as a generalized function of v ∈ V with coefficients meromorphic functions of z on U C /2iπΓ. In fact, as we will see, when f is in S H , the convergence of the series holds in the sense of the Fourier series of an is a locally constant function of v ∈ V reg with values in M H .
Note the covariance relation. For λ ∈ Λ, It is easy to compare A Λ (f )(z, v) when we change the lattice Λ.
This series is not absolutely convergent, but the oscillatory factor e 2iπnv insures the convergence in the distributional sense as a function of v.
We have Indeed, let us compute the L 2 -expansion of the periodic function v → e (v−[v])z 1−e z . By definition, this is We see in this one dimensional example that A Λ (f )(z, v) is a locally constant function of v.
In general, we have the following proposition.
We prove this by computing A Λ (f )(z, v) explicitly for a simple fraction f = f σ . Recall that the set of equations Φ eq is a subset of Λ. Let σ = {α 1 , α 2 , . . . , α r } be an element of B(Φ eq ). The elements α k belong to Λ. Let Q σ := ⊕ r k=1 [0, 1)α k be the semi-open parallelepiped spanned by σ. Definition 1.13. Let v be regular in V , and let σ ∈ B(Φ eq ) be a basis. Define T (v, σ) to be the set of elements λ ∈ Λ such that v + λ ∈ Q σ . This set depends only on the tope τ where v belongs, hence we denote it by T (τ, σ).

The dependance of
Example 1.15. We return to the Example 1.6, where Φ eq = {e 1 , e 2 , e 1 + e 2 }. To describe the function v → A Λ (f )(z, v) on V reg completely, it suffices to give its expression on each tope τ 1 and τ 2 which are depicted in Figure 1. This is true since any element of V reg can be translated to τ 1 or τ 2 by an element of Λ, and then one can use the covariance relation (1.9.1).
This operator is locally constant. We denote its value on τ by Z Λ (τ ): for any choice of v ∈ τ .
We now define a piecewise polynomial function of v associated to a function g(z) in R H .
First, the operator on M H given by multiplication by a function . We project back this function on S H using the projector R. Thus the map is a map from S H to S H depending on v. As S H is finite dimensional, we can take the trace of this operator, and thus obtain a function of v ∈ V reg . Let us record this definition.
on v is via e z,v , and the map R involves only the Taylor series of this function up to some order.
Thus we have associated to g ∈ R H (and Λ) a piecewise polynomial function P (H, Λ, g) on V reg .
It is easy to compare piecewise polynomial functions P (H, Λ, g) associated to different lattices. Let This follows immediately from Lemma 1.10.
Our next aim is to compute the piecewise polynomial function P (H, Λ, g) using residues. We need more definitions.
An ordered basis of Φ eq is a sequence [α 1 , α 2 , . . . , α r ] of elements of Φ eq such that the underlying set is in B(Φ eq ). We denote the set of ordered bases by − → B(Φ eq ).
. Then, to this data, one associates an iterated residue functional Res − → σ on R H as follows. For z ∈ U, Then a function f in R H can be expressed as a function f (z 1 , z 2 , . . . , z r ). We define The functional Res − → σ factors through the canonical projection R : A total order on Φ eq allows us to construct the set of Orlik-Solomon bases (see [4]), which provides diagonal basis of S H . However we will also use some other diagonal subsets.
Definition 1.19. Let g ∈ R H and τ a connected component of V reg . We denote by P (H, Λ, g, τ )(v) the polynomial function on V such that Hence, we may give a more explicit formula for the polynomial P (H, Λ, g, τ )(v) using a set − → D .
It is important to remark that the determination of a diagonal subset − → D depends essentially only on the system of hyperplanes H and not on the choice of Φ eq . The difficulties in writing an algorithm for P (H, Λ, g) lies in the description of a diagonal subset − → D , and also for each σ ∈ − → D , in the computation of A Λ (f σ ). The difficulty of this last computation depends on the lattice Λ. .
It may happen that even when the system Φ eq is not unimodular for the lattice Λ, we can choose − → D to consist of unimodular bases. In particular, using Proposition 1.20, we can give an explicit algorithm for computing the piecewise polynomial function P (H, Λ, g) for classical root systems in the form of a step polynomial. Let us define what this means.
Step( − → D ) the algebra of functions on V generated by the piecewise linear func- An element of the algebra Step( − → D ) will be called a step polynomial (associated to − → D ).
It is clear that a step polynomial is a periodic function on V , which is expressed by a polynomial formula on each tope.
consisting of unimodular basis (with respect to Λ). Then the piecewise polynomial function P (H, Λ, g) belongs to the algebra Proof. This is clear, as we have the formula (1.26.1) , 1.3. Multiple Bernoulli series. We return to our main object of study: the multiple Bernoulli series. Let V , Λ and H be as before. We denote by Γ ⊂ U the dual lattice to Λ, and by Γ reg (H The above series converges in the space of generalized functions on V . We state some obvious properties of B(H, Λ, g).
If we dilate a lattice Λ by ℓ, and if g is homogeneous of degree h, we clearly have With these two properties, we can compare B(H, Λ, g) over commensurable lattices. Definition 1.29. A generalized function b on V will be called piecewise polynomial relative to H and Λ if • the function b is locally L 1 , • for each tope τ in T (H, Λ), there exists a polynomial function b τ on V such that the restriction of b to τ coincides with the restriction of the polynomial b τ to τ .
As an L 1 -function is entirely determined by its restriction to V reg , we will not distinguish between piecewise polynomial generalized functions on V and piecewise polynomial functions on V reg as defined in the preceding section.
Be careful: the restriction to any tope τ of a piecewise polynomial generalized function b is polynomial. However, the converse is not true. For example the δ function of the lattice Λ restricts to 0 on any tope τ , but is not a piecewise polynomial generalized function, as it is not a locally L 1 -function.
Recall the definition of G H as given in Definition 1.2. If we multiply g by a polynomial p, the function Furthermore, if f ∈ G H , the generalized function B(H, Λ, f ) is a piecewise polynomial generalized function.
Let us emphasize on the subtle difference between the conditions Thus B(H, Λ, f )(v) is the constant function equal to −1 on any tope. However, it has some singular part δ Z , and is not locally L 1 .
In contrast, if f = 1 z , the generalized function Remark 1.32. It is interesting to understand the space of polynomials generated by the polynomial functions b τ = B(H, Λ, f, τ ), when τ runs over the topes, and the wall crossing formula between b τ 1 and b τ 2 when τ 1 and τ 2 are adjacent. We addressed some aspects of these theoretical questions in [3].
Consider the piecewise polynomial function P (H, Λ, f ) on V reg (H, Λ) as given in Definition 1.16.
For completeness, we give a proof of this theorem in the Appendix.
Our Maple program computes, given data H, Λ, f , where H is the hyperplane arrangement associated to a classical root system, a piecewise polynomial function on V in terms of step polynomials. Naturally, we can also evaluate this function at any point v ∈ V reg .
We return to the definition of multiple Bernoulli series in the way we introduced them in the introduction.
Let V be a vector space with a lattice Λ with dual lattice Γ. We considered Φ in the introduction as a list of elements of Λ. We then introduced the following definition. Let Γ reg (Φ) = {γ ∈ Γ; φ, γ = 0 for all φ ∈ Φ} and defined Consider H = ∪ k {φ k = 0} (some elements of the list Φ might define the same hyperplane) and g(z) = 1/ φ∈Φ φ, z , then We will also call the functions B(H, Λ, g)(v) multiple Bernoulli series.

Classical root systems
Let G be a simple, simply connected, compact Lie group of rank r with maximal torus T . We denote the Lie algebra of T and G by t and g respectively. Then the complexification h := t C is a Cartan subalgebra of g C . For be the set of roots; roots α ∈ R take imaginary values on t. We denote the root lattice by Q and its dual, the coweight lattice, byP .
For α ∈ R, there exists a unique element H α in [(g C ) α , (g C ) −α ] satisfying α, H α = 2; it is called the coroot associated to the root α. For any α ∈ R, iH α is in t, and for any α, β ∈ R, β(H α ) is integral. The lattice spanned by H α is called the coroot lattice and denoted byQ.
We denote by h R := α RH α , the real span of coroots. In this section we have V = h R , and its dual h * R is denoted by U as before. We follow the notation of Bourbaki for root data.
2.1. Diagonal subsets. To compute multiple Bernoulli series associated to classical root systems we need to construct explicit diagonal bases for the corresponding S H . Such bases can be constructed by an algorithmic procedure, based on Orlik-Solomon construction. However in some cases one can describe a diagonal subset − → D of − → B (Φ eq ) whose associated simple fractions form a basis for S H in a direct way, and that we present now.
2.1.1. The system of type A r . Let n = r + 1. We consider R n with standard basis {e 1 , e 2 , . . . , e n }. Let be the root system of type A and rank r.
Let {e i } be the dual basis to {e i } and and let H A r be the system of hyperplanes in U given by of positive coroots as equations of H A r . One way to find a diagonal basis of S H A r is as follows. Let Σ = [e 1 − e 2 , e 2 − e 3 , . . . , e r − e r+1 ] be the set of simple coroots. For a permutation w, we denote by Then − → σ w is an ordered basis associated to w, and the corresponding simple fraction is f w (z) := 1 r i=1 (z w(i) −z w(i+1) ) . Let W r be the subset of the Weyl group Σ r+1 of permutations of {e 1 , e 2 , . . . , e r+1 } leaving the last element e r+1 = e n fixed. Recall the following result (see for example Baldoni-Vergne [2] for a proof).
Proposition 2.1. The set − → D W consisting of ordered bases − → σ w for w ∈ W r is a diagonal subset of − → B(Φ eq (A r )).
We use the above basis in our Maple program.
We now give another interesting diagonal subset.

2.1.2.
Systems of type B r or C r . We consider V = R r with standard basis {e 1 , e 2 , . . . , e r }. Let B r = [±e i , ±(e i ± e j ), 1 ≤ i ≤ r, 1 ≤ i < j ≤ r] be the root system of type B and rank r.
Let C r = [±2e i , ±(e i ± e j ), 1 ≤ i ≤ r, 1 ≤ i < j ≤ r] be the root system of type C and rank r.
As roots of systems of type B and C are proportional, the system of hyperplanes in U = h * R defined by coroots of B and C are the same, and we denote it by H BC r . More precisely, let z = r i=1 z i e i in U, then the system of hyperplanes H BC r in U is given by We take the set We define similarly a flag basis − → σ of Φ eq (BC r ). This is a basis of the form − → σ = [α 1 , α 2 , . . . , α r ] of r elements of Φ eq (BC r ) so that α i = e i , or e i − e j or e i + e j with j < i. That is, α 1 = e 1 , α 2 = e 2 or e 2 − e 1 , or e 2 + e 1 , α 3 = e 3 or e 3 − e 2 , or e 3 + e 2 , or e 3 − e 1 , or e 3 + e 1 , etc. Clearly, there are (1)(3)(5) · · · (2r − 1) such sequences − → σ . Proof. We first prove, by induction on r, that simple fractions f b associated to a flag basis b generate S H BC r . We use the identities to reduce to the case where a simple fraction f b contains a linear form of type e r , or e r + e i or e r − e i in the denominator, but not any two at the same time. Then, by induction on r, we see that a simple fraction f b associated to flag basis b generates the space S H BC r . The dual property on the elements of − → D (BC r ) is evident.
Remark 2.4. Although the system Φ eq (BC r ) is not unimodular for the lattice Λ = ⊕Ze i , we see that any − → σ in the set − → D (BC r ) above is unimodular, so that the computation of Z Λ (τ )(f σ ) is easy.
2.1.3. The system of type D r . We consider V = R r with standard basis {e 1 , e 2 , . . . , e r }. Let be the root system of type D and rank r.
Let z = r i=1 z i e i in U. We consider the system of hyperplanes However, we did not find a nice diagonal basis for S H D r . Instead, we proceed as follows.
The set U H D r of regular elements for H D r contains U H BC r . Indeed, for any z in U H D r , we have z i ± z j = 0, but z i may equal zero.
We define the set Then, we have the following disjoint decomposition The set U k,r is clearly isomorphic to the open set U H BC r−1 in rank r − 1 via the map i k which inserts a zero coordinate in position k, and hence, . The above decomposition allows us to reduce calculations in system of type D to that of systems of type B or C.

2.2.
Calculations of multiple Bernoulli series for type A. We use the same notation as in Section 2.1.1.
Let Q A ⊂ U be the root lattice generated by A r , and P A ⊂ U be the weight lattice. Then P A is generated by Q A and e 1 − 1 r+1 (e 1 + e 2 + · · · + e r+1 ) and P A /Q A is of cardinality r + 1.
Let Γ be a lattice such that Q A ⊂ Γ ⊂ P A . We denote by Γ reg = Γ ∩ U H A r the set of regular elements in Γ.
Let Λ ⊂ V be the dual lattice to Γ, and let s = [s α ] be a list of exponents. Define where the set {H α , α > 0} is the set of positive coroots Φ eq (A r ).
If v ∈ V , If we use the diagonal basis − → D W for Φ eq (A r ) as defined in We now discuss two simple cases, where Γ is either the weight lattice P A , or the root lattice Q A .

2.2.1.
Bernoulli series for the weight lattice. If P A is the weight lattice, then the dual of P A is the coroot latticeQ A generated by simple coroots H α , and the system Φ eq (A r ) of equations (the positive coroots) is unimodular with respect toQ A .
Thus We assume all multiplicities s α = 1, and compute B( The simple coroots are H α 1 = e 1 − e 2 and H α 2 = e 2 − e 3 , and the remaining positive coroot is their sum H α 1 + H α 2 . The dual lattice has basis dual to H α 1 , H α 2 . Thus, for s = [1, 1, 1], . The symbol ′ above means that we sum over the integers m, n with mn(m + n) = 0. Denoting the fractional part of t with {t} ∈ [0, 1[ as before, we obtain that P We remark that the series (2.5.1) is not absolutely convergent, but the sum has a meaning and is a piecewise polynomial function.

2.2.2.
Bernoulli series for the root lattice. Let ξ = r j=1 (e j − e r+1 ). Then a system of representatives forP A /Q A consists of the elements λ j = j r+1 ξ, with j varying between 0 and r. Thus, using Formula (1.28.1), We obtain an expression for B(H A r ,P A , g A s ) in terms of the functions {( I v i ) + c/(r + 1)} where c are integers between 0 and r.
This is equal to where P is the piecewise polynomial function given in Equation (2.5.2).

2.3.
Calculation of multiple Bernoulli series for types C and B. We use the same notation as in Section 2.1.2.
We now consider the system H BC r of hyperplanes in U Let Λ be a lattice commensurable with ⊕Ze i , with dual lattice Γ. Denote simply by Γ reg = Γ reg (H BC r ). If g ∈ R H BC r , 2.3.1. Root system C r . Let P C be the weight lattice of the root system C r . We thus have the coroot latticeQ C = ⊕ r i=1 Ze i . Let s = [s α ] be a list of exponents and let g C s (z) = 1 α>0 H α , z sα . Here {H α , α > 0} are positive coroots of the system C r , which are explicitly The function B(H BC r ,Q C , g C s ) is a piecewise polynomial function on V of degree α s α . We use the diagonal basis constructed in Section 2.1.2 to compute it. Let us now compute the example given in the introduction, which corresponds to C 2 , and the exponent s = [2, 1, 1, 1] ordered in accordance with the order [2e 1 , 2e 2 , e 1 +e 2 , e 1 −e 2 ] of positive roots (so that coroots H α are in order [e 1 , e 2 , e 1 +e 2 , e 1 −e 2 ]). We express v = v 1 e 1 + v 2 e 2 and compute − n)) .
Consider for example the tope Then on τ 2 , the piecewise polynomial function B(H BC 2 ,Q C , g C s )(v) coincides with the polynomial If Clearly, as long coroots of B are twice the short coroots of C, and short coroots of B are long coroots of C, we have We then use the comparison formula with two lattices. Indeed, we have 2Q C ⊂Q B . The lattice 2Q C is of index 2 r−1 inQ B and a set of representatives is given, for example, by We then use Formulae (1.28.1) and (1.28.2). Since g C s is homogeneous of degree − α s α , we obtain where c 2 = 2 1≤i<j≤r s e i −e j +s e i +e j . In particular, if s = [m, . . . , m], then c 2 = 2 r(r−1)m . For example, for B 2 , we compute for v = v 1 e 1 +v 2 e 2 and multiplicities s = [2, 1, 1, 1] with respect to the order [e 1 − e 2 , e 2 , e 1 + e 2 , e 1 ] of roots, where the symbol ′ means that we sum over the m 1 , m 2 with We obtain where Q = B(H BC 2 ,Q C , g C s ) is given in Equation (2.5.4).
We obtain is a piecewise polynomial that is too long to be included here.

2.4.
Calculation of multiple Bernoulli series for type D. We follow the same notation as in Section 2.1.3.
Let We embed the list of roots of D r in to the list of roots of B r by writing the short roots e i of B r at the very end of the list. We denote by S = [s α , 0, . . . , 0] the list obtained from s by adjoining r zeros to its end. Then, by construction, we have . We now associate to the list s a list of exponents s k for the system B of rank r − 1. In s we eliminate the position corresponding to the roots e i ± e k for i < k, and e k ± e i for i > k. Then we assign the value s e i +e k + s e i −e k to the exponent corresponding to root e i of B r−1 for i < k , similarly we assign the value s e k +e i + s e k −e i to the exponent corresponding to the root e i of B r−1 for i > k.
We also let i k (v) to be the vector with r − 1 coordinates obtained from v = r i=1 v i e i by putting v k = 0. Let Γ = P D be the weight lattice of D andQ D the dual lattice generated by the coroots. Since P D is the weight lattice of the simply connected group Spin(2r), γ = r i=1 γ i e i is in P D if γ i ± γ j ∈ Z and P D = P B . Consider the intersection of P D with the hyperplane z k = 0. Then, we see that this intersection is isomorphic to the weight lattice of a system C r−1,k of type C, rank r − 1, embedded in C of rank r with simple roots {e 1 − e 2 , e 2 − e 3 , . . . , e k−1 − e k+1 , e k+1 − e k+2 , . . . , e r }.
Using the decomposition (2.4.1), we decompose the set of regular elements of the lattice P D as a disjoint union of the set of regular elements of the lattice P B and the set of regular elements of the lattice P C of rank r − 1.
In particular, if γ ∈ (P D ) reg and γ k = 0, then Thus, we can compute multiple Bernoulli series for a system of type D r by using computations for types B r and C r−1 with appropriate exponents. More explicitly,

Witten formula for volumes of moduli spaces of flat connections on surfaces
Let G be a simple, simply connected, compact Lie group of rank r with maximal torus T . For g 1 , g 2 ∈ G, we denote by [g 1 , the commutator of g 1 , g 2 . Let Σ be a compact connected oriented surface of genus g and let p := ∪ j {p j } be a set of s points on Σ. Let C := (C j ) be a set of s conjugacy classes in G. We consider the representation variety If the adjoint orbits C j are generic, this is an orbifold of dimension (2g − 2) dim G + s dim G/T . It parameterizes the set of flat G-valued connections on Σ − p, with holonomy around p j belonging to the conjugacy class C j modulo gauge equivalence. As shown by Atiyah-Bott [1], once a G-invariant inner product on g is chosen, the manifold M(G, g, s, C) carries a natural symplectic form, and Witten gave a formula for the volume of M(G, g, s, C) that we recall.
We use the notation of Section 2. We need some more definitions.
Let R + be a choice of positive roots; denote the highest root of R by θ. Let h + := {h ∈ h R ; α(h) ≥ 0 for all α ∈ R + } be the positive chamber (closed) in h R . Let A := {h ∈ h + ; θ(h) ≤ 1} be the fundamental alcove. An element of A is said to be regular if it lies strictly inside the alcove.
Let W = W (g C , h) be the Weyl group (identified with N G (T )/T ).
We now give the Witten formula.
Let a = {a 1 , a 2 , . . . , a s } be a set of regular elements in A ⊂ h + . Let C j be the adjoint orbit of exp(a j ); we denote the collection of orbits C j by C.
Consider the function on h * given by The above expression is always meaningful as a generalized function of the parameters a j . If s = 0, this formula has to be understood as Interchanging the sum and the product, N a (λ) may be expressed as Hence the function W (Φ(G), P, g, s)(a) can be expressed as a sum over W s with signs of Bernoulli series B(Φ 2g−2+s ,Q)( j w j a j ). Here, as before, Φ 2g−2+s means that each coroot in Φ is taken with multiplicity 2g − 2 + s.
As is well known, the series W (Φ(G), P, g, s)(a) computes the symplectic volume of M(G, g, s, C) up to a scalar factor, which we will give in the next section.
Let us now recall the normalization of the volume as the limit of the Verlinde formula. We need some more notation. Let ( | ) denote the G-invariant symmetric form on g C normalized such that (H θ |H θ ) = 2. We will use the same notation for the restricted form on h, and the induced form on h * . We call ( | ) the basic invariant form. It is positive definite on the real span h R , and negative definite on t.
Letȟ := ρ(H θ ) + 1 be the dual Coxeter number, where ρ is the half sum of positive roots. Let Q long ⊂ Q be the lattice spanned by long roots. The basic invariant form identifies h R and h R * ; under this isomorphism the coroot latticeQ is identified to Q long . Let q be the index of Q long in Q, and let f be the index of Q in P . Let Z = Z(G) denote the center of G.
For a positive integer ℓ, define the set P ℓ := {µ ∈ P ∩ h * + ; µ(H θ ) ≤ ℓ}. An element of P ℓ is said to be a weight of level ℓ. We denote by P ′ ℓ the subset of P ℓ consisting of elements µ satisfying µ(H θ ) < ℓ and µ(H α ) > 0 for any simple root α. By definition ofȟ, there is a bijection between sets P ℓ and P ′ ℓ+ȟ via µ → µ + ρ. Consider the maximal torus T of G with Lie algebra t. If t = exp X ∈ T , with X ∈ t, and α is a root (which takes imaginary values on t), we denote by e α (t) = e α,X . Let ∆(t) = α∈R (e α (t) − 1). An element of T is said to be regular if ∆(t) = 0. Denote by T ℓ the subgroup of elements t of T such that e α (t) is ℓ +ȟ root of unity for each long root α. We denote the set of regular elements in T ℓ by T reg ℓ . We now give the Verlinde formula.
We remark that if s i=1 λ i is not in the root lattice, then V (G, λ, g, ℓ) is zero.
Under the isomorphism given by the basic invariant form, an element a lying in A ⊂ h + defines an elementâ of h * + . We now consider a collection {a 1 , a 2 , . . . , a s } of rational elements in A, that is, each a j lies in the dense subset A ∩ (Q ⊗ Q) ⊂ A. We may choose ℓ large enough so that each λ j := ℓâ j is a weight; which then lies in P ℓ . We furthermore choose ℓ so that s j=1 λ j is in the root lattice and consider the space of conformal blocks V(G, g, s, λ, ℓ) associated to this collection λ = {λ 1 , λ 2 , . . . , λ s }. We can dilate simultaneously the weights λ j and the level ℓ by a factor k. Then, the function is a quasi-polynomial in k of degree m = dim(G)(g − 1) + s|R + |, the complex dimension of the moduli space M(G, g, s, C). The volume computes the highest term of this quasi-polynomial. More precisely, vol(M(G, g, s, C)) = lim k→∞ (ℓk) −m dim(V(G, g, s, [kℓâ 1 , . . . , kℓâ s ], kℓ). Proposition 3.1. Let a = {a 1 , · · · , a s } be a collection of regular rational elements in A. Let vol(G, g)(a) denote the symplectic volume of the moduli space M(G, g, s, C). Then, where p is the number of short positive roots of G, and ǫ G = 2 for any simple Lie group except G 2 , it is equal to 3 for G 2 .
We recall that for simply laced groups p = 0 since all roots are considered as long.
We now analyze the above formula as k gets large.
This expression explodes at each central vertex and the contribution from each, as k gets large, is (kℓ+ȟ) |R + | α>0 2iπ(α|µ+ρ) Also observe that, for z ∈ Z(G) both T reg ℓ and ∆(t) is invariant under t → tz. Moreover, since s i=1 λ i is in the root lattice by construction, χ V (kλ) (t) is also invariant. Therefore, we may add all these equal contributions from central vertices. (see also Remark 5.8. [15]). Hence, we get that the expression By virtue of the normalization in the basic invariant form, if α is a long root we have H α , γ = (α|γ); otherwise H α , γ = ǫ G (α|γ), where ǫ G = 2 for any simple Lie group except G 2 , it is equal to 3 for G 2 . Using also that the dimension of G is r + 2|R + |, and that |R + | = |Φ(G)|, we obtain lim k→∞ 1 (kℓ) m V (G, kλ, g, kℓ) as claimed.
Following the same line of arguments as in the proof of Proposition 3.1 we get that for s = 1 with λ = ℓâ lying in the root lattice, Hα∈Φ (2iπ H α , γ ) 2g−1 . That is, vol(G, g)(a) = (f q) g−1 |Z|ǫ Let us demonstrate this with an example. Example 3.3. We consider the moduli space of SU(2) bundles on a Riemann surface of genus g with one marking.
Let a = tH α be a regular element in A; in other words, 0 < t < 1/2. Letâ ∈ h * R denote the dual of a ∈ A under the isomorphism given by the basic invariant form. The Verlinde formula for SU(2) with marking λ = ℓâ = ℓtα ∈ P ℓ such that tℓ is a positive integer (that is, λ lies in the root lattice) is where c g,t (n) is the n th coefficient of the Taylor series expansion of e x(g−2t) x e x −1 2g−1 in x around zero, and B(p, t) denotes the p th Bernoulli polynomial in t. Clearly c g,t (0) = 1. In the expression for V (SU(2), tkℓα, g, kℓ) the highest term in k occurs when n = 0. Hence, We now calculate the volume using Equation (3.2.1).
where 2m is the real dimension of M(G, g, s, C), and IrrG denotes the set of irreducible representations of G. Above vol(G), vol(G/T ) are Riemannian volumes of G and G/T which we now express following Bourbaki (Ch. IX, pages 396-411): Choose a g-invariant scalar product. This determines a Lebesgue measure µ on g, via identification of g with R n by an orthonormal basis. Similarly let τ be the Lesbegue measure on t corresponding to the restriction of the scalar product on t. We can construct from µ and τ Haar measures µ G and µ T on G and T respectively.
Since we aim to compare the volume formula in Proposition 3.1 with that of Witten in Equation (3.3.1), we choose the normalized Killing form as the g-invariant scalar product in the above construction, as this was our choice in the previous section. Then, for this choice, with respect to µ G and µ T constructed as above, we get that .
Recall from the previous section that hence it takes positive values on a regular element t. Then, parametrizing irreducible representations of G with the cone of dominant weights P + , for C j the adjoint orbit of exp(a j ), we may write Observe that the summand above is invariant under the Weyl group (both the numerator and the denominator are anti-invariant by factor (sign(w)) s for a Weyl group element w). We get, Inserting the explicit expressions for the volume of G and G/T above into Equation (3.3.1), all (2π) factors cancel, and combining the terms we get vol(M (G, g, s, C) which is precisely the formula that we obtained in Proposition 3.1. Suppose that a is a regular element in A, in other words, 2a 1 − a 2 > 0 2a 2 − a 1 > 0 (in particular a 1 > 0 and a 2 > 0) and θ(a 1 H α 1 + a 2 H α 2 ) = a 1 + a 2 < 1.
We get, Example 4.4. With the notation of Example 4.3, we compute the volume of the moduli space of Spin(5) bundles on a Riemann surface of genus one and two markings. Let a = {a 1 , a 2 }, where a 1 and a 2 are regular elements in A. Write a 1 = t 1 e 1 + t 2 e 2 and a 2 = u 1 e 1 + u 2 e 2 . Then the function vol(B 2 , 1)(a) is a piecewise polynomial function of t 1 , t 2 , u 1 , u 2 . For example, choose v 1 = 1 2 e 1 + 1 5 e 2 , v 2 = 1 7 e 1 + 1 9 e 2 and consider τ (v) ⊂ A × A, the open set determined by the condition that a 1 + w(a 2 ) is in the same tope We compute some values.

More examples
In this section we will compute some instances of the Witten volume using the formula vol(G, g)(a) = W (Φ(G), P, g, s)(a)2 p(2g−2+s) (f q) g−1 |Z(G)|(−1) (g−1)|Φ(G)| |W | −1 as given in Proposition 3.1. We will denote by c vol the factor For convenience, we list values of the parameters in c vol for each type of classical Lie group in Table 1. 5.1. Tables of volumes of moduli spaces. We simply denote by vol(G, g) the Witten volume in the case of no marking, that is, when s = 0. We will list some values of vol(G, g) for classical Lie groups in Tables 2 and 3. We will also list some values of the factor c vol that we will need in Section 5.2 to compare our computations with other numerical results in literature.
Computations are very quick for rank less or equal to 4 (and relatively small genus). Beyond rank 5, computations cannot be made within a time limit of half-hour with our method.

Comparison results.
In this section we compare some of our computations of vol(G, g) with that of Komori-Matsumoto-Tsumura ( [7], [8], [9]). The setting is as follows.
As before, G is a simple, compact Lie group of rank r. We do not assume that G is simply-connected. Let L be the weight lattice of G. Let P be the weight lattice of the simply connected group covering G and let Q be its root lattice. Then, Q ⊂ L ⊂ P . Let P + be the 'cone' of dominant weights, and let L + = L ∩ P + .
Let s = [s α ] be a sequence of real variables indexed by the positive roots R + . For v ∈ h R , Komori-Matsumoto-Tsumura introduced If G is simply connected, then L = P , and we may denote ζ(s, v, G) by ζ(s, v, g), or for the Lie algebra g of type X r by ζ(s, v, X r ) as in [9]. The cone of dominant weights is the simplicial cone generated by fundamental weights ω 1 , ω 2 and ω 3 that are dual to simple coroots e 1 −e 2 , e 2 −e 3 and e 3 −e 4 respectively. Then, if we order the exponents s = [s i ] with respect to the order of the roots as given above, The series ζ(s, v, G) converges when the exponents s α are sufficiently large. It can be shown that ζ(s, v, G) can be continued as a meromorphic function of s. Let S = s α . Suppose s α are the same for all short roots, respectively for all long roots, and both are equal to positive even integers (that are not necessarily the same positive even integers). Then (2π) −S ζ(s, 0, G) is rational. Indeed, using the invariance of the sum under the Weyl group W , (2π) −S ζ(s, 0, G) is proportional to a Bernoulli series (with repetition of coroots in Φ matching the exponent data) which is obtained by summing over all the regular elements of the full lattice L. More precisely, where the series on the right hand side is a multiple Bernoulli series which has (in the case that it converges absolutely) rational value.
If all s α are equal to an even integer 2k, we denote the sequence s = [s α ] by s 2k . Then, for exponents s 2k , and G simply connected, we may compute ζ(s 2k , 0, G) using the Witten volume formula for g = k + 1, Above Φ(G) denotes the set of positive coroots as before. Thus we can use the values of the volume listed in the tables of the previous section to compute some instances of the series ζ(s 2k , 0, G).
We also give an example of D 4 with all exponents equal to 6 (that is k = 3 and s 6 = [6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6] ). It is also possible to compute ζ(s, 0, G) when the exponents in the list s = [s 1 , s 2 , s 3 , s 4 ] are different positive even integers for short and long roots. We conclude with one example of this kind.
Following [7], we want to demonstrate how the above series can be computed using the Bernoulli series B(H BC r ,Q C , g C s )(0) for the root system of type C r , where the exponents s = [s α ] are taken to be 0 for long positive roots, and 2k for short positive roots. Using the invariance of the sum under the Weyl group, which is of order 2 r r! for C r , we may write B(H BC r ,Q C , g C s )(0) = 2 r r! γ∈(P + C )reg 1 α>0 (2iπ H α , γ ) sα .

Appendix: Szenes formula
Let H be an arrangement of hyperplanes compatible with a lattice Λ. Let g ∈ R H . Consider This function (a generalized function on V ) coincide with a polynomial function B(H, Λ, g, τ ) on a tope τ (see Proposition 1.30). The piecewise polynomial function P (H, Λ, g) has been defined in Definition 1.16. Following Szenes [13], we prove the following formula. Here R : R H → S H is the total residue. and we can restrict φf to φ = 0. This operation kills the functions f having no poles of φ = 0.
This is easy to verify using for example a decomposition of f ′ with denominator on a set of independent hyperplanes (see Lemma 1.3).
The map res φ is well defined on S H , as elements in S H have at most a simple pole on φ = 0. It is easy to prove that we have the exact sequence Let v ∈ V reg (Λ, H). Its projection v 0 = p(v) belongs to V reg (Λ 0 , H 0 ).
Proof. We have If γ is such that φ, γ = 0, then the term f (2iπγ − z) has no pole on φ = 0. Thus we obtain, for z ∈ H φ , Let g ∈ R H ′ , and let g 0 be its restriction to H φ . Then the operator A(v, g) leaves S H ′ stable.
If F has at most a simple pole on φ = 0, then gF also has at most a simple pole on φ = 0, as g has no pole on φ = 0. Thus the maps in the diagram below are well defined. Its commutativity follows from Lemma 6.3. Lemma 6.4. Let g ∈ R H ′ . Then the following diagram is commutative.
We are now ready to prove Theorem 6.1 by induction on the number of hyperplanes in H. If there are less than r hyperplanes, then S H = {0}, the generalized function B(H, Λ, g) is supported on affine walls, so both sides of the equation of Theorem 6.1 vanish.
Assume that H consists of r independent hyperplanes intersecting on {0}. Changing the lattice Λ, we can eventually assume that Λ is the lattice generated by the equations φ k of the hyperplanes. Then, the theorem follows from Formula (6.1.2) in the one dimensional case.
Assume that H have more than r hyperplanes. Then by the Lemma 1.3, we can write a function in R H as a sum of functions g whose poles lie on an independent subset of hyperplanes of H, thus in number less or equal to r. Thus R H is linearly generated by functions g such that some equation φ ∈ Φ eq is not a pole of g. We consider such a couple (g, φ) and the arrangements H ′ and H 0 associated to φ by deletion and contraction. The function g is in R H ′ .
Let g 0 ∈ R H 0 be the restriction of g to H φ . Thus B(H 0 , Λ 0 , g 0 ) is a generalized function on H * φ = V /Rφ and p * B(H 0 , Λ 0 , g 0 ) is a function on V (constant in the direction φ).
Comparing with Proposition 6.5, we see by induction that Szenes formula holds.