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\begin{document}

\title{Multiple Bernoulli series  and volumes of moduli spaces 
of flat bundles over surfaces}

\author[vb]{Velleda Baldoni}
\ead{baldoni@axp.mat.uniroma2.it}

\author[ab]{Arzu Boysal}
\ead{arzu.boysal@boun.edu.tr}

\author[mv]{Mich\`ele Vergne}
\ead{vergne@math.jussieu.fr}

\address[vb]{Universita degli Studi di Roma Tor Vergata, Italy}
\address[ab]{Bo\u{g}azi\c{c}i University, Turkey}
\address[mv]{Institut de Math\'{e}matiques de Jussieu, France}

\begin{abstract}
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.
\end{abstract}

\begin{keyword}
multiple Bernoulli series \sep moduli spaces of flat connections \sep multiple zeta values
\end{keyword}

\maketitle

%{\small \tableofcontents}

\section*{Introduction}
Let $V$ be a finite dimensional real vector space, and $\Lambda$ a
lattice in $V$.  We denote the dual of $\Lambda$ by $\Gamma$.

We consider a finite sequence of vectors $\Phi$ lying in $\Lambda$,
and let $\Gamma_{\reg}(\Phi)=\{\gamma\in \Gamma |\; \ll \phi,\gamma
\rr \neq 0,\;\mbox{for all} \;\phi\in \Phi\}$ be the set of regular
elements in $\Gamma$ relative to $\Phi$.

In this paper we compute
\begin{equation}\label{eq:B}
\mathcal{B}(\Phi, \Lambda)(v)=\displaystyle\sum_{\gamma \in \Gamma_{\reg}(\Phi)}
\frac{e^{\ll 2i\pi v ,\gamma \rr}}{ \prod_{\phi \in \Phi} \ll 2i\pi
\phi,\gamma \rr}, \end{equation} a function  on the torus
$V/\Lambda$. This sum, if not absolutely convergent, has a meaning
as a generalized function. If $\Phi$ generates $V$, then
$\mathcal{B}(\Phi,\Lambda)$ is \textit{piecewise polynomial} (see \cite{sze1}).
%(cf. Proposition \ref{prop:polyn}).

For example, for $V=\R e_1\oplus \R e_2$ with standard lattice $\Lambda=\Z e_1\oplus \Z e_2$,
if we choose  $\Phi=[e_1,e_1,e_2, e_1+e_2,e_1-e_2]$, then
$$\mathcal B(\Phi,\Lambda)(v_1 e_1+v_2 e_2)=\sum'_{n_1,n_2} \frac{e^{2i\pi (v_1 n_1+v_2 n_2)}}{(2i\pi n_1)^2(2i\pi n_2) (2i\pi (n_1+n_2)) (2i\pi (n_1-n_2))},$$
where the summation $\sum'$ means that we sum only over the integers
$n_1$ and $n_2$ such that $n_1n_2(n_1+n_2)(n_1-n_2)\neq 0$. The
expression for $\mathcal B(\Phi,\Lambda)(v_1e_1+v_2 e_2)$ as a
piecewise polynomial function  of $v_1$ and $v_2$ (of degree $5$) is
given in Section \ref{section:clroot}, Equation (\ref{eq:BB2}).


We call $\mathcal{B}(\Phi, \Lambda)$ the {\it multiple Bernoulli series}
associated to $\Phi$ and $\Lambda$.  Multiple Bernoulli series have
been  extensively studied by A. Szenes (\cite{sze1},\cite{sze2}). They are natural  generalizations of Bernoulli series:
for $V=\R \omega$, $\Lambda=\Z \omega$ and $\Phi_k=[\omega,\omega,\ldots, \omega]$,
where  $\omega$ is repeated $k$ times with $k>0$, the function 
$$\mathcal{B}(\Phi_k,\Lambda)(t\omega)=\sum_{n \in \Z,\;n\neq 0} \frac{e^{2i\pi nt}}{(2i\pi n)^k}$$ is equal to 
$-\frac{1}{k!} B(k,\{t\})$ where $B(k,t)$ denotes the $k^{\text{th}}$ Bernoulli polynomial in
variable $t$, and $\{t\}=t-[t]$ is the fractional part of $t$.  If $k=2g$ and $t=0$,  due to the symmetry $n\to -n$,
$$\mathcal{B}(\Phi_{2g},\Lambda)(0)=2\frac{1}{(2i\pi)^{2g}}\zeta(2g).$$
From the residue theorem in one variable, for $k>0$,   $$\sum_{n\in \Z,\;n\neq 0} \frac{e^{2i\pi nt}}{
(2i\pi n)^k}={\rm Res}_{z=0}(
\frac{1}{z^k} e^{zt}\frac{1}{1-e^z}).$$
Szenes multidimensional residue formula (cf.~Theorem \ref{theo:main}) is the generalization  of this formula to higher
dimensions, and it is the tool that we use for computing
$\mathcal{B}(\Phi, \Lambda)(v)$ as a piecewise polynomial function.

A particular but crucial instance of multiple Bernoulli series is
when $\Lambda$ is the coroot lattice of a compact connected simple Lie group $G$, and  $\Phi$
is comprised of positive coroots of $G$.   The series
$\mathcal{B}(\Phi_{2g-2},\Lambda)$, where the argument $\Phi_{2g-2}$  refers to taking elements of $\Phi$ with
multiplicity $2g-2$, appeared in the work of E.~Witten (\cite{wi}, \S 3), where Witten shows that its value at $v=0$
(up to a scalar depending on $G$ and $g$) is the symplectic volume of the moduli space of flat
$G$-connections on a Riemann surface of genus $g$.
Similarly, for a collection of regular elements ${\bf v}=\{v_1,\ldots,v_s\}$ of the Cartan Lie algebra of $G$,
certain linear combinations of $\mathcal{B}(\Phi_{2g-2+s},\Lambda)$ at some particular values
(depending on ${\bf v}$) 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
$\bf v$. Then, its dependence on $\bf v$ is piecewise polynomial.

Multiple Bernoulli series have also been studied by P.E.~Gunnells and R.~Sczech (\cite{gs}) 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 \cite{gs}.  Yet, the techniques they use is a  generalization of the continued
fraction algorithm and it is different from ours.

Y. Komori, K. Matsumoto and H. Tsumura
(\cite{kmt0},\cite{kmt1},\cite{kmt2},\cite{kmt3}) studied the restriction of the
series (\ref{eq:B}), by summing it over the cone of dominant regular
weights of a semi-simple Lie group $G$, and defined  a function
$\zeta({\bf s}, v, G)$ (cf.~ Section \ref{comp}). They also obtained
relations between these functions over $\Q$. When $\Lambda$ is the
coroot lattice of a compact connected simple Lie group $G$ and  the
sequence $\Phi$ 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 $\mathcal{B}(\Phi,\Lambda)(0)$ over the full (regular)
weight lattice is just (up to multiplication by an appropriate power
of $(2\pi)$) Komori-Matsumoto-Tsumura  zeta function $\zeta({\bf
s},0,G)$. Thus, the value of $\zeta({\bf s},0,G)$  (up to a certain
power of  $(2\pi)$) is a rational number which can be computed
explicitly, we give examples of such computations.  As it is observed in \cite{kmt0},
some instances of $\zeta({\bf s}, v, G)$ also compute certain multiple zeta values.  
In the last part of the article we give various such computations of multiple zeta values using $\mathcal{B}(\Phi,\Lambda)$.


\medskip

Here is the outline of individual sections.

\medskip

In Section 1, we recall a formula due to A. Szenes, which allows an efficient computation of
$\mathcal{B}(\Phi,\Lambda)$.


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
\cite{kmt0}, \cite{kmt1}, \cite{kmt2} and \cite{kmt3}. 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 $\Phi$.
Following Komori-Matsumoto-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 slightly modified proof of
Szenes formula.


\subsection*{Acknowledgements}  Part of this work was completed when all three authors
were at Mathematisches Forschunginstitut Oberwolfach as a part of
Research in Pairs programme in February 2012. We would like to
express our gratitude to the institute for their hospitality.

The first author was partially supported by a PRIN$2009$ grant, the
second author was partially supported by Bo\u{g}azi\c{c}i University (B.U.)
Research Fund $5076$.  The third author wishes to thank Bo\u{g}azi\c{c}i
University for support of research visits.

We also thank Shrawan Kumar, and the referees for their suggestions.


\section{Szenes formula for multiple  Bernoulli  series}\label{sub:hyper}

\subsection{Functions on the complement of  hyperplanes}
In this subsection $U$ is an $r$-dimensional  complex vector space. We denote its dual by $V$.
If $\phi\in V$, we denote by $H_\phi=\{u\in U;\ll\phi,u\rr=0\}$ the hyperplane in $U$ determined by $\phi$.  

We recall briefly some structure theorems for the ring of
rational functions that are regular on the complement of a union of
hyperplanes \cite{brver1}.

Let $\CH=\{H_1,\ldots, H_N\}$ be  a  set of   
hyperplanes in $U$. Then, we may choose $\phi_k \in V$ such that $H_k=H_{\phi_k}$;  the element $\phi_k$ is
called an equation of $H_k$.  Clearly, an equation $\phi_k$ is not unique, it is determined up to a non-zero scalar multiple.
Consider  $U_{\mathcal H}:=\{u\in U; \langle \phi_k,u\rangle\neq
0 \,\, {\rm for \, all}\, k\,\},$  an open subset of $U$; an
element of $U_{\mathcal H}$ will be called {\it regular}.

%\begin{definition}
We denote by $S(V)$ the symmetric algebra of $V$  and identify it with
the ring of polynomial functions on $U$.  Let $\CR_{\mathcal H}$ denote the ring of rational
functions on $U$ that are regular on $U_\CH$, that is, the ring
generated by $S(V)$ together with inverses of the linear forms
$\phi_k$ defining $\CH$.
%\end{definition}

The ring $\mathcal D(U)$ of differential operators on $U$ with polynomial coefficients acts on $\CR_{\mathcal H}$.
In particular, $U$ operates on $\CR_\mathcal H$ by differentiation.
We denote by $\partial(U)\CR_\CH$ the subspace of $\CR_\mathcal H$ obtained by differentiation.

\medskip
In the particular case that $V$ is one dimensional and $\mathcal H=\{0\}$, the ring $\CR_\CH$ is the ring of Laurent polynomials $\C[z,z^{-1}]$, and the function $z^i$, for $i\neq -1$, is obtained as a derivative $\frac{d}{dz}\frac{1 }{i+1} z^{i+1}$.  Thus $\CR_\CH=\frac{d}{dz}\CR_\CH\oplus \C z^{-1}$.

If $f=\sum_{n} a_n z^n$ 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 $f$.
The linear form $\Res_{z=0}$ is characterized by the fact that it vanishes on $\frac{d}{dz} \CR_\CH$ and takes the value $a_{-1}$ on 
$\C z^{-1}$.
\medskip

By analogy to the one dimensional case demonstrated above, a linear functional on
$\CR_{\CH}$ vanishing on   $\partial(U)\CR_\CH$  will be called a
`residue'.

\medskip
Let us thus analyze the space   $\CR_{\CH}$ modulo  $\partial(U)\CR_\CH$.
\medskip

Consider  a set  $\Phi^{eq}:=\{\phi_1,\phi_2,\ldots, \phi_N\}$  of equations for $\mathcal H$. A subset
$\sigma$ of $\Phi^{eq}$ will be called a \textit{basis} if the elements $\phi_k$  in $\sigma$ form a basis of $V$.
We denote by $\mathfrak{B}(\Phi^{eq})$ the set of such subsets $\sigma$.
A subset  $\nu$ of $\Phi^{eq}$ will be called  \textit{generating} if  the elements $\phi_k$  in $\nu$ generate the vector space $V$.

\begin{definition}\label{def:simple} Let $\sigma:=\{\alpha_1,\alpha_2,\ldots,\alpha_r\} \in \mathfrak{B}(\Phi^{eq})$.  The  element
$$f_\sigma(z):=\frac{1}{\prod_{k=1}^r \alpha_k(z)}$$  of $\CR_{\mathcal H}$ is called a `simple fraction'.
We denote by $\CS_{\mathcal H}$ the subspace of $\CR_{\mathcal H}$ generated by the elements $f_\sigma, \, \sigma \in \mathfrak{B}(\Phi^{eq})$.
\end{definition}

Let $\nu=[\alpha_1,\ldots, \alpha_k]$ be  a sequence of $k$ elements of $\Phi^{eq}$ and ${\bf n}=[n_1,n_2,\ldots,n_k]$ be a sequence of positive integers.
We define
$$\theta(\nu,{\bf n})=\frac{1}{\alpha_1^{n_1}\cdots \alpha_{k}^{n_k}}.$$
We denote by ${\mathcal G}_{\mathcal H}$ the subspace of $\CR_{\mathcal H}$ generated by the elements
$\theta(\nu,\bf{n})$ where $\nu$  is generating.


As the notation suggests, the spaces $\CR_\CH$, $\CS_\CH$ and  ${\mathcal G}_{\mathcal H}$ depend only on $\CH$.
The term simple fraction comes from the fact that if $\sigma=\{\phi_1,\phi_2,\ldots,\phi_r\}$ is a basis, then we can choose coordinates $z_i$ on $U$ so that $\phi_i(z)=z_i$, so that for this system of coordinates
$f_\sigma(z)=\frac{1}{\prod_{i=1}^r z_i}$.

\medskip

We recall the following `partial fraction' decomposition result from \cite{brver1}.
\begin{lemma}\label{ind}
Let $\nu$ be a subset of $\Phi^{eq}$ generating a $t$ dimensional subspace of $V$.
 Then $\theta(\nu,{\bf n})$ may be written as a linear combination of elements
$\theta(\sigma,{\bf m})=\frac{1}{\alpha_{i_1}^{m_1}\cdots \alpha_{i_t}^{m_t}}$ where
$\sigma:=\{\alpha_{i_1}, \ldots,\alpha_{i_t}\}$ is a subset of $\nu$ consisting of $t$ independent elements
and ${\bf m}=\{m_1,\ldots, m_t\}$ a sequence of positive integers.
\end{lemma}

\begin{example}
$\displaystyle \frac{1}{z_1z_2(z_1+z_2)}=\frac{1}{z_1(z_1+z_2)^2}+\frac{1}{z_2(z_1+z_2)^2}.$
\end{example}

\begin{theorem}(Brion-Vergne \cite{brver1})
$$\CR_{\mathcal H}=\partial(U)\CR_{\mathcal H}\oplus \CS_{\mathcal H}.$$
\end{theorem}

The projector  ${\bf R}: \CR_{\mathcal H}\to \CS_{\mathcal H}$ will be called the
\textit{total residue}.  In view of this theorem, a residue is just a linear form on
$\CS_{\mathcal H}$.  

In the case that $\mathcal H$ is the set of hyperplanes
with equations the positive coroots of a simple compact Lie group
$G$, the dimension of $\CS_{\mathcal H}$ is given by the product of
exponents of $G$ \cite{proc}. In Section \ref{section:clroot}, we
will give an explicit basis for $\CS_{\mathcal H}$ 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.


\subsection{Szenes polynomial}
In this section and for the rest of the article, $V$ will denote a {\bf real} vector space of dimension $r$.

Let $U$ be the dual vector space of $V$. Let $\Lambda$ be a lattice in $V$ with dual lattice $\Gamma$ in $U$.

Let $\CH:=\{H_1,H_2,\ldots, H_N\}$ be a real arrangement of  hyperplanes  in $U$.
We say that $\Lambda$ and $\CH$ are \textit{compatible} if the hyperplanes in $\CH$ are rational with respect to
$\Lambda$, that is, they can be defined by equations $\phi_k\in \Lambda$.  If $\Lambda'$ is another lattice commensurable
with $\Lambda$, then   $\Lambda'$ and $\CH$ are also compatible.


Thus we now consider a lattice $\Lambda$ and a real arrangement of hyperplanes
$\CH=\{H_1,H_2,\ldots, H_N\}$ in $U$ rational
with respect to $\Lambda$.

We choose  $\Phi^{eq}=\{\phi_1,\phi_2,\ldots,\phi_N\}$, a set of defining equations for $\CH$, with each
$\phi_i$ in $\Lambda$. We sometimes refer to $\CH$ only via its set of equations $\Phi^{eq}$ and
write $\CH=\cup\{\phi_k=0\}$.

We denote  the complex arrangement defined by $\cup\{\phi_k=0\}$ in $U_{\C}$ with the same letter $\CH$, and 
denote by $U_\mathcal H=\{\prod_k \phi_k\neq 0\}$ the corresponding open subset of $U_\C$.

An \textit{admissible hyperplane} $W$ in $V$  (for the system $\CH$) is an hyperplane generated by $(r-1)$ linearly
independent elements $\phi_k$ of $\Phi^{eq}$. Such an hyperplane will also be called an (admissible) \textit{wall}.
An admissible \textit{affine} wall is a translate of a wall by an element of $\Lambda$.

An element $v\in V$ is called \textit{regular} for $(\CH,\Lambda)$  if $v$ is not on any affine wall
(we will simply say that $v$ is  regular).  The meaning of the word regular is thus different for elements $v\in V$ ($v$ is not on any affine wall) and $u\in U_\C$
($u$ is such that $\prod_k \langle \phi_k,u\rangle\neq 0$). However, it will be clear what regular means in the context.

A \textit{tope} $\tau$ is a connected component of the complement of all affine hyperplanes. Thus a tope $\tau$ is a
connected open subset of $V$ consisting of regular elements.
We denote the set of topes by $\mathcal{T}(\CH,\Lambda)$. As the notation indicates, $\mathcal{T}(\CH,\Lambda)$ does 
not depend on the choice of  equations  for $\CH$.

\begin{example}\label{ex:two}
Let $V=\R e_1\oplus \R e_2$  and  $\Lambda=\Z e_1 \oplus \Z e_2$.  Let $U$ be its dual with basis $\{e^1,e^2\}$.
We express $z\in U_\C$ as $z=z_1 e^1+z_2 e^2$, and consider  the set of hyperplanes $\CH=\{\{z_1=0\}, \{z_2=0\}, \{z_1+z_2=0\}\}$
with the set of equations $\Phi^{eq}=\{e_1,e_2, e_1+e_2\}$.
Figure \ref{a2} depicts topes associated to this pair.

\begin{figure}
\centering
\begin{minipage}[b]{0.45\linewidth}
\quad \quad\includegraphics[width=37mm]{tope2mod.mps}
 \caption{{\small Topes for Example \ref{ex:two}}}\label{a2}
\end{minipage}
\begin{minipage}[b]{0.45\linewidth} 
\quad \quad \includegraphics[width=37mm]{tope3mod.mps}
 \caption{{\small Topes for Example \ref{ex:btwo}}}\label{b2}
\end{minipage}
\end{figure}
\end{example}

\begin{example}\label{ex:btwo}
With $V$ and $\Lambda$ as above, we now consider the set of hyperplanes
$\CH=\{\{z_1=0\}, \{z_2=0\}, \{z_1+z_2=0\}, \{z_1-z_2=0\}\}$
with the set of equations $\Phi^{eq}=\{e_1,e_2, e_1+e_2, e_1-e_2\}$.
Figure \ref{b2} depicts topes associated to this pair.
\end{example}


We denote by $V_{reg}(\CH,\Lambda)$ (or simply $V_{reg}$) the set of ($\CH,\Lambda$) 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 $t\in \R$, we denote by $[t]$ the integral part of $t$, and by $\{t\}=t-[t]$ the fractional part of $t$.
If $\gamma \in \Gamma$ vanishes on an admissible hyperplane $W$, and $c$ is a constant, then the function
$v\to \{\langle\gamma,v\rangle+c\}$ is piecewise polynomial (piecewise linear) and is periodic with respect to $\Lambda$.
Szenes residue formula provides an  algorithm  to describe
 Bernoulli series in terms of these basic functions.


\begin{definition}\label{def:mh} Let $\mathcal M_{\mathcal H}$ be the space of functions $h/Q$ where $Q$ is a product of
linear forms belonging to $\Phi^{eq}$, and $h$  a holomorphic function defined in a neighborhood of $0$ in $U_\C$.

We define the space $\hat \CR_{\mathcal H}$ as the space of functions $\hat h /Q$ where
$\hat h=\sum_{k=0}^{\infty} P_k$ is a  formal power series and $Q$ is a product of linear forms belonging to
$\Phi^{eq}$.
\end{definition}

Taking the Taylor series $\hat h$ of $h$ at $0$ defines an
injective map from  $\mathcal M_{\mathcal H}$ to $\hat \CR_{\mathcal H}$.
The projector ${\bf R}$ from $\CR_{\mathcal H}$ to $\CS_{\mathcal H}$ extends to $\hat \CR_{\mathcal H}$.
Indeed ${\rm {\bf R}}$ vanishes outside the homogeneous components of degree $-r$ of the graded space $\CR_{\mathcal H}$.
Thus if $h/Q$ is an element in $\mathcal M_{\mathcal H}$, with $Q$ a product of $N$ elements of $\Phi^{eq}$, we take the Taylor series $[h]_{N-r}$ of $h$ up to order $N-r$, and define ${\rm {\bf R}}(\frac{h}{Q})={\rm {\bf R}} (\frac{[h]_{N-r}}{Q}).$
For example, the equality
$$\frac{e^{zt}}{e^z-1} = \frac{1}{z} (z\frac{e^{zt} }{e^z-1})$$
identifies  the function  $\frac{e^{zt}}{e^z-1}$ to an element of $\mathcal M_{\mathcal H}$ with ${\mathcal H}=\{0\}$.
Note that each homogeneous term of the Taylor  series expansion
$$z\frac{e^{zt}}{e^z-1}=\sum_{k=0}^{\infty} B(k, t)\frac{z^k}{k!},$$ where $B(k,t)$ is the $k^{\text{th}}$ Bernoulli
polynomial in $t$, is thus a polynomial in $t$.


Let $f\in \CR_{\CH}$, $z\in U_{\CH}$ and $\gamma\in \Gamma$.
Then if $z$ is small,   $2i\pi\gamma-z$ is still a regular element of $U_\C$. Now consider the series
$$S(f,z,v)= \sum_{\gamma\in \Gamma}  f(2i\pi\gamma-z) e^{ \langle v,2i\pi \gamma\rangle }.$$
In the case that $f$ above decreases sufficiently quickly at infinity, the series
$S(f,z,v)$ is absolutely convergent and defines a continuous function of $v$.
In general, as $f \in \CR_{\CH}$ is of at most polynomial growth,  $\sum_{\gamma\in \Gamma} f(2i\pi\gamma-z) e^{ \langle v,2i\pi \gamma\rangle }$
is the Fourier series of a  generalized function on  $V/\Lambda$.

Multiplying $S(f,z,v)$ by the exponential $e^{-\langle z,v\rangle}$ we introduce the following generalized function.

\begin{definition}
Let $f\in \CR_{\CH}$, $z\in U_{\CH}$ and small. We define the generalized function $\AV^{\Lambda}(f)(z,v)$ of $v$ by
$$\AV^{\Lambda}(f)(z,v)= \sum_{\gamma\in \Gamma}  f(2i\pi\gamma-z) e^{ \langle v,2i\pi \gamma-z\rangle }.$$
\end{definition}

The meaning of $\AV^{\Lambda}(f)$ is clear : average the function  $z\mapsto  f(-z)e^{-\langle v,z \rangle}$ over
$2i\pi \Gamma$ in order to obtain a function   on the complex torus $U_\C/2i\pi \Gamma$.
We  consider  $\AV^{\Lambda}(f)(z,v)$  as a  generalized function of $v\in V$ with coefficients meromorphic functions
of $z$ on $U_\C/2i\pi\Gamma$.  In fact, as we shall see in Proposition \ref{pro:eis_sh},
when $f$ is in $\CS_\CH$, the convergence of the series
$\sum_{\gamma\in \Gamma}  f(2i\pi\gamma-z) e^{ \la v,2i\pi \gamma \ra }$ holds in the sense of the Fourier series of an
$L^2$- periodic function of $v\in V/\Lambda$, and
$$v\mapsto e^{-\ll v,z\rr}\left( \sum_{\gamma\in \Gamma}  f(2i\pi\gamma-z) e^{ \langle v,2i\pi \gamma\rr }\right)=\AV^{\Lambda}(f)(z,v)$$
is a locally constant function of  $v\in V_{reg}$ with values in $\CM_{\CH}$.

%Thus, when $f\in \CS_\CH$, and $v$ varies in a tope $\tau$, the dependance of $v$ of the function
%%$$S(f,z,v)=e^{\langle z,v\rangle } \AV^{\Lambda}(f)(z,v)$$
%%is simply through the factor $e^{\langle z,v\rangle }$.
%%


Note the covariance relation. For $\lambda\in \Lambda$,

\begin{equation}\label{eq:covariance}
\AV^{\Lambda}(f)(z,v+\lambda)=e^{-\ll \lambda,z\rr}\AV^{\Lambda}(f)(z,v).
\end{equation}

It is easy to compare $\AV^{\Lambda}(f)(z,v)$ when we change the lattice $\Lambda$.


\begin{lemma}\label{lem: first compare}
Let $f\in \CR_\CH$. If $\Lambda^1\subset \Lambda^2$,  then
\begin{equation}\label{eq:SZlattices}
\AV^{\Lambda^2}(f)(z,v)=|\Lambda^2/\Lambda^1|^{-1}
\sum_{\lambda\in \Lambda^2/\Lambda^1}\AV^{\Lambda^1}(f)(z,v+\lambda).
\end{equation}
\end{lemma}

\begin{proof}
Denote the dual of $\Lambda^i$ by $\Gamma^i$.  Then,
$$\AV^{\Lambda^1}(f)(z,v+\lambda)= \displaystyle \sum_{\gamma\in \Gamma^1} f(2i\pi\gamma-z) e^{ \langle v+\lambda,2i\pi \gamma-z\rangle },$$
and the sum over $\lambda\in \Lambda^2/\Lambda^1$ of
$e^{\ll \lambda,2i\pi \gamma\rr}$ is zero except when $\gamma\in \Gamma^2$.
\end{proof}

\begin{example}\label{ex:dim1Eis}
Let $V=\R$, $\Lambda=\Z$, $z\in \C\setminus{\{0\}}$ and small, and $f(z)=\frac{1}{z}$. Then,
%For $v\in \R$
$$\AV^{\Lambda}(f)(z,v)=\sum_{n\in \Z}  \frac{e^{ v(2i\pi n-z)}}{(2i\pi n -z)}= e^{-zv}\sum_{n\in \Z}  \frac{e^{ 2i\pi n v}}{(2i\pi n -z)}.$$
This series is not absolutely convergent, but the  oscillatory factor $e^{2i\pi nv}$ insures the convergence in the distributional sense as a function of $v$.
The $L^2$-expansion of the periodic function
$v\mapsto \frac{e^{(v-[v])z}}{1-e^z}$ is
$$\sum_{n\in \Z} \left(\int_{0}^1 \frac{e^{(v-[v])z}}{1-e^z} e^{-2i\pi n v} dv\right) e^{2i\pi n v}=
\sum_{n\in \Z} \left(\int_{0}^1 \frac{e^{v(z-2i\pi n)}}{1-e^z} dv\right) e^{2i\pi n v}
$$
$$=\sum_{n\in \Z}  \frac{e^{ (z-2i\pi n)}-1}{(1-e^z)(z-2i\pi n)} e^{2i\pi n v}=\sum_{n\in \Z}  \frac{1}{(2i\pi n -z)} e^{2i\pi n v}.$$
Thus
%We can take the derivative with respect to $v$, and obtain
% $$\partial_v (\AV^{\Lambda}(f)(z,v))=\sum_{n\in \Z}  e^{ v(2i\pi n-z)}=e^{-vz}\delta_\Z(v).$$
%Thus, for $v$ in each open interval $]n,n+1[$, the derivative is $0$, so that the function is constant. Passing through an integer $n$, the constant value changes by
%$e^{-nz}$. More precisely, in this one  dimensional case, we obtain
\begin{equation}\label{eq:AV}
\AV^{\Lambda}(f)(z,v)=\frac{e^{-[v]z}}{1-e^z}.
\end{equation}
%(recall that  $[v]$ denotes the integral part of $v$).
\end{example}
We see in this one dimensional example  that $\AV^{\Lambda}(f)(z,v)$ is a locally constant function of $v$.
In general, we have the following proposition.

\begin{proposition}\label{pro:eis_sh}
If $f\in \CS_\CH$, the function
$v\to \AV^{\Lambda}(f)(z,v)$ is a locally constant function on $V_{reg}$, with values in $\CM_\CH$.
\end{proposition}


We prove the above proposition by computing
$\AV^{\Lambda}(f)(z,v)$ explicitly for a simple fraction $f=f_\sigma$.  Then the result will follow for any $f\in \CS_\CH$.

Recall that the set of equations $\Phi^{eq}$ is a subset  of  $\Lambda$.
Let $\sigma=\{\alpha_1,\alpha_2,\ldots,\alpha_r\}$ be an element of $\mathfrak{B}(\Phi^{eq})$, with $\alpha_k$ belonging 
to $\Lambda$. Let $Q_\sigma:=\oplus_{k=1}^r[0,1)\alpha_k$  be the semi-open parallelepiped spanned by $\sigma$.
For a regular element $v$ in $V$, we define
$\Trans(v,\sigma)$  to be the set of elements $\lambda\in \Lambda$ such that $v+\lambda\in Q_\sigma$.
%\end{definition}
This set depends only on the tope $\tau$ where $v$ belongs, hence we denote it by $\Trans(\tau,\sigma)$.

Let $\Lambda_\sigma$ be the sublattice  of $\Lambda$ generated by the elements in the basis $\sigma$. Then
the set $\Trans(\tau,\sigma)$ contains exactly $\Lambda/\Lambda_\sigma$ elements.

\begin{proposition}\label{pro:formulaEIS}
If $v\in \tau$ and $\sigma=\{\alpha_1,\alpha_2,\ldots,\alpha_r\} \in \mathfrak{B}(\Phi^{eq})$,
$$\AV^{\Lambda}(f_\sigma)(z,v)=\frac{1}{|\Lambda/\Lambda_{\sigma}|} \sum_{\lambda\in \Trans(\tau,\sigma)} \frac{e^{\langle \lambda,z\rangle }}{\prod_{i=1}^r (1-e^{\langle \alpha_i,z\rangle})}.$$
\end{proposition}

\begin{proof}
If $\Lambda=\Lambda_\sigma$, the formula reduces to the one  dimensional case. Otherwise, we use Lemma
\ref{lem: first compare} and the covariance relation (\ref{eq:covariance}).
\end{proof}


Thus the dependence of 
$\AV^{\Lambda}(f_\sigma)(z,v)$ on $v$ is only via the tope $\tau$ where it belongs.
Therefore, for any $f\in \CS_\CH$, the function $\AV^{\Lambda}(f)(z,v)$ is locally constant on $V_{reg}$.  
Hence the claim in Proposition \ref{pro:eis_sh}.

\begin{example}
We return to Example \ref{ex:two}.
To describe $\AV^{\Lambda}(f)(z,v)$ on $V_{reg}$, it suffices
to give its expression on topes $\tau_1$ and $\tau_2$ (see Figure \ref{a2}).  This is true since any
element of $V_{reg}$ can be translated to $\tau_1$ or $\tau_2$ by an element of $\Lambda$, and then one uses the
covariance relation (\ref{eq:covariance}). 

We choose $\sigma=\{e_1,e_1+e_2\}$ as a basis of $\Phi^{eq}$, and write $z=z_1e^1+z_2 e^2$.  Then
$f_\sigma(z)=\frac{1}{z_1(z_1+z_2)}$ is in $S_{\CH}$. If  $v\in \tau_1$, we have
$\AV^{\Lambda}(f_\sigma)(z,v)=\frac{e^{z_1}}{(1-e^{z_1})(1-e^{z_1+z_2})},$
while if $v\in \tau_2$, then
$\AV^{\Lambda}(f_\sigma)(z,v)=\frac{1}{(1-e^{z_1})(1-e^{z_1+z_2})}.$
\end{example}

\medskip

For $v\in V_{reg}$, denote by
$\SZ^{\Lambda}(v): \CS_{\mathcal H}\to \mathcal M_{\mathcal H}$
the map $$(\SZ^{\Lambda}(v)f)(z)=\AV^{\Lambda}(f)(z,v).$$
This operator is locally constant.
We denote its value on $\tau$ by $\SZ^{\Lambda}(\tau)$.  Hence
$(\SZ^{\Lambda}(\tau)f)(z)=\AV^{\Lambda}(f)(z,v)$
for any choice of $v\in \tau$.


We now define a  piecewise polynomial function of $v$ associated to a  function $g(z)$ in $\CR_\CH$.
%First, the operator  on  $\CM_\CH$   given by  multiplication by a function $h(z)$,  that is $f(z)\mapsto h(z)f(z)$,  is
%simply denoted  by $f\mapsto hf$.
If $v\in V$ and  $g\in \CR_{\mathcal H}$, then $g_v(z):=g(z)e^{\ll z,v\rr}$ is a function in $\CM_\CH$ depending on $v$.
Now suppose $v\in V_{reg}$, and consider the map $\CS_\CH\to \CM_\CH$ which associates to
$f\in \CS_\CH$ the function $g(z)e^{\ll z,v\rr}(\SZ^{\Lambda}(v)f)(z)$. We project back this function on $\CS_\CH$ using the projector
${\bf R}$. Thus the map
\begin{equation}\label{Sz}
f(z)\mapsto  {\bf R}\left(g(z)e^{\ll z,v\rr}
(\SZ^{\Lambda}(v)f)(z)\right)
\end{equation}
is a map from $\CS_\CH$ to $\CS_\CH$ depending on $v$.
As $\CS_\CH$ is finite dimensional, we can take the trace of this operator, and thus obtain a function of $v\in V_{reg}$.
Let us record this definition.


\begin{definition}\label{def:sz}
Let $g\in \CR_{\CH}$.
Define  the function $P(\CH,\Lambda,g)$  on $V_{reg}(\CH,\Lambda)$ by $$P(\CH,\Lambda,g)(v):=Tr_{\CS_\CH} \left({\bf R}\, g_v \,\SZ^{\Lambda}(v)\right).$$
\end{definition}
Let us see that $P(\CH,\Lambda,g)(v)$ is a polynomial function of $v$ on each tope $\tau$.
Indeed, to compute $P(\CH,\Lambda,g)(v)$ using (\ref{Sz}), we have to compute the total residue of  functions
$g(z)e^{\ll z,v\rr}\AV^{\Lambda}(f_i)(z,v)$ with $f_i$ varying over a basis of $\CS_\CH$.
If $v\in \tau$, then  $\AV^{\Lambda}(f_i)(z,v)=\SZ^{\Lambda}(\tau)f_i(z)$  is constant in $v$. So when $v$ stays in a
tope $\tau$,  the dependence of $g(z)e^{\ll z,v\rr}\AV^{\Lambda}(f_i)(z,v)=g(z)e^{\ll z,v\rr}(\SZ^{\Lambda}(\tau)f_i)(z)$
on $v$ is via  $e^{\ll z,v\rr}$, and the map $\bf{R}$ involves only the Taylor series of this function  up to some order.
Thus we have associated to $g\in \CR_\CH$ (and $\Lambda$) a piecewise polynomial function
$P(\CH,\Lambda,g)$   on $V_{reg}.$


It is easy to compare piecewise polynomial functions
$P(\CH,\Lambda,g)$  associated to different lattices:  Observe that if $\Lambda^1\subset \Lambda^2$,  then 
$V_{reg}(\CH,\Lambda^2)\subset V_{reg}(\CH,\Lambda^1)$. Then, using Lemma \ref{lem: first compare}, we have
\begin{equation}\label{eq:SZlattices2}
P(\mathcal H, \Lambda^2,g)(v)=|\Lambda^2/\Lambda^1|^{-1}
\sum_{\lambda\in \Lambda^2/\Lambda^1}P(\mathcal H, \Lambda^1,g)(v+\lambda).
\end{equation}

Our next aim  is to  compute the piecewise polynomial function
$P(\CH,\Lambda,g)$ using residues. We need more definitions.

An ordered basis of $\Phi^{eq}$ is a sequence
$[\alpha_1,\alpha_2,\ldots,\alpha_r]$ of elements of $\Phi^{eq}$ such that the underlying set is in
$\mathfrak{B}(\Phi^{eq})$. We denote the set of ordered bases of $\Phi^{eq}$ by ${\overrightarrow{\mathfrak{B}}}(\Phi^{eq})$.

Let  ${\overrightarrow{\sigma}}=[\alpha_1,\alpha_2,\ldots,\alpha_r] \in {\overrightarrow{\mathfrak{B}}}(\Phi^{eq})$.
Then, to this data, one associates an iterated residue
functional $\Res^{{\overrightarrow{\sigma}}}$ on $\CR_{\mathcal H}$  as follows. For $z\in U_\C$,
let $z_j=\ll z,\alpha_j\rr$. Then a function $f$ in $\CR_{\mathcal H}$ can be expressed as a function
$f(z_1,z_2,\ldots, z_r).$
We define
$$\Res^{{\overrightarrow{\sigma}}}(f):=\Res_{z_1=0}(\Res_{z_2=0}\cdots(\Res_{z_r=0}f(z_1,z_2,\ldots,z_r))\cdots).$$
Clearly $\Res^{{\overrightarrow{\sigma}}}(f_\sigma)=1$.  Moreover, the functional $\Res^{{\overrightarrow{\sigma}}}$ factors through the canonical projection ${\bf R}:\CR_\mathcal H \to
\CS_\mathcal H$, that is,
$\Res^{{\overrightarrow{\sigma}}}=\Res^{{\overrightarrow{\sigma}}} {\bf R}$.


\begin{definition}
A {\it diagonal subset of} ${\overrightarrow{\mathfrak{B}}}(\Phi^{eq})$  is a subset $\overrightarrow{\mathcal  D}$ of
${\overrightarrow{\mathfrak{B}}}(\Phi^{eq})$
such that the set of simple fractions  $f_\sigma$, ${\overrightarrow{\sigma}}\in \overrightarrow{\mathcal  D}$, forms a basis
of $\CS_{\mathcal H}$: $$\CS_{\mathcal H}=\oplus_{{\overrightarrow{\sigma}}\in \overrightarrow{\mathcal  D}} \C f_\sigma$$
and the dual basis to the basis $\{f_\sigma, {\overrightarrow{\sigma}} \in \overrightarrow{\mathcal  D}\}$ of $\CS_{\mathcal H}$ is the set of
linear forms $\Res^{{\overrightarrow{\sigma}}}$, that is, $\Res^{\overrightarrow{\tau}}(f_\sigma)=\delta_\sigma^\tau$,
for ${\overrightarrow{\sigma}},\overrightarrow{\tau} \in \overrightarrow{\mathcal  D}$.
\end{definition}

A total order  on $\Phi^{eq}$ allows us to construct the set  of Orlik-Solomon bases (see \cite{brver1}), which provides
diagonal basis of $\CS_{\CH}$. However we will also use some other diagonal subsets.

If $B:\CS_{\mathcal H}\to \mathcal
\CM_{\mathcal H}$ is an operator, the trace of the operator $A:={\bf R}B$ is thus $$Tr(A):=\sum_{{\overrightarrow{\sigma}}\in \overrightarrow{\mathcal  D}} \Res^{{\overrightarrow{\sigma}} }B f_\sigma.$$


\begin{definition}\label{def:localpoly}
Let $g\in \CR_\CH$, $\tau$ a connected component of $V_{reg}$, and $v\in \tau$.
We denote by
$P(\CH,\Lambda,g,\tau)(v)$
the
polynomial function on $V$  such that
$$P(\CH,\Lambda,g)(v)=P(\CH,\Lambda,g,\tau)(v).$$
\end{definition}

We may give a more explicit formula for the polynomial $P(\CH,\Lambda,g,\tau)(v)$ using a diagonal subset $\overrightarrow{\mathcal  D}$.



\begin{proposition}\label{prop:iteres}
Let $g\in \CM_{\CH}$ and $\tau\in \CT(\CH,\Lambda)$ be a tope.
Let $\overrightarrow{\mathcal  D}$ be a diagonal subset of ${\overrightarrow{\CB}}(\Phi^{eq})$.
Then
$$P(\CH,\Lambda,g,\tau)(v)=\sum_{\overrightarrow{\sigma} \in \overrightarrow{\mathcal  D}} \Res^{{\overrightarrow{\sigma}}} \Big( e^{\ll z,v\rr}g(z)  \SZ^{\Lambda}(\tau)(f_\sigma)(z)\Big).$$
\end{proposition}


Furthermore
$\SZ^{\Lambda}(\tau)(f_\sigma)(z)$  is given explicitly by
Proposition \ref{pro:formulaEIS}.
Thus, in principle, the above formula allows us to compute
$P(\CH,\Lambda,g)$.

\medskip

It is important to remark that the determination of a diagonal subset   $\overrightarrow{\mathcal  D}$ depends essentially
only on the system of hyperplanes $\CH$ and not on the choice of $\Phi^{eq}$.
The  difficulties in writing an algorithm for
$P(\CH,\Lambda,g)$ lies in the description of a diagonal subset $\overrightarrow{\mathcal  D}$, and also for each
$\sigma\in \overrightarrow{\mathcal  D}$, in the computation of $A^{\Lambda}(f_\sigma)$.
The difficulty of this  last computation depends on the lattice $\Lambda$.



\begin{definition}
Let $\Phi^{eq}\subset \Lambda$.
A basis $\sigma\in \mathfrak B(\Phi^{eq})$ is called unimodular with respect to $\Lambda$ if  $\Lambda_\sigma=\Lambda$ . A set $\Phi^{eq}$ is called unimodular,
if any basis $\sigma\in \mathfrak B(\Phi^{eq})$ is unimodular.
\end{definition}


%\begin{example}
%The set $\Phi^{eq}=\{ e_1, e_2,(e_1+e_2), (e_1-e_2)\}$ is contained in  $\Lambda=\Z e_1\oplus\Z e_2$.
%Then $\sigma=\{e_1+e_2,e_1-e_2\}$ belongs to $\mathfrak B(\Phi^{eq})$, and the index of $\Lambda_\sigma$ in
%$\Lambda$ is $2$. So $\Phi^{eq}$ is not unimodular.
%\end{example}

Let $\sigma=\{\alpha_1,\alpha_2,\ldots,\alpha_r\}$ be a  basis.
For $1\leq i \leq r$, the linear form    $v\to c_i^{\sigma}(v)$
is the coefficient of $v$ with respect to $\alpha_i$.
Now, consider the function  $\{t\}=t-[t]$; on each open interval $\tau=]n,n+1[$ it coincides with  the linear function  $t\mapsto (t-n)$.
We express $v= \sum_{i=1}^r c_i^{\sigma}(v)\alpha_i$, then $v-\sum_{i=1}^r [c_i^{\sigma}(v)]\alpha_i=\sum_{i=1}^r\{c_i^{\sigma}(v)\}\alpha_i$ is in $Q_\sigma$.
Thus, if $\sigma$ is a unimodular basis, the set $\Trans(\tau,\sigma)$ contains exactly the element $\lambda=-\sum_{i=1}^r [c_i^{\sigma}(v)]\alpha_i$ (which depends only on the tope $\tau$ where $v$ lies).


\begin{corollary}\label{cor:Step}
Let $\sigma$ be a unimodular basis in $\mathfrak B(\Phi^{eq})$. Let $v\in \tau$, and $\lambda=-\sum_{i=1}^r [c_i^{\sigma}(v)]\alpha_i$.  Then
$$\AV^{\Lambda}(f_\sigma)(z,v)= \frac{e^{\langle \lambda,z\rangle }}{\prod_{i=1}^r (1-e^{\langle \alpha_i,z\rangle})}.$$
\end{corollary}


It may happen that even when the system $\Phi^{eq}$ is not unimodular for  $\Lambda$, we can choose $\overrightarrow{\mathcal  D}$ to consist of unimodular bases.
In particular, using Proposition \ref{prop:iteres}, we can give an explicit algorithm for computing the piecewise 
polynomial function
$P(\CH,\Lambda,g)$ for classical root systems in the form of a `step polynomial'. Let us define what this means.


\begin{definition}
Let $\overrightarrow{\mathcal  D}$ be a subset of ${\overrightarrow{\CB}}(\Phi^{eq})$.
We denote by $Step(\overrightarrow{\mathcal  D})$ the algebra of functions on $V$ generated by the piecewise linear functions $v\to \{c_i^{\sigma}(v)\}$ with $\sigma$ running over
$\overrightarrow{\mathcal  D}$ and $1\leq i\leq r$.
An element of the algebra $Step(\overrightarrow{\mathcal  D})$ is called a step polynomial (associated to $\overrightarrow{\mathcal  D}$).
\end{definition}

It is clear that a step polynomial is a periodic function on $V$,
which is expressed by a polynomial formula on each tope.


\begin{proposition}
Suppose  $\overrightarrow{\mathcal  D}$ is a diagonal subset of ${\overrightarrow{\mathfrak B}}(\Phi^{eq})$ consisting of unimodular basis (with respect to  $\Lambda$).
Then, for $g\in {\mathcal G}_{\CH}$,  the piecewise polynomial function $P(\CH,\Lambda,g)$ belongs  to the algebra $Step(\overrightarrow{\mathcal  D})$.
\end{proposition}

\begin{proof}
This is clear, as we have the formula
\begin{equation}\label{steppoly}
 P(\CH,\Lambda,g)(v)=\sum_{\overrightarrow{\sigma} \in \overrightarrow{\mathcal  D}} \Res^{{\overrightarrow{\sigma}}} g(z) e^{\sum_{i=1}^r\{c_i^{\sigma}(v)\}\ll\alpha_i,z\rr}\frac{1}{\prod_{i=1}^r (1-e^{\langle \alpha_i,z\rangle})},
\end{equation}
and the dependence on $v$ is through the Taylor series expansion (in $z$) of $e^{\sum_{i=1}^r\{c_i^{\sigma}(v)\}\ll\alpha_i,z\rr}$ up to some order.
\end{proof}


\subsection{Multiple  Bernoulli series}\label{section:mbs}
We return to our main object of study: the multiple Bernoulli series.

Let $V$, $\Lambda$ and $\CH$ be as before.
We denote by $\Gamma\subset U$ the dual lattice  to $\Lambda$, and by
$\Gamma_{reg}(\CH)$ the set $\Gamma \cap U_{\CH}$.
If $\gamma \in \Gamma_{reg}(\CH)$, a function $g$ in $\CR_\CH$ is defined on $2i\pi \gamma$.


\begin{definition}
If $g\in \CR_{\mathcal H}$,  the generalized function  $\mathcal{B}(\mathcal H, \Lambda,g)(v)$ on $V$  is defined by
$$\mathcal{B}(\mathcal H, \Lambda,g)(v)=\sum_{\gamma \in \Gamma_{\reg}(\mathcal H)} g(2i\pi \gamma) e^{ 2i\pi \ll v ,\gamma \rr}.$$
\end{definition}

\noindent The above series converges in the space of generalized functions on $V$.

\medskip

We state some obvious properties of
$\mathcal{B}(\mathcal H, \Lambda,g)$, with which we can compare
it over commensurable lattices.

\begin{lemma}\label{lem:secondcompare}
If $\Lambda^1\subset \Lambda^2$, then
\begin{equation}\label{eq:lattices}
\mathcal{B}(\mathcal H, \Lambda^2,g)(v)=|\Lambda^2/\Lambda^1|^{-1}
\sum_{\lambda\in \Lambda^2/\Lambda^1}\mathcal{B}(\mathcal H, \Lambda^1,g)(v+\lambda).
\end{equation}
\end{lemma}

If we dilate a lattice $\Lambda$ by $\ell$, and if $g$ is homogeneous of degree $h$,
we clearly have
\begin{equation}\label{eq:dillattices}
\mathcal{B}(\mathcal H, \ell\Lambda
,g)(v)= \ell^{-h}
\mathcal{B}(\mathcal H, \Lambda,g)(\frac{v}{\ell}).
\end{equation}


\begin{definition}
A generalized function $b$  on $V$ is called {\it piecewise
polynomial} relative to $\CH$ and $\Lambda$ if it is locally $L^1$, and if, for any $\tau$ in $\mathcal{T}(\CH,\Lambda)$,  there exists a polynomial function $b^{\tau}$ on $V$ such that the restriction of $b$ to $\tau$
coincides with the restriction of the polynomial $b^{\tau}$ to $\tau$.
\end{definition}


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.
%It is not clear to us what is the `best description'  of a piecewise polynomial function.
%Indeed, we can either describe $b$  as a family of polynomial functions  $b(\tau)$ indexed by topes, or
%we can compute $b$ as a polynomial expression with respect to some basic piecewise polynomials.
One should be careful that, the restriction of a piecewise polynomial generalized function to any tope is polynomial, however, the converse is not true. For example, the $\delta$ function of the lattice $\Lambda$ restricts to $0$ on
any tope, but is not a piecewise polynomial generalized function, as it is not locally $L^1$.

\medskip

%Recall the definition of ${\mathcal G}_\CH$ as given in Definition \ref{def:simple}.
%If we multiply $g$ by a polynomial $p$, 

Any function $f$ in $\CR_\CH$ is of the form $pg$, with $g\in
{\mathcal G}_\CH$ and $p$ a polynomial. Moreover, the function
$v\mapsto \mathcal{B}(\mathcal H, \Lambda,pg)(v)$ is obtained from $\mathcal{B}(\mathcal H, \Lambda,g)(v)$ by differentiation (in the distribution sense).
Thus, the computation of
$\mathcal{B}(\mathcal H, \Lambda,f)$  for $f$ in $\CR_\CH$ can be reduced to that of
$\mathcal{B}(\mathcal H, \Lambda,g)$ for $g$ in $\mathcal G_\CH$.
The following proposition follows  from calculations in dimension one, Lemma \ref{ind} and
%the comparison 
formulae of Lemma \ref{lem:secondcompare}.



\begin{proposition}\label{prop:polyn}
If $f\in \CR_{\CH}$, the restriction of
$\mathcal{B}(\mathcal H,\Lambda,f)$ to any tope $\tau$ is given by a polynomial function.

Furthermore, if $f\in {\mathcal G}_\CH$,  the generalized function   $\mathcal{B}(\mathcal H,\Lambda,f)$ is a piecewise polynomial generalized function.
\end{proposition}

Let us emphasize on the subtle difference between  the conditions  $f\in \CR_{\CH}$ and $f\in {\mathcal G}_\CH$.

Consider $f=1$ in the one dimensional space $U_\C$ and
$\mathcal H=\{0\}$.  The function $f$ is not in ${\mathcal G}_{\CH}$.
For $v\in V$,  {\small $\mathcal{B}(\mathcal H,\Lambda,f)(v)=\sum_{n\neq 0}e^{2i\pi nv}=-1+\sum_{n\in \Z}e^{2i\pi nv},$} 
which, on any tope, is the constant function  equal to $-1$. However,
it has some singular part $\delta_{\Z}$, and it is not locally $L^1$.
In contrast, if $f=\tfrac{1}{z}$, the generalized function
$\mathcal{B}(\mathcal H,\Lambda,f)(v)=\sum_{n\neq 0}\tfrac{e^{2i\pi nv}}{2i\pi n}$ is locally $L^1$ and  equal to the piecewise polynomial
function $-B(1,\{v\})=1/2-\{v\}$ (see Figure \ref{berno}).

\begin{figure}
\begin{center}
  \includegraphics[width=93mm]{bern.mps}\\
  \caption{Graph of $\mathcal{B}(\{0\},\Z, 1/z)(v)= \frac{1}{2}-\{v\}$}\label{berno}
\end{center}
\end{figure}

\begin{definition}
Let $f\in \CR_{\CH}$.
Given a tope $\tau$ in $\mathcal T(\mathcal H,\Lambda)$, we denote by
$\mathcal{B}(\mathcal H,\Lambda,f,\tau)$
the polynomial function on $V$ which coincides with
$\mathcal{B}(\mathcal H,\Lambda,f)$ on the tope $\tau$.
\end{definition}

%\begin{remark}
%It is interesting to understand the space of polynomials generated by the polynomial functions
%$b^\tau=\mathcal{B}(\mathcal H,\Lambda,f,\tau)$, when $\tau$ runs over the topes, and the wall crossing formula between %$b^{\tau_1}$ and $b^{\tau_2}$ when
%$\tau_1$ and $\tau_2$ are adjacent.  We addressed some aspects of these theoretical questions in \cite{bover2}.
%\end{remark}

Recall the piecewise polynomial function
$P(\CH,\Lambda,f)$ on $V_{reg}(\CH,\Lambda)$ as given in Definition \ref{def:sz}.


\begin{theorem}(Szenes)\label{theo:main}
Let $f\in \CR_{\CH}$.
On $V_{\reg}(\CH,\Lambda)$, we have 
 $$\CB(\CH,\Lambda,f)= P(\CH,\Lambda,f).$$
\end{theorem}

%Thus for $v\in \tau$ the value of the Bernoulli series  at $v$ is  $P(\CH,\Lambda,f,\tau)(v)$.

For completeness, we give a proof of this theorem in the Appendix.

\medskip

Our Maple program  computes, given data $\mathcal H,\, \Lambda, \, f$,  where $\CH$ 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 $v\in V_{reg}$.

\medskip

We return to the definition of multiple Bernoulli series as given in the introduction: $V$ is a vector space with a lattice $\Lambda$, and $\Phi$ a {\bf list} of elements in $\Lambda$.  Associated to this data, we defined,
$$\mathcal{B}(\Phi, \Lambda)(v)=\sum_{\gamma \in \Gamma_{\reg}(\Phi)}  \frac{e^{\ll 2i\pi v ,\gamma \rr}}{
\prod_{\phi \in \Phi} \ll 2i\pi \phi,\gamma \rr},$$
where $\Gamma$ is the dual of $\Lambda$, and 
$\Gamma_{\reg}(\Phi)=\{\gamma\in \Gamma ; \ll \phi,\gamma \rr \neq
0\;\mbox{for all} \;\phi\in \Phi\}$ are its regular elements.

Consider $\CH=\cup_{\phi \in \Phi}\{\phi=0\}$
(some elements of the list $\Phi$ might define the same hyperplane)
and $g(z)=1/\prod_{\phi \in \Phi} \ll  \phi,z \rr$, then
$$\mathcal{B}(\Phi, \Lambda)(v)= \mathcal{B}(\mathcal H, \Lambda,g)(v).$$
We will also call the functions $\mathcal{B}(\mathcal H, \Lambda,g)(v)$
multiple Bernoulli series.


\section{Classical root systems}\label{section:clroot}
Let $G$ be a simple, connected, simply connected, compact Lie group of rank $r$
with maximal torus $T$.  We denote the Lie algebra of $T$ and $G$ by
$\frt$ and $\frg$ respectively. Then the complexification $\frh:=\frt_{\C}$ is a
Cartan subalgebra of $\frg_{\C}$.
%For $\alpha \in \frh^*$, define
%$(\frg_{\C})_{\alpha}=\{X \in \frg_{\C}; [H,X]=\ll \alpha,H\rr X\;\text{for all}\;H \in\frh\}$.
%If $\alpha \neq 0$ and $(\frg_{\C})_{\alpha}\neq 0$, then $\alpha$ is called a {\it root} of
%$\frh$ in $\frg_{\C}$. 
Let $R(\frg_{\C},\frh)\subset \frh^*$ be the set of roots; 
%roots $\alpha \in R$ take imaginary values on $\frt$.
we denote the root lattice by $Q$ and its dual, the coweight lattice, by $\check{P}$.
For  $\alpha \in R$, $H_{\alpha}$ denotes the associated {\it coroot}; the coroot lattice is denoted by 
$\check{Q}$.  The weight lattice is $P=\{ \lambda \in \frh^* ; \lambda(H_{\alpha}) \in
\mathbb{Z},\; \forall \alpha \in R\}$; a {\it regular} weight $\lambda\in P^{reg}$ is such that  $ \lambda(H_{\alpha}) \neq 0$ for all $H_\alpha$. 
We denote by $\frh_\R:=\sum_\alpha \R H_\alpha$ the real span of coroots.  

In this section we have $V=\mathfrak h_\R$, and
its dual $\frh_\R^*$ is denoted by $U$ as before. We follow the notation of Bourbaki for root data.

\subsection{Diagonal subsets}
To compute  multiple Bernoulli series associated to classical root systems we need to construct
explicit diagonal bases for the corresponding $\CS_{\mathcal H}$.
Such bases can be constructed by an algorithmic procedure, based on Orlik-Solomon construction \cite{os}.
However in some cases one can describe a diagonal subset $\overrightarrow{\mathcal  D}$ of 
${\overrightarrow{\CB}}(\Phi^{eq})$
whose associated simple fractions form a basis for $\CS_{\mathcal H}$ in a direct way.  We now demonstrate this.



\subsubsection{The system  of type $A_r$}\label{A}
Let $n=r+1$.  We consider $\R^{n}$ with standard basis $\{e_i\}$; denote the dual basis by $\{e^i\}$.
Let $A_r:=[(e_i-e_j);\, 1\leq i< j\leq n]$ be the root system of type $A$ and rank $r$, and 
$V=\{v=\sum_{i=1}^n v_i e^i;\, \sum_{i=1}^n v_i=0\}.$

Let $z=\sum_{i=1}^n z^i e_i$ be in $U_\C$ (hence $\sum_{i=1}^n z^i=0$)
and let $\CH_r^A$ be the system of hyperplanes in $U_\C$ given by $\CH_r^A=\cup_{1\leq i<j\leq n}\{z^i- z^j=0\}.$
We take the set
$\Phi^{eq}(A_r)=\{e^i-e^j; 1\leq i<j\leq n\}$ of positive coroots as equations of $\CH_r^A$.


One way to find a  diagonal basis of $\CS_{\CH_r^A}$ is as follows.

Let $\Sigma=[e^1-e^2,e^2-e^3,\ldots,e^{r}-e^{r+1}]$ be the set of simple coroots.
For a permutation $w$, we denote by $\overrightarrow{\sigma}_w=[e^{w(i)}-e^{w(i+1)},\, i=1,\ldots,r]$.
Then $\overrightarrow{\sigma}_w$ is an ordered basis associated to $w$, and the corresponding
simple fraction is $f_w(z):=\displaystyle \frac{1}{\prod_{i=1}^r (z^{w(i)}-z^{w(i+1)})}$.

Let $W_r$ be the subgroup of the Weyl group   $\Sigma_{r+1}$ of permutations of $\{e^1,e^2,\ldots, e^{r+1}\}$  leaving the
last element $e^{r+1}=e^n$ fixed.
Recall the following result (see for example Baldoni-Vergne \cite{balver} for a proof).


\begin{proposition}\label{pro:OW}
The set $\overrightarrow{\mathcal D}_W$  consisting of  ordered bases $\overrightarrow{\sigma}_w$  for
$w\in W_r$ is a diagonal subset of ${\overrightarrow {\mathfrak B}}(\Phi^{eq}(A_r))$.
\end{proposition}

We use the above basis in our Maple program.  We now give another interesting diagonal subset.

Consider  a sequence $ \overrightarrow{\sigma}=[\alpha_2,\alpha_3,\ldots,\alpha_n]$ where
$\alpha_i=e^i-e^j$ with $j<i$.
That is, $\alpha_2=e^2-e^1$, $\alpha_3=e^3-e^2$ or $e^3-e^1$, $\alpha_4=e^4-e^3$, or $e^4-e^2$, or $e^4-e^1$, etc.
Clearly,  $\overrightarrow{\sigma}$ is in ${\overrightarrow{\mathfrak B}}(\Phi^{eq}(A_r))$.
We call such $ \overrightarrow{\sigma}$ a {\textit flag basis}; there are $r!$ such sequences $\overrightarrow{\sigma}$.

\begin{lemma}
The set  $\overrightarrow{\mathcal  D}(A_r)$ consisting of flag bases is a diagonal subset of
${\overrightarrow{\mathfrak B}}(\Phi^{eq}(A_r))$.
\end{lemma}

We only need to prove that if $\overrightarrow{\sigma}$ and
$\overrightarrow{\tau}$ are two flag bases, then
$\Res^{\overrightarrow{\sigma}}f_\tau=0$ unless
$\overrightarrow{\sigma}=\overrightarrow{\tau}$. But this is evident.


\subsubsection{Systems of type $B_r$ or $C_r$}\label{B}
We consider $V=\R^r$ with standard basis $\{e^i\}$.
Let $B_r=[\pm e_i,\pm (e_i\pm e_j),\, 1\leq i \leq r,\,1\leq i<j\leq r]$ be the root system of type $B$
and rank $r$.

Let $C_r=[\pm 2 e_i,\pm (e_i\pm e_j),\,  1\leq i \leq r,\,  1\leq i<j\leq r]$
be the root system of type $C$ and rank $r$.

As roots of the systems of type $B$ and $C$ are proportional, the system of hyperplanes  in $U={\mathfrak h}_\R^*$ defined by coroots of $B$ and $C$ are the same,
and we denote that system (and the corresponding complex arrangement) by $\CH_r^{BC}$. More precisely, let $z=\sum_{i=1}^r z^i e_i$ in $U_\C$, then the system of hyperplanes
$\CH_r^{BC}$   in $U_\C$ is given by
$\CH_r^{BC}=\cup_{1\leq i<j\leq r}\{z^i\pm z^j=0\}\cup \cup_{1\leq i\leq r} \{z^i=0\}.$

We take the set
$\Phi^{eq}(BC_r)=\cup_{1\leq i<j\leq r}\{e^i\pm e^j=0\}\cup \cup_{1\leq i\leq r} \{e^i=0\}$
as equations of $\CH_r^{BC}$.

We now define a flag basis $\overrightarrow{\sigma}$ of $\Phi^{eq}(BC_r)$.
This is a basis of the form $\overrightarrow{\sigma}=[\alpha_1,\alpha_2,\ldots,\alpha_r]$ of $r$ elements of $\Phi^{eq}(BC_r)$  so that
$\alpha_i=e^i$,  or $e^i-e^j$  or $e^i+e^j$ with $j<i$.
That is, $\alpha_1=e^1$, $\alpha_2=e^2$ or $e^2-e^1$, or $e^2+e^1$,
 $\alpha_3=e^3$ or $e^3-e^2$, or $e^3+e^2$, or $e^3-e^1$, or $e^3+e^1$, etc.
Clearly, there are $1\cdot 3 \cdot 5\cdots(2r-1)$ such sequences $\overrightarrow{\sigma}$.
 
\begin{lemma}
The set  $\overrightarrow{\mathcal  D}(BC_r)$ consisting of flag bases  is a diagonal subset of ${\overrightarrow{\mathfrak B}}(\Phi^{eq}(BC_r))$.
\end{lemma}

\begin{proof}
We first prove, by induction on $r$, that simple fractions $f_b$ associated to a flag basis $b$ generate $\CS_{\CH_r^{BC}}$.
We use the identities
{\small $$\frac{1}{(x_r-x_i)}\frac{1}{(x_r+x_i)}=\frac{1}{(x_r+x_i)}\frac{1}{2x_r}+\frac{1}{(x_r-x_i)}\frac{1}{2x_r},$$}
{\small $$\frac{1}{x_r}\frac{1}{(x_r+x_i)}=-\frac{1}{(x_r+x_i)}\frac{1}{x_i}+\frac{1}{x_r}\frac{1}{x_i},\;\;\;
\frac{1}{x_r}\frac{1}{(x_r-x_i)}=\frac{1}{(x_r-x_i)}\frac{1}{x_i}-\frac{1}{x_r}\frac{1}{x_i},$$}
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 the simple fractions $f_b$ associated to flag basis $b$ generates the space $\CS_{\CH_r^{BC}}$.
The dual property  on the elements of  $\overrightarrow{\mathcal  D}(BC_r)$  is evident.
\end{proof}

\begin{remark}
Although the  system $\Phi^{eq}(BC_r)$ is not unimodular for the lattice $\Lambda=\oplus \Z e^i$, we see that any $ \overrightarrow{\sigma}$ in the set $\overrightarrow{\mathcal  D}(BC_r)$ above is unimodular, so that the computation of
$\SZ^{\Lambda}(\tau)(f_\sigma)$ is easy.
\end{remark}

\subsubsection{The system of type $D_r$}\label{D}
We consider $V=\R^r$ with standard basis $\{e^i\}$.
Let $D_r=[\pm (e_i\pm e_j);\, 1\leq i<j\leq r]$ be the root system of type $D$ and rank $r$.
Let $z=\sum_{i=1}^r z^i e_i$ in $U_\C$.
We consider the system of hyperplanes in $U_\C$
$$\CH_r^D=\cup_{1\leq i<j\leq r}\{ z^i\pm z^j=0\}.$$
The dimension of $\CS_{\CH_r^D}$ is known to be $1\cdot 3\cdot 5\cdots(2r-3)(r-1)$.
However, we did not find a nice diagonal basis for $\CS_{\CH_r^D}$.  We proceed as follows:
The set  $U_{\CH_r^D}$  of regular elements for $\CH_r^D$ contains $U_{\CH_r^{BC}}$.
Indeed, for any $z$ in $U_{\CH_r^D}$, we have $z^i\pm z^j\neq 0$, but $z^i$ may equal zero.
We define the set
$$U_{k,r}:=\{z^k=0,\, z^i\pm z^j \neq 0\; \text{for}\;1\leq i<j\leq r \; \text{and}\; z^i\neq 0 \;\text{for}\; 1\leq i\leq r, \,i\neq k\}.$$
Then, the following disjoint decomposition
$U_{\CH_r^D}=U_{\CH_r^{BC}}\bigcup \cup_{k=1}^r U_{k,r}$ holds.
The set $U_{k,r}$ is clearly isomorphic to the  set $U_{\CH^{BC}_{r-1}}$ in rank $r-1$ via the map
$i_k$ which inserts a zero coordinate in position $k$, and hence,
\begin{equation}\label{eq:disjoint}
U_{\CH_r^D}=U_{\CH_r^{BC}}\bigcup \cup_{k=1}^r i_k(U_{\CH^{BC}_{r-1}}).
\end{equation}
The above decomposition allows us to reduce calculations in systems of type $D$ to those of systems of type $B$ or $C$.

\subsection{Calculations of multiple Bernoulli series for type  $A$}
We use the same notation as in Section \ref{A}.

Let $Q_A\subset U$ be the root lattice generated by $A_r$,
and $P_A\subset U$ be the weight lattice. Then $P_A$  is  generated by $Q_A$ and 
$e_1-\frac{1}{r+1}(e_1+e_2+\cdots +e_{r+1})$.  The cardinality of $P_A/Q_A$ is  $r+1$.

Let $\Gamma$ be a lattice such that  $Q_A\subset \Gamma\subset P_A$.
We denote by $\Gamma_{reg}=\Gamma\cap U_{\CH_r^A}$ the set of regular elements in $\Gamma$.
Let $\Lambda\subset V$ be the dual lattice to $\Gamma$, and
let ${\bf s}=[s_{\alpha}]$ be a list of exponents.  Define
{\small $$g^A_{\bf s}(z)=\frac{1}{\prod_{\alpha>0}\ll H_\alpha,z\rr^{s_\alpha}},$$}
where the set $\{H_\alpha, \alpha>0\}$ is the set of positive coroots $\Phi^{eq}(A_r)$.
If $v\in V$,
$$\mathcal B(\CH_r^A,\Lambda, g^A_{\bf s})(v)=\sum_{\gamma\in \Gamma_{reg}} \frac{e^{2i\pi \ll v,\gamma\rr}}{\prod_{\alpha>0} (2i\pi \ll H_\alpha,\gamma \rr)^{s_\alpha}}.$$

If we use the diagonal basis $\overrightarrow{\mathcal  D}_W$ for $\Phi^{eq}(A_r)$ as defined in Proposition \ref{pro:OW}, then it consists of elements
$[e^{w(1)}-e^{w(2)},\ldots, e^{w(r)}-e^{r+1}]$ where $w$ is an element of $W_r$.
Thus if we express $v=\sum_{i=1}^r v_i (e^{i}-e^{r+1})$, the algebra $Step(\overrightarrow{\mathcal  D}_W)$ consists of functions $\{\sum_I v_i\}$ where $I$
runs over subsets of $\{1,2,\ldots, r\}$.


We now discuss two simple cases, where $\Gamma$ is either the weight lattice $P_A$, or the root lattice $Q_A$.

\subsubsection{Bernoulli series for the weight lattice}
The dual of the weight lattice $P_A$ is the coroot lattice $\check{Q}_A$
generated by simple coroots $H_\alpha$, and the system $\Phi^{eq}(A_r)$  of equations (the positive coroots) is unimodular with respect to $\check{Q}_A$.
Thus $\mathcal B(\CH_r^A,\check{Q}_A,g^A_{\bf s})(v)$ is a piecewise polynomial function of degree
$\sum_\alpha s_\alpha$ and lies in the algebra $Step(\overrightarrow{\mathcal  D}_W)$.

Our program  then gives
$\mathcal B(\CH_r^A,\check{Q}_A,g^A_{\bf s})(v)$ as a polynomial expression in  $\{\sum_I v_i\}$.
It also computes numerically the value of this function at any point $v$.


\begin{example}\label{a2exp1}
Consider the root system of type $A_2$. 
For ${\bf s}=[1,1,1]$,
%we compute $\mathcal B(\CH_2^A,\check{Q}_A, g^A_{\bf s})(v)$ 
%The simple coroots are $H_{\alpha_1}=e^1-e^2$ and $H_{\alpha_2}=e^2-e^3$, and the remaining positive coroot is their sum
%$H_{\alpha_1}+H_{\alpha_2}$.
%The dual lattice has basis dual to $H_{\alpha_1}$, $H_{\alpha_2}$.
{\small \begin{equation}\label{sumsone}
\mathcal B(\CH_2^A,\check{Q}_A, g^A_{\bf s})(v)=\sum_{\substack{m,n \in \Z \\ mn(m+n)\neq 0}}\frac{e^{2i\pi m v_1-2i\pi n v_3}}{(2i\pi m)(2i\pi n)(2i\pi (m+n))}.
\end{equation}}
where $v=v_1 e^1+ v_2 e^2+ v_3 e^3$ with $v_1+v_2+v_3=0$.
%The symbol $\sum'$ above means that we sum over the integers $m,n$ with $mn(m+n)\neq 0$.
Denoting the fractional part of $t$ with $\{t\}\in[0,1[$, we obtain
that $P(v_1,v_2,v_3):=\mathcal B(\CH_2^A,\check{Q}_A, g^A_{\bf s})(v)$ is equal to
{\small \begin{equation}\label{weight}
\frac{1}{6}
(\{v_2\}-\{v_1\})(\{v_1\}^2-3\{v_1+v_2\}\{v_1\}+\{v_2\}\{v_1\}+3\{v_1+v_2\}-1-3\{v_1+v_2\}\{v_2\}+\{v_2\}^2).
\end{equation}}
We remark that the series (\ref{sumsone}) is not absolutely convergent, but the sum has a meaning and is a piecewise polynomial function.
\end{example}

Let us give some numerical  examples.

Consider $A_2$ with ${\bf s}=[10,10,10]$, $v_1=v_2=0$.  Then, $\mathcal B(\CH_2^A,\check{Q}_A,g^A_{\bf s})(0)$ is
{\small $$\sum'_{m,n}\frac{1}{(2i\pi m)^{10}(2i\pi n)^{10}(2i\pi (m+n))^{10}}=
-{\frac {27739097}{4174671932121099276691439616000000}}.$$}
Consider $A_4$. 
We list the exponents with respect to the following order on the roots
$[e_1-e_2, e_1-e_3, e_1-e_4, e_1-e_5, e_2-e_3, e_2-e_4, e_2-e_5, e_3-e_4, e_3-e_5, e_4-e_5]$.
For ${\bf s}=[6,6,6,6,4,2,2,2,2,2]$ and $v=[0,0,0,0,0]$,
{\small $$\mathcal B(\CH_4^A,\check{Q}_A,g^A_{\bf s})(0)=\frac{1}{(2i\pi)^{38}}\times $$}{\small{$$\sum'\frac{1}{m_1^6m_2^4m_3^2m_4^2
(m_1+m_2)^6(m_1+m_2+m_3)^6(m_1+m_2+m_3+m_4)^6(m_2+m_3)^2(m_2+m_3+m_4)^2(m_3+m_4)^2}$$}}
{\small $$=\frac{66581757}{2081416538897698301902069565296214016000000000},$$}
while for  ${\bf s}=[4,4,4,4,4,4,4,4,4,4]$ and $v=[0,0,0,0,0]$, we obtain
{\small $$\mathcal B(\CH_4^A,\check{Q}_A,g^A_{\bf s})(0)=\frac{3998447009863}{19318834119102098604968210835862034086625280000000000}.$$}

\subsubsection{Bernoulli series for the root lattice}
Let $\xi=\sum_{j=1}^r (e^j-e^{r+1})$. Then a system of representatives
for $\check{P}_A/\check{Q}_A$ consists of the elements $\{\lambda_j=\frac{j}{r+1} \xi\;0\leq j \leq r\}$.
Using Formula (\ref{eq:lattices}),
$${\mathcal B}(\CH_r^A,\check{P}_A, g^A_{\bf s})(v)=\frac{1}{r+1}\sum_{j=0}^{r} {\mathcal B}(\CH_r^A ,\check{Q}_A, g^A_{\bf s})(v+\lambda_j).$$
Hence, we obtain  an expression for
${\mathcal B}(\CH_r^A,\check{P}_A, g^A_{\bf s})$ in terms of the functions $\{(\sum_I v_i)+c/(r+1)\}$ where $c$ are integers between $0$ and $r$.

\begin{example}
With the notation of  Example \ref{a2exp1}, we now compute
${\mathcal B}(\CH_2^A,\check{P}_A, g^A_{\bf s})(v)$ for $v=v_1 e^1+ v_2 e^2+ v_3 e^3$ with $v_1+v_2+v_3=0$.
We express  $v=v_1(e^1-e^2)-v_3(e^2-e^3)$.  For exponents ${\bf s}=[1,1,1]$,
{\small $${\mathcal B}(\CH_2^A,\check{P}_A, g^A_{\bf s})(v)=\sum_{\substack{m,n \in \Z \\(2m-n)(2n-m)(m+n)\neq 0}}\frac{e^{2i\pi m (v_1-v_2)+2i\pi n (v_2-v_3)}}
{(2i\pi (2m-n))(2i\pi (2n-m))(2i\pi (m+n))}$$}
{\small $$=\frac{1}{3}\left(P(v_1,v_2,v_3)+P(v_1+\frac{1}{3},v_2+\frac{1}{3},v_3-\frac{2}{3})+
P(v_1+\frac{2}{3},v_2+\frac{2}{3},v_3-\frac{4}{3})\right),$$}
where $P$ is the piecewise polynomial function given in Equation (\ref{weight}).
\end{example}

\subsection{Calculation of multiple Bernoulli series for types $B$ and $C$}
We use the same notation as in Section \ref{B}.

We now consider $ \CH_r^{BC}=\cup_{1\leq i\leq r}\{z^i=0\}\cup \cup_{1\leq i<j\leq r}\{z^i\pm z^j=0\}$, a
system of hyperplanes in $U_\C$.

Let $\Lambda$  be a lattice commensurable with $\oplus \Z e^i$,  with dual lattice $\Gamma$.
Denote simply by  $\Gamma_{reg}=\Gamma_\reg(\CH_r^{BC})$.
If $g\in \CR_{\CH_r^{BC}},$
\begin{equation}\label{typeC}
\mathcal B(\CH_r^{BC},\Lambda,g)(v)=\sum_{\gamma\in \Gamma_{\reg}}
 g(2i\pi \gamma)e^{2i\pi \ll v,\gamma\rr}.
\end{equation}

\subsubsection{Root system $C_r$}
Let $P_C$ be the weight lattice of the root system $C_r$.
We thus have the coroot lattice $\check{Q}_C=\oplus_{i=1}^r \Z e^i$.
Let ${\bf s}=[s_{\alpha}]$ be a list of exponents and let 
{\small $$g_{\bf s}^C(z)=\frac{1}{\prod_{\alpha>0}\ll H_\alpha,z\rr^{s_\alpha}}.$$ }
Here $\{H_\alpha, \; \alpha>0\}$ are positive coroots of the system $C_r$, which are explicitly
$\{ e^i, 1\leq i\leq r, (e^i\pm e^j), 1\leq i<j\leq r\}.$
Clearly, the function $g_{\bf s}^C$ belongs to $\CR_{\CH_r^{BC}}$.
If $v\in V$,
$$\mathcal B(\CH_r^{BC},\check{Q}_C, g_{\bf s}^C)(v)=\sum_{\gamma\in (P_C)_{reg}} \frac{e^{2i\pi \ll v,\gamma\rr}}{\prod_{\alpha>0} (2i\pi \ll H_\alpha,\gamma \rr)^{s_\alpha}}.$$
The function $\mathcal B(\CH_r^{BC},\check{Q}_C, g_{\bf s}^C)$ is a piecewise polynomial function on $V$ of degree
$\sum_\alpha s_\alpha$.
We  use the diagonal basis  constructed in Section \ref{B} to compute it.

Let us now compute the example given in the introduction, which corresponds to $C_2$ with exponents ${\bf s}=[2,1,1,1]$
ordered in accordance with the order $[2e_1,2 e_2,e_1+e_2,e_1-e_2]$ of positive roots 
(so that  $[e^1, e^2,e^1+e^2,e^1-e^2]$ is the corresponding order in positive coroots).
Express $v=v_1 e^1+v_2 e^2$, then $$\mathcal B(\CH_2^{BC},\check{Q}_C, g^C_{\bf s})(v)=
\sum'_{m,n}\frac{e^{2i\pi m v_1+2i\pi n v_2}}{(2i\pi m)^{2}(2i\pi n)(2i\pi (m+n))(2i\pi (m-n))}.$$
This piecewise polynomial function  $Q(v_1,v_2):=\mathcal B(\CH_2^{BC},\check{Q}_C, g^C_{\bf s})(v)$ is given by
{\small \begin{align}
&Q(v_1,v_2)=-\frac {1}{160}{\left\{-v_{{2}}+v_{{1}} \right\}}^{5}-\frac{1}{48}{\left\{v_{{1}} \right\} }^{2}
+\frac{1}{24}{\left\{ v_{{2}}+v_{{1}} \right\} }^{3} \left\{ v_{{2}}\right\} -\frac{1}{48}{\left\{ v_{{2}}+v_{{1}} \right\} }^{4} \left\{ v_{{2}} \right\} \nonumber \\
&+\frac{1}{24}{\left\{ v_{{1}} \right\} }^{3}-\frac{1}{48}{ \left\{ v_{{2}}+v_{{1}} \right\} }^{2} \left\{ v_{{2}} \right\} -{\frac {1}{960}}
\left\{ v_{{2}}+v_{{1}} \right\} +{\frac {1}{96}}{ \left\{ v_{{2}}+v_{{1}} \right\} }^{2}-{\frac {1}{96}}{ \left\{ v_{{2}}+v_{{1}} \right\} }^{3}
-\frac{1}{48}{\left\{ v_{{1}} \right\} }^{4}\nonumber \\
&-{\frac {1}{192}}{\left\{ v_{{2}}+v_{{1}}\right\}}^{4}+{\frac {1}{960}}\left\{-v_{{2}}+v_{{1}} \right\} +{\frac {1}{96}}\,{ \left\{ -v_{{2}}+v_{{1}}\right\} }^{2}-\frac{1}{32}{ \left\{ -v_{{2}}+v_{{1}} \right\} }^{3}
+{\frac {5}{192}}{\left\{ -v_{{2}}+v_{{1}} \right\} }^{4} \nonumber \\
&+\frac{1}{24}{\left\{ v_{{1}} \right\} }^{2}\left\{ v_{{2}} \right\} -\frac{1}{12}{ \left\{ v_{{1}} \right\} }^{3}\left\{ v_{{2}} \right\} 
+\frac{1}{24}{\left\{ v_{{1}} \right\} }^{4}\left\{ v_{{2}} \right\} +{\frac {1}{160}}\,{ \left\{ v_{{2}}+v_{{1}}\right\} }^{5}
+\frac{1}{24}{\left\{ -v_{{2}}+v_{{1}} \right\} }^{3}\left\{ v_{{2}} \right\} \nonumber \\
&-\frac{1}{48}{\left\{ -v_{{2}}+v_{{1}} \right\} }^{4} \left\{ v_{{2}} \right\} -\frac{1}{48}{ \left\{ -v_{{2}}+v_{{1}}\right\} }^{2} \left\{ v_{{2}} \right\}.
\label{eq:BB2}
\end{align}}


Let us demonstrate what happens  on a tope.   Figure \ref{b2} depicts topes associated to the pair
$\Phi^{eq}(BC_2)=\{e^1, e^2, e^1+e^2, e^1-e^2\}$ and
$\Lambda=\Z e^1 \oplus \Z e^2$.  For example, on $\tau_2=\{v_1>0,\, v_2>0,\, v_1>v_2,\, v_1+v_2<1\}$,
the piecewise polynomial function  $\mathcal B(\CH_2^{BC},\check{Q}_C, g^C_{\bf s})(v)$ coincides with the polynomial
\begin{align}
Q_{\tau_2}(v_1,v_2)=&\frac{1}{8}(-{\frac {1}{60}}v_{{2}}+\frac{1}{2}{v_{{1}}}^{2} v_{{2}}-{v_{{1}}}^{3} v_{{2}}+\frac{1}{6}{v_{{2}}}^{2}-v_{{1}}{v_{{2}}}^{2}+v_{{1}}
{v_{{2}}}^{3} +{v_{{1}}}^{2}{v_{{2}}}^{2}\nonumber \\
&-\frac{1}{6}{v_{{2}}}^{3}+\frac{1}{6}{v_{{2}}}^{4}+\frac{1}{2}{v_{{1}}}^{4}v_{{2}}-{v_{{1}}}^{2}{v_{{2}}}^{3}-{\frac {7}{30}}{v_{{2}}}^{5})
\end{align}
%If we compute $Q_{\tau_2}(v_1,v_2)$ for $v_1=\frac{1}{15},\ v_2=\frac{1}{30}$ we obtain $-\frac{276037}{5832000000}$.
%which is the same value that we obtain for $Q_{tau}(\frac{1}{15},\frac{1}{30})$ as it should be, since $(\frac{1}{15},%\frac{1}{30})\in \tau$

We give some more numerical examples with different exponents.
For example, we may compute with exponents ${\bf s}=[s_1,s_2,s_3,s_4]$ corresponding to the order $[2e_1,2 e_2,e_1+e_2,e_1-e_2]$ of roots, and $v=[v_1,v_2]$.
Then,
{\small \[\mathcal B(\CH_2^{BC},\check{Q}_C, g^C_{\bf s})(v_1e^1+v_2e^2)=\sum'_{m,n}\frac{e^{2i\pi mv_1+2i\pi n v_2}}
{(2i\pi m)^{s_1}(2i\pi n)^{s_2}
(2i\pi (m+n))^{s_3} (2i\pi (m-n))^{s_4}}\]}
\[=\left\{ \begin{array}{l@{\quad if \quad } l@{\quad and\quad} l}
\frac{810650239}{132316540312500}&  ${\bf s}=[2,2,1,1]$&$v=[1/5, 1/19]$\\
\frac{47036110438854761301636459941}{1529174429579197250943325345977126782238720}&${\bf s}=[2,3,4,5]$
& $v=[1/7, 1/17]$
\end{array}\right.\]


\subsubsection{Root system $B_r$}
Let $\Gamma=P_B$ be the weight lattice of the root system $B_r$, with its dual the coroot lattice $\check{Q}_B$.
Let  ${\bf s}=[s_{\alpha}]$ be a list of exponents. We define,
{\small $$g^B_{\bf s}(z)=\frac{1}{\prod_{\alpha>0}\ll H_\alpha,z\rr^{s_\alpha}},$$}
where $\{H_\alpha, \; \alpha>0\}$ are positive coroots of the system $B_r$.
If $v\in V$,
$$\mathcal B(\CH_r^{BC},\check{Q}_B, g_{\bf s}^B)(v)=\sum_{\gamma\in (P_B)_{reg}} \frac{e^{2i\pi \ll v,\gamma\rr}}{\prod_{\alpha>0} (2i\pi \ll H_\alpha,\gamma \rr)^{s_\alpha}}.$$


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
$g_{\bf s}^B=c_1 g_{\bf s}^C$ where $c_1=\dfrac{1}{2^{s_{2e_1}+\cdots+s_{2e_r}}}$.
Then $2 \check{Q}_C$ is a sublattice of $\check{Q}_B$ of index $2^{r-1}$, and a set  of representatives  is given, for example, by
$$F :=\{0,e^{i_1}+e^{i_2}+\cdots +e^{i_k}, 1\leq i_1< i_2<\cdots <i_k,\ k=2j,  \ 1\leq j\leq [r/2]\}.$$
We then use comparison Formulae (\ref{eq:lattices}) and (\ref{eq:dillattices}).
Since $g^C_{\bf s}$ is homogeneous of degree $- \sum_\alpha s_{\alpha}$, we obtain
$$\mathcal B(\CH_r^{BC},\check{Q}_B,g^B_{\bf s})(v)=
\frac{1}{2^{r-1}}c_2\left(\sum_{\lambda\in F}\mathcal B(\CH_r^{BC},\check{Q}_C,g^C_{\bf s})({ \frac{v+\lambda}{2}})\right)$$
where $c_2=2^{\sum_{1\leq i<j\leq r} s_{e_i-e_j}+s_{e_i+e_j}}$.
In particular, if  ${\bf s}=[m,\ldots,m]$, 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 exponents ${\bf s}=[2,1,1,1]$ ordered with respect to the order $[e_1-e_2, e_2, e_1+e_2, e_1]$ of roots,
$$(2i\pi)^5\mathcal B(\CH_2^{BC},\check{Q}_B,g^B_{\bf s})({v})=\sum'_{m_1,m_2}\frac{e^{2i\pi( \left( m_1+1/2\,m_2 \right) v_1+1/2\,m_2v_2
)}}{m_1^2m_2\left(2 m_1+\,m_2 \right)  \left( m_2+m_1
\right) } $$
where the symbol $\sum'$ means we sum over the integers $m_1,m_2$ with 
$\ \left( 2m_{{1}}+\,m_{{2}} \right)  \left( m_{{2}}+m_{{1}}\right) m_{{2}}m_{{1}}\neq 0$.
We obtain
$$\mathcal B(\CH_2^{BC},\check{Q}_B,g^B_{\bf s})({{v}})=2\left(Q(\frac{v_1}{2},\frac{v_2}{2})+Q(\frac{v_1+1}{2},\frac{v_2+1}{2})\right)$$
where $Q=\mathcal B(\CH_2^{BC},\check{Q}_C,g^C_{\bf s})$ is given in Equation (\ref{eq:BB2}).

In particular, for $u=[1/15,1/30]$ we obtain
$$\mathcal B(\CH_2^{BC},\check{Q}_B,g^B_{\bf s})({{u}})=\frac{-276037}{5832000000}.$$

For $B_3$,
we compute for $v=v_1 e^1+ v_2 e^2+ v_3 e^3$ and ${\bf s}=[1,1,1,1,1,1,1,1,1]:$
{\small $$(2i\pi)^{9}\mathcal B(\CH_3^{BC},\check {Q_B},g^B_{\bf s})({{v}})=$$}
{\tiny $$\sum'_{m_1,m_2,m_3}\frac{e^{2i\pi(\left(m_1+m_2+1/2\,m_3 \right)v_1+\left( m_2+1/2\,m_3 \right)v_2+1/2\,m_3v_3}}
{\left( 2m_1+2 m_2+m_3\right) \left(2m_2+\,m_3 \right) m_3 m_1 \left(m_1+2m_2+m_3\right)\left(m_1+m_2 \right)\left(m_1+m_2+m_3 \right) m_2\left(m_2+m_3 \right)}.$$}
We obtain
$$\mathcal B(\CH_3^{BC},\check {Q}_B,g^B_{\bf s})({v})=$$
{\small $$2^4\left(S(\frac{v_1}{2},\frac{v_2}{2},\frac{v_3}{2})+S(\frac{v_1+1}{2},\frac{v_2+1}{2},
\frac{v_3}{2})+S(\frac{v_1+1}{2},\frac{v_2}{2},\frac{v_3+1}{2})+S(\frac{v_1}{2},\frac{v_2+1}{2},\frac{v_3+1}{2})\right)$$}
where $S=\mathcal B(\CH_3^{BC},\check {Q}_C,g^C_{\bf s})$ is a piecewise polynomial that is too long to be included here.


\subsection{Calculation of multiple Bernoulli series for type $D$}
We follow the same notation as in Section \ref{D}.

Recall the system of hyperplanes $\mathcal{H}_r^D=\cup_{i,j}\{z^i\pm z^j=0 ,1\leq i<j\leq r\}$.  Let ${\bf s}=[s_\alpha]$ be a list of exponents;
the ordering of elements in the list ${\bf s}$ is taken to match the following ordering
$[e_1-e_2,e_1-e_3,\ldots,e_1-e_r,e_2-e_3,\ldots, e_1+e_2,\ldots,e_{r-1}+e_r]$ of positive roots of the system $D_r$.
We define {\small $$g^D_{\bf s}(z)=\frac{1}{\prod_{\alpha>0}\ll H_\alpha,z\rr^{s_\alpha}},$$}
where  $\{H_\alpha, \; \alpha>0\}$ are positive coroots of $D_r$.


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 ${\bf{S}}=[s_\alpha,0,\ldots, 0]$  the list obtained from  ${\bf s}$ by adjoining
$r$ zeros to its end.  Then, by construction, we have
$g^D_{\bf s}(z)=g^B_{\bf S}(z)$.

We now associate to the list $\bf{s}$  a list of exponents
${\bf{s}}_k$ for the system $B_{r-1}$. In $\bf{s}$ we
eliminate the position  corresponding to the roots $e_i\pm e_k$ for
$i<k$, and $e_k\pm e_i$ 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 $v=\sum_{i=1}^r v_i e^i$  by putting $v_k=0$.


Let $\Gamma=P_D$ be the weight lattice of $D$ and $\check{Q}_D$ the dual lattice generated by the coroots.
Since $P_D$ is the weight lattice of the simply connected group $\text{Spin}(2r)$,
$\gamma=\sum_{i=1}^r \gamma^i e_i$ is in $P_D$ if
$\gamma^i\pm \gamma^j\in \Z$ 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 $\{e_1-e_2,e_2-e_3,\ldots,e_{k-1}-e_{k+1},e_{k+1}-e_{k+2},\ldots, e_r\}.$


Using the decomposition (\ref{eq:disjoint}), 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 $\gamma \in (P_{D})_{reg}$ and $\gamma_k=0$, then
\[g^D_{\bf s}(\gamma)=\dfrac{1}{{\displaystyle \prod_{\gamma_i\pm \gamma_j \neq 0,\,\gamma_k=0}}
(\gamma_i- \gamma_j )^{s_{e_i-e_j}}(\gamma_i+ \gamma_j )^{s_{e_i+e_j}}}= c_kg^C_{{\bf s}_k}(i_k(\gamma))
\]
with $c_k=(-1)^{\sum_{j=k+1}^r{s_{e_k-e_j}}}.$
%and  $\hat{s}_{e_i-e_k}={s_{e_i-e_k}}$ if $i<k,$ $\hat{s}_{e_i-e_k}={s_{e_k-e_i}}$ otherwise.
In particular, if ${\bf s}=[m,\ldots,m]$, then $c_k=(-1)^{(r-k)m}$. 
Thus, we can compute multiple Bernoulli series for the system of type $D_r$ by using computations for types $B_r$  and
$C_{r-1}$ with appropriate exponents.  More explicitly,
\begin{align*}\mathcal{B}(\mathcal{H}_r^D, \check{Q}_D,g^D_{\bf s})({v})&=\sum_{\gamma \in (P_{D})_{reg}}  \frac{e^{ 2i\pi \ll v ,\gamma \rr}}{
\prod_{\alpha >0} (2i\pi \ll H_{\alpha},\gamma \rr)^{s_{\alpha}}}\\
%\[\sum_{\gamma=\sum \gamma_ie_i, \gamma_i\pm \gamma_j \neq 0}  \frac{e^{\ll 2i\pi v ,\gamma \rr}}{
%\prod_{1\leq i<j\leq n} ((2i\pi))^K (\gamma_i\pm \gamma_j )^{K} }=\]
%[\sum_{\gamma \in \check{P_B},}  \frac{e^{\ll 2i\pi v ,\gamma \rr}}{
%\prod_{1\leq i<j\leq n} ((2i\pi))^K (\gamma_i\pm \gamma_j )^{K} }=\]
&=\mathcal{B}(\mathcal{H}_r^{BC}, \check{Q}_B,g^B_{\bf S})({v})+
%\[=\sum_{ \gamma\in L_{\neq 0}}  \frac{e^{\ll 2i\pi v ,\gamma \rr}}{
%\prod_{1\leq i<j\leq n}  ((2i\pi)^{K}\gamma_i\pm \gamma_j )^{K} }+\]
%\[\sum_{k=1}^n(-1)^{n-k}\sum_{\gamma\in L_{ k}}  \frac{e^{\ll 2i\pi v ,\gamma \rr}}
%{\prod_{1\leq i<j\leq n \ i,j\neq k}  ((2i\pi)^{K}\gamma_i\pm \gamma_j )^{K} \prod_{1\leq i\leq n \  \ i \neq k}  (\gamma_i )^{2K} }\]
\sum_{k=1}^r c_k\mathcal{B}(\mathcal{H}_{r-1}^{BC}, {\check{Q}}_{C_{r-1,k}},g^C_{{\bf s}_k})({i_k(v)}).\end{align*}


\section{Witten formula for volumes of moduli spaces of flat connections on surfaces}
As in Section \ref{section:clroot}, let $G$ be a simple, connected, simply connected, compact Lie group of rank $r$
with maximal torus $T$.  For $g_1,g_2\in G$, we denote by
$[g_1,g_2]=g_1g_2 g_1^{-1} g_2^{-1}$ the commutator of $g_1$ and $g_2$.
Let $\Sigma$ be a compact connected oriented surface of genus $g$
and let $p:=\cup_j\{p_j\}$ be a set of  $s$ points on $\Sigma$. Let
$\mathcal C:=(\mathcal C_j)$ be a set of $s$ conjugacy classes in
$G$. We consider the representation variety
$$\mathcal M(G,g,s,\mathcal C):=\{(a,c)\in G^{2g}\times \mathcal C;\;\prod_{i=1}^g [a_{2i-1},a_{2i}]=\prod_{j=1}^s c_j\}/G.$$
If the adjoint orbits $\mathcal C_j$ are generic, $\mathcal M(G,g,s,\mathcal C)$ is an
orbifold of real dimension $(2g-2)\dim G+s\dim G/T$. It parameterizes the
set of flat $G$-valued connections on $\Sigma-p$, with holonomy
around $p_j$ belonging to the conjugacy class $\mathcal C_j$ modulo
gauge equivalence. As shown by Atiyah-Bott \cite{ab}, once a
$G$-invariant inner product on $\mathfrak g$ is chosen, the manifold
$\mathcal M(G,g,s,\mathcal C)$ carries a natural  symplectic form, and thus a natural volume form.
E.~Witten gave a formula for the volume of $\mathcal
M(G,g,s,\mathcal C)$ that we recall.


%We use the notation of Section \ref{section:clroot}.  We need some more definitions.

%For $\alpha \in \frh^*$, define
%$(\frg_{\C})_{\alpha}=\{X \in \frg_{\C}; [H,X]=\ll \alpha,H\rr X\;\text{for all}\;H \in\frh\}$.
%If $\alpha \neq 0$ and
%$(\frg_{\C})_{\alpha}\neq 0$, then $\alpha$ is called a {\it root} of
%$\frh$ in $\frg_{\C}$. Let $R=R(\frg_{\C},\frh)\subset \frh^*$ be
%the set of roots; denote the root lattice by $Q$. Roots $\alpha \in
%R$ take imaginary values on $\frt$.

%If $\alpha \in R$, there exists a unique element $H_{\alpha}$ in
%$[(\frg_{\C})_{\alpha},(\frg_{\C})_{-\alpha}]$ satisfying $\ll
%\alpha,H_\alpha \rr=2$; it is called the {\it coroot} associated to
%$\alpha$.  For each $\alpha$ in $R$, $i H_{\alpha}$ is in $\frt$.
%The lattice spanned by $\{H_{\alpha}, \,\alpha \in R\}$ is
%called the {\it coroot lattice} and denoted by $\check{Q}$.  We denote by
%$\frh_\R:=\sum_\alpha \R H_\alpha$, the real span of coroots.


Let $R^+$  be a choice of positive roots; denote the highest root of $R$ by $\theta$.
Let $\frh_{+} : =\{h\in \frh_\R;\; \alpha(h)\geq 0\;\text{for all}\;\alpha \in R^+\}$ be the positive chamber
(closed) in $\frh_\R$, and $\frA:=\{h\in \frh_{+} ;\; \theta(h)\leq 1\}$ be the fundamental alcove.
An element of $\frA$ is said to be {\it regular} if it lies strictly inside the alcove.
Let $W=W(\frg_{\C},\frh)$ be the Weyl group (identified with $N_G(T)/T$).


%$\Phi_m$ the sequence  $\Phi$ repeated $m$ times (multiplicity $m$).

%$\{\varpi_i \}_{1\leq i\leq r}$ fundamental weights, defined as the
%basis of $\frh^*$ dual to $\{H_{\alpha_i}\}_{1\leq i\leq r}$.

%Define the weight lattice $P=\{ \lambda \in \frh^* : \lambda(H_{\alpha}) \in
%\mathbb{Z},\; \forall \alpha \in R\}$. A {\it regular} weight
%$\lambda\in P^{reg}$ is such that  $ \lambda(H_{\alpha}) \neq 0$
%for all $H_\alpha$.
%Any element $\lambda$ of $P$ defines a character $e^\lambda$ of $T$ by $e^\lambda(\exp X)=\exp \lambda(X)$....OR $\exp 2i\pi\lambda(X)$?


%\begin{remark} We use complex roots (as opposed to real roots) which are
%complexification of infinitesimal roots of the pair $(\frg,\frt)$.
%Recall that for a real root $\mu:\frt \to \R$ the corresponding
%infinitesimal root is $2\pi i \mu$.
%
%Our motivation for this choice comes from our aim to
%compare two formulae (including the constants) of the symplectic volume of the moduli space; the first derived as the
%limit of the Verlinde formula which is stated in terms of complex roots, the second as given by Witten
%\cite{wi} working with infinitesimal roots.
%\end{remark}
%

We now give the Witten formula.


Let ${\bf a}=\{a_1,a_2,\ldots, a_s\}$ be a set  of regular elements in $\frA \subset \frh_+$.
Let $\mathcal C_j$ be the adjoint orbit of $\exp(a_j)$; we denote the collection of orbits $\mathcal C_j$ by $\mathcal C$.
Consider the function on $\frh^*$ given by
$$N_{\bf a}(\lambda)=\prod_{j=1}^s \sum_{w\in W} \varepsilon(w) e^{\ll wa_j,\lambda\rr}.$$

Let $\Phi=\Phi(G)$ be the list of positive coroots $H_\alpha$.  Define
\[
W(\Phi(G), P,g,s)({\bf a}):=\sum_{\gamma \in P^{reg}}\frac{N_{\bf a}(2i\pi \gamma)}{\prod_{H_\alpha \in \Phi}
(2i\pi \ll H_\alpha, \gamma\rr)^{2g-2+s}}.\]

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
\[
W(\Phi(G), P,g)=\sum_{\gamma \in P^{reg}}\frac{1}{\prod_{H_\alpha \in \Phi} (2i\pi \ll H_\alpha,
\gamma\rr)^{2g-2}}\]
which is meaningful if $g\geq 2$.


Interchanging the sum and the product, $N_{\bf a}(\lambda)$ may be expressed as
%$$N_{\bf a}(\lambda)=\sum_{w_{i_1}\in W}\sum_{w_{i_2}\in W} \cdots \sum_{w_{i_s}\in W}
%\varepsilon(w_{i_1})\varepsilon(w_{i_2})\cdots \varepsilon(w_{i_s})e^{\sum_{j=1}^s \ll w_{i_j}a_j,\lambda\rr}.$$
$$N_{\bf a}(\lambda)=\sum_{(w_1, w_2, \ldots, w_s)\in W^s}
\prod_{j=1}^s \varepsilon(w_j)e^{\sum_{j=1}^s \ll w_ja_j,\lambda\rr}.$$
Hence the function $W(\Phi(G), P,g,s)({\bf a})$ can be expressed as a sum over $W^s$ with signs of Bernoulli series
$\CB(\Phi_{2g-2+s},\check Q)(\sum_j w_j a_j)$.  Here, as before, $\Phi_{2g-2+s}$ means that each coroot in $\Phi$ is taken
with multiplicity $2g-2+s$.
As is well known, the series $W(\Phi(G), P,g,s)({\bf a})$ computes the
symplectic volume of $\mathcal M(G,g,s,\mathcal C)$ 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.
To demonstrate this we need some more notation.

%The coweight lattice $\check{P}:=\{h \in \frh:\; \alpha(h) \in \Z\;
%\text{for}\;\alpha \in R\}$.  Under the exponential map $\exp:\frh
%\to T ;\;\;H\mapsto \exp(2i\pi H)$, $\check{P}/\check{Q}$ is
%identified with


%Let $\check{P}_{\rm lg}=\{x \in \frh: \alpha(x) \in \Z \;{\rm
%for}\;\; \alpha\;\;{\rm long}\}$.

%Dual coxeter number $\check{h}:=\rho(H_\theta)+1$, $\rho$ the half
%sum of positive roots.

Let $(\;|\;)$ denote the $G$-invariant symmetric form  on $\frg_{\C}$ normalized such that
$(H_{\theta}|H_{\theta})=2$.  We will use the same notation for the
restricted form on $\frh$, and the induced form on $\frh^*$.
We call  $(\;|\;)$  the {\it basic invariant form}.  It is positive definite on 
$\frh_\R$, and negative definite on $\frt$.


Let $\check{h}:=\rho(H_\theta)+1$ be the dual Coxeter number, where $\rho$ is the half
sum of positive roots.   Let $Q_{long}\subset Q$ be the lattice spanned by long roots.
The basic invariant form identifies $\frh_\R$ and ${\frh_\R}^*$; under this isomorphism the coroot lattice $\check Q$ is identified to $Q_{long}$.
Let $q$ be the index of $Q_{long}$ in $Q$, and  let $f$ be the  index of $Q$ in $P$.
The center of $G$ is denoted by $Z=Z(G)$.

For a positive integer $\ell$, define the set \[P_\ell:=\{\mu \in P
\cap \frh_+^* ; \mu(H_{\theta})\leq \ell\}.\]
An element of $P_\ell$ is said to be a weight of {\it level} $\ell$.  We denote by $P'_\ell$ the subset of
$P_\ell$ consisting of elements $\mu$ satisfying $\mu(H_\theta)<\ell$ and $\mu(H_\alpha)>0$ for any simple root $\alpha$.
By definition of $\check{h}$, there is a bijection between sets $P_\ell$ and
$P'_{\ell+\check{h}}$ via $\mu \mapsto \mu+\rho$.

%The simply connected complex Lie group with Lie algebra $\frg_\C$ is the complexification $G_\C$ of the simply
%connected compact Lie group $G$.
%Then $T_\C$ is its maximal torus having $\frh$ as its Lie algebra.

Consider the maximal torus $T$ of $G$  with Lie algebra $\frt$. If $t=\exp X\in T$, with $X\in \frt$, and $\alpha$ is a
root (which takes imaginary values on $\frt$), we denote by $e^\alpha(t)=e^{\ll\alpha,X\rr}$.
Let $\Delta(t)=\prod_{\alpha \in R}(e^{\alpha}(t)-1)$.
An element $t$ of $T$ is said to be regular if $\Delta(t)\neq 0$.
Denote by $T_\ell$ the subgroup of elements $t$ of $T$ such that
$e^{\alpha}(t)$ is $\ell+\check{h}$ root of unity for each long root
$\alpha$.  We denote the set of regular elements in $T_\ell$ by $T^{\text{reg}}_\ell$.


We now give the Verlinde formula.


Consider a set $\underline \lambda=\{\lambda_1,\lambda_2,\ldots, \lambda_s\}$ with $\lambda_i \in P_\ell$.
Then to this collection ${\underline \lambda}$, the group $G$, and a nonnegative integer $g$,
is associated a vector space $\mathcal V(G,g,s, {\underline \lambda},\ell)$ (see \cite{tuy}), called the space of
conformal blocks, whose dimension is given by the Verlinde formula $V(G,\underline \lambda,g,\ell)$:
\[
V(G,\underline \lambda,g,\ell)=(fq)^{g-1}(\ell+\check{h})^{r(g-1)}\displaystyle \sum_{t \in
T_\ell^{\text{reg}}/W}\frac{\chi_{V({\underline \lambda})}(t)}{\Delta(t)^{g-1}}.\]
Above $r$ is the rank of $G$, $V(\underline \lambda)=V_{\lambda_1}\otimes V_{\lambda_2}\otimes \cdots \otimes V_{\lambda_s}$
where $V_{\lambda_i}$ denotes the simple $\frg$ module with highest weight $\lambda_i$, and
$\chi_{V_\lambda}$ denotes the character of $V_\lambda$.  By Weyl character formula
$\chi_{V_\lambda}=J(e^{\lambda+\rho})/J(e^\rho)$ where $J(e^\nu)=\displaystyle \sum_{w\in W} \varepsilon (w)e^{w\nu}$.


We remark that if $\sum_{i=1}^s \lambda_i$ is not in the root lattice, then $V(G,\underline \lambda,g,\ell)$ is zero.

Under the isomorphism given by the basic invariant form, an element $a$ lying in $\frA \subset \frh_+$ defines an
element $\hat a$ of $\mathfrak h_+^*$.
We now consider a collection $\{a_1,a_2,\ldots, a_s\}$ of rational elements in $\frA$, that is, each $a_j$ lies in the dense subset
$\frA \cap (\check{Q}\otimes \Q)\subset \frA$.
We may choose $\ell$ large enough so that each $\lambda_j:=\ell \hat a_j$ is a weight; which then lies in $P_{\ell}$.
We furthermore choose $\ell$  so that $\sum_{j=1}^s \lambda_j$ is in the root lattice and consider the space of
conformal blocks
$\mathcal V(G,g,s, {\bf \underline \lambda},\ell)$ associated to this collection
${\underline \lambda}=\{\lambda_1,\lambda_2,\ldots, \lambda_s\}$.
We can dilate simultaneously the weights $\lambda_j$ and the level $\ell$ by a factor $k$.
Then, the function
$$k\mapsto \dim(\mathcal V(G,g,s,[k\lambda_1,k\lambda_2,\cdots,k\lambda_s],k\ell))$$ is a quasi-polynomial in $k$ of degree
$m=\dim (G)(g-1)+s|R^+|$, the complex dimension of the moduli space $\mathcal M(G,g,s,\mathcal C)$.
The volume computes the highest term of this quasi-polynomial.
More precisely,
$$\vol(\mathcal M(G,g,s,\mathcal C))=\lim_{k\to \infty} (\ell k)^{-m} dim(\mathcal V(G,g,s, [k \ell \hat a_1,
\ldots,  k \ell \hat a_s], k\ell).$$


\begin{proposition}\label{P:factor}
Let ${\bf a}=\{a_1,\cdots, a_s\}$ be a  collection of regular rational elements in $\frA$.
Let $\vol(G,g)({\bf a})$ denote the symplectic volume of the moduli
space $\mathcal M(G,g,s,\mathcal C)$. Then,
\[\vol(G,g)({\bf a})=(fq)^{g-1}\frac{|Z|}{|W|}\epsilon_G^{p(2g-2+s)}(-1)^{(g-1)|\Phi(G)|}W(\Phi(G), P,g,s)({\bf a}),\]
where $p$ is the number of short positive roots of $G$, and
$\epsilon_G=2$ for any simple Lie group except for $G_2$, where $\epsilon_G=3$.
\end{proposition}

We recall that for simply laced groups $p=0$ since all roots are considered as long.

\begin{proof}
Choose $\ell$ so that each $\lambda_j:=\ell \hat a_j$  lies in $P_{\ell}$ and $\sum_{j=1}^s \lambda_j$ is a root.
Then,
\[
\vol(G,g)({\bf a})=\lim_{k \to
\infty}\frac{1}{(k\ell)^{m}}V(G, k \underline{\lambda}, g,k\ell),\] where
$m=\text{dim}(G)(g-1)+s|R^+|$ is the dimension of $\mathcal M(G,g,s,\mathcal C)$ and
\[V(G, k\underline \lambda, g,k\ell)=(fq)^{g-1}(k\ell+\check{h})^{r(g-1)}
\displaystyle \sum_{t \in T_{k\ell}^{\text{reg}}/W}\frac{\chi_{V(k\underline \lambda)}(t)}{\Delta(t)^{g-1}}.\]
An element $\mu \in P_\ell$ determines a unique regular element $h_\mu \in \frA$, the image of
$\frac{\mu+\rho}{\ell+\check{h}}$ under the identification given by the basic invariant form.
Denote the image of $h_\mu$ under the exponential map by $t_\mu \in T_\C$.
The set $\{t_\mu: \mu \in P_\ell\}$ form a set of representatives for $T_\ell^{\text{reg}}/W$.
Using also the bijection between sets $P_\ell$ and $P'_{\ell+\check{h}}$ given by $\mu \mapsto \mu+\rho$,
\[\begin{array}{ll}
V(G, k\underline \lambda, g,k\ell)&=(fq)^{g-1}(k\ell+\check{h})^{r(g-1)} \displaystyle
\sum_{\mu+\rho \in P'_{k\ell+\check{h}}}\frac{\chi_{V(k\underline \lambda)}(t_\mu)}
{\Delta(t_\mu)^{g-1}}\\
&=(fq)^{g-1}(k\ell+\check{h})^{r(g-1)}|W|^{-1}\displaystyle
\sum_{\mu+\rho \in W \cdot P'_{k\ell+\check{h}}}\frac{\prod_{j=1}^{s}J(e^{k\lambda_j+\rho})(t_\mu)}
{(J(e^\rho)(t_\mu))^s \Delta(t_\mu)^{g-1}}\\
&=\dfrac{(fq)^{g-1}}{|W|}(k\ell+\check{h})^{r(g-1)}(-1)^{|\Delta^+|(g-1)}\displaystyle
\sum_{\mu+\rho \in W \cdot P'_{k\ell+\check{h}}}\frac{\prod_{j=1}^{s}J(e^{k\lambda_j+\rho})(t_\mu)}
{(J(e^\rho)(t_\mu))^{2g-2+s}}.
\end{array}\]
The second line above follows from the fact that both $\chi_{V(\lambda)}(t)$
and $\Delta(t)$ are $W$-invariant.  The third line follows from the second by the identity
\begin{equation}\label{eq:delta}
\Delta(t)=J(e^\rho)(t)\overline{J(e^\rho)(t)}=(-1)^{|R^+|}(J(e^\rho)(t))^2.
\end{equation}


We now analyze $V(G, k\underline \lambda, g,k\ell)$ as $k$ gets large.


The expression $\prod_{j=1}^{s}J(e^{k\lambda_j+\rho})(t_\mu)$ in the summand is equal to 
\[\prod_{j=1}^{s}\displaystyle
\sum_{w\in W}\varepsilon(w)e^{w(k\lambda_j+\rho)}(t_\mu)
=\prod_{j=1}^{s}\displaystyle
\sum_{w\in W}\varepsilon(w)\exp\left( 2i\pi\left (\mu+\rho|\frac{w(k \ell\hat a_j+\rho)}{k\ell+\check h}\right)\right ).\]
%since $$e^{w(\lambda_j+\rho)}(t_\mu)=e^{\mu+\rho}(t_{w(\lambda_j+\rho)-\rho})
%=\exp\left( 2i\pi\left (\mu+\rho|\frac{w(k \ell\hat a_j+\rho)}{k\ell+\check h}\right)\right ).$$
Now as $k$ gets large, the expression $\exp\left( 2i\pi\left (\mu+\rho|\frac{w(k \ell\hat a_j+\rho)}{k\ell+\check h}\right)\right )$
approaches to $\exp(2i\pi(\mu+\rho|w\hat{a}_j))$. Observe also that the set $W \cdot
P'_{k\ell+\check{h}}$ approaches $P^{\text{reg}}$.  Denote an element
$\mu+\rho$ of this limiting set by $\gamma$.  Hence, $\prod_{j=1}^{s}J(e^{k\lambda_j+\rho})(t_\mu)$ approaches
$\prod_{j=1}^{s}\displaystyle
\sum_{w\in W}\varepsilon(w)e^{\ll 2i\pi\gamma,wa_j\rr}=N_{\bf a}(2i\pi \gamma)$.


Now we analyze the denominator of the summand, $$\displaystyle \frac{1}{J(e^\rho)(t_\mu)}
=\displaystyle \frac{1}{\prod_{\alpha>0}(e^{\alpha/2}(t_\mu)-e^{-\alpha/2}(t_\mu))}
=\displaystyle \frac{1}{\prod_{\alpha>0} 2i\sin(\pi\frac{(\alpha|\mu+\rho)}{k\ell+\check h})}.$$
This expression explodes at each central vertex and the contribution from each, as $k$ gets large, is
$\frac{(k\ell+\check h)^{|R^+|}}{\prod_{\alpha>0} 2i\pi(\alpha|\mu+\rho)}$.
Also observe that, for $z \in Z(G)$ both $T_\ell^{\text{reg}}$ and $\Delta(t)$ is invariant under $t\mapsto tz$.
Moreover, since $\sum_{i=1}^s \lambda_i$ is in the root lattice by construction, $\chi_{V(k\underline \lambda)}(t)$ is
also invariant.  Therefore, we may add all these equal contributions from central vertices.
(see also Remark 5.8. \cite{tw}).
Hence, we get that the expression
\[\displaystyle
\sum_{\mu+\rho \in W \cdot P'_{k\ell+\check{h}}}\frac{\prod_{j=1}^{s}J(e^{k\lambda_j+\rho})(t_\mu)}
{(J(e^\rho)(t_\mu))^{2g-2+s}}\] approaches
\[|Z(G)|(k\ell+\check{h})^{|R^+|(2g-2+s)}
\displaystyle \sum_{\gamma \in P^{\text{reg}}}\frac{N_{{\bf a}}(2i\pi \gamma)}
{\prod_{\alpha>0}(2i\pi(\alpha|\gamma))^{2g-2+s}}
.\]

By virtue of the normalization in the basic invariant form, if
$\alpha$ is a long root we have $\ll H_{\alpha},\gamma
\rr=(\alpha|\gamma)$; otherwise $\ll H_{\alpha},\gamma
\rr=\epsilon_G(\alpha|\gamma)$.
%where $\epsilon_G=2$ 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 $|R^+|=|\Phi(G)|$,
we obtain
\begin{multline*}\displaystyle \lim_{k \to
\infty}\frac{1}{(k\ell)^{m}}V(G, k\underline \lambda, g,k\ell)\\
=(fq)^{g-1}\frac{|Z(G)|}{|W|}(-1)^{|\Phi(G)|(g-1)}\epsilon_G^{p(2g-2+s)}W(\Phi(G),P,g,s)({\bf
a})\end{multline*} as claimed.
\end{proof}

\begin{remark}
In the case of one marking, the Verlinde formula reduces to
$$
V(G,\lambda,g,\ell)=(fq)^{g-1}(\ell+\check{h})^{r(g-1)}\displaystyle \sum_{t \in
T_\ell^{\text{reg}}}\frac{e^{\lambda}(t)}{D(t)\Delta(t)^{g-1}},$$
where $D(t)=\prod_{\alpha>0}(1-e^{-\alpha}(t))$.

Following the same line of arguments as in the proof of Proposition \ref{P:factor}
we get that for $s=1$ with $\lambda=\ell \hat a$ lying in the root lattice,
\begin{equation}\label{E:1mark}
\vol(G,g)(a)=(fq)^{g-1}|Z|\epsilon_G^{p(2g-1)}(-1)^{(g-1)|\Phi|}
\sum_{\gamma \in P^{\text{reg}}}\frac{e^{\ll a,2i\pi \gamma\rr}}
{\prod_{H_{\alpha}\in \Phi}(2i\pi\ll
H_{\alpha},\gamma\rr)^{2g-1}}.\end{equation} That is,
$$\vol(G,g)(a)=(fq)^{g-1}|Z|\epsilon_G^{p(2g-1)}(-1)^{(g-1)|\Phi|}\CB(\Phi_{2g-1},\check Q)(a).$$
\end{remark}

%Let us demonstrate this with an example.
%\begin{example}\label{ex:su2onemark}
%We consider the moduli space of $\SU(2)$ bundles on a Riemann surface of genus $g$ with one marking.

%Let $a=tH_\alpha $ be a regular element in $\mathfrak{A}$; in other words, $0<t<1/2$.
%Let $\hat{a} \in \frh_\R^*$ denote the dual of $a \in \frA$ under the isomorphism given by the basic invariant form.
%The Verlinde formula for $\SU(2)$ with marking
%$\lambda=\ell\hat{a}=\ell t\alpha \in P_\ell$ such that $t\ell$ is a positive
%integer (that is, $\lambda$ lies in the root lattice) is
%{\small \[V(\SU(2),t\ell\alpha,g,\ell)=2^{g-1}(\ell+2)^{g-1}2(-1)^{g}\sum_{n=0}^{\infty}c_{g,t}(n)(\ell+2)^{2g-1-n}
%\frac{B(2g-1-n,t)}{(2g-1-n)!},\]} where $c_{g,t}(n)$ is the $n^{th}$ coefficient of the Taylor series
%expansion of $e^{x(g-2t)}\left(\frac{x}{e^x-1}\right )^{2g-1}$ in $x$ around zero, and $B(p,t)$ denotes the
%$p^{\text{th}}$ Bernoulli polynomial in $t$.  Clearly $c_{g,t}(0)=1$.

%In the expression  for $V(\SU(2),tk\ell\alpha,g,k\ell)$ the highest term in $k$ occurs when $n=0$. Hence,
%\[
%\begin{array}{ll}
%\vol(\SU(2),g)(a)&=\displaystyle \lim_{k\to
%\infty}\frac{V(\SU(2),tk\ell\alpha,g,k\ell)}{(k\ell+2)^{3g-2}}\\
%&=2^{g-1}2(-1)^{g}c_{g,t}(0)\frac{B(2g-1,t)}{(2g-1)!}\\
%&=2^g(-1)^{g}\frac{B(2g-1,t)}{(2g-1)!}
%\end{array}\]

%We now calculate the volume using Equation (\ref{E:1mark}).

%We have $P=\Z \rho$ with $\ll \rho,H_\alpha \rr=1$.  In this case, with the notation of Proposition \ref{P:factor},
%$s=1$, $p=0$, $q=1$, $f=2$ and $|Z(SU(2))|=2$, hence $2^{p(2g-1)}(fq)^{g-1}|Z(G)|=2^g$.  Using the expression
%(\ref{E:1mark}) of the volume in the one marking case,
%{\small \[
%\begin{array}{ll}
%2^g(-1)^{(g-1)|\Phi|}
%\displaystyle \sum_{\gamma \in P^{\text{reg}}}\frac{e^{\ll a,2i\pi \gamma\rr}}
%{\prod_{H_{\alpha}\in \Phi}(2i\pi\ll H_{\alpha},\gamma\rr)^{2g-1}}&
%=2^g (-1)^{g-1}\displaystyle\sum_{n\neq 0}\frac{e^{2i\pi tn}}{(2i\pi n)^{2g-1}}\\
%&=2^g(-1)^g \displaystyle \frac{B(2g-1,t)}{(2g-1)!}. \end{array}\]}
%Clearly we get the same formula.
%\end{example}

\subsection{Volume of the moduli space as a function of the volume of $T$ and $G$}
Let us recall the formula for the symplectic volume  of the moduli
space $\mathcal M(G,g,s,\mathcal C)$ for a set of $s$ regular
conjugacy classes $\mathcal C=(\mathcal C_j)$ in $G$ as given by
E.~Witten (\cite{wi} equation 4.1.14),
{\small \begin{equation}\label{E:wformula}
\vol(\mathcal M(G,g,s,\mathcal C))=\frac{|Z(G)|}{(2\pi)^{2m}}\vol(G)^{2g-2}\vol(G/T)^s
\displaystyle \sum_{\lambda \in P^+}\frac{\prod_{j=1}^{s}[\chi_{V_{\lambda}}(\mathcal C_j)\sqrt{\Delta(\mathcal C_j)}]}
{\dim{V_\lambda}^{2g-2+s}}\end{equation}}
where $2m$ is the real dimension of $\mathcal M(G,g,s,\mathcal C)$,
and $P^+=P\cap \frh_+^*$ is the set of dominant weights parametrizing 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 $\frg$-invariant scalar product on $\frg$.  This determines a Lebesgue measure $\mu$ on $\frg$, via identification
of $\frg$ with $\R^n$ by an orthonormal basis.
Similarly let $\tau$ be the Lesbegue measure on $\frt$ corresponding to the restriction of the scalar product on $\frt$.
We can construct from $\mu$ and $\tau$ Haar measures $\mu_G$ and $\mu_T$
on $G$ and $T$ respectively.

Since we aim to compare the volume formula in Proposition \ref{P:factor} with that of Witten in Equation (\ref{E:wformula}),
we choose the normalized Killing form as the $\frg$-invariant scalar product in the above construction,
as this was our choice in the previous section.
Then, for this choice, with respect to $\mu_G$ and $\mu_T$ constructed as above, we get that
\[\vol(G)=(fq)^{1/2}(2\pi)^{|R^+|+r}\frac{1}{\prod_{\alpha>0}(\alpha|\rho)},\;\;\;
\vol(G/T)=\frac{(2\pi)^{|R^+|}}{\prod_{\alpha>0}{(\alpha|\rho)}}.\]


Observe that from formula (\ref{eq:delta}), $\Delta(t)$ takes positive values on a regular element $t$.
Then, for $\mathcal C_j$ the adjoint orbit of $\exp(a_j)$, we may write

$$\chi_{V_{\lambda}}(\mathcal C_j)=\frac{\displaystyle \sum_{w\in W}\varepsilon(w)e^{2i\pi\ll w(\lambda+\rho),a_j\rr}}
{i^{|R^+|}\sqrt{\Delta(\mathcal C_j)}}.$$

%\begin{remark}
%This is a remark for ourselves.  Above, the choice of square root of $-1$ is crucial.  If i don't choose this one
%Witten's formula differs from mine by factor of $-1$.  The root that i have chosen is also the choice that is done by
%Meinrenken-Woodward in their article titled `Moduli spaces of flat connections on 2 manifolds, cobordism, and Witten volume
%formulas'.  I understand from Witten that this choice has to do with choosing an orientation on the real manifold,
%and this is the compatible one, giving positive volume.
%\end{remark}

Let $d(\gamma)=\displaystyle \prod_{\alpha>0}\frac{\ll\gamma,H_\alpha\rr}{\ll\rho,H_\alpha\rr}$;
it computes the dimension of the irreducible representation $V_{\gamma-\rho}$. Then we have
\[\displaystyle \sum_{\lambda \in P^+}\frac{\prod_{j=1}^{s}\chi_{V_{\lambda}}({\mathcal C_j})\sqrt{\Delta({\mathcal C_j})}}
{\dim{V_\lambda}^{2g-2+s}}=\displaystyle \sum_{\lambda \in P^+}
\frac{\prod_{j=1}^{s}N_{a_j}(2i\pi(\lambda+\rho))}
{i^{s|R^+|}d(\lambda+\rho)^{2g-2+s}}.\]

Observe that the summand above is invariant under the Weyl group (both the numerator and the denominator are anti-invariant
by factor $(\text{sign}(w))^s$ for a Weyl group element $w$).  Using this invariance, we get
\[\displaystyle \sum_{\lambda \in P^+}\frac{\prod_{j=1}^{s}\chi_{V_{\lambda}}(\mathcal C_j)\sqrt{\Delta(\mathcal C_j)}}
{\dim{V_\lambda}^{2g-2+s}}=
\frac{1}{|W|}\displaystyle \sum_{\gamma \in P^{reg}}\frac{\prod_{j=1}^{s}N_{a_j}(2i\pi \gamma)}
{i^{s|R^+|d(\gamma)^{2g-2+s}}}.\]

Inserting  the expressions for the volume of $G$ and $G/T$ into Equation (\ref{E:wformula}),
all $(2\pi)$ factors cancel, and combining the terms we get
{\small \[\begin{array}{ll}
\vol(\mathcal M(G,g,s,\mathcal C))&=
\dfrac{|Z|}{|W|} (fq)^{g-1}
\dfrac{(\prod_{\alpha>0}\ll\rho,H_\alpha\rr)^{2g-2+s}}{(\prod_{\alpha>0}(\alpha|\rho))^{2g-2+s}}(-1)^{(g-1)|R^+|}
W(\Phi(G),P,g)(a)\\
&\\
&=\dfrac{|Z|}{|W|}
(fq)^{g-1}\epsilon_G^{p(2g-2+s)}(-1)^{(g-1)|R^+|}W(\Phi(G),P,g)(a),
\end{array}\]} which is precisely the formula that we obtained in
Proposition \ref{P:factor}.

\section{Various examples of volume calculations}
%\begin{example}
%With the notation of example \ref{ex:su2onemark} we compute the volume of the moduli space of $\SU(2)$ bundles on a
%Riemann surface of genus $g$ with $s$ markings ${\bf a}=\{a_j=t_j H_\alpha,\;j:1\cdots s\}$, all regular in $\frA$.  Using the expression
%in Proposition \ref{P:factor},
%\[\vol(\SU(2),g)({\bf a})=2^{g-1}2(-1)^{g-1}\frac{1}{2}\displaystyle \sum_{n\neq 0}
%\frac{\prod_{j=1}^s (e^{2i\pi n t_j}-e^{-2i\pi n t_j})}{(2i\pi n)^{2g-2+s}}\]
%\[=2^{g-1}(-1)^{g-1}\displaystyle \sum_{n\neq 0}\frac{\prod_{j=1}^s \sin(\pi n 2 t_j)\cdot (2i)^s}{(2i\pi n)^{2g-2+s}}\]
%\[=\frac{1}{2^{2g-2}}\frac{1}{\pi^{2g-2+s}}\displaystyle \sum_{n\neq 0}\frac{\prod_{j=1}^s \sin(\pi n 2 t_j)}{n^{2g-2+s}}\]
%\[=2\frac{1}{2^{g-1}}\frac{1}{\pi^{2g-2+s}}\displaystyle \sum_{n=1}^\infty \frac{\prod_{j=1}^s \sin(\pi n 2 t_j)}{n^{2g-2+s}},\]
%which is equation 3.18 in \cite{wi}.
%\end{example}
\begin{example}\label{exa2}
We now compute the volume of the moduli space of $\SU(3)$ bundles on a Riemann surface of genus one.

Simple roots are $\alpha_1=e_1-e_2$, $\alpha_2=e_2-e_3$, and fundamental weights are
$\omega_1=\frac{2e_1-e_2-e_3}{3}$,
$\omega_2=\frac{e_1+e_2-2e_3}{3}$.  The positive coroots are
\[\Phi(\SU(3))=\{H_{\alpha_1}=e^1-e^2,H_{\alpha_2}=e^2-e^3,H_{\alpha_1+\alpha_2}=e^1-e^3\}\]
and $P=\Z\omega_1\oplus \Z \omega_2$.  Let
$\gamma=n_1\omega_1+n_2\omega_2$.  Then $\gamma \in P^{\text{reg}}$
if and only if $n_1\neq 0$, $n_2\neq 0$ and $n_1+n_2 \neq 0$.  Now consider 
$a=a_1H_{\alpha_1}+a_2H_{\alpha_2}=a_1(e^1-e^3)+(a_2-a_1)(e^2-e^3) \in \frh_\R$.
Suppose that $a$ is a regular element in $\mathfrak{A}$, in other
words, $2a_1-a_2>0$, $2a_2-a_1>0$ 
(in particular $a_1>0$ and $a_2>0$)
and $\theta(a_1H_{\alpha_1}+a_2H_{\alpha_2})=a_1+a_2<1$.


We compute the volume using Formula (\ref{E:1mark}).
We have $s=1$, $p=0$, $q=1$, $f=3$ and $|Z(SU(3))|=3$; hence, for
$g=1$, $2^{p(2g-1)}(fq)^{g-1}|Z(G)|=3$. Then,
\[\begin{array}{ll}
\vol(\SU(3),1)(a)
%&=3\displaystyle\sum_{\gamma \in P_{reg}}\frac{e^{2i\pi \ll a,\gamma \rr}}{\prod_{H_\alpha \in \Phi} 2i\pi \ll H_\alpha, \gamma \rr}\\
 &=3\displaystyle\sum_{n_1\neq 0, n_2\neq 0, n_1+n_2 \neq 0}
\frac{e^{2i\pi(n_1a_1+n_2a_2)}}{(2i\pi n_1)(2i\pi
n_2)(2i\pi(n_1+n_2))}\\
& \\
&=\left\{\begin{array}{cl}
   -\frac{1}{2}(1+a_1-2a_2)(a_1-1+a_2)(2a_1-a_2) & a_1\leq a_2\\
   -1/2(a_1-2a_2)(a_1-1+a_2)(2a_1-1-a_2)   & a_1\geq a_2
       \end{array}\right.
\end{array}\]
\end{example}
\begin{example}
With the notation of Example~\ref{exa2}, we make similar
computations for $\SU(3)$ when genus $g=2$.

%We compute the volume employing the Formula (\ref{E:1mark}).
In this case, $s=1$, $p=0$, $q=1$, $f=3$, $|Z(\SU(3))|=3$, hence, for
$g=2$, $2^{s(2g-1)}(fq)^{g-1}|Z(G)|(-1)^{(g-1)|\Phi|}=-3^2$. Then,
{\small \[\begin{array}{ll}
\vol(\SU(3),2)(a)
%&=-9\displaystyle\sum_{\gamma \in P_{reg}}\frac{e^{2i\pi \ll a,\gamma \rr}}{\prod_{H_\alpha \in \Phi}(2i\pi \ll H_\alpha,\gamma\rr)^3}\\
&=-9\displaystyle \sum_{n_1\neq 0, n_2\neq 0, n_1+n_2 \neq 0}
\frac{e^{2i\pi(n_1a_1+n_2a_2)}}{(2i\pi n_1)^3(2i\pi
n_2)^3(2i\pi(n_1+n_2))^3}\\
&\\
&=\left\{\begin{array}{cl}
 1/40320 (a_2-2a_1)(a_2-1+a_1)(-1+2a_2-a_1)P_1 & a_1 \leq a_2\\
1/40320 (a_2+1-2a_1)(2a_2-a_1)(a_2-1+a_1)P_2 & a_1 \geq a_2
\end{array}\right.
\end{array}\]}
where the polynomials $P_1$ and $P_2$ above are too long to be included in here.
\end{example}
\begin{example}\label{exb2}
We now compute the volume of the moduli space of  $\text{Spin}(5)$ bundles on a Riemann surface of
genus $g=1$ with one marking.

Positive roots are
$\{\alpha_1+\alpha_2=e_1,\alpha_2=e_2,\theta=e_1+e_2,\alpha_1=e_1-e_2\}$,
with associated coroots $H_{e_1}=2e^1$, $H_{e_2}=2e^2$,
$H_{e_1-e_2}=e^1-e^2$, $H_{e_1+e_2}=e^1+e^2$.

Let $a=a_1H_{\alpha_1}+a_2H_{\alpha_2}$ be a regular element in $\frA$;
in other words, $a_1>a_2$, $2a_2>a_1$, $2a_2<1$. We can express $a$
as $a=t_1 e^1+t_2 e^2$ (with $t_1=a_1$ and $t_2=2a_2-a_1$), $t_1$
and $t_2$ satisfy $t_1>t_2,t_2>0, t_1+t_2<1$.
We calculate the volume for $B_2$ and genus $g=1$ employing the Formula (\ref{E:1mark}).
In this case, $s=1$, $p=2$, $q=2$, $f=2$, $|Z(\text{Spin}(5))|=2$. Hence, for
$g=1$, $2^{s(2g-1)}(fq)^{g-1}|Z(\text{Spin}(5))|(-1)^{(g-1)|\Phi|}=8$. We get,
\[\begin{array}{ll}
\vol(B_2,g=1)(a)&=\displaystyle \frac{1}{2}t_2(t_1-1)(t_1-1+t_2)(t_1-t_2)\\
&=(2a_2-a_1)(a_1-1)(-1+2a_2)(a_1-a_2).
\end{array}\]
\end{example}
\begin{example}
With the notation of Example \ref{exb2}, we compute the volume of the moduli space of $\text{Spin}(5)$ bundles
on a Riemann surface of genus one and two markings.

Let  ${\bf a}=\{a_1,a_2\}$, where $a_1$ and $a_2$ are  regular elements in $\frA$.
 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
$\vol(B_2,1)({\bf a})$ is a piecewise polynomial function of $t_1,t_2,u_1,u_2$.
For example, choose $v_1=\frac{1}{2}e^1+\frac{1}{5 }e^2,\  v_2=\frac{1}{7}e^1+\frac{1}{9} e^2 $  and consider   $\tau({\bf v})\subset \frA\times \frA$,
the open set determined by the condition that  $a_1+w(a_2)$ is  in the same tope as
$v_1+w(v_2)$ for each element $w$ in the Weyl group of $B_2$.
Then for ${\bf a}\in \tau({\bf v})$,  we have
{\small \begin{align*}
&\vol(B_2,1)({\bf a})=4 W(\Phi(B_2), P,1,2)({\bf a})\\
&=-{\frac {1}{60}}u_{{2}}{u_{{1}}}^{5}+\frac{1}{6}t_{{2}}u_{{2}}t_{{1}}{u_{{
1}}}^{3}-\frac{1}{6}t_{{2}}{u_{{2}}}^{3}t_{{1}}u_{{1}}+\frac{1}{6}{u_{{2}}}^{3}u_{
{1}}{t_{{2}}}^{2}-\frac{1}{6}u_{{2}}{t_{{2}}}^{2}{u_{{1}}}^{3}-\frac{1}{12}{u_{{2}}}^{3}u_{{1}}{t_{{2}}}^{3}\\
&+\frac{1}{12}u_{{2}}{t_{{2}}}^{3}{u_{{1}}}^{3}-{\frac {1}{60}}{u_{{2}}}^{5}t_{{1}}u_{{1}}+\frac{1}{12}t_{{2}}u_{{2}}{t_{{1}}}^{3}{u_{{1}}}^{3}-\frac{1}{4}t_{{2}}u_{{2}}{t_{{1}}}^{2}{u_{{1}}}^{3}+\frac{1}{4}
t_{{2}}{u_{{2}}}^{3}{t_{{1}}}^{2}u_{{1}}+\frac{1}{6}{t_{{2}}}^{2}u_{{2}}t_
{{1}}{u_{{1}}}^{3}\\
&-\frac{1}{6}{t_{{2}}}^{2}{u_{{2}}}^{3}t_{{1}}u_{{1}}
+{\frac {1}{60}}u_{{2}}t_{{1}}{u_{{1}}}^{5}-\frac{1}{12}{t_{{2}}}^{3}u_{{2}}t_{{1}}{u_{{1}}}^{3}+\frac{1}{12}{t_{{2}}}^{3}{u_{{2}}}^{3}t_{{1}}u_{{1}}
-\frac{1}{12}t_{{2}}{u_{{2}}}^{3}{t_{{1}}}^{3}u_{{1}}+{\frac{1}{60}}{u_{{2}}}^{5}u_{{1}}.
\end{align*}}
\end{example}
%We compute some values.
%\[ \vol(B_2,1)({\bf a})=\]
%\tiny{\[ \left\{ \begin{array}{l@{\quad if \quad } l@{\quad and\quad} l}
%\frac{141791}{372163703625}& a_1=\frac{1}{2}e_1+\frac{1}{5 }e_2& a_2=\frac{1}{7}e_1+\frac{1}{9} e_2\\
%\frac{1418037104720960397931}{3721637036250000000000000000}&a_1=(\frac{1}{2}+\frac{1}{10000})e_1+(\frac{1}{5 }+\frac{1}{100000})e_2
%& a_2=\frac{1}{7}e_1+\frac{1}{9} e_2
%\end{array}\right.
% \]}


\section{More examples}\label{japex}
In this section we compute some instances of volumes of moduli spaces of flat $G$-connections for classical simple Lie groups $G$ 
using the formula given in Proposition \ref{P:factor}.  We remark that in this case $\epsilon_G$ is always $2$.

We will denote by $c_{\vol}$ the factor
$$c_{\vol}:=2^{p(2g-2+s)}(fq)^{g-1}|Z(G)|(-1)^{(g-1)|\Phi(G)|}|W|^{-1}.$$
For convenience, we list values of the parameters in $c_{\vol}$ for each type of classical simple Lie group in Table
\ref{cosvalue}.

\begin{table}[]
\caption{Value tables}
\begin{center}
\begin{tabular}{||c|c|c|c|c|c||}
\hline 
$G$-type& $q$&$f$& $p$&$|Z(G)|$&$|W|$\\
\hline\hline
$A_r$&$1$&$r+1$&0&$r+1$&$(r+1)!$\\ \hline
$B_r$&2&2&$r$&2&$2^rr!$\\ \hline
$C_r$&$2^{r-1}$&2&$r(r-1)$&2&$2^rr!$\\  \hline
$D_r$&$1$&4&$0$&4&$2^{r-1}r!$\\ \hline
\end{tabular}
\end{center}
\label{cosvalue}
\end{table}

\subsection{Tables of  volumes of moduli spaces}
In the case of no marking ($s=0$) we denote the volume simply by $\vol(G,g)$.
We list some values of $\vol(G,g)$ for classical simple Lie groups in Tables \ref{Witten table A-D} and
\ref{Witten table B-C}.  We  also list some values of the factor $c_{\vol}$ that we need in Section \ref{comp} 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.

Here are some explicit running times regarding the volume tables: 

Type A, rank 4 and genus 4 running time less then 1 sec.

Type A, rank 5 and genus 2 running time less then 1 sec.

Type B, rank 4  and genus 3 running time less then 1 sec.

Type B, rank 5 and genus 3 running time 5 hours.

Type C and rank 4 and genus 4 running time less then 1 sec.

Type C, rank 5 and genus 2 less then 1 sec.

Type D rank 5 and genus 2 running time 2 sec.

Type D, rank 4 and  genus 4 running time  half an hour.

\medskip
All experiments were done with Maple 15 on a MacPro
with a Quad-Core intel Xeon machine running at
2.93GHz.

%\newpage

\vfill

\begin{sidewaystable}[p]
\vspace{0.5cm}\hspace{1cm}\scriptsize
\vspace{1cm}\hspace{2cm}\scriptsize
\begin{tabular}{|@{\barra}|c|c|c|c||}
\hline \hline
G&g&$\vol(G,g)$&$c_{\vol}$\\
\hline\hline
&2&$\frac{1}{20160}$&-$\frac{3}{2}$\\ \cline{3-4}
&3&$\frac{19}{41513472000}$&$\frac{9}{2}$\\ \cline{3-4}
&4&$\frac{1031}{189225711747072000}$&-$\frac{27}{2}$\\  \cline{3-4}
&5&$\frac{32293}{487445433460457472000000}$&$\frac{81}{2}$\\ \cline{3-4}
$A_2$&6&$\frac{27739097}{34359439770544026968653824000000}$&-$\frac{243}{2}$\\ \cline{3-4}
&7&$\frac{29835840687589}{3031957229004108930561012205092864000000000}$ &$\frac{729}{2}$\\ \cline{3-4}
&8&$\frac{71810708985991}{598678146610235332992855225968816553984000000000}$&-$ \frac{2187}{2}$\\ \cline{3-4}
&9&$\frac{221137132669842886663}{151246026314426013297816671756013269041872371712000000000000}$&$\frac{6561}{2}$\\ \cline{3-4}
&10&$\frac{7252062205115875364801443}{406913205782775738093852803149608268676912309073384308736000000000000}$&-$ \frac{19683}{2}$\\ \cline{3-4}

\hline \hline
&2&$\frac{23}{653837184000}$&$\frac{2}{3}$\\ \cline{3-4}
&3&$\frac{14081}{64814699109633048576000000}$&$\frac{8}{3}$\\ \cline{3-4}
$A_3$&4&$\frac{17634884778757}{11155091763154851836756404076509396992000000000}$&$\frac{32}{3}$\\  \cline{3-4}
&5&$\frac{4257463829989959473}{367215613941141638310663592914134282922173229170688000000000000}$&$\frac{128}{3}$\\ \cline{3-4}
\hline \hline
&2&$\frac{1}{27303661403504640000}$&$\frac{5}{24}$\\ \cline{3-4}
$A_4$&3&$\frac{3998447009863}{18546080754338014660769482402427552723160268800000000}$&$\frac{25}{24}$\\ \cline{3-4}
&4&$\frac{63897294036759565910707701677}{45362078256453588666977436687320229387024422032399906211071677239139256762368000000000000}$&$\frac{125}{24}$\\  \cline{3-4}
\hline \hline
$A_5$&2&$\frac{46511}{1266317152743400113929736683520000000}$&-$\frac{1}{20}$\\ \cline{3-4}
\hline \hline
&2&$\frac{1}{36}$&4\\ \cline{3-4}
&3&$\frac{1}{32400}$&16\\ \cline{3-4}
&4&$\frac{1}{14288400}$&64\\  \cline{3-4}
&5&$\frac{1}{5715360000}$& 256\\ \cline{3-4}
$D_2$&6&$\frac{1}{2240649734400}$&1024\\ \cline{3-4}
&7&$\frac{477481}{417483460137744000000}$& 4096\\ \cline{3-4}
&8&$\frac{1}{340802824602240000}$&16384\\ \cline{3-4}
&9&$\frac{13082689}{1737399167709235430400000000}$&65536\\ \cline{3-4}
&10&$\frac{1924313689}{99574446563452076311839744000000}$&262144\\ \cline{3-4}
\hline \hline
&2&$\frac{23}{653837184000}$&$\frac{2}{3}$\\ \cline{3-4}
$D_3$&3&$\frac{14081}{64814699109633048576000000}$&$\frac{8}{3}$\\ \cline{3-4}
&4&$\frac{17634884778757}{11155091763154851836756404076509396992000000000}$&$\frac{32}{3}$\\ \cline{3-4}
&5&$\frac{4257463829989959473}{367215613941141638310663592914134282922173229170688000000000000}$&$\frac{128}{3}$\\ \cline{3-4}
\hline \hline
&2&$\frac{68227}{1084047447508315948449792000000}$&$\frac{1}{12}$\\ \cline{3-4}
$D_4$&3&$\frac{3727283292300079}{4163527227475565552987044342329298418787470879468908707840000000000}$&$\frac{1}{3}$\\ \cline{3-4}
&4&$\frac{1}{10^{18}}\frac{5372550944533148798111597103943896132463}{401595287497255375910389668105337327730608192879514113863554560467009086512201613320691542243934208}$&$\frac{4}{3}$\\ \cline{3-4}
\hline \hline
$D_5$&2&$\frac{24136975469}{949559334622106350631397499004290699425805762560000000000}$&$\frac{1}{120}$\\ \cline{3-4}
\hline \hline
\end{tabular}
\caption{ \small{Witten volumes with $s=0$ type $A$, $D$}}
\label{Witten table A-D}
\end{sidewaystable}



\begin{sidewaystable}[p]
\vspace{0.5cm}\hspace{-2cm}\scriptsize
\vspace{1cm}\hspace{2cm}\scriptsize
\begin{tabular}{|@{\barra}||c|c|c|c||}
\hline \hline
G&g&$vol(G,g)$&$c_{\vol}$\\
\hline\hline
&2&$\frac{1}{604800}$&16\\ \cline{3-4}
&3&$\frac{4799}{444609285120000}$&1024\\ \cline{3-4}
&4&$\frac{830287}{1058640085457339793408000}$& 65536\\  \cline{3-4}
&5&$\frac{15426208283}{26882357138658593827179724800000000}$&4194304\\ \cline{3-4}
$B_2$&6&$\frac{116091757826572267}{276425625559496603207535068049765826560000000000}$&268435456\\ \cline{3-4}
&7&$\frac{89010391857578264959599577}{289587313199907652270725517206183101789728520798208000000000000}$ &17179869184\\ \cline{3-4}
&8&$\frac{11895139004463387894854541173}{52877369206125057710964488249573327181156553092261268684800000000000000}$&1099511627776\\ \cline{3-4}
&9&$\frac{633808384316154355255792820511421021}{3849636942831172456365873314390564274746568066276449044888380984262656000000000000000}$&70368744177664\\ \cline{3-4}
&10&$\frac{37272442737128198664590284425995639982427091128187}{309322223503982769661277518134242125943776840468695068262378581664562780524641356087296000000000000000000}$&4503599627370496\\ \cline{3-4}

\hline \hline
&2&$\frac{19}{4302395130249216000}$&$\frac{-32}{3}$\\ \cline{3-4}
&3&$\frac{82197390067823}{9138251172376454624670846219476498015846400000000}$&$\frac{8192}{3}$\\ \cline{3-4}
$B_3$&4&$\frac{4942776168078403026588037}{259372284831214187236355852194766861663874462695850253729180680192000000000000}$&$\frac{-2097152}{3}$\\  \cline{3-4}
&5&$\frac{354908282467357148490713188145792722330626811}{10^{17}87848969140024613480397739898042540441070542192393712407652560102158237674544102913901274865860608}$&$\frac{536870912}{3}$\\ \cline{3-4}
\hline \hline
$B_4$&2&$\frac{4999}{1099339586242016873556223477363507200000}$&$\frac{16}{3}$\\ \cline{3-4}
&3&$\frac{290525115705512122541476428433}{29110138980741515852319493982928220072033854458512007601770585360017103402900354236416000000000000000}$&$\frac{16384}{3}$\\ \cline{3-4}
\hline \hline
$B_5$&2&$\frac{673653727654631}{1145031992084337393955364755047070523026083511420277985481064448000000000000}$&$-\frac{32}{15}$\\ \cline{3-4}
\hline \hline
&2&$\frac{1}{604800}$&16\\ \cline{3-4}
&3&$\frac{479}{444609285120000}$&1024\\ \cline{3-4}
&4&$\frac{830287}{1058640085457339793408000}$&65536\\ \cline{3-4}
&5&$\frac{15426208283}{26882357138658593827179724800000000}$& 65536\\ \cline{3-4}
$C_2$&6&$\frac{116091757826572267}{276425625559496603207535068049765826560000000000}$& 268435456\\ \cline{3-4}
&7&$\frac{89010391857578264959599577}{289587313199907652270725517206183101789728520798208000000000000}$&17179869184\\ \cline{3-4}
&8&$\frac{11895139004463387894854541173}{52877369206125057710964488249573327181156553092261268684800000000000000}$& 1099511627776\\ \cline{3-4}
&9&$\frac{633808384316154355255792820511421021}{3849636942831172456365873314390564274746568066276449044888380984262656000000000000000}$&70368744177664\\ \cline{3-4}
\hline \hline
&10&$\frac{37272442737128198664590284425995639982427091128187}{309322223503982769661277518134242125943776840468695068262378581664562780524641356087296000000000000000000}$&4503599627370496\\ \cline{3-4}
\hline \hline

&2&$\frac{19}{33612461955072000}$&$\frac{-4096}{3}$\\ \cline{3-4}
$C_3$&3&$\frac{328842292593389}{2231018352630970367351280815301879398400000000}$&$\frac{134217728}{3}$\\ \cline{3-4}
&4&$\frac{79085160601187541918557593}{1978853491449082849398466889913687604247089101378252057870336000000000000}$&$\frac{ -4398046511104}{3}$\\ \cline{3-4}
&5&$\frac{22714138337998023353982263960846269191446635633}{2094482639790168129930442330790580283190501742181628046218217852167087499488451550338298675200000000000000000}$&$\frac{144115188075855872}{3}$\\ \cline{3-4}
\hline \hline
&2&$\frac{2668859}{2236612113936395006421353103360000000}$&$\frac{4194304}{3}$\\ \cline{3-4}
$C_4$&3&$\frac{297521088216082657235423644965697}{433774873327337441627971738322499693513421035684824103143372913599269142790144000000000000000}$&$\frac{1125899906842624}{3}$\\ \cline{3-4}
\hline \hline
$C_5$&2&$\frac{5395475362754527}{1066394142885075317654167073822645957605991057511677952000000000000}$&$\frac{-274877906944/15}{15}$\\ \cline{3-4}
\hline \hline
\end{tabular}
\caption{\small{Witten volumes with $s=0$ type $B$, $C$}}
\label{Witten table B-C}
\end{sidewaystable}


%We can also compute the value for $C_3$ and $g=6$, $C_4$ and $ g=4$, but the values is too long to be put in the table.
%$C_3, g=6:=\frac{907715382373293953380022951719974939549217359883260037441247}{308448834856298809146783986837335096849018498783010053926683173002375194700600447230485753482347281297795158851243212800000000000000000000}$
%$C_4,g=4:\frac{1375385291414363300098964738315680236692382190415485407848427673}{3425457990060264160597286183838711204634391772033830640239531020946782259530421474671832406384624545144768289039716273811767898931200000000000000000000000}{6274926367144615248599838564145895745790753013742408029118040007297016976753150208135805347561472}$

\newpage

\subsection{Comparison results}\label{comp}
In this section we compare some of our computations of $\vol(G,g)$ with that of Komori-Matsumoto-Tsumura
(\cite{kmt0},\cite{kmt1},\cite{kmt2},\cite{kmt3}). The setting is as follows.


Consider a simple, connected, compact Lie group $G$ of rank $r$.
Here 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, $Q\subset L\subset P$. We denote, as before, by $P^+$ the monoid of dominant weights.
Let $L^+=L\cap P^+$.


Let ${\bf s}=[s_\alpha]$ be  a sequence of real variables indexed by positive roots $R^+$.
For  $v\in \frh_\R$,  Komori-Matsumoto-Tsumura introduced
$$\zeta({\bf s},v,G)=\sum_{\gamma\in\rho+L^+} e^{2i\pi\ll v,\gamma\rr}\prod_{\alpha\in R^+}\frac{1}{\ll  \gamma,H_\alpha\rr^{s_\alpha}}.$$
If $G$ is simply connected, then $L=P$, and we may denote $\zeta({\bf s},v,G)$
by $\zeta({\bf s},v,\frg) $, or for the Lie algebra $\frg$ of type $X_r$ by $\zeta({\bf s},v,X_r) $ as in \cite{kmt2}.


\begin{example}
Consider the simply connected group $G=\SU(4)$; its positive roots are
$[e_1-e_2, e_2-e_3, e_3-e_4, e_1-e_3, e_2-e_4, e_1-e_4]$.

The monoid of dominant weights is freely generated by fundamental weights $\omega_1,\omega_2$ and
$\omega_3$ 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 ${\bf s}=[s_i]$ with respect to the order of the roots as given above,
{\small \begin{align*}
\zeta({\bf s},v,\SU(4))&=\zeta({\bf s},v,A_3)\\
&=\sum_{m_1=1}^{\infty} \sum_{m_2=1}^{\infty} \sum_{m_3=1}^{\infty}
\frac{e^{2i\pi \ll v,m_1\omega_1+m_2\omega_2+m_3\omega_3\rr}}{ m_1^{s_1} m_2^{s_2} m_3^{s_3} (m_1+m_2)^{s_4} (m_2+m_3)^{s_5} (m_1+m_2+m_3)^{s_6}}.
\end{align*}}
\end{example}

\medskip

The series $\zeta({\bf s},v,G)$ converges when the exponents $s_{\alpha}$ are sufficiently large.
%It can be shown that $\zeta({\bf s},v,G)$ can be continued as a meromorphic function of ${\bf s}$.
Let $S=\sum s_\alpha$.
Suppose $s_\alpha$ 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 $(2\pi)^{-S}\zeta({\bf s},0,G)$ is rational.
Indeed, using the invariance of the sum under the Weyl group $W$,
$(2\pi)^{-S}\zeta({\bf s},0,G)$ is proportional to a Bernoulli series (with repetition of coroots in $\Phi$
matching the exponent data) which is obtained by summing over
all the regular elements of the full lattice $L$.  More precisely,
\begin{equation}
\displaystyle \frac{\zeta({\bf s},0,G)}{(2\pi)^S}=|W|^{-1}{i^S}\sum_{\gamma\in L_{reg}} \frac{1}
{\prod_{\alpha\in R^+}  (2i\pi \ll \gamma,H_\alpha\rr)^{s_\alpha}},
\end{equation}
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 $s_\alpha$ are equal to an even integer $2k$, we denote the sequence ${\bf s}=[s_\alpha]$ by ${\bf s}_{2k}$.
Then, for exponents ${\bf s}_{2k}$, and $G$  simply connected,
we may compute $\zeta({\bf s}_{2k},0,G)$ using the Witten volume formula for $g=k+1$,
\begin{equation}\label{zeta}
\begin{split}
\zeta({\bf s}_{2k},0,G)=|W|^{-1}(2\pi)^{2k |R^+|}(-1)^{k|R^+|} W(\Phi(G), L ,k+1)(0)
\\
=|W|^{-1}(2\pi)^{2k |R^+|} (-1)^{k |R^+|}\frac{1}{c_{\vol}} \vol(G,k+1)({0}).\\
\end{split}
\end{equation}
%Above $\Phi(G)$ 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 $\zeta({\bf s}_{2k},0,G)$.
We now demonstrate some computations of $\zeta({\bf s}_{2k},0,G)$.


\subsubsection{Examples of type $A_r$}
Let $n=r+1$.  We consider the simply connected group $G=\SU(n)$.
If we write $N=|R^+|=\frac{n(n-1)}{2}$, then
$$\zeta({\bf s}_{2k},0,A_r)=(-1)^{kN}(2\pi)^{2kN}\frac{1}{n!}\frac{1}{c_{\vol}}\vol(\SU(n),k+1)(0),$$
where
$c_{\vol}=n^{k+1}(-1)^{k\frac{n(n-1)}{2}}\dfrac{1}{n!}$.
Thus we obtain the values of $\zeta({\bf s}_{2k},0,A_r)$ for $n=3,4,5,6$ using Table \ref{Witten table A-D}.

For instance, if $n=3$ (that is $r=2$), and $k=1$, then  we have $N=3$, $\vol(\SU(3),2)(0)=\frac{1}{20160}$ and
$c_{\vol}=-3/2$,
%first line of Table \ref{Witten table A-D},
and we obtain
$$\zeta({\bf s}_{2},0,A_2)=(2\pi)^{6}(-1)^3\frac{1}{3!}\frac{1}{9(-1)^3\frac{1}{3!}}\frac{1}{20160}
=\pi^{6}\frac{1}{2835}$$ as in \cite{kmt3} equation (7.11).


We give one other example whose parameters are not contained in the tables.
Consider $n=4$, $k=5$. Then, $N=6$ and
\begin{multline*}
\zeta({\bf s}_{2k},0,A_3)=(2\pi)^{60}\times\\
\frac{1393614066290742513412310095846}{58203152419058513584890890509712229288124323632762771449711578369140625}
\end{multline*}

\subsubsection{Examples of type  $B_r$, $C_r$ and $D_r$}
%We obtain Prop.~2.1 in \cite{kmt2}.
For root systems of type $B_r$ and $C_r$,  the number of positive
roots is $N=r^2$.
% and the order of the Weyl group is $|W|=r!2^r$. 
For example, for $B_r$ when all $s_\alpha=2k$,
$$\zeta({\bf s}_{2k},0,B_r)= \frac{1}{r!2^r}(2\pi)^{2kN}(-1)^{kN}\mathcal B(\CH_r^{BC},\check{Q}_B,g^B_{{\bf s}_{2k}})(0).$$
%$$\zeta({\bf s}_{2k},0,C_r))= \frac{1}{r!2^r}(2\pi)^{2kN}(-1)^{kN} \mathcal B(\CH_r^B,\check{P_C},g^C_{{\bf s}_{2k}})(0)$$

Explicitly for $C_2$, 
%fundamental weights are $e_1$ and $e_1+e_2$,and 
positive roots are $[e_1-e_2, 2e_2, 2 e_1, e_1+e_2]$.  We order the exponents with respect to the order given in the list of positive roots.  Then,
%We consider the multiple zeta series
$$\zeta([s_1,s_2,s_3,s_4],0,C_2)=\sum_{m=1}^\infty\sum_{n=1}^\infty \frac{1} {m^{s_1}n^{s_2}(m+n)^{s_3}(m+2n)^{s_4}},$$
In the particular case of  ${\bf {s_2}}=[2,2,2,2]$,  using Table \ref{Witten table B-C}, 
$$\zeta({\bf {s_2}},0,C_2)=\frac{1}{8}(-1)^4(2\pi)^{8}\frac{1}{16}\frac{1}{604800}=\frac {1}{302400}\,{\pi }^{8},$$
which is the equation (7.23) of \cite{kmt3}.

%For $B_2$ we compute similarly
%$\zeta([s_1,s_2,s_3,s_4],0,B_2)=$\zeta([s_1,s_2,s_3,s_4],0,C_2)=\sum_{m=1}^\infty\sum_{n=1}^\infty \frac{1} {m^{s_1}n^{s_2}(m+n)^{s_3}(m+2n)^{s_4}}$$

We also give an example of $D_4$ with all exponents equal to $6$ (that is
$k=3$ and ${\bf {s_6}}=[6,6,6,6,6,6,6,6,6,6,6,6]$ ).
{\small \begin{multline*}
\zeta({\bf s_6},0,D_4)= \pi^{72} \times \\
\frac{5372550944533148798111597103943896132463}{21770524158223250767856810653451043131130341521323218291199402843808716814637088000000000000000000}
\end{multline*}}


It is also possible to compute $\zeta({\bf s},0,G)$ when the exponents in the list ${\bf s}=[s_1,s_2,s_3,s_4]$ are
distinct positive even integers for short and long roots.  We conclude with one example of this kind.


Consider the list of exponents $[2,4,4,2]$ corresponding to the list of positive roots $[e_1-e_2, 2 e_2, 2 e_1, e_1+e_2]$
of $C_2$. Then,
$$\zeta([2,4,4,2],0, C_2)=\sum_{m=1}^\infty\sum_{n=1}^\infty \frac{1} {m^2n^4(m+n)^4(m+2n)^2}=\pi^{12}\frac{53}{6810804000},$$
which coincides with equation (4.30) of \cite{kmt3}.

\subsection{Some multiple zeta values}
Let $k$ be a positive integer. Consider the multiple zeta series
{\small $$\zeta_r(2k,2k,\ldots,2k):=
\sum_{m_1=1}^{\infty} \sum_{m_2=1}^{\infty}\cdots \sum_{m_r=1}^{\infty} \frac{1}{m_1^{2k}}
\frac{1}{(m_1+m_2)^{2k}}\cdots
\frac{1}{(m_1+m_2+\cdots+m_r)^{2k}}.$$}

Following \cite{kmt0}, we want to demonstrate how the above series can be computed
using the Bernoulli series ${\mathcal B}(\mathcal H_r^{BC},\check{Q}_C,g_{\bf s}^C)(0)$ for the root system of type
$C_r$, where the exponents
${\bf s}=[s_\alpha]$ are taken to be $0$ for long positive roots, and $2k$ for short
positive roots.
%This series of exponents is invariant by the action of the Weyl group.
Using the invariance of the sum under the Weyl group, which is of order $2^r r!$ for $C_r$,
$$\mathcal B(\CH_r^{BC},\check{Q}_C, g_{\bf s}^C)(0)=2^r r! \sum_{\gamma\in (P_C^+)_{reg}} \frac{1}{\prod_{\alpha>0} (2i\pi \ll H_\alpha,\gamma \rr)^{s_\alpha}}.$$

A dominant integral regular weight $\gamma \in {(P_C^+)}_{reg}$ is of the form $\gamma=\sum_{i=1}^r m_i \omega_i$
with $m_i\geq 1$ (where $\omega_i$ denote fundamental weights).  Recall that the root system of type $C_r$
admits $r$ long roots $\{{2e_i}\}_{1\leq i\leq r}$, with corresponding (short) coroots  $\{H_{2e_i}=e^i\}_{1\leq i\leq r}$.
If we express $H_{2e_i}=e^i=(e^i-e^{i+1})+ (e^{i+1}-e^{i+2})+\cdots+ e^r$, then
$\langle H_{2e_i},\gamma\rangle=m_i+m_{i+1}+\cdots +m_r.$
Thus,
$$\zeta_r(2k,2k,\ldots,2k)=(-1)^{kr}(2\pi)^{2k r}\frac{1}{2^r r!} \mathcal B(\CH_r^{BC},\check{Q}_C, g_{\bf s}^C)(0).$$
For example, $\zeta_2(4,4)=\dfrac{\pi^8}{113400}$, $\zeta_5(4,4,4,4,4)=\dfrac{\pi^{20}}{548828480360160000}$ and
$\zeta_5(6,6,6,6,6)=\dfrac{\pi^{30}}{1347828286825972065254765625}$.

\section{Appendix: Szenes formula}\label{szf}
Let $\CH$ be an arrangement of hyperplanes compatible with a lattice $\Lambda$.
Let $g\in \CR_{\CH}$.  Consider
$$\CB(\CH,\Lambda,g)(v)=\sum_{\gamma\in \Gamma_{reg}(\CH)}g(2i\pi\gamma) e^{2i\pi \la v,\gamma\ra}.$$

This function (a generalized function on $V$)  coincide with a polynomial function
$\CB(\CH,\Lambda,g,\tau)$ on a tope $\tau$ (see Proposition \ref{prop:polyn}).
The piecewise polynomial function $P(\CH,\Lambda,g)$ has been defined in  Definition \ref{def:sz}.
Following Szenes \cite{sze1}, we prove the following formula. 

\begin{theorem}(Szenes)\label{theo:mainformula}
Let $g\in \CR_{\CH}$.
On $V_{reg}(\CH,\Lambda)$ we have the equality
$$\CB(\CH,\Lambda,g)= P(\CH,\Lambda,g).$$
\end{theorem}



We recall that, for $f\in S_{\mathcal H}$,
$\SZ^{\Lambda}(v)(f)(z)=\sum_{\gamma\in \Gamma}  f(2i\pi\gamma-z) e^{\langle v,2i\pi\gamma-z\rangle }$,
and $P(\CH,\Lambda,g)(v)$ is the trace on $S_{\CH}$ of the operator  $A(v,g):\CS_\CH\to \CS_\CH$ defined by
\begin{equation}\label{eq:AVg}
f(z)\mapsto {\bf R}(e^{\ll z,v\rr}g(z) (\SZ^{\Lambda}(v)f)(z)).
\end{equation}
Here ${\bf R}: R_{\mathcal H}\to S_{\mathcal H}$ is the total residue.


We first consider the one dimensional case where $V=\R$, and $\Lambda=\Z$.
Here $\CH=\{0\}$, with equation $z=0$.
The topes are the intervals $]-n,n+1[$, and the space $\CS_\CH$ is one dimensional with basis $f_\sigma=\frac{1}{z}$.

Let $\tau=]0,1[$.
Assume $v\in \tau$ so that $[v]=0$.  If we consider $g(z)=\frac{1}{ z^k}$, the formula to be proven is
\begin{equation}\label{okone}
\sum_{n\neq 0}\frac{e^{2i\pi n v}}{(2i\pi n)^k}=\Res_{z=0} (\frac{1}{z^k} e^{zv}(\SZ^{\Lambda}(\tau)f_\sigma)(z)).
\end{equation}
As $\SZ^{\Lambda}(\tau)(f_\sigma)(z)=\frac{1}{1-e^z}$  (see Example \ref{ex:dim1Eis}),
we have  to verify that
$$\sum_{n\neq 0}\frac{e^{2i\pi n v}}{(2i\pi n)^k}=\Res_{z=0}( \frac{1}{z^k} e^{zv}\frac{1}{1-e^z}).$$


The poles of the function $\frac{1}{1-e^z}$ consist of the elements $2i\pi n$, with $n\in \Z$.  When $k\geq 0$,
the equality above follows from the residue theorem in one variable. If $k<0$, both sides vanish
(the left hand side gives a generalized function  supported on $\Z$, the right hand side has no poles).

Szenes formula generalizes this result in higher dimensions, which we aim to demonstrate below.


\begin{proof}
Our proof is a slightly modified version of Szenes's proof where we use the total residue as opposed to iterated residue.
\medskip


We first remark that using the comparison formula (\ref{eq:SZlattices2}) and those of Lemma \ref{lem:secondcompare} over
commensurable lattices, it suffices to prove the equality for any lattice $\Lambda$ (compatible with $\CH$) of our choice.
We will prove Theorem  \ref{theo:mainformula} by the standard `deletion-contraction' argument on arrangement of
hyperplanes.

Choose a set $\Phi^{eq}$ of equations for $\CH$. For $\phi\in \Phi^{eq}$,  we consider the following two arrangements:

$\bullet$  $\mathcal H'=\mathcal H\setminus H_\phi$.

$\bullet$ $\mathcal H_0=\{H\cap H_\phi, H\in \mathcal H'\}$, the trace of the arrangement $\mathcal H'$ on $H_\phi$.


Consider the vector space $V_0:=V/\R \phi$, let $p:V\to V_0$ be the projection.
The dual space $U_0$ of $V_0$ is the hyperplane $H_\phi$. We now compare the spaces $\CS_{\CH}$,
$\CS_{\CH_0}$ and $\CS_{\CH'}$.
\begin{definition}
We say that a function $f\in \CM_{\mathcal H}$ has at most  a simple pole along the hyperplane $\phi=0$ if
$\phi f \in \CM_{\mathcal H'}$.
In this case,  we define $res_\phi f\in \CM_{\mathcal H_0}$ by
$res_\phi f= (\phi f)|_{H_\phi}$
\end{definition}



In other words, the meromorphic function $f$ has at most a simple pole on $H_\phi$ if the denominator of $f$ contains
the factor $\phi$ at most once. Then we multiply $f$ by $\phi$, eliminating $\phi$ from the denominator of $f$, and we
can restrict $\phi f$ to $\phi=0$. This operation kills the functions $f$ having no poles of $\phi=0$.

If $f=\frac{1}{\phi}f'$ with $f'\in \CM_{\CH'}$, then
\begin{equation}\label{eq:resphi}
res_\phi {\rm {\bf R}} f={\rm {\bf R}} res_\phi f.
\end{equation}
This is easy to verify using for example a decomposition of $f'$ with denominator on a set of independent hyperplanes
(see Lemma \ref{ind}).


The map $res_\phi$ is well defined on $\CS_{\mathcal H}$, as elements in $\CS_{\mathcal H}$ have at most a simple pole on $\phi=0$.
It is easy  to prove that
we have the exact sequence
\begin{equation}\label{eq:exactseq}
\begin{CD} 0 @> >>\CS_{\mathcal H'}@>i>> \CS_{\mathcal H} @>res_\phi>> \CS_{\mathcal H_0}@>  >> 0 .\end{CD}
\end{equation}
Let $v\in V_{reg}(\Lambda,\CH)$. Its projection $v_0=p(v)$  belongs to $V_{reg}(\Lambda_0,\CH_0)$.

\begin{lemma}\label{lem:resSZ}
Let $v\in V_{reg}(\Lambda,\CH)$ and  $f\in \CS_{\mathcal H}$. Then
$$res_\phi \SZ^{\Lambda}(v)(f)= -\SZ^{\Lambda_0}(v_0)(res_\phi f),$$
with $v_0=p(v)$.
\end{lemma}
\begin{proof}
We have
$\SZ^{\Lambda}(v)(f)(z)=\sum_{\gamma\in \Gamma}  f(2i\pi\gamma-z) e^{\langle v,2i\pi\gamma-z\rangle }$.
If $\gamma$ is  such that $\ll \phi,\gamma\rr\neq 0$, then the term $f(2i\pi\gamma-z)$ has no pole on $\phi=0$. Thus we obtain, for $z\in H_\phi$,
\begin{eqnarray*}
res_\phi \SZ^{\Lambda}(v)(f)(z)&=&\sum_{\gamma\in \Gamma, \ll \gamma,\phi\rr =0} (\phi(z) f(2i\pi\gamma-z))|_{H_\phi} e^{ \langle v_0,2i\pi\gamma-z\rangle }\\
&=&-\sum_{\gamma\in \Gamma, \ll \gamma,\phi\rr=0}  \phi(2\pi \gamma-z) f(2i\pi\gamma-z)|_{H_\phi} e^{\langle v_0,2i\pi \gamma-z\rangle }.
\end{eqnarray*}
\end{proof}


\begin{lemma}
Let $g\in \CR_{\CH'}$.
Then the following diagram is commutative.
\begin{equation}\label{eq:cd}\begin{CD}
      0 @>>>  \CS_{\CH'} @>>> \CS_{\CH}  @>>> \CS_{\CH_0} @>>>  0 \\
      @. @VVA(v,g)V  @VVA(v,g)V  @VV-A(v_0,g_0)V @. \\
      0 @>>>  \CS_{\CH'} @>>>  \CS_{\CH} @>>> \CS_{\CH_0} @>>>  0
\end{CD}\end{equation}
\end{lemma}
\begin{proof}
Let $g\in \CR_{\CH'}$, and let $g_0$ be its restriction to $H_\phi$.
Then the  operator $A(v,g)$ leaves $\CS_{\CH'}$ stable. If $F$ has at most a simple pole on $\phi=0$, 
then $gF$ also has at most a simple pole on $\phi=0$, as $g$ has no pole on $\phi=0$. Thus maps in the above diagram are well defined. 
Its commutativity follows from Lemma \ref{lem:resSZ}.
\end{proof}


We are now ready to prove Theorem \ref{theo:mainformula} by induction on the number of hyperplanes in $\CH$.
If there are less than $r$ hyperplanes, then $\CS_\CH=\{0\}$, the generalized function $\CB(\CH,\Lambda,g)$ is
supported on affine walls, so both sides of the equation of Theorem \ref{theo:mainformula} vanish.


Assume that $\CH$ consists of $r$ independent hyperplanes.
Changing the lattice $\Lambda$, we may assume that $\Lambda$ is the lattice generated by the equations
$\phi_k$ of the hyperplanes.  Then, the theorem follows from Formula (\ref{okone}) in the
one dimensional case.


Assume that $\CH$ has more than  $r$ hyperplanes.
Then by Lemma \ref{ind}, we can write a function in $\CR_{\CH}$  as a sum of functions $g$ whose poles lie on an
independent subset of hyperplanes of $\CH$, thus in number less or equal to $r$.
Thus $\CR_\CH$ is linearly generated by functions $g$ such that  some equation $\phi\in \Phi^{eq}$  is not  a pole of $g$.
We consider such a couple $(g,\phi)$ and the arrangements $\CH'$ and $\CH_0$ associated to $\phi$ by deletion and
contraction.  The function $g$ is in $\CR_{\CH'}$.


Let $g_0 \in \CR_{\mathcal H_0}$ be the restriction of $g$ to $H_\phi$.  Thus $\mathcal{B}(\mathcal H_0,\Lambda_0,g_0)$ is
a generalized function on $H_\phi^*=V/\R \phi$ and $p^*\mathcal{B}(\mathcal H_0,\Lambda_0,g_0)$ is a function on $V$
(constant in the direction $\phi$).


We have the following recurrence relation for the function (eventually generalized)
$\mathcal{B}(\mathcal H,\Lambda,g)$ associated to an element $g\in
R_{\mathcal H'}$.

\begin{proposition} \label{prop:recug}
If $g\in \CR_{\mathcal H'}$, then
$$\mathcal{B}(\mathcal H,\Lambda,g)=\mathcal{B}(\mathcal H',\Lambda,g)-p^*
\mathcal{B}(\mathcal H_0,\Lambda_0,g_0).$$
\end{proposition}


This is  clear. Indeed the set $\Gamma_{reg}(\CH')$ is larger than $\Gamma_{reg}(\CH)$ as it may contain also elements
$\gamma$ with $\ll\gamma,\phi\rr=0$.
This additional summation gives rise to the term
$\mathcal{B}(\mathcal H_0,\Lambda_0,g_0).$


Let  $v\in V_{reg}(\CH,\Lambda)$.
As $P(\CH,\Lambda,g)(v)$ is the trace of the operator $A(v,g)$ defined in (\ref{eq:AVg}), the commutativity of the
diagram (\ref{eq:cd}) above implies that
$$P(\mathcal H,\Lambda,g)(v)=P(\mathcal H',\Lambda,g)(v)-
P(\mathcal H_0,\Lambda_0,g_0)(v_0).$$
Comparing with Proposition \ref{prop:recug}, we see by induction that Szenes formula holds.
\end{proof}

\medskip

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