Quadratic Weyl group multiple Dirichlet series of Type D_ 4^ (1)
Adrian Diaconu, Vicenţiu Paşol, and Alexandru A. Popa

In this paper and its sequel [30], we investigate the precise
relationship between the quadratic affine Weyl group multiple
Dirichlet series in the sense of [22, 16], and those defined
axiomatically by Whitehead [54] and [53]. In
particular, we show that the axiomatic quadratic Weyl group multiple
Dirichlet series of type
D_{\scalebox{1.1}{$\scriptscriptstyle 4$}}^{\scalebox{1.1}{$\scriptscriptstyle (1)$}}
over rational function fields of odd characteristic admits meromorphic
continuation to the interior of the corresponding complexified Tits
cone. We shall also determine the polar divisor of this function,
and compute the residue ateach of its poles. As a
consequence, we obtain an exact formula for a
weighted 4-th moment ofquadratic Dirichlet $L$-functions
over rational function fields; we shall also derive an asymptotic
formula for this weighted moment that is expected to generalize to any
global field.

§ INTRODUCTION

In this paper and its sequel [30], we investigate the precise
relationship between the quadratic affine Weyl group multiple
Dirichlet series in the sense of [22, 16], and those defined
axiomatically by Whitehead [54] and [53]. In
particular, we show that the axiomatic quadratic Weyl group multiple
Dirichlet series of type
D_{\scalebox{1.1}{$\scriptscriptstyle 4$}}^{\scalebox{1.1}{$\scriptscriptstyle (1)$}}
over rational function fields of odd characteristic admits meromorphic
continuation to the interior of the corresponding complexified Tits
cone. We shall also determine the polar divisor of this function,
andcompute the residue at each of its poles. As a
consequence, we obtain an exact formula for a
weighted 4-thmoment of quadratic Dirichlet $L$-functions
over rational function fields; we shall also derive an asymptotic
formula for this weighted moment that is expected to generalize to any
global field.

Before discussing our results in more detail, let us first recall some
basic facts about Weyl group multiple Dirichlet series associated
to classical root systems.

A Weyl group multiple Dirichlet series,
WMDS for short, is a Dirichlet series in several complex variables
attached to a root system satisfying a group of functional equations
isomorphic to the Weyl group of the root system; the number
of variables is precisely the rank of the root system. These objects
were initiallyintroduced on a case-by-case basis in the 1980's, when
the idea emerged that it could be useful to tie together a family of
related $L$-functions in one variable (e.g., the family of quadratic
Dirichlet $L$-functions) to create a double
Dirichlet series, which could be used to study the average
behavior of the original familyof $L$-functions; see
[39, 17, 37, 19, 33, 25]. Motivated by the problem
of understanding the asymptotics of moments of $L$-functions,
double Dirichlet series soon became multiple Dirichlet series
(see [18] and [27]),and since approximately
2003, the idea of how to construct WMDS attached to classical root
systems tookshape; see
[12, 13, 14, 15, 21, 22, 23]. It became clear that
a quadratic WMDS over a global field,
for exam-ple, associated to a (finite) reduced irreducible
root system
\Phi_{\scalebox{1.0}{$\scriptscriptstyle 0$}}
of rank $r$ has the form

\[
\sum_{n_{\scalebox{.75}{$\scriptscriptstyle 1$}},
\ldots, n_{\scalebox{.85}{$\scriptscriptstyle r$}}}
\frac{H(n_{\scriptscriptstyle 1}, \ldots,
n_{\scalebox{1.2}{$\scriptscriptstyle r$}};
m_{\scriptscriptstyle 1}, \ldots,
m_{\scalebox{1.2}{$\scriptscriptstyle r$}})
\Psi(n_{\scriptscriptstyle 1}, \ldots,
n_{\scalebox{1.2}{$\scriptscriptstyle r$}})}
{\prod |n_{\scriptscriptstyle i}|^{s_{\scalebox{.75}{$\scriptscriptstyle i$}}}}
\]

the sum being over $r$-tuples of representatives of non-zero
$S$-integers modulo $S$-units in the number field setting
(or $r$-tuples of effective divisors prime to $S$ in the function
field setting), for $S$ a large enough finiteset of places;
see, e.g., [23, 33]. The $r$-tuple
(m_{\scriptscriptstyle 1}, \ldots,
m_{\scalebox{1.2}{$\scriptscriptstyle r$}})
of non-zero $S$-integers (resp. effective divisorsprime to $S$) is a twisting parameter, and $|\cdot|$ denotes
the corresponding norm of an element. The function $H$ is the
most important part, giving the structure of the series, and
the function $\Psi$ is just a technical de-vice, varying over a
certain finite-dimensional vector space of complex-valued functions,
that makes the product $H\Psi$ well-defined.

The coefficients $H$ satisfy a twisted multiplicativity that reduces
their specification to the determination of the $p$-parts

\[
\sum_{k_{\scalebox{.75}{$\scriptscriptstyle 1$}},
\ldots, \, k_{\scalebox{.85}{$\scriptscriptstyle r$}}\, \ge \, 0}
H\big(p^{k_{\scalebox{.75}{$\scriptscriptstyle 1$}}}\!, \ldots,
p^{k_{\scalebox{.85}{$\scriptscriptstyle r$}}}\!;
p^{l_{\scalebox{.75}{$\scriptscriptstyle 1$}}}\!, \ldots,
p^{l_{\scalebox{.85}{$\scriptscriptstyle r$}}}\big)\,
p^{- \, k_{\scalebox{.75}{$\scriptscriptstyle 1$}}
s_{\scalebox{.75}{$\scriptscriptstyle 1$}} - \, \cdots \, - \,
k_{\scalebox{.85}{$\scriptscriptstyle r$}}
s_{\scalebox{.85}{$\scriptscriptstyle r$}}}
\]

odd primes $p$ and tuples
(l_{\scalebox{1.1}{$\scriptscriptstyle 1$}},\ldots,
l_{\scalebox{1.25}{$\scriptscriptstyle r$}})\in \mathbb{N}^{r};
see [21] and [23]. The $p$-parts are constructed so that
the resulting WMDS satisfies a group of functional equations isomorphic to
the Weyl group of $\Phi_{\sss 0}$;
the meromorphic continuation of this series to $\mathbb{C}^{r}$
is then essentially automatic. There are several equivalent
methods of representing the correct $p$-parts, namely,

* Definition by the “averaging method”, also known as the
Chinta-Gunnells method [22, 21, 23].

* Definition as spherical $p$-adic Whittaker functions [14, 15].

* Definition as sums over crystal bases [14, 46].

* Definition as partition functions of statistical-mechanical
lattice models [10, 11].

The Chinta-Gunnells method has been extended to the particular affine
root system
D_{\scalebox{1.1}{$\scriptscriptstyle 4$}}^{\scalebox{1.1}{$\scriptscriptstyle (1)$}}
in [16], and to any root system associated with a
symmetrizable Kac-Moody algebra in [42]. Moreover, a
Casselman-Shalika type formula for Whittaker functions on
metaplectic covers of Kac-Moody groups over non-archimedean local
fields has been recently established by Patnaik and Puskás
[49]. Despite these advances, we are quite far from a satisfactory theory
of multiple Dirichlet series in the general setting of Kac-Moody Lie
algebras and their Weyl groups. In a way or another, the main difficulties occurring in
the infinite-dimensional case are caused by the existence of imaginary roots. To be more precise,
let us consider, for example, the axiomatic multiple Dirichlet series associated to the fourth
moment of quadratic Dirichlet $L$-functions over rational function fields $\mathbb{F}_{\! q}(x)$ of odd
characteristics, see [28, 51, 54]; for simplicity, we shall assume throughout that $q\equiv 1 \!\pmod 4.$ For
\Re(s_{\scriptscriptstyle i}) >  1,
$i = 1, \ldots, 5,$ this series can be expressed as

\begin{equation} \label{eq: WMDS-affineD4-intro}
\sum_{\substack{d\; \mathrm{monic} \\ d = d_{\scalebox{.75}{$\scriptscriptstyle 0$}}^{}
d_{\scalebox{.75}{$\scriptscriptstyle 1$}}^{\scalebox{.8}{$\scriptscriptstyle 2$}}\\
d_{\scalebox{.75}{$\scriptscriptstyle 0$}} \;  \mathrm{monic \, \& \, square-free}}}
\frac{\prod_{i = 1}^{4} L\!\left(s_{\scriptscriptstyle i} + \tfrac{1}{2},
\chi_{d_{\scalebox{.8}{$\scriptscriptstyle 0$}}}\right)
\cdot P_{\scalebox{1.1}{$\scriptscriptstyle d$}}(\mathbf{s}\scalebox{1.}{$\scriptscriptstyle '$};
\chi_{d_{\scalebox{.8}{$\scriptscriptstyle 0$}}})}{|d|^{s_{\scalebox{.75}{$\scriptscriptstyle 5$}}}}
\end{equation}

P_{\scalebox{1.1}{$\scriptscriptstyle d$}}(\mathbf{s}\scalebox{1.}{$\scriptscriptstyle '$};
\chi_{d_{\scalebox{.8}{$\scriptscriptstyle 0$}}}),
$\mathbf{s}\scalebox{1.}{$\scriptscriptstyle '$} = (s_{\scriptscriptstyle 1}, \ldots, s_{\scriptscriptstyle 4}),$ are certain correction polynomials. Substitute
x_{\scalebox{1.1}{$\scriptscriptstyle i$}} \! = q^{- s_{\scalebox{.75}{$\scriptscriptstyle i$}} - \frac{1}{2}}
($i=1, \ldots,$ $5$), and denote the resulting function by $Z(\mathbf{x}; q),$ where
$\mathbf{x} = (x_{\scalebox{1.1}{$\scriptscriptstyle 1$}}, \ldots, x_{\scalebox{1.1}{$\scriptscriptstyle 5$}}).$
Then the function
Z(q^{\scalebox{.95}{$\scriptscriptstyle -1\slash 2$}}\mathbf{x}; q)
$and the Chinta-Gunnells average
Z_{\scriptscriptstyle W}^{\scriptscriptstyle \mathrm{CG}}(\mathbf{x}; \sqrt{q})
D_{\scalebox{1.1}{$\scriptscriptstyle 4$}}^{\scalebox{1.1}{$\scriptscriptstyle (1)$}}
satisfy the same group of functional equations.Consequently

\[
Z(q^{\scalebox{.95}{$\scriptscriptstyle -1\slash 2$}}\mathbf{x}; q)
= F(\mathbf{x}^{\scalebox{1.1}{$\scriptscriptstyle \delta$}})
Z_{\scriptscriptstyle W}^{\scriptscriptstyle \mathrm{CG}}(\mathbf{x}; \sqrt{q})
\]

for some function $F$ of one complex variable. In other words, the two functions differ by a power series
whose coefficients (apart from the constant term which is $1$) are supported on the positive affine imaginaryroots $n \delta,$ with $n \ge 1.$ The appearance of such correction factors is a common feature when dealing with
extensions of classical formulas to various infinite-dimensional settings, and pinning down these factors isusually quite a daunting task; see Macdonald's analogue [44] of Weyl's denominator identity for affine rootsystems,
Kac's generalization [41] of the same identity to symmetrizable Kac-Moody algebras, the affine
Gindikin-Karpelevich formula [7, 8], the affine Macdonald formula [45, 20, 9], and
Whittaker functions on $p$-adic loop groups [48, 49].

One of our main results is the following (Theorem <ref>):
— The factor $F(z)$ is given by

\[
F(z) = \prod_{n \, \ge \, 1} \!\left(1 - q  z^{2n - 1}\right)^{- 2}.
\]

In particular, the function $Z(q^{\scalebox{.95}{$\scriptscriptstyle -1\slash 2$}}\mathbf{x}; q)$ has
meromorphic continuation to
\Omega =\{\mathbf{x} \in \mathbb{C}^{5} : |\mathbf{x}^{\scalebox{1.1}{$\scriptscriptstyle \delta$}}| < 1 \},
and in this domain it satisfies a group of functional equations.

The function $Z_{\scriptscriptstyle
W}^{\scriptscriptstyle \mathrm{CG}}(\mathbf{x}; \sqrt{q})$ cannot
be meromorphically continued beyond the region $\Omega,$
andthe fact that $Z(q^{\scalebox{.95}{$\scriptscriptstyle -1\slash
2$}}\mathbf{x}; q)$ satisfies a group of functional
equations was already known by the work of Whitehead
[54]. The coefficients of the power series expansion of
Z_{\scriptscriptstyle W}^{\scriptscriptstyle \mathrm{CG}}(\mathbf{x}; \sqrt{q})
are polynomials in $u = \sqrt{q},$ and so
Z(u^{\scalebox{1.1}{$\scriptscriptstyle -1$}}\mathbf{x}; u^{\scalebox{1.1}{$\scriptscriptstyle 2$}})$
can be considered as a function of the additional complex variable $u.$ In particular, the function
Z(q^{\scalebox{.95}{$\scriptscriptstyle 1\slash 2$}}\mathbf{x}; q^{\scalebox{1.1}{$\scriptscriptstyle -1$}}),
obtained for $u = q^{\scalebox{1.1}{$\scriptscriptstyle -1\slash 2$}},$ gives – after substituting $q \to |p|$ –
the $p$-part of the global WMDS.

We have the following supplement to Theorem <ref>:
— Let
\Phi_{\scalebox{1.2}{$\scriptscriptstyle \mathrm{re}$}}^{\scalebox{1.2}{$\scriptscriptstyle +$}}
denote the set of positive affine real roots of a root system of type $D_{\scriptscriptstyle 4}^{\scriptscriptstyle (1)}\!.$
Then the singu-larities of $Z(q^{\scalebox{.95}{$\scriptscriptstyle -1\slash 2$}}\mathbf{x}; q)$ occur along the hypersurfaces $\mathbf{x}^{2\alpha} = q^{\scalebox{1.2}{$\scriptscriptstyle - 1$}},$ for
\alpha  \in \Phi_{\scalebox{.95}{$\scriptscriptstyle \mathrm{re}$}}^{\scalebox{.95}{$\scriptscriptstyle +$}}.
In particular, this function has a simple pole at
x_{\scalebox{1.1}{$\scriptscriptstyle 5$}} = q^{\scalebox{1.1}{$\scriptscriptstyle -1 \slash 2$}}
with residue

\[
\underset{x_{\scalebox{.75}{$\scriptscriptstyle 5$}} \to q^{\scalebox{.75}{$\scriptscriptstyle -\frac{1}{2}$}}}{\mathrm{Res}}
Z(q^{\scalebox{.95}{$\scriptscriptstyle -1\slash 2$}}\mathbf{x}; q)
= -\, \frac{q^{\scalebox{.9}{$\scriptscriptstyle -\frac{1}{2}$}}}{\left(P_{\phantom i}^{\scalebox{1.1}{$\scriptscriptstyle 2$}};
P_{\phantom i}^{\scalebox{1.1}{$\scriptscriptstyle 2$}}\right)_{\infty}
\!\left(q P_{\phantom i}^{\scalebox{1.1}{$\scriptscriptstyle 2$}};
P_{\phantom i}^{\scalebox{1.1}{$\scriptscriptstyle 2$}}\right)_{\infty}
\prod_{\scalebox{1.1}{$\scriptscriptstyle i = 1$}}^{\scalebox{1.1}{$\scriptscriptstyle 4$}}
\left(x_{\scalebox{1.1}{$\scriptscriptstyle i$}}^{\scalebox{1.1}{$\scriptscriptstyle 2$}};
P_{\phantom i}^{\scalebox{1.1}{$\scriptscriptstyle 2$}}\right)_{\infty}
\!\left(q x_{\scalebox{1.1}{$\scriptscriptstyle i$}}^{\scalebox{1.1}{$\scriptscriptstyle - 2$}}
P_{\phantom i}^{\scalebox{1.1}{$\scriptscriptstyle 2$}};
P_{\phantom i}^{\scalebox{1.1}{$\scriptscriptstyle 2$}}\right)_{\infty}
\cdot \;\,  \prod_{\scalebox{1.1}{$\scriptscriptstyle 1\, \le \, i \, < \, j \, \le \, 4$}}
\left(x_{\scalebox{1.1}{$\scriptscriptstyle i$}}
x_{\scalebox{1.1}{$\scriptscriptstyle j$}}; P\right)_{\infty}}
\]

P = (x_{\scalebox{1.1}{$\scriptscriptstyle 1$}}
x_{\scalebox{1.1}{$\scriptscriptstyle 2$}}
x_{\scalebox{1.1}{$\scriptscriptstyle 3$}}
x_{\scalebox{1.1}{$\scriptscriptstyle 4$}})\slash q,
and $(a; b)_{\infty} = \prod_{k\ge 0}\, (1 - ab^{\scalebox{1.1}{$\scriptscriptstyle k$}})$ is the $b$-Pochhammer symbol.

The residue at any other pole of this function can be obtained from the residue at
x_{\scalebox{1.1}{$\scriptscriptstyle 5$}} = q^{\scalebox{1.1}{$\scriptscriptstyle -1 \slash 2$}}
by applying a specific functional equation.

As a first application, we obtain the precise relationship between the function
$Z(q^{\scalebox{.95}{$\scriptscriptstyle -1\slash 2$}}\mathbf{x};q)$ and the Cassel-man-Shalika
formula [49] for unramified Whittaker functions on metaplectic covers of Kac-Moody
groups over non-archimedean local fields. With the notations and terminology of loc. cit., the authors prove that, for each dominant coweight $\lambda^{\scalebox{1.1}{$\scriptscriptstyle \vee$}},$ the value
\mathcal{W}(\varpi^{\lambda^{\scalebox{.9}{$\scriptscriptstyle \vee$}}})
of the metaplectic Whittaker function on the toral element $\varpi^{\lambda^{\scalebox{.9}{$\scriptscriptstyle \vee$}}},$ where $\varpi$ is a chosen uniformizer, is the $p$-adic specialization of the expression

\begin{equation} \label{eq: CS-formula}
v^{\langle \lambda^{\scalebox{.9}{$\scriptscriptstyle \vee$}}, \, \rho \rangle}
\widetilde{\fm}
\widetilde{\Delta}
\, \cdot \sum_{w \, \in \, W}
(-1)^{\scalebox{1.1}{$\scriptscriptstyle \ell(w)$}}
\Big(\prod_{\tilde{a}^{\scalebox{.8}{$\scriptscriptstyle \vee$}}
\in \, \widetilde{\Phi}^{^{\scalebox{.8}{$\scriptscriptstyle \vee$}}}\!(w)}
e^{-\tilde{a}^{\scalebox{.8}{$\scriptscriptstyle \vee$}}}\Big)w \, \mathbf{\star} \,
e^{\lambda^{\scalebox{.9}{$\scriptscriptstyle \vee$}}}.
\end{equation}

Here $\widetilde{\Phi}^{^{\scalebox{.8}{$\scriptscriptstyle \vee$}}}$ is the dual root system
constructed in [49] using a metaplectic structure $(\mathrm{Q}, n)$ on a root datum.The factor $\widetilde{\Delta}$ is defined by the formal infinite product

\[
\widetilde{\Delta} : =
\prod_{\tilde{a} \, \in \, \widetilde{\Phi}^{^{\scalebox{.95}{$\scriptscriptstyle +$}}}}
\!\left(\frac{1 - v e^{-\tilde{a}^{\scalebox{.8}{$\scriptscriptstyle \vee$}}}}
{1 - e^{-\tilde{a}^{\scalebox{.8}{$\scriptscriptstyle \vee$}}}} \right)^{\! m(\tilde{a})}
\]

where $m(\tilde{a})$ is the root-multiplicity of $\tilde{a}.$ When the root datum is associated with
an untwisted affine root system $\Phi$ of ADE type, the correction factor $\widetilde{\fm}$ is

\[
\widetilde{\fm} : = \mathrm{ct}(\widetilde{\Delta}^{\scalebox{1.1}{$\scriptscriptstyle -1$}})
= \prod_{i = 1}^{r} \prod_{j = 1}^{\infty}
\frac{1 - v^{\widetilde{m}_{\scalebox{.8}{$\scriptscriptstyle i$}}}e^{- j \tilde{\delta}}}
{1 - v^{\widetilde{m}_{\scalebox{.8}{$\scriptscriptstyle i$}} + 1}e^{- j \tilde{\delta}}}
\]

where $\{\widetilde{m}_{\scalebox{1.1}{$\scriptscriptstyle i$}}\}_{\scalebox{1.}{$\scriptscriptstyle i = 1, \ldots, r$}}$ is the set of exponents of the underlying finite root system to $\widetilde{\Phi},$ and $\tilde{\delta}$ is the minimal positive imaginary root of $\widetilde{\Phi}.$ For example, if $\Phi$ is of
$D_{\scalebox{1.1}{$\scriptscriptstyle 4$}}^{\scalebox{1.1}{$\scriptscriptstyle (1)$}}$ type, then $\widetilde{\Phi}$ is again of $D_{\scalebox{1.1}{$\scriptscriptstyle 4$}}^{\scalebox{1.1}{$\scriptscriptstyle (1)$}}$ type (see <cit.>), and if
\{\alpha_{\scriptscriptstyle i}\}_{\scriptscriptstyle i = 1, \ldots, 5}\subset
\Phi_{\scalebox{1.2}{$\scriptscriptstyle \mathrm{re}$}}^{\scalebox{1.2}{$\scriptscriptstyle +$}}
is the set of simple roots, $x_{\scalebox{1.1}{$\scriptscriptstyle i$}} : = e^{-\alpha_{\scalebox{.75}{$\scriptscriptstyle i$}}}\!,$ then

\[
\widetilde{\fm}(\mathbf{x}; v) = \prod_{i = 1}^{4} \prod_{j = 1}^{\infty}
\frac{1 - v^{\widetilde{m}_{\scalebox{.8}{$\scriptscriptstyle i$}}}\mathbf{x}^{2j \delta}}
{1 - v^{\widetilde{m}_{\scalebox{.8}{$\scriptscriptstyle i$}} + 1} \mathbf{x}^{2j \delta}},
\; \text{the exponents being $1, 3, 3, 5$}.
\]

If we specialize (<ref>) to the case when $\widetilde{G}$ is the metaplectic double cover of a simply connected affine Kac-Moody group $G$ of type
$D_{\scalebox{1.1}{$\scriptscriptstyle 4$}}^{\scalebox{1.1}{$\scriptscriptstyle (1)$}}$ over a non-archimedean local field, we get:
— If we define

\[
\tilde{\frak{c}}(\mathbf{x}; v) : = \prod_{n = 1}^{\infty} \!\left(1 - v\mathbf{x}^{n\delta}\right)^{\! 2}
\!\left(1 - v\mathbf{x}^{2n\delta} \right)^{\! 2} \;\, \text{and} \;\;\,
D(\mathbf{x}; v) : = \prod_{\alpha  \in \Phi_{\scalebox{.95}{$\scriptscriptstyle \mathrm{re}$}}^{\scalebox{.95}{$\scriptscriptstyle +$}}}
\!\left(1 - v\mathbf{x}^{2\alpha} \right)
\]

then the value of the unramified Whittaker function on the group $\widetilde{G}$
at the identity element is

\[
\mathcal{W}(1) =
(\widetilde{\fm} \tilde{\frak{c}} D)(\mathbf{x}; q^{\scalebox{1.1}{$\scriptscriptstyle -1$}})
Z(q^{\scalebox{.95}{$\scriptscriptstyle 1\slash 2$}}\mathbf{x}; q^{\scalebox{1.1}{$\scriptscriptstyle -1$}})
\]

where $q\equiv 1 \!\pmod 4$ is the size of the residue field.

A similar comparison result for metaplectic Whittaker
functions on simply laced affine Kac-Moody groups will appear in
[30]. In the finite-dimensional case, the equivalence between
the Chinta-Gunnells method and the representation of the
$p$-parts as spherical $p$-adic Whittaker functions was
estab-lished by McNamara [47].

Our main reason for singling out the study of WMDS of type
D_{\scalebox{1.1}{$\scriptscriptstyle 4$}}^{\scalebox{1.1}{$\scriptscriptstyle (1)$}}
concerns the fourth moment of quad-ratic Dirichlet
$L$-functions. Traditionally the moment problem for this family of
$L$-functions is asking for an asymptotic formula for
\sum_{\deg d \, = D} L\!\left(\tfrac{1}{2}, \chi_{d}\right)^{r}
($r \ge 1$) as $D \to \infty,$ the sum being over monic square-free
polynomials $d \in \mathbb{F}_{\! q}[x].$ A conjectural asymptotic
formula, for all $r,$ was first given by Andrade and Keating [3]
(see also [50]), and a refined version, exhibiting additional lower order
terms in the asymptotic formula, was recently proposed in
[31]. This conjecture is known only for $r \le 3,$ see
[34, 35, 26], and when$r = 4,$ a weaker
form of the asymptotic formula is known by the work
of Florea [36].

As an immediate consequence of Theorem <ref>, we have:
— Assuming $q\equiv 1 \!\pmod 4$ and $D \ge 1,$
we have the exact formula:

\[
\sum_{\substack{\deg d \, = D\\d\; \mathrm{monic}}} L\!\left(\tfrac{1}{2}, \chi_{d_{\scalebox{.8}{$\scriptscriptstyle 0$}}}\right)^{4}
\!P_{\scalebox{1.1}{$\scriptscriptstyle d$}}(\chi_{d_{\scalebox{.8}{$\scriptscriptstyle 0$}}})
= \mathrm{Coeff}_{\xi^{\scalebox{1.}{$\scriptscriptstyle D$}}}
\left[\, \prod_{n \, \ge \, 1} \!\left(1 - q  \xi^{4n - 2}\right)^{- 2}
\cdot Z_{\scriptscriptstyle W}^{\scriptscriptstyle
\mathrm{CG}}(\underline{1}, \xi; \sqrt{q})
\right]
\]

P_{\scalebox{1.1}{$\scriptscriptstyle d$}}(\chi_{d_{\scalebox{.8}{$\scriptscriptstyle 0$}}})
= P_{\scalebox{1.1}{$\scriptscriptstyle d$}}(0, \ldots, 0; \chi_{d_{\scalebox{.8}{$\scriptscriptstyle 0$}}})$
and $\underline{1}: = (1, 1, 1, 1).$ Here, for a monic polynomial
$d \in \mathbb{F}_{\! q}[x],$ we write
d = d_{\scalebox{1.1}{$\scriptscriptstyle 0$}}^{}
d_{\scalebox{1.1}{$\scriptscriptstyle 1$}}^{\scalebox{1.1}{$\scriptscriptstyle 2$}}
with $d_{\scalebox{1.1}{$\scriptscriptstyle 0$}}^{}$ monic and square-free.

While this result is probably special to rational function fields,
the following asymptotic formula is ex-pected to generalize to any
global field.
— For $D, N \ge 1$ and
(N + 1)^{\scalebox{1.1}{$\scriptscriptstyle -1$}} < \Theta
< N^{\scalebox{1.1}{$\scriptscriptstyle -1$}},
we have

\begin{equation} \label{eq: asymptotic-intro}
\sum_{\substack{\deg d \, = D\\d\; \mathrm{monic}}}
L\!\left(\tfrac{1}{2}, \chi_{d_{\scalebox{.8}{$\scriptscriptstyle 0$}}}\right)^{4}
\!P_{\scalebox{1.1}{$\scriptscriptstyle d$}}(\chi_{d_{\scalebox{.8}{$\scriptscriptstyle 0$}}})
\, = \sum_{n \, \le \, N} Q_{n}(D, q) q^{\scalebox{.8}{$\scriptscriptstyle \frac{D}{2n}$}}
\, + \, O_{\scalebox{1.1}{$\scriptscriptstyle \Theta$}, \, q}
\left(q^{\scalebox{.8}{$\scriptscriptstyle \frac{D\Theta}{2}$}}\right)
\end{equation}

where $Q_{n}(D, q)$ is a polynomial in $D$ of degree 10 if $n$ is odd
and of degree 7 if $n$ is even. Furthermore, if we let$\varrho = q^{-1\slash n} \in \mathbb{R},$ then the leading
coefficient of $Q_{n}(D, q)$ is given by an expression of the form

\[
(-1)^{\lfloor n\slash 2 \rfloor} \varrho^{3  \lfloor n\slash 2 \rfloor \cdot  \lfloor (n+1)\slash 2 \rfloor }
g_{\scalebox{1.1}{$\scriptscriptstyle n$}, \scalebox{1.}{$\scriptscriptstyle D$}}(\sqrt \varrho)
\]

g_{\scalebox{1.1}{$\scriptscriptstyle n$},
\scalebox{1.}{$\scriptscriptstyle D$}}\ne 0
is a power series in $\varrho^{\scalebox{1.1}{$\scriptscriptstyle 1\slash 2$}}$
with non-negative coefficients.

According to [31], a similar asymptotic formula should hold for
\sum_{\deg d \, = D} L\!\left(\tfrac{1}{2}, \chi_{d}\right)^{4}
summed only over square-free monics; it should be a consequence of
the analytic properties of
Z(q^{\scalebox{.95}{$\scriptscriptstyle -1\slash 2$}}\mathbf{x}; q)
(i.e., Theorem <ref> and Theorem <ref> above) and its twists, which shall be
addressed in a future work. However, from the ana-lytic
point of view, the asymptotic formula (<ref>)
should be sufficient. This is so since it has the correct order of
magnitude, and the coefficients
P_{\scalebox{1.1}{$\scriptscriptstyle d$}}(\chi_{d_{\scalebox{.8}{$\scriptscriptstyle 0$}}})
are, as we shall see, non-negative; when
d = d_{\scalebox{1.1}{$\scriptscriptstyle 0$}}
is square-free, we have
P_{\scalebox{1.1}{$\scriptscriptstyle d$}}(\chi_{d_{\scalebox{.8}{$\scriptscriptstyle 0$}}})
=  |d|^{\scalebox{1.1}{$\scriptscriptstyle - 1\slash 2$}},
which also explains the discrepancy in (<ref>)
by a factor of
q^{\scalebox{1.}{$\scriptscriptstyle D\slash 2$}}
(cf. <cit.>).

The study of the analytic properties of twisted versions of
Z(q^{\scalebox{.95}{$\scriptscriptstyle -1\slash 2$}}\mathbf{x}; q)
mentioned in the previous paragraph is also relevant to the problem of extending our results to
higher genus function fields. The main difficulty in dealing with both these general situations comes from the failure of the so-called local-to-global property whose complexity grows as the degrees of the twisting parameters increase. Recently, Friedlander [38] obtained an interesting explicit formula that can be interpreted as measuring the degree of failure of the local-to-global property of a (finite-dimensional) twisted Weyl group multiple Dirichlet series over rational function fields. We expect that an analogous formula will hold for affine Weyl group multiple Dirichlet series.

Using the $p$-parts
Z(p^{\scalebox{.95}{$\scriptscriptstyle 1\slash 2$}}\mathbf{x};
p^{\scalebox{1.1}{$\scriptscriptstyle -1$}}),
with $p$ a rational odd prime, one constructs (cf. <cit.>)
the ana-logue of (<ref>) over the
rationals. This series has the form

\begin{equation} \label{eq: WMDS-rational-intro}
Z(s_{\scriptscriptstyle 1}, \ldots, s_{\scriptscriptstyle 5};
\chi_{a_{\scalebox{.8}{$\scriptscriptstyle 2$}}}, \chi_{a_{\scalebox{.8}{$\scriptscriptstyle 1$}}}) \;
= \sum_{\substack{d  = d_{\scalebox{.75}{$\scriptscriptstyle 0$}}^{}
d_{\scalebox{.75}{$\scriptscriptstyle 1$}}^{\scalebox{.8}{$\scriptscriptstyle 2$}} \, \ge \, 1\\ (d, 2) = 1}}
\frac{\prod_{i = 1}^{4} L\!\left(s_{\scriptscriptstyle i} + \tfrac{1}{2},
\chi_{a_{\scalebox{.8}{$\scriptscriptstyle 1$}} d_{\scalebox{.8}{$\scriptscriptstyle 0$}}}\right)
\cdot \chi_{a_{\scalebox{.8}{$\scriptscriptstyle 2$}}}\!(d_{\scriptscriptstyle 0})
'$}; \chi_{a_{\scalebox{.8}{$\scriptscriptstyle 1$}}
d_{\scalebox{.8}{$\scriptscriptstyle 0$}}})}{d^{s_{\scalebox{.75}{$\scriptscriptstyle 5$}}}}
\end{equation}

a_{\scriptscriptstyle 1}, a_{\scriptscriptstyle 2} \in \{\pm 1,
\pm2 \}
and $\chi_{d},$ for $d\in \mathbb{Z}$ non-zero and
square-free, is the usual quadratic character; as
inthe function field case, this expression is
certainly valid when
\Re(s_{\scriptscriptstyle i}) >  1
($i = 1, \ldots, 5$). It has some initial continuation,
and in that region it satisfies a group $W$ of functional
equations isomorphic to a Weyl groupof type
D_{\scalebox{1.1}{$\scriptscriptstyle 4$}}^{\scalebox{1.1}{$\scriptscriptstyle (1)$}}
If $a_{\scriptscriptstyle 2} = 1,$ this function has a simple pole
at $s_{\scriptscriptstyle 5} = \frac{1}{2},$ and its residue can now
be explicitly com-puted as the infinite product
of zeta functions

\[
\prod_{p \ne 2} R\big(
p^{\scalebox{.9}{$\scriptscriptstyle - \frac{1}{2}$} - s_{\scalebox{.75}{$\scriptscriptstyle 1$}}}\!,
p^{\scalebox{.9}{$\scriptscriptstyle - \frac{1}{2}$} - s_{\scalebox{.75}{$\scriptscriptstyle 2$}}}\!,
p^{\scalebox{.9}{$\scriptscriptstyle - \frac{1}{2}$} - s_{\scalebox{.75}{$\scriptscriptstyle 3$}}}\!,
p^{\scalebox{.9}{$\scriptscriptstyle - \frac{1}{2}$} - s_{\scalebox{.75}{$\scriptscriptstyle 4$}}};
p^{\scalebox{.9}{$\scriptscriptstyle - \frac{1}{2}$}}\big)
\]

with $R(\underline{x};u)$
:= (x_{\scalebox{1.1}{$\scriptscriptstyle 1$}}, \ldots, x_{\scalebox{1.1}{$\scriptscriptstyle 4$}}))
given in Theorem <ref>; the residues at the other
simple poles (under the action of $W$) can be computed using the
functional equations.

Z(\mathbf{s}; \chi_{a_{\scalebox{.8}{$\scriptscriptstyle 2$}}},
\chi_{a_{\scalebox{.8}{$\scriptscriptstyle 1$}}})
(\mathbf{s} : = (s_{\scriptscriptstyle 1}, \ldots, s_{\scriptscriptstyle 5}))
admits meromorphic continuation to $\Re(\delta(\mathbf{s})) > 0,$ with
all its singularities contained in the set
\left\{w\big(s_{\scriptscriptstyle 5} = \frac{1}{2}\big)\right\}_{w \,\in \, W}.

Let us say some words about the proofs of our main results; the
proofs of Theorems <ref>-<ref> are (essentially) straightforward applications
of Theorems <ref> and <ref>. The key result in establishing these two
theorems is a new functional equation satisfied
by the Chinta-Gunnells average for affine root systems. This functional equation has no finite-dimensional analogue. To state this in the
D_{\scalebox{1.1}{$\scriptscriptstyle 4$}}^{\scalebox{1.1}{$\scriptscriptstyle (1)$}}
case, consider the average
Z_{\scriptscriptstyle W}(\mathbf{x}; u) = \sum_{w \, \in \, W} 1 \vert w(\mathbf{x}; u),
where $f \vert w$ is the Chinta-Gunnells action.
We will show in Proposition <ref> that,
as a function of the complex variables
x_{\scalebox{1.1}{$\scriptscriptstyle 1$}}, \ldots, x_{\scalebox{1.1}{$\scriptscriptstyle 5$}}, u,$
Z_{\scriptscriptstyle W}(\mathbf{x}; u)
is holomorphic for
|\mathbf{x}^{\scalebox{1.1}{$\scriptscriptstyle \delta$}}| < 1, u \in \mathbb{C}
except for the set of points for which $u\mathbf{x}^{\alpha}  \pm 1 = 0$
for some
\alpha \in \Phi_{\scalebox{1.2}{$\scriptscriptstyle \mathrm{re}$}}^{\scalebox{1.2}{$\scriptscriptstyle +$}}.
For$a, b \in \{\mathcal{e}, \mathcal{o}\},$ where $\mathcal{e}$
(resp. $\mathcal{o}$) stands for even (resp. odd), let
Z_{\scriptscriptstyle W}^{\scalebox{1.2}{$\scriptscriptstyle a, b$}}
denote the part of $Z_{\scriptscriptstyle W}$ which has parity
$a$ with respect to the involution
\varepsilon_{\scalebox{1.1}{$\scriptscriptstyle 5$}}(\underline{x}, x_{\scalebox{1.1}{$\scriptscriptstyle 5$}}) = (-\underline{x}, x_{\scalebox{1.1}{$\scriptscriptstyle 5$}})
and parity $b$ with respect to the involution
\varepsilon_{\scalebox{1.1}{$\scriptscriptstyle 1$}}(\underline{x},
x_{\scalebox{1.1}{$\scriptscriptstyle 5$}})
= (\underline{x}, -x_{\scalebox{1.1}{$\scriptscriptstyle 5$}}).
If we let
\mathbf{Z}  =
\!^{t}(Z_{\scriptscriptstyle W}^{\scalebox{1.2}{$\scriptscriptstyle \mathcal{e}, \mathcal{e}$}}\!,
Z_{\scriptscriptstyle W}^{\scalebox{1.2}{$\scriptscriptstyle \mathcal{e}, \mathcal{o}$}}\!,
Z_{\scriptscriptstyle W}^{\scalebox{1.2}{$\scriptscriptstyle \mathcal{o}, \mathcal{e}$}}),
then this vector function satisfies the functional equation

\begin{equation} \label{eq: fe-intro}
\mathbf{Z}(\mathbf{x}; u) = B(\mathbf{x}; u)
\mathbf{Z}\!\left(\mathbf{x}; u \mathbf{x}^{\delta}\right)
\end{equation}

for a 3 by 3 matrix $B(\mathbf{x}; u)$ with rational entries. In fact, the matrix

\[
\prod_{\substack{\alpha  \, \in \,
\Phi_{\scalebox{.95}{$\scriptscriptstyle \mathrm{re}$}}^{\scalebox{.95}{$\scriptscriptstyle +$}}
\\ \alpha \, < \, \delta}}
\!\left(1 - u^{2}\mathbf{x}^{2\alpha} \right)
\cdot B(\mathbf{x}; u)
\]

has polynomial entries in $\mathbf{x}$ and $u,$ and each entry of
$B(\mathbf{x}; u)$ is divisible by
\left(1 - u^{2}\mathbf{x}^{\delta}\right)^{\scalebox{1.1}{$\scriptscriptstyle 2$}}
see Theorem <ref>.(An explicit formula for
$B(\mathbf{x}; u)$ is obtained via the identity
B(\mathbf{x}; u) = A^{\scalebox{1.1}{$\scriptscriptstyle - 1$}}(\mathbf{x}; u\mathbf{x}^{\delta}),
with $A$ given in [1].)

The strategy to deduce Theorem <ref> and Theorem <ref>
from the functional equation (<ref>) goes as follows. The
divisibility of the matrix $B(\mathbf{x}; u)$ by
\left(1 - u^{2}\mathbf{x}^{\delta}\right)^{\scalebox{1.1}{$\scriptscriptstyle 2$}}
implies that the renormalization $\tilde{Z}_{\scriptscriptstyle W}$
of the Chinta-Gunnells average, defined by
(<ref>), and the function
$Z_{\scriptscriptstyle W}$ have the same singularities in $\Omega.$ We
then compute the residue of the function
$\tilde{Z}_{\scriptscriptstyle W}(\mathbf{x}; u)$ at
$x_{\scalebox{1.1}{$\scriptscriptstyle 5$}} = u^{\scalebox{1.1}{$\scriptscriptstyle -1$}}$
by using (<ref>) and the invariance of the average
$Z_{\scriptscriptstyle W}(\mathbf{x}; u)$ under the Weyl group,
which reduce the calculation to a simple application of the
classical Macdonald's identity [44]. The structure of this
residue (given by the formula in Theorem <ref>,
with $q$ replaced by $u^{\scalebox{1.1}{$\scriptscriptstyle 2$}}$) allows
us to construct a WMDS of type (<ref>),
\tilde{Z}_{\scriptscriptstyle W}(\mathbf{x};
|p|^{\scalebox{1.1}{$\scriptscriptstyle - 1\slash 2$}})
as the $p$-parts, which after substituting
x_{\scalebox{1.1}{$\scriptscriptstyle i$}} = q^{ - s_{\scalebox{.75}{$\scriptscriptstyle i$}}},
turns out to have the same residue as
\tilde{Z}_{\scriptscriptstyle W}(\mathbf{x}; \sqrt{q})
at the simple pole
x_{\scalebox{1.1}{$\scriptscriptstyle 5$}} = 1\slash \sqrt{q}.
Thus, by letting $\mathscr{Z}\!(\mathbf{x})$ denote
this WMDS in the variables
x_{\scalebox{1.1}{$\scriptscriptstyle i$}},
we deduce that
\mathscr{Z}\!(\mathbf{x}) = \tilde{Z}_{\scriptscriptstyle W}(\mathbf{x}; \sqrt{q}).
Finally, we note that the matrix
B(\mathbf{x}; u)\slash
\left(1 -
u^{2}\mathbf{x}^{\delta}\right)^{\scalebox{1.1}{$\scriptscriptstyle 2$}}
determines $\tilde{Z}_{\scriptscriptstyle W}(\mathbf{x}; u)$
recursively, see Lemma <ref>; this implies that thecoefficients of $\tilde{Z}_{\scriptscriptstyle W}(u\mathbf{x}; u)$
satisfy the dominance axiom [28, 54], hence
\tilde{Z}_{\scriptscriptstyle W}(\mathbf{x}; \sqrt{q})
= Z(q^{\scalebox{.95}{$\scriptscriptstyle -1\slash 2$}}\mathbf{x}; q),
and that the coefficients
P_{\scalebox{1.1}{$\scriptscriptstyle d$}}(\chi_{d_{\scalebox{.8}{$\scriptscriptstyle 0$}}})
in Theorem <ref> are non-negative.
Concluding remark.
By the analogy between number fields and function fields
of curves over finite fields,it is conceivable that the
WMDS (<ref>) satisfies some analogue
of the functional equation (<ref>). It is also quiteprobable that
the mechanism behind this additional symmetry will provide
a natural approach to prove the meromorphic continuation of
(<ref>) to the half-space
$\Re(\delta(\mathbf{s})) > 0$.
Acknowledgements. We would like to thank Manish M.
Patnaik for clarifying for us some aspects of his joint work with
Anna Puskás [49]. We would also like to thank Bogdan Ion and the anonymous referee
for useful comments and suggestions.
The authors were partially supported by the
CNCS-UEFISCDI grant PN-III-P4-ID-PCE-2020-2498. The first author was
also partially supported by a Simons Fellowship in Mathematics.

§ NOTATION

We denote by $\mathbb{N}$ the set of non-negative integers,
by $\mathbb{Z}$ the ring of (rational) integers, and by
$\mathbb{Q},$ $\mathbb{R},$ $\mathbb{C}$ the fields
of rational numbers, real numbers and complex numbers,

By $X \ll Y$ or $X = O(Y)$ we denote an inequality of
the form $|X| \le CY,$ for some constant $C.$ We refer to$C$ as the implied constant,
and its dependence on parameters that we wish to keep track of
(e.g., $\epsilon, q$) will be indicated by appropriate subscripts (e.g.,
$X \ll_{\epsilon, q} Y$ or $X = O_{\epsilon, q}(Y)$).

§ PRELIMINARIES

We begin by recalling some basic facts about affine root systems and
their Weyl groups, following closely the standard reference

Let $V$ be a finite-dimensional real vector space, equipped with a
positive definite symmetric scalar prod-uct $\langle x, y\rangle;$
we shall identify $V$ with its dual space $V^{\ast}$ via the scalar
product. Consider the vector space $\mathcal{F}$
of affine-linear functions on $V$ (i.e., the functions of the form
$f = f_{\scalebox{1.1}{$\scriptscriptstyle 0$}} + c\delta,$ where
$f_{\scalebox{1.1}{$\scriptscriptstyle 0$}}(x) = \langle Df, x\rangle$
is a linear functional on $V,$ $\delta$ is the constant function $1,$
and $c\in \mathbb{R}$). We define the positive semidefinite scalar product on
$\mathcal{F}$ by

\[
\langle f, g \rangle = \langle Df, Dg\rangle.
\]

The radical of this form is the one-dimensional subspace $\mathbb{R}\delta$ of
constant functions. For non-zero $x\in V,$ let$
x^{\scalebox{1.1}{$\scriptscriptstyle \vee$}}
\! = 2 x \slash \langle x, x\rangle,
and for each non-constant $f\in \mathcal{F},$ let
f^{\scalebox{1.1}{$\scriptscriptstyle \vee$}}
= 2 f \slash \langle f, f\rangle.

Let $f\in \mathcal{F}$ be non-constant. The orthogonal reflection
\sigma_{\!\scriptscriptstyle f}
in the affine hyperplane on which $f$ vanishes is given by

\[
\sigma_{\!\scriptscriptstyle f}(x) =
x - f^{\scalebox{1.1}{$\scriptscriptstyle \vee$}}\!(x)Df
= x - f(x)D f^{\scalebox{1.1}{$\scriptscriptstyle \vee$}}.
\]

The reflection
\sigma_{\!\scriptscriptstyle f}
acts on $\mathcal{F}$ by
\sigma_{\!\scriptscriptstyle f}(g)
= g \circ  \sigma_{\!\scriptscriptstyle f}.
For notational convenience, we shall write from now on
$\sigma_{\!\scriptscriptstyle f}g$instead of $\sigma_{\!\scriptscriptstyle f}(g).$

For each $v \in V,$ the translation $x \to x + v$ will be denoted by

§.§ Affine root systems

Let $\Phi_{\scalebox{1.0}{$\scriptscriptstyle 0$}}$ be a rank $r$
reduced irreducible root system, spanning a real vector space
$V,$ and let
W_{\scalebox{1.1}{$\scriptscriptstyle 0$}}
= W(\Phi_{\scalebox{1.0}{$\scriptscriptstyle 0$}})
denote the Weyl group of $\Phi_{\scalebox{1.0}{$\scriptscriptstyle 0$}}.$
The dual root system of
\Phi_{\scalebox{1.0}{$\scriptscriptstyle 0$}}
with respect to a $W_{\scalebox{1.1}{$\scriptscriptstyle 0$}}$-invariant
positive definite scalar product $\langle x, y\rangle$ on $V$ is
\Phi_{\scalebox{1.0}{$\scriptscriptstyle 0$}}^{\scalebox{1.1}{$\scriptscriptstyle \vee$}}
= \{\alpha^{\scalebox{1.1}{$\scriptscriptstyle \vee$}} :
\alpha \in \Phi_{\scalebox{1.0}{$\scriptscriptstyle 0$}}\}.

Regarding each
$\alpha \in \Phi_{\scalebox{1.0}{$\scriptscriptstyle 0$}}$ as a linear
function on $V$ (i.e.,$\alpha(x) = \langle \alpha, x\rangle$
for $x \in V$),

one gets the affine root system

\[
\Phi = \{\alpha + n\delta : \alpha \in
\Phi_{\scalebox{1.0}{$\scriptscriptstyle 0$}}, n\in \mathbb{Z}\}
\, \scalebox{1.85}{$\scriptscriptstyle \cup$}\,
\{m\delta\}_ {\scalebox{1.1}{$\scriptscriptstyle m \in \mathbb{Z}\setminus\{0\}$}}
\]

associated with $\Phi_{\scalebox{1.0}{$\scriptscriptstyle 0$}}.$ The
elements of the subset
\Phi_{\scalebox{1.3}{$\scriptscriptstyle \mathrm{re}$}} =
\{\alpha + n\delta : \alpha \in
\Phi_{\scalebox{1.0}{$\scriptscriptstyle 0$}}, n\in \mathbb{Z}\}
are called affine real roots, and the elements
$m\delta$ ($m \in \mathbb{Z}\setminus\{0\}$) are called
affine imaginary roots.

For each $\beta \in  \Phi_{\scalebox{1.3}{$\scriptscriptstyle \mathrm{re}$}},$
let $\sigma_{\!\scriptscriptstyle \beta}$
denote the orthogonal reflection in the affine hyperplane on which
$\beta$ vanishes. Explicitly, if $\beta = \alpha + n\delta$ then

\[
H_{\!\scalebox{1.1}{$\scriptscriptstyle \beta$}} =
\beta^{\scalebox{1.1}{$\scriptscriptstyle -1$}}(0)
= \{x \in V: \langle \alpha, x\rangle = - \, n\}
\;\, \mathrm{and} \;\,
\sigma_{\!\scriptscriptstyle \beta}(x)
= x - (\langle \alpha, x\rangle + n)
\alpha^{\scalebox{1.1}{$\scriptscriptstyle \vee$}}.
\]

The reflection $\sigma_{\!\scriptscriptstyle \beta}$
acts on $\Phi_{\scalebox{1.3}{$\scriptscriptstyle \mathrm{re}$}}$ by

\[
\sigma_{\!\scriptscriptstyle \beta}\beta'
=\, \beta' \circ \sigma_{\!\scriptscriptstyle \beta}
=\beta' - \langle\alpha', \alpha^{\scalebox{1.1}{$\scriptscriptstyle
\vee$}}\rangle\beta,
\quad \beta' = \alpha' + n'\delta \in \Phi_{\scalebox{1.3}{$\scriptscriptstyle \mathrm{re}$}}.
\]

The affine Weyl group $W$ of $\Phi$ is defined
to be the group of affine isometries of $V$ generated by all
reflec-tions $\sigma_{\!\scriptscriptstyle \beta},$
$\beta \in \Phi_{\scalebox{1.3}{$\scriptscriptstyle \mathrm{re}$}}.$ Note that, for each $\alpha \in
\Phi_{\scalebox{1.0}{$\scriptscriptstyle 0$}},$ the composition
\tau(\alpha^{\scalebox{1.1}{$\scriptscriptstyle \vee$}}): =
\sigma_{\!\scriptscriptstyle \alpha} \circ
\sigma_{\!\scriptscriptstyle \alpha + \delta}
sends an element $x \in V$ to $x +
\alpha^{\scalebox{1.1}{$\scriptscriptstyle \vee$}},$ that is,
$\tau(\alpha^{\scalebox{1.1}{$\scriptscriptstyle \vee$}})$ is
translation by $\alpha^{\scalebox{1.1}{$\scriptscriptstyle \vee$}}.$
Thus $W$ contains a subgroup of translations
$\tau(Q^{\scalebox{1.1}{$\scriptscriptstyle \vee$}})$ isomorphic to the
root lattice $Q^{\scalebox{1.1}{$\scriptscriptstyle \vee$}}$ of
\Phi_{\scalebox{1.0}{$\scriptscriptstyle 0$}}^{\scalebox{1.1}{$\scriptscriptstyle \vee$}},
and we have
W = W_{\scalebox{1.1}{$\scriptscriptstyle 0$}} \ltimes
\tau(Q^{\scalebox{1.1}{$\scriptscriptstyle \vee$}}).

Let $\alpha_{\scriptscriptstyle 1}, \ldots, \alpha_{\scriptscriptstyle
r}$ be a set of simple roots of
$\Phi_{\scalebox{1.0}{$\scriptscriptstyle 0$}},$ and let
0$}}^{\scalebox{1.1}{$\scriptscriptstyle +$}}$
(resp. $\Phi_{\scalebox{1.0}{$\scriptscriptstyle
0$}}^{\scalebox{1.1}{$\scriptscriptstyle -$}}$) be the set of
positive (resp. negative) roots determined by
$\alpha_{\scriptscriptstyle 1}, \ldots, \alpha_{\scriptscriptstyle
r}.$ If we let $\theta \in \Phi_{\scalebox{1.0}{$\scriptscriptstyle
0$}}^{\scalebox{1.1}{$\scriptscriptstyle +$}}$ denote the highest
root, then the affine roots $\alpha_{\scriptscriptstyle i}$ ($1
\le i \le r$) together with $\alpha_{\scriptscriptstyle 0}
= -\theta + \delta$ form a set of simple
roots for $\Phi.$

An affine real root $\beta$ is positive (resp. negative) relative
to the open $r$-simplex

\[
C  = \{x \in V : \text{$\alpha_{\scriptscriptstyle i}(x) >
0$ ($1\le i \le r$) and $(-\theta + \delta)(x) > 0$}\}
\]

if $\beta(x) > 0$ (resp. $\beta(x) < 0$) for all $x \in C.$ If we let
\Phi_{\scalebox{1.2}{$\scriptscriptstyle \mathrm{re}$}}^{\scalebox{1.2}{$\scriptscriptstyle +$}}
\Phi_{\scalebox{1.2}{$\scriptscriptstyle \mathrm{re}$}}^{\scalebox{1.2}{$\scriptscriptstyle -$}}
denote the set of positive (resp. negative)affine real roots, then
\Phi_{\scalebox{1.2}{$\scriptscriptstyle
\mathrm{re}$}}^{\scalebox{1.2}{$\scriptscriptstyle -$}}
= - \, \Phi_{\scalebox{1.2}{$\scriptscriptstyle
\mathrm{re}$}}^{\scalebox{1.2}{$\scriptscriptstyle +$}},
\Phi_{\scalebox{1.3}{$\scriptscriptstyle \mathrm{re}$}} =
\Phi_{\scalebox{1.2}{$\scriptscriptstyle
\mathrm{re}$}}^{\scalebox{1.2}{$\scriptscriptstyle +$}}
\, \scalebox{1.8}{$\scriptscriptstyle \cup$}\,
\Phi_{\scalebox{1.2}{$\scriptscriptstyle
\mathrm{re}$}}^{\scalebox{1.2}{$\scriptscriptstyle -$}}.
The set of positive affine real roots is

\[
\Phi_{\scalebox{1.2}{$\scriptscriptstyle \mathrm{re}$}}^{\scalebox{1.2}{$\scriptscriptstyle +$}}
= \{\alpha + (n + \chi(\alpha))\delta : \alpha \in \Phi_{\scalebox{1.0}{$\scriptscriptstyle 0$}},
n \in \mathbb{N}\}
\]

where $\chi$ is the characteristic function of
$\Phi_{\scalebox{1.0}{$\scriptscriptstyle 0$}}^{\scalebox{1.}{$\scriptscriptstyle  \minus$}}.$
An affine imaginary root $m\delta$ is positive or negative
according as $m > 0$ or $m < 0.$ If $\alpha$ and $\beta$ are distinct
affine roots then $\alpha < \beta$ will signify that
\beta - \alpha \in \sum_{i} \mathbb{N}\alpha_{\scriptscriptstyle i}.

The affine Weyl group $W$ is also a Coxeter group on the generators
\sigma_{\scriptscriptstyle i} = \sigma_{\!\scriptscriptstyle \alpha_{\scalebox{.55}{$\scriptscriptstyle i$}}}
($1 \le i \le r$), and
\sigma_{\scriptscriptstyle 0}
= \sigma_{\!\scriptscriptstyle -\theta + \delta},
subject to the relations $\sigma_{\scriptscriptstyle
i}^{\scalebox{1.0}{$\scriptscriptstyle 2$}} = 1$ for all $0\le i \le r ,$
(\sigma_{\scriptscriptstyle i}
\sigma_{\!\scriptscriptstyle j})^{m_{\scalebox{.75}{$\scriptscriptstyle ij$}}}
\! = 1
if $i \ne j$ and
m_{\scalebox{1.1}{$\scriptscriptstyle ij$}} < \infty.

Note that

\begin{equation} \label{eq: fixing-delta}
\text{$w\delta = \delta \circ w^{\scalebox{1.1}{$\scriptscriptstyle -1$}}
\! = \delta$ for all $w \in W.$}
\end{equation}

§.§ Extended Affine Weyl Group

With notations as before, let $P^{\scalebox{1.1}{$\scriptscriptstyle \vee$}}$
denote the coweight lattice of
\Phi_{\scalebox{1.0}{$\scriptscriptstyle 0$}},
that is,

\[
P^{\scalebox{1.1}{$\scriptscriptstyle \vee$}} =
\{\text{$\lambda \in V:
\langle \lambda, \alpha \rangle \in \mathbb{Z}$ for all
$\alpha \in \Phi_{\scalebox{1.0}{$\scriptscriptstyle 0$}}$}\}.
\]

The extended affine Weyl group is defined to be
\widetilde{W}: = W_{\scalebox{1.1}{$\scriptscriptstyle 0$}} \ltimes
\tau(P^{\scalebox{1.1}{$\scriptscriptstyle \vee$}}),
where $\tau(P^{\scalebox{1.1}{$\scriptscriptstyle \vee$}})$ is the
group of translations by elements of
$P^{\scalebox{1.1}{$\scriptscriptstyle \vee$}}.$
The extended affine Weyl group contains
$W$ as a normal subgroup, and the quotient
\widetilde{W} \slash W
\cong
P^{\scalebox{1.1}{$\scriptscriptstyle \vee$}} \slash
Q^{\scalebox{1.1}{$\scriptscriptstyle \vee$}}
is a finite (abelian) group.

The extended affine Weyl group acts on the set of affine roots. To
see this, take an element $w$ of $\widetilde{W},$ and express it as
w = w_{\scalebox{1.1}{$\scriptscriptstyle 0$}}
\tau(\lambda),
w_{\scalebox{1.1}{$\scriptscriptstyle 0$}} \in
W_{\scalebox{1.1}{$\scriptscriptstyle 0$}}
\lambda \in P^{\scalebox{1.1}{$\scriptscriptstyle \vee$}}.
If $\beta = \alpha + n\delta \in \Phi,$ then

\[
(w\beta)(x) = \beta(w^{\scalebox{1.1}{$\scriptscriptstyle -1$}}x)
= \langle w_{\scalebox{1.1}{$\scriptscriptstyle 0$}}\alpha, x \rangle + n
- \langle \lambda, \alpha \rangle \quad \;\, \text{(for $x \in V$)}.
\]

w\beta = w_{\scalebox{1.1}{$\scriptscriptstyle 0$}}\beta
- \langle \lambda, \alpha \rangle\delta \in \Phi
\langle \lambda, \alpha \rangle \in \mathbb{Z}.
It follows that $\widetilde{W}$ permutes the affine real roots
and fixesthe imaginary roots.

For an element $w \in W,$ we shall denote its length
(with respect to the generators $\sigma_{\scriptscriptstyle i}, 0
\le i \le r$) by $\ell(w).$ Ifwe let
\Phi(w) = \{\beta \in \Phi^{+} : w\beta \in \Phi^{-}\},
then $\ell(w) = |\Phi(w)|.$ The length function is extended to
$\widetilde{W}$ by the same formula. If
w_{\scalebox{1.1}{$\scriptscriptstyle 0$}} \in
W_{\scalebox{1.1}{$\scriptscriptstyle 0$}}
\lambda \in P^{\scalebox{1.1}{$\scriptscriptstyle \vee$}},

ℓ(w_1.1$\scriptscriptstyle 0$
= ∑_α∈Φ_1.0$\scriptscriptstyle 0$^1.1$\scriptscriptstyle +$
|⟨λ, α⟩+ χ(w_1.1$\scriptscriptstyle 0$α)|
where, as before, $\chi$ is the characteristic function of
0$}}^{\scalebox{1.}{$\scriptscriptstyle  \minus$}}.$

Let $O= \{w \in \widetilde{W} : \ell(w) = 0\}.$ The elements of $O$ permute the simple affine roots, and we have

\[
\widetilde{W} = W \rtimes O
\]

O \cong
\widetilde{W} \slash W
\cong
P^{\scalebox{1.1}{$\scriptscriptstyle \vee$}} \slash
Q^{\scalebox{1.1}{$\scriptscriptstyle \vee$}}.
In particular, $O$ is a finite abelian group.

§.§ The Chinta-Gunnells action

From now on we will assume that the affine root system $\Phi$ is
simply laced. Let $\alpha_{\scriptscriptstyle 0}, \ldots,
\alpha_{\scriptscriptstyle r}$ be the set of affine simple roots, where
\alpha_{\scriptscriptstyle 0}
= -\theta + \delta.
If two $\alpha_{\scriptscriptstyle i}$ and
$\alpha_{\scriptscriptstyle j}$ are connected in the Dynkin diagram of
$\Phi,$ we shall write $i \sim j.$ Define an action of $W$ on monomials
\mathbf{x}^{\beta} : =
\prod_{i = 0}^{r} x_{\scalebox{1.1}{$\scriptscriptstyle i$}}^{k_{\scalebox{.75}{$\scriptscriptstyle i$}}}
for $\beta = \sum_{i = 0}^{r } k_{\scriptscriptstyle i}\alpha_{\scriptscriptstyle i}$
in the root lattice of $\Phi$, by $w\xx^\b=\xx^{w^{-1}\b}$, which
corresponds to the contragredient action on roots.

We also have an action of $W$ on variables
\mathbf{x}
= (x_{\scalebox{1.1}{$\scriptscriptstyle 0$}},
\ldots, x_{\scalebox{1.1}{$\scriptscriptstyle r $}})
(w \mathbf{x}) _{\scriptscriptstyle j} = \mathbf{x}^{w^{\scalebox{.85}{$\scriptscriptstyle -1$}}\alpha_{\scalebox{.7}{$\scriptscriptstyle j$}}}\!,
that is (using that $\Phi$ is simply laced):

\[
(\sigma_{\scriptscriptstyle i}\mathbf{x})_{\scriptscriptstyle j} =
\begin{cases}
1\slash x_{\!\scalebox{1.1}{$\scriptscriptstyle j$}}
&\mbox{if $j = i$} \\
x_{\scalebox{1.1}{$\scriptscriptstyle i$}}
x_{\!\scalebox{1.1}{$\scriptscriptstyle j$}}
&\mbox{if $j \sim i$}  \\
x_{\!\scalebox{1.1}{$\scriptscriptstyle j$}} & \mbox{otherwise}.
\end{cases}
\]

Let $\varepsilon_{\scalebox{1.1}{$\scriptscriptstyle
i$}}\mathbf{x}$ be the involution defined by

\[
(\varepsilon_{\scalebox{1.1}{$\scriptscriptstyle i$}}\mathbf{x})_{\scalebox{1.1}{$\scriptscriptstyle j$}} =
\begin{cases}
- x_{\!\scalebox{1.1}{$\scriptscriptstyle j$}}
&\mbox{if $j \sim i$}  \\
x_{\!\scalebox{1.1}{$\scriptscriptstyle j$}} & \mbox{otherwise.}
\end{cases}
\]

Let $\mathbb{C}(\mathbf{x}, u)$ be the field of rational functions
in $x_{\sss 1}, \ldots, x_{\sss r}, u$, for an additional variable $u$.
With this notation, we define the action of a simple reflection
$\sigma_{\scriptscriptstyle i}$ on $f \in \mathbb{C}(\mathbf{x}, u)$ by

\[
f\vert\sigma_{\scriptscriptstyle i}(\mathbf{x}; u) =
f(\sigma_{\scriptscriptstyle i}\mathbf{x}; u)
J(x_{\scalebox{1.1}{$\scriptscriptstyle i$}}, 0)
+ f(\varepsilon_{\scriptscriptstyle i}\sigma_{\scriptscriptstyle i}\mathbf{x}; u)
J(x_{\scalebox{1.1}{$\scriptscriptstyle i$}}, 1),
%\quad f \in \mathbb{C}(\mathbf{x}, u)
\]

where, for $\varepsilon \in \{0, 1\},$

\begin{equation} \label{eq: function-J}
J(x, \varepsilon) = J(x, u, \varepsilon) =
\frac{x}{2}\left(\frac{u - x}{1 - ux} - (-1)^{\varepsilon}\right).
\end{equation}

One verifies as in <cit.> that this action extends to a
well-defined $W$-action on $\mathbb{C}(\mathbf{x}, u).$

To construct the analogue of the Chinta-Gunnells (average) function in
our context, let $\Delta(\mathbf{x})$ be defined by

\begin{equation} \label{eq: McD-denominator}
\Delta(\mathbf{x}) =
\prod_{n \, \ge \, 1}
\!\left(1 - \mathbf{x}^{2 n \delta}\right)^{r}
\, \cdot \prod_{\beta \, \in \,
\Phi_{\scalebox{.95}{$\scriptscriptstyle \mathrm{re}$}}^{\scalebox{.95}{$\scriptscriptstyle +$}}}
\!\left(1 - \mathbf{x}^{2\beta} \right).
\end{equation}

The product (<ref>) is absolutely convergent in
the region $|\mathbf{x}^{\scalebox{1.1}{$\scriptscriptstyle \delta$}}|
< 1$ of $\mathbb{C}^{r \scalebox{1.15}{$\scriptscriptstyle + 1$}},$
and by (<ref>), it satisfies the
transfor-mation formulas

\begin{equation} \label{eq: trans-form-McD-denominator}
\Delta(\mathbf{x}) = - x_{\scalebox{1.1}{$\scriptscriptstyle
i$}}^{\scalebox{1.1}{$\scriptscriptstyle 2$}}
\Delta(\sigma_{\scriptscriptstyle i}\mathbf{x})
\qquad (\text{for $i = 0, \ldots, r $}).
\end{equation}

The Chinta-Gunnells function can now be defined by
Z_{\scriptscriptstyle W}^{\scriptscriptstyle \mathrm{CG}}(\mathbf{x}; u)
= Z_{\scriptscriptstyle W}(\mathbf{x}; u)\slash \Delta(\mathbf{x}),

\begin{equation} \label{eq: average-zeta}
Z_{\scriptscriptstyle W}(\mathbf{x}; u) \,  : = \sum_{w \, \in \, W} 1 \vert w(\mathbf{x}; u).
\end{equation}

For affine root systems, however, there is a highly non-trivial
correction of
Z_{\scriptscriptstyle W}^{\scriptscriptstyle \mathrm{CG}}(\mathbf{x};
(corresponding to the affine imaginary roots) that one should take
into account. This issue will be addressed in Section
<ref>when the root system is $D_{\scriptscriptstyle 4}^{\scriptscriptstyle
(1)}\!,$ but for now let us concentrate on the function
$Z_{\scriptscriptstyle W}(\mathbf{x}; u)$ that we just

The following proposition (cf. <cit.>) gives
the largest possible region of convergence for the series (<ref>),
as a function of the complex variables
x_{\scalebox{1.1}{$\scriptscriptstyle 0$}}, \ldots,
x_{\scalebox{1.1}{$\scriptscriptstyle r $}}, u.

— For
x_{\scalebox{1.1}{$\scriptscriptstyle 0$}}, \ldots, x_{\scalebox{1.1}{$\scriptscriptstyle r $}}, u \in \mathbb{C}
such that $|\mathbf{x}^{\scalebox{1.1}{$\scriptscriptstyle \delta$}}| < 1,$
the series defining $Z_{\scriptscriptstyle W}(\mathbf{x}; u)$ converges absolutely and uniformly on every compact
subset away from the points for which $u\mathbf{x}^{\beta}  \pm 1 = 0$ for some
\beta \in \Phi_{\scalebox{1.2}{$\scriptscriptstyle \mathrm{re}$}}^{\scalebox{1.2}{$\scriptscriptstyle +$}}.
In addition, the function $Z_{\scriptscriptstyle W}(\mathbf{x}; u)$ is $W$– invariant under the Chinta-Gunnells action.

We shall follow closely [54]. Let $w$ be a fixed Weyl group element, and write
$w = \sigma_{\scriptscriptstyle i_{\scalebox{0.7}{$\scriptscriptstyle \ell$}}} \cdots \,
\sigma_{\scriptscriptstyle i_{\scalebox{0.7}{$\scriptscriptstyle 1$}}}$
in reduced form. It is well-known that the set
$\Phi(w) = \{\beta \in \Phi^{+} : w\beta \in \Phi^{-}\}$ is explicitly
given by the $\ell$ distinct positive roots

\[
\Phi(w) = \{\beta_{1} = \alpha_{\scriptscriptstyle i_{\scalebox{0.7}{$\scriptscriptstyle 1$}}},\,
\beta_{2} =  \sigma_{\scriptscriptstyle i_{\scalebox{0.7}{$\scriptscriptstyle 1$}}}\alpha_{\scriptscriptstyle i_{\scalebox{0.7}{$\scriptscriptstyle 2$}}}, \,
\beta_{3} =  \sigma_{\scriptscriptstyle i_{\scalebox{0.7}{$\scriptscriptstyle 1$}}}
\sigma_{\scriptscriptstyle i_{\scalebox{0.7}{$\scriptscriptstyle 2$}}}
\alpha_{\scriptscriptstyle i_{\scalebox{0.7}{$\scriptscriptstyle 3$}}}, \ldots, \,
\beta_{\ell} =  \sigma_{\scriptscriptstyle i_{\scalebox{0.7}{$\scriptscriptstyle 1$}}}
\sigma_{\scriptscriptstyle i_{\scalebox{0.7}{$\scriptscriptstyle 2$}}} \cdots\,
\sigma_{\scriptscriptstyle i_{\scalebox{0.7}{$\scriptscriptstyle \ell - $}
\scalebox{0.77}{$\scriptscriptstyle 1$}}}
\alpha_{\scriptscriptstyle i_{\scalebox{0.7}{$\scriptscriptstyle \ell$}}}
\}.\]

Using the fact that
f \vert w_{\scalebox{1.1}{$\scriptscriptstyle 1$}}\vert w_{\scalebox{1.1}{$\scriptscriptstyle 2$}}
= f \vert w_{\scalebox{1.1}{$\scriptscriptstyle 1$}}w_{\scalebox{1.1}{$\scriptscriptstyle 2$}},
one can verify by induction on $\ell$ the formula:

\begin{equation} \label{eq: 1-acted-by-w}
f \vert w(\mathbf{x}; u)
\; =  \sum_{\delta_{\scalebox{0.7}{$\scriptscriptstyle 1$}}, \ldots,
\delta_{\scalebox{0.7}{$\scriptscriptstyle \ell$}} \, \in \, \{0, 1\}}
f\!\left(w \e^{\sss \sum_{i=1}^\ell \dd_{\scalebox{0.7}{$\scriptscriptstyle i$}}
\beta_{\scalebox{0.7}{$\scriptscriptstyle i$}}} \xx;u \right)
\,\prod_{k = 1}^{\ell}
\langle \beta_{\scalebox{0.7}{$\scriptscriptstyle k$}},\, \sum_{i \, < \, k}
\delta_{\scalebox{0.7}{$\scriptscriptstyle i$}}
\beta_{\scalebox{0.7}{$\scriptscriptstyle i$}}\rangle}
\mathbf{x}^{\scriptscriptstyle \beta_{\scalebox{0.7}{$\scriptscriptstyle k$}}}\!,
\delta_{\scalebox{1.1}{$\scriptscriptstyle k$}} \right)
\end{equation}

\varepsilon^{\alpha} : =
\prod_{i} \varepsilon_{\scalebox{1.1}{$\scriptscriptstyle i$}}^{k_{\scalebox{.75}{$\scriptscriptstyle i$}}},
\alpha = \sum_{i} k_{\scriptscriptstyle i}\alpha_{\scriptscriptstyle i},
and $J$ is defined by (<ref>); we are interpreting the factor corresponding to $k = 1$ to be
J(\mathbf{x}^{\scriptscriptstyle \beta_{\scalebox{0.7}{$\scriptscriptstyle 1$}}}\!,
\delta_{\scalebox{1.1}{$\scriptscriptstyle 1$}}).

To estimate $1 \vert w(\mathbf{x}; u),$ express each root $\beta \in \Phi(w)$ as
$\beta = \alpha + (n + \chi(\alpha))\delta$ with
$\alpha \in \Phi_{\scalebox{1.0}{$\scriptscriptstyle 0$}},$
$n \in \mathbb{N},$ and $\chi$ the characteristic function of
$\Phi_{\scalebox{1.0}{$\scriptscriptstyle 0$}}^{\scalebox{1.}{$\scriptscriptstyle  \minus$}}.$
By assuming that these roots are as small as possible, one finds that

\[
\sum_{\alpha + (n + \chi(\alpha))\delta \, \in \,  \Phi(w)}
n + \chi(\alpha)
\, \ge  \, \frac{|\Phi_{\scalebox{1.0}{$\scriptscriptstyle 0$}}|
\left\lfloor \frac{\ell}{|\Phi_{\scalebox{1.0}{$\scriptscriptstyle 0$}}|}\right\rfloor
\left(\left\lfloor \frac{\ell}{|\Phi_{\scalebox{1.0}{$\scriptscriptstyle 0$}}|}\right\rfloor  - 1\right)}{2}}
\]

where $\lfloor x \rfloor$ denotes the integer part of $x,$ and
$|\Phi_{\scalebox{1.0}{$\scriptscriptstyle 0$}}|$ is the cardinality of
$\Phi_{\scalebox{1.0}{$\scriptscriptstyle 0$}}.$ One can also see that, for a positive affine real root
\beta = \alpha + (n + \chi(\alpha))\delta,

\[
\left|J\!\left(\mathbf{x}^{\beta}\!, \e \right)\right|  \le
|\mathbf{x}^{\scalebox{1.1}{$\scriptscriptstyle (n + \chi(\alpha))\delta$}}|
\, K(\mathbf{x}, u)\slash 2
\]

where we set

\[
K(\mathbf{x}, u) :=
\left(1 + |u|\right)
\max_{\alpha \, \in \, \Phi_{\scalebox{0.75}{$\scriptscriptstyle 0$}}}
\left\{|\mathbf{x}^{\alpha}|\left(1 + |\mathbf{x}^{\alpha}|\right)\right\}
\, \cdot \sup_{\beta \, \in \, \Phi_{\scalebox{.95}{$\scriptscriptstyle \mathrm{re}$}}^{\scalebox{.95}{$\scriptscriptstyle +$}}}
\left|1 - u\mathbf{x}^{\beta}\right|^{\sss -1};
\]

the supremum is finite since $|\mathbf{x}^{m\delta}| \to 0$ as $m \to \infty,$
and $u\mathbf{x}^{\beta} \ne \pm 1$ for all
\beta \in \Phi_{\scalebox{1.2}{$\scriptscriptstyle \mathrm{re}$}}^{\scalebox{1.2}{$\scriptscriptstyle +$}}.
It follows that

|1 |w(𝐱; u)| ≤|𝐱^1.1$\scriptscriptstyle \delta$|^1.05
$\scriptscriptstyle \frac{\ell^{\scalebox{.8}{$\scriptscriptstyle 2$}}}{2 |\Phi_{\scalebox{.65}{$\scriptscriptstyle 0$}}|}
- O(\ell)$K(𝐱, u)^ℓ
and thus

\begin{equation*}
\begin{split}
\sum_{w \, \in \, W} |1 \vert w(\mathbf{x}; u)| \, & \le \,
\sum_{\ell \, \ge \,  0}\sum_{\substack{w \, \in \, W\\ \ell(w) \, = \, \ell}}
|\mathbf{x}^{\scalebox{1.1}{$\scriptscriptstyle \delta$}}|^{\scalebox{1.05}
{$\scriptscriptstyle \frac{\ell^{\scalebox{.8}{$\scriptscriptstyle 2$}}}{2 |\Phi_{\scalebox{.65}{$\scriptscriptstyle 0$}}|}
- O(\ell)$}}K(\mathbf{x}, u)^{\ell}\\
& < \, \sum_{\ell \, \ge \,  0}
(r + 1)^{\ell} \, |\mathbf{x}^{\scalebox{1.1}{$\scriptscriptstyle \delta$}}|^{\scalebox{1.05}
{$\scriptscriptstyle \frac{\ell^{\scalebox{.8}{$\scriptscriptstyle 2$}}}{2 |\Phi_{\scalebox{.65}{$\scriptscriptstyle 0$}}|}
- O(\ell)$}}K(\mathbf{x}, u)^{\ell}
\end{split}
\end{equation*}

where, for the last inequality, we applied the trivial bound
\# \{w \in W : \ell(w)  = \ell \} \le r^{\ell - 1}(r + 1) < (r + 1)^{\ell}.
The last series converges since $|\mathbf{x}^{\scalebox{1.1}{$\scriptscriptstyle \delta$}}| < 1$ and
$u\mathbf{x}^{\beta} \ne \pm 1$ for all
\beta \in \Phi_{\scalebox{1.2}{$\scriptscriptstyle \mathrm{re}$}}^{\scalebox{1.2}{$\scriptscriptstyle +$}}.
This implies our first assertion.

Finally, for
x_{\scalebox{1.1}{$\scriptscriptstyle 0$}}, \ldots, x_{\scalebox{1.1}{$\scriptscriptstyle r $}}, u
in the region of absolute convergence, we have

\[
Z_{\scriptscriptstyle W} \vert w'(\mathbf{x}; u)
\, = \sum_{w \, \in \, W} 1 \vert ww'(\mathbf{x}; u)
= Z_{\scriptscriptstyle W}(\mathbf{x}; u)
\]

which completes the proof.

The singularities $u\mathbf{x}^{\beta}  \pm 1 = 0$ ($
\beta \in \Phi_{\scalebox{1.2}{$\scriptscriptstyle \mathrm{re}$}}^{\scalebox{1.2}{$\scriptscriptstyle +$}}
$) of the function $Z_{\scriptscriptstyle W}(\mathbf{x}; u)$ are at most simple poles, and the residues at these poles of a closely related function will be evaluated in [30] by generalizing the method
introduced in the present paper to arbitrary simply laced affine root systems.

We conclude this subsection by extending the Chinta-Gunnells action to
the group $\widetilde{W},$ which is done as follows.
An element $w \in \widetilde{W}$ acts on the multivariable
$\mathbf{x}$ by
(w \mathbf{x}) _{\scriptscriptstyle j}
= \mathbf{x}^{w^{\scalebox{.85}{$\scriptscriptstyle -1$}}\alpha_{\scalebox{.7}{$\scriptscriptstyle j$}}}\!,
and for $\eta \in O$ and a function $f(\mathbf{x}; u),$ we define

\[
f\vert \eta (\mathbf{x}; u) = f(\eta\mathbf{x}; u).
\]

This defines an action of $\widetilde{W} = W \rtimes O,$
and to check that it is well-defined, it suffices to check
its compati-bility with the commutation relations between the elements
of $O$ and the simple reflections $\sigma_{\scriptscriptstyle i}.$
Since $W$ is a normal subgroup of $\widetilde{W},$
it follows at once from <cit.> that, for
$\eta \in O,$ we have:
\eta \alpha_{\scriptscriptstyle i} = \alpha_{\scriptscriptstyle j}
if and only if
\eta \sigma_{\scriptscriptstyle i}
= \sigma_{\scriptscriptstyle j} \eta.
Moreover, one checks that

\[
\text{$\varepsilon^{\alpha} w
= w \varepsilon^{w^{\scalebox{.85}{$\scriptscriptstyle -1$}}\alpha}$
for all $\alpha$ in the root lattice and $w \in \widetilde{W}$}
\]

with $\varepsilon_{\scalebox{1.1}{$\scriptscriptstyle i$}}$ the sign
action defined at the beginning of this subsection. It is
easy to check that
f\vert \eta \vert \sigma_{\scriptscriptstyle i}
= f\vert \sigma_{\scriptscriptstyle j} \vert \eta,
for $\eta \in O$ and $i, j$ such that
\eta \alpha_{\scriptscriptstyle i} = \alpha_{\scriptscriptstyle j},
so the action is, indeed, well-defined.
One could also define the function
$Z_{\sss \wW}=\sum_{w\in \wW}1|w$, but it follows from the
definition of
the extended action that $Z_{\sss \wW}=n\cdot Z_{\sss W}$,
with $n$ the cardinality of $O$. For this reason, the extension of
the Chinta-Gunnells action to $\wW$ will in fact not be needed.
However, with no additional effort required, some
of the relations used in the proofs of our results will be stated
(for completeness) in the extended Weyl group.

§.§ Chinta-Gunnells averages of polynomials

We will also need the convergence and properties of the
Chinta-Gunnells average

\[
\ZWg:=\sum_{w \, \in \, W} g\vert w
\]

for any Laurent polynomial $g(\xx)$ with coefficients in
$\CC(u)$. Note that $Z_{\sss W, 1}=\ZW$.

— For any Laurent polynomial $g(\xx),$ there is
an integer $N = N_g\ge 0$ such that the series defining
$\xx^{\scalebox{1.1}{$\scriptscriptstyle N\dd$}} \ZWg(\xx)$
converges absolutely and uniformly on compacta in the
same region as $\ZW(\xx),$ and it is $W$-invariant under
the Chinta-Gunnells action.

By linearity, it is enough to consider the case
$g(\xx)=\xx^{\a}$ for
\a=a_{\sss 0} \a_{\sss 0}+\cdots + a_{\sss r} \a_{\sss r}
an element in theroot lattice of $\Phi.$
Using (<ref>) and (<ref>),
it also suffices to show that
|\xx^{w^{\scalebox{.85}{$\scriptscriptstyle -1$}} \a }| \ll |\xx^{\dde }|^{\sss -O(\ell)},
for the length $\ell$ of $w$ sufficiently large, say
\ell \ge \ell(w_{\sss 0})
for all $w_{\sss 0} \in W_{\sss 0}$; here we are taking
\xx = (x_{\sss 0}, \ldots, x_{\sss r})
in a compact set with $x_{\sss i} \ne 0$ for all $i.$

We decompose $w=w_{\sss 0} t,$ with $w_{\sss 0}\in W_{\sss 0}$ and
$t=\tau(\lam)$ a translation with $\lam\in P^\vee.$
Letting $\mu_{\sss i}\in P^\vee$ be the fun-damental coweights ($i=1,\ldots,r$), we can write
t^{\sss -1}=\prod_{i=1}^{r}
\tau(\mu_{\sss i})^{n_{\scalebox{.75}{$\scriptscriptstyle i$}}}
for some $n_{\sss i}\in \Z;$ we have

\[
\tau(\mu_{\sss i})\a_{\sss i}=\a_{\sss i}-\dd, \;\;
\tau(\mu_{\sss i})\a_{\sss 0}=\a_{\sss 0}+m_{\sss i} \dd \;\;\;\, \text{and} \;\;\;\,
\tau(\mu_{\sss i})\a_{\sss j}=\a_{\sss j} \;\text{$(j\ne 0, i)$}
\]

where $\theta=\sum_{i=1}^r m_{\sss i} \a_{\sss i}$ is the highest root
in $\Phi_{\sss 0}.$ It follows from (<ref>) that

\[
\sum_{i=1}^{r} |n_{\sss i}|  \le
\ell(t)\; =
\sum_{\alpha \in \Phi_{\scalebox{1.0}{$\scriptscriptstyle
0$}}^{\scalebox{1.1}{$\scriptscriptstyle +$}}} |\langle \lambda, \alpha \rangle|
\le \ell(w_{\sss 0}) + \ell \le 2\ell
\]

where the lower bound of $\ell(t)$ was obtained by retaining only the
terms corresponding to
\a = \a_{\sss i}\in \Phi_{\sss 0}^{\scalebox{1.1}{$\scriptscriptstyle +$}}.

On the other hand, if we write
w_{\sss 0}^{\sss -1}\a=\sum_{j=0}^{r} k_{\sss j} \a_{\sss j},
w^{\sss -1}\a=t^{\sss -1} w_{\sss 0}^{\sss -1} \a
= w_{\sss 0}^{\sss -1}\a + n\dd,
with $n$ given by

\[
n = \langle \lambda, w_{\sss 0}^{\sss -1}\a\rangle
= \sum_{i=1}^{r} (k_{\sss 0}m_{\sss i} - k_{\sss i})n_{\sss i}.
\]

\[
|n| \le \sum_{i=1}^{r} |k_{\sss 0}m_{\sss i} - k_{\sss i}||n_{\sss i}|
\le C\sum_{i=1}^{r} |n_{\sss i}| \le 2C\ell
\]

for a positive constant $C$ depending only upon $\Phi_{\sss 0}$ and
$\a.$ It follows that

\[
|\xx^{w^{\sss -1} \a}|
\le |\xx^{w_{\scalebox{.8}{$\scriptscriptstyle 0$}}^{\sss -1}\a}|
|\mathbf{x}^{\scalebox{1.1}{$\scriptscriptstyle \delta$}}|^{\scalebox{1.1}{$\scriptscriptstyle - |n|$}}
\ll |\mathbf{x}^{\scalebox{1.1}{$\scriptscriptstyle
\delta$}}|^{\scalebox{1.1}{$\scriptscriptstyle - 2C\ell$}}
\]

where the implied constant can be taken to be the maximum over
$\xx$ in the compact set and $w_{\sss 0}\in W_{\sss 0}$ of
|\xx^{w_{\scalebox{.8}{$\scriptscriptstyle 0$}}^{\sss -1}\a}|.
Accordingly $|g\vert w|$ satisfies an estimate of
type (<ref>). It is also clear that there are only finitely many
$w\in W$ such that $g\vert w$ has poles when $x_{i}= 0,$
which completes the proof.

For brevity, we state the following result only for the case
$D^{\sss (1)}_{\sss 4}$, the general case being considered in [30]. — Assume the root system $\Phi$ is affine
of type $D^{\sss (1)}_{\sss 4}.$ Then for every monomial $g,$
the function $\ZWg$ satisfies

\[
\ZWg(\xx) = C_{g}(\xx^{\scalebox{1.1}{$\scriptscriptstyle \delta$}})\ZW(\xx)
\]

where $C_{g}(x)$ is a Laurent polynomial with coefficients in
$\Z[u],$ which can be determined recursively in terms of $g.$

We use the labelling of simple roots for
$D^{\sss (1)}_{\sss 4}$ given in the beginning of the next section,
so that
\delta =
\alpha_{\scriptscriptstyle 1}
+ \alpha_{\scriptscriptstyle 2}
+ \alpha_{\scriptscriptstyle 3}
+ \alpha_{\scriptscriptstyle 4}
+ 2\alpha_{\scriptscriptstyle 5}.
We abbreviate $Z_g=\ZWg$ in this proof.

Let $g(\xx)=\xx^{\a}$ be a monomial with
\a = a_{\sss 1} \a_{\sss 1}+\cdots + a_{\sss 5} \a_{\sss 5}
with $a_{\sss i}\in\Z$. Then $g(\xx)$ is even (resp. odd) for theinvolution $\e_{\sss i}$ if and only if
$v_{\sss i}(g):=\sum_{j\sim i} a_{\sss j}$
is even (resp. odd). Note that in this case we have only two signfunctions, $\e_{\sss 1}=\e_{\sss i}$ for $i=1,\ldots, 4$, and
$\e_{\sss 5}$.

We use the following properties of the Chinta-Gunnells action.
If $g(\xx)$ is odd for the involution $\e_{\sss i}$, then

\[
g \vert \ss_{\sss i}=-x_{\sss i} \ss_{\sss i} g
\]

and if $g(\xx)$ is even for $\e_{\sss i}$, then

\[
[(x_{\sss i} - u) g]\vert \ss_{\sss i} = (u  - x_{\sss i})  \ss_{\sss i} g.
\]

By the previous lemma, we can act with $w\in W$ and sum
these relations over $W$ to obtain (replacing first $g$
by $g\slash x_{\sss i}$ in the even case):

Z_g =  - Z_x_.75$\scriptscriptstyle i$
ß_.75$\scriptscriptstyle i$ g
for $v_{\sss i}(g)$ odd

u Z_x_.75$\scriptscriptstyle i$
ß_.75$\scriptscriptstyle i$ g +
u Z_gx_.75$\scriptscriptstyle i$
- Z_x_.75$\scriptscriptstyle
i$^.75$\scriptscriptstyle 2$
ß_.75$\scriptscriptstyle i$ g
for $v_{\sss i}(g)$ even.

We now introduce an order on the monomials $g$.
Let $\db (g)$ be the $8$-tuple consisting of the differences:
±(a_1-a_2), ±(a_5-a_1-a_2),  ±(a_3+a_4-a_5),  ±(a_3-a_4)
ordered decreasingly. We order the monomials $g$
by the lexicographic order of the tuples $\db (g).$
Note that$g = \xx^{n\dd}$ if and only if  $\db(g) = (0,\ldots, 0),$
and that these are the smallest elements for the order
just defined.Using the relations above, we show that
$Z_{g}$ with $g\ne \xx^{n\dd}$ can be expressed
in terms of $Z_{g'}$ with $\db(g')<\db(g),$ which shows
recursively that $Z_{g}$ can be expressed in terms of
$Z_{\xx^{n\dd}} = \xx^{n\dd}Z_{1},$ with coefficients in $\Z[u].$

The differences in (<ref>) containing $a_{\sss i}$ are of type
$\pm (a_{\sss i}-b_{\sss i})$, $\pm(a_{\sss i}-c_{\sss i})$,
with $b_{\sss i}+c_{\sss i}=v_{\sss i}(g)$. For each index $i,$ we define
two differences:

\[
d_{\sss i}^{\sss (1)}=a_{\sss i}-b_{\sss i} \;\;\; \text{and} \;\;\;
d_{\sss i}^{\sss (2)}=c_{\sss i}-a_{\sss i}
\]

(fixing throughout a choice of $b_{\sss i}, c_{\sss i},$ e.g.,
$b_{\sss 1}=a_{\sss 2},$ $c_{\sss 1}=a_{\sss 5}-a_{\sss 2}$, etc.).
Since $\ss_{\sss i} g=g x_{\sss i}^{v_{\scalebox{.75}{$\scriptscriptstyle i$}}(g)-2a_{\scalebox{.75}{$\scriptscriptstyle i$}}}\!,$
and $b_{\sss i}+c_{\sss i}\linebreak =v_{\sss i}(g),$
the differences $\pm d_{\sss i}^{\sss (1)}$ and
$\pm d_{\sss i}^{\sss (2)}$ are switched for $\ss_{\sss i} g,$ and
the other four differences are the same for $g$ and $\ss_{\sss i} g,$ so $\db (\ss_{\sss i} g)=\db(g).$

If $g\ne \xx^{n\dd},$ then there exists an index $i$
with $d_{\sss i}^{\sss (1)} > d_{\sss i}^{\sss (2)};$ indeed,
if $d_{\sss i}^{\sss (1)}\le d_{\sss i}^{\sss (2)},$ that is
$2a_{\sss i}\le v_{\sss i}(g)$ for all $i,$ then
one must have equality for all $i,$ and so
$g = \xx^{n\dd}$ for some $n$, a contradiction.
We claim that the relations (<ref>) for this choice
of $i,$ express $Z_{g}$ in terms of $Z_{g'},$
with $\db (g') <\db (g),$ hence finishing the proof. Wehave $-d_{\sss i}^{\sss (2)}>-d_{\sss i}^{\sss (1)}$ as well, and
the pairs $(d_{\sss i}^{\sss (1)}, d_{\sss i}^{\sss (2)})$,
$(-d_{\sss i}^{\sss (2)}, -d_{\sss i}^{\sss (1)})$
for $x_{\sss i}\ss_{\sss i} g$ (resp. $x_{\sss i}^{\sss 2}\ss_{\sss i} g$) are

\[
(d_{\sss i}^{\sss (2)}+1,d_{i}^{\sss (1)}-1),\
( -d_{\sss i}^{\sss (1)}+1,-d_{i}^{\sss (2)}-1)
\;\;\;\; (\text{resp. } (d_{\sss i}^{\sss (2)}+2, d_{\sss i}^{\sss (1)}-2),\
(-d_{\sss i}^{\sss (1)}+2, -d_{\sss i}^{\sss (2)}-2)).
\]

It follows that
\db(g\slash x_{\sss i})=\db(x_{\sss i} \ss_{\sss i} g )<\db(g),$
unless $d_{\sss i}^{\sss (1)}=d_{\sss i}^{\sss (2)}+1,$
in which case $g=x_{\sss i} \ss_{\sss i} g$ and $Z_{g}=0.$
Simi-larly, if $v_{\sss i}(g)$ is even, we have
\db(x_{\sss i}^{\sss 2} \ss_{\sss i} g )<\db(g),$
unless $d_{\sss i}^{\sss (1)}=d_{\sss i}^{\sss (2)}+2,$ in which case
g=x_{\sss i}^{\sss 2} \ss_{\sss i} g$ and
$Z_{g}=u Z_{g\slash x_{\sss i}}$ with
$\db(g\slash x_{\sss i})<\db(g).$

The proof clearly gives an algorithm for computing recursively
the polynomial $C_{g}(x)$ in terms of $g.$

For $D_{\sss 4}=\la\ss_{\sss 2}, \ldots, \ss_{\sss 5} \ra$,
one can define the same 8-tuple $\db(g)$ for a monomial $g$ in
$x_{\sss 2},\ldots, x_{\sss 5}$, by setting $a_{\sss 1}=0$
in (<ref>). The above proof then shows that for all monomials
$g$, we have $Z_{\sss W, g}=p(u)Z_{\sss W, h},$ for a polynomial $p$
and a monomial
h=\prod x_{\sss i}^{a_{\scalebox{.75}{$\scriptscriptstyle i$}}}
such that $2a_{\sss i}\le v_{\sss i}(h)$ for $i=2, \ldots, 5$.
However, there are infinitely many such $h$,
for example $h = x_{\sss 2}^b x_{\sss 3}^a x_{\sss 4}^a x_{\sss 5}^{2a}$ for $2a\le b \le a$.
Therefore the previous proposition fails for finite Weyl groups.

When specializing $u=-1$, the quotient of the averages appearing in Proposition <ref>,
for an arbitrary simply laced affine root system $\Phi$, becomes

\[\frac{\ZWg}{\ZW}(\xx;-1)=\chi_\lam(\xx):=
\frac{\sum_{w\in W} (-1)^{\ell(w)} \xx^{w(\lam-\rho)+\rho}}
{\sum_{w\in W} (-1)^{\ell(w)} \xx^{\rho-w\rho}}
\]

where $g=\xx^\lam$, and $\rho=\om_{\sss 0}+\cdots+\om_{\sss r}$, with
$\om_{\sss i}$ the affine fundamental weights. If $\lam$ were an anti-dominant affine weight,
then $\chi_\lam$ would be the character of the infinite-dimensional representation of
the associated affine Kac-Moody Lie algebra with lowest weight $\lam$.
However, in our situation $\lam$ is an element of the affine root lattice, and letting $w_0\in W$ be such that
$w_0(\lam-\rho)=\mu$ is the unique anti-dominant affine weight in the $W$-orbit of $\lam-\rho$,
there are two possibilities. Either $\mu$ is singular, i.e., it is fixed by a simple reflection, and then $\chi_\lam=0$; or $\mu$ is regular, in which case one checks easily that it must be of the form $\mu=n\dd-\rho$ for some $n\in \Z$, and $\chi_\lam(\xx)=(-1)^{\ell(w_0)} \xx^{n\dd}$. We conclude that the specialization at $u=-1$
of the function $C_g(x)$ in Proposition <ref>  is either $0$ or $\pm x^n$, for some $n\in\Z$. For example,
when $\Phi$ is of type $D_{\sss 4}^{\sss (1)}$ and $g=x_{\sss 1}^2 x_{\sss 2}^2 x_{\sss 3}^2$, we have

\[C_g(x)=-u^4 x^{-1}+(u^2-1)^3 ,\; \text{and} \;\, C_g(x)|_{u=-1}=-x^{-1}.
\]

§ AN EXTRA FUNCTIONAL EQUATION

From now on we take the affine root system $\Phi$ to be of type
$D_{\scriptscriptstyle 4}^{\scriptscriptstyle (1)}\!,$
see <cit.>. The Dynkindiagram of
the finite root system $\Phi_{\scalebox{1.0}{$\scriptscriptstyle 0$}}
= D_{\scriptscriptstyle 4}$ is

\[
\underset{\text{Dynkin diagram of
$D_{\scriptscriptstyle 4}$}}{\begin{dynkinDiagram}[text style/.style={scale=1.,black},
edge length=1.25cm,
\drlap{#1}$}}}]D{4}
\end{dynkinDiagram}}
\]

and the Dynkin diagram of $D_{\scriptscriptstyle
4}^{\scriptscriptstyle (1)}$ is [extended]D4 with the
additional simple root denoted by $\alpha_{\scriptscriptstyle 1};$
note the shift inthe labeling of simple roots compared to
the previous section. The set of positive roots
\Phi_{\scalebox{1.0}{$\scriptscriptstyle 0$}}^{\scalebox{1.1}{$\scriptscriptstyle +$}}
is given explic-itly by

\[
\Phi_{\scalebox{1.0}{$\scriptscriptstyle 0$}}^{\scalebox{1.1}{$\scriptscriptstyle +$}}
= \{\alpha_{\scriptscriptstyle i},\;
\alpha_{\scriptscriptstyle 5},\;
\alpha_{\scriptscriptstyle i} +
\alpha_{\scriptscriptstyle 5},\;
\alpha_{\scriptscriptstyle i} +
\alpha_{\scriptscriptstyle j} +
\alpha_{\scriptscriptstyle 5},\;
\alpha_{\scriptscriptstyle 2} +
\alpha_{\scriptscriptstyle 3} +
\alpha_{\scriptscriptstyle 4} +
\alpha_{\scriptscriptstyle 5},\;
\alpha_{\scriptscriptstyle 2}
+ \alpha_{\scriptscriptstyle 3}
+ \alpha_{\scriptscriptstyle 4}
+ 2\alpha_{\scriptscriptstyle 5}
\}_{\scalebox{1.1}{$\scriptscriptstyle 2\, \le \, i\, \ne \, j\, \le\, 4$}};
\]

thus the highest root is
\theta =
\alpha_{\scriptscriptstyle 2}
+ \alpha_{\scriptscriptstyle 3}
+ \alpha_{\scriptscriptstyle 4}
+ 2\alpha_{\scriptscriptstyle 5},
\delta =
\alpha_{\scriptscriptstyle 1}
+ \alpha_{\scriptscriptstyle 2}
+ \alpha_{\scriptscriptstyle 3}
+ \alpha_{\scriptscriptstyle 4}
+ 2\alpha_{\scriptscriptstyle 5}.
The Weyl group of $\Phi$ is
W = \langle \sigma_{\scriptscriptstyle
i}\rangle_{\scalebox{.95}{$\scriptscriptstyle i = 1, \ldots, 5$}},
\sigma_{\scriptscriptstyle i} = \sigma_{\!\scriptscriptstyle
\alpha_{\scalebox{.55}{$\scriptscriptstyle i$}}}\!.

§.§ A cocycle associated with the Chinta-Gunnells action

For any function $f(\mathbf{x}),$ and $a, b \in \{\mathcal{e},
\mathcal{o}\},$ where $\mathcal{e}$ (resp. $\mathcal{o}$) stands for
even (resp. odd), let $f^{\scalebox{1.2}{$\scriptscriptstyle a, b$}}$
denote the part of $f$ which has parity $a$ with respect to the
involution $\e_{\sss 5}(\underline{x}, x_{\sss 5})
= (-\underline{x}, x_{\sss 5})$ and parity $b$ with respect tothe involution $\e_{\sss 1}(\underline{x}, x_{\sss 5})
= (\underline{x}, -x_{\sss 5}),$ where
$\underline{x}=(x_{\sss 1},\ldots, x_{\sss 4})$.
For the following lemma, it is convenient to write the
Chinta-Gunnells action as

x_iu- x_i/1-u x_i
f_i^+(ß_i)-x_i f_i^-(ß_i)
f_{\sss i}^{\pm}(\xx)=(f(\xx)\pm f(\e_{\sss i}\xx))\slash 2
are the even and odd components of $f(\xx)=f(\xx;u)$ with
respect to $\e_{\sss i}.$ — The Chinta-Gunnells action
of $w\in\wW$ preserves the subspace

\CC(\xx,u)_{\sss 0}:=\{f\in\CC(\xx,u) \mid f^{\oo}=0 \}.

Assume $f^{\oo} \!=0$. Then for all $i$
and $j\sim i$, we have

\[
(f\vert\ss_{\sss i})^{\oo}(\xx) =
[[-x_{\sss i} f_{\sss i}^{-}(\ss_{\sss i}\xx)]^-_{\sss i}]^-_{\sss j}
=-x_{\sss i} f^{\oo}(\ss_{\sss i}\xx)=0.
\]

Similarly, $(f\vert \eta)^{\oo} \!=0$ for all $\eta\in O.$

In particular, $(1\vert w)^{\oo} \! =0$ for all
$w\in \wW$, so $Z_{\scriptscriptstyle W}^{\oo} \equiv 0.$
We restrict henceforth the Chinta-Gunnells action
to the invariant subspace $\CC(\xx,u)_{\sss 0}.$

Letting $\bar{f}$ be the column vector

\[\bar{f}: = \!\, ^{t}\!\left(f^{\eee}, f^{\eo}, f^{\oe} \right)\]

we define a $3\times 3$ matrix $\Lambda_{w}(\xx)$ such that

f |w(𝐱)
= Λ_w(𝐱)
f̅(w𝐱)     (for $w \in \wW$)
with $\Lambda_{w}$ satisfying the $1$-cocycle relation
\Lambda_{w w\scalebox{.75}{$\scriptscriptstyle '$}}(\mathbf{x})
= \Lambda_{w\scalebox{.75}{$\scriptscriptstyle '$}}(\mathbf{x})
\Lambda_{w}(w{\scriptscriptstyle '} \mathbf{x})
for all $w, w{\scriptscriptstyle '} \in \wW.$
On the generators $\sigma_{\scriptscriptstyle i},$ this cocycle
is given by

\[
\Lambda_{\sigma_{\scalebox{0.75}{$\scriptscriptstyle i$}}}\!(\mathbf{x})
= \Lambda_{\sss 1}(x_{\scalebox{1.1}{$\scriptscriptstyle i$}}, u)
= - \, x_{\scalebox{1.1}{$\scriptscriptstyle i$}}\begin{pmatrix}
\frac{(1 - u^{\scalebox{.8}{$\scriptscriptstyle 2$}}) x_{\scalebox{.8}{$\scriptscriptstyle i$}}}
{1- u^{\scalebox{.8}{$\scriptscriptstyle 2$}}x_{\scalebox{.8}{$\scriptscriptstyle i$}}^{\scalebox{.8}{$\scriptscriptstyle 2$}}}
& 0 & - \frac{u (1 - x_{\scalebox{.8}{$\scriptscriptstyle i$}}^{\scalebox{.8}{$\scriptscriptstyle 2$}})}
{1 - u^{\scalebox{.8}{$\scriptscriptstyle 2$}}
x_{\scalebox{.8}{$\scriptscriptstyle i$}}^{\scalebox{.8}{$\scriptscriptstyle 2$}}} \\
0 & 1 & 0 \\
- \frac{u (1 - x_{\scalebox{.8}{$\scriptscriptstyle i$}}^{\scalebox{.8}{$\scriptscriptstyle 2$}})}
{1 - u^{\scalebox{.8}{$\scriptscriptstyle 2$}}
x_{\scalebox{.8}{$\scriptscriptstyle i$}}^{\scalebox{.8}{$\scriptscriptstyle 2$}}} & 0
& \frac{(1 - u^{\scalebox{.8}{$\scriptscriptstyle 2$}}) x_{\scalebox{.8}{$\scriptscriptstyle i$}}}
{1- u^{\scalebox{.8}{$\scriptscriptstyle 2$}}x_{\scalebox{.8}{$\scriptscriptstyle i$}}^{\scalebox{.8}{$\scriptscriptstyle 2$}}}  \\
\end{pmatrix}
\]

for $i = 1, \ldots, 4,$ and

\[
\Lambda_{\sigma_{\scalebox{0.75}{$\scriptscriptstyle 5$}}}\!(\mathbf{x})
= \Lambda_{\sss 2}(x_{\scalebox{1.1}{$\scriptscriptstyle 5$}}, u)
= - \, x_{\scalebox{1.1}{$\scriptscriptstyle 5$}}
\begin{pmatrix}
\frac{(1 - u^{\scalebox{.8}{$\scriptscriptstyle 2$}}) x_{\scalebox{.8}{$\scriptscriptstyle 5$}}}
{1- u^{\scalebox{.8}{$\scriptscriptstyle 2$}}x_{\scalebox{.8}{$\scriptscriptstyle 5$}}^{\scalebox{.8}{$\scriptscriptstyle 2$}}}
& - \frac{u (1 - x_{\scalebox{.8}{$\scriptscriptstyle 5$}}^{\scalebox{.8}{$\scriptscriptstyle 2$}})}
{1 - u^{\scalebox{.8}{$\scriptscriptstyle 2$}}
x_{\scalebox{.8}{$\scriptscriptstyle 5$}}^{\scalebox{.8}{$\scriptscriptstyle 2$}}}  & 0 \\
\hskip3pt
- \frac{u (1 - x_{\scalebox{.8}{$\scriptscriptstyle 5$}}^{\scalebox{.8}{$\scriptscriptstyle 2$}})}
{1 - u^{\scalebox{.8}{$\scriptscriptstyle 2$}}
x_{\scalebox{.8}{$\scriptscriptstyle 5$}}^{\scalebox{.8}{$\scriptscriptstyle 2$}}}
& \frac{(1 - u^{\scalebox{.8}{$\scriptscriptstyle 2$}}) x_{\scalebox{.8}{$\scriptscriptstyle 5$}}}
{1- u^{\scalebox{.8}{$\scriptscriptstyle 2$}}x_{\scalebox{.8}{$\scriptscriptstyle 5$}}^{\scalebox{.8}{$\scriptscriptstyle 2$}}} & 0 \\
0 & 0 & 1 \\
\end{pmatrix}.
\]

The group $O\subset \wW$ has order 4, and it is generated
by elements $\eta$ of order 2 with
$\eta\a_{\sss i}=\a_{\sss j}$ and $\eta \a_{\sss k}=\a_{\sss l}$
for $\{i, j, k, l\}$ a permutation of
$\{1, 2, 3, 4\}$. From (<ref>) we have that
$\Lambda_{\eta}$ is the identity matrix for $\eta\in O.$

We denote by $\ZZ$ the vector $\ov{\ZW}$.
The functional equation $\ZW=\ZW\vert w$ becomes:

(;u)=_w() (w;u)
for all $w\in \wW$, which follows from the cocycle relation of $\Lam_{w}$.

§.§ An extension of the Chinta-Gunnells action

Define the transformation $\tau$ acting on
$f\in\CC(\xx, u)$ by

\tau f(\xx; u) = f\!\left(\xx; u\slash\xx^{\dd}\right).

This transformation extends the natural action of
$w\in \wW$ given by
wf(\xx; u)=f(w^{\sss -1}\xx; u)
and we denote by $\wW\oplus \Z$ the group
generated by $\wW$ and $\tau^{a}$, $a\in\Z$,
which can be seen as a subgroup of $\End(\CC(\xx,u))$.
We show that there is an extension of the
Chinta-Gunnells action to this larger group
$\wW\oplus \Z$, from which we will derive an extra
functional equation for the vector function $\ZZ$.
As in the previous subsection, we willrestrict the action to
the space $\CC(\xx,u)_{\sss 0}$ of functions $f$ with $f^{\oo} \!=0.$

Similarly to the way the Chinta-Gunnells action is defined
in (<ref>), we look for an action of the form

τf^(;u)+A_(;u) τf^(;u) +
A_œ(;u) τf^œ(;u)
defined for $f\in\CC(\xx,u)_{\sss 0}$, for three
unknown functions $A_{\eee}$, $A_{\eo}$, $A_{\oe}\in\CC(\xx,u)$.
In order for this action to preserve the space
$\CC(\xx,u)_{\sss 0}$, we require that

\[
A_{\eee}^{\oo} \! =0,\quad A_{\eo}^{\oe} \! =0,\quad
A_{\oe}^{\eo} \! =0.\]

We also require this action to be compatible with the
Chinta-Gunnells action, namely

f|τ|w=f|w|τ    (for $w\in \wW$).

Formula (<ref>) can be written:

f|τ(;u)=A(;u) f̅(;u^)
for the $3\times 3$ matrix

A_    1/x_.75$\scriptscriptstyle 5$
x_5 A_    1/x_.75$\scriptscriptstyle 1$
x_1 A_œ

Defining $\Lambda_{\tau}:=A$,
condition (<ref>) is then equivalent to the fact that the
cocycle $\Lambda$ has a well-defined exten-sion to the monoid
$\wW\times \{ \tau^{n} \}_{n\ge 0} $ by
the $1$-cocycle relation, namely it satisfies
$\Lambda_{\tau w}=\Lambda_{w\tau}$ for all
$w\in \wW,$ that is,

_w(;u)A(w;u)=A(;u) _w(;u^).
We also require that $A$ is invertible,
so that $\Lambda_{\tau^{\sss -1}}$ is well-defined by the
cocycle relation. Therefore $f\vert \tau^{\sss -1}$ iswell-defined as well, by the analogue of (<ref>) written
for $\tau^{\sss -1}$, with $A$ replaced by $\Lambda_{\tau^{\sss -1}}$.

The main theorem of this section shows that there exists an extension
of the Chinta-Gunnells action to$\wW\oplus \Z$ as above, and
that $\ZZ$ satisfies an additional functional equation under the
transformation $\tau$.

There exists an invertible matrix $A(\xx;u)$ of the form (<ref>)
satisfying the following conditions:

* The cocycle relation (<ref>) is satisfied.

* There exists an element $a\in \CC\!\left(u,\xx^\dd\right)$ such that
the matrix $a\cdot A$ has polynomial entries in $\xx$ and $u$.

* The vector function $\ZZ(\xx;u)$ satisfies the functional

(;u)=A(;u) (; u^).

In [30], we will use results in [29] to show that a similar
result holds for arbitrary affine irreducible reduced root systems.

The matrix $A$ in the theorem is essentially unique;
more precisely, any invertible matrix $A$ ofthe form (<ref>)
that satisfies the cocycle relation is unique, up to multiplication by
elements in $\CC\!\left(u,\xx^\dd\right)$. Weomit the proof of this fact
since it is quite technical, and we will not need it in the sequel.

The relations (<ref>), for $w$ running through the generators $\ss_{\sss i}$ of $W$, reduce to a linear system of equations over the field $\Q(u)$,
with unknowns being the coefficients of the entries of $A$ (assuming that these entries are polynomials of bounded degree in $\xx$). Using MAGMA [4], we solved this sparse system assuming that the degree of each entry is at most 16 in the variables $\xx$ (in which case there are about 100,000 unknowns and about four times as many equations).
We found an explicit non-singular matrix
$A_{\sss 0}(\xx;u)$ of the form (<ref>) with polynomial
entries in $\xx,u$ satisfying (<ref>), and the interested
reader can find it in [1]. Thus we have the corresponding
extensions of the cocycle $\Lambda$, and of the
Chinta-Gunnells action, to all of $\wW\oplus\Z.$
By (<ref>) it follows that

\[
A_{\sss 0}(\xx;u)\ZZ\!\left(\xx;u\slash \xx^\dd\right)
=A_{\sss 0}(\xx;u) \cdot \sum_{w \, \in \, W}
\ov{1 \vert w}\!\left(\xx; u\slash \xx^\dd\right)
\; =\sum_{w\, \in \, W} \ov{1\vert \tau w}(\xx;u)
=\ov{Z_{\sss W, F}}(\xx;u)
\]

where $F = 1\vert \tau,$ and
Z_{\sss W, F}=\sum_{w\, \in \, W} F\vert w
The function $F$ is the sum of the
entries in the first column of $A_{\sss 0}$, so it is a
polynomial in $\xx$. By Proposition <ref>, it follows
that $Z_{\sss W, F}=a Z_{\sss W}$, with $a\in \CC\!\left(u,\xx^\dd\right)$,
and thus the matrix $A=A_{\sss 0}\slash a$ satisfies all
the conditions in the theorem. Given $A_{\sss 0}$, the constant $a$
can be determined explicitly using the algorithm in the
proof of Proposition <ref>. More explicitly,
the leading term among all the entries of $A_{\sss 0}$ is
$\xx^{16\dd}$ (it occurs in the $(2, 2)$ entry), and the
value of $a$ is given by

\[
a=u^{\sss 8} \xx^{\sss 5\dd} (\xx^{\sss \dd}-u^{\sss 2})^{\sss 3}
(\xx^{\sss \dd}-u^{\sss 4})(\xx^{\sss 3\dd}-u^{\sss 4}). \qedhere
\]

We shall also need the following information about $\Lam_{\tau^{\sss -1}}$.

— The vector function
$\mathbf{Z}(\mathbf{x}; u)$ satisfies the functional equation

\[
\mathbf{Z}(\mathbf{x}; u) = B(\mathbf{x}; u)
\mathbf{Z}\!\left(\mathbf{x}; u \mathbf{x}^{\delta}\right)
\]

for a 3 by 3 matrix $B(\mathbf{x}; u)$ satisfying, in addition,
the following two conditions:

* The matrix

\[
\prod_{\substack{\alpha  \, \in \,
\Phi_{\scalebox{.95}{$\scriptscriptstyle \mathrm{re}$}}^{\scalebox{.95}{$\scriptscriptstyle +$}}
\\ \alpha \, < \, \delta}}
\!\left(1 - u^{2}\mathbf{x}^{2\alpha} \right)
\cdot B(\mathbf{x}; u)
\]

has polynomial entries in $\mathbf{x}$ and $u.$

* Each entry of $B(\mathbf{x}; u)$ is divisible by
\left(1 - u^{2}\mathbf{x}^{\delta}\right)^{\scalebox{1.1}{$\scriptscriptstyle 2$}}\!.

B(\xx;u)=\Lambda_{\tau^{\sss -1}}(\xx;u)
= A^{\sss -1}\!\left(\xx; u\xx^{\dd}\right)
with $A$ from Theorem <ref>. The functional equation
fol-lows at once from the previous theorem, and the
other conditions follow from the explicit formula of $A,$
see [1].

We shall also need:

— The specialization
of the vector function $\mathbf{Z}$ to $u = 0$ is given by
\mathbf{Z}(\mathbf{x}; 0) = \, \!^{t}(\Delta(\mathbf{x}), 0, 0).

By induction on length, for $w \in W,$ we have
1 \vert w(\mathbf{x}; 0)
= (- 1)^{\scalebox{1.1}{$\scriptscriptstyle \ell(w)$}}
\prod_{\beta \, \in \, \Phi(w)} \mathbf{x}^{2\beta},
where we recall that$
\Phi(w) = \Phi^{+} \cap w^{\scalebox{1.1}{$\scriptscriptstyle -1$}}(\Phi^{-}).
Thus our assertion is just Macdonald's identity [44] in type
D_{\scalebox{1.1}{$\scriptscriptstyle 4$}}^{\scalebox{1.1}{$\scriptscriptstyle (1)$}}.$

Note that, by combining the functional equations (<ref>)
and (<ref>), we get

(; u) =Λ_wτ^k(;u)
(w; u ^k)
for all $w\in\wW$ and $k\in\Z$.

§ RENORMALIZATION

The correction of the Chinta-Gunnells average for the affine root
system $D_{\scriptscriptstyle 4}^{\scriptscriptstyle (1)}$ is given

\begin{equation} \label{eq: average-zeta-normalized}
\begin{split}
\tilde{Z}_{\scriptscriptstyle W}(\mathbf{x}; u)
& = \prod_{n \, \ge \, 1}
\!\left(1 - u^{2} \mathbf{x}^{(2n -
1)\delta}\right)^{- 2}
\cdot \, Z_{\scriptscriptstyle W}^{\scriptscriptstyle \mathrm{CG}}(\mathbf{x}; u)\\
& = \frac{1}{\Delta(\mathbf{x})\prod_{n \, \ge \, 1}
\!\left(1 - u^{2} \mathbf{x}^{(2n - 1)\delta}\right)^{\!\scalebox{1.1}{$\scriptscriptstyle 2$}}}
\, \cdot \sum_{w \, \in \, W} 1 \vert w(\mathbf{x}; u).
\end{split}
\end{equation}

As we shall see in Section <ref>, this function is directly
connected to a Weyl group multiple Dirichlet series associated with
the $4$-th moment of quadratic Dirichlet $L$-functions.

Letting $D(\mathbf{x}; u),$ the denominator of
$\tilde{Z}_{\scriptscriptstyle W}(\mathbf{x}; u),$ be defined by

\begin{equation} \label{eq: denominator}
D(\mathbf{x}; u)
: = \prod_{\alpha  \in
\Phi_{\scalebox{.95}{$\scriptscriptstyle \mathrm{re}$}}^{\scalebox{.95}{$\scriptscriptstyle +$}}}
\!\left(1 - u^{2}\mathbf{x}^{2\alpha} \right)
\end{equation}

we can now show the following:

The function $D\tilde{Z}_{\scriptscriptstyle W}(\mathbf{x}; u)$ is holomorphic in the region
$|\mathbf{x}^{\scalebox{1.1}{$\scriptscriptstyle \delta$}}| < 1.$

For $|\mathbf{x}^{\scalebox{1.1}{$\scriptscriptstyle \delta$}}| < 1,$ the absolutely convergent product
\Delta^{\scalebox{1.2}{$\scriptscriptstyle \mathrm{im}$}}(\mathbf{x}) : =
\prod_{n \, \ge \, 1}
\!\left(1 - \mathbf{x}^{2 n \delta}\right)^{\scalebox{1.1}{$\scriptscriptstyle 4$}}
is non-vanishing, and the divisibility (in the obvious sense) of $Z_{\scriptscriptstyle W}$ by
$\Delta^{\scalebox{1.2}{$\scriptscriptstyle \mathrm{re}$}}
=  \Delta \slash
\Delta^{\scalebox{1.2}{$\scriptscriptstyle \mathrm{im}$}}
in this region has already been discussed in <cit.>.

On the other hand, by (<ref>) and (<ref>), the function
$Z_{\scriptscriptstyle W}(\mathbf{x}; u)$ is a sum of rational functions whose denominators are
products of distinct factors of the form $1 - u^{2}\mathbf{x}^{2\alpha}$
\alpha \in \Phi_{\scalebox{1.2}{$\scriptscriptstyle \mathrm{re}$}}^{\scalebox{1.2}{$\scriptscriptstyle +$}}
It follows that $DZ_{\scriptscriptstyle W}\slash \Delta$ is holomorphicwhen
$|\mathbf{x}^{\scalebox{1.1}{$\scriptscriptstyle \delta$}}| < 1,$ and by Theorem <ref>, that
$(D\slash \Delta)\mathbf{Z}(\mathbf{x}; u)$ is divisible by
$\left(1 - u^{2}\mathbf{x}^{\delta}\right)^{\scalebox{1.1}{$\scriptscriptstyle 2$}}\!.$ Accordingly, the vector function
$(D\slash \Delta)\mathbf{Z}\!\left(\mathbf{x}; u \mathbf{x}^{\delta}\right)$ is divisible by
$\left(1 - u^{2}\mathbf{x}^{3\delta}\right)^{\scalebox{1.1}{$\scriptscriptstyle 2$}}\!,$ and by the functional equation

\[
\left(\frac{D}{\Delta}\mathbf{Z}\right)\!(\mathbf{x}; u) \; = \prod_{\substack{\alpha  \, \in \,
\Phi_{\scalebox{.95}{$\scriptscriptstyle \mathrm{re}$}}^{\scalebox{.95}{$\scriptscriptstyle +$}}
\\ \alpha \, < \, \delta}}
\!\left(1 - u^{2}\mathbf{x}^{2\alpha} \right)
\cdot B(\mathbf{x}; u)
\left(\frac{D}{\Delta}\mathbf{Z}\right)\!\left(\mathbf{x}; u \mathbf{x}^{\delta}\right)
\]

so does $(D\slash \Delta)\mathbf{Z}(\mathbf{x}; u).$ Proceeding by induction on $n,$
we see at once that $(D\slash \Delta)\mathbf{Z}(\mathbf{x}; u)$ is divisible by$
\left(1 - u^{2}\mathbf{x}^{(2n - 1)\delta}\right)^{\scalebox{1.1}{$\scriptscriptstyle 2$}}
for all $n \ge 1.$ Thus $DZ_{\scriptscriptstyle W}\slash \Delta$ is also divisible by the product
\prod_{n \, \ge \, 1}
\left(1 - u^{2}\mathbf{x}^{(2n - 1)\delta}\right)^{\scalebox{1.1}{$\scriptscriptstyle 2$}}\!,
which completes the proof.

Z_{\scriptscriptstyle W}
is $W$–invariant under the Chinta-Gunnells action, it follows from
(<ref>) and
the $W$–invariance ofthe
product over imaginary roots that $\tilde{Z}_{\scriptscriptstyle W}$
itself satisfies a functional equation with respect to each
$w \in W$. More precisely, we have:

= w
where we write $\|$ for the action defined on generators by

\[
f\|\ss_{\sss i}(\xx):=-\frac{1}{x_{\sss i}^{\sss 2}}
f\vert \ss_{\sss i}(\xx)
=\frac{1}{x_{\sss i}} f_{\sss i}^{-}(\ss_{\sss i}\xx) +
\frac{1-u\slash x_{\sss i}}{1-u x_{\sss i}} f_{\sss i}^{+}(\ss_{\sss i}\xx).
\]

It is clear that the subspace $\CC(\xx,u)_{\sss 0}$ defined in
Lemma <ref> is invariant under this action.
The functional equations satisfied by the vector $\tZZ:=\ov{\tZW}$ are:

(;u)=_w() (w;u)
where $\tL_{w}(\xx)=\tL_{w}(\xx;u)$ is the 3 by 3 matrix-cocycle such that

f  w(𝐱;u) =
for $w \in W$ and $f\in\CC(\xx,u)_{\sss 0}$.
On generators, $\tL_{w}(\xx)$ is given by
\tL_{\ss_{\scalebox{.75}{$\scriptscriptstyle i$}}}\!(\xx)
=-\frac{1}{x_{\sss i}^{\sss 2}} \Lam_{\ss_{\scalebox{.75}{$\scriptscriptstyle i$}}}\!(\xx)

The following lemma provides some structural properties of the
function $\tilde{Z}_{\scriptscriptstyle W}(\mathbf{x}; u)$ that will
be used to get some analytic information about the Weyl
group multiple Dirichlet series we will introduce in
Section <ref>.  — Set
\underline{x} = (x_{\scalebox{1.1}{$\scriptscriptstyle 1$}}, \ldots, x_{\scalebox{1.1}{$\scriptscriptstyle 4$}}),
\underline{k} = (k_{\scalebox{1.1}{$\scriptscriptstyle 1$}}, \ldots, k_{\scalebox{1.1}{$\scriptscriptstyle 4$}})
and $l = k_{\scalebox{1.1}{$\scriptscriptstyle 5$}}.$ Then we have:

* The function $\tilde{Z}_{\scriptscriptstyle W}(\mathbf{x}; u)$ can be written as

\begin{equation*}
\begin{split}
\tilde{Z}_{\scriptscriptstyle W}(\mathbf{x}; u) \, & = \,
\frac{\sum_{l - \mathrm{even}}
P_{\scalebox{.95}{$\scriptscriptstyle l$}}(\underline{x}; u)
x_{\scalebox{1.1}{$\scriptscriptstyle 5$}}^{\scalebox{1.1}{$\scriptscriptstyle l$}}}
{\prod_{j = 1}^{4} (1 - ux_{\!\scalebox{1.1}{$\scriptscriptstyle j$}})} \; +
\sum_{l - \mathrm{odd}} P_{\scalebox{.95}{$\scriptscriptstyle l$}}(\underline{x}; u)
x_{\scalebox{1.1}{$\scriptscriptstyle 5$}}^{\scalebox{1.1}{$\scriptscriptstyle l$}}\\
& = \, \frac{\sum_{|\underline{k}| - \mathrm{even}}
Q_{\scalebox{1.1}{$\scriptscriptstyle \underline{k}$}}(x_{\scalebox{1.1}{$\scriptscriptstyle 5$}}; u)
\underline{x}^{\scalebox{1.1}{$\scriptscriptstyle \underline{k}$}}}
{1 - u x_{\scalebox{1.1}{$\scriptscriptstyle 5$}}}  \;\;  +
\sum_{|\underline{k}| - \mathrm{odd}}
Q_{\scalebox{1.1}{$\scriptscriptstyle \underline{k}$}}(x_{\scalebox{1.1}{$\scriptscriptstyle 5$}}; u)
\underline{x}^{\scalebox{1.1}{$\scriptscriptstyle \underline{k}$}}
\end{split}
\end{equation*}

where $P_{\scalebox{.95}{$\scriptscriptstyle l$}}(\underline{x}; u)$ and
$Q_{\scalebox{1.1}{$\scriptscriptstyle \underline{k}$}}(x_{\scalebox{1.1}{$\scriptscriptstyle 5$}}; u)$ are
polynomials in $x_{\scalebox{1.1}{$\scriptscriptstyle 1$}}, \ldots, x_{\scalebox{1.1}{$\scriptscriptstyle 4$}}, u$
and $x_{\scalebox{1.1}{$\scriptscriptstyle 5$}}, u,$ respectively.
Here we set $|\underline{k}| = $$k_{\scalebox{1.1}{$\scriptscriptstyle 1$}} + \cdots + k_{\scalebox{1.1}{$\scriptscriptstyle 4$}}.$

* The power series obtained by expanding

\[
\sum_{l \, \ge \, 0} P_{\scalebox{.95}{$\scriptscriptstyle l$}}(\underline{x}; u)
x_{\scalebox{1.1}{$\scriptscriptstyle 5$}}^{\scalebox{1.1}{$\scriptscriptstyle l$}}
\]

is absolutely convergent for arbitrary
$\underline{x}\in \mathbb{C}^{\scriptscriptstyle 4},$ provided
$|x_{\scalebox{1.1}{$\scriptscriptstyle 5$}}|$ is sufficiently small, and the power series obtained by expanding

\[
\sum_{|\underline{k}| \, \ge \, 0}
Q_{\scalebox{1.1}{$\scriptscriptstyle \underline{k}$}}(x_{\scalebox{1.1}{$\scriptscriptstyle 5$}}; u)
\underline{x}^{\scalebox{1.1}{$\scriptscriptstyle \underline{k}$}}
\]

is absolutely convergent for any $x_{\scalebox{1.1}{$\scriptscriptstyle 5$}} \in \mathbb{C},$ provided all
$|x_{\scalebox{1.1}{$\scriptscriptstyle 1$}}|, \ldots, |x_{\scalebox{1.1}{$\scriptscriptstyle 4$}}|$ are sufficiently small.

* We have

\[
P_{\scriptscriptstyle 0}(\underline{x}; u)
= Q_{\scriptscriptstyle \underline{0}}(x_{\scalebox{1.1}{$\scriptscriptstyle 5$}}; u) \equiv 1
\]

where $\underline{0} = (0, \ldots, 0).

* The polynomials $P_{\scalebox{.95}{$\scriptscriptstyle l$}}(\underline{x}; u)$ are symmetric in
$\underline{x},$ and if $l$ is odd then $P_{\scalebox{.95}{$\scriptscriptstyle l$}}(\underline{x}; u)$ is even, i.e.,
P_{\scalebox{.95}{$\scriptscriptstyle l$}}(\underline{x}; u) = P_{\scalebox{.95}{$\scriptscriptstyle l$}}(-\underline{x}; u).

* We have the functional equations

\begin{equation} \label{eq: poly-P-Q-func-eq}
P_{\scalebox{.95}{$\scriptscriptstyle l$}}(x_{\scalebox{1.1}{$\scriptscriptstyle 1$}},
x_{\scalebox{1.1}{$\scriptscriptstyle 2$}},
x_{\scalebox{1.1}{$\scriptscriptstyle 3$}}, x_{\scalebox{1.1}{$\scriptscriptstyle 4$}}; u)
= x_{\scalebox{1.1}{$\scriptscriptstyle 1$}}^{\scriptscriptstyle l - \delta_{\scalebox{.75}{$\scriptscriptstyle l$}}}
P_{\scalebox{.95}{$\scriptscriptstyle l$}}\bigg(\frac{1}{x_{\scalebox{1.1}{$\scriptscriptstyle 1$}}},
x_{\scalebox{1.1}{$\scriptscriptstyle 2$}},
x_{\scalebox{1.1}{$\scriptscriptstyle 3$}},
x_{\scalebox{1.1}{$\scriptscriptstyle 4$}}; u\bigg)
\;\;\;\; \mathrm{and} \;\;\;\;
Q_{\scalebox{1.1}{$\scriptscriptstyle \underline{k}$}}(x_{\scalebox{1.1}{$\scriptscriptstyle 5$}}; u)
= x_{\scalebox{1.1}{$\scriptscriptstyle 5$}}^{\scriptscriptstyle |\underline{k}| - \delta_{\scalebox{.75}
{$\scriptscriptstyle |\underline{k}|$}}}
Q_{\scalebox{1.1}{$\scriptscriptstyle \underline{k}$}}\bigg(\frac{1}{x_{\scalebox{1.1}{$\scriptscriptstyle 5$}}}; u\bigg)
\end{equation}

with $\delta_{\scalebox{1.2}{$\scriptscriptstyle n$}} \! = 0$ or $1$ according as $n$ is even or odd.

For notational simplicity, we denote
$\tZ(\xx):=\tZW(\xx;u)$ in this proof (the variable $u$ being
fixed). The functional equation of $\tZ(\xx)$ can be
broken into its even and odd parts, according to $\e_{\sss i}$, as:

\[
\tZ^+_{\sss i}\!(\xx)=\frac{1-u\slash x_{\sss i}}{1-ux_{\sss i}}\tZ^+_{\sss i}\!(\ss_{\sss i}\xx),
\quad \tZ^-_{\sss i}\!(\xx)=\frac{1}{x_{\sss i}} \tZ^-_{\sss i}\!(\ss_{\sss i}\xx)
\]

for $i=1,\ldots, 5$. Note that $\tZ^\pm_{\sss i}=\tZ^\pm_{\sss 1}$ for
$i=1,\ldots, 4$. The even functional equations can
be expressed in terms of the functions

\[
G_{\sss 1}(\xx)=\prod_{i=1}^{4} (1-ux_{\sss i})
\tZ^+_{\sss 1}\!(\xx), \quad
G_{\sss 5}(\xx)=(1-ux_{\sss 5})
\tZ^+_{\sss 5}\!(\xx)
\]

as $G_{\sss 1}(\xx)=G_{\sss 1}(\ss_{\sss i}\xx)$ for $i=1, \ldots, 4$,
and $G_{\sss 5}(\xx)=G_{\sss 5}(\ss_{\sss 5}\xx)$.

By Theorem <ref>,
the function $D\tZ$ is holomorphic in
\Omega : =\{\mathbf{x} \in \mathbb{C}^{5} : | \mathbf{x}^{\scalebox{1.1}{$\scriptscriptstyle \delta$}}| < 1\},
and in this domain, it satisfies the functional equations

\[
D\tZ^+_{\sss i}\!(\xx) =
\frac{x_{\scalebox{1.2}{$\scriptscriptstyle i$}}
(1 + u x_{\scalebox{1.2}{$\scriptscriptstyle i$}})}
{u +  x_{\scalebox{1.2}{$\scriptscriptstyle i$}}}
D\tZ^+_{\sss i}\!(\ss_{\sss i}\xx), \quad
D\tilde{Z}^-_{\sss i}\!(\mathbf{x}) = - \,
\frac{x_{\scalebox{1.2}{$\scriptscriptstyle i$}}
(1 - u^{\scalebox{1.2}{$\scriptscriptstyle 2$}}
x_{\scalebox{1.2}{$\scriptscriptstyle i$}}^{\scalebox{1.2}{$\scriptscriptstyle 2$}})}
{u^{\scalebox{1.2}{$\scriptscriptstyle 2$}}
- x_{\scalebox{1.2}{$\scriptscriptstyle i$}}^{\scalebox{1.2}{$\scriptscriptstyle 2$}}}
D\tilde{Z}^-_{\sss i}\!(\ss_{\sss i}\xx )
\]

for $i=1,\ldots, 5$, which imply that
D\tilde{Z}^+_{\sss i}\!(\mathbf{x})
D\tilde{Z}^-_{\sss i}\!(\mathbf{x})
are divisible by $1 + u x_{\scalebox{1.2}{$\scriptscriptstyle i$}}$ and
1 - u^{\scalebox{1.2}{$\scriptscriptstyle 2$}}
x_{\scalebox{1.2}{$\scriptscriptstyle i$}}^{\scalebox{1.2}{$\scriptscriptstyle 2$}},
respectively. Thus the functions

\[
F_{\scalebox{1.1}{$\scriptscriptstyle 1$}}^+\!(\mathbf{x})\,  : =
\, \prod_{\substack{\alpha \in
\Phi_{\scalebox{.95}{$\scriptscriptstyle \mathrm{re}$}}^{\scalebox{.95}{$\scriptscriptstyle +$}} \\
\alpha \, \ne \, \alpha_{\scalebox{.7}{$\scriptscriptstyle 1$}}, \ldots,\alpha_{\scalebox{0.7}{$\scriptscriptstyle 4$}}}}
\!\left(1 - u^{2}\mathbf{x}^{2\alpha} \right)
\cdot G_{\sss 1}(\mathbf{x}),
\quad
F_{\scalebox{1.1}{$\scriptscriptstyle 1$}}^-\!(\mathbf{x})\,  : = \,
\prod_{\substack{\alpha  \in
\Phi_{\scalebox{.95}{$\scriptscriptstyle \mathrm{re}$}}^{\scalebox{.95}{$\scriptscriptstyle +$}} \\
\alpha \, \ne \, \alpha_{\scalebox{.7}{$\scriptscriptstyle 1$}}, \ldots, \alpha_{\scalebox{0.7}{$\scriptscriptstyle 4$}}}}
\!\left(1 - u^{2}\mathbf{x}^{2\alpha} \right)
\cdot \tilde{Z}^-_{\sss 1}\!(\mathbf{x})
\]

\[
H_{\scalebox{1.1}{$\scriptscriptstyle 5$}}^+\!(\mathbf{x})\,  : =
\, \prod_{\substack{\alpha  \in
\Phi_{\scalebox{.95}{$\scriptscriptstyle \mathrm{re}$}}^{\scalebox{.95}{$\scriptscriptstyle +$}} \\
\alpha \, \ne \, \alpha_{\scalebox{.7}{$\scriptscriptstyle 5$}}}}
\!\left(1 - u^{2}\mathbf{x}^{2\alpha} \right)
\cdot G_{\sss 5}(\mathbf{x}),
\quad
H_{\scalebox{1.1}{$\scriptscriptstyle 5$}}^-\!(\mathbf{x})\,  : =
\, \prod_{\substack{\alpha  \in
\Phi_{\scalebox{.95}{$\scriptscriptstyle \mathrm{re}$}}^{\scalebox{.95}{$\scriptscriptstyle +$}} \\
\alpha \, \ne \, \alpha_{\scalebox{.7}{$\scriptscriptstyle 5$}}}}
\!\left(1 - u^{2}\mathbf{x}^{2\alpha} \right)
\cdot \tilde{Z}^-_{\sss 5}\!(\mathbf{x})
\]

are still holomorphic in $\Omega.$ Notice that, for $i = 1, \ldots, 4,$
$F_{\scalebox{1.1}{$\scriptscriptstyle 1$}}^+\!(\mathbf{x})$ is
$\sigma_{\scriptscriptstyle i}$-invariant,
F_{\scalebox{1.1}{$\scriptscriptstyle 1$}}^-\!(\mathbf{x})
= x_{\scalebox{1.1}{$\scriptscriptstyle i$}}^{\scalebox{1.1}{$\scriptscriptstyle -1$}}
F_{\scalebox{1.1}{$\scriptscriptstyle 1$}}^-\!(\sigma_{\scriptscriptstyle i}\mathbf{x}),
and since $\sigma_{\scriptscriptstyle i}$ is just permuting the roots in
\Phi_{\scalebox{1.2}{$\scriptscriptstyle \mathrm{re}$}}^{\scalebox{1.2}{$\scriptscriptstyle +$}}
\setminus
\{\alpha_{\scalebox{1.1}{$\scriptscriptstyle 1$}}, \ldots, \alpha_{\scalebox{1.1}{$\scriptscriptstyle 4$}}\},
the product in the definition of
$F_{\scalebox{1.1}{$\scriptscriptstyle 1$}}^\pm$ is also $\sigma_{\scriptscriptstyle i}$-invariant. By expanding the inverse of this product, and
$F_{\scalebox{1.1}{$\scriptscriptstyle 1$}}^\pm\!(\mathbf{x})$ in
power series, we can write

\[
G_{\sss 1}(\xx)
\; = \sum_{l - \mathrm{even}} P_{\scalebox{.95}{$\scriptscriptstyle l$}}(\underline{x}; u)
x_{\scalebox{1.1}{$\scriptscriptstyle 5$}}^{\scalebox{1.1}{$\scriptscriptstyle l$}}, \quad
\tilde{Z}_{\scriptscriptstyle 1}^{-}\!(\mathbf{x})
\; = \sum_{l - \mathrm{odd}} P_{\scalebox{.95}{$\scriptscriptstyle l$}}(\underline{x}; u)
x_{\scalebox{1.1}{$\scriptscriptstyle 5$}}^{\scalebox{1.1}{$\scriptscriptstyle l$}};
\]

these expansions hold as long as $|\mathbf{x}^{\alpha}| < |u|^{\scalebox{1.1}{$\scriptscriptstyle -1$}}$ for all
\alpha \in \Phi_{\scalebox{1.2}{$\scriptscriptstyle \mathrm{re}$}}^{\scalebox{1.2}{$\scriptscriptstyle +$}}
\setminus
\{\alpha_{\scalebox{1.1}{$\scriptscriptstyle 1$}}, \ldots, \alpha_{\scalebox{1.1}{$\scriptscriptstyle 4$}}\}
(e.g., $\underline{x} \in \mathbb{C}^{\scriptscriptstyle 4}$ is arbitrary,
and $|x_{\scalebox{1.1}{$\scriptscriptstyle 5$}}|$ is sufficiently small),
which justifies the first part (i.e., the $P$-part) of 2. The functional
equation (<ref>) of the coefficients $P_{\scalebox{.95}{$\scriptscriptstyle l$}}(\underline{x}; u)$ now follows from the $\sigma_{\scriptscriptstyle 1}$-invariance
of $G_{\sss 1}$ and the functional equation
\tilde{Z}_{\sss 1}^{-}\!(\mathbf{x})
= x_{\scalebox{1.1}{$\scriptscriptstyle 1$}}^{\scalebox{1.1}{$\scriptscriptstyle -1$}}
\tilde{Z}_{\sss 1}^{-}\!(\sigma_{\scriptscriptstyle 1}\mathbf{x}).
It follows that
P_{\scalebox{.95}{$\scriptscriptstyle l$}}(\underline{x}; u)
are polynomials in
x_{\scalebox{1.1}{$\scriptscriptstyle 1$}}, \ldots,
x_{\scalebox{1.1}{$\scriptscriptstyle 4$}}, u,
and our assertions 4 follow from the fact that
$\tZ_{\sss 1}^-\!(\xx)=\tZ_{\sss 1}^-\!(\e_{\sss 5}\xx)$ (as $\tZ^{\oo}=0$),
and from the symmetry of
\tilde{Z}_{\sss 1}^{+}\!(\mathbf{x})
\tilde{Z}_{\sss 1}^{-}\!(\mathbf{x})
with respect to$
x_{\scalebox{1.1}{$\scriptscriptstyle 1$}}, \ldots, x_{\scalebox{1.1}{$\scriptscriptstyle 4$}}.

The $Q$-parts of 1, 2 and 5 follow from the same argument, applied
to $H_{\sss 5}^+$ and $H_{\sss 5}^-$.

Finally, notice that

\[
\tilde{Z}(\underline{x}, 0) =
\Delta(\underline{x}, 0)^{\scalebox{1.1}{$\scriptscriptstyle - 1$}}
\, \cdot \sum_{w \, \in \, \langle \sigma_{\scalebox{.75}{$\scriptscriptstyle i$}}
\rangle_{\scalebox{.75}{$\scriptscriptstyle 1\le i \le 4$}}} 1 \vert w(\underline{x}, 0)
= \prod_{j = 1}^{4} (1 - ux_{\!\scalebox{1.1}{$\scriptscriptstyle j$}})^{\scalebox{1.1}{$\scriptscriptstyle - 1$}}
\]

\[
\tilde{Z}(\underline{0}, x_{\scalebox{1.1}{$\scriptscriptstyle 5$}} ) =
\Delta(\underline{0}, x_{\scalebox{1.1}{$\scriptscriptstyle 5$}})^{\scalebox{1.1}{$\scriptscriptstyle - 1$}}
(1 + 1 \vert \sigma_{\scriptscriptstyle 5}(x_{\scalebox{1.1}{$\scriptscriptstyle 5$}} ))
= (1 - ux_{\scalebox{1.1}{$\scriptscriptstyle 5$}})^{\scalebox{1.1}{$\scriptscriptstyle - 1$}}
\]

which give 3. This completes the proof.

The extra functional equation satisfied by $\tZZ(\xx;u)$ is

(;u)=(;u) (;u^)
where $\tB(\xx;u)=B(\xx;u)\slash \big(1-u^{\sss 2}x^\dd\big)^{\sss 2},$
with the matrix $B(\xx;u)$ from Theorem <ref>. Using
this functional equation, we now show that $\tZZ(\xx;u)$ is
completely determined by $\tZZ(\xx;0)$ and $\tB(\xx;u).$
By contrast, thefunctional equations (<ref>) alone determine $\tZZ(\xx;u)$ only up to a power
series in $u$ and $\xx^\dd$ (see <cit.>).
Expand the vector function $\tZZ(\mathbf{x}; u)$ as

\[
\tZZ(\mathbf{x}; u)
= \tZZ_{\scalebox{1.1}{$\scriptscriptstyle 0$}}(\mathbf{x})
+ u\tZZ_{\scalebox{1.1}{$\scriptscriptstyle 1$}}(\mathbf{x})
+ u^{\scalebox{1.1}{$\scriptscriptstyle 2$}}\tZZ_{\scalebox{1.1}{$\scriptscriptstyle 2$}}(\mathbf{x}) + \cdots
\]

and let
\tB(\mathbf{x}; u) =
\tB_{\scalebox{1.1}{$\scriptscriptstyle 0$}}(\mathbf{x})
+ u\tB_{\scalebox{1.1}{$\scriptscriptstyle 1$}}(\mathbf{x}) + \cdots
$. Then we have

\[
\tZZ_{\scalebox{1.1}{$\scriptscriptstyle 0$}}(\mathbf{x})
\end{pmatrix},\quad
\tB_{\scalebox{1.1}{$\scriptscriptstyle 0$}}(\xx)
=\tB_{\scalebox{1.1}{$\scriptscriptstyle 0$}} \!:=
\begin{pmatrix}
1 & 0 & 0\\
0 & 0 & 0\\
0 & 0 & 0\\
\end{pmatrix}
\]

and the matrix $\tB(\mathbf{x}; u)$ determines $\tZZ(\mathbf{x}; u)$
recursively by

_n() = (I-^n _1.1$\scriptscriptstyle 0$)^-1
·∑_i=0^n-1 ^i_n-i() _i()
for $n\ge 1$, where $I$ is the identity matrix.

The expression for
\tZZ_{\scalebox{1.1}{$\scriptscriptstyle 0$}}(\mathbf{x})
follows at once from Lemma <ref>, and the
explicit formula of the matrix $B(\xx;u)$ yields
\tB_{\scalebox{1.1}{$\scriptscriptstyle 0$}}.
The recursion of $\tZZ_{n}(\xx)$ follows at once from (<ref>).

The following corollary will be needed in the proof of the main
theorem in Section <ref>.
— We have that

\[
\tilde{Z}_{\scriptscriptstyle W}(\mathbf{x}; u)
\!\!\!\pmod{u^{\scalebox{1.1}{$\scriptscriptstyle 2$}}}
= 1+ u(x_{\scalebox{1.1}{$\scriptscriptstyle 1$}}
+ x_{\scalebox{1.1}{$\scriptscriptstyle 2$}}
+ x_{\scalebox{1.1}{$\scriptscriptstyle 3$}}
+ x_{\scalebox{1.1}{$\scriptscriptstyle 4$}}
+ x_{\scalebox{1.1}{$\scriptscriptstyle 5$}}).
\]

From the explicit formula of the matrix $\tB(\mathbf{x}; u)$, one
finds that

\[
\tB_{\scalebox{1.1}{$\scriptscriptstyle 1$}}(\mathbf{x}) =
\begin{pmatrix}
0 & \mathbf{x}^{\delta}x_{\scalebox{1.1}{$\scriptscriptstyle 5$}}^{\scalebox{1.1}{$\scriptscriptstyle - 1$}} &
\mathbf{x}^{\delta}
\!\left(x_{\scalebox{1.1}{$\scriptscriptstyle 1$}}^{\scalebox{1.1}{$\scriptscriptstyle - 1$}}
+ x_{\scalebox{1.1}{$\scriptscriptstyle 2$}}^{\scalebox{1.1}{$\scriptscriptstyle - 1$}}
+ x_{\scalebox{1.1}{$\scriptscriptstyle 3$}}^{\scalebox{1.1}{$\scriptscriptstyle - 1$}}
+ x_{\scalebox{1.1}{$\scriptscriptstyle 4$}}^{\scalebox{1.1}{$\scriptscriptstyle - 1$}}\right)\\
x_{\scalebox{1.1}{$\scriptscriptstyle 5$}}^{} & 0 & 0\\
x_{\scalebox{1.1}{$\scriptscriptstyle 1$}}^{}
+ x_{\scalebox{1.1}{$\scriptscriptstyle 2$}}^{}
+ x_{\scalebox{1.1}{$\scriptscriptstyle 3$}}^{}
+ x_{\scalebox{1.1}{$\scriptscriptstyle 4$}}^{} & 0 & 0\\
\end{pmatrix}.
\]

The formula now follows from (<ref>) with $n=1$,
after multiplying by the row vector $(1, 1, 1).$

We shall also need a positivity result about the specializations
$P_{\scalebox{.95}{$\scriptscriptstyle l$}}(\u1;u)$ of the polynomials
in Lemma <ref>, with $\u1=(1,1,1,1)$.
— The polynomial
$P_{\scalebox{.95}{$\scriptscriptstyle l$}} (\u1; u)$ for $l$ odd,
and the power series
P_{\scalebox{.95}{$\scriptscriptstyle l$}}(\u1; u )\slash (1-u)^{4}
for $l$ even have non-negative coefficients.

By Proposition <ref>, the entries of the
matrix $\tB(\u1,x_{\sss 5}; u)$, as power series in
$x_{\sss 5}$ and $u$, have non-negative
coefficients. Thus the matrices $\tB_{n}(\u1, x_{\sss 5})$
have entries with non-negative coefficients, and by
in-duction using (<ref>), the same is true for the vectors $\tZZ_{n}(\u1, x_{\sss 5})$.
Accordingly, the function
\tZW(\underline{1},x_{\sss 5};u)
has non-negative coefficients when expanded as a power series,
and now Lemma <ref> finishes the proof.

§ RESIDUES

By Theorem <ref>, the
singularities of the function $\tilde{Z}_{\scriptscriptstyle W}(\mathbf{x}; u)$
can only occur at the zeros of the denominator $D(\mathbf{x}; u),$ and
thus $\tilde{Z}_{\scriptscriptstyle W}(\mathbf{x}; u)$ can only have
simple poles. Our goal is now to compute the residue of this function at each of
these poles.

The following theorem provides the key calculation:
— The residue of
$\tilde{Z}_{\scriptscriptstyle W}(\mathbf{x}; u)$ at
x_{\scalebox{1.1}{$\scriptscriptstyle 5$}} = u^{\scalebox{1.1}{$\scriptscriptstyle -1$}}
is given by the formula

\begin{equation*}
\begin{split}
R(\underline{x};u) : & = \lim_{x_{\scalebox{.75}{$\scriptscriptstyle 5$}} \to \scalebox{1.}{$\scriptscriptstyle 1\slash $}u} (1 - u x_{\scalebox{1.1}{$\scriptscriptstyle 5$}})\tilde{Z}_{\scriptscriptstyle W}(\mathbf{x}; u) \\
&= \frac{1}{\left(P_{\phantom i}^{\scalebox{1.1}{$\scriptscriptstyle 2$}};
P_{\phantom i}^{\scalebox{1.1}{$\scriptscriptstyle 2$}}\right)_{\infty}
\!\left(u_{\phantom i}^{\scalebox{1.1}{$\scriptscriptstyle 2$}}
P_{\phantom i}^{\scalebox{1.1}{$\scriptscriptstyle 2$}};
P_{\phantom i}^{\scalebox{1.1}{$\scriptscriptstyle 2$}}\right)_{\infty}
\prod_{\scalebox{1.1}{$\scriptscriptstyle i = 1$}}^{\scalebox{1.1}{$\scriptscriptstyle 4$}}
\left(x_{\scalebox{1.1}{$\scriptscriptstyle i$}}^{\scalebox{1.1}{$\scriptscriptstyle 2$}};
P_{\phantom i}^{\scalebox{1.1}{$\scriptscriptstyle 2$}}\right)_{\infty}
\!\left(u_{\phantom i}^{\scalebox{1.1}{$\scriptscriptstyle 2$}}
x_{\scalebox{1.1}{$\scriptscriptstyle i$}}^{\scalebox{1.1}{$\scriptscriptstyle - 2$}}
P_{\phantom i}^{\scalebox{1.1}{$\scriptscriptstyle 2$}};
P_{\phantom i}^{\scalebox{1.1}{$\scriptscriptstyle 2$}}\right)_{\infty}
\cdot \;\,  \prod_{\scalebox{1.1}{$\scriptscriptstyle 1\, \le \, i \, < \, j \, \le \, 4$}}
\left(x_{\scalebox{1.1}{$\scriptscriptstyle i$}}
x_{\scalebox{1.1}{$\scriptscriptstyle j$}}; P\right)_{\infty}}
\end{split}
\end{equation*}

P = (x_{\scalebox{1.1}{$\scriptscriptstyle 1$}}
x_{\scalebox{1.1}{$\scriptscriptstyle 2$}}
x_{\scalebox{1.1}{$\scriptscriptstyle 3$}}
x_{\scalebox{1.1}{$\scriptscriptstyle 4$}})\slash u^{\scalebox{1.1}{$\scriptscriptstyle 2$}}
and $(a; b)_{\infty} = \prod_{k\ge 0}\, (1 - ab^{\scalebox{1.1}{$\scriptscriptstyle k$}})$ is the
$b$-Pochhammer symbol.

The proof of this theorem will be given in the rest of this section.
Using results in [30], similar ideas can beused to prove
explicit formulas for multiple residues of the average zeta
function of arbitrary simply laced reduced irreducible affine root

To compute the residue at any other pole of
$\tilde{Z}_{\scriptscriptstyle W}(\mathbf{x}; u),$ let $\alpha$ be an
affine positive real root, and write it as$
\alpha = \sum_{i} n_{\scriptscriptstyle i}\alpha_{\scriptscriptstyle
in terms of simple roots. Let
w_{\scalebox{1.1}{$\scriptscriptstyle \alpha$}}
be an element of the Weyl group sending $\alpha$ to
$\alpha_{\scriptscriptstyle 5},$ and if $\alpha \ne
\alpha_{\scriptscriptstyle i},$ hence $n_{\scriptscriptstyle 5} \ge
1,$ let $\zeta$ be a
$2n_{\scriptscriptstyle 5}$-th root of $1$ in $\mathbb{C}.$
C_{\alpha,\,  \scalebox{1.1}{$\scriptscriptstyle \zeta$}}(\underline{x};u)
denote the limit

\[
C_{\alpha,\,  \scalebox{1.1}{$\scriptscriptstyle
\zeta$}}(\underline{x};u) : = \hskip-44pt
\lim_{\hskip44pt x_{\scalebox{.75}{$\scriptscriptstyle 5$}}\to
\zeta^{\scalebox{1.0}{$\scriptscriptstyle -1$}}(u
\mathbf{x}^{\alpha \scalebox{.75}{$\scriptscriptstyle '$}})^{\scalebox{1.0}{$\scriptscriptstyle -1$}\slash n_{\scalebox{.75}{$\scriptscriptstyle 5$}}}}
\!\left(1 -
\zeta u^{1\slash n_{\scalebox{.75}{$\scriptscriptstyle 5$}}}
\mathbf{x}^{\alpha \slash n_{\scalebox{.75}{$\scriptscriptstyle 5$}}}\right)
\!\tilde{Z}_{\scriptscriptstyle W}(\mathbf{x}; u)
\]

\alpha' : = \alpha - n_{\scriptscriptstyle
5}\alpha_{\scriptscriptstyle 5}
Here, we chose the principal branch of the complex logarithm, and so
C_{\alpha,\,  \scalebox{1.1}{$\scriptscriptstyle \zeta$}}(\underline{x};u)
is well-defined and analytic at least
when $x_{\scalebox{1.1}{$\scriptscriptstyle 1$}}, \ldots,
x_{\scalebox{1.1}{$\scriptscriptstyle 4$}}$ and $u$ are away from the
non-positive real axis.
— With notation as above, we have

\begin{equation} \label{eq: residue-general}
C_{\alpha,\,  \scalebox{1.1}{$\scriptscriptstyle \zeta$}}(\underline{x};u)
= \frac{1}{2n_{\scalebox{1.1}{$\scriptscriptstyle 5$}}}
\!\left.
\left\{R(w_{\scalebox{1.1}{$\scriptscriptstyle \alpha$}}\underline{x};u) \cdot
f_{w_{\scalebox{1.1}{$\scriptscriptstyle \alpha$}}}
\right\vert_{x_{\scalebox{.75}{$\scriptscriptstyle 5$}}\,=\,
\zeta^{\scalebox{1.0}{$\scriptscriptstyle -1$}}(u
\mathbf{x}^{\alpha \scalebox{.75}{$\scriptscriptstyle
'$}})^{\scalebox{1.0}{$\scriptscriptstyle -1$}\slash
n_{\scalebox{.75}{$\scriptscriptstyle 5$}}}}
\end{equation}

where $f_w=(1+ux_{\scalebox{1.1}{$\scriptscriptstyle 5$}})\| w$ and
w_{\scalebox{1.1}{$\scriptscriptstyle \alpha$}}
\underline{x}
represents the first four components of
w_{\scalebox{1.1}{$\scriptscriptstyle \alpha$}}
\mathbf{x}.

By applying the functional equation (<ref>) of the vector function $\tZZ =\ov{\tZW}$ corresponding to
$w=\ss_{\sss 5},$ one finds that

\[
\lim_{x_{\scalebox{.75}{$\scriptscriptstyle 5$}} \to e \scalebox{1.}{$\scriptscriptstyle /  $} u}
(1 -e u x_{\scalebox{1.1}{$\scriptscriptstyle 5$}}) \mathbf{\tilde{Z}}(\mathbf{x}; u)=
\frac 12 R(\underline{x};u)\cdot \, \!^{t}(1, e, 0)
\]

where $e\in \{-1,1\}$. Applying (<ref>) again, for
w=w_{\scalebox{1.1}{$\scriptscriptstyle \alpha$}},
we obtain

\[
\hskip-44pt \lim_{\hskip44pt x_{\scalebox{.75}{$\scriptscriptstyle 5$}}\to
\zeta^{\scalebox{1.0}{$\scriptscriptstyle -1$}}(u
\mathbf{x}^{\alpha \scalebox{.75}{$\scriptscriptstyle '$}})^{\scalebox{1.0}{$\scriptscriptstyle -1$}\slash n_{\scalebox{.75}{$\scriptscriptstyle 5$}}}}
\!\left(1 -
\zeta u^{1\slash n_{\scalebox{.75}{$\scriptscriptstyle 5$}}}
\mathbf{x}^{\alpha \slash n_{\scalebox{.75}{$\scriptscriptstyle 5$}}}\right)
\tZZ(\xx;u) = \frac{1}{2n_{\scalebox{1.1}{$\scriptscriptstyle 5$}}}
\!\left.
\left\{R(w_{\scalebox{1.1}{$\scriptscriptstyle \alpha$}}
\underline{x};u) \tL_{w_\a}\!(\xx)\right\}
\right\vert_{x_{\scalebox{.75}{$\scriptscriptstyle 5$}}\,=\,
\zeta^{\scalebox{1.0}{$\scriptscriptstyle -1$}}(u
\mathbf{x}^{\alpha \scalebox{.75}{$\scriptscriptstyle
'$}})^{\scalebox{1.0}{$\scriptscriptstyle -1$}\slash
n_{\scalebox{.75}{$\scriptscriptstyle 5$}}}} \cdot
\,\!^{t}(1, \zeta^{n_{\scalebox{.75}{$\scriptscriptstyle 5$}}}\!, 0).
\]

The formula in the lemma now follows using (<ref>),
after multiplying on the left by the vector $(1,1,1)$.

If $\alpha = \alpha_{\scriptscriptstyle i},$ for some $1\le i \le 4,$
the residue of $\tilde{Z}_{\scriptscriptstyle W}(\mathbf{x}; u)$ at
$x_{\scalebox{1.1}{$\scriptscriptstyle i$}} = 1\slash u$ can be
computed similarly from the functional equation (<ref>)
w_{\scalebox{1.1}{$\scriptscriptstyle \alpha$}} =
\sigma_{\scriptscriptstyle i}\sigma_{\scriptscriptstyle 5},
which sends the simple root $\alpha_{\scriptscriptstyle i}$ to
$\alpha_{\scriptscriptstyle 5}.$

§.§ Proof of Theorem <ref>

We start by recalling a result from [16], which identifies the
residue $R(\ux; u)$ up to a function depending only upon
P = (x_{\scalebox{1.1}{$\scriptscriptstyle 1$}}
x_{\scalebox{1.1}{$\scriptscriptstyle 2$}}
x_{\scalebox{1.1}{$\scriptscriptstyle 3$}}
x_{\scalebox{1.1}{$\scriptscriptstyle 4$}})\slash u^{\scalebox{1.1}{$\scriptscriptstyle 2$}}
and $u$.
— We have

\[
R(\ux; u) = f(P, u)\cdot
\frac{1}{\prod_{\scalebox{1.1}{$\scriptscriptstyle i = 1$}}^{\scalebox{1.1}{$\scriptscriptstyle 4$}}
\left(x_{\scalebox{1.1}{$\scriptscriptstyle i$}}^{\scalebox{1.1}{$\scriptscriptstyle 2$}};
P_{\phantom i}^{\scalebox{1.1}{$\scriptscriptstyle 2$}}\right)_{\infty}
\!\left(u_{\phantom i}^{\scalebox{1.1}{$\scriptscriptstyle 2$}}
x_{\scalebox{1.1}{$\scriptscriptstyle i$}}^{\scalebox{1.1}{$\scriptscriptstyle - 2$}}
P_{\phantom i}^{\scalebox{1.1}{$\scriptscriptstyle 2$}};
P_{\phantom i}^{\scalebox{1.1}{$\scriptscriptstyle 2$}}\right)_{\infty}
\cdot \;\,  \prod_{\scalebox{1.1}{$\scriptscriptstyle 1\, \le \, i \, < \, j \, \le \, 4$}}
\left(x_{\scalebox{1.1}{$\scriptscriptstyle i$}}
x_{\scalebox{1.1}{$\scriptscriptstyle j$}}; P\right)_{\infty}}
\]

where the function $f(P, u)$ is meromorphic in the region $|P|<1$,
with possible poles only when $P^{n}=\pm u^{\sss -2}$
forsome $n\ge 1$. In addition, if we set
\mathrm{z} :=
x_{\scalebox{1.1}{$\scriptscriptstyle 1$}}
x_{\scalebox{1.1}{$\scriptscriptstyle 2$}}
x_{\scalebox{1.1}{$\scriptscriptstyle 3$}}
x_{\scalebox{1.1}{$\scriptscriptstyle 4$}}
then for every $\epsilon > 0$, the function
f(u^{\scalebox{1.1}{$\scriptscriptstyle 2$}}\mathrm{z}, u)
is holomorphic inthe polydisc
|\mathrm{z}| < (1 + \epsilon)^{\sss -4},\, |u| < 1 + \epsilon.

For $|P|$ sufficiently small, the decomposition follows
from <cit.>. From
Theorem <ref>, itfollows that the function
\left.\left[D(\xx;u)\slash (1-u^{\sss 2}x_{\sss 5}^{\sss 2})\right]\right
\vert_{x_{\scalebox{.75}{$\scriptscriptstyle 5$}}=\scalebox{1.}{$\scriptscriptstyle 1\slash $}u} \cdot
is holomorphic in the region $|P|<1$, $u\ne 0$; when $|P|$ is small,
we can express this function as

\[
%\left. \frac{D(\xx;u)}{1-u^{\sss 2}x_{\sss 5}^{\sss 2}}\right|_{x_{\sss 5}=u^{\sss -1}} \cdot R(\ux;u)=
f(P,u) \left(P_{\phantom i}^{\scalebox{1.1}{$\scriptscriptstyle 2$}};
P_{\phantom i}^{\scalebox{1.1}{$\scriptscriptstyle 2$}}\right)_{\infty}
\!\left(u_{\phantom i}^{\scalebox{1.1}{$\scriptscriptstyle 4$}}
P_{\phantom i}^{\scalebox{1.1}{$\scriptscriptstyle 2$}};
P_{\phantom i}^{\scalebox{1.1}{$\scriptscriptstyle 2$}}\right)_{\infty}
\prod_{\scalebox{1.1}{$\scriptscriptstyle i = 1$}}^{\scalebox{1.1}{$\scriptscriptstyle 4$}}
\left(u_{\phantom i}^{\scalebox{1.1}{$\scriptscriptstyle 2$}}
x_{\scalebox{1.1}{$\scriptscriptstyle i$}}^{\scalebox{1.1}{$\scriptscriptstyle 2$}};
P_{\phantom i}^{\scalebox{1.1}{$\scriptscriptstyle 2$}}\right)_{\infty}
\!\left(u_{\phantom i}^{\scalebox{1.1}{$\scriptscriptstyle 4$}}
x_{\scalebox{1.1}{$\scriptscriptstyle i$}}^{\scalebox{1.1}{$\scriptscriptstyle - 2$}}
P_{\phantom i}^{\scalebox{1.1}{$\scriptscriptstyle 2$}};
P_{\phantom i}^{\scalebox{1.1}{$\scriptscriptstyle 2$}}\right)_{\infty}
\, \cdot \,  \prod_{\scalebox{1.1}{$\scriptscriptstyle 1\, \le \, i \, < \, j \, \le \, 4$}}
\left(-x_{\scalebox{1.1}{$\scriptscriptstyle i$}}
x_{\scalebox{1.1}{$\scriptscriptstyle j$}}; P\right)_{\infty}.
\]

However, the product of the Pochhammer symbols is holomorphic when
$|P|<1$, and if in addition we take$P^{n}\ne \pm u^{\sss -2}$ for all $n\ge
1$, we can choose $x_{\sss i}\in \CC$ $(i = 1, \ldots, 4)$
such that this product is non-zero. It follows that indeed
$f(P,u)$ is holomorphic for $|P|<1$ except for possible
poles when $P^{n}=\pm u^{\sss -2}$ for some $n\ge 1$,
and that the decomposition of $R(\ux; u)$ extends by
analytic continuation.

On the other hand, by Lemma <ref>,
part 1, and the functional equation
(<ref>) of the polynomials
Q_{\scalebox{1.1}{$\scriptscriptstyle \underline{k}$}}(x_{\scalebox{1.1}{$\scriptscriptstyle 5$}}; u)
we have

\begin{equation} \label{eq: holomorphicity-residue-u}
R(u\underline{x};u) \; =\sum_{|\underline{k}| \, \equiv \, 0 \!\!\!\!\pmod 2}
Q_{ \scalebox{1.1}{$\scriptscriptstyle \underline{k}$}}(u; u)
\underline{x}^{\scalebox{1.1}{$\scriptscriptstyle \underline{k}$}}
\end{equation}

which implies easily that
f(u^{\scalebox{1.1}{$\scriptscriptstyle 2$}}\mathrm{z}, u)
(as a function of the variables
\mathrm{z} =
x_{\scalebox{1.1}{$\scriptscriptstyle 1$}}
x_{\scalebox{1.1}{$\scriptscriptstyle 2$}}
x_{\scalebox{1.1}{$\scriptscriptstyle 3$}}
x_{\scalebox{1.1}{$\scriptscriptstyle 4$}}
and $u$) is holomorphic in the polydisc
< (1 + \epsilon)^{\sss -4},\, |u| < 1 + \epsilon.
This completes the proof.

Thus it remains to show that

f(P,u)=1/(P_i^1.1$\scriptscriptstyle 2$;
P_i^1.1$\scriptscriptstyle 2$)_∞
(u_i^1.1$\scriptscriptstyle 2$
P_i^1.1$\scriptscriptstyle 2$;
P_i^1.1$\scriptscriptstyle 2$)_∞.
The proof of this formula involves the following two steps:

(1) Formula (<ref>) holds for $u=-1$. This is an immediate consequence of Macdonald's formula.

(2) The function

\[
g(P,u)=f(P,u)\left(P_{\phantom i}^{\scalebox{1.1}{$\scriptscriptstyle 2$}}; P_{\phantom i}^{\scalebox{1.1}{$\scriptscriptstyle 2$}}\right)_{\infty}
\!\left(u_{\phantom i}^{\scalebox{1.1}{$\scriptscriptstyle 2$}}
P_{\phantom i}^{\scalebox{1.1}{$\scriptscriptstyle 2$}};
P_{\phantom i}^{\scalebox{1.1}{$\scriptscriptstyle 2$}}\right)_{\infty}
\]

is invariant under the transformation $(P,u)\mapsto (P, u\slash P).$
(It is here where we crucially use the extrafunctional equation
from <ref>.)

The proofs of (1) and (2) will be given in the next two subsections,
but for now let us show how they imply(<ref>).

By Proposition <ref>, the function
$g(u^{\scalebox{1.1}{$\scriptscriptstyle 2$}}\mathrm{z}, u)$
is holomorphic in the polydisc
|\mathrm{z}| < (1 + \epsilon)^{\sss -4},\, |u| < 1 + \epsilon
hencea normally convergent power series
in $\mathrm{z}$ and $u$. Substituting
\mathrm{z} \to P\slash u^{\scalebox{1.1}{$\scriptscriptstyle 2$}}
we can then write

\[
g(P,u) \; =\sum_{m, n\in \Z} a_{m,n} u^{m} P^{n}
\]

the series being normally convergent for, say
\epsilon \le |u| < 1 + \epsilon
|P| < |u|^{\scalebox{1.1}{$\scriptscriptstyle 2$}}(1 + \epsilon)^{\sss -4}
Notice that $a_{m,n}=0$ if $n < 0$. Since
$g(P,u) = g(P,uP)$, it follows at once (after
taking $u$ and $P$ such that $\epsilon \le |uP| < 1 + \epsilon$)
that$a_{m,n} = a_{m,n - m}$ for all $m,n \in \Z$.
Iterating, one finds that
for $k\in \Z$. Since $a_{m,n}=0$ for$n < 0$, it
follows that $a_{m,n} = 0$ if $m \ne 0$. Therefore
$g(P,u)$ is independent of $u$, and by (1), $g\equiv 1$.

§.§ The residue for $u=-1$

From the relation between $\ZW(\mathbf{x};u)$ and $\tZW(\mathbf{x}; u)$, we can write

\[
R(\ux;u)=\frac{1}{\left(u_{\phantom i}^{\scalebox{1.1}{$\scriptscriptstyle 2$}}
P_{\phantom i};
P_{\phantom i}^{\scalebox{1.1}{$\scriptscriptstyle 2$}}\right)_{\infty}^{\scalebox{1.1}{$\scriptscriptstyle 2$}}\cdot
\DD_{\sss 5}(\ux, 1\slash u)} \, \cdot
\lim_{x_{\scalebox{.75}{$\scriptscriptstyle 5$}} \to
\scalebox{1.}{$\scriptscriptstyle 1\slash $}u}
\frac{1-ux_{\sss 5}}{1-x_{\sss 5}^{\sss 2}}\ZW(\xx;u)
\]

where we put
\DD_{\sss 5}(\xx):=\DD(\xx)\slash (1-x_{\sss 5}^{\sss 2})
When $u=-1$, we see by induction on $\ell(w)$ that

1\vert w(\xx;-1)=(- 1)^{\scalebox{1.1}{$\scriptscriptstyle \ell(w)$}}
\prod_{\beta \, \in \, \Phi(w)} \mathbf{x}^{\beta}

and so $\ZW(\xx;-1)=F_{\sss MD}(\xx)$, where
F_{\sss MD}(\xx)=\sum_{w\in W} (- 1)^{\scalebox{1.1}{$\scriptscriptstyle \ell(w)$}}
\prod_{\beta \, \in \, \Phi(w)} \mathbf{x}^{\beta}
is the function studied by Macdonald [44]. Then

\[
\lim_{u \to -1}
\,\lim_{x_{\scalebox{.75}{$\scriptscriptstyle 5$}} \to
\scalebox{1.}{$\scriptscriptstyle 1\slash $}u}
\frac{1-ux_{\sss 5}}{1-x_{\sss 5}^{\sss 2}} \ZW(\xx;u)=
\frac 12 \ZW(\ux,-1;-1)=\frac{1}{2}F_{\sss MD} (\ux, -1)
\]

where in the first equality we interchanged the two limits;
this is justified since $\DD(\xx)$, and in particular
$1-x_{\sss 5}^{\sss 2}$, divides $\ZW(\xx;u)$
(see <cit.>), and so the function inside the
double limit is continuous at $(1,-1)$ as a function of
$z=ux_{\sss 5}$ and $u$. Consequently,

R(;-1)=1/2F_MD (, -1)/(P_i;
P_i^1.1$\scriptscriptstyle 2$)_∞^1.1$\scriptscriptstyle 2$·_5(, -1).
Using the list of roots of $D_{\sss 4}$ at the beginning
of Section <ref>, we have by Macdonald's formula [44]

\[
\begin{aligned}
F_{\sss MD}(\xx)=
&\prod_{i=1}^{4}\left(x_{\sss i}; \xx^{\dd}\right)_{\infty}
\left(x_{\sss i}^{\sss -1}\xx^{\dd}; \xx^{\dd}\right)_{\infty}
\left(x_{\sss i}x_{\sss 5}; \xx^{\dd}\right)_{\infty}
\left(x_{\sss i}^{\sss -1} x_{\sss 5}^{\sss -1}\xx^{\dd}; \xx^{\dd}\right)_{\infty}
\prod_{1\le i< j\le 4}\left(x_{\sss i}x_{\sss j}x_{\sss 5};
\xx^{\dd}\right)_{\infty} \\
\left(\xx^{\dd};\xx^{\dd}\right)^{\scalebox{1.1}{$\scriptscriptstyle 4$}}_{\infty}
\left(x_{\sss 5};\xx^{\dd}\right)_{\infty}
\left(x_{\sss 5}^{\sss -1}\xx^{\dd};\xx^{\dd}\right)_{\infty}.
\end{aligned}
\]

Also, $\DD(\xx)=F_{\sss MD} (x_{\sss 1}^{\sss 2}, \ldots, x_{\sss 5}^{\sss 2})$,
and one can easily check that (<ref>) matches the formula
in Theorem <ref> for$u=-1$.
This completes the proof of step (1) in <ref>.

§.§ Functional equations of the residue function

From the formula of the residue function in Proposition
<ref>, it is clear that
R(\underline{x};u) = R(-\underline{x};u).
Then by Lemma <ref>, part
1, it follows at once that

lim_x_.75$\scriptscriptstyle 5$ →1.$\scriptscriptstyle 1 \slash $u
(1-ux_5)(; u) = 1/2 R(x;u)·v_0
where we set $v_{\sss 0}= \, \!^{t}(1, 1, 0)$.

We need the extension of the cocycle $\tL(\xx;u)$ to $W\oplus \Z$,
as in Section <ref>, by
\tL_{\tau^{-1}}(\xx;u)=\tB(\xx;u)
with $\tB(\xx;u)$ defined as in the line following (<ref>), and

\tL_{\tau}(\xx;u)=\tA(\xx;u):=A(\xx;u)\cdot\left(1-u^{2}\slash \xx^{\dd}\right)^{2}

with $A(\xx;u)$ from Theorem <ref>. Then the functional
equation (<ref>) holds for $\tZZ(\xx;u)$,
with $\Lam(\xx;u)$ re-placed by $\tL(\xx;u)$.
As in Section <ref>, we have an action of $\tau$ extending the action $\|$
such that

fτ(;u)=(;u) ·f̅(;u^)
for all $f\in\CC(\xx,u)_{\sss 0}$.

— Let $w\in W$ be such that
$w \a_{\sss 5}=\a_{\sss 5}-\dd$. Then

\[
\cdot \!\left. R(w\underline{x};u\slash\xx^{\dd})\right \vert_{x_{\scalebox{.75}{$\scriptscriptstyle 5$}}=\scalebox{1.15}{$\scriptscriptstyle 1\slash u$}}
\]

\lambda_{w}(\ux;u)
=\left. [(1+ux_{\sss 5})\| \tau  w]^{\eee} (\xx;u)\right\vert_{x_{\scalebox{.75}{$\scriptscriptstyle 5$}}=\scalebox{1.15}{$\scriptscriptstyle 1\slash u$}}

Notice from (<ref>) that
[(1+ux_{\sss 5})\| \tau](\xx;u)
=\tA_{\eee}(\xx;u)+(ux_{\sss 5}\slash \xx^{\dd})\tA_{\eo}(\xx;u)$,
where $\tA_{\eee}$ (resp. $\tA_{\eo}$) is thesum of the entries
in the first (resp. second) column of the matrix $\tA(\xx;u)$.
w\xx=(w\ux, x_{\sss 5}\xx^{\dd})
the analogue of the functional equation (<ref>) for
$\tZZ(\xx;u),$ applied for $w\tau$, together with (<ref>) gives

\[
R(\ux;u)v_{\sss 0}= \!\left.\left\{R(w\underline{x};u\slash\xx^{\dd})
\tL_{w\tau}(\xx;u)\right\}\right\vert_{x_{\scalebox{.75}{$\scriptscriptstyle 5$}}=\scalebox{1.15}{$\scriptscriptstyle 1\slash u$}} \cdot \, v_{\sss 0}.
\]

Note that
\tL_{w\tau}(\xx;u)=\tL_{\tau}(\xx;u)\tL_{w}(\xx;u\slash \xx^{\dd})
does not have a pole at $x_{\sss 5}=1\slash u;$ by Theorem <ref>,
the ma-trix $\tL_{\tau}(\xx;u)$ does not have a pole there,
and since $\a_{\sss 5}+\dd \notin \Phi(w)$, neither does
$\tL_{w}(\xx;u\slash \xx ^{\dd})$. We thus ob-tain the functional
equation of $R(\ux;u)$, with $\lam_{w}(\ux;u)$ such that
\left. \tL_{w\tau} (\xx;u)
\right\vert_{x_{\scalebox{.75}{$\scriptscriptstyle 5$}}=\scalebox{1.15}{$\scriptscriptstyle 1\slash u$}} \cdot \,
v_{\sss 0}  =  \lambda_{w}(\ux;u)
v_{\sss 0}.
Com-bining (<ref>) and (<ref>), we get the formula for
$\lambda_{w}(\ux;u)$, and completes the proof.

By applying this proposition to the element
t=\ss_{\sss 1}\ss_{\sss 2}\ss_{\sss 3}\ss_{\sss 4}\ss_{\sss 5}$,
we obtain that

\[
R\left(u\slash x_{\sss 1},\ldots,u\slash x_{\sss 4}; u\slash P\right)
\]

with $\lambda_{t}(\ux;u)$ computed explicitly from the matrix
$A(\xx;u)$ in [1] as

\[
\lambda_{t}(\ux;u)=(1-u^{\sss 2})
\prod_{i=1}^{4} (1-u^{\sss 2}\slash x_{\sss i}^{\sss 2})
\prod_{1\le i<j\le 4} (1-u^{\sss 2}\slash x_{\sss i}x_{\sss j}).
\]

One can check easily that this factor matches the one from the same
functional equation applied to the right-hand side of the formula in
Theorem <ref>, and thus completing the proof
of step (2) in <ref>.

§ WMDS ASSOCIATED TO MOMENTS OF L-SERIES

By Theorem <ref> and the definition (<ref>) of
$D(\mathbf{x}; u),$ when $0 < u < 1,$ the function $\tilde{Z}_{\scriptscriptstyle W}(\mathbf{x}; u)$ is holomorphic for
$|x_{\scalebox{1.1}{$\scriptscriptstyle i$}}| < 1$ ($i = 1, \ldots,
5$), and thus, in this polydisc,

Z̃_W(𝐱; u) = 1 + ∑_k    0 a(k; u)𝐱^k
the sum being over tuples
\mathrm{k} =
(k_{\scalebox{1.1}{$\scriptscriptstyle 1$}}, \ldots, k_{\scalebox{1.1}{$\scriptscriptstyle 5$}})\in
\mathbb{N}^{5}\setminus \{\mathbf{0}\}.
The coefficients $a(\mathrm{k}; u)$ are polynomials in $u,$ and by Cauchy's inequalities,
for every $\epsilon > 0,$ we have the estimate
|a(\mathrm{k}; u)| \ll_{\epsilon} u^{- \epsilon |\mathrm{k}|}.

Before making the transition to multiple Dirichlet series,
let us briefly recall some facts about quadratic Dirichlet
$L$-functions in the rational function field setting.

Let $\mathbb{F}_{\! q}$ be a finite field of odd characteristic.
m \in \mathbb{F}_{\! q}[x],
$m \ne 0,$ set
|m| = q^{\deg m},
and for $d, m \in \mathbb{F}_{\! q}[x],$ $d \ne 0$ and
$m$ monic, let $\chi_{d}(m) = (d \slash m)$ denote
the quadratic symbol. We define $\chi_{d}(1) = 1,$ and if
$d \in \mathbb{F}_{\! q}^{\times},$we have $\chi_{d}(m) = \mathrm{sgn}(d)^{\deg m},$
for all non-constant $m \in \mathbb{F}_{\! q}[x],$
where $\mathrm{sgn}(d) = 1$ or $-1$ according as
$d \in (\mathbb{F}_{\! q}^{\times})^{\scalebox{1.1}{$\scriptscriptstyle 2$}}$
or not.

For $d$ square-free, the $L$-function associated
to the primitive character $\chi_{d}$ is defined by

\[
L(s, \chi_{d}) \;\; = \sum_{\substack{m \, \in \, \mathbb{F}_{\! q}[x] \\
m - \text{monic}}}
\chi_{d}(m) |m|^{-s}
\;\;\, =  \prod_{p - \text{monic \& irreducible}}
\big(1 - \chi_{d} (p)|p|^{-s}\big)^{\scalebox{1.1}{$\scriptscriptstyle -1$}}
\]

for complex $s,$ with $\Re(s) > 1.$ When $d$
is non-constant this $L$-function turns out to be a polynomial
in $q^{-s}$of degree $\deg d - 1,$ and if
$d \in \mathbb{F}_{\! q}^{\times},$

\[
L(s, \chi_{d}) = \frac{1}
{1 - \mathrm{sgn}(d) q^{{\scalebox{1.1}{$\scriptscriptstyle 1$}} - s}};
\]

d \in (\mathbb{F}_{\! q}^{\times})^{\scalebox{1.1}{$\scriptscriptstyle 2$}},
the $L$-function is just $\zeta(s)$–the zeta-function. In addition,
if one defines
\gamma_{\scalebox{1.1}{$\scriptscriptstyle q$}}(s, d)
\[
\gamma_{\scalebox{1.1}{$\scriptscriptstyle q$}}(s, d) =
q^{\frac{1}{2}{\scalebox{1.05}{$\scriptscriptstyle \left(3 \, + \, (-1)^{\deg d} \right)$}} \left(s - \frac{1}{2}\right)}
\big(1  -  \mathrm{sgn}(d) q^{- s} \big)^{\frac{1}{2}
\scalebox{1.1}{$\scriptscriptstyle \left(1 \, + \, (-1)^{\deg d}\right)$}}
\big(1 - \mathrm{sgn}(d) q^{s - \scalebox{1.1}{$\scriptscriptstyle 1$}} \big)^{-
\frac{1}{2} \scalebox{1.1}{$\scriptscriptstyle \left(1 \, + \, (-1)^{\deg d}\right)$}}
\]

then $L(s, \chi_{d})$ satisfies the functional equation

\[
L(s, \chi_{d}) =\gamma_{\scalebox{1.1}{$\scriptscriptstyle q$}}(s, d)
|d|^{\frac{1}{2} - s} L(1 - s, \chi_{d}).
\]

We now use the coefficients $a(\mathrm{k}; u)$ to build the relevant
family of multiple Dirichlet series.

Let $\mathbb{F}_{\! q}$ be a finite field of odd characteristic, and fix an element
\theta_{\scalebox{1.1}{$\scriptscriptstyle 0$}} \in \mathbb{F}_{\! q}^{\times}
\setminus (\mathbb{F}_{\! q}^{\times})^{\scalebox{1.1}{$\scriptscriptstyle 2$}}.
\mathbf{s} = (s_{\scriptscriptstyle 1}, \ldots, s_{\scriptscriptstyle 5})
a_{\scriptscriptstyle 1},  a_{\scriptscriptstyle 2}
\in \{1, \theta_{\scalebox{1.1}{$\scriptscriptstyle 0$}}\},
we define
\mathscr{W}\!(\mathbf{s};  a_{\scriptscriptstyle 2}, a_{\scriptscriptstyle 1})