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\newcommand{\RED}[1]{\textcolor{red}{#1}}
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\DeclareMathOperator{\ELL}{L}
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\usepackage{lipsum}

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\title{Permutation representations of classical Weyl groups on mod {\texorpdfstring{$q$}{q}} lattices}

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\author[Ryo Uchiumi]{\texorpdfstring{Ryo Uchiumi\thanks{\href{mailto:uchiumi.ryou.1xu@ecs.osaka-u.ac.jp}{ uchiumi.ryou.1xu@ecs.osaka-u.ac.jp}. The author was supported by JST SPRING, Grant Number JPMJSP2138.
}\addressmark{1}}{Ryo Uchiumi}}

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\address{\addressmark{1}Department of Mathematics, Osaka University, Toyonaka, Osaka 560-0043, Japan.}

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\abstract{

For a given linear action of a finite group on a lattice and a positive integer $q$, 
the $\!\!\!\!\mod{q}$ permutation representation is a quasi-polynomial 
in $q$.
In this paper, we compute the multiplicity of each irreducible representation in the $\!\!\!\!\mod{q}$ permutation representation of a classical Weyl group on the two types of lattices, generated by the standard basis and by coroots.
%Additionally, in the case of lattice generated by the standard basis, we give a representation of the multiplicity using integer partitions that characterize the irreducible character.
%In the case of coroot lattice, we show that the multiplicity is closely related to the Ehrhart polynomial of the fundamental alcove.

}

%% put your French abstract here, or comment this out if you don't have one


%% put your keywords here, or comment this out if you don't have them yet
\keywords{classical Weyl groups, lattices, mod $q$ permutation representations, quasi-polynomials, integer partitions}

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\begin{document}

\maketitle
%% note that you DO NOT have to put your abstract here -- it is generated by \maketitle and the \abstract and \resume commands above

\section{Introduction}
\subsection{Quasi-polynomials}

Let $R$ be a commutative ring.
A function $f : \mathbb{Z}_{>0} \to R$ is called a \textbf{quasi-polynomial} if there exists a positive integer $\widetilde{n} \in \mathbb{Z}_{>0}$ and polynomials $g_1(t),\ldots,g_{\widetilde{n}}(t) \in R[t]$ such that
\begin{align*}
    f(q) = g_r(q),\quad \text{if $q \equiv r \mod{\widetilde{n}} \quad (1 \leq r \leq \widetilde{n})$}.
\end{align*}
The positive integer $\widetilde{n}$ is called a \textbf{period} and each polynomial $g_r$ is called the \textbf{constituent} of $f$.
The quasi-polynomial $f$ has degree $d$ if all the constituents have degree $d$.
Moreover, the quasi-polynomial $f$ has the \textbf{gcd-property} if the polynomial $g_r$ depends on $r$ only through $\gcd\{\widetilde{n},r\}$.
In other words, $g_{r_1} = g_{r_2}$ if $\gcd\{\widetilde{n},r_1\} = \gcd\{\widetilde{n},r_2\}$.
Quasi-polynomials play an important role in many areas of mathematics and appear frequently, especially as counting functions.

\begin{example}[The Ehrhart quasi-polynomial]
    Let $\mathcal{P}$ be a rational polytope in $\mathbb{R}^\ell$.
    For $q \in \mathbb{Z}_{\geq 0}$, define
    \begin{align*}
        \ELL_{\mathcal{P}}(q) \ceq \#(q\mathcal{P} \cap \mathbb{Z}^\ell).
    \end{align*}
    Then $\ELL_{\mathcal{P}}(q)$ is a quasi-polynomial ({\cite[Theorem 3.23]{BeckRobins}}), known as the \textbf{Ehrhart quasi-polynomial}.
\end{example}

\begin{example}[The characteristic quasi-polynomial]
Let $L \simeq \mathbb{Z}^\ell$ be a lattice and $L^\vee \ceq \Hom_{\mathbb{Z}}
(L,\mathbb{Z})$ be the dual lattice.
Given $\alpha_1,\ldots,\alpha_n \in L^\vee$, 
we can associate a hyperplane arrangement $\mathscr{A} \ceq \{H_1,\ldots,H_n\}$ in $\mathbb{R}^\ell \simeq L \otimes \mathbb{R}$, where 
\begin{align*}
H_i \ceq \set{ x \in L\otimes \mathbb{R} \mid \alpha_i(x) = 0}.
\end{align*}
For a positive integer $q \in \mathbb{Z}_{>0}$, define the mod $q$ complement 
of the arrangement by 
\begin{align*}
\begin{split}
M(\mathscr{A},q) &\ceq (L/qL)^\ell\setminus\bigcup_{i=1}^\ell \overline{H}_i\\
&=\set{ \bar{x} \in L/qL \mid \alpha_i(x) \not\equiv 0 \mod{q}\quad  \text{for all $i \in \{1,\ldots,n\}$}}.
\end{split}
\end{align*}
It is known ({\cite[Theorem 2.4]{KamiyaTakemuraTerao}}) that 
\begin{align*}
\chi_{\text{quasi}}(\mathscr{A},q) \ceq \#M(\mathscr{A},q)
\end{align*}
is a quasi-polynomial with $\gcd$-property.
It is called a \textbf{characteristic quasi-polynomial}.
Furthermore, the first constituent of $\chi_{\text{quasi}}(\mathscr{A}, t)$ is equal to the \textbf{characteristic polynomial} $\chi(\mathscr{A},t)$ of $\mathscr{A}$, the most important invariant of $\mathscr{A}$.
\end{example}

\begin{example}[Equivariant Ehrhart theory]
    Let $L \simeq \mathbb{Z}^\ell$ be a lattice and let $\GAMMA$ be a finite group acting linearly on $L$.
    Suppose that $\mathcal{P}$ is a $\GAMMA$-invariant lattice polytope.
    For a positive integer $q \in \mathbb{Z}_{>0}$, the group acts on the lattice points $q\mathcal{P} \cap L$.
    Let $\chi_{q\mathcal{P}}$ denote the character of this permutation representation.
    Then the map
    \begin{align*}
        F : q \longmapsto \chi_{q\mathcal{P}}
    \end{align*}
    is a quasi-polynomial ({\cite[Theorem 5.7, Corollary 5.9]{Stapledon}}).
    It is an equivariant version of Ehrhart quasi-polynomials.
    In fact, for the identity element $1$ of $\GAMMA$, then $\chi_{q\mathcal{P}}(1) = \#(q\mathcal{P} \cap L)$, hence it is a generalization of Ehrhart theory.
    Furthermore, the multiplicity of a fixed irreducible character $\chi$ in $\chi_{q\mathcal{P}}$ is also a quasi-polynomial in $q$.
\end{example}

\subsection{{\texorpdfstring{$\MOD q$}{mod q}} permutation representations}

In \cite{UchiumiYoshinaga}, we consider $\MOD{q}$ permutation representations for linear finite group actions on lattices toward an equivariant version of characteristic quasi-polynomials.

Let $L$ be a lattice, and $\{\beta_1,\ldots,\beta_\ell\}$ be a $\mathbb{Z}$-basis of $L$, that is, $L = \mathbb{Z}\beta_1 \oplus  \cdots \oplus \mathbb{Z}\beta_\ell \simeq \mathbb{Z}^\ell$.
We identify an element $x = x_1\beta_1 + \cdots + x_\ell\beta_\ell$ of $L$ with the row vector $x = (x_1,\ldots,x_\ell)$ of $\mathbb{Z}^\ell$.

Let $\GAMMA$ be a finite group and let $\rho : \GAMMA \to \GL(L)$ be a group homomorphism.
Let us denote the representation matrix of $\rho(\gamma)$ by $R_\gamma$, and we consider the right multiplication, namely,
\begin{align*}
    \rho(\gamma) : L \longrightarrow L,\quad x \longmapsto xR_\gamma.
\end{align*}

For a positive integer $q \in \mathbb{Z}_{>0}$, define $\mathbb{Z}_q \ceq \mathbb{Z}/q\mathbb{Z}$.
We will consider the following \textbf{$\bm{q}$-reduction} of a vector $x = (x_1,\ldots,x_\ell) \in \mathbb{Z}^\ell$ and an integer matrix $A = (a_{ij})_{ij}$:
\begin{align*}
    [x]_q \ceq ([x_1]_q,\ldots,[x_\ell]_q) \in L_q^\ell,\qquad [A]_q \ceq \Bigl(
    [a_{ij}]_q\Bigr)_{ij},
\end{align*}
where $[z]_q = z + q\mathbb{Z} \in \mathbb{Z}/q\mathbb{Z}$ for $z \in \mathbb{Z}$.
Then, for a $\mathbb{Z}$-homomorphism $\varphi : \mathbb{Z}^\ell \to \mathbb{Z}^\ell$ represented by an integer matrix $A$, we can define the induced morphism $\varphi_q : \mathbb{Z}_q^\ell \to \mathbb{Z}_q^\ell$ by $x \longmapsto x[A]_q$.

Let $L_q \ceq L/qL \simeq \mathbb{Z}_q^\ell$.
The action of $\GAMMA$ on $L_q$ is induced by $\rho(\gamma)_q : L_q \longrightarrow L_q$.
Let $\chi_{L_q}$ denote the character of the permutation representation of $L_q$, and consider its irreducible decomposition:
\begin{align*}
    \chi_{L_q} = m(\chi_1;\, q) \cdot \chi_1 + \cdots + m(\chi_k;\, q) \cdot \chi_k,
\end{align*}
where $\{\chi_1,\ldots,\chi_k\}$ is the set of all irreducible characters of $\GAMMA$ and $m(\chi_i;\, q)$ denotes the multiplicity of $\chi_i$ in $\chi_{L_q}$.

In \cite{UchiumiYoshinaga}, 
%on the problem of how the permutation character $\chi_{L_q}$ depends on $q$, 
it is shown that the multiplicity $m(\chi;\, q)$ of the any irreducible character $\chi$ in $\chi_{L_q}$ is a quasi-polynomial in $q$ with $\gcd$-property (\cite[Corollary 2.2]{UchiumiYoshinaga}).
More explicity, 
    \begin{align}
        m(\chi;\, q) = \dfrac{1}{\#\GAMMA} \sum_{\gamma \in \GAMMA}\chi(\gamma) \left(
        \prod_{j = 1}^{r(\gamma)} \gcd\{e_{\gamma,j}, q\}
        \right) q^{\ell-r(\gamma)}, \label{s11}
    \end{align}
    where $r(\gamma) \ceq \rank(R_\gamma - I)$ with the identity matrix $I$ and $e_{\gamma,1},\ldots,e_{\gamma,r(\gamma)}$, with $e_{\gamma,1} \mid e_{\gamma,2} \mid \cdots \mid e_{\gamma,r(\gamma)}$, are the elementary divisors of the matrix $R_\gamma-I$.
Obviously, the map $q \longmapsto \chi_{L_q}$ is also a quasi-polynomial with $\gcd$-property.
Moreover, their quasi-polynomial have a period $\lcm{\set{e_{\gamma,r(\gamma)} | \gamma \in \GAMMA}}$ (\textbf{lcm-period}).

\begin{comment}
Define the \textbf{lcm-period} $\widetilde{n}$ by
\begin{align*}
    
\end{align*}
Then the lcm-period $\widetilde{n}$ is a period of the quasi-polynomial $m(\chi;\, q)$ for any irreducible character $\chi$.
In particular, for the trivial character $\bm{1}$ of $\GAMMA$, the quasi-polynomial $m(\bm{1};\, q)$ has the minimal period $\widetilde{n}$.
Therefore the minimal period of the quasi-polynomial $q \longmapsto \chi_{L_q}$ is equal to $\widetilde{n}$.
\end{comment}


\subsection{Classical Weyl groups}
In this paper, we compute $m(\chi;\, q)$ for each irreducible character $\chi$ when a group $\GAMMA$ is a classical Weyl group.
In this section, we recall classical Weyl groups and their conjugacy classes and irreducible characters.
See also {\cite{Carter, Weyl, Young}} for details.

For $\ell \in \mathbb{Z}_{>0}$, an \textbf{integer partition} of $\ell$ is an integer sequence $\lambda = (\lambda_1,\ldots,\lambda_t)$ satisfying $\lambda_1 \geq  \cdots \geq \lambda_t > 0$ and $\lambda_1 + \cdots + \lambda_t = \ell$.
We define $\ell(\lambda)$ as the length $t$ of $\lambda$ and $|\lambda| \ceq \lambda_1 + \cdots + \lambda_t$.
We consider that the empty sequence $\emptyset = ()$ to be an integer partition of $0$, with $\ell(\emptyset) = 0$ and $|\emptyset| = 0$.

%An integer partition $\lambda$ is often identified with a \textbf{Young diagram} with $\lambda_i$ boxes in the $i$-th row.
%Sometimes we consider an integer partition $\lambda$ to be a subset of $\mathbb{Z}^2$:
%\begin{align*}
%    \lambda = \set{(i,j) \in \mathbb{Z}^2 | 1 \leq i \leq \ell(\lambda),\ 1 \leq j \leq \lambda_i}.
%\end{align*}


\subsubsection{Type {\texorpdfstring{$A_{\ell-1}$}{Al-1}}}

The Weyl group of type $A_{\ell-1}$ is the symmetric group $\mathfrak{S}_\ell$.
The order of $\mathfrak{S}_\ell$ is $\#\mathfrak{S}_\ell = \ell!$.
Each element $\sigma \in \mathfrak{S}_\ell$ may be decomposed uniquely into a product of cyclic permutations up to the order of factors:
\begin{align*}
    \sigma = (i_{11} \cdots i_{1\lambda_1}) \cdots\cdots (i_{t1} \cdots i_{t\lambda_t}),\quad \lambda_1 \geq \cdots \geq \lambda_t.
\end{align*}
Then the integer partition $(\lambda_1,\ldots,\lambda_t)$ of $\ell$ is called the \textbf{cycle type} of $\sigma \in \mathfrak{S}_\ell$.
Two elements of $\mathfrak{S}_\ell$ are conjugate if and only if they have the same cycle type.
In other words, conjugacy classes of $\mathfrak{S}_\ell$ are parameterized by integer partitions of $\ell$.

Irreducible characters of $\mathfrak{S}_\ell$ are also parameterized by integer partition of $\ell$.
We denote the irreducible character defined by a integer partition $\lambda$ by $\chi^\lambda$.

\subsubsection{Type {\texorpdfstring{$B_\ell$}{Bl}} and {\texorpdfstring{$C_\ell$}{Cl}}}

The Weyl group of type $B_\ell$ and $C_\ell$ are the hyperoctahedral group $\mathfrak{H}_\ell$, isomorphic to the semidirect product $\mathfrak{S}_2^\ell \rtimes \mathfrak{S}_\ell$ (the wreath product $\mathfrak{S}_2 \wr \mathfrak{S}_\ell$).
The order of $\mathfrak{H}_\ell$ is $\#\mathfrak{H}_\ell = 2^\ell\ell!$.
The hyperoctahedral group $\mathfrak{H}_\ell$ can be regarded as a subgroup of the symmetric group $\mathfrak{S}_{I_\ell}$ of $I_\ell \ceq \{-\ell,\ldots,-1,1,\ldots,\ell\}$ by
\begin{align*}
    \mathfrak{H}_\ell = \set{\eta \in \mathfrak{S}_{I_\ell} | \text{$\eta(-i) = -\eta(i)$ for all $i \in I_\ell$}}.
\end{align*}
Especially, $\mathfrak{S}_2^\ell$ is to be regarded as a subgroup generalized by the sign transpositions $\tau_1,\ldots,\tau_\ell$:
\begin{align*}
\tau_i : \begin{cases*}
    \, j \,\longmapsto\, -j & if $\ j = i,\ -i$;\\
    \, j \,\longmapsto\, j & if $\ j \neq i,\ -i$
\end{cases*}\quad (1 \leq i \leq \ell).
\end{align*}


The hyperoctahedral group $\mathfrak{H}_\ell$ has two types of cyclic permutations, \textbf{even cyclic permutations} $(i_1 \cdots i_t)_+$ and \textbf{odd cyclic permutations} $(i_1 \cdots i_t)_-$:
\begin{align*}
    (i_1 \cdots i_t)_+ = \begin{pmatrix}
        i_1 & i_2 & \cdots &i_{t-1} & i_t\\
        i_2 & i_3 & \cdots & i_t & i_1
    \end{pmatrix},\qquad 
    (i_1 \cdots i_t)_- = \begin{pmatrix}
        i_1 & i_2 & \cdots & i_{t-1} & i_t\\
        i_2 & i_3 & \cdots & i_t & -i_1
    \end{pmatrix},
\end{align*}
where $i_1,\ldots,i_t \in I_\ell$ and they have pairwise different absolute values.
Note that an odd cyclic permutation $(i)_-$ of length $1$ is equal to the sign transposition $\tau_i$.
Each element $\eta \in \mathfrak{H}_\ell$ may be decomposed uniquely into a product of two types of cyclic permutations up to the order of factors:
\begin{align*}
    \eta = (i_{11} \cdots i_{1\lambda_1})_+ \cdots\cdots (i_{t1} \cdots i_{t\lambda_t})_+(j_{11} \cdots j_{1\mu_1})_- \cdots\cdots (j_{s1} \cdots j_{s\mu_s})_-,\\
    \lambda_1 \geq \cdots \geq \lambda_t,\quad \mu_1 \geq \cdots  \geq \mu_s.
\end{align*}
Then the pair of two integer partitions $\bigl((\lambda_1,\ldots,\lambda_t),\ (\mu_1,\ldots,\mu_s)\bigr)$ is called the \textbf{cycle type} of $\eta \in \mathfrak{H}_\ell$.
Two elements of $\mathfrak{H}_\ell$ are conjugate if and only if they have the same cycle type.
Hence conjugacy classes of $\mathfrak{H}_\ell$ are parameterized by the pair of integer partitions $(\lambda,\mu)$ with $|\lambda| + |\mu| = \ell$.
Note that an element $\eta \in \mathfrak{H}_\ell$ with cycle type $(\lambda,\emptyset)$ can be regarded as an element of $\mathfrak{S}_\ell$ with cycle type $\lambda$.
Hence we consider that $\mathfrak{S}_\ell$ is a subgroup of $\mathfrak{H}_\ell$.

Irreducible characters of $\mathfrak{H}_\ell$ are also parameterized by the pair of integer partitions $(\lambda,\mu)$ with $|\lambda| + |\mu| = \ell$.
Let $\chi^{\lambda,\mu}$ denote the irreducible character defined by $(\lambda,\mu)$.
Then the dimensions of $\chi^{\lambda,\mu}$, $\chi^\lambda$ and $\chi^\mu$ are connected by the following relation:
\begin{align}
    \dfrac{\chi^{\lambda,\mu}(1)}{\ell!} = \dfrac{\chi^\lambda(1)\chi^\mu(1)}{|\lambda|! \cdot |\mu|!}. \label{1.2}
\end{align}

\subsubsection{Type {\texorpdfstring{$D_\ell$}{Dl}}}

The Weyl group of type $D_\ell$ is a subgroup of index $2$ of the hyperoctahedral group $\mathfrak{H}_\ell$ denoted by $\widetilde{\mathfrak{H}}_\ell$.
An element $\eta \in \mathfrak{H}_\ell$ is in $\widetilde{\mathfrak{H}}_\ell$ if and only if $\eta$ has cycle type $(\lambda,\mu)$ such that $\ell(\mu)$ is even.
In other words, we can consider that 
\begin{align*}
    \widetilde{\mathfrak{H}}_\ell = \Set{ \eta \in \mathfrak{H}_\ell | \text{$\#\set{i \in [\ell] | \eta(i) < 0}$ is even} }.
\end{align*}

Let $C_{\lambda,\mu}$ denote the conjugacy class of $\mathfrak{H}_\ell$ parameterized by the pair of partitions $(\lambda,\mu)$.
Let $\widetilde{C}_{\lambda,\mu} \ceq C_{\lambda,\mu} \cap \widetilde{\mathfrak{H}}_\ell$.
If $(\lambda,\mu)$ satisfies 
\begin{align}
    \lambda_1 \equiv \cdots \equiv \lambda_{\ell(\lambda)} \equiv 0 \pmod{2} \quad \text{and} \quad \mu = \emptyset, \label{cond}
\end{align}
then $\widetilde{C}_{\lambda,\mu}$ is the union of two conjugacy classes of $\widetilde{\mathfrak{H}}_\ell$, which denote $\widetilde{C}_{\lambda,\mu}^1$ and $\widetilde{C}_{\lambda,\mu}^2$.
These classes have the following relation:
\begin{align}
\widetilde{C}_{\lambda,\mu}^1 = \tau_1^{-1}\widetilde{C}_{\lambda,\mu}^2\tau_1 = \set{ \tau_1^{-1}\gamma\tau_1 | \gamma \in \widetilde{C}_{\lambda,\mu}^2}. \label{1.4}
\end{align}
If $(\lambda,\mu)$ does not satisfy \eqref{cond}, then $\widetilde{C}_{\lambda,\mu}$ is also a conjugacy class of $\widetilde{\mathfrak{H}}_\ell$.
All conjugacy classes of $\widetilde{\mathfrak{H}}_\ell$ can be obtained in this way.

Let $(\lambda,\mu)$ be the pair of integer partitions satisfying $|\lambda| + |\mu| = \ell$.
Let $\widetilde{\chi}^{\lambda,\mu}$ denote the restriction to $\widetilde{\mathfrak{H}}_\ell$ of the irreducible character $\chi^{\lambda,\mu}$ of $\mathfrak{H}_\ell$.
If $\lambda \neq \mu$, then $\widetilde{\chi}^{\lambda,\mu}$ is also an irreducible character of $\widetilde{\mathfrak{H}}_\ell$, and furthermore $\widetilde{\chi}^{\lambda,\mu} = \widetilde{\chi}^{\mu,\lambda}$.
But if $\lambda = \mu$, then $\widetilde{\chi}^{\lambda,\lambda}$ is the sum of two irreducible characters of $\widetilde{\mathfrak{H}}_\ell$, which we denote as $\widetilde{\chi}^{\lambda,\lambda}_1$ and $\widetilde{\chi}^{\lambda,\lambda}_2$.
All irreducible characters of $\widetilde{\mathfrak{H}}_\ell$ can be obtained in this way.
Moreover, the characters $\chi^{\lambda,\mu}$, $\chi^{\mu,\lambda}$ and $\widetilde{\chi}^{\lambda,\mu}$ are connected by the following formula:
\begin{align}
    \dfrac{\chi^{\lambda,\mu}(\eta) + \chi^{\mu,\lambda}(\eta)}{2} = \begin{cases*}
        \widetilde{\chi}^{\lambda,\mu}(\eta) & if $\eta \in \widetilde{\mathfrak{H}}_\ell$;\\
        0 & if $\eta \not\in \widetilde{\mathfrak{H}}_\ell$.
    \end{cases*} \label{1.3}
\end{align}
The irreducible characters $\widetilde{\chi}^{\lambda,\lambda}_1$ and $\widetilde{\chi}^{\lambda,\lambda}_2$ are connected by the following relation:
\begin{align}
    \widetilde{\chi}^{\lambda,\lambda}_1(\eta) = \widetilde{\chi}^{\lambda,\lambda}_2(\tau_1^{-1}\eta\tau_1) \quad \text{for all $\eta \in \widetilde{\mathfrak{H}}_\ell$}. \label{1.6}
\end{align}
Note that $\eta$ and $\tau_1^{-1}\eta\tau_1$ are not necessarily conjugate as seen in \eqref{1.4}.


%=====================================%

\section{Lattices generated by the standard basis}

\subsection{Calculation results of the multiplicities}\label{sec2.1}

In this section, suppose that $L = \mathbb{Z}^\ell$.
Let $W$ be a Weyl group of type $A_{\ell-1}$, $B_\ell$, $C_\ell$ or $D_\ell$.
We compute the multiplicity $m(\chi;\, q)$ of each irreducible character $\chi$ in $\chi_{\mathbb{Z}^\ell_q}$.
Take a $\mathbb{Z}$-basis $\{e_1,\ldots,e_\ell\}$ of $\mathbb{Z}^\ell$, that is, $\mathbb{Z}^\ell = \mathbb{Z}e_1 \oplus \cdots \oplus \mathbb{Z}e_\ell$.
Then $W$ acts on $\mathbb{Z}^\ell$ by
\begin{align*}
w(e_i) = \begin{cases*}
    e_{w(i)} & if $w(i) > 0$;\\
    -e_{-w(i)} & if $w(i) < 0$
\end{cases*} \qquad (1 \leq i \leq \ell,\ w \in W).
\end{align*}
Obviously, $W$ also acts on $\mathbb{R}^\ell = \mathbb{R}e_1 \oplus \cdots \oplus \mathbb{R}e_\ell$ in the same way.
This is the natural action as a finite reflection group.

For $w \in W$, let $R_w$ be the representation matrix of $w$ and $r(w)$ denote the rank of matrix $R_w - I$, where $I$ is the identity matrix.

\begin{lemma}\label{l2.1}
%    Let $W$ be a classical Weyl group.
    For $w \in W$, the following values are all equal to $r(w)$:
    \begin{eenumerate}
        \item The number of eigenvalues of $w$ which are not equal to $1$ (counted with multiplicity); \label{l211}
        \item $\ell - \ell(\lambda)$; \label{l212}
        \item $\ell - \dim{(\mathbb{R}^\ell)^w}$; \label{l213}
        \item The minimum number of reflections required to express $w$ as a product of reflections, \label{l214}
    \end{eenumerate}
    where $(\lambda,\mu)$ is the cycle type of $w$ and $(\mathbb{R}^\ell)^w = \set{x \in \mathbb{R}^\ell | w(x) = x}$.
\end{lemma}

We obtain the elementary divisors $e_{w,1},\ldots,e_{w,r(w)}$ of $R_w-I$ by computed directly as follows:

\begin{lemma}\label{l2.2}
    Let $w \in W$ with cycle type $(\lambda,\mu)$.
    Then 
    \begin{align*}
        e_{w,j} = \begin{cases*}
            1 & {if} $\ j \leq r(w) - \ell(\mu)$;\\
            2 & {if} $\ j > r(w) - \ell(\mu)$
        \end{cases*}\quad (1 \leq j \leq r(w)).
    \end{align*}
\end{lemma}

Hence, using \eqref{s11}, $m(\chi;\, q)$ is obtained as follows:
\begin{theorem}\label{t2.3}
    For an irreducible character $\chi$ of $W$, we have
    \begin{align*}
        m(\chi;\, q) = \dfrac{1}{\#W}\sum_{w \in W}\chi(w) g(q)^{\ell(\mu_w)}q^{\ell(\lambda_w)},
    \end{align*}
    where $(\lambda_w,\mu_w)$ is the cycle type of $w$ and $g(q) \ceq \gcd\{2,q\}$.
\end{theorem}


Let $\bm{1}$ denote the trivial character of $W$.
By the Shephard--Todd formula (\cite{ShephardTodd}, see also \cite[Theorem 4.6]{OrlikSolomonTerao}), 
the first constituent $m(\bm{1};\, t)_1$ of the quasi-polynomial $m(\bm{1};\, t)$ can be expressed as
\begin{align}
m(\bm{1};\, t)_1 = \dfrac{1}{\#W}(t+m_1) \cdots (t+m_\ell), \label{st}
\end{align}
where $m_1,\ldots,m_\ell$ are the exponents of $W$.

Let $\delta$ be the determinant character of $W$.
By \eqref{st} and the reciprocity theorem (\cite[Theorem 2.10]{UchiumiYoshinaga}),
the first constituent $m(\delta;\, t)_1$ of the quasi-polynomial $m(\delta;\, t)$ can be expressed as
\begin{align*}
    m(\delta;\, t)_1 = \dfrac{1}{\#W}(t-m_1) \cdots (t-m_\ell).
\end{align*}
It is well known that the above polynomial on the right hand side is equal to the characteristic polynomial of the Coxeter arrangement of $W$ up to scalar (see \cite[Corollary 3.3]{OrlikSolomonTerao}).



\subsection{Factorization of the multiplicity}

Let $W$ be a classical Weyl group.
For a character $\chi$ of $W$, define $M_W(\chi;\, s,t) \in \mathbb{Z}[s,t]$ by
\begin{align*}
    M_W(\chi;\, s,t) \ceq \sum_{w \in W}\chi(w)s^{\ell(\mu_w)}t^{\ell(\lambda_w)},
\end{align*}
where $(\lambda_w,\mu_w)$ is the cycle type of $w$.
For simplicity, we denote $M_W(\chi;\, 1,t)$ (for $s = 1$) by $M_W(\chi;\, t)$.
Define a function $g(q) \ceq \gcd\{2,q\}$.
From \Cref{t2.3}, we have
\begin{align}
    m(\chi;\, q) = \dfrac{M_W\bigl(\chi;\, g(q),\, q\bigr)}{\#W}. \label{2.1}
\end{align}


Let $\lambda$ be an integer partition. Define a multiset $A_\lambda$ as
\begin{align*}
    A_\lambda \ceq \bigsqcup_{i=1}^{\ell(\lambda)}\set{i-j | 1 \leq j \leq \lambda_i}.
\end{align*}
%The multiset $A_\lambda$ is the collection of integers obtained by placing the number $i-j$ in the $(i,j)$-cell of Young diagram $\lambda$.


\subsubsection{Type {\texorpdfstring{$A_{\ell-1}$}{Al-1}}}

Suppose that $W = \mathfrak{S}_\ell$ (type $A_{\ell-1}$).
Let $\chi^\lambda$ denote the irreducible character parameterized by an integer partition $\lambda$.
Since $\ell(\mu_\sigma) = 0$ for all $\sigma \in \mathfrak{S}_\ell$, then $M_{\mathfrak{S}_\ell}({\chi^\lambda};\, t) = M_{\mathfrak{S}_\ell}({\chi^\lambda};\, s,t)$.
It is known that the factorization formula for $M_{\mathfrak{S}_\ell}(\chi^\lambda;\, t)$:

\begin{theorem}[Littlewood {\cite[p.\ 56]{Littlewood}}, Molchanov {\cite[Theorem 1]{Molchanov}}]
The roots of the polynomial $M_{\mathfrak{S}_\ell}(\chi^\lambda;\, t)$ are the numbers in $A_\lambda$:
    \begin{align*}
        M_{\mathfrak{S}_\ell}(\chi^\lambda;\, t) = \chi^\lambda(1) \prod_{a \in A_\lambda}(t-a) = \chi^\lambda(1)\prod_{i=1}^{\ell(\lambda)}\prod_{j = 1}^{\lambda_i}(t-i+j).
    \end{align*}
\end{theorem}

\begin{corollary}
    Let $\chi^\lambda$ be the irreducible character of $\mathfrak{S}_\ell$ parameterized by an integer partition $\lambda$.
    Then we have
    \begin{align*}
        m(\chi^\lambda;\, q) = \dfrac{\chi^\lambda(1)}{\ell!}\prod_{a \in A_\lambda}(q-a) = \dfrac{\chi^\lambda(1)}{\ell!}\prod_{i=1}^{\ell(\lambda)}\prod_{j = 1}^{\lambda_i}(q-i+j).
    \end{align*}
    Hence $m(\chi^{\lambda};\, q)$ is a quasi-polynomial with the minimal period $1$ (just a polynomial).
\end{corollary}




%\begin{example}
%    Let $\ell = 5$ and $\lambda = (2,2,1)$.
%    Then $\lambda^\ast = (3,2)$, and hence
%    \begin{align*}
%        A_\lambda : \begin{ytableau}
%            0 & -1 \\
%            1 & 0 \\
%            2
%        \end{ytableau}\, ,\qquad m(\chi^\lambda;\, q) = \dfrac{5q^2(q-1)(q+1)(q-2)}{120} = \dfrac{q^5 - 2q^4 - q^3 + 2q^2}{24}.
%    \end{align*}
%    Moreover $\lambda^\ast = (3,2)$, and hence
%    \begin{align*}
%    A_{\lambda^\ast} : \begin{ytableau}
%            0 & -1 & -2\\
%            1 & 0
%        \end{ytableau}\, ,\qquad m(\chi^{\lambda^\ast};\, -q) = \dfrac{5q^2(q-1)(q+1)(q+2)}{120} = \dfrac{q^5 + 2q^4 - q^3 - 2q^2}{24}.
%    \end{align*}
%\end{example}


\subsubsection{Type {\texorpdfstring{$B_\ell$}{Bl}} and {\texorpdfstring{$C_\ell$}{Cl}}}

Suppose that $W = \mathfrak{H}_\ell$ (type $B_\ell$ or $C_\ell$).
For the irreducible character $\chi^{\lambda,\mu}$, it is known by Young \cite{Young} in the following formula:
\begin{align}
M_{\mathfrak{H}_\ell}(\chi^{\lambda,\mu};\, s,t) = \dfrac{2^\ell\ell!}{|\lambda|!|\mu|!}M_{\mathfrak{S}_{|\lambda|}}\left(\chi^\lambda;\, \dfrac{t+s}{2}
\right)M_{\mathfrak{S}_{|\mu|}}\left(\chi^\mu;\, \dfrac{t-s}{2}
\right), \label{2.2}
\end{align}
where we consider that $M_{\mathfrak{S}_{|\emptyset|}}(\chi^\emptyset;\, \cdot) = 1$.
%Using the dimensional relation \eqref{1.2}, we have
%\begin{align}
%\begin{split}
%    M_{\mathfrak{H}_\ell}(\chi^{\lambda,\mu};\, s,t)
%    & = \dfrac{2^\ell \chi^{\lambda,\mu}(1)}{\chi^\lambda(1)\chi^\mu(1)}M_{\mathfrak{S}_{|\lambda|}}\left(\chi^\lambda;\, \dfrac{t+s}{2}
%\right)M_{\mathfrak{S}_{|\mu|}}\left(\chi^\mu;\, \dfrac{t-s}{2}
%\right)
%    &= \chi^{\lambda,\mu}(1)\prod^{{}^{\ }}_{a \in A_\lambda}(t+s-2a) \prod_{b \in A_\mu}(t-s-2b)\\
%    & = \chi^{\lambda,\mu}(1)\prod_{i=1}^{\ell(\lambda)}\prod_{j = 1}^{\lambda_i}\Bigl({t+s}-2(i-j)\Bigr)\prod_{i=1}^{\ell(\mu)}\prod_{j = 1}^{\mu_i}\Bigl({t-s}-2(i-j)\Bigr).
%\end{split}\label{2.3}
%\end{align}

\begin{theorem}\label{t2.8}
    Let $\chi^{\lambda,\mu}$ be the irreducible character of $\mathfrak{H}_\ell$ parameterized by $(\lambda,\mu)$.
    Then we have 
    \begin{align*}
        m(\chi^{\lambda,\mu};\, q) &= m\left(\chi^\lambda;\, \dfrac{q + g(q)}{2}\right) \cdot m\left(\chi^\mu;\, \dfrac{q - g(q)}{2}\right)\\
        & = \dfrac{\chi^{\lambda,\mu}(1)}{2^\ell\ell!}\prod_{a \in A_\lambda}(q+g(q)-2a) \prod_{b \in A_\mu}(q-g(q)-2b).
%        & = \dfrac{\chi^{\lambda,\mu}(1)}{2^\ell\ell!}\prod_{i=1}^{\ell(\lambda)}\prod_{j = 1}^{\lambda_i}\Bigl({q+g(q)}-2(i-j)\Bigr)\prod_{i=1}^{\ell(\mu)}\prod_{j = 1}^{\mu_i}\Bigl({q-g(q)}-2(i-j)\Bigr).
    \end{align*}
    Hence $m(\chi^{\lambda,\mu};\, q)$ is a quasi-polynomial with the minimal period $2$.
\end{theorem}
\begin{proof}
    It follows from equations \eqref{2.1}, \eqref{2.2} and the dimensional relation \eqref{1.2}.
\end{proof}



\subsubsection{Type {\texorpdfstring{$D_\ell$}{Dl}}}

Suppose that $W = \widetilde{\mathfrak{H}}_\ell$ (type $D_\ell$).
Let $(\lambda,\mu)$ be the pair of integer partitions satisfying $|\lambda| + |\mu| = \ell$.
From the equation \eqref{1.3}, we have
\begin{align}
\begin{split}
    M_{\widetilde{\mathfrak{H}}_\ell}(\widetilde{\chi}^{\lambda,\mu};\, s,t) &= \sum_{\eta \in \mathfrak{H}_\ell}\dfrac{\chi^{\lambda,\mu}(\eta) + \chi^{\mu,\lambda}(\eta)}{2}s^{\ell(\mu_\eta)}t^{\ell(\lambda_\eta)}
    = \dfrac{M_{\mathfrak{H}_\ell}(\chi^{\lambda,\mu};\, s,t) + M_{\mathfrak{H}_\ell}(\chi^{\mu,\lambda};\, s,t)}{2}.
\end{split} \label{2.4}
\end{align}

\begin{theorem}\label{t2.9}
    Let $\widetilde{\chi}^{\lambda,\mu}$ be the irreducible character of $\widetilde{\mathfrak{H}}_\ell$ parameterized by $(\lambda,\mu)$ satisfying $\lambda \neq \mu$.
    Then we have
    \begin{align*}
        m(\widetilde{\chi}^{\lambda,\mu};\, q) &= m(\chi^{\lambda,\mu};\, q) + m(\chi^{\mu,\lambda};\, q)\\
        &= m\left(\chi^\lambda;\, \dfrac{q + g(q)}{2}\right) \cdot m\left(\chi^\mu;\, \dfrac{q - g(q)}{2}\right)\\
        & \qquad \qquad \qquad \qquad + m\left(\chi^\mu;\, \dfrac{q + g(q)}{2}\right) \cdot m\left(\chi^\lambda;\, \dfrac{q - g(q)}{2}\right).
    \end{align*}
    Hence $m(\widetilde{\chi}^{\lambda,\mu};\, q)$ is a quasi-polynomial with the minimal period $2$.
\end{theorem}
\begin{proof}
    It follows from the equation \eqref{2.4} and \Cref{t2.8}.
%    For the case $g(q) = 1$, see also \RED{Molchanov}.
\end{proof}

Suppose that $\lambda = \mu$.
By the relation \eqref{1.6}, we have
\begin{align*}
    M_{\widetilde{\mathfrak{H}}_\ell}(\widetilde{\chi}^{\lambda,\lambda}_1;\, s,t) = M_{\widetilde{\mathfrak{H}}_\ell}(\widetilde{\chi}^{\lambda,\lambda}_2;\, s,t).
\end{align*}
Since $\widetilde{\chi}^{\lambda,\lambda} = \widetilde{\chi}^{\lambda,\lambda}_1 + \widetilde{\chi}^{\lambda,\lambda}_2$, then
\begin{align}
    M_{\widetilde{\mathfrak{H}}_\ell}(\widetilde{\chi}^{\lambda,\lambda}_i;\, s,t) = \dfrac{M_{\widetilde{\mathfrak{H}}_\ell}(\widetilde{\chi}^{\lambda,\lambda}_1;\, s,t) + M_{\widetilde{\mathfrak{H}}_\ell}(\widetilde{\chi}^{\lambda,\lambda}_2;\, s,t)}{2}
     = \dfrac{M_{\widetilde{\mathfrak{H}}_\ell}(\widetilde{\chi}^{\lambda,\lambda};\, s,t)}{2}. \label{2.5}
\end{align}
for $i \in \{1,2\}$.
\begin{theorem}
    Let $\widetilde{\chi}^{\lambda,\lambda}_1$ and $\widetilde{\chi}^{\lambda,\lambda}_2$ be the irreducible characters of $\widetilde{\mathfrak{H}}_\ell$ parameterized by $(\lambda,\lambda)$.
    Then we have
    \begin{align*}
        m(\widetilde{\chi}^{\lambda,\lambda}_1;\, q) = m(\widetilde{\chi}^{\lambda,\lambda}_2;\, q) = m(\chi^{\lambda,\lambda};\, q) = m\left(\chi^\lambda;\, \dfrac{q + g(q)}{2}\right) \cdot m\left(\chi^\lambda;\, \dfrac{q - g(q)}{2}\right).
    \end{align*}
\end{theorem}
\begin{proof}
    It follows from equations \eqref{2.4}, \eqref{2.5} and \Cref{t2.8}.
    %For the case $g(q) = 1$, see also \RED{Molchanov}.
\end{proof}



In the case of type $A_{\ell-1}$, $B_\ell$ and $C_\ell$, all the roots of each constituent of the quasi-polynomial $m(\chi;\, q)$ are integers.
However, it is not always the case in type $D_\ell$.
This phenomenon has also been observed in characteristic quasi-polynomials of the arrangements of root systems (see \cite[Example 3.5]{KamiyaTakemuraTeraoR}).

\begin{example}
    Let $\ell = 7$, $\lambda = (2,1,1)$ and $\mu = (3)$.
    Then 
    \begin{align*}
        m(\chi^{\lambda,\mu}; q) &= \begin{cases*}
            \dfrac{105(q+1)(q-1)(q+3)(q-3) \cdot (q+1)(q-1)(q+3)}{2^77!} & if $\gcd\{2,q\} = 1$;\vspace{2mm}\\
            \dfrac{105q(q+2)(q-2)(q+4) \cdot q(q+2)(q-2)}{2^77!} & if $\gcd\{2,q\} = 2$,
        \end{cases*}\\[8pt]
        m(\chi^{\mu,\lambda}; q) &= \begin{cases*}
            \dfrac{105(q+1)(q+3)(q+5) \cdot (q+1)(q-1)(q-3)(q-5)}{2^77!} & if $\gcd\{2,q\} = 1$;\vspace{2mm}\\
            \dfrac{105(q+2)(q+4)(q+6) \cdot q(q-2)(q-4)(q-6)}{2^77!} & if $\gcd\{2,q\} = 2$
        \end{cases*}
    \end{align*}
    and
    \begin{align*}
        m(\widetilde{\chi}^{\lambda,\mu};\, q) &= m(\chi^{\lambda,\mu};\, q) + m(\chi^{\mu,\lambda};\, q)\\
        &= \begin{cases*}
            \dfrac{105(q+1)^2(q-1)(q+3)(q-3)(q^2+q-14)}{2^67!} & if $\gcd\{2,q\} = 1$;\vspace{2mm}\\
            \dfrac{105q(q+2)(q-2)(q+4)(q^3-2q^2-20q+72)}{2^67!} & if $\gcd\{2,q\} = 2$.
        \end{cases*}
    \end{align*}
\end{example}
Note that $t^2+t-14$ and $t^3-2t^2-20t+72$ are irreducible polynomials over $\mathbb{Z}[t]$.

%=====================================%

\section{Coroot lattices}
Let $\PHI$ be a root system of type $A_{\ell-1}$, $B_\ell$, $C_\ell$ or $D_\ell$ and $W = W(\PHI)$ be the Weyl group of $\PHI$.
In this section, suppose that $L$ is the coroot lattice $\veeQ = \veeQ(\PHI)$ of $\PHI$:
\begin{align*}
    \veeQ = \veeQ(\PHI) = \sum_{\alpha \in \PHI}\mathbb{Z}\alpha^\vee,\quad \alpha^\vee \ceq \dfrac{2\alpha}{(\alpha,\alpha)}.
\end{align*}
We compute the multiplicity $m(\chi;\, q)$ of each irreducible character $\chi$ in $\chi_{\veeQ^\ell_q}$.
Especially in this case, $m(\bm{1};\, q)$ is the Ehrhart quasi-polynomial of the fundamental alcove of $\PHI$ with respect to $\veeQ$.
Haiman \cite[Section 7.4]{Haiman} details the case where $q$ is relatively prime to the Coxeter number.


\subsection{Type {\texorpdfstring{$A_{\ell-1}$}{Al-1}}}
Suppose that $\PHI$ is a root system of type $A_{\ell-1}$. Then $W = \mathfrak{S}_{\ell}$ and the coroot lattice $\veeQ$ can be expressed by
\begin{align*}
    \veeQ = \mathbb{Z}(e_1-e_2) \oplus \cdots \oplus \mathbb{Z}(e_{\ell-1} - e_\ell).
\end{align*}
Similar to \Cref{l2.1}, for $\sigma \in \mathfrak{S}_\ell$, $r(\sigma)$ is obtained as $r(\sigma) = \ell - \ell(\lambda)$, where $\lambda$ is the cycle type of $\sigma$.
We obtain the elementary divisors $e_{\sigma,1},\ldots,e_{\sigma,r(\sigma)}$ of $R_\sigma - I$ by computed directly as follows:
\begin{lemma}
    Let $\sigma \in \mathfrak{S}_\ell$ with cycle type $\lambda$.
    Then
    \begin{align*}
        e_{\sigma,1} = \cdots = e_{\sigma,r(\sigma)-1} = 1,\quad e_{\sigma,r(\sigma)} = \gcd\{\lambda_1,\ldots,\lambda_{\ell(\lambda)}\}.
    \end{align*}
\end{lemma}
\begin{theorem}
    Let $\chi^{\lambda}$ be the irreducible character of $\mathfrak{S}_\ell$.
    Then $m(\chi^\lambda;\, q)$ is a quasi-polynomial with minimal period $\ell$.
    In particular, if $\gcd\{\ell,q\} = 1$, then
    \begin{align*}
        m(\chi^\lambda;\, q) = \dfrac{q^{-1} M_{\mathfrak{S}_\ell}(\chi^\lambda;\, q)}{\ell!} = \dfrac{\chi^\lambda(1)}{\ell!}q^{-1}\prod_{a \in A_\lambda}(q-a).
    \end{align*}
\end{theorem}

\subsection{Type {\texorpdfstring{$B_\ell$}{Bl}}}
Suppose that $\PHI$ is a root system of type $B_\ell$. 
The coroot lattice $\veeQ$ can be expressed by 
\begin{align*}
    \veeQ = \mathbb{Z}(e_1-e_2) \oplus \cdots \oplus \mathbb{Z}(e_{\ell-1}-e_\ell) \oplus \mathbb{Z}(2e_\ell).
\end{align*}
\begin{lemma}\label{l3.3}
    Let $\eta \in \mathfrak{H}_\ell$ with cycle type $(\lambda,\mu)$.
    Then $r(\eta) = \ell - \ell(\lambda)$ and the following holds:
%    \begin{eenumerate}
%        \item Suppose that $\mu = \emptyset$.
            \begin{eenumerateii}
            %\item $r(\eta) = \ell - \ell(\lambda)$.
                \item If $\lambda$ has at least an odd part, then $e_{\eta,j} = \begin{cases*}
                    1 & if $\ j \leq r(\eta) - \ell(\mu) + 1$;\\
                    2 & if $\ j > r(\eta) - \ell(\mu) + 1$
                \end{cases*}\quad (1 \leq j \leq r(\eta))$.
                \item If $\lambda$ has all even parts and $\mu = \emptyset$, then $e_{\eta,j} = \begin{cases*}
                    1 &  {if} $\ j < r(\eta)$;\\
                    2 &  {if} $\ j = r(\eta)$.
                \end{cases*}$
%            \end{eenumeratei}
%        \item Suppose that $\mu \neq \emptyset$.
%            \begin{eenumerateii}
%                \item If $\lambda$ has odd, then $e_{\eta,j} =
%                \begin{cases*}
%                    1 & if $\ j \leq r(\eta) - \ell(\mu) + 1$;\\
%                    2 & if $\ j > r(\eta) - \ell(\mu) + 1$.
%                \end{cases*}$
                \item If $\lambda$ has all even parts and $\mu$ has all even or all odd parts, $ e_{\eta,j} = \begin{cases*}
                    1 & if $\ j \leq r(\eta) - \ell(\mu)$;\\
                    2 & if $\ j > r(\eta) - \ell(\mu)$.
                \end{cases*}$
                \item If $\lambda$ has all even parts, $\mu$ has both even and odd parts, then 
                \begin{align*}
                e_{\eta,j} = \begin{cases*}
                    1 & if $\ j \leq r(\eta) - \ell(\mu) + 1$;\\
                    2 & if $\ r(\eta) - \ell(\mu)+1 < j < r(\eta)$;\\
                    4 & if $\ j = r(\eta)$.
                \end{cases*}
                \end{align*}
            \end{eenumerateii}
%    \end{eenumerate}
\end{lemma}
\begin{theorem}
    Let $\chi$ be the irreducible character of $\mathfrak{H}_\ell$.
    Then $m(\chi;\, q)$ is a quasi-polynomial with minimal period 2 for $\ell = 2$ and 4 for $\ell \geq 3$.
\end{theorem}

\subsection{Type {\texorpdfstring{$C_\ell$}{Cl}}}
Suppose that $\PHI$ is a root system of type $C_\ell$.
The coroot lattice $\veeQ$ cen be expressed by 
\begin{align*}
    \veeQ = \mathbb{Z}(e_1-e_2) \oplus \cdots \oplus \mathbb{Z}(e_{\ell-1}-e_\ell) \oplus \mathbb{Z}e_\ell.
\end{align*}
Since $\veeQ$ is isomorphic to $\mathbb{Z}^\ell$, $r(\eta)$ and $e_{\eta,1},\ldots,e_{\eta,r(\eta)}$ are as in \Cref{l2.1} and \Cref{l2.2}.

\begin{theorem}
    Let $\chi$ be the irreducible character of $\mathfrak{H}_\ell$.
    Then $m(\chi;\, q)$ is a quasi-polynomial with minimal period 2.
\end{theorem}

\subsection{Type {\texorpdfstring{$D_\ell$}{Dl}}}
Suppose that $\PHI$ is a root system of type $D_\ell$.
The coroot lattice $\veeQ$ cen be expressed by 
\begin{align*}
    \veeQ = \mathbb{Z}(e_1-e_2) \oplus \cdots \oplus \mathbb{Z}(e_{\ell-1}-e_\ell) \oplus \mathbb{Z}(e_{\ell-1}+e_\ell).
\end{align*}
Since $\veeQ$ is isomorphic to the coroot lattice of type $B_\ell$, $r(\eta)$ and $e_{\eta,1},\ldots,e_{\eta,r(\eta)}$ are as in \Cref{l3.3}.

\begin{theorem}
    Let $\widetilde{\chi}$ be the irreducible character of $\widetilde{\mathfrak{H}}_\ell$.
    Then $m(\widetilde{\chi};\, q)$ is a quasi-polynomial with minimal period 2 for $\ell = 2$ and 4 for $\ell \geq 3$.
\end{theorem}

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