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\title{Lusztig \texorpdfstring{$q$}{q}-weight multiplicities and KR crystals}

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\author{\texorpdfstring{Hyeonjae Choi\thanks{\href{mailto:2pun4ani@snu.ac.kr}{2pun4ani@snu.ac.kr}. H. J. Choi is supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (No.0450-20240021)}\addressmark{1}, Donghyun Kim\thanks{\href{mailto:hyun920310@snu.ac.kr}{hyun920310@snu.ac.kr}. D.H. Kim was supported by NRF grants 2022R1I1A1A01070620.}\addressmark{1}, \and Seung Jin Lee\thanks{\href{mailto:lsjin@snu.ac.kr}{lsjin@snu.ac.kr}. S. J. Lee was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (No.0450-20240021)}\addressmark{1}}{Hyeonjae Choi, Donghyun Kim, and Seung Jin Lee}}

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\address{\addressmark{1}Department of Mathematical Sciences, Seoul National University, Seoul, Korea }

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\abstract{
Lusztig $q$-weight multiplicities extend the Kostka--Foulkes polynomials to a broader range of Lie types. 
In this work, we investigate these multiplicities through the framework of Kirillov--Reshetikhin crystals. 
Specifically, for type $C$ and type $B$ spin weights, we present a combinatorial formula for Lusztig $q$-weight multiplicities in terms of energy functions of Kirillov--Reshetikhin crystals, generalizing the charge statistic on semistandard Young tableaux for type $A$.
Additionally, we introduce level-restricted $q$-weight multiplicities for nonexceptional types, and prove its positivity by providing their combinatorial formulas.
}

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\keywords{Lusztig $q$-weight multiplicity, Kirillov--Reshetikhin crystals, Energy function, Level-restricted $q$-weight multiplicity, Kostka--Foulkes polynomial}

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\begin{document}

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\section{Introduction}

The \emph{Kostka--Foulkes polynomial} $K_{\lambda,\mu}(q)$ is the coefficient of the modified Hall--Littlewood polynomials when expressed in the Schur basis. 
One of the famous formulas for $K_{\lambda,\mu}(q)$ is due to Lascoux--Sch\"utzenberger  \cite{LasSch1978}:
\begin{equation}\label{eq: charge}
K_{\lambda,\mu}(q) = \sum_{T \in \SSYT(\lambda,\mu)} q^{\text{charge}(T)}
\end{equation}
where the sum is taken over all semistandard Young tableaux of shape $\lambda$ with weight $\mu$ and charge$(T)$ is a certain nonnegative integer associated to a semistandard Young tableau $T$.
The Kostka--Foulkes polynomial originates from geometry and representation theory, with connections to flag varieties, Springer fibers, and intersection homology \cite{Kir99}.
Due to its ubiquity and rich structure, studying Kostka--Foulkes polynomial has been an active area of research.
There exist several ways to generalize the Kostka--Foulkes polynomials, including Macdonald polynomials and Lusztig $q$-weight multiplicity. 
Lusztig $q$-weight multiplcity is a generalization of Kostka--Foulkes polynomials for any Lie type.
In this paper, we mainly consider Lusztig $q$-weight multiplicities beyond type $A$.

Let $\mathfrak{g}_n$ be a classical simple Lie algebra and let $L: R^{+}\rightarrow \mathbb{Z}$ be a function from the set of positive roots. 
For dominant weights $\lambda, \mu$, we define 
\begin{equation}\label{eq:q-analog}
    \operatorname{KL}^{\mathfrak{g}_n,L}_{\lambda,\mu}(q)=\sum_{w\in W}(-1)^w[e^{w(\lambda + \rho)-(\mu + \rho)}]\prod_{\alpha\in R^{+}}\frac{1}{1-q^{L(\alpha)}e^\alpha}
\end{equation}
where $[e^\beta]f$ denotes the coefficient of $e^\beta$ in $f$, $\rho = \frac{1}{2}\sum_{\alpha \in R^{+}}\alpha$, and $W$ is the Weyl group of $\mathfrak{g}_n$.
Notably, $\KL_{\lambda,\mu}^{\mathfrak{g}_n,L}(1)$ equals the weight multiplicity.
While, in general, $\KL^{\mathfrak{g}_n,L}_{\lambda,\mu}(q)$ is not a polynomial with nonnegative coefficients, we examine two cases where it is.
% polynomial in q

First, we take $L\equiv1$, i.e., $L(\alpha)=1$ for all $\alpha\in R^{+}$. 
Then $\KL^{\mathfrak{g}_n,L\equiv1}_{\lambda,\mu}(q)$, which we simply denote by $\KL^{\mathfrak{g}_n}_{\lambda,\mu}(q)$, is a famous \textit{Lusztig $q$-weight multiplicity} \cite{Lusztig1983}. 
Its nonnegativity follows from the theory of the affine Kazhdan--Lusztig polynomial.
Since the positivity of (affine) Kazhdan--Lusztig polynomials relies heavily on geometric machinery, specifically related to intersection homology theory, finding a combinatorial formula for these polynomials remains an important open problem. 

When $\mathfrak{g}_n$ is of type $A$, i.e., $\mathfrak{g}_n = A_{n-1}$, the Lusztig $q$-weight multiplicity $\KL^{A_{n-1}}_{\lambda,\mu}(q)$ recovers the Kostka--Foulkes polynomial, whose combinatorics are well understood via the charge formula by Lascoux and Sch\"utzenberger \eqref{eq: charge}.
The natural question to ask is whether there is a generalization of the charge statistic beyond type $A$ that provides a positive combinatorial formula for the Lusztig $q$-weight multiplicities.
This question has been a long-standing open problem for decades, and even when restricting to specific types, only partial results have been obtained. 
We provide a combinatorial formula for $\KL^{C_n}_{\lambda,\mu}(q)$ and $\operatorname{KL}^{B_n}_{\lambda,\mu}(q)$ for spin weights $\lambda$ and $\mu$, using the energy function in the column KR crystals (\Cref{thm: C lusztig} and \ref{thm: B lusztig}).
Note that the energy function in the column KR crystals for affine type $A$ is essentially the same as the charge statistic \cite{NY1997}.

We introduce another natural $q$-version of the weight multiplicity for nonexceptional types, which we call the \textit{level-restricted $q$-weight multiplicity}.
Consider a function $L_A$, where $L_A(\alpha) = 1$ if $\alpha$ is positive root of type $A$, i.e., $\alpha = \varepsilon_i - \varepsilon_j$ for $i<j$, and $L_A(\alpha)=0$ otherwise.
We denote the level-restricted $q$-weight multiplicity by $\KL^{\mathfrak{g}_n,L_A}_{\lambda,\mu}(q)$.
We provide a combinatorial formula for $\KL^{\mathfrak{g}_n,L_A}_{\lambda,\mu}(q)$ for nonexceptional types (\Cref{thm: level formula}). 
Notably, our findings refine the $X=K={^\infty }\KL$ theorem for tensor products of row KR crystals \cite{LS2007, S05} and provide new combinatorial objects for weight multiplicities.
%어케하지

The following table summarizes the main results:
\begin{table}[h]
    \centering
        \begin{tabular}{|c!{\vrule width 1.7pt}c|c|}
        \hline
        Type & Lusztig $q$-weight multiplicity & Level-restricted $q$-weight multiplicity\\\hline
        $B$&column KR crystal of type $D_{N+1}^{(2)} (\text{spin})$ &row KR crystal of type $D_{N+1}^{(2)}$\\\hline
        $C$&column KR crystal of type $B_{N}^{(1)}$&row KR crystal of type $C_{N}^{(1)}$\\\hline
        $D$&?&row KR crystal of type $B_{N}^{(1)}$\\\hline
        \end{tabular}
\end{table}

It is natural to try to complete the missing parts of the table. 
For Lusztig $q$-weight multiplicity in type $D$, we suspect that a formula related to the column KR crystal of affine type $C_N^{(1)}$ might exist. 
For non-spin weights in type $B$, we may need to step beyond the current framework of crystal theory, which is a big and fascinating challenge.

Another potential direction is to develop crystal structures on $\GSSOT$ or $\SSROT$ (see \Cref{combinatorial objects}), extending the work of Lee \cite{Lee2023}. 
Additionally, constructing a bijection between our new objects and existing combinatorial objects for weight multiplicities offers an interesting challenge for combinatorialists.

\section{Preliminaries}

\subsection{KR crystal}
Let $\mathfrak{g} \supset \mathfrak{g}' \supset \mathfrak{g}_0$ be an affine Kac--Moody algebra with index set $I = J \cup \{0\}$, where $\mathfrak{g}'$ is the derived subalgebra and $\mathfrak{g}_0$ is the simple Lie algebra. 
The corresponding quantized universal enveloping algebras are $U_q(\mathfrak{g}) \supset U_q'(\mathfrak{g}) \supset U_q(\mathfrak{g}_0)$.
For each pair $(r, s) \in J \times \mathbb{Z}_{\geq 0}$, there exist a finite-dimensional irreducible $U_q'(\mathfrak{g})$-module $W^{(r)}_s$, called the Kirillov--Reshetikhin (KR) module.
The KR modules $W^{(r)}_s$ have a crystal basis $B^{r,s}$, called the Kirillov--Reshetikhin (KR) crystals, for nonexceptional types \cite{OS2008}.

Let $\mathcal{C}$ be the category of tensor products of KR crystals.
The category $\mathcal{C}$ has the following remarkable properties \cite{LOS2012,LS2007}.
\begin{enumerate}
    \item For any $B_1,B_2 \in \mathcal{C}$, there exists a unique affine crystal isomorphism $R=R_{B_1,B_2} : B_2 \otimes B_1 \to B_1 \otimes B_2$, called the combinatorial R-matrix.
    \item Let $B=B_n\otimes \cdots \otimes B_1$ where $B_1, \dots , B_n \in \mathcal{C}$.
    There is a map $\overline{D}_B : B \to \mathbb{Z}$, called the energy function for $B\in\mathcal{C}$, such that $\overline{D}_B$ is constant on each $J$-component. 
    From now on, we will simply denote it as $\overline{D}$ instead of $\overline{D}_B$.
\end{enumerate}
In general, computing $\overline{D}$ is challenging, but when $B = (B^{1,1})^{\otimes n}$, the energy function is easy to compute.
Let
\begin{align*}
    a_n \otimes a_{n-1} \otimes \cdots \otimes a_1
\end{align*}
be an element in $(B^{1,1})^{\otimes n}$.
For the KR crystals of type $B_N^{(1)}$ and $C_N^{(1)}$, the energy function $\overline{D}$ is defined as $\overline{D}=\sum_{i=1}^{n-1} (n-i)H(a_{i+1},a_i)$.
For the KR crystals of type $B_N^{(1)}$, the function $H(b,a)$ is defined as follows:
\begin{align*}
    H(b,a) = \begin{cases}
    2 &\textrm{ if } a=1 \textrm{ and } b=\overline{1}\\
    1 &\textrm{ if } b \succ a \textrm{ and } (b,a)\neq (\overline{1},1) \\
    0 &\textrm{ if } b \preceq a
\end{cases}
\end{align*}
under the order $1\prec 2\prec \cdots  \prec \bar{2}\prec \bar{1}$.
For the KR crystals of type $C_N^{(1)}$, the function $H(b,a)$ is defined as follows:
\begin{align*}
    H(b,a) = \begin{cases}
    1 &\textrm{ if } b \succ a  \\
    0 &\textrm{ if } b \preceq a 
\end{cases}
\end{align*}
under the order $1\prec 2\prec  \cdots  \prec \bar{2}\prec \bar{1}$.

On the other hand, for the KR crystals of type $D_{N+1}^{(2)}$, the energy function $\overline{D}$ is defined as $\overline{D}=\sum_{i=1}^{n-1} 2(n-i)H(a_{i+1},a_i) + \vac$, where $\vac$ denotes the number of $\emptyset$ elements in $(B^{1,1})^{\otimes n}$.
The function $H(b,a)$ is defined as follows:
\begin{align*}
    H(b,a) = \begin{cases}
    1 &\textrm{ if } b \succ a \textrm{ or } a=b=\emptyset \\
    0 &\textrm{ if } b \preceq a 
\end{cases}
\end{align*}
under the order $\emptyset \prec 1\prec 2\prec 3 \prec \cdots  \prec \bar{3}\prec \bar{2}\prec \bar{1}$.
Note that for column KR crystals of affine type $A$, the energy function essentially coincides with the charge statistic \cite{NY1997}.

In \cite{Las1989}, Lascoux introduced the \textit{standardization map}, which maps SSYT to SYT while preserving the cocharge.
A generalized version of Lascoux's standardization map exists, mapping an element in $\otimes_i B^{a_i,b_i}$ to an element in $(B^{1,1})^{\otimes \sum_i a_ib_i}$ while preserving the coenergy function.
More details can be found in \cite{LOS2012, LS2007, S05}.

In this paper, we only consider row KR crystals, i.e., $B^{1,r}$, and column KR crystals, i.e., $B^{c,1}$.
For a nonnegative integer vector $\mu=(\mu_1,\mu_2,\dots,\mu_n)$, we employ the following notations:
\begin{equation*}
    B_{\mu}:=B^{1,\mu_n}\otimes\cdots\otimes B^{1,\mu_1}, \qquad B^{t}_{\mu}:=B^{{\mu_n},1}\otimes\cdots\otimes B^{{\mu_1},1}.
\end{equation*}
We say that $u \in B$ is \textit{classical highest weight element} if $e_i(b) = 0$ for all $i> 0$.
We define $\HW(B)$ to be the set of classical highest weight elements in $B$, and define $\HW(B,\lambda)=\{b\in \HW(B): \wt(b)=\lambda \}$. 




\subsection{Combinatorial objects and embedding into KR crystals}\label{combinatorial objects}

We introduce the combinatorial objects and embed them into KR crystals for certain types.

\begin{definition} Oscillating horizontal strip (ohs for short) $(\mu,\nu,\lambda)$ of length $r$ is a sequence of three partitions satisfying:
\begin{itemize}
\item $\nu / \mu$ and $\nu / \lambda$ are horizontal strips,
\item $|\nu / \mu|+|\nu / \lambda|=r$.
\end{itemize}
We say that ohs $(\mu,\nu,\lambda)$ is $g$-bounded if $\nu_1\leq g$. 
\end{definition}

\begin{definition} Generalized oscillating horizontal strip (gohs for short) $(\mu,\nu,\lambda)$ of length $r$ is a sequence of three partitions satisfying:
\begin{itemize}
\item $\nu / \mu$ and $\nu / \lambda$ are horizontal strips,
\item $|\nu / \mu|+|\nu / \lambda|=r\quad \text{or} \quad r-1$.
\end{itemize}
We say that gohs $(\mu,\nu,\lambda)$ is $g$-bounded if
\begin{itemize}
    \item $\nu_1\leq g$, if $|\nu / \mu|+|\nu / \lambda|=r$,
    \item $\nu_1+\frac{1}{2}\leq g$, if $|\nu / \mu|+|\nu / \lambda|=r-1$.
\end{itemize}
\end{definition}

\begin{definition} Reverse oscillating horizontal strip (rohs for short) $(\mu,\nu,\lambda)$ of length $r$ is a sequence of three partitions satisfying:
\begin{itemize}
\item $\mu / \nu$ and $\lambda / \nu$ are horizontal strips,
\item $|\mu / \nu|+|\lambda / \nu|=r$.
\end{itemize}
We say that rohs $(\mu,\nu,\lambda)$ is $g$-bounded if
\begin{equation*}
    \mu_1+(\lambda_1-\nu_1)+\max(\mu_2,\lambda_2)\leq 2g.
\end{equation*} 
\end{definition}

In any cases, we define the \emph{initial shape} denoted by $I(\mu,\nu,\lambda)$ to be $\mu$, and the \emph{final shape} denoted by $F(\mu,\nu,\lambda)$ to be $\lambda$.

A \textit{semistandard oscillating tableau} (SSOT for short) of shape $\lambda$ with weight $\mu$ is a sequence of ohs's $T=(T_1,\dots,T_n)$, where each $T_i$ is an ohs of length $\mu_i$, and the following conditions hold:
\begin{itemize}
    \item $I(T_1)=\emptyset$ and $F(T_n)=\lambda$,
    \item $F(T_i)=I(T_{i+1})$ for $1\leq i\leq n-1$.
\end{itemize} 
We define $c(T)$ to be the minimal number such that every $T_i$ is $c(T)$-bounded. 
%SSOTs vs SSOT??
The set of all SSOTs of shape $\lambda$ with weight $\mu$ is denoted by $\SSOT(\lambda,\mu)$, and we write $\SSOT_g(\lambda,\mu)$ for the subset of $\SSOT(\lambda,\mu)$ consisting of $\SSOT$s $T$ such that $c(T)\leq g$.
Similarly, we define a \textit{generalized semistandard oscillating tableau} (GSSOT for short) and a \textit{semistandard reverse oscillating tableau} (SSROT for short) using gohs and rohs, respectively.

For a length $n$ vector $\lambda=(\lambda_1,\lambda_2,\dots,\lambda_n)$ and a positive number $g$, we define the \textit{orthogonal complement} of $\lambda$ with respect to $g$, denoted by $\oc(\lambda,g)$, to be $(g-\lambda_n,\dots,g-\lambda_2,g-\lambda_1)$. 
We also define $\overline{\oc}(\lambda,g)$ to be $(g-\lambda_1,g-\lambda_2,\dots,g-\lambda_n)$. 
There is a bijection between King tableaux of shape $\lambda$ weight $\mu$ and $\SSOT(\oc(\lambda,g),\overline{\oc}(\mu,g)$) \cite{Lee2023}.

Given an ohs $(\mu,\nu,\lambda)$, we define $\cind(\mu,\nu,\lambda)$ to be
\begin{align*}
    \cind(\mu,\nu,\lambda):=\{i: \mu^{t}_i+1=\nu^{t}_i\}\cup \{\bar{i}: \lambda^{t}_i+1=\nu^{t}_i\}.
\end{align*}
We also define $\rind(\mu,\nu,\lambda)$ to be the multi-set consisting of $(\nu_i-\mu_i)$-many $i$'s and $(\nu_i-\lambda_i)$-many $\bar{i}$'s.
In other words, $\cind(\mu,\nu,\lambda)$ (respectively $\rind(\mu,\nu,\lambda)$) records column indices (respectively row indices) from the ohs. 
%??
We abuse the notation by identifying a set $\{v_1,\dots,v_n\}$ with the word $v_1\dots v_n$ where $v_1\prec \dots \prec v_n$. 
Here we employ the total order $\prec$ given by
\begin{equation*}
    1\prec 2\prec 3 \prec \cdots   \bar{3}\prec \bar{2}\prec \bar{1}.
\end{equation*}
For example, we identify the set $\{1,3,\bar{3}\}$ with the word $13\bar{3}$.

Now, we interpret the sets $\SSOT(\lambda,\mu)$, $\GSSOT(\lambda,\mu)$, and $\SSROT(\lambda,\mu)$ as sets of classical highest weights in specific KR crystals. 
We choose $N$ to be sufficiently large so that $N,\bar{N} \notin \cind(Y)$ and $N,\bar{N} \notin \rind(Y)$ for any ohs, gohs, or rohs $Y$ appearing in $\SSOT(\lambda,\mu)$, $\GSSOT(\lambda,\mu)$, or $\SSROT(\lambda,\mu)$.
\begin{lem}\label{lem:cind}
    \begin{enumerate}
        \item There exists a bijection 
        \begin{equation*}
            \phi_c: \SSOT(\lambda,\mu) \rightarrow \HW(B_\mu^t,\lambda^{t})
        \end{equation*}
        where $B_\mu^{t}$ is of type $B_N^{(1)}$.
        \item There exists a bijection 
        \begin{equation*}
            \phi_c: \GSSOT(\lambda,\mu) \rightarrow \HW(B_\mu^t,\lambda^{t})
        \end{equation*}
        where $B_\mu^{t}$ is of type $D_{N+1}^{(2)}$.
    \end{enumerate}
\end{lem}

\begin{lem}\label{lem: rind} 
    \begin{enumerate}
        \item There exists a bijection 
        \begin{equation*}
            \phi_r: \SSOT(\lambda,\mu) \rightarrow \HW(B_\mu,\lambda)
        \end{equation*}
        such that $c(T)=\varepsilon_0(\phi_r(T))$, where $B_\mu$ is of type $C_N^{(1)}$.
        \item There exists a bijection 
        \begin{equation*}
            \phi_r: \GSSOT(\lambda,\mu) \rightarrow \HW(B_\mu,\lambda)
        \end{equation*}
        such that $ 2c(T)=\varepsilon_0(\phi_r(T))$, where $B_\mu$ is of type $D_{N+1}^{(2)}$.
        \item There exists a bijection 
        \begin{equation*}
            \phi_r: \SSROT(\lambda,\mu) \rightarrow \HW(B_\mu,\lambda)
        \end{equation*}
        such that $2c(T)=\varepsilon_0(\phi_r(T))$, where $B_\mu$ is of type $B_{N}^{(1)}$.
    \end{enumerate}
    In the above, $\varepsilon_0(x)$ denotes the maximal number of times the $e_0$ operator can be applied to $x$.
\end{lem}

In other words, we can identify our combinatorial objects, i.e., $\SSOT$, $\GSSOT$, and $\SSROT$, with specific highest weights of KR crystals.
The map $\phi_c$ in Lemma $\ref{lem:cind}$ can be directly described using $\cind$ for an ohs, as demonstrated in the example below. 
However, the map $\phi_r$ requires a modification after applying $\rind$, which is not addressed here.

\begin{example}
    Let $\lambda = (2), \mu=(3,3)$. 
    Consider the following tableaux in $\SSOT(\lambda,\mu)$ and $\GSSOT(\lambda,\mu)$:
    \begin{align*}
        T_1 &= ((\emptyset,\ydiagram{2},\ydiagram{1}),(\ydiagram{1},\ydiagram{2,1},\ydiagram{2})) \in \SSOT(\lambda,\mu),\\
        T_2 &= ((\emptyset,\ydiagram{2},\ydiagram{2}),(\ydiagram{2},\ydiagram{2,1},\ydiagram{2})) \in \GSSOT(\lambda,\mu).
    \end{align*}
    For $T_1$, the $\cind$ of the first ohs is $\{1,2,\bar{2}\}$, and for the second ohs, it is $\{1,2,\bar{1}\}$.
    Due to the admissibility condition, $T_1$ maps to $[2] \otimes [1] \in B^{3,1}\otimes B^{3,1} $ via the map $\phi_c$. 
    Similarly, $T_2$ maps to $[\emptyset] \otimes [12] \in B^{3,1}\otimes B^{3,1} $ via the map $\phi_c$.
\end{example}


\section{Lusztig \texorpdfstring{$q$}{q}-weight multiplicity}
A partition $\mu=(\mu_1,\dots,\mu_\ell)$ is a weakly decreasing sequence of positive integers.
We say that $\mu$ is a partition of $n$, denoted by $\mu \vdash n$, if $|\mu| := \sum_{i} \mu_i = n$. 
We denote the length of $\mu$, i.e., the number of its parts, by $\ell(\mu)$.
We define $\Par_n$ to be the collection of partitions $\mu$ such that $\ell(\mu)\leq n$. 
Given $\mu\in \Par_n$, we naturally regard $\mu$ as a length $n$ vector by appending zeros if necessary. 
The set of dominant weights for type $C_n$ is $\Par_n$, and for type $B_n$, dominant weights consist of $\mu$ and $\mu^{\sharp}:=\mu+(\frac{1}{2})^n $ for $\mu\in \Par_n$.
We refer to $\mu^\sharp$ as a \emph{spin weight}.
    
\begin{thm}\label{thm: C lusztig}For $\lambda, \mu \in \Par_n$, we have
\begin{equation*}
    \KL^{C_n}_{\lambda,\mu}(q)=\sum_{T\in \SSOT_{g}(\oc(\lambda,g),\overline{\oc}(\mu,g))}q^{\overline{D}(\phi_c(T))}
\end{equation*}
where $g$ is a positive integer such that $g\geq \lambda_1$, and $\phi_c$ is the map in \Cref{lem:cind}. 
\end{thm}

\begin{rmk}\label{rmk: comb R}
Let $\alpha$ be any rearrangement of $\overline{\oc}(\mu,g)$.
Then,
\begin{equation*}
    \sum_{T\in \SSOT_{g}(\oc(\lambda,g),\overline{\oc}(\mu,g))}q^{\overline{D}(\phi_c(T))}=\sum_{T\in \SSOT_{g}(\oc(\lambda,g),\alpha)}q^{\overline{D}(\phi_c(T))}.
\end{equation*}
We choose $\alpha = \overline\oc(\mu,g)$ for the convenience of the proof.
Moreover, it can be shown that the right-hand side of \Cref{thm: C lusztig} is independent of the choice of $g\geq \lambda_1$.
\end{rmk}

\begin{example}\label{ex : C lusztig}
    Let $\lambda=(1,1,0,0), \mu = (0,0,0,0)$, and $g=1$. 
    Then, we have $\KL^{C_4}_{\lambda,\mu}(q) = q^6+q^4+q^2$.
    The set $\SSOT_1(\oc(\lambda,g),\overline{\oc}(\mu,g))$ consists of: 
    \begin{align*}
    T_1 &= ((\emptyset , \ydiagram{1}, \ydiagram{1}),(\ydiagram{1} , \ydiagram{1}, \emptyset),(\emptyset , \ydiagram{1}, \ydiagram{1}),(\ydiagram{1} , \ydiagram{1,1} , \ydiagram{1,1})),\\
    T_2 &= ((\emptyset , \ydiagram{1}, \ydiagram{1}),(\ydiagram{1} , \ydiagram{1,1}, \ydiagram{1,1}),(\ydiagram{1,1} , \ydiagram{1}, \ydiagram{1}),(\ydiagram{1} , \ydiagram{1,1} , \ydiagram{1,1})), \\
    T_3 &= ((\emptyset , \ydiagram{1}, \ydiagram{1}),(\ydiagram{1} , \ydiagram{1,1}, \ydiagram{1,1}),(\ydiagram{1,1} , \ydiagram{1,1}, \ydiagram{1,1,1}),(\ydiagram{1,1,1} , \ydiagram{1,1} , \ydiagram{1,1})).
    \end{align*}
     Via the map $\phi_c$, we embed $T_i$'s into $(B^{1,1})^{\otimes 4}$ of type $B_N^{(1)}$ as follows: 
     \begin{align*}
        \phi_c(T_1) = 1 \otimes 1 \otimes \bar{1} \otimes 1, \quad  
        \phi_c(T_2) = 1 \otimes \bar{1} \otimes 1 \otimes 1, \quad 
        \phi_c(T_3) = \bar{1} \otimes 1 \otimes 1 \otimes 1.
     \end{align*}
     The energy function values are:
     \begin{align*}
         \overline{D}(\phi_c(T_1)) = 6, \quad \overline{D}(\phi_c(T_2)) = 4, \quad \overline{D}(\phi_c(T_3)) = 2.
     \end{align*} 
     These values coincide with $\KL^{C_4}_{\lambda,\mu}(q) = q^6+q^4+q^2$.
\end{example}

For Lusztig $q$-weight multiplicities of type $B$, we introduce a $q,t$-generalization. 
The set of positive roots $R^{+}$ consists of $(\varepsilon_{i}\pm \varepsilon_{j})$'s for $i<j$ , referred to as long roots, and $\varepsilon_i$'s, referred to as short roots. 
Given dominant weights $\lambda, \mu$, we define $q,t$-weight multiplicities by
\begin{equation*}
    \KL^{B_n}_{\lambda,\mu}(q,t):=\sum_{w\in W}(-1)^w[e^{w(\lambda + \rho)-(\mu + \rho)}]\prod_{\substack{\alpha\in R^{+}\\ \text{$\alpha$ is a long root}}}\frac{1}{1-q e^\alpha}\prod_{\substack{\alpha\in R^{+}\\ \text{$\alpha$ is a short root}}}\frac{1}{1-t e^\alpha}.
\end{equation*}
The usual Lusztig $q$-weight multiplicities can be recovered by setting $t=q$. 
Consider $ B^{t}_{\mu} $ of type $ D_{N+1}^{(2)}$ and  $b = b_n \otimes \dots \otimes b_2 \otimes b_1 \in B^{t}_{\mu}$.
Recall that  $B^{r,1}$ of type  $D_{N+1}^{(2)}$ decomposes as $\cup_{0 \leq s \leq r} B(s w_1)$ as a classical crystal. 
For each $b_i \in B^{\mu_i,1}$, we define a \textit{vacancy}, denoted by $\vac(b_i)$, to be  $(\mu_i - s)$ if $b_i \in B(s w_1)$, i.e., $b_i$ corresponds to a word of length $s$.
Then, we define $\vac(b) := \sum_{i=1}^{n} \vac(b_i)$. 
Finally, the \textit{ $q,t$ -energy} of $ b $ is defined by
\begin{align*}
    \energy_{q,t}(b) := q^{\frac{\overline{D}(b) - \vac(b)}{2}} t^{\vac(b)}.
\end{align*}
    
\begin{thm}\label{thm: B lusztig}
    We have
    \begin{equation*}\KL^{B_n}_{\lambda^{\sharp},\mu^{\sharp}}(q,t)=\sum_{T\in \GSSOT_{g+\frac{1}{2}}(\oc(\lambda,g),\overline{\oc}(\mu,g))} \energy_{q,t}(\phi_c(T))
    \end{equation*}
    where $g$ is a positive integer such that $g\geq \lambda_1$ and $\phi_c$ is the map in \Cref{lem:cind}. 
\end{thm}

\begin{example}
    Let $\lambda=(1,1,1), \mu = (0,0,0)$, and $g=1$. 
    Then, we have $\KL^{B_3}_{\lambda^\sharp,\mu^\sharp}(q,t) = q^3t^3 + q^3t + q^2t + qt$.
    The set $\GSSOT_{1+\frac{1}{2}}(\oc(\lambda,g),\overline{\oc}(\mu,g))$ consists of:
    \begin{align*}
    T_1 &= ((\emptyset , \emptyset, \emptyset),(\emptyset , \emptyset, \emptyset),(\emptyset , \emptyset, \emptyset)),\\
    T_2 &= ((\emptyset , \emptyset,\emptyset),(\emptyset,\ydiagram{1},\ydiagram{1}),(\ydiagram{1},\emptyset,\emptyset)), \\
    T_3 &= ((\emptyset , \ydiagram{1},\ydiagram{1}),(\ydiagram{1},\ydiagram{1},\emptyset),(\emptyset , \emptyset, \emptyset)), \\
    T_4 &= ((\emptyset , \ydiagram{1},\ydiagram{1}),(\ydiagram{1},\ydiagram{1},\ydiagram{1}),(\ydiagram{1} , \ydiagram{1}, \emptyset)).
    \end{align*}
    Via the map $\phi_c$, we embed $T_i's$ into $(B^{1,1})^{\otimes 3}$ of type $D_{N+1}^{(2)}$ as follows:
   \begin{align*}
       \phi_c(T_1) = \emptyset \otimes \emptyset \otimes \emptyset, \quad
       \phi_c(T_2) = \bar{1} \otimes 1 \otimes \emptyset, \\ 
       \phi_c(T_3) = \emptyset \otimes \bar{1} \otimes 1, \quad
       \phi_c(T_4) = \bar{1} \otimes \emptyset \otimes 1.
   \end{align*}
    The $q,t$-energy values are:
    \begin{align*}
        \energy_{q,t}(\phi_c(T_1)) = q^3t^3, \quad 
        \energy_{q,t}(\phi_c(T_2)) = q^3t, \\ 
        \energy_{q,t}(\phi_c(T_3)) = q^2t, \quad
        \energy_{q,t}(\phi_c(T_4)) = qt.
    \end{align*}
    These values coincide with $\KL^{B_3}_{\lambda^\sharp,\mu^\sharp}(q,t) =  q^3t^3 + q^3t + q^2t + qt$.
\end{example}

%%%%% 점심
\subsection{Sketch of the proof of \texorpdfstring{\Cref{thm: C lusztig}}{Theorem 3.1}}

Given $\lambda,\mu \in \Par_n$ for $n\geq2$, we define $\lambda^{(1)}, \lambda^{(2)}, \dots, \lambda^{(n)}$ and ${\mu'}$ as follows:
\begin{align*}
    &\lambda^{(1)}=(\lambda_2,\ldots,\lambda_n),\hspace{45mm} {\mu'}=(\mu_2,\ldots,\mu_n),\\
    &\lambda^{(2)}=(\lambda_1+1,\lambda_3,\ldots,\lambda_n),\\
    &\lambda^{(3)}=(\lambda_1+1,\lambda_2+1,\lambda_4,\ldots,\lambda_n),\\
    &\hspace{25mm}\vdots\\
    &\lambda^{(n)}=(\lambda_1+1,\lambda_2+1,\ldots,\lambda_{n-1}+1).
\end{align*}

We define $\ROHS_{\leq m}(\lambda,r)$ to be the set of rohs $(\lambda, \tau, \nu)$ of length $r$ such that $\ell(\tau), \ell(\nu)$ $\leq m$.
As proved in \cite{Lecouvey2005}, the $(C_n,C_{n-1})$-recurrence is given as 
\begin{equation}\label{eq: C morris recurrence}
    \KL^{C_n}_{\lambda,\mu}(q)=\sum_{i\geq1}(-1)^{i-1}\sum_{\substack{r+2m=\lambda_i-\mu_1+1-i\\r,m\geq0}}q^{r+m}\sum_{(\lambda^{(i)},\tau,\nu)\in \operatorname{ROHS}_{\leq n-1}(\lambda^{(i)},r)}\KL^{C_{n-1}}_{\nu,{\mu'}}(q).
\end{equation}
Assuming that \Cref{thm: C lusztig} holds for $\KL^{C_{n-1}}_{\lambda,\mu}$, the right-hand side of \eqref{eq: C morris recurrence} becomes 
\begin{equation}\label{noname}
    \sum_{i\geq1}(-1)^{i-1}\sum_{\substack{r+2m=\lambda_i-\mu_1+1-i\\r,m\geq0}}q^{r+m}\sum_{(\lambda^{(i)},\tau,\nu)\in \operatorname{ROHS}_{\leq n-1}(\lambda^{(i)},r)}\sum_{T\in\SSOT_g(\oc(\nu,g),\overline{\oc}(\mu',g))}q^{\overline{D}(\phi_c(T))}
\end{equation}
where we choose a large enough $g$, say $g\geq \lambda_1+\lambda_2$, so that $g\geq \nu_1$ for any $\nu$ appearing in the summation in \eqref{noname}.


\begin{lem}\label{lem: add rohs}
Let $A^{(i)}$ be a set of pairs $(S,T)$ satisfying the following conditions:
\begin{itemize}
    \item $S\in \ROHS_{\leq n-1}(\lambda^{(i)},r)$ for some $r$ such that there exists $m\geq 0$ with $r+2m=\lambda_i-\mu_1+1-i$,
    \item $T\in\SSOT_g(\oc(\nu,g),\overline{\oc}(\mu',g))$ where $\nu=F(S)$.
\end{itemize}
Then, there exists a bijection 
\begin{equation*}
    \Phi^{(i)}: A^{(i)}\rightarrow \SSOT_{g}(\oc(\lambda^{(i)},g),\gamma^{(i)})
\end{equation*}
where $\gamma^{(i)}=(g-\mu_2,\dots,g-\mu_n,\lambda_i-\mu_1+1-i)$.
\end{lem}

\begin{lem}\label{lem: energy r+m}
    Keeping the notations in the previous lemma, let $(S,T)\in A^{(i)}$ such that $S\in \ROHS_{\leq n-1}(\lambda^{(i)},r)$.
    Then, we have
    \begin{equation*}
        \overline{D}(\phi_c(\Phi^{(i)}(S,T)))=r+m+\overline{D}(\phi_c(T))
    \end{equation*}
    where $m$ is an integer satisfying $r+2m=\lambda_i-\mu_1+1-i$.
\end{lem}
Using \Cref{lem: add rohs} and \ref{lem: energy r+m}, the equation \eqref{noname} simplifies to:
\begin{equation}\label{eq : morris C short}
     \sum_{i\geq1}(-1)^{i-1}\sum_{T\in\SSOT_{g}(\oc(\lambda^{(i)},g),\gamma^{(i)})}q^{\overline{D}(\phi_c(T))} .
\end{equation}
We now apply a sign-reversing involution to \eqref{eq : morris C short}, generalizing the argument presented in Zabrocki's thesis, which proves the result for type $A$ \cite{Zab1998}. 
\Cref{thm: B lusztig} can be proved using a similar argument.

\section{Level-restricted \texorpdfstring{$q$}{q}-weight multiplicity}

We provide a combinatorial formula for the level-restricted $q$-weight multiplicity in terms of level-restricted paths in KR crystals. 
For a nonnegative integer vector $\alpha=(\alpha_1,\dots,\alpha_n)$, we define $||\alpha||=\sum_{i=1}^{n}(i-1)\beta_i$, where $\beta$ is a rearrangement of $\alpha$ in weakly decreasing order. 

\begin{thm}\label{thm: level formula}
    We have the following formulas:
    \begin{align}\label{eq: tilted B}
        \KL^{B_n,L_A}_{\lambda,\mu}(q)&=\sum\limits_{\substack{T\in \GSSOT(\hat{\lambda},\bar{\mu})\\ c(T) \leq 2g }} q^{||\bar{\mu}||+|\bar{\mu}|-|\hat{\lambda}|-\lfloor\frac{\overline{D}(\phi_r(T))}{2}\rfloor},\\
        \label{eq: tilted C}
        \KL^{C_n,L_A}_{\lambda,\mu}(q)&=\sum\limits_{\substack{T\in \SSOT(\hat{\lambda},\bar{\mu})\\ c(T) \leq g }} q^{||\bar{\mu}||+|\bar{\mu}|-|\hat{\lambda}|-\overline{D}(\phi_r(T))},\\
        \KL^{D_n,L_A}_{\lambda,\mu}(q)&=\sum\limits_{\substack{T\in \SSROT(\hat{\lambda},\bar{\mu})\\ c(T) \leq 2g }} q^{||\bar{\mu}||+\frac{|\bar{\mu}|-|\hat{\lambda}|}{2}-\overline{D}(\phi_r(T))},\label{eq: tilted D}
    \end{align}
    where $g\in \frac{\mathbb{Z}_{\geq 0}}{2}$ such that $g\geq \lambda_1$, $\hat{\lambda}=\oc(\lambda,g)$, and $\bar{\mu}=\overline{\oc}(\mu,g)$.
\end{thm}
Note that the equations \eqref{eq: tilted B} and \eqref{eq: tilted D} also hold for spin weights.
When $\lambda$ and $\mu$ are spin weights, we choose $g$ such that $g \geq \lambda_1$ and $2g$ is odd. 
When $\lambda$ and $\mu$ are partitions, we choose $g$ such that $g \geq \lambda_1$ and $2g$ is even.  

\begin{example}
    Let $\lambda=(1,1,1,1), \mu = (0,0,0,0)$, and $g=1.5$. 
    Then, we have $\KL^{D_4,L_A}_{\lambda^\sharp,\mu^\sharp}(q) = q^4+q^2+1$. 
    The set $\SSROT(\oc(\lambda^\sharp,g),\overline{\oc}(\mu^\sharp,g))$ with $c(T) \leq 3 $ consists of: 
    \begin{align*}
    T_1 &= ((\emptyset , \emptyset, \ydiagram{1}),(\ydiagram{1} , \ydiagram{1}, \ydiagram{2}),(\ydiagram{2} , \ydiagram{1}, \ydiagram{1}),(\ydiagram{1},\emptyset,\emptyset)),\\
    T_2 &= ((\emptyset , \emptyset, \ydiagram{1}),(\ydiagram{1} , \ydiagram{1}, \ydiagram{1,1}),(\ydiagram{1,1} , \ydiagram{1}, \ydiagram{1}),(\ydiagram{1},\emptyset,\emptyset)),\\
    T_3 &= ((\emptyset , \emptyset, \ydiagram{1}),(\ydiagram{1} , \emptyset, \emptyset), (\emptyset , \emptyset, \ydiagram{1}),(\ydiagram{1} , \emptyset, \emptyset)).
    \end{align*}
    Via the map $\phi_r$, we embed $T_i$'s into $(B^{1,1})^{\otimes 4}$ of type $B_N^{(1)}$  as follows:
    \begin{align*}
        \phi_r(T_1) = \bar{1} \otimes \bar{1} \otimes 1 \otimes 1, \quad \phi_r(T_2) = \bar{1} \otimes \bar{2} \otimes 2 \otimes 1, \quad \phi_r(T_3) = \bar{1} \otimes 1 \otimes \bar{1} \otimes 1.
    \end{align*}
    The energy function values are:
    \begin{align*}
        \overline{D}(\phi_r(T_1)) = 4, \quad \overline{D}(\phi_r(T_2)) = 6, \quad \overline{D}(\phi_r(T_3)) = 8.
    \end{align*}
    Since $||\bar{\mu}||+\frac{|\bar{\mu}|-|\hat{\lambda}|}{2} = 8$, we obtain $q^4+q^2+1$, which coincides with $\KL^{D_4,L_A}_{\lambda^\sharp,\mu^\sharp}(q) = q^4+q^2+1$.
\end{example}


\begin{rmk}
    For any positive integer $k$, \Cref{thm: level formula} implies the following:
   \begin{align*}
        \KL^{B_n,L_A}_{\lambda+k^{n},\mu+k^{n}}(q)&=\sum\limits_{\substack{T\in \GSSOT(\hat{\lambda},\bar{\mu})\\ c(T) \leq 2g+2k }} q^{||\bar{\mu}||+|\bar{\mu}|-|\hat{\lambda}|-\lfloor\frac{\overline{D}(\phi_r(T))}{2}\rfloor},\\
        \KL^{C_n,L_A}_{\lambda+k^n,\mu+k^n}(q)&=\sum\limits_{\substack{T\in \SSOT(\hat{\lambda},\bar{\mu})\\ c(T) \leq g+k }} q^{||\bar{\mu}||+|\bar{\mu}|-|\hat{\lambda}|-\overline{D}(\phi_r(T))},\\
        \KL^{D_n,L_A}_{\lambda+k^n,\mu+k^n}(q)&=\sum\limits_{\substack{T\in \SSROT(\hat{\lambda},\bar{\mu})\\ c(T) \leq 2g+2k }} q^{||\bar{\mu}||+\frac{|\bar{\mu}|-|\hat{\lambda}|}{2}-\overline{D}(\phi_r(T))},
    \end{align*}
    where $\hat{\lambda}=\oc(\lambda,g)$ and $\bar{\mu}=\overline{\oc}(\mu,g)$. 
    When $k$ is sufficiently large, the right-hand sides becomes (up to a scalar) 
    \begin{equation*}
        \sum_{x\in \HW(B_{\bar{\mu}},\hat{\lambda})}q^{-\overline{D}(x)}
    \end{equation*}
    which is simply the 1-dim sum for certain tensor products of row KR crystals. 
    Therefore, \Cref{thm: level formula} can be regarded as a refinement of the $X=K= {^\infty}\KL$ theorem for tensor products of row KR crsytals \cite{LS2007,S05}. 
\end{rmk}



\acknowledgements{The authors are grateful to Mark Shimozono for Jae-Hoon Kwon for helpful comments.}

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