$t$-analogue of the $q$-characters of finite dimensional representations of quantum affine algebras %\\ {(Preliminary version: \today)}
Hiraku Nakajima
%\abstracts{ Frenkel-Reshetikhin introduced $q$-characters of finite dimensional representations of quantum affine algebras~\cite{FR}. % We give a combinatorial algorithm to compute them for all simple modules. % Our tool is $t$-analogue of the $q$-characters, which is similar to Kazhdan-Lusztig polynomials, and our algorithm has a resemblance with their definition. We need the theory of quiver varieties for the definition of $t$-analogues and the proof. But it appear only in the last section. % The rest of the paper is devoted to an explanation of the algorithm, which one can read without the knowledge about quiver varieties. % A proof is given only in part. A full proof will appear elsewhere.
math0009231.tex
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\begin{document}
%
\title[$t$-analogue of the $q$-characters]
{$t$-analogue of the $q$-characters of finite dimensional
representations of quantum affine algebras
%\\ {\rm \small (Preliminary version: \today)}
}
\author{Hiraku Nakajima}
\address{Department of Mathematics, Kyoto University, Kyoto 606-8502,
Japan
%\\Email:nakajima@kusm.kyoto-u.ac.jp}
}
\email{nakajima@kusm.kyoto-u.ac.jp}
\urladdr{http://www.kusm.kyoto-u.ac.jp/\textasciitilde nakajima}
\thanks{Supported by the Grant-in-aid
for Scientific Research (No.11740011), the Ministry of Education,
Japan.}
%
%\subjclass{Primary 17B37;
%Secondary 14D21, 14L30, 16G20, 33D80}
%
\begin{abstract}
%\abstracts{
Frenkel-Reshetikhin introduced $q$-characters of finite dimensional
representations of quantum affine algebras~\cite{FR}.
%
We give a combinatorial algorithm to compute them for all simple
modules.
%
Our tool is $t$-analogue of the $q$-characters, which is similar to
Kazhdan-Lusztig polynomials, and our algorithm has a resemblance with
their definition.

We need the theory of quiver varieties for the definition of
$t$-analogues and the proof. But it appear only in the last section.
%
The rest of the paper is devoted to an explanation of the algorithm,
which one can read without the knowledge about quiver varieties.
%
A proof is given only in part. A full proof will appear elsewhere.
\end{abstract}
%}
\maketitle

\section{The quantum loop algebra}

Let $\g$ be a simple Lie algebra of type ADE over $\C$, $\Lg = \g
\otimes \C[z,z^{-1}]$ be its loop algebra, and $\Ul$ be its quantum
universal enveloping algebra, or the quantum loop algebra for short.
It is a subquotient of the quantum affine algebra $\Ua$, i.e., without
central extension and degree operator.
%
Let $I$ be the set of simple roots, $P$ be the weight lattice, and
$P^*$ be its dual lattice (all for $\g$).
%
The algebra has the so-called Drinfeld's new realization: It is a
$\C(q)$-algebra with generators $q^h$, $e_{k,r}$, $f_{k,r}$, $h_{k,n}$
($h\in P^*$, $k\in I$, $r\in\Z$, $n\in \Z\setminus \{0\}$) with
certain relations (see e.g., \cite[12.2]{CP-book}).

The algebra $\Ul$ is a Hopf algebra, where the coproduct is defined
using the Drinfeld-Jimbo realization of $\Ul$. So a tensor product
$M\otimes_{\C(q)} M'$ of $\Ul$-modules $M$, $M'$ has a structure of a
$\Ul$-module.

Let $\Ule$ be its specialization at $q = \ve\in\C^*$.
%
For precise definition of the specialization, we first introduce an
integral form $\Uli$ of $\Ul$ and set $\Ule =
\Uli\otimes_{\Z[q,q^{-1}]} \C$, where $\Z[q,q^{-1}]\to \C$ is given by
$q^{\pm 1}\mapsto \ve^{\pm 1}$. See \cite{CP-roots} for detail. But we
assume $\ve$ is {\it not\/} a root of unity in this paper. So we just
replace $q$ by $\ve$ in the definition of $\Ul$.

The quantum loop algebra $\Ul$ contains the quantum enveloping algebra 
$\Uq$ for the finite dimensional Lie algebra $\g$ as a subalgebra. The 
specialization $\Ule$ contains the specialization $\Ue$ of $\Uq$.

\subsection{Finite dimensional representations of $\Ule$}

The algebra $\Ule$ contains a commutative subalgebra generated by
$q^h$, $h_{k,n}$ ($h\in P^*$, $k\in I$, $n\in\Z\setminus \{0\}$).
Let us introduce generating functions $\psi_k^\pm(z)$ ($k\in I$) by
\begin{equation*}
   \psi^{\pm}_k(z)
%  = \sum_{r=0}^\infty \psi^{\pm}_{k,\pm r} z^{\mp r}
  \defeq q^{\pm h_k}
   \exp\left(\pm (q-q^{-1})\sum_{m=1}^\infty h_{k,\pm m} z^{\mp m}\right).
\end{equation*}

A $\Ule$-module $M$ is called of {\it type $1$\/} if $M$ has a weight space
decomposition as a $\Ue$-module:
\begin{equation*}
  M = \bigoplus_{\lambda\in P} M(\lambda), \qquad
  M(\lambda) = \left\{ m\in M \left|\, q^h\ast m 
  = \ve^{\langle h,\lambda\rangle} m\right.\right\}.
\end{equation*}
We will only consider type $1$ modules in this paper.

A type $1$ module $M$ is an {\it l-highest weight module\/} ('{\it
l\/}' stands for the loop) if there exists a vector $m_0\in M$ such
that
\begin{gather*}
  e_{k,r}\ast m_0 = 0, \qquad \Ule^- \ast m_0 = M, \\
  \psi^{\pm}_{k}(z)\ast m_0 = \Psi^{\pm}_{k}(z) m_0
    \quad\text{for $k\in I$}
\end{gather*}
for some $\Psi^{\pm}_k(z)\in \C[[z^\mp]]$. The pair of
the $I$-tuple
$(\Psi^+(z),\Psi^-(z)) = (\Psi^+_k(z),\linebreak[0]
 \Psi^-_k(z))_{k\in I}\in
(\C[[z^\mp]]^I)^2$ is called the {\it l-highest weight\/} of $M$, and
$m_0$ is called the {\it l-highest weight vector}.

\begin{Theorem}[Chari-Pressley~\cite{CP-rep}]
\rom{(1)} Every finite-dimensional simple \linebreak[4]
$\Ule$-module of type $1$ is 
an {\it l\/}-highest weight module, and its {\it l\/}-highest weight
is given by
\begin{equation}\label{eq:hwt}
   \Psi^\pm_k(z) = \varepsilon^{\deg P_k}
  \left(\frac{P_k(\varepsilon^{-1}/z)}{P_k(\varepsilon/z)}\right)^{\pm}
\end{equation}
for some polynomials $P_k(u)\in\C[u]$ with $P_k(0) = 1$.
%
Here $\left(\ \right)^{\pm}
\in\C[[z^{\mp}]]$ denotes the expansion at $z = \infty$ and $0$
respectively.

\rom{(2)} Conversely, for given $P_k(u)$ as above, there exists a
finite-dimensional simple {\it l\/}-highest weight $\Ule$-module $M$
of type $1$ such that the {\it l\/}-highest weight is given by the
above formula.

Assigning to $M$ the $I$-tuple $P = (P_k)_{k\in I}\in \C[u]^I$
\rom($P_k(0) = 1$\rom) defines a bijection between the set of all
$P$'s and the set of isomorphism classes of finite-dimensional simple
$\Ule$-modules of type $1$.
\end{Theorem}

We denote by $L_P$ the simple $\Ule$-module associated to $P$.
We call $P$ the {\it Drinfeld polynomial}.
%
For the abuse of terminology, we also say `$P$ is the {\it
l\/}-highest weight of $L_P$'.

Since $\C\langle q^h$, $h_{k,n}\rangle$ is a commutative subalgebra
of $\Ule$, any $\Ule$-module $M$ decomposes into a direct sum $M =
\bigoplus M(\Psi^+,\Psi^-)$ of generalized eigenspaces, where
\begin{equation*}
\begin{split}
   & M(\Psi^+,\Psi^-) \\
   \defeq\; &  \left\{ m\in M \left|\,
     \text{$(\psi_k^\pm(z) - \Psi_k^\pm(z)\operatorname{Id})^N \ast m
     = 0$ for $k\in I$ and sufficiently large $N$}
   \right\}\right. ,
\end{split}
\end{equation*}
for $\Psi_k^\pm(z)\in\C[[z^\mp]]$. The pair of the $I$-tuple
$(\Psi^+,\Psi^-) = (\Psi^+_k, \Psi^-_k)_{k\in I}$ is called an
{\it l-weight}, and $M(\Psi^+,\Psi^-)$ is called an {\it l-weight
space\/} of $M$ if $M(\Psi^+,\Psi^-) \neq 0$.

\begin{Theorem}[Frenkel-Reshetikhin~\cite{FR}]
Any {\it l\/}-weight of any finite dimensional $\Ule$-module $M$ of
type $1$ has the following form
\begin{equation}\label{eq:lwt}
   \Psi_k^\pm(z) = \varepsilon^{\deg Q_k - \deg R_k}
   \left(\frac{Q_k(\varepsilon^{-1}/z) R_k(\varepsilon/z)}
     {Q_k(\varepsilon/z)R_k(\varepsilon^{-1}/z)}\right)^{\pm}
\end{equation}
for some polynomials
\[
  Q_k(u) = \prod_{i=1}^{s_k} (1 - a_{ki} u), \quad
  R_k(u) = \prod_{j=1}^{r_k} (1-b_{kj} u). 
\]
\end{Theorem}
Again for the abuse of terminology, we also say `$Q/R$ is an {\it
l\/}-weight of $M$'.
%
We denote the {\it l\/}-weight space $M(\Psi^+,\Psi^-)$ by $M(Q/R)$.

Frenkel-Reshetikhin~\cite{FR} defined the $q$-character of $M$ by
\begin{equation*}
  \chi_q(M) \defeq
  \sum_{Q/R}
  \dim M(Q/R)\; \prod_{k\in I}
  \prod_{i=1}^{s_k}\prod_{j=1}^{r_k} Y_{k,a_{ki}} Y_{k,b_{kj}}^{-1}.
\end{equation*}

\begin{Theorem}[Frenkel-Reshetikhin~\cite{FR}]\label{thm:FR}
\rom{(1)} $\chi_q$ defines an injective ring homomorphism from the
Grothendieck ring $\operatorname{Rep}\Ule$ of finite dimensional
$\Ule$-modules of type $1$ to $\Z[Y_{k,a}^\pm]_{k\in I,a\in\C^*}$ \rom(a
ring of Laurent polynomials in infinitely many variables\rom).

\rom{(2)} If we compose a map $Y_{k,a}^\pm \mapsto y_k^\pm$ \rom(forgetting
`spectral parameters'\rom), it gives the usual character of the
restriction of $M$ to a $\Ue$-module.
\end{Theorem}

\begin{Definition}
A monomial $\displaystyle \prod_{k\in I}
\prod_{i=1}^{s_k}\prod_{j=1}^{r_k} Y_{k,a_{ki}} Y_{k,b_{kj}}^{-1}$
appearing in the $q$-char\-ac\-ter $\chi_q$ is called {\it l-dominant\/}
if $r_k = 0$ for all $k$, i.e., a product of positive powers
of $Y_{k,c}$'s or $1$.
\end{Definition}

If $L_P$ is the simple $\Ule$-module with {\it l\/}-highest weight
$P$, its $q$-character contains an {\it l\/}-dominant monomial
corresponding to the {\it l\/}-highest weight. We denote it by
$m_P$. Its coefficient in $\chi_q(L_P)$ is $1$.

Since $\{ L_P \}_P$ forms a basis of $\operatorname{Rep}\Ule$, we have 
the following useful condition for the simplicity of a finite
dimensional $\Ule$-module $M$ of type $1$:
\begin{equation}
\label{eq:suff}
\text{
If $\chi_q(M)$ contains only one {\it l\/}-dominant term, then
$M$ is simple.
}
\end{equation}

\subsection{Example}
We give examples of $q$-characters.

If $\g = A_n$, we have an evaluation homomorphism
$\operatorname{ev}_a\colon \Ule \to \Ue$ corresponding to $\Lg \to
\g$; $z\mapsto a$ (Jimbo). Hence pullbacks of simple $\Ue$-modules are 
simple $\Ule$-modules.

\begin{Example}
Let $\g = A_1 = \algsl_2$ and $V$ be the $2$-dimensional simple
$\Ule$-module. Then the $q$-character of $M_a =
\operatorname{ev}_a(M)$ is given by\footnote{This can be checked
directly. But it also follows from \thmref{thm:ind} below.}
\begin{equation*}
   \chi_q(M_a)
   = Y_{1,a} + Y_{1,a\ve^2}^{-1}.
\end{equation*}
Since $\chi_q$ is a ring homomorphism, we have
\begin{equation*}
\begin{split}
   \chi_q(M_a\otimes M_b) &=
   \left(Y_{1,a} + Y_{1,a\ve^2}^{-1}\right)
   \left(Y_{1,b} + Y_{1,b\ve^2}^{-1}\right)\\
   &=
   Y_{1,a}Y_{1,b} + Y_{1,a\ve^2}^{-1}Y_{1,b} + Y_{1,a}Y_{1,b\ve^2}^{-1}
   + Y_{1,a\ve^2}^{-1}Y_{1,b\ve^2}^{-1}.
\end{split}
\end{equation*}
If $b\neq a\ve^2, a\ve^{-2}$, then $M_a\otimes M_b$ is simple by the
criterion~\eqref{eq:suff}.

If $b = a\ve^2$ or $a\ve^{-2}$, then the second or third term becomes
$1$. In fact, it is known that $M_a\otimes M_{a\ve^2}$ decomposes (in
$\operatorname{Rep}\Ule$) to a sum $M'_a\oplus M''$, where $M'$ is the
$3$-dimensional simple $\Ule$-module, and $M''$ is the trivial
module. Thus we have
\begin{equation*}
\begin{split}
  \chi_q(M_a\otimes M_{a\ve^2}) 
  & = \chi_q(M'_a) + \chi_q(M'') \\
  & = Y_{1,a}Y_{1,a\ve^2} + Y_{1,a}Y_{1,a\ve^4}^{-1}
   + Y_{1,a\ve^2}^{-1}Y_{1,a\ve^4}^{-1} + 1.
\end{split}
\end{equation*}
See also Examples~\ref{exm:A1''}, \ref{exm:A1}, \ref{exm:A1'}.
\end{Example}

\section{Standard modules}

\subsection{}

In \cite{Na-qaff} we defined a family of finite dimensional
$\Ule$-modules of type $1$ and called them {\it standard
modules}. They are parametrized by the $I$-tuples $P = (P_k)_{k\in
I}\in \C[u]^I$ exactly as simple modules. We denote by $M_P$
associated to $P$. The definition will be recalled in
\secref{sec:quiver}, but we give here their algebraic
identification due to Varagnolo-Vasserot~\cite{VV-std}.

\begin{Definition}
We say $L_P$ an {\it l-fundamental representation\/} if
\begin{equation*}
  P_k(u) =
  \begin{cases}
    1 - su  & \text{if $k = k_0$,} \\
    1 & \text{otherwise},
  \end{cases}
\end{equation*}
for some $s\in\C^*$ and $k_0\in I$.
%
We denote $L_P$ by $L(\Lambda_{k_0})_s$.
($\Lambda_k$ is the $k$-th fundamental weight of $\g$.)
\end{Definition}

For $s\in\C^*$ and a finite sequence
$(k_\alpha)_\alpha = (k_1, k_2, \dots)$ in $I$ and
a sequence $(n_\alpha)_\alpha = (n_1\ge n_2\ge \dots)$
of integers, we set
\begin{equation*}
   M(s; (k_\alpha)_\alpha, (n_\alpha)_\alpha) \defeq
   L(\Lambda_{k_1})_{\ve^{n_1}s}\otimes
   L(\Lambda_{k_2})_{\ve^{n_2}s}\otimes\cdots.
\end{equation*}
Note that $\Ule$ is {\it not\/} cocommutative Hopf algebra, so the
tensor product depends on the ordering of factors.

\begin{Theorem}[Varagnolo-Vasserot~\cite{VV-std}]\label{thm:VV}
\rom{(1)} A standard module $M$ is isomorphic to a module of the form
\begin{equation*}
\begin{split}
   & \bigotimes_{i} 
   M(s^i; (k^i_{\alpha_i})_{\alpha_i}, (n^i_{\alpha_i})_{\alpha_i}) \\
   =\; &
   M(s^1; (k^1_{\alpha_1})_{\alpha_1}, (n^1_{\alpha_1})_{\alpha_1})
   \otimes
   M(s^2; (k^2_{\alpha_2})_{\alpha_2}, (n^2_{\alpha_2})_{\alpha_2})
   \otimes
   \cdots
   \quad \text{\rom(finite tensor product\rom)}
\end{split}
\end{equation*}
such that $s^i/s^j\notin \ve^{\Z}$ for $i\neq j$ and
$n^i_1 \ge n^i_2 \ge \dots$ for each $i$.

\rom{(2)} The above tensor product is independent of the ordering of
the factors
\linebreak[4]
$M(s^i; (k^i_{\alpha_i})_{\alpha_i},(n^i_{\alpha_i})_{\alpha_i})$.

\rom{(3)} The $I$-tuple of polynomials $P$ corresponding to $M$ is the 
product of Drinfeld polynomials of {\it l\/}-fundamental representations
appearing as factors of $M$.
\end{Theorem}

Note that if $P$ is given, we can define a module $M$ of the above
form by decomposing $P$ into a product of Drinfeld polynomials of {\it
l\/}-fundamental representations. Thus we may denote the above module
by $M_P$.

The following properties of $M_P$ were shown in \cite{Na-qaff}:
\begin{enumerate}
\item $\{ M_P\}$ is a basis of $\operatorname{Rep}\Ule$.
\item $M_P$ is an {\it l\/}-highest weight module with {\it
l\/}-highest weight $P$ (i.e., given by \eqref{eq:hwt}).
\item $L_P$ is the unique simple quotient of $M_P$.
\item $M_P$ depends `continuously' on $P$ in a certain sense.
For example, $\dim M_P$ is independent of $P$.
\item for a generic $P$, $M_P \cong L_P$.
\end{enumerate}

Conjecturally $M_P$ is isomorphic to the specialization of the module
$V^{\max}(\lambda)$, introduced by Kashiwara~\cite{Kas}, and further
studied by Chari-Pressley~\cite{CP:Weyl}.

\section{$t$-analogues of $q$-characters}

A main tool in this paper is a $t$-analogue of the $q$-character:
\begin{equation*}
   \chi_{q,t}\colon \operatorname{Rep}\Ule \to
   \Z[t,t^{-1}][Y_{k,a}^\pm]_{k\in I,a\in\C^*}.
\end{equation*}
This is a homomorphism of additive groups, not of rings, and has the
property $\chi_{q,t=1} = \chi_q$.
%
We define $\chi_{q,t}$ for all standard modules $M_P$. Since 
$\{ M_P\}_P$ is a basis of $\operatorname{Rep}\Ule$, we can extend it 
linearly to any finite dimensional $\Ule$-modules.

For the definition we need geometric constructions of standard
modules, so we will postpone it to \subsecref{subsec:qchar}. We give
an alternative definition, which is conjecturally the same as the
geometric definition.

\subsection{A conjectural definition}

Let $M = M_P$ be a standard module, $Q/R$ be an {\it l\/}-weight of
$M$, $M(Q/R)$ be the corresponding {\it l\/}-weight space.
Define a filtration on $M(Q/R)$ by
\begin{gather*}
   0 = M^{-1}(Q/R) \subset M^0(Q/R)
   \subset M^1(Q/R) \subset \cdots \\
   M^n(Q/R)
   \defeq \bigcap_k \Ker(\psi_k^\pm(z)-\Psi_k^\pm(z)\id)^{n+1}.
\end{gather*}

\begin{Conjecture}
The $t$-analogue $\chi_{q,t}(M_P)$, defined geometrically in
\subsecref{subsec:qchar}, is equal to
\begin{equation*}
  \chi_{q,t}(M_P)
  = \sum_{Q/R}
  \sum_n t^{2n-d(Q/R,P)}
  \dim \left(M^{n}(Q/R)/M^{n-1}(Q/R)\right) \; m_{Q/R},
\end{equation*}
where $d(Q/R,P)$ is an integer \rom(determined explicitly from $Q/R$,
$P$ by \eqref{eq:dim} below\rom), and $m_{Q/R}$ is a monomial in
$Y_{k,a}^\pm$ corresponding to the {\it l\/}-weight space $M(Q/R)$.
\end{Conjecture}

This definition makes sense for any finite dimensional modules, but is
{\it not\/} well-defined on the Grothendiek group
$\operatorname{Rep}\Ule$. Thus the above does not hold for simple modules.

\subsection{}
A main result of this paper is a combinatorial algorithm for computing
$\chi_{q,t}(M_P)$ and $[M_P:L_Q]$. It is divided into three steps:
\begin{description}
\item[Step 1] Compute $\chi_{q,t}$ for all {\it l\/}-fundamental
repsentations.
\item[Step 2] Compute $\chi_{q,t}(M_P)$ for all standard modules
$M_P$.
\item[Step 3] Express the multiplicity $[M_P: L_Q]$ in terms of
$\chi_{q,t}(M_R)$ for various $R$.
\end{description}

Step 1 is a modification of Frenkel-Mukhin's algorithm~\cite{FM}
for computing $\chi_q$ of {\it l\/}-fundamental representations.
%
Step 2 is nothing but a study of $\chi_{q,t}$ of tensor products of
{\it l\/}-fundamental representations. Although $\chi_{q,t}$ is not a
ring homomorphism, $\chi_{q,t}$ of tensor products is given by a
simply modified multiplication.
%
For the proof we use an idea in \cite{Na-hom}.
%
Step 3 was essentially done in \cite{Na-qaff}.

\section{Step 3}

We start with Step 3. The algorithm is similar to the definition of
Kazhdan-Lusztig polynomials \cite{KL}. It is also similar to the
algorithm for computing the transition matrix between the
canonical basis and the PBW basis of type ADE \cite{Lu:can}.

\subsection{}
Let 
\[
   A_{k,a} \defeq Y_{k,a\ve} Y_{k,a\ve^{-1}}
   \prod_{l:l\neq k} Y_{l,a}^{c_{kl}},
\]
where $c_{kl}$ is the $(k,l)$-entry of the Cartan matrix.

\begin{Definition}
(1) Let $m$, $m'$ be monomials in $Y_{k,a}^\pm$ ($k\in I$,
$a\in\C^*$). We define an ordering $\le$ among monomials by
\begin{equation*}
   m\le m' \Longleftrightarrow
   \text{$\frac{m'}{m}$ is a monimial in $A_{k,a}^{-1}$ ($k\in I$,
   $a\in\C^*$)}.
\end{equation*}
Here a monomial in $A_{k,a}^{-1}$ means a product of nonnegative
powers of $A_{k,a}^{-1}$. It does not contain any factors $A_{k,a}$.

(2) If $\Psi^\pm, \Psi^{\prime\pm}$ are {\it l\/}-weights of finite
dimensional $\Ule$-modules, or $Q/R, Q'/R'$ are related to {\it
l\/}-weights by \eqref{eq:lwt}, we write
$\Psi^\pm\le \Psi^{\prime\pm}$, $Q/R \le Q'/R'$ if the corresponding
monomials $m$, $m'$ satisfy $m\le m'$.
\end{Definition}

Recall that $\chi_q(L_P)$ contains an {\it l\/}-dominant monomial
$m_P$ corresponding to the highest weight vector.
%
It is known that any monomial $m$ appearing $\chi_q(L_P)$,
$\chi_q(M_P)$ satisfies $m\le m_P$ (\cite[4.1]{FM},
\cite[13.5.2]{Na-qaff}).

Let
\begin{equation*}
   c_{QP}(t) \defeq \text{the coefficient of $m_Q$ in $\chi_{q,t}(M_P)$}.
\end{equation*}
Then $(c_{QP}(t))_{P,Q}$ is upper-triangular and $c_{PP}(t) = 1$ by
the above mentioned result.

Let $(c^{QP}(t))$ be the inverse matrix $(c_{QP}(t))^{-1}$. Let
\[
  u_{RP}(t) \defeq \sum_Q c^{RQ}(t^{-1}) c_{QP}(t).
\]
%
Let $\setbox5=\hbox{A}
\overline{\rule{0mm}{\ht5}\hspace*{\wd5}}$ be the involution on
$\Z[t,t^{-1}]$ given by $t^{\pm 1} \mapsto t^{\mp 1}$.

\begin{Lemma}[\protect{Lusztig~\cite[7.10]{Lu:can}}]
There exists a unique solution $Z_{QP}(t)\in\Z[t^{-1}]$ \rom($Q\le
P$\rom) of
\begin{gather}
   Z_{RP}(t) = \sum_{Q: R \le Q \le P} \overline{Z_{RQ}(t)} u_{QP}(t),
   \label{eq:Lus1}\\
   Z_{PP}(t) = 1, \quad Z_{QP}(t) \in t^{-1}\Z[t^{-1}]
   \;\text{for $Q < P$}. \label{eq:Lus2}
\end{gather}
\end{Lemma}

This lemma is proved by induction, and holds in a general
setting. Lusztig has been using this (or its variant) in many places.

\begin{Theorem}\label{thm:mul}
The multiplicity $[M_P : L_Q]$ of a simple module $L_Q$ in a standard
module $M_P$ is equal to
\(
   Z_{QP}(1)
\).
\end{Theorem}

The proof will be given in \subsecref{subsec:prf}.

\begin{Example}\label{exm:A1''}
Let $\g = A_1$, $M_a = \operatorname{ev}_a^*(M)$ where
$M$ is the $2$-dimensional simple $\Ue$-module as before.
%
By steps~1,2 explained below\footnote{or direct calculation for the
definition~\eqref{eq:tana}}
we have
\begin{equation}\label{eq:exp}
   \chi_{q,t}(M_{a\ve^2}\otimes M_a) = 
   Y_{1,a}Y_{1,a\ve^2} + Y_{1,a}Y_{1,a\ve^4}^{-1}
   + Y_{1,a\ve^2}^{-1}Y_{1,a\ve^4}^{-1} 
   + t^{-1} 1.
\end{equation}
Let $P(u) = (1-au)(1-a\ve^2 u)$ (i.e., $M_P = M_{a\ve^2}\otimes M_a$),
$Q(u) = 1$ (i.e. $M_Q = $ trivial module).
%
Then the above algorithm gives us
\(
   Z_{QP}(t) = t^{-1}.
\)
\end{Example}

\section{Step 1}


\subsection{Some definitions}

Let $M_P$ be a standard module. Let $m_P$ be the monomial
corresponding to the {\it l\/}-highest weight vector. Let $M_P(Q/R)$
be an {\it l\/}-weight space as before. We denote by $m_{Q/R}$ the
corresponding monomial. We define
\(
  w_{k,a}(P), v_{k,a}(Q/R,P)\in\Z_{\ge 0}, u_{k,a}(Q/R)\in\Z
\)
by
\begin{gather*}
   m_P = \prod_{k\in I, a\in\C^*} Y_{k,a}^{w_{k,a}(P)},
\\
   m_{Q/R} = m_P \prod_{k\in I, a\in\C^*} A_{k,a}^{-v_{k,a}(Q/R,P)}
   = \prod_{k\in I, a\in\C^*} Y_{k,a}^{u_{k,a}(Q/R)}.
\end{gather*}

Suppose two standard modules $M_{P^1}$, $M_{P^2}$ and {\it l\/}-weight
spaces $M_{P^1}(Q^1/R^1)\subset M_{P^1}$, $M_{P^2}(Q^2/R^2)\subset
M_{P^2}$ are given. We define
\begin{equation}
\begin{split}
   & d(Q^1/R^1,P^1; {Q^2/R^2},{P^2}) \\
   \defeq \; & \sum_{k,a} \left(
   v_{k,a}(Q^1/R^1,P^1) u_{k,a\ve^{-1}}(Q^2/R^2)
   + w_{k,a\ve}(P^1) v_{k,a}(Q^2/R^2,P^2)\right).
\end{split}
\end{equation}

We also define
\begin{equation}\label{eq:dim}
   d(Q/R,P) \defeq d(Q/R,P;Q/R,P)
%   \sum_{k,a}
%   v_{k,a}(Q/R,P)\left( u_{k,a\ve^{-1}}(Q/R) + w_{k,a\ve}(P)\right)
.
\end{equation}
We denote $d(Q^1/R^1,P^1; {Q^2/R^2},{P^2})$ also by
$d(m_{Q^1/R^1},m_{P^1}; m_{Q^2/R^2},m_{P^2})$.

We need the following modification of $\chi_{q,t}$. Write
$\chi_{q,t}(M_P) = \sum_m a_m(t)\; m$, where $m$ is a monomial and
$a_m(t)$ is its coefficient. Let
\begin{equation}\label{eq:modify}
   \widetilde{\chi_{q,t}}(M_P) \defeq
   \sum_m t^{d(m,m_P)} a_m(t)\; m,
\end{equation}
where $d(m,m_P)$ is defined in \eqref{eq:dim}.\footnote{In fact,
$d(m,m_P)$ is determined from $a_m(t)$ so that $t^{d(m,m_P)} a_m(t)$
is a polynomial in $t$ with nonzero constant term.}

\subsection{}
Frenkel-Mukhin \cite[5.1,5.2]{FM} proved that the image of the
$q$-character $\chi_q$ is contained in
\begin{equation*}
  \bigcap_{k\in I}
  \left( \Z[Y_{l,a}^\pm]_{l\neq k, a\in\C^*}
    \otimes\Z[Y_{k,b}(1+A_{k,b\ve}^{-1})]_{b\in\C^*}\right).
\end{equation*}

We have the $t$-analogue of this result, replacing
$(1+A_{k,b\ve}^{-1})^n$ by
\begin{equation*}
   %\boxed{1+A_{k,b\ve}^{-1}}^{n}_t
   \left(1+A_{k,b\ve}^{-1}\right)^{n}_t
   \defeq \sum_{r=0}^n t^{r(n-r)} \begin{bmatrix} n \\ r
  \end{bmatrix}_t A_{k,b\ve}^{-r},
\end{equation*}
where $\left[\begin{smallmatrix} n \\ r
  \end{smallmatrix}\right]_t$ is the $t$-binomial
coefficient.
%
More precisely, we have
\begin{Theorem}\label{thm:ind}
\rom{(1)} For each $k\in I$, $\widetilde{\chi_{q,t}}(M_P)$ is
expressed as a linear combination of
\begin{equation*}
   \prod_i Y_{k,b_i}^{n_i}\left(1+A_{k,b_i\ve}^{-1}\right)^{n_i}_t
   = 
   Y_{k,b_1}^{n_1}\left(1+A_{k,b_1\ve}^{-1}\right)^{n_1}_t
   Y_{k,b_2}^{n_2}\left(1+A_{k,b_2\ve}^{-1}\right)^{n_2}_t
   \cdots
\end{equation*}
with coefficients in
\(
%\begin{equation*}
   \Z[t][Y_{l,a}^\pm]_{l\neq k, a\in\C^*}
%\end{equation*}
\),
where $b_i\in\C^*$, $n_i\in \Z_{>0}$ with $b_i\neq b_j$ for
$i\neq j$.

\rom{(2)} If $L_P$ is an {\it l\/}-fundamental representation \rom(and hence
$M_P = L_P$\rom), then $\chi_{q,t}(M_P)$ contains no {\it l\/}-dominant
monomials other than $m_P$ and the condition above uniquely determines
$\chi_{q,t}(M_P)$. 
\end{Theorem}

\begin{Remark}
The statement (1) for $t=1$ was proved by
Frenkel-Mukhin~\cite{FM}. And the proof of (2) is the same for $t=1$
and the general case, as illustrated in the following examples. In
this sense, (2) should also be creditted to them.
\end{Remark}

\subsection{Graph}

We give few examples of $\chi_{q,t}$ of {\it l\/}-fundamental
representations determined by the above theorem.

We attach to each standard module $M_P$, an oriented colored graph
$\Gamma_P$. (It is a slight modification of the graph in
\cite[5.3]{FR}.) The vertices are monomials in
$\chi_{q,t}(M_P)$. We draw an colored edge $\xrightarrow{k,a}$ from
$m_1$ to $m_2$ if $m_2 = m_1 A_{k,a}^{-1}$. We also write the
multiplicity of the monomials in $\chi_{q,t}(M_P)$.

\begin{Example}\label{exm:A3}
Let $\g = A_3 = \algsl_4$ and $M_P = L(\Lambda_2)_1$. Then the corresponding 
graph $\Gamma_P$ is
\begin{equation*}
\begin{CD}
   Y_{2,1} @>{2,\ve}>> Y_{1,\ve}Y_{2,\ve^2}^{-1} Y_{3,\ve}
   @>{{1,\ve^2}}>> Y_{1,\ve^3}^{-1} Y_{3,\ve} @. @. @. \\
   @. @V{{3,\ve^2}}VV @VV{{3,\ve^2}}V @. @. \\
   @. Y_{1,\ve} Y_{3,\ve^3}^{-1} @>{{1,\ve^2}}>>
   Y_{1,\ve^3}^{-1} Y_{2,\ve^2} Y_{3,\ve^3}^{-1} @>{{2,\ve^3}}>>
   Y_{2,\ve^4}^{-1}.
\end{CD}
\end{equation*}
Let us explain how we determine this graph inductively. We start with
the {\it l\/}-highest weight $Y_{2,1}$. We know that its coefficient
is $1$. Applying \thmref{thm:ind}(1) with $k=2$, we get
$Y_{1,\ve}Y_{2,\ve^2}^{-1} Y_{3,\ve}$ with coefficient $1$. Then we
apply \thmref{thm:ind}(1) with $k=1$ to get $Y_{1,\ve^3}^{-1}
Y_{3,\ve}$. And so on. All multiplicities are $1$ in this case.
\end{Example}

For $\g = A_n$, it is known that the coefficients of
$\chi_{q,t}(L(\Lambda_k)_a)$ are all $1$.\footnote{
%
More generally, if the coefficients of $\alpha_k$ in the highest root
is $1$, then the same holds. This result easily follows from the
theory of quiver varieties. {\bf Exercise}: Check this using the above 
algorithm.
%
}
Thus $\chi_{q,t}(L(\Lambda_k)_a) = \chi_{q,t=1}(L(\Lambda_k)_a)$.

\begin{Example}\label{exm:D4}
Let $\g = D_4$ and $M_P = L(\Lambda_2)_{1}$. The graph $\Gamma_P$ is
Figure~\ref{fig:D4}.
\begin{figure}[p]
\begin{center}
\leavevmode
\begin{equation*}
\divide\dgARROWLENGTH by 3
\dgARROWPARTS=6
\begin{diagram}
  \node[3]{Y_{2,1}} \arrow{s,r}{2,\ve}
\\
  \node[3]{Y_{1,\ve}Y_{2,\ve^2}^{-1}Y_{3,\ve}Y_{4,\ve}}
  \arrow{sw,t,3}{3,\ve^2} \arrow{s,l,3}{1,\ve^2}
  \arrow{se,t,3}{4,\ve^2}
\\
  \node[2]{Y_{1,\ve} Y_{3,\ve^3}^{-1} Y_{4,\ve}}
  \arrow{s,l}{1,\ve^2} \arrow{se,b,1}{4,\ve^2}
  \node{Y_{1,\ve^3}^{-1} Y_{3,\ve} Y_{4,\ve}}
  \arrow{sw,b,1}{3,\ve^2}\arrow{se,b,1}{4,\ve^2}
  \node{Y_{1,\ve} Y_{3,\ve} Y_{4,\ve^3}^{-1}}
  \arrow{sw,b,1}{3,\ve^2} \arrow{s,r}{1,\ve^2}
\\
  \node[2]{Y_{1,\ve^3}^{-1} Y_{2,\ve^2} Y_{3,\ve^3}^{-1} Y_{4,\ve}}
  \arrow{sw,b,1}{2,\ve^3} \arrow{se,b,1}{4,\ve^2}
  \node{Y_{1,\ve} Y_{2,\ve^2} Y_{3,\ve^3}^{-1} Y_{4,\ve^3}^{-1}}
  \arrow{sw,b,1}{2,\ve^3}
  \arrow{s,r}{1,\ve^2}
  \node{Y_{1,\ve^3}^{-1} Y_{2,\ve^2} Y_{3,\ve} Y_{4,\ve^3}^{-1}}
  \arrow{sw,b,1}{3,\ve^2}
  \arrow{s,r}{2,\ve^3}
\\
  \node{Y_{2,\ve^4}^{-1}Y_{4,\ve}Y_{4,\ve^3}}
  \arrow{s,l}{4,\ve^4}
  \arrow{ese,b,1}{4,\ve^2}
  \node{Y_{1,\ve} Y_{1,\ve^3} Y_{2,\ve^4}^{-1}}
  \arrow{s,r,1}{1,\ve^4}
  \arrow{se,t,3}{1,\ve^2}
  \node{Y_{1,\ve^3}^{-1} Y_{2,\ve^2}^2 Y_{3,\ve^3}^{-1}
  Y_{4,\ve^3}^{-1}}
  \arrow{s,r}{2,\ve^3}
  \node{Y_{2,\ve^4}^{-1} Y_{3,\ve} Y_{3,\ve^3}}
  \arrow{sw,t,3}{3,\ve^2}
  \arrow{s,r}{3,\ve^4}
\\
  \node{Y_{4,\ve}Y_{4,\ve^5}^{-1}}
  \arrow{s,l}{4,\ve^2}
  \node{Y_{1,\ve}Y_{1,\ve^5}^{-1}}
  \arrow{s,l}{1,\ve^2}
  \node{[2]_t Y_{2,\ve^2}Y_{2,\ve^4}^{-1}}
  \arrow{s,r}{2,\ve^3}
  \node{Y_{3,\ve}Y_{3,\ve^5}^{-1}}
  \arrow{s,r}{3,\ve^2}
\\
  \node{Y_{2,\ve^2}Y_{4,\ve^3}^{-1}Y_{4,\ve^5}^{-1}}
  \arrow{se,b,1}{2,\ve^3}
  \node{Y_{1,\ve^3}^{-1}Y_{1,\ve^5}^{-1} Y_{2,\ve^2}}
  \arrow{se,b,1}{2,\ve^3}
  \node{Y_{1,\ve^3}Y_{2,\ve^4}^{-2}Y_{3,\ve^3}Y_{4,\ve^3}}
  \arrow{sw,b,1}{4,\ve^4}\arrow{s,r,2}{1,\ve^4}
  \arrow{se,t,3}{3,\ve^4}
  \node{Y_{2,\ve^2}Y_{3,\ve^3}^{-1}Y_{3,\ve^5}^{-1}}
  \arrow{s,r}{2,\ve^3}
\\
  \node[2]{Y_{1,\ve^3}  Y_{2,\ve^4}^{-1} Y_{3,\ve^3} Y_{4,\ve^5}^{-1}}
  \arrow{s,l}{1,\ve^4}
  \arrow{se,b,1}{3,\ve^4}
  \node{Y_{1,\ve^5}^{-1} Y_{2,\ve^4}^{-1} Y_{3,\ve^3} Y_{4,\ve^3}}
  \arrow{sw,b,1}{4,\ve^4}\arrow{se,b,1}{3,\ve^4}
  \node{Y_{1,\ve^3}  Y_{2,\ve^4}^{-1}Y_{3,\ve^5}^{-1}Y_{4,\ve^3}}
  \arrow{sw,b,1}{4,\ve^4} \arrow{s,r}{1,\ve^4}
\\
  \node[2]{Y_{1,\ve^5}^{-1} Y_{3,\ve^3} Y_{4,\ve^5}^{-1}}
  \arrow{se,b,2}{3,\ve^4}
  \node{Y_{1,\ve^3} Y_{3,\ve^5}^{-1} Y_{4,\ve^5}^{-1}}
  \arrow{s,r}{1,\ve^4}
  \node{Y_{1,\ve^5}^{-1} Y_{3,\ve^5}^{-1} Y_{4,\ve^3}}
  \arrow{sw,b,2}{4,\ve^4}
\\
  \node[3]{Y_{1,\ve^5}^{-1}Y_{2,\ve^4}Y_{3,\ve^5}^{-1}Y_{4,\ve^5}^{-1}}
  \arrow{s,r}{2,\ve^5}
\\
  \node[3]{Y_{2,\ve^6}^{-1}}
\end{diagram}
\end{equation*}
\caption{The graph for $L(\Lambda_2)_{1}$}
\label{fig:D4}
\end{center}
\end{figure}
It is known that the restriction of $M_P$ to a $\Ue$-module is a
direct sum of the adjoint representation and the trivial
representation. This fact is reflected in $\chi_{q,t}(M_P)$ where
$Y_{2,\ve^2} Y_{2,\ve^4}^{-1}$ has the coefficient $[2]_t$ and all
others has $1$. Note that the number of monomials is $28$, which is
the dimension of the adjoint representation. See also
Example~\ref{exm:D4'} below.
\end{Example}

Let us give a more complicated example.
\begin{Example}
Let $\g = A_2$ and $M_P 
= L(\Lambda_2)_{\ve}^{\otimes 2}\otimes L(\Lambda_1)_1$.
Although this is {\it not\/} an {\it l\/}-fundamental representation,
$\chi_{q,t}(M_P)$ has no {\it l\/}-dominant terms other than $m_P$,
so the condition
\thmref{thm:ind}(1) gives us $\chi_{q,t}$. The graph is
Figure~\ref{fig:A2}.

\begin{figure}[htbp]
\begin{center}
\leavevmode
\begin{equation*}
\divide\dgARROWLENGTH by 2
\dgARROWPARTS=6
\begin{diagram}
\node[2]{Y_{1,1} Y_{2,\ve}^2}
\arrow{sw,t,3}{1,\ve} \arrow{se,t,3}{2,\ve^2}
\\
\node{Y_{1,\ve^2}^{-1} Y_{2,\ve}^3}
\arrow{se,t,3}{2,\ve^2}
\node[2]{[2]_t Y_{1,1} Y_{1,\ve^2} Y_{2,\ve} Y_{2,\ve^3}^{-1}}
\arrow{wsw,t,2}{1,\ve^3}
\arrow{sw,b,2}{1,\ve}
\arrow{s,r,3}{2,\ve^2}
\\
\node{[2]_t Y_{1,1} Y_{1,\ve^4}^{-1} Y_{2,\ve}}
\arrow{s,l,3}{2,\ve^2}
\node{[3]_t Y_{2,\ve}^2 Y_{2,\ve^3}^{-1}}
\arrow{s,l,3}{2,\ve^2}
\node{Y_{1,1} Y_{1,\ve^2}^2 Y_{2,\ve^3}^{-2}}
\arrow{sw,b,1}{1,\ve}
\arrow{s,r,3}{1,\ve^3}
\\
\node{[2]_t Y_{1,\ve^2}^{-1} Y_{1,\ve^4}^{-1} Y_{2,\ve}^2}
\arrow{se,b,1}{2,\ve^2}
\node{[3]_t Y_{1,\ve^2} Y_{2,\ve} Y_{2,\ve^3}^{-2}}
\arrow{sw,b,1}{2,\ve^2}
\arrow{s,r,3}{1,\ve^3}
\node{[2]_t Y_{1,1} Y_{1,\ve^2} Y_{1,\ve^4}^{-1} Y_{2,\ve^3}^{-1}}
\arrow{sw,b,1}{1,\ve}
\arrow{s,r,3}{1,\ve^3}
\\
\node{Y_{1,\ve^2}^2 Y_{2,\ve^3}^{-3}}
\arrow{s,l,3}{2,\ve^2}
\node{([3]_t + 1)Y_{1,\ve^4}^{-1} Y_{2,\ve} Y_{2,\ve^3}^{-1}}
\arrow{sw,b,2}{2,\ve^2}
\node{Y_{1,1} Y_{1,\ve^4}^{-2}}
\arrow{s,r,3}{1,\ve}
\\
\node{[2]_t Y_{1,\ve^2} Y_{1,\ve^4}^{-1} Y_{2,\ve^3}^{-2}}
\arrow{se,b,2}{1,\ve^3}
\node[2]{Y_{1,\ve^2}^{-1} Y_{1,\ve^4}^{-2} Y_{2,\ve}}
\arrow{sw,b,2}{2,\ve^2}
\\
\node[2]{Y_{1,\ve^4}^{-2} Y_{2,\ve^3}^{-1}}
\end{diagram}
\end{equation*}
\caption{The graph for $L(\Lambda_2)_{\ve}^{\otimes 2}\otimes L(\Lambda_1)_1$}
\label{fig:A2}
\end{center}
\end{figure}
\end{Example}

\begin{Remark}
As we can see in above examples, the crystal graphs are subgraphs of
$\Gamma_P$. The set of vertices is the same, but the set of arrows is
smaller. We would like to discuss this further elsewhere.
\end{Remark}

\section{Step 2}

\subsection{}

Let $M_P = M(s^1; (k_{\alpha_1}^1), (n_{\alpha_1}^1)) \otimes
M(s^2; (k_{\alpha_2}^2), (n_{\alpha_2}^2)) \otimes \cdots$ be a standard
module with $s^i/s^j\notin \ve^{\Z}$ as in \thmref{thm:VV}.
\begin{Proposition}
We have
%\(
\begin{equation*}
   \chi_{q,t}(M_P) = \chi_{q,t}
   (M(s^1; (k_{\alpha_1}^1)_{\alpha_1}, (n_{\alpha_1}^1)_{\alpha_1}))
   \chi_{q,t}
   (M(s^2; (k_{\alpha_2}^2)_{\alpha_2}, (n_{\alpha_2}^2)_{\alpha_2})) \cdots
\end{equation*}
%\)
if $s^i /s^j\notin \ve^\Z$ for $i\neq j$.
\end{Proposition}

Thus it is enough to study
\begin{equation*}
  \chi_{q,t}(M(s; (k_\alpha)_\alpha, (n_\alpha)_\alpha))
  = \chi_{q,t}(L(\Lambda_{k_1})_{\ve^{n_1}s}\otimes
   L(\Lambda_{k_2})_{\ve^{n_2}s}\otimes\cdots).
\end{equation*}

Let
\begin{equation*}
   \chi_{q,t}(L(\Lambda_{k_\alpha})_{\ve^{n_\alpha}s}))
   = \sum_{r_\alpha} a_{m_\alpha,r_\alpha}(t) m_{\alpha,r_\alpha},
\end{equation*}
where $m_{\alpha,r_\alpha}$ is a monomial in
$Y_{k,a}^\pm$ and $a_{m_\alpha,r_\alpha}(t)\in\Z[t,t^{-1}]$ is its
coefficient.

If $t=1$, $\chi_{q,1}$ is a ring homomorphism, hence we have
\begin{equation*}
  \chi_{q,1}(M(a; (k_\alpha)_\alpha, (n_\alpha)_\alpha))
  = \sum_{r_1,r_2,\dots} \prod_\alpha
  a_{m_r,r_\alpha}(1) m_{\alpha,r_\alpha}.
\end{equation*}

\begin{Theorem}
Let $P^\alpha$ be the Drinfeld polynomial of
$L(\Lambda_{k_\alpha})_{\ve^{n_\alpha}s}$.
Then we have
\begin{equation*}
  \chi_{q,t}(M(a; (k_\alpha), (n_\alpha)))
  = \sum_{r_1,r_2,\dots} 
  t^{
  \sum_{\alpha, \beta} \pm d(m_{\alpha,r_\alpha}, m_{P^\alpha};
  m_{\beta,r_\beta},m_{P^\beta})}
  \prod_\alpha
  a_{m_r,r_\alpha}(t) m_{\alpha,r_\alpha},
\end{equation*}
where the sign for $d(m_{\alpha,r_\alpha}, m_{P^\alpha};
m_{\beta,r_\beta},m_{P^\beta})$ is $-$ if $\alpha \le \beta$ and $+$
otherwise.
\end{Theorem}

\begin{Example}\label{exm:A1}
For $\g = A_1$, we have
\[
  d(Y_{1,a\ve^2}^{-1}, Y_{1,a}; Y_{1,a}, Y_{1,a}) = 1,
\qquad d(Y_{1,a}, Y_{1,a}; Y_{1,a}^{-1}, Y_{1,a\ve^{-2}}) = 1
\]
and all others are $0$.
%
Then we get \eqref{eq:exp}.
%\begin{equation*}
%   \chi_{q,t}(M_{a\ve^2}\otimes M_a) = Y_{1,a} Y_{1,a\ve^2}
%   + Y_{1,a} Y_{1,a\ve^4}^{-1} + Y_{1,a\ve^2}^{-1} Y_{1,a\ve^4}^{-1}
%   + t^{-1} Y_{1,a\ve^2} Y_{1,a\ve^2}^{-1}.
%\end{equation*}

If $P = (1-au)^n$, we get
\begin{equation*}
  \chi_{q,t}(M_P) = \sum_{r=0}^n
  \begin{bmatrix} n \\ r\end{bmatrix}_t 
  Y_{1,a}^{n-r} Y_{1,a\ve^2}^{-r}
\end{equation*}
from $\chi_{q,t}(L(\Lambda_1)_a) = Y_{1,a} + Y_{1,a\ve^2}^{-1}$.
This also follows directly from the definition~\eqref{eq:tana}
below. The $t$-binomial coefficients appear as Poincar\'e polynomials
of Grassmann manifolds.
\end{Example}

\section{Restrition to $\Ue$}\label{sec:rest}

Finite dimensional simple $\Ue$-modules are classified by highest
weights.
%
Let $\operatorname{Res}M_P$ be the restriction of a standard module
$M_P$ to a $\Ue$-module. It decomposes into a sum of various
simple modules.
%
Once $\chi_{q}(M_P)$ is computed, the character of
$\operatorname{Res}M_P$ is given by replacing $Y^\pm_{k,a}$ by
$y_k^\pm$ (\thmref{thm:FR}(2)). Combining with the knowledge of
characters of simple finite dimensional $\Ue$-modules, we can
determine the multiplicity of simple modules in
$\operatorname{Res}M_P$.

Characters of simple finite dimensional $\Ue$-modules are the same as
that of simple $\g$-modules, hence are known. However, we express them
in terms of $\chi_{q,t}$ in this section.

\subsection{}
For a dominant weight $\bw = \sum w_k\Lambda_k$ we denote by $L_\bw$
the simple highest weight $\Ue$-module with the highest weight $\bw$.

We consider a standard module $M_P$ with $\deg P_k = w_k$.
%
By the `continuity' of $M_P$ on $P$, $\operatorname{Res}M_P$ depends
only on $w_k = \deg P_k$, and not on $P$ itself. Let us denote the
multiplicity of $L_{\bw'}$ in $\operatorname{Res}M_P$ by
$Z_{\bw',\bw}$, i.e.,
\begin{equation*}
   \operatorname{Res}M_P
    = \bigoplus_{\bw'} L_{\bw'}^{\oplus Z_{\bw',\bw}}.
\end{equation*}

We will give a formula expressing $Z_{\bw',\bw}$ in terms of
$\chi_{q,t}(M_P)$. Although we can give algorithm for arbitary
$P$ in principle, the following choice will make the formula simple.

Choose and fix orientations of edges in the Dynkin diagram. We define
integer $m(k)$ for each vertex $k$ so that $m(k) - m(l) = 1$ if we
have an oriented edge from $k$ to $l$, i.e., $k\to l$. Then we define
$P$ by
\begin{equation*}
   P_k(u) = (1 - u\ve^{m(k)})^{w_k}.
\end{equation*}

Let $\widetilde{\chi_{q,t}}(M_P)$ as in \eqref{eq:modify}.
%
Let $\widetilde{\chi_t}(M_P)\in\Z[t]\otimes \Z[t_k^\pm]_{k\in I}$ be
a $t$-analogue of the ordinary character which is obtained from
$\widetilde{\chi_{q,t}}(M_P)$ by sending $Y_{k,a}^\pm$ to $y_k^\pm$.

For another dominant weight $\bw' = \sum w'_k\Lambda_k$, let
\begin{equation*}
   c_{\bw',\bw}(t) \defeq \text{the coefficient of $\prod y_k^{w'_k}$ in
   $\widetilde{\chi_{t}}(M_P)$}.
\end{equation*}
The matrix $(c_{\bw',\bw}(t))_{\bw',\bw}$ is upper-triangular with
respect to the usual order on weights, and diagonal entries are all
$1$.

\begin{Theorem}\label{thm:wmul}
$c_{\bw',\bw}(0)$ is the weight multiplicity of $\bw'$ in the highest
weight module $L_\bw$ with the highest weight $\bw$.
\end{Theorem}

This is just a simple rephrasing of a main result in \cite{Na:1994,Na:1998}.
%
The proof will be given in \subsecref{subsec:wmul}.

Note that $c_{\bw',\bw}(1)$ gives the weight multiplicity of $\bw'$ in
$\operatorname{Res}M_P$ since $\widetilde{\chi_{t=1}}$ is the ordinary
character. Thus we have
\begin{equation*}
   c_{\bw'',\bw}(1) = \sum_{\bw'} c_{\bw'',\bw'}(0)\; Z_{\bw',\bw}.
\end{equation*}
This equation determines the multiplicity $Z_{\bw',\bw}$ only from the 
knowledge of $\chi_{q,t}$.

According to a conjecture of Lusztig~\cite{Lu:ferm} together with a
formula \eqref{eq:Poin} below, $c_{\bw',\bw}(t)$ should be written by
ferminonic form of Hatayama el al.~\cite{HKOTY}. More precisely, we
should have
%\begin{equation*}
\(
   \sum_{\bw'} c^{\bw'',\bw'}(0)\, c_{\bw',\bw}(t) = M(\bw,\bw'',t^2),
\)
%\end{equation*}
where $(c^{\bw'',\bw'}(0))$ is the inverse matrix of
$(c_{\bw'',\bw'}(0))$. See \cite{Lu:ferm} for the defintion of
$M(\bw,\bw',q)$.
%
Although this formula can be checked in many examples, the complexity
of the combinatorics prevent us from proving it in full generality.
%
Conjecturally $M(\bw,\bw',q=1)$ gives us the multiplicities of the
restriction of $M_P$ (Kirillov-Reshetikhin\footnote{In fact, they
consider more general modules, not necessarily standard modules.}).
Thus the conjecture is compatible with our result in this section.

\subsection{}
\begin{Example}\label{exm:A1'}
Let $\g = A_1$ and $\bw = 2\Lambda_1$. We take $P = (1-u)^2$ by the
above choice.
By Example~\ref{exm:A1}, we
have
\begin{equation*}
   \widetilde{\chi_{t}}(M_P) = y_1^2 + (1+t^2) + y_1^{-2}.
\end{equation*}
Thus $Z_{0,\bw} = 1$. Since $\operatorname{Res}(M_P) =
L_{\Lambda_1}\otimes L_{\Lambda_1} = L_{2\Lambda_1}\oplus L_0$, this is the
correct answer~!
\end{Example}

\begin{Example}
Let $\g = A_3$ and $\bw = \Lambda_2$. By Example~\ref{exm:A3} all the
coefficients of $\chi_{q,t}(L(\Lambda_2)_1)$ are $1$. Hence
$\operatorname{Res}M_P = \operatorname{Res} L(\Lambda_2)_1$ is simple
as a $\Ue$-module.
\end{Example}

\begin{Example}\label{exm:D4'}
Let $\g = D_4$, $\bw = \Lambda_2$, $\bw' = 0$. By Example~\ref{exm:D4} 
we have $c_{\bw',\bw}(t) = 4 + t^2$. Thus $Z_{\bw',\bw} = 1$, i.e.
$\operatorname{Res}(L(\Lambda_2)_1) = L_{\Lambda_2}\oplus L_0$.
\end{Example}


\section{Quiver varieties}\label{sec:quiver}

In this section, we give the definition of $\chi_{q,t}$ and prove
Theorems~\ref{thm:mul},~\ref{thm:wmul}.
%
As we mentioned, those proofs are essentially given in
\cite{Na:1994,Na:1998} and \cite{Na-qaff} respectively. The only
things we do here are translation of results into the language of
$\chi_{q,t}$.
%
We believe that this section gives good introductions to
\cite{Na:1994,Na:1998,Na-qaff}.

\subsection{}\label{subsec:classical}

Let $\bw = \sum w_k \Lambda_k$ ($w_k\in\Z_{\ge 0}$) be a dominant
weight of the finite dimensional Lie algebra $\g$.
In \cite{Na:1994,Na:1998,Na-qaff}, we have attached to each $\bw$, a
map $\pi\colon \Mw\to \M_0(\infty,\bw)$ with the following properties:
\begin{enumerate}
\item $\Mw$ is a finite disjoint union of nonsingular quasi-projective
  varieties of various dimensions.
\item $\M_0(\infty,\bw)$ is an affine algebraic variety.
\item $\pi$ is a projective morphism.
\item There exist actions of $G_\bw\times\C^*$ on $\Mw$ and
$\M_0(\infty,\bw)$ such that $\pi$ is equivariant.
\item $\M_0(\infty,\bw)$ is a cone, and the vertex (denoted by $0$) is
the unique fixed point of the $\C^*$-action (restriction of
$G_\bw\times\C^*$-action to the second factor).
\end{enumerate}
Here $G_\bw = \prod_{k\in I} \GL(w_k,\C)$.

We consider the fiber product
\begin{equation*}
   \Zw \defeq \M(\bw)\times_{\M_0(\infty,\bw)} \M(\bw).
\end{equation*}
The convolution product makes the (Borel-Moore) homology group
$H_*(\Zw,\C)$ into an associative (noncommutative) algebra.
One of main results in \cite{Na:1998} is a construction of a
surjective algebra homomorphism
\begin{equation*}
   \bU(\g) \to H_{\topdeg}(\Zw,\C),
\end{equation*}
where $\bU(\g)$ is the universal enveloping algebra of $\g$ ({\bf NB}:
not a `quantum' version). Here $H_{\topdeg}(\ )$ means the degree
$=\dim_\R\Zw$ part of the homology group. More precisely, we take
degree = dimension part on each connected components of $\Zw$, and
then make the direct sum. Note that the the dimension differs on
various components.

Let $\Law = \pi^{-1}(0)$. It is known that $\Mw$ has a holomorphic
symplectic form such that $\Law$ is a lagrangian subvariety. The
convolution makes $H_{\topdeg}(\Law,\C)$ (the top degree part of the
Borel-Moore homology group, in the same sense as above) into an
$H_{\topdeg}(\Zw,\C)$-module. It is a $\bU(\g)$-module by the above
homomorphism. By \cite[10.2]{Na:1998} it is the simple finite dimensional
$\bU(\g)$-module $L_\bw$ with highest weight $\bw$. And connected
components $\M(\bv,\bw)$ of $\Mw$ are parametrized by vectors $\bv =
\sum v_k \alpha_k$ ($\alpha_k$ is the $k$th simple root of $\g$) so that
\begin{equation*}
   H_{\topdeg}(\Law,\C) = \bigoplus_\bv
   H_{\topdeg}(\M(\bv,\bw)\cap\Law,\C)
\end{equation*}
is the weight space decomposition of the simple highest weight module
$L_\bw$, where $H_{\topdeg}(\M(\bv,\bw)\cap\Law,\C)$ has weight $\bw -
\bv$. In particular, $\bv=0$ corresponds to the highest weight vector.
In fact, $\M(0,\bw)$ is consisting of a single point.

The space $\M_0(\infty,\bw)$ has a stratification
\begin{equation*}
   \M_0(\infty,\bw) = \bigcup \Mreg(\bv,\bw),
\end{equation*}
where $\bv$ runs over the set of vectors such that $\bw-\bv$ is a
weight of $L_\bw$ which is {\it dominant\/} \cite[\S3]{Na:1994}.

\subsection{}

Let us give the $\Ul$-version of the construction of the previous
subsection.

We use the following notation: Let $R(G)$ denote the representation
ring of a linear algebraic group $G$. If $G$ acts a quasi-projective
variety $X$, $K^G(X)$ denotes the Grothendieck group of
$G$-equivariant coherent sheaves on $X$.

The representation ring $R(G_\bw\times\C^*)$ of $G_\bw\times\C^*$ is
isomorphic to the tensor product $R(G_\bw)\otimes_\Z
R(\C^*)$. Moreover, $R(\C^*)$ is isomorphic to $\Z[q,q^{-1}]$, where
$q$ is the canonical $1$-dimensional representation of $\C^*$.

The convolution makes the Grothendieck group
$K^{G_\bw\times\C^*}(\Zw)$ into a $R(G_\bw\times\C^*) =
R(G_\bw)[q,q^{-1}]$-algebra. One of main results in \cite{Na-qaff} is
a construction of an algebra homomorphism
\begin{equation*}
   \Uli\otimes_{\Z} R(G_\bw)\to
   K^{G_\bw\times\C^*}(\Zw)/\operatorname{torsion}.
\end{equation*}

By the equivariance of $\pi$, $\Law = \pi^{-1}(0)$ is invariant under
$G_\bw\times\C^*$. The convolution makes $K^{G_\bw\times\C^*}(\Law)$
into a $K^{G_\bw\times\C^*}(\Zw)$-module. Moreover, it is free of
finite rank over $R(G_\bw\times\C^*)$ \cite[\S7]{Na-qaff}. It is a
$\Uli\otimes_\Z R(G_\bw)$-module by the above homomorphism. By
\cite[\S13]{Na-qaff}, it contains a vector $m_0$ such that
\begin{equation}
\label{eq:unistd}
\begin{gathered}
  e_{k,r}\ast m_0 = 0, \qquad
  \left(\Uli^-\otimes_\Z R(G_\bw)\right) \ast m_0
      = K^{G_\bw\times\C^*}(\Law),
\\
  \psi^{\pm}_{k}(z)\ast m_0 
  = q^{w_k}
  \left(\frac{\Wedge_{-1/qz} q^{-1} W_k}
  {\Wedge_{-q/z} q^{-1} W_k}\right)^\pm \ast m_0
    \quad\text{for $k\in I$}.
\end{gathered}
\end{equation}
The right hand side of the third equation needs an explanation: First
$W_k$ is the vector representation of $\GL(w_k,\C)$, considered as a
$G_\bw\times\C^*$-module. Then $\Wedge_u V = \sum u^i \Wedge^i V$.
Since $\Wedge_{-q/z} q^{-1} W_k$ is $1 - (1/z) W_k + \dots$ ($1$ is
the trivial module), we can define $\left(\Wedge_{-q/z} q^{-1}
  W_k\right)^{-1}$ as a formal power series in $1/z$. This gives us
the case $(\ )^+$ of the above formula. In the case $(\ )^-$, we
expand as
\(
  \Wedge_{-q/z} q^{-1} W_k = 
  (-1/z)^{w_k}
  \left(\Wedge^{w_k} W_k - z \Wedge^{w_k-1} W_k +
    \cdots\right)
\).
Then $\Wedge^{w_k} W_k$ is an invertible element, we can also define
$\left(\Wedge_{-q/z} q^{-1} W_k\right)^{-1}$.
%
The vector $m_0$ is the canonical generator of
$K^{G_\bw\times\C^*}(\M(0,\bw))$. (Recall $\M(0,\bw)$ is a point.)


The module $K^{G_\bw\times\C^*}(\Law)$ should be considered as a
`universal' standard module since standard modules are obtained from
it by specializations as we explain now.

Let $a = (s,\ve)\in G_\bw\times\C^*$ be a semisimple element. It
defines a homomorphism $\chi_a\colon R(G_\bw\times\C^*)\to \C$ by
sending a representation to the value of the character at $a$. Then
\begin{equation}\label{eq:std}
   K^{G_\bw\times\C^*}(\Law) \otimes_{R(G_\bw\times\C^*)} \C
\end{equation}
is a module over $\Ule = \Uli\otimes_{\Z[q,q^{-1}]}\C$.
By \eqref{eq:unistd} it is a finite-dimensional {\it l\/}-highest
weight module. This is the {\it standard module\/} $M_P$, where
$P_k(u)  = \chi_a(\Wedge_{-u}\linebreak[0] q^{-1} W_k)$.
%
Note that the set of conjugacy classes of $a = (s,\ve)$ bijectively
corresponds to the set of $I$-tuple of polynomials $P$ with $\deg P_k = w_k$.

\subsection{}\label{subsec:qchar}

Let $A$ be the Zariski closure of $a^\Z$ in $G_\bw\times\C^*$. It is
an abelian reductive group. We have
$K^{G_\bw\times\C^*}(\Law)\otimes_{R(G_\bw\times\C^*)} R(A) \cong
K^A(\Law)$ \cite[\S7]{Na-qaff}. Since $\chi_a$ factors through $R(A)$,
the standard module $M_P$ is isomorphic to
$K^A(\Law)\otimes_{R(A)}\C$. By Thomason's localization theorem, it is
isomorphic to $K(\Law^A)\otimes_{\Z}\C$, where $\Law^A$ is the fixed
point set. Furthermore, it is isomorphic to $H_*(\Law^A,\C)$ via the
Chern character homomorphism \cite[\S7]{Na-qaff}.

Let $\pi^A\colon\Mw^A\to\M_0(\infty,\bw)^A$ be the restriction of the
map $\pi\colon\Mw\to\M_0(\infty,\bw)$ to the fixed point set. Let
$\Mw^A = \bigsqcup_\rho \M(\rho)$ be the decomposition into connected
components. Each $\M(\rho)$ is a nonsingular quasi-projective variety.
Then we have the direct sum decomposition
\begin{equation*}
   M_P \cong H_*(\Law^A,\C) \cong \bigoplus_\rho H_*(\M(\rho)\cap\Law,\C).
\end{equation*}
In \cite[\S13, \S14]{Na-qaff} we have shown that this is the {\it
l\/}-weight space decomposition of $M_P$. In particular, the index
$\rho$ can be considered as an {\it l\/}-weight of $M_P$. Thus we have
arrived at a geometric interpretation of $\chi_q$:
\begin{equation*}
   \chi_q(M_P) 
   = \sum_\rho \dim H_*(\M(\rho)\cap\Law,\C)\; m_\rho,
\end{equation*}
where $m_\rho$ is the monomial corresponding to the {\it l\/}-weight $\rho$.

Now we define the $t$-analogue $\chi_{q,t}$ by
\begin{equation}\label{eq:tana}
   \chi_{q,t}(M_P)
   \defeq
   \sum_\rho \sum_k \dim H_k(\M(\rho)\cap\Law,\C)\;
   t^{k-\dim_\C \M(\rho)} \, m_\rho.
\end{equation}

By \cite[\S14]{Na-qaff} we have a stratification
\begin{equation*}
   \M_0(\infty,\bw)^A = \bigcup_\rho \Mreg(\rho),
\end{equation*}
consisting of nonsingular locally closed subvarieties.
Here the index set $\{ \rho\}$ is the subset of the above index set
consisting of {\it l-dominant\/} {\it l\/}-weights.

\subsection{\protect{Proof of \thmref{thm:mul}}}\label{subsec:prf}

The {\it l\/}-highest weight $P$ is fixed throughout the proof. Thus
the dominant weight vector $\bw$ and the element $a =
(s,\ve)\in G_\bw$ are fixed.

We change the notation now. If $\rho$ corresponds to an {\it
l\/}-weight space $M_P(Q/R)$, we denote above $\M(\rho)$ by
$\M(Q/R,P)$. We also denote by $\Mreg(Q,P)$ for above $\Mreg(\rho)$ if
$\rho$ corresponds to an {\it l\/}-dominant {\it l\/}-weight $Q$. Thus
we have
\begin{equation*}
   \M(\bw)^A = \bigsqcup_{Q/R} \M(Q/R,P), \quad
   \M_0(\infty,\bw)^A = \bigcup_Q \Mreg(Q,P).   
\end{equation*}
In this notation $H_*(\M(P,P)\cap\Law,\C)$ is the {\it l\/}-highest
weight space. Since $\M(0,\bw)$ is a single point as we explained, we
have $\M(P,P) = \M(0,\bw)$. We also have $\Mreg(P,P) = \{ 0\}$.

\begin{Lemma}\label{lem:slice}
\rom{(1)} $\dim_\C \M(Q/R,P) = d(Q/R,P)$. $\dim_\C \Mreg(Q,P) = d(Q,P)$.

\rom{(2)} If $\Mreg(Q,P)\subset \overline{\Mreg(R,P)}$, then
$R\le Q$.

\rom{(3)} Choose $x\in\Mreg(Q,P)$. Then 
$(\pi^A)^{-1}(x)\cap\M(S/T,P)$ is isomorphic to
$\M(S/T,Q)\cap\Law$.
\end{Lemma}

\begin{proof}
(1) The first equation is the dimension formula
\cite[4.1.6]{Na-qaff}. The second equation follows from $\dim_\C \Mreg(Q,P) =
\dim_\C \M(Q,P)$, which is clear from the definition \cite[\S4]{Na-qaff}.

(2),(3) The results are known or trivial for $Q = P$. Now use the
transversal slice at $x\in\Mreg(Q,P)$ \cite[\S3]{Na-qaff} to reduce
a general case to this case.
\end{proof}

Let $D^b(\M_0(\infty,\bw)^A)$ be the bounded derived category of
complexes of sheaves such that cohomology sheaves are constant along
each stratum $\Mreg(Q,P)$. Let $IC(\Mreg(Q,P))$ be the intersection
homology complex associated with the constant local system
$\C_{\Mreg(Q,P)}$ on $\Mreg(Q,P)$. By using the transversal slice
\cite[\S3]{Na-qaff}, one can check that it is an object in
$D^b(\M_0(\infty,\bw)^A)$. Let $\C_{\M(Q/R,P)}$ be the constant local
system on $\M(Q/R,P)$. Then $\pi^A_*(\C_{\M(Q/R,P)})$ is an object of
$D^b(\M_0(\infty,\bw)^A)$ again by the transversal slice
argument. Using the decomposition theorem of
Beilinson-Bernstein-Deligne, we have shown that there exists an
isomorphism in $D^b(\M_0(\infty,\bw)^A)$:
\begin{equation}\label{eq:decomp}
   \pi^A_*(\C_{\M(R,P)}[\dim_\C\M(R,P)]) 
        \cong \bigoplus_{Q,k} L_{Q,k}(R,P)\otimes IC(\Mreg(Q,P))[k]
\end{equation}
for some vector space $L_{Q,k}(R,P)$ \cite[14.3.2]{Na-qaff}.
Since $\pi^A(\M(R,P))\subset\overline{\Mreg(R,P)}$ by
definition~\cite[\S4]{Na-qaff}, the summation runs over $Q\ge R$ by
\lemref{lem:slice}.
%
Let
\begin{equation*}
   L_{RQ}(t) \defeq
   \sum_{k} \dim L_{Q,k}(R,P)\, t^{-k}.
\end{equation*}
Applying the Verdier duality to the both hand side of
\eqref{eq:decomp} and using the self-duality of 
$\pi^A_*(\C_{\M(R,P)}[\dim_\C\M(R,P)])$ and $IC(\Mreg(Q,P))$, we find
$L_{RQ}(t) = \overline{L_{RQ}(t)}$.

Choose a point $x_Q$ from $\Mreg(Q,P)$ for each stratum. Let
$i_{x_Q}\colon \{ x_Q\} \to \M_0(\infty,\bw)^A$ denote the
inclusion. Consider
\begin{equation*}
   H^{k}(i_{x_Q}^! \pi^A_* \C_{\M(R,P)}[\dim \M(R,P)])
   = H_{\dim_\C \M(R,P)- k}((\pi^A)^{-1}(x_Q)\cap \M(R,P),\C).
\end{equation*}
By \lemref{lem:slice}(3) this is isomorphic to
\(
%\begin{equation*}
   H_{\dim_\C \M(R,P)- k}(\M(R,Q)\cap\Law,\C)
%\end{equation*}
\). Therefore we have
\begin{equation}\label{eq:C_RQ}
\begin{split}
   & \sum_k \dim H^{k}(i_{x_Q}^! \pi^A_* \C_{\M(R,P)}[\dim \M(R,P)])\,
   t^{\dim_\C \M(Q,P)-k}
\\
  = \; &
   \sum_d \dim H_{d}(\M(R,Q)\cap\Law,\C)\,
   t^{d + \dim_\C \M(Q,P) - \dim_\C \M(R,P)}
  = c_{RQ}(t),
\end{split}
\end{equation}
where we used $\dim_\C\M(R,P) - \dim_\C\M(Q,P) = \dim_\C\M(R,Q)$ in the
last equality.

By \cite[14.3.10]{Na-qaff}, we have
\begin{equation*}
   [M_Q:L_R] = \dim H^*(i_{x_Q}^! IC(\Mreg(R,P))).
\end{equation*}
(In fact, we defined the standard module $M_Q$ as
$H_*((\pi^A)^{-1}(x_Q),\C)$ in \cite[\S13]{Na-qaff}, which apriori
depends on $P$. Thus the definition coincides only when
$Q=P$. However, by using the transversal slice, we can show that the
right hand side is the same for both definitions. cf.\
\lemref{lem:slice}.)

Let
\begin{equation*}
   Z_{RQ}(t) \defeq
   \sum_k \dim H^k(i_{x_Q}^! IC(\Mreg(R,P)))\, t^{\dim\Mreg(Q,P)-k}.
\end{equation*}
We have $[M_Q:L_R] = Z_{RQ}(1)$. By the defining property of the
intersection homology, $Z_{RQ}(t)$ satisfies \eqref{eq:Lus2}.

Substituting \eqref{eq:decomp} into \eqref{eq:C_RQ}, we get
\begin{equation*}
   c_{SQ}(t) = \sum_{R} L_{SR}(t) Z_{RQ}(t).
\end{equation*}
Now $L_{SR}(t) = \overline{L_{SR}(t)}$ implies \eqref{eq:Lus1}.
This completes the proof of \thmref{thm:mul}.

\subsection{Proof of \protect{\thmref{thm:wmul}}}\label{subsec:wmul}

By the result explained in \subsecref{subsec:classical}, the weight
multiplicity of $\bw'$ in $L_\bw$ is equal to
\begin{equation*}
   \dim H_{\topdeg}(\M(\bw - \bw',\bw)\cap \Law,\C).
\end{equation*}
The assertion follows from more general formula
\begin{equation}\label{eq:Poin}
   \widetilde{\chi_{t}}(M_P)
   = \sum_{\bw'} 
   \sum_d H_d(\M(\bw-\bw',\bw)\cap \Law,\C)\,
      t^{\dim_\C \M(\bw-\bw',\bw) - d}\;
   \prod_k y_k^{w'_k}.
\end{equation}
Note that $\M(\bw-\bw',\bw)\cap \Law$ is a lagrangian subvariety in
$\M(\bw-\bw',\bw)$, so we have $\topdeg = \dim_\C\M(\bw-\bw',\bw)$.

In order to prove \eqref{eq:Poin}, we use \cite[5.7]{Na:1994}, where
the Betti numbers are given in terms of those of fixed point
components. It looks almost the same as above. However, there is one
significant difference. The $\C^*$-action used there is different from
our $\C^*$-action used here, defined in \cite[\S2]{Na-qaff}.
%
This is the reason why we choose $P$ and corresponding $a = (s,\ve)$
as explained in \secref{sec:rest}. Then $A = \overline{a^{\Z}}$ is
isomorphic to $\C^*$ and the action is the same as the $\C^*$-action
considered in \cite[\S5]{Na:1994}.

We decompose $\Mw^A = \bigsqcup \M(\rho)$ into connected components as
before. By \cite[5.7]{Na:1994} we have\footnote{
%
In fact, this formula even holds for general $P$ if we
replace $\dim_\C \M(\bw-\bw',\bw)-d$ by a suitable degree. However,
this degree shift is given by a complicated expression in $\rho$. So
our choice of $P$ is most economical.
%
}
\begin{equation*}
   \dim H_d(\M(\bw-\bw',\bw)\cap \Law,\C)
   = \sum_\rho \dim H_{\dim_\C \M(\bw-\bw',\bw)-d}(\M(\rho),\C),
\end{equation*}
where the summation runs over the set of
$\rho$ such that the corresponding monomial $m_\rho$ is sent to
$\prod_k y_k^{w'_k}$ after $Y_{k,a} \to y_k$.
The $\C^*$-action makes $\M_0(\infty,\bw)^{\C^*} = \{ 0\}$, so
$\M(\rho) = \M(\rho)\cap\Law$.
Hence the above expression coincides with the definition
of the coefficients $\widetilde{\chi_{t}}$.


\subsection*{Acknowledgement}
We would like to thank E.~Frenkel and E.~Mukhin for explanations of
their algorithm computing $q$-characters of fundamental
representations.

%\bibliographystyle{amsplain}
%\bibliography{qchar}

\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\begin{thebibliography}{10}

\bibitem{CP-book}
V.~Chari and A.~Pressley, 
\emph{A guide to quantum groups}, Cambridge University Press,
  Cambridge, 1994.

\bibitem{CP-rep}
\bysame, \emph{Quantum affine algebras and their representations},
  Representations of groups (Banff, AB, 1994), Amer. Math. Soc., Providence,
  RI, 1995, pp.~59--78.

\bibitem{CP-roots}
\bysame, \emph{Quantum affine algebras at roots of unity}, Represent. Theory
  \textbf{1} (1997), 280--328 (electronic).

\bibitem{CP:Weyl}
\bysame, \emph{{Integrable and {W}eyl modules for
  quantum affine $sl_2$}}, preprint, arXiv:math.QA/0007123.

\bibitem{FM}
E.~Frenkel and E.~Mukhin, \emph{{Combinatorics of q-characters of
  finite-dimensional representations of quantum affine algebras}},
preprint, arXiv:math.QA/9911112.

\bibitem{FR}
E.~Frenkel and N.~Reshetikhin, \emph{The $q$-characters of
representations of quantum affine algebras and deformations of
$\mathscr {W}$-algebras}, Recent developments in quantum affine
algebras and related topics (Raleigh, NC, 1998), Contemp. Math., 248,
Amer. Math. Soc., Providence, RI, 1999, pp.~163--205.

\bibitem{HKOTY}
G.~Hatayama, A.~Kuniba, M.~Okado, T.~Takagi and Y.~Yamada,
\emph{Remarks on fermionic formula}, Recent developments in quantum
affine algebras and related topics (Raleigh, NC, 1998),
Contemp. Math., 248, Amer. Math. Soc., Providence, RI, 1999,
pp.~243--291.

\bibitem{Kas}
M.~Kashiwara, \emph{Crystal bases of modified quantized enveloping
  algebra}, Duke Math. J. \textbf{73} (1994), no.~2, 383--413.

\bibitem{KL}
D.~Kazhdan and G.~Lusztig, \emph{Representations of {C}oxeter groups and
  {H}ecke algebras}, Invent. Math. \textbf{53} (1979), no.~2, 165--184.

\bibitem{Lu:can}
G.~Lusztig, \emph{Canonical bases arising from quantized enveloping algebras},
  J. Amer. Math. Soc. \textbf{3} (1990), no.~2, 447--498.

\bibitem{Lu:ferm}
\bysame, \emph{{{F}erminonic form and {B}etti numbers}}, preprint,
arXiv:math.QA/0005010.

\bibitem{Na:1994}
H.~Nakajima, \emph{Instantons on {A}{L}{E} spaces, quiver varieties, and
  {K}ac-{M}oody algebras}, Duke Math. J. \textbf{76} (1994), no.~2, 365--416.

\bibitem{Na-hom}
\bysame, \emph{Homology of moduli spaces instantons on {A}{L}{E} spaces~{I}},
J. Differential Geom. \textbf{40} (1994), 105--127.

\bibitem{Na:1998}
\bysame, \emph{Quiver varieties and {K}ac-{M}oody algebras}, Duke Math. J.
  \textbf{91} (1998), no.~3, 515--560.

\bibitem{Na-qaff}
\bysame, \emph{{Quiver varieties and finite dimensional representations
  of quantum affine algebras}}, preprint, arXiv:math.QA/9912158,
to appear in J. Amer. Math. Soc.

\bibitem{VV-std}
M.~Varagnolo and E.~Vasserot, \emph{{Standard modules of quantum affine
  algebras}}, preprint, arXiv:math.QA/0006084.

\end{thebibliography}

%\begin{figure}[p]
%\begin{center}
%\leavevmode
%\begin{equation*}
%\divide\dgARROWLENGTH by 2
%\dgARROWPARTS=6
%\begin{diagram}
%  \node[3]{Y_{2,1}} \arrow{s,r}{2,\ve}
%\\
%  \node[3]{Y_{1,\ve}Y_{2,\ve^2}^{-1}Y_{3,\ve}Y_{4,\ve}}
%  \arrow{sw,t,3}{3,\ve^2} \arrow{s,l,3}{1,\ve^2}
%  \arrow{se,t,3}{4,\ve^2}
%\\
%  \node[2]{Y_{1,\ve} Y_{3,\ve^3}^{-1} Y_{4,\ve}}
%  \arrow{s,l}{1,\ve^2} \arrow{se,b,1}{4,\ve^2}
%  \node{Y_{1,\ve^3}^{-1} Y_{3,\ve} Y_{4,\ve}}
%  \arrow{sw,b,1}{3,\ve^2}\arrow{se,b,1}{4,\ve^2}
%  \node{Y_{1,\ve} Y_{3,\ve} Y_{4,\ve^3}^{-1}}
%  \arrow{sw,b,1}{3,\ve^2} \arrow{s,r}{1,\ve^2}
%\\
%  \node[2]{Y_{1,\ve^3}^{-1} Y_{2,\ve^2} Y_{3,\ve^3}^{-1} Y_{4,\ve}}
%  \arrow{sw,b,1}{2,\ve^3} \arrow{se,b,1}{4,\ve^2}
%  \node{Y_{1,\ve} Y_{2,\ve^2} Y_{3,\ve^3}^{-1} Y_{4,\ve^3}^{-1}}
%  \arrow{sw,b,1}{2,\ve^3}
%  \arrow{s,r}{1,\ve^2}
%  \node{Y_{1,\ve^3}^{-1} Y_{2,\ve^2} Y_{3,\ve} Y_{4,\ve^3}^{-1}}
%  \arrow{sw,b,1}{3,\ve^2}
%  \arrow{s,r}{2,\ve^3}
%\\
%  \node{Y_{2,\ve^4}^{-1}Y_{4,\ve}Y_{4,\ve^3}}
%  \arrow{s,l}{4,\ve^4}
%  \arrow{ese,b,1}{4,\ve^2}
%  \node{Y_{1,\ve} Y_{1,\ve^3} Y_{2,\ve^4}^{-1}}
%  \arrow{s,r,1}{1,\ve^4}
%  \arrow{se,t,3}{1,\ve^2}
%  \node{Y_{1,\ve^3}^{-1} Y_{2,\ve^2}^2 Y_{3,\ve^3}^{-1}
%  Y_{4,\ve^3}^{-1}}
%  \arrow{s,r}{2,\ve^3}
%  \node{Y_{2,\ve^4}^{-1} Y_{3,\ve} Y_{3,\ve^3}}
%  \arrow{sw,t,3}{3,\ve^2}
%  \arrow{s,r}{3,\ve^4}
%\\
%  \node{Y_{4,\ve}Y_{4,\ve^5}^{-1}}
%  \arrow{s,l}{4,\ve^2}
%  \node{Y_{1,\ve}Y_{1,\ve^5}^{-1}}
%  \arrow{s,l}{1,\ve^2}
%  \node{[2]_t Y_{2,\ve^2}Y_{2,\ve^4}^{-1}}
%  \arrow{s,r}{2,\ve^3}
%  \node{Y_{3,\ve}Y_{3,\ve^5}^{-1}}
%  \arrow{s,r}{3,\ve^2}
%\\
%  \node{Y_{2,\ve^2}Y_{4,\ve^3}^{-1}Y_{4,\ve^5}^{-1}}
%  \arrow{se,b,1}{2,\ve^3}
%  \node{Y_{1,\ve^3}^{-1}Y_{1,\ve^5}^{-1} Y_{2,\ve^2}}
%  \arrow{se,b,1}{2,\ve^3}
%  \node{Y_{1,\ve^3}Y_{2,\ve^4}^{-2}Y_{3,\ve^3}Y_{4,\ve^3}}
%  \arrow{sw,b,1}{4,\ve^4}\arrow{s,r,2}{1,\ve^4}
%  \arrow{se,t,3}{3,\ve^4}
%  \node{Y_{2,\ve^2}Y_{3,\ve^3}^{-1}Y_{3,\ve^5}^{-1}}
%  \arrow{s,r}{2,\ve^3}
%\\
%  \node[2]{Y_{1,\ve^3}  Y_{2,\ve^4}^{-1} Y_{3,\ve^3} Y_{4,\ve^5}^{-1}}
%  \arrow{s,l}{1,\ve^4}
%  \arrow{se,b,1}{3,\ve^4}
%  \node{Y_{1,\ve^5}^{-1} Y_{2,\ve^4}^{-1} Y_{3,\ve^3} Y_{4,\ve^3}}
%  \arrow{sw,b,1}{4,\ve^4}\arrow{se,b,1}{3,\ve^4}
%  \node{Y_{1,\ve^3}  Y_{2,\ve^4}^{-1}Y_{3,\ve^5}^{-1}Y_{4,\ve^3}}
%  \arrow{sw,b,1}{4,\ve^4} \arrow{s,r}{1,\ve^4}
%\\
%  \node[2]{Y_{1,\ve^5}^{-1} Y_{3,\ve^3} Y_{4,\ve^5}^{-1}}
%  \arrow{se,b,2}{3,\ve^4}
%  \node{Y_{1,\ve^3} Y_{3,\ve^5}^{-1} Y_{4,\ve^5}^{-1}}
%  \arrow{s,r}{1,\ve^4}
%  \node{Y_{1,\ve^5}^{-1} Y_{3,\ve^5}^{-1} Y_{4,\ve^3}}
%  \arrow{sw,b,2}{4,\ve^4}
%\\
%  \node[3]{Y_{1,\ve^5}^{-1}Y_{2,\ve^4}Y_{3,\ve^5}^{-1}Y_{4,\ve^5}^{-1}}
%  \arrow{s,r}{2,\ve^5}
%\\
%  \node[3]{Y_{2,\ve^6}^{-1}}
%\end{diagram}
%\end{equation*}
%\caption{The graph for $L(\Lambda_2)_{1}$}
%\label{fig:D4}
%\end{center}
%\end{figure}

\end{document}

%%% Local Variables: 
%%% mode: latex
%%% TeX-command-default: "LaTeX2e"
%%% TeX-master: t
%%% End: 

math0009231.extracted_bib
\begin{thebibliography}{10}

\bibitem{CP-book}
V.~Chari and A.~Pressley, 
\emph{A guide to quantum groups}, Cambridge University Press,
  Cambridge, 1994.

\bibitem{CP-rep}
\bysame, \emph{Quantum affine algebras and their representations},
  Representations of groups (Banff, AB, 1994), Amer. Math. Soc., Providence,
  RI, 1995, pp.~59--78.

\bibitem{CP-roots}
\bysame, \emph{Quantum affine algebras at roots of unity}, Represent. Theory
  \textbf{1} (1997), 280--328 (electronic).

\bibitem{CP:Weyl}
\bysame, \emph{{Integrable and {W}eyl modules for
  quantum affine $sl_2$}}, preprint, arXiv:math.QA/0007123.

\bibitem{FM}
E.~Frenkel and E.~Mukhin, \emph{{Combinatorics of q-characters of
  finite-dimensional representations of quantum affine algebras}},
preprint, arXiv:math.QA/9911112.

\bibitem{FR}
E.~Frenkel and N.~Reshetikhin, \emph{The $q$-characters of
representations of quantum affine algebras and deformations of
$\mathscr {W}$-algebras}, Recent developments in quantum affine
algebras and related topics (Raleigh, NC, 1998), Contemp. Math., 248,
Amer. Math. Soc., Providence, RI, 1999, pp.~163--205.

\bibitem{HKOTY}
G.~Hatayama, A.~Kuniba, M.~Okado, T.~Takagi and Y.~Yamada,
\emph{Remarks on fermionic formula}, Recent developments in quantum
affine algebras and related topics (Raleigh, NC, 1998),
Contemp. Math., 248, Amer. Math. Soc., Providence, RI, 1999,
pp.~243--291.

\bibitem{Kas}
M.~Kashiwara, \emph{Crystal bases of modified quantized enveloping
  algebra}, Duke Math. J. \textbf{73} (1994), no.~2, 383--413.

\bibitem{KL}
D.~Kazhdan and G.~Lusztig, \emph{Representations of {C}oxeter groups and
  {H}ecke algebras}, Invent. Math. \textbf{53} (1979), no.~2, 165--184.

\bibitem{Lu:can}
G.~Lusztig, \emph{Canonical bases arising from quantized enveloping algebras},
  J. Amer. Math. Soc. \textbf{3} (1990), no.~2, 447--498.

\bibitem{Lu:ferm}
\bysame, \emph{{{F}erminonic form and {B}etti numbers}}, preprint,
arXiv:math.QA/0005010.

\bibitem{Na:1994}
H.~Nakajima, \emph{Instantons on {A}{L}{E} spaces, quiver varieties, and
  {K}ac-{M}oody algebras}, Duke Math. J. \textbf{76} (1994), no.~2, 365--416.

\bibitem{Na-hom}
\bysame, \emph{Homology of moduli spaces instantons on {A}{L}{E} spaces~{I}},
J. Differential Geom. \textbf{40} (1994), 105--127.

\bibitem{Na:1998}
\bysame, \emph{Quiver varieties and {K}ac-{M}oody algebras}, Duke Math. J.
  \textbf{91} (1998), no.~3, 515--560.

\bibitem{Na-qaff}
\bysame, \emph{{Quiver varieties and finite dimensional representations
  of quantum affine algebras}}, preprint, arXiv:math.QA/9912158,
to appear in J. Amer. Math. Soc.

\bibitem{VV-std}
M.~Varagnolo and E.~Vasserot, \emph{{Standard modules of quantum affine
  algebras}}, preprint, arXiv:math.QA/0006084.

\end{thebibliography}