\include{preamble}

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Speaker: Ryosuke Kodera

Title: Loewy series of Weyl modules for current Lie algebras

(joint work with Katsuyuki Naoi)

February 21, 2012

\section{Loewy series.}

Let $A$ be a $\bC$-algebra. Let $M$ be a $A$-module of finite length. The {\bf radical} of $M$ (denoted ${\rm rad}(M)$) is the smallest submodule such that the quotient is semisimple. This is also the intersection of all proper submodules. This gives us a filtration
\[
M \supset {\rm rad}(M) \supset {\rm rad}^2(M) \supset \cdots \supset 0
\]
which we call the {\bf radical series} of $M$. The {\bf socle} of $M$ (denoted ${\rm soc}(M)$) is defined to be the largest semisimple submodule of $M$. This is the sum of all simple submodules of $M$. Define ${\rm soc}^{k+1}(M)$ by ${\rm soc}^{k+1}(M) / {\rm soc}^k(M) = {\rm soc}(M / {\rm soc}^k(M))$. This gives us a filtration
\[
0 \subset {\rm soc}(M) \subset {\rm soc}^2(M) \subset \cdots \subset M
\]
which we call the {\bf socle series} of $M$.

Some properties:

\begin{compactenum}
\item The filtration length of both series coincide. We call this number the {\bf Loewy length} of $M$, and denote it by $\ell\ell(M)$. 

\item For any filtration $F^\bullet M$ of the form $M = F^0 M \supset F^1 M \supset \cdots \supset F^{\ell_F} M = 0$ such that $F^kM / F^{k+1} M$ is semisimple, we have $\ell_F \ge \ell\ell(M)$.
\end{compactenum}

\begin{definition}
$F^\bullet M$ is a {\bf Loewy series} of $M$ if $\ell_F = \ell\ell(M)$. $M$ is {\bf rigid} if the radical series and socle series of $M$ coincide. (This condition implies that $M$ has a unique Loewy series.)
\end{definition}

\begin{example}
Verma modules for complex semisimple Lie algebras are rigid. Let $M_w$ be the Verma module with highest weight $-w\rho -\rho$. Then $\ell\ell(M_w) = \ell(w) + 1$.
\end{example}

\section{Weyl modules.}

Let $\fg$ be a complex simple Lie algebra. Let $\fg = \fn_+ \oplus \fh \oplus \fn_-$ be a triangular decomposition and write $\fn_- = \langle f_i \mid i \in I \rangle$. The {\bf current Lie algebra} is $\fg[z] = \fg \otimes \bC[z]$. Then $U(\fg[z])$ is $\bZ_{\ge 0}$-graded and $U(\fg[z])_0 = U(\fg)$.

Let $M$ be a finite-dimensional graded $U(\fg[z])$-module. The {\bf grading filtration} $F^\bullet M$ is defined by $F^k M / F^{k+1} M \cong M_k$ (which is semisimple since $M$ is finite-dimensional).

\begin{lemma}[Beilinson--Ginzburg--Soergel]
\begin{compactenum}[\rm 1.]
\item If $M / {\rm rad}(M)$ is simple, then the radical series coincides with the grading filtration.
\item If ${\rm soc}(M)$ is simple, then the socle series coincides with the grading filtration.
\end{compactenum}
\end{lemma}

\begin{definition}
Let $\lambda \in P_+$ be a dominant weight. The Weyl module $W(\lambda)$ is generated by $v_\lambda$ with relations 
\begin{compactenum}
\item $\fn_+[z] v_\lambda = 0$, 
\item $h v_\lambda = \langle h, \lambda \rangle v_\lambda$ for $h \in \fh$,
\item $z \fh[z] v_\lambda = 0$,
\item $f_i^{\langle \alpha_i^\vee, \lambda \rangle + 1} v_\lambda = 0$ for $i \in I$. \qedhere
\end{compactenum}
\end{definition}

Some properties:
\begin{compactenum}
\item If we set $\deg v_\lambda = 0$, then $W(\lambda)$ is graded.
\item $W(\lambda)$ has a unique simple quotient $W(\lambda) \surjects W(\lambda)_0 = U(\fg) v_\lambda \cong V(\lambda)$ where $V(\lambda)$ is a highest weight module for $\fg$ (and $\fg[z]$ acts via the evaluation $z \mapsto 0$).
\item $W(\lambda)$ is finite-dimensional.
\item In particular, the radical series of $W(\lambda)$ coincides with the grading filtration.
\end{compactenum}

\begin{proposition}
If $\fg$ is of type ADE, then ${\rm soc}(W(\lambda))$ is simple.
\end{proposition}

\begin{proof}
By Fourier--Littelmann, $W(\lambda) \cong L_w(\Lambda)$ where $L_w(\Lambda) = U(\hat{\fb}) L(\Lambda)_{w \Lambda}$ is the Demazure module for $\hat{\fg}$ and $w \in \hat{W}$ and $\Lambda \in \hat{P}_+$ satisfy $w \Lambda = w_0 \lambda + \Lambda_0$. (Note that there is a natural map $\fg[z] \subset \fg[z,z^{-1}] \to \hat{g}$.) Therefore any nonzero submodule of $W(\lambda)$ contains $L(\Lambda)_\Lambda$, so it has a simple socle.
\end{proof}

\begin{example}
Set $\fg = \fsl_2$ and $\lambda = 3 \omega_1$. Then $W(3\omega_1)_0 = V(3\omega_1)$, $W(3\omega_1)_1 = V(\omega_1)$, and $W(3\omega_1)_2 = V(\omega_1)$. We take $w = s_1s_0s_1$ and $\Lambda = \Lambda_1 + 2\delta$.
\end{example}

\begin{theorem}
If $\fg$ is of type ADE, when $W(\lambda)$ is rigid.
\end{theorem}

\begin{remark}
For $\fg$ of type BCFG, $W(\lambda)$ is not rigid in general.
\end{remark}

\section{Quiver varieties.}

Let $\fg$ be of type ADE. Pick $\lambda \in P_+$ and $\alpha \in Q_+$ such that $V(\lambda)_{\lambda - \alpha} \ne 0$. Let $\fL(\alpha, \lambda)$ be  Nakajima's Lagrangian quiver variety.

\begin{example}
If $\fg = \fsl_2$, $\lambda = n\omega_1$ and $\alpha = r\alpha_1$, then $\fL(\alpha, \lambda) \cong \Gr(r,n)$.
\end{example}

There is an action of $U(\fg[z])$ on $\bigoplus_\alpha \rH_*(\fL(\alpha, \lambda))$ which was constructed by Varagnolo (based on Nakajima). Note that $\rH_{\rm odd}(\fL(\alpha, \lambda)) = 0$.

\begin{theorem}
There is an isomorphism of $U(\fg[z])$-modules $W(\lambda) \cong \bigoplus_\alpha \rH_*(\fL(\alpha, \lambda))$ and $W(\lambda)_k \cong \bigoplus_\alpha \rH_{2(\dim \fL(\alpha, \lambda) - k)}(\fL(\alpha, \lambda))$.
\end{theorem}

Let $B_w(\Lambda)$ be the Demazure crystal, which parametrizes a weight basis of $L_w(\Lambda)$. So we have a weight function ${\rm wt}_{\hat{P}} \colon B_w(\Lambda) \to \hat{P}$. For $b \in B_w(\Lambda)$, we have ${\rm wt}_{\hat{P}}(b) = {\rm wt}_P(b) + \Lambda_0 + D(b) \delta$ where $D(b) \in \bZ_{\ge 0}$.

The Poincar\'e polynomial for $\fL(\alpha, \lambda)$ can be expressed as
\[
\sum_{k=0}^{\dim \fL(\alpha, \lambda)} \dim \rH_{2(\dim \fL(\alpha, \lambda) - k)}(\fL(\alpha, \lambda)) t^k = \sum_{\substack{b \in B_w(\Lambda)\\ {\rm wt}_P(b) = \lambda - \alpha}} t^{D(b)}.
\]
By results of Fourier--Littelmann and Naoi, this can also be expressed as a one-dimensional sum (coming from solvable lattice models).

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