\include{preamble}

\begin{document}

Speaker: Syu Kato

Title: A homological study of Green polynomials

February 22, 2012

\section{Introduction.}

Let ${\rm Irr}(\fS_n)$ be the set of isomorphism classes of complex irreducible representations of $\fS_n$, which we index by $L_\lambda$ where $\lambda$ is a partition of $n$. The partitions $\lambda$ also index the orbits $\cO_\lambda$ in the nilpotent cone $\cN \subset \fgl_n$ via Jordan normal form. Hence we might expect a natural correspondence between ${\rm Irr}(\fS_n)$ and the nilpotent orbits, and this is given by the Springer correspondence.

Let $T \subset B \subset G = \GL(n,\bC)$ be a maximal torus and Borel subgroup, and let $\ft \subset \fb \subset \fg$ be their Lie algebras. Set $\fn = [\fb,\fb]$. Define $\tilde{\cN} = \{(gB, x) \in G/B \times \fg \mid x \in gng^{-1}\}$. There is a natural projection $\mu \colon \tilde{\cN} \to \cN$, which is the Springer resolution. We also have a natural projection ${\rm pr} \colon \tilde{\cN} \to G/B =: \cB$ which is a vector bundle.

Fix $x_\lambda \in \cO_\lambda$ and consider $\mu^{-1}(x_\lambda) \subset \tilde{\cN}$ and it maps injectively to its image in $\cB$ under ${\rm pr}$. We denote the image by $\cB_\lambda$, which is the Springer fiber of $x_\lambda$. Note that $\cB_\lambda$ is well-defined up to $G$-conjugation.

\begin{theorem}[Borel]
$\bC[\ft] \cong \rH^\bullet_G(\cB; \bC) \to \rH^\bullet(\cB; \bC)$ is surjective (here $\bC[\ft]] = \Sym(\ft^*)$ where $\deg(\ft^*) = 2$) and the kernel is $\langle \bC[\ft]_+^{\fS_n} \rangle$. 
\end{theorem}

Note that modding out by the above deal is the same as killing all trivial $\fS_n$-representations besides the one in degree 0. Furthermore, $\rH^\bullet(\cB; \bC)$ inherits an action of $\fS_n$.

\begin{theorem}[Springer, Hotta--Springer, De Concini--Procesi]
There is a natural map $\rH^\bullet(\cB; \bC) \to \rH^\bullet(\cB_\lambda; \bC)$ is
\begin{compactenum}[\rm 1.]
\item surjective (De Concini--Procesi)
\item There is a presentation of the kernel which gives $\rH^\bullet(\cB_\lambda; \bC)$ the structure of $\fS_n$-representation.
\item $\rH^{2 \dim \cB_\lambda}(\cB_\lambda; \bC) \cong L_\lambda$ as $\fS_n$-modules.
\item The assignment $\cO_\lambda \mapsto L_\lambda$ is a bijection (Springer correspondence)
\end{compactenum}
\end{theorem}

$\rH^\bullet(\cB_\lambda; \bC)$ gives characters of ``unipotent representations'', which is a subset of the irreducible representations of $\GL(n, \bF_q)$. Since this is a finite group, it has orthogonality relations for its characters.

Given partitions $\lambda, \mu$, define the {\bf Kostka polynomial} (or Green polynomials)
\[
K_{\lambda, \mu}(t) = \sum_{i \ge 0} [\rH^{2i}(\cB_\lambda; \bC) : L_\mu] t^i \in \bZ[t].
\]
Define $\Omega_{\lambda, \mu}$ to be the Poincar\'e polynomal (in $t$) of $\hom_{\fS_n}(L_\lambda \otimes L_\mu \otimes {\rm sgn}, \rH^\bullet(\cB))$. Also, define $K = (K_{\lambda, \mu})_{\lambda, \mu}$ and $\Omega = (\Omega_{\lambda, \mu})_{\lambda, \mu}$.

\begin{theorem}[Shoji]
There exists a diagonal square matrix $\Lambda$ such that $K^T \Lambda K = \Omega$ (orthogonality).
\end{theorem}

\begin{remark}
A generalization was given by Lusztig.
\end{remark}

\begin{theorem}[Borho--Mac Pherson]
$K_{\lambda, \mu}(t) = \begin{cases} t^{2\dim \cB_{\lambda}} & \text{if } \lambda = \mu\\
0 & \text{if } \lambda \not\le\mu \end{cases}$,
where $\le$ is dominance ordering, i.e., $\lambda \le \mu$ if and only if $\cO_\lambda \subset \ol{\cO_\mu}$.
\end{theorem}

In particular, since $K$ is upper-triangular and $\Lambda$ is diagonal, we can calculate them from $\Omega$.

Question: How can one integrate the perspectives of De Concini--Procesi and Shoki of the Springer correspondence?

\section{Reinterpretation of orthogonality relations.}

$\rH^\bullet(\cB_\lambda; \bC)$ is an $\fS_n$-module. But since $\cB_\lambda \subset \cB$, there is a natural action of $\ft^* = \rH^2(\cB; \bC)$ also. These two actions combine to give an action of $A = \bC[\fS_n] \ltimes \bC[\ft]$. We define $\deg(w) = 0$ for $w \in \fS_n$ and $\deg(x) = 2$ for $x \in \ft^*$. This equips $\rH^\bullet(\cB_\lambda; \bC)$ the structure of a graded $A$-module. The top of $\rH^\bullet(\cB_\lambda)$ is the trivial representation, and the socle of $\rH^\bullet(B_\lambda)$ is $L_\lambda$ in degree $2 \dim \cB_\lambda$.

Define $M_\lambda = \rH^{\rm BM}_\bullet(\cB_\lambda) = \bigoplus_{i \in \bZ} \rH^{\rm BM}_i(\cB_\lambda; \bC)$ which is a graded $A$-algebra. The top of $M_\lambda$ is $L_\lambda$ in degree 0. Let $P_\lambda = \bC[\ft] \otimes L_\lambda$ be the indecomposable projective cover.

The proper analogue of De Concini--Procesi is to have a surjection (thus far, a speculation) $P_\lambda \to M_\lambda \to 0$ for all $\lambda \vdash n$.

Let $\langle -, - \rangle_{\rm gEP} \colon K(A\text{-gmod}) \times K(A\text{-gmod}) \to \bZ[t,t^{-1}]$ be the graded Euler--Poincar\'e characteristic, defined by 
\[
\langle M,N \rangle_{\rm gEP} = \sum_{i \in \bZ}(-1)^i {\rm gdim}(\ext^i_A(M,N)).
\]

\begin{lemma}
$\langle P_\lambda, P_\mu \rangle_{\rm gEP} = \Delta {\rm gdim}(\hom_{\fS_n}(L_\lambda \otimes L_\mu, \rH^\bullet(\cB))$ where $\Delta = {\rm gdim}(\bC[\ft]^{\fS_n}) = ((1-t^4)(1-t^6) \cdots (1-t^{2n}))^{-1}$.
\end{lemma}

\begin{proof}
$\langle P_\lambda, P_\mu \rangle_{\rm gEP} = {\rm gdim}(\hom_A(P_\lambda, P_\mu)) = {\rm gdim}(\hom_{\fS_n}(L_\lambda, P_\mu))$.
\end{proof}

Note: The right hand side of the lemma is $\Omega$ up to $\Delta$ and the switch $t \mapsto t^{-1}$.

We define an isomorphism $A \cong A^{\rm op}$ via $w \in \fS_n \mapsto w^{-1}$ and $\ft^*$ is mapped to itself identically. If $M$ is a finite-dimensional graded $A$-module, then its graded dual $M^* = \bigoplus_{i \in \bZ} (M_i)^*$ is naturally an $A^{\rm op}$-module, and hence an $A$-module.

\begin{theorem}
$\langle M_\lambda, M^*_\mu \rangle_{\rm gEP} = 0$ unless $\lambda = \mu$.
\end{theorem}

\begin{proof}
Reinterpret Shoji's theorem using lemma plus some linear algebra.
\end{proof}

\begin{example}
Set $n=2$. Let $(1^2)$ be the trivial module and $(2)$ is the sign representation. Then $M_{(1,1)} = L_{1,1}[0] \oplus L_{2}[2]$ and $M_{(2)} = L_2[0]$. We have exact sequences
\begin{align*}
0 \to P_{1,1}(-4) \to P_{1,1} \to M_{1,1} \to 0\\
0 \to P_{1,1}(-2) \to P_2 \to M_2 \to 0.
\end{align*}
In particular, $\ext^i_A(M_{1,1}, M_2) = 0$ for all $i \ge 0$.
\end{example}

\section{Kostka systems.}

Let $W$ be a finite Coxeter group with reflection representation $\ft$. Let $A_W = \bC[W] \ltimes \bC[\ft]$ with $\deg(W) = 0$ and $\deg(\ft^*) = 2$. Then $A_w \surjects \bC[W]$ is a maximal semisimple quotient, which implies that $L_\chi$ (an irreducible $W$-module) are the irreducible graded $A$-modules up to shift. We define $\langle -,-\rangle_{\rm gEP}$ as before.

\begin{definition}[Phyla]
An ordered subdivision ${\rm Irr}(W) = \cO_1 \coprod \cO_2 \coprod \cdots \coprod \cO_N$ is called a {\bf phyla} $\cP$ of $W$. Given $\chi, \chi' \in {\rm Irr}(W)$, we define $\chi <_\cP \chi'$ if and only if $\chi \in \cO_i$ and $\chi' \in \cO_j$ with $i < j$, and $\chi \sim_\cP \chi'$ if and only if $\chi, \chi' \in \cO_j$.
\end{definition}

\begin{definition}[$\cP$-trace]
Fix $(W,\cP)$. For all $\chi \in {\rm Irr}(W)$, we define the {\bf $\cP$-trace} of $P_\chi$ as 
\[
P_{\chi,\cP} = P_\chi / \sum_{\substack{f \in \hom_{A_W}(P_{\chi'}, P_\chi)_{> 0}\\ \chi' \le_\cP \chi}} \im(f). \qedhere
\]
\end{definition}

\begin{definition}[Kostka system]
Fix $(W,\cP)$. Then $\{ K_\chi := P_{\chi,P} \}_{\chi \in {\rm Irr}(W)}$ is called a {\bf Kostka system} if $\langle K_\chi, K_{\chi'}^* \rangle_{\rm gEP} = 0$ unless $\chi \sim_\cP \chi'$. (This is actually a condition on $\cP$.)
\end{definition}

\begin{theorem}
$\{M_\lambda\}_{\lambda \vdash n}$ is a Kostka system where $\cP$ is an arbitrary refinement of the dominance ordering on partitions.
\begin{compactenum}[\rm 1.]
\item $\langle M_\lambda, M_\mu^* \rangle_{\rm gEP} = 0$ unless $\lambda = \mu$
\item The defining equations of each $M_\lambda$ is completely understood in a representation-theoretic way.
\end{compactenum}
In addition, we have $\ext^\bullet_A(M_\lambda, M_\mu) = 0$ unless $\lambda \le \mu$.
\end{theorem}

\begin{remark}
\begin{compactenum}[\rm 1.]
\item We can prove that all generalized Springer correspondences give Kostka systems.
\item Via this approach, the ``$a$-functions'' are not essential in Springer theory.
\item $\ext^\bullet_A(M_\lambda, M_\mu) = 0$ combined with Borho--Mac Pherson implies that $\langle M_\lambda, M_\mu^* \rangle_{\rm gEP} = 0$. \qedhere
\end{compactenum}
\end{remark}

\end{document}