Decomposition numbers and Kazhdan-Lusztig polynomials

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 26 June 2014

Abstract.

Introduction

This paper tries to figures out what the heck is going on in all these different Hecke algebra modular representation worlds.

Geometric constructions

Let $M$ and $N$ be $G\text{-varieties,}$ $\mu :M\to N$ a proper map, and let $ℱ$ be a pure perverse sheaf on $M\text{.}$ A transverse slice at $x\in N$ is a subvariety $V$ of $N$ containing $x$ with a ${ℂ}^{*}\text{-action}$ with strictly positive weights such that

(a)	$V$ is in generic position with respect to ${\mu }_{!}ℱ$ (why ${\mu }_{!}$ and not ${\mu }_{*}$??), i.e.$i_V^{!}{\mu }_{!}ℱ\left[2d\right]\left(d\right)⥲i_V^{*}{\mu }_{!}ℱ$ where $d=\text{codim}\left(V\right)$ and $i_V:V\hookrightarrow Y\text{.}$
(b)	There is a ${ℂ}^{*}\text{-equivariant}$ isomorphism ${ℂ}^r⥲V$ taking $0$ to $x$ such that $i_V^{!}{\mu }_{!}ℱ$ is ${ℂ}^{*}\text{-equivariant}$ for some given ${ℂ}^{*}\text{-action}$ on ${ℂ}^r\text{.}$

A $G\text{-equivariant}$ local system is a $G\text{-equivariant}$ locally constant sheaf. Let $N$ be a $G\text{-variety}$ and let $𝕆$ be a $G\text{-orbit}$ in $N\text{.}$ The orbit $𝕆$ can be identified with $G/G_x$ where $x\in 𝕆$ and $G_x$ is the stabilizer of $x\text{.}$ There is a homomorphism ${\pi }_1(𝕆,x)\to {\pi }_0(𝕆,x)=G_x/G_x^{\circ }$ and the representations of ${\pi }_1(𝕆,x)$ on the fiber of ${ℒ}_x$ of $G\text{-equivariant}$ local systems $ℒ$ are exactly the pullbacks of finite dimensional representations of $G_x/G_x^{\circ }$ to ${\pi }_1(𝕆,x)\text{.}$ In this way the irreducible $G\text{-equivariant}$ local systems on $𝕆$ can be indexed by (some of the) irreducible representations of $G_x/G_x^{\circ }$ [CGi1433132, Lemma 8.4.11].

[KS], [L1, 13.1(a)] Let $\mu :M\to N$ be a proper morphism between smooth connected varieties and let $$\stackrel{\sim }{M}=\left\{(x,\xi )\hspace{0.17em}\right|\hspace{0.17em}x\in M,\xi \in \text{ker}\left(T_{\mu \left(x\right)}^{*}N\stackrel{T_{\mu \left(x\right)}^{*}\mu }{\to }{T}_x^{*}M\right)\}\text{and}\stackrel{\sim }{\mu }:\stackrel{\sim }{M}⟶T^{*}N$$ be given by $(x,\xi )\mapsto \xi \text{.}$ Then the singular support (or characteristic variety) $SS\left({\mu }_{!}{𝒞}_M\right)$ of ${\mu }_{!}{𝒞}_M$ is a closed Lagrangian subvariety of $T^{*}N$ contained in $\text{im}\left(\stackrel{\sim }{\mu }\right)\text{.}$

The convolution algebra $H_{*}\left(Z\right)$

Let $M$ be a smooth not necessarily connected $G\text{-variety}$ and ${𝒞}_M$ the constant perverse sheaf on $M\text{.}$ Fix a $G\text{-equivariant}$ proper map $$\mu :M\to N\text{and set}Z=M{\times }_{N}M=\{(m_1,m_2)\in M\times M\hspace{0.17em}|\hspace{0.17em}\mu \left(m_1\right)=\mu \left(m_2\right)\},$$ where $N$ is a $G\text{-variety}$ consisting of finitely many orbits. Then $$H_{*}\left(Z\right)={\text{Ext}}_{D^b\left(N\right)}^{*}({\mu }_{*}{𝒞}_M,{\mu }_{*}{𝒞}_M)$$ is an algebra under sheaf theoretic convolution [CGi1433132, (8.6.6)] via the map $p_{13}:p_{12}^{-1}\left(M{\times }_{N}N\right)\cap p_{23}^{-1}\left(M{\times }_{N}M\right)\to M{\times }_{N}M$ where $p_{ij}:M{\times }_{N}M{\times }_{N}M\to M{\times }_{N}M$ is projection onto the $i\text{th}$ and $j\text{th}$ factors [CGi1433132, p. 116].

Indexing the simple $H_{*}\left(Z\right)$ modules

Assume that $N$ consists of finitely many $G\text{-orbits.}$ Then the $G\text{-orbits}$ of $N$ form an algebraic stratification of $N$ [CGi1433132, Cor. 3.2.24] and the decomposition theorem states that $${\mu }_{*}{𝒞}_M=\underset{k\in \mathbb{Z},(𝕆,\chi )}{⨁}{L}_{𝕆,\chi }\left(k\right)\otimes I{C}_{𝕆,\chi }\left[k\right],$$ where the sum is over pairs of a $G\text{-orbit}$ $𝕆$ and an irreducible $G$ equivariant local system $\chi$ on $𝕆\text{,}$ and $L_{𝕆,\chi }\left(k\right)$ are finite dimensional vector spaces [CGi1433132, Theorem 8.4.12]. Let $$\begin{array}{cc}{\displaystyle L_{𝕆,\chi }=\underset{k\in \mathbb{Z}}{⨁}{L}_{𝕆,\chi }\left(k\right)\text{so that}{\mu }_{*}{𝒞}_M\doteq \underset{𝕆,\chi }{⨁}{L}_{𝕆,\chi }\otimes I{C}_{𝕆,\chi },}& \text{(2.2)}\end{array}$$ where $\doteq$ is a linear isomorphism that does not necessarily preserve the gradings.

[CGi1433132, Theorem 8.6.12] The nonzero $L_{𝕆,\chi }$ in the decomposition ??? are the simple $H_{*}\left(Z\right)$ modules.

Standard modules

The automorphism of $Z=M{\times }_{N}M$ given by switching the two factors induces an involutive algebra antiautomorphism ${}^{*}:H_{*}\left(Z\right)\to H_{*}\left(Z\right)\text{.}$ If $ℳ$ is an $H_{*}\left(Z\right)$ module the contragredient module is the vector space $${ℳ}^{*}\text{with}\hspace{0.17em}{H}_{*}\left(Z\right)\hspace{0.17em}\text{action}\left(b\varphi \right)\left(m\right)=\varphi \left(b^{*}m\right),$$ for $b\in H_{*}\left(Z\right),$ $\varphi \in {ℳ}^{*}$ and $m\in ℳ\text{.}$

Let $x\in N$ and let $i_x:\left\{x\right\}\to N$ be the injection. Setting $M_x={\mu }^{-1}\left(x\right)$ and $m=\text{dim}\left(M\right),$ $${ℳ}_x=H_{*}\left(M_x\right)\cong H^{m-*}\left(i_x^{!}{\mu }_{*}{𝒞}_M\right)\text{and}{ℳ}_x^{*}=H^{*}\left(M_x\right)\cong H^{*-m}\left(i_x^{*}{\mu }_{*}{𝒞}_M\right)$$ are contragredient $H_{*}\left(Z\right)$ modules via the vector space isomorphism $H^i\left(M_x\right)\cong {\left(H_i\left(M_x\right)\right)}^{*}$ [CGi1433132, Cor. 8.6.25 and Prop. 8.6.15]. These are, respectively, the standard and costandard module for $H_{*}\left(Z\right)$ [CGi1433132, Prop. 8.6.15, 8.1.9, 8.1.11].

The functors $i_x^{!}$ and ${\left(i_x\right)}_{!}$ are adjoint functors and so the identity map in $\text{Hom}(i_x^{!}{\mu }_{*}{𝒞}_M,i_x^{!}{\mu }_{*}{𝒞}_M)$ gives rise to a morphism ${\left(i_x\right)}_{!}{i}_x^{!}\left({\mu }_{*}{𝒞}_M\right)\to {\mu }_{*}{𝒞}_M$ [CGi1433132, (8.3.18)]. Similary, there is a morphism ${\mu }_{*}{𝒞}_M\to {\left(i_x\right)}_{*}{i}_x^{*}\left({\mu }_{*}{𝒞}_M\right)$ and applying the functor $H^{*}$ to the sequence $${\left(i_x\right)}_{!}{i}_x^{!}{\mu }_{*}{𝒞}_M⟶{\mu }_{*}{𝒞}_M⟶{\left(i_x\right)}_{*}{i}_x^{*}{\mu }_{*}{𝒞}_M$$ gives [CGi1433132, (8.5.2)] $$H^{*}\left(i_x^{!}{\mu }_{*}{𝒞}_M\right)⟶H^{*}(N,{\mu }_{*}{𝒞}_M)⟶H^{*}\left(i_x^{*}{\mu }_{*}{𝒞}_M\right)\text{.}$$ This composition $\varphi :H_{*}\left(M_x\right)\to H^{*}\left(M_x\right)$ is an $H_{*}\left(Z\right)$ module homomorphism [CGi1433132, (8.6.18)] and it defines a $\stackrel{\sim }{H}\text{-contravariant}$ bilinear form on ${ℳ}_x=H_{*}\left(M_x\right)$ by $$⟨m_1,m_2⟩=\left(\varphi \left(m_1\right)\right)\left(m_2\right),m_1,m_2\in {ℳ}_x=H_{*}\left(M_x\right),$$ via the isomorphism $H^i\left(M_x\right)\cong {\left(H_i\left(M_x\right)\right)}^{*}$ [CGi1433132, Cor. 8.6.25].

In the decomposition of ${\mu }_{*}{𝒞}_M$ in ???, the local systems $\chi$ can be viewed as representations of the component group $Z_G\left(x\right)/Z_G{\left(x\right)}^{\circ }$ and this gives rise to $Z_G\left(x\right)/Z_G{\left(x\right)}^{\circ }$ actions on $H_{*}\left(M_x\right)$ and $H^{*}\left(M_x\right)$ which commute with the action of $H_{*}\left(Z\right)$ [CGi1433132, Lemma 3.5.3]. Take $\chi \text{-isotypic}$ components and define $${ℳ}_{x,\chi }=H_{*}{\left(M_x\right)}_{\chi }\text{and}{ℳ}_{x,\chi }^{*}=H^{*}{\left(M_x\right)}_{\chi },$$ for each irreducible representation $\chi$ of $Z_G\left(x\right)/Z_G{\left(x\right)}^{\circ },$ The map $\varphi :{ℳ}_x\to {ℳ}_x^{*}$ restricts to an $H_{*}\left(Z\right)\text{-homomorphism}$ $${\varphi }_{\chi }:H_{*}{\left(M_x\right)}_{\chi }⟶H^{*}{\left(M_x\right)}_{\chi }\text{.}$$ and defines a $H_{*}\left(Z\right)$ contravariant bilinear form on ${ℳ}_{x,\chi }\text{.}$

Decomposition numbers

Let $V$ be a transverse slice at $x\in N$ with ${ℂ}^{*}\text{-action}$ $\delta :{ℂ}^{*}\to GL\left(V\right)$ (assumed to have strictly positive weights). Let $i_V:V\to N$ be the inclusion and let $i:\left\{x\right\}\to V$ and $p:V\to \left\{x\right\}\text{.}$ Set $$𝒜=i_V^{!}{\mu }_{*}{𝒞}_M,{ℳ}_{x+t\delta }=H_{{ℂ}^{*}}^{*}\left(p_{!}𝒜\right)\text{and}{ℳ}_{x+t\delta }^{*}=H_{{ℂ}^{*}}^{*}\left(i^{*}𝒜\right)\text{.}$$

Let ${\mathbb{P}}_{\delta }V=(V-\left\{x\right\})/{ℂ}^{*}$ be the projective space corresponding to the ${ℂ}^{*}\text{-action}$ $\delta \text{.}$ Let $\widehat{𝒜}$ be the unique sheaf on ${\mathbb{P}}_{\delta }V$ whose pullback to $V-\left\{x\right\}$ is $𝒜\text{.}$ If $j:V\backslash \left\{0\right\}$ there is an exact triangle $$j_{!}{j}^{*}𝒜⟶𝒜⟶i_{!}{i}^{*}𝒜\stackrel{+1}{⟶},\text{in}\hspace{0.17em}{D}_{{ℂ}^{*}}\left(V\right),$$ which, after applying $p_{!}$ gives the exact triangle $$p_{!}{j}_{!}{j}^{*}𝒜⟶p_{!}𝒜⟶i^{*}𝒜\stackrel{+1}{⟶}\text{in}{D}_{{ℂ}^{*}}\left(\text{pt}\right),$$ Applying $H_{{ℂ}^{*}}^{*}$ gives the long exact sequence $$\cdots ⟶H^k\left(p_{!}{j}_{!}{j}^{*}𝒜\right)\stackrel{{\varphi }_k}{⟶}{H}_{{ℂ}^{*}}^k\left(p_{!}𝒜\right)⟶H_{{ℂ}^{*}}^k\left(i^{*}𝒜\right)⟶H^{k+1}\left(p_{!}{j}_{!}{j}^{*}𝒜\right)⟶\cdots \text{.}$$ If $n$ is large and $\alpha$ and $\beta$ are given by $${\mathbb{P}}^n\stackrel{\alpha }{⟵}(V\backslash \left\{0\right\}){\times }_{{ℂ}^{*}}({ℂ}^{n+1}\backslash \left\{0\right\})\stackrel{\stackrel{\beta }{\sim }}{⟶}{\mathbb{P}}_{\delta }V\times ({ℂ}^{N+1}\backslash \left\{0\right\}),$$ then, for $k$ smaller than $n\text{,}$ $$\begin{array}{ccc}{H}_{{ℂ}^{*}}^k\left(p_{!}{j}_{!}{j}^{*}𝒜\right)& =& H^k({\mathbb{P}}^N,{\alpha }_{!}{\beta }^{*}\left(\widehat{𝒜}ℂ\right))\\ & \cong & H^{k-1}({\mathbb{P}}_{\delta }V,𝒜)\otimes H^1({ℂ}^{n+1}\backslash \left\{0\right\},ℂ)=H^{k-1}({\mathbb{P}}_{\delta }V,\widehat{𝒜})\text{.}\end{array}$$ So the long exact sequence becomes $$\cdots ⟶H^{k-1}({\mathbb{P}}_{\delta }V,\widehat{𝒜})\stackrel{{\varphi }_k}{⟶}{H}_{{ℂ}^{*}}^k\left(p_{!}𝒜\right)⟶H_{{ℂ}^{*}}^k\left(i^{*}𝒜\right)⟶H^k({\mathbb{P}}_{\delta }V,\widehat{𝒜})⟶\cdots \text{.}$$ Since $𝒜,$ $i^{*}𝒜,$ $p_{!}𝒜$ and $\widehat{𝒜}$ are all pure of the same weight, $H^{k-1}(\mathbb{P}V,\widehat{𝒜})$ has weight $k-1$ and $H_{{ℂ}^{*}}^k\left(p_{!}𝒜\right)$ has weight $k\text{.}$ Thus, the map ${\varphi }_k$ is zero and there are short exact sequences $$0⟶H_{{ℂ}^{*}}^k\left(p_{!}𝒜\right)⟶H_{{ℂ}^{*}}^k\left(i^{*}𝒜\right)⟶H^k({\mathbb{P}}_{\delta }V,\widehat{𝒜})⟶0,$$ which provide an exact sequence $$0⟶{ℳ}_{x+t\delta }⟶{ℳ}_{x+t\delta }^{*}⟶H^{*}({\mathbb{P}}_{\delta }V,\widehat{𝒜})⟶0\text{.}$$ in the category of graded $ℂ\left[t\right]$ modules, where the $ℂ\left[t\right]$ actions are given by identifying $ℂ\left[t\right]$ with $H_{{ℂ}^{*}}^{*}\left(\text{pt}\right)\text{.}$

The filtration of ${ℳ}_{x+t\delta }$ given by $${ℳ}_{x+t\delta }={ℳ}_{x+t\delta }\left(0\right)\supseteq {ℳ}_{x+t\delta }\left(1\right)\supseteq \cdots \text{where}{ℳ}_{x+t\delta }\left(i\right)={ℳ}_{x+t\delta }\cap t^i\left({ℳ}_{x+t\delta }^{*}\right),$$ produces the Jantzen filtration of ${ℳ}_x,$ $${ℳ}_x={ℳ}_x^{\left(0\right)}\supseteq {ℳ}_x^{\left(1\right)}\supseteq \cdots ,\text{where}{ℳ}_x^{\left(i\right)}={ℳ}_{x+t\delta }\left(i\right)/t\left({ℳ}_{x+t\delta }\left(i\right)\right)\text{.}$$ Alternatively, the Jantzen filtration can be described in terms of the bilinear form ??? on ${ℳ}_{x+t\delta }$ since $${ℳ}_{x+t\delta }\left(j\right)=\{m\in {ℳ}_{x+t\delta }\hspace{0.17em}|\hspace{0.17em}⟨m,n⟩\in t^jℂ\left[t\right],\text{for all}\hspace{0.17em}n\in {ℳ}_{x+t\delta }\},\text{and}{ℳ}_x^{\left(j\right)}=\frac{{ℳ}_{x+t\delta }\left(j\right)}{t{ℳ}_{x+t\delta }\left(j\right)}\text{.}$$ The generalized decomposition numbers of the convolution algebra $H_{*}\left(Z\right)$ are the polynomials $$\begin{array}{cc}{\displaystyle {[{ℳ}_x:L_{𝕆,\chi }]}_t=\sum _k[{ℳ}_x^{\left(k\right)}/{ℳ}_x^{(k+1)}:L_{𝕆,\chi }]t^k\text{.}}& \text{(2.4)}\end{array}$$

The generalized decomposition numbers for the convolution algebra $H_{*}\left(Z\right)$ are given by $${[{ℳ}_x:L_{𝕆,\chi }]}_t=\sum _k{t}^k\text{dim}\left(H^k\left(i_x^{!}I{C}_{𝕆,\chi }\right)\right),$$ where $i_x^{!}I{C}_{𝕆,\chi }$ is the stalk at $x$ of the intersection cohomology sheaf $I{C}_{𝕆,\chi }$ appearing in the decomposition ???.

		Proof.
	Then $𝒜$ is a ${ℂ}^{*}$ equivariant pure perverse sheaf on $V\text{.}$ Let $i:\left\{x\right\}\to V$ and $p:V\to \left\{x\right\}$ the natural maps. Since $i^{!}$ and $p_{*}$ increase weights and $i^{*}$ and $p_{!}$ decrease weights, and since $i^{!}𝒜\cong p_{!}𝒜$ and $p_{*}𝒜\cong i^{*}𝒜$ it follows that $p_{!}𝒜$ and $i^{*}𝒜$ are pure. Thus $$p_{!}𝒜\cong \underset{j}{⨁}{H}^j\left(p_{!}𝒜\right)[-j]\text{and}{i}^{*}𝒜\cong \underset{j}{⨁}{H}^j\left(i^{*}𝒜\right)[-j],$$ in $D_{{ℂ}^{*}}\left(\text{pt}\right)\text{.}$ Identifying $H_{{ℂ}^{*}}^{*}\left(\text{pt}\right)$ with $ℂ\left[t\right],$ $$H_{{ℂ}^{*}}^{*}\left(p_{!}𝒜\right)=\underset{j}{⨁}{H}^j\left(p_{!}𝒜\right)[-j]\otimes ℂ\left[t\right]\text{and}{H}_{{ℂ}^{*}}^{*}\left(i^{*}𝒜\right)=\underset{j}{⨁}{H}^j\left(i^{*}𝒜\right)[-j]\otimes ℂ\left[t\right]\text{.}$$ Thus ${ℳ}_{x+t\delta }=H_{{ℂ}^{*}}^{*}\left(p_{!}𝒜\right)$ and ${ℳ}_{x+t\delta }^{*}=H_{{ℂ}^{*}}^{*}\left(i^{*}𝒜\right)$ are free $ℂ\left[t\right]$ modules and $${ℳ}_x=H^{*}\left(p_{!}𝒜\right)={ℳ}_{x+t\delta }/t{ℳ}_{x+t\delta }\text{and}{ℳ}_x^{*}=H^{*}\left(i^{*}𝒜\right)={ℳ}_{x+t\delta }^{*}/t{ℳ}_{x+t\delta }^{*}\text{.}$$ Next we want to claim that $${ℳ}_x^{\left(k\right)}/{ℳ}_x^{(k+1)}={\left({ℳ}_x\right)}^k=H^k\left(M_x\right)\text{.}$$ i.e. we must show that $$\frac{({ℳ}_{x+t\delta }\cap t^i{ℳ}_{x+t\delta }^{*})/t({ℳ}_{x+t\delta }\cap t^i{ℳ}_{x+t\delta }^{*})}{({ℳ}_{x+t\delta }\cap t^{i+1}{ℳ}_{x+t\delta }^{*})/t({ℳ}_{x+t\delta }\cap t^{i+1}{ℳ}_{x+t\delta }^{*})}=\frac{{ℳ}_{x+t\delta }^i}{t{ℳ}_{x+t\delta }}$$ Assuming this, the result follows from the decomposition theorem $$\begin{array}{c}{\displaystyle {\left({ℳ}_x\right)}^k={\left({ℳ}_{x+t\delta }\right)}^k/t{\left({ℳ}_{x+t\delta }\right)}^k=H^k\left(p_{!}𝒜\right)=H^k\left(i_x^{!}{\mu }_{*}{𝒞}_M\right)=\underset{𝕆,\chi }{⨁}{L}_{𝕆,\chi }\otimes H^k\left(i_x^{!}I{C}_{𝕆,\chi }\right)\text{and}}\\ {\displaystyle {\left({ℳ}_x^{*}\right)}^k={\left({ℳ}_{x+t\delta }^{*}\right)}^k/t{\left({ℳ}_{x+t\delta }^{*}\right)}^k=H^k\left(i^{*}𝒜\right)=H^k\left(i_x^{*}{\mu }_{*}{𝒞}_M\right)=\underset{𝕆,\chi }{⨁}{L}_{𝕆,\chi }\otimes H^k\left(i_x^{*}I{C}_{𝕆,\chi }\right)\text{.}}\end{array}$$ $\square$

Projective modules and reciprocity

By the decomposition theorem $$H_{*}\left(Z\right)\cong \left(\underset{(𝕆,\chi )}{⨁}\text{End}\left(L_{𝕆,\chi }\right)\right)⨁(\underset{(𝕆,\chi ),(𝕆\prime ,\chi \prime ),k>0}{⨁}{\text{Hom}}_{ℂ}(L_{𝕆,\chi },L_{𝕆\prime ,\chi \prime })\otimes {\text{Ext}}^k(I{C}_{𝕆,\chi },I{C}_{𝕆\prime ,\chi \prime }))\text{.}$$ Let $e_{𝕆,\chi }$ be a minimal idempotent in $\text{End}\left(L_{𝕆,\chi }\right)\text{.}$ Then $$P_{𝕆,\chi }=H_{*}\left(Z\right)e_{𝕆,\chi }\cong L_{𝕆,\chi }⨁(\underset{(𝕆\prime ,\chi \prime ),k>0}{⨁}{L}_{𝕆\prime ,\chi \prime }\otimes {\text{Ext}}_{D^b\left(N\right)}^k(I{C}_{𝕆,\chi },I{C}_{𝕆\prime ,\chi \prime }))$$ is the corresponding projective $H_{*}\left(Z\right)\text{-module.}$ Then $$[P_{𝕆,\chi }:L_{𝕆\prime ,\chi \prime }]=\text{dim}\hspace{0.17em}{\text{Ext}}^{*}({IC}_{𝕆,\chi },{IC}_{𝕆\prime ,\chi \prime }),$$ is the multiplicity of the simple module $L_{𝕆\prime ,\chi \prime }$ in the composition series of $P_{𝕆,\chi }\text{.}$ Let $$[M_{𝕆,\chi }:L_{𝕆\prime ,\chi \prime }]=\{\begin{array}{cc}{\displaystyle \sum _k\text{dim}\left(H^k{\left(i_x^{!}{IC}_{𝕆\prime ,\chi \prime }\right)}_{\chi }\right),}& \text{if}\hspace{0.17em}𝕆\prime \subseteq \overline{𝕆},\\ 0,& \text{otherwise.}\end{array}$$ be the multiplicity of the simple module $L_{𝕆\prime ,\chi \prime }$ in the composition series of $M_{𝕆,\chi },$ and let $$D_{\left(𝕆,\chi \right),\left(𝕆\prime ,\chi \prime \right)}=\{\begin{array}{cc}{\displaystyle \sum _k{(-1)}^k\text{dim}\hspace{0.17em}{H}^k(𝕆,{\left(\chi \prime \right)}^{*}\otimes \chi ),}& \text{if}\hspace{0.17em}𝕆=𝕆\prime ,\\ 0,& \text{otherwise.}\end{array}$$ be the Euler characteristic of $𝕆$ with coefficients in the local system ${\left(\chi \prime \right)}^{*}\otimes \chi$ where ${\left(\chi \prime \right)}^{*}$ is the dual local system to $\chi \prime \text{.}$ Then, with matrix notation, $$\begin{array}{cc}[P:L]=[M:L]D{[M:L]}^t,& \text{(2.6)}\end{array}$$ [CGi1433132, Theorem 8.7.5].

In the "examples"

(a)	In the category of finite dimensional??? representations of algebraic groups in characterstic $p\text{,}$ $P_{𝕆,\chi }$ is the indecomposable projective cover of $L_{𝕆,\chi },$ $M_{𝕆,\chi }$ are the Weyl modules the matrix $D$ is the identity and ??? is "???-reciprocity".
(b)	In the category of highest weight modules of a semisimple Lie algebra, $P_{𝕆,\chi }$ is the indecomposable projective cover of $L_{𝕆,\chi },$ $M_{𝕆,\chi }$ are the Verma modules, the matrix $D$ is the identity and ??? is "BGG-reciprocity".
(c)	In the category of finite dimensional representations of the affine Hecke algebra of type A, $\chi$ is always trivial, $M_{𝕆,\chi }$ are the standard modules, and the matrix $D$ is the identity.

Geometry of the Steinberg variety

Let $G$ be a semisimple connected algebraic group over $ℂ$ and let $𝔤$ be the Lie algebra of $G\text{.}$ The varieties $$ℬ=\left\{\text{Borel subalgebras of}\hspace{0.17em}𝔤\right\}\text{and}𝒩=\left\{\text{nilpotent elements of}\hspace{0.17em}𝔤\right\}$$ are the flag variety and the nilpotent cone, respectively. The Springer resolution (of the singularities of $𝒩\text{)}$ is the map $$\begin{array}{cccc}\mu :& \stackrel{\sim }{𝒩}& ⟶& 𝒩\\ & (x,𝔟)& ⟼& x,\end{array}\text{where}\stackrel{\sim }{𝒩}=\left\{(n,𝔟)\hspace{0.17em}\right|\hspace{0.17em}n\in 𝒩,𝔟\in ℬ,n\in 𝔟\}\text{.}$$ The Steinberg variety is $$Z=\stackrel{\sim }{𝒩}{\times }_{𝒩}\stackrel{\sim }{𝒩}=\left\{(x,𝔟,𝔟\prime )\hspace{0.17em}\right|\hspace{0.17em}x\in 𝒩,𝔟,𝔟\prime \in ℬ,x\in 𝔟,x\in 𝔟\prime \}\text{.}$$

Let $𝕆$ be the $G\text{-orbit}$ of an element $x\in 𝒩\text{.}$ Let $(x,y,h)$ be an ${𝔰𝔩}_2$ triple. The Lie algebra $𝔤$ decomposes into ${𝔰𝔩}_2$ modules under the adjoint action of this ${𝔰𝔩}_2$ triple and the various weight spaces of this ${𝔰𝔩}_2$ action are orthogonal with respect to the Killing form on $𝔤\text{.}$ Since $\text{ker}\left({\text{ad}}_y\right)$ is the span of the lowest weight vectors and $\text{im}\left({\text{ad}}_x\right)$ is the span of the non lowest weight vectors, $$T_x𝕆=[𝔤,x]=\text{im}\left({\text{ad}}_x\right)\text{and}{\left(T_x𝕆\right)}^{\perp }=Z_{𝔤}\left(y\right)=\text{ker}\left({\text{ad}}_y\right)$$ are complementary subspaces in $𝔤$ which are orthogonal with respect to the Killing form. Let $\gamma :{SL}_2\left(ℂ\right)\to G$ be the Jacobson-Morozov homomorphism corresponding to the ${𝔰𝔩}_2\text{-triple}$ $(x,y,h)\text{.}$ Setting $$t\circ (x+z)=x+t^2\text{Ad}\left(\gamma \left(\begin{array}{cc}{t}^{-1}& 0\\ 0& t\end{array}\right)\right)z=t^2\text{Ad}\left(\gamma \left(\begin{array}{cc}{t}^{-1}& 0\\ 0& t\end{array}\right)\right)(x+z),\text{for}\hspace{0.17em}t\in {ℂ}^{*},z\in Z_{𝔤}\left(y\right),$$ defines a ${ℂ}^{*}\text{-action}$ on $x+Z_{𝔤}\left(y\right)$ with strictly positive weights and [CGi1433132, 3.7.19] $$𝒩\cap (x+Z_{𝔤}\left(y\right))\text{is a transverse slice to}\hspace{0.17em}𝕆\hspace{0.17em}\text{in}\hspace{0.17em}𝒩\text{.}$$

Note that [LuBKT1, Prop. 11.10] says that $𝔤/Z_{𝔤}\left(y\right)$ has a nondegenerate symplectic form $(a,b)=⟨a,[y,b]⟩,$ where $⟨,⟩$ is the Killing form.

Let $q\in {ℂ}^{*},$ $s\in G$ semisimple, and define ${𝔤}^s=\{x\in 𝔤\hspace{0.17em}|\hspace{0.17em}\text{Ad}\left(s\right)x=x\},$ $$\begin{array}{ccc}{𝔤}_q^s& =& \{x\in 𝔤\hspace{0.17em}|\hspace{0.17em}\text{Ad}\left(s\right)x=qx\},\\ {𝔤}_{q^{-1}}^s& =& \{x\in 𝔤\hspace{0.17em}|\hspace{0.17em}\text{Ad}\left(s\right)x=q^{-1}x\},\end{array}\text{and}{𝔤}_{q^{\pm 1}}^s={𝔤}_q^s\oplus {𝔤}_{q^{-1}}^s\text{.}$$

Fix $x\in {𝔤}_q^s$ and let $𝕆$ be the $Z_G\left(s\right)\text{-orbit}$ of $x\text{.}$ As subspaces of ${𝔤}_q^s$ $$T_x𝕆=[{𝔤}^s,x]\subseteq {𝔤}_q^s\text{and}{\left(T_x𝕆\right)}^{\perp }=\{z\in {𝔤}_{q^{\pm 1}}^s\hspace{0.17em}|\hspace{0.17em}[z,y]=0\}\text{.}$$ and this identifies the conormal bundle $${𝒞}_{𝕆}=\{(x,z)\in 𝕆\times {𝔤}_{\pm }^{(s,q)}\hspace{0.17em}|\hspace{0.17em}[z,y]=0\}=𝕆\times ({𝔤}_{\pm }^{(s,q)}\cap Z_{𝔤}\left(y\right))$$ to the orbit $𝕆$ as a subspace of ${𝔤}_{\pm }^{(s,q)}\text{.}$

Let ${ℬ}^s=\{𝔟\in ℬ\hspace{0.17em}|\hspace{0.17em}\text{ln}\left(s\right)\in 𝔟\},$ $${\stackrel{\sim }{𝔤}}_q^s=\{(x,𝔟)\in {𝔤}_q^s\times {ℬ}^s\hspace{0.17em}|\hspace{0.17em}x\in 𝔟\}\text{and let}\mu :{\stackrel{\sim }{𝔤}}_q^s⟶{𝔤}_q^s$$ be the first projection.

(Compare [L2, Cor. 13.6]) Assume that $q^2\ne 1\text{.}$ If $L$ is a simple perverse sheaf on $M={𝔤}_q^s$ such that some shift $L\left[d\right]$ appears as a direct summand of ${\mu }_{*}{𝒞}_M$ then $SS\left(L\right)\subseteq {𝔤}_q^s\times 𝒩\text{.}$

		Proof.
	By ???, $SS\left({\mu }_{*}{𝒞}_M\right)$ is a closed Lagrangian subvariety of $T_x^{*}{𝔤}_q^s$ and is contained in $\text{im}\left(\stackrel{\sim }{\mu }\right)\text{.}$ Let us compute $\text{ker}\left(T_x^{*}\mu \right)\text{.}$ Fix an element $(x,𝔟)\in {\stackrel{\sim }{𝔤}}_q^s$ and let ${𝔟}^s={𝔤}^s\cap 𝔟,$ and ${𝔫}^s={𝔤}^s\cap 𝔫,$ where $𝔫$ is the unipotent radical of $𝔟\text{.}$ Then $$\begin{array}{c}{T}_{(x,𝔟)}{\stackrel{\sim }{𝔤}}_{𝔮}^s\text{is the kernel of}\begin{array}{ccc}{𝔤}_q^s\oplus ({𝔤}^s/{𝔟}^s)& \stackrel{\mathrm{\Phi }}{⟶}& {𝔤}_q^s/(𝔟\cap {𝔤}_q^s)\\ (z,p)& ⟼& z+[x,p]\end{array},\\ T_{(x,𝔟)}^{*}{\stackrel{\sim }{𝔤}}_q^s\text{is the cokernel of}\begin{array}{ccc}𝔟\cap {𝔤}_{q^{-1}}^s& \stackrel{{\mathrm{\Phi }}^t}{⟶}& {𝔤}_{q^{-1}}^s\oplus {𝔫}^s\\ \xi & ⟼& (\xi ,-[x,\chi ])\end{array}\text{.}\end{array}$$ To justify this let us show that ${\mathrm{\Phi }}^t$ is naturally the transpose of the map $\mathrm{\Phi }\text{.}$ With respect to the Killing form, ${𝔤}_{q^{-1}}^s$ is the dual space to ${𝔤}_q^s$ and ${𝔫}^s$ is the dual space to ${𝔤}^s/{𝔟}^s\text{.}$ If $x\in 𝔟\cap {𝔤}_q^s$ and $\xi \in {𝔤}_{q^{-1}}^s\cap 𝔟$ then both ${\text{ad}}_x$ and ${\text{ad}}_{\xi }$ raise weights with respect to the Borel subalgebra $𝔟\text{.}$ So ${\text{ad}}_x{\text{ad}}_{\xi }$ is nilpotent. So $⟨x,\xi ⟩=\text{Tr}\left({\text{ad}}_x{\text{ad}}_{\xi }\right)=0\text{.}$ Since $\text{dim}(𝔟\cap {𝔤}_q^s)+\text{dim}(𝔟\cap {𝔤}_{q^{-1}}^s)=\text{dim}\left({𝔤}_q^s\right)$ it follows that $𝔟\cap {𝔤}_q^s$ is dual to ${𝔤}_q^s/(𝔟\cap {𝔤}_q^s)\text{.}$ Finally, the equality $$⟨z+[x,p],\xi ⟩=⟨z,\xi ⟩+⟨[x,p],\xi ⟩=⟨z,\chi ⟩+⟨p,-[x,\xi ]⟩,$$ demonstrates that $\mathrm{\Phi }$ and ${\mathrm{\Phi }}^t$ are transpose maps. Using the nondegenerate form $⟨,⟩:{𝔤}_q^s\times {𝔤}_{q^{-1}}^s\to ℂ$ to identify ${𝔤}_{q^{-1}}^s$ with $T_x^{*}{𝔤}_q^s,$ $$\begin{array}{ccc}{𝔤}_{q^{-1}}^s& \stackrel{T_x^{*}\mu }{⟶}& T_{(x,𝔟)}^{*}{\stackrel{\sim }{𝔤}}_q^s\\ \xi & ⟼& (\xi ,0)\hspace{0.17em}\text{mod}\hspace{0.17em}\text{im}\left({\mathrm{\Phi }}^t\right)\end{array}\text{and}\text{ker}\hspace{0.17em}{T}_x^{*}\mu =\{\xi \in 𝔟\cap {𝔤}_{q^{-1}}^s\hspace{0.17em}|\hspace{0.17em}[x,\xi ]=0\}\text{.}$$ If $\xi \in \text{ker}\hspace{0.17em}{T}_x^{*}\mu$ then there is a Borel subalgebra $b\in {ℬ}^s$ such that $\xi \in 𝔟$ and $h=\text{log}\left(s\right)\in 𝔟\text{.}$ The element $h$ must be in the 0 weight space of $𝔤$ relative to the Borel subalgebra $𝔟$ and $\xi$ must lie in the positive part of $𝔤$ with respect to $𝔟$ since $h$ acts on $\xi$ by nonzero scalars. Thus ${\text{ad}}_{\xi }$ is nilpotent and $\xi \in 𝒩\text{.}$ So $$\text{im}\left(\stackrel{\sim }{\mu }\right)=\{(x,\xi )\in {𝔤}_{q^{-1}}^s\times {𝔤}_q^s\hspace{0.17em}|\hspace{0.17em}\xi \in 𝒩,[x,\xi ]=0,\hspace{0.17em}x,\xi \in 𝔟\hspace{0.17em}\text{for some}\hspace{0.17em}𝔟\in {ℬ}^s\}\text{.}$$ So $SS\left({\mu }_{*}{𝒞}_M\right)\subseteq {𝔤}_q^s\times 𝒩\text{.}$ $\square$

[Gro1994-2, Theorem 2] Assume that $q\ne 1\text{.}$ Consider $(s,x,\chi )$ be an indexing triple. Let $(x,y,\text{log}\left(s\right))$ be an ${𝔰𝔩}_2$ triple and let $𝕆$ be the $Z_G\left(s\right)$ orbit of $x\text{.}$ Then $$SS\left({IC}_{𝕆,\chi }\right)\subseteq {𝔤}_q^s\times 𝒩\text{if and only if}{Z}_{𝔤}\left(y\right)\cap {𝔤}_q^s\subseteq 𝒩\text{.}$$

[Lus1989-2, §7] Let $q\in {ℂ}^{*},$ $q\ne 1\text{.}$ The following are equivalent.

(a)	${\displaystyle \sum _{w\in W}{q}^{\ell \left(w\right)}}\ne 0\text{.}$
(b)	In the reflection representation of $W\text{,}$ $\text{det}(q-w)\ne 0$ for all $w\in W\text{.}$
(c)	For any semisimple element $s\in G\text{,}$ ${𝔤}_q^s=\{x\in 𝔤\hspace{0.17em}|\hspace{0.17em}\text{Ad}\left(s\right)x=qx\}$ consists entirely of nilpotent elements.

The Steinberg variety in Type A

Consider an infinite page of graph paper (i.e. ${\mathbb{Z}}^2\text{)}$ with diagonals labeled (lower left to upper right) by the elements of $\mathbb{Z}\text{.}$ $$\text{PICTURE}$$ A segment is a finite sequence of consecutive boxes in a row of this page of graph paper, or, more precisely, an equivalence class of pairs of integers $(k\prime ,k)$ where the equivalence relation is $(k\prime ,k)\sim (k\prime +\ell ,k+\ell )\text{.}$ $$\text{PICTURE}$$ Consider a book of pages of graph paper with page numbers labeled by the cosets of $q$ in ${ℂ}^{*}\text{.}$ The diagonals of each page are labeled by $\mathbb{Z}$ and the content of a box $b$ is $$q^{2c\left(b\right)}=\left(\text{page number}\right)q^{2\left(\text{diagonal number}\right)}\text{.}$$ A multisegment is a finite multiset of segments in this book.

Let $q\in {ℂ}^{*}$ and let $s$ be a semisimple element in ${GL}_n\left(ℂ\right)\text{.}$ Let $x\in 𝒩$ such that $\text{Ad}\left(s\right)x=q^{2}x\text{.}$ Up to conjugation by elements of ${GL}_n\left(ℂ\right)$ one can assume that $s$ is a diagonal matrix and that $n$ is a direct sum of Jordan blocks. Thus $n$ can be specified by the sizes ${\lambda }_i$ of its Jordan blocks and $s\text{,}$ by its diagonal entries $s_b\text{.}$ Thus the pair $(s,n)$ corresponds to a multisegment where the rows have lengths ${\lambda }_i$ and contents $s_b\text{.}$

Alternatively, one can think of this multisegment as a decomposition of the vector space $V$ as an module for the ${𝔰𝔩}_2\text{-triple}$ $(x,y,h),$ where $h=\text{log}\left(s\right)\text{.}$ The lengths of the segments are the dimensions of the ${𝔰𝔩}_2\text{-submodules}$ and the contents of the boxes are the weights of $s\text{.}$

An aperiodic multisegment is a multisegment which does not contain any box of height $\ell$ or, more precisely, if, for each $k\prime \le k,$ at least one of the pairs $(k\prime ,k),(k\prime +1,k+1),\dots ,(k\prime +\ell -1,k+\ell -1)$ is not in the multisegment.

[L2, Proposition 15.5] Let $G={GL}_n\left(ℂ\right)\text{.}$ The pairs $(s,x)$ where $s\in G$ is semisimple $x\in {𝒩}^{(s,q)}$ and $Z_{𝔤}\left(y\right)\cap {𝔤}_q^s\subseteq 𝒩,$ for an ${𝔰𝔩}_2\text{-triple}$ $(x,y,\text{log}\left(s\right))$ are indexed by the aperiodic multisegments.

		Proof.
	Let $s\in G\text{,}$ $x\in {𝒩}^{(s,q)}$ and let $(x,y,\text{log}\left(s\right))$ be an ${𝔰𝔩}_2\text{-triple.}$ Assume that $z\in Z_{𝔤}\left(y\right)\cap {𝔤}_q^s\text{.}$ If $z:V\to V$ is not nilpotent and let $z=z_s+z_n$ be the Jordan decomposition of $z\text{.}$ Then $z_s\in Z_{𝔤}\left(y\right)\cap {𝔤}_q^s,$ $V=\text{im}\left(z_s\right)\oplus \text{ker}\left(z_s\right),$ and $z$ is an invertible transformation on $\text{im}\left(z_s\right)\text{.}$ The decomposition of $V$ is $s$ stable since $z\in {𝔤}_q^s\text{????,}$ and $y$ stable since $z_s$ commutes with $y\text{.}$ So the number of segments on $\text{im}\left(z_s\right)$ which begin at $k\prime$ and end at $k$ is the same as the number of segments that begin at $k\prime -1$ and end at $k-1\text{.}$ Conversely, if the segment is periodic and contains a box of length $\ell$ then it is easy to construct $z\in Z_{𝔤}\left(y\right)\cap {𝔤}_q^s$ which is not nilpotent by construct a map "perpendicular" to $y$ on the periodic part. $\square$

Kazhdan-Lusztig polynomials

Let $W$ be a finite Weyl group and let $\ell \left(w\right)$ be the length of $w\in W\text{.}$ The Iwahori-Hecke algebra $H$ is the algebra with basis $T_w\text{,}$ $w\in W\text{,}$ and multiplication determined by the relations $$\begin{array}{cc}{T}_w{T}_{w\prime }=T_{ww\prime },\text{if}\hspace{0.17em}\ell \left(ww\prime \right)=\ell \left(w\right)+\ell \left(w\prime \right),\text{and}{T}_{s_i}^2=v^{-2}+(v^{-2}-1)T_{s_i},& \text{(3.5)}\end{array}$$ for simple reflections $s_i$ in $W\text{.}$ Setting $H_w=v^{\ell \left(w\right)}{T}_w,$ for $w\in W\text{,}$ we have $H_{s_i}^2=1+(v^{-1}-v)H_{s_i}\text{.}$ Define a $ℂ\text{-linear}$ involution $\bar{\phantom{A}}:H\to H$ by $$\begin{array}{cc}\bar{v}=v^{-1}\text{and}\overline{H_x}={\left(H_{x^{-1}}\right)}^{-1}\text{.}& \text{(3.6)}\end{array}$$ The Kazhdan-Lusztig polynomials ${\stackrel{\sim }{P}}_{y,x}\left(v\right)$ for $W$ are defined by finding the unique elements ${\underset{\_}{H}}_x,$ $x\in W\text{,}$ of $H$ such that $$\overline{{\underset{\_}{H}}_x}={\underset{\_}{H}}_x,\text{and}{\underset{\_}{H}}_x=\sum _{y\in W}{\stackrel{\sim }{P}}_{y,x}\left(v\right)H_y\text{with}{\stackrel{\sim }{P}}_{y,x}\in v\mathbb{Z}\left[v\right]\text{.}$$ These are renormalized versions of the original Kazhdan-Lusztig polynomials $P_{y,x}\left(q\right)$ defined in [KL1]: If $q=v^{-2}$ then ${\stackrel{\sim }{P}}_{y,x}\left(v\right)=v^{\ell \left(x\right)-\ell \left(y\right)}{P}_{y,x}\left(q\right)\text{.}$

If $W$ is replaced by an affine Weyl group $\stackrel{\sim }{W}$ then ??? defines the affine Hecke algebra $\stackrel{\sim }{H}$ and the resulting polynomials ${\stackrel{\sim }{P}}_{y,x}\left(v\right),$ $y,x\in \stackrel{\sim }{W},$ are the Kazhdan-Lusztig polynomials for the affine Weyl group $\stackrel{\sim }{W}\text{.}$

Let $\stackrel{\sim }{W}$ be an affine Weyl group and let $W$ be the corresponding finite Weyl group. Let ${𝒜}^{+}$ be the set of alcoves for the hyperplane arrangement of $\stackrel{\sim }{W}$ which are in the dominant chamber. The minimal length representatives of cosets in $W\backslash \stackrel{\sim }{W}$ are in bijection with the alcoves in the dominant chamber via $x\to x{A}^{+},$ where $A^{+}$ is the fundamental alcove for the affine Weyl group. The "sign" representation of $H$ is given by $H_s\to -v$ (i.e. $T_s\to -1\text{)}$ and the $\stackrel{\sim }{H}$ module $${\text{Ind}}_H^{\stackrel{\sim }{H}}\left(\text{sgn}\right)=\stackrel{\sim }{H}{\otimes }_H\text{sgn},$$ has basis $N_A=N_{x{A}^{+}}=N_x=H_x\otimes 1,$ indexed by the alcoves $A$ in ${𝒜}^{+}\text{.}$ Define an involution on $𝒩$ by ??? and $$\overline{N_1}=N_1\text{and}\overline{hn}=\bar{h}\bar{n},\text{for}\hspace{0.17em}n\in 𝒩\hspace{0.17em}\text{and}\hspace{0.17em}h\in \stackrel{\sim }{H}\text{.}$$ For each $A\in {𝒜}^{+}$ there is a unique element ${\underset{\_}{N}}_A\in 𝒩$ such that $$\overline{{\underset{\_}{N}}_A}={\underset{\_}{N}}_A,\text{and}{\underset{\_}{N}}_A=\sum _{B\in 𝒜}{n}_{B,A}\left(v\right)N_B\text{with}{n}_{B,A}\left(v\right)\in v\mathbb{Z}\left[v\right]\text{.}$$ The polynomials $n_{B,A}\left(v\right)$ are the parabolic Kazhdan-Lusztig polynomials for $\stackrel{\sim }{H}$ corresponding to the sign representation of $H\text{.}$ Soergel [Soe1998, Prop 3.4] proves the following identity of Deodhar [Dd], $$\begin{array}{cc}{\displaystyle n_{y,x}\left(v\right)=\sum _{z\in W}{(-v)}^{\ell \left(z\right)}{\stackrel{\sim }{P}}_{zy,x}\left(v\right)\text{.}}& \text{(3.7)}\end{array}$$

Schubert varieties

Let $B$ be the Borel subgroup consisting of upper triangular matrices in ${GL}_n\left(ℂ\right)\text{.}$ Let $V={ℂ}^n$ and let ${\epsilon }_i=(0,\dots ,0,1,0,\dots ,0)$ be the vector with $1$ in the $i\text{th}$ entry and all other entries $0\text{.}$ The group $B$ is the stabilizer of the fixed flag $$V=V_0\supseteq V_1\supseteq \cdots \supseteq V_n=0,\text{where}{V}_i=\text{span-}\{{\epsilon }_1,\dots ,{\epsilon }_i\},$$ and the coset space $G/B$ can be identified with the flag variety $$ℬ=\{V=V^1\supseteq V^2\supseteq \cdots \supseteq V^n\supseteq 0\hspace{0.17em}|\hspace{0.17em}\text{dim}(V^i/V^{i+1})=1\}$$ of flags in $V\text{.}$ The $B$ orbits $X_w=BwB$ on $G/B$ are indexed by the permutations $w$ in the symmetric group $S_n$ and the closures $\overline{X_w}$ are the Schubert varieties in $G/B\text{.}$

Let ${\nu }_1,\dots ,{\nu }_k\in {\mathbb{Z}}_{\ge 0}$ such that ${\nu }_1+\cdots +{\nu }_k=n\text{.}$ The partial flag variety $$\{V=V^1\supseteq V^2\supseteq \cdots \supseteq V^k\supseteq 0\hspace{0.17em}|\hspace{0.17em}\text{dim}(V^i/V^{i+1})={\nu }_i\},$$ is naturally identified with the coset space $G/P_{\nu }$ where $P_{\nu }$ is the parabolic subgroup $$P_{\nu }\subseteq {GL}_n\left(ℂ\right)\text{with Levi subgroup}{GL}_{{\nu }_1}\times \cdots \times {GL}_{{\nu }_k}\text{.}$$ The $P_{\nu }$ orbits $X_M$ on $G/P_{\nu }$ are indexed by the matrices $$\begin{array}{cc}M=\left(m_{ij}\right)\text{with}{m}_{ij}\in {\mathbb{Z}}_{\ge 0}\text{and}\text{row and column sums}\hspace{0.17em}{\nu }_1,{\nu }_2,\dots ,{\nu }_k,& \text{(3.8)}\end{array}$$ or, equivalently, by the double cosets $$S_M=\{w\in S_n\hspace{0.17em}|\hspace{0.17em}\text{Card}(w{B}_i\cap B_j)=m_{ij}\},$$ in $S_{\nu }\backslash S_n/S_{\nu },$ where $S_{\nu }=S_{{\nu }_1}\times \cdots S_{{\nu }_k}\text{.}$ The Schubert variety $\overline{X_M}$ in $G/P_{\nu }$ is the closure of the orbit $X_M\text{.}$ The double coset $S_M$ contains a unique longest element $w_M\text{.}$

[KLu1980, Theorem 4.3] Let $M$ and $N$ be matrices of the form ??? and let $i_N^{!}{IC}_M$ be the stalk at a point of the $P_{\nu }\text{-orbit}$ $X_N\subseteq {GL}_n/P_{\nu }$ of the intersection cohomology sheaf ${IC}_M$ of the Schubert variety $\overline{X_M}\text{.}$ Then $$\sum _{i\ge 0}{q}^{i/2}\text{dim}\left(H^i\left(i_N^{!}{IC}_M\right)\right)=P_{w_N,w_M}\left(q\right),$$ where $P_{y,w}\left(q\right)$ is the Kazhdan-Lusztig polynomial for the symmetric group $S_n$ and $w_M$ denotes the longest element of the double coset $S_M$ in ???.

Affine Schubert varieties

Let $S_n$ be the symmetric group. The affine symmetric group ${\stackrel{\sim }{S}}_n$ is the group $${\stackrel{\sim }{S}}_n={\mathbb{Z}}^n\times \text{????}{S}_n=\left\{t_{\lambda }w\hspace{0.17em}\right|\hspace{0.17em}\lambda =({\lambda }_1,\dots ,{\lambda }_n)\in {\mathbb{Z}}^n,w\in S_n\},$$ with multiplication given by $$t_{\lambda }w{t}_{\mu }v=t_{\lambda }{t}_{w\mu }wv=t_{\lambda +w\mu }wv,\text{where}w\mu =w({\mu }_1,\dots ,{\mu }_n)=({\mu }_{w\left(1\right)},\dots ,{\mu }_{w\left(n\right)})\text{.}$$ Let ${\epsilon }_i=(0,\dots ,0,1,0,\dots ,0)$ be the element of ${\mathbb{Z}}^n$ with $1$ in the $i\text{th}$ position and all other entries $0,$ and define $$\omega =t_{{\epsilon }_n}(1,2,\dots ,n),s_0=t_{{\epsilon }_1-{\epsilon }_n}(1,n)\text{and}{s}_i=(i,i+1),1\le i\le n-1\text{.}$$

Let $𝔽$ be a field. If $V$ is an $n$ dimensional vector space over $𝔽\left(\left(t\right)\right)$ then $G=GL\left(V\right)\cong {GL}_n\left(𝔽\left(\left(t\right)\right)\right)\text{.}$ A lattice is a free $𝔽\left[\left[t\right]\right]$ submodule of rank $n$ in $V\text{.}$ Fix a sequence of lattices, $$L_1\supseteq L_2\supseteq \cdots \supseteq L_{\ell -1}\supseteq t{L}_1\text{such that}{\text{dim}}_{𝔽}(L_i/L_{i+1})=1,1\le i\le \ell -1\text{.}$$ The stabilizer of this sequence in $GL\left(L_1\right)\cong {GL}_n\left(𝔽\left[\left[t\right]\right]\right)$ is an Iwahori subgroup of ${GL}_n\left(𝔽\left(\left(t\right)\right)\right)\text{.}$ The coset space $G/I$ can be identified with the set of sequences of lattices $$\{M_1\supseteq M_2\supseteq \cdots \supseteq M_{\ell -1}\supseteq t{M}_1\hspace{0.17em}|\hspace{0.17em}{\text{dim}}_{𝔽}(M_i/M_{i+1})=1\}\text{.}$$ The $I$ orbits $X_w$ on $G/I$ are in bijection with the double cosets in $I\backslash G/I$ and are indexed by the elements of the affine symmetric group ${\stackrel{\sim }{S}}_n\text{.}$ The affine Schubert varieties are the closures $\overline{X_w}$ of the orbits $X_w$ in $G/I\text{.}$ The orbits $$\begin{array}{ccc}{X}_{s_i}& =& \{L_1\supseteq \cdots \supseteq L_{i-1}\supseteq M_i\supseteq L_{i+1}\supseteq \cdots \supseteq L_{\ell -1}\supseteq t{L}_1\hspace{0.17em}|\hspace{0.17em}{M}_i\ne L_i,{\text{dim}}_{𝔽}(M_i/L_{i+1})=1\},\\ X_{s_0}& =& \{M_1\supseteq L_2\supseteq \cdots \supseteq L_{\ell -1}\supseteq t{M}_1\hspace{0.17em}|\hspace{0.17em}{M}_1\ne L_1,{\text{dim}}_{𝔽}(M_1/L_2)=1\},\\ X_{\omega }& =& \{L_2\supseteq L_3\supseteq \cdots \supseteq L_{\ell -1}\supseteq t{L}_1\supseteq t{L}_2\},\end{array}$$ have $\text{dim}\left(X_{s_i}\right)=1$ and $\text{dim}\left(X_{\omega }\right)=0,$ respectively.

[KLu1980, Theorem 5.5] If $y,w\in {\stackrel{\sim }{S}}_n$ let $i_y^{!}{IC}_w$ be the stalk of the intersection cohomology sheaf of the Schubert variety $\overline{X_w}$ at a point of the orbit $X_y\text{.}$ Then $$\sum _{i\ge 0}{q}^{i/2}\text{dim}\left(H^i\left(i_y^{!}{IC}_w\right)\right)=P_{y,w}\left(q\right),$$ where $P_{y,w}\left(q\right)$ is the Kazhdan-Lusztig polynomial for the affine Hecke algebra $\stackrel{\sim }{H}$ of the affine symmetric group ${\stackrel{\sim }{S}}_n\text{.}$

Quiver varieties

Let $A_{\infty }$ (resp. ${\stackrel{\sim }{A}}_{\ell -1}\text{)}$ be the graph with vertices indexed by $\mathbb{Z}$ (resp. $\mathbb{Z}/\ell \mathbb{Z}\text{)}$ and edges $i\to i-1,$ $i\in \mathbb{Z}$ (resp. $i\in \mathbb{Z}/\ell \mathbb{Z}\text{).}$ $$\text{PICTURE}$$ An $A_{\infty }\text{-graded}$ vector space (or $\mathbb{Z}\text{-graded}$ vector space) $V$ is a collection ${\left(V_i\right)}_{i\in \mathbb{Z}}$ of finite dimensional vector spaces indexed by the vertices of $A_{\infty }\text{.}$ If $V$ is an $A_{\infty }\text{-graded}$ vector space let $$E_V^{-}=\underset{i\to i-1}{⨁}\text{Hom}(V_i,V_{i-1})\text{and}{G}_V=\prod _{i\in \mathbb{Z}}GL\left(V_i\right),$$ where the direct sum is over the edges of $A_{\infty }$ and the product is over the vertices. Then $E_V^{-}$ is a variety with a $G_V$ action. A representation of the quiver $A_{\infty }$ is an element of $E_V^{-}$ for some $A_{\infty }\text{-graded}$ vector space $V$ and the isomorphism classes of representations are the $G_V$ orbits in $E_V^{-}\text{.}$

One makes analogous definitions of the representations of the quiver ${\stackrel{\sim }{A}}_{\ell -1}\text{.}$ A nilpotent representation of ${\stackrel{\sim }{A}}_{\ell -1}$ is a representation $(x_{i\to i-1}:V_i\to V_{i-1})$ of ${\stackrel{\sim }{A}}_{\ell -1}$ such that, for all $k\in \mathbb{Z},$ there is an $n$ such that $x_{k+n\to k+n-1}\cdots x_{k+2\to k+1}{x}_{k+1\to k}=0\text{.}$

Let $V$ be an $n$ dimensional vector space and $G=GL\left(V\right)\text{.}$ Let $q\in {ℂ}^{*},$ and suppose that $s\in GL\left(V\right)$ is a semisimple element such that all eigenvalues of $s$ of the form $q^{2i},$ $i\in \mathbb{Z}/\ell \mathbb{Z}\text{.}$ Then $s$ induces a $\mathbb{Z}/\ell \mathbb{Z}$ grading on $V$ given by $$V_j=\left(q^{2j}\hspace{0.17em}\text{eigenspace of}\hspace{0.17em}s\right)\text{and elements of}{𝔤}_q^s=\{x\in 𝔤\hspace{0.17em}|\hspace{0.17em}\text{Ad}\left(s\right)x=qx\}$$ are exactly the nilpotent representations of the quiver ${\stackrel{\sim }{A}}_{\ell -1}$ on the graded vector space $V\text{.}$ The group $Z_G\left(s\right)$ is naturally identified with $G_V$ and thus, the variety $${𝔤}_{-}^{(s,q)}\hspace{0.17em}\text{with its}\hspace{0.17em}{Z}_G\left(s\right)\hspace{0.17em}\text{action}\text{be identified with}{E}_V\hspace{0.17em}\text{with its}\hspace{0.17em}{G}_V\hspace{0.17em}\text{action.}$$

If $x=(x_{i\to i+1}:V_i\to V_{i+1})$ is a representation of $A_{\infty }$ on $V$ let $$r_{ii}={\nu }_i,\text{and}{r}_{ij}=\text{rank}\left(x_{j-1\to j}\cdots x_{i\to i+1}\right),\text{for}\hspace{0.17em}i<j,$$ and define a matrix $$M\left(x\right)=\left(m_{ij}\right)\text{by}{m}_{ij}=\{\begin{array}{cc}0,& \text{if}\hspace{0.17em}j<i-1,\\ r_{i-1,i},& \text{if}\hspace{0.17em}j=i-1,\\ r_{ij}-r_{i-1,j}-r_{i,j-1}+r_{i-1,j+1},& \text{if}\hspace{0.17em}i\le j\text{.}\end{array}$$ This gives a bijection between representations of $A_{\infty }$ with vector space ${\oplus }_j{V}_j$ and matrices $M=\left(m_{ij}\right)$ as in ???

Let $P_{\nu }$ be the stabilizer of $$U_1\supseteq U_2\supseteq \cdots \supseteq U_k,\text{where}{U}_i=V_1\oplus \cdots \oplus V_i\text{.}$$

[Zel1985, Theorem 1] Let ${\left({\nu }_i\right)}_{i\in \mathbb{Z}}$ be nonnegative integers such that $\sum {\nu }_i=n$ and let $P_{\nu }\subseteq {GL}_n\left(ℂ\right)$ be as in ???.

(a)	The map $x\to M\left(x\right)$ defined by ??? is a bijection between the (isomorphism classes of) representations $x$ of $A_{\infty }$ such that $\text{dim}\left(V_i\right)={\nu }_i$ and matrices $M=\left(m_{ij}\right)$ as in ??? which satisfy $m_{ij}=0$ for $j<i-1\text{.}$
(b)	The map which sends $x$ to the flag $V^1\subseteq V^2\subseteq \cdots \subseteq V^k,$ where $$V^i=\{(v_1,\dots ,v_k)\in V_1\oplus \cdots \oplus V_k\hspace{0.17em}|\hspace{0.17em}x{v}_j=v_{j+1}\hspace{0.17em}\text{for}\hspace{0.17em}j\ge i\},$$ is an isomorphism between $E_V$ and an open subvariety of $G/P_{\nu }\text{.}$
(c)	The closure $\overline{{𝕆}_x}$ of the $G_V$ orbit of $x$ in $E_V$ is locally isomorphic to the Schubert variety $X_{M\left(x\right)}$ in $G/P_{\delta }\text{.}$

Combinatorial reading of word $w_{M\left(x\right)}$ from multisegment.

[L1, Theorem 11.3, Corollary 11.6]

(a)	The closure $\overline{{𝕆}_x}$ of a $G_V$ orbit ${𝕆}_x$ of a nilpotent representation $x$ in $E_V$ is locally isomorphic to the closure of an affine Schubert variety of type ${\stackrel{\sim }{A}}_{n-1},$ where $n={\nu }_1+\dots +{\nu }_k\text{.}$
(b)	The local intersection cohomology of the closure of any stratum of $E_V$ vanishes in odd degrees and has Poincaré polynomials given by some of the Kazhdan-Lusztig polynomials for the affine Weyl group of type ${\stackrel{\sim }{A}}_{n-1}\text{.}$

		Proof.
	Let $V_i$ be a sequence of vector spaces over $𝔽$ labeled by the vertices of ${\stackrel{\sim }{A}}_{\ell -1}\text{.}$ Let ${\nu }_i=\text{dim}\left(V_i\right),$ $n={\nu }_1+\cdots +{\nu }_{\ell },$ and let ${\stackrel{\sim }{S}}_{\nu }$ be the parabolic subgroup of the affine symmetric group ${\stackrel{\sim }{S}}_n$ given by $${\stackrel{\sim }{S}}_{\nu }=\text{??????????}$$ If $V=𝔽\left(\left(t\right)\right){\otimes }_{𝔽}\left({\oplus }_i{V}_i\right)$ then $GL\left(V\right)$ can be identified with ${GL}_n\left(𝔽\left(\left(t\right)\right)\right)$ be choosing an $𝔽\left(\left(t\right)\right)$ basis of $V\text{.}$ A lattice is a free $𝔽\left[\left[t\right]\right]$ submodule of rank $n$ in $V\text{.}$ Let $I_{\nu }$ be the stabilizer in ${GL}_n\left(𝔽\left(\left(t\right)\right)\right)$ of the fixed sequence of $𝔽\left[\left[t\right]\right]$ lattices $$L_1\supseteq L_2\supseteq \cdots \supseteq L_{\ell }\supseteq t{L}_1\text{given by}{L}_i=\underset{j=i}{\overset{\ell }{⨁}}{V}_j\oplus \left(\underset{j=1}{\overset{\ell }{⨁}}\underset{h\ge 1}{⨁}{t}^j{V}_j\right)\text{.}$$ Then $\text{dim}(L_i/L_{i+1})={\nu }_i$ and $\text{dim}(L_{\ell }/t{L}_1)={\nu }_{\ell }\text{.}$ The group $I_{\nu }$ has finitely many orbits on the set $$Z=\{M_1\supseteq \cdots \supseteq M_{\ell }\supseteq t{M}_1\hspace{0.17em}|\hspace{0.17em}{M}_i\subseteq L_i,\hspace{0.17em}\text{dim}(L_i/M_i)=d_i\},$$ and the closure of each of these orbits is an affine Schubert variety. An element $M_1\supseteq \cdots \supseteq M_{\ell }\supseteq t{M}_1$ of $Z$ determines a nilpotent representation of the quiver ${\stackrel{\sim }{A}}_{\ell -1}\text{,}$ $$L_1/M_1⟵L_2/M_2⟵\cdots ⟵L_{\ell }/M_{\ell }\stackrel{t}{⟵}{L}_1/M_1,$$ where the last map is multiplication by $t$ and the other maps are inclusions. Conversely, a nilpotent representation $x$ of ${\stackrel{\sim }{A}}_{\ell -1}$ on the vector spaces ${\left(V_j\right)}_{1\le j\le \ell }$ determines an element of the subset $$Z\prime =\{M_1\supseteq \cdots \supseteq M_{\ell }\supseteq t{M}_1\hspace{0.17em}|\hspace{0.17em}{M}_i\subseteq L_i,\text{dim}(L_i/M_i)=d_i,M_i\oplus V_i=L_i\},$$ of $Z$ via $$M_i=\{\sum _{1\le i<j\le \ell }(x_{0,j\to i}\left(p_{0j}\right)-p_{0j})+\sum _{j=1}^{\ell }\sum _{h\ge 1}(x_{h,j\to i}\left(p_{hj}\right)-t^h{p}_{hj})\hspace{0.17em}|\hspace{0.17em}{p}_{hj}\in V_j\}$$ where the map $x_{h,j\to i}$ is the product $x_{i+1\to i}\cdots x_{\ell \to \ell -1}{x}_{1\to \ell }{x}_{2\to 1}\cdots x_{j\to j-1}$ such that $x_{1\to \ell }$ appears $h$ times. The subset $Z\prime$ is an open dense subvariety of $Z$ which is isomorphic to the variety of all nilpotent representations of the quiver ${\stackrel{\sim }{A}}_{\ell -1}\text{.}$ The fibers of the map from $Z$ to isomorphism classes of nilpotent representations of ${\stackrel{\sim }{A}}_{\ell }$ are the $G\text{-orbits}$ of $Z$ and the intersections of these orbits with $Z\prime$ are the $G\text{-orbits}$ of $Z\prime \text{.}$ $\square$

Affine Hecke algebras

Following [Lus1998, 1.4] define the affine Hecke algebra $\stackrel{\sim }{H}$ as the algebra given by generators ${\stackrel{\sim }{T}}_1,\dots ,{\stackrel{\sim }{T}}_n,$ and $X^{\lambda },$ $\lambda \in P,$ with relations $\underset{\underset{m_{ij}}{⏟}}{{\stackrel{\sim }{T}}_i{\stackrel{\sim }{T}}_j{\stackrel{\sim }{T}}_i\cdots }=\underset{\underset{m_{ij}}{⏟}}{{\stackrel{\sim }{T}}_j{\stackrel{\sim }{T}}_i{\stackrel{\sim }{T}}_j\cdots },$ where $\pi /m_{ij}$ is the angle between $H_{{\alpha }_i}$ and $H_{{\alpha }_j},$
${\stackrel{\sim }{T}}_i^2=(q-q^{-1}){\stackrel{\sim }{T}}_i+1,$ $1\le i\le n,$
$X^{\lambda }{X}^{\mu }=X^{\lambda +\mu },$ for $\lambda ,\mu \in P,$
${\stackrel{\sim }{T}}_i{X}^{\lambda }=X^{s_i\lambda }{\stackrel{\sim }{T}}_i+(q-q-\prime ){\displaystyle \frac{X^{\lambda }-X^{s_i\lambda }}{1-X^{-{\alpha }_i}}},$ for $\lambda \in P,$ $1\le i\le n\text{.}$

Let $T=\text{Hom}(X,{ℂ}^{*})$ be the set of group homomorphisms from the group $X=\left\{X^{\lambda }\hspace{0.17em}\right|\hspace{0.17em}\lambda \in P\}$ to $q$ in ${ℂ}^{*}\text{.}$ An $\stackrel{\sim }{H}\text{-module}$ $M$ has central character $s\in T$ if $$pm=s\left(p\right)m,\text{for all}\hspace{0.17em}p\in ℂ{\left[X\right]}^W=Z\left(\stackrel{\sim }{H}\right)\text{, and}\hspace{0.17em}m\in M,$$ where the action of $s\in T$ on $ℂ\left[X\right]$ is the linear extension of its action on $X\text{.}$ If it exists, the central character of a module $M$ is defined uniquely up to the action of $W$ on $T\text{.}$ It follows from Dixmier's version of Schur's lemma [Wa] that every element of $Z\left(\stackrel{\sim }{H}\right)$ acts on an irreducible $\stackrel{\sim }{H}$ module $M$ by a constant. Thus, every irreducible $\stackrel{\sim }{H}\text{-module}$ has a central character.

Indexing irreducible representations

Fix $s\in G$ semisimple and $q\in {ℂ}^{*}\text{.}$ Let ${ℬ}^s=\{𝔟\in ℬ\hspace{0.17em}|\hspace{0.17em}\text{Ad}\left(s\right)𝔟=𝔟\},$ $${𝒩}^{(s,q)}=\{x\in 𝒩\hspace{0.17em}|\hspace{0.17em}\text{Ad}\left(s\right)x=q^{2}x\},{\stackrel{\sim }{𝒩}}^{(s,q)}=\{(x,𝔟)\in {𝒩}^{(s,q)}\times {ℬ}^s\hspace{0.17em}|\hspace{0.17em}x\in 𝔟\},$$ and let $$\begin{array}{cccc}\mu :& {\stackrel{\sim }{𝒩}}^{(s,q)}& ⟶& {𝒩}^{(s,q)}\\ & (x,𝔟)& ⟼& x\end{array}\text{and}{Z}^{(s,q)}={\stackrel{\sim }{𝒩}}^{(s,q)}{\times }_{{𝒩}^{(s,q)}}{\stackrel{\sim }{𝒩}}^{(s,q)}\text{.}$$ The following theorem says that the convolution algebra $H_{*}\left(Z^{(s,q)}\right)$ is a (very nice!) homomorphic image of the affine Hecke algebra $\stackrel{\sim }{H}\text{.}$

[CGi1433132, Prop. 8.1.5]. Fix $q\in {ℂ}^{*}$ and $s\in G$ semisimple. Define a one dimensional representation ${ℂ}_{s,q}=ℂv_{s,q}$ of the center $Z\left(\stackrel{\sim }{H}\right)$ of the affine Hecke algebra $\stackrel{\sim }{H}$ by $$p{v}_{s,q}=s\left(p\right)v_{s,q},\text{for all}\hspace{0.17em}p\in ℂ{\left[X\right]}^W=Z\left(\stackrel{\sim }{H}\right)\text{.}$$ Then $$H_{*}\left(Z^{(s,q)}\right)\cong {ℂ}_{s,q}{\otimes }_{Z\left(\stackrel{\sim }{H}\right)}\stackrel{\sim }{H}\text{.}$$

[Gro1994-2], [CGi1433132, Remark 8.8.8] Assume $q\ne 1\text{.}$ The irreducible $\stackrel{\sim }{H}$ modules are indexed by triples $(s,x,\chi )$ such that $Z_{𝔤}\left(y\right)\cap {𝔤}_{-}^{(s,q)}\subseteq 𝒩$ where $(x,y,\text{log}\left(s\right))$ is an ${𝔰𝔩}_2$ triple, and $\chi$ appears in $H^{*}\left({ℬ}_{s,x}\right)$ with nonzero multiplicity.

		Proof.
	Theorem ??? says that the nonzero $L_{𝕆,\chi }$ in the decomposition $${\mu }_{*}{𝒞}_{{\stackrel{\sim }{𝒩}}^{(s,q)}}·=\underset{𝕆,\chi }{⨁}{L}_{𝕆,\chi }\otimes {IC}_{𝕆,\chi }$$ are the simple $\stackrel{\sim }{H}$ modules with central character $s\text{.}$ Suppose that $𝕆,\chi$ satisfy the conditions in the statement of the theorem. Let $s=s_0{s}_1$ be the Langlands decomposition of $s$ and let $P$ be the corresponding Kazhdan-Lusztig-Langlands parabolic subgroup of $G\text{.}$ Let ${𝒫}_{s,x}=\{𝔟\in {ℬ}_{s,x}\hspace{0.17em}|\hspace{0.17em}𝔟\subset 𝔭\},$ where $𝔭$ is the Lie algebra of $P\text{,}$ and define $$\widehat{𝕆}=\text{union of components of}\hspace{0.17em}{\stackrel{\sim }{𝒩}}^{(s,q)}\hspace{0.17em}\text{with nonempty intersection with}\hspace{0.17em}{𝒫}_{s,x}\text{.}$$ Then $\widehat{𝕆}$ is a smooth subvariety of ${\stackrel{\sim }{𝒩}}^{(s,q)}$ and the injection $\widehat{𝕆}\hookrightarrow {\stackrel{\sim }{𝒩}}^{(s,q)}$ and the decomposition theorem applied to $\mu :\widehat{𝕆}\to \overline{𝕆}$ gives $${\mu }_{*}{𝒞}_M=\left({\mu }_{*}{𝒞}_{\widehat{𝕆}}\right)\oplus A\doteq (\underset{𝕆\prime ,\chi \prime }{⨁}{\hat{L}}_{𝕆\prime ,\chi \prime }\otimes {IC}_{𝕆\prime ,\chi \prime })\oplus B\oplus A,$$ where $A$ is supported on components of ${\stackrel{\sim }{𝒩}}^{(s,q)}$ disjoint from $\widehat{𝕆}$ and $B$ is supported on $\overline{𝕆}/𝕆\text{.}$ Taking the stalk at $x\in 𝕆$ gives $$H_{*}\left({\widehat{ℬ}}_{s,x}\right)\doteq \sum _{\chi \prime }{\hat{L}}_{𝕆,\chi \prime }\otimes \chi \prime ,\text{since}IC(𝕆,\chi \prime ){|}_{𝕆}=\chi \left[\text{dim}\hspace{0.17em}𝕆\right]\text{.}$$ By [CGi1433132, Prop. 8.8.2] $\chi$ appears in $H^{*}\left({\widehat{ℬ}}_{s,x}\right)$ with nonzero multiplicity and so ${\hat{L}}_{𝕆,\chi }$ is nonzero. So $L_{𝕆,\chi }$ is nonzero. $\square$

Standard modules

Using the map $\pi :{𝒩}^{(s,q)}\to ℬ$ given by projection onto the second factor, $$M_x={\mu }^{-1}\left(n\right)\text{is identified with}{ℬ}_{s,x}=\{𝔟\in ℬ\hspace{0.17em}|\hspace{0.17em}\text{Ad}\left(s\right)𝔟=𝔟,x\in 𝔟\},$$ for all $n\in {𝒩}^{(s,q)}$ [CGi1433132, 8.1.7]. Then $${ℳ}_{s,n,\chi }=H_{*}{\left({ℬ}_{s,n}\right)}_{\chi }\doteq H^{-*}{\left(i_x^{!}{\mu }_{*}{𝒞}_{{\stackrel{\sim }{𝒩}}^{(s,q)}}\right)}_{\chi }\text{and}{ℳ}_{s,n,\chi }^{*}=H^{*}{\left({ℬ}_{s,n}\right)}_{\chi }\doteq H^{*}{\left(i_x^{*}{\mu }_{*}{𝒞}_{{\stackrel{\sim }{𝒩}}^{(s,q)}}\right)}_{\chi }$$ are the standard and costandard modules for $H_{*}\left(Z^{(s,q)}\right),$ respectively [CGi1433132, Prop. 8.6.15, 8.1.9, 8.1.11].

Decomposition numbers

The transverse slices for this situation seem to be given on [CGi1433132, p. 172]. See also [L2, Prop. 10.6].

The involutive antiautomorphism of $\stackrel{\sim }{H}$ induced by switching the two factors of the Steinberg variety $𝒵=\stackrel{\sim }{𝒩}{\times }_{𝒩}\stackrel{\sim }{𝒩}$ is given by $${\stackrel{\sim }{T}}_i^{*}={\stackrel{\sim }{T}}_i,\text{and}{\left(X^{\lambda }\right)}^{*}=X^{\lambda },\lambda \in P,$$ and the bilinear form $⟨,⟩:M_{x+t\delta }\times M_{x+t\delta }\to ℂ\left[t\right]$ is contravariant $$⟨h{m}_1,m_2⟩=⟨m_1,h^{*}{m}_2⟩,\text{for}\hspace{0.17em}{m}_1,m_2\in {ℳ}_{x+t\delta },\hspace{0.17em}h\in \stackrel{\sim }{H}\text{.}$$ [Lus1998, Lemma 12.12].

The general method of ??? produces a Jantzen filtration $${ℳ}_{s,x,\chi }={ℳ}_{s,x,\chi }^{\left(0\right)}\supseteq {ℳ}_{s,x,\chi }^{\left(1\right)}\supseteq \cdots$$ of ${ℳ}_{s,x,\chi }$ and the generalized decomposition numbers for the affine Hecke algebra $\stackrel{\sim }{H}\left(q\right)$ are the polynomials $$\begin{array}{cc}{\displaystyle {[{ℳ}_{s,x,\chi }:L_{s,x\prime ,\chi }]}_t=\sum _k[{ℳ}_{s,x,\chi }^{\left(k\right)}/{ℳ}_{s,x,\chi }^{(k+1)}:L_{s,x\prime ,\chi }]t^k\text{.}}& \text{(4.3)}\end{array}$$ Theorem ??? applied to this situation yields the following refinement of [CGi1433132, Theorem 8.6.23].

[Gin1987-2, Theorem 2.6.2] and [Gro1994-2]. $${[{ℳ}_{s,x,\chi }:L_{s,x\prime ,\chi }]}_t=\sum _k{t}^k\text{dim}\hspace{0.17em}{H}^k\left(i_x^{!}{IC}_{x\prime ,\chi }\right)\text{.}$$

Type A affine Hecke algebras

The affine Hecke algebra of type A is the algebra $\stackrel{\sim }{H}$ given by generators $X^{{\epsilon }_1},T_1,\dots ,T_{n-1}$ and relations

(a)	$T_i{T}_{i+1}{T}_i=T_{i+1}{T}_i{T}_{i+1},$ for $1\le i\le n-2,$
(b)	$T_i{T}_j=T_j{T}_i,$ if $|i-j|>1,$
(c)	$T_i^2=(q-q^{-1})T_i+1,$ for $1\le i\le n-1,$
(d)	$X^{{\epsilon }_1}{T}_i=T_i{X}^{{\epsilon }_1},$ if $i>1,$
(e)	$X^{{\epsilon }_1}{T}_1{X}^{{\epsilon }_1}{T}_1=T_1{X}^{{\epsilon }_1}{T}_1{X}^{{\epsilon }_1}\text{.}$

Let $$L=\sum _{i=1}^n\mathbb{Z}{\epsilon }_i,\text{and}{X}^{{\epsilon }_i}=T_{i-1}\cdots T_2{T}_1{X}^{{\epsilon }_1}{T}_1{T}_2\cdots T_{i-1},\text{for}\hspace{0.17em}1\le i\le n,$$ and define $X^{\lambda }={\left(X^{{\epsilon }_1}\right)}^{{\lambda }_1}\cdots {\left(X^{{\epsilon }_n}\right)}^{{\lambda }_n},$ for $\lambda ={\lambda }_1{\epsilon }_1+\cdots +{\lambda }_n{\epsilon }_n$ in $L\text{.}$ If $w$ is in the symmetric group $S_n$ define $T_w=T_{i_1}\cdots T_{i_p}$ for a reduced expression $w=s_{i_1}\cdots s_{i_p}$ of $w$ in terms of the simple reflections $s_i=(i,i+1),$ $1\le i\le n-1\text{.}$

[Reference???]

(a)	The elements $X^{{\epsilon }_1},\dots ,X^{{\epsilon }_n}$ all commute with each other.
(b)	The elements $X^{\lambda }{T}_w,$ $\lambda \in L,$ $w\in S_n,$ are a basis of $\stackrel{\sim }{H}\text{.}$
(c)	The center $Z\left(\stackrel{\sim }{H}\right)$ is the ring of symmetric functions $ℂ{[X^{\pm {\epsilon }_1},\dots ,X^{\pm {\epsilon }_n}]}^{S_n}\text{.}$

An $\stackrel{\sim }{H}$ module $M$ has central character $s=(s_1,\dots ,s_n)\in {\left({ℂ}^{*}\right)}^n$ if $$p(X^{{\epsilon }_1},\dots ,X^{{\epsilon }_n})m=p(s_1,\dots ,s_n)m,\text{for all}\hspace{0.17em}p\in ℂ{[X^{\pm {\epsilon }_1},\dots ,X^{\pm {\epsilon }_n}]}^{S_n}=Z\left(\stackrel{\sim }{H}\right)\text{.}$$ The central character of a module $M$ is unique up to the $S_n$ action permuting the coordinates of $s\text{.}$

The aperiodic multisegments with $n$ boxes index the irreducible representations of the affine Hecke algebra ${\stackrel{\sim }{H}}_n$ of type $A\text{.}$

		Proof.
	In type $A$ the component group $G_x/G_x^{\circ }$ for a point $x\in {𝒩}^{(s,q)}$ is always $\left\{1\right\}$ and so the local system $\chi$ is always trivial and so we may simply write $${\mu }_{*}{𝒞}_{{\stackrel{\sim }{𝒩}}^{(s,q)}}\doteq \underset{𝕆}{⨁}{L}_{𝕆}\otimes {IC}_{𝕆},$$ where the sum is over $Z_G\left(s\right)$ orbits $𝕆$ in ${𝒩}^{(s,q)}\text{.}$ Thus, by Theorem ??? the irreducible $\stackrel{\sim }{H}$ modules are indexed by pairs $(s,x)$ such that $Z_g\left(y\right)\cap {𝔤}_q^s\subseteq 𝒩$ for an ${𝔰𝔩}_2\text{-triple}$ $(x,y,\text{log}\left(s\right))\text{.}$ By Theorem ???, these ${𝔰𝔩}_2$ triples are indexed by aperiodic multisegments. $\square$

Let ${\lambda }_1,\dots ,{\lambda }_m$ be the sizes of the Jordan blocks of $x$ and let $r_j={\lambda }_1+\dots +{\lambda }_j\text{.}$ Let ${\stackrel{\sim }{H}}_{\lambda }$ be the "parabolic" subalgebra of $\stackrel{\sim }{H}$ generated by $X^{{\epsilon }_1},\dots ,X^{{\epsilon }_n}$ and $T_i\text{,}$ $i\notin \{r_1,\dots ,r_m\}\text{.}$ Define a one dimensional representation ${ℂ}_{s,x}=ℂv_{s,x}$ of ${\stackrel{\sim }{H}}_{\lambda }$ by $$X^{{\epsilon }_k}{v}_{s,x}=s_k{v}_{s,x},1\le k\le n,\text{and}{T}_i{v}_{s,x}=q{v}_{s,x},\text{for}\hspace{0.17em}i\notin \{r_1,\dots ,r_m\}\text{.}$$ Then the standard module ${ℳ}_{s,x}$ of $\stackrel{\sim }{H}$ is given by $${ℳ}_{s,x}\cong \stackrel{\sim }{H}{\otimes }_{{\stackrel{\sim }{H}}_{\lambda }}{ℂ}_{s,x}\text{.}$$

From Theorem ??? we obtain the following corollary (a refinement of [Zel1985, Cor. 1]). An alternative way of obtaining this result is explained in [OR].

Let $\stackrel{\sim }{H}$ be the affine Hecke algebra corresponding to ${GL}_n\left(ℂ\right)\text{.}$

(a)	If $q$ is not a root of $1,$ the generalized decomposition numbers for the affine Hecke algebra are $${[M_{s,x}:L_{s,x\prime }]}_t=P_{w_x,w_{x\prime }}\left(t\right),$$ where $P_{u,w}\left(t\right)$ are the Kazhdan-Lusztig polynomials of the symmetric group $S_n\text{.}$
(2)	If $q$ is a root of unity the generalized decomposition numbers for the affine Hecke algebra are $${[M_{s,x}:L_{s,x\prime }]}_t=P_{{\stackrel{\sim }{w}}_x,{\stackrel{\sim }{w}}_{x\prime }}\left(t\right),$$ where $P_{u,w}\left(t\right)$ are the Kazhdan-Lusztig polynomials of the affine symmetric group ${\stackrel{\sim }{S}}_n\text{.}$

Put in a converse to this last theorem.

Cyclotomic Hecke algebras

Let $𝔽$ be a field, $u_1,\dots ,u_r\in 𝔽$ and $q\in {𝔽}^{*}\text{.}$ The cyclotomic Hecke algebra $H_{r,1,n}(u_1,\dots ,u_r;q)$ is the affine Hecke algebra with the additional relation $$(X^{{\epsilon }_1}-u_1)(X^{{\epsilon }_2}-u_2)\cdots (X^{{\epsilon }_1}-u_r)=0\text{.}$$ Note that $H_{r,1,n}(u_1,\dots ,u_r;q)=H_{r,1,n}(u_{\pi \left(1\right)},\dots ,u_{\pi \left(r\right)};q)$ for any permutation $\pi \text{.}$

Indexing the simple modules

A partition $\lambda =(\lambda \ge {\lambda }_2\ge \dots )$ is $p\text{-restricted}$ if ${\lambda }_i-{\lambda }_{i+1}<p$ for all $i\ge 1\text{.}$

[Mat1998, Theorem 4.8 and Theorem 3.7] Let $p=\text{char}\left(𝔽\right)\text{.}$ Let $u_1,\dots ,u_r\in k$ and $q\in k^{*},$ and consider the cyclotomic Hecke algebra $H_{r,1,n}$ with these parameters.

(a)

Assume $q\ne 1,$ $u_1,\dots ,u_k$ are nonzero and $u_{k+1}=\cdots =u_r=0\text{.}$ Then the simple $H_{r,1,n}$ modules $D^{\mu }$ are indexed by multipartitions $\mu =({\mu }^{\left(1\right)},\dots ,{\mu }^{\left(r\right)})$ such that (1) $\bar{\mu }=({\mu }^{\left(1\right)},\dots ,{\mu }^{\left(k\right)})$ is a Kleshchev multipartition, (2) ${\mu }^{(k+1)}=\cdots ={\mu }^{(k-1)}=\varnothing$ (3) ${\mu }^{\left(r\right)}$ is $p$ restricted.

(b)

If $q=1$ the simple $H_{r,1,n}\text{-modules}$ $D^{\mu }$ are indexed by multipartitions $\mu =({\mu }^{\left(1\right)},\dots ,{\mu }^{\left(r\right)})$ such that (1) Each ${\mu }^{\left(i\right)},$ $1\le i\le r,$ is $p$ restricted, (2) ${\mu }^{\left(i\right)}=\varnothing$ if there exists $j>i$ such that $u_j=u_i\text{.}$

Decomposition numbers

[Ar2] The algebra $H_{r,1,n}$ is semisimple if and only if

(a)	$u_i\ne u_j{q}^{2k}$ for $i\ne j,$ $k\in \mathbb{Z},$ $\left|k\right|<n,$
(b)	$q^2$ is not a primitive $\ell \text{th}$ root of unity for any $2\le \ell \le n\text{.}$

In the generic case, when the algebra $H_{r,1,n}=H_{r,1,n}(u_1,\dots ,u_r;q)$ is semisimple the irreducible representations $S^{\lambda }$ are indexed by $r\text{-tuples}$ $\lambda =({\lambda }^{\left(0\right)},\dots ,{\lambda }^{\left(r\right)})$ of partitions with $n$ boxes total. One can construct an integral form ${}_{𝒜}{H}_{r,1,n}$ of the Ariki-Koike algebra and an integral form ${}_{𝒜}{S}^{\lambda }$ of the module $S^{\lambda }$ so that the module $$S_{\lambda }=𝔽{\otimes }_{𝒜}{}_{𝒜}{S}^{\lambda },$$ can be defined for the algebra $H_{r,1,n}$ for any choice of $u_1,\dots ,u_r\in 𝔽,$ $q\in {𝔽}^{*}\text{.}$ This module is called the Specht module for the algebra $H_{r,1,n}$ (though it ought to be called a Young module).

There is a unique $H_{r,1,n}$ contravariant form on $S^{\lambda },$ $${⟨,⟩}_t:S_t^{\lambda }\times S_t^{\lambda }\to ℂ\left[t\right]\text{such that}⟨v_{\lambda },v_{\lambda }⟩=1,$$ where $v_{\lambda }$ is the vector indexed by the row reading tableau of shape $\lambda \text{.}$ Letting $$S^{\lambda }\left(j\right)=\{m\in S_t^{\lambda }\hspace{0.17em}|\hspace{0.17em}⟨m,n⟩\in t^jℂ\left[t\right],\hspace{0.17em}\text{for all}\hspace{0.17em}n\in S_t^{\lambda }\},$$ the Jantzen filtration of $S^{\lambda }$ is $$S^{\lambda }={\left(S^{\lambda }\right)}^{\left(0\right)}\supseteq {\left(S^{\lambda }\right)}^{\left(1\right)}\supseteq \cdots \text{where}{\left(S^{\lambda }\right)}^{\left(j\right)}=\frac{S^{\lambda }\left(j\right)}{t{S}^{\lambda }\left(j\right)}\text{.}$$

The generalized decomposition numbers of the Ariki-Koike algebras are the polynomials $$\begin{array}{cc}{\displaystyle {[S^{\lambda }:D^{\mu }]}_t=\sum _k[\frac{{\left(S^{\lambda }\right)}^{\left(k\right)}}{{\left(S^{\lambda }\right)}^{(k+1)}}:D^{\mu }]t^k\text{.}}& \text{(5.3)}\end{array}$$

The generalized decomposition number for the Ariki-Koike algebra are given by KL polynomials. $${[S^{\lambda }:D^{\mu }]}_t=n_{\text{??}}\left(t\right),$$ where $n_{BA}\left(t\right)$ is the parabolic Kazhdan-Lusztig polynomial of the affine Hecke algebra corresponding to the sign representation of $H\text{.}$

Iwahori-Hecke algebras

The Iwahori-Hecke algebra of type A is the algebra $H_n\left(q\right)=H_{1,1,n}\text{.}$ From Theorems ??? and ???

(a)	$H_n\left(q\right)$ is semisimple if and only if $q^2$ is not a primitive $\ell \text{th}$ root of $1$ for $2\le \ell \le n\text{.}$
(b)	The simple $H_{}\left(q\right)$ modules $D^{\mu }$ are indexed by the $\ell$ regular partitions with $n$ boxes.

A partition $\mu$ is $\ell \text{-regular}$ if it is an aperiodic multisegment. The irreducible representations $D^{\mu }$ of the Iwahori-Hecke algebra $H_n\left(q\right)$ where $q$ is a primitive $\ell \text{th}$ root of unity are indexed by the $\ell$ regular partitions $\mu$ with $n$ boxes. The Specht modules $S^{\lambda }$ are indexed by the partitions $\lambda$ of $n\text{.}$

(LLT conjecture) Let $d_{\lambda \mu }\left(t\right)$ be the coefficients in the expansion of the canonical basis of the Fock space in terms of the PBW basis. If $\lambda$ is a partition and $\mu$ is an $\ell$ regular partition then $$d_{\lambda \mu }\left(t\right)={[S^{\lambda }:D^{\mu }]}_t,$$ where $S^{\lambda }$ is the Specht module and $D^{\mu }$ is the irreducible module for the Iwahori-Hecke algebra of type A when $q$ is a primitive $\ell \text{th}$ root of unity.

Connection to the quantum group

The proof of the following theorem is a "direct calculation" which verifies that the $i\text{-induction}$ and $i\text{-restriction}$ functors act on Specht modules in the same way that the $E_i$ and $F_i$ operators act on the multipartition basis of the Fock space.

[Ari1996] The $i\text{-induction}$ and $i\text{-restriction}$ functors make the direct sum of the Grothendieck groups of the categories of finite dimensional modules of the Ariki-Koike algebras $H_{r,1,n}(q^{a_1},\dots ,q^{a_r}),$ $n\ge 0,$ into an irreducible highest weight $U_v\left({\widehat{𝔰𝔩}}_{\ell }\right)$ module of highest weight $\mathrm{\Lambda }={\mathrm{\Lambda }}_{a_1}+\cdots +{\mathrm{\Lambda }}_{a_r},$ where ${\mathrm{\Lambda }}_i$ are the fundamental weights of $U_v\left({\widehat{𝔰𝔩}}_{\ell }\right)\text{.}$

A similar calculation verifies that the $i\text{-induction}$ and $i\text{-restriction}$ functors act on standard modules in the same way that the $E_i$ and $F_i$ operators act on the PBW basis of $U^{-}$ for the quantum group of type $A_{\infty }$ or type ${\stackrel{\sim }{A}}_{\ell -1}\text{.}$

It is interesting to note that every standard module for ${\stackrel{\sim }{H}}_n$ is a Specht module for some choice of parameters for $H_{r,1,n}\text{.}$

[Ari1996] The $i\text{-induction}$ and $i\text{-restriction}$ functors make the direct sum of the Grothendieck groups of the categories of finite dimensional modules of the affine Hecke algebras ${\stackrel{\sim }{H}}_n$ of type A into a $U_v\left({\widehat{𝔰𝔩}}_{\ell }\right)$ module isomorphic to $U^{-}\text{.}$

Under this identification the simple modules form the canonical basis of $L\left(\mathrm{\Lambda }\right)$ and the standard modules form the "PBW" basis. There is an inner product on the quantum group which should "become" the inner product/Jantzen filtrations on modules for the affine Hecke algebra.

The canonical basis and Kazhdan-Lusztig polynomials

By [Lus1998, Cor. 13.6] and [Lus1998, Cor 15.6] the canonical basis of $U^{-}=U_q\left({\widehat{𝔰𝔩}}_{\ell }\right)$ is indexed by a subset of the aperiodic multisegments. On the other hand the PBW basis of $U^{-}$ is indexed by all positive integral linear combinations of the positive roots (elements in the nonnegative part of the root lattice), and these correspond exactly (HOW DO WE DO THIS?? what do elements in the positive root lattice of ${\widehat{𝔰𝔩}}_{\ell }$ look like?) to the description of aperiodic multisegments as sums of indecomposable representations of the quiver. This shows that the canonical basis is indexed by the set of aperiodic multisegments.

By [Lus1989-2, Cor. 10.7] the transition matrix between the canonical basis of the negative part of the quantum group $U_q\left({\widehat{𝔰𝔩}}_{\ell }\right)$ and the PBW basis is given by the Poincaré polynomials of the intersection cohomology sheaves for the quiver variety. Theorem ??? shows that these polynomials are Kazhdan-Lusztig polynomials for the affine symmetric groups ${\stackrel{\sim }{S}}_n\text{.}$ The correspondence is made precise and summarized in the following theorem.

Define the canonical basis of $U_v\left({\widehat{𝔰𝔩}}_{\ell }\right)$ here.

(a)

Let $\lambda$ and $\mu$ be multisegments and let $E^{\lambda }$ and $B^{\mu }$ be the corresponding elements of the PBW-basis and the canonical basis of $U_v^{-}\left({𝔤𝔩}_{\infty }\right),$ respectively. Then $$E^{\lambda }=\sum _{\mu }{P}_{w^{\lambda },w^{\mu }}\left(t\right)B_{\mu },$$ where $P_{y,x}$ are the Kazhdan-Lusztig poylnomials of the symmetric group ${\stackrel{\sim }{S}}_n\text{,}$ $n=\left|\lambda \right|\text{.}$

(b)

Let $\lambda$ and $\mu$ be aperiodic multisegments and let $E^{\lambda }$ and $B^{\mu }$ be the corresponding elements of the PBW-basis and the canonical basis of $U_v^{-}\left({\widehat{𝔰𝔩}}_{\ell }\right),$ respectively. Then $$E^{\lambda }=\sum _{\mu }{P}_{w^{\lambda },w^{\mu }}\left(t\right)B_{\mu },$$ where $P_{y,x}$ are the Kazhdan-Lusztig poylnomials of the affine Weyl group ${\stackrel{\sim }{S}}_n\text{,}$ $n=\left|\lambda \right|\text{.}$

By [Lus1998, Cor. 11.8] the canonical basis "descends" to the modules $L\left(\mathrm{\Lambda }\right)\text{.}$

Canonical basis inside the Fock space

Assume that $q$ is an $\ell \text{th}$ root of unity and $\mathrm{\Lambda }={\mathrm{\Lambda }}_{a_1}+\cdots +{\mathrm{\Lambda }}_{a_r}$ where $u_1=q^{2a_1},\dots ,u_r=q^{2a_r}$ are the parameters of $H_{r,1,n}\text{.}$ The generalized Fock space $ℱ\left(\mathrm{\Lambda }\right)$ is the $U_v\left({\widehat{𝔰𝔩}}_{\ell }\right)$ module given by $$ℱ\left(\mathrm{\Lambda }\right)=\underset{\lambda =({\lambda }^{\left(1\right)},\dots ,{\lambda }^{\left(r\right)})}{⨁}ℂ\left(v\right)\lambda ,$$ with basis indexed by multipartitions with $r$ components and with $U_v\left({\widehat{𝔰𝔩}}_{\ell }\right)$ action $$\begin{array}{ccc}\begin{array}{ccc}{K}_{h_i}\lambda & =& v^{{\left|{\lambda }^{+}\right|}_i-{\left|{\lambda }^{-}\right|}_i}\lambda ,\\ E_i\lambda & =& {\displaystyle \sum _{c(\lambda /\mu )=i}{v}^{{\left|{\lambda }^{-}\right|}_{i,>(\lambda /\mu )}-{\left|{\lambda }^{+}\right|}_{i,>(\lambda /\mu )}}\mu ,}\end{array}& & \begin{array}{ccc}{K}_d\lambda & =& v^{-C_{\mathrm{\Lambda }}\left(\lambda \right)}\lambda ,\\ F_i\lambda & =& {\displaystyle \sum _{c(\mu /\lambda )=i}{v}^{{\left|{\lambda }^{+}\right|}_{i,<(\mu /\lambda )}-{\left|{\lambda }^{-}\right|}_{i,<(\mu /\lambda )}}\mu ,}\end{array}\end{array}$$ where $$\begin{array}{ccc}{\left|{\lambda }^{+}\right|}_i& =& \left(\text{\# addable boxes of content}\hspace{0.17em}i\hspace{0.17em}\text{of}\hspace{0.17em}\lambda \right)\\ {\left|{\lambda }^{+}\right|}_{i,>b}& =& \left(\text{\# addable boxes of content}\hspace{0.17em}i\hspace{0.17em}\text{of}\hspace{0.17em}\lambda \hspace{0.17em}\text{above}\hspace{0.17em}b\right)\\ {\left|{\lambda }^{+}\right|}_{i,<b}& =& \left(\text{\# addable boxes of content}\hspace{0.17em}i\hspace{0.17em}\text{of}\hspace{0.17em}\lambda \hspace{0.17em}\text{below}\hspace{0.17em}b\right)\\ \phantom{A}\\ {\left|{\lambda }^{-}\right|}_i& =& \left(\text{\# removable boxes of content}\hspace{0.17em}i\hspace{0.17em}\text{of}\hspace{0.17em}\lambda \right)\\ {\left|{\lambda }^{-}\right|}_{i,>b}& =& \left(\text{\# removable boxes of content}\hspace{0.17em}i\hspace{0.17em}\text{of}\hspace{0.17em}\lambda \hspace{0.17em}\text{above}\hspace{0.17em}b\right)\\ {\left|{\lambda }^{-}\right|}_{i,<b}& =& \left(\text{\# removable boxes of content}\hspace{0.17em}i\hspace{0.17em}\text{of}\hspace{0.17em}\lambda \hspace{0.17em}\text{below}\hspace{0.17em}b\right)\\ \phantom{A}\\ C_u\left(\lambda \right)& =& {\displaystyle \sum _i\left(\text{\# of boxes of content}\hspace{0.17em}{u}_i\hspace{0.17em}\text{of}\hspace{0.17em}{\lambda }^{\left(i\right)}\right)}\end{array}$$ The irreducible highest weight module $L\left(\mathrm{\Lambda }\right)$ is the submodule of $ℱ\left(\mathrm{\Lambda }\right)$ generated by the highest weight vector $\varnothing$ (of weight $\mathrm{\Lambda }\text{),}$ $$L\left(\mathrm{\Lambda }\right)=U_v\left({\widehat{𝔰𝔩}}_{\ell }\right)\varnothing \text{.}$$ Define a $ℂ\text{-linear}$ involution on $L\left(\mathrm{\Lambda }\right)$ by $$\bar{v}=v^{-1},\overline{K_h}=K_{-h},\overline{E_i}=E_i,\overline{F_i}=F_i,\text{and}\overline{u\varnothing }=\bar{u}\varnothing ,$$ for $u\in U_v\left({\widehat{𝔰𝔩}}_{\ell }\right)\text{.}$ The canonical basis $B_{\mu }$ of $L\left(\mathrm{\Lambda }\right)$ is determined by the conditions $$\overline{B_{\mu }}=B_{\mu },\text{and}{B}_{\mu }=\mu +\sum _{\lambda >\mu }{b}_{\lambda \mu }\left(v\right)\lambda ,\text{with}{b}_{\lambda \mu }\in v\mathbb{Z}\left[v\right]\text{.}$$

The canonical basis of $L\left(\mathrm{\Lambda }\right)$ corresponds to the simple modules of the algebras $H_{r,1,n}$ where the parameters in these algebras are determined by $\mathrm{\Lambda }\text{.}$ The Verma module $M\left(\mathrm{\Lambda }\right)$ has canonical basis in correspondence with the canonical basis of $U^{-}$ since $M\left(\mathrm{\Lambda }\right)\cong U^{-}{v}_{\mathrm{\Lambda }}\text{.}$ The canonical basis of $M\left(\mathrm{\Lambda }\right)$ corresponds to the simple modules of the affine Hecke algebras ${\stackrel{\sim }{H}}_n$ and the PBW basis corresponds to the standard modules of ${\stackrel{\sim }{H}}_n\text{.}$ Viewing $L\left(\mathrm{\Lambda }\right)$ as a quotient $M\left(\mathrm{\Lambda }\right)/I$ corresponds to restricting from ${\stackrel{\sim }{H}}_n$ modules to $H_{r,1,n}\text{-modules.}$ Under this restriction the PBW basis corresponds to the Specht modules of $H_{r,1,n}\text{.}$ This gives an indexing of the Specht modules by aperiodic multisegments, and this corresponds (CHECK THIS) to the Jimbo indexing of the canonical basis of $L\left(\mathrm{\Lambda }\right)\text{.}$

A cylindrical multipartition is a multipartition $\lambda =({\lambda }^{\left(1\right)},\dots ,{\lambda }^{\left(r\right)})$ such that for each $1\le s\le r,$ $${\lambda }^{\left(s\right)}\subseteq {\lambda }^{(s+1)},\text{as configurations on the same page.}$$ A cylindrical multipartition is $\ell \text{-restricted}$ if it is an aperiodic multisegment.

In this case the top element of the crystal graph is $\varnothing =(\varnothing ,\dots ,\varnothing )\text{.}$ The rule for adding boxes is the same as in the Kleshchev case except that order of reading the addable and removable nodes is different.

The canonical basis of $L\left(\mathrm{\Lambda }\right)$ is indexed by the cylindrical multisemgments.

The isomorphism $L\left(\mathrm{\Lambda }\right)\cong L\left({\omega }_{i_1}\right)+\cdots +L\left({\omega }_{i_k}\right)$ gives the correpondence between the Jimbo parametrization by aperiodic multisegments and the parametrization of the nodes of the crystal graph by Kleshchev multipartitions.

Let $\lambda =({\lambda }^{\left(1\right)},\dots ,{\lambda }^{\left(r\right)})$ be a multipartition. In $\lambda ,$ successively pair each unpaired addable node of content $c$ with the nearest higher unpaired removable node of content $c$ (if it exists) until there are no more possible pairings. If it exists, a good node of content $c$ is the highest unpaired node of content $c$ after pairing the addable and removable nodes of content $c\text{.}$ Highest means that the node has maximal content among unpaired nodes in the partition ${\lambda }^{\left(i\right)}$ with $i$ minimal. It would be best to define an ordering on the boxes??? A Kleshchev multipartition $\lambda =({\lambda }^{\left(1\right)},\dots ,{\lambda }^{\left(r\right)})$ is an $r\text{-tuple}$ of partitions such that one obtains $\varnothing =(\varnothing ,\dots ,\varnothing )$ after consecutively removing good nodes. This definition is obtained by thinking of the crystal graph of $L\left(\mathrm{\Lambda }\right)$ as $$L\left(\mathrm{\Lambda }\right)=L\left({\mathrm{\Lambda }}_{i_1}\right)\otimes \cdots \otimes L\left({\mathrm{\Lambda }}_{i_r}\right)\text{.}$$

The canonical basis $B_{\mu }$ of $L\left(\mathrm{\Lambda }\right)$ is indexed by the Kleshchev multipartitions $\mu =({\mu }^{\left(1\right)},\dots ,{\mu }^{\left(r\right)})\text{.}$

Let $u_1=q^{a_1},\dots ,u_r=q^{a_r}$ be the parameters of the Ariki-Koike algebra $H_{r,1,n}\text{.}$ Assume that the $(a_1,a_2,\dots ,a_r)$ are integers (we can always reduce to this case).

Goodman and Wenzl

Goodman and Wenzl prove "directly" that the polynomials which describe the canonical basis are the same as parabolic KL polynomials by doing the appropriate matching up and then showing that they are both computed by the same algorithm. The matching up is given as follows.

Consider the Fock space module $L\left({\mathrm{\Lambda }}_0\right)$ for the quantum group $U_v\left({\widehat{𝔰𝔩}}_{\ell }\right)\text{.}$ The canonical basis of this module is indexed by $\ell$ regular partitions $\mu \text{.}$ The truncated Fock space has basis labeled by the $\ell$ regular partitions $\mu$ with $\ell \left(\mu \right)\le k\text{.}$ Identify these with dominant integral weights of ${𝔤𝔩}_k$ by letting $$\lambda =\sum _{i=1}^k({\lambda }_i-{\lambda }_{i+1}){\omega }_i,\text{where}{\omega }_i={\epsilon }_1+\cdots +{\epsilon }_i\text{.}$$ The affine Weyl group ${\stackrel{\sim }{S}}_k$ of type $A_{k-1}$ is the reflection group defined by the hyperplane arrangement $$H_{\alpha +j\ell \delta }=\{x\in {𝔥}^{*}\hspace{0.17em}|\hspace{0.17em}⟨x,{\alpha }^{\vee }⟩=j\ell \},\alpha >0,j\in \mathbb{Z}\text{.}$$ in the space ${𝔥}^{*}=\text{span}\{{\epsilon }_1,\dots ,{\epsilon }_k\}\text{.}$ Let $\stackrel{\sim }{H}$ and $H$ be the affine and Iwahori-Hecke algebras corresponding to ${\stackrel{\sim }{S}}_k$ and $S_k\text{,}$ respectively.

[GWe1999] Let $n_{B,A}\left(b\right)$ be the parabolic Kazhdan-Lusztig polynomials for $\stackrel{\sim }{H}$ coming from the sign representation of $H$ as given in ????. Let $\lambda$ be a partition??? and $\mu$ an $\ell$ regular partition. Then $${[S^{\lambda }:D^{\mu }]}_t=\{\begin{array}{cc}{n}_{a(\lambda +\rho ),a(\mu +\rho )}\left(t\right),& \text{if}\hspace{0.17em}\lambda \hspace{0.17em}\text{is in the}\hspace{0.17em}{\stackrel{\sim }{S}}_k\hspace{0.17em}\text{orbit of}\hspace{0.17em}\stackrel{\sim }{\mu },\\ 0,& \text{otherwise,}\end{array}$$ and $a(\lambda +\rho )$ is the alcove which contains $\lambda +\rho$ in its closure and which lies on the positive side of all the hyperplanes containing $\mu \text{.}$

Kac-Moody Lie algebras

Let $𝔤$ be a $\mathbb{Z}\text{-graded}$ Lie algebra such that ${𝔤}_0$ is reductive and $𝔤$ is semisimple for $\text{ad}{𝔤}_0\text{.}$

Let $𝒪$ be the category of $\mathbb{Z}$ graded $𝔤$ modules which are locally finite for $𝔟={𝔤}_{\ge 0}$ and semisimple for ${𝔤}_0\text{.}$ Let $\mathrm{\Lambda }$ be set of isomorphism classes of irreducible finite dimensional $\mathbb{Z}\text{-graded}$ ${𝔤}_0\text{.}$ modules and for $E\in \mathrm{\Lambda }$ define $$\mathrm{\Delta }\left(E\right)=U𝔤{\otimes }_{U𝔟}E,\text{and}\mathrm{\nabla }\left(E\right)={\text{Hom}}_{g\le 0}(U𝔤,E),$$ A tilting module is a module $T$ which admits a $\mathrm{\Delta }$ flag and a $\mathrm{\nabla }$ flag. Define $$\begin{array}{ccc}L\left(E\right)& =& \text{the unique simple quotient (i.e. the head) of}\hspace{0.17em}\mathrm{\Delta }\left(E\right)\text{.}\\ & =& \text{the unique simple submodule (i.e. the socle) of}\hspace{0.17em}\mathrm{\nabla }\left(E\right),\text{and}\\ T\left(E\right)& =& \text{the tilting module with parameter}\hspace{0.17em}E,\end{array}$$ where the tilting module with parameter $E$ is the unique indecomposable module such that

(a)	${\text{Ext}}_{𝒪}^1(\mathrm{\Delta }\left(F\right),T)=0$ for all $F\in \mathrm{\Lambda },$
(b)	$T\left(E\right)$ has a $\mathrm{\Delta }\text{-flag}$ $0=T_0\subseteq T_1\subseteq \cdots$ with $T_1=\mathrm{\Delta }\left(E\right)\text{.}$

Suppose, in addition, that ${𝔤}_0$ contains an element $\partial$ such that $[\partial ,X]=iX$ for all $i\in \mathbb{Z}$ and $X\in {𝔤}_i\text{.}$ Let $\overline{𝒪}$ be the category of all $𝔤\text{-modules}$ which are locally finite for $𝔟$ and semisimple for ${𝔤}_0\text{.}$ This is the category $𝒪$ "without the grading". Let $\bar{\mathrm{\Lambda }}$ be the set of isomorphism classes of finite dimensional irreducible representations of ${𝔤}_0\text{.}$

If $𝔤$ is a Kac-Moody algebra then $${𝔤}_0=𝔥,{𝔤}_1=\underset{i=0}{\overset{n}{⨁}}{𝔤}_{{\alpha }_i}\text{and}\bar{\mathrm{\Lambda }}={𝔥}^{*}\text{.}$$ The Kazhdan-Lusztig conjecture for the case when $𝔤$ is symmetrizable was established by Kashiwara.

[Kas90] Assume $𝔤$ is symmetrizable and let $\lambda \in \bar{\mathrm{\Lambda }}={𝔥}^{*}$ be such that $⟨\lambda +\rho ,{\alpha }^{\vee }⟩\in {\mathbb{Z}}_{>0}\text{.}$ Then $$[\mathrm{\Delta }(x\circ \lambda ):L(y\circ \lambda )]=P_{x,y}\left(1\right),$$ where $P_{x,y}\left(t\right)$ is the Kazhdan-Lusztig polynomial for the Weyl group corresponding to $𝔤\text{.}$

Let $J$ be a subset of the simple roots and let ${𝔤}_J$ be the corresponding "parabolic" subalgebra of $𝔤\text{.}$ Let ${\mathrm{\Delta }}^J\left(\lambda \right)=\mathrm{\Delta }\left(\lambda \right)$ for $\lambda$ a dominant integral weight for ${𝔤}_J\text{.}$

[Soe1998, Prop. 7.5] Let $𝔤$ be symmetrizable. Let $\lambda \in {𝔥}^{*}$ be such that $⟨\lambda ,{\alpha }^{\vee }⟩\in {\mathbb{Z}}_{\ge 0}$ for all simple roots $\alpha \text{.}$ Then $$[{\mathrm{\Lambda }}^J(x\circ \lambda ):L\left(\nu \right)]=\{\begin{array}{cc}{n}_{x,y}\left(1\right),& \text{if}\hspace{0.17em}\nu =y\circ \lambda \hspace{0.17em}\text{with}\hspace{0.17em}y\in W_J,\\ 0,& \text{otherwise.}\end{array}$$

		Proof.
	Let $L\left(\lambda \right)$ be the finite dimensional irreducible representation of ${𝔤}_J$ of highest weight $\lambda \text{.}$ Let $M\left(\mu \right)$ be the Verma module for ${𝔤}_J$ of highest weight $\mu \text{.}$ Applying the functor $U𝔤{\otimes }_{U𝔟}$ to the BGG-resolution $$0⟶{𝒞}_{\ell \left(w_J\right)}⟶\cdots ⟶{𝒞}_1⟶{𝒞}_0⟶L\left(\lambda \right)⟶0,\text{where}{𝒞}_k=\underset{\ell \left(w\right)=k}{⨁}M(w\circ \lambda ),$$ gives an exact sequence $$0⟶U𝔤{\otimes }_{U𝔟}{𝒞}_{\ell \left(w_0\right)}⟶\cdots ⟶U𝔤{\otimes }_{U𝔟}{𝒞}_1⟶U𝔤{\otimes }_{U𝔟}{𝒞}_0⟶{\mathrm{\Delta }}^J\left(\lambda \right)⟶0,$$ where ${\mathrm{\Delta }}^J\left(\lambda \right)$ is $\mathrm{\Delta }\left(\lambda \right)$ for $\lambda$ a dominant integral weight for ${𝔤}_J\text{.}$ Thus, for all $\nu \in {𝔥}^{*}$ and all dominant integral weights $\lambda$ for ${𝔤}_J\text{,}$ $$[{\mathrm{\Delta }}^J\left(\lambda \right):L\left(\nu \right)]=\sum _{w\in W}{\left(-1\right)}^{\ell \left(w\right)}[\mathrm{\Delta }(w\circ \lambda ):L\left(\nu \right)],$$ The result now follows from ??? and ???. $\square$

Quantum groups

This introductory material is taken from [CP, § 11.2].

Let $\xi$ be a primitive $\ell \text{th}$ root of unity where $\ell$ is odd and greater than $d_i$ for all $i\text{.}$ All representations are complex representations.

Let $V_q\left(\lambda \right)$ be the irreducible $U_q\text{-module}$ with highest weight $\lambda \in P$ and highest weight vector $v_{\lambda }\text{.}$ Let $V_{𝒜}\left(\lambda \right)$ be the $U_{𝒜}^{\text{res}}\text{-module}$ generated by $v_{\lambda }$ and define the $${\mathrm{\Delta }}_{\xi }^{\text{res}}\left(\lambda \right)=V_{𝒜}^{\text{res}}\left(\lambda \right){\otimes }_{𝒜}ℂ,\text{Weyl module,}$$ via the homomorphism $𝒜\to ℂ$ given by $q\to \xi \text{.}$ Define $$V_{\xi }^{\text{res}}\left(\lambda \right)=\text{the unique simple quotient of}\hspace{0.17em}{W}_{\xi }^{\text{res}}\left(\lambda \right)\text{.}$$

[CP, Theorem 11.2.8] The finite dimensional simple $U_{\xi }^{\text{res}}\text{-modules}$ of type $1$ are $V_{\xi }^{\text{res}}\left(\lambda \right),$ $\lambda \in P^{+}\text{.}$

Let $F:U_{\xi }^{\text{res}}\to U$ be the Frobenius map. Let $V\left(\lambda \right)$ denote the simple $U\text{-module}$ of highest weight $\lambda$ let $F^{*}\left(V\left(\lambda \right)\right)$ be its pullback.

[CP, 11.2.9 and 11.2.11] (Tensor product theorem) Let $\lambda \in P^{+}$ and write $\lambda ={\lambda }_0+\ell {\lambda }_1$ with ${\lambda }_0,{\lambda }_1\in P^{+},$ and $⟨{\lambda }_0,{\alpha }_i^{\vee }⟩<\ell$ for $1\le i\le n\text{.}$ Then $$V_{\xi }^{\text{res}}\left(\lambda \right)\cong V_{\xi }^{\text{res}}\left({\lambda }_0\right)\otimes F^{*}\left(V\left(\lambda \right)\right),\text{and}V{\left(\lambda \right)}^F\cong V_{\xi }^{\text{res}}\left(\ell \lambda \right)\text{.}$$

The following theorem was conjectured by Lusztig [Lu??] and proved by Kazhdan-Lusztig via the passage to the category of representations of affine Lie algebras of level $-h-\ell$ via Theorem ??? below.

Let $\lambda \in P^{+}\text{.}$ Then $$[W_{\xi }^{\text{res}}\left(w\prime \lambda \right):V_{\xi }^{\text{res}}\left(\lambda \right)]={(-1)}^{\ell \left(w\prime w_{\lambda }\right)}{P}_{w\prime ,w_{\lambda }}\left(1\right),$$ where $w_{\lambda }\in \stackrel{\sim }{W}$ is of minimal length such that $w_{\lambda }^{-1}\lambda$ is in $A^{+}$ and $P_{y,x}$ is the Kazhdan-Lusztig polynomial for the affine Weyl group $\stackrel{\sim }{W}\text{.}$

Let $𝔤\prime$ be a finite dimensional complex semisimple Lie algebra. Let $$ℒ𝔤\prime =𝔤\prime {\otimes }_{ℂ}ℂ[t,t^{-1}],\stackrel{\sim }{𝔤}=ℒ𝔤\prime \oplus ℂz,\text{and}𝔤=\stackrel{\sim }{𝔤}\oplus ℂ\partial \text{.}$$ Then $ℒ𝔤\prime$ is the loop algebra and $𝔤$ is an affine Kac-Moody Lie algebra.

[KLu1980, I-IV] For many $\ell \in ℂ-{\mathbb{Q}}_{\le 0}$ there is an equivalence of categories $${\stackrel{\sim }{𝒪}}^e(K=-h-\ell )\cong U_{\zeta }{\text{-mod}}^{e,1}\text{.}$$ where ${\stackrel{\sim }{𝒪}}^e(K=-h-\ell )$ is the category of $\stackrel{\sim }{𝔤}$ modules $M$ of finite length such that

(a)	$K=2hz$ acts by the value $-h-\ell ,$
(b)	${\stackrel{\sim }{𝔤}}_{\ge 0}$ acts locally finitely, and
(c)	${\stackrel{\sim }{𝔤}}_{\ge 0}$ acts locally nilpotently,

and $U_{\zeta }{\text{-mod}}^{e,1}$ is the category of finite dimensional type 1 representations of the quantum group $U_q\left(𝔤\prime \right)$ where $q=e^{-i\pi /\ell }\text{.}$

Let $ℬ$ be the principal block (the smallest direct summand containing the trivial representation) of the category of finite dimensional $U_{\zeta }\left(𝔤\prime \right)$ modules where $\zeta$ is a primitive $\ell \text{th}$ root of unity. The simple objects $L_A$ of $ℬ$ are indexed by the set ${𝒜}^{+}$ of alcoves in the dominant chamber. By the Kazhdan-Lusztig equivalence the block $ℬ$ is equivalent to the block of $\overline{𝒪}$ containing $L\left(\mu \right)$ with $$\mu =\frac{-(h+\ell )}{2h}\gamma ,\text{where}\gamma =2\rho -2{\rho }_f,$$ where $\rho \in {𝔥}^{*}$ is such that $⟨\rho ,{\alpha }_i^{\vee }⟩=1$ for all $0\le i\le n,$ and ${\rho }_f$ is the half sum of the positive roots of ${𝔤}_0=𝔤\prime \oplus ℂz\oplus \mathbb{Z}\partial \text{.}$

[Soergel] Let $\stackrel{\sim }{W}$ be the affine Weyl group of the Weyl group $W$ of $𝔤\prime$ and let $\stackrel{\sim }{H}$ and $H$ be the corresponding affine and Iwahori-Hecke algebras, respectively. Let $T_A$ be the unique indecomposable tilting module in $ℬ$ which admits a $\mathrm{\nabla }$ flag ending with a surjection $T_A\to {\mathrm{\nabla }}_A\text{.}$ Then $$[T_A:{\mathrm{\nabla }}_B]=n_{B,A}\left(1\right),$$ where $n_{B,A}\left(v\right)$ is the parabolic Kazhdan-Lusztig polynomial for $\stackrel{\sim }{H}$ corresponding to the sign representation of $H\text{.}$

Algebraic groups in characteristic $p$

Let $𝔽$ be a field of characteristic $p$ and let $G$ be a semisimple split simply connected algebraic group over $𝔽\text{.}$ Let $B$ be a Borel subgroup of $G$ and $T$ a maximal torus.

Let $\lambda \in X\left(T\right)\text{.}$ The Weyl module associated to $\lambda$ is $$H^0\left(\lambda \right)=H^0(G/B,\lambda )={\text{Ind}}_B^G{𝔽}_{\lambda }\text{.}$$ The simple module are $L\left(\lambda \right)$ where $L\left(\lambda \right)$ is the unique simple submodule of $H^0\left(\lambda \right)\text{.}$

[Soe2000, Theorem 1.2] Let $i_x^{!}{IC}_y$ be the stalk of the intersection homology sheaf on the Schubert variety $\overline{X_w}$ at a point of the Schubert cell $X_x\text{.}$ Then $$[H^0(st+\rho ):L(st+y\rho )]=\sum _i{\text{dim}}_k\left(H^k\left(i_x^{!}{IC}_y\right)\right)\text{.}$$

This is part of the Lusztig conjecture [Lu7].

Erdmann's ${GL}_n$ to $S_n$ decomposition number conversion

Let $K$ be an algebraically closed field of characteristic $p\text{.}$

Let ${GL}_n\left(K\right)$ be the general linear group with entries in $K\text{.}$ Let $$\begin{array}{ccc}\mathrm{\Delta }\left(\lambda \right)& =& \text{Weyl module with highest weight}\hspace{0.17em}\lambda ,\\ \mathrm{\nabla }\left(\lambda \right)& =& \text{the contravariant dual of}\hspace{0.17em}\mathrm{\Delta }\left(\lambda \right),\\ T\left(\lambda \right)& =& \text{the indecomposable tilting module with highest weight}\hspace{0.17em}\lambda ,\\ L\left(\lambda \right)& =& \text{simple}\hspace{0.17em}{GL}_n\left(K\right)\text{-module of highest weight}\hspace{0.17em}\lambda \text{.}\end{array}$$ The modules $\mathrm{\Delta }\left(\lambda \right)$ are the reductions mod $p$ of the characteristic $0$ irreducible representations of ${GL}_n\left(ℂ\right)\text{.}$ The $L\left(\lambda \right)$ are the simple ${GL}_n\left(K\right)$ modules. The simple module $L\left(\lambda \right)$ is the unique module in the head of $\mathrm{\Delta }\left(\lambda \right)$ and the unique module in the socle of $\mathrm{\nabla }\left(\lambda \right)\text{.}$ The tilting module $T\left(\lambda \right)$ is the unique indecomposable module of highest weight $\lambda$ which has both a $\mathrm{\Delta }$ and $\mathrm{\nabla }$ filtration.

(a)	$[T\left(\lambda \right):\mathrm{\Delta }\left(\mu \right)]=[\mathrm{\Delta }\left(\mu \prime \right):L\left(\lambda \prime \right)]\text{.}$ for arbitrary heighest weights, what does conjugation mean in this context?
(b)	(Steinberg's tensor product theorem) $\mathrm{\Delta }{\left(\mu \right)}^F\otimes \mathrm{\Delta }\left((p-1)\delta \right)\cong \mathrm{\Delta }(p\mu +(p-1)\delta )\text{.}$
(c)	$T{\left(\mu \right)}^F\otimes \mathrm{\Delta }\left((p-1)\delta \right)\cong T(p\mu +(p-1)\delta )\text{.}$

A $p\text{-regular}$ partition is a partition $\mu$ that does not have $p$ equal parts. Let $S_r$ be the symmetric group. The irreducible representations of the symmetric group in characteristic $0$ are indexed by the partitions $\mu$ with $r$ boxes. Their reductions mod $p$ are the Specht modules $S^{\mu }\text{.}$ The irreducible $K{S}_r$ modules $D^{\mu }$ are indexed by the $p$ regular partitions $\mu$ with $r$ boxes.

[Gre1980, Theorem 6.6g] and [Er, Proposition 2.3]

(a)	Let $\lambda ,\mu ⊢r,$ $\ell \left(\lambda \right)\le n,$ $\ell \left(\mu \right)\le n,$ such that $\lambda$ is $p$ regular. $$[T\left(\lambda \right):\mathrm{\Delta }\left(\mu \right)]=[S^{\mu }:D^{\lambda }]\text{.}$$
(b)	Let $\lambda$ and $\mu$ be partitions of $r$ with length $\le n\text{.}$ Then $$[\mathrm{\Delta }\left(\mu \right):L\left(\lambda \right)]=[S^{p\mu \prime +(p-1)\delta }:D^{p\lambda \prime +(p-1)\delta }]\text{.}$$

		Proof.
	(a) Apply the Schur functor. (b) is a consequence of the following equality. $$\begin{array}{ccc}[\mathrm{\Delta }\left(\mu \right):L\left(\lambda \right)]& =& [T\left(\lambda \prime \right),\mathrm{\Delta }\left(\mu \prime \right)]\\ & =& [T(p\lambda \prime +(p-1)\delta ):\mathrm{\Delta }(p\mu \prime +(p-1)\delta )]=[S^{p\mu \prime +(p-1)\delta }:D^{p\lambda \prime +(p-1)\delta }]\text{.}\end{array}$$ $\square$

References

[Ari1996] S. Ariki, On the decomposition numbers of the Hecke algebra of $G(m,1,n),$ J. Math. Kyoto Univ. 36 (1996), 789–808.

[AMa2000] S. Ariki and A. Mathas, On the number of simple modules of the Hecke algebras of type $G(r,p,n),$ Math. Zeitschrift 233 (2000), no. 3, 601–623.

[Erd1996????] K. Erdmann, Decomposition numbers for symmetric groups and composition factors of Weyl modules, J. Algebra 180 (1996), 316-320.

[FLO1999] O. Foda, B. Leclerc, M. Okado, J.-Y. Thibon, T. Welsh, Branching functions of $A_{n-1}^{\left(1\right)}$ and Jantzen-Seitz problem for Ariki-Koike algebras, Adv. Math. 141 (1999), 322-365.

[GWe1999] F. Goodman and H. Wenzl, Crystal bases of quantum affine algebras and affine Kazhdan-Lusztig polynomials, Int. Math. Res. Not. 5 (1999), 251-275.

[Gro1994-2] I. Grojnowski, Representations of affine Hecke algebras (and affine quantum ${\text{GL}}_n$) at roots of unity International Math. Res. Notices 5 (1994), 215-217.

[Gro1996] I. Grojnowski, Jantzen filtrations, unpublished notes, 1996.

[Gin1987-2] V. Ginzburg, Geometrical aspects of representation theory, Proc. Int. Cong. Math. (Berkeley 1986), Vol 1, Amer. Math. Soc. 1987, 840-848.

[Gre1980] J.A. Green, Polynomial representations of ${GL}_n,$ Lecture Notes in Mathematics 830, Springer-Verlag, New York, 1980.

[JMM1991] M. Jimbo, K. Misra, T. Miwa, M. Okado, Combinatorics of representations of $U_q\left(\widehat{𝔰𝔩}\left(n\right)\right)$ at $q=0$, Comm. Math. Phys. 36 (1991), 543-566.

[KLu1980] D. Kazhdan and G. Lusztig, Schubert Varieties and Poincare Duality, Proc. Symp. Pure Math. 36 (1980), 185-203. MR84g:14054

[KSc1985] M. Kashiwara and P. Schapira, Microlocal study of sheaves, Astérisque 128 (1985).

[LLT1996] A. Lascoux, B. Leclerc, J.-Y. Thibon, Hecke algebras at roots of unity and crystal bases of quantum affine algebras, Comm. Math. Phys. 181 (1996), 205-263.

[Lus1981] G. Lusztig, Green polynomials and singularities of nilpotent classes, Adv. in Math. 42 (1981), 169-178. MR83c:20059

[Lus1990] G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc 3 (1990), 447-498.

[Lus1991] G. Lusztig, Quivers, perverse sheaves and quantized enveloping algebras, J. Amer. Math. Soc 4 (1991), 365-421.

[Lus1989-2] G. Lusztig, Representations of affine Hecke algebras, in Orbites unipotentes et représentations, Astérisque 171-172 (1989), 73-89.

[Lus1998] G. Lusztig, Bases in equivariant K-theory, Rep. Theory 2 (1998), 298-369.

[Lus1987-3] G. Lusztig, Some problems in the representation theory of finite Chevalley groups, Proc. Symp. Pure Math. 37 Amer. Math. Soc. (1987), 313-317.

[Mat1998] A. Mathas, Simple modules of Ariki-Koike algebras, Proc. Symp. Pure Math. Amer. Math. Soc. 63 (1998), 383-396.

[Soe1998] W. Soergel, Character formulas for tilting modules over Kac-Moody algebras, Representation Theory 0 (1998), 432-448.

[Soe2000] W. Soergel, On the relation between intersection cohomology and representation theory in positive characteristic, Journal of Pure and Applied Algebra 152 (2000), 311-335.

[Zel1985] A. Zelevinsky, Two remarks on graded nilpotent classes, Russ. Math. Surv. 40 (1985), 249-250.

Proof of Theorem ???

[Lus1989-2, §7] Let $q\in {ℂ}^{*},$ $q\ne 1\text{.}$ The following are equivalent.

(a)	$\sum _{w\in W}{q}^{\ell \left(w\right)}\ne 0\text{.}$
(b)	In the reflection representation of $W\text{,}$ $\text{det}(q-w)\ne 0$ for all $w\in W\text{.}$
(c)	For any semisimple element $s\in G\text{,}$ ${𝔤}_{-}^{(s,q)}=\{x\in 𝔤\hspace{0.17em}|\hspace{0.17em}\text{Ad}\left(s\right)x=qx\}$ consists entirely of nilpotent elements.

		Proof.
	We may assume that $G$ is semisimple. It is well known that, as polynomials in $\mathbb{Z}\left[q\right],$ $\text{det}(q-w)$ divides $$\left(\sum _{w\in W}{q}^{\ell \left(y\right)}\right){(q-1)}^r,$$ where $r$ is the rank of $W\text{.}$ Hence (a) implies (b). It is also well known that $$\left|W\right|=\sum _{w\in W}{(-1)}^{\ell \left(w\right)}\left(\sum _{y\in W}{q}^{\ell \left(y\right)}\right){(q-1)}^r\text{det}{(q-w)}^{-1}\text{.}$$ Hence (b) implies (a). Assume that (b) doesn't hold. Then we can find a maximal torus $T$ of $G$ with Lie algebra $𝔥$ and an element $\dot{w}\in N\left(T\right)$ such that $(q-\text{Ad}\left(\dot{w}\right))\xi =0$ for some $\xi \in 𝔥-\left\{0\right\}\text{.}$ We may assume that $\dot{w}$ is of finite order, hence semisimple. Hence (c) does not hold. So (c) implies (b). Assume now that (c) doesn't hold. Let $s\in G$ be a semisimple element and $\xi$ a non-nilpotent element such that $\text{Ad}\left(s\right)\xi =q\xi \text{.}$ The same identity is then satisfied by the semisimple part of $\xi$ so that we can assume that $\xi$ is semisimple and nonzero. Let $G\prime =\{g\in G\hspace{0.17em}|\hspace{0.17em}\text{Ad}\left(g\right)\xi \in {ℂ}^{*}\xi \}$ and let $\psi :G\prime \to {ℂ}^{*}$ be the homomorphism defined by $\varphi \left(g\right)=\lambda$ where $\text{Ad}\left(g\right)\xi =\lambda \xi \text{.}$ If $\text{Ad}\left(g\right)\xi =\lambda \xi$ with $\lambda$ not a root of $1$ the $\xi$ is clearly nilpotent, a contradiction. Hence the image of $\psi$ contains only roots of $1\text{.}$ Being a closed subgroup of ${ℂ}^{*}\text{,}$ the image of $\psi$ must be finite. Since the centralizer $Z_G\left(\xi \right)$ is connected we have $\text{ker}\hspace{0.17em}\psi =Z_G\left(\xi \right)={\left(G\prime \right)}^{\circ }\text{.}$ Hence ${\psi }^{-1}\left(q\right)$ is a connected component of $G\prime ,$ so it contains some element of finite order. Hence we can assume that $s$ has finite order. Let $\gamma$ be the space of all maximal tori of $Z_G\left(\xi \right)\text{.}$ It is well known that $\gamma$ has the same rational homology with compact support as an affine space. Now $s$ acts on $\gamma$ by conjugation. By the fixed point theorem it follows that ${\gamma }^s\ne 0$ so that there exists a maximal torus $T$ of $Z_G\left(\xi \right)$ normalized by $s\text{.}$ Let $𝔥$ be the Lie algebra of $T\text{.}$ Then $\xi \in 𝔥$ and $\text{Ad}\left(s\right):𝔥\to 𝔥$ has $\xi$ as a $q\text{-eigenvector.}$ Hence $\text{det}(q-\text{Ad}\left(s\right),𝔥)=0\text{.}$ But $\text{Ad}\left(s\right)$ acts on $𝔥$ as an element of the Weyl group of $T$ and we see that (b) doesn't hold. Thus (b) implies (c). $\square$

(1)	Note that [CGi1433132, Lemma 8.5.3] is Schur's lemma for $IC$ sheaves. For [CGi1433132] $\pi :N\to M,$ $x\in N$ and $M_x={\pi }^{-1}\left(x\right)\text{.}$ In fact!, What Grojnowski does seems to be buried in the proof of [CGi1433132, Theorem 8.6.23].
(2)	Note that [CGi1433132, p. 463] says that "the question whether $M_{𝕆,\chi }$ has a unique simple quotient remains open, as far as we know".

Remarks on quiver representations

If $k\prime \le k$ are two integers then the vector spaces $$V{(k\prime ,k)}_i=\{\begin{array}{cc}ℂ,& \text{if}\hspace{0.17em}k\prime \le i\le k,\\ 0,& \text{otherwise,}\end{array}\text{and the maps}{n}_{i\to i-1}=\text{id},\text{if}\hspace{0.17em}k\le i-1,i\le k,$$ form an indecomposable representation of $A_{\infty }\text{.}$ The segments index the indecomposable representations $V(k\prime ,k)$ of ${\stackrel{\sim }{A}}_{\infty }$ and, since every representation of $A_{\infty }$ is a direct sum of indecomposable representations the multisegments index the (isomorphism classes of) representations of ${\stackrel{\sim }{A}}_{\infty }\text{.}$

The segments index the indecomposable representations $V(k\prime ,k)$ of ${\stackrel{\sim }{A}}_{\ell -1}\text{.}$ Every representation of $A_{\ell }$ is a direct sum of indecomposable representations and so the multisegments index the isomorphism classes of representations of ${\stackrel{\sim }{A}}_{\ell -1}\text{.}$

If $k\prime \le k$ are two integers then the vector spaces $$V{(k\prime ,k)}_i\text{with basis}\left\{b_{ih}\hspace{0.17em}\right|\hspace{0.17em}k\prime \le h\le k,h=i\hspace{0.17em}\text{mod}\hspace{0.17em}\ell \}\text{and the maps}{x}_{i\to i-1}\left(b_{i,h}\right)=b_{i-1,h-1}$$ form an indecomposable representation of ${\stackrel{\sim }{A}}_{\ell -1}\text{.}$

A segment is a finite sequence of consecutive boxes in a row of this page of graph paper, or, more precisely, an equivalence class of pairs of integers $(k\prime ,k)$ where the equivalence relation is $(k\prime ,k)\sim (k\prime +\ell ,k+\ell )\text{.}$ The segments index the indecomposable representations of the quiver ${\stackrel{\sim }{A}}_{\ell }\text{.}$

Notes and References

These notes are from /Work2004/Dell_Laptop/Unpublished/GRT/GRT12.6.00.tex

Research supported in part by National Science Foundation grant DMS-9622985.

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