Connecting decomposition numbers

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.

Kazhdan-Lusztig polynomials and canonical bases

2.1.1 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 $$\begin{array}{cc}{T}_w{T}_{w\prime }=T_{ww\prime },\text{if}\hspace{0.17em}\ell \left(ww\prime \right)=\ell \left(w\right)+\ell \left(w\prime \right),\text{and}{T}_{s_i}^2=v^{-2}+(v^{-2}-1)T_{s_i},& \text{(2.1)}\end{array}$$ 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 $$\begin{array}{cc}\bar{v}=v^{-1}\text{and}\overline{H_x}={\left(H_{x^{-1}}\right)}^{-1}\text{.}& \text{(2.2)}\end{array}$$ 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 ${\stackrel{\sim }{P}}_{y,x}\left(v\right)=v^{\ell \left(x\right)-\ell \left(y\right)}{P}_{y,x}\left(q\right)\text{.}$ Note that $$P_{w\prime ,w}\left(v\right)=0\text{unless}w\prime \le w\text{.}$$

2.1.2 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 $\stackrel{\sim }{W}\text{.}$

2.1.3 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], $$\begin{array}{cc}{\displaystyle n_{y,x}\left(v\right)=\sum _{z\in W}{(-v)}^{\ell \left(z\right)}{\stackrel{\sim }{P}}_{zy,x}\left(v\right)\text{.}}& \text{(2.3)}\end{array}$$

The canonical basis

2.2.1 Define $${\epsilon }_i={\mathrm{\Lambda }}_i-{\mathrm{\Lambda }}_{i-1},1\le i\le \ell ,{\alpha }_0={\epsilon }_{\ell }-{\epsilon }_1+\delta ,\text{and}{\alpha }_i={\epsilon }_i-{\epsilon }_{i+1},1\le i\le \ell -1\text{.}$$

2.2.2 The negative part of the quantum group associated to the graph is the algebra defined by generators $F_i,$ indexed by the nodes of the graph, and by relations $$\sum _{p=0}^{-a_{ij}+1}\frac{[-a_{ij}+1]!}{\left[p\right]![-a_{ij}+1-p]!}{F}_j^p{F}_i{F}_j^{-a_{ij}+1-p}=0,\text{for}\hspace{0.17em}i\ne j,$$ where $-a_{ij}$ is the number of edges connecting $i$ and $j$ in the graph.

2.2.3 These positive roots for $U\left({\widehat{𝔰𝔩}}_{\ell }\right)$ are $$\begin{array}{ccc}{\epsilon }_i-{\epsilon }_j,& & 1\le i<j\le \ell ,\\ {\epsilon }_i-{\epsilon }_j+k\delta ,i\ne j& & 1\le i,j\le \ell ,k\in {\mathbb{Z}}_{>0},\\ k\delta ,& & k\in {\mathbb{Z}}_{>0}\text{.}\end{array}$$ These roots can be identified with pairs of integers $(a,b),$ where $a<b$ and $0\le b\le \ell -1$ via $${\epsilon }_a-{\epsilon }_b={\epsilon }_{a\hspace{0.17em}\text{mod}\hspace{0.17em}\ell }-{\epsilon }_b+k\delta ,\text{where}\hspace{0.17em}a=-(k+1)\ell +i\hspace{0.17em}\text{with}\hspace{0.17em}1\le i\le \ell \text{.}$$ The algebra $U^{-}$ has a PBW indexed by nonegative linear combinations of positive roots and these linear combinations correspond to the description of aperiodic multisegments as sums of indecomposable representations of the quiver.

2.2.4 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 the set of aperiodic multisegments comparing dimensions (of graded components) allows one to conclude that the canonical basis is indexed by the set of aperiodic multisegments.

2.2.5 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.

If $c=\sum _{\alpha >0}{c}_{\alpha }\alpha \in Q^{+}$ and let $V_c$ be the corresponding sum of indecomposable representations of the quiver and let ${𝕆}_c$ denote the corresponding $G_V$ orbit in $E_V=E_d\text{.}$ Let $$E_i^c=\prod _{\alpha >0}{E}_{\alpha }^{\left(c_{\alpha }\right)},$$ where $E_{\alpha }^{\left(k\right)}=T_{i_1}\cdots T_{i_p}{E}_{{\alpha }_{i_p+1}}^{\left(k\right)},$ $E_i^{\left(k\right)}=E_{{\alpha }_i}^k/\left[k\right]!$ and the order of the product and the definition of the $E_{\alpha }^{\left(k\right)}$ are with respect to a fixed choice of reduced decomposition of $w_0\text{.}$

[Lus1989-2, Cor. 10.7(a)] Let ${𝕆}_c$ be an orbit in $E_d\text{.}$ Let $IC\left({\mu }_{*}{𝒞}_{c\prime }\right)$ be the intersection cohomology complex of the closure of ${𝕆}_{c\prime }\text{.}$ Then $$b_i^c=\sum _{c\prime }{P}_{c,c\prime }{E}_i^{c\prime },$$ where $P_{c,c\prime }=v^{d\left(c\prime \right)-d\left(c\right)}\sum _a\text{dim}\left(H^{2a}\left(IC\left(i_f^{!}{\mu }_{*}{𝒞}_{c\prime }\right)\right)\right)$ for $f\in {𝕆}_c\text{.}$

CHECK THIS WITH LUSZTIG'S BOOK. Alternatively, [Lus1998, 2.2] we may define $$L_{i,a}={\mu }_{!}{𝒞}_{{\stackrel{\sim }{ℱ}}_{i,a}}$$

2.2.6 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.

Let $V$ be a $\mathrm{\Gamma }\text{-graded}$ vector space. Define $V_j=t^k{V}_i$ for $j=k\ell +i$ with $1\le i\le \ell ,$ and fix the sequence of lattices $$L_1\supseteq \cdots \supseteq L_{\ell }\supseteq t{L}_1,\text{given by}{L}_i=\underset{j>i}{⨁}{V}_j\text{.}$$ Let $x$ be a nilpotent representation of the quiver on the graded vector space $V\text{.}$ Define the flag $M_1\supseteq \cdots \supseteq M_{\ell }\supseteq t{M}_1$ by $$M_i=\{\sum _{j>i}{x}_{j\to i}\left(v_j\right)-v_j\hspace{0.17em}|\hspace{0.17em}{v}_j\in V_j\}$$ This map has image in the subvariety of the affine flag variety given by $$Z\prime =\{M_1\supseteq \cdots \cdots M_{\ell }\supseteq t{M}_1\hspace{0.17em}|\hspace{0.17em}{M}_i\subseteq L_i,\text{dim}(L_i/M_i)=d_i,M_i\oplus V_i=L_i\}\text{.}$$

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

2.2.8

(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 ${\stackrel{\sim }{S}}_n\text{,}$ $n=\left|\lambda \right|\text{.}$

2.2.9 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 $L\left(\mathrm{\Lambda }\right)\text{.}$

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).

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.

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 algorithm is essentially that given by adding a single box at a time (applying a sequence of $i$ induction functors) and computing the canonical basis inductively. 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 $$\begin{array}{c}{\displaystyle \lambda =\sum _{i=1}^k({\lambda }_i-{\lambda }_{i+1}){\omega }_i,\text{where}{\omega }_i={\epsilon }_1+\cdots +{\epsilon }_i\text{.}}\\ {\displaystyle \rho =(k-1){\epsilon }_1+(k-2){\epsilon }_2+\cdots +{\epsilon }_{k-1}+0{\epsilon }_k\text{.}}\end{array}$$ Associate partitions to weights of ${𝔰𝔩}_k$ by $$\stackrel{\sim }{\lambda }=\sum _{i=1}^{k-1}({\lambda }_i-{\lambda }_{i+1}){\omega }_i$$

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. Let ${𝒜}^{+}$ be the set of alcoves in the dominant chamber and consider the module ${\text{Ind}}_H^{\stackrel{\sim }{H}}\left(\text{sgn}\right){\otimes }_H\stackrel{\sim }{H}$ with basis $S_k\backslash {\stackrel{\sim }{S}}_k$ so that the action is given by $$N_A{C}_s=\{\begin{array}{cc}{N}_{As}+v{N}_A,& \text{if}\hspace{0.17em}As\in {𝒜}^{+}\hspace{0.17em}\text{and}\hspace{0.17em}As>A,\\ N_{As}+v^{-1}{N}_A,& \text{if}\hspace{0.17em}As\in {𝒜}^{+}\hspace{0.17em}\text{and}\hspace{0.17em}As<A,\\ 0,& \text{if}\hspace{0.17em}As\notin {𝒜}^{+}\text{.}\end{array}$$ where the action of $x\in {\stackrel{\sim }{S}}_n$ on alcoves $\left\{w{A}^{+}\hspace{0.17em}\right|\hspace{0.17em}w\in {\stackrel{\sim }{S}}_n\}$ is given by $Ax=\left(w{A}^{+}\right)x=wx{A}^{+}\text{.}$

The elements of ${\stackrel{\sim }{S}}_n$ are $t_{\lambda }w,$ $\lambda \in {\mathbb{Z}}^n$ and $w\in S_n\text{.}$ The nonextended affine Weyl group is $$\left\{t_{\lambda }w\hspace{0.17em}\right|\hspace{0.17em}w\in S_n,{\lambda }_1+\cdots {\lambda }_k=0\}$$ Thus $S_k\backslash {\stackrel{\sim }{S}}_k$ has coset representatives $t_{\lambda },$ $\lambda \in {\mathbb{Z}}^n\text{.}$ We need to identify these elements with alcoves. Note that $t_{\lambda }$ acts as translation by $\ell {\lambda }_1{\epsilon }_1+\cdots +\ell {\lambda }_k{\epsilon }_k$ and $w$ permutes the coordinates.

If $\lambda \in P^{+}$ then $$\lambda =\left(t_{\ell {\lambda }_1}w\right){\lambda }_0,\text{where}\hspace{0.17em}{\lambda }_1\in \ell Q\cap P^{+}\hspace{0.17em}\text{and}\hspace{0.17em}{\lambda }_0\in A^{+}\text{.}$$ Note that translating $\lambda$ by $-\ell {\lambda }_1$ with $\lambda \in Q\cap P^{+}$ corresponds to removing $\ell$ boxes from a row of $\ell$ and placing them on a lower row. Since $\ell {\lambda }_1\in P^{+}$ we must remove more boxes from the first row than the second, more from the second than the third, etc. The resulting sequence should be in the "fundamental hexagon" i.e. $|{\lambda }_i-{\lambda }_{i+1}|\le \ell \text{.}$ Then a permutation of this will put the result in the fundamental alcove and yield ${\lambda }_0\text{.}$

A partition is $\ell \text{-regular}$ if it has no more than $\ell -1$ rows of any given length. (The conjugate condition, that $\lambda$ has no more than $\ell -1$ columns of any given length, is called $p\text{-restricted,}$ and says that $\lambda$ is in the compact torus $\sum _i(\mathbb{Z}/\ell \mathbb{Z}){\omega }_i\text{.)}$

[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 $\mu \text{.}$

Question: In what sense is the $p\text{-core}$ of a partition the fundamental alcove representative of the $\stackrel{\sim }{W}$ orbit of $\lambda$ under the dot action?

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 $\lambda \in P,$ $1\le i\le n\text{.}$

[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.

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 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 $A\text{.}$

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 $$\begin{array}{cc}{\displaystyle {[S^{\lambda }:D^{\mu }]}_t=\sum _k[\frac{{\left(S^{\lambda }\right)}^{\left(k\right)}}{{\left(S^{\lambda }\right)}^{(k+1)}}:D^{\mu }]t^k\text{.}}& \text{(3.3)}\end{array}$$

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.

The algebra $U^{-}$ is generated by the $f_i\text{,}$ $i\in \mathbb{Z}/\ell \mathbb{Z},$ with Serre relations (WHICH ARE???).

[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 $U_v\left({\widehat{𝔰𝔩}}_{\ell }\right)\text{.}$

		Proof.
	According to [Ari1996, Lemma 4.2] $$F^{s,x}{F}_i=\sum c_{s,x}^{s\prime x\prime }{F}^{s\prime ,x\prime },$$ where $$c_{s,x}^{s\prime ,x\prime }=\text{(\# ways to add a box of content}\hspace{0.17em}i\hspace{0.17em}\text{at the end of a row of}\hspace{0.17em}x\hspace{0.17em}\text{and get}\hspace{0.17em}x\prime \text{.}$$ $\square$

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.

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 $$0⟶{𝒞}_{\ell \left(w_J\right)}⟶\cdots ⟶{𝒞}_1⟶{𝒞}_0⟶L\left(\lambda \right)⟶0,\text{where}{𝒞}_k=\underset{\underset{\ell \left(w\right)=k}{w\in W_J}}{⨁}M(w\circ \lambda ),$$ gives an exact sequence $$0⟶U𝔤{\otimes }_{U𝔟}{𝒞}_{\ell \left(w_0\right)}⟶\cdots ⟶U𝔤{\otimes }_{U𝔟}{𝒞}_1⟶U𝔤{\otimes }_{U𝔟}{𝒞}_0⟶\mathrm{\Delta }\left(\lambda \right)⟶0,$$ for $\lambda$ a dominant integral weight for ${𝔤}_J\text{.}$ Thus, for all $\nu \in {𝔥}^{*}$ and $\lambda \in {𝔥}^{*}$ such that $⟨\lambda ,{\alpha }_i^{\vee }⟩\in {\mathbb{Z}}_{\ge 0}$ for ${\alpha }_i\in J$ $$[\mathrm{\Delta }\left(\lambda \right):L\left(\nu \right)]=\sum _{w\in W}{\left(-1\right)}^{\ell \left(w\right)}[\mathrm{\Delta }(w\circ \lambda ):L\left(\nu \right)],$$ 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 [Lu9?] and proved by Kazhdan-Lusztig via the passage to the category of representations of affine Lie algebras of level $-h-\ell$ via Theorem ??? below.

Assume that $𝔤$ is type $A\text{,}$ $D$ or $E$ and let $\theta$ be the highest root. Let $$A^{-}=\{\lambda \in {𝔥}^{*}\hspace{0.17em}|\hspace{0.17em}⟨\lambda +\rho ,\theta ⟩\ge -\ell ,⟨\lambda ,{\alpha }_i⟩\le -1,1\le i\le n\}$$ be the chamber bounded by the hyperplanes $$⟨\lambda +\rho ,\theta ⟩=-\ell ,⟨\lambda ,{\alpha }_i⟩=-1,1\le i\le n,$$ and let ${\stackrel{\sim }{W}}_{\ell }$ be the corresponding affine Weyl group. Let $w_{\lambda }$ be of minimal length such that $w_{\lambda }^{-1}\lambda \in A^{-}\text{.}$

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 $\stackrel{\sim }{W}\text{.}$

[Soe1998, §7] explains that (at least within the Grothendieck ring of the principal block of the category of finite dimensional $U_{\xi }$ modules) this formula can be inverted and written in the form $$\sum _A{n}_{B,\hat{A}}\left(1\right)L_A={\mathrm{\nabla }}_C,$$ The principal block is the smallest direct summand of the category of finite dimensional $U_{\xi }$ modules which contains the trivial representation. Here $L_A$ denotes the simple and ${\mathrm{\nabla }}_C$ is the module $W_{\xi }^{\text{res}}\left(\lambda \right)\text{.}$

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].

Let $K$ be an algebraically closed field of characteristic $p\text{.}$ Let $G$ be a reductive affine algebraic group and let $B$ a Borel subgroup and $T\subset B$ a maximal torus. Let $W$ be the Weyl group, $X=X\left(T\right)$ the weight lattice and $X^{+}$ the set of dominant integral weights. If $\lambda \in X$ Let $$\begin{array}{ccc}\mathrm{\nabla }\left(\lambda \right)& =& {\text{Ind}}_B^G\left(K_{\lambda }\right),\\ \mathrm{\Delta }\left(\lambda \right)& =& \mathrm{\nabla }{(-w_0\lambda )}^{*}=\text{the Weyl module of highest weight}\hspace{0.17em}\lambda ,\\ L\left(\lambda \right)& =& \text{soc}\left(\mathrm{\nabla }\left(\lambda \right)\right)=\text{head}\left(\mathrm{\Delta }\left(\lambda \right)\right)\text{.}\\ T\left(\lambda \right)& =& \text{indecomposable tilting module of highest weight}\hspace{0.17em}\lambda ,\\ P\left(\lambda \right)& =& \text{the projective cover of}\hspace{0.17em}L\left(\lambda \right),\end{array}$$

The simple $G\text{-modules}$ are $\left\{L\left(\lambda \right)\hspace{0.17em}\right|\hspace{0.17em}\lambda \in X^{+}\}\text{.}$

If $(p-1)\rho \in X^{+}$ The Steinberg module is $$\mathrm{\nabla }\left((p-1)\rho \right)=L\left((p-1)\rho \right)=\mathrm{\Delta }\left((p-1)\rho \right)=T\left((p-1)\rho \right)\text{.}$$ Let $F:G\to G$ be a Frobenius map.

(Steinberg tensor product theorem) [Jz, II (3.19)] $$\mathrm{\nabla }\left((p-1)\rho \right)\otimes \mathrm{\nabla }{\left(\mu \right)}^F\cong \mathrm{\nabla }((p-1)\rho +p\mu )\text{.}$$

[Don1993, Prop. 2.1] Let $\mu \in X^{+}$ and let $\lambda \in X^{+}$ be such that $⟨\lambda {\alpha }_i^{\vee }⟩\le p-1,$ for $1\le i\le n\text{.}$ Then $$T(2(p-1)\rho +w_0\lambda )\otimes T{\left(\mu \right)}^F\cong T(2(p-1)\rho +w_0\lambda +p\mu )\text{.}$$

The general linear group

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)	[Er, Lemma 2.1b] $T{\left(\mu \right)}^F\otimes \mathrm{\Delta }\left((p-1)\delta \right)\cong T(p\mu +(p-1)\delta )\text{.}$

		Proof.
	(c) Note that $T\left((p-1)\rho \right)\cong \mathrm{\Delta }\left((p-1)\rho \right)\cong L\left((p-1)\rho \right)$ and apply Donkin's result [Don1993, Proposition 2.1] $\square$

The symmetric group

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

The irreducible representations of the symmetric group in characterstic $0$ are indexed by the partitions $\mu$ with $r$ boxes. Their reductions mod $p$ are the Specht modules $S^{\mu }\text{.}$

A $p\text{-regular}$ partition is a partition $\mu$ that does not have $p$ equal parts.

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. $$\begin{array}{ccc}[\mathrm{\Delta }\left(\mu \right):L\left(\lambda \right)]& =& [T\left(\lambda \prime \right),\mathrm{\Delta }\left(\mu \prime \right)]\\ & =& [T(p\lambda \prime +(p-1)\delta ):\mathrm{\Delta }(p\mu \prime +(p-1)\delta )]=[S^{p\mu \prime +(p-1)\delta }:D^{p\lambda \prime +(p-1)\delta }]\text{.}\end{array}$$ $\square$

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Notes and References

These notes are from /Work2004/Dell_Laptop/Unpublished/Decompnos/conndecomp8.11.00.tex

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

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