Almost semisimple algebras

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

Last update: 11 May 2012

Convolution algebras

The decomposition theorem

Let $M$ be a smooth $G-$variety and let $N$ be a $G-$variety with finitely many $G-$orbits such that the orbit decomposition is an algebraic stratification of $N,$ $$N=\underset{\phi }{⨆}G{x}_{\phi },and\mu :M\to N$$ is a $G-$equivariant projective morphism. Let ${𝒞}_M$ be the constant perverse sheaf on $M.$ The decomposition theorem [CG, 8.4.12] says that $${\mu }_{*}{𝒞}_M=\underset{\genfrac{}{}{0ex}{}{i\in \mathbb{Z}}{\lambda =(\phi ,\chi )\in \hat{M}}}{⨁}L(\lambda ,i)\otimes I{C}^{\lambda }\left[i\right]\doteq \underset{\lambda \in \hat{M}}{⨁}L\left(\lambda \right)\otimes I{C}^{\lambda },whereL\left(\lambda \right)=\underset{i\in \mathbb{Z}}{⨁}L(\lambda ,i),$$ ${\mu }_{*}$ is the derived functor of sheaf theoretic direct image, $\lambda$ runs over the indexes of the intersection cohomology complexes $I{C}^{\lambda },$ $L\left(\lambda \right)$ are finite dimensional vector spaces, and $\&eqdot;$ indicates an equality up to shifts in the derived category.

Convolution algebras

Let $\mu :M\to N$ be a proper map. The convolution algebra is $$A={\mathrm{Ext}}_{D^b\left(N\right)}^{*}({\mu }_{*}{𝒞}_M,{\mu }_{*}{𝒞}_M)=\underset{k\in \mathbb{Z}}{⨁}{\mathrm{Ext}}^k({\mu }_{*}{𝒞}_M,{\mu }_{*}{𝒞}_M).$$ The decomposition theorem for ${\mu }_{*}{𝒞}_M$ induces a decomposition of $A.$ Since the intersection cohomology complexes $I{C}_{\phi }$ are the simple objects in the category of perverse sheaves, $${\mathrm{Ext}}_{D^b\left(N\right)}^0(I{C}^{\lambda },I{C}^{\mu })={\delta }_{\lambda \mu }ℂ,and{\mathrm{Ext}}_{D^b\left(N\right)}^k(I{C}^{\lambda },I{C}^{\mu })=0,for\; k\in {\mathbb{Z}}_{<0},$$ and the decomposition of $A$ simplifies to $$A=\underset{\lambda \in \hat{M}}{⨁}{\mathrm{End}}_{ℂ}\left(L\left(\lambda \right)\right)\oplus \left(\underset{k\in {\mathbb{Z}}_{>0}}{⨁}\left(\underset{\lambda ,\mu \in \hat{M}}{⨁}{\mathrm{Hom}}_{ℂ}(L\left(\lambda \right),L\left(\mu \right))\otimes {\mathrm{Ext}}_{D^b\left(N\right)}^k(I{C}^{\lambda },I{C}^{\mu })\right)\right).$$ In the context there is a good theory of projective, standard and simple modules, and their decomposition matrices satisfy a BGG reciprocity. View elements of $A$ as sums $$\sum _{\lambda ,\mu }\sum _{P\in \hat{L}\left(\lambda \right),Q\in \hat{L}\left(\mu \right)}{c}_{PQ}^{\lambda \mu }{a}_{PQ}^{\lambda \mu }where{c}_{PQ}^{\lambda \mu }\in ℂ,and{a}_{PQ}^{\lambda \mu }\in \underset{k>0}{⨁}{\mathrm{Ext}}_{D^b\left(N\right)}^k(I{C}^{\lambda },I{C}^{\mu }).$$ The algebra $A$ is completely controlled by the dimensions of the $L\left(\lambda \right)$ and the multiplication in $$A_{\mathrm{basic}}={\mathrm{Ext}}^{*}(IC,IC)whereIC=\underset{\lambda \in \hat{M}}{⨁}I{C}^{\lambda },$$ an algebra which has all one dimensional simple modules. The radical filtration of $A$ is $${\mathrm{Rad}}^l\left(A\right)=\underset{\lambda ,\mu \in \hat{M}}{⨁}{\mathrm{Hom}}_{ℂ}(L\left(\lambda \right),L\left(\mu \right))\otimes \left(\underset{k\in {\mathbb{Z}}_{\ge l}}{⨁}{\mathrm{Ext}}_{D^b\left(N\right)}^k(I{C}^{\lambda },I{C}^{\mu })\right)$$ and the nonzero $$L\left(\lambda \right) \; are\; the\; simple\; A-modules.$$

Projective modules

Let $e^{\lambda }$ be a minimal idempotent in $\underset{\mu }{⨁}\mathrm{End}\left(L\left(\mu \right)\right).$ Then $$P\left(\lambda \right)=A{e}^{\lambda }=L\left(\lambda \right)\oplus \left(\underset{\genfrac{}{}{0ex}{}{k>0}{\mu }}{⨁}L\left(\mu \right)\otimes {\mathrm{Ext}}_{D^b\left(N\right)}^k(I{C}^{\mu },I{C}^{\lambda })\right)$$ is the projective cover of the simple $A-$module $L\left(\lambda \right).$ Define an $A-$module filtration $$P\left(\lambda \right)\supseteq P{\left(\lambda \right)}^{\left(1\right)}\supseteq P{\left(\lambda \right)}^{\left(2\right)}\supseteq \cdots$$ by $$P{\left(\lambda \right)}^{\left(m\right)}=\underset{\genfrac{}{}{0ex}{}{k\ge m}{\mu }}{⨁}L\left(\mu \right)\otimes {\mathrm{Ext}}_{D^b\left(N\right)}^k(I{C}^{\mu },I{C}^{\lambda }).$$ Then $$L\left(\lambda \right)=P\left(\lambda \right)/P{\left(\lambda \right)}^{\left(1\right)}and\mathrm{gr}\left(P\left(\lambda \right)\right)is\; a\; semisimple\; A-module.$$ Thus the multiplicity of the simple $A-$module $L\left(\mu \right)$ in a composition series of $P\left(\lambda \right)$ is $$[P\left(\lambda \right):L\left(\mu \right)]=\dim \left({\mathrm{Ext}}^{*}(I{C}_{𝕆,\chi },I{C}_{𝕆\prime ,\chi \prime })\right)=\sum _{k\ge 0}\dim \left({\mathrm{Ext}}_{D^b\left(N\right)}^k(I{C}^{\mu },I{C}^{\lambda })\right).$$

Standard and costandard modules

Let $\lambda =(\phi ,\chi ),$ $$x\in {𝕆}^{\phi },and\; let{i}_x:\left\{x\right\}\hookrightarrow Nbe\; the\; injection.$$ Then $i_x^{!}{\mu }_{*}{𝒞}_M$ is the stalk of ${\mu }_{*}{𝒞}_M$ at $x$ and the Yoneda product makes $$\begin{array}{rcl}{\Delta }^{\phi }& =& H^{*}\left(i_x^{!}{𝒞}_M\right)={\mathrm{Hom}}_{D^b\left(\left\{x\right\}\right)}(ℂ,i_x^{!}{\mu }_{*}{𝒞}_M[*])={\mathrm{Hom}}_{D^b\left(N\right)}({\left(i_x\right)}_{!}ℂ[-*],{\mu }_{*}{𝒞}_M),and,\\ {\nabla }^{\phi }& =& H^{*}\left(i_x^{*}{𝒞}_M\right)=H^{*}(\left\{x\right\},i_x^{*}{\mu }_{*}{𝒞}_M)={\mathrm{Hom}}_{D^b\left(\left\{x\right\}\right)}(ℂ,i_x^{!}{\mu }_{*}{𝒞}_M[*])={\mathrm{Hom}}_{D^b\left(N\right)}({\left(i_x\right)}_{!}ℂ[-*],{\mu }_{*}{𝒞}_M),\end{array}$$ into right $A-$modules. The action of an element $a\in {\mathrm{Ext}}^k({\mu }_{*}{𝒞}_M,{\mu }_{*}{𝒞}_M)={\mathrm{Hom}}_{D^b\left(N\right)}({\mu }_{*}{𝒞}_M,{\mu }_{*}{𝒞}_M\left[k\right])$ sends $$H^{*}(\left\{x\right\},i_x^{!}{\mu }_{*}{𝒞}_M)\to H^{*+k}(\left\{x\right\},i_x^{!}{\mu }_{*}{𝒞}_M).$$

A $G-$equivariant local system is a $G-$equivariant locally constant sheaf. The orbit ${𝕆}^{\phi }$ can be identified with $G/G_x$ where $G_x$ is the stabilizer of $x.$ ${\pi }_0({𝕆}^{\phi },x)=G_x/G_x^{°}$ where $G_x^{°}$ is the connected component of the identity in $G_x.$ There is a homomorphism ${\pi }_1({𝕆}^{\phi },x)\to {\pi }_0({𝕆}^{\phi },x)=G_x/G_x^{°}$ and the representations of ${\pi }_1({𝕆}^{\phi },x)$ on the fibers ${ℒ}_x$ of $G-$equivariant local systems $ℒ$ are exactly the pullbacks of finite dimensional representations of $C=G_x/G_x^{°}$ to ${\pi }_1({𝕆}^{\phi },x).$ In this way the irreducible $G-$equivariant local systems on ${𝕆}^{\phi }$ can be indexed by (some of the) irreducible representations of $G_x/G_x^{°}$ [CG, Lemma 8.4.11]. There is an action of $C=G_x/G_x^{°}$ on ${\Delta }^{\phi }$ which commutes with the action of $A.$ Similar arguments apply to ${\nabla }^{\phi }.$ As $(A,C)$ bimodules, $${\Delta }^{\phi }=\underset{\chi \in \hat{C}}{⨁}\Delta (\phi ,\chi )\otimes \chi and{\nabla }^{\phi }=\underset{\chi \in \hat{C}}{⨁}\nabla (\phi ,\chi )\otimes \chi ,$$ and the standard and costandard $A-$modules are $$\Delta \left(\lambda \right)=\Delta (\phi ,\chi )and\nabla \left(\lambda \right)=\nabla (\phi ,\chi ).$$ Using the decomposition theorem $$\Delta \left(\lambda \right)=H^{*}{\left(i_x^{!}{𝒞}_M\right)}_{\chi }=\underset{\genfrac{}{}{0ex}{}{k\in \mathbb{Z}}{\mu }}{⨁}L\left(\mu \right)\otimes H^k{\left(i_x^{!}I{C}^{\mu }\right)}_{\chi },$$ where the subscript $\chi$ denotes the $\chi -$isotypic component. Define a filtration Then $\Delta {\left(\lambda \right)}^{\left(m\right)}$ is an $A-$module and $\mathrm{gr}\left(\Delta \left(\lambda \right)\right)$ is a semisimple $A-$module. This (and a similar argument for $\nabla \left(\lambda \right)$) show that the multiplicity of the simple $A-$module $L\left(\mu \right)$ in composition series of $\Delta \left(\lambda \right)$ and $\nabla \left(\lambda \right)$ are $$[\Delta \left(\lambda \right):L\left(\mu \right)]=\sum _k\dim \left(H^k{\left(i_x^{!}I{C}^{\mu }\right)}_{\chi }\right)and[\nabla \left(\lambda \right):L\left(\mu \right)]=\sum _k\dim \left(H^k{\left(i_x^{*}I{C}^{\mu }\right)}_{\chi }\right).$$ Define the standard KL-polynomial and the costandard KL-polynomial of $A$ to be $$P_{\lambda \mu }^{\Delta }\left(t\right)=\sum _k{t}^k\dim \left(H^k{\left(i_x^{!}I{C}^{\mu }\right)}_{\chi }\right)and{P}_{\lambda \mu }^{\nabla }\left(t\right)=\sum _k{t}^k\dim \left(H^k{\left(i_x^{*}I{C}^{\mu }\right)}_{\chi }\right),$$ respectively. Then ??? says that $$[\Delta \left(\lambda \right):L\left(\mu \right)]=P_{\lambda \mu }^{\Delta }\left(1\right)and[\nabla \left(\lambda \right):L\left(\mu \right)]=P_{\lambda \mu }^{*}\left(1\right).$$ These identities are analogues of the original Kazhdan-Lusztig conjecture describing the multiplicities of simple $𝔤-$modules in Verma modules.

The contravariant form

Note that there is a canonical homomorphism $$\Delta \left(\lambda \right)\stackrel{c_{\lambda }}{\to }\nabla \left(\lambda \right)$$ coming from applying the functor $H^{*}$ to the composition $${\left(i_x\right)}_{!}{\left(i_x\right)}^{!}{\mu }_{*}{𝒞}_M\to {\mu }_{*}{𝒞}_M\to {\left(i_x\right)}_{*}{\left(i_x\right)}^{*}{\mu }_{*}{𝒞}_M,$$ where the two maps arise from the canonical adjoint functor maps. Use the map $c_{\lambda }$ to define a bilinear form on $\Delta \left(\lambda \right)$ by $$\begin{array}{rccc}⟨,⟩:& \Delta \left(\lambda \right)\otimes \Delta \left(\lambda \right)& \to & ℂ\\ & m_1\otimes m_2& \mapsto & m_1\cap c_{\lambda }\left(m_2\right)\end{array}$$ Then $$L\left(\lambda \right)=\Delta \left(\lambda \right)/\mathrm{Rad}\left(⟨,⟩\right).$$

Contragradient modules

There is an involutive automorphism ${\phantom{A}}^t:A\to A$ on $A$ (coming from switching the two factors in $Z=M{\times }_{N}M$). If $M$ is an $A-$module the contragradient module is $$M^{*}={\mathrm{Hom}}_{ℂ}(M,ℂ)with\left(a\psi \right)\left(m\right)=\psi \left(a^t\left(m\right)\right),for\; a\in A, \psi \in M^{*}, \; and\; m\in M.$$ Then $$\nabla \left(\lambda \right)\cong \Delta {\left(\lambda \right)}^{*}.$$

Reciprocity

If $\lambda =(\phi ,\rho )$ define $$d_{\lambda }={\dim }_{ℂ}\left({𝕆}^{\phi }\right),and\; assume\; that{\mathrm{Ext}}_{D^b\left(N\right)}^{d_{\psi }+d_{\phi }+k}(I{C}^{\phi },I{C}^{\psi })=0,for\; all\; odd\; k.$$ Then $$\begin{array}{rcl}[P\left(\lambda \right):L\left(\mu \right)]& =& \sum _k\dim {\mathrm{Ext}}_{D^b\left(N\right)}^k(I{C}^{\lambda },I{C}^{\mu })\\ & =& \sum _k\dim {\mathrm{Ext}}_{D^b\left(N\right)}^{d_{\lambda }+d_{\mu }+k}(I{C}^{\lambda },I{C}^{\mu })\\ & =& \sum _k{(-1)}^k\dim {\mathrm{Ext}}_{D^b\left(N\right)}^{d_{\lambda }+d_{\phi }+k}(I{C}^{\lambda },I{C}^{\mu })\\ & =& {(-1)}^{d_{\phi }+d_{\psi }}\sum _{𝕆}\chi (𝕆,i_{𝕆}^{!}I{C}_{\phi }^{\vee }\stackrel{!}{\otimes }{i}_{𝕆}^{!}I{C}_{\psi })\\ & =& {(-1)}^{d_{\phi }+d_{\psi }}\sum _{𝕆}\chi \left(𝕆,{(-1)}^{d_{\phi }}\sum _{\alpha ,k}[{\mathscr{H}}^k{i}_{𝕆}^{!}\left(I{C}_{\phi }^{\vee }\right):\alpha ]\alpha \stackrel{!}{\otimes }{(-1)}^{d_{\psi }}\sum _{\beta ,l}[{\mathscr{H}}^l{i}_{𝕆}^{!}\left(I{C}_{\psi }\right):\beta ]\beta \right)\\ & =& \sum _{𝕆,\alpha ,\beta }\chi \left(𝕆,\sum _k[{\mathscr{H}}^k{i}_{𝕆}^{!}\left(I{C}_{\phi }\right):{\alpha }^{*}]\alpha \stackrel{!}{\otimes }\sum _l[{\mathscr{H}}^l{i}_{𝕆}^{!}\left(I{C}_{\psi }\right):\beta ]\beta \right)\\ & =& \sum _{\alpha ,\beta }\sum _k\dim {\mathscr{H}}^k\left(i_{\alpha }^{!}I{C}_{\phi }\right)\left(\sum _{𝕆}\chi (𝕆,{\alpha }^{*}\stackrel{!}{\otimes }\beta )\right)\sum _l\dim {\mathscr{H}}^l\left(i_{\beta }^{!}I{C}_{\psi }\right)\\ & =& \sum _{\alpha ,\beta }[{ℳ}_{\alpha }^{!}:L_{\phi }]\left(\sum _{𝕆}\chi (𝕆,{\alpha }^{*}\otimes \beta )\right)[{ℳ}_{\beta }^{!}:L_{\psi }]\\ & =& \sum _{\alpha ,\beta }{P}_{\phi \alpha }\left(1\right)D_{\alpha \beta }{P}_{\psi \beta }\left(1\right)\\ & =& {\left(PD{P}^t\right)}_{\phi \psi },\end{array}$$ where

1. the third equality follows from the vanishing of Ext groups in odd degrees,
2. $\chi$ denotes the Euler characteristic,
3. $P$ is the matrix $(P_{\phi \alpha }\left(1\right),$ and
4. $D$ is the matrix $\left(\sum _{𝕆}\chi (𝕆,{\alpha }^{*}\otimes \beta )\right).$

This identity is the "BGG reciprocity" for the algebra $A.$

The Steinberg variety

Let $x\in N$ and define $$Z=M{\times }_{N}M=\{(m_1,m_2)\in M\times M|\mu \left(m_1\right)=\mu \left(m_2\right)\}and{M}_x={\mu }^{-1}\left(x\right).$$ There are commutative diagrams $$\begin{array}{c} Z=M×NM M×M N=NΔ N×N ι Δ μ12 μ1×μ2 \end{array}and\begin{array}{c} Mx M {x} N ι ix μ μ \end{array}$$ which (via base change) provide isomorphisms $$\begin{array}{rcl}{H}_{*}\left(Z\right)& =& {\mathrm{Hom}}_{D^b\left(Z_{12}\right)}({ℂ}_{Z_{12}},{\left({ℂ}_{Z_{12}}[*]\right)}^{\vee })\\ & =& {\mathrm{Hom}}_{D^b\left(Z_{12}\right)}({\mu }_{12}^{*}{ℂ}_N,{\iota }^{!}{𝒞}_{M_1\times M_2}[m_1+m_2][-*])\\ & =& {\mathrm{Hom}}_{D^b\left(N\right)}({ℂ}_N,{\left({\mu }_{12}\right)}_{*}{\iota }^{!}{𝒞}_{M_1\times M_2}[m_1+m_2-*])\\ & =& {\mathrm{Hom}}_{D^b\left(N\right)}({ℂ}_N,{\Delta }^{!}{({\mu }_1\times {\mu }_2)}_{*}({𝒞}_{M_1}⊠{𝒞}_{M_2})[m_1+m_2-*])\\ & =& {\mathrm{Hom}}_{D^b\left(N\right)}({ℂ}_N,{\Delta }^{!}({\left({\mu }_1\right)}_{*}{𝒞}_{M_1}⊠{\left({\mu }_2\right)}_{*}{𝒞}_{M_2})[m_1+m_2-*])\\ & =& {\mathrm{Ext}}_{D^b\left(N\right)}^{m_1+m_2-*}({\left({\mu }_1\right)}_{*}{𝒞}_{M_1},{\left({\mu }_2\right)}_{*}{𝒞}_{M_2}),\\ & & \\ H_{*}\left(M_x\right)& =& {\mathrm{Hom}}_{D^b\left(M_x\right)}({ℂ}_{M_x},{\left({ℂ}_{M_x}[*]\right)}^{\vee })\\ & =& {\mathrm{Hom}}_{D^b\left(M_x\right)}({\mu }^{*}{ℂ}_{\left\{x\right\}},{\left(\left({\iota }^{*}{ℂ}_M\right)[*]\right)}^{\vee })\\ & =& {\mathrm{Hom}}_{D^b\left(\left\{x\right\}\right)}({ℂ}_{\left\{x\right\}},{\mu }_{*}\left({\iota }^{!}{ℂ}_M\left[2m\right]\right)[-*])\\ & =& {\mathrm{Hom}}_{D^b\left(\left\{x\right\}\right)}({ℂ}_{\left\{x\right\}},i_x^{!}{\mu }_{*}{𝒞}_M[m-*])\\ & =& H^{m-*}\left(i_x^{!}{\mu }_{*}{𝒞}_M\right),\end{array}$$ and $$\begin{array}{rcl}{H}^{*}\left(M_x\right)& =& {\mathrm{Hom}}_{D^b\left(M_x\right)}({ℂ}_{M_x},{ℂ}_{M_x}[*])\\ & =& {\mathrm{Hom}}_{D^b\left(M_x\right)}({\mu }^{*}{ℂ}_{\left\{x\right\}},{ℂ}_{M_x}[*])\\ & =& {\mathrm{Hom}}_{D^b\left(\left\{x\right\}\right)}({ℂ}_{\left\{x\right\}},{\mu }_{*}{ℂ}_{M_x}[*])\\ & =& {\mathrm{Hom}}_{D^b\left(\left\{x\right\}\right)}({ℂ}_{\left\{x\right\}},{\mu }_{!}{\iota }^{*}{ℂ}_M[*])\\ & =& {\mathrm{Hom}}_{D^b\left(\left\{x\right\}\right)}({ℂ}_{\left\{x\right\}},i_x^{*}{\mu }_{!}{ℂ}_M[*])\\ & =& {\mathrm{Hom}}_{D^b\left(\left\{x\right\}\right)}({ℂ}_{\left\{x\right\}},i_x^{*}{\mu }_{*}{𝒞}_M[*-m])\\ & =& H^{*-m}\left(i_x^{*}{\mu }_{*}{𝒞}_M\right).\end{array}$$

The category $D^b\left(N\right)$

The category ${\mathrm{Comp}}^b\left(\mathrm{Sh}\left(N\right)\right)$ is the category of all finite complexes $$A=(0\to A^{-m}\to A^{-m+1}\to \cdots \to A^{n-1}\to A^n\to 0),m,n\in {\mathbb{Z}}_{\gg 0},$$ of sheaves on $N$ with morphisms being morphisms of complexes which commute with the differentials. The $j^{\mathrm{th}}$ cohomology sheaf of $A$ is $${\mathscr{H}}^j\left(A\right)=\frac{\ker (A^j\to A^{j+1})}{\mathrm{im}(A^{j-1}\to A^j)}.$$ A morphism in ${\mathrm{Comp}}^b\left(\mathrm{Sh}\left(N\right)\right)$ is a quasi-isomorphism if it induces isomorphisms on cohomology. The category $D^b\left(\mathrm{Sh}\left(N\right)\right)$ is the category ${\mathrm{Comp}}^b\left(\mathrm{Sh}\left(N\right)\right)$ with additional morhpisms obtained by formally inverting all quasi-isomorphisms.

Assume that $N$ is a $G-$variety with a finite number of orbits such that the $G-$orbit decomposition $$N=\underset{\phi }{⨆}{𝕆}^{\phi }is\; an\; algebraic\; stratification\; of\; X.$$ A constructible sheaf is a sheaf that is locally constant on strata of $N.$ A constructible complex is a complex such that all of its cohomology sheaves are constructible.

The derived category of bounded constructible complexes of sheaves on $N$ is the full subcategory $D^b\left(N\right)$ of $D^b\left(\mathrm{Sh}\left(N\right)\right)$ consisting of constructible complexes. Full means that the morphisms in $D^b\left(N\right)$ are the same as those in $D^b\left(\mathrm{Sh}\left(N\right)\right).$

The shift functor $\left[i\right]:D^b\left(N\right)\to D^b\left(N\right)$ is the functor that shifts all complexes by $i.$

The Verdier duality functor ${\phantom{A}}^{\vee }:D^b\left(N\right)\to D^b\left(N\right)$ is defined by requiring $${\mathrm{Hom}}_{D^b\left(N\right)}(A_1,A_2\left[i\right])={\mathrm{Hom}}_{D^b\left(N\right)}({\Delta }^{*}(A_1⊠A_2^{\vee })[-i],{ℂ}_N\left[2{\dim }_{ℂ}N\right]),for\; all\; i\in \mathbb{Z},$$ where $\Delta :N\to N\times N$ is the diagonal map.

The Verdier duality functor satisfies the properties $${\left(A^{\vee }\right)}^{\vee }=A,{\left(A\left[i\right]\right)}^{\vee }=A^{\vee }[-i],and{\mathrm{Hom}}_{D^b\left(N\right)}(A_1,A_2)={\mathrm{Hom}}_{D^b\left(N\right)}(A_2^{\vee },A_1^{\vee }).$$ Define $$\begin{array}{rcll}{\mathrm{Ext}}_{D^b\left(X\right)}^k(A_1,A_2)& =& {\mathrm{Hom}}_{D^b\left(X\right)}(A_1,A_2\left[k\right]),& \\ H^k\left(A\right)& =& H^k(X,A)={\mathrm{Hom}}_{D^b\left(X\right)}({ℂ}_X,A\left[k\right]),& the\mathit{\text{hypercohomology}}of\; A\in D^b\left(N\right),\\ H^k\left(A\right)& =& {\mathrm{Hom}}_{D^b\left(N\right)}({ℂ}_N,{ℂ}_N\left[k\right]),& the\mathit{\text{cohomology}}of\; N,\\ H_k\left(N\right)& =& {\mathrm{Hom}}_{D^b\left(N\right)}({ℂ}_N,{\left({ℂ}_N\left[k\right]\right)}^{\vee }),& the\mathit{\text{Borel-Moore homology}}of\; N,\\ {𝔻}_X& =& {ℂ}_X^{\vee },& the\mathit{\text{dualizing complex,}}\end{array}$$ respectively. The Yoneda product $${\mathrm{Ext}}_{D^b\left(N\right)}^p(A_1,A_2)\times {\mathrm{Ext}}_{D^b\left(N\right)}^q(A_2,A_3)\to {\mathrm{Ext}}_{D^b\left(N\right)}^{p+q}(A_1,A_3)$$ is given by $${\mathrm{Hom}}_{D^b\left(N\right)}(A_1,A_2\left[p\right])\times {\mathrm{Hom}}_{D^b\left(N\right)}(A_2\left[p\right],A_3[p+q])\to {\mathrm{Hom}}_{D^b\left(N\right)}(A_1,A_3[p+q]),$$ using the canonical identification ${\mathrm{Hom}}_{D^b\left(N\right)}(A_2,A_3\left[q\right])\cong {\mathrm{Hom}}_{D^b\left(N\right)}(A_2\left[p\right],A_3[p+q]).$

If $f:X\to Y$ is a morphism define $$\begin{array}{rcl}{f}_{*}& =& derived\; functor\; of\; sheaf\; theoretic\; direct\; image,\\ f^{*}& =& derived\; functor\; of\; sheaf\; theoretic\; inverse\; image,\end{array}$$ $$f^{!}A={\left(f^{*}{A}^{\vee }\right)}^{\vee }, \; for\; A\in D^b\left(Y\right),and{f}_{!}A={\left(f_{*}{A}^{\vee }\right)}^{\vee }, \; for\; A\in D^b\left(X\right).$$ Then $$\begin{array}{rcl}{\mathrm{Hom}}_{D^b\left(X\right)}(f^{*}{A}_1,A_2)& =& {\mathrm{Hom}}_{D^b\left(Y\right)}(A_1,f_{*}{A}_2),and,\\ {\mathrm{Hom}}_{D^b\left(X\right)}(A_2,f^{!}{A}_1)& =& {\mathrm{Hom}}_{D^b\left(Y\right)}(f_{!}{A}_2,A_1).\end{array}$$ If $f:X\to Z$ and $g:Y\to Z$ define the base change formula as $$\begin{array}{c} X×ZY Y X Z π2 f π1 g \end{array}{g}^{!}{f}_{*}A={\left({\pi }_2\right)}_{*}{\pi }_1^{!}A,for\; A\in D^b\left(X\right),$$ where $X{\times }_{Z}Y=\{(x,y)\in X\times Y|f\left(x\right)=g\left(y\right)\}.$

The category of perverse sheaves on $X$ is a full subcategory of $D^b\left(X\right)$ which is abelian. The simple objects in the category of perverse sheaves are the intersection cohomology complexes $$I{C}_{\phi }indexed\; by\; pairs\phi =(𝕆,\chi ),$$ where $𝕆$ is a $G-$orbit on $X$ and $\chi$ is an irreducible local system on $X.$ By ???, the local systems $\chi$ on $𝕆$ can be identified with (some of the) representations of the component group $Z_G\left(x\right)/Z_G{\left(x\right)}^{°}$ where $x$ is a point in $𝕆.$ If $X$ is smooth the constant perverse sheaf ${𝒞}_X$ on $X$ is given by $${\phantom{|}{𝒞}_X|}_{X_i}={ℂ}_{X_i}\left[{\dim }_{ℂ}{X}_i\right],$$ on the irreducible components of $X.$ Since the intersection cohomology complexes $I{C}_{\phi }$ are the simple objects in the category of perverse sheaves, $${\mathrm{Ext}}_{D^b\left(N\right)}^0(I{C}_{\phi },I{C}_{\psi })=ℂ\cdot {\delta }_{\phi \psi }and{\mathrm{Ext}}_{D^b\left(N\right)}^k(I{C}_{\phi },I{C}_{\psi })=0,if\; k>0.$$

Dlab-Ringel algebras

Let $C$ and $D$ be rings, $$\begin{array}{ll}L,& a\; (C,D) \; bimodule,\\ R,& a\; (D,C) \; bimodule,\end{array}and\epsilon :L{\otimes }_{D}R\to C,$$ a $(C,C)$ bimodule homomorphism. Define an algebra $$A=C\oplus D\oplus L\oplus R\oplus R{\otimes }_{C}L$$ and product determined by the multiplication in $C$ and $D,$ the module structure of $R$ and $L$ and the additional relations $$cr=0,dl=0,rd=0,lc=0,and(r_1\otimes l_1)(r_2\otimes l_2)=r_1\otimes \epsilon (l_1\otimes r_2)l_2.$$ Let

1. $e_C$ be the image of the identity of $C$ in $A,$ and
2. $e_D$ be the image of the identity of $D$ in $A.$

Then, if $e=e_C$ then $$1=e_C+e_D,\begin{array}{rclrcl}C& =& e_{C}A{e}_C,& L& =& e_{C}A{e}_D,\\ R& =& e_{D}A{e}_C,& D\prime & =& e_{D}A{e}_D,\end{array}$$ so that $$A=\{\phantom{|}\left(\begin{array}{cc}c& l\\ r& d\prime \end{array}\right)|c\in C,l\in L,r\in R,d\prime \in D\prime \}$$ with matrix multiplication. Then

1. $e_{D}A{e}_D=D+R{\otimes }_{C}L$ is a subring of $A,$ and
2. $R{\otimes }_{C}L$ is an ideal in $e_{D}A{e}_D,$ and
3. $R{\otimes }_{C}L=e_{D}A{e}_{C}A{e}_D.$

Structure of $Z\left(\epsilon \right)$

Let $$\epsilon :L{\otimes }_{D}R\to Cbe\; a\; (C,C) \; bimodule\; homomorphism.$$ The left radical $L\left(\epsilon \right)$ and the right radical $R\left(\epsilon \right)$ of $\epsilon$ are defined by $$\begin{array}{rcl}L\left(\epsilon \right)& =& \{l\in L|\epsilon (l\otimes r)\in \mathrm{Rad}\left(C\right),for\; allr\in R\},\\ R\left(\epsilon \right)& =& \{r\in R|\epsilon (l\otimes r)\in \mathrm{Rad}\left(C\right),for\; alll\in L\}.\end{array}$$ The map $\epsilon$ is nondegenerate if $\mathrm{Rad}\left(C\right)=0, L\left(\epsilon \right)=0,$ and $R\left(\epsilon \right)=0.$ Let $$\begin{array}{rcl}\stackrel{\_}{C}& =& C/\mathrm{Rad}\left(C\right),\\ \stackrel{\_}{L}& =& L/L\left(\epsilon \right),\\ \stackrel{\_}{R}& =& R/R\left(\epsilon \right),\end{array}and\begin{array}{rrcl}\phi :& R{\otimes }_{C}L& \to & \stackrel{\_}{R}{\otimes }_{\stackrel{\_}{C}}\stackrel{\_}{L}\\ & \stackrel{\_}{r}\otimes \stackrel{\_}{l}& \mapsto & \stackrel{\_}{r\otimes l}.\end{array}$$ Then $\ker \phi$ is generated by $R{\otimes }_{C}L\left(\epsilon \right)$ and $R\left(\epsilon \right){\otimes }_{C}L,$ and we have that $\ker \phi \cdot R\subseteq R\left(\epsilon \right)$ and $L\cdot \ker \phi \subseteq L\left(\epsilon \right).$ Then $$I=\mathrm{Rad}\left(C\right)+L\left(\epsilon \right)+R\left(\epsilon \right)+\ker \phi is\; a\; nilpotent\; ideal\; of\; A\left(\epsilon \right),$$ and $$\frac{A\left(\epsilon \right)}{I}\cong A\left(\stackrel{\_}{\epsilon }\right)where\; the\; map\begin{array}{rrcl}\stackrel{\_}{\epsilon }:& \stackrel{\_}{L}{\otimes }_D\stackrel{\_}{R}& \to & \stackrel{\_}{C}\\ & l\otimes r& \mapsto & \stackrel{\_}{l}\otimes \stackrel{\_}{r}\end{array}$$ is a nondegenerate $(\stackrel{\_}{C},\stackrel{\_}{C})$ bimodule homomorphism.

If $\epsilon :L{\otimes }_{D}R\to C$ is nondegenerate and $R$ is a projective $C-$module then there is a $(D,C)$ bimodule isomorphism $$\begin{array}{rrcc}\tau :& R& \stackrel{\sim }{\to }& L^{*}\\ & r& \mapsto & \begin{array}{rrcl}{\lambda }_r:& L& \to & C\\ & l& \mapsto & \epsilon (l\otimes r)\end{array}\end{array}so\; that\epsilon =\mathrm{ev}\circ (\mathrm{id}\otimes \tau )$$ and $$A\left(\epsilon \right)\cong A\left({\mathrm{ev}}_L\right).$$

If $C,D,L,R$ are finite dimensional vector spaces over $𝔽$ and $D=𝔽$ then $$\epsilon ={\epsilon }_0\oplus {\mathrm{ev}}_P:(L_0\oplus P^{*}){\otimes }_D(R_0\oplus P)\to C,$$ with $P$ projective and $\mathrm{im}{\epsilon }_0\subseteq \mathrm{Rad}\left(C\right).$

If $\epsilon ={\epsilon }_0\oplus {\mathrm{ev}}_P$ with $P$ finitely generated and projective then $$\begin{array}{rcl}A\left(\epsilon \right)-mod& \stackrel{\sim }{\to }& A\left({\epsilon }_0\right)-mod\\ M& \mapsto & eM\end{array}wheree=1-\sum _i{p}_i\otimes {\alpha }_i.$$

If $\mathrm{im}\epsilon \subseteq \mathrm{Rad}\left(C\right)$ then $$\mathrm{Rad}\left(A\left({\epsilon }_0\right)\right)=I=\mathrm{Rad}\left(C\right)\oplus \mathrm{Rad}\left(D\right)\oplus L_0\oplus R_0\oplus R_0{\otimes }_C{L}_0$$ and $$\frac{A\left({\epsilon }_0\right)}{\mathrm{Rad}\left(A\left({\epsilon }_0\right)\right)}\cong \frac{C}{\mathrm{Rad}\left(C\right)}\oplus \frac{D}{\mathrm{Rad}\left(D\right)}.$$

The module category of $Z\left(\epsilon \right)$

Let $𝒞$ and $𝒟$ be categories $$F:𝒞\to 𝒟andG:𝒞\to 𝒟be\; functors,\; andF\stackrel{\epsilon }{\to }G,$$ a natural transformation. Define a category $𝒜$ with

1. Objects: $(M,V;\begin{array}{c} FM GM V εM m n \end{array}), \; where\; M\in 𝒞, V\in 𝒟, \; and\; m,n\in \mathrm{Mor}\left(𝒟\right),$
2. Morphisms: $(f,g)$ with $f\in \mathrm{Mor}\left(𝒞\right), g\in \mathrm{Mor}\left(𝒟\right)$ such that $$\begin{array}{c} FM GM V εM m n FM′ GM′ V′ εM′ m′ n′ \end{array}commutes.$$

A fundamental case is when $𝒟$ is the category of vector spaces over $𝔽.$

The equivalence between the category $𝒜$ and the module category of $Z\left(\epsilon \right)$ is given by letting $𝒞=C-mod$ and $𝒟=D-mod$ and $$\begin{array}{rrcl}F:& 𝒞& \to & 𝒟\\ & M& \mapsto & R{\otimes }_{C}M\end{array}and\begin{array}{rrcl}G:& 𝒞& \to & 𝒟\\ & M& \mapsto & {\mathrm{Hom}}_C(L,M)\end{array}$$ where the $D-$action on ${\mathrm{Hom}}_C(L,M)$ is given by $$\left(d\phi \right)\left(l\right)=\phi \left(ld\right),for\; d\in D, l\in L, \; and\; \phi \in {\mathrm{Hom}}_C(L,M).$$ Then let $\epsilon :F\to G$ be the natural transformation given by $$\begin{array}{rccc}\epsilon :& F& \to & G\\ & R{\otimes }_{C}M& \stackrel{{\epsilon }_M}{\to }& {\mathrm{Hom}}_C(L,M)\\ & r\otimes m& \mapsto & \begin{array}{rccc}\tau :& L& \to & M\\ & l& \mapsto & \epsilon (l\otimes r)m.\end{array}\end{array}$$ Then $$\begin{array}{rcl}𝒜& \stackrel{\sim }{\to }& A-mod\\ (X,Y,\rho ,\lambda )& \leftrightarrow & M\end{array}whereX=eM,Y=(1-e)M,$$ and the $L-$action and $R-$action on $M$ define $\rho$ and $\lambda$ via $$ly=\left(\lambda \left(y\right)\right)\left(l\right)andrx=\rho (r\otimes x),for\; l\in L, r\in R, x\in X, \; and\; y\in Y.$$ Note that $$lx=0 \; and\; ry=0,for\; l\in L, r\in R, x\in X, y\in Y,$$ and $$R{\otimes }_{C}X=\begin{array}{c} FX GX Y εX ρ λ \end{array}={\mathrm{Hom}}_C(L,X)$$ commutes.

Macpherson-Vilonen

Let $X$ be a Thom-Mather stratified space with a fixed stratification such that all strata have even codimension. Let $$P\left(X\right) \; be\; the\; category\; of\; perverse\; sheaves\; on$ X.$$$ Let $S$ be a closed stratum such that $S$ is contractible and let $$\iota :X-S\hookrightarrow X,$$ be the inclusion. Let $$j:L-K\hookrightarrow L,where\begin{array}{rcl}L& =& the\; link\; of\; S\\ \cup |& & \\ K& =& perverse\; link\; of\; S, \; a\; closed\; subset\; of\; L.\end{array}$$ Let $$\begin{array}{rrcl}F:& P(X-S)& \to & \left\{vector\; spaces\right\}\\ & P& \mapsto & {ℍ}^{-d-1}(K;P)\end{array}$$ and $$\begin{array}{rrcl}G:& P(X-S)& \to & \left\{vector\; spaces\right\}\\ & P& \mapsto & {ℍ}^{-d}(L,K;P)={ℍ}^{-d}(L,j_{!}P{|}_{L-K}).\end{array}$$ Let $𝒜$ be the corresponding category as in the previous section. Then the map $$\begin{array}{ccc}P\left(X\right)& \stackrel{\sim }{\to }& 𝒜\\ Q& \mapsto & \left({\phantom{|}Q|}_{X-S},\begin{array}{c} ℍ-d-1 (K,Q) ℍ-d (L,K;Q) ℍ-d (𝔻,K;Q) εX ρ λ \end{array}\right)\end{array}$$ is an equivalence of categories, where ${\phantom{|}Q|}_{X-S}={\iota }^{*}Q,$ and ${\epsilon }_Q$ is the coboundary homomorphism in the long exact sequence for the pair $L,K.$ What is $𝔻$????

Examples.

1. The flag variety.
2. The nilpotent cone.

Quasihereditary algebras

Let $𝔽$ be a field. A separable algebra over $𝔽$ is an algebra $A$ such that $$\frac{A}{\mathrm{Rad}\left(A\right)}\cong \underset{\lambda }{⨁}ϵ\hat{A}{M}_{d_{\lambda }}\left(𝔽\right).$$ Two algebras $A$ and $B$ are Morita equivalent if $\mathrm{Mod}-A$ is equivalent to $\mathrm{Mod}-B$ (Check this in Gelfand-Manin).

A ring $A$ is semiprimary if there is a nilpotent ideal $\mathrm{Rad}\left(A\right)$ such that $A/\mathrm{Rad}\left(A\right)$ is semisimple artinian. Note: If $A$ is finite dimensional then $A$ is semiprimary.

A hereditary ring is a ring $A$ such that every submodule of a projective module is projective.

A hereditary ideal is an ideal $J$ such that

1. $J$ is projective as a right $A-$module,
2. $J^2=J,$ and
3. $J\mathrm{Rad}\left(A\right)J=0.$

Note: $J^2=J$ if and only if there is an idempotent $e\in A$ with $J=AeA.$

A quasihereditary ring is a semiprimary ring $A$ with a chain of ideals $$0=J_0\subseteq J_1\subseteq \cdots \subseteq J_m=Asuch\; that\frac{J_l}{J_{l-1}} \; is\; a\; hereditary\; ideal\; of\; \frac{A}{J_{l-1}}$$ for each $1\le l\le m-1.$

Let $A$ be a quasihereditary algebra $$0=J_0\subseteq J_1\subseteq \cdots \subseteq J_m=A.$$ Let $e$ be an idempotent in $A$ such that $$J_{m-1}=AeAandeA(1-e)\subseteq \mathrm{Rad}\left(A\right).$$ Let $$C=eAeandD=\frac{A}{AeA}=\frac{A}{J_{m-1}}$$ and $${}_{C}L_D=eA(1-e)and{}_{C}R_D=(1-e)Ae$$ and let $$\begin{array}{rrcl}\epsilon :& L{\otimes }_{D}R& \to & C\\ & l\otimes r& \mapsto & lr.\end{array}$$ Assume $D$ is a separable $k-$algebra. Then

1. $D+(1-e)AeA(1-e)=(1-e)A(1-e),$
2. $A=C\left(\epsilon \right),$
3. $C$ is quasihereditary with heredity chain $$0=I_0\subseteq \cdots \subseteq I_{m-1}=C,where{I}_l=e{J}_{l}e.$$

Highest weight categories

Let $A$ be a finite dimensional algebra and let $\hat{A}$ be an index set for $$\begin{array}{lll}& L\left(\lambda \right),& the\; simple\; A-modules.\\ & & \\ Let& P\left(\lambda \right)& be\; the\; projective\; cover\; of\; L\left(\lambda \right), \; and\\ & I\left(\lambda \right)& the\; injective\; hull\; of\; L\left(\lambda \right).\\ & & \\ Let& \le & be\; a\; partial\; order\; on\; \hat{A}.\\ & & \\ Let& \nabla \left(\lambda \right)& be\; the\; largest\; subobject\; of\; I\left(\lambda \right) \; with\; composition\; factors\; L\left(\mu \right) \; with\; \mu \le \lambda ,\\ & \Delta \left(\lambda \right)& be\; the\; largest\; quotient\; of\; P\left(\lambda \right) \; with\; composition\; factors\; L\left(\mu \right) \; with\; \mu \le \lambda .\end{array}$$ Then $𝒜=A-mod$ is a highest weight category if $P(\lambda )$ has a filtration $$0=P{\left(\lambda \right)}^{\left(m\right)}\subseteq \cdots \subseteq P{\left(\lambda \right)}^{\left(1\right)}\subseteq P\left(\lambda \right),$$ with $$\frac{P\left(\lambda \right)}{P{\left(\lambda \right)}^{\left(1\right)}}\cong \Delta \left(\lambda \right)and\frac{P{\left(\lambda \right)}^{\left(k\right)}}{P{\left(\lambda \right)}^{(k+1)}}\cong \Delta \left(\mu \right),with\; \mu <\lambda ,$$ for $1\le k\le m-1.$

Highest weight categories satisfy BGG-reciprocity, $$[I\left(\lambda \right):\nabla \left(\mu \right)]=[\Delta \left(\mu \right):L\left(\lambda \right)].$$

		Proof.
	Since $${\mathrm{Ext}}^1(\Delta \left(\lambda \right),\nabla \left(\mu \right))=0and\mathrm{Hom}(\Delta \left(\lambda \right),\nabla \left(\mu \right))=\{\begin{array}{ll}\mathrm{End}\left(L\left(\mu \right)\right),& if\; \lambda =\mu ,\\ 0,& if\; \lambda \ne \mu ,\end{array}\phantom{\}}$$ if follows that $$\mathrm{Hom}(\Delta \left(\lambda \right),M)=\left(number\; of\nabla \left(\lambda \right)in\; a\nabla -filtration\; ofM\right).$$ Thus $$[I\left(\mu \right):\nabla \left(\lambda \right)]=\frac{\dim \left(\mathrm{Hom}(\Delta \left(\lambda \right),I\left(\mu \right))\right)}{\dim \left(\mathrm{End}\left(L\left(\lambda \right)\right)\right)}=[\Delta \left(\lambda \right):L\left(\mu \right)].$$ How does this proof compare to the proof for convolution algebras in Chriss and Ginzburg? $\square$

Examples of highest weight categories.

1. $G=G\left(\stackrel{\_}{𝔽}\right), 𝒜$ the category of finite dimensional rational $G-$modules, and $\nabla \left(\lambda \right)=H^0(G/B,{ℒ}_{\lambda }),$
2. $𝒜$ the category $𝒪,$ and $\nabla \left(\lambda \right)=M{\left(\lambda \right)}^{\vee }.$

Vogan, Irreducible characters of semisimple Lie groups II; The Kazhdan-Lusztig conjectures $$P_{yw}=\sum _i{q}^i\dim \left({\mathrm{Ext}}^{l\left(w\right)-l\left(y\right)-2i}(M_y,L_w)\right),for\; y\le w.$$

Let $A$ be a finite dimensional algebra and let $𝒜=A-$mod. Then $𝒜$ is a highest weight category if and only if $A$ is a quasihereditary algebra.

		Proof.
	⇒) Assume $𝒜$ is a highest weight category. Let $\lambda$ be a maximal weight and let $$P\left(\lambda \right)=A{e}_{\lambda }andJA{e}_{\lambda }A.$$ Then $J$ is projective as a left $A-$module, $${\mathrm{Hom}}_A(J,A/J)=0,J\cdot \mathrm{Rad}\left(J\right)=0.$$ So $J$ is a hereditary ideal. Finally, $(A/J)-$mod is a highest weight category with $\widehat{(A/J)}=\hat{A}-\left\{\lambda \right\}.$ ⇐) Assume $A$ is a quasihereditary algebra, $$0=J_0\subseteq J_1\subseteq \cdots \subseteq J_m=A.$$ Define $\lambda <\mu$ if $$L\left(\lambda \right) \; appears\; in\; \frac{J_i/J_{i-1}}{\mathrm{Rad}(J_i/J_{i-1})}andL\left(\mu \right) \; appears\; in\; \frac{J_j/J_{j-1}}{\mathrm{Rad}(J_j/J_{j-1})},$$ with $i<j.$ Suppose $i$ is (the unique integer) such that $L\left(\lambda \right)$ appears in $(J_i/J_{i-1})/\mathrm{Rad}(J_i/J_{i-1})$ and let $$\Delta \left(\lambda \right) \; be\; the\; projective\; cover\; of\; L\left(\lambda \right), \; as\; an\; A/J_{i-1} \; module.$$ Then $L\left(\lambda \right)$ is the simple head of $A\left(\lambda \right)$ and, since $J_{i-1}\cdot \mathrm{Rad}(A/J_{i-1})\cdot J_{i-1}=0,$ all other composition factors of $A\left(\lambda \right)$ are lower. If $L\left(\lambda \right)$ is a simple $A-$module then there is an idempotent $e_{\lambda }\in A$ such that $P\left(\lambda \right)=A{e}_{\lambda }$ ($e_{\lambda }$ is a minimal idempotent). Then $$0=J_0{e}_{\lambda }\subseteq J_1{e}_{\lambda }\subseteq \cdots \subseteq J_m{e}_{\lambda }=A{e}_{\lambda }=P\left(\lambda \right)$$ is a good filtration of $P\left(\lambda \right).$ $\square$

Duals and Projectives

Let $L$ be a $C-$module and let $$Z={\mathrm{End}}_C\left(L\right)$$ so that $L$ is a $(C,Z)$ bimodule. The dual module to $L$ is the $(Z,C)$ bimodule $$L^{*}={\mathrm{Hom}}_C(L,C).$$ The evaluation map is the $(C,C)$ bimodule homomorphism $$\begin{array}{rrcl}\mathrm{ev}:& L{\otimes }_Z{L}^{*}& \to & C\\ & l\otimes \lambda & \mapsto & \lambda \left(l\right)\end{array}$$ and the centralizer map is the $(Z,Z)$ bimodule homomorphism $$\begin{array}{rrcc}\xi :& L^{*}{\otimes }_{C}L& \to & Z\\ & \lambda \otimes l& \mapsto & \begin{array}{rrcl}{z}_{\lambda ,l}:& L& \to & L\\ & m& \mapsto & \lambda \left(m\right)l.\end{array}\end{array}$$ Recall that [Bou, Alg. II §4.2 Cor.]

1. $L$ is a projective $C-$module if and only if $1\in \mathrm{im}\xi ,$
2. If $L$ is a projective $C-$module then $\xi$ is injective,
3. If $L$ is a finitely generated projective $C-$module then $\xi$ is bijective,
4. If $L$ is a finitely generated free module then $${\xi }^{-1}\left(z\right)=\sum _i{b}_i^{*}\otimes z\left(b_i\right),$$ where $\{b_1,...,b_d\}$ is a basis of $L$ and $\{b_1^{*},...,b_d^{*}\}$ is the dual basis in $M^{*}.$

Statement (a) says that $L$ is projective if and only if there exist $b_i\in L$ and $b_i^{*}\in L^{*}$ such that $$if\; l\in L \; then\; l=\sum _i{b}_i^{*}\left(l\right)b_i,so\; that\; \xi \left(\sum _i{b}_i^{*}\otimes b_i\right)=1.$$

Cellular algebras

A cellular algebra is an algebra $A$ with $$\begin{array}{ccc}a\; basis& \{\phantom{|}{a}_{ST}^{\lambda }|\lambda \in \hat{A},S,T\in {\hat{A}}^{\lambda }\}& \\ an\; involutive\; antihomomorphism& {\phantom{A}}^{*}:A\to A,& and\\ a\; partial\; order& \le on\hat{A}& \end{array}$$ such that

1. ${\left(a_{ST}^{\lambda }\right)}^{*}=a_{TS}^{\lambda },$
2. If $A(<\lambda )=\mathrm{span-}\{\phantom{|}{a}_{ST}^{\mu }|\mu <\lambda \}$ then $$a{a}_{ST}^{\lambda }=\sum _{Q\in {\hat{A}}^{\lambda }}{A}^{\lambda }{\left(a\right)}_{QT}{a}_{QT}^{\lambda }\mathrm{mod}A(<\lambda ),for\; alla\in A.$$

Applying the involution ${\phantom{A}}^{*}$ to (b) and using (a) gives that $$a_{TS}^{\lambda }{a}^{*}=\sum _{Q\in {\hat{A}}^{\lambda }}{A}^{\lambda }{\left(a\right)}_{QS}{a}_{TQ}^{\lambda }\mathrm{mod}A(<\lambda ),for\; alla\in A.$$

The concept of a cellular algebra is not really the "right" one. The "right" one comes from the structure of a convolution algebra whenever the decomposition theorem holds [CG, 8.6.9].

Peter Webb's generalized reciprocity

Let $𝔬$ be a complete discrete valuation ring, $k=𝔬/𝔭$ its residue field and let ${}_{𝔬}A$ be an algebra over $𝔬,$ $$\begin{array}{ccccc}k& \leftarrow & 𝔬& \to & 𝕂\\ {}_{k}A& \leftarrow & {}_{𝔬}A& \to & {}_{𝕂}A\end{array}$$

The diagram $$\begin{array}{c} K0 (A𝕂 ) G0 (A𝕂 ) K0 (Ak ) G0 (Ak ) cA cλ e=Dt D \end{array}$$ commutes, where $e$ is defined by lifting idempotents. Furthermore $e=D^t.$

		Proof.
	If $P$ is projective, $U$ any finitely generated module, put $$⟨P,U⟩=\dim \mathrm{Hom}(P,U).$$ This is well defined on $K_0\left({}_{𝕂}A\right)\times G_0\left({}_{𝕂}A\right)$ and $K_0\left({}_{k}A\right)\times G_0\left({}_{k}A\right).$ Then $$e\left(P\right)=𝕂{\otimes }_{𝔬}\hat{P},wherek{\otimes }_{𝔬}\hat{P}=P.$$ $\square$

Let $U_0$ be a $𝔬-$form of $U$ and let $P$ be projective. Then ${\mathrm{Hom}}_{{}_{𝔬}A}(\hat{P},U_0)$ is an $𝔬-$lattice in ${\mathrm{Hom}}_{{}_{𝕂}A}(K{\otimes }_{𝔬}\hat{P},U)$ and the morphism ${\mathrm{Hom}}_{{}_{𝔬}A}(\hat{P},U_0)\to {\mathrm{Hom}}_{{}_{k}A}(P,U_0/𝔭U_0)$ is reduction mod $𝔭.$

$\dim {\mathrm{Hom}}_{{}_{𝕂}A}(K{\otimes }_{𝔬}\hat{P},U)={\mathrm{rank}}_{𝔬}{\mathrm{Hom}}_{{}_{𝔬}A}(\hat{P},U_0)=\dim {\mathrm{Hom}}_{{}_{k}A}(P,U_0/𝔭U_0).$

This shows that $e$ and $D$ are the transpose of each other with respect to the forms. The diagram commutes from the definition of $e.$

The Cartan matrix $$C_{{}_{k}A}=D{C}_{{}_{𝕂}A}{D}^t$$ where $C_{{}_{𝕂}A}$ is the Cartan matrix of $A.$

If ${}_{𝕂}A$ is semisimple then $C_{{}_{𝕂}A}=\mathrm{id}.$

The category $𝒪$

Let $U$ be a $\mathbb{Z}$ graded algebra with

1. $U_0$ reductive,
2. $U_i$ finite dimensional,
3. $U$ semisimple under the adjoint action.

The category $𝒪$ is the category of $\mathbb{Z}$ graded $U$ modules which are

1. $U_0$ semisimple, and
2. $U_{\ge 0}$ locally finite.

Define $${𝒪}_{\le n}=\{M\in 𝒪|M_i=0ifi>n\}.$$

Standard and costandard modules

Let ${\hat{U}}_0$ be an index set for the finite dimensional $\mathbb{Z}-$graded $U_0$ modules. The Verma module or standard module and the coVerma module or costandard module are given by $$\Delta \left(\lambda \right)=U{\otimes }_{U_{\ge 0}}{U}_0^{\lambda }and\nabla \left(\lambda \right)={\mathrm{Hom}}_{U_{\le 0}}(U,U_0^{\lambda }),for\; \lambda \in {\hat{U}}_0.$$ Let $M\in 𝒪.$ A $\Delta -$flag for $M$ is an increasing filtration $$0=M^{\left(0\right)}\subseteq M^{\left(1\right)}\subseteq M^{\left(2\right)}\subseteq \cdots such\; thatM=\bigcup _i{M}^{\left(i\right)},$$ and, for each $i\ge 1,$ $M^{\left(i\right)}/M^{(i-1)}\cong \Delta \left({\lambda }^{\left(i\right)}\right)$ for some ${\lambda }^{\left(i\right)}\in {\hat{U}}_0.$

1. $\Delta \left(\lambda \right)$ has simple head $L\left(\lambda \right).$
2. $\nabla \left(\lambda \right)$ has simple socle $L\left(\lambda \right).$
3. $\left\{L\left(\lambda \right)\right|\lambda \in {\hat{U}}_0\}$ are the simple objects in $𝒪.$

1. $\Delta \left(\lambda \right)$ is the projective cover of $L\left(\lambda \right)$ in ${𝒪}_{\le \left|\lambda \right|}.$
2. $\nabla \left(\lambda \right)$ is the injective hull of $L\left(\lambda \right)$ in ${𝒪}_{\le \left|\lambda \right|}.$
3. ${\mathrm{Hom}}_{𝒪}(\Delta \left(\mu \right),\nabla \left(\lambda \right))=\{\begin{array}{ll}0,& if\lambda \ne \mu ,\\ ℂ,& if\lambda =\mu .\end{array}\phantom{\}}$
4. ${\mathrm{Ext}}_{𝒪}^1(\Delta \left(\mu \right),\nabla \left(\lambda \right))=0.$

Projectives

If $K=⨁K_i$ is a $\mathbb{Z}$ graded $U_{\ge 0}$ module define $${\tau }_{\ge n}=\frac{K}{\underset{i>n}{⨁}{K}_i}=\underset{i\le n}{⨁}{K}_i.$$ If $\lambda \in {\hat{U}}_0$ define $$Q=U{\otimes }_{U_{\ge 0}}{\tau }_{\le n}\left(U_{\ge 0}{\otimes }_{U_0}{U}_0^{\lambda }\right),$$ and let $P_{\le n}\left(\lambda \right)$ be an indecomposable summand of $Q$ which has $L\left(\lambda \right)$ as a quotient and define $K_{m,n},$ for $m\ge n$ by the exact sequence $$0\to K_{m,n}\to P_{\le m}\left(\lambda \right)\to P_{\le n}\left(\lambda \right)\to 0.$$

1. $Q$ is projective and $Q\to L\left(\lambda \right)\to 0.$
2. $P_{\le n}\left(\lambda \right)$ is a projective cover of $L\left(\lambda \right)$ in ${𝒪}_{\le n}.$
3. $P_{\le n}\left(\lambda \right)$ has a $\Delta$ flag.
4. $K_{m,n}$ has a $\Delta$ flag.
5. $L\left(\lambda \right)$ has a projective cover in $P\left(\lambda \right)$ in $𝒪$ if and only if the projective system $P_{\le m}\left(\lambda \right)\to P_{\le n}\left(\lambda \right)$ stabilizes, in which case $$P\left(\lambda \right)\cong P_{\le n}\left(\lambda \right),for\; n\gg 0.$$

Injective module

Tilting modules

Let $\lambda \in {\hat{U}}_0.$ A tilting module is a module that has both a $\Delta$ flag and a $\nabla$ flag.

There is a unique indecomposable tilting module $T\left(\lambda \right)$ of highest weight $\lambda .$

Blocks

Define $\ge$ on ${\hat{U}}_0$ by $$\mu \ge \lambda if[\Delta \left(\mu \right):L\left(\lambda \right)]\ne 0or[\nabla \left(\mu \right):L\left(\lambda \right)]\ne 0.$$ Let $\left[\lambda \right]$ denote the equivalence class of $\lambda$ with respect to the equivalence relation generated by $\ge .$ Define $${𝒪}^{\left[\lambda \right]}=\{M\in 𝒪|if[M:L\left(\mu \right)]\ne 0then\mu \in \left[\lambda \right]\},$$ and for $M\in 𝒪$ define $$M^{\left[\lambda \right]}=U(\sum \mathrm{im}\left(P_{\le n}\left(\lambda \right)\stackrel{\phi }{\to }M\right)),$$ the submodule of $M$ generated by the images of morphisms $\phi :P_{\le n}\left(\lambda \right)\to M.$

$$𝒪=⨁{𝒪}^{\left[\lambda \right]}andM=⨁M^{\left[\lambda \right]},for\; M\in 𝒪.$$

Multiplicities

Let $𝒜$ be an abelian category and let $L$ be simple. Let $m\in 𝒜.$ The multiplicity of $L$ in $M$ is $$[M:L]=\underset{F}{\sup }\mathrm{Card}\left\{i\right|F_{i}M/F_{i+1}M\cong L\},$$ where the supremum is over all (finite) filtrations of $M.$ $$If\; 0\to M\prime \to M\to M\prime \prime \to 0 \; is\; exact\; then\; [M:L]=[M\prime :L]+[M\prime \prime :L].$$

If $M\in {𝒪}_{\le n}$ and $N\in 𝒪$ with a $\Delta$-flag then $$[M:L\left(\lambda \right)]=\dim {\mathrm{Hom}}_{𝒪}(P_{\le n}\left(\lambda \right),M)and[N:\Delta \left(\mu \right)]=\dim \mathrm{Hom}(N,\nabla \left(\mu \right)).$$ Thus $$[P_{\le n}\left(\lambda \right):\Delta \left(\mu \right)]=[\nabla \left(\mu \right):L\left(\lambda \right)],for\lambda ,\mu \in {\hat{U}}_{0}andn\ge \max \{\left|\lambda \right|,\left|\mu \right|\}.$$

The category ${𝒪}_{\mathrm{int}}$

Start with $U=U_{<0}{U}_0{U}_{>0}.$ $${𝒪}_{\mathrm{int}}=\{M\in U-\mathrm{mod}|M\in U_0^{\mathrm{ss}},M\in U_{>0}^{\mathrm{nilp}},M\in U_{<0}^{\mathrm{nilp}}\}.$$

Finite dimensional algebras

Let $A$ be a finite dimensional algebra.

1. The projective indecomposables are $Ae$ for a minimal idempotent $e$ of $A.$
2. The simples $L\left(\lambda \right)$ are the simple heads of the projective indecomposables $P\left(\lambda \right).$
3. The blocks are $Az$ for a minimal central idempotent $z$ of $A.$
4. The Cartan matrix is $$[P\left(\lambda \right):L\left(\mu \right)].$$

Temperley-Lieb algebras

Computation of the ${\epsilon }_{\sigma }^{\gamma }$

The quantum dimensions of the finite dimensional simple $U_q{\mathrm{𝔰𝔩}}_2$ modules are $${\dim }_q\left(L(k-2j)\right)=\prod _{b\in (k-j)}\frac{[2+c\left(b\right)]}{\left[h\left(b\right)\right]}=\prod _{i=0}^{k-j-1}\frac{[2+i]}{[k-j-\mathrm{i}]}=[k-j+1]=\left[\dim \left(L(k-2j)\right)\right].$$ As a $(U_q{\mathrm{𝔰𝔩}}_2,T{L}_k\left(n\right))$ bimodule $$V^{\otimes k}\cong \sum _{j=0}^{\lfloor \frac{k}{2}\rfloor }L(k-2j)\otimes T{L}_k^{(k-j,j)}.$$ Thus $${\mathrm{tr}}_q\left(b\right)=\sum _{j=0}^{\lfloor \frac{k}{2}\rfloor }{\dim }_q\left(L(k-2j)\right){\chi }_{T{L}_k}^{(k-j,j)}\left(b\right),for\; b\in T{L}_k\left(n\right),$$ and $${\mathrm{tr}}_q\left(a_{\genfrac{}{}{0ex}{}{ZX}{\sigma }}\right)={\delta }_{ZX}{\dim }_q\left(L\left(\sigma \right)\right)and{\mathrm{tr}}_q\left(b_{\genfrac{}{}{0ex}{}{ZX}{\genfrac{}{}{0ex}{}{\sigma \mu }{\gamma }}}\right)={\delta }_{\genfrac{}{}{0ex}{}{ZX}{\sigma \mu }}{\dim }_q\left(L\left(\gamma \right)\right).$$ If $a\in A$ then $$\begin{array}{rcl}{\mathrm{tr}}_q\left(a{e}_k\right)& =& {\mathrm{tr}}_q\left(a\right){\mathrm{tr}}_q\left(e_k\right)=n{\mathrm{tr}}_q\left(a\right),and\; so\\ {\mathrm{tr}}_q\left({\epsilon }_1\left(b\right)\right)& =& \frac{1}{n}{\mathrm{tr}}_q\left({\epsilon }_1\left(b\right)e_k\right)=\frac{1}{n}{\mathrm{tr}}_q\left(e_{k}b{e}_k\right)=\frac{1}{n}{\mathrm{tr}}_q\left(b{e}_k^2\right)\\ & =& {\mathrm{tr}}_q\left(b{e}_k\right)={\mathrm{tr}}_q\left(b(T_k-q)\right)=(z-q){\mathrm{tr}}_q\left(b\right)=(\frac{q^2}{n}-q){\mathrm{tr}}_q\left(b\right)=\frac{1}{n}{\mathrm{tr}}_q\left(b\right).\end{array}$$ So $$\frac{1}{n}{\dim }_q\left(L\left(\gamma \right)\right)=\frac{1}{n}{\mathrm{tr}}_q\left(b_{\genfrac{}{}{0ex}{}{ZX}{\genfrac{}{}{0ex}{}{\sigma \mu }{\gamma }}}\right)={\mathrm{tr}}_q\left({\epsilon }_1\left(b_{\genfrac{}{}{0ex}{}{ZX}{\genfrac{}{}{0ex}{}{\sigma \mu }{\gamma }}}\right)\right)={\mathrm{tr}}_q\left({\epsilon }_{\sigma }^{\gamma }{a}_{\genfrac{}{}{0ex}{}{ZX}{\sigma }}\right)={\epsilon }_{\sigma }^{\gamma }{\dim }_q\left(L\left(\sigma \right)\right).$$ Thus $${\epsilon }_{\sigma }^{\gamma }=\frac{\left[\dim \left(L\left(\gamma \right)\right)\right]}{n\cdot \left[\dim \left(L\sigma )\right)\right]}.$$

Generators and relations

The Temperley-Lieb algebra, $ℂT_k\left(n\right),$ is the algebra over $ℂ$ given by generators $E_1,E_2,...,E_{k-1}$ and relations $$\begin{array}{rcll}{E}_i{E}_j& =& E_j{E}_i,& if|i-j|>1,\\ E_i{E}_{i\pm 1}{E}_i& =& E_i,& and\\ E_i^2& =& n{E}_i.& \end{array}$$

If $$\left[2\right]=q+q^{-1}=nthenq=\frac{1}{2}(n+\sqrt{n^2-4}),q^{-1}=\frac{1}{2}(n-\sqrt{n^2-4}),$$ since $q^2-nq+1=0.$ Then $$\left[k\right]=\frac{q^k-q^{-k}}{q-q^{-1}}=\frac{1}{2^{k-1}}\sum _{m=1}^{\frac{k+1}{2}}\left(\genfrac{}{}{0ex}{}{k}{2m-1}\right)n^{k-2m+1}{(n^2-4)}^{m-1}.$$ The problem with this expression is that it is not clear that $\left[k\right]$ is a polynomial in $n$ with integer coefficients (which alternate in sign?).

The Iwahori-Hecke algebra $H_k\left(q\right)$ is the algebra over $ℂ$ with generators $T_1,T_2,...,T_{k-1}$ and relations $$\begin{array}{rcll}{T}_i{T}_j& =& T_j{T}_i,& if|i-j|>1,\\ T_i{T}_{i\pm 1}{T}_i& =& T_{i+1}{T}_i{T}_{i+1},& if2\le i\le k-1,\\ T_i^2& =& (q-q^{-1})T_i+1.& \end{array}$$ There is a surjective algebra homomorphism $$\phi :H_k\left(q\right)\to T_k\left(n\right)given\; by\phi \left(T_i\right)=E_i-q^{-1}and\phi (q+q^{-1})=n,$$ with $$\ker \phi =⟨T_i{T}_{i+1}{T}_i+T_i{T}_{i+1}+T_{i+1}{T}_i+T_i+T_{i+1}+1⟩.$$ Composing with the surjective homomorphism $$\begin{array}{rcl}{\tilde{H}}_k\left(q\right)& \to & H_k\left(q\right)\\ X^{{\epsilon }_i}& \mapsto & T_{i-1}\cdots T_2{T}_1^1{T}_2\cdots T_{i-1}\\ T_i& \mapsto & T_i\end{array}$$

Murphy elements

Let us write $$T_i=E_i-q^{-1},so\; that{X}^{{\epsilon }_1}=1,and{X}^{{\epsilon }_i}=T_{i-1}{X}^{{\epsilon }_{i-1}}{T}_{i-1}$$ in the Temperley-Lieb algebra. Then define $m_1,...,m_k$ by $$m_1=0and(q-q^{-1})m_j=q^{i-2}{X}^{{\epsilon }_i}-q^{i-4}{X}^{{\epsilon }_{i-1}}for\; 2\le i\le k.$$ Solving for $X^{{\epsilon }_i}$ in terms of the $m_i$ gives $$X^{{\epsilon }_i}=(q-q^{-1})(q^{-(i-2)}{m}_i+q^{-(i-2+1)}{m}_{i-1}+\cdots +q^{-(2i-4)}{m}_2)+q^{-2(i-1)},$$ from which one obtains $$q^{(k-2)}(X^{{\epsilon }_1}+X^{{\epsilon }_2}+\cdots +X^{{\epsilon }_k})-q\left[k\right]=(q-q^{-1})(m_k+\left[2\right]m_{k-1}+\cdots +[k-1]m_2).$$ Using the definition of $X^{{\epsilon }_i}$ and substituting for $X^{{\epsilon }_{i-1}}$ in terms of the $m_i$ gives $$\begin{array}{rcl}(q-q^{-1})m_i& =& q^{i-2}{X}^{{\epsilon }_i}-q^{i-4}{X}^{{\epsilon }_{i-1}}\\ & =& q^{i-2}(E_{i-1}-q^{-1})X^{{\epsilon }_{i-1}}(E_{i-1}-q^{-1})-q^{i-4}{X}^{{\epsilon }_{i-1}}\\ & =& q^{i-2}{E}_{i-1}{X}^{{\epsilon }_{i-1}}{E}_{i-1}-q^{i-3}(E_{i-1}{X}^{{\epsilon }_{i-1}}+X^{{\epsilon }_{i-1}}{E}_{i-1})\\ & =& q^{i-2}{E}_{i-1}((q-q^{-1})(q^{-(i-3)}{m}_i+q^{-(i-3+1)}{m}_{i-1}+\cdots +q^{-(2i-6)}{m}_2)+q^{-2(i-2)})E_{i-1}\\ & & -q^{i-3}{E}_{i-1}((q-q^{-1})(q^{-(i-3)}{m}_i+q^{-(i-3+1)}{m}_{i-1}+\cdots +q^{-(2i-6)}{m}_2)+q^{-2(i-2)})\\ & & -q^{i-3}((q-q^{-1})(q^{-(i-3)}{m}_i+q^{-(i-3+1)}{m}_{i-1}+\cdots +q^{-(2i-6)}{m}_2)+q^{-2(i-2)})E_{i-1}\\ & =& q^{i-2}(q-q^{-1})q^{-(i-3)}{E}_{i-1}{m}_{i-1}{E}_{i-1}-q^{i-3}(q-q^{-1})q^{-(i-3)}(E_{i-1}{m}_{i-1}+m_{i-1}{E}_{i-1})\\ & & +q^{i-2}(q+q^{-1})E_{i-1}((q-q^{-1})(q^{-(i-3)}{m}_i+q^{-(i-3+1)}{m}_{i-1}+\cdots +q^{-(2i-6)}{m}_2)+q^{-2(i-2)})\\ & & -2q^{i-3}{E}_{i-1}((q-q^{-1})(q^{-(i-3)}{m}_i+q^{-(i-3+1)}{m}_{i-1}+\cdots +q^{-(2i-6)}{m}_2)+q^{-2(i-2)})\\ & =& q^{i-2}(q-q^{-1})q^{-(i-3)}{E}_{i-1}{m}_{i-1}{E}_{i-1}-q^{i-3}(q-q^{-1})q^{-(i-3)}(E_{i-1}{m}_{i-1}+m_{i-1}{E}_{i-1})\\ & & +q^{i-2}(q-q^{-1})E_{i-1}((q-q^{-1})(q^{-(i-3)}{m}_i+q^{-(i-3+1)}{m}_{i-1}+\cdots +q^{-(2i-6)}{m}_2)+q^{-2(i-2)})\end{array}$$ since $E_{i-1}$ commutes with $m_2,m_3,...,m_{i-1}.$ Thus $$\begin{array}{rcl}{m}_i& =& q^{-(i-2)}{E}_{i-1}+q{E}_{i-1}{m}_{i-1}{E}_{i-1}-(E_{i-1}{m}_{i-1}+m_{i-1}{E}_{i-1})\\ & & +(q-q^{-1})(m_{i-2}+q^{-1}{m}_{i-3}+q^{-2}{m}_{i-4}+\cdots +q^{-(i-4)}{m}_2)E_{i-1}.\end{array}$$ It seems to me that this formula provides the easiest way to compute $m_i$ in terms of the $E$s. I would not be too worried about the coefficients of $E_1{E}_4$ and $E_2{E}_4$ in $m_4$ looking strange. One expects diagrams that are equal to their own flip to act a bit differently in $m_k.$ Note also that $$\left[3\right]-1=\frac{\left[4\right]}{\left[2\right]}and\left[3\right]+1={\left[2\right]}^2,$$ so these are pretty nice $q-$versions of 2. Let's have a look at $m_6$ and see if we can get an induction going. It might help to categorize the terms according to what their flip is to see where the next level is coming from.

For $n$ such that $ℂT_k\left(n\right)$ is semisimple, the simple $T_k\left(n\right)$ are indexed by partitions in the set $${\hat{T}}_k=\{\lambda ⊢k|\lambda has\; at\; most\; two\; columns\}.$$ The irreducible $ℂT_k\left(n\right)$ modules have seminormal basis $$\left\{v_T\right|Tis\; a\; standard\; tableau\; of\; shape\lambda \}$$ and $$X^{{\epsilon }_i}{v}_T=q^{2c\left(T\left(i\right)\right)}{v}_T.$$ Since $c\left(T\left(i\right)\right)=c\left(T(i-1)\right)-1$ if the boxes $T\left(i\right)$ and $T(i-1)$ are in the same column and $c\left(T\left(i\right)\right)+c\left(T(i-1)\right)=3-i$ if the boxes $T\left(i\right)$ and $T(i-1)$ are in different columns it follows that $$m_i{v}_T=\frac{q^{i-2}{q}^{2c\left(T\left(i\right)\right)}-q^{i-4}{q}^{2c\left(T(i-1)\right)}}{q-q^{-1}}=c_T\left(i\right)v_T,$$ where $$c_T\left(i\right)=\{\begin{array}{ll}0,& ifT\left(i\right)andT(i-1)are\; in\; the\; same\; column,\\ [i-2+2c\left(T\left(i\right)\right)],& ifT\left(i\right)andT(i-1)are\; in\; different\; columns.\end{array}\phantom{\}}$$

Now we want to define pseudomatrix units in $ℂT_k\left(n\right)$ according to the left and right eigenspaces of the $m_i.$ Let $$p_{ST}\in L_S\cap R_T,$$ normalized so that the coefficients are in $\mathbb{Z}\left[n\right]$ with greatest common divisor 1. Then $$\begin{array}{rcl}{p}_{ST}{p}_{UV}& =& {\gamma }_T{\delta }_{UV}{p}_{SV},\\ p_{ST}& =& \sum _{S^{+},T^{+}}{c}_{S^{+}{T}^{+}}{p}_{S^{+}{T}^{+}},\\ p_{ST}{e}_k{p}_{UV}& =& {\beta }_{T^{-}}{\delta }_{T^{-}{U}^{-}}{p}_{p^{+}{T}^{+}},\\ e_{k+1}{p}_{ST}{e}_{k+1}& =& {\epsilon }_{S^{+}{T}^{+}}{\delta }_{S\left(k\right)T\left(k\right)}{p}_{ST}{e}_{k+1}.\end{array}$$

Examples

Let's start with generic $n.$ Here $$e_{ST}=\frac{\left[a\right]}{\left[b\right]}{e}_{S^{-}{U}^{-}}{E}_{k-1}{e}_{U^{-}{T}^{-}}.$$ Then $$E_k=\sum \frac{\left[b\right]}{\left[a\right]}{e}_{ST}and{m}_k=\sum {\mu }_k\left(S\right)e_{SS},$$ where the first sum is over all pairs $(S,T)$ such that $S=T$ or $S$ and $T$ differ at the $k-1$st level.

In $ℂT_2\left(n\right)$ let $$\left(\begin{array}{cc}{p}_{12,12}& \\ & p_{\genfrac{}{}{0ex}{}{1}{2}\genfrac{}{}{0ex}{}{,}{}\genfrac{}{}{0ex}{}{1}{2}}\end{array}\right)=\left(\begin{array}{cc}\left[2\right]e_{12,12}& \\ & \left[2\right]e_{\genfrac{}{}{0ex}{}{1}{2}\genfrac{}{}{0ex}{}{,}{}\genfrac{}{}{0ex}{}{1}{2}}\end{array}\right)$$ In $ℂT_3\left(n\right)$ let $$\left(\begin{array}{ccc}{p}_{\begin{array}{cc}1& 2\\ 3& \end{array}\genfrac{}{}{0ex}{}{,}{}\begin{array}{cc}1& 2\\ 3& \end{array}}& p_{\begin{array}{cc}1& 2\\ 3& \end{array}\genfrac{}{}{0ex}{}{,}{}\begin{array}{cc}1& 3\\ 2& \end{array}}& \\ p_{\begin{array}{cc}1& 3\\ 2& \end{array}\genfrac{}{}{0ex}{}{,}{}\begin{array}{cc}1& 2\\ 3& \end{array}}& p_{\begin{array}{cc}1& 3\\ 2& \end{array}\genfrac{}{}{0ex}{}{,}{}\begin{array}{cc}1& 3\\ 2& \end{array}}& \\ & & p_{\begin{array}{c}1\\ 2\\ 3\end{array}\genfrac{}{}{0ex}{}{,}{}\begin{array}{c}1\\ 2\\ 3\end{array}}\end{array}\right)=\left(\begin{array}{ccc}\left[2\right]e_{\begin{array}{cc}1& 2\\ 3& \end{array}\genfrac{}{}{0ex}{}{,}{}\begin{array}{cc}1& 2\\ 3& \end{array}}& \left[3\right]\left[2\right]e_{\begin{array}{cc}1& 2\\ 3& \end{array}\genfrac{}{}{0ex}{}{,}{}\begin{array}{cc}1& 3\\ 2& \end{array}}& \\ \left[2\right]e_{\begin{array}{cc}1& 3\\ 2& \end{array}\genfrac{}{}{0ex}{}{,}{}\begin{array}{cc}1& 2\\ 3& \end{array}}& \left[3\right]\left[2\right]e_{\begin{array}{cc}1& 3\\ 2& \end{array}\genfrac{}{}{0ex}{}{,}{}\begin{array}{cc}1& 3\\ 2& \end{array}}& \\ & & \left[3\right]e_{\begin{array}{c}1\\ 2\\ 3\end{array}\genfrac{}{}{0ex}{}{,}{}\begin{array}{c}1\\ 2\\ 3\end{array}}\end{array}\right)$$ In $ℂT_4\left(n\right)$ let $$\left(\begin{array}{cccccc}{p}_{\begin{array}{cc}1& 2\\ 3& 4\end{array}\genfrac{}{}{0ex}{}{,}{}\begin{array}{cc}1& 2\\ 3& 4\end{array}}& p_{\begin{array}{cc}1& 2\\ 3& 4\end{array}\genfrac{}{}{0ex}{}{,}{}\begin{array}{cc}1& 3\\ 2& 4\end{array}}& & & & \\ p_{\begin{array}{cc}1& 3\\ 2& 4\end{array}\genfrac{}{}{0ex}{}{,}{}\begin{array}{cc}1& 2\\ 3& 4\end{array}}& p_{\begin{array}{cc}1& 3\\ 2& 4\end{array}\genfrac{}{}{0ex}{}{,}{}\begin{array}{cc}1& 3\\ 2& 4\end{array}}& & & & \\ & & p_{\begin{array}{cc}1& 2\\ 3& \\ 3& \end{array}\genfrac{}{}{0ex}{}{,}{}\begin{array}{cc}1& 2\\ 3& \\ 3& \end{array}}& p_{\begin{array}{cc}1& 2\\ 3& \\ 3& \end{array}\genfrac{}{}{0ex}{}{,}{}\begin{array}{cc}1& 3\\ 2& \\ 2& \end{array}}& p_{\begin{array}{cc}1& 2\\ 3& \\ 3& \end{array}\genfrac{}{}{0ex}{}{,}{}\begin{array}{cc}1& 4\\ 2& \\ 2& \end{array}}& \\ & & p_{\begin{array}{cc}1& 3\\ 2& \\ 2& \end{array}\genfrac{}{}{0ex}{}{,}{}\begin{array}{cc}1& 2\\ 3& \\ 3& \end{array}}& p_{\begin{array}{cc}1& 3\\ 2& \\ 2& \end{array}\genfrac{}{}{0ex}{}{,}{}\begin{array}{cc}1& 3\\ 2& \\ 2& \end{array}}& p_{\begin{array}{cc}1& 3\\ 2& \\ 2& \end{array}\genfrac{}{}{0ex}{}{,}{}\begin{array}{cc}1& 4\\ 2& \\ 2& \end{array}}& \\ & & p_{\begin{array}{cc}1& 4\\ 2& \\ 2& \end{array}\genfrac{}{}{0ex}{}{,}{}\begin{array}{cc}1& 2\\ 3& \\ 3& \end{array}}& p_{\begin{array}{cc}1& 4\\ 2& \\ 2& \end{array}\genfrac{}{}{0ex}{}{,}{}\begin{array}{cc}1& 3\\ 2& \\ 2& \end{array}}& p_{\begin{array}{cc}1& 4\\ 2& \\ 2& \end{array}\genfrac{}{}{0ex}{}{,}{}\begin{array}{cc}1& 4\\ 2& \\ 2& \end{array}}& \\ & & & & & \\ & & & & & \\ & & & & & p_{\begin{array}{c}1\\ 2\\ 3\\ 4\end{array}\genfrac{}{}{0ex}{}{,}{}\begin{array}{c}1\\ 2\\ 3\\ 4\end{array}}\end{array}\right)=\left(\begin{array}{cccccc}{\left[2\right]}^2{e}_{\begin{array}{cc}1& 2\\ 3& 4\end{array}\genfrac{}{}{0ex}{}{,}{}\begin{array}{cc}1& 2\\ 3& 4\end{array}}& {\left[2\right]}^2{e}_{\begin{array}{cc}1& 2\\ 3& 4\end{array}\genfrac{}{}{0ex}{}{,}{}\begin{array}{cc}1& 3\\ 2& 4\end{array}}& & & & \\ {\left[2\right]}^2{e}_{\begin{array}{cc}1& 3\\ 2& 4\end{array}\genfrac{}{}{0ex}{}{,}{}\begin{array}{cc}1& 2\\ 3& 4\end{array}}& {\left[2\right]}^2{e}_{\begin{array}{cc}1& 3\\ 2& 4\end{array}\genfrac{}{}{0ex}{}{,}{}\begin{array}{cc}1& 3\\ 2& 4\end{array}}& & & & \\ & & \left[3\right]{\left[2\right]}^2{e}_{\begin{array}{cc}1& 2\\ 3& \\ 3& \end{array}\genfrac{}{}{0ex}{}{,}{}\begin{array}{cc}1& 2\\ 3& \\ 3& \end{array}}& \left[3\right]{\left[2\right]}^2{e}_{\begin{array}{cc}1& 2\\ 3& \\ 3& \end{array}\genfrac{}{}{0ex}{}{,}{}\begin{array}{cc}1& 3\\ 2& \\ 2& \end{array}}& \left[3\right]{\left[2\right]}^2{e}_{\begin{array}{cc}1& 2\\ 3& \\ 3& \end{array}\genfrac{}{}{0ex}{}{,}{}\begin{array}{cc}1& 4\\ 2& \\ 2& \end{array}}& \\ & & \left[3\right]{\left[2\right]}^2{e}_{\begin{array}{cc}1& 3\\ 2& \\ 2& \end{array}\genfrac{}{}{0ex}{}{,}{}\begin{array}{cc}1& 2\\ 3& \\ 3& \end{array}}& \left[3\right]{\left[2\right]}^2{e}_{\begin{array}{cc}1& 3\\ 2& \\ 2& \end{array}\genfrac{}{}{0ex}{}{,}{}\begin{array}{cc}1& 3\\ 2& \\ 2& \end{array}}& \left[3\right]{\left[2\right]}^2{e}_{\begin{array}{cc}1& 3\\ 2& \\ 2& \end{array}\genfrac{}{}{0ex}{}{,}{}\begin{array}{cc}1& 4\\ 2& \\ 2& \end{array}}& \\ & & \left[3\right]{\left[2\right]}^2{e}_{\begin{array}{cc}1& 4\\ 2& \\ 2& \end{array}\genfrac{}{}{0ex}{}{,}{}\begin{array}{cc}1& 2\\ 3& \\ 3& \end{array}}& \left[3\right]{\left[2\right]}^2{e}_{\begin{array}{cc}1& 4\\ 2& \\ 2& \end{array}\genfrac{}{}{0ex}{}{,}{}\begin{array}{cc}1& 3\\ 2& \\ 2& \end{array}}& \left[3\right]{\left[2\right]}^2{e}_{\begin{array}{cc}1& 4\\ 2& \\ 2& \end{array}\genfrac{}{}{0ex}{}{,}{}\begin{array}{cc}1& 4\\ 2& \\ 2& \end{array}}& \\ & & & & & \\ & & & & & \\ & & & & & \left[4\right]\left[3\right]\left[2\right]e_{\begin{array}{c}1\\ 2\\ 3\\ 4\end{array}\genfrac{}{}{0ex}{}{,}{}\begin{array}{c}1\\ 2\\ 3\\ 4\end{array}}\end{array}\right)$$

The special value $n=\pm \sqrt{2},$ i.e. when $\left[4\right]=0.$
Then $$p_{\begin{array}{cc}1& 4\\ 2& \\ 2& \end{array}\genfrac{}{}{0ex}{}{,}{}\begin{array}{ccc}1& 4& 3\\ 2& & \end{array}}=p_{\begin{array}{c}1\\ 2\\ 3\\ 3\end{array}\genfrac{}{}{0ex}{}{,}{}\begin{array}{c}1\\ 2\\ 3\\ 3\end{array}}and\; we\; let{p}_{\begin{array}{cc}1& 4\\ 2& \\ 2& \end{array}\genfrac{}{}{0ex}{}{,}{}\begin{array}{cc}1& 4\\ 2& \\ 2& \end{array}}^{\left(2\right)}=1-e_{\begin{array}{cc}1& 2\\ 3& \end{array}\genfrac{}{}{0ex}{}{,}{}\begin{array}{cc}1& 2\\ 3& \end{array}}.$$ In this basis $$\mathrm{Rad}\left(ℂT_4\right)=\mathrm{span}\left(\begin{array}{cccccc}0& 0& & & & \\ 0& 0& & & & \\ & & 0& 0& 1& \\ & & 0& 0& 1& \\ & & 1& 1& 1& \\ & & & & & 0\end{array}\right)and{\mathrm{Rad}}^2\left(ℂT_4\right)=\mathrm{span}\left(\begin{array}{cccccc}0& 0& & & & \\ 0& 0& & & & \\ & & 0& 0& 0& \\ & & 0& 0& 0& \\ & & 0& 0& 1& \\ & & & & & 0\end{array}\right)$$ $$ℂT_1=\left\{\left(a\right)\right\}=\left\{\left(\begin{array}{cc}a& \\ & a\end{array}\right)\right\}=\left\{\left(\begin{array}{ccc}a& 0& \\ 0& a& \\ & & a\end{array}\right)\right\}$$ and $$ℂT_2=\left\{\left(\begin{array}{cc}{a}_1& \\ & a_2\end{array}\right)\right\}=\left\{\left(\begin{array}{ccc}{a}_1& 0& \\ 0& a_2& \\ & & a_2\end{array}\right)\right\}$$ and $$ℂT_3=\left\{\left(\begin{array}{ccc}{a}_{11}& a_{12}& \\ a_{21}& a_{22}& \\ & & a_3\end{array}\right)\right\}=\left\{\left(\begin{array}{cccccc}{a}_{11}& a_{12}& & & & \\ a_{21}& a_{22}& & & & \\ & & a_{11}& a_{12}& 0& \\ & & a_{21}& a_{22}& 0& \\ & & 0& 0& a_3& \\ & & & & & a_3\end{array}\right)\right\}.$$

The special value $n=\pm 1,$ i.e. when $\left[3\right]=0.$
Then $$p_{\begin{array}{cc}1& 3\\ 2& \end{array}\genfrac{}{}{0ex}{}{,}{}\begin{array}{cc}1& 3\\ 2& \end{array}}=p_{\begin{array}{c}1\\ 2\\ 3\end{array}\genfrac{}{}{0ex}{}{,}{}\begin{array}{c}1\\ 2\\ 3\end{array}}and\; we\; let{p}_{\begin{array}{cc}1& 3\\ 2& \end{array}\genfrac{}{}{0ex}{}{,}{}\begin{array}{cc}1& 3\\ 2& \end{array}}^{\left(2\right)}=1-e_{\begin{array}{cc}1& 2\\ 3& \end{array}\genfrac{}{}{0ex}{}{,}{}\begin{array}{cc}1& 2\\ 3& \end{array}}.$$ In this basis $$\mathrm{Rad}\left(ℂT_3\right)=\mathrm{span}\left(\begin{array}{ccc}0& 1& \\ 1& 1& \\ & & 0\end{array}\right)and{\mathrm{Rad}}^2\left(ℂT_3\right)=\mathrm{span}\left(\begin{array}{ccc}0& 0& \\ 1& 0& \\ & & 0\end{array}\right).$$ Then $$E_1=\left(\begin{array}{ccc}1& 0& \\ 0& 0& \\ & & 0\end{array}\right),E_2=\left(\begin{array}{ccc}1& 1& \\ 1& 1& \\ & & 0\end{array}\right),1=\left(\begin{array}{ccc}1& 0& \\ 0& 0& \\ & & 1\end{array}\right),$$ $$m_2=\left(\begin{array}{ccc}1& 0& \\ 0& 0& \\ & & 0\end{array}\right),m_3=\left(\begin{array}{ccc}-1& 0& \\ 0& 1& \\ & & 0\end{array}\right).$$ $$ℂT_1=\left\{\left(\begin{array}{cc}a& \\ & a\end{array}\right)\right\}=\left\{\left(\begin{array}{ccc}a& 0& \\ 0& 0& \\ & & a\end{array}\right)\right\}$$ and $$ℂT_2=\left\{\left(\begin{array}{cc}{a}_1& \\ & a_2\end{array}\right)\right\}=\left\{\left(\begin{array}{ccc}{a}_1& 0& \\ 0& 0& \\ & & a_2\end{array}\right)\right\}.$$

The special value $n=0,$ i.e. when $\left[2\right]=0.$
Then $$p_{12,12}=p_{\genfrac{}{}{0ex}{}{1}{2}\genfrac{}{}{0ex}{}{,}{}\genfrac{}{}{0ex}{}{1}{2}}and\; we\; let{p}_{12,12}^{\left(2\right)}=1.$$ In the basis $$\left(\begin{array}{cc}{p}_{12,12}& \\ & p_{12,12}^{\left(2\right)}\end{array}\right)$$ $$e_1=\left(\begin{array}{cc}1& \\ & 0\end{array}\right),m_2=\left(\begin{array}{cc}1& \\ & 0\end{array}\right),and\mathrm{Rad}\left(ℂT_2\right)=\mathrm{span}\left(\begin{array}{cc}1& \\ & 0\end{array}\right).$$ With respect to this basis there is a new matrix $$ℰ=\left(\begin{array}{cc}{e}_2{p}_{12,12}^2{e}_2& e_2{p}_{12,12}{p}_{12,12}^{\left(2\right)}{e}_2\\ e_2{p}_{12,12}^{\left(2\right)}{p}_{12,12}{e}_2& e_2{\left(p_{12,12}^{\left(2\right)}\right)}^2{e}_2\end{array}\right)=\left(\begin{array}{cc}n& 1\\ 1& n\end{array}\right)=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),$$ which is not diagonal. In $ℂT_3$ the basis elements %%%%%%%%%%%%%%%% $$\left(\begin{array}{ccc}{p}_{\begin{array}{cc}1& 2\\ 3& \end{array}\genfrac{}{}{0ex}{}{,}{}\begin{array}{cc}1& 2\\ 3& \end{array}}^{\left(2\right)}& p_{\begin{array}{cc}1& 2\\ 3& \end{array}\genfrac{}{}{0ex}{}{,}{}\begin{array}{cc}1& 3\\ 2& \end{array}}& \\ p_{\begin{array}{cc}1& 3\\ 2& \end{array}\genfrac{}{}{0ex}{}{,}{}\begin{array}{cc}1& 2\\ 3& \end{array}}^{\left(2\right)}& p_{\begin{array}{cc}1& 3\\ 2& \end{array}\genfrac{}{}{0ex}{}{,}{}\begin{array}{cc}1& 3\\ 2& \end{array}}^{\left(2\right)}& \\ & & p_{\begin{array}{c}1\\ 2\\ 3\end{array}\genfrac{}{}{0ex}{}{,}{}\begin{array}{c}1\\ 2\\ 3\end{array}}\end{array}\right)=\left(\begin{array}{ccc}{p}_{12,12}{e}_2{p}_{12,12}^{\left(2\right)}& p_{12,12}^{\left(2\right)}{e}_2{p}_{12,12}^{\left(2\right)}& \\ p_{12,12}{e}_2{p}_{12,12}& p_{12,12}{e}_2{p}_{12,12}^{\left(2\right)}& \\ & & 1-p_{\begin{array}{cc}1& 2\\ 3& \end{array}\genfrac{}{}{0ex}{}{,}{}\begin{array}{cc}1& 2\\ 3& \end{array}}^{\left(2\right)}-p_{\begin{array}{cc}1& 3\\ 2& \end{array}\genfrac{}{}{0ex}{}{,}{}\begin{array}{cc}1& 3\\ 2& \end{array}}^{\left(2\right)}\end{array}\right)$$ form a set of matrix units. In this basis $$E_1=\left(\begin{array}{ccc}0& 1& \\ 0& 0& \\ & & 0\end{array}\right),E_2=\left(\begin{array}{ccc}0& 0& \\ 1& 0& \\ & & 0\end{array}\right),1=\left(\begin{array}{ccc}1& 0& \\ 0& 1& \\ & & 1\end{array}\right),$$ $$m_2=\left(\begin{array}{ccc}0& 1& \\ 0& 0& \\ & & 0\end{array}\right),m_3=\left(\begin{array}{ccc}-1& 0& \\ 0& -1& \\ & & 0\end{array}\right),$$ $$ℂT_1=\left\{\left(a\right)\right\}=\left\{\left(\begin{array}{cc}0& \\ & a\end{array}\right)\right\}=\left\{\left(\begin{array}{ccc}a& 0& \\ 0& a& \\ & & a\end{array}\right)\right\}$$ and $$ℂT_2=\left\{\left(\begin{array}{cc}{a}_2& \\ & a_1\end{array}\right)\right\}=\left\{\left(\begin{array}{ccc}{a}_1& a_2& \\ 0& a_1& \\ & & a_1\end{array}\right)\right\}.$$

References

[GW1] F. Goodman and H. Wenzl, The Temperley-Lieb algebra at roots of unity, Pacific J. Math. 161 (1993), no. 2, 307-334.

[GL1] J. Graham and G. Lehrer, Diagram algebras, Hecke algebras and decomposition numbers at roots of unity, Ann. Sci. École Norm. Sup. (4) 36 (2003), no. 4, 479-524.

[GL2] J. Graham and G. Lehrer, The two-step nilpotent representations of the extended affine Hecke algebra of type A, Compositio Math. 133 (2002), no. 2, 173-197.

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