Geometric Representation Theory

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

Last update: 26 June 2014

Abstract.

Introduction

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

Geometric constructions

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

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

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

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

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

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

Assume that $X$ is a $G\text{-variety}$ with a finite number of orbits such that the decomposition $X=\bigsqcup 𝕆$ into $G\text{-orbits}$ is an algebraic stratification of $X\text{.}$ A constructible sheaf is a sheaf that is locally constant on strata of $X\text{.}$ 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 $X$ is the full subcategory $D^b\left(X\right)$ of $D^b\left(\text{Sh}\left(X\right)\right)$ consisting of constructible complexes. Full means that the morphisms in $D^b\left(X\right)$ are the same as those in $D^b\left(\text{Sh}\left(X\right)\right)\text{.}$ Let $\left[i\right]:D^b\left(X\right)\to D^b\left(X\right)$ denote the functor that shifts all complexes by $i\text{.}$ The Verdier duality functor ${}^{\vee }:D^b\left(X\right)\to D^b\left(X\right)$ is defined by requiring $${\text{Hom}}_{D^b\left(X\right)}(A_1,A_2\left[i\right])={\text{Hom}}_{D^b\left(X\right)}({\mathrm{\Delta }}^{*}(A_1⊠A_2^{\vee })[-i],{ℂ}_X\left[2{\text{dim}}_{ℂ}\hspace{0.17em}X\right]),\text{for all}\hspace{0.17em}i\in \mathbb{Z}\text{, where}$$ $\mathrm{\Delta }:X\to X\times X$ is the diagonal map. The Verdier duality functor satisfies the properties $${\left(A^{\vee }\right)}^{\vee }=A,{\left(A\left[i\right]\right)}^{\vee }=A^{\vee }[-i],\text{and}{\text{Hom}}_{D^b\left(X\right)}(A_1,A_2)={\text{Hom}}_{D^b\left(X\right)}(A_2^{\vee },A_1^{\vee })\text{.}$$ If $f:X\to Y$ is a morphism define $$\begin{array}{ccc}{f}_{*}& =& \text{derived functor of sheaf theoretic direct image,}\\ f^{*}& =& \text{derived functor of sheaf theoretic inverse image,}\end{array}$$ $$f^{!}A={\left(f^{*}{A}^{\vee }\right)}^{\vee },\hspace{0.17em}\text{for}\hspace{0.17em}A\in D^b\left(Y\right),\text{and}{f}_{!}A={\left(f_{*}{A}^{\vee }\right)}^{\vee },\hspace{0.17em}\text{for}\hspace{0.17em}A\in D^b\left(X\right)\text{.}$$ Then $$\begin{array}{ccc}{\text{Hom}}_{D^b\left(X\right)}(f^{*}{A}_1,A_2)& =& {\text{Hom}}_{D^b\left(Y\right)}(A_1,f_{*}{A}_2),\text{and}\\ {\text{Hom}}_{D^b\left(X\right)}(A_2,f^{!}{A}_1)& =& {\text{Hom}}_{D^b\left(Y\right)}(f_{!}{A}_2,A_1)\text{.}\end{array}$$ If $f:X\to Z$ and $g:Y\to Z$ define the base change formula is $$\begin{array}{ccc}X{\times }_{Z}Y& \stackrel{{\pi }_2}{⟶}& Y\\ \downarrow {\pi }_1& & \downarrow g\\ X& \stackrel{f}{⟶}& Z\end{array}{g}^{!}{f}_{*}A={\left({\pi }_2\right)}_{*}{\pi }_1^{!}A,\text{for}\hspace{0.17em}A\in D^b\left(X\right),$$ where $X{\times }_{Z}Y=\{(x,y)\in X\times Y\hspace{0.17em}|\hspace{0.17em}f\left(x\right)=g\left(y\right)\}\text{.}$

The set ${\text{Ext}}_{D^b\left(X\right)}^k(A_1,A_2),$ the hypercohomology $H^{*}\left(A\right)=H^{*}(X,A)$ of a complex $A\in D^b\left(X\right),$ the cohomology $H^{*}\left(X\right)$ of $X\text{,}$ the Borel-Moore homology $H_{*}\left(X\right)$ of $X\text{,}$ and the dualizing complex ${𝔻}_X$ are defined by $$\begin{array}{ccc}{\text{Ext}}_{D^b\left(X\right)}^k(A_1,A_2)& =& {\text{Hom}}_{D^b\left(X\right)}(A_1,A_2\left[k\right]),\\ H^k\left(A\right)=H^k(X,A)& =& {\text{Hom}}_{D^b\left(X\right)}({ℂ}_X,A\left[k\right]),\\ H^k\left(X\right)& =& {\text{Hom}}_{D^b\left(X\right)}({ℂ}_X,{ℂ}_X\left[k\right]),\\ H_k\left(X\right)& =& {\text{Hom}}_{D^b\left(X\right)}({ℂ}_X,{\left({ℂ}_X\left[k\right]\right)}^{\vee }),\\ {𝔻}_X& =& {ℂ}_X^{\vee },\end{array}$$ respectively. The Yoneda product $${\text{Ext}}_{D^b\left(N\right)}^p(A_1,A_2)\times {\text{Ext}}_{D^b\left(N\right)}^q(A_2,A_3)⟶{\text{Ext}}_{D^b\left(N\right)}^{p+q}(A_1,A_3)$$ is given by $${\text{Hom}}_{D^b\left(N\right)}(A_1,A_2\left[p\right])\times {\text{Hom}}_{D^b\left(N\right)}(A_2\left[p\right],A_3[p+q])⟶{\text{Hom}}_{D^b\left(N\right)}(A_1,A_3[p+q]),$$ using the canonical identification ${\text{Hom}}_{D^b\left(N\right)}(A_2,A_3\left[q\right])\cong {\text{Hom}}_{D^b\left(N\right)}(A_2\left[p\right],A_3[p+q])\text{.}$

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 $$I{C}_{\varphi }\text{indexed by pairs}\varphi =(𝕆,\chi ),$$ where $𝕆$ is a $G\text{-orbit}$ on $X$ and $\chi$ is an irreducible local system on $X\text{.}$ By ???, the local systems $\chi$ on $𝕆$ can be identified with (some of the) representations of the component group $Z_G\left(x\right)/Z_G{\left(x\right)}^{\circ }$ where $x$ is a point in $𝕆\text{.}$ If $X$ is smooth the constant perverse sheaf ${𝒞}_X$ on $X$ is given by $${𝒞}_X{|}_{X_i}={ℂ}_{X_i}\left[{\text{dim}}_{ℂ}\hspace{0.17em}{X}_i\right],$$ on the irreducible components of $X\text{.}$ Since the intersection cohomology complexes $I{C}_{\varphi }$ are the simple objects of the category of perverse sheaves, $${\text{Ext}}_{D^b\left(N\right)}^0(I{C}_{\varphi },I{C}_{\psi })=ℂ·{\delta }_{\varphi \psi }\text{and}{\text{Ext}}_{D^b\left(N\right)}^k(I{C}_{\varphi },I{C}_{\psi })=0,\text{if}\hspace{0.17em}k>0\text{.}$$

The decomposition theorem

Let $M$ be a smooth $G\text{-variety}$ and let $N$ be a $G\text{-variety}$ with finitely many $G\text{-orbits}$ such that the orbit decomposition is an algebraic stratification of $N\text{,}$ $$N=\bigsqcup 𝕆,\text{and}\mu :M⟶N$$ is a $G\text{-equivariant}$ projective morphism. Let ${𝒞}_M$ be the constant perverse sheaf on $M\text{.}$ The decomposition theorem [CGi1433132, 8.4.12] says that $${\mu }_{*}{𝒞}_M=\underset{\underset{\varphi =(𝕆,\chi )}{i\in \mathbb{Z}}}{⨁}{L}_{\varphi }\left(i\right)\otimes I{C}_{\varphi }\left[i\right]\doteq ⨁L_{\varphi }\otimes I{C}_{\varphi },$$ where ${\mu }_{*}$ is the derived functor of sheaf theoretic direct image, $\varphi$ runs over the indexes of the intersection cohomology complexes $I{C}_{\varphi },$ $L_{\varphi }\left(i\right)$ are finite dimensional vector spaces, $\doteq$ indicates an equality up to shifts in the derived category, and $$L_{\varphi }=\underset{i\in \mathbb{Z}}{⨁}{L}_{\varphi }\left(i\right)\text{.}$$

The convolution algebra

Let $\mu :M\to N$ be a proper map. Using the Yoneda product to define a multiplication, the convolution algebra is the associative algebra defined by $$A={\text{Ext}}_{D^b\left(N\right)}^{*}({\mu }_{*}{𝒞}_M,{\mu }_{*}{𝒞}_M)=\sum _{k\in \mathbb{Z}}{\text{Ext}}_{D^b\left(N\right)}^k({\mu }_{*}{𝒞}_M,{\mu }_{*}{𝒞}_M)\text{.}$$ The decomposition theorem gives $$\begin{array}{ccc}A& =& {\displaystyle \underset{k\in \mathbb{Z}}{⨁}\underset{i,j,\varphi ,\psi }{⨁}{\text{Hom}}_{ℂ}(L_{\varphi }\left(i\right),L_{\psi }\left(j\right))\otimes {\text{Ext}}_{D^b\left(N\right)}^k(I{C}_{\varphi }\left[i\right],I{C}_{\psi }\left[j\right])}\\ & =& {\displaystyle \underset{k\in \mathbb{Z}}{⨁}\underset{i,j,\varphi ,\psi }{⨁}{\text{Hom}}_{ℂ}(L_{\varphi }\left(i\right),L_{\psi }\left(j\right))\otimes {\text{Ext}}_{D^b\left(N\right)}^{k+j-i}(I{C}_{\varphi },I{C}_{\psi })}\\ & \doteq & {\displaystyle \underset{k\ge 0}{⨁}\underset{\varphi ,\psi }{⨁}{\text{Hom}}_{ℂ}(L_{\varphi },L_{\psi })\otimes {\text{Ext}}_{D^b\left(N\right)}^k(I{C}_{\varphi },I{C}_{\psi })}\\ & =& {\displaystyle \left(\underset{\varphi }{⨁}{\text{End}}_{ℂ}\left(L_{\varphi }\right)\right)⨁(\underset{k>0}{⨁}\underset{\varphi ,\psi }{⨁}{\text{Hom}}_{ℂ}(L_{\varphi },L_{\psi })\otimes {\text{Ext}}_{D^b\left(N\right)}^k(I{C}_{\varphi },I{C}_{\psi }))\text{.}}\end{array}$$ Since the product only raises degrees on the ${\text{Ext}}_{D^b\left(N\right)}^k$ it follows that the right hand sum is an ideal of $A\text{.}$ This ideal is nilpotent since the complexes are bounded. The quotient of $A$ by this ideal is the left hand sum, which is a semisimple algebra since it is a direct sum of matrix algebras. It follows that $$\begin{array}{cc}\begin{array}{c}{\displaystyle \text{rad}\hspace{0.17em}A=(\underset{k>0}{⨁}\underset{\varphi ,\psi }{⨁}{\text{Hom}}_{ℂ}(L_{\varphi },L_{\psi })\otimes {\text{Ext}}_{D^b\left(N\right)}^k(I{C}_{\varphi },I{C}_{\psi })),\text{and}}\\ \text{the nonzero}{L}_{\varphi }\text{are the simple}\hspace{0.17em}A\text{-modules.}\end{array}& \text{(2.2)}\end{array}$$

Projective modules and reciprocity

Let $e_{\psi }$ be a minimal idempotent in $\underset{\varphi }{⨁}\text{End}\left(L_{\varphi }\right)\text{.}$ Then $$P_{\psi }=A{e}_{\psi }=L_{\psi }⨁(\underset{\underset{\varphi }{k>0}}{⨁}{L}_{\varphi }\otimes {\text{Ext}}_{D^b\left(N\right)}^k(I{C}_{\varphi },I{C}_{\psi }))$$ is the projective cover of the simple $A\text{-module}$ $L_{\psi }\text{.}$ Define an $A\text{-module}$ filtration $P_{\psi }\supseteq F^1\supseteq F^2\supseteq \cdots$ by $$F^m=\underset{\underset{\varphi }{k\ge m}}{⨁}{L}_{\varphi }\otimes {\text{Ext}}_{D^b\left(N\right)}^k(I{C}_{\varphi },I{C}_{\psi })\text{.}$$ Then $$L_{\psi }=P_{\psi }/F^1\text{and}{\text{gr}}_F{P}_{\psi }\text{is a semisimple}\hspace{0.17em}A\text{-module.}$$ Thus the multiplicity of the simple $A\text{-module}$ $L_{\varphi }$ in a composition series of $P_{\psi }$ is $$[P_{\psi }:L_{\varphi }]=\text{dim}\hspace{0.17em}{\text{Ext}}^{*}(I{C}_{𝕆\chi },I{C}_{𝕆\prime ,\chi \prime })=\sum _{k\ge 0}\text{dim}\hspace{0.17em}{\text{Ext}}_{D^b\left(N\right)}^k(I{C}_{\varphi },I{C}_{\psi })\text{.}$$

Standard and costandard modules

Let $\psi =(𝕆,\chi )$ and let $x\in 𝕆\text{.}$ The local system $\chi$ on $𝕆$ can be identified with a representation of the component group $Z_G\left(x\right)/Z_G{\left(x\right)}^{\circ },$ where $Z_G\left(x\right)$ is the centralizer of $x$ in $G$ and $Z_G{\left(x\right)}^{\circ }$ is the connected component of the identity in $Z_G\left(x\right)\text{.}$ Let $i_x:\left\{x\right\}\hookrightarrow N$ be the injection. Then $i_x^{!}{\mu }_{*}{𝒞}_M$ is the stalk of ${\mu }_{*}{𝒞}_M$ at $x$ and the Yoneda product makes $$\begin{array}{ccc}{H}^{*}\left(i_x^{!}{𝒞}_M\right)& =& {\text{Hom}}_{D^b\left(\left\{x\right\}\right)}(ℂ,i_x^{!}{\mu }_{*}{𝒞}_M[*])={\text{Hom}}_{D^b\left(N\right)}({\left(i_x\right)}_{!}ℂ[-*],{\mu }_{*}{𝒞}_M),\\ H^{*}\left(i_x^{*}{𝒞}_M\right)& =& H^{*}(\left\{x\right\},i_x^{*}{\mu }_{*}{𝒞}_M)={\text{Hom}}_{D^b\left(\left\{x\right\}\right)}(𝔻,i_x^{!}{\mu }_{*}{𝒞}_M[*])={\text{Hom}}_{D^b\left(N\right)}({\left(i_x\right)}_{!}ℂ[-*],{\mu }_{*}{𝒞}_M),\end{array}$$ into a right $A\text{-module.}$ The action of an element $a\in {\text{Ext}}^k({\mu }_{*}{𝒞}_M,{\mu }_{*}{𝒞}_M)={\text{Hom}}_{D^b\left(N\right)}({\mu }_{*}{𝒞}_M,{\mu }_{*}{𝒞}_M\left[k\right])$ sends $$H^{*}(\left\{x\right\},i_x^{!}{\mu }_{*}{𝒞}_M)⟶H^{*+k}(\left\{x\right\},i_x^{!}{\mu }_{*}{𝒞}_M)\text{.}$$ There is an action of $Z_G\left(x\right)/Z_G{\left(x\right)}^{\circ }$ on $H^{*}\left(i_x^{!}{\mu }_{*}{𝒞}_M\right)$ which commutes with the action of $A\text{.}$ Thus the $\chi \text{-isotypic}$ component $H^{*}{\left(i_x^{!}{\mu }_{*}{𝒞}_M\right)}_{\chi }$ of $H^{*}\left(i_x^{!}{\mu }_{*}{𝒞}_M\right)$ is an $A\text{-module.}$ Similar arguments apply to $H^{*}\left(i_x^{*}{\mu }_{*}{𝒞}_M\right)\text{.}$ Thus we can make the following definition: If $\psi =(𝕆,\chi )$ the standard and costandard $A\text{-modules}$ are $${ℳ}_{\psi }^{!}=H^{*}{\left(i_x^{!}{\mu }_{*}{𝒞}_M\right)}_{\chi }\text{and}{ℳ}_{\psi }^{*}=H^{*}{\left(i_x^{*}{\mu }_{*}{𝒞}_M\right)}_{\chi }\text{.}$$ Using the decomposition theorem $${ℳ}_{\psi }^{!}=H^{*}{\left(i_x^{!}{𝒞}_M\right)}_{\chi }=\underset{\underset{\varphi }{k\in \mathbb{Z}}}{⨁}{L}_{\varphi }\otimes H^k{\left(i_x^{!}I{C}_{\varphi }\right)}_{\chi }\text{.}$$ Define a filtration $H^{*}{\left(i_x^{!}{𝒞}_M\right)}_{\chi }\supseteq F^1\supseteq F^2\supseteq \cdots$ by $$F^m=\underset{j\ge m}{⨁}\underset{\varphi }{⨁}{L}_{\varphi }\otimes H^j\left(i_x^{!}I{C}_{\varphi }\right)\text{.}$$ Then $F^m$ is an $A\text{-module}$ and ${\text{gr}}_F\left({ℳ}_{\psi }^{!}\right)$ is a semisimple $A\text{-module.}$ This (and a similar argument for $H^{*}\left(i_x^{*}{\mu }_{*}{𝒞}_M\right)\text{)}$ show that the multiplicity of the simple $A\text{-module}$ $L_{\varphi }$ in composition series of ${ℳ}_{\psi }^{!}$ and ${ℳ}_{\psi }^{*}$ are $$[{ℳ}_{\psi }^{!}:L_{\varphi }]=\sum _k\text{dim}\hspace{0.17em}{H}^k{\left(i_x^{!}I{C}_{\varphi }\right)}_{\chi }\text{and}[{ℳ}_{\psi }^{*}:L_{\varphi }]=\sum _k\text{dim}\hspace{0.17em}{H}^k{\left(i_x^{*}I{C}_{\varphi }\right)}_{\chi }\text{.}$$ Define the standard KL-polynomial and the costandard KL-polynomial of $A$ to be $$P_{\varphi \psi }^{!}\left(q\right)=\sum _k{q}^k\hspace{0.17em}\text{dim}\hspace{0.17em}{H}^k{\left(i_x^{!}I{C}_{\varphi }\right)}_{\chi }\text{and}{P}_{\varphi \psi }^{*}\left(q\right)=\sum _k{q}^k\hspace{0.17em}\text{dim}\hspace{0.17em}{H}^k{\left(i_x^{*}I{C}_{\varphi }\right)}_{\chi },$$ respectively. Then ??? says that $$[{ℳ}_{\psi }^{!}:L_{\varphi }]=P_{\varphi \psi }^{!}\left(1\right)\text{and}[{ℳ}_{\psi }^{!}:L_{\varphi }]=P_{\varphi \psi }^{*}\left(1\right)\text{.}$$ These identities are analogues of the original Kazhdan-Lusztig conjecture describing the multiplicities of simple $𝔤\text{-modules}$ in Verma modules.

Reciprocity

Let $d_{\varphi }={\text{dim}}_{ℂ}\hspace{0.17em}{𝕆}_{\varphi }$ and $$\text{assume that}{\text{Ext}}_{D^b\left(N\right)}^{d_{\psi }+d_{\varphi }+k}(I{C}_{\varphi },I{C}_{\psi })=0,\text{for all odd}\hspace{0.17em}k\text{.}$$ Then $$\begin{array}{ccc}[P_{\psi }:L_{\varphi }]& =& {\displaystyle \sum _k\text{dim}\hspace{0.17em}{\text{Ext}}_{D^b\left(N\right)}^k(I{C}_{\varphi },I{C}_{\psi })}\\ & =& {\displaystyle \sum _k\text{dim}\hspace{0.17em}{\text{Ext}}_{D^b\left(N\right)}^{d_{\varphi }+d_{\psi }+k}(I{C}_{\varphi },I{C}_{\psi })}\\ & =& {\displaystyle \sum _k{(-1)}^k\hspace{0.17em}\text{dim}\hspace{0.17em}{\text{Ext}}_{D^b\left(N\right)}^{d_{\varphi }+d_{\psi }+k}(I{C}_{\varphi },I{C}_{\psi })}\\ & =& {\displaystyle {(-1)}^{d_{\varphi }+d_{\psi }}\sum _{𝕆}\chi (𝕆,i_{𝕆}^{!}I{C}_{\varphi }^{\vee }\stackrel{!}{\otimes }{i}_{𝕆}^{!}I{C}_{\psi })}\\ & =& {\displaystyle {(-1)}^{d_{\varphi }+d_{\psi }}\sum _{𝕆}\chi (𝕆,{(-1)}^{d_{\varphi }}\sum _{\alpha ,k}[{\mathscr{H}}^k{i}_{𝕆}^{!}\left(I{C}_{\varphi }^{\vee }\right):\alpha ]\alpha \stackrel{!}{\otimes }{(-1)}^{d_{\psi }}\sum _{\beta ,\ell }[{\mathscr{H}}^{\ell }{i}_{𝕆}^{!}\left(I{C}_{\psi }\right):\beta ]\beta )}\\ & =& {\displaystyle \sum _{𝕆,\alpha ,\beta }\chi (𝕆,\sum _k[{\mathscr{H}}^k{i}_{𝕆}^{!}\left(I{C}_{\varphi }\right):{\alpha }^{*}]\alpha \stackrel{!}{\otimes }{(-1)}^{d_{\psi }}\sum _{\ell }[{\mathscr{H}}^{\ell }{i}_{𝕆}^{!}\left(I{C}_{\psi }\right):\beta ]\beta )}\\ & =& {\displaystyle \sum _{\alpha ,\beta }\sum _k\text{dim}\hspace{0.17em}{\mathscr{H}}^k\left(i_{\alpha }^{!}I{C}_{\varphi }\right)\left(\sum _{𝕆}\chi (𝕆,{\alpha }^{*}\stackrel{!}{\otimes }\beta )\right)\sum _{\ell }\text{dim}\hspace{0.17em}{\mathscr{H}}^{\ell }\left(i_{\beta }^{!}I{C}_{\psi }\right)}\\ & =& {\displaystyle \sum _{\alpha ,\beta }[{ℳ}_{\alpha }^{!}:L_{\varphi }]\left(\sum _{𝕆}\chi (𝕆,{\alpha }^{*}\otimes \beta )\right)[{ℳ}_{\beta }^{!}:L_{\psi }]}\\ & =& {\displaystyle \sum _{\alpha ,\beta }{P}_{\varphi \alpha }\left(1\right)D_{\alpha \beta }{P}_{\psi \beta }\left(1\right)}\\ & =& {\displaystyle {\left(PD{P}^t\right)}_{\varphi \psi },}\end{array}$$ 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 $\left(P_{\varphi \alpha }\left(1\right)\right),$ and
(4)	$D$ is the matrix $\left(\sum _{𝕆}\chi (𝕆,{\alpha }^{*}\otimes \beta )\right)\text{.}$

This identity is the "BGG reciprocity" for the algebra $A\text{.}$

Borel-Moore homology

Recall that the Borel-Moore homology is defined by $$H_i\left(X\right)=H^{-i}(X,{𝔻}_X)={\text{Hom}}_{D^b\left(X\right)}({ℂ}_X,{ℂ}_X{\left[i\right]}^{\vee })\text{.}$$ Let $f:X\to Y$ be a proper map. Then, since $f_{*}=f_{!},$ $H^{*}(Y,f_{*}ℱ)=H^{*}(X,ℱ),$ $f^{!}{𝔻}_Y={𝔻}_X$ and there is a canonical map $f_{!}{f}^{!}{𝔻}_Y\to {𝔻}_Y$ there is a pushforward $$f_{*}:H_{*}\left(X\right)⟶H_{*}\left(Y\right),$$ coming from the map $H_{*}\left(X\right)=H^{*}(X,{𝔻}_X)=H^{*}(X,f^{!}{𝔻}_Y)=H^{*}(Y,f_{*}{f}^{!}{𝔻}_Y)=H^{*}(Y,f_{!}{f}^{!}{𝔻}_Y)\to H^{*}(Y,{𝔻}_Y)=H_{*}\left(Y\right)\text{.}$ If $f:X\to Y$ is a flat map with smooth fibers of complex dimension $d$ then $f^{!}=f^{*}\left[2d\right]$ and there is a canonical map ${𝔻}_X\to f_{*}{f}^{*}{𝔻}_X$ which induces the pullback $$f^{*}:H_{*}\left(Y\right)⟶H_{*+d}\left(X\right)$$ via $H_{*}\left(Y\right)=H^{-*}(Y,{𝔻}_Y)\to H^{-*}(Y,f_{*}{f}^{*}{𝔻}_Y)=H^{-*}(Y,f_{*}{f}^{!}{𝔻}_Y[-2d])=H^{-*}(X,F^{!}{𝔻}_Y[-2d])=H^{-*-d}(X,f^{!}{𝔻}_Y)=H^{-*-d}(X,{𝔻}_X)=H_{*+d}\left(X\right)\text{.}$ If $X$ is a smooth variety, $i:Y\hookrightarrow X$ is a codimension $d$ subvariety and $j:Z\hookrightarrow X$ is a closed subvariety the pullback $$\begin{array}{ccc}Y\cap Z& \stackrel{\iota }{⟶}& Z\\ \downarrow j& & \downarrow j\\ Y& \stackrel{i}{⟶}& X\end{array}{\iota }^{*}:H_{*}\left(Z\right)⟶H_{*-d}(Y\cap Z)$$ is defined by $H_{*}\left(Z\right)=H^{-i}({ℂ}_Z,{𝔻}_Z)=H^{-i}({ℂ}_Z,j^{!}{𝔻}_X)\to H^{-i}({ℂ}_Z,j^{!}{i}_{*}{i}^{*}{𝔻}_X)=H^{-i}({ℂ}_Z,{\iota }_{*}{j}^{!}{i}^{*}{ℂ}_X\left[2{\text{dim}}_{ℂ}\hspace{0.17em}X\right])=H^{-i}({\iota }^{*}{ℂ}_Z,j^{!}{𝔻}_Y\left[2d\right])=H^{-i}({ℂ}_{Y\cap Z},{𝔻}_{Y\cap Z}\left[2d\right])=H_{i-2d}(Y\cap Z)\text{.}$

For arbitrary $CW\text{-complexes}$ $M_1$ and $M_2$ let $$⊠:H_{*}\left(M_1\right)\otimes H_{*}\left(M_2\right)⟶H_{*}(M_1\times M_2)$$ be the Kunneth isomorphism for Borel-Moore homology [CGi1433132, 2.6.19]. Let $M$ be a smooth oriented manifold and let $Z$ and $\stackrel{\sim }{Z}$ be closed subsets of $M\text{.}$ Let $m={\text{dim}}_{ℝ}\hspace{0.17em}M$ and let $\mathrm{\Delta }:M\to M\times M$ be the diagonal imbedding. The cup product $$H_i\left(Z\right)\otimes H_j\left(\stackrel{\sim }{Z}\right)\stackrel{\cap }{⟶}{H}_{i+j-m}(Z\cap \stackrel{\sim }{Z})\text{is given by}c\cap \stackrel{\sim }{c}={\mathrm{\Delta }}^{*}(c⊠\stackrel{\sim }{c})\text{.}$$

Let $M_1\text{,}$ $M_2$ and $M_3$ be connected oriented ${ℂ}^{\infty }$ manifolds. Let $p_{ij}:M_1\times M_2\times M_3\to M_i\times M_j$ be the canonical projections and fix closed subsets $$\begin{array}{c}{Z}_{12}\subseteq M_1\times M_2\text{and}{Z}_{23}\subseteq M_2\times M_3\text{such that}\\ p_{13}:p_{12}^{-1}\left(Z_{12}\right)\cap p_{23}^{-1}\left(Z_{23}\right)⟶M_1\times M_3\text{is proper.}\end{array}$$ This map has image $Z_{12}\circ Z_{23}$ as described in ???. The convolution in Borel-Moore homology $$H_{*}\left(Z_{12}\right)\otimes H_{*}\left(Z_{23}\right)\stackrel{*}{⟶}{H}_{*}(Z_{12}\circ Z_{23})\text{is given by}{c}_{12}*c_{23}={\left(p_{13}\right)}_{*}(p_{12}^{*}{c}_{12}\cap p_{23}^{*}{c}_{23})\text{.}$$

Equivariant $K\text{-theory}$

Let $X$ be a quasiprojective variety with a linear algebrac group $G$ acting on $X\text{.}$ The equivariant $K\text{-theory}$ of $X$ is $$K^G\left(X\right)=\text{Grothendieck group of the category of}\hspace{0.17em}G\text{-equivariant coherent sheaves on}\hspace{0.17em}X\text{.}$$ If $X$ and $Y$ are $G\text{-varieties}$ and $f:X\to Y$ is a $G\text{-equivariant}$ morphism define the pullback $$f^{*}:K^G\left(Y\right)\to K^G\left(X\right)\text{by}{f}^{*}\left[ℱ\right]=\left[f^{*}ℱ\right],$$ where $f^{*}:\text{Sh}\left(Y\right)\to \text{Sh}\left(X\right)$ is the pullback functor on sheaves. By definition $f^{*}ℱ={𝒪}_Y{\otimes }_{{𝒪}_X}ℱ,$ and this functor is not always flat, which means that the definition above is not well defined on $K\text{-groups,}$ and so if $f$ is not flat it is necessary to be more careful and define $f^{*}\left[ℱ\right]=\sum {(-1)}^i{\text{Tor}}_i^{{𝒪}_X}({𝒪}_Y,ℱ)$ where ${\text{Tor}}_i^{{𝒪}_X}({𝒪}_Y,ℱ)={\mathscr{H}}^i\left(f_{*}{𝒪}_Y{\otimes }_{{𝒪}_X}{F}^{•}\right)$ for a finite $G\text{-equivariant}$ locally free resolution $F^{•}$ of $ℱ\text{.}$ If $X$ and $Y$ are quasiprojective $G\text{-varieties}$ and $f:X\to Y$ is a proper morphism then define the pushforward $$f_{*}:K^G\left(X\right)\to K^G\left(Y\right)\text{by}{f}_{*}\left[ℱ\right]=\sum _i{(-1)}^i\left[R^i{f}_{*}ℱ\right],$$ where the inner $f_{*}:\text{Sh}\left(X\right)\to \text{Sh}\left(Y\right)$ is the direct image functor on sheaves and $R^i$ denotes the $i\text{th}$ derived functor.

Let ${\text{Coh}}^G\left(X\right)$ denote the category of $G\text{-equivariant}$ coherent sheaves on $X\text{.}$ If $Z$ and $\stackrel{\sim }{Z}$ are $G\text{-varieties}$ define $$⊠:{\text{Coh}}^G\left(Z\right)\otimes {\text{Coh}}^G\left(\stackrel{\sim }{Z}\right)⟶{\text{Coh}}^G(Z\times \stackrel{\sim }{Z})\text{by}ℱ⊠\stackrel{\sim }{ℱ}=p_Z^{*}ℱ{\otimes }_{{𝒪}_{Z\times \stackrel{\sim }{Z}}}{p}_{\stackrel{\sim }{Z}}^{*}\stackrel{\sim }{ℱ},$$ where $p_Z:Z\times \stackrel{\sim }{Z}\to Z$ and $p_{\stackrel{\sim }{Z}}:Z\times \stackrel{\sim }{Z}\to \stackrel{\sim }{Z}$ are the projections. Let $M$ be smooth and let $\mathrm{\Delta }:M\to M\times M$ be the diagonal imbedding. Let $Z$ and $\stackrel{\sim }{Z}$ be closed $G\text{-invariant}$ subvarieties of $M$ and define the tensor product $$K^G\left(Z\right)\otimes K^G\left(\stackrel{\sim }{Z}\right)\stackrel{\otimes }{⟶}{K}^G(Z\cap \stackrel{\sim }{Z})\text{by}\left[ℱ\right]\otimes \left[\stackrel{\sim }{ℱ}\right]={\mathrm{\Delta }}^{*}(ℱ⊠\stackrel{\sim }{ℱ})\text{.}$$ Let $M_1\text{,}$ $M_2$ and $M_3$ be smooth quasiprojective $G\text{-varieties.}$ Let $p_{ij}:M_1\times M_2\times M_3\to M_i\times M_j$ be the canonical projections and fix $G\text{-stable}$ closed subvarieties $$Z_{12}\subseteq M_1\times M_2\text{and}{Z}_{23}\subseteq M_2\times M_3\text{such that}{p}_{13}:p_{12}^{-1}\left(Z_{12}\right)\cap p_{23}^{-1}\left(Z_{23}\right)⟶M_1\times M_3\text{is proper.}$$ This map has image $$Z_{12}\circ Z_{23}=\{(m_1,m_3)\in M_1\times M_3\hspace{0.17em}|\hspace{0.17em}(m_1,m_2)\in Z_{12}\hspace{0.17em}\text{and}\hspace{0.17em}(m_2,m_3)\in Z_{23}\hspace{0.17em}\text{for some}\hspace{0.17em}{m}_2\in M_2\}\text{.}$$ The convolution in equivariant $K\text{-theory}$ $$K^G\left(Z_{12}\right)\otimes K^G\left(Z_{23}\right)\stackrel{*}{⟶}{K}^G(Z_{12}\circ Z_{23})\text{is given by}{ℱ}_{12}*{ℱ}_{23}={\left(p_{13}\right)}_{*}(p_{12}^{*}{ℱ}_{12}\otimes p_{23}^{*}{ℱ}_{23})\text{.}$$

Example 1. Consider our setup $\mu :M\to N$ and let $Z=M{\times }_{N}M=\{(m_1,m_2)\in M\times M\hspace{0.17em}|\hspace{0.17em}\mu \left(m_1\right)=\mu \left(m_2\right)\}\text{.}$ Then $$\begin{array}{ccc}Z\circ Z& =& \left\{(m_1,m_3)\hspace{0.17em}\right|\hspace{0.17em}\mu \left(m_1\right)=\mu \left(m_2\right)=\mu \left(m_3\right)\hspace{0.17em}\text{for some}\hspace{0.17em}{m}_2\in M\}\\ & =& Z\text{.}\end{array}$$ In this case convolution is a product $$K^G\left(Z\right)\otimes K^G\left(Z\right)\stackrel{*}{⟶}{K}^G\left(Z\right)\text{.}$$

Example 2. If $x\in N$ and $M_x={\mu }^{-1}\left(x\right)$ then $$M_x=\left\{(m_2,m_3)\hspace{0.17em}\right|\hspace{0.17em}{m}_2=m_3\hspace{0.17em}\text{and}\hspace{0.17em}\mu \left(m_2\right)=x\}\text{.}$$ Then $$Z\circ M_x=\left\{(m_1,m_3)\hspace{0.17em}\right|\hspace{0.17em}\mu \left(m_1\right)=\mu \left(2\right)=\mu \left(m_3\right)=x\}=M_x$$ and so convolution $$K^G\left(Z\right)\otimes K^G\left(M_x\right)\stackrel{*}{⟶}{K}^G\left(M_x\right)$$ makes $M_x$ into an $K^G\left(Z\right)\text{-module.}$

Chern character and Riemann-Roch

The (homological) Chern character is a $\mathbb{Z}\text{-linear}$ map ${\text{ch}}_{*}:K\left(X\right)\to H_{*}\left(X\right)$ such that

(1)	${\text{ch}}_{*}\left(\left[{𝒪}_X\right]\right)=\left[X\right]+\text{lower degree terms}\text{,}$ where ${𝒪}_X$ is the stucture sheaf of $X\text{,}$ $\left[X\right]$ is the fundamental class of $X\text{,}$ and lower degree terms consists of a sum of homogeneous elements of $H_{*}\left(X\right)$ of degree $<2{\text{dim}}_{ℂ}\hspace{0.17em}X\text{.}$
(2)	If $M$ is smooth, $U\subseteq M$ is a Zariski open set, and $X\subseteq M$ is closed, and $f:X\cap U\hookrightarrow X$ is the inclusion, then $${\text{ch}}_{*}\left(f^{*}\left[ℱ\right]\right)=f^{*}\left({\text{ch}}_{*}\left[ℱ\right]\right),\text{for}\hspace{0.17em}ℱ\in K\left(X\right)$$
(2)	If $X$ and $Y$ are subvarieties of smooth varieties $M$ and $N$ respectively and $f:X\to Y$ is a proper map then $$T{d}_N·f_{*}\left({\text{ch}}_{*}ℱ\right)=f_{*}(T{d}_M·{\text{ch}}_{*}ℱ),\text{for}\hspace{0.17em}ℱ\in K\left(X\right),$$ where $T{d}_M$ denotes the Todd class of $M$ in $H^{*}\left(M\right)\text{.}$
(3)	If $Z$ and $\stackrel{\sim }{Z}$ are closed subvarieties of a smooth variety $M$ then $${\text{ch}}_{*}(\left[ℱ\right]\otimes \left[\stackrel{\sim }{ℱ}\right])={\text{ch}}_{*}\left(\left[ℱ\right]\right)\cap {\text{ch}}_{*}\left(\left[\stackrel{\sim }{𝒢}\right]\right)\text{.}$$
(4)	Let $Z\subseteq M_1\times M_2$ and define $$RR:K\left(X\right)⟶H_{*}\left(Z\right)\text{by}RR\left(\left[ℱ\right]\right)=(1⊠T{d}_{M_2})\cup {\text{ch}}_{*}\left(\left[ℱ\right]\right)\text{.}$$ Let $M_1\text{,}$ $M_2$ and $M_3$ be smooth varieties and let $Z_{12}\subseteq M_1\times M_2$ and $Z_{23}\subseteq M_2\times M_3$ be closed subvarieties so that convolution is defined. Then $$RR\left({ℱ}_{12}\right)*RR\left({ℱ}_{23}\right)=RR({ℱ}_{12}*{ℱ}_{23}),$$ where the product on the left is convolution in $K\text{-theory}$ and the product on the right is convolution in Borel-Moore homology.

Convolution algebras and Borel-Moore homology

Define $$Z=M{\times }_{N}M=\{(m_1,m_2)\in M\times M\hspace{0.17em}|\hspace{0.17em}\mu \left(m_1\right)=\mu \left(m_2\right)\}\text{.}$$ Then we have a commutative diagram $$\begin{array}{ccc}Z=M{\times }_{N}M& \stackrel{\iota }{⟶}& M\times M\\ \downarrow {\mu }_{12}& & \downarrow {\mu }_1\times {\mu }_2\\ N=N_{\mathrm{\Delta }}& \stackrel{\mathrm{\Delta }}{⟶}& N\times N\end{array}$$ which (via base change) provides an isomorphism $$\begin{array}{ccc}{H}_{*}\left(Z\right)& =& {\text{Hom}}_{D^b\left(Z_{12}\right)}({ℂ}_{Z_{12}},{\left({ℂ}_{Z_{12}}[*]\right)}^{\vee })\\ & =& {\text{Hom}}_{D^b\left(Z_{12}\right)}({\mu }_{12}^{*}{ℂ}_N,{\iota }^{!}{𝒞}_{M_1\times M_2}[m_1+m_2][-*])\\ & =& {\text{Hom}}_{D^b\left(N\right)}({ℂ}_N,{\left({\mu }_{12}\right)}_{*}{\iota }^{!}{𝒞}_{M_1\times M_2}[m_1+m_2-*])\\ & =& {\text{Hom}}_{D^b\left(N\right)}({ℂ}_N,{\mathrm{\Delta }}^{!}{({\mu }_1\times {\mu }_2)}_{*}({𝒞}_{M_1}⊠{𝒞}_{M_2})[m_1+m_2-*])\\ & =& {\text{Hom}}_{D^b\left(N\right)}({ℂ}_N,{\mathrm{\Delta }}^{!}({\left({\mu }_1\right)}_{*}{𝒞}_{M_1}⊠{\left({\mu }_2\right)}_{*}{𝒞}_{M_2})[m_1+m_2-*])\\ & =& {\text{Ext}}_{D^b\left(N\right)}^{m_1+m_2-*}({\left({\mu }_1\right)}_{*}{𝒞}_{M_1},{\left({\mu }_2\right)}_{*}{𝒞}_{M_2})\text{.}\end{array}$$ Let $x\in N$ and define $M_x={\mu }^{-1}\left(x\right)\text{.}$ Then the commutative diagram $$\begin{array}{ccc}{M}_x& \stackrel{\iota }{⟶}& M\\ \downarrow \mu & & \downarrow \mu \\ \left\{x\right\}& \stackrel{i_x}{⟶}& N\end{array}$$ induces (via base change) an isomorphism $$\begin{array}{ccc}{H}_{*}\left(M_x\right)& =& {\text{Hom}}_{D^b\left(M_x\right)}({ℂ}_{M_x},{\left({ℂ}_{M_x}[*]\right)}^{\vee })\\ & =& {\text{Hom}}_{D^b\left(M_x\right)}({\mu }^{*}{ℂ}_{\left\{x\right\}},{\left(\left({\iota }^{*}{ℂ}_M\right)[*]\right)}^{\vee })\\ & =& {\text{Hom}}_{D^b\left(\left\{x\right\}\right)}({ℂ}_{\left\{x\right\}},{\mu }_{*}\left({\iota }^{!}{ℂ}_M\left[2m\right]\right)[-*])\\ & =& {\text{Hom}}_{D^b\left(\left\{x\right\}\right)}({ℂ}_{\left\{x\right\}},i_x^{!}{\mu }_{*}{𝒞}_M[m-*])\\ & =& H^{m-*}\left(i_x^{!}{\mu }_{*}{𝒞}_M\right)\end{array}$$ and another isomorphism $$\begin{array}{ccc}{H}^{*}\left(M_x\right)& =& {\text{Hom}}_{D^b\left(M_x\right)}({ℂ}_{M_x},{ℂ}_{M_x}[*])\\ & =& {\text{Hom}}_{D^b\left(M_x\right)}({\mu }^{*}{ℂ}_{\left\{x\right\}},{ℂ}_{M_x}[*])\\ & =& {\text{Hom}}_{D^b\left(\left\{x\right\}\right)}({ℂ}_{\left\{x\right\}},{\mu }_{*}{ℂ}_{M_x}[*])\\ & =& {\text{Hom}}_{D^b\left(\left\{x\right\}\right)}({ℂ}_{\left\{x\right\}},{\mu }_{!}{\iota }^{*}{ℂ}_M[*])\\ & =& {\text{Hom}}_{D^b\left(\left\{x\right\}\right)}({ℂ}_{\left\{x\right\}},i_x^{*}{\mu }_{!}{ℂ}_M[*])\\ & =& {\text{Hom}}_{D^b\left(\left\{x\right\}\right)}({ℂ}_{\left\{x\right\}},i_x^{*}{\mu }_{*}{𝒞}_M[*-m])\\ & =& H^{*-m}\left(i_x^{*}{\mu }_{*}{𝒞}_M\right)\text{.}\end{array}$$

Weights in equivariant $K\text{-theory}$

Let $T$ be an algebraic torus. Let $M$ be a smooth quasiprojective variety with an action of $T\text{.}$ Let $$M^T=\left\{T\text{-fixed points of}\hspace{0.17em}M\right\},\text{and}\iota :M^T\hookrightarrow M$$ be the inclusion. Then $$N=T_{M^T}M=\text{the normal bundle of}\hspace{0.17em}{M}^T\hspace{0.17em}\text{in}\hspace{0.17em}M$$ has an action of $T$ on the fibers. Thus the normal bundle $N$ can be decomposed $$N=\underset{\lambda }{⨁}{N}_{\lambda },$$ according to the irreducible representations $\lambda$ of $T\text{.}$ Let $${\lambda }_T=\sum _{i\ge 0}{(-1)}^i\left({\wedge }^i{N}^{\vee }\right)=\underset{\lambda }{⨂}\left(\sum _i{(-1)}^i{\lambda }^i\left({\mathrm{\Lambda }}^i{N}_{\lambda }^{\vee }\right)\right)\in K^T\left(M^T\right)=R\left(T\right)\otimes K\left(M^T\right)\text{.}$$ Then then composition ${\iota }^{*}{\iota }^{*}:K^T\left(M^T\right)\to K^T\left(M^T\right)$ is equal to multiplication by the element ${\lambda }_T,$ $$\begin{array}{ccc}{K}^T\left(M^T\right)& \stackrel{{\iota }_{*}}{⟶}& K^T\left(M\right)\\ \multicolumn{2}{c}{{\lambda }_T\searrow }& \downarrow {\iota }^{*}\\ & & K^T\left(M^T\right)\end{array}$$ An element $t\in T$ is regular if $${\lambda }_t=\text{the evaluation of}\hspace{0.17em}{\lambda }_T\hspace{0.17em}\text{at}\hspace{0.17em}t,$$ obtained by evaluating all elements of $R\left(T\right)$ at $t\text{,}$ is an invertible element of $ℂ\otimes K\left(M^T\right)\text{.}$ Equivalently, an element $t\in T$ is regular if $$\text{(a)}\hspace{0.17em}\hspace{0.17em}{M}^t=M^T,\text{or, if}\text{(b)}\hspace{0.17em}\hspace{0.17em}\lambda \left(t\right)\ne 1\hspace{0.17em}\text{for all}\hspace{0.17em}\lambda \hspace{0.17em}\text{such that}\hspace{0.17em}{N}_{\lambda }\ne 0\text{.}$$ For a regular element $t\in T$ define $${\text{res}}_t:K^T{\left(M\right)}_t⟶ℂ\otimes K\left(M^T\right),\text{by}\hspace{0.17em}{\text{res}}_t={\left({\iota }_{*}\right)}^{-1}={\left({\lambda }_t\right)}^{-1}{\iota }^{*}\text{.}$$ By [CGi1433132, 5.11.7], if $f:X\to Y$ is a $T\text{-equivariant}$ proper morphism and if $t$ is an element which is both $X\text{-regular}$ and $Y\text{-regular,}$ then the following diagram commutes: $$\begin{array}{ccc}{K}^T\left(X\right)& \stackrel{f_{*}}{⟶}& K^T\left(Y\right)\\ \downarrow {\text{res}}_t& & \downarrow {\text{res}}_t\\ K\left(X^T\right)& \stackrel{f_{*}}{⟶}& K\left(Y^T\right)\end{array}$$ Thus, if $Y$ is a point, then ${\text{pt}}^T=\text{pt}$ and $$\begin{array}{ccc}{\text{res}}_t{f}_{*}ℱ& =& {\displaystyle \sum _i{(-1)}^i{\text{res}}_t{H}^i(X,ℱ)}\\ & =& {\displaystyle \sum _i{(-1)}^i\text{Tr}(t,H^i(X,ℱ)),\text{and}}\\ f_{*}{\text{res}}_tℱ& =& {\displaystyle \sum _i{(-1)}^i\text{Tr}(t,H^i(X^T,\left({\lambda }_t^{-1}\right)\left(ℱ{|}_{X^T}\right)))\text{.}}\end{array}$$ If $X^T$ is a collection of disjoint points and $ℱ$ is the constant sheaf then $$\sum _i{(-1)}^i\text{Tr}(t,H^i\left(X\right))={\text{res}}_t{f}_{*}{ℂ}_X=f_{*}{\text{res}}_t{ℂ}_X=\sum _i{(-1)}^i\text{Tr}(t,H^i(X^T,\left({\lambda }_t^{-1}\right){ℂ}_{X^T}))=\text{Card}\left(X^T\right),$$ which is the Grothendieck-Lefschetz trace formula. In Carter [Ca] p. 504, the Grothendieck trace formula is given as $$\left|X^{F^n}\right|=\sum _{i=0}^{2d}{(-1)}^i\text{Tr}(F^n,H^i\left(X\right)),$$ where $F$ is Frobenius on $X$ over ${𝔽}_q\text{.}$

Monodromy filtration on $\mathrm{\Psi }$

Let $\mathrm{\Psi }$ be an object of an abelian category and let $q$ be a nilpotent endomorphism of $\mathrm{\Psi }\text{.}$ By [Del1980, Prop. 1.6.1] there exists a unique finite increasing filtration $$\cdots \subseteq {\mathrm{\Psi }}^{\{i-1\}}\subseteq {\mathrm{\Psi }}^{\left\{i\right\}}\subseteq \cdots$$ of $\mathrm{\Psi }$ that $q{\mathrm{\Psi }}^{\left\{i\right\}}\subseteq {\mathrm{\Psi }}^{\{i-2\}}$ and $q^k$ induces an isomorphism $$q^k:{\text{Gr}}_k\mathrm{\Psi }\stackrel{\sim }{⟶}{\text{Gr}}_{-k}\mathrm{\Psi }\text{.}$$ If $\mathrm{\Psi }$ is a vector space and $q$ is a nilpotent endomorphism in Jordan form then for each Jordan block of $q$ there is a basis $\{e_{-d},e_{-d-2},\dots ,e_{d-2},e_d\}$ such that $q{e}_i=e_{i-2}\text{.}$ Then let ${\mathrm{\Psi }}^{\left\{k\right\}}=\text{span}\left\{e_i\hspace{0.17em}\right|\hspace{0.17em}i\le k\}\text{.}$ If $\mathrm{\Psi }$ is considered as an ${𝔰𝔩}_2\text{-module}$ where $q$ acts as $f$ in an ${𝔰𝔩}_2\text{-triple,}$ then ${\mathrm{\Psi }}^{\left\{k\right\}}$ is the span of all weight spaces which have weights $\le k\text{.}$

Jantzen filtrations on $\text{ker}\hspace{0.17em}q$ and $\text{coker}\hspace{0.17em}q$

Consider $\mathrm{\Psi }$ as above with a nilpotent action of $q\text{.}$ Define an increasing filtrations on $${ℳ}^{!}=\text{ker}\hspace{0.17em}q\text{and}{ℳ}^{*}=\text{coker}\hspace{0.17em}q$$ by $${\left({ℳ}^{!}\right)}^{\left(i\right)}=\{\begin{array}{cc}\text{ker}\hspace{0.17em}q\cap \text{im}\hspace{0.17em}{q}^{-i},& \text{for}\hspace{0.17em}i\le 0,\\ \text{ker}\hspace{0.17em}q,& \text{for}\hspace{0.17em}i\ge 0,\end{array}\text{and}{ℳ}_{*}^{\left(i\right)}=\frac{\text{ker}\hspace{0.17em}{q}^i+\text{im}\hspace{0.17em}q}{\text{im}\hspace{0.17em}q}\text{.}$$ Conceptually, ${ℳ}^{!}=\text{ker}\hspace{0.17em}q$ is the span of the lowest weight vectors in $\mathrm{\Psi }$ and ${ℳ}_{*}$ is the span of the highest weight vectors in $\mathrm{\Psi }\text{.}$ The Jantzen filtrations are the filtrations on these subspaces coming from the weights of $\mathrm{\Psi }$ as an ${𝔰𝔩}_2\text{-module.}$ $$\text{PICTURE OF}\hspace{0.17em}\mathrm{\Psi }$$

The weight filtration

Let $X_0$ be a scheme of finite type over ${𝔽}_q$ and let ${ℱ}_0$ be a ${\bar{\mathbb{Q}}}_{\ell }$ sheaf on $X_0\text{.}$ The Frobenius map is $$\begin{array}{cccc}{\text{Fr}}_q:& {\overline{𝔽}}_q& ⟶& {\overline{𝔽}}_q\\ & x& ⟼& x^q\end{array}$$ The sheaf ${ℱ}_0$ on $X_0$ defines a sheaf $ℱ$ on $X$ with an action of a Frobenius map $F_q^{*}:{\text{Fr}}_q^{*}ℱ\stackrel{\sim }{⟶}ℱ\text{.}$ The set of points of $X$ defined over ${𝔽}_{q^n}$ is $$X^{F^n}=\left\{F^n\hspace{0.17em}\text{fixed points of}\hspace{0.17em}X\right\}=\left\{\text{points of}\hspace{0.17em}X\hspace{0.17em}\text{defined over}\hspace{0.17em}{𝔽}_{q^n}\right\}$$ and $F^n$ acts on the fibers ${ℱ}_x$ for each $x\in X^{F^n}\text{.}$

A ${\bar{\mathbb{Q}}}_{\ell }$ sheaf ${ℱ}_0$ on $X_0$ is pointwise pure of weight $w$ $\text{(}w\in \mathbb{Z}\text{)}$ if, for all $n$ and all $x\in X^{F^n},$ the eigenvalues of $F^n$ on ${ℱ}_x$ are algebraic numbers all of whose complex conjugates have absolute value ${\left(q^n\right)}^{w/2}\text{.}$ The sheaf ${ℱ}_0$ is mixed if it has a finite filtration with pointwise pure quotients. If $K\in D^b\left(X_0\right)$ then

(a)	$K\in D^b\left(X_0\right)$ is mixed if all its cohomology sheaves ${\mathscr{H}}^{k}K$ are mixed.
(b)	$K$ has weights $\le w,$ if, for each $i\text{,}$ ${\mathscr{H}}^{i}K$ has weights $\le w+i\text{.}$ We write $K\in D_{\le w}^b\left(X_0\right)\text{.}$
(c)	$K$ has weights $\ge w$ if $K^{\vee }$ has weights $\le w\text{.}$ We write $K\in D_{\ge w}^b\left(X_0\right)\text{.}$
(d)	$K$ is pure of weight $w$ if $K\in D_{\le w}^b\left(X_0\right)\cap D_{\ge w}^b\left(X_0\right)\text{.}$

The weight structure of a complex can be determined locally [BBD1982, Prop. 5.1.9]

(a)	$K\in D_{\le w}^b$ if and only if $i_x^{*}K$ has weights $\le w$ for each closed point $x\in X_0,$ $i_x:\left\{x\right\}\hookrightarrow X_0\text{.}$
(b)	$K\in D_{\ge w}^b$ if and only if $i_x^{!}K$ has weights $\ge w$ for each closed point $x\in X_0,$ $i_x:\left\{x\right\}\hookrightarrow X_0\text{.}$

Let $f:X_0\to Y_0$ be a separated morphism. Then $$\begin{array}{cccc}{f}_{!}:& D_{\le w}^b\left(X_0\right)& ⟶& D_{\le w}^b\left(Y_0\right)\\ f^{*}:& D_{\le w}^b\left(Y_0\right)& ⟶& D_{\le w}^b\left(X_0\right)\\ f^{!}:& D_{\ge w}^b\left(Y_0\right)& ⟶& D_{\ge w}^b\left(X_0\right)\\ f_{*}:& D_{\ge w}^b\left(X_0\right)& ⟶& D_{\ge w}^b\left(Y_0\right)\\ \otimes :& D_{\le w}^b\times D_{\le w\prime }^b& ⟶& D_{\le w+w\prime }^b\\ \text{Ext}:& D_{\le w}^b\times D_{\ge w\prime }^b& ⟶& D_{\ge -w+w\prime }^b\\ {}^{\vee }:& D_{\le w}^b& ⟶& D_{\ge -w}^b\end{array}$$ Note that the first statement is the main theorem of [Del1980].

Let ${ℱ}_0$ be a mixed perverse sheaf on $X_0\text{.}$ Define the weight filtration of ${ℱ}_0,$ $$\cdots {ℱ}_0^{\left(i\right)}\subseteq {ℱ}_0^{(i+1)}\subseteq \cdots ,$$ of ${ℱ}_0$ by setting ${ℱ}_0^{\left(i\right)}\subseteq {ℱ}_0$ to be such that $${ℱ}_0^{\left(i\right)}\hspace{0.17em}\text{has simple subquotients in}\hspace{0.17em}{D}_{\le i}^b\left(X_0\right)\text{and}{ℱ}_0/{ℱ}_0^{\left(i\right)}\hspace{0.17em}\text{has simple subquotients in}\hspace{0.17em}{D}_{>i}^b\left(X_0\right)\text{.}$$ By [BBD1982, Thm. 5.3.5], the weight filtration is the unique increasing filtration of ${ℱ}_0$ such that $${\text{Gr}}_i{ℱ}_0\hspace{0.17em}\text{is pure of weight}\hspace{0.17em}i\text{.}$$ Note that the proof of the existence of the weight filtration uses the stability properties in (???).

A perverse sheaf on $X$ is an object $K\in D_c^b(X,{\bar{\mathbb{Q}}}_{\ell })$ such that for every stratum $S$ $$H^j{i}_S^{*}K=0,\text{for}\hspace{0.17em}j>-\frac{1}{2}\text{dim}\left(S\right),\text{and}{H}^j{i}_S^{!}K=0,\text{for}\hspace{0.17em}j<-\frac{1}{2}\text{dim}\left(S\right),$$ where $i_S:S\hookrightarrow X$ is the inclusion.

(0)	The category of perverse sheaves on $X$ is artinian and noetherian. Every object has finite length.
(a)	[BBD1982, 5.3.4] Every simple mixed perverse sheaf ${ℱ}_0$ on $X_0$ is pure.
(b)	If ${ℱ}_0$ on $X_0$ is a pure perverse sheaf then $ℱ$ on $X$ is semisimple.
(c)	If ${ℱ}_0$ on $X_0$ is an indecomposable pure perverse sheaf then $${ℱ}_0\cong S_0\otimes E_n$$ where $S_0$ is simple and $E_n$ has fiber $\text{span-}\{e_1,\dots ,e_n\}$ with $F\text{-action}$ given by the regular unipotent element in ${GL}_n\text{.}$ $\text{(}{E}_n$ is indecomposable of weight $0\text{.)}$
(d)	Any mixed perverse sheaf $A$ has a finite increasing filtration $W$ by weights such that each ${\text{gr}}_W^{i}A$ is pure. Every morphism is compatible with these filtrations.
(e)	Every pure indecomposable perverse sheaf is of the form $S_0\otimes E$ where $S_0$ is a simple perverse sheaf and $E$ is a finite rank cyclic $ℂ\left[t\right]\text{-module.}$
(g)	The simple perverse sheaves $ℱ$ on $X$ are $$j_{!*}L\left[\text{dim}\hspace{0.17em}U\right],\text{where}{j}_{*!}ℱ=\text{Im}({}^p{j}_{!}ℱ\to {}^p{j}_{*}ℱ),$$ $j:U\hookrightarrow X$ with $U$ smooth, connected and of the same dimension in all irreducible components, and $L$ is an irreducible ${\bar{\mathbb{Q}}}_{\ell }\text{-sheaf}$ on $U\text{.}$

		Proof.
	(a) There exists $j:U_0\hookrightarrow X_0$ lisse connected of dimension $d$ and a lisse sheaf $L_0$ on $U_0$ such that ${ℱ}_0=j_{!*}\left(L_0\left[d\right]\right)$ (by [BBD1982, 4.3.1]). One may retract $U_0$ [BBD1982, 4.3.2]. The lisse sheaf $L_0$ is irreducible, and it follows from the definitions that one can take it to be pure. One may take $U_0$ affine and it only remains to apply [BBD1982, 5.3.2]. (c) The proof is identical to the proof of [Del1980, 3.4.1(iii)]. Let $$ℱ\prime =\text{direct sum of the simple perverse sheaves in}\hspace{0.17em}ℱ\text{.}$$ We want to show that $ℱ\prime =ℱ\text{.}$ Let ${ℱ}_0^{″}={ℱ}_0/{ℱ}_0^{\prime }$ $\text{(}{ℱ}_0^{\prime }\subseteq ℱ$ since $ℱ\prime$ is stable under Frobenius). The (image of) the extension $$0⟶{ℱ}_0^{\prime }⟶{ℱ}_0⟶{ℱ}_0^{″}⟶0$$ is $0$ in ${\text{Ext}}^1(ℱ″,ℱ\prime )$ by [BBD1982, 5.1.15 (iii)] (which follows readily from the stabilities) and the fact that ${ℱ}_0^{\prime }$ and ${ℱ}_0^{″}$ are pure of the same weight. Thus $$ℱ\cong ℱ\prime \oplus ℱ″\text{.}$$ If $ℱ″$ is nonzero then it has a simple subobject, but this contradicts the maximality of $ℱ″\text{.}$ So $ℱ\prime =ℱ\text{.}$ $\square$

Nearby cycles and vanishing cycles

Let $R$ be the ring of a Henselian discrete valuation,
$K$ the field of fractions of $R\text{,}$
$\bar{K}$ the algebraic closure of $K\text{,}$
$k=R/𝔪$ the residue field of $R\text{,}$
$\bar{k}$ the algebraic closure of $k\text{.}$ Let $$S=\text{Spec}\left(R\right),$$ which Deligne [Del1980] calls a "trait". Then $s:\text{spec}\hspace{0.17em}k\to S$ is the closed point of $S\text{,}$
$\eta :\text{spec}\hspace{0.17em}K\to S$ is the geometric point of $S\text{,}$
$\bar{s}:\text{spec}\hspace{0.17em}\bar{k}\to S$ is the localization of $s$ in $S\text{,}$
$\bar{\eta }:\text{spec}\hspace{0.17em}\bar{k}\to S$ is the geometric point localized at $\eta \text{.}$ Let $$f:X⟶S$$ be a (finitely generated, quasiprojective, flat) scheme over $R\text{.}$ Then $X_s=X{\times }_S\text{spec}\left(k\right)$ is the special fiber,
$X_{\eta }=X{\times }_S\text{spec}\left(K\right),$ and
$X_{\bar{\eta }}=X{\times }_S\text{spec}\left(\bar{K}\right)$ is the generic fiber. Let $ℱ$ be a sheaf on $X\text{.}$ Then we have the following diagram where $${ℱ}_s=i^{*}ℱ,{ℱ}_{\eta }=j^{*}ℱ,{ℱ}_{\bar{\eta }}=k^{*}{j}^{*}ℱ\text{.}$$ The nearby cycles complex is $${\mathrm{\Psi }}_fℱ=i^{*}R{j}_{*}{k}_{*}{k}^{*}{j}^{*}ℱ\text{.}$$ The adjunction morphism $ℱ\to R{j}_{*}{k}_{*}{k}^{*}{j}^{*}ℱ$ gives a morphism $${\theta }_f:i^{*}⟶{\mathrm{\Psi }}_f\text{.}$$ The vanishing cycles complex is $${\Phi }_fℱ=\text{cone}(\theta :i^{*}ℱ\to {\mathrm{\Psi }}_fℱ)\text{.}$$ $$\begin{array}{ccc}& & {\mathrm{\Phi }}_fℱ\\ & \searrow & & \nearrow \\ i^{*}ℱ& \stackrel{{\theta }_f}{⟶}& {\mathrm{\Psi }}_fℱ\end{array}$$ which, explicitly, is the complex $${\mathrm{\Phi }}_fℱ=(\cdots \to 0\to {ℱ}_s\stackrel{{\theta }_f}{⟶}{\left({\mathrm{\Psi }}_fℱ\right)}^0\to {\left({\mathrm{\Psi }}_fℱ\right)}^1\to \cdots ),$$ where ${\left({\mathrm{\Psi }}_fℱ\right)}^0$ occurs at the $0^{\text{th}}$ spot. The hypercohomology of ${\mathrm{\Phi }}_fℱ$ describes the deviation between the cohomology of the generic fiber and the special fiber.

Example. Let $S={ℂ}^1,$
$\text{spec}\left(k\right)=0,$
$\text{spec}\left(K\right)={ℂ}^1\backslash \left\{0\right\},$
$\text{spec}\left(\bar{K}\right)=ℂ,$ which is the universal cover of ${ℂ}^1\backslash \left\{0\right\}$ via the projection $p:ℂ\to {ℂ}^1\backslash \left\{0\right\}$ given by $p\left(z\right)=e^z\text{.}$ Let $X$ be a smooth complex algebraic variety. Then we have the diagram $$\begin{array}{ccccccc}{f}^{-1}\left(0\right)& \stackrel{i}{⟶}& X& \stackrel{j}{⟵}& f^{-1}({ℂ}^1\backslash \left\{0\right\})& \stackrel{k}{⟵}& {\stackrel{\sim }{X}}^{*}\\ \downarrow f& & \downarrow f& & \downarrow f& & \downarrow \\ \left\{0\right\}& \hookrightarrow & {ℂ}^1& \hookleftarrow & {ℂ}^1\backslash \left\{0\right\}& \stackrel{p}{⟵}& ℂ\end{array}$$ Note that $f\in \mathrm{\Gamma }(X,{𝒪}_X),$
$Y=f^{-1}\left(0\right)$ is the subvariety of $X$ defined by the single equation $f=0,$
${\stackrel{\sim }{X}}^{*}=X{\times }_{ℂ}ℂ$ is a fiber product via the maps $p$ and $f$ and $j\circ k:{\stackrel{\sim }{X}}^{*}\to X$ is "projection onto the first factor".

Frobenius, the Galois group and Monodromy

The inertia group is the kernel $I$ in $$0⟶I⟶\text{Gal}(\bar{K}/K)⟶\text{Gal}(\bar{k}/k)⟶0\text{.}$$ Suppose $k={𝔽}_q,$ $q=p^n\text{.}$ The Frobenius automorphism is the element of $\text{Gal}({\overline{𝔽}}_q/{𝔽}_q)$ given by $$\begin{array}{ccc}{\overline{𝔽}}_q& ⟶& {\overline{𝔽}}_q\\ x& ⟼& x^q\end{array}$$ The geometric Frobenius $F$ is the inverse of the Frobenius automorphism. The Weil group is the subgroup of $\text{Gal}({\overline{𝔽}}_q/{𝔽}_q)$ given by $$W({\overline{𝔽}}_q/{𝔽}_q)=\left\{F^n\hspace{0.17em}\right|\hspace{0.17em}n\in \mathbb{Z}\}\text{.}$$ Identify $W({\overline{𝔽}}_q/{𝔽}_q)=\mathbb{Z}$ $\text{Gal}({\overline{𝔽}}_q/{𝔽}_q)\hat{\mathbb{Z}},$ the profinite completion of $\mathbb{Z}\text{.}$ The Weil group $W(\bar{K}/K)$ is the inverse image of $\text{Gal}(\bar{k}/k)$ in $\text{Gal}(\bar{K}/K)\text{.}$ There is an exact sequence $$O\to P\to I\stackrel{t}{⟶}\hat{Z}{Z}_{\left(p\prime \right)}\left(1\right)\to 0$$ with $${\hat{Z}}_{\left(p\prime \right)}\left(1\right)=\prod _{\ell \ne p}{\mathbb{Z}}_{\ell }\left(1\right),t=\prod _{\ell \ne p}{t}_{\ell },$$ so that the products are over all primes $\ell \ne p,$
$P$ is a $\text{pro-}p$ group,
$t_{\ell }:I\to {\mathbb{Z}}_{\ell }\left(1\right)$ is the $\ell \text{-component}$ of the map $t\text{,}$
$\text{ker}\hspace{0.17em}{t}_{\ell }$ is a profinite group of order prime to $\ell \text{.}$

Let $(\rho ,V)$ be an $\ell \text{-adic}$ representation of $\text{Gal}(\bar{K}/K)\text{.}$ The local monodromy group is the image $\rho \left(I\right)\text{.}$ The logarithm of the unipotent part of the monodromy is the unique nilpotent endomorphism $$N:V\left(1\right)\to V\text{such that}\rho \left(\sigma \right)=\text{exp}\left(t_{\ell }\left(\sigma \right)N\right),\text{for}\hspace{0.17em}\sigma \in I_1,$$ where $I_1$ is a finite index subgroup of $I\text{.}$ Note that $N$ commutes with the action of $W(\bar{k}/k)\text{.}$

Let $ℱ$ be a ${\bar{\mathbb{Q}}}_{\ell }$ sheaf on $X\text{.}$ Then ${ℱ}_{\bar{\eta }}$ is an $\ell \text{-adic}$ representation of $W(\bar{K}/K)\text{.}$ The sheaf ${ℱ}_0$ on $X_0$ is pointwise pure of weight $\beta$ if, for every $x\in X_0,$ $F$ acts on ${ℱ}_x$ with eigenvalues of absolute value $q^{\beta }\text{.}$

[Del1980,1.8.1 and 1.8.4] Assume that ${ℱ}_0$ is pointwise pure of weight $\beta \text{.}$

(a)	The eigenvalues of $F$ on ${ℱ}_s$ are of absolute value $q^{\alpha }$ with $\alpha \le \beta \text{.}$
(b)	Let $M$ be the filtration induced by the action of the logarithm of the unipotent part of the modnoromy on ${ℱ}_{0\eta }\text{.}$ The representation of $W(\bar{k}/k)$ on ${\text{Gr}}_i^M\left({ℱ}_{0\eta }\right)$ is pure of weight $\beta +i\text{.}$

The Langlands classification

Let $W$ be an index set for the $G$ orbits on $N\text{.}$ Define a partial order on $W$ by $$w_1\le w_2\text{if}{𝕆}_{w_1}\subseteq \overline{{𝕆}_{w_2}}\text{.}$$ The lower order ideals in this partial order correspond to the closed $G\text{-invariant}$ subsets of $N\text{.}$

Let $𝒞$ be an abelian category with a fixed collection of subcategories ${𝒞}_J,$ indexed by the lower order ideals $J$ in $W\text{.}$ For any lower order ideals $J_1\subseteq J_2$ let $${𝒞}_{J_2\backslash J_1}=\frac{{𝒞}_{J_2}}{{𝒞}_{J_1}}$$ be the corresponding quotient category. The category $𝒞$ is $W\text{-stratified}$ if

(a)	${𝒞}_{J_1\cap J_2}={𝒞}_{J_1}\cap {𝒞}_{J_2},$
(b)	${𝒞}_{J_1\cup J_2}$ is the smallest subcategory that contains ${𝒞}_{J_1}$ and ${𝒞}_{J_2},$
(c)	If $J_1\subseteq J_2$ then ${𝒞}_{J_1}\subseteq {𝒞}_{J_2},$
(d)	If $J_1\subseteq J_2$ then the quotient map $$j_{J_2\backslash J_1}^{*}:{𝒞}_{J_2}⟶{𝒞}_{J_2\backslash J_1}\text{has left and right adjoints}{j}_{(J_2\backslash J_1)!}\text{and}{j}_{(J_2\backslash J_1)*}\text{.}$$
(e)	imbedding condition see [BBe1993, 2.6.4.(iii)].

Think of $𝒞$ as the category of modules associated to $X$ and think of ${𝒞}_J$ as the subcategory of modules supported on the closed $G\text{-invariant}$ subset of $X$ corresponding to $J\text{.}$ Define $$j_{i!}\text{and}{j}_{i*}\text{to be the left and right adjoints to}{j}_i^{*}:{𝒞}_{\bar{i}}\to {𝒞}_i\text{.}$$ For each pair $(i,L)$ where $L$ is an irreducible object of ${𝒞}_i$ define the corresponding standard and costandard objects of $𝒞$ by $${ℳ}_{(i,L)}^{!}=j_{i!}\left(L\right)\text{and}{ℳ}_{(i,L)}^{*}=j_{i*}\left(L\right),$$ respectively. Then $$j_{i!*}:{𝒞}_i\to {𝒞}_{\bar{i}}\text{given by}{j}_{i!*}=\text{Im}(j_{i!}\to j_{i*})$$ transforms irreducible objects into irreducible ones. If any object of $𝒞$ has finite length (or, equivalently, all objects of the ${𝒞}_i$ have finite length) then the irreducible objects of $𝒞$ are $$j_{i!*}\left(L\right)\text{running over all pairs}\hspace{0.17em}(i,L),$$ where $L$ is an irreducible object in ${𝒞}_i\text{.}$

References

[BBe1993] A. Beilinson and J. Bernstein, A proof of Jantzen conjectures, in I.M. Gelfand seminar, S. Gelfand and S. Gindikin eds., Adv. in Soviet Math. 16 Part 1 (1993), 1–50.

[BBD1982] A. Beilinson, J. Bernstein, and P. Deligne, Faisceaux pervers, in Analyse et topologie sur les espaces singuliers (I), Astérisque 100, Soc. Math. France, 1982.

[CGi1433132] N. Chriss and V. Ginzburg, Representation theory and complex geometry, Birkhäuser Boston, Inc., Boston, MA, 1997. x+495 pp. ISBN: 0-8176-3792-3, MR1433132 and MR2838836.

[Del1980] P. Deligne, La Conjectture de Weil. II, Publ. Math. IHES 52 (1980), 137-252.

[FKi1988] E. Freitag, R. Kiehl, Etale cohomology and the Weil conjecture, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3 Folge, Band 13, Springer-Verlag, 1988.

Notes and References

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

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

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