%Strg c t p fuer pdflatex \documentclass[12pt,german,xcolor=dvipsnames]{beamer} \usepackage{graphicx} \usepackage{amsmath} \usepackage{rotating} \usepackage[german]{babel} \usepackage{enumerate} \usepackage{xspace} %\usepackage{showkeys} \usepackage{young} \usepackage{makeidx} \usepackage{newlfont} \usepackage{amssymb} \usepackage{amsxtra} \usepackage{amsfonts} \usepackage{amscd} \usepackage{mathrsfs} \usepackage{stmaryrd} \usepackage{psfrag} \usepackage{color} \usepackage[latin1]{inputenc} \usepackage{times} \usepackage[mathscr]{euscript} \usepackage[all]{xy} \usepackage[active]{srcltx} \input epsf \DeclareMathAlphabet{\mathpzc}{OT1}{pzc}{m}{it} \setlength{\arraycolsep}{2pt} \input{newcommands} \title[]{~\\[2ex] The Hecke Algebra and its Categorification} \author[]{Wolfgang Soergel} \institute[]{\inst{} Mathematisches Institut\\ Universit\"at Freiburg\\[4ex] \vspace*{.9cm} %\textcolor{red}{\\ allgemeine Fragen/Hinweise hier positionieren} } \date[]{\small \hbox{Januar 2015}} \beamersetuncovermixins{\opaqueness<1>{25}}{\opaqueness<2->{15}} % ================================================== % ================================================== \begin{document} % ================================================== % ================================================== \titlepage \begin{frame} The formalism of Hecke algebras\\[3mm] \begin{itemize} \item $G$ finite group, $(\DZ G,\ast)$ group ring \\[2mm]\item $B\subset G$ subgroup, $^B(\DZ G)^B$ the $B$-biinvariant functions, stable under $\ast$ and under $\ast_B\pdef \ast/|B|$ \\[2mm]\item $\mathcal H=\mathcal H(G,B)\pdef (^B(\DZ G)^B,\ast_B) $ the {\bf Hecke algebra} \\[2mm]\item Unit element the characteristic function $\underline B=1_{\mathcal H}$ of $B$ \\[2mm]\item $\DZ$-basis of $\mathcal H$ are the characteristic functions $\underline D$ for $D$ runnig over all double cosets \\[2mm]\item For $V$ a $G$-module $V^B$ is an $\mathcal H$-module \end{itemize} \end{frame} \begin{frame} The algebra of Hecke operators\\[3mm] \begin{itemize} \item $G=\op{GL}(2;\DR)^+$ acts on the upper half plane $\mathbb H^+$ \\[2mm] \item $G=\op{GL}(2;\DR)^+$ acts on $\mathcal O^{\op{an}}(\mathbb H^+)(\diff z)^{\otimes k}$ \\[2mm]\item Take $B=\op{SL}(2;\DZ)$ \\[2mm]\item $\mathcal H(G,B)$ acts on $\left(\mathcal O^{\op{an}}(\mathbb H^+)(\diff z)^{\otimes k}\right)^{\op{SL}(2;\DZ)}$ \\[2mm]\item These are the classical Hecke operators acting on modular functions \\[2mm]\item Sure these groups are infinite. Then should use the more general definition $$\cal{H} (G,B) \pdef \op{End}^{G}_\DZ(\op{prod}^{G}_{B} \DZ)^{\op{opp}}$$ \end{itemize} \end{frame} \begin{frame} Towards the Iwahori-Matumoto Hecke algebras\\[3mm] \begin{itemize} \item Take $G=\op{GL}(n;\mathbb F_q)$ and $ B$ upper triangular matrices \\[2mm]\item Bruhat decomposition $G=\bigsqcup_{x\in W}BxB$ for $W= \mathcal S_n$ the permutation matrices \\[2mm]\item The $T_x\pdef \underline{BxB}$ form a basis of the Hecke algebra \end{itemize} \end{frame} \begin{frame} For $s\in S\pdef\{(i,i+1)\text{ transposition}\mid 1\leq i1$ off the diagonal \\[2mm]\item For $x\in W$ put $l(x)$ minimal number of $s\in S$ needed to express $x$ \\[2mm]\item Any finite group $W$ generated by reflections is always part of a Coxeter system $(W,S)$ \end{itemize} \end{frame} \begin{frame} \begin{itemize} \item For any Coxeter System $(W,S)$ there is a Hecke algebra $\mathcal H=\mathcal H(W,S)\pdef \bigoplus_{x\in W} \DZ[q] T_x$ with $$T_s^2=(q-1)T_s +q\text{ for }s\in S$$ $$T_xT_y=T_{xy}\text{ if }l(x)+l(y)=l(xy)$$ \\[2mm]\item Called {\bf Iwahori-Matsumoto} Hecke algebra \\[2mm]\item Specializes to group ring $\DZ W$ for $q\mapsto 1$ \end{itemize} \end{frame} \begin{frame} Alternative description of the Hecke algebra $\mathcal H(W,S)$ as $\DZ[q]$-ringalgebra with generators $T_s$ for $s\in S$ subject to the relations $T_s^2=(q-1)T_s +q$ and braid relations $$T_sT_t\ldots =T_tT_s\ldots$$ with $m(s,t)$ factors on both sides \end{frame} \begin{frame} Categorification of Hecke algebra $\mathcal H(W,S)$ by bimodules\\[3mm] \begin{itemize} \item Let $(W,S)$ be a Coxeter system \\[2mm]\item Choose a representation $W\looparrowright V$ wich is \begin{itemize} \item finite dimensional \\[2mm]\item over an infinite field $k$ with $\op{char}k\neq 2$ \\[2mm]\item exactly the conjugates $t=wsw^{-1}$ of elements of $s\in S$ have fixed point spaces of codimension one \end{itemize} \item Call such a representation {\bf reflection faithful} \\[2mm]\item Typical example: Symmetric group $\mathcal S_n$ permuting the coordinates of $k^n$ \end{itemize} \end{frame} \begin{frame} Categorification of Hecke algebra $\mathcal H(W,S)$ by bimodules\\[3mm] \begin{itemize} \item Choose $W\looparrowright V$ reflection faithful representation \\[2mm]\item Put $R\pdef \mathcal O(V)$ a polynomial ring \\[2mm]\item Let $R\op{-Mod_\DZ-}R$ be the category of $\DZ$-graded $R$-bimodules or more precisely $R\otimes_kR$-modules \\[2mm]\item Let $$R\op{-Modbf_\DZ-}R$$ be the subcategory of graded bifinite bimodules \\[2mm]\item Bifinite means finitely generated from the left and from the right \end{itemize} \end{frame} \begin{frame} Categorification of Hecke algebra $\mathcal H(W,S)$ by bimodules\\[3mm] \begin{itemize} \item Let $\langle R\op{-Modbf_\DZ-}R\rangle $ be the split Grothendieck group \\[2mm]\item It becomes a ring under $\otimes_R$ \\[2mm]\item {\bf Categorification Theorem:} There is exactly one ring homomorphism $$\mathcal E :\mathcal H \ra \langle R\op{-Modbf_\DZ-}R\rangle$$ such that we have $\mathcal E (T_{s}+1) = \langle R \otimes_{R^{s}} R\rangle \; \forall s \in \mathcal S$ and $\mathcal E (q) = \langle R\langle -1\rangle\rangle$ \\[2mm]\item Notation $(M\langle n\rangle)_i=M_{i+n}$ for grading shift \end{itemize} \end{frame} \begin{frame} Sketch of proof of bimodule-categorification\\[3mm] \begin{itemize} \item Recall quadratic relation $T_s^2=(q-1)T_s +q$ \\[2mm]\item Rewrite to $(T_s+1)^2=(q+1)(T_s+1)$ \\[2mm]\item Need $\langle R \otimes_{R^{s}} R\rangle^{2}= \langle R\langle -1\rangle \oplus R\rangle\langle R \otimes_{R^{s}} R\rangle$ \\[2mm]\item $( R \otimes_{R^{s}} R)\otimes_R( R \otimes_{R^{s}} R)\cong ( R\langle -1\rangle \oplus R)\otimes_R( R \otimes_{R^{s}} R)$ \\[2mm]\item $ R \otimes_{R^{s}} R \otimes_{R^{s}} R\cong (R \otimes_{R^{s}} R)\langle -1\rangle \oplus (R \otimes_{R^{s}} R)$ \\[2mm]\item Follows from recalling in the middle left\\ $R=\alpha R^s\oplus R^s\cong R^s\langle -1\rangle\oplus R^s$ \\with $\alpha\in V^\ast$ equation of $V^s$ \\[2mm]\item So only need to check braid relations for bimodules \\[2mm]\item Need only to argue for dihedral groups. Omitted. \end{itemize} \end{frame} \begin{frame} Categorification of Kazhdan-Lusztig basis\\[3mm] \begin{itemize} \item Extend scalars in Hecke algebra $\mathcal H$ from $\DZ[q]$ to $\DZ[v,v^{-1}]$ by $q=v^{-2}$ \\[2mm]\item Kazhdan-Lusztig constructed a canonical basis $(C_x)_{x\in W}$ of $\mathcal H_v$ as a $\DZ[v,v^{-1}]$-module \\[2mm]\item Regrade $R$ to sit only in even degrees to get categorification map $\mathcal E: \mathcal H_v\ra \langle R\op{-Modbf_\DZ-}R\rangle$ \\[2mm]\item {\bf Indecomposable Bimodule Theorem:} There exist indecomposable bimodules $B_x\in R\op{-Modbf_\DZ-}R$ such that $\mathcal E(C_x)=\langle B_x\rangle$ \\[2mm]\item In words: The elements of the Kazhdan-Lusztig canonical basis correspond under the categorification theorem to indecomposable bimodules \end{itemize} \end{frame} \begin{frame} Definition of Kazhdan-Lusztig basis\\[3mm] \begin{itemize} \item Put $H_x=v^{l(x)}T_x$ \\[2mm]\item $C_x \in H_x +\sum_y v\DZ[v]H_y$ and $d(C_x)=C_x$ is selfdual, uniquely determines the canonical basis element $C_x$ \\[2mm]\item Duality $d:\mathcal H_v\ra \mathcal H_v$ the unique ring automorphism, which fixes $H_s+v$ for $s\in S$ and maps $d:v\mapsto v^{-1}$ \\[2mm]\item In particular $C_s=H_s+v$ for $s\in S$ a simple reflection \end{itemize} \end{frame} \begin{frame} Discussion of categorification of KL-basis \begin{itemize} \item Take simple reflections $s,t,\ldots,u\in S$ \\[2mm]\item Form the bimodules $R\otimes_{R^s}R\otimes_{R^t}R\ldots\otimes_{R^u}R$ \\[2mm]\item Krull-Schmid decompose those bimodules: Get very special indecomposable bimodules $B_x$ categorifying the Kazhdan-Lusztig basis \\[2mm]\item Call the graded bimodules $R\otimes_{R^s}R\otimes_{R^t}R\ldots\otimes_{R^u}R$ and all you get from them by taking finite direct sums, direct summands and grading shifts {\bf special bimodules} and denote the monoidal category of those $$R\op{-SMod_\DZ-}R$$ Its indecomposables are precisely the $B_x\langle n\rangle$. \end{itemize} \end{frame} \begin{frame} Positivity Corollaries of categorification\\[3mm] \begin{itemize} \item $C_x C_y\in \sum_z \DN[v,v^{-1}] C_z$ since $B_x\otimes_R B_y$ is an actual bimodule, decomposes as $$B_x\otimes_R B_y=\bigoplus_{z,n} B_z\langle n\rangle^{m(z,n)}$$ \\[2mm]\item $C_x=\sum_y P_{x,y}(v)H_y$ with $P_{x,y}(v)\in \DZ[v]$ the {\bf Kazhdan-Lusztig polynomials} \\[2mm]\item Coefficients of Kazhdan-Lusztig-Polynomials are non-negative, since they can be interpreted as $\op{rk} \op{Hom}_{R-R}(\mathcal O(\Gamma(x)), B_y)$ \\[2mm]\item Here $\Gamma(x)\subset V\times V$ is the graph of $x$ and $\mathcal O(\Gamma(x))$ the regular functions on $\Gamma(x)$, a quotient of $\mathcal O(V\times V)=R\otimes R$. Put another way, $\mathcal O(\Gamma(x))=R$ as left $R$-module with the right $R$-action twisted by $x$ \end{itemize} \end{frame} \begin{frame} \begin{itemize} \item Example: $w_\circ\in W$ longest element of finite reflection group. $C_{w_\circ}=v^{l(w_\circ)}\sum_{x\in W}T_x=\sum_{x\in W}v^{l(w_\circ)-l(x)}H_x$ $$B_{w_\circ}=\mathcal O\left(\bigcup_{x\in W}\Gamma(x)\right)$$ is the bimodule of all regular functions on the union of the graphs of all Weyl group elements $\Gamma(x)\subset V\times V$ \\[2mm]\item In general $B_x$ is still supported on $\bigcup_{y\leq x}\Gamma(y)$ \\[2mm]\item If $C_{x}=\sum_{y\leq x}v^{l(x)-l(y)}H_y$, then $B_{x}=\mathcal O\left(\bigcup_{y\leq x}\Gamma(y)\right)$ \end{itemize} \end{frame} \begin{frame} Application to representation theory\\[3mm] \begin{itemize} \item $\mathfrak g$ a semisimple complex Lie algebra, $Z\subset U(\mathfrak g)$ the center of its enveloping algebra \\[2mm]\item $\mathcal M\subset \mathfrak g\op{-Mod}$ the category of all representations of $\mathfrak g$ locally finite under $Z$ \\[2mm]\item $\mathcal P$ the category of all functors $\mathcal M\ra \mathcal M$ isomorphic to a direct summand of some functor $E\otimes_\DC$ for $E$ finite dimensional representation, so-called {\bf projective functors} \\[2mm]\item Equivalence of categories between \{indecomposable projective functors starting and ending with the trivial central character\} and $\{\hat B_x\mid x\in W\}\subset \hat R \op{-Mod-}\hat R$ \end{itemize} \end{frame} \begin{frame} Application to representation theory, variant\\[3mm] \begin{itemize} \item $\mathcal O_\circ\subset \mathfrak g\op{-Mod}$ principal block of BGG-category $\mathcal O$ \\[2mm]\item Equivalence of categories between \{indecomposable projectives of $\mathcal O_\circ$\} and $\{B_x\otimes_R\DC\mid x\in W\}\subset R \op{-Mod}$ \\[2mm]\item Gives new proof of KL-conjecture on Jordan-H"older multiplicities of Verma modules \\[2mm]\item $$\op{Der}^{\op{b}}(\mathcal O_\circ)\cong \op{Hot}^{\op{b}}(\op{proj}\mathcal O_\circ)\cong \op{Hot}^{\op{b}}(R\op{-SMod})$$ for $R\op{-SMod}\subset R\op{-Mod}$ the subcategory of all $B\otimes_R\DC$ for $B\in R\op{-SMod-}R$ \\[2mm]\item Can define graded version $\mathcal O_\circ^\DZ$ of $\mathcal O_\circ$ formally such that $\op{proj}\mathcal O_\circ^\DZ= R\op{-SMod_\DZ}$ \end{itemize} \end{frame} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{frame} Categorification of $\DN$ \begin{itemize} \item $k$ a field \\[2mm]\item $\op{dim}:\op{Modf}_k\ra \DN$ ``decategorification'' \\[2mm]\item Multiplication corresponds to tensor product $$\op{dim}(V\otimes W)=(\op{dim}V) (\op{dim} W)$$ \end{itemize} \end{frame} \begin{frame} Categorification of $\op{Ens}(X,\DN)=\op{Maps}(X,\DN)$ for $X$ a set \begin{itemize} \item $k$ a field and $\op{Mod}_k/X\supset\op{Modf}_k/X $ sheaves on the discrete set $X$ alias families $(\mathcal F_x)_{x\in X}$ of vector spaces respectively finitely generated vector spaces \\[2mm]\item $\op{Dim}:\op{Modf}_k/X\ra \op{Ens}(X,\DN)$ ``decategorification'' \\[2mm]\item Multiplication corresponds to tensor product $$\op{Dim}(\mathcal F\otimes \mathcal G) =(\op{Dim}\mathcal F) (\op{Dim} \mathcal G)$$ \end{itemize} \end{frame} \begin{frame} Categorification of maps \begin{itemize} \item $f:X\ra Y$ map of finite sets leads to morphisms $$ \op{Ens} (X, \mathbb N) \begin{array}{c}f_{!} \\[-1ex]\longrightarrow \\[-1ex] \longleftarrow \\[-1ex] f^\ast \end{array} \op{Ens} (Y,\mathbb N) $$ called pull-back and integration along the fibres \\[2mm]\item $|X|1= c_!c^\ast 1$ for $c:X\ra\op{pt}$ constant map \end{itemize} \end{frame} \begin{frame} \begin{itemize} \item $f:X\ra Y$ map of finite sets leads to functors \begin{displaymath} \op{Modf}_k/X \begin{array}{c}f_{!} \\[-1ex]\longrightarrow \\[-1ex] \longleftarrow \\[-1ex] f^\ast \end{array} \op{Modf}_k/Y \end{displaymath} called pull-back and integration along the fibres \\[2mm]\item $(f^\ast \mathcal G)_x\pdef \mathcal G_{f(x)}$ and $(f_! \mathcal F)_y\pdef \bigoplus_{x \in f^{-1} (y)} \mathcal F_x$ \\[2mm]\item Commutative diagrams \begin{displaymath} \begin{array}{ccc} \op{Modf}_k/X & \begin{array} {c}f_{!}\\[-1ex]\longrightarrow \\[-1ex] \longleftarrow \\[-1ex] f^\ast \end{array} & \op{Modf}_k/Y\\ {\scriptstyle \op{Dim}} \downarrow & & \downarrow {\scriptstyle \op{Dim}}\\ \op{Ens}(X, \mathbb N)& \begin{array} {c}f_{!}\\[-1ex]\longrightarrow \\[-1ex] \longleftarrow \\[-1ex] f^\ast \end{array} & \op{Ens} (Y, \mathbb N) \end{array} \end{displaymath} \end{itemize} \end{frame} \begin{frame} Grothendieck function-sheaf correspondence \begin{itemize} \item To $X_\circ$ variety over $\mathbb F_q$ and $\ell$ prime $\neq\op{char}\mathbb F_q$ associate $\op{Der}^c (X_\circ ; \mathbb Q_l)$ triangulated $\DQ_\ell$-category \\[2mm]\item Called \glqq cohomologically constructible complexes of \'etale sheaves on $X_\circ$\grqq\ \\[2mm]\item Define map $$\op{Tr}:\op{Der}^c (X_\circ ; \mathbb Q_l)\ra \op{Ens}(X_\circ (\mathbb F_{q}), \mathbb Q_l)$$ \\[2mm]\item $ \op{Tr} (\mathcal F_\circ ) : x \mapsto \sum_i (-1)^i \op{Tr}(\op{F}_g^\ast \mid \mathcal H^i \mathcal F_x) $ with $\mathcal F \pdef \mathcal F_\circ \times_{\mathbb F_q} \mathbb F$ sheaf on $X \pdef X_\circ \times_{\mathbb F_q} \mathbb F$ and $\op{F}_g$ Frobenius \end{itemize} \end{frame} \begin{frame} Grothendieck function-sheaf correspondence \begin{itemize} \item To $f:X_\circ\ra Y_\circ$ morphism of varieties over $\mathbb F_q$ associate triangulated functors $f_!, f^\ast$ fitting into a commutative diagram \begin{displaymath} \begin{array}{ccc} \op{Der}^c (X_\circ ; \mathbb Q_l) & \begin{array} {c}f_{!}\\[-1ex]\longrightarrow \\[-1ex] \longleftarrow \\[-1ex] f^\ast \end{array} & \op{Der}^c (Y_\circ ; \mathbb Q_l)\\ {\scriptstyle \op{Tr}} \downarrow & & \downarrow {\scriptstyle \op{Tr}}\\ \op{Ens}(X_\circ (\mathbb F_{q}), \mathbb Q_l)& \begin{array} {c}f_{!}\\[-1ex]\longrightarrow \\[-1ex] \longleftarrow \\[-1ex] f^\ast \end{array} & \op{Ens} (Y_\circ (\mathbb F_{q}), \mathbb Q_l) \end{array} \end{displaymath} \\[2mm]\item For $c:X_\circ\ra \op{pt}_\circ$ this specializes to $|X_\circ(\mathbb F_q)|1= c_!c^\ast 1=c_!c^\ast \op{Tr}(\DQ_l)= \op{Tr}(c_!c^\ast\DQ_l)=$\\ $=\sum_i (-1)^i \op{tr} (\op{F}_g | {\op{H}}^i_c (X; \mathbb Q_l))$ Grothendieck-Lefschetz \end{itemize} \end{frame} \begin{frame} \begin{itemize} \item Let $G$ be a finite group. The multiplication in the group ring could for $f,g\in \op{Ens}(G,\DZ)$ be written as $$f\ast g=\op{mult}_! ((\op{pr}_1^\ast f)(\op{pr}_2^\ast g))$$ with $\op{pr}_1,\op{pr}_2,\op{mult}: G\times G\ra G$ the projections and the multiplication. \\[2mm]\item A natural candidate for the categorification of the group ring in case $G=G_\circ(\mathbb F_q)$ is thus $\op{Der}^c (G_\circ ; \mathbb Q_l)$ with the convolution functor $$\mathcal F\ast \mathcal G\pdef \op{mult}_! ((\op{pr}_1^\ast \mathcal F)\otimes(\op{pr}_2^\ast \mathcal G))$$ \end{itemize} \end{frame} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{frame} \begin{itemize} \item Recall $G=\op{GL}(n;\mathbb F_q)$ and $ B$ upper triangular matrices and $$\mathcal H_q=(^B\op{Ens}(G,\DZ)^B, \ast/|B|)$$ functions on the group, $B$-invariant from both sides \\[2mm]\item So a natural categorification ought to be some both sides equivariant derived category of \'{e}tale sheaves $$\op{Der}^c_{B_\circ\times B_\circ} (G_\circ ; \mathbb Q_l)$$ \\[2mm]\item Let's be a bit less perfect and try for the usual topology version of the equivariant derived category $$\op{Der}^c_{B\times B} (G ; \mathbb Q)$$ with $G=\op{GL}(n;\mathbb C)$ and metric topology \end{itemize} \end{frame} \begin{frame} First discuss equivariant cohomology \begin{itemize} \item $G\looparrowright X$ topological group acting on topological space \\[2mm]\item ${\op{H}}^\ast(X/G)$ not a good concept \\[2mm]\item For $f:X\ra Y$ is a morphism of $G$-spaces, which is a fibration with contractible fibers, need not have ${\op{H}}^\ast(Y/G)\sira{\op{H}}^\ast(X/G)$ \\[2mm]\item Example: $\DR\sra\op{pt}$ with $\DZ$-action \\[2mm]\item Better concept ${\op{H}}^\ast_G(X)\pdef {\op{H}}^\ast({\op{E}}G\times_GX)$ equivariant cohomology \\[2mm]\item ${\op{E}}G$ contractible with topologially free $G$-action, the universal bundle over the classifying space \end{itemize} \end{frame} \begin{frame} Examples for equivariant cohomology \begin{itemize} \item ${\op{H}}^\ast_G(X)\pdef {\op{H}}^\ast({\op{E}}G\times_GX)$ \\[2mm]\item $G\looparrowright X$ topological group acting freely on topological space, then ${\op{H}}^\ast_G(X)={\op{H}}^\ast(X/G)$ \\[2mm]\item ${\op{H}}^\ast_G(\op{pt})= {\op{H}}^\ast({\op{E}}G\times_G\op{pt})= {\op{H}}^\ast({\op{E}}G/G)={\op{H}}^\ast({\op{B}}G)$ the ring of characteristic classes \\[2mm]\item ${\op{H}}^\ast_{\DC^\times}(\op{pt})={\op{H}}^\ast(\DP^\infty\DC)= \DZ[t]$ with $\op{deg}t=2$ \\[2mm]\item ${\op{H}}^\ast_{B}(\op{pt})=\DZ[t_1,\ldots,t_n]$ with $\op{deg}t_i=2$ for $B\subset\op{GL}(n;\DC)$ upper triangular matrices \\[2mm]\item For $P\ra X$ a principal $G$-bundle, pullback ${\op{H}}^\ast_{G}(\op{pt})\ra {\op{H}}^\ast_{G}(P)={\op{H}}^\ast(P/G)= {\op{H}}^\ast(X)$ gives its characteristic classes \end{itemize} \end{frame} \begin{frame} Derived category for $X$ a topological space \begin{itemize} \item $\op{Der}(X)=\op{Der}(\op{Ab}/X)$ derived category of abelian sheaves on $X$ \\[2mm]\item $f:X\ra Y$ continous map of locally compact Hausdorff spaces gives triangulated functors $f_!: \op{Der}(X)\ra \op{Der}(Y)$ and $f^\ast: \op{Der}(Y)\ra \op{Der}(X)$ \\[2mm]\item For $c:X\ra\op{pt}$ constant map, get $c_!c^\ast \DZ={\op{H}}^\ast_c(X)$ \\[2mm]\item $c^\ast \DZ\defp \underline{X}$ the constant sheaf on $X$ \\[2mm]\item $\op{Der}_X(\underline{X}, \underline{X}[\ast])={\op{H}}^\ast(X)$ the cohomology ring of $X$ \\[2mm]\item $\op{Der}_X(\underline{X}, \mathcal F[\ast])={\op{H}}^\ast(X;\mathcal F) =\mathbb H\mathcal F$ (hyper)cohomology of the sheaf(complex) $\mathcal F$ \\[2mm]\item $\mathbb H\mathcal F$ is a $ {\op{H}}^\ast(X)$-module \end{itemize} \end{frame} \begin{frame} Equivariant derived category \begin{itemize} \item $G\looparrowright X$ topological space with $G$-action \\[2mm]\item $\op{Der}_G(X)=\{\mathcal F\in \op{Der}({\op{E}}G\times_GX)| \begin{array}[t]{l} \exists \mathcal G\in \op{Der}(X)\text{ such}\\ \text{that } p^\ast\mathcal F\cong q^\ast\mathcal G\} \end{array}$ with ${\op{E}}G\times_G X\stackrel{p}{\leftarrow} {\op{E}}G\times X\stackrel{q}{\ra}X$ \\[2mm]\item For $\mathcal F\in \op{Der}_G(X)$ get $\mathbb H_G\mathcal F\in {\op{H}}^\ast_G(X)\op{-Mod}$ \\[2mm]\item $f^\ast$ and $f_!$ for equivariant maps of locally compact Hausdorff spaces \\[2mm]\item $\op{Der}_G(X)=\op{Der}(X/G)$ in the case of a topologically free action \\[2mm]\item $\op{Der}_G(\op{pt})\subset \op{dgDer-}({\op{H}}^\ast_G(\op{pt}),d=0)$ for $G$ a complex connected algebraic group \\[2mm]\item $\op{Der}_B(\op{pt})\subset \op{dgDer-}\DZ[t_1,\ldots,t_n]$ \end{itemize} \end{frame} \begin{frame} The natural categorification of the Hecke algebra \begin{itemize} \item Again $G=\op{GL}(n;\DC)$ with $B$ the upper triangular matrices \\[2mm]\item The natural categorification of the Hecke algebra $\mathcal H=(^B(\DZ G)^B,\ast_B)$ is the constructible equivariant derived category with convolution $$(\op{Der}_{B\times B}^c(G),\ast_B)$$ \\[2mm]\item The convolution is $$\mathcal F\ast_B\mathcal G\pdef \op{mult}_!\op{desc} ((\op{pr}_1^\ast \mathcal F)\otimes (\op{pr}_2^\ast \mathcal G))$$ $\op{pr}_i:G\times G\ra G$\\ $\op{desc}: \op{Der}_{B\times B\times B\times B}^c(G\times G)\ra \op{Der}_{B\times B}^c(G\times_B G)$\\ $\op{mult}:G\times_B G\ra G$ \end{itemize} \end{frame} \begin{frame} Now need intersection cohomology \begin{itemize} \item For $X\As \DP^n\DC$ a smooth irreducible complex projective algebraic variety the cohomology ${\op{H}}^\ast(X)$ has remarkable properties: \begin{itemize} \item Poincar\'{e} duality \item Hard Lefschetz \item Hodge Diamond \item Positivities \end{itemize} \item For $X\As \DP^n\DC$ an non-smooth irreducible complex projective algebraic variety {\bf intersection cohomology} ${\op{IH}}^\ast(X)$ continues to have these properties \\[2mm]\item For $X$ smooth, ${\op{IH}}^\ast(X)={\op{H}}^\ast(X)$ \\[2mm]\item In general ${\op{IH}}^\ast(X)$ is an ${\op{H}}^\ast(X)$-module \end{itemize} \end{frame} \begin{frame} Intersection cohomology complex \begin{itemize} \item For $X$ irreducible complex algebraic variety can still define intersection cohomology ${\op{IH}}^\ast(X)$ \\[2mm]\item Formally ${\op{IH}}^\ast(X)=\mathbb H \mathcal{IC}_X$ for $\mathcal{IC}_X\in \op{Der}(X)$ the {\bf intersection cohomology complex} \\[2mm]\item Aside: For $\mathcal D$-modules have the Riemann-Hilbert correspondence, a fully faithful triangulated functor $$\op{RH}:\op{Der}_{\op{hol, reg}}^{\op{b}}(\mathcal D_X\op{-Mod}^{\op{qc}}) \hra \op{Der}(X)$$ \\[2mm]\item The unique simple $\mathcal D_X$-module restricting to $\mathcal O_U$ on any open smooth subset $U\co X$ gets mapped by RH to $\mathcal{IC}_X$ \end{itemize} \end{frame} \begin{frame} \begin{itemize} \item Back to $G=\op{GL}(n;\DC)\supset B$ with $G=\bigsqcup_{x\in W}BxB$ for $W= \mathcal S_n$ the permutation matrices \\[2mm]\item Consider $\mathcal{IC}_x\defp i_{x!}\mathcal{IC}_{\overline{BxB}}$ for $i_x: \overline{BxB}\hra G$ intersection cohomology complex of Schubert variety \\[2mm]\item All finite direct sums of shifts of $\mathcal{IC}_x$ form an additive subcategory $\op{Der}^{ss}_{B\times B}(G)\subset \op{Der}_{B\times B}(G)$ of ``perversely semisimple complexes'' \\[2mm]\item This subcategory is even stable under convolution, due to the so-called decomposition theorem \\[2mm]\item {\bf Theorem:} The functor of hypercohomology gives an equivalence of monoidal categories $$ \begin{array}{cccc} \mathbb H_{B\times B}: &(\op{Der}^{ss}_{B\times B}(G),\ast_B)&\sirra& (R\op{-SMod_\DZ-}R,\otimes_R)\\[2mm] &\mathcal{IC}_x&\mapsto&B_x[\op{dim}B]\\ \end{array} $$ \end{itemize} \end{frame} \begin{frame} Here $\mathbb H_{B\times B}: (\op{Der}^{ss}_{B\times B}(G),\ast_B)\sirra (R\op{-SMod_\DZ-}R,\otimes_R)$ is defined using the identifications $$ \begin{array}{ccc} {\op{H}}^\ast_{B\times B}(\op{pt})&\sra&{\op{H}}^\ast_{B\times B}(G)\\[2mm] \wr\da&&\da\wr\\[2mm] \mathcal O(V\times V)&\sra&\mathcal O\left(\bigcup_{x\in W}\Gamma(x)\right)\\[2mm] \wr\da&&\da\wr\\[2mm] R\otimes R&\sra& R\otimes_{R^W} R \end{array} $$ for $V=\op{Lie}T$ and $T\subset B$ a maximal torus and degrees on $\mathcal O$ doubled to match cohomological degrees. \end{frame} \begin{frame} \frametitle{COMMERCIAL FOR TWO THEOREMS 6} %\frametitle{\large\bf\red COMMERCIAL FOR THEOREM 6} \begin{itemize} \item in [S, \emph{Universelle$\ldots$}, Math. Ann. \textbf{284} (1989)] tdo-case \item in [S, \emph{The prime\dots}, Math. Z. \textbf{204} (1990)] general \end{itemize} $G$ be a connected complex affine algebraic group,\\ $B$ a closed subgroup, $X=G/B$ the homogeneous space,\\ $n=\op{dim}X$ its dimension, $x\in G/B$ the natural base point,\\ $V,W$ finite dimensional rational representations of $B$,\\ $ \mathcal V,\mathcal W$ the sheaves of sections of the associated bundles.\\[2mm] Then the action leads to an $G$-equivariant isomorphism $$\Gamma(X; \op{Dif}(\mathcal V,\mathcal W))\sira \op{Hom}_\DC(\op{H}^n_x(X;\mathcal V), \op{H}^n_x(X;\mathcal W))^{G\op{-alg}}_B $$ \begin{itemize} \item Have $U(\mathfrak g)\otimes_{U(\mathfrak b)}\left(V\otimes_\DC\bigwedge^{\op{max}}( \mathfrak g/\mathfrak b)\right)\sira \op{H}^n_x(X;\mathcal V)$ \item Can replace $G\op{-alg}$ by $\mathfrak g\op{-finite}$ if $G$ is simply connected \item $B$-compatibility is automatic for $B$ connected \end{itemize} \end{frame} \begin{frame} Summing up: $$ \begin{array}{ccccc} \langle \op{Der}^{ss}_{B\times B}(G),\ast_B\rangle&\sira& \langle R\op{-SMod_\DZ-}R,\otimes_R\rangle &\stackrel{\sim}{\leftarrow}& \mathcal H_v\\[2mm] \mathcal{IC}_x[-\op{dim}B]&\mapsto&B_x&\leftmapsto& C_x\\[2mm] \text{intersection}&&\text{special}&&\text{canonical}\\ \text{cohomology}&&\text{bimodule}&&\text{basis}\\ \end{array} $$ Original motivation: Sheaf-function-correspondence $$ \begin{array}{ccc} (\op{Der}_{B_\circ\times B_\circ}(G_\circ;\mathbb Q_\ell),\ast_{B_\circ})&\ra& \mathcal H_q\\[2mm] \mathcal{IC}_x&\mapsto& v^?C_x \end{array} $$ This was the starting point of Kazhdan-Lusztig \end{frame} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{frame} More categorification of the Hecke algebra \begin{itemize} \item Given $X$ a complex algebraic variety can define variant $\op{MDer}(X)$ of $\op{Der}(X)$ with functors $f^\ast, f_!$ as before such that $\op{MDer}(\op{pt})=\op{Der}(\DC\op{-Modf}_\DZ)$ \item Joint with Matthias Wendt, Rahbar Virk, work in progress \item Based on new progress in motives by Ayoub, Cisinski-Deglise, Drew,\dots \item Variant of Hodge theory \end{itemize} \end{frame} \begin{frame} Our old equivalence can be upgraded further to $$\begin{array}{ccc} (\op{Der}^{ss}_{B\times B}(G),\ast_B)&\sirra& (R\op{-SMod_\DZ-}R,\otimes_R)\\[2mm] \wr\da &&\da\\[2mm] ( \op{MDer}_{B\times B}(G)_{w=0},\ast_B)&&\da\\[2mm] \da&&\da\\[2mm] ( \op{MDer}_{B\times B}(G),\ast_B)&\sirra& (\op{Hot}^{\op{b}}( R\op{-SMod_\DZ-}R),\otimes_R) \end{array}$$ \end{frame} \begin{frame} Back to knot invariants (variation on Webster-Williamson) \begin{Bild} \includegraphics[width=0.45\textwidth]{SkriptenBilder/BildSiCo} \qquad\includegraphics[width=0.4\textwidth]{SkriptenBilder/BildSiCo0002}\\ $i_{s!}\underline{BsB}\pdef T_s^!$\hspace{3cm}$i_{s*}\underline{BsB}\pdef T_s^*$ \end{Bild} \begin{itemize} \item Take $s=(a+1,a+2)\in \mathcal S_n=W$ the transposition \item $T_s^!,T_s^*\in \op{MDer}_{B\times B}(G)$ \item Recall $\underline{BsB}\mapsto s$ under $\mathcal H\mapsto \DZ W$ given by $q\mapsto 1$ \end{itemize} \end{frame} \begin{frame} \begin{itemize} \item Given a braid $Z$, scan it from the top and convolve corresponding $T_s^!,T_s^*$ with $\ast\pdef \ast_B$ to get an object $M(Z)\in \op{MDer}_{B\times B}(G)$ \item For $M(Z)$ to be well-defined, use braid relations $T_s^!\ast T_t^!\ast T_s^! \cong T_t^!\ast T_s^!\ast T_t^!$ for $sts=tst$ and similarly for $st=ts$ and $!$ replaced by $*$ \item These are geometrically clear, since $BsB\times_BBtB\times_BBsB \sira BstsB$ by multiplication, so $T_s^!\ast T_t^!\ast T_s^!\cong i_{sts!}\underline{BstsB}=i_{tst!}\underline{BtstB}\cong T_t^!\ast T_s^!\ast T_t^!$ etc \item Also need $T_s^!\ast T_s^*\cong T_s^*\ast T_s^! \cong i_{e!} \underline{B}=i_{e*} \underline{B}$ unit object Calculation, but not so hard: only on $\DP^1\DC$ \end{itemize} \end{frame} \begin{frame} Calculation in bimodules \begin{itemize} \item $ \op{MDer}_{B\times B}(G)\sirra\op{Hot}^{\op{b}}( R\op{-SMod_\DZ-}R)$ \item $T^!_s$ maps to $\ldots\ra 0\ra R\otimes_{R^s}R\sra R\ra 0\ra\ldots$ multiplication map \item $T^*_s$ maps to $\ldots\ra 0\ra R\hra R\otimes_{R^s}R\ra 0\ra\ldots$ \begin{itemize} \item Geometrically, need $\mathcal O(\Gamma(e))\hra \mathcal O(\Gamma(e)\cup\Gamma(s))$ \item Given by choosing linear function on $V\times V$, whose zero set intersects $\Gamma(e)\cup\Gamma(s)$ precisely in $\Gamma(s)$ \item Multiply a function on $\Gamma(e)$ with this linear function and extend by zero to $\Gamma(e)\cup\Gamma(s)$ \end{itemize} \item $M(Z)$ corresponds to $B(Z)\in \op{Hot}^{\op{b}}( R\op{-SMod_\DZ-}R)$ the tensor product of these elementary complexes \end{itemize} \end{frame} \begin{frame} To get an invariant of the knot $K(Z)$ obtained closing the braid $Z$ procede as follows: \begin{itemize} \item Take at each stage of the bimodule complex $\ldots\ra B(Z)^q\ra B(Z)^{q+1}\ra\ldots $ of bimodules the Hochschild homology \item Get for each $j$ a complex of (graded) vector spaces $\ldots\ra \op{HH}_j(B(Z)^q)\ra \op{HH}_jB(Z)^{q+1})\ra\ldots $ \item Take its cohomology groups $\mathcal H^q(\op{HH}_j(B(Z)^\ast)$ \item This is {\bf Khovanov's triply graded knot homology:} \begin{itemize} \item Choosen and fixed degree $j$ of Hochschild homology \item Degree $q$ of cohomology of the resulting complex \item Internal degree, the bimodules beeing graded \end{itemize} \item It categorifies the HOMFLYPT polynomial, which can be gotten as some Euler characteristic \end{itemize} \end{frame} \begin{frame} I am still lacking full geometric understanding of why this has to give a knot invariant. Webster-Williamson seem to understand it better. And the construction of $\op{MDer}$ is very recent. \end{frame} \begin{frame} Recall relation to representation theory\\[3mm] \begin{itemize} \item $\mathfrak g$ a semisimple complex Lie algebra, $Z\subset U(\mathfrak g)$ the center of its enveloping algebra \\[2mm]\item $\mathcal M\subset \mathfrak g\op{-Mod}$ the category of all representations of $\mathfrak g$ locally finite under $Z$ \\[2mm]\item $\mathcal P$ the category of all functors $\mathcal M\ra \mathcal M$ with split embedding in some functor $E\otimes_\DC$ for $\op{dim}_\DC E<\infty$ \item $\mathcal M=\bigsqcap_{\chi\in\op{Max}Z}\mathcal M_\chi$ and $\mathcal P=\bigsqcap_{\chi,\psi\in\op{Max}Z}\;\;{_\psi}\mathcal P_\chi$ \item Equivalence of monoidal categories for $\chi=\op{Ann}_Z\DC$ $$\hat{\mathbb V}:{_\chi}\mathcal P_{\chi} \sirra \hat S \op{-SMod-}\hat S$$ \item $\mathfrak g\supset\mathfrak b\supset\mathfrak h $ as usual, $S\pdef U (\mathfrak h)$ polynomial ring \end{itemize} \end{frame} \begin{frame} Construction of $\hat{\mathbb V}:{_\chi}\mathcal P_{\chi} \sirra \hat S \op{-SMod-}\hat S$ \begin{itemize} \item Abbreviate $U\pdef U(\mathfrak g)$, recall $\chi=Z^+=\op{Ann}_Z\DC$ \item $U /U \chi^n$ form an inverse system in $\mathcal M_\chi$ \item They also are of finite length as a $U $-bimodules \item For $P\in {_\chi}\mathcal P_{\chi}$ still $P(U /U \chi^n)$ naturally is a bimodule \item The $P(U /U \chi^n)$ are {\bf Harish-Chandra bimodules}:\\ By definition, these are the bimodules of finite length, which are in addition locally finite for the adjoint action of $\mathfrak g$. \item Call $\op{HCH}$ the category of Harish-Chandra bimodules \item $_\chi\!\op{HCH}_\chi$ has a unique simple object $L$ of maximal Gelfand-Kirillov dimension \item There is an exact functor $\mathbb V:{_\chi\!\op{HCH}_\chi}\ra \DC\op{-Modf}$ with $L\mapsto\DC$ and killing the other simples. It is essentially unique. \end{itemize} \end{frame} \begin{frame} Construction of $\hat{\mathbb V}: {_\chi}\mathcal P_{\chi} \sirra \hat S\op{-SMod-}\hat S$, continued \begin{itemize} \item By functoriality, our exact functor $\mathbb V$ is even a functor $\mathbb V:{_\chi\!\op{HCH}_\chi}\ra Z\op{-Modf-}Z$ \item Looking closer, our exact functor $\mathbb V$ is even a functor $\mathbb V:{_\chi\!\op{HCH}_\chi}\ra \hat Z\op{-Modf-}\hat Z$ for $\hat Z=Z_\chi^\wedge$ \item Set $$\hat{\mathbb V} P\pdef \varprojlim_n \mathbb V(P(U/U\chi^n))$$ \item Use natural isomorphism $\hat Z\sira \hat S$ induced by unnormalized Harish-Chandra isomorphism $Z\sira S^{(W\cdot)}\subset S$ with $S=\mathcal O(\mathfrak h^\ast)$ and $W$-action shifted to fix $-\rho$ determined by $\DC_{-2\rho}\cong \bigwedge^{\op{max}}(\mathfrak g/\mathfrak b)$ over $\mathfrak h$\dots \end{itemize} \end{frame} \begin{frame} \begin{itemize} \item Consider ${_\chi\!\op{HCH}^n_\chi}\pdef\{M\in {_\chi\!\op{HCH}_\chi}\mid M\chi^n=0\}$ \item Has enough projectives: The $P(U/U\chi^n)$ for $P\in{_\chi}\mathcal P_{\chi}$ \item Get by the above also equivalence $\mathbb V: \op{proj}({_\chi\!\op{HCH}^n_\chi})\sirra S\op{-SMod-}S/(S^+)^n$ \item In the case $n=1$ have ${_\chi\!\op{HCH}^1_\chi}\sirra \mathcal O_\circ$ equivalence with principal block of BGG-category by tensoring with dominant Verma $\otimes_U\Delta(0)$ \item Proof of KL-conjectures using bimodules: \end{itemize} $$ \begin{array}{ccccccc} &P_x&\mapsto &B_x\otimes_S\DC&\leftmapsto&B_x&\\[2mm] Q\in&\op{proj}( \mathcal O_\circ)&\sirra& S\op{-SMod}&\leftarrow & S\op{-SMod_\DZ-}S&\ni B_x\\ \da\;\;\;&\da&&&&\da&\;\;\da\\ \sum_y (Q:\Delta_y)y\in&\DZ[W]&\longleftarrow&\stackrel{v=1}{\longleftarrow} &\longleftarrow&\mathcal H_v&\ni C_x \end{array} $$ $$\RA \;\sum_y (P_x:\Delta_y)y=C_x(1)\;\RA\; [\Delta_y:L_x]=(P_x:\Delta_y)=P_{yx}(1)$$ \end{frame} \begin{frame} Graded versions and Koszul duality \begin{itemize} \item Construct $\DZ$-graded version $\mathcal O_\circ^\DZ$ of $\mathcal O_\circ$ by declaring $\op{proj}( \mathcal O_\circ^\DZ)= S\op{-SMod}_\DZ$ \item Then $\sum_i[\Delta^\DZ_y:L^\DZ_x\langle i\rangle]v^i= \sum_i(P^\DZ_x:\Delta^\DZ_y\langle i\rangle)v^i=P_{yx}(v)$ \item Characterization in joint recent work with Rottmaier: $\mathcal O_\circ^\DZ$ is ``the essentially unique $\DZ$-graded version of the artinian category $\mathcal O_\circ$ compatible with the action of the center'' \item Deduce $\op{Hot}^{\op{b}}(\op{proj}( \mathcal O_\circ^\DZ))= \op{Hot}^{\op{b}}(S\op{-SMod}_\DZ)$ \item Thus get Koszul duality $K$ triangulated functor $$ \begin{array}{cccccc} \op{Der}^{\op{b}}( \mathcal O_\circ^\DZ)&\sirra &\op{Hot}^{\op{b}}(S\op{-SMod}_\DZ)&\sirra&\op{MDer}_{N\times B}(G)\\ K\da&&&&\da\wr \\ \op{Der}^{\op{b}}( \mathcal O_\circ^\DZ)&\stackrel{\sim}{\leftarrow}&\stackrel{\sim}{\leftarrow}&\stackrel{\sim}{\leftarrow}&\op{MDer}_{N}(G/B)\\ \da &&&&\da \\ \op{Der}^{\op{b}}( \mathcal O_\circ)&\stackrel{\sim}{\leftarrow}&\op{Der}^{\op{b}}_{N}(\mathcal D_{G/B}\op{-Mod}^{\op{qc}})&\stackrel{\sim}{\leftarrow}&\op{Der}_{N}(G/B) \end{array} $$ \end{itemize} \end{frame} \begin{frame} Kozsul duality $K$ preceded by $\mathcal O$-duality $d$, properties: \begin{itemize} \item $Kd:\op{Der}^{\op{b}}( \mathcal O_\circ^\DZ)\ra \op{Der}^{\op{b}}( \mathcal O_\circ^\DZ)$ triangulated contravariant \item $\Delta_x^\DZ\mapsto \Delta_{w_\circ x}^\DZ$ \item $L_x^\DZ\mapsto P_{w_\circ x}^\DZ$ \item $P_x^\DZ\mapsto L_{w_\circ x}^\DZ$ \item $Kd(M[n])\cong (KdM)[-n]$ \item $Kd(M\langle n\rangle)\cong (KdM)[ n]\langle n\rangle$ \item Funny formulas $\sum_i\op{dim}\op{Ext}^i_{\mathcal O}(\Delta_x, L_y)= [\Delta_{w_\circ x}:L_{w_\circ y}]$ \item $Kd$ gives $\op{Der}(\Delta^\DZ_x, L^\DZ_y[i]\langle j\rangle)= \op{Der}(P^\DZ_{w_\circ y}[-i+j]\langle j\rangle,\Delta^\DZ_{w_\circ x})$ \item This explains these funny formulas \end{itemize} \end{frame} \begin{frame} Other things on Koszul duality \begin{itemize} \item Variant exchanging parabolic and singular category $\mathcal O$ \item Variant from parabolic-singular to singular-parabolic \item BGG-resolution of simple Verma corresponds to Verma flag of antidominant projective \item More natural from Langlands philosophy point of view \end{itemize} \end{frame} \begin{frame} Variant for Harish-Chandra modules \begin{itemize} \item Consider ${\overline{\op{HCH}}}$ the category of $U$-bimodules $M$ such that every vector is killed by some $\chi^n$ from right and left and $\{v\in M\mid \chi v=0\}$ is of finite length \item Has enough injectives and finite homological dimension \item Using $\mathbb V$ and some duality get contravariant equivalence $ \op{inj}\overline{\op{HCH}}\sirra S\op{-SMod-}S$ \item Define {\bf $\DZ$-graded version} $\overline{\op{HCH}}_\DZ$ of $\overline{\op{HCH}}$ by declaring $ \op{inj}\overline{\op{HCH}}_\DZ\pdef ( S\op{-SMod_\DZ-}S)^{\op{opp}}$ \item Deduce $ \op{Hot}^{\op{b}}(\op{inj}\overline{\op{HCH}}_\DZ)\sirra \op{Hot}^{\op{b}}( S\op{-SMod_\DZ-}S)^{\op{opp}}$ \item Get $ \op{Der}^{\op{b}}(\overline{\op{HCH}}_\DZ)\sirra \op{MDer}_{B^\vee\times B^\vee}(G^\vee)^{\op{opp}}$ Koszul duality \item Need dual group $G^\vee$ since $S=\mathcal O(\mathfrak h^\ast)$ but $R=\mathcal O(\mathfrak h)$ \end{itemize} \end{frame} \begin{frame} Koszul duality for real groups (conjectural) \begin{itemize} \item $\overline{\op{HCH}}$ is a category of Harish-Chandra modules for $\op{GL}(n;\DC)$ viewed as a real Lie group \item Similar things seem to work for real Lie group \item Take complex reductive connected algebraic group $G$ with an antiholomorphic involution $\bar \gamma$ fixing a Borel subgroup $B\subset G$ \item $\theta$ holomorphic extension of Cartan involution \item Conjecture contravariant equivalence $$ \op{MDer}_{B^\vee}^{\op{bc}}{\op{Z}}^1_{\gamma}(\Gamma;G^\vee) \;\sirra\; \op{MDer}_{\lfloor B;1\rfloor}^{\op{cfb}}{\op{Z}}^1_{\theta}(\Gamma;G)^{\op{opp}}$$ \end{itemize} \end{frame} \end{document} %%% Local Variables: %%% mode: pdflatex %%% End: