%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} \newcommand{\wt}{\tilde} \newcommand{\wh}{\hat} \newcommand{\td}{\tilde} \newcommand{\eps}{\varepsilon} \title[]{~\\[2ex] Graded Representation Categories and Motives} \author[]{Wolfgang Soergel} \institute[]{\inst{} Mathematisches Institut\\ Universit\"at Freiburg\\[4ex] \vspace*{.9cm} %\textcolor{red}{\\ allgemeine Fragen/Hinweise hier positionieren} } \date[]{\small \hbox{Juni 2015}} \beamersetuncovermixins{\opaqueness<1>{25}}{\opaqueness<2->{15}} % ================================================== % ================================================== \begin{document} % ================================================== % ================================================== \titlepage \begin{frame} \begin{itemize} \item $\mathfrak g\supset \mathfrak b$ a semisimple complex Lie algebra with a Borel \\[2mm]\item $\mathcal O^\infty\subset \mathfrak g\op{-Mod}$ the category of all finite length representations of $\mathfrak g$ locally finite under $\mathfrak b$ \\[2mm]\item {\bf Theorem:} [S,Rottmaier] There is an essentially unique $\DZ$-graded cover of the artinian category $\mathcal O^\infty$ compatible with the action of the center \\[2mm]\item Artinian category: Abelian category in which each object has finite length \end{itemize} \end{frame} \begin{frame} \begin{itemize} \item {\bf $\DZ$-graded cover} of artinian category $\mathcal A$: Quadruple $(\wt{\mathcal A},[1], v, \eps)$ with \begin{itemize} \item $\wt{\mathcal A}$ abelian category \item $[1]:\wt{\mathcal A}\sira \wt{\mathcal A}$ strict automorphism ``shift the grading'' \item $v: \wt{\mathcal A} \rightarrow \mathcal A$ exact functor ``forget the grading'' \item $\eps: v \overset{\sim}{\Rightarrow} v[1]$ isotransformation \end{itemize} such that the following hold: \begin{itemize} \item The pair $(v, \eps)$ induces an isomorphisms $$\bigoplus_{i \in \mathbb{Z}} \wt{\mathcal A}(M, N[i]) \overset{\sim}{\rightarrow} \mathcal A(vM, vN)\quad \forall M, N \in \wt{\mathcal A}$$ \item For every epi $M\sra vN$ exists epi $vP\sra M$ such that the composition $vP\ra vN$ comes from $P\ra N$ \end{itemize} \end{itemize} \end{frame} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{frame} Standard example:\\[3mm] \begin{itemize} \item $A$ a left-artinian ring with a $\DZ$-grading \\[2mm]\item $\mathcal A$ the category of artinian $A$-modules \\[2mm]\item $\tilde{\mathcal A}$ the category of artinian $\DZ$-graded $A$-modules \\[2mm]\item $[1]$ shift of grading \\[2mm]\item $v$ forget grading \\[2mm]\item $\eps$ the identification between ``first shift the grading, then forget it'' and ``forget it right away'' \\[6mm]\item {\bf Theorem:} There is an essentially unique {\color{Red}$\DZ$-graded cover of the artinian category $\mathcal O^\infty$} compatible with the action of the center \end{itemize} \end{frame} \begin{frame} \begin{itemize} \item {\bf Theorem:} There is an essentially unique $\DZ$-graded cover of the artinian category $\mathcal O^\infty$ {\color{Red}compatible with the action of the center} \\[2mm]\item $Z\subset {\op{U}}(\mathfrak g)$ center of enveloping algebra \\[2mm]\item For $\chi\in \op{Max} Z$ construct natural $\DZ$-grading on $Z/\chi^n$\\[2mm] \begin{itemize} \item $Z\sira S^W$ Harish-Chandra isomorphism \item Take $\lambda\in \op{Max} S$ with isotropy group $W_\lambda$ \item $Y=Y(\lambda)\pdef(S)^{W_\lambda} \subset S$ the $W_\lambda$-invariants \item Grading on $S$ corresponding to $\DC^\times$-action contracting to $\lambda$ induces grading on $Y$ \item $\chi=\lambda\cap Z$ and $\mu=\lambda\cap Y$ \item $Z^\wedge_\chi\overset{\sim}{\rightarrow} (S^\wedge_\lambda)^{W_\lambda} \overset{\sim}{\leftarrow}Y^\wedge_\mu$ leading to $Z/\chi^n\overset{\sim}{\rightarrow} Y/\mu^n$ \item This grading on $Z/\chi^n$ independent of $\lambda$ \end{itemize} \end{itemize} \end{frame} \begin{frame}\begin{itemize} \item {\bf Theorem:} There is an essentially unique $\DZ$-graded cover of $\mathcal O^\infty$ {\color{Red}compatible with the action of the center} \item Call a $\DZ$-graded cover of $\mathcal O^\infty$ {\bf compatible with the action of the center} if for every graded object $M$ and $\chi\in\op{Max}Z$ such that $\chi^n (v M)=0$ the action $$Z/\chi^n\ra \op{End}_{\mathfrak g}(vM)$$ is homogeneous up to doubling degrees \end{itemize} \end{frame} \begin{frame} \begin{itemize} \item {\bf Theorem:} There is an {\color{Red}essentially unique} $\DZ$-graded cover of $\mathcal O^\infty$ compatible with the action of the center \item \\[2mm] $(\wt{\mathcal A},\td v, \td \eps)$ and $(\wh{\mathcal A}, \hat{v}, \hat{\eps})$ graded covers of $\mathcal A$ \item\\[2mm] A {\bf cover-equivalence} is a triple $(F, \pi,\epsilon)$ where \begin{itemize} \item $F: \wt{\mathcal A} \rightarrow \wh{\mathcal A}$ additive functor \item $\epsilon :[1]F \overset{\sim}{\Rightarrow} F[1]$ isotransformation \item $\pi: \hat{v} F \overset{\sim}{\Rightarrow} \td v$ isotransformation \end{itemize} such that commutes: \[ \begin{xy} \xymatrix{ \hat{v} [1] F \ar@{=>}[d]^\wr_{\hat{\epsilon}} \ar@{=>}[r]^{\epsilon}_{\sim} & \hat{v} F [1] \ar@{=>} [r]^{\pi}_{\sim} & \td v [1] \ar@{=>}[d]^{\td \epsilon}_\wr \\ \hat{v} F \ar@{=>}[rr]^{\pi}_{\sim} & & \td v } \end{xy} \] \item Two covers are said to be {\bf cover-equivalent} iff there is a cover-equi\-valence from one to the other \item This is an equivalence relation \end{itemize} \end{frame} \begin{frame} Standard example: Let $A$ a left-artinian ring with two $\DZ$-gradings. Then \\[6mm] the $ \DZ$-graded covers $\wt{A}\operatorname{-Modf}^{\DZ}$ and $\wh{A}\operatorname{-Modf}^{\DZ}$ of $A\operatorname{-Modf}$ are cover-equivalent\\[3mm] iff \\[3mm] there exists a $\DZ$-grading on the abelian group $A$ making it a graded $\wh{A}$-$\wt{A}$-bimodule $\;{^{\wh{}}\!\!A^{\wt{}}}$ \end{frame} \begin{frame} \begin{itemize} \item {\bf Theorem:} There exists a $\DZ$-graded cover of $\mathcal O^\infty$ compatible with the action of the center, and any two such $\DZ$-graded covers are cover-equivalent \item \\[3mm] The same holds for the BGG-category $\mathcal O$ \item \\[3mm] The same holds for the category of finite length Harish-Chandra modules for a complex reductive group \item \\[3mm] I expect almost the same to hold for the category of finite length Harish-Chandra modules for a real reductive group \end{itemize} \end{frame} \begin{frame} Sketch of proof in the case of BGG-category $\mathcal O$ \begin{itemize} \item Call $Q\in A\op{-mod}$ {\bf bicentralizing} if and only if the obvious map is an isomorphism $$ A \overset{\sim}{\rightarrow} \op{End}_{\op{End}_A Q} Q $$ \item For the artinian rings $A$ describing blocks of category $\mathcal O$, the modules $Q$ corresponding to the antidominant projective are bicentralizing by K"onig-Xi \item Given gradings $\hat A$ and $\tilde A$ on $A$ find compatible gradings $\hat Q$ and $\tilde Q$ on $Q$ \item From $Z\sra \op{End}_AQ$ get $$A\sira \op{Hom}_{\op{End}_A Q} (\hat Q,\tilde Q)=\op{Hom}_{Z} (\hat Q,\tilde Q)$$ \item By compatibility with center this gives the graded $\hat A$-$\tilde A$-bimodule we need \end{itemize} \end{frame} \begin{frame} {\bf Tilting Theorem:} [Keller, Rickard,\dots] Let $\mathcal A$ be an abelian category and $T\in\op{Hot}\mathcal A$ a ``tilting complex'', i.e. a complex such that $$\op{Hot}_{\mathcal A}(T,T[n])\sira \op{Der}_{\mathcal A}(T,T[n])\;\forall n$$ and all these spaces vanish for $n\neq 0$. Then the embedding of the additive subcategory $\op{add}(T)$ generated by $T$ can be extended to a fully faithful triangulated functor $$\op{Hot}^{\op{b}}(\op{add}(T))\stackrel{\sim}{\hra} \op{Der}{\mathcal A}$$ \begin{description} \item[Idea of proof:] View a complex of complexes as a double complex and take the total complex. \item[Problem:] In our complex of complexes, $d^2$ is only homotopic to zero and we may have $d^2\neq 0$. \item[Solution:] Use the special assumptions on $T$. \end{description} \end{frame} \begin{frame} $E =\op{End}_{\mathcal{A}} T$ is naturally a dg-Ring. By d\'evissage \begin{displaymath} \langle T \rangle_\Delta^{\op{Der}}\stackrel{\approx}{\leftarrow} \langle T \rangle_\Delta^{\op{Hot}} \sirra \op{dgFrei-} E \end{displaymath} $$\op{dgFrei-} E\sirra \op{dgFrei-} Z \stackrel{\approx}{\leftarrow}\op{dgFrei-} H=\langle H\rangle_\Delta \stackrel{\approx}{\leftarrow} \op{Hot}^{\op{b}}(\op{add}(H)) $$ $\op{Hot}_{\mathcal A}(T, ):\op{Hot}(\mathcal A)\ra \op{Mod-}H$ induces $\op{add}(T)\sira \op{add}(H)$ Together $$\langle T \rangle_\Delta^{\op{Der}}\sirra \op{Hot}^{\op{b}}(\op{add}(H)) \sirra \op{Hot}^{\op{b}}(\op{add}(T))$$ \end{frame} \begin{frame} {\bf Tilting Theorem, Variant:} Let $\mathcal A$ be an abelian category and $(T_i)_{i\in I}$ a ``tilting family'', i.e. a family of complexes such that $$\op{Hot}_{\mathcal A}(T_i,T_j[n])\sira \op{Der}_{\mathcal A}(T_i,T_j[n])\;\forall n\in\DN, i,j\in I$$ and all these spaces vanish for $n\neq 0$. Then the embedding of the additive subcategory $\op{add}(T_i\mid i\in I)$ generated by the $T_i$ can be extended to a fully faithful triangulated functor $$\op{Hot}^{\op{b}}(\op{add}(T_i\mid i\in I))\stackrel{\sim}{\hra} \op{Der}{\mathcal A}$$ \end{frame} \begin{frame} In joint work with Matthias Wendt (helped by Drew and Deglise) we use a new six-functor formalism for complex varieties \begin{itemize} \item To $X$ a complex variety associate a triangulated complex tensor category $(\op{MDer} (X), \otimes)$ refining constructible sheaves \\[2mm]\item To $f:X\ra Y$ a morphism of complex varieties associate triangulated functors $f_!,f^!,f_\ast, f^\ast$ between $\op{MDer} (X)$ and $\op{MDer} (Y)$ \\[2mm]\item This admits a weight structure alias co-t-structure due to work of H\'{e}bert \\[2mm]\item Satisfies the axiomatics of a fibered multicategory over correspondences [Fritz H"ormann] \end{itemize} \end{frame} \begin{frame} \begin{itemize} \item Denote by $\underline X\in \op{MDer} (X)$ the tensor unit \\[2mm]\item Denote by $\op{var}$ the one-point variety and by $\op{fin}:X\ra \op{var}$ the morphism to it. Get $\underline X=\op{fin}^\ast \underline{\op{var}}$. \\[2mm]\item The cone on $\underline{\op{var}}\ra \op{fin}_\ast \underline{\DC^\times}=\op{fin}_\ast\op{fin}^\ast \underline{\op{var}}$ is denoted $\DC(-1)[-1]$ and shifting and dualizing it gives the {\bf Tate object} $\DC(1)$ \\[2mm]\item Have, say by tilting, fully faithful embedding of triangulated tensor categories $$\op{Der}^{\op{b}}(\DC\op{-Modf}^\DZ)\stackrel{\sim}{\hra} \op{MDer}(\op{var})$$ with $\DC(1)\mapsto \DC(1)$. Here on the left $(1)$ means shift of internal grading, as opposed to homological grading. \\[2mm]\item Call its image $\op{MTDer}(\op{var})$ and the objects in there {\bf mixed Tate objects} \end{itemize} \end{frame} \begin{frame} Let $G\supset B$ be a connected complex reductive group with a Borel subgroup. Define $$\op{MTDer}_{(B)}(G/B)\subset\op{MDer}(G/B) $$ to consist of all objects $\mathcal F$ with $j^\ast\mathcal F\in \op{fin}^\ast\op{MTDer}(\op{var})$ for any embedding $j:S\hra X$ of a Bruhat cell. Call the objects of this full triangulated subcategory {\bf stratified mixed Tate}.\\[6mm] {\bf Theorem:} [S, Wendt] The stratified mixed Tate objects of weight zero form a tilting family, thus we get an equivalence of triangulated categories $$\op{Hot}^{\op{b}}(\op{MTDer}_{(B)}(G/B)_{w=0})\sirra \op{MTDer}_{(B)}(G/B)$$\\[3mm] \emph{Proof:} For the one-point flag variety this is new progress on motives. The rest of the proof consists in recovering old arguments. Similar results hold more generally for ``affinely Whitney-stratified varieties, whose weight zero objects are pointwise pure''. \end{frame} \begin{frame} On the other hand, taking total cohomology gives a fully faithful functor $$\mathbb H:\op{MTDer}_{(B)}(G/B)_{w=0}\stackrel{\sim}{\hra} {\op{H}}(G/B)\op{-Mod}^\DZ$$ If we call its image ${\op{H}}(G/B)\op{-SMod}^\DZ_{\op{ev}}$, we get in total an equivalence $$\op{Hot}^{\op{b}}({\op{H}}(G/B)\op{-SMod}^\DZ_{\op{ev}})\sirra \op{MTDer}_{(B)}(G/B)$$ \end{frame} \begin{frame} Similarly, in joint work with Wendt and Virk, we get an equivalence $$\op{Hot}^{\op{b}}(R\op{-SMod^\DZ_{\op{ev}}-}R)\sirra \op{MTDer}_{B}(G/B)=\op{MTDer}_{B\times B}(G)$$ Here, $R=\mathcal O(\op{Lie}T)$ is a polynomial ring with even grading, and the category $R\op{-SMod^\DZ-}R$ of special bimodules consists of all sums of summands of shifts of tensor products of the form $$R\otimes_{R^s} R\otimes_{R^t} R\ldots \otimes_{R^u} R $$ for $s,t,\ldots,u$ simple reflections. The Rouqier complexes $R\otimes_{R^s} R\sra R$ and $R\hra R\otimes_{R^s} R$ correspond to $j_\ast\underline{BsB}$ and $j_!\underline{BsB}$. Their braid relations as well as the construction of Khovanov homology get more transparent this way. \end{frame} \begin{frame} \begin{itemize} \item Can construct $\DZ$-graded version of principal block of $\mathcal O$ as $\op{MTPer}_{(B)}(G/B)$ \\[2mm]\item Here for $(X,\mathcal S)$ stratified complex variety, put $$\op{MTDer}_{\mathcal S}(X)\subset\op{MDer}_{\mathcal S}(X) $$ to consist of objects $\mathcal F$ with $j^\ast\mathcal F\in \op{fin}^\ast\op{MTDer}(\op{var})$ for any embedding $j:S\hra X$ of a stratum. Call them {\bf stratified mixed Tate}. \item For \glqq affinely Whitney-Tate stratified variety'', can put perverse t-structure to get $\op{MTPer}_{\mathcal S}(X)\subset \op{MTDer}_{\mathcal S}(X)$ \end{itemize} \end{frame} \begin{frame} \begin{itemize} \item{\bf Theorem:} Let $(X,\mathcal S)$ be an affinely Whitney-Tate stratified complex variety and assume pointwise purity of weight zero objects. Then \glqq tilting'' gives equivalences\end{itemize}\\[2mm] $$ \op{Hot}^{\op{b}}(\op{MTDer}_{\mathcal{S}}(X)_{w=0}) \stackrel{\approx}{\rightarrow} \op{MTDer}_{\mathcal{S}}(X)\stackrel{\approx}{\leftarrow} \op{Der}^{\op{b}}(\op{MTPer}_{\mathcal{S}}(X)) $$ \begin{itemize}\item Specializes to Koszul duality for category $\mathcal O$ \item Similar things can be done for real groups: Work in progress with Virk and Wendt \end{itemize} \end{frame} \begin{frame} Koszul duality for real groups\\[3mm] \begin{itemize} \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 $$ \op{MTDer}_{B^\vee}^{\op{bc}}{\op{Z}}^1_{\gamma}(\Gamma;G^\vee) \;\sirra\; \op{MTDer}_{\lfloor B;1\rfloor}^{\op{cfb}}{\op{Z}}^1_{\theta}(\Gamma;G)^{\op{opp}}$$ \item On the right use monodromic sheaves with unipotent monodromy and suitable finiteness conditions \item Reinterprets work of Adams, Barbasch, Vogan \item In part also joint work with Bernstein \end{itemize} \end{frame} \begin{frame} {\bf Thanks!} \end{frame} \end{document} %%% Local Variables: %%% mode: pdflatex %%% End: