%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] Equivariant Motives in Representation Theory} \author[]{Wolfgang Soergel} \institute[]{\inst{} Mathematisches Institut\\ Universit\"at Freiburg\\[4ex] \vspace*{.9cm} %\textcolor{red}{\\ allgemeine Fragen/Hinweise hier positionieren} } \date[]{\small \hbox{September 2015}} \beamersetuncovermixins{\opaqueness<1>{25}}{\opaqueness<2->{15}} % ================================================== % ================================================== \begin{document} % ================================================== % ================================================== \titlepage \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 $$\op{Hot}^{\op{b}}(\op{add}(T))\stackrel{\sim}{\hra} \op{Der}{\mathcal A}$$ \pause \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} {\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\pause To $X$ a complex variety associate a triangulated complex tensor category $(\op{MDer} (X), \otimes)$ refining constructible sheaves \\[2mm]\item\pause 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\pause This admits a weight structure alias co-t-structure due to work of H\'{e}bert \end{itemize} \end{frame} \end{document} \begin{frame} \begin{itemize} \item Denote by $\underline X\in \op{MDer} (X)$ the tensor unit \\[2mm]\item\pause 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\pause 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\pause 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]\pause {\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] \pause \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 $$\mathbb H:\op{MTDer}_{(B)}(G/B)_{w=0}\stackrel{\sim}{\hra} {\op{H}}(G/B)\op{-Mod}^\DZ$$\pause 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 $$\op{Hot}^{\op{b}}(R\op{-SMod^\DZ_{\op{ev}}-}R)\sirra \op{MTDer}_{B}(G/B)=\op{MTDer}_{B\times B}(G)$$\pause 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. \pause 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} Koszul duality for real groups\\[3mm] \begin{itemize} \item %Take complex reductive connected algebraic group Take an antiholomorphic involution $\bar \gamma$ of $G$ fixing our Borel subgroup $B\subset G$ \item\pause Let $\theta$ be the holomorphic extension of Cartan involution \item\pause 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\pause On the right use monodromic sheaves with unipotent monodromy and suitable finiteness conditions \item\pause 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: