\section{Abandoned attempts} \subsection{Koszul duality by grading shift} \begin{Bemerkungl}[\textbf{Ubiquity of dg-modules with an additional grading}] Suppose $\mathcal I$ is an additive category on which a group $\Gamma$ acts and suppose $P\in\op{Ket}_{\mathcal I}$ is a complex. Then $$\textstyle R\pdef \op{Ket}_{\mathcal I}^{\op{dg},\Gamma}(P)\pdef \bigoplus_{\gamma\in\Gamma} \op{Ket}_{\mathcal I}^{\op{dg}}(P, \langle\gamma\rangle P)$$ is a dg-ring with an additional $\Gamma$-grading and in an analogous way as in the ungraded case \ref{Ubdg} we get an equivalence of triangulated categories\label{Ubdgg} $$\op{Hot}_{\mathcal I}\supset\langle \langle \gamma\rangle P \mid \gamma\in \Gamma\rangle^{\op{Hot}}_\Delta\sirra \langle \langle \gamma\rangle R \mid \gamma\in \Gamma\rangle^{\op{dgHot}}_\Delta \subset \op{dgHot}_{-R}^\Gamma$$ Here on the right we consider the dg-category of $\Gamma$-graded right dg-modules $\op{Mod}_{-R}^{{\op{dg}},\Gamma}$ over $R$ and the corresponding homotopy category $\op{dgHot}_{-R}^\Gamma$, a triangulated category, and inside the full triangulated subcategory generated by $R$ itself and its shifts by $\gamma\in\Gamma$. Formally $(\langle\gamma\rangle M)^\kappa\pdef M^{\kappa+\gamma}$. \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Ubiquity of derived dg-modules with an additional grading}] Now suppose $\mathcal A$ is an abelian category on which a group $\Gamma$ acts and suppose $T\in\op{Ket}_{\mathcal A}$ is a complex with the property that the quotient functor induces isomorphisms\label{dgU} $$\op{Hot}_{\mathcal A}(T,\langle\gamma\rangle[n]T)\sira \op{Der}_{\mathcal A}(T,\langle\gamma\rangle[n]T)$$ for all $n\in \DZ,\gamma\in\Gamma$. We then say the family of complexes $\langle\gamma\rangle T$ is {\bf quis\-end\-un\-fol\-ded}. Then as in the ungraded case \ref{udDG} but now for $ R\pdef \op{Ket}_{\mathcal A}^{\op{dg},\Gamma}(T)$ we get a $\Gamma$-graded dg-ring and we get an equivalence of triangulated categories $$\op{Der}_{\mathcal A}\supset \langle \langle \gamma\rangle T \mid \gamma\in \Gamma\rangle^{\op{Der}}_\Delta\sirra \langle \langle \gamma\rangle R \mid \gamma\in \Gamma\rangle^{\op{dgDer}}_\Delta \subset \op{dgDer}_{-R}^\Gamma$$ Here on the right hand side we mean the localization $\op{dgDer}_{-R}^\Gamma$ of $\op{dgHot}_{-R}^\Gamma$ at quasiisomorphisms. \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Restriction with an additional grading}] Now let $\Gamma$ be an abelian group and $R$ be a differential graded ring with an additional $\Gamma$-grading. We will need the case $\Gamma=\DZ$, but for clarity let us keep the different copies of $\DZ$ separated for the time being. The same as in \ref{resqu} can be done for differential graded and in addition $\Gamma$-graded $R$-modules and gives us for any quasiisomorphism $R\qri S$ of $\Gamma$-graded dg-rings an equivalence of triangulated categories\label{reQI} $$\op{dgDer}_{-S}^\Gamma\supset \langle \langle \gamma\rangle S \mid \gamma\in \Gamma\rangle^{\op{dgDer}}_\Delta\sirra \langle \langle \gamma\rangle R \mid \gamma\in \Gamma\rangle^{\op{dgDer}}_\Delta\subset \op{dgDer}_{-R}^\Gamma$$ \end{Bemerkungl} \begin{Beispielex}[\textbf{Case of usual $\Gamma$-graded ring}] Any $\Gamma$-graded ring $R$ is a dg-ring concentrated in homological degree zero with trivial differential. In this case $$\op{dgDer}_{-R}^\Gamma=\op{Der}(\op{Mod}^{\Gamma}_{-R})$$ is the derived category of the abelian category of $\Gamma$-graded right $R$-modules. \end{Beispielex} \begin{Bemerkungl}[\textbf{Assembling everything to get Koszul duality \ref{koDU}}] Let $B$ be a Koszul ring as in \ref{KOz} and assume in addition $B^0=k$ is a field. We view $B$ as a $\Gamma$-graded differential graded ring with $\Gamma=\DZ$, concentrated in homological degree zero.\label{pfKO} Let $P\sra B^0$ be a projective resolution in $B\op{-Modfg}^\Gamma$. Put $R\pdef \op{Mod}_{B}^{\op{dg},\Gamma}(P)$, a $\Gamma$-graded dg-ring. Then we get equivalences of triangulated categories $$\begin{array}{llll} \op{Der}^{\op{b}}(B\op{-Modfg}^\Gamma)&\sirra&\op{Der}^{\op{b}}_{B\op{-Modfg}^\Gamma}(B\op{-Mod}^\Gamma)&\text{mostly by \eref{LUK}{TD},}\\[1mm] &\silla&\langle \langle \gamma\rangle B^0\mid \gamma\in\Gamma \rangle_\Delta^{\op{Der}}&\text{since $B^0$ is a field,}\\[1mm] &\sirra& \langle \langle \gamma\rangle P\mid \gamma\in\Gamma \rangle_\Delta^{\op{Der}}&\text{since $B^0\cong P$ in $\op{Der}$},\\[1mm] &\sirra& \langle \langle \gamma\rangle R\mid \gamma\in\Gamma \rangle_\Delta^{\op{dgHot}}&\text{by derived ubiquity \ref{dgU}}.\\[1mm] \end{array} $$ Here the last triangulated category is considered to be generated in $\op{dgHot}^{\Gamma}_{-R}$. The complex $P$ is living in bidgrees $(\gamma, n)$ for $n\leq 0$, so in the lower halfplane with the differentials going up vertically, and it can easily be chosen to live in the lower right quadrant and with additional care in its part on or above the antidiagonal, in formulas $\gamma+n\geq 0$. Now by Koszulity of $B$ the homology of $R$ is concentrated in the \glqq antidiagonal bidegrees\grqq\ $(\gamma,n)$ with $\gamma+n=0$ and, as there are no negative extensions, with $n\geq 0$, so the antidiagonal in the upper left quadrant. The main new trick now is to {\bf shear the bigrading}. Namely, we declare the homogeneous part of bidegree $(\gamma,n)$ to sit from now on and in new notation in bidegree $(\!(\gamma, n+\gamma)\!)$. So after shearing the homology will in our case sit on the negative horizontal coordinate axis. Let us denote this new $\Gamma$-graded dg-ring $S$. By Koszulity, its homology is now concentrated in homological degree zero and we thus have quasiisomorphisms $S\qli \mathcal Z^0S\qri \mathcal H^0S=E^{(-)}$ where $E^{(-)}$ is our extension algebra $E$ but with all degrees made negative. So we can go on with our chain of equivalences and get $$\begin{array}{llll} \langle \langle \gamma\rangle R\mid \gamma\in\Gamma \rangle_\Delta^{\op{dgHot}}&\sirra& \langle \langle \gamma\rangle S\mid \gamma\in\Gamma \rangle_\Delta^{\op{dgHot}}&\text{shearing the bigrading,}\\[2mm] &\sirra& \langle \langle \gamma\rangle \mathcal Z^0S\mid \gamma\in\Gamma \rangle_\Delta^{\op{dgHot}}&\text{by $\mathcal Z^0S\qri S$ and \ref{reQI},}\\[2mm] &\silla& \langle \langle \gamma\rangle E^{(-)}\mid \gamma\in\Gamma \rangle_\Delta^{\op{dgHot}}&\text{by $\mathcal Z^0B\qri E^{(-)}$ and \ref{reQI},}\\[2mm] &\sirra& \op{Der}^{\op{b}}(\op{Modfg^{\Gamma}-}E^{(-)})&\text{discussed in the sequel,}\\[2mm] &\sirra& \op{Der}^{\op{b}}(\op{Modfg^{\Gamma}-}E)&\text{negativating the $\Gamma$-grading.} \end{array} $$ For the last step we again use $E^0=B^0$ is a field and need a priori that $E$ is of finite homological dimension. With more trickery we can see that the finite homological dimension of $E$ also follows from our assumptions, but we won't need these details. We see the simple object $B^0$ goes to the projective object $E$ and $\langle \gamma\rangle B^0$ goes to $\langle -\gamma\rangle[\gamma]E$. \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Proof of Koszul duality \ref{koDU} if $B^0$ is a product of fields}] If instead $B^0=\bigoplus_{x\in W} k1_x$ is a product of finitely many copies of a field $k$, we can repeat all the arguments above with the twist to start with $\langle \langle \gamma\rangle B^01_x| \gamma, x \rangle_\Delta^{\op{Der}}$, take $P_x\sra B^01_x$ a projective resolution and then keep the $1_x$ all the way through, changing sides to $1_x R$ and ending up with $\langle \langle \gamma\rangle 1_xE| \gamma, x \rangle_\Delta^{\op{Der}}$. Here we find on objects\label{kdso} $$\langle \gamma\rangle B^01_x \;\mapsto\; \langle -\gamma\rangle [\gamma] 1_xE$$ \end{Bemerkungl} \subsection{Kazhdan-Lusztig-Vermutungen, Peng?} \begin{Bemerkungl}[\textbf{Die Kazhdan-Lusztig-Vermutungen}] Seien $(\mathfrak g\supset\mathfrak h, R^+)$ eine halbeinfache komplexe Liealgebra mit einer Cartan'schen und einem System positiver Wurzeln. In \cite{KL-C} wird die Vermutung ausgesprochen, da"s die Jordan-H"older-Multiplizit"aten der Vermamoduln gegeben werden durch die Identit"aten $$ [\Delta(y\cdot 0):{\op{L}}(x\cdot 0)]= h_{y,x}^{v=1}$$ Es ist von vornherein klar, da"s gilt $[\Delta(y\cdot 0):{\op{L}}(\lambda)]=0$ f"ur $\lambda\not\in W\cdot 0$, da in diesem Fall verschiedene maximale Ideale des Zentrums der Einh"ullenden die jeweiligen Moduln annullieren. \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Die Kazhdan-Lusztig-Vermutungen f"ur Erweiterungsr"aume}] Seien $(\mathfrak g\supset\mathfrak h, R^+)$ eine halbeinfache komplexe Liealgebra mit einer Cartan'schen und einem System positiver Wurzeln. Bezeichne $w=w_0\in W$ das l"angste Element de Weylgruppe. Es ist eine Involution $w^2=e$. Wir verwenden im folgenden die Notationen $L^{x} \pdef L (xw \cdot 0), \Delta^{x} \pdef \Delta(xw \cdot 0), \nabla^{x} \pdef \nabla(xw \cdot 0)$, so da"s zum Beispiel $L^e=\Delta^e=\Delta(-2\rho)$ ein einfacher Vermamodul ist. Wir schreiben $\op{Ext}^{i}\pdef \op{Ext}^{i}_{\cal{O}}$. In diesen Notationen lautet eine feinere Formulierung der Kazhdan-Lusztig-Vermutungen, die wir in \ref{KLV} ausf"uhrlicher diskutieren, da"s f"ur alle $x\in W$ gilt $$\underline{H}_{x}=\sum_{i,y} \op{dim} \op{Ext}^{i} (\Delta^{y}, L^{x})\; v^{i}H_{y} $$ Wir vereinbaren die Notation $\op{Ext}(\Delta,L)$ f"ur die $(W\times W)$-Matrix mit Eintr"agen in $\DZ[v]$ gegeben durch $$\op{Ext}(\Delta,L)_{y,x}=\sum_{i} \op{dim} \op{Ext}^{i} (\Delta^{y}, L^{x})\; v^i$$ Geometrisch beschreibt diese Matrix die Dimensionen der Halme der Schnittkohomologiekomplexe der Schubertvariet"aten, dazu sp"ater mehr. F"ur ${\op{H}}$ die Matrix der Kazhdan-Lusztig-Polynome mit Eintr"agen ${\op{H}}_{y,x}=h_{y,x}$ bedeuten die Kazhdan-Lusztig-Vermutungen f"ur Erweiterungen also die Identit"at $$\op{Ext}(\Delta,L)={\op{H}}$$ \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Multiplizit"atenvermutung, Variante}] Erkl"aren wir die Jordan-H"older-Matrix als die Matrix $\op{J}$ mit Eintr"agen ${\op{J}}_{x,y}\pdef [\Delta^y:{\op{L}}^x]$, so erh"alt die Multiplizit"atenvermutung die Gestalt einer Matrixidentit"at $${\op{J}} ={\op{U}} ({\op{H}}^{v=1})^\top{\op{U}}$$ f"ur ${\op{U}}$ f"ur die\label{MuVa} Matrix des Umindizierens ${\op{U}}_{x,y}\pdef\delta_{x,yw}$. Wie bei jeder Permutationsmatrix ist die Inverse von ${\op{U}}$ ihre Transponierte und wegen $w^2=e$ fallen diese zusammen, in Formeln ${\op{U}}={\op{U}}^\top={\op{U}}^{-1}$. \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Jordan-H"older-Matrix als inverse Erweiterungsmatrix}] Ich erinnere an \ref{JHEx}. Seien $(\mathfrak g\supset\mathfrak h, R^+)$ eine halbeinfache komplexe Liealgebra mit einer Cartan'schen und einem System positiver Wurzeln. F"ur beliebiges $\lambda\in\frak{h}^\ast$ gilt in der Grothendieckgruppe der Kategorie $\cal{O}$ die Formel $$[L(\lambda)] = \sum_{i,\mu} (-1)^{i} \op{dim}_{\Bbb{C}} \op{Ext}^{i}_{\cal{O}} (\Delta(\mu), L(\lambda)) [\Delta(\mu)]$$ Das folgt aus $\op{dim}\op{Ext}^{i}_{\cal{O}} (\Delta(\mu), \nabla(\lambda))=\delta_{\lambda,\mu}\delta_{i,0}$ und daraus, da"s die Jordan-H"older-Multiplizit"aten $[\nabla(\lambda):{\op{L}}(\mu)]= [\Delta(\lambda):{\op{L}}(\mu)]$ "ubereinstimmen. Das ist kein tiefliegendes Resultat und spezialisiert zu $$[L^x] = \sum_{i,y} (-1)^{i} \op{dim}_{\Bbb{C}} \op{Ext}^{i}_{\cal{O}} (\Delta^y, L^x) [\Delta^y]= \sum_{y} {\op{Ext}(\Delta,L)}_{y,x}^{v=-1} [\Delta^y]$$ Zusammen mit der Matrix ${\op{J}}$ der Jordan-H"older-Multiplizit"aten mit den Eintr"agen ${\op{J}}_{x,y}\pdef [\Delta^y:{\op{L}}^x]$, f"ur die in der Grothendieckgruppe $[\mathcal O]$ gilt $[\Delta^y]=\sum_x{\op{J}}_{x,y}[{\op{L}}^x]$, finden wir die Identit"at\label{jhM} $${\op{J}}\circ (\op{Ext}(\Delta,L)^{v=-1})= \op{I}$$ von $\DZ$-wertigen $(W\times W)$-Matrizen. Die Matrix ${\op{J}}$ der Jordan-H"older-Mul\-ti\-pli\-zi\-t"a\-ten ist also invers zu Matrix $\op{Ext}(\Delta,L)$ der Erweiterungen von Vermamoduln durch einfache h"ochste Gewichtsmoduln, ausgewertet bei $v=-1$. \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Inversionsformeln f"ur Kazhdan-Lusztig-Polynome}] Die Inversionsformeln f"ur Kazhdan-Lusztig-Polynome aus \cite{KL-C}, vergleiche \eref{invv}{SPW}, lauten in unserer Notation $$ \sum_z (-1)^{l(x)+l(z)} h_{z,x} h_{z w, y w}=\delta_{x,y} $$ mit der Notation $w=w_0$ f"ur das l"angste Element der Weylgruppe. Nun wissen wir aus der Kombinatorik, da"s in $h_{z,x}$ die Variable $v$ nur in Potenzen der Parit"at $l(x)+l(z)$ mit von Null verschiedenen Koeffizienten auftritt, so da"s gilt $h_{z,x}(-v)= (-1)^{l(x)+l(z)} h_{z,x}(v)$. Die Inversionsformeln sagen uns mithin, da"s wir die inverse Matrix zur Matrix $\op{H}$ der Kazhdan-Lusztig-Polynome erhalten, indem wir die Zeilen und Spalten umindizieren mit $x$ statt $xw$, diese Matrix zus"atzlich transponieren und darin $v$ durch $(-v)$ substituieren, in Formeln\label{invF} $${\op{U}} {\op{H}}^\top {\op{U}}\circ ({\op{H}}^{v=-v})= \op{I}$$ mit der Umindizierungsmatrix ${\op{U}}$ aus \ref{MuVa}. \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Erweiterungsvermutung impliziert Multiplizit"atsvermutung}] Spezialisieren wir die Inversionsformel \ref{invF} bei $v=1$, so erhalten wir $${\op{U}} ({\op{H}}^{v=1})^\top {\op{U}}\circ ({\op{H}}^{v=-1})= \op{I}$$ Zusammen mit der vergleichsweise allgemeinen Identit"at ${\op{J}}\circ (\op{Ext}(\Delta,L)^{v=-1})= \op{I}$ aus \ref{jhM} sehen wir so, da"s aus $\op{Ext}(\Delta,L)={\op{H}}$ folgt ${\op{J}}={\op{U}} ({\op{H}}^{v=1})^\top {\op{U}}$, da"s also die Erweiterungsvermutung die Multiplizit"atenvermutung impliziert. \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Category $\mathcal O_0$ as a module category}] The principal block $\mathcal O_0\subset\mathcal O$ has the simples $L_x\pdef L(x\cdot 0)=L^{xw}$ and they have indecomposable projective covers $P_x$. As explained in \ref{AKO} we get an equivalence of categories $$\op{Hom}_{\mathfrak g}(P,\;):\mathcal O_0 \sirra \op{Modf-}A$$ for $A\pdef \op{End}_{\mathfrak g}(P)$. This ring comes with pairwise orthogonal idempotents $1_x$ given by the projections onto the factors of the direct sum.\label{KtoM} In more detail we find $$\begin{array}{lll}\op{dim}_\DC 1_xA1_y&=& \op{dim}_\DC\op{Hom}(P_y, P_x)\\ &=& [P_x:L_y]\\ &=&\sum_z(P_x:\Delta_z)[\Delta_z:L_y]\\ &=&\sum_z[\Delta_z:L_x][\Delta_z:L_y] \end{array}$$ In terms of our Jordan-H"older matrix ${\op{J}}_{x,y}\pdef [\Delta^y:{\op{L}}^x]$ we get $\op{dim}_\DC 1_{xw}A1_{yw}= \sum_z {\op{J}}_{x,z} {\op{J}}_{y,z}= {\op{J}} {\op{J}}^\top$ and thus $$\op{dim}_\DC 1_{x}A1_{y}= ({\op{U}}{\op{J}} {\op{J}}^\top {\op{U}})_{x,y}=(({\op{H}}^{v=1})^\top({\op{H}}^{v=1}))_{x,y}$$ \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Testing for Koszulity}] Let us imagine $A$ could be equipped with an $\DN$-grading making it a Koszul ring. Then we would have $A^0=\bigoplus \DC 1_x$ and $E=\op{Ext}^*_{\op{A}^{\op{opp}}}(A^0,A^0)\cong \op{Ext}^n(L, L)$ with $L=\bigoplus_x L_x$ as $L\mapsto A^0$ under our equivalence $\mathcal O_0 \sirra \op{Modf-}A$ above. To compute $\op{dim}_\DC 1_xE^n1_y$ we can use geometrical arguments and abbreviate $\op{ext}^n\pdef \op{dim}_\DC\op{Ext}^n_{\mathcal O}$ and find $$\begin{array}{lll}\op{dim}_\DC 1_xE^n1_y&=& \op{ext}^n(L_y, L_x)\\ &=&\sum_{i+j=n,\;z\in W}\op{ext}^i(L_y,\nabla_z) \op{ext}^j(\Delta_z, L_x)\\ &=&\sum_{i+j=n,\;z\in W}\op{ext}^i(\Delta_z,L_y) \op{ext}^j(\Delta_z, L_x) \end{array}$$ and thus $$\sum_n (\op{dim}_\DC 1_xE^n1_y)v^n= ({\op{U}}{\op{E}}^\top{\op{E}}{\op{U}})_{x,y}= ({\op{U}}{\op{H}}^\top{\op{H}}{\op{U}})_{x,y}$$ with the extension version of the Kazhdan-Lusztig conjectures for the second equality. The numerical Koszulity criterion $\op{I}={\op{P}}_E(-t)^\top \;{\op{P}}_A(t)$ discussed in \ref{nEkd} would thus follow from the inversion formula ${\op{U}} {\op{H}}^\top {\op{U}}\circ ({\op{H}}^{v=-v})= \op{I}$, if $A$ had an $\DN$-grading with $\sum_n (\op{dim}_\DC 1_xA^n1_y)v^n= ({\op{H}}^\top {\op{H}})_{x,y}$. This looks plausible, since evaluating at $v=1$ leads to a correct formula by \ref{KtoM}. \nichtfinal{Highest weight and parity vanishing ausreichend?} In addition it looks plausible that there might exist an isomorphism of graded rings $A\sira E$ mapping $1_x$ to $1_{xw}$, written out an isomorphism $$\textstyle\op{End}(\bigoplus_x P^x)\sira \op{Ext}^*_{\mathcal O}(\bigoplus_x L_x, \bigoplus_x L_x)$$ This was conjectured by Beilinson and Ginzburg \cite{BGi} and proven in \cite{So-A}. \end{Bemerkungl} \newpage \section{Trying to write a program for MSRI} I propose to assume for the main lectures of this summer school that the only reductive group is $G=\op{GL}_n$. \subsection{Some representation theory} A big bunch of the theory consists of reductions of problems of representation theory to Weyl group combinatorics. One way to think of it is that a representation of $\op{GL}_n$ behaves like a bunch of representations of copies of $\mathfrak{sl}_2$ bound together with very little interaction. \begin{enumerate} \item Simple finite dimensional representations of $\mathfrak{sl}_2(\DC)$; \item Maximal toral subalgebra, root space decomposition, Weyl group: Easy for the general linear group; \item Enveloping algebra, Verma modules, center of the enveloping algebra, Harish-Chandra isomorphism, Weyl character formula; \item Category $\mathcal O$. The principal block in the case of $\mathfrak{sl}_2(\DC)$. Translation functors. Projective functors; \item Simple highest weight modules and their characters. BGG-reciprocity; \item Harish-Chandra modules. Harish-Chandra modules for complex groups and category $\mathcal O$; \end{enumerate} \subsection{Some combinatorics} \begin{enumerate} \item Coxeter Systems and realization as reflection groups; \item Bruhat decomposition and Bruhat order; \item Hecke algebras; \item Kazhdan-Lusztig-Polynomials; \end{enumerate} \subsection{Some homological algebra} I wonder to what extent we can assume the participants to have experience with derived categories. \begin{enumerate} \item The derived category dg-modules; \item Tilting and Koszul duality; \end{enumerate} \subsection{Some geometry} I wounder to what extent we can assume the participants to have experience with sheaf cohomology. \begin{enumerate} \item The equivariant derived category and its six functors; \item The categorified Hecke algebra; \end{enumerate} \subsection{Some possible goals} \begin{enumerate} \item Projective functors? Bernstein-Gelfand. \item Koszul duality for category $\mathcal O$, in the form $L\pdef \bigoplus_{x\in W} L(x\cdot 0)$ sum of all simples of the principal block $\mathcal O_0$ and $A\pdef \op{Ext}^*_{{\mathcal O}_0}(L,L)$, then $$\mathcal O_0\cong A\op{-Modf}$$ To understand the statement, need: \begin{enumerate} \item Category $\mathcal O$; \item Extensions in abelian categories and their Yoneda-product; \item The enveloping algebra and its center; \item Verma modules, simple highest weight modules; \end{enumerate} To understand the motivation, need: \begin{enumerate} \item Kazhdan-Lusztig polynomials and their inversion formulas; \item Kazhdan-Lusztig conjectures $[M(x\cdot 0):L(y\cdot 0)]$; \item BGG-reciprocity, so projective objects in $\mathcal O$, they admit a Verma flag. Projective covers and their unicity. Close by are translation functors and how they categorify the Weyl group. \end{enumerate} \item Explain the proof of Koszul duality for category $\mathcal O$. Need: \begin{enumerate} \item Harish-Chandra-Isomorphism for the center of the enveloping algebra; \item Translation functors and wall-crossing; \item Hypercohomology; \item Intersection cohomology complexes; \item Cohomology ring of the flag variety, Borel isomorphism. Will later come out easily in the study of equivariant derived categories, just claim it here; \item Six-functor-formalism. Well, four functors mostly do the job, bit six is much better; \item Localization (to understand extensions of simple objects in $\mathcal O$); \item Some spectral sequence arguments (to deduce a limit by parity va\-ni\-shing); \item Poincar\'e duality for the flag variety; \end{enumerate} \item Formulation of main conjecture, original form: Some equivariant Ext-Algebra describes some category of Harish-Chandra-modules $$\mathcal M_\chi\cong \op{Ext}^*_{G^\vee}X(\chi)\op{-Modfnil}$$ To understand the statement, need: \begin{enumerate} \item Equivariant derived category; \item Equivariant intersection cohomology sheaves; \item Harish-Chandra-Modules; \item The enveloping algebra and its center; \end{enumerate} To understand the motivation, need: \begin{enumerate} \item Some local Langlands philosophy; \item Some motivation for Harish-Chandra-modules; \end{enumerate} \item Some examples for real groups, mostly $\op{SL}(2;\DR)$ and tori; \item The example of complex groups. In this case, we can construct equivalences $$\mathcal M_0 \cong \mathcal O_0^\infty \qquad \op{Ext}^*_{B^\vee}(G^\vee/B^\vee) \cong \op{Ext}^*_{G^\vee}X(0)$$ and $\mathcal O_0^n$ has enough projectives and we have fully faithful functors to some category of special bimodules. Need: \begin{enumerate} \item Equivariant derived category; \item Equivariant intersection cohomology sheaves; \item equivariant six functors; \item Equivalences from Harish-Chandra-Modules to category $\mathcal O$; \end{enumerate} \item New interpretation in terms of motivic derived categories. Tilting formalism. Need: \begin{enumerate} \item More homological algebra: Derived category of a dg-ring. Functors by dg-bimodules; \item Motivic variant of equivariant derived category. \end{enumerate} \end{enumerate} \section{Program for KAZUAL, May 2024} %\subsection{General ideas} %I propose to approach the theme in the way it was %discovered historically, with the notable %exception of $\mathcal D$-modules, which I propose to %leave out altogether. %Ich habe 5 Doppelstunden zu je 90 Minuten und am Schluß 75 Minuten. \subsection{Outlines of my talks} Somewhere: Proof of KL-conjecture by Koszul duality without $\mathcal D$-modules. Peng last lecture. \begin{enumerate} \item Kozsul selfduality for category $\mathcal O$. How much of the general Koszul duality formalism? Proof needs and interprets inversion formulas for KL polynomials. \item Equivariant derived category. Start with equivariant sheaves. Mostly assume sheaves, derived categories, six-functor-formalism. Give description over a point and description of functors there as a dgDer of a $\DZ$-graded ring with differential zero. Deduce cohomology ring of flag variety, equivariant cohomology ring of flag variety. Relation to characteristic classes. \item Main conjecture for real reductive groups. Assume ABV-parameters known. Assume dual group known. Discuss tori. Discuss $\op{SL}(2;\DR)$ in some detail. \item Discuss the case of complex reductive groups, in particular $\op{SL}(2;\DC)$. \item Motivic equivariant categories, $\DZ$-graded versions, categorical interpretation of inversion formulas. \item \end{enumerate} \subsection{Proof of Kazhdan-Lusztig conjectures by Koszul duality} \begin{Bemerkungl} First we need to show that $\mathbb V: \mathcal O_0\ra C\op{-Mod}$ is fully faithful on projectives and for any simple reflection $s$ and any $M\in \mathcal O_0$ there exists an isomorphism $\mathbb V \theta_s M\cong C\otimes_{C^s}\mathbb V M$ with $\theta_s$ the wall crossing functor. In addition $\mathbb V(M(0))\cong \DC$ for the projective Verma module $M(0)\in \mathcal O_0$. \end{Bemerkungl} \begin{Bemerkungl} Next we remark $\mathbb H: \op{Der}(G/B)\ra C\op{-Mod}^\DZ$ is faithful on perversely semisimple complexes constructible along the Bruhat stratification. I denote the full subcategory consisting of these objects by $\op{Der}_{(B)}(G/B)^{\op{ss}}$. Furthermore we remark that for any simple reflection $s$ and $\pi_s: G/B\sra G/P_s$ the projection we have $$\mathbb H\pi_s^*\pi_{s*}\mathcal F\cong C\otimes_{C^s}\mathbb H\mathcal F$$ for any $\mathcal F$, at least for $\mathcal F\in \op{Der}_{(B)}(G/B)^{\op{ss}}$, maybe even in complete generality, I don't know by heart. We also know $\mathbb H \mathcal L^e\cong \DC$ for $\mathcal L^e$ the scyscraper sheaf at the one point cell $B/B$. Inductively we find $$\mathbb V \theta_s\ldots \theta_t M(0)\cong \mathbb H \pi_s^*\pi_{s*}\ldots \pi_t^*\pi_{t*}\mathcal L^e$$ as $C$-modules for any simple reflections $s,\ldots,t$. Now it is relatively easy, given other simple reflections $b,\ldots, c,$ to calculate using BGG-reciprocity the dimension of $\op{Hom}_{\mathfrak g}(\theta_s\ldots \theta_t M(0), \theta_b\ldots \theta_c M(0))$ and similarily the dimension of $\bigoplus_q\op{Der}_{G/B}([q]\pi_s^*\pi_{s*}\ldots \pi_t^*\pi_{t*}\mathcal L^e, \pi_b^*\pi_{b*}\ldots \pi_c^*\pi_{c*}\mathcal L^e)$ and show these dimensions are equal. We know $\mathbb V$ is fully faithful and $\mathbb H$ is faithful, so this dimension check shows $\mathbb H$ is also fully faithful on all objects of the type considered here. There are also geometric arguments to prove this, they are more natural and general, but less easy to reach from where we are. \end{Bemerkungl} \begin{Bemerkungl} We put $\mathcal L^x\pdef i_*\mathcal{IC}(\overline{BxB/B}) \in \op{Der}_{(B)}(G/B)^{\op{ss}}$. Then we know $\mathbb H \mathcal L^e\cong \DC$. If $xs>x$ in the Bruhat order, then furthermore for some finite multiplicities $m(y,x,s)$ we have $$[1]\pi_s^*\pi_{s*}\mathcal L^{x}\cong \mathcal L^{xs}\oplus \bigoplus_{l(y)\leq l(x)} ( \mathcal L^y)^{\oplus m(y,x,s)}$$ We deduce $\mathbb H$ is fully faithful on $\op{Der}_{(B)}(G/B)^{\op{ss}}$, since the objects of this category are just direct sums of direct summands of shifts of the objects $\pi_s^*\pi_{s*}\ldots \pi_t^*\pi_{t*}\mathcal L^e$ treated already above. We deduce all $\mathbb H\mathcal L^{y}$ are indecomposable as ungraded $C$-modules, since our simple perverse sheaves have no negative extensions and just scalars as endomorphisms and all extensions of positive degree are nilpotent and form a maximal ideal in the endomorphism ring. We further deduce isomorphisms of graded $C$-modules $$[1]C\otimes_{C^s}\mathbb H \mathcal L^{x}\cong[1]\mathbb H \pi_s^*\pi_{s*}\mathcal L^{x}\cong \mathbb H\mathcal L^{xs}\oplus \bigoplus_{l(y)\leq l(x)} ( \mathbb H\mathcal L^y)^{\oplus m(y,x,s)}$$ \end{Bemerkungl} \begin{Bemerkungl} Next we need to know that if $l(xs)>l(x)$, then for some finite multiplicities $n(y,x,s)\leq 0$ we have $$\theta_sP(x\cdot 0)\cong P(xs\cdot 0)\oplus \bigoplus_{l(y)\leq l(x)} P(y\cdot 0)^{\oplus n(y,x,s)}$$ This is easily seen on Verma flag multiplicities. Applying $\mathbb V$ we find isomorphisms of ungraded $C$-modules $$C\otimes_{C^s}\mathbb V P(x\cdot 0)\cong \mathbb V\theta_sP(x\cdot 0)\cong \mathbb V P(xs\cdot 0)\oplus \bigoplus_{l(y)\leq l(x)} \mathbb V P(y\cdot 0)^{\oplus n(y,x,s)}$$ By Krull-Schmid and induction we obtain for all $x\in W$ an isomorphism of $C$-modules $\mathbb V P(x\cdot 0)\cong \mathbb H\mathcal L^x$ and equalities $ n(y,x,s)= m(y,x,s)$. From there the Kazhdan-Lusztig conjectures follow without difficulty. \end{Bemerkungl} \subsection{Some superfluous arguments} \begin{Bemerkungl} Now we remark that if $M,N$ are finite dimensional modules over a finite dimensional $\DZ$-graded $\DC$-ringalgebra $C$ and $M\oplus N$ as well as $N$ admit a grading making them graded $C$-modules and $N$ is indecomposable, then $M$ admits such a grading as well. Indeed, consider $\op{in}:N\hra M\oplus N$ and $\op{pr}: M\oplus N\sra N$ be the obvious injection and projection. Our gradings lead to a decomposition into homogeneous components $\op{in}=\sum \op{in}_\nu$ und $\op{pr} = \sum \op{pr}_\nu$ and we have $\op{id}_N = \sum_\nu \op{pr}_\nu \circ \op{in}_{-\nu}$. By assumption the endomorphismen ring of $N$ is local, so the nonunits form a twosided ideal and there has to exist $\nu$ such that $\op{pr}_\nu \circ \op{in}_{-\nu}$ is an automorphism of $N$ homogeneous of degree zero. Let $u$ be its inverse. Then $\op{in}_{-\nu}$ is splitting the projection $u \circ \op{pr}_\nu : (M\oplus N) \twoheadrightarrow N$ and $K\pdef \op{ker} (u \circ \op{pr}_\nu)\subset M\oplus N$ is a homogeneous submodule such that $K\oplus N\cong M\oplus N$ as graded $C$-modules. Forgetting the grading, we deduce from\label{tdso} Krull-Schmid we necessarily have $K\cong M$. \end{Bemerkungl} \begin{Bemerkungl} Now by induction on $l(x)$ we show all $\mathbb V P(x\cdot 0)$ admit a $\DZ$-grading compatible with the obvious $\DZ$-grading of $C$. Indeed, for $l(x)=0$ we have $\mathbb V P(x\cdot 0)\cong \mathbb V M( 0) \cong \DC$ and the statement is clear. But if $\mathbb V P(y\cdot 0)$ admits a compatible $\DZ$-grading for $l(y)\leq l(x)$ and $s$ is a simple reflection with $l(xs)>l(x)$, then by induction $\mathbb V \theta_s P(x\cdot 0)\cong C\otimes_{C^s}\mathbb V P(x\cdot 0)$ admits a compatible grading and all its indecomposable summands $\mathbb V P(y\cdot 0)$ with the only possible exception of $\mathbb V P(xs\cdot 0)$ also admit a compatible grading and chipping them off one by one using the previous considerations \ref{tdso} we deduce the only direct summand $\mathbb V P(xs\cdot 0)$ where we didn't know it yet also admits a compatible grading. \end{Bemerkungl} \subsection{The dual group, ABV-spaces: Cunningham} \begin{Bemerkungl}[\textbf{The dual group}] I recall the classification of connected compact Lie groups. A free abelien group $X$ of finite rank will be called a {\bf lattice}. A finite subgroup $W\subset \op{Aut}X$ generated be reflections will be called a {\bf finite lattice reflection group}. Given a lattice reflection $s$, an element $\alpha\in X$ will be called a {\bf root for $s$}, if we have $s\lambda-\lambda\in \DZ\alpha\;\forall \lambda\in X$. By a {\bf stable choice of roots} $R$ for a lattice reflection group we mean a $W$-stable subset $R\subset X$ consisting of exactly two roots for every reflection $s\in W$. Now the classification of compact connected Lie groups tells us we have a bijection on isomorphism classes $$\begin{array}{ccc} \left\{ \begin{array}{c} \text{Connected compact}\\ \text{Lie groups} \end{array} \right\} & \overset{\sim}{\ra} & \left\{ \begin{array}{c} \text{Finite lattice reflection groups}\\ \text{with stable choice of roots} \end{array} \right\} \\[6mm] K&\mapsto&W\looparrowright \frak{X}(T)\supset R \end{array}$$ for one and any maximal torus $T\subset K$. Here $W\pdef {\op{N}}_KT/T$ is the normalizer of the maximal torus divided by the torus itself, the so-called {\bf Weyl group}, and $R\pdef {\op{P}}_T(\op{Lie}_\DC K)\backslash \{0\}$ is the set of nonzero weights of the maximal torus on the complexified Lie algebra, the so-called {\bf root system}. The classification of connected reductive algebraic groups over an algebraically closed ground field is the same, only the maximal tori are now to be understood in an algebraic way. Now a lattice reflection group will also act as a lattice reflection group on the dual lattice $X^\vee$ and for every root $\alpha$ of a lattice reflection $s$ we may define its {\bf coroot} $\alpha^\vee\in X^\vee$ by $s\lambda=\lambda-\langle \lambda,\alpha^\vee\rangle \alpha$. This way to $(W\looparrowright X \supset R)$ as above we can associate the {\bf dual datum} $(W\looparrowright X^\vee \supset R^\vee)$. The {\bf dual group} for a connected reductive complex algebraic group $G$ is then defined as the connected reductive complex algebraic group $G^\vee$ corresponding to the dual datum. \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Case of $\DR^\times$}] We let $\bar\gamma:\DC^\times \ra \DC^\times $ be complex conjugation. Then Hilbert 90 says ${\op{H}}^1_{\bar \gamma}(\Gamma;\DC^\times)$ has just one element \end{Bemerkungl} \begin{Bemerkungl} Suppose $G$ is a connected reductive complex algebraic group and $\bar\gamma$ is an antiholomorphic involution stabilizing a Borel $B\subset G$.\label{ndfccK} %\label{ndfcc} As usual, $\bar\gamma$ induces an automorphism of the character lattice $\mathfrak X(B)\pdef \op{GrpVar}_\DC(B,\DC^\times)$ by putting $\chi^\gamma(b)\pdef \overline{\chi(\bar \gamma b)}$. This is an automorphism of the based root datum for $G\supset B$ and induces a holomorphic automorphism $\gamma$ of the dual group $G^\vee$, which stabilises the distinguished Borel $B^\vee\subset G^\vee$ corresponding to the positive coroots. \end{Bemerkungl} \begin{enumerate} \item Bedeutung de-Rham-Algebra \eref{pkDR}{TSF}. \item Hauptvermutung \eref{ndfcc}{Koszul}. \end{enumerate} %%% Local Variables: %%% mode: latex %%% TeX-master: "XXKAZ" %%% End: