\section{First lecture on Koszul duality, July 3} \subsection{Autoduality for category $\mathcal O$} \begin{Bemerkungl} Let $(\mathfrak g\supset\mathfrak h, R^+)$ be a semisimple complex Lie algebra with a Cartan and a system of positive roots. We recall category $\mathcal O$. It contains the Verma modules $\Delta(\lambda)$, their simple quotients $\op{L}(\lambda)$ and their $\mathcal O$-projective covers $\op{P}(\lambda)$. \end{Bemerkungl} \begin{Bemerkungl} Let $Z\pdef{\op{Z}}(\op{U}(\mathfrak g))$ be the center of the enveloping algebra. Denote by $Z^+\pdef \op{Ann}_Z\DC$ the annihilator of the trivial onedimensional representation. The full subcategory $$\mathcal O_0\pdef \{M\in \mathcal O\mid \exists n \text{ such that }(Z^+)^n M=0\}$$ is called the {\bf principal block of category $\mathcal O$}. Its simple objects are the ${\op{L}}(x\cdot 0)$ for $x\in W$. We abbreviate ${\op{L}}(x\cdot 0), \Delta(x\cdot 0), {\op{P}}(x\cdot 0)$ as $L^x, \Delta^x, P^x$. \end{Bemerkungl} \begin{Theorem}[\textbf{Autoduality isomorphism for $\mathcal O_0$}] Given $(\mathfrak g\supset\mathfrak h, R^+)$ a semisimple complex Lie algebra with a Cartan and a system of positive roots there is an isomorphism of $\DC$-ringalgebras\label{SDO} $$\textstyle \op{End}_{\mathcal O_0}(\bigoplus_{x\in W}{\op{P}}^x) \sira \op{Ext}_{\mathcal O_0}^*(\bigoplus_{x\in W}{\op{L}}^x,\bigoplus_{x\in W}{\op{L}}^x)$$ \end{Theorem} \begin{Bemerkungl} This was conjectured in \cite{BGi} and proven in \cite{So-A}. Both sides are of finite dimension over $\DC$. \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Images of projectors}] Denote by $1_x$ the projektors onto the summands. Our autoduality isomorphism has the additional property $1_x\mapsto 1_{xw_0}$ for $w_0$ the longest element of the Weyl group. It will turn out helpful to introduce the notation $P_x\pdef P^{xw_0}$. Then we get an isomorphism $$\textstyle \op{End}_{\mathcal O_0}(\bigoplus_{x\in W}{\op{P}}_x) \sira \op{Ext}_{\mathcal O_0}^*(\bigoplus_{x\in W}{\op{L}}^x, \bigoplus_{x\in W}{\op{L}}^x)$$ with $1_x\mapsto 1_x$ for the obvious projectors on both sides. \end{Bemerkungl} \begin{Proposition}[\textbf{Abelian category as category of modules}] Let $\mathcal A$ be an abelian category, in which every object has finite length. Assume there exists a projective objekt $P\in \mathcal A$ surjecting on every simple object. Then we have an equivalence of categories\label{AKM} $$\op{Hom}_{\mathcal A}(P, \;): \mathcal A \sirra \op{Modfl-}\op{End}_{\mathcal A}(P)$$ \end{Proposition} \begin{Bemerkungw} For $A\pdef \op{End}_{\mathcal O_0}(\bigoplus_{x\in W}{\op{P}}_x)$ we get an equivalence of categories $\mathcal O_0\sirra \op{Modfl-}A$. Together with the autoduality isomorphism \ref{SDO} we get an equivalence of categories\label{AKO} $$\mathcal O_0\sirra \op{Modfl-}\op{Ext}_{\mathcal O_0}^*(L,L)$$ with the abbreviation $L\pdef \bigoplus_{x\in W}{\op{L}}^x$. This equivalence is our starting point to formulate a conjecture on \glqq Koszul-duality for real reductive groups\grqq. \end{Bemerkungw} \begin{Bemerkungl}[\textbf{Rewriting the autoduality isomorphism}] We choose an autoduality isomorphism $A\sira \op{Ext}_{\mathcal O_0}^*(L,L)$ as in \ref{SDO} and give $A$ the induced grading $A=\bigoplus_{n\geq 0} A^n$. We thus have $A^0=A/A^{>0}=\bigoplus \DC 1_x$ and this is the direct sum of all simple $A$-modules, up to isomorphism. Thus we have $L\mapsto A^0$ under our equivalence, in formulas $$\begin{array}{ccc} \mathcal O_0&\sirra& \op{Modfl-}A\\ L&\mapsto &A^0 \end{array} $$ Thus our equivalence of abelian categories leads to an isomorphism\label{vDs} $\op{Ext}^*(L,L)\sira \op{Ext}_{-A}(A^0,A^0)$ and the autoduality isomorphism leads to an isomorphism of graded rings $$A\sira \op{Ext}_{-A}^*(A^0,A^0)$$ Recall we have $1_x\mapsto 1_{xw_0}$ for this isomorphism. \end{Bemerkungl} \subsubsection*{Exercises} \begin{Ubungex} Just once check the case of $\mathfrak{sl}_2$. Recall how $\mathcal O_0$ is equivalent in this case to finite dimensional representations of a quiver with two dots, two arrows going back and forth between them and ane relation, namely composing the two arrows in on way is zero. The functor maps an object $M\in \mathcal O_0$ to the pair of vector spaces $M_{-\alpha}, M_0$ with arrows given by $E$ and $F$ from the Lie algebra. \end{Ubungex} \subsection{Koszul duality for Koszul rings} \begin{Bemerkungl} Given a vector space $V$ of finite dimension $\op{dim}_kV=n<\infty$ over a field $k$ the {\bf Koszul complex} is an exact complex $$ \textstyle\bigwedge^n V \otimes {\op{S}}V\hra \bigwedge^{n-1} V \otimes {\op{S}}V \ra \ldots \ra V \otimes {\op{S}}V\ra {\op{S}}V \sra k$$ We may see it as a projective resolution $P^{-n}\hra P^{-n+1}\ldots\ra P^{-1}\ra P^0\sra k$ of the augmentation module $k={\op{S}}^0V$ in the category of right ${\op{S}}V$-modules. It gives isomorphisms $(\bigwedge^i V)^*\sira \op{Ext}^i_{{\op{S}}V}(k,k)$. \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Ext Algebra}] To compute the Ext algebra with its Yoneda product, we consider the partial evaluations $(\bigwedge^i V^*)\otimes \bigwedge^j V \ra \bigwedge^{j-i} V$ with negative wedges understood to be zero. They give us maps $(\bigwedge^i V^*)\ra \op{Hom}_{ {\op{S}}V}(P,P[i])$ and even isomorphisms $(\bigwedge^i V^*)\sira \op{Hot}_{ {\op{S}}V}(P,P[i])$ and we see these fit together to an isomorphism of $\DZ$-graded algebras\label{KdKr} $$\textstyle\bigwedge V^*\sira \bigoplus_i \op{Ext}^i_{ {\op{S}}V}(k,k)$$ \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Reinterpretation of the Koszul complex}] Now let us view the Koszul complex as a bigraded vector space in the positive quadrant with components $\bigwedge^iV\otimes {\op{S}}^jV$ and with a differential of bidegree $(-1,1)$, whose cohomology is concentrated in the bidegree $(0,0)$. By summing along the verticals we obtain our projective resolution of the ${\op{S}}V$-module $k$ above. By rather summing along the horizontals and adding $k$ up front we obtain on the other hand an exact complex of $\bigwedge V^*$-modules $$\textstyle k\hra \bigwedge V\otimes {\op{S}}^0V \ra \bigwedge V\otimes {\op{S}}^1V\ra \bigwedge V\otimes {\op{S}}^2V\ra\ldots$$ This can be seen as an injective resolution $k\hra I^0\ra I^1\ra\ldots$ of the augmentation module $k$ of $\bigwedge V^*$ leading in the same way as above to an algebra isomorphism\label{rKc} $$\textstyle{\op{S}}V\sira \bigoplus_j \op{Ext}^j_{ \bigwedge V^*}(k,k)$$ \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Ext-Algebra of an $\DN$-graded ring and examples}] Given any $\DN$-graded ring $B$ we denote by $E(B)$ the $\DN$-graded ring $$E(B)\pdef \op{Ext}^*_{B}(B^0, B^0)$$ For any finite dimensional vector space $V$ over a field $k$ we constructed what in this notation are isomorphisms $E( {\op{S}}V)\sira \bigwedge V^*$ and $E( \bigwedge V^*)\sira {\op{S}}V$ in \ref{KdKr} and \ref{rKc}. For the $\DN$-graded ring $A$, whose finite length right modules are equivalent to $\mathcal O_0$, we find by \ref{vDs} $$E(A^{\op{opp}})\cong A$$ \end{Bemerkungl} \begin{Bemerkungwex} The common framework which can accomodate all these examples is the theory of {\bf Koszul rings}. The case of the symmetric algebra discussed here is the archetypical example of a Koszul ring and its dual, the case of the ring $A$ describing the principal block $\mathcal O_0$ of category $\mathcal O$ as right $A$-modules being a most surprising example of a Koszul ring isomorphic to its own opposed dual or as we say an {\bf autodual Koszul ring}. \end{Bemerkungwex} \begin{Definition} A nonnegatively graded ring $B$ is called {\bf Koszul}, if $B^0$ is semisimple and in the abelian category $B\op{-Mod}^\DZ$ of $\DZ$-graded $B$-modules we have\label{KOz} $$\op{Ext}^i(B^0, B^0\langle -j\rangle)=0\;\; \text{ unless }i=j.$$ Here $\langle j\rangle$ means an internal, non-homological grading shift normalized so that $B^0\langle -j\rangle$ is concentrated in the internal degree $j$. \end{Definition} \begin{Beispielex} The symmetric algebra and the exterior algebra over a finite dimensional vector space are Koszul rings by what we saw already. \end{Beispielex} \begin{Bemerkungl} In fact, we only need the case $B^0$ is a finite product of copies of $\DC$. \end{Bemerkungl} \begin{Theorem}[\textbf{Koszul duality \cite{BGSo}}] Let $B$ be a Koszul ring and assume it is finitely generated as a $B^0$-module from the left as well as from the right. Assume in addition $E=E(B)\pdef \op{Ext}^*_{B}(B^0, B^0)$ is right noetherian. Then there exists an equivalence of triangulated categories\label{koDU} $$\op{Der}^{\op{b}}(B\op{-Modfg}^\DZ)\sirra \op{Der}^{\op{b}}(\op{Modfg^\DZ-}E)$$ \end{Theorem} \begin{proof} Needs some theory, will de done in \ref{pfKO}. We there give the proof under the additional assumptions $B^0$ is a finite product of copies of a field and $E$ is of finite homological dimension. \end{proof} \begin{Korollar}[\textbf{Koszul duality for the symmetric algebra}] Let $V$ be a finite dimensional vector space. Then there exists an equivalence of triangulated categories $$\textstyle \op{Der}^{\op{b}}({\op{S}}V\op{-Modfg}^\DZ)\sirra \op{Der}^{\op{b}}(\bigwedge V^*\op{-Modfg}^\DZ)$$ \end{Korollar} \begin{Bemerkungl} In algebraic geometry this is quite helpful. In fact, the quotient category ${\op{S}}V\op{-Modfg}^\DZ/\{\text{finite dimensional graded modules}\}$ is the category of coherent $\mathcal O$-modules on the projective space $\mathbb P V^*$. %\nichtfinal{Pr"azisieren!} \end{Bemerkungl} \begin{Bemerkungl} There are funny things on Koszul duality \cite{BGSo} I will not discuss in any detail. If $B$ is a \glqq left finite\grqq\ Koszul ring, meaning all $B^i$ are finitely generated left $B^0$-modules, then the same is true for $E(B)$ and $E(E(B))=B$ canonically. Koszul rings are always quadratic and for a left finite Koszul ring the Koszul dual is the opposed ring of the quadratic dual $E^{\op{opp}}=B^!$. \end{Bemerkungl} \subsection{Numerical Koszulity criterion} \begin{Bemerkungl}[\textbf{Numerical evidence for Koszulity}] Suppose we have a field $k$ and a $\DN$-graded $k$-ringalgebra $B=\bigoplus_{i\geq 0} B^i$ with $B^0=k$ such that all $B^i$ are finite dimensional. Suppose $B$ is Koszul. A minimal projective resolution of $B^0=k$ will take the form $$\ldots \ra B\otimes_k V^2\ra B\otimes_k V^1\ra B\otimes_k V^0\sra k$$ with all $V^j$ finite dimensional vector spaces over $k$ put in degree $j$ and isomorphisms $\op{Ext}^j_B(B^0,B^0)\cong (V^j)^*$. The alternating sum of dimensions in the homogeneous parts of this complex leads to $$\begin{array}{lll}\delta_{n,0}&=&\sum_{i+j=n}(-1)^j(\op{dim}B^i)(\op{dim}V^j) \\&=&\sum_{i+j=n}(-1)^j(\op{dim}B^i)(\op{dim}\op{Ext}^j_B(B^0,B^0)) \end{array} $$ We abbreviate $E=E(B)=\op{Ext}^*_B(B^0,B^0)$ and can rewrite this as an equation of generating functions $$1= \left(\sum_{j}(\op{dim}B^j)t^j\right)\left(\sum_{i}(\op{dim}E^i)(-t)^i\right)={\op{P}}_B(t){\op{P}}_E(-t)$$ with the so-called {\bf Poincar\'e series} of our $\DN$-graded algebras defined by the big brackets. The was half the proof of the following theorem. \end{Bemerkungl} \begin{Theorem}[\textbf{Numerical Koszulity criterion}] Suppose given a field $k$ and an $\DN$-graded $k$-ringalgebra $B=\bigoplus_{i\geq 0} B^i$ with $B^0=k$ and all $B^i$ finite dimensional. Then $B$ is Koszul if and only if we have $${\op{P}}_B(t){\op{P}}_E(-t)=1$$ \end{Theorem} \begin{proof} One implication was proven already. The other implication is proven by turning the argumentation around. Details can be found in \cite{BGSo}. \end{proof} \begin{Bemerkungl}[\textbf{Numerical evidence for Koszulity, variant}] Now let us assume $k=B^0=E^0=F 1_x\oplus \ldots\oplus F 1_y$ is no longer a field but rather a finite product of copies of a field $F$ for $1_x,\ldots ,1_y$ pairwise orthogonal idempotents indexed by a finite set $W$. We ask $F$ to act the same from the left and the right and $B$ is Koszul with finite dimensional homogeneous components. Then everything remains true, but the equation reads now $$\delta_{x,z}\delta_{n,0}=\sum_{i+j=n,\; y\in W}(-1)^j(\op{dim} 1_xB^i1_y)(\op{dim}_\DC1_zE^j1_y)$$ (remark $(1_yV^j1_z)^*=1_zE^j 1_y$) or put as an equation of generating functions, which are now $(W\times W)$-matrices with entries $\sum (\op{dim}_\DC1_xB^i1_y)t^i$ for ${\op{P}}_B$ and analogous for ${\op{P}}_E$, we get the identity matrix as the product\label{nEkd} $${\op{P}}_B(t) {\op{P}}_E(-t)^\top =\op{I}$$ \end{Bemerkungl} \begin{Theorem}[\textbf{Numerical Koszulity criterion, variant}] Suppose given a field $F$ and an $\DN$-graded $F$-ringalgebra $B=\bigoplus_{i\geq 0} B^i$ with $B^0=F 1_x\oplus \ldots\oplus F 1_y$ and all $B^i$ finite dimensional. Then $B$ is Koszul if and only if we have\label{ncKv} $${\op{P}}_B(t){\op{P}}_E(-t)^\top=\op{I}$$ \end{Theorem} \begin{proof} One implication was proven already. The other implication is proven by turning the argumentation around. Details can be found in \cite{BGSo}. \end{proof} \begin{Bemerkungl}[\textbf{Motivation of Koszul duality by inversion formulas}] Knowing BGG-reciprocity, the KL-conjectures compute $\op{dim}1_xA1_y$ for $A$ describing $\mathcal O_0$. Knowing more geometry, they also compute $\op{dim}1_xE(A)1_y$. Then the inversion formulas for KL-polynomials then show the numerical Koszulity criterion at $t=1$. This was a central motivation for Beilinson and Ginzburg. \end{Bemerkungl} \begin{Ubungex} Check the claims for $\mathfrak{sl}_2$. Here the Ext-Algebra was computed by hand already. \end{Ubungex} \newpage \section{Second lecture on Koszul duality, July 5} \subsection{Equivariant cohomology} \begin{Bemerkungl}[\textbf{Prerequisites}] To develop the equivariant version of sheaf cohomology, we need in addition to sheaf cohomology itself only two things: \begin{description} \item[Contractible pullback:] Pulling back along a fibre bundle with contractible fibre induces isomorphisms on sheaf cohomology. So in formulas, if $\pi:X\ra Y$ is a continous map of topological spaces and there is a contractible space $F$ such that $Y$ can be covered by open subsets $U\co Y$ with the property that there exist homeomorphisms $h:U\times F\sira \pi^{-1}U$ with $\pi h=\op{pr}_U$, then $\pi^*:{\op{H}}^q(X;M)\sira {\op{H}}^q(Y;M)$ for any abelian group $M$. \item[Contractible free space:] For any topological group $G$ there exists a contractible free $G$-space ${\op{E}}G$, i.e.\ a contractible $G$-space ${\op{E}}G$ such that $\pi: {\op{E}}G\ra {\op{E}}G/G$ is a fibre bundle with fibre $G$. In more detail we might call this property \glqq topologically free\grqq, but we abbreviate to free in this class. We can obtain such a thing by a constuction of Milnor. Every continous group homomorphism $\phi:G\ra H$ then even induces an equivariant continous map $\phi:{\op{E}}G\ra {\op{E}}H$. \end{description} \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Equivariant cohomology}] With these prerequisites, the general theory can be considered an exercise. \begin{enumerate} \item Product of two $G$-spaces is free if one of the factors is. \item Put ${\op{H}}^q_G(X)\pdef {\op{H}}^q_G({\op{E}}G\times_{/G} X)$. Discuss why the choice of the free contractible $G$-space ${\op{E}}G$ is irrelevant. \item \textbf{Trivial group acting:} For $G=1$ get $\op{pr}^*:{\op{H}}^q(X)\sira {\op{H}}^q({\op{E}}1\times_{/1}X)={\op{H}}_1^q(X)$. Omit the $1$. \item \textbf{Equivariant pullback:} Get for $(\phi\acts f):G\acts X\ra H\acts Y$ a pullback $(\phi\acts f)^*:{\op{H}}^q_H(Y)\ra {\op{H}}^q_G(X)$. \item\textbf{Cup product:} We should define \glqq equivariant multipullback\grqq\ for families $(\phi_\rho\acts f_\rho):G\acts X\ra H_\rho\acts Y_\rho$ of equivariant maps indexed by $1\leq \rho\leq r$ with any $r\geq 0$ as multilinear maps $${\op{H}}^{q_1}_{H_1}(Y_1)\times \ldots \times{\op{H}}^{q_r}_{H_r}(Y_r) \ra {\op{H}}^q_G(X)$$ with $q=q_1+\ldots +q_r$, but since we even didn't do this carefully in the nonequivariant situation, I cannot and will not dwelve into this. This encodes the cup product, which may be seen as multipullback along twice the identity, and its compatibilities. \item \textbf{Free quotient isomorphism:} If $G$ acts freely on $X$ we have $$(1\acts q)^*:{\op{H}}_1^q(X/G)\sira {\op{H}}^q_G(X)$$ By the rule for the trivial group acting, we have ${\op{H}}_1^q(X/G)={\op{H}}^q(X/G)$. \item For $G=\DZ$ can take $E=\DR$ with translation action. So we get $${\op{H}}^q_\DZ(\op{top})={\op{H}}^q(\DR\times_{/\DZ}\op{top}) ={\op{H}}^q(S^1)$$ \item Given a principal $G$-bundle $\pi: X\ra Y$ alias a free $G$-space $X$ with projection $\pi:X/G\sira Y$ we get homomorphisms of abelian groups $${\op{H}}^q_G(\op{top})\ra {\op{H}}^q_G(X)\sila {\op{H}}^q(X/G)\sila {\op{H}}^q(Y)$$ This homomorphism $C_X:{\op{H}}^q_G(\op{top})\ra {\op{H}}^q(Y)$ is called the {\bf characteristic homomorphism} of $X$, the images of distinguished elements of ${\op{H}}^q_G(\op{top})$ are called {\bf characteristic classes}. \item For $G=\DC^\times$ we may take $E=\DC^\infty\backslash 0$ and get $${\op{H}}^q_{\DC^\times}(\op{top}) ={\op{H}}^q((\DC^\infty\backslash 0)\times_{/\DC^\times}\op{top}) ={\op{H}}^q(\mathbb P^\infty\DC)$$ This is a polynomial ring $\DZ[t]$ with $t\in {\op{H}}^2(\mathbb P^\infty\DC)$. The image of $t$ under the characteristic homomorphism of a principal $\DC^\times$-bundle is called its {\bf first Chern class}. If we have a complex line bundle instead and want its Chern class, we first remove its zero section to get a principal $\DC^\times$-bundle. \item Maybe as true exercises:\begin{enumerate} \item {\textbf{Equivariant contractible pullback isomorphism:}} Given $f:X\ra Y$ a $G$-equivariant map, which is a fibration with contractible fibres, pull-back gives isomorphisms ${\op{H}}^q_G(Y)\sira {\op{H}}^q_G(X)$. \item If $G$ is contractible, ${\op{H}}^q(X)={\op{H}}_1^q(X)\sira {\op{H}}_G^q(X)$ under pullback. \item {\textbf{Ignoring contractible nonacting normal subgroups:}} If $N\subset G$ is a contractible normal subgroup acting freely on $G$ and $X$ is a $G/N$-space, ${\op{H}}^q_{G/N}(X)\sira {\op{H}}_G^q(X)$ under pullback. This generalizes the preceding point. \item {\textbf{Generalized free quotient isomorphism:}} If If $K\subset G$ is a normal subgroup acting freely on $G$ and $X$, then we have ${\op{H}}^q_{G/K}(X/K)\sira {\op{H}}_G^q(X)$ under pullback. \item {\textbf{Induction isomorphism:}} If $G\subset H$ is a subgroup acting freely on $H$, ${\op{H}}^q_{H}(H/G)\sira {\op{H}}_G^q(\op{top})$ under pullback and even ${\op{H}}^q_{H}(H\times_{/G}X)\sira {\op{H}}_G^q(X)$ for any $G$-space $X$. \item {\textbf{Restricting to subgroups with contractible quotient:}} If If $K\subset G$ is a subgroup acting freely on $G$ such that $G/K$ is contractible, then we have ${\op{H}}^q_{G}(X)\sira {\op{H}}_K^q(X)$ under pullback. \end{enumerate} \end{enumerate} \end{Bemerkungl} \subsection{Equivariant de-Rham complex} \begin{Bemerkungl} I hope to have convinced everyone that the general theory is sort of easy. More serious is the problem to actually calculate examples beyond $S^1$ and $\DZ$, say describe $${\op{H}}_G^q(\op{top})$$ for $G$ a Lie group. At least for real coefficients and $G$ connected this can be done by an equivariant version of the de-Rham complex due to Cartan. For $G$ discrete on the other hand, ${\op{H}}_G^q(\op{top})$ is called {\bf group cohomology} and is a whole business in itself. For $G$ a Galois group with its Krull topology, we get what is called {\bf Galois cohomology} and again this is a whole business in itself. This is mostly considered for discrete-continous representations of the Galois group in question and will turn out a special case of equivariant cohomology with coefficients in an equivariant sheaf, we will be getting there. \end{Bemerkungl} \begin{Bemerkungl} \begin{enumerate} \item In the same way sheaf cohomology with $\DR$-coefficients of a smooth paracompact manifold can be calculated by the de-Rham complex, equivariant cohommology with $\DR$-coefficients of a smooth paracompact manifold $X$ acted upon by a connected compact Lie group $G$ can be calculated be the so-called equivariant de-Rham complex. \item This complex is $$\Omega^*(G{\ssearrow}X)\pdef\big(\Omega^*(X)\otimes_\DR \mathcal O^{\op{db}}_\DR(\mathfrak g)\big)^G$$ Here $\mathcal O^{\op{db}}_\DR(\mathfrak g)\subset \op{Ens}(\mathfrak g,\DR)$ denotes the sub-$\DR$-ringalgebra generated by the linear maps $\mathfrak g\ra \DR$ and graded in a way these linear maps have degree two, thus doubling the usual grading. \item As differential on $\Omega^*(G{\ssearrow}X)$ we take $$\diff_G(\omega\otimes f)\pdef (\diff\omega)\otimes f - \sum_{\rho} i_{\xi_\rho}\omega \otimes \xi_\rho^\top f$$ for $\xi_1,\ldots,\xi_r$ a basis of $\mathfrak g$ and $\xi_\rho^\top$ the corresponding dual basis alias coordinate on $\mathfrak g$. Furthermore we let $\xi=\xi^X$ denote the vector field of the infinitesimal action of $\xi$ %nach \ref{VFoU} and $i_\xi\omega$ the partial insertion.% \ref{ixi}. This in the later summands lowers the degree of $\omega$ by one but increases the degree of the function by two, since we use the doubled grading, so indeed our differential gets the total degree up by one. \item The complex is easily seen to be functorial in $G\acts X$. The equivariant de-Rham theorem gives us for a compact connected Lie group $K$ and a smooth paracompact $K$-manifold $X$ a natural isomorphism $$\mathcal H^q\Omega^*(K\acts X)\sira {\op{H}}_K^q(X;\DR)$$ \item The most difficult part in proving the comparision isomorphism is to show that for a smooth principal $G$-bundle $P\ra X$ pulling back forms gives isomorphisms $$\mathcal H^q\Omega^*(1\acts X)\sira \mathcal H^q\Omega^*(P\acts E)$$ I will not write out this proof here. \item For a connected compact Lie group $K$ acting on the one-point manifold we get in particular isomorphisms $${\op{H}}^*_K(\op{top};\DR)\sira \mathcal O^{\op{db}}_\DR(\op{Lie}K)^K$$ with the doubled grading on the right, so ${\op{H}}^q_K(\op{top};\DR)=0$ for $q$ odd. We call this graded $\DR$-ringalgebra sometimes $\mathcal A_K$. \item For example we find an isomorphism ${\op{H}}^*_{S^1}(\op{top};\DR)\sira \DR[t]$ with $\op{deg}t=2$. Pulling back by the surjection with contractible kernel $\DC^\times\sra S^1$ we get an isomorphism ${\op{H}}^*_{S^1}(\op{top};\DR)\sira {\op{H}}^*_{\DC^\times}(\op{top};\DR)$ and this fits with what we found before. \item For a homogeneous space $K/T$ we recall ${\op{H}}^*_{T}(\op{top})\sira {\op{H}}^*_{K}(K/T)$. If $K$ is compact connected and $T$ is a maximal torus, we recall the Chevalley isomorphism $\mathcal O_\DR(\op{Lie} K)^K\sira \mathcal O_\DR(\op{Lie} T)^W$ by restriction. \item Special case $K=\op{U}(n)\subset T$ diagonal unitary matrices. Weyl group is the symmetric group permuting the diagonal entries. Then $\mathcal O_\DR(\op{Lie} T)^W=\DR[t_1,\ldots,t_n]^{\mathcal S_n}$ is the ring of symmetric polynomials. The elementary symmetric polynomials correspond to elements $c_i\in {\op{H}}^{2i}_{\op{U}(n)}(\op{top};\DR)$ which in turn lead to Chern classes. \item Given a connected compact Lie group $K$ with two connected compact subgroups $L,M$ we will explain later in more detail how to get an isomorphism $${\op{H}}^*_{L\times M}(K;\DR)\sira \mathcal A_L\otimes_{\mathcal A_K}^{\op{L}} \mathcal A_M$$ for the action of $L$ on the left and of $M$ on the right. The case $L=K$ we did already, then ${\op{H}}^*_{K\times M}(K;\DR)\cong {\op{H}}^*_{K}(K/M;\DR)\sira {\op{H}}^*_{M}(\op{top};\DR)$ under pulling back. If a derived tensor product is needed, things are a bit more tricky then usual since we have to tensor in $\op{dgDer-}(\mathcal A_K,d=0)$. \item We will mostly work with complex connected reductive groups $G$ instead of compact connected Lie groups. Remark $\op{GL}(n;\DC)/\op{U}(n)$ is contractible by the either the Iwasawa decomposition $\op{U}(n)\times A\times N\sira \op{GL}(n;\DC)$ or the Cartan decomposition. Similarly, $G/K$ is contractible for $K\subset G(\DC)$ a compact real form alias a maximal compact subgroup. Thus $${\op{H}}^*_{G(\DC)}(\op{top};\DR)\sira {\op{H}}^*_{K}(\op{top};\DR)$$ by restriction. Changing to complex coefficients we find $${\op{H}}^*_{G(\DC)}(\op{top};\DC)\sira \mathcal O^{\op{db}}(\op{Lie}G)^G\sira\mathcal O^{\op{db}}(\op{Lie}T)^W$$ now for complex regular functions on the complex Lie algebra of $G$ or on the complex Lie algebra of a maximal torus $T\subset G$, always for $G$ a complex connected reductive algebraic group. \item Take $T\subset B\subset G$ a complex connected reductive group with a Borel and a maximal torus. We know $B/T$ is contractible. We put $S\pdef \mathcal O^{\op{db}}(\op{Lie} T)$, it is the symmetric algebra on the dual if the Lie algebra with doubled grading. For the cohomology of the flag variety we find $${\op{H}}^*(G/B;\DC)= {\op{H}}_{1\times B}^*(G;\DC)= {\op{H}}_{1\times T}^*(G;\DC)= \mathcal A_1\otimes_{\mathcal A_G}\mathcal A_T=\DC\otimes_{S^W} S= S/(S^+)^WS$$ For the $T$-equivariant cohomology of the flag variety we find similarly $${\op{H}}^*_T(G/B;\DC)= {\op{H}}_{B\times B}^*(G;\DC)= {\op{H}}_{T\times T}^*(G;\DC)= \mathcal A_T\otimes_{\mathcal A_G}\mathcal A_T=S\otimes_{S^W} S$$ \item For the ordinary cohomology of $G$ we find ${\op{H}}^*_{1\times 1}(G;\DC)\cong \DC\otimes_{S^W}^{\op{L}}\DC$ to be interpreted in $\op{dgDer}$. We will come back to this later. \item Exercises. \begin{enumerate} \item We may in the same way compute cohomology of projective space and try to see we get back what we know already. \item We may compute cohomology of Grassmannians, as well as their equivariant cohomology say for a maximal torus. \item We may try to understand why $S\otimes_{S^W}S\sira \mathcal O(\bigcup_{w\in W}\Gamma(w))$ can be seen as the regular functions on the union of the graphs of the Weyl group elements acting on $\op{Lie}T$. I think the map is easy, it would work for any finite group. That it is an isomorphism is too difficult for an exercise, but the case of $\op{SL}(2;\DC)$ is accessible and instructive. \end{enumerate} \end{enumerate} \end{Bemerkungl} \subsection{Equivariant sheaves} We start with equivariant sheaves before coming to the equivariant derived categories. I follow a geometric path which seems the most natural to me. In my experience most mathematicians already knowing the subject find this approach confusing, maybe since it is so different from the usual path. \begin{Bemerkungl} Given $G{{\ssearrow}} X$ a topological space $X$ with the action of a topological monoid $G$ we define an\label{gAQm} {\bf $G$-equivariant sheaf of sets on $X$}\index{"aquivariant!Garbe} to be a sheaf of sets $\mathcal F\in\op{Ens}_{/X}$ along with a continous action $G\times \bar{\mathcal F}\ra \bar{\mathcal F}$ of our monoid on its \'etale space such that the projection $\bar{\mathcal F}\ra X$ is equivariant. The category of $G$-equivariant sheaves of sets on $X$ will be denoted $$\op{Ens}_{/G {\ssearrow} X}$$ \end{Bemerkungl} \begin{Beispielex}[\textbf{Equivariant sheaves on the one-point space}] In case of a discrete monoid acting on a one-point space the functor of global sections is an equivalence $$\op{Ens}_{/G {\ssearrow} {\op{top}}}\sira G\op{-Ens}$$ to the category of $G$-sets. In case of a topological monoid $G$ we get the same with $G\op{-Ens}$\index{Ens@$G\op{-Ens}$ Kategorie der $G$-Mengen} meaning discrete sets $F$ with a continous $G$-action, so $G\times F\ra F$ should be continous. If $G$ is a topological group admitting a connected neighbourhood of $1\in G$, pullback from the quotient by the component of the identity element gives an isomorphism of\label{Gpt} categories $$(G/G^\circ)\op{-Ens}\sira G\op{-Ens}$$ \end{Beispielex} \begin{Bemerkungl}[\textbf{Equivariant abelian sheaves}] Let $G{{\ssearrow}} X$ be a topological space acted upon by a topological monoid. We define an equivariant sheaf of abelien groups to be sheaf of abelian groups ${\cal{F}}$ with a continous action of $G$ an its \'etale space such that\label{gAQmg} foor all $g \in G$ and $ x \in X$ the map $g: {\cal{F}}_{x} \ra {\cal{F}}_{gx}$ induced on the stalks is a group homomorphism. We denote this category by\index{Ab@$\op{Ab}_{/G {\ssearrow} X}$ equivariante abelschen Garben auf $X$} $$\op{Ab}_{/G {\ssearrow} X}$$ \end{Bemerkungl} \begin{Bemerkungl} This definition is quite ad-hoc. To give a more conceptual definition, we should explain what is a morphism of abelian sheaves over a continous map of topological spaces and then define a $G$-equivariant abelian sheaf as an abelian sheaf with a morphism $\underline G\boxtimes \mathcal F\ra \mathcal F$ over $G\times X\ra X$ for $\underline G=\DZ_G$ the constant sheaf on $G$, and ask of this morphism the \glqq obvious\grqq\ properties. To make this precise in a satisfying way however I would need the framework of \glqq multifibrations\grqq\ and this would take us too far. \end{Bemerkungl} \begin{Beispielex}[\textbf{Equivariant abelian sheaves on the one-point space}] As in the case of set-sheaves we get an isomorphism of categories $$\op{Ab}_{/G {\ssearrow} {\op{top}}}\sira G\op{-Ab}$$ In case $G$ is a topological group and there exists a connected neigbourhood of the identity, pullback gives an isomorphism of categories $$(G/G^\circ)\op{-Ab}\sira G\op{-Ab}$$ \end{Beispielex} \begin{Bemerkungl} A {\bf morphism $(G{\ssearrow} X)\ra(H{\ssearrow} Y)$ of spaces with action}\label{MTopo} is a pair $(\phi ,f)$ consisting of a continous monoid homomorphism $\phi : G \ra H$ and a continous map $f:X\ra Y$ such that $f(gx)=\phi(g)f(x)\;\forall g,x$. We denote such pairs by $\phi {\ssearrow} f$. \end{Bemerkungl} \begin{Bemerkungl} Given a morphism $(\phi {\ssearrow} f):(G{\ssearrow} X)\ra(H{\ssearrow} Y)$ and an equivariant sheaf $\mathcal G\in \op{Ens}_{/H{\ssearrow} Y}$ we construct the {\bf equivariant pullback}\index{Garbe!"aquivariant zur"uckgezogene} $$f^*\mathcal G=(\phi{\ssearrow} f)^*\mathcal G$$ by taking the pullback sheaf $\op{\acute{e}t}(f^*\mathcal G)= X\times_Y \bar{\mathcal G}$ and equip it with the obvious $G$-action. In the same way we construct equivariant pullback for abelian sheaves $$(\phi{\ssearrow} f)^*:\op{Ab}_{/H{\ssearrow} Y}\ra \op{Ab}_{/G{\ssearrow} X}$$ \end{Bemerkungl} \begin{Bemerkungl} We call a topological space $X$ with the action of a topological group $G$ {\bf topologically free} or to simplify just {\bf free}, if it admits a cover by open $G$-stable subsets $U\co X$ which are equivariantly isomorphic to $G$-spaces of the form $G\times W$ for some topological space $W$. Equivalent is the condition that $X\ra X/G$ is a principal $G$-bundle. \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Forget trivial group acting}] Sure forgetting the action of the trivial group we get $$\op{Ens}_{/1{\ssearrow}X}\sirra \op{Ens}_{/ X}$$ \end{Bemerkungl} \begin{Satzex}[\textbf{Quotient equivalence}] Given a topological group $G$ and a free $G$-space $X$\label{GQR} equivariant pullback by $\phi{\ssearrow}f$ for $f:X\sra G\backslash X$ and $\phi:G\sra 1$ is an equivalence of categories $$\op{Ens}_{/1{\ssearrow}(X/G)}\sirra \op{Ens}_{/G{\ssearrow} X}$$ \end{Satzex} \begin{Satzex}[\textbf{Induction equivalence}] Let $G$ be a topological group, $H \subset G$ a subgroup acting freely on $G$ and $Y$ an $H$-space. Then pullback along $H{\ssearrow} Y\ra G{\ssearrow} (G \times_{/H} Y)$ gives an equivalence of categories $$\op{Ens}_{/G{\ssearrow} (G \times_{/H} Y)} \sirra \op{Ens}_{/H{\ssearrow} Y}$$ \end{Satzex} \begin{Bemerkungl} A quasiinverse can be given in \'etale spaces as ${\bar{\cal F}} \mapsto G \times_{/H}{\bar{\cal F}}$. \end{Bemerkungl} \begin{Beispielex}[\textbf{Equivariant sheaves on homogeneneous spaces}] Take in the induction equivalence $Y = {\op{top}}$. Recalling the description \ref{Gpt} of equivariant sheaves on the one-point space and the induction equivalence \ref{ProE} we get for any subgroup $H\subset G$ of a topological group acting freely and such in $H$ the identity has a connected neighbourhood an equivalence\label{eqHG} $$(H/H^{\circ})\op{-Ab}\sirra \op{Ab}_{/G{\ssearrow}(G/H)} $$ In particular all $G$-equivariant sheaves on $G/H$ are locally constant in this case. \end{Beispielex} \newpage \section{Third lecture on Koszul duality, July 8} \subsection{Equivariant derived category} \begin{Bemerkungl} Given a topological space $X$ let $\op{Der}(X)=\op{Der}_{/X}\pdef\op{Der}(\op{Ab}_{/X})$ denote the derived category of the abelian category of abelian sheaves on $X$. What follows I mostly learned from \cite{BeLu}. \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Strong contractible pullback}] To discuss quivariant derived categories, we need the following strenghtening of contractible pullback: Given a fiber bundle $f:X\ra Y$ with contractible fibres, for any $\cal{F} \in \op{Der} (\op{Ab}_{/Y})$ the unit of the adjunction is an isomorphism $$\eta:\cal{F} \sira f_{\ast}f^{\ast}\cal{F}$$ We will not discuss the proof. \end{Bemerkungl} \begin{Definition} Given a topological group $G$ and a $G$-space $X$ we define the \defnoind{$G$-equivariant derived category of $X$}\index{"aquivariant!derivierte Kategorie} $$ \op{Der}_{G{\ssearrow}} (X) \pdef\{\mathcal F\in \op{Der} ({\op{E}}G\times_{/G} X)\mid \exists \mathcal G \in \op{Der} (X) \text{ such that }{\op{quot}}^*\mathcal F\cong{\op{pr}}_X^*\mathcal G\} $$ Here $\op{quot}:{\op{E}}G\times X \ra {\op{E}}G\times_{/G} X$ and $\op{pr}_X:{\op{E}}G\times X \ra X$ are the obvious. This inherits a triangulation and a truncation structure from $\op{Der} ({\op{E}}G\times_{/G} X)$. \end{Definition} \begin{Bemerkungl} Sometimes we also use the notation $\op{Der}_{G{\ssearrow}} (X)= \op{Der}_{/G{\ssearrow}X}$. In general we prefer to note $\mathcal C(M,N)$ the space of morphisms from an object $M$ to an object $N$ in a category $\mathcal C$. \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Equivariant derived category for a given resolution}] Instead of $\op{pr}_X:{\op{EG}}\times X\ra X$ we could take any $G$-equivariant map $p:P\ra X$ which is a fibre bundle with contractible fibres and free $G$-action on its total space, and we will get a canonically isomorphic triangulated category $$ \op{Der}_{G{\ssearrow}} (X,P) \pdef\{\mathcal F\in \op{Der} (P/G)\mid \exists \mathcal G \in \op{Der} (X) \text{ such that }q^*\mathcal F\cong p^*\mathcal G\} $$ with the notation $q:P\ra P/G$ for the quotient map. \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Relation to equivariant cohomology}] Given a topological group $G$ and a $G$-space $X$ and an abelian group $M$ the constant sheaf on $P\pdef {\op{E}}G\times_{/G}X$ with fibre $M$ is an object of the equivariant derived category. We denote this object by $M_{G\sacts X}\in \op{Der}_{/G{\ssearrow}X} $ and get isomorphisms $$\op{Der}_{/G{\ssearrow}X}(M_X,M_X[q])\sira {\op{H}}^q( P/G;M)\sira{\op{H}}^q_G( P;M) \sila {\op{H}}^q_G(X;M)$$ by definition, free quotient isomorphism and equivariant contractible pullback isomorphism. This leads to an isomorphism between the space of morphisms from $M_X$ to $M_X[q]$ in the equivariant derived category and the $q$-th equivariant cohomology of $X$ with coefficients in $M$. \end{Bemerkungl} \begin{Proposition}[\textbf{Equivariant sheaves as usual sheaves}] Given $G$ a topological group and $X$ a $G$-space and $p:P\pdef {\op{E}}G\times X\ra X$\label{AgGg} we get equivalences of categories $$\op{Ens}_{/G{\ssearrow}X}\sirra \{\mathcal F\in \op{Ens}_{/(P/G)}\mid\exists \mathcal G\in \op{Ens}_{/X}\text{ mit }\op{quot}^* \mathcal F\cong p^* \mathcal G\}$$ for $\op{quot}: P \ra P/G$ the quotient map via $\bar{\mathcal E} \mapsto (P\times_X\bar{\mathcal E})/G$ on the \'etale spaces. Same for abelian sheaves. \end{Proposition} \begin{proof}[Proof] First we check our functor is reasonably defined. Given an \'etale $G$-equi\-va\-ri\-ant map of free $G$-spaces the map induced on the orbit spaces is also etale. %, see eg \eref{QeT}{TG}. Thus with $P\times_X\bar{\mathcal E}\ra P$ also $ (P\times_X\bar{\mathcal E})/G\ra P/G$ is \'etale. Thus our functor is reasonably defined. We get a quasiinverse starting with the remark that the unit of adjunction is an isotransform $\op{id}\siRa p_*p^*$ and thus we get back $\mathcal G$ from $\mathcal F$ as $\mathcal G=p_{*} \op{quot}^*\mathcal F$. This shows $\bar{\mathcal G}$ admits a continous action of the dicretized variant $G^{\delta}$ of $G$, which under pullback gives the action of $G$ on $\overline{\op{quot}^* \mathcal F}\cong P\times_X \bar{\mathcal G}$ as a sheaf on $P$. But since the pulled back action is continous with respect to the true topology on $G$, the action on $\bar{\mathcal G}$ has to be continous as well, since it fits into a commutative diagram $$\begin{array}{ccc} G\times (P\times_X \bar{\mathcal G}) &\ra& P\times_X \bar{\mathcal G}\\ \da&&\da\\ G\times \bar{\mathcal G} &\ra& \bar{\mathcal G} \end{array} $$ and the left vertical is final as a map of topological spaces. \end{proof} \begin{Bemerkungl}[\textbf{Equivariant sheaves and equivariant derived category}] Let $G$ be a topological group and $X$ a $G$-space. Proposition \ref{AgGg} gives us an equivalence $$\op{Ab}_{/G{\ssearrow}X}\sirra\{\mathcal F\in \op{Der}_{G{\ssearrow}} (X)\mid \mathcal H^i\mathcal F\neq 0\;\RA\; i= 0\}$$ This even leads to a triangulated functor $\op{Der}(\op{Ab}_{/G{\ssearrow}X})\ra \op{Der}_{/G{\ssearrow}X}$ which how\-ever is in general not an equivalence of categories. This already fails for $S^1\acts\op{top}$. \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Equivariant cohomology with equivariant coefficients}] Given an equivariant sheaf $\mathcal F\in \op{Ab}_{/G{\ssearrow}X}$ we put $${\op{H}}^q_G(X;\mathcal F)\pdef \op{Der}_{/G{\ssearrow}X}(\DZ_{G{\ssearrow}X}, \mathcal F[q])$$ This can also be interpreted as ${\op{H}}^q_G({\op{E}}G\times_{/G}X;{\op{E}}G\times_{/G}\bar{\mathcal F})$ for the sheaf on the Borel construction given by its \'etale space. \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Trivial group acting}] In case of the trivial group, we find that $\op{quot}$ is an homeomorphism and $p=\op{pr}_X:{\op{E}}1\times X\ra X$ is a fibre bundle with contractible fibres, so pulling back along $p$ gives an equivalence $p^*:\op{Der}(X)\sirra \op{Der}_{1{\ssearrow}} (X)$ with quasiinverse $p_*$. \end{Bemerkungl} \subsection{Equivariant derived functors} \begin{Bemerkungl}[\textbf{Equivariant pullback}] We get for $(\phi\acts f):G\acts X\ra H\acts Y$ a pullback functor $(\phi\acts f)^*:\op{Der}_{H{\ssearrow}Y}\ra \op{Der}_{G{\ssearrow}X}$. \end{Bemerkungl} \begin{Bemerkungl} We should more generally define \glqq equivariant multipullback\grqq\ for families $(\phi_\rho\acts f_\rho):G\acts X\ra H_\rho\acts Y_\rho$ of equivariant maps indexed by $1\leq \rho\leq r$ with any $r\geq 0$ as functors $\op{Der}_{H_1{\ssearrow}Y_1}\times \ldots \times \op{Der}_{H_r{\ssearrow}Y_r}\ra \op{Der}_{G{\ssearrow}X}$, but since we even didn't do this carefully in the nonequivariant situation, I cannot and will not dwelve into this. The multipullback with twice the identity is usually denoted $\otimes$ or $\otimes^{\op{L}}$. The multipullback with the empty family $r=0$ is the constant sheaf $\DZ_{G{\ssearrow}X}$. \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Equivariant pusforward}] If $H$ is openlocally contractible, meaning every neighborhood of every point contains a contractible open neighborhood of this same point, the equivariant pullback along $(\phi\acts f):G\acts X\ra H\acts Y$ admits a right adjoint $$(\phi\acts f)_*:\op{Der}_{/G{\ssearrow}X}\ra \op{Der}_{/H{\ssearrow}Y}$$ and $\mathcal F\otimes$ in $\op{Der}_{/H{\ssearrow}Y}$ admits a right adjoint, the internal hom to be denoted $\mathcal F{\Rrightarrow}$ or classically $\op{R}\!{\mathcal H}om(\mathcal F,\;)$. \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Deduce cohomology multipullback}] Let $f$ be a morphism. Transitivity of multipullback on $(f,f)\circ (\op{id}\curlywedge\op{id})= f\curlywedge f= (\op{id}\curlywedge\op{id})\circ f$ gives isomorphisms $f^*(\mathcal F\otimes \mathcal G) \sira (f^*\mathcal F)\otimes (f^*\mathcal G)$. Adjunction gives $\mathcal F\otimes \mathcal G \ra f_*((f^*\mathcal F)\otimes (f^*\mathcal G))$. Applying to $\mathcal F=f_*\mathcal E$ and $\mathcal G=f_*\mathcal H$ gives $f_*\mathcal E\otimes f_*\mathcal H \ra f_*((f^*f_*\mathcal E)\otimes (f^*f_*\mathcal H))$. Using the counit of adjunction gives $$f_*\mathcal E\otimes f_*\mathcal H \ra f_*(\mathcal E\otimes \mathcal H)$$ Now let two morphisms $f_1, f_2$ start at the same space. Let $l$ always denote the empty family. Then $(l_1,l_2)\circ (f_1,f_2)=l$ gives us using transitivity of multipullback an isomorphism $\DZ_{X}\sira f^*_1\DZ_{Y_1}\otimes f^*_2\DZ_{Y_2}$. Pushing down to a point with $a:X\ra \op{top}$ and combining with what we already found we get $$a_*\DZ_{X}\sira a_*(f^*_1\DZ_{Y_1}\otimes f^*_2\DZ_{Y_2})\leftarrow a_*f^*_1\DZ_{Y_1}\otimes a_*f^*_2\DZ_{Y_2}$$ Now let $b_\rho:Y_\rho\ra \op{top}$ so we can write $a=b_\rho f_\rho$ and using the unit of the adjunction we get $b_{\rho*}\DZ_{Y_\rho}\ra b_{\rho*}f_{\rho*}f_\rho^*\DZ_{Y_\rho} \sira a_*f^*_\rho\DZ_{Y_\rho}$. Putting everything together we found a morphism $$a_*\DZ_{X}\leftarrow b_{1*}\DZ_{Y_1}\otimes b_{2*}\DZ_{Y_2}$$ in the derived category of abelian sheaves on a point. This in turn induces on homology bilinear maps $\mathcal H^{p}( b_{1*}\DZ_{Y_1})\times \mathcal H^{q}( b_{2*}\DZ_{Y_2})\ra \mathcal H^{p+q}( a_{*}\DZ_{X})$ alias the looked-for bilinear maps $${\op{H}}^{p}( Y_1)\times {\op{H}}^{q}(Y_2)\ra {\op{H}}^{p+q}(X)$$ \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Deduce equivariant cohomology multipullback}] With equivariant cohomology it's all the same, only notation may be more elaborate. With ringed spaces, it's again all the same. We will not go beyond constant coefficient rings, mostly the complex numbers. \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Equivariant shriek functors}] Given $(\op{id}\acts f):G\acts X\ra G\acts Y$ with $X,Y$ locally compact Hausdorff spaces with $\op{Ab}_{/X}$ of finite cohomological dimension we can also construct an equivariant version, for which we use the notations $$(\op{id}_G\acts f)_!=(G\acts f)_!=f_!:\op{Der}_{/G{\ssearrow}X}\ra \op{Der}_{/G{\ssearrow}Y}$$ of proper push-forward. If in $G$ is a Lie group, this admits a right adjoint $(G\acts f)^!$. \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Equivariant intersection cohomology}] As long as we do not change the group all these functors behave like their nonequivariant counterparts discussed in Peng's class. This behaviour is called a \glqq six-functor-formalism\grqq. In particular we have $G$-equivariant perverse sheaves and the equivariant intersection cohomology sheaf of a complex variety acted upon by a Lie group. \end{Bemerkungl} \subsection{Spaces with finitely many orbits} \begin{Bemerkungl}[\textbf{Parameter set}] Let a complex algebraic group $G$ act on a variety $X$ with finitely many orbits. The irreducible $G$-equivariant perverse sheaves on $X$ with coefficients in a field $\mathbb K$ are parametrized by the data $$\op{Par}_G(X)\pdef \left\{(Y,\tau)\left|\begin{array}{ll} Y\subset X\text{ is a $G$-orbit,}\\\tau\text{ is an irreducible $G$-equivariant}\\ \text{local system of $\mathbb K$-spaces on $Y$.} \end{array} \right.\right\}$$ Here a local system is understood as a locally constant sheaf of finite dimensional vector spaces over $\mathbb K$. We have $G/G_y\sira Y$ for any point $y\in Y$ and $G_y$ its isotropy group. By what we discussed before we thus get for any $y\in Y$ a bijection $$\left\{\begin{array}{c}\text{Irreps of the component group}\\ \text{$G_y/G_y^\circ$ in $\mathbb K$-vector spaces} \end{array} \right\} \sira \left\{\begin{array}{c}\text{irreducible $G$-equivariant}\\ \text{ local systems on $Y$} \end{array} \right\}$$ The irreducible perverse sheaves are then given in two notation schemes as $$\begin{array}{ccl} \op{Par}_G(X)&\sira &\op{irr}(\op{Per}_G(X))\\[2mm] \pi= (Y,\tau)&\mapsto&\mathcal L^\pi =i_*{\mathcal I}{\mathcal C}(\bar Y;\tau)=j_{!*}\tau[\op{dim}Y] \end{array}$$ for $i:\bar Y\hra X$ and $j:Y\hra X$ the inclusions. \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Equivariant Ext-algebra}] Let a complex algebraic group $G$ act on a variety $X$ with finitely many orbits. Let $\mathcal L\pdef \bigoplus\mathcal L^\pi$ be the direct sum of all irreducible perverse sheaves. We put $$\op{Ext}^*_G(X)\pdef \op{Der}_{/G\sacts X}(\mathcal L, \mathcal L[*])$$ and call this the {\bf equivariant Ext-algebra}. It is a nonnegatively graded $\mathbb K$-ring\-al\-ge\-bra. Its degree-zero part $\op{Ext}^0_G(X)$ has a basis consisting of the pairwise orthogonal idempotents $1_\pi$ for $\pi\in \op{Par}_G(X)$. \end{Bemerkungl} \begin{Beispielex}[\textbf{$\mathcal O_0$ as modules for equivariant Ext-algebra}] Let $\mathfrak g$ be a complex semisimple Lie algebra, $G$ a complex connected semisimple algebraic group with $\mathfrak g=\op{Lie}G$ and $G\supset B\supset N$ a Borel and its unipotent radical. It is known there is an equivalence of categories $\mathcal O_0\sirra \op{Per}_N(G/B)$. It is known that in this case the general functor to be discussed later is fully faithful $$\op{Der}^{\op{b}}(\op{Per}_N(G/B))\vra \op{Der}_N(G/B)$$ This is a consequence of $N$ as well as all $N$-orbits being contractible. In \ref{vDs} we found $\mathcal O_0\sirra \op{Modfl-}A$ for $A=\op{Ext}^*_{\mathcal O_0}(L,L)$. Geometrically this can be rewritten as an equivalence $$\mathcal O_0\sirra \op{Modfl-}\op{Ext}^*_N(G/B)$$ This is the rephrasing of the autoduality of $\mathcal O_0$ which conjecturally admits a generalization to representations of reductive groups over local fields, as will be discussed in the remaining lectures. \end{Beispielex} \subsection{Some equivalences} \begin{Bemerkungl}[\textbf{Induction equivalence}] Given $\iota: G\subset H$ a subgroup of a topological group acting freely and $G\acts X$ a $G$-space the pullback by $(\iota\acts i)$ for the obvious inclusion $i:X\hra H\times_{/G}X$ is an equivalence $$(\iota\acts i)^*:\op{Der}_{H\sacts}(H\times_{/G}X)\sirra \op{Der}_{G\sacts}(X)$$ \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Contractible group acting}] Given $N$ a contractible topological group and $N\acts X$ an $N$-space the pullback by $(\iota\acts \op{id})$ for the obvious inclusion $\iota:1\hra N$ is fully faithful $$(\iota\acts {\op{id}})^*:\op{Der}_{N\sacts}(X)\vra \op{Der}_{1\sacts}(X)$$ Sure as we discussed already $\op{Der}_{1\sacts}(X)=\op{Der}(X)$ canonically. \end{Bemerkungl} \newpage \section{Fourth lecture on Koszul duality, July 10} \subsection{Representations of real reductive groups} \begin{Bemerkungl} A continous representation of a topological group $G$ on a topological complex vector space $V$ is a group homomorphism $G\ra \op{GL}(V)$ such that the induced map $G\times V\ra V$ is continous. \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Harish-Chandra modules}] To study continuos representations of real reductive Lie groups $G$, one may reduce lots of questions to the study of the purely algebraic \glqq Harish-Chandra modules\grqq. To define those, let $\mathfrak g\pdef \op{Lie} G$ be the Lie algebra and $K\subset G$ a compact subgroup. A $\mathfrak g$-$K$-module is a complex vector space $H$ with a structure of representation of the Lie algebra $\mathfrak g$ and a locally finite action of $K$ continous on any finite dimensional $K$-stable subspace compatible in the the way that \begin{enumerate} \item The action $\mathfrak g\times H\ra H$ is $K$-equivariant for the adjoint action of $K$ on $\mathfrak g$; \item The differential of the $K$-action to an action of $\op{Lie}K$ coincides with the restriction of the action of $\mathfrak g$ to $\op{Lie}K$. \end{enumerate} To stress how algebraic this condition is, we may pass to the complexification $K_\DC$ of $K$, a complex algebraic group containing $K$ as an abstract subgroup with the property, that restriction ist an isomorphism of categories $$\{\text{Algebraic representations of $K_\DC$}\}\sira \{\text{Our special representations of $K$}\}$$ A {\bf Harish-Chandra module} is defined to be a $\mathfrak g$-$K$-module of finite length for $K\subset G$ a maximal compact subgroup. Those maximal compact subgroups exist and are unique up to conjugation. \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Motivation for Harish-Chandra modules}] Let $G$ be a real reductive group and $K\subset G$ a maximal compact sunbgroup. We mention two results to illustrate how close Harish-Chandra modules are to true representations. First, given an irreducible unitary representation on a Hilbert space $V$ the subspace of $K$-finite vectors $V_K\subset V$ has a natural $\mathfrak g$-action and we get in this way by a theorem of Harish-Chandra an injection on isomorphism classes $$\{\text{Simple unitary representations of $G$}\}\hra \{\text{Simple $\mathfrak g$-$K$-modules}\}$$ Second, taking $K$-finite vectors gives by a theorem of Casselman an equivalence of categories $$\{\text{Smooth finite length representations of $G$}\}\sirra \{\text{Finite length $\mathfrak g$-$K$-modules}\}$$ We denote this category by $\mathcal M(G)$. \end{Bemerkungl} \subsection{Koszul duality conjecture} \begin{Bemerkungl}[\textbf{Parameter space of Adams-Barbasch-Vogan}] Start with a connected complex reductive algebraic group $G$ with an antiholomorphic involution $\bar \gamma$ fixing some Borel subgroup $B$. Put $\mathfrak g\pdef \op{Lie}_\DC G$ and let $Z\subset \op{U}(\mathfrak g)$ be the center of the enveloping algebra. Fix an infinitesimal character $\chi\in \op{Max}Z$.\label{ndfcM} Let $(G^\vee\supset B^\vee, \gamma)$ be the Langlands dual, a connected complex reductive algebraic group $G^\vee$ with a distinguished Borel subgroup $B^\vee$ and a holomorphic involution $ \gamma$ stabilizing $B^\vee$. Then by \cite{ABV} %Meistenteils Kopiert aus \ref{ndfc} XXKoszul we have the Adams-Barbasch-Vogan parameter space $$G^\vee{\ssearrow}X(\chi)$$ This is a $G^\vee$-variety $X(\chi)$ acted upon with finitely many orbits. It is a variant of the Langlands parameter space. We will discuss it later in more detail. \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Koszul duality and Langlands philosophy}] Let $(G,\bar\gamma)$ be a connected complex reductive algebraic group with an antiholomorphic involution fixing a Borel subgroup $B$. Among other things \cite{ABV} construct a bijection $$\op{Par}_{G^\vee}(X(\chi))\sira \bigsqcup_{\delta\in {\op{H}}^1_{\bar\gamma}(\Gamma;G)} \op{irr}\mathcal M_\chi (G(\DR;\delta))$$ between their variant of Langlands parameters and the set of irreducible Harish-Chandra modules killed by $\chi$ of a bunch of real forms of $G$ discussed in \ref{reF}. In addition they prove that Vogan's inversion formulas \cite{Vo4} for Kazhdan-Lusztig-Vogan polynomials can be rephrased as saying the $\op{IC}$-matrix of $G^\vee{\ssearrow} X(\chi)$ specialized at $v=1$ is inverse transpose up to signs to the Jordan-H"older multiplicity matrix $\op{JH}$ giving simple subquotients of standard representations. Motivated by this and by Koszul duality for $\mathcal O_0$ in \cite{So-A}, I conjectured in \cite{So-L} there should be an equivalence of categories $$\op{Ext}_{G^\vee}^*(X(\chi))\op{-Nil}\sirra \bigoplus_{\delta\in {\op{H}}^1_{\bar\gamma}(\Gamma;G)} \mathcal M^\infty_\chi (G(\DR;\delta))$$ I am aware of very interesting progress of Nadler-BenZvi as well as Scholze towards a proof, and of recent partial results \cite{BezVi}, \cite{LaRo}, but to the best of my knowledge the conjecture as it stands is still open. Let us discuss the notation used here. On the right, it's the categories of Harish-Chandra modules of the real forms $G(\DR;\delta)$ annihilated by some power of $\chi$. On the left, it is one of our geometric extension algebras. Finally, $\op{Nil}$ means \glqq finite dimensional modules killed by all homogeneous elements of sufficiently high degree\grqq. This is a categorification of \cite{ABV}, but at the same time I specialized the setup of \cite{ABV} somewhat to simplify. The conjecture as stated should hold analogously in the full generality of \cite{ABV} and analogously for other local fields like $\mathbb Q_p$. \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Relation to ABV-parametrization}] Given an $\DN$-graded $\DC$-algebra $A=\bigoplus_{i\geq 0}A^i$ with $A^0=\bigoplus_{\pi\in \op{Par}} \DC 1_\pi$ it is clear the irreducible objects of $A\op{-Nil}$ are precisely the $A^01_\pi$ for $\pi\in \op{Par}$. Our conjectured equivalence should give back the bijection of \cite{ABV}. \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Nonabelian group cohomology}] Given a group $\Gamma$ acting on another group $A$ the isomorphism classes of $\Gamma$-equivariant $A$-torsors $X$ are classified by the pointed set ${\op{H}}^1(\Gamma;A)$. For simplicity of notation, we assume $\Gamma$ acts from the right and use exponential notation for this action. We will just need the case $\Gamma=\{1,s\}$ is a two-element group and just define ad-hoc $${\op{Z}}^1(\Gamma;A)\pdef \{a\in A\mid a a^s=1\}$$ with the $A$-action given by $b*a\pdef ba(b^s)^{-1}$. Then ${\op{H}}^1(\Gamma;A)\pdef {\op{Z}}^1(\Gamma;A)/(A*)$ is defined to be the set of $A$-orbits with distinguished element the orbit of $a=1$. To recall which involution $s$ of $A$ was used here we sometimes write ${\op{Z}}^1_s(\Gamma;A)$ and ${\op{H}}^1_s(\Gamma;A)$. \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Relation to equivariant cohomology}] If $A$ is abelian, this coincides with the equivariant cohomology ${\op{H}}^1_\Gamma(\op{top};A)$ of $A$ considered as a $\Gamma$-equivariant abelian sheaf on the one-point space. The generalization to not necessarily abelian coefficients works only for ${\op{H}}^q$ with $0\leq q\leq 1$ and ${\op{H}}^1$ will only be a pointed set and have no group structure anymore. \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Group cohomology and real forms}] Any $g\in {\op{Z}}^1_{\bar\gamma}(\Gamma;G)$ above gives rise to an antiholomorphic involution $ (\op{int}g)\circ\bar \gamma$. Furthermore the corresponding real form $G(\DC)^{(\op{int}g)\circ\bar \gamma}$ depends only on the image $\delta\in {\op{H}}^1_{\bar\gamma}(\Gamma;G)$ of $g$ in the cohomology set. We denote it by $$G(\DR;\delta)$$ This construction does neither give an injection nor a surjection of ${\op{H}}^1_{\bar\gamma}(\Gamma;G)$ to the set of real forms up to isomorphism. Rather the real forms are classified by\label{reF} ${\op{H}}^1_{\bar\gamma}(\Gamma;\op{Aut}G)$ and we precomposed this with $\op{int}:G\ra\op{Aut}G$, but le us not go into this in any detail. \end{Bemerkungl} \subsection{Examples for tori} \begin{Bemerkungl}[\textbf{Parameter spaces for tori}] Suppose $G=T\cong \DC^\times\times \ldots\times \DC^\times$ is a torus. Let $\mathfrak X(T)\pdef \op{GrpVar}_\DC(T,\DC^\times)$ be its character lattice, a free abelian group of finite rank. Then $T^\vee \pdef\mathfrak X(T)\otimes_\DZ \DC^\times$ is called the {\bf dual torus}. It comes with an obvious isomorphism between its character lattice and the dual of the character lattice of the original torus $$\mathfrak X(T)^\vee\sira \mathfrak X(T^\vee)$$ Any antiholomorphic involution $\bar\gamma$ on $T$ gives an involution $\gamma: \mathfrak X(T)\ra \mathfrak X(T)$ by $$\lambda^\gamma\pdef \text{(complex conjugation)}\circ \lambda\circ \bar\gamma$$ This induces an involution $\gamma$ on the dual lattice $\mathfrak X(T)^\vee$ and thus a holomorphic involution $\gamma$ on the dual torus $T^\vee$. In this case the ABV parameter space for trivial central character specializes to $$X(Z^+)={\op{Z}}^1_\gamma(\Gamma;T^\vee)$$ with the *-action of $T^\vee$ as discussed above. \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Testing the conjecture for $\DR^\times$}] We try the case $(G,\bar\gamma)=(\DC^\times, z\mapsto\bar z)$. The set ${\op{H}}^1_{\bar\gamma}(\Gamma;\DC^\times)$ has just one element be Hilbert 90 or explicit calculation, so we are looking at Harish-Chandra modules for just one real form with corresponding Lie group $\DR^\times$. We find an equivalence of categories $$(\DC[x]\times \DC[x])\op{-Nil}\;\sirra \; \mathcal M^\infty_{Z^+}(\DR^\times)$$ Indeed, as a Lie group $\DR^\times\cong \{\pm 1\}\times (\DR,+)$ and the category of continous finite dimensional representations of $(\DR,+)$ is equivalent to the category of finite dimensional complex vector spaces with a distinguished endomorphism, the infinitesimal generator, and being killed by a power of $Z^+$ amounts to the infinitesimal generator being nilpotent. On the geometric side, we find $T^\vee=\DC^\times$ with the involution $\gamma=\op{id}$ and $$X(Z^+)= {\op{Z}}^1_{\gamma}(\Gamma;T^\vee) =\{z\in \DC^\times\mid z^2=1\} =\{\pm 1\}$$ consists of two points acted upon by $\DC^\times$. There are two simple equivariant perverse sheaves, namely the constant sheaves $\mathcal L^+$ and $\mathcal L^-$ on each one of the two points. There are no Ext between the two, but we find $$\op{Ext}^*_{\DC^\times}(\mathcal L^+,\mathcal L^+)=\op{H}^*_{\DC^\times}(\op{top};\DC)= \op{H}^*({\op{B}}\DC^\times;\DC)=\op{H}^*(\mathbb P^\infty\DC;\DC)=\DC[x]$$ is the cohomology ring of the classifying space, a polynomial ring with generator of degree $\op{deg}x=2$. So in this case at least we indeed found an equivalence of categories $$\op{Ext}^*_{\DC^\times}(X(Z^+))\op{-Nil}\;\sirra\; \mathcal M^\infty_{Z^+}(\DR^\times)$$ \end{Bemerkungl} \begin{Bemerkungl} Let me discuss a split torus $T(\DR;\bar\gamma)\cong \DR^\times\times\ldots\times \DR^\times$. Then the parameter space is $T^\vee(\DC)\cong \DC^\times\times\ldots\times \DC^\times$ acting on a discrete set $X(Z^+)$ and we have canonically $$\op{Lie}T^\vee(\DC)\sila \mathfrak X(T)\otimes_\DZ \DC\sira (\op{Lie}T(\DC))^* \sira(\op{Lie}_\DC T(\DR;\bar\gamma))^*$$ and thus we also have canonically $${\op{H}}^*_{T^\vee(\DC)}(\op{top})\sila \mathcal O(\op{Lie}T^\vee(\DC))\sira {\op{S}}(\op{Lie}_\DC T(\DR;\bar\gamma)) \sira {\op{U}}(\op{Lie}_\DC T(\DR;\bar\gamma))$$ In this way, it is natural to identify ${\op{H}}^*_{T^\vee(\DC)}(\op{top})\op{-Nil}$ with $ {\op{U}}(\op{Lie}_\DC T(\DR;\bar\gamma))\op{-Nil}$ which in our case are just the finite dimensional representations of $T(\DR;\bar\gamma)^\circ$ killed by some power of $Z^+$. \end{Bemerkungl} \begin{Ubungex} Check the conjecture for the real tori $S^1$ and $\DC^\times$. All real tori are products of some copies of these two tori along with some copies of $\DR^\times$ treated already. \end{Ubungex} \subsection{Parameters for trivial central character} \begin{Bemerkungl} Suppose $G$ is complex connected reductive with $(G,G)$ simply connected. Let $\bar\gamma$ be an antiholomorphic involution stabilizing a Borel $B\subset G$. Then the ABV-parameter space for trivial central character specializes to $$X(Z^+)=G^\vee\times_{/B^\vee}Z^1_{\gamma}(\Gamma;G^\vee(\DC))$$ We recall the bijection $Z^1_{\gamma}(\Gamma;G^\vee(\DC))\sira \{a\in G^\vee(\DC)\mid a a^\gamma=1\}$ and the action of $b\in G^\vee(\DC)$ being $b*a\pdef ba(b^\gamma)^{-1}$. \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Induction equivalence}] For $G\supset B$ a topological group with a subgroup acting freely on it and a $B$-space $X$ pulling back is an equivalence $$\op{Der}_{G\acts}( G\times_{/B}X)\sira \op{Der}_{B\acts}(X)$$ \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Induction isomorphism for equivariant Ext algebra}] In the situation of algebraic groups acting with finitely many orbits on an algebraic variety we easily see that the perverse truncation structures correspond under the induction equivalence up to shift and we deduce an isomorphism of graded rings $$\op{Ext}_G( G\times_{/B}X)\sira \op{Ext}_B(X)$$ So in particular we find above $${\op{Ext}}_{G^\vee}X(Z^+)\cong {\op{Ext}}_{B^\vee}Z^1_{\gamma}(\Gamma;G^\vee(\DC))$$ \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Example of complex groups}] Let $G$ be a reductive connected complex algebraic group. Then $G(\DC)=H(\DR;\bar\gamma)$ with $H=G\times G$ and $\bar \gamma(x,y)\pdef (\bar s(y),\bar s(x))$ for $\bar s$ a split antiholomorphic involution of $G(\DC)$. If $\bar s$ stabilizes the Borel $B(\DC)$, then $\bar\gamma$ stabilizes $B(\DC)\times B(\DC)$. In this case $\gamma:H^\vee\ra H^\vee$ just switches the factors in $H^\vee=G^\vee\times G^\vee$ and $$Z^1_\gamma(\Gamma; H^\vee)=\{(x,y)\in G^\vee\times G^\vee \mid (x,y)(y,x)=(1,1)\}$$ and we get a bijection $G^\vee\sira Z^1_\gamma(\Gamma; H^\vee)$ by $x\mapsto (x,x^{-1})$. The action of $(a,b)\in H^\vee$ is given by $(a,b)(x,x^{-1})(b^{-1},a^{-1})$ and corresponds to $(a,b)x=axb^{-1}$. So in the end we find $${\op{Ext}}_{H^\vee}X(Z^+)\cong {\op{Ext}}_{B^\vee\times B^\vee}(G^\vee)\cong {\op{Ext}}_{B^\vee}(G^\vee/B^\vee)$$ for the action of $B^\vee$ on $G^\vee$ by left and right multiplication and the last isomorphism by the induction equivalence. Now I think Peng explained or will explain an equivalence of categories $$\tilde{\mathcal O}_0\sira {\op{Ext}}_{B^\vee}(G^\vee/B^\vee)\op{-Nil}$$ On the other hand work of Bernstein-Gelfand \cite{BG} gives an equivalence of categories $\mathcal M^\infty_{Z^+}(G(\DC))\sira \tilde{\mathcal O}_0$ from Harish-Chandra modules for $G(\DC)$ with trivial central character to $\tilde{\mathcal O}_0$. Finally, the real forms to consider are parametrized by ${\op{H}}^1_{\bar\gamma}(\Gamma;H(\DC))$, so let us compute that. We find ${\op{Z}}^1_{\bar\gamma}(\Gamma;H(\DC))= \{(x,y)\mid (x,y)(y^{\bar s}, x^{\bar s})=(1,1)\}$ alias $xy^{\bar s}=1$ so we get $G(\DC)\sira {\op{Z}}^1_{\bar\gamma}(\Gamma;H(\DC))$ by $x\mapsto (x, (x^{\bar s})^{-1})$ and the action of $(a,b)\in H(\DC)$ is by $(a,b)x=ax(b^{\bar s})^{-1}$ and clearly there is only one orbit even if we let only pairs $(a,1)$ act. \end{Bemerkungl} \subsection{The case $\op{SL}(2;\DR)$} Well, let us see how much time there is. I think this is very good for exercise sessions, following \cite{So-L}. \newpage \section{Fifth lecture on Koszul duality, July 11} \subsection{Algebra} \begin{Bemerkungl}[\textbf{$\op{Ab}$-categories}] Let us define an $\op{Ab}$-category to be a category together with an addition on each space of morphisms such that they all morphism spaces become abelien groups and compositions of morphisms become bilinear maps. \end{Bemerkungl} \begin{Beispielex} An $\op{Ab}$-category with only one object is a ring. An $\op{Ab}$-category with finite products or equivalently finite coproducts is called an additive category. Given a category with finite products, an $\op{Ab}$-structure is unique if it exists. \end{Beispielex} \begin{Bemerkungl}[\textbf{Ubiquity of modules}] Given an additive category $\mathcal I$ and an object $P\in \mathcal I$ with $R\pdef \op{End}_{\mathcal I}(P)$ its endomorphism ring the functor $\op{Hom}_{\mathcal I}(P,\:): \mathcal I\ra \op{Mod-}R$ induces an equivalence of categories $$\mathcal I\supset \langle P\rangle_{\oplus}^{\mathcal I}\sirra \langle R\rangle_{\oplus} \subset \op{Mod-}R$$ Indeed, homomorphisms $P^{\oplus n}\ra P^{\oplus m}$ are just matrices of homomorphisms $P\ra P$ alias matrices with entries in $R$ and homomorphisms $R^{\oplus n}\ra R^{\oplus m}$ of right $R$-modules are the same. We denote by $\op{Frei-}R\pdef \langle R\rangle_{\oplus}$ the category of these free $R$-modules for finite sets of free generators and rewrite this as an equivalence $$\langle P\rangle_{\oplus}^{\mathcal I}\sirra \op{Frei-}\op{End}_{\mathcal I}(P)$$ \end{Bemerkungl} \subsection{Differential graded algebra} \begin{Bemerkungl} To get a deeper and less phenomenological understanding of Koszul duality, we need to dive deeper into homological algebra. What follows I learned from \cite{BeLu} and Keller \cite{Kel}. The roots can be traced back to Verdier's thesis published in SGA $4\frac{1}{2}$. \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Differential graded abelian groups}] Let $\op{dgAb}$ denote the category of differential graded abelian groups alias chain complexes. Given objects $X_1,\ldots,X_r,Y$ define a multimorphism $X_1\curlyvee\ldots\curlyvee X_r\ra Y$ to be, for quickness sake, a chain map $X_1\otimes\ldots\otimes X_r\ra Y$. In case $r=0$ define a multimorphism $\curlywedge \ra Y$ to be an element of $Y^0$. Given a finite family of objects $(X_i)_{i\in I}$ define a multimap $(X_i)_{i\in I} \ra Y$ as a thing associating to each order on $I$ a map as above in such a way, that changing the order introduces signs in the usual way. We can check that then the composition of multimaps is associative. From that point on we stop worrying about signs. We have two multifunctors $$\mathcal Z^0,\mathcal H^0:\op{dgAb}\ra \op{Ab}$$ In addition we have a multifunctor of total homology $\mathcal H:\op{dgAb}\ra \op{dgAb}$ landing in the multicategory of complexes with zero differential, which we also denote $\op{sgAb}$. We also call its objects supergraded abelian groups. \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Differential graded categories}] A $\op{dgAb}$-category or as shorthand dg-category $\mathcal C$ consists of objects and for any two objects $X,Y$ a morphism object $\mathcal C(X,Y)\in \op{dgAb}$ and for any three objects $X,Y,Z$ a composition rule $\mathcal C(X,Y)\curlyvee \mathcal C(Y,Z)\ra \mathcal C(X,Z)$ in form of a multimorphism such the composition is associative and for any $X$ there exists an identity $\curlyvee\ra \mathcal C(X)\pdef \mathcal C(X,X)$ satisfying the usual axioms. Given a dg-category we can form the two $\op{Ab}$-categories $\mathcal Z^0\mathcal C$ and $\mathcal H^0\mathcal C$. \end{Bemerkungl} \begin{Bemerkungl}[\textbf{dg-category of a dg-ring}] Let $(R,d)$ be a differential graded ring. This is by definition a complex along with morphisms of complexes $R\otimes R\ra R$ and $\DZ[0]\ra R$ satisfying the usual associativity and unit conditions. Consider the category $$\op{dgMod}_{-R}$$ of differential graded right $R$-modules. Objects are complexes $(M,d)$ with an action $M\otimes R\ra M$ satisfying the usual conditions. Morphisms are the obvious. We construct in addition a dg-category $\op{Mod}^{\op{dg}}_{-R}$ with the same objects be letting $$\op{Mod}_{-R}^{\op{dg}}(M,N)\subset \op{Hom}(M,N)$$ to be the subcomplex of the homomorphism complex of all morphism of various degrees commuting with the right action of homogeneous elements of $R$. To avoid confusion, I prefer to denote the homomorphism complex $$(M{\Rrightarrow}N)=(M{\Rrightarrow}_{\op{dgAb}}N)=\op{Hom}(M,N)$$ It is the internal Hom of $\op{dgAb}$, but no time to dwelve into this. %and our subcomplex %$$(M{\Rrightarrow}_{-R}N) \subset (M{\Rrightarrow}N)$$ %is the egalisator of the two morphisms %of complexes $(M{\Rrightarrow}N)\ra ((M\otimes R){\Rrightarrow}N)$ %given by precomposing with the action on $M$ and by %tensoring with the identity on $R$ followed by %postcomposing with the action on $N$. By definition we get $\op{dgMod}_{-R}=\mathcal Z^0\op{Mod}_{-R}^{\op{dg}}$. \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Homotopy category and derived category of a dg-ring}] Given a dg-ring $R$ we define a new $\op{Ab}$-category $$\op{dgHot}_{-R}\pdef \mathcal H^0\op{Mod}_{-R}^{\op{dg}}$$ This is called the {\bf homotopy category} of right dg-$R$-modules. It has a natural structure of triangulated category just in the same way as usual homotopy categories of complexes and can be localized at quasiisomorphisms to give a triangulated category $$\op{dgDer}_{-R}\pdef (\op{dgHot}_{-R})_{\op{qis}}$$ The localization functor $Q:\op{dgHot}_{-R}\ra \op{dgDer}_{-R}$ is fully faithful on the full subcategory with shifted copies of $R$ as their objects and thus is fully faithful on the triangulated subcategory generated by $R$ and therefore induces an isomorphism on the full triangulated subcategories $\langle R\rangle^{\op{dgHot}}_\Delta\sira \langle R\rangle^{\op{dgDer}}_\Delta$ generated by these objects. We denote them $$\op{dgFrot-}R\pdef \langle R\rangle^{\op{dgHot}}_\Delta=\langle R\rangle^{\op{dgDer}}_\Delta$$ Evaluating at $1\in R$ and passing to the quotient furthermore gives isomorphisms $\op{dgDer}_{-R}(R,M)\sila \op{dgHot}_{-R}(R,M)\sira \mathcal H^0 M$. \end{Bemerkungl} \begin{Beispielex}[\textbf{Case of usual ring}] Any ring $R$ is a dg-ring concentrated in homological degree zero with trivial differential. Then $\op{dgDer}_{-R}=\op{Der}(\op{Mod-}R)$ is the usual derived category of the abelian category of right $R$-modules. \end{Beispielex} \begin{Bemerkungl}[\textbf{Ubiquity of dg-modules}] Assume $\mathcal I$ is an additive category. Then the category $\op{dg}\mathcal I=\op{Ket}_{\mathcal I}$ of complexes in $\mathcal I$ can be upgraded to a dg-category $\op{Ket}^{\op{dg}}_{\mathcal I}$ in the obvious way. Given $P\in \op{Ket}_{\mathcal I}$ a complex in $\mathcal I$ we can consider its endomorphism complex $R\pdef \op{Ket}^{\op{dg}}_{\mathcal I}(P)$, a differential graded ring. We get a dg-functor\label{Ubdg} $$ \op{Ket}^{\op{dg}}_{\mathcal I}(P,\;): \op{Ket}^{\op{dg}}_{\mathcal I} \ra \op{Mod}^{\op{dg}}_{-R}$$ It is dg-fully faithful on the full additive subcategory generated by the $P[n]$, basically since we have $R\sira \op{Mod}^{\op{dg}}_{-R}(R)$ by left multiplication. This functor passes down to the homotopy categories and we get a functor $$ \op{Ket}^{\op{dg}}_{\mathcal I}(P,\;): \op{Hot}_{\mathcal I} \ra \op{dgHot}_{-R}$$ It is by \glqq dévissage\grqq\ fully faithful on the full triangulated subcategory generated by $P$ and induces an equivalence of triangulated categories $$\langle P\rangle_\Delta^{\op{Hot}}\sirra \op{dgFrot-}R$$ \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Extension of scalars}] For any homomorphism of dg-rings $R\ra S$ we have the triangulated extension of scalars $\otimes_RS:\op{dgHot-}R\ra \op{dgHot-}S$ with $R\mapsto S$. This induces a functor on homotopy free objects $\otimes_RS:\op{dgFrot-}R\ra \op{dgFrot-}S$. We also have the derived functor $\otimes^{\op{L}}_RS:\op{dgDer-}R\ra \op{dgDer-}S$. If our homomorphism is a quasiisomorphism $R\qri S$, then it obviously induces an equivalence $$\otimes_RS:\op{dgFrot-}R\sirra \op{dgFrot-}S$$ \end{Bemerkungl} \begin{Bemerkungl} If our homomorphism is a quasiisomorphism $R\qri S$, you even get an equivalence $\op{dgDer}_{-S}\sirra \op{dgDer}_{-R}$ taking more care with infinities, but this needs much more work, see say 22.37 in \cite{Stacks}. \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Realization, weight complex and tilting}] Let $\mathcal I$ be an additive category and $ T\in \op{Ket}_{\mathcal I}$ a complex and $R\pdef \op{Ket}_{\mathcal I}^{\op{dg}}(T)$ its dg-endomorphism dg-ring. In this case we get an equivalence $\langle T\rangle_{\oplus}^{\op{Hot}}\sirra \op{Frei-}\mathcal H^0R$ by what we saw at the very beginning of this lecture. Now for any dg-ring $R$ we have morphisms of dg-rings $$\mathcal H^0R \leftarrow (\mathcal Z^0R\oplus R^{<0})\ra R$$ and by what we saw before get a diagram of triangulated functors $$\begin{array}{ccccccc} \op{Der}(\op{Mod-}\mathcal H^0R)&&&&&&\\ \cup&&&&&&\\ \op{Hot}^{\op{b}}(\op{Frei-}\mathcal H^0R)&= &\op{dgFrot-}\mathcal H^0R&\leftarrow & \op{dgFrot-}(\mathcal Z^0R\oplus R^{<0})&\ra &\op{dgFrot-}R\\ \ua{\scriptstyle \wr\wr}&&&&&&\ua{\scriptstyle \wr\wr}\\ \op{Hot}^{\op{b}}(\langle T\rangle_{\oplus}^{\op{Hot}}) &&& &&&\langle T\rangle_{\Delta}^{\op{Hot}} \end{array} $$ If we assume in addition $\mathcal H^nR=0$ for $n\neq 0$, then both morphisms of dg-rings are quasiisomorphisms and our whole diagram consists of equivalences of triangulated categories and we get an equivalence $$ \op{Hot}^{\op{b}}(\langle T\rangle_{\oplus}^{\op{Hot}}) \sirra \langle T\rangle_{\Delta}^{\op{Hot}}$$ If we only assume in addition $\mathcal H^nR=0$ for $n< 0$, only going to the left is a quasiisomorphism and we get a functor from left to right. It is some sort of \glqq taking the total of a double complex\grqq, but beware the traps. If we only assume in addition $\mathcal H^nR=0$ for $n> 0$, its the other way round, we get a functor from right to left. \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Ubiquity of derived dg-modules}] Assume now $\mathcal A$ is an abelian category and $T\in \op{Ket}_{\mathcal A}$ is a complex in $\mathcal A$ with the property that the quotient functor induces isomorphisms\label{udDG} $$\op{Hot}_{\mathcal A}(T,[n]T)\sira \op{Der}_{\mathcal A}(T,[n]T)\quad\forall n\in\DZ$$ For example this is the case if $T$ is a complex of injectives bounded below or a complex of projectives bounded above. We call a complex $T$ with this property {\bf quis-end-unfolded}, meaning unfolded with respect to endomorphisms and localization at quasiisomorphisms. Then we can replace above in the upper indices $\op{Hot}$ by $\op{Der}$, since clearly localization then induces an equivalence $\langle T\rangle_{\Delta}^{\op{Hot}}\sirra \langle T\rangle_{\Delta}^{\op{Der}}$ and a forteriori $\langle T\rangle_{\oplus}^{\op{Hot}}\sirra \langle T\rangle_{\oplus}^{\op{Der}}$. If in addition $\op{Der}_{\mathcal A}(T,[n]T)=0$ for $n\neq 0$, we call such $T$ {\bf tilting}, we obtain an equivalence $$ \op{Hot}^{\op{b}}(\langle T\rangle_{\oplus}^{\op{Der}}) \sirra \langle T\rangle_{\Delta}^{\op{Der}}$$ \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Variants}] There are variants of these arguments replacing $T$ with a family of objects $(T_i)_{i\in I}$. In case of a finite family, we can get them very easily by taking $T\pdef \bigoplus T_i$ and equipping $R$ with the pairwise orthogonal idempotents $1_i$ and take everywhere triangulated subcategories generated bs the $T_i$ or the $1_iR$ and so on. In case of an infinite family, we should interpret $R$ itself as a dg-category with objects $I$, no time for this. \end{Bemerkungl} \begin{Bemerkungl} If we let $T$ be a family of resolutions of perverse sheaves by complexes of injective sheaves bounded below, since perverse sheaves have no negative Ext, we get a functor $\op{Hot}^{\op{b}}(\op{Perv}_X)\ra \op{Der}(\op{Ab}_{/X})$. This passes to $$\op{Der}^{\op{b}}(\op{Perv}_X)\ra \op{Der}(\op{Ab}_{/X})$$ and gives the realization functor of \cite{BBD}. The other case $n>0\RA \mathcal H^nR=0$ leads to the weight complex functor of Bondarko. \end{Bemerkungl} \subsection{Formality and derived category} \begin{Bemerkungl}[\textbf{Constant homology complexes on manifolds}] Let $X$ be a paracompact smooth manifold with de-Rham complex $\Omega^*(X)$. This is a dg-ringalgebra over $\DR$. I want to construct an equivalence $$\op{Der}(\DR\op{-Mod}_{/X})\supset \langle \DR_X\rangle_\Delta\sirra \op{dgFrot-}\Omega^*(X)$$ By what we saw before, if $\DR_X\hra \mathcal I^\lhd$ is an injective resolution of the constant sheaf and $R\pdef \op{End}^{\op{dg}}(\mathcal I^\lhd)$ its endomorphism complex we get an equivalence $$\op{Der}(\DR\op{-Mod}_{/X})\supset \langle \DR_X\rangle_\Delta\sirra\op{dgFrot-}R$$ Our injective resolution factors over the de-Rham resolution as $\Bbb{R}_X \ra \Omega^{\ast}_{X} \overset{c}{\ra} \mathcal I^{*}$ for some chain map $c$. Now the trick is to check that $\op{Ket}^{\op{dg}}_{\DR_X}(\Omega_X^*,\mathcal I^*)$ is an $R$-$\Omega^*(X)$-dg-bimodule homologically free with basis $c$ from both sides. Then use the general statement below. \end{Bemerkungl} \begin{Proposition}[\textbf{Equivalences by bimodules}] Let $A,B$ be dg-rings and let $D$ be an $A$-$B$-dg-bimodule with $d\in \mathcal H^0D$ such that $\mathcal H D$ is a free $\mathcal H A$-module with basis $d$ as well as a free right $\mathcal H B$-module with basis $d$. Then $\otimes_A^{\op{L}}D:\op{dgDer}_{-A}\ra \op{dgDer}_{-B}$ induces an equivalence $$\op{dgFrot-}A\sirra \op{dgFrot-}B$$ \end{Proposition} \begin{Bemerkungl}[\textbf{Constant homology complexes on compact K"ahler manifolds}] In case $X$ is compact K"ahler and for complex coefficients some Hodge theory tricks \cite{DGMS} give us a complex dg-ringalgebra $E$ and quasiismorphisms of dg-ringalgebras $$\Omega^*(X;\DC)\qli E\qri {\op{H}}^*(X;\DC)$$ Using the Lemma again, we get even an equivalence $$\op{Der}(\DC\op{-Mod}_{/X})\supset \langle \DC_X\rangle_\Delta^{\op{Der}} \sirra\op{dgFrot-} {\op{H}}^*(X)$$ for the cohomology ring viewed as a dg-ring with trivial differential. \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Equivariant derived category of one-point space}] By essentially the same method in \cite{BeLu} the authors establish for any compact connected Lie group $K$ an equivalence of triangulated categories $$\op{Der}(K\sacts\op{top})\supset \langle \DC_{K\sacts\op{top}}\rangle_\Delta \sirra \langle \mathcal A_K\rangle_\Delta=\op{dgFrot-}\mathcal A_K \subset \op{dgDer}_{-\mathcal A_K}$$ where $\mathcal A_K={\op{H}}^*_K(\op{top};\DC)$ is considered as a dg-ring with zero differential. For the following we need still from \cite{BeLu} the bounded below variant $$\op{Der}(K\sacts\op{top})\supset \langle \DC_{K\sacts\op{top}}\rangle^+_\Delta \sirra \langle \mathcal A_K\rangle^+_\Delta\subset \op{dgDer}_{-\mathcal A_K}$$ with the notation meaning on the right what is generated by $\bigoplus_{q\leq 0}\mathcal A_K[q]$ and similarly on the other side. In addition, for any homomorphism $\phi: K\ra L$ of connected compact Lie groups they then establish commutative diagrams \begin{displaymath} \xymatrix{ \langle \DC_{L\sacts\op{top}}\rangle^+_\Delta \ar[d]_-{(\phi\sacts \op{id})^*} \ar[rr]^-{\approx} && \langle \mathcal A_{L}\rangle^+_\Delta \ar[d]^-{\otimes^{\op{L}}_{\mathcal A_L}\mathcal A_K} \\ \langle \DC_{K\sacts\op{top}}\rangle^+_\Delta \ar@{=>}[urr]^{\sim}\ar[rr]^-{\approx} && \langle \mathcal A_{K}\rangle^+_\Delta } \qquad \xymatrix{ \langle \DC_{L\sacts\op{top}}\rangle^+_\Delta \ar[rr]^-{\approx}\ar@{=>}[drr]^{\sim} && \langle \mathcal A_{L}\rangle^+_\Delta \\ \langle \DC_{K\sacts\op{top}}\rangle^+_\Delta\ar[rr]^-{\approx} \ar[u]^-{(\phi\sacts \op{id})_*} && \langle \mathcal A_{K}\rangle^+_\Delta \ar[u]_-{\op{res}_{\mathcal A_K}^{\mathcal A_L}} } \end{displaymath} \end{Bemerkungl} \begin{Beispielex}[\textbf{Equivariant cohomology of homogeneous spaces}] Put $\iota:G\hra H$. Let $\underline{G\acts X}\pdef \DC_{G\sacts X}$ denote the constant sheaf. The obvious isomorphism $(\iota\acts \bar 1)^*\underline{H\acts(H/G)}\sira \underline{G\acts\op{top}}$ gives, since by the induction equivalence the adjoint has to be a quasiinverse, an isomorphism $$(\iota\acts \bar 1)_*\underline{G\acts\op{top}}\sira \underline{H\acts(H/G)}$$ Since $(\iota\acts \op{id})=(\op{id}\acts a)\circ (\iota\acts \bar 1)$ for $a:H/G\ra \op{top}$ the constant map we deduce an isomorphism $$(\iota\acts \op{id})_*\underline{G\acts\op{top}}\sira (\op{id}\acts a)_*\underline{H\acts (H/G)}$$ Now we know for the equivalence $\langle \underline{{G\acts\op{top}}}\rangle^+_\Delta \sirra \langle \mathcal A_G\rangle^+_\Delta$ that $\underline{{G\acts\op{top}}}\mapsto \mathcal A_G$. Thus by the above we find $$(\iota\acts \op{id})_*\underline{G\acts\op{top}}\mapsto \op{res}_{\mathcal A_G}^{\mathcal A_H}\mathcal A_G$$ which together with the above reproves ${\op{H}}^q_G(\op{top})\cong {\op{H}}^q_H(H/G)$. Now restricting again the action of $H$ to a connected closed subgroup $I\subset H$ we find $$(\op{id}\acts a)_*\underline{I\acts (H/G)}\mapsto \mathcal A_I\otimes^{\op{L}}_{\mathcal A_H}\mathcal A_G$$ If $\mathcal A_I$ or $\mathcal A_G$ is free over $\mathcal A_H$, we finally get $${\op{H}}^*_I(H/G)\sira \mathcal A_I\otimes_{\mathcal A_H}\mathcal A_G$$ \end{Beispielex} \begin{Beispielex} Let us compute the cohomology of the circle group $S^1$. We find in general ${\op{H}}^*(G;\DC)={\op{H}}^*_{1\times 1}(G;\DC)= \DC\otimes_{\mathcal A_G}^{\op{L}}\DC$ to be understood in $\op{dgDer-}\mathcal A_G$ and in case $G=S^1$ we get $\DC\sira \op{Keg}((t\cdot):\DC[t][-2]\ra \DC[t])$ with $\op{deg}t=2$. Thus $$\DC\otimes_{\DC[t]}^{\op{L}}\DC\sira \op{Keg}(0:\DC[-2]\ra \DC)$$ which gives back what we know already. In case of a more general connected compact Lie group, $\mathcal A_G$ will be a polynomial ring with generators in even degrees. A suitable resolution of $\DC$ can be obtained by tensoring over $\DC$ resultions of the type considered above and we obtain that ${\op{H}}^*(G;\DC)$ is an exterior algebra in generators of suitable odd degrees. \end{Beispielex} \subsection{Koszul duality} \begin{Bemerkungl} I am not good with the history of what is called Koszul duality here, but \cite{BGS} is certainly an important part of it. Now recall from \ref{koDU} and \ref{KOz} what we want to prove. \end{Bemerkungl} \begin{Theorem}[\textbf{Koszul duality \cite{BGSo}}] Let $B$ be a Koszul ring and assume it is finitely generated as a $B^0$-module from the left as well as from the right. Assume in addition $E=E(B)\pdef \op{Ext}^*_{B}(B^0, B^0)$ is right noetherian. Then there exists an equivalence of triangulated categories $$K:\op{Der}^{\op{b}}(B\op{-Modfg}^\DZ)\sirra \op{Der}^{\op{b}}(\op{Modfg^\DZ-}E)$$ with $K(M\langle j\rangle)=(KM)\langle -j\rangle[j]$. \end{Theorem} \begin{Definition} A nonnegatively graded ring $B$ is called {\bf Koszul}, if $B^0$ is semisimple and in the abelian category $B\op{-Mod}^\DZ$ of $\DZ$-graded $B$-modules we have $$\op{Ext}^i(B^0, B^0\langle -j\rangle)=0\;\; \text{ unless }i=j.$$ Here $\langle j\rangle$ means an internal, nonhomological grading shift normalized so that $B^0\langle -j\rangle$ is concentrated in the internal degree $j$. \end{Definition} \begin{proof}[Proof] For simplicity assume $B^0=\DC$. By definition, $B$ being a Koszul ring means the family of all $B^0[i]\langle -i\rangle$ is a tilting family in $\op{Der}(B\op{-Mod}^\DZ)$. We deduce with tilting for infinite families an equivalence $$ \op{Hot}^{\op{b}}(\langle B^0[i]\langle -i\rangle\rangle_{\oplus}^{\op{Der}}) \sirra \langle \langle B^0[i]\langle -i\rangle\rangle_{\Delta}^{\op{Der}}$$ On the right, we get $\op{Der}^{\op{b}}(B\op{-Modfg}^\DZ)$ by our finiteness assumptions on $B$. On the left, we check that we have a fully faithful functor $$\langle B^0[i]\langle -i\rangle\rangle_{\oplus}^{\op{Der}}\vra \op{Modfg^\DZ-}E$$ given on the objects $B^0[i]\langle -i\rangle$ by $B^0[i]\langle -i\rangle\mapsto E\langle i\rangle$ and on their morphisms by mapping $e: B^0[i]\langle -i\rangle\ra B^0[j]\langle -j\rangle$ alias $e\in \op{Ext}^{j-i}_{B\op{-Mod}^\DZ}(B^0, B^0\langle i-j\rangle)$ alias $e\in \op{Ext}^{j-i}_B(B^0, B^0)$ alias $e\in E^{j-i}$ to left multiplication $(e\cdot): E\langle i\rangle \ra E\langle j\rangle$. It easily extends to finite direct sums of such objects to give an equivalence $$\langle B^0[i]\langle -i\rangle\rangle_{\oplus}^{\op{Der}}\sirra \langle E\langle i\rangle\rangle_{\oplus}\subset \op{Mod^\DZ-}E$$ and thus an equivalence $$\op{Hot}^{\op{b}}(\langle B^0[i]\langle -i\rangle\rangle_{\oplus}^{\op{Der}})\sirra \op{Hot}^{\op{b}}(\langle E\langle i\rangle\rangle_{\oplus})\subset \op{Hot}^{\op{b}}(\op{Mod^\DZ-}E)$$ But now the homotopy category of projective bounded complexes sits fully faithfully in the derived category $\op{Hot}^{\op{b}}(\langle E\langle i\rangle\rangle_{\oplus})\vra \op{Der}(\op{Mod^\DZ-}E)$ and by the finiteness conditions we asked for we even get an equivalence \begin{displaymath}\op{Hot}^{\op{b}}(\langle E\langle i\rangle\rangle_{\oplus})\sirra \op{Der}^{\op{b}}(\op{Modfg^\DZ-}E) \end{displaymath} To determine the effect of the internal grading shift, we calculate $(B^\circ [i]\langle -i\rangle)\langle j\rangle= (B^\circ [i-j]\langle j -i\rangle)[ j]$ which gets mapped to $(E\langle i -j\rangle)[ j]=(E\langle i\rangle)\langle -j\rangle[ j]$. \end{proof} \begin{Bemerkungl}[\textbf{Koszulity for category $\mathcal O_0$}] Let $A\pdef \op{End}_{\mathfrak g}(\bigoplus_{x\in W}{\op{P}}_x)$ be our ring describing the principal block $\mathcal O_0$ and choose a grading and an autoduality isomorphism $$A\sira E(A^{\op{opp}})$$ of graded rings as in \ref{vDs} with $1_x\mapsto 1_{xw_0}$ in degree zero. To prove $B\pdef A^{\op{opp}}$ is Koszul we apply the numerical Koszulity criterion in its variant \ref{ncKv}. Geometric arguments to be discussed later allow us to express the generating functions ${\op{P}}_B$ and ${\op{P}}_{E(B)}$ by Kazhdan-Lusztig polynomials. The inversion formulas for Kazhdan-Lusztig polynomials then show ${\op{P}}_B(t){\op{P}}_{E(B)}(-t)^\top=\op{I}$ and this does the trick. \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Koszul duality for category $\mathcal O_0$}] (\cite{BGi}, proved in \cite{BGSo}.) Let $A\pdef \op{End}_{\mathfrak g}(\bigoplus_{x\in W}{\op{P}}_x)$ be our ring describing the principal block $\mathcal O_0$ with $1_x$ the projector onto $P_x$ and choose a grading and an isomorphism $$A\sira E(A^{\op{opp}})$$ of graded rings as in \ref{vDs} with $1_x\mapsto 1_{xw_0}$ in degree zero as before. We now know $B\pdef A^{\op{opp}}$ is Koszul and by Koszul duality \ref{koDU} we get an equivalence of triangulated categories $$K: \op{Der}^{\op{b}}(\op{Modfg^\DZ-}A)\sirra \op{Der}^{\op{b}}(\op{Modfg^\DZ-}A)$$ We know an equivalence $\mathcal O_0\sirra \op{Modfg-}A$ with $P_x\mapsto 1_xA$ and $L_x\mapsto 1_xA^0$. Let us define $$\mathcal O_0^\DZ\pdef \op{Modfg^\DZ-}A$$ to be the {\bf $\DZ$-graded version of $\mathcal O_0$}. The simple objects of $\mathcal O_0^\DZ$ are the $L_x^\DZ\pdef 1_xA^0$ and their shifts, so in total all the $\langle \gamma\rangle L_x^\DZ$ for $x\in W$ and $\gamma\in \DZ$. Their projective covers are the $P_x^\DZ\pdef 1_xA$ and their shifts. Under Koszul duality by \ref{kdso} we find $$K:\langle \gamma\rangle L_{xw_0}^\DZ \;\mapsto\; \langle -\gamma\rangle [\gamma] P_x^\DZ$$ Now assume $K:I\mapsto L_{x}^\DZ$ for some unknown object $I$ of the derived category. We deduce $\op{Der}(\langle \gamma\rangle [n]L_{yw_0}^\DZ,I)\cong \op{Der}(\langle -\gamma\rangle [n+\gamma]P_{y}^\DZ,L_{x}^\DZ)$ is onedimensional for $x=y$ and $n=\gamma=0$ and zero else. Some fiddling shows $I$ must be the injective hull $I=I_{xw_0}^\DZ$ of $L_{xw_0}^\DZ$ and we get $$K: \langle \gamma\rangle I_{xw_0}^\DZ \;\mapsto\; \langle -\gamma\rangle [\gamma] L_x^\DZ$$ This much can also be shown in the generality of Koszul rings. Special to our situation is the interpolation $$K: \langle \gamma\rangle \nabla_{xw_0}^\DZ \;\mapsto\; \langle -\gamma\rangle [\gamma] \Delta_x^\DZ$$ of the above statements, so dual Vermas go to Vermas. We deduce $$\op{Ext}^n_{\mathcal O^\DZ_0}(L_{xw_0}^\DZ, \langle \gamma\rangle \nabla_{yw_0}^\DZ) = \op{Der}(L_{xw_0}^\DZ, [n]\langle \gamma\rangle \nabla_{yw_0}^\DZ) \sira \op{Der}(P_{x}^\DZ, [n+\gamma]\langle -\gamma\rangle \Delta_{y}^\DZ)$$ so that this is only nonzero in case $n+\gamma=0$ and then taking dimensions we find $$\op{dim}\op{Ext}^n_{\mathcal O_0}(L_{xw_0}, \nabla_{yw_0}) =\op{dim}\op{Ext}^n_{\mathcal O^\DZ_0}(L_{xw_0}^\DZ, \langle -n\rangle \nabla_{yw_0}^\DZ) = [\langle n\rangle \Delta_{y}^\DZ: L_{x}^\DZ]$$ The left hand side is known to be given by some Kazhdan-Lusztig polynomials, which when evaluated at $1$ and multiplied by $(-1)^n$ give the inverse transpose to the matrix of multiplicities $ [ \Delta_{yw_0}^\DZ: L_{xw_0}^\DZ]$. In this way we see the whole theory can be interpreted as a categorification of the inversion formulas for Kazhdan-Lusztig polynomials and the graded structure can be interpreted as explaining how the coefficients of Kazhdan-Lusztig polynomials are actual multiplicities. \end{Bemerkungl} \newpage \section{Sixth lecture on Koszul duality, July 12} \subsection{Motivic methods} \begin{Bemerkungl} As you can see in \cite{BGi} this was part of Beilinson's vision from the very beginning. Now new developments in the theory of motives made it possible to get a simplified view on it, if you are willing to accept on faith these new methods, as in my own case by lack of competence. \end{Bemerkungl} \begin{Bemerkungl} As explained in \cite{SVW,HoLi}, it is possible to construct in many situations so-called \glqq $\DZ$-graded versions\grqq\ of equivariant derived categories by artificially killing all extensions between Tate motives. I will mostly cite \cite{SVW} only since I know it better. \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Mixed things}] Let $G\acts X$ be an affine complex algebraic group acting on a complex variety with finitely many orbits. Under the additional \glqq Whitney-Tate\grqq\ condition WT, which is satisfied notably in the case of the parameter spaces of ABV, we construct in \cite{SVW} by motivic methods a \glqq mixed Tate version\grqq\ $\op{MTDer}_{G\sacts}(X)$ of the equivariant derived category. This is a triangulated category with an additional automorphism $(1/2)$ called the \glqq root of Tate twist\grqq. We also construct a triangulated functor $$v:\op{MTDer}_{G\sacts}(X) \ra \op{Der}_{G\sacts}(X)$$ along with isomorphisms $v\mathcal F(1/2)\sira v\mathcal F$ such that $$\bigoplus_{i\in \DZ} \op{MTDer}_{G\sacts}(\mathcal F,\mathcal G(i/2)) \sira\op{Der}_{G\sacts}(v\mathcal F,v\mathcal G)$$ for any two objects $\mathcal F,\mathcal G$ of our new categories. Our new categories admit a \glqq weight structure\grqq\ and any simple perverse object $\mathcal L^\pi\in \op{Per}_{G\sacts}(X)$ admits a unique lift of \glqq weight zero\grqq\ $\mathcal L^\pi\in \op{MTDer}_{G\sacts}(X)_{\op{wt}=0}$. Furthermore $$\op{MTDer}_{G\sacts}(X)_{\op{wt}=0}= \big\langle \mathcal L^\pi[i](i/2)\mid i\in \DZ,\pi\in \op{Par}_{G}(X)\big\rangle_\oplus$$ \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Pointwise purity and tilting}] Let $G\acts X$ be an affine complex algebraic group acting on a complex variety with finitely many orbits. In addition to the Whitney-Tate-condition WT, let us ask for the \glqq pointwise purity\grqq\ condition PP, which also is satisfied notably in the case of the parameter spaces of ABV. Assuming WTPP, we can show $$\op{MTDer}_{G\sacts}(\mathcal L^\pi,\mathcal L^\tau[i](j/2))=0\quad\text{ unless }i=j.$$ In other words, $\op{MTDer}_{G\sacts}(X)_{\op{wt}=0}$ is a tilting family, and since the relevant finiteness conditions are satisfied, we find that tilting gives a triangulated equivalence $$\op{Hot}^{\op{b}}(\op{MTDer}_{G\sacts}(X)_{\op{wt}=0})\sirra \op{MTDer}_{G\sacts}(X)$$ \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Moral}] I see this as a wonderfully explicit way to explain the \glqq unreasonable effectiveness of intersection cohomology in representation theory\grqq: The intersection cohomology understands the whole equivariant derived category, which is such a natural object, that its appearence doesn't need further justification. Well, and if you still insist on further justification, I might invoke Grothendiek's function-sheaf correspondence. From my point of view, this is the essence of the appearence of Koszul duality in representation theory. \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Significance of equivariant Ext-algebra}] In the same way as discussed in the proof of Koszul duality for Koszul rings, we obtain under WTPP an equivalence $$\op{MTDer}_{G\sacts}(X)_{\op{wt}=0}\sirra \op{Frei^\DZ-}\op{Ext}_G(X)$$ where we see $E\pdef \op{Ext}_G(X)$ as a $\DZ$-graded ring with a decomposition $1=\sum 1_\pi$ into pairwise orthogonal idempotents and denote by $\op{Frei^\DZ-}E$ in such a case $\langle 1_\pi E\langle i\rangle\rangle_{\oplus}\subset \op{Mod^\DZ-}E$. Since the homotopy category of projective objects is fully faithful in the derived category, we find the upper row of the diagram \begin{displaymath} \xymatrix{\op{Hot}^{\op{b}}(\op{MTDer}_{G\sacts}(X)_{\op{wt}=0})\ar[d]^\approx\ar[r]^-\approx& \op{Hot}^{\op{b}}(\op{Frei^\DZ-}E)\ar[d]^\approx\ar@{^{(}->}[r]^-\sim& \op{Der}(\op{Mod^\DZ-}E)\\ \op{MTDer}_{G\sacts}(X)\ar@{-->}[r]^-\approx_-K&\op{Der}^{\op{b}}(\op{Modfg^\DZ-}E)\ar[ru] } \end{displaymath} where the left vertical comes from before and the right vertical to be an equivalence needs some finiteness conditions satisfied in our situations. Since morally $(i/2)=\langle -i\rangle$ we get for $K$ the lower horizontal $K(M(i/2))=(KM)\langle i\rangle[-i]$. \end{Bemerkungl} \begin{Beispielex}[\textbf{Groups acting on the one-point variety}] We find an equivalence $\op{MTDer}_{\sacts \DC^\times}(\op{var}) \sirra \op{Der}^{\op{b}}(\op{Modfg^\DZ-}\DC[t])$ with $t$ a free variable of degree $\op{deg}t=2$ for the multiplicative group $\DC^\times$ acting on the one-point variety $\op{var}$. This is to be compared with the equivalence $$\op{Der}(S^1\acts\op{top})\supset \langle \DC_{S^1\sacts\op{top}}\rangle_\Delta \sirra \langle \DC[t]\rangle_\Delta=\op{dgFrot-}\DC[t] \subset \op{dgDer}_{-\DC[t]}$$ from \cite{BeLu} we discussed before. Similarly, we get for $G$ any connected complex algebraic group $$\op{MTDer}_{\sacts G}(\op{var}) \sirra \op{Der}^{\op{b}}(\op{Modfg^\DZ-}\mathcal A_G)$$ In a way, the end result looks nicer, but there is quite some hidden work going into the construction of $\op{MTDer}$. \end{Beispielex} \begin{Beispielex}[\textbf{Cohomology of a group}] Together with the obvious extensions of the change-of-group results of \cite{BeLu} also proven in \cite{SVW}, we see we can indeed calculate the cohomology of $G$ as an ordinary torsion group $\DC\otimes_{\mathcal A_G}^{\op{L}}\DC$ in the category of graded $\mathcal A_G$-modules, if we take afterwards the total of this bigraded space in the correct way. More precisely consider the diagram $$\begin{array}{ccccc} 1\acts \op{var}&\ra &G\acts G&\leftarrow &1\acts G\\ \parallel&&\da&&\da\\ 1\acts \op{var}&\ra &G\acts \op{var}&\leftarrow& 1\acts \op{var} \end{array}$$ of spaces with action. Pushing forward the constant sheaf under the left upper horizontal we again get a constant sheaf, as pulling back in an (induction) equivalence, maps constant to constant and in adjoint, hence quasiisomorphic. Now in the left square pushing forward in the horizontals commutes with pushing down in the verticals. Similarily, in the right square, pulling back in the horizontals, a change of group, commutes with pushing down in the verticals. In the end we see that extending the group action and restricting again to the trivial group in the lower horizontal gives the cohoomology of the space $G$. Now under Koszul duality extension of the group action under $G\ra H$ corresponds to restriction from $\mathcal A_H$ to $\mathcal A_G$ and restriction of the group action corresponds to $\otimes^{\op{L}}_{\mathcal A_H}\mathcal A_G$. Thus under Koszul duality $$K:\op{fin}_*\underline G \mapsto \DC\otimes_{\mathcal A_G}^{\op{L}}\DC$$ This is Koszul duality over a one-point variety with no group acting. To get ${\op{H}}^i(G)$, we have to add all parts of double degree $(i, j/2)$ on the left, thus all parts of double degree $(i-j, j)$ on the right for the first degree homological, the second internal. In case $G=\DC^\times $ we find $\mathcal A_G=\DC[t]$ with $\op{deg}(t)=2$ and $\DC\otimes_{\DC[t]}^{\op{L}}\DC = \DC^{(0,0)}\oplus \DC^{(-1,2)}$ is twodimensional sitting in bidegrees $(0,0)$ and $(-1,2)=(1-2,2)$ and we find again the cohomology of $\DC^\times$ as we know and love it. \end{Beispielex} \begin{Bemerkungl}[\textbf{Categorification of the Hecke algebra}] We take $G\supset B$ a complex connected reductive group and get equivalences $$\op{MTDer}_{(B\times B)\sacts}(G)_{\op{wt}=0}\sirra\op{Der}^{\op{c}}_{(B\times B)\sacts}(G)_{\op{ss}} \sirra S\op{-SMod^\DZ-}S$$ to special bimodules by our forgetting functor $v$ and hypercohomology, as discussed by Peng with $S\pdef \mathcal A_B$ a polynomial ring with its variables in degree two. We take the bounded homotopy catgories and together with tilting deduce an equivalence $$\op{MTDer}_{(B\times B)\sacts}(G) \sirra \op{Hot}^{\op{b}}(S\op{-SMod^\DZ-}S)$$ We can even show the geometric convolution $*_B$ corresponds to $\otimes_S$ here. Now the right hand bimodule side is more accessible to direct calculations and the left hand geometric side is what morally one might want to call a categorification of the Hecke algebra. Some things are also easier to check on the geometric side. Consider for example on the bimodule side the so-called Rouqier complexes $(S\otimes_{S^t} S[0]\ra S)$ and $(S\ra S\otimes_{S^t} S[0])$ for simple reflections $t$. They correspond on the geometric side to $j_!\underline{BtB}$ and $j_*\underline{BtB}$. Now it is known that if $st\ldots p=w$ is a shortest expression of some element $w\in W$, then the multiplication map is an isomorphism $$BsB\times_{/B}BtB\ldots \times_{/B} BpB\sira BwB$$ and geometric convolution will give us $j_!\underline{BsB}\ast_B j_!\underline{BtB}\ldots \ast_{B} j_!\underline{BpB}\sira j_!\underline{BwB}$. This explains why Rouquier complexes satisfy braid relations and proves it for cristallographic Coxeter groups. Rouquier proved this even for arbitrary Coxeter groups by explicit algebraic calculations. I bet he was motivated by these ideas, which were difficult to turn into a real proof at the time. \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Knot homology}] Some of you might now this gives a nice access to Khovanov's knot homology. Namely, given a braid $b$ in $n$ strands, construct a complex of special bimodules $F(b)$ for the symmetric group $S_n$ by tensoring Rouquier complexes, take one for overcrossing and the oher for undercrossing. Then take the complexes $${\op{HH}}_i(F(b))$$ and their homology $\mathcal H^j({\op{HH}}_iF(b))$ and show this only depends on the oriented link obtained by closing the braid $b$. Since our bimodules have interal grading, this is a triply graded vector space, the triply graded link homology. \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Categorifying inversion formulas for KLV-polynomials}] Let us get back to our main conjecture. It said $$\mathcal M_\chi(G)\cong \op{Nil-}(\op{Ext}_{G^\vee}X(\chi))$$ where on the left we have to consider representations of a whole bunch of real forms, as explained by \cite{ABV} in all detail. Let us put $E\pdef \op{Ext}_{G^\vee}X(\chi)$. Sure enough, $G^\vee\acts X(\chi)$ has the WTPP property, so we have, as explained under the heading of significance of the Ext-algebra, the Koszul duality equivalence in the horizontal of the diagram \begin{displaymath} \xymatrix{ \op{MTDer}_{G^\vee\sacts}(X(\chi))\ar[r]^-\approx_-K &\op{Der}^{\op{b}}(\op{Modfg^\DZ-}E)\ar[r]&\op{Der}^{\op{b}}(\op{Modfg-}E)\\ &\op{Modfd^\DZ-}E\ar[r]\ar[u]&\op{Nil-}E\ar[u] } \end{displaymath} By construction $K:\mathcal L^\pi\mapsto 1_\pi E$. Introduce the $\DZ[v,v^{-1}]$-valued Euler characteristic form in the upper middle by $$\op{Eu}(M,N)\pdef \sum_{i,j}(-1)^iv^j\op{dim}_\DC \op{Der}(M,N[i]\langle j\rangle)$$ for some variable $v$. If we let $L_\pi\pdef 1_\pi E^0$ correspond to the irreducible representations, we get $$\op{Eu}( K\mathcal L^\pi,L_\tau)= \op{Eu}( 1_\pi E,L_\tau)=\delta_{\pi,\tau}$$ Now let us hope that the standard representation $M_\pi$ having $L_\pi$ as a unique simple subobject also lifts to a $\DZ$-graded object $M_\pi\in \op{Modfd^\DZ-}E$. This follows from \cite{SVW} at least in the case of trivial central character. On the other hand consider the standard object $\mathcal M^\pi\pdef j_*\tau[\op{dim}Y]\in \op{MTDer}$, please excuse the accidents of notation $j:Y\hra X(\chi)$ and $\pi=(Y,\tau)$, so $j\neq j$ and $\tau\neq \tau$ locally. I conjectured in \cite{So-L} what can now be written $$\op{Eu}( K\mathcal M^\pi,M_\tau)= \delta_{\pi,\tau}$$ and want to explain how this would categorify the inversion formulas for KLV-polynomials. \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Interlude on inversion formulas from linear algebra class}] Recall the linear algebra class. Given a linear map $f:V\ra W$ of finite dimensional vector spaces and ordered bases $\mathcal A, \mathcal B$ of $V,W$ and $f^\ttop:W^*\ra V^*$ the transposed map, then for the matrices we have $$_{\mathcal A^\ttop}[f^\ttop]_{\mathcal B^\ttop}=(_{\mathcal B}[f]_{\mathcal A})^\ttop$$ For $V=W$ and $f=\op{id}$ we get the inversion formula $$_{\mathcal A^\ttop}[\op{id}]_{\mathcal B^\ttop}=((_{\mathcal A}[\op{id}]_{\mathcal B})^\ttop)^{-1}$$ In words, the transition matrix between dual bases is inverse transpose to the transition matrix of the original bases. \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Inversion formulas for KLV-polynomials}] Well, this was a good warm-up. Now write in the respective Grothendieck groups $$\begin{array}{lll} [M_\pi]&=& \sum [M_\pi:L_\tau\langle j\rangle]\;[L_\tau\langle j\rangle] \\[2mm] [\mathcal M^\rho]&=& \sum [\mathcal M^\rho:\mathcal L^\sigma(i/2)]\;[\mathcal L^\sigma(i/2)] \end{array} $$ Pairing the left hand sides moved with $K$ and the right hand sides using the Euler pairing $\op{Eu}$ we find that our conjecture implies $$\begin{array}{lll} \delta_{\pi,\rho}&=& \sum \;[\mathcal M^\rho:\mathcal L^\sigma(i/2)]\;[M_\pi:L_\tau\langle j\rangle]\op{Eu}(K(\mathcal L^\sigma(i/2)), L_\tau\langle j\rangle)\\[2mm] &=& \sum (-1)^i[\mathcal M^\rho:\mathcal L^\tau(i/2)]\;[M_\pi:L_\tau\langle i\rangle] \end{array}$$ since $K(\mathcal L^\sigma(i/2))=(K\mathcal L^\sigma)[-i]\langle i\rangle$. These finally are Vogan's inversion formulas. \end{Bemerkungl} \begin{Bemerkungl}[\textbf{Eberhardt's K-theory version}] As explained in \cite{EbK} 1.6, what Peng explained can be understood as a variant of an equivalence $$\op{DK}(B\backslash G/B;\DQ)\sirra \op{D}_{\op{mon}}(U^\vee\backslash G^\vee(\DC)/U^\vee;\DQ)$$ between K-motives on the finite Hecke stack and monodromic sheaves on the Langlands dual. The proof follows the bimodule ideas. Letting $\theta$ be the holomorphic Cartan involution corresponding to a real form, the version for real reductive groups then would be, that is the claim and conjecture, $$\op{DK}([B\backslash G/B]^{h\theta})\sirra \op{D}_{\op{mon}}([U^\vee\backslash G^\vee/U^\vee]^{h\theta^\vee}(\DC))$$ with the upper index $h\theta$ meaning homotopy fixpoints. I am fascinated and in fact believe this is the way to go, although I hardly understand what it all means. For the next summer school, but not to be explained by me! \end{Bemerkungl} \newpage %%% Local Variables: %%% mode: latex %%% TeX-master: "XXKAZ" %%% End: