\section{Trying to write a program for MSRI}
I propose to assume for the main lectures of this
summer school that the only
reductive group is $G=\op{GL}_n$.

\subsection{Some representation theory}
A big bunch of the theory consists of reductions
of problems of representation theory to Weyl group
combinatorics. One way to think of it is that
a representation of $\op{GL}_n$ behaves like
a bunch of representations of copies of $\mathfrak{sl}_2$
bound together with very little interaction.
\begin{enumerate}
\item
  Simple finite dimensional representations of $\mathfrak{sl}_2(\DC)$; 
\item
 Maximal toral subalgebra, root space decomposition,
  Weyl group: Easy for the general linear group;
\item
  Enveloping algebra, Verma modules, center of the enveloping
  algebra, Harish-Chandra isomorphism, Weyl character formula;
\item
  Category $\mathcal O$. The principal block in the case of
   $\mathfrak{sl}_2(\DC)$. Translation functors. Projective functors;
\item
  Simple highest weight modules and their characters. BGG-reciprocity;
\item
  Harish-Chandra modules. Harish-Chandra modules for complex groups
  and category $\mathcal O$; 
\end{enumerate}

\subsection{Some combinatorics}
\begin{enumerate}
\item Coxeter Systems and realization as reflection groups;
\item Bruhat decomposition and Bruhat order;
\item Hecke algebras;
\item Kazhdan-Lusztig-Polynomials;
  
\end{enumerate}

\subsection{Some homological algebra}
I wonder to what extent we can assume the
participants to have experience
with derived categories. 
\begin{enumerate}
\item The derived category dg-modules;
\item Tilting and Koszul duality;
\end{enumerate}

\subsection{Some geometry}
I wounder to what extent we can assume the
participants to have experience
with sheaf cohomology.
\begin{enumerate}
\item The equivariant derived category and its six functors;
\item The categorified Hecke algebra; 
\end{enumerate}

\subsection{Some possible goals}
\begin{enumerate}
\item
 Koszul duality for category $\mathcal O$,
  in the form $L\pdef \bigoplus_{x\in W} L(x\cdot 0)$ sum of
  all simples of the principal block $\mathcal O_0$ and $A\pdef
  \op{Ext}^*_{{\mathcal O}_0}(L,L)$, then $$\mathcal O_0\cong A\op{-Modf}$$
  To understand the statement, need:
   \begin{enumerate}
  \item Category $\mathcal O$;
    \item Extensions in abelian categories and their Yoneda-product;
    \item  The enveloping algebra and its center;
    \item Verma modules, simple highest weight modules;
   \end{enumerate}
   To understand the motivation, need:
   \begin{enumerate}
     \item Kazhdan-Lusztig polynomials and their inversion formulas; 
   \item Kazhdan-Lusztig conjectures
     $[M(x\cdot 0):L(y\cdot 0)]$;
   \item BGG-reciprocity, so projective objects in $\mathcal O$,
     they admit a Verma flag.  Projective covers and their unicity. Close by are translation functors and
     how they categorify the Weyl group.
   \end{enumerate}
 \item
   Explain the proof of Koszul duality for category $\mathcal O$. 
   Need:
   \begin{enumerate}
   \item
     Harish-Chandra-Isomorphism for the center of the enveloping algebra;
   \item
     Translation functors and wall-crossing;
   \item
     Hypercohomology;
   \item
     Intersection cohomology complexes;
   \item
     Cohomology ring of the flag variety, Borel isomorphism.
     Will later come out easily in the study of equivariant derived
     categories, just claim it here;
   \item Six-functor-formalism. Well, four functors mostly do the job,
     bit six is much better;
   \item
     Localization (to understand extensions of simple objects in $\mathcal O$);
   \item
     Some spectral sequence arguments (to deduce a limit by parity va\-ni\-shing);
   \item
     Poincar\'e duality for the flag variety;
   \end{enumerate}

   
\item Formulation of main conjecture, original form:
  Some equivariant Ext-Algebra describes some category of Harish-Chandra-modules
  $$\mathcal M_\chi\cong \op{Ext}^*_{G^\vee}X(\chi)\op{-Modfnil}$$
  To understand the statement, need:
  \begin{enumerate}
  \item  Equivariant derived category;
    \item Equivariant
  intersection cohomology sheaves;
  \item  Harish-Chandra-Modules;
  \item  The enveloping algebra and its center;
  \end{enumerate}
 To understand the motivation, need:
  \begin{enumerate}
  \item
    Some local Langlands philosophy;
  \item
    Some motivation for Harish-Chandra-modules;
  \end{enumerate}
\item Some examples for real groups, mostly $\op{SL}(2;\DR)$ and tori;
  
\item 
  The example of complex groups.
  In this case, we can construct equivalences
  $$\mathcal M_0 \cong \mathcal O_0^\infty \qquad  \op{Ext}^*_{B^\vee}(G^\vee/B^\vee) \cong \op{Ext}^*_{G^\vee}X(0)$$
  and $\mathcal O_0^n$ has enough projectives and we
  have fully faithful functors to some category of special bimodules. 
  Need:
  \begin{enumerate}
  \item  Equivariant derived category;
    \item Equivariant
      intersection cohomology sheaves;
      \item equivariant six functors;
  \item  Equivalences from Harish-Chandra-Modules to category $\mathcal O$;
  \end{enumerate}

\item
  New interpretation in terms of motivic derived categories.
  Tilting formalism. Need:
   \begin{enumerate}
   \item More homological algebra: Derived category of a dg-ring.
     Functors by dg-bimodules;
     \item Motivic variant of equivariant derived category. 
\end{enumerate}
 
\end{enumerate}

  

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