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\title[]{~\\[2ex]
What is modular category $\mathcal O$?}
\author[]{Wolfgang Soergel}
\institute[]{\inst{}
   Mathematisches Insitut\\
  Universit\"at Freiburg\\[4ex]

\vspace*{.9cm}
%\textcolor{red}{\\ allgemeine Fragen/Hinweise hier positionieren}
  }

\date[]{\small \hbox{27. M\"arz 2017}}

\beamersetuncovermixins{\opaqueness<1>{25}}{\opaqueness<2->{15}}
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\begin{document}

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\titlepage
\begin{frame}
   Modular category $\mathcal O$
is a subquotient of the category
of rational representations of a
    connected  reductive algebraic group in positive characteristic,
   that behaves very much like usual category $\mathcal O$.
       
    %  I invented it as a baby example to study typical problems of modular
%      representation theory in a particularily easy case.

  \end{frame}

\begin{frame}
   Modular category $\mathcal O$
is a {\color{Red}subquotient} of the category
of rational representations of a
    connected  reductive algebraic group in positive characteristic,
   that behaves very much like usual category $\mathcal O$.
       
    %  I invented it as a baby example to study typical problems of modular
%      representation theory in a particularily easy case.

  \end{frame}
% ---------- ---------- ---------- ----------
%\section{ Die Lusztig-Vermutung} %\"uber Charaktere einfacher Darstellungen algebraischer Gruppen
% ======== ======== ======== SECTION ========

  \begin{frame}
Given an abelian category $\mathcal A$ and a set of objects 
$\mathcal K\subset \mathcal A$ there always exists an exact functor
$F:\mathcal A\ra \mathcal A/\mathcal K$ 
to  an abelian category    $\mathcal A/\mathcal K$ such that 
$F$ annihilates all objects of $\mathcal K$ and that every exact functor
$G:\mathcal A\ra\mathcal B$ of abelian categories annihilating all objects
of $\mathcal K$ factorizes uniquely over $F$. 
We call $\mathcal A/\mathcal K$ the {\bf quotient category}.
\begin{displaymath}
 \xymatrix{
\mathcal A\ar[dr]^G\ar[r]^F  &\mathcal A/\mathcal K\ar@{-->}[d]\\
&\mathcal B
}
\end{displaymath}
  \end{frame}

\begin{frame}
   Modular category $\mathcal O$
is a subquotient of the category
of {\color{Red}rational representations} of a
    connected  reductive algebraic group in positive characteristic,
   that behaves very much like usual category $\mathcal O$.
       
    %  I invented it as a baby example to study typical problems of modular
%      representation theory in a particularily easy case.

  \end{frame}


% \begin{frame}
%  For affine algebraic groups $G\supset B$ the  restriction admits a
% right adjoint, induction
%  %  F"ur affine algebraische Gruppen $G\supset B$ hat das Restringieren einen
% %   Rechtsadjungierten, das Induzieren
% $$
% G\operatorname{-Mod} \begin{array}{c}
%   \overset{\operatorname{res}}{\longrightarrow}\\[-1ex]
%   \underset{\operatorname{ind}}{\longleftarrow}
% \end{array}  B\operatorname{-Mod}
% $$
% \begin{displaymath}
%   \begin{array}{ccl}
%     \operatorname{ind}^G_B V &=&\{f : G \rightarrow V\mid f \text{ algebraic
%       $B$-equivariant}\}\\[1ex]
%     & =&\{\text{ algebraic sections in } G \times_B V \twoheadrightarrow G/B\}
%   \end{array}
% \end{displaymath} 
% \end{frame}

\begin{frame}
From now on:\\[2ex]\begin{itemize} 
\pause
\item 
$ k=\bar k$ algebraically closed field\\[2ex]
\pause\item $G\supset B$  connected affine algebraic group over $k$ with a
  Borel, for example $G=\op{GL}(r;k)\supset B$ upper triangular matrices\\[2ex]
\pause\item
$\mathfrak X=\mathfrak X(B)\pdef \{\lambda:B\ra k^\times\mid \lambda\text{ homomorphism}\}$\\
the weight lattice\\[2ex]
% \pause\item
% $\nabla (\lambda)\pdef\operatorname{ind}^G_B k_\lambda $ induced representation
% of $\lambda\in \mathfrak X$\\[2ex]
\pause\item
$\mathfrak X^+\pdef 
\left\{\lambda\in \mathfrak X\left| 
 \begin{array}{ll}\exists L\text{ simple representation of }G\\
\text{with }\op{Hom}_B(L,k_\lambda)\neq 0
\end{array}\right\}\right.
$
\end{itemize}

% \vspace{4ex}
% Beispiel:
% \begin{itemize} \item
%  $G=\op{GL}(r;k)\supset B$ obere Dreiecksmatrizen
% \end{itemize}
\end{frame}




\begin{frame}
 \begin{picture}(300,150)(10,-90) 
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\put(47, 77){$\mathfrak X^+$ dominant weights}
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\pause %4
\put(47,-40){$\mathfrak X^+\sira\;\;\{\text{simple
 representations
of }G\}$}
\put(49,-55){$\lambda\;\;\mapsto \;\;L(\lambda)
 $ 
the simple representation $L$ with}
\put(100,-70){highest weight $\lambda$ alias $\op{Hom}_B(L,k_\lambda)\neq 0$}

% \pause
%  \put(49,-95){$\nabla (\lambda) $ described by the Weyl
%    character formula }
% \pause
% \put(49,-110){For $\operatorname{char} k =0$ we have $L (\lambda)
%   = \nabla (\lambda)$ 





\end{picture}
\end{frame}





% \end{frame}
%----------------------------------------------------------------------------
%--------------------------------------------------------------------



\begin{frame}
   Modular category $\mathcal O$
is a subquotient of the category
of rational representations of a
    connected  reductive algebraic group 
{\color{Red}in positive characteristic},
   that behaves very much like usual category $\mathcal O$.
       
    %  I invented it as a baby example to study typical problems of modular
%      representation theory in a particularily easy case.

  \end{frame}




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 affine Weyl group $\mathcal W \pdef W \ltimes \langle R \rangle$}


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\put(0,-55){$\rho$ half the sum of positive roots}

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 \pause %5

\put(0,-90){{\bf Linkage principle:} The representation category decomposes}
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\put(0,-140){Concentrate on principal block 
$G\op{-Prin}\pdef
 G\op{-Modf}_{(\mathcal
  W\cdot_p 0)}$}




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\begin{picture}(300,150)(10,-90) 




\put(0,-40){}


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\put(10,-55){The highest weights of simples of $G\op{-Prin}$}






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\put(0,-66){\circle{7}}\put(0,-66){\circle{9}}
\put(10,-70){Take $\op{st}\pdef (p-1)\rho$ the Steinberg point}

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\put(0,-81){\circle{8}}\put(0,-81){\circle*{4}}
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  in $\mathcal W\cdot_p0$}


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\put(10,-100){The other highest weights of simples of $\mathcal A\pdef G\op{-Prin}_{\leq \op{st} + \lambda}$}
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%\put(190,70){\circle{7}}
 

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G\op{-Prin}_{\not\geq \op{st} + w_\circ \lambda}$}
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\put(130,-135){ $\mathcal O\pdef \mathcal A/
\mathcal K$}



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 \put(0,-40){Modular category $\mathcal O$ is a subquotient
 of the category of} 
\put(0,-55){rational representations of a    connected  reductive} 
\put(0,-70){algebraic group in positive characteristic,
   {\color{Red}that behaves}}
\put(0,-85){\color{Red}very much like usual category $\mathcal O$:}
\put(0,-105){(1) Simple objects are parametrized by the finite Weyl group;}
\put(0,-120){(2) It's a highest weight category ($\nabla$-objects, 
BGG-reciprocity);}
\pause
\put(0,-140){Modular $\mathcal O$ is relevant, since
$[\nabla(\nu):\bar L(\mu)]=[\op{ind}_B^Gk_\nu:L(\mu)]$}
%    \item 
%    \end{itemize}
    %  I invented it as a baby example to study typical problems of modular
%      representation theory in a particularily easy case.
\end{picture}
  \end{frame}


  \begin{frame}
 It is even much more similar to usual category $\mathcal O$:
 \begin{itemize}
 \item It admits wall crossing functors with analogous effect on standard
  objects;\pause
\item It admits an exact functor $\mathbb V: \mathcal O\ra C\op{-Mod}$
fully faithful on injectives. 
\item
Here $C$ is the nilfibre algebra 
(formerly called coinvariant algebra) 
with coefficients in $k$, so
\begin{center}
  $C\pdef S/S(S^+)^W\text{ for } S\pdef\op{Sym}(\op{Hom}_\DZ(\mathfrak X,k));$
\end{center}
\pause
\item
The images of the indecomposable injectives are the indecomposable
summands occurring in $C$-modules of the form
$$C\otimes_{C^r} C\otimes_{C^s}\ldots\otimes_{C^t}( C/C^+)$$
for arbitrary simple reflections $r,s,\ldots,t\in W$. 
 \end{itemize}
  \end{frame}

  \begin{frame}
Significance of modular category $\mathcal O$:
    \begin{itemize}
    \item Lusztig's conjecture would imply that
length of the Krull-Schmidt decomposition of these tensor products 
is independent of the characteristic
of $k$.\pause
\item
Consider $G=\op{GL}(n;k)$
with $C$ the quotient of the polynomial ring $k[X_1,\ldots,X_n]$
by the ideal generated by the symmetric polynomials of positive degree.  
Williamson showed that for $n=4m+7$
the number of summands will not yet be stable for any prime 
factor $p$ of the Fibonacci numbers $F_m, F_{m+1}$.\pause
\item
In this way one obtains counterexamples to Lusztig's conjecture
for $\op{GL}(n;k)$ with $\op{char} k$ not bounded by any multiple of $n$.
This is sometimes cited as {\bf Williamson torsion explosion}.
    \end{itemize}
  \end{frame}

  \begin{frame}
    Some names and history:
    \begin{itemize}
    \item I made up modular category $\mathcal O$ in 2000 in the hope to
          prove Lusztig's conjecture in a particularily easy case.\pause
     \item
Williamson used it to quite drastically disprove  Lusztig's conjecture
 last year.\pause
\item
There are also manifold 
relations to suitable categories of sheaves with modular coefficients on
the flag manifold and the dual flag manifold. State of the art is
 work of this year by Achard and Riche. 
    \end{itemize}
\pause
\begin{center}
  \Huge THANK YOU!
\end{center}

  \end{frame}


\end{document}














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 the alcoves of  $x \cdot_p 0,$ $y \cdot_p 0$ put }
\put(0,-55){ $ L_{A}\pdef L(x \cdot_p 0)$, $\nabla_B\pdef \nabla(y \cdot_p 0)$
and $ m_{B,A}\pdef P_{w_\circ y,w_\circ x}$}
\pause\put(0,-80){{\bf Lusztig conjecture:} For $A$ in the fundamental box
  should have}

\put(70,-105){$
 [L _A]= 
\sum_{B}  (-1)^{d(A,B)}  m_{B,A} (1)\; [\nabla_B] 
$}

\pause
\put(0,-135){$d(A,B)$  number of reflecting hyperplanes separating $A$ from $B$}
\pause
\put(0,-150){But what are the Kazhdan-Lusztig polynomials $m_{B,A}\in\DZ[v,v^{-1}]$?}


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over the set
   
 }
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%=\sum_B m_{B,A}B


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\end{document}

\pause %5

\end{frame}

\begin{frame}
Lusztig-Vermutung: Gegeben $x\in \mathcal W^+$ gilt
\begin{eqnarray*}
 [L (x \cdot_p 0)]= 
\sum_{y\in \mathcal W^+}  (-1)^{l(x)+l(y)}  P_{yw_\circ,xw_\circ} (1)\; [\nabla (y \cdot_p 0)] 
\end{eqnarray*}
 \begin{itemize}

\item  $\mathcal W^+\pdef\{x\mid 0\leq \langle x \cdot_p 0, \alpha^\vee
  \rangle \leq (p+1)(p+1-h)\forall \alpha\in R^+\}$
\item   $w_\circ \in W$ das l"angste Element
\item   $h=\op{max}\{\langle\rho,\alpha^\vee\rangle+1\mid \alpha\in R^+\}$ die
  Coxeterzahl


 \end{itemize}
\end{frame}
% \begin{frame}%7
%   \begin{itemize}
%   \item Affine Weylgruppe $\mathcal W = W \ltimes \langle R \rangle$ 
% \item Operation
%     als affine Spiegelungsgruppe  $\mathfrak X$
% \item 
%     Neue Operation $w \cdot_p \lambda \pdef p w p^{-1} (\lambda + \rho)
%     -\rho$
% \item Neue Spiegelebenen:\\ $\{\lambda\in\mathfrak X\otimes_\DZ\DR\mid \langle\lambda
%   +\rho,\alpha^\vee\rangle= pn\}$ f"ur $\alpha\in R$ und $n\in \DZ$

%    \end{itemize}
% \end{frame}
















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% \end{picture}
% \end{frame}





\begin{frame}
\begin{displaymath}
 \begin{array}{ccc}
\left\{ \begin{array}{c}\text{Einfache}\\
\text{Darstellungen}\\
\text{von }G \end{array} \right\}
&\overset{\sim}{\rightarrow} & \hspace{5ex}\mathfrak X^+ 
% \left\{ \begin{array}{c}\lambda : B \rightarrow
%     k^\times\text{ mit}\\
%  \nabla (\lambda) :=\operatorname{ind}^G_B k_\lambda \neq 0 \end{array} \right\}
% \\[5ex]
% L &\mapsto & \left( \begin{array}{c} \text{Das $\lambda$ mit }\\ \operatorname{Hom}_B (k_\lambda, L) \neq 0 \end{array}\right)
\\[3ex]
L (\lambda):= \operatorname{soc} \nabla(\lambda) & \mapsfrom & \hspace{5ex}\lambda
\end{array}
\end{displaymath}
\begin{itemize}
\item\pause $\nabla (\lambda) $ wird beschrieben durch Weyl'sche
  Charakter\-formel. 
\item\pause 
  Die Matrix der Jordan-H"older-Multiplizit"aten $[\nabla (\lambda):L(\mu)]$
  ist invertierbar.
\item\pause
Aus den $[\nabla (\lambda):L(\mu)]$ lassen sich die irreduziblen Charaktere berechnen!
\end{itemize}

\end{frame}






\begin{frame}
\begin{itemize}
\item 
Im Fall $G=\op{SL}(2;k)$ ist $B=\left\{{*\;0 \choose *\;*}\right\}$ eine
Borel'sche. \pause
\item F"ur $\varepsilon:\left\{{t\;0 \choose *\;*}\right\} \mapsto t$ 
 habe  $\nabla(n\varepsilon)\cong k[X,Y]^{(n)}$.\pause
\end{itemize}
Falls $\op{char}k=p>0$:
\begin{itemize}
\item $L(n\varepsilon)=\nabla(n\varepsilon)$ falls $n<p$.\pause
\item $L(p\varepsilon)\cong kX^p+kY^p\subsetneq \nabla(p\varepsilon)$. \pause
\item
Habe kurze exakte Sequenz
% $$kX^p+kY^p\hra  k[X,Y]^{(p)}\sra  k[X,Y]^{(p-2)}$$\pause
% $$L(p\varepsilon)\hra \nabla(p\varepsilon)\sra L((p-2)\varepsilon)$$\pause
$$\begin{array}{ccccc}
  kX^p+kY^p&\hra & k[X,Y]^{(p)}&\sra & k[X,Y]^{(p-2)}\pause\\[2mm]
L(p\varepsilon)&\hra &\nabla(p\varepsilon)&\sra& L((p-2)\varepsilon)\end{array}$$\pause

$[\nabla(p\varepsilon):L(\mu)]=1$ f"ur $\mu=p\varepsilon$ und $\mu= (p-2)\varepsilon$\\
$[\nabla(p\varepsilon):L(\mu)]=0$ sonst\end{itemize}

\end{frame}









\begin{frame}
\begin{displaymath}
 \begin{array}{ccc}
\left\{ \begin{array}{c}\text{Einfache}\\
\text{Darstellungen}\\
\text{von }G \end{array} \right\}
&\overset{\sim}{\rightarrow} & \mathfrak X^+ 
% \left\{ \begin{array}{c}\lambda : B \rightarrow
%     k^\times\text{ mit}\\
%  \nabla (\lambda) :=\operatorname{ind}^G_B k_\lambda \neq 0 \end{array} \right\}
\\[5ex]
L &\mapsto & \left( \begin{array}{c} \text{Das $\lambda$ mit }\\ \operatorname{Hom}_B (k_\lambda, L) \neq 0 \end{array}\right)\\[3ex]
L (\lambda):= \operatorname{soc} \nabla(\lambda) & \mapsfrom & \lambda
\end{array}
\end{displaymath}
\begin{itemize}
\item\pause $\nabla (\lambda) $ wird beschrieben durch Weyl'sche
  Charakter\-formel. \pause F"ur $\operatorname{char} k =0$ gilt $L (\lambda) = \nabla (\lambda)$.
\item\pause 
  Die Matrix der Jordan-H"older-Multiplizit"aten $[\nabla (\lambda):L(\mu)]$
  ist invertierbar.
\item\pause
Aus den $[\nabla (\lambda):L(\mu)]$ lassen sich die irreduziblen Charaktere berechnen!
\end{itemize}

\end{frame}


\begin{frame}
Lusztig-Vermutung: Gegeben $x\in \mathcal W^+$ gilt
\begin{eqnarray*}
 [L (x \cdot_p 0)]= 
\sum_{y\in \mathcal W^+}  (-1)^{l(x)+l(y)}  P_{yw_\circ,xw_\circ} (1)\; [\nabla (y \cdot_p 0)] 
\end{eqnarray*}
 \begin{itemize}

\item  $\mathcal W^+\pdef\{x\mid 0\leq \langle x \cdot_p 0, \alpha^\vee
  \rangle \leq (p+1)(p+1-h)\forall \alpha\in R^+\}$
\item   $w_\circ \in W$ das l"angste Element
\item   $h=\op{max}\{\langle\rho,\alpha^\vee\rangle+1\mid \alpha\in R^+\}$ die
  Coxeterzahl


 \end{itemize}
\end{frame}
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\frametitle{Bild H}%Bild aus Buch

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\end{document}




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