Noncommutative Counterparts of the Springer Resolution
Roman Bezrukavnikov
%{Sheaves may be coherent, but what about you trying %to say something to that girl?} % Springer resolution of the set of nilpotent elements in a semisimple Lie algebra plays a central role in geometric representation theory. A new structure on this variety has arisen in several representation theoretic constructions, such as the (local) geometric Langlands duality and modular representation theory. It is also related to some algebro-geometric problems, such as the derived equivalence conjecture and description of T.~Bridgeland's space of stability conditions. The structure can be described as a noncommutative counterpart of the resolution, or as a $t$-structure on the derived category of the resolution. The intriguing %(in our view) fact that the same $t$-structure appears in these seemingly disparate subjects has strong technical consequences for modular representation theory. % (Lusztig's conjectures). %And it allows to Koszul Koszul??
math0604445.tex
\documentclass{amsart}[12pt,epsf,amscd,amstex,amssymb]

%%%%%%%%%%%%%%%%
%%%%%%%%%% Use this for your addresse(s)
%%%%%%%%%% they will appear at the end of your article
%%%%%%%%%% use one \contact[]{} for each author

%\contact[bezrukav@math.mit.edu]{77 Massachusetts ave, MIT office
%2-284, Cambridge MA 02139 USA}

%\contact[e-mail address]{2. Author(s)'s address}
%\contact[e-mail address]{3. Author(s)'s address}

%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Commands and environments
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%\numberwithin{equation}{section}


\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{conjecture}[theorem]{Conjecture}

\newtheorem*{coro}{Corollary} %%%% for unnumbered statements


\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{example}[theorem]{Example}


\newcommand{\Ext}{\operatorname{Ext}}
\newcommand{\Hom}{\operatorname{Hom}}
\newcommand{\Ker}{\operatorname{Ker}}
\newcommand{\End}{\operatorname{End}}
\newcommand{\Tor}{\operatorname{Tor}}
\newcommand{\Alc}{\operatorname{Alc}}
\newcommand{\rmod}{\operatorname{mod}}

\newcommand{\tii}{\widetilde}

\newcommand{\kk}{{\bf k}}


\newcommand{\bI}{{\bf I}}
\newcommand{\bLG}{{\bf ^LG}}



\newcommand{\GO}{^L{\bf G}_O}

\newcommand{\bp}{{\bf p}}

\newcommand{\cal}{\mathcal}

\newcommand{\Qlbar}{{\overline {{\mathbb Q}_l} }}

\newcommand{\A}{{\mathcal A}}

\renewcommand{\AA}{{\mathcal A}}

\newcommand{\BB}{{\mathcal B}}
\newcommand{\DD}{{\mathcal D}}
\newcommand{\EE}{{\mathcal E}}
\newcommand{\MM}{{\mathcal M}}
\newcommand{\PP}{{\mathcal P}}
\renewcommand{\SS}{{\mathcal S}}
\newcommand{\VV}{{\mathcal V}}

\newcommand{\C}{{\mathcal C}}

\newcommand{\OO}{{\mathcal O}}
\newcommand{\E}{{\mathcal E}}

\newcommand{\N}{{\mathcal N}}
\newcommand{\Nt}{{\widetilde{\N}}}
\renewcommand{\gg}{{\mathfrak g}}
\newcommand{\hh}{{\mathfrak h}}
\newcommand{\pp}{{\mathfrak p}}
\newcommand{\gt}{{\tii{\mathfrak g}}}
\newcommand{\At}{\tii{A}}
\newcommand{\de}{\partial}

\newcommand{\tw}{ {}^{(1)} }               %Frobenius twist

\newcommand{\imbed}{\hookrightarrow}
\newcommand{\iso}{{\tii \longrightarrow}}
\newcommand{\isol}{{\tii \longleftarrow}}
\newcommand{\To}{\longrightarrow}

\newcommand{\Gm}{{{\mathbb G}_m}}

\newcommand{\al}{\alpha}

\newcommand{\la}{\lambda}
\newcommand{\La}{\Lambda}
\newcommand{\Ga}{\Gamma}

\newcommand{\Lotimes}{\overset{\rm L}{\otimes}}

\newcommand{\oplusl}{\bigoplus\limits}
\newcommand{\cupl}{\bigcup\limits}
\newcommand{\suml}{\sum\limits}


\newcommand{\bA}{{\bf A}}
\newcommand{\bS}{{\bf S}}
\newcommand{\bY}{{\bf Y}}
\newcommand{\bpr}{{\bf pr}}

%%%%%%%%%%% from prop

\newcommand{\Gr}{{ {\cal G}r}}
\newcommand{\Fl}{{ {\cal F}\ell}}
\newcommand{\Flf}{{\cal{B}}}



\newcommand{\bl}{<<} %{{\bf \langle}}
\newcommand{\br}{>>} % {{\bf \rangle}}

%\newcommand{\To}{\longrightarrow}
%\newcommand{\iso}{{\widetilde \longrightarrow}}
%\newcommand{\isol}{{\widetilde \longleftarrow}}
%\newcommand{\imbed}{\hookrightarrow}
\newcommand{\sur}{\twoheadrightarrow}
\newcommand{\<}{\langle}
\renewcommand{\>}{\rangle}

\newsymbol\ltimes 226E
%\newsymbol\rtimes 226F
%\newsymbol\boxtimes 1202

\def\square{\hbox{\vrule\vbox{\hrule\phantom{o}\hrule}\vrule}}
\newcommand{\epf}{\square}

\newcommand{\Vbar}{{\bar V}}
\newcommand{\Wbar}{{\bar W}}
\newcommand{\Hbar}{{\bar H}}

\renewcommand{\P}{{\mathcal P}}
\newcommand{\Pmixed}{\P^{mix}}
\newcommand{\Pmixedf}{\P^{mix}_f}
\newcommand{\fP}{{^f{\mathcal P}}}
\newcommand{\Pf}{{{\mathcal P}^f}}
\newcommand{\fPf}{{^f{\mathcal P}^f}}

\newcommand{\Ntil}{{\tilde{\mathcal N}}}
\newcommand{\Ntt}{{\tilde{\tilde{\mathcal N}}}}

\newcommand{\wtil}{{\tilde w}}

\newcommand{\St}{{St}}
\newcommand{\Stt}{{\widetilde{St}}}


\newcommand{\fW}{{^f W}}
\newcommand{\Wf}{{W^f}}
\newcommand{\fWf}{{^fW^f}}


\newcommand{\ho}{{{\frak h}^0}}
\newcommand{\h}{{\frak h}}

\renewcommand{\t}{{\frak t}}
\renewcommand{\b}{{\frak b}}
\renewcommand{\u}{{\frak u}}
\renewcommand{\c}{{\underline c}}

\newcommand{\M}{{\mathcal M}}
\newcommand{\Mgo}{{\frak M}}




\newcommand{\B}{\Flf}


\newcommand{\gcheck}{{\g\check{\ }}}
\newcommand{\gchecktil}{{\tilde\g\check{\ }}}

\newcommand{\gj}{{\frak j}}
\newcommand{\gp}{{\frak p}}
%\newcommand{\g}{{\frak g}}
%\newcommand{\p}{{\frak p}}
\newcommand{\Lg}{{^L{\frak g}}}
\newcommand{\LG}{{^L{G}}}
\newcommand{\LB}{{^L{B}}}

\newcommand{\Lgtil}{{^L\tilde{\frak g}}}

\newcommand{\pit}{{\widetilde \pi}}

\renewcommand{\L}{{\mathcal L}}
\newcommand{\LL}{{\mathcal L}}
\renewcommand{\O}{{\mathcal O}}
\newcommand{\Ohat}{{\hat\O}}
\newcommand{\Olambda}{{\O(\lambda)}}
\newcommand{\R}{{\mathcal R}}
\newcommand{\F}{{\mathcal F}}
\newcommand{\G}{{\mathcal G}}

\newcommand{\ff}{{\mathfrak f}}


\newcommand{\Coheq}{Coh^\LG}

\newcommand{\cons}{{\underline{{\Bbb Q}_l}}}

\newcommand{\J}{{\mathcal J}}
\renewcommand{\H}{{\mathcal H}}


\newcommand{\bq}{{\bf q}}


\newcommand{\unb}{{\underline b}}

\newcommand{\bbF}{{\mathbb F}}
\newcommand{\bFbar}{{\bar{\Bbb F}}}
\newcommand{\Gcheck}{{G\check{\ }}}

\newcommand{\bG}{{\bf G}}

\newcommand{\Feq}{F}

\newcommand{\Zet}{{\mathbb Z}}
\newcommand{\Pn}{{\mathbb P}^n}
\newcommand{\Aone}{{\mathbb A}^1}
\newcommand{\Atwo}{{\mathbb A}^2}
\newcommand{\Pone}{{\mathbb P}^1}

\newcommand{\Fq}{{\mathbb F}_q}

\newcommand{\bF}{{\mathbb F}}


\newcommand{\bbA}{{\mathbb A}}

\newcommand{\Ql}{{  {\mathbb Q}_l       }}
\newcommand{\Qlb}{{ \bar{\mathbb Q}_l   }}
\newcommand{\Qu}{{\mathbb Q}}
\newcommand{\Ce}{{\mathbb C}}
\newcommand{\RE}{{\mathbb R}}
\newcommand{\bbS}{{\mathbb S}}


\newcommand{\D}{{\mathfrak D}}

\newcommand{\bu}{\bullet}

\renewcommand{\k}{{\bf k}}


\newcommand{\nab}{\nabla}
\newcommand{\nabtil}{\tilde\nabla}
\newcommand{\del}{\Delta}

\newcommand{\codim}{{\rm co}\dim}


\title[Noncommutative
 Springer Resolution]{Noncommutative
Counterparts of the Springer Resolution}
%[My former wife's friends]{My former wife's friends and other

\author[Roman Bezrukavnikov]{Roman Bezrukavnikov}
%\thanks{The author
%is partially supported by NSF grant DMS-0505466 and Sloan Foundation grant
%0965-300-L216}}

%\author[Like someone else was once]{I want to be a person
%like someone else was once\thanks{The author is
% grateful, and greasy tears of gratitude
%stream down his plumpy cheeks.
%\newline The author was supported by an unidentified
%spiritual force, according to some sources.}}

\begin{document}



\begin{abstract}
%{\bf Sheaves may be coherent, but what about you trying
%to say something to that girl?}
%
Springer resolution of the set of nilpotent elements in a semisimple
Lie algebra plays a central role in geometric representation theory.
A new  structure on this variety has arisen in
several  representation theoretic constructions,
such as the (local) geometric Langlands duality and  modular
representation theory. It is also related to some algebro-geometric
problems, such as the derived equivalence conjecture and description of
 T.~Bridgeland's space of stability
conditions. The structure can be described as a noncommutative
counterpart of the resolution, or as a $t$-structure on the derived
category of the resolution.
 The intriguing %(in our view)
 fact that the same $t$-structure
appears in these seemingly disparate subjects has strong technical
consequences for  modular representation theory.
% (Lusztig's conjectures).
%And it allows to Koszul Koszul??
\end{abstract}

%\begin{classification}
%Primary: 17B50, %modular Lie algebras
%18F99, %categories and geometry
 %20G15, Linear groups over arbitrary fields
%20G05. %representation theory
% Secondary: 20G42, %quantum groups
%22E67. %loop groups
%\end{classification}

%\begin{classification}
%Primary -- blank; Secondary -- some glimpses of hope.
%\end{classification}

%\begin{keywords}
%Derived categories of sheaves and modules, modular Lie algebras,
%local geometric Langlands duality.
%\end{keywords}

%\begin{keywords}
%beep. beep. beep-beep-beep. [censored]
%\end{keywords}


\maketitle

\tableofcontents

\section{Introduction}

Springer resolution
 of the variety of nilpotent elements in a
semi-simple Lie algebra
%, given by the moment map from the cotangent bundle
%of a generalized flag variety to the nilpotent set,
 is
ubiquitous in geometric representation theory. In this article we
show that, besides of this well-known resolution of singularities,
the  variety of nilpotents, as well as some other closely related varieties,
admits a particular {\em noncommutative resolution of singularities},
which arises in different representation theoretic and
algebro-geometric constructions. Here by a noncommutative resolution
of a singular variety $Y$ we mean, following, e.g., \cite{vdB}, a
coherent sheaf of associative $\OO_Y$ algebras satisfying certain
natural conditions, and defined up to a Morita equivalence.

 The constructions are related to such subjects as:
the (local) geometric Langlands duality program
and categorification of representation theory of affine Hecke algebras,
  representation theory of modular Lie algebras and quantum enveloping
algebras at roots of unity, Bridgeland's theory of stability conditions
on triangulated categories, and categorical MacKay
correspondence and generalizations.


\medskip



Let  $G$ be a semi-simple adjoint algebraic group, $\gg$ be its Lie
algebra and $\N\subset \gg$ be the variety of nilpotent elements.
Let $\BB$ be the variety of Borel subalgebras in $\gg$, also known
as the flag variety of $G$, and $\Nt=T^*(\BB)$ be the cotangent
bundle to $\BB$. The Springer resolution is the moment map
$\pi:\Nt\to \N$.

 Our noncommutative resolution $A$ of
$\N$ comes with an equivalence
 between the derived category $D(A)$ of modules over $A$ and the derived
 category $D(\Nt)$ of coherent sheaves on  $\Nt$.
%  (this provides an example for the "noncommutative derived equivalence" conjecture of
% \cite{vdB}).
Thus $A$ is determined uniquely up to Morita
equivalence
 by the
 $t$-structure on $D(\Nt)$ induced by the equivalence, i.e., by the
 image of the subcategory of $A$ modules in $D(A)$ under the
 equivalence. We will call this $t$-structure the {\em
 exotic $t$-structure} and objects of its heart {\em exotic sheaves}.
 Thus an exotic sheaf is a complex of coherent sheaves on $\Nt$ which corresponds to
 an $A$-module under the equivalence $D(A)\cong D(\Nt)$.

 A closely related  data first appeared in \cite{AB}, which can be
 considered as a contribution to a local version of the geometric Langlands duality
 program \cite{BD},  \cite{GaiICM}, \cite{Edik}.
 A typical result of geometric Langlands duality
   is an equivalence between
 some derived category of constructible sheaves on a variety related
 to $\LG$ bundles on a curve $C$ and derived category of coherent sheaves on a variety
 related to $G$ local systems on $C$; here $G$ and $\LG$ are reductive groups,
  which are dual in the sense of Langlands. In the
 local version of the theory the curve $C$ is a punctured formal
 disc $\mathbb D$. The role of the moduli stack of $\LG$  bundles is played by a
homogeneous space for the group $\LG((t))$, where $\LG((t))$ stands
for the group  of maps from $\mathbb D$ to $\LG$ (also
known as the formal loop %, or arch,
 group).
  An example of such a homogeneous space is the {\em affine flag variety} $\Fl$ of $\LG$.
   For an appropriate choice of the category of constructible
 sheaves, the variety related to $G$ local systems turns out to be
 $\Nt$, or rather the quotient stack $\Nt/G$ of $\Nt$ by the natural action of $G$.
  An equivalence between the derived category of $G$-equivariant coherent sheaves on $\Nt$
  and a certain triangulated category of constructible sheaves on
  $\Fl$ is proved in \cite{AB}.
The image of the subcategory of perverse sheaves on $\Fl$ under this
equivalence turns out to consist of {\em equivariant exotic
sheaves}, which are closely related to exotic sheaves (see section
\ref{mutasect} below).

%  It allows one to
% transport the $t$-structure of perverse sheaves on the $\LG$ side
% to coherent sheaves on $\Nt$, and the result coincides with a version of the
% exotic $t$-structure (see section \ref{ } for more details).


Another construction leading to exotic sheaves is related to modular
representation theory.

 In the second half of the
20th century various geometric methods for representation theory of
semi-simple Lie algebras over characteristic zero fields have been
developed. One of the culminating points is the Localization Theorem
% of Beilinson
%and Bernstein
\cite{BeBe}, %and Brylinsky and Kashiwara
\cite{BrylKa}, motivated by a conjecture by Kazhdan and Lusztig,
which provides an equivalence between the category of modules over a
semi-simple Lie algebra $\gg$ with a fixed (integral regular)
central character and the category of $D$-modules on the flag
variety $\BB$.
 In the paper
\cite{BMR}I, motivated by Lusztig's extension \cite{LKthII}II of
Kazhdan-Lusztig conjectures to the modular setting, we provide a
similar result for semi-simple Lie algebras over algebraically
closed fields of positive characteristic. More precisely, we
establish
 a derived localization theorem, which is an equivalence between the derived
category of appropriately defined $D$-modules (called crystalline,
or PD $D$-modules) on a flag variety and the derived category of Lie
algebra  modules, where a part of the center, the so-called
Harish-Chandra center, acts by a fixed character.

Furthermore, in the case of positive characteristic %coefficients
there is a close relationship between crystalline $D$-modules on a
smooth variety $X$ and coherent sheaves on the cotangent space
$T^*X$ \cite{BMR}I, \cite{OV}.
%For example, crystalline $D$-modules on the affine line
%$\bbA^1$ are just modules over the Weyl algebra, which is generated
%by two elements $x,y$ subject to the relation $xy-yx=1$. If we work
%over a characteristic $p$ field $\kk$, then $x^p$ and $y^p$ lie in
%the center of this algebra; moreover, the Weyl algebra turns out to
%be an Azumaya algebra over $\kk[x^p,y^p]$. A similar fact holds true
%for any smooth variety over $\kk$.
The algebra of crystalline differential operators has a huge center
provided by invariant polynomials of the $p$-curvature of a $D$-module.
This allows to view the differential operators
 as a sheaf of algebras on %the Frobenius twist of
 the cotangent bundle.  This algebra turns out to be an Azumaya algebra.
In the case of the flag variety this Azumaya algebra splits on the
formal neighborhood of each Springer fiber. Thus the derived
localization theorem yields a full embedding from the category of
finite dimensional $\gg$-modules with a fixed (integral regular)
action of the Harish-Chandra center into the derived category of
coherent sheaves on $\Nt$. It turns out that the image of this
embedding consists precisely of exotic sheaves with proper support.
A similar relation is expected between exotic sheaves over a field
of characteristic zero and representations of the quantum Kac--De
Concini  enveloping algebra at a root of unity \cite{KDC}, and also
with some class of $\widehat{\Lg}$ modules at the critical level
 (cf. \cite{Kobi} and \cite{Edik_Denis} respectively); here
$\widehat{\Lg}$ stands for the affine Kac-Moody algebra
corresponding to the Langlands dual algebra $\Lg$.

Thus exotic sheaves are related, on the one hand, to perverse sheaves on
the affine flag variety for the dual group, and on the other hand,
to modular Lie algebra representations.
Comparison of these  two connections allows one to
apply the known deep results about weights of Frobenius acting on Ext's
between irreducible perverse sheaves to numerical questions about
modular representations, thereby providing a strategy for a proof of
Lusztig's conjectures from \cite{LKthII}II.
 The conjectures relate the classes of irreducible
$\gg$-modules %with a fixed central character
  to elements of the {\em
canonical basis} in the Borel-Moore homology of a Springer fiber;
thus our work provides a categorification of the canonical bases in
(co)homology of Springer fibers. See also Remark \ref{cohotilt}
 for an application
to representations of quantum groups.
% In fact, most of the results have
%been obtained by unraveling formulas in Lusztig's Theorems and
%Conjectures.


I also would like to point out some parallels between exotic sheaves
and objects
 arising in the work of algebraic geometers studying
derived categories of coherent sheaves on algebraic varieties.
Exotic sheaves can be described in terms of a certain action of the
affine braid group $B_{aff}$ of $\LG$ on $D(\Nt)$. This description
can be reformulated in terms
 a  map from the set of alcoves (connected components of
the complement to affine coroot hyperplanes in the dual space to the
Cartan algebra of $\gg$ over $\RE$) to the set of $t$-structures on
$D(\Nt)$. %, satisfying certain axioms.
 A similar data has been used by
Bridgeland in \cite{BrK3} to construct a component in
the space of stability conditions \cite{Brstab},
on the derived categories of coherent sheaves on certain varieties.
See also Examples \ref{flop}, \ref{subr} below.

The appearance of the affine braid group, which can be interpreted
as the fundamental group of the set of regular semisimple conjugacy
classes in the dual group $\LG(\Ce)$, suggests a possibility that
the structures described above admit a natural interpretation via
homological mirror duality, which would identify our derived
category of coherent sheaves with a certain Fukaya type category,
where the action of the affine braid group arises from monodromy of
some family over the space of  regular semisimple conjugacy classes
in $\LG(\Ce)$.




Another connection to algebraic geometry is provided by \cite{BeKal}
and \cite{Kal}. As has been noted above, the derived localization
theorem can be interpreted as a construction of a noncommutative
resolution of the nilpotent cone $\N$ using crystalline differential
operators in positive characteristic. It turns out that for more
general resolutions of singularities, which carry an algebraic
symplectic form, a non-commutative resolution can be constructed by
a similar procedure. The construction
 involves
quantizing the algebraic symplectic variety in characteristic
$p$, and relating modules
over the quantization to coherent sheaves.
It has been carried out in
\cite{BeKal}
 for crepant resolutions of  quotients
$V/\Gamma$, where $V$ is a vector space equipped with a symplectic
form, and $\Gamma$ is a finite subgroup in $Sp(V)$; this yields a
particular case of the so-called {\em categorical MacKay
correspondence}. The particular case when $\Gamma$ is the symmetric
group on $n$ letters acting on $V=(\Atwo)^n$ is related to representations
of the rational Cherednik algebra \cite{BFG}.
In Kaledin's work \cite{Kal}
 the construction is generalized to more general
symplectic resolutions of singularities.

\medskip

In the remainder of the text  we explain some of these contexts  (in
the order which is roughly inverse to the above) in some detail.



This is text is a mixture of an exposition of published results and
announcement of yet unpublished ones; statements for which no reference
is provided, and which are not well-known, are to appear in a future
publication.

%\medskip

{\bf{Notations and conventions}} Throughout the text we work over an
algebraically closed field $\kk$; when a semi-simple group $G$ is
involved, we assume that characteristic of $\kk$ is zero or exceeds
the Coxeter number of $G$.

For an algebraic variety $X$ we let $\O_X$ denote the structure
sheaf, and $D(X)=D^b(Coh_X)$ be the bounded derived category of
coherent sheaves on $X$. Given an action of an algebraic group $H$
on $X$ we write $Coh^H(X)$ for the category of $H$-equivariant
coherent sheaves; given a  coherent
sheaf of associative $\O_X$ algebras  we let $Coh(X,\A)$ be the
 category of sheaves of coherent $\A$ modules; if $\A$ is $H$-equivariant
for an algebraic group $H$ acting on $X$, we let $Coh^H(X,\A)$ be
the category of $H$-equivariant sheaves of coherent $\A$-modules.
We write $D(X)$, $D^H(X)$, $D(\A)$, $D^H(\A)$ for the bounded
derived category of $Coh(X)$, $Coh^H(X)$, $Coh(X,\A)$, $Coh^H(X,\A)$
respectively, and $K(X)$, $K^H(X)$, $K(\A)$, $K^H(\A)$ for the
corresponding Grothendieck groups. In particular, there notations
apply for an algebra $A$ finite over the center of finite type.

%For an associative ring $R$ we let $R^{op}$ be the opposite ring.
%
 The
functors of pull-back, push-forward etc. between categories of
sheaves are understood to be the derived functors.

\medskip

{\bf Acknowledgements} I thank my coauthors S.~Arkhipov,
V.~Ginzburg, D.~Kaledin, I.~Mirkovi\' c, V.~Ostrik and D.~Rumynin
for their contribution to the joint results, without which they
would have never been accomplished.
%, and also for the permission to quote these results.
%
The project described here was conceived during IAS Special Year in
Representation Theory (98/99) led by G.~Lusztig. Most of the results
have been obtained by unraveling the formulas in Lusztig's papers,
thus they owe their existence to him. I have learned Lusztig's
results and many other things from M.~Finkelberg. I have also
benefitted a lot from ideas of I.~Mirkovi\' c and his generosity in
sharing them. Conversation with many people were very helpful, the
incomplete list includes  A.~Beilinson, V.~Drinfeld, D.~Gaitsgory,
V.~Ginzburg, V.~Ostrik.
%
I am very grateful to all these people.
%
Finally, I thank M.~Finkelberg, D.~Kazhdan and I.~Mirkovi\' c for
reading a preliminary version of this text and making helpful
comments and suggestions.

\section{Noncommutative resolutions and braid group actions}
\label{se2}


\subsection{Braid group actions and noncommutative Springer resolution}


%\subsubsection{Generalities}
Though the motivation for the study of our main object comes from
applications to representation theory, we first describe it in the
language of  algebraic geometry. We briefly recall some ideas
 from \cite{BonOr}, \cite{Bridge}, \cite{vdB}.




Let $Z$ be a singular algebraic
 variety. We refer, e.g., to
\cite{vdB} for the notion of a {\em crepant} resolution;
 it is easy to see that resolutions $\pi$, $\pit$ described above
 are crepant.

  By a {\em noncommutative resolution}
%(NC resolution for short)
\cite{vdB}\footnote{The definition in {\em loc. cit.} is wider, we
use a version convenient for our exposition.}
one means a coherent torsion free sheaf $A$ of associative $\O_Z$
algebras, which is generically a sheaf of matrix algebras and
has finite homological dimension. There exists also a notion
of a noncommutative crepant resolution, see \cite{vdB}.
It has been conjectured in {\em loc. cit.} that any two crepant
 resolutions, commutative or not, are derived equivalent, in particular,
for any crepant resolution $X\to Z$ and any noncommutative
resolution $A$ of $Y$ we have an
equivalence $D(X)\cong D(A)$. % we will
%refer to this conjecture. % as {\em the NC derived equivalence
%conjecture}. It says, in particular, that for any crepant resolution
%$X\to Z$ and any noncommutative resolution $A$ of $Y$ we have an
%equivalence $D(X)\cong D(A)$.

%In this article we argue that the Grothendieck-Springer resolution
%maps $\pi$, $\pit'$ come together with  natural and useful
%noncommutative counterparts. More presicely,



\subsubsection{The set-up}
Notations $G$, $\gg$, $\BB$, $\pi:\Nt\to \N$ has been defined in the
Introduction. Recall that $\Nt=T^*(\BB)$ parametrizes pairs
$(\b,x)$, where $\b\in \BB$ is a Borel subalgebra, and $x$ is the
element in the nilpotent radical of $\b$. The Springer map
$\pi:\Nt\to \N$ is given by $\pi:(\b,x)\mapsto x$.
 It is embedded in the
{\em Grothendieck simultaneous resolution} $\pit: \gt\to \gg$, where
$\gt$ is the variety of pairs $(\b,x)$, $\b \in \BB$, $x\in \b$, and
$\pit:(\b,x)\mapsto x$. The variety $\gt$ is smooth, and the map
$\pit$ is proper and generically finite of degree $|W|$, where $W$
is the Weyl group. It factors as a composition of a resolution of
singularities $\pit': \gt\to \gg\times_{\hh/W} \hh$ and the finite
projection $\gg\times_{\hh/W} \hh\to \gg$; here $\hh$ is the Cartan
algebra of $\gg$.
 Let $\gg^{reg}\subset \gg$ denote the subspace of regular
(not necessarily semi-simple) elements, and $\gt^{reg}$ be the
preimage of $\gg^{reg}$ in $\gt$; then $\pit'$ induces an
isomorphism $\gt^{reg} \cong \gg^{reg}\times_{\hh/W}\hh$.

 Much of representation theory of $G$ or $\gg$ is in one way or
another related to the geometry of these spaces and maps.


\subsubsection{Affine braid group action}
For a characterization of our noncommutative resolution
we need to introduce some notations.

Let $\Lambda$ be the root lattice of $G$.
% and $Q\subset \La$ be the
%root lattice.
%Thus $\La=Pic^G(\BB)$ can be identified with the group
%of isomorphism classes of $G$-equivariant line bundles on $\BB$, and
%$Q= Pic^{G^{ad}}(\BB)$, where $G^{ad}=G/Center$ is the adjoint
%group.
 For $\la\in \La$ we will write $\O(\la)$ for the
corresponding $G$-equivariant line  bundle on $\BB$, and we set
$\F(\la)=\F\otimes _{\OO_{\BB}}\O(\la)$ if $\F\in D(X)$ for some $X$
mapping to $\BB$.

Let $W$ be the Weyl group, and set $W_{aff}=W\ltimes \La$.
%$W_{aff}'= W\ltimes \La$.
Then $W$, $W_{aff}$
%act naturally in the space $\hh^*$, and they are
%generated by reflections, thus they
 are Coxeter groups. Notice that
%the simple reflections in $W_{aff}$ are reflections at affine {\em
%coroot} hyperplanes, thus
$W_{aff}$ is the affine Weyl group of the
{\em Langlands dual} group $^LG$.
%; the group $W_{aff}'$ is the
%corresponding extended affine Weyl group.
% The following claim is
%offered to the reader as an exercise.

It was mentioned above that $\gt^{reg}\cong \gg\times_{\hh/W}\hh$;
thus $W$ acts on this space via its action on the second factor.
%\begin{lemma}
The formulas $\La\owns \la:\F \mapsto \F(\la)$, $W\owns w:\F \mapsto
w^*(\F)$ are easily shown to define an action\footnote{Throughout
the paper by an action of a group on a category I mean  {\em a weak
action}, i.e., a homomorphism to the group of isomorphism classes of
autoequivalences. I believe that in all the examples in this text a
finer structure can be established, though I have not studied this
question.} of $W_{aff}$ on the category of coherent sheaves on
$\gt^{reg}$.


The  characterization of our ``noncommutative Springer resolution''
%(as well as many previously known construction, e.g., various
%constructions of the Springer representation \cite{CG},
%\cite{Greenb}, \cite{BorhoMcp})
  relies on the possibility to extend
this action to a weaker structure on the whole of  $\gt$. To
describe this weaker structure recall that to each Coxeter group
one can associate an Artin braid group; let
$B_{aff}$ denote the group corresponding to $W_{aff}$.
 It admits a  topological interpretation, as the fundamental group
of the space of regular semi-simple conjugacy classes in the
universal cover of the dual group $\LG(\Ce)$.
For $w\in W_{aff}$ consider the minimal decomposition of $w$ as a product
of simple reflection, and take the product of corresponding generators
of $B_{aff}$. This product is well known to be independent on the choice
of the decomposition of $w$, thus we get a map $W_{aff} \to B_{aff}$
which is one-sided inverse  to the canonical surjection $B_{aff}
\to W_{aff}$. We denote this map by $w\mapsto \tii w$. The map is not a
homomorphism, however, we have  $\tii{uv}=\tii u \cdot \tii
v$ for any $u,v\in W_{aff}$ such that $\ell(uv)=\ell(u)+\ell(v)$,
where $\ell(w)$ denotes the length of the minimal decomposition of
$w$.
%
%$$B_{aff}=\pi_1(^LH^{reg}_{sc}(\Ce))=\pi_1((\hh_\Ce^* - \cupl_{\al\in
%\Delta_{aff}} H_\al)/W_{aff}),$$ where $^L H_{sc}$ is the Cartan
%group of the universal cover of the Langlands dual group,
%$^LH_{sc}^{reg}\subset ^LH_{sc}$ is the set of regular elements,
%$\Delta_{aff}$ denotes the set of affine coroots,  and for $\al\in
%\Delta_{aff}$ we let $H_\al\subset \hh^*$  be the corresponding
%affine hyperplane.
 Let $B_{aff}^+\subset B_{aff}$ be the sub-monoid
generated by $\tii w$, $w\in W_{aff}$.
% where $s$ runs over the simple
% as a product of simple reflections. The
%length functions admits a
%natural extension to $W_{aff}'$ and one can describe $B_{aff}'$ by
%generators $\tii w$, $w\in W_{aff}'$ and relations as above.


For a simple reflection $s_\al\in W$ let $\bS
_\al\subset \gt^2$ be the closure
of the graph of $s_\al$ acting on $\gt^{reg}$. We let $S_\al$
denote the intersection of $\bS_\al$ with $\Nt^2$.
Let $\bpr^\al_i:\bS\to \gt$, $pr^\al_i:S\to \Nt$, where $i=1,2$,
be the projections.

Let $\La^+\subset \La$ be the set of dominant weights in $\La$.

For a scheme $Y$ over $\gg$ we set $\tii Y=\Nt\times _\gg Y$, $\tii
\bY=\gt \times _\gg Y$.

\begin{theorem}\label{bract}
a) There exists an (obviously unique) action of $B_{aff}'$
on $D(\gt)$, $D(\Nt)$ such that
 for $\la \in \La^+\subset \La \subset W_{aff}'$
we have $\tii \la:\F \mapsto \F(\la)$
and for a simple reflection
$ s_\al\in W$ we have $\tii s_\al:\F \mapsto (\bpr_1^\al)^*(\bpr_2^\al)_*
\F$ (respectively,
 $\tii s_\al:\F \mapsto (pr_1^\al)^*(pr_2^\al)_*
\F$).

b) This action induces an action on $D(\tii \bY)$, $D(\tii Y)$ for
any scheme $Y$ over $\gg$ such that $\Tor_i^{\O_\gg}(\O_\gt,
\O_Y)=0$, respectively $\Tor_i^{\O_\gg}(\O_\Nt, \O_Y)=0$, for $i>0$.
\end{theorem}

{\em Comment on the proof.}
The Theorem can be deduced from material of either section
 \ref{se3} or \ref{se5} below. The most direct proof relies on
the result of \cite{Rina}.



\begin{remark}\label{tr_slice}
 An example of $Y$ satisfying the assumptions of the
Theorem is given by a transversal slice to a nilpotent orbit. In
particular, if $Y$ is a transversal slice to a subregular orbit,
then $\Nt\times _\gg Y$ is well known to be the minimal resolution
of a simple surface singularity. The affine braid group action in
this case coincides with the one constructed by Bridgeland in
\cite{BrK3}III.
\end{remark}

\begin{remark}\label{heck}
The induced action of $B_{aff}$ on the Grothendieck group
$K(\Nt)$ factors through $W_{aff}$. If one passes to the category
of sheaves equivariant with respect to the multiplicative group,
acting by dilations in the fibers of the projection
$\Nt\to \BB$, then the induced action factors through the {\em
affine Hecke algebra} $\H$, cf. discussion after Theorem \ref{equiv}.
Furthermore, this construction
yields an action of $\H$ on the Grothendieck group
$K(\pi^{-1}(e))$ for each $e\in \N$; these $\H$ modules are called
the standard $\H$-modules.
Thus the Theorem provides a {\em categorification} of the
standard modules for the affine Hecke algebra.
\end{remark}

%$K$-group is isomorphic to cohomology, recover Springer
%representation extended to the affine Weyl group. Below we will
%introduce an additional $\Gm$ action, and see that the braid group
%action on the $\Gm$-equivariant categories yields an action of the
%{\em affine Hecke algebra} on the $K$-group, which coincides with
%the standard construction of affine Hecke algebra action \cite{ }.
The next result, which plays an important technical role in the
proofs, is a categorical counterpart of the quadratic relation in
the affine Hecke algebra, see discussion after Theorem \ref{equiv}.
% (or rather its less precise
%form $\tii s_\al ^2 + C \tii s_\al -q=0$ for some constant $C$).
%\end{remark}

\begin{proposition}
For every simple reflection $s_\al\in W_{aff}$
% $\tii s_\alpha$ of $B_{aff}$
and every $\F\in \D$ we have a (canonical) isomorphism in the quotient
category $\D/\langle \F\rangle$
$$\tii s_\alpha(\F)\cong \tii s_\alpha^{-1}(\F)\mod \langle \F\rangle.$$
Here $\langle \F\rangle$ denotes the full triangulated subcategory
generated by $\F$.
\end{proposition}

%{\em Sketch of proof.} When $\al$ is a finite coroot, so that the action
%of $\tii s_\al$ is given by an explicit correspondence, the statement is
%easy to check directly. To finish the proof one observes that the element
%$\tii s_{\al_0}$, where $\al_0$ is the affine coroot,
% is conjugate to an element $\tii s_\al$ for some finite coroot $\al$.

\subsubsection{The $t$-structure and the noncommutative resolution}

We will describe certain noncommutative resolutions $A$, $\bA$ of
$\N$, $\gg\times_{\hh/W} \hh$ respectively, together with
equivalences $D(A)\cong D(\Nt)$, $D(\bA)\cong D(\gt)$, and show how
they appear in representation theory.  Such data is uniquely
determined by the $t$-structures on $D(\Nt)$, $D(\gt)$, which are
the images of the tautological $t$-structures on $D(A)$, $D(\bA)$.



\begin{definition} Let $D$ be a triangulated category equipped with an
action of $B_{aff}$.  A $t$-structure $(D^{<0}, \, D^{\geq 0})$ on
$D$ will be called {\em braid right exact} if any
%for any  $w\in W_{aff}$
%the element $\tii w\in B_{aff}$
$b \in B_{aff}^+$ sends $D^{<0}$ to $D^{<0}$.
\end{definition}


\begin{theorem}\label{tstr}
a) Let $X$ be either $\tii Y$ or $\tii \bY$, where $Y\to \gg$ is as
in  Theorem \ref{bract}.


The category $D(X)$ admits a  unique
$t$-structure which is

i) braid right exact, and

ii)  compatible with the standard $t$-structure on
the derived category of vector spaces under the functor of derived global
sections $R\Gamma$.


%b) For any $Y$ as in Theorem \ref{bract} the above $t$-structure
%induces a $t$-structure on $D(\tii \bY)$ (respectively,
%$D(\tii Y)$).

b) There exists a vector bundle $\E_X$ on $X$, such that the functor
$\F\mapsto R\Hom(\E, \F)$ is an equivalence between $D(X)$ and
$D(A_X)$, sending the $t$-structure described in (a) to the
tautological $t$-structure on $D(A_X)$;  here $A_X=\End(\E_X)^{op}$,
where the upper index denotes the opposite ring.

 Moreover, there exists a
vector bundle $\E=\E_\gt$ on $\gt$, such that for any $X$
we can take $\E_X$ to be the
pull-back of $\E$ to $X$.

\end{theorem}

%\begin{remark}
%Property a(ii) can be replaced by the following equivalent
%requirement: the structure sheaf $\O$ is a projective object in the
%heart of the $t$-structure.
%\end{remark}

\begin{remark}
It is clear from the definitions that if $X$ is smooth, then $A_X$
is a noncommutative resolution of $Y\times _\gg \N$ or $Y\times
_{\hh/W}\hh$. In particular, for $Y=\gg$ we get
 $A=A_\Nt$, $\bA = A_{\gt}$, which
are the promised noncommutative resolutions of $\N$,
$\gg\times_{\hh/W}\hh$.
\end{remark}


We will call the $t$-structures described in Theorem \ref{tstr} the
{\em exotic} $t$-structures, the objects of their heart will be
called exotic sheaves.
%\footnote{In \cite{Hum} terminology was
%slightly different}

% We will denote the pull-back of $\E$ to
%$\tii \bY$, $\tii Y$ by $\E_{ \bY}$, $\E_{ Y}$ respectively, and set
%$A_Y=\End(\E_{\tii Y})^{op}$, $\bA_Y=\End(\tii \bY)^{op}$. Thus
%$A_Y$, $\bA_Y$ is a noncommutative resolution of $Y\cap \N$,
%$Y\times _{\hh/W}\hh$ respectively, provided that $\Nt_Y$, $\gt_Y$ is
%an ordinary resolution.





\begin{example}\label{flop}
Let $G=SL(2)$, thus $\Nt$ is the total space of the line bundle
$\O(-2)$ on $\Pone$, and $\gt$ is the total space of the vector
bundle $\O_\Pone (-1)\oplus \O_\Pone (-1)$. In this case we can set
$\E\cong \O_\gt\oplus \O_\gt(1)$.

This $t$-structure on $D(\gt)$ appeared in Bridgeland's
 proof of the derived equivalence conjecture for varieties of
dimension three  \cite{Brflop}. More precisely, for a flop of
three-folds $X,X'\mapsto Y$ Bridgeland constructs some
noncommutative resolution
 of $Y$ which is derived equivalent to
both $X$ and $X'$. The simplest example of a three-fold flop is as
follows: $X=X'=\gt$, $Y=\gg\times _{\hh/W}\hh$ and the two maps
$X,\, X'\to Y$ are $\pit'$ and $\pit''=\iota \circ \pit'$, where
$\iota$ is an involution of $\gg\times_{\hh/W} \hh$ given by
$(x,h)\mapsto (x,-h)$. The $t$-structure on $D(\gt)$ given by
Bridgeland's construction applied to this flop turns out to coincide
with the $t$-structure provided by Theorem \ref{tstr}.

\end{example}

\begin{example}\label{subr}
Let $Y$ be a transversal slice to the subregular orbit.  Thus $Y$ is
isomorphic to the quotient $\Atwo/\Gamma$ for some finite subgroup
$\Gamma\subset SL(2)$. The fiber product $X=\Nt\times_\gg Y$ is the
minimal resolution of $Y$. It is well known that there exists a
natural equivalence $D(X)\cong D^\Gamma(\Atwo)$. The exotic
$t$-structure coincides with the one induced from the tautological
$t$-structure on $D^\Gamma (\Atwo)$. Thus $A_X$ is Morita equivalent
to the smash product algebra $\Gamma \# \O(\Atwo)$. This
$t$-structure appears also in \cite{BrK3}II.
\end{example}

%\bigskip

%A proof of the Theorem will be discussed later in section \ref{ }.

%\subsubsection{$DG$-version}\label{dgsusu}
%We would like to mention the following variation of Theorem
%\ref{tstr}.


%\begin{proposition}\label{dgprop} Let $Y$ be an affine finite type scheme mapping
%to $\g$. Assume that the image of $Y$ in $\g$ is (set theoretically)
%contained in $\N$.
%\end{proposition}

%\begin{example} $Y=Spec(\kk)$ mapping to zero. Koszul duality
%\end{example}
\subsubsection{Parabolic version}\label{parvar}
One can also consider the partial flag varieties $\PP=G/P$, where
$P\subset G$ is a parabolic subgroup; thus $\PP$ parameterizes
parabolic subalgebras $\pp\subset \gg$ of a given type. There exist
parabolic versions of the Grothendieck-Springer spaces:
$\gt_\PP=\{\pp\in \PP, \, x\in \pp \}$ and $\Nt_\PP=T^*(\PP)$. We
have a proper map $\pi_\PP:\gt \to \gt_\PP$, $(gB,x)\mapsto (gP,x)$.
Also, the projection $G/B\to G/P$
induces a closed embedding $\iota_\PP%=d\,pr_\PP^\BB
:\BB\times _\PP \Nt_\PP\imbed \Nt$; we let $pr_\BB^\PP$ denote the
projection $\BB\times _\PP \Nt_\PP \to \Nt_\PP$.

The following result easily follows from the results of
\cite{BMR}II.

\begin{theorem}
a) There exists a unique  $t$-structure on $D(\gt_\PP)$, whose heart
contains the image of exotic sheaves under the functor $R\pi_{\PP*}:
D(\gt) \to D(\gt_\PP)$. %This $t$-structure is nice.

b) There exists a unique  $t$-structure on $D(\Nt_\PP)$, such that
for any object $\F$ in its heart the object
$(\iota_{\PP*}(pr^\BB_\PP)^* \F)(\rho)$ is an exotic sheaf.
% This $t$-structure is nice.
\end{theorem}

One also has induced nice $t$-structures on $D(Y\times _\gg
\gt_\PP)$, $D(Y\times _\gg \Nt_\PP)$ for $Y$ satisfying a Tor
vanishing condition; we omit the details to save space.

\begin{example}
Let $G=SL(n+1)$ and $\PP=\Pn$. The heart of the $t$-structure on
$\Nt_\PP=T^*\Pn$ has a projective generator $\oplusl_{i=0}^n
\O_{T^*\Pn}(-i)$. The heart of the $t$-structure on $\gt_{\Pn}$ has
a projective generator $\oplusl_{i=0}^n \O_{\gt_\PP}(i)$.
%{\bf is this right?} --- who cares?
 \end{example}


\subsubsection{Reformulation in terms of $t$-structure assigned to
alcoves} %Recall that $\Delta_{aff}$ denotes the set of affine
%coroots.
A connected component of the complement to the coroot
hyperplanes $H_\al$ %, $\al \in \Delta_{aff}$
 in the dual space to {\em real} Cartan algebra
$\hh^*_\RE$ is called an alcove; in particular, the {\em fundamental
alcove} $A_0$ is the locus of points where all positive coroots take
value between zero and one. Let $\Alc$ be the set of alcoves. For
$A_1, \, A_2\in \Alc$ we will say that $A_1$ lies above $A_2$ if for
any positive coroot $\check\alpha$ and $n\in \Zet$, such that the
affine hyperplane $H_{\check\al,n}=\{\la,\ |\, \langle \check
\alpha, \la\rangle =n \}$
 separates $A_1$ and $A_2$, $A_1$ lies above $H_{\al,n}$, while $A_2$ lies
below $H_{\al,n}$, i.e. for $\mu\in A_2$, $\la\in A_1$ we have $\<
\al,\mu\> < n < \<\al,\la\>$ .


\begin{lemma}
There exists a unique map $\Alc\times \Alc\to B_{aff}$,
 $(A_1,A_2)\mapsto b_{A_1,A_2}$, such that

i) $b_{A_2, A_3} b_{A_1,A_2}=b_{A_1A_3}$ for any $A_1,A_2,A_3\in \Alc$.

ii) $b_{A_1,A_2}=\tii w$, provided that $A_2$
lies above $A_1$. Here $w\in W_{aff}$ is such that $w(A_1)=A_2$.
\end{lemma}

%\begin{remark}
%The element $b_{A_1,A_2}$ admits the following topological description.
%Notice that a homotopy class of a path connecting two alcoves determines
%an element in $B_{aff}$, because
% each alcove is contractible, and alcoves are permuted transitively by
%$W_{aff}$.

%Let $\la\in \hh^*$ be a positive weight. Then the subspace $-i\la
%+\hh^*_\RE$ does not intersect any of the affine coroot
%hyperplanes. For two points $x\in A_1$, $y\in A_2$ consider
%the path from $x$ to $y$ which is a composition of the following
%three paths: $t\mapsto x- it\la$, any path from $x-it\la$ to $y-it\la$
%in $-i\la+\hh_\RE$ and the path $t\mapsto y -i (1-t)\la$.
%Then $b_{A_1,A_2}$ is represented by this path.
%\end{remark}

%Another exercise we offer the reader is to prove that
The following result is equivalent to Theorem  \ref{tstr}.

%Set $\D=D(\tii Y)$ or $D(\tii \bY)$, where $Y$ is as in Theorem \ref{bract}.

\begin{theorem}\label{tstralc}
Let $X=\tii Y$ or $\tii \bY$, where $Y$ is as in Theorem
\ref{bract}.


There exists a unique collection of $t$-structures indexed by
alcoves, $(D^{\leq 0}_A(X),\, D^{>0}_A(X))$  such that:

0) (Normalization)
 The derived global sections  functor $R\Gamma$ is $t$-exact with
respect to the $t$-structure corresponding to $A_0$.

1) (Compatibility with the braid action)
 The action of the element $b_{A_1,A_2}$ sends the $t$-structure
corresponding to $A_1$ to the $t$-structure corresponding to $A_2$.


2) (Monotonicity)
 If $A_1$ lies above $A_2$, then $D^{>0}_{A_1}(X)\supset D^{>0}_{A_2}(X)$.

%\bigskip

%All these $t$-structures are nice.
\end{theorem}

\begin{remark}
The exotic $t$-structure described in Theorem \ref{tstr}
is the one attached to the fundamental alcove $A_0$ by the construction of
 Theorem \ref{tstralc}.
\end{remark}

\begin{remark}
The data described in Theorem \ref{tstralc} resembles the one
obtained by Bridgeland in the course of description of the manifold
of stability conditions on some derived categories of coherent
sheaves. To enhance this point we mention a positivity property of
the $t$-structure $ (D^{\leq 0}_A(X),\, D^{>0}_A(X))$; such
properties play a role in the definition of stability conditions
\cite{Brstab}.
\end{remark}

It is easy to show that %under some technical assumptions
%\footnote{It
%suffices to assume that $X$ admits a projective morphism to an affine
%variety.}
%on an algebraic variety $X$, satisfied for $\Nt$ and $\gt$,
% any nice $t$-structure on $D(X)$
each of the above $t$-structures  induces a $t$-structure on the
full subcategory $D^f(X)\subset D(X)$ consisting of complexes whose
cohomology sheaves have proper support. Let $\A_A = D^{\leq 0}_A(X)
\cap D^{\geq 0}_A(X)$ be the heart of the $t$-structure, and set
$\A_A^f=\A_A\cap D^f(X)$. It is easy to show that $\A_A^f$ consists
of objects {\em of finite length} in $\A_A$.

Assume that $\kk=\Ce$ and $X$ is smooth. Recall that for a smooth
complex variety $X$
 we have the Chern character map $K(D^f(X))\to
H_*^{BM}(X)$, where $H_*^{BM}$ stands for the Borel-Moore homology
of the corresponding complex variety endowed with the classical
topology.  We have a perfect pairing between cohomology and
Borel-Moore homology.

We have a well-known identification $\hh^*%\cong H^2(\BB)=H^2(\Nt)
=H^2(\BB)$.
\begin{proposition}\label{pospos}
For $A\in \Alc$, $\F\in \A_A^f$, $\F\ne 0$ and $x\in A\subset
\hh^*_\RE \subset  H^2(\BB)$ we have
$$\< ch(\F), pr^* (\exp (x)) \> >0,$$
where $exp(x)=1+x+\frac{x^2}{2}+\cdots +\frac{x^{\dim \BB}}{(\dim
\BB)!}$, and $pr$ stands for the projection $X\to \BB$.
\end{proposition}

Finally, we describe compatibility of our $t$-structures with duality.

Let $\SS$ denote the Grothendieck-Serre duality functor.

\begin{proposition}\label{minusA}
$\SS$ sends $\A_A^f$ to $\A_{-A}^f$, where $-A$ denotes the alcove
opposite to $A$.
\end{proposition}



\subsection{Equivariant category and mutations of exceptional
sets}\label{mutasect}
 The categories $D(\Nt)$, $D(\gt)$ have
equivariant versions $D^G(\Nt)$, $D^G(\gt)$. It turns out that these
equivariant categories carry  $t$-structures which are, on the one
hand, closely related to the above $t$-structures on non-equivariant
categories, and, on the other hand, admit a direct description in
terms of generating exceptional sets in a triangulated category.

{\em Until the end of \ref{neqgr} we assume that $char(\kk)=0$.}

\subsubsection{Exceptional sets and mutations}
Recall that an ordered set of objects  $\nab=\{ \nab^i,\;  i\in I\}$
in a triangulated category is called {\it exceptional} if we have
$\Hom^\bu (\nab^i, \nab^j)=0$ for $i<j$;
 $\Hom ^{n}(\nab^i,\nab^i)=0$ for $n\ne 0$,
 and $\End(\nab^i)=\k$.
A set  $\del=\{ \del_i,\;  i\in I\}$ of objects is called {\em dual}
to  $\nab$ if $\Hom^\bu(\del_i,\nab^i)=\k$, and
$\Hom^\bu(\del_i,\nab^j)=0$ for $i\ne j$; it is exceptional provided
$\nab$ is, where the order on $\del$ is defined to be opposite to
that on $\nab$.
%
 Let  $\nab$, $\del$ be two dual
 exceptional sets which generate a triangulated category $\D$; assume that $\{j\ |\
j\leq i\}$ is finite for every $i\in I$. Then  there exists a unique
$t$-structure $(\D^{\geq 0}, \D^{<0})$ on $\D$, such that
$\nab\subset \D^{\geq 0}$; $\del\subset \D^{\leq 0}$. This
construction is closely related to the definition of a perverse
sheaf, see \cite{Hum} for details.

%
%For example, the "perverse" $t$-structures on the derived category
%of constructible sheaves smooth with respect to a given
%stratification \cite{BBD}
% arises this way  (at least if the stratum are contractible
%trivial), cf. \cite{Humphr}, Remark ... .
%
 Let $(I,\preceq)$
be an ordered set, and $\nab^i\in \D$, $i\in I$ be an exceptional
set. Let $\leq$ be another order on $I$; we assume that $\{j\ |\
j\leq i\}$ is finite for every $i\in I$. We let $\D_{\leq i}$ be the
full triangulated subcategory generated by  $ \nab^j$, $j\leq i$,
and similarly for  $\D_{< i}$. Then for $i\in I$ there exists a
unique (up to a unique isomorphism) object $\nab^i_{mut}$ such that
$\nab^i_{mut}\in \D_{\leq i}\cap \D_{<i}^\perp$, and
$\nab^i_{mut}\cong \nab^i\mod \D_{<i}$ (see e.g. \cite{Hum}).
 The objects $\nab^i_{mut}$ form an exceptional set indexed by
$(I,\leq)$.

% We have $\D_{\leq i} =\bl \{ \nab^j_{mut}\ |\ j\leq i\} \br$, $\bl
%\nab\br=\bl \nab_{mut} \br$.



 We will say that the exceptional set $(\nab^i_{mut})$ is
 the $ \leq$ mutation of $(\nab^i)$. This construction is
related, cf. \cite{Hum}, to the action of the braid group on the set
of exceptional sets in a given triangulated category constructed in
\cite{BoKa}, this action is also called the action  by mutations.


\subsubsection{Exceptional sets in $D^G(\Nt)$}
Recall the standard partial order $\preceq$ on the set $\La$ of
weights of $G$, which is given by: $\la \preceq \mu$ if $\mu-\la$ is
a sum of positive roots. Then
 line bundles $\O_\Nt(\la)$ generate $D^G(\Nt)$, and
we have $\Hom^\bu(\O(\la), \O(\mu))=0$ unless $\mu \preceq \la$ and
$\Hom^\bu(\O(\la), \O(\la))=\k$ \cite{Hum}. Thus   for any complete
order on $\La$ compatible with the partial order $\preceq$, the set
of objects $\O(\lambda)$
 indexed by $\Lambda$ with this order is
an  exceptional set generating $D^G(\Nt)$.



We now introduce another partial
 ordering $\leq$ on $\Lambda$. To this end, recall
the 2-sided Bruhat partial order on the %extended
affine Weyl group $W_{aff}$. For $\lambda\in \Lambda$ let
$w_\lambda$ be the minimal length representative of the coset
$W\lambda\subset W_{aff}$.
 We set
$\mu\leq \lambda$ if $w_\mu$ precedes $w_\lambda$ in the Bruhat
order.

%
We fix a complete order
% $\preceq_{compl}$ on $\Lambda$ compatible
%with $\preceq$ and another complete order
%
 $\leq_{compl}$ on $\Lambda$ compatible
with $\leq$; we assume that $\{\mu \ |\ \mu \leq_{compl} \la \}$ is
finite for any $\la$.
%
We define the exceptional set $\nab_\lambda$ to be the $
\leq_{compl}$ mutation of the set $\O(\lambda)  $. It follows from
the above that $\nab^\la$ is an exceptional set generating
$D^G(\Nt)$. We define the {\em equivariant exotic $t$-structure} to
be the $t$-structure of the exceptional set $\nab^\la$, the objects
in the heart will be called equivariant exotic sheaves.

We now state compatibility between exotic and equivariant exotic
$t$-structures. Roughly speaking, over an orbit in $\N$ of
codimension $2d$ they differ by a shift by $d$. To state this
property more precisely,
 we need to recall the {\em perverse coherent
$t$-structure} \cite{izvrat}. Let $H$ be an algebraic group (assumed
for simplicity of statements connected) acting on an algebraic
variety $X$. Let $\bp$ be a function, called the perversity
function, from the set of $H$-invariant points of the scheme $X$
%(i.e., the set of irreducible $H$-invariant subvarieties
%in $X$)
 to $\Zet$. We assume that $\bp$ is
{\em strictly %!
 monotone and comonotone}, i.e. for points $x,\,y$, such that $x$ lies in the closure of $y$
we have $\bp(y) < \bp(x) < \bp(y)+\dim (y) -\dim (x)$. Then one can
define the perverse $t$-structure on $D^H(X)$, which
% If, moreover, $\bp$
%is {\em strictly } monotone and comonotone, i.e. for points $x,\,y$
%such that $x$ lies in the closure of $y$ we have $\bp(y) < \bp(x) <
%\bp(y)+\dim (y) -\dim (x)$, then this $t$-structure
 shares some
properties with  perverse $t$-structure on the derived category of
constructible sheaves \cite{BBD}. For example, each perverse
coherent sheaf (i.e., object in the heart of the $t$-structure) has
finite length, and irreducible objects are in bijection with pairs
$(O,\LL)$, where $O\subset X$ is an $H$-orbit, and $\LL$ is an
irreducible $H$-equivariant vector bundle on $O$. In particular, if
the action is such that all orbits have even dimension, then the
perversity function $\bp(x)=\frac{\codim x}{2}$, called the middle
perversity, is strictly monotone and comonotone. It is well-known
that the adjoint action of a semi-simple group $G$ on the nil-cone
$\N$ has even dimensional orbits.

This construction works also for the category $D^H(A)$, where $A$ is
a coherent sheaf of associative $\O_X$ algebras equivariant under
$H$. %; the sheaf $A$ is assumed to be Cohen-Macaulay . %then the perverse t-structure is compatible
%with the forgetful functor.



\begin{proposition}\label{pervrA}
There exists a $G$-equivariant vector bundle $\E$ on $\Nt$, such
that $\E$, with the $G$-equivariant structure forgotten, is a
projective generator for the heart of the exotic $t$-structure.

We have an equivalence $\F\mapsto R\Hom(\E, \F)$ between $D^G(\Nt)$
and $D^G(A)$, where $A=End(\E)^{op}$. Under this equivalence the
equivariant exotic $t$-structure corresponds to the perverse
coherent $t$-structure of the middle perversity.
\end{proposition}

%{\em Comments on proofs} This follows from \cite{Humph} ...


\subsection{Grading on exotic sheaves and canonical bases}

\subsubsection{Graded equivariant category
 and positivity by Frobenius weights}
We proceed to state a deep property of exotic sheaves related to an
additional grading on the Ext spaces between them. Recall the
current assumption that $char (\kk)=0$.

Consider the category $D^{G\times \Gm}(\Nt)$, where $\Gm$ acts on
$\Nt$ by $t:x\mapsto t^2 x$. For $d\in \Zet$ let $\F\mapsto \F(d)$
denote twisting by the $d$-th power of the tautological character of
$\Gm$.

%We let $\O_\BB(\la)$ be the $G\times \Gm$ equivariant line bundles
%on $\BB$ (where $\Gm$ acts trivially), which is sent to $\O(\la)$ by
%the forgetful functor $Coh^{G\times \Gm}(\BB) \to Coh^G(\BB)$, and
%where $\Gm$ acts trivially, %by the $d$-th power of the tautological character;
%and let $\O(\la,0)=\O_\Nt(\la, 0)$ be its pull-back to $\Nt$. We set
%$\O(\la,d)=\O(\la,0)(d)$.

%For $\la\in \La$ let $\delta_\la$ be the minimal length of an
%element $w\in W$, such that $w(\la)\in \La^+$.
We refer to \cite{Hum} for an elementary description of a canonical
lifting  $\tii \del_\la$, $\tii \nab^\la$ of $\del_\la$, $\nab^\la$
to $D^{G\times \Gm}$.
% by requiring that $\tii \del_\la$,
%$\tii \nab^\la$ are isomorphic to $\O(\la, \delta_\la)$ modulo the
%subcategory generated by $\O(\mu, d)$, $d \in \Zet$, $\mu <_{compl}
%\la$.
 This also fixes a lifting $\tii L$ of each irreducible
equivariant exotic sheaf $L$ to $D^{G\times \Gm}$.



\begin{theorem}\label{dpst}  For irreducible exotic equivariant
sheaves $L_1$, $L_2$ we have

 $\Ext^i(\tii L_1, \tii L_2(d))=0$ for
$d\leq 0$ and all $i$.

\end{theorem}

%An additional compatibility. Let $O\subset \N$ be an orbit. For an
%irreducible representation $V$ of the centralizer of a point $e\in
%O$ we can define a $G$-equivariant bundle $\VV$  and a $G\times
%\Gm$-equivariant bundle $\tii \VV$ on $O$. Here $\VV$ is
%characterized by the requirement that the fiber of $\VV$ at $e$ is
%$Z$-equivariantly isomorphic to $V$; to define $\tii \VV$ fix a
%homomorphism $\varphi:SL(2)\to G$, such that $d\varphi$ sends the
%standard upper triangular generator of $sl(2)$ to $e$, then we get
%an action of $\Gm$ on $\g$.  Then we get an action $a_\varphi$ of
%the multiplicative group $\Gm$ on $\g$ given $a_\varphi(t):x\mapsto
%t^2\cdot ad (\varphi (diag(t^{-1},t))) x$. This action fixes $e$;
%then $\tii \VV$ is characterized by the requirement that the image
%of $a_\varphi$ acts trivially on the fiber $\VV_e$, while the
%forgetful functor sends $\tii \VV$ to $\VV$.

\begin{remark}
The Theorem follows from results of  \cite{B} on relation between
 exotic
sheaves and perverse sheaves on the affine flag manifold of the dual
group, see also Proposition \ref{qFr} below. They allow to deduce
the Theorem from Gabbers's Theorem \cite{BBD} on positivity of
weights of Frobenius action on Ext's between pure perverse sheaves
of the same weight. Thus it is the least elementary of the results
mentioned so far in this text.

The motivation for the Theorem is its consequence below, which shows
(in most cases) that classes of exotic sheaves form a {\em canonical
basis} in the Grothendieck group. This is parallel to the proof of
the Kazhdan-Lusztig conjecture: according to Soergel, cf.
\cite{SoerICM}, the latter is equivalent to the statement that for a
certain explicitly defined graded version of
Bernstein-Gel'fand-Gel'fand category $O$ the grading on $\Ext^1$
between irreducible objects has vanishing components of non-positive
degrees. The only known way to prove this vanishing is to identify
category $O$ with a category of perverse sheaves or Hodge
$D$-modules, and use Gabber's Theorem or its Hodge theoretic
analogue.
\end{remark}

\begin{remark}\label{cohotilt}
Another application of Theorem \ref{dpst} is explained in
\cite{Hum}. Together with {\em Koszul duality} formalism of
\cite{BGS} it allows to show that equivariant exotic sheaves control
cohomology of quantum groups at a root of unity with coefficients in
a tilting module.
\end{remark}

%\subsection{Symplectic resolutions of (quotient) singularities}
%{\bf Probably goes away}

% One possible proof of Theorems \ref{bract}, \ref{tstr} relies on
% the study of $D$-modules in positive characteristic, see ... below.
%This inspired the method to approach another case of derived
%equivalence conjecture.



%Quotient singularity, Kaledin.

\subsubsection{Non-equivariant graded category and canonical
bases}\label{neqgr} We fix $X=\Nt$. Recall the category
$\A^f=\A_{A_0}^f\subset \A$ of exotic sheaves of finite length. It
is easy to see that $\A^f = \oplusl_{e\in \N} \A_e$, where
$\A_e=\A\cap \D_e$, and $\D_e\subset D(\Nt)$ is the full subcategory
of complexes whose cohomology sheaves are set-theoretically
supported on $\BB_e=\pi^{-1}(e)$. We have $K(\A_e)\cong K(\BB_e).$
Furthermore, the Chern character map provides an isomorphism
$K(\BB_e)_F\cong
H_*^{BM}(\BB_e)_F,$ %=(H^*(\BB_e))^*$,
 where $F$ denotes the coefficient
field of characteristic zero ($\Ce$ or $\Qlbar$), see, e.g,
 \cite{BMR}I.
% in the case
%$char(\kk)=0$ the latter isomorphism follows from \cite{DLP}, and
%the general case is a formal consequence

The classes of irreducible objects form a basis in $K(\A_e)$. We
proceed to explain the properties of the category, which are needed
to relate this basis to {\em the canonical bases} in
$H_*^{BM}(\BB_e)$. The definition of the latter is due to Lusztig
\cite{LKthII}II, and follows the example of Kashiwara's
characterization of crystal bases \cite{Kashiwa}. More precisely,
Lusztig suggested a way to characterize a basis in
$H_*^{BM}(\BB_e)$, and conjectured that a basis satisfying his
axioms exists; he showed that it is then unique (up to a sign). We
will not recall Lusztig's characterization in detail; instead we
describe its structure and  explain the properties of exotic
sheaves, which imply (modulo a technicality, which is easy to check
in many cases)
 that Lusztig's axioms are satisfied by the basis
of irreducible exotic sheaves.

\medskip

%\subsubsection{Graded version} % and the affine Hecke algebra action}
 % and canonical lifting of irreducibles}
One can find a homomorphism $\varphi:SL(2)\to G$, such that
$d\varphi$ sends the standard upper triangular generator of $sl(2)$
%matrix with nonzero entry equal to one
to $e$. Then we get an action $a_\varphi$ of the multiplicative
group $\Gm$ on $\gg$ given $a_\varphi(t):x\mapsto t^2\cdot ad
(\varphi (diag(t^{-1},t))) x$. This action fixes $e$.
%; we can (and will) assume that $Y$ is
%invariant under $a_\varphi$.

 We let
$\D_e^{\Gm}\subset D^{\Gm}(\Nt)$ %=D^{\Gm}_{\BB_e}(\Nt)$ (or  $\gt$).
be the full subcategory of complexes, which are set theoretically
supported on $\pi^{-1}(e)$. Twisting by the tautological character
of $\Gm$ defines an auto-equivalence of this category, which we
denote by $\F\mapsto \F(1)$.
%
%It is easy to show that
The exotic $t$-structure is inherited by the $\Gm$-equivariant
category; we let $\A_e^{gr}$ denote the heart of the latter. It is
easy to see that the forgetful functor $\D_e^{\Gm} \to \D_e$ sends
$Irr(\A_e^{gr})$ to  $Irr(\A_e)$, where $Irr$ stands for the set of
isomorphism classes of irreducible objects. This gives a bijection
$Irr(\A_e^{gr})/\Zet\cong Irr(\A_e)$, where $\Zet$ acts on
$Irr(\A_e^{gr})$ by $\F\mapsto \F(n)$. We also have
%\begin{equation}\label{isKgr}
$K(\A_e^{gr})\cong K^{\Gm}(\BB_e)\cong K(\BB_e)[v,v^{-1}],$
%\end{equation}
where multiplication by $v$ corresponds to twisting by the
tautological character of $\Gm$.


%\subsubsection{Canonical lifting of irreducible
%exotic sheaves and canonical basis}

 The canonical basis in $K(\BB_e)[v,v^{-1}]$ is characterized (up to a sign) by two
properties: {\em invariance under an involution} and {\em asymptotic
orthogonality} \cite{LKthII}II. These are reflected, respectively,
in categorical properties (i) and (ii) in the next Theorem.

%\subsubsection{Grading and its positivity}

%At this point we would like to mention a seemingly innocuous
%statement, which has deep consequences to representation theory.

% Thus $Y$ and $\tii Y$
%acquire a $\Gm$ action.

%The deepest property of our noncommutative resolution is summarized
%in (part b of) the following.


Notice that the action of $B_{aff}$ on $\D_e$ is inherited by
$\D_e^{\Gm}$. Recall that $\SS$ is the Grothendieck-Serre duality.

In view of Theorem \ref{tstralc} and Proposition \ref{minusA}, the
contravariant auto-equivalence $\tii w_o \circ \SS$ is $t$-exact
with respect to the $t$-structure corresponding to the fundamental
alcove $A_0$, hence it permutes irreducible objects of $\A_{A_0}$;
here $w_o\in W$ is the long element.

\begin{theorem}\label{cordpst} %Assume that $char (\kk)=0$.
There exists a canonical section of the map $Irr(\A_e^{gr})\to Irr
(\A_e)$, $L\mapsto \tii L$, such that

i) The image of the section is invariant  under every automorphism
of $G$ which is identity on the image of $\varphi _e$, and also
under $\tii w_o \circ \SS$.

ii) $\Ext^1_{\bar \A_e^{gr}}(\tii L_1, \tii L_2(i))=0$ for $i\leq 0$
and any $L_1, \, L_2\in Irr(\A_e)$; here $\bar \A_e^{gr}\subset
\A_e$ is the full subcategory of objects where the ideal of the
point $e$ in $\O(\gg)$ acts by zero.
\end{theorem}

{\em Comments on the proof.} The Theorem can be deduced formally
from Theorem \ref{dpst} and Proposition \ref{pervrA}. Thus its proof
relies on ideas of geometric Langlands duality used in \cite{AB},
and on Gabber's Theorem (see comments after Theorem \ref{dpst}).





\begin{corollary}\label{suppid}
%a) Suppose that an endomorphism $\Upsilon$ of  $K^{\Gm} (\BB_e)$ is
%a linear combination (with coefficients in $\Ce$) of maps induced by
%automorphisms of $G$ acting by identity on the image of $\varphi_e$.
%Suppose, moreover, that the induced endomorphism $\bar \Upsilon$ of
%$K(\BB_e)$ equals the map induced by $w_o\circ \SS$. Then for every
%$L\in Irr(\A_e)$ the class of $\tii L$ is invariant under the
%composition $\Upsilon \circ \tii w_0 \tii \SS$.

%b) For $L_1, L_2\in Irr(\A_e)$ we have $([\tii L_1]\ |\ [\tii
%L_2])-\delta _{L_1,L_2}\in v\Zet[[v]]$.
Suppose that the involution $\tii \beta$ defined in \cite{LKthII}II,~\S 5.11
 induces
identity on the specialization at $q=1$. Then Conjecture 5.12
of {\em loc. cit.}, except, possibly, 5.12(g), holds;
moreover, the signed basis ${\bf B}_{\BB_e}^\pm$,
 whose existence is conjectured
in {\em loc. cit.},
 is formed by the classes of the objects $\tii
L$, where $L$ runs over  irreducible objects in $\A_e$.

\end{corollary}

%We will call elements of a basis satisfying the properties of the
%Corollary a {\em canonical basis}. A slight modification of the
%arguments of Lusztig \cite{LKthII}II shows that a canonical basis is
%unique up to multiplication of some of its elements by $-1$.

\begin{remark}
%In \cite{LKthII}II Lusztig defines a certain involution $\Upsilon$. It
%is clear from the definition that it satisfies the first condition
%of the Corollary. It probably also satisfies the second one; this is
%easy to show in many cases, e.g. for $e=0$.

%Assuming that Lusztig's involution satisfies our requirement, the
%Corollary implies Conjectures ... .. .. in \cite{LKthII}II.
The assumptions of the Corollary are easy to check in many cases,
e.g., if the nilpotent element $e$ is regular in a Levi subalgebra.
\end{remark}

%\subsubsection{Involutions}
%Our goal here is to write down an explicitly computable involution
%$\iota$  on $H_*^{BM}[v,v^{-1}]$, which sends $v$ to $v^{-1}$ and
%preserves the classes of irreducible objects in $\A_e^{gr}$ up to
%multiplication by a power of $v$.


%\subsubsection{Asymptotic orthogonality}

\begin{remark} In fact, in \cite{LKthII}II Lusztig works with sheaves which are also equivariant under
a maximal torus in the centralizer of $e$. We omit this version here
to simplify notations, treating this set-up does not involve new
ideas.
\end{remark}


\begin{remark}
Validity of Conjecture 5.12(g) of \cite{LKthII}II is related to the
following
 question. Let $Y\subset \gg$ be a (Slodowy) transversal slice to a
$G$ orbit in $\N$, and $X=\Nt\times _\gg Y$. Let $A_X$ be as in
Theorem \ref{tstr}. One can show that $A$ can be endowed with a
natural grading; moreover, Theorem \ref{cordpst} is equivalent to
the fact that this grading can be chosen so that the graded
components of negative degree vanish, while the component of degree
zero is semi-simple. The question is whether the resulting graded
algebra is Koszul. If $e$ is subregular, then the positive answer is
easy to prove.
\end{remark}


\subsubsection{Independence of the (large) prime}
It is not hard to show that (co)homology of the Springer fiber is
independent of the ground field $\kk$, i.e. we have canonical
isomorphisms $H_\bu ^{BM}(\BB_e^\kk)\cong H_\bu^{BM}(\BB_e^\Ce)$,
where the upper index denotes the ground field, and $H_\bu^{BM}$
stands for $l$-adic Borel-Moore homology, $l\ne char(\kk)$.

 The definition of
the exotic $t$-structure is not specific to a particular ground
field.
 This allows one to prove the following.


\begin{proposition}\label{indep}
For all but finitely many prime numbers $p$ the following is true.
The classes in $H_\bu^{BM}(\BB_e^\kk)=H_\bu^{BM}(\BB_e^\Ce)$ of
irreducible exotic sheaves over $\kk$ of characteristic $p$ coincide
with the classes of irreducible exotic sheaves over $\Ce$.
\end{proposition}


%\subsubsection{Grading and its positivity}

\section{$D$-modules %and deformation quantization
in positive characteristic and localization Theorem}\label{se3}
\subsection{Generalities on crystalline
 $D$-modules in positive characteristic}
\subsubsection{Definition and description of the center} Let
$X$ be a smooth variety over the field $\kk$.

%\subsubsection{Frobenius center and Azumaya property}
The sheaf $\DD=\DD_X$ of {\em crystalline differential operators}
(or differential operators without divided powers, or PD
differential operators) on $X$ is defined as the enveloping of the
tangent Lie algebroid, i.e.,  for an affine open $U\subset X$ the
algebra $\DD(U)$   contains the subalgebra $\O$ of functions, has an
$\O$-submodule identified with the Lie algebra of vector fields
$Vect(U)$ on $U$, and these subspaces generate $D(U)$ subject to
relations $\xi_1\xi_2-\xi_2\xi_1=[\xi_1,\xi_2]\in Vect(U)$ for
$\xi_1, \xi_2\in Vect(U)$, and  $\xi \cdot f -f\cdot \xi =\xi(f)$
for $\xi\in Vect(U)$ and $f\in \O(U)$.

If $char(\kk)=0$, then $\DD_X$ is the  familiar sheaf of differential
operators. From now on assume that $\kk$ is of characteristic $p>0$.
Then $\DD_X$ shares some features with the characteristic zero case;
%more familiar differential operators
%on smooth algebraic varieties in characteristic zero.
 for example,
$\DD_X$ carries an increasing filtration ``by order of a
differential operator'', and the associated graded $gr(\DD_X)\cong
\OO_{T^*X}$ canonically. On the other hand, some phenomena are
special to the characteristic $p$ setting. We have an action map
$\DD_X\to End(\OO_X)$,  which is not injective, unlike in the case
of a characteristic zero.
% (thus the term ``differential operators''
%for sections of $\DD_X$ is not quite accurate; still, we will  use
%it occasionally)
 For example, if $X={\mathbb A}^1=Spec(k[x])$, the
section $\de_x^p\ne 0$ of $\DD_X$ acts by zero on $\OO$. Also,
$\DD_X$ has a huge center; for example, if $X={\mathbb
A}^n=Spec(k[x_1,\dots,x_n])$, then $x_i^p$ and $\de_{x_i}^p$ are
readily seen to generate the center $Z(\DD_{{\mathbb A}^n})$ freely.
More generally, for any $X$ the center $Z(\DD_X)$
 is freely generated by elements
of the form $f^p$, $f\in \OO_X$ and $\xi^p-\xi^{[p]}$, $\xi \in
Vect_X$, where $\xi^{[p]}$ is the {\em restricted power} of the
vector field $\xi$; it is characterized by
$Lie_{\xi^{[p]}}(f)=Lie_\xi^p(f)$ for
 $f\in \OO_X$,
where $Lie$ stands for the Lie derivative. The center $Z(\DD_X)$ is
canonically isomorphic to the sheaf of rings $\OO_{T^*X\tw}$ where
the super-index $(1)$ stands for Frobenius twist.\footnote{Recall
that Frobenius twist of a variety $X$ over a perfect field $\kk$ is
defined to be isomorphic to $X$ as an abstract scheme, with the
$\kk$-linear structure twisted by Frobenius.
%while the action of $t\in \kk$ on $\O_{X\tw}$ is defined to the
%action of $t^p$ on $\O_X={\O_X\tw}
Not only $X\cong X\tw$ as abstract schemes, but also $X\cong X\tw$
as $\kk$-schemes, provided that $X$ is defined over $\bF_p$. For this
reason we will sometimes identify $X$ with $X\tw$ and omit Frobenius
twist from notation.}
%
% $X^{(1)}=X\times
%_{Spec(\kk)}Spec(\kk)$ where the map $Spec(\kk)\to Spec(\kk)$ is the
%Frobenius map $x\mapsto x^p$.
 Thus $\DD_X$ can be considered as a
quasi-coherent sheaf of
 algebras on $T^*X\tw$.

\subsubsection{Azumaya property} Recall that an {\em Azumaya algebra
}
 on a scheme $X$
is a locally free sheaf $\AA$ of associative $\OO_X$ algebras, such
that the fiber of $\AA$ at every geometric point is isomorphic to a
matrix algebra. % (see e.g. \cite{Milne}).
The following fundamental observation is due to Mirkovi\' c and
Rumynin, though a weak form of it can be traced to an earlier work
\cite{hren}.

\begin{theorem}\cite{BMR}I
 $\DD_X$ is an Azumaya algebra of rank $p^{2\dim (X)}$ on $T^*X^{(1)}$.
 \end{theorem}


See  \cite{OV}, \cite{BelKa} for generalizations and applications.

%\begin{remark}
%For example, for $X=\bbA^n$  the Theorem readily reduces to the
%following elementary claim: {\em for an algebraically closed field
%$\kk$ the associative unital algebra with generators $x_1,\dots,
%x_n; \xi_1\dots \xi_n$ and relations $\xi_i x_j-x_j\xi_i
%=\delta_{i,j},\ x_i^p=\alpha_i$, $\xi_i^p=\beta_i$ for some
%$\al_i,\beta_i\in\kk$ is isomorphic to the matrix algebra
%$Mat_{p^n}(\kk)$.}
%\end{remark}


Recall  that two Azumaya algebras $\AA$, $\AA'$ are called equivalent (we then
write $\AA\sim \AA'$) if they are Morita equivalent, i.e. if there
exists a coherent locally projective sheaf $\MM$ of $\AA-\AA'$
bimodules, such that
 $\AA'\iso End(\MM)^{op}$; we will then say that $\MM$ provides
an equivalence between $\AA$ and $\AA'$.
 In particular, an Azumaya algebra $\AA$ is {\em split} if $\AA\sim \OO_X$;
this happens iff $\AA\cong  End(\EE)$ for a vector bundle $\EE$.
 For
two equivalent Azumaya algebras $\AA$, $\AA'$ we have an equivalence
of categories of modules $Coh(X,\AA)\cong Coh(X,\AA')$, depending on
the choice of a bimodule providing the equivalence between $\AA$ and
$\AA'$; in particular, for a split Azumaya algebra we have
$Coh(X,\AA)\cong Coh(X)$.

%The set of equivalence classes admits an injective map to the etale
%cohomology group $H^2(\OO^\times_X)$ \cite{Milne}.
%;the class of split Azumaya algebra goes to the identity element.

For a smooth variety $X$ over a positive characteristic field, the
Azumaya algebra $\DD_X$ is not split unless $\dim (X)=0$. However,
it is split on the zero section, see \cite{OV} for more information.
%One can deduce that it splits on the formal neighborhood of the zero
%section provided that $H^2(T^*X,\OO)=0$.

We will also need a twisted  version of differential operators. If
$\LL$ is a line bundle on $X$, then one can consider the sheaf
$\DD^\LL=\DD_X^\LL$ of differential operators in $\LL$.
 A similar argument shows
that this is also an Azumaya algebra over $T^*X\tw$; moreover, we
have a canonical equivalence
\begin{equation}\label{twisteq}
\DD_X\sim D^\LL_X, \end{equation}
 given by the bimodule
$D_X\otimes_{\O(X)} \LL^{-1}$.

\begin{remark}\label{praz}
 Notice that if $\LL=\LL_0^p=Fr^*(\LL_0)$ for some line
bundle $\LL_0$, then we have a canonical isomorphism $\DD^\LL_X\cong
\DD_X$; however, the above equivalence $\DD^\LL_X\cong \DD_X$ is not
identity, but rather tensor product
 {\em over the
ring $\O_X^p=\O_{X\tw}$} with the line bundle $\LL_0\tw$.
\end{remark}

\subsection{Crystalline operators on $\BB$}
We now consider $X=\BB$. We abbreviate $\DD_\BB^{\O(\la)}=\DD^\la$.

\subsubsection{Splitting the Azumaya algebra}\label{split_sect}
%It is
%known %(see, e.g., \cite{KumarLaur})
%that higher cohomology of
%$\O_{\Nt}$ vanish.
It was mentioned above that  $\DD_\BB$ splits on the zero section.
In fact, we have the following stronger statement.

\begin{theorem}\label{spl}

%Let $X=G/P$ and $\pi_P:T^*(G/P)\to \gg^*=\gg$ be the moment map.

a)  There exists an Azumaya algebra $\AA$ on $\N\tw$, such that
$\DD^{-\rho}\cong \pi^*(\AA)$.

b) For any $\la$ we have an equivalence of Azumaya algebras on $\N\tw$:
$\DD^\la\sim \pi^*(\AA)$.

c) $\DD^\la(\BB)$ is split on the formal neighborhood of every fiber of
$\pi$.
\end{theorem}
{\em Sketch of proof.}  (a) reduces to
irreducibility of baby Verma modules with highest weight $-\rho$, which
follows from \cite{BrGo}.
 It implies, moreover,
that the statement holds for $\A$ being the quotient of the
enveloping algebra  $U(\gg)$ by the central ideal corresponding to
$-\rho$.
 (b) follows from (a) in view of the equivalence
\eqref{twisteq}. Finally, (c) follows from (b), since every Azumaya
algebra over a complete local ring with an algebraically closed
residue field  splits.

\medskip
% For an algebraic
%variety $X$ let $Coh^f(X)\subset Coh(X)$ denote the subcategory
%of sheaves with proper support.

Let $\DD^\la-\rmod^f\subset Coh(\Nt\tw, \DD^\la)$
  the full subcategory
of sheaves, whose support (which is a subvariety in $\Nt\tw$) is
proper. Let $Coh^f(\Nt)\subset Coh(\Nt)$ be the full subcategory of
sheaves with proper support.


\begin{corollary}\label{eqccoh}
For every $\la\in \La$ we have an equivalence $ \DD^\la-\rmod^f
\cong Coh^f(\Nt)$.
\end{corollary}

\proof Since the target of $\pi$ is affine, a subscheme $Z$ in
$\Nt\tw$ is proper iff it lies in a finite union of nilpotent
neighborhoods of Springer fibers. Thus the claim follows from
Theorem \ref{spl}(b).

\medskip

For each $\la\in \La$ and $e\in \N$
we fix  the splitting bundle $\EE^\la_e$ for $\DD^\la$ on the formal
neighborhood of $\pi^{-1}(e)$ as follows. For $\la=-\rho$ we let $\EE_e^\la$
be the pull-back under $\pi$
of a splitting bundle for the Azumaya algebra $\A$ on the formal neighborhood of $e$ in
$\N\tw$. For a general $\la$ we get $\EE^\la$
from $\EE^{-\rho}$ by applying the canonical equivalence \eqref{twisteq}
%\footnote{We
% would like to warn the reader that most of the time
%we consider $\EE^\la_e$, $\EE^\rho_e$ as sheaves on $\Nt\tw$
%(or rather its formal completion at $\pi^{-1}(e)$), and there seems to be
%no simple way to relate these two sheaves.}
between $\DD^{-\rho}$ and $\DD^\la$; thus $\EE^\la=\E^{-\rho}\otimes _{\O_\BB}
\O(\la+\rho)$.

We let $\Feq_\la$ denote the resulting equivalence between
$\DD^\la-\rmod^f$ and $Coh^f(\Nt)$. Notice that for $\la'=\la+p\mu$
the sheaves of algebras $\DD^\la$ and $ \DD^{\la'}$ are canonically
identified; however, the equivalences $\Feq_\la$ and $\Feq_{\la'}$
are different, cf. Remark \ref{praz}.



%\begin{remark} Have variants: infinitesimally twisted $\DD$-modules and
%the extended cotangent bundle, $G/P$.
%\end{remark}
%\subsubsection{Variant: infinitesimally twisted $\DD$-modules and
%the extended cotangent bundle}

%\subsection{Derived equivalences and noncommutative resolutions}

\subsubsection{Derived localization in positive
characteristic}\label{derloc} Let $U=U(\gg)$ be the enveloping
algebra.

Assume first that $char(\kk)=0$. Recall the famous Localization
Theorem \cite{BeBe}, \cite{BrylKa}, which provides an equivalence
$U^\la-\rmod\cong \DD^\la-\rmod(\BB)$, where $\la$ is a dominant
integral weight, $\DD^\la-\rmod$ denotes the corresponding twisted
$D$-modules category, and $U^\la-\rmod$ is the category of
$\gg$-modules with central character corresponding to $\la$. For two
 integral weights $\la, \, \mu$ the categories $\DD^\mu-\rmod$ and
$\DD^\la-\rmod$ can be identified by means of the equivalence
$T^\la_\mu:\F \mapsto \F\otimes \O(\la-\mu)$. If $\la, \, \mu$ are
dominant, then the global sections functors intertwine this
equivalence with  the translation functor, which provides an
equivalence $U^\la-\rmod\cong U^\mu-\rmod$.

Assume now that $\mu$ is integral regular, thus
$\mu=w(\la+\rho)-\rho$ for some dominant integral $\la$, $w\in W$.
Then the functor of global sections on $D_\mu-\rmod$ is no longer
exact; however, it follows from \cite{BeBeder} that the derived
functor $R\Ga=R\Ga_\mu:D^b(\DD^\mu-\rmod)\to D^b(U^\la-\rmod)$ is
still an equivalence. The triangle formed by the three equivalences
$R\Ga_\mu$, $T^\la_\mu$, $R\Gamma_\la$ does not commute. Thus we get
an auto-equivalence  $R_w$ of $D^b(\DD^\la-\rmod)$,
$R_w=R\Ga_\la^{-1} \circ R\Ga_\mu\circ T_\la^\mu$. In \cite{BeBeder}
it is shown that $R_w$ can be described by an explicit
correspondence, which makes it natural to call $R_w$ the {\em Radon
transform}, or {\em the intertwining functor}. Moreover, the
assignment $\tii w\mapsto R_w$ extends to an action of the Artin
braid group $B$ attached to $G$ on $D^b(U^\la-\rmod)$.

A part of this picture can be generalized to characteristic $p$.


The obvious characteristic $p$
 analogue of the above equivalence of abelian categories
%$U^\la-\rmod \cong \DD^\la-\rmod$
 does not hold for any integral
$\la$. Indeed, it is well known that for any coherent sheaf $\F$ on
the (Frobenius twist  of) a smooth variety the sheaf $Fr^*(\F)$
carries a flat connection; in particular, so does the sheaf
$Fr^*(L)=L^{\otimes p}$, where
 $L$ is a line bundle.
 Thus for $\F\in \DD^\la-\rmod$ we have $\F\otimes L^p\in
\DD^\la-\rmod$. If $L$ is anti-ample and the support of $\F$ is
projective of positive dimension, then  some of the higher derived
functors $R^i\Gamma(\F\otimes L^{dp})\ne 0$ for large $d$.

However, we do have an analogue of the "derived" localization
Theorem. From now on assume that $char(\kk)=p>0.$

The center $Z$ of $U$ contains the subalgebra $Z_{HC} =U^G\cong
Sym(\hh)^W$, which we call the Harish-Chandra center. We have a
natural map $\La/p\La \to \hh^*/W$, $\la \mapsto d \la\mod W$. Thus
every $\la\in \La$ defines a maximal ideal of $Z_{HC}$. We let
$U^\la=U\otimes_{Z_{HC}} \kk$ denote the corresponding central
reduction. Notice that the set of weights  $\mu\in\La$, such that
the quotients $U^\la$ and $U^\mu$ of $U$ coincide,
 is precisely the $W_{aff}$-orbit of $\la$ with respect
to the  action $w\bu \la = p\, w(\frac{\la+\rho}{p})-\rho$.
We will say that $\la\in \La$ is $p$-regular if the stabilizer in $W$
of $\la+p\La\in \La/p\La$ is trivial.

We also have another central subalgebra $Z_{Fr}\subset U$, called
the Frobenius center. It is generated by expressions of the form
$x^p-x^{[p]}$, $x\in \gg$, where the {\em restricted power} map
$x\mapsto x^{[p]}$ is characterized by $ad(x^{[p]})=ad(x)^p$. Thus
maximal ideals of $Z_{Fr}$ are in bijection with points of
$\gg^*\cong \gg$.

Let $U^\la-\rmod$ denote the category of finitely generated $U^\la$-modules,
and let $U^\la-\rmod^f\subset U^\la-\rmod$ be the full subcategory of finite
length modules.

For a pair $\la\in \La$, $e\in \gg^*$  let $U^\la_{\hat e}-\rmod$ be
the category of finitely generated $U^\la$-modules, which are killed
by some power of the maximal ideal of $e$ in $Z_{Fr}$. This category
is zero unless $e\in \N$. We also have $U^\la-\rmod^f =\oplusl_{e\in
\N} U^\la_{\hat e}-\rmod$.


Let $\DD^\la_{\hat e}-\rmod \subset \DD^\la-\rmod$ be the full
subcategory of objects which are supported on a nilpotent
neighborhood of $\pi^{-1}(e)$; here we think of $\DD^\la$ modules as
sheaves on $\Nt\tw$ with an additional structure.


\begin{theorem}\label{BMRth} \cite{BMR}I a) For every $\la\in \La$
we have a natural isomorphism $\Ga(\DD^\la)\cong U^\la$.

b) If $\la\in \La$ is $p$-regular, then the derived global sections
functor provides an equivalence $R\Ga_\la: D(\DD^\la)  \iso
D(U^\la)$. It restricts to equivalences $D^b(\DD^\la\rmod^f)\cong
D^b(U^\la-\rmod^f)$, $D^b(\DD^\la_{\hat e}-\rmod)\cong
D^b(U^\la_{\hat e}-\rmod)$.
\end{theorem}

\begin{remark}\label{dgRem}
This Theorem has several versions and generalization. One can work
with the more general categories of twisted $D$-modules, thereby
obtaining a category of modules over an Azumaya algebra on the
formal
 neighborhood of $\Nt$
in $\gt$, or more general subschemes or formal completions of $\gt$.
For singular weights $\la$ there is a version of the Theorem that
relates derived categories of modules to sheaves on (the
neighborhoods of)  {\em parabolic Springer fiber} \cite{BMR}II.
Another construction works with differential operators on a partial
flag variety $G/P$ for a parabolic subgroup $P\subset G$, {\em loc.
cit.}, cf. also subsection \ref{parvar} above.

For a scheme $Y$ mapping to $\gg$ and satisfying the Tor vanishing
conditions of Theorem \ref{bract} we have an equivalence between the
derived category of modules over  Azumaya algebras on $\tii \bY$,
$\tii Y$ obtained as pull-back of the algebra of (twisted)
differential operators and derived category of modules over the
algebra of global sections.

If $Y$ is a transversal slice to a nilpotent orbit, then the algebra
of global section of the Azumaya algebra on $\tii Y$ is probably
related to Premet's quantization of Slodowy slices and generalized
Whittaker $D$-modules, see \cite{Premet}, \cite{GinGan}.

There exists a generalization of this result for $Y$ not satisfying
the $\Tor$ vanishing condition. It involves coherent sheaves on the
{\em differential graded} scheme, which is the derived fiber product
of $Y$ and $\gt$ over $\gg$. The particular case $Y=\{0\}$ is
closely related to the description of the derived category of the
principal block in representations of a quantum group at a root of
unity provided by \cite{ABG}.

\end{remark}


%In the statement of part (b) of the next Proposition we identify the category
%$\DD^\la-\rmod$ for  $\la\in \La$ with the category $\DD^0-\rmod$
%by means of the equivalence \eqref{twisteq}.
%$T_{\la}^{0}$.

\begin{proposition}
a) A weight $\la\in \La$ is $p$-regular iff $\frac{\la+\rho}{p}$ lies
in some alcove.

b) The $t$-structure on $\DD^0-\rmod$ induced by the equivalence
$R\Ga^\la\circ T_0^\la$ for a $p$-regular $\la$ depends only on the
alcove of $\frac{\rho+\la}{p}$ (see beginning of section
\ref{derloc} for notation).
\end{proposition}

Thus we get a collection of $t$-structure on $\DD^0-\rmod$ indexed
by
 alcoves; we denote the $t$-structure attached to $A\in \Alc$ by $D^{<0}_A(\DD)$, $D^{\geq 0}_A(\DD)$.
The following properties of the collection follow from \cite{BMR}.

%We now fix $\la_0\in\La$, such that  $\frac{\rho+\la_0}{p}$.
%For $\la\in \La$ we identify  the category
%$\DD^\la-\rmod$ with  $\DD^{\la_0}-\rmod$
%by means of
%$T_{\la}^{\la_0}$.

\begin{theorem}\label{abcd}
a) Let $A_1$, $A_2$ be two alcoves. If $A_1$ lies above $A_2$, then
$D^{> 0}_{A_1}(\DD)\supset D^{> 0}_{A_2}(\DD)$.

b) There exists an action of $B_{aff}$ on $D(\DD)$, such that the following
holds.
Let $\la\in \La$, $\frac{\la+\rho}{p}=w(\frac{\rho}{p})$ for $w\in W_{aff}$,
and let $A$ be the alcove of $\frac{\la+\rho}{p}$.
Then $R\Ga_\la\cong R\Ga_0\circ b_{A_0,A_1}$.
Thus $b_{A_0,A_1}$ sends the $t$-structure
 $D^{<0}_{A_0}(\DD)$, $D^{\geq 0}_{A_0}(\DD)$ to the $t$-structure
 $D^{<0}_A(\DD)$, $D^{\geq 0}_A(\DD)$.

c) The restriction of the $B_{aff}$ action to $D^b(\DD^\la-mod^f)
\cong D^b(Coh^f(\Nt))$ coincides with the restriction of the
action from Theorem \ref{bract}.

d) The derived global sections functor $R\Gamma$ on
$D^b(Coh^f(\Nt))$ is $t$-exact with respect to the $t$-structure
induced from $( D^{<0}_{A_0}(\DD)$, $D^{\geq 0}_{A_0}(\DD))$ via the
equivalence $\Feq_0$ (see the end of section \ref{split_sect} for
notation).

\end{theorem}

\begin{corollary}\label{AU}
The $t$-structure on $Coh^f(\Nt)$ induced by the exotic
$t$-structure coincides with the one induced from
$(D^{<0}_{A_0}(\DD)$, $D^{\geq 0}_{A_0}(\DD))$ via the equivalence
$\Feq_0$ (see the end of \ref{split_sect}).

In particular, we have a Morita equivalence
 $\A_e\sim U-\rmod_{\hat
e}^\la$ for every $p$-regular $\la\in \La$.
\end{corollary}

The Corollary follows by comparing Theorem \ref{abcd} with Theorem
\ref{tstralc}.

\begin{corollary}\label{Ue}
a) Let $U_e^\la$ denote the specialization of the enveloping algebra
$U(\gg)$ at the central character corresponding to  $e\in \N$ and a
regular integral weight $\la$. Then we have a canonical isomorphism
$K(U_e^\la)_F\cong H_\bu^{BM}(\BB_e)_F$, where $F$ is a field of
characteristic zero.

b) The images of the irreducible modules under this isomorphism is
independent of the base field $\kk$, except for a finite number of
values of the characteristic.

\end{corollary}

Part (a) of the Corollary follows directly from Theorem \ref{BMRth}
(cf. the discussion preceding Proposition \ref{pospos}), while part
(b) follows from  Corollary \ref{AU} and Proposition \ref{indep}.

\begin{remark}
For $e=0$ part (a) of the  Proposition is standard, and part (b) can
be deduced from \cite{AJS}. Our method uses the principal tool of
\cite{AJS}, namely, the reflection functors, in the disguise of the
braid group action; geometry of the Springer map is the new
ingredient.
\end{remark}

In fact, we have the following stronger, though more difficult
statement. We will say that a basis in $H_\bu^{BM}(\BB_e)$  is
canonical if it is the image of a basis in the equivariant
Grothendieck group $K^{\Gm}(\BB_e)$, satisfying Lusztig's axioms
\cite{LKthII}II,  under forgetting the equivariance composed with
the Chern character map. According to a result of \cite{LKthII}II
such a basis is unique up to multiplication of some of its elements
by $-1$, if it exists.

\begin{corollary}  Enforce the assumption of Corollary \ref{suppid}.
Then for almost all $p=char(\kk)$ the isomorphism
$K^0(U-mod_e^0)_F\cong H_*^{BM}(\BB_e)_F$ of Corollary \ref{Ue}(a)
sends classes of irreducible objects to elements of a canonical
basis. Thus Conjecture 17.2 of \cite{LKthII}II holds in this case.
\end{corollary}

This Corollary is immediate from Corollary \ref{suppid}  together
with Theorem \ref{abcd}(d). Thus its proof, unlike the proof of
Corollary \ref{Ue}, relies on Gabber's Theorem \cite{BBD} and ideas
of local geometric Langlands duality, on which the results of
\cite{AB} are based.

\begin{remark}
The particular case $e=0$ of the Conjecture 17.2 of \cite{LKthII}II
is well known to imply the previous Lusztig conjectures
\cite{LConj}, which describe characters of algebraic groups in
finite characteristics. Lusztig's program for a proof of these
conjectures has been carried out by several authors. An alternative
proof is given in \cite{ABG}.

Notice that the strategy of proof of this conjecture outlined above
does not use quantum groups.
\end{remark}

%\subsubsection{Numerical consequences}

%\begin{corollary}
%If an affine $Y$ maps $\gg$, and $Tor_{>0}^{\OO_\gg}(\OO_Y, \OO_\Nt)
%$ vanishes. Then the sheaf of algebras $pr_2^*(\DD^\la)$ on
%$Y\times_\gg \Nt$  is derived affine.
%\end{corollary}

%Example. Fix $\chi\in \N$, let $O_\chi=G(\chi)$ Slodowy slices.



%\subsection{Quantization} Reduce



\section{Perverse sheaves on  affine flags of the dual group
(local geometric Langlands).}\label{se5}

%In this section we assume that $\kk$ is of characteristic zero.

\subsection{Generalities on geometric Langlands duality}
 Recall that $\LG$ is the
group dual to $G$ in the sense of Langlands. Several good
 surveys of geometric Langlands duality program has appeared
recently \cite{Edik1}, \cite{Edik}, \cite{GaiICM},
% see also \cite{BD},
 so I will only
briefly recall the set-up.

The geometric Langlands duality is a categorification of the classical
Langlands duality for functional fields. The latter seeks to attach
an automorphic form to a homomorphism from a version of the
 Galois group to $\LG$. In other words, the problem
is to provide
a spectral decomposition for Hecke operators acting in the space of
 automorphic functions, and relate the space of spectral parameters
to homomorphisms of the Galois group  to the dual group. As was
probably first observed by A.~Weil, %\cite{Weil},
 in the case of
 a functional field
the automorphic space in question is the set of isomorphism classes
of $G$-bundles, possibly with an additional level structure, on an
algebraic curve over $\Fq$.  Thus it is the set of $\Fq$ points of
the corresponding moduli space (stack).

Passage to the geometric duality theory is based on the following
variation of Grothendieck's sheaf-function correspondence principle.
The variation says that for an algebraic variety (or stack) over
$\Fq$ a natural categorification of the space of functions on the
set $X(\Fq)$ is the derived category of $l$-adic sheaves on $X$.
Thus the objective of the geometric duality theory is a spectral
decomposition of the  derived category of $l$-adic sheaves on a
moduli space of $G$-bundles, where the space of spectral parameters
is identified with the space of $\LG$ local systems. It is a
non-trivial, and not completely solved, problem to assign a formal
meaning to the previous sentence; however, in some cases it amounts
to an equivalence between the $l$-adic derived category of the
moduli stack and the derived category of coherent sheaves on a stack
mapping to the stack of local systems.

The above formulations referred to a more developed {\em global}
version of the theory. However, the classical Langlands conjectures
have both a global and a local version. The global one provides a
conjectural classification of automorphic representations of the
group of adele points of a reductive group over a global field, i.e.
either a number field, or the field of rational functions on a curve
over a finite field. The local one describes all irreducible
representations of a reductive group over a local field; recall that
a functional local field is a field of formal Laurent series
$\Fq((t))$. The geometric theory studies the derived category of
$l$-adic sheaves on
 homogeneous spaces of the {\em formal loop group} $\bLG((t))$ of the
 dual group $\LG$. The latter is a group ind-scheme over $\Fq$,
 whose group of $\Fq$ points is identified with $\LG(\Fq((t)))$.
The results are expected to link such $l$-adic derived categories to
coherent sheaves on spaces related to $G$-local systems on the
puncture formal disc, cf. \cite{Edik_Denis}.
% See \cite{Ginzb}, \cite{MV}, and also
%

\subsection{Results of \cite{AB}, \cite{B}, \cite{Betobe}}
Some particular results of this type have been achieved in {\em loc.
cit.} % \cite{AB}, \cite{B}, \cite{Betobe}.
%To state them we need some notation.

\subsubsection{Statement of a result}
Recall that the Iwahori subgroup $I\subset \LG((\Fq(t)))$ consists
of those maps from the punctured formal disc to $\LG$, which can be
extended to a map from the whole disc, so that the image of the
closed point is contained in a fixed Borel subgroup $\LB\subset
\LG$.

We have a  group subscheme (pro-algebraic group) $\bI\subset
\bLG((t))$, such that  $\bI(\Fq)=I$. The affine flag space $\Fl$ of
$\LG$ is the homogeneous space $\bLG((t))/\bI$. It is  an
ind-algebraic variety such that $\Fl(\Fq)=\LG(\Fq((t)))/I$.
 The
group $\bI$ acts on $\Fl$. The orbits of this action, called affine
Schubert cells, are in bijection with the affine Weyl group
$W_{aff}$.

Let $\P$ denote the category of perverse sheaves on $\Fl$, which are
equivariant with respect to the prounipotent radical of $\bI$. Let
$\P^I\subset \P$ be the full subcategory of $\bI$ equivariant
sheaves.

Let $^{nf}\P^I\subset \P^I$ be the Serre subcategory generated by
irreducible objects, corresponding to those $w\in W_{aff}$, which
are not the minimal length representatives of a left $W$ coset. Let
$^f\P=\P/^{nf}\P$ be the Serre quotient category.


\begin{remark}
To clarify the definition of $^f\P$ we remark  that this category
can be also described as the category of Iwahori-Whittaker sheaves
\cite{AB}. Thus it is related to the Whittaker model, which is one
of the main tools in representation theory of reductive groups over
local and global fields.
\end{remark}

\begin{theorem}\label{equiv}
a) \cite{Betobe} We have a canonical equivalence $D^b(\P)\cong
D^G(\gt\times _\gg \Nt)$.

b) \cite{AB} We have a canonical equivalence $D^b(^f\P)\cong
D^G(\Nt)$. The image of $^f\P$ under this equivalence consists of
equivariant exotic sheaves.

\end{theorem}

%\subsubsection{Motivation and comments on the proof}
The Theorem is motivated by the  known isomorphisms of
Grothendieck groups; the question of possibility of such (or
similar) equivalence has been raised, e.g., by V.~Ginzburg, see
Introduction to \cite{CG}. More precisely, the Grothendieck group of
the two categories appearing in Theorem \ref{equiv}(a) are
isomorphic to the
 group algebra of the affine Weyl group of $\LG$. A more
 interesting version of the isomorphism is obtained by replacing the
 categories by their graded version: $D^G(\gt\times _\gg \Nt)$ is
 replaced by $D^{G\times \Gm}(\gt\times \Nt)$, while the definition
 of the graded version of $\P$ is more subtle
 (cf. Proposition \ref{qFr} below
 and also \cite{BGS}). The corresponding Grothendieck groups turn
 out to be isomorphic to {\em the  affine Hecke algebra},
 see \cite{CG}, \cite{LKthII}I.

Similarly, the  Grothendieck groups of both categories appearing in
Theorem \ref{equiv}(b) are identified with the {\em anti-spherical}
module over the extended affine Weyl group, while in the graded
version of the theory we get the anti-spherical module over the
affine Hecke algebra. Here by the anti-spherical module we mean the
induction of the sign representation from the finite Weyl group
(respectively, Hecke algebra) to the affine one.

I would like to emphasize that this isomorphism of the two
realizations of the affine Hecke algebra is the key step in the
proof of classification of its representations due to Kazhdan and
Lusztig \cite{KL} (see also \cite{CG}), which establishes a
particular case of the local Langlands conjecture. This is another
illustration of the relation of Theorem \ref{equiv} to local
Langlands duality.

\medskip

The proof of Theorem \ref{equiv} builds on previously known
constructions of categories related to $G$ in terms of perverse
sheaves on homogeneous spaces for $\bLG((t))$. The first important
result is {\em the geometric Satake isomorphism} \cite{Ginzb},
\cite{MV}, \cite{BD}, which identifies the tensor category $Rep(G)$
of algebraic representations with the category of perverse sheaves
on the affine Grassmannian $\Gr=\bLG((t))/\GO$ equivariant with
respect to $\GO$. Here $\GO\subset \bLG((t))$ is the group
subscheme, such that $\GO(\Fq)$ consists of maps which extend to the
non-punctured disc. Furthermore, Gaitsgory \cite{Zen} used this
result to provide a categorification of the description of the {\em
center} of the affine Hecke algebra. Using some ideas of
I.~Mirkovi\' c we observe that the so-called Wakimoto sheaves
provide a categorification of the maximal abelian subalgebra in the
affine Hecke algebra due to Bernstein, see, e.g., \cite{CG},
\cite{LKthII}I.
 The maximal
projective object in the category of sheaves on the finite
dimensional flag variety  of $\LG$ smooth along the Schubert
stratification, which plays a central role in Soergel's description
of category $O$, cf. \cite{SoerICM}, is
  a categorification of the
$q$ anti-symmetrizer (an element of the finite Hecke algebra, which
acts by zero in all irreducible representation except for the sign
representation). Under the equivalence of Theorem \ref{equiv}(a) it
corresponds to the structure sheaf of $\gt\times _\gg \Nt$. A
combination of these ingredients yields a proof of the Theorem.


\subsubsection{Possible generalizations}
It is natural to ask if the multiplication in the affine Hecke
algebra corresponds to a monoidal structure on the derived
categories of coherent sheaves and constructible sheaves appearing
in Theorem \ref{equiv}(a). In order to get such a monoidal
structure, we need to replace the categories defined above  by
closely related ones with the same Grothendieck group. One way to do
it is as follows. Let $\bI'$ be the pro-unipotent radical of $\bI$.
Let $\P'$ be the category of perverse sheaves on "the basic affine
space" $\bLG((t))/\bI'$, which are $\bI$-monodromic with unipotent
monodromy. Then convolution provides the derived category $D^b(\P')$
with a monoidal structure. Notice that this monoidal category does
not have a unit object, though this can be repaired by adding some
pro-objects to the category, the unit object is then the free
pro-unipotent local system on $\bI/\bI'\subset \bLG((t))/\bI'$.

On the dual side we consider the category $D^G(\gt\times _\gg\gt)$.
Then one can show that convolution provides this category with a
monoidal structure. Let $Coh^G(\gt\times _\gg\gt)'\subset
Coh^G(\gt\times_\gg \gt)$ denote the full subcategory of complexes,
whose cohomology sheaves are set-theoretically supported on the
preimage of $\N\subset \gg$. A standard argument shows that it
yields a full embedding of derived categories $D^G(\gt\times
_\gg\gt)':=D^b (Coh^G(\gt\times _\gg\gt)')$ into  $D^G(\gt\times_\gg
\gt)$. The full subcategory $D^G(\gt\times _\gg \gt)'\subset D^G(\gt
\times _\gg \gt)$ is closed under the convolution product, though it
does not contain the unit object $\delta_*(\O)$, where
$\delta:\gt\to \gt\times _\gg \gt$ is the diagonal embedding.

It is easy to see that the push-forward (respectively, pull-back)
functors $Coh^G(\gt\times _\gg \Nt)\to Coh^G(\gt\times_\gg \gt)'$,
$\P\to \P'$ induce isomorphisms of Grothendieck groups.

\begin{theorem} \cite{Betobe}
We have a natural monoidal equivalence $D(\P')\cong D^G(\gt\times
_\gg \gt)'$.
\end{theorem}

%See \cite{EdikDenis} for an application (??).

\begin{remark}
Another version of Theorem \ref{equiv} links the monoidal $\bI$
equivariant derived category to the monoidal derived category of
$G$-equivariant coherent sheaves on the fiber square of $\Nt$ over
$\gg$. An additional subtlety in this case is related to
nonvanishing of  $\Tor_{>0}^{\O_\gg}(\O_\Nt, \O_\Nt)$. One actually
has to take these $\Tor$ groups into account by working with the
{\em derived fiber product}, which is a differential-graded scheme,
rather than an ordinary scheme. This issue does not arise in the
other settings mentioned above, because
$\Tor_{>0}^{\O_\gg}(\O_\gt,\O_\gt)=0=\Tor_{>0}^{\O_\gg}(\O_\gt,\O_\Nt)$.
However, one has to work with differential graded schemes in order
to define the convolution product on $D^G(\gt\times _\gg\gt)$.
\end{remark}


\subsubsection{Relation to the material of section \ref{se2}}
Many of the constructions from section \ref{se2} are motivated by
the equivalences of Theorem \ref{equiv}.
% We now explain this point in more detail.

%First of all, notice that
For example, the categories $D(\P)$, $D(\P')$ carry a natural
$B_{aff}$ action by {\em Radon transforms}, cf. beginning of section
\ref{derloc}, where a similar structure for a finite dimensional
flag variety is mentioned. To define the action we recall that the
$\bLG((t))$ orbits on $\Fl^2$ are indexed by the affine Weyl group.
% (we assume for simplicity the $\LG$ is simply-connected).
If $\Fl_w^2$ is the orbit corresponding to $w\in W_{aff}$, and
$pr_i^w:\Fl_w^2\to \Fl$ are the projections, where $i=1,2$ then we
define a functor $R_w:D(\P)\to D(\P)$ by $R_w(\F)=pr^w_{2*}
pr_1^{w*}(\F)$. Then we have an action of $B_{aff}$ on $D(\P)$,
$D(^f\P)$, such that $\tii w\mapsto R_w$. Under the equivalences of
Theorem \ref{equiv}(b) this action corresponds to the action
described in section \ref{se2}.

%Also, the description of equivariant exotic $t$-structure via
%mutations of exceptional sets (see section \ref{mutasect}) turns
%into a standard description of the perverse $t$-structure on
%constructible sheaves under the equivalence of Theorem
%\ref{equiv}(b) (cf. \cite{izvrat}).


Finally, I would like to quote the statement that allows to link the
grading on $\Ext$ spaces appearing in  Theorem \ref{dpst} to
Frobenius weights, thus providing a way to prove Theorem \ref{dpst}.
%
To state it we introduce the following notation.
Let $\Phi$ be either of the two equivalences appearing in Theorem
\ref{equiv}. Let $Fr$ be the autoequivalence of the corresponding
category of constructible sheaf, sending a sheaf to its pull-back
under the Frobenius morphism.
 Let $\bq$ be an automorphism of either $\Nt$ or $\gt\times _\gg \Nt$
given by $(\b,x)\mapsto (\b, qx)$ or $(\b_1,\b_2,x)\mapsto (\b_1,
\b_2, qx)$ respectively; here $q$ stands for the cardinality of the
base finite field $\Fq$.

\begin{proposition}\label{qFr} (cf. \cite{AB})
 We have a canonical isomorphism $\Phi\circ
\bq^*\cong Fr\circ \Phi$.
\end{proposition}
%\subsection{Sketch of proof of Theorem \ref{dpst}.}






\begin{thebibliography}{1}

\bibitem{Rina} R.~Anno, {\em Spherical functors and braid group actions,}
in preparation.

\bibitem{AJS}
H.H.~Andersen, J.C. Jantzen, W. Soergel
{\em Representations of quantum groups at a $p$th root of unity
and of semisimple groups in characteristic $p$: independence of $p$,}
 Ast\' erisque {\bf 220} (1994), 321 pp.

\bibitem{AB} S.~Arkhipov S.,  R.~Bezrukavnikov,
\emph{Perverse sheaves on affine flags and Langlands dual group,} electronic
preprint, math.RT/0201073, to appear in Israel Math. J.


\bibitem{ABG} S. Arkhipov,  R. Bezrukavnikov,  V. Ginzburg
\emph{Quantum Groups, the loop
Grassmannian, and the Springer resolution,}
J. Amer. Math. Soc.
{\bf 17} (2004), no. 3, 595--678

\bibitem{BeBe} A.~Beilinson, J.~Bernstein,
\emph{Localisation de $g$-modules,}
 (French) C. R. Acad. Sci. Paris S\' er. I Math. 292 (1981), no. 1, 15--18.




\bibitem{BeBeder}
A. Beilinson, A. Bernstein,
{\em
A generalization of Casselman's submodule theorem,}
  Representation theory of reductive groups (Park City, Utah, 1982),  35--52,
 Progr. Math., 40, Birkh\" auser Boston, Boston, MA, 1983.




\bibitem{BBD} A.~Beilinson, J.~Bernstein, P.~Deligne,
\emph{Faisceaux pervers,}
  5--171, Ast\' erisque, {\bf 100}, Soc. Math. France, Paris, 1982.



\bibitem{BD} A. Beilinson, V. Drinfeld,
{\em
Quantization of Hitchin's Integrable System and Hecke Eigensheaves},
preprint
available at http://www.math.uchicago.edu/$\sim$arinkin/langlands/


\bibitem{BGS}
A. Beilinson, V. Ginzburg, W. Soergel,
\emph{Koszul duality patterns in representation theory,}
J. Amer. Math. Soc. {\bf 9} (1996), no. 2, 473--527.


\bibitem{BelKa}
A. Belov-Kanel, M. Kontsevich, {\em
Automorphisms of the Weyl algebra}
Lett. Math. Phys. {\bf 74} (2005), no. 2, 181--199.




\bibitem{vdB} M.~van den Bergh, {\em Noncommutative crepant resolutions,}
The legacy of Niels Henrik Abel, 749--770, Springer, Berlin, 2004.



\bibitem{B}  R. Bezrukavnikov
{\em Perverse sheaves on affine flags and nilpotent cone
 of the Langlands dual group,} electronic preprint
math.RT/0201256,  to appear in  Israel J. Math.



\bibitem{izvrat} R. Bezrukavnikov,
 {\em Perverse coherent sheaves (after Deligne),}
electronic preprint,
math.AG/0005152.

%\bibitem{nilpokon} R. Bezrukavnikov, {\em Quasi-exceptional sets and
%equivariant coherent sheaves on the nilpotent cone,}
%Represent. Theory {\bf 7} (2003), 1-18.

\bibitem{Hum} R. Bezrukavnikov, {\em Cohomology of tilting modules over
 quantum groups and $t$-structures on derived categories of coherent sheaves,}
to appear in Inv. Math.

\bibitem{Betobe}
  R. Bezrukavnikov, {\em An equivalence between two categorical
realization of the affine Hecke algebra,} in preparation.



\bibitem{BFG}  R. Bezrukavnikov, M. Finkelberg, V. Ginzburg.
{\em Cherednik algebras and Hilbert schemes in characteristic p, }
 electronic preprint, math.RT/0312474, to appear in Represent. Theory.



\bibitem{BeKal} R.  Bezrukavnikov, D. Kaledin
{\em McKay equivalence for symplectic resolutions of quotient
singularities,}
Tr. Mat. Inst. Steklova {\bf 246} (2004),
Algebr. Geom. Metody, Svyazi i Prilozh., 20--42;
 translation in Proc. Steklov Inst. Math. 2004, no. 3 (246), 13--33.



\bibitem{BeKalqua} R. Bezrukavnikov, D. Kaledin
{\em Fedosov quantization in positive characteristic}, electronic preprint,
math.AG/0501247.

\bibitem{BMR}  R.  Bezrukavnikov, I.~Mirkovic, D.~Rumynin,
{\em Localization of modules for a semisimple Lie algebra in prime
characteristic,} electronic preprint math.RT/0205144, to appear in
Annals  of Math;
%\bibitem{BM} R.  Bezrukavnikov, I.~Mirkovic, D.~Rumynin,
{\em Singular localization and intertwining functors for
semisimple Lie algebras in prime characteristic,}  electronic
preprint math.RT/0602075, submitted to Nagoya Math. J.

%\bibitem{BO}  Bezrukavnikov Ostrik

\bibitem{BoKa} A. Bondal, M. Kapranov,
{\it Representable functors, Serre functors, and mutations,} Izv.
Ak. Nauk {\bf 35} (1990), no. 3, 519--541.

\bibitem{BonOr}  A. Bondal,
D. Orlov,
{\em Derived categories of coherent sheaves,}
 Proceedings of the International Congress of Mathematicians,
 Vol. II (Beijing, 2002), 47--56, Higher Ed. Press, Beijing, 2002.

\bibitem{Bridge} T.~Bridgeland  {\em Derived categories of coherent sheaves}
report of ICM 2006, available as an electronic preprint math.AG/0602129.

\bibitem{Brstab} T. Bridgeland, {\em Stabilities on triangulated categories,}
electronic preprint math.AG/0212237, to appear in Ann. Math.

\bibitem{BrK3} T. Bridgeland, \emph{Stability conditions on K3 surfaces,}
electronic preprint math.AG/0307164;
%
%\bibitem{BrP2} T. Bridgeland,
{\em  Stability conditions on a non-compact Calabi-Yau threefold,}
electronic preprint math.AG/0509048;
%
%\bibitem{Brbr} T. Bridgeland,
{\em
 Stability conditions and Kleinian singularities,} electronic preprint
math.AG/0508257.


\bibitem{Brflop} T. Bridgeland
{\em Flops and derived categories,} Invent. Math.
{\bf 147} (2002), no. 3, 613--632.


\bibitem{BrGo} K. Brown, I. Gordon,
{\it  The ramification of centres: Lie algebras in positive characteristic
 and quantised enveloping algebras,}  Math. Z.  {\bf 238}  (2001),
 no. 4, 733--779.

\bibitem{BrylKa}
J.-L. Brylinski, M. Kashiwara,
{\em Kazhdan-Lusztig conjecture and holonomic systems,}
Invent. Math. {\bf 64} (1981), no. 3, 387--410.

\bibitem{CG} N. Chriss, V. Ginzburg
{\em Representation theory and complex geometry,}
Birkh\" auser Boston, Inc., Boston, MA, 1997.


%\bibitem{DLP}
%C. De Concini, G. Lusztig,  C. Procesi,
%{\em
% Homology of the zero-set of a nilpotent vector field on a flag manifold,}
%J. Amer. Math. Soc.  {\bf 1}  (1988),  no. 1, 15--34.

\bibitem{Edik1} E. Frenkel, {\em
Recent advances in the Langlands program,}
Bull. Amer. Math. Soc. (N.S.) {\bf 41} (2004), no. 2, 151--184.

\bibitem{Edik} E. Frenkel,
{\em Lectures on the Langlands Program and Conformal Field Theory,}
electronic preprint hep-th/0512172.



\bibitem{Edik_Denis} E. Frenkel, D. Gaitsgory,
{\em Local geometric Langlands correspondence and affine Kac-Moody
 algebras,} electronic preprint math.RT/0508382.



\bibitem{Zen}
D. Gaitsgory,
\emph{Construction of central elements in the affine Hecke algebra
via nearby cycles,}
 Invent. Math.  {\bf 144}  (2001),  no. 2, 253--280.

\bibitem{GaiICM} D. Gaitsgory, {\em
Geometric Langlands correspondence for ${\rm GL}\sb n$,}
 Proceedings of the International Congress of Mathematicians,
 Vol. II (Beijing, 2002), 571--582, Higher Ed. Press, Beijing, 2002.

%\bibitem{GinICM} Ginzburg ICM

\bibitem{Ginzb} V. Ginzburg,
{\em Perverse sheaves on a Loop group and Langlands' duality,}
 electronic preprint alg-geom/9511007.

\bibitem{GinGan} W.-L. Gan, V. Ginzburg
{\em Quantization of Slodowy slices,}
 Int. Math. Res. Not.  {\bf 5} (2002) 243--255.




\bibitem{hren}
W. H\" urlimann,
{\em Sur le groupe de Brauer d'un anneau de polyn\^ omes en
caract\' eristique $p$
et la th\' eorie des invariants,} in
``The Brauer group'' (Sem., Les Plans-sur-Bex, 1980), pp. 229--274,
Lecture Notes in Math. {\bf 844},
Springer, Berlin (1981).


\bibitem{KDC} C. De Concini, V. Kac,
{\em Representations of quantum groups at roots of $1$,}
 Operator algebras, unitary representations, enveloping algebras,
 and invariant theory (Paris, 1989), 471--506, Progr. Math.,
 92, Birkh�user Boston, Boston, MA, 1990.

\bibitem{Kal} D. Kaledin,
{\em Derived equivalences by quantization,} electronic preprint
math.AG/0504584.




\bibitem{Kashiwa} M.Kashiwara,
{\em On crystal bases of the q-analogue of universal enveloping
algebras,} Duke {\bf 63} (1991), 465--516.


\bibitem{KL} D. Kazhdan, G. Lusztig,
{\em Proof of the Deligne-Langlands conjecture for Hecke algebras,}
 Invent. Math. {\bf 87} (1987), no. 1, 153--215.

\bibitem{Kobi}  E.~Backelin, K.~Kremnitzer,
{\em Localization for quantum groups at a root of unity,}
  electronic preprint math.RT/0407048.


%\bibitem{KumarLaur} Kumar Lauritzen


\bibitem{MV}  I. Mirkovi\' c, K. Vilonen
{\em Perverse Sheaves on affine Grassmannians and Langlands Duality,}
 electronic preprint math.AG/9911050.




%\bibitem{Lam} Laumon about Langlands

%\bibitem{LuHeck} G. Lusztig, {\em
%Representations of affine Hecke algebras,}
% Orbites unipotentes et repr�sentations, II. Ast\' erisque
%{\bf 171-172}
% (1989), 73--84.

\bibitem{LKthII}
G.Lusztig,
{\em Bases in equivariant $K$-theory},
Represent. Theory {\bf 2} (1998), 298--369.
{\em Bases in equivariant $K$-theory II},
Represent. Theory {\bf 3} (1999), 281-353.

\bibitem{LConj} G. Lusztig, {\em Some problems in the representation
theory of finite Chevalley groups,} Proc. Symp. Pure Math., 37
(1980), pp 313--317.

%\bibitem{LuBerngen} Lusztig, Bernstein's generators for H aff

%\bibitem{Milne}
%J.S.Milne,
%{\em
%Etale cohomology}.
%Princeton Math. Series, 33.
%Princeton U. Press (1980).


\bibitem{OV} A. Ogus, V. Vologodsky,
{\em Nonabelian Hodge Theory in Characteristic p,} preprint
math.AG/0507476.



\bibitem{Premet} A. Premet
{\em Special transverse slices and their enveloping algebras,}
With an appendix by Serge Skryabin,
Adv. Math. {\bf 170} (2002), no. 1, 1--55.

%\bibitem{Riche} Riche

%\bibitem{ST} Seidel, Thomas

%\bibitem{Soer} W. Soergel, {\em
%Kategorie $\mathcal O$, perverse Garben und Moduln \" uber den Koinvarianten
%zur Weylgruppe,} J. Amer. Math. Soc. {\bf 3} (1990), no. 2, 421--445.

\bibitem{SoerICM} W. Soergel, {\em Gradings on representation categories,}
 Proceedings of the International Congress of Mathematicians, 
(Z\" urich, 1994), 800--806, Birkh\" auser, Basel, 1995.

%\bibitem{Weil} Weil
\end{thebibliography}

\end{document}







\section{Sketches of proofs}
{\bf Do I want this section?}

Statement of quadratic relations.

Uniqueness.

Braid relations via exceptional sets.



CY trick.

Deformation theory arguments -- reduction of existence to char p.

OR: using exceptional sets again.

Rigidity and




\subsection{Some details on Geometric Langlands}

I thought I'd something more to say.

\end{document}

\subsubsection{Equivalences with coherent sheaves}

Combining the previous section with \ref{split_sect} we get an
equivalence with coherent sheaves.

Also get $D^b(Coh(\Nt))\cong D^b(A-\rmod^{fg})$ for some $A$ etc.

Lifting to characteristic zero by deformation theory.

\subsection{Generalizations}

Kaledin and MacKay.

Bridgeland's stabilities (??).



\section{(Local) Geometric Langlands}
The same derived categories of coherent sheaves (or rather their
equivariant versions) appear in quite a different construction, leading,
however, to a closely related t-structure. To explain it we need to
recall briefly some ideas of the Geometric Langlands Duality theory
(see [Gaitsgory], [BD] for more details).

\subsubsection{Generalities on Geometric Langlands Duality}
Geometric Langlands Duality program (due mostly to Beilinson and
Drinfeld) has its roots in the classical Langlands duality
conjectures, more precisely, in Drinfeld's proof \cite{Dr} of a
particular case of these conjectures.  The classical Langlands
duality  theory has two parts: the global and the local ones. The
global (respectively, local) theory starts with a reductive group
$\bG$ over a global field $\ff$ (respectively, a local field $F$).
The conjectures relate homomorphisms of (a variant of) the absolute
Galois groups $Gal(\ff)$, $Gal(F)$ into the Langlands dual group
$\Gcheck$ to, respectively, irreducible automorphic representations
of the adele group $\bG(A_\ff)$ (where $A_\ff$ is the ring of
adeles), or all irreducible (continuous) representations of the
group $\bG(F)$.

Assume that $\ff$, $F$ are functional fields, i.e., $\ff$ is the
field of rational functions on a reduced irreducible  curve $C$ over
a finite field and $F\cong \bbF_q((t))$. Then a homomorphism
$Gal(\ff)\to \Gcheck$ can be interpreted as a $\Gcheck$ local system
on an open curve $C_0\subset C$, while a representation of $Gal(F)$
is a local system on the punctured formal disc.

Also, the quotients of the automorphic space $\bG(A_\ff)/\bG(\ff)$,
and of $\bG(F)$ by an appropriately chosen open compact subgroup
$U\subset \bG(A_\ff)$ (respectively, $U=U_1\times U_2\subset
\bG(F)^2$) can be interpreted as the set of $\bbF_q$ points of a
certain ind-algebraic stack over $\bbF_q$. Grothendieck's
sheaf-functions correspondence principle predicts that for an
$\bbF_q$ scheme (or stack) $X$ an "interesting" function $f$ on the
set $X(\bbF_q)$ with values in a characteristic zero field should
appear as a trace of Frobenius function, i.e., it should have the
form $x\mapsto Tr(Frob, \F_x)$ for an etale sheaf (or complex of
sheaves) $\F$ on $X$.
 In Geometric Langlands Duality theory one hopes to obtain the
statements about the spaces of functions on $U\backslash
\bG(A_\ff)/\bG(\ff)$, $U\backslash G(F)$ predicted by the classical
Langlands conjecture in the case of a functional field as
consequences of some richer statements about the category of etale
sheaves on the corresponding scheme (or stack).

The classical Langlands conjectures imply a spectral decomposition
of the space of functions on $U\backslash \bG(A_\ff)/\bG(\ff)$ or on
$U\backslash G(F)$, where the set of spectral parameters is
identified with the set of Galois representations into $\Gcheck$.
The geometric statement is expected to provide a "spectral
decomposition" for the (derived) category of etale sheaves on the
corresponding moduli stack of local systems (see [Dennis cat over a
stack] for the notion of an {\em abelian} category over a stack; one
expects that similar (though subtler) techniques  can be used to
define the notion of a triangulated (or rather differential graded)
category over a stack).

A typical example of an abelian category over a (Noetherian) stack
$X$ is the category $Coh(X,\AA)$ of sheaves of modules over a
coherent sheaf of associative $\OO_X$ algebras on $X$.



Notice that a non-commutative resolution discussed above was also
defined as a sheaf of non-commutative algebras satisfying some
properties; thus it is perhaps not surprising that the two subjects
are related.



xxxxxxxxxxxxxxxxxx


 Geometric Langlands philosophy see [Gaitsgory ICM
talk and papers] -- equivalences.


\section{Possible direct descriptions of the $t$-structure}

Mutations of the exceptional set in $D^b(Coh^G(\Nt))$.

Bridgeland's stabilities (?)



\bigskip

Plan.

Introduction.


 Generalities on NC resolution with derived equivalences.

t-structures arising in char p
 and braid group actions. relation to
Bridgeland stability. Applications to modulo reps.

Local geometric Langlands: same braid group action etc.

Concluding discussion: Frenkel-Gaitsgory; mirror symmetry???

\bigskip

Our examples fit in the following framework.
 Recall the definition of an NC resolution from [Bondal Orlov]
{\bf do we need it?}

Crepant (NC) resolutions and conjectured equivalences.

Replacing $X$ by an NC (affine) resolution -- derived affinity --
$t$-structure.

section: {Characteristic $p$ constructions}

$D$-modules in char $p$; Azumaya algebra,

Derived $D$-affinity;

Splitting;

t-structure.

Induced t-structure on base changes with vanishing Tor's.

Question -- how to make it independent of $p$ (Example of $P^1$).

Braid positivity.

Cor. Andersen-Jantzen-Soergel.

Digression -- relation to Bridgeland's stability work. Maybe mention
Seidel-Thomas and mirror symmetry?

subsection: {Numerical questions}

Lusztig conjectures. Discussion of what's needed -- Positivity
Lemma, cf. Humphr.

section: {Geometric Langlands}

Two realizations of $H_{aff}$. Categorification. (Copy from grant
proposal or paper introduction)

Claim. Braid positivity holds. (B/c convolution with $!, *$)

\end{document}

 Let $k$ be an algebraically closed field,
 $\pi:X\to Y $ be a projective morphism of quasi-projective
varieties over $k$ where $Y$ is affine (the latter assumption is not
essential and made to simplify the exposition). We will only be
interested in the situation when  $X$ is smooth (though this may not
be necessary for some of the definitions). Let $D(X)= D^b(Coh_X)$ be
the bounded derived category of coherent sheaves on $X$. We will say
that $\pi$ is {\it NC-affine} if there exists a finite
$\O(Y)$-algebra $A$ and an equivalence of triangulated categories
\begin{equation}\label{equiv}
\phi:D^b(A-\rmod)\iso D(X)
\end{equation} compatible with the $\O(Y)$ linear
structure on the categories; %should we require more?
 here $A-\rmod$ denotes the category of finitely generated $A$-modules.



Examples of such a situation have recently become the object of
attention of several researchers; some general conjectures have been
made [???].





\subsubsection{} An equivalence \eqref{equiv} yields a preferred
object $\E_\phi\in D(X)$,  namely the image of the free rank 1
module under $\phi$. This object is {\it tilting}, i.e. satisfies
$Ext^i(\E_\phi,\E_\phi)=0$ for $i\ne 0$. Moreover, it is a generator
for $D(X)$, i.e. the thick subcategory generated by $\E_\phi$ is the
whole  $D(X)$ (notice that the algebra $A$ appearing above
necessarily has finite homological dimension, so the thick
subcategory in $D^b(A-\rmod)$ generated by the free module is the
whole category). Conversely, given a tilting generator $\E$ of
$D(X)$ we get an equivalence $\phi_\E:D^b(A_\E-\rmod)\iso D(X)$ where
$A_\E=End(\E)$, and $\phi^{-1}:V\mapsto RHom(\E,V)$ (cf. e.g. [??]).

In all examples below the tilting generator $\E_\phi$ will in fact
be a {\it locally free} sheaf on $Y$.

%\medskip


\subsubsection{}
On the other hand, equivalence  \eqref{equiv} defines a {\it
t-structure} on the triangulated category $D(X)$, the tautological
$t$-structure on $D^b(A-\rmod)$ transported by means of $\phi$. The
object $\E_\phi$ is a projective generator for the heart of this
$t$-structure. Though the $t$-structure does not determine the pair
$(A,\phi)$ uniquely, it does determine $A$  up to Morita
equivalence.

  This  t-structure on $D(X)$ will be our primary object of consideration.
(Notice that the property of $\E_\phi$ to be localy free does not
depend on the choice of a projective generator).



\subsubsection{}
Let us now describe our class of examples. Let $G$ be a semi-simple
algebraic group over $k$; let $\gg$ be its Lie algebra and $\N\subset
\gg$ be the nilpotent cone. We assume that $char(k)>h$ or
$char(k)=0$,
 where $h$ is the Coxeter number
of $\gg$. We fix a nilpotent element $\chi\in \N$, and let
$S_\chi\subset \gg$ be the (Slodowy) transversal slice to the orbit
$G(\chi)$ (see [??]); we also set $Y'_\chi=Y_\chi\cap \N$.




\end{document}

\section{Differential operators and exotic $t$-structures}
\subsubsection{Differential operators and Azumaya algebras} Let $X$
be a smooth variety over a perfect field $k$. If $char(k)=0$ we let
$D_X$ be the sheaf of differential operators. If  $char(k)=p>0$ we
let $D_X$ be the sheaf of "crystalline" differential operators on
$X$ (differential operators without divided powers, PD differential
operators in the terminology of Ogus), cf. [BMR].

(Part (a) of) the following fundamental observation belongs to
Mirkovic and Rumynin, see [BMR] for details.

{\bf Theorem.} {\em Assume that $char(k)=p>0$.

a) $D_X$ contains a central subalgebra canonically isomorphic to the
algebra $\O_{T^*X\tw}$ (the latter is viewed as a sheaf of algebras
on $X$ by means of the obvious affine projection  $T^*X\tw\to X\tw$)
(this is the "$p$-curvature" construction). Thus $D_X$ can be viewed
as a sheaf of algebras on $T^*X\tw$. This sheaf is an Azumaya
algebra of rank $p^{2\dim X}$
 on the Frobenius twist of the cotangent bundle
$T^*X\tw$.

b) If $L$ is a line bundle on $X$ then the sheaf $D_X(L)$ of
crystalline differential operators on $L$ is also an Azumaya algebra
on $T^*X\tw$, which is equivalent to $D_X$ (more precisely, $D_X(L)$
is an inner twist of $D_X$). }

\subsubsection{Splitting on the zero section}\label{zersec}
Recall that an Azumaya algebra is called split if it is isomorphic
to $End(V)$ for a vector bundle $V$. If this is the case we will
call $V$ a splitting vector bundle. $V$ is defined uniquely up to
twist by a line bundle.

%[Misha: Hyperkahler quotient, critical q-n of Hilb.]



\end{document}

\section{Introduction}








(Part (a) of)
the following fundamental observation belongs to Mirkovic and Rumynin,
see [BMR] for details.

{\bf Theorem.} {\em Assume that $char(k)=p>0$.

a) $D_X$ contains a central subalgebra canonically isomorphic to
the algebra $\O_{T^*X\tw}$ (the latter is viewed as a sheaf of algebras
on $X$ by means of the obvious affine projection  $T^*X\tw\to X\tw$)
(this is the "$p$-curvature" construction). Thus $D_X$ can be viewed
as a sheaf of algebras on $T^*X\tw$. This sheaf
is an Azumaya algebra of rank $p^{2\dim X}$
 on the Frobenius twist of the cotangent bundle
$T^*X\tw$.

b) If $L$ is a line bundle on $X$ then the sheaf $D_X(L)$ of
crystalline differential operators on $L$ is also an Azumaya algebra
on $T^*X\tw$, which is equivalent to $D_X$ (more precisely, $D_X(L)$
is an inner twist of $D_X$).
}

\subsubsection{Splitting on the zero section}\label{zersec}
Recall that an Azumaya algebra is called split if it
is isomorphic to $End(V)$ for a vector bundle $V$. If this is the case
we will call $V$ a splitting vector bundle. $V$ is defined uniquely up to
twist by a line bundle.

{\bf Claim.} {\em
The restriction of the Azumaya algebra $D_X(L)$ to the zero section
$X\tw\overset{\iota_0}{\imbed} T^*X\tw$ is split; more precisely, we have
$\iota_0^*(D_X(L))=End (Fr_*(L))$, where $Fr:X\to X\tw$ is the Frobenius.

Moreover, assume\footnote{This is
a very strong condition, which holds for $X=\Pn$, but does not hold
for any other projective homogeneous space.}
 that $X$ and $Fr_X$ are lifted to $Z/p^2$. Then\footnote{This result
is due to Vadim Vologodsky.} $D_X(L)$ is split on the formal
neighborhood $X_\infty\tw$ of the zero section in $T^*X\tw$;
more precisely, we have $D_X(L)|_{X_\infty}\cong
End(pr^* Fr_* (L))|_{X_\infty}$,
where $pr:T^*X\tw\to X\tw$ is the projection.}

\subsubsection{Variant.} \label{vartor} (see \cite{BMR}I, \S 2.3)
 Let $T=\Gm\otimes \Lambda$ be a torus, and $\t$ be its Lie algebra;
 let $\Xt$ be a $T$-torsor
on $X$. Then we get a sheaf $\Dt_X$ of twisted (crystalline) differential
operators. The sheaf $\Dt_X$ defines an Azumaya algebra of rank $p^{2\dim X}$
on $\tcT X\tw \times _{{\t^*}\tw}\t^*$, where
$\tcT X$ is the total space of the extension
$0\to \Omega^1 \to \widetilde \Omega^1 \to \Lambda\otimes \O\to 0$
associated to $\Xt$; the projection $\tcT X\tw\to {\t^*}\tw$ is the
tautological one; and the map $\t^*\to {\t^*}\tw$ is the Artin-Schreier
map given on linear functions by $t\mapsto t^p-t^{[p]}$, $t\in \t$.

\medskip

We will only deal with the restriction of the Azumaya algebra $\Dt_X$
to the formal neighborhood of the closed subvariety $T^*X\tw
\overset{i_\lambda}{\imbed}
\tcT X\tw \times _{{\t^*}\tw}\t^* $,
where $i_\lambda$ is the  imbedding of the preimage of $d\lambda
\in \t^*$ for a character $\lambda:T\to \Gm$. This formal neighborhood is
canonically isomorphic to the formal neighborhood of $T^*X\tw$
in $\tcT X\tw$ (the preimage of zero under the projection $\tcT X\to
\Lambda \otimes \Ga$).

Modules over $\Dt_X$ supported on an infinitesimal neighborhood
of the image of $i_0$ will be called {\it infinitesimally twisted}
$D_X(L)$-modules (with respect to the torsor $\Xt$).


\subsection{Derived $D$-affinity}
 Let $G$ be a simple (simply-connected) algebraic group over an algebraically
closed field $k$; $\gg$ be its Lie algebra. We fix
 a parabolic $P\subset G$, and set $X=\P=G/P$,
  $\Nt^P=T^*(X)$
 (when $P=B$ is a Borel subgroup we also use the notation $\B=\P$).

Recall that a smooth variety $X$ over a field $k$,   $char(k)=0$ is
called {\it $D$-affine} if the functor of global sections provides
an equivalence between the categories of $D$-modules on $X$, and
modules over the algebra of global differential operators
$\Gamma(X,D_X)$. A Theorem of Beilinson and Bernsetin asserts that
$X_P$ is $D$-affine (and these are the only known examples of
$D$-affine projective varieties (?)); moreover, for $P=B$ the
algebra of global sections is identified with a quotient of the
enveloping algebra $U(\gg)$ by a central character.

The obvious analogue of this notion for crystalline differential operators
in positive characteristic is vacuous: for a line bundle $L$ on $X$
the line bundle $L^p$ has a structure of a $D_X$-module, but can have nonzero
higher cohomology.

However, let us say that $X$ is {\it derived $D$-affine} if the derived
functor $R\Gamma: D^b(D_X-\rmod) \to D^b(Mod(\Gamma(X,D_X)))$ is an equivalence
of triangulated categories (in obvious notations).

The following result (for $P=B$; the general case is similar) is proved in
[BMR].






We appreciate if your contribution comprises between 10 and 25 pages.
The text should be addressed to a general mathematical audience; the
background and the ideas behind the results should therefore be suitably
explained.

Authors are requested to use  standard \LaTeX\ and the class file
\begin{verbatim}
icmart.cls
\end{verbatim}
This file is essentially `article.cls', slightly changed, loading amsmath,
amsfonts, amssymb, latexsym, and with amsthm.sty included. It sets the page size to
\begin{verbatim}
\textheight=548pt
\textwidth=357pt
\end{verbatim}
The \TeX\ source file should begin
with
\begin{verbatim}
\documentclass{icmart}
\end{verbatim}
No personal style files should be used.
Each paper should contain   the  2000 Mathematics Subject Classification.
We would very much appreciate it if you could avoid one-letter
lower case newly defined commands like
\begin{verbatim}
\def\e{\varepsilon}
\end{verbatim}
since this can interfere with conversion of your article to Times fonts later.


\section{Some rules}

In order to achieve a uniform appearance of all the contributions, we encourage
you to to observe the following rules when preparing your article.


\subsection{Displayed formulas} If you have displayed formulas consisting of more than one line
we would prefer if you use
\begin{verbatim}\begin{align}...\end{align}\end{verbatim}
 instead of
\begin{verbatim}\begin{eqnarray}...\end{eqnarray}\end{verbatim}
(respectively the  starred forms) since the former yields a better spacing. Compare:
\begin{align}
A&=  f(x_i)= F'(x)
\\
B&=  g(x_i)= G'(x)
\end{align}
\begin{eqnarray}
A&=& f(x_i)= F'(x)
\\
B&=& g(x_i)= G'(x)
\end{eqnarray}
In case you do not want the numbering for every line, type
\begin{verbatim}\nonumber\end{verbatim}
at the end of the line where you do not want a number.
\begin{align}
A&=  f(x_i)= F'(x)\nonumber
\\
B&=  g(x_i)= G'(x)
\end{align}
If you want a number for the complete block, this works:
\begin{verbatim}
\begin{equation}\begin{split}...\end{split}\end{equation}
\end{verbatim}
\begin{equation}
\begin{split}
A&=  f(x_i)= F'(x)
\\
B&=  g(x_i)= G'(x)
\end{split}
\end{equation}
If you prefer numbering of equations in the form (2.1), (2.2),\dots, add the line
\begin{verbatim}
\numberwithin{equation}{section}
\end{verbatim}
to the preamble of your document.


\subsection{Theorems and alike}

For theorems, lemmas, definitions, etc.\ use the standard syntax.

\begin{verbatim}
\begin{theorem}...\end{theorem}
\end{verbatim}

Put optional arguments into square brackets (``Main Theorem, [5]'' in the example below).

\begin{theorem}[Main Theorem, \cite{Sch}]
If a knot $K$ has Seifert form $V_K$ and its
Alexander polynomial is not $1$, then there is an infinite family
$\{K_i\}$ of non-concordant knots such that each $K_i$ has Seifert
form $V_K$.
\end{theorem}

\begin{definition}
A \emph{preference order}  (or \emph{preference relation}) on $\mathcal X$ is a binary
relation $\succ$ with the following two properties.
\begin{enumerate}

\item \emph{Asymmetry:} If $x\succ y$, then $y\not\succ x$.

\item \emph{Negative transitivity}: If $x\succ y$ and $z\in{\mathcal X}$, then
either $x\succ z$ or $z\succ y$ or both must hold.

\end{enumerate}
\end{definition}

In this example file,   enumerations of theorems,
lemmas definitions, etc.\ appear consecutively.
If you want separate numbering (Theorem 2.1, Definition 2.1) change e.g.\
\begin{verbatim}
\newtheorem[theorem]{definition}
\end{verbatim}
to
\begin{verbatim}
\newtheorem{definition}{Definition}[section]
\end{verbatim}

If you want a statement unnumbered, just define
\begin{verbatim}
\newtheorem*{coro}{Corollary}
\end{verbatim}
to obtain

\begin{coro}
If all the coefficients of {\rm(A.2)} are entire
functions, then all local solutions of {\rm(A.2)} admit a
meromorphic continuation over the whole complex plane~$\mathbb{C}$.
\end{coro}

For a proof, use

\begin{verbatim}
\begin{proof}...\end{proof}
\end{verbatim}

An end-of-proof sign $\Box$ is set automatically.

\begin{proof} This finishes the proof of the corollary.
\end{proof}


\subsection{Operator names}

There are several \TeX-commands setting things automatically upright like $\det$, $\sin$,\dots\,.
If you need operators not predefined, simply define e.g.
\begin{verbatim}
\newcommand{\Hom}{\operatorname{Hom}}
\newcommand{\Ker}{\operatorname{Ker}}
\end{verbatim}
and then use
\begin{verbatim}
\Hom, \Ker
\end{verbatim}
to obtain
$$
\varphi \in \Hom(G/H) \Longrightarrow \Ker(\varphi) \not= \{0\}.
$$
It is desirable that abbreviated mathematical expressions standing for ``words''
appear in roman (upright) typeface.

\section{References}

It follows a list of references showing you the style in which
books and journal articles should be listed.

\renewcommand{\refname}{}    %%%% for this example
\vspace*{-36pt}              %%%% file only!


\frenchspacing
\begin{thebibliography}{7}

\bibitem{And} Anderson, M. T.,  Geometric aspects of the AdS/CFT
correspondence.
In \emph{AdS/CFT Correspondence: Einstein Metrics and their Conformal
Boundaries}
(ed. by Olivier Biquard).
IRMA Lect. Math. Theor. Phys. 8, European Math. Soc. Publishing House,
Z\"urich 2005, 1--31.

\bibitem{BaRh} Babu{\v{s}}ka, I.,  Rheinboldt,  W. C.,
Error Estimates for Adaptive Finite Element Computations,
\emph{SIAM J. Numer. Anal.}  \textbf{15} (1978), 736--754.

\bibitem{FrQu}
Freedman, M. H.,  Quinn, F., \emph{Topology of 4-manifolds}.
Princeton Mathematical Series~39, Princeton University
Press, Princeton, NJ, 1990.

\bibitem{Ma} Mal'cev, A. I., On homomorphisms onto finite groups.
\emph{Ivanov. Gos. Ped. Inst. Ucen. Zap.} \textbf{18} (1958), 49--60.

\bibitem{Sch} Schweizer, M., Hedging of options in a general
semimartingale model. Diss. ETH Z\"urich No. 8615, 1988.

\bibitem{Vig} Vigneras, M. F., Induced representations of reductive
$p$-adic groups
in characteristic $\ell \neq p$. \emph{Selecta Math.} (N.S) \textbf{4}
(1998), 549--623.

\end{thebibliography}


\section{Deadline and where to send the file}


We ask you to send your contribution (tex, dvi, pdf as well as eps files for illustrations)
\textbf{no later} than December 31, 2005 by ftp to the following account:

\smallskip\noindent
Hostname: ftp.ems-ph.org

\smallskip\noindent
Username: icm2006

\smallskip\noindent
Passwort: pfpw4eph

\smallskip\noindent
Folder: submissions

\bigskip\noindent
This would help us for timely production.







\end{document}





In this section we discuss another way to obtain a related
t-structure.

Let $\H$ denote an affine Hecke algebra. Properties of $\H$, in
particular its representation theory had received much attention
during the last two-three decades. One of the sources of interest
comes from the fact, that representations of $\H$ (for the parameter
specialized to a prime power $p^n$)
  correspond bijectively to
 representations of a semisimple $p$-adic group.
Thus classification of such representations of $\H$, achieved by
Kazhdan and Lusztig in \cite{KL}, constitutes a special case of the
local Langlands conjecture (the so-called unramified case).

A large part of representation theory of $\H$ is based on the
existence of two essentially different geometric models for this
algebra.

The first one is obtained as follows. If $G$ is the semisimple
group, $F$ is a non-Archimedian local field with residue field
$\bbF_{q}$, and $I\subset G(F)$ is the Iwahori subgroup, then we can
think of the Hecke algebra (or rather of its specialization at $q$)
   as of $I$-invariant
compactly supported functions on $G(F)/I$. The latter space
 is the set of $\bbF_q$ points of  an ind-algebraic variety
 over  $\bbF_q$;
this variety is known as the affine flag variety of $G$, and will be
denote by $\Fl$. Following the Grothendieck's ``sheaf-function
correspondence'' idea, one should work instead of $I$-invariant
functions on $G(F)/I$ with  the (complexes of)
  etale sheaves on $\Fl$, constant
on the orbits of the Iwahori subgroup. (Passage back to functions is
then made
 by
taking traces of the Frobenius operator acting on stalks of a
sheaf). One then  defines a subcategory of this category, whose
Grothendieck group is identified with $\H$, the product coming from
the convolution of sheaves. This geometry underlines the existence
of the Kazhdan-Lusztig bases in $\H$, and related theories of cells
in affine Weyl groups and asymptotic Hecke algebras, developed in
depth by Lusztig in a series of papers \cite{cells1},
 \cite{cells2}, \cite{cells3}, \cite{cells4}.

Another way to connect $\H_q$ to geometry is via the Steinberg
variety $St$ of the Langlands dual group $\Gcheck$.  One of the
theorems in this direction says that $\H$ is naturally isomorphic to
the Grothendieck group of the category of coherent sheaves on $St$,
equivariant under the action of $\Gcheck$ (see \cite{CG} or
\cite{K1}). This type of result is used essentially in the
Kazhdan-Lusztig classification \cite{KL} of irreducible
representations of $\H$ (see also exposition in \cite{CG}). This
point of view on $\H$
 is also taken in Lusztig's recent articles \cite{K1}, \cite{K2}.


To understand the interplay
 between the two theories (and obtain new results in
either of them) it is desirable to find a relation between the two
geometric categories, which both have $\H$ as their Grothendieck
group. This question had been raised before, see e.g. introduction
to \cite{CG}, or \cite{GK}. An answer to this question had been
given by the author in a (yet unpublished) joint work with
S.~Arkhipov \cite{AB}.


\subsection{Statement of the main result of \cite{AB}}

%\bigskip

To state the  result we will need some notations.

Let $G$ be a semi-simple split algebraic group. The corresponding
 affine Hecke algebra $\H$ is an algebra  over
the ring of Laurent polynomials $A=k[t,t^{-1}]$, where $k$ is the
coefficient ring. (If $F$ is a non-Archimedian local field  with
residue field $\bbF_q$, then
 the ring of $k$-valued functions on the $p$-adic group $G(F)$,
biinvariant under the Iwahori subgroup $I$ with convolution as
multiplication coincides with $\H/(t-q)$.)

Let $\gcheck$ be the Lie algebra of the Langlands dual group
$\Gcheck$, and $\N\subset \gcheck$ be the subvariety of nilpotent
elements. Let $\pi :\gchecktil\to \gcheck$ be the Grothendieck
simultaneous resolution. Thus $\gchecktil$ is a vector bundle over
$\B$, the flag variety of $\Gcheck$, whose fiber over a point $x\in
\B$ is identified with the corresponding Borel subalgebra $\b\subset
\gcheck$. It contains the cotangent bundle to the flag variety $\B$
as a subbundle; this subbundle is denoted by $\Ntil$ (since
$\pi|_\Ntil$ is a resolution of singularities of $\N$). Let $\Ntt$
be the {\it scheme-theoretic} preimage of $\N$ under $\pi$. Thus
$\Ntt$ is a scheme with nilpotents, the ring of rational functions
on $\Ntt$ having length $\#W_f$, the order of the Weyl group. The
variety
 $\Ntil$ is the maximal reduced subscheme of $\Ntt$.
We define the ``thickened'' Steinberg variety by $\Stt=\Ntt\times_\N
\Ntil$. Thus $\Stt$ is also a scheme with nilpotents, and its
maximal reduced subvariety is the Steinberg variety. The group
$\Gcheck$ acts on the above  varieties  in a natural way. For a
$\Gcheck$-variety $X$ let $Coh^\Gcheck (X)$ be the category of
$\Gcheck$-equivariant coherent sheaves on $X$.

On the other hand, let  $\Db(\Fl)$  be the derived category of
$l$-adic sheaves on the affine flag variety $\Fl_{\bFbar_q}$, smooth
along the stratification by the $I$-orbits; and let $\P\subset
\Db(\Fl)$ be the subcategory of perverse sheaves \cite{BBD}.
 (The subindex $\bFbar_q$ denotes extension of scalars to the
algebraic closure of $\bbF_q$). Also, let $\Db_{mix}(\Fl)$ denote
the category of mixed $l$-adic complexes with the same smoothness
condition
 (satisfying an additional technical assumption);
% namely we require that cohomology of a stalk of a complex under
% consideration is isomorphic to a sum of tensor powers of a fixed square
% root of the Tate twist);
and $\Pmixed$ be the subcategory of perverse sheaves.
 There are known
equivalences $\Db(\P)= \Db(\Fl)$, $\Db(\Pmixed)=\Db_{mix}(\Fl)$. It
is also well-known that the Grothendieck group
$K^0(\Pmixed)=K^0(\Db_{mix}(\Fl))$ is canonically isomorphic to
$\H$; here multiplication by the formal parameter $v$ corresponds to
the  square root of the
 Tate twist  functor $X\mapsto X(\frac{1}{2})$ on $\Pmixed$.


We assume that our basic local field $F$ has positive
characteristic, i.e. $F\cong \bbF_q((\epsilon))$.

For $\Gcheck$ and related varieties we  take $\bar{\Bbb Q}_l$ as the
ground field.

\begin{theorem}\label{equi} \cite{AB}
 There exists a natural equivalence of triangulated categories
\begin{equation}\label{eq}
\Db(\P)\cong \Db(Coh^\Gcheck(\Stt));
\end{equation}
\begin{equation}\label{eqmixed}
\Db(\Pmixed)\cong \Db(Coh^{\Gcheck\times G_m}(\Stt)),
\end{equation}
where $G_m$ is the multiplicative group, acting on $\Stt$ by
dilating the nilpotent element. Under \eqref{eqmixed} the
 functor $X\mapsto X(\frac{1}{2})$ corresponds to twisting
of a coherent sheaf by the tautological character of $G_m$.
\end{theorem}

The proof of Theorem \ref{equi} is based  on a remarkable
construction by   Gaitsgory (following an idea of Beilinson) the
so-called geometric
 realization of the center of the affine Hecke algebra
(see \cite{KGB}). In  that paper an action of the category of
representations of $\Gcheck$ on  $\Pmixed$ (and hence on
$\Db(\Pmixed)$) is constructed. On the level of Grothendieck groups
this action corresponds to multiplication by  elements of the center
of $\H$ (whose explicit description goes back to  J.~Bernstein).



*******************************

\subsubsection{Differential operators and Azumaya algebras} Let $X$
be a smooth variety over a perfect field $k$. If $char(k)=0$ we let
$D_X$ be the sheaf of differential operators. If  $char(k)=p>0$ we
let $D_X$ be the sheaf of "crystalline" differential operators on
$X$ (differential operators without divided powers, PD differential
operators in the terminology of Ogus), cf. [BMR].

(Part (a) of) the following fundamental observation belongs to
Mirkovic and Rumynin, see [BMR] for details.

{\bf Theorem.} {\em Assume that $char(k)=p>0$.

a) $D_X$ contains a central subalgebra canonically isomorphic to the
algebra $\O_{T^*X\tw}$ (the latter is viewed as a sheaf of algebras
on $X$ by means of the obvious affine projection  $T^*X\tw\to X\tw$)
(this is the "$p$-curvature" construction). Thus $D_X$ can be viewed
as a sheaf of algebras on $T^*X\tw$. This sheaf is an Azumaya
algebra of rank $p^{2\dim X}$
 on the Frobenius twist of the cotangent bundle
$T^*X\tw$.

b) If $L$ is a line bundle on $X$ then the sheaf $D_X(L)$ of
crystalline differential operators on $L$ is also an Azumaya algebra
on $T^*X\tw$, which is equivalent to $D_X$ (more precisely, $D_X(L)$
is an inner twist of $D_X$). }

\subsubsection{Splitting on the zero section}\label{zersec}
Recall that an Azumaya algebra is called split if it is isomorphic
to $End(V)$ for a vector bundle $V$. If this is the case we will
call $V$ a splitting vector bundle. $V$ is defined uniquely up to
twist by a line bundle.

**********************************************88



\section{Modular representation theory}
 We let $\k$ be an algebraically closed field of
characteristic $p$; in this section we discuss applications of
exotic sheaves to representation theory of $\gg$ over $\k$. We start
by recalling some basic information on the structure of such
representations.

We assume that $p$ is not less than the Coxeter number.


Let  $U= U(\gg)$ be the enveloping algebra of $\gg$. Let $Z$ be the
center of $U$. Then $Z$ is generated by its two subalgebras: the
Harish-Chandra center $Z_{HC}=U^G\cong Sym (\hh)^W$, and the
Frobenius center $Z_{Fr}\cong Sym(\gg\tw)$, where $\tw$ denotes the
Frobenius twist. The subalgebra $Z_{Fr}$ is generated by expressions
of the form $x^p-x^{[p]}$, $x\in \gg$, where the {\em restricted
power} map $x\mapsto x^{[p]}$ is characterized by
$ad(x^{[p]}=ad(x)^p$. Thus $Spec(Z)\subset \hh^*/W\times
(\gg^*)\tw$. For a pair $\la\in \hh^*$, $\chi\in \gg^*$ we let
$U-\rmod^\la_\chi$ be the category of finitely generated
$\gg$-modules, which are killed by some power of the kernel of the
corresponding character of $Z$.

We have a natural map $\La/p\La \to \hh^*$, $\la \mapsto d \la$. We
will say that an element $\la\in \hh^*$ is integral if it lies in
the image of this map. We will say that it is regular if its
stabilizer in $W$ is trivial. The study of $U-\rmod_\chi^\la$
reduces \cite{ } to the case when $\la$ is integral and regular.

For an integral $\la$ we have $U-\rmod_\chi^\la\ne 0$ iff $\chi\in
\N$.

\begin{theorem} a) There exists a canonical equivalence of
triangulated categories $D^b(U-\rmod_\chi^\la)\cong
D^b(Coh_{\BB_\chi}(\gt))$.

b) This equivalence sends modules to exotic sheaves.
\end{theorem}

Action of $B_{aff}$ corresponds to reflection functors.

\begin{remark} In the notations of \cite{BMR}I this is equivalence
... .

We also have a parabolic variant.

\end{remark}



%\subsubsection{Numerical corollaries and Lusztig's conjectures}


\begin{corollary} (conjectured by Lusztig)
The number of irreducible  representations in a regular block of the
category of $u_\chi$ modules equals the sum of Betti numbers of the
Springer fiber $\BB_\chi$.
\end{corollary}

Since the definition of exotic sheaves was uniform, we have

\begin{corollary} Independence of $p$.
\end{corollary}

Recall [AJS] in some setting did it. Can be done by combining their
methods with geometry. They used {\em reflection functors}. We will
rather use their close relatives: {\em intertwining functors}.
Recall from [BB2] etc. how intertwining functors work (generators
$I_\al$ fit  into distinguished triangles with reflection functors
$R_\al$, intertwine localizations on different levels.

In fact, Lusztig has suggested a much more detailed conjectural
picture.

Further discussion in this section is about a forthcoming joint work
with Mirkovi\' c and (partly) Rumynin.

Characterized. Invariance under involution reduces (at least in most
cases) reduces to.

Almost orthonormality follows from ... above.

%Use the slices. Need to show the "slice algebra" from the Example in
%the previous section can be arranged to be positively graded.

%Which can be deduced from the next section.



%All constructions depended on $p$, would like to  prove
%independence.

%Affine braid group action.

%Unique braid positive $t$-structure.

XXXXXXXXXXXXXXXXXXXXXXXXXXxxx


Hi again,

Well, I'll try to describe what fits in an e-mail now, and then I
elaborate as needed when we talk.

1) Let me describe the starting point of establishing the monoidal
equivalence between some category of constructible complexes and
 $D=D(Coh(\tilde \gg\times_\gg \tilde g))$. Notice that the affine closure
 of the above variety is $g\times _{\hh/W} h\times _{h/W} h$, where
$\h$ is the (abstract) Cartan. Also, the affine quotient by $G$ (the
spectrum of global $G$-invariant functions) is $\h\times _{\h/W}
\h$. The functor "G-invariants in the global section" from $D$ to
$DCoh (\h\times_{\h/W} \h)$ is (I think) monoidal; in particlar, in
$D$ we have a full monoidal subcategory whose objects are direct
sums of several copies of the structure sheaf $O$. It is equivalent
as a monoidal category to $Coh_{fr}(\h\times _{\h/W} \h)$ -- the
category of free $\O$-modules on the fiber square.

I start by constructing a monoidal full embedding from
$Coh_{fr}(\h\times _{\h/W} \h)$ to the category of perverse sheaves
basically by guessing the (pro)object (called $\Xi$, because it
generalizes Beilinson's maximal extension) where $O$ should go, and
then looking at how this thing convolves with itself. This can be
done, on the one hand, just by showing that the convolution
$\Xi*\Xi$ is abstractly isomorphic to the sum of a few copies of
$\Xi$, and that left convolution with $\Xi$ commutes with the right
action of $O(\h)$. On the other hand, the functor represented by
$\Xi$ has a geometric meaning (generic vanishing cycles at 0), and
there is (or at least should be) a way to use this to build a
monoidal structure on this functor going in the opposite direction;
restricting it to the category of sums of copies of $\Xi$ we get the
inverse equivalence with $Coh(h\times_{h/W} h)$.


Taking this together with Dennis's central sheaves (or functors) we
get a full monoidal subcategory in perverse sheaves, which is
monoidally  equivalent to   $Coh^G_{fr}(\gg \times _{h/W} h
\times_{h/W} h)$ (fullness applies to the derived category, i.e.
there are no higher Ext's in the category of perverse sheaves).

1.5) This is definitely a side topic, but I can't resist mentioning
it. There is a strange relation between the category I've just
described (equivalent to $Coh^G_{fr}(\gg \times _{h/W} h
\times_{h/W} h)$) and a version of our category. Namely, if we
consider $T\times T$ (rather than $G\times G$) equivariant Ext's
between two irreducible objects $IC_\lambda$, $IC_\mu$ in our Satake
category they turn out to be isomorphic to Hom between
$V_\lambda\otimes O$ and $V_\mu\otimes O$ where $O=O(\gg \times
_{h/W} h \times_{h/W} h)$, and $V_\lambda$, $V_\mu$ are irr. reps of
the dual group, while Hom is taken in the $G$-equivariant category.
This can then be reinterpreted as Hom between objects in the
category of (pro)perverse sheaves described in (1). The homological
grading corresponds to the grading by Frobenius weight.

2) Now, to answer the question of what is the image of the structure
sheaf of the dg scheme in the monoidal category, I recall the
mnemonic rule that replacing Iwahori monodromicity condition on one
side by the $G(O)$ equivariance condition amounts in the coherent
description to taking fiber product with the zero section  over
$\tilde \gg$, and forgetting a copy of the flag space. Moreover, the
averaging functor for perverse sheaves corresponds to the obvious
restrict -- push-forward functor, _up to a $\rho$-shift.

Applying this to $\tilde \gg\times _\gg \tilde \gg$ we get a
description of the dg-structure sheaf as a double averaging of
something. The first averaging applied to the pro-sheaf $\Xi$ yields
some explicit perverse sheaf (of length 3 for sl(2)), and then the
next averaging gives some complex. This description is explicit, but
doesn't show there is anything particularly nice about it.

3) It may help to recall how the monoidal structure on the functor
from the Satake category to coherent sheaves taken in the most naive
sense is constructed. It is parallel (Koszul dual) to the 2nd
approach to constructing the monoidal structure in (1) above. We
just observe that the global cohomology functor is monoidal, this
yields a monoidal structure on the inverse equivalence, going from
perverse sheaves to coherent ones.

Well, hope it makes some sense.

Bests,
 Roman.

 XXXXXXXXXXXXXXXXXXXXXX


a) The vector bundle $\E _{ \Sigma_e}$ admits a $\Gm$-equivariant
structure.

b) This $\Gm$-equivariant structure can be chosen so that the
induced grading on $A_{\Sigma_e}=\End(\E_{ \Sigma_e})^{op}$ has the
following property: the components of negative weights vanish, and
the degree zero component is semi-simple.
\end{theorem}

Part (a) can be derived from Ext vanishing and rigidity, but

XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX


XXXXXXXXXXXXXXXXXXXXX




To motivate the following we have to recall the appearance of the
spaces $\Nt$, $\gt$ in the representation theory of $p$-adic groups.
%more precisely, in  the proof of a particular case of the local
%Langlands conjecture.

%For a variety $X$ we will write $K(X)$ for the Grothendieck group of
%the category of coherent sheaves on $X$; given an action of an
%algebraic group $H$ on $X$ we write $K^H(X)$ for the Grothendieck
%group of $H$-equivariant coherent sheaves.



 We will use the following general
observation. Assume that $X %@>{\pi}>>
\overset{\pi}{\To} Y$ is a proper morphism
 where $X$ is smooth
 %(OVER WHAT? : a commutative noetherian ring $R$
%of finite global dimension ?).
 Set $D(X\to Y)=D^b(Coh_{X\times_Y
X}(X\times X))$;
 then  the convolution
provides the category $D(X\to Y)$ with a monoidal structure. This
monoidal category acts by convolution
 on the category
$D^b(Coh(X))$; and more generally it acts on $D^b(Coh_{X\times_Y
Z}(X))$ whenever $Z\subset Y$ is a locally closed subscheme. Thus
the Grothendieck group $K(X\times_Y X)=K(Coh_{X\times_Y X}(X\times
X))$ acquires an algebra structure; the same applies to the
equivariant categories once compatible actions of an algebraic group
on $X$ and $Y$ are given.
%Under
%the additional condition $Tor_{>0}^{\O_Y}(\O_X,\O_X)=0$ the category
%$D^b(Coh(X\times_Y X))$ is  monoidal; it
%  acts on $D^b(Coh(X\times_Y
%Z))$ provided $Tor_{>0}^{\O_Y}(\O_Z,\O_X)=0$. (The construction
%works in general for the dg-schemes % $X\Ltimes_Y X,\ X\Ltimes_Y Z$
%and the assumptions guarantee that these are schemes in order to
%avoid the dg-setting.)



%By a {\em weak geometric action} of a group $G$ on $D(X)$ over $Y$
%we will understand a homomorphism from $G$ to the monoid of
%isomorphism classes of objects of the monoidal category $D^b(X\to
%Y)$. Thus a weak geometric action induces an action of $G$ on
%$D^b(Coh(X\times_Y Z))$ for any $Z$ as above.

%In fact, we have a weak geometric action of $B_{aff}$ on  $\gt$ over
%$\gg$. This gives a module of $W_{aff}'$. Now let $\H$ denote an
%affine Hecke algebra. Then we have $\H=K^{G\times \Gm}(\gt\times_\g
%\gt)$.


Notice that the action of $B_{aff}$ on $\D_e$ is inherited by
$\D_e^{\Gm}$. Moreover, the induced action of $B_{aff}$ on the
Grothendieck group $K^{\Gm}(\BB_e)=K(\D_e^{\Gm})$ induces an action
of {\em the affine Hecke algebra $\H$}. Recall that $\H$ is a ring
over $\Ce[v,v^{-1}]$ whose specialization at $v=1$ is identified
with the group algebra $\Ce[W_{aff}]$. The action of $\H$ on
$K^{\Gm}(\BB_e)$ is such that $v:[\F]\mapsto [\F(1)]$.


Recall that the {\em affine Hecke algebra} $\H$ is a deformation of
the group algebra $\Zet[W_{aff}]$; it is  a quotient of the group
algebra $\Zet[B_{aff}]$ by the relations $(\tii s_\al -1)(\tii
s_\al+q)=0$. Thus $\H$ is an algebra over $\Zet[q,q^{-1}]$, and its
specialization at $q=1$ is identified with $\Zet[W_{aff}]$. Then we
have canonical algebra isomorphisms [??], [??]

\begin{equation}\label{He}
\begin{split}
\Zet[W]\cong K^G(\gt\times_\g \gt) \\
\H\cong K^{G\times \Gm}(\gt\times_\g\gt).
\end{split}
\end{equation}

$\H$  plays an important role in representation theory of
semi-simple groups over $p$-adic fields. More precisely, let $F$ be
a non-Archimedian local field with residue field $\bbF_{p^a}$. Let
$\LG=^L{\bf G}(F)$ be the dual group over $F$. Let $I\subset \LG(F)$
is the Iwahori subgroup. Then the algebra of $I$ bi-invariant
compactly supported functions  on $\LG$ is canonically isomorphic to
$\H/(q-p^a)$. This allows to prove that the category of $\H/(q-p^a)$
modules is equivalent to the category of (smooth) $\LG$-modules
which are generated by Iwahori fixed vectors. {\bf not over $\Zet$
though}. Thus a classification of such representations constitutes a
particular case of the local Langlands conjecture. This particular
case has been proved by Kazhdan and Lusztig \cite{KL} (cf. also
\cite{ }, Ginzburg congress, Chriss, etc). I would like to emphasize
that the key step in the proof is the realization \eqref{He} of $\H$
-- this is not the most difficult technical step, but it is the step
where the dual group appears. I now explain briefly a
"categorification" of these isomorphisms, which follows the general
pattern of the geometric Langlands duality theory, see [BD], [Denis
cong], [Edik].




 We can think of the Hecke algebra (or rather
of its specialization at $p^a$)
   as of $I$-invariant
compactly supported functions on $\LG/I$. The latter space
 is the set of $\bF_{p^a}$ points of  an ind-algebraic variety
 over  $\bF_{p^a}$;
this variety is known as the affine flag variety of $\LG$, and will
be denote by $\Fl$. Following the Grothendieck's ``sheaf-function
correspondence'' idea, one expects that an interesting space of
functions on the set of points of an $\bF_q$ variety (or stack)
underlies an interesting category of etale sheaves (or $l$-adic
complexes). Notice that a function can be obtained from an $l$-adic
complex by taking the alternating sums of traces of Frobenius on
stalks of the sheaf.

In fact, if $\P$ is the category of perverse sheaves on
$\Fl_{\bar\bF_{p^a}}$ equivariant with respect to the Iwahori group,
then the Grothendieck group $K(\P)$ is isomorphic to
$\Zet[W_{aff}]$, while the deformation $\H$ of $\Zet[W_{aff}]$ is
closely related to the Grothendieck group of $I$-equivaraint Weil
perverse sheaves on $\Fl$.

Similarly, the $\Zet[W]=K^G(\Nt\times _\g \Nt)$-module  $K^{G}(\Nt)$
is isomorphic to the Grothendieck group of a certain Serre quotient
$^f \P$ of $\P$ (see [AB]).

A natural question is whether these isomorphisms of Grothendieck
groups can be upgraded to equivalences of categories. Several
results of this kind have been obtained \cite{B}, \cite{AB},
\cite{Betobe}.

Here we mention only one of them, which is essential to the above
applications.

To state it, recall that for an $\bF_{p^a}$ scheme $X$ the Frobenius
morphism yields a functor $Fr^*$ from the category of etale sheaves
(or complexes) on $X_{\bar\bF_{p^a}}$ to itself.

Let also $\bq:\Nt\to \Nt$ denote the map $(\b,x)\mapsto (\b, p^ax)$.


\begin{theorem} [AB]
a) There exists an equivalence of triangulated categories $D^b(^f
\P)\cong D^G(\Nt)$.

b) It intertwines the endo-functors $Fr^*$ and $\bq^*$.
\end{theorem}

{\em Comments on the proof.} The Theorem relies on the construction
of .... . State enough compatibilities, so that the functor is
forced upon us. We start with geometric Satake. \cite{Zen} provides
categorification of the center. Notice in the coherent have this
automorphism -- Monodromy. Filtration (or rather highest vector) --
Wakimoto. That's enough to .. .

Some compatibilities:

\begin{proposition}
Braid group action ?? .
\end{proposition}

Get an functor $D^{G\times \Gm}(\Nt)$ to Weil complexes.

\begin{proposition}
a) Compatibility with the equivariant exotic $t$-structure.

b) $\tii \del_\la$, $\tii \nab^\la$ go to (co)standard sheaves, of
weight zero on the generic stratum in the support.
\end{proposition}

math0604445.extracted_bib
\begin{thebibliography}{1}

\bibitem{Rina} R.~Anno, {\em Spherical functors and braid group actions,}
in preparation.

\bibitem{AJS}
H.H.~Andersen, J.C. Jantzen, W. Soergel
{\em Representations of quantum groups at a $p$th root of unity
and of semisimple groups in characteristic $p$: independence of $p$,}
 Ast\' erisque {\bf 220} (1994), 321 pp.

\bibitem{AB} S.~Arkhipov S.,  R.~Bezrukavnikov,
\emph{Perverse sheaves on affine flags and Langlands dual group,} electronic
preprint, math.RT/0201073, to appear in Israel Math. J.


\bibitem{ABG} S. Arkhipov,  R. Bezrukavnikov,  V. Ginzburg
\emph{Quantum Groups, the loop
Grassmannian, and the Springer resolution,}
J. Amer. Math. Soc.
{\bf 17} (2004), no. 3, 595--678

\bibitem{BeBe} A.~Beilinson, J.~Bernstein,
\emph{Localisation de $g$-modules,}
 (French) C. R. Acad. Sci. Paris S\' er. I Math. 292 (1981), no. 1, 15--18.




\bibitem{BeBeder}
A. Beilinson, A. Bernstein,
{\em
A generalization of Casselman's submodule theorem,}
  Representation theory of reductive groups (Park City, Utah, 1982),  35--52,
 Progr. Math., 40, Birkh\" auser Boston, Boston, MA, 1983.




\bibitem{BBD} A.~Beilinson, J.~Bernstein, P.~Deligne,
\emph{Faisceaux pervers,}
  5--171, Ast\' erisque, {\bf 100}, Soc. Math. France, Paris, 1982.



\bibitem{BD} A. Beilinson, V. Drinfeld,
{\em
Quantization of Hitchin's Integrable System and Hecke Eigensheaves},
preprint
available at http://www.math.uchicago.edu/$\sim$arinkin/langlands/


\bibitem{BGS}
A. Beilinson, V. Ginzburg, W. Soergel,
\emph{Koszul duality patterns in representation theory,}
J. Amer. Math. Soc. {\bf 9} (1996), no. 2, 473--527.


\bibitem{BelKa}
A. Belov-Kanel, M. Kontsevich, {\em
Automorphisms of the Weyl algebra}
Lett. Math. Phys. {\bf 74} (2005), no. 2, 181--199.




\bibitem{vdB} M.~van den Bergh, {\em Noncommutative crepant resolutions,}
The legacy of Niels Henrik Abel, 749--770, Springer, Berlin, 2004.



\bibitem{B}  R. Bezrukavnikov
{\em Perverse sheaves on affine flags and nilpotent cone
 of the Langlands dual group,} electronic preprint
math.RT/0201256,  to appear in  Israel J. Math.



\bibitem{izvrat} R. Bezrukavnikov,
 {\em Perverse coherent sheaves (after Deligne),}
electronic preprint,
math.AG/0005152.

%\bibitem{nilpokon} R. Bezrukavnikov, {\em Quasi-exceptional sets and
%equivariant coherent sheaves on the nilpotent cone,}
%Represent. Theory {\bf 7} (2003), 1-18.

\bibitem{Hum} R. Bezrukavnikov, {\em Cohomology of tilting modules over
 quantum groups and $t$-structures on derived categories of coherent sheaves,}
to appear in Inv. Math.

\bibitem{Betobe}
  R. Bezrukavnikov, {\em An equivalence between two categorical
realization of the affine Hecke algebra,} in preparation.



\bibitem{BFG}  R. Bezrukavnikov, M. Finkelberg, V. Ginzburg.
{\em Cherednik algebras and Hilbert schemes in characteristic p, }
 electronic preprint, math.RT/0312474, to appear in Represent. Theory.



\bibitem{BeKal} R.  Bezrukavnikov, D. Kaledin
{\em McKay equivalence for symplectic resolutions of quotient
singularities,}
Tr. Mat. Inst. Steklova {\bf 246} (2004),
Algebr. Geom. Metody, Svyazi i Prilozh., 20--42;
 translation in Proc. Steklov Inst. Math. 2004, no. 3 (246), 13--33.



\bibitem{BeKalqua} R. Bezrukavnikov, D. Kaledin
{\em Fedosov quantization in positive characteristic}, electronic preprint,
math.AG/0501247.

\bibitem{BMR}  R.  Bezrukavnikov, I.~Mirkovic, D.~Rumynin,
{\em Localization of modules for a semisimple Lie algebra in prime
characteristic,} electronic preprint math.RT/0205144, to appear in
Annals  of Math;
%\bibitem{BM} R.  Bezrukavnikov, I.~Mirkovic, D.~Rumynin,
{\em Singular localization and intertwining functors for
semisimple Lie algebras in prime characteristic,}  electronic
preprint math.RT/0602075, submitted to Nagoya Math. J.

%\bibitem{BO}  Bezrukavnikov Ostrik

\bibitem{BoKa} A. Bondal, M. Kapranov,
{\it Representable functors, Serre functors, and mutations,} Izv.
Ak. Nauk {\bf 35} (1990), no. 3, 519--541.

\bibitem{BonOr}  A. Bondal,
D. Orlov,
{\em Derived categories of coherent sheaves,}
 Proceedings of the International Congress of Mathematicians,
 Vol. II (Beijing, 2002), 47--56, Higher Ed. Press, Beijing, 2002.

\bibitem{Bridge} T.~Bridgeland  {\em Derived categories of coherent sheaves}
report of ICM 2006, available as an electronic preprint math.AG/0602129.

\bibitem{Brstab} T. Bridgeland, {\em Stabilities on triangulated categories,}
electronic preprint math.AG/0212237, to appear in Ann. Math.

\bibitem{BrK3} T. Bridgeland, \emph{Stability conditions on K3 surfaces,}
electronic preprint math.AG/0307164;
%
%\bibitem{BrP2} T. Bridgeland,
{\em  Stability conditions on a non-compact Calabi-Yau threefold,}
electronic preprint math.AG/0509048;
%
%\bibitem{Brbr} T. Bridgeland,
{\em
 Stability conditions and Kleinian singularities,} electronic preprint
math.AG/0508257.


\bibitem{Brflop} T. Bridgeland
{\em Flops and derived categories,} Invent. Math.
{\bf 147} (2002), no. 3, 613--632.


\bibitem{BrGo} K. Brown, I. Gordon,
{\it  The ramification of centres: Lie algebras in positive characteristic
 and quantised enveloping algebras,}  Math. Z.  {\bf 238}  (2001),
 no. 4, 733--779.

\bibitem{BrylKa}
J.-L. Brylinski, M. Kashiwara,
{\em Kazhdan-Lusztig conjecture and holonomic systems,}
Invent. Math. {\bf 64} (1981), no. 3, 387--410.

\bibitem{CG} N. Chriss, V. Ginzburg
{\em Representation theory and complex geometry,}
Birkh\" auser Boston, Inc., Boston, MA, 1997.


%\bibitem{DLP}
%C. De Concini, G. Lusztig,  C. Procesi,
%{\em
% Homology of the zero-set of a nilpotent vector field on a flag manifold,}
%J. Amer. Math. Soc.  {\bf 1}  (1988),  no. 1, 15--34.

\bibitem{Edik1} E. Frenkel, {\em
Recent advances in the Langlands program,}
Bull. Amer. Math. Soc. (N.S.) {\bf 41} (2004), no. 2, 151--184.

\bibitem{Edik} E. Frenkel,
{\em Lectures on the Langlands Program and Conformal Field Theory,}
electronic preprint hep-th/0512172.



\bibitem{Edik_Denis} E. Frenkel, D. Gaitsgory,
{\em Local geometric Langlands correspondence and affine Kac-Moody
 algebras,} electronic preprint math.RT/0508382.



\bibitem{Zen}
D. Gaitsgory,
\emph{Construction of central elements in the affine Hecke algebra
via nearby cycles,}
 Invent. Math.  {\bf 144}  (2001),  no. 2, 253--280.

\bibitem{GaiICM} D. Gaitsgory, {\em
Geometric Langlands correspondence for ${\rm GL}\sb n$,}
 Proceedings of the International Congress of Mathematicians,
 Vol. II (Beijing, 2002), 571--582, Higher Ed. Press, Beijing, 2002.

%\bibitem{GinICM} Ginzburg ICM

\bibitem{Ginzb} V. Ginzburg,
{\em Perverse sheaves on a Loop group and Langlands' duality,}
 electronic preprint alg-geom/9511007.

\bibitem{GinGan} W.-L. Gan, V. Ginzburg
{\em Quantization of Slodowy slices,}
 Int. Math. Res. Not.  {\bf 5} (2002) 243--255.




\bibitem{hren}
W. H\" urlimann,
{\em Sur le groupe de Brauer d'un anneau de polyn\^ omes en
caract\' eristique $p$
et la th\' eorie des invariants,} in
``The Brauer group'' (Sem., Les Plans-sur-Bex, 1980), pp. 229--274,
Lecture Notes in Math. {\bf 844},
Springer, Berlin (1981).


\bibitem{KDC} C. De Concini, V. Kac,
{\em Representations of quantum groups at roots of $1$,}
 Operator algebras, unitary representations, enveloping algebras,
 and invariant theory (Paris, 1989), 471--506, Progr. Math.,
 92, Birkh�user Boston, Boston, MA, 1990.

\bibitem{Kal} D. Kaledin,
{\em Derived equivalences by quantization,} electronic preprint
math.AG/0504584.




\bibitem{Kashiwa} M.Kashiwara,
{\em On crystal bases of the q-analogue of universal enveloping
algebras,} Duke {\bf 63} (1991), 465--516.


\bibitem{KL} D. Kazhdan, G. Lusztig,
{\em Proof of the Deligne-Langlands conjecture for Hecke algebras,}
 Invent. Math. {\bf 87} (1987), no. 1, 153--215.

\bibitem{Kobi}  E.~Backelin, K.~Kremnitzer,
{\em Localization for quantum groups at a root of unity,}
  electronic preprint math.RT/0407048.


%\bibitem{KumarLaur} Kumar Lauritzen


\bibitem{MV}  I. Mirkovi\' c, K. Vilonen
{\em Perverse Sheaves on affine Grassmannians and Langlands Duality,}
 electronic preprint math.AG/9911050.




%\bibitem{Lam} Laumon about Langlands

%\bibitem{LuHeck} G. Lusztig, {\em
%Representations of affine Hecke algebras,}
% Orbites unipotentes et repr�sentations, II. Ast\' erisque
%{\bf 171-172}
% (1989), 73--84.

\bibitem{LKthII}
G.Lusztig,
{\em Bases in equivariant $K$-theory},
Represent. Theory {\bf 2} (1998), 298--369.
{\em Bases in equivariant $K$-theory II},
Represent. Theory {\bf 3} (1999), 281-353.

\bibitem{LConj} G. Lusztig, {\em Some problems in the representation
theory of finite Chevalley groups,} Proc. Symp. Pure Math., 37
(1980), pp 313--317.

%\bibitem{LuBerngen} Lusztig, Bernstein's generators for H aff

%\bibitem{Milne}
%J.S.Milne,
%{\em
%Etale cohomology}.
%Princeton Math. Series, 33.
%Princeton U. Press (1980).


\bibitem{OV} A. Ogus, V. Vologodsky,
{\em Nonabelian Hodge Theory in Characteristic p,} preprint
math.AG/0507476.



\bibitem{Premet} A. Premet
{\em Special transverse slices and their enveloping algebras,}
With an appendix by Serge Skryabin,
Adv. Math. {\bf 170} (2002), no. 1, 1--55.

%\bibitem{Riche} Riche

%\bibitem{ST} Seidel, Thomas

%\bibitem{Soer} W. Soergel, {\em
%Kategorie $\mathcal O$, perverse Garben und Moduln \" uber den Koinvarianten
%zur Weylgruppe,} J. Amer. Math. Soc. {\bf 3} (1990), no. 2, 421--445.

\bibitem{SoerICM} W. Soergel, {\em Gradings on representation categories,}
 Proceedings of the International Congress of Mathematicians, 
(Z\" urich, 1994), 800--806, Birkh\" auser, Basel, 1995.

%\bibitem{Weil} Weil
\end{thebibliography}