Wreath Macdonald polynomials and categorical McKay correspondence
Roman Bezrukavnikov
Michael Finkelberg
Mark Haiman has reduced Macdonald positivity conjecture to a statement about geometry of the Hilbert scheme of points on the plane, and formulated a generalization of the conjectures where the symmetric group is replaced by the wreath product $\fS_n\ltimes (\Zet/r\Zet)^n$. He has proven the original conjecture by establishing the geometric statement about the Hilbert scheme, as a byproduct he obtained a derived equivalence between coherent sheaves on the Hilbert scheme and coherent sheaves on the orbifold quotient of ${\mathbb A}^{2n}$ by the symmetric group $\fS_n$. A short proof of a similar derived equivalence for any symplectic quotient singularity has been obtained by the first author and Kaledin \cite{BK} via quantization in positive characteristic. In the present note we show the properties of the derived equivalence which imply the generalized Macdonald positivity for wreath products.
1208.3696.tex
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\author{Roman Bezrukavnikov, Michael Finkelberg}
\title[Wreath Macdonald polynomials]{Wreath Macdonald polynomials and categorical McKay
correspondence}

\begin{document}


\begin{abstract}
Mark Haiman has reduced Macdonald positivity conjecture to a statement about
geometry of the Hilbert scheme of points on the plane, and formulated a generalization
of the conjectures where the symmetric group is replaced by the wreath product
$\fS_n\ltimes (\Zet/r\Zet)^n$. He has proven the original conjecture
by establishing  the geometric statement about
the Hilbert scheme, as a byproduct he obtained a derived equivalence
between coherent sheaves on the Hilbert scheme and coherent sheaves
on the orbifold quotient of ${\mathbb A}^{2n}$ by the symmetric group $\fS_n$.

A short proof of a similar derived equivalence for any symplectic quotient singularity has been obtained by the first author and Kaledin
\cite{BK} via quantization in positive characteristic. In the present note we show
the properties of the derived equivalence which imply the generalized Macdonald
positivity for wreath products.

 \end{abstract}

\maketitle %\footnote{Note to the author: everyone has his own psychological
%problems,  some at least make an effort not to put them on display.}


\centerline{\em{To the memory of Andrei Zelevinsky.}}

\tableofcontents

\section{Introduction}  The celebrated Macdonald positivity conjecture
asserts that the entries of the matrix expressing the transformed
Macdonald polynomials
via Schur polynomials have non-negative coefficients.

The following approach to the conjecture was suggested by M.~Haiman.

Let $\on{Hilb}^n(\Atwo)$ be the Hilbert scheme of $n$ points on the plane.
The action of the two-dimensional torus $T=\Gm^2$ on $\Atwo$ induces an action
of $T$ on $\on{Hilb}^n(\Atwo)$. Let $\Lambda$ be the ring of symmetric polynomials.
According to Haiman's program, one can
identify the space $\Lambda[q^{\pm1},t^{\pm1}]$ appearing in Macdonald positivity conjecture with
 the Grothendieck group $K^0(\Coh^T(\on{Hilb}^n(\Atwo)))$ in such a way that
   the modified Macdonald
polynomials correspond to classes of sky-scrapers at the fixed points of $T$,
 while Schur functions correspond to classes of indecomposable summands components of the {\em Procesi bundle}. The latter is a certain vector bundle on
$\on{Hilb}^n(\Atwo)$ of rank $n!$
 carrying an action of the symmetric group and a compatible action
 of the ring of polynomials in $2n$ variables.
 An explicit construction of this bundle is the crucial and laborious ingredient in Haiman's work.
 As a byproduct of this construction Haiman obtained
an  equivalence of derived categories:
\begin{equation}\label{dereq}
D^b(\Coh(\on{Hilb}^n(\Atwo))\cong D^b(\Coh^{\fS_n}({\mathbb A}^{2n}))
\end{equation}
and an  isomorphism between $\on{Hilb}^n(\Atwo)$ and the Hilbert quotient of
$\BA^{2n}$ by the action of $\fS_n$.


The existence of a rank $n!$ bundle with an action of $\fS_n$ and of the polynomial ring
which induces a  derived equivalence \eqref{dereq}
 has been proven  in another, shorter way in \cite{BK}
via quantization in positive characteristic.
%More precisely {\em loc. cit.} proves the following: {\bf insert Thm}.


The goal of the present note is to demonstrate that the information provided by
\cite{BK} is sufficient to deduce the positivity conjecture, bypassing the explicit
construction of Procesi bundle.
\footnote{Notice however that we do not propose an alternative proof of
the isomorphism between $\on{Hilb}^n(\Atwo)$
 and the Hilbert quotient of $\BA^{2n}$, which
 is one of the equivalent statements
 of the $n!$ conjecture. Existence of such an isomorphism is equivalent to the fact that each fiber
 of the rank $n!$ vector bundle is generated by an $\fS_n$ invariant vector
 under the action of the polynomial ring.
  We do not know how to deduce this property of the vector bundle by our
 methods.}
 Furthermore, Haiman~\cite{H} has suggested an extension of the
conjecture to the setting where $\fS_n$ is replaced by the wreath
product $\fS_n\ltimes (\Zet/r\Zet)^n$. We also deduce that
 previously unknown   conjecture.

\medskip

Let us formulate our result in more detail.
We consider the cyclic quiver $Q$ with the set of vertices
$I=\{i_0,\ldots,i_{r-1}\}\cong \BZ/r\BZ$.
Our basic field $\sk$ will be either $\BC$ or
$\overline\BF_p$ with $p\gg0$. Given $I$-graded vector spaces $W,V$ of dimension vectors
$\lambda_0=(1,0,\ldots,0),\ \nu=(v_0,\ldots,v_{r-1})$, we consider the Nakajima
quiver varieties $\fM(W,V)\stackrel{\pi}{\longrightarrow}\fM_0(W,V)$ over $\sk$.
It is well known (see e.g.~\cite[Section~7.2.3]{H} or~\cite[Lemma~7.8]{G}) that
$\fM_0(W,V)\cong\BA^{2n}/\Gamma_n$ where $\Gamma_n$ is the wreath product
$\BZ/r\BZ\wr\fS_n$, and $n$ is defined as follows. Let $\delta$ stand for
the dimension vector $(1,\ldots,1)$. Then we have a unique decomposition
$\nu=\nu_0+n\delta$ where $\nu_0$ is the content vector of a certain $r$-core
partition. Moreover, $\fM(W,V)\cong Y_{\Gamma,\nu}$, where $Y_{\Gamma,\nu}$
is a certain connected
component of the fixed point set $(\on{Hilb}^m(\BA^2))^\Gamma$ of a Hilbert
scheme of points on $\BA^2$. Here $m=v_0+\ldots+v_{r-1}$, and $\Gamma=\BZ/r\BZ$.
Finally, $\pi$ is a symplectic semismall resolution of singularities.

According to the result of \cite{BK} we have an equivalence of derived
categories $D^b(\Coh( Y_{\Gamma,\nu}))\cong D^b(\Coh^{\Gamma_n}(\BA^{2n}))$
provided that characteristic of $\sk$ is zero or sufficiently large.
Let $\CE\in D^b(\Coh(Y_{\Gamma,\nu}))$ be the image of $\CO\otimes\sk[\Gamma_n]\in
\Coh^{\Gamma_n}(\BA^{2n})$ under the equivalence constructed in \cite{BK}.
Then it is shown in {\em loc. cit.} that $\CE$ is a vector bundle.
It automatically carries an action of $\Gamma_n$, thus it can canonically
be written as $\CE=\oplus \rho\otimes \CE_\rho$ where $\rho$ runs over the set of irreducible
representations of $\Gamma_n$ and $\CE_\rho$ is a vector bundle on $Y_{\Gamma,\nu}$.

Furthermore, $\CE$ carries an action of the polynomial algebra
$\sk[x_1,\dots, x_n,
y_1,\dots, y_n]=\CO(\BA^{2n})$. Let $\overline\CE_\rho$ be the image of $\CE_\rho$
in the sheaf of coinvariants $\CE/(\sum\limits_{i=1}^n x_i \CE)$,
and  $\overline\CE{}'_\rho$ be the image of $\CE_\rho$ in $\CE/(\sum\limits_{i=1}^n y_i \CE)$.

Our main result is

\begin{thm}
\label{unique}
If $\rho=\rho_\blambda$ for a multi-partition $\blambda$, then the set of $T$-fixed points
in the support of $\overline\CE_\rho$ consists of
$\blambda$ and some $\bmu$ satisfying $\bmu\preceq\blambda$, while the
set of $T$-fixed points in the support of
$\overline\CE{}'_\rho$ consists of $\blambda$ and some $\bmu$ satisfyying
$\bmu\succeq\blambda$.
\end{thm}

Here we used notations and conventions introduced below in
subsection~\ref{components}.

The proof of the Theorem appears in subsection~\ref{210}. The next
statement follows from Corollary \ref{rom}.

\begin{cor}
Conjecture \cite[7.2.19]{H} holds.
\end{cor}



\medskip

{\bf Acknowledgements.} We are grateful to V.~Ginzburg and I.~Gordon  for some very
useful discussions. Special thanks go  to I.~Losev for expert
critical comments which contributed to our understanding of the material and influenced
the exposition.

This work was conceived when the authors enjoyed the
hospitality and support of IAS at HUJI in the Fall 2010.
M.F. was partially supported by the RFBR grants 12-01-00944, 12-01-33101,
the National Research University Higher School of Economics' Academic Fund
award No.12-09-0062 and
the AG Laboratory HSE, RF government grant, ag. 11.G34.31.0023.
This study was carried out within the National Research University Higher
School of Economics
Academic Fund Program in 2012-2013, research grant No. 11-01-0017.
This study comprises research findings from the ``Representation Theory
in Geometry and in Mathematical Physics" carried out within The
National Research University Higher School of Economics' Academic Fund Program
in 2012, grant No 12-05-0014. R.B. was partially supported by NSF grant DMS-1102434.

%{\bf PREZHDE CHEM PROPOET PETUX TRIZHDY OTRECHESH'SJA!}




\section{Localization of modules over the invariants in the Weyl algebra}
\label{2}

\subsection{Notations}
\label{Notations}
From now on we assume that in case $\sk=\overline\BF_p$, the characteristic
$p$ is bigger than $|\Gamma_n|$. In that case, for an algebraic variety $X$
over $\sk$ we will denote by $X^{(1)}$ its Frobenius twist, and by
$\on{Fr}:\ X\to X^{(1)}$ the Frobenius morphism.
Note that in the presentation
$\fM_0(W,V)\cong\BA^{2n}/\Gamma_n$ the vector space $U=\BA^{2n}$ carries a
$\Gamma_n$-invariant symplectic form $\omega$. It gives rise to the Weyl
algebra $\sW_\sk$ acted upon by $\Gamma_n$. According to~\cite[Lemma~6.1]{BK},
for large $p$
the $\sW_\sk^{\Gamma_n}-\sW_\sk\rtimes\sk[\Gamma_n]$-bimodule $\sW_\sk$ gives rise
to the Morita equivalence $\sW_\sk^{\Gamma_n}\sim\sW_\sk\rtimes\sk[\Gamma_n]$.
We will denote $Y_{\Gamma,\nu}$ by $Y$ for short, and if we want to stress
the base field, we will denote it by $Y_\sk$.

In case $\sk=\overline\BF_p$, the Frobenius twist $Y^{(1)}$ carries a sheaf
$\sO$ of Azumaya algebras such that $H^i(Y^{(1)},\sO)=0$ for $i>0$, and
$H^0(Y^{(1)},\sO)=\sW_\sk^{\Gamma_n}$~(\cite[Lemma~6.2]{BK}). From the point
of view of~\cite{EGO}, $\sO$ is the localization of the spherical subalgebra
of the symplectic reflection algebra $H_0$ of $(\Gamma_n,U)$
(rational Cherednik algebra) for the zero value of parameters.

We choose a $\Gamma_n$-invariant Lagrangian vector subspace $L\subset U$.
We will denote the categorical quotient $L/\Gamma_n\subset U/\Gamma_n$
by $\fL_0\subset\fM_0(W,V)$. We will denote the scheme-theoretic preimage
$\pi^{-1}(\fL_0)\subset\fM(W,V)=Y$ by $\fL\subset Y$. The completions of our
schemes along their subschemes will be denoted like $\widehat{Y}_\fL$.
%$\widehat{Y}_{\pi^{-1}(0)}$.
Note that the dilation action of $\Gm$ on $\Hom(V_i,V_{i\pm1})$
descends to the $\Gm$-action
on $\fM_0(W,V)$, and then lifts to the $\Gm$-action on $Y$.

We will also need another commuting action of $\Gm$ on $Y$: the hyperbolic one, preserving
the symplectic form on $Y$. To distinguish it from the dilation action above,
we will denote this copy of $\Gm$ by ${\mathbb G}_m^h$.
So if a point $y$ of $Y$ is
represented by a quadruple $(B_i\in\Hom(V_i,V_{i+1}),\ B'_i\in\Hom(V_i,V_{i-1}),\
x_0\in V_0,\ x_0^*\in V_0^*)$, and $c\in{\mathbb G}_m^h$,
then $cy$ is represented
by $(cB_i,c^{-1}B'_i,x_0,x_0^*)$.
%Note that this hyperbolic action gives rise
%to a choice of

\subsection{Splitting bundles for  Azumaya algebras}\label{sec22}

\begin{lem}\label{zeroth}
a) There exists an Azumaya algebra $\CA$ on $({\mathbb A}^{2n})^{(1)}/\Gamma_n$,
such that the pull-back of $\CA$ to $Y^{(1)}$ is Morita equivalent to $\sO$,
while its pull-back to $({\mathbb A}^{2n})^{(1)}$ is $\Gamma_n$-equivariantly
Morita equivalent to $\sW$.

b) The Azumaya algebra $\CA$ splits on the formal neighborhood of $L^{(1)}/\Gamma_n$.
\end{lem}


\proofpt  (a) is shown in \cite[Proposition 6.5]{BK}.

To check (b) we first check that $\CA|_{L^{(1)}/\Gamma_n}$ splits.
Since $L^{(1)}/\Gamma_n$ is smooth, in view of \cite[Corollary IV.2.6]{Mln}
(which says that the Brauer group of a regular scheme injects into the Brauer group of a dense
open subscheme) it suffices to check that the restriction of $\CA$ to
a nonempty open subset in $L^{(1)}/\Gamma_n$ splits. Let $L_0\subset L$
be the union of free $\Gamma_n$ orbits. Then splitting of $\CA|_{L^{(1)}_0/\Gamma_n}$
amounts to a $\Gamma_n$-equivariant splitting of $\sW_\sk$ on $L^{(1)}_0$.
This is provided by
the module $\CO(L)$ equipped with the natural action of $\sW_\sk$, where
elements in $L$ act by multiplication by linear functions and elements in
a fixed $\Gamma_n$-invariant Lagrangian complement $'L$ act by derivatives.
Now we see  by induction that pull-back of $\CA$ to the $n$-th infinitesimal neighborhood
of $L^{(1)}/\Gamma_n$ splits; moreover, given a choice of a splitting on the $n$-th
infinitesimal neighborhood, there exists a compatible splitting on the $(n+1)$st one, which
clearly implies statement~(b).
The compatible splitting on the $(n+1)$st neighborhood exists since the obstruction lies in the second cohomology of $\CO$
on the affine variety $L^{(1)}/\Gamma_n$. $\square$

\begin{cor}
\label{first}
a) The Azumaya algebra $\sW_\sk$ splits $\Gamma_n$-equivariantly on
$\widehat{U}{}^{(1)}_{L^{(1)}}$;
$\widehat{U}{}^{(1)}_0$.

b)
 The Azumaya algebra $\sO$ splits  on
$\widehat{Y}{}^{(1)}_{\fL^{(1)}}$. $\square$
%$\widehat{Y}{}^{(1)}_{\pi^{-1}(0)^{(1)}}$.

%c) The splitting bundles in (a) can be equipped with a
%$\Gmo\times (\Gmo)^h$ equivariant structure.
\end{cor}


Now we have the following chain of equivalences of derived categories,
cf.~\cite[Remark~6.8 and the proof of~Theorem~6.7]{BK}:
\begin{multline}
\label{mult}
%D^b\Coh(Y^{(1)})\cong
 D^b\Coh(\widehat{Y}{}^{(1)}_{\fL^{(1)}})\cong
D^b(\sO|_{\widehat{Y}{}^{(1)}_{\fL^{(1)}}}-\Mod)\underset{R\Gamma}{\iso}
D^b(\sW_\sk^{\Gamma_n}|_{\widehat\fM_0(W,V){}^{(1)}_{\fL_0^{(1)}}}-\Mod)\\
\cong
D^b(\sW_\sk\rtimes\sk[\Gamma_n]|_{\widehat{U}{}^{(1)}_{L^{(1)}}}-\Mod)\cong
D^b\Coh^{\Gamma_n}(\widehat{U}{}^{(1)}_{L^{(1)}}).
%\cong D^b\Coh^{\times\Gamma_n}(U^{(1)})
%
%D^b\Coh^\Gmo(Y^{(1)})\cong
%D^b\Coh^\Gmo(\widehat{Y}{}^{(1)}_{\pi^{-1}(0)^{(1)}})\cong
%D^b(\sO|_{\widehat{Y}{}^{(1)}_{\pi^{-1}(0)^{(1)}}}-\Mod^\Gmo)\iso\\
%D^b(\sW_\sk^{\Gamma_n}|_{\widehat\fM_0(W,V){}^{(1)}_0}-\Mod^\Gmo)\cong
%D^b(\sW_\sk\rtimes\sk[\Gamma_n]|_{\widehat{U}{}^{(1)}_0}-\Mod^\Gmo)\cong\\
%D^b\Coh^{\Gmo\times\Gamma_n}(\widehat{U}{}^{(1)}_0)\cong
%D^b\Coh^{\Gmo\times\Gamma_n}(U^{(1)})
\end{multline}
Notice that the first (respectively, the last) equivalence depends on the choice of a splitting bundle
whose existence is guaranteed by Corollary \ref{first}(b) (respectively, a).
We denote those splitting bundles by $\CE_\CL$, $\CE_L$ respectively.

\begin{prop}\label{22}
There exists an equivalence of triangulated categories
$$\Phi_\sk:D^b\Coh^{\Gmo\times(\Gmo)^h\times \Gamma_n}(U^{(1)})\iso
D^b\Coh^{\Gmo\times (\Gmo)^h}(Y^{(1)}),$$
which is compatible with the composed equivalence in \eqref{mult}
for an appropriate choice of the splitting bundles in Corollary
\ref{first}.
\end{prop}

\proofpt Let $\CA$ be as in Lemma \ref{zeroth},
and let $\CE_{res}$  be a splitting
bundle for $\sO \otimes _{\CO(Y^{(1)})} \pi^*(\CA^{op})$.
Then it is shown in \cite{BK} that the algebra $\End (\CE_{res})^{op}$ is Morita
equivalent to $\Gamma_n \# \CO({\mathbb A}^{2n})$ and
the functor $\F\mapsto R\Hom(\CE_{res},\F)$ provides an equivalence
\begin{equation}\label{eqBK}
D^b(\Coh(Y^{(1)}))\iso D^b(\End(\CE_{res})^{op}-\Mod_{fg})\cong \Coh^{\Gamma_n}(
({\mathbb A}^{2n})^{(1)}).
\end{equation}

Here the first equivalence is given by $\F\mapsto R\Hom(\CE_{res},\F)$. To describe
the second equivalence  we need to fix a $\Gamma_n$-equivariant
splitting bundle $\CE_{orb}$ for $\sW_\sk$.
Then we get:
\begin{multline*}
\End (\CE_{res})=\Gamma (\sO) \otimes _{\CO({\mathbb A}^{2n}/\Gamma_n)^{(1)} }
\Gamma(\CA)=\sW^{\Gamma_n}\otimes  _{\CO({\mathbb A}^{2n}/\Gamma_n)^{(1)}}
\Gamma(\CA)\sim \\
  \Gamma_n\# \sW   \otimes _{\CO({\mathbb A}^{2n}/\Gamma_n)^{(1)}}
\Gamma(\CA)=\Gamma_n\# \End_{\CO({\mathbb A}^{2n})^{(1)} }(\CE_{orb})\sim
\Gamma_n \#  \CO(({\mathbb A}^{2n})^{(1)}).
\end{multline*}
Here the first Morita equivalence is given by the bimodule
$\sW\otimes  _{\CO({\mathbb A}^{2n}/\Gamma_n)^{(1)}}
\Gamma(\CA)$, while the second one comes from the equivalence between
the categories of $\Gamma_n$-equivariant coherent sheaves on $({\mathbb A}^{2n})^{(1)}$
and  $\Gamma_n$-equivariant modules over $\End_{\CO({\mathbb A}^{2n})^{(1)} }(\CE_{orb})$
given by $\F\mapsto \F\otimes_{\CO} \CE_{orb}$, the latter tensor product being equipped
with the diagonal $\Gamma_n$-action.


Thus equivalence \eqref{eqBK} depends on the choice of splitting bundles
$\CE_{res}$, $\CE_{orb}$ in Corollary \ref{first}, which we assume to be fixed
from now on.  That choice determines (up to an isomorphism) a choice of splitting bundles
$\CE_L$, $\CE_{\CL}$ in the following manner.
By Lemma \ref{zeroth}(b) Azumaya algebra $\CA$ splits on the formal neighborhood
of $L^{(1)}/\Gamma_n$. The splitting bundle on that neighborhood is determined
uniquely up to twisting by a line bundle on the formal neighborhood;
%$\widehat{({\mathbb A}^{2n})^{(1)}/\Gamma_n}_{L^{(1)}/\Gamma_n}$;
since
every such bundle is easily seen to be trivial, the splitting is unique up to an isomorphism.
A diagram chase shows that under those choices the equivalences
 \eqref{mult} and \eqref{eqBK} are compatible.



%\footnote{The equivalence
%in \cite{BK} is based on a particular choice of a quantization whose algebra of global
%sections  is isomorphic to $\sW_\sk$. This choice may differ from the one
%used in the present paper. However, the argument of \cite{BK} applies
%to any such choice.}



We  claim that $\CE_{res}$ carries a canonical $\Gmo\times(\Gmo)^h$-equivariant structure.
This follows from (the proof of) \cite[Proposition 4.3(i)]{BK}.

%{\em  we sketch the argument
%for the reader's convenience -- or do we? }
%Let $Y_0$ be the maximal open subset such that $\pi$ induces an isomorphism between
%$Y_0$ and its image in ${\mathbb A}^{2n}/\Gamma_n$.

%XXX

The $\Gmo\times (\Gmo)^h$-equivariant structure on $\CE_{res}$
induces a bigrading on the ring $\End(\CE_{res})$, it is easy to see
that the category of bigraded modules is canonically equivalent
to $\Coh^{\Gamma_n\times \Gmo\times (\Gmo)^h}({\mathbb A}^{2n})$.

Thus we get equivalences
\begin{equation}\label{eqbigr}
D^b(\Coh^{ \Gmo\times (\Gmo)^h}(Y^{(1)}))\cong D^b(\End(\CE_{res})^{op}-\Mod^{\mathrm{bigr}}
_{\mathrm{fg}})\cong
D^b(\Coh^{\Gamma_n\times \Gmo\times (\Gmo)^h}({\mathbb A}^{2n})^{(1)}),
\end{equation}
where $\Mod^{\mathrm{bigr}}
_{\mathrm{fg}}$ denotes the category of finitely generated bigraded modules.

It is clear from the definition that the constructed equivalence is compatible
with \eqref{eqBK}, hence with \eqref{mult}.
$\square$


 We will use the same notation $\Phi_\sk$
  for the induced compatible equivalence on categories with
 less equivariance:
 $D^b\Coh^{\Gamma_n}(\widehat{U}{}^{(1)}_{L^{(1)}}) \iso
D^b\Coh (\widehat{Y}{}^{(1)}_{\fL^{(1)}})$ and
$D^b\Coh^{\Gamma_n\times \Gmo}(\widehat{U}{}^{(1)}_{L^{(1)}}) \iso
D^b\Coh^{\Gmo} (\widehat{Y}{}^{(1)}_{\fL^{(1)}})$.


\begin{rem}
One can construct in a $\Gmo\times (\Gmo)^h$-equivariant bundle on $Y^{(1)}$
providing an equivalence $\Phi_\sk$ in a different way which is probably better suited for
generalization to other symplectic resolutions of singularities. Instead of
Lemma \ref{zeroth} one can use Corollary \ref{first}(b) to get a
tilting bundle on the formal neighborhood of $\pi^{-1}(0)$ in $Y^{(1)}$.
That bundle can be equipped with a $\Gmo\times (\Gmo)^h$-equivariant structure
by Proposition \ref{vologod} in the Appendix, which then can be shown to come
from a uniquely defined vector bundle on $Y^{(1)}$ (cf. \cite[Lemma 2.12]{BK}).
\end{rem}

\begin{lem}\label{lem_comm}
a) Assume that the splitting bundles $\CE_{res}$, $\CE_{\CL}$
and $\CE_{orb}$, $\CE_L$ are chosen compatibly as in the proof of Proposition
\ref{22}.

Then the following diagram is commutative:
$$
\begin{CD}
D^b\Coh^\Gmo_{\fL^{(1)}} (Y^{(1)})@>{\Phi_\sk^{-1}}>>
D^b\Coh^{\Gmo\times\Gamma_n}_{L^{(1)}}(U^{(1)})\\
@V{\F\mapsto R\Gamma(\F\otimes \CE_\CL)}VV @V{\F\mapsto \F\otimes \CE_L}VV \\
D^b(\sW_\sk^{\Gamma_n} -\Mod^\Gmo_{\fL_0^{(1)}})
@>{M\mapsto \sW_\sk\otimes _{\sW_\sk^{\Gamma_n} } M}>>
D^b(\sW_\sk\rtimes\sk[\Gamma_n] -\Mod^\Gmo_{L^{(1)}}),
\end{CD}
$$
where lower indices denote the set theoretic support condition on the coherent
sheaf/module/sheaf of modules.

b) For any choice of a $\Gamma_n$-equivariant splitting
$\bar{\CE}_L$ for $\sW_\sk$ on $L^{(1)}$
there exists a splitting bundle $\CE_{orb}$ compatible with $\bar{\CE}_L$.
\end{lem}

\proofpt  a) follows by diagram chase.
To check (b) notice that any two $\Gamma_n$-equivariant
splitting bundles for $\sW_\sk$ on $L^{(1)}$ differ by a twist
by a $\Gamma_n$-equivariant line bundle. It is easy to see that
such the isomorphism class of such a line bundle is determined
by the character of $\Gamma_n$ by which it acts on the fiber of the line
bundle at zero. Thus we can start with an arbitrary $\CE_{orb}$, after possibly
twisting it by a character of $\Gamma_n$ we will achieve the
 desired compatibility.
$\square$

Notice that $\sW|_{L^{(1)}}$ has a standard $\Gamma_n$-equivariant
splitting provided by the $\sW$ module $\CO(L)$ where elements in $L\subset U\cong U^*$
act by multiplication by linear functions and elements in a fixed 
$\Gamma_n$-invariant
complement to $L$ act by derivatives.
By Lemma \ref{lem_comm}(b) there exists a splitting bundle $\CE_{orb}$ which is compatible
with that equivariant splitting of $\sW|_{L^{(1)}}$, from now on we assume that
the equivalence are defined using a choice of $\CE_{orb}$ satisfying that compatibility.

\medskip

Given an irreducible $\sk[\Gamma_n]$-module $\rho$, we denote by
$\CF_\rho$ the coherent sheaf $\Phi_\sk(\CO_{L^{(1)}}\otimes\rho)$.

\subsection{Lagrangian components}
\label{components}
We now fix the
%or is it "a"?
  Lagrangian linear subspace $L\subset U$ given by the condition of nilpotency of
the compositions of $B_i$'s.
%{\bf I don't understand -- isn't the set $B_i=0$ Lagrangian?}
According to~\cite[Proposition~7.2.18]{H} or~\cite[Lemma~5.1]{G},
the ${\mathbb G}_m^h$-fixed points $Y^{{\mathbb G}_m^h}_{\Gamma,\nu}$
are naturally labeled by $r$-partitions
$\bmu=(\mu^{(1)},\ldots,\mu^{(r)})$ of total size $|\bmu|=\sum_i|\mu^{(i)}|=n$
(recall that $\nu=\nu_0+n\delta$). We will abuse notation by using the same
letter for a multi-partition and for the corresponding fixed point.

Now the Lagrangian subvariety $\fL\subset Y$ is a union of locally closed
Lagrangian subvarieties $\fL_\bmu^\circ$ labeled by the set $\CP(r,n)$ of
$r$-partitions of total
size $n$. Each piece $\fL_\bmu^\circ$ is isomorphic to the affine space $\BA^n$,
and is defined as the attracting set of the corresponding
${\mathbb G}_m^h$-fixed point. We define $\fL_\bmu$ as the closure of
$\fL_\bmu^\circ$. I.~Gordon~\cite[Section~5.4]{G} has defined a partial order
on $\CP(r,n)$ generated by the rule $\bmu\precg\blambda$ if $\bmu\ne\blambda$
and $\fL_\blambda\cap\fL_\bmu^\circ\ne\emptyset$. Moreover, he has described this
order combinatorially in~\cite[Sections~6,7]{G}. In order to formulate the
description, recall
the bijection $\tau_{\mathbf s}$~\cite[6.2]{G},~\cite[7.2.17]{H} between the
set $\CP(r,n)$ of multipartitions, and the set
$\CP_{\nu_0}(|\nu_0|+rn)$ of usual partitions of size $|\nu_0|+rn$ having
$r$-core $\nu_0$. Then according to~\cite[Proposition~7.10]{G}, we have
$\bmu\precg\blambda\ \Rightarrow\ \bmu\precc\blambda\ \Leftrightarrow\
^t\tau_{\mathbf s}(^t\bmu)\lhd\
^t\tau_{\mathbf s}(^t\blambda)$, that is, one transposed partition dominates the
other.

\subsection{Hamiltonian reduction}
\label{G3}
In this subsection we recall some results of~\cite{GG},~\cite{G1},~\cite{L}.
We denote by $R(Q,\nu)$ the vector space
$V_0\oplus\bigoplus_{i\in\BZ/r\BZ}\Hom(V_i,V_{i+1})$, and we denote by $R(DQ,\nu)$
the cotangent vector space $R(Q,\nu)\oplus R(Q,\nu)^*$ (representations of the
double quiver). We denote by $\sW_\sk(\nu)$ the Weyl algebra of the symplectic
vector space $R(DQ,\nu)$, that is the algebra of differential operators
$\CalD_\sk(\nu):=\CalD_\sk(R(Q,\nu))$. The symplectic action of
$G_\nu=\prod_{i\in I}GL(V_i)$ on $R(DQ,V)$ gives rise to the map
$\tau:\ \fg_\nu:=\on{Lie}G_\nu\to\sW_\sk(\nu)$. For $\chi=(\chi_i)_{i\in I}$ such
that $\chi\cdot\nu:=\sum_{i\in I}\chi_iv_i=0$ we set
$\tau_\chi(\xi):=\tau(\xi)-\sum_{i\in I}\chi_i\on{Tr}\xi_i$ where
$\xi=(\xi_i)_{i\in I}\in\fg$.

The quantum Hamiltonian reduction is defined as
$\sW_\sk(\nu)\red_\chi G_\nu=\CalD_\sk(\nu)\red_\chi G_\nu:=
\left(\CalD_\sk(\nu)/\CalD_\sk(\nu)\cdot
\tau_\chi(\fg_\nu)\right)^{G_\nu}$. For certain $\chi(\nu)\in\BQ^I$,
we have a canonical isomorphism
$\eta:\ \sW_\sk(\nu)\red_{\chi(\nu)}G_\nu\iso\sW_\sk^{\Gamma_n}$.
In case $\nu_0=0$, that is $\nu=n\delta$, this isomorphism
is proved in~\cite[Theorem~1.4]{G1},~\cite[Theorem~6.2.1]{L};
in this case $\chi(n\delta)_{i_0}=\frac{1-r}{r}$, while
$\chi(n\delta)_i=\frac{1}{r}$ for $i\ne i_0$. For general $\nu$ this isomorphism
follows from the argument in~\cite[Section~6.4]{L}: the action of the affine
Weyl group by the reflection functors establishes a canonical isomorphism of
$\sW_\sk(\nu)\red_{\chi(\nu)}G_\nu$ and $\sW_\sk(n\delta)\red_{\chi(n\delta)}G_{n\delta}$.
We fix such $\chi(\nu)$ as in {\em loc. cit.}



\subsection{BFG localization}
\label{BFG}

We will recall the notations and results of~\cite[Section~3]{bfgi}.
For any algebraic variety $X$ over $\sk=\overline{\mathbb F}_p$
we denote by $X^{(1)}$ its Frobenius
twist, and we denote by $\on{Fr}:\ X\to X^{(1)}$ the Frobenius morphism.
For a connected linear algebraic group $A$ over $\sk$ we have an exact
sequence of groups $1\to A_1\to A\stackrel{\on{Fr}}{\longrightarrow}A^{(1)}\to1$
where $A_1$ stands for the Frobenius kernel. The Lie algebra $\fa$ of $A$
is equipped with a natural structure of $p$-Lie algebra, and its universal
enveloping algebra $U(\fa)$ contains the $p$-center $\fZ(\fa)$.
We denote by ${\mathbb X}^*(\fa)$ the lattice of characters of $\fa$ of the
type $\chi=\on{dlog}f$ where $f:\ A\to{\mathbb G}_m$ is an algebraic character
of $A$. For such a character $\chi$ we denote by $I_\chi$ the kernel of the
corresponding homomorphism $U(\fa)\to\sk$. We set
$I_\chi^{(1)}:=I_\chi\cap\fZ(\fa)$, a maximal ideal of $\fZ(\fa)$.
Note that $I_\chi^{(1)}=I_0^{(1)}$ (see~\cite[(3.2.4) and~3.3]{bfgi}).
We denote by $\fu(\fa)$ the quotient of $U(\fa)$ by the two-sided ideal
generated by $I^{(1)}_\chi=I^{(1)}_0$. The image of $I_\chi$ in $\fu(\fa)$
is denoted by $\fri_\chi\subset\fu(\fa)$.

Now we take $A=G_\nu=\prod_{i\in\BZ/r\BZ}GL(V_i)$, and we denote its
Lie algebra by $\fg_\nu$. The $p$-center $\fZ_\nu$ of the Weyl algebra
$\CalD_\sk(\nu)$ equals $\sk[R(DQ,\nu)^{(1)}]$. Thus we may localize the
$\fZ_\nu$-module $\CalD_\sk(\nu)$ to $R(DQ,\nu)^{(1)}$ and view it as a sheaf
of algebras $\CalD_\sk(\nu)^{(1)}$.
We have the moment map $\mu^{(1)}:\ R(DQ,\nu)^{(1)} \to[\fg_\nu^*]^{(1)}$.
The quotient $\CalD_{\sk,\chi}(\nu)^{(1)}:=
\CalD_\sk(\nu)^{(1)}/(\CalD_\sk(\nu)^{(1)}\cdot I^{(1)}_\chi)$ is just
the restriction of $\CalD_\sk(\nu)^{(1)}$ to the scheme-theoretic zero-fiber
$\sM(W,V)^{(1)}$
of the moment map (and is independent of $\chi\in{\mathbb X}^*(\fg_\nu)$).
We consider
$\CE_\chi:=(\CalD_{\sk,\chi}(\nu)^{(1)}/\CalD_{\sk,\chi}(\nu)^{(1)}\cdot
\fri_\chi)^{G_{\nu,1}}$: a $G_\nu^{(1)}$-equivariant sheaf on
$\sM(W,V)^{(1)}$.

We restrict $\CE_\chi$ to the open subset $\sM(W,V)^{(1),s}\subset
\sM(W,V)^{(1)}$ of stable points. The action of $G_\nu^{(1)}$ on
$\sM(W,V)^{(1),s}$ is free; the projection $\on{pr}:\ \sM(W,V)^{(1),s}\to
\fM(W,V)^{(1)}$ is a $G_\nu^{(1)}$-torsor. We set
\begin{equation}
\label{locp}
\CA_\chi:=
\on{pr}_*(\CE_\chi|_{\sM(W,V)^{(1),s}})^{G_\nu^{(1)}},\ \on{and\ put}\
\sA_\chi:=\Gamma(\fM(W,V)^{(1)},\CA_\chi).
\end{equation}

Similarly to~\cite[Section~7]{bfgi}, we have a canonical isomorphism
$\sA_\chi\cong\sW_\sk(\nu)\red_\chi G_\nu=\CalD_\sk(\nu)\red_\chi G_\nu$.
We have $\CA_{\chi(\nu)}\cong\sO$ of~\ref{Notations}.

If we choose another
stability condition $\zeta$ (from an open chamber, i.e. not on a wall)
in the construction of $\fM(W,V)$, we obtain another
quiver variety $\fM_\zeta(W,V)$ with a sheaf of Azumaya algebras $\CA^\zeta_\chi$
on $\fM_\zeta(W,V)^{(1)}$ such that $\Gamma(\fM_\zeta(W,V)^{(1)},\CA^\zeta_\chi)=
\CalD_\sk(\nu)\red_\chi G_\nu$.
If $V=V^{n\delta}$, i.e. $\dim V_i=n$ for any $i\in\BZ/r\BZ$,
for an appropriate choice of the stability condition we have
$\fM_\zeta(W,V^{n\delta})=Y_{\Gamma,\nu}$. Then, as discussed in Section~\ref{G3},
$\sW_\sk^{\Gamma_n}=\CalD_\sk(n\delta)\red_{\chi(n\delta)} G_{n\delta}=
\Gamma(Y_{\Gamma,\nu}^{(1)},\CA^\zeta_{\chi(n\delta)})$.

%Moreover, given a $\CalD_\sk(\CX,n\chi(\nu)_{i_0})$-module $M$, the Hamiltonian
%reduction as above gives rise to the $\CA_{\chi(\nu)}$-module $\CH(M)$.
%and we have canonically $\BH(M)\cong\Gamma(Y_{\Gamma,\nu}^{(1)},\CH(M))$.

%Now let us denote by $\overset{\circ}{U}\subset U=\BA^{2n}$ the open subspace
%formed by all
%the points that belong to not more than one reflection hyperplane. We denote by
%$\overset{\circ}\fM_0(W,V)\subset\fM_0(W,V)=\BA^{2n}/\Gamma_n$
%the open subspace
%$\overset{\circ}{U}/\Gamma_n$. Finally, we denote by
%$\overset{\circ}{Y}_{\Gamma,\nu}=
%\overset{\circ}\fM(W,V)$ the preimage $\pi^{-1}(\overset{\circ}\fM_0(W,V))$.
%Then the birational isomorphism between $Y_{\Gamma,\nu}$ and
%$Y_{\Gamma,n\delta}$
%is {\em regular} on $\overset{\circ}{Y}_{\Gamma,\nu}$ and identifies it with
%$\overset{\circ}{Y}_{\Gamma,n\delta}$. This common open piece will be denoted
%by $\overset{\circ}{Y}$. Its complement has codimension 2 in $Y_{\Gamma,\nu}$.
%???The Azumaya algebras $\CA_{\chi(\nu)}$ and $\CA_{\chi(n\delta)}$ coincide on
%$\overset{\circ}{Y}{}^{(1)}$ being localizations of the same algebra
%$\sW_\sk^{\Gamma_n}$. Conversely, this coincidence implies the isomorphism of
%global sections
%$\Gamma(Y_{\Gamma,\nu}^{(1)},\CA_{\chi(\nu)})\cong
%\Gamma(\overset{\circ}{Y}{}^{(1)},\CA_{\chi(\nu)})\cong
%\Gamma(\overset{\circ}{Y}{}^{(1)},\CA_{\chi(n\delta)})\cong
%\Gamma(Y_{\Gamma,n\delta}^{(1)},\CA_{\chi(n\delta)})$???.

According to~\cite[Theorem~7.2.4]{bfgi}, the functor of global sections
$R\Gamma:\ D^b(\CA_{\chi(\nu)}-\on{mod})\to D^b(\sA_{\chi(\nu)}-\on{mod})=
D^b(\sW_\sk^{\Gamma_n}-\on{mod})$ is an equivalence of categories.
We denote the quasiinverse equivalence by $M\mapsto\ ^{\on{loc}}M$ (localization).



\subsection{Verma modules}
\label{irr O}
If $\sk=\BC$, all the irreducible modules over $\sW_\BC\rtimes\BC[\Gamma_n]$
in the category $\CO$ are of the form $\sV_\bmu:=\BC[L]\otimes\rho_\bmu$ where
$\bmu$ is an $r$-multipartition of $n$, and $\rho_\bmu$ is the corresponding
irreducible $\BC[\Gamma_n]$-module. Under the Morita equivalence of
$\sW_\BC\rtimes\BC[\Gamma_n]$ and $\sW_\BC^{\Gamma_n}$, the Verma module
$\sV_\bmu$ goes to $\sV_\bmu^{\Gamma_n}:=\BC[L]^{\rho_\bmu}$
(the isotypic component).

If $\sk=\overline\BF_p$, we keep the same notation for the (spherical)
Verma module $\sV_\bmu^{\Gamma_n}:=\sk[L]^{\rho_\bmu}$.

\begin{lem}
\label{exact}
$^{\on{loc}}\sV_\bmu^{\Gamma_n}\in\CA_{\chi(\nu)}-\on{mod}\subset
D^b(\CA_{\chi(\nu)}-\on{mod})$ is an $\CA_{\chi(\nu)}$-module (as opposed to a
complex of modules).
\end{lem}

\proofpt
We choose a complementary $\Gamma_n$-invariant Lagrangian subspace $'L:\
'L\oplus L=U$. The composed projection $Y_{\Gamma,\nu}\to\BA^{2n}/\Gamma_n\to\ 'L/\Gamma_n=\BA^n/\Gamma_n$
is flat since all the fibers are of the same dimension $n$. Hence a
sequence $e_1,\ldots,e_n$ of generators of $\sk[\BA^n/\Gamma_n]$ is a
regular sequence for $\CO_{Y_{\Gamma,\nu}}$. Hence the sequence
$e_1^p,\ldots,e_n^p$ is a regular sequence for $\on{Fr}_*\CO_{Y_{\Gamma,\nu}}$ on $Y_{\Gamma,\nu}^{(1)}$. By Nakayama Lemma, $e_1^p,\ldots,e_n^p$ form a regular sequence for the Azumaya algebra
$\CA_{\chi(\nu)}$ on $Y_{\Gamma,\nu}^{(1)}$. According to Corollary~\ref{first},
on the formal neighbourhood of the central fiber, $\CA_{\chi(\nu)}\simeq
\End(E)$ splits. Since $E$ contains $\CO_{Y_{\Gamma,\nu}^{(1)}}$ as a direct
summand, $\End(E)$ contains $E$ as a direct summand, hence $e_1^p,\ldots,e_n^p$ forms a regular sequence for $E$ as well.
\qed




\subsection{Localization of the family of Cherednik algebras}
\label{loca}
%The irreducible components of the Lagrangian subvariety
%$\fL\subset Y_{\Gamma,\nu}$ are parametrized by $r$-multipartitions of $n$
%in two ways. The first one, due to Gordon, was described
%in~Section~\ref{components};
%the second one, as the ``leading components'' of the support of localization of
%Verma modules, was described in~Section~\ref{irr O}.
%In order to identify the two parametrizations we set
%$\sk=\overline\BF_p$, and recall the following construction.

We include the algebras $\sW_\sk^{\Gamma_n}\subset\sW_\sk\rtimes\sk[\Gamma_n]$
into the $(r+1)$-parametric family of (spherical) Cherednik algebras
$e\bH e\subset\bH$, see e.g.~\cite[6.1]{L}. The parameters are denoted by
$\hbar,c_0,c_1,\ldots,c_{r-1}$ (in {\em loc. cit.} $c_0$ is denoted $k$).
The case of Weyl algebra corresponds to
$\hbar=1,c_0=c_1=\ldots=c_{r-1}=0$. The (spherical) Verma module
$\sV_{\bmu}^{\Gamma_n}$ (resp. $\sV_{\bmu}$) is included into the
$(r+1)$-parametric family of $e\bH e$- (resp. $\bH$)-modules
$\bV_{\bmu}^{\Gamma_n}$ (resp. $\bV_{\bmu}$). If $\sk=\overline\BF_p$, there
is a smooth family $\bY$ of algebraic varieties over the base $\BA^{r+1}$ with
coordinates $\hbar,c_0,c_1,\ldots,c_{r-1}$ (with a few affine hyperplanes in
$c_0,c_1,\ldots,c_{r-1}$ removed), and the flat sheaves of algebras
$\bA\subset\bB$ on $\bY^{(1)}$ with the following properties.

(a) The fiber $\bY_{(1,0,\ldots,0)}$ equals $Y$;

(b) The fiber $\bY_{(\hbar,\underline{c})}$ for general $(\underline{c})$
is the smooth affine cyclic
Calogero-Moser space $\on{Spec}Z_{\underline{c}^p-\hbar^{p-1}\underline{c}}$,
see~\cite[3.10]{G},~\cite{G0}; in particular, $\bY_{(0,\underline{c})}$
equals $\on{Spec}Z_{\underline{c}^p}$, while $\bY_{(1,\underline{c})}$
equals $\on{Spec}Z_{\underline{c}^p-\underline{c}}$.


%(c) The fiber $\bY_{1,k,\underline{c}}$ for general $(k,\underline{c})$ equals
%the smooth affine cyclic Calogero-Moser space
%$\on{Spec}Z_{k^p-k,\underline{c}^p-\underline{c}}$.

(c) The fiber $\bA_{(1,0,\ldots,0)}$ equals $\CA_{\chi(\nu)}$ (notations
of~Section~\ref{G3}), the fiber
$\bB_{(1,0,\ldots,0)}$ is an Azumaya algebra $\CB_{\chi(\nu)}$ such that
$\Gamma(Y^{(1)},\CB_{\chi(\nu)})\cong\sW_\sk\rtimes\sk[\Gamma_n]$;

(d) The fiber $\bA_{(0,\underline{c})}$ for general $(\underline{c})$ equals
$\on{Fr}_*\CO_{\bY_{(0,\underline{c})}}$;

(e) $\Gamma(\bY^{(1)}_{(\hbar,\underline{c})},\bA_{(\hbar,\underline{c})})=
e\bH_{(\hbar,\underline{c})}e$, and
$\Gamma(\bY^{(1)}_{(\hbar,\underline{c})},\bB_{(\hbar,\underline{c})})=
\bH_{(\hbar,\underline{c})}$.

The construction of $\bY,\bA,\bB$ is similar to the one in~\cite{bfgi} for
$r=1$. In particular, $\bB=\underline{\on{End}}_\bA(\ ^{\on{loc}}e\bH)$,
cf.~Section~7.3 of {\em loc. cit.}
The property (b) follows from~\cite[Theorem~1.1]{T}.

Now the module $\bV_{\bmu}^{\Gamma_n}$ localizes to the $\bA$-module
$^{\on{loc}}\bV_{\bmu}^{\Gamma_n}$, and the module $\bV_{\bmu}$
localizes to the $\bB$-module
$^{\on{loc}}\bV_{\bmu}$.



\subsection{Supports of Verma modules}
\label{supp}

\begin{prop}
\label{benz}
The support
$\on{supp}(\ ^{\on{loc}}\sV_{\bmu}^{\Gamma_n})=
\on{supp}(\ ^{\on{loc}}\sV_{\bmu})$ consists of $\fL_{\bmu}^{(1)}$ and
some $\fL_\blambda^{(1)}$ such that $\blambda\precc\bmu$.
%$\blambda\precg\bmu$ (and hence $\blambda\precc\bmu$).
\end{prop}

\proofpt
We study the support of the fiber
$^{\on{loc}}\sV_{\bmu,(\hbar,\underline{c})}$ in
$\on{Spec}Z_{\underline{c}^p-\hbar^{p-1}\underline{c}}^{(1)}$ for general
$(\underline{c})$. The hyperbolic ${\mathbb G}_m^h$ of
Section~\ref{components}
acts on $\bY$, and the ${\mathbb G}_m^h$-fixed points in a general fiber
are uniformly numbered by the set of $r$-partitions of $n$, according
to~\cite[3.8--3.10]{G}. As in Section~\ref{components}, we define
$\fL^\circ_{\blambda,\underline{c}}\subset\on{Spec}Z_{\underline{c}}$
as the attracting set of the corresponding ${\mathbb G}_m^h$-fixed point.
Contrary to the situation of Section~\ref{components}, for a general
$(\underline{c})$, the Lagrangian subvariety
$\fL^\circ_{\blambda,\underline{c}}\subset\on{Spec}Z_{\underline{c}}$
is closed, i.e. equals its closure $\fL_{\blambda,\underline{c}}$.

As $\underline{c}$ goes to $\underline{0}$, the Lagrangian subvariety
$\fL_{\bmu,\underline{c}}$ degenerates into the (closed) Lagrangian
subvariety contained in $\fL_{\bmu}\cup\bigcup_{\blambda\precc\bmu}\fL_\blambda$.
In effect, let $\bL_\bmu\subset\bY$ be the closed irreducible subvariety of
$\bY$ whose fiber over general $(\hbar,\ul{c})$ is
$\fL_{\bmu,\ul{c}^p-\hbar^{p-1}\ul{c}}$. The following lemma is well known to the experts (e.g. to B.~Webster).

\begin{lem}
\label{loseu}
The intersection $\bL_\bmu\cap Y$ contains
$\fL_{\bmu}\cup\bigcup_{\blambda\precg\bmu}\fL_\blambda$ and is contained in
$\fL_{\bmu}\cup\bigcup_{\blambda\precc\bmu}\fL_\blambda$.
\end{lem}

\proofpt
The intersection $\bL_\bmu\cap Y$ is Lagrangian in $Y$ since it is
contained in $\pi^{-1}\fL_0=\fL$.
Since $\bL_\bmu\cap Y$ is closed and contains $\fL_\bmu$, it must contain
$\fL_{\bmu}\cup\bigcup_{\blambda\precg\bmu}\fL_\blambda$. Conversely, if
$\fL_\blambda\subset\bL_\bmu\cap Y$, then $\blambda\precc\bmu$. In effect,
%if $\bnu\!\not\!\!\!\precg\bmu$,
%then there is a neighbourhood $U_\bnu\subset\bY$ of the
%fixed point $\bnu$ such that no point $x\in U_\bnu$ flows to $\bmu$ under
%the action of $\BG_m^h$. \qed
let us consider first the case $r=1$ (no cyclic group). Then both the
geometric and combinatorial orders are nothing but the dominance order on
partitions.
The parameter $\underline{c}$ is just one number $c$, and $\bY$ is just the
family degenerating the Calogero-Moser space into the Hilbert scheme.
The equivariant Borel-Moore homology groups $H^{\BG_m^h}_{BM}(\bY_{(\hbar,c)})$
are canonically identified for all the fibers of $\bY$. The fundamental
classes of $\fL_{\mu,c^p-\hbar^{p-1}c}$ form a basis in $H^{\BG_m^h}_{BM}(\bY_{(\hbar,c)})$.
For $c^p-\hbar^{p-1}c\ne0$, when $\fL_{\mu,c^p-\hbar^{p-1}c}$ is closed, its
fundamental class in the localized equivariant Borel-Moore homology is
proportional to the class of the fixed point $\mu$. On the other hand,
under the above identification, this class equals a class in
$H^{\BG_m^h}_{BM}(Y)$ supported in $\bL_\mu\cap Y$ (and proportional to the
fixed point class). Now under the well known
identification of $H^{\BG_m^h}_{BM}(Y)$ with the degree $n$ part of the Fock
space of symmetric functions, the fixed point classes correspond to the
Schur functions, while the fundamental classes of $\fL_\lambda$ correspond
to the monomial functions (see e.g.~\cite{nak}). The transition matrix between
these two bases is upper triangular in the dominance order, hence we are done.

For general $r$, recall that $Y_{\Gamma,\nu}$ is a connected component of the
fixed point set $(\on{Hilb}^m(\BA^2))^\Gamma$ where $m=v_0+\ldots+v_{r-1}$.
A fixed point $\blambda\in Y_{\Gamma,\nu}$ goes to the fixed point
$^t\tau_{\mathbf s}(\blambda)$ (a partition of $m$; notations of~\ref{components})
of $\on{Hilb}^m(\BA^2)$. If $\blambda$ lies in $\bL_\bmu\cap Y$, then
$^t\tau_{\mathbf s}(\blambda)$ lies in $\bL_{^t\tau_{\mathbf s}(\bmu)}\cap
\on{Hilb}^m(\BA^2)$, hence $^t\tau_{\mathbf s}(\blambda)\lhd\
{}^t\tau_{\mathbf s}(\bmu)$, that is $\blambda\preceqc\bmu$. \qed



Going back to the proof of the proposition,
the desired embedding $\on{supp}(\ ^{\on{loc}}\sV_{\bmu})\subset\fL_{\bmu}^{(1)}\cup
\bigcup_{\blambda\precc\bmu}\fL_\blambda^{(1)}$ follows from Lemma~\ref{exact} and
the next

\begin{lem}
\label{benzrou}
For a general $(\underline{c})$, the support
$\on{supp}(\ ^{\on{loc}}\sV_{\bmu,(\hbar,\underline{c})})$ equals
$\fL_{\bmu,(\underline{c}^p-\hbar^{p-1}\underline{c})}^{(1)}$.
\end{lem}

\proofpt As the Verma module $\sV_{\bmu,(\hbar,\underline{c})}$
is indecomposable,
the support $\on{supp}(\ ^{\on{loc}}\sV_{\bmu,(\hbar,\underline{c})})$ must lie
at exactly one component
$\fL_{\blambda,(\underline{c}^p-\hbar^{p-1}\underline{c})}^{(1)}$,
and we must prove that
$\blambda=\bmu$. It follows that $\sV_{\bmu,(\hbar,\underline{c})}$ possesses
a unique ${\mathbb G}_m^h$-equivariant irreducible quotient, supported at
the point $\blambda$ of the spectrum of the $p$-center of the Cherednik algebra
$\bH_{\hbar,\underline{c}}$. On the other hand, $\sV_{\bmu,(0,\underline{c})}$
is the fiber over a prime over $p$ in $\on{Spec}\BZ[\sqrt[r]{1}]$
of the family of Verma modules over the family of Cherednik algebras.
Recall the family $\bM(\bmu)$ of baby Verma modules over the family of Cherednik
algebras over $\on{Spec}\BC\to\on{Spec}\BZ[\sqrt[r]{1}]$ introduced
in~\cite[4.1]{G0}.
This family of Cherednik algebras (resp. baby Verma modules) extends to one over
$\on{Spec}\BZ[\sqrt[r]{1}]$, and its fiber over a prime over $p$ coincides
with the family of Section~\ref{loca} (see~\ref{loca}.(e)).
Now $\sV_{\bmu,(0,\underline{c})}$
surjects onto the baby Verma module $M(\bmu)_{\underline{c},\sk}$. It follows
that $\blambda=\bmu$. This completes the proof of the lemma along with the
proof of the proposition. \qed

The proposition is proved. \qed


%\subsection{More on Lagrangian components}
%\label{more}
%In this section we describe the orbit $\CN^{\nu_0}_\bmu$ explicitly.
%This description is not used in the rest of the paper.

%We start with a case $\nu=n\delta$, i.e. $\nu_0=0$.
%Given an $r$-partition $\bmu=(\mu^{(1)},\ldots,\mu^{(r)})$ we describe an
%irreducible component $\fL'_\bmu\subset\fL$. To this end we first describe
%a locally closed subvariety $\CN_\bmu\subset R(Q,\nu)$. A point of $\CN_\bmu$
%is a collection $(x_0\in V_0,\ B_i\in\Hom(V_i,V_{i+1}),\ 0\leq i\leq r-1)$.
%If we forget $x_0$, we get a representation of a cyclic quiver, and our
%first condition on $\CN_\bmu$ is that this representation is nilpotent, i.e.
%$(B_{r-1}B_{r-2}\cdots B_0)^n=0$. It is well known (see e.g.~\cite{Lu}) that
%there are finitely many isomorphism classes of nilpotent representations of the
%cyclic quiver of a given dimension. They are direct sums of indecomposable
%nilpotent representations denoted by $[i,j]$ where $i\leq j\in\BZ$, and we set
%$[i+rk,j+rk]=[i,j]$ for any $k\in\BZ$. The total dimension of $[i,j]$ equals
%$j-i+1$. A point $(x_0\in V_0,\ B_i\in\Hom(V_i,V_{i+1}),\ 0\leq i\leq r-1)$ lies
%in $\CN_\bmu$ if $x_0$ is general, and the isomorphism class of
%$(B_i\in\Hom(V_i,V_{i+1}))$ is $\bigoplus\limits_{1\leq k\leq r}^lT^{(k)}_l$.
%Here $T^{(r)}_l=[0,(\mu^{(r)}_l-1)r-1]\oplus[0,0]\oplus[1,1]\oplus\ldots\oplus
%[r-1,r-1]$, and for $1\leq k\leq r-1$ we have $T^{(k)}_l=[0,(\mu^{(k)}_l-1)r+k]
%\oplus[k+1,k+1]\oplus[k+2,k+2]\oplus\ldots\oplus[r-1,r-1]$, where
%$\mu^{(k)}=(\mu^{(k)}_1,\mu^{(k)}_2,\ldots)$. Finally, it is easy to see that
%the conormal bundle $T^*_{\CN_\bmu}R(Q,\nu)\subset R(DQ,\nu)$ has a nonempty
%intersection with the stable locus $\sM(W,V)^s$ of the zero level of the
%moment map, hence this intersection projects to an open subvariety of certain
%irreducible component of $\fL\subset\fM(W,V)$. The latter component is our
%desired $\fL'_\bmu$.

%At the other extreme, if $n=0$, that is $\nu=\nu_0$, then
%$\fM(W,V)$ is just one point, and hence there is a unique nilpotent
%$G_{\nu_0}$-orbit $N^{\nu_0}$ in the space of representations of nilpotent
%quiver of dimension $\nu_0$ with the following property: if $\CN^{\nu_0}$ is the
%product of this orbit and $V_0$, then
%$T^*_{\CN^{\nu_0}}R(Q,\nu_0)\subset R(DQ,\nu_0)$ has a nonempty
%intersection with the stable locus $\sM(W,V)^s$ of the zero level of the
%moment map.

%Now let us consider the case of general $\nu=\nu_0+n\delta$.
%For an $r$-partition $\bmu=(\mu^{(1)},\ldots,\mu^{(r)})$ of $n$, we define
%$\CN^{\nu_0}_\bmu\subset R(Q,\nu)$ as follows: a point
%$(x_0\in V_0,\ B_i\in\Hom(V_i,V_{i+1}),\ 0\leq i\leq r-1)$ lies
%in $\CN^{\nu_0}_\bmu$ if it is a direct sum of a point in $\CN_\bmu$ and a
%point in $N^{\nu_0}$. Once again, it is easy to see that
%the conormal bundle $T^*_{\CN_\bmu^{\nu_0}}R(Q,\nu)\subset R(DQ,\nu)$ has a nonempty
%intersection with the stable locus $\sM(W,V)^s$ of the zero level of the
%moment map, hence this intersection projects to an open subvariety of certain
%irreducible component of $\fL\subset\fM(W,V)$. The latter component is our
%desired $\fL'_\bmu$.

%\begin{lem}
%\label{iain}
%$\fL'_\bmu=\fL_\bmu$.
%\end{lem}

%\proofpt
%For $r=1$, $Y$ is just the Hilbert scheme of $n$ points on $\BA^2$, and the
%result is well known. The general case follows when we view $Y$ as a fixed
%point set component in $\on{Hilb}(\BA^2)$.
%\qed






\subsection{Procesi bundle}
\label{Proc}
Under the equivalence $\Phi_\sk$ of~(\ref{mult}) the structure sheaf
$\CO_{U^{(1)}}\rtimes\sk[\Gamma_n]\in D^b\Coh^{\Gm\times\Gamma_n}(U^{(1)})$ goes to
$\CE\in D^b\Coh^\Gm(Y^{(1)})$, to be called the {\em Procesi vector bundle}.
It splits into the direct sum $\CE=\bigoplus\CE_\blambda\otimes\rho_\blambda^*$
over the set of
$r$-partitions of $n$ numbering the irreducible $\sk[\Gamma_n]$-representations;
under the above equivalence $\CE_\blambda$ corresponds to
$\CO_{U^{(1)}}\otimes\rho_\blambda\in D^b\Coh^{\Gm\times\Gamma_n}(U^{(1)})$.

Recall that for an irreducible $\sk[\Gamma_n]$-module $\rho_\blambda$ we
denote by $\CF_\blambda$ the coherent sheaf
$\Phi_\sk(\CO_{L^{(1)}}\otimes\rho_\blambda)$.

We denote by $'L\subset U$ the $\Gamma_n$-invariant Lagrangian complement to
$L$ in $U$. Repeating the definitions of~Section~\ref{components} with $L$
replaced by $'L$ we obtain the Lagrangian components $'\fL_\bmu\subset Y$
numbered by the $r$-partitions of $n$. Note that the adjacency order on these
components is the inverse of the order in {\em loc. cit.}:
$'\fL_\blambda\cap\ '\fL_\bmu^\circ\ne\emptyset$ iff $\blambda\preceqg\bmu$.

We define the opposite Verma modules over $\sW_\sk\rtimes\sk[\Gamma_n]$ as
$'\sV_\bmu:=\sk[\ 'L]\otimes\rho_{^t\!\bmu}$. Under the Morita equivalence they
correspond to the $\sW_\sk^{\Gamma_n}$-modules
$'\sV_\bmu^{\Gamma_n}:=\sk[\ 'L]^{\rho_{^t\!\bmu}}$.
We have $\Ext^\bullet(\sV_\blambda,\ '\sV_\bmu)=0$ iff $\blambda\ne\bmu$,
and $\Ext^n(\sV_\blambda,\ '\sV_\blambda)\simeq\sk$, since
$\Ext_{k[U]}^\bullet(\sk[L],\sk[\ 'L])$ is a one-dimensional $\sk$-vector
space in degree $n$, and $\Gamma_n$ acts on it via the sign representation
$\det L$, and $\rho_\bmu\otimes\det L\simeq\rho_{^t\!\bmu}$.

Also, we define the sheaf
$'\CF_\bmu:=\Phi_\sk(\CO_{'L^{(1)}}\otimes\rho_{^t\!\bmu})$.

\begin{prop}
\label{renamed}
We have $\on{supp}\CF_\blambda=\on{supp}(\ ^{\on{loc}}\sV_\blambda)$.
\end{prop}

\proofpt Recall that the $\Gamma_n$ equivariant splitting bundle
$\CE_{orb}$ used in the definition of the equivalence $\Phi_\sk$ is assumed
to satisfy a compatibility stated after the proof of Lemma \ref{lem_comm} (end
of section \ref{sec22}).
It is clear that a Verma module $\sV_\blambda$ is isomorphic to
$\CO(L)\otimes _{\CO_{L^{(1)} } } (\CO_{L^{(1)}}\otimes\rho_{\blambda})$.
Thus the Proposition follows from the commutative diagram in Lemma \ref{lem_comm}(a).
%
%Note that the restriction of the Azumaya algebra $\sW_\sk|_{L^{(1)}}$ splits
%$\Gamma_n$-equivariantly, and equals $\End(\on{Fr}_*\CO_L)$. Looking at the
%definition of the equivalence $\Phi_\sk$ in~(\ref{mult}), we see that
%$\on{supp}\CF_\blambda$ equals the support of the localization of the
%$\sW_\sk\rtimes\sk[\Gamma_n]$-module $\sk[L]\otimes\rho_\blambda=\sV_\blambda$.
\qed

\begin{cor}
\label{benzroukao}
If the Lagrangian component $\fL^{(1)}_\bmu$ lies in the support of
$\CF_\blambda$, then $\bmu\preceqc\blambda$. If the Lagrangian component
$'\fL^{(1)}_\bmu$ lies in the support of
$'\CF_\blambda$, then $\bmu\succeqc\blambda$.
\end{cor}

\proofpt
We  argued in the proof
of~Proposition~\ref{benz} that the support of the localized Verma module
$^{\on{loc}}\sV_\blambda$ consists of $\fL^{(1)}_\blambda$ and some smaller
Lagrangian components $\fL^{(1)}_\bmu,\ \bmu\precc\blambda$. Thus Corollary
follows from Proposition \ref{renamed}.

The second part is proved similarly.
\qed

\subsection{Proof of Theorem~\ref{unique}}
\label{210}
Note that the composed projection
$Y_{\Gamma,\nu}\to\BA^{2n}/\Gamma_n\to\BA^n/\Gamma_n$ is flat since all the
fibers are of the same dimension $n$. It follows that
$\overline\CE_{\rho_\bmu}\cong \CF_\bmu$ and $\overline\CE{}'_{\rho_\bmu}\cong\ '\CF_\bmu$;
the required properties of $\CF_\bmu$, $\ '\CF_\bmu$ have been established
in~Corollary~\ref{benzroukao}. \qed


\subsection{Wreath Macdonald polynomials}
\label{wreath}
The equivalence $\Phi_\sk$ extends to the same named equivalence
$D^b\Coh^T(U^{(1)})\iso D^b\Coh^T(Y^{(1)})$ where $T$ is the 2-dimensional
torus with coordinates $(q,t)$ acting on $\BA^2$ such that $\Gm$ is the
diagonal subtorus $\Gm=\{(q,q)\}$, while
${\mathbb G}^h_m$ is the antidiagonal subtorus $\{(q,q^{-1})\}$.
We keep the same name for the
induced isomorphism of the $K$-groups of the categories in question.
We identify the equivariant $K$-group $K^T(pt)$ with $\BZ[q^{\pm1},t^{\pm1}]$.
We denote by $\iota_\blambda^*:\ K^T(Y^{(1)})\to K^T(pt)$ the fiber at the
torus fixed point $\blambda\in Y^{(1)}$.

Note that $\iota_\bmu^*\CE$ carries an action of $\Gamma_n$ since
$\CE$ carries a fiberwise action of $\Gamma_n$ by construction. Thus
$[\iota_\bmu^*\CE]\in\BZ[q^{\pm1},t^{\pm1}][K(\Gamma_n)]$.

\begin{cor}
\label{rom}
$\iota_\bmu^*[\CE]\otimes\sum_i(-q)^i[\Lambda^i(\ 'L^{(1)})]\in
\BZ[q^{\pm1},t^{\pm1}][\rho_\blambda,\ ^t\blambda\succeqc\bmu]$; also,\\
$\iota_\bmu^*[\CE]\otimes\sum_i(-t)^i[\Lambda^i(L^{(1)})]\in
\BZ[q^{\pm1},t^{\pm1}][\rho_\blambda,\ ^t\blambda\succeqc\ ^t\bmu]$.
\end{cor}

\proof (cf.~\cite[Proof of~Proposition~5.4.2]{H}).
The Koszul resolution shows that
$[\CO_{'L^{(1)}}]=[\CO_{U^{(1)}}]\otimes\sum_i(-t)^i[\Lambda^i(L^{(1)})]$.
We have $\operatorname{Tor}^{\CO_{L^{(1)}}}(\iota^*_\bmu(\CE),\sk)=
\operatorname{Tor}^{\CO_{U^{(1)}}}(\iota^*_\bmu(\CE\otimes\CO_{'L^{(1)}}),\sk)$.
By the local duality, the second statement of the corollary is equivalent to
$\operatorname{Ext}_{\CO_{U^{(1)}}\rtimes\sk[\Gamma_n]}(\CO_{'L^{(1)}}\otimes
\rho_\blambda,\iota^*_\bmu(\CE))=0$ unless $\blambda\succeqc\ ^t\bmu\
\Leftrightarrow\ ^t\blambda\preceqc\bmu$. Since $\Phi_\sk(\iota^*_\bmu(\CE))$ is
the skyscraper sheaf at the point $\bmu\in Y$, the latter Ext-vanishing
follows from $\operatorname{supp}\Phi_\sk(\CO_{'L^{(1)}}\otimes\rho_\blambda)
%\subset\bigcup_{\bmu\succeqg\ ^t\blambda}\ '\fL_\bmu
\subset\bigcup_{\bmu\succeqc\ ^t\blambda}\ '\fL_\bmu$.

The first statement of the corollary is proved similarly.
\qed

\section{Erratum to \cite{BK}}

We use this opportunity to correct some mistakes in \cite{BK}.


%Lemma 2.10
Proof of Lemma 2.12 on p. 8: the action of ${\mathbb G}_m$ on $X_R$ induces
an action of the formal completion of ${\mathbb G}_m$ on the formal completion
${\mathcal X}$, cf. Appendix
(section \ref{volgd}). So references to ${\mathbb G}_m$ equivariant structures
on the bundle on $\mathcal X$ in the proof should be replaced by ones
to equivariance with respect to the formal completion of ${\mathbb G}_m$.

Proposition 3.8 on p. 12: ``restriction to the generic fiber of the formal scheme $\hat{X}$"
should be removed, the statement of the Proposition should read as follows:
``The localization $\O_h\otimes_{\sk[[h]]}\sk((h)) $ is a sheaf of Azumaya algebras over
the sheaf of commutative rings $\O_{\hat{X}}\otimes _{\sk[[h]]}\sk((h))$."



The most essential correction\footnote{We thank Ivan Losev for pointing this out.}
 is for section 5.1.3 on p. 20 which provides the central step
in the proof of Proposition 5.3. It is claimed  there that the  quantization
of Kleinian singularity whose existence is claimed in  Proposition 5.3
is obtained from the Weyl algebra of the vector space $M$ by Hamiltionian reduction,
using the quantum moment map described in Example 5.7. This is false; e.g. in the example
$\Gamma=\{\pm 1\}$ acting on the two dimensional symplectic vector space, (the fiber at $h=1$ of)  the quantization
arising from Example 5.7 is identified with  the enveloping algebra $U(sl(2,\sk))$ reduced
at the singular Harish-Chandra central character $-\rho=-1$, while 
$\Gamma$-invariants
in the Weyl algebra are isomorphic to the reduction of $U(sl(2,\sk))$ by the central
character $-\frac{1}{2}$.
%[To see this consider the standard embedding of $sl(2)$ in the Weyl algebra
The argument breaks down as the displayed isomorphism
${\mathcal R}_h \cong \O_h\otimes_{\O_{X_0}^p}\otimes \O_{V_0}^p$ does not hold,
because, contrary to the mistaken statement in \cite{BK},
the commutative ring $\O(V^{(1)})[[h]]$ does not act on $\CR_h$ in  a natural way.

In order to get the quantization claimed in Proposition 5.3 one has to use a different
quantum moment map, which differs from the one in \cite[Example 5.7]{BK}
by the map $\chi(\nu)\cdot 1$, where $\chi(\nu)$ is the character
 of the Lie algebra $\g_\nu$ introduced in section \ref{G3}.
This follows from the result of \cite{CBH} or more general results of
\cite{G1} and \cite{L},
as recalled in section \ref{G3}.





\section{Appendix: Equivariant structure on rigid vector bundles}
\label{volgd}

\centerline{\bf {by Vadim Vologodsky}}
\vskip 6pt



\subsection{Rigid vector bundles}
Let $\TTT=(\Gm)^d$ be a $d$-dimensional torus.
We begin with the following result.

\begin{lem}
\label{odin}
Let $X$ be a proper (not necessarily smooth) scheme over an algebraicaly
closed field $\sk$, $E$ a vector bundle over $X$ such
that $\Ext^1(E;E)=0$. Assume that $X$ is endowed with a
$\TTT$-action  $m: \TTT \times X \to X$.
Then the action of  $\TTT$ lifts to $E$, i.e.
$E$ has a $\TTT$-equivariant structure.
\end{lem}
\proofpt First, we claim that $E$ is $\TTT$-invariant, that is
$m^* E \simeq p_2^* E$  locally with respect to $\TTT$.

Indeed, since $\Ext^1(E;E)=0$, the claim is true if we replace
$\TTT$ by its formal completion at $1$.
Now, let us consider the vector bundle $F= \Hom(m^* E, p_2^* E)$.
Since $X$ is proper, $p_{1 *} F$ is a coherent sheaf on  $\TTT$.
Theorem on Formal Functions implies that the
formal completion of $p_{1 *}F$
 at $1$ is a vector bundle. Therefore $p_{1 *}F$ is a vector bundle over
some
open neighborhood $U$ of $1$ in  $\TTT$. Further, shrinking
$U$, we may assume that there exists a section $s$ of   $p_{1 *}F$ over
$U$ which is equal to $\Id \in (p_{1 *}F)_1$ at the point $1\in  \TTT$.
It implies that   $m^* E \simeq p_2^* E$ over some open  neighborhood of
$1$. Since the subgroup of $\TTT$ generated by an open neighborhood
of $1$ coincides with $\TTT$, the claim follows.

We conclude that there exists an extension of affine algebraic groups

$$1\to \Aut(E) \to G \to  \TTT \to 1$$
such that the action of $G$ on $X$ (through  $\TTT$) lifts to $E$.

Note that the group $\Aut(E)$ is smooth and connected (since it is an open subscheme
of the  affine space $\End(E)$). Hence the lemma follows from the next one.

\begin{lem}
\label{dva} For a smooth  connected group $H$,
any extension
  $$1\to H \to G \stackrel{h}{\to}  \TTT \to 1$$
splits in the weak sense, i.e. there exists
a homomorphism $i:\TTT \to   G$ such that $h\circ i = \Id$.
\end{lem}
\proofpt The result is well known. We include the argument for the
readers' convenience.

Any affine algebraic group is a semidirect product of a reductive group
and a unipotent one. Therefore it is enough to prove the lemma in the following
two cases.

1) $H$ is a unipotent  group.

In this case the lemma follows from the structural result cited above:
if $G= U\times \TTT$ is the semidirect factorization, the morphism
$h$ factors through the second factor and, since the kernel of $h$ is
connected, there exists a splitting.

2) $H$ is  a reductive group.

In this case, $\rk(G)=\rk(H)+\dim (\TTT)$. Therefore, the intersection of a
maximal torus  $T\subset G$ with $H$ is a toral subgroup of $H$
of dimension $\rk(H)$. Hence its  connected component is a Cartan subgroup of $H$;
the intersection $T\cap H$ is contained in the centralizer of that Cartan subgroup,
hence it coincides with the Cartan subgroup. Thus, the lemma reduces
 to the obvious case when $G$ is a torus.
\qed


Let
${\mathcal X}\to{\mathcal Y}$ be a morphism. Let $y_l$ (resp. $\hat y$) be
the $l$-th infinitesimal neighborhood (resp. the formal neighborhood) of a closed  point $y\in{\mathcal Y}$, and let  ${\mathcal X}_{y_l}$ (resp. ${\mathcal X}_{\hat y}$)
be the preimage of $y_l$ (resp. $\hat y$)  in $\mathcal X$.  Assume that  ${\mathcal X}_{y_l}$, ($l=1,2,\cdots$), are endowed with compatible $\TTT$-actions
$m_l:\ \TTT\times{\mathcal X}_{y_l}\to{\mathcal X}_{y_l}$\footnote{In particular,  ${\mathcal X}_{\hat y}$ is equipped with an action of the formal completion $\hat  \TTT$ of $  \TTT$.}.  A $\TTT$-equivariant structure on
a vector bundle $E$ over  ${\mathcal X}_{\hat y}$ is a compatible system of $\TTT$-equivariant structures on $E_l:= E_{|{\mathcal X}_{y_l}}$. Equivalently,  this is  an action of     $\hat  \TTT$ on $E$ lifting the $\hat  \TTT$-action on
 ${\mathcal X}_{\hat y}$ such that for each $l$ the $\hat  \TTT$-action on $E_l$ can be extended to an action of $ \TTT$.
\begin{prop}
\label{vologod}
Assume that
${\mathcal X}\to{\mathcal Y}$ is proper and $\Ext^1(E,E)=0$. Then $E$ admits a  $\TTT$-equivariant structure.
\end{prop}


\proofpt
We consider the projective system of algebraic groups
$H_l=\Aut(E_l)$. Since
$\End_\sk(E_l)$ is a finite-dimensional $\sk$-vector space,
the projective system above
satisfies the Mittag-Leffler condition, i.e.
for any integer $l$ there exists an integer $N\geq l$ such that
$$\im( H_k \to H_l) =\im( H_N \to H_l), \quad\forall k>N.$$
Similarly, using that $\Ext^1(E,E)=0$ and  that $Ext^1(E_l, E_l)$ is finite-dimensional, we conclude that
$$\im( Ext^1(E_N, E_N)\to Ext^1(E_l, E_l))=0,$$
for $N$ large enough.

%Denote by $\hat H= \underset  \longleftarrow  \lim H_l(=\Aut(E) )$ the projective limit regarded as a pro-algebraic  group ,
Arguing as in the proof of Lemma ~\ref{odin},
we construct an extension of pro-algebraic groups
$$1\to \Aut(E) \to G\to\TTT\to1.$$
For a scheme $S$, $G(S)$ is the group of pairs $(t: S \to \TTT, \theta)$, where $\theta$ is a compatible system of isomorphisms between $(m_l \circ (t\times Id))^* E_l $ and the pullback of $E_l$ on $S\times  {\mathcal X}_{y_l}$ with respect to the  second projection.
 The Mittag-Leffler condition ensures that this extension
splits in the weak sense.
\qed









%We keep the notation and assumptions of Proposition ~\ref{vologod}.
%Let $\widetilde{\mathcal X}_{\hat y}$ stand for
%${\mathcal X}_{\hat y}\times{\mathbb A}^1$.
%Let $\widetilde E$ be a vector bundle on $\widetilde{\mathcal X}_{\hat y}$
%such that ${\widetilde E}|_{{\mathcal X}_{\hat y}}=E$. Assume in addition
%that $\widetilde E$ is equivariant with respect to a {\em new}
%$\TTT$-action $h:\ \TTT\times\widetilde{\mathcal X}_{\hat y}\to
%\widetilde{\mathcal X}_{\hat y}$, trivial along
%${\mathcal X}_{\hat y}$ and standard along ${\mathbb A}^1$.
%We will say that $\widetilde E$ is $h$-equivariant, as opposed to the
%$m$-equivariance of Proposition ~\ref{vologod}
%(the $\TTT$-action $m:\ \TTT\times\widetilde{\mathcal X}_{\hat y}\to
%\widetilde{\mathcal X}_{\hat y}$ is the old one along ${\mathcal X}_{\hat y}$
%and trivial along ${\mathbb A}^1$).
%Then automatically we have $\Ext^1({\widetilde E},{\widetilde E})=0$.

%\begin{prop}
%\label{volgdsky}
%The $m$-equivariant structure on $E$ (of Proposition ~\ref{vologod})
%can be extended to an $m$-equivariant structure on $\widetilde E$.
%\end{prop}

%The proof is similar to the one of ~\ref{vologod}.












\begin{thebibliography}{99}

%\bibitem{BEG} Yu.~Berest, P.~Etingof, V.~Ginzburg, {\it Cherednik algebras
%and differential operators on quasi-invariants},  Duke Math. J. {\bf 118}
%(2003),  279--337.

%\bibitem{BC} Y.~Berest and O.~Chalykh, {\it Quasi-invariants of complex
%reflection groups}, Comp. Math. {\bf 147} (2011), 965--1002.

\bibitem{BK} R.~Bezrukavnikov, D.~Kaledin, {\em McKay equivalence for
symplectic resolutions of quotient singularities}, Proc. Steklov Inst. Math.
{\bf 246}, no. 3 (2004), 13--33.

\bibitem{bfgi} R.~Bezrukavnikov, M.~Finkelberg, V.~Ginzburg, {\em Cherednik
algebras and Hilbert schemes in characteristic $p$}, Representation Theory
{\bf 10} (2006), 245--298.


\bibitem{CBH} W. Crawley-Boevey, M. Holland, {\em
Noncommutative deformations of Kleinian singularities,}
Duke Math. J. {\bf 92} (1998), no. 3, 605�635.

\bibitem{EGO} P.~Etingof, W.~L.~Gan, V.~Ginzburg, A.~Oblomkov,
{\em Harish-Chandra homomorphisms and symplectic reflection algebras for
wreath-products}, Publ. Math. Inst. Hautes \'Etudes Sci. {\bf 105} (2007),
91--155.

\bibitem{GG} W.~L.~Gan, V.~Ginzburg, {\em Almost-commuting variety,
$\mathcal D$-modules, and Cherednik algebras}, with an appendix by Ginzburg.
Int. Math. Res. Pap. IMRP {\bf 2006}, 26439, 1--54.

\bibitem{G0} I.~Gordon, {\em Baby Verma modules for rational Cherednik
algebras}, Bull. London Math. Soc. {\bf 35} (2003), 321--336.

\bibitem{G1} I.~Gordon, {\em A remark on rational Cherednik algebras and
differential operators on the cyclic quiver}, Glasg. Math. J. {\bf 48} (2006),
145--160.

\bibitem{G} I.~Gordon, {\em Quiver varieties, category $\CO$ for rational
Cherednik algebras, and Hecke algebras},
Int. Math. Res. Pap. IMRP {\bf 2008}, no. 3, Art. ID rpn006, 69 pp.

\bibitem{H0} M.~Haiman, {\em Hilbert schemes, polygraphs and the Macdonald positivity conjecture,} J. Amer. Math. Soc. {\bf 14} (2001), no. 4, 941--1006.

\bibitem{H} M.~Haiman, {\em Combinatorics, symmetric functions, and Hilbert
schemes}, in: Current developments in mathematics, 2002, Int. Press,
Somerville, MA (2003), 39--111.

%\bibitem{sraco} I. Losev, {\it Completions of symplectic reflection algebras},
%Selecta Math., {\bf 18} (2012), no. 1, 179--251.

\bibitem{L} I.~Losev, {\em Isomorphisms of Quantizations via Quantization of
Resolutions},
Advances in Mathematics {\bf 231} (2012), pp. 1216-1270.
%arXiv/1010.3182.

\bibitem{Mln} J.-S. Milne, {\em \' Etale cohomology,} Princeton Math Serier {\bf 33},
Princeton U. Press (1980).

\bibitem{nak} H.~Nakajima, {\em Jack polynomials and Hilbert schemes of points
on surfaces}, arXiv:alg-geom/9610021.

%\bibitem{Lu} G.~Lusztig, {\em Aperiodicity in quantum affine $gl_n$},
%Asian J. Math. {\bf 3} (1999), no. 1, 147--177.

\bibitem{T} R.~Travkin, {\em Quantum geometric Langlands correspondence in
positive characteristic: the $GL_N$ case}, arXiv/1110.5707.


\end{thebibliography}



\footnotesize{ {\bf R.B.}: Department of Mathematics, Massachusetts Institute
of Technology,\\ Cambridge MA 02139, USA and IITP;\\ %\hphantom{x}\ab\,
{\tt bezrukav@math.mit.edu}}



\footnotesize{ {\bf M.F.}: IMU, IITP, and
National Research University Higher School of Economics,\\ Department of
Mathematics, 20 Myasnitskaya st, Moscow 101000 Russia;\\ %\hphantom{x}\ab\,
{\tt fnklberg@gmail.com}}

%\footnotesize{ {\bf I.L.}: Department of Mathematics, Northeastern University,
%Boston MA 02115, USA;\\ %\hphantom{x}\ab\,
%{\tt i.loseu@neu.edu}}

\footnotesize{ {\bf V.V.}: Department of Mathematics, Oregon University,
Eugene OR 97403, USA;\\ %\hphantom{x}\ab\,
{\tt vvologod@uoregon.edu}}



\end{document}

TO DO: more precise reference to [11] (Milne) in proof of Lemma 2.1.

Erratum to [BK]

Support in Prop 2.6.

1208.3696.extracted_bib
\begin{thebibliography}{99}

%\bibitem{BEG} Yu.~Berest, P.~Etingof, V.~Ginzburg, {\it Cherednik algebras
%and differential operators on quasi-invariants},  Duke Math. J. {\bf 118}
%(2003),  279--337.

%\bibitem{BC} Y.~Berest and O.~Chalykh, {\it Quasi-invariants of complex
%reflection groups}, Comp. Math. {\bf 147} (2011), 965--1002.

\bibitem{BK} R.~Bezrukavnikov, D.~Kaledin, {\em McKay equivalence for
symplectic resolutions of quotient singularities}, Proc. Steklov Inst. Math.
{\bf 246}, no. 3 (2004), 13--33.

\bibitem{bfgi} R.~Bezrukavnikov, M.~Finkelberg, V.~Ginzburg, {\em Cherednik
algebras and Hilbert schemes in characteristic $p$}, Representation Theory
{\bf 10} (2006), 245--298.


\bibitem{CBH} W. Crawley-Boevey, M. Holland, {\em
Noncommutative deformations of Kleinian singularities,}
Duke Math. J. {\bf 92} (1998), no. 3, 605�635.

\bibitem{EGO} P.~Etingof, W.~L.~Gan, V.~Ginzburg, A.~Oblomkov,
{\em Harish-Chandra homomorphisms and symplectic reflection algebras for
wreath-products}, Publ. Math. Inst. Hautes \'Etudes Sci. {\bf 105} (2007),
91--155.

\bibitem{GG} W.~L.~Gan, V.~Ginzburg, {\em Almost-commuting variety,
$\mathcal D$-modules, and Cherednik algebras}, with an appendix by Ginzburg.
Int. Math. Res. Pap. IMRP {\bf 2006}, 26439, 1--54.

\bibitem{G0} I.~Gordon, {\em Baby Verma modules for rational Cherednik
algebras}, Bull. London Math. Soc. {\bf 35} (2003), 321--336.

\bibitem{G1} I.~Gordon, {\em A remark on rational Cherednik algebras and
differential operators on the cyclic quiver}, Glasg. Math. J. {\bf 48} (2006),
145--160.

\bibitem{G} I.~Gordon, {\em Quiver varieties, category $\CO$ for rational
Cherednik algebras, and Hecke algebras},
Int. Math. Res. Pap. IMRP {\bf 2008}, no. 3, Art. ID rpn006, 69 pp.

\bibitem{H0} M.~Haiman, {\em Hilbert schemes, polygraphs and the Macdonald positivity conjecture,} J. Amer. Math. Soc. {\bf 14} (2001), no. 4, 941--1006.

\bibitem{H} M.~Haiman, {\em Combinatorics, symmetric functions, and Hilbert
schemes}, in: Current developments in mathematics, 2002, Int. Press,
Somerville, MA (2003), 39--111.

%\bibitem{sraco} I. Losev, {\it Completions of symplectic reflection algebras},
%Selecta Math., {\bf 18} (2012), no. 1, 179--251.

\bibitem{L} I.~Losev, {\em Isomorphisms of Quantizations via Quantization of
Resolutions},
Advances in Mathematics {\bf 231} (2012), pp. 1216-1270.
%arXiv/1010.3182.

\bibitem{Mln} J.-S. Milne, {\em \' Etale cohomology,} Princeton Math Serier {\bf 33},
Princeton U. Press (1980).

\bibitem{nak} H.~Nakajima, {\em Jack polynomials and Hilbert schemes of points
on surfaces}, arXiv:alg-geom/9610021.

%\bibitem{Lu} G.~Lusztig, {\em Aperiodicity in quantum affine $gl_n$},
%Asian J. Math. {\bf 3} (1999), no. 1, 147--177.

\bibitem{T} R.~Travkin, {\em Quantum geometric Langlands correspondence in
positive characteristic: the $GL_N$ case}, arXiv/1110.5707.


\end{thebibliography}