Representations of symplectic reflection algebras and resolutions of deformations of symplectic quotient singularities
Iain Gordon and S. Paul Smith
We give an equivalence of triangulated categories between the derived category of finitely generated representations of symplectic reflection algebras associated with wreath products (with parameter $t=0$) and the derived category of coherent sheaves on a crepant resolution of the spectrum of the centre of these algebras.
math0310187.tex
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\begin{document}

\pagenumbering{arabic}

\title[symplectic reflection algebras]{Representations of symplectic reflection algebras and resolutions of deformations of symplectic quotient singularities}

\author{Iain Gordon and S. Paul Smith}
\address{Department of Mathematics, University of Glasgow, Glasgow G12 8QW}
\email{ig@maths.gla.ac.uk}
\address{ Department of Mathematics, Box 354350, University of Washington, Seattle, WA 98195}
\email{smith@math.washington.edu}

\thanks{2000 Mathematics Subject Classification: 14E15, 16Rxx, 16S38, 18E30.\\ The first author was partially supported by the Nuffield Foundation
grant NAL/00625/G and by the University of Washington's Milliman
Fund. The second author was partially supported by an NSF grant
DMS-0070560. Both authors are grateful to the Edinburgh
Mathematical Society and the Leverhulme Trust for support.}
\begin{abstract}
We give an equivalence of triangulated categories between the
derived category of finitely generated representations of
symplectic reflection algebras associated with wreath products
(with parameter $t=0$) and the derived category of coherent
sheaves on a crepant resolution of the spectrum of the centre of
these algebras.
\end{abstract}
\maketitle

\section{Introduction}
\label{intro}
\subsection{}
In this paper we take a step towards a geometric understanding of the representation theory of certain symplectic reflection algebras (with parameter $t=0$). A number of papers have shown that this representation theory is closely related to the singularities of the centre of these algebras and to resolutions of these singularities, \cite{EG}, \cite{BG}, \cite{GK}. Here we make such a relationship precise by proving that the category of finitely generated modules is derived equivalent to the category of coherent sheaves on an appropriate desingularisation.

 A long term goal in this project is to find character formulae for simple modules, generalising the work \cite{FG} and \cite{gor} in which Kostka polynomials appear. A simple consequence of the derived equivalence is a geometric interpretation for the number of simple modules of a symplectic reflection algebra with given central character. A closer analysis will undoubtedly reveal more.

\subsection{}
Let us summarise our results.
Let $\Gamma$ be a non-trivial finite subgroup of $\SL(2,\CC)$ and $n$
a positive integer.
 Let $\hc$ be the symplectic reflection algebra (with parameter $t=0$) for the wreath product $\Gamma_n = S_n\ltimes \Gamma^n$. (All undefined notation and definitions can be found later in the paper.) The spectrum of the centre of this algebra, $\xc = \spec \zc$, is a deformation of the symplectic quotient singularity $\CC^{2n}/\Gamma_n$. Our principal
result is the following.

\begin{theorem}
\label{main.thm}
There is a crepant resolution $\pi_{\mathbf{c}} : Y_\mathbf{c}
\longrightarrow \xc$ such that there is an equivalence of
triangulated categories
$$D^b(\md \hc) \longrightarrow D^b(\coh Y_{\mathbf{c}}),$$
between the bounded derived category of finitely generated $\hc$--modules and the bounded derived category of coherent sheaves on $Y_\mathbf{c}$. \end{theorem}

\subsection{}
In the special case $\mathbf{c}=\mathbf{0}$ the variety
$X_{\mathbf{c}}$ is the orbit space $\mathbb{C}^{2n}/\Gamma_n$, so
the above theorem includes the results of \cite{KV} on Kleinian
singularities ($n=1$) and the observation of \cite[Section
4.4]{wang1} (for general $n$). The proof we give here, however, is
by deformation from the $\mathbf{c}=\mathbf{0}$ case and so
depends on these results. To prove the equivalence, we use the
methods of \cite{BKR}, which were adapted to a mildly
non--commutative situation in \cite{vdb2}, together with results
from \cite{hai} and \cite{hai2}.

\subsection{}
For $x \in \xc$, let $\mathfrak{m}_x$ be the corresponding maximal ideal of $\zc$. The simple modules of the finite dimensional algebra  $\hc/\mathfrak{m}_x\hc$ are the simple $\hc$--modules  with central character $x$. The following corollary is a straightforward consequence of the above theorem.

\begin{corollary}
Let $\pi_{\mathbf{c}} :Y_{\mathbf{c}}\longrightarrow \xc$ be the crepant resolution in Theorem \ref{main.thm}. For all $x\in \xc$, there is an isomorphism of Grothendieck groups $K(\hc/\mathfrak{m}_x\hc) \cong K(\pi_{\mathbf{c}}^{-1}(x))$.
\end{corollary}

\subsection{}
In the case $n=1$ this recovers known results on the simple modules of deformed preprojective algebras, whilst for $\mathbf{c}=0$ the content is essentially the (generalised) McKay correspondence, \cite{kal}.

\subsection{} Symplectic reflection algebras also exist for finite Coxeter groups, $W$.
However, \cite[Theorem 1.1]{GK} shows that the only the orbits
spaces $\mathbb{C}^{2n}/W$ admitting crepant resolutions are for
$W$ of type $A$ and $B$, that is $W = S_n$ or $W = S_n\ltimes
(\mathbb{Z}/2\mathbb{Z})^n$. Thus there is no analogue of Theorem
\ref{main.thm} valid for all finite Coxeter groups; our result is
as general as we can expect.

\subsection{}
The paper is organised as follows. In Section \ref{sra} we recall
the definition and basic properties of symplectic reflection
algebras. A discussion of non--commutative crepant resolutions and
derived equivalences is given in Section \ref{nc}. Section
\ref{ss} presents some results on deformations of semi--small
morphisms and their relation to crepant resolutions and derived
equivalences. Finally, in Section \ref{main}, we prove our main
result and discuss the application to counting simple modules.

\section{Symplectic reflection algebras}
\label{sra}
\subsection{}
Let $\tilde{\omega}$ be the standard symplectic form on
$\mathbb{C}^2$, $\Gamma$ a finite subgroup of $SL(2,\mathbb{C})$
and $n$ a positive integer. The wreath product $\Gamma_n
\equiv S_n\ltimes \Gamma^n$ acts on $V\equiv (\mathbb{C}^2)^n$,
preserving the symplectic form $\omega \equiv \tilde{\omega}^n$.

\subsection{} Recall that $\gamma \in \Gamma_n$
acting on $V$ is called a \textit{symplectic
reflection} if $\dim (1-\gamma)(V) = 2$. The set of all symplectic
reflections is denoted by $\mathcal{S}$. Let $\mathbf{c}$ be a
$\CC$--valued function on $\mathcal{S}$, constant on conjugacy
classes ($\gamma \mapsto c_{\gamma}$). Given $\gamma\in
\mathcal{S}$ define the form $\omega_{\gamma}$ on $V$ to have radical
$\ker (1-\gamma)$ and to be the restriction of $\omega$ on the
$(1-\gamma)(V)$.
\subsection{} The \textit{symplectic reflection algebra} $\hc$ is
the $\CC$--algebra, defined as the quotient of the skew group ring $TV\ast \Gamma_n$
by the relations
$$ x\otimes y - y\otimes x = \sum_{\gamma\in \mathcal{S}}
c_{\gamma} \omega_{\gamma}(x,y)\gamma,$$ for all $x,y\in V$.

\begin{remark} Usually, symplectic reflection algebras depend on a
further parameter $t\in \CC$, \cite[Section 1]{EG}. The definition above is
the case $t=0$.
\end{remark}

\subsection{} \label{ff} There is an increasing $\mathbb{N}$--filtration on
$\hc$, obtained by setting $F^0\hc = \CC\Gamma_n$, $F^1 = V\otimes
\CC\Gamma_n + \CC\Gamma_n$, and $F^i = (F^1)^i$. By the PBW
theorem, \cite[Theorem 1.3]{EG}, $\gr \hc \cong \CC[V]\ast \Gamma_n$ (where we have used
$\omega$ to identify the $\Gamma_n$--spaces $V$ and $V^*$). In
particular, a non--zero $\mathbf{c}$ yields a flat family of
symplectic reflection algebras $H_{u\mathbf{c}}$ over $\CC[u]$.

Each $\hc$ is a prime noetherian ring because its associated graded ring has these
properties \cite[Theorem 1.3]{EG}, \cite[Theorem 3.17]{Mont},
and \cite[Prop. 1.6.6, Theorem 1.6.9]{McCRob}.

\subsection{}
Let $\zc$ denote the centre of $\hc$, and set $\xc = \spec \zc$. It is known that $\zc$ is a finitely generated
$\CC$-algebra, and that $\hc$ is a finite $\zc$--module, \cite[Theorem 1.5 and Theorem 3.1]{EG}. By a lemma of
Dixmier, it follows that all simple $\hc$--modules are finite
dimensional. In fact $|\Gamma_n|$ is the strict upper bound for
the dimension of simple $\hc$--modules, \cite[Proposition 3.8]{EG}. We
therefore have a map
$$\chi : \textsf{Simp} (\hc) \longrightarrow \xc$$ which sends a
simple module $S$ to its central character $\chi(S)\in \xc$.

\subsection{}
The algebra $\zc$ has a Poisson bracket, making $\xc$ a Poisson
variety.
The following subsets of $X_{\mathbf{c}}$ are the same:
\begin{enumerate}
\item{}
the locus where the form is non-degenerate;
\item{}
its non-singular locus, $\textsf{Sm}(\xc)$ \cite[Theorem 7.8]{BG};
\item{}
the Azumaya locus of $\hc$, that is $\{\chi(S) : S$ is a simple
module of maximal dimension$\}$ \cite[Theorem 1.7]{EG};
\item{}
$\{x \in \xc : \chi^{-1}(x) \text{ is a singleton}\}$ \cite[Theorem 3.7] {EG}.
\end{enumerate}
Moreover, by \cite[Theorem 3.7]{EG}, if $x\in \textsf{Sm}(\xc)$,
then the unique simple $\hc$-module having central character $x$
is isomorphic to $\CC\Gamma_n$ as a $\CC\Gamma_n$-module.

\subsection{}
Let $e\in \CC\Gamma_n$ be the symmetrising
idempotent $|\Gamma_n|^{-1}\sum_{\gamma\in\Gamma_n} \gamma$.
The map $\zc \longrightarrow e\hc e$, $z\mapsto ze$, is an
isomorphism, \cite[Theorem 3.1]{EG}.
Thus, following \ref{ff}, there is a flat family of
commutative algebras $Z_{u\mathbf{c}}$ over $\CC[u]$. We set
$X_{u\mathbf{c}}= \spec Z_{u\mathbf{c}}$.

\subsection{}
\label{az}
For later use we need the following lemma.
\begin{lemma} For non--zero $\mathbf{c}$, the following sets are the same
\begin{enumerate}
\item the smooth locus, $\textsf{Sm}(X_{u\mathbf{c}})$;
\item the Azumaya locus of $H_{u\mathbf{c}}$;
\item the central characters of simple $\hc$-modules that are isomorphic to the
regular representation of $\Gamma_n$ as $\Gamma_n$-modules.
\end{enumerate}
\end{lemma}
\begin{proof}
As the generic simple module for $H_{u\mathbf{c}}$ has dimension
$|\Gamma_n|$ it follows that the Azumaya locus of
$H_{u\mathbf{c}}$ is the union of the Azumaya loci for
$H_{\lambda\mathbf{c}}$ for $\lambda \in \CC$. This proves the
equivalence of (1) and (3).

Since the associated graded ring of $H_{u\mathbf{c}}$ is
$\CC[V\oplus \CC]\ast \Gamma_n$, it follows from \cite[Proposition
1.6.6, Theorem 1.6.9 and Corollary 7.6.18]{McCRob} and
\cite[Theorem 3.8]{BrGoo} that the equivalence of (1) and (2)
follows if the non--Azumaya locus has codimension at least 2.
This, however, is clear as for each $\lambda\in \CC$ the Azumaya locus
of $H_{\lambda\mathbf{c}}$ has codimension at least 2 in
$X_{\lambda\mathbf{c}}$.
\end{proof}

\subsection{}
\label{norm} Observe as a consequence of the above proof that
$Z_{u\mathbf{c}}$ is normal. Indeed, $Z_{u\mathbf{c}}$ is Cohen--Macaulay (in
fact Gorenstein) since $eH_{u\mathbf{c}}e$ has an associated graded ring
isomorphic to $\CC [V \oplus \CC]^{\Gamma_n}$, \cite[Theorem 3.3]{EG}.
Combining this with the smoothness of $X_{u\mathbf{c}}$ in codimension one,
which is proved above, shows that $Z_{u\mathbf{c}}$ is normal, \cite[Theorem
2.2.11]{CG}.

\section{Non--commutative crepant resolutions}
\label{nc}
\subsection{}
Let $X$ be a Gorenstein $\CC$--variety.
A resolution of singularities $f: Y\longrightarrow
X$ is said to be \textit{crepant} if $f^*\omega_X = \omega_Y$.
Crepant resolutions are a generalisation of the notion of a
minimal resolution in two dimensions. However, crepant
resolutions need not exist, and need not be unique when they do
exist.

\subsection{}
\label{sympcrep} In the setup of Section \ref{sra} we could take
$X=\xc$. This is a Gorenstein variety, \cite[Theorem 1.5(i)]{EG}.
Moreover, since the Poisson form on $\xc$ is symplectic on
$\textsf{Sm}(\xc)$, the canonical bundle $\omega_{\xc}$ is
trivial. Thus any crepant resolution of $\xc$ has trivial
canonical class.
\begin{remark}
In fact, $\yc$ is a crepant resolution of $\xc$ if and only if
$\yc$ is a symplectic variety whose form agrees with that of $\xc$
on $\textsf{Sm}(\xc)$, \cite[Proposition 3.2]{kal0}
\end{remark}

\subsection{}
For the following definitions see \cite[Sections 3 and 4]{vdb2}. Throughout $R$ denotes a commutative noetherian domain over $\CC$. A module--finite $R$--algebra $A$ is
\textit{homologically homogeneous} if, for all $\mathfrak{p} \in \spec R$,
$\gldim A_{\mathfrak{p}} = \Kdim R_{\mathfrak{p}}$ and
$A_{\mathfrak{p}}$ is maximal Cohen--Macaulay.
A \textit{non--commutative crepant resolution} of $R$ is
a homologically homogeneous $R$--algebra of the form $A =
\edo_R(M)$, where $M$ is a reflexive $R$--module.

A justification for this definition of non--commutative crepant resolution is given in \cite[Section 4]{vdb2}.

\subsection{}
\label{stab}
We recall some material from \cite[Sections 3 and 4]{K} and
\cite[Section 6]{vdb2}. Set $X= \spec R$.
Let $A$
be an $R$--algebra that is finitely generated as an $R$--module, and let
$(e_i)_{i=1, \ldots ,p}$ be pairwise orthogonal idempotents in
$A$ such that $1= \sum_i e_i$.

For a map $R\longrightarrow K$ with $K$ a field and $V$ a finite
dimensional $A\otimes_R K$--module, we write $\underline{\dim} V =
(\dim_K e_iV)_i \in \mathbb{Z}^p$.

Pick $\lambda \in \homo_{\mathbb{Z}}(\mathbb{Z}^p,\mathbb{R})$ and let $\alpha = \underline{\dim}
V$.
We say that a finite dimensional $A \otimes_R K$-module
$V$ is \textit{stable} (respectively, \textit{semi-stable})
with respect to $\lambda$ if $\lambda (\alpha) = 0$ and
for every proper $A \otimes_RK$-submodule
$W$ of $V$ we have $\lambda (\underline{\dim} W) < 0$
(resp., $\lambda(\underline{\dim} W) \le 0$).

Our definition of stability is different from that in \cite{vdb2}:
where we have $\lambda(\underline{\dim} W)<0$ Van den Bergh has
$\lambda(\underline{\dim} W) >0$. Of course, the difference is
only cosmetic because we can pass back and forth between the two
notions by replacing $\lambda$ by $-\lambda$. The reason for this
difference is that later on in Section \ref{cluster} we want our
notion of stability to coincide with the notion that Haiman uses
in \cite{hai}.


We say that $\lambda$ is \textit{generic} (for $\alpha)$ if all semi-stable
representations of dimension vector $\alpha$ are stable.
There is a generic $\lambda$ if and only if $\alpha$ is indivisible, meaning that the
greatest common divisor of the $\alpha_i$s is 1.
The condition $\lambda  (\beta) \neq 0$ for all $0 < \beta
< \alpha$ ensures $\lambda$ is generic.

\subsection{}

Let $T$ be an $R$-scheme.
A \textit{family} of $A$-modules
of dimension $\alpha$
parametrised by $T$
is a locally free sheaf ${\mathcal F}$ of
${\mathcal O}_T$-modules
together with an $R$-algebra homomorphism $\phi:A \to \End_T
{\mathcal F}$
such that $e_i{\mathcal F}$ has constant rank $\alpha_i$ for all $i$.
We say that ${\mathcal F}$ is
{\it semi-stable} (resp., {\it stable}) if for every every field $K$
and every morphism $\xi:\Spec K \to T$,  $\xi^*{\mathcal F}$
is a semi-stable (resp., stable) $A \otimes_R K$-module.
Two families $({\mathcal F},\phi)$ and $({\mathcal F}',\phi')$
are \textit{equivalent} if there is an invertible
${\mathcal O}_T$-module ${\mathcal L}$ and an isomorphism $\psi:
{\mathcal F} \to {\mathcal F}' \otimes {\mathcal L}$ such that
the diagram
$$
\begin{CD}
A @>{\phi}>> \End {\mathcal F}
\\
@V{\phi'}VV @VV{\psiol}V
\\
\End {\mathcal F}' @>{\sim}>{\id \otimes {\mathcal L}}>
\End {\mathcal F}' \otimes {\mathcal L}
\end{CD}
$$
commutes; in this diagram $\psiol(\nu)=\psi\nu\psi^{-1}$.

A family $({\mathcal U},\rho)$ parametrised by $W$ is
{\it universal}
if for every $R$-scheme $T$ and every family $({\mathcal F},\phi)$
over $T$ there is a unique morphism $\xi:T \to W$ such that
$(\xi^*{\mathcal U},\xi^* \rho)$ is equivalent to
$({\mathcal F},\phi)$; here $\xi^*  \rho$ denotes
the composition $A \to \End {\mathcal U} \to \End \xi^*
{\mathcal U}$.


Proposition \ref{prop.vdb} below says that in suitable situations
there is a universal family. We also express this by saying that $W$ is
a fine moduli space for families of $A$-modules of dimension $\alpha$
and call ${\mathcal U}$ the {\it universal family}.




\subsection{}
\label{repres}

%Fix $\alpha\in \mathbb{Z}^p$ and suppose that
%$\lambda$ is generic for $\alpha$.

Define a functor
$\mathbf{R}^s: \hbox{$R$-schemes} \longrightarrow \Sets$ by
$$
T \mapsto \{\hbox{equiv. classes of families of
$\lambda$--stable $A$--modules over $T$ with dimension $\alpha$}\}.
$$

\begin{prop}
\label{prop.vdb}
\cite[Proposition 6.2.1]{vdb2}
Suppose that $\lambda$ is generic for $\alpha$. Then {\bf R}${}^s$
is represented by a closed subscheme $W^s \subset \PP_X^N$.
\end{prop}

We write $f:W^s \to X$ for the structure morphism, and
${\mathcal B}$ for the universal family
of $\lambda$-stable $A$-modules of dimension $\alpha$.

If $U$ is either an open or closed subscheme of $X$ then the
representing scheme for $U$ is $f^{-1}(U)$ and its universal family is
${\mathcal B} \big\vert{}_U$ (cf. the sentence after \cite[Lemma 6.2.2]{vdb2}).


\subsection{} \label{trivspace}
It is shown in \cite[Lemma 6.2.3]{vdb2} that in the
case $A=\mat_{\sum\alpha_i}(R)$ the map $W^s\longrightarrow X$ is
an isomorphism.

\subsection{}
\label{der}


Assume $A=\End_R M$ is non--commutative crepant resolution of $R$.
Suppose we have an $R$-module decomposition $M = \oplus_{i=1}^p M_i$ and
let $e_1,\ldots,e_p$ be the projections onto the $M_i$s viewed as
idempotents in $A$.
Define
$$
\alpha_i:=\rank M_i=\rank e_iM.
$$
Suppose that $\lambda$ is generic for $\alpha$. Let
$f: W^s\longrightarrow X$ and $\mathcal{B}$ be as in \ref{repres}.

Let $U \subset X$ be the locus where $M$ is locally free. It
follows from \ref{trivspace} that $f^{-1}(U)\longrightarrow U$
is an isomorphism. Let $Y$ be the closure of $f^{-1}(U)$; this is
the unique irreducible component of $W^s$ mapping birationally onto $X$.
We continue to denote the restriction of $f$ to
$Y$ by $f$. Let $\mathcal{G}$ be the restriction of $\mathcal{B}$
to $Y$.

There is a pair of adjoint functors between $D^b(\coh Y)$ and
$D^b(\md A)$:
\begin{eqnarray*}
\Phi: D^b(\coh Y)\longrightarrow D^b(\md A) : C \mapsto
\mathbf{R}\Gamma(C\otimes^{\mathbf{L}}_{\mathcal{O}_Y} \mathcal{G}) \\
\Psi : D^b(\md A) \longrightarrow D^b(\coh Y) : D \mapsto
D\otimes^{\mathbf{L}}_A\mathcal{G}^*.
\end{eqnarray*}
The following theorem follows \cite{BKR} closely.
\begin{theorem}\cite[6.3.1]{vdb2}
Assume that for every point $x\in X$,
$$
\dim (Y\times_X Y)\times_X \spec \mathcal{O}_{X,x}  \le \codim x +1.
$$
Then $f: Y \longrightarrow X$ is
a crepant resolution of $X$ and $\Phi$ and $\Psi$ are inverse
equivalences.
\end{theorem}

\subsection{}
\label{fibreres}
Let $x\in X$.
We write $D^b_x(\coh Y)$ and $D^b_x(\md A)$ for the full
subcategories of $D^b(\coh Y)$ and $D^b(\md A)$ consisting of
complexes supported on $f^{-1}(x)$ and on $x$ respectively.
Since $\Phi$ and $\Psi$ are functors over $X$ they restrict to
equivalences between $D^b_x(\coh Y)$ and $D^b_x(\md A)$,
\cite[9.1]{BKR} and \cite[6.6]{vdb2}.

\subsection{}

Fix a non--zero $\mathbf{c}$.
The following lemma allows us to apply the machinery in this section.

\begin{lemma}
\label{lem.nc.crep}
The algebra $\hc$ is a non--commutative crepant resolution of $\xc$.
\end{lemma}
\begin{proof}
By \cite[Theorem 15.(ii), (iii)]{EG} $\hc e$ is a finitely generated, reflexive,
Cohen--Macaulay $\hc$--module. Furthermore, by \cite[Theorem 1.5(iv
)]{EG} we have $\hc \cong \edo_{e\hc e}(\hc e)$.
By \cite[Corollary 6.18]{McCRob}, $\gldim \hc < \infty$.
Thus $\hc$ is a non--commutative crepant resolution of
$e\hc e \cong \zc$ by \cite[Lemma 4.2]{vdb2}.
\end{proof}

Let $H_{u\mathbf{c}}$ be the flat family of symplectic reflection algebras
defined in \ref{ff}. The arguments used to prove Lemma \ref{lem.nc.crep}
all extend to $H_{u\mathbf{c}}$, showing that $H_{u\mathbf{c}}$
is a non-commutative crepant resolution of $X_{u\mathbf{c}}$.

\subsection{}
\label{sect.apply-to-Hc}

If we set $R=eH_{u\mathbf{c}}e$, $M=H_{u\mathbf{c}}e$, and
$A=H_{u\mathbf{c}}$, we are in the situation of subsection
\ref{der}. Let $(e_i)_{i=1, \ldots ,p}\in \CC\Gamma_n$ be the
central orthogonal idempotents corresponding to the irreducible
representations, labelled so that $e_1$ corresponds to the trivial
representation. Thus $e_1=e$. By the PBW theorem for $\hc$ and
$H_{u\mathbf{c}}$ we can consider the $e_i$s as elements of $\hc$
and $H_{u\mathbf{c}}$. If we set $M_i=e_iM$ there is a
decomposition $H_{u\mathbf{c}}e=M=\oplus_{i=1}^p M_i$ as in
Section 3.8. Let $\alpha_i=\rank e_i H_{uc}e$ and write
$\alpha=(\alpha_i)_{1 \le i \le p} \in \ZZ^p$.

Let $E_i$ be the simple $\CC\Gamma_n$-module corresponding to $e_i$.
As remarked in the proof of \cite[Lemma 2.24]{EG},
$e_iH_{u\mathbf{c}}e \cong
\Hom_{\Gamma_n}(E_i,H_{u\mathbf{c}}e)$
so
$$
\alpha_i=\rank_{Z_{u\mathbf{c}}} e_iH_{u\mathbf{c}}e = \dim E_i.
$$
In particular, $\alpha_1=1$ so $\alpha$ is indivisible, and there are many
maps $\lambda:\ZZ^p \to \RR$ such that $\lambda(\alpha)=0$ and
$\lambda(\beta)>0$ for all $ 0<\beta<\alpha$.
For each such $\lambda$ there is a moduli space for $\lambda$--stable
$H_{u\mathbf{c}}$--modules having dimension $\alpha$ (equivalently, that are
isomorphic to $\CC\Gamma_n$ as $\CC\Gamma_n$-modules), and
each such moduli space has a unique irreducible component that maps
birationally to $X_{u\mathbf{c}}$.




\section{Semi--small maps}
\label{ss}
\subsection{}
\label{defss}
A proper birational map $f:Y \longrightarrow X$ between irreducible
varieties is \textit{semi-small} if
$$
2 \codim_Y Z \geq \codim_X f(Z)
$$
for all irreducible subvarieties $Z \subset Y$.
Note that if $f$ is semi--small the above inequality holds for all (not necessarily irreducible) subvarieties of $Y$. It is a theorem of Verbitsky, \cite[Theorem 2.8]{ver}, and Kaledin, \cite[Proposition 4.4]{kal0}, that any crepant resolution of $V/\Gamma_n$ is semi--small.
\subsection{}
\label{sathyp}

The following result relates semi--small maps to the hypothesis
of Theorem \ref{der}.
\begin{lemma}
Suppose that $f: Y\longrightarrow X$ is a semi--small morphism
between irreducible varieties of finite type over a field $k$. Then
$$
\dim (Y\times_X Y)\times_X \spec \mathcal{O}_{X,x} \leq \codim x +1
$$
for every point $x\in X$.
\end{lemma}
\begin{proof}
It is well-known (see, e.g., \cite[Proposition 2.1.1 and Remark 2.1.2]{dCM})
that semi-smallness of $f$ is equivalent to the condition that every
irreducible component of $Y \times_X Y$ has dimension at most $\dim
X$, so it suffices to prove that if $Z$ is an irreducible variety
of dimension $\le \dim X$ and $g:Z \to X$ a morphism,
then $\dim Z \times_X \Spec {\mathcal O}_{X,x} \le \codim x$ for every
point $x \in X$.

This reduces to the affine case.
We need to prove the following: if $R \to S$ is a homomorphism
between two domains that are finitely generated $k$-algebras such that
$\Kdim S \le \Kdim R$, then
$\Kdim S \otimes_R R_\fp \le \Kdim R_\fp$ for all $\fp \in \Spec R$.

Let $\Rol$ denote the image of $R$ in $S$, and let $\fq \in \Spec R$
denote the kernel of $R \to S$.
The hypotheses on $R$ and $S$ are such that they, their
localizations, and the homomorphic images of these are catenary.

Write $d$ for $\Kdim S-\Kdim \Rol$.
Since the Krull dimensions of $\Rol$ and $S$ are equal to the
transcendence degrees over $k$ of their fraction fields,
we can write  $S=\Rol[x_1,\ldots,x_d,\ldots,x_n]$ where
$\{x_1,\ldots,x_d\}$ is algebraically independent over $\Rol$,
and $x_{d+1},\ldots,x_n$ are algebraic over $\Rol[x_1,\ldots,x_d]$.

Write $\Rol_{\fp}$ for $\Rol \otimes_R R_{\fp}$. Then
$S \otimes_R R_\fp = \Rol_{\fp}[x_1,\ldots,x_n]$. The
Krull dimension of an extension $C[x]$ is equal to
either $\Kdim C$ or $\Kdim C +1$
depending on whether $x$ is algebraic over $C$ or not,
so an induction argument
shows that $\Kdim S \otimes_R R_\fp \le \Kdim \Rol_{\fp} +d$.
Thus
\begin{align*}
\Kdim S \otimes_R R_\fp
& \le \Kdim \Rol_{\fp} + \Kdim S - \Kdim \Rol
\\
& \le \Kdim \Rol_{\fp} + \Kdim S- \Kdim R + \hgt \fq,
\end{align*}
where $\hgt \fq$ denotes the height of $\fq$.
But $\hgt \fq \le \hgt \fq R_{\fp}$ so
$$
\Kdim S \otimes_R R_\fp
%& \le \Kdim \Rol_{\fp} + \Kdim S- \Kdim R + \hgt \fq
%\\
 \le \Kdim \Rol_{\fp} + \Kdim S- \Kdim R + \hgt \fq R_{\fp}
= \Kdim R_{\fp} + \Kdim S- \Kdim R,
$$
and the result follows because $\Kdim S \le \Kdim R$.
\end{proof}

\subsection{}
\label{defdim}
For the rest of the section we will be interested in schemes of finite type over $\CC$ with a $\CC^*$-action. All morphisms will be
$\CC^*$-equivariant. We always consider $\CC$ as to have the multiplicative action of $\CC^*$.
In case $f: X\longrightarrow \CC$ is a $\CC^*$--equivariant morphism, we will denote the fibres $f^{-1}(s)$ by $X_s$. Note that $X_s \cong X_1$ for all non--zero $s\in \CC$.

We say that an affine variety $X$ has an \textit{expanding} $\CC^*$--action if the corresponding $\mathbb{Z}$--grading $\CC[X] = \oplus_{i\in\mathbb{Z}} \CC[X]_i$ is concentrated in non--negative degrees.

\subsection{} The following lemma can be compared with \cite[II.4.2, Satz 2]{kraft}.
\begin{lemma}
\label{lem.kraft}
Let $X$ be an irreducible variety and  $f:X \longrightarrow \CC$ be a $\CC^*$-equivariant morphism.
Let $Z \subset X_s$ be an irreducible subvariety. Then either $\overline{\CC^* Z}\cap X_0$ is empty or
$$
\dim
(\overline{\CC^* Z} \cap X_0)
=\dim Z.
$$

Furthermore, if $X$ is an affine variety with expanding $\CC^*$--action, then $\overline{\CC^* Z}\cap X_0$ is non--empty.

\end{lemma}
\begin{proof}
Suppose $\overline{\CC^* Z}\cap X_0$ is non--empty. Since $\overline{\CC^*Z}$ is irreducible and  $\dim (\overline{\CC^*Z}) = \dim Z +1$, we have $\dim(\overline{\CC^*Z}\cap X_0)\leq \dim Z$. On the other hand,
the dimension of the fibers of the restriction $f|_{\overline{\CC^*Z}}:
\overline{\CC^* Z} \to \CC$ is minimal on a dense open
set of $\mathbb{C}$. Since the $\CC^*$--action identifies the fibres of this map over non--zero elements of $\CC$, we see that the minimal fibre dimension is bounded below by $\dim Z$, as required.

Now assume that $X$ is an affine variety with expanding $\CC^*$--action. Let $I$ be the ideal of $\mathbb{C}[X]$ annihilating $Z$. Then the ideal corresponding to $\overline{\CC^*Z}\cap X_0$ is $\gr I$, the ideal consisting of leading terms of elements of $I$, \cite[II.4.2, Satz 3]{kraft}. In particular, as $I$ is proper, so too is $\gr I$. Thus $\overline{\CC^*Z}\cap X_0$ is non--empty.
\end{proof}

\subsection{}
We need a simple lemma.
\label{push}
\begin{lemma}
\label{lem.good.pi}
Suppose there is a commutative diagram of $\CC^*$-equivariant morphisms
$$
\setlength{\unitlength}{1mm}
\begin{picture}(25,15)
\put(0,13){$Y$}
\put(4,15){\vector(1,0){20}}
\put(25,13){$X$}
\put(14,16){$\scriptstyle\pi$}
\put(3,11){\vector(1,-1){7}}
\put(24,11){\vector(-1,-1){7}}
\put(12,0){$\CC$}
\end{picture}
$$
where $\pi$ is proper.
If $Z \subset Y_s$ is an irreducible subvariety, then
\begin{equation}
\label{eqn.good.pi}
\pi(\overline{\CC^* Z} \cap Y_0) =
\overline{\CC^* \pi(Z)} \cap X_0.
\end{equation}
In particular, if $\overline{\CC^* Z} \cap Y_0$ is non--empty, then $\dim (\overline{\CC^* Z} \cap Y_0) = \dim Z$ and $\dim \pi(\overline{\CC^* Z} \cap Y_0) = \dim \pi(Z)$.
\end{lemma}
\begin{proof}
First we show that the right-hand side of (\ref{eqn.good.pi}) is contained
in the left-hand side.
Let $x \in \overline{\CC^* \pi(Z)} \cap X_0$. Certainly, $\CC^*\pi(Z)=\pi (\CC^* Z)
\subset \pi(\overline{\CC^* Z})$; but the last term is closed because $\pi$
is proper, so $\overline{\pi(\CC^*Z)} \subset \pi(\overline{\CC^* Z})$.
Hence $x =\pi(y)$ for some $y \in \overline{\CC^* Z}$; but $x \in X_0$, so $y \in
Y_0$, whence $x \in \pi(\overline{\CC^* Z} \cap Y_0)$.

The proof that the left-hand side of (\ref{eqn.good.pi}) is contained
in the right-hand side does not depend on the hypothesis that $\pi$
is proper:
For any subsets $W$ and $W'$ of $Y$, $\pi(W \cap W') \subset \pi(W) \cap \pi(W')$,
and $\pi(\overline{W}) \subset \overline{\pi(W)}$. Thus $\pi(\overline{W} \cap W') \subset \pi(\overline{W})
\cap \pi(W') \subset\overline{\pi(W)} \cap \pi (W')$. Now apply this with
$W=\CC^* Z$ and $W'=Y_0$.

Under the non--emptiness hypothesis, the equality of dimensions follows from Lemma \ref{defdim}.
\end{proof}

\subsection{}
The following result allows us to deform semi--small morphisms.
\begin{lemma}
\label{lem.good.pi.2}
Let $X$ be an affine variety with expanding $\CC^*$--action and suppose that we have a commutative diagram of $\CC^*$-equivariant morphisms
$$
\setlength{\unitlength}{1mm}
\begin{picture}(25,15)
\put(0,13){$Y$}
\put(4,15){\vector(1,0){20}}
\put(25,13){$X$}
\put(14,16){$\scriptstyle\pi$}
\put(3,11){\vector(1,-1){7}}
\put(24,11){\vector(-1,-1){7}}
\put(12,0){$\CC$}
\end{picture}
$$
where $\pi$ is proper. Assume that $\dim Y_0 = \dim Y_s$ and $\dim X_0 = \dim X_s$ for all $s\in\CC$. If $\pi_0: Y_0 \longrightarrow X_0$ is semismall, so too is $\pi_s:Y_s \longrightarrow X_s$ for
all $s$.
\end{lemma}
\begin{proof}
Let $Z \subset Y_s$ be an irreducible subvariety. Set $Z_0:=\overline{\CC^* Z} \cap Y_0$. As $X$ has an expanding action, $\pi(Z_0) = \overline{\CC^*\pi(Z)}\cap X_0$ is non--empty by Lemmas \ref{defdim} and \ref{push}. Thus the semi--small hypothesis shows
\begin{equation}
\label{eqn.1/2-small}
2 \dim Y_0 - 2 \dim Z_0 \ge \dim X_0 - \dim \pi(Z_0).
\end{equation}
By Lemma \ref{push}, we may replace $\dim Z_0$ and $\dim \pi(Z_0)$ in this
inequality by $\dim Z$ and $\dim \pi(Z)$. Now the lemma follows by replacing $\dim Y_0$ and $\dim X_0$ in this inequality by $\dim Y_s$ and $\dim X_s$.
\end{proof}

\section{Application}
\label{main}
\subsection{}
\label{cluster}
A representation of $H_{\mathbf{0}} = \CC[V]\ast \Gamma_n$ is
called a $\Gamma_n$--\textit{constellation} if its restriction to
$\CC \Gamma_n$ is isomorphic to the regular representation.
A constellation $M$ is a \textit{cluster} if it is generated as a
$\CC[V]$ module by $M^{\Gamma_n}$, the copy of the
trivial representation it contains.

We write $K(\Gamma_n)$ for the Grothendieck group of $\CC\Gamma_n$ and
$\alpha$ for the class of regular representation.
Let $\lambda : K(\Gamma_n) \longrightarrow \mathbb{R}$ be a linear
function such that $\lambda(\alpha)=0$.
Following \ref{stab}, a constellation $M$ is $\lambda$
(semi--)stable if for every proper $H_{\mathbf{0}}$--submodule
$N\subset M$ we have $\lambda (N) (\leq ) <0$.
For generic $\lambda$ there is a moduli space of
$\lambda$--stable $\Gamma_n$--constellations, a projective scheme
over $\spec Z_{\mathbf{0}} = V/\Gamma_n$.

If $\lambda$ is chosen so that $\lambda(\alpha)=0$ and, for each simple
$\Gamma_n$-module $S$,
$$
\lambda(S)=\begin{cases}
    1 & \text{if $S$ is trivial}
    \\
    < 0 & \text{if $S$ is not trivial}
\end{cases}
$$
then a constellation is $\lambda$-stable if and only if it is
a cluster.

\subsection{}
\label{haith} The following construction was given in
\cite[Corollaries 3 and 4]{wang2}. Let $X_{\Gamma}$ be the minimal
resolution of the Kleinian singularity $\CC^2/\Gamma$. We have
maps
\begin{eqnarray*} \hil^n(X_{\Gamma}) \longrightarrow \sym^n(X_{\Gamma}) \longrightarrow \sym^n(\CC^2/\Gamma) \cong V/\Gamma_n, \end{eqnarray*}
where the first map is the Hilbert--Chow map, \cite[Chapter
1]{nakbook}, and the second arises from functoriality. Since
$X_{\Gamma}$ is symplectic, so too is $\hil^n(X_{\Gamma})$ and
thus the composition is a crepant resolution of $V/\Gamma_n$, see
\ref{sympcrep}.

\subsection{}
Let $\hil^{S_n}(X_{\Gamma}^n)$ denote the $S_n$--Hilbert scheme of
Ito and Nakamura, \cite[Introduction and Sect. 8.2]{in}. The
following result is due to Haiman: we include our own outline of
the proof for the reader's benefit.

\begin{theorem}\cite[Section 7.2.3]{hai2}, \cite{hai3} There is an isomorphism between
$\hil^n(X_{\Gamma})$ and $\hil^{S_n}(X_{\Gamma}^n)$. In particular,
there exists $\lambda$ such that $\hil^n(X_{\Gamma})$ is a moduli space of $\lambda$--stable $\Gamma_n$ constellations.
\end{theorem}
\begin{proof} The isomorphism follows from the $n!$--conjecture applied to
the smooth surface $X_{\Gamma}$, \cite[Sect. 5.2]{hai} and \cite{hai2}.
There is a  commutative diagram of $S_n$-equivariant morphisms
\[
\begin{CD} \Sigma @> p >> X_{\Gamma}^n \\
@V q VV @VVV \\
\hil^{S_n}(X^n_{\Gamma}) @>>> V/\Gamma_n
\end{CD}
\]
in which $\Sigma$ is the reduced fibre product and
$q$ is finite and flat.

It is well-known that $X_{\Gamma}$ is a fine moduli space for
$\Gamma$-clusters on $\CC^2$. We write ${\mathcal B}$ for the
locally free sheaf on $X_\Gamma$ that is the
universal family of $\Gamma$-clusters.
The obvious permutation action makes ${\mathcal B}^{\boxtimes n}$
an $S_n$-equivariant sheaf on $X_{\Gamma}^n$.
Let $\mathcal{P}$ denote
%the bundle on $\hil^n (X_{\Gamma})$ corresponding to
$q_*p^*\mathcal{B}^{\boxtimes n}$.
Because $q$ and $p$ are $S_n$-equivariant ${\mathcal P}$ is an
$S_n$-equivariant sheaf on $\hil^{S_n}(X^n_{\Gamma})$. Since
the $S_n$-action on  $\hil^{S_n}(X^n_{\Gamma})$ is trivial this means
that $S_n$ acts as automorphisms of ${\mathcal P}$.

The ring homomorphism $\CC[x,y]*\Gamma \to \End {\mathcal B}$
induces homomorphisms $\CC[x,y]^{\otimes n} * \Gamma^n \to \End
p^* {\mathcal B}^{\boxtimes n}$ and $\CC[V] * \Gamma^n \to \End
q_*p^* {\mathcal B}^{\boxtimes n} = \End {\mathcal P}$. Combining
the last of these with the $S_n$-action produces a ring
homomorphism $\CC[V]*\Gamma_n \to \End{\mathcal P}$.

Consider $\lambda:K(\Gamma_n) \to \RR$ of the form
$$\lambda (M) = C \rho (M|_{\Gamma^n}) + \sigma (M|_{S_n})$$
where $\rho:K(\Gamma^n) \to \RR$ and $\sigma:K(S_n) \to \RR$ are
such that stable constellations are clusters. It can be shown that
for a suitable choice of $C \gg 1$, the geometric fibers
$\mathcal{P}(x):=\mathcal{P}/\fm_x\mathcal{P}$ of ${\mathcal P}$
are $\lambda$-stable $\Gamma_n$--constellations, and hence  that
$\mathcal{P}$ is a family of $\lambda$--stable
$\Gamma_n$--constellations.

The fixed points subsheaf $\mathcal{P}^{\Gamma_{n-1}}$ is the
universal family of $\CC[x,y]*\Gamma$-modules whose fibres have
$n$ copies of the regular representation of $\Gamma$ \cite[Prop.
7.2.12]{hai2}.

Let $M_{\lambda}$ be the moduli space of $\lambda$--stable
$\Gamma_n$--constellations and ${\mathcal S}$ the
universal family on it of $\lambda$-stable
$\CC[V]*\Gamma_n$-constellations.

The homomorphisms $\phi: \CC[V]*\Gamma_n \to \End {\mathcal P}$
and $\psi:\CC[V]*\Gamma_n \to \End {\mathcal S}$ restrict to
homomorphisms $\phi':\CC[x,y]*\Gamma \to \End {\mathcal
P}^{\Gamma_{n-1}}$ and $\psi':\CC[x,y]*\Gamma \to \End {\mathcal
S}^{\Gamma_{n-1}}$.


Since $\mathcal{P}$ is a family of $\lambda$--stable
$\Gamma_n$--constellations, there is a morphism
$f:  \hil^n(X_{\Gamma}) \longrightarrow M_{\lambda}$
such that $f^*\mathcal{S} \cong \mathcal{P}$ and $\phi=f^* \psi$.
Thus $\phi'=f^*  \psi'$.
Similarly, by the universal property of
${\mathcal P}^{\Gamma_{n-1}}$, there exists
$g: M_{\lambda} \longrightarrow \hil^n(X_{\Gamma})$
such that $g^*\mathcal{P}^{\Gamma_{n-1}} \cong \mathcal{S}^{\Gamma_{n-1}}$
and $\phi'=g^* \psi'$.

Both $M_{\lambda}$ and $\hil^n(X_{\Gamma})$ have trivial $\Gamma_n$-action so
$f$ and $g$ are automatically $\Gamma_n$--equivariant. Therefore
$$
(gf)^*\mathcal{P}^{\Gamma_{n-1}} \cong f^*(\mathcal{S}^{\Gamma_{n-1}})
\cong \mathcal{P}^{\Gamma_{n-1}}.
$$
Notice too that $f^*g^*  \psi'=\psi'$.
Since $\hil^n(X_{\Gamma})$ is a fine moduli space with universal
family $\mathcal{P}^{\Gamma_{n-1}}$, it follows that $gf=\id$.

There is a non-empty open subset $U$ of $V/\Gamma_n$ such that the natural maps
$\alpha:\hil^n X_\Gamma \to V/\Gamma_n$ and
$\beta:M_\lambda \to V/\Gamma_n$ restrict to
isomorphisms $\alpha^{-1}(U) \to U$ and $\beta^{-1}(U) \to U$.
The closure $Y_\lambda$ of $\beta^{-1}(U)$ is the unique irreducible
component of $M_\lambda$ that maps birationally to $V/\Gamma_n$.
Since $f$ and $g$ are morphisms of $V/\Gamma_n$-schemes and $fg=\id$ they
restrict to mutually inverse isomorphisms between $\alpha^{-1}(U)$
and $\beta^{-1}(U)$
and hence between their closures. But $\hil^n X_\Gamma$ is irreducible, so
$f$ and $g$ yield an isomorphism $\hil^n X_\Gamma \cong Y_\lambda$.
\end{proof}

\subsection{}
\label{haith2}

As noted in \cite[Section 4.4]{wang1} and \cite{hai2} the previous
theorem, together with the main result in \cite{BKR}, has the
following important consequence.

\begin{corollary}
\label{cor.haiman}
The derived categories $D^b(\coh \hil^n(X_{\Gamma}))$ and
$D^b(\md H_{\mathbf{0}})$ are equivalent.
\end{corollary}
\begin{proof}
This follows from Theorem \ref{haith} since the resolution
$\hil^n(X_{\Gamma})\longrightarrow V/\Gamma_n$ is crepant,
hence semismall by \ref{defss}, and so, using Lemma \ref{sathyp},
satisfies the hypothesis of Theorem \ref{der}.
\end{proof}

\subsection{}
\label{sect.perturb}
Set $\Theta = \{ \mu :K(\Gamma_n) \longrightarrow \mathbb{R} :
\mu (\alpha) = 0 \}$. Let $\lambda$ be the element in $\Theta$ given
by Theorem \ref{haith}. Define $\Theta_{\lambda}^+ = \{ 0\neq
M\subset \CC\Gamma_n : \lambda(M) >0 \}$ and
$\Theta_{\lambda}^- = \{ 0 \neq
M \subset \CC\Gamma_n : \lambda(M)<0\}$. If $M$ is a proper
$\Gamma_n$--submodule of the regular representation such that
$\lambda (M) = 0$ we can perturb $\lambda$ to $\mu$ so that
$\Theta_{\lambda}^+\cup \{ M\} \subseteq \Theta_{\mu}^+ $ and
$\Theta_{\lambda}^- \subseteq \Theta_{\mu}^-$.
The $\lambda$--stable constellations are the same as
the $\mu$--stable constellations
since $\lambda$ is generic.
Notice that every $\mu$-semistable constellation is stable.
Thus, without loss of generality, we may replace $\lambda$ by $\mu$
and assume that $\lambda(M) \neq 0$ for all proper
subrepresentations of a $\lambda$-stable constellation $M$.


\subsection{}

Let $H_{u\mathbf{c}}$ be the flat family of symplectic reflection
algebras defined in \ref{ff}.

We will now use Van den Bergh's result in Theorem \ref{der} to
extend Corollary \ref{cor.haiman} to the deformations $Y_{\mathbf{c}}
\to X_{\mathbf{c}}$ where $Y_{\mathbf{c}}$ is a suitable moduli space of
$H_{\mathbf{c}}$-modules.

In Section \ref{sect.apply-to-Hc}
 $\alpha$ denoted the element of $\ZZ^p$ defined by
$$
\alpha_i=\rank_{Z_{u\mathbf{c}}} e_iH_{u\mathbf{c}}e = \dim E_i
$$
where $E_i$ is the irreducible representation of $\Gamma_n$
corresponding to the central idempotent $e_i \in \CC\Gamma_n
\subset H_{\mathbf{c}}$. Therefore under the isomorphism
$K(\Gamma_n) \to \ZZ^p$, $[E_i] \mapsto
(\delta_{1i},\ldots,\delta_{ni})$, we have $[\CC\Gamma_n] \mapsto
\alpha$. In particular, the use of $\alpha$ in the last few
subsections is compatible with the use of $\alpha$ in Section
\ref{sect.apply-to-Hc}.

Let $\lambda:K(\Gamma_n) \to \RR$ be generic for $\alpha$. Let $W$
be the moduli space, as constructed in Section \ref{nc}, of
$\lambda$--stable $H_{u\mathbf{c}}$--modules isomorphic to
$\CC\Gamma_n$ (equivalently, of dimension $\alpha$), and let $Y$
be the irreducible component of $W$ that maps birationally to
$\textsf{Sm}(X_{u\mathbf{c}})$.

There is a natural $\CC^*$-equivariant map $W \to \CC$ and its restriction to
$Y$ fits into the following commutative diagram in the $\mathbb{C}^*$--equivariant
category
$$
\setlength{\unitlength}{1mm}
\begin{picture}(25,15)
\put(0,13){$Y$}
\put(4,15){\vector(1,0){20}}
\put(25,13){$X$}
\put(14,16){$\scriptstyle\pi$}
\put(3,11){\vector(1,-1){7}}
\put(2,6){$f$}
\put(24,11){\vector(-1,-1){7}}
\put(21,6){$g$}
\put(12,0){$\CC$}
\end{picture}
$$
The horizontal arrow is obtained by taking the central character.

\begin{theorem}
Keep the above notation.
\begin{enumerate}
\item The fibre $Y_{\mathbf{c}} := f^{-1}(1)$ is a crepant resolution of $\xc = \spec \zc$.
\item There is an equivalence of categories between $D^b (\coh Y_{\mathbf{c}})$ and $D^b(\md \hc)$.
\end{enumerate}
\end{theorem}
\begin{proof}
For $\tau\in \CC$, we write $W_{\tau}$ for the fiber of $W \to \CC$ over $\tau$;
we also define $Y_{\tau} = f^{-1}(\tau)$ and $X_{\tau} = g^{-1}(\tau)$.
By Theorem \ref{der} and Lemma \ref{sathyp}, it is enough to show that $Y_1$
is the irreducible component of $W_1$ that is birational to $\textsf{Sm}(X_1)$,
and that the restriction of $\pi$ to a morphism $Y_1\longrightarrow X_1$ is
semi--small.

The variety $Y_{\tau}$ is a moduli space of $\lambda$--stable
$H_{\tau\mathbf{c}}$--modules of dimension
$\alpha=\underline{\dim} (\CC\Gamma_n)$.
By Lemma \ref{az}, $Y_{\tau}$ contains the irreducible component of
$W_{\tau}$ that maps birationally to $\textsf{Sm}{X_{\tau}}$. In particular,
$\dim Y_{\tau} \geq  \dim X_{\tau}$; in fact, these dimensions
are equal because $Y$ is irreducible of dimension $\dim X = \dim X_{\tau}+1$.
If $\tau$ is non--zero, then $Y_{\tau}$ is irreducible since
$\overline{\CC^*Y_{\tau}}$ is a closed subset of Y of the same dimension as
$Y$. On the other hand, if $\tau = 0$ then \cite[Section 8]{BKR} combined with Corollary \ref{haith2} shows that this particular irreducible component is a connected component of the moduli space $W_0$.
By \ref{norm}, $X$ is normal, so Zariski's main theorem implies that $Y_{0}$
is connected \cite[Corollary III.11.4]{Ha}, and we see that $Y_{0}$ is also
irreducible. We deduce that for any $\tau$, $Y_\tau$ is the irreducible
component of $W_\tau$ that is birational to $\textsf{Sm}(X_{\tau})$.

The semi--smallness follows from Lemma \ref{lem.good.pi.2} since the restriction of $\pi$ to $Y_{0}\longrightarrow X_{0}$ is semi--small, being the crepant resolution of Theorem \ref{haith}.
\end{proof}

\subsection{}
\label{nosimples}
Let $\mathfrak{m}_x$ be the maximal ideal of $\zc$ corresponding to $x\in \xc$.
The simple $\hc$-modules with central character $x$ are precisely the
simple modules of the finite dimensional algebra $\hc/\mathfrak{m}_x\hc$.
\begin{corollary}
Let $\pi_{\mathbf{c}} : Y_{\mathbf{c}} \longrightarrow \xc$ be the crepant resolution above. There is an equivalence of
triangulated categories between $D^b_x(\coh Y_{\mathbf{c}})$ and $D^b_x(\md \hc)$. In particular there is an isomorphism between the Grothendieck groups $K(\pi_{\mathbf{c}}^{-1}(x))$ and $K(\hc/\mathfrak{m}_x\hc)$.
\end{corollary}
\begin{proof}
The first sentence has already been noted in \ref{fibreres}. By
devissage, the Grothendieck groups of $D^b_x(\coh Y_{\mathbf{c}})$
and $D^b_x(\md \hc)$ are isomorphic to
$K(\pi_{\mathbf{c}}^{-1}(x))$ and $K(\hc/\mathfrak{m}_x\hc)$
respectively, thus confirming the second sentence.
\end{proof}

\subsection{}
When $n=1$ the varieties $\xc$ are deformations of Kleinian singularities
$\CC^2/\Gamma$ and any crepant resolution coincides with the minimal resolution.
Hence, by the McKay correspondence, the $K$--theory of the fibre $f^{-1}(x)$ is completely determined by the type of orbifold singularity at $x\in \xc$. Indeed, suppose the singularity at $x\in \xc$ is locally of the form $\CC^2/G$ for some finite subgroup $G$ of $SL_2(\CC)$. Then the rank of $K(f^{-1}(x))$ equals the number of irreducible representations of $G$. On the other hand, the algebras $\hc$ are deformed preprojective algebras. These algebras also depend only on the type of orbifold singularity at $x\in \xc$, since there is a ''slice" theorem which reduces the representation theory to the case of the point $0\in \CC^2/G$, \cite[Corollary 4.10]{CB}. Thus the rank of $K(\hc /\mathfrak{m}_x\hc)$ also equals the number of irreducible representations of $G$, as expected.

Of course, in the case of arbitrary $n$ but $\mathbf{c}=0$, the corollary (which is an immediate consequence of Haiman's work) recovers the generalised McKay correspondence proved by Kaledin, \cite{kal}, for the orbifold singularities appearing locally in $V/\Gamma_n$.

\subsection{}
Corollary \ref{nosimples} gives us a geometric description for the number of simple modules in $\hc/\mathfrak{m}_x\hc$. In practice this is not immediately applicable as we have no geometric understanding of $Y_{\mathbf{c}}$. There is, however, evidence to suggest that the following question has a positive answer:

\begin{question} Let $V$ have Gorenstein singularities and let $v\in V$. Is $\rank K(f^{-1}(v))$ independent of the choice of crepant resolution $f: \tilde{V}\longrightarrow V$?
\end{question}

Indeed, \cite[Proposition 6.3.2]{denloe} shows that the mixed Hodge polynomial (of Borel--Moore homology) of $f^{-1}(x)$ is independent of the choice of resolution. Thus if the homology groups of the fibres are spanned by algebraic cycles (as seems reasonable given the results of \cite{kal} and the comments below), the answer is ``yes''. Furthermore, \cite[Section 5]{BO} conjectures that all crepant resolutions of $X$ have equivalent bounded derived categories of coherent sheaves. Confirmation of this conjecture would also give a positive answer.

In the particular case of $\xc$, it is possible to show that there
is a crepant resolution which can be described as a quiver
variety, \cite{nak}, generalising the $\mathbf{c} = \mathbf{0}$
case in \cite[Sections 1.3 and 1.4]{wang3} and the generic
$\mathbf{c}$ case in \cite[Section 11]{EG}. The $K$--theory of the
fibres has been studied, and is related to weight spaces of
integrable representations of Kac--Moody Lie algebras. Hence it is
reasonable to expect the number of simple
$\hc/\mathfrak{m}_x\hc$--modules also has this interesting
description. We will return to this in future work.

\begin{thebibliography}{10}

\bibitem{BO} A. Bondal and D. Orlov, Derived categories of coherent sheaves, {\it preprint}, AG/0206295.

\bibitem{BKR}
T. Bridgeland, A. King, and M. Reid,
The McKay correspondence as an equivalence of derived categories,
{\it J. Amer. Math. Soc.,} {\bf 14} (2001) 535--554.

\bibitem{BrGoo}
K.A.Brown and K.R.Goodearl, Homological aspects of noetherian PI Hopf algebras and irreducible modules of maximal dimension, {\it J.Algebra}, {\bf 198} (1997), 240--265.

\bibitem{BG}
K.A. Brown and I. Gordon, Poisson orders, representation theory and symplectic reflection algebras, {\it J. Reine Angew. Math.}, {\bf 559} (2003), 193--216.

\bibitem{CG}
N. Chriss and V. Ginzburg, Representation theory and complex geometry, Birkh\"{a}user, Boston, 1997.

\bibitem{CB}
W. Crawley-Boevey, Normality of Marsden-Weinstein reductions for
representations of quivers, {\it Math. Ann.,} {\bf 325} (2003),
55--79.

\bibitem{dCM}
M. de Cataldo and L. Migliorini,
The Chow motive of semismall resolutions, {\it preprint}, AG/0204067.

\bibitem{denloe}
J. Denef and F. Loeser, Germs of arcs on singular algebraic varieties and motivic integration, {\it Invent. Math.}, {\bf 135} (1999), 201--232.

\bibitem{EG}
P. Etingof and V. Ginzburg, Symplectic reflection algebras,
Calogero-Moser space, and deformed Harish-Chandra homomorphism,
{\it Invent. Math.,} {\bf 147} (2002) 243--348.

\bibitem{FG}
M. Finkelberg and V. Ginzburg, Calogero--Moser space and Kostka polynomials, {\it Adv.Math.}, {\bf 172} (2003), 137--150.

\bibitem{GK} V. Ginzburg and D. Kaledin, Poisson deformations of symplectic quotient singularities, {\it preprint},
AG/0212279, to appear {\it Adv.Math.}

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

\bibitem{hai} M. Haiman, Hilbert schemes, polygraphs and the
Macdonald positivity conjecture, {\it J. Amer. Math. Soc.}, {\bf
14} (4) (2001), 941--1006.

\bibitem{hai2} M. Haiman, Combinatorics, symmetric functions, and
Hilbert schemes, {\it preprint} (2003) to appear {\it Current
Developments in Mathematics, Proceedings of a conference at
Harvard University, November 2002}


\bibitem{hai3} M. Haiman, Cores, quivers and $n!$ conjectures, {\it in
preparation}

\bibitem{Ha}
R. Hartshorne, {\it Algebraic Geometry,} Graduate Texts in
Mathematics, no. 52, Springer-Verlag, 1977.

\bibitem{in} Y. Ito and I. Nakamura, McKay correspondence and Hilbert schemes, {\it Proc. Japan Acad. Ser. A Math. Sci.}, {\bf 72} (1996), 135--138.

\bibitem{kal0} D. Kaledin, Dynkin diagrams and crepant resolutions of quotient singularities, {\it preprint}, AG/9903157, to appear {\it Selecta Math.}

\bibitem{kal} D. Kaledin, McKay correspondence for symplectic quotient singularities, {\it Invent.Math.}, {\bf 148} (2002), no.1, 151--175.

\bibitem{KV}
M. Kapranov and E. Vasserot,
Kleinian singularities, derived categories and Hall algebras,
{\it Math. Ann.,} {\bf 316} (2000) 565--576.

\bibitem{K}
A.D. King,
Moduli of representations of finite--dimensional algebras,
{\it  Quart. J.  Math. Oxford Ser. (2),} {\bf 45} (1994) 515--530.

\bibitem{kraft}  H. Kraft, {\it Geometrische Methoden in der Invariantentheorie}, Aspects of Mathematics,
 D1. Friedr. Vieweg und Sohn, Braunschweig, 1984.

\bibitem{McCRob} J.C.McConnell and J.C.Robson, {\it Noncommutative Noetherian Rings}, Graduate studies in mathematics, 30 { American Mathematical Society}, 2001.

\bibitem{Mont}
S.M. Montgomery, {\em Fixed Rings of Finite Automorphism Groups of
Associative Rings,} Lecture Notes in Mathematics 818,
Springer-Verlag, 1980.


\bibitem{nak} H. Nakajima, Quiver varieties and Kac-Moody
algebras, \newblock {\it Duke Math. J.}, {\bf 91} (1998), no. 3, 515--560.

\bibitem{nakbook} H. Nakajima, {\it Lectures on Hilbert schemes of points on surfaces}, American Mathematical Society, Providence, RI, 1999.

\bibitem{vdb2} M. Van den Bergh, Non--commutative crepant
resolutions, {\it preprint}, RA/0211064, to appear {\it
Proceedings of the Abel Bicentennial Conference}.

\bibitem{ver} M.Verbitsky, Holomorphic symplectic geometry and orbifold singularities, {\it Asian J.Math.}, {\bf 4} (2000), no.3, 553--563.

\bibitem{wang2} W.Wang, Equivariant $K$-theory, wreath products, and Heisenberg algebra, {\it Duke Math. J.}, {\bf 103} (2000), no. 1, 1--23.

\bibitem{wang3} W.Wang, Hilbert schemes, wreath products, and the McKay
correspondence, \textit{preprint}, math.AG/9912104

\bibitem{wang1} W.Wang, Algebraic structures behind Hilbert schemes and wreath products, in \textit{Recent developments in infinite-dimensional Lie algebras and conformal field theory} (Charlottesville, VA, 2000), 271--295, Contemp. Math., 297, Amer. Math. Soc., 2002.

\end{thebibliography}

\end{document}

math0310187.extracted_bib
\begin{thebibliography}{10}

\bibitem{BO} A. Bondal and D. Orlov, Derived categories of coherent sheaves, {\it preprint}, AG/0206295.

\bibitem{BKR}
T. Bridgeland, A. King, and M. Reid,
The McKay correspondence as an equivalence of derived categories,
{\it J. Amer. Math. Soc.,} {\bf 14} (2001) 535--554.

\bibitem{BrGoo}
K.A.Brown and K.R.Goodearl, Homological aspects of noetherian PI Hopf algebras and irreducible modules of maximal dimension, {\it J.Algebra}, {\bf 198} (1997), 240--265.

\bibitem{BG}
K.A. Brown and I. Gordon, Poisson orders, representation theory and symplectic reflection algebras, {\it J. Reine Angew. Math.}, {\bf 559} (2003), 193--216.

\bibitem{CG}
N. Chriss and V. Ginzburg, Representation theory and complex geometry, Birkh\"{a}user, Boston, 1997.

\bibitem{CB}
W. Crawley-Boevey, Normality of Marsden-Weinstein reductions for
representations of quivers, {\it Math. Ann.,} {\bf 325} (2003),
55--79.

\bibitem{dCM}
M. de Cataldo and L. Migliorini,
The Chow motive of semismall resolutions, {\it preprint}, AG/0204067.

\bibitem{denloe}
J. Denef and F. Loeser, Germs of arcs on singular algebraic varieties and motivic integration, {\it Invent. Math.}, {\bf 135} (1999), 201--232.

\bibitem{EG}
P. Etingof and V. Ginzburg, Symplectic reflection algebras,
Calogero-Moser space, and deformed Harish-Chandra homomorphism,
{\it Invent. Math.,} {\bf 147} (2002) 243--348.

\bibitem{FG}
M. Finkelberg and V. Ginzburg, Calogero--Moser space and Kostka polynomials, {\it Adv.Math.}, {\bf 172} (2003), 137--150.

\bibitem{GK} V. Ginzburg and D. Kaledin, Poisson deformations of symplectic quotient singularities, {\it preprint},
AG/0212279, to appear {\it Adv.Math.}

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

\bibitem{hai} M. Haiman, Hilbert schemes, polygraphs and the
Macdonald positivity conjecture, {\it J. Amer. Math. Soc.}, {\bf
14} (4) (2001), 941--1006.

\bibitem{hai2} M. Haiman, Combinatorics, symmetric functions, and
Hilbert schemes, {\it preprint} (2003) to appear {\it Current
Developments in Mathematics, Proceedings of a conference at
Harvard University, November 2002}


\bibitem{hai3} M. Haiman, Cores, quivers and $n!$ conjectures, {\it in
preparation}

\bibitem{Ha}
R. Hartshorne, {\it Algebraic Geometry,} Graduate Texts in
Mathematics, no. 52, Springer-Verlag, 1977.

\bibitem{in} Y. Ito and I. Nakamura, McKay correspondence and Hilbert schemes, {\it Proc. Japan Acad. Ser. A Math. Sci.}, {\bf 72} (1996), 135--138.

\bibitem{kal0} D. Kaledin, Dynkin diagrams and crepant resolutions of quotient singularities, {\it preprint}, AG/9903157, to appear {\it Selecta Math.}

\bibitem{kal} D. Kaledin, McKay correspondence for symplectic quotient singularities, {\it Invent.Math.}, {\bf 148} (2002), no.1, 151--175.

\bibitem{KV}
M. Kapranov and E. Vasserot,
Kleinian singularities, derived categories and Hall algebras,
{\it Math. Ann.,} {\bf 316} (2000) 565--576.

\bibitem{K}
A.D. King,
Moduli of representations of finite--dimensional algebras,
{\it  Quart. J.  Math. Oxford Ser. (2),} {\bf 45} (1994) 515--530.

\bibitem{kraft}  H. Kraft, {\it Geometrische Methoden in der Invariantentheorie}, Aspects of Mathematics,
 D1. Friedr. Vieweg und Sohn, Braunschweig, 1984.

\bibitem{McCRob} J.C.McConnell and J.C.Robson, {\it Noncommutative Noetherian Rings}, Graduate studies in mathematics, 30 { American Mathematical Society}, 2001.

\bibitem{Mont}
S.M. Montgomery, {\em Fixed Rings of Finite Automorphism Groups of
Associative Rings,} Lecture Notes in Mathematics 818,
Springer-Verlag, 1980.


\bibitem{nak} H. Nakajima, Quiver varieties and Kac-Moody
algebras, \newblock {\it Duke Math. J.}, {\bf 91} (1998), no. 3, 515--560.

\bibitem{nakbook} H. Nakajima, {\it Lectures on Hilbert schemes of points on surfaces}, American Mathematical Society, Providence, RI, 1999.

\bibitem{vdb2} M. Van den Bergh, Non--commutative crepant
resolutions, {\it preprint}, RA/0211064, to appear {\it
Proceedings of the Abel Bicentennial Conference}.

\bibitem{ver} M.Verbitsky, Holomorphic symplectic geometry and orbifold singularities, {\it Asian J.Math.}, {\bf 4} (2000), no.3, 553--563.

\bibitem{wang2} W.Wang, Equivariant $K$-theory, wreath products, and Heisenberg algebra, {\it Duke Math. J.}, {\bf 103} (2000), no. 1, 1--23.

\bibitem{wang3} W.Wang, Hilbert schemes, wreath products, and the McKay
correspondence, \textit{preprint}, math.AG/9912104

\bibitem{wang1} W.Wang, Algebraic structures behind Hilbert schemes and wreath products, in \textit{Recent developments in infinite-dimensional Lie algebras and conformal field theory} (Charlottesville, VA, 2000), 271--295, Contemp. Math., 297, Amer. Math. Soc., 2002.

\end{thebibliography}