We extend two well-known results on primitive ideals in enveloping algebras of semisimple Lie algebras, the {Irreducibility theorem} for associated varieties and {Duflo theorem} on primitive ideals, to much wider classes of algebras. Our general version of Irreducibility theorem says that if $A$ is a positively filtered associative algebra such that $\gr{A}$ is a commutative Poisson algebra with {finitely many} symplectic leaves, then the associated variety of any primitive ideal in $A$ is the closure of a single connected symplectic leaf. Our general version of Duflo theorem says that if $A$ is an algebra with a `triangular structure', see \S2, then any primitive ideal in $A$ is the annihilator of a simple highest weight module. Applications to Symplectic reflection algebras and Cherednik algebras are discussed.
math0202079.tex
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\begin{document}
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\centerline{\Large {\bf ON  PRIMITIVE  IDEALS}}

\vskip 6mm
\centerline{\large {\sc Victor Ginzburg}}

\bigskip
\begin{abstract}
We extend two well-known results
on primitive ideals in enveloping algebras of semisimple Lie algebras,
 the {\it Irreducibility 
theorem} for associated varieties and {\it Duflo theorem}
on primitive ideals,
to  much wider classes of algebras. Our general version of Irreducibility
theorem says that if $A$ is a positively filtered associative
algebra such that $\gr{A}$ is a commutative Poisson algebra
with  {\it finitely many} symplectic leaves, then the associated
variety of any primitive ideal in $A$ is the closure of
a single connected symplectic leaf. Our  general version of  Duflo theorem
says that if $A$ is an algebra with a `triangular
structure', see \S2,
  then  any primitive ideal in $A$ is the
annihilator of a simple highest weight module. 
Applications to Symplectic reflection algebras
and Cherednik algebras are discussed.
\end{abstract}

\bigskip
\section{Introduction}\setcounter{equation}{0}
Let $A$ be an associative, not necessarily
commutative, algebra.
Recall that a two-sided ideal $I\subsetneq A$ is called {\it primitive}
if it is the annihilator of a simple left $A$-module.
Primitive ideals often play a role similar to
the one played by
maximal ideals in the case of commutative algebras.
The set of all primitive ideals in $A$, equipped with
Jacobson topology, is called the {\it primitive spectrum}
of $A$. It is a
fundamental invariant of $A$ analogous to the affine scheme
$\Spec A$ in the  commutative case.

A complete understanding of the primitive spectrum
is only available in some `very special'
cases. Our motivation in this paper
comes from one such case where $A=\U\g$ is the universal enveloping 
algebra of a complex semisimple Lie algebra $\g$.
Classification of  primitive ideals in $\U\g$
was one of the central themes of representation theory
during the 70--80's, and it is
nowadays  well understood, cf. [J1], [J2] and references therein. Yet, 
details of the classification and the methods involved
still appear to be quite complicated. 
%They exploit the full force
%of a large number of deep  results on
%Harish-Chandra modules,
%Kazhdan-Lusztig polynomials, perverse sheaves, etc. 


Our goal is to show that,
quite surprisingly, two very important results on
 primitive ideals in $\U\g$ have natural generalizations
to much wider classes of algebras. The first result is
{\it Duflo theorem} [Du], saying that any primitive ideal  in $\U\g$
is the
annihilator of a simple {\it highest weight} module.
We generalize this to arbitrary algebras $A$ that have a
triangular structure, i.e. a pair of `opposite'
subalgebras $A^\pm$ such that $A$ is a finite type
$(A^+\td A^-)$-bimodule, satisfying certain additional 
technical conditions, see \S2.
 Our approach is based
on the notion of Jacquet functor borrowed from representation theory,
and on Gabber's {\it Separation theorem} [Ga].

The second result is a
generalization of  the {\it Irreducibility theorem}  saying that 
the associated variety of a  primitive ideal in
$\U\g$ is irreducible,
specifically, it is the
closure of a nilpotent conjugacy class in $\g$. The
latter theorem was first partially proved (by a case-by-case
argument) in [BoBr], and in a more conceptual way in [KT]
and [J2] (independently), using
many earlier deep results due to Joseph, Gabber,
Lusztig, Vogan and others. Our 
argument is an adaptation of a much more
direct proof discovered subsequently by Vogan [Vo],
combined with some recent results by Brown-Gordon [BrGo].

We got involved into these matters
because of our interest in 
primitive ideals in Symplectic reflection algebras,
a new important class
of associative algebras introduced recently in [EG]
in connection with various   works
 in combinatorics, completely integrable systems, and
generalized McKay correspondence for orbifolds $\C^n/\G$.
In  \S6, we will describe 
all possible subvarieties in  $\C^n/\G$ that may arise as associated
varieties of primitive ideals in a Symplectic reflection algebra.

An important special class of   Symplectic reflection algebras
comes from Coxeter groups of finite root systems.
The algebras in question  have been 
called {\it rational 
Cherednik algebras},  see [EG], [DO], [BEG],
since they
appeared as a
`rational' degeneration of  {\it double-affine 
Hecke algebras} introduced by Cherednik [Ch].
Thus, the latter may (and should) be  thought of as a deformation
of the former. From this point of view,
representation theory of the
rational Cherednik  algebra  is perhaps
 `more basic' than (or at least should be studied before)
 that  of the double-affine 
Hecke algebra in the same sense as 
the representation theory of semisimple Lie algebras
is  `more basic'  than that  of the corresponding
quantum groups. 

`Experimental evidence' suggests
that, although rational Cherednik  algebras are quite far from
enveloping algebras,
 their representation theory 
shares many features of  representation theory
of semisimple Lie algebras, see [BEG].
Thus, classification of
primitive ideals in  Cherednik algebras
seems to be a 
challenging problem.
The results of the present  paper provide a first step in
this direction. 

In more detail,  let $\sH_c$ be the
 Cherednik algebra  associated to the Weyl group $W$ of
a root system in a vector space $\h$.
There is an
analogue
 of the Bernstein-Gelfand-Gelfand
 category of highest weight modules for $\sH_c$, to
be denoted  $\oo_c$, see [BEG].
The isomorphism classes of simple objects in  $\oo_c$ 
are known to be
parametrized by the set $\irrep(W)$ of irreducible representations
of the group $W$. Given $\sigma \in \irrep(W)$,
let $L_\sigma$  denote  the corresponding simple object of  $\oo_c$,
and $I_\sigma:=\ann(L_\sigma)\sset \sH_c$ denote its annihilator.
The assignment $\Theta: \sigma\mapsto I_\sigma$ gives a map from
$\irrep(W)$ to the set ${\sf{Prim}}(\sH_c)$ of primitive ideals in $\sH_c$,
and our general version of Duflo theorem implies that
the map $\Theta$ is {\it surjective}, in particular,
 ${\sf{Prim}}(\sH_c)$ is a finite set. Moreover, our Irreducibility
theorem implies that $\var(I_\sigma)$, the associated
variety of $I_\sigma,$ is the closure of $W$-saturation
of a certain symplectic leaf in $\h\times\h^*$.
This leads to
two important open problems
\begin{problem}[Classification of primitive ideals]\label{prob}
\vi Describe the fibers of the map
$\Theta:\irrep(W)\too
{\sf{Prim}}(\sH_c) $
in a more
direct way.

\vii For each  $\sigma \in \irrep(W)$, describe 
the associated variety of $I_\sigma$, in particular,
for which pairs  $\sigma,\tau \in \irrep(W)$ does one have:
$\var(I_\sigma)=\var(I_\tau)$ ?
\end{problem}
\begin{remark}
The reader should keep in mind that the Cherednik algebra
$\sH_c$ depends on complex parameter(s) `$c$', which is an
analogue of  `central character' in the Lie algebra case.
The solution to Problem \ref{prob} should depend on the value
of  `$c$' in an essential way.
\end{remark}

As a step towards Problem \ref{prob}(ii),
we will compare, in \S6,
the associated variety of
$L_\sigma$ with that of its  annihilator $I_\sigma$.
Specifically, we show (Theorem \ref{levi})
that the associated variety of
any simple highest weight $\sH_c$-module $L$
equals $\var(\ann L)\;\cap\;(\h\times\{0\}),$
the intersection of the associated
variety of the annihilator of $L$ (in $\sH_c$) with
the  Lagrangian subspace $\h\times\{0\}\sset \h\times\h^*$.\medskip


\noindent
{\bf Acknowledgments.} 
          {\footnotesize{     It is a pleasure to acknowledge my
debt to Ofer Gabber for many
 extremely helpful
remarks, and for a very careful reading of the manuscript.}}


\section{Main results}
\setcounter{equation}{0}

Throughout the paper we work with associative algebras over $\C$,
the field of complex numbers.
We write $\otimes=\otimes_\C$.


    

                                                   
Let $A$ be a unital $\C$-algebra  equipped
with
a multiplicative increasing filtration:
$0=A_{-1}\subset A_0\subset A_1\subset\ldots,\,$
such that $A_i\cdot A_j\subset A_{i+j}$, and
$\cup_{j\geq 1}\,A_j=A\,,\,1\in A_0$.
Furthermore, we assume that
$\gr A = \bigoplus_j\,A_j/A_{j-1}$,
the associated graded algebra,
is a finitely generated commutative  algebra without zero-divisors,
in other words $\gr A \simeq\C[X]$ is
 the coordinate ring of a (reduced)
irreducible  affine
algebraic variety $X$. For any (say, left) ideal $I\sset A$,
the filtration on $A$ induces one on $I$, so that
$\gr I$ becomes an ideal in $\gr A$.
We write $\var(I)\sset X$ for the zero variety
of the ideal $\gr I$, usually referred to as
the associated variety of $I$.


Let $\ell$ be the maximal integer (or $\infty$, if $A$ is commutative) such that,
for all $i,j\geq 0$ and $a_i\in A_i,\,a_j\in A_j,$ one has:
$a_i\cd a_j-a_i\cd a_j \in A_{i+j-\ell}$.\footnote{Note that $\ell\geq 1$
since  $\gr A $ is commutative.}
It is well-known that the assignment: $a_i,a_j \mapsto
[a_i,a_j]\,\text{\sl mod}\;A_{i+j-\ell-1}$ descends
to 
a canonical Poisson bracket on $\gr A$ that
makes
 $X$ a 
Poisson algebraic variety. If  $X$ is smooth, then one may view
$X$ as a complex-analytic manifold equipped with a holomorphic
Poisson structure. For each point $x\in X$ one defines
$\sS_x$, the {\it symplectic leaf} through $x$, to be the
set of points that could be reached from $x$  by going along
Hamiltonian flows. 

If  $X$ is not necessarily smooth,
let $\sing(X)$ denote the singular locus of $X$,
and for any $k\geq 1$ define inductively:
$\sing^k(X):=\sing\bigl(\sing^{k-1}(X)\bigr)$.
We get a finite partition $X=\bigsqcup_{k=0}^n\;X^k,$
where the strata $X^k:=\sing^{k-1}(X)\ssminus\sing^k(X)$
are smooth analytic varieties (by definition we put
$X^0= X\ssminus \sing(X)$).
It is known, cf. e.g., [BrGo], [Po],
that each $X^k$ inherits a  Poisson structure,
so for any point $x\in X^k$ there is a well-defined
 symplectic leaf $\sS_x\sset X^k$.
This way one defines symplectic leaves on an
arbitrary Poisson algebraic variety. 

In general, each symplectic leaf is a connected
smooth analytic (but not necessarily algebraic)
subset in $X$. However, if the algebraic variety $X$ consists of
finitely many symplectic leaves only, then it was shown
in [BrGo] that each leaf is a smooth irreducible
locally-closed algebraic subvariety in
$X$, and partition into  symplectic leaves
gives an algebraic stratification of $X$.

Our first main result reads

\begin{theorem}[Irreducibility theorem]\label{irreducibility}
Assume that the Poisson variety $\Spec(\gr{A})$
has only finitely many symplectic leaves. Then,
for any primitive ideal $I\sset A$,
the variety $\var(I)$ is the closure of 
a {\sl single} (connected) symplectic leaf.
\end{theorem}

In the classical case of a semisimple Lie
algebra $\g$, given a primitive ideal
$I\sset\U\g$, let $\cI:=I\cap \cZ$ be the intersection
of $I$ with  $\cZ$, the center of $\U\g$.
A standard argument based on a version of Schur lemma, see e.g. [Di],
implies that $\cI$ is a maximal ideal in $\cZ$.
We let $A:= \U\g/\cI\cdot\U\g$ be equipped with the
filtration induced by the standard increasing filtration on $\U\g$.
The algebraic variety
$\Spec(\gr{A})$ is known to coincide with the 
{\it nilpotent variety} in $\g$. The latter variety
is partitioned into finitely many conjugacy classes.
These turn out to be  exactly the symplectic leaves.
Thus, our Theorem becomes  the well-known result first
 proved in [BoBr, Theorem 6.5] for classical simple
Lie algebras, and in [J2, Theorem 3.10] and [KT, Proposition 11]
in general
(in the Lie algebra  case our argument reduces to the proof in [Vo],
and doesn't give anything new).
\medskip

To formulate our second main result, 
fix a  unital associative $\C$-algebra  $A$, and given
 $a\in A$ write $\ad a\mapsto [a,x]:=a\cd x-x\cd a$
for the adjoint  action of $a$ on $A$. 
Put $\Z_+=\{1,2,\ldots\},$ and $\Z_-=-\Z_+$.


Now, let $A^\pm \subset A$ be two  (non-unital) finitely-generated
subalgebras in $A$ and   $\bh\in A$, such that 
\smallskip

\noindent
${\sf{(T0)}}\quad$ \parbox[t]{137mm}{$A$ is 
finitely generated as an $A^+\otimes (A^-)\opp$-module,
i.e.,
as an $(A^+\td A^-)$-bimodule.}

 \noindent
${\sf{(T1)}}\quad$ \parbox[t]{137mm}{The action of $\ad\bh$ on $A$ 
is diagonalizable; the subalgebras $A^\pm$ are $\ad\bh$-stable.}

\noindent
${\sf{(T2)}}\quad$ \parbox[t]{137mm}{All the eigenvalues of
$\ad\bh$-action on  $A^\pm$ are in $\Z_\pm$ \enspace
(`$+$' for $A^+$, and `$-$' for $A^-$).}

We call the data $(A^\pm,\bh)$ a  {\it commutative triangular structure}
on $A$ if conditions (T0)-(T2) hold and, moreover,
both subalgebras  $A^\pm$ are commutative.
We will also consider non-commutative subalgebras  $A^\pm$.
In that  case, we will assume further that
the algebra $A$ is   equipped
with
a multiplicative increasing filtration:
$0=A_{-1}\subset A_0\subset A_1\subset\ldots.\,$
We write $\gr A$ for  the associated graded algebra
(which is not assumed to be commutative, in general),
and endow $A^\pm$ with induced filtrations.

We will say
that the data $(A^\pm,\bh)$
 gives a {\it non-commutative triangular structure} on $A$ if,
in addition to (T0)-(T2),
the following holds: 
\smallskip

\noindent
${\sf{(T0')}}\quad$  \parbox[t]{137mm}{$\gr A$ is a
finitely generated $(\gr A^+\td \gr A^-)$-bimodule.}

\noindent
${\sf{(T1')}}\quad$ \parbox[t]{137mm}{We have:
 $\bh\in A_1$, moreover, the image
of $\bh$ in $A_1/A_0$ is a central element in $\gr A$.}

\noindent
${\sf{(T3)}}\quad$ \parbox[t]{137mm}{The subalgebras
$\gr A^\pm$ are
commutative.}

\noindent
${\sf{(T4)}}\quad$ \parbox[t]{137mm}{The algebras $\gr A^\pm$
are generated by the subspace $\grs{0}A^\pm\oplus\grs{1}A^\pm$,
the corresponding degree $\leq 1$ components.
 Moreover, $\dim(\grs{i}A^\pm) <\infty$. }
\smallskip


Clearly, $(T0')$ implies $(T0)$.
Conditions $(T3)$-$(T4)$ imply that the subspaces
$A_1^\pm$ form finite dimensional Lie subalgebras in $A$
(with respect to the commutator bracket). We denote them
by  $\n^\pm=A_1^\pm$. It follows, since $\gr A^\pm$ is
generated by the corresponding degree $\leq 1$ components,
that the algebra $A^\pm$ is generated by the subspace
$A_1^\pm$, hence it is the quotient of the augmentation
ideal in  $\U\n^\pm$ by some
other  $\ad\bh$-stable two-sided ideal
in the enveloping algebra $\U\n^\pm$.
Condition $(T1')$ implies that,
for each $i\geq 1,$ the space $A_i$ is 
$\ad\bh$-stable.
Further, the Lie algebras $\n^\pm$ are
necessarily nilpotent, and also
 $1\not\in A^\pm$ (unless $A=0$), because of $(T2)$.


Given a non-unital algebra $\A$ and an $\A$-module $M$,
we say that $\A$-action on
$M$ is locally-nilpotent if, for any $m\in M,$ there exists a large enough
integer $n=n(m)\gg 0$ such that: $a_1\cdot \ldots\cdot a_n\cdot m=0\,,\,
\forall a_1,\ldots,a_n\in\A,$
and similarly for right $\A$-modules.

\begin{definition}\label{oo}
Given an algebra $A$ with triangular structure,
let $\ou$, resp. $\od$,
be the category of  finitely-generated
left, resp. right, $A$-modules $M$ such that
the 
$A^-$-action, resp. $A^+$-action,  on $M$ is locally-nilpotent.
\end{definition}



Recall that a two-sided ideal  $I\subsetneq A$ is called prime,
if for any two-sided ideals $J_1,J_2\sset A$ we have:
$\,J_1\cdot J_2\sset I\enspace\Longrightarrow\enspace
J_1\sset I\enspace\text{or}\enspace J_2\sset I.$
Our second main result reads

\begin{theorem}[Generalized Duflo theorem]\label{duflo}
Let $(A^\pm, \bh)$ be a (commutative or non-commutative)
triangular structure on
$A$.
Assume that  the  category $\ou$ (or
$\od$) has finitely many isomorphism classes of simple objects.
Then the following conditions
on a two-sided ideal $I\subsetneq A$ are equivalent:

\vi $I$ is a primitive ideal;

\vii $I$ is a prime ideal;

\viii $I$ is the left annihilator
of a simple object of $\ou$.

\iv $I$ is the right annihilator
of a simple object of $\od$.
\end{theorem}


In the classical case, one considers a semisimple Lie
algebra $\g$ with a fixed triangular decomposition:
$\g=\n^+\oplus\h\oplus\n^-$. Given a primitive ideal
$I\sset\U\g$, we define $A:= \U\g/\cI\cdot\U\g,$ 
where $\cI:=I\cap \cZ$ as above. 
Let
$A^\pm$ be the image  in $A$ of $\n^\pm\cd\U\n^\pm$.
The Harish-Chandra homomorphism: $\cZ\into\sym\h,$
combined with
Poincar\'e-Birkhoff-Witt theorem imply easily
that $A$ is a finitely generated  $(A^+\td A^-)$-bimodule,
see e.g. [Di]. We equip $A$ with an increasing filtration
induced by the standard  increasing filtration on $\U\g$.
Further, if $\bh\in\h$ denotes the sum of positive coroots,
then the $\ad\bh$-action on $\U\n^\pm$ is known to 
be diagonal with eigenvalues in $\Z_\pm$. 
The data  $(A^\pm,\bh)$
gives a non-commutative triangular structure on $A$.
The categories $\ou$ and $\od$ are both
equivalent, in this
case, to the standard category $\oo$ (with fixed
central character corresponding to $\cI$) as defined
by Bernstein-Gelfand-Gelfand. Simple objects of the
latter category are well-known
to be parametrized by a (necessarly finite) Weyl group orbit,
so that Theorem \ref{duflo} applies.
This gives a new proof (cf. also [J2, n.2.4] for a similar idea)
of the original Duflo theorem~[Du].

\begin{remark} In the special case $A=\U\g$ above,
our proof  of   Duflo theorem is 
 perhaps not much simpler than the earlier
proofs, see [Du], [BG], [J1],
because of its heavy dependence on the  difficult 
Separation theorem due to
Gabber [Ga]. One
advantage of our approach however is that,
unlike all other proofs, it requires very little
information about representation theory of 
the algebra~$A$.
\end{remark}

Simultaneously with Theorem \ref{duflo} we will also prove

\begin{theorem}\label{duflo2} Under the assumptions of
Theorem \ref{duflo}, the algebra $A$ has finite length
as an $A\otimes A\opp$-module.
\end{theorem}


We say that a $\bh$-action  on a vector space $M$
 is locally-finite, if any element $m\in M$ is contained
in a $\bh$-stable finite dimensional vector subspace.

Our last important result reads

\begin{theorem}\label{criterion}
Let $(A^\pm, \bh)$ be a triangular structure on $A$. Then

\vi The $\bh$-action on any object $M$ of either $\ou$ or $\od$ is
locally-finite, and
all (generalized) eigenspaces of this $\bh$-action  are
finite-dimensional.

\vii The following two properties of the category $\ou,$ resp.  $\od$,
are equivalent
\vskip -7mm
\begin{equation}\label{cat}
\begin{array}{ll}
{\hskip -6mm}\bullet& \text{\it The  category  has only finitely
many (isomorphism classes of) simple objects.}\\
{\hskip -6mm}\bullet& \text{\it  Any object  of the category has finite length.}
\end{array}
\end{equation}
\vskip -4mm

\viii If properties \eqref{cat} hold for  $\ou$ or  $\od$, then both
categories $\ou\,,\,\od$,
have enough projectives and injectives.
\end{theorem}




\section{Proof of Theorem \ref{irreducibility}}
\setcounter{equation}{0}
We 
need to recall some results due to Brown-Gordon [BrGo].

Let $B$ be a finitely generated Poisson algebra,
with  Poisson  bracket $\br$.
An ideal $J\sset B$ (with respect to the commutative product) 
is called a {\it Poisson ideal} if
it is also an ideal with respect to the Lie bracket
$\br$, i.e. if $\{J,B\}\sset J$. 
For the following
standard result of commutative algebra, see e.g. \cite[Theorem 4.5]{CO},
or \cite[\S3.3.2]{Di}.

\begin{lemma}\label{dixmier}
If $J$ is a  Poisson ideal then so are both its radical
and all the associated  prime ideals in $B$.
\hfill\qed
\end{lemma}


Write $\fm_x$ for the maximal  ideal in $B$ corresponding to
a  closed point $x\in \Spec B$, and recall the notation
$\sS_x\sset \Spec B$ for the symplectic leaf through $x$.
Following \cite{BrGo}, given any ideal $J\sset B$ we let $\pp(J)$
denote the maximal Poisson ideal contained in $J$.
If $J$ is prime then so is  $\pp(J)$, and one has

\begin{proposition}[\cite{BrGo}, Prop.\,3.7]\label{finite_leaves}
Assume the variety $\Spec B$ consists of finitely many
symplectic leaves. Then, for any closed point $x\in \Spec B$,
the leaf $\sS_x$ coincides with the regular locus of the
zero variety of the ideal $\pp(\fm_x)$.
Thus, $\sS_x$ is a smooth connected locally-closed subvariety
in  $\Spec B$.
\hfill\qed
\end{proposition}

Recall  that the Poisson structure on $B$ restricts
to a (non-degenerate) symplectic structure on each symplectic
leaf. In particular, all symplectic leaves
have even (complex) dimension. Hence, from Lemma 
\ref{dixmier} and Proposition \ref{finite_leaves}
we deduce
\begin{corollary}\label{even_dim}
If the variety $\Spec B$ consists of finitely many
symplectic leaves, then every irreducible
component of the zero variety of any
Poisson ideal $J\sset B$ is the closure  of a
symplectic leaf, in particular, it is even-dimensional.
\hfill\qed
\end{corollary}




Next, let $A$ be an associative filtered $\C$-algebra
as in Theorem \ref{irreducibility}. 
We recall

\begin{lemma}[\cite{BoKr}, Korollar 3.6]\label{GKdim}
If $I\sset J \sset A$ are two-sided ideals, $I\neq J$ and moreover $I$
is prime, then: $\dim\var(J)< \dim\var(I)$.
\hfill\qed
\end{lemma}


Given a finitely generated $A$-module $M$, choose a 
{\it good} filtration (cf. e.g. [Be]) on $M$
and write $\Supp(\gr M)$ for the support
of the corresponding associated graded $\gr A$-module,
 a reduced algebraic subvariety in $\Spec(\gr A)$.
It is well-known (due to Bernstein \cite{Be}) that this subvariety
 is independent of the choice of 
a good filtration on $M$, and it will be denoted
$\gsup M$ below. Note that if
$I\sset A$ is a left ideal, then in the special
case: $M=A/I$ we have by definition
$\gsup(A/I)=\var(I)$.



Assume from now on that $\gr{A}$ is a Poisson algebra with finitely many
symplectic leaves.
It is straightforward to
verify that, for any two-sided ideal $I\sset A$,
the associated graded  ideal, $\gr{I},$ is a Poisson ideal in $\gr{A}$.
Thus by Corollary \ref{even_dim}, every irreducible
component of  $\var(I)$, the zero variety  of  $\gr{I}$,
is the closure  of a
symplectic leaf.
\smallskip

\noindent
{\bf Proof of Theorem \ref{irreducibility}:\;}
Let $I$ be a primitive ideal. We've just seen that 
proving the Theorem  amounts to showing
that  $\var(I)$ is irreducible. To prove this,
we mimic the argument in ([Vo], \S\S3-4). Put 
$M=A/I$, viewed as an $A$-bimodule, and let $X:= \Spec(\gr{A})$.
The filtration on $A$ induces a filtration
on $M$, and we  view $\gr{M}= 
\gr{A}/\gr{I}$ as a finitely generated $\gr{A}$-module.
Let $\mm$ denote the coherent sheaf on  $X$
corresponding to this  $\gr{A}$-module.
By definition we have: $\Supp\mm=\var(I)$,
and we must show that  this variety is irreducible.
Pick an irreducible component of   $\Supp\mm$ of maximal
dimension, $\dim(\Supp\mm)$. 
Corollary \ref{even_dim} says that there exists a 
 symplectic leaf $\sS\sset X$ such that  this irreducible component
is $\overline{\sS}$,
the Zariski closure of $\sS$.
Furthermore, by Lemma \ref{dixmier}, any imbedded component\footnote{
here  $\var(I)$ is viewed  as a scheme rather than as a reduced variety.}
of $\var(I)$
 (corresponding to an associated prime of $\gr I$)
 that has non-empty intersection with $\sS$ must
contain the whole of $\sS$.



A key point is that, since all  symplectic leaves
are even dimensional, we have
\begin{equation}\label{codim2}
\dim\bigl(\overline{\sS}\ssminus\sS\bigr)
\leq \dim\sS -2 = \dim(\Supp\mm) -2\,.
\end{equation}
The idea of the proof of Irreducibility Theorem given in ([Vo], \S4), that we will
follow closely, is to introduce a certain new filtration, $F_\bullet,$ on $M=A/I$
defined "locally" in terms of the stratum $\sS$, and such that
$\gr^{F}M$ gives rise to a coherent sheaf on  $\sS$. Then, since the
boundary of  $\sS$ has codimension two by \eqref{codim2},
the global sections of $\gr^{F}M$ over $\sS$ necessarily form
a {\it finitely generated} $\gr A$-module which is, moreover,
automatically supported on $\overline{\sS}$. Hence, the filtration
$F$ is good, and one obtains $\var(I)=\gsup(A/I)=\Supp(\gr^{F}A/I)
=\overline{\sS}$.


We proceed to a more detailed exposition.
First, Proposition \ref{finite_leaves} implies that
$\sS$ is Zariski-open in $\overline{\sS}$, hence also in $\Supp\mm$.
Therefore, $Z:=(\Supp\mm)\ssminus \sS$ is a Zariski closed subset in
 $X$. Thus $X\ssminus Z$ is a Zariski open
subset  in  $X$. 
We  may choose a smaller {\it affine} Zariski open
subset $U\subset X\ssminus Z$ such that $(\Supp\mm)\cap U= \sS$.
Write $\jmath: U \into X$
for the corresponding open imbedding.
Set $\mm_{_U}:=\mm|_U$, a coherent sheaf on $U$.
By construction, the leaf $\sS$ is closed in $U,$ and $\Supp\, \mm_{_U}=
\sS$. Hence, $\Supp(\jmath_*\mm_{_U})=
\overline{\sS}.$ Therefore,
 (\ref{codim2}) yields
$$
\dim\bigl(\Supp(\jmath_*\mm_{_U})\;\cap\;
(X\ssminus U)\bigr) \leq\dim(\Supp\,\mm_{_U}) -2\,.
$$
A standard result of algebraic geometry now says that
$\jmath_*\mm_{_U}$ is a coherent sheaf on $X$. Therefore, we have:
\begin{equation}\label{coh}
\G(U\,,\,\mm_{_U})=\G(X\,,\,\jmath_*\mm_{_U})
\quad\text{is a finitely generated $\gr{A}$-module}.
\end{equation}

We are going to apply microlocal techniques --
pioneered by Gabber [Ga1] --
 equivalent to the argument in
([Vo], \S3), that has been
 also inspired by Gabber. To this end, observe  that the
variety $X$ has a cone-structure, that is
a contracting $\C^*$-action induced by the 
(non-negative) grading on $\gr{A}$. Clearly,
$\var(I)$, hence $\sS\,,\,Z$,  are $\C^*$-stable 
subvarieties in~$X$. 

For any non-zero homogeneous element $f\in \gr{A}$,
let $\UU_{\!_{^f}}$
denote   the complement in
$X$ of the divisor $f=0$. Thus, $\UU_{\!_{^f}}$
is an affine Zariski open $\C^*$-stable subset of $X$. 
Let $\wh{A}_{\UU_{\!_{^f}}}$
 be     the formal micro-localization of
$A$ at $\UU_{\!_{^f}}$,   see e.g.  ([Gi], \S1.3).
Thus, $\wh{A}_{\UU_{\!_{^f}}}$    is a complete $\Z$-filtered ring 
such that $\gr(\wh{A}_{\UU_{\!_{^f}}}) =\C[\UU_{\!_{^f}}],$ 
the ring of regular functions
on the affine
open set $\UU_{\!_{^f}}$. The assignment:
$\UU_{\!_{^f}}\longmapsto   \wh{A}_{\UU_{\!_{^f}}}$ extends to
a sheaf $\UU\mapsto \wh{A}_{\UU}$
in
 the topology of Zariski open cone-subsets
in $X\ssminus\{0\}$. For any
Zariski open cone-subset $\UU$, we have an imbedding
$\gr(\wh{A}_{\UU}) \into \C[\UU],$
which becomes an equality if $\UU$ is affine.
There is
a canonical algebra imbedding $A\into \wh{A}_\UU$,
 which is strictly compatible with the filtrations
and such that the induced map
$\gr A \too \gr(\wh{A}_\UU)$ gets identified  with
restriction of functions: $\gr A=\C[X]\too\C[\UU].$

Further, given a finitely generated $A$-module $L$,
we let $\wh{L}_{\UU_{\!_{^f}}}=\wh{A}_{\UU_{\!_{^f}}}\otimes_A L$ denote its
formal micro-localization, a finitely generated
$\wh{A}_{\UU_{\!_{^f}}}$-module. The assignment: $\UU_{\!_{^f}}\mapsto \wh{L}_{\UU_{\!_{^f}}}$
extends to  a sheaf $\wh{L}$ in
 the topology of Zariski open cone-subsets
in $X\ssminus\{0\}$. One has a  canonical sheaf isomorphism
$\wh{L}\simeq \wh{A}\otimes_A L$, in particular, an isomorphism $\Gamma(\UU,
\wh{L})
\simeq \Gamma(\UU,
\wh{A})\otimes_A L$, for any {\it affine} open cone-subset
$\UU$. 

We now set $\UU:=U$, the affine Zariski open
subset in $X\ssminus
Z$ that has been chosen earlier, cf. \eqref{coh},
and which we now additionally assume to be a cone-subset.
We
apply the formal micro-localization at $U$ to our
$A$-module $M=A/I$, to get the left
$\wh{A}_U$-module $\wh{M}_U=\wh{A}_U\otimes_A (A/I)=
\wh{A}_U/\wh{A}_U\cd I$. We note that since $M$ has
also a right $A$-action (the ideal $I$ is a
two-sided ideal) the left $\wh{A}_U$-module $\wh{M}_U$
acquires a canonical  $\wh{A}_U$-bimodule structure.
Equivalently, one can get the same bimodule
by applying the formal micro-localization procedure
to $M$ viewed as a left $A\otimes A\opp$-module
(from this point of view, $\wh{M}_U$ is `supported' on the 
diagonal  $U \sset U\times U$,
since $M$, viewed as a $A\otimes A\opp$-module,
is clearly  `supported' on the 
diagonal  $X\sset X\times X$).
Further, we have a canonical
$A$-bimodule map
$$
i:\;M\too \wh{A}_U\otimes_A M=\wh{M}_U\,.
$$
Let $K$ denote the kernel of $i$ and $\overline{M}$ denote
the image of $i$. Both  $K$ and $\overline{M}$
are $A$-bimodules again, in particular,
 the preimage of $K$ under the projection
$A\onto A/I=M$ is clearly a two-sided ideal,
say $J\sset A$. Thus, we have the following diagram
\begin{equation}\label{micro_imbed}
{
\diagram
0 \rto & J/I\ddouble\rto & A/I\ddouble\rto&A/J\ddouble\rto&0\\
      & K\rto & M\rto &\overline{M}&\!\!\!\! \!\!\!\!\! \!\!\!
\stackrel{\imath}{\into}\;\wh{M}_U=\wh{A}_U\otimes_A M\,,
\hphantom{xx}
\enddiagram
}
\end{equation}
where we have factored the map $i$ as a composition:
$M\onto \overline{M}\stackrel{\imath}{\into}\wh{M}_U$.
The latter factorization shows that the imbedding
$\imath$ induces an isomorphism:
$\wh{A}_U\otimes_A\overline{M}\iso\wh{A}_U\otimes_A M$
 of the corresponding micro-localizations.
The  micro-localization functor being exact,
this yields $\wh{K}_U=\wh{A}_U\otimes_A~K=0$.
Hence $(\gsup K)\cap U=\emptyset$.


We claim that $K=0$. If not,
then the two-sided ideal  $J \sset A$, see (\ref{micro_imbed}),
properly contains $I$. Hence, Lemma \ref{GKdim}
yields $\dim(\gsup A/J) <\dim(\gsup A/I)$.
But since $\overline{\sS}$ is a component
of $\gsup A/I$ of maximal dimension,
we get: $\dim\overline{\sS}>\dim(\gsup A/J)$.
Therefore, the equality of sets:
$\gsup A/I= (\gsup J/I)\;\cup\;(\gsup A/J),$
which follows from 
the short exact sequence in the top row of (\ref{micro_imbed}),
cannot be achieved unless $\sS\sset \gsup J/I= 
\gsup K$. This last inclusion contradicts
the conclusion of the previous paragraph,
saying that $(\gsup K)\cap U=\emptyset$.
 Thus, $K=0$ and our claim follows.

To proceed further, observe that
the module $\wh{M}_U$ comes equipped with
a natural increasing filtration $F_\bullet\wh{M}_U$
obtained, essentially, by localizing the standard filtration on $M=A/I$.
Moreover, for the associated graded module corresponding
to the  filtration  $F_\bullet\wh{M}_U$ one has
a canonical  isomorphism $\gr^{\!F}(\wh{M}_U)
\iso \G(U\,,\,\mm_{_U}),\,$ since
 $U$ is  affine. We see that 
$\gr^{\!F}(\wh{M}_U)$ 
is a finitely generated $\gr{A}$-module, due to (\ref{coh}).
Further, the increasing filtration on $\wh{M}_U$
induces a filtration $F_\bullet{M}=
M\;\cap\;F_\bullet\wh{M}_U$ via the  imbedding $i: M\into \wh{M}_U$
(we have shown that $i$ is injective).
Then $\gr^{\!F}M$ is a $\gr{A}$-submodule of 
 $\gr^{\!F}(\wh{M}_U)$, hence is also finitely
generated. Therefore the filtration
$F_\bullet{M}$ is a {\it good} filtration on $M$.
This yields $\gsup M =\Supp\bigl(\gr^{\!F}M\bigr)$,
and we deduce
$$
\sS\;\sset\;\var(I)\;=\;\gsup M\;=\;\Supp\bigl(\gr^{\!F}M\bigr)
\;\sset\;\Supp\bigl(\gr^{\!F}\wh{M}_U\bigr)\;\sset\;
\Supp\bigl( \jmath_*\mm_U\bigr)\; \sset\; \overline{\sS}\,.
$$
It follows that $\var(I)=\overline{\sS},$
and  the Theorem is proved.\hfill\qed

\section{Basic properties of categories $\ou$ and $\od$}
\setcounter{equation}{0}
In this section we prove a few technical lemmas that will be of use
later. 

Given a 
 $\bh$-action on a vector space
$V$ 
 and $\mu\in\C$, for the generalized $\mu$-eigenspace of $\bh$
 write $V_\mu:=
\{v\in V\enspace|\enspace (\bh-\mu\cdot\id)^N\cdot
v=0\enspace\text{for}\enspace
 N\gg 0\}.$ If the action
on $V$ is locally-finite then there is a (possibly infinite)
direct sum decomposition $V=\oplus_{\mu\in\C}\;V_\mu.$

Let $(A^\pm,\bh)$ be a
triangular structure on $A$.
We consider the following statements about an $A$-module $M$:
\begin{equation}\label{item}
{
\parbox[t]{135mm}{
{\sf 1.}\quad
$M$ {\it is finitely generated over $A^+$. Furthermore, for any
good filtration  on $M$ \hphantom{x}\;\;\quad as an $A$-module, 
$\gr M$ is
finitely-generated over $\gr(A^+)$}.\newline
{\sf 2.}\quad{\it The $\bh$-action on $M$ is locally-finite
and each generalized $\bh$-eigenspace $M_\mu$ is \hphantom{x}\;\;\quad
finite-dimensional.}\newline
{\sf 3.}\quad{\it 
The real parts of the eigenvalues of $\bh$-action in $M$
are all bounded from below.}
}}
\end{equation}
\noindent
 It will be convenient below
to introduce the notation $\,\bA^\pm:=A^\pm +\C\cdot 1 \subseteq
A$ and, for $m\geq 1$,  put  $(A^-)^m:=
A^-\cdot\ldots\cdot A^-$ ($m$ factors).

\begin{lemma}\label{fin1} 
Let $M$ be an $A$-module generated by a finite-dimensional
subspace $M_0\sset M$ such that, for some $m\geq 1$, we have:
 $(A^-)^m\cdot M_0=0.$ Then properties \eqref{item}(1)-(3) hold.
\end{lemma}

\begin{proof}  We start with statement \eqref{item}(2).
Properties (T0)-(T1) imply that there exists an $\ad\bh$-stable
finite-dimensional
$\C$-vector subspace $E\subset A$ such that $A= \bA^+\cdot E\cdot
\bA^-$.
Furthermore, looking at the weights of  $\ad\bh$-action on $A$,
for the centralizer of $\bh$ in $A$ we find:
\begin{equation}\label{ofer}
A^{\ad\bh}\enspace \bigl(:= \{a\in A\enspace\big|\enspace [a,\bh]=0\}\bigr)\;=\;
\sum\nolimits_{\beta,\gamma\in\C}\;
(\bA^+)_{-\beta-\gamma}\cdot E_\beta\cdot(\bA^-)_\gamma\,,
\end{equation}
where $(-)_\beta$ denotes $\ad\bh$-eigenspace in $(-)$ with the eigenvalue $\beta$.

Observe further that, for any integer $m>0$, the space $(A^-)^m$
 has finite codimension in $A^-$
since
a similar result holds for the enveloping algebra
of a (finite-dimensional) nilpotent Lie algebra. It follows that, for
all $\gamma\in\C$ except possibly a finite set, one has:
$(\bA^-)_\gamma \sset (A^-)^m$. Further, the algebra $A^+$ being finitely-generated,
each eigenspace $(A^+)_\alpha$ is finite-dimensional,
by (T3). Therefore, \eqref{ofer} implies 
that there exist finite-dimensional subspaces $B_m\subset A^\bh$
such that
$$A^{\ad\bh} = B_m \,+\, A^{\ad\bh}\cap\bigl(\bA^+\cdot E\cdot
(A^-)^m\bigr)\quad,\quad
m=1,2,\ldots \,.$$
Hence,  since $(A^-)^m\cdot M_0=0$, we get
$A^{\ad\bh}\cdot M_0=B_m\cdot  M_0,$  is a finite dimensional space.
The subspace $A^{\ad\bh}\cdot M_0\sset M$ being  clearly $\bh$-stable, we conclude
that
$M$ is generated as an $A$-module by the  $\bh$-stable
finite dimensional subspace $A^{\ad\bh}\cdot M_0\sset M$. From
$M= A\cdot M_0\sset
A\cdot (A^{\ad\bh}\cdot M_0)$ we deduce that  the  $\bh$-action on $M$ is locally-finite,
since $\ad\bh$-action on $A$ is locally-finite. 

To prove statement \eqref{item}(3), 
choose  $B$, a complementary subspace  to $(A^-)^m$
in $\bA^-$
so that $\bA^-=B\oplus (A^-)^m$. Then we
have: $M= A\cdot M_0= \bA^+\cdot E\cdot\bigl(B\oplus (A^-)^m\bigr)
\cdot M_0=\bA^+\cdot E\cdot B\cdot M_0.$
Further, the space $E\cdot B\cdot M_0$ is
 finite-dimensional, and the action of
$A^+$ may only increase  the real parts of the eigenvalues of
$\bh$-action in $M$. It follows that
 the real parts of the eigenvalues of $\bh$-action in $M$
are all bounded from below, and \eqref{item}(3) is proved.


Next, since each $\ad\bh$-eigenspace in $A^+$ is
finite-dimensional,
we deduce from the equality $M=\bA^+\cdot (E\cdot B\cdot M_0)$ that
 each $\bh$-eigenspace in $M$ is
finite-dimensional. This completes the proof of \eqref{item}(2).
The same  equality clearly implies that $M$  is
finitely generated over $A^+$. 

To complete the proof of \eqref{item}(1)
we recall, see e.g. \cite[Corollary 2.3.19]{CG},
 the following well-known fact:
if for {\it some} good filtration
on $M$, the associated graded module $\gr M$ is finitely-generated over
$\gr A^+$, then the same holds for {\it any} good filtration
on $M$. Thus, we may prove the result for a particular
  good filtration on $M$ of our choice. To this end, observe
that $M$ is, by definition, a quotient of the direct sum
of finitely many copies of the $A$-module $A/A\cdot(A^-)^m$.
We equip $A/A\cdot(A^-)^m$ with the filtration inherited from
the standard filtration on $A$. Further,  equip the direct sum
with the direct sum filtration, and choose the filtration on
$M$ to be the induced quotient-filtration.
This way, we see that $\gr M$ is a quotient 
 of the direct sum
of finitely many copies of the $\gr A$-module 
$(\gr A)\big/(\gr A)\cdot\gr (A^-)^m$.
Thus, we are reduced to showing that $(\gr A)\big/(\gr A)\cdot(\gr
A^-)^m$
is a finitely generated $\gr A^+$-module.
But this is clear from property $(T0')$,
 saying that $\gr A$ is a finitely generated $(\gr A^+)\td(\gr
A^-)$-bimodule.~\end{proof}

Recall the
categories $\ou$ and $\od$ from Definition \ref{oo}. 


\begin{corollary}\label{fin2} An $A$-module $M$ is 
 generated by a finite-dimensional
subspace $M_0\sset M$ such that
 $(A^-)^m\cdot M_0=0$, for some $m\geq 1,$ if and only if $M\in\ou$.
\end{corollary}
\begin{proof} Assume that $M$ is generated by $M_0$, as above.
Then Lemma \ref{fin1} implies that statement
\eqref{item}(3) holds. Hence,
since the $A^-$-action decreases  the real parts of the eigenvalues of
$\bh$-action in $M$, we conclude that the  $A^-$-action
on $M$ is locally-nilpotent. 

Conversely, any object of $M\in\ou$ is generated by
a finite-dimensional
subspace $M_0\sset M$, and the action of  $A^-$ on $M$
is  locally-nilpotent. Hence, there exists $m\gg 0$ such
that $(A^-)^m\cdot M_0=0.$
\end{proof}

\begin{corollary}\label{fin3} The category $\ou$ is abelian, and
properties \eqref{item}(1)-(3) hold for
any object $M\in\ou$.
\end{corollary}
\begin{proof} The last claim is immediate from 
   Lemma \ref{fin1} and
Corollary \ref{fin2}. In particular, we conclude that any object $M\in\ou$
is finitely generated over $A^+$, due to property \eqref{item}(1).
Now, $A^+$, being a quotient of
$\n^+\cdot\U\n^+$, is a Noetherian algebra. Therefore, any $A$-submodule
of $M$ belongs to $\ou$. Hence, the category $\ou$ is abelian.
\end{proof}

\begin{lemma}\label{isoclass}
The $A$-module $A/A\cd\n^-$ is an object of $\ou$, and any
simple object of $\ou$ is isomorphic to a quotient of  $A/A\cd\n^-$.
\end{lemma}
\begin{proof} The module  $A/A\cd\n^-$ is generated over $A$ by a single vector
(= coset of $1\in A$),
which is annihilated by the algebra $A^-$. Hence, Corollary \ref{fin2}
yields $A/A\cd\n^-\in \ou.$

Now,
given a simple object $L\in \ou$,
we may choose a non-zero element
$v\in L$ in the generalized $\bh$-eigenspace
corresponding to an eigenvalue with minimal possible real part.
Commutation relations show that the action
of any element of $\n^-$
decreases the  real part of the eigenvalue. Hence, we must have:
$\n^-\cdot v= 0$.
Therefore, the assignment: $a \mapsto a(v)$
gives a well-defined $A$-module map
$A/A\cdot\n^- \to  L$. This map is nonzero, hence, surjective
since $L$ is simple.~\end{proof}

\begin{remark}
Analogues of the above results
 hold also for the category
$\od$.
$\lozenge$
\end{remark}
\smallskip


Given a (possibly infinite dimensional) vector space $V$
with  locally
finite $\bh$-action such that all  $\bh$-eigenspaces $V_\la$
are finite-dimensional,
let $V^\star= \bigoplus_\la\;(V_\la)^*\sset \Hom_{_\C}(V,\C)$ denote the direct sum of 
the spaces dual to $V_\la$'s. If $V$ is a left
$A$-module, then  $V^\star$ is an $A$-stable subspace
in the right $A$-module $\Hom_\C(V,\C)$
(= linear dual of $V$). The right  $A$-module $V^\star$ is
called the  {\it restricted dual} of $V$.
There is
 a canonical  $A$-module isomorphism: $V\iso (V^\star)^\star$.


\begin{lemma}\label{dual}
Assume that any object of either $\ou$ or $\od$ has finite length.
Then the restricted duality functor
$M\mapsto M^\star$
induces anti-equivalences: $\ou \,
\stackrel{_\sim}{\longleftrightarrow}\, \od.$~\qed
\end{lemma}
\begin{proof} 
An object  has
finite length if and only if it satisfies both ascending and descending
chain conditions.
Restricted duality clearly interchanges 
these two conditions. Thus, $M^\star$  has
finite length if and only if so does $M$.


Further, it is clear that
the real parts of the eigenvalues of $\bh$-action in $M$
are bounded from below if and only if 
the real parts of the eigenvalues of $\bh$-action in $M^\star$
are bounded from above. Thus, it follows
by Corollary \ref{fin2} and Lemma \ref{fin1} that   $M\in \ou$
 if and only if $M^\star\in\od$.
\end{proof}

\begin{corollary}\label{both}
The equivalent 
conditions \eqref{cat} hold for the category $\ou$ if and only if they
hold for  the category $\od.$~\qed
\end{corollary}



\noindent
{\bf Proof of Theorem \ref{criterion}.\;} Part (i) of the
Theorem 
is immediate from
Corollary \ref{fin2} and Lemma \ref{fin1}. 
To prove part (ii), 
 fix $M\in \ou$ and a simple object  $L\in \ou$. We claim that
 there exists an integer
$a({M:L})$ such that the following holds:
\vskip -9mm
\begin{align}\label{text}
&\text{\it For any  descending, resp. ascending, chain of
subobjects: $M=M^0\supset M^1\supset \ldots\supset M^N$,}\nonumber\\ 
&\text{\it  let
$I$ be the set formed by the indices `$i$' such that $M^i/M^{i+1}$
contains  $L$ as}\\
&\text{\it a subquotient. Then, one has}:
 \,\#I\leq a({M:L}).\nonumber
\end{align}
To prove the claim, choose  $\mu\in\C$
such that the eigenspace $L_\mu$ is non-zero.
Then, we clearly have
$$\dim M_\mu\, =\, \sum\nolimits_{i=1}^N\,\, \dim(M^i/M^{i+1})_\mu
\,\leq\, \sum\nolimits_{i\in I}\,\, \dim(M^i/M^{i+1})_\mu\,\leq\, \#I\cdot \dim
L_\mu\,.
$$
Since $\dim L_\mu\neq 0$, and $\dim M_\mu <\infty$
by part (i) of the Proposition, we deduce:
$\#I\leq \dim M_\mu/\dim L_\mu$. Thus, we can take $a({M:L})$ to
be the integral part of
$\,(1+\dim M_\mu/\dim L_\mu).$

Now, assume that   $\{L_1,\ldots,L_d\}$ is  a complete {\it (finite)} collection
of isomorphism classes of simple objects of $\ou$. To prove that
an object $M\in\ou$ has finite length,
one has to verify both ascending and descending chain conditions
for $M$. Consider, for instance, a descending chain of
subobjects:
$M=M^0\supsetneq M^1\supsetneq \ldots\supsetneq M^N$.
We must show that
the length of this chain
is bounded from above by a certain number $a=a(M)$ which is independent 
of the chain.
But this is clear from Claim \ref{text}, since for each $i=1,\ldots,N,$
the non-zero object $M^i/M^{i+1}$ must contain
at least one of the simple objects $L_r\,,\,r=1,\ldots,d,$
as a subquotient. Therefore, $N\leq \sum_{r=1}^d\,a({M:L_r})$,
and we may put $a(M):=\sum_{r=1}^d\,a({M:L_r})$.

To complete the proof of part (ii) of the Theorem,
assume any object of $\ou$ has finite length.
Then, the module $A/A\cdot\n^-$ has finite length by   Lemma
\ref{isoclass},
hence it follows from the Lemma that the number
of isomorphism classes of simple objects in $\ou$ is finite.

To prove part (iii), write again  $\{L_1,\ldots,L_d\}$ for  a complete  collection
of isomorphism classes of simple objects of $\ou$. We know that for each
$r=1,\ldots,d$
there is a real number $\lambda_r\in\mathbb{R}$ such that the real
parts of $\bh$-eigenvalues in $L_r$ are all $\geq \lambda_r$.
It follows  that, for   {\it any} object $M\in\ou$, the real
parts of $\bh$-eigenvalues in $M$
 are all $\geq {\mathtt{min}}(\lambda_1,\ldots,\lambda_d).$
Thus, since $A^-$-action decreases  the real
parts of $\bh$-eigenvalues by at least $1$, we observe
that, for each $\mu\in \C$ there exists an integer $n(\mu)>0$ such
that we have:
\begin{equation}\label{nmu}
(A^-)^{n(\mu)}\cdot M_\mu=0\quad,\quad\forall M\in \ou\,.
\end{equation}

Given  $\mu\in \C$, for each $k=1,2,\ldots,$ we put
$$
%\begin{equation}\label{J}
J(\mu,k):= A\cdot (\bh-\mu)^k + A\cdot(A^-)^{n(\mu)} \,.
$$
%\end{equation}
Clearly  $J(\mu,1)\supseteq J(\mu,2)\supseteq \ldots, $ is a
 descending chain of left ideals in $A$.
We claim that this chain stabilizes, i.e., there exists $\ell\gg 0$
such that we have: $J(\mu,\ell)=J(\mu,\ell+1)$ $=\ldots.$ 
To prove the claim, observe first
that the left $A$-module
$A/A\cdot(A^-)^{n(\mu)}$ is an object of $\ou$,
by Corollary \ref{fin2}. Hence, this object has finite length,
by part (ii) of the Theorem. 
Now, for
 $k=1,2,\ldots,$ let $N^k\subset A/A\cdot(A^-)^{n(\mu)}$ be 
an $A$-submodule generated by the coset $(\bh-\mu)^k\;{\sf
{mod}}\,\bigl(A\cdot(A^-)^{n(\mu)}\bigr).$
Clearly $N^1\supseteq N^2\supseteq \ldots.$ We conclude that this
chain of submodules must stabilize, hence there exists $\ell\gg 0$
such that the canonical  projections $
\bigl(A/A\cdot(A^-)^{n(\mu)}\bigr)\big/N^\ell\twoheadleftarrow
\bigl(A/A\cdot(A^-)^{n(\mu)}\bigr)\big/N^{\ell+1}\twoheadleftarrow\ldots$
are all bijections. But these projections are nothing but the
 canonical  projections $A/J(\mu,\ell)\twoheadleftarrow
A/J(\mu,\ell+1)\twoheadleftarrow\ldots,$
and our stabilization claim follows.

We put $P(\mu):= A/J(\mu,\ell)$. The $A$-module $P(\mu)$ is
a quotient of $A/A\cdot(A^-)^{n(\mu)}$, hence,  an object of $\ou$.
We claim\footnote{
this observation has been used earlier, see [Gu].} that 
$P(\mu)$ is a projective in $\ou$. To see this,
let $M\in \ou$ and fix $v\in M_\mu$. Then $(A^-)^{n(\mu)}\cdot v=0$,
due to \eqref{nmu}.
Furthermore, there exists $k$ large enough, such that
$(\bh-\mu)^k\cdot v=0$. Therefore, $J(\mu,k)\cdot v=0$,
hence $J(\mu,\ell)\cdot v=0$, by the  stabilization claim proved above. Therefore,
the assignment $a\mapsto a(v)$
descends to a well-defined  $A$-module map $\varphi_v: P(\mu)=A/J(\mu,\ell)
\too M$.
Thus, we get a canonical isomorphism
\begin{equation}\label{functor}
 M_\mu \iso \Hom_A\bigl(P(\mu), M)\quad,\quad v\longmapsto
\varphi_v\,.
\end{equation}
We conclude that the functor: $\ou\too \text{\it Vector
spaces}\;,\;
M\mapsto M_\mu$, which is clearly exact,
 is represented by the object $P(\mu)$. Hence,
$P(\mu)$ is projective.

To complete the proof, let $L\in\ou$ be a simple object.
Find $\mu\in \C$ such that $L_\mu\neq 0$. Then 
\eqref{functor} implies that there is a nonzero $A$-module
map $P(\mu)\to L$. This map is surjective, since $L$ is simple.
Thus, we have shown that any 
 simple object in $\ou$ is isomorphic to a quotient of 
$P(\mu)$, for an appropriate $\mu\in\C$.
 It follows that the category $\ou$ has enough projectives.
Similarly, one proves that  the category $\od$ has enough projectives
(using Lemma \ref{dual}). But then the equivalence of
Lemma \ref{dual} implies that  category $\ou$ has enough injectives,
as well.
\qed\medskip 


Now, fix an associative algebra $A$  with triangular
 structure $(A^\pm,\bh)$.
Let $\oa$ be the category of  finitely-generated
left $A\otimes A\opp$-modules $M$ such that the action
on $M$ of the subalgebras
$A^-\otimes 1\sset A\otimes A\opp$ and $1\otimes (A^+)\opp\sset A\otimes A\opp$
are both locally-nilpotent.


\begin{lemma}\label{triv}
If conditions \eqref{cat} (i.e., the assumptions of  Duflo Theorem) hold
then

\vi The category $\oa$ is  abelian and any  object $M\in \oa$ has finite length.

\vii Any  simple object
of $\oa$ has the form
$L= L'\boxtimes L'',$
where $L'$ is a simple object in $\ou$
and $L''$  is a simple object in~$\od$.


\viii
The action of the element
$\bh\otimes 1 + 1\otimes\bh$ on any object $M\in \oa$
is  locally-finite.
\end{lemma}


\begin{proof} First, an argument  very similar
to the proof of Lemma \ref{fin1} shows that analogues of
statements \eqref{item}(1)-(3) hold for any object
of $\oa$. Specifically, we leave to the reader to adapt the arguments that
have been used above to prove
that 
\begin{itemize}
\item The   action of the element
$\bh\otimes 1 + 1\otimes\bh$ on any object $M\in \oa$
is  locally-finite.
\item Each 
 generalized eigenspace of the $
(\bh\otimes 1 + 1\otimes\bh)$-action in  $M\in \oa$
 is finite dimensional, and
 the real parts of the  generalized eigenvalues are all  bounded from
below.
\item For any good filtration on an
 $A\otimes A\opp$-module
 $M\in \oa$, the module
$\gr M$ is finitely-generated over $\gr(A^+)\otimes \gr(A^-)\opp$.
\end{itemize}

Thus, part (iii) of the Lemma follows.
We now prove part (ii). Fix a simple object $L\in \oa$.
Choose a non-zero element
$v\in L$ in the generalized $\bh$-eigenspace
corresponding to an eigenvalue with minimal possible real part.
Commutation relations show that the action
of any element of either $\n^-\otimes 1$ or $1\otimes \n^+$
decreases the real part of the  eigenvalue. Hence, we must have:
$(\n^-\otimes 1)v= 0$ and $(1\otimes \n^+)v= 0$.
Therefore, the assignment
$\dis a'(v')\,\boxtimes\, a''(v'')
\;\longmapsto \;(a'\otimes a'')(v)$
gives a well-defined $A\otimes A\opp$-module map
$\pi: L'\boxtimes L'' \too L$,
where we put
$L':= A/A\cd\n^-$ and  $L'':=A\opp/A\opp\cd\n^+$,
and where $v'\in L'\,,\, v''\in L''$ denote the  canonical
generators (corresponding to the coset of $1\in
A$). The map $\pi$ is nonzero, hence surjective,
since $L$ is simple.


Further, we know
  that
$L'\in \ou$ and $L''\in \od$ (by Lemma \ref{isoclass} and its
$\od$-analogue).
By our assumptions, all objects of $\ou$ and $\od$
have finite length, cf. Corollary \ref{both}. Therefore both $L'$  and $L''$
have finite Jordan-H\"older
series with simple subquotients, say,
$L'_1,\ldots, L'_p$, and $L''_1,\ldots, L''_q,$ respectively.
It is clear from Jacobson Density theorem that,
for any $i\in [1,p]\,,\,j\in [1,q],$
the
$A\otimes A\opp$-module $L'_i\boxtimes L''_j$ is simple.
Hence,  $L'\boxtimes L''$  also has a finite Jordan-H\"older
series (as an
$A\otimes A\opp$-module) with simple subquotients of the form
$L'_i\boxtimes L''_j.$
It follows that $L$, being a simple quotient of
 $L'\boxtimes L''$, must be isomorphic to some
simple subquotient of the Jordan-H\"older
series for $L'\boxtimes L''$, that is to some  $L'_i\boxtimes L''_j$.
Part (ii) follows.

We see from part (ii) that the category $\oa$ has finitely many
isomorphism classes of simple objects.  
Then, mimicing the proof of Theorem \ref{criterion}(ii),
one deduces that any object in $\oa$ has finite length.
This completes the proof of part (i).
\end{proof}





\section{Jacquet functor and Proof of Theorem \ref{duflo}}
\setcounter{equation}{0}
We review the construction of Jacquet functor following 
[Ca] and [CO].

Let $A$ be an associative algebra and 
${\mathfrak  r}\sset A$ a 
finite dimensional Lie subalgebra (with respect to the
commutator bracket). Assume further that ${\mathfrak
r}$ has the form of a semi-direct product
  ${\n} \rtimes \C \cd \delta$
such that

\noindent
$\vi\quad$ The adjoint  action of ${\n}$ on $A$
 is locally-nilpotent and

\noindent
$\vii\quad$
$\ad\delta$-action on $A$ is locally-finite, and
all eigenvalues of $\ad\delta$-action on $\n$ are in $\Z_+$.


Note that the second condition implies
that $\n$ is a nilpotent Lie algebra.
Let $\U_+$ denote the associative subalgebra  (without unit) in $A$
generated by $\n$. Thus, $\U_+$ is a quotient
of  the augmentation ideal in the enveloping
algebra $\U\n$. 



For each $k\geq 1$, let $\U_+^k$ be the $k$-th power of $\U_+$.
These powers form a descending
chain of subalgebras $\U_+\supset \U_+^2 \supset\ldots,$ such that
$\cap_{k\geq 1}\,\U_+^k=0$. Note that condition (i) above implies
that, for any $a\in A$, there exists $n=n(a)\in\Z$ such that
\begin{equation}\label{continuous}
a\cdot\U_+^k\sset \U_+^{k+n}\cdot A\quad,\quad\forall k\gg 0\,.
\end{equation}
Recall further that for any left $A$-module $M$,
the linear dual $M^*=\Hom_{_\C}(M,\C)$ has a natural
right $A$-module structure.
Given a {\it left} $A$-module $M$ we define
\begin{align}\label{Jdual}
\j^\dag(M)\,:=\,\lim_{\stackrel{\longrightarrow}{k}}\,& \bigl(M/\U_+^k\cd
 M\bigr)^*\\
&=\{m^*\in M^*\;\mid\; m^*\bigl(\U_+^k\cdot M\bigr)=0\enspace
\text{for some}\enspace k=k(m^*)\gg 0\}\,.\nonumber
\end{align}
Formula (\ref{continuous}) insures that $\j^\dag(M)$ is an
$A$-submodule of the right $A$-module $M^*$. Moreover,
it is clear that the action on $\j^\dag(M)$
of the Lie subalgebra $\n\sset A$ is locally nilpotent.
Thus, we get
a functor $\j^\dag:\,\text{\it left $A$-modules}\too
\text{\it right $A$-modules, locally-nilpotent relative to $\n$}$.

It is convenient to get the following slightly different interpretation
of this functor. Observe that formula (\ref{continuous}) 
implies
 that the multiplication map: $ A\stackrel{a\cdot}{\too}A$
(by any element   $a\in A$)
is continuous in the topology on $A$ defined by
the set $\{\U_+^k\cdot A\}_{k\geq 1}$ of fundamental neighborhoods of
zero. Thus, the completion
$\hA=\lim\limits_{\stackrel{\longleftarrow}{k}}\, A/\U_+^k\cd A$
acquires the structure of a complete topological algebra.
There is a canonical algebra map:
$A\to \hA$ with dense image.
Similarly, for any left $A$-module $M$, the completion
\begin{equation}\label{Jhat}
{\widehat \j}(M)
= \lim_{\stackrel{\longleftarrow}{k}}\;
M/ \U_+^k\cd M
\end{equation}
acquires the structure of a complete topological 
left $\hA$-module. 
A non-commutative analogue of the
standard Artin-Rees lemma implies that, on the category of
finitely-generated
$A$-modules, the  functor
$M\mapsto {\widehat  \j}(M)$ is  exact.
Furthermore, if $M$ is finitely generated over
$A$, then  one has
a canonical surjective morphism  $\hA\otimes_{_A}M\onto
{\widehat  \j}(M)$, see \cite[ch.10]{AM}.
It is clear that $\j^\dag(M)$ is
nothing but the {\it continuous dual} of
${\widehat  \j}(M)$.

The  object $\widehat \j(M)$
is too
large to
be finitely generated over $A$. Using the element
$\bh\in A$, that has not played any role so far,
one  may do better. 

\noindent
{\bf Notation.\;} Given $\la\in \C$ and a locally-finite
$\bh$-action on a vector space $V$, we write
$[V]_\la $ for the generalized 
$\delta$-eigenspace  corresponding to the eigenvalue $\la$.
We will often view $A$-modules as either
$\U\r$- or $\U\n$-modules via the homomorphism
$\U\r\to A$ induced by the imbedding $\r\into A$.

Let $\Amod$ be the category of left $A$-modules which
are finitely generated as $\U_+$-modules. This is
an abelian category since $\U_+$, being a quotient of
the augmentation ideal in $\U\n$, is Noetherian.
Notice that for any $  M\in \Amod$ and any $k\geq 1$, 
the space $M/ \U_+^k\cd M$ is finite  dimensional.
 The action of   $\delta\in A$ on $M$
induces a $\delta$-action on each finite dimensional space in the
inverse system
\begin{equation}
\label{inv_sys}
M/\U_+\cd M\;
\longleftarrow\; M/\U_+^2\cd M \;   \longleftarrow
\;M/\U_+^3\cd M\; \longleftarrow \;\ldots
\end{equation}
One deduces from the positivity of $\;\ad\delta$-eigenvalues on ${\n}$
that, for each $\lambda \in \C$, the corresponding inverse system
of generalized 
$\la$-eigenspaces: $\,\big[M/\U_+^k\cd M\big]_\lambda$
 {\sl stabilizes},
i.e.
the projections in \eqref{inv_sys} induce isomorphisms:
$\displaystyle \,\big[M/\U_+^{k+1}\cd M\big]_\lambda
\iso\big[M/\U_+^k\cd M\big]_\lambda,\,$ for all $k$ sufficiently large.
\begin{definition}\label{jacquet}
 Let
 $\j(M)=\bigoplus_{\la\in\C}\; \big[\lim\limits_{\stackrel{\longleftarrow}{k}}\;
M/ \U_+^k\cd M\big]_\lambda$ be the 
direct sum of all
such ``stable'' generalized $\delta$-eigenspaces,
to be called the Jacquet module for $M$.
\end {definition}
It is clear that $\j(M)$ may be identified with
the direct sum of all generalized 
$\delta$-eigenspaces in $\widehat \j(M)$. Thus, we have


\noindent
$(1)\quad$ $\j(M)$ is an $A$-submodule in $\widehat \j(M)$ which
is dense in $\widehat \j(M)$ in the $\U_+$-adic topology.



\noindent
$(2)\quad$ $\delta$-action on $\j(M)$ is locally finite, with
finite-dimensional
generalized eigenspaces.

\noindent
$(3)\quad$  $\j(M)$ is finitely-generated over $\U_+$, that is,
$\j(M)\in \Amod$.

\noindent
$(4)\quad$ The set of $\delta$-eigenvalues in $\j(M)$ is
bounded from below by some constant $
\lambda(M) \in \C$.

\noindent
$(5)\quad$ $\j^\dag(M)=\bigl(\j(M))^\star$ is the  restricted dual of  $\j(M)$.
\smallskip

\noindent
Properties (1) and  (2) above follow from the stabilization of
eigen-spaces in the inverse system (\ref{inv_sys}).
To prove (3), let $N_0\subsetneq N_1\subsetneq 
\ldots $ be a strictly  ascending chain of $\U_+$-submodules
in $\j(M)$.
Property (2) implies that, for each $i\ge 0$, there exists $\la(i)\in\C$
such that $\big[N_i\big]_{\la(i)} \neq \big[N_{i+1}\big]_{\la(i)}$.
It follows that for the corresponding closures in
$\widehat \j(M)$ we have:
$\overline{N}_i\neq \overline{N}_{i+1}$.
Thus, $\overline{N}_0\subsetneq 
\overline{N}_1\subsetneq 
\ldots $ is  a strictly ascending chain of $\widehat \U_+$-submodules
in $\widehat \j(M)$. But this contradicts the fact
that $\widehat \j(M)$ is a  Noetherian  $\widehat \U_+$-module
(the latter holds since
$M$ is finitely-generated
over $\U_+$, hence $\widehat \j(M)$ is finitely-generated
over $\widehat \U_+$). Thus, $\j(M)$ must be a finitely-generated
$\U_+$-module, and (3) is proved.
Property (4) follows from (3), and property (5) is clear from
(2) and (4). 
\smallskip


Since taking (generalized) 
eigenspaces is an exact functor, we conclude that the assignment
 $M\mapsto \j(M)$ gives an exact functor $\j: \Amod\to\Amod,$
called the Jacquet functor.
\smallskip



Next, let $(A^\pm,\bh)$ be a 
triangular structure on $A$.
We have  the corresponding categories  $\ou,\od$.
We also consider the category $\Amod$ for
the subalgebra $\U_+:=A^+.$

\begin{lemma}\label{duality}
 \vi The  Jacquet functor gives
an exact functor $\j:\;\; \Amod\too \ou$. 

\vii If conditions \eqref{cat} hold, then 
the functor $\j^\dag$ gives an
exact functor:\linebreak
 $\Amod\too \od$ and, moreover, $\j^\dag(M)=\bigl(\j(M)\bigr)^\star,$
for any $M\in\Amod$.
\end{lemma}

\begin{proof} To prove (i) we note that for any $M\in \Amod,$
the $A^-$-action on  $\j(M)$ is 
 locally-nilpotent since
the set of the real parts of $\delta$-eigenvalues in 
$\j(M)$ is
bounded from below by some constant $\lambda(M) \in \mathbb{R}$.
We know also that $\j(M)$ is 
 finitely-generated over $A^+$ (property (3) above), hence, over $A$.
Thus, $\j(M)\in\ou$,
and (i) is proved. 

Now, Lemma \ref{dual} yields
$\bigl(\j(M)\bigr)^\star\in \od$. But  property (5) above
says: 
 $\j^\dag(M)\simeq\bigl(\j(M)\bigr)^\star.$
This
completes the proof of part (ii).~\end{proof}
\smallskip


Associated  with the triangular
 structure $(A^\pm,\bh)$ on $A$, we have the subalgebra
$A^+\otimes (A^-)\opp\sset A\otimes A\opp$.
We introduce an abelian category $\AAA$  of 
left $A\otimes
A\opp$-modules
as follows.
If the  triangular
 structure is {\it commutative}, we let $\AAA$ be the category of
 $A\otimes A\opp$-modules
which are finitely generated over the subalgebra
$A^+\otimes (A^-)\opp$. Thus, an object of
$\AAA$ is an $(A\td A)$-bimodule, finitely-generated
as an $(A^+\td A^-)$-bimodule.


If  the  triangular
 structure on $A$ is {\it non-commutative}, then 
the imbedding $A^+\otimes (A^-)\opp$
$\into A\otimes  A\opp$ induces a
 graded algebra homomorphism:
$\gr A^+ \otimes \gr A^- \into\gr(A\otimes A\opp) $.
We define  $\AAA$ to be the category of 
finitely-generated left $A\otimes A\opp$-modules $M$ 
such that $\gr M$ is a finitely
generated   $(\gr A^+)\otimes (\gr A^-)$-module,
for any {\it good} increasing
filtration on $M$. 
In particular, any object of $\AAA$ is a  finitely-generated 
$A^+\otimes (A^-)\opp$-module.

Recall  that, given a non-commutative triangular structure on $A$,
 we have the Lie subalgebras $\n^\pm\sset A^\pm$.
In case of a {\it commutative} triangular structure  on $A$,
we let $\n^\pm$ be any $\bh$-stable finite-dimensional
subspace of $A^\pm$ that generates  $A^\pm$ as an associative
algebra. We then regard  $\n^\pm$ as abelian  Lie subalgebras in $A$.
Thus, for  any  (commutative or  non-commutative) triangular structure on
$A$, we may consider the Lie algebra
 $\,\n^+\oplus\n^-=
\n^+\otimes 1 + 1\otimes \n^-\sset A\otimes A\opp$.
Therefore,
 there is a well-defined Jacquet
functor on the category $\AAA\,$ (relative
to the Lie algebra $\n^+\oplus\n^-\sset  A\otimes A\opp,$
and the element $\bh\otimes 1 + 1\otimes\bh$).
Furthermore, one mimics the proof of Lemma \ref{duality}
to show an analogue of the Lemma, saying that the Jacquet
functor is exact and takes the category 
$\AAA$ into $\oa$. Thus, we get an exact  functor $\j:\, \AAA\too\oa.$


\begin{proposition}
\label{faithful} 
\vi The Jacquet functor $\j:  \AAA\too\oa$
is {\sl faithful}, that is we have:
$\,\,\dis M\in \AAA\,\,\text{and}\,\,
M \neq 0 \;\; \Longrightarrow \;\; \j(M) \neq 0.
$

\vii If  conditions \ref{cat} hold for either $\ou$ or $\od$,
then  any object of the category\linebreak $\AAA$ has finite length.
\end{proposition}

\begin{remark} There is also a `one-sided' analogue of this Proposition
formulated as follows.
Assume, for concreteness, that the  triangular 
structure on $A$ is  non-commutative. We
 define $\AM$ to be 
 the category of 
finitely-generated left $A$-modules $M$
such that $\gr M$ is a finitely
generated   $(\gr A^+)$-module,
for any {\it good} increasing
filtration on $M$.
In particular, any object of $\AM$ is a  finitely-generated 
$A^+$-module,
but the category  $\AM$ is, in general, {\it smaller}
than the category $\Amod$ for $\U_+=A^+$. 
Similarly to  Proposition \ref{faithful}
one proves
\begin{proposition}
\label{faithful2}
\vi The Jacquet functor gives a {\sl faithful}
 functor $\j:  \AM\to\ou$.


\vii  If  conditions \ref{cat} hold for $\ou$
then any object of the category $\AM$ has finite length.\qed
\end{proposition}
\end{remark}
\medskip

Our proof of Proposition \ref{faithful} (and  Proposition \ref{faithful2})
will be based on  Gabber's Separation theorem, which we now
recall, cf. [Ga] for more details.

Let $\r=\n\rtimes\C\cd\bh$ be a solvable Lie
algebra such that
the adjoint $\bh$-action on $\n$ is diagonal
with all the eigenvalues in $\Z_+$,
 as at the beginning of this section.
Write $\U_\bullet\r$ and $\U_\bullet\n$
for the standard increasing filtrations on the corresponding
enveloping algebras. Thus, $\gr(\U\n)= \sym{\n}$.
Let $M$ be a left $\U\r$-module.
The following is ([Ga], Theorem 1):

\begin{theorem}[Separation theorem]
\label{gabber}  
Let $M$ be a non-zero $\U{\mathfrak r}$-module, and
  $\{M_j\}_{j \geq 0}$ an 
increasing filtration on $M$ compatible with
 $\U{\mathfrak r}$-action (i.e.,
$\U_i\r\cdot M_j\sset M_{i+j}\,,\,\forall i,j\geq 0$).
If  $\gr M$ is finitely generated {\sl
  over $\gr(\U{\n})$} (not only over $\gr(\U{\r})$),
then: $\,{\n} \cdot M \neq M$.
\end{theorem}

\begin{remark}
If  ${\n}$ is abelian, then   the Separation theorem simplifies,
and becomes the following  standard 
result  in Commutative Algebra (cf. e.g. [SW]):
{\it If $M$ is an $\U\r$-module which is finitely-generated
over the subalgebra $\U\n$, then: $\,{\n} \cdot M \neq M$}
(no filtration is needed in this case).
To prove this,
we claim first that the point $0
\in \Spec(\sym{\n})$ belongs to $\Supp(M)$. If not, then there is a
polynomial $P \in \sym{\n}$ that vanishes on $\Supp\ M$ and such that
$P(0) \neq 0$. Replacing $P$ by its high enough
power we may achieve that $P$
annihilates $M$, i.e. $P \in \ann(M) \subset \sym{\n}$. But the space
$\ann(M)$ is clearly stable under the adjoint $\delta$-action on
$\sym{\n}$. Moreover, since $P = P(0) + P_1$, where $P_1 \in
 {\n}\!\cdot\!(\sym{\n})$, and all weights of $ad \delta$-action 
on ${\n}\!\cdot\!\sym{\n}$ are
strictly positive, we deduce from $P \in \ann(M)$ that $P(0)$, $P_1 \in
\ann(M)$. Since $P(0) \neq 0$ this yields $1 \in \ann(M)$, a
contradiction. Thus we have proved $0 \in \Supp \ M$.
 But then $M/\n\cd M$, the geometric fiber of $M$ at $0$,
is non-zero, due to Nakayama lemma.
Thus, $M\neq \n\cd M$. $\enspace\Box$
\end{remark}
\smallskip

Here is an example\footnote{kindly communicated to me
by O. Gabber} showing that
the separation result may fail, in general, if (for $\n$ non-commutative)  the
assumption of  Theorem \ref{gabber}:   `$\gr M$ {\it is finitely generated
  over} $\gr(\U{\n})$' is replaced by the
weaker assumption:  `$M$  {\it is finitely generated
  over} $\U{\n}$'.
\smallskip

\noindent
{\bf Example.\;}
Let $\n$ be the 3-dimensional Heisenberg Lie algebra with basis
$x,y,z,$ where $z$ is central and $[x,y]=z$. Define
a semi-direct product $\r=\n\rtimes\C\cd\bh$
by the commutation relations: $[\bh,x]=x\,,\,[\bh,y]=y,$
and $[\bh, z]=2z$. 
Further, let $D$ be the associative algebra
 of polynomial differential operators on the 2-plane
$\BA:=\C^2,$ with coordinates `$y,z$'.
The assignment:
$$x\mapsto z\frac{\partial}{\partial y}\enspace,
\enspace y\mapsto y\enspace,
\enspace z\mapsto z\enspace,\enspace\bh\mapsto 2z\frac{\partial}{\partial z}
+y\frac{\partial}{\partial y}\,.
$$
extends to a Lie algebra imbedding: $\r\into D.$ 
We set $f(y,z):= y\cdot z -1$. Thus $f\in \C[y,z]$ is
 a polynomial, and
 the equation $f=0$ defines a hyperbola
 $X\sset
\BA$ such that

\noindent
$\bullet\quad$ The curve $X$ is smooth, irreducible, and does not intersect the axis $z=0$, and

\noindent
$\bullet\quad$ The restriction to $X$ of the projection $\BA\onto \C$
along the $y$-axis gives an \'etale morphism:
$X\to \C$.

Set $U:=\BA\ssminus X$, an affine Zariski open subset in $\C^2$.
The vector space $M:=\C[U]\big/\C[\BA]= \C[\BA][\frac{1}{f}]\big/\C[\BA]$ has
a natural $D$-module structure (making $M$ a simple holonomic
$\dd$-module on the plane $\BA$, with support $X$). It is straightforward to verify that
the action of $\frac{\partial}{\partial y}$ on $M$
induces isomorphisms:
$$\mbox{$\frac{1}{f^n}$}\cdot\C[y,z]\Big/\mbox{$\frac{1}{f^{n-1}}$}\cdot\C[y,z] \iso
\mbox{$\frac{1}{f^{n+1}}$}\cdot\C[y,z]\Big/\mbox{$\frac{1}{f^n}$}\cdot\C[y,z]
\quad,\quad\forall n=1,2,\ldots\,.
$$
It follows easily that $M=\U\n\cdot[\frac{1}{f}],$ i.e.,
the class $[\frac{1}{f}]\in
\C[\BA][\frac{1}{f}]\big/\C[\BA]$ 
generates $M$ over the subalgebra $\U\n\sset D$.

On the other hand, it is clear that $z\cdot M=M$,
i.e., the separation property {\sl fails} for $M$
(in this case,  $\gr(M)$ is not
finitely generated over $\gr(\U\n)$). $\lozenge$
\medskip

\noindent
{\sl Proof of Proposition \ref{faithful}.\;} The argument is quite standard.
Set $\n=\n^+\oplus\n^-$, a Lie subalgebra
in $A\otimes A\opp$, and write $\U_+$ for the
augmentation ideal in $\U(\n^+\oplus\n^-)$.
The algebra $A^+\otimes (A^-)\opp$ is a quotient of $\U_+$.
Hence we may (and will) view any $A\otimes A\opp$-module $M$
as an $\U_+$-module.
We have a  functor
$\widehat \j$ on $A\otimes A\opp$-modules
defined as  the completion with respect to the
subalgebra $A^+\otimes (A^-)\opp$.
In view of the identifications above, 
for any $M\in \AAA$, we can write:
${\widehat \j(M)}= \lim\limits_{\stackrel{\longleftarrow}{k}}\;
M/ \U_+^k\cd M$.
The kernel of the canonical
map $M\to \widehat \j(M)$ is clearly equal to
$K=\bigcap_{k\geq 0}\, \U_+^k M$.
It is routine to verify
 that $K$ 
 is an $A\otimes A\opp$-submodule in $M$.
Moreover, a non-commutative version of Artin-Rees
lemma, see [AM], [Ca], implies that
$\U_+\cdot K=K$ or, equivalently,
that $\n\cdot K=K$.

We would like to apply the Separation 
theorem to conclude that
$K=0$. To  verify
the conditions of that Theorem,
observe that $K \in \AAA$,
as a submodule of $M\in \AAA$.
Choose a good
filtration $K_\bullet$
on $K$. Then, since $\bh\in A_1,$ for any $i\in \Z,$ we
get:
$
(\bh\otimes 1 + 1\otimes\bh)\cdot K_i\sset K_{i+1}.$
We see that all the conditions of the Separation theorem  \ref{gabber} 
hold. Hence  $K=0$, as we wanted.

Thus, the canonical map $M \to 
{\widehat \j(M)}$ is injective.
 Hence, $M\in \AAA$
$\&\;M\neq 0$
implies ${\widehat \j(M)}\neq 0$.
We conclude that $\j(M)$, being a dense subspace in 
${\widehat \j(M)}$, is also non-zero,
and part (i) follows.

To prove (ii), it suffices to  verify 
the descending chain condition for
any object $M\in\AAA$. Let
$M=M^0\supsetneq M^1\supsetneq  M^2\supsetneq  \ldots$ be a descending chain
of submodules in $M$.
Using  the exactness of
Jacquet functor and part (i), from
 $M^i/M^{i+1}\neq 0\,,\,
\forall i,$ we deduce:
$\j(M^i)/\j(M^{i+1})=\j(M^i/M^{i+1})\neq 0$.
Hence,
$\j(M^0)\supsetneq  \j(M^1)\supsetneq  \j(M^2)\supsetneq   \ldots$ is a 
strictly  descending chain
of submodules in $\j(M)$. 
This contradicts the fact
that $\j(M)$, being an object of $\oa$,
must have finite length by Lemma
\ref{triv}.
Claim (ii) follows.~\qed\medskip






\noindent
{\bf
Proof of Theorem \ref{duflo2}.\;}
Conditions $(T0)$, $(T0')$ imply
that $A\in \AAA$, as an $(A\td A)$-bimodule.
Thus, Theorem \ref{duflo2} follows
directly from Proposition \ref{faithful}(ii).\qed
\smallskip





\noindent
{\bf
Proof of Theorem \ref{duflo}.\;} The implication
(i)$\,\Longrightarrow\,$(ii) is standard.

To prove  (ii)$\,\Longrightarrow\,$(iii),
fix a prime ideal $I\sset A$. As we have observed above,
 $A\in \AAA$, hence  $A/I\in \AAA$. Therefore,
 we may apply the Jacquet functor
to get an object $\j(A/I)\in \oa$. 



Given an $A\otimes A\opp$-module, resp. $(A\td A)$-bimodule,
$M$, we write $\lann(M)$ for the annihilator of $M$ in the
subalgebra $A\otimes 1 \sset A\otimes A\opp$.
The proof of  Proposition \ref{faithful}
shows that $A/I$ injects into $\widehat{\j}(A/I)$
and, moreover, the image of $A/I$ is dense in $\widehat{\j}(A/I)$.
Thus we  have
$I=\lann(A/I)=\lann\bigl(\widehat{\j}(A/I)\bigr).$
Furthermore, $\lann\bigl(\widehat{\j}(A/I)\bigr)=
\lann\bigl({\j}(A/I)\bigr),$ for $\j(A/I)$ is dense
in $\widehat{\j}(A/I)$. Thus,
$\lann\bigl({\j}(A/I)\bigr)=I$ is a prime 
ideal in $A$.\footnote{It was pointed out by a referee that
a similar observation is contained in [J2, n.2.4].}



Now,  any object of $\oa$, in particular,
$M=\j(A/I),$  has finite length by Lemma \ref{triv}.
Let $0=M_0\sset M_1\sset M_2\sset\ldots\sset M_n=M,$
be its Jordan-H\"older series.
For each $i=1,2,\ldots,n,$ put $J_i:=\lann(M_i/M_{i-1}).$
Then, by a standard argument, see e.g. [BG], one proves:
$I=J_m,$ for some $m\in [1,n]$. 

We provide more details for the reader's convenience.
First, since $J_i(M_i)\sset M_{i-1}$ we deduce:
%\begin{align*}
$$
J_1\cdot\ldots\cdot J_{n-1}\cdot J_n(M)\;\;\sset\;\;
J_1\cdot\ldots\cdot J_{n-1}(M_{n-1})\;\;\sset\;\;\ldots\;\;\sset\;\;
J_1(M_1)\;=\;0.
$$
Hence, the ideal $J_1\cdot\ldots\cdot J_n$ annihilates $M$,
and therefore $J_1\cdot\ldots\cdot J_n\sset \lann(M)$.
But, $\lann(M)=I$ being prime, the inclusion above implies
that there exists an $m$ such that $J_m\sset I$.
On the other hand,
 since $M_i/M_{i-1}$ is a subquotient of $M$,
 for any $i=1,2,\ldots,n,$ one clearly has an opposite
inclusion $I=\lann(M)\sset\lann(M_i/M_{i-1})=J_i$.
Thus, we obtain $I=J_m=\lann(M_m/M_{m-1})$, as claimed.

By construction,
 $M_m/M_{m-1}$ is a simple object
of $\oa$. Lemma \ref{triv} yields\linebreak
$M_m/M_{m-1}= L'\boxtimes L'',$
where $L'\in \ou$ and $L''\in \od$ are some simple objects.
Further, it is clear that $\lann(L'\boxtimes L'')=
\lann(L')$. Thus, we deduce that
$\,I=\lann(M_m/M_{m-1})=$
$\lann(L'\boxtimes L'')=\lann(L')\,$ is the annihilator
of a simple object of $\ou$. Part (iii) follows.

Part (iv) is equivalent to (iii) by duality.
The implication
(iii)$\,\Longrightarrow\,$(i) is trivial.
\qed






\section{Applications to Symplectic reflection algebras}
\setcounter{equation}{0}

Let $(V,\om)$ be a finite dimensional
symplectic vector space, and $\G\sset Sp(V,\om)$
a finite subgroup of symplectic automorphisms of $V$.
An element $s\in \G$ is called  a {\sl symplectic
reflection} if $\;\;\rk(\id-s)=2.$
Let $S$ denote the set of symplectic reflections in $\G.$
The group $\G$ acts on $S$ by conjugation.
For each $s \in S$, there is an $\om$-orthogonal direct sum decomposition
$V= {\mathtt{Image}}(\id-s) \oplus \Ker(\id-s)$,
and we write $\om_s$ for the (possibly degenerate)
skew-symmetric form on $V$ which coincides with $\om$
on ${\mathtt{Image}}(\id-s)$, and has $\Ker(\id-s)$ as the
 radical.

Write $\C\G$ for
the group algebra of $\G$, and  $(TV)\#\G$ 
 for the smash-product of the tensor algebra $TV$ with
$\C\G$. From now on we identify $V$ with $V^*$ via the symplectic form,
hence the Symmetric algebra on $V$ gets identified with
 $\C[V]$, the polynomial algebra on $V$.

In [EG], we have introduced a class of associative algebras
 $\sH_{t,c}(V,\om,\G)$, called Symplectic
reflection algebras, as follows.
Fix an ${\mathtt{Ad}^{\,}}\G$-invariant function $c: S \to \C\,,\, s\mapsto
c_s,$ and a complex number $t\in\C$.
Let  $\kappa: V\times V \to \C\G$ be a
skew-symmetric $\C$-bilinear pairing given 
by the formula:
$$\kappa(x,y) = t \cdot \om(x,y)\cdot 1 + \sum\nolimits_{s\in S}\; c_s\cdot
\om_s(x,y)\cdot s\quad,\quad\forall x,y \in V\,.$$
We define the Symplectic
reflection algebra with parameters $(t,c)$ by
\begin{equation}\label{rel}
\sH_{t,c}(V,\om,\G)
\; :=\; (TV\#\G)/I\langle x\otimes y - y\otimes x - \kappa(x,y) \in
T^2V\, \oplus\, \C\G\rangle_{\,x,y \in V}\,\,,
\end{equation}
where 
 $I\langle\ldots\rangle $ stands for the two-sided ideal in
$(TV)\#\G$ generated by the indicated set.
 Thus, $\sH_{t,c}(V,\om,\G)$
is an associative algebra which may be thought of as a
deformation of $\C[V]\#\G$ (= cross-product of $\C[V]$
with $\G$). Clearly,  $\sH_{t,c}(V,\om,\G)$ contains $\C\G$
  as a subalgebra. Furthermore,
there is a natural increasing filtration on
 $\sH_{t,c}(V,\om,\G)$ such that $\C\G$ has filtration degree zero,
and elements of $V\sset TV$ are assigned  filtration degree one.
It has been proved in  [EG] that 
the corresponding associated graded  algebra is described
by the following canonical  graded  algebra isomorphism:
\begin{equation}\label{H_filt}
\gr\sH_{t,c}(V,\om,\G)\;\simeq\;\C[V]\#\G \qquad\text{({
\sl{Poincar\'e-Birkhoff-Witt property}})}.
\end{equation}


Write $\e=\frac{1}{\sharp\G}\sum_{g\in\G}\,g\in \C\G$
for the averaging idempotent, viewed as an element in
$\sH_{t,c}(V,\om,\G)$.
We let ${\sf{eHe}}:=\e\cdot\sH_{t,c}(V,\om,\G)\cdot\e$
be the {\it spherical subalgebra} in  $\sH_{t,c}(V,\om,\G)$.
The increasing filtration on
 $\sH_{t,c}(V,\om,\G)$ induces a filtration
on the  spherical subalgebra, and from (\ref{H_filt}) one finds that
$\gr({\sf{eHe}})=\C[V]^\G=\C[V/\G],$ the algebra
of $\G$-invariant polynomials on $V$. 
The canonical  Poisson structure on the commutative
algebra $\gr({\sf{eHe}})$ makes $V/\G$ a  Poisson variety.
It is
easy to see, cf. ([EG], Theorem 1.6),
 that 
this Poisson structure on $V/\G$
equals
the $t$-multiple of  the standard one induced
by 
the symplectic structure on $V$
(for $t=0$ the spherical subalgebra ${\sf{eHe}}$ becomes commutative,
and the corresponding Poisson structure on
$\gr({\sf{eHe}})=\C[V/\G]$ reduces to zero).


\noindent
{\bf{Symplectic leaves in $V/\G$}}.\,
The  leaves
of the standard Poisson structure on $\C[V/\G]$ 
are     described
as follows, see  [BrGo]. Let $\Isot$ denote the
(finite) set of all subgroups $G\sset \G$ that occur as 
isotropy groups of points in $V$.
Given a  subgroup $G\sset \G$,
write $V^G\sset V$ for the vector subspace of
$G$-fixed points. Let 
$V_G$ denote the
image of $V^G$ under the projection $V\onto V/\G$,
and let $\overset{_\circ}{V}_G= V_G\ssminus\bigcup'_{_{G'\in\Isot}}\; 
\left(V_G\cap V_{G'}\right),$
be the complement in $V_G$ of the union of
all {\it proper} subsets in  $V_G$ of the form $V_G\cap V_{G'}$
(here 
the symbol $\cup'$ indicates that the union is taken
over proper subsets only). 
It is easy to see that 
for each subgroup  $G\in \Isot$, the set
$\overset{_\circ}{V}{}^{_G}$ equals
the image of the
subset $\overset{_\circ}{V}{}^{_G}\sset 
V^G$ formed by the points whose isotropy group equals
$G$. If $N_G$ denotes the normalizer of $G$ in $\G$,
then the group $N_G/G$ acts freely on $\overset{_\circ}{V}{}^{_G}$
and we have $\overset{_\circ}{V}_G\simeq
\overset{_\circ}{V}{}^{_G}\big/(N_G/G)$.
It follows that each set $\overset{_\circ}{V}_G$ is
a smooth connected locally-closed
subvariety of $V/\G$ and one shows that these varieties
 are exactly the  symplectic leaves in
$V/\G$, see e.g., [BrGo]. Hence, there are only finitely
many  symplectic leaves. 

    From the Irreducibility theorem \ref{irreducibility}
we deduce

\begin{corollary}\label{symplectic_refl}
Let   $\sH_{t,c}(V,\om,\G)$ be a  Symplectic
reflection algebra, and $t\neq 0$.
Then, for any primitive ideal $I$ in
the spherical subalgebra $\e\cd\sH_{t,c}(V,\om,\G)\cd\e$,
the variety $\var(I)$ has the form $V_G$, for a certain
subgroup $G\in\Isot$.
\end{corollary}


We remark that Theorem \ref{irreducibility}
is not applicable to the Symplectic reflection
algebra $\sH_{t,c}(V,\om,\G)$ itself since
$\gr\sH_{t,c}(V,\om,\G)=
\C[V]\#\G,$ is a non-commutative algebra.
Note however, that 
finitely generated $\C[V]\#\G$-modules may be naturally
identified with $\G$-equivariant coherent sheaves on
$V$. The support of such a sheaf is a  $\G$-stable
subvariety in $V$.
In particular, given a two-sided ideal $I\sset \sH_{t,c}(V,\om,\G)$
one can view $\gr\bigl(\sH_{t,c}(V,\om,\G)/I\bigr)$
as a finitely generated $\C[V]\#\G$-module, via (\ref{H_filt}).
Let $\var(I)\sset V$ stand for its support.

Using the technique of {\it Poisson orders} developed in [BrGo],
one can refine our argument to obtain the following

\begin{proposition}\label{symplectic_refl2}
For  any primitive ideal $I\sset\sH_{t,c}(V,\om,\G)$,
the variety $\var(I)$ is the $\G$-saturation
of vector subspace $V^G$, for a certain
subgroup $G\in\Isot$.\hfill\qed
\end{proposition}

\noindent
{\bf Rational Cherednik algebras.}\,
Let $ W $ be a finite Coxeter group in a
complex vector space $\h$.
Thus, $\h$ is the complexification
of a  real Euclidean  vector space and $W$ is
generated by a finite set $S\subset W$ of
reflections $s\in S$ with respect to certain
hyperplanes $\{H_s\}_{s\in S}$
in that  Euclidean  space.


For each $s\in S$, we choose 
a nonzero
linear function $\alpha_{s}\in\h^*$ which vanishes on $H_s$
(called the {\it positive root} corresponding to $s$),
and let $\alpha_s^{\vee}=2(\alpha_s, -)/(\alpha_s,\alpha_s) \in \h $
be the corresponding coroot.
The group $W$ acts   on the set $S$,
and  diagonally  on $V=\h\oplus\h^*$,
 by conjugation. We  equip  the space
 $V=\h\oplus\h^*$  with the canonical symplectic
structure of the cotangent bundle.

 Following [EG], to each 
 $W$-invariant function $c: S \to \C\,,\, c\mapsto c_s,$
 one associates the Symplecic reflection algebra $\sH_c:=\sH_{1,c}(V,\om,W)$,
 called  the {\it rational Cherednik algebra}.
The algebra $\sH_c$
is generated
by the vector spaces $\h$, $\h^*,$ and the set
$W,$ with defining relations  
(cf.  formula (1.15) of [EG] for $t=1$) given
by
\begin{equation}
\label{defrel}
\begin{array}{lll}\displaystyle
&{}_{_{\vphantom{x}}}w\cd x\cd w^{-1}= w(x)\;\;,\;\;
w\cd y\cd w^{-1}= w(y)\,,&\;
\forall y\in \h\,,\,x\in \h^*\,,\,w\in W\break\medskip\\
&{}^{^{\vphantom{x}}}{}_{_{\vphantom{x}}}x_1\cd x_2 = 
 x_2\cd x_1\;\enspace,\;\enspace
y_1\cd y_2=y_2\cd y_1\,, &\;
\forall y_1,y_2\in \h,\;x_1\,,\,x_2 \in \h^*\,\break\medskip\\
&{}^{^{\vphantom{x}}}y\cd x-x\cd y = \langle y,x\rangle
+\sum\limits_{^{_{s \in S}}}\,
c_s\cd\langle y,\alpha_s\rangle
\langle\alpha_s^\vee,x\rangle \cd s\,,&\; \forall y\in
\h\,,\,x\in \h^*\,.
\end{array}
\end{equation}
 Thus, the elements  $x\in\h^*$ generate a 
subalgebra $\C[\h]\subset \sH_c$ of polynomial functions on $\h$,
 the elements  $y\in\h$ generate a 
subalgebra $\C[\h^*]\subset \sH_c$,
and the elements $w\in W$
span  a copy of the group
algebra $\C W$ sitting naturally inside $\sH_c$.
Furthermore, the Poincar\'e-Birkhoff-Witt property yields
  a `triangular 
decomposition' $\,\sH_c\simeq \C[\h^*]\otimes\C[W]\otimes\C[\h],$
see ([EG], Corollary 4.4). We put $A^+:=
\C^{^{_{>0}}}[\h]^W,$ and
$A^-:=
\C^{^{_{>0}}}[\h^*]^W$. Further, in  [BEG] we have constructed
an element $\bh\in \sH_c$ such that the data $(A^\pm,\bh)$ 
gives a commutative triangular structure on the
rational Cherednik algebra $\sH_c$.

The categories
$\ou, \od$ (see Definition \ref{oo}) for  $\sH_c$ have only finitely
many (isomorphism classes of) simple objects, by [DO].
Thus Theorem \ref{duflo} applies (note that we are using
here only the easy, purely commutative,
 part of Gabber's Separation theorem, since
$A^\pm$ are commutative algebras) and we get
\begin{corollary}\label{cher} Both
Duflo theorem and Theorem \ref{duflo2}
hold for rational Cherednik algebras.
\end{corollary}

In [BEG] we  introduced a category $\O$ to be the
category of finitely-generated left $\sH_c$-modules $M$
such that the $A^-$-action on $M$ is locally-finite.
It is clear from the triangular
decomposition $\,\sH_c\simeq \C[\h^*]\otimes\C[W]\otimes\C[\h]$
that in the notation of \S4 we have:
$\ou\sset \O\sset \Amm$. Note that the inclusion
$\ou\sset \O$ is {\it strict} since
we do not require the  locally-finite
$A^-$-action to be  locally-nilpotent.
We have also introduced in [BEG] a category $\HC$ of Harish-Chandra
bimodules over the Cherednik algebra $\sH_c$,
and showed that (in our present notation)
any object of $\HC$ belongs to $\AAA$.
Thus, from Propositions \ref{faithful}-\ref{faithful2}
we deduce
\begin{corollary}
\label{cher2}
Any object of the categories $\O$ and $\HC$ has finite length.
\hfill\qed
\end{corollary}

\begin{remark} Notice that in [BEG] we were only able to prove that
any object  of the category $\ou$ has finite length, but the
technique in loc. cit. was insufficient to
prove Corollary \ref{cher2}
in full generality.
\end{remark}
\smallskip

As has been explained before Proposition \ref{symplectic_refl2},
for any $\sH_c$-module, hence for an object $M\in \O$,
one
can define a $W$-stable subvariety
$\gsup{M}\sset \h\oplus\h^*$.
The  $A^-$-action on $M$  being locally-finite,
it follows, since $A^-= \C^{^{_{>0}}}[\h^*]^W,$
that $\gsup{M}\sset \h\oplus \{0\}$.
Next, identify $\h^*$ with $\h$ via the invariant form and observe
that, since $W$ is finite,  the isotropy group  of a point $(y,x)\in  \h\oplus \h^*=
\h\oplus \h$ coincides with  the isotropy group of the
generic linear combination $t_1\cdot y + t_2\cdot x\,,\, t_1,t_2\in \C$.
Therefore we see that, in the notation of 
Theorem \ref{symplectic_refl}, 
the set $\Isotw$ is exactly the set ${\mathtt{Parab}}(W)$ of all parabolic
subgroups in $W$, cf. [Hu].
\begin{theorem}
\label{levi}
For  any simple object $M\in \O$,
the variety $\gsup{M}$ is the $W$-saturation
of vector subspace $\h^G\oplus \{0\}$, for a certain
subgroup $G\in {\mathtt{Parab}}(W)$.
\end{theorem}

\noindent
{\sl Sketch of Proof.\;} We adapt the known argument used
in the case of highest weight modules over $\U\g$,
the enveloping algebra of a semisimple Lie algebra.
To this end, write $\ann M\sset \sH_c$ for the annihilator
of $M\in \O$. Thus, $\ann M$ is a primitive ideal in $\sH_c$.


Introduce the notation $\Lambda:=\gsup M\sset \h\oplus\h^*.$
Clearly, $\La\sset \gsup(\sH_c/\ann M )$. 
Further, by Proposition \ref{symplectic_refl2} 
there exists a subgroup $G\in {\mathtt{Parab}}(W)$ such that
$\gsup(\sH_c/\ann M )=W\cdot (\h\oplus\h^*)^G$.
Therefore we get:
\begin{equation}\label{pf1}
\La\sset (\h\oplus\{0\})\;\cap\;W\cdot (\h\oplus\h^*)^G
=W\cdot (\h^G\oplus\{0\})
\end{equation}

Next, we claim that 
\begin{equation}\label{pf2}
\dim \La =\dim (\h^G)\,.
\end{equation}
To prove this, choose a finite-dimensional
$\C[\h^*]^W$-stable subspace $M^0\sset M$ that generates
$M$ over the subalgebra $\C[\h]^W$. For each
$i\geq 0$, set $M_i= (\sH_c)_i\cdot M^0$, where $\{(\sH_c)_i\}_{i\geq 0}$
is the standard increasing filtration on the Cherednik algebra $\sH_c$,
cf. [EG]. This gives a filtration on $M$ compatible with
that on $\sH_c$ and such that $\gr M$ is
finitely-generated over the subalgebra $\gr\C[\h]^W \sset
\gr{\sH_c}.$ Hence, the restriction to $\gsup M\sset \Spec(\gr{\sH_c})$
of the canonical projection:
$\Spec(\gr{\sH_c})\onto \Spec(\C[\h]^W)$ is a finite morphism.
Therefore the dimension of the variety $\gsup M$ is unaffected
by replacing the algebra $\sH_c$ by the algebra $\C[\h]^W$.
Now, following Joseph [J3], we define
an increasing chain of $\C[\h]^W$-submodules
 $D_k(M)\sset \End_{_\C}M\,,\,
k=0,1,\ldots,$ inductively as follows
$$ D_0(M)=\C[\h]^W\,,
\quad\text{and}\quad
D_k(M)=\{u \in \End_{_\C}M\;\;\big|\;\;
[u\,,\,\C[\h]^W]\sset D_{k-1}(M)\}\,.
$$
Set $D(M):= \bigcup_{k\geq 0}\,D_k(M).$
Joseph showed that $D(M)$ is an associative subalgebra
in $\End_{_\C}M$ (which should be thought of as
an algebra of  \,`differential operators' on $M$).
Moreover, he proved  in ([J3], Lemma 2.3)
that $\dim\gsup D(M)\,\leq\, 2\dim(\gsup M)$.

Observe further  that the
 action in $M$ of an element $a\in \sH_c$ gives
an endomorphism $\hat{a}\in \End_{_\C}M$,
and the assignment $a\mapsto \hat{a}$ gives an 
algebra imbedding: $\sH_c/\ann M \into \End_{_\C}M$.
The adjoint action of the subalgebra $\C[\h]^W$
on $\sH_c$ being locally-nilpotent, see [BEG],
the image of the embedding: $\sH_c/\ann M \into  \End_{_\C}M$
is contained in $D(M)$.
      From this, following Joseph [J3], one derives:
\begin{equation}\label{joseph}
\dim\gsup(\sH_c/\ann M )\;\leq\; \dim\gsup D(M)\; \leq\; 2\dim(\gsup M)\,.
\end{equation}
Hence, using \eqref{pf1} and writing `$\dim(-)$' instead of
`$\dim\gsup(-)$', we deduce
$$
\dim (\h\oplus\h^*)^G \;=\;
\dim(\sH_c/\ann M )\; \leq\; 2\dim M\;
=\; 2\dim\La\; \leq\; 2\dim\h^G\; =\;\dim (\h\oplus\h^*)^G \,.
$$
It follows that
 all the inequalities above must be equalities, and
\eqref{pf2} is proved. In particular, there exists 
 $\La^\circ\sset (\h^G\oplus\{0\})$, an irreducible
component of $\La$ such that $\dim 
\La^\circ =\dim(\h^G\oplus\{0\})$. This dimension equality yields:
$\La^\circ =\h^G\oplus\{0\}.$

We claim next that {\it all} irreducible components
of $\La= \gsup M$ have the same dimension, i.e.,
the following version of
 Gabber's equi-dimensionality theorem holds for the
Cherednik algebra 
\begin{proposition}\label{equi}
For any simple $\sH_c$-module $M$, the variety
$\gsup M$ is equidimensional.
\end{proposition}

To prove this, we recall that 
the equi-dimensionality theorem is, as explained e.g. in 
[Gi], a formal
consequence of Gabber-Kashiwara theorem. 
The proof of the latter  theorem given,
e.g. in ([Gi], p.342-345) works for any 
filtered
algebra $\A$ such that $\gr\A$ is the coordinate ring
of a {\it smooth} affine algebraic variety.
Now, in our present situation,
we have: $\gr \sH_c = \C[\h\oplus\h^*]\#W$,
which is not a  commutative algebra.
However,
the formal microlocalization
construction  can be carried out with respect to any
multiplicative set $S\sset \A\,,\, (0\not\in S),$ for
any algebra $\A,$ provided
 the principal symbols of the elements of $S$
belong to the center of $\gr\A$,
see ([Gi], footnote on p. 337).
In our case we have a central subalgebra:
$\C[\h\oplus\h^*]^W \sset \gr \sH_c.$
Furthermore, the  algebra $\gr \sH_c$
 has {\it finite} homological dimension,
as a cross-product of  the polynomial
algebra $\C[\h\oplus\h^*]$ with a finite group.
Going through the proof of   Gabber-Kashiwara theorem
given in ([Gi], pp. 342--345) one sees that 
 the finiteness of homological dimension of $\sH_c,$
plus the existence of
the formal microlocalization with  respect to
 elements whose symbols belong to
$\C[\h\oplus\h^*]^W\ssminus\{0\}$, is enough
to conclude 
that the proof of   Gabber-Kashiwara theorem
applies to the algebra $\sH_c$
as well.
This implies Proposition \ref{equi}.

      From 
Proposition  \ref{equi} we deduce
 that all irreducible components
of the variety $\La=\gsup M$ have the same dimension, which is
equal to $\dim \h^G$. But then the inclusion in \eqref{pf1}
forces each irreducible component to be 
a $W$-translate of $\h^G$, and  Theorem \ref{levi} follows.
\qed
\medskip

\begin{remark}
It was pointed out to us by A. Yekutieli, that
Gabber's equi-dimensionality theorem also
holds for the
spherical subalgebra  ${\sf{eHe}}$ in any Symplectic reflection
algebra
${\sf{H}}=\sH_{t,c}(V,\om,\G)$, i.e. one
has\end{remark}
\begin{proposition} Let $\sH_{t,c}(V,\om,\G)$
be a  Symplectic reflection
algebra. Then,
for any simple ${\sf{eHe}}$-module $M$, the variety
$\gsup M$ is equidimensional.
\end{proposition}

\noindent
{\sl Sketch of proof.\;} Recall that  PBW-proprty \eqref{H_filt}
implies that
$\gr({\sf{eHe}})=\C[V]^\G=\C[V/\G].$ We see that
 $\Spec\bigl(\gr({\sf{eHe}})\bigr) \simeq V/\G$ is not a smooth variety.
 Nonetheless, $ V/\G$
 is a Gorenstein variety, and
homological duality formalism
is known  to  work for Gorenstein varieties
as nicely as for smooth varieties. In particular, it follows from  [YZ]
that
the bimodule ${\sf{eHe}}$ is (up to shift) a  rigid dualizing complex
for the algebra  ${\sf{eHe}}$
(in the sense of [YZ]). Therefore, the proof
of   Gabber-Kashiwara theorem still goes through, cf. \cite[Purity~theorem]{YZ}.
\qed

\footnotesize{
\begin{thebibliography}{APKa}
\bibitem[AM]{AM} M. Atiyah, I. Macdonald:
{\it Introduction to commutative algebra.} Addison-Wesley,
 Reading, Ma. 1969.

\bibitem[BEG]{BEG} Yu. Berest, P. Etingof, and V.  Ginzburg:
{\it Cherednik algebras and differential operators on quasi-invariants.}
{\tt{arXiv:math.QA/0111005}}, \, to appear in Duke Math. J.

\bibitem[Be]{Be} J. Bernstein: {\it 
Modules over a ring of differential operators. An investigation of the
fundamental solutions of equations with constant coefficients.}
 (Russian) Funkcional. Anal. Appl. {\bf 5} (1971), 1--16. 


\bibitem[BG]{BG} J. Bernstein, S. Gelfand:
{\it Tensor products of finite- and infinite-dimensional
representations of semisimple Lie algebras.} Compositio Math. {\bf 41}
 (1980),  245--285. 
\bibitem[BoBr]{BoBr} W. Borho, J.-L. Brylinski: {\it
Differential operators on homogeneous spaces.} I.
{\sl Irreducibility of the associated variety for
 annihilators of induced modules.} Invent. Math.  {\bf 69} (1982),
437--476.

\bibitem[BoKr]{BoKr} W. Borho, H. Kraft: {\it
\"Uber die Gelfand-Kirillov-Dimension.} Math. Ann. {\bf 220} (1976),
 1--24. 

\bibitem[BrGo]{BrGo} K. Brown, I. Gordon: {\it
Poisson orders, symplectic reflection algebras and representation
theory.}
{\tt{arXiv:math.RT/020104}}.
\bibitem[Ca]{Ca} W. Casselman: {\it 
Jacquet modules for real reductive groups.}
 Proceedings of the ICM (Helsinki, 1978), pp. 557--563, 
Acad. Sci. Fennica, Helsinki, 1980.
\bibitem[CO]{CO} W. Casselman, M. Osborne:
{\it The restriction of admissible representations to $\n$.}
Mathem. Ann. {\bf 233} (1978), 193--198.

\bibitem[Ch]{Ch} I. Cherednik: {\it Double affine Hecke
 algebras,
Knizhnik-Zamolodchikov equations, and Macdonald operators,}
IMRN (Duke math.J.) {\bf 9} (1992), 171-180.


\bibitem[CG]{CG} N. Chriss, V. Ginzburg: {\it
Representation theory and complex geometry.} 
Birkh\"auser Boston, 1997.

\bibitem[Di]{Di} J. Dixmier: {\it 
Enveloping algebras.} Graduate
Studies in Math., {\bf 11} A.M.S., Providence,
RI, 1996.




\bibitem[Du]{Du} M. Duflo: {\it Sur la classification des id\'eaux primitifs dans l'alg\`ebre enveloppante d'une
alg\`ebre de Lie semi-simple.}
 Ann. of Math. {\bf  105} (1977),  107--120.
\bibitem[DO]{DO} C. Dunkl, E. Opdam:
{\it Dunkl operators for complex reflection groups.}
{\tt{arXiv:math.RT/0108185}}.

\bibitem[EG]{EG} P. Etingof, V. Ginzburg:
{\it Symplectic reflection algebras, 
Calogero-Moser space, and deformed Harish-Chandra
   homomorphism.} 
Invent. Math. {\bf 147} (2002), 243-348,
[ {\tt{arXiv:math.AG/0011114}} ].

\bibitem[Ga]{Ga} O. Gabber: {\it A separation theorem and Harish-Chandra
modules.}
Unpublished manuscript 1981.
\bibitem[Ga1]{Ga1} O. Gabber: {\it 
The integrability of the characteristic variety. }
Amer. J. Math. {\bf 103} (1981),  445--468. 

\bibitem[Gi]{Gi} V. Ginzburg: {\it 
Characteristic varieties and vanishing cycles.}
 Invent. Math. {\bf 84} (1986), 327--402.

\bibitem[Gu]{Gu} N. Guay: \textit{Projective modules in the category $\oo$ for the
Cherednik algebra.} Preprint, University of Chicago, December 2001.


\bibitem[Hu]{hum} J. Humphreys: \textit{Reflection groups and Coxeter groups.}
Cambridge Studies in Advanced Mathematics, {\bf 29} 
Cambridge University Press, Cambridge, 1990.

\bibitem[J1]{J1} A. Joseph: {\it 
Primitive ideals in enveloping algebras.}
{\sl Proceedings of the ICM}
 (Warsaw, 1983), 403--414, PWN, Warsaw, 1984.

\bibitem[J2]{J2}  A. Joseph: {\it
On the associated variety of a primitive ideal.}
 J. Algebra {\bf 93} (1985),
509--523.

\bibitem[J3]{J3}  A. Joseph: {\it
Gelfand-Kirillov dimension for the annihilators of simple quotients of
Verma modules.} J. London Math. Soc. {\bf  18} (1978),  50--60.

\bibitem[KT]{KT} M. Kashiwara, T. Tanisaki: {\it 
The characteristic cycles of holonomic systems on a flag manifold
related to the Weyl group algebra.}
 Invent. Math. {\bf 77} (1984), 185--198.

\bibitem[Po]{Po} A. Polishchuk: {\it 
 Algebraic geometry of Poisson brackets.}
 J. Math. Sci. {\bf 84} (1997),  1413--1444.

\bibitem[SW]{SW}
J.T. Stafford, N.R. Wallach: {\it 
The restriction of admissible modules to parabolic subalgebras. }
Trans. Amer. Math. Soc. {\bf 272} (1982),  333--350.
 
\bibitem[Vo]{Vo} D. Vogan: {\it 
Associated varieties and unipotent representations. Harmonic analysis
on reductive groups.} 315--388, Progr. Math.,
{\bf  101}, Birkh\"auser Boston, Boston, MA,
1991.
%\bibitem[Vo2]{Vo2} D. Vogan: {\it 
%Gelfand-Kirillov dimension for Harish-Chandra modules.}
% Invent. Math. {\bf 48} (1978), 75--98.

\bibitem[YZ]{YZ} A. Yekutieli, J. Zhang: {\it
Rings with Auslander dualizing complexes.}
 J. Algebra {\bf 213} (1999),  1--51.

\end{thebibliography}
}
%\vskip 1cm



\footnotesize{
Department of Mathematics, University of Chicago, 
Chicago IL
60637, USA;\\ 
\hphantom{x}\quad\, {\tt ginzburg@math.uchicago.edu}}



\end{document}


math.RT/0202079





math0202079.extracted_bib
\begin{thebibliography}{APKa}
\bibitem[AM]{AM} M. Atiyah, I. Macdonald:
{\it Introduction to commutative algebra.} Addison-Wesley,
 Reading, Ma. 1969.

\bibitem[BEG]{BEG} Yu. Berest, P. Etingof, and V.  Ginzburg:
{\it Cherednik algebras and differential operators on quasi-invariants.}
{\tt{arXiv:math.QA/0111005}}, \, to appear in Duke Math. J.

\bibitem[Be]{Be} J. Bernstein: {\it 
Modules over a ring of differential operators. An investigation of the
fundamental solutions of equations with constant coefficients.}
 (Russian) Funkcional. Anal. Appl. {\bf 5} (1971), 1--16. 


\bibitem[BG]{BG} J. Bernstein, S. Gelfand:
{\it Tensor products of finite- and infinite-dimensional
representations of semisimple Lie algebras.} Compositio Math. {\bf 41}
 (1980),  245--285. 
\bibitem[BoBr]{BoBr} W. Borho, J.-L. Brylinski: {\it
Differential operators on homogeneous spaces.} I.
{\sl Irreducibility of the associated variety for
 annihilators of induced modules.} Invent. Math.  {\bf 69} (1982),
437--476.

\bibitem[BoKr]{BoKr} W. Borho, H. Kraft: {\it
\"Uber die Gelfand-Kirillov-Dimension.} Math. Ann. {\bf 220} (1976),
 1--24. 

\bibitem[BrGo]{BrGo} K. Brown, I. Gordon: {\it
Poisson orders, symplectic reflection algebras and representation
theory.}
{\tt{arXiv:math.RT/020104}}.
\bibitem[Ca]{Ca} W. Casselman: {\it 
Jacquet modules for real reductive groups.}
 Proceedings of the ICM (Helsinki, 1978), pp. 557--563, 
Acad. Sci. Fennica, Helsinki, 1980.
\bibitem[CO]{CO} W. Casselman, M. Osborne:
{\it The restriction of admissible representations to $\n$.}
Mathem. Ann. {\bf 233} (1978), 193--198.

\bibitem[Ch]{Ch} I. Cherednik: {\it Double affine Hecke
 algebras,
Knizhnik-Zamolodchikov equations, and Macdonald operators,}
IMRN (Duke math.J.) {\bf 9} (1992), 171-180.


\bibitem[CG]{CG} N. Chriss, V. Ginzburg: {\it
Representation theory and complex geometry.} 
Birkh\"auser Boston, 1997.

\bibitem[Di]{Di} J. Dixmier: {\it 
Enveloping algebras.} Graduate
Studies in Math., {\bf 11} A.M.S., Providence,
RI, 1996.




\bibitem[Du]{Du} M. Duflo: {\it Sur la classification des id\'eaux primitifs dans l'alg\`ebre enveloppante d'une
alg\`ebre de Lie semi-simple.}
 Ann. of Math. {\bf  105} (1977),  107--120.
\bibitem[DO]{DO} C. Dunkl, E. Opdam:
{\it Dunkl operators for complex reflection groups.}
{\tt{arXiv:math.RT/0108185}}.

\bibitem[EG]{EG} P. Etingof, V. Ginzburg:
{\it Symplectic reflection algebras, 
Calogero-Moser space, and deformed Harish-Chandra
   homomorphism.} 
Invent. Math. {\bf 147} (2002), 243-348,
[ {\tt{arXiv:math.AG/0011114}} ].

\bibitem[Ga]{Ga} O. Gabber: {\it A separation theorem and Harish-Chandra
modules.}
Unpublished manuscript 1981.
\bibitem[Ga1]{Ga1} O. Gabber: {\it 
The integrability of the characteristic variety. }
Amer. J. Math. {\bf 103} (1981),  445--468. 

\bibitem[Gi]{Gi} V. Ginzburg: {\it 
Characteristic varieties and vanishing cycles.}
 Invent. Math. {\bf 84} (1986), 327--402.

\bibitem[Gu]{Gu} N. Guay: \textit{Projective modules in the category $\oo$ for the
Cherednik algebra.} Preprint, University of Chicago, December 2001.


\bibitem[Hu]{hum} J. Humphreys: \textit{Reflection groups and Coxeter groups.}
Cambridge Studies in Advanced Mathematics, {\bf 29} 
Cambridge University Press, Cambridge, 1990.

\bibitem[J1]{J1} A. Joseph: {\it 
Primitive ideals in enveloping algebras.}
{\sl Proceedings of the ICM}
 (Warsaw, 1983), 403--414, PWN, Warsaw, 1984.

\bibitem[J2]{J2}  A. Joseph: {\it
On the associated variety of a primitive ideal.}
 J. Algebra {\bf 93} (1985),
509--523.

\bibitem[J3]{J3}  A. Joseph: {\it
Gelfand-Kirillov dimension for the annihilators of simple quotients of
Verma modules.} J. London Math. Soc. {\bf  18} (1978),  50--60.

\bibitem[KT]{KT} M. Kashiwara, T. Tanisaki: {\it 
The characteristic cycles of holonomic systems on a flag manifold
related to the Weyl group algebra.}
 Invent. Math. {\bf 77} (1984), 185--198.

\bibitem[Po]{Po} A. Polishchuk: {\it 
 Algebraic geometry of Poisson brackets.}
 J. Math. Sci. {\bf 84} (1997),  1413--1444.

\bibitem[SW]{SW}
J.T. Stafford, N.R. Wallach: {\it 
The restriction of admissible modules to parabolic subalgebras. }
Trans. Amer. Math. Soc. {\bf 272} (1982),  333--350.
 
\bibitem[Vo]{Vo} D. Vogan: {\it 
Associated varieties and unipotent representations. Harmonic analysis
on reductive groups.} 315--388, Progr. Math.,
{\bf  101}, Birkh\"auser Boston, Boston, MA,
1991.
%\bibitem[Vo2]{Vo2} D. Vogan: {\it 
%Gelfand-Kirillov dimension for Harish-Chandra modules.}
% Invent. Math. {\bf 48} (1978), 75--98.

\bibitem[YZ]{YZ} A. Yekutieli, J. Zhang: {\it
Rings with Auslander dualizing complexes.}
 J. Algebra {\bf 213} (1999),  1--51.

\end{thebibliography}