An analogue of the Kac-Weisfeiler conjecture

0.18 MB·English

by  Akaki Tikaradze

AN ANALOGUE OF THE KAC-WEISFEILER
CONJECTURE
2
AKAKITIKARADZE
1
0
2
Abstract. In this paper we discuss an analogue of the Kac-Weisfeiler

n conjectureforacertainclassofalmostcommutativealgebras. Inpartic-
a ular,weprovetheKac-WeisfeilertypestatementforrationalCherednik
J
algebras.

4
2

]
T 1. introduction
R
Throughout k = k¯ we be an algebraically closed field of characteristic
.
h
p > 2. Let A be an affine k-algebra which if finite over its center Z(A). In
t
a this setting one is interested in studying simple modules, in particular their
m
dimensions. By Schur’s lemma all simple A-modules are finite dimensional
[ and they are paramentrized by the corresponding characters of Z(A) : we

4 have a surjective map with finite fibers {Irr} → SpecZ(A), from the set
v of isomorphism classes of simple A-modules to the set of characters of χ ∈
7
Z(A). For each character χ ∈ SpecZ(A), we will denote by A the algebra
8 χ
3 A ⊗Z(A) k, where k is viewed as a Z(A) module via χ. We would like
2 to study the largest power of p that divides dimensions of all simple A -
χ
.
7 modules. Letusdenotethisnumberbyi(χ).Wewouldliketorelatefunction
0 i : SpecZ(A) → Z to geometry of SpecZ(A). More specifically, let ∪S =
+ i
0
SpecZ(A) be the smooth stratification of SpecZ(A), and for χ ∈ SpecZ(A)
1
: denote by s(χ) the dimension of the smooth stratum containing χ.
v
Motivated bytheKac-Weisfeiler conjecture[KW],whichisnowatheorem
i
X of Premet[P], it is tempting to state the following.
r
a Conjecture 1. Supposed that A is a nonnegatively filtered k-algebra, such
that grA is a finitely generated commutative domain over k. Assume that
(grA)p ⊂ grZ(A) and that SpecgrA is a union of finitely many symplectic
leaves and is a Cohen-Macaulay variety, then for any central character χ ∈
SpecZ(A), we have i(χ) ≥ 1s(χ).
2
Letusexplainhow doestheabove relate totheKac-Weisfeiler conjecture.
LetgbeaLiealgebraofasemisimplesimply-connectedalgebraicgroupG
over k.Thengiven thatpislargeenough,wehaveVeldkamp’stheorem([V])
describingthecenter of theenveloping algebra of g: Z(ug) = Zp(g)⊗Zp∩ZHC
Z where Z is the p-center, generated by elements gp −g[p],g ∈ g, and
HC p
Z = UgG is the Harish-Chandra part of the center. A character of Z(Ug)
HC
can bethought of as a pair (χ,λ),χ ∈ g∗,λ : Z → k. PutA= Ug/ker(λ).
HC
1
2 AKAKITIKARADZE
Then A inherits the filtration from Ug, and grA = k[N], where N is the
nilpotent cone of g∗. Thus, grAis a Cohen-Macaulay domain, and SpecgrA
is a union of finitely many symplectic leaves. Thus, assumptions of the
conjecture are met. We have that χ ∈ SpecZ(A). Now it follows that
i(χ) = dimGχ. Thus Conjecture 1 in this case says that any simple A-
1
module affording character χ has dimension divisible by p2dimGχ, which is
the statement of the Kac-Weisfeiler conjecture.
As a supporting evidence for the above conjecture, we will show it for χ
in the smooth locus of SpecZ(A) (Proposition 3.2). Also, we will show that
for all but finite χ ∈ SpecZ(A) any irreducible representation affording χ
has dimension divisible by p (Corollary 3.2).
Following the approach of Premet and Skryabin [PS], we prove the Kac-
Weisfeiler type statement (which is weaker than Conjecture 1) for a large
class of filtered algebras which includes rational Cherednik algebras (Corol-
lary 4.1).
2. Codimensions of Poisson ideals
We start by recalling the definition of algebraic symplectic leaves.
Definition 2.1. LetAbeaPoissonalgebraoverk.Thenaclosedsymplectic
leaf of SpecA is a closed subvariaty defined by a prime Poisson ideal I
such that the Poisson variety SpecA/I is a symplectic variety. A closed
symplectic leaf of a Poisson variety X is a closed subvariety Z ⊂ X such
thatforany affineopensubsetSpecA⊂ X,Z∩SpecAis aclosed symplectic
leaf of SpecA. Finally, an algebraic symplectic leaf of a Poisson variety X
is a closed symplectic leaf of an open subvariety of X.
We will recall the definition of Poisson orders by Brown-Gordon [BGo]
Definition 2.2. A Poisson order is a pair of an affine k-algebra A and its
central subalgebra Z , such that A is finitely generated moduleover Z , and
0 0
Z is a Poisson algebra, A is a Poisson Z -module such that for a ∈ Z , the
0 0 0
Poisson bracket {a,} is a derivation of A. In this case A is called Poisson
Z -order.
0
Definition 2.3. Let A be a Poisson Z -order. A Poisson A-module is a left
0
A-module M equipped with a k-bilinear map {,} :Z ⊗M → M such that
0
{{a,a′},m} = {a,{a′,m}}−{a′,{a,m}}
{a,bm} = {a,b}m+b{a,m}
for all a,a′ ∈ Z ,b ∈ A,m ∈ M.
0
Let A be an associative k-algebra, and let Z be its central subalgebra.
AssumemoreoverthatAisafinitelygeneratedZ-module. LetL ⊂DerZ(A)
be a Z-submodule which contains all inner derivations and is closed under
thecommutator bracket andtakingtothep-thpower. ThusLisarestricted
AN ANALOGUE OF THE KAC-WEISFEILER CONJECTURE 3
Lie subalgebraof DerZ(A). Let DL(A) denote the following algebra. DL′ (A)
is generated by L as an algebra over A subject to the following relations:
i a−ai = l(a),i i −i i = i ,
l l l1 l2 l2 l1 [l1,l2]
(il1)p = il1p,a ∈ A,l1,l2 ∈ L,iada−a = 0
It is immediate that if L is the set of all inner derivations, then D′ (A) = A.
L
Given a Poisson Z -order we will define algebra D′ (A) as follows. Let
0 Z0
L ⊂ Derk(A) be the restricted Lie subalgebra of Derk(A) generated by all
{a,−},a ∈ Z and all inner derivations. Then D′ (A) denotes D′ (A).
0 Z0 L
We will be primarily interested in two-sided Poisson ideals of A, i.e. two
sided ideals I ⊂ A, such that {z,a} ∈ I for all z ∈ Z,a ∈ I. Clearly given
such an ideal I, both I,A/I are D′ (A)-modules. The following is very
Z
0
standard
Lemma 2.1. Let A be a Poisson Z -order. Then A embeds in D′ (A) and
0 Z0
D′ (A) is finite over its central subalgebra (Z )p.
Z0 0
Proof. Let J is the kernel of the map A → D′ (A). Since A is naturally a
Z
0
D′ (A)-module, we have that JA= 0, so J = 0. It is immediate that (Z )p
Z0 0
is central in DZ′ 0(A), and since Derk(A) is a finite (Z0)p-module, we get
that D′ (A) is finite over (Z )p.
Z0 0
(cid:3)
Let X be a variety over k. Recall the definition of the quasi-coherent
sheaf of algebras of the crystalline differential operators D(X). Locally it is
generated by O and vector fields TX subject to the condition
X
[ξ,f]= ξ(f),ξ ·ξ −ξ ·ξ = [ξ ,ξ ],
1 2 2 1 1 2
where f,ξ and ξ are local sections of O ,TX respectively.
1 2 X
Given an affine Poisson k-algebra S, We will denote D′ (S) simply by
S
D′(S). Also, given a Poisson ideal I ⊂ S we will denote by D′(S,I) the
quotient of D′(S) by the two sided ideal generated by the image of I in
D′(S). Finally, for any closed point y ∈ SpecS/I, D′(S,I) will denote the
y
quotient of D′(S,I) by the two-sided ideal generated by the image of mp
y
(where m denotes the maximal ideal corresponding to y).
y
Clearly, D′(S,I) is a finite dimensional algebra over k.
y
WewillusethefollowingresultofBezrukavnikov-Mirkovic-Rumynin[BMR]
Theorem 2.1 ([BMR] Theorem 2.2.3). Let X be a smooth variety over k,
then the ring of crystalline differential operators D(X) is an Azumaya sheaf
of algebras over T∗X(1) (the Frobenius twist of the cotangent bundle of X).
Corollary 2.1. Let B is a finitely generated Poisson domain over k, such
that SpecB is a symplectic variety, then D′(B) is isomorphic to the matrix
y
algebra of dimension p2dimB over k.
4 AKAKITIKARADZE
Proof. SinceSpecB is asymplectic variety, it follows that D′(B) is isomor-
y
phic to the stalk of D(X) at (y,0) ∈T∗X(1). Therefore, by Theorem 2.1 we
are done.
(cid:3)
Recall that for any closed point y of a Poisson variety X, we denote
by d(y) the dimension of the symplectic leaf of X which contains y. The
following will be crucial.
Theorem 2.2. Let A be a Poisson S-order, and let y ∈ SpecS be a closed
point. Then any finite dimensional Poisson A-module which as an S-module
is supported on y has dimension divisible by pd(y).
Proof. Let us put d(y) = d. For a Poisson algebra S, we will denote by
Der′(S) ⊂Der(S) thek-span of all derivations of the forma{b,−},a,b ∈ S.
Let Y ⊂ SpecS bethe symplectic leaf containing y. Let I bea Poisson ideal
corresponding to Y. Let f ∈ S be an element which does not vanish on y
and vanishes on Y −Y. Let us put S′ = S , then D′(S′,I) = D′(S,I) and
f y y
SpecS′ containstheclosedsymplecticleafthroughy.AnyPoissonS-module
M supported on y is D′(S′) -module. Consider a descending filtration M ⊃
y
IM ⊃ I2M ··· ⊃ 0. Since I ⊂ m , we have IlM = 0 for some l. Each
y
quotient IjM/Ij+1M is a module over D(S′,I) . Therefore, it suffices to
y
prove that any finite dimensional D(S′,I) -module has dimension divisible
y
by pd(y).
Denote S′/I by B. Thus, SpecB is a symplectic variety. We have a
natural projection j : D′(S′,I) → D′(B) . Since the localization of B at
y y
m is a regular local ring with the residue field k, we have that B/mp =
y y
k[x ,··· ,x ]/mp, where m = (x ,··· ,x ). Remark that B/(m )p is the
1 d y 1 d y
image of S′ in D′(S,I) . We will denote the images of x ,··· ,x in D′(S,I)
y 1 d
again by x ,··· ,x . Let y ,··· ,y ∈ Der′(B/mp) be such that
1 d 1 d y
[y ,x ] = δ ,[y ,y ]= 0.
i j ij i j
Thus, D′(B)y is generated by Pai<py1a1···ydad as a free left (or right)
p
B/m -module. Let ξ ,··· ,ξ be any lifts of y ,··· ,y in the image of
y 1 d 1 d
Der′(S′) in D′(S,I) . Notice that [ξ ,x ] = δ . Let us denote by J a k-
y i j ij
subalgebra of D′(S,I) generated by elements of Der′(S) whose images are
y
in I. Then J is an ideal of the Lie algebra D′(S,I)p and [B/mp,J] = 0. De-
y y
notebyN ⊂D′(S,I)py thek-span ofelements of theformξ1a1···ξnad,ai < p,
so dimN = pd. For any ξ ∈ Der′(S), its image in D′(B) can be written
y
as d b y for some b ∈ B/m p. Thus ξ − b ξ ∈ J, so Der′(S) ⊂
Pi=1 i i i y P i i
NJ(B/m p). Therefore, D′(S,I)p = NJ(B/m p).
y y y
Let V be a simple D′(S,I)p-module. Let v 6= 0,v ∈ V be such that
y
m v = 0. Then m Jv = 0, so NJv = V. Now we claim that if v ,··· ,v ∈
y y 1 k
Jv are linearly independent and k n v = 0,n ∈ N, then n = 0 for
Pi=1 i i i i
all i. Indeed, we have that [x ,n ]v = ∂niv = 0. Proceeding by
Pi j i i Pi ∂ξj i
AN ANALOGUE OF THE KAC-WEISFEILER CONJECTURE 5
induction on the total degrees of n in ξ ,··· ,ξ , we are done. Therefore,
i 1 n
dimV = dimN dim(Jv) is a multiple of pd.
(cid:3)
Proposition 2.1. Let M be a finite dimensional Poisson module over a
Poisson S-order. Then dimM is divisible by inf{pd(y,y ∈ Supp(M) ⊂
SpecS}.
Proof. Let us write M = ⊕ M , where M is the submodule of
y∈Supp(M) y y
elements of M supported on y. Obsereve that M is actually a Poisson sub-
y
module of M. Indeed, Let a ∈ M , then (m )pia = 0, for some i. Therefore,
y y
for any a ∈ S, (m )pi{a,m} = 0, since {a,mpi}= 0. Now applying Theorem
y y
2.2 to each M , we are done. (cid:3)
y
Recall the following standard definition.
Definition 2.4. A quantization of a Poisson S-order A is an ~-complete
flat associative k[[~]]-algebra A′, such that A= A′/~A′ and Poisson bracket
of S is induced from the commutator bracket of S′.
Proposition 2.2. Let A′ be a quantization of a Poisson S-order A. Let
J ⊂ A′[~−1] be a two-sided ideal of finite codimension over k((~)), then
dimk((~)) A′[~−1]/J is a multiple of inf (pd(y),y ∈ suppA/((A′∩J)/~).
Proof. Let us put J′ = J ∩ A′, then A′/J′ is a free k[[~]]-module and
A′[~−1]/J = A′/J′⊗k[[~]]k((~)).Thereforedimk((~))A′[~−1]/J = dimk[[~]]A′/J′.
LetusputJ′/~ = N ⊂ A.ThenN isaPoissonsubmoduleofinA,andA′/J′
is a quantization of A/N. Therefore, dimk[[~]]A′/J′ = dimkA/N. Thus, ap-
plying Corollary 2.1 to A/N we are done.
(cid:3)
3. Estimates for filtered algebras
Let A be an associative k-algebra equipped with a positive filtration by
k-subspaces A ⊂ A ··· ⊂ A ··· ,A A ⊂ A ,A = ∪A . Recall that
0 1 n n m n+m n
the center of the associated graded algebra grA = ⊕A /A becomes
n n−1
equipped with the natural Poisson bracket and grA is a Poisson module
over it. Indeed, let d be the largest integer such that [a,b] ⊂ A
n+m−d
for any b ∈ A ,a ∈ A ,gra ∈ Z(grA). Then one puts {gra,grb} =
m n
[a,b]/A ∈ (grA) .
n+m−d−1 n+m−d−1
In this section we apply results from the previous section to representa-
tions of certain class of filtered affine k-algebras.
RecallforafilteredalgebraAtheconstructionoftheReesalgebraR(A) =
⊕ A ~n ⊂ A[~], where ~ is an indeterminate. Then
n n
R(A)/~R(A) = grA,R(A)/(~−λ)R(A) = A,R(A)[~−1] = A[~,~−1],
for any λ ∈ k,λ 6= 0. Since Z(A) inherits filtration from A, we have
R(Z(A)) = Z(R(A)). WewillfixtheembeddingA ⊂ R(A)[~−1] =A[~,~−1].
6 AKAKITIKARADZE
More generally, if Z ⊂ Z(A) is a subalgebra such that grA is finite over
0
grZ , then R(A) is finite over R(Z ).
0 0
We will need the following standard fact.
Proposition 3.1. Let M be a nonegatively filtered module over a nonnega-
tivelyfiltered commutative k-algebra B, suchthat grM is a finitely generated
Cohen-Macaulay module over grB and grB is a finitely generated algebra
over k. Then both M,R(M) are Cohen-Macaulay modules over grB,R(B)
respectively.
We will recall the proof for the convenience of the reader.
Proof. Letuschoosealgebraicallyindependenthomogeneouselementsx ,··· ,x ∈
1 n
grB, such that grB is finite over k[x ,··· ,x ]. Then grM is a finitely gen-
1 n
erated k[x ,··· ,x ]-module, and since it is a Cohen-Macaulay module it is
1 n
a projective (by Lemma 2.2, 2.4 [BBG]), and hence by the Quillen-Suslin
theorem a free k[x ,··· ,x ]-module. Let y ··· ,y ∈ grM be a homoge-
1 n 1 m
neous basis of grM over k[x ,··· ,x ]. Let a ,··· ,a ,b ,···b be the any
1 n 1 n 1 m
lifts of x ,··· ,x ,y ,··· ,y in B,M respectively. Then it follows immedi-
1 n 1 m
ately that a ,··· ,a are algebraically independent and M,R(M) is a free
1 n
B,R(B)-module with basis b ,··· ,b ; b ~degbi respectively
1 n i
(cid:3)
Proposition 3.2. Suppose that A is a nonnegatively filtered k-algebra, such
that grA an affine commutative Cohen-Macaulay domain. Suppose also that
SpecgrA consists of finitely many symplectic leaves. If (grA)p ⊂ grZ(A)
then for any character χ which belongs to the smooth locus of SpecZ(A), a
1
simple A-module affording χ has dimension p2dimA.
Proof. Applying [[T] Lemma 2.4], we get that grZ(A) = (grA)p. By [[T]
Theorem 2.3], the complement of the Azumaya locus of Z(A) has codi-
1
mension 2, and the largest possibledimension of A-module is p2dimgrA. Let
U ⊂ SpecZ(A) bethesmooth locus of SpecZ(A). We claim that A is alo-
|U
cally free sheaf over U. Indeed, by the assumption grA is a Cohen-Macaulay
module over (grA)p. But then by 3.1 A is a Cohen-Macaulay module over
Z(A), since grZ(A) = (grA)p. So A| is a Cohen-Macualay over U, and
U
since U is nonsingular, we get that A| is locally free over U. To summarize,
U
A| is locally free and its Azumaya locus has complement of codimension
U
≥ 2. Therefore, A| is an Azumaya algebra over U by [[BG] Lemma 3.6].
U
(cid:3)
From now on in this section we will follow very closely Premet and
Skryabin [PS]. We will use the following result from [PS].
Lemma 3.1([PS]Lemma2.3). LetAbe finite and projective overitscentral
affine subalgebra Z0, and let L ⊂ DerZ0(A) be a restricted Lie subalgebra
of derivations. Then for any i ≥ 0, the set of characters χ ∈ SpecZ , such
0
that A has an L-stable two sided ideal of codimension i is closed.
χ
AN ANALOGUE OF THE KAC-WEISFEILER CONJECTURE 7
Ofcourse,L-stabletwo-sidedidealofAisthesameasaD′ (A)-submodule
L
of A.
From now on in this section, by L ⊂ DerR(Z0)(R(A)) we will always
denote the restricted Lie algebra generated by all inner derivations and by
~−dad(a), for all a ∈ R(A), such that a/~ ∈ Z(grA). In this setting, we
have the following
Lemma 3.2. D′ (R(A))/~ = D′(grA) and D′ (R(A))[~−1]= A[~,~−1].
L L
Proof. First equality is immediate. The second equality follows from the
fact that L consists of inner derivations in Der(A[~,~−1]).
(cid:3)
We will use the following
Assumption 1. Let A be a positively filtered k-algebra, such that Z(grA)
is finitely generated over k and grA is a finitely generated module over
Z(grA). Moreover, assume that there in a positive integer n, such that
(Z(grA))pn ⊂ grZ(A).
Inwhatfollows, wewillusethefollowingstandardnotations. Forasubset
W of an affine variety X, we will denote by I(W) the reduced ideal of zeros
of W in O(X), and for an ideal I ⊂ O(X),V(I) will denote the set of zeros
of I in X.
We have the following
Theorem 3.1. Let algebra A satisfy the assumption 1. Let Z ⊂ Z(A) be a
0
subalgebra such that R(A) is finitely generated projective module over R(Z ).
0
Let W ⊂ SpecZ be a closed subset consisting of points χ ∈ W such that
0
A has a two sided ideal of codimension pij where p does not divide j, then
χ
for any y′ ∈ V(grI(W)) ⊂ SpecgrZ0, there is y ∈ SpecZ(grA) which is in
the preimage of y′ in SpecZ(grA) → SpecgrZ , such that d(y) ≤ i.
0
Proof. Recall that R(z )[~−1] = Z [~,~−1]. We again denote by L the Lie
0 0
subalgebraofDer(R(A))asinabovelemma. SpecZ ⋉(A1−0) ֒→ SpecR(Z ).
0 0
We a hve a map ρ : Z(grA) → HH1(R(A)) defined as ρ(a) = 1 ad(a′),
~d
where a′ ∈ p−1(a),a ∈ Z(grA). and p : R(A) → R(A)/~ = grA is the quo-
tient map. Let X denote the set of all closed points χ ∈ SpecR(Z0) such
that R(A) has a two sided L-stable ideal of codimension pij. By lemma
χ
3.1, X is closed. We claim that X ∩ {~ = 0} is the set of all characters
χ : grZ0 → k such that there is a Poisson ideal in grA containing ker(χ)
of codimension pij. Indeed, for two-sided ideal I ⊂ Gr to be closed under
the map {a,} for all a ∈ Z(grA) is the same as ideal p−1(I) being L-stable.
Similarly, X∩{~ 6= 0 consists of characters χ : Z (A)[~,~−1]→ k such that
0
A has a two-sided ideal of codimension pij. Indeed, given a two sided ideal
χ
I ⊂ A which contains ker(χ) ∩ Z (A) ideal p−1(I) is L-stable, since ~ is
0
invertible on R(A)/p−1(I), where p : R(A) → R(A)/(~−χ(~)) = A is the
quotient map.
8 AKAKITIKARADZE
Therefore, W ×(A1−0) ⊂ X, In particular, W ×(A1−0)∩(~ = 0) ⊂ X.
But W ×(A1−0)∩(~= 0) is precisely V(grI(W)). Thus for any character
χ ∈ V(gr(I(W))) Algebra (grA)/m (grA) has a Poisson ideal of codimen-
χ
sion pij, where m is the maximal ideal of grZ ⊂ (grA)p corresponding
χ 0
to χ.. Let us choose such an ideal J ⊂ (grA)/m p. Let {y ,i ∈ I} be the
χ i
(finite) preimage of χ under the map SpecgrA → SpecgrZ . Then grA/J
0
is a Poisson Z(grA)- module supported on {y }. Applying Corollary 2.1, we
i
are done,
(cid:3)
Recall that for a character χ, i(χ) denotes the largest power of p which
divides all dimensions of irreducible representations affording χ.
Corollary 3.1. Let A satisfy the assumption 1. Let Z be a subalgebra of
0
Z(A)suchthatR(A)isafinitelygeneratedprojective module overR(Z ).Let
0
G ⊂ Aut(R(A),R(Z )) be a subgroup of the group of k[~]-algebra automor-
0
phisms of R(A) which preserve R(Z ). Then, for any χ ∈SpecZ , and any
0 0
y ∈ V(grI(Gχ)) ⊂ grZ , we have i(χ) ≥ infd(y′), where y′ ∈ SpecZ(grA)
0
runs through preimages of y under the map SpecZ(grA) → SpecgrZ .
0
Proof. Let W ⊂ SpecZ be a set of characters χ′ such that i(χ′) = i(χ).
0
So χ ∈ W and W is closed by Lemma 3.1. Therefore Gχ ⊂ W, and by
Corollary 2.1, we are done.
(cid:3)
Corollary 3.2. Let A be satisfy the assumption 1. Then if SpecZ(grA)
is a union of algebraic symplectic leaves, and if SpecZ(grA) has at most
one (the origin) zero dimensional symplectic leaf, then all but finitely many
irreducible representations of A have dimension divisible by p.
Proof. Recallthatgivenafinitedimensionalk-alebraS,allsimpleS-modules
have dimensions divisible by p if and only if 1 ∈ [S,S]. Therefore, we are
required to show that the support of 1 ∈ A/[A,A] on SpecZ(A) is a fi-
nite set. Let Y be the support of 1 ∈ D′ (R(A)/[D′ (R(A),D′ (R(A)] on
L L L
SpecZ(D′ (R(A)). We have that Y ∩{~ 6= 0} is X ×(A−{0}), where X is
L
the support of 1∈ A/[A,A] on SpecZ(A). On the other hand, Y ∩{~ = 0}
is the support of 1 ∈ D′(grA)/[D′(grA),D′(grA)] in SpecZ(D′(grA)). By
the following lemma, the letter set is finite, so X has to be finite and we are
done.
Lemma3.3. Undertheassumptionofthetheorem, 1 ∈ D′(grA)/[D′(grA),D′(grA)]
is supported on the origin of SpecZ(grA)p.
Proof. Indeed, if y ∈ SpecZ(grA)p,y 6= 0 (where 0 denotes the origin of
SpecZ(grA)), then all representations of D′(grA))/m p are divisible by
y
pd(y)(as in the proof of Theorem 2.2. But since d(y) > 0, we conclude that
allsimpleD′(grA))/m p-moduleshave dimensionsdivisiblebyp.Therefore,
y
1∈ D′(grA))/[D′(grA),D′(grA)]+m pD′(grA) so y ∈/ Supp(1).
y
(cid:3)
AN ANALOGUE OF THE KAC-WEISFEILER CONJECTURE 9
(cid:3)
Inparticular,theassumptionsoftheabovearesatisfiedwhenSpecZ(grA)
consists of finitely many symplectic leaves. This result can be thought of as
a positive characteristic analogue of a result due to Etingof-Schedler [ES],
whichstatesthatifAisanonnegativelyfilteredalgebraoverCsuchthatgrA
isfiniteoveritscenterandSpecZ(grA)isaunionoffinitelymanysymplectic
leaves, then A has a finitely many nonisomorphic finite dimensional simple
modules.
4. Applications to rational Cherednik algebras
At first, we will recall the decomposition of V/W into symplectic leaves
[[BGo], Proposition 7.4], where V is a symplectic vector space over k and
W ⊂ Sp(V) is a finite group such that p does not divide |W|. Given a
subgroupH ⊂ W denotebyVH thesetofallv ∈ VH suchthatthestabilizer
0
of v is H. Then we have the decomposition of V/W into symplectic leaves
V/W = ∪VH/W,whereH rangesthroughallconjugacyclassesofsubgroups
0
of W. In particular, for v ∈ V/W dimension of the symplectic leaf through
v is dimVWv, where W is (a conjugacy class) the stabilizer of v.
v
Recall that for a class function c : W → k, Etingof-Ginzburg defined a
symplectic reflection algebra H (W,V). This algebra comes equipped with
c
the natural filtration such that grH (W,V) = k[W] ⋉ SymV, (the PBW
c
property, [EG] Theorem 1.3).
Proposition 4.1. Let W ⊂ Sp(V) be a finite group. If p does not divide
|W|, then for a symplectic reflection algebra H (W,V), there are only finitely
c
many irreducible representations whose dimension is not divisible by p.
Proof. Thecenter of grH = k[W]⋉SymV is (SymV)Γ, so grH (W,V) is
1,c c
finite over its center. Also, by a theorem of Etingof ([BGF], Theorem 9.1.1)
grZ(H ) = ((SymV)Γ)p. Since as explained above, Spec(SymV)Γ = V/W
1,c
is a union of finitely many symplectic leaves, all the assumptions of 3.2 are
satisfied and we are done.
(cid:3)
ImportantclassofsymplecticreflectionalgebrasconsistsofRationalChered-
nik algebras. Let us recall their definition.
Let h be a finite dimensional vector space over k. Given a finite group
W ⊂ GL(h), let S ⊂ Γ be the set of pseudo-reflections (recall that s ∈ W
is a pseudo-reflection if Im(Id − s) 1-dimensional). Let α ∈ h∗ be the
s
generator of Im(s−1)|h∗ and α′s ∈ h be the generator of Im(Id−s) such
that (α ,α′) = 2. Let c : S → k be a W-invariant function. The rational
s s
Cherednikalgebra,H (W,h),asintroducedbyEtingofandGinzburg[EG],is
c
thequotientoftheskewgroupalgebraofthetensoralgebra,k[W]⋉T(h⊕h∗),
by the ideal generated by the relations
[x,x′]= 0,[y,y′]= 0,[y,x] = (y,x)−Xcs(y,αs)(x,α′s)s,
10 AKAKITIKARADZE
x,x′ ∈ h,y,y′ ∈ h∗.
There is a standard filtration on H given by setting degx = 1,degy =
c
1,deg(g) = 0 for all x ∈ V,y ∈ h∗,g ∈ Γ. The PBW property of H (W,h)
c
saysthatgrH = kΓ⋉Sym(h⊕h∗). Whencisidentically 0thenH (W,h) =
c 0
k⋉D(h), where D(h) is the ring of (crystalline) differential operators on h.
Algebra H (W,h) has a distinguished central subalgebra Z = Z′ ⊗Z”,
c 0
whereZ′(respectively Z”)denotes ((Symh∗)W)p(((Symh)W)p [[BC], Propo-
sition 4.2].
For a rational Cherednik algebra H (W,h) we have the spherical subal-
c
gebra B = eH (W,h)e (e = 1 g ∈ W). Algebra B inherits a filtration
c c |W|P c
from Hc(W,h) and grBc = (Sym(h⊕h∗))W. Then eZ0 ⊂Z(Bc)
′
Corollary 4.1. Assume that p does not divide |W|. Let χ : eZ → k be
a central character. Let us identify χ : e((Symh∗)W)p → k with a point
in h/W. Let W be the stabilizer in W of a preimage of χ in h. Then all
χ
irreducible representations of B affording χ have dimensions divisible by
c
hWχ
p .
Proof. We need to prove that given a character µ : eZ = eZ′ ⊗eZ” → k,
0
such that µ| : eZ′ = χ, then any B -module affording µ has dimension
c
divisible by phWχ. Let us write µ = (χ,χ′), χ′ ∈ SpeceZ”.
Since, Sym(h⊕h∗)W = grB is a Cohen-Macaulay algebra, it is a Cohen-
c
Macaulay module over greZ , hence by Proposition 3.1 R(B ) is a Cohen-
0 1,c
Macauly module over R(eZ ). However, R(eZ ) is a polynomial algebra,
0 0
therefore R(B ) is a projective (actually free by the Quillen-Suslin the-
1,c
orem) R(eZ )-module. eZ is a polynomial algebra, it follows that B is
0 0 c
projective (actually free) over eZ . We have an action of G on H (W,h)
0 m c
which preserves B corresponding to the grading with deg(x)= 1,deg(y) =
c
−1,deg(g) = 0,g ∈ Γ = 0,x ∈ h,y ∈ h∗. This action preserves eZ . There-
0
fore we may apply 3.1. We need to understandV(grI(G µ)) ⊂ SpecegrZ
m 0
and its preimage in SpecgrB .
c
We haveI(G µ)∩eZ′ = I(G χ).ThengrI(G µ)∩eZ′ = grI(G χ).In-
m m m m
deed,clearly grI(G χ) ⊂ grI(G µ)∩greZ′.Supposethatf ∈ grI(G µ)∩
m m m
eZ′. Therefore, there is g ∈ greZ” such that degg < degf and f + g ∈
I(G µ). Thus, f(tχ) + g(t−1χ′,tχ) = 0,t ∈ k∗. But the letter is a Lau-
m
rent polynomial with leading term tdegff(χ). Hence f(χ) = 0, so f ∈
grI(G χ). Therefore p(V(grI(G µ)) = V(grI(G χ)) = kχ, where p :
m m m
Spec(eZ′⊗Z”)→ SpeceZ′ is the projection. Thuswe conclude that thereis
χ”∈ SpeceZ”, such that Let (χ”,χ) ∈ V(grI(G µ). Then if v ∈ (h×h∗)W
m
isapreimageof (χ”,χ)underthemapSpecgrB =(h×h∗)W → Spec(eZ′⊗
c
eZ”), then the projection of v its projection on h/W is χ.
We conclude that W is a subgroup of W , so (h×h∗)Wχ ⊂ (h×h∗)Wv.
v χ
Therefore,Usingthedescriptionofsymplecticleaves of(h⊕h∗)/W discussed
above, we conclude d(v) ≥ 2dimhWχ. So, applying 3.1 H (W,h)-module
c