Cherednik algebras and differential operators on quasi-invariants

0.56 MB·English

by  Yuri Berest

CHEREDNIK ALGEBRAS AND DIFFERENTIAL
OPERATORS ON QUASI-INVARIANTS
Yuri Berest, Pavel Etingof and Victor Ginzburg
Abstract
Wedevelop representation theoryofthe rationalCherednikalgebraHc associatedto afinite Coxeter
0 group W in a vector space h, and a parameter ‘c’. We use it to show that, for integral values of ‘c’,
1 thealgebraHc issimpleandMoritaequivalent toD(h)#W,thecrossproductofW withthealgebra
0 ofpolynomialdifferentialoperators onh.
2 Chalykh,Feigin,andVeselov[CV],[FV],introducedanalgebra,Qc,ofquasi-invariantpolynomials
n on h, such that C[h]W ⊂ Qc ⊂ C[h]. We prove that the algebra D(Qc) of differential operators on
a quasi-invariantsisasimplealgebra,Moritaequivalent toD(h). ThesubalgebraD(Qc)W ⊂D(Qc)of
J W-invariant operators turns out to be isomorphic to the spherical subalgebra eHce ⊂Hc. We show
6 thatD(Qc)isgenerated,asanalgebra,byQc andits‘Fourierdual’Q♭c,andthatD(Qc)isarankone
projectiveQc⊗Q♭c-module(viamultiplication-actiononD(Qc)onoppositesides).
]
A Table of Contents
Q 1. Introduction
2. Standard modules over the rational Cherednik algebra
h. 3. Harish-Chandra Hc-bimodules
t 4. The spherical subalgebra eHce
a 5. A trace on the Cherednik algebra
m 6. The eHce-module structure on quasi-invariants
7. Differential operators on quasi-invariants
[
8. Translation functors and Morita equivalence

6 9. Applications of the shift operator
v 10. Appendix: A filtration on differential operators
5
0
0 1 Introduction
1
1
1 Let W be a finite Coxeter group in a complex vector space h, and R ⊂ h∗ the corresponding
0 set of roots. To each W-invariant function c : R → C, c 7→ c , one can attach an associative
α
/
h algebraHc,calledtherational Cherednik algebra. Thisisaveryinterestingalgebratiedtoexciting
t works in combinatorics, completely integrable systems, and generalized McKay correspondence.
a
m Historically, the rational Cherednik algebra appeared as a ‘rational’ degeneration of the double-
affine Hecke algebraintroducedbyCherednik[Ch2]. Thus,thelattermay(andshould)bethought
:
v of as a deformation of the former. From this point of view, representation theory of the rational
i Cherednik algebra is perhaps ‘more basic’ than (or at least should be studied before) that of
X
the double-affine Hecke algebra in the same sense as the representation theory of semisimple Lie
r
a algebras is ‘more basic’ than that of the corresponding quantum groups. Thus, one of our goals is
to begin a systematic study of H representation theory.
c
Given α∈h∗, write α∨ ∈h for the coroot, and s ∈GL(h) for the reflection corresponding to
α
α. Recall from [EG], that the rational Cherednik algebra H (which was denoted H in [EG]) is
c 1,c
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
w·x·w−1 =w(x) , w·y·w−1 =w(y), ∀y ∈h, x∈h∗, w ∈W
x ·x =x ·x , y ·y =y ·y , ∀y ,y ∈h, x , x ∈h∗
1 2 2 1 1 2 2 1 1 2 1 2 (1.1)
y·x−x·y =hy,xi− c ·hy,αihα∨,xi·s , ∀y ∈h, x∈h∗.
α α
α∈RP/{±1}
1
Thus, the elements x ∈ h∗ generate a subalgebra C[h] ⊂ H of polynomial functions on h, the
c
elements y ∈ h generate a subalgebra C[h∗] ⊂ H , and the elements w ∈ W span a copy of the
c
group algebra CW sitting naturally inside H . Furthermore, it has been shown by Cherednik, see
c
also [EG], that multiplication in H induces a vector space isomorphism:
c
C[h] ⊗ CW ⊗ C[h∗] −∼→ H (Poincar´e-Birkhoff-Witt isomorphism for H ). (1.2)
C C c c
The name for the isomorphism above comes from its analogy with the well-known isomorphism:
U(n )⊗ U(h)⊗ U(n ) −∼→ U(g)fortheenvelopingalgebraofacomplexsemisimpleLiealgebra
− C C +
g with triangular decomposition: g=n +h+n .
+ −
The Poincar´e-Birkhoff-Witt isomorphism for H allows one to introduce a category O of
modules over the algebra H similar to the categocry O of highest weight modules over UHc(g),
c g
consideredby Bernstein-Gelfand-Gelfand,see[BGG].The categoryO splits upintoadirectsum
of its subcategories O (λ¯), one for each λ¯ ∈ h∗/W. We will be mHacinly concerned below with
the category O (0), wHchich is most interesting among all O (λ¯)’s. The category O (0) is the
Cherednik algebHcra counterpart of the subcategory O (χ) ⊂ HOc corresponding, in theHcBernstein-
g g
Gelfand-Gelfand setting1, to a fixed character χ of the center of U(g).
Theisomorphism(1.2)showsthatthegroupalgebraCW playsaroleofthesubalgebraU(h)⊂
U(g). Thus, following the classical construction due to Verma, to each irreducible representation
τ ∈Irrep(W)onecanassociatea‘standard’moduleM(τ)∈O (0),ananalogueofVermamodule.
Hc
It is easy to show, see [DO], that eachstandard module M(τ) has a unique simple quotient, L(τ).
Furthermore,anyobjectofthecategoryO (0)hasfinitelength,andthecollection{L(τ)}
isacompletecollectionofisomorphismclaHscsesofsimpleobjectsofO (0). Thus,thesimpτle∈Iorrbepj(eWct)s
inO (0)areparametrizedbythesetIrrep(W),whilethesimpleobjHecctsinO (χ)areparametrized
Hc g
(for regular χ) by elements of the Weyl group W itself.
The structureofthe categoryO (0) depends cruciallyonthe value ofthe parameter‘c’. The
Hc
categoryissemisimpleforalmostall‘c’,see[OR],inwhichcaseM(τ)=L(τ),foranyτ. However,
for a certain set of ”singular” values of ‘c’, the multiplicities [M(τ) : L(σ)] are unknown. These
multiplicities are Cherednik algebra analogues of Kazhdan-Lusztig type multiplicities for affine
Hecke algebras, see [CG], and they are expected to be provided by some Intersection cohomology.
In this paper we are concerned with the case of integral values of ‘c’, which is, in a sense,
intermediatebetweenthetwoextremecasesabove. Wewillsee,althoughitisnotaprioriobvious2,
that, for any such ‘c’, the category O (0) is semisimple. Further, we apply H -representation
Hc c
theory to show that, for integral ‘c’, the algebra H is simple and Morita equivalent to D(h)#W,
c
the cross product of W with the algebra D(h) of polynomial differential operators on h.
Our motivation to study the case of integral ‘c’ comes also from an interesting connection
with the theory of Calogero-Moser integrable systems and the theory of differential operators on
singular algebraic varieties. In more detail, let C[h]W ⊂ C[h], be the subalgebra of W-invariant
polynomials. Foreachnon-negativeintegralvalue of‘c’, Chalykh,Feigin, andVeselov[CV], [FV],
have introduced an algebra Q of so-called W-quasi-invariant polynomials, such that C[h]W ⊂
c
Q ⊂ C[h]. We use representation theory of Cherednik algebras to study differential operators
c
on quasi-invariants, that is, the algebra D(Q ) of differential operators on the singular variety
c
SpecQ . The Calogero-Moser differential operator may be viewed as an analogue of the second
c
order Laplacian on the variety SpecQ .
c
Itturnsoutthatthe algebraD(Q )isalmostasnice asthe algebraofdifferentialoperatorson
c
asmoothvariety. Specifically,weprovethatthe algebraD(Q )isMoritaequivalentto thealgebra
c
1 We emphasize that it is the parameter ‘c’, and not λ¯ ∈ h∗/W that plays the role of central character in
representationtheoryofHc. Theparameterλ∈h∗plays,inourpresentsituation,theroleof‘Whittakercharacter’,
seeremarkafterCorollary4.5.
2Seeremarkattheendof§2,andalso[OR].
2
ofpolynomialdifferentialoperatorsonthevectorspaceh. ThesubalgebraD(Q )W ofW-invariant
c
differential operators will be shown to be isomorphic to the spherical subalgebra eH e. Further,
c
D(Q )isasimplealgebraequippedwithanaturalinvolution,ananalogueoftheFouriertransform
c
for differential operators on SpecQ . The algebra D(Q ) contains Q and its ‘Fourier dual’, Q♭,
c c c c
as two maximal commutative subalgebras. We show that D(Q ) is generated, as an algebra, by
c
Q and Q♭, and that D(Q ) is a rank one projective Q ⊗Q♭-module (under multiplication-action
c c c c c
on D(Q ) on opposite sides).
c
The rings of differential operators on general singular algebraic varieties typically have rather
unpleasantbehavior(e.g. arenot Noetherian, cf. [BGG1]). The questionof simplicity andMorita
equivalenceofsuchringshasbeenstudiedbyseveralauthors,see[BW],[Sm],[Mu],[SS],[HS],[CS],
[VdB], (and also [MvdB] for a result similar in spirit to ours). The varieties SpecQ studied in
c
the present paper seem to be natural generalizationsof the one-dimensionalexamples constructed
in [Sm] and [Mu].
The paper is organized as follows. Sections 2,3,5 are devoted to ‘pure’ representation theory
of Cherednik algebras. In §2 we exploit an idea due to Opdam relating highest weight modules
over the Cherednik algebra to finite dimensional representations of the Hecke algebra H (e2πic);
W
thus §2 has some overlap with [DO] and [OR]. In section 3 a new and quite useful notion of a
Harish-Chandra H -bimodule is introduced. In §4 we prove a Cherednik algebra counterpart of
c
an important theorem (due to Levasseur-Stafford [LS]), saying that the spherical subalgebra in
H is generated, for all regular values of ‘c’, by its two invariant commutative subalgebras, C[h]W
c
and C[h∗]W. This result has numerous applications. In §5 we compute a trace on the Cherednik
algebra of type A, and use it to derive some applications to finite-dimensional representations of
H . Quasi-invariants are introduced in §6, and the structure of the algebra D(Q ) is studied in
c c
detail in §§7,9. The structure of D(Q ) is in turn exploited in section 8 to get further results in
c
representationtheoryofCherednikalgebras,e.g.,togiveanexplicitconstructionofsimpleHarish-
Chandra bimodules. These results of §8 bear some resemblance with the technique of translation
functors, a well-knownandverypowerfultool in representationtheory ofsemisimple Lie algebras.
Acknowledgments. We are very grateful to E. Opdam and R. Rouquier for generously sharing their ideas
withus,andformakingtheresultsof[OR]availabletousbeforeitspublication. Theseresultsplayacrucialrolein
ourargumentsand,toagreatextent,hastriggeredthepresentwork. WewouldliketothankT.Staffordformany
interesting comments and other useful information that was quite essential for us. The first author was partially
supported by the NSF grant DMS 00-71792 and A. P. Sloan Research Fellowship; the work of the second author
waspartlyconducted fortheClayMathematicsInstituteandpartiallysupportedbytheNSFgrantDMS-9988796.
2 Standard modules over the rational Cherednik algebra
Mostoftheresultsofthissection(inparticular,Theorem2.2andLemma2.10)areduetoOpdam-
Rouquier [OR], and are reproduced here for the reader’s convenience only.
Fix a finite Coxeter group W in a complex vector space h. Thus, h is the complexification of
a real Euclidean vector space and W is generated by reflections with respect to a certain finite
set {H } of hyperplanes in that Euclidean space. We write (·, ·) for the complex bilinear form
α
on h extending the Euclidean inner product. For each hyperplane, H , we choose nonzero linear
α
functions ±α ∈ h∗ which vanish on H . The set R ⊂ h∗ of all such linear functions is called
α
the set of roots. Write s ∈ GL(h) for the reflection corresponding to α ∈ R, and α∨ ∈ h for
α
the corresponding coroot, a vector such that s (α∨) = −α∨. The lengths of roots and coroots
α
are normalized so that hα,α∨i = 2. We make a choice of the set R of positive roots so that
+
3
R = R ⊔ (−R ). The W-action on h induces canonical actions on the symmetric algebras
+ +
Sym(h)=C[h∗] and Sym(h∗)=C[h].
Let hreg denote the complement to the union of all the reflection hyperplanes, i.e., the comple-
ment to the zero set of the discriminant polynomial δ = α∈C[h]. Given a C[h]W-module
α∈R+
M,wewillwriteM| forC[hreg]W⊗ M,thelocalizQationtohreg/W. NotethatifM isaC[h]-
hreg/W C[h]W
moduleviewedasaC[h]W-modulebyrestrictionofscalars,thenM| isaC[hreg]-modulewhich
hreg/W
coincides with M|hreg :=C[hreg]⊗C[h] M. In particular, Hc|hreg = C[hreg]⊗C[h] Hc, the localization of
the left regular H -module, acquires a natural algebra structure, hence an H -bimodule structure,
c c
suchthattheimbeddingH ֒→ C[hreg]⊗ H becomesanalgebramap. Alternatively,thealgebra
c C[h] c
Hc|hreg is obtained by Ore localization of Hc with respect to the multiplicative set {δk}k=1,2,....
The group W acts freely on hreg. Let B be the braid group of W, that is the fundamental
W
group of the variety hreg/W. Fix a point ∗ ∈ hreg inside a Weyl chamber in h, and for each simple
reflections ∈W,letT betheclassinB =π (hreg, ∗)correspondingtoastraightpathfromthe
α α W 1
point ∗ to the point s (∗) with an inserted little semi-circle (oriented counter-clockwise) around
α
the hyperplane α = 0. Given a W-invariant function q : R → C×, α 7→ q , let H (q) be
α W
the Hecke algebra. This algebra is obtained by taking the quotient of C[B ] by the relations
W
(T −1)(T +q ) = 0, one for each simple reflection s . It is known that dimH (q) = |W|, for
α α α α W
any function q :R→C×.
Definition 2.1. A W-invariant function q : R → C× is said to be regular if the Hecke algebra
H (q) is semisimple. Write Reg for the set of regular W-invariant functions q.
W
Since H (1)=CW, we see that q =1 is a regular function. Moreover,the set Reg is a dense
W
Zariski open subset in the set of all W-invariant functions q :R→C× (the latter set is naturally
identified with (C∗)l, where l is the number of W-orbits in R). The complement of Reg has real
codimension ≥ 2, therefore Reg is a connected set. Hence, using rigidity of semisimple algebras
one proves that H (q)≃CW, for any q ∈Reg.
W
We define standard modules over the Cherednik algebra H , see (1.1), as follows. Fix λ∈h∗,
c
andletW bethestabilizerofλinW . LetC[h∗]#W bethecross-productofW withthepolyno-
λ λ λ
mialalgebra. SendingP ∈C[h∗]toP(λ)·1yieldsanalgebrahomomorphism: C[h∗]#W ։ CW .
λ λ
Given τ ∈ Irrep(W ), an irreducible representation of W , we write λ#τ for the representation
λ λ
of C[h∗]#W obtained via the pull-back by the homomorphism above. Then we set M(λ,τ) :=
λ
IndHc (λ#τ). The module M(λ,τ) is called a standard module. In particular, if λ = 0 we
C[h∗]#Wλ
have W =W, in which case we write M(0,τ)=M(τ) for the corresponding standard module.
λ
LetC[R]W denotethevectorspaceofallW-invariantfunctionsc:R→C,andC[R]W ⊂C[R]W
reg
the set of functions c ∈ C[R]W such that exp(2πic) ∈ Reg, i.e., such that the Hecke algebra
H (e2πic) is semisimple. The goal of this section is to prove the following result due to Opdam-
W
Rouquier in the key special case λ=0.
Theorem 2.2 ([OR],[GGOR]). If c ∈ C[R]W then, for any λ ∈ h∗ and τ ∈ Irrep(W ), the
λ
reg
standard module M(λ,τ) is a simple H -module.
c
Since exp(2πic)=1∈Reg for any integral valued function c, Theorem 2.2 yields
Corollary 2.3. All standard modules, M(λ,τ), are simple, for any c∈Z[R]W.
By analogy with representation theory of semisimple Lie algebras we introduce the following
Definition 2.4. LetO bethecategoryoffinitely-generatedH -modulesM,suchthattheaction
Hc c
on M of the subalgebra C[h∗]⊂H is locally finite, i.e., dim C[h∗]·m<∞, for any m∈M.
c C
4
WesaythatanobjectM ∈O hastypeλ¯ ∈h∗/W =Spec(C[h∗]W)if,foranyP ∈C[h∗]W,the
actiononM ofthe element P −PHc(λ¯)∈H is locally nilpotent. Let O (λ¯) be the full subcategory
of modules having type λ¯. Then, by a rocutine argument, cf. e.g. [DHic], one obtains a direct sum
decomposition: O = O (λ¯).
Hc λ¯∈h∗/W Hc
L
Lemma 2.5. (i) Any object M ∈ O is finitely generated over the subalgebra C[h] ⊂ H , in
particular, O is an abelian category.Hc c
Hc
(ii)For any λ∈h∗ and τ ∈Irrep(W ), we have M(λ,τ)∈O .
λ Hc
(iii)For any M ∈O , there exists a nonzero homomorphism M(λ,τ)→M, for certain λ∈h∗
Hc
and τ ∈Irrep(W ).
λ
(iv) Every object of the category O (0) has finite length.
Hc
Remark. Using Lemma 2.5(iv), it is proven in [Gi] that, more generally, every object of the
category O also has finite length.
Hc
To prove Lemma 2.5, we need some notation. Let {x } and {y } be a pair of dual bases of h∗
i i
and h, respectively. We view h∗ and h as subspaces in H , and let h = 1 (x y +y x ) ∈ H
c 2 i i i i i c
denote the canonical element, which is independent of the choice of the bPases. The element h
satisfies the following commutation relations:
h·x=x·(h+1) , ∀x∈h∗, and h·y =y·(h−1) , ∀y ∈h. (2.6)
Toprovethefirstoftheseformulas,usethenotation[a,b]:=a·b−b·a. Givenx∈h∗,wecalculate
1 1
[h,x]= (x y +y x ), x = x ·[y ,x]+[y ,x]·x
i i i i i i i i
2 2
(cid:2)Xi (cid:3) Xi (cid:0) (cid:1)
1 c
= x ·hy ,xi+hy ,xi·x − α x ·hy ,αihα∨,xis +hy ,αihα∨,xis ·x
i i i i i i α i α i
2 2
Xi (cid:16)(cid:0) (cid:1) αX∈R+ (cid:0) (cid:1)(cid:17)
c
= x ·hy ,xi − α ·hα∨,xi· hy ,αix ·s +hy ,αis ·x
i i i i α i α i
2
Xi (cid:0) (cid:1) αX∈R+ (cid:16)Xi (cid:0) (cid:1)(cid:17)
c
=x− α ·hα∨,xi· α·s +s ·α , (2.7)
α α
Xα∈R+ 2 (cid:0) (cid:1)
where in the last equality we have used the identities hy ,xi·x = x, and hy ,αi·x = α.
i i i i i i
Since (sα)−1·α·sα =sα(α)=−α, we find P P
α·s +s ·α=s ·(s )−1·α·s +s ·α=s ·(−α)+s ·α=0.
α α α α α α α α
Thus, each term in the sum on the last line of (2.7) vanishes, and we deduce: [h,x] = x. This
proves the first identity in (2.6); the second one is proved similarly.
LetV beabimoduleoveranassociativealgebraA. Foranya∈A,wehaveanadjointa-action
on V given by ada : v 7→ av−va. We say that the adjoint action of A on V is locally nilpotent
if, for any v ∈V, there exists an integer n≥0 such that ada0◦ada1◦ada2◦ ... ◦adan(v)=0, for
any a ,...,a ∈ A. Let n(v) be the smallest among such integers n ≥ 0, to be referred to as the
0 n
order of nilpotency of v.
Lemma 2.8. Let A be a finitely generated commutative algebra and V an A-bimodule, such that
the adjoint action of A on V is locally nilpotent. Then for any v ∈ V, the space AvA is finitely
generated both as a left and as a right A-module.
5
Proof. We proceed by induction in the order of nilpotency of v. If n(v) =0, the result is clear,
since AvA=Av. So, we have to prove the statement for n(v)=m, knowing it for n(v)=m−1.
Fix v ∈ V of nilpotency order m. Let a ,...,a be generators of A, and put: u = [a ,v], i =
1 d i i
1,...,d. We have: A·v·A = A·v·a ·...·a . By commuting v with a , we get A·v·a ·a ·...
i1 iN i1 i1 i2
·aiN ⊂A·v·ai2·...·aid +A·uPi1·A. Now, continuing like this (i.e. interchanging v with ai2 etc.), we
get A·v·A ⊂A·v+ d A·u ·A. By the induction assumption, the module on the right is finitely
i=1 i
generated. Thus, byPthe Hilbert-Noether lemma, so is the module AvA. (cid:3)
Proof of Lemma 2.5. (i) Let M ⊂ M be the finite dimensional vector space spanned by a finite
0
set of generators. Then H ·M = M. On the other hand, the space, M′ := C[h∗]·M is finite
c 0 0 0
dimensional, since the action of C[h∗] is locally finite. So M =H ·M =C[h]·CW ·C[h∗]·M =
c 0 0
C[h]·CW ·M′. But CW ·M′ is finite dimensional, so we are done.
0 0
(ii) The H -module M(λ,τ) is generated by the W-stable finite dimensional subspace E :=
c
CW ·τ ⊂ M(λ,τ). Hence, given v ∈ M(λ,τ), there exists a finite dimensional subspace B ⊂ H
c
such that C[h∗]·v ⊂ C[h∗]·B ·E. Moreover, since C[h∗] is finite over C[h∗]W, we may find B
large enough, so that C[h∗] ·E ⊆ C[h∗]W ·B ·E. But C[h∗]W · B ⊂ C[h∗]W · B · C[h∗]W. By
Lemma 2.8 applied to V = H , there exists a finite dimensional subspace B′ ⊂ H , such that
c c
C[h∗]W·B·C[h∗]W ⊂B′·C[h∗]W.Hence, C[h∗]W·B·E ⊂C[h∗]W·B·C[h∗]W·E ⊂B′·C[h∗]W·E.
The latter space is finite dimensional since C[h∗]W ·E is finite dimensional, and (ii) follows.
(iii) The action of the subalgebra C[h∗] ⊂ H on M ∈ O being locally finite, it follows that
c Hc
M contains a vector annihilated by the maximal ideal J of the algebra C[h∗] corresponding to a
point λ ∈ h∗. The elements of M which are annihilated by J clearly form an W -stable vector
λ
space. Hence, this vector space contains a simple W -module σ. Any vector in this σ gives rise to
λ
a nonzero element of HomC[h∗]#Wλ(λ#σ,M)=HomHc M(λ,σ),M .
To prove (iv), we observe that the h-action on a(cid:0)ny standard(cid:1)module M(τ) is diagonal, with
finite dimensional eigenspaces. It follows that, for any τ ∈ Irrep(W), the multiplicity of each
simple object of O (0) in M(τ) is bounded from above by the dimension of the corresponding
h-eigenspace. FurtHhcer, the category O (0) has only finitely many simple objects, and therefore,
Hc
each M(τ) has a finite Jordan-Ho¨lder series.
We say that an object N ∈ O (0) is a highest weight object if it is a quotient of a standard
Hc
module. It follows from the paragraphabove that any highest weight object has finite length.
To prove that an arbirary object M ∈ O (0) has finite length it suffices to show that M has
Hc
a finite filtration by subobjects 0 = F ⊂ F ⊂ ... ⊂ F = M, such that successive quotients
0 1 n
F /F arehighestweightobjects. Toconstructsuchafiltration,weusethefactthatM contains
i i−1
a nonzero highest weight submodule N , by part (iii) of the Lemma. Put M := M/N , which is
1 1 1
also an object of O (0). If M =N we are done; if not, then M also contains a nonzero highest
Hc 1 1
weightobjectN .We setM =M /N , anditeratethe procedure. Thus,foreachi≥1,wegetan
2 2 1 2
object M which is a quotient of M. Let F :=ker(M ։ M ). Clearly, F ∈O (0), and we have
i i i i Hc
anincreasingchainofsubobjectsinM: 0=F ⊂F ⊂.... Since M isaNoetherianC[h]-module,
0 1
this sequence must stabilize. This means that at some step l, the object N cannot be chosen,
l+1
which means that M = 0. Thus, the {F } provide a finite filtration with quotients being highest
l i
weight objects, and the result follows.
Remark. The argument above proves also that if for some c ∈ C[R]W and λ¯ ∈ h∗/W, all the
standardmodules M(λ,τ) aresimple, theneveryobjectofO (λ¯)has finite length. As we willsee
Hc
below, these conditions always hold, for instance, provided c∈C[R]W.
reg
Corollary. The action ofh on anyobject M ∈O (0)is locally finite, that is dim(C[h]·m)<∞,
Hc
for any m∈M.
6
We need to recall a few important results about Dunkl operators. According to Chered-
nik, see [EG] Proposition 4.5, the algebra H has a faithful ”Dunkl representation”, an injec-
c
tive algebra homomorphism Θ : H ֒→ D(hreg)#W. This morphism extends by C[hreg]-linearity
c
to a map: Hc|hreg = C[hreg]⊗C[h] Hc −→ D(hreg)#W, which gives an algebra isomorphism Θ :
Hc|hreg −∼→ D(hreg)#W (surjectivityisclearsincethesetofDunkloperatorsgeneratesD(hreg)#W
over the subalgebra C[hreg]#W). Now, for any Hc-module M, the localization, M|hreg, has a natu-
ralHc|hreg-modulestructure,therefore,aD(hreg)#W-modulestructure,viatheDunklisomorphism
above. Thus, since hreg is affine, one can view M|hreg as a W-equivariant D-module on hreg.
ObservefurtherthattheD-moduleonhreg arisingfromanyobjectM ∈O isfinitelygenerated
Hc
overthesubalgebraC[hreg],byLemma2.5(i). Hence,M|hreg,viewedasaD-moduleonhreg,mustbe
avectorbundlewithflatconnection. Inparticular,thestandardmoduleM(τ)isafreeC[h]-module
ofrankdim(τ), asfollowsfromPoincar´e-Birkhoff-Wittisomorphism(1.2). Thus M(τ)|hreg,viewed
as a D-module on hreg, is the trivial vector bundle C[hreg]⊗τ equipped with a flat connection.
The following well-known result plays a crucial role in this paper.
Proposition 2.9. (i) (Dunkl [D], Cherednik [Ch]) The connection on C[hreg]⊗τ arising from the
standard module M(τ) is the Knizhnik-Zamolodchikov connection with values in τ.
(ii) (Opdam [O]) The monodromy representation of the fundamental group B = π (hreg/W)
W 1
corresponding to this connection factors through the Hecke algebra H (e2πic).
W
Proof. Although this is a known result, we will give a short proof for reader’s convenience. Let
us write down a system of linear differential equations which defines the horisontalsections of the
corresponding connection. For this purpose, consider the generating subspace τ ⊂ M(τ)|hreg. We
have yv = 0 for y ∈ h and v ∈ τ. Therefore (since the isomorphism Θ : Hc|hreg → D(hreg)#W is
defined via Dunkl operators), we have
hα,yi hα,yi
∂ v = y− c (s −1) v = c (1−s )v
y α α α α
(cid:20) Xα∈R+ α (cid:21) Xα∈R+ α
In the trivialization M(τ)|hreg ≃C[hreg]⊗τ, the above formulas for the Dunkl operators equip the
trivial bundle C[hreg]⊗τ with the flat connection
dα
∇=d+ c ⊗(Id −s ) : C[hreg]⊗τ −→ Ω1(hreg)⊗τ.
α τ α
Xα∈R+ α
This vector bundle with connection ∇ is W-equivariant, hence, descends to hreg/W. The corre-
sponding flat sections are (multivalued) functions f :hreg/W →τ satisfying the holonomic system
of differential equations: ∂ f = c hα,yi(1 − s )f. An elementary theory of ordinary
y α∈R+ α α α
differential equations says that, foPr generic c, the image of any element Tα in the monodromy
representation of this system satisfies the equation (Tα−1)(Tα+e2πicα) = 0. By continuity, the
same holds for all c, see [O] for more details. (cid:3)
In general, given a W-equivariant vector bundle M on hreg with flat connection, the germs of
horizontal holomorphic sections of M form a locally constant sheaf on hreg/W. Let Mon(M) be
the corresponding monodromy representation of the fundamental group π (hreg/W,∗) in the fiber
1
over ∗, where ∗ is some fixed point in hreg/W. The assignment: M 7→ Mon(M) gives a functor
from the category of W-equivariant vector bundles on hreg with flat connection to the category of
finite dimensional representations of the group π (hreg/W,∗)=B .
1 W
The following result is due to Opdam-Rouquier.
7
Lemma 2.10 ([OR],[GGOR]). Let N be an object of O which is torsion-free over the subalge-
Hc
binrjaecCti[vhe],⊂forHac.nyTMhen∈tOhe.canonical map: HomHc(M,N) → HomBW(cid:0)Mon(M|hreg),Mon(N|hreg)(cid:1) is
Hc
Proof. (borrowed from [OR],[GGOR]). By assumption, the module N contains no C[h]-submodule
supported on a proper subset of h. It follows that if f : M →N is an H -module morphism such
c
thattheinducedmorphism: M|hreg →N|hreg haszeroimage,thenf itselfhasthezeroimage,thatis
f =0. Thus, for any M ∈OHc, the canonical map i1 :HomHc(M,N)→HomHc|hreg M|hreg,N|hreg
is injective. (cid:0) (cid:1)
Now, as has been explained earlier, we may regard M := M|hreg and N := N|hreg as W-
equivariant vector bundles on hreg with flat connections. Hence, the map of the Lemma can be
factored as a composition
HomHc(M,N) ֒→i1 HomHc|hreg M|hreg,N|hreg
(cid:0) (cid:1)
global W-invariant horizontal
=HomD(hreg)#W(M,N) −∼→ (cid:26) sections of (M∗⊗N) (cid:27)
֒→i2 Mon(M∗⊗N) BW =HomBW Mon(M|hreg),Mon(N|hreg) .
(cid:0) (cid:1) (cid:0) (cid:1)
We already know that i is injective. Further, any D(hreg)#W-module morphism between
1
vectorbundleswithflatconnectionsisgivenbyahorizontalsectionoftheHom-bundle. Thelatter
is nothing but a global horizontal section of M∗⊗N, hence i is injective. Finally, assigning to a
2
horizontal section its value in the fiber at a given point ∗ ∈ hreg/W gives an injection (which is a
bijection, if the connection has regular singularities), and the Lemma follows. (cid:3)
The following corollary of the above results and its proof were communicated to us by E.
Opdam.
Corollary 2.11 ([OR],[GGOR]). If c∈C[R]W, then all standard modules M(τ) are irreducible.
reg
Proof. Fix τ ∈ Irrep(W), and given c ∈ C[R]W write Monτ(c) := Mon M(τ)|hreg for the corre-
spondingmonodromyrepresentationoftheHeckealgebraHW(e2πic). The(cid:0)assignmen(cid:1)tc7→Monτ(c)
gives a continuous function on the set C[R]W. Furthermore,this set is connected and contains the
reg
point c = 0, for which we have: Mon (0) = τ. Since, H (q) ≃ CW for all q ∈ Reg, a simple
τ W
deformation argument shows that Mon (c) is a simple H (e2πic)-module, for any c ∈ C[R]W.
τ W
reg
Since M(τ) is free over C[h], hence torsion-free, Lemma 2.10 yields: dimHom M(τ),M(τ) =1.
Hc
Next, fix two non-isomorphic W-modules τ,σ ∈Irrep(W). A similar argum(cid:0)ent shows tha(cid:1)t, for
each c ∈C[R]W, the H (e2πic)-modules Mon (c) and Mon (c) are simple and non-isomorphic to
W τ σ
reg
eachother. Thus,Hom (Mon (c),Mon (c))=0. Hence,Lemma2.10yields: Hom M(σ),M(τ)
BW σ τ Hc
=0. Thus, we conclude: dimHom M(σ),M(τ) =dimHom (τ,σ), ∀σ,τ ∈Irrep(W(cid:0)). (cid:1)
Hc W
Now, assume M(τ) is not a s(cid:0)imple H -mo(cid:1)dule, and let M ⊂ M(τ) be a proper nonzero
c
submodule. Then, Lemma 2.5(iii) says that there exists λ ∈ h∗ and σ ∈ Irrep(W ) such that
λ
Hom M(λ,σ),M 6= 0. Furthermore, the direct sum decomposition O = O (λ¯)
Hc Hc λ¯∈h∗/W Hc
implie(cid:0)s that λ mu(cid:1)st be zero, so that M(λ,σ) = M(σ) and σ ∈ Irrep(W). HenLce, the dimension
of Hom M(σ),M(τ) is at least 1 if σ 6= τ, and is at least 2 if σ = τ. But this contradicts the
Hc
dimensio(cid:0)n equality: d(cid:1)imHom M(σ),M(τ) = dimHom (τ,σ) proved earlier. The contradiction
Hc W
shows that M(τ) is simple. (cid:0) (cid:1) (cid:3)
8
The module M(λ,τ) can be identified, as a vector space, with C[h]⊗IndW τ, via the PBW-
Wλ
theorem. We fix such an identification. Thus, the standard increasing filtration on C[h] by degree
of polynomials gives rise to an increasing filtration F M(λ,τ) on M(λ,τ), such that F M(λ,τ)=
• 0
IndW τ. It is clear that operators corresponding to elements y ∈h preserve this filtration.
Wλ
Lemma 2.12. If c∈C[R]W, then any h-weight vector in M(λ,τ) belongs to F M(λ,τ).
0
reg
Proof. The proof is based on a continuity argumentand on the fact that the result clearly holds
true, provided M(λ,τ) is a simple H -module, e.g., for λ=0.
c
First,weformtheinducedmoduleX(λ)=IndHc λ. Wecan(andwill)canonicallyidentify,via
C[h∗]
thePBW-theorem,theunderlyingvectorspaceofthemoduleX(λ)withC[h]⊗CW. Asabove,the
standardincreasingfiltrationonC[h]by degreeofpolynomialsgivesriseto anincreasingfiltration
F X(λ), such that F X(λ)=CW.
• 0
Let WeightX(λ) be the sum of all the h-weight subspaces of X(λ). It is clear that, for any
λ ∈ h∗, the space WeightX(λ) contains F X(λ). Further, one has an H -module direct sum de-
0 c
composition: X(λ) = τ∗⊗M(λ,τ), where τ∗ stands for the representation dual to
τ∈Irrep(Wλ)
τ. Therefore, to proveLthe Lemma it suffices to show that every h-weight vector in X(λ) belongs
to F X(λ)=CW.
0
Fix λ∈h∗, and for any t∈C, view WeightX(t·λ) as a subspace in C[h]⊗CW (note that the
latter is independent of t∈C). For any integer d≥0, consider the set
U = t∈C WeightX(t·λ) ∩ F X(t·λ) = F X(t·λ) .
d d 0
(cid:8) (cid:12) (cid:0) (cid:1) (cid:9)
It is clear that C =U ⊃U ⊃(cid:12).... Observe that by Corollary 2.11, M(τ) is a simple H -module,
0 1 c
for any τ ∈ Irrep(W). It follows that for t = 0 one has WeightX(t·λ) = F X(t·λ). Hence, each
0
setU containsthe pointt=0andisZariskiopen,byanelementarycontinuityargument. Hence,
d
for each d≥0, the set CrU is a proper Zariski closed, hence finite, subset of C. It follows that
d
the set∪ (CrU )is a unionofa countable family offinite sets, hence a countablesubset inC.
d≥0 d
Let T be the set of all t ∈ C such that WeightX(t·λ) = F X(t·λ). By definition we have
0
T =∩ U .Therefore,CrT =∪ (CrU )isacountableset. Further,thiscountablesetmust
d≥0 d d≥0 d
be invariant under dilations, since the assignment x 7→ t·x, y 7→ t−1·y, t ∈ C∗, x ∈ h∗, y ∈ h,
gives a C∗-action on H by algebra automorphisms. Finally, we know that 0∈T, i.e., 06∈CrT.
c
These properties force the set CrT to be empty, hence T =C, and we are done. (cid:3)
Proof of Theorem 2.2. Let N be a nonzero submodule of M(λ,τ). Thus, N ∈ O . Hence, N
Hc
containsanonzeroh-weightvectorv. Lemma2.12yieldsv ∈F M(λ,τ). ButF M(λ,τ)=IndW τ
0 0 Wλ
is an irreducible C[h∗]#W-module, so v ∈F M(λ,τ)⊂N implies N =M(λ,τ).
0
Remark. Ithasbeenshownin[DO]thateachstandardmoduleM(τ), τ ∈Irrep(W),hasaunique
simple quotientL(τ). Moreover,any objectof the categoryO (0) has finite length, and the finite
set{L(τ)} is acompletecollectionofthe isomorphismHcclassesofsimple objectsofO (0).
Further, itτi∈sIrprerpo(Wve)d in [Gu] and [OR] that the category O (0) has enough projectives, i.e.,Hceach
moduleL(τ)hasanindecomposableprojectivecoverP(τ)∈HcO (0). Moreover,anyprojectiveP ∈
O (0) has a finite increasing filtration by H -submodules: 0=HcF ⊂F ⊂...⊂F =P, such that
Hc c 0 1 l
F /F ≃ M(σ ), for some standard modules M(σ ), σ ∈ Irrep(W), i = 1,...,l. Furthermore,
i i−1 i i i
for any τ,σ ∈Irrep(W), Guay [Gu], and Opdam-Rouquier [OR], proved the following analogue of
Brauer-Bernstein-Gelfand-Gelfand type reciprocity formula [P(τ) : M(σ)] = [M(σ) : L(τ)], see
[BGG]. This formula implies, in particular, that the category O (0) is semisimple if and only if
Hc
all the standard modules M(τ), τ ∈ Irrep(W), are simple, i.e. if and only if M(τ) = L(τ), ∀τ ∈
Irrep(W). It would be interesting to obtain similar results for the category O (λ¯), with λ¯ 6=0.
Hc
9
3 Harish-Chandra H -bimodules
c
One of the goals of this section is to prove
Theorem 3.1. If c∈C[R]W, then H is a simple algebra.
c
reg
The proof of the Theorem will be based on the concept of a Harish-Chandra bimodule which
we now introduce.
Let V be an H -bimodule, i.e., a left H ⊗Hop-module. For any x ∈ H we have an adjoint
c c c c
x-action on V given by adx:v 7→xv−vx.
Definition 3.2. A finitely generated H ⊗(H )op-module V is called a Harish-Chandra bimodule
c c
if, for any x∈C[h]W or x∈C[h∗]W, the adx-action on V is locally nilpotent.
Here are the first elementary results about Harish-Chandra bimodules.
Lemma 3.3. (i) Harish-Chandra bimodules form a full abelian subcategory in the category of
H -bimodules; this subcategory is stable under extensions.
c
(ii) Any Harish-Chandra bimodule is finitely-generated over the subalgebra C[h]W ⊗C[h∗]W ⊂
H ⊗(H )op, with C[h∗]W acting on the right and C[h]W on the left. Similarly, it is also finitely-
c c
generated over the subalgebra C[h∗]W ⊗C[h]W ⊂H ⊗(H )op;
c c
(iii)Any Harish-Chandra bimodule is finitely-generated as a left, resp. right, H - module;
c
(iv)For any maximal ideal J ⊂C[h∗] and any Harish-Chandra bimodule V, we have V/V·J ∈
O , as a left module.
Hc
(v) The algebra H is a Harish-Chandra H -bimodule.
c c
Proof. The algebra H has an increasing filtration F H such that gr(H ) = C[h×h∗]#W,
c • c c
see [EG]. Hence H is a Noetherian algebra, for this is clearly true for the algebra gr(H ) =
c c
C[h×h∗]#W. Part (i) of the Lemma follows.
To prove (ii), observe that the filtration on H gives rise to the tensor product filtration:
c
(H ⊗Hop) := F (H )⊗F (Hop), on the algebra H ⊗Hop. Choose a finite-dimensional
c c p p=i+j i c j c c c
subspaceV0 ⊂VPgeneratingV asanHc⊗Hocp-moduleand,foreachp≥0,putVp :=(Hc⊗Hocp)p·V0.
ThisisagoodfiltrationonV,andonecanregardgr(V),theassociatedgradedspace,asaW×W-
equivariantfinitely-generatedmodule overC[h ×h ×h∗×h∗], wherethe subscripts‘1,2’indicate
1 2 1 2
the corresponding copy of h. Then, for any homogeneous element P ∈C[h]W, the adP-action on
V corresponds to the action of the element Pleft⊗1−1⊗Pright ∈ C[h ]W ⊗C[h ]W on gr(V).
1 2
The (local) nilpotency of the adP-action on a Harish-Chandra module thus implies that, if P is
homogeneousofdegree>0,thentheactionoftheelementPleft⊗1−1⊗PrightongrV isnilpotent.
Hence, the supportof grV, viewedasa C[h ×h ×h∗×h∗]-module, is containedin the zero locus
1 2 1 2
of the polynomial Pleft⊗1−1⊗Pright ∈C[h ×h ×h∗×h∗]. This way,for any Harish-Chandra
1 2 1 2
module V, we deduce the following upper bound on the set supp(grV)⊂h ×h ×h∗×h∗:
1 2 1 2
supp(grV) ⊂ ∪ Graph(w) × ∪ Graph(y) , (3.4)
w∈W y∈W
(cid:16) (cid:17) (cid:16) (cid:17)
where Graph(w) denotes the graph in h ×h , resp. h∗×h∗, of the w-action map: x7→w(x).
1 2 1 2
In particular, the restriction of the composite map: h ×h ×h∗ ×h∗ → h ×h∗ → h /W ×
1 2 1 2 1 1 1
h∗/W to supp(grV) is a finite map. Therefore, gr(V) is finitely generated over the subalgebra
1
C[h]W ⊗C[h∗]W, hence the same holds for V itself. Part (ii) follows.
To prove (iii) we observe that (3.4) implies also that the restriction to supp(grV) of the
corresponding projection: h ×h ×h∗×h∗ → h ×h∗ is a finite map. We deduce similarly that
1 2 1 2 1 2
10