HARISH-CHANDRA BIMODULES FOR QUANTIZATIONS OF TYPE A KLEINIAN

SINGULARITIES

JOSE SIMENTAL

Abstract. We compute the number of irreducible Harish-Chandra bimodules over algebras quantizing Kleiniansingularities of type A. Our methods are based on the interplay between two equivalent realizations of thesealgebras: as a spherical rational Cherednik algebra for Z`+1; and as a central reduction of the finite W-algebraassociated to sl`+1 at a subregular nilpotent element. We use this result to give a classification of the irre-ducible fully supported Harish-Chandra bimodules for rational Cherednik algebras associated to any complexreflection group W .

1. Introduction

In this paper we study Harish-Chandra bimodules over quantizations of type-A Kleinian singularities.Recall that the Kleinian singularity of type A` is the singular surface C2/Γ`+1, where Γ`+1 is a subgroupof SL2(C) that is cyclic of order ` + 1. Algebras quantizing C[C2/Γ`+1] have been extensively studied inthe literature and they admit many equivalent descriptions: as algebras generalizing U(sl2), [Ho, Sm]; asquantum hamiltonian reductions, [EGGO, Go, L5]; as spherical rational Cherednik algebras, [CBH, EG]; oras central reductions of finite W-algebras, [P, L5]. It is the latter two approaches that we use in the presentpaper. From these approaches, Harish-Chandra bimodules have been studied, for example, in [BEG, L4, Si]from the Cherednik algebra perspective, and [Gi2, L2] from the W-algebra perspective. In particular, in[Si] the author obtained a reduction result that allows us to reduce the classification of irreducible, fullysupported Harish-Chandra bimodules over any rational Cherednik algebra to the case of cyclic groups, whichis precisely the case covered in this paper. Thus, the present paper finishes the general classification result,see Theorem 5.1.

Let us give a general definition of a Harish-Chandra bimodule following [BL, Section 5]. Let A, A′ beZ≥0-filtered algebras quantizing the same graded Poisson algebra A. Then, an A-A′-bimodule B is said tobe Harish-Chandra (shortly, HC) if it can be equipped with a bimodule filtration in such a way that grB is afinitely generated A-module, meaning that the left and right actions of A on grB coincide. This generalizesthe well-known case where A = A′ = U(g), see e.g. [BG], and this more general notion has been studied in avariety of settings, see e.g. [BL, BPW, Gi2, L2, L4, L7, O, Si, Sp]. In particular, Harish-Chandra bimoduleshave been instrumental in studying the representation theory of finite W-algebras, [Gi2, L2], and they havebeen used in [L7] to produce derived equivalences between different categories O for rational Cherednikalgebras. For these types of algebras, every HC bimodule has finite length, see e.g. [Gi, Section 5]. It is aninteresting problem, then, to classify the irreducible HC bimodules. Progress towards this has been made in[L8] on the W-algebra side and in [L4, Si] on the Cherednik algebra side. Neither of these works, however,covers the case we treat in the present paper.

Our first main result is a classification of irreducible fully supported HC bimodules over algebras quantizingthe Kleinian singularity C2/Z`+1. Such an algebra depends on a parameter k, which is a collection of complexnumbers k0 = 0, k1, . . . , k` ∈ C, see Subsection 2.1. Let us denote the algebra associated to k0, . . . , k` by Ak.Using a natural exponential formula, see Subsection 2.2, we have associated a collection of nonzero complexnumbers, q0 = 1, q1, . . . , q` ∈ C×, which determine the number and structure of irreducible fully supportedHC Ak-bimodules. More precisely, let us denote simply by q the multiset q := q0, q1, . . . , q`, and considerthe set X := i ∈ 1, . . . , ` + 1 : exp(2π

√−1i/(` + 1))qj ∈ q with the same multiplicity as qj for all j =

0, . . . , `. Now let m := minX. It is easy to see that m divides ` + 1. So set p := (` + 1)/m, and consider

the subgroup W = 〈sp〉, where s ∈ Z`+1 is a generator.

1

2 JOSE SIMENTAL

Theorem 1.1. The number of irreducible, fully supported HC Ak-bimodules is m = |Z`+1/W |. Moreover,the quotient category of all HC Ak-bimodules modulo the full subcategory of HC bimodules without proper

support is equivalent, as a monoidal category, to the category of representations of Z`+1/W .

Using Theorem 1.1 we can prove a similar statement for all rational Cherednik algebras. The rationalCherednik algebra Hk(W ) associated to the complex reflection group W depends on a parameter k whichis a collection of complex numbers. Using a sort of exponentiation we get a collection of nonzero complex

numbers q, that we can use to form a normal subgroup W ⊆ W in a spirit similar to the one formed inTheorem 1.1. Again, we have that the quotient category of all HC Hk-bimodules modulo those bimodules

with proper support is equivalent, as a monoidal category, to the category of representations of W/W , seeSubsection 5.1.

Let us sketch the structure of the paper as well as our method of proof. First of all, using the representationtheory of rational Cherednik algebras, in particular the Knizhnik-Zamolodchikov (KZ) functor, we canprovide an upper bound on the number of irreducible HC bimodules with full support for algebras quantizingKleinian singularities, this is done in Section 2 of the paper. We remark that this bound is basically aconsequence of the results in [Si]. Sections 3 and 4 are devoted to proving that this upper bound is also alower bound, and hence provides a precise count. In Section 3 we review known material on finite W-algebras,including a relationship between HC bimodules for the W-algebras and for enveloping algebras, [Gi2, L2].This section contains no new results. In Section 4 we use the representation theory of finite W-algebras toshow that our upper bound is also a lower bound, this is done purely combinatorially by studying weightsof sl`+1. Finally, in Section 5 we give a precise description of the category of fully supported HC bimodulesfor any rational Cherednik algebra. This is based on restriction functors from [L4], as well as localizationtechniques from [Si] and results from the present paper. At the beginning of each section, we describe itscontent in more detail.

Remark 1.2. Let us remark that a classification of irreducible HC Ak-bimodules has been carried out, inthe case the parameter k is regular (i.e., the algebra Ak is simple) in [Sp]. There, the realization of Ak asan algebra generalizing U(sl2) plays a crucial role, see [Sp, Chapter 5].

2. Rational Cherednik Algebras

The first realization of quantizations of C[C2/Γ`+1] we need is that of a spherical rational Cherednikalgebra associated to the action of the complex reflection group Z`+1 on the vector space R = C. In orderto study the representation theory of the spherical rational Cherednik algebra it is more convenient to studyfirst that of the full rational Cherednik algebra and this is what we are going to do. First, in Subsection2.1 we introduce the rational Cherednik algebra Hc := Hc(W,R) in the general case of a complex reflectiongroup W and give a few of its basic properties. In Subsection 2.2 we define a category O of modules overHc and, following [GGOR], we define a functor KZ from category Oc to the category of modules over acertain Hecke algebra. In Subsection 2.3 we review the connection between the category O and the study ofHarish-Chandra bimodules, and use this connection in Subsection 2.4 to give an upper bound on the numberof irreducible HC Hc-bimodules with full support in the case W = Z`+1, this is the only partially new resultin this section and it is based on a result previously obtained in [Sp, Si]. In Subsection 2.5 we explain howthe constructions for the full Cherednik algebra have analogs for the spherical Cherednik algebra. Finally,in Subsection 2.6 we explore some functors basically introduced in [BC] that will allow us to make somecomputations easier.

2.1. Definitions. Let W be a complex reflection group and R its reflection representation. Let us denote byS ⊆W the set of reflections. Pick a collection of complex numbers cs ∈ C : s ∈ S subject to the conditioncs = cwsw−1 for w ∈ W . For each reflection s ∈ S, let αs ∈ R∗ be an eigenvector of s with eigenvalueλs 6= 1, and let α∨s ∈ R be an eigenvector with eigenvalue λ−1

s . We remark that αs, α∨s are determined up to

multiplication by a nonzero scalar, and we partially resolve this ambiguity by requiring that 〈αs, α∨s 〉 = 2.

Definition 2.1. The rational Cherednik algebra Hc := Hc(W,R) is the quotient of the smash-productalgebra T (R⊕R∗)#W by the relations:

(1) (a) [y, y′] = 0 (b) [x, x′] = 0 (c) [y, x] = 〈y, x〉 −∑

s∈S〈y, αs〉〈α∨s , x〉css

HARISH-CHANDRA BIMODULES FOR QUANTIZATIONS OF TYPE A KLEINIAN SINGULARITIES 3

for y, y′ ∈ R, x, x′ ∈ R∗,

We remark that the algebras CW,C[R],C[R∗] sit naturally as subalgebras of Hc. Moreover, multiplicationgives a vector space isomorphism C[R] ⊗ CW ⊗ C[R∗] −→ Hc, [EG, Theorem 1.3]. This is equivalent tosaying that, under the filtration defined by putting R,R∗ in degree 1 and W in degree 0, we have grHc =C[R⊕R∗]#W . It is clear from the relations that Hc may be graded with degR = −1,degW = 0,degR∗ = 1.This grading is inner. Define the Euler element as follows. Pick a basis yi of R and let xi be the dual basisin R∗. Then,

eu :=∑i

xiyi +dimR

2−∑s∈S

2cs1− λs

s

It is easy to check that [eu, w] = 0, [eu, x] = x, [eu, y] = −y for w ∈ W,x ∈ R∗, y ∈ R, so that the adjointaction of eu does define the desired grading on Hc.

It will be more convenient to work with a different parametrization of the algebra Hc, as follows. Letus denote by A the set of reflection hyperplanes in R. For each hyperplane V ∈ A, its pointwise stabilizerWV ⊆ W is cyclic, of order say `V . Let sV ∈ WV be a generator such that det(sV |R) = exp(2π

√−1/`V ) =

λ−1sV

=: ηV . We may and will assume that αs = αsV , α∨s = α∨sV for all s ∈ WV ∩ S = WV \ 1. We will

denote these elements by αV ∈ R∗, α∨V ∈ R, respectively.

For each i = 0, . . . , `V , consider the element ei,V := `−1V

∑`V −1j=0 η−ijV sjV , this is an idempotent in the group

algebra CWV . For each hyperplane V ∈ A, pick a collection of complex numbers k0,V = 0, k1,V , . . . , k`V −1,V .We requite that ki,V = ki,V ′ for all i = 0, . . . , `V − 1 = `V ′ − 1 if V , V ′ are W -conjugate. Let us denote by kthe collection ki,V : V ∈ A/W, i = 0, . . . , `V − 1. Define the algebra Hk similarly to Hc with the relation(1)(c) replaced by

[y, x] = 〈y, x〉 − 1

2

∑V ∈A〈y, αV 〉〈α∨V , x〉

`V −1∑i=0

(ki,V − ki−1,V )ei,V

where k−1,V = k`V −1,V . Note that the ki,V ’s are defined up to a common summand and this is why wealways assume that k0,V = 0. We remark that Hc = Hk where the parameter k is computed from c asfollows:

(2) ki,V =∑

s∈WV ∩S

2cs1− λs

(λ−is − 1)

Similarly, starting from k we can compute c as follows. For s ∈ S, let Vs ∈ A be the reflection hyperplaneof s, i.e. Vs = ker(idV −s). Then,

(3) cs =1

2`Vs

`Vs−1∑j=0

(kj,Vs − kj−1,Vs)λ−js

The reason why the k-parametrization is better for our purposes will become apparent later, see inparticular Subsections 2.2 and 3.4.

2.2. Category O. Let us proceed to the definition of the category Ok. All results in this subsection may befound in [GGOR]. By definition, we say that a module M ∈ Hk -mod belongs to category Ok if the followingtwo conditions are satisfied.

(O1) M is finitely generated.(O2) The action of R ⊆ Hk on M is locally nilpotent.

In particular, if M ∈ Ok then M is finitely generated over C[R]. Also, we remark that every finitedimensional module belongs to category Ok, this follows because the grading on Hk defined in Subsection2.1 is inner. Let us construct Verma (or standard) modules. Consider an irreducible representation S of W .We let R act on S by 0, so that S becomes a C[R∗]#W -module. Now define

∆(S) := IndHkC[R∗]#W S = Hk ⊗C[R∗]#W S

4 JOSE SIMENTAL

We remark that, thanks to the PBW property, ∆(S) = C[R]⊗ S as a vector space. It is very easy to showthat ∆(S) has a unique irreducible quotient, which we denote by L(S). The L(S) form a complete andirredundant list of irreducible objects in Ok. We remark that the action of eu on every module M ∈ Okis locally finite with finite dimensional generalized eigenspaces. It follows, in particular, that every moduleM ∈ Ok has finite length, see e.g. [GGOR, Section 2.5].

Let us proceed to the definition of the KZ functor. Let δ =∏V ∈A αV ∈ C[R]. We remark that some

power of δ, say δm, is W -invariant. In particular, the adjoint action of δm on Hk is locally nilpotentand so we can form the noncommutative localization Hk[δ

−1]. This algebra is isomorphic, via the Dunkl-Opdam representation, to D(Rreg)#W , see [GGOR, Subsection 5.2]. Here, D(Rreg) denotes the algebra of(algebraic) global differential operators on Rreg := R \

⋃V ∈A V . So, for M ∈ Ok, the localization M [δ−1]

becomes a D(Rreg)#W -module that is finitely generated over C[Rreg]. Thus, M [δ−1] is a W -equivariantvector bundle on Rreg with a flat connection. Moreover, this connection has regular singularities. NoweM [δ−1] is a local system with regular singularities on Rreg/W . Here e ∈ CW denotes the trivial idempotent.So taking the flat sections functor we get that (eM [δ−1])∇ is a representation of π1(Rreg/W ). We abbreviateKZ(M) := (eM [δ−1])∇. It turns out that the image of the KZ functor factors through the representationcategory of a certain quotient of Cπ1(Rreg/W ) that we explain next.

Fix a point y ∈ Rreg. For each reflection hyperplane V ∈ A, let tV ∈ π1(Rreg/W ) be represented by apath from y to sV y that is a straight line with an inserted arc of length ηV around V (a generator of themonodromy around H in the terminology of [BMR].) The elements tV V ∈A form a system of generators ofπ1(Rreg/W ). For i = 0, . . . , `V − 1 define:

qV,i := exp

(2π√−1(kV,i − i)`V

)note, in particular, that qV,i = qV ′,i is V , V ′ are W -conjugate, and that qV,0 = 1 for every V ∈ A. Wedenote by q the collection qV,i : V ∈ A/W, i = 0, . . . , `H − 1 of nonzero complex numbers. Define theHecke algebra

Hq(W ) := Cπ1(Rreg/W )

/((`V −1∏i=0

(tV − qV,i)

)V ∈A

)Then, we have that KZ(Ok) = Hq -mod ⊆ Cπ1(Rreg/W ) -mod, see [L6]. Moreover, KZ is a quotient

functor that identifies Ok/O0k∼= Hq -mod, where O0

k is the full subcategory of category Ok consisting ofmodules whose set-theoretic support as C[R]-modules is properly contained in R, [GGOR, Theorem 5.14]

To finish this subsection, let us remark that the KZ functor is fully faithful on projective objects, see e.g.[GGOR, Theorem 5.3]. It follows, in particular, that KZ induces a bijection between the blocks of categoryOk and those of Hq -mod see, for example, Corollary 5.18 in loc. cit.

2.3. Harish-Chandra bimodules. Let us explore connections between the category Ok and Harish-Chandra Hk-bimodules. First, let us define what these are in this context. We remark that if k, k′ areparameters for the Cherednik algebras Hk, Hk′ , the algebras C[R]W ,C[R∗]W sit naturally inside both Hk

and Hk′ .

Definition 2.2 ([BEG], Definition 3.2). An Hk-Hk′-bimodule B is said to be Harish-Chandra (shortly, HC)if the following conditions are satisfied.

(HC1) B is finitely generated.(HC2) The adjoint actions of C[R]W and C[R∗]W on B are locally nilpotent.

We denote the category of HC Hk-Hk′-bimodules by HC(Hk-Hk′). Note that this is an abelian category.

Let us see how this relates to the definition of a HC bimodule given in the introduction. Recall thatboth algebras Hk, Hk′ are filtered, with associated graded grHk = grHk′ = C[R ⊕ R∗]#W . The centerof this algebra coincides with the graded Poisson algebra C[R ⊕ R∗]W . It is shown in [L4, Subsection 5.4]that an Hk-Hk′-bimodule B is HC if and only if it can be equipped with a bimodule filtration in such away that grB is a finitely generated C[R ⊕ R∗]W -module. So we may define the singular support of B,SS(B) ⊆ (R⊕R∗)/W , as the set-theoretic support of grB as a C[R⊕R∗]W -module. As usual, this does not

HARISH-CHANDRA BIMODULES FOR QUANTIZATIONS OF TYPE A KLEINIAN SINGULARITIES 5

depend on the choice of the filtration. We remark that SS(B) is a union of symplectic leaves of the Poissonvariety (R ⊕ R∗)/W . In this paper, we are mostly interested in those bimodules whose singular supportcoincides with the entire variety (R⊕R∗)/W .

We remark that, when k = k′ the category HC(Hk) := HC(Hk-Hk′) is actually monoidal, and that aHC Hk-Hk′-bimodule is finitely generated both as a left Hk-module and as a right Hk′-module, see [L7,Proposition 3.1]. As in the case of category O, thanks to the existence of a grading element it follows thatevery finite dimensional Hk-Hk′-bimodule is HC. Another example of a HC Hk-bimodules is the regularbimodule Hk. Further examples of HC Hk-Hk′-bimodules are provided by the following construction.

Definition 2.3. Let M ∈ Ok′ N ∈ Ok. A map f ∈ HomC(M,N) is said to be locally finite if the adjointaction of C[R]W and C[R∗]W on f is nilpotent. The set of locally finite maps is denoted by Homfin(M,N).Note that this is an Hk-Hk′-sub-bimodule of HomC(M,N).

Note that, by definition, Homfin(M,N) is the union of its HC (= finitely generated) sub-bimodules. Evenmore is true, the bimodule Homfin(M,N) is actually HC, see [L4, Proposition 5.7.1]. So we can produceHC bimodules starting from modules in category O.

Lemma 2.4. Let B be an irreducible HC Hk-Hk′-bimodule with full singular support. Then, there exists aunique irreducible representation S of W satisfying the following conditions

(i) Lk(S) has full support as a C[R]-module.(ii) B = Soc(Homfin(∆k′(triv), Lk(S))).

Where triv denotes the trivial representation of the group W

Proof. By [Si, Proposition 4.3], there exists an irreducible representation S of W such that B is a sub-bimodule of Homfin(∆k′(triv), Lk(S)). The fact that Lk(S) has full support follows directly from [Si, Lemma2.5]. Now the D(Rreg/W )-bimodule e Homfin(∆k′(triv), Lk(S))[δ−1]e is isomorphic to the bimodule ofdifferential maps Diff(C[Rreg/W ], eLk(S)[δ−1]) which is irreducible, [Si, Corollary 4.5 and Lemma 4.2].It follows that Soc(Homfin(∆k′(triv), Lk(S))) contains at most one summand with full singular support.Now, since Lk(S) is irreducible, for every nonzero subbimodule B′ of Homfin(∆k′(triv), Lk(S)) we have anepimorphism B′ ⊗Hk′ ∆k(triv) Lk(S). Thanks to [Si, Lemma 2.4] this implies that any sub-bimodule ofHomfin(∆k′(triv), Lk(S)) has full support. It follows that we must have B = Soc(Homfin(∆k′(triv), Lk(S))).Let us now show that S is unique. By [Si, Lemma 4.8] we have that KZk(B ⊗Hk′ ∆k′(triv)) = KZk(Lk(S)).So the result follows because KZk(Lk(S)) 6∼= KZk(Lk(S

′)) if Lk(S), Lk(S′) are nonisomorphic irreducibles

with full support.

We remark that the proof of Lemma 2.4 also shows that the bimodule Homfin(∆k′(triv), Lk(S)) is eitherzero or has an irreducible socle. So we get that the number of non-isomorphic irreducible HC Hk-Hk′-bimodules with full support coincides with the number of irreducible representations S of W such thatLk(S) has full support and Homfin(∆k′(triv), Lk(S)) is nonzero. This is the number we are going to find,first in the special case where W = Z`+1 and then in the general case.

2.4. Cyclic groups. Now we specialize to the case W = Z`+1, acting on R = C. Here, there is a singlereflection hyperplane, namely 0, so the rational Cherednik algebra Hk depends on the parameters k0 =0, k1, . . . , k` ∈ C. We fix a generator s ∈ Z`+1 that acts by multiplication by η := exp(2π

√−1/(`+ 1)) on R,

so that s acts on R∗ by η−1. We denote by ei := 1`+1

∑`j=0 η

−ijsj the idempotent in the group algebra CZ`+1

defining the 1-dimensional representation Ei, i = 0, . . . , `. In particular, s acts on Ei by multiplication byηi, and E0 = triv, the trivial representation.

Let y ∈ R and x ∈ R∗ be such that 〈x, y〉 = 1. In particular, C[R] = C[x], while C[R∗] = C[y]. Thanksto the PBW property of Hk, as a vector space we have that ∆(Ei) = C[x], with the action given byx.xj = xj+1, s.xj = ηi−jxj the action of y can be recovered uniquely from these formulas and y.1 = 0. Itfollows that Li := L(Ei) is either finite dimensional or Li = ∆(Ei).

The Hecke algebra Hq := Hq(Z`+1) is the quotient of the polynomial algebra C[t] by the ideal generatedby a polynomial of degree `+ 1. More precisely, denoting qi := exp(2π

√−1(ki − i)/(`+ 1)) we have

Hq = C[t]

/∏i=0

(t− qi)

6 JOSE SIMENTAL

We remark that qi is precisely the scalar by which t acts on the 1-dimensional representation KZ(∆(Ei))of Hq. It follows that the blocks of category Ok are parametrized by distinct elements in the multisetq = q0 = 1, q1, . . . , q`. For Q ∈ q, we will denote by BQ the block corresponding to Q, the simple Libelongs to BQ if and only if qi = Q. Note that the block BQ contains a unique irreducible module with fullsupport, that is, a unique irreducible Verma module.

Recall that we are trying to find #i = 0, 1, . . . , ` : ∆(Ei) = Li,Homfin(∆(E0),∆(Ei)) 6= 0. An upperbound can be obtained using [Si, Lemma 4.8].

Lemma 2.5. Assume that k0 = 0. Let i ∈ 0, . . . , ` be such that Homfin(∆(E0),∆(Ei)) 6= 0. Then,multiplication by qi induces a permutation on the (multi-)set of Hecke parameters q = q0 = 1, q1, . . . , q`.

Proof. Assume that i ∈ 0, . . . , ` is as in the statement of lemma. According to Lemma 4.8 in [Si], forevery module M ∈ Hq -mod, the C[t, t−1]-module KZ(∆(Ei)) ⊗C M factors through Hq. Now, KZ(∆(Ei))is the 1-dimensional Hq-module where t acts by qi. This already tells us that, for every p ∈ q0, . . . , q`,qip ∈ q0, . . . , q` and we need to check that they have the same multiplicity. For p ∈ q0, . . . , q` let m(p)be the multiplicity of p in the multiset of Hecke parameters, and let M ∈ Hq -mod be the indecomposablemodule of dimension m(p) where t acts with generalized eigenvalue p. So KZ(∆(Ei)) ⊗C M ∈ Hq -mod isan indecomposable module of dimension m(p) where t acts with generalized eigenvalue qip. Then, we getm(p) ≤ m(qip) ≤ m(q2

i p) ≤ . . . . Since q is finite, there exists a > 0 such that qap = p. So all inequalitiesabove are equalities and we are done.

Corollary 2.6. Assume Homfin(∆(E0),∆(Ei)) 6= 0. Then qi is a (not necessarily primitive) (`+ 1)-root ofunity.

Proof. From the previous lemma, it is clear that qi is a root of unity. Now let n ∈ Z>0 be minimal withqni = 1. Define an equivalence relation on q0, . . . , q` by qj1 ∼ qj2 if there exists a ∈ Z with qai qj1 = qj2 . Picka representative qj1 , . . . , qjb from each equivalence class. Then, it is clear that `+1 = nm(qj1)+ · · ·+nm(qjb).Thus, n divides `+ 1.

Let us now express the previous results in terms of the blocks of category Ok. First of all, we remarkthat if B ∈ HC(Hk) and M ∈ Ok then B ⊗Hk M ∈ Ok, see e.g. [L7, Proposition 3.1]. Now, let B be anirreducible HC Hk-bimodule with full support, B ⊆ Ok be a block and ∆(Ej) ∈ B be the unique irreducibleVerma module in B, in particular, B = Bqj .

Lemma 2.7. The following is true.

(i) The module B ⊗Hk ∆(Ej) has a simple head.(ii) The head of B ⊗Hk ∆(Ej) has full support. In particular, it is isomorphic to ∆(Em) for some m.

(iii) Any two of ∆(Ej),∆(Em) and B completely determine the other, that is:(iiia) If B′ is an irreducible HC Hk-bimodule with full support such that B ⊗Hk ∆(Ej) surjects onto

∆(Em), then B ∼= B′.(iiib) If ∆(Ej′) is an irreducible Verma module such that B ⊗Hk ∆(Ej′) surjects onto ∆(Em), then

j = j′.(iiic) If ∆(Em′) is an irreducible Verma module such that B ⊗Hk ∆(Ej) surjects onto ∆(Em′), then

m = m′.

Proof. Recall that B = Soc(Homfin(∆(E0),∆(En))) for some unique irreducible Verma module ∆(En), seeLemma 2.4. Thanks to [Si, Lemma 4.8], we have that KZ(B ⊗Hk ∆(Ej)) = KZ(En) ⊗ KZ(Ej). This isa 1-dimensional (= irreducible) Hq-module, where t acts with eigenvalue qnqj . This already tells us thatB ⊗Hk ∆(Ej) can have at most one irreducible subquotient with full support. But by [Si, Lemma 2.5], anyirreducible quotient of B⊗Hk ∆(Ej) has full support. Assertions (i) and (ii) follow. Now note that we musthave qnqj = qm. In other words, we must have that B = Soc(Homfin(∆(E0), S)), where S is the uniqueirreducible Verma module in the block Bqmq−1

j, and (iiia) follows. Similarly, (iiib) follows from the equation

qj = qmq−1n . Finally, (iiic) is an immediate consequence of (i) and (ii).

Lemma 2.8. Denote by IrrHC(Hk) the set of irreducible HC Hk-bimodules with full support, and by Bk theset of blocks of Ok. Then, the following is true.

HARISH-CHANDRA BIMODULES FOR QUANTIZATIONS OF TYPE A KLEINIAN SINGULARITIES 7

(i) There exists an embedding Υ : IrrHC(Hk) → S(Bk), where the latter set is that of permutations onBk.

(ii) For every B ∈ IrrHC(Hk), Υ(B) is completely determined by its action on any element of Bk.(iii) The tensor product • ⊗Hk • induces a group structure on IrrHCHk. With this group structure, Υ is

a group homomorphism.(iv) With the structure from (iii), IrrHC(Hk) is isomorphic to a subgroup of Z`+1.

Proof. (i) and (ii). Define Υ as follows. Let B ∈ IrrHC(Hk), let BQ ∈ Bk be a block of Ok, and let ∆(Ej) bethe unique irreducible Verma module in BQ. Then, the head of B⊗Hk ∆(Ej) is an irreducible Verma module,Lemma 2.7(ii), which therefore determines a block BQ′ ∈ Bk. Thus, by definition Υ(B)(BQ) := BQ′ and we

get a map IrrHC(Hk) → Maps(Bk), where the latter is the set of maps from Bk to itself (not necessarilybijections.) The fact that the image is in S(Bk) follows from Lemma 2.5. Now (iiia) in Lemma 2.7 showsboth that Υ is indeed injective, and assertion (ii).

(iii). Let us remark that if B1, B2 ∈ HC(Hk) then B1 ⊗Hk B2 ∈ HC(Hk), see e.g. [L7, Proposition

3.1]. Now assume that B1, B2 ∈ IrrHC(Hk). We claim that B1 ⊗Hk B2 has a unique irreducible subquotient

with full support. Indeed, let B ∈ IrrHC(Hk) be a subquotient of B1 ⊗Hk B2. We may assume that Bj =Soc(Homfin(∆(E0),∆(Eij ))), j = 1, 2. In particular, we have KZ(B1 ⊗Hk B2 ⊗Hk ∆(E0)) = KZ(∆(Ei1))⊗KZ(∆(Ei2)), which is a 1-dimensional Hq-module and therefore must be equal to KZ(∆(Ep)) for someirreducible Verma module ∆(Ep). Thus, we must have B = Soc(Homfin(∆(E0),∆(Ep))). The fact thatB1 ⊗Hk B2 indeed has a subquotient with full support follows from KZ(B1 ⊗Hk B2 ⊗Hk ∆(E0)) 6= 0 and[Si, Lemma 2.5]. Our claim follows. So we see that the tensor product does induce a product structure onIrrHC(Hk). By construction, Υ is a semigroup homomorphism. Finally, (iv) follows from Corollary 2.6.

Remark 2.9. The fact that the tensor product induces a group structure on IrrHC(Hk) also follows, somewhatmore indirectly, from results in [L4, Section 3]. Indeed, there a functor is constructed that identifies, as amonoidal category, the quotient category HC(Hk) of all HC Hk-bimodules modulo the full subcategory ofbimodules without full suport (= finite-dimensional bimodules) with the category of representations of aquotient of the group Z`+1.

So we have an upper bound on the number of irreducible infinite dimensional HC Hk-bimodules, whichis given in terms of the parameter q for the Hecke algebra Hq. Let us be explicit about this. Consider theset X := i : ηiqj is a parameter of Hq of the same multiplicity as qj for all j = 0, 1, . . . , ` ⊆ 1, . . . , `+ 1,and let m ∈ X be its minimal element (note that `+ 1 ∈ X.) Similarly to the proof of Corollary 2.6 we cansee that m is actually a divisor of ` + 1. Let p = (` + 1)/m. In particular, |X| = p. Then, the number ofirreducible infinite dimensional HC Hk-bimodules cannot exceed p.

Theorem 2.10. Using the notation of the previous paragraph, we have that | IrrHC(Hk)| = |X|.

In order to prove Theorem 2.10, we will explicitly construct an element in S(Bk) that is in the image ofΥ and whose order coincides with |X|. To do this, we will use the theory of finite W-algebras. But first, wehave to put our results on the context of the spherical subalgebra e0Hke0.

2.5. Spherical Cherednik algebra. In this Subsection we keep assuming that W = Z`+1. Recall the

notation e0 = 1`+1

∑`i=0 s

i, the trivial idempotent in CZ`+1. As an element of Hk, e0 is also idempotent

and so we have the (non-unital) subalgebra Ak := e0Hke0. This algebra inherits a filtration from the oneon Hk, and grAk = C[C2/Z`+1], so this is an algebra quantizing the Kleinian singularity C2/Z`+1. There isalso a notion of category O for the algebra Ak. In fact, under the functor Hk -mod → Ak -mod,M 7→ e0M

we simply have Osphk = e0Ok. However, for our purposes we will need an internal definition of the category

Osphk that was introduced in [GL] and that we explain next.

First of all, we define the spherical Euler element eusph := e0eu = e0eue0 ∈ Ak. This is a grading elementfor Ak, with integral eigenvalues, so we have Ak =

⊕i∈ZAk(i). We have some distinguished subalgebras:

Ak(< 0) :=⊕

i<0Ak(i), Ak(≤ 0) :=⊕

i≤0Ak(i). Similarly, we define the algebras Ak(> 0), Ak(≥ 0). An

Ak-module M is in category Osphk if:

(Osph1) M is finitely generated.(Osph2) eusph acts on M locally finitely.

8 JOSE SIMENTAL

(Osph3) Ak(< 0) acts on M locally nilpotently.

We also have a notion of Verma modules in categoryOsphk , although it is slightly more complicated. First ofall, we define the Cartan subquotient A0

k := Ak(0)/(Ak(0)∩AkAk(< 0)) = Ak(≤ 0)/(Ak(≤ 0)∩AkAk(< 0)).Now take an irreducible A0

k-module S, which we may view as an Ak(≤ 0)-module by means of an epimorphismAk(≤ 0) A0

k. Then, we define the spherical Verma module

∆sph(S) := IndAkAk(≤0)(S) = Ak ⊗Ak(≤0) S.

We remark that ∆sph(S) has a unique simple quotient, which we denote Lsph(S), and the Lsph(S) form

a complete and irredundant list of irreducible modules in Osphk , see e.g. [GL, Section 3.1].

In our case, the Cartan subquotient A0k has quite a simple structure. To see this, let us denote u :=

e0xm, v := e0y

m, w := e0xy ∈ Ak. Since grAk = C[C2/Z`+1] = C[p, q, r]/(r`+1 − pq), with gru = p, gr v =q, grw = r, we have that u, v, w generate the algebra Ak. Moreover, for some polynomial fk(t) ∈ C[t] ofdegree `+ 1 we have that

Ak = C〈u, v, w〉/([w, u] = (`+ 1)u, [w, v] = −(`+ 1)v, uv = fk(w), vu = fk(w + 1))

see e.g. [CBH, Ho]. The grading on Ak given by deg u = `+1,deg v = −(`+1), degw = 0 coincides with thegrading induced from the adjoint eusph-action. It now easily follows that A0

k = C[w]/(fk(w)). In particular,every irreducible A0

k-module is 1-dimensional.

Lemma 2.11. For every irreducible A0-module S, there exists i ∈ 0, . . . , ` such that ∆sph(S) ∼= e0∆(Ei).Thus, the term Verma Ak-module is unambiguous.

Proof. For i ∈ 0, . . . , `, consider the Ak-module e0∆(Ei). Note that, as a vector space, we have e0∆(Ei) =⊕m≥0 Cxi+m(`+1). The lowest weight space Cxi is annihilated by Ak(< 0) and it is stable under the A(0)-

action. So it is a 1-dimensional, hence irreducible, A0k-module, say Si. Thus, we have a nonzero map

∆sph(Si)→ e0∆(Ei) which is the identity on the lowest weight space. Thus, it is an isomorphism of C[x`+1]-(and hence of Ak-)modules. We have thus shown that for every i ∈ 0, . . . , ` there exists a (necessarilyunique) A0

k-irreducible module Si with ∆sph(Si) ∼= e0∆(Ei). The result now follows because, thanks to the

equality Osphk = e0Ok, the number of nonisomorphic irreducible A0k-modules coincides with the number of

isomorphism classes in e0∆(Ei) : i = 0, . . . , ` (and they coincide with the rank of the Grothendieck group

of Osphk .)

For later use, let us record the scalar by which eusph acts on the lowest weight space of the Verma modulee0∆(Ei). Recalling that we have an isomorphism ∆(Ei) = C[x], we may identify this lowest weight spacewith Cxi. The element eu acts on this space by the scalar:

(4)1

2+ i−

∑j=1

2cj1− η−j

ηij =1

2+ i− 1

`+ 1

∑j=0

kj + k`−i

We also have a notion of Harish-Chandra bimodules for the algebra Ak. In fact, we can use the sameDefinition 2.2 because [y`+1, e0] = [x`+1, e0] = 0. Equivalently, an Ak-bimodule B is HC if it can be equippedwith a bimodule filtration in such a way that grB is a finitely generated C[C2/Z`+1]-module, meaning thatthe right and left actions coincide. The equivalence between these two definitions is basically establishedin [L4, Section 5.4], so this notion does coincide with the definition given in the introduction to this paper.The functor Hk -bimod→ Ak -bimod, B 7→ e0Be0 preserves the categories of HC bimodules.

We call a parameter k spherical if Hk = Hke0Hk, that is, if the algebras Ak and Hk are Morita equivalent,and aspherical otherwise. We remark that k being spherical is equivalent to the algebra Ak having finiteglobal dimension, and this happens if and only if the polynomial fk(t) has no repeated roots. The set ofaspherical parameters was determined in [CBH]. We have that k is aspherical if and only if there existu ∈ 0, 1, . . . , ` and an integer p with 1 ≤ p ≤ u such that

p = ku − ku−p

HARISH-CHANDRA BIMODULES FOR QUANTIZATIONS OF TYPE A KLEINIAN SINGULARITIES 9

for example, the parameter k with ki = i for all i = 0, . . . , `+ 1 is aspherical. If the parameter k is spherical,

then the natural functors Ok → Ospherk and HC(Hk)→ HC(Ak) are actually equivalences.

2.6. Shift functors. Let us say that two parameters k, k′ have integral difference ifki−k′i`+1 ∈ Z + α for all

i = 0, . . . , ` and a constant α which is independent of i - under our assumption that k0 = 0, k′0 = 0 weget α = 0. Note that this is equivalent to requiring that there exists a constant β ∈ C× such that for thecorresponding Hecke parameters q and q′ we have qi = βq′i for every i. Using the results of [Go, Section 4],see also [BC, Section 5.4], if k, k′ have integral difference we may define a functor HC(Ak)→ HC(Ak′) thatinduces an equivalence between the categories HC(Ak) and HC(Ak′) that are defined by modding out thesubcategories of finite dimensional bimodules, compare with the proof of Proposition 5.1 in [Si].

Thanks to the description of aspherical parameters given in the previous subsection, we see that, given aparameter k we can always choose k′ such that k′ is spherical and k, k′ have integral difference. Since in thispaper we are interested in computing the number of irreducibles of HC(Hk), we will always assume that ourparameter is spherical.

3. Finite W-algebras

We now describe another way to quantize the algebra C[C2/Z`+1] that is more Lie-theoretic in nature. Afinite W-algebra W is an algebra associated to a semisimple Lie algebra g and a nilpotent element e ∈ g.When e = 0, we have W = U(g), the universal enveloping algebra; while when e is a regular nilpotentthen W = Z, the center of the universal enveloping algebra of g. It turns out that when e is a subregularnilpotent inside sl`+1, central reductions of the finite W-algebraW are algebras quantizing C[C2/Z`+1]. Thissection is organized as follows. First, in Subsection 3.1 we give a construction of the algebra W following[GG]. In Subsection 3.2 we describe a category of modules over the algebra W that is analogous to (andin the case of interest for us essentially coincides with) the category Osph. In Subsection 3.3 we reviewthe definition of a HC bimodule over the finite W-algebra and the construction of a functor from [Gi2, L2]relating HC bimodules over the W-algebra with a more classical notion of a HC bimodule over the universalenveloping algebra U(g). Finally, in Subsection 3.4 we review results of [L5] on the isomorphism betweencentral reductions of finite W-algebras and spherical rational Cherednik algebras. We remark that this entiresection contains no new results.

3.1. Construction. Let us sketch a way to construct the algebra W. Let G be a reductive algebraicgroup with Lie algebra g. We follow [GG], see also [P2, W]. Complete the nilpotent element e ∈ g toan sl2-triple (e, f, h). The adjoint action of h induces a grading g =

⊕i∈Z g(i) and the space g(−1) is

symplectic with symplectic form ω(x, y) = (e, [x, y]), where (·, ·) is a nondegenerate invariant bilinear formon g (e.g. the Killing form.) So we can take a lagrangian subspace l ⊆ g(−1) and form the nilpotent algebram := l ⊕

⊕i<−1 g(i). Note that, since l is lagrangian, χ := (e, ·) is a character of m. Now let Iχ ⊆ U(g) be

the left U(g)-ideal generated by x− χ(x) : x ∈ m. We define W := EndU(g)(U(g)/Iχ)opp = (U(g)/Iχ)adm.It is checked in [GG, Theorem 4.1] that the algebra W does not depend, up to a distinguished isomorphism,on the choice of the lagrangian space l.

The algebra W is filtered with the so-called Kazhdan filtration that is obtained from the usual (PBW)filtration of the universal enveloping algebra U(g) and the grading on the latter algebra that is induced bythe adjoint action of h ∈ U(g), see e.g. [GG, Section 4]. Namely, let us denote by U(g)≤i the PBW filtrationof U(g), and let U(g)≤i(j) := x ∈ U(g)≤i : [h, x] = jx, so that U(g)≤i =

⊕j U(g)≤i(j). The Kazhdan

filtration is now defined to be Fn U(g) :=∑

j+2i≤n U(g)≤i(j). This induces a filtration on W ⊆ U(g)/Iχ.

Under this filtration, we have that grW = C[S ], where S := e+ ker[f, ·] is the Slodowy slice associated tothe nilpotent e ∈ g.

We remark that the center of W is isomorphic to the center Z of U(g). Indeed, since Z ⊆ U(g)adm itis clear that we have a map Z → W. According to an argument of Ginzburg, see e.g. [P2, Question 5.1],this map is an isomorphism onto its image and identifies Z with the center of W. So, if we denote by h aCartan subalgebra of g and by W the Weyl group of g, thanks to the Harish-Chandra isomorphism we haveZ ∼→ C[h∗]W . Let us explain our conventions on the Harish-Chandra isomorphism. Under a decompositiong = n− ⊕ h ⊕ n+, we have the vector space decomposition U(g) = S(h) ⊕ (n−U(g) + U(g)n+). Then, ourHarish-Chandra isomorphism is the restriction to Z of the projection to the first factor, followed by the map

10 JOSE SIMENTAL

S(h)→ S(h), f(λ) 7→ f(λ− ρ), where ρ denotes, as usual, half the sum of positive roots. In particular, theimage of the Harish-Chandra homomorphism Z → S(h) falls into the invariants for the usual W -action onS(h) - not the dot-action. It follows that for λ ∈ h∗/W we may define the central reduction Wλ. Underthe induced filtration from that on W we have that grWλ = C[S ∩ N ] where N denotes the nilpotentcone in g. As we will see in Subsection 3.4 these central reductions, for the special case of a subregularnilpotent element in sl`+1, are isomorphic to the algebras Ak from the previous section, with an explicitcorrespondence between the parameters λ and k. But before, let us examine the representation theory ofthe algebra W.

3.2. Category OW . We continue with the setup from the previous subsection, and review the definition ofthe category O for the W-algebra introduced in [BGK]. Let Q := ZG(e, h, f) and T ⊆ Q a maximal torus.Let t ⊆ g denote the Lie algebra of T . Note that we have a embedding t → W, this follows because t isannihilated by (e, ·) and m is t-stable. Pick θ ∈ X∗(T ) = Hom(C×, T ) ⊆ t, this will be an analog of the Eulerelement. We have a decomposition W =

⊕n∈ZW(n) with respect to the eigenvalues of ad θ. Analogously

to Subsection 2.5 we define the following subalgebras of W:

W(< 0) :=⊕n<0

W(n) W(≤ 0) :=⊕n≤0

W(n).

Now we can define category OW . By definition, a W-module M belongs to category OW if

(a) M is finitely generated.(b) θ acts on M locally finitely.(c) W(< 0) acts on M locally nilpotently.

We also have a Cartan subquotient, W0 := W(≤ 0)/(W(≤ 0) ∩WW(< 0)) and Verma modules ∆W(S)for S an irreducible representation ofW0. We remark that the category OW depends on the choice of θ, butwe are going to ignore θ from the notation. For λ ∈ h∗/W , we also have a category OWλ that is, by definition,

the full subcategory of categoryOW whereW acts with central character determined by λ, cf. Subsection 3.1.

In [L3] it is proven that the category OW is equivalent to a certain full subcategory of U(g), the so-calledcategory of generalized Whittaker modules, see Theorem 4.1 in loc. cit. In the case when e ∈ g is regular in acertain Levi subalgebra, this category coincides with the category studied in [Ba, MS], see [L3, Proposition4.2]. Since we are going to use only a very special case of this result, we defer a more detailed expositionuntil Subsection 4.1.

3.3. Functors between categories of HC bimodules. Now we introduce a category of HC bimodulesfor the algebra W. A W-bimodule B is said to be HC if it can be equipped with a filtration in such away that grB is a finitely generated C[S ]-module. We have an analogous notion for bimodules over theenveloping algebra U(g). We remark that a U(g)-bimodule B is HC if and only if it is finitely generated andthe ad g action on B is locally finite, see e.g. [BG]. In particular, for a finite dimensional g-module V wemay consider the U(g) bimodule ΦV := V ⊗ U(g) with action given by:

ξ(v ⊗ f) = ξ.v ⊗ f + v ⊗ ξf, (v ⊗ f)ξ = v ⊗ fξ, v ∈ V, f ∈ U(g), ξ ∈ g.

The bimodule ΦV is HC. Moreover, for every HC U(g)-bimodule B there exists a suitable finite dimensionalbimodule V and an epimorphism ΦV B. Note that, if M is a left U(g)-module then we have thatΦV ⊗U(g) M = V ⊗C M .

A nice relationship between the categories of HC U(g)- and W-bimodules was discovered in [Gi2, L2].Namely, there exists a functor •† : HC(U(g)) → HC(W) satisfying the following conditions (actually, thestatement in [L2] is stronger, as it tells us that the image falls in the category of Q-equivariant HC W-bimodules where Q = ZG(e, f, h) acts naturally on W, but we will not need this):

(1) •† is exact and intertwines the tensor products.(2) (U(g))† =W.(3) For λ, µ ∈ h∗/W , let λHC(U(g))µ be the full subcategory of HC U(g)-bimodules that factor throughU(g)λ ⊗ U(g)opp

µ , and let λHC(W)µ have a similar meaning for the W-algebra W. Then, •† maps

λHC(U(g))µ to λHC(W)µ.

HARISH-CHANDRA BIMODULES FOR QUANTIZATIONS OF TYPE A KLEINIAN SINGULARITIES 11

Let us describe the construction of this functor following [L2]. This is based on another realization of Wby means of deformation quantization. In order to do so, we consider the homogenized enveloping algebraU~(g) := T (g)/(u ⊗ v − v ⊗ u − ~[u, v], u, v ∈ g). An advantage of this algebra is that now we have analgebra map U~(g) S(g), ~ 7→ 0. So let mχ ⊆ S(g) = C[g∗] be the maximal ideal corresponding tothe element χ = (e, ·) ∈ g∗, and Mχ ⊆ U~(g) its pullback under the epimorphism U~(g) S(g). Thenwe can consider the completion U∧~ of U~(g) at Mχ, this is a complete C[[~]]-algebra. According to thedecomposition theorem, [L, Theorem 3.3.1] we have an isomorphism of topological C[[~]]-algebras:

(5) U∧~∼=→ A~(V )∧⊗C[[~]]W∧~

where:

• A~(V )∧ is the completion at 0 of the homogenized Weyl algebra of a vector space V .• W∧~ is a C[[~]]-algebra equipped with a C×-action satisfying (W∧~ )C×−l.f./(~ − 1) ∼= W as filtered

algebras, where •C×−l.f. stands for the subalgebra of locally finite elements.

We remark that the isomorphism (5) is C×-equivariant and that the algebra W∧~ can be realized in termsof Fedosov quantization, see e.g. [L, 2.2].

The functor •† is now constructed as follows. Start with a HC U(g)-bimodule B. Pick a good filtration onB and consider the Rees bimodule R~(B) =: B~, this is a U~(g)-bimodule. We can consider the completionB∧~ = U∧~ ⊗U~(g) B~ which is a bimodule over A~(V )∧⊗C[[~]]W∧~ by means of the isomorphism (5). So theelements annihilated by the adjoint action of V form a bimodule over the algebra W∧~ , we denote this

bimodule by (B∧~ )adV . We remark that (B∧~ )adV carries a compatible action of C×, this follows because theisomorphism (5) is C×-equivariant. Finally, define

B† := ((B∧~ )adV )C×−l.f./(~− 1),

this is a functor we need. The proof that •† satisfies properties (1), (2) and (3) above is in [L2, Section3.4]. We remark that this construction coincides with the quantum Hamiltonian reduction constructionintroduced by Ginzburg in [Gi2], this is established in [L2, Subsection 3.5].

3.4. W-algebras vs. Spherical Cherednik algebras. Now we tie together the Cherednik and W-algebrasides of the picture. The results in this section come from [L5] in the form we need them, see also [EGGO, Go].Recall the spherical rational Cherednik algebra Ak = e0Hke0 from the previous section. According to [L5,Theorems 5.3.1 and 6.2.2] there is a filtered algebra isomorphism between Ak and a certain central reductionWλ of the W-algebra W associated to a subregular nilpotent e := e(`,1) ∈ sl`+1, where λ ∈ h is computed as

follows. Let π1, . . . , π` ∈ h∗ be the fundamental weights. Then, λ =∑`

i=1 λiπi, where

(6) λi =1

`+ 1(1− ki + ki−1), i = 1, . . . , `

We remark that the filtered isomorphism ϕ : Ak → Wλ constructed in [L5] induces the identity atthe associated graded level. This has the following consequence. Note that grAk = C[C2/Z`+1] =

C[x, y, z]/(z`+1−xy) has C[C2/Z`+1]0 = C,C[C2/Z`+1]1 = 0 and C[C2/Z`+1]2 = C. Since eusph ∈ A≤2k \A

≤1k

and θ ∈ W≤2λ \W

≤1λ , we have that there exist n ∈ Z, α ∈ C such that, ϕ(eusph) = nθ+α. So, up to a change

of sign of θ, we have that the pullback under the isomorphism ϕ identifies the corresponding categories O.For a specific choice of θ, we will determine the values of n and α in Subsection 4.3, see Lemma 4.3.

Now, since the isomorphism Ak →Wλ is filtered, we have that the notions of a HC Ak- andWλ-bimodulecoincide. So our strategy will be as follows. We have an upper bound in terms of the parameter q for theHecke algebra Hq. We have interpreted this in terms of bijections between the blocks of category Ok. Wehave seen that the categories Ok and OWλ coincide up to a choice of the cocharacter θ ∈ X∗(T ). So we needto produce a HC Wλ-bimodule that induces a desired bijection between the blocks of category O. We willsee that there exists a finite dimensional representation V of sl`+1 such that a quotient of the W-bimodule(ΦV )† satisfies this requirement.

12 JOSE SIMENTAL

4. Proof of the main result.

Now we prove our main result following the strategy above. In order to do this, first we review resultsrelating the category OWλ to the usual BGG category O, which we denote Og. This is done in two steps.

First, we have an equivalence between the category OWλ and the category of generalized Whittaker modulesWhitχ that we define in Subsection 4.1, this is due to Losev, [L3]. Then we have a functor from the BGGcategory O to the category Whitχ that was studied by Backelin in [Ba] and which we review in Subsection4.2. A nice feature of this functor is that it commutes with the multiplication by a finite dimensional module.Subsection 4.3 is not strictly necessary but will motivate some of the constructions we use, its goal is tocompare the labels between Verma modules in the category Ok and OWλ (or, equivalently, Whitχ,) we dothis by looking at the action of the Euler element in the lowest weight space of a Verma module. We giveour main argument in Subsection 4.4.

4.1. The Whittaker category and Losev’s functor. Recall the category OWλ of modules over the finiteW-algebra W. In [L3] it is shown that this category is equivalent to a certain subcategory of the categoryof U(g)-modules. Let us describe this category in the case of interest for us.

So letG = SL`+1, g = sl`+1 and e = e(`,1) ∈ g be a subregular nilpotent. We may assume that e is in Jordanform. Fix an sl2-triple (e, f, h) and note that ZG(e, f, h) is a 1-dimensional torus, say T . Up to conjugation,we may assume that T = diag(a, a, . . . , a, a−`) : a ∈ C× ∼= C×. As before, we have an embeddingLie(T ) =: t → W, T acts on W and its differential coincides with the adjoint action of t on W. Now letθ ∈ X∗(T ), the cocharacter lattice of T . Embedding X∗(T ) in t we may take e.g. θ = diag(−1,−1, . . . ,−1, `).Taking now L := ZG(θ), l := Lie(L) we have

L =

(A 00 det(A)−1

): A ∈ GL`

∼= GL`, l =

(A 00 − tr(A)

): A ∈ gl`

∼= gl`

Note that we have that l is stable under the adjoint action of h, and that, under the grading given by adh,l(−1) = 0. The algebra m :=

⊕i≤−2 l(i) is identified with the algebra of strictly lower triangular matrices

in gl`. Now denoting by g>0 the sum of eigenspaces for the action of ad θ with positive eigenvalues, wehave that n := m ⊕ g>0 coincides with the algebra of strictly lower triangular matrices in sl`+1. So we geta Cartan decomposition sl`+1 = n ⊕ h ⊕ n−, where n is the algebra of strictly lower triangular matrices; hconsists of diagonal matrices; and n− is the subalgebra of strictly upper triangular matrices. Denoting by

χ := (e, ·), we have that χ((xij)) =∑`+1

i−1 xi+1,i. In particular, χ([n, n]) = 0, so χ|n is a character.We are now ready to define the category of Whittaker modules. We say that a module M ∈ U(g) -mod

belongs to Whitχ if:

(Wh1) M is finitely generated.(Wh2) n ⊆ g acts on M with generalized eigencharacter χ.(Wh3) M decomposes into the direct sum of its generalized Z-eigenspaces.

For λ ∈ h∗/W , we denote by Whitχ(λ) the full subcategory of Whitχ consisting of objects where Zacts with character λ. We remark that this is not a Serre subcategory of Whitχ. We also remark that, ifB ∈ λHC(U(g))λ and N ∈ Whitχ(λ) then B ⊗U(g) N ∈ Whitχ(λ).

There is also a notion of Verma modules for the category Whitχ(λ), these were introduced in [MS]. Toconstruct them, we need some preliminaries. First, let Ω denote the set of roots with respect to the Cartandecomposition g = n⊕h⊕n−, that is, Ω = εi− εj : i, j = 1, . . . , `+1, i 6= j, where εi ∈ gl∗`+1 is the functionthat takes the (i, i)-th entry of a matrix. The set of simple roots is Π := εi − εi+1 : i = 1, . . . , `. Let pχ bethe parabolic subalgebra of sl`+1 determined by the set of roots Π′ := α ∈ Π : χ|gα 6= 0. Explicitly:

pχ =

∗ ∗ · · · ∗ 0∗ ∗ · · · ∗ 0...

.... . .

... 0∗ ∗ · · · ∗ ∗

Now let pχ = l⊕ nχ be the adh-stable Levi decomposition of pχ. We have:

HARISH-CHANDRA BIMODULES FOR QUANTIZATIONS OF TYPE A KLEINIAN SINGULARITIES 13

nχ =

(0 0∗ 0

): ∗ ∈ Cn

∼= Cn

where Cn is considered as an abelian Lie algebra. Now let bχ = b∩ gχ, nχ = n∩ gχ. Here, b = n⊕ h. Again,χ([nχ, nχ]) = 0, so χ|nχ is a character. It is easy to see that χ|nχ is nondegenerate. We will denote by Cχthe 1-dimensional representation of nχ determined by χ. Note that the center of U(l) may be identified with

S(h)W′, where W ′ is the Weyl group of Π′, that is, S` ⊆ S`+1. So, for λ ∈ h∗ we get via the Harish-Chandra

isomorphism a maximal ideal I(λ) in the center of U(l). We have the induced module:

Y (λ) := U(l)/I(λ)U(l)⊗U(nχ) CχThis is an l-module, that might be considered as a pχ-module by letting nχ act by 0. So we have that the

Verma module ∆Whit(λ) is the induced g-module:

∆Whit(λ) := U(g)⊗U(pχ) Y (λ).

The following is Proposition 2.1 in [MS].

Proposition 4.1. The following is true.

(1) The module ∆Whit(λ) belongs to Whitχ(λ).(2) The modules ∆Whit(λ) and ∆Whit(µ) are isomorphic if and only if λ and µ belong to the same S`-orbit.(3) The module ∆Whit(λ) admits a unique irreducible quotient, LWhit(λ). Moreover, LWhit(λ) ∼= LWhit(µ)

if and only if µ and λ belong to the same S`-orbit, and the LWhit(λ) form a complete list of irreduciblemodules in Whitχ.

We remark that, thanks to Proposition 4.1, the irreducible modules in Whitχ(λ) are parametrized byS`-orbits in S`+1λ. In particular, when λ ∈ h∗ is regular there are ` + 1 of them. It turns out that thecategory Whitχ(λ) is equivalent to the category OWλ , this is the main result of [L3]. More precisely, we havethe following result.

Theorem 4.2. The categories Whitχ(λ) and OWλ are equivalent. Moreover, an equivalence Ψ : Whitχ(λ)→OWλ satisfies the following.

(1) For µ ∈ S`+1λ, there exists an irreducible representation S of W0 with Ψ(∆Whit(µ)) = ∆W(S). Inother words, this equivalence sends Verma modules to Verma modules.

(2) For B ∈ λ HC(U(g))λ and M ∈ Whitχ(λ) we have a natural isomorphism Ψ(B ⊗U(g) M) ∼= B† ⊗WΨ(M).

The statement (1) of the previous theorem is part of [L3, Theorem 4.1], while (2) follows easily by theconstruction of the functor Ψ, see Lemma 5.3 in loc. cit.

4.2. The BGG category O and Backelin’s functor. Now we relate the category of Whittaker modulesWhitχ(λ) with the classical BGG category O of U(g)-modules, following [Ba]. For the sake of completeness,let us define the latter category. An U(g)-module M is said to be in category Og if the following conditionsare satisfied.

(BGG1) M is finitely generated.(BGG2) M is the direct sum of its h-weight spaces.(BGG3) M is n-locally nilpotent.

As before, for a weight λ ∈ h∗ we denote by Ogλ the full subcategory of Og where Z acts with character

induced by λ. There is a well-known notion of Verma modules in category Og, see e.g. [Hu, Section 1.3], wewill denote these by ∆g(λ). We remark that, thanks to our convention on the Harish-Chandra isomorphism,U(g) acts on ∆g(λ) with central character determined by λ + ρ, that is, ∆g(λ) ∈ Og

λ+ρ. In [Ba], Backelin

defines a Jacquet functor Γ : Og → Whitχ. This functor satisfies the following.

(1) It is exact, [Ba, Lemma 3.2].(2) It sends Verma modules to Verma modules. More precisely, Γ(∆g(λ)) = ∆Whit(λ+ ρ), [Ba, Proposi-

tion 6.9]. Note that it follows that for λ ∈ h∗, Γ sends Ogλ to Whitχ(λ).

(3) For a finite dimensional g-module V and M ∈ Og, Γ(V ⊗M) = V ⊗ Γ(M), this follows easily fromthe definition of Γ, see e.g. the proof of Proposition 6.9 in [Ba].

14 JOSE SIMENTAL

4.3. Correspondence between Vermas. In this subsection, we find an explicit correspondence betweenthe Cherednik Verma modules e0∆(Ei) and the W-algebra Verma modules ∆W(S) under the isomorphism

ϕ : Ak → Wλ where, recall, λ =∑`

i=1 λiπi with λi = 1`+1(1 − ki + ki−1). Thanks to Theorem 4.2, the

W-algebra Verma modules have the form Ψ(∆Whit(wλ)) for w ∈ S`+1. Our strategy will be to compare theaction of θ on the lowest weight space of Ψ(∆Whit(wλ)) with (4), which gave us a formula for the action ofeusph on the lowest weight space of e0∆(Ei).

Recall that we have fixed the elements e = e(`,1) and θ = diag(−1,−1, . . . , `). Now, upon the identification

h = h∗ = x ∈ C`+1 :∑xi = 0 as above, write λ = (λ1, . . . , λ`, λ`+1). According to [MS, Proof of Lemma

2.2], the scalar by which θ acts on the lowest weight space of Ψ(∆Whit(si,`+1λ)) is (` + 1)λi. An easycalculation shows that this is:

(7)1

`+ 1

i−1∑j=1

−j(1− kj + kj−1) +`+1∑j=i

(`+ 1− j)(1− kj + kj−1)

= 1− i+`

2− 1

`+ 1

∑j=0

kj + ki−1

Note that, if λ ∈ h∗ is regular (equivalently, if k is spherical) then all these numbers are distinct. Thus,comparing (7) and (4) we see the following.

Lemma 4.3. Under the isomorphism ϕ : Ak → Wλ we have that ϕ(eusph) = θ + `+12 . Moreover, we have

that e0∆(Ei) ∼= ϕ∗Ψ(∆Whit(s(`+1−i,`+1)λ)), where ϕ∗ : OWλ → Osphk is the pullback functor.

Let us see a consequence of Lemma 4.3. First of all, note that we have an action of S`+1 on the set ofparameters k for the spherical Cherednik algebra that is induced from the action of S` on h∗, for σ ∈ S`+1 we

will denote by σ ? k the parameter associated to σλ, we have an isomorphism Akϕk−→Wλ =Wσλ

ϕ−1σ?k−→ Aσ?k,

see [GL, Section 4] for a more general discussion. Thanks to Lemma 4.3, the pullback under this isomorphism

of e0∆sphσ?k(Ei) is e0∆sph

k (Eσ(i)).

4.4. Constructing a HC bimodule. We are now ready to prove our main theorem. So let k be a Cherednikparameter such that the Hecke algebra Hq has the form:

C[t]

/m−1∏i=0

p−1∏j=0

(t− ηmjβi)

where:

• η = exp(2π√−1/(`+ 1)).

• m ∈ Z divides (`+ 1) and mp = `+ 1.• β0 = 1, β1, . . . , βm−1 ∈ C× are not necessarily distinct.

Thanks to the shift functors, see Subsection 2.6, we may assume that k has the form:

k0 = 0 kp = a1 . . . k(m−1)p = am−1

k1 = m+ 1 kp+1 = a1 +m+ 1 . . . k(m−1)p+1 = am−1 +m+ 1k2 = 2(m+ 1) kp+2 = a1 + 2(m+ 1) . . . k(m−1)p+2 = am−1 + 2(m+ 1)...

... . . .kp−1 = (p− 1)(m+ 1) k2p−1 = a1 + (p− 1)(m+ 1) . . . k` = am−1 + (p− 1)(m+ 1)

for some a1, . . . , am−1 ∈ C. In particular, the associated parameter for the W-algebra is λ =∑`

i=1 λiπiwhere

λi =

(`+ 1)−1(1− aj + aj−1 + (p− 1)(m+ 1)) i = jp, j = 1, . . . ,m− 1

(`+ 1)−1(−m) else

We remark that we can shift the ai’s by multiples of `+ 1 so that λ− ρ is a dominant weight. It is easyto see that σ(λ)− λ = (σ(λ)− ρ− (λ− ρ)) is an integral weight, where σ is the following element of S`+1:

HARISH-CHANDRA BIMODULES FOR QUANTIZATIONS OF TYPE A KLEINIAN SINGULARITIES 15

(1 2 . . . p− 1 p p+ 1 p+ 2 . . . 2p 2p+ 1 . . . (m− 1)p (m− 1)p+ 1 . . . ` `+ 1p 1 . . . p− 2 p− 1 2p p+ 1 . . . 2p− 1 3p . . . `+ 1 (m− 1)p . . . `− 1 `

)Let us remark that σ is suggested by Lemma 4.3. Indeed, we want a bijection between blocks such that

Bqi 7→ Bηmqi , where each block is labeled by the corresponding Hecke eigenvalue, and this is precisely whatσ does. Since λ−ρ is assumed to be dominant, thanks to [Hu, Remark 3.9] we have that there exists a finitedimensional sl`+1-module V and a nonzero map f : V ⊗∆g(λ − ρ) → ∆g(σλ − ρ). In particular, we get anonzero map Γ(f) : V ⊗∆Whit(λ)→ ∆Whit(σλ). Recall that V ⊗∆Whit(λ) = ΦV ⊗U(g) ∆Whit(λ). So we get a

nonzero map (ΦV )†⊗W Ψ(∆Whit(λ))→ Ψ(∆Whit(σλ)). Let I(λ) ⊆ Z ⊆ W be the maximal ideal determined

by λ so that, in particular, (ΦV )λ† := (ΦV )†/(I(λ)(ΦV )† + (ΦV )†I(λ)) is a Wλ-bimodule. Note that we still

have a nonzero map (ΦV )λ† ⊗WλΨ(∆Whit(λ)) → Ψ(∆Whit(σλ)). Now ∆Whit(σλ) = ∆Whit(s(`,`+1)λ). Thus,

applying the pullback under isomorphism ϕ : Ak →Wλ we obtain thanks to Lemma 4.3 a nonzero map:

ϕ∗((ΦV )λ† )⊗Ak e0∆(E0)→ e0∆(E1)

In particular, Homfin(e0∆(E0), e0∆(E1)) 6= 0. This implies that Homfin(∆(E0),∆(E1)) 6= 0. But ∆(E0)belongs to the block of category Ok indexed by 1, while ∆(E1) belongs to the block indexed by ηm. SoSoc(Homfin(∆(E0),∆(E1))) ∈ IrrHC(Hk) has order (`+ 1)/m = p, cf. Lemma 2.8. This concludes the proofof Theorem 2.10.

Remark 4.4. We remark that the proof of Theorem 2.10 actually shows a bit more. Indeed, for a block Bqiin category Ok let ∆qi ∈ Bqi be the Verma module whose length coincides with the multiplicity of qi in theset of Hecke parameters. Then, under the assumptions of Theorem 2.10, our proof shows that there exists a-not necessarily irreducible- HC Hk-bimodule B with B ⊗∆qi = ∆ηmqi.

5. Application to general rational Cherednik algebras

In this section we present a classification of irreducible, fully supported HC bimodules over general rationalCherednik algebras that were introduced in Section 2. To do so, first we recall preliminary results on HCbimodules, in particular restriction functors from [L4] and a reduction result obtained by the author in [Si].After this, we are able to prove our main classification result, Theorem 5.1 which is a consequence of theaforementioned reduction result and Theorem 2.10.

5.1. Harish-Chandra bimodules. Let Hk = Hk(W,R), see Section 2 for the notation. Recall that anHk-bimodule B is said to be HC if it is finitely generated and for every a ∈ C[R]W ∪ C[R∗]W the adjointaction of a on B is locally nilpotent or, equivalently, [L4, Subsection 5.4], B is HC if it can be equipped witha filtration in such a way that grB is a finitely generated Z(C[R ⊕ R∗]#W ) = C[R ⊕ R∗]W -module. As inSection 2, we have the notion of the (singular) support of B, SS(B) := supp(grB). Let us remark that, inthe case W = Z`+1 and R = C that we considered in Section 2, an irreducible bimodule B has full supportif and only if it is infinite dimensional.

To formulate our main theorem, we need some notation. First, let us denote by HC(Hk) the quotient ofthe category of all HC Hk-bimodules by the full subcategory consisting of bimodules with proper support.We will see that this category is equivalent to the category of representations of a certain quotient of the

group W . Let W be the subgroup of W determined as follows. For each reflection hyperplane V ∈ A considerthe set XV := i ∈ 1, . . . , `V : ηiV qV,j is a parameter of Hq(WV ) of the same multipicity as qV,j for all j =

0, . . . , `V − 1 where, recall, ηV = exp(2π√−1/`V ). Note that `V ∈ XV . Now let mV := minXV . Similarly

to Subsection 2.4, mV divides `V , and let pV := `V /mV . Then, W := 〈spVV : V ∈ A〉. Thanks to the

conjugation invariance of the parameter k, W is actually a normal subgroup of W . The following is ourmain result.

Theorem 5.1. The category HC(Hk) is equivalent, as a monoidal category, to the category of representations

of the group W/W .

We remark that the fact that HC(Hk) is equivalent to a category of the form (W/W ) -rep for a certain

normal subgroup W of W is an immediate consequence of [L4, Theorem 3.4.6(5)]. The fact that W ⊆ W

16 JOSE SIMENTAL

is, basically, [Si, Lemma 4.8]. We also remark that, in the case W = Z`+1, Theorem 5.1 is a reformulationof Theorem 2.10, we will elaborate on both of these remarks in the following subsection. The proof ofTheorem 5.1 will be given in the next subsection after we explain some preliminary results, which are basedon restriction functors introduced in [L4].

5.2. Reduction to rank-1 reflection groups. Recall that we denote by HC(Hk) the quotient of thecategory of all HC Hk-bimodules by the full subcategory consisting of bimodules with proper support. In[L4, Section 3] it is shown that there exists a functor •† : HC(Hk) → W -rep that identifies the category

HC(Hk) with a full subcategory of W -rep closed under taking direct summands and tensor products. We willwork in slight more generality. First of all, let us describe the possible supports of HC bimodules, these areclosures of symplectic leaves in (R⊕R∗)/W which may be described as follows. Let W ′ ⊆W be a parabolicsubgroup, that is, the stabilizer of a point in R. Then, the set LW ′ := π(y, x) ∈ R ⊕ R∗ : W(y,x) = W ′ is

a symplectic leaf, where π : R ⊕R∗ → (R ⊕R∗)/W denotes the projection. Its closure is LW ′ = π(y, x) ∈R⊕R∗ : W ′ ⊆W(y,x). We remark that LW ′ = LW ′′ provided that W ′ and W ′′ are W -conjugate.

For a parabolic subgroup W ′ ⊆ W , let HCLW (Hk) denote the full subcategory of HC bimodules whose

support is contained in LW . Define similarly HC∂LW (Hk), where ∂LW = LW \LW , this is a Serre subcategory

in HCLW (Hk). Finally, define HCLW := HCLW (Hk)/HC∂LW (Hk). For example, HC(Hk) = HCL1(Hk).

Now, there exists a decomposition R = RW ′ ⊕ RW′, where RW ′ is a unique W ′-invariant complement

to RW′. So we may consider the rational Cherednik algebra Hk(W

′, RW ′), where the parameter k here is

determined as follows. Let H ′ ⊆ RW ′ be a reflection hyperplane. Then, H := H ′ ⊕ RW ′ is a reflectionhyperplane for the action of W and, moreover, WH = W ′H′ , so abusing the notation we denote by k theparameter determined by that of the algebra Hk(W,R). We remark that we have an action of NW (W ′)on Hk(W

′, RW ′) by algebra automorphisms, the action of W ′ ⊆ NW (W ′) coincides with the usual adjointaction. Denoting ΞW ′ := NW (W ′), a HC Hk(W

′, RW ′) is said to be ΞW ′-equivariant if it is equippedwith a compatible NW (W ′)-action in such a way that the action of W ′ ⊆ NW (W ′) coincides with theadjoint action, w.x = wxw−1. We denote the category of ΞW ′-equivariant HC Hk(W

′, RW ′)-bimodules byHCΞW ′ (Hk(W

′, RW ′)).

Proposition 5.2 ([L4], Subsection 3.6). For a parabolic subgroup W ′ ⊆ W , there exists a functor •†WW ′

:

HC(Hk)→ HCΞW ′ (Hk(W′, RW ′)) satisfying the following properties.

(1) •†WW ′

is exact and intertwines the tensor products.

(2) For a HC Hk-bimodule B, B†WW ′

is finite dimensional if and only if SS(B) ⊆ LW ′, and it is zero if

and only if SS(B) ⊆ ∂LW ′.(3) The restriction of •†W

W ′to HCL(Hk) is a quotient functor that identifies the quotient category HCL(Hk)

with a full subcategory of HCΞW ′0 (Hk(W

′, RW ′)) that is closed under subquotients and tensor prod-

ucts. Here, HCΞW ′0 (Hk(W

′, RW ′)) denotes the category of finite-dimensional ΞW ′-equivariant HCHk(W

′, RW ′)-bimodules.(4) For parabolic subgroups W ′′ ⊆W ′ ⊆W , we have an isomorphism of functors FWW ′•†W

W ′′∼= •†W ′

W ′′FW

•†WW ′

. Here, FW : HCΞW ′ (Hk(W′, RW ′)) → HC(Hk(W

′, RW ′)) is the forgetful functor that forgets

the ΞW ′-equivariant structure; and FWW ′ : HCΞW ′′ (Hk(W′′, RW ′′))→ HCNW ′ (W

′′)/W ′′(Hk(W′′, RW ′′))

is the functor restricting the ΞW ′′-equivariant structure to a NW ′′(W′)/W ′′-equivariant structure.

We remark that the embedding •† = •†W1 : HC(Hk) → W -rep mentioned above is a special case of

Proposition 5.2(3) when W ′ = 1. So we need to compute the image of •†. The following is a consequenceof [Si, Theorem 3.10].

Proposition 5.3. Let M be a representation of W . Assume that for all rank-1 parabolic subgroups W ′ ⊆Wthere exists a HC Hk(W

′, RW ′)-bimodule BW ′ such that (BW ′)†W ′1= ResWW ′(M). Then, M belongs to the

image of •†W1.

Since the image of •†W1 is closed under direct summands and tensor products, we have that it is of the

form W/W -rep for some normal subgroup W of W . Using a simple counting argument, Theorem 2.10

HARISH-CHANDRA BIMODULES FOR QUANTIZATIONS OF TYPE A KLEINIAN SINGULARITIES 17

implies that, in the case W = Z`+1, W = W , so Theorem 5.1 for cyclic groups follows. From here it follows,

thanks to Proposition 5.2(4), that W ⊆W for any complex reflection group W .

Proof of Theorem 5.1. We now complete the proof of Theorem 5.1. Recall that we need to computethe image of •†W1 , and we have seen that this has the form W/W -rep for some normal subgroup W ⊆ W .

Moreover, we have seen that W ⊆ W . So we need to check the other inclusion. Let M ∈ (W/W ) -rep.

Note that, by the definition of W , for every rank-1 (= cyclic) parabolic subgroup W ′ of W , we have that

ResWW ′(M) ∈ (W ′/W ′) -rep. Since Theorem 5.1 is valid for cyclic groups, we have that there exists a HC

Hk(W′, RW ′)-bimodule BW ′ such that (BW ′)†W ′1

= ResWW ′(M). So the hypotheses of Proposition 5.3 are

satisfied. It follows that M = B†W1for some HC Hk-bimodule B. This finishes the proof of Theorem 5.1.

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Department of Mathematics. Northeastern University. Boston, MA 02115. USA.E-mail address: simentalrodriguez.j@husky.neu.edu