On Harish-Chandra bimodules for rational Cherednik ...josesr/thesis.pdf · On Harish-Chandra...

On Harish-Chandra bimodules for rational Cherednik algebras

by Jose Simental Rodrıguez

B. S. Universidad Nacional Autonoma de MexicoM. S. Ohio University

A dissertation submitted to

The Faculty ofthe College of Science ofNortheastern University

in partial fulfillment of the requirementsfor the degree of Doctor of Philosophy

April 18, 2017

Dissertation directed by

Ivan LoseuProfessor of Mathematics

Dedication

A mi mama, mi papa, la Fer y el Kbto.

ii

Acknowledgments

First and foremost, I would like to express my deepest gratitude to my advisor, Professor

Ivan Losev. Not only he proposed the problem that gives rise to this work and gave countless

suggestions during many meetings, he also made sure I learned a lot of interesting Mathe-

matics during my time at Northeastern. Besides that, he always showed tremendous support

and encouragement throughout the past five years.

I would also like to thank The Boss, Professor Pavel Etingof, for his generous sharing of ideas

and contagious passion for Mathematics, which he always expresses with splendid sense of

humor and palpable gusto.

Thanks to Professor Valerio Toledano Laredo for his interest in my work and his great

lectures. Thanks to Professor Emanuele Macrı for always being a friendly face and agreeing

to be in my committee, even if him being in Paris makes this a nontrivial task.

Thanks also go to my academic siblings Alexey, Boris, Dmytro, Huijun and Seth for many

mathematical discussions, most of which had not much to do with the contents of this

dissertation. Special thanks go to Seth for carefully reading parts of this work and giving

very helpful suggestions. Also, thanks to Vassilis for being a living encyclopedia of abstract

nonsense, which helped me in some parts of this work.

I must also thank Andras, Brian, Pengshuai and Simone for making of our office a nice place

to work, even if it has no windows and it gets too cold during weekends.

The past five years in Boston were not entirely Mathematics, and for that I would like to

thank all, my friends, who helped me keeping my sanity. Many thanks to Adolfo, Andras,

iii

Arturo, Brian, Chen, Darcy, Enrico, Floran, Gouri, Jonier, Mayra, Monika, Rahul, Saif,

Sensei, Simone, Tong, Vassilis, Whitney and all other people that are not mentioned here.

Special mention goes to The Grading party, formerly Good fibrations, for all those Fridays

that became an indispensable part of my life.

Muito obrigado ao Jonier. Eu nao sei se ha palavras para expressar o efeito que tu causou

na minha vida, mas acho que contigo as palavras sobram.

Finalmente, mas no al final, agradezco a toda mi familia por todo el apoyo y amor que me

han dado, no solamente durante los ultimos cinco anos. Saben que son correspondidos, y

que esto es por y para ustedes.

Jose Simental Rodrıguez

Northeastern University

April 2017

iv

Abstract of Dissertation

We study Harish-Chandra bimodules for rational Cherednik algebras Hc(W ) associated to

a complex reflection group W and parameter c. Our results allow to partially reduce the

study of these to smaller algebras. We use this to classify those pairs of parameters (c, c′) for

which there exist Harish-Chandra bimodules with full support, and we give a description of

the category of all Harish-Chandra bimodules modulo those bimodules with proper support.

When W is the symmetric group, we produce an embedding from the category of Harish-

Chandra Hc-bimodules to the category Oc, prove that its image is closed under subquotients,

and find the irreducibles in its image. Finally, when W is of cyclotomic type we produce a

duality in the category of Harish-Chandra bimodules. We do this in the more general setting

of quantized quiver varieties. Our methods are based on localization techniques, the study

of partial KZ functors, the action of Namikawa-Weyl groups and restriction functors.

v

Table of Contents

Dedication ii

Acknowledgments iii

Abstract of Dissertation v

Table of Contents vi

Disclaimer x

Chapter 1 Introduction 1

1.1 Harish-Chandra bimodules with full support . . . . . . . . . . . . . . . . . . 2

1.2 Harish-Chandra bimodules for rational Cherednik algebras of type A . . . . 5

1.3 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4 Structure of the dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Chapter 2 Rational Cherednik Algebras 12

2.1 Rational Cherednik Algebras. . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.1 Definition and examples. . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.2 Dunkl Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.3 Filtrations and gradings . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.1.4 Spherical rational Cherednik algebra . . . . . . . . . . . . . . . . . . 15

vi

2.1.5 Homogeneous rational Cherednik algebras . . . . . . . . . . . . . . . 16

2.1.6 Sheafification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2 Category Oc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.2 Verma modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2.3 Supports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3 The Knizhnik-Zamolodchikov functor . . . . . . . . . . . . . . . . . . . . . . 20

2.3.1 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3.2 The cyclic group case . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3.3 Hecke algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3.4 KZ functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3.5 Integral parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3.6 Regular parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.4 Restriction functors for category O . . . . . . . . . . . . . . . . . . . . . . . 25

2.4.1 Restriction functors for Hecke algebras . . . . . . . . . . . . . . . . . 25

2.4.2 Bezrukavnikov-Etingof isomorphisms of completions . . . . . . . . . . 26

2.4.3 Construction of the restriction functor . . . . . . . . . . . . . . . . . 27

Chapter 3 Preliminaries on Harish-Chandra bimodules 30

3.1 Definition and basic results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.1.1 Harish-Chandra bimodules for the spherical subalgebras . . . . . . . 32

3.1.2 Alternative definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2 Singular supports and annihilators . . . . . . . . . . . . . . . . . . . . . . . 35

3.2.1 Singular supports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2.2 Filtration by supports . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3 Restriction functors for HC bimodules, I: naive construction . . . . . . . . . 38

3.3.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

vii

3.3.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.3.3 Locally finite maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.4 Restriction functors for HC bimodules, II: equivariance. . . . . . . . . . . . . 45

3.4.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.4.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Chapter 4 Reduction to corank 1. 52

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.2 Technical lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.2.1 Supports and symplectic leaves . . . . . . . . . . . . . . . . . . . . . 53

4.2.2 Annihilators and liftings . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.3 Main result for homogeneous algebras . . . . . . . . . . . . . . . . . . . . . . 57

4.4 Specializing parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Chapter 5 Bimodules with full support. 63

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.2 Localization of Harish-Chandra bimodules . . . . . . . . . . . . . . . . . . . 65

5.2.1 Bimodules of differential maps . . . . . . . . . . . . . . . . . . . . . . 65

5.2.2 Harish-Chandra bimodules and bimodules of differential maps . . . . 66

5.2.3 Harish-Chandra bimodules and the KZ functor. . . . . . . . . . . . . 68

5.3 The Namikawa-Weyl group . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.3.1 A reparametrization of Hc . . . . . . . . . . . . . . . . . . . . . . . . 69

5.3.2 Finite W-algebras and rational Cherednik algebras of cyclic groups . 71

5.3.3 The Namikawa-Weyl group . . . . . . . . . . . . . . . . . . . . . . . . 73

5.3.4 Equivalences from the Namikawa-Weyl group . . . . . . . . . . . . . 74

5.3.5 Action on the set of Hecke parameters . . . . . . . . . . . . . . . . . 76

5.4 Subgroup Wc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

viii

5.4.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.4.2 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

Chapter 6 Type A 87

6.1 Introduction and preliminary results. . . . . . . . . . . . . . . . . . . . . . . 87

6.1.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

6.1.2 Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6.2 Functor Φc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6.2.1 Case c = r/n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6.2.2 General case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.3 Irreducible Harish-Chandra bimodules. . . . . . . . . . . . . . . . . . . . . . 101

6.3.1 Results of Wilcox and consequences . . . . . . . . . . . . . . . . . . . 101

6.3.2 Bimodules with minimal support. . . . . . . . . . . . . . . . . . . . . 103

6.3.3 Description of irreducibles . . . . . . . . . . . . . . . . . . . . . . . . 105

6.4 Two-parametric case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6.4.1 Proof of Theorem 6.1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6.4.2 Bimodules with minimal support . . . . . . . . . . . . . . . . . . . . 109

Chapter 7 Duality 113

7.1 Nakajima quiver varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

7.1.1 Representation spaces and moment maps. . . . . . . . . . . . . . . . 114

7.1.2 G.I.T. quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

7.1.3 Universal quiver varieties . . . . . . . . . . . . . . . . . . . . . . . . . 118

7.1.4 Isomorphisms between quiver varieties . . . . . . . . . . . . . . . . . 119

7.2 Quantizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

7.2.1 Quantizations of M00 . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

7.2.2 Quantizations of Mθ0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

ix

7.2.3 The period map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

7.2.4 Isomorphisms of quantizations . . . . . . . . . . . . . . . . . . . . . . 124

7.3 Harish-Chandra bimodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

7.3.1 Harish-Chandra bimodules: algebra level . . . . . . . . . . . . . . . . 125

7.3.2 Harish-Chandra bimodules: sheaf level . . . . . . . . . . . . . . . . . 126

7.3.3 Localization theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 128

7.3.4 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

7.4 Connection to rational Cherednik algebras . . . . . . . . . . . . . . . . . . . 130

Bibliography 132

x

Disclaimer

I hereby declare that the work in this thesis is that of the candidate alone, except where

indicated in the text, and as described below.

Chapter 2 contains preliminary material and does not contain original results of the candi-

date.

Chapter 3 contains both original results of the candidate as well as results of other authors,

as indicated in the text.

Chapters 4-6 are part of work of the author to be published as Harish-Chandra bimodules

over rational Cherednik algebras. 27 pages. Submitted. Available at https://arxiv.org/

abs/1409.5465

Chapter 7 contains both original results of the candidate, part of an ongoing work, and

results of other authors, as indicated in the text.

xi

https://arxiv.org/abs/1409.5465


Chapter 1

Introduction

In this work we study the representation theory of rational Cherednik algebras. Rational

Cherednik algebras form an interesting class of algebras associated to the action of a complex

reflection group W and its reflection representation R, a precise definition is Definition 2.1.1.

The rational Cherednik algebra depends on a parameter c, which is a conjugation invariant

function c : S → C, where S is the set of reflections of W . One of the reasons why this

algebra is interesting is that it admits a filtration whose associated graded is the smash-

product algebra C[R⊕R∗]#W , and so one can use the algebra Hc to produce quantizations

of the algebra of invariants C[R⊕R∗]W , see [EG, CBH], among others.

A consequence of Hc having a filtration with associated graded C[R⊕R∗]#W , together with

the explicit presentation of Hc given in Definition 2.1.1, is that we have three subalgebras

in Hc, namely C[R],C[R∗] and CW and, moreover, as a vector space we have Hc = C[R]⊗

CW ⊗ C[R∗]. So we can think of CW as a “Cartan subalgebra” and of C[R],C[R∗] as

positive and negative “Levi subalgebras” of Hc, respectively. From this point of view,

the representation theory of rational Cherednik algebras has many similarities to that of

semisimple Lie algebras. For example, one has a category Oc, first defined in [GGOR] and

that has been extensively studied in recent years, see, for example, [BEG2, BE, EGL, ES,

Gi2, GL, L5, L6, L7, R, RSVV, Sh, SV, V, Wi], to name just a few.

1

In [BEG], a notion of Harish-Chandra bimodules for rational Cherednik algebras was in-

troduced. The definition is similar to that of Harish-Chandra bimodules for universal en-

veloping algebras that arise in the study of projective functors, see [BG]. In particular, a

Harish-Chandra (Hc, Hc′)-bimodule gives a functor B ⊗Hc′ • : Oc′ → Oc. These functors

have been used in [L7] to produce derived equivalenes between different categories Oc. We

remark, however, that as opposed to the Lie algebra setting, the functor of multiplying by a

Harish-Chandra bimodule is not in general exact. Let us also remark that, while in the Lie

algebra setting the categories of Harish-Chandra bimodules and category O are very similar,

cf. [BG, Section 5], this no longer needs to be the case in the Cherednik algebra setting, see

Section 6.1.1.

It is known, thanks to [Gi2, L3], that the category of Harish-Chandra (Hc, Hc′)-bimodules is

an abelian category with finitely many simples and such that every object has finite length.

An interesting problem, then, is to describe the category HC(Hc, Hc′) of Harish-Chandra

(Hc, Hc′)-bimodules. In this work, we obtain several partial results towards this description.

As a first step we show in Corollary 3.3.14 that the category HC(Hc, Hc′) is equivalent to

the category of representations of a finite-dimensional algebra. In the remainder of this

introduction, we will give an overview of our results, as well as of previously known results

on the structure of the category HC(Hc, Hc′).

1.1 Harish-Chandra bimodules with full support

Let us say that a parameter c : S → W is regular if the algebra Hc is simple, cf. Section 2.3.6.

For example, it is known that if W is a Coxeter group every integral parameter c : S → Z

is regular. In [BEG], Berest-Etingof-Ginzburg studied the category HC(Hc, Hc′) in the case

where W is a Coxeter group and c, c′ are integral, in connection to the algebra of differential

operators on c-quasi-invariants for the action of W on R. In particular, they showed that

when c is integral, then the algebra of differential operators on c-quasi-invariants is naturally

2

a bimodule over the spherical rational Cherednik algebra Ac := eHce, where e ∈ CW is the

trivial idempotent, and they used this to produce the following result.

Theorem 1.1.1 ([BEG]). Assume W is a Coxeter group and c, c′ : S → Z are parame-

ters. Then, the category HC(Hc, Hc′) is equivalent to the category of representations of W .

Moreover, if c = c′ then this is an equivalence of monoidal categories.

Theorem 1.1.1 was later extended by Spencer, [Sp], by considering the more general case in

which c, c′ are regular, not necessarily integral.

Theorem 1.1.2 ([Sp]). Let W be a Coxeter group and c, c′ : S → C regular parameters.

The following is true.

1. The category HC(Hc, Hc′) is nonzero if and only if there exists a character ε : W → C×

such that c− εc′ is integral.

2. Assume that c − εc′ is integral for a character c′ : W → C×. Then, HC(Hc, Hc′) is

equivalent to the category of representations of W/W ′, where W ′ := 〈s ∈ S : c(s) 6∈ Z〉.

If c = c′, this is an equivalence of monoidal categories.

Here, we extend Theorem 1.1.2 in two directions. Firstly, we remove the restriction that the

parameters c, c′ are regular, at the cost of only looking at a certain quotient of the category

HC(Hc, Hc′), which we are going to denote by HC(Hc, Hc′), see Chapter 5 for a precise

definition of HC(Hc, Hc′). We remark that, if either c or c′ is regular, then HC(Hc, Hc′) =

HC(Hc, Hc′).

Secondly, we remove the restriction that W is a Coxeter group. In fact, in our result W

can be any complex reflection group. In this setting we still have a notion of integral

parameters, although it is more complicated, see Section 2.3.5. However, the condition of

having a character ε : W → C× with c− εc′ being integral turns out to be sufficient, but not

necessary, to guarantee that the category HC(Hc, Hc′) is nonzero.

3

To find necessary and sufficient conditions for HC(Hc, Hc′) to be nonzero we need to introduce

the action of a certain group on the space of parameters for the Cherednik algebra. This

is the Namikawa-Weyl group, which is a product of symmetric groups, one for each orbit of

reflection hyperplanes H ⊆ R. This group coincides with the usual Namikawa-Weyl group

for a symplectic resolution of (R⊕R∗)/W when the latter variety admits such a resolution, cf.

[Nam]. In general, we can take the Namikawa-Weyl group of a Q-factorial terminalization,

which always exists, see [L10]. Here, we introduce the Namikawa-Weyl group, as well as its

action on the space of parameters, by more elementary methods, see Section 5.3.

Theorem 1.1.3. Let W be a complex reflection group, and c, c′ : S → C parameters. Then,

the following holds.

1. The category HC(Hc, Hc′) is nonzero if and only if we can get from c to c′ by a sequence

of integral translations and actions of elements of the Namikawa-Weyl group.

2. If HC(Hc, Hc′) is nonzero, then it is equivalent to the category of representations of

W/Wc, for an explicitly defined normal subgroup Wc ⊆ W . If c = c′, this is an

equivalence of monoidal categories.

Let us remark that the condition of getting from c to c′ by a sequence of integral translations

and elements of the Namikawa-Weyl group is not complicated to check, and it actually

expresses a relationship between the Hecke parameters q(c), q(c′) associated to c and c′,

respectively. The parameters q(c), q(c′) are a collection of nonzero complex numbers that

are determined from c, c′ via an exponential formula, see Section 2.3.4. We elaborate on

the connection between the parameters q(c), q(c′) and the condition from Theorem 1.1.3 in

Section 5.3.5.

We also mention that the reason why the action of the Namikawa-Weyl group does not appear

in Theorems 1.1.1, 1.1.2 is that in the case of a Coxeter group, the action of an element of

the Namikawa-Weyl group can be obtained by multiplying by a character, followed by an

4

integral translation. Similarly, the reason why multiplication by a character does not appear

in our Theorem 1.1.3 is that multiplication by a character is obtained by applying an element

of the Namikawa-Weyl group followed by an integral translation. However, when W is not

a Coxeter group, the Namikawa-Weyl group is bigger than the group of characters. We

elaborate on this in Section 5.3.5.

The definition of the subgroup Wc ⊆ W is explicit, but technical, see Section 5.4. Here, we

just mention that this subgroup satisfies the following properties.

1. It is a complex reflection group (but not, in general, a parabolic subgroup of W ).

2. Wc = Wc′ if either c − c′ is integral, or c and c′ belong to the same Namikawa-Weyl

group orbit.

3. Wc = {1} if and only if c belongs to the Namikawa-Weyl group orbit of an integral

parameter.

4. Generically, Wc = W .

Note, in particular, that the structure (and even the number of irreducibles) of HC(Hc, Hc)

for regular c really depends on the parameter c, which is in contrast with category Oc, which

is equivalent to the category of representations of W whenever c is regular.

1.2 Harish-Chandra bimodules for rational Cherednik

algebras of type A

When W = Sn, acting on its reflection representation R = {(x1, . . . , xn) ∈ Cn :∑n

i=1 xi = 0},

we obtain information on the category HC(Hc, Hc′), not just its quotient HC. First of all,

recall that HC(Hc, Hc′) = HC(Hc, Hc′) if one of c, c′ is regular, so we will assume that both c

and c′ are singular. It is known that the set of singular parameters is given by those nonzero

5

rational numbers r/m where the fraction r/m is irreducible and 1 < m ≤ n. We will denote

Hc(Sn) =: Hc(n).

Theorem 1.2.1. There is a functor Φc : HC(Hc(n), Hc(n))→ Oc(n), which is a fully faithful

embedding whose image is closed under subquotients.

The functor Φc is similar in spirit to the one used in [BG, Section 5], namely, it is taking

tensor product with a projective Verma module. We remark, however, that Theorem 1.2.1

does not generalize beyond type A. Indeed, in Section 6.1.1 we show via an example that if

W is, say, a cyclic group, then there cannot exist a functor as the one in Theorem 1.2.1 for

arbitrary c.

Of course, the natural question now is to describe the image of the functor Φc. We cannot do

this in general, but we can describe the irreducible modules in the image of Φc. We remark,

however, that the image of Φc is not closed under extensions, so knowing which irreducibles

it contains is not enough to describe it.

Let us describe the irreducible modules in the image of Φc. The irreducible modules in

category Oc are parametrized by partitions of n. If λ is such a partition, it is easy to see

that there exists a unique way to express λ as λ = mµ+ν, where the partition ν is m-regular,

that is, no two consecutive parts of ν differ by more than m−1. Let us say that an m-regular

partition ν is m-trivial if ν = ((m− 1), . . . , (m− 1), d, 0, . . . ) for some 0 ≤ d < m− 1.

Theorem 1.2.2. Assume c = r/m > 0. Let λ be a partition of n, and decompose it as

λ = mµ+ ν, where ν is m-regular. Then, the irreducible module Lc(λ) corresponding to the

partition λ belongs to the image of Φc if and only if ν is m-trivial.

Now assume c, c′ are different, singular parameters. So c = r/m, c′ = r′/m′.

Lemma 1.2.3. The category HC(Hc, Hc′) is zero unless m = m′.

Thanks to the previous lemma, we can assume that c = r/m, c′ = r′/m. Let us set ` :=

bn/mc We construct a filtration {0} = HC(Hc, Hc′(n))`+1 ⊆ HC(Hc(n), Hc′(n))` ⊆ · · · ⊆

6

HC(Hc(n), Hc′(n))1 ⊆ HC(Hc(n), Hc′(n))0 by Serre subcategories. Let us set HC(Hc(n), Hc′(n))i :=

HC(Hc(n), Hc′(n))i/HC(Hc(n), Hc′(n))i+1 for i = 0, . . . `. Then, we show the following re-

sult.

Theorem 1.2.4. Let i ∈ {0, 1, . . . , `}.

1. Assume n− im 6= 0. Then, HC(Hc(n), Hc′(n))i 6= 0 if and only if either c+ c′ or c− c′

is an integer. If this is the case, then HC(Hc(n), Hc′(n))i is equivalent to the category

of representations of Si, and this is an equivalence of monoidal categories when c = c′.

2. Assume n = im. Then, HC(Hc(n), Hc′(n))i is equivalent to the category of representa-

tions of Si. If c = c′, this is an equivalence of monoidal categories.

Let us remark that the filtration mentioned above is the filtration by supports that has

already been constructed by Losev, [L3], in a more general setting that we will review in

Section 3.2.2 . What is new here is the description of the subquotients.

1.3 Duality

Our last main result is a description of duality for certain types of rational Cherednik alge-

bras, namely, those associated to the groups G(`, 1, n) := Snn (Z/`Z)n, viewed as the group

of monomial (n × n)-matrices whose nonzero entries are `-roots of 1. In fact, we construct

this duality in a more general setting: quantizations of quiver varieties.

Quantized quiver varieties fall into the more general setting of quantizations of symplectic

resolutions, see [BPW, BLPW]. Let us assume that X is a smooth, symplectic variety with

a C×-action rescaling the symplectic form and such that the natural map X → X0 is a

resolution of singularities, where X0 is the affine variety Spec(C[X]). By a quantization of

X, we mean a filtered sheaf of algebras A on the conical topology (where open = Zariski-

7

open + C×-stable) of X, together with a Poisson isomorphism ι : grA → SX1. The sheaf A

is supposed to satisfy some technical assumptions, see Section 7.2.2. A quantization of X0

is a filtered algebra A together with an isomorphism of Poisson algebras grA→ C[X].

Let us remark that, if A is a quantization of X, then A := Γ(X,A) is a quantization

of X0. We have functors of localization and global sections Γ(X, ·) : A -mod → A -mod,

Loc : A -mod → A -mod. These functors, however, are not always an equivalence. For

example, for X0 we can take the nilpotent cone in a semisimple Lie algebra g, and X is

the Steinberg variety. The quantizations of X are naturally parametrized by weights. The

Beilinson-Bernstein localization theorem tells us that the functors of localization and global

sections are inverse equivalences precisely when the weight is regular and dominant. If the

global sections and localization functors are an equivalence, we say that localization holds

at A.

For quantizations A,A′ of X, with A := Γ(X,A), A′ := Γ(X,A′), we can define categories

of Harish-Chandra (A,A′)-bimodules, as wel as of Harish-Chandra (A,A′)-bimodules, see

Section 7.3. Let us denote by HC(A,A′) the category of Harish-Chandra (A,A′)-bimodules,

and by HC(A,A′)-the category of Harish-Chandra (A,A′)-bimodules. We have functors

Γ : HC(A,A′) ↔ HC(A,A′) : Loc which are quasi-inverse equivalences when localization

holds both at A and (A′)opp.

The category HC(A,A) has a natural duality: the homological duality. By standard homo-

logical algebra results, for a Harish-ChandraA-bimodule B, the complex RHomA -bimod(B,A)

has nonzero homology only at degree − dimX. So there is a natural duality HC(A,A) →

HC(A,A)opp. Of course, by taking global sections we would like to have a duality HC(A,A)→

HC(A,A)opp. But localization does not need to hold at A and Aopp simultaneously.

Luckily, when X is a smooth Nakajima quiver variety there is a way to fix this. In fact,

there is an isomorphism A→ Aopp for any quantization of X0. A small technical issue is that

1following [BPW, BLPW], to avoid confusions with category O, we denote the structure sheaf of a varietyX by SX

8

this isomorphism does not induce the identity at the associated graded level. So we need to

slightly generalize the definition of a Harish-Chandra bimodule to take this into account.

Let us state the duality theorem, adapted for type A rational Cherednik algebras.

Theorem 1.3.1. Let c ∈ C be not a negative rational number. Let N ∈ Z be such that

−c+N 6∈ Q<0. Then, there is an equivalence of categories

D : HC(Hc(n), Hc(n))→ HC(H−c+N(n), H−c+N(n))opp

The restriction c 6∈ Q<0 is not very important and, in fact, it can be removed, see Corollary

7.4.2.

1.4 Structure of the dissertation

Let us now briefly mention how is this work structured.

In Chapter 2, we recall basic definitions and properties of rational Cherednik algebras and

their categories Oc. In particular, we recall the connection of the category Oc to Hecke

algebras via the KZ functor, and the Bezrukavnikov-Etingof parabolic restriction functors.

This chapter contains no new results.

In Chapter 3, we recall some basic known facts about the category of Harish-Chandra bi-

modules, as well as proving some new results. We recall the original definition due to

Berest-Etingof-Ginzburg, [BEG2], as well as a very useful equivalent definition due to Losev,

[L3]. We also recall Losev’s restriction functors, first in their naive (but easier) version,

and then in their equivariant version, which is the version that will be most useful for

us. In this chapter, we prove some new results concerning the structure of the category of

Harish-Chandra bimodules. For example, we show that the tensor product of two irreducible

bimodules vanishes unless the two bimodules have the same support, see Section 3.2 for the

definition of the singular support of a Harish-Chandra bimodule. We also show that the

9

category of Harish-Chandra bimodules is equivalent to the category of representations of a

finite-dimensional algebra. We would like to remark that, while this result is new, all the

ingredients to prove it are already in the literature, specifically in work of Ginzburg, [Gi2],

and Losev, [L3, L7].

Chapter 4 is technical. Its main purpose is to find sufficient conditions for an equivariant

bimodule to be in the image of the equivariant restriction functor. This will be our main

tool to produce new Harish-Chandra bimodules, see Theorem 4.1.1.

Chapter 5 studies the category HC. There are two main ingredients in this chapter. One

of them is localization to the regular locus, which relates Harish-Chandra bimodules to D-

modules on Rreg/W and therefore to the KZ functor and Hecke algebras. The other one is

the Namikawa-Weyl group action. We introduce this group by means of a certain isomor-

phism of the spherical rational Cherednik algebras for cyclic groups and central reductions

of the finite W-algebra associated to the subregular nilpotent element in sl`. After carefully

studying the action of the Namikawa-Weyl group on the set of parameters, we prove the

results mentioned in Section 1.1 of this introdution.

Chapter 6 deals with rational Cherednik algebras of symmetric groups. There, we prove

the results mentioned in Section 1.2. To prove an equivalence between the category of

Harish-Chandra bimodules and a full subcategory of category Oc, our main ingredients are

the restriction functors for Harish-Chandra bimodules, even in their naive (non-equivariant)

version. So the first thing we do is to study Harish-Chandra bimodules in the easiest non-

trivial case, namely, when c = r/n (for the symmetric group Sn). We find all the irreducible

bimodules in the category HC(Hr/n(n), Hr/n(n)), and after this we proceed to produce a de-

sired equivalence. To find the image of this functor, our main ingredients are Theorem 4.1.1,

10

together with results of Wilcox, [Wi], on the support of irreducible modules in category O,

and a remarkable symmetry result of Calaque-Enriquez-Etingof, [CEE], see also [EGL].

Finally, in Chapter 7, we produce the duality mentioned in Section 1.3. In order to do this,

we briefly recall the definition of Nakajima quiver varieties, as well as their quantizations.

We also recall results of [BPW, BLPW] regarding localization theorems and quantizations

of line bundles. After giving all necessary preliminaries, we proceed to define the duality for

Harish-Chandra bimodules over quantized quiver varieties, and its particular case of type A

rational Cherednik algebras.

11

Chapter 2

Rational Cherednik Algebras

2.1 Rational Cherednik Algebras.

2.1.1 Definition and examples.

Let W be a complex reflection group, with reflection representation R and set of reflections

S ⊆ W . For each reflection s ∈ S, let αs ∈ R∗ be an eigenvector with eigenvalue λs 6= 1.

In particular, the reflection hyperplane of s is given by Γs := {αs = 0} ⊆ R. Also, let

α∨s ∈ R be an eigenvector of s with eigenvalue λ−1s . We remark that α∨s , αs are unique up

to multiplication by a nonzero scalar, and we normalize so that 〈αs, α∨s 〉 = 2. To define the

rational Cherednik algebra, we need a conjugation-invariant function c : S → C.

Definition 2.1.1 (Etingof and Ginzburg, [EG]). The rational Cherednik algebra Hc :=

Hc(W,R) is the quotient of the smash product algebra T (R ⊕ R∗)#W , where T (•) denotes

the tensor algebra, by the relations:

[x, x′] = 0, [y, y′] = 0 x, x′ ∈ R∗, y, y′ ∈ R

[y, x] = 〈y, x〉 −∑

s∈S c(s)〈αs, y〉〈x, α∨s 〉s x ∈ R∗, y ∈ R.(2.1)

Let us give a few examples of rational Cherednik algebras.

12

Example 2.1.2. Assume the function c is constant 0. Then, H0 = D(R)#W , where D(R)

denotes the algebra of differential operators on R.

Example 2.1.3. Assume W = Sn is the symmetric group, acting on Cn by permuting the

coordinates. Then, c may be thought of as a single complex number and Hc(n) := Hc(Sn,Cn)

is the algebra

Hc(n) =C〈x1, . . . , xn, y1, . . . , yn〉#Sn

[xi, xj] = 0 = [yi, yj], [yi, xj] = c(ij), i 6= j [yi, xi] = 1− c∑

i 6=j(ij)

where (ij) ∈ Sn denotes the transposition i↔ j.

Example 2.1.4. Now let W = Sn again, acting on its (n − 1)-dimensional reflection rep-

resentation R := {(x1, . . . , xn) ∈ Cn :∑n

i=1 xi = 0}. Then we have that a presentation for

Hc(n) := Hc(Sn, R) is given by

C〈x1, . . . , xn, y1, . . . , yn〉#Sn∑xi = 0 =

∑yi, [xi, xj] = 0 = [yi, yj], [yi, xj] = c(ij)− 1/n, i 6= j, [yi, xi] = 1− 1/n− c

∑i 6=j(ij)

Note that Hc(n) is a the subalgebra of Hc(n) generated by xi := xi − 1n

∑nj=1 xj, yi :=

yi − 1n

∑nj=1 yj, i = 1, . . . , n, and Sn. Furthermore, if X = 1

n

∑nj=1 xj ∈ Hc(n) and

Y = 1n

∑nj=1 yj ∈ Hc(n), then the algebra generated by X, Y in Hc(n) is isomorphic to

the algebra of differential operators on C and, moreover,

Hc(n) ∼= Hc(n)⊗D(C)

Example 2.1.5. Now let W = Z/`Z, with generator s which acts on R = C by multiplication

by η := exp(2π√−1/`). Denote ci := c(si), i = 1, . . . , `− 1. Then we have that Hc(W,C) is

given by

C〈x, y, s〉s` = 1, sx = η−1xs, sy = ηys, [y, x] = 1−

∑`−1i=1 cis

i

2.1.2 Dunkl Operators

Let us mention that the rational Cherednik algebra Hc := Hc(W,R) can be explicitly realized

as a subalgebra of the smash-product algebra D(Rreg)#W , where Rreg is the principal open

13

set of R where the W -action is free. To produce a desired algebra we need the concept of

Dunkl operators. Note that Rreg = R \ {δ = 0}, where δ :=∏

s∈S αs ∈ S(R∗).

Definition 2.1.6 (See [Du]). Let c : S → C be a conjugation-invariant function. Then, for

y ∈ R, we have the Dunkl operator

Dy := ∂y −∑s∈S

2c(s)〈αs, y〉(1− λs)αs

(1− s)

Theorem 2.1.7 (See e.g. Chapter 7, [Et]). The rational Cherednik algebra Hc is isomorphic

to the subalgebra of D(Rreg)#W generated by W,R∗ and the Dunkl operators.

We remark that the hardest part of the theorem is proving that Dunkl operators actually

commute. This is due to Dunkl, see e.g. [Du, Et].

Note that the action of Dy ∈ D(Rreg)#W on C[Rreg] preserves the space C[R] as, for any

function f ∈ C[R], (1− s)f is divisible by αs. In particular, it follows that C[R] is naturally

a module over Hc. We call this the polynomial representation of Hc.

2.1.3 Filtrations and gradings

The algebra Hc comes equipped with several filtrations that we now mention. First of

all, we have the Bernstein filtration, Hc =⋃m≥0B

mHc, which is given by degW = 0,

degR = degR∗ = 1.

Theorem 2.1.8 (PBW Theorem for Rational Cherednik Algebras, [EG]). We have grBHc =

C[R⊕R∗]#W . Equivalently, the multiplication in Hc induces a vector space isomorphism

S(R∗)⊗ CW ⊗ S(R)∼=−→ Hc

We also have the geometric filtration, Hc =⋃m≥0G

mHc, which is induced by the eponymous

filtration on D(Rreg)#W . Here, we have deg(R) = 1 while deg(W ) = deg(R∗) = 0. It follows

from the PBW theorem that we also have an isomorphism grGHc = C[R ⊕ R∗]#W , where

in the latter algebra we take the grading with deg(R∗) = deg(W ) = 0, deg(R) = 1.

14

We remark that the Bernstein and geometric filtrations are related. To explain how this is so,

we need a grading on the rational Cherednik algebra. It is easy to see from the relations that

setting deg(R) = −1, deg(R∗) = 1, deg(W ) = 0 induces a grading on the rational Cherednik

algebra. It is then easy to see (say from the PBW decomposition) that

BmHc =⊕k

{a ∈ GkHc : deg(a) = m− 2k}

The grading on Hc described above is actually inner. To see that this is the case, pick a

basis {yi} of R with dual basis {xi}. Define the deformed Euler element :

eu :=∑

xiyi +dimR

2−∑s∈S

2c(s)

1− λss ∈ Hc

Note that eu is actually independent of the basis {yi}. It is an easy calculation to check that

[eu, w] = 0 for w ∈ W , that [eu, x] = x for x ∈ R∗, and that [eu, y] = −y for y ∈ R.

2.1.4 Spherical rational Cherednik algebra

Let e := 1|W |∑

w∈W w be the trivial idempotent of CW . We define the spherical rational

Cherednik algebra to be Ac := eHce. While Ac is a subalgebra of Hc, they do not have the

same unit: the unit of Ac coincides with e. Note that Ac inherits both filtrations from Hc,

and we have that the associated graded of Ac under any of these filtrations coincides with

e(C[R⊕R∗]#W )e = C[R⊕R∗]W (of course, the grading on C[R⊕R∗]W depends on which

filtration on Ac one takes).

Often, but not always, the algebras Hc and Ac are Morita equivalent via the (Hc, Ac)-

bimodule Hce. When this is the case, we will say that the parameter c is spherical. Otherwise,

c will be called aspherical.

Example 2.1.9. Consider the rational Cherednik algebra Hc(n) associated to the symmetric

group on n letters. Then, the aspherical locus is (−1, 0) ∩ {mr

: 1 < r ≤ n, gcd(m; r) = 1},

see e.g [BE, Corollary 4.2]

15

2.1.5 Homogeneous rational Cherednik algebras

Below, we will also need to consider a homogeneous version of the rational Cherednik alge-

bras. let S =⊔ri=1 Si be the decomposition of S into conjugacy classes, and let ~, c1, . . . , cr

be independent variables. For s ∈ Si, define c(s) := ci. Let c be the vector space with basis

~, c1, . . . , cr. Then, H is the S(c)-algebra defined by generators and relations analogous to

those of the previous subsection, with the commutation relation between R and R∗ replaced

by

[y, x] = ~〈y, x〉 −∑s∈S

c(s)〈αs, y〉〈x, α∨s 〉s,

note that the algebra H is graded, with W,R∗ in degree 0 and ~, c in degree 1. We remark

that H is a flat S(c)-algebra, and that H/cH = C[h⊕ h∗]#W - this is just a reformulation

of the PBW theorem.

Let us explain why the algebra H is important for us. Let R~(Hc) denote the Rees algebra of

Hc with respect to the geometric filtration. We then have a graded surjection H � R~(Hc),

which is given by w 7→ w, x 7→ x, y 7→ ~y, ci 7→ ~ci, where w ∈ W , x ∈ R∗, y ∈ R and

ci := c(s) for s ∈ Si. Thus, we can pass from filtered Hc-modules to graded H-modules by

means of the Rees construction.

2.1.6 Sheafification

We remark that the construction of the rational Cherednik algebra can be sheafified over

R/W , cf. [Et2]. First of all, note that if f ∈ C[R]W is a W -invariant function, then the

adjoint action of f on Hc is locally nilpotent. It follows, in particular, that the localization

Hc[f−1] is well-defined. By the PBW theorem, this algebra coincides with the tensor product

C[R/W ][f−1] ⊗C[R/W ] Hc. So we have a sheaf of algebras Hc := SR/W ⊗C[R/W ] Hc, where

SR/W denotes the structure sheaf of R/W . Thanks to the Dunkl presentation of Hc, it

follows that for an open set U ⊆ R/W , the algebra of sections Hc(U) is generated by

C[π−1(U)], W , and Dunkl operators, where π : R→ R/W denotes the projection.

16

Note that we can also sheafify the spherical rational Cherednik algebra to R/W , we denote

the sheafification by Ac. We can also sheafify the homogeneous algebra H to get a sheaf

of S(c)-algebras H on R/W . We will use this more geometric point of view on rational

Cherednik algebras when defining restriction functors.

2.2 Category Oc

2.2.1 Definition

We now review the category O for the algebra Hc, following [GGOR]. Recall the PBW

decomposition Hc = S(R∗) ⊗ CW ⊗ S(R). We then have the following definition, which is

analogous to that of (the principal block of) category O for a semisimple Lie algebra g.

Definition 2.2.1 ([GGOR]). A module M ∈ Hc -mod is said to be in category Oc if the

following conditions are satisfied.

1. M is finitely generated.

2. R acts on M by locally nilpotent endomorphisms.

For example, if M ∈ Hc -mod is finite-dimensional, then M ∈ Oc, this follows from the

existence of the deformed Euler element eu.

Example 2.2.2. Assume c = 0, so that Hc = D(R)#W . Then, O0 is the category of W -

equivariant D(R)-modules whose Fourier transform is supported at 0 ∈ R. By Kashiwara’s

Lemma, this is equivalent to the category of representations of W .

Example 2.2.3. Recal that via the Dunkl embedding we have the polynomial representation

C[R] of Hc. It is easy to see that C[R] ∈ Oc.

17

2.2.2 Verma modules

We will now construct, for every irreducible representation τ of W , a module ∆c(τ) ∈ Oc.

First, let us note that the algebra S(R)#W is a subalgebra of Hc. If τ is an irreducible

representation of W , then we can consider it as a representation of S(R)#W by letting R

act by 0.

Definition 2.2.4 ([GGOR]). Let τ be an irreducible representation of W , which we consider

a representation of S(R)#W as in the above paragraph. We define the Verma module ∆c(τ)

to be the induced module

∆c(τ) := IndHcS(R)#W τ = Hc ⊗S(R)#W τ = S(R∗)⊗ τ

where the last equality is just as vector spaces.

Note that ∆c(triv) is precisely the polynomial representation of Hc. We remark that ∆c(τ)

is naturally graded by the action of eu, with eigenvalues being of the form cτ + i, i ≥ 0 for

some cτ ∈ C, and the eigenspace with eigenvalue cτ coincides with τ ⊗ 1 ⊆ τ ⊗ S(R∗) . It

easily follows that ∆c(τ) admits a unique irreducible quotient, which we denote by Lc(τ).

It is also not hard to see that {L(τ) : τ ∈ W -irrep} forms a complete and irredundant

collection of irreducible modules in category Oc. Let us also mention that category Oc is

a finite length category and, moreover, it is a highest weight category, in particular it has

enough projectives. We remark that a consequence of this is that taking Verma modules gives

an isomorphism between Grothendieck groups. More explicitly, consider the Grothendieck

groups [W -rep] := C⊗K0(W -rep) and [Oc]. Then, the map [τ ] 7→ [∆c(τ)] is an isomorphism.

2.2.3 Supports

It is clear from the definition that a module in category Oc is finitely generated over S(R∗) =

C[R]. Thus, we may define its support supp(M) ⊆ R as the set-theoretic support of M as a

coherent sheaf over R. In other words, it is the zero set of the radical of the annihilator of M

18

in C[R]. Since M is, in particular, a W -equivariant C[R]-module, supp(M) is a W -stable,

closed subvariety of R. Even more is true. For a subgroup W ′ ⊆ W , let XW ′ := {b ∈ R :

Wb is W-conjugate to W ′}. Note that XW ′ is a locally closed subvariety of R, and the XW ′

form a stratification of W when W ′ runs over the conjugacy classes of parabolic subgroups

of W where, recall, a subgroup W ′ ⊆ W is called parabolic if it is the stabilizer of a point

in R.

Proposition 2.2.5 (Proposition 3.2, [BE]). For any module M ∈ Oc, supp(M) is a union

of sets of the form XW ′. Moreover, if M is irreducible, then supp(M) = XW ′ for some

parabolic subgroup W ′ ⊆ W .

For a parabolic subgroup W ′ ⊆ W , let us denote by Oc,W ′ the full Serre subcategory of Oc

consisting of modules whose support is contained in XW ′ . Let us also denote by Otorc,W ′ the

full subcategory consisting of modules whose support is contained in ∂XW ′ := XW ′ \ XW ′ .

This is a Serre subcategory of Oc,W ′ and we can form the quotient

O◦c,W ′ := Oc,W ′/Otorc,W ′

For example, if W ′ = {1}, then Oc,W ′ = Oc, and Otorc,W ′ consists of modules with proper

support. We will see a description of the category O◦c,{1} in the next section. On the

other extreme, Oc,W = O◦c,W consists of the category of finite-dimensional modules in Oc.

Let us remark that the number of irreducible objects in O◦c,W ′ coincides with the number of

irreducible modules inOc whose support coincides with XW . Let us now define the associated

graded category:

grOc :=⊕W ′⊆W

O◦c,W ′

Note that the number of irreducible objects of Oc and grOc coincide. Let us also remark

that, for Weil generic c, we have Oc = grOc = Oc,{1}.

19

2.3 The Knizhnik-Zamolodchikov functor

2.3.1 Localization

In this section, we follow Section 5 of [GGOR]. Recall that we denote δ :=∏s∈S

αs. This is

a W -semiinvariant element of C[R]. Also recall that we denote by Rreg the principal open

set in R determined by δ. Note that Rreg = R \⋃s∈S Γs coincides with the locus where

the W -action is free. From the Dunkl embedding it is clear that the operator [δ, ·] is locally

nilpotent on Hc, so the localization Hc[δ−1] makes sense. Moreover, it is not hard to see that

the Dunkl embedding induces an isomorphism Hc[δ−1] ∼= D(Rreg)#W .

We then have a functor [δ−1] : Oc −→ (D(Rreg)#W ) -mod, M 7→ M [δ−1]. Note that this

functor annihilates every module whose support is contained in the zero set of δ. Thanks to

Proposition 2.2.5, the localization functor kills all modules in Oc whose support is properly

contained in R, that is Otorc,{1}. We also remark that, since every module in Oc is finitely

generated over C[R], the image of the localization functor is contained in the category of

W -equivariant local systems on Rreg. Since W -acts freely on Rreg, taking W -invariants gives

an equivalence of the latter category with that of local systems on the quotient Rreg/W .

Taking then flat sections, we get a finite-dimensional representation of the braid group,

BW := π1(Rreg/W ).

Oc LocW (Rreg) Loc(Rreg/W ) BW -repM 7→M [δ−1] L7→eL L7→L∇

It turns out that the image of the composite functor Oc → BW -rep falls inside the category

of modules of a certain quotient of the group algebra CBW . To motivate this definition, we

are going to start with the case when W = Z/`Z acting on its reflection representation C as

in Example 2.1.5. Note that, here, Rreg = C×, while BW = Z.

20

2.3.2 The cyclic group case

Let W = Z/`Z, with generator s acting on C by multiplication by η := exp(2π√−1/`). For

each i = 0, . . . , ` − 1, let Ei be the 1-dimensional representation of W where s acts by ηi.

In particular, E0 = triv, the trivial representation. Note that, if x ∈ R∗, y ∈ R are such

that 〈x, y〉 = 1, then ∆c(Ei) = C[x], with action given by x.xm = xm+1, s.xm = ηi−mxm, the

action of y can be recovered uniquely from these formulas and y.1 = 0.

Now, ∆c(Ei)[δ−1] = C[x, x−1]⊗Ei. Since y annihilates Ei, we have that the Dunkl operator

also annihilates Ei. It follows that ∆c(Ei) = C[x, x−1] with connection given by

∇ = d−`−1∑m=1

2cm1− η−m

(ηim − 1)dx

x

where cm := c(sm). It follows that, if we define

ki :=`−1∑m=1

2cm1− η−m

(ηim − 1

)then [(∆c(Ei))[δ

−1]W ]∇ is the 1-dimensional C[t, t−1]-module where t acts by the scalar

qi := exp

(2π√−1(ki − i)`

)and, moreover, the image of the composite functor Oc → BW -rep factors through the algebra

Hq(W ) := C[t]/∏`−1

i=0(t − qi). This is the simplest case of a so-called Hecke algebra, which

we define next.

2.3.3 Hecke algebras

Recall that we denote BW := π1(Rreg/W ) the braid group associated to W . The group BW

admits a system of generators indexed by the set A of reflection hyperplanes on R. For each

Γ ∈ A, the pointwise stabilizer WΓ is cyclic, of order say `Γ. Let sΓ ∈ S ∩WΓ be the element

with determinant exp(2π√−1/`Γ), and let TΓ be a generator of the monodromy around Γ

such that a lift of TΓ to Rreg is represented by a path from x0 to sΓ(x0), see [BMR, Appendix

1] for a precise definition. The set {TΓ}Γ∈A is a generating set for the group BW .

21

To define the Hecke algebra, for each reflection hyperplane Γ ∈ A, fix nonzero complex

numbers qΓ,0, . . . , qΓ,`Γ−1, in such a way that if Γ,Γ′ are W -conjugate then qΓ,i = qΓ′,i for

each i = 0, . . . , `Γ − 1 = `Γ′ − 1. We denote this collection of complex numbers by q.

Definition 2.3.1 ([BMR]). The Hecke algebra Hq(W ) is the quotient of the group algebra

CBW by the relations∏`Γ−1

i=0 (TΓ − qΓ,i), one for each Γ ∈ A.

For example, setting qΓ,i = exp(2π√−1i/`Γ) we recover the group algebra CW .

2.3.4 KZ functor

Let us go back to the rational Cherednik algebra Hc := Hc(W ). In [GGOR] it is shown

that the image of the functor Oc → BW -rep falls inside the category of finite dimensional

Hq(W )-modules, where the parameter q explicitly depends on c as follows. Recall that for

each reflection s ∈ S, λs denotes the unique non-trivial eigenvalue for the action of s on R∗.

For each reflection hyperplane Γ ∈ A, define

kΓ,i :=∑

s∈S∩WΓ

2c(s)

1− λs(λ−is − 1), i = 0, . . . , `Γ − 1. (2.2)

Note that kΓ,i depends only on the conjugacy class of Γ, and that kΓ,0 = 0. Now the

parameter q is computed as follows:

qΓ,i := exp(2π√−1(kΓ,i − i)/`Γ). (2.3)

Note that qΓ,0 = 1. For example, if c = 0 then Hq(W ) = CW . We call the functor

Oc → Hq -mod the Knizhnik-Zamolodchikov (shortly, KZ) functor, and denote it KZ : Oc →

Hq -mod.

Theorem 2.3.2 ([GGOR], [L5]). The functor KZ : Oc → Hq -mod is exact and induces an

equivalence O◦c,{1} ∼= Hq -mod.

22

For example, let τ be a 1-dimensional representation of W . For a hyperplane Γ ∈ A, let

sΓ be a generator of WΓ with λ−1sΓ

= exp(2π√−1/`Γ). Denote by EΓ,i the 1-dimensional

representation of WΓ where sΓ acts by λ−isΓ . In particular, we have that ResWWΓτ = EΓ,τ(Γ) for

some τ(Γ) ∈ {0, . . . , `Γ− 1}. Then, we have that KZ(∆(τ)) is the 1-dimensional Hq-module

where TΓ acts by qΓ,Γ(τ). In particular, KZ(triv) is the trivial representation of Hq, that is,

the representation of Hq where all TΓ’s act by 1.

2.3.5 Integral parameters

Let us define the notion of integral parameters following [L7]. Let p∗ := c/~ ∼= C|S/W |,

a vector space with basis c1, . . . , cr. We remark that the set of parameters ‘c’ for the

Cherednik algebra Hc can be naturally identified with p, the dual of p∗. So we can view kΓ,i

as an element of p∗, its value on a parameter c ∈ p is given by the formula (2.2). Define p∗Z

to be the Z-lattice inside p∗ spanned by elements `−1Γ kΓ,i, and let pZ ⊆ p be the dual lattice.

The lattice pZ consists of all parameters c such that qΓ,i = η−iΓ , where ηΓ := exp(2π√−1/`Γ),

so Hq = CW . Moreover, we have that c− c′ ∈ pZ if and only if qΓ,i = q′Γ,i for every Γ ∈ A,

i = 0, . . . , `Γ− 1 and thus the set of parameters for the Hecke algebra can be identified with

p/pZ, see [L7, Subsection 2.6]. For example, if W is a Coxeter group, then pZ coincides with

the set of parameters for which c(s) ∈ Z for all s ∈ S.

Let us give a spanning set for pZ. First, we need to introduce some notation. For each

Γ ∈ A, the set of characters of WΓ is identified with Z/`ΓZ, an isomorphism Z/`ΓZ∼=−→

Hom(WΓ,C×) is given by m 7→ (s 7→ det(s)m). We have a morphism Hom(W,C×) −→∏Γ∈A/W Hom(WΓ,C×), given by restriction. According to [R, Subsection 3.3.1], this is an

isomorphism. Thus, we have a correspondence between 1-dimensional characters χ of W and

|A/W |-tuples of integers (mΓ) with 0 ≤ mΓ ≤ `Γ − 1. So, to a character χ ∈ Hom(W,C×)

associated to the tuple (mΓ) we assign χ ∈ p, given by

23

χ(kΓ,i) =

`Γ if i ≥ `−mΓ

0 if i < `−mΓ.

Clearly, χ ∈ pZ, and the elements χ form a spanning set for pZ.

Let us explain the reason why we are interested in the elements χ. According to [BC,

Proposition 5.6], for each parameter c ∈ χ we have an algebra isomorphism Ac ∼= eχHc+χeχ,

where eχ := 1|W |∑

w∈W χ(w−1)w denotes the idempotent in CW corresponding to χ. We will

use this below, see Definition 3.1.5.

Example 2.3.3. Let us see what the lattice pZ is when W is a simply laced Weyl group.

In this case, we have that `Γ = 2 and all the reflection hyperplanes are conjugate. The

parameter c may be thought of as a single complex number, so we have p = C. For every

reflection hyperplane Γ we have kΓ,0 = 0, kΓ,1 = −2c. It follows that pZ = Z ⊆ C. By the

same calculation, for a real reflection group W , pZ coincides with the set of parameters c

such that c(s) ∈ Z for every reflection s ∈ W .

2.3.6 Regular parameters

Recall that, if c ∈ pZ, the Hecke algebra Hq(W ) is equal to the group algebra CW . In

particular, KZ : Oc → Hq is a quotient functor between two categories that have the same

number of irreducible objects, and is therefore an equivalence. Moreover, a consequence of the

highest weight structure on Oc is that, for c outside of a countable collection of hyperplanes

in pZ, the category Oc is semisimple with irreducible objects given by the Verma modules

∆c(τ), τ ∈ W -irrep.

Definition 2.3.4. We say that a parameter c ∈ p is regular if category Oc is semisimple.

Otherwise, we say that c is singular.

Theorem 2.3.5 ([BE, GGOR, V]). Let c ∈ pZ. Then, the following are equivalent.

24

1. c is regular.

2. The algebra Hc is simple.

3. Every module M ∈ Oc has full support.

4. The Hecke algebra Hq is isomorphic to the group algebra CW .

5. The KZ functor KZ : Oc → Hq is an equivalence of categories.

Note that it follows from Theorem 2.3.5 that every regular parameter is spherical. Also note

that, if c and c′ are regular, we have that Oc ∼= Oc′ , as both categories are equivalent to the

category of representations of W . As we will see, this is no longer the case for Harish-Chandra

bimodules.

Example 2.3.6 ([DJO]). Let W = Sn acting on its reflection representation Cn−1. Then,

a parameter c ∈ p = C is singular if and only if c is a non-integral rational number of the

form r/m with m ≤ n.

2.4 Restriction functors for category O

2.4.1 Restriction functors for Hecke algebras

Let us remark that, if b ∈ R and W ′ := Wb is a parabolic subgroup of W , then there

is a natural inclusion of algebras ι : Hq(W′) ↪→ Hq(W ) where, abusing the notation, we

denote by q the restriction of q to S ∩W ′. This allows us to define a restriction functor,

HResWW ′ := ι∗ : Hq(W ) -mod → Hq(W′) -mod. The map ι : Hq(W

′) ↪→ Hq(W ) is actually

induced from an inclusion BW ′ ↪→ BW , see for example [BMR, Section 4] for details.

25

2.4.2 Bezrukavnikov-Etingof isomorphisms of completions

There is also a restriction functor on the level of category O, [BE]. This functor depends

on the choice of a point b ∈ R whose stabilizer Wb coincides with W ′. For distinct b, b′ with

this property, the functors are isomorphic (but not in a canonical way) so we will just denote

this functor by ResWW ′ . Let us describe the construction of this functor.

Recall the sheafification Hc of the rational Cherednik algebra. Let H∧bc (= C[R/W ][b]⊗C[R/W ]

Hc) be the algebra of sections of the sheaf Hc on the formal neighborhood of [b] ∈ R/W .

Note that H∧bc is generated by the algebra C[R]∧Wb, W , and Dunkl operators.

Bezrukavnikov and Etingof showed in [BE, Theorem 3.2] that H∧bc is isomorphic to a ma-

trix algebra of size |W/W ′| with coefficients in Hc(W′, R)∧0. Let us give a more precise

statement. We will need the centralizer construction, which we describe next. Let H ⊆ G

be finite groups, and let C be an algebra containing CH. In particular, both CG and C

are H-modules, where H acts on both algebras by multiplication on the left. Consider

HomH(CG,C), which is a free C-module of rank |G/H|. We define the centralizer algebra

to be

Z(H,G,C) := EndC(HomH(CG,C))

Of course, Z(H,G,C) ∼= Mat|G/H|(C), but this isomorphism is not canonical. There is,

however, a canonical way to recover C from Z(H,G,C), as follows. Let e(H) ∈ Z(H,G,A)

be defined by:

[(e(H).f)(g)] =

f(g) if g ∈ H,

0 otherwise

then e(H) ∈ Z(H,G,A) is an idempotent, and e(H)Z(H,G,A)e(H) ∼= A.

Theorem 2.4.1 ([BE]). Let b ∈ R and W ′ := Wb be the stabilizer of b. Then, there is an

isomorphism

26

θb : H∧bc∼=−→ Z(W ′,W,Hc(W

′, R)∧0)

where we abuse the notation and also denote by c the restriction c|S∩W ′. This isomorphism

is given by

[θb(w).f ](u) = f(wu), w ∈ W

[θb(x)f ](u) = (x+ 〈ux, b〉)f(u) x ∈ R∗

[θb(y).f ](u) = u(y)f(w) +∑

s∈S\W ′2cs

1−λs〈αs,w(y)〉αs+〈αs,b〉(f(su)− f(u)) y ∈ R

Let us give an intuitive way to see why Theorem 2.4.1 should hold. First of all, recall

that the algebra H∧bc is generated by C[R]Wb, W and Dunkl operators. Now, we have that

C[R]∧Wb =⊕

b′∈WbC[R]∧b′, and note that the algebra generated by C[R]∧b, Wb ⊆ W and

Dunkl operators is isomorphic to Hc(Wb, R)∧b - here, we are using that elements of the

form 1/αs for s ∈ Wb belong to C[R]∧b. From here, it is easy to see that H∧bc should

be a matrix algebra of size |W/W ′| with coefficients in Hc(W′, R)∧b, and the centralizer

algebra construction is simply an invariant way of saying this. Finally, we remark that

Hc(W′, R)∧b ∼= Hc(W

′, R)∧0.

2.4.3 Construction of the restriction functor

Let us now proceed to the construction of the restriction functor. First of all, we have a

completion functor •∧b : Oc → H∧bc -mod, M 7→ C[R/W ]∧[b] ⊗C[R/W ] M .

Definition 2.4.2. Let O∧bc ⊆ H∧bc denote the category of all H∧bc -modules which are finitely

generated over C[R]Wb.

Note that we have •∧b : Oc → O∧bc . Now, for N ∈ O∧bc , the pushforward (θb)∗N lies

in Z(W ′,W,Hc(W′, R)∧0) -mod, so thanks to the discussion above we have e(W ′)(θb)∗N ∈

Hc(W′, R)∧0 -mod. Moreover, we have e(W ′)(θb)∗(N) ∈ Oc(W ′, R)∧0.

27

Lemma 2.4.3 (Theorem 2.3, [BE]). The completion functor •∧b : Oc → O∧bc has a right

adjoint Eb : O∧bc → Oc which is given by taking the space of R-locally nilpotent vectors. If

b = 0, the functors •∧0 and E0 are quasi-inverse equivalences of categories.

Thanks to the previous lemma, E0(e(W ′)(θb)∗N) is a module in category O for the algebra

Hc(W′, R). To get a module in category O for Hc(W

′, RW ′), we take the set of elements

that are annihilated by RW ′ ⊆ Hc(W′, R). We remark that this functor Oc(W ′, R) →

Oc(W ′, RW ′) is an equivalence of categories. So we have the functor Resb, which is con-

structed as the composition of the several functors we have just defined.

Oc(W,R) Oc(W,R)∧b Oc(W ′, R)∧0 Oc(W ′, R)

Oc(W ′, RW ′)

Resb

M 7→M∧b e(W ′)(θb)∗ E0

∼=

Let us remark that the functor Resb does not depend on b, up to an isomorphism, which

is not canonical. Since this is not going to be of great importance for us, we will simply

denote ResWW ′ := Resb. Let us remark that ResWW ′ admits a right adjoint functor, IndWW ′ .

This is constructed by taking the quasi-inverse of all functors in the diagram above except

the completion functor •∧b. Here, we take the right adjoint Eb. Just as for the restriction

functor, the induction functor depends non-canonically on b. Note also that both IndWW ′ and

ResWW ′ are exact functors.

Theorem 2.4.4. The following is true.

1. IndWW ′ is also a left adjoint of ResWW ′. In particular, IndWW ′ and ResWW ′ preserve projective

and injective objects, [L4]

2. ResWW ′(M) 6= 0 if and only if XW ′ ⊆ SS(M).

3. ResWW ′(M) is finite-dimensional if and only if SS(M) = XW ′, [BE].

28

4. At the level of Grothendieck groups, under the identification [W -rep]∼=−→ [Oc], [τ ] 7→

[∆c(τ)], the map [ResWW ′ ] : [Oc(W,R)]→ [Oc(W ′, R)] gets identified with the restriction

functor for finite groups, [resWW ′ ] : [W -rep]→ [W ′ -rep], [BE].

5. The functor ResWW ′ preserves the categories of modules with a standard filtration, [Sh].

6. The restriction functor intertwines the KZ functors, that is, we have KZ ◦ ResWW ′ =

HResWW ′ ◦KZ, where KZ denotes the KZ functor from category Oc(W ′, RW ′) to the

category of finite-dimensional representations of Hq(W′), [Sh].

7. The induction functor IndWW ′ does not kill nonzero modules, [SV].

Let us give two easy consequences of the previous result that will be needed later.

Lemma 2.4.5. Let Pc ∈ Oc be a projective generator, and W ′ a parabolic subgroup of W .

Then, ResWW ′(Pc) ∈ Oc(W ′) is a projective generator.

Proof. The module ResWW ′(Pc) is projective since ResWW ′ admits an exact biadjoint functor.

Now let M ∈ Oc(W ′) be irreducible. Then HomOc(W ′)(ResWW ′(Pc),M) is naturally isomorphic

to HomOc(W )(Pc, IndWW ′(M)), which is nonzero since IndWW ′(M) is nonzero. We are done.

Lemma 2.4.6. Let τ be a 1-dimensional representation of W . Then, for any parabolic

subgroup W ′ ⊆ W , ResWW ′(∆c(τ)) = ∆c(resWW ′(τ)).

Proof. By (5) of Theorem 2.4.4, ResWW ′(∆c(τ)) has a standard filtration. Since τ is 1-

dimensional, resWW ′(τ) is an irreducible representation of W ′. The result now follows from

(4) of Theorem 2.4.4.

29

Chapter 3

Preliminaries on Harish-Chandra

bimodules

In this chapter, we start studying Harish-Chandra bimodules. These are a special class

of bimodules over rational Cherednik algebras that were introduced in [BEG] associated

to the study of the space of quasi-invariants for the action of a real reflection group on

its reflection representation. Harish-Chandra bimodules naturally provide functors between

distinct categories O which are analogous to the projective functors in Lie theory, and this is

a reason to study Harish-Chandra bimodules. Let us remark, however, that unlike projective

functors, the functors induced by Harish-Chandra bimodules are in general not exact. Here,

we will see that the category of Harish-Chandra bimodules is equivalent to the category of

representations of a finite-dimensional algebra, and so it is an interesting category on its own

right.

3.1 Definition and basic results

Fix a complex reflection group W , and parameters c, c′ ∈ p to form the rational Cherednik

algebras Hc := Hc(W,R), Hc′(W,R). Note that the algebras C[R]W ,C[R∗]W are subalgebras

30

of both Hc and Hc′ . Thus, if B is a (Hc, Hc′)-bimodule and a ∈ C[R]W ∪ C[R∗]W , we can

consider the adjoint action of a on B, ad(a) : B → B, b 7→ ab− ba.

Definition 3.1.1 ([BEG]). Let B be a (Hc, Hc′)-bimodule. We say that B is Harish-Chandra

(shortly, HC) if the following conditions are satisfied.

(HC1) B is finitely generated as a bimodule.

(HC2) For any a ∈ C[R]W ∪ C[R∗]W , ad(a) : B → B is locally nilpotent.

We denote the category of HC (Hc, Hc′)-bimodules by HC(Hc, Hc′). Since both Hc, Hc′ are

noetherian, it follows that the category HC(Hc, Hc′) is a full Serre subcategory of the category

(Hc, Hc′)-bimod.

Example 3.1.2. The regular bimodule Hc is a Harish-Chandra Hc-bimodule.

The following are basic results about the category of HC bimodules.

Proposition 3.1.3 (Lemma 3.3, [BEG]). (1) Any B ∈ HC(Hc, Hc′) is finitely generated

as a left Hc-module, as a right Hc′-module, and as a C[R]W ⊗ C[R∗]W -module (here,

C[R]W is considered inside Hc, while C[R∗]W is considered inside Hc′).

(2) If B ∈ HC(Hc, Hc′), B′ ∈ HC(Hc′ , Hc′′) then B ⊗Hc′ B′ ∈ HC(Hc, Hc′′).

(3) If B ∈ HC(Hc, Hc′) and M ∈ Oc′, then B ⊗Hc′ M ∈ Oc.

Proof. Let us show (1). Equip Hc, Hc′ with the Bernstein filtration. It is easy to see that we

can find a bimodule filtration on B such that gr(B) is a finitely generated C[R ⊕ R∗]#W -

bimodule, and for every element a ∈ C[R]W ∪ C[R∗]W , the adjoint action of a on grB

is nilpotent (rather than just locally nilpotent). Since, in particular, gr(B) is a finitely

generated C[R⊕R∗]-bimodule, this implies that gr(B) is finitely generated:

(a) As a left C[R⊕R∗]#W -module,

31

(b) As a right C[R⊕R∗]#W -module,

(c) And as a C[R]W ⊗ C[R∗]W -module, where the action of C[R]W is on the left and that

of C[R∗]W is on the right.

this implies (1). For (2), note that (1) implies that B ⊗Hc′ B′ is finitely generated as a

(Hc, Hc′′)-bimodule, while the fact that the adjoint action of C[R]W ∪ C[R∗]W is locally

nilpotent is an easy exercise. Finally, for (3), note that (1) implies again that B ⊗Hc′ M

is finitely generated over Hc. It is easy to see that the action of the augmentation ideal

C[R∗]W+ ⊆ C[R∗]W is locally nilpotent. In particular, this implies that for every v ∈ B⊗Hc′M ,

the C[R∗]W -module C[R∗]Wv is finite-dimensional and supported at 0. But then the same

is true for the C[R∗]-module C[R∗]v. It follows that the action of R on B ⊗Hc′ M is locally

nilpotent, so B ⊗Hc′ M ∈ Oc.

From (3) of Proposition 3.1.3 it follows that taking tensor product with a bimodule B ∈

HC(Hc, Hc′) gives a right exact functor B ⊗Hc′ • : Oc′ → Oc, while from (2) of the same

proposition it follows that the composition of two such functors is again of the same form.

We will use the functors B ⊗Hc′ • extensively in this work.

3.1.1 Harish-Chandra bimodules for the spherical subalgebras

Recall that Ac, Ac′ denote the spherical subalgebras of Hc, Hc′ , respectively. Note that we

have C[R]W ,C[R∗]W ⊆ Ac, Ac′ , so we can give the definition of a HC (Ac, Ac′)-bimodule

completely analogously to Definition 3.1.1. We will denote by HC(Ac, Ac′) the category

of HC (Ac, Ac′)-bimodules. Statements (1) and (2) of Proposition 3.1.3 remain valid upon

replacing Hc, Hc′ with Ac, Ac′ . Moreover, the functors

(Hc, Hc′) -bimod→ (Ac, Ac′) -bimod (Ac, Ac′) -bimod→ (Hc, Hc′) -bimod

B 7→ eBe B′ 7→ Hc ⊗Ac B ⊗Ac′ Hc′

32

preserve the categories of HC bimodules, and they induce inverse equivalences HC(Hc, Hc′)∼=↔

HC(Ac, Ac′) provided both c and c′ are spherical.

Let us give examples of Harish-Chandra bimodules over the spherical subalgebras. Let

χ ∈ Hom(W,C×) be a 1-dimensional character and consider the parameter χ ∈ p that was

constructed in Section 2.3.5. Recall that we have an isomorphism Ac ∼= eχHc+χeχ. Thus,

the space eHc+χeχ is an (Ac+χ, Ac)-bimodule.

Proposition 3.1.4 ([BC, BL, L7]). eHc+χeχ ∈ HC(Ac+χ, Ac).

Proof. Let us remark that the isomorphism Ac → eχHc+χeχ is constructed from a filtered

automorphism on D(Rreg)#W , where on the latter algebra we take the geometric filtration.

It follows that this isomorphism preserves the filtration given by deg(R) = 1, deg(R∗) =

degW = 0. Moreover, the associated graded morphism grAc → gr eχHc+χeχ = C[R⊕R∗]W

is the identity and, under the induced bimodule filtration on eHec+χ, the associated graded

gr eHec+χ is the space of semi-invariants C[R⊕R∗]W,χ−1. Note that the left and right actions

of grAc, grAc′ on gr eHc+χeχ coincide. Since deg(C[R]W ) = 0, this already shows that the

adjoint action of C[R]W on eHc+χeχ is locally nilpotent.

Now we show that the adjoint action of C[R∗]W on eHc+χeχ is locally nilpotent. Let us abuse

the notation and denote by ad eu : eHc+χeχ → eHc+χeχ the operator x 7→ euc+χx − xeuc.

Viewing the space eHc+χeχ inside D(Rreg)#W , it is easy to see that the ad eu action is

diagonalizable. Now we twist the filtration on eHc+χeχ by this action:

G′i(eHc+χeχ) :=⊕k

Gk(eHc+χeχ) ∩ {a ∈ eHc+χeχ : ad(eu)(a) = (i− k)a}

Note that now the left and right actions of C[R∗]W preserve the filtration, while the associated

graded does not change. It follows as in the previous paragraph that the adjoint action of

C[R∗]W is locally nilpotent. Thus, eHc+χeχ is HC.

Definition 3.1.5 ([BC]). Let χ ∈ Hom(W,C×) be a 1-dimensional character. For c ∈ p we

33

define the shift bimodule

Bc,c+χ := Hc+χ ⊗Ac+χ eHc+χeχ ⊗Ac Hc ∈ HC(Hc+χ, Hc)

and the shift functor Φc,c+χ := Bc,c+χ ⊗Hc • : Oc → Oc+χ.

3.1.2 Alternative definition

An alternative definition of a HC bimodule was found by Losev in [L3]. One direction of

the following result follows similarly to the proof of Proposition 3.1.4, while the other one is

considerably more technical. We refer the reader to [L3] for a proof.

Theorem 3.1.6 (Section 5.4, [L3]). Let B ∈ (Hc, Hc′) -bimod. Then, the following are

equivalent.

1. B is HC.

2. There exists a bimodule filtration on B, to be called a good filtration, such that the

following conditions are satisfied.

(a) grB is a finitely generated bimodule over C[R⊕R∗]#W .

(b) The left and right actions of C[R⊕R∗]W = Z(C[R⊕R∗]#W ) on grB coincide.

Let us remark that in (2) of Theorem 3.1.6 we can take either the Bernstein or the geometric

filtration on Hc, Hc′ , this follows since we can get one filtration from the other by twisting

with the adjoint Euler action, see e.g. Section 2 in [L7].

Thanks to Theorem 3.1.6 we can give a definition of HC bimodules for the homogeneous

Cherednik algebra H. Let B be a HC (Hc, Hc′)-bimodule with a good filtration, where

we take the geometric filtration on Hc and Hc′ . Then, the Rees bimodule R~(B) is a

(R~(Hc),R~(Hc′))-bimodule. Recall from Section 2.1.5 that both algebras R~(Hc),R~(Hc′)

are quotients of H. In particular, R~(B) is a H-bimodule. The following definition is then

tailored so that R~(B) is a HC H-bimodule.

34

Definition 3.1.7 ([L3]). Let B be a graded H-bimodule. We say that B is HC if the following

conditions are satisfied.

(i) B is finitely generated as a H-bimodule.

(ii) The left and right actions of ~ on B coincide.

(iii) B is flat as a C[~]-module.

(iv) The left and right actions of Z(H/~H) on B/~B coincide.

3.2 Singular supports and annihilators

3.2.1 Singular supports

Let B be a HC (Hc, Hc′)-bimodule, equipped with a good filtration with respect to the

geometric filtrations on Hc, Hc′ . We can consider grB to be a C[R ⊕ R∗]W -module. Note

that, while the module grB itself depends on the chosen good filtration, its set-theoretic

support (as a coherent sheaf on (R⊕R∗)/W ) does not.

Definition 3.2.1. We define the singular support of a HC (Hc, Hc′)-bimodule B to be the

set-theoretic support of the C[R ⊕ R∗]W -module grB, where the associated graded is taken

with respect to any good filtration on B.

SS(B) := supp(grB) ⊆ (R⊕R∗)/W

Similarly, for a HC H-bimodule B, we define its singular support

SS(B) := supp(B/cB) ⊆ (R⊕R∗)/W

For a (Hc, Hc′)-bimodule B, we denote by LAnn(B) ⊆ Hc the left annihilator and by

RAnn(B) ⊆ Hc′ the right annihilator.

35

Lemma 3.2.2 ([L8]). Let B be a HC (Hc, Hc′)-bimodule. Then SS(B) = SS(Hc/LAnn(B)) =

SS(Hc′/RAnn(B)), where Hc/LAnn(B) (resp. Hc/RAnn(B)) is viewed as a HC (Hc, Hc)-

(resp. (Hc′ , Hc′)-)bimodule.

Proof. Let us deal with the statement for the left annihilator, the other statement is analo-

gous. First of all, since a good filtration on B is compatible with the filtration on Hc which

induces a good filtration on Hc/LAnn(B), it follows easily that SS(B) ⊆ SS(Hc/LAnn(B)).

To show the other inclusion, consider the (Hc, Hc)-bimodule B := HomHc′(B,B). Note that

a good filtration on B induces a filtration on B, by setting

F iB := {ϕ ∈ B : ϕ(F jB) ⊆ F i+jB for all j}

we remark that this filtration is exhausting because B is finitely generated as a right Hc′-

module, and it is compatible with the filtration on Hc because so is the filtration on B. Note

that, upon restricting to C[R⊕R∗]W , we have

gr B ⊆ HomC[R⊕R∗]W (grB, grB)

from where it follows that B is HC, that F is a good filtration, and that SS(B) ⊆ SS(B). Now

note that we have an inclusion Hc/LAnn(B) → B, which is obviously a map of (Hc, Hc)-

bimodules. The result follows.

Lemma 3.2.3. Let B be an irreducible HC (Hc, Hc′)-bimodule and M ∈ Oc be irreducible.

Then:

(i) B ⊗Hc′ M = 0 unless Ann(M) = RAnn(B).

(ii) If B⊗Hc′M 6= 0, then the annihilator of every irreducible quotient of B⊗HcM coincides

with LAnn(B) and, moreover, Ann(B ⊗Hc′ M) = LAnn(B).

Proof. Assume B ⊗Hc′ M 6= 0, and let N be an irreducible quotient of B ⊗Hc′ M . Note

that, since B is irreducible, we have an inclusion B ↪→ HomC(M,N). Let us first check that

36

RAnn(B) = Ann(M), which will show (i). Note that⋂f∈B ker(f) is a proper submodule

of M . Since M is irreducible, we must have⋂f∈B ker(f) = 0. Now, if a ∈ RAnn(B),

then aM ⊆⋂f∈B ker(f), so a ∈ Ann(M). The other inclusion follows from the inclusion

Ann(M) ⊆ RAnn(HomC(M,N)), which is clear. We are done with (i).

To show (ii) we must show, first, that LAnn(B) = Ann(N). Observe that∑

f∈B f(M) is a

nonzero submodule of N , so we must have∑

f∈B f(M) = N . From here it follows similarly

to the previous paragraph that LAnn(B) = Ann(N). To prove the last statement of the

lemma, note that we clearly have LAnn(B) ⊆ Ann(B⊗Hc′ M). On the other hand, by what

we just proved Ann(B⊗Hc′M) ⊆ Ann(N) = LAnn(B). So LAnn(B) = Ann(B⊗Hc′M).

Let us rephrase the previous lemma in terms of supports. First of all, let us remark that

the C[R ⊕ R∗]W -module grB is a Poisson module, that is, it is equipped with a Poisson

bracket {·, ·} : C[R⊕R∗]⊗ grB → grB satisfying the obvious compatibility conditions with

the Poisson bracket on C[R ⊕ R∗]W . This is a standard result. It follows that the singular

support of B is a union of symplectic leaves of (R ⊕ R∗)/W . We will need the following

result of [BrGo].

Lemma 3.2.4. The symplectic leaves of (R⊕R∗)/W are in bijection with conjugacy classes

of parabolic subgroups of W . Namely, for every parabolic subgroup W ′ ⊆ W , the set LW ′ :=

π{(x, y) ∈ R ⊕ R∗ : W(x,y) = W ′} is a symplectic leaf, where π : R ⊕ R∗ → (R ⊕ R∗)/W is

the quotient map.

Recall now, that for an irreducible module M ∈ Oc′ , supp(M) = XW ′ for a parabolic

subgroup W ′ ⊆ W . It follows from Lemma 3.2.3 that if B is an irreducible HC bimodule

such that B ⊗Hc M 6= 0, then SS(B) = LW ′ . In particular, if B is an irreducible HC

bimodule such that B ⊗Hc′ M 6= 0 for some finite-dimensional M ∈ Oc′ , then B itself is

finite-dimensional.

37

3.2.2 Filtration by supports

The notion of singular support allows us to define a filtration on the category HC(Hc, Hc′),

as follows. Let L ⊆ (R ⊕ R∗)/W be a symplectic leaf. We define the full Serre subcategory

HCL(Hc, Hc′) ⊆ HC(Hc, Hc′) to be that whose objects are HC (Hc, Hc′)-bimodules B with

SS(B) ⊆ L. Similarly, we define HC∂L(Hc, Hc′), where ∂L := L\L. Note that HC∂L(Hc, Hc′)

is a Serre subcategory of HCL(Hc, Hc′) and we can form the quotient

HCL(Hc, Hc′) := HCL(Hc, Hc′)/HC∂L(Hc, Hc′)

For example, when L = L{1} is the dense symplectic leaf, then HCL(Hc, Hc′) is the quotient

of the category HC(Hc, Hc′) by the Serre subcategory consisting of bimodules with proper

singular support. On the other extreme, when L = LW = {0}, then HC0(Hc, Hc′) is the

category of finite-dimensional bimodules. Define the associated graded category

gr HC(Hc, Hc′) :=⊕L

HCL(Hc, Hc′)

Note that the irreducible objects of HCL(Hc, Hc′) are in bijective correspondence with the

simple HC bimodules whose singular support coincides with L. In particular, the number of

irreducibles in gr HC(Hc, Hc′) is no greater than the number of irreducibles in HC(Hc, Hc′).

In the sequel, we will see that these numbers actually coincide (and are finite).

3.3 Restriction functors for HC bimodules, I: naive

construction

3.3.1 Construction

Let B be a HC (Hc, Hc′)-bimodule. Note that, by its very definition, for any principal open

set U ⊆ R/W , the localization C[U ] ⊗C[R/W ] B is actually a (Hc(U),Hc′(U))-bimodule, cf.

Section 2.1.6. So we can sheafify the bimodule B to get a (Hc,Hc′)-bimodule B.

38

Now let b ∈ R, with stabilizer Wb, and let B∧b be the sections of B on the formal neigh-

borhood of [b] ∈ R/W . In other words, we have B∧b = C[R/W ]∧[b] ⊗C[R/W ] B. Thanks to

the considerations of the previous paragraph, this is a (H∧bc , H∧bc′ )-bimodule. So we can take

the pushforward with respect to the Bezrukavnikov-Etingof isomorphism θb, and we get a

(Z(Wb,W,Hc(Wb), R)∧0, Z(Wb,W,Hc′(Wb, R)∧0))-bimodule (θb)∗(B∧b). Recall the idempo-

tent e(Wb) ∈ Z(Wb,W,Hc(Wb, R)∧0). Abusing the notation, we also denote by e(Wb) a simi-

larly defined idempotent in the algebra Z(Wb,W,Hc′(Wb, R)∧0). So e(Wb)B∧be(Wb) becomes

a (Hc(Wb, R)∧0, Hc′(Wb, R)∧0)-bimodule. Recall now the decomposition R = RWb⊕RWb and

a similar decomposition for R∗. Let B†b be the subspace of e(Wb)B∧be(Wb) consisting of

vectors v satisfying:

• They commute with RWb , (R∗)Wb , that is, xv = vx, yv = vy for x ∈ (R∗)Wb , y ∈ RWb .

• For every a ∈ C[RWb]Wb ∪ C[R∗Wb

]Wb ad(a)N(v) = 0 for N � 0.

The assignment B 7→ B†b is functorial, and B†b is a (Hc(Wb, RWb), Hc′(Wb, RWb

))-bimodule.

Theorem 3.3.1 (Section 3, [L3]). The assignment B 7→ B†b defines a functor

•†b : HC(Hc, Hc′)→ HC(Hc(Wb, RWb), Hc′(Wb, RWb

)).

This functor is exact and admits a right adjoint

•†b : HC(Hc(Wb, RWb), Hc′(Wb, RWb

))→ HC(Hc, Hc′),

where the latter category is the ind-completion of HC(Hc, Hc′),

Let us remark that the objects of HC(Hc, Hc′) are (Hc, Hc′)-bimodules which are the union

of their HC (= finitely generated) sub-bimodules. For the sake of completeness, let us give

a sketch of the construction of the functor •†b . Note first that the algebra Hc(Wb, R) is

naturally isomorphic to the tensor product D(RWb)⊗Hc(Wb, RWb), this follows easily from

the decomposition R = RWb ⊕RWb. So the completion of this algebra at 0 is Hc(Wb, R)∧0 =

39

D((RWb)∧0)⊗Hc(Wb, RWb)∧0. It follows that, if B ∈ HC(Hc(Wb, RWb

), Hc′(Wb, RWb)), then

D((RWb)∧0) ⊗ B∧0 is a (Hc(Wb, R)∧0, Hc′(Wb, R)∧0)-bimodule. Recall that the category of

(Hc(Wb, R)∧0, Hc′(Wb, R)∧0)-bimodules is equivalent to that of (H∧bc , H∧bc′ )-bimodules. Let us

denote by B′ the bimodule corresponding to D((RWb)∧0)⊗B∧0 under this equivalence. Now

B†b is the subspace of B′ consisting of vectors on which C[R]W ,C[R∗]W act locally nilpotently.

We clearly have that B†b ∈ HC(Hc, Hc′). In Corollary 3.3.15, we will see that the image of •†b

is actually contained in the smaller category HC(Hc, Hc′), that is, B†b is finitely generated

for every B ∈ HC(Hc(Wb, RWb), Hc′(Wb, RWb

)). Let us remark that this will be a formal

consequence of properties of the categories of HC bimodules and, in particular, we do not

need the explicit description of the functor •†b to prove it.

3.3.2 Properties

Important properties of the functor •†b are summarized in the following result. Recall that

symplectic leaves of (R⊕R∗)/W are labeled by conjugacy classes of parabolic subgroups of

W , and we denote by LW ′ the symplectic leaf corresponding to (the conjugacy class of) the

parabolic subgroup W ′. Similarly, the symplectic leaves of (RWb⊕ R∗Wb

)/Wb are labeled by

conjugacy classes of parabolic subgroups of Wb, and we denote these by LWb

W ′ .

Lemma 3.3.2 ([L3]). The following is true.

1. Assume that SS(B) =⋃ni=1 LWi

. Then, LWb

W ′ ⊆ SS(B†b) if and only if W ′ is conjugate

(in W ) to one of W1, . . . ,Wn.

2. If SS(B) ⊆ LWbthen B†b is finite-dimensional. Moreover, B†b = 0 if SS(B) ( LWb

.

3. If B1 ∈ HC(Hc, Hc′), B2 ∈ HC(Hc′ , Hc′′) then we have a natural isomorphism (B1⊗Hc′

B2)†b∼=→ B1,†b ⊗Hc′ (Wb,RWb ) B2,†b.

4. If M ∈ Oc′, then we have a natural isomorphism ResWWb(B⊗Hc′ M)

∼=→ B†b ⊗Hc′ (Wb,RWb )

ResWWb(M).

40

5. Assume B ∈ HC(Hc(Wb, RWb), Hc′(Wb, RWb

)) is finite-dimensional. Then B†b is finitely

generated, that is, B†b ∈ HC(Hc, Hc′).

3.3.3 Locally finite maps

Let us now give a way to construct HC bimodules. Consider modules N ∈ Oc, M ∈ Oc′ .

Then, HomC(M,N) is a (Hc, Hc′)-bimodule.

Definition 3.3.3 ([BEG]). By Homfin(M,N) we denote the (Hc, Hc′)-sub-bimodule of the

bimodule HomC(M,N) consisting of all those vectors that are locally nilpotent under the

adjoint action of C[R]W ∪ C[R∗]W .

Clearly, Homfin(M,N) is the direct limit (= union) of its HC sub-bimodules. We have, in

fact, that Homfin(M,N) is HC.

Lemma 3.3.4 (Proposition 5.7.1 in [L3]). For any M ∈ Oc′ , N ∈ Oc, the (Hc, Hc′)-bimodule

Homfin(M,N) is finitely generated, and so it is HC.

We will use the following result, which is [L3, Lemma 5.7.2]. Alternatively, it follows from

Lemma 3.2.3.

Lemma 3.3.5. Let M ∈ Oc′, N ∈ Oc be irreducible. Then, Homfin(M,N) = 0 unless

supp(M) = supp(N).

For us, the bimodules of locally finite maps are important because, in fact, every irreducible

HC (Hc, Hc′)-bimodule can be embedded into a bimodule of locally finite maps.

Lemma 3.3.6 (Lemma 3.10 in [L7]). Let Pc′ ∈ Oc′ be a projective generator, and let B ∈

HC(Hc, Hc′) be a nonzero HC bimodule. Then, B ⊗Hc′ Pc′ 6= 0.

Proof. Let W ′ be a parabolic subgroup such that LW ′ is dense in SS(B), and let b ∈ R be such

that Wb = W ′. Then, B†b is a finite-dimensional (Hc(Wb, RWb), Hc′(Wb, RWb

))-bimodule, and

41

therefore we can find an irreducible finite-dimensionalHc′(Wb, RWb)-module, sayN , such that

B†b⊗Hc′ (W ′)N 6= 0. Thanks to Lemma 2.4.5, we have B†b⊗Hc(W ′)ResWW ′(Pc′) � B†b⊗Hc′ (W ′)N .

So 0 6= B†b ⊗Hc(W ′) ResWW ′(Pc′)∼= ResWW ′(B ⊗Hc′ Pc′). We are done.

Corollary 3.3.7. Let B be an irreducible HC (Hc, Hc′)-bimodule. Then, there exist irre-

ducible modules M ∈ Oc′, N ∈ Oc and a monomorphism B ↪→ Homfin(M,N).

Proof. By Lemma 3.3.6 there exists an irreducible module M ∈ Oc′ with B ⊗Hc′ M 6= 0.

Since the latter module is in category Oc, there exists an irreducible module N ∈ Oc and a

nonzero map f : B⊗Hc′M → N . Then, v 7→ (m 7→ f(v⊗Hc′m)) defines a nonzero morphism

B → Homfin(M,N).

Corollary 3.3.8. Let B1 be an irreducible HC (Hc, Hc′)-bimodule, and B2 an irreducible HC

(Hc′ , Hc′′)-bimodule. Then, B1 ⊗Hc′ B2 = 0 unless SS(B1) = SS(B2).

Proof. Assume that SS(B1) 6= SS(B2), and denote B := B1 ⊗Hc′ B2. First, we assume

that SS(B1) 6⊆ SS(B2). By Lemma 3.3.6 it is enough to show that B ⊗Hc′′ N = 0 for

all irreducible modules N ∈ Oc′′ . If B2 ⊗Hc′′ N = 0 we are done. So we may assume

that B2 ⊗Hc′′ N 6= 0. By Lemma 3.2.3 this implies that Ann(B2 ⊗Hc′′ N) = LAnn(B2).

If B1 ⊗Hc′ (B2 ⊗Hc′′ N) 6= 0, then B1 ⊗Hc′ M 6= 0 for some irreducible subquotient M of

B2 ⊗Hc′′ N . So RAnn(B1) = Ann(M) ⊇ Ann(B2 ⊗Hc N) = LAnn(B2). Thus, SS(B1) =

SS(Hc′/RAnn(B1)) ⊆ SS(Hc′/LAnn(B2)) = SS(B2), a contradiction with our assumption.

We conclude that B1 ⊗Hc′ B2 = 0.

Now assume that SS(B2) 6⊆ SS(B1). Let copp be defined by copp(s) := −c(s−1). Then, it

is easy to check that we have an isomorphism Hc(W,R) → Hcopp(W,R∗)opp given by x 7→

x, y 7→ y, w 7→ w−1, x ∈ R∗, y ∈ R,w ∈ W . We get an equivalence ρc,c′ : (Hc, Hc′) -bimod→

(H(c′)opp(W,R∗), Hcopp(W,R∗)) -bimod. Similarly, we get equivalences ρc′,c′′ , ρc,c′′ . Note that

these equivalences preserve the categories of HC bimodules as well as the support of a HC

bimodule. We have that ρc,c′′(B1 ⊗Hc′ B2) = ρc′,c′′(B2) ⊗H(c′)opp (W,R∗) ρc,c′(B1). Thus, the

result in this case follows from the previous paragraph.

42

Corollary 3.3.9. Let B be an irreducible HC (Hc, Hc′)-bimodule. Then, there exists a

parabolic subgroup W ′ ⊆ W such that SS(B) = LW ′.

Proof. Let M ∈ Oc′ be an irreducible module such that B⊗Hc′M 6= 0. By Proposition 2.2.5

there exists a parabolic subgroup W ′ ⊆ W such that supp(M) = XW ′ . Now it follows from

Lemma 3.2.3 that SS(B) = LW ′ .

Corollary 3.3.10. The following holds.

1. The irreducible objects in HC(Hc, Hc′) and gr HC(Hc, Hc′) are in bijective correspon-

dence.

2. Let W ′ be a parabolic subgroup of W , and assume HCLW ′ (Hc, Hc′) 6= 0. Then, O◦c,W ′

and O◦c′,W ′ are both nonzero.

3. We have HCLW ′ (Hc, Hc) 6= 0 if and only if O◦c,W ′ 6= 0.

We will see now that HC(Hc, Hc′) has a finite number of irreducible objects. We will use

the following result due to Ginzburg, [Gi2, Corollary 6.7] and, independently, Losev, [L8,

Section 4.1].

Proposition 3.3.11. Every object in HC(Hc, Hc′) has finite length.

Corollary 3.3.12. The category HC(Hc, Hc′) has finitely many irreducible objects.

Proof. Recall that every irreducible HC (Hc, Hc′)-bimodule is contained in a bimodule of the

form Homfin(M,N) where M ∈ Oc′ , N ∈ Oc are irreducible modules with the same support.

By Proposition 3.3.11, Homfin(M,N) has finite length. This, together with the fact that

the categories O have finitely many irreducible modules, give the desired result.

Let us now see that, in fact, the category HC(Hc, Hc′) is equivalent to the category of

finite-dimensional representations of a finite-dimensional algebra. By Proposition 3.3.11 and

Corollary 3.3.12, it is enough to check that the category HC(Hc, Hc′) has enough injective

objects. In order to do so, we will use the following result.

43

Lemma 3.3.13 (Lemma 3.9, [L7]). Let Pc′ ∈ Oc′ be projective. Then, the functor •⊗Hc′ Pc′ :

HC(Hc, Hc′)→ Oc is exact.

Corollary 3.3.14. The category HC(Hc, Hc′) has enough injective objects. In particular,

it is equivalent to the category of finite-dimensional representations of a finite-dimensional

algebra.

Proof. Let Pc′ be a projective generator of Oc′ . Thanks to Lemma 3.3.13, the functor

• ⊗Hc′ Pc′ is exact. Note that the right adjoint of this functor is precisely Homfin(Pc′ , •) :

Oc → HC(Hc, Hc′). Being a functor whose left adjoint is exact, the functor Homfin(Pc′ , •)

sends injective objects to injective objects.

Now let B be an irreducible HC (Hc, Hc′)-bimodule. Recall that there exist simple M ∈

Oc′ , N ∈ Oc, such that B ↪→ Homfin(M,N). In particular, B ↪→ Homfin(M,E), where E ∈

Oc is an injective module containing N . But now there exists n > 0 such that P nc′ �M , and

therefore B ↪→ Homfin(M,E) ↪→ Homfin(Pnc′ , E). By the observation in the first paragraph

of this proof, the latter bimodule is injective. Thus, every irreducible HC(Hc, Hc′)-bimodule

may be embedded in an injective object. It follows from Proposition 3.3.11 that HC(Hc, Hc′)

has enough injectives.

Corollary 3.3.15. Let W ′ ⊆ W be a parabolic subgroup and b ∈ R such that Wb = W ′.

Let B ∈ HC(Hc(W′, RW ′), Hc′(W

′, RW ′)). Then, B†b is finitely generated. In particular, we

have •†b : HC(Hc(W′, RW ′), Hc′(W

′, RW ′))→ HC(Hc, Hc′).

Proof. Note that, since HC(Hc, Hc′) has finitely many simples and enough projective ob-

jects, it is enough to show that HomHC(P,B†) is finite-dimensional for every projective

P ∈ HC(Hc, Hc′). This is immediate by adjunction.

In the sequel, we will need a much subtler version of the restriction functors. The main new

feature of the upgraded restriction functors is that they see a certain equivariance on the

image, which is not possible to see using the constructions in this section.

44

3.4 Restriction functors for HC bimodules, II: equiv-

ariance.

We will need an enhanced version of restriction functors for HC bimodules, introduced

in [L3, Section 3], the exposition here follows [L7, Section 3]. The enhancement comes

from an equivariance on the target category, which we now explain. Let W ⊆ W be a

parabolic subgroup, NW (W ) its normalizer, and Ξ := NW (W )/W . Recall that we have the

decomposition R := RW ⊕ RW , and note that the normalizer NW (W ) preserves RW . It

follows that NW (W ) acts on Hc := Hc(W,RW ) by algebra automorphisms, in such a way

that the action of W ⊆ NW (W ) coincides with the adjoint action, w : x 7→ wxw−1.

Definition 3.4.1. A Ξ-equivariant HC (Hc, Hc′)-bimodule is a bimodule B ∈ HC(Hc, Hc′)

together with an action of NW (W )-action on B such that

• The structure map Hc ⊗B ⊗Hc′ → B is NW (W )-equivariant.

• For w ∈ W , b ∈ B, we have w.b = wbw−1.

Let us denote by HCΞ(Hc, Hc′) the category of Ξ-equivariant HC (Hc, Hc′)-bimodules, and

by HCΞ0 (Hc, Hc′) the subcategory consisting of finite-dimensional equivariant bimodules.

3.4.1 Construction

Here, we follow [L7]. We work with the homogeneous Cherednik algebra H. Let us denote

Rreg−W := R \⋃s 6∈W

Γs = {δW 6= 0}

where δW :=∏

s 6∈W αs. Note that Rreg−W/W is the etale locus of the projection R/W →

R/W . The space C[Rreg−W/W ]⊗C[R/W ] Hc is naturally an algebra, which can be identified

with the subalgebra of D(Rreg−W/W ×R/W R)#W generated by C[Rreg−W/W ×R/W R],

CW , and the Dunkl operators. We denote this algebra by Hc,reg−W . The same proof of [BE,

Lemma 3.2] gives the following.

45

Lemma 3.4.2. There is an isomorphism

Θ : Hc,reg−W → Z(W,W,C[Rreg−W/W ]⊗C[R/W ] Hc(W,R)) (3.1)

Let us give a geometric intuition for the previous lemma. Note that there is a W -equivariant

isomorphism

Rreg−W/W ×R/W R∼=→

⊔w∈W/W wRreg−W ⊆ W/W ×R

(Wx,wx) 7→ (wW,wx)(3.2)

Let us denote by X the variety⊔w∈W/W wRreg−W . So we can think of Hc,reg−W as Hc(W,X ),

the rational Cherednik algebra associated to the action of W on the variety X , see e.g. [Et2],

[Wi, Section 2] for generalities on rational Cherednik algebras associated to the action of

a complex reflection group on a smooth algebraic variety (not necessarily a vector space),

in this work we will only use varieties that are disjoint unions of Zariski open sets inside a

vector space. Similarly, we think of C[Rreg−W/W ] ⊗C[R/W ] Hc(W,R) as being the rational

Cherednik algebra Hc(W,Rreg−W ). The isomorphism in Lemma 3.4.2 is an invariant way of

expressing the decomposition in (3.2).

Now let L be the projection to Rreg−W/W of {x ∈ R : Wx = W} ⊆ Rreg−W . Note that

L is closed in Rreg−W , so we can look at its formal neighborhood L. Denote H∧Lc,reg−W :=

C[Rreg−W/W ]∧L ⊗C[R/W ] Hc, which is naturally an algebra and the isomorphism in Lemma

3.4.2 can be restricted to an isomorphism

Θ : H∧Lc,reg−W∼=→ Z(W,W,C[Rreg−W/W ]∧L ⊗C[R/W ] Hc(W,R)). (3.3)

The isomorphism in (3.3) will take the role of the isomorphism Θb in the definition of the re-

striction functors. For technical reasons, let us introduce these functors for the homogeneous

Cherednik algebras H. We remark that the isomorphisms (3.1), (3.3) are still valid at the

level of these algebras, provided we define H(W,R) as a C[c]-algebra, even if the defining

relations do not involve all the variables c1, . . . , cn.

46

Now let B be a HC H-bimodule. Consider the space B∧Lreg−W := C[Rreg−W ]∧L ⊗C[R/W ] B. A

priori, this is only a (H∧Lreg−W ,H)-bimodule.

Lemma 3.4.3 (Lemma 3.6.3, [L3]). There is a unique right multiplication map making

B∧Lreg−W a H∧Lreg−W -bimodule such that the commutator [C[Rreg−W/W ]∧L,B∧Lreg−W ] is contained

in ~B∧Lreg−W .

Proof. Consider the projection ηW : Rreg−W/W → R/W , which is etale. Note that it

restricts to a covering ηW : L → ηW (L) with Galois group Ξ. So the formal neighborhood

(R/W )∧ηW (L) is a quotient by the action of Ξ on the formal neighborhood (Rreg−W/W )∧L.

Recall now that we may sheafify B and H, to get a sheaf B:= SR/W ⊗C[R/W ] B of bi-

modules over H. In particular, the restriction B|(R/W )

∧ηW (L) is a sheaf of bimodules over

H|(R/W )

∧ηW (L) . Note that B∧Lreg−W = C[Rreg−W/W ]∧L ⊗C[R/W ]∧ηW (L) Γ(B|

(R/W )∧ηW (L)), and a

similar equation holds for HLreg−W .

Now note that, since C[R/W ] ⊆ H lives in degree 0, the sheaf B is graded, and the grading

is bounded below. In particular, Γ(B|(R/W )

∧ηW (L)) is a graded Γ(H|(R/W )

∧ηW (L))-bimodule,

with the grading bounded below. Note also that B∧Lreg−W is a graded (H∧Lreg−W ,H)-bimodule

with the grading bounded below.

By the first paragraph of this proof, we can find a free basis f1, . . . , fk of C[Rreg−W/W ]∧L over

C[R/W ]∧ηW (L) and elements aij ∈ C[R/W ]∧ηW (L), i = 1, . . . , k, j = 0, . . . , k − 1, such that

Pi(fi) := fki +ai,k−1fk−1i + · · ·+ai,0 = 0, but P ′i (fi) = kfk−1

i + (k− 1)ai,k−1fk−2i + · · ·+ai,1 is

invertible in C[Rreg−W/W ]∧L. In particular, for every x ∈ B∧Lreg−W , xPi(fi) = 0. Assume, for

a moment, that we have already defined a right multiplication with the properties required

by the statement of the lemma. Then it is easy to see that we must have

0 = [Pi(fi), x] = P ′i (fi)[fi, x] +Qi(fi, x) + F (fi, ai,j, x) (3.4)

where Qi(fi, x) := fk−1i [ai,k−1, x] + · · · + fi[ai,1, x] + [ai,0, x], and F (fi, ai,j, x) is an expres-

sion involving several brackets and products. Note that Qi(fi, x), F (fi, ai,j, x) ∈ ~BLreg−W .

47

Thanks to (3.4), we must have [fi, x] = −P ′i (fi)−1(Qi(fi, x) + F (fi, ai,j, x)) ∈ ~B∧Lreg−W .

This shows uniqueness of the right product map. Existence is shown by induction, starting

from the fact that C[Rreg−W/W ]∧L must commute with the lowest degree component of

B∧Lreg−W .

Note that, actually, B∧Lreg−W becomes a Ξ-equivariant H∧Lreg−W -bimodule, the action ofNW (W )

comes from the adjoint action on B and the action on C[Rreg−W/W ]∧L. Now we recall the

isomorphism (3.3) in its homogeneous version. Since Rreg−W is stable under the action of

NW (W ), the idempotent e(W ) ∈ Z(W,W,HLreg−W ) is NW (W )-invariant. It follows that

e(W )B∧Lreg−W e(W ) is a Ξ-equivariant H(W,R)∧Lreg−W -bimodule. According to [L7], there ex-

ists a, unique up to isomorphism, Ξ-equivariant H-bimodule B† such that

e(W )B∧Lreg−W e(W ) = C[L ×RW/W ]∧L ⊗C[L×RW /W ] (D~(L)⊗C[~] B†)

Let us be more precise. First, note that we have an isomorphism H(W,R)∧Lreg−W = D~(L)⊗C[~]

H∧0 that is induced from the natural isomorphism H(W,R) = D~(RW ) ⊗C[~] H. Define a

functor G : HCΞ(H) → HCΞ(H(W,R)∧Lreg−W ) by G(B) := C[L × Rreg−W/W ]∧L ⊗C[L×RW /W ]

(D~(L) ⊗C[~] B). Note that G is a full embedding, with left inverse F given by first taking

the centralizer of D~(L) and then taking elements which are locally finite with respect to

the adjoint action of the Euler element. Then, e(W )B∧Lreg−W e(W ) is in the image of G, and

B† := F(e(W )B∧Lreg−W e(W )). Thus, we have.

Theorem 3.4.4 ([L3, L7]). The assignment B 7→ B† gives a functor HC(H)→ HCΞ(H).

Let us now proceed to construct the restriction functors for specialized parameters. Let

B ∈ HC(Hc, Hc′). Find a good filtration for B and consider the Rees bimodule, R~(B). This

is a HC H-bimodule, so we can consider R~(B)†. Then, B† := R~(B)†/(~− 1). According to

[L3, Subsection 3.9], B† does not depend, up to a distinguished isomorphism, on the chosen

good filtration. Thus, we get a functor •† : HC(Hc, Hc′)→ HCΞ(Hc, Hc′).

48

Theorem 3.4.5 (Theorem 3.4.6, [L3]). Let W ⊆ W be a parabolic subgroup. The following

is true.

1. Let b ∈ R be such that Wb = W . Then, the functors •† and F ◦ •†b are isomorphic,

where F : HCΞ(Hc, Hc′)→ HC(Hc, Hc′) is the functor that forgets the Ξ-equivariance.

2. Let L := LW so that, in particular, •†|HCL(Hc,Hc′ )factors through HCL(Hc, Hc′). Then

•† induces an equivalence between HCL(Hc, Hc) and a full subcategory of HCΞ0 (Hc, Hc′)

that is closed under taking subquotients.

3. The functor •† : HC(Hc, Hc′)→ HCΞ(Hc, Hc′) admits a right adjoint •† : HCΞ(Hc, Hc′).

4. There is a functor embedding of •† ◦ F into •†b.

5. For B ∈ HCL(Hc, Hc′), the kernel and cokernel of the adjunction morphism B → (B†)†

belong to HC∂L(Hc, Hc′).

3.4.2 Applications

Semisimplicity of the head

Let us see some applications of the restriction functors. The first one of these tells us that,

if L = L{1} is the dense symplectic leaf, then HCL(Hc, Hc) is semisimple. Let us denote

HC(Hc, Hc) := HCL{1}(Hc, Hc).

Proposition 3.4.6. The category HC(Hc, Hc) is semisimple.

Proof. Here we take the restriction functor for W = {1}. Note that Ξ = W , and Hc = C, so

HCΞ0 (Hc, Hc) is precisely the category of finite dimensional representations of W . So thanks

to Theorem 3.4.5, HC(Hc, Hc) can be embedded as a full subcategory of the category of

representations of W . Moreover, this subcategory is closed under subquotients (= direct

summands) and tensor products. It follows that HC(Hc, Hc) is equivalent to the category

49

of representations of W/N for a normal subgroup N ⊆ W . In particular, it is a semisimple

category.

In Chapter 5, we will find an explicit description of the subgroup N that appears in the proof

of Proposition 3.4.6, see Section 5.4. We will also see, Corollary 5.3.10, that HC(Hc, Hc′) is

semisimple for different parameters c, c′.

Injectivity of the regular bimodule

Recall that the category HC(Hc, Hc) has enough injectives.

Proposition 3.4.7. The regular bimodule Hc is injective in the category of HC Hc-bimodules.

Proof. In view of Proposition 3.3.11, we need to show that Ext(B,Hc) = 0 for any irreducible

HC bimodule B, where Ext denotes Ext1Hc -bimod. We separate in two cases.

Case 1: B has proper support. This case is due to Bezrukavnikov-Losev and it is contained in

an old version of the paper [BL]. We provide a proof for convenience of the reader. Consider

an exact sequence 0 → Hc → X → B → 0. Let L ⊆ (R ⊕ R∗)/W be the open symplectic

leaf, and consider the corresponding restriction functor •†. Note that B† = 0. Since the

restriction functor is exact, we must then have ((Hc)†)† = (X†)

†. We have the adjunction

map X → ((Hc)†)†. The latter bimodule admits a filtration whose associated graded is

contained in ((C[R⊕R∗]#W )†)†. By construction, ((C[R⊕R∗]#W )†)

† is the global sections

of the restriction of C[R ⊕ R∗]#W to L. But the complement of this leaf has codimension

2. Hence, ((C[R ⊕ R∗]#W )†)† = C[R ⊕ R∗]#W , and this implies that ((Hc)†)

† = Hc. Now

the adjunction map X → Hc is a splitting of the exact sequence 0→ Hc → X → B → 0.

Case 2: B has full support. Assume 0 → Hcϕ−→ X −→ B → 0 is an exact sequence. Pick

again the dense symplectic leaf L ⊆ (R⊕R∗)/W and consider the corresponding restriction

functor •†. We have an exact sequence 0 → (Hc)† → X† → B† → 0. Since the category

of Ξ-equivariant Harish-Chandra Hc-bimodules is semisimple, Proposition 3.4.6, this exact

sequence splits, X† = (Hc)† ⊕B†. Now, recall from the previous case that (Hc,†)† = Hc, and

50

that we have the adjunction morphism X → (X†)† = Hc ⊕ (B†)

†. By [L3, Theorem 3.7.3],

the kernel of this morphism is a HC bimodule with proper support, so the morphism must

be injective. Thus, we can consider X ⊆ Hc⊕ (B†)†, and ϕ = (ϕ1, ϕ2), where ϕ1 : Hc → Hc,

ϕ2 : Hc → (B†)†. Let us remark that, since the center of Hc is trivial, every nonzero

endomorphism of Hc is an automorphism. So, if ϕ1 6= 0, we can find a splitting for ϕ. Thus,

we may assume ϕ1 = 0, and ϕ2 : Hc → (B†)† is an inclusion.

Now recall that we have the adjunction morphism B → (B†)†. Since B is irreducible, this is

actually an injection. The cokernel of this morphism is a HC bimodule with proper support,

cf. Theorem 3.4.5. Thus, we must have B ⊆ ϕ2(Hc), so B is isomorphic to an ideal of Hc.

But this implies that B† = (Hc)†. So the exact sequence 0→ Hc → X → B → 0 induces an

inclusion X ⊆ (X†)† = Hc ⊕Hc and, reasoning as in the previous paragraph, we can find a

splitting for ϕ. Thus, Ext(Hc, B) = 0.

51

Chapter 4

Reduction to corank 1.

4.1 Introduction

Let W ⊆ W be a parabolic subgroup, and let L ⊆ (R ⊕ R∗)/W be the corresponding

symplectic leaf. Recall that, if we denote Ξ := NW (W )/W , then we have the restriction

functor

•† : HCL(Hc, Hc′) ↪→ HCΞ0 (Hc, Hc′)

A natural question, then, is to describe the image of this functor or, equivalently, of the

functor •† : HCL(Hc, Hc′) → HCΞ0 (Hc, Hc′). We would like to know, for example, the

number of irreducible objects in the image. In this chapter, we reduce this question to the

case where W sits inside W in corank 1, that is, when codimR(RW ) = 1, equivalently, when

W is a maximal parabolic subgroup of W . More precisely, we prove the following result.

Theorem 4.1.1. Let B ∈ HCΞ0 (Hc, Hc′). Assume that, for every parabolic subgroup W ′ ⊆ W

containing W in corank 1, there exists a HC (H ′c, H′c′)-bimodule BW ′ such that (BW ′)†W ′W

= B.

Here, H ′c = Hc(W′, RW ′) and the NW ′(W )/W -equivariant structure on B is restricted from

the Ξ-equivariant structure. Then, there exists a HC (Hc, Hc′)-bimodule B with B†WW = B.

The proof of Theorem 4.1.1 passes through its homogeneous version, which is not surprising

52

giving the construction of the functor •†. In the first two sections of this chapter we give

preliminary technical results that will go into the proof of Theorem 4.1.1. In particular, we

study supports and annihilators of bimodules over localized Cherednik algebras. Then, we

state and prove a variant of the homogeneous version of Theorem 4.1.1. After that, we use

the Rees construction to get Theorem 4.1.1.

4.2 Technical lemmas

4.2.1 Supports and symplectic leaves

Let W ⊆ W be a parabolic subgroup. Recall the W -equivariant isomorphism

Rreg−W/W ×R/W R∼=−→

⊔w∈W/W

wRreg−W ⊆ W/W ×R (4.1)

Also recall that we denote by X the variety⊔w∈W/W wRreg−W ′ . So Hreg−W = H(W,X ).

Note that X × R∗ = T ∗X is a symplectic algebraic variety, and the action of W on T ∗X is

by symplectomorphisms. So (T ∗X )/W is a Poisson variety. Moreover, it follows from the

isomorphism (4.1) that X/W = Rreg−W/W ′ and (T ∗X )/W = (T ∗Rreg−W )/W . Let us define

a HC Hreg−W -bimodule in a manner completely analogous to Definition 3.1.7. As before,

for a HC Hreg−W -bimodule B, its support SS(B) ⊆ (T ∗Rreg−W )/W is a union of symplectic

leaves.

We can describe the symplectic leaves inside (T ∗Rreg−W )/W using the results in [BrGo,

7.4]. We remark that [BrGo] works with actions on a vector space, but the proofs work

in our setting. Namely, let W ′′ ⊆ W be a parabolic subgroup. Let LWW ′′ := πRreg−W ({x ∈

T ∗Rreg−W : W x = W ′′}), where πRreg−W : T ∗Rreg−W → (T ∗Rreg−W )/W is the quotient

by the W -action. Note that LWW ′′ depends only on the conjugacy class of W ′′ in W . The

symplectic leaves in (T ∗Rreg−W )/W are precisely the sets LWW ′′ , where W ′′ ⊆ W is a parabolic

subgroup. Note that LWW ′′ =⊔W ′′⊆W ′′′⊆W L

WW ′′′ . It follows that the unique closed leaf inside

53

(T ∗Rreg−W )/W is LWW . We remark that LWW = RWreg × (R∗)W , where RW ′

reg := {b ∈ R : Wb =

W}.

Similarly, we can think of Hreg−W as being the rational Cherednik algebra H(W,Rreg−W )

(recall, we are taking all variables ~, c1, . . . , cn even if the defining relations do not involve all

of them). The singular support of a HC Hreg−W -bimodule B is again a union of symplectic

leaves of (T ∗Rreg−W )/W . Let us describe a relation between supports of HC Hreg−W and

Hreg−W -bimodules. Note that C[X ] =⊕

w∈W/W C[wRreg−W ]. We may think of the idempo-

tent e(W ) ∈ Z(W,W,Hreg−W ) ∼= Hreg−W introduced in Section 2.4 as being the primitive

idempotent in C[X ] ⊆ Hreg−W corresponding to the direct summand C[Rreg−W ]. It follows

that for a HC Hreg−W -bimodule B, SS(B) = SS(e(W )Be(W )).

The description of the singular support of a HC bimodule has the following consequence for

the support of the elements of B/cB.

Lemma 4.2.1. Let B be a HC Hreg−W -bimodule. Consider B/cB as a C[X/W ]⊗C[R∗/W ]-

module where, recall, X =⊔w∈W/W wRreg−W , and X/W = Rreg−W/W . Then, for every

nonzero element v ∈ B/cB, its support Xv ⊆ Rreg−W/W ×R∗/W contains RWreg × (R∗)W .

In view of the description of the symplectic leaves inside (X × R∗)/W , Lemma 4.2.1 is a

consequence of the following result.

Lemma 4.2.2. Let A be a commutative, Noetherian Poisson algebra, and let M be a finitely

generated Poisson A-module. Then, for every element m ∈ M , its set-theoretic support

Xm ⊆ Spec(A) is a Poisson subvariety.

Proof. First of all, let I ⊆ A be any ideal. For k ≥ 0, let MIk := {n ∈ M : Ikn = 0}. Note

that MIk ⊆ MIk+1 , so that M(I) :=⋃k≥0MIk is a submodule of M . We claim that it is a

Poisson submodule. Takem ∈MIk and a ∈ A. Let a1, . . . , a2k ∈ I, so that a1 · · · a2km = 0. It

follows that 0 = {a, a1 . . . a2km} = a1 · · · a2k{a,m}+{a, a1 · · · a2k}m. Thanks to the Leibiniz

identity again, {a, a1 · · · a2k} = a1 · · · ak{a, ak+1 · · · a2k}+ak+1 · · · a2k{a, a1 · · · ak} ∈ Ik. Since

54

m ∈MIk , this implies that a1 · · · a2k{a,m} = 0. Thus, {A,m} ⊆MI2k . So M(I) is a Poisson

submodule and thus supp(M(I)) ⊆ Spec(A) is a Poisson subvariety.

Now specialize to the case where I = AnnA(m). Since A is Noetherian and M is finitely

generated, M(I) = MIk for some k > 0. Then, Ik ⊆ Ann(M(I)). On the other hand, since

m ∈ M(I) and I = AnnA(m), Ann(M(I)) ⊆ I. So√

Ann(M(I)) =√I and the result

follows.

Let us remark that, thanks to the correspondence between supports of HC Hreg−W - and

Hreg−W -bimodules, we get from Lemma 4.2.1 the following result.

Corollary 4.2.3. Let B be a HC Hreg−W -bimodule. Consider B/cB as a C[Rreg−W/W ] ⊗

C[R∗/W ]-module. Then, for every nonzero v ∈ B/cB, its support Xv ⊆ Rreg−W/W ×R∗/W

contains RWreg × (R∗)W .

4.2.2 Annihilators and liftings

We will describe the annihilator of a HC Hreg−W -bimodule as a left C[Rreg−W/W ]-module.

In order to do so, we need the following finiteness result.

Lemma 4.2.4. Let B be a HC Hreg−W -bimodule. Then, B is finitely generated over the al-

gebra C[Rreg−W/W ][c]⊗S(c)C[R∗/W ][c]opp. Similarly, B is finitely generated over the algebra

C[R∗/W ][c]⊗S(c)C[Rreg−W/W ][c]opp, where the superscript opp means that the corresponding

algebra acts on the right.

Proof. Since B is HC, we have that B/cB is a module over C[X×R∗]W , which is the center of

the algebra Hreg−W/cHreg−W . This latter algebra is finite over its center, so B/cB is a finitely

generated module over C[X ×R∗]W . Now, the natural map (X ×R∗)/W −→ X/W ×R∗/W

is finite, so B/cB is a finitely generated module over C[X ]W ⊗ C[R∗]W . Let v1, . . . , vm be

generators of B/cB under the action of C[X ]W⊗C[R∗]W . We can assume that these elements

are homogeneous with respect to the grading on B/cB inherited from the one on B. Let

55

v1, . . . , vm be homogeneous lifts of v1, . . . , vm. It is now standard to see that v1, . . . , vm are

generators of B under the action of C[X ]W [c]⊗S(c) C[R∗]W [c]opp.

Lemma 4.2.5. Let B be a HC Hreg−W -bimodule. Assume that SS(B) = LWW is the minimal

symplectic leaf. Then, as a (left or right) C[Rreg−W/W ]-module, B is annihilated by a power

of the ideal I of functions vanishing on LW ⊆ Rreg−W/W where LW := {x ∈ R : Wx = W}.

Proof. First, we show that any element in B is annihilated by a large enough power of I.

Recall that LWW = RWreg × (R∗)W

′. In particular, LW × 0 ⊆ LWW . It follows by our assumption

on SS(B) that In ⊆ C[Rreg−W/W ] ⊆ C[Rreg−W/W ] ⊗ C[R∗/W ] ⊆ C[X × R∗]W annihilates

B/cB for n� 0. So for any i ∈ Z, InBi ⊆ cBi−1. Now the claim follows because the grading

on B is bounded below.

Now let v1, . . . , vm be generators of B as a C[Rreg−W/W ][c]⊗S(c) C[R∗/W ][c]opp-module, and

let N � 0 be such that INvi = 0 for all i = 1, . . . ,m. It is easy to see that INB = 0. We

are done.

Let us discuss some consequences of Lemma 4.2.5. Recall that we have a natural action of

the group NW (W ) on the algebra Hreg−W by algebra automorphisms, in such a way that

the action of W ⊆ NW (W ) concides with the adjoint action. Recall also that we denote

Ξ = NW (W )/W . The map ηW : Rreg−W/W → R/W is etale and it restricts to a covering

ηW : L → ηW (L) = L/Ξ with Galois group Ξ, where L := LW . This implies that the formal

neighborhood (R/W )∧ηW (L) may be identified with the quotient by the action of Ξ on the

formal neighborhood (Rreg−W/W )∧L. Now let B be a Ξ-equivariant HC Hreg−W -bimodule

supported on LWW . Thanks to Lemma 4.2.5, B may be thought of as a quasi-coherent sheaf

on an infinitesimal neighborhood of L ⊆ Rreg−W/W . Thus, the space of invariants BΞ is a

quasi-coherent sheaf on an infinitesimal neighborhood of ηW (L) ⊆ R/W , and we may think

of it as a quasi-coherent sheaf on WRreg−W/W .

We claim that, moreover, B = C[Rreg−W/W ] ⊗C[R/W ] BΞ. Recall that the map ηW :

(Rreg−W/W )∧L → (WRreg−W/W )∧ηW (L) is the quotient by the free Ξ-action on the for-

56

mal neighborhood of L, and C[WRreg−W/W ]∧ηW (L) may be identified with the algebra of

Ξ invariants in C[Rreg−W/W ]∧L. So the desired equality will follow if we show that the

right-hand side is equal to C[Rreg−W/W ]∧L⊗C[WRreg−W /W ]∧ηW (L) BΞ. But this is clear by our

description of the annihilator of B (and of BΞ).

4.3 Main result for homogeneous algebras

We are now ready to state our main result. Let W be a parabolic subgroup of W , and let B

be a Ξ = NW (W )/W -equivariant HC Hreg−W -bimodule. We require that SS(B) = LWW , the

minimal symplectic leaf in Rreg−W/W . In particular, this implies thanks to Lemma 4.2.5 that

B may be considered as a HC H∧Lreg−W -bimodule. Recall that we denote ηW : Rreg−W/W →

R/W . From the previous subsection it follows that ηW∗(BΞ) is an H-bimodule satisfying

(ηW∗(BΞ))∧Lreg−W = B. However, ηW∗(B

Ξ) need not be finitely generated over H so it is not,

in general, a HC H-bimodule. Similarly, if W ′ is a parabolic subgroup of W containing W ,

(ηW∗(BΞ))∧LW ′reg−W ′ does not need to be a HC H

∧LW ′reg−W ′-bimodule. However, we can further

localize to the punctured formal neighborhood L×W ′ . The algebra of functions of LW ′ is a

localization of the algebra C[Rreg−W ′/W ′]∧LW ′ at a W ′-invariant subset, and so we have the

algebra HLW ′ := C[LW ′ ] ⊗C[Rreg−W ′/W ′]∧LW ′ H∧LW”

reg−W ′ . Similarly, we can form the bimodule

(ηW∗(BΞ))L×W ′

, which is now a HC HL×W ′-bimodule.

Theorem 4.3.1. Let B be a Ξ-equivariant HC Hreg−W -bimodule. Assume that SS(B) = LWW

and that for all parabolic subgroups W ′ with W ⊆ W ′ in corank 1, there is a HC H∧LW ′reg−W ′-

bimodule BW ′ whose localization to L×W ′ concides with (ηW∗(B

Ξ))L×W ′. Then, there exists a

HC H-bimodule B such that B = Breg−W .

The proof of Theorem 4.3.1 is inspired by [L5, Section 3], where a similar result is shown at

the level of category O (for the stratum corresponding to the dense symplectic leaf.) The

strategy is as follows. We will define a bimodule that is coherent over H|U , where U is

57

an open subset in R/W whose complement has codimension 2. Then, we can take global

sections. This open subset will be the image in R/W of

Rsr−W :=⋃

W⊆W ′in corank 1

Rreg−W ′ .

It is clear that the complement of Rsr−W in R has codimension 2. Moreover, RW ∩Rsr−W is

an open subset of RW whose complement has codimension at least 2. Indeed, we have

RW ∩Rsr−W = RW \⋃

s,s′ 6∈WΓs∩RW 6=Γs′∩RW

Γs ∩ Γs′ . (4.2)

The way to get a desired bimodule is as follows. First, for each parabolic subgroup W ′

containing W in corank 1, we will construct a HC Hreg−W ′-bimodule with the property that

its lift to Rreg−W/W coincides with B. Then we will get our bimodule by, roughly speaking,

glueing the bimodules defined over Rreg−W ′/W ′.

Proof of Theorem 4.3.1. Part 1: Constructing HC Hreg−W ′-bimodules. For each parabolic

subgroup W ′ containing W in corank 1, let BW ′ be a HC H∧LW ′reg−W ′-bimodule that localizes

to (ηW∗(BΞ))L×W ′

. Note that we may assume that BW ′ ⊆ ηW∗(BΞ)L×W ′

, if this is not the case

we can just replace BW ′ by its quotient by the maximal sub-bimodule that is killed by the

localization, we can find such a sub-bimodule because BW ′ is a finitely generated bimodule

over the noetherian algebra H∧LW ′reg−W ′ .

On the other hand, let ηW ′ : Rreg−W ′/W ′ → R/W be the natural projection, and consider

η∗W ′(ηW∗(BΞ)) = C[Rreg−W ′/W ′] ⊗C[R/W ] BΞ. The inclusion C[Rreg−W ′/W ′] ↪→ C[L

×W ′ ] in-

duces a map η∗W ′ηW∗(BΞ)→ ηW∗(B

Ξ)L×W ′that we claim to be injective. Indeed, this follows

because inside (R ⊕ R∗)/W we have LW ′ ⊆ LW and the singular support of every finitely

generated H-sub-bimodule of ηW∗(BΞ) (which is the union of its HC sub-bimodules) con-

tains LW . The claim is now a consequence of the fact that LWW is the minimal symplectic

leaf inside T ∗XW/W , cf. Subsection 4.2.1.

58

Define BW ′ := (η∗W ′(ηW∗(BΞ)))∩BW ′ ⊆ ηW∗(B

Ξ)L×W ′. Note that this is an Hreg−W ′-bimodule.

Let us see that it is finitely generated. By a suitable straightforward adaptation of Lemma

4.2.4, ηW∗(BΞ)L×W ′

is finitely generated over the algebra C[L×W ′ ][c] ⊗S(c) C[R∗]W [c]opp. Note

that BW ′ is a C[LW ′ ][c]⊗S(c)C[R∗]W [c]opp-lattice inside of (ηW∗(BΞ))L×W ′

. So what we need to

show is that η∗W ′(ηW∗(BΞ))∩L is finitely generated over C[Rreg−W ′/W ′][c]⊗S(c)C[R∗/W ][c]opp

for some lattice L. We can produce such a lattice as follows. Again thanks to Lemma

4.2.4, B is finitely generated over C[Rreg−W/W ][c] ⊗S(c) C[R∗/W ][c]opp, so we have an epi-

morphism Υ : (C[Rreg−W/W ][c] ⊗S(c) C[R∗/W ][c]opp)⊕n → B, which in turn induces an

epimorphism Υ : (C[L×W ′ ][c] ⊗S(c) C[R∗/W ][c]opp)⊕n → ηW∗(B

Ξ)L×W ′. We take L to be the

image of the restriction of Υ to (C[LW ′ ][c] ⊗S(c) C[R∗/W ][c]opp)⊕n. This is clearly a lat-

tice. Since C[Rreg−W ′/W ′] ∩ C[LW ′ ] = C[Rreg−W ′/W ′] we have that L ∩ η∗W ′(ηW∗(BΞ))

coincides with the intersection of η∗W ′(ηW∗(BΞ)) with the image of the restriction of Υ to

(C[Rreg−W ′/W ′][c] ⊗S(c) C[R∗/W ][c]opp)⊕n. So L ∩ η∗W ′(ηW∗(BΞ)) is finitely generated over

C[Rreg−W ′/W ′][c] ⊗S(c) C[R∗/W ][c]opp. Note that it follows that BW ′ is a HC Hreg−W ′-

bimodule with SS(BW ′) = LW ′W .

It remains to show that (BW ′)reg−W = B. Since B = ηW∗(BΞ)reg−W = (ηW∗(B

Ξ)reg−W ′)reg−W

it is enough to check that the lift of BW ′ toRreg−W/W coincides with that of (ηW∗(B

Ξ))reg−W ′ .

By definition, BW ′ ⊆ (ηW∗(BΞ))reg−W ′ ⊆ (ηW∗(B

Ξ))L×W ′. Now, for every b ∈ (ηW∗(B

Ξ))L×W ′

there exists f ∈ C[LW ′ ] vanishing on LW ′ with fb ∈ BW ′ . If, moreover, b ∈ (ηW∗(BΞ))reg−W ′

then f ∈ C[Rreg−W ′/W ′] and fb ∈ BW ′ . This implies the desired result.

Part 2: Glueing. First we will define a sheaf on R, then we will take W -invariants to pass

to R/W . For each parabolic subgroup W ′ containing W in corank 1, let πW ′ : Rreg−W ′ →

Rreg−W ′/W ′ be the projection, and ιW ′ : Rreg−W ′ → R the inclusion. So we can consider

ιW ′∗π∗W ′BW ′ . We will take the intersection of these sheaves, so we need to find a sheaf

containing all of them. Since Rreg−W ′ ⊆ Rsr−W for all W ′, this will be a sheaf defined on

59

Rsr−W . So let π : R → R/W and ι : Rsr−W → R be the natural projection and inclusion,

respectively. By construction, viewing ηW∗(BΞ) as a quasicoherent sheaf on R/W , we may

think of ιW ′∗π∗W ′BW ′ as being contained inside of ι∗π

∗ηW∗(BΞ). So the intersection

B :=⋂

W⊆W ′in corank 1

ιW ′∗π∗W ′BW ′ .

makes sense and is a sheaf on Rsr−W .

Note that W acts naturally on ι∗π∗ηW∗(B

Ξ). Now notice that, for a parabolic subgroup W ′ ⊆

W and w ∈ W , we have a canonical, graded isomorphism Hreg−W ′ ∼= Hreg−wW ′w−1 . Indeed,

recall that Hreg−W ′ is the rational Cherednik algebra for the action of W on XW ′ , a disjoint

union of Zariski open subsets of R, cf. Subsection 4.2.1. It is clear that XW ′ = XwW ′w−1

and the isomorphism between the algebras follows. So tracing back the construction, we see

that we can pick our bimodules BLW ′ in such a way that, for w ∈ W , w(ιW ′∗π∗W ′BW ′) =

ιwW ′w−1∗π∗wW ′w−1BwW ′w−1 . So B is W -stable. Finally, define

B := (π∗B)W ,

where π : R → R/W is the projection. We claim that B is stable under the bimodule

action of H|π(Rsr−W ). To see this first note that, by definition, B = π∗B ∩ ∗BΞ, where

: Rreg−W → R/W is the projection. Each one of the bimodules on the right-hand side of

the previous equality is stable under the (left or right) action of H|π(Rsr−W ). So B is also

H|π(Rsr−W )-stable.

Now set B := Γ(π(Rsr−W ), B). We have that B is a H-bimodule. We claim that it is HC.

First of all, since B ⊆ ∗BΞ, we have that B is C[~]-flat. It is also clear that B/~B is a

module (rather than a bimodule) over Z(H/~H) and that B is graded. So, to finish the

claim that B is HC, we need to show that it is finitely generated. We will show that, in fact,

B/cB is finitely generated over the algebra C[R/W ] ⊗ C[R∗/W ] ⊆ C[(R ⊕ R∗)/W ]. The

following is an easy consequence of [Wi, Lemma 3.6].

60

Lemma 4.3.2. Let X be an affine Noetherian scheme, and let U ⊆ X be an open subset

of X whose complement has codimension at least 2. Let M be a coherent sheaf on U , and

assume that the support of any global section m ∈ Γ(U,M) contains an irreducible component

of U . Then, Γ(U,M) is finitely generated over C[X].

Proof. By [Wi, Lemma 3.6], we get that Γ(U,M)/IΓ(U,M) is finitely generated over the

algebra C[X]/IC[X], where I is the nilradical of C[X]. The result follows.

Note that we can look at B/cB as a coherent sheaf on an infinitesimal neighborhood U

of π(Rsr−W ∩ RW ) × R∗/W , this follows from our assumptions on the singular support

of B, SS(B) = LWW , the construction of B ⊆ ∗BΞ and Lemma 4.2.5. This infinitesimal

neighborhood may be regarded as an open set inside an infinitesimal neighborhood X of

π(Rsr−W )×R∗/W . Now, it follows from Lemma 4.2.1 that the support of any global section

m ∈ Γ(U, B/cB) contains π(RWreg) ×W (R∗)W/W . Then, it follows from Lemma 4.3.2 that

B/cB is finitely generated over C[X]. In particular, it is finitely generated over C[R/W ]⊗

C[R∗/W ], this follows because the codimension of the complement of RW−sr in R is 2. Then,

B is a HC H-bimodule. By construction, Breg−W = B. This finishes the proof of Theorem

4.3.1. �

Let us remark one important feature of the bimodule B we have constructed: it has no

sub-bimodules whose singular support is properly contained inside LW . Indeed, this follows

from Corollary 4.2.3 and the fact that B ⊆ BΞ.

4.4 Specializing parameters.

Let c, c′ : S → C be conjugation invariant functions. Recall that R~(Hc), R~(Hc′) are quo-

tients of H and that, if B is a HC Hc-Hc′-bimodule with a good filtration, then R~(B)

is a HC H-bimodule. An analogous result holds for HC (Hc,reg−W , Hc′,reg−W )-bimodules.

We remark that, since R∗ is in degree 0, the Rees construction commutes with localiza-

61

tion: for an affine, open subset U ⊆ R/W , R~(B)|U = R~(B|U). We also remark that

the Bezrukavnikov-Etingof isomorphisms hold in the specialized setting. So we can take

a Ξ-equivariant HC (Hc,reg−W , Hc′,reg−W )-bimodule B such that SS(B) = LWW and for ev-

ery parabolic subgroup W ′ containing W in corank 1, the (Hc(W′, R)L×W ′

, Hc′(W′, R)L×W ′

)-

bimodule BΞ

L×W ′is the localization of a HC Hc(W

′, R)LW ′ -Hc′(W′, R)LW ′ -bimodule. By The-

orem 4.3.1, we can find a HC H-bimodule B that lifts to R~(B). Since BLW ′ is a bimodule

over R~(Hc(W′, R)LW ′ )- R~(Hc′(W

′, R)LW ′ ), we see that the bimodule B factors through

R~(Hc)- R~(Hc′), so B/(~ − 1)B is a HC Hc-Hc′-bimodule that lifts to B. We summarize

this discussion in the following theorem, which is a specialized version of Theorem 4.3.1.

Theorem 4.4.1. Let W be a parabolic subgroup of W , and let B be a Ξ-equivariant HC

Hc,reg−W -Hc′,reg−W -bimodule, where c, c′ ∈ C[S] are conjugation invariant functions. Assume

that SS(B) = LWW and that for all minimal parabolic subgroups W ′ ⊆ W containing W , the

bimodule (ηW∗(BΞ))L×W ′

is the localization of a HC Hc(W′, R)LW ′ -Hc′(W

′, R)LW ′ -bimodule.

Then, there exists a HC Hc-Hc′-bimodule B such that Breg−W = B.

Proof of Theorem 4.1.1. Recall the functor G defined in Section 3.4.1. Since B is finite

dimensional, SS(G(R~(B))) = LWW . Our assumptions on B imply, since G is a fully faithful

embedding, that G(R~(B)) satisfies the conditions of Theorem 4.4.1. So we can find a HC H-

bimodule B with Breg−W = G(R~(B)). Note that since SS(G(R~(B))) = LWW , (Breg−W )L =

Breg−W . Thus, by the construction of the restriction functor, B†WW = R~B. It remains to

put B := B/(~− 1)B. �

62

Chapter 5

Bimodules with full support.

5.1 Introduction

In this chapter, we give a description of the category HC(Hc, Hc′). Recall that this is the

quotient of the category of all HC (Hc, Hc′)-bimodules by the sull subcategory consisting of

bimodules with proper support. Let us remark that, when W is a Coxeter group and c ∈ pZ

then, [BEG2], we have

HC(Hc, Hc′) ∼=

W -rep, c′ ∈ pZ

0 else

Let us remark that, in this case, HC = HC, and that the above equivalence is an equivalence

of monoidal categories when c = c′. The construction of [BEG2] is based on the study of the

module of c-quasi-invariants when c is integral. This notion can be generalized to the case

of an arbitrary complex reflection group, see [BC], but it only makes sense for parameters in

pZ. Here, we take a different approach and use bimodules of locally finite maps and the KZ

functor. We generalize the result of [BEG2] by constructing a normal subgroup Wc ⊆ W

satisfying the following properties.

1. Wc = {1} if and only if Hq = CW .

63

2. Wc = W for c outside of a countable collection of hyperplanes.

3. Wc = Wc′ provided Hq = Hq′ .

Let us remark that, if W is a Coxeter group, then the condition Hq = CW simply means

that c ∈ pZ. For a more general complex reflection group, we have to take into consideration

a certain symmetric group action, the Namikawa-Weyl group of W . We will elaborate on

this on Section 5.3.

Theorem 5.1.1. For any c ∈ p, we have an equivalence of monoidal categories HC(Hc, Hc) ∼=

(W/Wc) -rep.

We also provide a description of the category HC(Hc, Hc′) when c 6= c′. In fact, in the case

when this category is nonzero, we have HC(Hc′ , Hc′) ∼= HC(Hc, Hc′) ∼= HC(Hc, Hc). The

Namikawa-Weyl group acts on the space of parameters, and we have the following result.

Theorem 5.1.2. The category HC(Hc, Hc′) is nonzero if and only if we can get from c to c′

by integral translations and the action of elements of the Namikawa-Weyl group. Moreover, if

this is the case, then the categories HC(Hc, Hc′), HC(Hc′ , Hc), HC(Hc, Hc) and HC(Hc′ , Hc′)

are all equivalent.

This chapter is structured as follows. In Section 5.2, we study fully supported Harish-

Chandra bimodules via the KZ functor. In particular, we obtain a description of irreducible

HC bimodules with full support by means of bimodules of locally finite maps, and obtain

some restrictions on when can this be nonzero. We remark that the main result of this section,

Lemma 5.2.7, has already appeared in the literature, with basically the same proof, on a

slightly less general version, see [Sp]. In Section 5.3 we construct an action of a product of

symmetric groups on the space p that has several good properties, in particular, it preserves

categories of HC bimodules. To define this action we need the theory of finite W-algebras,

and this section includes a few facts about it. In Section 5.3 we also prove Theorem 5.1.2.

Finally, in Section 5.4 we construct the group Wc and prove Theorem 5.1.1.

64

5.2 Localization of Harish-Chandra bimodules

5.2.1 Bimodules of differential maps

We are going to use localization to the regular locus to relate the (Hc, Hc′)-bimodules of

locally finite maps with the D(Rreg)/W -bimodules of differential maps. So let us recall some

basic facts about the latter.

First, we recall Grothendieck’s definition of differential operators. Let X be a smooth,

affine algebraic variety. If M and N are C[X]-modules, then the space of C[X]-differential

operators is a subspace of HomC(M,N), defined via an increasing filtration Diff(M,N) =⋃n≥0 Diff(M,N)n, where the components Diff(M,N)n are inductively defined as follows:

Diff(M,N)−1 := 0,

Diff(M,N)n+1 := {f ∈ HomC(M,N) : [a, f ] ∈ Diff(M,N)n for all a ∈ C[X]}.

If M,N are D(X)-modules, then Diff(M,N) is a D(X)-bimodule. We remark that if N is a

local system then we have a D(X)-bimodule isomorphism, N ⊗C[X] D(X) ∼= Diff(C[X], N),

where C[X] is equipped with the trivial flat connection and the flat connection on N ⊗C[X]

D(X) is as in [HTT, Proposition 1.2.9]. An explicit isomorphism is given by n ⊗C[X] d 7→

(f 7→ d(f)n). Note that this implies that Diff(C[X], N) is finitely generated both as a right

and as a left D(X)-module whenever N is a local system. As a right D-module, an explicit

set of generators is n1 ⊗C[X] 1, . . . , ni ⊗C[X] 1, where n1, . . . , ni are generators of the C[X]-

module N . This set also generates N ⊗C[X] D(X) as a left D(X)-module. The following

lemma describes a basic property of the bimodules of the form Diff(C[X], N) whenever N is

a local system.

Lemma 5.2.1. Let N be an irreducible local system on the smooth, affine algebraic variety

X. Then, the D(X)-bimodule Diff(C[X], N) is irreducible.

65

5.2.2 Harish-Chandra bimodules and bimodules of differential maps

Recall that, if B is an irreducible HC (Hc, Hc′)-bimodule, then there exist irreducible modules

M ∈ Oc′ , N ∈ Oc, such that B can be embedded into Homfin(M,N). Moreover, the supports

of M and N coincide and, if supp(M) = supp(N) = XW ′ for some parabolic subgroup

W ′ ⊆ W then SS(B) = LW ′ . When SS(B) = (R⊕R∗)/W = L{1}, we see that the choice of

M can be arbitrary.

Lemma 5.2.2. Let B ∈ HC(Hc, Hc′) be an irreducible bimodule. Assume that SS(B) =

(R ⊕ R∗)/W . Let M ∈ Oc′ be a (not necessarily irreducible) module with full support.

Then, there exists N ∈ Oc irreducible with full support, such that B can be embedded into

Homfin(M,N).

Proof. It is enough to check that B ⊗Hc′ M 6= 0. We have that ResW{1}(B ⊗Hc′ M) =

B†W{1} ⊗C ResW{1}(M) 6= 0, since the rational Cherednik algebra of the group {1} is simply C.

We are done.

In particular, for M we can take the polynomial representation ∆c′(triv). We remark that

∆c′(triv) has an irreducible socle, say Sc′(W ), and this is the unique subquotient of ∆c′(triv)

that has full support. These claims follow from the fact that KZ(∆c′(triv)) = C, the trivial

representation of the Hecke algebra Hq′ . In particular, e∆c′(triv)[δ−1] = C[Rreg/W ] is the

trivial local system where, recall, δ :=∏

s∈S αs and e ∈ CW is the trivial idempotent.

Note that for an irreducible module N ∈ Oc, we have that eHomfin(∆c′(triv), N)[δ−1]e is a

D(Rreg/W )-bimodule. We claim that this bimodule is isomorphic to Diff(C[Rreg/W ], eN [δ−1])

whenever the former bimodule is nonzero. This claim follows from Lemma 5.2.1 and the fol-

lowing result.

Lemma 5.2.3. Let N ∈ Oc be an irreducible module with full support. For any standard

module ∆c′(τ), the bimodule eHomfin(∆c′(τ), N)[δ−1]e is isomorphic to a sub-bimodule of

Diff(e∆c′(τ)[δ−1], eN [δ−1]).

66

Proof. Set M := ∆c′(τ). Let f ∈ Homfin(M,N). Since for some m, δm is W -invariant, we

have that (ad(eδm))kf = 0, so for every x ∈M ,

k∑j=0

(−1)j(kj

)(eδm)(k−j)f((eδm)jx) = 0.

Then, since M is free as a C[R]-module we can extend f to eM [δ−1] by

f(δ−mx) = −(eδm)−kk∑j=1

(−1)j(kj

)(eδm)(k−j)f((eδm)(j−1)x).

To see that this actually defines an inclusion, assume that f 6= 0. Then, f(x) 6= 0 for some

element x ∈ M . Since N is torsion-free (see e.g. [GGOR, Proposition 5.21]), the element

f(x) is not a zero divisor. This implies that the image of f in Diff(e∆c′(τ)[δ−1], eN [δ−1]) is

nonzero.

Corollary 5.2.4. Let N ∈ Oc be an irreducible bimodule with full support. Assume that

Homfin(∆c′(triv), N) 6= 0. Then, eHomfin(∆c′(triv), N)[δ−1]e is the bimodule of differential

maps Diff(C[Rreg/W ], eN [δ−1]).

Using the ideas in the proof of Lemma 5.2.3, together with the fact that the regular bimodule

Hc is injective in the category of HC bimodules, cf. Proposition 3.4.7, we prove the following

result.

Corollary 5.2.5. For any c ∈ p, we have an isomorphism of the regular bimodule Hc with

the space of locally finite maps Homfin(∆c(triv),∆c(triv)).

Proof. Reasoning as in the proof of Lemma 5.2.3, we have that Homfin(∆c(triv),∆c(triv))

is a HC Hc-bimodule whose localization to Rreg is an irreducible D(Rreg)#W -bimodule. So

Homfin(∆c(triv),∆c(triv)) contains a unique irreducible bimodule with full support. It is

easy to see that any subbimodule of Homfin(∆c(triv),∆c(triv)) has full support, so this bi-

module has an irreducible socle. In particular, it is indecomposable.

On the other hand, we have a natural map Hc → Homfin(∆c(triv),∆c(triv)), x 7→ (m 7→

67

xm). Since the representation ∆c(triv) is faithful, this is an inclusion. Then, by Proposition

3.4.7, we have thatHc must be isomorphic to a direct summand of Homfin(∆c(triv),∆c(triv)).

By the previous paragraph, we must have Hc∼= Homfin(∆c(triv),∆c(triv)).

Remark 5.2.6. We remark that the isomorphism in Corollary 5.2.5 is also an algebra

isomorphism with respect to the composition structure on Homfin(∆c(triv),∆c(triv)). This

generalizes [BEG, Proposition 8.10 (i)].

5.2.3 Harish-Chandra bimodules and the KZ functor.

Since for any HC (Hc, Hc′)-bimodule B and any module M ∈ Oc′ , the module B ⊗Hc′ M is

a module in category Oc, it makes sense to ask what is the image of a module of the form

B⊗Hc′M under the KZ functor. In this subsection, we answer this question when B has the

form Homfin(∆c′(triv),M) for an irreducible module with full support M ∈ Oc. Namely, we

have the following result.

Lemma 5.2.7. Let c, c′ ∈ pZ be parameters and consider the rational Cherednik algebras

Hc, Hc′. Let q, q′ be the associated sets of parameters for the Hecke algebras Hq,Hq′, so that

we have KZc : Oc → Hq -mod, KZc′ : Oc′ → Hq′ -mod. Let M ∈ Oc be an irreducible module

with full support. Assume that Homfin(∆c′(triv),M) 6= 0. Then, for every finite dimensional

module N ∈ Hq′ -mod, the π1(Rreg/W )-module KZc(M)⊗C N factors through Hq.

Proof. We show that for every N ∈ Oc′ :

KZc(Homfin(∆c′(triv),M)⊗Hc′ N) = KZc(M)⊗C KZc′(N).

Since Hq′ -mod is a quotient of Oc′ via the KZ functor, this implies the result. Now, by

Lemma 5.2.3 the localization to Rreg/W (this means, first localize to Rreg and then take W -

invariants) of Homfin(∆c′(triv),M)⊗Hc′ N is Diff(C[Rreg/W ], eM [δ−1])⊗D(Rreg/W ) eN [δ−1].

Since eM [δ−1] is a local system on Rreg/W , Diff(C[Rreg/W ], eM [δ−1]) ∼= eM [δ−1]⊗C[Rreg/W ]

68

D(Rreg/W ). Then,

e(Homfin(∆c′(triv),M)⊗Hc′ N)[δ−1] = eM [δ−1]⊗C[Rreg/W ] D(Rreg/W )⊗D(Rreg/W ) eN [δ−1]

= eM [δ−1]⊗C[Rreg/W ] eN [δ−1].

By [HTT, Proposition 4.7.8], the flat sections (eM [δ−1] ⊗C[Rreg/W ] eN [δ−1])∇ coincide with

(eM [δ−1])∇⊗C (eN [δ−1])∇, with diagonal action of the braid group π1(Rreg/W ). The lemma

is proved.

Remark 5.2.8. Recall that ∆c′(triv) has an irreducible socle, which we denote by Sc′(W ).

Moreover KZ(Sc′(W )) = KZ(∆c′(triv)). Then, Lemma 5.2.7 holds, with the same proof, if

we substitute ∆c′(triv) by Sc′(W ). We will mostly use this form of the lemma.

Lemma 5.2.7 gives necessary conditions on a module N (or, rather, on KZc(N)) for the

existence of a nonzero locally finite map in Homfin(∆c′(triv), N). We will see that these

conditions are also sufficient. The proof of this is based on the shift bimodules of Section

3.1.1 and the action of a certain product of symmetric groups on the space of parameters p.

In the next section, we introduce this action.

5.3 The Namikawa-Weyl group

5.3.1 A reparametrization of Hc

In the sequel, a reparametrization of the rational Cherednik algebra Hc by parameters which

are more amenabe with the KZ functor will be convenient. Let us, first, recall some notation.

By A we denote the set of reflection hyperplanes in R, and for each reflection hyperplane Γ

we denote by WΓ its stablizer, this is a cyclic group of order `Γ. Let sΓ ∈ WΓ be a generator

with det(sΓ|R) = exp(2π√−1/`Γ) =: ηΓ. We may and will assume that αs = αsΓ , α∨s = α∨sΓ

for every s ∈ WΓ ∩ S. We will denote these elements by αΓ ∈ R∗, α∨Γ ∈ R, respectively.

69

Now, for each i = 0, . . . , `Γ − 1 we have the idempotent

ei,Γ :=1

`Γ

`Γ−1∑j=0

η−ijΓ sjΓ ∈ CWΓ

so that, for example, e0,Γ ∈ CWΓ is the trivial idempotent. For each reflection hyperplane

Γ ∈ A, pick a collection of numbers kΓ,0, . . . , kΓ,`Γ−1 such that kΓ,i = kΓ′,i for every i =

0, . . . , `Γ − 1 = `Γ′ − 1 if Γ,Γ′ are in the same W -orbit. Then define the algebra Hk by

generators and relations similar to (2.1), with the last relation replaced by

[y, x] = 〈y, x〉 − 1

2

∑Γ∈A

〈y, αΓ〉〈α∨Γ, x〉`Γ−1∑i=0

(kΓ,i − kΓ,i−1)ei,Γ (5.1)

where kΓ,−1 := kΓ,`Γ−1. We remark that Hk = Hc, where the parameter c : S → C is

recovered from k as follows. For each reflection s ∈ S, let Γs be the reflection hyperplane of

s, that is, Γs = ker(idR−s). Then,

c(s) =1

2`Γs

`Γs−1∑j=0

(kΓs,j − kΓs,j−1)λ−js (5.2)

Note that the kΓ,i are only defined up to a common summand. In this work, we will always

assume that kΓ,0 = 0. In this case, we can recover the parameter k from the parameter c via

(2.2), that is,

kΓ,i =∑

s∈S∩WΓ

2c(s)

1− λs(λ−is − 1), i = 0, . . . , `Γ − 1.

We will still denote the space of k-parameters by p. In particular, we have the notion

of integral parameters: a parameter k is integral if the corresponding c-parameter is, or

equivalently, if kΓ,i/`Γ ∈ Z for every Γ ∈ A, i = 0, . . . , `Γ − 1, cf. Section 2.3.5.

Remark 5.3.1. Note that for k, k′ ∈ p we have k − k′ ∈ pZ if and only if, for each Γ ∈ A

and i = 0, . . . , `Γ − 1, we have

q(k)Γ,i = exp

(2π√−1(kΓ,i − i)`Γ

)= exp

(2π√−1(k′Γ,i − i)`Γ

)= q(k′)Γ,i

Below, we will extensively use this without further mention.

70

For the rest of this section, we will work with the k-parameters. The reason for this will

become clear in Section 5.3.2, cf. Theorem 5.3.2.

5.3.2 Finite W-algebras and rational Cherednik algebras of cyclic

groups

Consider the cyclic group W = Z/`Z with generator s acting on R = C by multiplication

by exp(2π√−1/`). For each collection of complex numbers k0 = 0, k1, . . . , k`−1 we have the

rational Cherednik algebra Hk, and its sperical subalgebra Ak = e0Hke0.

It turns out that the spherical subalgebra Ak is isomorphic to a finite W-algebra. We are

not going to give a full definition of a finite W-algebra here, we are simply going to state

its main properties that will be important for us. For more on W-algebras, the reader can

consult the surveys [L, Wa].

A finite W-algebra is a filtered associative algebra U(g, e) associated to a complex semisimple

Lie algebra g and a nilpotent element e ∈ g. When e = 0, then U(g, e) = U(g), the universal

enveloping algebra of g. On the other hand, when e ∈ g is a regular nilpotent, U(g, e) = z,

the center of U(g) which, by the Harish-Chandra isomorphism, is isomorphic to the algebra

of W (g)-invariant functions on the Cartan subalgebra, C[h]W (g), where W (g) denotes the

Weyl group of g. In general, U(g, e) falls between these cases: it is an associative algebra

whose center coincides with z. In particular, for λ ∈ h we can consider the central reduction

U(g, e)λ = U(g, e)/Iλ U(g, e), where Iλ ⊆ z = C[h]W (g) is the ideal of all functions vanishing

on [λ] ∈ h/W (g). Obviously, U(g, e)λ = U(g, e)w(λ) for every w ∈ W (g).

Now let g = sl`, the Lie algebra of traceless (` × `)-matrices. Let e := e(`−1,1) ∈ sl` be the

subregular nipotent, that is, the nilpotent element that has a Jordan block of size (`−1) and a

Jordan block of size 1. Identify the Cartan subalgebra h with {(x1, . . . , x`) ∈ C` :∑xi = 0}.

Let αi := εi − εi+1, i = 1, . . . , ` − 1 be the simple roots, where εi is the i-th coordinate

function, and let π1, . . . , π`−1 be the corresponding fundamental weights.

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Theorem 5.3.2 ([L2]). The algebra Ak is isomorphic, as a filtered algebra, to the algebra

U(sl`, e(`−1,1))λ where

λ =`−1∑i=1

λiπi, λi =1

`(1− ki + ki−1) (5.3)

We denote this isomorphism by ϕk : Ak → U(sl`, e(`−1,1))λ.

A consequence of Theorem 5.3.2 is that the symmetric group S` acts on the space of param-

eters k, in such a way that there is a filtered isomorphism between Ak and Aσ(k) for every

σ ∈ S`. Indeed, we can just define σ(k) to be the parameter associated to σ(λ) and we have

the filtered isomorphism

Ak U(sl`, e(`−1,1))λ U(sl`, e(`−1,1))σ(λ) Aσ(k)ϕk

ϕ−1σ(k)

Since these are filtered isomorphisms, an easy consequence of the definitions is the following

result.

Lemma 5.3.3. Let W = Z/`Z be a cyclic group, with reflection representation R. Then,

for every σ ∈ S` transfer of the structure gives equivalences of categories

HC(Ak, Ak) ∼= HC(Ak, Aσ(k)) ∼= HC(Aσ(k), Aσ(k)) ∼= HC(Aσ(k), Ak)

which preserve the support of a bimodule.

Example 5.3.4. Let us consider the group W = Z/2Z. In this case, we may think of the

parameter k as a single complex number. Then, the associated parameter for the finite W-

algebra is (1 − k)ρ/2, where ρ = α/2 and α is the positive root of the root system A1. Let

σ ∈ Nam be the unique non-trivial element. Then, we have σ(k) = −k + 2. Since in this

case k = −2c, in terms of the c-parameter we have σ(c) = −c− 1.

Note that, in general, the equivalences in Lemma 5.3.3 do not lift to equivalences between

categories of HC Hk-bimodules. They do, however, if we only consider the categories of

bimodules with full support. In this case, we have an equivalence

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HC(Hk, Hk) ∼= HC(Hk, Hσ(k)) ∼= HC(Hσ(k), Hσ(k)) ∼= HC(Hσ(k), Hk) (5.4)

We would like to have similar equivalences for every complex reflection group, not just cyclic

groups. This is what we are going to achieve next. So, first, we need to find a proper

substitute for the group S`.

5.3.3 The Namikawa-Weyl group

Let W be a complex reflection group, R its reflection representation, A the set of reflection

hyperplanes. Remember that for each reflection hyperplane Γ ∈ A, its stabilizer is a cyclic

group, of order say `Γ.

Definition 5.3.5. The Namikawa-Weyl group of W is the group

Nam := Nam(W ) :=∏

Γ∈A/W

S`Γ

Let us remark that Nam(W ) acts on the space of parameters k for the rational Cherednik

algebra Hk(W ). Indeed, for Γ ∈ A, the factor S`Γ acts on the subset {kΓ,0, . . . , kΓ,`Γ−1} as in

the previous subsection, and leaves {kΓ′,0, . . . , kΓ′,`Γ′−1} fixed if Γ and Γ′ are not in the same

W -orbit.

Example 5.3.6. Let W be the Weyl group of an irreducible root system. Then, Nam =

Z/2Z, if the root system is simply-laced, or Z/2Z × Z/2Z, if the root system is of type Bn,

F4 or G2. The action of Nam on the space of parameters is given by Example 5.3.4.

Remark 5.3.7. When W = Sn or, more generally, when W is the wreath-product Sn n

(Z/`Z)n, the variety (R ⊕ R∗)/W admits a symplectic resolution and Nam coincides with

the Namikawa-Weyl group of the symplectic resolution, see [BPW, Section 2.2]. For general

W , the variety (R⊕R∗)/W does not admit a symplectic resolution. It does, however, admit

a Q-factorial terminalization, the notion of the Namikawa-Weyl group can still be extended

to this setting, and this notion still coincides with Definition 5.3.5, see [L10, Section 2.3].

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5.3.4 Equivalences from the Namikawa-Weyl group

We now generalize (5.4) for every complex reflection group, where the role of the symmetric

group is now played by the Namikawa-Wey group Nam(W ). We start with the following

preparatory lemma.

Lemma 5.3.8. Let W be a complex reflection group, Nam := Nam(W ) its Namikawa-Weyl

group, and k a parameter for the Cherednik algebra. Then, for every σ ∈ Nam, the category

HC(Hk, Hσ(k)) is nonzero.

Proof. We use Theorem 4.1.1, with W = {1}, so Ξ = W and

HCΞ0 (Hk(W ), Hσ(k)(W )) = W -rep

Let Γ ∈ A be a reflection hyperplane and WΓ its setwise stabilizer. By (5.4), the cate-

gory HC(Hk(WΓ, RWΓ), Hσ(k)(WΓ, RWΓ

)) is nonzero. So we can find an irreducible (= 1-

dimensional) representation τΓ of WΓ such that Homfin(Sσ(k)(WΓ),∆k(τΓ)) is nonzero where,

recall, Sσ(k)(WΓ) denotes the socle of the polynomial representation ∆σ(k)(trivWΓ). Note that,

by the W -invariance of the parameter k, we may and will assume that τΓ = τΓ′ if Γ,Γ′ are

in the same W -orbit.

Now, we have that Hom(W,C×) ∼=∏

Γ∈A/W Hom(WΓ,C×), cf. [R]. So our choice of 1-

dimensional representations {τΓ : Γ ∈ A} determines a 1-dimensional representation τ

of W . We claim that Homfin(Sσ(k)(W ),∆k(τ)) 6= 0. To see this, we will use Theorem

4.1.1. Assume for the moment that for every reflection hyperplane Γ ∈ A, the bimod-

ule of locally finite maps Homfin(Sσ(k)(WΓ),ResWWΓ(∆k(τ))) is nonzero. Then, the restriction

Homfin(Sσ(k)(WΓ),ResWWΓ(∆k(τ)))†WΓ

{1}is a nonzero sub-bimodule of the 1-dimensional bimod-

ule HomC(C,KZk(∆k(τ))), so the conditions of Theorem 4.1.1 are satisfied and we get a HC

(Hk, Hσ(k))-bimodule B that localizes to HomC(C,KZk(∆k(τ))). Using Lemma 5.2.7 it is

easy to see that, up to subquotients with proper support, B = Homfin(Sσ(k)(W ),∆c(τ)).

74

So what we need to show now is that, for every Γ ∈ A, Homfin(Sσ(k)(WΓ),ResWWΓ∆k(τ)) is

nonzero. But this follows immediately from Lemma 2.4.6.

We remark that, with the same technique as in the proof of Lemma 5.3.8, we can show the

following result.

Lemma 5.3.9. Assume that the category HC(Hk, Hk′) is nonzero. Then, there exists a 1-

dimensional character τ of W such that Homfin(Sk′(W ),∆k(τ)) 6= 0 where, recall Sk′(W ) ∈

Ok′ is the irreducible module that gets sent to the trivial representation under the KZ functor.

Proof. Assume that HC(Hk, Hk′) 6= 0. Then, there exists an irreducible module N ∈

Ok with full support such that Homfin(Sk′(W ), N) 6= 0. Now, for each reflection hyper-

plane Γ ∈ A, consider the pointwise stabilizer WΓ ⊆ W . This is a cyclic group. Note

that, since Homfin(Sk(W ), N) 6= 0, we have that Homfin(ResWWΓ(Sk(W )),ResWWΓ

(N)) 6= 0,

this follows from Lemma 3.3.2 (4). Since KZ commutes with restriction, we have that

Sk′(WΓ) is the unique subquotient of ResWWΓ(Sk(W )) with full support, which implies that

Homfin(Sk′(WΓ),ResWWΓ(N)) 6= 0. Now, in category O for the rational Cherednik alge-

bra of WΓ, for every irreducible representation (= 1-dimensional character) τ of WΓ, we

have that either Lk(τ) = ∆k(τ), or Lk(τ) is finite dimensional. Since ResWWΓ(N) has

full support, we conclude that there exists an irreducible representation τΓ of WΓ with

Homfin(Sk′(WΓ),∆k(τΓ)) 6= 0. We remark that we can take τΓ = τΓ′ if Γ,Γ′ ∈ A are con-

jugate, this follows from the conjugation invariance of k. Now proceed as in the proof of

Lemma 5.3.8.

Corollary 5.3.10. Assume that HC(Hk, Hk′) 6= 0. Then, the categories HC(Hk, Hk′) and

HC(Hk′ , Hk′) are equivalent. Moreover, they are equivalent to the category of representations

of W/W ′ for some normal subgroup W ′ of W .

Proof. Let B be a HC (Hk, Hk′)-bimodule with full support. By the previous lemma, we

may assume that B = Homfin(Sk′(W ),∆k(τ)) for a 1-dimensional character τ of W , so that

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eB[δ−1]e = Diff(C[Rreg/W ], N). Here, N = e∆k(τ)[δ−1] is a rank 1 local system on Rreg/W .

Then, the tensor product functor eB[δ−1]e ⊗D(Rreg/W ) • induces a self-equivalence in the

category of D(Rreg/W )-bimodules. Indeed, this follows because eB[δ−1]e = N ⊗C[Rreg/W ]

D(Rreg/W ) and N is a line bundle on Rreg/W . This implies that B⊗Hk′ • : HC(Hk′ , Hk′)→

HC(Hk, Hk′) induces an equivalence between HC(Hk′ , Hk′) and HC(Hk, Hk′). The last as-

sertion is immediate from Proposition 3.4.6.

Corollary 5.3.11. Assume that either k′ = σ(k) for some element σ ∈ Nam, or that

k− k′ ∈ pZ. Then, the categories HC(Hk, Hk), HC(Hk, Hk′), HC(Hk′ , Hk) and HC(Hk′ , Hk′)

are all equivalent.

Proof. Let us do the first case. That HC(Hk, Hk) ∼= HC(Hk, Hk′) and HC(Hk′ , Hk′) ∼=

HC(Hk′ , Hk) follow immediately from Lemma 5.3.8 and Corollary 5.3.10. Note that these

equivalences come from tensoring with a HC bimodule, and this becomes an equivalence

after localizing to Rreg. Let B1 ∈ HC(Hk, Hk′) and B2 ∈ HC(Hk′ , Hk) be such bimodules.

Then, B1⊗Hk •⊗Hk B2 : HC(Hk, Hk)→ HC(Hk′ , Hk′) induce an equivalence HC(Hk, Hk)→

HC(Hk′ , Hk′). The case k − k′ ∈ pZ is similar, instead of Lemma 5.3.8 we have to use the

shift bimodules, cf. Definition 3.1.5.

5.3.5 Action on the set of Hecke parameters

Let k be a parameter for the rational Cherednik algebra Hk. Recall that we are always

assuming that kΓ,0 = 0 for every reflection hyperplane Γ, and under this assumption the

Hecke parameter q := q(k) is simply given by

qΓ,i = exp

(2π√−1(kΓ,i − i)`Γ

)(5.5)

For σ ∈ Nam, we will find the parameter q(σ(k)) in terms of q(k). Since both the action of the

Namikawa-Weyl group and the calculation of the Hecke parameters are done by restricting

76

to a single reflection hyperplane Γ, it is enough to do this in the case of a cyclic group

W = Z/`Z.

Now let i = 1, . . . , ` − 1 and let σi := (i, i + 1) ∈ S`. For k = (k0 = 0, k1, . . . , k`−1), we are

going to calculate the parameter σi(k) = (k′0 = 0, k′1, . . . , k′`−1).

First, assume that i 6= 1, `−1. Recall that σi(πj) = πj, if j 6= i, and σi(πi) = πi−1−πi+πi+1.

An easy calculation now shows that

k′j =

kj, j 6= i, i− 1

ki − 1, j = i− 1

ki−1 + 1, j = i

Note that a similar formula holds for i = `− 1, here we use that σ`−1(πj) = πj if j 6= `− 1,

σ`−1(π`−1) = π`−2 − π`−1. Note that it follows that

q(k′)j =

q(k)j, j 6= i− 1, i

q(k)i, j = i− 1

q(k)i−1 j = 1

For the case i = 1, we have σ1(π1) = π2 − π1, while σ1(πj) = πj if j 6= 1. In this case, we

have k′0 = 0, k′1 = 2− k1, and k′j = 1 + kj − k1 if j 6= 0, 1. Thus, we get

q(k′)j =

1, j = 0

q(k)−11 j = 1

q(k)jq(k)−11 j 6= 0, 1

Let us denote by S{2,...,`} the group of permutations of the set {2, . . . , `}. Of course, S{2,...,`}

is isomorphic to S`−1, an isomorphism S{2,...,`} → S`−1, σ 7→ σ is given by σ(i) = σ(i+1)−1,

i = 1, . . . , `− 1.

This discussion has the following consequence.

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Lemma 5.3.12. Let k = (k0 = 0, k1, . . . , k`−1}), and let σ ∈ S`. Then, q(σ(k)) is determined

by the following:

• q(σ(k))0 = 1.

• q(σ(k))i = q(k)σ(i) if σ ∈ S{2,...,`}, i = 1, . . . , `− 1.

• q(σ(k))i = q(k)−11 q(k)i, if σ = s1, i = 1, . . . , `− 1.

Finally, let us remark that there is also an action of the group of characters Hom(W,C×) on

the set of parameters p in such a way that Hk is isomorphic to Hε(k) for any ε ∈ Hom(W,C×).

Since this action is, again, defined by restricting to a single reflection hyperplane, let us

define it only in the case when W is a cyclic group, say W = Z/`Z = 〈s : s` = 1〉. In this

case, the group of characters is identified with Z/`Z, j 7→ (s 7→ ηj), η := exp(2π√−1/`).

We denote the character s 7→ ηj by εj. Then, we have for k = (k0 = 0, k1, . . . , k`−1),

εj(k)i = ki−j − k`−j, where the subscripts are taken modulo `. An isomorphism Hk → Hεk

is given by x 7→ x, y 7→ y, w 7→ ε(w)w, x ∈ R∗, y ∈ R, w ∈ W .

Lemma 5.3.13. Let ε : W → C× be a 1-dimensional character, and let k be a parameter.

Then, there exist σ ∈ Nam(W ) and k′ ∈ pZ such that ε(k) = σ(k) + k′.

Proof. It is again enough to show this when W = Z/`Z is a symmetric group. Assume ε = εj

for some j = 0, . . . `− 1. Then, we have

q(εj(k))i = exp

(2π√−1(ki−j − k`−j − i)

`

)= exp

(2π√−1(ki−j − i+ j)

`

)exp

(2π√−1(−k`−j − j

`)

)= q(k)i−jq(k)−1

`−j

(5.6)

Thanks to Lemma 5.3.12, we can find an element σ ∈ Nam such that q(εj(k))i = q(σ(k))i

for every i = 0, . . . , `− 1. But this means that εj(k)− σ(k) ∈ pZ. We are done.

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Example 5.3.14. Take W = Z/2Z. Then, the action of the only nontrivial character ε of

Z2 is given by ε(k) = −k = σ(k) + 2, cf. Example 5.3.4, and the integral k-parameters are

precisely the even numbers, cf. Section 2.3.5. In the c-parameters, we have ε(c) = −c =

σ(c) + 1, and here the lattice of integral parameters are the integers.

We are ready to show for which pairs of parameters k, k′ we have HC(Hk, Hk′) 6= 0. Recall

that, if this is the case, then we have HC(Hk, Hk′) ∼= HC(Hk′ , Hk′), cf. Corollary 5.3.10.

Thus, the following result will reduce the description of the category HC(Hk, Hk′) to the

case where k = k′. We will study this in Section 5.4.

Theorem 5.3.15. The category HC(Hk, Hk′) is nonzero if and only if there exists a sequence

k0 = k, k1, . . . , km = k′, where ki can be obtained from ki−1 from an integral translation or

by applying an element of the Namikawa-Weyl group for every i = 1, . . . ,m.

Proof. First, assume that such a sequence exist. Then, thanks to Corollary 5.3.11, the

categories HC(Hki−1 , Hki) are nonzero for every i = 1, . . .m. In particular, we can find

Bi ∈ HC(Hki−1 , Hki) such that eB[δ−1]e ∼= N i⊗D(Rreg/W )D(Rreg/W ) for a line bundle N i on

Rreg/W with a flat connection, cf. Lemma 5.3.9. Then, B1 ⊗Hk1 B2 ⊗Hk2 · · · ⊗Hkm−1 B

m ∈

HC(Hk, Hk′) is not killed upon localization to Rreg, and thus HC(Hk, Hk′) 6= 0.

Now assume HC(Hk, Hk′) 6= 0. Thanks to Lemma 5.3.9, we can find a character ε : W → C×

such that Homfin(∆k′(triv),∆k(ε)) 6= 0. Under the equivalence ϕ∗ : Ok → Oε−1(k), coming

from the isomorphism ϕ : Hk → Hε−1(k) we have ϕ∗(∆k(ε)) = ∆ε−1(k)(triv). Thus, Theorem

5.3.15 is a consequence of Lemma 5.3.13 and the following result.

Lemma 5.3.16. Assume Homfin(∆k′(triv),∆k(triv)) 6= 0. Then, there exists σ ∈ Nam such

that σ(k′)− k ∈ pZ.

Proof. First of all, note that a parameter k is integral if and only if k|WΓ∈ pZ(WΓ) for

every reflection hyperplane Γ ∈ A. Also, since ResWWΓ(∆?(triv)) = ∆?(triv(WΓ)) where

? = k, k′, cf. Lemma 2.4.6, we have that Homfin(∆k′(triv(WΓ)),∆k(WΓ)) 6= 0 for every

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Γ ∈ A. Finally, since the action of the Namikawa-Weyl group is defined by restricting to

stabilizers of reflection hyperplanes, we may assume that W is a cyclic group Z/`Z.

So assume Homfin(∆k′(triv),∆k(triv)) 6= 0 and W is a cyclic group. Since KZk(∆k(triv)) =

C, the trivial representation of Hq(k), Lemma 5.2.7 implies that Hq(k)′ -mod ⊆ Hq(k) -mod

as full subcategories of C[t, t−1] -mod. But Hq(k),Hq(k)′ are commutative algebras of the

same dimension. This implies that Hq(k) = Hq(k′), in other words, the numbers q(k)0 =

1, q(k)1, . . . , q(k)`−1 and q(k′)0 = 1, q(k′)1, . . . , q(k′)`−1 coincide up to a permutation of the

indices that fixes 0. Now the lemma is an immediate consequence of Lemma 5.3.12 and the

definition of having integral difference.

Let us examine the condition of Theorem 5.3.15 in the case of a Coxeter group W . In this

case, the stabilizer of every reflection hyperplane has order 2, and we have that the Namikawa-

Weyl group is Z|A/W |2 . We can think of a k-parameter as a |A/W |-tuple of complex numbers.

The action of an element of the Namikawa-Weyl group has the effect of changing the sign and

adding even integers to some components, cf. Example 5.3.4. Adding an integral parameters

amount to adding even integers in every component. Now for every reflection hyperplane Γ,

let sΓ ∈ W be the reflection such that Γ = ker(id−sΓ). In terms of the c-parameters, we

have kΓ,0 = 0, kΓ,1 = 2c(sΓ). Thus, translating to the c-parameters we have.

Corollary 5.3.17. Let W be a Coxeter group, and let c, c′ ∈ p. Then, HC(Hc, Hc′) 6= 0 if

and only if there exists a character ε : W → C× such that c−εc′ : S → C is an integer-valued

function.

Finally, let us mention a few words on how to check the condition in Theorem 5.3.15. Recall

that two parameters k and k′ have integral difference if and only if q(k)Γ,i = q(k′)Γ,i for

every i = 0, . . . , `Γ − 1. It follows from Lemma 5.3.12 that the parameters k, k′ satisfy the

condition in Theorem 5.3.15 if and only if, for every reflection hyperplane Γ ∈ A there exists

i0 ∈ {0, 1, . . . , `Γ − 1} such that

80

{q(k′)Γ,0, . . . , q(k′)Γ,`Γ−1} = {q(k)−1

Γ,i0q(k)Γ,0, . . . , q(k)−1

Γ,i0q(k)Γ,`Γ−1} (5.7)

as multisets.

Let us also mention that, in the statement of Theorem 5.3.15, we may always assume that

m ≤ 2. This is a consequence of the following result.

Lemma 5.3.18. Let k ∈ p, σ ∈ Nam and k′ ∈ pZ. Then

(1) There exists k1 ∈ pZ such that σ(k) + k′ = σ(k + k1).

(2) There exists k2 ∈ pZ such that σ(k + k′) = σ(k) + k2.

Proof. Since (2) is a formal consequence of (1), we only need to show (1). Note that we have

q(σ(k) + k′) = q(σ(k)). It follows that σ−1(σ(k) + k′) ∈ k + pZ. We are done.

Corollary 5.3.19. The following are equivalent.

1. The category HC(Hk, Hk′) is nonzero.

2. There exist σ ∈ Nam, k1 ∈ pZ such that k′ = σ(k) + k1.

3. There exist σ′ ∈ Nam, k2 ∈ pZ such that k′ = σ′(k + k2).

4. For every reflection hyperplane Γ ∈ A, there exists i0 ∈ {0, 1, . . . , `Γ − 1} such that

(5.7) holds.

5.4 Subgroup Wc

5.4.1 Definition

For the rest of this section, it will be more convenient to return to the ‘c-parametrization’

of the rational Cherednik algebra. Of course, we still have an action of the Namikawa-Weyl

81

group, as well as of the group of characters, on the space of parameters, and Lemmas 5.3.12,

5.3.13 remain valid.

Recall that the category HC(Hc, Hc) is equivalent to the category of representations of W/W ′

for some normal subgroup W ′ ⊆ W . Here, we describe the group W ′. To motivate our

description, we first look at the case where W is a cyclic group.

So assume W = Z/`Z, with generator s. The Hecke algebra Hq(c) is the quotient of the poly-

nomial algebra C[t] by the ideal generated by the polynomial (t− 1)∏`−1

i=1(t− q(c)i). We re-

mark that q(c)i is the scalar by which t acts on KZ(Ci), where Ci is the irreducible representa-

tion of W where s acts by multiplication by exp(2π√−1i/`). Now, if Homfin(∆(triv),∆(Ci))

is nonzero then, thanks to Lemma 5.2.7, multiplication by q(c)i induces a map q(c)→ q(c),

where we think of q(c) as a multiset q(c) = {q(c)0 = 1, q(c)1, . . . , q(c)`−1}. It is not hard to

see that this map is actually a bijection, i.e. it preserves multiplicities. In particular, q(c)i

is an `-root of 1.

So set η := exp(2π√−1/`). Note that the group W acts on the set of Hecke parameters,

the element si acts on a multiset q′ = {q′0, . . . , q′`−1} by multiplying each element by ηi. The

stabilizer of q(c), the Hecke parameter associated to the Cherednik parameter c, is cyclic, so

it is generated by sm, where m divides `, say mp = `. By definition, Wc := 〈sp〉. Note that

for generic c we have that m = `, so Wc = W .

Example 5.4.1. Let W = Z/4Z, with generator s acting on C by s 7→√−1. Let the

parameter c be given by c(s) = 0, c(s2) = −1/2, c(s3) = 0. Then, we have k0 = 0, k1 =

1, k2 = 0, k3 = 1, so {q0, q1, q2, q3} = {1, 1,−1,−1} and Wc = 〈s2〉.

Let us generalize the definition of Wc for the case where W is any complex reflection group.

Set q := q(c). Fix a reflection hyperplane Γ ∈ A. Let ηΓ := exp(2π√−1/`Γ), and sΓ ∈ WΓ be

the unique element with det(sΓ|R) = ηΓ, the element sΓ is a generator of WΓ. Now consider

the set XΓ := {i ∈ {1, . . . , `Γ} : ηiΓqΓ,j ∈ {qΓ,0 = 1, . . . , qΓ,`Γ−1} with the same multiplicity as

qΓ,j for every j = 0, . . . , `Γ − 1}. For example, `Γ ∈ XΓ. Now let mΓ := minXΓ. It is

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clear that mΓ is a divisor of `Γ, say mΓpΓ = `Γ. We define Wc := 〈spΓ

Γ : Γ ∈ A〉 ⊆ W . By

definition, this is a reflection group. Note that the conjugation invariance of c implies that

Wc is a normal subgroup of W .

Note that Wc = {1} if and only if mΓ = 1 for every reflection hyperplane Γ ∈ A. This

happens if and only if {qΓ,0 = 1, . . . , qΓ,`Γ−1} = {1, ηΓ, . . . , η`Γ−1Γ }, that is, if and only if

c ∈ Nam(pZ). On the other hand Wc = W if and only if mΓ = `Γ for every Γ ∈ A, and this

is a generic condition.

Example 5.4.2. Let W be a Coxeter group, so that the setwise stabilizer of every reflection

hyperplane is cyclic of order 2. For a reflection hyperplane Γ, let sΓ be a reflection such that

Γ = ker(id−sΓ). Note that the group Wc will be generated by those sΓ for which mΓ = 2.

We have that mΓ = 1 if and only if {−1,−qΓ,1} = {1, qΓ,1}, that is, if and only if qΓ,1 = −1.

Since qΓ,1 = − exp(2π√−1c(sΓ)), we get Wc = 〈s : c(s) 6∈ Z〉.

It is clear that Wc = Wc′ provided there exists σ ∈ Nam such that c− σ(c′) ∈ pZ, as integral

translations do not affect the Hecke parameter q. Let us check that the subgroup Wc is also

stable under the action of the Namikawa-Weyl group.

Lemma 5.4.3. Let c ∈ p and let σ ∈ Nam. Then, Wc = Wσ(c).

Proof. It is enough to check this when W is a cyclic group, so let W = Z/`Z = 〈s : s` = 1〉,

and η := exp(2π√−1/`). Assume that Wc = 〈sp〉 with mp = `. This means that there exist

Q0 = 1, Q1, . . . , Qm−1 ∈ C× such that

q(c) = {Qjηmi : j = 0, . . . ,m− 1, i = 0, . . . , p− 1}

It is enough to check that Wc = Wσi(c) for i = 1, . . . , ` − 1, where, recall, σi = (i, i + 1) ∈

S` = Nam(W ). Thanks to Lemma 5.3.12, q(c) = q(σi(c)) as a multi-set for i = 2, . . . , `− 1,

so that in this case we have Wc = Wσi(c). For σ1, Lemma 5.3.12 implies that there exist

j0 ∈ {0, . . . ,m− 1}, i0 ∈ {0, . . . , p− 1} such that

q(σ1(c)) = {Q−1j0Qjη

m(i−i0):j=0,...,m−1,i=0,...,p−1}

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Setting Q′j := Q−1j0Qj, we get that q(σ1(c)) = {Q′jηmi : j = 0, . . . ,m − 1, i = 0, . . . , p − 1},

so Wσ1(c) = 〈sp′〉 with p′ dividing p. But since c = σ1σ1(c), we also have that p divides p′.

Since p, p′ ∈ {1, . . . , `}, this implies that Wc = 〈sp〉 = Wσ1(c).

5.4.2 Main result

Theorem 5.4.4. The category HC(c, c) is equivalent, as a monoidal category, to the category

of representations of W/Wc.

To prove Theorem 5.4.4, we will check that there exists a parameter c′ ∈ p and an element

σ ∈ Nam such that σ(c)− c′ ∈ pZ and the algebra Hc′(W ) decomposes as Hc(Wc)#WcW , for

some parameter c ∈ C[S ∩Wc]Wc which is naturally computed from c′. Since σ(c)− c′ ∈ pZ,

the categories HC(c, c) and HC(c′, c′) are equivalent, cf. Corollary 5.3.11. The result will

now follow if we check that Hc(Wc) has a unique irreducible HC bimodule with full support.

Assume, for the moment, that the parameter c is such that, for Γ ∈ A, c(siΓ) = 0 unless

i = pΓ, 2pΓ, . . . , (mΓ − 1)pΓ. Then, it is clear from the relations (2.1) that the subalgebra

of Hc generated by R,R∗ and Wc is isomorphic to Hc(Wc), where c simply denotes the

restriction of the parameter c to Wc. So Hc is generated by Hc(Wc) and W . Moreover,

the subalgebra Hc(Wc) is stable under the adjoint action of W , this follows because Wc is

a normal subgroup of W . It follows that Hc∼= Hc(Wc)#WcW , where the latter algebra

is Hc(Wc) ⊗Wc CW with product defined analogously to the smash-product algebra, using

the action of W on Hc(Wc). Thus, HC Hc-bimodules with full support correspond to W -

equivariant HC Hc(Wc)-bimodules with full support, where the action of Wc ⊆ W coincides

with that coming from the inclusion Wc ⊆ Hc(Wc).

Denote q := q(c) and let us now examine the Hecke parameters qΓ,i, still under the assumption

that c(siΓ) = 0 unless i = pΓ, 2pΓ, . . . , (mΓ − 1)pΓ. It follows easily from (2.2) that kΓ,i =

kΓ,i+mΓfor all i. But then it follows that:

qΓ,i+mΓ= exp(2π

√−1(kΓ,i − i−mΓ)/`Γ) = η−mΓ

Γ qΓ,i

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Note that, given numbers QΓ,0 = 1, QΓ,1, . . . , QΓ,mΓ−1∈ C× we can always find a parameter

c ∈ p with c(siΓ) = 0 unless i is a multiple of pΓ and such that qΓ,i = QΓ,i. This implies the

following.

Lemma 5.4.5. Let c ∈ p be a parameter, and let mΓ, pΓ have the same meaning as in

Section 5.4.1. Then, there exists a parameter c′ ∈ p such that for every Γ ∈ A, c′(siΓ) = 0

unless i = pΓ, 2pΓ, . . . , (mΓ − 1)pΓ and Hq(c) = Hq(c′). In particular, σ(c)− c′ ∈ pZ for some

σ ∈ Nam, and so the categories HC(Hc, Hc) and HC(Hc′ , Hc′) are equivalent.

Proof of Theorem 5.4.4. Thanks to Lemmas 5.4.5, 5.4.3 and Corollary 5.3.11, we may as-

sume that c(siΓ) = 0 unless i is a multiple of pΓ, i.e. that Hc∼= Hc(Wc)#WcW . We claim

now that Hc(Wc) has a unique irreducible HC bimodule with full support. For Γ ∈ A, let

qΓ,0

= 1, . . . , qΓ,mΓ−1

be the parameters for the Hecke algebra Hq(Wc) associated to c, and

denote ηΓ

:= exp(2π√−1/mΓ) = ηpΓ

Γ . We also denote m′Γ := min{i ∈ {1, . . . ,mΓ} : ηiΓq

Γ,j∈

qΓ

with the same multiplicity as qΓ,j

for every j = 0, . . . ,mΓ− 1}. Thanks to Lemma 5.2.7,

our claim will follow if we check the following.

Claim: For every hyperplane Γ ∈ A, m′Γ = mΓ.

We proceed by contradiction. Assume there exists 0 < i < mΓ such that, for every j =

0, . . . ,mΓ − 1, ηiΓq

Γ,jis in the multiset q

Γwith the same multiplicity as q

Γ,j. Note that we

have

qΓ,j

= exp

(2π√−1(ki − i)mΓ

)= qpΓ

Γ,j

Thus, ηiΓq

Γ,j∈ q

Γimplies that ηiΓqΓ,j ∈ {qΓ,0 = 1, . . . , qΓ,mΓ−1

}, with the same multiplicity as

qΓ,j. But

qΓ = {qΓ,0, . . . , qΓ,mΓ−1, ηmΓqΓ,0, . . . , η

mΓqΓ,mΓ−1, . . . , η(pΓ−1)mΓqΓ,0, . . . , η

(pΓ−1)mΓqΓ,mΓ−1}.

Thus, we see that ηiΓqΓ,j ∈ q with the same multiplicity as qΓ,j for every j = 0, . . . , `Γ − 1.

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This contradicts the choice of mΓ. Thus, Hc(Wc) has a unique irreducible HC bimodule with

full support. Since Hc = Hc(Wc)#WcW , this proves Theorem 5.4.4. �

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Chapter 6

Type A

6.1 Introduction and preliminary results.

6.1.1 Main results

We now turn our attention to type A, that is, W = Sn, with reflection representation R =

{(x1, . . . , xn) ∈ Cn :∑xi = 0}. Throughout this chapter, we denote Hc(n) := Hc(Sn, R).

Similary, we denote Hq(n) := Hq(Sn), the Hecke algebra associated to Sn with parameter

q ∈ C×. More explicitly, Hq(n) is the algebra with generators T1, . . . , Tn−1 and relations

Ti+1TiTi+1 = TiTi+1Ti, i = 1, . . . , n− 2;

TiTj = TjTi, |i− j| > 1;

(Ti − 1)(Ti + q) = 0, i = 1, . . . , n− 1

Note the change in the sign of q, in particular, if q = exp(2π√−1c) then the KZ functor

for Oc has its image in the category of Hq-modules. There are two main results in this

chapter. The first one of them is an embedding of the category of Harish-Chandra bimodules

HC(Hc(n), Hc(n)) into category Oc, for any parameter c ∈ C.

Theorem 6.1.1. Assume c 6∈ R<0, and consider the functor Φc := Φ∆c(triv) : HC(Hc, Hc)→

Oc, B 7→ B ⊗Hc ∆c(triv). Then

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1. The functor Φc is a fully faithful embedding whose image is closed under subquotients.

2. The functor Φc preserves supports in the sense that, for a parabolic subgroup W ′ ⊆ Sn,

we have LW ′ ⊆ SS(B) if and only if XW ′ ⊆ supp(Φc(B)). In particular, it sends a

finite-dimensional bimodule to a finite-dimensional module.

Let us remark that the restriction c 6∈ R<0 is not very important. Indeed, we have an

isomorphism Hc(n)→ H−c(n), which implies that in this case Theorem 6.1.1 remains valid

upon substituting the trivial representation with the sign representation.

We also remark that Theorem 6.1.1 does not generalize to an arbitrary complex reflection

group. For example, let W = Z/4Z = 〈s : s4 = 1〉 with parameter given by c(s) =

0, c(s2) = −1/2, c(s3) = 0. Then, there are two irreducible bimodules with full support in

HC(Hc, Hc), cf. Example 5.4.1, and there are 4 irreducible finite-dimensional bimodules, this

follows because category Oc has 2 irreducible finite-dimensional modules. Then, there are 6

irreducible bimodules in HC(Hc, Hc), so there cannot exist an embedding HC(Hc, Hc)→ Oc

with the properties of the one in Theorem 6.1.1.

Of course, a question that comes after Theorem 6.1.1 is to describe the image of the functor

Φc. Recall from Example 2.3.6 that the singular locus of Sn consists of those rational numbers

that can be written in the form r/m, gcd(r;m) = 1, 1 < m ≤ n. The following result is from

[BEG]. Alternatively, it can be easily deduced from the results in Chapter 4 of this work,

more precisely Theorem 5.1.1, cf. Example 5.4.2.

Proposition 6.1.2. Assume c ∈ C is regular. Then, Φc : HC(Hc, Hc) → Oc is an equiva-

lence if c ∈ Z. If c 6∈ Z, then HC(Hc, Hc) is equivalent to the category of finite-dimensional

vector spaces, and the only irreducible in the image of Φc is ∆c(triv).

To explain what happens in the singular case, we need some notation regarding partitions.

A partition λ is a non-increasing sequence λ = (λ1, λ2, · · · , ) of non-negative integers that is

eventually 0. The size of the partition λ is |λ| :=∑

i≥0 λi, this is a non-negative integer. We

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write λ ` |λ| to indicate that λ is a partition of size |λ|. We add partitions componentwise,

(λ+ µ)i = λi + µi. Similarly, we can multiply a partition by a positive integer in an obvious

way.

For a positive integer m, we say that a partition λ is m-regular if λi − λi+1 < m for every

i = 1, 2, . . . . It is easy to see from the division algorithm that for every partition λ, there exist

unique partitions µ, ν such that λ = mµ + ν and ν is m-regular. We say that an m-regular

partition ν is m-trivial if ν = ((m − 1), (m − 1), . . . , (m − 1), d, 0, ...) with 0 ≤ d ≤ m − 1,

where the number of components equal to m− 1 may be zero. Obviously, for each positive

integer k there exists a unique m-trivial partition ν with |ν| = k.

Now assume that c ∈ C is positive and singular, so c = r/m > 0, gcd(r;m) = 1, 1 < m ≤ n.

For each partition λ ` n, we will also denote by λ the irreducible representation of the

symmetric group Sn labeled by the partition λ. So we have the irreducible representation

Lc(λ) ∈ Oc.

Theorem 6.1.3. Let c = r/m > 0, gcd(r;m) = 1, 1 < m ≤ n. Let λ ` n be a partition, and

let λ = mµ + ν be its decomposition so that ν is m-regular. Then, Lc(λ) ∈ im(Φc) if and

only if ν is m-trivial.

Note that, while Theorem 6.1.3 describes the irreducible objects in the image of Φc, it does

not describe the image of Φc. Indeed, we will see through an example that the image of Φc

is, in general, not closed under extensions, see Example 6.2.4.

Let us now turn out our attention to the study of the category HC(Hc, Hc′) for distinct

c, c′ ∈ C. We remark that if either c or c′ is regular, then HC(Hc, Hc′) = HC(Hc, Hc′) and

in this case the description of HC(Hc, Hc′) follows from Chapter 4, cf. Corollary 5.3.17,

Example 5.4.2. So we will assume that c, c′ are singular.

Theorem 6.1.4. Let c = r/m, c′ = r′/m′, gcd(r,m) = gcd(r′,m′) = 1, 1 < m,m′ ≤ n.

1. If HC(Hc, Hc′) 6= 0, then m = m′.

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2. Assume m = m′ is not a divisor of n. Then, HC(Hc, Hc′) 6= 0 if and only if either c+c′

or c−c′ is an integer. In this case, for each symplectic leaf L ⊆ (R⊕R∗)/Sn, there is a

category equivalence HCL(Hc, Hc′) ∼= HCL(Hc, Hc) ∼= HCL(Hc′ , Hc) ∼= HCL(Hc′ , Hc′).

Note that if m = m′ is a divisor of n, Theorem 6.1.4 fails. Indeed, we can take for example

m = m′ = n. In this case, both algebras Hc and Hc′ have a finite-dimensional module, so

HC(Hc, Hc′) 6= 0, regardless of the integrality of c− c′ or c + c′. For this case, see Theorem

6.4.3.

6.1.2 Preliminary results

Now we gather some results on the structure of the algebra Hc(n) and of category Oc that we

will use in our arguments. It is known, cf. [BE, Example 3.25], [L3, Theorem 5.8.1], that the

algebra Hc := Hc(n) is simple unless c = r/m with r,m ∈ Z, gcd(r;m) = 1 and 1 < m ≤ n.

In this case, [L3, Theorem 5.8.1 (2)], the algebra Hc has bn/mc proper nonzero two-sided

ideals that are linearly ordered by inclusion, say J1 ⊂ J2 ⊂ · · · ⊂ Jbn/mc. Moreover, J 2i = Ji

for any i = 1, . . . , bn/mc. We set J0 := {0}, Jbn/mc+1 := Hc.

The classification of two-sided ideals gives a characterization of the possible supports of

HC bimodules. For i = 0, 1, . . . , bn/mc, consider the subgroup S×im ⊆ Sn, and consider

the set Xi := {x ∈ R ⊕ R∗ : Wx = S×im }. Let Li be the image of Xi under the natural

projection R ⊕ R∗ → (R ⊕ R∗)/Sn. This is a symplectic leaf. The support of Hc/Ji is

Li. Now recall Lemma 3.2.2, that says that for a HC (Hc, Hc′)-bimodule B, SS(B) =

SS(Hc/LAnn(B)) = SS(Hc′/RAnn(B)). This already implies (1) of Theorem 6.1.4, namely,

that HC(Hc, Hc′) = 0 unless c, c′ have the same denominator when expressed as an irreducible

fraction. Thus, throughout the rest of this chapter we will assume that c = r/m, c′ = r′/m,

with gcd(r;m) = gcd(r′;m) = 1 and 1 < m ≤ n.

We now give a description of the supports of irreducible modules in Oc. For every i =

0, 1, . . . , bn/mc, let X ′i = {(x1, . . . , xn) ∈ Cn :∑xi = 0, x1 = x2 = · · · = xm, xm+1 = · · · =

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x2m, . . . , x(i−1)m+1 = · · · = xim}, and let Xi be the union of the Sn translates of X ′i, so that

R = X0 ⊃ X1 ⊃ · · · ⊃ Xbn/mc. Then, [BE, Example 3.25], [Wi, Theorem 3.9], any module

in category Oc is supported on one of the Xi. Denote by Oic the full subcategory of Oc

consisting of all modules whose support is contained in Xi, and Obn/mc+1c = 0. Note that

each Oic is a Serre subcategory of Oc. Let us explain a description of the category Oic/Oi+1c

obtained in [Wi]. Let p := n− im, q := exp(2π√−1c) and consider the Hecke algebra Hq(p).

Then [Wi, Theorem 1.8] tells us that the category Oic/Oi+1c is equivalent to the category of

finite dimensional modules over the algebra CSi ⊗ Hq(p). In particular, for c = r/n and

i = 1, we have that the category of finite-dimensional Hc(n)-modules is equivalent to the

category of vector space. This is already in [BEG2].

Let us recall how [Wi, Theorem 1.8] is proved, as this will be important for our arguments. So

let i and p be as in the previous paragraph. Consider the subgroup S×im ⊆ Sn. Let R := {x ∈

R : S×im ⊆ StabSn(x)}(= X ′i) and Rreg := {x ∈ R : StabSn(x) = S×im }. Then, Wilcox proves

that we have a localization functor, Loci : Oi → D(Rreg)#(Si×Sp)-mod, M 7→ C[Rreg]⊗C[R]

M that factors throughOi/Oi+1 and that identifies this quotient category with a subcategory

of the category of (Si × Sp)-equivariant D(Rreg)-modules with regular singularities. Then,

he checks that under the Riemann-Hilbert correspondence that identifies the latter category

with the category of finite dimensional representations of π1(Rreg/(Si × Sp)), the image

of Oic/Oi+1c gets identified with the subcategory (CSi ⊗ Hq(p))-mod of π1(Rreg/(Si × Sp))-

rep where, recall, q = exp(2π√−1c). We denote by KZi : Oi → (CSi ⊗ Hq(p))-mod the

composition of the localization functor Loci with the Riemann-Hilbert correspondence.

This construction has the following consequence for HC bimodules. Let S ∈ Oc be the

irreducible module supported on Xi that gets sent to the trivial CSi ⊗Hq(p)-module under

KZi so that, in particular, Loci(S) = C[Rreg]. Then, the proofs in Section 5.2.3 can be carried

out in this setting and we see that, whenever T is a simple module with Homfin(S, T ) 6= 0,

and N is another simple module in Oi, then KZi(Homfin(S, T )⊗HcN) = KZi(S)⊗C KZi(T ).

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Lemma 6.1.5. Let S ∈ Oic′ be the irreducible module satisfying KZic′(S) = C, the trivial

CSi⊗Hq(p)-module. Let T ∈ Oc (necessarily supported on Xi) be a simple module satisfying

Homfin(S, T ) 6= 0. Then, for every M ∈ CSi ⊗Hq′(p) -mod, the π1(Rreg/(Si × Sp))-module

KZi(T )⊗C M factors through CSi ⊗Hq(p).

We will see, Lemma 6.2.5, that in fact every irreducible HC (Hc, Hc′)-bimodule supported

on Li has the form Homfin(S, T ), where S ∈ Oic′ is the module such that KZic′ is the

trivial representation of CSi ⊗Hq′(p), and T is such that KZic(T ) = λ ⊗ C, where λ is any

representation of Si, and C is the trivial representation of Hq(p), i.e., that on which all Ti

act by 1.

Let us show now that the category HCLi(Hc, Hc) is actually semisimple for i = 0, . . . , bn/mc.

Consider the subgroup W := S×im ⊆ Sn. Note that Ξ := NSn(W )/W may be identified

with Si × Sn−mi. Now recall the restriction functor •†SnW : HCLi(Hc, Hc) → HCΞ0 (Hc, Hc),

that identifies the source category with a full subcategory of the target category closed

under taking subquotients and tensor products, cf. Section 3.4. Here, Hc is the rational

Cherednik algebra of W . But the algebra Hc is isomorphic to Hc(m)⊗i, and this algebra

has a unique irreducible finite-dimensional module, hence also a unique irreducible finite-

dimensional bimodule. So HCΞ0 (Hc, Hc)

∼= Ξ -rep, and similarly to the proof of Proposition

3.4.6 we get the following result.

Lemma 6.1.6. Let c = r/m, gcd(r;m) = 1, 1 < m ≤ n. Then, for every i = 0, . . . , bn/mc,

the category HCLi(Hc, Hc) is equivalent to the category of representations of Ξ/N for some

normal subgroup N ⊆ Ξ. In particular, it is semisimple.

In Theorem 6.3.8, we will see that, upon identifying Ξ = Si × Sn−mi, the normal subgroup

N in Lemma 6.1.6 is precisely the factor Sn−mi, and so HCLi(Hc, Hc) ∼= Si -rep.

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6.2 Functor Φc

6.2.1 Case c = r/n

We now start studying the functor Φc := • ⊗Hc ∆c(triv) : HC(Hc, Hc) → Oc where c > 0 is

a singular parameter. We start with the easiest case, which is when c = r/n, gcd(r;n) = 1.

In this case, thanks to Theorem 5.1.1, we have that there exists a unique irreducible HC

Hc-bimodule with full support, which necessarily has to coincide with the unique non-trivial

proper two-sided ideal J of Hc. Since there is a unique irreducible finite-dimensional Hc-

module, cf. [BEG2], we have that there is a unique irreducible finite-dimensional HC Hc-

bimodule, which is M := Hc/J . Note that these are all the irreducible HC Hc-bimodules,

this follows from the description of the possible supports of HC bimodules that was recalled

in Section 6.1.2. Our first task will be to compute the indecomposable bimodules. After

that, we will prove Theorem 6.1.1 in this case.

Indecomposable bimodules

Note that the bimodule J does not have self-extensions, this follows because HC(Hc, Hc) is a

semisimple category, Theorem 5.1.1. Also note that, since the category of finite-dimensional

Hc-modules is semisimple, cf. [BEG2, Proposition 1.12], M does not have self-extensions.

It is clear that Ext(M,J ) 6= 0, as the regular bimodule Hc is a non-split extension of J by

M . Here and for the rest of this section, we denote Ext := Ext1Hc -bimod. On the other hand,

Bezrukavnikov and Losev construct in [BL, Section 7.6] a non-split extension of M by J ,

the so-called double wall-crossing bimodule D.

Our goal now is to show that M,J , Hc and D form a complete list of indecomposable HC

Hc-bimodules. This is a consequence of the following result.

Proposition 6.2.1. The following is true:

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(i). Ext(Hc,M) = 0. (ii). Ext(M,Hc) = 0. (iii). dim(Ext(M,J )) = 1.

(iv). Ext(Hc,J ) = 0. (v). Ext(J , Hc) = 0. (vi). Ext(M,D) = 0.

(vii). Ext(D,J ) = 0. (viii) Ext(D,M) = 0. (ix) Ext(J , D) = 0.

(x) dim(Ext(J ,M)) = 1.

Proof. We show that (i) holds more generally, namely, we have the following result.

Lemma 6.2.2. Let Hc be any rational Cherednik algebra of type A (we do not put restrictions

on the parameter c), and let M be an irreducible Harish-Chandra Hc-bimodule with minimal

support. Then, Ext(Hc,M) = 0.

Proof. We know that Ext•(Hc,M) = HH•(Hc,M), where HH• denotes Hochschild cohomol-

ogy, so we need to compute HH1(Hc,M). It is well known that this is the space of outer

derivations (i.e. the space of derivations modulo the space of inner derivations). Now, let

δ : Hc → M be a derivation. Since J 2 = J , Subsection 6.1, the Leibniz rule implies that

δ(J ) = 0, so δ factors through the quotient algebra Hc/J . This implies that HH1(Hc,M) =

HH1(Hc/J ,M) (note that M is an Hc/J -bimodule since RAnn(M) =LAnn(M) = J , so

this last Hochschild cohomology does make sense). Now, both Hc/J and M are irreducible

HC bimodules with minimal support. Recall from Lemma 6.1.6 that the category of HC

bimodules with minimal support is semisimple. Then, Ext(Hc/J ,M) = 0, which implies

that HH1(Hc/J ,M) = ExtHc/J -bimod(Hc/J ,M) = 0.

Then (i) is a special case of Lemma 6.2.2. Note that (ii) and (v) are consequences of

Proposition 3.4.7. Now (iii) is a consequence of (ii): we have a long exact sequence

0→ Hom(M,J )→ Hom(M,Hc)→ Hom(M,M)→ Ext(M,J )→ Ext(M,Hc)→ · · ·

Now, both Hom(M,Hc) and Ext(M,Hc) are 0, so Hom(M,M) → Ext(M,J ) must be an

isomorphism and the claim follows. Again using long exact sequences, we can see that

dim(Ext(Hc,J )) = − dim(Hom(J ,J ))+dim(Ext(M,J )) = 0, so (iv) is proved. Statements

(vi), (ix) are consequences of the following.

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Lemma 6.2.3. Assume c = r/n > 0, with gcd(r;n) = 1. The bimodule D is injective in the

category of HC Hc(n)-bimodules.

Proof. Here we make use of the functor Φc and its right adjoint, G := Homfin(∆c(triv), •).

Note that the object ∆c(triv) is projective in Oc so, thanks to Lemma 3.3.13, Φc is exact

and Homfin(∆c(triv), •) sends injectives to injectives. So the lemma will follow if we find an

injective object E ∈ Oc with D ∼= G(E).

First, we remark that ∆c(triv) has a unique nonzero proper submodule, say I, see [BEG2,

Theorem 1.3]. The costandard module ∇c(triv) is injective in category Oc, it has a unique

proper submodule isomorphic to Lc(triv) and ∇c(triv)/Lc(triv) ∼= I, all of these properties

follow from the construction of ∇c(triv), see e.g. [GGOR, Subsection 2.3]. So G(∇c(triv)) is

injective and containsG(Lc(triv)) = Homfin(∆c(triv), Lc(triv)) = Homfin(Lc(triv), Lc(triv)) =

M . It follows that we have an injection D ↪→ G(∇c(triv)). Note, however, that we have an

exact sequence

0→ G(Lc(triv))→ G(∇c(triv))→ G(I) (6.1)

Now, we have that G(I) ⊆ G(∆c(triv)) = Hc, cf. Corollary 5.2.5. It is easy to see that

the inclusion G(I) ⊆ G(∆c(triv)) is proper, so we must have G(I) = J . Thanks to the

exact equence (6.1), we conclude that the composition length of G(∇c(triv)) is at most 2.

So D ∼= Homfin(∆c(triv),∇c(triv)) and is therefore injective.

Now we show that Ext(D,J ) = 0. Assume we have a short exact sequence

0→ J → Xπ→ D → 0 (6.2)

Consider the induced exact sequence 0→ J → π−1(M)→M → 0. So either π−1(M) = Hc

or π−1(M) = J ⊕M . If π−1(M) = Hc, then the exact sequence 0→ π−1(M)→ X → J → 0

gives X = Hc⊕J , cf. (iv), which contradicts the existence of the exact sequence (6.2). Then,

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we must have π−1(M) = J ⊕M . Using again the exact sequence 0→ π−1(M)→ X → J →

0, we get that X = J ⊕ V , where V is an extension of M by J . Then (6.2) forces V = D

and the sequence splits.

The proof of (viii) is similar: say that we have a short exact sequence

0→M → X → D (6.3)

So we see that Soc(X) = M ⊕M and we have an exact sequence

0→M ⊕M → X → J → 0

An extension of M ⊕M by J must be of the form B ⊕M , where B is an extension of M

by J . Using the short exact sequence (6.3) we see that X ∼= D⊕M . Finally, (x) is an easy

consequence of the previous statements.

Note that the previous proposition implies that both Hc and D are injective-projective in

the category HC(Hc, Hc). The injective hull of J coincides with the projective cover of M ,

which is Hc, while D is both the injective hull of M and the projective cover of J . In other

words, the category HC(Hc(n), Hc(n)) is equivalent to the category of representations of the

quiver

• •α

β

with relations αβ = βα = 0. It follows, in particular, that the homological dimension of

HC(Hc, Hc) is infinite.

Proof of Theorem 6.1.1, case c = r/n

We now show Theorem 6.1.1 in the case where c = r/n > 0, gcd(r;n) = 1. Since we know

the indecomposable HC bimodules and the Krull-Schmidt theorem holds in the category

HC(Hc, Hc) (this is an easy consequence of Corollary 3.3.14) we can actually do this “by

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hand”. First of all, note that the functor Φc is faithful. Indeed, Φc(M) = Lc(triv), Φc(J ) = I

and, since M an J are the only irreducible HC bimodules up to isomorphism, it follows that

Φ(B) 6= 0 for any HC (Hc, Hc)-bimodule B. Faithfulness of Φc now follows from the general

fact that an exact functor that does not kill any nonzero object is faithful. By standard

properties of adjunctions, it follows that the unit of adjunction B → GΦc(B) is injective for

every B ∈ HC(Hc, Hc).

Now, we have that Φc(Hc) = ∆c(triv), from where it follows by exactness that Φc(J ) = I,

Φc(M) = Lc(triv). The module Φc(D) is an extension of Lc(triv) by I. This extension is

non-split since D embeds into G(Φc(D)). So Φc(D) = ∇c(triv). Note that it follows that the

unit of adjunction idHC(Hc,Hc) → GΦc is an isomorphism, so Φc is a fully faithful embedding.

The claim about its image being closed under subquotients follows immediately from our

computations, as well as the claim about the supports.

Example 6.2.4. Let W = S2, c = 1/2. So there are two irreducible modules in category

Oc, Lc(triv), which is finite-dimensional, and Lc(sign), which coincides with the socle of

∆c(triv). Both irreducibles are in the image of Φc. However, Φc is not an equivalence, for

Oc being a highest weight category has finite homological dimension, while we have seen that

HC(Hc, Hc) has infinite homological dimension. Alternatively, this claim can be seen from

the fact that Oc is equivalent to the principal block of category O(sl2). This implies that the

image of Φc in this case is not closed under extensions.

6.2.2 General case

Φc is faithful.

Let us show that Φc is faithful in the general case. Since Φc is exact, it is enough to show

that Φc(B) 6= 0 for any irreducible HC bimodule B.

Lemma 6.2.5. Let c = r/m > 0, 1 < m ≤ n, gcd(r;m) = 1. Let B ∈ HC(Hc, Hc). Then,

Φc(B) = B ⊗Hc ∆c(triv) is nonzero.

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Proof. Let i ∈ {0, . . . , bn/mc} be such that SS(B) = Li. If i = 0, then the result follows

from Lemma 5.2.2. So we may assume i > 0. Let W := S×im ⊆ Sn, so that B†SnWis finite-

dimensional. Now Hc = Hc(m)⊗i, so thanks to [BEG2] ∆c(trivW ) has a finite-dimensional

quotient. So B†SnW⊗ ∆c(trivW ) is nonzero. But this is ResWW (B ⊗Hc ∆c(triv)), see Lemmas

2.4.6 and 3.3.2. We are done.

It follows that Φc is faithful.

Φc is full.

Now we show that Φc is full. In order to do this, we will show that the unit map idHC(Hc,Hc) →

GΦc is an isomorphism where, recall, G : Oc → HC(Hc, Hc) is Homfin(∆c(triv), •). Since Φc

is faithful, the unit is always injective.

Lemma 6.2.6. Let B ∈ HC(Hc, Hc). Then, the adjunction map B → Homfin(∆c(triv), B⊗Hc

∆c(triv)) is an isomorphism.

Proof. Denote N := Φc(B) ∈ Oc. Since Φc is exact and faithful, to check that the map B →

Homfin(∆c(triv), N) is an isomorphism, it is enough to check that Homfin(∆c(triv), N)⊗HcN

is isomorphic to N . Note that, since N is in the image of Φc, we have an epimorphism

Homfin(∆c(triv), N) � N . We show that its kernel is zero. To do so, let ` := bn/mc and

let W := S×`m ⊆ Sn, and consider ResSnW . This functor is exact and, by our choice of `, does

not kill nonzero modules. So it is enough to check that the induced epimorphism

ResSnW (Homfin(∆c(triv), N)⊗Hc ∆c(triv)) � ResSnW (N) (6.4)

is an isomorphism, i.e. it is injective. Recall that we have ResSnW (Homfin(∆c(triv), N) ⊗Hc

∆c(triv)) = Homfin(∆c(triv), N)†SnW⊗Hc

∆c(trivW ). By construction of the restriction func-

tor, we have that Homfin(∆c(triv), N)†SnWembeds into Homfin(∆c(trivW ),ResSnW (N)), so by

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exactness of the functor • ⊗Hc∆c(trivW ), it is enough to show that the morphism

Homfin(∆c(trivW ),ResSnW (N))⊗Hc∆c(trivW )→ ResSnW (N)

is an isomorphism. Since Hc = Hc(m)⊗`, this follows by the results of Section 6.2.1, provided

we show that ResSnW (N) belongs to the image of the functor • ⊗Hc∆c(trivW ). This follows

from (6.4) and results of Section 6.2.1, namely, that for c = r/m the image of Φc is closed

under subquotients. We are done.

Remark 6.2.7. Note that it follows from the proof of Lemma 6.2.6 that, whenever N ∈ Oc

is such that there exists a HC bimodule B and an epimorphism B⊗Hc ∆c(triv) then we have

Homfin(∆c(triv), N)†SnW= Homfin(∆c(trivW ),ResSnW (N))

where, recall, W = S×`m ⊆ Sn, where c = r/m > 0 and ` := bn/mc.

The image of Φc is closed under subquotients

Let us now show that the image of the functor Φc is closed under subquotients. First, we

show that it is closed under quotients.

Lemma 6.2.8. Let M := B ⊗Hc ∆c(triv) for some B ∈ HC(Hc, Hc), and assume that

f : M � N is an epimorphism. Then, there exists a quotient B′ of B such that B′ ⊗Hc

∆c(triv) = N .

Proof. Note that by Lemma 6.2.6 we may asumme that B = Homfin(∆c(triv),M). Consider

the induced morphism

B = Homfin(∆c(triv),M)f→ Homfin(∆c(triv), N)

and let us denote by B′ its image. So we have a map B′ ⊗Hc ∆c(triv) → N which is an

epimorphism since f is an epimorphism. Let us show that it is injective. So let W := S×`m ⊆

Sn where ` := bn/mc. It is enough to check that the induced epimorphism ResSnW (B′ ⊗Hc

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∆c(triv)) → ResSnW (N) is an isomorphism. This follows exactly as in the proof of Lemma

6.2.6.

Let us now show that the image of Φc is closed under submodules. So let M = B⊗Hc∆c(triv)

and N ′ ⊆ M . Let N = M/N ′. By Lemma 6.2.8 we know that there exists a quotient B′ of

B such that B′ ⊗Hc ∆c(triv) = N . If we let B′′ be the kernel of the epimorphism B → B′,

exactness of Φc shows that N ′ = B′′ ⊗Hc ∆c(triv).

Finally, we remark that (2) of Theorem 6.1.1 is a consequence of Lemma 3.2.3. This finishes

the proof of Theorem 6.1.1. Let us state an easy consequence.

Corollary 6.2.9. Assume c 6∈ R<0. For any B ∈ HC(Hc, Hc), there is an order-preserving

bijection between sub-bimodules of B and submodules of Φc(B). In particular, there is

an order-preserving bijection between the set of ideals of Hc and the set of submodules of

∆c(triv), which is given by J 7→ J∆c(triv).

The statement about ideals in Hc and submodules of the polynomial representation ∆c(triv)

is not new. The set of ideals ofHc was calculated by Losev in [L3], while the set of submodules

of the ∆c(triv) was calculated by Etingof-Stoica in [ES].

Another consequence of Theorem 6.1.1 is that we can give an alternative description of the

double wall-crossing bimodule D of [BL, Section 7]. There, the double wall-crossing bimodule

is constructed by taking the derived tensor product of two wall-crossing bimodules (one that

crosses a wall and one that goes back) and observing that the homology of this derived

tensor product is concentrated in degree 0. The main property of the double wall-crossing

bimodule is that it is indecomposable and has a composition series in which the composition

factors are those appearing in the regular bimodule Hc, but the order is the opposite, see

[BL, Theorem 7.7].

Proposition 6.2.10. Let c 6∈ R<0. Then, the double wall-crossing bimodule D is isomorphic

to Homfin(∆c(triv),∇c(triv)). If c ∈ R<0 the result is still valid after replacing triv with sign.

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Proof. The proposition is only interesting when c is singular, so we assume c = r/m > 0,

gcd(r;m) = 1, 1 < m ≤ n. The bimodule Homfin(∆c(triv),∇c(triv)) is injective and its socle

coincides with Hc/Jbn/mc. So there exists a monomorphism D → Homfin(∆c(triv),∇c(triv)).

That this is an isomorphism now follows because the composition length of the bimodule

Homfin(∆c(triv),∇c(triv)) is at most the composition length of Hc, which coincides with the

composition length of D.

Corollary 6.2.11. The double wall-crossing bimodule D is injective in HC(Hc(n), Hc(n)).

6.3 Irreducible Harish-Chandra bimodules.

In this section, we compute the irreducible modules in the image of the functor Φc. Before, let

us fix some notation. Recall that we are taking c = r/m > 0, with 1 < m ≤ n, gcd(r;m) = 1,

and ` := bn/mb. It follow from results in [ES] that for every i = 0, . . . , `, the polynomial

representation ∆c(triv) has a unique irreducible subquotient supported on Xi, let us denote

this irreducible module by Si. In fact, we have Si = Ji+1∆c(triv)/Ji∆c(triv) where, recall,

{0} = J0 ⊆ J1 ⊆ · · · ⊆ Jbn/mc+1 = Hc are the ideals in the rational Cherednik algebra

Hc. It follows from Theorem 6.1.1 and Lemma 3.2.3 that every irreducible HC Hc-bimodule

has the form Homfin(Si, L) for some i = 0, . . . , `, where L ∈ Oc is an irreducible module in

category Oc supported on Xi.

6.3.1 Results of Wilcox and consequences

Let us recall more precisely the results of Wilcox [Wi], that says that the quotient category

Oic/Oi+1c is equivalent to the category of representations of CSi⊗Hq(p), where p = n−mi and

q = exp(2π√−1c). We are interested in a more concrete description of this, namely, which

irreducible modules in category Oc have a given support, and, for an irreducible L ∈ Oc

supported on Xi, what is its image under the functor KZi : Oic → (CSi ⊗Hq(p)) -mod.

101

First, recall that irreducible modules over the Hecke algebra Hq(p) are parametrized by m-

regular partitions of p, see for example [M, Chapter 3]. We denote by Dν the irreducible

representation of Hq(p) corresponding to the m-regular partition ν. The representation Dν

is trivial (i.e., it is 1-dimensional and all Ti act by 1) precisely when ν is the unique m-trivial

partition of p, this is an easy special case of the LLT algorithm, cf. [M, Chapter 6].

Now let λ be a partition of n, and decompose it as λ = mµ+ ν, where ν is m-regular. The

following is part of [Wi, Theorem 1.8].

Proposition 6.3.1. Assume c = r/m > 0, gcd(r;m) = 1, 1 < m ≤ n. The support

of the irreducible module Lc(λ) is X |µ|. Moreover, the functor KZ|ν| sends Lc(λ) to the

representation µ⊗Dν of S|µ| ⊗Hq(n−m|µ|).

Together with Lemma 6.1.5, Proposition 6.3.1 already implies one direction of Theorem

6.1.3.

Corollary 6.3.2. Let c = r/m > 0, gcd(r;m) = 1, 1 < m ≤ n. Let λ be a partition of n, and

let λ = mµ+ν be it decomposition so that ν is m-regular. Assume that Homfin(S|µ|, Lc(λ)) 6=

0 (equivalently, Lc(λ) ∈ im(Φc)). Then, ν is m-trivial.

Proof. According to Lemma 6.1.5, for every representation N ∈ (S|µ|⊗Hq(n−m|µ|)) -mod,

the π1(Rreg/(S|µ|× Sn−m|µ|))-representation N ⊗KZ|µ|(Lc(λ)) factors through S|µ|⊗Hq(n−

m|µ|). But this can only happen if Dν is the trivial representation of Hq(n−m|µ|).

We will see that, in fact, the converse of the previous result holds, thus proving Theorem 6.1.3.

In order to do this, we are going to count the number of irreducible objects of HCLi(Hc, Hc).

In particular, we will see that it equals the number of partitions of i. This will show our

desired result.

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6.3.2 Bimodules with minimal support.

We give a complete description of the category of HC Hc(n)-bimodules with minimal support

when the parameter c has the form c = r/m, for m a divisor of n. In particular, we show

that it is equivalent to the category of representations of Sn/m. Throughout this section, we

denote ` := n/m. For convenience, we assume first that c > 0, we will deal with the case

c < 0 at the end of this section. The following is our main result in this section.

Proposition 6.3.3. Let c := r/m, where gcd(r;m) = 1 and m is a divisor of n. Let

` := n/m. Then, the category HCL`(Hc(n), Hc(n)) of HC Hc-bimodules with minimal support

is equivalent, as a monoidal category, to the category of representations of S`.

The proof of Proposition 6.3.3 will be done by induction on r. The proof for the case r = 1

is based on a remarkable symmetry result obtained in [CEE], see also [EGL]. There is a

symmetry of parameters for the simple quotients of spherical rational Cherednik algebras,

namely, for positive integers n,N (not necessarily coprime) consider the Cherednik algebras

HN/n(n) and Hn/N(N), with maximal ideals Jmax and J ′max, respectively. Both parameters

are spherical so eJmaxe, e′J ′maxe′ are the maximal ideals of the spherical Cherednik alge-

bras AN/n(n) and An/N(N), respectively. Here, e ∈ CSn and e′ ∈ CSN denote the trivial

idempotents in their respective group algebras.

Proposition 6.3.4 (Proposition 9.5, [CEE] and Proposition 7.7 [EGL]). There is an iso-

morphism between the algebras AN/n(n)/eJmaxe and An/N(N)/e′J ′maxe′. Moreover, an iso-

morphism maps (the images of) the subalgebras C[Rn]Sn ,C[R∗n]Sn to (the images of) the

subalgebras C[RN ]SN , C[R∗N ]SN , respectively.

Since HC Ac-bimodules with minimal support are precisely the ones whose annihilator is the

maximal ideal in Ac, we have the following easy consequence of Proposition 6.3.4.

Proposition 6.3.5. The isomorphism AN/n(n)/eJmaxe ∼= An/N(N)/e′J ′maxe′ induces a ten-

sor equivalence between the categories of minimally supported HC AN/n(n)-bimodules and

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minimally supported HC An/N(N)-bimodules.

Now, the parameter c = r/m > 0, with gcd(r;m) = 1 and m` = n for k ∈ Z>1, is spherical

for the rational Cherednik algebra associated to Sn. Then, Proposition 6.3.5 has the following

consequence.

Corollary 6.3.6. The categories of minimally supported HC Hr/m(n)-bimodules and mini-

mally supported HC Hm/r(r`)-bimodules are equivalent as monoidal categories.

Now the case r = 1 of Proposition 6.3.3 is an easy consequence of Corollary 6.3.6 and

Theorem 5.4.4, that asserts that the category of HC Hm(k)-bimodules is equivalent, as a

monoidal category, to the category of representations of S`, see also [BEG, Theorem 8.5].

To complete the proof of Proposition 6.3.3 we use an inductive argument for which we will

need the theory of shift functors for rational Cherednik algebras, cf. Definition 3.1.5, which

in type A originally appeared in [GS, Section 3]. Consider the (Ac+1(n), Ac(n))-bimodule

Qc+1c := eHc+1(n)esignδ. Here, esign denotes the sign idempotent, esign = 1

n!

∑σ∈Sn sign(σ)σ.

The bimodule Qc+1c is HC, cf. Proposition 3.1.4. The functor F : Ac(n)-mod→ Ac+1(n)-mod

given by F (M) = Qc+1c ⊗Ac M is then an equivalence of categories, see [BE, Corollary 4.3].

A quasi-inverse functor is given by tensoring with the (Ac+1(n), Ac(n))-bimodule P c+1c :=

δ−1esignHc+1(n)e, see Section 3 in [GS] (we remark that [GS] assumes that c 6∈ 12

+ Z, an

assumption that was later removed in [BE, Corollary 4.3]). The bimodule P c+1c is also HC.

It then follows that we have an equivalence of monoidal categories F : HC(Ac(n), Ac(n))→

HC(Ac+1(n), Ac+1(n)), F (B) = Qc+1c ⊗Ac(n)B⊗Ac(n)P

c+1c . Clearly, this equivalence preserves

the filtrations of the categories of HC bimodules by the support.

We now proceed to finish the proof of Proposition 6.3.3. So let r,m, n, ` be as in the statement

of that proposition. We work over spherical subalgebras, and we make the following inductive

assumption:

For every 0 < r′ < r and every m′, `′ ∈ Z>0 with gcd(r′,m′) = 1, the category of minimally

supported bimodules HCL`′ (Ar′/m′(m′`′), Ar′/m′(m

′`′)) is equivalent, as a monoidal category,

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to the category of representations of S`′.

Clearly, Proposition 6.3.5, together with Theorem 5.4.4, give the base of induction. Now,

using Proposition 6.3.5 again, we have that the categories HCL`(Ar/m(n), Ar/m(n)) and

HCL`(Am/r(`r), Am/r(`r)) are equivalent as monoidal categories. Using shift functors, we

get a tensor equivalence between HCL`(Am/r(`r), Am/r(`r)) and HCL`(Ar′/r(`r), Ar′/r(`r)),

where 0 < r′ < r. By our inductive assumption, this is tensor equivalent to S`-rep. Propo-

sition 6.3.3 now follows by sphericity, since we are assuming our parameter c is positive.

The description of the irreducible objects in HCL`(Hr/m(n), Hr/m(n)) follows at once from

Theorem 6.1.1. We have that S` = Lc(m triv`) = Lc(triv), while the irreducible modules in

O`c are those of the form Lc(mλ) for a partition λ ` `. Thus, the irreducible HC bimodules

with minimal support have the form Homfin(Lc(triv), Lc(mλ)), λ ` `.

Remark 6.3.7. We remark that, while {Homfin(L(triv), L(mλ)) : λ ` `} forms a com-

plete and irredundant list of irreducible HC bimodules with minimal support, we have that

Homfin(L(mµ), L(mλ)) 6= 0 where λ, µ are any partitions of `. This follows from, for exam-

ple, Theorem 8.16 in [BEG], which gives a description of Homfin(L(mµ), L(mλ)) as a direct

sum of bimodules of the form Homfin(L(triv), L(mξ)).

To finish this subsection, let us explain what happens when we have c = r/m < 0, with

gcd(r;m) = 1 and m divides n, say `m = n. In this case, the category HCL`(Hc(n), Hc(n))

is still equivalent to the category of representations of S`. This follows because there is

an equivalence HCL`(Hc(n), Hc(n)) ∼= HCL`(H−c(n), H−c(n)) induced by an isomorphism

Hc(n)→ H−c(n), mapping R∗ 3 x 7→ x, R 3 y 7→ y, Sn 3 σ 7→ sign(σ)σ.

6.3.3 Description of irreducibles

We use the results of the previous subsection and Theorem 4.1.1 to give a classification of

all irreducible HC Hc(n)-bimodules where we assume that c has the form c = r/m > 0, with

1 < m ≤ n and gcd(r;m) = 1. The following is the main result of this section.

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Theorem 6.3.8. Let c = r/m > 0, with 1 < m ≤ n, gcd(r;m) = 1, and let i =

1, . . . , bn/mc. Then, the category HCLi(Hc(n), Hc(n)) is equivalent, as a monoidal category,

to the category of representations of Si.

Before proceeding to the proof of Theorem 6.3.8 we set W := S×im ⊆ Sn and describe

the objects in the category HCΞ0 (Hc, Hc) where, recall, Ξ = NSn(W )/W . This category

is equivalent to the category of representations of Ξ = Sn−mi × Si, this follows because

the algebra Hc has a unique irreducible finite dimensional bimodule (that does not admit

non-trivial self-extensions). This bimodule is B := HomC(Lc(trivW ), Lc(trivW )), this is a

consequence of the fact that L(trivW ) is the unique irreducible finite dimensional module

over the algebra Hc. Moreover, since c = r/m and W = S×im , we have that Hc = Hc(m)⊗i,

and B = B⊗i, where B is the unique irreducible finite dimensional bimodule over Hc(m),

so B admits a Ξ-equivariant structure, where Si permutes the tensor factors and Sn−mi acts

trivially. Under the equivalence HCΞ0 (Hc, Hc) → (Si × Sn−mi)-rep, B corresponds to the

trivial representation. So we have the following result.

Lemma 6.3.9. The irreducible objects in HCΞ0 (Hc) have the form B⊗ ξ, where ξ runs over

the set of irreducible representations of Si × Sn−mi, which acts diagonally. The irreducibles

where Sn−mi acts trivially correspond precisely to those representations B⊗ξ where ξ factors

through Si.

Proof of Theorem 6.3.8. We need to check that, if ξ is an irreducible representation of Si,

then the equivariant bimodule B ⊗ ξ belongs to the image of •†SnW . By Theorem 4.1.1, for

every parabolic subgroup W ′ containing W in corank 1 we need to produce a HC Hc(W′)-

bimodule B′ with B′†W ′W= B⊗ ξ, with the restricted NW ′(W )/W -equivariant structure. The

subgroups W ′ have three different types. Either W ′ ∼= S×(i−2)m × S2m, W ′ ∼= S

×(i−1)m × Sm+1

or W ′ ∼= S×im × S2.

Case 1. W ′ ∼= S×(i−2)m × S2m. So that NW ′(W )/W ∼= S2 acting on Hc = Hr/m(m)⊗i by

permuting two of the tensor factors. Thanks to the results of Section 6.3.2 the functor •†W ′W :

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HCL(Hc(W′), Hc(W

′))→ HCS20 (Hc, Hc) is essentially surjective, were HCL(Hc(W

′), Hc(W′))

denotes the category of minimally supported HC Hc(W′)-bimodules. So we can certainly

find a bimodule B′ with B′†W ′W= B ⊗ ξ.

Case 2. W ′ ∼= S×(i−1)m × Sm+1, so that NW ′(W )/W ∼= {1}. Thus, what we have to check

here is that B belongs to the image of the functor •†W ′W . But this follows because the image

of •†W ′W is closed under sub-bimodules.

Case 3. W ′ ∼= Sim × S2, so that NW ′(W )/W ∼= S2, acting trivially on Hc. Thanks to our

assumptions on ξ, S2 also acts trivially on ξ. It also acts trivially on B. So we need to

check that B, with trivial action of S2, belongs to the image of •†W ′W . Upon the identification

HCS20 (Hc, Hc) → S2-rep, B corresponds to the trivial representation. The image of the

restriction functor is closed under tensor products and sub-bimodules. Since the trivial

representation of S2 is contained in S⊗2 for any representation S of S2, the result follows.

Now the proof of Theorem 6.3.8 follows since, by Corollary 6.3.2, the number of irreducible

objects in HCLi(Hc(n), Hc(n)) is no greater than the number of irreducible representations

of Si. �

Note that Theorem 6.3.8 and Corollary 6.3.2 imply Theorem 6.1.3. As a consequence, we

have that the irreducible HC Hc(n)-bimodules have the form Homfin(∆c(triv), Lc(mµ+ ν)),

where µ and ν are partitions such that m|µ|+ |ν| = n and ν is m-trivial.

6.4 Two-parametric case.

6.4.1 Proof of Theorem 6.1.4

We study the category HC(Hc(n), Hc′(n)) when the parameters c, c′ are distinct. First,

we remark that if either c or c′ is a regular parameter, then every HC (Hc, Hc′)-bimodule

has full support, and this case has been described in Theorem 5.4.4. So we may assume

that both parameters c, c′ are singular. Since for a HC bimodule B we have SS(B) =

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SS(Hc/LAnn(B)) = SS(Hc′/RAnn(B)), the description of the two-sided ideals of Hc given

in Section 6.1, together with Theorem 5.4.4 imply that a necessary condition for HC(Hc, Hc′)

to be nonzero is that c and c′ have the same denominator when expressed as irreducible

fractions. Then, throughout this subsection we assume that c = r/m, c′ = r′/m, gcd(r;m) =

gcd(r′;m) = 1, 1 < m ≤ n.

Recall that, for i = 1, . . . , bn/mc we have the functor KZic′ : Oic′ → (CSi⊗Hq′(Sn−mi))-mod,

where q′ = exp(2π√−1c′). LetN = Sic′ ∈ Oic′ be the irreducible module with KZic′(N) = triv,

this is the unique irreducible subquotient of ∆c′(τ) supported on Xi, where τ = triv is c′ > 0

and τ = sign if c′ < 0. Then, reasoning completely analogously to the proof of Lemma 6.2.5,

we have that every irreducible HC (Hc(n), Hc′(n))-bimodule supported on the closure of the

symplectic leaf Li is contained in a bimodule of the form Homfin(Sic′ ,M) for an irreducible

module M ∈ Oic and, moreover, that whenever Homfin(Sic′ ,M) is nonzero then, for every

module L ∈ (CSi ⊗Hq′(n))-mod the Bi × Bn−mi-module KZi(M)⊗C L factors through the

algebra CSi ⊗Hq(n−mi).

Proposition 6.4.1. Let i ∈ {1, . . . , bn/mc} and assume that n−mi 6= 0. Then, the category

HCLi(Hc, Hc′) is 0 unless c− c′ ∈ Z or c+ c′ ∈ Z.

Proof. Assume HCLi(Hc, Hc′) 6= 0. Let M ∈ Oic be irreducible such that Homfin(Sic′ ,M) 6= 0.

Let KZi(M) = M1⊗M2, whereM1 is an irreducible representation of Si andM2 an irreducible

representation ofHq(n−mi). Then, for every representation L = L1⊗L2 of CSi⊗Hq′(n−mi),

we have that the module (L1 ⊗M1)⊗ (L2 ⊗M2) is a representation of CSi ⊗Hq′(n−mi).

If M2 is the trivial representation of Hq(n −mi), this implies that {1,−q′} = {1,−q} and

therefore c− c′ ∈ Z. Otherwise, we get {−q, qq′} = {1,−q}, so c+ c′ ∈ Z.

Now assume that c′ = c + k, with c > 0 and k ∈ Z>0. Then, using shift functors we

have an equivalence of categories HC(Hc, Hc) ∼= HC(Hc, Hc′) ∼= HC(Hc′ , Hc′) preserving the

filtration by supports so that, in particular, they descend to equivalences HCLi(Hc, Hc) ∼=

HCLi(Hc, Hc′) ∼= HCLi(Hc′ , Hc′). Since we have an isomorphism Hc′(n) → H−c′(n) fixing

108

the subalgebras C[R]Sn ,C[R∗]Sn , we also have an equivalence HC(Hc, H−c−k) ∼= HC(Hc, Hc)

preserving the filtration by supports. Similar results hold if c < 0 and k ∈ Z<0.

Assume now that c = c′ + 1, with −1 < c′ < 0. In this case, the shift functor is not an

equivalence. However, it does induce a derived equivalence cPc′ ⊗LHc′ • : Db(Oc′)→ Db(Oc),

see e.g. [GL, Section 5]. It follows that, if we denote by DbHC(Hc, Hc′) the subcategory of

Db((Hc, Hc′)-bimod) consisting of complexes with HC homology, then we have a derived

equivalence cPc′ ⊗LHc • : DbHC(Hc′ , Hc) → Db

HC(Hc, Hc). Thus, the categories HC(Hc, Hc)

and HC(Hc, Hc′) have the same number of irreducibles. We remark here that an analogous

of Lemma 6.1.5 implies that the category HCLi(Hc, Hc′) is embedded, using the restriction

functor, in the category of representations of Si. Hence, HCLi(Hc, Hc′) ∼= Si-rep for i =

1, . . . , bn/mc. The same holds for the category HCLi(Hc′ , Hc).

Note that Theorem 6.1.4 follows from Proposition 6.4.1 together with the discussion above.

6.4.2 Bimodules with minimal support

We now come to the case that is not covered by Proposition 6.4.1. Note that this proposition

may fail when m divides n and we look at the categories of bimodules with minimal support.

As an easy example, if c = r/n, c′ = r′/n, gcd(r;n) = gcd(r′;n) = 1, then HC0(Hc, Hc′) 6= 0,

so it does not matter whether c+ c′ or c− c′ are integers.

So assume m divides n, say n = m`, 1 < m ≤ n. Let c = r/m, c′ = r′/m as irreducible

fractions. We have the following result.

Proposition 6.4.2. The category HCL`(Hc(n), Hc′(n)) is equivalent to the category of rep-

resentations of S`.

Note that in Proposition 6.4.2 we do not impose any other conditions on c and c′. We just

require that they are expressed as irreducible fractions with the same denominator which is

a factor of n, with quotient `.

109

Proof. We proceed in several steps.

Step 1. We remark that, using the isomorphisms Hc(n) → H−c(n), we may assume that

both c, c′ are positive. Moreover, using shift functors, we may assume that 0 < c, c′ < 1. So

we have 0 < r, r′ < m ≤ n. Since both c and c′ are positive, they are spherical. So we can

work over the spherical subalgebras Ac(n), Ac′(n).

Step 2. Let us introduce the following notation. For a positive integer N , set RN =

{(z1, . . . , zN) ∈ CN :∑N

i=1 zi = 0}. This is, of course, the reflection representation of SN .

Let x1, . . . , xN be the coordinate functions on CN , and for a positive integer k let pk,N(x) =

xr1 + · · ·+ xrN . So the invariant algebra C[RN ]SN is generated by p2,N(x), . . . , pN,N(x). Simi-

larly, the invariant algebra C[R∗N ]SN is generated by p2,N(y), · · · , pN,N(y).

Step 3. For c > 0, let us denote by Ac(n) the quotient of the spherical subalgebra Ac(n)

by its unique maximal ideal. Recall the isomorphism ϕN,M : AM/N(N) → AN/M(M) from

Proposition 6.3.4. It is known that ϕN,M maps pk,N(x) 7→ (N/M)pk,M(x), while pk,N(y) 7→

(M/N)k−1pk,M(y). Both of these assertions follow from [CEE, Section 8], see also [EGL,

Section 7].

Step 4. In Steps 4-8 we are going to produce an equivalence from the category of minimally

supported bimodules HCL`(Ac(n), Ac′(n)) to HC(AN1(`), AN2(`)) for some integers N1, N2 ∈

Z>0, Proposition 6.4.2 follows from here. First of all note that, since we are taking bimodules

with minimal support, HCL`(Ac(n), Ac′(n)) = HCL`(Ac(n), Ac′(n)), so we are going to think

of objects in HCL`(Ac(n), Ac′(n)) as honest bimodules. So let B ∈ HCL`(Ar/m(n), Ar′/m(n)).

Since B has minimal support it is, in particular, a (Ar/m(n), Ar′/m(n))-bimodule. Using the

isomorphisms ϕN,`r, ϕN,`r′ , we may think of B as an (Am/r(`r), Am/r′(`r′))-bimodule, equiv-

alently, as an (Am/r(`r), Am/r′(`r′))-bimodule whose left (resp. right) annihilator coincides

with the maximal ideal of Am/r(`r) (resp. of Am/r′(`r′)). By Step 3, the following operators

110

act locally nilpotently on B:

ak(x) : b 7→ (m/r)pk,`r(x)b− (m/r′)bpk,`r′(x) (6.5)

dk(y) : b 7→ (r/m)k−1pk,`r(y)b− (r′/m)k−1bpk,`r′(y) (6.6)

Step 5. Now let m = rk1 + m1, with 0 ≤ m1 < r, k1 ∈ Z. So we have the shift

(Am1/r(`r), Am/r(`r))-bimodule, say Pm1/r,m/r(`r). Consider B′ := Pm1/r,mr(`r) ⊗Am/r(`r) B,

which is an (Am1/r(`r), Am/r′(`r′))-bimodule . We claim that the operators (6.5), (6.6) act lo-

cally nilpotently onB′. For (6.5), this follows because ak(x)(b1⊗b2) = ad((m/r)pk,`r(x))(b1)⊗

b2 − b1 ⊗ ak(x)(b2) and the operator ad((m/r)pk,`r(x)) acts locally nilpotently on Pm1/r,mr .

The reasoning for (6.6) is the same. Now let m′ = r′k′1 + m′1 be division with remain-

der, and consider the shift (Am/r′(`r′), Am′1/r′(`r

′))-bimodule Pm/r′,m′1/r′(`r′). Let B1 :=

B′⊗Am/r′ (`r′) Pm/r′,m′1/r′(`r′). This is an (Am1/r(`r), Am′1/r′(`r

′))-bimodule. It is clear that its

left (resp. right) annihilator is the maximal ideal in Am1/r(`r) (resp. Am′1/r′(`r′)), that it is

finitely generated as a left or right module, and that the operators (6.5), (6.6) act locally

nilpotently on B1.

Step 6. Now we use the isomorphisms ϕ`r,`m1 and ϕ`r′,`m′1 to view B1 as bimoule over

(Ar/m1(`m1), Ar′/m′1(`m′1)). Note that the operators that act locally nilpotently on B1 now

are

a1k(x) : b 7→ (m/m1)pk,`m1(x)b− (m/m′1)bpk,`m′1(x) (6.7)

d1k(y) : b 7→ (m1/m)k−1pk,`m1(y)b− (m′1/m)k−1bpk,`m′1(y) (6.8)

And we repeat the same procedure of multiplying by shift bimodules on the left and right,

to get an (Ar1/m1(`m1), Ar′1/m′1(`m′1))-bimodule B2, which is finitely generated as a left or

right module, and on which the operators (6.7), (6.8) act locally nilpotently.

Step 7. Continuing with this procedure, since gcd(r,m) = 1 = gcd(r′,m), the Euclidean

algorithm tells us that we are going to get to a (AN1(`), AN2(`))-bimodule B, where N1, N2

111

are integers, B is finitely generated as either a left or right module, and the operators

b 7→ mpk,`(x)b−mbpk,`(x) (6.9)

b 7→ (1/m)k−1pk,`(y)b− (1/m)k−1bpk,`(y) (6.10)

act locally nilpotently. From (6.9), it follows that C[R`]S` acts locally nilpotently on B. From

(6.10) it follows that C[R∗` ]S` acts locally nilpotently on B, too. Thus, B ∈ HC(AN1(`), AN2(`)).

Step 8. It is clear that everything we have done in Steps 4-7 can be reversed. So we get a

category equivalence HCL`(Ac(n), Ac′(n)) ∼= HC(AN1(`), AN2(`)). Since all our parameters

are positive, hence spherical, we get HCL`(Hc(n), Hc′(n)) ∼= HC(HN1(`), HN2(`)). The latter

category is equivalent to the category of representations of S`, this follows from Theorem

5.1.1. We are done.

Let us summarize our results in the two-parametric setting in the following theorem.

Theorem 6.4.3. Let c = r/m, c′ = r′/m be such that gcd(r;m) = 1 = gcd(r′;m′), and

1 < m ≤ n. Then.

1. Let i = 0, . . . , bn/mc. Then, HCLi(Hc(n), Hc′(n)) = 0 unless c− c′ ∈ Z or c + c′ ∈ Z.

If c − c′ ∈ Z or c + c′ ∈ Z, then HCLi(Hc(n), Hc′(n)) is equivalent to the category of

representations of Si.

2. If n−mi = 0, then HCLi(Hc(n), Hc′(n)) is equivalent to the category of representations

of Si, without further restrictions on the parameters c, c′.

Note that, in particular, if c = r/m, c′ = r′/m with m` = n but neither c − c′ nor c + c′ is

an integer, then HC(Hc(n), Hc′(n)) = HCLi(Hc(n), Hc′(n)).

112

Chapter 7

Duality

In this chapter, we construct a duality on the category of Harish-Chandra bimodules for type

A (or, more generally, cyclotomic) rational Cherednik algebras. Our construction, however,

is done at the level of algebras quantizing Nakajima quiver varieties. These algebras are

special cases of the algebras that are studied in [BPW, BLPW], that is, they arise as the

global sections of a quantization of a smooth conical variety. The representation theory

of quantized quiver varieties has been studied in more detail in [BL, L9], where Harish-

Chandra bimodules play a crucial role. The main result in this chapter is the construction

of a duality functor in the category of Harish-Chandra bimodules. To obtain this functor

we need to introduce categories of twisted HC bimodules. The twist here is provided by an

automorphism of the quiver variety.

Since the spherical subalgebra of a cyclotomic rational Cherednik algebra is a special case

of a quantized quiver variety, cf. [EGGO, Go, L2, O], we obtain, in particular, a duality

in the category of HC bimodules for cyclotomic Cherednik algebras. For type A Cherednik

algebras, we show that this duality fixes every irreducible module.

113

7.1 Nakajima quiver varieties

In this section, we define Nakajima quiver varieties. These are defined as the G.I.T. Hamil-

tonian reduction of a reductive group acting on a representation space of a quiver. Thus,

first we study this action and the respective moment map. Later, we study the G.I.T. prop-

erties of the action, giving in particular a concrete description of the semistable points (for a

particular choice of stability condition). After that, we define both the affine and projective

Nakajima quiver varieties, as well as their universal deformations. This section does not con-

tain new results. Nakajima quiver varieties were first studied in [Nak], and our exposition

here mostly follows [Nak2, Gi3].

7.1.1 Representation spaces and moment maps.

Let Q = (Q0, Q1, s, t) be a quiver: Q0 is the set of vertices; Q1 the set of arrows; and

s, t : Q1 → Q0 the maps that to each arrow assign its starting and terminating vertex,

respectively. We will consider the co-framed quiver:

Q♥ := (Q♥0 = Q0 tQ′0, Q♥1 := Q1 tQ′1, s♥, t♥)

where Q′0 := {k′ : k ∈ Q0} is a copy of Q0; Q′1 := {αi : i ∈ Q0} is a set of arrows indexed by

Q0; s♥|Q1 = s, s(αk) = k, t♥|Q1 = t, and t♥(αk) = k′. In pedestrian terms, the quiver Q♥

is obtained from Q by attaching to each vertex k ∈ Q a coframing vertex i′ with an arrow

k → k′.

Now let v,w ∈ Z<0 be dimension vectors. We can consider the space of representations:

R(v,w) := Rep(Q♥, (v,w)) =⊕α∈Q1

Mat(vs(α),vt(α))⊕⊕k∈Q0

Mat(vk,wk)

We will denote an element of R(v,w) by (Xα, ik)α∈Q1,k∈Q0 .Note that the reductive group

G := GL(v) :=∏

k∈Q0GL(vk) acts on the space R(v,w) by changing basis. For g =

114

(gk)k∈Q0 ∈ G we get

g.(Xα, ik) = (gt(α)Xαg−1s(α), ikg

−1k )

We take the induced action of G on T ∗R. Since this is an induced action, it is Hamiltonian,

let us describe the moment map. First, we will describe this action in linear-algebraic terms.

For any m > 0 we have a GL(n)-equivariant isomorphism Mat(n,m)∗ ∼= Mat(m,n) which is

given by the trace form. Thus, we have a G-equivariant identification of T ∗R with the space

of representations of the double quiver Q♥

T ∗R =⊕α∈Q1

(Mat(vs(α),vt(α))⊕Mat(vt(α),vs(α))

)⊕⊕k∈Q0

(Mat(vk,wk)⊕Mat(wk,vk))

We will denote an element of T ∗R by (Xα, Yα, ik, jk). Thus, the action of G is given by

g.(Xα, Yα, ik, jk) = (gt(α)Xαg−1s(α), gs(α)Yαg

−1t(α), ikg

−1k , gkjk)

And the moment map is given by

µ : T ∗R→ gl(v)∗ = gl(v) =: g, (Xα, Yα, ik, jk) 7→∑α∈Q1

(YαXα −XαYα)−∑k∈Q0

jkik (7.1)

Equation (7.1) should be interpreted as follows. First, we have identified gl(v) with its dual

via the trace form. We have that µ(Xα, Yα, ik, jk) is a (Q0)-tuple of matrices. So what

Equation 7.1 says is that, for k ∈ Q0 in the k-th position we have

∑α∈Q1

s(α)=k

YαXα −∑α∈Q1

t(α)=k

XαYα − jkik

The dual to this map is the comoment map µ∗ : g → C[T ∗R]. Since the moment map µ is

G-equivariant, for every λ ∈ (g∗)G the group G acts on µ−1(λ). For λ ∈ (g∗)G, we will be

interested in the affine variety

115

M0λ :=M0

λ(v,w) := µ−1(λ)//G = Spec(C[µ−1(λ)]G) = Spec([C[T ∗R]/({µ∗(ξ)−〈λ, ξ〉 : ξ ∈ g})]G)

Note that the variety M00 comes equipped with a C×-action induced by the C× action on

T ∗R by dilations. In particular, the Poisson bracket on M00 has degree −2. In general, M0

λ

is singular. In some cases, we can construct resolutions of singularities by looking at G.I.T.

quotients of T ∗R. This is what we will do next.

7.1.2 G.I.T. quotients

Note that the group of characters X(GL(v)) may be identified with ZQ0 , to θ = (θk)k∈Q0 we

associate the character (gk) 7→∏

k∈Q0det(gk)

θk . For θ ∈ Q0, we have the graded algebra of

semi-invariants

C[T ∗R]θ−si :=⊕n≥0

C[T ∗R]nθ (7.2)

where, recall, C[T ∗R]nθ := {f ∈ C[T ∗R] : f(g−1x) = θn(g)f(x) for every x ∈ T ∗R}.

Recall that a point x ∈ T ∗R is said to be θ-semistable if there exist n > 0 and f ∈ C[T ∗R]nθ

such that f(x) 6= 0. Let us describe the semistable points with respect to certain stability

conditions. First of all, for a Q♥-representation x = (Xα, Yα, ik, jk) ∈ T ∗R = Rep(Q♥, (v,w))

we denote by x = (Xα, Yα) ∈ Rep(Q, (v)) the representation of Q that is obtained by

forgetting the framing and coframing maps.

Proposition 7.1.1 (Proposition 5.1.5, [Gi3]). 1. Assume θ ∈ ZQ0

>0. Then, a representa-

tion x = (Xα, Yα, ik, jk) ∈ T ∗R is θ-semistable if and only if the only subrepresentation

of x which contains (im(jk))k∈Q0 is the entire representation x.

2. Assume θ ∈ ZQ0

<0. Then, a representation x = (Xα, Yα, ik, jk) ∈ T ∗R is θ-semistable if

and only if the only subrepresentation of x contained in (ker(ik))k∈Q0 is the 0 represen-

tation.

116

We will denote θ+ := (1, 1, . . . , 1) and θ− := −θ+. In particular, θ+ falls under (1) of the

previous proposition, while θ− falls under (2).

Now we proceed to define the G.I.T. Hamiltonian reduction of T ∗R by the action of G. First

of all, obviously G acts on µ−1(λ) for λ ∈ (g∗)G. So, similarly to (7.2) we may define the

algebra of semi-invariants C[µ−1(λ)]θ−si, and the set of θ-semistable points µ−1(λ)θ−ss. We

remark that µ−1(λ)θ−ss = (T ∗R)θ−ss ∩ µ−1(λ), this is a consequence of the Hilbert-Mumford

criterion. Now we can define the G.I.T. Hamiltonian reduction

Mθλ := Proj(C[µ−1(λ)]θ−si) = µ−1(λ)θ−ss//G (7.3)

Let us remark, first, that the formalism of Hamiltonian reduction implies that the symplectic

structure on T ∗R gives Mθλ the structure of an algebraic Poisson variety. We also remark

that the 0th graded component of the algebra C[µ−1(λ)]θ−si is precisely C[µ−1(λ)]G, so we

have a projective morphism $ : Mθλ → M0

λ. We will state sufficient conditions for this

morphism to be a resolution of singularities.

Of course, for $ : Mθλ →M0

λ to be a resolution of singularities we need, first, that Mθλ is

smooth. The varietyMθλ will be smooth and symplectic provided the G-action on µ−1(λ)θ−ss

is free. When this happens, we will say that the pair (θ, λ) is generic. The following result

gives a sufficient condition for a pair (θ, λ) to be generic. We denote by g(Q) the Kac-Moody

algebra associated to the quiver Q. By αi, we mean the simple root of g(Q) corresponding

to the vertex i ∈ Q0.

Proposition 7.1.2. The pair (θ, λ) is generic if there is no v′ ∈ ZQ0

≥0 such that:

1. Componentwise, v′ ≤ v.

2.∑

i∈Q0v′iα

i is a root of g(Q).

3.∑

i∈Q0v′iλi =

∑i∈Q0

v′iθi = 0.

117

For example, both θ+ and θ− are generic, in the sense that (θ±, λ) is generic for any λ ∈ (g∗)G.

This will be important for us later.

From now on, we assume that the pair (θ, λ) is generic. Let us denote by Mλ the affine

variety Spec[Γ(Mθλ,SMθλ )]. In particular, we have a projective morphism ρ :Mθ

λ →Mλ.

Proposition 7.1.3 (Proposition 2.3, [BL]). The map ρ is a resolution of singularities. More-

over, the variety Mλ is independent of the stability condition θ.

So the map $ will be a resolution of singularities provided we have Mλ = M0λ (assuming

(θ, λ) is generic). This is the case when the moment map µ is flat, [BL, Proposition 2.5].

Sufficient conditions for this to happen were found by Crawley-Boevey in [CB] see, for

example, Theorem 1.1 in loc. cit. A consequence of this is that µ is flat whenever Q is a

finite or affine quiver and ν :=∑

i∈Q0(wiωi − viαi) is a dominant weight for g(Q), where

ωi, αi denote the fundamental weights and simple roots for g(Q), respectively. In general,

all constructions that follow remain valid if we replace the variety M0λ by Mλ.

7.1.3 Universal quiver varieties

Now assume that the stability condition θ is such that (θ, λ) is a generic pair for every λ.

For example, we can take θ = θ+ or θ−. In this case, the action of G on µ−1((g∗)G)θ−ss is

free, and so the “universal” quiver varieties

Mθp := µ−1((g∗)G)θ−ss//G, M0

p := µ−1((g∗)G)//G, Mp := Spec(Γ(Mθp,SMθ

p))

are smooth and symplectic. We remark that Mθp is a scheme over p := (g∗)G and its

specialization to λ ∈ p coincides withMθλ. In particular,Mθ

p is a deformation ofMθ0 over p.

Note that we have an action of C× onMθp that restricts to the usual action onMθ

0 on the fiber

over 0. On the other hand, Namikawa, cf. [Nam], has proved that the variety Mθ0 admits

a universal deformation M θ over the space H2DR(Mθ

0,C). Let us see the relation between

the universal deformation M θ and the universal quiver varietyMθp. We have a natural map

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p → H2DR(Mθ

0,C) given as follows. Let χ ∈ X(G) be a character. Consider the line bundle

Vχ on T ∗R, which is trivial as a line bundle and with a G-action given by χ−1. Since Vχ is

G-equivariant, its restriction Vχ|θ−ssµ−1(0) descends to a line bundle S(χ) on Mθ0. For example,

the line bundle S(θ) is ample by definition. This defines a map ι : X(G) → H2DR(Mθ

0,C),

χ 7→ c1(S(χ)) (the first Chern class) which extends by linearity to ι : p→ H2DR(Mθ

0,C). We

will assume the following.

Assumption 7.1.4. The map ι : p → H2DR(Mθ

0,C) is an isomorphism and therefore the

universal deformation M θ of Mθ0 coincides with Mθ

p.

We remark that Assumption 7.1.4 is known to be true when Q is a finite or affine quiver,

and it is conjectured to hold in all cases, [BPW]. If we do not make Assumption 7.1.4 we

simply have Mθ0 = p×H2

DR(Mθ0,C) M θ.

7.1.4 Isomorphisms between quiver varieties

Let us remark that we have the following C×-equivariant symplectomorphism of T ∗R

Υ : T ∗R→ T ∗R

(Xα, Yα, ik, jk) 7→ (−Y tα, X

tα,−jtk, itk)

(7.4)

where •t denotes matrix transposition. Note, however, that this map is not G-equivariant.

Rather, we have that Υ(g.x) = (gt)−1Υ(x). This implies the following.

Lemma 7.1.5. 1. For any character θ ∈ X(G), the map Υ induces a graded isomorphism

Υ∗ : C[T ∗R]θ−si → C[T ∗R]−θ−si.

2. The following diagram commutes

T ∗R Υ //

µ

��

T ∗R

µ

��g

ξ 7→−ξt // g

119

Corollary 7.1.6. For any λ ∈ (g∗)G, we have an induced, graded isomorphism

C[µ−1(λ)]θ−si → C[µ−1(−λ)]−θ−si

and consequently we have an isomorphism of projective varieties Mθλ → M−θ

−λ. These iso-

morphisms glue together to an isomorphism Mθp → M−θ

p such that the following diagram

commutes

Mθp

//

��

M−θp

��p

λ 7→−λ // p

Let us remark that Υ also induces a C×-equivariant Poisson automorphism of the affine

quiver variety Υ : M00 → M0

0. We will denote by Υ∗ : C[M00] → C[M0

0] the induced

automorphism on its algebra of functions.

7.2 Quantizations

Let us proceed to quantizations of the quiver varieties M00,Mθ

0 and Mθp. Since M0

0 is an

affine variety, its quantizations are straightforward to define. It is slightly harder to define

quantizations of the varieties Mθ0 and Mθ

p. We follow, mostly, [BL]. The only new result of

this section is in Section 7.2.4, see (7.7), but is not really original.

7.2.1 Quantizations of M00

Recall that M00 is an affine, Poisson variety with a C×-action. Moreover, note that the

Poisson bracket has degree −2 with respect to the C×-action.

Definition 7.2.1. A quantization of M00 is a pair (A, ι) where

1. A is an associative, filtered algebra A =⋃i≥0Ai such that, for a ∈ Ai, b ∈ Aj,

[a, b] ∈ Ai+j−2 (in particular, this implies that grA is equipped with a Poisson bracket

of degree −2.)

120

2. ι : grA → C[M00] is an isomorphism of graded Poisson algebras.

By an isomorphism of quantizations we mean a filtered isomorphism f : A → A′ that

induces the identity on C[M00] on the associated graded level. Next, we review how to get

quantizations of the M00 using quantum Hamiltonian reduction.

Quantum Hamiltonian reduction: algebra level

Let us, first, describe the general case. Then we will specialize to our situation with quiver

varieties.

Let G be a reductive algebraic group acting on an algebra A by algebra automorphisms.

In particular, the Lie algebra g acts on A by derivations. For ξ ∈ g, let us denote by

ξA : A → A the corresponding derivation. We say that a map Φ : g → A is a quantum

comoment map if it is G-equivariant and, for ξ ∈ g, ξA = [Φ(g), ·]. Note, in particular,

that Φ([ξ, η]) = [Φ(ξ),Φ(η)]. So the quantum comoment map extends to an algebra map

Φ : U(g)→ A.

For a character λ ∈ (g∗)G, we consider the ideal Iλ := U(g){ξ − 〈λ, ξ〉 : ξ ∈ g}. Now we

define the quantum Hamiltonian reduction

Aλ := A///λG := [A/AΦ(Iλ)]G

We remark that Aλ has an algebra structure induced from the algebra structure on Aλ.

Moreover, if we denote by Mλ the cyclic A-module A/AΦ(Iλ) then Aλ = EndA(Mλ)opp.

As in the classical case, we may define a “universal” quantum Hamiltonian reduction, as

follows. Denote P := g/[g, g](∼= (g∗)G). Consider the ideal I := U(g)[g, g] ⊆ U(g). The

universal quantum Hamiltonian reduction is

AP := [A/AΦ(I)]G

we remark that this is an S(P)-algebra, that is, the image of P under the natural map

121

P → AP is contained in the center of AP. Under the natural identification P ∼= (g∗)G, we

have that Aλ coincides with the specialization of AP at λ.

Let us now specialize to the case of Nakajima quiver varieties. Here, the algebra A is the

algebra of global differential operators on R, D(R). The quantum comoment map is given as

follows: the action of G on R induces a map g→ VectR, ξ 7→ ξR. But VectR ⊆ D(R), so we

may consider this as a map to D(R), and this is a quantum comoment map. For λ ∈ (g∗)G,

we will denote by Aλ the quantum Hamiltonian reduction of D(R) at λ. If the moment map

µ is flat, this is a quantization of M00.

7.2.2 Quantizations of Mθ0

Now we want to define quantizations of the variety Mθ0. Since this is not an affine variety,

a quantization of it will not be specified by a single algebra of functions. Rather, we need

a sheaf on Mθ0 whose associated graded coincides with the structure sheaf SMθ

0. Note,

however, that the last sentence does not make sense as stated: for an open set U ⊆ Mθ0,

the algebra Γ(U,SMθ0) is only naturally graded when U is C×-stable. So, before, we need to

change the topology.

Definition 7.2.2. The conical topology on Mθ0 is that topology whose open set are the

Zariski open C×-stable subsets of Mθ0.

We remark that every point x ∈ Mθ0 has a Zariski open neighborhood which is affine and

C×-stable, this is known as Sumihiro’s theorem, [Su]. So the conical topology is still fine

enough for most purposes. A quantization of Mθ0 will then be a filtered sheaf of algebras

in the conical topology whose associated graded coincides with SMθ0. For technical reasons,

this sheaf of algebras is supposed to satisfy some conditions which we now make precise.

Definition 7.2.3. A quantization of Mθ0 is a pair (Aθ, ι) where Aθ is a filtered sheaf of

algebras in the conical topology and ι : grAθ →Mθ0 is a Poisson isomorphism, such that the

filtration on Aθ satisfies the following conditions.

122

1. It is complete, meaning that every Cauchy sequence in the topology determined by

the filtration converges. More explicitly, we require that for every sequence {xi}∞i=1 of

sections of Aθ such that for every n ∈ Z there exist i0 such that xi − xj ∈ (Aθ)≤n for

i, j > i0, there exists a section x ∈ Aθ such that for every m ∈ Z there exist j0 such

that x− xi ∈ (Aθ)≤m for every i > j0.

2. It is separated, meaning that⋂i∈Z(Aθ)≤i = 0. Equivalently, the limit x of the previous

paragraph is unique.

Quantum Hamiltonian reduction: sheaf level

Quantizations of Mθ0 may be produced similarly those of M0

0. Instead of using the algebra

D(R), we use the microlocalization of the sheaf of differential operators on R. This is a sheaf

DR on the conical topology of T ∗R, where the action of C× is by dilations on the cotangent

fibers, see, for example, [Gi], [K] for details on microlocalization. The global sections of DR

coincide with D(R).

Now let f ∈ C[T ∗R] be an nθ-semiinvariant element, where n > 0. Then, have the open set

Xf ⊆Mθ0 that is µ−1(0) ∩ {f 6= 0}//G. That the open sets Xf form a base of the topology

ofMθ0 follows by the definition of Proj. It is easy to see that if, moreover, f is homogeneous

with respect to the C×-action on T ∗R, then Xf is open in the conical topology, and the sets

Xf form a basis for the conical topology of Mθ0.

So, for λ ∈ (g∗)G we define the sheaf Dθλ by setting, on an open set Xf :

Dθλ(Xf ) := DR((T ∗R)f )///λG

It is possible to see that Dθλ defined in this way is a quantization of Mθ. Let us denote by

Dλ the algebra of global sections of Dθλ. This is a quantization ofM0. If the moment map µ

is flat, then we actually have that Dλ = Aλ. For proofs of these statements, see [BL, Section

2].

123

Let us remark that we also have the notion of quantizations of the universal quiver variety

MθP. This is a sheaf of C[p]-algebras in the conical topology (recall that we have an action of

C× on Mθp) satisfying conditions analogous to those of Definition 7.2.3, see [BPW, Section

3].

7.2.3 The period map

Isomorphism classes of quantizations ofMθ0 andMθ

p have been parametrized in [BK], [L2] by

the vector spaces H2DR(Mθ

0,C) and H2DR(Mθ

p/p,C), respectively. For λ ∈ H2DR(Mθ

0,C) = p,

we denote by Aθλ the quantization of Mθ0 with period λ. We remark that it is not the case

that Aθλ coincides with Dθλ, for this we would have to take a symmetrized quantum comoment

map, see [BPW, Section 3.4], [L2, Section 3.2].

The quantization ofMθ0 (or ofMθ

p) with period 0 is called the canonical quantization. It is

characterized by the fact that it is isomorphic, as a quantization, to its opposite, this is [L2,

Corollary 2.3.3]. In fact we have, Aθ−λ = (Aθλ)opp, cf. [L2].

7.2.4 Isomorphisms of quantizations

Now let A θ be the canonical quantization of the universal deformation M θ (= Mθp by

Assumption 7.1.4) of Mθ0. Recall that A θ is characterized by it being isomorphic to its

opposite. This implies, in particular, that we have an isomorphism Υ∗A −θ ∼= A θ, where

Υ : Mθp → M−θ

p is the isomorphism introduced in Subsection 7.1.4. Thus, we have an

induced isomorphism

Γ(M θ,A θ)∼=→ Γ(M−θ,A −θ) (7.5)

We remark, however, that (7.5) is not an isomorphism of C[p]-algebras. Rather, it induces

the automorphism on C[p] given by f(λ) 7→ f(−λ).

124

Now let λ ∈ p be a period of quantization. According to [BLPW], we have an isomorphism

of quantizations of M00:

Aλ = Γ(Mθ0,Aθλ) ∼= Γ(M θ,A θ)/Iλ (7.6)

where Iλ is the ideal generated by the maximal ideal of λ, mλ ⊆ C[p] ⊆ Γ(M θ,A θ). It

follows from (7.5) and (7.6) that we have an isomorphism

Φλ : Aλ∼=→ A−λ (7.7)

This is a filtered isomorphism that is, however, not an isomorphism of quantizations. Indeed,

it follows by construction that the associated graded of (7.7) coincides with the isomorphism

Υ∗ : C[M00]→ C[M0

0] constructed in Subsection 7.1.4.

7.3 Harish-Chandra bimodules

In this section we proceed to define Harish-Chandra bimodules. The definition is, basically,

the same as with rational Cherednik algebras. However, for technical reasons (see (7.7))

we define a wider class of bimodules, which we call twisted HC bimodules. The twist here

is provided by a C×-equivariant automorphism of M00. When this automorphism is the

identity, we recover the usual definition of HC bimodules.

7.3.1 Harish-Chandra bimodules: algebra level

Definition 7.3.1. Let A,A′ be filtered quantizations of the same graded Poisson algebra A,

and let f : A→ A be a graded Poisson automorphism of A. We say that a (A,A′)-bimodule

B is f -twisted Harish-Chandra (shortly, f -HC) if it admits a bimodule filtration such that:

1. grB is a finitely generated A-bimodule.

2. For any a ∈ A, b ∈ grB, ab = bf(a).

125

We denote the category of f -HC (A,A′)-bimodules by fHC(A,A′).

We remark that, when f is the identity, we recover the usual category of HC bimodules that

is considered in [BL, BLPW]. We will abbreviate HC(A,A′) := idHC(A,A′), and call these

bimodules simply HC. The following proposition is clear.

Proposition 7.3.2. 1. The category fHC(A,A′) is a Serre subcategory of the category

of all (A,A′)-bimodules.

2. Every f -HC (A,A′)-bimodule B is finitely generated as a left A-module and as a right

A′-module.

3. The tensor product ⊗A′ gives a bifunctor:

gHC(A,A′)× fHC(A′,A′′)→ g◦fHC(A,A′′)

Let us specialize to the case where A = Aλ is a quantization of the Nakajima quiver variety

M00. In this case, we write fHC(λ, λ′) := fHC(Aλ,Aλ′). The following result is clear.

Proposition 7.3.3. Twisting the right action by the isomorphism Φ−µ (see (7.7)) gives a

category equivalence fHC(λ, µ)∼=→ Υ∗◦fHC(λ,−µ). Similarly, twisting the left action by the

isomorphism Φλ gives a category equivalence fHC(λ, µ)∼=→ f◦Υ∗HC(−λ, µ).

7.3.2 Harish-Chandra bimodules: sheaf level

We proceed to give the definitions of sheaf-theoretic HC bimodules. For the sake of con-

creteness, we will work with the varietiesMθ0, although the definitions can be stated for any

symplectic resolution. So let Aθλ, Aθµ be quantizations ofMθ0. Note that the external tensor

product Aθλ � (Aθµ)opp is a quantization of the product Mθ0 ×Mθ

0.

Now let f : M00 → M0

0 be a C×-equivariant Poisson automorphism. Let Σf ⊆ M00 ×M0

0

denote the graph of f , with its reduced scheme structure, and let Sf be the scheme-theoretic

preimage of Σf under the natural mapMθ0×Mθ

0 �M00×M0

0. For example, when f is the

126

identity then Σf is just the diagonal and Sf is the usual Steinberg variety. So we call Sf the

f -Steinberg variety.

Definition 7.3.4. An Aθλ � (Aθµ)opp-module B is said to be f -twisted Harish-Chandra

(shortly, f -HC) if it admits a filtration such that grB is coherent and scheme-theoretically

supported on the f -Steinberg variety Sf . We denote this category of f -HC Aθλ � (Aθµ)opp-

modules by fHC(λ, µ).

Note that, by definition, dimSf = dim Σf = dimM00 = (1/2) dim(Mθ

0 ×Mθ0). By standard

results in homological duality (see, for example, [BL, Section 4.2]) we get the following.

Proposition 7.3.5. The functor D : B 7→ ExtdimM00

Aθλ�(Aθµ)opp(B,Aθλ � (Aθµ)opp) gives an equiva-

lence:

D : fHC(λ, µ)∼=−→ f−1HC(µ, λ)opp

Proof. Standard results in homological duality say that the functor D gives an equivalence

fHC(Aθλ� (Aθµ)opp)→ fHC((Aθλ)opp �Aθµ). We have now to compose with the isomorphism

(Aθλ)opp�Aθµ → Aθµ�(Aθλ)opp. At the associated graded level, this induces the automorphism

of Mθ0 ×Mθ

0 that interchanges the factors. So the image of Sf is Sf−1 and the result is

proved.

Quantization of line bundles

Let us give an example of a Harish-ChandraHC(λ, λ+θ)-bimodule. Recall that on the variety

Mθ0 we have the C×-equivariant ample line bundle S(θ). According to [BPW, Section 5.1],

there exists a unique (Aθλ�(Aθλ+θ)opp)-module λAθλ+θ admitting a filtration whose associated

graded coincides with the pushforward of the line bundle S(θ) to Mθ0 × Mθ

0 under the

diagonal embedding. Thus, λAθλ+θ ∈ HC(λ, λ + θ). The following proposition now follows

from uniqueness of the quantizations of line bundles and from the fact that, it being a

quantization of a line bundle, taking tensor product with λAθλ+θ does not affect the support

of a module.

127

Proposition 7.3.6. For any n,m ∈ Z, taking tensor product with quantizations of line

bundles gives equivalences

fHC(λ+ nθ, µ)← fHC(λ, µ)→ fHC(λ, µ+mθ)

7.3.3 Localization theorems

Note that we have the global sections and localization functors

Γθλ : Aθλ -mod↔ Aλ -mod : Locθλ

When these Γθλ and Locθλ are quasi-inverse equivalences of categories, we say that abelian

localization holds at λ. In general, it is not easy to find the locus where abelian localization

holds. However, the following result, [BPW], tells us that abelian localization holds for λ

sufficiently dominant.

Theorem 7.3.7 (Corollary B.1, [BPW]). Let λ ∈ p. Then, abelian localization holds at

λ+ nθ for n� 0.

Since Sf is defined to be precisely the scheme-theoretic preimage of the graph Σf , [BLPW,

Proposition 2.13] implies.

Proposition 7.3.8. Global sections and localization induce functors

Γ : fHC(λ, µ)↔ f∗HC(λ, µ) : Loc

moreover, if abelian localization holds at λ and −µ, these are quasi-inverse equivalences of

categories.

7.3.4 Duality

We use our previous work to construct a duality functor between categories of twisted Harish-

Chandra bimodules. These functors will be constructed as composition of several equiva-

lences we have already seen. The duality step happens at the level of sheaves, this is just

128

the homological duality given by Proposition 7.3.5. To pass from sheaves to bimodules we

use, of course, localization theorems. The problem here is that, in general, localization does

not hold at λ and −λ simultaneously. So we need to first use an equivalence provided by

Proposition 7.3.3. This will introduce a twist by Υ that will be cancelled at the end using

an equivalence of the same form.

Theorem 7.3.9. Let λ be a period of quantization such that abelian localization holds at λ.

Then, for any C×-equivariant Poisson automorphism f :M00 →M0

0 and n� 0 there is an

equivalence of categories

D : f∗HC(λ, λ)∼=−→ (f∗)−1

HC(λ− nθ, λ− nθ)opp

Proof. As we have said above, this equivalence is just a composition of several equivalences

that we have introduced before. Let us list these.

(1) f∗HC(λ, λ)∼=−→ Υ∗◦f∗HC(λ,−λ).

(2) Υ∗◦f∗HC(λ,−λ)∼=−→ f◦ΥHC(λ,−λ).

(3) f◦ΥHC(λ,−λ)∼=−→ Υ◦f−1HC(−λ, λ)opp.

(4) Υ◦f−1HC(−λ, λ)opp∼=−→ Υ◦f−1HC(−λ+ nθ, λ− nθ)opp.

(5) Υ◦f−1HC(−λ+ nθ, λ− nθ)opp∼=−→ (f∗)−1◦Υ∗HC(−λ+ nθ, λ− nθ)opp.

(6) (f∗)−1◦Υ∗HC(−λ+ nθ, λ− nθ)opp∼=−→ (f∗)−1

HC(λ− nθ, λ− nθ)opp.

Equivalences (1) and (6) are provided by Proposition 7.3.3; the equivalence (2) is simply

the localization theorem, cf. Proposition 7.3.8. The localization theorem also provides the

equivalence (5): here, we need to take n big enough so that localization will hold at −λ+nθ,

cf. Theorem 7.3.7. The equivalence (4) is given by the translation equivalences of Proposition

7.3.6. Finally, the duality (3) is the homological duality of Proposition 7.3.5.

129

Remark 7.3.10. Theorem 7.3.9 holds, with the same proof, if we take two parameters

λ, µ such that localization holds at λ and −µ. In this case, we obtain an equivalence D :

HC(λ, µ)→ HC(µ− nθ, λ− nθ)opp, for n� 0.

7.4 Connection to rational Cherednik algebras

Let us remark that the spherical rational Cherednik algebras of the groups G(`, 1, n) =

Snn (Z/`Z)n can be realized as quantized quiver varieties. Let Q be the cyclic quiver with `

vertices (in particular, if ` = 1, then Q is a point with a loop). Pick a vertex in Q and call it

0 (so this is the extending vertex of the affine type A Dynkin diagram underlying Q). For a

dimension vector v, we take the vector n(1, 1, . . . , 1) (note that (1, 1, . . . , 1) is the affine root

for the affine type A Dynkin diagram underlying Q) while for a dimension vector w we the

vector that takes the value 1 at the vertex 0 and 0 everywhere else. For example, for ` = 1

the quiver Q♥ is

n 1

Let us denote by A := e(H/(~−1))e. This is a C[p′]-algebra, where we now denote the space

of parameters for the Cherednik algebra by p′. Let us denote by P the set of parameters for

the quantum Hamiltonian reduction, P = gv/[gv, gv].

Theorem 7.4.1 ([EGGO, Go, L2, O]). There is a filtered isomorphism A → AP, which

induces a linear isomorphism between the spaces of parameters ω : p′ → P. In particular,

for every c ∈ p′ we have that the algebras Ac and Aω(c) are isomorphic as filtered algebras.

Let us give a description of the map ω in the case of rational Cherednik algebra of type A.

Upon identifying p′ = Cc P = Cz, this map is simply c 7→ −z − 1, see [L2]. In particular,

we have that the categories HC(Ac, Ac′) and HC(A−c−1, A−c′−1) are equivalent.

130

Let us now say that, for c ∈ C, abelian localization holds for the rational Cherednik algebra

Ac if it holds for λ, where λ is such that Aλ = A−c. It is known, see [GS, Sections 5 and

6] and [EG, Corollary 4.2], that abelian localization fails at c if and only if c = −r/m, with

1 < m ≤ n and 1 ≤ r < m. Note that this is the case if and only if c is aspherical. Let

us denote the locus of aspherical parameters for the type A Cherednik algebra by A. Thus,

setting f = id in Theorem 7.3.9 we get the following result.

Corollary 7.4.2. Let c, c′ ∈ C and let N ∈ Z be such that (−1)1−χA(c)c+N, (−1)1−χA(c′)c′+

N 6∈ A, where χA is the indicator function of A (for example, take N > |<(c)|, |<(c′)|).

Then, there is an equivalence of categories

D : HC(Hc(n), Hc′(n))→ HC(H(−1)χ1−A(c′)c′+N

(n), H(−1)1−χA(c)c+N(n))opp.

Proof. If neither c nor c′ are aspherical, then this is clear from Theorem 7.3.9, see Re-

mark 7.3.10. In the case where one of them is aspherical, say c′, first use the equivalence

HC(Hc(n), Hc′(n))→ HC(H−c(n), Hc′(n)) given by the isomorphism Hc → H−c.

Let us now compare the double wall-crossing bimodule D with the dual of the regular

bimodule Hc. So let N be sufficiently big so that localization holds at c and N − c. We will

denote by DN−c(n) the double wall-crossing bimodule for the algebra HN−c(n).

Proposition 7.4.3. Let c = r/m, gcd(r;m) = 1, 1 < m ≤ n and c 6∈ (−1, 0). Let N ∈ Z

be such that −c + N 6∈ (−1, 0), and consider the duality functor D : HC(Hc(n), Hc(n)) →

HC(H−c+N(n), H−c+N(n)). Then, D(Dc(n)) = H−c+N(n).

Proof. First of all, note that the duality preserves supports. Since both Hc(n) and H−c+N(n)

have a unique irreducible HC bimodule with full support, cf. Theorem 5.4.4, the duality has

to send the unique irreducible bimodule with full support over Hc(n) to the unique irreducible

fully supported bimodule over H−c+N(n). Now, D(Dc(n)) will be an H−c+N(n)-bimodule

whose socle coincides with the unique minimal ideal of H−c+N(n). Since H−c+N(n) is the

131

injective hull of its unique minimal ideal, we have an embedding D(Dc(n)) → H−c+N(n).

The result now follows by comparing the composition lengths.

Of course, the previous proposition has its corresponding result when c is aspherical. Here,

we need to take N such that localization holds at −c and N + c (note that N = 1 suffices).

Then, we have the duality functor D : (Hc(n), Hc(n)) → D(Hc+N(n), Hc+N(n))opp, and we

get D(Dc(n)) = Hc+N(n).

Corollary 7.4.4. For any c ∈ C, the regular bimodule Hc(n) and the double wall-crossing

bimodule Dc(n) are injective-projective in the category HC(Hc(n), Hc(n)).

Proof. We have seen that both Hc(n) and Dc(n) are injective in HC(Hc(n), Hc(n)), see

Proposition 3.4.7 and Corollary 6.2.11. The result now follows immediately from Proposition

7.4.3.

132

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