On category for the rational O Cherednik algebra of the complex

reflection group (Z/!Z) Sn !

by Richard Thomas Robert Vale

A thesis submitted to the Faculty of Information and Mathematical Sciences at the University of Glasgow for the degree of Doctor of Philosophy

December 2006

c Richard Vale 2006 " 2 Acknowledgements

I would like to thank my supervisors Prof. K. A. Brown and Prof. I. Gordon for giving me a vast amount of help and suggesting many problems and their solutions. I would also like to thank Prof. T. Lenegan of the University of Edinburgh for helping me to obtain funding to visit Chicago in April 2006, and for arranging for me to visit Edinburgh in July 2006. I would like to thank Q. Gashi, V. Ginzburg and I. Gordon for their hospitality during my visit to Chicago. I would like to thank those mathematicians with whom I have had valuable conversations or correspondence, in particular S. Ariki, O. Chalykh, and C. Stroppel. I would like to thank many of my colleagues for their friendship. I acknowledge that my studies were funded by the Engineering and Physical Sciences Research Council.

i Statement

This thesis is submitted in accordance with the regulations for the degree of Doctor of Philosophy in the University of Glasgow. Chapters 1 and 2 cover notation, definitions and known results (apart from Theorem 2.1). Chapters 3, 4 and 5 are the author’s original work except where stated otherwise. Some of the results therein will appear in [70] and [69].

ii Summary

The topic of this thesis is the rational Cherednik algebra of the complex reflection group (Z/!Z) Sn, ! and in particular the category of modules over the Cherednik algebra. The structure of the thesis O is as follows. In Chapter 1, we begin by recalling some standard results which will be used in the text. These include some basic notions of algebraic geometry, the representation spaces of quivers, and quotients and blocks of abelian categories. We also recall some very basic results of invariant theory of finite groups and the corresponding skew group algebras. We then introduce symplectic reflection algebras in Section 1.2, and define Cherednik algebras of complex reflection groups as a special case. After recalling the most important ring-theoretic properties of Cherednik algebras and their spherical subalgebras in Proposition 1.18 and Proposition 1.19, we introduce category , which will be our main object of study. We then explain how to construct the KZ functor from O category to the category of finite-dimensional modules over the Hecke algebra of W . Next, in O Section 1.4, we specialise to the case of the group W = (Z/!Z) Sn = G(!, 1, n), and describe some ! of the main facts about the Hecke algebra of this group (which is also known as the Ariki-Koike algebra). We end Chapter 1 by defining an isomorphism which is an analogue in the G(!, 1, n) case of the Fourier automorphism of the Cherednik algebra in the Coxeter case (Theorem 1.40). In Chapter 2, we give a proof that for any complex reflection group W , simplicity of the Cherednik algebra of W is equivalent to semisimplicity of category (Theorem 2.1). In the case O of W = G(!, 1, n), we recall that semisimplicity of category is equivalent to semisimplicity of the O Ariki-Koike algebra (Theorem 2.4). We begin Chapter 3 by recalling in Section 3.1 some of the known results on finite-dimensional representations of the rational Cherednik algebra of W = G(!, 1, n). We then prove a new result which says that when the KZ functor satisfies a condition called separating simples, we can com-

iii pletely describe the structure of category (Theorem 3.7). We prove that separating simples is O equivalent to the Hecke algebra having one less simple module than the group algebra CW (Theo- rem 3.23). We then prove in Section 3.4 that this property determines the Ariki-Koike algebra up to isomorphism. This chapter has been submitted for publication in [69]. In Chapter 4, we study shift functors for the Cherednik algebra of W = G(!, 1, n). First we prove a shift relation, Theorem 4.1, for the spherical subalgebra, and use it to construct the Heckman- Opdam shift functors. In Section 4.1, we give some conditions under which the Heckman-Opdam shift functors are equivalences, and we prove that when they are equivalences, the Heckman-Opdam shift functors commute. We then turn in Section 4.2 to another notion of shift functor, which we call the Boyarchenko-Gordon shift functor. This functor is only well-defined when a hypothesis (Hypothesis 4.48) holds. We prove that the hypothesis holds in the case n = 1, and conjecture that it holds in general. Under Hypothesis 4.48, we construct the Boyarchenko-Gordon shift functor, and prove that it coincides with a product of Heckman-Opdam shift functors, provided that all of the functors in the product are equivalences (Theorem 4.62). In Chapter 5, we give an application of the results of the previous chapters, by proving an analogue for G(!, 1, n) of a result of Gordon [34] on the diagonal coinvariants of Coxeter groups. This chapter has been accepted for publication in [70].

iv Introduction

The aim of this thesis is to investigate the category of modules for the rational Cherednik algebra. O This category has many interesting properties; it is large enough to contain information about the structure of the algebra itself, and yet small enough to be understood using finite combinatorics. There is also a conjectural relationship between category and categories of coherent sheaves on O the resolutions of a certain symplectic singularity, coming from the fact that the Cherednik algebra can be viewed as a deformation of this singularity. Furthermore, category is an analogue in the O Cherednik algebra theory of a similar category of modules for the enveloping algebra of a finite- dimensional Lie algebra, and as such there are many techniques in existence which can be brought to bear on the problem of trying to find out how it behaves. Rational Cherednik algebras are a special case of symplectic reflection algebras. Symplectic reflection algebras were introduced by Etingof and Ginzburg, [26]. Given a symplectic vector space, V and a finite subgroup U Sp(V ), the symplectic reflection algebras H are a family ⊂ t,c of deformations of the skew group algebra C[V ] U. They were motivated by the representation ∗ theory, geometry and integrable systems which are related to these algebras. There is a dichotomy in the theory according to whether the parameter t is 0 or 1. In the Cherednik algebra case, t is taken to be 1. A good reference for symplectic reflection algebras in the t = 0 case is [26]. In the Cherednik algebra case, one takes V = h h where h is the reflection representation ⊕ ∗ of a complex reflection group W , and takes U to be the subgroup of Sp(V ) defined by the natural action of W on h h . The symplectic reflection algebra associated to (h h , U) is called the ⊕ ∗ ⊕ ∗ rational Cherednik algebra of W (we often refer to it just as the Cherednik algebra of W ). The

Cherednik algebra Hk(W ) depends on a vector of complex parameters k, and when k is taken to be

0, the Cherednik algebra reduces to the algebra An(C) W , where An(C) denotes the Weyl algebra ∗ with n = dim(h). In general, the Cherednik algebra is still very similar to the Weyl algebra,

v since it may be identified with the subalgebra of End(C[h]) generated by C[h], W and a family of commuting differential-difference operators k : v h called Dunkl operators. The Dunkl {∇v ∈ } operators were introduced in the Coxeter case in [22] and in the general case in [23]. Although the Dunkl operators commute, they have a difference term which explicitly depends on the action on W . Thus, the Cherednik algebra is more sensitive to the W –action than the Weyl algebra. The PBW Theorem of Etingof and Ginzburg [26] implies that the Cherednik algebra has a vector space decomposition into three parts

Hk(W ) = C[h] CW C[h∗]. ∼ ⊗ ⊗

This is reminiscent of the decomposition of the enveloping algebra of a semisimple Lie algebra g as (g) = (n+) (h) (n ), and it is possible to define an analogue of the BGG category U U ⊗ U ⊗ U − O (see [56]) in the Cherednik algebra situation. This category was first introduced in [6] and then studied in some detail in [33]. As in the Lie case, category is a highest-weight category, meaning O that it is combinatorially well-behaved. An important fact about category is the existence of a functor O

KZ : mod O → H − where denotes the Hecke algebra of W , a finite-dimensional algebra which can be viewed as H a deformed version of the group algebra of W , depending on complex parameters which are the exponentials of the parameters k. The functor KZ exists because, after a suitable localisation, the Cherednik algebra becomes the skew group algebra of W over the algebra of differential operators on an open set hreg h. Thus, a module for the Cherednik algebra may be regarded as a W –equivariant ⊂ D–module on hreg. This D–module gives a vector bundle with a flat connection, which in turn gives a finite-dimensional representation of the fundamental group of hreg/W . This representation turns out to factor through the Hecke algebra. The construction is explained in detail in Section 1.29. We exploit the KZ functor in Chapter 3. Passing back and forth between the Cherednik algebra and Hecke algebra allows us to prove theorems about both of these objects. The Hecke algebra is an interesting object in its own right, and in the G(!, 1, n) case it was introduced in the paper [2] by Ariki and Koike. A lot is known about the representation theory of this algebra, and we are able to use this to describe the block structure and composition multiplicities of category in the O case where is close to being semisimple (Theorem 3.7). In the other direction, we are also able O vi to describe the Ariki-Koike algebra in the almost-semisimple case, using the Cherednik algebra techniques. The results of Chapter 4 are motivated by a more geometric viewpoint. Just as the Cherednik algebra may be regarded as a deformation of C[h h∗] W , it has a subalgebra which may be regarded ⊕ ∗ W as a deformation of the fixed ring C[h h∗] , or equivalently, as a noncommutative version of the ⊕ singularity (h h )/W . This subalgebra (the so-called spherical subalgebra) is just eH e where ⊕ ∗ k 1 e = W w W w is the symmetrising idempotent. The work of Gordon and Stafford [36], [37] | | ∈ shows th!at, in the case where W = Sn, the spherical subalgebra is related to the geometry of the resolution of singularities of (h h )/W . However, it is not enough just to study eH e for one ⊕ ∗ k value of k, rather it is necessary to consider all integer shifts of k at once, to construct a so-called

Z-algebra out of the eHke. For this purpose, Gordon and Stafford used Heckman-Opdam shift functors, which are functors eH e Mod eH e Mod, k − → k! − where the parameters k$ are obtained from k by an integer shift. In Section 4.1, we define analogues of these functors for the group G(!, 1, n). The situation here is more complicated, since the Chered- nik algebra of G(!, 1, n) depends on ! parameters rather than just one. We give some conditions under which these functors can be shown to be equivalences. Category is an important tool in O proving these results. In the second part of Chapter 4, we turn to another kind of shift functor which has been defined by Gordon [35], following Boyarchenko [10]. This functor relies on a description of eHke as a factor of the ring of differential operators on the representation space of a quiver (this is known as a Hamiltonian reduction, see for example [30]). The quiver here is a cyclic quiver, with one vertex and one edge attached, which we denote by Q . Let G be the base-change group of the quiver Q ∞ ∞ and let ε be the dimension vector of Q with 1 at the extra vertex and n at each vertex of the ∞ cycle. Then Gordon’s description of eHke is D(Rep(Q , ε))G eHke = ∞ ∼ Ik for some ideal Ik. Unfortunately, in our case this description is not quite complete, since it depends on a certain homomorphism called the radial part map having the correct image. We are at least able to show that this holds for W = Z/!Z. Assuming it holds in general (which we conjecture to be true), we construct our own version of the Boyarchenko-Gordon shift functors. We then in

vii Section 4.3 study the question of when these coincide with the Heckman-Opdam shift functors. We are able to show that in some cases, they do coincide (Theorem 4.62). This result is useful because the Z–algebra associated to the Boyarchenko-Gordon shift functors is related, via an associated graded construction, to one associated with a Nakajima quiver variety [58]. In some cases, these quiver varieties are resolutions of the singularity (h h )/W . On the other hand, if the Boyarchenko- ⊕ ∗ Gordon shift functors are known to be equivalences, one can use the Z–algebra theory of [36] to show that there is an explicit relationship between the category of coherent sheaves on the quiver variety and the category of finitely-generated eHke–modules. Thus, it is important to know when the Boyarchenko-Gordon and Heckman-Opdam shift functors coincide, since the Boyarchenko-Gordon shift functors have the correct associated graded properties, whereas it is in the Heckman-Opdam case that we have a better chance of showing that these functors are equivalences. It should be possible to go further and study coherent sheaves on the quiver variety using the Cherednik algebra in the manner of [37]. However, we do not pursue this line of research in this thesis. In Chapter 5, we use the results of the earlier chapters to prove a theorem (Theorem 5.2) about the diagonal coinvariants of G(!, 1, n). This theorem follows a proof of Gordon [34]. It does not use the Boyarchenko-Gordon shift functors and therefore does not rely on Hypothesis 4.48. Theorem 5.2 states that the diagonal coinvariant ring

C[h h∗] ⊕ W C[h h∗] , ⊕ + - has a quotient with good combinatorial properties. It is a nice application of Cherednik algebra techniques to prove a result in commutative algebra. The structure of this thesis is as follows. In Chapter 1 we give a list of basic definitions, then define symplectic reflection algebras and rational Cherednik algebras, giving their basic properties. We discuss the case of the group G(!, 1, n) in detail. In Chapter 2, we study the generic case, in which the Cherednik algebra is simple and category is semisimple. In Chapter 3, we present our O results on the almost-semisimple case. In Chapter 4, we define and study the Heckman-Opdam and Boyarchenko-Gordon shift functors. In order to define the Heckman-Opdam shift functors, we use Poisson geometry to prove a shift relation, Theorem 4.1, following [6] and [7]. Finally, Chapter 5 contains a proof of Theorem 5.2 on diagonal coinvariants.

viii Contents

Statement ii

Summary iii

Introduction v

1 The rational Cherednik algebra 1 1.1 Notation and conventions ...... 1 1.2 Symplectic reflection algebras and the PBW theorem ...... 8 1.3 The rational Cherednik algebra ...... 9 1.3.1 The spherical subalgebra ...... 11

1.3.2 Representation theory of Hk ...... 12 1.3.3 Ideals of H and category ...... 16 k O 1.3.4 The Dunkl representation ...... 16 1.3.5 The Knizhnik-Zamolodchikov functor ...... 17 1.3.6 Double centraliser property ...... 20 1.3.7 Twisting by a linear character ...... 21 1.3.8 Absence of self-extensions ...... 21 1.4 The case of W = G(!, 1, n) ...... 22 1.4.1 The rational Cherednik algebra of G(!, 1, n) ...... 22 1.4.2 Representation theory of the Ariki-Koike algebra ...... 24 1.4.3 Fourier map ...... 27

ix 2 The semisimple case 29 2.1 The semisimple case ...... 29 2.2 Remarks ...... 33

3 The almost-semisimple case 34 3.1 Finite-dimensional modules ...... 34 3.2 The main theorem ...... 37 3.3 Proof of Theorem 3.7 ...... 39 3.3.1 Proof of parts (1) and (2) ...... 41 3.3.2 Blocks ...... 42 3.3.3 Proof of parts (3) and (4) ...... 46 3.3.4 Proof of part (5) ...... 47 3.3.5 Characterisations of separating simples ...... 48 3.4 The Ariki-Koike algebra in the almost-semisimple case ...... 50

4 Shift functors 55 4.1 The Heckman-Opdam shift functors ...... 56 4.1.1 A shift relation ...... 56 4.1.2 The shift functors ...... 64 4.1.3 The semisimple and almost-semisimple cases ...... 66 4.1.4 The asymptotic parameter case ...... 67 4.1.5 Shift functors on category ...... 74 O 4.1.6 A commutativity property ...... 75 4.2 The Boyarchenko-Gordon shift functors ...... 78 4.2.1 Gordon’s construction ...... 79 4.2.2 The radial part map ...... 81 4.2.3 A shift functor ...... 85 4.3 Comparison of the shift functors ...... 87

4.3.1 A remark on Z–algebras ...... 92 4.3.2 Nakajima quiver varieties ...... 94

x 5 Diagonal coinvariants 98 5.1 A quotient ring of the diagonal coinvariants ...... 100

Bibliography 104

xi Chapter 1

The rational Cherednik algebra

1.1 Notation and conventions

The purpose of this section is to set out some notational conventions which will be used throughout the text, and also to supply some definitions which will be used in the proofs of certain results in this thesis.

Noncommutative rings

A good reference for standard notions in noncommutative algebra is [55]. In this thesis, we will work exclusively over the field of complex numbers C. An algebra for us will mean a C-algebra, associative and unital, and we do not require that a subalgebra of an algebra shares the same identity element. For an algebra A with subalgebra B, we will say that A is finite over B to mean that A is a finitely generated left B-module. If A is a Noetherian domain we denote by Frac(A) the division ring of fractions of A. For a C-algebra A, a filtration is by definition an increasing collection of subspaces = i : i Z with a = 0 for all a < 0 and ∞ i = A, such that F {F ∈ } F ∪i=0F i j i+j . If a i i 1 then we say that a has degree deg(a) = i with respect to the F F ⊂ F ∈ F \ F − filtration . We will denote the associated graded algebra with respect to by gr A, or simply by F F F grA if there is no risk of confusion. For an algebra A we write GK dim A for the Gelfand-Kirillov dimension of A; see [47] for the definition and basic properties of this dimension. By convention, we use A Mod to denote the category of all left modules over an algebra A, and − A mod to denote the full subcategory of finitely generated left A–modules. We use the notation −

1 Mod A and mod A for the analogous categories of right A–modules. A good reference for − − standard notions of homological algebra is the book [63]. We will require the notions of Auslander- regularity and Cohen-Macaulayness for a noncommutative ring. We give the definitions of these notions following [47, Section 12.9]. We do not discuss these properties in detail since we require them only to apply some technical theorems of noncommutative algebra (see Lemma 4.38).

Definition 1.1. An algebra R is Auslander-Gorenstein if RR and RR have finite injective dimen- sion, and given integers 0 i < j and a finitely generated left or right R–module M, together with ≤ j i a submodule N of ExtR(M, R), we have ExtR(N, R) = 0.

Definition 1.2. An algebra R is Auslander-regular if R is Auslander-Gorenstein and has finite global dimension.

Definition 1.3. An algebra R is Cohen-Macaulay if for all finitely generated R–modules M, we have GK dim(R) = GK dim(M) + min j 0 : Extj (M, R) = 0 . { ≥ R 1 }

Geometry

For basic definitions of algebraic geometry, a good reference is [42]. All varieties in this thesis are defined over C. For a Zariski-closed subset V of Cn, we write C[V ] for the coordinate ring of V , that is, the ring C[x1, . . . , xn]/I where I is the ideal of functions which vanish on V . Given a finitely generated commutative domain A, we write maxspec(A) for the set of maximal ideals of A, which is an affine variety. Given an affine variety V , we denote by D(V ) the ring of differential operators of

C[V ], as defined in [55, Chapter 15]. We use the term “D–module on V ” as a synonym for “D(V )– module”. We will often use the fact that if V is a smooth affine variety and f C[V ] 0 , then ∈ \ { } 1 D(Vf ) is isomorphic to the localization D(V )[f − ], where Vf denotes the open subset of points x V where f(x) = 0. This follows from [55, Theorem 15.1.25] combined with [55, Corollary ∈ 1 15.5.6]. Given an algebraic variety V , which is not assumed to be affine, a vector bundle on V may be regarded as a locally free sheaf of –modules, where denotes the coordinate sheaf of V . For OV OV the fundamental properties of vector bundles on algebraic varieties, see [48, Chapter 1]. Given a vector bundle B on V , a connection on B is a –module map OV : B Ω1 B ∇ → ⊗OV 2 satisfying for any affine open U V , (fb) = df b + f (b) for all f (U) and all b B(U). ⊂ ∇ ⊗ ∇ ∈ OV ∈ Here Ω1 denotes the cotangent sheaf of V (the sheaf of sections of the cotangent bundle T V V ). ∗ → Given a vector field v on V , a vector bundle B, and a connection on B, we may define a ∇ map of sheaves : B B by setting (b) := (b)(v). The connection is said to be flat if ∇v → ∇v ∇ ∇ [ , ] = for all vector fields v and w on V . Information about the basic properties of ∇v ∇w ∇[v,w] connections may be found in [19, I.2]. There is an analogous definition of connection on a vector bundle over a complex manifold. In order to construct the KZ functor in Section 1.29, we will require the notion of a connection having regular singularities. As this has a lengthy definition, and it will not be used elsewhere in this thesis, we refer the reader to [19, II, Chapter 4] for the definition.

Quivers

One example of an affine variety that will be used in this thesis is the space of all representations of a quiver Q with a given dimension vector.

Definition 1.4. A quiver Q is by definition a 4–tuple (Q0, Q1, h, t) where:

Q is a finite set, called the set of vertices of Q. • 0 Q is a finite set, called the set of arrows of Q. • 1 h, t : Q Q are functions. • 1 → 0 A quiver should be viewed as a finite directed graph, where h, t : Q Q are the functions 1 → 0 that assign to an arrow its head and tail respectively.

Given a quiver Q, a dimension vector for Q is a function Q0 Z 0. Given a dimension vector → ≥ α for Q, a representation of Q of dimension vector α is a map which assigns to each x Q a ∈ 0 vector space V of dimension α(x), and to each a Q a linear map X : V V . The space x ∈ 1 a t(a) → h(a) Rep(Q, α) of representations of a quiver Q with dimension vector α may be identified with the affine variety

Rep(Q, α) = Mat(α(t(a)) α(h(a)), C). × a Q1 "∈ The algebraic group GL(α) = GL(V ) acts on Rep(Q, α) via (g X) = g X g 1 for x Q0 x a h(a) a t−(a) ∈ · a Q1. Every point of Rep(Q, α#) is fixed by the normal subgroup C∗ = (λid, . . . , λid) : λ C∗ ∈ ∼ { ∈ } 3 under this action, so this gives an action of the group G(α) = GL(α)/C∗ on Rep(Q, α). Two representations A, B Rep(Q, α) are said to be isomorphic if and only if they lie in the same orbit ∈ of G(α).

Category theory

Nearly all the categories considered in this thesis will be subcategories of A Mod for some algebra − A. Furthermore, most of the categories in this thesis will be abelian. A reference for the basic properties of abelian categories is [28]. We will require the construction of the quotient of an abelian category by a Serre subcategory, which we now describe, following the exposition in [62].

Definition 1.5. Let be an abelian category. An abelian subcategory of is said to be Serre if C A C for every short exact sequence

0 X$ X X$$ 0 → → → → in , we have X if and only if X , X . C ∈ A $ $$ ∈ A If is an abelian category and is a Serre subcategory, let Σ denote the class of morphisms C A ⊂ C f in with ker(f) and cok(f) . Then we define the quotient / to be an additive category C ∈ A ∈ A C A / together with a functor Q : / such that Q(f) is invertible for all f Σ, and satisfying C A C → C A ∈ the following universal property: for any additive category and any additive functor F : such that F (f) is invertible D C → D for all f Σ, there exists a unique functor / making the following diagram commute. ∈ C A → D Q / / C C A {{ F {{ {{  }{{ D By [62, I, Theorem 14.1], we have the following proposition.

Proposition 1.6. If is an abelian category such that the class of subobjects of each object of C C is a set, and is a Serre subcategory of , then the quotient category / exists. It has objects A C C A ob( / ) = ob( ) C A C and morphisms

s Hom / (X, Y ) = limZ X Hom(Z, Y ). C A s→Σ ∈ 4 The quotient functor Q is defined to be the identity on objects, and for a morphism f : X Y one → s defines Q(f) to be (f s)s Σ, Z X ◦ ∈ → A proof that the above definition really does give a well-defined pair ( / , Q) satisfying the C A universal property may be found in [62, Theorem 14.1, Chapter 1]. In particular, the quotient functor Q : / is essentially surjective on objects, ie. every object of / is isomorphic to C → C A C A the image under Q of an object of . Furthermore, it follows from [62, Exercise 7.3] that / is C C A an abelian category and Q is an exact functor. If is an abelian category then we may partition the set S of simple objects of into equivalence A A 1 classes called blocks. We define a relation on the set S by L1 L2 if Ext (L1, L2) = 0. Taking ∼ ∼ A 1 the reflexive, symmetric and transitive closure of this relation gives an equivalence relation which we also denote by . The equivalence classes of S under are called the blocks of . Furthermore, ∼ ∼ A if an object M of has a finite composition series all of whose factors belong to the block B, then A we say that M belongs to the block B. Of course, in general not every object of need belong to A a block. The following proposition is standard, but we were not able to find a convenient reference for it. Recall that a composition series of an object M in an abelian category is by definition a chain A 0 = M0 M1 Mn = M of subobjects of M such that Mi/Mi 1 is simple for each i. We ⊂ ⊂ · · · ⊂ − say that the Jordan-Ho¨lder theorem holds in if whenever an object M in has two composition A A series 0 = M M M = M and 0 = M M M = M, then m = n and the 0 ⊂ 1 ⊂ · · · ⊂ n 0$ ⊂ 1$ ⊂ · · · ⊂ m$ same composition factors appear in the two composition series with the same multiplicities.

Proposition 1.7. Let be an abelian category in which the Jordan-H¨older theorem holds and let M A be an indecomposable object of . Suppose M has a composition series 0 M M = M. A ⊂ 1 ⊂ · · · ⊂ n Then all the composition factors Mi/Mi 1 belong to the same block of . − A Proof. Let 1, 2, . . . , n = I J be a partition into disjoint subsets such that { } ∪ 1 1 Ext (Mi/Mi 1, Mj/Mj 1) = Ext (Mj/Mj 1, Mi/Mi 1) = 0 A − − A − − if i I, j J. We first show that we may take I = 1, 2, . . . a for some a. Indeed, suppose there ∈ ∈ { } is j J, i I with j < i. Then after relabelling, we may assume that j J, j + 1 I. The exact ∈ ∈ ∈ ∈ sequence M M M 0 j j+1 j+1 0 → Mj 1 → Mj 1 → Mj → − − 5 must split, so we may define a new composition series 0 M M M = M of M ⊂ 1$ ⊂ 2$ ⊂ · · · ⊂ n$ with j I, j + 1 J. Therefore, we may assume I = 1, . . . , a , J = a + 1, a + 2, . . . , n for ∈ ∈ { } { } some a. Now fix r with a + 1 r n. Using the long exact sequence for Ext1, we may show ≤ ≤ 1 by induction on t that Ext (Mt, Mr/Mr 1) = 0, 1 t a. Indeed, for t a, if we have shown A − ≤ ≤ ≤ 1 Ext (Mt 1, Mr/Mr 1) = 0, then since 0 Mt 1 Mt Mt/Mt 1 0 is exact, we get an exact A − − → − → → − → sequence

1 1 1 Ext (Mt 1, Mr/Mr 1) Ext (Mt, Mr/Mr 1) Ext (Mt/Mt 1, Mr/Mr 1), A − − ← A − ← A − −

1 which shows Ext (Mt, Mr/Mr 1) = 0 since t I. A − ∈ 1 Then, using the same method, we may show by induction on r that Ext (Ma, Mr/Ma) = 0 for A a + 1 r n. Therefore, M = M M/M , which implies that either I or J is empty, since M is ≤ ≤ a ⊕ a indecomposable.

If A is a finite-dimensional algebra, the blocks of A may also be defined as follows. Write 1 A as a sum of pairwise orthogonal central idempotents 1 = e + + e with n maximal. ∈ 1 · · · n This corresponds to a decomposition A = n Ae of A into indecomposable subalgebras which are ⊕i=1 i known as the blocks of A (see [13, I.16.I]). The blocks of the algebra A are in bijection with the blocks of the category = A mod because every simple left A–module S satisfies e S = S for some A − i 1 1 unique ei. Therefore, if S, T belong to distinct blocks of A then ExtA(S, T ) = ExtA(T, S) = 0, while if S, T belong to the same block of A then S and T are composition factors of the indecomposable module Ae and hence belong to the same block of by Proposition 1.7. i A

Invariants and skew group algebras

For a finite group W , we write Irrep(W ) for the set of irreducible representations of W over C. We refer to the character of a one-dimensional representation as a linear character of W . Let V be a finite-dimensional complex vector space and let W be a finite subgroup of the general linear group GL(V ). Then W acts naturally on the coordinate ring C[V ] of V , which is just the polynomial ring C[x1, x2, . . . , xn] where n = dim(V ), and we can consider the ring of polynomial invariants C[V ]W . Then C[V ]W may be viewed as the coordinate ring of the orbit space V/W . See [4, Remark, page 8].

6 Definition 1.8. A linear map s : h h is called a complex reflection if s fixes a hyperplane H → pointwise, and s has finite multiplicative order. A finite subgroup of GL(h) generated by complex reflections is called a complex reflection group.

The complex reflection groups were classified by Shepherd and Todd in [66]. They fall into one infinite family and 34 exceptional cases. Complex reflection groups are important because of the second part of the following theorem.

Theorem 1.9. Let V be a finite-dimensional complex vector space and W a finite subgroup of GL(V ).

1. The algebra C[V ]W is a finitely generated domain and C[V ] is a finite C[V ]W –module.

2. C[V ]W is smooth (ie. has finite homological dimension) if and only if W is a complex reflection group.

Proof. It is obvious that C[V ]W is a domain since C[V ]W C[V ]. The Hilbert-Noether theorem ⊂ [4, Theorem 1.3.1] states that C[V ]W is finitely generated and C[V ] is a finite C[V ]W –module. Furthermore, C[V ]W has finite global dimension if and only if it is a polynomial ring, by Serre’s converse to Hilbert’s syzygy theorem [4, Corollary 4.2.3] and [4, Theorem 6.2.2(b)]. But C[V ]W is a polynomial ring if and only if W is generated by complex reflections, by [4, Theorem 7.2.1].

The algebra C[V ]W is contained in a larger, noncommutative algebra called the skew group algebra C[V ] W . Such an algebra may be defined whenever a finite group W acts on an algebra ∗ A, and we describe it in this more general context.

Definition 1.10. Let A be an algebra and W a finite group which acts faithfully on A by algebra automorphisms. The skew group algebra A W may be defined as the free A–module with basis given ∗ by the elements of W , equipped with the multiplication a w a w = a w (a )w w for a , a A 1 1 · 2 2 1 1 2 1 2 1 2 ∈ and w , w W . 1 2 ∈ Some properties of A are inherited by A W . ∗ Proposition 1.11. Let A be an algebra and W a finite group which acts faithfully on A by algebra automorphisms. If A is Noetherian then so is A W . If A is simple and the group of units in A is ∗ central in A, then A W is simple. ∗ 7 Proof. We have that A W is Noetherian by [61, Proposition 1.6], and the second part follows ∗ from [55, Proposition 7.8.12].

In the case A = C[V ], the ring of invariants and the skew group algebra are related as follows.

Proposition 1.12. Let V be a finite-dimensional complex vector space and W a finite subgroup of 1 GL(V ). Let e = W w W w CW denote the symmetrising idempotent of W . Then there is an | | ∈ ∈ algebra isomorphism ! W C[V ] e(C[V ] W )e → ∗ defined by z ze. 5→

1 Proof. If f C[V ] then efe = W w W w(f) e. ∈ | | ∈ $ ! % 1.2 Symplectic reflection algebras and the PBW theorem

The aim of this chapter is to introduce the rational Cherednik algebra. Rational Cherednik algebras are a special case of symplectic reflection algebras, which were introduced in [26]. We will present the basic facts about symplectic reflection algebras, following the exposition of [26] and [12]. Let

V be an n–dimensional complex vector space equipped with a symplectic form ω : V V C, ⊗ → that is, a bilinear form on V which is non-degenerate and skew-symmetric. The group of all linear transformations γ : V V such that ω(γx, γy) = ω(x, y) for all x, y is called the symplectic group → Sp(V ) of V .

Definition 1.13. A linear transformation γ : V V is called a symplectic reflection if rank(1 → − γ) = 2.

Let Γ ! Sp(V ) be a finite subgroup of Sp(V ) generated by symplectic reflections. Then the triple (V, ω, Γ) is called a symplectic triple. We may associate an algebra to a symplectic triple as follows. First, let T (V ) = i V be the tensor algebra of V . Then form the skew group algebra ⊕i∞=0 ⊗ T (V ) Γ. Now choose a bilinear map θ : V V CΓ taking values in the group algebra. Define Iθ ∗ ⊗ → to be the 2–sided ideal of T (V ) Γ generated by all the elements of the form x y y x θ(x, y) ∗ ⊗ − ⊗ − for x, y V . Finally, define an algebra ∈

H := T (V ) Γ/I . θ ∗ θ

8 Let us note that if θ = 0 then H = Sym(V ) Γ, but that otherwise, H need not be isomorphic θ ∗ θ to Sym(V ) Γ. However, there is in general a relationship between H and Sym(V ) Γ, since we ∗ θ ∗ may define a filtration on Hθ as follows. Let F 1(Hθ) = 0, F0(Hθ) = CΓ, F1(Hθ) = V + CΓ and − F (H ) = (F (H ))i for all i 1. Then consider the associated graded algebra i θ 1 θ ≥

grF (Hθ) := i∞=0(Fi(Hθ)/Fi 1(Hθ)). ⊕ −

Because the image of x y y x in gr (H ) is zero for all x, y V , there is a natural map ⊗ − ⊗ F θ ∈

ρ : Sym(V ) Γ gr (H ). ∗ → F θ

Definition 1.14. If ρ is an isomorphism, then Hθ is said to have the PBW property.

Note that if Hθ has the PBW property then Hθ has a vector space basis given by the expressions a γ where a is an ordered monomial in a basis of V , and γ Γ. That is, H is isomorphic to · ∈ θ Sym(V ) CΓ as vector spaces. ⊗C The following result is of fundamental importance.

Theorem 1.15. [26, Theorem 1.3] Let (V, ω, Γ) be a symplectic triple. Let S be the set of symplectic reflections in Γ. For s S, let ω be the symplectic form on V with radical rad ω = ker(1 s) and ∈ s s − with ωs im(1 s) = ω. Suppose there is a t C and a conjugation-invariant function c : S C such | − ∈ → that

θ(x, y) = tω(x, y)1 + csωs(x, y)s s S &∈ for all x, y V . Then the PBW property holds for H . ∈ θ

If the PBW property holds, then Hθ is said to be a symplectic reflection algebra.

1.3 The rational Cherednik algebra

The rational Cherednik algebras arise as a special case of symplectic reflection algebras. Let h be a finite-dimensional complex vector space and let W be a finite group acting faithfully on h, with the action being generated by complex reflections. In this thesis we will be mostly interested in the group G(!, 1, n) to be defined below. However, the following theory holds for any complex reflection group.

9 Since W acts on h, we have the contragredient action on h and hence an action of W on h h . ∗ ⊕ ∗ Now define a symplectic form ω on h h by ω(a + f, b + g) = f(b) g(a). Then it is easy to ⊕ ∗ − see that ω is nondegenerate. Denote the action map W GL(h h ) by w wˆ. Then wˆ is → ⊕ ∗ 5→ a symplectic reflection if and only if w is a complex reflection. So the group Wˆ = wˆ w W { | ∈ } is generated by symplectic reflections and therefore (h h , ω, Wˆ ) is a symplectic triple. We now ⊕ ∗ construct a symplectic reflection algebra from this triple. Each complex reflection w in W has by definition a reflecting hyperplane H. Let be the set A of reflecting hyperplanes of W . For H , let W be the set of elements of w which fix every ∈ A H element of H. Then each WH is a cyclic group and the set of complex reflections of W is H WH . ∪ ∈A Let nH = WH and let vH h be chosen so that CvH is a WH –stable complement to H. Also, let | | ∈ α h be a linear functional with kernel H. Let S denote the set of complex reflections in W . H ∈ ∗ For s S, x h and y h, we have in the notation of Theorem 1.15 ∈ ∈ ∗ ∈

αH (y)x(vH ) ω(x, y) = y(x), ωsˆ(x, y) = . αH (vH ) ˆ Therefore, we may define a symplectic reflection algebra T (h h∗) W /Iθ, where we choose t C ⊕ ∗ ∈ and a function S C, w cw with cγ 1wγ = cw for all w S and γ W , and set → 5→ − ∈ ∈

αH (y)x(vH ) θ(x, y) = t y(x)1 + cwwˆ. · αH (vH ) H w W 1 &∈A ∈ &H \{ }

It will be more convenient later on to take a basis of WH given by the idempotents eH,i = 1 i n w W (det w) w. We fix the parameter t = 1, and define complex parameters kH,i C such H ∈ H ∈ that! n 1 H − c w = n (k k )e w H H,i+1 − H,i H,i w W i=0 &∈ H & where kH,0 = kH,nH := 0. We therefore have

n 1 H − c = (det w)i(k k ) w H,i+1 − H,i &i=0 for all w W . Since Wˆ = W , the symplectic reflection algebra we have constructed may be ∈ H ∼ defined as follows.

Notation 1.16. Write C[h] = Sym(h∗) for the algebra of polynomial functions on h. Similarly,

C[h∗] = Sym(h).

10 Definition 1.17. Let W be a complex reflection group acting faithfully on a vector space h. Let k = kH,i H , 0 i nH be a set of complex parameters with kH,0 = kH,n 1 := 0 and such { | ∈ A ≤ ≤ } H − that k = k for all H , all 1 i n 1, and all w W . Then the rational Cherednik w(H),i H,i ∈ A ≤ ≤ H − ∈ algebra H (W ) associated to (h, W, k = k ) is the algebra generated by h, h and W subject to k { H,i} ∗ the relations:

[x, x$] = 0, [y, y$] = 0

1 1 wxw− = w(x), wyw− = w(y)

nH 1 αH (y)x(vH ) − i [y, x] = y(x) + (kH,i+1 kH,i)(det w) w αH (vH ) − H w W i=0 &∈A &∈ H & for all x, x h , all y, y h and all w W . $ ∈ ∗ $ ∈ ∈ Let us fix a basis y , . . . y of h and the dual basis x , . . . x of h . The rational Cherednik { 1 n} { 1 n} ∗ algebra is, by construction, a special case of a symplectic reflection algebra, and we therefore have the following proposition.

Proposition 1.18. The rational Cherednik algebra Hk has the following properties:

1. Hk has a filtration with h, h∗ in degree 1 and W in degree 0, and grHk = C[h h∗] W . ∼ ⊕ ∗

2. A basis of Hk is given by the set

p q γ : p is an ordered monomial in the x , q is an ordered monomial in the y , γ W . { · · i i ∈ }

In other words, as vector spaces Hk = C[h∗] C[h] CW via the multiplication map. ∼ ⊗ ⊗

3. The ring Hk is Noetherian and has finite homological dimension.

Proof. The first two parts follow from the PBW theorem 1.15. The third part is a general fact about symplectic reflection algebras, contained in [12, Theorem 4.4].

1.3.1 The spherical subalgebra

There is a certain important subalgebra of Hk called the spherical subalgebra. Let e CW be the ∈ 1 idempotent e = W w W w. Then eHke is a filtered algebra with a filtration inherited from Hk | | ∈ W and gr(eHke) = e C[!h h∗] W e = C[h h∗] . This is a domain, so eHke is a domain. We have ⊕ ∗ ⊕ the following proposition.

11 Proposition 1.19. [26, Theorem 1.5] The spherical subalgebra eHke is a Noetherian domain which is Auslander-Gorenstein and Cohen-Macaulay.

Proof. We have shown that eHke is a domain. The same argument shows that it is Noetherian, since W C[h h∗] is Noetherian. In order to show that eHke is Auslander-Gorenstein, we use [8, Theorem ⊕ 3.9], which states that in this situation, eHke is Auslander-Gorenstein if gr(eHke) is Auslander- W Gorenstein. But gr(eHke) = C[h h∗] , which is an Auslander-Gorenstein ring by Watanabe’s ⊕ Theorem (see [4, Theorem 4.6.2]), since Wˆ is a subgroup of SL(h h ). Furthermore, a theorem ⊕ ∗ W of Hochster and Eagon given in [4, Theorem 4.3.6] implies that C[h h∗] is Cohen-Macaulay. It ⊕ then follows from the proof of [8, Theorem 3.9], combined with [47, Proposition 6.6], that eHke is Cohen-Macaulay.

1.3.2 Representation theory of Hk

We now introduce the topic of this thesis: the representation theory of the rational Cherednik algebra. The first step is to define some modules which are the analogues of Verma modules in Lie theory. Given a representation τ of W , we may make τ into a C[h∗] W –module by declaring that ∗ h C[h∗] acts by zero. Since C[h∗] W Hk, we may form the induced module. ⊂ ∗ ⊂

Definition 1.20. The standard module associated to τ is the Hk–module

M(τ) := IndHk (τ) = H (τ). [h ] W k C[h∗] W C ∗ ∗ ⊗ ∗

Notice that Proposition 1.18 implies that M(τ) is isomorphic as a W –module to C[h] τ. In ⊗ particular, M(τ) is a free C[h]–module. From now on, we will assume that τ is irreducible. Now we explain how to give M(τ) a grading. Recall that we have fixed a basis y of h and { i} the dual basis x of h . Let eu = x y . Define an element { i} ∗ i i i n 1 ! H − z = nH kH,ieH,i H i=1 &∈A & where eH,i are the idempotents defined above. Then z is a central element of CW . Let h = eu z. − Then by [33, (4), Section 3.1], we have [h, y] = y for all y h and [h, x] = x for x h and − ∈ ∈ ∗ [h, w] = 0 for all w W . Since z Z(CW ), z acts on an irreducible W –representation τ by a ∈ ∈ scalar λ. For i Z, let ∈ M(τ) = x M(τ) : hx = (i λ)x . i { ∈ − } 12 Then we have the following lemma due to Dunkl and Opdam.

Lemma 1.21. [23] The grading induced by h has the following properties.

1. For each i 0, let C[h]i denote the homogeneous polynomials of degree i. Then M(τ)i = ≥ C[h]i τ. ⊗ 2. Every submodule of M(τ) inherits this grading. That is, if N M(τ) is any H –submodule, ⊂ k then x N : hx = (i λ)x = N M(τ)i for all i Z. { ∈ − } ∩ ∈ 3. M(τ) has a unique maximal submodule R(τ), and hence a unique simple quotient L(τ).

Proof. Since [h, x] = x for all x h∗, we see that [h, p] = ip if p C[h]i. But h (p τ) = ∈ ∈ · ⊗ [h, p] τ pz(1 τ) since each y acts on τ by zero. This proves the first part. The second part ⊗ − ⊗ i is [23, 2.27]. The third part is [23, Corollary 2.28].

The simple quotients L(τ) of the M(τ) for τ Irrep(W ) are the building blocks of a very ∈ important category of Hk–modules which we now describe.

If M is a Hk–module and α C, define the α h–weight subspace of M to be the generalised h– ∈ eigenspace of M with eigenvalue α, that is, the set x M : (h α)N x = 0 for sufficiently large N . { ∈ − } Definition 1.22. Define a category (H ), “category ”, to be the full subcategory of the category O k O of Hk–modules consisting of those Hk–modules M such that

1. M is finitely generated;

2. h Hk acts locally nilpotently on M. That is, if y h and m M then there exists N N ⊂ ∈ ∈ ∈ such that yN m = 0;

3. M is the sum of its h–weight subspaces.

Remark 1.23. It will be shown below in the proof of Theorem 1.25 that part (3) of Definition 1.22 actually follows from parts (1) and (2). We choose to include (3) as part of the definition because of its extreme importance in the study of category . O We collect together the basic properties of category and the modules M(τ) in the following O proposition.

13 Proposition 1.24. Write = (H ). O O k 1. is an abelian category, closed under extensions, subobjects and quotients. O 2. For all τ Irrep(W ), the standard module M(τ) belongs to . ∈ O 3. The set of simple objects of is L(τ) : τ Irrep(W ) . O { ∈ } 4. Every object M of has a composition series of the form O

M = Mk Mk 1 M1 M0 = 0 ⊃ − ⊃ · · · ⊃ ⊃

with Mi/Mi 1 = L(τi) for some τi Irrep(W ), 1 i k. We write [M : L(τ)] for the − ∼ ∈ ≤ ≤ composition multiplicity of L(τ) in M.

5. Let cτ C be the scalar by which z acts on τ. Then if σ = τ and [M(τ) : L(σ)] = 0, then ∈ 1 1 cτ cσ N. − ∈ 6. For all τ, [M(τ) : L(τ)] = 1.

7. M(τ) is an indecomposable Hk–module

8. M(τ) ∼= M(σ) if and only if L(τ) ∼= L(σ) if and only if τ ∼= σ as W –modules.

9. For each τ Irrep(W ), let (τ) denote the submodule of Hom [h ] W (Hk, τ) consisting of ∈ ∇ C ∗ ∗ those elements m for which dim(C[h] m) < . Then (τ) and L(τ) is the unique · ∞ ∇ ∈ O simple submodule of (τ) and [ (τ) : L(τ)] = 1. ∇ ∇ 10. Every simple object L(τ) of has a projective cover P (τ). Furthermore, P (τ) has a series O of submodules 0 = P P P = P (τ) 0 ⊂ 1 ⊂ · · · ⊂ n

such that Pi/Pi 1 is a standard module M(τi) for each i, Pn/Pn 1 = M(τ), and the BGG − − reciprocity formula [P (τ) : M(σ)] = [M(σ) : L(τ)] holds for all σ, τ Irrep(W ). ∈ 11. Dually, every L(τ) has an injective envelope I(τ) and I(τ) has a series of submodules

0 = I I I = I(τ) 0 ⊂ 1 ⊂ · · · ⊂ n

such that Ii/Ii 1 is a costandard module (τi) for all i, I1/I0 = (τ), and the BGG reciprocity − ∇ ∇ formula [I(τ) : (σ)] = [ (σ) : L(τ)] holds for all τ and σ. ∇ ∇ 14 Proof.

1. Follows from [33, Theorem 2.19].

2. This is [33, Lemma 2.3].

3. See [33, Proposition 2.11].

4. See [33, Corollary 2.16].

5. See [23, (32)].

6. See [23, (31)].

7. This is [23, Corollary 2.28].

8. If M(τ) ∼= M(σ) then L(τ) ∼= L(σ) since L(τ) is the unique simple quotient of M(τ). If L(τ) ∼= L(σ) then let λ be the eigenvalue of z on τ. Since L(τ) is a quotient of M(τ), the set x L(τ) : hx = λx equals 1 τ. This is W –stable since h commutes with W , so there { ∈ − } ⊗ is an isomorphism of W –modules τ ∼= σ.

9. The module (τ) is defined in [33, Section 2.3.1]. The statement that L(τ) is the unique ∇ simple submodule of (τ) is given in [33, Section 2.5.1]. ∇ 10. The statement about projective covers is [33, Corollary 2.10]. By [33, 2.6.2], there is a reciprocity formula [P (τ) : M(σ)] = [ (σ) : L(τ)]. But by [33, Proposition 3.3], we have ∇ [ (σ)] = [M(σ)] in the Grothendieck group K ( ), which yields the desired reciprocity ∇ 0 O formula.

11. It is shown in [33, Theorem 2.19] that is a highest-weight category in the sense of Cline- O Parshall-Scott [17]. The existence of a filtration of the given form follows from [17, Definition 3.1 (c)], while the BGG reciprocity is [17, Theorem 3.11] combined with the above-noted fact that [M(σ)] = [ (σ)] for all σ Irrep(W ). ∇ ∈

The modules (σ) appearing in the proof of Proposition 1.24 are called costandard modules ∇ We work mostly with standard modules in this thesis, and will use costandard modules only in the proof of Theorem 3.26.

15 1.3.3 Ideals of H and category k O One reason why category is useful is that it can be related to the ideal structure of H by the O k following theorem, which will be used in the next chapter to determine when Hk is simple.

Theorem 1.25 (Ginzburg). Let I be a proper ideal of Hk. Then I annihilates a nonzero object of category . O + Proof. The proof follows from [31, Theorem 2.3]. In the notation of [31], take A = Hk, A = C[h]+

(meaning the element of positive degree in C[h]), A− = C[h∗]+. Then (A±, h) is a commutative triangular structure on A in the sense of [31]. In [31, Definition 2.2], a category is defined as ↑O the category of finitely generated A–modules such that the action of A− is locally nilpotent. But in [31, Theorem 2.5], it is proved that if M ↑ and m M then dim(C[h] m) < , and hence ∈ O ∈ · ∞ m belongs to a sum of generalised h–eigenspaces. Thus, the category of Definition 1.22 is the O same as . Now by [31, Theorem 2.3], any primitive ideal of A annihilates a nonzero object of ↑O . But any nonzero ideal is contained in a primitive ideal. ↑O

1.3.4 The Dunkl representation

Of particular interest is the module M(triv) where triv denotes the trivial representation of W .

Since C[h] triv = C[h] as vector spaces, we get an action of Hk on C[h]. This action may be given ⊗ ∼ by the following homomorphism Hk End (C[h]). → C

x x x h∗ 5→ ∈ w w w W 5→ ∈ nH 1 αH (y) − y y = ∂y + nH kH,ieH,i y h 5→ ∇ αH ∈ H i=1 &∈A & where w denotes the endomorphism p w(p) = p w 1, and x denotes the endomorphism p xp. 5→ ◦ − 5→ Also, if y = α y then ∂ := α ∂ . i i i y i i ∂xi

The oper!ators y are called!Dunkl operators, and this map is called the Dunkl representation. ∇ 1 Write δ = H αH . Then the Dunkl operators may be viewed as elements of D(h)[δ− ] W . ∈A ∗ reg reg 1 Write h =#h H H = x h : δ(x) = 0 . Then D(h ) = D(h)[δ− ] by [55, Theorem 15.1.25, \ ∪ ∈A { ∈ 1 } Corollary 15.5.6]. So the Dunkl representation defines a homomorphism

p : H D(hreg) W. k → ∗

16 Proposition 1.26. The set δn of powers of δ forms a left and right Ore set in H . { }n∞=0 k

Proof. We must show that, given z Hk and N N, there exists M N and t Hk with ∈ ∈ ∈ ∈ δM z = tδN . By the PBW Theorem 1.15, it suffices to show this when z is a monomial in x , { i} y and w. But δ commutes with each x , and every w W acts by a scalar on δ, so it suffices { i} i ∈ M r N to check that, for each 1 i n and r N, there is an M with δ y Hkδ . But the ≤ ≤ ∈ i ∈ M M 1 commutation relations of Definition 1.17 imply that [yi, δ ] (C[h] W )δ − for all M, and ∈ ∗ therefore δN+ryr H δN as required. A similar proof shows that the powers of δ form a right Ore i ∈ k set as well.

1 Because of Proposition 1.26, we may form the localisation Hk[δ− ], and for any Hk–module M, 1 we may consider the localisation M[δ ]. We often write these localisations as H reg and M reg − k|h |h respectively.

Proposition 1.27. [33, Theorem 5.6] The map p is injective and the localization p reg : H reg |h k|h → D(hreg) W is an algebra isomorphism. ∗ Note that Proposition 1.27 says that we may regard H as the subalgebra of D(hreg) W k ∗ generated by C[h], W and the Dunkl operators.

1.3.5 The Knizhnik-Zamolodchikov functor

The Dunkl representation provides a link between the Cherdnik algebra and differential operators. This may be used to construct a functor KZ from category to a category of modules over a certain O finite-dimensional algebra, which is one of the most powerful tools for studying category , and O will be put to extensive use in the next chapter. We now give a description of how to define this functor. 1 Let M be an object of and consider the localization M[δ ] = M reg . This is a module O − |h reg for H reg , which is isomorphic to D(h ) W by Proposition 1.27. So M reg may be regarded k|h ∗ |h reg W reg W as a W –equivariant D(h )–module. The fixed point set M reg is then a D(h ) –module. The |h reg W reg 1 W algebras D(h ) and D(h /W ) are isomorphic , so M reg may be regarded as a D–module |h on hreg/W , which is a smooth complex variety. By [9, VI, 1.6, 1.7], there is an equivalence of categories between the category of D–modules on hreg/W and the category of vector bundles on

1This follows from [49, Th´eor`eme 4.2].

17 hreg/W endowed with a flat connection. We show that the connection associated with any object M of has regular singularities. Indeed, by [33, Proposition 5.7], the connection associated to O M(τ) has regular singularities. Now, [19, Proposition 4.6] states that if V V V is an exact $ → → $$ sequence of vector bundles with flat connection on a smooth complex manifold and V $, V $$ have regular singularities, then so does V . From the sequence M(τ) L(τ) 0 and the fact that the → → functors of localization to hreg and taking W –invariants are exact (because W is a finite group), we get that the connection associated to L(τ) has regular singularities. Hence, the connection associated to any object M of has regular singularities as well. Now by [9, IV, 7.1.1], there is an O equivalence of categories between the category of algebraic vector bundles on hreg/W endowed with a flat connection with regular singularities, and the category of holomorphic vector bundles on the complex manifold hreg/W endowed with a flat connection. Finally, by [9, IV, 1.1] and [19, Corollaire 1.4], there is an equivalence of categories between the category of holomorphic vector bundles on hreg/W endowed with a flat connection, and the category of finite-dimensional representations of the fundamental group π (hreg/W, ), where is any choice of basepoint. In this way, we obtain an 1 ∗ ∗ exact functor π (hreg/W, ) mod. The group π (hreg/W, ) is called the braid group B of O → 1 ∗ − 1 ∗ W W .

Now, following [33, 5.2.5], we define an algebra to be the quotient of CBW by the relations H nH 1 − j 2πik (T 1) (T det(s)− e− H,j ) = 0 (1.1) − − j'=1 where H , s W is the reflection about H with eigenvalue e2πi/nH , and T is an s–generator ∈ A ∈ of the monodromy around H in the parlance of [11, Section 4.C]. Note that the sign of kH,j differs from that given in [33, Section 5.2.5], because there is a sign error in [33], as remarked in [64, Section 5.2.1].

By [33, Theorem 5.13], the functor CBW mod constructed above factors through , and O → − H in this way we obtain an exact functor

KZ : mod O → H − f where mod denotes the category of finite-dimensional –modules. H − f H Definition 1.28. We call the algebra = (W ) the Hecke algebra of W with parameters k. H Hk We list the most important properties of the KZ functor in the following proposition.

18 Proposition 1.29. There exists an exact functor (the Knizhnik-Zamolodchikov functor) KZ : O → mod . Denote by the full subcategory of consisting of those objects M such that H − f Otor O M reg = 0. Then KZ factorises as |h

Q KZ KZ : / 0 mod O −→ O Otor−−→H − f where Q is the quotient functor. Furthermore, the following properties hold.

1. KZ0 is an equivalence.

i j 2. If kH,i kH,j + − / Z for all H and all i = j, then for any M and any τ Irrep(W ), − nH ∈ ∈ A 1 ∈ O ∈ we have a bijection

Hom (M, M(τ)) Hom (KZ(M), KZ(M(τ))). O → H

Proof. The first part is [33, Theorem 5.14]. The second part follows from [33, Proposition 5.9], which states that, under the given hypotheses,

∼ Hom (M, M(τ)) Hom / tor (QM, QM(τ)) O → O O where Q : / is the quotient functor. Since KZ is an equivalence, we get part (2) of the O → O Otor 0 proposition.

Note that since Hk is Noetherian, for any two finitely generated Hk–modules M, N, there is a

reg ∼ reg reg natural isomorphism HomHk (M, N) h HomH reg (M h , N h ) (see for example [63, 3.84]). | → k|h | | It is easy to check using this fact that the full subcategory of H reg –modules whose objects are k|h those modules of the form M reg for some M satisfies the universal property of / . |h ∈ O O Otor Therefore, we get the following corollary of [33, Proposition 5.9].

i j Corollary 1.30. Suppose kH,i kH,j + − / Z for all H and all i = j. Then for any M − nH ∈ ∈ A 1 ∈ O and any τ Irrep(W ), we have a bijection ∈

Hom (M, M(τ)) ∼ HomH reg (M hreg , M(τ) hreg ). O → k|h | |

19 1.3.6 Double centraliser property

One important property of category which follows from the existence of the KZ functor is the O so-called double centraliser property. This states that category is the category of modules over O a certain finite-dimensional algebra, and also that is the endomorphism ring of an object of H category . The double centraliser property is not known to hold for all complex reflection groups O since it requires that the dimension of the Hecke algebra is W . This is known to be true in all H | | cases, except for some of the exceptional complex reflection groups, for which it is conjectured to be true (see [33, Remark 5.12]). The double centraliser property is the following pair of theorems.

Theorem 1.31. [33, Theorem 5.15] Let W be a complex reflection group, k a vector of complex parameters for the Cherednik algebra of W , and the associated Hecke algebra. Suppose that Hk dim( ) = W . For each τ Irrep(W ), let P (τ) be the projective cover of L(τ) in the category Hk | | ∈ = (H (W )). Define an object PKZ by O O k ∈ O

PKZ = τ Irrep(W ) dim KZ(L(τ)) P (τ). ⊕ ∈ ·

Then there is an isomorphism of algebras

opp k = End (PKZ) . H ∼ O

Dually, we have the following theorem.

Theorem 1.32. [33, Theorem 5.16] Let W be a complex reflection group, k a vector of complex parameters for the Cherednik algebra of W , and the associated Hecke algebra. Suppose that Hk dim( ) = W . Suppose X is a projective generator for the category (for example, X can Hk | | ∈ O O be taken to be τ Irrep(W )P (τ)). Then there is an equivalence of categories ⊕ ∈

= End (KZ(X))opp mod. O ∼ Hk −

20 1.3.7 Twisting by a linear character

Following [33, Section 5.4.1], let ζ be a linear character of W (that is, a group homomorphism

W C∗), and define an automorphism of T (h h∗) W via → ⊕ ∗

x x x h∗ 5→ ∈ y y y h 5→ ∈ w ζ(w) w w W. 5→ · ∈

From the generators and relations 1.17, it is easy to check that this defines a map H H k → ζ(k) u(H) where ζ(k) is defined as follows. There is some u(H) such that ζ = det− . Then |WH |WH

ζ(k) = k k H,i H,u(H)+i − H,u(H) for 1 i n 1, where the subscripts are taken modulo n . By [33, 5.4.1], this isomorphism ≤ ≤ H − H H H gives an equivalence of categories F : (H ) (H ) which satisfies F V (τ) = k → ζ(k) ζ O k → O ζ(k) ζ V (τ ζ 1) where V stands for any of the symbols M, L or P . Furthermore, if (W ) denotes the ⊗ − Hk Hecke algebra associated to (H ) via the KZ functor, then there is a commutative diagram O k

Fζ (H ) / (H ) O k O ζ(k)

KZ KZ

 Gζ  (W ) mod / ζ(k)(W ) mod Hk − H − where G is an equivalence induced by an isomorphism of algebras (W ) = (W ). ζ Hk ∼ Hζ(k)

1.3.8 Absence of self-extensions

Before specialising to the case W = G(!, 1, n), we give one more useful property of category for O a general complex reflection group W , namely the absence of self-extensions of simple objects. We give the proof, which is identical to the proof given in [7] in the symmetric group case.

Theorem 1.33. [7, Proposition 1.12] For any τ Irrep(W ), Ext1 (L(τ), L(τ)) = 0. ∈ O

21 i π Proof. Suppose there is a short exact sequence 0 L(τ) N L(τ) 0. Then the set of → → → → generalised h–eigenvalues of N is equal to the set of generalised h–eigenvalues of L(τ). Consider the lowest weight space N0 N, that is, if z acts on τ by λ C, then ⊂ ∈ N := x N : (h + λ)kx = 0 for sufficiently large k . 0 { ∈ } If x N and y h, then yx is a generalised h–eigenvector with a lower eigenvalue than x, since ∈ 0 ∈ [h, y] = y. Therefore, yx = 0. So hN = 0. Now choose a 0 = v i(1 τ) N and a v N − 0 1 ∈ ⊗ ⊂ 0 $ ∈ 0 with 0 = π(v ) 1 τ. Then H v is a quotient of M(τ), since hv = 0. If H v has length > 1 then 1 $ ∈ ⊗ k $ $ k $ Hkv$ = N and so M(τ) surjects onto N, which contradicts [M(τ) : L(τ)] = 1. Therefore, Hkv$ is simple. Also, H v = im(i) is simple, since it is a nonzero quotient of L(τ). Then if H v H v = 0 k k ∩ k $ 1 then H v = H v and so 0 = π(v ) π(H v) = 0, a contradiction. Therefore, H v H v = 0 k k $ 1 $ ∈ k k ∩ k $ which forces N = H v H v = im(i) H v , and the sequence splits. k ⊕ k $ ⊕ k $

1.4 The case of W = G(!, 1, n)

Let us introduce the infinite family of complex reflection groups G(!, p, n). Let !, n 1 and p ! be ≥ | natural numbers. Then G(!, 1, n) may be defined to be the set of all n n complex matrices with × exactly one nonzero entry in each row and column, such that the nonzero entries are powers of 2πi n ε := e # . In this way, G(!, 1, n) has a natural defining representation h = C . Note that G(!, 1, n) n is isomorphic to the group (Z/!Z) Sn = (Z/!Z) ! Sn. This representation is irreducible if ! > 1. ! The group G(1, 1, n) is the symmetric group of degree n, and h = Cn is its natural representation.

The group G(2, 1, n) is the Weyl group of type Bn. If ! > 2 then G(!, 1, n) is not a Coxeter group. Further, define G(!, p, n) to be the normal subgroup of G(!, 1, n) consisting of those matrices in G(!, 1, n) such that the pth power of the product of the nonzero entries is 1.

1.4.1 The rational Cherednik algebra of G(!, 1, n)

From now on, we take W = G(!, 1, n). Let y , y , . . . , y denote the standard basis of h and { 1 2 n} x , . . . , x the dual basis of h . Then the complex reflections in W may be described as follows. { 1 n} ∗ First, for 1 i n and 1 t ! 1, we have an element st W defined by ≤ ≤ ≤ ≤ − i ∈ t t si(yi) = ε yi st(y ) = y j = i i j j 1

22 and for 1 i < j n and 0 t ! 1 we have an element σ(t) defined by ≤ ≤ ≤ ≤ − ij (t) t σij (yi) = ε− yj (t) t σij (yj) = ε yi σ(t)(y ) = y k = i, j ij k k 1 The set of complex reflections of W is st : 1 i n, 1 t ! 1 σ(t) : i < j, 0 t ! 1 . { i ≤ ≤ ≤ ≤ − } ∪ { ij ≤ ≤ − } The reflecting hyperplane of st is H = v h : x (v) = 0 while the reflecting hyperplane of σ(t) is i i { ∈ i } ij I = v h : x (v) = ε tx (v) . For each reflecting hyperplane H, we choose a linear functional i,j,t { ∈ i − j } α and a vector v as in Section 1.3. We can choose α = x , α = x εtx , v = y and H H Hi i Ii,j,t i − j Hi i v = y ε ty . The hyperplanes H belong to a single W –orbit C while C = I form Ii,j,t i − − j i s σ { i,j,t} another W –orbit. We have nCs = ! and nCσ = 2. We write κ00 = kIi,j,t,1 for all i, j, t and κj = kHi,j for all i, where κ0 = κ( = 0. Then the rational Cherednik algebra Hκ = Hκ(W ) of W is the quotient of the C–algebra T (h h∗) W by the relations [x, x$] = 0 for x, x$ h∗, [y, y$] = 0 for ⊕ ∗ ∈ y, y h, together with the commutation relations $ ∈ n ( 1 ( 1 − − [y, x] = y(x) + y(x )x(y ) (κ κ ) εrjsr i i j+1 − j i r=0 &i=1 &j=0 & ( 1 − t t (t) + κ y(x ε x )x(y ε− y )σ (1.2) 00 i − j i − j ij 1 i

( 1 ( 1 ( 1 − − − [y , x ] = 1 + (κ κ ) εprsr + κ σ(t) 1 i n i i p+1 − p i 00 ir ≤ ≤ p=0 r=0 r=i t=0 & & &+ & ( 1 − [y , x ] = κ εtσ(t) i = j. (1.3) i j − 00 ji 1 &t=0 For the algebra Hκ = Hκ(G(!, 1, n)) we have all the theory of the preceding sections: category = , the Dunkl representation and the KZ functor mod. We now identify the algebra O Oκ O → H − 23 through which the KZ functor factors in the W = G(!, 1, n) case. Recall that this algebra is the H quotient of CBW by the relations (1.1) of Section 1.3.5.

Definition 1.34. [2] Let !, n be positive integers and let q, u1, u2, . . . , u( C be complex numbers. ∈

The Ariki-Koike algebra with parameters (q, u1, . . . , u() is the algebra generated by Ts, Tt2 , . . . , Ttn subject to the relations:

T T T T T T T T = 0 s t2 s t2 − t2 s t2 s [T , T ] = 0 i 3 s ti ≥ T T T T T T = 0 2 i n 1 ti ti+1 ti − ti+1 ti ti+1 ≤ ≤ − [T , T ] = 0 i j > 1 ti tj | − | (T 1)(T + q) = 0 2 i n ti − ti ≤ ≤ ( (T 1) (T u ) = 0 s − s − j j'=2 From the braid diagram in [11, Table 1], we see that the Hecke algebra through which H 2πiκ00 KZκ factors is the Ariki-Koike algebra with parameters (q, u1, . . . , u( 1) where q := e and − u := ε j+1e 2πiκj 1 for 1 j !, where as before, ε = e2πi/(. Note in particular that q = 0 and j − − − ≤ ≤ 1 uj = 0 for all j. We denote this algebra by (q,u1,...,u# 1)(G(!, 1, n)). The Ariki-Koike algebra will 1 H − be our main tool for obtaining information about category , so we now describe its representation O theory.

1.4.2 Representation theory of the Ariki-Koike algebra

A good reference for the Ariki-Koike algebra is the survey article [54]. We remark that is a H finite-dimensional algebra of dimension n!!n = W , by [54, Theorem 2.2]. If we put q = 1 and | | i 1 ui = ε − for all i in Definition 1.34, then the Ariki-Koike algebra reduces to the group algebra CW . Thus, may be regarded as a deformation of CW . H Let n be a a positive integer. A partition of n is by definition a sequence of nonnegative integers λ λ λ with λ = n. We ignore trailing zeroes, so that the partition 1 " 2 " · · · " t i i λ1 λ2 λt is identified wit!h λ1 λ2 λt 0 0 . Each partition has a " " · · · " " " · · · " " " " · · · th Young diagram, which is a left-justified array of n boxes in the plane, with λi boxes in the i row.

24 For example, the Young diagram

corresponds to the partition 5 3 2 of 10. For a partition λ of n, we write λ = n = λ . " " | | i i We identify λ with its Young diagram and label the rows of the Young diagram from the top!down and the columns from left to right. The box in the (i, j) position is called the (i, j)–node of λ. For example, in the Young diagram of 5 " 3 " 2, a λ = b the box labelled a is the (1, 2)–node of λ, and b is the (3, 1)–node. In general, if x is the (i, j)–node, we write row(x) = i and col(x) = j. A multipartition of n with ! parts is an !–tuple of partitions λ = (λ(1), λ(2), . . . , λ(()) with ( λ(i) = n. We may identify a multipartition λ with an !–tuple of Young diagrams, and hence i=1 | | r!egard λ as a subset of N N N. A node (i, j, k) of λ is defined to be a node (i, j) of λ(k) for some × × k. More generally, a node is any element of N N N. × × By [54], for each multipartition λ of n with ! parts, the Ariki-Koike algebra has a Specht H module Sλ. These modules are the cell modules with respect to a certain cellular basis of . H Each Specht module has a quotient Dλ which is either 0 or absolutely irreducible, and the set Dλ : Dλ = 0 is a complete set of pairwise nonisomorphic simple –modules (see [54, Theorem { 1 } H 3.12]). We will need a parametrisation of this set. There are two different parametrisations, depending on whether q = 1 or q = 1. 1 Lemma 1.35. [52, Theorem 3.7] If q = 1 and λ is a multipartition of n with ! parts, then Dλ = 0 1 (s) if and only if λ = ∅ whenever s < t and us = ut.

If q = 1 then the description, due to Ariki and stated in [54, Theorem 3.24] is more complicated. 1 The nonzero Dλ are in bijection with the set of Kleshchev multipartitions, which we now describe. Given a multipartition λ, the residue of a node x in row i and column j of λ(k) is defined to be u qj i. A node x in λ with residue a is called a removable a–node if λ x is a multipartition. A k − \ { } node x not in λ with residue a is called an addable a–node if λ x is a multipartition. ∪ { } Say a node y λ(r) is below a node x λ(k) if either r > k, or r = k and row(y) > row(x). ∈ ∈

25 A removable a–node x is called normal if whenever x$ is an addable a–node below x then there are more removable a–nodes between x and x$ than there are addable a–nodes. The highest normal a node in λ is called the good a–node. The set of Kleshchev multiparitions with ! parts is defined inductively as follows: the empty partition (∅, . . . , ∅) is Kleshchev, and otherwise λ is Kleshchev if and only if there is some a C ∈ and a good a–node x λ such that λ x is Kleshchev. ∈ \ { } Lemma 1.36. Suppose q = 1. Then the set Dλ : Dλ = 0 of nonisomorphic irreducible 1 { 1 } (G(!, 1, n))–modules is in bijection with the set of Kleshchev multipartitions of n with ! H(q,u1,...,u#) parts.

Example 1.37. We now give an example to illustrate the above definitions. Consider the multi- partition λ of n = 2 defined by λ = Ø . $ % The addable nodes of λ are the nodes labelled x, y, z on the following diagram. The node labelled r is the only removable node. r x z y ( ) Let q be a non-root of unity and suppose u1 = u2 = q. Then the residues of the nodes of λ and the addable nodes of λ are given by the following diagram.

q q2 q3 q 1 ( ) The node r is a removable q2–node. There is no addable q2–node below r, and so r is a good q2– node. Thus, λ is Kleshchev if and only if µ := Ø is Kleshchev. But µ has one removable node, and this node has residue q, and there is$an adda%ble node below it with residue q, and no removable nodes lie between them. Thus, the unique removable node of µ cannot be a good node. Therefore, µ is not Kleshchev and so λ is not Kleshchev.

By [54, Theorem 3.13], the Ariki-Koike algebra is semisimple if and only if Sλ = Dλ for all λ. Furthermore, semisimplicity can be expressed entirely in terms of the parameters thanks to the following semisimplicity criterion.

26 Theorem 1.38. [1, Main Theorem] The algebra = (G(!, 1, n)) is semisimple if and H H(q,u1,...,u#) only if d=( 1 − [n] ! (u qdu ) = 0, q i − j 1 i=j d= (+1 '+ '− where [n] ! = n (1 + q + + qj 1). q j=1 · · · − # Finally we need a description of the blocks of . This is given in [51, Corollary 2.16]. Define H the content cont(λ) of a multipartition λ to be the multiset of residues of λ, ie. the set of residues counted according to multiplicity. Then by [51, Lemma 2.2], all the composition factors of a Specht module Sλ for belong to the same block, so we can say a Specht module belongs to a block B if H all its composition factors belong to B.

Theorem 1.39. [51, Corollary 2.16] Let λ and µ be multipartitions of n with ! parts. If q = 1, 1 two Specht modules Sλ and Sµ of belong to the same block if and only if cont(λ) = cont(µ). H

1.4.3 Fourier map

We close this chapter with an analogue for W = G(!, 1, n) of the so-called Fourier automorphism of the Cherednik algebra of Sn, which will be needed in the proof of Lemma 4.39. It is easy to see that (0) the group W is generated by the elements s1 and σi := σi 1,i, 2 i n. In fact, by [2, Proposition − ≤ ≤ 2.1], W has a presentation as the abstract group generated by symbols s and σ , 2 i n, subject 1 i ≤ ≤ to the relations

( s1 = 1

s1σ2s1σ2 = σ2s1σ2s1

σ2 = 1 1 i n 1 i ≤ ≤ − σ σ σ = σ σ σ 2 i n 1 i i+1 i i+1 i i+1 ≤ ≤ − σ σ = σ σ i j > 1 i j j i | − | s σ = σ s i 3. (1.4) 1 i i 1 ≥

p p Calculating with the defining representation of W shows that these relations imply s1σ2s1σ2 = p p σ s σ s for all integers p. It follows that there is an automorphism (sic) W ∼ W defined on the 2 1 2 1 → 1 (k) k (0) k generators by s s− and σ σ for all i. Note that since σ = s σ s− , this automorphism 1 5→ 1 i 5→ i ij i ij i

27 (k) ( k) sends σij to σij− for all i, j, k. This can be extended to a map

ψ : T (h h∗) W T (h h∗) W ⊕ ∗ → ⊕ ∗ by setting ψ(y ) = x and ψ(x ) = y . This map descends to the Cherednik algebra, and by i i i − i checking the relations (1.3) of Section 1.4, we get the following theorem (which has not appeared before in the literature).

( Theorem 1.40. Let κ = (κ00, κ1, . . . , κ( 1) C and define κ = (κ00, κ1, κ1 κ( 1, κ1 κ( 2, . . . , κ1 − ∈ − − − − − κ2). Then there is an algebra isomorphism

ψ : H H κ → κ given on the generators as above. In particular, ψ(C[h]) = C[h∗] and ψ(C[h∗]) = C[h].

If ! = 1 or ! = 2 then κ = κ, and ψ coincides with the Fourier automorphism of Hκ(W ) defined in [33, Remark 4.6]. Otherwise we may write ψ = ψ : H H and since κ = κ, we get κ κ → κ 2 (ψκψκ) = idHκ .

28 Chapter 2

The semisimple case

In this short chapter, we consider the Cherednik algebra Hκ of the group W = G(!, 1, n). We determine when Hκ is simple and show that the simplicity of the ring Hκ is equivalent to the semisimplicity of the category (recall that an abelian category is said to be semisimple if and Oκ A only if every short exact sequence in splits). A In fact, we will prove that simplicity of the Cherednik algebra Hk(W ) is equivalent to semisim- plicity of the Hecke algebra (W ) for any complex reflection group W and parameters k such that Hk dim = W (recall that by [33, Remark 5.12], this is known to hold for all complex reflection Hk | | groups apart from some of the exceptional types, and is conjectured to hold for these also).

2.1 The semisimple case

Let W be a complex reflection group and let k be a vector of parameters for the Cherednik algebra reg of W . By Proposition 1.27, H reg is isomorphic to D(h ) W , a simple ring. So H is in some k|h ∗ k sense close to being simple. Before we begin the study of the more interesting cases where Hk is not simple, it is helpful to know precisely when Hk is a simple ring. We will answer this in the G(!, 1, n) case with the following theorem.

Theorem 2.1. Let W be a complex reflection group and k a vector of complex parameters for the Cherednik algebra of W . Write H = H (W ), = (H (W )) and = (W ), the Hecke k k O O k Hk Hk algebra associated to k. Suppose dim = W . Hk | | Then the following are equivalent.

29 1. Hk is a simple ring.

2. Category is a semisimple category. Ok 3. The Hecke algebra is semisimple. Hk

( In particular, if W = G(!, 1, n) and κ C , then Hκ is a simple ring if and only if κ is semisimple ∈ O if and only if the Ariki-Koike algebra is semisimple. Hκ We begin with a lemma first proved in [6, Remark, page 9].

Lemma 2.2. [6] Let W be an arbitrary complex reflection group and k a vector of complex parameters for the Cherednik algebra. Category = (H ) is semisimple if and only if M(τ) = O O k L(τ) for all τ Irrep(W ). ∈ Proof. Suppose M(τ) = L(τ) for all τ. By [33, Corollary 2.10], the projective cover P (τ) of L(τ) has a filtration 0 = Q0 Q1 Qd = P (τ) with Qi/Qi 1 a standard module for all i, ⊂ ⊂ · · · ⊂ − and Qd/Qd 1 = M(τ). By BGG reciprocity, [P (τ) : M(σ)] = [M(σ) : L(τ)] for all τ, σ. Thus, − [P (τ) : M(σ)] = 0 if τ = σ, while [P (τ) : M(τ)] = 1. So P (τ) = M(τ) = L(τ). Therefore, all the 1 ∼ ∼ L(τ) are projective and, since every object of has a finite filtration by L(τ), every object of is O O projective. So is semisimple. O Conversely, suppose is semisimple. Then M(τ) = L(τ) R(τ) for all τ. But M(τ) is O ⊕ indecomposable, so M(τ) = L(τ) for all τ.

Our next lemma relates semisimplicity of to localisation. Recall that is the full subcat- O Otor egory of objects M in with M reg = 0. O |h Lemma 2.3. Let W be an arbitrary complex reflection group and k a vector of complex parameters. Category = (H ) is semisimple if and only if = 0. O O k Otor Proof. Suppose is semisimple. Then by Lemma 2.2, L(τ) = M(τ) for all τ. So L(τ) is a free O reg C[h]–module for all τ, and so L(τ) hreg = C[h ] [h] L(τ) = 0 for all τ. Therefore, M hreg = 0 for | ⊗C 1 | 1 all nonzero M in . O Conversely, suppose = 0. Then L(τ) reg = 0 for all τ. Now apply [33, Proposition 5.21], Otor |h 1 which states that if L(τ) reg = 0 then L(τ) is a submodule of a standard module. By Proposition |h 1 1.24, we may choose an ordering τ < τ < < τ on Irrep(W ) such that [M(τ ) : L(τ )] = 0 only 1 2 · · · m i j 1 30 if i j. We show by induction on this ordering that M(τ) = L(τ) for all τ. First, consider L(τ ). ≤ 1 Since L(τ ) reg = 0, L(τ ) M(τ ) for some i. So i 1, therefore i = 1 and L(τ ) M(τ ). But 1 |h 1 1 ⊂ i ≤ 1 ⊂ 1 [M(τ1) : L(τ1)] = 1 and L(τ1) is a quotient of M(τ1). So we conclude that M(τ1) = L(τ1). Now consider L(τ ). Then L(τ ) M(τ ) for some i, so i 2. But i = 1 since we have already shown 2 2 ⊂ i ≤ 1 that the only composition factor of M(τ1) is L(τ1). Therefore i = 2 and M(τ2) = L(τ2). Continuing inductively, we get M(τ) = L(τ) for all τ. Therefore, is semisimple by Lemma 2.2. O We can now prove that parts (2) and (3) of Theorem 2.1 are equivalent. This was first proved in [35] by a different method.

Lemma 2.4. Let W be a complex reflection group and let k be a vector of parameters for the Cherednik algebra of W . Let H (W ) be the Cherednik algebra of W and let = (H (W )). Let k O O k be the Hecke algebra associated to k. Suppose dim = W . Then is a semisimple category Hk Hk | | O if and only if is a semisimple algebra. Hk Proof. Suppose is semisimple. Then = 0 by Lemma 2.3, and it follows that the functor KZ O Otor 0 is an equivalence mod. So mod is a semisimple category and therefore is a O → Hk − Hk − Hk semisimple algebra. Conversely, suppose is semisimple. By the hypothesis, dim( ) = W , and so we may Hk Hk | | apply Theorem 1.32 to get that

= End (KZ(X))opp mod O ∼ Hk − for some object KZ(X) of mod. Since is semisimple, KZ(X) is a direct sum of simple modules, Hk− Hk and hence End (KZ(X)) is a direct product of matrix algebras Matri (C), so is a semisimple algebra Hk by Wedderburn’s theorem. Therefore, End (KZ(X))opp mod is a semisimple category, so is Hk − O also semisimple.

Lemma 2.5. Let W be a complex reflection group and k a vector of complex parameters for the rational Cherednik algebra of W . The rational Cherednik algebra Hk = Hk(W ) is a simple ring if and only if = (H (W )) is a semisimple category. O O k Proof. Suppose H is a simple ring. Let τ Irrep(W ). Then L(τ) is a nonzero H –module, so k ∈ k the left annihilator of L(τ) is a proper two-sided ideal of Hk, and is therefore zero. We show that L(τ) reg = 0. If not, then L(τ) reg = 0. But L(τ) = M(τ)/R(τ), so as a vector space |h 1 |h 31 L(τ) = (C[h] τ)/R(τ). Let v τ be nonzero, and consider 1 v + R(τ) L(τ). Then there is ⊗ ∈ ⊗ ∈ some t with δt(1 v) + R(τ) = R(τ), and therefore if f C[h] then δt(f v) = 0 in L(τ). Also, we ⊗ ∈ ⊗ t t may as well take t to be a multiple of H WH , so that δ is W –invariant. Then δ annihilates ∈A | | L(τ), a contradiction. Therefore, L(τ)#hreg = 0. Therefore, is semisimple by Lemma 2.3. | 1 O Finally, if is semisimple, we show that H is simple. Let I H be a proper two-sided ideal. O k ⊂ k Then I annihilates a simple object L(τ) of by Theorem 1.25. But L(τ) = M(τ) by Lemma 2.2. O So IM(τ) = 0. Therefore, I reg annihilates M(τ) reg = 0. So I reg = H reg and since H reg |h |h 1 |h 1 k|h k|h is a simple ring, we get I reg = 0. But I H and, by the PBW Theorem, H is a torsionfree |h ⊂ k k C[h]–module, so I hreg = 0 implies I = 0 as required. |

Proof of Theorem 2.1

Theorem 2.1 follows immediately from Lemma 2.4 combined with Lemma 2.5. In the case of W = G(!, 1, n), it is known that dim = W by [54, Theorem 2.2], so Theorem 2.1 applies to W . H | | Example 2.6 (the cyclic case). Some insight into the Cherednik algebra of G(!, 1, n) can be obtained by studying the simplest case, namely when n = 1. In this case, all calculations can be done explicitly.

We let W = G(!, 1, 1) = Z/!Z = s , acting on h = C via s(y) = εy where y is a basis of ∼ , - { } h. Let κ1, . . . , κ( 1 C. From the relations (1.2), we see that the Cherednik algebra Hκ(W ) is − ∈ generated by three elements x, y, s with the relations

1 1 sxs− = ε− x

1 sys− = εy ( 1 − [y, x] = 1 + (κ κ )!e (2.1) p+1 − p p p=0 & 1 ( 1 pj j where ep = ( j−=0 ε s .

The group!W has ! irreducible representations which we denote τi, 0 i ! 1. We take ≤ ≤ − τi = Cei. We wish to consider the standard modules M(τi). The module M(τi) is isomorphic to a C[x] as a vector space and has a basis x τi : a 0 . From the commutation relation (2.1) and { ⊗ ≥ } induction, we obtain ( 1 a 1 a − − a 1 [y, x ] = a + (κ κ )! e x − .  p+1 − p p+r &p=0 &r=0   32 It follows that the action of y on xa τ M(τ ) is given by ⊗ i ∈ i

a a 1 y (x τ ) = (a + !(κ κ ))x − τ , (2.2) · ⊗ i i+a − i ⊗ i where the subscript i of κi is taken modulo !. Since any Hκ–submodule of M(τi) is a C[x]– submodule, we get that any proper quotient of M(τi) must be finite-dimensional, and furthermore from equation (2.2), we see that M(τ ) has a proper quotient if and only if a + !(κ κ ) = 0 for i i+a − i some a 1. ≥ Now we describe the Hecke algebra of W . The braid group of W is by definition the fundamental group π1(C∗/W ) ∼= π1(C∗) = Z, so the Hecke algebra has one generator T , and using the relations (1.1), may be described as C[T ] = H ( (T u ) j=1 − j (j 1) 2πiκ where uj = ε− − ε− j for all j. This al#gebra is semisimple unless ut+r = ut for some t and r some ! > r > 0. This happens if and only if κt+r κt + Z. So !(κt+r κt)+ r α! = 0 for some − ( ∈ − − α Z. If r α! > 0 then by equation (2.2), there is a finite-dimensional simple module in category ∈ − . If r α! < 0 then !(κt+r+(( r) κt+r) + (α 1)! + (! r) = 0, and so again by equation (2.2), O − − − − − category contains a finite-dimensional module. So we see that M(τ) = L(τ) for all τ Irrep(W ) O ∈ if and only if the algebra is semisimple. This confirms Lemma 2.4. H

2.2 Remarks

1. Let W = G(!, 1, n). From the semisimplicity criterion 1.38, we get that Hκ is simple if and only if

i 2πiκi j 2πi(κj +cκ00) [n] 2πiκ ! ε− e− ε− e− = 0. e 00 − 1 i=j, ( c ( + −'≤ ≤ $ % ( In particular, the set of κ C such that Hκ is simple, is the complement of a countable union ∈ of hyperplanes, and hence is a dense subset of C( in the Euclidean topology (such parameters are sometimes said to be Weil generic).

2. Our proof of Lemma 2.5 is original. The implication semisimple = H simple was Hk ⇒ k proved in [6, Theorem 3.1]

3. The calculations of Example 2.6 are standard. See for example [16, Section 2.2].

33 Chapter 3

The almost-semisimple case

3.1 Finite-dimensional modules

( In this chapter, we take W = G(!, 1, n). For κ C , we write as usual Hκ for the Cherednik ∈ algebra, for the Ariki-Koike algebra, for the category (H ), and KZ for the KZ functor. Hκ Oκ O κ κ reg We have seen in Chapter 2 that H is almost always simple. Furthermore, H reg = D(h ) κ κ|h ∼ ∗ W = (D(h) W )[δ 1], which is a simple ring by Proposition 1.11. So H is close to being simple. ∗ − κ However, it is an interesting fact that not only is Hκ not always simple, in fact, it can have finite- dimensional modules. These modules are of great interest and we will give one application of them in Chapter 5. Much of the research on Cherednik algebras to date has involved constructing and classifying their finite-dimensional modules, see for example [7], [15], [20] and [34]. The study of finite-dimensional modules can be reduced to category because of the following well-known O theorem.

Theorem 3.1. Every finite-dimensional H –module belongs to category . κ O

Proof. A finite-dimensional Hκ–module M is clearly a direct sum of generalised h–eigenspaces. Since [h, y] = y for all y h, if W denotes the generalised eigenspace of M with eigenvalue α, − ∈ α then hWα Wα 1. But the set of the real parts of the generalised eigenvalues of h on a finite- ⊂ − dimensional module is bounded below, so each y h must act locally nilpotently, as required. ∈ Another useful fact about finite-dimensional modules is that they are not “seen” by the KZ functor.

34 Proposition 3.2. If L is a finite-dimensional Hκ–module then KZ(L) = 0.

Proof. Suppose L is a finite-dimensional Hκ–module. Then the left annihilator of L is nonzero, since H is an infinite-dimensional algebra. So there is some x H with xL = 0. Therefore, κ ∈ κ reg annD(hreg) W (L hreg ) = 0. But D(h ) W is a simple ring by Proposition 1.11. So L hreg = 0 and ∗ | 1 ∗ | hence KZ(L) = 0.

In the W = G(!, 1, n) case, a family of finite-dimensional modules has been constructed by Chmutova and Etingof, which will be very useful to us. We summarise the results we need in the following theorem.

Theorem 3.3. [16] Suppose !(n 1)κ00 + !κs = s + t! < 0 for some t Z and some 1 s !, − − ∈ ≤ ≤ and suppose [n] ! = 0 where q = e2πiκ00 . Then there exists a finite-dimensional quotient Y˜ of q 1 c M(triv) and, as graded W –modules, Y˜ = U n where r := s t! > 0 and U is the representation c ∼ r⊗ − r r 1 C[u]/(u ) of s1 = Z/!Z with s1u = ε− u, while Sn acts by permuting the factors of the tensor , - ∼ product U n. In particular, dim(Y˜ ) = (s t!)n. r⊗ c − Proof. By the definition of the Cherednik algebra in [16, Section 2.1, Section 4.1], the parameters k and cj of [16] are related to κ00, κi by

κ = k 00 − r 1 ( 1 2 − − at κ = ε− c . r − ! t a=0 t=1 & & Translating our parameters into the language of [16], we therefore get

( 1 ts − 1 ε− !(n 1)k + 2 ct − t = r − 1 ε− &t=1 − where r = s !t is a positive integer of the form (p 1)! + s for some nonnegative integer p and − − some 1 s ! 1. Then we have the module Y˜ defined in [16, Theorem 4.2] which is a quotient ≤ ≤ − c of M(triv). Furthermore, since [n] ! = 0, we may apply [16, Theorem 4.3] to conclude that Y˜ is q 1 c ˜ n finite-dimensional, and by [16, Theorem 4.3 (ii)] and [16, Theorem 4.2] we get that Yc ∼= Ur⊗ .

In the case of W = Sn, the rational Cherednik algebra depends on only one parameter c = κ00, and the finite-dimensional modules have been completely classified in the paper [7]. The result is as follows.

35 Theorem 3.4. [7] Suppose the rational Cherednik algebra Hc(Sn) has a finite-dimensional module. Then the following hold.

1. There are r, n N with (r, n) = 1 and c = r/n. ∈ ± 2. For some linear character χ of S , L(χ) is finite-dimensional, and if τ = χ then dim L(τ) = n 1 . ∞ 3. Category splits as = ss, where is generated by L( kh χ), 0 k n, and O O O∧ ⊕ O O∧ ∧ ⊗ ≤ ≤ ss is a semisimple category generated by the other simple objects. O 4. The composition multiplicities in the nontrivial block are O∧

1 if j = i, i + 1 [M( ih χ) : L( jh χ)] = ∧ ⊗ ∧ ⊗  0 otherwise

 Proof. Parts 1 and 2 follow from [7, Theorem 1.2]. Part 3 is [7, Theorem 1.3 (iii)]. Part 4 is [7, Corollary 3.10].

We cannot hope that Theorem 3.4 will hold in the case of G(!, 1, n) for any !. For example, consider the case of the cyclic group W = G(!, 1, 1). We use the notation of Example 2.6. We have the standard module M(τ ), 0 i ! 1. Equation (2.2) shows that if we take κ = i/! i ≤ ≤ − i − for 1 i ! 2, then in fact H (G(!, 1, 1)) has ! 1 finite-dimensional simple modules and just ≤ ≤ − κ − one infinite dimensional simple. If, however, we insist that there is exactly one finite-dimensional simple module in category , then we have the following analogue of Theorem 3.4. The proof of O the following proposition can be viewed as a toy version of the proof of Theorem 3.7 to follow.

Proposition 3.5. Suppose W = G(!, 1, 1) = Z/!Z and there is exactly one finite-dimensional simple module in the category = (H (W )). Then the unique finite-dimensional simple module O O κ is L(τ ) for some i, and there exists a = 0 such that there is an exact sequence i 1

0 M(τ ) = L(τ ) M(τ ) L(τ ) 0, → a+i a+i → i → i → implying that L(τ ) and L(τ ) belong to the same block. Furthermore, for each j = i, i+a, L(τ ) i a+i 1 { j } is a semisimple block of . O

36 Proof. In the notation of Example 2.6, if L(τ ) is finite-dimensional then we must have y (xa τ ) = 0 i · ⊗ i for some a 1. The generator s of W acts on xa τ via the scalar ε (i+a). Let R(τ ) denote ≥ ⊗ i − i the unique maximal submodule of M(τi). Then we have a map M(τi+a) # R(τi) defined by 1 τ xa τ . Since we assumed that L(τ ) was infinite-dimensional, M(τ ) must be ⊗ i+a 5→ ⊗ i i+a i+a simple, and so this map is an isomorphism.

In order to prove the statement about the blocks, note from Example 2.6 that if u1, . . . , u( denote the parameters in the Hecke algebra , then two of the u must coincide, since there is H i a finite-dimensional simple module in category . Furthermore, no more than two of the u can O i coincide, or else there would be more than one finite-dimensional simple module, by Equation (2.2). Relabelling the parameters, let us say u = u := u, and u, u , u , . . . , u are all distinct. Then 1 2 { 3 4 (} the block decomposition of is H C[T ] C[T ] = ( . H ∼ (T u)2 × ×i=3 (T u ) − ( − i ) Therefore, has ! 1 blocks. By [33, Corollary 5.18], there is a bijection between blocks of and H − O blocks of . Since we have already shown that one block of contains at least two simples, all the H O other blocks must be singletons. Furthermore, these blocks are semisimple by Theorem 1.33.

3.2 The main theorem

We return to the case of W = G(!, 1, n). We have seen in Section 2.2 that is semisimple for Oκ almost all values of κ. Now let us consider the case where is not semisimple. It turns out Oκ that if = is close to being semisimple then we can still completely describe the structure of Oκ O category , and yet category can contain finite-dimensional modules. Note that by Lemma 2.3, O O is semisimple if any only if KZ is an equivalence. Motivated by this, we look for conditions under O which KZ is close to being an equivalence. We make the following definition.

Definition 3.6. We say that the KZ functor separates simples if whenever S # T are simple objects of , then KZ(S) # KZ(T ). O Most of this chapter will be devoted to the proof of the following theorem.

Theorem 3.7. Suppose ! > 1 and KZ separates simples. Then either is semisimple, or the O following hold:

37 1. There exists a linear character χ of W such that L(χ) is finite-dimensional and all the other simple objects in are infinite-dimensional. O 2. There exists a positive integer r not divisible by !, such that dimL(χ) = rn.

3. Let s N be the residue of r modulo !, 1 s ! 1. Then there is a representation hs of ∈ ≤ ≤ − W with dim h = dim h such that if τ / ih χ 0 i n , then M(τ) = L(τ). s ∈ {∧ s ⊗ | ≤ ≤ } 4. = ss where is generated by the L( ih χ) and ss is a semisimple category O O∧ ⊕ O O∧ ∧ s ⊗ O generated by the other simple objects.

5. The composition multiplicities in are O∧

1 if j = i, i + 1 [M( ih χ) : L( jh χ)] = ∧ s ⊗ ∧ s ⊗  0 otherwise

 Before proving Theorem 3.7, we make some remarks. Theorem 3.7 is an analogue for G(!, 1, n) of Theorem 3.4. Although the methods we use for proving Theorem 3.7 are similar to those of [7], we have to use different arguments to get round the problem that in the G(!, 1, n) case, it is not known whether the functor KZ takes standard modules M(λ) in to the corresponding Specht O modules Sλ for , ie. we do not know an analogue of [7, Lemma 3.2]. We also have to do some H work to calculate the blocks of the Hecke algebra at the parameters that we are interested in. One reason why Theorem 3.7 is of interest is that it gives a source of examples of choices of κ such that there is a finite-dimensional object in category , and yet category is completely O O understood. We will later, in Theorem 3.23, give examples of choices of parameters such that KZ separates simples.

Remark 3.8. It would be interesting to know whether, as in the ! = 1 and n = 1 cases, KZ is guaranteed to separate simples whenever there is just one finite-dimensional simple object in category . We cannot prove this, but note that Theorem 3.7 is true in the n = 1 case, because in O this case either L(τ) = M(τ) or L(τ) is finite-dimensional. So in the n = 1 case, if KZ(L(τ)) = 0 1 then L(τ) = M(τ). Hence if KZ separates simples then there can be at most one finite-dimensional simple object in category , and Theorem 3.7 reduces to Proposition 3.5. Therefore, in the proof O of Theorem 3.7, we may assume that n > 1.

38 3.3 Proof of Theorem 3.7

We begin with a lemma. As usual, write for the Ariki-Koike algebra defined in Definition 1.34, H with parameters q = e2πiκ00 and u = ε (i 1)e 2πiκi 1 , 1 i !. i − − − − ≤ ≤ Lemma 3.9. Suppose that KZ separates simples. Then has at least Irrep(W ) 1 simple modules. H | |− Proof. Suppose KZ separates simples. For any simple object S of , KZ(S) is either 0 or simple O because KZ induces an equivalence / mod, and if S is simple then either S reg = 0 or O Otor → H − |h S reg is simple. Furthermore, the simple objects KZ(S) are pairwise nonisomorphic. If KZ(S) = 0 for |h some S then since KZ separates simples, we must have KZ(T ) = 0 for all simple T # S. Therefore, 1 has at least Irrep(W ) 1 simple modules. H | | −

Note that when has Irrep(W ) simple modules, we must have L(τ) reg = 0 for all τ, and so H | | |h 1 = 0, and so is semisimple by Lemma 2.3. From now on, we will assume that we are not in Otor O the semisimple case. We prove a series of lemmas which will give (at least in principle) a description of all the possible values of the parameters such that has Irrep(W ) 1 simple modules. First, H | | − we show that q = 1. 1 Lemma 3.10. Suppose that has Irrep(W ) 1 simple modules. Then q = 1. H | | − 1 Proof. Suppose q = 1. Then by Lemma 1.35, since is not semisimple, there must be some s < t H with us = ut. Under the assumption that n > 1 and ! > 1, there are at least three multipartitions λ with λ(s) = ∅. Hence, by Lemma 1.35, there are at least three Dλ which are zero and so 1 H cannot have Irrep(W ) 1 simple modules. So q = 1. | | − 1 Therefore, the simple –modules are in bijection with Kleshchev multipartitions by Lemma H 1.36. Now, Ariki’s semisimplicity criterion (Theorem 1.38) tells us that [n]q! i

39 labelled λ in the diagram below is an addable node below µ with the same residue as µ, and there are no removable nodes between them. k boxes in row µ 2λ 34 5 .

Hence, ρk is not Kleshchev and therefore ρn, a row of n boxes, is not Kleshchev. So we may define multipartitions λ1 = (ρn, ∅, . . . , ∅) and λ2 = (∅, ρn, ∅, . . . , ∅), neither of which is Kleshchev (here we use the hypothesis that ! > 1). This contradicts the fact that there is only one non-Kleshchev multipartition, and so [n] ! = 0. q 1 Therefore, there exist integers 1 i, j ! and n < c < n such that u = qcu . Writing what ≤ ≤ − i j this means in terms of the κi, we get

!(κj κi) !cκ00 (i j) !Z. (3.1) − − − − ∈ The next step is to show that c = n 1. | | − Lemma 3.12. Suppose has Irrep(W ) 1 simple modules. Then there exist 1 i = j ! such H | | − ≤ 1 ≤ n 1 that ui = q − uj.

Proof. Redefining c if necessary, we have that there are i < j with qcu = u . Either c 0 or c 0. i j ≥ ≤ Consider the case c 0. In this case, let ρ be a row of c + 1 boxes, and take a multipartition ≥ c+1 th τ with ρc+1 as its i part and ∅ everywhere else. If c < n 1 then consider two multipartitions − th defined as follows: λ is the multipartition of n whose i part is ρn and µ is the multipartition of n whose ith part is n 1 boxes in row −

2 34 5 .

Then τ is not Kleshchev, and so λ is clearly not Kleshchev. Also, µ is not Kleshchev. (To see this, consider the (1, c+ 1)–node of µ(i). This node cannot be a good node in any multiparition obtained by deleting some nodes from µ. This is because there is an addable node below it with the same residue, namely the unique node that can be added to the empty diagram µ(j). The only removable node between them is the (2, 1)–node of µ(i). But this node cannot have residue qc because we have already shown that qc+1 = 1). Hence there are two non-Kleshchev multipartitions, which 1 contradicts our hypothesis. So c = n 1. − 40 In the c 0 case, we take γ to be a column of c + 1 boxes, and do a similar argument to ≤ c+1 − show that c = (n 1). − − The above argument shows that the multiplicative order of q must be at least 2n 1. − Lemma 3.13. Suppose has Irrep(W ) 1 simple modules. Then qk = 1, 1 k 2n 2. H | | − 1 ≤ ≤ −

n+a n 1 Proof. Indeed, suppose q = 1 where a is a nonnegative integer. Then if q − ui = uj for some i, j, we get q a 1u = u . But the above argument in the c 0 case shows that a 1 n or − − i j ≤ − − ≤ − else we would have more than one non-Kleshchev multipartition.

3.3.1 Proof of parts (1) and (2)

Now we may rewrite our condition (3.1) on the parameters as

!(κ κ ) + ( 1)a!(n 1)κ = (i j) + !t j − i − − 00 − for some a 0, 1 and some t Z. Note that (i j) + !t cannot be zero because 1 i, j ! and ∈ { } ∈ − ≤ ≤ i = j. If it is positive, multiply through by 1 (possibly interchanging the roles of i and j, and 1 − changing a), so assume that (i j) + !t < 0. Now we apply one of the twists from Section 1.3.7. − (t) a (t) Consider the linear character of W which sends σrs to ( 1) σrs for all r, s, t, and which sends s − m i to ε− sm. Explicitly checking with the set of generators and relations of W given in Equation 1.4 of Section 1.4.3 shows that this is a well-defined character of W . Now by Section 1.3.7, we have an isomorphism of Cherednik algebras ψ : H H where κ = ( 1)aκ and κ = κ κ . The κ → κ! 0$ 0 − 00 u$ u+i − i twist ψ induces an auotequivalence of category which preserves the dimension of the objects. O Our new parameters satisfy

!κj$ i + !(n 1)κ0$ 0 = (i j) + !t < 0. (3.2) − − −

Now we are in a position where we can use Theorem 3.3. There is a finite-dimensional quotient Y˜c of Mκ! (triv). Therefore, Lκ! (triv) is finite-dimensional. By Section 1.3.7, twisting by ψ sends Lκ(χ) n to Lκ! (triv) for some linear character χ of W . Furthermore, dim Lκ(χ) = dim Lκ! (triv) = r by

Theorem 3.3. Since Lκ(χ) is finite-dimensional, KZκ(Lκ(χ)) = 0 by Proposition 3.2, and therefore KZ (L (τ)) = 0 for τ = χ, by our assumption that KZ separates simples. Therefore L (τ) is κ κ 1 1 κ infinite-dimensional if τ = χ. We have proved parts (1) and (2) of Theorem 3.7. 1

41 3.3.2 Blocks

To proceed further, it is necessary to calculate the blocks of the Hecke algebra.

Standing assumption 3.14. We have parameters q and u1, . . . , u( for the Hecke algebra. We are assuming that there is exactly 1 non-Kleshchev multipartition, and we have already shown that qn 1u = u for some i = j. − i j 1 First, we prove the following lemma.

Lemma 3.15. Under assumption 3.14, if k = i, j then for each t = k, we have u /u = qc for any 1 1 k t 1 n < c < n. − Proof. Suppose u = qcu for some n < c < n. If t = i, j then it would follow from the earlier k t − 1 calculations that there is another non-Kleshchev multipartiton, so we need only consider the case where t = i or t = j. Suppose i < j. If t = i then suppose there is n < c < n with u = qcu , and − k i n 1 uj = q − ui. Suppose k > j. If c < 0 then there are at least two non-Kleshchev multipartitions: th one is the multipartition whose only nontrivial part is a row ρn of n boxes in the i position, th and the other is the multipartition whose only nontrivial part is a column γn of n boxes in the i position. Similarly, if k < j then there are at least two non-Kleshchev multipartitions. On the other hand, if c 0 then u = qcu = qc (n 1)u and hence there exists a non-Kleshchev multipartition ≥ k i − − j which is ∅ except in the jth position, and one which is ∅ except in the ith position. Similarly, if t = j, we reach the same conclusion, and so such a c cannot exist. Similar arguments deal with the i > j case.

Recall from Theorem 1.39 that if α and β are multipartitions then the Specht modules Sα and Sβ belong to the same block if and only if cont(α) = cont(β). The next lemma is needed to study the content of a multipartition.

Lemma 3.16. Under assumption 3.14, let α = (α(1), α(2), . . . , α(()) be a multipartition of n. Then cont(α(r)) cont(α(s)) = ∅ for all r = s. ∩ 1 Proof. By Lemma 3.15 and our assumption that qn 1u = u , we get that for all r, s, u /u = qc − i j r s 1 for any (n 1) < c < n 1. Now, if the residue of some node x in α(r) is equal to the residue of − − − some other node y in α(s), then

col(x) row(x) col(y) row(y) urq − = usq − .

42 But if t := col(x) + row(y) row(x) col(y) then u /u = qt. However, t n 2 and t (n 2), − − s r ! − ≥ − − a contradiction.

The next lemma is useful in determining a multipartition from its content.

Lemma 3.17. Under assumption 3.14, if α and β are multipartitions of n and 1 k !, then ≤ ≤ cont(α(k)) = cont(β(k)) implies α(k) = β(k).

Proof. We show that if two nodes of α(k) have the same residue, then they lie on the same diagonal. It will follow that the multiplicity of a residue in cont(α) is equal to the length of the corresponding diagonal of α. The same is true of β. Thus under the hypothesis, the Young diagrams α and β have diagonals of the same lengths, so they are equal. (k) j i j i Suppose then that nodes (i, j) and (i$, j$) in α have the same residue. Then ukq − = ukq !− ! . Thus qj i j!+i! = 1 and therefore if j i = j i then either z := j i j + i n or z n. But − − − 1 $ − $ − − $ $ ≥ ≤ − 2 j + i , j + i n + 1 and so z cannot be either n or n. Therefore, z = 0 and j i = j i . ≤ $ $ ≤ ≥ ≤ − − $ − $ In other words, (i, j) and (i$, j$) lie on the same diagonal.

We are finally in a position to calculate the blocks of the Hecke algebra. In order to determine the blocks of , we first note that if ρ denotes a row of length a and γ a column of length b, H a b th th then we may define a multipartition λa to have ρa in the i place and γn a in the j place. For − example, if ! = 3, n = 3, i = 3, j = 2 then

λ0 = ∅ ∅ , λ1 = ∅ , λ2 = ( ∅ ) , λ3 = ( ∅ ∅ ). $ % 6 7 Then if qn 1u = u , then cont(λ ) = u qx 0 x n 1 and hence all the λ belong to the − i j a { i | ≤ ≤ − } a same block. It remains to show that if α, β are multipartitions and one of them is not of the form

λa, then they belong to distinct blocks.

Claim 1. Under assumption 3.14, if α and β are multipartitions of n and cont(α) = cont(β) and

α, β are not both of the form λa, then α = β.

Our aim is now to prove Claim 1, so we suppose that we have two multipartitions α = (α(1), . . . , α(()) and β = (β(1), . . . , β(()) with cont(α) = cont(β). We will show that if k = i, j 1 then α(k) = β(k).

(k) (t) Lemma 3.18. Let k = i, j. If x cont(α ) then x / t=kcont(β ). 1 ∈ ∈ ∪ + 43 Proof. There is a unique integer b with n + 1 b n 1 such that x = u qb. We consider the − ≤ ≤ − k cases b 0 and b 0 separately. In the case b 0, we now prove by induction that x / cont(β(t)) ≥ ≤ ≥ ∈ for any t = k. The proof for b 0 is very similar, so we omit it. 1 ≤ For the base step, suppose b = 0. Then u cont(α(k)). Hence u is a residue of β. If k ∈ k u cont(β(t)) where t = k then u = u qc r for some column c and row r of β(t). But clearly k ∈ 1 k t − (t) n < c r < n which contradicts Lemma 3.15. Therefore x = uk / t=kcont(β ) and so − − ∈ ∪ + u cont(β(k)). k ∈ Now we do the inductive step. Suppose b > 0. Suppose u qb is a residue of β(t) with t = k. Then k 1 u qb = u qc r for some c, r. So u /u = qc r b. Since c r < n and b > 0, we have c r b < n. k t − k t − − − − − So by Lemma 3.15, c r b n. Therefore, r n + c b n + 1 b. But β(t) contains at least − − ≤ − ≥ − ≥ − r boxes, by definition of r. So β(t) n + 1 b. | | ≥ − b (k) Next, we note that since ukq is the residue of a node in α , this node must lie on the diagonal containing (1, b + 1). So there are at least b + 1 boxes in the first row of α(k) and hence there is a (k) b 1 (k) node in the first row of α with residue ukq − . By induction on b, this is also a residue of β . So there is a box in column b and row 1 of β(k). Therefore, β(k) b. So β β(k) + β(t) n + 1, | | ≥ | | ≥ | | | | ≥ a contradiction.

It follows from Lemma 3.18 that if cont(α) = cont(β) then cont(α(k)) = cont(β(k)) for all k = i, j. Then applying Lemma 3.17, we get α(k) = β(k). It remains to deal with α(i) and α(j). The 1 proof of this case will be very similar to Lemma 3.18, but slightly more complicated. Given multipartitions α = (α(1), . . . , α(()) and β = (β(1), . . . , β(()), with cont(α) = cont(β), let (i) (j) a1 be the length of the first row of α and a2 be the length of the first column of α and define b1, b2 similarly for β. First we prove a technical lemma.

Lemma 3.19. Under assumption 3.14, suppose a + a < n. Then u qa1 / cont(α). 1 2 i ∈

Proof. First, we show that u qa1 / cont(α(k)) when k = i, j. So let k = i, j and suppose there i ∈ 1 1 (k) a1 (k) is a node of α with residue uiq . Say this node lies in column c and row r of α . Then u qa1 = u qc r. So u /u = qa1 (c r). We show that a (c r) lies between n and n. If i k − i k − − 1 − − − a (c r) n then c + n r + a n, a contradiction. While if a (c r) n then 1 − − ≥ ≤ 1 ≤ 1 − − ≤ − c n + a + r n + 1, a contradiction. So n < a (c r) < n, which violates Lemma 3.15. ≥ 1 ≥ − 1 − − a1 (k) Hence, uiq is not a residue of α .

44 a1 (i) Next, we show that uiq is not a residue of α . If it is, then there is a node in column c and row r of α(i) whose residue is u qa1 = u qc r. So qa1 (c r) = 1. So by Lemma 3.13, if a (c r) = 0 i i − − − 1 − − 1 then either a (c r) 2n 1 or a (c r) (2n 1). If a (c r) (2n 1) then 1 − − ≥ − 1 − − ≤ − − 1 − − ≤ − − 2n a +r 1+2n c, which is impossible. If a (c r) 2n 1 then c+2n a +r+1 n+2, ≤ 1 − ≤ 1 − − ≥ − ≤ 1 ≤ which is impossible if n > 1. Therefore, a = c r. But c a and r 1, so this is also impossible. 1 − ≤ 1 ≥ a1 (i) Therefore, uiq cannot be a residue of α .

a1 (j) The argument that uiq is not a residue of α is very similar. We use the fact that a1 < n a . − 2 The Claim 1 follows from the next lemma. We use the same notation as Section 3.19.

Lemma 3.20. Under assumption 3.14, if a + a < n then if x cont(α(i)) then x / cont(β(j)). 1 2 ∈ ∈ Proof. By Lemma 3.18, cont(α(k)) = cont(β(k)) for k = i, j. Therefore, by Lemma 3.16, we get 1 cont(α(i)) cont(α(j)) = cont(β(i)) cont(β(j)). This is a disjoint union. ∪ ∪ If x cont(α(i)) then x = u qb for some b with n + 1 b n 1. As in the proof of Lemma ∈ i − ≤ ≤ − 3.18, we consider the cases b 0 and b 0 separately. We give the proof only for the b 0 case. ≥ ≤ ≥ The proof is by induction on b. (i) (j) For the base step, if b = 0 then ui is a residue of α . If this is a residue of β , then it has the form u = u qn 1qc r for some c, r. So qn 1+c r = 1. Now, n 1+c r 0. If n 1+c r 2n 1 i i − − − − − − ≥ − − ≥ − then c r n which is impossible. So n 1 + c r = 0. Hence, c = 1, r = n, and β(j) must be a − ≥ − − column of n boxes. But then cont(β(j)) = u qn 1, u qn 2, . . . , u q, u . Since 0 a < n, we have { i − i − i i} ≤ 1 u qa1 cont(β(j)) = cont(β) = cont(α), which contradicts Lemma 3.19. Therefore u must be a i ∈ i residue of β(i), which proves the base step. b (i) b For the inductive step, suppose b > 0 and uiq is a residue of α . If uiq is a residue of a node in column c and row r of β(j), then u qb = u qn 1qc r. So qc r+n 1 b = 1. Since c r < n and i i − − − − − − b > 0, we have c r b < n. So c r b + n 1 < 2n 1. Therefore, either c r b + n 1 = 0 − − − − − − − − − or c r b + n 1 (2n 1). If the latter holds then c + 3n r + b + 2 2n + 1 since we may − − − ≤ − − ≤ ≤ take b n 1. Hence 1 + n c + n 1, a contradiction. We therefore get c r b + n 1 = 0. So ≤ − ≤ ≤ − − − r n b. But β(j) has at least r nodes. Therefore, β(j) n b and has at least n b rows. But ≥ − | | ≥ − − since u qb cont(α(i)), we get u qb 1 cont(α(i)), as in the proof of Lemma 3.18. By induction on i ∈ i − ∈ b, u qb 1 cont(β(i)). So, as in the proof of Lemma 3.18, there is a box in row 1 and column b of i − ∈ β(i). Therefore, β(i) b and β(i) has at least b columns. So β = λ in the notation of Section 1. | | ≥ b 45 Therefore cont(β) = u , qu , . . . , qn 1u . So u qa1 cont(β) = cont(α). This contradicts Lemma { i i − i} i ∈ b (i) 3.19. Therefore, uiq must be a residue of β and this proves the inductive step.

Now we prove Claim 1. Suppose we have a multipartition α not of the form λ . Suppose β = α. a 1 We show that cont(α) = cont(β). Indeed, if β = λ for any b, then by Lemmas 3.18 and 3.20, 1 1 b cont(α(k)) = cont(β(k)) for all k. Therefore, by Lemma 3.17, α(k) = β(k) for all k, so α = β, a contradiction. On the other hand, if β = λ for some b, then u qa1 cont(β) cont(α) by Lemma b i ∈ \ 3.19. So cont(α) = cont(β). 1 Therefore, Sα is the unique Specht module in its block. Furthermore, Sλa 0 a n form a { | ≤ ≤ } block, by the same reasoning.

3.3.3 Proof of parts (3) and (4)

By Theorem 1.39, we get that there is one block of the Hecke algebra containing n + 1 of the Specht modules, and all the other blocks are singletons. Hence, there are Irrep(W ) n blocks. | | − By [33, Corollary 5.18], the blocks of are in bijection with blocks of and hence also has O H O Irrep(W ) n blocks. We work in the category (H ). Now by [16, Theorem 2.3], there is a | | − O κ! BGG-resolution of Y˜c, ie. an exact sequence

0 Y˜ M(triv) M(h ) M( nh ) 0 (3.3) ← c ← ← s ← · · · ← ∧ s ← where hs is a certain n–dimensional irreducible representation of W . By Proposition 1.24 parts (5) and (6), there is an ordering on Irrep(W ) such that the matrix whose entries are the composition multiplicities [M(τ) : L(σ)] is upper triangular with ones on the diagonal. Hence, it has an inverse with integer entries, and it follows from the fact that the classes [L(µ)] form a basis of the Grothendieck group K ( ), that the classes [M(τ)] form a basis of the Grothendieck group 0 O K ( ) as well. Therefore, none of the maps in this sequence (3.3) can be zero, or else there would 0 O be a nontrivial linear relation amongst the [M( ih )], contradicting the fact that [M(µ)] : µ ∧ s { ∈ Irrep(W ) is a basis of K ( ). By Proposition 1.7, for any τ, all the composition factors of M(τ) } 0 O belong to the same block, and hence all the L( ih ) belong to the same block. There are n + 1 ∧ s simples in this block and hence by counting we see that all the other blocks must be singletons. Using the fact that simple objects in have no self-extensions (Theorem 1.33), we get that these O blocks are semisimple. Translating back to category (H ), we get parts (3) and (4) of Theorem O κ 3.7.

46 In order to prove part (5) of Theorem 3.7, we require the following easy lemma.

Lemma 3.21. The module Y˜c is isomorphic to L(triv).

Proof. Since Y˜c is finite-dimensional, its only composition factor can be L(triv), by part (1) of ˜ Theorem 3.7. But M(triv) # Yc and [M(triv) : L(triv)] = 1.

3.3.4 Proof of part (5)

It remains to compute the composition multiplicities in the one nontrivial block . Again we work O∧ in the category (H ). [33, Proposition 5.21(ii)] tells us that each L( ih ), i > 0 is a submodule O κ! ∧ s of a standard module. Write L = L( ih ) and M = M( ih ). Let R be the radical of M . We i ∧ s i ∧ s i i cannot have a nonzero map L M if j > i for the following reason. i → j

Lemma 3.22. If j > i then [Mj : Li] = 0.

Proof. The argument is based on [34, Lemma 4.2].

j i Recall Proposition 1.24, part (5), which states that [Mj : Li] = 0 only if c hs c hs N, 1 ∧ − ∧ ∈ nH 1 where cτ denotes the scalar by which the element z := H i=1− nH kH,ieH,i CW acts on ∈A ∈ i the irreducible representation τ of W . We calculate c hs!, 0 i !n. In our situation, ∧ ≤ ≤ ( 1 n ( 1 ( 1 − (k) − − rt t z = κ$ (1 σ ) + κ$ ε s . 00 − ab r u &a i then c hs c hs / N. − ∧ − ∧ ∈

So L1 is a submodule either of M0 or M1. It can’t be a submodule of M1 because [M1 : L1] = 1, so L 3 M . So L 3 R . But by Lemma 3.21, we have Y˜ = L(triv). Hence Y˜ is simple and 1 → 0 1 → 0 c ∼ c so R = ker(M Y˜ ) = im(M M ) is a quotient of M . Hence [R : L ] = 1. If we had 0 0 → c 1 → 0 1 0 1 [R : L ] = 0 for some i > 1 then R would have L as a quotient for some i > 1. Therefore, so would 0 i 1 0 i M . But M has a unique simple quotient L . Therefore, it is impossible to have [R : L ] = 0 for 1 1 1 0 i 1 i > 1 and we conclude that R0 = L1.

47 We have shown that the composition factors of M0 are L0 and L1. To conclude the argument, we show by induction that the composition factors of Mi are Li and Li+1. Consider first Li+1. Then, by [33, Proposition 5.21(ii)], L is a submodule of some M . We cannot have j i+1, and i+1 j ≥ by induction, we cannot have j < i. Hence, L is a submodule of M and so L 3 R . Now i+1 i i+1 → i Ri = ker(Mi Mi 1) by induction and so Ri is a quotient of Mi+1. Therefore, [Ri : Li+1] = 1. If → − there was a j > i + 1 with [R : L ] = 0 then we would have that for some j > i + 1, L would be a i j 1 j quotient of Ri and hence a quotient of Mi+1, contradicting the fact that Mi+1 has a unique simple quotient. Therefore, Ri = Li+1 and we are done. This proves part (5) of Theorem 3.7.

3.3.5 Characterisations of separating simples

Now that we have completed the proof of Theorem 3.7, let us turn our attention to the question of when KZ separates simples, and whether it is possible to choose κ such that KZ separates simples.

Theorem 3.23. The following are equivalent

1. KZ separates simples.

2. If q, u , . . . , u are the parameters of the Ariki-Koike algebra , then (q+1) (u u ) = 0, 1 ( H i

# τ Irrep(W ) : L(τ) reg = 0 Irrep(W ) 1. { ∈ |h 1 } ≥ | | −

3. The algebra has at least Irrep(W ) 1 nonisomorphic simple modules. H | | −

Proof. First, we show that (2) implies (1). We must show that if L(σ) reg = L(τ) reg = 0 then |h ∼ |h 1 σ = τ. Suppose then that L(σ) reg = L(τ) reg = 0. By [33, Proposition 5.21(ii)], there exists a |h ∼ |h 1 standard module M(λ) such that L(σ) 3 M(λ). Let t = dimHom(L(σ), M(λ)). Then M(λ) must → have t submodules isomorphic to L(σ), because the only automorphisms of L(σ) are the scalars. Therefore, L(σ) t M(λ) and M(λ) has no submodule isomorphic to L(σ) (t+1). Now since ⊕ ⊂ ⊕ L(σ) reg = L(τ) reg , we have Hom(L(τ) reg , M(λ) reg ) = Hom(L(σ) reg , M(λ) reg ) = 0 and hence |h ∼ |h |h |h |h |h 1 by Corollary 1.30, Hom(L(τ), M(λ)) = 0 (using the condition on the parameters). Therefore, M(λ) 1 t has a submodule isomorphic to L(τ) and hence a submodule isomorphic to L(τ) + L(σ)⊕ . This t sum must be direct if L(σ) # L(τ), hence M(λ) has a submodule L(τ) L(σ)⊕ and M(λ) hreg has ⊕ |

48 t (t+1) a submodule L(τ) reg L(σ) ⊕reg = L(σ) ⊕reg . Therefore, dim(Hom(L(σ) reg , M(λ) reg )) t + 1 |h ⊕ |h |h |h |h ≥ and so dim(Hom(L(σ), M(λ))) t + 1, a contradiction. So L(σ) = L(τ) and hence σ = τ. ≥ ∼ Next, (1) implies (3) by Lemma 3.9. Finally, to show (3) implies (2), note that under the hypothesis that has Irrep(W ) 1 H | | − simple modules, it has already been shown in Lemma 3.11 that [n] ! = 0, hence q = 1 since we q 1 1 − assume n 2, and in Lemma 3.15 that u = u for all i = j, so the condition on the parameters ≥ i 1 j 1 holds. Furthermore, by the essential surjectivity of KZ, if has Irrep(W ) 1 simple modules then, H | |− because KZ is essentially surjective on objects and exact, there are at least Irrep(W ) 1 of the | | − L(τ) with KZ(L(τ)) = 0 and hence with L(τ) reg = 0. 1 |h 1 Having characterised when it is possible for KZ to separate simples, let us turn our attention to constructing examples of choices of the parameters κ so that KZ separates simples.

Theorem 3.24. Let κ00 / Q, and suppose there is some 1 b ! 1 with ∈ ≤ ≤ −

!κ + !(n 1)κ = b + !t < 0 b − 00 − for some t Z. Suppose further that if i = j and i, j = 1, b + 1 then ui / qZuj. Then KZ ∈ 1 { } 1 { } ∈ separates simples.

Proof. Recall that u = 1 and u = ε (j 1)e 2πiκj 1 for j 2. Then the condition on the 1 j − − − − ≥ n 1 parameters says that ub+1 = q − . It follows that the multipartition

λ = ( , ∅, ∅, . . . , ∅ ) · · · n boxes

4 52 3 (1) n 1 is not Kleshchev, because the box at the right hand end of the row λ has residue q − and is not a normal node. Next, we show that all the other Dλ are nonzero. By [3, 1.2], Dλ will be (λ(1),λ(b+1)) nonzero if and only if D is a nonzero module for the Ariki-Koike algebra of Z2 Sn, with ! parameters q and u = 1, u = qn 1. Since q = 1, this is the case if and only if the multipartition 1 2 − 1 (λ(1), λ(b+1)) is a Kleshchev multipartition of the integer λ(1) + λ(b+1) n. So it suffices to show | | | | ≤ , ∅ that every bipartition (λ, µ) = · · · with λ + µ = t n is Kleshchev. This is a 1 8 n boxes 9 | | | | ≤ straightforward induction argument: if µ = ∅ then the rightmost node of the bottom row of µ is a 4 52 1 3 good node and so may be removed. This reduces us to the case where µ = ∅. But then the same procedure may be applied to λ, proving that (λ, µ) is Kleshchev.

49 We have shown that has Irrep(W ) 1 nonisomorphic irreducible modules, and so KZ separates H | |− simples.

Note that in the situation of Theorem 3.24, it follows from Theorem 3.3 that the module L(triv) is finite-dimensional, so L(triv) must be the unique finite-dimensional simple module in . If χ is an O arbitrary linear character of W , then we may twist the parameters by χ to obtain new parameters

κ$. By Section 1.3.7, twisting by χ induces an isomorphism of Ariki-Koike algebras and so by

Theorem 3.23, KZκ! separates simples. This shows the following.

Corollary 3.25. For every linear character χ of W and every r > 0 with ! $ r, there is a choice of parameters κ such that KZκ separates simples and the unique finite-dimensional simple module in category is L(χ) of dimension rn. O

3.4 The Ariki-Koike algebra in the almost-semisimple case

We close this chapter by using the facts proved about category in Theorem 3.7 to prove a O theorem about the Hecke algebra which does not mention the Cherednik algebra in its hypothesis or conclusion. This theorem is an example of a general philosophy suggested by Rouquier in [64] of using the Cherednik algebra and the KZ functor as a tool to prove theorems about Hecke algebras. We know that is semisimple if and only if the number of irreducible modules Irrep( ) Hκ | Hκ | of κ equals the number of irreducible modules of CW , and that in this case κ = CW . So H H ∼ the property of having Irrep(W ) simple modules determines the algebra up to isomorphism. | | Hκ We show that the property of having Irrep(W ) 1 simple modules also determines up to | | − Hκ isomorphism.

Theorem 3.26. Suppose and are Ariki-Koike algebras corresponding to some parameters Hκ Hµ ( κ, µ C and that Irrep( κ) = Irrep( µ) = Irrep(W ) 1. Then there is an isomorphism of ∈ | H | | H | | | − algebras = . Hκ ∼ Hµ opp Proof. By Theorem 1.31, there is an algebra isomorphism κ = End (PKZ) where H ∼ O

PKZ = dim KZ(L(τ))P (τ). τ Irrep(W ) ∈ " Here, P (τ) is the projective cover of L(τ). The strategy of the proof is to calculate PKZ in the case where KZκ separates simples, and show that its endomorphism ring can be written in a way that

50 does not depend on κ. We work in the category = and write KZ = KZ , M(τ) = M (τ), O Oκ κ κ and so forth. By Theorem 3.7, there is a linear representation χ of W with = ss, where O O∧ ⊕ O is the subcategory of generated by L( ih χ) : 0 i n . Let λi = ih χ and let O∧ O { ∧ s ⊗ ≤ ≤ } ∧ s ⊗ S = λi : 0 i n . Write M = M(λi), L = L(λi) and P = P (λi) (the projective cover of L ). { ≤ ≤ } i i i i For σ, τ Irrep(W ), we have in general ∈

dim Hom(P (σ), P (τ)) = [P (τ) : L(σ)]

= [P (τ) : M(γ)][M(γ) : L(σ)] γ & = [M(γ) : L(τ)][M(γ) : L(σ)] γ & = [M(γ) : L(τ)][M(γ) : L(σ)] + [M(γ) : L(τ)][M(γ) : L(σ)]. γ S γ /S &∈ &∈ If γ / S then M(γ) = L(γ), so we get ∈ n dim Hom(P (σ), P (τ)) = [Mi : L(τ)][Mi : L(σ)] + δγτ δγσ. i=0 γ /S & &∈ Now, if σ / S or τ / S, this sum must be δ . Otherwise, σ, τ S and so σ = λa, τ = λb for some ∈ ∈ στ ∈ a, b. We get n a b dim Hom(P (λ ), P (λ )) = [Mi : La][Mi : Lb] &i=0 which equals 2 if a = b and 1 if a b = 1 and 0 otherwise. So we get | − |

2 if σ = τ S  ∈  1 if σ = τ / S dim Hom(P (σ), P (τ)) =  ∈  1 if σ, τ = λa, λa+1 , 0 a n 1  { } { } ≤ ≤ −  0 otherwise    The ring End (PKZ) is a matrix algebra with entries in the various Hom-spaces Hom(P (σ), P (τ)). O We calculate the multiplication relations between basis elements of the Hom(P (σ), P (τ)) and show that these relations do not depend on κ. It will follow that the structure constants of End (PKZ) do O not depend on κ, which will prove the theorem provided that the multiplicity of each P (τ) in PKZ n 1 is also independent of κ. But in our situation PKZ = τ /S(dim τ) P (τ) 1 i n i −1 Pi since ⊕ ∈ · ⊕ ⊕ ≤ ≤ − $ 6 7 % 51 n 1 1 dimC KZ(Li) = i −1 as a vector space . By BGG reciprocity, we have [Pi : Mi] = [Mi : Li] = 1 = − i i 1 [Mi 1 : Li] = [P6i : M7 i 1], and [Pi : M(σ)] = [M(σ) : Li] = 0 if σ = λ , λ − . Therefore, the factors − − 1 in any filtration of Pi by standard modules are Mi and Mi 1. But by [33, Corollary 2.10], Pi has − a filtration by standard modules with M as the top factor, so P may be described as P = Mi , i i i Mi 1 − 0 1 2 1 2 1 meaning that there is a series 0 = Pi Pi Pi = Pi with Pi = Mi 1 and Pi /Pi = Mi. We may ⊂ ⊂ ∼ − ∼ write the resulting composition series of Pi as

Li

Li+1 Pi = Li 1 − Li

This description of P makes it easy to write down the nontrivial maps P P . i i → i First, there are two obvious maps P P , namely the identity map id and the map ξ which i → i i i 2 is projection onto the bottom composition factor Li followed by inclusion. Note that ξi = 0 and 2 therefore End (Pi) = C[ξi]/(ξi ), since we have already shown that dim Hom(Pi, Pi) = 2. O Next, we describe the map P P . This is a map Mi Mi+1 . So we may construct a i i+1 Mi 1 Mi → − → map fi,i+1 : Pi Pi+1 by factoring out the copy of Mi 1 and then embedding Mi in Pi+1. This → − map is nonzero, so Hom(Pi, Pi+1) = Cfi,i+1, 1 i n 1. ≤ ≤ − Now we describe the map Pi Pi 1, n i 2. By [33, Proposition 5.2.1 (ii)], Pi Li is → − ≥ ≥ ⊃ i injective and therefore Pi contains the injective envelope Ii = I(λ ) of Li. Therefore, since Pi is indecomposable, P = I . Now, recall that category contains a costandard module (τ) L(τ) i i O ∇ ⊃ for every τ Irrep(W ), with [ (τ)] = [M(τ)] in K ( ). Write = (λi). Then L ∈ ∇ 0 O ∇i ∇ i ⊂ ∇i Li+1 by Proposition 1.24, so i has a composition series of the form i = . Furthermore, by ∇ ∇ Li i 1 Proposition 1.24, i Ii and Ii has a filtration by costandard modules of the form Ii = ∇ − ∇ ⊂ ∇i i 1 i 2 Since I = P , to get a map ∇ − = P P = ∇ − , we may factor out the copy of and then i i i i i 1 i 1 i ∇ → − ∇ − ∇ embed i 1 in Pi 1. This gives a nonzero map fi,i 1, and therefore Hom(Pi, Pi 1) = Cfi,i 1. In ∇ − − − − − particular, this shows that the image of fi,i 1 has length 2. − 1 this can be readily shown using the following argument: since for any τ Irrep(W ), M(τ) = C[h] τ as C[h]– ∈ ∼ ⊗ modules, we get that the vector bundle associated to M(τ) on hreg/W has rank dim(τ). Therefore, dim KZ(M(τ)) = dim(τ) for all τ. Now recall from Theorem 3.7 that the composition factors of Mi are Li and Li+1. Write di = n j i n j i n j i n dim KZ(Li). Then di = ( 1) − (di + di+1) where dn+1 := 0. So di = ( 1) − dim Mi = ( 1) − = j=i − j=i − j=i − i n 1 i −1 . ! ! ! 6 7 − 6 7 52 Now we calculate multiplication relations between the various fi,i+1, fi,i 1 and ξi. First, it is − immediate from the definitions that ξi+1fi,i+1 = fi,i+1ξi = 0. We need to do a little more work

i 1 to show that the same holds for fi,i 1. Take the description of Ii as Ii = ∇ − . Then Ii has a − ∇i composition series

Li

Li 1 Ii = − Li+1

Li So there is a map ζ : I I defined by projection onto the bottom composition factor L followed i i → i i by the embedding Li 3 Ii. Clearly, ζifi 1,i = fi 1,iζi 1 = 0. But since Pi = Ii, we may regard ζi → − − − 2 2 as a map Pi Pi. Therefore, there are a, b C with ζi = aidi + bξi. Since ζ = 0, we get a = 0 → ∈ i and hence ζi is a nonzero multiple of ξi. This shows that ξifi 1,i = fi 1,iξi 1 = 0. − − − Finally, we need to calculate fi+1,ifi,i+1 and fi 1,ifi,i 1. Consider first fi 1,ifi,i 1. By the − − − − definition of fi,i 1 above, we have [im(fi,i 1) : Li] = 0. Hence, im(fi,i 1) cannot be contained − − 1 − in the submodule of Pi 1 isomorphic to Mi 2, and therefore fi 1,ifi,i 1 must be nonzero. Since − − − − fi 1,ifi,i 1ξi = 0, fi 1,ifi,i 1 must be a nonzero multiple of ξi. Let us replace ξi by fi 1,ifi,i 1. So − − − − − − we may assume that fi 1,ifi,i 1 = ξi, and this does not change any of the relations which have − − already been calculated. Now consider fi+1,ifi,i+1. We show that this composition is nonzero.

Indeed, the image im(fi,i+1) has composition factors Li and Li+1. If fi+1,ifi,i+1 were zero, then we would get that im(fi+1,i) could only have composition factors Li+1 and Li+2. But we have shown that im(f ) has length 2, and [P : L ] = 0, a contradiction. Therefore, f f = 0 and so i+1,i i i+2 i+1,i i,i+1 1 there is a nonzero bi,i+1 C, n 1 i 1, such that ∈ − ≥ ≥

fi+1,ifi,i+1 = bi,i+1ξi = bi,i+1fi 1,ifi,i 1. − −

It remains to do some rescaling. Let

1 ξi$ = ξi, 1 i n b12b23 bi 1,i ≤ ≤ · · · − fi,$ i 1 = fi,i 1 2 i n − − ≤ ≤ 1 f $ = f 1 i n 1. i,i+1 b b b i,i+1 ≤ ≤ − 12 23 · · · i,i+1

53 Then we have the following relations:

ξi$fi$ 1,i = fi$ 1,iξi$ 1 = 0 − − −

ξi$+1fi,$ i+1 = fi,$ i+1ξi$ = 0

fi$ 1,ifi,$ i 1 = fi$+1,ifi,$ i+1 = ξi$. (3.4) − −

These are the only nontrivial relations between the various Hom(P (σ), P (τ)). This shows that we may choose a basis of Hom(P (σ), P (τ)) for each σ, τ such that the composition relations between the basis elements are independent of κ. Hence, we may choose a basis of the algebra End (PKZ) O such that the structure constants are independent of κ. This proves the theorem.

Remark 3.27. By variations on the arguments given in the above proof, it is possible to show that

1 j = i + 1, i 1 1 − dimC Ext (Li, Lj) = O  0 otherwise

 Li and so the composition series of P may be written more symmetrically as P = I = Li 1 Li+1 . i i i − ⊕ Li Note that since Theorem 3.7 implies that the Ariki-Koike algebra has Irrep(W ) n blocks, by | | − n n 1 counting we get that the algebra Bn := End ( i=1 i −1 Pi) is a block of the Ariki-Koike algebra. O ⊕ − From the relations (3.4), it is clear that Bn is indepen6den7t both of κ and !. So we have the following corollary.

(i Corollary 3.28. Let !1, !2 > 1 and for i = 1, 2 let κi C and suppose κi (G(!i, 1, n)) has ∈ H Irrep(G(! , 1, n)) 1 simple modules. Then the unique nonsemisimple blocks of (G(! , 1, n)) | i | − Hκ1 1 and (G(! , 1, n)) are isomorphic algebras. Hκ2 2

Remark 3.29. The representation theory of the algebra Bn is described in [7, 5.3] and [25, 3.2].

54 Chapter 4

Shift functors

( In this chapter, we take W = G(!, 1, n). For κ C , we write as usual Hκ for the Cherednik ∈ algebra, for the Ariki-Koike algebra, for the category (H ), and KZ for the KZ functor. Hκ Oκ O κ κ The aim of this chapter is to study certain relationships between category and for Oκ Oκ! κ = κ . Recall that the parameters for the associated Ariki-Koike algebra are the exponentials 1 $ of the κi. Therefore, if κ$ = κi + ai for all i = 00, 1, 2, . . . , ! 1 for some ai Z, then the i − ∈ Ariki-Koike algebras and are equal. It turns out that in some cases this isomorphism of Hκ Hκ! Ariki-Koike algebras extends to an equivalence = . In this chapter, we consider two different Oκ ∼ Oκ! notions of shift functor. Both of these are really defined as functors eH e Mod eH e Mod, κ − → κ! − but we will show in Section 4.1.5 that in some circumstances these can be extended to functors H Mod H Mod and then give functors . First, in Section 4.1, we consider the κ − → κ! − Oκ → Oκ! Heckman-Opdam shift functors. These functors appear to have been first defined in [7, Lemma 4.7]. Our aim is to give some conditions on the parameters which guarantee that these functors are equivalences. Second, in Section 4.2, we consider a different notion of shift functor, which we call the Boyarchenko-Gordon shift functor. We are not able to construct these functors in all cases, but we will show that they exist provided a hypothesis (Hypothesis 4.48) holds. Having defined the Boyarchenko-Gordon shift functors, we address in Section 4.3 the question of whether the two notions of shift functor coincide.

55 4.1 The Heckman-Opdam shift functors

In this section we will define and study the Heckman-Opdam shift functors for G(!, 1, n), largely ( following [36]. For each value of κ = (κ00, κ1, . . . , κ( 1) C , we will define functors − ∈ F a : eH e Mod eH e Mod κ κ[a] − → κ − where κ[a] is obtained from κ by incrementing the values of some of the parameters by integers. We are interested in when these functors are equivalences, since in such cases, they give a powerful tool for studying category . We will show that there are two cases of interest in which the functors O a Fκ are equivalences. The first case is when the KZ functor separates simples. The second case is when the parameters are “asymptotic”, in a sense to be defined below.

4.1.1 A shift relation

The theorem which allows the Heckman-Opdam shift functors to be constructed is the following so-called shift relation.

( ( Proposition 4.1. Let κ = (κ00, κ1, . . . , κ( 1) C and define κ[a] C by κ[a]00 = κ00 + 1, − ∈ ∈ κ[a] = κ + 1 for 1 i a, and κ[a] = κ for a + 1 i ! 1. Let θ be the Dunkl representation i i ≤ ≤ i i ≤ ≤ − κ of Hκ and θκ[a] be the Dunkl representation of Hκ[a]. Then there is an equality of subsets of D(hreg) W ∗ 1 eθκ[a](Hκ[a])e = eµa− θκ(Hκ)µae, where µ = ( n x )a (x( x(). a i=1 i i 1. We remark that a much more general shift relation, which holds for all complex reflection groups, appears in unpublished work of Berest and Chalykh [5]. Our proof of Proposition 4.1 will follow the argument of [7, Proposition

4.6] in the ! = 1 case. The strategy is to prove Proposition 4.1 first in the case where Hκ is simple, and then to extend to all values of κ using a specialisation argument.

( Notation 4.2. Say κ C is regular if Hκ is simple. ∈ 1 In the proof of Proposition 4.1, we fix a and write e− := µaeµ− (we have e− CW because a ∈ 1 µa is a W –semiinvariant). So we wish to show that θκ[a](eHκ[a]e) = µa− θκ(e−Hκe−)µa.

56 In the regular case, we will first show that eHκe is generated as an algebra by the subset W W C[h] e C[h∗] e. The proof will require the notion of a Poisson bracket, which we now explain. ∪ Definition 4.3. Let R be a commutative algebra. A Poisson bracket on R is a bilinear map , : R R R which is a Lie bracket and which satisfies the identity {− −} × → xy, z = x y, z + y x, z { } { } { } for all x, y, z R. We refer to (R, , ) as a Poisson algebra. ∈ {− −} A standard example of a Poisson algebra can be obtained as follows. Let R = iR be a ∪i∞=0F i i 1 j j 1 filtered algebra and suppose that gr R is commutative. For x R/ − R and y R/ − R, F ∈ F F ∈ F F let xˆ iR, yˆ jR be lifts of x and y respectively. Then define ∈ F ∈ F

i+j 2 x, y := [xˆ, yˆ] + − R. { } F Then it is routine to check that , defines a Poisson bracket on R. {− −} A concrete example of a Poisson algebra is the following. Let (V, ω) be a symplectic vector space and let W be a finite subgroup of Sp(V ). For x, y V , set x, y = ω(x, y). Then , ∈ { } {− −} may be extended to C[V ] in a natural way, and this gives a Poisson bracket. The only hard part to check is the Jacobi identity, but it follows from the axioms for a Poisson algebra that one only needs to check the Jacobi identity on a set of algebra generators of C[V ]. This bracket also induces W a bracket on C[V ] , which we denote by , ω. {− −} We require the notion of the degree of a Poisson bracket.

Definition 4.4. Let R = R be a graded commutative algebra with a Poisson bracket , . ⊕i∞=0 i {− −} We say that , has degree k if for all i and j and for all x R and all y R , we have {− −} ∈ i ∈ j x, y Ri+j+k, and furthermore there exist i, j with x Ri, y Rj and x, y / Ri+j+k 1. { } ∈ ∈ ∈ { } ∈ − For more information on Poisson brackets, we refer the reader to the book [71].

Notation 4.5. Let X = (a, b) : a h, b h . By convention, we will generally write X = h h { ∈ ∈ ∗} ⊕ ∗ when we consider X as a representation of W , as in Chapter 1, and X = h h when we are × ∗ thinking of X as an algebraic variety.

In the proof of Proposition 4.1, we will consider two filtrations on Hκ. One is the filtration F defined in Section 1.2. The other is a filtration with 0 = C[h] W and i = (C[h] + h)i W for F F ∗ F ∗ 57 i 1. In other words, we place the elements of h and W in filtration degree 0, and the elements ≥ ∗ of h in filtration degree 1. We require a lemma concerning the filtration . F Lemma 4.6.

gr (Hκ) = C[h h∗] W. F ∼ × ∗ Proof. If x h and y h, then [y, x] 0, so there is a natural map ∈ ∗ ∈ ∈ F

φ : C[h h∗] W # gr (Hκ) × ∗ F which is clearly surjective. We wish to show that φ is injective. Let α, β Zn be multiindices ∈ !0 α α1 α β β1 βn α β and write x for x x n and y for y yn . Suppose an element λ x y w ker(φ) 1 · · · n 1 · · · αβw ∈ where the sum runs over α, β Zn and w W .Then for a fixed nat!ural number a, writing ∈ !0 ∈ α β α β a 1 β := i βi, we have β =a α,w λαβwx y w ker(φ) and so β =a α,w λαβwx y w − . | | | | ∈ | | ∈ F But th!is contradicts th!e PBW!theorem. So ker(φ) = 0 and φ is a!n isomo!rphism.

W The filtration induces a filtration on eHκe, and gr (eHκe) = eC[h h∗] W e = C[h h∗] . F F ∼ × ∗ ∼ × W In this way, we get a Poisson bracket on C[h h∗] . Similarly, e−Hκe− also defines a Poisson × W bracket on C[h h∗] . × Also, recall that h h is a symplectic vector space with symplectic form ω((a, α), (b, β)) = × ∗ W β(a) α(b). This defines a third Poisson bracket , ω on C[h h∗] . − {− −} × W Lemma 4.7. The Poisson bracket on C[h h∗] induced from the isomorphism gr (eHκe) = × F ∼ W W C[h h∗] coincides with the bracket defined by gr (e−Hκe−) = C[h h∗] , and furthermore, both × F ∼ × of these brackets coincide with the natural bracket , . {− −}ω W Proof. The algebra C[h h∗] is graded by polynomial degree. Note that this grading is not × W the grading inherited from the isomorphism C[h h∗] = gr (eHκe). The idea of the proof is to × ∼ F calculate the degree of the three Poisson brackets with respect to the grading by polynomial degree. W Let p, q be homogeneous polynomials in C[h h∗] . Then, using the notation of the proof of × n Lemma 4.6, there are scalars λαβ, α, β Z 0 such that ∈ ≥ α β p = λαβx y . α + β =deg(p) | | | &| Let pˆ eH e be the sum of noncommutative monomials ∈ κ α β pˆ = λαβx y e. α + β =deg(p) | | | &| 58 Then p is the image of pˆ in gr (eHκe). Similarly, we may define qˆ. The defining relations of the F Cherednik algebra show that [pˆ, qˆ] F deg(p)+deg(q) 2(H ) and hence p, q has polynomial degree ∈ − κ { } deg(p) + deg(q) 2. Therefore, the degree of the bracket , induced from eH e is 2. ≤ − {− −} κ ≤ − To show that this bracket has degree exactly 2, we calculate − n n x(, x y . { i j j} &i=1 &j=1 In the Cherednik algebra, we can compute

n n n n ( ( [ xi , xjyj] = [xi , xjyj] &i=1 &j=1 &i=1 &j=1 n n ( 1 − b ( 1 b = xi xj[xi, yj]xi− − &i=1 &j=1 &b=0 A tedious calculation using the relations 1.3 then yields

n n n ( ( [ xi , xjyj]e = ! xi e. 8− 9 &i=1 &j=1 &i=1 Thus, multiplying on the left and right by e, we get that n n n x(, x y = ! x(. { i j j} − i &i=1 &j=1 &i=1 So , has degree exactly 2. Since h = Cn is an irreducible representation of G(!, 1, n), {− −} − it now follows from [26, Theorem 2.23] that , is a scalar multiple of , . But since {− −} {− −}ω n x(, n x y = ! n x(, we must have , = , as required. { i=1 i j=1 j j}ω − i=1 i {− −} {− −}ω !The pr!oof that the bracke!t induced from e−Hκe− also coincides with , ω is identical. {− −} We need one more definition before proceeding with the proof of Proposition 4.1.

Definition 4.8. If S R where R is a Poisson algebra, then we say that the Poisson subalgebra ⊂ of R Poisson-generated by S is the smallest subalgebra of R which contains S and which is closed under the Poisson bracket.

Our next lemma is based on [6, Lemma 4.7]. We follow the proof of [6] very closely, except we have to make some minor modifications since W is not a Coxeter group. Note that if R is a Poisson algebra which is a domain, then the Poisson structure on R can be extended to any localisation of R in a natural way. For s, x R, define s 1, x = s 2 s, x . In ∈ { − } − − { } 59 W this way, the Poisson structure on C[h h∗] considered in Lemma 4.7 induces a Poisson structure × reg W on C[h h∗] , which we also denote by , . × {− −}

reg W reg W W Lemma 4.9. The algebra C[h h∗] is Poisson-generated by C[h ] C[h∗] . × ∪

reg W Proof. Let R := C[h h∗] and let A be the Poisson subalgebra of R Poisson-generated by × reg W W W W C[h ] C[h∗] . Since C[h h∗] = C[h] C[h∗], we have that C[h h∗] is a finite C[h] C[h∗] – ∪ × ⊗ × ⊗ reg reg W W module, by Theorem 1.9. So C[h h∗] is a finite C[h ] C[h∗] –module and it follows that R × ⊗ reg W W is also a finite C[h ] C[h∗] –module. Therefore, R is a finite A–module and R is a Noetherian ⊗ reg W W reg W W C[h ] C[h∗] –module. Since A is a C[h ] C[h∗] –submodule, it follows that A is a ⊗ ⊗ reg W W Noetherian C[h ] C[h∗] –module and in particular, A is a Noetherian ring. Furthermore, A ⊗ is a domain since A R. ⊂ In geometric language, we have an affine variety Y = (hreg h )/W and another affine variety × ∗ Y := maxspec(A). The inclusion A 3 R induces a map f : Y Y . This is a finite map since $ → → $ R is a finite A–module, hence f is surjective by [24, Corollary 9.3]. We aim to show that f is an isomorphism. First, we show that f is injective and then that f is an isomorphism. reg Before beginning the proof, we note that the Poisson bracket on C[h h∗] satisfies f, yi = × { } ∂f for all f C[hreg]. Furthermore, let L := (! 1)(! 2) and define D : A A by D = − ∂xi ∈ − − → n x(, . Then D( 2( n y() is a nonzero scalar multiple of n xLy2. Hence, n xLy2 { i=1 i −} − j=1 j i=1 i i i=1 i i ∈ A!. ! ! ! Now we show that f is injective. Let (q , p ), i = 1, 2, be points of hreg h and denote by [q , p ] i i × ∗ i i their images in (hreg h )/W . We have f([q , p ]) = f([q , p ]) if and only if a(q , p ) = a(q , p ) × ∗ 1 1 2 2 1 1 2 2 reg W W for all a A. If this holds, then since A C[h ] C[h∗] , we have q2 = wq1, p2 = up1 for ∈ ⊃ ∪ some u, w W . So [q , p ] = [q , w 1up ]. Thus it suffices to show that if (q, p) hreg h and W ∈ 2 2 1 − 1 ∈ × ∗ p denotes the stabiliser of p in W , then there exists a A such that the values a(q, gp) are distinct ∈ as g runs over a set of coset representatives W/Wp of Wp in W . Let , be a W –invariant Hermitian inner product on h such that the basis y is or- ,− −- { i} thonormal, and use the same notation to denote a W –invariant Hermitian inner product on h∗ such that the basis x is orthonormal. We may choose z h such that the inner products { i} ∈ ∗ z, gp are distinct for distinct gp. Write z = zixi and gp = (gp)ixi with zi, (gp)i C, so that , - ∈ reg z, gp = zi(gp)i where the bar stands for co!mplex conjugatio!n. Now, since h (a1, . . . , an) , - ⊂ { ∈ n C ai = 0!for all i , we have qi = 0 for all i (where qi are the coordinates of q with respect to the | 1 } 1

60 dual basis of x ). Therefore, there is a well-defined linear functional { i}

L reg reg zˆ = (q− z )x h∗ = T ∗(h ) = T ∗(h /W ). i i i ∈ q ∼ q &i (Here, we used the fact that T (hreg) = T (hreg/W ) since W acts freely on hreg). Choose b q∗ ∼ q∗ ∈ reg W ∂b L C[h ] such that dbq = zˆ, ie. q = q− zi for 1 i n. Let ∂xi | i ≤ ≤

1 L 2 L ∂b a = b, xi yi = xi yi A. { 2 } − ∂xi ∈ &i &i L ∂b Then a(q, gp) = q (gp)i q = (gp)izi, which are distinct for distinct gp. So f is − i i ∂xi | − i injective. ! ! reg W Now we show that df is injective on tangent spaces. Let a1, . . . , an C[h ] be chosen so ∈ reg ∂ak that (dai)q 1 i n is a basis for h∗ = T ∗(h /W ). Note that (dak)q = xi( q), so { | ≤ ≤ } q i ∂xi | ∂ak 1 L 2 L ∂ai det( q) = 0. For 1 i n, define bi := ai, x y = x yj A. ! ∂xi | 1 ≤ ≤ { 2 j j j } − j j ∂xj ∈ reg Then for (q, p) h h∗, ! ! ∈ × ∂ak (dak)(q,p) = xi( q) ∂xi | &i and, for 1 k, i n, there exist scalars α such that ≤ ≤ ki

L ∂ak reg (dbk)(q,p) = αkixi + yi(qi q) T(∗q,p)(h h∗/W ) = h∗ h. ∂xi | ∈ × ⊕ &i &i Therefore, the vectors (da ) (db ) , 1 k n are linearly independent in h h, since { k (q,p)} ∪ { k (q,p)} ≤ ≤ ∗ ⊕

∂ak 0 ∂xi q L ∂ak 2 det | = ( qi )(det( q)) = 0.  L ∂ak  ∂x | 1 q q i i ∗ i ∂xi | '   Hence, if (q, p) Y then T Y is spanned by (da) a A so f induces a surjection on cotan- ∈ (∗q,p) { (q,p)| ∈ } gent spaces, hence an injection on tangent spaces. So by [41, Theorem 14.9], f is an isomorphism. So the inclusion A 3 R is an isomorphism, hence is surjective. So A = R. → Now we require the following lemma due to Levasseur-Stafford.

Lemma 4.10. [50, Lemma 9] Let R S be two Noetherian domains such that S is simple and is ⊂ a finite left and right R–module. Suppose Frac(R) = Frac(S). Then R = S.

Our next lemma is exactly [6, Theorem 4.6]. The proof is identical to the proof in [6], but for completeness we give the argument anyway.

61 W Lemma 4.11. [6, Theorem 4.6] If κ is regular then eHκe is generated as an algebra by C[h] e ∪ W W W C[h∗] e and e−Hκe− is generated as an algebra by C[h] e− C[h∗] e−. ∪

Proof. We give the proof for eHκe, the proof for e−Hκe− being the same but with e replaced by W W e−. Let S = eHκe and R the subalgebra of S generated by C[h] e C[h∗] e. Then with respect to ∪ W W the filtration F on Hκ which gives h, h∗ degree 1, we get gr(S) = C[h h∗] . Now, C[h h∗] is a × × W W finite C[h] C[h∗] –module, and so gr(S) is a finite gr(R)–module. So S is a finite left and right ⊗ R–module. Furthermore, S is a Noetherian domain and S is simple by [6, Lemma 4.1]. Clearly, W W gr(R) is a domain. Also, since gr(S) is a Noetherian C[h] C[h∗] –module, so is gr(R). Hence, ⊗ gr(R) is a Noetherian ring and therefore so is R. In order to apply Lemma 4.10, it remains to show that Frac(R) = Frac(S).

Now we use the second filtration on Hκ. Recall the element δ = H αH from Chapter F ∈A 2( 2( 1. The element δ C[h] is W –invariant, so we may localise R and S at#δ . Write R hreg/W and ∈ | S reg for these localisations. It suffices to show that R reg = S reg , and since the filtration |h /W |h /W |h /W 1 induces a nonnegative filtration on S reg , it suffices to show that gr(R reg ) = gr(S reg ). F |h /W |h /W |h /W 2( 2( reg We have gr(S hreg/W ) = gr(eHκ[δ− ]e). By [36, Lemma 6.8], this equals egr(Hκ[δ− ])e = eC[h | ∼ × reg W reg W h∗] W e = C[h h∗] . The algebra C[h h∗] inherits a Poisson bracket from this construc- ∗ ∼ × × tion. By Lemma 4.7, this bracket coincides with the standard bracket. Furthermore, gr(R reg ) |h /W reg W reg W W is a Poisson subalgebra of C[h h∗] containing C[h ] and C[h∗] . So by Lemma 4.9, × gr(R reg ) = gr(S reg ). So R = S as required. |h /W |h /W We are now in a position to prove Proposition 4.1 in the case when κ is regular. Recall that θ : H D(hreg) W and θ : H D(hreg) W denote the Dunkl rep- κ κ → ∗ κ[a] κ[a] → ∗ κ κ[a] W resentations. Write T for the Dunkl operator θκ(yi) and similarly for T . If f C[h] then i i ∈ 1 W θκ(f) = θκ[a](f) = f so eθκ[a](f)e = eµ− θκ(f)µae trivially. Now suppose f C[h∗] . Let a ∈ g C[h]W . Then g is a symmetric polynomial in x( , . . . , x( . We have ∈ 1 n

1 ( ( 1 a a ( ( µ− θ (f)µ (g) = (x x )− ( x )− θ (f)( x ) ( (x x )g) a κ a i − j i κ i i − j 'i

= θκ[a](f)(g) by [23, Prop 3.26].

1This is why we have to use the filtration and not F . F

62 Therefore, if we denote by res(D) the restriction of a W –invariant map D : C[h] C[h] to → W 1 C[h] , then we have shown that res(µ− θκ(f)µa) = res(θκ[a](f)). But if D : C[h] C[h] then a → W 1 eDe(g) = e(D av)(g) where av : C[h] C[h] denotes the averaging map. So res(µ− θκ(f)µa) = ◦ → a 1 1 res(θκ[a](f)) implies that µa− e−θκ(f)e−µa = eµa− θκ(f)µae = eθκ[a](f)e. By Lemma 4.11, we obtain

1 µa− θκ(e−fe−)µa = θκ[a](efe) for all f H . By the PBW theorem, eH e is spanned by the W –invariant noncommutative ∈ κ κ polynomials p(x, y) in the variables x1, . . . , xn and y1, . . . , yn. We have shown the following.

Lemma 4.12. Let p(x, y) be any W –invariant noncommutative polynomial in the variables x1, . . . , xn and y1, . . . , yn. Then for regular κ, we have

κ 1 κ[a] ep(x, T )e = µa− e−p(x, T )e−µa, where p(x, T κ) D(hreg) W denotes p with the Dunkl operator T κ substituted for y , 1 i n. ∈ ∗ i i ≤ ≤ It remains to extend to the case of non-regular κ. This part of the argument is based on an argument of Berest and Chalykh [5]. Let κ00,κ1, . . . ,κ( 1 be commuting indeterminates and let − reg κ D = C[κ00, . . . ,κ( 1] (D(h ) W ). Define Dunkl operators Ti D by the usual formula for the − ⊗C ∗ ∈ Dunkl operators from Section 1.3.4, but with the parameters κ replaced by the indeterminates κ. reg W reg Let bλ be a C–basis of D(h ) = e(D(h ) W )e. Then there are aλ(κ), aλ$ (κ) C[κ00, . . . ,κ( 1] { } ∼ ∗ ∈ − with

κ ep(x, Ti )e = aλ(κ)bλ &λ 1 κ[a] eµa− p(x, Ti )µae = aλ$ (κ)bλ &λ where κ[a] = κ + 1 and κ[a] = κ + 1, 1 i a and κ[a] = κ otherwise. Now by Lemma 00 00 i i ≤ ≤ i i 4.12, for all λ we have aλ(κ) = aλ$ (κ) for all regular κ. Therefore, aλ and aλ$ are equal on a dense ( subset of C , so they are equal everywhere. So aλ(κ) = aλ$ (κ). This proves the following theorem.

Theorem 4.13. Let p(x, y) be any W –invariant noncommutative polynomial in the variables ( x1, . . . , xn and y1, . . . , yn. Then for all κ C , we have ∈ κ 1 κ[a] ep(x, T )e = µa− e−p(x, T )e−µa, where p(x, T κ) D(hreg) W denotes p with the Dunkl operator T κ substituted for y , 1 i n. ∈ ∗ i i ≤ ≤ 63 Since eHκe is generated by W –invariant noncommutative polynomials in the xi’s and yi’s, Proposition 4.1 follows immediately from Theorem 4.13.

4.1.2 The shift functors

We use the following notation.

Notation 4.14. In general, we have a parameter shift p : κ κ[a] for each 0 a ! 1. Clearly a 5→ ≤ ≤ − pa has an inverse p a which we denote by κ κ[ a]. If a1, a2, . . . ak are integers with the same − 5→ − sign, then we will write κ[a a a ] as shorthand for ((κ[a ])[a ] )[a ]. 1 2 · · · k 1 2 · · · k

For the rest of this chapter, we will identify θκ(Hκ) with Hκ, so that the Cherednik algebra Hκ will be regarded as the subset of D(hreg) W generated by C[h], W and the Dunkl operators. We ∗ also write Uκ = eθκ(Hκ)e = θκ(eHκe). With this notation, Proposition 4.1 reads

1 Uκ[a] = eµa− Hκµae.

1 We write ea− := µaeµa− . Then recall that ea− is an idempotent in CW , since µa is a W –semiinvariant, so e H , and it follows that a− ∈ κ eHκµae is a U U –bimodule. κ − κ[a]

Definition 4.15. The Heckman-Opdam shift functor F a associated to κ C( and 0 a ! 1 κ ∈ ≤ ≤ − is the functor U Mod U Mod defined by κ[a] − → κ −

M eH µ e M. 5→ κ a ⊗Uκ[a]

We may also define a related functor F a : U Mod U Mod by F a(M) = eµ 1H e M. κ− κ − → κ[a] − κ− a− κ ⊗Uκ Remark 4.16. Note that the functors F a and F a also give functors U mod U mod. κ κ− κ[a] − ↔ κ − 1 This is because eµaHκe and eµa− Hκe are finitely-generated right eHκe–modules. Finite generation follows by considering the associated graded modules with respect to the order filtration . For F W example, gr (eµaHκe) = (µaC[h h∗]) . Since µa is a W –semiinvariant, Cµa is a one-dimensional F ⊕ W W,χ 1 representation of W , say with character χ. Then (µaC[h h∗]) = µaC[h h∗] − , which is a ⊕ ⊕ W finitely-generated C[h h∗] –module by Theorem 1.9. ⊕

64 a We are interested in the question of when the functor Fκ is an equivalence of categories (note a that we cannot hope that Fκ will always be an equivalence, as shown, for example, in [36, Section 3.14]). The first step in addressing this question is to write the functor in another way. Using Proposition 4.1, we have an isomorphism ψ : U e H e defined by ψ(x) = µ xµ 1. Write κ[a] → a− κ a− a a− U := e H e . There is a twist functor G : M M ψ from U Mod to U Mod defined κ− a− κ a− ψ 5→ κ[a] − κ− − ψ a by M = µaM. We may rewrite Fκ (M) as

F a(M) = eH µ e M κ κ a ⊗Uκ[a] 1 = eHκµaeµa− 1 µaM ⊗µaeHκ[a]eµa− ψ = eHκe− M a ⊗Uκ− ψ = Pe Q G (M) ◦ ea− ◦ where Q : U − Mod Hκ Mod is defined by Q = Hκe− ( ) and Pe : Hκ Mod ea− κ − → − ea− a ⊗Uκ− − − → U Mod is defined by P = e( ). κ − e − a ψ ψ Therefore, F = Pe Q G . The functor G is an equivalence because ψ is an isomorphism. κ ◦ ea− ◦ We turn to the problem of showing that P and Q are equivalences. The following standard e ea− lemma is very useful to us.

Lemma 4.17. Let A be an algebra and x A an idempotent such that AxA = A. Then the ∈ functors P : M xM and Q : N Ax N are inverse equivalences between A Mod and x 5→ x 5→ ⊗xAx − xAx Mod. − Proof. We only check on objects, since checking that certain maps are natural etc. is straightfor- ward. Clearly P Q N = N for all N. It remains to show that if M A mod then Ax xM = x x ∈ − ⊗xAx ∼ M. The multiplication map Ax xM M is surjective because AxM = AxAM = AM = M, ⊗xAx → so we need only show that it is injective. Suppose a A, b M and a xb = 0. We need to i ∈ i ∈ i i show that aix xbi = 0. Since AxA = A, we have 1 = wixzi fo!r some wi, zi A. Then ⊗ ∈ aix xb!i = wjxzjaix xbi = wjx xzjaixbi!= wjx xzj aixbi = 0. ⊗ i j ⊗ j ⊗ i j ⊗ i

! a! ! ! ! a ! ! We have F = Pe Q Gψ. Similarly, we may write F − = Gψ 1 e−( ) Hκe eHκe ( ). κ ◦ ea− ◦ κ − ◦ a − ◦ ⊗ − Therefore, Lemma 4.17 gives us the following result.

a a Lemma 4.18. The functors Fκ and Fκ− are equivalences whenever HκeHκ = Hκ = Hκea−Hκ. a a Furthermore, if these conditions hold then Fκ and Fκ− are quasi-inverses.

65 Now, following [36], we relate the conditions of Lemma 4.18 to category . From Theorem 1.25 O we get the following.

Lemma 4.19. Let eχ CW be an idempotent corresponding to a linear character χ of W . Then ∈ ( for any choice of parameter κ C , HκeχHκ = Hκ if and only if eχLκ(τ) = 0 for all τ Irrep(W ). ∈ 1 ∈ Proof. By Theorem 1.25, if H e H = H , then e would annihilate some simple object of category κ χ κ 1 κ χ . O

4.1.3 The semisimple and almost-semisimple cases

The easiest case in which the Heckman-Opdam shift functors are equivalences is the case where category is semisimple. We know by Theorem 2.1 that this happens if and only if H is simple. O κ But if H is simple then H xH = H for any nonzero x H . This proves the following theorem. κ κ κ κ ∈ κ Theorem 4.20. Suppose is semisimple. Then the Heckman-Opdam shift functors F a and F a Oκ κ κ− are equivalences.

a The next situation in which we wish to check that Fκ is an equivalence is the case when the a KZ functor separates simples. By [36, Remark 3.14], it is not true that Fκ is always an equivalence whenever KZκ separates simples, so it will be necessary to impose a further condition. We first need a lemma, which shows that the condition of Lemma 4.19 will be satisfied for all τ such that

L(τ) reg = 0. |h 1 Lemma 4.21. Suppose M is an object of and KZ(M) = 0. Then for any linear character χ of O 1 W , if e denotes the corresponding idempotent, then e M = 0. χ χ 1 reg Proof. Consider the two-sided ideal I = H e H . Localising to h , I reg is a nonzero ideal of κ χ κ |h H reg , which is a simple ring. Therefore, I reg = H reg and therefore I contains a power of κ|h |h κ|h k k the element δ = H αH . Say δ I. Then if eχM = 0 then δ M = 0 and M hreg = 0 so ∈A ∈ | KZ(M) = 0. #

a Therefore, if KZκ separates simples, then Fκ will be an equivalence provided that if L is the unique finite-dimensional simple object in , then eL = 0 and e L = 0. Oκ 1 a− 1

Theorem 4.22. Suppose KZκ separates simples, and suppose further that the unique finite-dimensional simple object L in the category has dimension dim(L) > (n!)n. Then eL = 0 and e L = 0. Oκ 1 a− 1 66 Proof. We will show in fact that the unique finite-dimensional simple object L in category O contains a copy of every linear representation of W if dim(L) > (n!)n. By Theorem 3.7, the dimension of L is of the form rn = ((p 1)! + s)n for some p 1 and some 1 s ! 1. − ≥ ≤ ≤ − So our hypothesis on the dimension of L says that p n + 1. Now recall that L is constructed ≥ as the module Y˜c of [16, Section 4.1]. By Theorem 3.3, there exists a linear character χ of W r n such that L is isomorphic as a W –module to χ (C[u]/(u ))⊗ , where the generators of W act ⊗ ( t j jt j as follows: the generator s1 with s1 = 1 acts by s1(u ) = ε− u , while Sn acts by permuting the factors of the tensor product. Now consider an arbitrary linear character ζ of W . Such a character acts on s by ε t for some t, and takes all the transpositions σ(0) either to σ(0) or σ(0) (ie. its 1 − ij ij − ij restriction to the symmetric group is either the trivial or the sign representation). Consider the t t+( t+(n 1)( elements u , u , . . . , u − C[u]. Provided r n!, these are guaranteed to remain linearly ∈ ≥ r t+(i 1)( independent in C[u]/(u ). Write ai = u − , 1 i n. Then consider the element ≤ ≤

a := sgn(σ)ba a a , σ(1) ⊗ σ(2) ⊗ · · · ⊗ σ(n) σ Sn &∈ where b = 0 if ζ is the trivial representation of S and b = 1 if it is the sign representation. Then |Sn n Ca is a copy of the representation corresponding to χ ζ, so eχ ζ L = 0. Since ζ was arbitrary, ⊗ ⊗ 1 this shows that L contains a copy of every linear representation of W .

Corollary 4.23. If KZ separates simples and the unique finite-dimensional simple object L in κ Oκ n a a has dimension dim(L) > (n!) , then Fκ and Fκ− are quasi-inverse equivalences.

4.1.4 The asymptotic parameter case

a Now let us turn to the other situation in which we can prove that Fκ is an equivalence. This is the situation where the parameters are asymptotic, in the sense of Definition 4.24 below.

Definition 4.24. Say the parameter tuple κ = (κ00, κ1, . . . , κ( 1) is asymptotic if κ00 R and − ∈ κi R for all i, and the following two conditions hold. ∈ 1 κ < n(n + 1) 00 −2

κi κi 1 nκ00 − − ≥ − for 0 i ! 2, where as usual we put κ0 = 0 and κ 1 = κ( 1. ≤ ≤ − − −

67 The point of this definition is that Theorem 4.26 below shows that when the parameters are asymptotic, category is equivalent to the category of modules over an algebra called the cy- O clotomic q–Schur algebra. We will exploit the combinatorics of the category of modules over the q–Schur algebra in order to prove that the Heckman-Opdam shift functors are equivalences. Note that, in this section, we will freely identify a multipartition of n with ! parts with the corresponding representation of W .

We begin by giving the definition of the cyclotomic q–Schur algebra. Let q, u1, . . . , u( be nonzero complex numbers. Then we have the Ariki-Koike algebra of Definition 1.34. Recall that this H algebra is generated by elements T and T , 2 i n. By [54, Section 2.2], the subalgebra of s ti ≤ ≤ generated by T , . . . , T is isomorphic to the Hecke algebra of the group S , as defined in [53, H t2 tn n Chapter 1], with parameter q. Let s1, . . . , sn 1 be the usual generators of the symmetric group, so − that s = (i, i + 1) is the transposition. For a word w in these generators, let w = s s s be a i i1 i2 · · · ik reduced expression in the terminology of [53]. Then define Tw := Tti 1 Tti 1 . The element 1− · · · k− ∈ H T is well-defined by [54, Section 2.2]. Furthermore, for k = 1, . . . , n, define an element L by w k ∈ H L := q1 kT T T T T . Now for a multipartition λ = (λ(1), . . . , λ(()) of n with ! parts, k − tk · · · t2 s t2 · · · tk define i 1 (r) d − λ r=1 | | m = ( T ) ! (L u ) λ w  j − i  w S i=2 j=1 &∈ λ ' '   where S = S (1) S (2) S (#) , and S (i) denotes the group of permutations of the Young λ λ × λ × · · · × λ λ diagram λ(i) which stabilise each row of λ(i).

Definition 4.25. The cyclotomic q–Schur algebra = (q, u1, . . . , u( 1) is the algebra S S −

End mλ H  H λ Π# "∈ n   ( where Πn denotes the set of mulipartitions of n with ! parts.

By [54, Theorem 4.13, Theorem 4.14], for each λ Π( there is a module W λ for called a Weyl ∈ n S module. Each Weyl module has a quotient F λ which is absolutely irreducible, and furthermore, F λ : λ Π( is a complete set of nonisomorphic irreducible –modules. { ∈ n} S We have the following theorem due to Rouquier.

Theorem 4.26. [64, Theorem 6.8] Suppose that the parameters are asymptotic and (q+1) (u i

Lemma 4.27. Suppose the parameters are asymptotic and (q + 1) (u u ) = 0. Suppose λ, i

Lemma 4.28. [36, Corollary 4.6] Suppose the parameters are asymptotic and (q + 1) (u i

Proof. The proof is by induction on µ. We usually write λ ! µ for λ !dom µ. Suppose that [M(µ) : L(λ)] = 0 with λ = µ, and the conclusion of the lemma holds for all ν < µ. The 1 1 module L(λ) has a projective cover P (λ), by Theorem 1.24. Since L(λ) is a composition factor of M(µ), there is a submodule A of M(µ) such that L(λ) 3 M(µ)/A. Therefore, we get a map → P (λ) M(µ)/A, and the projectivity of P (λ) yields a map P (λ) M(µ), which is certainly → → nonzero. Now by [33, Corollary 2.10], there is a filtration of P (λ) of the form

P (λ) = A A A = 0 0 ⊃ 1 ⊃ · · · ⊃ t

69 where A /A = M(λ ) for some λ . Let K be the kernel of the map P (λ) M(µ) and choose i t t+1 t t → so that (K + A )/K = 0 and (K + A )/K = 0. This gives a map i 1 i+1

ψ : M(λ ) = A /A (A + K)/K 3 P (λ)/K M(µ). i i i+1 → i → →

This map is nonzero, and it follows that λ µ by Lemma 4.27. Now, if λ = µ then we have i ≤ i a nonzero map ψ : M(µ) M(µ). Such a ψ must be surjective. Indeed, if we write µ for → the irreducible representation of W corresponding to µ, then for some nonzero v µ we have ∈ 1 v M(µ). This vector generates M(µ) as a H –module, so ψ(1 v) must be nonzero. But ⊗ ∈ κ ⊗ ψ(1 v) must belong to the lowest weight space of M(µ) for the h–action, which is 1 µ. Any ⊗ ⊗ nonzero vector in 1 µ generates the whole of M(µ), and therefore ψ must be surjective. It follows ⊗ that the map P (λ)/K M(µ) is surjective, and therefore so is the map L(λ) M(µ)/A. So L(λ) → → is a simple quotient of M(µ). Therefore, L(λ) ∼= L(µ) so λ = µ, a contradiction. This shows that

λi < µ. Now by the BGG reciprocity, [P (λ) : M(λ )] = [M(λ ) : L(λ)] = 0. So by induction, λ λ i i 1 ≤ i and therefore λ < µ.

Given a Hκ–module M and a scalar α C, we have the weight space Wα(M), which is defined ∈ to be the generalised eigenspace for the action of h with eigenvalue α. Because h commutes with

W , the space Wα(M) is W –stable for all α. Following [36, Section 3.10], we define the graded

Poincar´e series of a Hκ–module M to be the expression

α p(M, v, W ) := v [Wα(M) : λ][λ] α C λ Irrep(W ) &∈ ∈ & where v is an indeterminate, and the [λ] denote the isomorphism classes of irreducible representa- tions of W . We also make the following definition.

Definition 4.29. If W ! GL(h) is a complex reflection group acting faithfully on a finite-dimensional complex vector space h, denote by C[h]W the ideal of C[h] generated by the invariant polynomials , + - of positive degree. Then the ring of coinvariants of W is the ring

C[h] . C[h]W , + - th W This ring is graded by polynomial degree and we denote the i graded component by (C[h]/ C[h] )i. , + - 70 Proposition 4.30. [36, Proposition 3.10] The graded Poinar´e series of M(λ) is

c µ Gµ(v)[λ µ] p(M(λ), v, W ) = v− λ ⊗ , n (1 vi() ! i=1 − where Gµ(v) is the polynomial in v defined by #

C[h] i Gµ(v) = : µ v . [h]W i 0 >( C + )i ? &≥ , - Proof. First of all, the lowest weight space of M(λ) is 1 λ. The eigenvalue of h on this space is ⊗ just the eigenvalue of z, because h = x y z. In our notation, this is c . − i i i − − λ As a graded W –module, M(λ) = (!C[h] λ)[ cλ] where [ cλ] denotes a grading shift. So it ⊗ − − i remains to compute the graded Poincar´e series i,τ v [C[h]i : τ][τ] of C[h]. By the proof of [68, Theorem 2.2], there is a decomposition as graded!W –modules,

W C[h] C[h] = C[h] , ⊗ C[h]W , + - and the graded Poincar´e series of C[h] is the product of the graded Poincar´e series of the two factors. It is known (see [11, Table 1]) that the degrees of a set of algebraically independent homogeneous generators of C[h]W are !, 2!, . . . , n!, which says that the graded Poincar´e series of [h]W is n (1 vi() 1. On the other hand, the graded Poincare series of C[h] is G (v) by C i=1 − [h]W µ µ − 0C + 1 definition#. !

Remark 4.31. The polynomials Gµ(v) are referred to as fake degrees and appear in work of Opdam [60].

Now we explain how Lemma 4.27 and Proposition 4.30 enable us to prove that the shift functors a are equivalences. Recall from Lemma 4.19 that, to prove that Fκ is an equivalence, it suffices to show that eL (τ) = 0 and e L (τ) = 0 for all τ. That is, no simple object of the category is κ 1 a− κ 1 Oκ annihilated by e or ea−. Suppose we have a linear character χ of W and an M(τ). Then the composition factors of M(τ) are L(τ) and L(µ) for some set of µ < τ. The generalised eigenvalues of h on L(µ) are a subset of χ those on M(µ). Let w(τ) denote the set of weights (generalised h–eigenvalues) of M(τ) and let dτ χ χ denote inf d w(τ) [W (M(τ)) : χ] = 0 . If we can show that dτ < dµ for all µ < τ, then we { ∈ | d 1 } will be done, because there will be a copy of χ in M(τ) which cannot lie in the radical R(τ), and hence from the exact sequence 0 R(τ) M(τ) L(τ) 0, there will be a copy of χ in L(τ). → → → → 71 Proposition 4.32. Let χ be a one-dimensional representation of W and let µ < τ be two repre- sentations of W . Then χ χ dτ < dµ.

Proof. By [68, Theorem 5.3], the polynomials Gµ(v) may be explicitly written as follows. Fix a (1) (2) (() ( 1 (i) multipartition µ = (µ , µ , . . . , µ ) with ! parts. Define r(µ) = − (i 1) µ . Furthermore, i=1 − | | ( ( 2( n( define (v )n = (1 v )(1 v ) (1 v ), and if η is a partition w!ith parts η1, η2, . . . (ie. ηi is the − − · · · − length of the ith row of the Young diagram of η), define n(η) = (i 1)η , the partition statistic. i − i Also, define a polynomial !

H (v) = (1 vhooklength(i,j)), η − (i,j) a'box in η where hooklength(i, j) is by definition the cardinality of the set of boxes which lie to the right of or below the box (i, j). Then [68, Theorem 5.3] states that

( 1 (n(µ(i)) r(µ) ( − v Gµ(v) = v (v )n ( . Hµ(i) (v ) 'i=0 Therefore,

( 1 (i) − (n(λ ) cµ r(λ) ( v 1 p(M(µ), v, W ) = v− v (v )n ( n i( [λ µ]. Hλ(i) (v ) i=1(1 v ) ⊗ &λ 'i=0 − # Now, the linear character χ appears in λ µ if and only if λ = µ χ. The term (v() cancels with ⊗ ∗ ⊗ n n (1 vi() = (v() , and by taking the leading term, we see that the smallest weight of a copy of i=1 − n #χ in M(µ) is ( 1 χ − (i) d = c + r(χ µ∗) + ! n((χ µ∗) ). µ − µ ⊗ ⊗ &i=0 Therefore, we wish to show that if µ < τ then

(i) (i) c + r(χ τ ∗) + ! n((χ τ ∗) ) < c + r(χ µ∗) + ! n((χ µ∗) ). − τ ⊗ ⊗ − µ ⊗ ⊗ &i &i Since n( ) and r( ) are nonnegative, this will certainly hold if we can show that − −

c c > r(θ) + ! n(θ(i)) (4.1) τ − µ &i for every multipartition θ.

72 To verify Equation 4.1, we use the calculations in [64, Sections 6.3, 6.4]. In the notation of [64], Rou h = κ00, hi = κ i and his cχτ is the negative of our cτ . Let us write cτ = cτ = cχτ for the − − − parameter appearing in [64].

By [64, Proposition 6.4], if h 0 and hi hi 1 nh for 1 i ! 1 and µ < τ, then ≤ − − ≥ − ≤ ≤ − cRou cRou. We want to show that the difference can be made large enough so that Equation 4.1 µ ≥ τ holds. Given µ < τ, there are µ , µ , . . . , µ such that µ = µ < µ < < µ = τ and for each j, 1 2 t 1 2 · · · t there is no multipartition ξ with µi < ξ < µi+1. Then cτ cµ = j(cµj cµj 1 ) cµt cµt 1 . So − − − ≥ − − we may assume that there is no µ$ with µ < µ$ < τ. Now, [64, Lem!ma 6.3] gives three possible case in which τ > µ and there is no other multipartition strictly between them. The cases are labelled (a), (b) and (c). Then using the proof of [64, Proposition 6.4], we have:

in case (c), there are positive integers i < i$ with

Rou Rou c c = c c = !h(i$ i + 1) !h τ − µ µ − τ − − ≥ − because h < 0. (s) (s) in case (b), there is an s with τi > τi+1 and

c c = cRou cRou = !h(τ (s) τ (s) ) !h τ − µ µ − τ − i − i+1 ≥ − again because h < 0. (s) in case (a), let ps be the number of nonzero parts of µ . Then there is an s such that

Rou Rou (s+1) cτ cµ = cµ cτ = !(hs hs 1) + !h(ps + µ1 1) !nh + !h(n 1) !h − − − − − ≥ − − ≥ − where we used the fact that the parameters are asymptotic, and that p + µ(s+1) µ = n. s 1 ≤ i | i| n(n+1) Thus in all cases cτ cµ !h = !κ00 > ! since the parameters ar!e assumed to − ≥ − − 2 be asymptotic. To verify Equation 4.1, it remains to show that, for all multipartitions θ, r(θ) + ! n(θ(i)) ! n(n+1) . The maximum possible value of r(θ) = (i 1) θ(i) is (! 1)n, while i ≤ 2 − | | − (i) (i) th!e maximum possible value of i n(θ ) is attained when θ is !a column of length n for exactly (j) (i) one i, and the other θ are ∅.!Therefore, the maximum value of r(θ) + ! i n(θ ) is attained when θ = (∅, ∅, , ∅, γn), where γn denotes a column of n boxes, and this m!aximum value equals · · · (! 1)n + ! 1 n(n 1) = ! n(n+1) . This verifies Equation 4.1 and finishes the proof. − 2 − 2 From the remarks proceeding Proposition 4.32, we get the following theorem.

73 Theorem 4.33. Suppose the parameter κ is asymptotic and (q + 1) (u u ) = 0. Then the i

4.1.5 Shift functors on category O We may construct functors Sa : H Mod H Mod which are very similar to the functors κ κ[a] − → κ − κ Fa . They may be defined by Sa(M) = H µ e (eM). κ κ a ⊗eHκ[a]e Note that eSa(M) = F a(eM). The functor Sa will be an equivalence provided M eM and κ κ κ 5→ H µ e ( ) are both equivalences. By Lemma 4.18, Sa will therefore be an equivalence if κ a ⊗eHκ[a]e − κ Hκea−Hκ = Hκ and Hκ[a]eHκ[a] = Hκ[a]. Parts (1), (2) and (3) of the following theorem follow from the proofs of Theorem 4.20, 4.22 and 4.33 respectively.

a Theorem 4.34. The functor Sκ is an equivalence in any of the following cases:

1. is semisimple. Oκ 2. The functor KZ separates simples, and eL = 0, where L is the unique finite-dimensional κ 1 simple object in , and furthermore, e L = 0 where L is the unique finite-dimensional Oκ[a] a− $ 1 $ simple object in category . Oκ 3. The parameters κ and κ[a] are asymptotic and (q + 1) (u u ) = 0. i

74 a It is also possible to prove various other properties of the functors Sκ that hold when they are equivalences. However, we will not use these properties in the sequel, and their proofs are identical to the proofs of [36] for the ! = 1 case, so we omit them.

4.1.6 A commutativity property

a Finally, we turn to an obvious question about the functors Fκ . If we have two integers a, b with 0 a, b ! 1 then κ[ab] = κ[ba]. This suggests to ask the question whether any route from κ[ab] ≤ ≤ − to κ gives the same shift functor, ie. if we shift first to κ[a] and then to κ, is this the same as shifting first to κ[b] and then to κ? We will show that this is the case when the functors involved are all known to be equivalences. Although the theorem is original, the argument we use is due to Ginzburg-Gordon-Stafford [32]. We start with a lemma.

Lemma 4.36. Suppose eH µ e induces a Morita equivalence U Mod U Mod and that κ a κ[a] − → κ − eH µ e induces an equivalence U Mod U Mod. Let P = eH µ e and P = eH µ e. κ[a] b κ[ab]− → κ[a]− 1 κ[a] b 2 κ a Let T = P P . Then the multiplication map 2 ⊗Uκ[a] 1 m : T = P P P P 2 ⊗ 1 → 2 1 is an isomorphism of U U –bimodules. κ − κ[ab] Proof. Clearly m is surjective, so we need only show that it is injective. Suppose m( a b ) = 0. i ⊗ i Then since by the Morita theorem, each of the modules P1, P2 is a progenerator, so is!their product reg W reg W P P . The map m becomes, on localising by δ, the natural map D(h ) reg W D(h ) 2 ⊗ 1 ⊗D(h ) → reg W D(h ) , which is an isomorphism. So ( a b ) reg = 0. But since P P is projective on i ⊗ i |h 2 ⊗ 1 both sides, it is a submodule of a free C[h]!–module. Therefore, ai bi = 0 as required. ⊗ ! The next argument is due to [32], and uses Lemma 4.38 below.

Definition 4.37. [55, Section 5.1.7] Let R be a ring. A left R–module M is said to be torsionless if for every m M 0 there exists α : M R with α(m) = 0. A torsionless left R–module M ∈ \ { } → 1 is reflexive if the natural map

M M ∗∗ = Hom (Hom (M, R), R) → R R is an isomorphism of left R–modules.

75 Note that the class of reflexive modules is closed under direct sums and direct summands. Therefore, every projective R–module is reflexive. If R is a Noetherian domain and Frac(R) denotes the quotient division ring of R, then for a left R–module M Frac(R), the right R–module M := Hom (M, R) may be identified with ⊂ ∗ R q Frac(R) : Mq R (see [55, Section 5.1.8]). Therefore, M may be naturally identified with { ∈ ⊂ } ∗∗ a subset of Frac(R). We will need to use the following Lemma due to Gabber.

Lemma 4.38. [67, Theorem 2.2] Suppose k is a field and R is a prime Noetherian ring which is finitely-generated as a k–algebra and is Auslander-regular and Cohen-Macaulay.

If M is a finitely-generated left R–submodule of Frac(R) then M ∗∗ is the unique largest left R–submodule M of Frac(R) such that M M and GK dim(M /M) GK dim(R) 2. $ ⊂ $ $ ≤ − Now we give Stafford’s argument. First, we give a bound on the GK dimension of a proper quotient of the spherical subalgebra.

( Lemma 4.39 (Stafford). Let κ C . Let I be a nonzero two-sided ideal of Uκ. Then GK dim(Uκ/I) ∈ ≤ 2n 2. − reg W reg W Proof. Since U reg = D(h ) is a simple ring, I reg = D(h ) . Hence, I contains a power κ|h ∼ |h of δ. Now recall the Fourier map ψ from Theorem 1.40. This map ψ takes δ to an element

δ C[h∗] Hκ. Considering ψ(I), we get that I contains some power of δ as well. Now with ∈ ⊂ W respect to the filtration F on Uκ from Section 1.15, grUκ = C[h h∗] which is a finite module ∼ × W W W M W N over C[h] C[h∗] . Hence gr(Uκ/I) is a finite module over C[h] /(δ ) C[h∗] /(δ ) for some ⊗ ⊗ M, N. We show that this algebra has Gelfand-Kirillov dimension 2n 2. First ≤ − W W W W C[h] C[h∗] C[h] C[h∗] GK dim GK dim + GK dim (δM ) ⊗ N ≤ (δM ) N 8 (δ ) 9 ( ) 8 (δ ) 9 W W by [47, Lemma 3.10]. Next, GK dim(C[h] ) = GK dim(C[h∗] ) = dim(h∗/W ) by [47, Theo- rem 4.5]. Since W is a finite group, this dimension is n. By [47, Proposition 3.15], we have W M W N W M GK dim(C[h] /(δ )) n 1 and similarly GK dim(C[h∗] /(δ )) n 1. So GK dim(C[h] /(δ ) ≤ − ≤ − ⊗ W N C[h∗] /(δ )) 2n 2. Applying [47, Lemma 4.3] now gives GK dim(gr(Uκ/I)) 2n 2. But ≤ − ≤ − by [47, Proposition 6.6], we have GK dim(U /I) = GK dim(gr(U /I)) 2n 2 as required. κ κ ≤ − ( Lemma 4.40. [32] Let κ , κ$ C be any two parameter tuples, and let R = x Uκ xUκ Uκ . ∈ { ∈ ! | ⊂ ! }

Suppose Uκ! has finite homological dimension. Then if Q is a nonzero reflexive left ideal of Uκ! with QU Q, then Q = R. κ ⊂ 76 Proof. By definition, Q R, so we may consider the U U –bimodule R/Q. Since Q = 0 and ⊂ κ! − κ 1 reg W D(h ) = U reg = U reg is a simple ring, we have (R/Q) reg = 0. κ|h κ! |h |h We show that R/Q is a Noetherian right Uκ–module. It suffices to show that R is a Noetherian right U –module. Take the filtration on D(hreg)W by order of differential operators. This κ F induces the filtration on Uκ and Uκ! which was considered in Lemma 4.6 above. Then gr (R) F F is a submodule of gr (Uκ ), which is a Noetherian gr (Uκ)–module since gr (Uκ ) = gr (Uκ). So F ! F F ! F N R is a Noetherian Uκ–module and so is R/Q. This shows that R/Q = i=1 viUκ for some vi.

Therefore, the fact that (R/Q) reg = 0 implies that R/Q is annihilated on t!he left by a sufficiently |h 2( high power of δ Uκ . So I := annU (R/Q) = 0. ∈ ! κ! 1 Therefore, by Lemma 4.39, the GK dimension of U /I is less than or equal to 2n 2. But R/Q κ! − is a finitely generated U /I–module and so GKdim(R/Q) 2n 2 by [47, Proposition 5.1]. κ! ≤ −

Next Uκ! is a Cohen-Macaulay and Auslander-Gorenstein algebra by Proposition 1.19. By assumption, Uκ! has finite homological dimension. Hence, by Definition 1.2, Uκ! is Auslander- regular and Cohen-Macaulay. Furthermore, Uκ! is a prime ring since it is a domain. Now we may apply Lemma 4.38 and conclude that R Q = Q. So R = Q. ⊂ ∗∗

Remark 4.41. Note that the hypothesis that Uκ! has finite homological dimension holds whenever

Hκ! eHκ! = Hκ! , since in this case, Hκ! is Morita equivalent to Uκ! , and Hκ! always has finite homological dimension.

Now we are in a position to prove the commutativity result.

a b b a Theorem 4.42. Suppose Fκ , Fκ[a], Fκ and Fκ[b] are equivalences, and that Uκ has finite homological dimension. Then the following diagram commutes

b Fκ[a] U Mod / U Mod κ[ab] − κ[a] −

a a Fκ[b] Fκ

b  Fκ  Uκ[b] Mod / U Mod − κ − Or, more succintly, the Heckman-Opdam shift functors commute.

Proof. By Lemma 4.36, we have eH µ e eH µ e = eH µ eH µ e and eH µ e eH µ e = κ a ⊗ κ[a] b κ a κ[a] b κ b ⊗ κ[b] a

77 eHκµbeHκ[b]µae. So we show that we have an equality of bimodules

eHκµaeHκ[a]µbe = eHκµbeHκ[b]µae.

By Lemma 4.36, both of these bimodules are projective on both sides, so they are both reflexive.

We show that they are both contained in Uκ. We have

eH µ eH µ e eH µ eH e κ a κ[a] b ⊂ κ a κ[a] 1 = eHκµaeµa− Hκµae by Proposition 4.1

= eHκea−Hκµae eH e ⊂ κ

Similarly, eH µ eH µ e eH e. So eH µ eH µ e and eH µ eH µ e are left ideals of eH e κ b κ[b] a ⊂ κ κ a κ[a] b κ b κ[b] a κ which are both reflexive and both stable under right multiplication by eHκ[ab]e. Hence by Lemma 4.40, they are both equal to x U xU U . { ∈ κ | κ[ab] ⊂ κ} Remark 4.43. Theorem 4.34 may be viewed as an analogue for the group G(!, 1, n) of [7, Theorem

4.8] for W = Sn, where the condition that KZ separates simples is analogous to the condition 1 c + Z 0, while the condition that c is sufficiently large is analogous to the parameters κ being ∈ h ≥ | | asymptotic. It is known that there are many other situations in which and are equivalent, Oκ Oκ! for κ = κ$. Indeed, define a partial order on Irrep(W ) by λ <κ µ if and only if cλ cµ Z>0. 1 − ∈ ( Suppose (q +1) (ui uj) = 0. Then [64, Theorem 5.5] states that, for τ Z , κ and κ+τ are i

4.2 The Boyarchenko-Gordon shift functors

Now we consider a second notion of shift functor. This comes from a homomorphism constructed by Gordon [35], following Boyarchenko [10]. Write D(V ) for the ring of differential operators on an affine variety V . The idea is to define a quiver Q with a dimension vector ε, such that there ∞ is an isomorphism D(Rep(Q , ε)) G ∞ U (4.2) I → κ ( κ ) 78 where G = G(ε) is the base-change group of the quiver Q, and Iκ is an ideal of D(Rep(Q , ε)) ∞ which depends on the parameters κ. With this description of Uκ, it is easy to construct a functor U Mod U Mod for some shifted values of the parameters κ . We refer to the functors defined κ− → κ! − $ in this way as the Boyarchenko-Gordon shift functors. In [35, Section 4.5], it is asked whether the Boyarchenko-Gordon and Heckman-Opdam shift functors coincide. Our aim is to investigate this question. The first step is to explain the isomorphism (4.2). This ismorphism comes from a map

G D(Rep(Q , ε)) Uκ ∞ → called the radial part map. Such a map was constructed in [35], following Oblomkov [59]. In our situation it is convenient to alter slightly the definition of the radial part map that was given in [35] and [59]. We will therefore explain in full how to define this map. Unfortunately, the radial part map is by definition a map D(Rep(Q , ε))G D(hreg)W , and it ∞ → is not clear that the image of our version of the radial part map will always be equal to Uκ (although we conjecture that this is so). Our results will therefore depend on the hypothesis that the image of the radial part map does coincide with Uκ (Hypothesis 4.48). We will prove the hypothesis holds when n = 1, so that our results at least apply to the case of W a cyclic group.

4.2.1 Gordon’s construction

We now quote some of the main results of [35]. Choose once for all n, ! 1 and define some quivers ≥ as follows. Let Q be the quiver with ! vertices labelled 0, 1, . . . , ! 1 with cyclic orientation 0 1 − ← ← · · · ← (! 1) 0. Let Q be the quiver whose set of vertices is 0, 1, . . . ! 1, and whose arrows − ← ∞ { − ∞} consist of the arrows of Q together with one extra arrow 0. Let δ be the dimension vector ∞ → of Q with δi = 1 for all i, and let ε be the dimension vector of Q with ε = 1 and εi = nδi for ∞ ∞ i = . For example, if ! = 4 then 1 ∞ 1 1 ? ?? ? ??  ??  ??  ?  ? Q = 0 2 , Q = / 0 2 _??  ∞ ∞ _??  ??  ??  ?   ?   3 3 and δi = 1, i = 0, 1, 2, 3, while ε = 1, εi = n, i = 0, 1, 2, 3. ∞ 79 We consider the space RQ := Rep(Q , ε) of representations of Q with dimension vector ε. ∞ ∞ ∞ The points of this space have the form

(X0, X1, . . . , X( 1, v) −

n where Xi are n n matrices and v C . Here, Xj is considered to be attached to the arrow j j+1 × ∈ → ( 1 ( 1 and v to the arrow 0. We write G for the group GL(ε) = ( − GLn C∗)/C∗ = − GLn, ∞ → i=0 × ∼ i=0 which acts naturally on Rep(Q , ε) by change of basis. Explicit#ly, for (g0, . . . g( 1) G#, we have ∞ − ∈

1 1 1 (g0, . . . g( 1) (X0, X1, . . . X( 1, v) = (g0X0g1− , g1X1g2− , . . . , g( 1X( 1g0− , g0v). − · − − −

Contained within RQ is the space Rep(Q, nδ) which also has a natural action of G which is the re- ∞ striction of the action on RQ . We identify Rep(Q, nδ) with the subspace (X0, X1, . . . , X( 1, 0) ∞ { − } ⊂ Rep(Q , ε). ∞ We may relate these representation spaces to the space h = Cn with its natural action of ˆ W = G(!, 1, n) by defining a subset := (X, X, . . . , X) : X = diag(x1, . . . xn) for some xi C S { ∈ } ⊂ Rep(Q, nδ). Then ˆ is isomorphic to h, and if we embed W in G by sending a permutation matrix S 1 1 1 1 2 (( 1) σ to (σ− , σ− , . . . , σ− ) and a diagonal matrix s to (1, s− , s− , . . . , s− − ), then the embedding ˆ 3 Rep(Q, nδ) is W –equivariant. We denote by the image of hreg under this embedding. We S → S loosely write = hreg. S Let T∆ denote the subgroup of G consisting of the elements (T, T, . . . , T ) where T is a diagonal matrix. The action map π : G hreg G hreg Rep(Q, nδ) induces a map (G/T ) hreg × → · ⊂ ∆ × → Rep(Q, nδ). Here, G/T∆ denotes the set of right cosets of T∆ in G, which can be given the structure of an affine variety by [27, Theorem 11.4.4]. The group W normalises T . Hence, W acts on (G/T ) hreg via w [gT , x] = [gT w 1, wx]. ∆ ∆ × · ∆ ∆ − Therefore, π induces a map

π : (G/T ) hreg Rep(Q, nδ) ∆ ×W → where (G/T ) hreg denotes the quotient space of (G/T ) hreg by the W –action. ∆ ×W ∆ × Proposition 4.44. [35, Lemma 2.2] The image of π is an open set, which we denote Rep(Q, nδ)reg, and π induces an isomorphism

ω : (G/T ) hreg ∼ Rep(Q, nδ)reg. ∆ ×W →

80 In [35, Section 2.5], this setup is extended to Rep(Q , ε) = Rep(Q, nδ) Cn as follows. Let ∞ × := (1, 1, . . . 1) = hreg, a subset of RQ , and define S∞ S × ∼ ∞

reg n 1 U $ = ([gT∆, x], v) ((G/T∆) W h ) C : g0− v is a cyclic vector for diag(x) . ∞ { ∈ × × }

Let U = (ω idCn )(U $ ). The open subset U RQ is useful because of the following ∞ × ∞ ∞ ⊂ ∞ proposition.

Proposition 4.45. [35, Lemma 2.5] The G–action on U is free, and the projection U hreg/W ∞ ∞ → is a principal G–bundle.

4.2.2 The radial part map

We now explain how to define the radial part map, following an idea shown to the author by Ginzburg. Recall that we wish to construct a map D(RQ )G D(hreg)W = D(hreg/W ). One ∞ → way of doing this is to start with a function f C[hreg]W and extend f to Rep(Q, nδ)reg Cn via ∈ ×

f ([gT∆, x], v) := f(x) for all g G and all v Cn. The function f makes sense1 because of Proposition 4.44. By ∈ ∈ restriction to the open set U , we may apply any differential operator D D(RQ ) to f . Since ∞ ∈ ∞ = hreg, we may therefore define a differential operator D on hreg by D(f) = D(f ) . The map S∞ ∼ |S∞ D D induces a map D(RQ )G D(hreg/W ) which is clearly a homomorphism of algebras. 5→ ∞ → The image of this map does not yet depend on κ, so we twist by something that depends on κ. We now describe this twist. 1 ( 1 ( 1 Define Γi = κ( i κ( i+1 , 1 i ! 1 and let Γ0 = i=−1 Γi = κ1 + − . Define a − − − − ( ≤ ≤ − − − ( function s on RQ by ! ∞

n 1 s(X0, X1, . . . , X( 1, v) = v X0X1 X( 1v (X0X1 X( 1) − v, − ∧ · · · − ∧ · · · ∧ · · · − so that s(X0, X1, X( 1, v) = 0 if and only if v is a cyclic vector for X0X1 X( 1. For 1 i ! · · · − 1 · · · − ≤ ≤ − 1, we also define functions si by si(X0, X1, . . . X( 1, v) = det(XiXi+1 X( 1)s(X0, X1, . . . , X( 1, v). − · · · − − Define ( 1 − Γi κ00+Γ0 1 ζκ := si s − . 'i=1 1The symbol f should be read “f antlers”.

81 Then ζκ is not a well-defined function on RQ , but for any point p , there exists a small ∞ ∈ S∞ Euclidean neighbourhood of p such that there is a well-defined branch of ζκ in this neighbourhood. 1 Also, for any D D(RQ ), the conjugation ζκ− D ζκ makes sense as a differential operator ∈ ∞ ◦ ◦ ( 1 on the open set of points where s − s = 0. Note that since the Γ have been chosen to sum to i=1 i 1 i zero, ζκ may also be expressed as # n i Γj κ00 1 ζκ = det(Xi) j=1 s − . ! 'i=1 Definition 4.46. The radial part map associated to κ is the map

G reg W Rκ : D(RQ ) D(h ) ∞ → given for D D(RQ )G and f C[hreg]W by ∈ ∞ ∈

1 Rκ(D)(f) = ζκ− Dζκ(f ) . |S∞

It is useful to note that Rκ preserves the order filtration on differential operators. Here, the order filtration on D(RQ )G is by definition the filtration induced from the order filtration on ∞ D(RQ ), while the order filtration on D(hreg)W = D(hreg/W ) is the natural order filtration. ∞

G Lemma 4.47. If D D(RQ ) is a differential operator of order n, then the order of Rκ(D) is ∈ ∞ at most n.

Proof. We may ignore the conjugation by ζκ, since conjugation does not change the order of a differential operator. Write R for the map D(RQ )G D(hreg)W defined by R(D)(f) = D(f ) . ∞ → |S∞ Let g C[hreg]W and let D D(RQ )G be a differential operator of order n. Then we must show ∈ ∈ ∞ that [R(D), g] has order n 1. By definition, for f C[hreg]W , we have ≤ − ∈

[R(D), g](f) = R(D)(gf) g R(D)(f) − · = D(g f ) g D(f ) |S∞ − |S∞ · |S∞ = [D, g ](f ) |S∞

So [R(D), g] = R([D, g ]). By induction on the order of D, we get the result.

Now we introduce our hypothesis.

82 G reg W Hypothesis 4.48. The image of the map Rκ : D(RQ ) D(h ) is the spherical subalgebra ∞ → Uκ = eHκe.

In [59, Theorem 2.5], Oblmokov has proved a similar statement to Hypothesis 4.48 for the quiver Q rather than Q . However, Oblomkov’s methods do not seem to work in our case. ∞ We now present some evidence for believing Hypothesis 4.48 to be true.

Theorem 4.49. If ! = 1 then Hypothesis 4.48 holds.

n κ00 1 Proof. In the ! = 1 case, the group G is GLn and RQ = gln C . Also, ζκ reduces to s − . In ∞ × the notation of [32], we have c = κ so [32] shows that ImR = eH e. − 00 κ κ Hypothesis 4.48 also holds in the n = 1 case. We give the argument for this in full, following very closely [59, Proposition 3.3].

Theorem 4.50 (Oblomkov). If n = 1 then Hypothesis 4.48 holds.

reg Proof. In the n = 1 case, we have W = Z/!. The defining representation h is C, and h = C∗. (+1 ( The space RQ is isomorphic to C , and G = (C∗) . We continue to write points of RQ as ∞ ∞ (+1 (X0, X1, . . . , X( 1, v) C . We write xi for the coordinate function corresponding to Xi and ν for − ∈ the coordinate function corresponding to v. Recall from Example 2.6 that the rational Cherednik 1 algebra eHκe is generated by three elements x, y, s, where s(x) = ε− x and s(y) = εy. We may ( 1 ignore the parameter κ00 and regard κ = (κ1, . . . , κ( 1) as an element of C − . With respect to the − W /( ( ( standard filtration on eHκe, we have gr(eHκe) = C[h h∗] = C[x, y]Z = C[x , y , xy]. Therefore, ⊕ ( ( in order to show that ImRκ contains eHκe, it suffices to show that x e, y e and xye all lie in the image. We do this by an explicit calculation. (+1 ( )# /( The radial part map Rκ : D(C ) C∗ D(C∗)Z may be written explicitly as →

( 1 i # 1 i − Γj i=−1 j=1 Γj j=1 Γ0 1 Rκ(D)(f(x)) = x− D( xi ν − f ) . ! ! ! · |S∞ 'i=1 G ( 1 ( We have Rκ(F ) = F for all F C[RQ ] , so Rκ( i=−0 xi) = x . |S∞ ∈ ∞ ( ( Now we show that y e lies in the image of Rκ. Via#the Dunkl embedding, y is regarded as the ( d 1 ( 1 1 ( 1 ij j Dunkl operator , where = + − !κ e , where e := − ε s . A calculation gives ∇ ∇ dx x i=1 i i i ( j=0 d 1 ei = ( + !κi+1)ei+1 for all i, and so! ! ∇ dx x

( d 1 d 1 d 1 d e e = ( + !κ1)( + !κ2) ( + !κ( 1) . ∇ dx x dx x · · · dx x − dx 83 r ( 1 r ( ∂ ∂ ∂ This operator maps a monomial x to r − (!κ +r i)x − . Now consider Rκ( ). i=1 ( i ∂x0 ∂x1 ∂x# 1 − − · · · − r r/( r/( r/( If f = x then f = x0 x1 x( 1, so # · · · − ( 1 − r/(+ i Γ ∂ ∂ ∂ 1 ∂ ∂ ∂ j=1 j r/( Γ0 1 Rκ( )(f) = ζκ− xi x0 ν − ∂x0 ∂x1 · · · ∂x( 1 ∂x0 ∂x1 · · · ∂x( 1 8 ! 9@ − − i=1 @ ' @S∞ ( 1 i @ − r ( @ = r/! (r/! + Γj)x − 'i=1 &j=1 ( 1 i 1 − r ( = r (r + ! Γ )x − . !( j 'i=1 &j=1 i ∂ ∂ ∂ 1 ( But by definition of Γ , ! Γ = !κ i, and so R ( ) = # y e. j j=1 j ( i κ ∂x0 ∂x1 ∂x# 1 ( − − · · · − Finally, we show that !xy lies in the image of Rκ. The element exye eHκe corresponds under ∈ d r r the Dunkl embedding to the differential operator x dx , which maps x to rx . In a similar way to the above calculation, we have

( 1 ( 1 i i − ∂ − Γj +r/( 1 r 1 j=1 − Γ0 1 Rκ( xi )(x ) = ζκ− (r/! + Γj)xixi ν − ∂xi ! |S∞ &i=0 &i=0 &j=1 ( 1 i − r = (r/! + Γj)x &i=0 &j=1 ( 1 i r − r = rx + Γjx . &i=1 &j=1 ( 1 ∂ ( 1 i Hence, Rκ( − xi − Γj 1) = xye as required. This shows that ImRκ eHκe. i=0 ∂xi − i=1 j=1 · ⊃ To obtain t!he reverse incl!usion,!we follow Oblomkov’s argument. Take the filtration by order of reg W differential operators on D(RQ ) and D(h ) . The map Rκ preserves this filtration by Lemma ∞ 4.47. The associated graded map

# grR G (C∗) κ 1 Z/( reg W grD(RQ ) = C[x0, . . . , x( 1, y0, . . . y( 1, ν, ω] C[x± , y] = grD(h ) ∞ − − −→ 1 is induced by the map which maps xi to x, yi to ( y and ν, ω to 0. Note also that the Dunkl ( ( 1 /( embedding identifies gr(eHκe) with C[x , y , xy] C[x± , y]Z . We now define an ideal Iκ of ⊂ D(RQ )G as follows: I is the 2–sided ideal generated by the elements x ∂ x ∂ Γ , κ i ∂xi i 1 ∂xi 1 i ∞ − − − − 1 i ! 1, together with ν ∂ (Γ 1). Then by similar calculations to the above, we get that ≤ ≤ − ∂ν − 0 − I ker R , and grI = ker grR , because κ ⊂ κ κ κ # ( 1 ( 1 (C∗) C[x0, . . . , x( 1, y0, . . . , y( 1, ν, ω] − − ( ( − − = C[ xi, yi, x0y0] ∼ C[x , y , xy]. ( xiyi = xjyj 1 i,j ( 1, νω = 0) −→ { } ≤ ≤ − 'i=0 'i=0 84 We obtain grImR = Im(grR ) = gr(eH e), which together with ImR eH e, shows that κ κ κ κ ⊃ κ ImRκ = eHκe as required.

As stated above, we have not been able to show that Hypothesis 4.48 holds for general !, n and κ. For the rest of this chapter, we make the standing assumption that we have chosen !, n and κ so that Hypothesis 4.48 holds.

Remark 4.51. In order to prove Hypothesis 4.48 in general, it would suffice to show that imR κ ⊂ Uκ. An associated graded argument like that at the end of Theorem 4.50 would then complete the proof, using the fact that Gan has shown [29, Theorem 3] that the associated graded map of

Rκ (with respect to the order filtration) would be an isomorphism. The method of [59, Lemma

3.1] may be used to show that imRκ is contained in the localisation (Uκ) n # , but this does not i=1 xi appear to be enough. #

4.2.3 A shift functor

Before we can construct the shift functor, we must calculate the kernel of the map Rκ. Since G acts on C[RQ ], we get an action of the Lie algebra g of G by differential operators on C[RQ ] ∞ ∞ which we denote by g X τˆ(X). This action is given by f d eXt f. We get X(ζ ) = < 5→ 5→ dt t=0 · κ ( 1 − Γ Tr(X ) (Γ + κ 1)Tr(X ) ζ . We define a character@χ of g by − i=1 i i − 0 00 − 0 κ @ κ $ % ! ( 1 − χ (X) = (Γ + κ 1)Tr(X ) + Γ Tr(X ). κ 0 00 − 0 i i &i=1 G Then we get that (D(RQ )(τˆ + χκ)(g)) ker Rκ. Because G is a reductive group, the functor ∞ ⊂ of taking G–invariants is exact. We write Rκ to denote the induced map

D(RQ ) G ∞ # eHκe. D(RQ )(τˆ + χκ)(g) ( ∞ ) We wish to show that this map is an isomorphism. In order to show this, we require the concept of the moment map µ : T ∗RQ g∗. This map, which appears in work of Crawley-Boevey [18] and ∞ → Holland [43], arises from the fact that the cotangent bundle T ∗RQ is a symplectic vector space ∞ equipped with a Hamiltonian G–action, but we define µ via the following explicit formula: a point of T ∗RQ may be regarded as a tuple of the form ∞

(X0, . . . , X( 1, Y0, . . . , Y( 1, v, w) − − 85 n n where Xi, Yi are n n matrices and v C , w (C )∗. Then from [35, Section 2.6], we have × ∈ ∈

µ(X0, . . . , X( 1, Y0, . . . , Y( 1, v, w) = − −

(Y0X0 X( 1Y( 1 + vw, Y1X1 X0Y0, . . . , Y( 1X( 1 X( 2Y( 2). − − − − − − − − −

Theorem 4.52. [35] Under Hypothesis 4.48, the map Rκ is an isomorphism of algebras.

Proof. By Hypothesis 4.48, Rκ is a surjection, so it suffices to show that ker(Rκ) = (D(RQ )(τˆ + ∞ G χκ)(g)) . To show this, we use Gelfand-Kirillov dimension.

Recall that the Bernstein filtration on the Weyl algebra Ar is defined as the filtration with the coordinate functions xi and derivations ∂i in degree 1. See [43, Section 2.1]. D(RQ ) G The space ∞ inherits a filtration from the Bernstein filtration on D(RQ ), D(RQ )(τˆ+χκ)(g) ∞ ∞ $ % D(RQ ) G and we denote the associated graded algebra with respect to this filtration by gr ∞ . B D(RQ )(τˆ+χκ)(g) ∞ By [43, Proposition 2.4], we have $ %

G D(RQ ) 1 G grB ∞ = C[µ− (0)] , D(RQ )(τˆ + χκ)(g) ( ∞ ) where µ is the moment map. (In fact, [43, Proposition 2.4] is stated with respect to the order filtration. But, as remarked in [43, Section 2.1] and [10, Section 3.1], the proposition is still true 1 G with the order filtration replaced by the Bernstein filtration). By [35, Theorem 2.6], C[µ− (0)] is G D(RQ ) a domain of Gelfand-Kirillov dimension 2n. So ∞ is also a domain of Gelfand- D(RQ )(τˆ+χκ)(g) ∞ Kirillov dimension 2n by [47, Proposition 6.6]. $Furthermore, wit%h respect to the filtration F of W Section 1.15, grUκ = C[h h∗] and so GK dim(Uκ) = 2n as well. Therefore, Rκ is a surjection from ⊕ a domain of GK-dimension 2n to an algebra of GK-dimension 2n. It follows from [47, Proposition

3.15] that Rκ must be an isomorphism.

Given a vector space V with a G–action and a linear character θ of G, we denote by V G,θ the space of (G, θ)–semiinvariants V G,θ = x V : g(x) = θ(g)x for all g G . { ∈ ∈ } Following [35], given a character θ of G, we now consider the space

G,θ θ D(RQ ) Bκ := ∞ D(RQ )(τˆ + χκ)(g) ( ∞ ) of (G, θ)–semiinvariants.

We may consider θ to be an element of Z(, since the character group of G is isomorphic to

( ( 1 θi Z via (θ0, . . . , θ( 1) (g i=−0 det(gi) ). Assuming Hypothesis 4.48, Theorem ?? holds, and − 5→ 5→ # 86 θ so the space Bκ is clearly a right Uκ–module. Also , for X D(RQ ) and X g, we have ∈ G ∞ ∈ θ D(RQ ) τˆ(X)D = [τˆ(X), D] + Dτˆ(X), so B is a left ∞ –module. We calculate the κ D(RQ )(τˆ+χκ dθ)(g) ∞ − ( $ % κ$ C such that χκ dθ = χκ . From the definition of χκ we get ∈ − !

κ$ + Γ$ 1 = κ + Γ 1 θ 00 0 − 00 0 − − 0 Γ$ = Γ θ , i 1. i i − i "

( i Using − !Γ = !κ (! i) we obtain j=1 j i − − ! ( 1 − κ$ = κ θ 00 00 − i &i=0 ( i − κ$ = κ θ , i 1 (4.3) i i − j " &j=0 So Bθ is a U U –bimodule. κ κ! − κ Definition 4.53. We call the functor U Mod U Mod defined by Bθ the Boyarchenko- κ − → κ! − κ ⊗ − Gordon shift functor.

Remark 4.54. Note that of course the Boyarchenko-Gordon shift functor is only well-defined when Hypothesis 4.48 holds.

4.3 Comparison of the shift functors

We now come to the main aim of this chapter. We wish to prove that under certain conditions the Boyarchenko-Gordon shift functor coincides with one of the Heckman-Opdam shift functors defined θ reg W above. The idea of the proof is to modify the embedding Rκ to get a map from Bκ to D(h ) which is a map of U U –bimodules, and then to show that in certain circumstances the image κ! − κ of this map is precisely the bimodule defining a Heckman-Opdam shift functor. We can only show this for some values of κ and θ as explained below. ( ( θ G,θ reg W Given θ Z and κ C , the first step is to define a map Pκ : D(RQ ) D(h ) . ∈ ∈ ∞ → Since s(g p) = det(g0)s(p) and si(g p) = det(gi)s(p) for all p RQ and all g G, the function · · ∈ ∞ ∈ ( 1 θi θ0 G,θ G ξ := i=−1 si s is (G, θ)–semiinvariant. Therefore, if D D(RQ ) then Dξ D(U) , − ∈ ∞ ∈ ( 1 where#U is the open set of points where s − si = 0 (we must restrict to U because the · i=1 1 # 87 θ are allowed to be negative), so we may set P θ(D) := R (Dξ) D(hreg)W . If X g then i κ κ! ∈ ∈ (τˆ(X) + χ (X))ξ ker R , so P θ induces a map κ ∈ κ! κ Pθ : Bθ D(hreg)W . κ κ → Proposition 4.55. The map Pθ is a map of U U –bimodules. That is, R (E)P θ(D)R (F ) = κ κ! − κ κ! κ κ θ G G,θ Pκ (EDF ) for all E, F D(RQ ) and all D D(RQ ) . ∈ ∞ ∈ ∞ θ Proof. It is clear that Pκ is a left Uκ! –module map, so it suffices to show that Rκ! (Dξ)Rκ(F ) = G,θ G reg W Rκ (DF ξ) for all D D(RQ ) and all F D(RQ ) . But if f C[h ] then ! ∈ ∞ ∈ ∞ ∈

1 Rκ! (Dξ)(f) = ζκ− D(ζκ! ξf ) ! |S∞ 1 = ξ (ζκ! ξ)− D(ζκ! ξf ) |S∞ |S∞ 1 = ξ ζκ− D(ζκf ) |S∞ |S∞

= Rκ(ξD)(f)

Therefore, Rκ! (Dξ)Rκ(F ) = Rκ(ξD)Rκ(F ) = Rκ(ξDF ) = Rκ! (DF ξ) as required.

In order to compare the Boyarchenko-Gordon and Heckman-Opdam shift functors, we first specialise to the case θ = ε , where ε is the character of G with (ε ) = δ . In this case, the − a a a i ai shift (4.3) is

κ0$ 0 = κ00 + 1

κ$ = κ + 1 1 i ! a i i ≤ ≤ − κ$ = κ ! a < i ! 1. i i − ≤ − This is the same as the shift κ κ[! a] in the notation of Notation 4.14. We wish to show that 5→ − εa the map Pκ− gives an isomorphism

εa εa 1 Pκ− : Bκ− eµ(− aHκe → −

n ( a ( ( where µ( a := ( i=1 xi) − i

88 Lemma 4.56. For all κ C(, ∈

εa 1 (imPκ− )∗∗ = (eµ(− aHκe)∗∗. −

1 1 Proof. If θ = εi, i 1, then ξ = si− while if θ = ε0 then ξ = s− . So if D D(RQ ) then − ≥ − ∈ ∞ 1 εa 1 G ξ− D D(RQ ), and Pκ− (ξ− D) = Rκ(D) for all D D(RQ ) , since Pκ(D) = Rκ(ξD) by ∈ ∞ ∈ ∞ the proof of Proposition 4.55. Therefore, imP εa U . Taking duals with respect to the right κ− ⊃ κ U –module structure, we get (imP εa ) x Frac(U ) : U x U = U . So (imP εa ) is a κ κ− ∗ ⊂ { ∈ κ κ ⊂ κ} κ κ− ∗ Uκ Uκ[( a]–bisubmodule of Uκ. But for any module X, X∗ is a reflexive module by [55, Section − − εa 1 5.1.7]. So by Lemma 4.40, (imPκ− )∗ = x Uκ : xUκ[( a] Uκ . Similarly, (eµ(− aHκe)∗ is a { ∈ − ⊂ } − εa 1 εa reflexive Uκ Uκ[( a]–bisubmdoule of Uκ. So (imPκ− )∗ = (eµ(− aHκe)∗ = (Bκ− )∗. This gives the − − − ∼ result.

1 Lemma 4.57. Suppose eµ(− aHκe is a reflexive right Uκ–module. Then the image of the map − εa εa reg W 1 Pκ− : Bκ− D(h ) is contained in eµ(− aHκe. → −

1 εa 1 Proof. If eµ(− aHκe is reflexive, then Lemma 4.56 yields a map (Bκ− )∗∗ eµ(− aHκe. The map − → − P εa is the composition of this map with the natural map B εa (B εa ) . κ− κ− → κ− ∗∗

εa 1 Now we can prove that the image of Pκ− is eµ(− aHκe for regular κ. −

Theorem 4.58. Suppose κ C( is regular. Then ∈

θ 1 imPκ = eµ(− aHκe. −

εa Proof. If κ is regular then Hκ is simple, hence Uκ is simple by [6, Lemma 4.1]. Since imPκ−

εa reg W εa 1 contains Uκ, we get imPκ− hreg = D(h ) . But imPκ− eµ(− aHκe by Lemma 4.57 (since | ⊂ − 1 eµ(− aHκe is reflexive by Theorem 4.20). So − 1 eµ(− aHκe − = 0. εa imPκ− @ @hreg @ 1 @ Now, we note that eµ(− aHκe is a Noetherian left @Uκ[( a]–module. This follows by considering the − − associated graded modules with respect to the filtration F on Hκ, as in Remark 4.16. Hence, 1 eµ#− aHκe − εa is a Noetherian left Uκ[( a]–module, and so is finitely generated. If vi denotes a finite imPκ− − { } N set of generators, then we may choose N N large enough such that viδ = 0 for all i. Therefore, ∈

89 1 1 eµ#− aHκe eµ#− aHκe some power of δ annihilates − εa on the right. So annMod Uκ − εa = 0 and therefore, imPκ− − imPκ− 1 ( ) since Uκ is a simple ring, we get

1 eµ(− aHκe − = 0, εa imPκ− as required.

We may extend Theorem 4.58 from regular κ to a larger class of κ by considering the associated

εa graded map of Pκ− . Consider the filtration by order of differential operators on D(RQ ) and ∞ reg θ D(h ). The filtration on D(RQ ) induces a filtration on Bκ for any θ, which we also refer to ∞ as the filtration by order. It is clear from Lemma 4.47 that Rκ respects the filtration by order. Therefore, so does the map Pθ : Bθ D(hreg)W . κ κ → 1 Now suppose that eµ(− aHκe is a rexflexive right Uκ–module (this happens for example if − 1 εa εa eµ(− aHκe induces a Morita equivalence). Then by Lemma 4.57,there is a map Pκ− : Bκ− − → 1 eµ(− aHκe. − ( 1 Lemma 4.59. Suppose κ C is chosen so that eµ(− aHκe is a rexflexive right Uκ–module. Then ∈ − εa 1 with respect to the order filtrations on Bκ− and eµ(− aHκe, the map −

εa εa 1 grPκ− : grBκ− gr(eµ(− aHκe) → − is independent of κ.

1 1 εa 1 G, εa Proof. By [36, Lemma 6.8], gr(eµ(− aHκe) = eµ(− aC[h h∗]e, while grBκ− = C[µ− (0)] − − − × by [35, Lemma 4.1]. Furthermore, suppose D D(RQ )G is a differential operator of order n. ∈ ∞ Then for any f C[hreg]W , we have ∈

εa 1 Pκ− (D)(f) = ζκ−[( a]D(ξζκ[( a]f ) − − |S∞ 1 = ζκ−[( a][D, ξζκ[( a]](f ) + ξ D(f ) . − − |S∞ |S∞ |S∞ Modulo differential operators of order n 1, this is just D (f ξ D(f ) ), which is ≤ − 5→ 5→ |S∞ |S∞ independent of κ.

( 1 Theorem 4.60. Suppose κ C is chosen so that eµ(− aHκe is a reflexive right Uκ–module.Then ∈ −

εa εa 1 Pκ− : Bκ− eµ(− aHκe → − is a surjection of Uκ[( a] Uκ–bimodules. − − 90 ( εa εa εa Proof. Let µ C be regular. Then grP− = grP− by Lemma 4.59. But grP− is surjective, ∈ κ µ µ εa since Pµ− is graded-surjective by Theorem 4.58 combined with [65, Corollary 4.5]. Therefore,

εa εa grPκ− is also surjective, so Pκ− is surjective, as required.

We close this section by explaining how to extend Theorem 4.60 from the character ε to an − a arbitrary θ with θ 0 for all i. First of all, we consider the case θ = ε ε where 0 a, b ! 1. i ≤ − a − b ≤ ≤ − 1 1 Lemma 4.61. Suppose eµ(− aHκ[( b]e and eµ(− bHκe both induce Morita equivalences. Then − − −

εa εb 1 1 imPκ− − = eµ(− aHκ[( b]eµ(− bHκe. − − − Proof. By the proof of Lemma 4.36, the multiplication map

1 1 1 1 m : eµ(− aHκ[( b]e eµ(− bHκe eµ(− aHκ[( b]eµ(− bHκe − − ⊗ − → − − − 1 1 is an isomorphism. This implies that eµ(− aHκ[( b]eµ(− bHκe is a reflexive module on both sides, so − − − we have

εa εb εa εb εa εb 1 1 1 1 Pκ− − : Bκ− − (Bκ− − )∗∗ (eµ(− aHκ[( b]eµ(− bHκe)∗∗ = eµ(− aHκ[( b]eµ(− bHκe. → → − − − − − −

εa εb 1 1 Therefore, imPκ− − eµ(− aHκ[( b]eµ(− bHκe. There is a commutative diagram ⊂ − − −

εa εb εa εb Bκ−[( b] Bκ− / Bκ− − − ⊗

ε ε a − b εa εb Pκ−[# b] Pκ Pκ− − − ⊗

  1 1 m 1 1 eµ(− aHκ[( b]e eµ−( bHκe / eµ(− aHκ[( b]eµ(− bHκe − − ⊗ − − − −

εa εb εa εb where the map Bκ−[( b] Bκ− Bκ− − is the multiplication map induced by the multiplication − ⊗ → εa εb of D(RQ ). The map Pκ−[( b] Pκ− is surjective by Theorem 4.60. Also, m has been shown to ∞ − ⊗ εa εb be an isomorphism. Therefore, Pκ− − is surjective, which shows that it is an isomorphism.

( N Now suppose we are given an arbitrary θ Z with θi 0 for all i. Write θ = k=1 ε( ik . ∈ ≤ − − θ Define κ[θ] = κ[i1i2 iN ]. Let F denote the composition ! · · · κ

θ iN iN 1 i2 i1 F := F − F − − F − F − κ κ[i1i2 iN 1] κ[i1i2 iN 2] κ[i1] κ ··· − ◦ ··· − ◦ · · · ◦

ik 1 which is a functor Uκ Mod U Mod. The functor F − is by definition eµ− H e κ[θ] κ[i1 ik 1] ik κ[i1 ik 1] − → − ··· − ··· − ⊗ 1 θ . Let A = eµ− H e, 1 k N and let A = A A A . k ik κ[i1 ik 1] κ N 1 N 2 1 − ··· − ≤ ≤ − − · · · 91 Theorem 4.62. Suppose Ak induces an equivalence for each k. Then the image of

Pθ : Bθ D(hreg)W κ κ →

θ is Aκ.

Proof. The argument is the same as the proof of Lemma 4.61.

4.3.1 A remark on Z–algebras

In this section, we give an interpretation of Theorem 4.62 in terms of Z–algebras, as promised in the introduction. In the definition of Z–algebras, it is necessary to drop our assumption that all algebras have to have an identity element.

Definition 4.63. [36, Section 5.1] A Z–algebra is an algebra (which is not assumed to have a unit) of the form

B = Bij i j 0 ≥"≥ where B B B for all i j k and B B = 0 if j = a. Furthermore, each of the subalgebras ij jk ⊂ ik ≥ ≥ ij ak 1 B is required to have a unit element 1 such that 1 b = b 1 = b for all b B for any j. ii i i ij ij j ij ij ∈ ij

A Z–algebra may be thought of as an algebra of infinite lower triangular matrices, with the diagonal entries being the algebras Bii.

Example 4.64. If ∞ S = Si "i=0 ˆ ˆ ˆ is a graded commutative ring, then there is a Z–algebra S associated to S, given by S = i j 0Sij ⊕ ≥ ≥ where Sˆij = Si j for all i, j. −

We consider some categories of modules over a Z–algebra B, following [36, Section 5.1]. First, let B grmod denote the category of Noetherian graded left B–modules M = M such that − ⊕i∞=0 i B M M for all i, j, and B M = 0 if j = k. The homomorphisms from M to N are B–module ij j ⊂ i ik j 1 maps f such that f(M ) N for all k. k ⊂ k Say M B grmod is bounded if M = 0 for all but finitely many n. Denote by B tors the ∈ − n − subcategory of B grmod consisting of those M which are bounded. (Note that in [36], B tors is − − taken to be the subcategory of B grmod consisting of those M which are the direct limit of their − 92 bounded submodules. But in the Noetherian case, this is the same as being bounded, as remarked in the proof of [10, Theorem 4.4]). Then B tors is a Serre subcategory of B grmod, and we − − may form the quotient category B grmod B qgr := − . − B tors − ( ( 1 Let θ Z with θi 0 for all i. Regard θ as a character of G = − GLn. Recall the moment ∈ ≤ i=0 map µ : T ∗RQ g from Section 4.2. Consider the graded ring # ∞ →

∞ 1 G,θi S := C[µ− (0)] . "i=0 Let Sˆ be the Z–algebra associated to this ring, in the notation of Example 4.64. We have the following theorem. Both the theorem and its proof are modelled very closely on the main theorem of [36].

i j θ − ( Theorem 4.65. Suppose Hypothesis 4.48 holds and that P j is injective for all i j. Let κ C κ[θ ] ≥ ∈ i j θ − be regular and define a Z–algebra Bκ by (Bκ)ij = B j for i j 0. Define another Z–algebra κ[θ ] ≥ ≥ grB by (grB ) = gr(B ) for all i j 0, where the associated graded space is taken with κ κ ij κ ij ≥ ≥ respect to the order filtration. Then there are equivalences of categories

1. U mod = B qgr. κ − ∼ κ −

2. grB qgr = Sˆ qgr. κ − ∼ −

Proof. The proof is an application of results of [36]. By Theorem 4.62, Bκ is a Morita Z–algebra in the sense of [36, Section 5.4]. Therefore, there is an equivalence of categories (B ) mod κ 00 − → θ 1 G,θ Bκ qgr by [36, Lemma 5.5]. The second part follows from the fact that grB = C[µ− (0)] for − κ i j θ − any θ, by [35, Section 4.1]. The maps induced by this isomorphism on the grBκ are the natural ˆ multiplications, by [36, Lemma 6.7]. So grBκ ∼= S as Z–algebras.

Remark 4.66. The algebra S has a projective variety Proj(S) associated to it. By [58, 3.i], Proj(S) is the Nakajima quiver variety Mθ with stability condition θ. By [58, 3.iii], there is a natural map π : Mθ µ 1(0)//G, where µ 1(0)//G denotes the affine space whose coordinate → − − 1 G 1 G W ring is C[µ− (0)] . By [18, Theorem 1.1], C[µ− (0)] is isomorphic to C[h h∗] . Thus, there ⊕ is a map π : Mθ (h h )/W . For some θ, this is known to be a resolution of singularities → ⊕ ∗ (see [39, Proposition 7.2.7]). Furthermore, the category Sˆ qgr may be identified with the category −

93 coh Proj(S) of coherent sheaves on Mθ, by [42, Exercise 5.9] 1. Thus Theorem 4.65 gives a − relationship between the geometry of resolutions of (h h )/W and the Cherednik algebra. ⊕ ∗

4.3.2 Nakajima quiver varieties

θ In view of Theorem 4.65, we now wish to address the question of when the map Pκ is injective. θ θ We have already shown that, under Hypothesis 4.48, Pκ has image Aκ provided κ is regular and θ 0 for all i. Therefore, if we can show that Pθ is injective, then Pθ will be an isomorphism of i ≤ κ κ θ θ Bκ with Aκ, and the two notions of shift functor will coincide. In the previous section we have introduced the Nakajima quiver variety

θ 1 G,θi M := Proj ∞ C[µ− (0)] . ⊕i=0 $ % We aim to prove the following theorem.

Theorem 4.67. Suppose θi < 0 for all i. Then

Pθ : Bθ D(hreg)W κ κ → is injective.

In order to prove Theorem 4.67, we will require a series of lemmas. Let us first note that, by [38, Lemma 4.3], Mθ is smooth if ( θ = 0 and θ + θ = m ( θ for any 1 i < j ! 1 i=0 i 1 i · · · j 1 − i=0 i ≤ ≤ − θ and any (n 1) m (n !1). In particular, if θi < 0 for all 0! i ! 1, then M is smooth. − − ≤ ≤ − ≤ ≤ − Also, we see that if θi < 0 for all i, then both Theorem 4.67 and Theorem 4.62 apply, and for θ such θ, Pκ is an isomorphism. By [46, Section 2], Mθ may be identified with the set of closed orbits of G on the set

1 ss 1 1 G,θi µ− (0) := x µ− (0) : f ∞ C[µ− (0)] with f(x) = 0 θ { ∈ ∃ ∈ ∪i=1 1 }

1 1 s of θ–semistable points of µ− (0). We may also consider the set of θ–stable points µ− (0)θ of µ 1(0), which is the set of points x µ 1(0)ss such that the stabiliser of x is finite and there exists − ∈ − θ 1 G,θi 1 f ∞ C[µ− (0)] such that the orbit of x is a closed subset of p µ− (0) : f(p) = 0 . Our ∈ ∪i=1 { ∈ 1 } 1 This requires that S be generated by S1 as an S0–algebra. In our case, this is true because the proof of 1 G, ε 1 G, ε Theorem 4.36 combined with [36, Lemma 6.7] implies that the multiplication map C[µ− (0)] − a C[µ− (0)] − b ⊗ → 1 G, ε ε C[µ− (0)] − a− b is surjective for all a, b.

94 first goal is to show that the sets of stable and semistable points coincide. This result is based on an argument of Cassens and Slodowy [14].

We may identify T ∗RQ with Rep(Q , ε) where Q is the double of the quiver Q (that is, ∞ ∞ ∞ ∞ the quiver obtained from Q by adding an edge in the opposite direction for each edge of Q ). ∞ ∞ Similarly, we have the double Q of the cyclic quiver Q. For a representation A of Q of dimension ∞ ( 1 vector α, we write θ(A) = i∞=0 αiθi, where θ := n j−=0 θj. Then by [46, Proposition 3.1], we ∞ − have that a representation!A of Q is semistable if a!nd only if θ(A) = 0 and θ(M) 0 for all ∞ ≥ M A, while A is stable if and only if A is semistable, and θ(M) > 0 for all 0 = M A with ⊂ 1 ⊂ M = A. 1 Lemma 4.68. If θ < 0 for all 0 i ! 1, then µ 1(0)ss = µ 1(0)s. i ≤ ≤ − − θ − θ 1 Proof. Let V be a θ–semistable representation of Q that lies in µ− (0). Suppose V is not stable. ∞ Then there exists a submodule U V such that θ(U) = 0 and U = 0, V . We define a representation ≤ 1 N of Q by N = U if U = 0 and N = V/U otherwise. In either case, θ(N) = 0 and N = 0, so ∞ ∞ ∞ N = 0 by the choice of θ, a contradiction.

1 The variety µ− (0) is not irreducible. By [30, Theorem 3.3.3], it has a decomposition

1 µ− (0) = M0 ∪ · · · ∪ Mn into irreducible components, each of which satisfies G( ) (this follows from the explicit Mi ⊂ Mi description of given in [30]). Our next aim is to show that the semistable points of µ 1(0) are Mi − all contained in a single . Mi Lemma 4.69. Suppose Mθ is smooth. Then there is some i with 0 i n with µ 1(0)ss . ≤ ≤ − θ ⊂ Mi Proof. Since µ 1(0)ss µ 1(0), we have µ 1(0)ss = n (µ 1(0)ss ). Let π : µ 1(0)ss Mθ − θ ⊂ − − θ ∪i=0 − θ ∩ Mi − θ → 1 ss 1 s be the quotient map. Since µ− (0)θ = µ− (0)θ, it follows from geometric invariant theory (see for 1 ss example [57, Definition 0.6, Theorem 1.10]) that π takes closed G–invariant subsets of µ− (0)θ to closed subsets of Mθ. So we have

θ 1 ss M = π(µ− (0) ). θ ∩ Mi Ai Since we have seen that Mθ is smooth, and Mθ is known to be connected by [38, Section 3.9], we see that Mθ is irreducible. Hence there exists i with

θ 1 ss M = π(µ− (0) ). θ ∩ Mi 95 Let x µ 1(0)ss. Then there exists y µ 1(0)ss with π(x) = π(y). So by the standard ∈ − θ ∈ Mi ∩ − θ geometric invariant theory description of the quotient Mθ (see [46, Section 2]), we get Gx Gy = ∅. ∩ 1 1 ss But Gx and Gy are closed orbits in µ− (0)θ , since x and y are stable by Lemma 4.68. Therefore, Gx = Gy and so x G( ) = as required. ∈ Mi Mi We spend the rest of this subsection proving Theorem 4.67. Let D D(RQ )G,θ and suppose ∈ ∞ θ that Pκ (D) = 0. Then Rκ(Dξ) = 0. Considering the restriction of Dξ to V := U U , we have ∩ ∞

1 G reg ζ− Dξζ ker(R : D(V ) D(h /W )). κ κ ∈ →

Therefore by [65, Corollary 4.5], we get

1 G ζ− Dξζ (D(V )τˆ(g)) . κ κ ∈

G,θ G,θ Therefore, D D(RQ ) (D(V )(τˆ + χκ)(g)) . So it suffices to show that ∈ ∞ ∩

G,θ G,θ (D(RQ )(τˆ + χκ)(g)) = (D(RQ ) D(V )(τˆ + χκ)(g)) . ∞ ∞ ∩

In order to prove the equality, we consider the filtration by order on D(RQ ), which induces a ∞ filtration on D(V ), the ring of differential operators on an open subset of RQ . Since D(RQ )(τˆ+ ∞ ∞ χκ)(g) is clearly contained in D(RQ ) D(V )(τˆ + χκ)(g), it is enough to show that ∞ ∩

G,θ G,θ gr(D(RQ )(τˆ + χκ)(g)) gr(D(RQ ) D(V )(τˆ + χκ)(g)) . ∞ ⊃ ∞ ∩

G,θ G,θ The right hand side is contained in gr(D(RQ )) gr(D(V )(τˆ + χκ)(g)) , so it suffices to show ∞ ∩ that G,θ G,θ (C[T ∗RQ ]µ∗(g)) (C[T ∗RQ ] C[T ∗V ]µ∗(g)) . (4.4) ∞ ⊃ ∞ ∩

The right hand side of (4.4) consists of regular (G, θ)–semiinvariant functions on T ∗RQ which ∞ vanish on T V µ 1(0). The cotangent bundle T V of V is just the set of points (X, Y, i, j) ∗ ∩ − ∗ ∈ 1 Rep(Q , ε) such that (X, i) V . In particular, T ∗V µ− (0) contains the point (X, 0, i, 0) for ∞ ∈ ∩ each (X, i) V , and hence is a nonempty open subset of µ 1(0). If p T V , then since θ < 0 for ∈ − ∈ ∗ i ( 1 θi θ0 all i, i=−1 si− s− is a regular (G, θ)–semiinvariant function on T ∗RQ with s(p) = 0. Therefore, ∞ 1 1 ss 1 1 ss by de#finition of µ− (0) , we get T ∗V µ− (0) µ− (0) . By Lemma 4.69, there is an i with θ ∩ ⊂ θ M µ 1(0)ss . So T V µ 1(0) is an open subset of the irreducible variety and therefore − θ ⊂ Mi ∗ ∩ − Mi any (G, θ)–semiinvariant function that vanishes on T V µ 1(0) must vanish on and hence ∗ ∩ − Mi 96 on µ 1(0)ss as well. But every (G, θ)–semiinvariant function vanishes on µ 1(0) µ 1(0)ss by − θ − \ − θ 1 definition. So any such function vanishes on the whole of µ− (0). Now by [35, Theorem 2.6], the ring 1 C[T ∗RQ ] C[µ− (0)] = ∞ C[T ∗RQ ]µ∗(g) ∞ is known to be reduced, and so it follows from Hilbert’s Nullstellensatz that any function that 1 vanishes on µ− (0) belongs to C[T ∗RQ ]µ∗(g), as required. This verifies (4.4) and completes the ∞ proof of Theorem 4.67.

97 Chapter 5

Diagonal coinvariants

The purpose of this chapter is to give an application of the results of the previous chapters. We will prove a result in classical invariant theory by using the Cherednik algebra. The fact that this is a purely commutative result indicates the usefulness of the Cherednik algebra. Let W be a complex reflection group acting faithfully on a finite-dimensional complex vector space h. Recall from Definition 4.29 that we may define the ring of coinvariants

C[h] CW := . C[h]W , + - It is known that CW is a finite–dimensional algebra isomorphic to CW as a W –module (see [45, Theorem 24-1]). There is interest in analogues of this construction with the representation h h ⊕ ∗ in place of h, see for example [40]. The ring

C[h h∗] DW := ⊕ W C[h h∗] , ⊕ + - is called the ring of diagonal coinvariants of W . The ring DW has a natural grading with deg(h∗) = 1 and deg(h) = 1. The following result was conjectured by Haiman in [40] and proved in Gordon [34]: − Theorem 5.1. [34] Let W be a finite Coxeter group of rank n with Coxeter number h and sign representation ε. Then there exists a W –stable quotient ring RW of DW with the properties:

n 1. dim(RW ) = (h + 1)

2. R is graded with Hilbert series t hn/2(1 + t + + th)n W − · · ·

W 3. The image of C[h] in RW is C[h]/ C[h] , + - 98 4. The character χ of the W –module R ε is χ(w) = (h + 1)dim ker(1 w) W ⊗ − In [34], a method for generalising this result to the groups W = G(!, 1, n) was outlined. It is the aim of the present chapter to carry this out, using the theory developed in the earlier chapters. The following result will be proved:

Theorem 5.2. Let W = G(!, 1, n) where ! > 1 and let h be the reflection representation of W .

Then there exists a W –stable quotient ring SW of DW with the properties:

n 1. dim(SW ) = (!n + 1)

n ((n) (n n 2. S is graded with Hilbert series t− − 2 (1 + t + + t ) W · · · W 3. The image of C[h] in SW is C[h]/ C[h] , + - 4. The character χ of S nh as a W –module is χ(w) = (!n + 1)dim ker(1 w) W ⊗ ∧ ∗ −

Theorem 5.1 is proved by obtaining RW as the associated graded module of a finite–dimensional module over the rational Cherednik algebra of W . The properties of this module are derived by studying the category for the rational Cherednik algebra. This is also the method that will be O used to prove Theorem 5.2.

Choose κ00, κ1, . . . , κ( 1 C such that − ∈ !κ + !(n 1)κ = 1 !n 1 − 00 − − and KZκ separates simples. Such a choice is possible by Theorem 3.24. Consider the parameters

κ[1], where recall we put κ[1]00 = κ00 + 1, κ[1]1 = κ1 + 1 and κ[1]i = κi for i > 1. Then

!κ[1] + !(n 1)κ[1] = 1. 1 − 00 − Applying Theorem 3.3, we get that there is a one-dimensional module L (triv) in , and as κ[1] Oκ[1] a W –module, L (triv) is trivial, so eL (triv) = 0. Recall the shift functor S1 : κ[1] κ[1] 1 κ Oκ[1] → Oκ constructed in Section 4.1.5. Since !κ + !(n 1)κ = 1 !n, we see from Theorem 3.3 that the 1 − 00 − − unique finite-dimensional simple object in the category has dimension (1 + !n)n. So by Lemma Oκ 1 4.22, e−L (triv) = 0. Therefore, Theorem 4.34 tells us that S is an equivalence. Therefore, 1 κ 1 κ S1(L (triv)) is a finite-dimensional simple module in . But KZ separates simples, so by κ κ[1] Oκ κ Theorem 3.7 we have 1 Sκ(Lκ[1](triv)) = Lκ(triv).

99 5.1 A quotient ring of the diagonal coinvariants

We follow the proof of [34, Section 5] to obtain the desired ring SW of Theorem 5.2. Choose κ as 1 above, and write L = Lκ(triv). Since L = Sκ(Lκ[1](triv)), we may write L as

ψ L = Hκe1− e H e (eLκ[1](triv)) (5.1) ⊗ 1− κ 1−

ψ where ( ) denotes twisting by the isomorphism eH e e−H e− coming from Proposition 4.1 − κ[1] → 1 κ 1 1 n ( ( (recall that e− = µ eµ− where µ = x (x x ) = δ). Write Λ for the one-dimensional 1 1 1 1 i=1 i · i

Lemma 5.3. There is a surjection of C[h h∗] W –modules: ⊕ ∗

ψ C[h h∗] W e1− [h h ]W grΛ # grL. ⊕ ∗ ⊗C ⊕ ∗

Proof. This follows from taking the associated graded of equation (5.1) and applying [36, Lemma 6.7, Lemma 6.8].

W The definition of the shift isomorphism from Proposition 4.1 implies that C[h h∗] acts on ⊕ + ψ n n grΛ by 0. Also, Ce− is the representation h∗ of W . Hence SW := grL h is a quotient 1 ∧ ⊗ ∧ ring of DW . Most of the properties of SW listed in Theorem 5.2 are consequences of the following theorem. Recall that D is graded with deg(h) = 1 and deg(h ) = 1 and that this grading is W − ∗ W –stable. In general, if M is a Z–graded module and χk : w tr(w M ) is the character of the 5→ | k th i k graded piece then the graded character of M is defined to be the formal power series i χit . ! Theorem 5.4. The graded character of grL = S nh is W ⊗ ∧ ∗ (n+1 n det (1 t w) n (( ) h∗ w t− − 2 | − 5→ det (1 tw) |h∗ − Proof. Recall the element h H from Chapter 1. By the proof of Lemma 3.3.4, z acts by 0 on ∈ κ the trivial representation triv = 0h of W and hence h also acts by 0 on the trivial representation ∧ ∗ triv. Hence, the eigenvalue of h on the subspace C[h]d triv M(triv) is d. ⊗ ⊂ By Theorem 3.3, the representation Lκ(triv) of Hκ is isomorphic as a graded W –module to n (n+1 1 U ⊗ where U(n+1 = C[u]/(u ), regarded as a representation of Z/!Z = s1 via s1(u) = ε− u, (n+1 , - and where S acts by permuting the factors of the tensor product. Since ui 0 i !n is a basis n { | ≤ ≤ } 100 of U , we may define distinct basis elements a by a := u((i 1)+1, 1 i n. Then the element (n+1 i i − ≤ ≤ nh v := σ Sn sgn(σ)aσ(1) aσ(2) aσ(n) affords the representation ∗ of W , and lies in degree ∈ ⊗ ⊗· · ·⊗ ∧ n + !n !(i 1) = n + ! n . Hence, hv = (n + ! n )v. i=1 − 2 2 n W!e define h$ = h n 6 !7 Hκ. We now cal6cu7late the graded character of L = Lκ(triv) with − − 2 ∈ 1 respect to the h$–eigenspace6s .7 By [16, Thm 2.3], Lκ(triv) admits a BGG-resolution:

2 n 0 L (triv) M(triv) M(h∗) M( h∗) M( h∗) 0. ← κ ← ← ← ∧ ← · · · ← ∧ ←

Hence, in the Grothendieck group of , O n i i [L (triv)] = ( 1) [M( h∗)] κ − ∧ &i=0 Now, again by the proof of Lemma 3.3.4, z acts by i(!n + 1) on ih , and hence the lowest − ∧ ∗ eigenvalue of h on M( ih ) is i(!n + 1) n ! n . Therefore, the graded character of M( ih ) is $ ∧ ∗ − − 2 ∧ ∗ n χ (w)ti(#n+1) n (( ) ih (n+1 i i((n+1) t 2 ∧ ∗ . But det (1 t w) = ( 1) χ i (w)t (this follows readily from − − det (1 tw) h∗ 6 7 i h∗ |h∗ − | − − ∧ diagonalising w) which gives the graded character!of Lκ(triv) with respect to the h$–eigenspaces as

n (n+1 n (( ) det h∗ (1 t w) w t− − 2 | − . 5→ det (1 tw) |h∗ −

By the definition of the diagonal coinvariant ring DW , there is a unique copy of the trivial rep- resentation in D , which lies in degree 0, and hence a unique copy of nh in S nh , and W ∧ ∗ W ⊗ ∧ ∗ hence a unique copy of nh in L (triv) (since S nh = grL, which is isomorphic to L as a ∧ ∗ κ W ⊗ ∧ ∗ W –module). The unique copy of nh in L (triv) must be spanned by v. But h v = 0 and hence h ∧ ∗ κ $ $ n must act by 0 on the element e− 1 L which affords the unique copy of h in L. But because 1 ⊗ ∈ ∧ ∗ the grading induced by h on L gives e− 1 degree 0 and x h degree 1, and y h degree 1, $ 1 ⊗ ∈ ∗ ∈ − we see that grL has the same graded character as Lκ(triv), which proves the theorem.

Proof of Theorem 5.2 (3)

This is similar to a proof in [34, Section 5]. Let us recall what it means for a finite-dimensional graded algebra to satisfy Poincar´e duality.

1 the point of the proof is that, while equation 5.1 says that L and Lκ(triv) are isomorphic as W –modules, we must compare them as graded W –modules.

101 N Definition 5.5. Let A = Ai be a finite-dimensional graded C–algebra with Ai = 0, i > N. ⊕i=0 Then A is said to satisfy Poincar´e duality if dim(A ) = 1 and for all 0 r N, the multiplication N ≤ ≤ pairing

Ar AN r AN × − → is nonsingular (that is, if x Ar and xAN r = 0 then x = 0). ∈ − It is well-known (see for example [44]) that the ring of coinvariants A = C[h]/ C[h]W satisfies , + - Poincar´e duality. Therefore the highest degree graded component of A, which by [45, Theorem 20-3] lies in degree (d 1) where the d are the degrees of the fundamental invariants of W , is an ideal i i − i of A which is!contained in every nonzero ideal. This ideal is called the socle of A. In the case of W = G(!, 1, n) the degrees are !, 2!, . . . !n by [11, Table 1], so the socle lies in degree 1 !n(n+1) n. 2 − ψ W The image of C[h] in SW corresponds to the subspace C[h]e− Λ of L. If p C[h] e− then 1 ⊗ ∈ + 1 ψ ψ ψ p Λ = e−pe− Λ = e− e−pe−Λ = e− epe Λ = 0. ⊗ 1 1 ⊗ 1 ⊗ 1 1 1 ⊗ ·

W ψ W Thus the ideal generated by C[h] annihilates e− Λ . On the other hand, the quotient C[h]/ C[h] + 1 ⊗ , + - contains a unique (up to scalar) element of maximal degree 1 !n(n + 1) n, say q. The space Cq 2 − W ψ is the socle of C[h]/ C[h] . We claim qe− Λ = 0. By the PBW theorem, any element of , + - 1 ⊗ 1 Hκ can be written as a sum of terms of the form p wp+ where p C[h∗], p+ C[h] and − − ∈ ∈ ψ w W . Since p and w do not increase degree, it would follow if qe1− Λ were zero, then L ∈ − ⊗ could have no subspace in degree 1 !n(n + 1) n. But the Hilbert series of L has highest order 2 − (n2 n ( 1 n(n 1) 1 (n(n+1) n ψ ψ term t − − 2 − = t 2 − . Thus qe− Λ is non–zero and C[h]e− Λ is isomorphic to 1 ⊗ 1 ⊗ W ψ (C[h]/ C[h] )e− Λ . This proves Theorem 5.2 (3). , + - 1 ⊗ $ In the case W = Bn = G(2, 1, n), Gordon has previously constructed a ring RW in [34], having the same properties as SW (see Theorem 5.1). As mentioned above, our proof is modelled on [34], and we can check that in the Bn case RW and SW coincide.

Corollary 5.6. If W = Bn then RW = SW .

Proof. Haiman, in [39, Conjecture 7.25], constructs a ring R˜W which is the quotient of DW by the largest ideal J such that J (D ) = 0. He states that R˜ has the same Hilbert series as R ∩ W ε W W and is isomorphic to RW as a W –module. He proves that these conditions imply that R˜W = RW .

But the ring SW also has the same Hilbert series as RW and is isomorphic to RW as a W –module.

Hence, R˜W = SW by the same proof. Therefore, SW = RW .

102 Remark 5.7. We close with a remark about the meaning of Theorem 5.2. Note that the diagonal coinvariant ring DW is in fact a bigraded ring, with h in degree (1, 0) and h∗ in degree (0, 1). It is an interesting problem to calculate the Frobenius series of DW and its quotient SW , that is, the series, i j dim((SW )(i,j))p q (i,j) 2 &∈Z where p and q are indeterminates. Theorem 5.2 would follow from a description of the Frobenius series and bigraded character of SW analogous to that given in the ! = 1 case in [39, Theorem 4.2.5], 1 since the Hilbert series can be obtained from the Frobenius series by setting q = p− . It is not clear how the Frobenius series of SW could be calculated using Cherednik algebra methods. Nevertheless, the existence and nice combinatorial properties of the ring SW may be seen as evidence that an analogue of the combinatorial part of Haiman’s n! theorem may hold for the groups G(!, 1, n).

103 Bibliography

[1] S. Ariki. On the semi-simplicity of the Hecke algebra of (Z/rZ) S . J. Algebra, 169(1):216–225, ! n 1994.

[2] S. Ariki and K. Koike. A Hecke algebra of (Z/rZ) S and construction of its irreducible ! n representations. Adv. Math., 106(2):216–243, 1994.

[3] S. Ariki and A. Mathas. The number of simple modules of the Hecke algebras of type G(r, 1, n). Math. Z., 233(3):601–623, 2000.

[4] D. J. Benson. Polynomial invariants of finite groups, volume 190 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1993.

[5] Y. Berest and O. Chalykh. Quasi-invariants of complex reflection groups. In preparation.

[6] Y. Berest, P. Etingof, and V. Ginzburg. Cherednik algebras and differential operators on quasi-invariants. Duke Math. J., 118(2):279–337, 2003.

[7] Y. Berest, P. Etingof, and V. Ginzburg. Finite-dimensional representations of rational Chered- nik algebras. Int. Math. Res. Not., (19):1053–1088, 2003.

[8] J.-E. Bjo¨rk. The Auslander condition on Noetherian rings. In S´eminaire d’Alg`ebre Paul Dubreil et Marie-Paul Malliavin, 39`eme Ann´ee (Paris, 1987/1988), volume 1404 of Lecture Notes in Math., pages 137–173. Springer, Berlin, 1989.

[9] A. Borel, P.-P. Grivel, B. Kaup, A. Haefliger, B. Malgrange, and F. Ehlers. Algebraic D- modules, volume 2 of Perspectives in Mathematics. Academic Press Inc., Boston, MA, 1987.

[10] M. Boyarchenko. Quantization of minimal resolutions of Kleinian singularities. Adv. Math., to appear. arXiv:math/0505165.

104 [11] M. Brou´e, G. Malle, and R. Rouquier. Complex reflection groups, braid groups, Hecke algebras. J. Reine Angew. Math., 500:127–190, 1998.

[12] K. A. Brown. Symplectic reflection algebras. In Proceedings of the All Ireland Algebra Days, 2001 (Belfast), number 50, pages 27–49, 2002.

[13] K. A. Brown and K. R. Goodearl. Lectures on algebraic quantum groups. Advanced Courses in Mathematics. CRM Barcelona. Birkh¨auser Verlag, Basel, 2002.

[14] Heiko Cassens and Peter Slodowy. On Kleinian singularities and quivers. In Singularities (Oberwolfach, 1996), volume 162 of Progr. Math., pages 263–288. Birkha¨user, Basel, 1998.

[15] T. Chmutova. Representations of the rational Cherednik algebras of dihedral type. J. Algebra, to appear. arXiv:math.RT/0405383.

[16] T. Chmutova and P. Etingof. On some representations of the rational Cherednik algebra. Represent. Theory, 7:641–650 (electronic), 2003.

[17] E. Cline, B. Parshall, and L. Scott. Finite-dimensional algebras and highest weight categories. J. Reine Angew. Math., 391:85–99, 1988.

[18] W. Crawley-Boevey. Decomposition of Marsden-Weinstein reductions for representations of quivers. Compositio Math., 130(2):225–239, 2002.

[19] P. Deligne. E´quations diff´erentielles a` points singuliers r´eguliers. Springer-Verlag, Berlin, 1970. Lecture Notes in Mathematics, Vol. 163.

[20] C. Dez´el´ee. Repr´esentations de dimension finie de l’alg`ebre de Cherednik rationnelle. Bull. Soc. Math. France, 131(4):465–482, 2003.

[21] R. Dipper, G. James, and A. Mathas. Cyclotomic q-Schur algebras. Math. Z., 229(3):385–416, 1998.

[22] C. F. Dunkl. Differential-difference operators associated to reflection groups. Trans. Amer. Math. Soc., 311(1):167–183, 1989.

[23] C. F. Dunkl and E. M. Opdam. Dunkl operators for complex reflection groups. Proc. London Math. Soc. (3), 86(1):70–108, 2003.

105 [24] D. Eisenbud. Commutative algebra with a view toward algebraic geometry, volume 150 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1995.

[25] K. Erdmann and D. K. Nakano. Representation type of Hecke algebras of type A. Trans. Amer. Math. Soc., 354(1):275–285 (electronic), 2002.

[26] P. Etingof and V. Ginzburg. Symplectic reflection algebras, Calogero-Moser space, and de- formed Harish-Chandra homomorphism. Invent. Math., 147(2):243–348, 2002.

[27] Walter Ferrer Santos and Alvaro Rittatore. Actions and invariants of algebraic groups, volume 269 of Pure and Applied Mathematics (Boca Raton). Chapman & Hall/CRC, Boca Raton, FL, 2005.

[28] P. J. Freyd. Abelian categories. Repr. Theory Appl. Categ., (3):1–190 (electronic), 2003.

[29] Wee Liang Gan. Chevalley restriction theorem for the cyclic quiver. Manuscripta Math., 121(1):131–134, 2006.

[30] Wee Liang Gan and Victor Ginzburg. Almost-commuting variety, -modules, and Cherednik D algebras. IMRP Int. Math. Res. Pap., pages 26439, 1–54, 2006. With an appendix by Ginzburg.

[31] V. Ginzburg. On primitive ideals. Selecta Math. (N.S.), 9(3):379–407, 2003.

[32] V. Ginzburg, I. Gordon, and J. T. Stafford. In preparation.

[33] V. Ginzburg, N. Guay, E. Opdam, and R. Rouquier. On the category for rational Cherednik O algebras. Invent. Math., 154(3):617–651, 2003.

[34] I. Gordon. On the quotient ring by diagonal invariants. Invent. Math., 153(3):503–518, 2003.

[35] I. Gordon. A remark on rational Cherednik algebras and differential operators on the cyclic quiver. Glasg. Math. J., 48(1):145–160, 2006.

[36] I. Gordon and J. T. Stafford. Rational Cherednik algebras and Hilbert schemes. Adv. Math., 198(1):222–274, 2005.

[37] I. Gordon and J. T. Stafford. Rational Cherednik algebras and Hilbert schemes. II. Represen- tations and sheaves. Duke Math. J., 132(1):73–135, 2006.

106 [38] Iain Gordon. Quiver varieties, category for rational Cherednik algebras, and Hecke algebras, O 2007. arXiv.org:math/0703150.

[39] M. Haiman. Combinatorics, symmetric functions, and Hilbert schemes. In Current develop- ments in mathematics, 2002, pages 39–111. Int. Press, Somerville, MA, 2003.

[40] Mark D. Haiman. Conjectures on the quotient ring by diagonal invariants. J. Algebraic Combin., 3(1):17–76, 1994.

[41] J. Harris. Algebraic geometry. A first course, volume 133 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1992.

[42] R. Hartshorne. Algebraic geometry. Springer-Verlag, New York, 1977. Graduate Texts in Mathematics, No. 52.

[43] M. P. Holland. Quantization of the Marsden-Weinstein reduction for extended Dynkin quivers. Ann. Sci. E´cole Norm. Sup. (4), 32(6):813–834, 1999.

[44] R. Kane. Poincar´e duality and the ring of coinvariants. Canad. Math. Bull., 37(1):82–88, 1994.

[45] R. Kane. Reflection groups and invariant theory. CMS Books in Mathematics/Ouvrages de Math´ematiques de la SMC, 5. Springer-Verlag, New York, 2001.

[46] A. D. King. Moduli of representations of finite-dimensional algebras. Quart. J. Math. Oxford Ser. (2), 45(180):515–530, 1994.

[47] G. R. Krause and T. H. Lenagan. Growth of algebras and Gelfand-Kirillov dimension, vol- ume 22 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, revised edition, 2000.

[48] J. Le Potier. Lectures on vector bundles, volume 54 of Cambridge Studies in Advanced Math- ematics. Cambridge University Press, Cambridge, 1997. Translated by A. Maciocia.

[49] T. Levasseur. Rel`evements d’op´erateurs diff´erentiels sur les anneaux d’invariants. In Opera- tor algebras, unitary representations, enveloping algebras, and invariant theory (Paris, 1989), volume 92 of Progr. Math., pages 449–470. Birkha¨user Boston, Boston, MA, 1990.

107 [50] T. Levasseur and J. T. Stafford. Invariant differential operators and an homomorphism of Harish-Chandra. J. Amer. Math. Soc., 8(2):365–372, 1995.

[51] S. Lyle and A. Mathas. Blocks of affine and cyclotomic Hecke algebras. arXiv:math.RT/0607451.

[52] A. Mathas. Simple modules of Ariki-Koike algebras. In Group representations: cohomology, group actions and topology (Seattle, WA, 1996), volume 63 of Proc. Sympos. Pure Math., pages 383–396. Amer. Math. Soc., Providence, RI, 1998.

[53] A. Mathas. Iwahori-Hecke algebras and Schur algebras of the symmetric group, volume 15 of University Lecture Series. American Mathematical Society, Providence, RI, 1999.

[54] A. Mathas. The representation theory of the Ariki-Koike and cyclotomic q-Schur algebras. In Representation theory of algebraic groups and quantum groups, volume 40 of Adv. Stud. Pure Math., pages 261–320. Math. Soc. Japan, Tokyo, 2004.

[55] J. C. McConnell and J. C. Robson. Noncommutative Noetherian rings. Pure and Applied Mathematics (New York). John Wiley & Sons Ltd., Chichester, 1987. With the cooperation of L. W. Small, A Wiley-Interscience Publication.

[56] R. V. Moody and A. Pianzola. Lie algebras with triangular decompositions. Canadian Math- ematical Society Series of Monographs and Advanced Texts. John Wiley & Sons Inc., New York, 1995. , A Wiley-Interscience Publication.

[57] David Mumford. Geometric invariant theory. Ergebnisse der Mathematik und ihrer Grenzge- biete, Neue Folge, Band 34. Springer-Verlag, Berlin, 1965.

[58] Hiraku Nakajima. Quiver varieties and Kac-Moody algebras. Duke Math. J., 91(3):515–560, 1998.

[59] A. Oblomkov. Deformed Harish-Chandra homomorphism for the cyclic quiver. arXiv:math/0504395.

[60] E. M. Opdam. Complex reflection groups and fake degrees. arxiv:math/9808026.

[61] D. S. Passman. Infinite crossed products, volume 135 of Pure and Applied Mathematics. Aca- demic Press Inc., Boston, MA, 1989.

108 [62] N. Popescu and L. Popescu. Theory of categories. Martinus Nijhoff Publishers, The Hague, 1979.

[63] J. J. Rotman. An introduction to homological algebra, volume 85 of Pure and Applied Mathe- matics. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1979.

[64] R. Rouquier. q-Schur algebras and complex reflection groups I. arXiv:math.RT/0509252.

[65] Gerald W. Schwarz. Lifting differential operators from orbit spaces. Ann. Sci. E´cole Norm. Sup. (4), 28(3):253–305, 1995.

[66] G. C. Shephard and J. A. Todd. Finite unitary reflection groups. Canadian J. Math., 6:274– 304, 1954.

[67] J. T. Stafford. Auslander-regular algebras and maximal orders. J. London Math. Soc. (2), 50(2):276–292, 1994.

[68] J. R. Stembridge. On the eigenvalues of representations of reflection groups and wreath prod- ucts. Pacific J. Math., 140(2):353–396, 1989.

[69] R. Vale. On category for the rational Cherednik algebra of G(m, 1, n): the almost semisimple O case. Submitted. arXiv.org:math/0606523.

[70] R. Vale. Rational Cherednik algebras and diagonal coinvariants of G(m, p, n). J. Algebra, to appear. arXiv:math.RT/0505416.

[71] P. Vanhaecke. Integrable systems in the realm of algebraic geometry, volume 1638 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1996.

109