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On Category O for the Rational Cherednik Algebra of the Complex Reflection Group

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On Category O for the Rational Cherednik Algebra of the Complex Reflection Group 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.
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