Properties of Cherednik algebras and graded Hecke algebras

142 Pages·2006·0.78 MB·English

Properties of Cherednik algebras
and graded Hecke algebras
by
Katrin Eva Gehles
A thesis submitted to the
Faculty of Information and Mathematical Sciences
at the University of Glasgow
for the degree of
Doctor of Philosophy
April 2006
(cid:13)c K E Gehles 2006
i
For my family
Acknowledgements
Above all I am grateful to my supervisors Ken A. Brown and Iain Gordon for their con-
tinuous assistance and encouragement. Their guidance and advice has been invaluable.
Myspecialthanksgotomyfamilyfortheirloveandsupport, whichkeptmegrounded.
I also wish to thank all the postgraduate students and friends I met along the way, most
notably Mo, without whom my stay in Glasgow could not have been as enjoyable.
Finally, I thank EPSRC and the University of Glasgow for their financial support
throughout my studies.
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Summary
The objects of investigation in this thesis are four distinct but related types of algebras,
namely the Cherednik algebras and graded Hecke algebras of the title. There are two
aspects which connect these four kinds of algebras. Firstly, they are all non-commutative
deformations of some skew group algebras. Secondly, symplectic reflection algebras ap-
pearinallcasesascertainspecialisationsordegenerationsofthealgebrasinquestion. Our
interest lies in examining ring-theoretic properties of these algebras with a view towards
applications to geometric questions.
InChapter1wepresentsomebasicdefinitions andbackgroundmaterial, whichwewill
refer to in the remainder of this document.
Chapter 2 is dedicated to introducing the three types of algebras that are subsumed
underthenameCherednikalgebras. ThesearethedoubleaffineHeckealgebra,thetrigono-
metric double affine Hecke algebra and the rational Cherednik algebra, see [Che04] for an
overview. We begin a ring-theoretic study of the double affine Hecke algebra and the
trigonometric double affine Hecke algebra. In particular, we provide at least partial an-
swers to the questions whether these rings are noetherian and prime. We also investigate
their global dimensions. Cherednik algebras all display a dichotomy in their behaviour
depending on the specialisation of certain parameters. The parameter specialisations for
whichtheCherednikalgebrasarefinitelygeneratedovertheircentresandthusPIalgebras
are of particular interest to us, because in these cases connections to algebraic geometry
come to the forefront. Therefore, we investigate the ring-theoretic properties of the PI
Cherednik algebras in more detail and provide faithful representations for these algebras.
Our description of these representations is very explicit and makes statements by [Obl04]
precise. From these representations we obtain embeddings of the PI Cherednik algebras
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into skew group algebras.
The second aim of our study of Cherednik algebras is to understand the connections
between the PI cases of these three kinds of algebras. In Chapter 3 we concentrate on the
Cherednik algebras attached to the root system of type A . In this example we are able
1
to set up a framework of relations between the PI Cherednik algebras. The processes of
degeneration and completion, which make up this framework, are described in sufficient
detailtoenableustotransferacrossgeometricinformationfromthePIrationalCherednik
algebra to the PI double affine Hecke algebra. Following Lusztig’s work on affine Hecke
algebras in [Lus89] we first consider the process of degeneration: using certain filtrations
of the Cherednik algebras and the corresponding associated graded algebras one can pass
from the double affine Hecke algebra via the trigonometric double affine Hecke algebra to
the rational Cherednik algebra. Secondly, again extending the work in [Lus89], we show
that the PI Cherednik algebras of type A are isomorphic after completing at suitable
1
ideals. These isomorphisms turn out to be isomorphisms between the completions of the
skew group algebras into which we embedded the PI Cherednik algebras in the previous
chapter. Chapter 3 concludes with an explanation of how this framework of degeneration
and completion can be used to answer geometric questions about these PI algebras.
Finally, Chapter 4 contains our work on graded Hecke algebras, which were defined
by [Dri86] and [RS03]. We show that, similarly to the Cherednik algebras, graded Hecke
algebras are finitely generated over their centres precisely when certain parameters are
specialised to zero. Graded Hecke algebras can be viewed as generalisations of symplectic
reflection algebras and our result generalises work by Etingof and Ginzburg for symplectic
reflection algebras, [EG02]. After deriving some of the basic properties of graded Hecke
algebras we introduce the theory of PBW deformations to motivate the definition of these
algebras. We then proceed by examining the spherical subalgebra of a graded Hecke
algebra, which plays a crucial role in proving our main theorem. In our treatment we
follow the strategy used in [EG02] to study symplectic reflection algebras. The concept
of a Poisson bracket and the theory of orbit varieties feature in the proofs of further
preliminary results. The understanding we gained from considering graded Hecke algebras
as PBW deformations then allows us to deduce the main theorem.
Statement
This thesis is submitted in accordance with regulations for the degree of Doctor of Philos-
ophy in the University of Glasgow. No part of this thesis has previously been submitted
by me for a degree at this or any other university.
Chapter 1 covers definitions and basic results. Sections 3.2, 3.3, 4.3 - 4.5 consist of
the author’s original work. Further original work by the author is contained in Chapter 2
unless referenced otherwise.
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Contents
Acknowledgements ii
Summary iii
Statement v
Introduction 1
1 Notation and preliminaries 12
1.1 Localisations of non-commutative rings . . . . . . . . . . . . . . . . . . . . . 12
1.2 Associated graded algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3 PI algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4 Smoothness and Azumaya algebras . . . . . . . . . . . . . . . . . . . . . . . 17
1.5 Completions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.6 Skew group algebras and invariant rings . . . . . . . . . . . . . . . . . . . . 19
2 Cherednik algebras 23
2.1 The double affine Hecke algebra . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.1.1 Definition and first properties . . . . . . . . . . . . . . . . . . . . . . 24
2.1.2 The specialisation q = 1 . . . . . . . . . . . . . . . . . . . . . . . . . 33
e
2.2 The trigonometric double affine Hecke algebra . . . . . . . . . . . . . . . . . 40
2.2.1 Definition and first properties . . . . . . . . . . . . . . . . . . . . . . 40
2.2.2 The specialisation q = 0 . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.3 The rational Cherednik algebra . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.3.1 Definition and first properties . . . . . . . . . . . . . . . . . . . . . . 57
2.3.2 The specialisation q = 0 . . . . . . . . . . . . . . . . . . . . . . . . . 59
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CONTENTS vii
3 Equivalences of PI Cherednik algebras of type A 63
1
3.1 The root system of type A . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
1
3.1.1 The DAHA of type A . . . . . . . . . . . . . . . . . . . . . . . . . . 64
1
3.1.2 The trigonometric DAHA of type A . . . . . . . . . . . . . . . . . . 65
1
3.1.3 The RCA of type A . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
1
3.2 Degenerations for type A . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
1
3.2.1 From the DAHA to the trigonometric DAHA . . . . . . . . . . . . . 67
3.2.2 From the trigonometric DAHA to the RCA . . . . . . . . . . . . . . 70
3.3 Completions for type A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
1
3.3.1 FromtheRCAtothetrigonometricDAHA-atidealscorresponding
to the origin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.3.2 From the RCA to the trigonometric DAHA - at arbitrary maximal
ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.3.3 From the trigonometric DAHA to the DAHA - at ideals correspond-
ing to the origin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.3.4 From the trigonometric DAHAto the DAHA- at arbitrarymaximal
ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.3.5 Geometric application . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4 Graded Hecke algebras 93
4.1 Definition and first properties . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.2 PBW deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.3 The spherical subalgebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.4 Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.5 Proof of the main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
A Embeddings for PI Cherednik algebras of type A 128
1
References 130
Introduction
Our research into the properties of Cherednik algebras and graded Hecke algebras was
inspired by results of first Etingof and Ginzburg in [EG02], then Gordon in [Gor03a], and
finallyOblomkovin[Obl04]. Inthesepaperstherepresentationtheoryofnon-commutative
algebras is employed to tackle questions in algebraic geometry.
Symplectic reflection algebras
Traditionally, if G is a finite group acting on a smooth complex affine algebraic variety
X, then the orbit variety X/G can be studied via its coordinate ring C[X]G, the ring
of G-invariant regular functions on X. This study is part of geometric invariant theory.
One of the natural questions to ask is whether the variety X/G is smooth. When X is
a vector space and G acts linearly, the Chevalley-Shephard-Todd theorem implies that
X/G is smooth if and only if G is generated by complex reflections in its action on X.
If the variety X/G is singular, a possible next step is to find a smooth deformation of
the singularities. Recently, for example in [EG02], researchers have investigated the skew
group algebra C[X]∗G in order to study the G-equivariant geometry of X. Here C[X]
denotes the coordinate ring of X. The centre of C[X]∗G is precisely the ring C[X]G. In
the case of a symplectic vector space V over C and G ⊆ GL(V) a group that preserves the
symplectic form on V, Etingof and Ginzburg study non-commutative deformations of the
algebra C[V]∗G to great effect in [EG02]. They call the deformation algebras that occur
symplectic reflection algebras (SRA). In some cases their techniques indeed lead to the
discovery of smooth deformations of the singular symplectic varieties V/G. This work is
extended in [Gor03a] and produces an almost complete classification of the cases in which
a smooth deformation exists. There has also been progress on making the relationship
between the deformations and desingularisations of V/G precise, see for example [GS04].
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CONTENTS 2
The algebras that we will explore in the following all demonstrate similar behaviour to
SRAs. In fact, Cherednik algebras and graded Hecke algebras are all directly related to
SRAs. Some of these algebras are generalisations or special cases of SRAs, whereas others
can be degenerated to a particular kind of SRA. Therefore we will expand a little on SRAs
before we examine the actual subjects of this thesis.
An SRA, say H, is a filtered algebra and is defined in such a way that grH ∼= C[V]∗G,
where grH denotes the associated graded algebra of H. This fact is called the PBW
property of SRAs. SRAs depend on some deformation parameters and the PBW result
ensures that the algebras do not collapse for any specialisations of these parameters. The
characteristics of SRAs differ vastly depending on the specialisations of the deformation
parameters, oneofwhichwedenotebyq. Inparticular, ifq = 0theSRAhasalargecentre
and the algebra is finitely generated over its centre. It is an old result that an algebra
which is finitely generated over its centre satisfies a polynomial identity. Therefore we
call this specialisation the PI case. When the parameter q is specialised to other values,
however, the centres of the SRAs are trivial. Thus it is the PI case in which questions
about the geometric structure of the variety corresponding to the centre of an SRA arise.
See [Bro02] for a survey of these results.
Akeyresultin[EG02]-withaviewtonon-commutativegeometry-isthatthecentreof
aPISRAisadeformationofthecoordinateringofthevarietyV/G. Theresultsin[EG02]
and [Gor03a] use representation theory to determine whether or not this deformation is
smooth. Firstly, if an SRA H is finitely generated over its centre, it can be shown, using
a generalised version of Schur’s lemma, that all simple H-modules are finite dimensional.
Secondly, there exists an upper bound for the dimensions of the simple H-modules and
the maximal dimension of a simple H-module is called the PI degree of H. Etingof and
Ginzburg showed in [EG02] that this number equals |G|, the order of the group. Finally,
if there exists a simple H-module of dimension less than |G|, then the algebra H is not
Azumaya. In the case of a PI SRA it is shown in [EG02] that the algebra H is Azumaya
if and only if the variety corresponding to its centre is smooth.
We will see that the Cherednik algebras and graded Hecke algebras are not only also
non-commutative deformations of skew group algebras, but they display the same di-
chotomy of behaviour for distinct specialisations of the deformation parameters as the
SRAs. Moreover, PBW results, analogous to the one for SRAs, exist in all our cases.