Contemporary Mathematics

Volume 583, 2012

http://dx.doi.org/10.1090/conm/583/11571

Dunkl operators and quasi-invariants of complex reflection

groups

Yuri Berest and Peter Samuelson

Abstract. In these notes we give an introduction to representation theory

of rational Cherednik algebras associated to complex reflection groups. We

discuss applications of this theory in the study of finite-dimensional represen-

tations of the Hecke algebras and polynomial quasi-invariants of these groups.

1. Introduction

These are lecture notes of a minicourse given by the first author at the summer

school on Quantization at the University of Notre Dame in June 2011. The notes

were written up and expanded by the second author who took the liberty of adding

a few interesting results and proofs from the literature. In a broad sense, our goal

is to give an introduction to representation theory of rational Cherednik algebras

and some of its recent applications. More specifically, we focus on the two concepts

featuring in the title (Dunkl operators and quasi-invariants) and explain the relation

between them. The course was originally designed for graduate students and non-

experts in representation theory. In these notes, we tried to preserve an informal

style, even at the expense of making imprecise claims and sacrificing rigor.

The interested reader may find more details and proofs in the following ref-

erences. The original papers on representation theory of the rational Cherednik

algebras are [EG02b], [BEG03a, BEG03b], [DO03] and [GGOR03]; surveys

of various aspects of this theory can be found in [Rou05], [Eti07], [Gor08] and

[Gor10]. The quasi-invariants of Coxeter (real reflection) groups first appeared in

the work of O. Chalykh and A. Veselov [CV90, CV93] (see also [VSC93]); the

link to the rational Cherednik algebras associated to these groups was established

in [BEG03a]; various results and applications of Coxeter quasi-invariants can be

found in [EG02a], [FV02], [GW04], [GW06], [BM08]; for a readable survey, we

refer to [ES03].

The notion of a quasi-invariant for a general complex reflection group was

introduced in [BC11]. This last paper extends the results of [BEG03a], unifies

the proofs and gives new applications of quasi-invariants in representation theory

and noncommutative algebra. It is the main reference for the present lectures.

2010 Mathematics Subject Classification. Primary 16S38; Secondary 14A22, 17B45.

The first author was supported in part by NSF Grant DMS 09-01570.

c 2012 American Mathematical Society

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