Dunkl Operators and Quasi-invariants of Complex Reflection Groups

TL;DR

This paper introduces Dunkl operators and quasi-invariants within the context of complex reflection groups, providing an accessible overview of their relation and applications in the representation theory of rational Cherednik algebras.

Contribution

It offers an accessible introduction to Dunkl operators and quasi-invariants, highlighting their connection and recent applications in the representation theory of rational Cherednik algebras.

Findings

Relation between Dunkl operators and quasi-invariants explained

Applications to representation theory of rational Cherednik algebras discussed

Lecture notes aimed at graduate students and nonexperts

Abstract

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 nonexperts in representation theory. In these notes, we tried to preserve an informal style, even at the expense of making imprecise claims and sacrificing rigor.