Dunkl operators for complex reflection groups

TL;DR

This paper introduces Dunkl operators for complex reflection groups, exploring their algebraic properties, deformations, and singular parameters, with implications for polynomial modules and Cherednik algebras.

Contribution

It defines Dunkl operators for complex reflection groups and analyzes their associated algebraic structures and singular parameters, extending previous real reflection group theory.

Findings

Defined Dunkl operators for complex reflection groups.

Described singular parameters explicitly for G(m,p,N).

Connected Dunkl operators to polynomial modules and Cherednik algebras.

Abstract

Dunkl operators for complex reflection groups are defined in this paper. These commuting operators give rise to a parametrized family of deformations of the polynomial De Rham complex. This leads to the study of the polynomial ring as a module over the "rational Cherednik algebra", and a natural contravariant form on this module. In the case of the imprimitive complex reflection groups G(m,p,N), the set of singular parameters in the parameter family of these structures is described explicitly, using the theory of nonsymmetric Jack polynomials.