Dunkl Operators for Arbitrary Finite Groups

TL;DR

This paper extends Dunkl operators from finite Coxeter groups to arbitrary finite groups using non-commutative geometry, introducing quantum principal bundles and cyclic Dunkl operators with notable properties.

Contribution

It generalizes Dunkl operators to all finite groups via non-commutative geometry, including quantum principal bundles and cyclic Dunkl connections.

Findings

Dunkl operators are generalized to arbitrary finite groups.

Introduction of cyclic Dunkl connections and operators.

Establishment of the zero curvature property.

Abstract

The Dunkl operators associated to a necessarily finite Coxeter group acting on a Euclidean space are generalized to any finite group using the techniques of non-commutative geometry, as introduced by the authors to view the usual Dunkl operators as covariant derivatives in a quantum principal bundle with a quantum connection. The definitions of Dunkl operators and their corresponding Dunkl connections are generalized to quantum principal bundles over quantum spaces which possess a classical finite structure group. We introduce cyclic Dunkl connections and their cyclic Dunkl operators. Then we establish a number of interesting properties of these structures, including the characteristic zero curvature property. Particular attention is given to the example of complex reflection groups, and their naturally generalized siblings called groups of Coxeter type.