Dunkl operators for arbitrary finite groups

Abstract

The Dunkl operators associated with 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.