Dunkl Operators as Covariant Derivatives in a Quantum Principal Bundle

TL;DR

This paper provides a geometric interpretation of Dunkl operators as covariant derivatives on a quantum principal bundle associated with Coxeter groups, offering new insights into their properties and commutativity.

Contribution

It constructs a quantum principal bundle for Coxeter groups and identifies Dunkl operators as covariant derivatives, linking harmonic analysis with geometric structures.

Findings

Dunkl operators are covariant derivatives in a quantum principal bundle.

The connection on this bundle has zero curvature.

A new proof of Dunkl operators' commutativity is provided.

Abstract

A quantum principal bundle is constructed for every Coxeter group acting on a finite-dimensional Euclidean space $E$, and then a connection is also defined on this bundle. The covariant derivatives associated to this connection are the Dunkl operators, originally introduced as part of a program to generalize harmonic analysis in Euclidean spaces. This gives us a new, geometric way of viewing the Dunkl operators. In particular, we present a new proof of the commutativity of these operators among themselves as a consequence of a geometric property, namely, that the connection has curvature zero.