From Dirac to Dunkl Operators through Symmetry Reduction

Abstract

This paper presents a geometric and analytic derivation of Dirac-Dunkl operators as symmetry reductions of the flat Dirac operator on Euclidean space. Starting from the standard Dirac operator, we restrict to a fundamental Weyl chamber of a finite Coxeter group equipped with the Heckman-Opdam measure, and determine the necessary drift and reflection corrections that ensure formal skew-adjointness under this weighted geometry. This procedure naturally reproduces the Dunkl operators as the unique first-order deformations compatible with reflection symmetry, whose Clifford contraction defines the Dirac-Dunkl operator and whose square yields the Dunkl Laplacian. We then extend the construction to include arbitrary unitary representations of the reflection group, obtaining representation-dependent Dirac-Dunkl operators that act on spinor- or matrix-valued functions. In the scalar and sign representations, these operators recover respectively the bosonic and fermionic Calogero-Moser systems, while higher-dimensional representations give rise to multi-component spin-Calogero models. The resulting framework unifies analytic, geometric, and representation-theoretic aspects of Dirac and Dunkl operators under a single symmetry-reduction principle.