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.