Finite-dimensional representations of the symmetry algebra of the dihedral Dunkl--Dirac operator

TL;DR

This paper studies the symmetry algebra of the Dunkl--Dirac operator, classifies its finite-dimensional irreducible representations for certain root systems, and constructs polynomial solutions as explicit examples.

Contribution

It provides a detailed classification of finite-dimensional representations of the symmetry algebra associated with the Dunkl--Dirac operator for reducible root systems of rank three.

Findings

All irreducible finite-dimensional representations identified.

Conditions for unitarity established.

Polynomial solutions constructed explicitly.

Abstract

The Dunkl--Dirac operator is a deformation of the Dirac operator by means of Dunkl derivatives. We investigate the symmetry algebra generated by the elements supercommuting with the Dunkl--Dirac operator and its dual symbol. This symmetry algebra is realised inside the tensor product of a Clifford algebra and a rational Cherednik algebra associated with a reflection group or root system. For reducible root systems of rank three, we determine all the irreducible finite-dimensional representations and conditions for unitarity. Polynomial solutions of the Dunkl--Dirac equation are given as a realisation of one family of such irreducible unitary representations.