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

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.