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.

v3 40p. Final version accepted in J. Algebra. See v2 for proof of Thm 4.1

References in corpus (3)

• Deformations of unitary Howe dual pairs
• An exceptional symmetry algebra for the 3D Dirac-Dunkl operator
• Supercentralizers for deformations of the Pin osp dual pair

Cited by in corpus (2)

• Generalised symmetries and bases for Dunkl monogenics
• The dihedral Dunkl--Dirac symmetry algebra with negative Clifford signature