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.

arXiv commentv3 40p. Final version accepted in J. Algebra. See v2 for proof of Thm 4.1

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Cited by in corpus (2)

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