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

Hendrik De Bie; Alexis Langlois-Rémillard; Roy Oste; Joris Van der Jeugt

doi:10.1016/j.jalgebra.2021.09.025

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arXiv · 2010.03381

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

Hendrik De Bie·Alexis Langlois-Rémillard·Roy Oste·Joris Van der Jeugt

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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.

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BibTeX RIS

Hendrik De Bie, Alexis Langlois-Rémillard, Roy Oste, Joris Van der Jeugt. 2020-10-07. Finite-dimensional representations of the symmetry algebra of the dihedral Dunkl--Dirac operator. https://doi.org/10.1016/j.jalgebra.2021.09.025

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