The total angular momentum algebra related to the S3 Dunkl Dirac equation

We consider the symmetry algebra generated by the total angular momentum operators, appearing as constants of motion of the S3 Dunkl Dirac equation. The latter is a deformation of the Dirac equation by means of Dunkl operators, in our case associated to the root system A2, with corresponding Weyl group S3, the symmetric group on three elements. The explicit form of the symmetry algebra in this case is a one-parameter deformation of the classical total angular momentum algebra so(3), incorporating elements of S3. This was obtained using recent results on the symmetry algebra for a class of Dirac operators, containing in particular the Dirac-Dunkl operator for arbitrary root system. For this symmetry algebra, we classify all finite-dimensional, irreducible representations and determine the conditions for the representations to be unitarizable. The class of unitary irreducible representations admits a natural realization acting on a representation space of eigenfunctions of the Dirac Hamiltonian. Using a Cauchy-Kowalevski extension theorem we obtain explicit expressions for these eigenfunctions in terms of Jacobi polynomials.