Dunkl operators and a family of realizations of osp(1|2)

Abstract

In this paper, a family of radial deformations of the realization of the Lie superalgebra osp(1|2) in the theory of Dunkl operators is obtained. This leads to a Dirac operator depending on 3 parameters. Several function theoretical aspects of this operator are studied, such as the associated measure, the related Laguerre polynomials and the related Fourier transform. For special values of the parameters, it is possible to construct the kernel of the Fourier transform explicitly, as well as the related intertwining operator.

arXiv comment28 pages, some small changes, accepted in Trans. Amer. Math. Soc

References in corpus (1)

• The inversion formula and holomorphic extension of the minimal representation of the conformal group

Cited by in corpus (6)

• The Dunkl oscillator in the plane II : representations of the symmetry algebra
• A Dirac-Dunkl equation on $S^2$ and the Bannai-Ito algebra
• Orthogonality of Hermite polynomials in superspace and Mehler type formulae
• On the Clifford-Fourier transform
• Algebraic approach to slice monogenic functions
• The Howe duality and polynomial solutions for the symplectic Dirac operator