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

De Bie, H.; Ørsted, B.; Somberg, P.; Souček, V.

doi:10.1090/S0002-9947-2012-05608-X

Dunkl operators and a family of realizations of $\mathfrak {osp}(1\vert 2)$
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by H. De Bie, B. Ørsted, P. Somberg and V. Souček PDF

Trans. Amer. Math. Soc. 364 (2012), 3875-3902 Request permission

Abstract:

In this paper, a family of radial deformations of the realization of the Lie superalgebra $\mathfrak {osp}(1|2)$ 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.

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