Dunkl operators and a family of realizations of \(\mathfrak{osp}(1|2)\) (Q2839959)

The present paper is devoted to radial deformations of the realization of the Lie superalgebra \(\mathfrak{osp}(1|2)\) in the theory of Dunkl operators. In Section 2, the necessary background on Dunkl operators is given. In Section 3, a new family of Dirac operators is obtained. It is shown that they generate the Lie superalgebra \(\mathfrak{osp}(1|2)\). Theorem 1 states that the operators of the form NEWLINENEWLINE\[NEWLINE \mathbf D =r^{1-a/2}\mathcal D_k+br^{-a/2-1}\underline{x}+cr^{-a/2-1}\underline{x} E, NEWLINE\]NEWLINE NEWLINEwhere \(b\), \(c\) are arbitrary complex numbers, and \(\underline{x}=\sum_{i=1}^me_ix_i\), \( E=\sum_{i=1}^mx_i\partial_{x_i}\), \(r^2=\sum_{i=1}^mx^2_i\), generate a Lie superalgebra, which is isomorphic to \(\mathfrak{osp}(1|2)\). The square of \(\mathbf D\) is explicitly calculated (Subsection 3.1), and the measure related to \(\mathbf D\) is determined (Subsection 3.2). In Subsection 4.1, the Fisher decomposition and the dual pair related to \(\mathbf D\) are obtained. In Subsection 4.2, the Laguerre polynomials and functions related to \(\mathbf D\) are introduced. A new class of Fourier transforms, which are related to \(\mathbf D\) is introduced in Section 5. In case that \(\mathbf D^2\) is scalar the kernel of the associated Fourier transform is determined. The very special case where \(a=-2\) is discussed in Subsection 5.2.