Howe duality in Dunkl superspace (Q625937)

Clifford analysis is a generalization to higher dimension of the theory of holomorphic functions in the complex plane where the role of the Cauchy-Riemann operator is taken over by the Dirac operator. Superspaces are spaces with not only commuting but also anti-commuting variables, while Dunkl operators are perturbations of the usual partial derivatives by a finite reflection group. In this paper, the Fischer decomposition is established for solutions of the super Dunkl Dirac operator. Two explicit expressions for the components of the Fischer decomposition are given. The result is general without restrictions on multiplicity functions or on super dimensions. The Fischer decomposition provides a module for the Howe dual pair \(G \times {\mathbf{osp}}(1|2)\) on the space of spinor valued polynomials with \(G\) the Coxeter group (or the finite reflection group) and \({\mathbf{osp}}(1|2)\) a finite-dimensional Lie supersubalgebra, while the generators of the Lie superspace reveal the naturality of the Fischer decomposition. The obtained result is basic for the theory of Clifford Dunkl superspaces since many key facts in the theory are consequences of the Fischer decomposition as shown in the classical Clifford analysis case.