Howe duality in Dunkl superspace
type: Item
instance of: scholarly article
title: Howe duality in Dunkl superspace
zbMATH Open document ID: 1213.30087
published in: Science China. Mathematics
publication date: 2011-02-25T00:00:00Z
review text: 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.
reviewed by: Nele De Schepper
Mathematics Subject Classification ID: 30G35
Mathematics Subject Classification ID: 31A30
Mathematics Subject Classification ID: 33C52
Mathematics Subject Classification ID: 58C50
zbMATH DE Number: 5857743
zbMATH Keywords: Dunkl operators
zbMATH Keywords: superspace
zbMATH Keywords: Fischer decomposition
zbMATH Keywords: Dirac operator
author: Guangbin Ren
MaRDI profile type: Publication
full work available at URL: s11425-010-4063-y
OpenAlex ID: W1984333901
cites work: The Howe dual pair in Hermitean Clifford analysis
cites work: Clifford analysis for finite reflection groups
cites work: Spherical harmonics and integration in superspace
cites work: Fourier transform and related integral transforms in superspace
cites work: Schrödinger equation with delta potential in superspace
cites work: A Clifford analysis approach to superspace
cites work: Correct rules for Clifford calculus on superspace
cites work: Q3517489
cites work: Fundamental solutions for the super Laplace and Dirac operators and all their natural powers
cites work: Q4003806
cites work: Calogero-Moser-Sutherland models. Workshop, Centre de Recherches Mathématique, Montréal, Canada, March 10--15, 1997
cites work: Orthogonal polynomials of several variables
cites work: Q3995205
cites work: Q4663733
cites work: Q4400626
cites work: Q3998721
cites work: Almansi‐type theorems in Clifford analysis
cites work: The Howe duality for the Dunkl version of the Dirac operator
cites work: Almansi decomposition for Dunkl operators
cites work: Almansi decompositions for polyharmonic, polyheat, and polywave functions
cites work: Segal-Bargmann transforms associated with finite Coxeter groups
DOI: 10.1007/S11425-010-4063-Y
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description: scientific article; zbMATH DE number 5857743