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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Dunkl operators and a family of realizations of $ \mathfrak{osp}(1\vert2)$


Authors: H. De Bie, B. Ørsted, P. Somberg and V. Souček
Journal: Trans. Amer. Math. Soc. 364 (2012), 3875-3902
MSC (2010): Primary 33C52, 30G35, 43A32
Published electronically: February 27, 2012
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Abstract: In this paper, a family of radial deformations of the realization of the Lie superalgebra $ \mathfrak{osp}(1\vert 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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Additional Information

H. De Bie
Affiliation: Department of Mathematical Analysis, Ghent University, Krijgslaan 281, 9000 Gent, Belgium
Email: Hendrik.DeBie@UGent.be

B. Ørsted
Affiliation: Department of Mathematical Sciences, University of Aarhus, Building 530, Ny Munkegade, DK 8000, Aarhus C, Denmark
Email: orsted@imf.au.dk

P. Somberg
Affiliation: Mathematical Institute of Charles University, Sokolovská 83, 186 75 Praha, Czech Republic
Email: somberg@karlin.mff.cuni.cz

V. Souček
Affiliation: Mathematical Institute of Charles University, Sokolovská 83, 186 75 Praha, Czech Republic
Email: soucek@karlin.mff.cuni.cz

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05608-X
PII: S 0002-9947(2012)05608-X
Keywords: Dunkl operators, Clifford analysis, generalized Fourier transform, Laguerre polynomials, Kelvin transform
Received by editor(s): February 12, 2010
Received by editor(s) in revised form: March 30, 2011
Published electronically: February 27, 2012
Additional Notes: The first author is a Postdoctoral Fellow of the Research Foundation - Flanders (FWO) and a Courtesy Research Associate at the University of Oregon.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.