The Howe duality for the Dunkl version of the Dirac operator

Bent Ørsted; Petr Somberg; Vladimir Soucek

doi:10.1007/s00006-009-0166-3

The Howe duality for the Dunkl version of the Dirac operator

Bent Ørsted, Petr Somberg, Vladimir Soucek

Publikation: Bidrag til tidsskrift/Konferencebidrag i tidsskrift /Bidrag til avis › Tidsskriftartikel › Forskning › peer review

Abstract

The classical Fischer decomposition of polynomials on Euclidean space makes it possible to express any polynomial as a sum of harmonic polynomials multiplied by powers of |x|². A deformation of the Laplace operator was recently introduced by Ch.F. Dunkl. It has the property that the symmetry with respect to the orthogonal group is broken to a finite subgroup generated by reflections (a Coxeter group). It was shown by B. Ørsted and S. Ben Said that there is a deformation of the Fischer decomposition for polynomials with respect to the Dunkl harmonic functions. In Clifford analysis, a Dunkl version of the Dirac operator was introduced and studied by P. Cerejeiras, U. Kähler and G. Ren. The aim of the article is to describe an analogue of the Fischer decomposition for solutions of the Dunkl Dirac operator. The main methods used are coming from representation theory, in particular, from ideas connected with Howe dual pairs.

Originalsprog	Engelsk
Tidsskrift	Advances in Applied Clifford Algebras
Vol/bind	19
Nummer	2
Sider (fra-til)	403-415
Antal sider	13
ISSN	0188-7009
DOI	https://doi.org/10.1007/s00006-009-0166-3
Status	Udgivet - 2009

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10.1007/s00006-009-0166-3

Citationsformater

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abstract = "The classical Fischer decomposition of polynomials on Euclidean space makes it possible to express any polynomial as a sum of harmonic polynomials multiplied by powers of |x|2. A deformation of the Laplace operator was recently introduced by Ch.F. Dunkl. It has the property that the symmetry with respect to the orthogonal group is broken to a finite subgroup generated by reflections (a Coxeter group). It was shown by B. {\O}rsted and S. Ben Said that there is a deformation of the Fischer decomposition for polynomials with respect to the Dunkl harmonic functions. In Clifford analysis, a Dunkl version of the Dirac operator was introduced and studied by P. Cerejeiras, U. K{\"a}hler and G. Ren. The aim of the article is to describe an analogue of the Fischer decomposition for solutions of the Dunkl Dirac operator. The main methods used are coming from representation theory, in particular, from ideas connected with Howe dual pairs.",

author = "Bent {\O}rsted and Petr Somberg and Vladimir Soucek",

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T1 - The Howe duality for the Dunkl version of the Dirac operator

AU - Ørsted, Bent

AU - Somberg, Petr

AU - Soucek, Vladimir

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N2 - The classical Fischer decomposition of polynomials on Euclidean space makes it possible to express any polynomial as a sum of harmonic polynomials multiplied by powers of |x|2. A deformation of the Laplace operator was recently introduced by Ch.F. Dunkl. It has the property that the symmetry with respect to the orthogonal group is broken to a finite subgroup generated by reflections (a Coxeter group). It was shown by B. Ørsted and S. Ben Said that there is a deformation of the Fischer decomposition for polynomials with respect to the Dunkl harmonic functions. In Clifford analysis, a Dunkl version of the Dirac operator was introduced and studied by P. Cerejeiras, U. Kähler and G. Ren. The aim of the article is to describe an analogue of the Fischer decomposition for solutions of the Dunkl Dirac operator. The main methods used are coming from representation theory, in particular, from ideas connected with Howe dual pairs.

AB - The classical Fischer decomposition of polynomials on Euclidean space makes it possible to express any polynomial as a sum of harmonic polynomials multiplied by powers of |x|2. A deformation of the Laplace operator was recently introduced by Ch.F. Dunkl. It has the property that the symmetry with respect to the orthogonal group is broken to a finite subgroup generated by reflections (a Coxeter group). It was shown by B. Ørsted and S. Ben Said that there is a deformation of the Fischer decomposition for polynomials with respect to the Dunkl harmonic functions. In Clifford analysis, a Dunkl version of the Dirac operator was introduced and studied by P. Cerejeiras, U. Kähler and G. Ren. The aim of the article is to describe an analogue of the Fischer decomposition for solutions of the Dunkl Dirac operator. The main methods used are coming from representation theory, in particular, from ideas connected with Howe dual pairs.

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DO - 10.1007/s00006-009-0166-3

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JO - Advances in Applied Clifford Algebras

JF - Advances in Applied Clifford Algebras

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The Howe duality for the Dunkl version of the Dirac operator

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