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June, 2010 The Howe dual pair in Hermitean Clifford analysis
Fred Brackx , Hennie De Schepper , David Eelbode , Vladimir Souček
Rev. Mat. Iberoamericana 26(2): 449-479 (June, 2010).


Clifford analysis offers a higher dimensional function theory studying the null solutions of the rotation invariant, vector valued, first order Dirac operator $\partial$. In the more recent branch Hermitean Clifford analysis, this rotational invariance has been broken by introducing a complex structure $J$ on Euclidean space and a corresponding second Dirac operator $\partial_J$, leading to the system of equations $\partial f = 0 = \partial_J f$ expressing so-called Hermitean monogenicity. The invariance of this system is reduced to the unitary group. In this paper we show that this choice of equations is fully justified. Indeed, constructing the Howe dual for the action of the unitary group on the space of all spinor valued polynomials, the generators of the resulting Lie superalgebra reveal the natural set of equations to be considered in this context, which exactly coincide with the chosen ones.


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Fred Brackx . Hennie De Schepper . David Eelbode . Vladimir Souček . "The Howe dual pair in Hermitean Clifford analysis." Rev. Mat. Iberoamericana 26 (2) 449 - 479, June, 2010.


Published: June, 2010
First available in Project Euclid: 4 June 2010

zbMATH: 1201.30061
MathSciNet: MR2677004

Primary: 15A66 , 22E46 , 30G35‎

Keywords: Hermitean Clifford analysis , Howe dual pair

Rights: Copyright © 2010 Departamento de Matemáticas, Universidad Autónoma de Madrid

Vol.26 • No. 2 • June, 2010
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