Revista Matemática Iberoamericana

The Howe dual pair in Hermitean Clifford analysis

Fred Brackx , Hennie De Schepper , David Eelbode , and Vladimir Souček

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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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Rev. Mat. Iberoamericana, Volume 26, Number 2 (2010), 449-479.

First available in Project Euclid: 4 June 2010

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Primary: 22E46: Semisimple Lie groups and their representations 30G35: Functions of hypercomplex variables and generalized variables 15A66: Clifford algebras, spinors

Hermitean Clifford analysis Howe dual pair


Brackx, Fred; De Schepper, Hennie; Eelbode, David; Souček, Vladimir. The Howe dual pair in Hermitean Clifford analysis. Rev. Mat. Iberoamericana 26 (2010), no. 2, 449--479.

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