Revista Matemática Iberoamericana

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Volume 26, Issue 2, 2010, pp. 449–479
DOI: 10.4171/RMI/606

Published online: 2010-08-31

The Howe dual pair in Hermitean Clifford analysis

Fred Brackx[1], Hennie De Schepper[2], David Eelbode[3] and Vladimír Souček[4]

(1) Gent University, Belgium
(2) Ghent University, Gent, Belgium
(3) Universiteit Antwerpen, Belgium
(4) Charles University, Prague, Czech Republic

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.

Keywords: Hermitean Clifford analysis, Howe dual pair

Brackx Fred, De Schepper Hennie, Eelbode David, Souček Vladimír: The Howe dual pair in Hermitean Clifford analysis. Rev. Mat. Iberoam. 26 (2010), 449-479. doi: 10.4171/RMI/606