The Howe dual pair in Hermitean Clifford analysis. (English) Zbl 1201.30061

In this paper, the classical Clifford analysis is additionally furnished with a complex structure. The Riemannian setting is now replaced by Kähler geometry. Unfortunately, the rotation invariance of the null solutions of the Dirac operator is not longer valid. The authors study a new associated Dirac operator \(\partial_J\). With Hermitian analysis, simultaneously null solutions of both operators \(\partial\) and \(\partial_J\) are considered. In this way, the rotation invaraince reduces to \(U(n)\)-invariance. The main interest of the paper lies in the consideration of the Howe dual pair with respect to Hermitean Clifford analysis. Spinor valued polynomials play an important role. Fischer decompositions for Hermitian monogenic functions are re-classified.


30G35 Functions of hypercomplex variables and generalized variables
15A66 Clifford algebras, spinors
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