Explicit Formulae for the Hermitean Monogenic Projection Operator

Abstract

Hermitean Clifford analysis is a recent branch of Clifford analysis, refining the standard Euclidean case. It focusses on the simultaneous null solutions, called Hermitean monogenic functions, of two complex Dirac operators which are invariant under the action of the unitary group. As in harmonic analysis and in Euclidean Clifford analysis, a Fischer decomposition for homogeneous polynomials has been obtained also in the Hermitean Clifford setting, its structure however being far more subtle, as a consequence of underlying group symmetries involved. We present explicit formulae for the Hermitean monogenic component in this Fischer decomposition, resulting from the action of a projection operator on the considered polynomial.