Fischer decomposition for ‐monogenics in quaternionic Clifford analysis

Abstract

Spaces of spinor‐valued homogeneous polynomials, and in particular spaces of spinor‐valued spherical harmonics, are decomposed in terms of irreducible representations of the symplectic group Sp(p). These Fischer decompositions involve spaces of homogeneous, so‐called ‐monogenic polynomials, the Lie super algebra being the Howe dual partner to the symplectic group Sp(p). In order to obtain Sp(p)‐irreducibility, this new concept of ‐monogenicity has to be introduced as a refinement of quaternionic monogenicity; it is defined by means of the four quaternionic Dirac operators, a scalar Euler operator underlying the notion of symplectic harmonicity and a multiplicative Clifford algebra operator P underlying the decomposition of spinor space into symplectic cells. These operators and P, and their Hermitian conjugates, arise naturally when constructing the Howe dual pair , the action of which will make the Fischer decomposition multiplicity free. Copyright © 2016 John Wiley & Sons, Ltd.