Fischer decomposition in symplectic harmonic analysis

Abstract

In the framework of quaternionic Clifford analysis in Euclidean space \(\mathbb {R}^{4p}\), which constitutes a refinement of Euclidean and Hermitian Clifford analysis, the Fischer decomposition of the space of complex valued polynomials is obtained in terms of spaces of so-called (adjoint) symplectic spherical harmonics, which are irreducible modules for the symplectic group Sp\((p)\). Its Howe dual partner is determined to be \(\mathfrak {sl}(2,\mathbb {C}) \oplus \mathfrak {sl}(2,\mathbb {C}) = \mathfrak {so}(4,\mathbb {C})\).