Fundaments of Quaternionic Clifford Analysis III: 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 $\mathfrak{sl}(2,\mathbb{C})\oplus \mathfrak{sl}(2,\mathbb{C})=\mathfrak{so}(4,\mathbb{C})$.

Cited by in corpus (1)

• The Cauchy Integral Formula in Hermitian, Quaternionic and osp(4|2) Clifford Analysis