Fundaments of Quaternionic Clifford Analysis III: Fischer Decomposition in Symplectic Harmonic Analysis

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}$)$.