In this paper we develop the Hermitian refinement of symplectic Clifford analysis, by introducing a complex structure J on the canonical symplectic manifold (R 2n , ω 0 ). This gives rise to two symplectic Dirac operators D s and D t (in the sense of Habermann [8]), leading to a u(n)-invariant system of equations on R 2n . We discuss the solution space for this system, culminating in a Fischer decomposition for the space of polynomials on R 2n with values in the symplectic spinors. To make this decomposition explicit, we will construct the associated embedding factors using a transvector algebra.

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In this paper we consider (polynomial) solution spaces for the symplectic Dirac operator (with a focus on 1-homogeneous solutions). This space forms an infinite-dimensional representation space for the symplectic Lie algebra sp(2m). Because so(m) ⊂ sp(2m), this leads to a branching problem which generalises the classical Fischer decomposition in harmonic analysis. Due to the infinite nature of the solution spaces for the symplectic Dirac operators, this is a non-trivial question: both the summands appearing in the decomposition and their explicit embedding factors will be determined in terms of a suitable Mickelsson-Zhelobenko algebra.