Branching symplectic monogenics using a Mickelsson--Zhelobenko algebra

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 $\mathfrak{s}\mathfrak{p}(2m)$. Because $\mathfrak{s}\mathfrak{o}(m)\subset \mathfrak{s}\mathfrak{p}(2m)$, 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.