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 \(\mathfrak {sp}(2m)\) . Because \(\mathfrak {so}(m)\subset \mathfrak {sp}(2m)\) , 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.