The Orthogonal Branching Problem for Symplectic Monogenics

Abstract

In this paper we study the $$\mathfrak {sp}(2m)$$ -invariant Dirac operator $$D_s$$ which acts on symplectic spinors, from an orthogonal point of view. By this we mean that we will focus on the subalgebra $$\mathfrak {so}(m) \subset \mathfrak {sp}(2m)$$ , as this will allow us to derive branching rules for the space of 1-homogeneous polynomial solutions for the operator $$D_s$$ (hence generalising the classical Fischer decomposition in harmonic analysis for a vector variable in $${\mathbb {R}}^m$$ ). To arrive at this result we use techniques from representation theory, including the transvector algebra $${\mathcal {Z}}(\mathfrak {sp}(4),\mathfrak {so}(4))$$ and tensor products of Verma modules.