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.

在本文中，我们从正交的角度研究了作用于辛旋量的\(\mathfrak {sp}(2m)\) -不变量狄拉克算子\(D_s\) 。我们的意思是我们将关注子代数\(\mathfrak {so}(m) \subset \mathfrak {sp}(2m)\)，因为这将允许我们推导出 1-齐次空间的分支规则算子\(D_s\)的多项式解（因此在\({\mathbb {R}}^m\)中的向量变量的谐波分析中推广经典 Fischer 分解）。为了得出这个结果，我们使用了表示论中的技术，包括转向量代数\({\mathcal {Z}}(\mathfrak {sp}(4),\mathfrak {so}(4))\)和 Verma 模块的张量积。