The orthogonal branching problem for symplectic monogenics

TL;DR

This paper investigates the branching rules of the symplectic Dirac operator acting on symplectic spinors, using representation theory techniques to generalize classical harmonic analysis decompositions.

Contribution

It introduces a new approach to derive branching rules for symplectic Dirac operators via orthogonal subalgebra analysis and representation theory methods.

Findings

Derived branching rules for symplectic Dirac operator solutions

Generalized Fischer decomposition in the symplectic setting

Applied transvector algebra techniques to representation theory

Abstract

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