Branching symplectic monogenics using a Mickelsson--Zhelobenko algebra

TL;DR

This paper investigates the decomposition of polynomial solutions to the symplectic Dirac operator, using Mickelsson-Zhelobenko algebra to explicitly determine the structure of these infinite-dimensional representation spaces.

Contribution

It introduces a novel approach to the branching problem for symplectic Dirac solutions via Mickelsson-Zhelobenko algebra, generalizing classical harmonic analysis decompositions.

Findings

Explicit decomposition of symplectic Dirac solution spaces.

Determination of embedding factors using Mickelsson-Zhelobenko algebra.

Extension of Fischer decomposition in the symplectic setting.

Abstract

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.