Branching symplectic monogenics using a Mickelsson--Zhelobenko algebra

2023/01/12 by David Eelbode, Eelbode, David, Güner Muarem +1
Mathematics · #00A00 #15A66 #17B10 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Finite Group Theory Research #Representation Theory (math.RT) #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.2301.05066

openalex publication_date 2023/01/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 \mathfraksp(2m). Because \mathfrakso(m)⊂ \mathfraksp(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.

Related

• The orthogonal branching problem for symplectic monogenics
• The Howe duality and polynomial solutions for the symplectic Dirac operator
• Models for irreducible representations of the symplectic algebra using Dirac-type operators
• A Hermitian refinement of symplectic Clifford analysis
• Basic aspects of symplectic Clifford analysis for the symplectic Dirac operator