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Published in:
2024 | OriginalPaper | Chapter
Branching Symplectic Monogenics Using a Mickelsson–Zhelobenko Algebra
Authors : David Eelbode, Guner Muarem
Published in: Operator and Matrix Theory, Function Spaces, and Applications
Publisher: Springer Nature Switzerland
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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.