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.
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Acknowledgements
The author G.M. was supported by the FWO-EoS project G0H4518N.
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Eelbode, D., Muarem, G. (2024). Branching Symplectic Monogenics Using a Mickelsson–Zhelobenko Algebra. In: Ptak, M., Woerdeman, H.J., Wojtylak, M. (eds) Operator and Matrix Theory, Function Spaces, and Applications. IWOTA 2022. Operator Theory: Advances and Applications, vol 295. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-50613-0_6
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DOI: https://doi.org/10.1007/978-3-031-50613-0_6
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