A Hermitian refinement of symplectic Clifford analysis

2023/09/15 by David Eelbode, Eelbode, David, Güner Muarem +1 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic and Geometric Analysis #Complex Variables (math.CV) #FOS: Mathematics #Representation Theory (math.RT) #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.2309.08749

openalex publication_date 2023/09/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we develop the Hermitian refinement of symplectic Clifford analysis, by introducing a complex structure \mathbbJ on the canonical symplectic manifold (\mathbb R²ⁿ,ω₀). This gives rise to two symplectic Dirac operators D_s and D_t (in the sense of Habermann), leading to a \mathfraku(n)-invariant system of equations on ℝ²ⁿ. We discuss the solution space for this system, culminating in a Fischer decomposition for the space of polynomials on \mathbb R²ⁿ with values in the symplectic spinors. To make this decomposition explicit, we will construct the associated embedding factors using a transvector algebra.

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