A Hermitian refinement of symplectic Clifford analysis

TL;DR

This paper introduces a Hermitian refinement of symplectic Clifford analysis by incorporating a complex structure, leading to new Dirac operators, invariant systems, and a Fischer decomposition for polynomial solutions.

Contribution

It develops a Hermitian refinement of symplectic Clifford analysis with new Dirac operators and explicit Fischer decomposition using transvector algebra.

Findings

Introduction of complex structure $ extbf{J}$ on symplectic manifold

Development of $ extbf{D}_s$ and $ extbf{D}_t$ Dirac operators

Explicit Fischer decomposition for polynomial solutions

Abstract

In this paper we develop the Hermitian refinement of symplectic Clifford analysis, by introducing a complex structure $\mathbb{J}$ on the canonical symplectic manifold $(\mathbb {R}^{2n},\omega_0)$. This gives rise to two symplectic Dirac operators $D_s$ and $D_t$ (in the sense of Habermann), leading to a $\mathfrak{u}(n)$-invariant system of equations on $\mathbb{R}^{2n}$. We discuss the solution space for this system, culminating in a Fischer decomposition for the space of polynomials on $\mathbb {R}^{2n}$ with values in the symplectic spinors. To make this decomposition explicit, we will construct the associated embedding factors using a transvector algebra.