A Hermitian refinement of symplectic Clifford analysis

In this paper, we develop the Hermitian refinement of symplectic Clifford analysis, by introducing a complex structure J on the canonical symplectic manifold (R2n,ω0)$$ \left({\mathrm{\mathbb{R}}}^{2n},{\omega}_0\right) $$. This gives rise to two symplectic Dirac operators Ds$$ {D}_s $$ and Dt$$ {D}_t $$ (in the sense of Habermann), leading to a u(n)$$ \mathfrak{u}(n) $$‐invariant system of equations on R2n$$ {\mathrm{\mathbb{R}}}^{2n} $$. We discuss the solution space for this system, culminating in a Fischer decomposition for the space of (harmonic) polynomials on R2n$$ {\mathrm{\mathbb{R}}}^{2n} $$ with values in the symplectic spinors. To make this decomposition explicit, we will construct the associated embedding factors using a transvector algebra.