The Howe duality and polynomial solutions for the symplectic Dirac operator

We study various aspects of the metaplectic Howe duality realized by Fischer decomposition for the metaplectic representation space of polynomials on $\mathbb{R}^{2n}$ valued in the Segal-Shale-Weil representation. As a consequence, we determine symplectic monogenics, i.e., the space of polynomial solutions of the symplectic Dirac operator.