  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.
