Dirac operators on Weil representations I

Let G be the metaplectic double cover of the group of four-by-four real symplectic matrices. Let $frg$ be the complexified Lie algebra of G. Let $W_0$ and $W_1$ be the Harish-Chandra modules of the even and odd Weil representations of G, respectively. We find the Dirac cohomology of $W_0$ and $W_1$ with respect to the Dirac operator corresponding to a maximal compact subalgebra of $frg$, and then also with respect to the Kostant\u27s cubic Dirac operator corresponding to a compact Cartan subalgebra of $frg$. The results can be considered as examples illustrating the main results of [11]