On the lattice model of the Weil representation

Let F be a local field. In the case of F being the real field, Pierre Cartier constructed Heisenberg-Weil representations of a Heisenberg group in families using non-self-dual lattices. This result was later reformulated by Jae-Hyun Yang in another paper. We extend this family of representations to a representation of a Jacobi subgroup by incorporating a rational Metaplectic group. In the case of F being a non-archimedean local field of odd residual characteristic or a finite unramified extension of the field of 2-adic numbers, we obtain similar results for a Jacobi group by incorporating a Metaplectic group.