The invariants of the Weil representation of ${\mathrm{S}\mathrm{L}}_2(\mathbb{Z})$

The transformation behaviour of the vector valued theta function of a positive-definite even lattice under the metaplectic group ${\mathrm{M}\mathrm{p}}_2(\mathbb{Z})$ is described by the Weil representation. We show that the invariants of this representation are induced from $5$ fundamental invariants. As an application we give simple generating sets for Jacobi forms of singular weight.