Restriction of the metaplectic representation over a $p$-adic field to an anisotropic torus

Abstract

In this article, we examine the restriction of the metaplectic representation $\pi$ over a $p$-adic field $k$, $p\ne 2$, of zero characteristic to an isotropic torus $S$ contained in the symplectic group. First we give necessary and sufficient conditions on the momentum map in order that $S$ be admissible, that is ${\pi }_{|S}$ decomposes with finite multiplicities. Let us say that a torus contained in the symplectic group is irreducible if its action on the symplectic space is irreducible over $k$. Then we examine the case when $S$ is a proper subtorus of a maximal irreducible torus $T$ in the symplectic group and give sufficient conditions on $T$ in order that $S$ never be admissible. When these conditions are not satisfied, we give examples of admissible proper tori of a maximal irreducible torus. Finally, for any admissible subtorus $S$ of a certain type of maximal irreducible torus, we compute the multiplicity of the unitary characters of $S$ appearing into ${\pi }_{|S}$. We also show that the multiplicity of such a character is equal to the volume of the symplectic reduction of the inverse image under the momentum map of a linear form associated to it.