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

TL;DR

This paper studies how the metaplectic representation over a p-adic field restricts to certain tori, providing conditions for admissibility and calculating multiplicities of characters.

Contribution

It offers necessary and sufficient conditions for the restriction to be admissible and computes multiplicities for characters of admissible tori in the symplectic group.

Findings

Conditions for admissibility of tori based on the momentum map.

Examples of admissible tori within maximal irreducible tori.

Multiplicity of characters equals the volume of a symplectic reduction.

Abstract

In this article, we examine the restriction of the metaplectic representation $\pi$ over a $p$-adic field $k$, $p\neq2$, 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 $\pi_{\vert S}$ 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 $\pi_{\vert S}$. 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.