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

Khemais Maktouf, Pierre Torasso

TL;DR

The paper analyzes how the Weil (metaplectic) representation restricted to an anisotropic torus in a $p$-adic symplectic group decomposes. It proves that admissibility is equivalent to geometric properties of the momentum map $\phi$, and provides a detailed description of restrictions to maximal irreducible tori, including when subtori are admissible and how ramification affects multiplicities. In admissible cases, character multiplicities are expressed as volumes of symplectic reductions $S\backslash\phi^{-1}(g)$, bridging representation theory and symplectic geometry. The work extends real-case intuition to non-archimedean settings, offering explicit criteria and volume formulas that depend on ramification data and group-theoretic structures.

Abstract

In this article, we examine the restriction of the metaplectic representation $π$ 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 $π_{\vert S}$ 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 $π_{\vert S}$. 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.

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

TL;DR

The paper analyzes how the Weil (metaplectic) representation restricted to an anisotropic torus in a -adic symplectic group decomposes. It proves that admissibility is equivalent to geometric properties of the momentum map , and provides a detailed description of restrictions to maximal irreducible tori, including when subtori are admissible and how ramification affects multiplicities. In admissible cases, character multiplicities are expressed as volumes of symplectic reductions , bridging representation theory and symplectic geometry. The work extends real-case intuition to non-archimedean settings, offering explicit criteria and volume formulas that depend on ramification data and group-theoretic structures.

Abstract

In this article, we examine the restriction of the metaplectic representation over a -adic field , , of zero characteristic to an isotropic torus contained in the symplectic group. First we give necessary and sufficient conditions on the momentum map in order that be admissible, that is 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 . Then we examine the case when is a proper subtorus of a maximal irreducible torus in the symplectic group and give sufficient conditions on in order that 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 of a certain type of maximal irreducible torus, we compute the multiplicity of the unitary characters of appearing into . 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.

Paper Structure

This paper contains 7 sections, 63 theorems, 153 equations.

Table of Contents

  1. Generalities
  2. Restriction of the Weil representation to maximal irreducible tori
  3. Momentum map and admissibility of anisotropic tori
  4. Subtori of maximal irreducible tori
  5. Multiplicity of characters and geometry of the momentum map
  6. Appendix
  7. Appendix

Key Result

Lemma 2.1

a) Suppose that $k"$ is unramified over $k'$. Then one has $T=\mu_{q'+1}\times T_{1}$. Moreover, $(T_{j})_{j\in\mathbb{N}}$ is a strictly decreasing sequence of open subgroups of $T$ and a basis of neighbourhoods of $1$. b) Suppose that $k"$ is ramified over $k'$. Then one has $T=\{1,-1\}\times T_{1

Theorems & Definitions (72)

  • Definition 2.1
  • Lemma 2.1
  • Proposition 2.1
  • Lemma 2.2
  • Corollary 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Theorem 2.1
  • Proposition 2.4
  • Definition 3.1
  • ...and 62 more