Restriction of the metaplectic representation over a $p$-adic field to an anisotropic torus
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arXiv
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| Format: | Preprint |
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2025
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| author | Maktouf, Khemais Torasso, Pierre |
| author_facet | Maktouf, Khemais Torasso, Pierre |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_05317 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Restriction of the metaplectic representation over a $p$-adic field to an anisotropic torus Maktouf, Khemais Torasso, Pierre Representation Theory 22E50 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. |
| title | Restriction of the metaplectic representation over a $p$-adic field to an anisotropic torus |
| topic | Representation Theory 22E50 |
| url | https://arxiv.org/abs/2512.05317 |