Distinguished Representations for $\rm{SL}_n(D)$ where $D$ is a quaternion division algebra over a $p$-adic field
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arXiv
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2025
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| _version_ | 1866909451642994688 |
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| author | Choiy, Kwangho Patel, Shiv Prakash |
| author_facet | Choiy, Kwangho Patel, Shiv Prakash |
| contents | Let $D$ be a quaternion division algebra over a non-archimedean local field $F$ of characteristic zero. Let $E/F$ be a quadratic extension and $\rm{SL}_{n}^{*}(E) = {\rm{GL}}_{n}(E) \cap \rm{SL}_{n}(D)$. We study distinguished representations of $\rm{SL}_{n}(D)$ by the subgroup $\rm{SL}_{n}^{*}(E)$. Let $π$ be an irreducible admissible representation of $\rm{SL}_{n}(D)$ which is distinguished by $\rm{SL}_{n}^{*}(E)$. We give a multiplicity formula, i.e. a formula for the dimension of the $\mathbb{C}$-vector space ${\rm{Hom}}_{\rm{SL}_{n}^{*}(E)} (π, \mathbbm{1})$, where $\mathbbm{1}$ denotes the trivial representation of $\rm{SL}_{n}^{*}(E)$. This work is a non-split inner form analog of a work by Anandavardhanan-Prasad which gives a multiplicity formula for $\rm{SL}_{n}(F)$-distinguished irreducible admissible representation of $\rm{SL}_{n}(E)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_04500 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Distinguished Representations for $\rm{SL}_n(D)$ where $D$ is a quaternion division algebra over a $p$-adic field Choiy, Kwangho Patel, Shiv Prakash Representation Theory Number Theory 22E50, 11S37, 20G25, 22E35 Let $D$ be a quaternion division algebra over a non-archimedean local field $F$ of characteristic zero. Let $E/F$ be a quadratic extension and $\rm{SL}_{n}^{*}(E) = {\rm{GL}}_{n}(E) \cap \rm{SL}_{n}(D)$. We study distinguished representations of $\rm{SL}_{n}(D)$ by the subgroup $\rm{SL}_{n}^{*}(E)$. Let $π$ be an irreducible admissible representation of $\rm{SL}_{n}(D)$ which is distinguished by $\rm{SL}_{n}^{*}(E)$. We give a multiplicity formula, i.e. a formula for the dimension of the $\mathbb{C}$-vector space ${\rm{Hom}}_{\rm{SL}_{n}^{*}(E)} (π, \mathbbm{1})$, where $\mathbbm{1}$ denotes the trivial representation of $\rm{SL}_{n}^{*}(E)$. This work is a non-split inner form analog of a work by Anandavardhanan-Prasad which gives a multiplicity formula for $\rm{SL}_{n}(F)$-distinguished irreducible admissible representation of $\rm{SL}_{n}(E)$. |
| title | Distinguished Representations for $\rm{SL}_n(D)$ where $D$ is a quaternion division algebra over a $p$-adic field |
| topic | Representation Theory Number Theory 22E50, 11S37, 20G25, 22E35 |
| url | https://arxiv.org/abs/2501.04500 |