Distinguished representations for $\rm{SL}(n,F)$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912712090451968 |
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| author | Choiy, Kwangho Patel, Shiv Prakash |
| author_facet | Choiy, Kwangho Patel, Shiv Prakash |
| contents | Let $F$ be a finite field, and let $\mathbb{E}$ be either a quadratic field extension $E/F$ or the split algebra $F \oplus F$. We study distinguished representations of $\rm{SL}_{2n}(F)$ by the subgroup $H_{\flat} := \rm{SL}_{2n}(F) \cap \rm{GL}_{n}(\mathbb{E})$, which is a variation of the work of Anandavardhanan and Prasad on distinguished representations of $\rm{SL}_{n}(\mathbb{E})$ by the subgroup $\rm{SL}_n(F)$. This is in a similar framework of our earlier work of a $p$-adic non-split variation of Anandavardhanan-Prasad over finite fields. We give a formula for the dimension of the complex vector space $\rm{Hom}_{H_{\flat}}(π_{\flat}, 1)$ in terms of certain characters of $F^{\times}$, where $π_{\flat}$ is an irreducible representation which is also distinguished by $H_{\flat}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_12299 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Distinguished representations for $\rm{SL}(n,F)$ Choiy, Kwangho Patel, Shiv Prakash Representation Theory 11F70, 20G25 Let $F$ be a finite field, and let $\mathbb{E}$ be either a quadratic field extension $E/F$ or the split algebra $F \oplus F$. We study distinguished representations of $\rm{SL}_{2n}(F)$ by the subgroup $H_{\flat} := \rm{SL}_{2n}(F) \cap \rm{GL}_{n}(\mathbb{E})$, which is a variation of the work of Anandavardhanan and Prasad on distinguished representations of $\rm{SL}_{n}(\mathbb{E})$ by the subgroup $\rm{SL}_n(F)$. This is in a similar framework of our earlier work of a $p$-adic non-split variation of Anandavardhanan-Prasad over finite fields. We give a formula for the dimension of the complex vector space $\rm{Hom}_{H_{\flat}}(π_{\flat}, 1)$ in terms of certain characters of $F^{\times}$, where $π_{\flat}$ is an irreducible representation which is also distinguished by $H_{\flat}$. |
| title | Distinguished representations for $\rm{SL}(n,F)$ |
| topic | Representation Theory 11F70, 20G25 |
| url | https://arxiv.org/abs/2511.12299 |