Symplectic period for a representation of $GL_n(D)$
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866910458429046784 |
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| author | Sharma, Hariom Verma, Mahendra Kumar |
| author_facet | Sharma, Hariom Verma, Mahendra Kumar |
| contents | Let $D$ be a quaternion division algebra over a non-archimedean local field $K$ of characteristic zero, and let $Sp_n(D)$ be the unique non-split inner form of the symplectic group $Sp_{2n}(K)$. This paper classifies the irreducible admissible representations of $GL_{n}(D)$ with a symplectic period for $n = 3$ and $4$, i.e., those irreducible admissible representations $(π, V)$ of $GL_{n}(D)$ which have a linear functional $l$ on $V$ such that $l(π(h)v) = l(v)$ for all $v \in V$ and $h \in Sp_n(D)$. Our results also contain all unitary representations having a symplectic period, as stated in Prasad's conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_11674 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Symplectic period for a representation of $GL_n(D)$ Sharma, Hariom Verma, Mahendra Kumar Representation Theory 22E50, 11F70 Let $D$ be a quaternion division algebra over a non-archimedean local field $K$ of characteristic zero, and let $Sp_n(D)$ be the unique non-split inner form of the symplectic group $Sp_{2n}(K)$. This paper classifies the irreducible admissible representations of $GL_{n}(D)$ with a symplectic period for $n = 3$ and $4$, i.e., those irreducible admissible representations $(π, V)$ of $GL_{n}(D)$ which have a linear functional $l$ on $V$ such that $l(π(h)v) = l(v)$ for all $v \in V$ and $h \in Sp_n(D)$. Our results also contain all unitary representations having a symplectic period, as stated in Prasad's conjecture. |
| title | Symplectic period for a representation of $GL_n(D)$ |
| topic | Representation Theory 22E50, 11F70 |
| url | https://arxiv.org/abs/2311.11674 |