Supercuspidal representations of $\mathrm{GL}_{n}(F)$ distinguished by an orthogonal involution
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866912162235023360 |
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| author | Zou, Jiandi |
| author_facet | Zou, Jiandi |
| contents | Let $F$ be a non-archimedean locally compact field of residue characteristic $p\neq2$, let $G=\mathrm{GL}_{n}(F)$ and let $H$ be an orthogonal subgroup of $G$. For $π$ a complex smooth supercuspidal representation of $G$, we give a full characterization for the distinguished space $\mathrm{Hom}_{H}(π,1)$ being non-zero and we further study its dimension as a complex vector space, which generalizes a similar result of Hakim for tame supercuspidal representations. As a corollary, the embeddings of $π$ in the space of smooth functions on the set of symmetric matrices in $G$, as a complex vector space, is non-zero and of dimension four, if and only if the central character of $π$ evaluating at $-1$ is $1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2011_07349 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Supercuspidal representations of $\mathrm{GL}_{n}(F)$ distinguished by an orthogonal involution Zou, Jiandi Representation Theory Let $F$ be a non-archimedean locally compact field of residue characteristic $p\neq2$, let $G=\mathrm{GL}_{n}(F)$ and let $H$ be an orthogonal subgroup of $G$. For $π$ a complex smooth supercuspidal representation of $G$, we give a full characterization for the distinguished space $\mathrm{Hom}_{H}(π,1)$ being non-zero and we further study its dimension as a complex vector space, which generalizes a similar result of Hakim for tame supercuspidal representations. As a corollary, the embeddings of $π$ in the space of smooth functions on the set of symmetric matrices in $G$, as a complex vector space, is non-zero and of dimension four, if and only if the central character of $π$ evaluating at $-1$ is $1$. |
| title | Supercuspidal representations of $\mathrm{GL}_{n}(F)$ distinguished by an orthogonal involution |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2011.07349 |