Supercuspidal representations of $\mathrm{GL}_{n}(F)$ distinguished by a unitary involution
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arXiv
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| Format: | Preprint |
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2019
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| _version_ | 1866915071694733312 |
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| author | Zou, Jiandi |
| author_facet | Zou, Jiandi |
| contents | Let $F/F_{0}$ be a quadratic extension of non-archimedean locally compact fields of residue characteristic $p\neq 2$. Let $R$ be an algebraically closed field of characteristic different from $p$. For $π$ a supercuspidal representation of $G=\mathrm{GL}_{n}(F)$ over $R$ and $G^τ$ a unitary group in $n$ variables contained in $G$, we prove that $π$ is distinguished by $G^τ$ if and only if $π$ is Galois invariant. When $R=\mathbb{C}$ and $F$ is a $p$-adic field, this result first as a conjecture proposed by Jacquet was proved in 2010's by Feigon-Lapid-Offen by using global method. Our proof is local which works for both complex case and $l$-modular case with $l\neq p$. We further study the dimension of $\mathrm{Hom}_{G^τ}(π,1)$ and show that it is at most one. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1909_10450 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Supercuspidal representations of $\mathrm{GL}_{n}(F)$ distinguished by a unitary involution Zou, Jiandi Representation Theory Number Theory Let $F/F_{0}$ be a quadratic extension of non-archimedean locally compact fields of residue characteristic $p\neq 2$. Let $R$ be an algebraically closed field of characteristic different from $p$. For $π$ a supercuspidal representation of $G=\mathrm{GL}_{n}(F)$ over $R$ and $G^τ$ a unitary group in $n$ variables contained in $G$, we prove that $π$ is distinguished by $G^τ$ if and only if $π$ is Galois invariant. When $R=\mathbb{C}$ and $F$ is a $p$-adic field, this result first as a conjecture proposed by Jacquet was proved in 2010's by Feigon-Lapid-Offen by using global method. Our proof is local which works for both complex case and $l$-modular case with $l\neq p$. We further study the dimension of $\mathrm{Hom}_{G^τ}(π,1)$ and show that it is at most one. |
| title | Supercuspidal representations of $\mathrm{GL}_{n}(F)$ distinguished by a unitary involution |
| topic | Representation Theory Number Theory |
| url | https://arxiv.org/abs/1909.10450 |