The sign of linear periods
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866914871786864640 |
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| author | Anandavardhanan, U. K. Lu, Hengfei Matringe, Nadir Sécherre, Vincent Yang, Chang |
| author_facet | Anandavardhanan, U. K. Lu, Hengfei Matringe, Nadir Sécherre, Vincent Yang, Chang |
| contents | Let $G$ be a group with subgroup $H$, and let $(π,V)$ be a complex representation of $G$. The natural action of the normalizer $N$ of $H$ in $G$ on the space $\mathrm{Hom}_H(π,\mathbb{C})$ of $H$-invariant linear forms on $V$, provides a representation $χ_π$ of $N$ trivial on $H$, which is a character when $\mathrm{Hom}_H(π,\mathbb{C})$ is one dimensional. If moreover $G$ is a reductive group over a local field, and $π$ is smooth irreducible, it is an interesting problem to express $χ_π$ in terms of the possibly conjectural Langlands parameter $ϕ_π$ of $π$. In this paper we consider the following situation: $G=\mathrm{GL}_m(D)$ for $D$ a central division algebra of dimension $d^2$ over a local field $F$ of characteristic zero, $H$ is the centralizer of a non central element $δ\in G$ such that $δ^2$ is in the center of $G$, and $π$ has generic Jacquet-Langlands transfer to $\mathrm{GL}_{md}(F)$. In this setting the space $\mathrm{Hom}_H(π,\mathbb{C})$ is at most one dimensional. When $\mathrm{Hom}_H(π,\mathbb{C})\simeq \mathbb{C}$ and $H\neq N$, we prove that the value of the $χ_π$ on the non trivial class of $\frac{N}{H}$ is $(-1)^mε(ϕ_π)$ where $ε(ϕ_π)$ is the root number of $ϕ_π$. Along the way we extend many useful multiplicity one results for linear and Shalika models to the case of non split $G$. When $F$ is $p$-adic we also classify standard modules with linear periods and Shalika models, which are new results even when $D=F$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_12106 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The sign of linear periods Anandavardhanan, U. K. Lu, Hengfei Matringe, Nadir Sécherre, Vincent Yang, Chang Representation Theory 22E50, 11F70 Let $G$ be a group with subgroup $H$, and let $(π,V)$ be a complex representation of $G$. The natural action of the normalizer $N$ of $H$ in $G$ on the space $\mathrm{Hom}_H(π,\mathbb{C})$ of $H$-invariant linear forms on $V$, provides a representation $χ_π$ of $N$ trivial on $H$, which is a character when $\mathrm{Hom}_H(π,\mathbb{C})$ is one dimensional. If moreover $G$ is a reductive group over a local field, and $π$ is smooth irreducible, it is an interesting problem to express $χ_π$ in terms of the possibly conjectural Langlands parameter $ϕ_π$ of $π$. In this paper we consider the following situation: $G=\mathrm{GL}_m(D)$ for $D$ a central division algebra of dimension $d^2$ over a local field $F$ of characteristic zero, $H$ is the centralizer of a non central element $δ\in G$ such that $δ^2$ is in the center of $G$, and $π$ has generic Jacquet-Langlands transfer to $\mathrm{GL}_{md}(F)$. In this setting the space $\mathrm{Hom}_H(π,\mathbb{C})$ is at most one dimensional. When $\mathrm{Hom}_H(π,\mathbb{C})\simeq \mathbb{C}$ and $H\neq N$, we prove that the value of the $χ_π$ on the non trivial class of $\frac{N}{H}$ is $(-1)^mε(ϕ_π)$ where $ε(ϕ_π)$ is the root number of $ϕ_π$. Along the way we extend many useful multiplicity one results for linear and Shalika models to the case of non split $G$. When $F$ is $p$-adic we also classify standard modules with linear periods and Shalika models, which are new results even when $D=F$. |
| title | The sign of linear periods |
| topic | Representation Theory 22E50, 11F70 |
| url | https://arxiv.org/abs/2402.12106 |