On dual groups of symmetric varieties and distinguished representations of $p$-adic groups
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866908859314995200 |
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| author | Takeda, Shuichiro |
| author_facet | Takeda, Shuichiro |
| contents | Let $X=H\backslash G$ be a spherical variety over a $p$-adic field. Assume $G$ is split. Let $\widehat{G}$ be the Langlands dual group of $G$. There is a complex group $\widehat{G}_X$ whose root datum is the little Weyl group of $X$. It was proposed by Sakellaridis-Venkatesh and fully proven by Knop and Schalke that there is a homomorphism $\widehatφ_X:\widehat{G}_X\times\operatorname{SL}_2(\mathbb{C})\to \widehat{G}$. Conjecturally, this detects the $H$-distinguished representations of $G$.
In this strictly utilitarian note, assuming $X$ is a symmetric variety, we give a more conceptual way of constructing the homomorphism $\widehatφ_X:\widehat{G}_X\times\operatorname{SL}_2(\mathbb{C})\to \widehat{G}$, and make a few conjectures on how $\widehatφ_X$ is related to $H$-distinguished representations of $G$ by using various known examples and conjectures, especially in the framework of the theory of Kato-Takano and Lagier on relative cuspidality and relative square integrability. We will also show that the local Langlands parameter of the trivial representation of $G$ factors through $\widehatφ_X$ for any symmetric variety $X=H\backslash G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_15800 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On dual groups of symmetric varieties and distinguished representations of $p$-adic groups Takeda, Shuichiro Representation Theory Number Theory Let $X=H\backslash G$ be a spherical variety over a $p$-adic field. Assume $G$ is split. Let $\widehat{G}$ be the Langlands dual group of $G$. There is a complex group $\widehat{G}_X$ whose root datum is the little Weyl group of $X$. It was proposed by Sakellaridis-Venkatesh and fully proven by Knop and Schalke that there is a homomorphism $\widehatφ_X:\widehat{G}_X\times\operatorname{SL}_2(\mathbb{C})\to \widehat{G}$. Conjecturally, this detects the $H$-distinguished representations of $G$. In this strictly utilitarian note, assuming $X$ is a symmetric variety, we give a more conceptual way of constructing the homomorphism $\widehatφ_X:\widehat{G}_X\times\operatorname{SL}_2(\mathbb{C})\to \widehat{G}$, and make a few conjectures on how $\widehatφ_X$ is related to $H$-distinguished representations of $G$ by using various known examples and conjectures, especially in the framework of the theory of Kato-Takano and Lagier on relative cuspidality and relative square integrability. We will also show that the local Langlands parameter of the trivial representation of $G$ factors through $\widehatφ_X$ for any symmetric variety $X=H\backslash G$. |
| title | On dual groups of symmetric varieties and distinguished representations of $p$-adic groups |
| topic | Representation Theory Number Theory |
| url | https://arxiv.org/abs/2308.15800 |