Distinction of the Steinberg representation with respect to a symmetric pair
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866917794146156544 |
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| author | Wang, Chuijia Zou, Jiandi |
| author_facet | Wang, Chuijia Zou, Jiandi |
| contents | Let $K$ be a non-archimedean local field of residual characteristic $p\neq 2$. Let $G$ be a connected reductive group over $K$, let $θ$ be an involution of $G$ over $K$, and let $H$ be the connected component of $θ$-fixed subgroup of $G$ over $K$. By realizing the Steinberg representation of $G$ as the $G$-space of complex smooth harmonic cochains following the idea of Broussous--Courtès, we study its space of distinction by $H$ as a finite dimensional complex vector space. We give an upper bound of the dimension, and under certain conditions, we show that the upper bound is sharp by explicitly constructing a basis using the technique of Poincaré series. Finally, we apply our general theory to the case where $G$ is a general linear group and $H$ a special orthogonal subgroup, which leads to a complete classification result. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_03247 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Distinction of the Steinberg representation with respect to a symmetric pair Wang, Chuijia Zou, Jiandi Representation Theory Number Theory 22E50, 20E42, 11F70 Let $K$ be a non-archimedean local field of residual characteristic $p\neq 2$. Let $G$ be a connected reductive group over $K$, let $θ$ be an involution of $G$ over $K$, and let $H$ be the connected component of $θ$-fixed subgroup of $G$ over $K$. By realizing the Steinberg representation of $G$ as the $G$-space of complex smooth harmonic cochains following the idea of Broussous--Courtès, we study its space of distinction by $H$ as a finite dimensional complex vector space. We give an upper bound of the dimension, and under certain conditions, we show that the upper bound is sharp by explicitly constructing a basis using the technique of Poincaré series. Finally, we apply our general theory to the case where $G$ is a general linear group and $H$ a special orthogonal subgroup, which leads to a complete classification result. |
| title | Distinction of the Steinberg representation with respect to a symmetric pair |
| topic | Representation Theory Number Theory 22E50, 20E42, 11F70 |
| url | https://arxiv.org/abs/2410.03247 |