On smooth-group actions on reductive groups and spherical buildings
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866912278283026432 |
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| author | Adler, Jeffrey D. Lansky, Joshua M. Spice, Loren |
| author_facet | Adler, Jeffrey D. Lansky, Joshua M. Spice, Loren |
| contents | Let $k$ be a field, and suppose that $Γ$ is a smooth $k$-group that acts on a connected, reductive $k$-group $\widetilde G$. Let $G$ denote the maximal smooth, connected subgroup of the group of $Γ$-fixed points in $\widetilde G$. Under fairly general conditions, we show that $G$ is a reductive $k$-group, and that the image of the functorial embedding $\mathscr{S}(G) \longrightarrow \mathscr{S}(\widetilde G)$ of spherical buildings is the set of ``$Γ$-fixed points in $\mathscr{S}(\widetilde G)$'', in a suitable sense. In particular, we do not need to assume that $Γ$ has order relatively prime to the characteristic of $k$ (nor even that $Γ$ is finite), nor that the action of $Γ$ preserves a Borel-torus pair in $\widetilde G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_00183 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On smooth-group actions on reductive groups and spherical buildings Adler, Jeffrey D. Lansky, Joshua M. Spice, Loren Representation Theory Algebraic Geometry Group Theory 20G07, 20G15, 14L30, 20E42 Let $k$ be a field, and suppose that $Γ$ is a smooth $k$-group that acts on a connected, reductive $k$-group $\widetilde G$. Let $G$ denote the maximal smooth, connected subgroup of the group of $Γ$-fixed points in $\widetilde G$. Under fairly general conditions, we show that $G$ is a reductive $k$-group, and that the image of the functorial embedding $\mathscr{S}(G) \longrightarrow \mathscr{S}(\widetilde G)$ of spherical buildings is the set of ``$Γ$-fixed points in $\mathscr{S}(\widetilde G)$'', in a suitable sense. In particular, we do not need to assume that $Γ$ has order relatively prime to the characteristic of $k$ (nor even that $Γ$ is finite), nor that the action of $Γ$ preserves a Borel-torus pair in $\widetilde G$. |
| title | On smooth-group actions on reductive groups and spherical buildings |
| topic | Representation Theory Algebraic Geometry Group Theory 20G07, 20G15, 14L30, 20E42 |
| url | https://arxiv.org/abs/2503.00183 |