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| Natura: | Preprint |
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2023
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| Accesso online: | https://arxiv.org/abs/2309.09049 |
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| _version_ | 1866910602850467840 |
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| author | DeBacker, Stephen |
| author_facet | DeBacker, Stephen |
| contents | Suppose $k$ is a nonarchimedean local field, $K$ is a maximally unramified extension of $k$, and $\mathbf{G}$ is a connected reductive $k$-group. If $\mathbf{T}$ is a $K$-minisotropic maximal $k$-torus in $\mathbf{G}$, then we use Bruhat-Tits theory to describe the stable classes in the $\mathbf{G}$-orbit of $\mathbf{T}$, the rational classes in the $\mathbf{G}$-orbit of $\mathbf{T}$, and the $k$-embeddings, up to rational conjugacy, into $\mathbf{G}$ of $\mathbf{T}$. We also provide, via Bruhat-Tits theory, a complete and explicit description of: the rational conjugacy classes of $K$-minisotropic maximal tame $k$-tori in $\mathbf{G}$; the stable classes of $K$-minisotropic maximal tame $k$-tori in $\mathbf{G}$; and the $k$-embeddings, up to rational conjugacy, into $\mathbf{G}$ of a $K$-minisotropic maximal tame $k$-torus of $\mathbf{G}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_09049 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Totally Ramified Maximal Tori and Bruhat-Tits theory DeBacker, Stephen Representation Theory Group Theory Primary 20G25, Secondary 22E35 Suppose $k$ is a nonarchimedean local field, $K$ is a maximally unramified extension of $k$, and $\mathbf{G}$ is a connected reductive $k$-group. If $\mathbf{T}$ is a $K$-minisotropic maximal $k$-torus in $\mathbf{G}$, then we use Bruhat-Tits theory to describe the stable classes in the $\mathbf{G}$-orbit of $\mathbf{T}$, the rational classes in the $\mathbf{G}$-orbit of $\mathbf{T}$, and the $k$-embeddings, up to rational conjugacy, into $\mathbf{G}$ of $\mathbf{T}$. We also provide, via Bruhat-Tits theory, a complete and explicit description of: the rational conjugacy classes of $K$-minisotropic maximal tame $k$-tori in $\mathbf{G}$; the stable classes of $K$-minisotropic maximal tame $k$-tori in $\mathbf{G}$; and the $k$-embeddings, up to rational conjugacy, into $\mathbf{G}$ of a $K$-minisotropic maximal tame $k$-torus of $\mathbf{G}$. |
| title | Totally Ramified Maximal Tori and Bruhat-Tits theory |
| topic | Representation Theory Group Theory Primary 20G25, Secondary 22E35 |
| url | https://arxiv.org/abs/2309.09049 |