Wonderful embedding for group schemes in the Bruhat--Tits theory
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915348627849216 |
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| author | Li, Shang |
| author_facet | Li, Shang |
| contents | For a reductive group $G$ over a discretely valued Henselian field $k$, using valuations of root datum and concave functions, the Bruhat--Tits theory defines an important class of open bounded subgroups of $G(k)$ which are essential objects in representation theory and arithmetic geometry. Moreover, these subgroups are uniquely determined by smooth affine group schemes whose generic fibers are $G$ over the ring of integers of $k$. To study these group schemes, when $G$ is adjoint and quasi-split, we systematically construct wonderful embedding for these group schemes which are uniquely determined by a big cell structure. The way that we construct our wonderful embedding is different from classical methods in the sense that we avoid embedding a group scheme into an ambient space and taking closure. We use an intrinsic and functorial method which is a variant of Artin--Weil method of birational group laws. Beyond the quasi-split case, our wonderful embedding is constructed by étale descent. Moreover our wonderful embedding behaves in a similar way to the classical wonderful compactification of $G$. Our results can serve as a bridge between the theory of wonderful compactifications and the Bruhat--Tits theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_12777 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Wonderful embedding for group schemes in the Bruhat--Tits theory Li, Shang Algebraic Geometry For a reductive group $G$ over a discretely valued Henselian field $k$, using valuations of root datum and concave functions, the Bruhat--Tits theory defines an important class of open bounded subgroups of $G(k)$ which are essential objects in representation theory and arithmetic geometry. Moreover, these subgroups are uniquely determined by smooth affine group schemes whose generic fibers are $G$ over the ring of integers of $k$. To study these group schemes, when $G$ is adjoint and quasi-split, we systematically construct wonderful embedding for these group schemes which are uniquely determined by a big cell structure. The way that we construct our wonderful embedding is different from classical methods in the sense that we avoid embedding a group scheme into an ambient space and taking closure. We use an intrinsic and functorial method which is a variant of Artin--Weil method of birational group laws. Beyond the quasi-split case, our wonderful embedding is constructed by étale descent. Moreover our wonderful embedding behaves in a similar way to the classical wonderful compactification of $G$. Our results can serve as a bridge between the theory of wonderful compactifications and the Bruhat--Tits theory. |
| title | Wonderful embedding for group schemes in the Bruhat--Tits theory |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2505.12777 |