Tits groups of affine Weyl groups
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866929385107357696 |
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| author | Ganapathy, Radhika |
| author_facet | Ganapathy, Radhika |
| contents | Let $G$ be a connected, reductive group over a non-archimedean local field $F$. Let $\breve F$ be the completion of the maximal unramified extension of $F$ contained in a separable closure $F_s$. In this article, we construct a Tits group of the affine Weyl group of $G(F)$ when the derived subgroup of $G_{\breve F}$ does not contain a simple factor of unitary type. If $G$ is a quasi-split ramified odd unitary group, we show that there always exist representatives in $G(F)$ of affine simple reflections that satisfy Coxeter relations (which is weaker than asking for the existence of a Tits group). If $G = U_{2r}, r \geq 3,$ is a quasi-split ramified even unitary group, we show that there don't even exist representatives in $G(F)$ of the affine simple reflections that satisfy Coxeter relations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_08976 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Tits groups of affine Weyl groups Ganapathy, Radhika Representation Theory 22E50, 20C08 Let $G$ be a connected, reductive group over a non-archimedean local field $F$. Let $\breve F$ be the completion of the maximal unramified extension of $F$ contained in a separable closure $F_s$. In this article, we construct a Tits group of the affine Weyl group of $G(F)$ when the derived subgroup of $G_{\breve F}$ does not contain a simple factor of unitary type. If $G$ is a quasi-split ramified odd unitary group, we show that there always exist representatives in $G(F)$ of affine simple reflections that satisfy Coxeter relations (which is weaker than asking for the existence of a Tits group). If $G = U_{2r}, r \geq 3,$ is a quasi-split ramified even unitary group, we show that there don't even exist representatives in $G(F)$ of the affine simple reflections that satisfy Coxeter relations. |
| title | Tits groups of affine Weyl groups |
| topic | Representation Theory 22E50, 20C08 |
| url | https://arxiv.org/abs/2406.08976 |