Tits groups of affine Weyl groups
Abstract
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.
- Publication:
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arXiv e-prints
- Pub Date:
- June 2024
- DOI:
- 10.48550/arXiv.2406.08976
- arXiv:
- arXiv:2406.08976
- Bibcode:
- 2024arXiv240608976G
- Keywords:
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- Mathematics - Representation Theory;
- 22E50;
- 20C08
- E-Print:
- arXiv admin note: text overlap with arXiv:2107.01768