Tits groups of affine Weyl groups

TL;DR

This paper constructs Tits groups for affine Weyl groups associated with certain reductive groups over local fields, highlighting cases where such groups exist or do not exist based on the group's structure.

Contribution

It introduces a construction of Tits groups for affine Weyl groups of reductive groups over local fields under specific conditions, and analyzes the existence of Coxeter relation representatives in unitary groups.

Findings

Tits groups are constructed for affine Weyl groups when the derived subgroup condition is met.

For quasi-split ramified odd unitary groups, Coxeter relation representatives always exist in G(F).

For even unitary groups, such representatives do not exist in G(F).

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.