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.