Let $G$ a connected reductive split group over $k=\bar{k}$, $(B,T)$ a split Borel pair. Let $F:=k((t)))$. Let $\tilde{W}$ the extended Weyl group, $\tilde{W}=N_{G}(T(F))/T(O)$.

By Iwasawa decomposition,

$G(F)=\coprod\limits_{w\in\tilde{W}}IwI$,

where $I$ is the Iwahori subgroup associated to $B$.

We denote by $\overline{IwI}/I$ the closure of $IwI/I$ in $G(F)/I$. In general, this closure is singular and if we assume that $G$ is semsimple and simply connected, then we know that $\tilde{W}$ is generated by simple roots and we can solve the singularities with a Demazure resolution by writing $w=s_{1}\dots s_{l}$ with $s_{i}$ simple reflexions.

Nevertheless, when $G$ is only split conneceted reductive, then $\tilde{W}$ is no more generated by simple reflexions. My question is then how do we solve the singularities of $\overline{IwI}/I$?