affine weyl group and affine schubert cells

Question

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 $\overline{IwI}/I$?

Answer 1

If $G$ isn't simply connected, then $G(F)$ isn't connected, so $G(F)/I$ isn't connected. Your question is about resolving those $\overline{IwI}/I$ that live in components other than the one containing the basepoint $I/I$.

On such a component $C$, there will still be a closed $I$-orbit $X_C$, automatically smooth. (In the familiar case, $X_C = \{ I/I \}$.) Generalize the Bott-Samelson-Demazure-Hansen resolution to $P_{\alpha_1} \times^I \cdots \times^I P_{\alpha_k} \times^I X_C \to C$. The LHS is smooth, and can be made to resolve any choice of $\overline{IwI}/I$, through suitable choice of $\vec \alpha$.