Let G a semisimple simply connected group over an algebraically closed field.

Let $Gr:= G(k((t))/G(k[[t]])$ be the affine grassmanian. It admits a stratification indexed by the dominant cocaracter

$Gr=\coprod\limits_{\lambda\in X_{*}(T)^{+}} G(k[[t]])t^{\lambda}G(k[[t]])$

Let $\overline{Gr}^{\lambda}$ be the closure of a strata. The question is concerning the resolution of sigularity.

I Know that we have to pass to a certain strata $\overline{IwI}\subset G(k((t)))/I$ where I is the Iwahori, and then solve the strata.

But first, how this $w$ is related to $\lambda$ and second if we consider the strata $\overline{Kt^{\lambda}I}$, do we have a proper birational map to $\overline{Gr}^{\lambda}$ and how can we solve $\overline{Kt^{\lambda}I}$?