resolution of strata of the affine grassmanian

Question

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 $\overline{Kt^{\lambda}I}$?

Answer 1

More generally, let $\overline{BwP}/P$ be an orbit closure in a partial flag manifold. If $w$ is minimal in the coset $w W_P$, then $\overline{BwB}/B \to \overline{BwP}/P$ is birational, so it suffices to resolve $\overline{BwB}/B$. We can do that with a Bott-Samelson-Demazure-Hansen manifold, constructed from a reduced word for $w$. One place to read about those is the book [Brion-Kumar] on Frobenius splitting.

In your case, $B=I$ and $P=G(k[[t]])$, $W/W_P$ is the coweight lattice $\Lambda$, and $W = W_{finite} \ltimes \Lambda$. So $w$ is the minimal representative of the coset of $\lambda$.