MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

share|cite|improve this question
up vote 4 down vote accepted

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 $\lambda$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.