# Strata of the Affine Grassmannian

Let $G$ be a connected, simply-connected complex semisimple linear algebraic group, and denote by $\mathcal{G}$ its affine Grassmannian. Fix a maximal torus $T\subseteq G$. We know that $\mathcal{G}$ has a natural stratification $\{\mathcal{G}^{\lambda}\}_{\lambda\in\Lambda^+}$, where $\Lambda^+$ is the collection of dominant coweights of $T$. I am looking for a detailed formulation of the result that $\mathcal{G}^{\lambda}$ is an affine bundle over a partial flag variety. I would appreciate any and all references.

## 2 Answers

The general statement to know is, if $\mathbb G_m$ acts on a smooth complete scheme $X$, then each Białynicki-Birula stratum is an affine bundle over its fixed-point set. This is in the original paper [B-B].

It's tempting to try to apply it to the ind-scheme $\mathcal G$, but that's infinite-dimensional, and more importantly, not ind-smooth.

To actually apply it, start with $\mathcal G^\lambda$, take its closure, and resolve the singularities thereby introduced equivariantly w.r.t. the loop rotation action. Use that equivariance argue that $\mathcal G^\lambda$ is a B-B stratum of either its closure or the resolution.

• This seems like it's making things too complicated. If you look at the Iwahori orbit through $t^\lambda$, this is an affine space, and the orbit of $G(\mathcal{O})$ is just the induction from the Iwahori up to the whole group, and thus clearly an affine space bundle over $G(\mathcal{O})/I\cong G/B$. – Ben Webster Nov 20 '13 at 3:14

One can also argue as follows:

$G(\mathcal{O})$ operates transitivly on each stratum $\mathcal{G}^\lambda$. The isotropy group at $t^\lambda$ is $P^a_\lambda:= (t^\lambda)^{-1}G(\mathcal{O})t^\lambda\cap G(\mathcal{O})$. Let $P_\lambda$ denote the parabolic subgroup of $G$ with Levi factor $\lambda$. Then the evaluation at zero homomorphism $ev_0:G(\mathcal{O})\to G$ restricts to an epimorphism $P^a_\lambda \to P_\lambda$ with pro-unipotent kernel $U$. More over $ev_0$ induces a map $G(\mathcal{O})/P_\lambda^a\to G/P_\lambda$ with fiber $(\ker ev_0)/U$ which is easily seen to be affine.