TITLE: Strata of the Affine Grassmannian
QUESTION [7 upvotes]: 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.

REPLY [5 votes]: 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.