Let $G$ be a connected, simplyconnected 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.
The general statement to know is, if $\mathbb G_m$ acts on a smooth complete scheme $X$, then each BiałynickiBirula stratum is an affine bundle over its fixedpoint set. This is in the original paper [BB].
It's tempting to try to apply it to the indscheme $\mathcal G$, but that's infinitedimensional, and more importantly, not indsmooth.
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 BB stratum of either its closure or the resolution.

4$\begingroup$ 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 $G(\mathcal{O})/I\cong G/B$. $\endgroup$ – 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 prounipotent 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.