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.
Affine Grassmannian – Understanding Strata
ag.algebraic-geometryreference-requestrt.representation-theory
Best Answer
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.