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This question is closely related to Peter Crooks question. Strata of the Affine Grassmannian

Let $G$ be a complex reductive group, $\mathcal{K}:= \mathbb{C}((t))$, $\mathcal{O}:= \mathbb{C}[[t]]$ and $Gr=G(\mathcal{K})/G(\mathcal{O})$ be the corresponding affine grassmannian.

The $G(\mathcal{O})$-orbits on $Gr$ may be indexed by the integral dominant coweights $P^+$. For $\lambda\in P^+$ it is well known that for every the $G(\mathcal{O})$-orbit $G(\mathcal{O}) [\lambda]= G(\mathcal{O}) \lambda G(\mathcal{O})/G(\mathcal{O})$ there exists an equivariant affine bundle $G(\mathcal{O}) [\lambda]\to G/P_\lambda$ where $P_\lambda\subset G$ is the parabolic subgroup of $G$ with Levi-factor $\lambda$.

Is there also an equivariant vector bundle $G(\mathcal{O}) [\lambda]\to G/P_\lambda$?

I think I read this somewhere but I can't remember where.

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up vote 7 down vote accepted

Yes, though it's more naturally an affine bundle than a vector one. One good way to think about it is the existence of loop rotation on both $G(\mathcal{O})$ and $G(\mathcal{K})$, that is, an action of $\mathbb{C^*}$ induced by the action on $\mathbb{C}((t))$ that gives $t$ weight 1. Under this action, if we consider $g\in G(\mathcal{O})$, its limit $\lim_{s\to 0}s\cdot g$ exists and lies in $G(\mathbb{C})$. In particular, the induced $\mathbb{C}^*$ on $Gr$ preserves $G(\mathcal{O})$ orbits, and these are even closed under taking $\lim_{s\to 0}s\cdot g$, which must lie in the orbit $G[t^\lambda]$, since $[t^\lambda]$ is fixed by this action. The map you're looking for is precisely $g\mapsto \lim_{s\to 0}s\cdot g$, and the fiber is the orbit of $[t^\lambda]$ under the "unipotent radical" of $G(\mathcal{O})$ (which is a finite dimensional affine space).

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