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.