Let $G$ be a reductive group and $LG$ be the algebraic loop group of $G$; i.e. $LG(k) = G( k((t)) )$. There is a fair amount of literature on the affine Grassmanian $LG(k)/G(k[[t]])$ and its Picard group. I'm looking for reference that discusses $LG(k)/G(k[t^{-1}])$. For example, what is its Picard group? Does it have a nice stratification indexed by elements of the affine Weyl group?

Acutally I am interested in $LG(k)/G(k[t^{-1}])$ but I'm even more interested in $LG(k)/\mathcal{B}^-$ where $\mathcal{B}^-$ is the subgroup of elements of $G(k[t^{-1}])$ that map to $B^- \subset G$ under the map induced from $k[t^{-1}] \to k[t^{-1}]/(t^{-1}) = k$; here $B^-$ is a Borel subgroup of $G$