Two dimensional versions of the affine Grassmannian and their geometric description

For a smooth affine algebraic group $G$ over an algebraically closed field, we consider several two-variables generalizations of the affine Grassmannian $G(!(t)!)/G[![t]!]$, given by quotients of the double loop group $G(!(x)!)(!(y)!)$. We prove that they are representable by ind-schemes if $G$ is solvable. Given a smooth surface $X$ and a flag of subschemes of $X$, we provide a geometric interpretation of the two-variables Grassmannians, in terms of bundles and trivialisation data defined on appropriate loci in $X$, which depend on the flag.

💡 Research Summary

The paper studies two‑dimensional analogues of the affine Grassmannian for a smooth affine algebraic group (G) over an algebraically closed field (k). The classical affine Grassmannian is the fppf sheaf (Gr_G = LG/JG) where (LG(R)=G(R((t)))) and (JG(R)=G(R