Two dimensional versions of the affine Grassmannian and their geometric description

TL;DR

This work extends the affine Grassmannian to two variables by studying quotients of the double loop group $LLG$ by natural subgroups, defining several 2D Grassmannians and proving their ind-scheme representability for solvable $G$. It then builds geometric counterparts on a smooth surface $X$ with a flag $(D,Z)$ and develops a fibre-functor framework to interpret these objects as moduli of bundles with trivializations on prescribed loci, establishing comparisons with the corresponding quotient Grassmannians in low-dimensional models such as $X= ext{A}^2$. Key results include Theorem A and Theorem B, which provide ind-representability and canonical identifications in special cases, and Theorem C with corollaries that extend these identifications to arbitrary $X$ via reduction to affine patches. The paper also outlines future directions, including potential 2D Geometric Satake, and suggests extending the geometric jet Grassmannian to varying flags in a broader program. Overall, it develops a concrete, flag-aware two-dimensional generalization of the Beilinson–Drinfeld Grassmannian with explicit ind-scheme models in the solvable setting and a geometric interpretation on surfaces.

Abstract

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.

Theorems & Definitions (67)

• Remark 1.1
• Lemma 1.2
• Corollary 1.3
• proof
• Remark 1.4
• Definition 1.5
• Remark 1.6
• Theorem 1.7
• Remark 1.8
• Lemma 1.9