Drinfeld-Gaitsgory-Vinberg interpolation Grassmannian and geometric Satake equivalence (with appendix by Dennis Gaitsgory)

TL;DR

This paper proves a conjecture relating the cohomology of intersections of semiinfinite orbits in the affine Grassmannian to the universal enveloping algebra of a Lie algebra, advancing the understanding of geometric Satake correspondence.

Contribution

It confirms Schieder's conjecture linking cohomology with Lie algebra structures and constructs an action of the Schieder bialgebra on the geometric Satake fiber functor.

Findings

Proved Schieder's conjecture for reductive groups.

Constructed an action of Schieder bialgebra on the Satake fiber functor.

Proposed a conjectural generalization for Kac-Moody Lie algebras.

Abstract

Let $G$ be a reductive complex algebraic group. We fix a pair of opposite Borel subgroups and consider the corresponding semiinfinite orbits in the affine Grassmannian $Gr_G$. We prove Simon Schieder's conjecture identifying his bialgebra formed by the top compactly supported cohomology of the intersections of opposite semiinfinite orbits with $U(\check{\mathfrak n})$ (the universal enveloping algebra of the positive nilpotent subalgebra of the Langlands dual Lie algebra $\check{\mathfrak g}$). To this end we construct an action of Schieder bialgebra on the geometric Satake fiber functor. We propose a conjectural construction of Schieder bialgebra for an arbitrary symmetric Kac-Moody Lie algebra in terms of Coulomb branch of the corresponding quiver gauge theory.