Drinfeld–Gaitsgory–Vinberg interpolation Grassmannian and geometric Satake equivalence

Let G be a reductive complex algebraic group. We fix a pair of opposite Borel subgroups and consider the corresponding semi-infinite 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 semi-infinite orbits with U(n^∨) (the universal enveloping algebra of the positive nilpotent subalgebra of the Langlands dual Lie algebra 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.