Twisted geometric Satake equivalence via gerbes on the factorizable grassmannian

Abstract

We prove a generalization of the twisted geometric Satake equivalence of Finkelberg--Lysenko in the context of the factorizable grassmannian of a reductive group G relative to a smooth curve X, similar to Gaitsgory's generalization in "On de Jong's conjecture" of the Satake equivalence itself. In order to express the dependence of this result on factorizability, we consider a certain 2-category of gerbes on this grassmannian, which we call "sf gerbes"; the theorem then concerns perverse sheaves twisted by such gerbes. This notion of twisting differs from, but in special cases is equivalent to, that considered by Finkelberg--Lysenko. We give a classification of these gerbes in terms of quadratic forms on the coweight lattice of G and gerbes on X. In addition, we introduce the technical device of universal local acyclicity (ULA) into the proof of the Satake equivalence, using it in combination with Beilinson's nearby cycles gluing of perverse sheaves to reduce all of the main constructions in the Mirkovic--Vilonen proof of the equivalence to the ULA case, where they have simple descriptions whose properties have simple proofs. We also give a geometric proof that the category of spherical perverse sheaves on the ordinary affine grassmannian is semisimple.