On the geometric Satake equivalence for Kac-Moody groups

Abstract

This article establishes a geometric Satake equivalence for affine Kac-Moody groups as an equivalence of abelian semisimple categories over algebraically closed fields. We define a well-behaved category of equivariant sheaves on the double affine grassmannian \Gr_{G}, seen as a infty-stack, that we equip with a t-structure. We obtain an Braden's hyperbolic localization theorem for such a stack and prove that the constant term functor is t-exact using dimension estimates for affine MV-cycles. We then deduce the sought-for equivalence and prove that the IC-complexes match with the irreducible highest weight representations of the Langlands dual group G^{\vee}.