On Koszul duality for Kac-Moody groups

Abstract

For any Kac-Moody group G with Borel B, we give a monoidal equivalence between the derived category of B-equivariant mixed complexes on the flag variety G/B and (a certain completion of) the derived category of G[superscript V]-monodromic mixed complexes on the enhanced flag variety G[superscript V]/U[superscript V], here G[superscript V] is the Langlands dual of G. We also prove variants of this equivalence, one of which is the equivalence between the derived category of U-equivariant mixed complexes on the partial flag variety G/P and a certain ``Whittaker model'' category of mixed complexes on G[superscript V]/B[superscript V]. In all these equivalences, intersection cohomology sheaves correspond to (free-monodromic) tilting sheaves. Our results generalize the Koszul duality patterns for reductive groups in [BGS96].