On Koszul duality for Kac-Moody groups

For any Kac-Moody group G G with Borel B B , we give a monoidal equivalence between the derived category of B B -equivariant mixed complexes on the flag variety G / B G/B and (a certain completion of) the derived category of G ∨ G^ -monodromic mixed complexes on the enhanced flag variety G ∨ / U ∨ G^ /U^ , here G ∨ G^ is the Langlands dual of G G . We also prove variants of this equivalence, one of which is the equivalence between the derived category of U U -equivariant mixed complexes on the partial flag variety G / P G/P and a certain “Whittaker model” category of mixed complexes on G ∨ / B ∨ G^ /B^ . 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 ].