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 $B^{\vee }$-monodromic mixed complexes on the enhanced flag variety $G^{\vee }/U^{\vee }$, here $G^{\vee }$ 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 certain "Whittaker model" category of mixed complexes on $G^{\vee }/B^{\vee }$. In all these equivalences, intersection cohomology sheaves correspond to (free-monodromic) tilting sheaves. Our results generalize the Koszul duality patterns for reductive groups in the work of Beilinson, Ginzburg and Soergel .

92 pages; with appendices by Zhiwei Yun; final version

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