Universal Koszul duality for Kac–Moody groups

Abstract

We prove a monoidal equivalence, called universal Koszul duality, between genuine equivariant K-motives on a Kac–Moody flag variety and constructible monodromic sheaves on its Langlands dual. The equivalence is obtained by a Soergel-theoretic description of both sides which extends results for finite-dimensional flag varieties by Taylor and the first author. Universal Koszul duality bundles together a whole family of equivalences for each point of a maximal torus. At the identity, it recovers an ungraded version of Beilinson, Ginzburg and Soergel’s and Bezrukavnikov and Yun’s Koszul duality for equivariant and unipotently monodromic sheaves. It also generalizes Soergel-theoretic descriptions for monodromic categories on finite-dimensional flag varieties by Lusztig and Yun, Gouttard and the second author. For affine Kac–Moody groups, our work sheds new light on the conjectured quantum Satake equivalences by Cautis and Kamnitzer as well as Gaitsgory. On our way, we establish foundations on six functors for reduced K-motives and introduce a formalism of constructible monodromic sheaves.