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Universal Koszul Duality for Kac-Moody Groups

Jens Niklas Eberhardt, Arnaud Eteve

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-Soergel's and Bezrukavnikov-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-Yun, Gouttard and the second author. For affine Kac-Moody groups, our work sheds new light on the conjectured quantum Satake equivalences by Cautis-Kamnitzer and Gaitsgory. On our way, we establish foundations on six functors for reduced K-motives and introduce a formalism of constructible monodromic sheaves.

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-Soergel's and Bezrukavnikov-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-Yun, Gouttard and the second author. For affine Kac-Moody groups, our work sheds new light on the conjectured quantum Satake equivalences by Cautis-Kamnitzer and Gaitsgory. On our way, we establish foundations on six functors for reduced K-motives and introduce a formalism of constructible monodromic sheaves.
Paper Structure (50 sections, 66 theorems, 129 equations)

This paper contains 50 sections, 66 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Genuine equivariant $K$-motives
  4. $K$-theoretic Hecke category
  5. Monodromic sheaves
  6. Monodromic Hecke category
  7. Universal Koszul duality
  8. Specialization
  9. Further directions
  10. Parabolic/Whittaker duality
  11. Quantum Satake equivalence
  12. Structure
  13. Notations
  14. Acknowledgements
  15. Kac--Moody groups and $K$-theory Soergel bimodules
  16. ...and 35 more sections

Key Result

Theorem 1

There is an equivalence of monoidal categories between the $K$-theoretic Hecke category and the categoryThroughout this paper, we work with $\infty$-categories. On the level of homotopy categories, the target of the equivalence is the bounded homotopy category of chain complexes. of bounded chain complexes of $K$-theoretic Soergel bimodules asso

Theorems & Definitions (136)

  • Theorem : \ref{['thm:soergelktheoreticheckecategory']}
  • Theorem : \ref{['lemCharacterizationMonodromic']}
  • Definition : \ref{['def:monodromicSheaves']}
  • Theorem : \ref{['thm:soergelmonodromicheckecategory']}
  • Theorem : Universal Koszul Duality, \ref{['thm:main']}
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Lemma 2.4
  • ...and 126 more