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K-theory Soergel Bimodules

Jens Niklas Eberhardt

Abstract

We initiate the study of K-theory Soergel bimodules-a K-theory analog of classical Soergel bimodules. Classical Soergel bimodules can be seen as a completed and infinitesimal version of their new K-theoretic analog. We show that morphisms of K-theory Soergel bimodules can be described geometrically in terms of equivariant K-theoretic correspondences between Bott-Samelson varieties. We thereby obtain a natural categorification of K-theory Soergel bimodules in terms of equivariant coherent sheaves. We introduce a formalism of stratified equivariant K-motives on varieties with an affine stratification, which is a K-theoretric analog of the equivariant derived category of Bernstein-Lunts. We show that Bruhat-stratified torus-equivariant K-motives on flag varieties can be described in terms of chain complexes of K-theory Soergel bimodules. Moreover, we propose conjectures regarding an equivariant/monodromic Koszul duality for flag varieties and the quantum K-theoretic Satake.

K-theory Soergel Bimodules

Abstract

We initiate the study of K-theory Soergel bimodules-a K-theory analog of classical Soergel bimodules. Classical Soergel bimodules can be seen as a completed and infinitesimal version of their new K-theoretic analog. We show that morphisms of K-theory Soergel bimodules can be described geometrically in terms of equivariant K-theoretic correspondences between Bott-Samelson varieties. We thereby obtain a natural categorification of K-theory Soergel bimodules in terms of equivariant coherent sheaves. We introduce a formalism of stratified equivariant K-motives on varieties with an affine stratification, which is a K-theoretric analog of the equivariant derived category of Bernstein-Lunts. We show that Bruhat-stratified torus-equivariant K-motives on flag varieties can be described in terms of chain complexes of K-theory Soergel bimodules. Moreover, we propose conjectures regarding an equivariant/monodromic Koszul duality for flag varieties and the quantum K-theoretic Satake.
Paper Structure (32 sections, 17 theorems, 60 equations)

This paper contains 32 sections, 17 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Soergel bimodules
  3. $K$-theory Soergel bimodules
  4. Atiyah--Segal completion theorem
  5. Correspondences and Erweiterungssatz
  6. $K$-motives on flag varieties
  7. Further Directions
  8. Categorification of $K$-theory Soergel bimodules
  9. Diagrammatic Calculus and Algebraic Properties
  10. Equivariant/Monodromic duality
  11. Quantum $K$-theoretic Satake
  12. Structure of the paper
  13. Acknowledgements
  14. Preliminaries on equivariant $K$-theory and $K$-motives
  15. Definition
  16. ...and 17 more sections

Key Result

Theorem 1

Let $\mathbf{x}, \mathbf{y}$ be sequences of simple reflections. Then convolution induces an isomorphism

Theorems & Definitions (42)

  • Theorem : Theorem \ref{['thm:erweiterungssatzconvolutionktheory']}
  • Theorem : Corollary \ref{['cor:kmotivesviacomplexesofsoergelbimodules']}
  • Conjecture 1.1: Ungraded, uncompleted equivariant/monodromic Koszul duality
  • Conjecture 1.2
  • Remark 2.1
  • Remark 2.2
  • Proposition 3.1
  • proof
  • Definition 3.2
  • Proposition 3.3
  • ...and 32 more