K‐theory Soergel bimodules

Abstract

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