K‐theory Soergel bimodules.

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