Hecke algebra action on twisted motivic Chern classes and K-theoretic stable envelopes

Let G be a linear semisimple algebraic group and B its Borel subgroup. Let T subset of B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {T}}\subset B$$\end{document} be the maximal torus. We study the inductive construction of Bott-Samelson varieties to obtain recursive formulas for the twisted motivic Chern classes of Schubert cells in G/B. To this end we introduce two families of operators acting on the equivariant K-theory KT(G/B)[y]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text {K}}_{\mathbb {T}}(G/B)[y]$$\end{document}, the right and left Demazure-Lusztig operators depending on a parameter. The twisted motivic Chern classes coincide (up to normalization) with the K-theoretic stable envelopes. Our results imply wall-crossing formulas for a change of the weight chamber and slope parameters. The right and left operators generate a twisted double Hecke algebra. We show that in the type A this algebra acts on the Laurent polynomials. This action is a natural lift of the action on KT(G/B)[y]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text {K}}_{\mathbb {T}}(G/B)[y]$$\end{document} with respect to the Kirwan map. We show that the left and right twisted Demazure-Lusztig operators provide a recursion for twisted motivic Chern classes of matrix Schubert varieties.