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

Abstract

Let G be a linear semisimple algebraic group and B its Borel subgroup. Let \({\mathbb {T}}\subset B\) 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 \({\text {K}}_{\mathbb {T}}(G/B)[y]\), 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 \({\text {K}}_{\mathbb {T}}(G/B)[y]\) 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.