Motivic Chern classes of Schubert cells, Hecke algebras, and applications to Casselman's problem

TL;DR

This paper computes motivic Chern classes of Schubert cells in flag varieties, links them to Hecke algebra actions, and applies these results to solve conjectures related to Casselman's problem in representation theory.

Contribution

It introduces a recursive method to compute motivic Chern classes using Hecke algebra operators and connects these classes to Iwahori invariants and Casselman's basis, providing new proofs and applications.

Findings

Motivic Chern classes are determined recursively by Demazure-Lusztig operators.

The classes are shown to be equivalent to K-theoretic stable envelopes.

The results prove conjectures on transition matrix factorizations and holomorphy.

Abstract

Motivic Chern classes are elements in the K-theory of an algebraic variety $X$, depending on an extra parameter $y$. They are determined by functoriality and a normalization property for smooth $X$. In this paper we calculate the motivic Chern classes of Schubert cells in the (equivariant) K-theory of flag manifolds $G/B$. We show that the motivic class of a Schubert cell is determined recursively by the Demazure-Lusztig operators in the Hecke algebra of the Weyl group of $G$, starting from the class of a point. The resulting classes are conjectured to satisfy a positivity property. We use the recursions to give a new proof that they are equivalent to certain K-theoretic stable envelopes recently defined by Okounkov and collaborators, thus recovering results of Feh\'er, Rim\'anyi and Weber. The Hecke algebra action on the K-theory of the Langlands dual flag manifold matches the Hecke action on the Iwahori invariants of the principal series representation associated to an unramified character for a group over a nonarchimedean local field. This gives a correspondence identifying the duals of the motivic Chern classes to the standard basis in the Iwahori invariants, and the fixed point basis to Casselman's basis. We apply this correspondence to prove two conjectures of Bump, Nakasuji and Naruse concerning factorizations and holomorphy properties of the coefficients in the transition matrix between the standard and the Casselman's basis.