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

TL;DR

The paper develops a comprehensive framework linking motivic Chern classes of Schubert cells to Demazure-Lusztig operators in equivariant K-theory, proving a DL-recursion that determines MC_y from a point and establishing a Hecke-duality with dual motivic classes. It then connects these motivic objects to K-theoretic stable envelopes, showing that MC_y are pull-backs of stable envelopes and that there is a canonical isomorphism between Iwahori invariants of principal series representations and localized K_T(G/B) via the DL action. Using this dictionary, the authors resolve conjectures of Bump, Nakasuji and Naruse about coefficients in transition matrices between standard and Casselman bases, including factorization and holomorphy properties. The work thus provides a robust geometric and representation-theoretic bridge between Schubert calculus, Hecke algebras, and p-adic representations, with implications for Casselman’s problem and Langlands-type correspondences in the Iwahori setting.

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 Fehér, Rimányi 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.

Theorems & Definitions (95)

• Theorem 1.1
• Conjecture 1: Positivity Conjecture
• Theorem 1.2
• Theorem 1.3: Geometric Bump-Nakasuji-Naruse Conjecture
• Theorem 1.4
• Proposition 2.1
• Definition 3.1
• Remark 3.2
• Lemma 3.3
• proof