Wall-crossings and a categorification of $K$-theory stable bases of the Springer resolution

TL;DR

This paper explores the relationship between $K$-theory stable bases of the Springer resolution, their transformations via affine Hecke algebra operators, and their categorification through Lie algebra Verma modules, revealing connections to quantum cohomology monodromy.

Contribution

It establishes a categorification of $K$-theory stable bases using Verma modules and links wall-crossing matrices to quantum cohomology monodromy.

Findings

Change of alcoves operators are Demazure-Lusztig operators.

Stable bases are categorified by Verma modules.

Wall-crossing matrices match quantum cohomology monodromy.

Abstract

We compare the $K$-theory stable bases of the Springer resolution associated to different affine Weyl alcoves. We prove that (up to relabelling) the change of alcoves operators are given by the Demazure-Lusztig operators in the affine Hecke algebra. We then show that these bases are categorified by the Verma modules of the Lie algebra, under the localization of Lie algebras in positive characteristic of Bezrukavnikov, Mirkovi\'c, and Rumynin. As an application, we prove that the wall-crossing matrices of the $K$-theory stable bases coincide with the monodromy matrices of the quantum cohomology of the Springer resolution.