  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.
