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

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, Mirković, 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.

V2: 31 pages; exposition changes throughout

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• Trigonometric weight functions as K-theoretic stable envelope maps for the cotangent bundle of a flag variety
• Shadows of characteristic cycles, Verma modules, and positivity of Chern-Schwartz-MacPherson classes of Schubert cells
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Cited by in corpus (2)

• Stable bases of the Springer resolution and representation theory
• On equivariant oriented cohomology of Bott-Samelson varieties