Wall-crossings and a categorification of K-theory stable bases of the Springer resolution. (6th November 2021)

• Record Type: Journal Article
• Authors: Su, Changjian; Zhao, Gufang; Zhong, Changlong
• Is Part Of: Compositio mathematica. Volume 157:Number 11(2021)
• Page Start: 2341
• Page End: 2376
• Publication Date: 2021-11-06
• Subjects: stable basis -- Springer resolution -- Hecke algebra -- Verma module -- wall-crossing; 19L47 -- 20C08 -- 14M15 -- 17B50; Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X21007533
• Languages: English
• ISSNs: 0010-437X

Abstract

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.