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

Part of: Special varieties Representation theory of groups Lie algebras and Lie superalgebras Topological $K$-theory

Published online by Cambridge University Press:  06 October 2021

Changjian Su ,
Gufang Zhao  and
Changlong Zhong
Changjian Su
Affiliation:
Department of Mathematics, University of Toronto, 40 St. George Street, Room 6290, Toronto, ONM5S 2E4, Canadachangjiansu@gmail.com
Gufang Zhao
Affiliation:
School of Mathematics and Statistics, The University of Melbourne, 813 Swanston Street, Parkville, VIC3010, Australiagufangz@unimelb.edu.au
Changlong Zhong
Affiliation:
State University of New York at Albany, 1400 Washington Avenue, ES 110, Albany, NY12222, USAczhong@albany.edu
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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.

Keywords

stable basisSpringer resolutionHecke algebraVerma modulewall-crossing

MSC classification

Primary: 19L47: Equivariant $K$-theory 20C08: Hecke algebras and their representations 14M15: Grassmannians, Schubert varieties, flag manifolds
Secondary: 17B50: Modular Lie (super)algebras
Type
Research Article
Information
Compositio Mathematica , Volume 157 , Issue 11 , November 2021 , pp. 2341 - 2376
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Copyright
© 2021 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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