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Geometric properties of the Kazhdan–Lusztig Schubert basis

Cristian Lenart, Changjian Su, Kirill Zainoulline and Changlong Zhong

Vol. 17 (2023), No. 2, 435–464
Abstract

We study classes determined by the Kazhdan–Lusztig basis of the Hecke algebra in the K-theory and hyperbolic cohomology theory of flag varieties. We first show that, in K-theory, the two different choices of Kazhdan–Lusztig bases produce dual bases, one of which can be interpreted as characteristic classes of the intersection homology mixed Hodge modules. In equivariant hyperbolic cohomology, we show that if the Schubert variety is smooth, then the class it determines coincides with the class of the Kazhdan–Lusztig basis; this property was known as the smoothness conjecture. For Grassmannians, we prove that the classes of the Kazhdan–Lusztig basis coincide with the classes determined by Zelevinsky’s small resolutions. These properties of the so-called KL Schubert basis show that it is the closest existing analogue to the Schubert basis for hyperbolic cohomology; the latter is a very useful testbed for more general elliptic cohomologies.

Keywords
Schubert calculus, flag variety, $K$-theory, hyperbolic cohomology, Hecke algebra, Kazhdan–Lusztig Schubert basis
Mathematical Subject Classification
Primary: 14M15, 55N20
Secondary: 05E99, 19L47, 20C08
Milestones
Received: 22 July 2021
Revised: 9 February 2022
Accepted: 4 April 2022
Published: 24 March 2023
Authors
Cristian Lenart
Department of Mathematics and Statistics
State University of New York at Albany
Albany, NY
United States
Changjian Su
Department of Mathematics
University of Toronto
Toronto, ON
Canada
Yau Mathematical Sciences Center
Tsinghua University
Beijing
China
Kirill Zainoulline
Department of Mathematics and Statistics
University of Ottawa
Ottawa, ON
Canada
Changlong Zhong
Department of Mathematics and Statistics
State University of New York at Albany
Albany, NY
United States

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