Download this article
Download this article For screen
For printing
Recent Issues

Volume 18
Issue 7, 1221–1401
Issue 6, 1039–1219
Issue 5, 847–1038
Issue 4, 631–846
Issue 3, 409–629
Issue 2, 209–408
Issue 1, 1–208

Volume 17, 12 issues

Volume 16, 10 issues

Volume 15, 10 issues

Volume 14, 10 issues

Volume 13, 10 issues

Volume 12, 10 issues

Volume 11, 10 issues

Volume 10, 10 issues

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
About the journal
Ethics and policies
Peer-review process
Submission guidelines
Submission form
Editorial board
Editors' interests
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Author index
To appear
Other MSP journals
Geometric properties of the Kazhdan–Lusztig Schubert basis

Cristian Lenart, Changjian Su, Kirill Zainoulline and Changlong Zhong

Vol. 17 (2023), No. 2, 435–464

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.

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

Open Access made possible by participating institutions via Subscribe to Open.