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PARABOLIC KAZHDAN–LUSZTIG BASIS, SCHUBERT CLASSES, AND EQUIVARIANT ORIENTED COHOMOLOGY

Part of: (Co)homology theory Topological $K$-theory Algebraic combinatorics Homology and cohomology theories Special varieties

Published online by Cambridge University Press:  11 February 2019

Cristian Lenart ,
Kirill Zainoulline Open the ORCID record for Kirill Zainoulline [Opens in a new window]  and
Changlong Zhong
Cristian Lenart
Affiliation:
Department of Mathematics and Statistics, State University of New York at Albany, Albany, NY12222, USA (clenart@albany.edu; czhong@albany.edu) URL: http://www.albany.edu/∼lenart/; http://www.albany.edu/∼cz954339/
Kirill Zainoulline
Affiliation:
Department of Mathematics and Statistics, University of Ottawa, 150 Louis-Pasteur, Ottawa, ON, K1N 6N5, Canada (kirill@uottawa.ca) URL: http://mysite.science.uottawa.ca/kzaynull/
Changlong Zhong
Affiliation:
Department of Mathematics and Statistics, State University of New York at Albany, Albany, NY12222, USA (clenart@albany.edu; czhong@albany.edu) URL: http://www.albany.edu/∼lenart/; http://www.albany.edu/∼cz954339/
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Abstract

We study the equivariant oriented cohomology ring $\mathtt{h}_{T}(G/P)$ of partial flag varieties using the moment map approach. We define the right Hecke action on this cohomology ring, and then prove that the respective Bott–Samelson classes in $\mathtt{h}_{T}(G/P)$ can be obtained by applying this action to the fundamental class of the identity point, hence generalizing previously known results of Chow groups by Brion, Knutson, Peterson, Tymoczko and others. Our main result concerns the equivariant oriented cohomology theory $\mathfrak{h}$ corresponding to the 2-parameter Todd genus. We give a new interpretation of Deodhar’s parabolic Kazhdan–Lusztig basis, i.e., we realize it as some cohomology classes (the parabolic Kazhdan–Lusztig (KL) Schubert classes) in $\mathfrak{h}_{T}(G/P)$. We make a positivity conjecture, and a conjecture about the relationship of such classes with smoothness of Schubert varieties. We also prove the latter in several special cases.

Keywords

Schubert calculuselliptic cohomologyflag varietyHecke algebraparabolic Kazhdan–Lusztig basis

MSC classification

Primary: 14M15: Grassmannians, Schubert varieties, flag manifolds 14F43: Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
Secondary: 55N20: Generalized (extraordinary) homology and cohomology theories 55N22: Bordism and cobordism theories, formal group laws 19L47: Equivariant $K$-theory 05E99: None of the above, but in this section
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 19 , Issue 6 , November 2020 , pp. 1889 - 1929
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Copyright
© Cambridge University Press 2019

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