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

Published online by Cambridge University Press:  11 February 2019

Cristian Lenart
Affiliation:
Department of Mathematics and Statistics, State University of New York at Albany, Albany, NY12222, USA ([email protected]; [email protected]) 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 ([email protected]) 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 ([email protected]; [email protected]) URL: http://www.albany.edu/∼lenart/; http://www.albany.edu/∼cz954339/

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.

Type
Research Article
Copyright
© Cambridge University Press 2019

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