Left Demazure–Lusztig Operators on Equivariant (Quantum) Cohomology and K-Theory

Abstract We study the Demazure–Lusztig operators induced by the left multiplication on partial flag manifolds $G/P$. We prove that they generate the Chern–Schwartz–MacPherson classes of Schubert cells (in equivariant cohomology), respectively their motivic Chern classes (in equivariant K-theory), in...

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Bibliographic Details
Published in:International mathematics research notices 2022-08, Vol.2022 (16), p.12096-12147
Main Authors: Mihalcea, Leonardo C, Naruse, Hiroshi, Su, Changjian
Format: Article
Language:English
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Summary:Abstract We study the Demazure–Lusztig operators induced by the left multiplication on partial flag manifolds $G/P$. We prove that they generate the Chern–Schwartz–MacPherson classes of Schubert cells (in equivariant cohomology), respectively their motivic Chern classes (in equivariant K-theory), in any partial flag manifold. Along the way, we advertise many properties of the left and right divided difference operators in cohomology and K-theory and their actions on Schubert classes. We apply this to construct left divided difference operators in equivariant quantum cohomology, and equivariant quantum K-theory, generating Schubert classes and satisfying a Leibniz rule compatible with the quantum product.
ISSN:1073-7928
1687-0247
DOI:10.1093/imrn/rnab049