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 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.

v2: added a new section in the Appendix on the symmetry of left vs right operators; added some references and minor stylistic changes

References in corpus (2)

• Trigonometric weight functions as K-theoretic stable envelope maps for the cotangent bundle of a flag variety
• A Schubert calculus recurrence from the noncomplex W-action on G/B

Cited by in corpus (7)

• From motivic Chern classes of Schubert cells to their Hirzebruch and CSM classes
• Hook formula for Coxeter groups via the twisted group ring
• Bumpless pipe dreams meet Puzzles
• Chern Classes of Open Projected Richardson Varieties and of Affine Schubert Cells
• Automorphisms of the Quantum Cohomology of the Springer Resolution and Applications
• Chevalley formulae for the motivic Chern classes of Schubert cells and for the stable envelopes
• Geometric properties of the Kazhdan-Lusztig Schubert basis