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.

arXiv commentv2: 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 (10)

• Double Schubert polynomials for the classical groups
• Equivariant Chern classes of singular algebraic varieties with group actions
• Trigonometric weight functions as K-theoretic stable envelope maps for the cotangent bundle of a flag variety
• Motivic Chern classes and K-theoretic stable envelopes
• Quantum K-theory of Grassmannians
• Chern-Schwartz-MacPherson classes for Schubert cells in flag manifolds
• A Schubert calculus recurrence from the noncomplex W-action on G/B
• Equivariant K-theory of affine flag manifolds and affine Grothendieck polynomials
• Loop structure on equivariant $K$-theory of semi-infinite flag manifolds
• Whittaker functions from motivic Chern classes

Cited by in corpus (8)

• Shadows of characteristic cycles, Verma modules, and positivity of Chern-Schwartz-MacPherson classes of Schubert cells
• 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
• 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
• Chern Classes of Open Projected Richardson Varieties and of Affine Schubert Cells
• Geometric properties of the Kazhdan-Lusztig Schubert basis