Left Demazure-Lusztig operators on equivariant (quantum) cohomology and K theory

TL;DR

The paper develops a comprehensive, geometry-driven framework for left Demazure-Lusztig operators acting on equivariant cohomology and K-theory of flag varieties, including partial flags $G/P$, and extends these constructions to quantum analogues. It proves that left DL operators recursively generate Schubert- and Peierls-type bases via actions on Chern-Schwartz-MacPherson classes and motivic Chern classes, and establishes analogous recursion formulas for Segre motivic classes, with dualities and R-matrix interpretations. The work then extends the left-operator formalism to equivariant quantum cohomology and quantum K-theory, showing that left Weyl-group multiplication induces automorphisms of the quantum rings and yields Leibniz rules for quantum products, enabling quantum Schubert calculus and potential double-quantum Schubert polynomials. An appendix connects these operators to convolution constructions, verifying their equivalence with classical Demazure operators and illustrating symmetry with right operators, thereby unifying classical and quantum pictures. Overall, the paper provides a robust toolkit for computing and understanding Schubert classes and their motivic and quantum refinements across (equivariant) cohomology and K-theory, with links to stable envelopes and geometric representation 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.

Theorems & Definitions (64)

• Theorem 1.1: Theorem \ref{['thm:csmleft']}
• Theorem 1.2: Theorem \ref{['thm:LDLMCP']}
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Corollary 3.3
• proof
• Lemma 3.4
• Theorem 4.1