[Paper Review] Left Demazure-Lusztig operators on equivariant (quantum) cohomology and K theory

Leonardo C. Mihalcea, Hiroshi Naruse+1 more|arXiv (Cornell University)|Aug 28, 2020

Advanced Algebra and Geometry22 references4 citations

TL;DR

This paper introduces left Demazure-Lusztig operators acting on equivariant (quantum) cohomology and K-theory of partial flag manifolds $G/P$, proving they recursively generate Schubert classes via actions induced by left multiplication by Weyl group elements. The key contribution is establishing that these operators generate Chern-Schwartz-MacPherson and motivic Chern classes in equivariant cohomology and K-theory, respectively, and extend naturally to quantum cohomology and K-theory with a Leibniz rule compatible with the quantum product.

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.

Motivation & Objective

• To define and study left Demazure-Lusztig operators on equivariant cohomology and K-theory of partial flag manifolds $G/P$.
• To show that these operators generate Chern-Schwartz-MacPherson classes in equivariant cohomology and motivic Chern classes in equivariant K-theory.
• To extend the left operator framework to equivariant quantum cohomology and K-theory, preserving compatibility with the quantum product.
• To establish a geometric and algebraic framework for left actions that are compatible with Schubert class recursion and Hecke algebra structures.

Proposed method

• The paper defines left divided difference operators via left multiplication by Weyl group elements on $G/P$, inducing automorphisms on equivariant cohomology and K-theory rings.
• It constructs left Demazure-Lusztig operators as deformations of left divided difference operators, compatible with the quantum product in equivariant quantum cohomology and K-theory.
• The authors use localization isomorphisms $\overline{\Psi}_P: K_T(G/P) \xrightarrow{\sim} R(T) \otimes_{R(G)} R(T)^{W_P}$ to relate K-theory classes to representations and compute actions via character formulas.
• They prove commutative diagrams linking left/right actions, localization, and push-pull operations, showing consistency with known constructions in K-theory and equivariant cohomology.
• The method relies on the fact that left multiplication induces ring automorphisms on equivariant quantum cohomology and K-theory, enabling recursive generation of Schubert classes.
• The approach connects to convolution operators and Harada-Landweber-Sjamaar models, providing a geometric realization of the left Hecke algebra action.

Experimental results

Research questions

• RQ1 How do left Demazure-Lusztig operators act on equivariant cohomology and K-theory of partial flag manifolds $G/P$?
• RQ2 Can these operators generate Chern-Schwartz-MacPherson and motivic Chern classes of Schubert cells in $G/P$?
• RQ3 Do the left operators extend to equivariant quantum cohomology and K-theory while preserving the Leibniz rule with respect to the quantum product?
• RQ4 How do the left and right actions of the Weyl group relate geometrically and algebraically in the context of equivariant cohomology and K-theory?
• RQ5 What is the precise algebraic structure of the left action in terms of degenerate and full Hecke algebras?

Key findings

• Left Demazure-Lusztig operators generate the Chern-Schwartz-MacPherson classes of Schubert cells in equivariant cohomology of $G/P$.
• The same operators generate the motivic Chern classes of Schubert cells in equivariant K-theory of $G/P$.
• The left operators extend to equivariant quantum cohomology and K-theory, satisfying a Leibniz rule compatible with the quantum product.
• The left action on $K_T(G/P)$ is isomorphic to the action of $1 \otimes \partial'_i$ under the localization isomorphism $\overline{\Psi}_P$, linking it to representation theory.
• The left and right actions commute with the respective localization and push-pull maps, as shown via commutative diagrams involving $\pi_*$ and $\pi^*$.
• The left Demazure operators coincide with convolution operators and match the models of Harada-Landweber-Sjamaar, establishing consistency across geometric and algebraic frameworks.