Equivariant oriented cohomology of flag varieties

TL;DR

This paper connects the algebraic structures of equivariant oriented cohomology rings of flag varieties with geometric operators, providing a geometric justification for previously algebraically defined objects.

Contribution

It bridges algebraic and geometric perspectives by matching algebraic operators with geometric push-forward and pull-back maps in equivariant cohomology.

Findings

Identifies the equivariant cohomology ring with a dual coalgebra based on root data.

Shows how geometric operators correspond to algebraic structures.

Uses fixed point localization to relate cohomology rings to pointwise data.

Abstract

Given an equivariant oriented cohomology theory $h$, a split reductive group $G$, a maximal torus $T$ in $G$, and a parabolic subgroup $P$ containing $T$, we explain how the $T$-equivariant oriented cohomology ring $h_T(G/P)$ can be identified with the dual of a coalgebra defined using exclusively the root datum of $(G,T)$, a set of simple roots defining $P$ and the formal group law of $h$. In two papers [Push-pull operators on the formal affine Demazure algebra and its dual, arXiv:1312.0019] and [A coproduct structure on the formal affine Demazure algebra, arXiv:1209.1676], we studied the properties of this dual and of some related operators by algebraic and combinatorial methods, without any reference to geometry. The present paper can be viewed as a companion paper, that justifies all the definitions of the algebraic objects and operators by explaining how to match them to equivariant oriented cohomology rings endowed with operators constructed using push-forwards and pull-backs along geometric morphisms. Our main tool is the pull-back to the $T$-fixed points of $G/P$ which injects the cohomology ring in question into a direct product of a finite number of copies of the $T$-equivariant oriented cohomology of a point.