Grassmannians and the equivariant cohomology of isotropy actions

TL;DR

This paper presents a simplified approach to computing the equivariant cohomology of Grassmannians and isotropy actions, building on GKM methods and applying a model from Kapovitch for connected Lie groups.

Contribution

It introduces a less involved, more accessible method for calculating equivariant cohomology rings of homogeneous spaces, simplifying previous GKM-based approaches.

Findings

Recovered the equivariant cohomology of Grassmannians using the new method.

Provided a streamlined proof approach that reduces complexity.

Enhanced understanding of isotropy actions on Grassmannians.

Abstract

Recent work of Chen He has determined through GKM methods the Borel equivariant cohomology with rational coefficients of the isotropy action on a real Grassmannian and an real oriented Grassmannian through GKM methods. In this expository note, we propound a less involved approach, due essentially to Vitali Kapovitch, to computing equivariant cohomology rings $H^*_K(G/H)$ for $G,K,H$ connected Lie groups, and apply it to recover the equivariant cohomology of the Grassmannians. The bulk is setup and commentary; once one believes in the model, the proof itself is under a page.