Equivariant Cohomology of Rationally Smooth Group Embeddings

TL;DR

This paper computes the equivariant cohomology rings of rationally smooth projective embeddings of reductive groups, linking algebraic and combinatorial data, and characterizes when these rings relate to toric varieties.

Contribution

It provides a detailed description of the equivariant cohomology of these embeddings and characterizes when the cohomology is derived from associated toric varieties.

Findings

Explicit formulas for equivariant cohomology rings.

Characterization of embeddings with cohomology from toric restrictions.

Connection between cohomology, roots, idempotents, and monoid data.

Abstract

We describe the equivariant cohomology ring of rationally smooth projective embeddings of reductive groups. These embeddings are the projectivizations of reductive monoids. Our main result describes their equivariant cohomology in terms of roots, idempotents, and underlying monoid data. Also, we characterize those embeddings whose equivariant cohomology ring is obtained via restriction to the associated toric variety. Such characterization is given in terms of the closed orbits.