Spaces of subgroups of toral groups

TL;DR

This paper investigates the structure of conjugacy classes of subgroups within toral parts of compact Lie groups, exploring invariants as sheaves and applying findings to rational G-equivariant cohomology theories, especially for rank 2 groups.

Contribution

It introduces a detailed analysis of the space of toral subgroup conjugacy classes and their invariants, advancing algebraic models for rational G-equivariant cohomology.

Findings

Explicit description for all toral subgroups of rank 2 groups

Invariants of subgroups form sheaves over the conjugacy class space

Framework supports algebraic modeling of rational G-equivariant cohomology

Abstract

We study the space of conjugacy classes of subgroups of a compact Lie group G whose identity component is a torus, and consider how various invariants of subgroups behave as sheaves over this space. This feeds in to the author's programme to give algebraic models of rational G-equivariant cohomology theories. The methods are illustrated by making the outcome explicit for all toral subgroups of compact connected rank 2 groups.