Cohomology of the spaces of commuting elements in Lie groups of rank two

TL;DR

This paper computes the cohomology rings of spaces of commuting pairs in rank-two simple Lie groups, extending Baird's work on invariants of Weyl groups to specific cases.

Contribution

It explicitly determines the cohomology rings of Hom$(\mathbb{Z}^2,G)$ for rank-two simple Lie groups, building on Baird's invariance results.

Findings

Cohomology rings for Hom$(\mathbb{Z}^2,G)$ are explicitly computed.

Results extend Baird's invariance approach to rank-two cases.

Provides algebraic descriptions of commuting element spaces in Lie groups.

Abstract

Let $G$ be the classical group, and let Hom$(\mathbb{Z}^m,G)$ denote the space of commuting $m$-tuples in $G$. Baird proved that the cohomology of Hom$(\mathbb{Z}^m,G)$ is identified with a certain ring of invariants of the Weyl group of $G$. In this paper by using the result of Baird we give the cohomology ring of Hom$(\mathbb{Z}^2,G)$ for simple Lie group $G$ of rank 2.