Commuting tuples in reductive groups and their maximal compact subgroups

Abstract

Let $G$ be a reductive algebraic group and $K\subset G$ a maximal compact subgroup. We consider the representation spaces $Hom\left({\mathbb{Z}}^k,K\right)$ and $Hom\left({\mathbb{Z}}^k,G\right)$ with the topology induced from an embedding into $K^k$ and $G^k$, respectively. The goal of this paper is to prove that $Hom\left({\mathbb{Z}}^k,K\right)$ is a strong deformation retract of $Hom\left({\mathbb{Z}}^k,G\right)$.