The topology of nilpotent representations in reductive groups and their maximal compact subgroups

Abstract

Let $G$ be a complex reductive linear algebraic group and let $K\subset G$ be a maximal compact subgroup. Given a nilpotent group $\Gamma$ generated by $r$ elements, we consider the representation spaces $Hom\left(\Gamma ,G\right)$ and $Hom\left(\Gamma ,K\right)$ with the natural topology induced from an embedding into $G^r$ and $K^r$ respectively. The goal of this paper is to prove that there is a strong deformation retraction of $Hom\left(\Gamma ,G\right)$ onto $Hom\left(\Gamma ,K\right)$. We also obtain a strong deformation retraction of the geometric invariant theory quotient $Hom\left(\Gamma ,G\right)∕∕G$ onto the ordinary quotient $Hom\left(\Gamma ,K\right)∕K$.