Let $K$ be a compact Lie group, $G$ be its complexication, and $F$ be any finitely generated Abelian group. We prove that the conjugation orbit space $\operatorname{Hom}(F,K)/K$ is a strong deformation retract of the GIT conjugation orbit space $\operatorname{Hom}(F,G)/G$. As a corollary, we determine necessary and sufficient conditions for $\operatorname{Hom}(F,G)/G$ to be irreducible when $G$ is connected and semisimple, and $F$ is free Abelian. This is joint work with C. Florentino.