The description of the space of commuting elements in a compact Lie group is an interesting algebro-geometric problem with applications in Mathematical Physics, notably in Supersymmetric Yang Mills theories.

When the Lie group is complex reductive, this space is the character variety of a free abelian group. Let \(K\) be a compact Lie group (not necessarily connected) and \(G\) be its complexification. We consider, more generally, an arbitrary finitely generated abelian group \(A\), and show that the conjugation orbit space $\mathrm{Hom}(A,K)/K$ is a strong deformation retract of the character variety $\mathrm{Hom}(A,G)/G$.

As a Corollary, in the case when \(G\) is connected and semisimple, we obtain necessary and sufficient conditions for $\mathrm{Hom}(A,G)/G$ to be irreducible. This is also related to an interesting open problem about irreducibility of the variety of \(k\) tuples of \(n\) by \(n\) commuting matrices.