Topological components of spaces of commuting elements in connected nilpotent Lie groups

We study the homotopy type of spaces of commuting elements in connected nilpotent Lie groups, via almost commuting elements in their Lie algebras. We give a necessary and sufficient condition on the fundamental group of such a Lie group $G$ to ensure $\mathrm{H}\mathrm{o}\mathrm{m}({\mathbb{Z}}^k,G)$ is path-connected. In particular for the reduced upper unitriangular groups and the reduced generalized Heisenberg groups, $\mathrm{H}\mathrm{o}\mathrm{m}({\mathbb{Z}}^k,G)$ is not path-connected, and we compute the homotopy type of its path-connected components in terms of Stiefel manifolds and the maximal torus of $G$.