Abstract

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 $\mathrm{Hom}<!--FUNCTION\; APPLICATION-->({\mathbb{Z}}^k,G)$ is path-connected. In particular for the reduced upper unitriangular groups and the reduced generalized Heisenberg groups, $\mathrm{Hom}<!--FUNCTION\; APPLICATION-->({\mathbb{Z}}^k,G)$ 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 $G$.

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