Hilbert-Poincare series for spaces of commuting elements in Lie groups

In this article we study the homology of spaces ${rm Hom}(mathbb{Z}^n,G)$ of ordered pairwise commuting $n$-tuples in a Lie group $G$. We give an explicit formula for the Poincare series of these spaces in terms of invariants of the Weyl group of $G$. By work of Bergeron and Silberman, our results also apply to ${rm Hom}(F_n/Gamma_n^m,G)$, where the subgroups $Gamma_n^m$ are the terms in the descending central series of the free group $F_n$. Finally, we show that there is a stable equivalence between the space ${rm Comm}(G)$ studied by Cohen-Stafa and its nilpotent analogues.