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

Abstract

In this article we study the homology of spaces $\mathrm{Hom}({\mathbb{Z}}^n,G)$ 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 $\mathrm{Hom}(F_n/{\Gamma }_n^m,G)$, 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 $\mathrm{Comm}(G)$ studied by Cohen-Stafa and its nilpotent analogues.

20 pages, journal version