Hilbert–Poincaré series for spaces of commuting elements in Lie groups

Abstract

In this article we study the homology of spaces Hom(Zn,G) of ordered pairwise commuting n-tuples in a Lie group G. We give an explicit formula for the Poincaré 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 Hom(Fn/Γmn,G) , where the subgroups Γmn are the terms in the descending central series of the free group Fn . Finally, we show that there is a stable equivalence between the space Comm(G) studied by Cohen–Stafa and its nilpotent analogues.