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

## TL;DR

This paper derives explicit formulas for the homology of spaces of commuting elements in Lie groups, connecting algebraic invariants with topological properties, and explores stable equivalences with nilpotent analogues.

## Contribution

It provides a new explicit formula for the Poincare series of these spaces based on Weyl group invariants, extending previous work to broader classes of groups.

## Key findings

- Explicit Poincare series formula in terms of Weyl group invariants.
- Application of results to spaces of homomorphisms from quotients of free groups.
- Establishment of stable equivalence between commuting spaces and their nilpotent analogues.

## Abstract

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.