Torsion in the space of commuting elements in a Lie group

Let G be a compact connected Lie group, and let Hom(Z(m), G) be the space of pairwise commuting m-tuples in G. We study the problem of which primes p Hom(Z(m), G)1, the connected component of Hom(Z(m), G) containing the element (1, . . . ,1), has p-torsion in homology. We will prove that Hom(Z(m), G)(1) for m = 2 has p-torsion in homology if and only if p divides the order of the Weyl group of G for G = SU(n) and some exceptional groups. We will also compute the top homology of Hom(Z(m), G)(1) and show that Hom(Z(m), G)(1) always has 2-torsion in homology whenever G is simply-connected and simple. Our computation is based on a new homotopy decomposition of Hom(Z(m), G)(1), which is of independent interest and enables us to connect torsion in homology to the combinatorics of the Weyl group.