  The main purpose of this paper is to introduce a method to stabilize certain
spaces of homomorphisms from finitely generated free abelian groups to a Lie
group $G$, namely $Hom(\mathbb Z^n,G)$. We show that this stabilized space of
homomorphisms decomposes after suspending once with summands which can be
reassembled, in a sense to be made precise below, into the individual spaces
$Hom(\mathbb Z^n,G)$ after suspending once. To prove this decomposition, a
stable decomposition of an equivariant function space is also developed. One
main result is that the topological space of all commuting elements in a
compact Lie group is homotopy equivalent to an equivariant function space after
inverting the order of the Weyl group. In addition, the homology of the
stabilized space admits a very simple description in terms of the tensor
algebra generated by the reduced homology of a maximal torus in favorable
cases. The stabilized space also allows the description of the additive reduced
homology of the individual spaces $Hom(\mathbb Z^n,G)$, with the order of the
Weyl group inverted.
