Homological stability for spaces of commuting elements in Lie groups

In this paper we study homological stability for spaces ${\rm Hom}($\mathbb{Z}$^n,G)$ofpairwisecommuting$n$-tuplesinaLiegroup$G$. We prove that for each $n⩾1$, these spaces satisfy rational homological stability as $G$ ranges through any of the classical sequences of compact, connected Lie groups, or their complexifications. We prove similar results for rational equivariant homology, for character varieties, and for the infinite-dimensional analogues of these spaces, $\mathrm{C}\mathrm{o}\mathrm{m}\mathrm{m}(G)$ and ${\rm B_{com}} G$, introduced by Cohen-Stafa and Adem-Cohen-Torres-Giese respectively. In addition, we show that the rational homology of the space of unordered commuting $n$-tuples in a fixed group $G$ stabilizes as $n$ increases. Our proofs use the theory of representation stability - in particular, the theory of ${\mathrm{F}\mathrm{I}}_W$-modules developed by Church-Ellenberg-Farb and Wilson. In all of the these results, we obtain specific bounds on the stable range, and we show that the homology isomorphisms are induced by maps of spaces.