Configuration spaces of commuting elements

In this article we introduce the space of configurations of commuting elements in a topological group and show that it satisfies rational homological stability for the sequences of unitary, special unitary and symplectic groups. We also prove that it satisfies cohomological rational representation stability with respect to the number of elements in the tuple for finite products of such groups, in particular cohomological rational stability for the space of unordered configurations of commuting elements. Finally we present some computations of cohomology in the unstable range.