Spaces of homomorphisms, formality and Hochschild homology

Let G be a discrete group. The topological category of finite dimensional unitary representations of G is symmetric monoidal under direct sum and has an associated E_∞-space Kᵈᵉᶠ(G). We show that if G and A are finitely generated groups and A is abelian, then Kᵈᵉᶠ(G× A) Kᵈᵉᶠ(G)⊗ A as E_∞-spaces, where A is the Pontryagin dual of A. We deduce a homology stability result for the homomorphism varieties Hom(G× Zʳ,U(n)) using the local-to-global principle for homology stability of Kupers--Miller. For a finitely generated free group F and a field k of characteristic zero, we show that the singular k-chains in Kᵈᵉᶠ(F) are formal as an E_∞-k-algebra. Using this we describe the equivariant homology of Hom(F × A,U(n)) for every n in terms of higher Hochschild homology of an explicitly determined commutative k-algebra. As an example we show that Hom(F× Zʳ,U(2)) is $U(2)$-equivariantly formal for every r and we compute the Poincar{é} polynomial.