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 K^def (G). We show that if G and A are finitely generated groups and A is abelian, then K^def (G A) K^def (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 K^def (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 Poincaré polynomial.