Let G be a reductive affine algebraic group defined over a field k of characteristic zero. We study the cotangent complex of the derived G–representation scheme DRep G (X) of a pointed connected topological space X. We use an (algebraic version of) unstable Adams spectral sequence relating the cotangent homology of DRep G (X) to the representation homology HR∗(X, G):= π∗ O [DRep G (X)] to prove some vanishing theorems for groups and geometrically interesting spaces. Our examples include virtually free groups, Riemann surfaces, link complements in ℝ 3 and generalized lens spaces. In particular, for any finitely generated virtually free group Γ, we show that HR i (B Γ, G)= 0 for all i> 0. For a closed Riemann surface Σ g of genus g≥ 1, we have HR i (Σ g, G)= 0 for all i> dim G. The sharp vanishing bounds for Σ g actually depend on the genus: we conjecture that if g= 1, then HR i (Σ g, G)= 0 for i> r a n k G, and if g≥ 2, then HR i (Σ g, G)= 0 for i> dim Z (G), where Z (G) is the center of G. We prove these bounds locally on the smooth locus of the representation scheme Rep G [π 1 (Σ g)] in the case of complex connected reductive groups. One important consequence of our results is the existence of a well-defined K–theoretic virtual fundamental class for DRep G (X) in the sense of Ciocan-Fontanine and Kapranov (Geom. Topol. 13 (2009) 1779–1804). We give a new “Tor formula” for this class in terms of functor homology.