Vanishing theorems for representation homology and the derived cotangent complex

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) := pi 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 Gamma, we show that HRi (B Gamma, G) = 0 for all i > 0. For a closed Riemann surface Sigma(g) of genus g >= 1, we have HRi (Sigma(g), G) = 0 for all i > dim G. The sharp vanishing bounds for Sigma(g) actually depend on the genus: we conjecture that if g = 1, then HRi (Sigma(g), G) = 0 for i > rank G, and if g >= 2, then HRi (Sigma(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)[pi(1) (Sigma(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.