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. In this paper, we study the cotangent complex of the derived $G$-representation scheme ${\rm DRep}_G(X)$ of a pointed connected topological space $X$. We use an (algebraic version of) unstable Adams spectral sequence relating the cotangent homology of ${\rm DRep}_G(X)$ to the representation homology ${\rm HR}_*(X,G) := \pi_*{\mathcal O}[{\rm DRep}_G(X)]$ to prove some vanishing theorems for groups and geometrically interesting spaces. Our examples include virtually free groups, Riemann surfaces, link complements in ${\mathbb R}^3$ and generalized lens spaces. In particular, for any f.g. virtually free group $\Gamma$, we show that $\, {\rm HR}_i({\rm B}\Gamma, G) = 0 \,$ for all $i > 0$. For a closed Riemann surface $\Sigma_g$ of genus $g \ge 1$, we have $\, {\rm HR}_i(\Sigma_g, G) = 0 \,$ for all $i > \dim G$. The sharp vanishing bounds for $\Sigma_g$ depend actually on the genus: we conjecture that if $g = 1$, then $\, {\rm HR}_i(\Sigma_g, G) = 0 \,$ for $i > {\rm rank}\,G$, and if $g \ge 2$, then $\, {\rm HR}_i(\Sigma_g, G) = 0 \,$ for $i > \dim\,{\mathcal Z}(G) \,$, where ${\mathcal Z}(G)$ is the center of $G$. We prove these bounds locally on the smooth locus of the representation scheme ${\rm Rep}_G[\pi_1(\Sigma_g)]\,$ 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 ${\rm DRep}_G(X)$ in the sense of Ciocan-Fontanine and Kapranov. We give a new `Tor formula' for this class in terms of functor homology.