  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.
