We study the derived representation scheme DRepg( a) parametrizing the representations of a Lie algebra a in a reductive Lie algebra g. We define two canonical maps Trg(a): HC(r) •(a) → H• [DRepg(a)]G and phig(a) : H•[DRep g(a)]G → H•[DReph( a)]W , called the Drinfeld trace and the derived Harish-Chandra homomorphism, respectively. The Drinfeld trace is defined on the r-th Hodge component of the cyclic homology of the universal enveloping algebra Ua of the Lie algebra a and depends on the choice of a G-invariant polynomial P epsilon Symr( g*)G on the Lie algebra g. The Harish Chandra homomorphism phig( a) is a graded algebra homomorphism extending to representation homology the natural restriction map k[Rep g(a)]G → k[Reph(a)] W , where h ⊂ g is a Cartan subalgebra of g and W is the associated Weyl group. We give general formulas for these maps in terms of Chern---Simons forms. As a consequence, we show that, if a is an abelian Lie algebra, the composite map phi g(a) • Tr g(a) is given by a canonical differential operator defined on differential forms on A = Sym( a) and depending only on the Cartan data (h, W, P), where P epsilon Sym( h*)W. We derive a combinatorial formula for this operator that plays a key role in the study of derived commuting schemes in [4].