For a finite-dimensional reductive Lie algebra g, we will
introduce and study a derived representation scheme DRep(A,g) parametrizing
the representations of a given Lie algebra A in g. We relate the homology
of DRep(A,g) to the classical (Chevalley-Eilenberg) cohomology of current
Lie algebras. This allows us to construct a canonical map F from
DRep(A,g)^G to DRep(A,h)^W, relating the G-invariant part of representation
homology of A in g to the W-invariant part of representation homology of A
in a Cartan subalgebra of g. We call this map the derived Harish-Chandra
homomorphism as it is a natural homological extension of the classical
Harish-Chandra restriction map. We conjecture that if A is a two
dimensional abelian Lie algebra, then F is a quasi-isomorphism for any
finite-dimensional reductive Lie algebra g. We provide some evidence for
this conjecture and explain the relation to the strong Macdonald
conjectures proposed by P.Hanlon and B.Feigin in the late 80s and recently
proved (in full generality) by S. Fishel, I. Grojnowski and C. Teleman.?