Abstract. We study the derived representation scheme DRep n (A) parametrizing the n-dimensional representations of an associative algebra A over a field of characteristic zero. We show that the homology of DRep n (A) is isomorphic to the Chevalley-Eilenberg homology of the current Lie coalgebra gl * n (C) defined over a Koszul dual coalgebra of A. This gives a conceptual explanation to main results of [BKR] and [BR], relating them (via Koszul duality) to classical theorems on (co)homology of current Lie algebras gl n (A). We extend the above isomorphism to representation schemes of Lie algebras: for a finite-dimensional reductive Lie algebra g, we define the derived affine scheme DRep g (a) parametrizing the representations (in g) of a Lie algebra a; we show that the homology of DRep g (a) is isomorphic to the Chevalley-Eilenberg homology of the Lie coalgebra g * (C), where C is a cocommutative DG coalgebra Koszul dual to the Lie algebra a. We construct a canonical DG algebra map Φg(a) : DRep g (a) G → DRep h (a) W , relating the G-invariant part of representation homology of a Lie algebra a in g to the Winvariant 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, for a two-dimensional abelian Lie algebra a, the derived Harish-Chandra homomorphism is a quasi-isomorphism. We provide some evidence for this conjecture, including proofs for gl 2 and sl 2 as well as for gl n , sln, son and sp 2n in the inductive limit as n → ∞. For any complex reductive Lie algebra g, we compute the Euler characteristic of DRep g (a) G in terms of matrix integrals over G and compare it to the Euler characteristic of DRep h (a) W . This yields an interesting combinatorial identity, which we prove for gl n and sln (for all n). Our identity is analogous to the classical Macdonald identity, and our quasi-isomorphism conjecture is analogous to the strong Macdonald conjecture proposed in [H1, F] and proved in [FGT]. We explain this analogy by giving a new homological interpretation of Macdonald's conjectures in terms of derived representation schemes, parallel to our Harish-Chandra quasi-isomorphism conjecture.

Abstract. This paper is a sequel to [4], where we study the derived representation scheme DRep g (a) parametrizing the representations of a Lie algebra a in a reductive Lie algebra g. In [4], we constructed two canonical maps Trg(a) : HC (r)W called the Drinfeld trace and the derived Harish-Chandra homomorphism, respectively. In this paper, we give an explicit formula for the Drinfeld trace in terms of Chern-Simons forms. As a consequence, we show that, if a is an abelian Lie algebra, the composite map Φg(a) • Trg(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 ∈ Sym(h * ) W . We prove a combinatorial formula for this operator that plays an important role in the study of derived commuting schemes in [4].

In this short note we show that representation and character varieties of discrete groups can be viewed as tensor products of suitable functors over the PROP of cocommutative Hopf algebras. Such view point has several interesting applications. First, it gives a straightforward way of deriving the functor sending a discrete group to the functions on its representation variety, which leads to representation homology. Second, using a suitable deformation of the functors involved in this construction, one can obtain deformations of the representation and character varieties for the fundamental groups of 3-manifolds, and could lead to better understanding of quantum representations of mapping class groups.

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