We introduce a derived representation scheme associated with a quiver, which may be thought of as a derived version of a Nakajima variety. We exhibit an explicit model for the derived representation scheme as a Koszul complex and by doing so we show that it has vanishing higher homology if and only if the moment map defining the corresponding Nakajima variety is flat. In this case we prove a comparison theorem relating isotypical components of the representation scheme to equivariant K-theoretic classes of tautological bundles on the Nakajima variety. As a corollary of this result we obtain some integral formulas present in the mathematical and physical literature since a few years, such as the formula for Nekrasov partition function for the moduli space of framed instantons on S 4 . On the technical side we extend the theory of relative derived representation schemes by introducing derived partial character schemes associated with reductive subgroups of the general linear group and constructing an equivariant version of the derived representation functor for algebras with a rational action of an algebraic torus. Contents 1. Introduction 1 2. Derived representation schemes of an algebra 6 3. The case of Nakajima quiver varieties 19 4. Main results 26 5. Examples 33 Appendix A. Projective model structure on T-equivariant dg-algebras 38 Appendix B. Representation theory of G = G v 41 References 42

In this paper we propose a procedure for a noncommutative derived Poisson reduction, in the spirit of the Kontsevich-Rosenberg principle: "a noncommutative structure of some kind on A should give an analogous commutative structure on all schemes Rep n (A)". We use double Poisson structures as noncommutative Poisson structures and noncommutative Hamiltonian spaces -as first introduced by M. Van den Bergh -to define (derived) zero loci of Hamiltonian actions and a noncommutative Chevalley-Eilenberg and BRST constructions, showing how we recover the corresponding commutative constructions using the representation functor. 2.1. Graded objects 2.2. Multi-brackets on differential graded algebras 2.3. Double Poisson brackets 2.4. Building new double Poisson structures from old 3. Derived noncommutative Poisson reduction 3.1. Crash course in noncommutative geometry 3.2. Natural double Poisson structure on cotangent bundles 3.3. Noncommutative Hamiltonian spaces 3.4. Noncommutative Chevalley-Eilenberg and BRST 4. Representation schemes 4.1. Representation schemes of double Poisson algebras 4.2. Hamiltonian spaces 4.3. Chevalley-Eilenberg and BRST 5. Some homological computations and examples 5.1. Computation of the Chevalley-Eilenberg (co)homology 5.2. Path algebras of quivers 5.3. The scheme of commuting matrices and similar

We introduce a derived representation scheme associated with a quiver, which may be thought of as a derived version of a Nakajima variety. We exhibit an explicit model for the derived representation scheme as a Koszul complex and by doing so we show that it has vanishing higher homology if and only if the moment map defining the corresponding Nakajima variety is flat. In this case we prove a comparison theorem relating isotypical components of the representation scheme to equivariant K-theoretic classes of tautological bundles on the Nakajima variety. As a corollary of this result we obtain some integral formulas present in the mathematical and physical literature since a few years, such as the formula for Nekrasov partition function for the moduli space of framed instantons on $$S^4$$ S 4 . On the technical side we extend the theory of relative derived representation schemes by introducing derived partial character schemes associated with reductive subgroups of the general linear group and constructing an equivariant version of the derived representation functor for algebras with a rational action of an algebraic torus.