On the Reducedness of Quiver Schemes

In this paper we prove that a quiver scheme in characteristic zero is reduced if the moment map is flat. We use the reducedness result to show that the equivariant integration formula computes the K-theoretic Nekrasov partition function of five dimensional $\mathcal{N}=2$ quiver gauge theories when the moment map is flat. We also give an explicit characterization of flatness of moment map for finite and affine type A Dynkin quivers with framings. As an application, we give a refinement of a theorem of Ivan Losev on the relation between quantized Nakajima quiver variety in type A and parabolic finite W-algebra in type A.