  We show that a semi-stable moduli space of representations of an acyclic
quiver can be identified with two GIT quotients by reductive groups. One of a
quiver Grassmannian of a projective representation, the other of a quiver
Grassmannian of an injective representation. This recovers as special cases the
classical Gelfand-MacPherson correspondence and its generalization by Hu and
Kim to bipartite quivers, as well as the Zelevinsky map for a quiver of Dynkin
type A with the linear orientation.
