  We prove a conjecture of Lusztig on a microlocal characterization of his
perverse sheaves. For any finite quiver without loops, an equivariant simple
perverse sheaf on the variety of quiver representations is a Lusztig's perverse
sheaf if and only if its singular support is contained in Lusztig's Lagrangian
variety, that is, the variety of nilpotent representations of the preprojective
algebra of the quiver.
