  In this paper we study the variety of one dimensional representations of a
finite $W$-algebra attached to a classical Lie algebra, giving a precise
description of the dimensions of the irreducible components. We apply this to
prove a conjecture of Losev describing the image of his orbit method map. In
order to do so we first establish new Yangian-type presentations of
semiclassical limits of the $W$-algebras attached to distinguished nilpotent
elements in classical Lie algebras, using Dirac reduction.
