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In this paper we study the variety of one dimensional representations of afinite $W$-algebra attached to a classical Lie algebra, giving a precisedescription of the dimensions of the irreducible components. We apply this toprove a conjecture of Losev describing the image of his orbit method map. Inorder to do so we first establish new Yangian-type presentations ofsemiclassical limits of the $W$-algebras attached to distinguished nilpotentelements in classical Lie algebras, using Dirac reduction.

One dimensional representations of finite $W$-algebras, Dirac reduction and the orbit method