  We consider finite W-algebras U(g,e) associated to even multiplicity
nilpotent elements in classical Lie algebras. We give a classification of
finite dimensional irreducible U(g,e)-modules with integral central character
in terms of the highest weight theory for finite W-algebras. As a corollary, we
obtain a parametrization of primitive ideals of U(g) with associated variety
the closure of the adjoint orbit of e and integral central character.
