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

Abstract

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.