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

TL;DR

This paper investigates one-dimensional representations of finite W-algebras associated with classical Lie algebras, describing their irreducible components and confirming a conjecture related to the orbit method using Dirac reduction and Yangian-type presentations.

Contribution

It provides a detailed description of the irreducible components of these representations and proves Losev's conjecture by establishing new presentations via Dirac reduction.

Findings

Dimensions of irreducible components are precisely described.

Confirmed Losev's conjecture on the orbit method map.

Established new Yangian-type presentations of semiclassical limits.

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.