Finite dimensional irreducible representations of finite W-algebras associated to even multiplicity nilpotent orbits in classical Lie algebras

TL;DR

This paper classifies finite-dimensional irreducible modules of finite W-algebras linked to even multiplicity nilpotent orbits in classical Lie algebras, connecting representation theory with primitive ideals.

Contribution

It provides a highest weight classification of modules and parametrizes primitive ideals with specific associated varieties.

Findings

Classification of finite-dimensional irreducible modules

Parametrization of primitive ideals with given associated variety

Connection between W-algebra modules and primitive ideals

Abstract

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.