Highest weight theory for finite W-algebras

TL;DR

This paper introduces a highest weight theory framework for finite W-algebras, aiming to classify finite-dimensional irreducible modules and proposing conjectures based on type A results.

Contribution

It defines Verma module analogues for finite W-algebras and formulates conjectures for nilpotent orbits of standard Levi type.

Findings

Defined Verma modules for finite W-algebras

Proposed conjectures for classification in type A

First step towards classifying irreducible modules

Abstract

We define analogues of Verma modules for finite W-algebras. By the usual ideas of highest weight theory, this is a first step towards the classification of finite dimensional irreducible modules. Motivated by known results in type A, we then formulate some precise conjectures in the case of nilpotent orbits of standard Levi type.