On changing highest weight theories for finite W-algebras

TL;DR

This paper explores how to change the highest weight theories for finite W-algebras by constructing isomorphisms between different parameterizations, especially for classical Lie types, aiding classification of modules.

Contribution

It provides a method to relate different highest weight parameterizations of finite W-algebras for types A, C, and D, enhancing understanding of their module classifications.

Findings

Constructs isomorphisms between parameter sets for different parabolic choices in type A.

Extends isomorphism construction to types C and D with even nilpotent elements.

Facilitates classification of finite-dimensional modules for these W-algebras.

Abstract

A highest weight theory for a finite W-algebra U(g,e) was developed in [BGK]. This leads to a strategy for classifying the irreducible finite dimensional U(g,e)-modules. The highest weight theory depends on the choice of a parabolic subalgebra of g leading to different parameterizations of the finite dimensional irreducible U(g,e)-modules. We explain how to construct an isomorphism preserving bijection between the parameterizing sets for different choices of parabolic subalgebra when g is of type A, or when g is of types C or D and e is an even multiplicity nilpotent element