On the structure of the category O for W-algebras

TL;DR

This paper investigates the structure of the category O for finite W-algebras associated with semisimple Lie algebras, establishing an equivalence with certain g-module categories, especially for principal Levi type nilpotent elements.

Contribution

It proves an equivalence between the category O for W-algebras and specific categories of g-modules, extending understanding of their structure and relationships.

Findings

Established an equivalence of categories for W-algebras and g-modules.

Connected the category O for W-algebras to generalized Whittaker modules.

Provided insights into the structure when e is of principal Levi type.

Abstract

W-algebra (of finite type) W is a certain associative algebra associated with a semisimple Lie algebra, say g, and its nilpotent element, say e. The goal of this paper is to study the category O for W introduced by Brundan, Goodwin and Kleshchev. We establish an equivalence of this category with certain category of g-modules. In the case when e is of principal Levi type (this is always so when g is of type A) the category of g-modules in interest is the category of generalized Whittaker modules introduced McDowel and studied by Milicic-Soergel and Backelin.