Restriction to Levi subalgebras and generalization of the category O

TL;DR

This paper introduces a family of categories generalizing the BGG category for reductive Lie algebras, classifies simple modules within these, and identifies cases where the categories are semisimple.

Contribution

It defines new categories extending the BGG category and classifies simple modules, revealing semisimplicity in certain cases.

Findings

Classification of simple modules in the new categories

Identification of semisimple categories among the generalizations

Extension of the BGG category framework

Abstract

The category of all modules over a reductive complex Lie algebra is wild, and therefore it is useful to study full subcategories. For instance, Bernstein, Gelfand and Gelfand introduced a category of modules which provides a natural setting for highest weight modules. In this paper, we define a family of categories which generalizes the BGG category, and we classify the simple modules for a subfamily. As a consequence, we show that some of the obtained categories are semisimple.