Restriction to Levi subalgebras and generalization of the category $𝒪$

Tomasini, Guillaume

Annales de l'Institut Fourier, Tome 63 (2013) no. 1, pp. 37-88.

L’étude de la catégorie de tous les modules sur une algèbre de Lie complexe semi-simple est un problème réputé très difficile. Il est donc utile pour approcher ce problème de se restreindre à des sous-catégories pleines. Ainsi, Bernstein, Gelfand et Gelfand ont introduit une catégorie de modules qui fournit un cadre naturel pour étudier les modules de plus haut poids. Dans cet article, nous définissons une famille de catégories qui généralise la catégorie BGG et nous étudions les modules irréductibles pour une certaine sous-famille. Comme corollaire, nous montrons que certaines de ces catégories sont semi-simples.

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.

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