Classification of irreducible weight modules

Mathieu, Olivier

Annales de l'Institut Fourier, Tome 50 (2000) no. 2, pp. 537-592

Let $𝔤$ be a reductive Lie algebra and let $𝔥$ be a Cartan subalgebra. A $𝔤$-module $M$ is called a weighted module if and only if $M={\oplus }_{\lambda }{M}_{\lambda }$, where each weight space $M_{\lambda }$ is finite dimensional. The main result of the paper is the classification of all simple weight $𝔤$-modules. Further, we show that their characters can be deduced from characters of simple modules in category $𝒪$.

Soit $𝔤$ une algèbre de Lie réductive et soit $𝔥$ une sous-algèbre de Cartan. Un $𝔤$-module $M$ est dit module de poids si et seulement si il admet une décomposition $M={\oplus }_{\lambda }{M}_{\lambda }$, où chaque espace de poids $M_{\lambda }$ est de dimension finie. Notre résultat principal est la classification de tous les $𝔤$-modules de poids simples. Également, leurs caractères sont déduits de formules des caractères des modules simples de la catégorie $𝒪$.

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