On bounded generalized Harish-Chandra modules
[Sur les modules de Harish-Chandra bornés généralisés]
Annales de l'Institut Fourier, Tome 62 (2012) no. 2, pp. 477-496.

Soient 𝔤 une algèbre de Lie réductive complexe et 𝔨𝔤 une sous-algèbre réductive. On dit qu’un (𝔤,𝔨) module M est borné si les 𝔨-multiplicités de M sont uniformément bornées. Dans cet article, nous commençons une étude générale des (𝔤,𝔨)-modules bornés. Nous donnons une condition forte pour qu’une sous-algèbre 𝔨 soit bornée, c’est-à-dire qu’il existe un (𝔤,𝔨)-module simple borné de dimension infinie (Corollaire 4.6) puis nous établissons une condition suffisante pour qu’une sous-algèbre 𝔨 soit bornée (Theorème 5.1). Nous pouvons alors classifier les sous-algèbres réductives bornées maximales de 𝔤=sl(n).

Let 𝔤 be a complex reductive Lie algebra and 𝔨𝔤 be any reductive in 𝔤 subalgebra. We call a (𝔤,𝔨)-module M bounded if the 𝔨-multiplicities of M are uniformly bounded. In this paper we initiate a general study of simple bounded (𝔤,𝔨)-modules. We prove a strong necessary condition for a subalgebra 𝔨 to be bounded (Corollary 4.6), i.e. to admit an infinite-dimensional simple bounded (𝔤,𝔨)-module, and then establish a sufficient condition for a subalgebra 𝔨 to be bounded (Theorem 5.1). As a result we are able to classify the maximal bounded reductive subalgebras of 𝔤=sl(n).

DOI : https://doi.org/10.5802/aif.2685
Classification : 17B10,  22E46
Mots clés : module de Harish-Chandra généralisé, (𝔤,𝔨)-module borné
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Penkov, Ivan; Serganova, Vera. On bounded generalized Harish-Chandra modules. Annales de l'Institut Fourier, Tome 62 (2012) no. 2, pp. 477-496. doi : 10.5802/aif.2685. http://archive.numdam.org/articles/10.5802/aif.2685/

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