On bounded generalized Harish-Chandra modules

Penkov, Ivan; Serganova, Vera

doi:10.5802/aif.2685

On bounded generalized Harish-Chandra modules
[Sur les modules de Harish-Chandra bornés généralisés]

Penkov, Ivan ¹ ; Serganova, Vera ²

¹ Jacobs University Bremen School of Engineering and Science Campus Ring 1 28759 Bremen (Germany)
² University of California Berkeley Department of Mathematics Berkeley CA 94720 (USA)

Annales de l'Institut Fourier, Tome 62 (2012) no. 2, pp. 477-496.

Résumé
Abstract

Soient $𝔤$ une algèbre de Lie réductive complexe et $𝔨 \subset 𝔤$ 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 $𝔨 \subset 𝔤$ 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 $𝔤 = sl (n)$ .

MR Zbl

DOI : 10.5802/aif.2685

Classification : 17B10, 22E46
Keywords: Generalized Harish-Chandra module, bounded $(\mathfrak{g},\mathfrak{k})$-module
Mot clés : module de Harish-Chandra généralisé, $(\mathfrak{g},\mathfrak{k})$-module borné

Affiliations des auteurs :

Penkov, Ivan ¹ ; Serganova, Vera ²

¹ Jacobs University Bremen School of Engineering and Science Campus Ring 1 28759 Bremen (Germany)
² University of California Berkeley Department of Mathematics Berkeley CA 94720 (USA)

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     title = {On bounded generalized {Harish-Chandra} modules},
     journal = {Annales de l'Institut Fourier},
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     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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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. https://aif.centre-mersenne.org/articles/10.5802/aif.2685/

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