On bounded generalized Harish-Chandra modules

[Sur les modules de Harish-Chandra bornés généralisés]

Penkov, Ivan ; Serganova, Vera

Annales de l'Institut Fourier, Tome 62 (2012), p. 477-496 / Harvested from Numdam

Résumé

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

Abstract

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

Détail

Publié le : 2012-01-01

MR 2985507 | Zbl 1281.17010

DOI : https://doi.org/10.5802/aif.2685
Classification:  17B10,  22E46
Mots clés: module de Harish-Chandra généralisé, $(\mathfrak{g},\mathfrak{k})$-module borné

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