On bounded generalized Harish-Chandra modules
Annales de l'Institut Fourier, Volume 62 (2012) no. 2, pp. 477-496.

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).

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).

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

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