On bounded generalized Harish-Chandra modules
Annales de l'Institut Fourier, Volume 62 (2012) no. 2, p. 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 : https://doi.org/10.5802/aif.2685
Classification:  17B10,  22E46
Keywords: Generalized Harish-Chandra module, bounded (𝔤,𝔨)-module
@article{AIF_2012__62_2_477_0,
     author = {Penkov, Ivan and Serganova, Vera},
     title = {On bounded generalized Harish-Chandra modules},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {62},
     number = {2},
     year = {2012},
     pages = {477-496},
     doi = {10.5802/aif.2685},
     mrnumber = {2985507},
     zbl = {1281.17010},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2012__62_2_477_0}
}
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/item/AIF_2012__62_2_477_0/

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