On bounded generalized Harish-Chandra modules

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 $𝔤=\mathrm{sl}\left(n\right)$.

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 $𝔤=\mathrm{sl}\left(n\right)$.

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