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Représentations irréductibles bornées des groupes de Lie exponentiels

Published online by Cambridge University Press:  20 November 2018

J. Ludwig  and
C. Molitor-Braun
J. Ludwig
Affiliation:
Département de Mathématiques, Université de Metz, Ile de Saulcy, F-57045 Metz cedex 1, France. courriel: ludwig@poncelet.sciences.univ-metz.fr
C. Molitor-Braun
Affiliation:
Séminaire de mathématique, Centre Universitaire de Luxembourg, 162A, Avenue de la Faïencerie, L-1511 Luxembourg, Luxembourg. courriel: molitor@cu.lu
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Abstract

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Let $G$ be a solvable exponential Lie group. We characterize all the continuous topologically irreducible bounded representations $(T,\mathcal{U})$ of $G$ on a Banach space $\mathcal{U}$ by giving a $G$-orbit in ${{n}^{*}}$ ($\mathfrak{n}$ being the nilradical of $\mathfrak{g}$), a topologically irreducible representation of ${{L}^{1}}({{\mathbb{R}}^{n}},\,\,\omega )$ , for a certain weight $\omega $ and a certain $n\,\in \,\mathbb{N}$, and a topologically simple extension norm. If $G$ is not symmetric, i.e., if the weight $\omega $ is exponential, we get a new type of representations which are fundamentally different from the induced representations.

Résumé

Résumé

Soit $G$ un groupe de Lie résoluble exponentiel. Nous caractérisons toutes les représentations $(T,\mathcal{U})$ continues bornées topologiquement irréductibles de $G$ dans un espace de Banach $\mathcal{U}$ à l’aide d’une $G$-orbite dans ${{n}^{*}}$ ($\mathfrak{n}$ étant le radical nilpotent de $\mathfrak{g}$), d’une représentation topologiquement irréductible de ${{L}^{1}}({{\mathbb{R}}^{n}},\,\,\omega )$, pour un certain poids $\omega $ et un certain $n\,\in \,\mathbb{N}$, d’une norme d’extension topologiquement simple. Si $G$ n’est pas symétrique, c. à d. si le poids $\omega $ est exponentiel, nous obtenons un nouveau type de représentations qui sont fondamentalement différentes des représentations induites.

Keywords

43A20groupe de Lie résoluble exponentielreprésentation bornée topologiquement irréductibleorbitenorme d’extensionsous-espace invariantidéal premieridéal primitif
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 53 , Issue 5 , 01 October 2001 , pp. 944 - 978
Copyright
Copyright © Canadian Mathematical Society 2001

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