Une condition nécessaire et suffisante d'existence de la série discrète d'un groupe localement compact

Mohamed Akkouchi , Allai Bakali , Samir Kabbaj
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Abstract

Let G be a topological locally compact group. The aim of this Note is a contribution to the study of the existence problem for square integrable continuous and unitary representations for G. One of our main results (Theorem 6.2) will give a necessary and sufficient condition for the existence of the discrete series for G. Our approach is based on the notions of units and bounded elements in L2(G) introduced by R. Godement in [6]. We perform a study of these notions. A particular attention is paid to the case of pure units. We associate to each pure unit a transform called Plancherel transform. We characterize the pure units with the use of their Plancherel transforms. We develop new methods giving a new proof to the well known theorem of Bargmann (see [3]) in the case of Lorentz groups, the Harish-Chandra theorem (see [5]) in the case of semi-simple Lie groups and a well known theorem of Duflo and Moore (see [4]) in the case of general nonunimodular locally compact groups. Our methods allow us to give an explicit expression of the formal operator introduced in [4] by Duflo and Moore.

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局部紧群离散级数存在的充分必要条件
设G是一个拓扑局部紧群。本文的目的是对G的平方可积连续酉表示的存在性问题的研究做出贡献。我们的一个主要结果(定理6.2)将给出G的离散级数存在的充分必要条件。我们的方法是基于R. Godement在[6]中引入的L2(G)中的单位和有界元素的概念。我们对这些概念进行了研究。特别注意的是纯单位的情况。我们给每个纯单元关联一个称为Plancherel变换的变换。我们用它们的Plancherel变换来描述纯单位。我们发展了新的方法,对洛伦兹群的著名的巴格曼定理(见[3]),半简单李群的harsh - chandra定理(见[5])和一般非单模局部紧群的著名的Duflo和Moore定理(见[4])给出了新的证明。我们的方法允许我们给出Duflo和Moore在[4]中引入的形式算子的显式表达式。
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