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

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 L 2( 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.