Duflo conjecture for solvable Lie groups

Let G be an exponential solvable Lie group, g its Lie algebra and pi a unitary irreducible representation of G which is square integrable modulo the center, associated by the Kirillov-Bemat map (Auslander and Moore, 1966; Bernat et al., 1972 11,2]) to a G-orbit Omega. Let H be a closed connected subgroup of G with Lie algebra h and p : g* -> h* the restriction map. We say that the representation pi is H-admissible if its restriction to the subgroup H splits in irreducible representations with finite multiplicities. We shall prove the following conjecture due to Duflo: The representation pi is H-admissible, if and only if, the restriction of p to Omega is proper on the range p(Omega). In the case at hand, these two conditions are equivalent to g = h + 3, where 3 is the center of g. (C) 2010 Publie par Elsevier Masson SAS pour l'Academie des sciences.