Duflo conjecture for solvable Lie groups [La conjecture de Duflo pour les groupes résolubles exponentiels]

摘要

Let G be an exponential solvable Lie group, g its Lie algebra and π a unitary irreducible representation of G which is square integrable modulo the center, associated by the Kirillov-Bernat map (Auslander and Moore, 1966; Bernat et al., 1972 [1,2]) to a G-orbit Ω. 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 π 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 π is H-admissible, if and only if, the restriction of p to Ω is proper on the range p(Ω). In the case at hand, these two conditions are equivalent to g=h+z, where z is the center of g.