Duflo conjecture for solvable group with 2-step unipotent radical

We study Duflo’s conjecture on admissible restrictions of unitary representations in the setting of solvable Lie groups. More precisely, we consider a connected, simply connected solvable Lie group G = S ⋉U, where the unipotent radical U is 2-step nilpotent, and a closed connected subgroup H = R ⋉ uH. For a strongly regular coadjoint orbit Ωg, we investigate the relation between the H-admissibility of the associated representation πg and the geometric properties of the moment map pg,h: Ωg → h∗.Duflo’s conjecture predicts that πg|H is H-admissible if and only if the moment map is weakly proper, and that multiplicities admit a geometric interpretation via symplectic reduction. This conjecture has been established in several cases, including compact groups, reductive pairs, and certain low-rank situations. In the solvable setting, it was proved by Kouki in the exponential case and for semidirect products of a torus with the Heisenberg group.In this paper, we extend these results to solvable groups with 2-step unipotent radical. Inspired by the Heisenberg case, we introduce a geometric condition PA(Ωg) expressed in terms of the transverse symplectic structure and the weights of the torus action.We prove that this condition is equivalent to both the properness of the moment map and the H-admissibility of πg, thus providing a positive answer to Duflo’s conjecture in this setting. Furthermore, we obtain an explicit formula for multiplicities in terms of a partition function associated with the weights.