Mixed representations of exponential solvable Lie groups

Let G be an exponential solvable Lie group, H and A two closed connected subgroups of G and sigma a unitary and irreducible representation of H. We prove the orbital spectrum formula of the Up-Down representation rho(G, H, A, sigma) = Ind(H)(G)sigma(\A). When G is nilpotent, the multiplicities of such representation turns out to be uniformly infinite or finite and bounded. A necessary and sufficient condition for the finiteness of the multiplicities is given. The same results are obtained when G is exponential solvable the group, H and A are invariant.