A variant of the Frobenius reciprocity for restricted representations on nilpotent Lie groups

Let G be a nilpotent connected and simply connected Lie. group, K an analytic subgroup of G and pi a unitary and irreducible representation of G. We study in this paper a variant of the Fobenius reciprocity for the restricted representation pi(vertical bar K) of pi on K. It consists in proving that generically, the multiplicity of any isotopic component involved in the central canonical disintegration of pi(vertical bar K) coincides with the dimension of a certain space of generalized tempered distributions which are semi-invariant under the action of a subgroup of K. This problem was considered and partially solved in our previous work (1).