Multiplicity one for representations corresponding to spherical distribution vectors of class p

In this paper one considers a unimodular second countable locally compact group G and the homogeneous space X := H/G, where H is a closed unimodular subgroup of G. Over X complex vector bundles are considered such that H acts on the fibers by a unitary representation with closed image. The natural action of G on the space of square integrable sections is unitary and possesses an integral decomposition in so-called spherical distributions of class ρ. The uniqueness of this decomposition can be characterized by a number of equivalent properties. Uniqueness is shown to hold for a class of semidirect products.