Unitary representations of semi-direct product groups with infinite dimensional Abelian normal subgroup

We generalize Mackey's theory of induced representations to groups which are semidirect products of a locally compact group H by an infinite dimensional Abelian group A. A theorem on inducing in stages is proved and conditions ensuring conservation of the irreducibility of the representation through the inducing process are given. When A is a nuclear space or the Hilbert space A_O of a Gelfand triplet A_O ? A ? A′_O, a cylindrical ergodic measure on the dual space A′ of A (respectively A′_O) is associated to each irreducible representation of G, and it is proved that the representation is induced when the measure is concentrated on an orbit. In the last part, A is supposed to be a Hilbert space and sufficient conditions are given for the preceding ergodic measures to be concentrated on an orbit.