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

Abstract

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 AO of a Gelfand triplet AO ⊂ 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.