On the Dual Topology of a Class of Cartan Motion Groups

Let (G, K) be a compact Riemannian symmetric pair, and let G 0 be the associated Cartan motion group.Under some assumptions on the pair (G, K) , we give a precise description of the set ( G 0 ) gen of all equivalence classes of generic irreducible unitary representations of G 0 .We also determine the topology of the space (g 0 /G 0 ) gen of generic admissible coadjoint orbits of G 0 and we show that the bijection between ( G 0 ) gen and (g 0 /G 0 ) gen is a homeomorphism.Furthermore, in the case where the pair (G, K) has rank one, we prove that the unitary dual G 0 is homeomorphic to the space g 0 /G 0 of all admissible coadjoint orbits of G 0 .