Lipsman mapping and dual topology of semidirect products

We consider the semidirect product G = K proportional to V where K is a connected compact Lie group acting by automorphisms on a finite dimensional real vector space V equipped with an inner product <,> . We denote by (G) over cap the unitary dual of G (note that we identify each representation pi is an element of (G) over cap to its classes [pi]) and by g(double dagger)/G the space of admissible coadjoint orbits, where g is the Lie algebra of G. It was pointed out by Lipsman that the correspondence between g(double dagger)/G and (G) over cap is bijective. Under some assumption on G, we prove that the Lipsman mapping Theta : g(double dagger)/G -> (G) over cap O bar right Arrow pi(O) is a homeomorphism.