Another proof for the continuity of the Lipsman mapping

UDC 515.1We consider the semidirect product G = K ⋉ 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 〈 , 〉 . By G ^ we denote the unitary dual of G and by ‡ / G the space of admissible coadjoint orbits, where is the Lie algebra of G . It was pointed out by Lipsman that the correspondence between G ^ and ‡ / G is bijective. Under some assumption on G , we give another proof for the continuity of the orbit mapping (Lipsman mapping) Θ : ‡ / G - → G ^ .