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Let $G_n=U(n)\ltimes {\mathbb H}_n $ be the semi-direct product of theunitary group acting by automorphisms on the Heisenberg group ${\mathbb H}_n$.According to Lipsman, the unitary dual $\widehat {G_n} $ of $G_n $ is in one toone correspondence with the space of admissible coadjoint orbits $\mathfrakg_n^\ddagger /G_n $ of $G_n $. In this paper, we determine the topology of thespace $\mathfrak g_n^\ddagger /G_n $ and we show that the correspondence with$\widehat {G_n} $ is a homeomorphism.

On the dual topology of the groups U(n)xH_n