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

Let $G_n=U(n)⋉{\mathbb{H}}_n$ be the semi-direct product of the unitary 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 to one correspondence with the space of admissible coadjoint orbits $\mathfrak g_n^\ddagger /G_n $of$G_n $. In this paper, we determine the topology of the space ${\mathfrak{g}}_n^{‡}\mathrm{/}{G}_n$ and we show that the correspondence with $\widehat{G_n}$ is a homeomorphism.