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

## TL;DR

This paper characterizes the topology of the space of admissible coadjoint orbits for a specific semi-direct product group involving unitary and Heisenberg groups, establishing a homeomorphism with the unitary dual.

## Contribution

It explicitly determines the topology of the admissible coadjoint orbit space and proves the homeomorphism with the unitary dual for the group G_n.

## Key findings

- The topology of the admissible coadjoint orbit space is explicitly determined.
- The correspondence between the dual space and the orbit space is a homeomorphism.
- Provides a detailed topological description of the dual space for G_n.

## Abstract

Let $G_n=U(n)\ltimes {\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^\ddagger /G_n $ and we show that the correspondence with $\widehat {G_n} $ is a homeomorphism.