Orbit method and dual topology for certain Lie groups

Abstract

Let ${\mathbb {H}}_d$ ($d\in {\mathbb {N}}^*$) be the $(2d+1)$-dimensional Heisenberg group and let K be a compact connected subgroup of $\text{Aut}({\mathbb {H}}_d)$ acting in the usual way on ${\mathbb {H}}_d.$ In this work, we define the semidirect product $G\,{:}{=}\,K\ltimes {\mathbb {H}}_d,$ for which $(K,{\mathbb {H}}_d)$ is a Gelfand pair. It is well-known in the representation theory of Lie groups (theory of orbit method) that the unitary dual ${\widehat{G}}$ of G is in one to one correspondence with the space of admissible coadjoint orbits ${\mathfrak {g}}^\ddagger /G$ (see, [19]).  The aim of this paper, is to give a nice description of the topology of ${\mathfrak {g}}^\ddagger /G$ and we show that the dual topology of G could be read from the quotient topology of ${\mathfrak {g}}^\ddagger /G.$ More precisely, we prove that the Kirillov–Lipsman’s (orbit mapping) bijection

$$\begin{aligned} {\widehat{G}} \simeq {\mathfrak {g}}^\ddagger /G \end{aligned}$$

is a homeomorphism. This is a generalization of the previous result obtained in [12].