Kirillov-Lipsman orbit method of a class of Gelfand pairs

Abstract Let $G:=K\ltimes N$ be the semidirect product with Lie algebra $\mathfrak{g},$ where $N$ is a connected and simply connected nilpotent Lie group, and $K$ is a subgroup of the automorphism group, $Aut(N),$ of $N.$ We say that the pair $(K,N)$ is a nilpotent Gelfand pair when the set $L_K^1(N)$ of integrable $K$-invariant functions on $N$ forms an abelian algebra under convolution.According to Lipsman, the unitary dual $\widehat{G}$ of $G$ is in one-to-one correspondence with the space of admissible coadjoint orbits $\mathfrak{g}^\ddag/G$ of $G.$ Under some assumptions on the pair $(K,N)$ we will show in this work, that the quotient topology of $\mathfrak{g}^\ddag/G$ could be read from the usual topology of some parameters space $\mathcal{P},$ called Mackey's parameters space (will be defined below in section 3). Furthermore, we show that the Kirillov-Lipsman bijection $$\widehat{G}\simeq\mathfrak{g}^\ddagger/G$$is a homeomorphism for a class of Lie groups associated with the nilpotent Gelfand pairs $(K,N).$