Corwin-Greenleaf multiplicity function for compact extensions of the Heisenberg group

Let $\mathbb{H}_n$ be the $(2n+1)$-dimensional Heisenberg group and $K$ a closed subgroup of $U(n)$ acting on $\mathbb{H}_n$ by automorphisms such that $(K,\mathbb{H}_n)$ is a Gelfand pair. Let $G=K\ltimes\mathbb{H}_n$ be the semidirect product of $K$ and $\mathbb{H}_n$. Let $\mathfrak{g}\supset\mathfrak{k}$ be the respective Lie algebras of $G$ and $K$, and $\operatorname{pr}: \mathfrak{g}^{*}\longrightarrow\mathfrak{k}^{*}$ the natural projection. For coadjoint orbits $\mathcal{O}^{G}\subset\mathfrak{g}^{*}$ and $\mathcal{O}^{K}\subset\mathfrak{k}^{*}$, we denote by $n\big(\mathcal{O}^{G},\mathcal{O}^{K}\big)$ the number of $K$-orbits in $\mathcal{O}^{G}\cap \operatorname{pr}^{-1}(\mathcal{O}^{K})$, which is called the Corwin-Greenleaf multiplicity function. In this paper, we give two sufficient conditions on $\mathcal{O}^G$ in order that $$n\big(\mathcal{O}^G,\mathcal{O}^K\big)\leq 1\:\:\text{for any $K$-coadjoint orbit}\:\:\mathcal{O}^{K}\subset\mathfrak{k}^{*}.$$ For $K=U(n)$, assuming furthermore that $\mathcal{O}^{G}$ and $\mathcal{O}^{K}$ are admissible and denoting respectively by $\pi$ and $\tau$ their corresponding irreducible unitary representations, we also discuss the relationship between $n\big(\mathcal{O}^G,\mathcal{O}^K\big)$ and the multiplicity $m(\pi,\tau)$ of $\tau$ in the restriction of $\pi$ to $K$. Especially, we study in Theorem 4 the case where $n(\mathcal{O}^{G},\mathcal{O}^{K})\neq m(\pi,\tau)$. This inequality is interesting because we expect the equality as the naming of the Corwin-Greenleaf multiplicity function suggests.