Let $G=K\ltimes\mathbb{ R}^n$, where $K$ is a compact connected subgroup of $O(n)$ acting on $\mathbb{ R}^n$ by rotations. Let $\mathfrak{ g}\supset\mathfrak{ k}$ be the respective Lie algebras of $G$ and $K$, and $pr: \mathfrak{ g}^{ *}\longrightarrow\mathfrak{ k}^{ *}$ the natural projection. For admissible coadjoint orbits $\mathcal{ O}^{ G}\subset\mathfrak{ g}^{ *}$ and $\mathcal{ O}^{ K}\subset\mathfrak{ k}^{ *}$, we denote by $n(\mathcal{ O}^{ G},\mathcal{ O}^{ K})$ the number of $K$-orbits in $\mathcal{ O}^{ G}\cap pr^{ -1}(\mathcal{ O}^{ K})$, which is called the Corwin-Greenleaf multiplicity function. Let $\pi\in\widehat{ G}$ and $\tau\in\widehat{ K}$ be the unitary representations corresponding, respectively, to $\mathcal{ O}^G$ and $\mathcal{ O}^K$ by the orbit method. In this paper, we investigate the relationship between $n(\mathcal{ O}^G,\mathcal{ O}^K)$ and the multiplicity $m(\pi,\tau)$ of $\tau$ in the restriction of $\pi$ to $K$. If $\pi$ is infinite-dimensional and the associated little group is connected, we show that $n(\mathcal{ O}^G,\mathcal{ O}^K)\neq 0$ if and only if $m(\pi,\tau)\neq 0$. Furthermore, for $K=SO(n)$, $n\geq 3$, we give a sufficient condition on the representations $\pi$ and $\tau$ in order that $n(\mathcal{ O}^G,\mathcal{ O}^K)=m(\pi,\tau)$.
Categories: math.RT