Corwin-Greenleaf multiplicity function for compact extensions of $\mathbb{R}^n$

Abstract

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 $n(\mathcal{O}^G,\mathcal{O}^K)=m(\pi,\tau)$.