Corwin–Greenleaf multiplicity function for compact extensions of ℝn

Abstract

Let G = K ⋉ ℝn, where K is a compact connected subgroup of O(n) acting on ℝn by rotations. Let 𝔤 ⊃ 𝔨 be the respective Lie algebras of G and K, and pr : 𝔤* → 𝔨* the natural projection. For admissible coadjoint orbits [Formula: see text] and [Formula: see text], we denote by [Formula: see text] the number of K-orbits in [Formula: see text], which is called the Corwin–Greenleaf multiplicity function. Let π ∈ Ĝ and [Formula: see text] be the unitary representations corresponding, respectively, to [Formula: see text] and [Formula: see text] by the orbit method. In this paper, we investigate the relationship between [Formula: see text] and the multiplicity m(π, τ) of τ in the restriction of π to K. If π is infinite-dimensional and the associated little group is connected, we show that [Formula: see text] if and only if m(π, τ) ≠ 0. Furthermore, for K = SO(n), n ≥ 3, we give a sufficient condition on the representations π and τ in order that [Formula: see text].