Corwin–Greenleaf multiplicity function of a class of Lie groups

Let [Formula: see text] be a simply connected nilpotent Lie group, and let [Formula: see text] be a connected compact subgroup of the automorphism group, [Formula: see text] of [Formula: see text] Let [Formula: see text] be the semidirect product (of [Formula: see text] and [Formula: see text]). Let [Formula: see text] be the respective Lie algebras of [Formula: see text] and [Formula: see text] and [Formula: see text] be the natural projection. It was pointed out by Lipsman, that the unitary dual [Formula: see text] of [Formula: see text] is in one-to-one correspondence with the space of admissible coadjoint orbits [Formula: see text] (see [R. L. Lipsman, Orbit theory and harmonic analysis on Lie groups with co-compact nilradical, J. Math. Pures Appl. 59 (1980) 337–374]). Let [Formula: see text] be a generic representation of [Formula: see text] and let [Formula: see text] To these representations we associate, respectively, the admissible coadjoint orbit [Formula: see text] and [Formula: see text] (via the Lipsman’s correspondence). We denote by [Formula: see text] the number of [Formula: see text]-orbits in [Formula: see text] which is called the Corwin–Greenleaf multiplicity function. The Kirillov–Lipsman’s orbit method suggests that the multiplicity [Formula: see text] of an irreducible [Formula: see text]-module [Formula: see text] occurring in the restriction [Formula: see text] could be read from the coadjoint action of [Formula: see text] on [Formula: see text] Under some assumptions on the pair [Formula: see text] we prove that for a class of generic representations [Formula: see text] one has [Formula: see text] Moreover, we show that the Corwin–Greenleaf multiplicity function is bounded ([Formula: see text]) for a special class of subgroups of [Formula: see text] Finally, we give a necessary and sufficient conditions to obtain a nonzero multiplicity ([Formula: see text]).