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).