Branching rule and coadjoint orbit for Heisenberg Gelfand pairs

Abstract

Let [Formula: see text] be a closed connected subgroup of the unitary group [Formula: see text] and let [Formula: see text] be the [Formula: see text]-dimensional Heisenberg group ([Formula: see text]). We consider the semidirect product [Formula: see text] such that [Formula: see text] is a Gelfand pair. 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 [10]). 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] In this paper, we show that [Formula: see text] For the special case [Formula: see text] (maximal torus in the unitary group [Formula: see text]) and [Formula: see text], we prove that [Formula: see text]