Branching rule and coadjoint orbit for Heisenberg Gelfand pairs

Let K be a closed connected subgroup of the unitary group U(d), d is an element of N-& lowast; and let H-V be the (2d + 1)-dimensional Heisenberg group (V ? C-d). We consider the semidirect product G = K (sic) H-V, such that (K, H-V) is a Gelfand pair. Let g superset of k be the respective Lie algebras of G and K and q : g(& lowast;) -> k(& lowast;) be the natural projection. It was pointed out by Lipsman, that the unitary dual Gb of G is in one-to-one correspondence with the space of admissible coadjoint orbits g & Dagger;/G (see [10]). Let pi is an element of (G) over cap be a generic representation of G and let tau is an element of (K) over cap. To these representations we associate, respectively, the admissible coadjoint orbit O-G subset of g(& lowast;) and O-K subset of k(& lowast;) (via the Lipsman's correspondence). We denote by chi(O-G, O-K) the number of K-orbits in O-G & cap; (q-1)(O-K), which is called the Corwin- Greenleaf multiplicity function. The Kirillov-Lipsman's orbit method suggests that the multiplicity m(pi)(tau) of an irreducible K-module tau occurring in the restriction pi|(K) could be read from the coadjoint action of K on O-G & cap; (q-1)(O-K). In this paper, we show that m(pi) (tau) (sic) 0 -> chi(O-G, O-K) = (sic) 0. For the special case K = T-d (maximal torus in the unitary group U(d)) and G(d) := T-d (sic) H-d, we prove that chi(O-d(G), O-d(T)) <= 1.