Corwin–Greenleaf multiplicity function of a class of Lie groups.

Let N be a simply connected nilpotent Lie group, and let K be a connected compact subgroup of the automorphism group, A u t (N) , of N. Let G : = K ⋉ N be the semidirect product (of K and N). Let ⊃ be the respective Lie algebras of G and K and q : ∗ → ∗ be the natural projection. It was pointed out by Lipsman, that the unitary dual G ̂ of G is in one-to-one correspondence with the space of admissible coadjoint orbits ‡ / G (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 π ∈ G ̂ be a generic representation of G and let τ ∈ K ̂. To these representations we associate, respectively, the admissible coadjoint orbit G ⊂ ∗ and K ⊂ ∗ (via the Lipsman's correspondence). We denote by χ ( G , K) the number of K -orbits in G ∩ q − 1 ( K) , which is called the Corwin–Greenleaf multiplicity function. The Kirillov–Lipsman's orbit method suggests that the multiplicity m π (τ) of an irreducible K -module τ occurring in the restriction π | K could be read from the coadjoint action of K on G ∩ q − 1 ( K). Under some assumptions on the pair (K , N) , we prove that for a class of generic representations π ∈ G ̂ , one has m π (τ) ≠ 0 ⇒ χ ( G , K) ≠ 0. Moreover, we show that the Corwin–Greenleaf multiplicity function is bounded (≤ 1) for a special class of subgroups of G. Finally, we give a necessary and sufficient conditions to obtain a nonzero multiplicity ( m π (τ λ) ≠ 0). [ABSTRACT FROM AUTHOR]