Let G ⊃ H be Lie groups, g ⊃ h their Lie algebras, and pr : g* → h* the natural projection. For coadjoint orbits OG ⊂ g* and OH ⊂ h*, we denote by n(OG,OH) the number of H-orbits in the intersection OG ∩ pr-1(OH), which is known as the Corwin-Greenleaf multiplicity function. In the spirit of the orbit method due to Kirillov and Konstant, one expects that n(OG,OH) coincides with the multiplicity of τ ∈ Ĥ occurring in an irreducible unitary representation π of G when restricted to H, if π is ''attached'' to OG and τ is ''attached'' to OH. Results in this direction have been established for nilpotent Lie groups and certain solvable groups; however, very few attempts have been made so far for semisimple Lie groups.
This paper treats the case where (G,H) is a semisimple symmetric pair. In this setting, the Corwin-Greenleaf multiplicity function n(OG,OH) may become greater than one, or even worse, may take infinity. We give a sufficient condition on the coadjoint orbit OG in g* in order that
n(OG,OH) ≤ 1 for any coadjoint orbit OH ⊂ h*.The results here are motivated by a recent multiplicity-free theorem of branching laws of unitary representations obtained by one of the authors.
© Toshiyuki Kobayashi