T. Kobayashi and S. Nasrin,
Multiplicity one theorem in the orbit method,
Amer. Math. Soc. Transl., Advances in the Mathematical Sciences, Series 2 210 (2003), 161-169,
Special volume in memory of Professor F. Karpelevič.

Let G ⊃ H be Lie groups, g ⊃ h their Lie algebras, and pr : g* → h* the natural projection. For coadjoint orbits O^G ⊂ g* and O^H ⊂ h*, we denote by n(O^G,O^H) the number of H-orbits in the intersection O^G ∩ pr^-1(O^H), 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(O^G,O^H) 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 O^G and τ is ''attached'' to O^H. 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(O^G,O^H) may become greater than one, or even worse, may take infinity. We give a sufficient condition on the coadjoint orbit O^G in g* in order that

n(O^G,O^H) ≤ 1 for any coadjoint orbit O^H ⊂ h*. The results here are motivated by a recent multiplicity-free theorem of branching laws of unitary representations obtained by one of the authors.

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© Toshiyuki Kobayashi