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.

Amer. Math. Soc. Transl., Advances in the Mathematical Sciences, Series 2

Special volume in memory of Professor F. Karpelevič.

LetG⊃Hbe Lie groups, g ⊃ h their Lie algebras, and pr : g* → h* the natural projection. For coadjoint orbitsO^{G}⊂ g* andO^{H}⊂ h*, we denote byn(O^{G},O^{H}) the number ofH-orbits in the intersectionO^{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 thatn(O^{G},O^{H}) coincides with the multiplicity of τ ∈Ĥoccurring in an irreducible unitary representation π ofGwhen restricted toH, if π is ''attached'' toO^{G}and τ is ''attached'' toO^{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 functionn(O^{G},O^{H}) may become greater than one, or even worse, may take infinity. We give a sufficient condition on the coadjoint orbitO^{G}in g* in order thatThe results here are motivated by a recent n(O^{G},O^{H}) ≤ 1 for any coadjoint orbitO^{H}⊂ h*.multiplicity-freetheorem of branching laws of unitary representations obtained by one of the authors.

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