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Multiplicity one theorem in the orbit method. (English) Zbl 1048.22006
Gindikin, S. G. (ed.), Lie groups and symmetric spaces. In memory of F. I. Karpelevich. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3472-X/hbk). Transl., Ser. 2, Am. Math. Soc. 210(54), 161-169 (2003).
From the abstract: Let $$G\supset H$$ be Lie groups, $$\mathfrak g\supset\mathfrak h$$ their Lie algebras, and $$\text{pr}:\mathfrak g^*\to\mathfrak h^*$$ the natural projection. For coadjoint orbits $$\mathcal O^G\subset\mathfrak g^*$$ and $$\mathcal O^H\subset\mathfrak h^*$$, we denote by $$n(\mathcal O^G,\mathcal O^H)$$ the number of $$H$$-orbits in the intersection $$\mathcal O^G\cap\text{pr}^{-1}(\mathcal O^H)$$, which is known as the Corwin-Greenleaf multiplicity function. In the spirit of the orbit method due to Kirillov and Konstant, one expects that $$n(\mathcal O^G,\mathcal O^H)$$ coincides with the multiplicity of $$\tau\in\widehat H$$ occurring in an irreducible unitary representation $$\pi$$ of $$G$$ when restricted to $$H$$, if $$\pi$$ is “attached” to $$\mathcal O^G$$ and $$\tau$$ is “attached” to $$\mathcal O^H$$. 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 $$n(\mathcal O^G,\mathcal O^H)$$ may become greater than one, or even worse, may take infinity. We give a sufficient condition on the coadjoint orbit $$\mathcal O^G$$ in $$\mathfrak g^*$$ in order that $n(\mathcal O^G,\mathcal O^H)\leq1\text{ for any coadjoint orbit }\mathcal O^H\subset\mathfrak h^*\,.$ The results here are motivated by a recent multiplicity-free theorem of branching laws of unitary representations obtained by one of the authors.
For the entire collection see [Zbl 1022.00008].

##### MSC:
 22E46 Semisimple Lie groups and their representations 32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects) 22E60 Lie algebras of Lie groups
##### Keywords:
orbit method; Corwin–Greenleaf multiplicity