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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
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