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Corwin-Greenleaf multiplicity functions for Hermitian symmetric spaces. (English) Zbl 1161.22008

Let \(G\) be a Lie group and \(H\) a closed subgroup of \(G\) with Lie algebras \(g\) and \(h\), respectively. There is a natural projection map \(pr:g^{*}\rightarrow h^{*}\), where \(g^{*}\) and \(h^{*}\) are the vector duals of \(g\) and \(h\), respectively. Given a co-adjoint orbit \({\mathcal O}^{G}\) of \(G\) and a co-adjoint orbit \({\mathcal O}^{H}\) of \(H\), the number \(n({\mathcal O}^{G},{\mathcal O}^{H})\) of \(H\)-orbits in the intersection \({\mathcal O}^{G}\cap pr^{-1}({\mathcal O}^{H})\) of \({\mathcal O}^{G}\) with the inverse image of \({\mathcal O}^{H}\) defines the so-called Corwin-Greenleaf multiplicity function.
In the paper under review, the author investigates relationships between this function, geometric properties of co-adjoint orbits and multiplicity formulas in the case where the group \(G\) is of Hermitian type. The main results proved by the author are a boundedness theorem and a non-vanishing criterion for Corwin-Greenleaf multiplicity functions. The proof is based on explicit formulas for the restriction of representations (branching laws) due to T. Kobayashi.
Reviewer: Salah Mehdi (Metz)

MSC:

22E46 Semisimple Lie groups and their representations
22E60 Lie algebras of Lie groups
32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
53C35 Differential geometry of symmetric spaces
81S10 Geometry and quantization, symplectic methods
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References:

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