Open Access
July 2008 Corwin–Greenleaf multiplicity functions for Hermitian symmetric spaces
Salma Nasrin
Proc. Japan Acad. Ser. A Math. Sci. 84(7): 97-100 (July 2008). DOI: 10.3792/pjaa.84.97


Kobayashi’s multiplicity-free theorem asserts that irreducible unitary highest weight representations $\pi$ are multiplicity-free when restricted to any symmetric pairs if $\pi$ is of scalar type. The Hua–Kostant–Schmid–Kobayashi branching laws embody this abstract theorem with explicit irreducible decomposition formulas of holomorphic discrete series representations with respect to symmetric pairs. In this paper, we study the ‘classical limit’ (geometry of coadjoint orbits) of a special case of these representation theoretic theorems in the spirit of the Kirillov–Kostant–Duflo orbit method. \\ First, we consider the Corwin–Greenleaf multiplicity function $n (\mathcal{O}^{G},\,\mathcal{O}^{K})$ for Hermitian symmetric spaces $G/K$. The first main theorem is that $n(\mathcal{O}^{G},\,\mathcal{O}^{K}) \le 1$ for any $G$-coadjoint orbit $\mathcal{O}^{G}$ and any $K$-coadjoint orbit $\mathcal{O}^{K}$ if $\mathcal{O}^{G} \cap \sqrt{-1} ([\mathfrak{k}, \mathfrak{k}] + \mathfrak{p})^{\bot} \ne \emptyset$. Here, $\mathfrak{g} = \mathfrak{k} + \mathfrak{p}$ is the Cartan decomposition of the Lie algebra $\mathfrak{g}$ of $G$. The second main theorem is a necessary and sufficient condition for $n (\mathcal{O}^{G},\,\mathcal{O}^{K}) \ne 0$ by means of strongly orthogonal roots.


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Salma Nasrin. "Corwin–Greenleaf multiplicity functions for Hermitian symmetric spaces." Proc. Japan Acad. Ser. A Math. Sci. 84 (7) 97 - 100, July 2008.


Published: July 2008
First available in Project Euclid: 17 July 2008

zbMATH: 1161.22008
MathSciNet: MR2450059
Digital Object Identifier: 10.3792/pjaa.84.97

Primary: 22E46
Secondary: 22E60 , 32M15 , 53C35 , 81S10

Keywords: branching law , Corwin–Greenleaf multiplicity function , Hermitian symmetric space , highest weight representations , Kobayashi’s multiplicity-free theorem , orbit method

Rights: Copyright © 2008 The Japan Academy

Vol.84 • No. 7 • July 2008
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