## Proceedings of the Japan Academy, Series A, Mathematical Sciences

### Corwin–Greenleaf multiplicity functions for Hermitian symmetric spaces

Salma Nasrin

#### Abstract

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.

#### Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 84, Number 7 (2008), 97-100.

Dates
First available in Project Euclid: 17 July 2008

https://projecteuclid.org/euclid.pja/1216308249

Digital Object Identifier
doi:10.3792/pjaa.84.97

Mathematical Reviews number (MathSciNet)
MR2450059

Zentralblatt MATH identifier
1161.22008

#### Citation

Nasrin, Salma. Corwin–Greenleaf multiplicity functions for Hermitian symmetric spaces. Proc. Japan Acad. Ser. A Math. Sci. 84 (2008), no. 7, 97--100. doi:10.3792/pjaa.84.97. https://projecteuclid.org/euclid.pja/1216308249

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