Proceedings of the Japan Academy, Series A, Mathematical Sciences

Corwin–Greenleaf multiplicity functions for Hermitian symmetric spaces

Salma Nasrin

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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

Permanent link to this document
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

Subjects
Primary: 22E46: Semisimple Lie groups and their representations
Secondary: 22E60: Lie algebras of Lie groups {For the algebraic theory of Lie algebras, see 17Bxx} 32M15: Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras [See also 22E10, 22E40, 53C35, 57T15] 53C35: Symmetric spaces [See also 32M15, 57T15] 81S10: Geometry and quantization, symplectic methods [See also 53D50]

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

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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