Geometry of coadjoint orbits and multiplicity-one branching laws for symmetric pairs

TL;DR

This paper explores the geometric structure of coadjoint orbits in semisimple Lie groups and establishes conditions under which certain multiplicities in representation restrictions are zero or one, confirming predictions of the orbit philosophy.

Contribution

It proves that for symmetric pairs of holomorphic type, the intersection count of specific coadjoint orbits is at most one, and characterizes orbits with nonzero intersection, advancing the orbit method in representation theory.

Findings

The intersection count is either zero or one for symmetric pairs of holomorphic type.

Identifies which coadjoint orbits have nonzero intersection with the projected orbits.

Results align with the orbit philosophy and classical limits of multiplicity-free branching laws.

Abstract

Consider the restriction of an irreducible unitary representation $\pi$ of a Lie group $G$ to its subgroup $H$. Kirillov's revolutionary idea on the orbit method suggests that the multiplicity of an irreducible $H$-module $\nu$ occurring in the restriction $\pi|_H$ could be read from the coadjoint action of $H$ on $O^G \cap pr^{-1}(O^H)$ provided $\pi$ and $\nu$ are "geometric quantizations" of a $G$-coadjoint orbit $O^G$ and an $H$-coadjoint orbit $O^H$,respectively, where $pr: \sqrt{-1} g^{\ast} \to \sqrt{-1} h^{\ast}$ is the projection dual to the inclusion $h \subset g$ of Lie algebras. Such results were previously established by Kirillov, Corwin and Greenleaf for nilpotent Lie groups. In this article, we highlight specific elliptic orbits $O^G$ of a semisimple Lie group $G$ corresponding to highest weight modules of scalar type. We prove that the Corwin--Greenleaf number $\sharp(O^G \cap pr^{-1}(O^H))/H$ is either zero or one for any $H$-coadjoint orbit $O^H$, whenever $(G,H)$ is a symmetric pair of holomorphic type. Furthermore, we determine the coadjoint orbits $O^H$ with nonzero Corwin-Greenleaf number. Our results coincide with the prediction of the orbit philosophy, and can be seen as "classical limits" of the multiplicity-free branching laws of holomorphic discrete series representations (T.Kobayashi [Progr.Math.2007]).