Geometry of Coadjoint Orbits and Multiplicity-one Branching Laws for Symmetric Pairs

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]).