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

Consider the restriction of an irreducible unitary representation $pi$ of a Lie group $G$ to its subgroup $H$. Kirillovs revolutionary idea on the orbit method suggests that the multiplicity of an irreducible $H$-module $ u$ 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 $ u$ 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]).