  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]).
