Consider the restriction of an irreducible unitary representation π 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 ν occurring in the restriction π|H could be read from the coadjoint action of H on OG pr-1(OH), provided π and ν are 'geometric quantizations of a G-coadjoint orbit OG and an H-coadjoint orbit OH, respectively, where pr sqrt(-1) ->sqrt(-1) is the projection dual to the inclusion 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 OG of a semisimple Lie group G corresponding to highest weight modules of scalar type. We prove that the Corwin-Greenleaf number (OGpr-1(OH))/H is either zero or one for any H-coadjoint orbit OH, whenever (G,H) is a symmetric pair of holomorphic type. Furthermore, we determine the coadjoint orbits OH 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]).
© Toshiyuki Kobayashi