Abstract
Suppose an irreducible unitary representation π of a Lie group G is obtained as a geometric quantization of a coadjoint orbit [Formula: see text] in the Kirillov–Kostant–Duflo orbit philosophy. Let H be a closed subgroup of G, and we compare the following two conditions. (1) The restriction π|H is discretely decomposable in the sense of Kobayashi. (2) The momentum map [Formula: see text] is proper. In this article, we prove that (1) is equivalent to (2) when π is any holomorphic discrete series representation of scalar type of a semisimple Lie group G and (G, H) is any symmetric pair.
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