Abstract
Let K be a compact connected semi-simple Lic group, let G = KC, and let G = K AN be an Iwasawa decomposition. To a given K-invariant Kaehler structure w on G/N, there corresponds a pre-quantum line bundle L on G/N. Following a suggestion of A.S. Schwarz, in a joint paper with V. Guillemin, we studied its holomorphic sections O(L) as a K-representation space. We defined a K-invariant L 2 -structure on O(L), and let H w ⊂ O(L) denote the space of square-integrable holomorphic sections. Then H w is a unitary K-representation space, but not all unitary irreducible K-representations occur as subrepresentations of H w . This paper serves as a continuation of that work, by generalizing the space considered. Let B be a Borel subgroup containing N, with commutator subgroup (B, B) = N. Instead of working with G/N = G/(B,B), we consider G/(P,P), for all parabolic subgroups P containing B. We carry out a similar construction, and recover in H w the unitary irreducible K-representations previously missing. As a result, we use these holomorphic sections to construct a model for K: a unitary K-representation in which every irreducible K-representation occurs with multiplicity one. © 1997 American Mathematical Society.