Kaehler structures on K-C/(P,P)

Abstract

Let K be a compact connected semi-simple Lie group, let G = K-C, and let G = KAN be an Iwasawa decomposition. To a given K-invariant Kaehler structure omega 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-omega subset of O(L) denote the space of square-integrable holomorphic sections. Then H-omega is a unitary K-representation space, but not all unitary irreducible K-representations occur as subrepresentations of H-omega. 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-omega 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.