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Kaehler structures on
Authors:Meng-Kiat Chuah
Affiliation:Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan
Abstract:Let $K$ be a compact connected semi-simple Lie group, let $G = K_{mathbf C}$, 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 ${mathbf L}$ on $G/N$. Following a suggestion of A.S. Schwarz, in a joint paper with V. Guillemin, we studied its holomorphic sections ${mathcal O}({mathbf L})$ as a $K$-representation space. We defined a $K$-invariant $L^2$-structure on ${mathcal O}({mathbf L})$, and let $H_omega subset {mathcal O}({mathbf L})$ 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.

Keywords:Lie group   Kaehler   line bundle
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