Kaehler structures on

Author:
Meng-Kiat Chuah

Journal:
Trans. Amer. Math. Soc. **349** (1997), 3373-3390

MSC (1991):
Primary 53C55

DOI:
https://doi.org/10.1090/S0002-9947-97-01840-0

MathSciNet review:
1401766

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

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Additional Information

**Meng-Kiat Chuah**

Affiliation:
Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan

Email:
chuah@math.nctu.edu.tw

DOI:
https://doi.org/10.1090/S0002-9947-97-01840-0

Keywords:
Lie group,
Kaehler,
line bundle

Additional Notes:
The author was supported in part by NSC85-2121-M-009017

Article copyright:
© Copyright 1997
American Mathematical Society