Kaehler structures on $K_{\mathbf C}/(P,P)$
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- by Meng-Kiat Chuah PDF
- Trans. Amer. Math. Soc. 349 (1997), 3373-3390 Request permission
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.References
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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
- Additional Notes: The author was supported in part by NSC85-2121-M-009017
- © Copyright 1997 American Mathematical Society
- 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