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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Kaehler structures on $K_{\mathbf C}/(P,P)$

Author: Meng-Kiat Chuah
Journal: Trans. Amer. Math. Soc. 349 (1997), 3373-3390
MSC (1991): Primary 53C55
MathSciNet review: 1401766
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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 [Enhancements On Off] (What's this?)

  • 1. R. Abraham and J. Marsden, Foundations of Mechanics, 2nd. ed., Addison-Wesley, 1985. MR 81e:58025
  • 2. T. Brocker and T. tom Dieck, Representations of compact Lie groups, Springer-Verlag, N.Y. 1985. MR 86i:22023
  • 3. C. Chevalley, Theory of Lie Groups, Princeton U. Press, Princeton 1946. MR 7:412c
  • 4. M.K. Chuah and V. Guillemin, Kaehler structures on $K_{\mathbf C}/N ,$ Contemporary Math. 154 : The Penrose transform and analytic cohomology in representation theory (1993), 181-195. MR 94k:22028
  • 5. V. Guillemin and S. Sternberg, Geometric quantization and multiplicities of group representations, Invent. Math. 67 (1982), 515-538. MR 83m:58040
  • 6. V. Guillemin and S. Sternberg, Symplectic techniques in physics, Cambridge U. Press, Cambridge 1990. MR 91d:58073
  • 7. I.M. Gelfand and A. Zelevinski, Models of representations of classical groups and their hidden symmetries, Funct. Anal. Appl. 18 (1984), 183-198. MR 86i:22024
  • 8. S. Helgason, Differential Geometry, Lie groups, and symmetric spaces, Academic Press, 1978. MR 80k:53081
  • 9. S. Helgason, Groups and Geometric Analysis, Academic Press, 1984. MR 86c:22017
  • 10. A. Knapp, Representation Theory of Semisimple Groups, Princeton U. Press, Princeton 1986. MR 87j:22022
  • 11. B. Kostant, Quantization and unitary representations, Lecture Notes in Math. 170, Springer 1970, 87-208. MR 45:3638
  • 12. H.S. La, P. Nelson, A.S. Schwarz, Virasoro Model Space, Comm. Math. Phys. 134 (1990), 539-554. MR 92c:22041

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Additional Information

Meng-Kiat Chuah
Affiliation: Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan

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

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