Kaehler structures on

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 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.

**1.**Ralph Abraham and Jerrold E. Marsden,*Foundations of mechanics*, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1978. Second edition, revised and enlarged; With the assistance of Tudor Raţiu and Richard Cushman. MR**515141****2.**Theodor Bröcker and Tammo tom Dieck,*Representations of compact Lie groups*, Graduate Texts in Mathematics, vol. 98, Springer-Verlag, New York, 1985. MR**781344****3.**Claude Chevalley,*Theory of Lie Groups. I*, Princeton Mathematical Series, vol. 8, Princeton University Press, Princeton, N. J., 1946. MR**0015396****4.**Meng-Kiat Chuah and Victor Guillemin,*Kaehler structures on 𝐾_{𝐶}/𝑁*, The Penrose transform and analytic cohomology in representation theory (South Hadley, MA, 1992) Contemp. Math., vol. 154, Amer. Math. Soc., Providence, RI, 1993, pp. 181–195. MR**1246384**, 10.1090/conm/154/01363**5.**V. Guillemin and S. Sternberg,*Geometric quantization and multiplicities of group representations*, Invent. Math.**67**(1982), no. 3, 515–538. MR**664118**, 10.1007/BF01398934**6.**Victor Guillemin and Shlomo Sternberg,*Symplectic techniques in physics*, 2nd ed., Cambridge University Press, Cambridge, 1990. MR**1066693****7.**I. M. Gel′fand and A. V. Zelevinskiĭ,*Models of representations of classical groups and their hidden symmetries*, Funktsional. Anal. i Prilozhen.**18**(1984), no. 3, 14–31 (Russian). MR**757246****8.**Sigurdur Helgason,*Differential geometry, Lie groups, and symmetric spaces*, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR**514561****9.**Sigurdur Helgason,*Groups and geometric analysis*, Pure and Applied Mathematics, vol. 113, Academic Press, Inc., Orlando, FL, 1984. Integral geometry, invariant differential operators, and spherical functions. MR**754767****10.**Anthony W. Knapp,*Representation theory of semisimple groups*, Princeton Mathematical Series, vol. 36, Princeton University Press, Princeton, NJ, 1986. An overview based on examples. MR**855239****11.**Bertram Kostant,*Quantization and unitary representations. I. Prequantization*, Lectures in modern analysis and applications, III, Springer, Berlin, 1970, pp. 87–208. Lecture Notes in Math., Vol. 170. MR**0294568****12.**HoSeong La, Philip Nelson, and A. S. Schwarz,*Virasoro model space*, Comm. Math. Phys.**134**(1990), no. 3, 539–554. MR**1086743**

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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:
http://dx.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