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

 
 

 

Simply transitive groups and Kähler structures on homogeneous Siegel domains


Author: Josef Dorfmeister
Journal: Trans. Amer. Math. Soc. 288 (1985), 293-305
MSC: Primary 32M10; Secondary 53C55
DOI: https://doi.org/10.1090/S0002-9947-1985-0773062-6
MathSciNet review: 773062
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Abstract: We determine the Lie algebras of all simply transitive groups of automorphisms of a homogeneous Siegel domain $ D$ as modifications of standard normal $ j$-algebras. We show that the Lie algebra of all automorphisms of $ D$ is a "complete isometry algebra in standard position". This implies that $ D$ carries a riemannian metric $ \tilde g$ with nonpositive sectional curvature satisfying Lie $ \operatorname{Iso}(D,\tilde g) = \operatorname{Lie}\; \operatorname{Aut}\,$   D. We determine all Kähler metrics $ f$ on $ D$ for which the group $ \operatorname{Aut}(D,f)$ of holomorphic isometries acts transitively. We prove that in this case $ \operatorname{Aut}(D,f)$ contains a simply transitive split solvable subgroup.

The results of this paper are used to prove the fundamental conjecture for homogeneous Kähler manifolds admitting a solvable transitive group of automorphisms.


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  • [1] R. Azencott and E. N. Wilson, Homogeneous manifolds with negative curvature. I, Trans. Amer. Math. Soc. 215 (1976), 323-362. MR 0394507 (52:15308)
  • [2] -, Homogeneous manifolds with negative curvature. II, Mem. Amer. Math. Soc. No. 178 (1976). MR 0426002 (54:13951)
  • [3] N. Bourbaki, Groupes et algèbres de Lie, Chapitre I, Hermann, Paris, 1971. MR 0271276 (42:6159)
  • [4] J. E. D'Atri and J. Dorfmeister, The isotropy representation for homogeneous Siegel domains, Pacific J. Math. (to appear). MR 810773 (87a:32026)
  • [5] J. Dorfmeister, Zur Konstruktion Homogener Kegel, Math. Ann. 216 (1975), 79-96. MR 0382737 (52:3619)
  • [6] -, Inductive construction of homogeneous cones, Trans. Amer. Math. Soc. 252 (1979), 321-349. MR 534125 (80h:32059)
  • [7] -, Algebraic description of homogeneous cones, Trans. Amer. Math. Soc. 255 (1979), 61-89. MR 542871 (80j:53049)
  • [8] -, Homogene Siegel-Gebiete, Habilitationsschrift, Münster, 1978.
  • [9] -, Homogeneous Siegel domains, Nagoya Math. J. 86 (1982), 39-83. MR 661219 (84b:32042)
  • [10] -, Homogeneous Kähler manifolds admitting a transitive solvable group of automorphisms, Ann. Sci. École Norm. Sup. (to appear).
  • [11] J. Dorfmeister and M. Koecher, Reguläre Kegel, Jber. Deutsch. Math.-Verein. 81 (1979), 109-151. MR 544691 (80i:17019)
  • [12] S. G. Gindikin, Analysis in homogeneous domains, Russian Math. Surveys 19 (1964), 1-89. MR 0171941 (30:2167)
  • [13] S. G. Gindikin and E. B. Vinberg, Kählerian manifolds admitting a transitive solvable automorphism group, Math. Sb. 74 (1967), 333-351. MR 0224114 (36:7161)
  • [14] S. G. Gindikin, I. I. Piatetskii-Shapiro and E. B. Vinberg, Homogeneous Kähler manifolds, C.I.M.E. Edizione Cremonese, Roma, 1967.
  • [15] I. I. Piatetskii-Shapiro, Bounded homogeneous domains in $ n$-dimensional complex space, Amer. Math. Soc. Transl. 43 (1964), 299-320.
  • [16] -, Automorphic functions and the geometry of classical domains, Gordon and Breach, New York and London, 1969. MR 0252690 (40:5908)
  • [17] J. L. Koszul, Sur la forme hermitienne canonique des espaces homogènes complexes, Canad. J. Math. 7 (1955), 562-576. MR 0077879 (17:1109a)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1985-0773062-6
Keywords: Homogeneous bounded domain, Kähler metric, solvable transitive group of automorphisms
Article copyright: © Copyright 1985 American Mathematical Society

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