Abstract
LetG be a connected, simply-connected, real semisimple Lie group andK a maximal compactly embedded subgroup ofG such thatD=G/K is a hermitian symmetric space. Consider the principal fiber bundleM=G/K s →G/K, whereK s is the semisimple part ofK=K s ·Z 0 K andZ 0 K is the connected center ofK. The natural action ofG onM extends to an action ofG 1=G×Z 0 K . We prove as the main result thatM is weakly symmetric with respect toG 1 and complex conjugation. In the case whereD is an irreducible classical bounded symmetric domain andG is a classical matrix Lie group under a suitable quotient, we provide an explicit construction ofM=D×S 1 and determine a one-parameter family of Riemannian metrics Ω onM invariant underG 1. Furthermore,M is irreducible with respect to Ω. As a result, this provides new examples of weakly symmetric spaces that are nonsymmetric, including those already discovered by Selberg (cf. [M]) for the symplectic case and Berndt and Vanhecke [BV1] for the rank-one case.
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[BV1] J. Berndt and L. Vanhecke,Geometry of weakly symmetric spaces, J. Math. Soc. Japan48 (1996), 745–760.
[BV2] J. Berndt and L. Vanhecke,Commutativity and weak symmetry on Riemannian manifolds, Preprint 1996.
[BKV] J. Berndt, O. Kowalski and L. Vanhecke,Geodesics in weakly symmetric spaces, Ann. Global Anal. Geom.15 (1997), 153–156.
[BPV] J. Berndt, F. Prüfer, L. Vanhecke,Symmetric-like Riemannian manifolds and geodesic symmetries, Proc. Roy. Soc. Edinburgh Sect. A125 (1995), 265–282.
[BRV] J. Berndt, F. Ricci and L. Vanhecke,Weakly symmetric groups of Heisenberg type, Preprint 1995.
[BTV] J. Berndt, P. Tondeur and L. Vanhecke, Examples of weakly symmetric spaces in contact geometry, Boll. Un. Mat. Ital. (7) 10, Suppl. fasc. (1006), to appear.
[F] M. Flensted-Jensen,Spherical functions on a simply connected semisimple Lie group, Amer. J. Math.99 (1977), 341–361.
[GV1] J. C. González-Dávila and L. Vanhecke,New examples of weakly symmetric spaces, Monatsh. Math., to appear.
[GV2] J. C. González-Dávila and L. Vanhecke,A new class of weakly symmetric spaces, Preprint 1995.
[He] S. Helgason,Differential Geometry, Lie Groups, and Symmetric Spaces, Pure and Applied Mathematics80, Academic Press, 1978.
[Hu] L. K. Hua,Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains, AMS Translations of Mathematical Monographs6, 1963.
[HW] R. Herb and J. A. Wolf,Wave packets for the relative discrete series I. The holomorphic case, J. Funct. Anal.73:1 (1987), 1–37.
[M] H. Maaß,Siegel's Modular Forms and Dirichlet Series, Lecture Notes in Mathematics216, Springer-Verlag, 1971.
[Na] S. Nagai,Weakly symmetric spaces in complex and quaternionic space forms, Arch. Math.65 (1995), 342–351.
[Ng] H. Nguyêñ,Weakly Symmetric Spaces and Bounded Symmetric Domains, Thesis, University of California, 1996.
[O] A. Onishchik,Topology of Transitive Transformation Groups, J. A. Barth Verlag, Leipzig e.a., 1994.
[Sa] I. Satake,Algebraic Structures of Symmetric Domains, RIMS Publ. Math. Soc. Japan14, 1980.
[Se] A. Selberg,Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc.20 (1956), 47–87.
[Sz] Z. I. Szabo,Spectral geometry for operator families on Riemannian man ifolds, Proc. Symp. Pure Math.54 (1993), 615–665.
[Ti] J. A. Tirao,Square integrable representations of semisimple Lie groups, Trans. AMS190 (1974), 57–75.
[TW] Y. L. Tong and S. P. Wang,Harmonic forms dual to geodesic cycles in quotients of SU(p, 1), Math. Ann.258 (1982), 289–318.
[W] J. A. Wolf,Fine Structure of Hermitian Symmetric Spaces, Symmetric Spaces, Marcel Dekker, 1972.
[Z] W. Ziller,Weakly symmetric spaces, Topics in Geometry: In Memory of Joseph D' Atri, Progress in Nonlinear Diff. Eqs.20 (1996), 355–368.
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Research partially supported by an NSF grant. The author wishes to thank the International Erwin Schroedinger Institute for its hospitality during the preparation of this paper.
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Nguyêñ, H. Weakly symmetric spaces and bounded symmetric domains. Transformation Groups 2, 351–374 (1997). https://doi.org/10.1007/BF01234540
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DOI: https://doi.org/10.1007/BF01234540