## Abstract

Let G be a connected, simply-connected, real semisimple Lie group and K a maximal compactly embedded subgroup of G such that D = G/K is a hermitian symmetric space. Consider the principal fiber bundle M = G/K_{s} → G/K, where K_{s} is the semisimple part of K = K_{s} · Z^{0}_{K} and Z^{0}_{K} is the connected center of K. The natural action of G on M extends to an action of G^{1} = G × Z^{0}_{K}. We prove as the main result that M is weakly symmetric with respect to G^{1} and complex conjugation. In the case where D is an irreducible classical bounded symmetric domain and G is a classical matrix Lie group under a suitable quotient, we provide an explicit construction of M = D × S^{1} and determine a one-parameter family of Riemannian metrics Ω on M invariant under G^{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.

Original language | English (US) |
---|---|

Pages (from-to) | 351-374 |

Number of pages | 24 |

Journal | Transformation Groups |

Volume | 2 |

Issue number | 4 |

State | Published - Dec 1 1997 |

## All Science Journal Classification (ASJC) codes

- Algebra and Number Theory
- Geometry and Topology