Weakly symmetric spaces and bounded symmetric domains

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 · Z⁰_K and Z⁰_K is the connected center of K. The natural action of G on M extends to an action of G¹ = G × Z⁰_K. We prove as the main result that M is weakly symmetric with respect to G¹ 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 × S¹ and determine a one-parameter family of Riemannian metrics Ω on M invariant under G¹. 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.