Weakly symmetric spaces and bounded symmetric domains

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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/Ks → G/K, where Ks is the semisimple part of K = Ks · Z0K and Z0K is the connected center of K. The natural action of G on M extends to an action of G1 = G × Z0K. 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.

Original languageEnglish (US)
Pages (from-to)351-374
Number of pages24
JournalTransformation Groups
Volume2
Issue number4
StatePublished - Dec 1 1997

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory
  • Geometry and Topology

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