Geometry of the Shilov boundary of a bounded symmetric domain. (English) Zbl 1175.32011

The first part is a survey of the theory of bounded symmetric domains. This theory is presented along two main approaches: as special cases of Riemannian symmetric spaces of noncompact type on one hand, as unit balls in positive Hermitian Jordan triple systems on the other hand.
In the second part, some results obtained during the last years by the author (alone or in collaboration) are presented. An invariant for triples in the Shilov boundary of such a domain is constructed. It generalizes an invariant constructed by E. Cartan for the unit sphere in \(\mathbb C^2\) and also the triple Maslov index on the Lagrangian manifold.
The paper contains no proofs (except for a few proofs that are sketched), but appropriate references are given.


32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
53C35 Differential geometry of symmetric spaces