Geometry of the Shilov Boundary of Bounded Symmetric Domains

The first part will be devoted to a presentation of several approaches to bounded symmetric domains. • Hermitian symmetric spaces of the noncompact type In this perspective, M = G/K is a Riemannian symmetric space of the noncompact type, together with an appropriate complex struc-ture. If p is the tangent space at the origin o = eK, then its com-plexification pC splits as p+ ⊕ p − (holomorphic/antiholomorphic tangent space). One major result is the Harish Chandra imbbed-ding: M can be analytically embedded as the unit ball for some complex norm in p+. • Bounded symmetric domains Here one considers a bounded domain D in some complex vector space V ' CN, such that, for each point z ∈ D there exists an involutive holomorphic transformation sz of D which admits z as isolated fixed point. The introduction of the Bergmann metric give