Pseudo-Hermitian Symmetric Spaces of Tube Type

Abstract

Let Ω be an open connected cone in a real vector space V ≃ ℝn. One defines G(Ω) = {g ∈ GL(n,∝) ∣ gΩ = Ω}. The cone Ω is said to be homogeneous if the group G(Ω) acts transitively on Ω. For the beginning let us assume that Ω is convex and that \(\bar \Omega \) is pointed (this means that \(\bar \Omega \) ∩ (}\(\bar \Omega \)) = {0}). The convex cone Ω is said to be selfdual if there exists a positive inner product on V such that Ω✶ = Ω, where the open dual cone Ω✶ is defined by $$G(\Omega ) = \{ g \in GL(n,\mathbb{R}|g\Omega = \Omega \} .$$ The open convex cone Ω is said to be symmetric if it is homogenous and selfdaul. Let us recall the connection between symmetric convex cones and Jordan algebras. A Jordan algebra V is a vector space equipped with a product, i.e., a bilinear map V × V → V such that (J1) xy = xy, (J2) x(x 2 ) = x 2 (xy).