This article is part of an ongoing project aiming at the connections between
causal structures on homogeneous spaces, Algebraic Quantum Field Theory (AQFT),
modular theory of operator algebras and unitary representations of Lie groups.
In this article we concentrate on non-compactly causal symmetric space $G/H$.
This class contains the de Sitter space but also other spaces with invariant
partial ordering.
The central ingredient is an Euler element h in the Lie algebra of \fg. We
define three different kinds of wedge domains depending on h and the causal
structure on G/H. Our main result is that the connected component containing
the base point eH of those seemingly different domains all agree. Furthermore
we discuss the connectedness of those wedge domains. We show that each of those
spaces has a natural extension to a non-compactly causal symmetric space of the
form G_\C/G^c where G^c is certain real form of the complexification G_\$ of G.
As G_\C/G^c is non-compactly causal it also comes with the three types of wedge
domains. Our results says that the intersection of those domains with $G/H$
agrees with the wedge domains in G/H.
