  Motivated by construction in Algebraic Quantum Field Theory we introduce
wedge domains in compactly causal symmetric spaces M=G/H, which includes in
particular anti de Sitter space in all dimensions and its coverings. Our wedge
domains generalize Rindler wedges in Minkowski space. The key geometric
structure we use is the modular flow on M defined by an Euler element in the
Lie algebra of G. Our main geometric result asserts that three seemingly
different characterizations of these domains coincide: the positivity domain of
the modular vector field; the domain specified by a KMS like analytic extension
condition for the modular flow; and the domain specified by a polar
decomposition in terms of certain cones.
  In the second half of the article we show that our wedge domains share
important properties with wedge domains in Minkowski space. If G is semisimple,
there exist unitary representations of G and isotone covariant nets of real
subspaces defined for any open subset of M, which assign to connected
components of the wedge domains a standard subspace whose modular group
corresponds to the modular flow on M. This corresponds to the
Bisognano--Wichmann property in Quantum Field Theory. We also show that the set
of G-translates of the connected components of the wedge domain provides a
geometric realization of the abstract wedge space introduced by the first
author and V. Morinelli.
