Given a globally causal homogeneous manifold \(M\) and an invariant differential operator \(D\) on \(M\), the author gives sufficient conditions for \(M\) and \(D\) to have a fundamental solution. More explicitly, \(M\) is a causal variety if there is a family \((C_x)_{x\in M}\) of closed connexe cones with nonempty interior set for which each \(C_x\) is contained in the tangent space \(T_xM\) of \(M\) at \(x\). The variety \(M\) is said to be a causal homogeneous variety if it is homogeneous (\(M=G/H\), \(G\) a connexe Lie group and \(H\) a closed subgroup of \(G\)) and it is a causal variety, where the family \((C_x)\) is \(G\)-invariant by the left action of \(G\) over \(M\). This causal homogeneous structure is global if there is no nontrivial closed causal curve. If \(M\) is a symmetric space with involutive automorphism \(\sigma\), \(\mathfrak g= \text{Lie} (G)\) has a decomposition \(\mathfrak g=\mathfrak h + \mathfrak q\) into eigenspaces associated to the eigenvalues 1 and \(-1\), respectively. On the other hand, there is a Cartan decomposition of \(\mathfrak g =\mathfrak t +\mathfrak p\) corresponding to a Cartan involution that commutes with \(\sigma\). The global invariant causal structure of \(M\) is given by a cone \(C\subset \mathfrak q\) invariant under Ad(\(H\)). Let \(\mathfrak a\) be a Cartan subspace in \(\mathfrak p \cap\mathfrak q\). By the Harish-Chandra homomorphism every left invariant differential operator \(D\) can be identified with a \(W\)-invariant polynomial \(\gamma_D \in S(\mathfrak a)^W\), where \(W\) is the Weyl group of the restricted root system \(\Delta(\mathfrak g, \mathfrak a)\) associated to \(\mathfrak a\). Let \(\Omega \in \mathfrak a^*\) be the interior set of the dual cone of \(C\cap\mathfrak a\). An invariant differential operator \(D\) is hyperbolic with respect to \(\Omega\) if there exists \(\lambda_o\in \mathfrak a^*\) such that \(\gamma_D(\lambda) \neq 0\) for every \(\lambda\) in \(\lambda_o+\Omega+i\mathfrak a^*\subset \mathfrak a^*_\mathbb{C}\). A fundamental solution of \(D\) over \(C^\infty_c(\mathfrak a)\) is said to be causal if its support is contained in the cone \(\exp(C\cap\mathfrak a)\). In this situation the author conjectures that an invariant differential operator \(D\) has a causal fundamental solution if it is hyperbolic with respect to \(\Omega\). The principal result of the paper is the proof of the conjecture for the following cases: (i) \(M=G/H\) is a causal symmetric space, where \(G\) is a complexification of \(H\) and \(\mathfrak h=\text{Lie} (H)\) is an hermitian simple Lie algebra (i.e. \(M\) is of Ol’shanskii type). (ii) \(M\) is a causal symmetric space of rank one.