Real orbits of complex spherical homogeneous spaces: the split case

We identify the \(G(\mathbb R)\)-orbits of the real locus \(X(\mathbb R)\) of any spherical complex variety \(X\) defined over \(\mathbb R\) and homogeneous under a split connected reductive group \(G\) defined also over \(\mathbb R\). This is done by introducing some reflection operators on the set of real Borel orbits of \(X(\mathbb R)\). We thus investigate the existence problem for an action of the Weyl group of \(G\) on the set of real Borel orbits of \(X(\mathbb R)\). In particular, we determine the varieties \(X\) for which these operators define an action of the very little Weyl group of \(X\) on the set of open real Borel orbits of \(X(\mathbb R)\). This enables us to give a parametrization of the \(G(\mathbb R)\)-orbits of \(X(\mathbb R)\) in terms of the orbits of this new action.