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.