Real Orbits of Complex Spherical Homogeneous Spaces: the Split Case

Abstract

Abstract We introduce some reflection operators on the set of real Borel 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 ${\mathbb {R}}$. 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 the above-mentioned reflection 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 for such $X({\mathbb {R}})$ in terms of the orbits of this new action.