On the little Weyl group of a real spherical space

In the present paper we further the study of the compression cone of a real spherical homogeneous space Z= G/ H. In particular we provide a geometric construction of the little Weyl group of Z introduced recently by Knop and Krötz. Our technique is based on a fine analysis of limits of conjugates of the subalgebra Lie (H) along one-parameter subgroups in the Grassmannian of subspaces of Lie (G). The little Weyl group is obtained as a finite reflection group generated by the reflections in the walls of the compression cone.