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 $Z=G\mathrm{/}H$. In particular we provide a geometric construction of the little Weyl group of $Z$ introduced recently by Knop and Kr"otz. Our technique is based on a fine analysis of limits of conjugates of the subalgebra $\mathrm{L}\mathrm{i}\mathrm{e}(H)$ along one-parameter subgroups in the Grassmannian of subspaces of $\mathrm{L}\mathrm{i}\mathrm{e}(G)$. The little Weyl group is obtained as a finite reflection group generated by the reflections in the walls of the compression cone.