Let Z be an algebraic homogeneous space $$Z=G/H$$Z=G/H attached to real reductive Lie group $$G$$G. We assume that Z is real spherical, i.e., minimal parabolic subgroups have open orbits on Z. For such spaces, we investigate their large scale geometry and provide a polar decomposition. This is obtained from the existence of simple compactifications of Z which is established in this paper.
- Polar decomposition
- Spherical spaces