Simple compactifications and polar decomposition of homogeneous real   spherical spaces
  Let Z be an algebraic homogeneous space Z=G/H attached to real reductive Lie group 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. 
Friedrich Knop, Bernhard Kr\"otz, Eitan Sayag, Henrik Schlichtkrull
math.RT