Relationship between orbit decomposition on the flag varieties and multiplicities of induced representations

Let G be a real reductive Lie group and H a closed subgroup. T. Kobayashi and T. Oshima established a finiteness criterion of multiplicities of irreducible G-modules occurring in the regular representation C1ðG=HÞ by a geometric condition, referred to as real sphericity, namely, H has an open orbit on the real flag variety G=P . This note discusses a refinement of their theorem by replacing a minimal parabolic subgroup P with a general parabolic subgroup Q of G, where a careful analysis is required because the finiteness of the number of H-orbits on the partial flag variety G=Q is not equivalent to the existence of H-open orbit on G=Q.