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

Abstract

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 $C^{\infty}(G/H)$ by a geometric condition, referred to as \textit{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 $G/Q$.