A generalization of the Kobayashi-Oshima uniformly bounded multiplicity theorem

Let $P$ be a minimal parabolic subgroup of a real reductive Lie group $G$ and $H$ a closed subgroup of $G$. Then it is proved by T. Kobayashi and T. Oshima that the regular representation $C^{\infty}(G/H)$ contains each irreducible representation of $G$ at most finitely many times if the number of $H$-orbits on $G/P$ is finite. Moreover, they also proved that the multiplicities are uniformly bounded if the number of $H_{\mathbb C}$-orbits on $G_{\mathbb C}/B$ is finite, where $G_{\mathbb C}, H_{\mathbb C}$ are complexifications of $G, H$, respectively, and $B$ is a Borel subgroup of $G_{\mathbb C}$. In this article, we prove that the multiplicities of the representations of $G$ induced from a parabolic subgroup $Q$ in the regular representation on $G/H$ are uniformly bounded if the number of $H_{\mathbb C}$-orbits on $G_{\mathbb C}/Q_{\mathbb C}$ is finite. For the proof of this claim, we also show the uniform boundedness of the dimensions of the spaces of group invariant hyperfunctions using the theory of holonomic ${\mathcal D}_{X}$-modules.