  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.
