  Following the ideas of Ginzburg, for a subgroup $K$ of a connected reductive
$\mathbb{R}$-group $G$ we introduce the notion of $K$-admissible $D$-modules on
a homogeneous $G$-variety $Z$. We show that $K$-admissible $D$-modules are
regular holonomic when $K$ and $Z$ are absolutely spherical. This framework
includes: (i) the relative characters attached to two spherical subgroups $H_1$
and $H_2$, provided that the twisting character $\chi_i$ factors through the
maximal reductive quotient of $H_i$, for $i = 1, 2$; (ii) localization on $Z$
of Harish-Chandra modules; (iii) the generalized matrix coefficients when
$K(\mathbb{R})$ is maximal compact. This complements the holonomicity proven by
Aizenbud--Gourevitch--Minchenko. The use of regularity is illustrated by a
crude estimate on the growth of $K$-admissible distributions which based on
tools from subanalytic geometry.
