  Let $\X\simeq G/K$ be a Riemannian symmetric space of non-compact type,
$\widetilde \X$ its Oshima compactification, and $(\pi,\mathrm{C}(\widetilde
\X))$ the regular representation of $G$ on $\widetilde \X$. We study integral
operators on $\widetilde \X$ of the form $\pi(f)$, where $f$ is a rapidly
falling function on $G$, and characterize them within the framework of
pseudodifferential operators, describing the singular nature of their kernels.
In particular, we consider the holomorphic semigroup generated by a strongly
elliptic operator associated to the representation $\pi$, as well as its
resolvent, and describe the asymptotic behavior of the corresponding semigroup
and resolvent kernels.
