Integral operators on the Oshima compactification of a Riemannian symmetric space of non-compact type. Microlocal analysis and kernel asymptotics

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.