Integral operators on the Oshima compactification of a Riemannian symmetric space of non-compact type. Regularized traces and characters

Consider a Riemannian symmetric space $X= G/K$ of non-compact type, where $G$ denotes a connected, real, semi-simple Lie group with finite center, and $K$ a maximal compact subgroup of $G$. Let $\widetilde X$ be its Oshima compactification, and $(\pi,\mathrm{C}(\widetilde X))$ the regular representation of $G$ on $\widetilde X$. In this paper, a regularized trace for the convolution operators $\pi(f)$ is defined, yielding a distribution on $G$ which can be interpreted as global character of $\pi$. In case that $f$ has compact support in a certain set of transversal elements, this distribution is a locally integrable function, and given by a fixed point formula analogous to the formula for the global character of an induced representation of $G$.