Consider a Riemannian symmetric space of non-compact type, where G is a connected, real, semisimple Lie group, and K a maximal compact subgroup. Let be its Oshima compactification, and the left-regular representation of G on . In this paper, we examine the convolution operators for rapidly decaying functions f on G, and characterize them within the framework of totally characteristic pseudodifferential operators, describing the singular nature of their kernels. As a consequence, we obtain asymptotics for heat and resolvent kernels associated to strongly elliptic operators on . As a further application, a regularized trace for the operators can be defined, yielding a distribution on G which can be interpreted as a global character of π, and is given by a fixed point formula analogous to the Atiyah–Bott character formula for an induced representation of G.