Integral operators on the Oshima compactification of a Riemannian symmetric space

Consider a Riemannian symmetric space X=G/KX=G/K of non-compact type, where G is a connected, real, semisimple Lie group, and K   a maximal compact subgroup. Let X˜ be its Oshima compactification, and (π,C(X˜)) the left-regular representation of G   on X˜. In this paper, we examine the convolution operators π(f)π(f) 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 X˜. As a further application, a regularized trace for the operators π(f)π(f) 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.