Invariant Integral Operators on the Oshima Compactification of a Riemannian Symmetric Space: Kernel Asymptotics and Regularized Traces

Abstract

Let $\mathbb{X}$ be a Riemannian symmetric space of non-compact type. Then $\mathbb{X}$ is isomorphic to G∕K, where G is a connected, real, semisimple Lie group, and K a maximal compact subgroup. Consider further the Oshima compactification $\widetilde{\mathbb{X}}$ of $\mathbb{X}$ [8], which is a simply connected, closed, real-analytic manifold carrying an analytic G-action. The orbital decomposition of $\widetilde{\mathbb{X}}$ is of normal crossing type, and the open orbits are isomorphic to G∕K, the number of them being equal to 2^l, where l denotes the rank of G∕K.