Fundamental solution for (Delta - lambda_z)^n on a symmetric space G/K

Abstract

We determine a fundamental solution for the differential operator (Delta - lambda_z)^n on the Riemannian symmetric space G/K, where G is any complex semi-simple Lie group, and K is a maximal compact subgroup. We develop a global zonal spherical Sobolev theory, which enables us to use the harmonic analysis of spherical functions to obtain an integral representation for the solution. Then we obtain an explicit expression for the fundmantal solution, which allows relatively easy estimation of its behavior in the eigenvalue parameter lambda_z, with an eye towards further applications to automorphic forms involving asociated Poincare series.