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

TL;DR

This paper derives an explicit fundamental solution for a high-order differential operator on complex semi-simple Lie group symmetric spaces, utilizing harmonic analysis and spherical functions, with implications for automorphic forms.

Contribution

It develops a global spherical Sobolev theory and provides an explicit integral representation for the fundamental solution of (Delta - lambda_z)^n on G/K.

Findings

Explicit fundamental solution derived

Integral representation obtained

Behavior estimation in eigenvalue parameter achieved

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.