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

2011/04/21 by Amy T. DeCelles, Amy DeCelles, DeCelles, Amy · 1 citation
Mathematics · Physics and Astronomy · #22E46 #33C52 #46F12 #58J40 #Advanced Algebra and Geometry #FOS: Mathematics #Geometry and complex manifolds #Nonlinear Waves and Solitons #Number Theory (math.NT) #Primary: 43A85 #Representation Theory (math.RT) #Secondary: 43A90 #math.NT #math.RT #msc:22E46 #msc:33C52 #msc:43A85 #msc:43A90 #msc:46F12 #msc:58J40

paper · pdf · doi:10.48550/arxiv.1104.4313

Revised proof of Prop 2.1 and 2.2; 17 pages, results from the author's PhD thesis (University of Minnesota, 2011) under the direction of Paul Garrett

openalex publication_date 2011/04/21 · arxiv created 2012/06/13 · arxiv updated 2012/06/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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