The residues of the resolvent on Damek-Ricci spaces

Abstract

We determine the poles and residues of the resolvent kernel of the Laplacian on a Damek-Ricci space S . S. We show that all poles are simple and the residues define convolution operators of finite rank. This generalizes a result of Guillopé-Zworski for the real hyperbolic n n -space. If S S corresponds to a symmetric space of negative curvature G / K G/K , the image of each residue is a g c {\frak g}_c -module with a specific highest weight. We compute the dimension by the Weyl dimension formula.