The residues of the resolvent on Damek-Ricci spaces

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