Resonances for the Laplacian on Riemannian symmetric spaces: the case of SL(3, R)/SO(3)

Abstract

We show that the resolvent of the Laplacian on SL(3, R)/SO(3) can be lifted to a meromorphic function on a Riemann surface which is a branched covering of C. The poles of this function are called the resonances of the Laplacian. We determine all resonances and show that the corresponding residue operators are given by convolution with spherical functions parameterized by the resonances. The ranges of these operators are infinite dimensional irreducible SL(3, R)-representations whose Langlands parameters can also be read off from the corresponding resonances. Alternatively, they are given by the differential equations which determine the image of the Poisson transform associated with the resonance