Resonances for the Laplacian on products of two rank one Riemannian symmetric spaces

Abstract

Abstract. Let X = X 1 ×X 2 be a direct product of two rank-one Riemannian symmetric spaces of the noncompact type. We show that when at least one of the two spaces is isomorphic to a real hyperbolic space of odd dimension, the resolvent of the Laplacian of X can be lifted to a holomorphic function on a Riemann surface which is a branched covering of C. In all other cases, the resolvent of the Laplacian of X admits a singular meromorphic lift. 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 finite dimensional and explicitly realized as direct sums of finite-dimensional irreducible spherical representations of the group of the isometries of X.

On the Connection between Spherical Laplace Transform and Non-Euclidean Fourier Analysis

“…which follows from (17) and from the uniqueness of the analytic continuation. The properties mentioned above imply that f (τ) is of the form:…”

“…This allowed them to prove the conjecture that every non-compactly causal symmetric space occurs as a component of a distinguished boundary of some complex crown [15]. Finally, it is worth recalling that the question of relating the harmonic analysis of different real forms of a complex symmetric space has been studied also in the context of scattering theory and resonances [16][17][18][19][20].…”

Residue representations -- the rank one case

“…Partial results were obtained by Mazzeo and Vasy [MV05] and Strohmaier [Str05]. Complete results for most of the rank-two cases were proved in a series of papers by Hilgert, Pasquale and Przebinda [HPP16,HPP17b,HPP17a]. For the Laplacian acting on line bundles over complex hyperbolic spaces, the resonances has been completely determined by Will in [Wil03].…”