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

TL;DR

This paper investigates the resonances of the Laplacian on products of two rank-one symmetric spaces, revealing their structure, explicit determination, and the nature of associated residue operators, with special cases involving hyperbolic spaces.

Contribution

It provides a complete characterization of the Laplacian's resonances on these spaces, including explicit formulas and the structure of residue operators, extending previous understanding of spectral properties.

Findings

Resonances are explicitly determined for all cases.

Residue operators are finite-dimensional and correspond to spherical representations.

Holomorphic and meromorphic lifts of the resolvent are classified based on space types.

Abstract

Let $X=X_1 \times X_2$ 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 $\mathbb C$. 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 $X$.