Resonances for the Laplacian: the cases $BC_2$ and $C_2$ (except $SO_0(p,2)$ with $p>2$ odd)

TL;DR

This paper studies the spectral properties of the Laplacian on certain symmetric spaces with root systems $BC_2$ and $C_2$, explicitly locating resonances and describing the structure of associated residue operators.

Contribution

It reduces the analysis of the Laplacian's resolvent to a product of rank-one spaces and explicitly characterizes the meromorphic continuation and resonances for these cases.

Findings

Resonances are located on a branched Riemann surface.

The resolvent admits a meromorphic continuation with finite-rank residues.

Residue operators decompose into finite-dimensional spherical representations.

Abstract

Let $X=G/K$ be a Riemannian symmetric space of the noncompact type and restricted root system $BC_2$ or $C_2$ (except $G=SO_0(p,2)$ with $p>2$ odd). The analysis of the meromorphic continuation of the resolvent of the Laplacian of $X$ is reduced from the analysis of the same problem for a direct product of two isomorphic rank-one Riemannian symmetric spaces of the noncompact type which are not isomorphic to real hyperbolic spaces. We prove that the resolvent of the Laplacian of $X$ can be lifted to a meromorphic function on a Riemann surface which is a branched covering of the complex plane. Its poles, that is the resonances of the Laplacian, are explicitly located on this Riemann surface. The residue operators at the resonances have finite rank. Their images are finite direct sums of finite-dimensional irreducible spherical representations of $G$.