Resonances for the Laplacian on Riemannian symmetric spaces: the case of SL(3,$\mathbb{R}$)/SO(3)

TL;DR

This paper characterizes the resonances of the Laplacian on SL(3, R)/SO(3), explicitly determines their locations, and describes the associated residue operators and their representation-theoretic properties.

Contribution

It provides a complete description of the resonances for the Laplacian on SL(3, R)/SO(3), including their residues, representations, and unitarizability, advancing understanding of spectral theory on symmetric spaces.

Findings

All resonances are explicitly determined.

Residue operators are convolution with spherical functions.

One resonance corresponds to a unitarizable representation.

Abstract

We show that the resolvent of the Laplacian on SL(3,$\mathbb{R}$)/SO(3) can be lifted to a meromorphic function on a Riemann surface which is a branched covering of $\mathbb{C}$. 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,$\mathbb{R}$)-representations. We determine their Langlands parameters and wave front sets. Also, we show that precisely one of these representations is unitarizable. Alternatively, they are given by the differential equations which determine the image of the Poisson transform associated with the resonance.