Resonances of the Laplace operator on homogeneous vector bundles on symmetric spaces of real rank-one

TL;DR

This paper investigates the resonances of the Laplacian on homogeneous vector bundles over rank-one symmetric spaces, identifying their properties, representations, and unitarizability, thus advancing understanding of spectral theory in geometric analysis.

Contribution

It provides a comprehensive analysis of Laplacian resonances on vector bundles over rank-one symmetric spaces, including explicit determination of resonance representations and their properties.

Findings

Resonances are explicitly determined for the Laplacian on these bundles.

Resonance representations are all irreducible under certain conditions.

The paper identifies Langlands parameters, wave front sets, and unitarizability of the resonances.

Abstract

We study the resonances of the Laplacian acting on the compactly supported sections of a homogeneous vector bundle over a Riemannian symmetric space of the non-compact type. The symmetric space is assumed to have rank-one but the irreducible representation $\tau$ of $K$ defining the vector bundle is arbitrary. We determine the resonances. Under the additional assumption that $\tau$ occurs in the spherical principal series, we determine the resonance representations. They are all irreducible. We find their Langlands parameters, their wave front sets and determine which of them are unitarizable.