Residue representations -- the rank one case

TL;DR

This paper develops an algorithm to analyze residue representations associated with Laplacian resonances on rank-one symmetric spaces, determining their irreducibility, Langlands parameters, and geometric properties.

Contribution

It introduces a novel algorithm for classifying residue representations in rank-one symmetric spaces, including their irreducibility and parameters.

Findings

Algorithm successfully determines irreducibility of residue representations.

Application to classical real rank-one Lie groups.

Provides explicit analysis of Laplacian on p-forms.

Abstract

With each resonance of the Laplacian acting on the compactly supported sections of a homogeneous vector bundle over a Riemannian symmetric space of the non-compact type, One can associate a residue representation. The purpose of this paper is to study them. The symmetric space is assumed to have rank-one but the irreducible representation {\tau} of K defining the vector bundle is arbitrary. We give an algorithm that aims at determining if these representations are irreducible, finding their Langlands parameters, their Gelfand-Kirillov dimensions and wave front sets. As an example, we apply this algorithm to the Laplacian of the p-forms in the cases of all the classical real rank-one Lie groups.