Explicit Formulas from the Continuous Spectrum

TL;DR

This paper investigates how the continuous spectrum influences the Selberg trace formula for rank-2 groups, providing explicit formulas in the specific case of SL(3,R) and SL(3,Z).

Contribution

It introduces explicit formulas connecting the continuous spectrum to the Selberg trace formula in the rank-2 setting, focusing on SL(3,R) and SL(3,Z).

Findings

Explicit formulas derived for the continuous spectrum's role.

Insights into the spectral decomposition in rank-2 cases.

Foundational step for generalizations in spectral theory.

Abstract

The purpose of this note is to announce the results of our investigation into the role played by the continuous spectrum in the development of the Selberg trace formula vis-\`a-vis a pair $(G,\Gamma)$. For the sake of simplicity, we shall restrict ourselves to a "rank-2" situation, a case in point being when $G = \textbf{SL}(3,\mathbb{R}), \ \Gamma = \textbf{SL}(3,\mathbb{Z})$. Full details (in all generality) will appear elsewhere.