The Selberg trace formula revisited

TL;DR

This paper introduces a novel approach to the spectral analysis of the Selberg trace formula using Plancherel decomposition of asymptotically finite functions, aiming to generalize to a relative trace formula.

Contribution

It develops a new method for analyzing the spectral side of the Selberg trace formula based on Plancherel decomposition, with potential extensions to invariant and relative trace formulas.

Findings

Proposes a new spectral approach to the Selberg trace formula.

Introduces the concept of Plancherel decomposition for asymptotically finite functions.

Lays groundwork for a generalized relative trace formula.

Abstract

A new approach to the Selberg trace formula, and more precisely to its spectral side, is developed. The approach relies on a notion of "Plancherel decomposition" of "asymptotically finite functions", and may generalize to obtain a general relative trace formula. This is an incomplete first version that will be complemented by an account of the invariant trace formula.