Isolation of the Cuspidal Spectrum: the Function Field Case

TL;DR

This paper extends methods for isolating the cuspidal automorphic spectrum using multipliers of the Schwartz algebra from number fields to the function field case, aiding in spectral analysis and comparison of orbital integrals.

Contribution

It proves the analogous result for isolating the cuspidal spectrum in the function field setting, building on recent techniques used for number fields.

Findings

Successfully isolates the cuspidal spectrum in the function field case.

Provides tools for spectral analysis and comparison of orbital integrals in function fields.

Extends the applicability of Schwartz algebra multipliers to new mathematical settings.

Abstract

Isolating cuspidal automorphic representations from the whole automorphic spectrum is a basic problem in the trace formula approach. For example, matrix coefficients of supercupidal representations can be used as test functions for this, which kills the continuous spectrum, but also a large class of cuspidal automorphic representations. For the case of number fields, multipliers of the Schwartz algebra is used in the recent work [3] to isolate all cuspidal spectrum which provide enough test functions and suitable for the comparison of orbital integrals. These multipliers are then applied to the proof of the Gan-Gross-Prasad conjecture for unitary groups [3,2]. In this article, we prove similar result on isolating the cuspidal spectrum in [3] for the function field case.