Isolation of the cuspidal spectrum: The function field case

Abstract

Isolating cuspidal automorphic representations from the whole automorphic spectrum is a basic problem in the trace formula approach. For example, matrix coefficients of supercuspidal representations can be used as test functions for this. However, they kill a large class of interesting cuspidal automorphic representations. For the case of number fields, multipliers of the Schwartz algebra are used in the recent work (see Beuzart-Plessis et al. (2019)) to isolate all the cuspidal spectrum. In particular, they are suitable for the comparison of orbital integrals. These multipliers are then applied to the proof of the Gan-Gross-Prasad conjecture for unitary groups (see Beuzart-Plessis et al. (2019, 2020)). In this article, we prove the similar result on isolating the cuspidal spectrum in Beuzart-Plessis et al. (2019) for the function field case.