Isolation of the cuspidal spectrum:The function field case

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.(2021))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.(2021,2022)).In this article,we prove the similar result on isolating the cuspidal spectrum in Beuzart-Plessis et al.(2021)for the function field case.