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

Isolation of the Cuspidal Spectrum: the Function Field Case