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.
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Acknowledgements
The first author was supported by National Natural Science Foundation of China (Grant No. 11971254). The second author was supported by National Natural Science Foundation of China (Grant No. 11501382). The authors thank Professor Ye Tian for his consistent encouragement, and thank Professor Yifeng Liu for reading the preliminary version of their manuscript. The authors also thank the anonymous referees for both the careful reading of their manuscript, and the very helpful comments and suggestions.
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Cai, L., Xu, B. Isolation of the cuspidal spectrum: The function field case. Sci. China Math. 65, 1331–1342 (2022). https://doi.org/10.1007/s11425-020-1851-y
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DOI: https://doi.org/10.1007/s11425-020-1851-y