Relative Functoriality and Functional Equations via Trace Formulas
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Langlands’ functoriality principle predicts deep relations between the local and automorphic spectra of different reductive groups. This has been generalized by the relative Langlands program to include spherical varieties, among which reductive groups are special cases. In the philosophy of Langlands’ “beyond endoscopy” program, these relations should be expressed as comparisons between different trace formulas, with the insertion of appropriate L-functions. The insertion of L-functions calls for one more goal to be achieved: the study of their functional equations via trace formulas.
The goal of this article is to demonstrate this program through examples, indicating a local-to-global approach as in the project of endoscopy. Here, scalar transfer factors are replaced by “transfer operators” or “Hankel transforms” which are nice enough (typically, expressible in terms of usual Fourier transforms) that they can be used, in principle, to prove global comparisons (in the form of Poisson summation formulas). Some of these examples have already appeared in the literature; for others, the proofs will appear elsewhere.
KeywordsTrace formula Langlands program Beyond endoscopy
Mathematics Subject Classification (2010)11F70
Most of the calculations in this paper were performed while visiting the University of Chicago during the winter and spring quarters of 2017. I am grateful to Ngô Bao Châu for the invitation, and for numerous conversations and references, which made this paper possible. His ideas permeate the paper. The paper was finished during my stay at the Institute for Advanced Study in the Fall of 2017.
This work was supported by the NSF grant DMS-1502270 and by a stipend to the IAS from the Charles Simonyi Endowment.
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