Relative functoriality and functional equations via trace formulas

TL;DR

This paper explores the use of trace formulas and transfer operators to study functoriality and functional equations in the Langlands program, providing examples and a local-to-global approach.

Contribution

It introduces a novel framework using transfer operators and Hankel transforms to analyze functional equations via trace formulas within the relative Langlands program.

Findings

Demonstrates the use of transfer operators in functional equations

Provides examples of trace formula comparisons

Proposes a local-to-global approach for the Langlands program

Abstract

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.