Relative Functoriality and Functional Equations via Trace Formulas

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.

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Notes

  1. 1.

    If the map \(X(F)\times X(F)\to \mathfrak {X}(F)\) is not surjective on points, this means that there are non-trivial G-torsors R which contribute F-points to \(\mathfrak {X}\). Then, (1.3) should be replaced by

    $$\bigoplus\limits_{\alpha} \mathcal S\left( X^{\alpha}\times X^{\alpha}\right) \to \mathcal M\left( \mathfrak{C}_{X}\right), $$

    with α ranging over all isomorphism classes of G-torsors Rα, and Xα := X ×GRα, a Gα-space, where Gα is the inner form AutG(Rα). For example, for X = T∖PGL2, Xα includes X as well as the quotient TPD×, where D is the quaternion division algebras over F, and PD× is the quotient of its multiplicative group by the center. For simplicity of exposition, we will ignore such cases in this paper, although the particular case of a non-split torus in PGL2 has already been studied in [26, 32].

  2. 2.

    This convention on local gamma factors was misstated in [32, Lemma 2.12], where ψ− 1 instead of ψ was used to define Fourier transforms. As a result, ψ should be replaced by ψ− 1 in all gamma factors of [26, 32].

  3. 3.

    In the Archimedean case, the Schwartz spaces are nuclear Fréchet spaces, hence their completed tensor product; in the non-Archimedean cases, the completion should be ignored.

  4. 4.

    For the topological properties of extended Schwartz spaces and the notion of holomorphic sections, I point the reader to [26, Appendix A].

  5. 5.

    That is, when the spaces are defined over a global field k, one can choose the base point to be defined over k and the local Haar measures to factorize a Tamagawa measure. This includes the Haar measure on AX used in the definition of Mellin transform, which induces the Haar–Plancherel measure dχ on \(\widehat {A_{X}}\) that appears in the statement of the theorem.

  6. 6.

    For example, for G = GL2, \(\psi (-e^{-\alpha _{1}})\) denotes the function \(\left (\begin {array}{ll} a & * \\ &d \end {array}\right )\mapsto \psi \left (-\frac {d}{a}\right )\).

  7. 7.

    We use the same symbol for the “half-density” basic vector as for the “measure” basic vector; their quotient is simply a half-density \((\delta (t) dt)^{\frac {1}{2}}\) on NG/ /N, according to the integration formula (3.2). (Remember that dt denotes a Haar measure on the torus A, here.)

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Acknowledgements

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.

Funding

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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Correspondence to Yiannis Sakellaridis.

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Lecture at the Annual Meeting 2017 of the Vietnam Institute for Advanced Study in Mathematics

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Sakellaridis, Y. Relative Functoriality and Functional Equations via Trace Formulas. Acta Math Vietnam 44, 351–389 (2019). https://doi.org/10.1007/s40306-018-0295-7

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Keywords

  • Trace formula
  • Langlands program
  • Beyond endoscopy

Mathematics Subject Classification (2010)

  • 11F70