Periodic Reporting for period 1 - RELANTRA (Relative Langlands Functoriality, Trace Formulas and Harmonic Analysis)
Berichtszeitraum: 2022-09-01 bis 2025-02-28
The Langlands program is a web of vast and far-reaching conjectures connecting seemingly distinct
areas of mathematics that are number theory and representation theory. At the heart of this program
lies an important principle called functoriality, that postulates the existence of deep relations between
the automorphic representations of different groups, as well as related central analytic objects called
automorphic L-functions. Recently, a new and particularly promising way to look at these notions,
and that has come to be called the relative Langlands program, has emerged. It essentially consists in
replacing groups by certain homogeneous spaces and to study their automorphic or local spectra.
As for the usual Langlands program, trace formulas are essential tools in the relative setting both
to tackle new conjectures than to deepen our understanding of the underlying principles. A main
theme of this proposal would be to make fundamental new contributions to the development of these
central objects in the local setting notably by: (1) Studying systematically the spectral expansions of
certain simple versions especially in the presence of an outer automorphism (twisted trace formula) (2)
Developing far-reaching local relative trace formulas for general spherical varieties making in particular
original new connections to the geometry of cotangent bundles. These progress would then be applied
to establish new and important instances of relative Langlands correspondences/functorialities. In
a slightly different but related direction, we also aim to study and develop other important tools of
harmonic analysis in a relative context, including Plancherel formulas and new kind of Paley-Wiener
theorems, with possible applications to new global comparison of trace formulas and factorization of
automorphic periods.
areas of mathematics that are number theory and representation theory. At the heart of this program
lies an important principle called functoriality, that postulates the existence of deep relations between
the automorphic representations of different groups, as well as related central analytic objects called
automorphic L-functions. Recently, a new and particularly promising way to look at these notions,
and that has come to be called the relative Langlands program, has emerged. It essentially consists in
replacing groups by certain homogeneous spaces and to study their automorphic or local spectra.
As for the usual Langlands program, trace formulas are essential tools in the relative setting both
to tackle new conjectures than to deepen our understanding of the underlying principles. A main
theme of this proposal would be to make fundamental new contributions to the development of these
central objects in the local setting notably by: (1) Studying systematically the spectral expansions of
certain simple versions especially in the presence of an outer automorphism (twisted trace formula) (2)
Developing far-reaching local relative trace formulas for general spherical varieties making in particular
original new connections to the geometry of cotangent bundles. These progress would then be applied
to establish new and important instances of relative Langlands correspondences/functorialities. In
a slightly different but related direction, we also aim to study and develop other important tools of
harmonic analysis in a relative context, including Plancherel formulas and new kind of Paley-Wiener
theorems, with possible applications to new global comparison of trace formulas and factorization of
automorphic periods.
So far, this project has lead to the following main scientific achievements: 1) The complete resolution of a conjecture of Gan-Gross-Prasad relating global Bessel periods on unitary groups to central values of Rankin-Selberg L-functions; 2) The proof of a similar conjecture for the so-called Fourier-Jacobi periods; 3) The development of new kind of local twisted trace formulas in a very general setting which already had applications to a conjecture of Prasad on the description of the discrete spectrum of certain local symmetric spaces named "galoisian" in terms of the local Langlands correspondence for classical groups; 4) Important progress towards the development of a general Plancherel formula for real spherical spaces à la Sakellaridis-Venkatesh.
Other important progress have been made thanks to this project on fundamental questions in harmonic analysis on reductive groups over non-archimedean local fields and are currently in the process of being finalized. These concern more precisely: 5) A uniform proof for all classical groups of a conjecture of Hiraga-Ichino-Ikeda relating formal degrees of square-integrable representations to their adjoint gamma factors; 6) The partial resolution of a long-standing conjecture on the local integrability of irreducible characters over local fields of positive characteristics, extending som foundational results of Harish-Chandra to this setting.
Other important progress have been made thanks to this project on fundamental questions in harmonic analysis on reductive groups over non-archimedean local fields and are currently in the process of being finalized. These concern more precisely: 5) A uniform proof for all classical groups of a conjecture of Hiraga-Ichino-Ikeda relating formal degrees of square-integrable representations to their adjoint gamma factors; 6) The partial resolution of a long-standing conjecture on the local integrability of irreducible characters over local fields of positive characteristics, extending som foundational results of Harish-Chandra to this setting.
The aforementioned resolution of the Gan-Gross-Prasad conjectures for Bessel and Fourier-Jacobi periods on unitary groups should have important impact on the field, both for the methods developed at this occasion which might be useful to tackle similar open problems than for arithmetic applications related to special values of L-functions.
The newly developed local twisted relative trace formulas provide the first example of such distributions outside of the group case that have been established in the full expected generality; i.e. without any restrictions on the space of test functions. This has required introducing a new kind of regularization process and can be seen as an important step towards the obtention of completely general local trace formulas for spherical varieties. This work was however so far limited to the special case of symmetric varieties and even there the results are not yet complete: the geometric expansions of these trace formulas were obtained under an additional technical condition, that the variety is 'coregular', and we didn't compute the full spectral expansion yet. We plan to continue working on these problems, in particular the development of a complete spectral expansion for symmetric varieties and the extension of the regularization process to arbitrary spehrical varieties during the second half of the project.
The newly developed local twisted relative trace formulas provide the first example of such distributions outside of the group case that have been established in the full expected generality; i.e. without any restrictions on the space of test functions. This has required introducing a new kind of regularization process and can be seen as an important step towards the obtention of completely general local trace formulas for spherical varieties. This work was however so far limited to the special case of symmetric varieties and even there the results are not yet complete: the geometric expansions of these trace formulas were obtained under an additional technical condition, that the variety is 'coregular', and we didn't compute the full spectral expansion yet. We plan to continue working on these problems, in particular the development of a complete spectral expansion for symmetric varieties and the extension of the regularization process to arbitrary spehrical varieties during the second half of the project.