Relative Langlands Functoriality, Trace Formulas and Harmonic Analysis

Introduction

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, which postulates the existence of deep relations between the automorphic representations of different groups, as well as related central analytic objects called automorphic L-functions.

The Relative Langlands Program

Recently, a new and particularly promising way to look at these notions, known as the relative Langlands program, has emerged. It essentially consists of replacing groups by certain homogeneous spaces and studying their automorphic or local spectra.

Importance of Trace Formulas

As with the usual Langlands program, trace formulas are essential tools in the relative setting, both to tackle new conjectures and to deepen our understanding of the underlying principles.

Main Contributions

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 progressions would then be applied to establish new and important instances of relative Langlands correspondences/functorialities.

Additional Directions

In a slightly different but related direction, I also aim to study and develop other important tools of harmonic analysis in a relative context, including Plancherel formulas and new kinds of Paley-Wiener theorems. This may have possible applications to new global comparisons of trace formulas and factorization of automorphic periods.