p-adic Groups, Representations, and the Langlands Program

ongoing

My objectives consist of laying new foundations for the representation theory of p-adic groups and making significant progress on the local, global and relative Langlands program. The Langlands program is a far-reaching collection of conjectures that relate different areas of mathematics including number theory and representation theory. Work in this area has also lead to the resolution of other major conjectures including Fermat's Last Theorem. A fundamental problem on the representation theory side is the construction of all (irreducible, smooth, complex) representations of certain matrix groups, called p-adic groups. Despite much progress in the past 40 years, we still know surprisingly little about these representations in the general setting. My first main objective is the construction of all (supercuspidal) representations in full generality. This will form the foundation for the future of the representation theory of p-adic groups and have a plethora of applications also beyond this area. Solving this problem will involve tackling all the complications that arise in the non-tame case compared to the tame case. I will then demonstrate the power of this result beyond the representation theory of p-adic groups by making significant contributions to the - global Langlands program. This will be achieved by constructing congruences between automorphic forms based on the existence of enough suitable (omni-)supercuspidal types for p-adic groups. - relative Langlands program. I will prove finite multiplicity of the representations occurring in the space of function on a spherical variety by combining my results about the shape of representations with properties of the moment map. Finally, I will use my insights to advance the explicit local Langlands correspondence by proving that the most-general construction to date, which treats non-singular representations, satisfies all required properties and suggesting a correspondence beyond non-singular representations.