Lab: Clifton Cunningham, Andrew Fiori, James Mracek Ahmed Moussaoui and Bin Xu
A geometric approach to Arthur packets for p-adic groups
Colloquially known as the Grand Unified Theory of Mathematics, the significance and centrality of the Langlands Program in the landscape of mathematics cannot be overestimated. The Langlands Programme as we now know it was framed as a research agenda in the late 1960s. At the same time that the Nobel Prize was being awarded to Murray Gell-Mann ``for his contributions and discoveries concerning the classification of elementary particles and their interactions,'' the Langlands Program predicted that fundamental properties of the elementary particles of arithmetic, L-functions, should be reflected in automorphic representations of algebraic groups.
One of the pillars of the Langlands program is a conjecture, known as the Langlands Correspondence, which predicts the precise form of the relation between Galois representations and automorphic representations and is crucial to attaching complete L-functions to automorphic representations. In 2013 this conjecture was proved for a large class of groups which includes the classical groups, by James Arthur relying on the work of many mathematicians, including Colette Moeglin and Jean-Loup Waldspurger and Fields Medalist Ngô Bảo Châu.
The precise form of the Langlands Correspondence is difficult to calculate and, in full generality, quite delicate even to state, relying on numerous interrelated canonical bijections. In very rough terms, it arranges representations of the group G(F), where F is a non-Archimedean local field, into so-called `packets', indexed by the Langlands parameters, namely, representations of a group, closely related to the Galois group of F, with the image in the Langlands dual group of G. These packets are the so-called L-packets; describing them is a wide open problem.
One can think of partitioning the set of suitable representations of G(F) into L-packets as a description of the spectrum of G -- indeed, this is the generalization of Fourier analysis to a non-abelian locally compact group. However, this is where it matters what one means by the `suitable': it turns out that the natural class of representations that arise in the L-packets is not precisely the class of unitary representations of G(F) which one needs to understand in harmonic analysis, e.g., to have a Plancherel formula. James Arthur proposed a modification both to the set of parameters, defining what is now called Arthur parameters, and to the packets, which became Arthur packets; a complete description of Arthur packets may be thought of as the ultimate understanding of the harmonic analysis of G(F).
In the early 1990s, David Vogan introduced the idea of using microlocal analysis on a moduli space of Langlands parameters to study Arthur packets for p-adic groups. This idea was intended to parallel a construction for Real groups developed in conjunction with Jeffrey Adams and Dan Barbasch in their 1992 book _The Langlands Classification and Irreducible Characters for Real Reductive Groups_. Vogan's idea for p-adic groups was presented in the form of conjectures that appeared in his 1993 paper on the _Local Langlands Correspondence_. In broad strokes, the idea is to use a microlocal study of singularities that appear in a moduli space of Langlands parameters with shared ``infinitesimal parameter'' to illuminate local Arthur packets. However, until James Arthur produced a purely local description of Arthur packets for p-adic groups in his 2013 book on _The Endoscopic Classification of Representations: Orthogonal and Symplectic Groups_, it was difficult to test -- let alone verify -- Vogan's conjectures. They remain open conjectures to this date.
This project is aimed at proving Vogan's conjectures for p-adic groups. The idea is simple enough: adapt the 1992 book _The Langlands Classification and Irreducible Characters of Real Groups_ by Jeff Adams, Dan Barbash and David Vogan to p-adic groups, following the direction charted by David Vogan in his 1993 paper _The Local Langlands Conjecture_. Our first result in this direction is _Arthur packets for p-adic groups by way of microlocal vanishing cycles of perverse sheaves, with examples_, by Cunningham, Fiori, Moussaoui, Mrack and Xu, to appear Memoirs of the American Mathematical Society in 2021, available on the arxiv in the meantime. In this paper we lay the foundation for our approach to this problem and use purely geometric techniques to define what we call ABV-packets; Vogan's conjectures product that every (pure) Arthur packet is an ABV-packet; in fact, ABV-packets are a more general notion. In our first paper we also gather supporting evidence and illustrate our methods by calculating many examples of Arthur packets and, independently, ABV-packets and showing that they coincide in the cases we consider.
This project was suggested to Clifton by James Arthur at the BIRS workshop The Future of Trace Formulas in December 2014. We began working on this project in September 2015 when Ahmed Moussaoui, Andrew Fiori and Bin Xu joined forces with Clifton Cunningham. James Mracek's PhD thesis, jointly supervised by Clifton Cunningham and Lisa Jeffreys, also fits into this project.
