The Arithmetic Geometry and Representation Theory of the Local Langlands Correspondence

The Langlands program is a far-reaching framework in modern mathematics connecting two seemingly unrelated types of objects. These are Galois representations, which encode the symmetries of polynomials, and certain analytic functions known as automorphic forms. The relationship between these objects has traditionally been studied using techniques from representation theory. Recently, however, a groundbreaking geometric perspective introduced by Fargues and Scholze has opened up new avenues of investigation, allowing mathematicians to apply powerful tools from algebraic geometry. Despite its elegance, this geometric approach remains inexplicit and only partially understood in relation to classical representation-theoretic results. In this project, the PI will develop a new and explicit theory, of cuspidal vector bundles, that will provide a means to connect these two approaches and enhance understanding of each. Beyond advancing mathematical knowledge, the project will provide training opportunities for graduate and undergraduate students and contribute to the mathematical community through workshops and conferences.

The formulation of the categorical local Langlands conjecture of Fargues and Scholze was a major breakthrough in the Langlands program for p-adic groups. However, this theory is quite inexplicit and its connections to classical representation-theoretic results are not well understood. The PI will investigate the representation-theoretic consequences of the geometric formulation of the local Langlands correspondence, as developed by Fargues and Scholze. The first main objective is to construct a category of cuspidal sheaves on the stack of L-parameters. The PI will prove this category is semisimple, with simple objects given by irreducible cuspidal vector bundles. This will yield geometric interpretations and generalizations of the combinatorial data appearing in Moeglin's work on supercuspidal enhanced Langlands parameters for classical groups, the conjectures of Aubert–Moussaoui–Solleveld, and Lusztig’s classification of cuspidal local systems. The second major goal is to construct the Galois-side analogues of Hecke eigensheaves associated to Arthur parameters. These eigensheaves are conjecturally related to the cohomology of local Shimura varieties. Drawing on techniques from linear Koszul duality, the PI will define these eigensheaves and propose new local-global compatibility conjectures linking them to the cohomology of Igusa stacks. Additionally, a detailed study of their stalks will extend the PI’s prior work with Oi on endoscopic character identities for generalized L-packets.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

Award Number: 2502131
Principal Investigator: Alexander Bertoloni Meli
Funds Obligated: $146,000
State: MA