The refined Arthur--Langlands conjectures beyond the supercuspidal case

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This project investigates facets of the Langlands program -- a fundamental area of Mathematics bridging number theory, geometry, and analysis, and examining in a systematic way the symmetries inherent in solutions of algebraic equations. By relating these symmetries to geometry, the program creates a powerful way of obtaining arithmetic information. The project is particularly concerned with obtaining such information in an algorithmically computable way. Deep and explicit knowledge of the arithmetic of numbers is essential for many technological advances over the last decades that we take for granted today, including encryption and digital security. Part of this work involves mentoring of young scientists.More precisely, the project aims at deepening our understanding of the refined Arthur-Langlands conjectures for general connected and disconnected reductive groups, in particular the internal structure of local and global packets and the character identities satisfied by them. The study of disconnected groups, while of interest in its own right, is also essential for the study of connected groups. The special case of disconnected groups over the real numbers will serve as a first step. An important role in obtaining the refined conjectures beyond the case of discrete parameters is played by self-intertwining operators acting on parabolically induced representations, and the project seeks to obtain natural normalizations of these and a precise statement of the intertwining relation. The project will also construct non-generic Arthur packets for local and global unitary groups, and obtain a description of the full discrete automorphic spectrum of global unitary groups.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.