On the refined Arthur-Langlands conjectures

This page contains some informations about the mini-course in Orsay by Tasho Kaletha, entitled ``On the refined Arthur-Langlands conjectures''.

There will be two lectures, on Friday June 9th and Friday June 16th, 10 am - 12 am, in room 113-115 (building 425, first floor).

Abstract : The Arthur-Langlands conjectures provide a description of the discrete automorphic representations of connected reductive groups defined over global fields, as well as of the irreducible admissible representations of such groups defined over local fields. The crude forms of these conjectures partition the set of representations into packets, while the refined forms describe individual representations. When the group in question is quasi-split, the refined form of these conjectures has been known for a long time and important special cases have recently been proved. For non-quasi-split groups (for example classical groups over division algebras), only a crude form of the conjectures was known until recently. In this mini-course I will present a precise formulation of the refined local and global conjectures for arbitrary connected reductive groups. It is based on the construction of certain Galois gerbes defined over local and global fields and the study of their cohomology. In the case of the local conjecture, I will also discuss the relationship between various points of view, involving Vogan's pure inner forms, the Adams-Barbasch-Vogan strong real forms, Kottwitz's isocrystals with additional structure, and the Arthur-Shelstad mediating functions and spectral transfer factors. Depending on time and interest, I can discuss results towards these conjectures, such as the explicit construction of the local correspondence for supercuspidal parameters under some conditions, and special cases of the global correspondence for classical groups.