Local Langlands and Springer Correspondences

Abstract

This notes are the written version of a course given by the author at the workshop "Representation theory of p-adic groups" that was held at IISER Pune, India, in July 2017 (to appear in Representations of p-adic groups: Contributions from Pune, Progress in Math., Birkhäuser). They give notably an overview of results obtained jointly with Ahmed Moussaoui and Maarten Solleveld on the local Langlands correspondence, focusing on the links of the latter with both the generalized Springer correspondence and the geometric conjecture, the so-called ABPS Conjecture, introduced in collaboration with Paul Baum, Roger Plymen and Maarten Solleveld. Langlands parameters correspond by the LLC to the irreducible supercuspidal representations of G. The validity of this conjecture is proved for representations with unipotent reduction of the group G of the F-rational points of any connected reductive algebraic group which splits over an unramified extension of F in [FOS, Theorem 2] (when G is simple of adjoint type it is a special case of [Lus4], [Lus5]), for the Deligne-Lusztig depth-zero supercuspidal representations (as a consequence of [DeRe]), and also for general linear groups and split classical p-adic groups (any representation) (see [Mou1]), for inner forms of linear groups and of special linear groups, and for quasi-split unitary p-adic groups (any representation) (see [AMS1, § 6]).