ON FINITE LENGTH SMOOTH REPRESENTATIONS OF $p$-ADIC GENERAL LINEAR GROUPS (Automorphic form, automorphic $L$-functions and related topics)

Abstract

We survey qualitative aspects of the study of the decomposition of finite-length smooth representations of the groups GLn(F), for a p-adic field F, with emphasis on techniques that have been developing in the recent years. We state general goals and questions on decomposition of parabolic induction of irreducible representations, and review applications for branching laws such as the local Gan-Gross-Prasad program. We give a flavor of the categorical links, and their possible applications, between the p-adic setting and the representation theory of quiver Hecke algebras of type A. Finally, we review, as a case study, the recent RSK classification of irreducible representations, introduced by the author with Lapid.