Mod p representations of GL_n of p-adic fields

Abstract

Let F be a finite extension of Q_p.  A mod p local Langlands correspondence is known for GL_2(Q_p) and believed to exist between mod p
representations of the absolute Galois group of F and certain mod p
representations of GL_n(F) for arbitrary F and n.  A major obstacle to the study of the correspondence is that the supersingular mod p representations of GL_n(F), which by recent work of Herzig are known to be the building blocks of the mod p representation theory of this group, are very poorly understood. After an introduction to the mod p representation theory of GL_n(F), we will discuss recently discovered obstacles to understanding the supersingular representations, as well as what is known.