The classification of irreducible admissible mod p representations of a p-adic GL_n

TL;DR

This paper classifies irreducible admissible mod p representations of p-adic GL_n, establishing a link between supersingular and supercuspidal representations, and extends previous work from n=2 to general split reductive groups.

Contribution

It introduces a new classification framework for mod p representations of p-adic groups using the mod p Satake transform, generalizing prior results for GL_2 to higher dimensions.

Findings

Supersingular representations are equivalent to supercuspidal representations.

Provides a classification of irreducible admissible smooth representations of GL_n(F).

Extends classification results to general split reductive groups under certain hypotheses.

Abstract

Let F be a finite extension of Q_p. Using the mod p Satake transform, we define what it means for an irreducible admissible smooth representation of an F-split p-adic reductive group over \bar F_p to be supersingular. We then give the classification of irreducible admissible smooth GL_n(F)-representations over \bar F_p in terms of supersingular representations. As a consequence we deduce that supersingular is the same as supercuspidal. These results generalise the work of Barthel-Livne for n = 2. For general split reductive groups we obtain similar results under stronger hypotheses.