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

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.