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

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

2010-05-11

arxiv.org/abs/1005.1713v2

Invent. Math. 186 (2011), no. 2, 373-434

Mathematics

Number Theory

55 pages, to appear in Inventiones Mathematicae

Scientific paper

10.1007/s00222-011-0321-z

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.

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