, Volume 186, Issue 2, pp 373-434
Date: 18 Mar 2011

The classification of irreducible admissible mod p representations of a p-adic GL n

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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  \(\overline{ \mathbb{F}}_{p}\) to be supersingular. We then give the classification of irreducible admissible smooth GL n (F)-representations over  \(\overline{ \mathbb{F}}_{p}\) in terms of supersingular representations. As a consequence we deduce that supersingular is the same as supercuspidal. These results generalise the work of Barthel–Livné for n=2. For general split reductive groups we obtain similar results under stronger hypotheses.

Partially supported by NSF grant DMS-0902044.