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

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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 \(\overline{ \mathbb{F}}_{p}\) 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.

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  1. Department of Mathematics, 2033 Sheridan Road, Evanston, IL, 60208-2730, USA

    Florian Herzig

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  1. Florian Herzig
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Correspondence to Florian Herzig.

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Partially supported by NSF grant DMS-0902044.

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Herzig, F. The classification of irreducible admissible mod p representations of a p-adic GL n . Invent. math. 186, 373–434 (2011). https://doi.org/10.1007/s00222-011-0321-z

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