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A classification of irreducible admissible mod $p$ representations of $p$-adic reductive groups

Authors: N. Abe, G. Henniart, F. Herzig and M.-F. Vignéras
Journal: J. Amer. Math. Soc. 30 (2017), 495-559
MSC (2010): Primary 22E50
Published electronically: June 14, 2016
MathSciNet review: 3600042
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Abstract: Let $F$ be a locally compact non-archimedean field, $p$ its residue characteristic, and $\textbf {G}$ a connected reductive group over $F$. Let $C$ be an algebraically closed field of characteristic $p$. We give a complete classification of irreducible admissible $C$-representations of $G=\mathbf {G}(F)$, in terms of supercuspidal $C$-representations of the Levi subgroups of $G$, and parabolic induction. Thus we push to their natural conclusion the ideas of the third author, who treated the case $\mathbf {G}=\mathrm {GL}_m$, as further expanded by the first author, who treated split groups $\mathbf {G}$. As in the split case, we first get a classification in terms of supersingular representations of Levi subgroups, and as a consequence show that supersingularity is the same as supercuspidality.

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Additional Information

N. Abe
Affiliation: Creative Research Institution (CRIS), Hokkaido University, N21, W10, Kita-ku, Sapporo, Hokkaido 001-0021, Japan
MR Author ID: 858099

G. Henniart
Affiliation: Laboratoire de Mathématiques d’Orsay, Université de Paris-Sud, Orsay cedex F-91405, France; CNRS, Orsay cedex F-91405, France
MR Author ID: 84385

F. Herzig
Affiliation: Department of Mathematics, University of Toronto, 40 St. George Street, Room 6290, Toronto, Ontario M5S 2E4, Canada
MR Author ID: 876324

M.-F. Vignéras
Affiliation: Institut de Mathématiques de Jussieu, 175 rue du Chevaleret, Paris 75013, France

Keywords: Modulo $p$ representations, reductive groups, admissible, supercuspidal, supersingular
Received by editor(s): December 6, 2014
Received by editor(s) in revised form: January 23, 2016, and May 3, 2016
Published electronically: June 14, 2016
Additional Notes: The first author was supported by JSPS KAKENHI Grant 26707001.
The third author was partially supported by a Sloan Fellowship and an NSERC grant.
Article copyright: © Copyright 2016 American Mathematical Society