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

Abe, N. ¹   ; Henniart, G. ²   ; Herzig, F. ³   ; Vignéras, M.-F. ⁴

¹ Creative Research Institution (CRIS), Hokkaido University, N21, W10, Kita-ku, Sapporo, Hokkaido 001-0021, Japan
² Laboratoire de Mathématiques d’Orsay, Université de Paris-Sud, Orsay cedex F-91405, France; CNRS, Orsay cedex F-91405, France
³ Department of Mathematics, University of Toronto, 40 St. George Street, Room 6290, Toronto, Ontario M5S 2E4, Canada
⁴ Institut de Mathématiques de Jussieu, 175 rue du Chevaleret, Paris 75013, France

Journal of the American Mathematical Society, Tome 30 (2017) no. 2, pp. 495-559

Résumé

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.

Détail

DOI : 10.1090/jams/862

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