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

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.