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

Let F F be a locally compact non-archimedean field, p p its residue characteristic, and G \textbf {G} a connected reductive group over F F . Let C C be an algebraically closed field of characteristic p p . We give a complete classification of irreducible admissible C C -representations of G = G ( F ) G=\mathbf {G}(F) , in terms of supercuspidal C C -representations of the Levi subgroups of G G , and parabolic induction. Thus we push to their natural conclusion the ideas of the third author, who treated the case G = G L m \mathbf {G}=\mathrm {GL}_m , as further expanded by the first author, who treated split groups G \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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