We prove that any connected reductive group of semisimple $F$-rank 1 over a $p$-adic field admits an irreducible admissible supersingular mod-$p$ representation. This establishes one of the missing cases in Vign\'eras' existence proof for general reductive groups.
Extensions for supersingular representations of $GL_2(Q_p)$
Coefficient systems and supersingular representations of $GL_2(F)$
Modulo $p$ parabolic induction of pro-$p$-Iwahori Hecke algebra
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