Supersingular representations of rank 1 groups

TL;DR

This paper proves the existence of irreducible admissible supersingular mod-$p$ representations for connected reductive groups of semisimple $F$-rank 1 over $p$-adic fields, filling a key gap in Vignéras' theory.

Contribution

It establishes the existence of supersingular representations for rank 1 groups, completing a crucial case in the broader theory of mod-$p$ representations.

Findings

Confirmed existence of supersingular representations for rank 1 groups.

Fills a missing case in Vignéras' proof for general reductive groups.

Advances understanding of mod-$p$ representation theory for p-adic groups.

Abstract

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.