Existence of supersingular representations of p-adic reductive groups

TL;DR

This paper proves that certain p-adic groups, excluding some specific types, admit irreducible admissible supercuspidal (supersingular) representations over any field of characteristic p, expanding understanding of their representation theory.

Contribution

It establishes the existence of supersingular representations for a broad class of p-adic groups, excluding specific known cases, over any characteristic p field.

Findings

Existence of supersingular representations for most p-adic groups.

Excludes groups isomorphic to PGL(m,D) and certain ramified unitary groups.

Applicable over any field of characteristic p.

Abstract

Let G be a connected simple adjoint p-adic group not isomorphic to a projective linear group PGL(m,D) of a division algebra D, or an adjoint ramified unitary group of a split hermitian form in 3 variables. We prove that G admits an irreducible admissible supercuspidal (i.e. supersingular) representation over any field of characteristic p.