# On the existence of admissible supersingular representations of $p$-adic reductive groups

## TL;DR

This paper proves that for any connected reductive group over a finite extension of er, there exists an irreducible admissible supersingular representation over a field of characteristic p, confirming their existence.

## Contribution

It establishes the existence of irreducible admissible supersingular representations for all such groups, a fundamental question in p-adic representation theory.

## Key findings

- Existence of supersingular representations for all connected reductive groups over finite extensions of er.
- Construction of irreducible admissible supersingular representations over characteristic p fields.
- Advances understanding of the representation theory of p-adic groups.

## Abstract

Suppose that $\mathbf{G}$ is a connected reductive group over a finite extension $F/\mathbb{Q}_p$, and that $C$ is a field of characteristic $p$. We prove that the group $\mathbf{G}(F)$ admits an irreducible admissible supercuspidal, or equivalently supersingular, representation over $C$.