Representations of a $p$-adic group in characteristic $p$

TL;DR

This paper extends the classification of irreducible admissible representations of a p-adic group and modules of the pro-p Iwahori Hecke algebra to fields of characteristic p that are not algebraically closed.

Contribution

It generalizes existing classifications from algebraically closed fields to arbitrary fields of characteristic p.

Findings

Classifications hold over non-algebraically closed fields.

Supersingular representations remain central to the classification.

Results unify understanding of representations over different fields.

Abstract

Let $F$ be a locally compact non-archimedean field of residue characteristic $p$, $\textbf{G}$ a connected reductive group over $F$, and $R$ a field of characteristic $p$. When $R$ is algebraically closed, the irreducible admissible $R$-representations of $G=\textbf{G}(F)$ are classified in term of supersingular $R$-representations of the Levi subgroups of $G$ and parabolic induction; there is a similar classification for the simple modules of the pro-$p$ Iwahori Hecke $R$-algebra. In this paper, we show that both classifications hold true when $R$ is not algebraically closed.