A note on the admissibility of smooth simple $RG$-modules

TL;DR

This paper proves that all smooth irreducible representations of a p-adic reductive group over certain rings are admissible, leveraging recent finiteness results for Hecke algebras.

Contribution

It establishes the admissibility of smooth irreducible R-linear representations of p-adic groups, extending the understanding of their structure over more general rings.

Findings

All smooth irreducible R-linear representations are admissible.

Uses finiteness results of Hecke algebras to prove admissibility.

Extends admissibility results to broader algebraic settings.

Abstract

Let $G$ be a $p$-adic reductive group and $R$ be a noetherian Jacobson $\mathbb{Z}[1/p]$-algebra. In this note, we show that every smooth irreducible $R$-linear representation of $G$ is admissible using the finiteness result of Dat, Helm, Kurinczuk and Moss for Hecke algebras over $R$.