# Mod-$p$ maximal compact inductions do not have irreducible admissible subrepresentations

## TL;DR

This paper proves that mod-$p$ maximal compact inductions in p-adic split reductive groups lack irreducible admissible subrepresentations, clarifying the structure of these representations in the mod-$p$ setting.

## Contribution

It establishes a new non-existence result for irreducible admissible subrepresentations within mod-$p$ maximal compact inductions.

## Key findings

- Maximal compact inductions in mod-$p$ setting have no irreducible admissible subrepresentations.
- Provides insight into the structure of mod-$p$ representations of p-adic groups.
- Clarifies limitations of irreducibility in the context of mod-$p$ representation theory.

## Abstract

Let $p$ be a prime number. We show in this short note that mod-$p$ maximal compact inductions of a $p$-adic split reductive group do not have irreducible admissible subrepresentations.