On the $\ell$-modular composition factors of the Steinberg representation

TL;DR

This paper investigates the structure of the Steinberg representation over fields of prime characteristic, proving the simplicity of its socle and determining its composition length for certain groups.

Contribution

It provides a new proof of the socle's simplicity using Hecke algebras and identifies the simple socle within principal series representations, also computing composition lengths for specific groups.

Findings

Socle of Steinberg representation is always simple.

Identified the simple socle among principal series representations.

Determined composition length for $ ext{GL}_n(q)$ and classical groups with linear primes.

Abstract

Let $G$ be a finite group of Lie type and $\St_k$ be the Steinberg representation of $G$, defined over a field $k$. We are interested in the case where $k$ has prime characteristic~$\ell$ and $\St_k$ is reducible. Tinberg has shown that the socle of $\St_k$ is always simple. We give a new proof of this result in terms of the Hecke algebra of $G$ with respect to a Borel subgroup and show how to identify the simple socle of $\St_k$ among the principal series representations of~$G$. Furthermore, we determine the composition length of $\St_k$ when $G=\GL_n(q)$ or $G$ is a finite classical group and $\ell$ is a so-called linear prime.