Abstract induced modules for reductive algebraic groups with Frobenius maps

TL;DR

This paper investigates the structure of induced modules for reductive algebraic groups over finite fields, establishing conditions for their composition series and classifying their irreducible factors.

Contribution

It provides new criteria for the existence of finite-length composition series of induced modules and classifies all their composition factors in certain cases.

Findings

Finite-length composition series exist when char(𝕂) ≠ char(𝔽_q).

Necessary and sufficient conditions for composition series with rational characters.

Classification of all composition factors when series exist.

Abstract

Let ${\bf G}$ be a connected reductive algebraic group defined over a finite field $\mathbb{F}_q$ of $q$ elements, and ${\bf B}$ be a Borel subgroup of ${\bf G}$ defined over $\mathbb{F}_q$. Let $\Bbbk$ be a field and we assume that $\Bbbk=\bar{\mathbb{F}}_q $ when $\text{char}\ \Bbbk=\text{char} \ \mathbb{F}_q$. We show that the abstract induced module $\mathbb{M}(\theta)=\Bbbk{\bf G}\otimes_{\Bbbk{\bf B}}\theta$ (here $\Bbbk{\bf H}$ is the group algebra of ${\bf H}$ over the field $\Bbbk$ and $\theta$ is a character of ${\bf B}$ over $\Bbbk$) has a composition series (of finite length) if $\text{char}\ \Bbbk\ne \text{char} \ \mathbb{F}_q$. In the case $\Bbbk=\bar{\mathbb{F}}_q$ and $\theta$ is a rational character, we give a necessary and sufficient condition for the existence of a composition series (of finite length) of $\mathbb{M}(\theta)$. We determine all the composition factors whenever a composition series exists. Thus we obtain a large class of abstract infinite-dimensional irreducible $\Bbbk{\bf G}$-modules.