Abstract induced modules for reductive algebraic groups with Frobenius maps

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.