Abstract-Induced Modules for Reductive Algebraic Groups With Frobenius Maps

Abstract

Abstract Let ${\textbf{G}}$ be a connected reductive algebraic group defined over a finite field $\mathbb{F}_q$ of $q$ elements and $\textbf{B}$ be a Borel subgroup of ${\textbf{G}}$ defined over $\mathbb{F}_q$. Let $\mathbb{k}$ be a field and we assume that $\mathbb{k}=\bar{\mathbb{F}}_q $ when $\textrm{char}\ \mathbb{k}=\textrm{char} \ \mathbb{F}_q$. We show that the abstract-induced module $\mathbb{M}(\theta )=\mathbb{k}{\textbf{G}}\otimes _{\mathbb{k}\textbf{B}}\theta $ (here $\mathbb{k}\textbf{H}$ is the group algebra of $\textbf{H}$ over the field $\mathbb{k}$ and $\theta $ is a character of $\textbf{B}$ over $\mathbb{k}$) has a composition series (of finite length) if $\textrm{char}\ \mathbb{k}\ne \textrm{char} \ \mathbb{F}_q$. In the case $\mathbb{k}=\bar{\mathbb{F}}_q$ 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 $\mathbb{k}{\textbf{G}}$-modules.