Abstract-Induced Modules for Reductive Algebraic Groups With Frobenius Maps

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