Search

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. [ABSTRACT FROM AUTHOR]