Abstract irreducible representations of reductive algebraic groups with Borel-stable line

Let p be a prime number and \(\Bbbk =\bar{{\mathbb {F}}}_p\) $\mathbb{k}={\overline{\mathbb{F}}}_p$ , the algebraic closure of the finite field \({\mathbb {F}}_p\) ${\mathbb{F}}_p$ of p elements. Let \(\textbf{G}\) $\mathbf{G}$ be a connected reductive group defined over \({\mathbb {F}}_p\) ${\mathbb{F}}_p$ and \(\textbf{B}\) $\mathbf{B}$ be a Borel subgroup of \(\textbf{G}\) $\mathbf{G}$ (not necessarily defined over \({\mathbb {F}}_p\) ${\mathbb{F}}_p$ ). Let \(\Bbbk \textbf{H}\) $\mathbb{k}\mathbf{H}$ be the group algebra of a group \(\textbf{H}\) $\mathbf{H}$ over \(\Bbbk \) $\mathbb{k}$ . We show that for each (one-dimensional) character \(\theta \) $\theta$ of \(\textbf{B}\) $\mathbf{B}$ (not necessarily rational), there is a unique (up to isomorphism) irreducible \(\Bbbk \textbf{G}\) $\mathbb{k}\mathbf{G}$ -module \({\mathbb {L}}(\theta )\) $\mathbb{L}(\theta )$ containing \(\theta \) $\theta$ as a \(\Bbbk \textbf{B}\) $\mathbb{k}\mathbf{B}$ -submodule, and moreover, \({\mathbb {L}}(\theta )\) $\mathbb{L}(\theta )$ is isomorphic to a parabolic induction from a finite-dimensional irreducible \(\Bbbk \textbf{L}\) $\mathbb{k}\mathbf{L}$ -module for some Levi subgroup \(\textbf{L}\) $\mathbf{L}$ of \(\textbf{G}\) $\mathbf{G}$ . Thus, we have classified and constructed all (abstract) irreducible \(\Bbbk \textbf{G}\) $\mathbb{k}\mathbf{G}$ -modules with \(\textbf{B}\) $\mathbf{B}$ -stable line (i.e. an one-dimensional \(\Bbbk \textbf{B}\) $\mathbb{k}\mathbf{B}$ -submodule). As a byproduct, we give a new proof of a result of Borel and Tits on the classification of finite-dimensional irreducible \(\Bbbk \textbf{G}\) $\mathbb{k}\mathbf{G}$ -modules.