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