Irreducible Modules of Reductive Groups with Borel-stable Line

Abstract

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

23 pages

References in corpus (2)

• Some Infinite Dimensional Representations of Reductive Groups with Frobenius Maps
• On the principal series representations of semisimple groups with Frobenius maps