Irreducible Modules of Reductive Groups with Borel-stable Line

TL;DR

This paper classifies all irreducible modules of a reductive group over an algebraically closed field of characteristic p that contain a Borel-stable line, using parabolic induction and providing a new proof of a classical classification result.

Contribution

It provides a complete classification and construction of irreducible modules with Borel-stable lines for reductive groups over algebraically closed fields of characteristic p, including a new proof of a known classification.

Findings

Unique irreducible modules containing a given Borel character

Modules are isomorphic to parabolic inductions from Levi subgroups

New proof of Borel and Tits classification of irreducible modules

Abstract

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.