Some Non Quasi-finite irreducible Modules of Semisimple Groups with Frobenius Maps

TL;DR

This paper investigates specific induced modules of semisimple algebraic groups over algebraic closures of finite fields, demonstrating the existence of irreducible, non-quasi-finite submodules under certain conditions.

Contribution

It identifies conditions under which certain submodules of induced modules are irreducible and non quasi-finite, extending previous work on module structures of algebraic groups.

Findings

Certain submodules are irreducible when the character is antidominant and non-trivial.

These submodules are shown to be non quasi-finite.

The results extend understanding of module irreducibility in algebraic group representations.

Abstract

This paper is the continuation of \cite{CXY}. Let ${\bf G}$ be a simply connected semisimple algebraic group over $\Bbbk=\bar{\mathbb{F}}_q$, the algebraically closure of $\mathbb{F}_q$ (the finite field with $q=p^e$ elements), and $F$ be the standard Frobenius map. Let ${\bf B}$ be an $F$-stable Borel subgroup and ${\bf T}$ an $F$-stable maximal torus contained in ${\bf B}$. This paper studies the original induced module $\op{Ind}_{\bf B}^{\bf G}\lambda=\Bbbk{\bf G}\otimes_{\Bbbk{\bf B}}\lambda$ (here $\Bbbk{\bf H}$ is the group algebra of the group ${\bf H}$, and $\lambda$ is a rational character of ${\bf T}$ regarded as a ${\bf B}$-module). We show that if $\lambda$ is antidominant and not trivial, then certain submodule of $\op{Ind}_{\bf B}^{\bf G}\lambda$ is irreducible and non quasi-finite.