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

Abstract

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

arXiv commentA stronger result which cover this paper is originated by the same author, so I withdraw this paper