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

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 $\op{Ind}{\bf B}^{\bf G}\lambda=\Bbbk{\bf G}\otimes{\Bbbk{\bf B}}\lambda$(here$\Bbbk{\bf H}$isthegroupalgebraofthegroup${\bf H}$,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 $\text{op}{Ind}_{\mathbf{B}}^{\mathbf{G}}\lambda$ is irreducible and non quasi-finite.