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

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.