  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.
