On the principal series representations of semisimple groups with Frobenius maps

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 ${\bf B}$. Set $G_{q^r}={\bf G}^{F^r}$ and $B_{q^r}={\bf B}^{F^r}$ for any $r>0$. 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 regular and dominant, then there is a surjective ${\bf G}$-module homomorphism $\op{Ind}_{B_{q^r}}^{\bf G}\lambda\rightarrow \op{St}\otimes L(\lambda)$ for any $r>0$, where $\op{St}$ is the infinite dimensional Steinberg module defined by Nanhua Xi. As a consequence, we show that $\op{Ind}_{\bf B}^{\bf G}\lambda$ is irreducible if $\lambda$ if and only if $\lambda$ is regular and antidominant. Moreover, for ${\bf G}=SL_2(\bar{\mathbb{F}}_q)$ and $0<\lambda<p$, we show that $\op{Ind}_{\bf B}^{\bf G}\lambda$ have infinite many composition factors with each finite dimensional. Consequently, we find certain $\lambda$ for which $\op{Ind}_{\bf B}^{\bf G}\lambda$ has an infinite submodule filtration for the general ${\bf G}$.