On the principal series representations of semisimple groups with Frobenius maps

Abstract

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 $\mathbf{B}$. Set $G_{q^r}={\mathbf{G}}^{F^r}$ and $B_{q^r}={\mathbf{B}}^{F^r}$ for any $r>0$. 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 regular and dominant, then there is a surjective $\mathbf{G}$-module homomorphism $\op{Ind}_{B_{q^r}}^{\bf G}λ\rightarrow \op{St}\otimes L(λ)$ for any $r>0$, where $\op{St}$ is the infinite dimensional Steinberg module defined by Nanhua Xi. As a consequence, we show that $\op{Ind}_{\bf B}^{\bf G}λ$ is irreducible if $\lambda$ if and only if $\lambda$ is regular and antidominant. Moreover, for $\mathbf{G}=S{L}_2({\bar{\mathbb{F}}}_q)$ and $0<\lambda <p$, we show that $\op{Ind}_{\bf B}^{\bf G}λ$ have infinite many composition factors with each finite dimensional. Consequently, we find certain $\lambda$ for which $\op{Ind}_{\bf B}^{\bf G}λ$ has an infinite submodule filtration for the general $\mathbf{G}$.

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

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