On the principal series representations of semisimple groups with Frobenius maps

TL;DR

This paper investigates principal series representations of semisimple algebraic groups over algebraic closures of finite fields, establishing conditions for irreducibility and analyzing composition factors, including infinite filtrations in specific cases.

Contribution

It provides new criteria for the irreducibility of induced modules and explores their structure, including infinite composition factors and submodule filtrations in the context of Frobenius maps.

Findings

Irreducibility of induced modules characterized by regular and antidominant weights.

Existence of infinite composition factors in certain induced modules for SL_2.

Construction of infinite submodule filtrations in general cases.

Abstract

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 ${\bf G}$.