The Submodule Structure of Spherical Principal Series Module of Reductive Groups with Frobenius Maps in Cross Characteristic

Xiaoyu Chen

Published 2017-07-09Version 1

Let ${\bf G}$ be a connected reductive group over $\bar{\mathbb{F}}_q$, the algebraically closure of $\mathbb{F}_q$ (the finite field with $q=p^e$ elements), with the standard Frobenius map $F$. Let ${\bf B}$ be an $F$-stable Borel subgroup. Let $\Bbbk$ be a field of characteristic $r\neq p$. In this paper, we completely determine the composition factors of the induced module $\op{Ind}_{\bf B}^{\bf G}\op{tr}=\Bbbk{\bf G}\otimes_{\Bbbk{\bf B}}\op{tr}$ (here $\Bbbk{\bf H}$ is the group algebra of the group ${\bf H}$, and $\op{tr}$ is the trivial ${\bf B}$-module). In particular, we find a new family of infinite dimensional irreducible abstract representations of ${\bf G}$.