The Decomposition of Permutation Module for Infinite Chevalley Groups

Abstract

Let $\mathbf{G}$ be a connected reductive group defined over ${\mathbb{F}}_q$, the finite field with $q$ elements. Let $\mathbf{B}$ be an Borel subgroup defined over ${\mathbb{F}}_q$. In this paper, we completely determine the composition factors of the induced module $\mathbb{M}(\op{tr})=\Bbbk{\bf G}\otimes_{\Bbbk{\bf B}}\op{tr}$ ($\op{tr}$ is the trivial $\mathbf{B}$-module) for any field $\mathbb{k}$.

Accepted by Science China Mathematics