The Decomposition of Permutation Module for Infinite Chevalley Groups

Abstract: Let ${\bf G}$ be a connected reductive group defined over $\mathbb{F}q$, the finite field with $q$ elements. Let ${\bf B}$ 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 ${\bf B}$-module) for any field $\Bbbk$.