Chevalley groups over $\mathbb{Z}$: A representation-theoretic approach

Abstract

Let $G(\mathbb{Q})$ be a simply connected Chevalley group over $\mathbb{Q}$ corresponding to a simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$. Let $V$ be a finite dimensional faithful highest weight $\mathfrak{g}$-module and let $V_{\mathbb{Z}}$ be a Chevalley $\mathbb{Z}$-form of $V$. Let $\Gamma (\mathbb{Z})$ be the subgroup of $G(\mathbb{Q})$ that preserves $V_{\mathbb{Z}}$ and let $G(\mathbb{Z})$ be the group of $\mathbb{Z}$-points of $G(\mathbb{Q})$. Then $G(\mathbb{Q})$ is \emph{integral} if $G(\mathbb{Z})=\Gamma (\mathbb{Z})$. Chevalley's original work constructs a scheme-theoretic integral form of $G(\mathbb{Q})$ which equals $\Gamma (\mathbb{Z})$. Here we give a representation-theoretic proof of integrality of $G(\mathbb{Q})$ using only the action of $G(\mathbb{Q})$ on $V$, rather than the language of group schemes. We discuss the challenges and open problems that arise in trying to extend this to a proof of integrality for Kac-Moody groups over $\mathbb{Q}$.