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 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 \({{\mathbb {Q}}}\) .
