Chevalley groups over Z: a representation-theoretic approach

Let G(Q) be a simply connected Chevalley group over Q corresponding to a simple Lie algebra g over C. Let V be a finite-dimensional faithful highest weight g-module and let VZ be a Chevalley Z-form of V. Let Gamma(Z) be the subgroup of G( Q) that preserves VZ and let G(Z) be the group of Z-points of G(Q). Then G(Q) is integral if G(Z) = Gamma(Z). Chevalley's original work constructs a scheme-theoretic integral form of G(Q) which equals Gamma(Z). Here we give a representation-theoretic proof of integrality of G(Q) using only the action of G(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 Q.