On linear representations of Chevalley groups over commutative rings

Let G be the universal Chevalley-Demazure group scheme corresponding to a reduced irreducible root system of rank at least 2, and let R be a commutative ring. We analyze the linear representations : G(R)(+) -> GL(n) (K) over an algebraically closed field K of the elementary subgroup G(R)(+) subset of G(R). Our main result is that under certain conditions, any such representation has a standard description; that is, there exist a commutative finite-dimensional K-algebra B, a ring homomorphism f: R -> B with Zariski-dense image, and a morphism of algebraic groups Sigma: G(B) -> GL(n) (K) such that coincides with Sigma degrees F on a suitable finite index subgroup Gamma subset of G(R)(+), where F : G(R)(+) -> G(B)(+) is the group homomorphism induced by f. In particular, this confirms a conjecture of Borel and Tits for Chevalley groups over a field of characteristic zero.