On linear representations of Chevalley groups over commutative rings

Abstract

Let $G$ be the universal Chevalley-Demazure group scheme corresponding to a reduced irreducible root system of rank $\ge 2$, and let $R$ be a commutative ring. We analyze the linear representations $\rho :G(R{)}^{+}\to G{L}_n(K)$ over an algebraically closed field $K$ of the elementary subgroup $G(R{)}^{+}\subset G(R).$ Our main result is that under certain conditions, any such representation has a standard description, i.e. there exists a commutative finite-dimensional $K$-algebra $B$, a ring homomorphism $f:R\to B$ with Zariski-dense image, and a morphism of algebraic groups $\sigma :G(B)\to G{L}_n(K)$ such that $\rho$ coincides with $\sigma \circ F$ on a suitable finite index subgroup $\Gamma \subset G(R{)}^{+},$ where $F:G(R{)}^{+}\to G(B{)}^{+}$ 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.

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