On linear representations of Chevalley groups over commutative rings

TL;DR

This paper proves that linear representations of elementary Chevalley groups over rings are essentially standard, confirming a conjecture of Borel and Tits in characteristic zero fields, with implications for understanding their structure.

Contribution

It establishes a standard description for all such representations, linking them to algebraic group morphisms over finite-dimensional algebras, thus extending known classification results.

Findings

Representations are described via algebraic group morphisms over finite-dimensional algebras.

Confirms Borel-Tits conjecture for Chevalley groups over characteristic zero fields.

Provides a structural classification of elementary Chevalley group representations over rings.

Abstract

Let $G$ be the universal Chevalley-Demazure group scheme corresponding to a reduced irreducible root system of rank $\geq 2$, and let $R$ be a commutative ring. We analyze the linear representations $\rho \colon G(R)^+ \to GL_n (K)$ 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 $f \colon R \to B$ with Zariski-dense image, and a morphism of algebraic groups $\sigma \colon G(B) \to GL_n (K)$ such that $\rho$ coincides with $\sigma \circ F$ on a suitable finite index subgroup $\Gamma \subset G(R)^+,$ where $F \colon G(R)^+ \to G(B)^+$ 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.