On abstract homomorphisms of Chevalley groups over the coordinate rings of affine curves

TL;DR

This paper proves a rigidity property for abstract representations of Chevalley groups over rings including affine curve coordinate rings, showing that finite-dimensional representations are standard, extending known results beyond arithmetic groups.

Contribution

It establishes a general rigidity theorem for Chevalley groups over a broad class of rings, including coordinate rings of affine curves, using a new verification of condition (Z).

Findings

Finite-dimensional representations of SL_n(Z[X]) are standard.

First unconditional rigidity result for finitely generated linear groups beyond arithmetic groups.

Develops a new approach verifying condition (Z) over rings of interest.

Abstract

The goal of this paper is to establish a general rigidity statement for abstract representations of elementary subgroups of Chevalley groups of rank at least 2 over a class of commutative rings that includes the localizations of 1-generated rings and the coordinate rings of affine curves. This is achieved by developing the approach introduced in our previous work, and in particular by verifying condition (Z) over the class of rings at hand. Our main result implies, for example, that any finite-dimensional representation of SL_n(Z[X]) (for n at least 3) over an algebraically closed field of characteristic 0 has a standard description, yielding thereby the first unconditional rigidity statement for finitely generated linear groups other than arithmetic groups/lattices.