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The goal of this paper is to establish a general rigidity statement forabstract representations of elementary subgroups of Chevalley groups of rank atleast 2 over a class of commutative rings that includes the localizations of1-generated rings and the coordinate rings of affine curves. This is achievedby developing the approach introduced in our previous work, and in particularby verifying condition (Z) over the class of rings at hand. Our main resultimplies, 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 astandard description, yielding thereby the first unconditional rigiditystatement for finitely generated linear groups other than arithmeticgroups/lattices.

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