Characters of algebraic groups over number fields

Let k be a number field, G an algebraic group defined over k, and G(k) the group of k-rational points in G. We determine the set of functions on G(k) which are of positive type and conjugation invariant, under the assumption that G(k) is generated by its unipotent elements. An essential step in the proof is the classification of the G(k)-invariant ergodic probability measures on an adelic solenoid naturally associated to G(k); this last result is deduced from Ratner's measure rigidity theorem for homogeneous spaces of S-adic Lie groups.