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