Characters of algebraic groups over number fields (Q2694785)

Summary: 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; this appears to be the first application of Ratner's theorems in the context of operator algebras.