Characters of algebraic groups over number fields

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

[学科分类] 神经科学

[关键词] Algebraic groups;characters of discrete groups;von Neumann algebras;invariant random subgroups;Ratner’s theory