2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/64978We generalize a theorem of Tate and show that the second cohomology of the Weil group of a global or local field with coefficients in $\C^*$ (or more generally, with coefficients in the complex points of a tori over $\C$) vanish, where the cohomology groups are defined using measurable cochains in the sense of Moore. We recover a theorem of Labesse proving that admissible homomorphisms of a Weil group to the Langlands dual of a reductive group can be lifted to an extension of the Langlands dual group by a tori.To appear in Compositio Mathematica. Minor editing, and a conjecture in the earlier version on the vanishing of the higher cohomology groups of Weil groups is falseNumber Theory11R34 (Primary) 22E55 (Secondary)text