On the vanishing of the measurable Schur cohomology groups of Weil groups

Abstract

We 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 an algebraic torus over C) vanish, where the cohomology groups are defined using measurable cochains in the sense of Moore. We recover a theorem of Labesse stating that the admissible homomorphisms of a Weil group to the Langlands dual group of a reductive group can be lifted to an extension of the Langlands dual group by a torus.