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 $\mathbb{C}^*$ (or, more generally, with coefficients in the complex points of an algebraic torus over $\mathbb{C}$) 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.