Let $W\_p$ be the Weil group of $\mathbf{Q}\_p$. What is the Galois cohomology group $H^2(W\_p,\mathbf{C}^{\times} )$ (with trivial action)? Is it zero, or something huge and complicated?


(This group comes up, at least for me, when you want to compare two Weil group representations whose projectivizations agree.)



Is H^2(W_p,C^times) well-known?
It is known that $H^2(W, C^\times)$ is trivial, when $W$ is the Weil group of a global or local field, with the trivial action on $C^\times$, and the cohomology is taken in the sense of Moore (measurable cochains). This is the main result of C.S. Rajan, "On the vanishing of the measurable cohomology groups of Weil groups", Compositio Math. 140 (2004) 84-98 (also easy to find online). 


Is this well-known? I don't know. I only found this paper recently, and hopefully now it will become better-known!


