Galois representations arising from some compact Shimura varieties

Our aim is to establish some new cases of the global Langlands correspondence for GLm. Along the way we obtain a new result on the description of the cohomology of some compact Shimura varieties. Let F be a CM field with complex conjugation c and be a cuspidal automorphic representation of GLm(AF ). Suppose that c and that is cohomological. A very mild condition on is imposed if m is even. We prove that for each prime l there exists a continuous semisimple representation R l () : Gal(F /F ) GLm(Q l ) such that and R l () correspond via the local Langlands correspondence (established by Harris-Taylor and Henniart) at every finite place w l of F ("local-global compatibility"). We also obtain several additional properties of R l () and prove the Ramanujan-Petersson conjecture for . This improves the previous results obtained by Clozel, Kottwitz, Harris-Taylor and Taylor-Yoshida, where it was assumed in addition that is square integrable at a finite place. It is worth noting that the mild condition on in our theorem is removed by a p-adic deformation argument, thanks to Chenevier-Harris.