Galois representations arising from some compact Shimura varieties

Our aim is to establish some new cases of the global Langlands cor-respondence for GLm. Along the way we obtain a new result on the de-scription of the cohomology of some compact Shimura varieties. Let F be a CM field with complex conjugation c and Π be a cuspidal automorphic representation of GLm(AF). Suppose that Π ∨ ' Π ◦ c and that Π ∞ is cohomological. A very mild condition on Π ∞ is imposed if m is even. We prove that for each prime l there exists a continuous semisimple representa-tion Rl(Π) : Gal(F/F) → GLm(Ql) such that Π and Rl(Π) correspond via the local Langlands correspondence (established by Harris-Taylor and Hen-niart) at every finite place w- l of F (“local-global compatibility”). We also obtain several additional properties of Rl(Π) and prove the Ramanujan