Using a level-raising argument (and a result of Larsen on the image of Galois representations in compatible systems), we prove that for any automorphic representation for U(3), the l-adic Galois representation which is attached to by the work of Blasius and Rogawski is the one expected by local Langlands correspondance at every finite place (at least up to semi-simplification and for a density one set of primes l). We rely on the work of Harris and Taylor, who have proved the same results (for U(n)) assuming the base change of is square-integrable at one place. As a corollary, every automorphic representation which is tempered at an infinite number of places is tempered at all places.
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Joël Bellaïche, Sur la compatibilité entre les correspondances de Langlands locale et globale pour U(3). Comment. Math. Helv. 81 (2006), no. 2, pp. 449–470DOI 10.4171/CMH/58