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Représentations galoisiennes associées aux représentations automorphes autoduales de $$GL(n)$$. (Galois representations associated with self-dual automorphic representations of $$GL(n)$$). (French) Zbl 0739.11020
To certain automorphic representations of $$GL(n,A_ F)$$, $$F$$ a totally real number field, $$\ell$$-adic Galois representations are attached relating eigenvalues of Hecke operators to eigenvalues of Frobenius. This is done by first transporting the representation of $$GL(n,A_ F)$$ to an appropriate unitary group $$U$$. This involves base change from $$GL(n)$$ over $$F$$ to $$GL(n)$$ over a totally imaginary quadratic extension $$F_ c$$ of $$F$$, correspondence between the latter and the multiplicative group $$D^*$$ of a division algebra over $$F$$ with involution of the second kind and descent from $$D^*$$ to the unitary group $$U$$. Then Kottwitz’ results are applied. Essential is the hypothesis that the original representation $$\pi$$ of $$GL(n,A_ F)$$ be self-dual.
The result implies the Ramanujan conjecture at almost all places for $$\pi$$. There is an analogous result for a $$CM$$-field.

##### MSC:
 11F70 Representation-theoretic methods; automorphic representations over local and global fields
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