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.


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