Irreducibility of automorphic Galois representations of $GL\left(n\right)$, $n$ at most $5$

Calegari, Frank; Gee, Toby

Annales de l'Institut Fourier, Tome 63 (2013) no. 5, pp. 1881-1912.

Nous prouvons l’irréductibilité pour $n$ inférieur ou égal à $5$ des représentations galoisiennes $l$-adiques associées aux représentations automorphes cuspidales algébriques et régulières de ${\mathrm{GL}}_n$ sur un corps totalement réel qui sont autoduales à torsion près. Nous prouvons également l’irréductibilité des représentations galoisiennes modulo $l$ pour presque tout $l$, et nous montrons l’indépendance en $l$ de l’algèbre de Lie de la clôture Zariskienne de la représentation $l$-adique.

Let $\pi$ be a regular, algebraic, essentially self-dual cuspidal automorphic representation of ${\mathrm{GL}}_n\left({𝔸}_F\right)$, where $F$ is a totally real field and $n$ is at most $5$. We show that for all primes $l$, the $l$-adic Galois representations associated to $\pi$ are irreducible, and for all but finitely many primes $l$, the mod $l$ Galois representations associated to $\pi$ are also irreducible. We also show that the Lie algebras of the Zariski closures of the $l$-adic representations are independent of $l$.

Classification : 11F80, 11R39
Mots clés : Galois representations, automorphic representations, représentations galoisiennes, représentations automorphes

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