Irreducibility of automorphic Galois representations of $GL\left(n\right)$, $n$ at most $5$
[Irréductibilité des représentations galoisiennes associées à certaines représentations automorphes de $GL\left(n\right)$ pour $n$ inférieur ou égal à $5$]
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 $l$.

DOI : https://doi.org/10.5802/aif.2817
Classification : 11F80,  11R39
Mots clés : Galois representations, automorphic representations, représentations galoisiennes, représentations automorphes
@article{AIF_2013__63_5_1881_0,
author = {Calegari, Frank and Gee, Toby},
title = {Irreducibility of automorphic Galois representations of $GL(n)$, $n$ at most $5$},
journal = {Annales de l'Institut Fourier},
pages = {1881--1912},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {63},
number = {5},
year = {2013},
doi = {10.5802/aif.2817},
mrnumber = {3186511},
zbl = {1286.11084},
language = {en},
url = {http://archive.numdam.org/articles/10.5802/aif.2817/}
}
Calegari, Frank; Gee, Toby. Irreducibility of automorphic Galois representations of $GL(n)$, $n$ at most $5$. Annales de l'Institut Fourier, Tome 63 (2013) no. 5, pp. 1881-1912. doi : 10.5802/aif.2817. http://archive.numdam.org/articles/10.5802/aif.2817/

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