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If $G$ is a unitary group in $n$ variables over $\mathbb Q$, attached to an hermitian form for an imaginary quadratic extension $E/\mathbb Q$ and if we suppose that the hermitian form is definite over $\mathbb R$ (which implies that $G$ is compact over $\mathbb R$), then it is now known that to any automorphic representation of $G(\mathbb A_{\mathbb Q})$ is attached a continuous Galois representation $\rho_\pi$ of the absolute Galois group $G_E$ over a finite extension of $ \mathbb Q_p$ ($p$ any prime). It is also expected that $\rho_\pi$ is absolutely irreducible provided $\pi$ is stable (that is that its base change to $GL_n(\mathbb A_E)$ is cuspidal). But

When is the absolute irreducibility of $\rho_\pi$ currently known ?

I am interested altogether in published results, preprints not yet published, announcements at seminar as well. I vaguely remember that a very general result, true for any $n$, was announced a few years ago by I don't remember whom (though a corner of my mind seems to believe it was a former student of R. Taylor), but I can't find it on the internet. I am aware of many results for low $n$, $n \leq 6$ or $7$, some old, some relatively recent, but is there a result valuable for general $n$.

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  • $\begingroup$ There is a density result for general n, and in the more general CM situation, in the paper "Automorphy and irreducibility of some $\ell$-adic representations" of Patrikis and Taylor (available on R. Taylor website, see Theorem D). I suspect this is not the result you believe you once heard, though. $\endgroup$
    – tkr
    May 13, 2014 at 22:33
  • $\begingroup$ There was a preprint a few years ago, not by a former student of R. Taylor, but by a former student of someone closely associated, that claimed such a result, but it had a mistake. This might be what you're thinking of; the BLGGT/PT results are the best I know of for general n. $\endgroup$
    – TSG
    May 14, 2014 at 5:29
  • $\begingroup$ Thanks @tkr et TSG. The claim mentioned by TSG is probably the one I remembered. $\endgroup$
    – Joël
    May 14, 2014 at 13:10

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