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 $n$.
