Sur la fonctorialité, pour $\mathrm{GL}(4)$, donnée par le carré extérieur

Let $k$ be a number field. Henry H. Kim has established the exterior square transfer for $\mathrm{GL}(4)$, which attaches to any cuspidal automorphic representation $\Pi$ of $\mathrm{GL}(4,\mathbb A_k)$ an automorphic representation $\Pi$ of $\mathrm{GL}(6,\mathbb A_k)$. At a finite place $v$ of $k$, the local component $\rho_v$ of $\rho$ gives, via the Langlands correspondence, a degree 4 representation $\sigma_v$ of the Weil–Deligne group of $k_v$. Then $\Pi$ is the unique isobaric automorphic representation of $\mathrm{GL}(6,\mathbb A_k)$ such that, whenever $\rho_v$ is unramified, $\Pi_v$ corresponds, via the Langlands correspondence, to the exterior square $\Lambda^2\sigma_v$ of $\sigma_v$. Kim proves that $\Pi_v$ corresponds to $\Lambda^2\sigma_v$ even when $\rho_v$ is ramified, except possibly if $v$ is above 2 or 3 and $\rho_v$ is cuspidal. We complete Kim's work in showing that $\Pi_v$ corresponds to $\Lambda^2\sigma_v$ at all finite places $v$ of $k$.