Let $\Sigma$ be an $n$-dimensional representation of the global Weil group $W_F$ for a number field $F$, and $\Pi$ an automorphic representation of $\operatorname{GL}_n(\mathbb A_F)$. Suppose that at every place $v$ of $F$ where $\Sigma_v$ and $\Pi_v$ are both unramified, we have $\Pi_v \leftrightarrow \Sigma_v$ under the local Langlands correspondence.

In Guy Henniart's proof on the local Langlands correspondence, he claims that $\operatorname{Det} \Sigma$ then corresponds to the central character of $\Pi$ under the isomorphism $W_F^{\operatorname{ab}} \cong \mathbb A_F^{\ast}/F^{\ast}$ of global class field theory.

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Why should this be the case? It is true that at almost all places $v$, $\operatorname{Det} \Sigma_v$ identifies with the central character of $\Pi_v$ under the isomorphism $F_v^{\ast} \cong W_{F_v}^{\operatorname{ab}}$. We have for $a = (a_v) \in \mathbb A_F^{\ast}$,

$$\varpi_{\Pi}(a) = \prod\limits_v \varpi_{\Pi_v}(a_v)$$

so it seems like something could go wrong with $\varpi_{\Pi}$ at the ramified places.


Now that I have thought about it a little more, I think this just comes down to the following fact: If $\chi = \otimes \chi_v$ is a character of $\mathbb A_F^{\ast}/F^{\ast}$, and $S$ is a finite set of places of $F$, then $\chi$ is completely determined by the characters $\chi_v : v \not\in S$. This is because if $\eta = \otimes_v \eta_v$ is another character of $\mathbb A_F^{\ast}/F^{\ast}$ with $\eta_v = \chi_v$ for all $v \not\in S$, then $\chi \eta^{-1}$ defines a character of $\prod\limits_{v \in S} k_v^{\ast}$ which is trivial on the diagonal embedding $\Delta$ of $k^{\ast}$. But $\Delta$ is dense in this product by weak approximation, so in fact $\chi = \eta$ on $\mathbb A_F^{\ast}$.

Now $\operatorname{Det} \circ \Sigma$ corresponds to a character of $\mathbb A_F^{\ast}/F^{\ast}$ which agrees at almost all places with the central character of $\Pi$ (also trivial on $F^{\ast}$), so they must be the same character.


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