The local Langlands correspondence $\text{rec}$ for $\text{GL}_{n}$ itself is not Galois equivariant (i.e. invariant under automorphisms of its field of definition) but rather its twist by contragradient and an unramified character is. In Carayol's article "Non-abelian Lubin-Tate theory" in Ann Arbor proceedings, this twist, which I will call $r_{\ell}$, is just referred as "Hecke correspondence," which makes me feel like there is something more about it. Is there a deeper reason to believe why not the local Langlands correspondence but its slight twist is the one that is Galois equivariant? More generally, is there an a priori reason why $r_{\ell}$ is the one that appears in geometric constructions?

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    $\begingroup$ You may want to add link to the article and at least one top level tag. $\endgroup$ Jan 8 '16 at 14:55

The reason $r_\ell$ appears in geometric constructions involving Shimura varieties is because Hecke operators act on the cohomology of a Shimura variety with coefficients in $\mathbb{Q}$; that's the source of the good behavior of $r_\ell$ with respect to automorphisms of the coefficient field.


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