Suppose that $E/F$ is a unramified extension of local fields of characteristic zero. Let $G = GL_n$. Then it is well-known (due to Clozel?) that base change of tempered representations from $G(F)$ to $G(E)$ holds.

Question: does the same result hold in the case of characteristic $p > 0$?


EDIT: As Olivier says, this actually seems to follow immediately from LLC for function fields. Thanks!

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    $\begingroup$ Doesn't that follow from the proof of the local Langlands correspondence for function fields (by Laumon, Rappoport and Stuhler)? $\endgroup$ – Olivier Feb 24 '13 at 13:40

Maybe I should reproduce my comment as an answer, in order for MO not to treat the question has unanswered.

The full local Langlands correspondence is a theorem of G.Laumon, M.Rappoport and U.Stuhler in $\mathcal D$-elliptic sheaves and the Langlands correspondence (Invent. Math. 113).

Here is the Mathscinet review:


The result for $n=2$ is due to Deligne.


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