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Deligne-Lusztig theory is an important tool in understanding the depth zero representations of $p$-adic groups. Is there an analogue of Deligne-Lusztig theory that helps in understanding positive depth representations of $p$-adic groups? If the answer is no in general, how about in the case of $GL(n,F)$, where $F$ is a $p$-adic field?

The reason why I ask is as follows: Teruyoshi Yoshida has released a paper recently ("On non-abelian Lubin-Tate theory via vanishing cycles") where he compares the cohomology of a Deligne-Lusztig variety for $GL(n,F)$ with the cohomology that arises from non-abelian Lubin-Tate theory (the latter of which I have no understanding of). It seems that because of this comparison, he is then able to give a proof of the local Langlands correspondence for $GL(n,F)$ for depth zero representations (possibly up to "twisting").

So I was wondering if one could try to do the same thing that he does in the case of positive depth representations of $GL(n,F)$ (i.e. compare two different cohomologies, one of which is given by non-abelian Lubin-Tate theory, and the other of which is given by some analogue of Deligne-Lusztig theory, if there is such a thing for $GL(n,F)$)

Sincerely,

Moshe Adrian

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You should look at Jared Weinstein's preprints on the arxiv --- the program that you suggest is exactly what he is doing: arxiv.org/find/math/1/au:+Weinstein_J/0/1/0/all/0/1 –  Emerton Dec 14 '10 at 16:57
    
Dear Emerton, Thank you for your comment, this is very helpful. Moshe –  Moshe Adrian Dec 14 '10 at 20:40
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