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 $GL(n,F)$)