Dear community,

In light of the recent work of DeBacker/Reeder on the depth zero local Langlands correspondence, I was wondering if there is an attempt to "geometrize" the depth zero local Langlands correspondence.

In particular, in Teruyoshi Yoshida's thesis, one can see a glimpse of this for $GL(n,F)$, where $F$ is a $p$-adic field. Namely : Suppose $k$ is the residue field of $F$. Let $w$ be the cyclic permutation $(1 \ 2 \ 3 \ ... \ n)$ in the Weyl group $S_n$ of $GL(n,F)$. Let $\widetilde{Y_w}$ be the Deligne-Lusztig variety associated to $w$, and denote by $H^*(\widetilde{Y_w})$ the alternating sum of the cohomologies $H_c^i(\widetilde{Y_w}, \overline{\mathbb{Q}_{\ell}})$. Let $T_w(k) = k_n^*$ be the elliptic torus in $GL(n,k)$, where $k_n$ is the degree $n$ extension of $k$.

Since $T_w(k)$ and $GL(n,k)$ act on cohomology, $H^*(\widetilde{Y_w})$ is an element of the Grothendieck group of $GL(n,k) \times T_w(k)$-modules. There is a canonical surjection $I_F \rightarrow k_n^* = T_w(k)$, where $I_F$ is the inertia subgroup of the Weil group of $F$. Therefore, we may pull back the $GL(n,k) \times T_w(k)$ action on $H^*(\widetilde{Y_w})$ to an action of $GL(n,k) \times I_F$.

By Deligne-Lusztig Theory, as $GL(n,k) \times T_w(k)$-representations,

$$H_c^{n-1}(\widetilde{Y_w}, \overline{\mathbb{Q}_{\ell}})^{cusp} = $$

$$\displaystyle\sum_{\theta \in C} \pi_{\theta} \otimes \theta$$

where $C$ denotes the set of all characters of $k_n^*$ that don't factor through the norm map $k_n^* \rightarrow k_m^*$ for any integer $m$ such that $m \neq n$ and $m$ divides $n$, and where cusp denotes "cuspidal part". Here, $\pi_{\theta}$ is the irreducible cuspidal representation of $GL(n,k)$ associated to the torus $T_w(k)$ and the character $\theta$ of $T_w(k)$.

One of Yoshida's main theorems is that in this decomposition $$\displaystyle\sum_{\theta \in C} \pi_{\theta} \otimes \theta,$$ the correspondence $\theta \leftrightarrow \pi_{\theta}$ is indeed the depth zero local Langlands correspondence for $GL(n,k)$, ''up to twisting'' (this twist is unimportant for my question), by comparing with Harris-Taylor.

So my question is : Has anyone tried to generalize this to more general groups, but still working only in depth zero local Langlands? One could try to do this, and then compare to the recent work of DeBacker/Reeder (they write down a fairly general depth zero local Langlands correspondence). In other words, has anyone tried to realize depth zero local Langlands in the cohomology of Deligne-Lusztig varieties outside of the case $GL(n,F)$, which Yoshida did?

A priori the above idea for $GL(n)$ won't work on the nose for other reductive groups since the tori that arise in other reductive groups vary considerably, but something similar might. One would possibly want to try to pull back the action of $T_w(k)$ on cohomology to the inertia group $I_F$ in a more general setting, where now $T_w(k)$ is a more general torus in a more general reductive group. Then, one could compare to DeBacker/Reeder.

I took a look at the case of unramified $U(3)$, and it seems that things will work quite nicely.

My other question is : It might turn out that what I'm proposing is an easy check if one understands DeBacker/Reeder and Deligne-Lusztig enough to write this down in general. If so, then is my original question even interesting? It would basically say that Deligne-Lusztig theory is very naturally compatible with local Langlands correspondence, but the hard work is really in DeBacker/Reeder and Deligne-Lusztig, and putting everything together might not be difficult. Is the original question interesting regardless of whether or not it is difficult to answer?

Sincerely,

Moshe Adrian