Geometric construction of depth zero local Langlands correspondence

Question

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

Answer 1

Yoshida considers the Lubin-Tate tower in his geometric realization of the depth zero supercuspidals for $GL(n)$. For unitary groups, I'm sure that the answer to your question will be found in a similar analysis for the corresponding Rapoport-Zink spaces. I don't believe this has been done yet, but it ought to be done soon -- I'm an interested party, as is (at least) Tasho Kaletha at Princeton. In the meantime, I have a reinterpretation of Yoshida's thesis that uses the $p$-adic period mappings out of the Lubin-Tate tower in an essential way. I've just posted some notes yesterday (!) here. The bit about DL varieties appears at the very end. I resort to using coordinates and deriving (as Yoshida does) an explicit equation for the DL variety, but with a little more work I bet a more abstract approach can be found. Since the theory of $p$-adic period maps for more general RZ spaces has all been worked out, I bet this approach can be used for unitary groups as well.

Answer 2

Teruyoshi Yoshida responded to my question by e-mail and he is ok with my posting his response on mathoverflow :

"Dear Moshe,

thanks for your interest - yes it would be very interesting to do this with more general Rapoport-Zink spaces, but i) I haven't been successful in finding an intrinsic moduli interpretation of my model for tame Lubin-Tate space, hence the difficulty in generalizing to other groups ii) the so-called Drinfeld level structures do not seem to give nice integral models for the Rapoport-Zink spaces with deeper levels. In spite of these obstacles in arithmetic-geometry, it would be interesting to investigate the cohomology for other RZ spaces (there are works by Ito-Mieda, Shin, Strauch etc). Feel free to quote my email in mathoverflow.

very best, Teruyoshi"

Answer 3

Jared and community,

Thank you for your responses! I agree, I think one should be able to do things without explicit equations for the DL varieties.

I was also thinking along a different line than your line of attack. Here is a concrete example of how things might go (and I'm not sure if everything that follows is completely correct) : Consider the unramified p-adic group $U(3,F)$ over the p-adic field $F$. Let $k$ be the residue field of $F$. There are two natural types of depth zero supercuspidal Langlands parameters $$\phi_i : W_F \rightarrow GL(3,\mathbb{C}) \rtimes \mathbb{Z} / 2 \mathbb{Z}$$ for $U(3,F)$, where $W_F$ is the Weil group. In both cases, the image of $I_F$ lands in $\hat{T}$, the dual maximal torus, where $I_F$ is the inertia group. Moreover, if $\Phi_F$ denotes Frobenius, $\phi_1(\Phi_F) = {}^{(13)} w$ and $\phi_2(\Phi_F) = {}^{(12)} w$, where ${}^{(13)} w$ denotes the Weyl group element of $GL(3,\mathbb{C})$ that switches the first and third entries of the maximal torus, and ${}^{(12)} w$ is the Weyl group element that switches the first and second entries of the maximal torus.

Consider a Langlands parameter of type $\phi_1$ (what follows will also work in an analogous way for parameters of type $\phi_2$). For ease of notation, set $w := {}^{(13)} w$. Let $\widetilde{Y_w}$ denote the DL variety associated to $w$. Associated to $w$ is also a twisted torus $T_w$. Unwinding definitions, one gets that $T_w(k) = k_2^1 \times k_2^1 \times k_2^1$, where $k_2$ denotes the degree 2 extension of $k$, and $k_2^1$ denotes the group of norm $1$ elements from $k_2$ to $k$. This torus splits over $k_2$, and the $k_2$-points are given by $T_w(k_2) = k_2^* \times k_2^* \times k_2^*$.

The twisted torus $T_w(k)$ and $U(3,k)$ both act on $H_c^*(\widetilde{Y_w})$. As $U(3,k) \times T_w(k)$-representations, we have that $H_c^*(\widetilde{Y_w})$ contains

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

where $C$ denotes the set of all characters of $T_w(k)$ in general position, and where $\pi_{\theta}$ is the representation of $U(3,k)$ associated to the torus $T_w(k)$ and the character $\theta$ of $T_w(k)$. The question is to get a $U(3,k) \times I_F$-action on $H_c^*(\widetilde{Y_w})$ so that we can get a local Langlands correspondence (and in the case of $U(3,F)$, it turns out that we don't even need the full Weil group). Note that we have the norm map on tori : $$N : T_w(k_2) \rightarrow T_w(k)$$ Finally, as in the example of $GL(n,F)$ above (which Yoshida considers), one can consider the canonical surjection $$\alpha : I_F \rightarrow k_2^*$$ We may now consider the maps $$\alpha_1 : I_F \rightarrow T_w(k_2) = k_2^* \times k_2^* \times k_2^*$$ $$x \mapsto (\alpha(x), 1, 1)$$

$$\alpha_2 : I_F \rightarrow T_w(k_2) = k_2^* \times k_2^* \times k_2^*$$ $$x \mapsto (1, \alpha(x), 1)$$

$$\alpha_3 : I_F \rightarrow T_w(k_2) = k_2^* \times k_2^* \times k_2^*$$ $$x \mapsto (1, 1, \alpha(x))$$

We may pull back the action of $T_w(k)$ on $H_c^*(\widetilde{Y_w})$ to $I_F$ by the maps $\alpha_i \circ N$. In particular, form the direct sum $$ \displaystyle\bigoplus_{i=1}^3 (\alpha_i \circ N)$$

This direct sum is an $I_F$-action on $H_c^*(\widetilde{Y_w})$, which is what we sought. If $\theta$ is a character of $T_w(k)$ in general position, denote by $\eta_{\theta}$ the pullback of $\theta$ to $I_F$ by $ \displaystyle\bigoplus_{i=1}^3 (\alpha_i \circ N)$. Via this action, $H_c^*(\widetilde{Y_w})$, as a $U(3,k) \times I_F$-module, contains

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

With a fair amount of work (essentially in progress by Joshua Lansky and myself, and by results of Joshua Lansky and Jeffrey Adler), one can show that $\pi_{\theta} \leftrightarrow \eta_{\theta}$ "is the local Langlands correspondence". To be precise, I would have to say something about L-packets (since the L-packet of this Langlands parameter consists of more than one representation), but I will brush that under the rug for now. With much less work (I think), one can show that $\pi_{\theta} \leftrightarrow \eta_{\theta}$ agrees with the correspondence of DeBacker/Reeder (in particular, DeBacker/Reeder's correspondence is therefore correct for $U(3,F)$). Again, I am ignoring the entire L-packet for now (but essentially, you can conjugate $\theta$ around to various isomorphic tori in the group and get an L-packet of representations, as in DeBacker/Reeder). Moreover, since we are dealing with $U(3,F)$, it turns out that we don't even need to worry about ``twisting'', as we did with $GL(n,F)$.

This above construction is analogous to the $GL(n,F)$ example. In the case of $GL(n,F)$, one could pull the torus action back to $I_F$ using the norm map as above, take the direct sum of the analogous $\alpha_i \circ N$, and get the local Langlands correspondence ``up to twisting''. So really (as far as I can tell) this is what is being done in the case of $GL(n,F)$ as well.

It doesn't seem completely unreasonable that a similar construction might be able to be carried out in more generality (maybe this is wishful thinking) .