## Questions on mod-p representations of p-adic groups

As the title says. There are more questions than answers in this subject, and for my own sake I decided to collect some of them here. I reserve the right to add to this list in the future, and I encourage readers to leave additional questions in the comments!

In this post, $F$ denotes a finite extension of $\mathbf{Q}_p$, $G$ denotes (the $F$-points of) a p-adic reductive group, $P=MU \subset G$ a parabolic with Levi decomposition. Let $\mathrm{Rep}(G)$ be the abelian category of smooth representations of $G$ on $\overline{\mathbf{F}_p}$-vector spaces, and let $D(G)$ be its derived category. On this category of representations, there are various finiteness conditions one can consider: admissible, locally admissible, finitely generated, finitely presented. These conditions will be abbreviated in obvious ways. Any admissible representation $\pi$ has a (Gelfand-Kirillov) dimension $\mathrm{dim}_G \pi$, which is an integer in the interval $[0,\dim G]$.

First, some basic structural questions.

Q1. (Emerton-Gee) Do finitely presented representations form an abelian subcategory of $\mathrm{Rep}(G)$? This reduces to: is the kernel of a map between finitely presented representations also finitely presented? Yes for $\mathrm{SL}_2(F)$ (Shotton). For higher rank groups there is some negative evidence (Timmins).

Q2. (Emerton) Is any finitely generated admissible representation necessarily of finite length? Yes for tori, and for $\mathrm{GL}_2(\mathbf{Q}_p)$ and related groups (Emerton).

Q3. If yes to Q2., then locally admissible representations form a locally finite abelian category, so they admit a block decomposition (Gabriel). Is this block decomposition interesting in general?

Q4. Is the evident functor $\alpha_G: D^+(\mathrm{Rep}(G)_{\mathrm{l.adm}}) \to D(G)$ fully faithful? Note that $\mathrm{Rep}(G)_{\mathrm{l.adm}}$ has enough injectives (Emerton). This reduces to: does $\mathrm{Ext}^i(A,B)$ vanish for all $i>0$, all locally admissible reps. $A$, and all locally admissible reps. $B$ which are injective in $\mathrm{Rep}(G)_{\mathrm{l.adm}}$?

Q5. Which integers can possibly occur as the dimension of an irreducible admissible representation?

Q5′. Does every irreducible admissible representation have dimension $\leq \dim \gamma_G$? Here $\gamma_G$ is the dimension of the flag variety associated with the quasisplit form of $G$, so e.g. $\gamma_G = [F:\mathbf{Q}_p]n(n-1)/2$ for $G=\mathrm{GL}_n(F)$. This question reduces to the case of supersingular representations. Yes for $\mathrm{GL}_2(\mathbf{Q}_p)$ and related groups.

Next, recall the derived duality functor $\mathcal{S}_G : D(G) \to D(G)$, which is given by derived internal hom towards the trivial representation. This restricts to a self-equivalence on $D^b_{\mathrm{adm}}(G)$ satisfying biduality. If $\pi$ is an admissible rep in degree zero, then $\mathcal{S}^i_G(\pi) = H^i(\mathcal{S}_G(\pi))$ agrees with the i’th derived dual of $\pi$ defined by Kohlhaase (H.-Mann) – moreover, it is admissible of dimension $\leq i$, with equality in degree $\dim_G \pi$, and it vanishes for $i > \dim_G \pi$.

Q6. Does $\mathcal{S}^i_G(-)$ preserve the property of being admissible of finite length?

Q7. If $\pi$ is irreducible admissible and supersingular, can some $\mathcal{S}^i_G(\pi)$ admit a non-supersingular subquotient?

Recall that an admissible $\pi$ is Cohen-Macaulay if $\mathcal{S}^i_G(\pi)$ is nonzero for a single $i$.

Q8. Are there examples of irreducible admissible supersingular representations which are not Cohen-Macaulay?

Q9. Does the AHHV classification imply a clean classification of irreducible admissible Cohen-Macaulay representations in terms of supersingular Cohen-Macaulay representations of Levi subgroups?

Next, recall that parabolic induction induces a t-exact functor $\mathrm{Ind}_P^G : D(M) \to D(G)$ which preserves (local) admissibility. This functor has a right adjoint $\mathbf{R}_G^P$ (easy) and a left adjoint $\mathbf{L}_G^P$ (Heyer). Heyer showed that $\mathbf{L}_G^P$ preserves $D^b_{\mathrm{adm}}$. It is also true that $\mathbf{R}_G^P$ preserves $D^b_{\mathrm{adm}}$, and in fact that $\mathbf{R}_G^P \cong \chi_P \otimes \mathcal{S}_M \circ \mathbf{L}_G^P \circ \mathcal{S}_G [-\dim U]$ as functors on $D^b_{\mathrm{adm}}(G)$ (H.).

Q10. Is there some a priori interplay between the functors $\mathbf{L}_G^P$ and $\mathbf{R}_G^P$ and GK dimension? Note that $\mathrm{Ind}_P^G$ increases the dimension exactly by $\dim U$.

Q11. Writing $R\mathrm{Ord}_{\overline{P}}: D^+(\mathrm{Rep}(G)_{\mathrm{l.adm}}) \to D^+(\mathrm{Rep}(M)_{\mathrm{l.adm}})$ for Emerton’s derived ordinary parts functor, is it true that $\alpha_M \circ R\mathrm{Ord}_{\overline{P}} \cong \mathbf{R}_G^P \circ \alpha_G$?

Q12. Does second adjointness hold, in the sense that $\mathbf{L}_G^P \cong \mathbf{R}_G^{\overline{P}}[\dim U] \otimes \chi_P$ as functors on $D^b_{\mathrm{adm}}(G)$? This is equivalent to asking for an isomorphism $\mathcal{S}_M \circ \mathbf{L}_G^P \cong \mathbf{L}_G^{\overline{P}} \circ \mathcal{S}_G$ as functors on $D^b_{\mathrm{adm}}(G)$. Note that second adjointness cannot hold on all of $D(G)$, because $\mathbf{R}_G^{\overline{P}}$ does not preserve arbitrary direct sums (Abe-Henniart-Vigneras), hence cannot be a left adjoint.

Q13. Does some form of the geometric lemma hold, describing $\mathbf{L}_{G}^{P'} \circ \mathrm{Ind}_{P}^{G}$ as an iterated extension of functors of the form $\mathrm{Ind}_{?}^{M'} \circ \mathrm{Weyl\,twist} \circ \mathbf{L}_M^{?}[\mathrm{shift}]$?

Parabolic induction is one of the most basic operations in the representation theory of p-adic groups. In the classical setting of complex coefficients, parabolic induction has a left adjoint given by the Jacquet module, and also a right adjoint given – miraculously – by the (twisted) Jacquet module for the opposite parabolic. Here all three functors are exact. One then defines supercuspidal representations as those which are killed by all Jacquet module functors, or equivalently as those which don’t occur in any parabolic induction.

With mod $p$ coefficients, parabolic induction is still exact, so it easily passes to a functor on derived categories. More precisely, fix $G$ a p-adic reductive group, and let $D(G)$ be the derived category of the category of smooth $\overline{\mathbf{F}_p}$ representations of $G$. Let $P=MU \subset G$ be any parabolic subgroup. Then the usual operation of parabolic induction upgrades to a t-exact functor $\mathrm{Ind}_P^G:D(M) \to D(G)$ which preserves $D^b_{\mathrm{adm}}$. By general nonsense, $\mathrm{Ind}_P^G$ commutes with all direct sums, and hence admits a right adjoint $\mathbf{R}_{G}^{P}: D(G) \to D(M)$. Much less obviously, a recent theorem of Heyer shows that $\mathrm{Ind}_P^G$ commutes with direct products, and hence admits a left adjoint $\mathbf{L}_{G}^{P}: D(G) \to D(M)$. Moreover, Heyer also shows that $\mathbf{L}_{G}^{P}$ preserves $D^{b}_{\mathrm{adm}}$, and computes its values in some examples.

Exercise. Show that $\mathbf{R}_{G}^{P}$ restricted to $D^{b}_{\mathrm{adm}}(G)$ satisfies the isomorphism $\mathbf{R}_{G}^{P} \cong (\mathcal{S}_M \circ \mathbf{L}_{G}^{P} \circ \mathcal{S}_G)[-\dim U] \otimes \chi_P$, where $\mathcal{S}_G$ is Kohlhaase’s derived duality funtor and $\chi_P:M \to \mathbf{F}_p^\times$ is the integral modulus character. Deduce that $\mathbf{R}_{G}^{P}$ preserves $D^{b}_{\mathrm{adm}}$.

Now, if you start with an irreducible representation $\pi$ in degree zero, it is formal that $\mathbf{R}_{G}^{P}(\pi)$ resp. $\mathbf{L}_{G}^{P}(\pi)$ will be concentrated in nonnegative resp. nonpositive degrees, and $H^0$ of it is something explicit: $H^0 \mathbf{L}_{G}^{P}(\pi)$ is basically the (naive) Jacquet module, and $H^0 \mathbf{R}_{G}^{P}(\pi)$ is Emerton’s functor $\mathrm{Ord}_{\overline{P}}(\pi)$ of ordinary parts. In particular, when $\pi$ is supersingular, both of these things vanish in degree zero. But of course, they might be nonzero in other degrees, since $\mathbf{L}_{G}^{P}(\pi)$ and $\mathbf{R}_{G}^{P}(\pi)$ are not t-exact.

In the special case where $G=\mathrm{GL}_2(\mathbf{Q}_p)$ and $P=B$ is the Borel, Heyer showed that $\mathbf{L}_{G}^{B}(\pi)$ vanishes identically for any irreducible supersingular representation $\pi$, and the above exercise then implies that also $\mathbf{R}_{G}^{B}(\pi)$ vanishes identically. However, if there’s one thing we’ve learned in recent years, it’s that p-adic Langlands is only simple for $\mathrm{GL}_2(\mathbf{Q}_p)$ – for every other group, the whole story is completely different.

Theorem (Yongquan Hu). If $G=\mathrm{GL}_2(\mathbf{Q}_{p^2})$, there are plenty of irreducible admissible supersingular representations $\pi$ such that $\mathbf{L}_{G}^{B}(\pi)$ and $\mathbf{R}_{G}^{B}(\pi)$ are both nonzero!

This is actually immediate from Corollary 1.2 here and basic adjunctions.

On further reflection, it is probably true that “most” supersingular representations of a given group have the property that some $\mathbf{L}_{G}^{P}(-)$ or $\mathbf{R}_{G}^{P}(-)$ is nonzero. If you believe in some version of the mod p Langlands correspondence, this is reflected in the fact that “most” mod p Galois representations are reducible (e.g., they are Zariski-dense in the Emerton-Gee stack).

Question. Is it true that “second adjointness” holds in this setting, in the sense that $\mathbf{L}_{G}^{P} \cong \mathbf{R}_{G}^{\overline{P}}[\dim U] \otimes \chi_P$ as functors on on $D^b_{\mathrm{adm}}(G)$, or even on all of $D(G)$?

One can check by hand that this isomorphism is OK on irreps of $\mathrm{GL}_2(\mathbf{Q}_{p})$ using the calculations in Heyer’s paper and the exercise above. If this question has an affirmative answer, then $\mathbf{L}_{G}^{P}(\pi)$ is necessarily concentrated in degrees $[1-\dim U,-1]$ for any supersingular $\pi$, and similarly for $\mathbf{R}_{G}^{P}(\pi)$. In particular, in the setting of Hu’s example above, we would get that $\mathbf{L}_{G}^{B}(\pi)$ is concentrated in degree $-1$, and $\mathbf{R}_{G}^{B}(\pi)$ is concentrated in degree $1$.

## Report from Oberwolfach

In the first week of February, I attended an Oberwolfach workshop on Nonarchimedean geometry and applications. It was a great pleasure to attend a conference in person after such a long period of isolation. Thank you to the organizers for making this week so enjoyable! As usual, here are some scattered recollections from the workshop.

• Due to the hybrid nature of the workshop, the talks on several days didn’t start until 2 pm, presumably to accommodate participants in North America, with the final talk scheduled after dinner. This left a huge swath of unscheduled time, from 9 am until 2 pm, which actually turned out to be kind of great. I don’t know if it was the hunger for in-person interaction after 2 years of isolation, but people really seemed to take full advantage of this free time for vigorous discussion and collaboration. I actually liked this schedule better than the usual schedule.
• Best talks: Johannes Nicaise, Lucas Mann, Piotr Achinger, Yujie Xu, Ben Heuer.
• Worst talk: [redacted]
• Categorical Langlands for GL1 = Langlands for mice.
• Random Question 1 (via PS): Let $U \subset X$ be an open immersion of an affine scheme into a smooth projective variety. Is the complement $X \smallsetminus U$ an ample divisor on $X$?
• Random Question 1′ (via DC): Let $U \subset V$ be an open immersion of affine schemes. Is $U$ the nonvanishing locus of a section of an ample line bundle on $V$?
• One recurring theme throughout the week was the p-adic Simpson correspondence, with excellent talks from Ben Heuer and Matti Würthen. Here something quite amusing happened: in an informal conversation on Tuesday, Ben explained the complicated status of p-adic Simpon to me, and stressed that its most optimistic conjectural form isn’t actually written down, because no one wants to be the one to make a false conjecture. But then Matti Würthen did explicitly conjecture this exact statement in his Friday lecture! For the record, the hope is that for a smooth projective variety $X/ \mathbf{C}_p$, there is an equivalence of categories from $\mathbf{C}_p$-representations of $\pi_1^{\mathrm{et}}(X)$ towards semistable Higgs bundles on $X$ with vanishing Chern classes. As Ben stressed to me, this definitely fails if $\mathbf{C}_p$ is replaced with a larger algebraically closed nonarchimedean field. Hmm…
• One consequence of COVID measures is that seating for meals was not randomized as usual, but rather was fixed for the entire week. I was assigned to eat with Torsten Wedhorn, Bogdan Zavyalov, and François Loeser. This ended up being a really pleasant group to eat with! I already knew Torsten and Bogdan fairly well, but I’d never spoken with François before, and it turns out he’s a totally charming and delightful person. It was especially wonderful to hear him talk about his astonishing achievements in ultra long distance running. One memorable quote: “The first night without sleep is no problem. The second night is… interesting. And the third night… well… this I cannot recommend.”
• My relationship with Hochschild cohomology has gone from “???” to a vague understanding and a desire to learn more. Thanks to DC for some helpful explanations!
• There was much discussion among the younger participants about what Fargues’s categorical local Langlands conjecture should look like with mod-p coefficients (i.e. in the $\ell=p$ setting). Of course on the spectral side, one expects to see some category of quasicoherent or ind-coherent sheaves on the special fiber of the Emerton-Gee stack for $\hat{G}$. On the automorphic side, one should have some category of mod-p sheaves on $\mathrm{Bun}_G$, and the correct category should fall out of the general formalism developed by Mann in his thesis. One tantalizing fact, sketched out during some of these conversations, is that $\mathrm{Bun}_G$ is definitely $p$-cohomologically smooth (in a precise sense), not of dimension 0 as in the $\ell \neq p$ case, but of dimension equal to the dimension of $G(\mathbf{Q}_p)$ as a $p$-adic Lie group. It is surely no coincidence that this matches the expected dimension of the Emerton-Gee stack for $\hat{G}$.

Another tantalizing observation: the relationship between the Emerton-Gee stack and Wang-Erickson’s stack of Galois representations is perfectly analogous to the difference between the stacks $\mathrm{LocSys}_G$ and $\mathrm{LocSys}_{G}^{\mathrm{restr}}$ appearing in AGKRRV.

On the other hand, it also became clear that most of the analysis in Fargues-Scholze cannot carry over naively to the setting of p-adic coefficients, and that many of the crucial tools developed in their paper simply won’t help here. In particular, the magic charts $\pi_b: \mathcal{M}_b \to \mathrm{Bun}_G$ used by FS, which are $\ell$-cohomologically smooth for all primes $\ell \neq p$, are definitely NOT $p$-cohomologically smooth. This already fails for $G=\mathrm{GL}_2$. Likewise, their “strict Henselian” property should fail badly. New ideas are very much required!

## Tales from the dropbox

As today’s service to the number theory entertainment complex, I could write an ill-advised rant about the abc conjecture situation. Instead, here’s another dropbox link. This time it’s some notes I wrote while trying to understand Kohlhaase’s paper Smooth duality in natural characteristicWarning: These are only about 40% done. The only original content (so far) is Proposition 1.6 and Theorem 1.12.

## Elliptic curves over Q(i) are potentially automorphic

This spectacular theorem was announced by Richard Taylor on Thursday, in a lecture at the joint meetings.  Taylor credited this result and others to Allen-Calegari-Caraiani-Gee-Helm-Le Hung-Newton-Scholze-Taylor-Thorne (!), as an outcome of the (not so) secret mini-conference which took place at the IAS this fall.  The key new input here is work in progress of Caraiani-Scholze on the cohomology of non-compact unitary Shimura varieties, which can be leveraged to check (at least in some cases) the most difficult hypothesis in the Calegari-Geraghty method: local-global compatibility at l=p for torsion classes.

The slides from my talk can be found here. Naturally I managed to say “diamond” a bunch of times.