## 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 $1$.