## 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 $\mathrm{Ind}_{?}^{M'} \circ \mathrm{Weyl\,twist} \circ \mathbf{L}_M^{?}[\mathrm{shift}]$?