Tag: Vigneras

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}]?

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