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, denotes a finite extension of , denotes (the -points of) a p-adic reductive group, a parabolic with Levi decomposition. Let be the abelian category of smooth representations of on -vector spaces, and let 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 has a (Gelfand-Kirillov) dimension , which is an integer in the interval .
First, some basic structural questions.
Q1. (Emerton-Gee) Do finitely presented representations form an abelian subcategory of ? This reduces to: is the kernel of a map between finitely presented representations also finitely presented? Yes for (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 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 fully faithful? Note that has enough injectives (Emerton). This reduces to: does vanish for all , all locally admissible reps. , and all locally admissible reps. which are injective in ?
Q5. Which integers can possibly occur as the dimension of an irreducible admissible representation?
Q5′. Does every irreducible admissible representation have dimension ? Here is the dimension of the flag variety associated with the quasisplit form of , so e.g. for . This question reduces to the case of supersingular representations. Yes for and related groups.
Next, recall the derived duality functor , which is given by derived internal hom towards the trivial representation. This restricts to a self-equivalence on satisfying biduality. If is an admissible rep in degree zero, then agrees with the i’th derived dual of defined by Kohlhaase (H.-Mann) – moreover, it is admissible of dimension , with equality in degree , and it vanishes for .
Q6. Does preserve the property of being admissible of finite length?
Q7. If is irreducible admissible and supersingular, can some admit a non-supersingular subquotient?
Recall that an admissible is Cohen-Macaulay if is nonzero for a single .
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 which preserves (local) admissibility. This functor has a right adjoint (easy) and a left adjoint (Heyer). Heyer showed that preserves . It is also true that preserves , and in fact that as functors on (H.).
Q10. Is there some a priori interplay between the functors and and GK dimension? Note that increases the dimension exactly by .
Q11. Writing for Emerton’s derived ordinary parts functor, is it true that ?
Q12. Does second adjointness hold, in the sense that as functors on ? This is equivalent to asking for an isomorphism as functors on . Note that second adjointness cannot hold on all of , because 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 as an iterated extension of functors of the form ?