## Bontje, 8/25/2007-5/8/2023

Our beloved dog Bontje died yesterday. During her long and joyous life, she travelled widely, lived in countless cities on three continents, ate like royalty, barked at many eminent mathematicians, charmed strangers everywhere with her huge personality, and was deeply loved by J. and myself. We miss her terribly.

## Distinguished affinoids

Fix a complete nonarchimedean field $K$ equipped with a fixed norm, with residue field $k$. Let $A$ be a $K$-affinoid algebra in the sense of classical rigid geometry. Here’s a funny definition I learned recently.

Definition. A surjection $\alpha : T_{n,K} \twoheadrightarrow A$ is distinguished if the associated residue norm $|\cdot|_\alpha$ equals the supremum seminorm $|\cdot|_{\mathrm{sup}}$. A $K$-affinoid algebra $A$ is distinguished if it admits a distinguished surjection from a Tate algebra.

Being distinguished imposes some obvious conditions on $A$: since the supremum seminorm is a norm iff $A$ is reduced, it certainly it implies
1) $A$ is reduced.
Since any residue norm takes values in $|K|$, it also implies
2) $|A|_{\mathrm{sup}} = |K|$.

If $K$ is stable (which holds if $K$ is discretely valued or algebraically closed), then the converse is true: 1) and 2) imply that $A$ is distinguished. Since 2) is automatic for $K$ algebraically closed, we see that any reduced $K$-affinoid is distinguished if $K$ is algebraically closed. It is also true that if $\alpha: T_{n,K} \to A$ is a distinguished surjection, then $\alpha^\circ: T_{n,K}^{\circ} \to A^\circ$ is surjective. Moreover, if $A$ satisfies 2), or $K$ is not discretely valued, then a surjection $\alpha: T_{n,K} \to A$ is distinguished iff $\alpha^\circ$ is surjective. Either way, if $A$ is distinguished then $A^\circ$ is a tft $K^\circ$-algebra.

All of this can be found in section 6.4.3 of BGR.

Question 1. If $A$ is reduced, is there a finite extension $L/K$ such that $A\otimes_K L$ is distinguished as an $L$-affinoid algebra?

This should be easy if it’s true. I didn’t think much about it.

Now suppose $A$ is distinguished, and let $\tilde{A} = A^{\circ} / A^{\circ \circ}$ be its reduction to a finite type $k$-algebra. As usual we have the specialization map $\mathrm{sp}: \mathrm{Sp}A \to \mathrm{Spec} \tilde{A}$. It is not hard to see that if $D(f) \subset \mathrm{Spec} \tilde{A}$ is a principal open, then $\mathrm{sp}^{-1}D(f)$ is a Laurent domain in $\mathrm{Sp}A$. Much less obvious is that for any open affine $U \subset \mathrm{Spec} \tilde{A}$, the preimage $\mathrm{sp}^{-1}U$ is an affinoid subdomain such that $A_U=\mathcal{O}(\mathrm{sp}^{-1}U)$ is distinguished and $\widetilde{A_U} = \mathcal{O}_{\mathrm{Spec} \tilde{A}}(U)$. This is buried in a paper of Bosch.

Loosely following Bosch, let us say an affinoid subdomain $V \subset \mathrm{Sp}A$ is formal if it can be realized as $\mathrm{sp}^{-1}U$ for some open affine $U \subset \mathrm{Spec} \tilde{A}$. Now let $X$ be a reduced quasicompact separated rigid space over $K$. Let us say a finite covering by open affinoids $U_1=\mathrm{Sp}A_1,\dots,U_n= \mathrm{Sp}A_n \subset X$ is a formal cover if
1) all $A_i$ are distinguished, and
2) for each $(i,j)$, the intersection $\mathrm{Sp}A_{ij}=U_{ij} := U_i \cap U_j$, which is automatically affinoid, is a formal affinoid subdomain in $U_i$ and in $U_j$.

This is a very clean kind of affinoid cover: we can immediately build a formal model for $X$ by gluing the tft formal affines $\mathrm{Spf}(A_i^\circ)$ along their common formal affine opens $\mathrm{Spf}(A_{ij}^\circ)$. Moreover, the special fiber of this formal model is just the gluing of the schemes $\mathrm{Spec}\widetilde{A_i}$ along the affine opens $\mathrm{Spec}\widetilde{A_{ij}}$.

Question 2. For $X$ a reduced qc separated rigid space over $K$, is there a finite extension $L/K$ such that $X_L$ admits a formal affinoid cover?

## When is it supercuspidal?

Today I want to talk about an amazing theorem of Moeglin, which I learned from WTG.

Let $F/\mathbf{Q}_p$ be a finite extension, $G=\mathrm{SO}_{2n+1}$ the split odd special orthogonal group over $F$, $G'$ its unique inner form. By work of Arthur and Moeglin, there is a natural bijection between discrete series representations of $G$ or $G'$, and pairs $(\phi,\chi)$ where $\phi: W_F \times \mathrm{SL}_2 \to \mathrm{Sp}_{2n}$ is a discrete L-parameter and $\chi$ is a character of the centralizer group $A_\phi$. In this setting, $\phi$ is discrete if it is the sum of $m$ pairwise-distinct irreducible representations $\phi_i = \sigma_i \boxtimes [d_i] : W_F \times \mathrm{SL}_2 \to \mathrm{Sp}_{2n_i}$ with $\sum_{1 \leq i \leq m} n_i = n$. Here $[d]:\mathrm{SL}_2 \to \mathrm{SL}_d$ is the usual d-1st symmetric power representation. The associated centralizer group $A_\phi$ is of the form $\{ \pm 1\}^m$, and has a canonical basis indexed by the irreducible summands $\phi_i$. Given $(\phi,\chi)$, let $\pi(\phi,\chi)$ be the associated discrete series representation. Note that $\pi(\phi,\chi)$ is a representation of $G$ if $\chi$ is trivial on the evident subgroup $\{ \pm 1 \} = Z(\mathrm{Sp}_{2n}) \subset A_\phi$, and is a representation of $G'$ otherwise. This splits the representations up evenly: for $\phi$ fixed, there are $2^m$ possible $\chi$‘s, and we get a Vogan L-packet $\Pi_\phi = \Pi_\phi(G) \cup \Pi_\phi(G')$ where $\Pi_\phi(G)$ and $\Pi_\phi(G')$ each contain $2^{m-1}$ elements.

Question. When is $\pi(\phi,\chi)$ a supercuspidal representation?

For $n=2$ I had previously memorized the (already complicated) answer to this question, so you can imagine my pleasure when I learned that Moeglin found a simple conceptual criterion which works in general! To state her theorem, we need a little vocabulary.

Definition. A discrete parameter $\phi=\oplus_i \phi_i$ is without gaps if for every $\sigma \boxtimes [d]$ occurring among the $\phi_i$‘s with $d \geq 3$, then also $\sigma \boxtimes [d-2]$ occurs among the $\phi_i$‘s.

Definition. Suppose $\phi$ is without gaps. A character $\chi$ of the component group is alternating if for every pair $\sigma \boxtimes [d]$ and $\sigma \boxtimes [d-2]$ (with $d \geq 3$) occurring among the $\phi_i$‘s, $\chi(\sigma \boxtimes [d]) = - \chi (\sigma \boxtimes [d-2])$. Moreover we require that on every summand of the form $\sigma \boxtimes [2]$, we have $\chi(\sigma \boxtimes [2])=-1.$

Theorem (Moeglin). The representation $\pi(\phi,\chi)$ is supercuspidal iff $\phi$ is without gaps and $\chi$ is alternating.

Example 0. By definition, $\phi$ is supercuspidal if $d_i =1$ for all summands. In this case, $\phi$ is (vacuously) without gaps and every $\chi$ is (vacuously) alernating, so $\Pi_\phi$ consists entirely of supercuspidal representations. The converse – if $\Pi_\phi$ consists only of supercuspidals then necessarily $\phi$ is supercuspidal – is also immediate!

Example 1. Let $\sigma_2, \sigma_2':W_F \to \mathrm{SL}_2$ be distinct supercuspidal parameters. Then $\phi = \sigma_2 \oplus \sigma_2' \oplus \sigma_2' \boxtimes [3]:W_F \times \mathrm{SL}_2 \to \mathrm{Sp}_{10}$ is a discrete parameter without gaps, with component group of size 8. It is easy to see that four of the possible $\chi$‘s are alternating, and two of these are trivial on the center of $\mathrm{Sp}_{10}$. Thus, the packets $\Pi_\phi(\mathrm{SO}_{11})$ and $\Pi_\phi(\mathrm{SO}_{11}')$ each contain four elements, two of which are supercuspidal and two of which are non-supercuspidal.

Example 2. Let $\sigma_2:W_F \to \mathrm{SL}_2$ and $\sigma_3:W_F \to \mathrm{O}_3$ be supercuspidal parameters. Then $\phi = \sigma_2 \oplus \sigma_3 \boxtimes [2] \oplus \sigma_3 \boxtimes [4]:W_F \times \mathrm{SL}_2 \to \mathrm{Sp}_{20}$ is a discrete parameter without gaps, with component group again of size 8. Now only two of the possible $\chi$‘s are alternating, and one of these is trivial on the center of $\mathrm{Sp}_{20}$. Thus, the packets $\Pi_\phi(\mathrm{SO}_{21})$ and $\Pi_\phi(\mathrm{SO}_{21}')$ each contain four elements, one of which is supercuspidal and three of which are non-supercuspidal.

Example 3. Let $\tau:W_F \to \{ \pm 1 \}$ be a nontrivial character. Then $\phi = 1 \boxtimes [2] \oplus \tau \boxtimes [2]:W_F \times \mathrm{SL}_2 \to \mathrm{Sp}_{4}$ is a discrete parameter (vacuously) without gaps, with component group of size 4. Now only one of the possible $\chi$‘s is alternating, and it is trivial on the center of $\mathrm{Sp}_{4}$. Thus, the packets $\Pi_\phi(\mathrm{SO}_{5})$ and $\Pi_\phi(\mathrm{SO}_{5}')$ each contain two elements, with $\Pi_\phi(\mathrm{SO}_{5})$ containing one supercuspidal and $\Pi_\phi(\mathrm{SO}_{5}')$ containing no supercuspidals.

More generally, if $\phi$ is without gaps and all $d_i$‘s are even, then only one $\chi$ is alternating, so the packet $\Pi_\phi$ contains a single supercuspidal representation (which may be a representation of $G$ or $G'$ – both possibilities occur) swimming in a sea of discrete series representations.

References:

Colette Moeglin, Classification des series discretes pour certains groupes classiques p-adiques, 2006

Bin Xu, On the cuspidal support of discrete series for p-adic quasisplit Sp(N) and SO(N), 2015

## Report from Oberwolfach

Recently returned from a workshop on “Arithmetic of Shimura varieties.”

• The organizers did a great job choosing the speakers. For one thing, the overlap between speakers this time and speakers at the previous edition of this event was a singleton set, which I think is a reasonable choice. Moreover, the majority of the speakers were junior people, which is also totally reasonable. It was great to hear what everyone is doing.
• Best talks: Ana Caraiani, Teruhisa Koshikawa, Keerthi Madapusi, Sug Woo Shin, Joao Lourenco
• Best chaotic talk with amazingly strong theorems: Ian Gleason
• SWS has advised a very disproportionate number of Alexanders.
• The usual hike wasn’t possible, due to snow in the mountains. Ah well. Instead we hiked along a path parallel to the road. But there was still cake.
• The food was slightly better than usual: they didn’t serve the notorious bread casserole, and one dinner (the polenta thing) was actually really good.
• “I mean, you know Ben. He’s pretty unflappable. But, yeah… [redacted], uh… flaps him.”
• During the workshop, LM and I hit upon a conceptual explanation for Bernstein-Zelevinsky duality, which works both for group representations and for sheaves on $\mathrm{Bun}_G$, even when $\ell=p$! More on this later.
• “What was the motivation for this conjecture?” “The motivation was that it is true.”
• Some young people have extremely weird expectations for how the postdoc job market should work.
• The notion of “genericity” in various guises, and its relevance for controlling the cohomology of local and global Shimura varieties, was very much in the air. This came up in Caraiani and Koshikawa’s talks, and also in my (prepared but undelivered, see the first bullet above) talk. My handwritten notes are here, and may be of some interest. Conjectures 3 and 5, in particular, seem quite fun.
• Had some interesting conversations with VL about nearby cycles and related topics. Here’s a concrete question: can the results in this paper be adapted to etale cohomology? There are definite obstructions in positive characteristic related to Artin-Schreier sheaves, but in characteristic zero it should be ok.
• During the workshop, Ishimoto posted a beautiful paper completing Arthur’s results for inner forms of odd special orthogonal groups, at least for generic discrete parameters. I was vaguely sure for several years that this was the (only) missing ingredient in proving compatibility of the Fargues-Scholze LLC and the Arthur(-Ishimoto) LLC for $\mathrm{SO}_{2n+1}$ and its unique inner form. After reading this paper, and with some key assists from SWS and WTG, I now see how to prove this compatibility (at least over unramified extensions $F/\mathbf{Q}_p$ with $p>2$). It shouldn’t even take many pages to write down!
• On a related note, shortly before the workshop, Li-Huerta posted his amazing results comparing Genestier-Lafforgue and Fargues-Scholze in all generality!

As always, Oberwolfach remains one of my favorite places to do mathematics. Thank you to the organizers for putting together a wonderful workshop!

## Postdoc position at NUS

I’m looking to hire a postdoc here at NUS! This position is for two years, with the possibility of renewal for a third year, and carries no teaching duties. Ideally you will collaborate with me, but the job comes with near-total freedom to pursue your research. I also want to emphasize that Singapore is a beautiful country, with friendly people and amazing food, and it’s hard to imagine anyone regretting coming here for a few years.

Serious applicants, whose research interests are compatible with mine, are encouraged to apply via Mathjobs here. Although there is a quasi-official deadline of Jan. 31, in reality the position will stay open until I hire a suitable candidate, so late applications are welcome too.

## 2022 in review

Number of intercontinental moves: 1

Number of referee reports written: 6

Number of quick opinions written: Maybe 9 or 10?

Number of recommendation letters written: 6

Number of papers finished: 3

Percentage of those papers which use $\infty$-categories: 100%

Do I understand $\infty$-categories yet: of course not.

Number of papers accepted: 1

Number of papers rejected after a 20 month review process and a favorable referee report, on the basis of lazy and intellectually dishonest additional’ quick opinions: 1

Will I ever submit to that journal again: I doubt it!

Number of conferences attended: 4

Number of conferences attended which served cakes decorated in gold leaf: 1

Coolest things I read carefully this year:
-DC’s amazing new approach to Poincare duality: why struggle to define a trace map when you can define a cotrace map with elegance and panache? See here, and also here.
-DHKM’s beautiful and creative use of the Fargues-Scholze technology to settle a bunch of notorious foundational questions about modular representations of p-adic groups.
-LM’s thesis. Brutal and overwhelming, in the best possible way.

Did I get COVID this year: yes.

Pitchfork-style rating of my COVID experience: 4.3/10

Do I honestly feel, in this moment, like I had any good ideas this year: meh.

Research goals for 2023:
-figure out how to really’ picture the Drinfeld compactification of $\mathrm{Bun}_P$.
-formulate a precise categorical p-adic local Langlands conjecture.
-[10+ further items omitted due to vagueness]

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

## Small announcements

• After two years and many excellent talks, we’ve decided to officially end the RAMpAGe seminar. Bhargav Bhatt gave the very first talk, way back in June 2020 – now Bhargav has generously agreed to also give the final talk, on August 10 at 12 noon EST. See you there!
• In December I’ll move to a tenured position at the National University of Singapore. To my colleagues in Asia – I’m really looking forward to travelling more easily around the region, beginning new collaborations, and engaging with the arithmetic geometry community!

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 $1$.