## Report from Oberwolfach

In the first week of February, I attended an Oberwolfach workshop on Nonarchimedean geometry and applications. It was a great pleasure to attend a conference in person after such a long period of isolation. Thank you to the organizers for making this week so enjoyable! As usual, here are some scattered recollections from the workshop.

• Due to the hybrid nature of the workshop, the talks on several days didn’t start until 2 pm, presumably to accommodate participants in North America, with the final talk scheduled after dinner. This left a huge swath of unscheduled time, from 9 am until 2 pm, which actually turned out to be kind of great. I don’t know if it was the hunger for in-person interaction after 2 years of isolation, but people really seemed to take full advantage of this free time for vigorous discussion and collaboration. I actually liked this schedule better than the usual schedule.
• Best talks: Johannes Nicaise, Lucas Mann, Piotr Achinger, Yujie Xu, Ben Heuer.
• Worst talk: [redacted]
• Categorical Langlands for GL1 = Langlands for mice.
• Random Question 1 (via PS): Let $U \subset X$ be an open immersion of an affine scheme into a smooth projective variety. Is the complement $X \smallsetminus U$ an ample divisor on $X$?
• Random Question 1′ (via DC): Let $U \subset V$ be an open immersion of affine schemes. Is $U$ the nonvanishing locus of a section of an ample line bundle on $V$?
• One recurring theme throughout the week was the p-adic Simpson correspondence, with excellent talks from Ben Heuer and Matti Würthen. Here something quite amusing happened: in an informal conversation on Tuesday, Ben explained the complicated status of p-adic Simpon to me, and stressed that its most optimistic conjectural form isn’t actually written down, because no one wants to be the one to make a false conjecture. But then Matti Würthen did explicitly conjecture this exact statement in his Friday lecture! For the record, the hope is that for a smooth projective variety $X/ \mathbf{C}_p$, there is an equivalence of categories from $\mathbf{C}_p$-representations of $\pi_1^{\mathrm{et}}(X)$ towards semistable Higgs bundles on $X$ with vanishing Chern classes. As Ben stressed to me, this definitely fails if $\mathbf{C}_p$ is replaced with a larger algebraically closed nonarchimedean field. Hmm…
• One consequence of COVID measures is that seating for meals was not randomized as usual, but rather was fixed for the entire week. I was assigned to eat with Torsten Wedhorn, Bogdan Zavyalov, and François Loeser. This ended up being a really pleasant group to eat with! I already knew Torsten and Bogdan fairly well, but I’d never spoken with François before, and it turns out he’s a totally charming and delightful person. It was especially wonderful to hear him talk about his astonishing achievements in ultra long distance running. One memorable quote: “The first night without sleep is no problem. The second night is… interesting. And the third night… well… this I cannot recommend.”
• My relationship with Hochschild cohomology has gone from “???” to a vague understanding and a desire to learn more. Thanks to DC for some helpful explanations!
• There was much discussion among the younger participants about what Fargues’s categorical local Langlands conjecture should look like with mod-p coefficients (i.e. in the $\ell=p$ setting). Of course on the spectral side, one expects to see some category of quasicoherent or ind-coherent sheaves on the special fiber of the Emerton-Gee stack for $\hat{G}$. On the automorphic side, one should have some category of mod-p sheaves on $\mathrm{Bun}_G$, and the correct category should fall out of the general formalism developed by Mann in his thesis. One tantalizing fact, sketched out during some of these conversations, is that $\mathrm{Bun}_G$ is definitely $p$-cohomologically smooth (in a precise sense), not of dimension 0 as in the $\ell \neq p$ case, but of dimension equal to the dimension of $G(\mathbf{Q}_p)$ as a $p$-adic Lie group. It is surely no coincidence that this matches the expected dimension of the Emerton-Gee stack for $\hat{G}$.

Another tantalizing observation: the relationship between the Emerton-Gee stack and Wang-Erickson’s stack of Galois representations is perfectly analogous to the difference between the stacks $\mathrm{LocSys}_G$ and $\mathrm{LocSys}_{G}^{\mathrm{restr}}$ appearing in AGKRRV.

On the other hand, it also became clear that most of the analysis in Fargues-Scholze cannot carry over naively to the setting of p-adic coefficients, and that many of the crucial tools developed in their paper simply won’t help here. In particular, the magic charts $\pi_b: \mathcal{M}_b \to \mathrm{Bun}_G$ used by FS, which are $\ell$-cohomologically smooth for all primes $\ell \neq p$, are definitely NOT $p$-cohomologically smooth. This already fails for $G=\mathrm{GL}_2$. Likewise, their “strict Henselian” property should fail badly. New ideas are very much required!

## Fall roundup

Apologies for the lack of blogging. This has been an unusually busy fall.

• My student Linus Hamann has a website! Please go there and check out his beautiful preprints, especially his paper on comparing local Langlands correspondences for GSp4.
• There have been a lot of great papers this year, but I was especially struck by these gorgeous ideas from Teruhisa Koshikawa. Readers might recall that the seminal Caraiani-Scholze papers contain a fun part (p-adic geometry of Shimura varieties and their Hodge-Tate fibers, semiperversity of Hodge-Tate pushforwards) and a not fun part (arguments with the twisted stable trace formula and Shin’s stable trace formula for Igusa varieties). Koshikawa completely eliminates the not fun part, replacing it with an extremely clever use of the Fargues-Scholze machinery. Even in the setting of the CS papers, Koshikawa’s main theorem is stronger; moreover, his technique opens the door to a wide generalization of the CS vanishing results beyond the specific unitary Shimura varieties they treated. (Note for ambitious readers: The problem of working out these generalizations has already been “taken” by specific people.)
• Eagle-eyed readers of H.-Kaletha-Weinstein might’ve noticed that the entire paper depends crucially on a non-existent preprint cited as [GHW]. As discussed in a previous post, the point of GHW is to construct the functor $Rf_!$ in etale cohomology for certain stacky maps of Artin v-stacks, by adapting some machinery of Liu-Zheng which they built to solve the analogous problem in the setting of Artin stacks. Since the above-mentioned papers of Hamann and Koshikawa both depend directly on HKW, and thus indirectly on GHW, I’ve felt some increased pressure recently* to actually produce this paper!
However, I think this pressure helped push me past the final points of confusion in this project, and I’m pleased to report that after nearly 4 years of struggle, the details of GHW have finally come together. I’m cautiously optimistic that the paper will be publicly available within a few months. The arguments are an infernal mixture of delicate p-adic geometry and general $\infty$-categorical constructions. Actually, this is the most intense and frustrating project I’ve ever worked on. It will be good to finish it.
• As always, David Roberts offers a voice of clarity against the nonsense burbling out from the IUT cultists.

*Both from myself and from the referee for HKW.

## Comparing local Langlands correspondences

At least six people have independently asked me some variant of the question:

What are the prospects for showing that the Fargues-Scholze construction of L-parameters is compatible with other constructions of the local Langlands correspondence?

In this post I’ll briefly lay out the answer as I see it.

For reductive groups $G$ over finite extensions $F/\mathbf{Q}_p$, the situation is complicated, since the status of LLC is complicated.

1. $\mathrm{GL}_n$ and $D_{1/n}^{\times}$. Compatibility for these groups is known and already proved in Fargues-Scholze, and follows from the realization of local Langlands and local Jacquet-Langlands in the cohomology of the Lubin-Tate tower.
2. Any inner form of $\mathrm{GL}_n$. Compatibility here is Theorem 1.0.3 in H.-Kaletha-Weinstein.
3. $\mathrm{SL}_n$ and inner forms. Compatibility should follow from the previous two points, but I guess it’s not completely trivial. Someone should write it down.
4. $\mathrm{GSp}_4$ and $\mathrm{Sp}_4$, and their unique inner forms. Compatibility for these groups has been proved by my student Linus Hamann. His preprint should be available very soon, and I’ll write a detailed blog post about it at that time. The arguments here rely on a number of special features of the group $\mathrm{GSp}_4$.
5. Split $\mathrm{SO}_{2n+1}$ and closely related groups. Partial results here are definitely possible by extending Hamann’s arguments, but it’s not clear to me whether complete results can be expected. I’ll say more about this when I write about Hamann’s paper.
6. Unitary groups. Partial results should be possible by combining some aspects of Hamann’s methods with recent works of Nguyen and Bertoloni-Meli–Nguyen.
7. $\mathrm{GSp}_{2n}$ and $\mathrm{Sp}_{2n}$ and their inner forms, $n>2$. This seems out of reach.
8. Even special orthogonal groups. I’m frankly confused about what’s going on here. Is there even an unambiguous LLC? In any case, this also seems hard.
9. Exceptional groups. There’s no “other” LLC here. Go home. (OK, for $G_2$ there’s a very cool recent paper of Harris-Khare-Thorne.)
10. General groups splitting over a tame extension, $p$ not too small. Here Kaletha has given a general construction which attaches a supercuspidal L-packet to any supercuspidal L-parameter. Compatibility of this construction with Fargues-Scholze might be approachable by purely local methods, but it seems to require substantial new ideas. An extremely weak partial result – constancy on Kaletha’s packets of the FS map from reps to L-parameters – is probably within reach, using the main results in H.-Kaletha-Weinstein.

The key point in many of the above situations is the following. Let’s say a group $G$ is accessible if it admits a geometric conjugacy class of minuscule cocharacters $\mu$ such that
1. The pair $(G,\mu)$ is totally Hodge-Newton reducible in the sense of Chen-Fargues-Shen.
2. Any L-parameter $\varphi: W_F \to \phantom{}^L G$ can be recovered up to isomorphism from the composition $r_{\mu} \circ \varphi$. (In practice one asks for slightly weaker versions of this.)
3. The local Shimura varieties attached to the local Shimura datum $(G,\mu,b)$ (with $b \in B(G,\mu)$ the unique basic element) uniformize the basic locus in a global Shimura variety of abelian type.

For groups satisfying this condition, there is hope. Very roughly, condition 2. implies that the FS construction is incarnated in the cohomology of a single local Shimura variety, whose cohomology can also be tightly related to the cohomology of a global Shimura variety using conditions 1. and 3. One then needs to know enough about the cohomology of these global Shimura varieties, namely that it realizes the “other” LLC you care about. Of course, this short outline veils substantial technical difficulties.

It turns out that $\mathrm{GL}_{n}$, $\mathrm{GU}_n$, $\mathrm{GSp}_4$, and $\mathrm{SO}_{2n+1}$ are all accessible, and this accounts for the definitive results in scenarios 1.-4. above and my optimism in scenarios 5.-6. On the other hand, $\mathrm{GSp}_{2n}$ is not accessible for $n>2$, and neither is $\mathrm{SO}_{2n}$ for $n>3$, and no exceptional groups are accessible. Hence my pessimism in scenarios 7.-9.

For reductive groups over finite extensions $F/\mathbf{F}_{p}((t))$, the situation is completely different. Here Genestier-Lafforgue have constructed a local Langlands correspondence for all groups, uniquely characterized by its compatibility with V. Lafforgue’s construction of global Langlands parameters. It is an extremely attractive problem to compare the Genestier-Lafforgue LLC with the Fargues-Scholze LLC. This should absolutely be within reach! After all, both constructions are realized in the cohomology of moduli spaces of shtukas, so the only “real” task should be to physically relate the moduli spaces of shtukas used by GL with those used by FS. This is probably not trivial: the spaces used by FS are local and totally canonical, while those used by GL seem to depend on a globalization and some auxiliary choices in a messy way. Nevertheless, I’d be surprised if this comparison is still an open problem two years from now.

## H.-Kaletha-Weinstein study guide and FAQ

Tasho Kaletha, Jared Weinstein and I have just posted a joint paper, On the Kottwitz conjecture for local shtuka spaces. This is a heavily revised and rewritten version of a preprint by Tasho and Jared which has been available since 2017 (henceforth “KW”).

Is the main result here the same as in KW?
More or less, yes. The main theorem confirms the Kottwitz conjecture for the cohomology of moduli spaces of local shtukas, ignoring the Weil group action and allowing for an “error term” consisting of a non-elliptic virtual representation.

Is the rough idea of the proof still the same as in KW?
Yes: the idea is still to compute the cohomology by applying a suitable Lefschetz-Verdier trace formula, and explicitly computing the “local terms” at all of the elliptic fixed points.

OK, what’s new, then?
Well, basically all of the details are different. First and foremost, the discussion of the relevant Lefschetz-Verdier trace formula for v-stacks has been completely rewritten, using Lu-Zheng’s magical formalism of symmetric monoidal 2-categories of cohomological correspondences. This leads to huge conceptual simplifications, both in the statements and the proofs. Actually, we go beyond the formalism of Lu-Zheng, introducing certain 2-categories of “based cohomological correspondences”. This formalism play a crucial role in the proof of Theorem 4.5.3, which is a key ingredient in the proof of the main theorem. It’s also hard to imagine proving Theorem 4.6.1 without this formalism.

Secondly, a key calculation of local terms on the $B_{\mathrm{dR}}$-affine Grassmannian, Theorem 5.1.3, now has a completely different and correct (!) proof, via a degeneration to a similar calculation on the Witt vector affine Grassmannian. This degeneration argument relies on the observation that the local terms appearing in the Lefschetz-Verdier trace formula are compatible with any base change, Proposition 5.3.1; this seems to be a new observation even for schemes. (My first proof of Proposition 5.3.1 involved checking the commutativity of 500 diagrams, but eventually I realized that it follows immediately from the Lu-Zheng formalism!) Once we’ve degenerated to the Witt vector affine Grassmannian, we make an intriguing global-to-local argument using the trace formula and known properties of the weight functors in geometric Satake, and relying crucially on a recent result of Varshavsky.

There are some other notable improvements:

• All assumptions in KW of the form “assume that some representation admits an invariant $\overline{\mathbf{Z}_{\ell}}$-lattice” have now been eliminated. This is far from formal, since we definitely don’t have a 6-functor formalism with $\overline{\mathbf{Q}_{\ell}}$-coefficients at hand. Instead, the idea is to reduce “by a continuity argument” to the case where things do admit lattices. The key ingredient here is Theorem 6.5.4, formerly a conjecture of Taylor, which should have lots of other applications.
• The final part of the main theorem, giving practical criteria for the error term to vanish, is new.
• Theorem 1.0.3 is new.

On a practical note, the Fargues-Scholze paper is now finished and available, so we can properly build on the powerful machinery and results in that paper.

Finally, the margins are no longer gigantic.

Sure. The main Theorem 1.0.2/6.5.1 in the paper is essentially a combination of two separate results, Theorem 6.5.2 and Theorem 3.2.9. Theorem 3.2.9 is conditional on the refined local Langlands correspondence, and the material needed to formulate and prove this result is developed in Chapter 3 and Appendix A. The bulk of the paper (sections 4-6 and Appendices B-C) is devoted to building up enough material to prove Theorem 6.5.2.

Each of sections 3, 4, and 5 is fairly self-contained, although section 5 depends slightly on section 4. Section 6 builds heavily on sections 4 and 5, and also on [FS21]. Section 5 is actually devoted to proving a single result, Theorem 5.1.3 (and its reinterpretation Theorem 5.1.4), and can be treated as a black box.

One thing you could do is read the introduction, then jump to Proposition 6.4.7 and work your way backwards as needed until all the notation makes sense. This proposition is the technical heart of the paper. The key inputs into the proof of Proposition 6.4.7 are Theorem 5.1.4, Theorem 4.5.3, and Corollary 4.3.8. The use of “the trace formula” in this whole argument is actually somewhat hidden: it is entirely encoded in the equality sign labelled “Cor. 4.3.8” at the bottom of p. 66.

From Proposition 6.4.7, Theorem 6.4.9 (a weak form of Theorem 6.5.2) follows quickly, and then Theorem 6.5.2 follows from Theorem 6.4.9 by a continuity argument which relies crucially on Theorem 6.5.4.

This sounds cool, but you didn’t punt any key ingredients into a separate and currently nonexistent paper, did you?
I’m glad you asked. In working out the trace formula formalism in enough generality to give a conceptual proof of the main theorem, it became clear at some point that we needed the existence and expected properties of the functors $Rf_!$ and $Rf^!$ in etale cohomology for “smooth-locally nice” maps $f$ between Artin v-stacks. For maps which are actually “nice”, and in particular non-stacky, these functors were constructed by Scholze, but extending them to the setting of stacky maps is not formal. Already in the analogous setting of schemes versus Artin stacks, these functors for Artin stacks were only constructed recently by Liu-Zheng, making heavy use of $\infty$-categories. In a forthcoming paper, Jared and I together with my postdoc Dan Gulotta will explain how to construct $Rf_!$ and $Rf^!$ for smooth-locally nice maps of Artin v-stacks. The arguments will very closely follow the arguments of Liu-Zheng, with the exception of one key non-formal piece of input (which took me three years of intermittent work).

Why do you only prove an explicit formula for the virtual character of $\mathrm{Mant}_{b,\mu}(\rho)$ restricted to elliptic elements of $G(F)$? What’s so special about elliptic elements in this context?
The fixed points of elliptic elements lie in the “admissible locus” inside the relevant closed Schubert cell. This is discussed in detail in the introduction.

Fine, but maybe there’s still an obvious explicit formula for the virtual character of $\mathrm{Mant}_{b,\mu}(\rho)$ at any strongly regular semisimple element.
Here’s an example to illustrate why I think this is a hard problem. I’ll be slightly heuristic about the details; if you want more precision, please leave a comment.

Let’s take $G= \mathrm{GL}_2$ and $\mu=(1,0)$, so we’re in the Lubin-Tate/Drinfeld setting with $G_b(F)=D^\times$ the units in the quaternion algebra over $F$. Let $\rho$ be the trivial representation of $D^\times$. Then $\mathrm{Mant}_{b,\mu}(\rho) = \mathrm{St}-\mathbf{1}$ as a virtual representation of $G(F)$, by an old calculation of Schneider-Stuhler. Note that $\mathrm{St}+\mathbf{1}$ is a principal series representation, hence non-elliptic, so the virtual character of $\mathrm{St}-\mathbf{1}$ on elliptic elements of $G(F)$ is the constant function $-2$. This matches perfectly with the fact that any elliptic $g\in G(F)$ has two fixed points in $\mathbf{P}^1$, both contained in $\Omega^{1}$, and the “naive” local terms of the relevant sheaf $j_!\mathbf{Z}_{\ell}[1]$ at both these points are $-1$. Here $j:\Omega^1 \to \mathbf{P}^1$ is the evident open immersion.

On the other hand, if $g \in G(F)$ is regular semisimple and nonelliptic, then it’s conjugate to some $t=\mathrm{diag}(t_1,t_2)$ with $t_1 \neq t_2 \in F^\times$. In this case there are still two fixed points, but they both lie in the “boundary” $\mathbf{P}^1-\Omega^{1}$. Since $j_!\mathbf{Z}_{\ell}[1]$ restricted to the boundary is identically zero, its naive local terms at both fixed points are zero. Nevertheless, the trace formula still works, so the contribution of the true local terms at these two points must add up to the virtual character of $\mathrm{St}-\mathbf{1}$ evaluated at $t$. This character value can be computed explicitly by van Dijk’s formula, and turns out to be $-2+\tfrac{|t_1| + |t_2|}{|t_1-t_2|}$. So this slightly strange expression needs to emerge from the sum of these two local terms.

Putting these observations together, we see that in this example, the true local terms do not equal the naive local terms at the non-elliptic fixed points. Morally, the analysis in Section 3 of the paper can and should be interpreted as an unravelling of naive local terms (at elliptic fixed points). Hopefully this makes the difficulty clear.

Can you give an example where the error term in Theorem 1.0.2 is nonzero?
Sure, look at the two-dimensional Lubin-Tate local Shimura datum and take $\rho$ trivial again, so $\rho \in \Pi_{\phi}(G_b)$ with $\phi$ the Steinberg parameter. Then $\mathrm{Mant}_{b,\mu}(\rho) = \mathrm{St}-\mathbf{1}$ as discussed in the previous answer, while the sum on the right-hand side in Theorem 1.0.2 is just $2 \mathrm{St}$. So in this case $\mathrm{err}= -(\mathrm{St}+\mathbf{1})$ is, up to sign, a reducible principal series representation.

It would be very interesting to formulate an extension of the Kottwitz conjecture which covered all discrete L-parameters. Since the Kottwitz conjecture is best regarded as a reflection of the Hecke eigensheaf property in Fargues’s conjecture, this is closely related to understanding Fargues’s conjecture for general discrete parameters.

It seems like the Kottwitz conjecture should also make sense for local shtukas over Laurent series fields. Why doesn’t the paper cover this case too?
Good question! The short answer is that all of the p-adic geometry should work out uniformly over Laurent series fields and over finite extensions of $\mathbf{Q}_p$, and indeed Fargues-Scholze handles these two situations uniformly. However, the literature on local Langlands and harmonic analysis for reductive groups over positive characteristic local fields case is pretty thin. This is the main reason we stuck to the mixed characteristic case. For more on the state of the literature, take a look at section 1.6 in Kal16a.

Do the methods of this paper give any information about $\mathrm{Mant}_{b,\mu}$ for non-basic b?
Yes: If $b$ is non-basic, or $b$ is basic and $\rho$ is parabolically induced, our methods can be applied to prove that $\mathrm{Mant}_{b,\mu}(\rho)$ is always a virtual combination of representations induced from proper parabolic subgroups of $G$. This is a soft form of the Harris-Viehmann conjecture. The details of this argument will be given elsewhere (in the sense of Deligne-Mumford).

Any surprising subtleties?
Lemma 6.5.5 was a bracing experience. My first “proof” was absolute nonsense, due to some serious misconceptions I had about harmonic analysis on p-adic groups – big thanks to JT for setting me straight. My second proof didn’t work because I cited a result in a famous article (>100 mathscinet citations) whose proof turns out to be fallacious – big thanks to JFD for setting me straight. Tasho and I devised the current proof, which in hindsight is actually a very natural argument. (I was hoping, apparently unreasonably, for a shortcut.)

Closing thoughts:
-At least for me, the proof of Theorem 6.5.4 is a very striking illustration of the raw Wagnerian power of the machinery developed in [FS21]. This result was totally out of reach a few years ago, but with the machinery of [FS21] in hand, the proof is less than a page! Note that this argument crucially uses the $D_{\mathrm{lis}}$ and Hecke operator formalism of [FS21] with general coefficient rings.
-Let me again emphasize how heavily the paper depends on the recent preprints [LZ20], [Var20], and (of course) [FS21].
-The paper depends, more significantly than most, on a choice of isomorphism $\mathbf{C} \simeq \overline{\mathbf{Q}_{\ell}}$. It’s naturally of interest to wonder how everything depends on this choice. Imai has formulated a conjectural answer to this question. Of course, “independence of $\ell$” results in the etale cohomology of diamonds are probably very hard.

## Remarks on Fargues-Scholze, part 2

Today I want to talk about section VII.3 in the manuscript. Here they define and study a functor $f_{\natural}$ on solid sheaves which is left adjoint to the usual pullback functor. But why is this called relative homology?

To explain this name, you have to remember that in the usual formalism of etale cohomology, homology is realized as the compactly supported cohomology of the dualizing complex. Symbolically, if $X$ is a variety with structure map $f:X \to \mathrm{Spec} k$, then the homology of $X$ is given by $Rf_! Rf^! \Lambda$. Now, you might ask whether this works in families: if $f:X \to Y$ is some map of varieties, maybe I can find some complex on $Y$ whose stalk at $y$ realizes the homology of $X_y$? If the constant sheaf is $f$-ULA, then formation of $Rf^! \Lambda$ commutes with any base change, and exactly the same formula works, but in general there is no naive sheaf with this property.

The punchline now is that $f_{\natural} \Lambda$ does have this property: when the constant sheaf is $f$-ULA (e.g. if $Y$ is a point) it agrees with $Rf_! Rf^! \Lambda$ by Proposition VII.5.2, and its formation commutes with arbitrary base change, so it really does give a complex on $Y$ whose stalks realize the homology of the fibers of $f$. The only twist is that $f_{\natural}\Lambda$ is a solid sheaf in general, not a classical etale sheaf.

## Remarks on Fargues-Scholze

The Fargues-Scholze geometrization paper is available! In this post, and probably some future posts also, I’ll make some random comments on this paper. These won’t mean anything unless you’ve read (at the very least) the beautifully written introduction to the paper. To be clear, I have nothing of substance to say about the “big picture” – these will be purely technical remarks.

First of all, at the bottom of p. 324, one finds the slightly cryptic claim that although there are no general $i_!$ functors in the $D_{lis}$ setting (for $i$ a locally closed immersion), one can define functors $i^{b}_!$ in the $D_{lis}$ setting, where $i^b: \mathrm{Bun}_{G}^b \to \mathrm{Bun}_G$ is the inclusion of any Harder-Narasimhan stratum into $\mathrm{Bun}_G$. This is stated without proof. However, if you read carefully, you will notice that these lisse $i^{b}_!$ functors are actually used in the paper, in the proof of Proposition VII.7.6. So maybe it’s worth saying something about how to construct them.

To build $i^{b}_!$ in the $D_{lis}$ setting, factor $i^b$ as the composition $\mathrm{Bun}_{G}^b \overset{i}{\to} \mathrm{Bun}_{G}^{ \leq b} \overset{j}{\to} \mathrm{Bun}_G$. Here $\mathrm{Bun}_{G}^{ \leq b}$ is the open substack of bundles which are “more semistable” than $\mathcal{E}_b$. It will also be convenient to write $\mathrm{Bun}_{G}^{ < b} = \mathrm{Bun}_{G}^{ \leq b} - \mathrm{Bun}_{G}^{ b}$. Note that $i$ is a closed immersion, and $j$ is an open immersion, so $j_! = j_{\natural}$ clearly preserves $D_{lis}$. The subtlety is in making sense of $i_!$, since then we can write $i_{!}^{b} = j_! i_!$ as usual.

For $i_!$, we need the local chart $\pi_{b}: \mathcal{M}_b \to \mathrm{Bun}_{G}^{\leq b}$ and its punctured version $\pi_{b}^\circ : \mathcal{M}_{b}^\circ = \mathcal{M}_{b} \times_{\mathrm{Bun}_{G}^{\leq b}} \mathrm{Bun}_{G}^{< b} \to \mathrm{Bun}_{G}^{\leq b}$. Recall that these charts also come with compatible maps $q_b: \mathcal{M}_b \to [\ast / G_b(E)]$ and $q_b^{\circ}: \mathcal{M}_{b}^{\circ} \to [\ast / G_b(E)]$. Then for any $A \in D_{lis}(\mathrm{Bun}_{G}^{b},\Lambda) \cong D_{lis}([\ast / G_b(E)],\Lambda)$, the correct definition turns out to be

$i_! A = \mathrm{Cone}(\pi_{b \natural}^{\circ} q_{b}^{\circ \ast}A \to \pi_{b \natural} q_{b}^{\ast}A)\,\,\,\,(1)$.

The point here is that in the lisse world, the only pushforward functors which come for free are the functors $f_{\natural}$ for cohomologically smooth maps $f$. Since $\pi_{b}$ and $\pi_{b}^{\circ}$ are cohomologically smooth – one of the hardest theorems in the paper! – the above construction preserves $D_{lis}$. Moreover, it’s easy to check that the formula above has the right properties. Indeed, the *-restriction of the RHS of (1) to $\mathrm{Bun}_{G}^{b}$ is just $A$, by Proposition VII.7.2, while its complementary restriction to $\mathrm{Bun}_{G}^{ < b}$ clearly vanishes.