6/6 On the local Langlands conjectures for reductive groups over p-adic fields
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Scholze, Peter
Consider a reductive group G over a p-adic field F. The local Langlands conjecture relates the irreducible smooth representations of G(F) with the set of (local) L-parameters, which are maps from the Weil group of F to the L-group of G; refinements of the conjecture relate the fibres of this map with the automorphism group of the L-parameter. Based on ideas from V. Lafforgue's work in the global function field case, I outlined a strategy for attaching (semisimple) L-parameters to irreducible smooth representations of G(F) in my 2014 Berkeley course. At the same time and place, L. Fargues formulated a conjecture relating the local Langlands conjecture with a geometric Langlands conjecture on the Fargues-Fontaine curve. The goal of this course will be to discuss some of the developments since then. On the foundational side, this concerns basics on the etale cohomology of diamonds including smooth and proper base change and Poincare duality, leading up to a good notion of "constructible" sheaves on the stack of G-bundles on the Fargues-Fontaine curve. On the applied side, this concerns the construction of (semisimple) L-parameters, the conjecture of Harris (as modified by Viehmann) on the cohomology of non-basic Rapoport-Zink spaces, and the conjecture of Kottwitz on the cohomology of basic Rapoport-Zink spaces.
 So let's recall the setup from yesterday, no not yesterday, Wednesday. So we have G or Qp a reductive group. And at some point I was assumed for simplicity that it's quasi-split. Also, I fixed some algebraically closed base field. So I worked in a geometric setting. And I was considering the stack bun G. I defined on the category of perfectoid spaces over K. Taking any test object S to group out of G torsors on XS, where this was a relative fact from 10 curve. And so I stated last time a theorem to the effect that in this suitable interpretation of these words, this is whose all-tint stack of dimension 0. And so then I fixed my coefficients. So coefficients were Oe mod L to the n where E over Q also find an extension. Of course L is not p. And so we had a variable reductive function. So this category of chiefs of lambda modules on bun G to itself. Thanks. I believe that conservative asks this question about Gavir. So it maps it into the dualizing complex of bun G. And then we had the definition that F is reflexive if F maps isomorphically to its double-redued dual. And this was our substitute for the notion of constructible sheaves on the sky where the usual notion of constructibility would not be the good thing to consider. And the main theorem I stated last time, which still had a star because it still depends on one conjecture, says the following that you can completely understand when a sheave is reflexive. So F is G at bun G. And there is reflexive if and only if. For all points and as I said last time points on the stack are classified by Codd-Witzel set B of G of isocrystals and all comological degrees I. So this B sum of corresponds to geometric point XB bar to bun G. For all such guys and all integers, if I look at the stock of this complex really at XB bar, I can look at all the homology groups. These are all representations of the corresponding automorphism group of this B, which is this, well at least of G's quasi-split, it's this inner form of the levy of G, GB of QP, if this isn't a misaligned representation. And so today I want to explain the proof of this theorem. And so the proof is by induction on bun G, which is somehow bounded by some fixed B. Okay, so it's somehow starts proving a similar result just on the same stable locus and then you go further and further out into the non-stable locus, one stratum at a time. And so the abstract theorem behind this stack has this great set of points. Okay, the question is whether there is some special case of a more general theorem applying to more general states and I don't think so. I think it's really something very special for bun G. Yeah, so I proved the same theorem for all open subsets of bun G. And what about things which are in subsets, a tolerable bun G? Things which are tolerable bun G might be too... So for things which are very close to bun G it might still be okay, but... Okay, I can tell. Certainly the proof makes use of very special properties of bun G. So you start on the same stable locus. What I've already said last time implicitly is that it's a following theorem which for GLN is 2 2 kth volume and u. For general G, 2 2 5, it says that if you look at bun G and then you look at the same stable locus, then this decomposes into a disjoint union over all basic B of a classifying space for the corresponding inner form of G. Okay. And so we have to understand what is the ad of such a guy is. And for this let me state the following proposition. So in fact it falls in the following generality that if you have H in some GLN QP, a closed-stop group, so for example G of QP for any linear group G or an open-subgroup zero or whatever, then you can try to understand what this de-ed of the point mod H is. So the proposition is that as it de-ed of the point mod H coefficients in lambda is really just the derived category of smooth H representations on lambda modules. Okay, RER was late. Smooth H representations on lambda modules. And this is the equivalent. And it was a fixed choice of H measure. The aduality gets identified with smooth duality. And which takes any pi in absolute to the homomorphisms from pi into lambda, but then it takes a smooth part of this. And then this passes to the derived category. I mean this sounds very reasonable, but actually verifying this is a little bit, there's some trouble, but okay, you can do it. In particular, the fact that you really get the smooth part here only is related to some finiteness results for the homology of smooth diamonds. So there is something hidden, some work hidden in the statement. Also it looks very reasonable. Not true that the B-in categories are different. Which are B-in categories? I mean this is not a priori the derived category of an B-in category, although somehow a posterior. It is here. Because here and here not in the setup where, so if this was some diamond, some locally spatial diamond, then this N-test some find comodical dimension, then this the ad-wood automatically be the derived category of a tall sheath. But here we're not in the setup. In particular, sheaths on this are somehow, oh, you have to first find a smooth cover of this guy by a diamond and so on. A priori unraveling what this really is, this is a bit of a trouble. Anyway, once you have checked this, you find that in particular if you try to understand what reflexive sheaths are, then those are the ones where passing through the double-smooth dual is an isomorphism. This reflexive if and only if for I and Z, I locate H, I of F at a somewhat geometric point, which then becomes a representation of H. That's because it can translate to the same question about smooth representations. And this factor is exact, so understanding the radiality, you can understand it in each degree individually. And then the condition that the double-smooth dual is the same thing, means that on all the fixed vectors under open subgroups, it must be finite dimensional. It must be reflexive lambda module, which means it's finite. There is no restriction on degrees. So that's good. So now for, so, so, finishes in a stable case. So now we need to, so for the induction, let's fix some B and B of G, not same as stable, not basic. And so then we have this inclusion from some B, bun G, which is less than B, into bun G, which is less than equal to B. And let me also already give a name to the complementary closed. We have some I, bun G, B, meaning equal to B. And so we assume that the theorem is known, I mean really the obvious variant of the theorem, known for bun G less than B. Okay, so, so what do I have to do? So we need to prove the following results. One is a finiteness result, which says the following. So assume you have some F, which lives on bun G less than B, and it's reflexive. So I take, push this forward to the whole guy and then restrict to the new closed stratum. So then I get a sheaf on bun G equal to B. And we need to know that this is reflexive. And you know that it is etal, in de etal by general single-signal. No, because all functors are by definition, functors on de etal. So you modify them to be de etal. Right, for non-fazer compact immersions I said that I define this R.J. Lohr star to be the edge joint of J. Upper star. So R.J. Lohr star, if I did it on V-sheafs, would not lie in the de etal, but I just reflected back into the de etal. By using the right adjoint to the inclusion. This means it's complicated to calculate it. You must do it this way. But circumvutes with smooth space change. And the second thing you need to prove is a duality result. And if F lies on this de etal bun G, less than B, less than reflexive. Okay, formally the statements look very similar now, but I will try to explain that the first thing really is a fineness result, whereas the second thing is some kind of local duality result. That will become, I think, clearer when I go into the proofs. So why is this what we want to know? So that's why does this imply, why does this give the induction step? Well, you already understand reflexive shoes on the two strata. So we understand reflexive shoes on the strata. And this bun G equal to B is this point moduloose group, which has these kind of automorphisms of EV, where this was an extension of this periodic group, GB of QP, by a connected group. And so you can prove the usual thing that connected groups don't actually interfere with the Alladek sheaf theory. So actually the de etal on this bun G equal to B lambda is equivalent to de etal of just the point mod GB of QP lambda. And so this means that on the strata you again can apply the results it says that these are given by smooth representation. Do you mean a connected group has no homology or something? Yeah, it's a connected group has no homology. Yeah, so it has no homology. You don't use the semi direct order structure, you just need an extension. Do I need the semi direct product structure? I would think I just need an extension. I have a semi direct product in case I need it. So we understand on the strata. So then say for the theorem we need to prove two directions. So which direction should I do first? So assume first that say G in de etal bun G is equal to B lambda has admissible stocks in the sense of theorem. And then we have a triangle like so. And this is reflexive by induction. And then the whole thing is reflexive by two. This is reflexive by the case of bun G equal to B. And then the whole thing is reflexive as is proper to closed immersion. And we had seen that inclusions by proper maps preserve flexibility. So extensions of reflexive are reflexive and so we see that G is reflexive. And so conversely, assume that G is reflexive. So what do we want to see? We want to see it has good stocks. And so you do the same thing. So the left thing, so same exact sequence implies that this thing is reflexive. Same triangle. But then because this is a closed immersion, this actually implies that I opposite of G is reflexive. I mean so on the open part, we already know that it must have good stocks admissible stocks. And so this means it has admissible stock. So for the argument of hardness, it didn't quite appear, but it will be a step in getting to the duality. But so because of this identification of reflexive sheaves here again with admissible representations of Jb of Qp, this statement about this reflexivity is really just saying that something is an admissible representation. So in this sense, it's really a finiteness result. Whereas this reflexivity on this stack here is a much more geometric statement. Much more like an actual duality statement. And so, right, I think I moved the wrong page here. So we want to do these things and for this we use a chart to compute anything. So this chart is MB, which was an MB tilde, much G of Qp, y sub pi B. And so where MB, so B, so I said this somewhere, but let me now make say it explicitly again. I assume that G is quasi-stimulus implicitly. So then B is actually the image of some BM, BM basic, where M is contained in P, contained in G, this is a levy. And this is a parabolic encoding some slope filtration. And what is MB tilde that classifies P twosers? E plus an isomorphism of the reduction to M with the bundle corresponding to this basic element. Then this maps via the GB of Qp, which is the same thing as the MBM Qp twosers. To MB, which is a set of P twosers, E such that E times Pm is fiber-wise isomorphic, this point-wise isomorphic to EN. And this maps to bungee, E while sending E to the pushout. And the conjecture, which is this conjecture that explains this star, its main theorem is that this map from MB to bungee is equal to B, which is open to bungee is L-comologicalism. And we can prove this for G or N. And MB itself is also M code. And MB itself is also a smooth out instead, as a consequence. And because the map is smooth and bungee itself is smooth, this guy needs to be smooth. But this is actually unconjectural, so you can directly show that this is smooth. So now we have the following diagram that this maps to the bungee. The pre-imature I will denote by MbCirc, EefI, bungee B, let's call this I prime. What you get here is a fiber is actually just a point modulo Gb of Qp. So effectively what you're doing over this stratum is you fix the splitting of the hard aneurysm infiltration. And then you get this as the automorphism group. And you also have on top of this a similar diagram. We can call this point XB, it's just a copy of spark K. So this start to be always on the left and close start to on the right. Is that conflicting with standard usage? Yeah, this is the open part, this is the close part. I mean I use I and J in the usual way, I think. All right. So we have this picture. And so let's start with the finiteness. So by using the conjecture, smooth space change plus some unraveling of the formalism, plus, this part of the unraveling of the formalism, that this Mb tilde is some kind of strictly Hanselian space. So it behaves a little bit like power series over an algebraic closed field. What you can prove is the following, that if you look at the sections on point K, where I say K and GB of Qp is an open pro-P subgroup, pro-P just for simplicity essentially, but makes things slightly easier. So I mean I fixed here my F and D at all, bun G less than D lambda, and I should put the pole back on the pi B here. So if I pull it back under pi B here, I get a shift here, and then whether I do push forward here and pull back here or upstairs, this doesn't matter. And so I want to understand what happens if I push forward here and pull back here, and then I want to know that this is an admissible representation here. So what I want to know is that these kinds of global sections, if I go to the point mod K, that this is finite in each degree. And you can compute this just as a homology of this open part mod K. So we call that MB tilde over MB was this torsion in this group, so you can in particular divide by this group. That's still some kind of stacky object anyway. And with coefficients, where does this do reflexive? So the homology of this guy, this is called pi B tilde, the star takes a K variance, where pi B tilde is a composite projection, not smooth. So this description is better because here we still have a reflexive sheath. Here we're pulling back too far, this is not reflexive anymore. What's nice about this formula is that it tells you that if you do this thing, you push forward and you pull back and you know that what you get is just some smooth representation of the script JB of QP. This smooth representation is just this thing here, because we've identified the K in variance for any K with the K in variance here. So what you want to prove really is that this homology here is an admissible representation of this group. What is this thing with the not smooth just you have the conjecture that the map is smooth? Well, okay mate. What is pi B? Pi B is smooth, not pi B tilde. So the torsion is not smooth? Yeah, so coefficients under the profaned group are not smooth because the fibers are not smooth, profaned sets are not smooth. There was this other thing that if you have something smooth and then you divide, then the quotient is still smooth over the base. But that's a different statement. Okay, so the torsions under profaned group are not? No. Okay, so this gives you a way of computing what you want to compute. And so now we need to prove, so we need to show that this is finite. In each homological degree. And so this, there are two ingredients. So by the way, when you write K acting on this gamma, there is required some foundation. So in general if a group acts on the top, it's not the script group acting on the top, so you need some to know that it takes the other definition. Okay, so I will use two things. Well, actually no. I claim that in each homological degree, the map is an injection, as I can just say, and the image will always be just the invariance. Forget about any topology of this group acting on this set. I mean, I think it's just. And is it also a coherent couple? Everything inside is a, okay. Let me say what these objects are. So the open part of this nb tilde is actually a quasi-compact separated spatial, well spatial implies quasi-compact diamond. Whereas this quotient here would again be a stack. In the sense, this is nicer here. And so this means that you can compute commodgy, I also find it dimensional. So compute commodgy as a direct limit of chess commodgy over finite. Over covers, over top covers. With finitely many. If it was not quasi-compact, then to do this, you would have to use infinitely many things. And then if you write down the chess complex, it's infinite products and they become very big. But if you use this information carefully. But why is it not hypercovered for homology? Okay, so there's this thing that f might be unbounded, but actually for the statement, you can reduce to the case where f is bounded. Because you know by induction that canonical truncations preserve reflexivity. By induction, canonical truncations preserve reflexivity because the condition of being reflexive is a condition of homology sheaves. And so this means and each individual homology group of the sky by some finite dimensionality only depends on some bounded piece of the complex. So you can reduce the case where this is actually just a sheave. In which case you are allowed to just use a direct limit of usual chess commodgy. I don't know what I'm saying, but there is a result when the coherent top was the homology is completely general by hypercover and not by... If you want you can also do hyper. No, I do have an analog of Arthens theorem on joints of Ansel ring that allow you to do check homology or... So maybe, yeah, so maybe let's say hyper. It's all hypercovered. But still will find many terms in each degree. And also you know that this sheave f is not too big. For example, it has countable dimensional stocks. And then you have to sit down and do it carefully. But it implies that all the homology groups, they are at most countable. So in particular you have to check that there are not too many etal things and so on. But the critical part that makes everything work is the quality of the complexity here. You have a countable co-final system. Yeah, you have a countable co-final system of such a class. And then the second thing is you apply punk radially. So because f is reflexive, so in particular it's the dual of something else, of some f prime. And then this implies that actually the homology of this guy with coefficients in this guy is a dual of the compact support combo of this guy. And obviously the thing of which is a dual of in two numbers. And of a organic complaint because I'm using a complex support for something which is not representable. The way to get around this issue is to not try to do this directly for the projection. Absolutely, but for the map to the point mod k. And then you use the description of sheaves on the point mod k. So you can justify this. But this means that actually the co-mology is a dual space. You mean as lambda module? If you have something dual space which is countable then it must be finite. One can also do this argument in a slight variance when one doesn't need this countability here, but you can argue slightly differently. But then you need that everything is stable on the pullback to pull-finance sets. But anyway, that's the essential idea that on the one hand it's automatically dual and on the other hand by this quasi-compacity it's somewhat more of a direct limit nature. And so playing this off against each other implies finiteness. And so the other part is reflexivity. So duality. So again we start with our f which lives on this less than b part. And we want to prove that the extension by zero is reflexive. So let's try to unravel what this means. So what is the dual of J lower streak? Well by the general formalism that's R J lower star of the dual. So we have an exact triangle that J lower streak of the dual of f, R J lower star of the dual of f, maps to the stuff concentrated at the point for the push forward. And so by the way this is a step where we come here in a second where we really already need to know that this kind of gadget here is finite to know that this extra term that we get here is not too big. And then we apply duality again. So then first of all the order of the terms change. And so here in the middle we get the double dual of f which is the dual of f. What do we get on the right? On the right we get the dual of this guy. The dual again turns the J lower streak into the R J lower star. And then we get this off the double dual but that's just R J lower star of f. I'm hoping I'm doing this right. And so then here the duality commutes with the I lower star. And so what we get here is the I lower star of the dual of this guy. And we have our natural map here from J lower streak. Let's call this a star. So what is clear is that if I pull back this map under J then this is an isomorphism. For example because then this term vanishes, this term will be f again. And this term vanishes and it's a term that vanishes. So this term vanishes and it's a term that vanishes. So in other words I need to prove that these two things become isomorphic after I pull back under I. Now there was a shift equivalently. So this I upper star isomorphic to the dual of what I was getting for the dual sheave. So this is kind of local from gradiality. And let me actually go one step further and then we have some break. And to prove a finiteness we figured out how to compute this procedure. Maybe instead we do it here and then if we do it there then the sections are given by these guys. So we're going to get to one step further. You have to prove the following. Gv of Qp and open for p subgroup. This guy, so this is the left hand side. So it's isomorphic to the dual of the sections for the dual of pi upper star f. Well maybe just destroy some new shifts. So in other words this is a punker radiality on this stack which is this open part modulo. You also need that the map is correct. And so let me say one last thing before the break. If you want to prove the statement for potentially unbounded f there's again a way to formally reduce to the case that f is concentrated in one degree. So we can assume that f is concentrated in one degree. Okay and so then I will explain how to prove this after the break. No there's no, it's not a bounded graph category sorry. No no there's potentially unbounded. I mean what you do in this case is that you see that, well if you formulate the statement for the dual then both statements take direct filter direct limits to limits. And so you can reduce to the thing where you're bounded, comlogically bounded above. But then by some finite dimensionality you can forget about stuff which is too far on the other side. Anyway you can do this. So a break for 15 minutes. I should have stressed that. I mean there's one commutation of duality with J Loas Schrieg which is a 10 transforms J Loas Schrieg into J Loas Starr. But this other junction one might know is actually failed. So it's not in general true that this R J Loas Starr would be taken by duality J Loas Schrieg. That's what you want to prove and it's, for constructible sheaves you know it's true but only because you know by duality for constructible sheaves. Okay so, wait wait wait. This time it's not the R here. Okay so, actually something slightly funny happens here with this Poincare duality. Namely it's a Poincare duality where the pairing is an odd degree. So let me actually tell you about an analog. So it is that for punctured spectrum. Right, it's the analog of the punctured spectrum exactly. So I assume you have an A2 over K and you have the origin. And let's say M is a strict handzolization of A2 at X and then it has a close strautom which is X and it has a punctured spectrum at CERC. And then, well if you get inspired by the complex picture, okay C, then you would think that this is a very small neighborhood of this point and so it's like a three square and then there is a coordinate which gives you a distance to X. So this has a three dimensional Poincare duality. Because for homology this factor doesn't matter. So that's the kind of Poincare duality which we want to prove. So we have like a strict handzolization of our space at this point. And then there is some homotopically trivial direction which gives you the distance to the origin and then you have some kind of Milner sphere around this point or whatever it's called. Which is this compact manifold of one real dimension less. But now we do a trick and so let me explain the trick in this example first. So assume you have a self map which is contracting towards zero. For example in characteristic P, you might take XY goes to X to the P. And I assume it's actually an automorphism. So let me also assume that F is an automorphism. So for example after passing to the perfection. Which doesn't really matter as regards et alchromology. So then what I can do is I can divide the punctured spectrum by F to the Z. Where you would imagine that in the analogous situation what you're somehow doing is you're contracting in regards to the distance you're contracting to the origin. And so the idea is that this is something like this but then on this distance factor you're somehow folding this together with multiplicative period P. So this would be in a three times as one. And this satisfies some four dimensional concrete or OT. And one can, if one knows finiteness of homology for M-Circ. One can deduce the three dimensional duality from the four dimensional duality for M-Circ mod F to the N times Z where N is sufficiently large. So the idea is that the homology of this quotient is some of the group homology of Z acting on the homology here. But then if N is sufficiently divisible, the homology is finite, this will actually act trivially on homology. And so the homology of this quotient will actually just be direct sum of two copies of the homology of M-Circ. And then you can deduce the desired duality here. So in algebraic geometry it's a slightly tricky to make this work. You can actually do something like this if you replace a strict generalization by this formal completion and then pass to edict spaces. So then this M-circ is some edict space itself and you can make sense of this quotient and it will behave like a proper smooth object. And so you can also make sense of these objects in our case. So what you do is following. In our situation, we pick some kind of UP. It's an element of GB of QP which is central. X is positive slope on the unipotent radical of P. So let me try to say an example what I mean. So if G is GO2 and B corresponds to this bundle O of 1 plus O, then GB is this levy, GO1 times GO1, sitting inside of the borrel. And then I look at the element P, 1 in this guy. And so then I can take the quotient by the action of UP. So the idea is that UP is such an operator as this F up there. This is an analog of this operator F up there that is contracting MB tilde all towards the origin. But if you remove the origin, then it's acting freely. It can take the quotient, so the action is free. And now the map to the base is proper and smooth. Let me give you an example of what's happening here. In this GL2 example. What was this MB SERC? I said last time that this was the edX spectrum of this long series field, modulo profile net group. It was a free action. So this is a quasi-compact space, but the map to the point is not quasi-compact because whenever a base changes to a non-argimedian field, it becomes a punctured open unit disk modulo profile net group. But in these coordinates, UP corresponds to the map which takes t to t to the p. And so if I take this guy, mod u p to the z, then it's this guy modulo for b to the z and then also modulo profile net group. Forget about this profile net group, it's not the important part of what I'm trying to say. So you get this quotient. And so again there is this crucial difference of perspectives between looking at these just as a diamond or looking at the structure morphism. If you look at this just as an object, this is very bad. It's not quasi-separated. Just a sheaf. Perfectoid space and characteristic p. It's not quasi-separated because if you take a quasi-compact object over it like this edX spectrum and then takes a fiber product, you get z-many copies of it because you're dividing by the z actually. But if you look at it relatively, no, it's still quasi-compact but not quasi-separated. But this map is wonderful. It's quasi-compact, quasi-separated. It's even proper and smooth. It's relatively represented by diamonds, by proper and smooth. Because whenever you base change the picture to an algebraically closed field, what you get is a punctured open unit disk modulo-seaction of Rubinius, which is a wonderful object. So, in the form of the base change to some spaC, I get punctured open unit disk modulo-seaction of Rubinius. This is a wonderfully nice object. And because it is proper and smooth, we can apply Bronco-Aduality on this space. We can just use a formalism plus our assumption that f is reflexive. We get this kind of even-dimensional Bronco-Aduality on this stack. And then, okay, let me write the argument on again. Let's say you need compatibility in the end. You get some point, right? Do I write any compatibility with the map that you got? Yes, there is, to check some compatibility of maps. But so if n is sufficiently divisible, the homology of this quotient and it's pulled back here, is the same as, I mean, up to the nz, and it's trivially on the homology of this guy by finiteness. Because if you have any automorphism of a finite object, then some big power of it will be trivial. And then once this happens, this means that the homology, by some, who should say, a spectra sequence, the homology of this quotient is just two copies of the homology of this guy. So you have got a spectra sequence, but then you need to know that it's... I mean, it acts trivially in the derived category. I want that the action is trivially in the derived category. And then you need to check that the even-dimensional Bronco-Aduality, which you get on this space, really comes from two odd-dimensional Bronco-Adualities, relating this with the dual of this and the other way around. So that's why I wanted to say about the proof of the main theorem. Okay, so let me make some concluding remarks about a few things that one can deduce from the formalism. So this is some of the continuing discussion of the first half of the first lecture. So, and for this, I would like to pass to El Alic coefficient. And so give me some kind of... let me give you some kind of... I don't want to claim that I can make sense of this in as good a way as would be desirable, but you can make sense of some objects. So you can just make sense of some due reflexive bungee with OE coefficients. That's essentially by passing to the limit of the story modulo powers of L. So if you would have done everything in an infinity category, and you actually passed to the limit, there's a way to do it without infinity categories. Then I can just formally set... And so there's also... this is a canonical object that... where any possible definition would give you... reasonable definition should give you the same answer in the end. And for the next thing, that's not true. I define this with E coefficients to be just the idempotent completion. There is a problem in the limit process because then you go from OE modulo powers of L to... I don't want to do this just under the derived category. I don't take a limit of derived categories. If I do a limit, I do it of infinity categories. No, but I'm saying that because of infinitely many homology groups and infinitely tall dimensions, when you pass from OE modulo to L to the smaller powers, you don't necessarily go from a reflexive guy to a reflexive guy when you take modulo... when you pass from one ring to another because you have infinitely many possible homologies to give them. And then you take toes with a smaller ring and you can have many things on the left and on the right and then you will have many toes so you will get possibly real... So you can work with bounded maybe in a reflexive. Bounded... let me do bounded. Okay. And maybe I also want... Okay. The item potent completion of the same terms with E. And then I don't claim that this satisfies any kind of descent anymore or that you can check. Yeah, I mean this is some kind of... not good thing maybe, but... And... You need the item potent completion. You don't need your... Yeah, let me do the item potent completion because I might want to use some operators, some explosion operators to cut an object into pieces. So I do want an item potent completion. But is it necessary? You ask... yeah, I think it's necessary. Because for DPC of NIC, schemes are not necessary. It's already item potent. No, but I would expect that you can have representations over OE of such periodic groups, which are somehow... which have some congruences, but then generically they decompose. And then if you want to somehow get decomposition that you have generically, if you want to get it integrally, I don't think you can do it. I think you need an item potent completion there. At least for the part of the drive categories I'm eventually interested in. Confidence at this. And so let's now fix an R parameter. Let's fix the R parameter. When can I define a full subcategory? Which is somehow the phi part of this guy? So there is also a question like in elatic shields, that you can have no probianus eigenvalues which are not elatic units, then you need values. I don't know if this occurs in this... Well, I'm working on an algebraically closed field right now. So right at the beginning I fix an algebraically closed space field and I'm working on the space field all the time. But this W of Qp, so you have the... Yes, so it might be... well, I can't define this for anything. It might be that it's just empty. But I mean this real group, it's not related to... The fact that this real group appears here has nothing to do with this algebraically closed space field that I had. You were asking if the full value element is not kind of not elatic unit. So yeah, so it's probably true that if I have a guy here whose image is not compact in some sense, then what I'm just defining just now will be empty. But I can just still define it, okay? Lg is a Langmuir's Julia. Lg is a Langmuir's Julia. Okay, high part. By the condition that all excursion operators, as I defined them in the first lecture, are given by phi or are determined by... And so let me discuss what one would expect about this category. So let me assume that this phi is actually a cuspital. So assume that the group S phi, which is a centralizer in the dual group of phi, is finite. So phi is cuspital. This implicitly also says that G is... Well, the central trowel has no split part. Okay, so GLM would be disallowed for this state. G hat is a dual group. So Lg is the symmetric product of G hat with a real group. I guess... So in this case, you would expect... and these expectations are results that I would believe to be within reach, okay? You would expect that all these reflexive guys in this component are concentrated on the same unstable locus. So actually, what this guy is, this lump, Ql bar, this phi part, it should just be a direct... Well, it doesn't matter, there's only finite number. So under this assumption, it's automatically true that there are only finitely many basic elements, only finitely many connected components. Direct sum overall, basic b of the derived category of... derived category with admissible homologies of this group Jb of Qp, all these inner forms, and then you take a phi part of that in the sense that you only allow those admissible representations. All of those constituents have this fixed alparameter phi. Where the alparameter is as defined in the first lecture. So this means that this thing which is a priori defined in terms of some very complicated geometry boils down to some very concrete representation theory. So why would I expect that this can be proved? So if you would have a sheaf which has this alparameter but lives on some other stratum, then this other stratum is given by the automorphism group there as an inner form of a levy. And you would expect that if you have representation of an inner form of a levy that then the associated alparameter should be induced, which is wheeled out by this hospitality condition. So you only have to prove some compatibility with parabolic induction to get this statement. And so that's the first expectation you would have. And the second expectation you would have is that all HEC operators are T-exact for the standard T structure. Well, it doesn't really matter, right? Because these are just points here. So the perverse is the same as the usual T structure. But you are speaking about just... Well, on this part, some of the perverse and the usual T structure are the same. It's certainly not true that HEC operators are exact for any kind of T structure on the folder-rived category. You need to make some assumption. And for these costful parameters, it doesn't matter which one you take. And so you would actually, this way, get an action... But was perverse ever defined actually? Perverse was never defined anyway. So you would have to talk about something that's not this. Right now everything is a little undefined, maybe. Let me try to explain this picture. So you actually get some kind of action where you have algebraic representations. So the algebra times... Okay, let me call C the category for phi to be the direct sum of all B and B of G basic. So the category of admissible QL bar representations of GB of QP takes a phi part. So it's some kind of representation theory at a category attached to this costful parameter phi. And so this would mean that whenever you have a representation of the R group, you can act on this category. And you get another object of this category. But actually you get an object with a V-group action. Oops, what was this? And the actions continues. Okay. And this action satisfies some conditions. These are objects of C phi together with the action of W. Right. So continuous means that it's like in the most of presentations that every vector is the object. Every vector is an open stabilizer. Well, after all these might intertwine the allylic action, the allylic part of the V-group with the allylic topology of the QL bar. That will actually not happen. And so this gives you some purely category-serialistic setup here, where you have a category. You have some category of representations of an algebraic group acting on it, where after you act, you get an extra action of some other group and satisfying some competibilities. And you can write down all the structures there is. And then there's the following theorem, which seems to be well known to the experts in geometric Langlunds. For example, Gatesquery and Vincent LaForghens on this seem to know this. It seems to be well known, but anyway, somebody who actually wrote down a proof that I understood this, you know, as unshoots. So, yeah, so I assume that S-phi is finite. I'm just writing a new blackboard, I guess. And then there is actually a tensor category of representations of the centralized group S-phi on this corresponding isotopic component. Which on, under these assumptions, under these expectations, would actually be an action on this category, C-phi. Such that the action up there, actually this action is unique if you write down correctly all the competibilities. It's given by the following procedure. So, if you have a representation of the L-group, so in particular, you get a representation of S-phi times the V-group. Because you can restrict via phi to a representation of the V-group, but then this action of the V-group commutes with S-phi by the definition of S-phi. So, you get a representation of the product group. Which is the same thing as representations of S-phi, which are V-Q-P-equivariant. And then if you have a representation of Q-L-bar, or S-phi, then you have this action. So, you get an endomorphism of this reflex subcategory. What do you mean by action of the tensor category? Just on the usual derivative category or there is some higher... No, no, on the usual derivative category, just on the category. So, this means there is some rise of opposites and some compatibility. Right. So, for every object, you get an endomorphism, and then you have two objects, then the action of the tensor product is the composite of the actions. And when you have three, you have a compatibility. But it's only on the drive category, so it stops at some point. Let me not try to remember at which point. So, this means that you can describe the action of the L-group actually in terms of the action of this S-phi and the L-parameter. And so, this actually... if you unravel what this means, this is the form of the Kotwitz conjecture. So, this action of the representations of the L-group, this encodes the homology of R-rpxing spaces and it will take isotopic components under some guys in this file part. And then, this computes the output of this function in terms of the action of this category of representations of S-phi, which is doing some combinatorial thing like realizing some juckey-Languards correspondence. So, it takes one representation with L-parameter-phi and produces another representation. So, this gives you the representation theoretic part of the picture. And then, it also tells you how the Wehr-group acts, namely, well, it's also given by this procedure. This is the form of Kotwitz's conjecture on the homology of basic R-rpxing spaces. Well, this is the form of the Kotwitz conjecture, except that it doesn't really tell you what this action is. So, proving the Kotwitz conjecture would then come down to actually understanding what this action is. So, let me say one last word about this. And so, there's a following conjecture, which is essentially due to Kallator, except, of course, it didn't have this formalism to state it. But Kallator conjectured exactly how this category is C-phi or the irreducible objects of C-phi look like. Okay, so let me assume for this that G-square is split. So then, what Kallator effectively conjectures is that this category C-phi is equivalent to the category of representations of S-phi, where here's a trivial representation gets mapped to use a unique generic representation, and fix a bit of data. So, unique generic representation of G of Qp, which is the one which corresponds to one sort of trivial element, B. And then, in general, he has a way to parameterize all the irreducible representations of all these, what he calls extended pure inner forms, all these JB of QPs. And they should be given by certain representations of S-phi, which have a certain behavior. And, well, if you combine this over all possible different B's, then they should be parameterized by all possible irreducible representations of S-phi. And so, it relates irreducible objects on both sides, but actually this should be in a equivalent of categories. And this should also be true. In other words, GQP equal to what? G1 of Qp. So, I mean, if you take for B the trivial element, then, uh, compatibility with, say, irreducible S-phi. So, in other words, there should be this unique generic representation. And then this action of the representations of S-phi would give you a whole function from the category of representations of S-phi into here. And this should be in equivalence. But, so this isomorphism, it depends on the choice of vitica data, but some of the actions of the representations of S-phi on it, they don't, don't not depend on the choice of vitica data. The unique generic representation here, so some of the basis element here depends on the choice of vitica data, but the kind of module structure over the representation of S-phi do not depend on it. So, some base.3 version of this conjecture, some of the, this category, some of three of rank one over the representations of S-phi. Okay, and, uh, what Carl Leder really made much more precise is the, he said, he wrote down some explicit endoscopic character relations for this action. Uh, expected endoscopic. And so, this is actually also something that seems to be in reach. So, there is work in progress of Carl Leder and Weinstein, whether he used the left-shed-straight formula, and check that it is compatible, and check that the bungee picture is compatible. With these endoscopic character relations. So, the way they phrase the argument is that they assume that Carl Leder's conjecture is known in his form, that the irreducible objects up here in the expected way and satisfy the expected endoscopic character relations, which Carl Leder has proved in a wide variety of cases, and then they prove that what you see in the coromalgy of these basic-warp-wadding spaces really is realizing this correspondence of representations. And so, let me just say that this generalizes work of faultings and trough for the limited case of Farg for some unitary groups. So, there are faultings and trough proved by some purely local methods using the left-shed-straight formula in Li-Touare, that in the coromalgy of these limited spaces you realize the Jacqui-Languines correspondence, and this is some large generalization of this fact. And so, my time is up. Thank you very much. Very nice lectures. Any questions or comments? I mean in this middle backboard, of course, there is also the action of the real group, right? Whereas in the lower blackboard, there is no action of the real group, right? Well, some of this theorem tells you that the action of the real group is already as expected. The action comes down to determining what happens for the periodic groups. And for example, for the Li-Touare, which you can't quite apply, so if you do pgl or something, prove a version of the theorem which allows a center. The theorem would already tell you that if you plug in some super-custable representation of gln, what you get is some representation of the division algebra tensored with the correct L-parameter. And then if you combine this tensored with the L-parameter, that is defined by these methods. If you combine this with Falkingston's and Stroff paper, which tells you that you see the correct Jacqui-Languines correspondence, basically you see that you get the correct thing, but actually the output of this is that what you see here in this picture is really the usual local L-Languines correspondence. I mean, are you using some global thing to determine this? No, this is purely local argument, but I'm using that there is a local characterization of local L-Languines for gln. That is what's realized in the Li-Touare. Well, I mean, yes, but that's what it is, a local L-Languines correspondence is approved by global methods.
 So let's recall the setup from yesterday, no not yesterday, Wednesday.  So we have G or Qp a reductive group.  And at some point I was assumed for simplicity that it's quasi-split.  Also, I fixed some algebraically closed base field.  So I worked in a geometric setting.  And I was considering the stack bun G.  I defined on the category of perfectoid spaces over K.  Taking any test object S to group out of G torsors on XS, where this was a relative fact from 10 curve.  And so I stated last time a theorem to the effect that in this suitable interpretation of these words, this is whose all-tint stack of dimension 0.  And so then I fixed my coefficients.  So coefficients were Oe mod L to the n where E over Q also find an extension.  Of course L is not p.  And so we had a variable reductive function.  So this category of chiefs of lambda modules on bun G to itself.  Thanks.  I believe that conservative asks this question about Gavir.  So it maps it into the dualizing complex of bun G.  And then we had the definition that F is reflexive if F maps isomorphically to its double-redued dual.  And this was our substitute for the notion of constructible sheaves on the sky where the usual notion of constructibility would not be the good thing to consider.  And the main theorem I stated last time, which still had a star because it still depends on one conjecture, says the following that you can completely understand when a sheave is reflexive.  So F is G at bun G.  And there is reflexive if and only if.  For all points and as I said last time points on the stack are classified by Codd-Witzel set B of G of isocrystals and all comological degrees I.  So this B sum of corresponds to geometric point XB bar to bun G.  For all such guys and all integers, if I look at the stock of this complex really at XB bar,  I can look at all the homology groups.  These are all representations of the corresponding automorphism group of this B, which is this, well at least of G's quasi-split,  it's this inner form of the levy of G, GB of QP, if this isn't a misaligned representation.  And so today I want to explain the proof of this theorem.  And so the proof is by induction on bun G, which is somehow bounded by some fixed B.  Okay, so it's somehow starts proving a similar result just on the same stable locus and then you go further and further out into the non-stable locus, one stratum at a time.  And so the abstract theorem behind this stack has this great set of points.  Okay, the question is whether there is some special case of a more general theorem applying to more general states and I don't think so.  I think it's really something very special for bun G.  Yeah, so I proved the same theorem for all open subsets of bun G.  And what about things which are in subsets, a tolerable bun G?  Things which are tolerable bun G might be too...  So for things which are very close to bun G it might still be okay, but...  Okay, I can tell.  Certainly the proof makes use of very special properties of bun G.  So you start on the same stable locus.  What I've already said last time implicitly is that it's a following theorem which for GLN is 2 2 kth volume and u.  For general G, 2 2 5, it says that if you look at bun G and then you look at the same stable locus,  then this decomposes into a disjoint union over all basic B of a classifying space for the corresponding inner form of G.  Okay.  And so we have to understand what is the ad of such a guy is.  And for this let me state the following proposition.  So in fact it falls in the following generality that if you have H in some GLN QP, a closed-stop group,  so for example G of QP for any linear group G or an open-subgroup zero or whatever,  then you can try to understand what this de-ed of the point mod H is.  So the proposition is that as it de-ed of the point mod H coefficients in lambda  is really just the derived category of smooth H representations on lambda modules.  Okay, RER was late.  Smooth H representations on lambda modules.  And this is the equivalent.  And it was a fixed choice of H measure.  The aduality gets identified with smooth duality.  And which takes any pi in absolute to the homomorphisms from pi into lambda, but then it takes a smooth part of this.  And then this passes to the derived category.  I mean this sounds very reasonable, but actually verifying this is a little bit, there's some trouble, but okay, you can do it.  In particular, the fact that you really get the smooth part here only is related to some finiteness results for the homology of smooth diamonds.  So there is something hidden, some work hidden in the statement.  Also it looks very reasonable.  Not true that the B-in categories are different.  Which are B-in categories?  I mean this is not a priori the derived category of an B-in category, although somehow a posterior.  It is here.  Because here and here not in the setup where, so if this was some diamond, some locally spatial diamond, then this N-test some find comodical dimension, then this the ad-wood automatically be the derived category of a tall sheath.  But here we're not in the setup.  In particular, sheaths on this are somehow, oh, you have to first find a smooth cover of this guy by a diamond and so on.  A priori unraveling what this really is, this is a bit of a trouble.  Anyway, once you have checked this, you find that in particular if you try to understand what reflexive sheaths are, then those are the ones where passing through the double-smooth dual is an isomorphism.  This reflexive if and only if for I and Z, I locate H, I of F at a somewhat geometric point, which then becomes a representation of H.  That's because it can translate to the same question about smooth representations.  And this factor is exact, so understanding the radiality, you can understand it in each degree individually.  And then the condition that the double-smooth dual is the same thing, means that on all the fixed vectors under open subgroups, it must be finite dimensional.  It must be reflexive lambda module, which means it's finite.  There is no restriction on degrees.  So that's good.  So now for, so, so, finishes in a stable case.  So now we need to, so for the induction, let's fix some B and B of G, not same as stable, not basic.  And so then we have this inclusion from some B, bun G, which is less than B, into bun G, which is less than equal to B.  And let me also already give a name to the complementary closed. We have some I, bun G, B, meaning equal to B.  And so we assume that the theorem is known, I mean really the obvious variant of the theorem, known for bun G less than B.  Okay, so, so what do I have to do?  So we need to prove the following results.  One is a finiteness result, which says the following.  So assume you have some F, which lives on bun G less than B, and it's reflexive.  So I take, push this forward to the whole guy and then restrict to the new closed stratum.  So then I get a sheaf on bun G equal to B.  And we need to know that this is reflexive.  And you know that it is etal, in de etal by general single-signal.  No, because all functors are by definition, functors on de etal.  So you modify them to be de etal.  Right, for non-fazer compact immersions I said that I define this R.J. Lohr star to be the edge joint of J. Upper star.  So R.J. Lohr star, if I did it on V-sheafs, would not lie in the de etal, but I just reflected back into the de etal.  By using the right adjoint to the inclusion.  This means it's complicated to calculate it.  You must do it this way.  But circumvutes with smooth space change.  And the second thing you need to prove is a duality result.  And if F lies on this de etal bun G, less than B, less than reflexive.  Okay, formally the statements look very similar now, but I will try to explain that the first thing really is a fineness result, whereas the second thing is some kind of local duality result.  That will become, I think, clearer when I go into the proofs.  So why is this what we want to know?  So that's why does this imply, why does this give the induction step?  Well, you already understand reflexive shoes on the two strata.  So we understand reflexive shoes on the strata.  And this bun G equal to B is this point moduloose group, which has these kind of automorphisms of EV, where this was an extension of this periodic group, GB of QP, by a connected group.  And so you can prove the usual thing that connected groups don't actually interfere with the Alladek sheaf theory.  So actually the de etal on this bun G equal to B lambda is equivalent to de etal of just the point mod GB of QP lambda.  And so this means that on the strata you again can apply the results it says that these are given by smooth representation.  Do you mean a connected group has no homology or something?  Yeah, it's a connected group has no homology.  Yeah, so it has no homology.  You don't use the semi direct order structure, you just need an extension.  Do I need the semi direct product structure? I would think I just need an extension.  I have a semi direct product in case I need it.  So we understand on the strata.  So then say for the theorem we need to prove two directions. So which direction should I do first?  So assume first that say G in de etal bun G is equal to B lambda has admissible stocks in the sense of theorem.  And then we have a triangle like so.  And this is reflexive by induction. And then the whole thing is reflexive by two.  This is reflexive by the case of bun G equal to B.  And then the whole thing is reflexive as is proper to closed immersion.  And we had seen that inclusions by proper maps preserve flexibility.  So extensions of reflexive are reflexive and so we see that G is reflexive.  And so conversely, assume that G is reflexive.  So what do we want to see? We want to see it has good stocks.  And so you do the same thing. So the left thing, so same exact sequence implies that this thing is reflexive.  Same triangle.  But then because this is a closed immersion, this actually implies that I opposite of G is reflexive.  I mean so on the open part, we already know that it must have good stocks admissible stocks.  And so this means it has admissible stock.  So for the argument of hardness, it didn't quite appear, but it will be a step in getting to the duality.  But so because of this identification of reflexive sheaves here again with admissible representations of Jb of Qp,  this statement about this reflexivity is really just saying that something is an admissible representation.  So in this sense, it's really a finiteness result.  Whereas this reflexivity on this stack here is a much more geometric statement.  Much more like an actual duality statement.  And so, right, I think I moved the wrong page here.  So we want to do these things and for this we use a chart to compute anything.  So this chart is MB, which was an MB tilde, much G of Qp, y sub pi B.  And so where MB, so B, so I said this somewhere, but let me now make say it explicitly again.  I assume that G is quasi-stimulus implicitly.  So then B is actually the image of some BM, BM basic, where M is contained in P, contained in G, this is a levy.  And this is a parabolic encoding some slope filtration.  And what is MB tilde that classifies P twosers?  E plus an isomorphism of the reduction to M with the bundle corresponding to this basic element.  Then this maps via the GB of Qp, which is the same thing as the MBM Qp twosers.  To MB, which is a set of P twosers, E such that E times Pm is fiber-wise isomorphic, this point-wise isomorphic to EN.  And this maps to bungee, E while sending E to the pushout.  And the conjecture, which is this conjecture that explains this star, its main theorem is that this map from MB to bungee is equal to B, which is open to bungee is L-comologicalism.  And we can prove this for G or N.  And MB itself is also M code.  And MB itself is also a smooth out instead, as a consequence.  And because the map is smooth and bungee itself is smooth, this guy needs to be smooth.  But this is actually unconjectural, so you can directly show that this is smooth.  So now we have the following diagram that this maps to the bungee.  The pre-imature I will denote by MbCirc, EefI, bungee B, let's call this I prime.  What you get here is a fiber is actually just a point modulo Gb of Qp.  So effectively what you're doing over this stratum is you fix the splitting of the hard aneurysm infiltration.  And then you get this as the automorphism group.  And you also have on top of this a similar diagram.  We can call this point XB, it's just a copy of spark K.  So this start to be always on the left and close start to on the right.  Is that conflicting with standard usage?  Yeah, this is the open part, this is the close part.  I mean I use I and J in the usual way, I think.  All right.  So we have this picture.  And so let's start with the finiteness.  So by using the conjecture, smooth space change plus some unraveling of the formalism,  plus, this part of the unraveling of the formalism, that this Mb tilde is some kind of strictly Hanselian space.  So it behaves a little bit like power series over an algebraic closed field.  What you can prove is the following, that if you look at the sections on point K, where I say K and GB of Qp is an open pro-P subgroup,  pro-P just for simplicity essentially, but makes things slightly easier.  So I mean I fixed here my F and D at all, bun G less than D lambda,  and I should put the pole back on the pi B here.  So if I pull it back under pi B here, I get a shift here, and then whether I do push forward here and pull back here or upstairs,  this doesn't matter.  And so I want to understand what happens if I push forward here and pull back here, and then I want to know that this is an admissible representation here.  So what I want to know is that these kinds of global sections, if I go to the point mod K, that this is finite in each degree.  And you can compute this just as a homology of this open part mod K.  So we call that MB tilde over MB was this torsion in this group, so you can in particular divide by this group.  That's still some kind of stacky object anyway.  And with coefficients, where does this do reflexive?  So the homology of this guy, this is called pi B tilde, the star takes a K variance, where pi B tilde is a composite projection, not smooth.  So this description is better because here we still have a reflexive sheath.  Here we're pulling back too far, this is not reflexive anymore.  What's nice about this formula is that it tells you that if you do this thing, you push forward and you pull back and you know that what you get is just some smooth representation of the script JB of QP.  This smooth representation is just this thing here, because we've identified the K in variance for any K with the K in variance here.  So what you want to prove really is that this homology here is an admissible representation of this group.  What is this thing with the not smooth just you have the conjecture that the map is smooth?  Well, okay mate.  What is pi B?  Pi B is smooth, not pi B tilde.  So the torsion is not smooth?  Yeah, so coefficients under the profaned group are not smooth because the fibers are not smooth, profaned sets are not smooth.  There was this other thing that if you have something smooth and then you divide, then the quotient is still smooth over the base.  But that's a different statement.  Okay, so the torsions under profaned group are not?  No.  Okay, so this gives you a way of computing what you want to compute.  And so now we need to prove, so we need to show that this is finite.  In each homological degree.  And so this, there are two ingredients.  So by the way, when you write K acting on this gamma, there is required some foundation.  So in general if a group acts on the top, it's not the script group acting on the top, so you need some to know that it takes the other definition.  Okay, so I will use two things.  Well, actually no.  I claim that in each homological degree, the map is an injection, as I can just say, and the image will always be just the invariance.  Forget about any topology of this group acting on this set.  I mean, I think it's just.  And is it also a coherent couple?  Everything inside is a, okay.  Let me say what these objects are.  So the open part of this nb tilde is actually a quasi-compact separated spatial, well spatial implies quasi-compact diamond.  Whereas this quotient here would again be a stack.  In the sense, this is nicer here.  And so this means that you can compute  commodgy, I also find it dimensional.  So compute commodgy as a direct limit of chess commodgy over finite.  Over covers, over top covers.  With finitely many.  If it was not quasi-compact, then to do this, you would have to use infinitely many things.  And then if you write down the chess complex, it's infinite products and they become very big.  But if you use this information carefully.  But why is it not hypercovered for homology?  Okay, so there's this thing that f might be unbounded, but actually for the statement, you can reduce to the case where f is bounded.  Because you know by induction that canonical truncations preserve reflexivity.  By induction, canonical truncations preserve reflexivity because the condition of being reflexive is a condition of homology sheaves.  And so this means and each individual homology group of the sky by some finite dimensionality only depends on some bounded piece of the complex.  So you can reduce the case where this is actually just a sheave.  In which case you are allowed to just use a direct limit of usual chess commodgy.  I don't know what I'm saying, but there is a result when the coherent top was the homology is completely general by hypercover and not by...  If you want you can also do hyper.  No, I do have an analog of Arthens theorem on joints of Ansel ring that allow you to do check homology or...  So maybe, yeah, so maybe let's say hyper.  It's all hypercovered.  But still will find many terms in each degree.  And also you know that this sheave f is not too big.  For example, it has countable dimensional stocks.  And then you have to sit down and do it carefully.  But it implies that all the homology groups, they are at most countable.  So in particular you have to check that there are not too many etal things and so on.  But the critical part that makes everything work is the quality of the complexity here.  You have a countable co-final system.  Yeah, you have a countable co-final system of such a class.  And then the second thing is you apply punk radially.  So because f is reflexive,  so in particular it's the dual of something else, of some f prime.  And then this implies that actually the homology of this guy with coefficients in this guy  is a dual of the compact support combo of this guy.  And obviously the thing of which is a dual of in two numbers.  And of a organic complaint because I'm using a complex support for something which is not representable.  The way to get around this issue is to not try to do this directly for the projection.  Absolutely, but for the map to the point mod k.  And then you use the description of sheaves on the point mod k.  So you can justify this.  But this means that actually the co-mology is a dual space.  You mean as lambda module?  If you have something dual space which is countable then it must be finite.  One can also do this argument in a slight variance when one doesn't need this countability here,  but you can argue slightly differently.  But then you need that everything is stable on the pullback to pull-finance sets.  But anyway, that's the essential idea that on the one hand it's automatically dual  and on the other hand by this quasi-compacity it's somewhat more of a direct limit nature.  And so playing this off against each other implies finiteness.  And so the other part is reflexivity.  So duality.  So again we start with our f which lives on this less than b part.  And we want to prove that the extension by zero is reflexive.  So let's try to unravel what this means.  So what is the dual of J lower streak?  Well by the general formalism that's R J lower star of the dual.  So we have an exact triangle that J lower streak of the dual of f,  R J lower star of the dual of f,  maps to the stuff concentrated at the point for the push forward.  And so by the way this is a step where we come here in a second  where we really already need to know that this kind of gadget here is finite  to know that this extra term that we get here is not too big.  And then we apply duality again.  So then first of all the order of the terms change.  And so here in the middle we get the double dual of f which is the dual of f.  What do we get on the right?  On the right we get the dual of this guy.  The dual again turns the J lower streak into the R J lower star.  And then we get this off the double dual but that's just R J lower star of f.  I'm hoping I'm doing this right.  And so then here the duality commutes with the I lower star.  And so what we get here is the I lower star of the dual of this guy.  And we have our natural map here from J lower streak.  Let's call this a star.  So what is clear is that if I pull back this map under J then this is an isomorphism.  For example because then this term vanishes, this term will be f again.  And this term vanishes and it's a term that vanishes.  So this term vanishes and it's a term that vanishes.  So in other words I need to prove that these two things become isomorphic after I pull back under I.  Now there was a shift equivalently.  So this I upper star isomorphic to the dual of what I was getting for the dual sheave.  So this is kind of local from gradiality.  And let me actually go one step further and then we have some break.  And to prove a finiteness we figured out how to compute this procedure.  Maybe instead we do it here and then if we do it there then the sections are given by these guys.  So we're going to get to one step further. You have to prove the following.  Gv of Qp and open for p subgroup.  This guy, so this is the left hand side.  So it's isomorphic to the dual of the sections for the dual of pi upper star f.  Well maybe just destroy some new shifts.  So in other words this is a punker radiality on this stack which is this open part modulo.  You also need that the map is correct.  And so let me say one last thing before the break.  If you want to prove the statement for potentially unbounded f there's again a way to formally reduce to the case that f is concentrated in one degree.  So we can assume that f is concentrated in one degree.  Okay and so then I will explain how to prove this after the break.  No there's no, it's not a bounded graph category sorry.  No no there's potentially unbounded. I mean what you do in this case is that you see that,  well if you formulate the statement for the dual then both statements take direct filter direct limits to limits.  And so you can reduce to the thing where you're bounded,  comlogically bounded above.  But then by some finite dimensionality you can forget about stuff which is too far on the other side.  Anyway you can do this.  So a break for 15 minutes.  I should have stressed that. I mean there's one commutation of duality with J Loas Schrieg which is a 10 transforms J Loas Schrieg into J Loas Starr.  But this other junction one might know is actually failed.  So it's not in general true that this R J Loas Starr would be taken by duality J Loas Schrieg.  That's what you want to prove and it's, for constructible sheaves you know it's true but only because you know by duality for constructible sheaves.  Okay so, wait wait wait.  This time it's not the R here.  Okay so, actually something slightly funny happens here with this Poincare duality.  Namely it's a Poincare duality where the pairing is an odd degree.  So let me actually tell you about an analog.  So it is that for punctured spectrum.  Right, it's the analog of the punctured spectrum exactly.  So I assume you have an A2 over K and you have the origin.  And let's say M is a strict handzolization of A2 at X and then it has a close strautom which is X and it has a punctured spectrum at CERC.  And then, well if you get inspired by the complex picture, okay C, then you would think that this is a very small neighborhood of this point and so it's like a three square and then there is a coordinate which gives you a distance to X.  So this has a three dimensional Poincare duality.  Because for homology this factor doesn't matter.  So that's the kind of Poincare duality which we want to prove. So we have like a strict handzolization of our space at this point.  And then there is some homotopically trivial direction which gives you the distance to the origin and then you have some kind of Milner sphere around this point or whatever it's called.  Which is this compact manifold of one real dimension less.  But now we do a trick and so let me explain the trick in this example first.  So assume you have a self map which is contracting towards zero.  For example in characteristic P, you might take XY goes to X to the P.  And I assume it's actually an automorphism. So let me also assume that F is an automorphism.  So for example after passing to the perfection.  Which doesn't really matter as regards et alchromology.  So then what I can do is I can divide the punctured spectrum by F to the Z.  Where you would imagine that in the analogous situation what you're somehow doing is you're contracting in regards to the distance you're contracting to the origin.  And so the idea is that this is something like this but then on this distance factor you're somehow folding this together with multiplicative period P.  So this would be in a three times as one.  And this satisfies some four dimensional concrete or OT.  And one can, if one knows finiteness of homology for M-Circ.  One can deduce the three dimensional duality from the four dimensional duality for M-Circ mod F to the N times Z where N is sufficiently large.  So the idea is that the homology of this quotient is some of the group homology of Z acting on the homology here.  But then if N is sufficiently divisible, the homology is finite, this will actually act trivially on homology.  And so the homology of this quotient will actually just be direct sum of two copies of the homology of M-Circ.  And then you can deduce the desired duality here.  So in algebraic geometry it's a slightly tricky to make this work.  You can actually do something like this if you replace a strict generalization by this formal completion and then pass to edict spaces.  So then this M-circ is some edict space itself and you can make sense of this quotient and it will behave like a proper smooth object.  And so you can also make sense of these objects in our case.  So what you do is following.  In our situation, we pick some kind of UP.  It's an element of GB of QP which is central.  X is positive slope on the unipotent radical of P.  So let me try to say an example what I mean.  So if G is GO2 and B corresponds to this bundle O of 1 plus O, then GB is this levy, GO1 times GO1, sitting inside of the borrel.  And then I look at the element P, 1 in this guy.  And so then I can take the quotient by the action of UP.  So the idea is that UP is such an operator as this F up there.  This is an analog of this operator F up there that is contracting MB tilde all towards the origin.  But if you remove the origin, then it's acting freely.  It can take the quotient, so the action is free.  And now the map to the base is proper and smooth.  Let me give you an example of what's happening here.  In this GL2 example.  What was this MB SERC?  I said last time that this was the edX spectrum of this long series field, modulo profile net group.  It was a free action.  So this is a quasi-compact space, but the map to the point is not quasi-compact because whenever a base changes to a non-argimedian field,  it becomes a punctured open unit disk modulo profile net group.  But in these coordinates, UP corresponds to the map which takes t to t to the p.  And so if I take this guy, mod u p to the z,  then it's this guy modulo for b to the z and then also modulo profile net group.  Forget about this profile net group, it's not the important part of what I'm trying to say.  So you get this quotient.  And so again there is this crucial difference of perspectives between looking at these just as a diamond or looking at the structure morphism.  If you look at this just as an object, this is very bad. It's not quasi-separated.  Just a sheaf.  Perfectoid space and characteristic p.  It's not quasi-separated because if you take a quasi-compact object over it like this edX spectrum and then takes a fiber product,  you get z-many copies of it because you're dividing by the z actually.  But if you look at it relatively,  no, it's still quasi-compact but not quasi-separated.  But this map is wonderful. It's quasi-compact, quasi-separated. It's even proper and smooth.  It's relatively represented by diamonds, by proper and smooth.  Because whenever you base change the picture to an algebraically closed field,  what you get is a punctured open unit disk modulo-seaction of Rubinius, which is a wonderful object.  So, in the form of the base change to some spaC, I get punctured open unit disk modulo-seaction of Rubinius.  This is a wonderfully nice object.  And because it is proper and smooth, we can apply Bronco-Aduality on this space.  We can just use a formalism plus our assumption that f is reflexive.  We get this kind of even-dimensional Bronco-Aduality on this stack.  And then, okay, let me write the argument on again.  Let's say you need compatibility in the end. You get some point, right?  Do I write any compatibility with the map that you got?  Yes, there is, to check some compatibility of maps.  But so if n is sufficiently divisible, the homology of this quotient  and it's pulled back here, is the same as, I mean, up to the nz,  and it's trivially on the homology of this guy by finiteness.  Because if you have any automorphism of a finite object, then some big power of it will be trivial.  And then once this happens, this means that the homology,  by some, who should say, a spectra sequence, the homology of this quotient  is just two copies of the homology of this guy.  So you have got a spectra sequence, but then you need to know that it's...  I mean, it acts trivially in the derived category.  I want that the action is trivially in the derived category.  And then you need to check that the even-dimensional Bronco-Aduality,  which you get on this space, really comes from two odd-dimensional Bronco-Adualities,  relating this with the dual of this and the other way around.  So that's why I wanted to say about the proof of the main theorem.  Okay, so let me make some concluding remarks about a few things that one can deduce from the formalism.  So this is some of the continuing discussion of the first half of the first lecture.  So, and for this, I would like to pass to El Alic coefficient.  And so give me some kind of... let me give you some kind of...  I don't want to claim that I can make sense of this in as good a way as would be desirable,  but you can make sense of some objects. So you can just make sense of some due reflexive bungee with OE coefficients.  That's essentially by passing to the limit of the story modulo powers of L.  So if you would have done everything in an infinity category,  and you actually passed to the limit, there's a way to do it without infinity categories.  Then I can just formally set...  And so there's also... this is a canonical object that...  where any possible definition would give you... reasonable definition should give you the same answer in the end.  And for the next thing, that's not true.  I define this with E coefficients to be just the idempotent completion.  There is a problem in the limit process because then you go from OE modulo powers of L to...  I don't want to do this just under the derived category.  I don't take a limit of derived categories.  If I do a limit, I do it of infinity categories.  No, but I'm saying that because of infinitely many homology groups and infinitely tall dimensions,  when you pass from OE modulo to L to the smaller powers,  you don't necessarily go from a reflexive guy to a reflexive guy when you take modulo...  when you pass from one ring to another because you have infinitely many possible homologies to give them.  And then you take toes with a smaller ring and you can have many things on the left and on the right  and then you will have many toes so you will get possibly real...  So you can work with bounded maybe in a reflexive.  Bounded... let me do bounded.  Okay.  And maybe I also want...  Okay.  The item potent completion of the same terms with E.  And then I don't claim that this satisfies any kind of descent anymore or that you can check.  Yeah, I mean this is some kind of... not good thing maybe, but...  And...  You need the item potent completion.  You don't need your...  Yeah, let me do the item potent completion because I might want to use some operators,  some explosion operators to cut an object into pieces.  So I do want an item potent completion.  But is it necessary?  You ask... yeah, I think it's necessary.  Because for DPC of NIC, schemes are not necessary.  It's already item potent.  No, but I would expect that you can have representations over OE of such periodic groups,  which are somehow... which have some congruences, but then generically they decompose.  And then if you want to somehow get decomposition that you have generically,  if you want to get it integrally, I don't think you can do it.  I think you need an item potent completion there.  At least for the part of the drive categories I'm eventually interested in.  Confidence at this.  And so let's now fix an R parameter.  Let's fix the R parameter.  When can I define a full subcategory?  Which is somehow the phi part of this guy?  So there is also a question like in elatic shields,  that you can have no probianus eigenvalues which are not elatic units,  then you need values.  I don't know if this occurs in this...  Well, I'm working on an algebraically closed field right now.  So right at the beginning I fix an algebraically closed space field  and I'm working on the space field all the time.  But this W of Qp, so you have the...  Yes, so it might be... well, I can't define this for anything.  It might be that it's just empty.  But I mean this real group, it's not related to...  The fact that this real group appears here has nothing to do  with this algebraically closed space field that I had.  You were asking if the full value element is not kind of not elatic unit.  So yeah, so it's probably true that if I have a guy here  whose image is not compact in some sense,  then what I'm just defining just now will be empty.  But I can just still define it, okay?  Lg is a Langmuir's Julia.  Lg is a Langmuir's Julia.  Okay, high part.  By the condition that all excursion operators,  as I defined them in the first lecture,  are given by phi or are determined by...  And so let me discuss what one would expect about this category.  So let me assume that this phi is actually a cuspital.  So assume that the group S phi,  which is a centralizer in the dual group of phi, is finite.  So phi is cuspital.  This implicitly also says that G is...  Well, the central trowel has no split part.  Okay, so GLM would be disallowed for this state.  G hat is a dual group.  So Lg is the symmetric product of G hat with a real group.  I guess...  So in this case, you would expect...  and these expectations are results that I would believe to be within reach, okay?  You would expect that all these reflexive guys in this component  are concentrated on the same unstable locus.  So actually, what this guy is, this lump, Ql bar, this phi part,  it should just be a direct...  Well, it doesn't matter, there's only finite number.  So under this assumption, it's automatically true  that there are only finitely many basic elements,  only finitely many connected components.  Direct sum overall, basic b of the derived category  of...  derived category with admissible homologies of this group Jb of Qp,  all these inner forms, and then you take a phi part of that  in the sense that you only allow those admissible representations.  All of those constituents have this fixed alparameter phi.  Where the alparameter is as defined in the first lecture.  So this means that this thing which is a priori defined in terms of some very complicated geometry  boils down to some very concrete representation theory.  So why would I expect that this can be proved?  So if you would have a sheaf which has this alparameter  but lives on some other stratum, then this other stratum is given by  the automorphism group there as an inner form of a levy.  And you would expect that if you have representation of an inner form of a levy  that then the associated alparameter should be induced,  which is wheeled out by this hospitality condition.  So you only have to prove some compatibility with parabolic induction to get this statement.  And so that's the first expectation you would have.  And the second expectation you would have is that all HEC operators are T-exact for the standard T structure.  Well, it doesn't really matter, right? Because these are just points here.  So the perverse is the same as the usual T structure.  But you are speaking about just...  Well, on this part, some of the perverse and the usual T structure are the same.  It's certainly not true that HEC operators are exact for any kind of T structure on the folder-rived category.  You need to make some assumption.  And for these costful parameters, it doesn't matter which one you take.  And so you would actually, this way, get an action...  But was perverse ever defined actually?  Perverse was never defined anyway.  So you would have to talk about something that's not this.  Right now everything is a little undefined, maybe.  Let me try to explain this picture.  So you actually get some kind of action where you have algebraic representations.  So the algebra times...  Okay, let me call C the category for phi to be the direct sum of all B and B of G basic.  So the category of admissible QL bar representations of GB of QP takes a phi part.  So it's some kind of representation theory at a category attached to this costful parameter phi.  And so this would mean that whenever you have a representation of the R group, you can act on this category.  And you get another object of this category.  But actually you get an object with a V-group action.  Oops, what was this?  And the actions continues.  Okay. And this action satisfies some conditions.  These are objects of C phi together with the action of W.  Right.  So continuous means that it's like in the most of presentations that every vector is the object.  Every vector is an open stabilizer.  Well, after all these might intertwine the allylic action, the allylic part of the V-group with the allylic topology of the QL bar.  That will actually not happen.  And so this gives you some purely category-serialistic setup here, where you have a category.  You have some category of representations of an algebraic group acting on it, where after you act, you get an extra action of some other group and satisfying some competibilities.  And you can write down all the structures there is.  And then there's the following theorem, which seems to be well known to the experts in geometric Langlunds.  For example, Gatesquery and Vincent LaForghens on this seem to know this.  It seems to be well known, but anyway, somebody who actually wrote down a proof that I understood this, you know, as unshoots.  So, yeah, so I assume that S-phi is finite.  I'm just writing a new blackboard, I guess.  And then there is actually a tensor category of representations of the centralized group S-phi on this corresponding isotopic component.  Which on, under these assumptions, under these expectations, would actually be an action on this category, C-phi.  Such that the action up there, actually this action is unique if you write down correctly all the competibilities.  It's given by the following procedure.  So, if you have a representation of the L-group, so in particular, you get a representation of S-phi times the V-group.  Because you can restrict via phi to a representation of the V-group, but then this action of the V-group commutes with S-phi by the definition of S-phi.  So, you get a representation of the product group.  Which is the same thing as representations of S-phi, which are V-Q-P-equivariant.  And then if you have a representation of Q-L-bar, or S-phi, then you have this action.  So, you get an endomorphism of this reflex subcategory.  What do you mean by action of the tensor category? Just on the usual derivative category or there is some higher...  No, no, on the usual derivative category, just on the category.  So, this means there is some rise of opposites and some compatibility.  Right. So, for every object, you get an endomorphism, and then you have two objects, then the action of the tensor product is the composite of the actions.  And when you have three, you have a compatibility.  But it's only on the drive category, so it stops at some point.  Let me not try to remember at which point.  So, this means that you can describe the action of the L-group actually in terms of the action of this S-phi and the L-parameter.  And so, this actually... if you unravel what this means, this is the form of the Kotwitz conjecture.  So, this action of the representations of the L-group, this encodes the homology of R-rpxing spaces and it will take isotopic components under some guys in this file part.  And then, this computes the output of this function in terms of the action of this category of representations of S-phi,  which is doing some combinatorial thing like realizing some juckey-Languards correspondence.  So, it takes one representation with L-parameter-phi and produces another representation.  So, this gives you the representation theoretic part of the picture.  And then, it also tells you how the Wehr-group acts, namely, well, it's also given by this procedure.  This is the form of Kotwitz's conjecture on the homology of basic R-rpxing spaces.  Well, this is the form of the Kotwitz conjecture, except that it doesn't really tell you what this action is.  So, proving the Kotwitz conjecture would then come down to actually understanding what this action is.  So, let me say one last word about this.  And so, there's a following conjecture, which is essentially due to Kallator, except, of course, it didn't have this formalism to state it.  But Kallator conjectured exactly how this category is C-phi or the irreducible objects of C-phi look like.  Okay, so let me assume for this that G-square is split.  So then, what Kallator effectively conjectures is that this category C-phi is equivalent to the category of representations of S-phi,  where here's a trivial representation gets mapped to use a unique generic representation, and fix a bit of data.  So, unique generic representation of G of Qp, which is the one which corresponds to one sort of trivial element, B.  And then, in general, he has a way to parameterize all the irreducible representations of all these, what he calls extended pure inner forms, all these JB of QPs.  And they should be given by certain representations of S-phi, which have a certain behavior.  And, well, if you combine this over all possible different B's, then they should be parameterized by all possible irreducible representations of S-phi.  And so, it relates irreducible objects on both sides, but actually this should be in a equivalent of categories.  And this should also be true.  In other words, GQP equal to what?  G1 of Qp. So, I mean, if you take for B the trivial element,  then, uh, compatibility with, say, irreducible S-phi.  So, in other words, there should be this unique generic representation.  And then this action of the representations of S-phi would give you a whole function from the category of representations of S-phi into here.  And this should be in equivalence.  But, so this isomorphism, it depends on the choice of vitica data,  but some of the actions of the representations of S-phi on it, they don't, don't not depend on the choice of vitica data.  The unique generic representation here, so some of the basis element here depends on the choice of vitica data,  but the kind of module structure over the representation of S-phi do not depend on it.  So, some base.3 version of this conjecture, some of the, this category, some of three of rank one over the representations of S-phi.  Okay, and, uh, what Carl Leder really made much more precise is the, he said,  he wrote down some explicit endoscopic character relations for this action.  Uh, expected endoscopic.  And so, this is actually also something that seems to be in reach.  So, there is work in progress of Carl Leder and Weinstein,  whether he used the left-shed-straight formula, and check that it is compatible, and check that the bungee picture is compatible.  With these endoscopic character relations.  So, the way they phrase the argument is that they assume that Carl Leder's conjecture is known in his form,  that the irreducible objects up here in the expected way and satisfy the expected endoscopic character relations,  which Carl Leder has proved in a wide variety of cases, and then they prove that what you see in the coromalgy of these basic-warp-wadding spaces  really is realizing this correspondence of representations.  And so, let me just say that this generalizes work of faultings and trough for the limited case of Farg for some unitary groups.  So, there are faultings and trough proved by some purely local methods using the left-shed-straight formula in Li-Touare,  that in the coromalgy of these limited spaces you realize the Jacqui-Languines correspondence, and this is some large generalization of this fact.  And so, my time is up.  Thank you very much. Very nice lectures. Any questions or comments?  I mean in this middle backboard, of course, there is also the action of the real group, right?  Whereas in the lower blackboard, there is no action of the real group, right?  Well, some of this theorem tells you that the action of the real group is already as expected.  The action comes down to determining what happens for the periodic groups.  And for example, for the Li-Touare, which you can't quite apply, so if you do pgl or something,  prove a version of the theorem which allows a center.  The theorem would already tell you that if you plug in some super-custable representation of gln,  what you get is some representation of the division algebra tensored with the correct L-parameter.  And then if you combine this tensored with the L-parameter, that is defined by these methods.  If you combine this with Falkingston's and Stroff paper, which tells you that you see the correct Jacqui-Languines correspondence,  basically you see that you get the correct thing, but actually the output of this is that what you see here in this picture  is really the usual local L-Languines correspondence.  I mean, are you using some global thing to determine this?  No, this is purely local argument, but I'm using that there is a local characterization of local L-Languines for gln.  That is what's realized in the Li-Touare.  Well, I mean, yes, but that's what it is, a local L-Languines correspondence is approved by global methods.
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