 So, let's recall the setup from yesterday, no, not yesterday, Wednesday, so we got 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. Work in a geometric setting. I was considering the stack bun G. I defined on the category of perfectoid spaces over K, taking any test object S to the group point of G torsors on XS, where this was a relative fact from 10-curve. And so, as they had last time a theorem to the effect that in the suitable interpretation of these words, this is whose R-team stack of dimension zero. Okay, and so then I fixed my coefficients. So, coefficients were OE mod L to the N, where E over Q are also finite extension. Of course, L is not P. And so, we had a vergeduality function from this category of chiefs of lambda modules on bun G to itself. Can't believe that Konsevich asked this question how it governs. So, it naps it into the dualizing complex of bun G. And then we had the definition that F is reflexive if F naps isomorphically to its W-reduced rule. And this was our substitute for the notion of constructible chiefs 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 chief is reflexive. So, if and only if for all points, and as I said last time, points on the stack are classified by Kottwitz's set B of G of isocrystals and all comological degrees I. So, this B sum of corresponds to a geometric point XB bar to bun G for such guys and all integers. If I look at the stock of this complex really at XB bar, then it's a complex so 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 quarter splits it's this inner form of the levy of G, GB of QP if this isn't admissible representation. Okay, and so today I want to explain the proof of this theorem. And so the proof is by by induction on bun G, which is somehow bounded by some fixed B. Okay, so you start proving a similar result just on the same unstable locus and then you go further and further out into the non-same unstable locus, one stratum at a time. And so, okay, the question is whether there is some special case of a more general theorem applying to more general sex 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 some sense atarable bun G? Things which are atarable bun G. So for things which are very close to bun G, it might still be okay, but okay, I can't tell. Certainly the proof makes use of very special properties of bun G. So you start on the same stable locus. And so what I've already said last time inclusively is that it is following theorem, which for GLN is 2, 2, 3, and U. For general G, 2 to 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 overall basic B of a classifying space for the corresponding inner form of G. And so we have to understand what this D add of such a guy is. And for this, let me state the following proposition. So in fact, it holds in the following generality that if you have H in some GLN QP, close subgroup, so for example G of QP for any linear group G or an open subgroup thereof or whatever, then you can try to understand what this D add of the point mod H is. And so the proposition is that the D add of the point mod H to coefficients in lambda is really just the derived category of smooth H representations on lambda modules. The RER was late. Smooth H representations on lambda modules. And under this equivalence and with a fixed choice of high measure, the idea duality gets identified with smooth duality and which takes any pi in absolute to the homomorphisms from pi into lambda, but then you take the 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 is 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 obedient categories are identified. Which obedient categories? I mean, this is not a priori the derived category of an obedient category, although somehow posterior it is here because here and we're not in the setup where, so if this was some diamond, some locally spatial diamond, then this n-test some pharmacological dimension, then this d-ad would automatically be the derived category of etals sheaves, but here we're not in the setup. In particular, sheaves on this are somehow, oh, you have to first find a smooth cover of this guy by a diamond and so on. I mean, a priori unraveling what this really is, this is a bit of a trouble. Anyway, once you've checked this, you find that in particular, if you try to understand what reflexive sheaves are, then those are the ones where passing through the double dual or double smooth dual is an isomorphism. So 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 factors under open subgroups, it must be finite dimensional, it must be reflexive London module, which means it's finite. There's no restriction on degrees. There's no restriction on degrees, right? Okay, so that's good. What do I want to say next? So that somehow deals with the same unstable locus. Okay, so now for, so, so finishes the same 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 that may 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, which is reflexive. Then if 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 still reflexive. And you know that it is a tau, in d at tau, by general single experiment. No, because all functions are by definition functions on d at tau. So you modify them to be d at tau if they are. Right, for non-partic compact immersions, I said that I define this rj lower star to be the edge joint of j upper star. So rj lower star, if I did it on v sheafs, would not lie in the d at tau, but I just reflected back into the d at tau. They're using the right adjoint to the inclusion. This means it's complicated to calculate it. This means it's complicated to calculate it, yeah. But you must do it this way. But still commutes with smooth space change. And the second thing you need to prove is a duality result. So again, if f lies on this d at tau, by general single experiment. Okay, formally the statements look very similar now, but I will try to explain that the first thing really is a final 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 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, bungee less than b by induction. And this bungee equal to b is this point moduloose group, which are these kind of automorphisms of eb, 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 analytic sheaf theory. So actually the d at tau on this bungee equal to b lambda is equivalent to d at tau of just the point mod qb of qp lambda. And so this means that on the stratum you again can apply the results it says that these are given by smooth representation. Do you mean the connected group has no homology or something? Right, the connected group has no homology. I thought yeah, I connected kind of some kind of unipotent group here. Yeah, so it has no homology. All right. You don't use the semi-direct water structure, you just need an extension. Do I need the semi-direct product structure? I would think I just need an extension. Okay, I have a semi-direct product in case I need it. Okay. Okay, so so we understand on the stratum. 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 d at tau bungee less or equal to b lambda has admissible stalks 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 bungee equal to b. And then the whole thing is reflexive as i is proper. It's a closed immersion. And we have seen that inclusions by proper maps preserve reflexivity. And so extensions of reflexive sheaves are reflexive and so we see that g is reflexive. And so conversely assume that g is reflexive. Okay, so what do we want to see? We want to see it has good stalks. Right. And so you do the same thing. So the left thing, so okay, so same exact sequence implies that this thing is reflexive, the 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 stalks, admissible stalks. And so this means the admissible stalk. So for the argument of finiteness 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 the Jb of Qp, this statement about this reflexivity is really just saying that something is an implicit representation. So in this sense, it's really a finiteness result. Whereas this reflexivity on this stack here is a much more geometric statement. It's much more like an actual reality statement. And so single the wrong page here. So we want to do these things. And for this we use a chart. So to compute anything, charts, mb, which was an mb tilde, 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 for simplicity. So then b is actually the image of some bm and bm basic, where m is contained in p, it's contained in g, so this is a levy. And this is a parabolic encoding some slope filtration. And what is mb tilde that classifies p-torsors e plus an isomorphism of the reduction to m with the bundle corresponding to this basic element. Then this maps wire the gb of qp, which is the same thing as the mbm qp torsor to mb, which is a set of p-torsors. e such that e times pm is fiber-wise isomorphic, this point-wise isomorphic. And then this maps to bun g 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 bun g is equal to b, which is open in bun g is l-chromologically smooth. And we can prove this for geoland. And mb itself is also a smooth outing stick as a consequence. And because the map is smooth and bun g itself is smooth, this guy needs to be smooth. But this is actually unconjectural, so you can directly show that this is smooth. Okay, so now we have to find diagrams that this maps to bun g is equal to b. There's some j tilde. Let's call this j prime here actually. The pre-mature I will denote by mc, mb-circ, here you have i, bun gb. 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 aneurysma infiltration. And then you get this as the automorphism group. And you also have on top of this a similar diagram. You can call this point xb. It's just a copy of spark k. The open star two is always on the left and the closed part to the right. Is that conflicting with standard usage? Yeah, this is the open part, this is the closed 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, why is this part of the unraveling of the formalism, that this mb tilde is some kind of strictly Hanzillion space. So it behaves a little bit like power series over an algebraically closed field. What you can prove is the following, that if you look at the sections on point mod k where I say k and gb of qp is an open propy subgroup. Propy just for simplicity essentially, but it makes things slightly easier. So I mean I fix here my f and d et al bun g less than d lambda and I should put the pullback under 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 cohomology 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. With coefficients where this is the reflexive. A different way of writing the same thing is actually to write the cohomology of this guy. This calls pi b tilde if the star takes a k variance. Where pi b tilde is a composite. 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. But what's nice about this formula is that it tells you that it's actually 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. The smooth representation is just this thing here because we've identified the k invariance for any k with the k invariance here. So what you want to prove really is that this is that this cohomology here is an admissible representation. What is the thing with the not smooth just you write the conjecture that the map is well okay mate well pi b is smooth yeah not pi b tilde. Ah so the the torso is not smooth. Yeah so pro pro project quotients on the profite group are not smooth because profite sets the fibers are not smooth. Profite 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 torsos and the profite groups are not. No okay so you 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 common logic degree and so this uh there are two two ingredients. So by the way the when you write k acting on this r gamma there is required some foundation so in general if a group acts on the top it's a non-descript group acting on the torsos 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 common logic degree the map is an injection this 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 so everything inside is okay let me say what these objects are. So the open part of this mb tilde is actually a quasi-compact separated spatial well spatial implies quasi-compact but diamond whereas this quotient here would again be a stack in this sense this is nicer here and so this means that you can compute commonology also finite dimensional compute commonology as a direct limit of Czech commonologies over finite over covers or it's our covers it's 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 Czech complex there's some infinite products and they can become very big but if you use this information carefully 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 conditional commonology sheaves and so this means and each individual commonology group of the sky by some finite dimensionality only depends on some bounded piece of the complex so you can reduce to the case where this is actually just a sheaf in which case you are allowed to just use a direct limit of usual Czech but i'm saying it there is a result when the coherent top was the commonology is completely general by hypercovering if you want you can also do hyper you know i do have an analog of outings theorem on joints of an zel ring that allows you to do Czech so maybe yeah so maybe let's say hyper over it's all hyper covers but still will finally make terms in each degree and also you know that this sheaf 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 commonology 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 ever makes everything work is a quasi-compacity here you have a co-finance a countable co-finance system yeah you have a countable co-final system of such things and then the second thing is you apply from grade wellity so because f is reflexive so in particular it's a dual of something else of some f prime and then this implies that actually the commonology of this guy with coefficients in this guy is a dual the r-home of the complex support commonology of this guy of the thing of which is a dual off into lambda and Ofer will again complain because I'm using a complex support commonology 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 sheaths on the point mod k so you can justify this but this means that actually the commonology is a dual space as lambda module the structure theorem for lambda modules right and then I mean if you have something the dual space which is countable then must be finite okay one can also do this argument and slight variance or one doesn't need this countability here and but argue slightly differently but then you need that everything is stable on the pullback to pull finite sets but anyway so 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 these off against each other implies finiteness all right and so the other part is reflexivity so duality so again we start with our f which lives on this less mp part and we want to prove that the extension by zero so let's try to unravel what this means so what is the dual of j lower shriek well by the general formalism that's r j lower star of the dual and so and so we have an exact we have a triangle that j lower shriek of the dual of f that's r j lower star of the dual of f maps to um the stuff concentrated at the point for the push forward and so by the way this is a step will become clear in a second where we really already need to know that this kind of gadget here is finite to know that this extra 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 okay so here in the middle we get the double dual f which is a dual what do we get on the right on the right we get the dual of this guy the dual again turns the j lower shriek 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 and so then here's 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 shriek okay so what is clear is that if i pull back this map under j then this is nice for example because then this term vanishes this term will be f again this map is a nice morphism this map is a nice morphism then this map is so it remains to check it on the complementary close so i upper stuff but if i pull this back under i then it's zero because i took the extension by zero so i need to prove that this is zero if i pull it back under i so in other words i need to prove that these two things become isomorphic after i pull back under i uh with a shift equivalently if i do this the i upper star then it's isomorphic to the dual of what i was getting for the dual sheave which is some kind of local from gradiality and let me actually go one step further and then we have to break and the proof of finiteness we figured out how to compute this procedure maybe instead we do it here and then if you do it there then the sections are given by these guys so going yet one step further you have to prove the following that for all k in gv of qp and open for p subgroup the sections of this guy so this is the left hand side it's isomorphic to the dual of the sections for the dual of pi upper star f maybe just destroy some new shifts so in words so in other words this is a ponker radiality on this deck which is this open part modular you also need that the map is the correct map and so uh let me say one last thing before the break um 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 f is concentrated okay and so then i will explain how to prove this after the break no there's no it's not a bounded draft category sorry no no there's it'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 directly filter direct limits to limits and so you can reduce to the thing where you bound it com logically bounded above but then by some finite dimensionality you can forget about stuff which is too far on the other side and anyway you can do this so break for 15 minutes so maybe i should have stressed that i mean there's one commutation of duality was j lower shriek which is a 10 transforms j lower shriek into j lower star but this other junction one might know it's actually failed so you it's not in general true that this r j lower star would be taken by duality j lower shriek that's what you want to prove and it's some of force for constructible sheeps you know it's true but only because you know by duality for constructible sheeps um okay so uh wait wait wait this time it's not the r here okay so actually something slightly funny happens here with this punker ideology namely it's a punker ideology whether pairing is an odd degree so let me actually tell you about an analog so it is that for punctured spectrum right the analog was a punctured spectrum exactly so uh so i assume you have an a2 over a k and you have the origin and uh let's say m is a strict generalization of a2 at x and then it has a closed stratum which is x and it has a punctured spectrum and sir and then well if you are get inspired by it's like a complex picture okay see 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 the distance to to x and so this has a three-dimensional punker ideology because for homology this factor doesn't matter so that's the kind of punker ideology which we want to prove so we have like the strict generalization of our space at this point and then there's somehow a 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 um which is this compact manifold of one real dimension less and but now we do a trick and so let me explain the trick uh in this example first so assume you have a self map which is contracting towards zero uh for example on characteristic p uh you might take x y goes to x to the p and assume it's actually an automorphism so let me also assume that f is an automorphism uh so for example after passing to the perfection rectified which doesn't really matter as regards etalc homology 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 here is you're contracting uh in regards to the distance you're contracting to the origin 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 punctured reality okay and um one can uh if one knows finiteness of chronology for m-circ one can deduce the three dimensional duality uh from the four dimensional duality in circumference f to the n times z where n is sufficiently large so the idea is that the chronology of this quotient is some of the group chronology of z acting on the chronology here but then if n is sufficiently divisible um this it's a chronology as finite this will actually act trivially on chronology and so the chronology of this quotient will actually just be direct sum of two copies of the chronology of m-circ and then you can deduce the desired geology here all right some hope for any fault and so in algebraic geometry it's a slightly tricky to make this work you can I think actually do something like this if you replace a strict generalization by this formal completion and then pass to attic spaces so then this m-circ is some attic 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 the following so in our situation fix some kind of up what just called up it's an element of gb of qp which is central that is some of x was positive slope on the important radical so the let me try to say an example what I mean so if g is g o2 and b corresponds to this bundle of 1 plus o then gb is this levy g o1 times g o1 sitting inside of the borrel and then I look at the element p comma 1 in this guy okay 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 take some power of it is an analog of this operator f up there that is contracting mb tilde all towards the origin and 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 g o2 example still what was this mb cirq I said last time that this was the attic spectrum of this long series field modular pro for net group so this is a quasi compact space but the map to the point is not quasi compact because whenever I base change this to some non-archimedean field it becomes a punctured open unit disk modular pro for net group but in these coordinates the p corresponds to the map which takes t to t to the p and so if I take this guy module p to the z then it's this guy modular Frobenus to the z and then also modular profiler but just forget about this profiler 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 on the perfecto it's basic characteristic p it's not quasi separated because if you take a quasi compact object over it like this attic spectrum and then takes a fiber product you get the 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 um but this map is wonderful it's quasi compact quasi separated it's even proper and smooth represented by it's relatively represented by diamonds by proper and smooth guys because whenever you base change the picture to an algebraically closed field what you get is a punctured open unit disk modular the action of Frobenus which is a wonderful object of the base change in some spa c I get punctured open unit disk modular Frobenus and so and this is a wonderfully nice object and so and because it is proper and smooth we can apply puncture duality on this space so if you then just use a formalism plus our assumption that f is reflexive we can we get this kind of even dimensional puncture duality on this stick and then get it right the argument on again to check some compatibility of maps um but so if n is sufficiently divisible uh the cosmology this quotient uh is the same as I mean uh u p to the n z x trivially on the cosmology of this guy uh 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 uh this means that the cosmology by some uh who should say a spectral sequence the cosmology of this quotient is just two copies of the cosmology of you have got a spectral sequence but then you will need to know that it there is I mean it acts trivially some in the derived category okay I want that the action is trivially in the derived category and then you think it really splits as it acts and okay and then you need to check that the even dimensional puncture duality which you get on this space really comes from two odd dimensional puncture dualities 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 zero okay so let me make some concluding remarks about a few things that one can deduce from the formalism so this is somehow continuing the discussion uh of the first half of the first lecture so and for this I would like to pass to ally coefficients 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 uh some dereflexive 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 she could literally pass to the limit um there's a way to do it without infinity categories um then I can just formally set and so there's also no Thomas this is a canonical object that um where any possible definition would give you reasonable definition should give you the same answer in the end so for the next thing that's not true okay I define this with e coefficients to be just the idempotent completion nobody there is a problem in the limit process because then you go from oe modulo powers of l to one i don't want to do this just under the derived category yeah 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 that because of infinitely many comology groups and infinitely tau dimension when you pass from oe mod l to them to oe mod l to the smaller powers okay you don't necessarily go from a reflexive guy to a reflexive guy when you take mod when you pass from one ring to another because you have infinitely many possible comologies to given them and then you take tau's with a smaller ring and you can have many things on the left and on the right and then you will have many tau's that so you will get possibly will look okay so you can focus bounded maybe in a reflexive bounded let me do bounded okay and maybe I also want okay the idempotent completion of the single tensor 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 yeah let me do the idempotent completion because I might want to use some operators some excursion operators to cut to cut an object into pieces so I do want an idempotent completion but is it necessary is it you ask but yeah I think it's necessary I think because for DPC of nice no because schemes it's not necessary to it's already idempotent that is your any 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 that you compose and then if you want to somehow get the decomposition that you have generically if you want to get it integrally I don't think you can do it so I think you need an idempotent completion there um okay so I at least for the part of the drive categories I'm eventually interested in I'm confident that this is defined um and so let's now fix an L parameter fix same as simple parameter one can define a full subcategory which is somehow the five part what's this guy so there is also a question like in ellatic shift that you can have no for bernieus eigenvalues which are not ellatic units generally values I don't know if this occurs in this well I'm working over an algebraically closed field right now so right at the beginning I fix an algebraically closed base field and I'm working over the space field all the time but this w of q p so you have the uh yes so it might be well I can define this for anything it might be that it's just empty but I mean this real group it's not related to uh the fact that this very group appears here has nothing to do with this algebraically closed base field that I had so I'm asking if the for bernieus element is not kind of not ellatic unit so yeah so it's probably true that if I have a guy here whose image is not compact in some sense uh then what I'm just defining just now will be empty but I can just still define it okay lg is a Langland's Julian uh five part um by the condition that all excursion operators as I define them in the first lecture are given by phi or are determined by and so let me discuss what one what what what one would expect about this category uh 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 this implicitly also says that g is well the central trawler says no split part okay so glm would be disallowed for g hat uh the dual group so lg is the semi direct product of g hat was available so in this case you would expect and these expectations are results that I would believe to be within reach okay um you would expect that all these reflexive guys in this component are concentrated on the semi-stable locus so actually um what this guy is this lump actual bar it's five part it should just be uh 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 and direct sum overall basic b uh of the derived category of derived category with admissible homologies of this group jb of qp of these 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 l-parameter phi where the l-parameter is as defined in the first lecture so this means that this thing which is a priority find in terms of some very complicated geometry boils down to some very concrete representation theory why would I expect that this can be proved so if you would have a sheath which has this l-parameter but lives on some other stratum then this other stratum is given by an inner the automorphism group there is 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 l-parameter should be induced which is wheeled out by this hospitality condition so you would only have to prove some compatibility with parabolic induction to get this statement so that's the first expectation you would have and the second expectation you would have is that uh all hack operators are t-exact for the standard t-structure that is not perverse but but well it doesn't really matter right because these are just just have points here so the perverse is the same as the usual t-structure ah but you are speaking about just well on this part and on this part some of the perverse and the usual t-structure are the same so um it's certainly not true that hack operators are except for any kind of t-structure on the fold arrived category you need to make some assumption and for these cuspial parameters it doesn't matter which one we take and so you would actually this way get an action um right now everything is a little undefined maybe let me try to explain this picture um so you actually get some kind of action uh we have here the algebraic representations of the outgroup times okay let me call c the category for phi to be the direct sum of all b and b of g basic so category of admissible ql bar representations of gb of qp takes a phi part so it's some kind of representation theory at the category attached to this cuspital parameter phi and so this would mean that whenever you have a representation of the outgroup you can act on this category and you get another object of this category but actually you get an object with a wave group action and the action is continuous okay uh 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 like in the most representations that is right that every vector is an open stabilizer right every vector is an open stabilizer well after a while it might it might intertwine the allylic action the allylic part of the wave group with the allylic topology of this ql bar that will actually not happen and so um this gives you some purely category theoretic uh 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 structure there is and then there's the following theorem uh which seems to be well known to the experts in geometric Langlund so for example Gatesbury and Vincent LaForgue and so on they seem to know this um it seems to be well known but anyway uh somebody who actually wrote down a proof that I understood this you want to answer it um so yeah so assume that s phi is fine idea then there is actually an of the tensor category of representations of the centralized group s phi uh on this corresponding isotopic component which on under these assumptions uh under these expectations would actually be an action on this category uh c phi um such that uh 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 real group because you can restrict via phi to a representation of the real group but then this action of the real group commutes with s phi by the definition of s phi so you get a representation of the product group uh so which is the same thing as representations of s phi which are vqp a covariant and then if you have a representation of ql bar of s phi then you have this action so you get an endomorphism of this reflex subcategory so what do you mean by action of the tensile category just uh on the usual the rank category there is some higher no no on the usual the right category just on the category so this means there is some there are some 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 query when you have three you have a compatibility but it's only on the drive category so it just stops at some point okay let me not try to remember at which point um okay 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 do unravel what this means this is the form of the cotwitz conjecture so this action of the representations of the l group this encodes the cosmology of wrap up what zing spaces and you then take isotopic components under some guys in this file part and then this computes the output of this function in terms of uh the action of this category of representations of s phi which is doing some combinatorial thing like realizing some jucky length correspondence this takes one uh representation with l-point-meter phi and produces another representation so this gives you the representation theoretic part of the picture and then it also tells you how the real group acts namely well it's also given there's a form of cotwitz conjecture on the cosmology of basic wrap up zing space well this is the form of the cotwitz conjecture except that uh it doesn't really tell you what this action is so proving the cotwitz 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 in this well which is essentially due to caleta except uh of course it didn't have this formalism to state it but caleta uh conjectured exactly how this category c phi or the irreducible objects of c phi look like um okay so let me assume for this a g-square z-split so then what caleta 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 uh and fix the later caleta unique generic representation of g of qp which is the one which corresponds to a 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 qp's and they should be given by certain representations of s phi which have a certain behavior um and well if you combine this of all possible different b's then they should be parameterized by all possible irreducible representations of the s phi and so he relates irreducible objects on both sides but actually this should be an equivalent of categories and and this should also be true uh what do you do at the gqp equal to what? g1 of qp so i mean if you take for b the trivial element so then uh comparatively with the irrepressible object 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 an equivalence so this isomorphism it depends on the choice of vitica data but some of the action 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 point three version of this conjecture some of that 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 and so so this is actually also something that seems to be in reach so there is work in progress of Carl Leder and Weinstein where they use the left shed trace formula and checks 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 satisfies 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 coromology of these basic robot thing spaces really is realizing this correspondence of representations and so let me just say that this generalizes uh work of fountains and straw for the limintate case of pharx for some unitary groups pharx and unitary groups i think u 1 n minus 1 right and also meeder had some results of gs for gsp4 i believe so there are fountains and straw proved by some purely local methods using the left shed trace formula and limintate or drain fuel tower as that in the coromology of these limintate spaces you realize the jucky langans correspondence and this is some large generalization of this and so my time is up so thank you very much very nice lectures are there any questions or comments i mean in this middle backboard of course there's also the action to deal with whereas in the lower blackboard there's no action to deal with well some as this serum tells you that the action of the very group is already as expected and everything comes down to determine and what happens on the for the periodic groups and for example for the limintate tower uh which you can't quite apply so if you pgl or something uh or prove a version of the serum which allows the center um the serum would already tell you that if you plug in some super customer representation of gln what you get is some representation of the division algebra tensor with the correct l parameter and then if you combine this or tensor with the l parameter that is defined by these methods um if you combine this with fountains and straw paper which tells you that you see the correct jucky langans correspondence basically see that you get the correct thing but actually the output of this is that actually what you see here in this picture is really the usual local langans correspondence using some global thing to determine this no this is purely local this argument but i'm using that uh there is a local characterization of local langans for gln that is what's realized in the limintate tower well i mean when yes but that's all that it is a local langans correspondence is proved by global methods