 Last talk before the weekend. I hope you're not yet too tired Right, so As in my last talk, so I have a connected reductive truth over QP or QP could really be any Be any non-accommodation local field, but for concreteness QP And so then last time I introduced this Steak bungee which was a steak on this category of perfectoid spaces over Fp bar and And it was sending any such perfectoid space to the G bundles on XS And I was discussing some of its geometric properties and then on the dual side On the length dual side, which I didn't talk about last time So there we have some prime. I fixed some prime L not equal to P because I will look at some kind of eliotic sheeps and At some point on my talk. I need to assume that this is not too small and I Mean in our paper we call this very good, which sounds like a very strong condition I just want to point out. It's really not a strong condition. So if GS GLN and this is really all L And if G is classical in the sense that has been used at this conference, then it's just means that L is not too so And You can also just I mean I'm always working with integral coefficients the L and if you work with QL coefficients You can just also completely forget about this So on the dual side we we have the dual group G head Living over the L and I guess it comes with its action of gamma, which I recall was The absolute color group of QP for me Right and so then I Gave a talk this morning where he explains that there is this modular space of L parameters Which were in the end somewhat denoted as some of the space of one co cycles from from the way group of QP to the dual group One co cycles was suspect to to this action and then up to G head conjugation and These were I mean I Like I like to advocate that when you do algebra should always replace all your topology things by condense things Anyways, and then when you do that, I mean you don't have to worry about the continuity to just do the obvious thing so So some of the technical problems that That's spent some time discussing somewhat just completely absent in the world of condensed mathematics And then let me state a theorem that's probably discussed in more detail in Jump on so that's next lecture. Just want to point that out. So this is some kind of reasonable scheme It's a district on union actually and this this John Union of Fn schemes of finite type Was the L it's flat and the local complete intersection I guess the English is a complete intersection, but and you can also say what its dimension is its dimension Equal to the dimension of G which is a mention of G head. I Mean dimension relative dimension, I think And Throughout my talk, I will pretend That the district on union wouldn't be there Actually, otherwise, there's something kind of very small technical Changes, I would have to make that don't really play any role So basically if you would fix an open subgroup of the wild energy that extravially this would literally be true and then as you Makes a subgroup of the wild inertia smaller and smaller some more and more connected components arise and Okay, there's some abound ramifications. So to say It's a minor thing So I guess I will sometimes just call this and speck are using notation from John Franco that's talk Okay, so so so back to the I Want to talk about LAX sheaths on so I mean one of the main constructions really in our papers Is that they exist a reasonable category some kind of derived category of LAX sheaths on bungee? I mean, that's maybe that's or it makes this part of the serum They exist there is such a reasonable category This actually took us a long time to really do this with CL coefficients It's much easier if you work with Z mod L to the end coefficients for some n Okay, you can also do it with the L coefficients Now the passage to the L coefficients is not as easy summer as an Algebraic geometry where you can just formally in some sense takes the inverse limit overall and In that case you would get some kind of Banach space representations of LAX Banach space representations of your groups But you don't want Banach space representations You want just one that the L pretends like it's discrete and so this it's not so easy, but it works Such that So what I said last time should be true is true So some are D of bungee is some are stratified into the categories for the strata and so the technical way to phrase it Is that it's to say that it has a semi-octagonal decomposition So some are great pieces and derived categories on all the strata bungee B so There's an enumerated by the elements of B and B of G which enumerated the strata of bungee and on each stratum You really just get a category derived category of smooth GB of Qp representations over the L And I will just denote Probably abbreviates this here to D of GB Qp and I might introduce includes the coefficients are I might not Good so What else it's a nice thing so It's a derived category right so so it better be something like a triangulated category and Let's just say triangulated In some point, I guess it will be upgraded to this higher categorical land and become a stable infinity category and it's it's One of these good guys. It's actually has kind of some nice nice generating class So it's what's called compactly generated So this means that there is a class of objects. That's about generalized the whole category under cold limits And we're humming out of these guys well Is a stable infinity category setting I could be would be allowed that humming out of these commutes with all cold limits In the triangulated setting I have to say it commutes with all direct sums And you can even characterize the complex objects so an object in here is compact if and only if it has bounded support finite support and the restriction of a to each stratum which you then can consider as just a complex of representations is compact Well here it just means that it's Yeah, it's basically a financially presented object except you might have to be slightly careful over zeal of how you say it's a Some the compact generators here would be to compactly induce Say the trivial representation The case a pro p subgroup Or prime to L at least Okay, so in representation series. This is very important notion of it from a representation being finally generated This somehow captured by this notion of a compact object in this category and you can also translate some other things into Into this setting of bungee. So for example, there's one important Duality on smooth representations. I'm not sure if it was talked about here at the summer school There's a so-called parents and say Levinsky duality Which is a self-duality of all the finally Generate representations and this is actually also defined on bungee. So technically I could say that This is what's called a self-dual guy The category self-dual so there's a kind of duality functor and this corresponds to the brain sense of Levinsky duality Let me out. I mean these are not really relevant for the rest of the talk Let me just also include another very important notion Which is the notion of an admissible representation that this can also be Phrased on bungee somehow corresponds to a very natural notion on a sheath That's Mr. Representations that correspond to the so-called ULA sheaths University locally a cyclic. Yeah, sorry for all be Stable Sorry again Yeah, it's a somewhat a split in some sense same also gonna decomposition Yeah, because I mean when you're when you are so the question was whether the same also going to the composition So I have some extra adjoins and it has yeah So they need to start the cataclysm and re-embeds in the big category either by J. Lois star by J. Lois Schrieg functor How the compact objects internally compact I Don't think so. I mean it is not true at all in representation somehow because if you tend the two compact objects And this is a huge thing. All right So there's a reasonable category That some are contains all the smooth representations that you care about and not just for G But for for example for all of its extended pure in the forms. They are all living there together But and all many notions from representation theory can somehow be generalized to the setting of bungee so some kind of generalized representation theory setting I Mean should maybe point out that often when you summer past Pass to a geometric version of something then it's just really connected to the previous setting by some kind of analogy But here it's really there's literally fully faithful functor from like the cascading representation series And so let me now face the main conjecture Which is this can be regarded as a refinement of this categorical local England's conjecture that was talked about already a bit and I guess for this conjecture I should assume that G is quasi-split assume G is quasi-split and To fix the vitica data You have G and the fixable L you can do that because you're in this plaza split case And it has a unipotent radical and then you fix this non degenerate character On this unipotent radical which I guess can go to Let's say it goes to the vectors of L bar cross is and jump from so that's talk I think that they have been some talks with what someone explains that if you really want paid them down a precise local language Correspondence that becomes necessary to fix vitica data and so on and so the same is somewhat true here I need to fix the vitica data to really write down a precise local language correspondence And so the conjecture is that then that if you make this choice Then there exists a canonical equivalence and sometimes people object to such formulations, but let me do us over this Between two things so one one thing is this thing we we care about D of bungee and now I guess I need to extend my scalars to this field right the other thing down here and the other thing is bounded sorry let me Pass to the compact objects in here That's often denoted by such an upper omega So because this category is compactly generated all the information is already contained in the compact objects anyways Your reducibles are in there and so on so it's not much of an issue and then the dual side should be Basically is a bounded derived category of coherent sheaths on the on the dual side on On the stack of our parameters. I guess implicitly I'm again base changing this here to the bitvectors of F L bar And there's a small modification I have to take you have to ask for again for this new important singular support conditions that you might have seen a couple of times come up in Sam Raskin's talk Let me just say that you can ignore this if work over If you invert L So if you work with ql coefficients that of the L coefficients All right, that's the conjecture so It's mostly because I need a vitica character Which needs all the people are woods of unity and implicitly I guess I've probably chosen a square root of p Maybe I didn't because I might have chosen a different action on the gov and g head Let's say I did but then it's also contained in there and Let me just I mean this is pretty like let's just say say Already say something about what this should do. So one thing it should do It should send the structure sheath here to the vitica sheath This is a condition we've seen come up a couple of times. So this is what you get by Extending by zero from the stratum for for bond one It's a compact induction from you Qp to gqp psi Wait j1 Let me write this down here j1 or generally jb will be the inclusion from some stratum into And I regard this representation here as a As a sheath on on on the stratum for equal to one, okay, and yeah, so I should say that As I said before so some of this contains the category of representations of g of Qp in particular But it also includes the one for all b basic Or all b but for b basic in particular you see all the extended pure inner forms appearing here and So even if I assume that you when I'm somewhat interested in the local angels correspondence for a group which is not quasi splits And you might think this conjecture doesn't say anything about that But actually it does because it might still appear here as an extended pure in a form and basically you can up to Modifying the center a little bit you can reach all all inner forms in this way So it really says that even if I assumed in the beginning that use quality split it really says something about all all groups and Some are taking care of these canonical parameterizations as in like maybe touch the call it as work how to how to do this for inner forms, okay That's the conjecture Let me contrast this with what we can prove so far The main serum is something a little related but slightly different I one thing you can observe in this picture here is that I Mean this side is something where you can tend those things together if you have two career in chiefs You can take the tensor product and you might be a little bit careful with staying bounded if you take derived and the products But basically here you can act by tendering with things and so you would expect that you can also act By the same category somewhere on the other side and this is what we construct so there exists a canonical action of Perth and now there's this unfortunate circumstance that perfect has been used for different things So now it means perfect complexes These are basically like one of career in chiefs except You can really represent them by finite complexes of vector bundles, which is automatic if the scheme is smooth But not otherwise So you take the same stack and this acts on I mean this in this case I can even do it with the L coefficients I mean this This really works for all G you don't have to and without vitica datum Just there and if you want you can either put the omega here. You don't doesn't matter So this is known as a spectral action. Okay, so this can be Constructed this structure. Yeah, so and this can be used actually to make this conjecture a much more precise Maybe you can really ask that this should be Should be linear over this Equivalent should be linear with respect to this action and this basically determines This I mean forget a bit pretenses actually was smooth So generically this guy is smooth. So on a very large part. This doesn't make much of a difference then this is basically Perth and So, you know where the structure she goes But then everything else you just get get by acting why the perfect was a perfect complex on it So, you know where any perfect complex goes under this equivalence and then the conjecture is really just that this Functor that you've constructed is an equivalence of categories. I should say that Our paper someone not really not the first one which constructs such a spectral action So we are really like looking at what people in geometric Languines have done and then just translate whatever they've done So maybe the first paper that did something like this in the similar setting is this paper of Nadler and dune on Betty geometric Languines and then Like these six authors are rinking Gates, Kerry, Kastan, Ruskin, Rosenblum, Varshavsky Let's somehow explain how that argument should work in general Yeah, all right. So this is our main theorem and this might look a bit abstract So let me deduce a couple of much more concrete consequences from this. It's an action, right? So in particular The unit object here should go to the unit object here Okay, so unit object is a structure she fears the unit object here is here I mean it's yet goes to the identity functor here, but it's a functor So the endomorphisms of the unit object here should go to the endomorphisms of the unit identity functor here It's a bit of a brain fuck, but It certainly implies that there is a canonical map From From the endomorphisms of the structure sheath Towards the endomorphisms of the identity functor What is this Well, this is precisely the endomorphism of structure sheath up precisely the global sections of the structure sheath, which is precisely Rg head so this appeared also in drum form so that's talk This here is some of the Bernstein Center of the category of sheaves on bungee But just because you somehow have the category of representation sitting fully facefully inside there It's certainly maps to the Bernstein Center of the group G of Qp or of any of its you know forms whatever But now G was general anyway, so let's not worry about this some particular you get a map as was promised in Dutch talk from This ring of invariant functions on the stack of Alparameters towards the Bernstein Center of your group and One can use us to do something even more like concrete and which is really what we're maybe What we've maybe been setting out to do To construct Alparameters, so at least same as simple Alparameters Let's say K over the L of some algebraic closed field So basically I want to handle simultaneously the cases that K is either FL bar or QL bar and Let's say we have an urge smooth smooth representations with K coefficients and supplies an irreducible smooth K representation Our group G of Qp then we get the same as simple Alparameter by Pi from G out of L So that's it's not a homomorphism. It's a one core cycle in general Or you might also write as a homomorphism to the L group if you want it up to L conjugation Sorry K in our paper we call this field L, but I thought it's confusing Because they're also L parameters So where is this coming from? Well, so we have this map from Rg head towards this Bernstein Center, right? But the Bernstein Center that acts on any representation. So it certainly maps to the endomorphisms of Pi and the pi of some are K linear There's also an homomorphism over K, but by Schur's lemma This is actually K. It's actually you don't quite need irreducible. You're sure irreducible would be good enough So you get a map from Rg head to K so in other words we get a K point of the spectrum of the sky and This is really just a different. I mean this is Basically by definition It's a coarse modular space. You take the space of co-cycled and then pass to the course quotient by G head and Again that alluded to like geometric points of such a thing They are precisely like say my simple conjugacy classes in here. So they exactly say my simple parameters So we get a completely uniform construction of air parameters That just works for any group over any in our community in the local field The question was whether we can see that this alpha meter is invariant under passage to the Brinsk and Silvinsky dual and G Yes, we can see that. I mean, maybe there's some dual that I mean I guess there's some expected behavior for the altimeter and that that works. You can also see that's behave Yeah, so we approve a proposition that with many many basic operations is compatible So it was passage to the smooths dual with passage to character twists passage to central characters parabolic induction Yeah, and some of the Torah, it's the right thing and for GLN. It's the right thing. That's basically what we proved. So There's some of the question does this agree with other local language correspondences and I mean, obviously should there's obviously the right thing But It's easy to see that for Torah. It's okay and for GLN. You somehow know that Now by someone Harris Taylor, you know that the local language correspondence is realized in the common module of limited towers and limited towers There are examples of such local Shemur varieties which Jared explained Realized by modifications of vector ones on the far front-end curve So it's very very much in our setting and we can use this to prove that for GLN. At least we get the right thing It's also true for inner forms of GLN There the issue was that it When we when someone Fargan I wrote our paper, I guess it wasn't really known that for any inner form of GLN The jacket as easy the local language correspondence is realized in the homology of some local Shemur variety It was only somehow characterized this local language correspondence by the Jackie Langland's character identities, but then Hans and Carlita Weinstein Proved Some cases of the Codwitz conjecture that summer so in some sense they showed that in this geometry of bungee in on this local Shemur varieties the core character identities are realized by studying some left shit straight formula on these things and so Using this they could deduce it for inner forms of GLN and more generally the argument should show some I show that More or less that if you can do it for the quasi-split thing then you can do it for inner forms There are some other isolated cases where people have done something for GSP for Lino's Harmon has proved something But in general it still seems like an open problem It's even I think at the time of Open whether it agrees with the construction of Genesee and Lafauce in the function field case where their construction is of extremely similar nature and you would expect you can just compare the geometry but It's not obvious that you can do that So the question was about this map for GLN. So for GLN Helm proved or Helm and Moss proved that this is an isomorphism over ZL and I think one main input into the argument is really the existence of the map Which is somewhat supplied by our argument But I didn't check how much you can simplify the argument So I don't think that's a completely formal argument that once you have the map it's an isomorphism Okay, I mean if you have maps both vice and of a QL there is a right thing then obviously it's an isomorphism. So yeah, okay another question Those inner forms Yes, but okay, we proved some of that Okay, so that's that All right So let me say a little bit about the construction of this Spectral action And let me just for simplicity assume that she is split we went to great troubles not doing this assumption in the paper So so then let me actually recall something from Sam Raskin's talk So he was implicitly Why explicitly I'm not quite sure staying something that So here we want to construct an action of this category or in something else and we will Reinterpret this in terms of some other data. He was having something where he was He wanted to construct some drill felt sheaf was which was basically a similar very similar object So he was reinterpreting what it means to give an object and here in terms of some other data And I want to recall that first so in Sam Raskin's talk the following was claimed that Assuming you have the following data the data of the functor from representations of the dual group to the eye representations of Like the veil version say of the pi 1 of a curve There was no suddenly usual curve over of q just for a second to the eye and then in some sense Functorial in the finite set I in a set I sense I don't want to make precise right now so So why would you ever consider such data? Well, this is precisely the data that comes from the core module of modular space of stucas after conjuring theorem so Exactly the data produced in congeal is talked Why so what did congeal do? See she fixed some finite set I which was some parameterizing the legs and she fixed the representation of the dual group to the eye Which was giving here the precise modular space of stucas at some of the bound on the legs and then Whenever you have such a thing then you can write down a modular space of stucas You can take its core module and then you get some sheath on eye copies of the curve And then she was spending a lot of time showing that this is actually an in-lease sheath And so actually maybe let me just not write Let me write some kind of derived category of representations of this will say ql bar coefficients and then she showed that actually what you're getting is just the representation of pi 1 veil to the eye on a ql bar vector space and then there's some of some compatibility some of it if some legs collide and it's really again a modular space of stucas with one leg less and then The commodity doesn't actually change. So this is what's encoded in this Functorality, I didn't specify and then the serum is that this is actually the same amount of data as an object a classic complex of classic Korean sheaths on Like the stack of L parameters so homomorphism from the way group To the new group and so So you get this dream fear and the thing it's somehow characterized by is that If you apply this functor to any representation V Then this would be the same thing as taking the global section. So let me raise our gamma on the stack of the dream felt sheath Tensored with V applied to the universal Representation so in other words you can get the coromology of all the modular space of stucas Which is exactly what's encoded in this these funny functars in terms of taking this funny sheath drift and tensored with some so this was a formula that appeared at the end of this talk just When I was empty, so you just get the space of automatic forms here and you can forget about this factor And that was also more or less appearing in met amethyst talk in some analog question more varieties and And then some of the extra thing he said was that this guy should actually be just some random sheath, but then Some of the extra input he claimed was that this should actually be just a dualizing complex of this step okay, but anyway, so there's this translation between constructing something on the stack of L parameters and Having such such amounts of data Okay, and so the similar thing will be true for us so there's oops So here's a serum Here's a serum that again is from my papers partner So this is a serum that So we work quite hard to do this with the L coefficients and is in this theorem some of the assumption that L is very good comes in So let's say we have some I guess I have to say the word zeal linear stable infinity category now So this one the applications will just be this D of bungee Item potent complete yes, so whenever you have an item potent and the morphism of an object you can Split split the object according to the important As I probably will be the maybe the compact objects in there Never mind so then Giving an action Of this category of perfect complexes. I guess now that I my group of splits is really just homomorphisms On see Is a covalent to giving well, let me Not erase the main serum the main conjecture For all finite sets I an action of The category of grab G add to the I so that's as is here So representation over zeal These will actually be the the ones I'm fine projective module. So there's some rather concrete category on The end of morphisms so it maps to the end of morphisms of C But you don't actually want just an amorphism But it should also be a representation of your group to the I is here So you want some let me just write to the Wqp to the I this means Wq to the I aqua variant objects So any any V in here? that's to some TV which is from C and To see but actually doesn't just go to see because the aquarium so it actually gives you Wqp to the I aqua variant objects And so this should be in What should to satisfy this should be exact Zl linear And one idle and take all this data and then you have to say it's functorial in I okay This is the data you have to produce Why should you ever be able to have to produce such data well You want to have the category of perfect complexes on here acting But this is a quotient stack. So in particular it maps to like the point much he had and there is a Like in particular vector months as the representations of G check So if you have representation check you can pull it back to the second then act and this is basically what this would do If you just have one V But okay, you should also be able to do this if you tend to a few of them together But the point is that it should actually become from here and over here This vector one suddenly get new endomorphism So basically given by this algebra R and this new endomorphism there in some sense and precisely encoded by getting this extra V-group aqua variants here Okay It's just trivially They're trivially on here So it's just It's just an open so this category is just the category of an object on in here plus it Plus an action of the very group Let me not try to worry about what what's the function in I is because I would get confused But I mean here and here is a model I use just a standard representation right and it goes to the identity and defunctor here Well, this is a monoidal category and so you can take limits a model categories Okay, so and so how do we construct this so construction is that TV is it's just such a heck operator as they always exist in geometric plugins and I will come back to this in just a second How they're constructed. So basically in geometric language using cat cooperators. You always have these Functors from here to here to somewhat I copies of your curve and then in here you have a full subcategory Where in the curve directions the sheath doesn't actually change So instead of like a varying sheath you some of the sheath is always constant But when you pass around a loop in your in your curve, you pick up a monodic representation So I'm with a vague group and yeah Hi, one of your curve of the 5 from 10 curve is some a big group of Qp So If you're locally constant the curve to actually learn then in this category And so the claim is that the second right actually always factors through here. And so these are your heck operators Okay, this will be how we do it So so let me be more precise So let me actually Come back now again to a rather concrete example So let's assume that Let's say we offer only one leg one Let's also say that the group is just you too So the dual group is also to you and these just a standard representation Then you can actually write down the heck operator So then here we have the stack of ranked two vector bundles. Let me just write this as fun, too Then over this you have some Hacker stack Which would in principle depend on all the decorations, but let me just not write decorations and this should map to bond two times the curve and I said previously the curve doesn't exist. So this is a bit fishy and I will explain what I should actually write that But let me first write down what the second stack is The parameter is two vector bundles for the two projections a point of the curve and the map so e and e prime rank two vector bundles x the point of the curve and We won't size in a second and f is a modification of these two vector bundles so it should be an isomorphism away from this point x and and Then it should have a prescribed behavior at the point x which is encoded in this standard representation choice of the standard representation here And it's actually just gets these very simple Modifications where you just get an embedding of vector bundles e to e prime There's a core kernel is a is a degree one skyscraper sheath at x So if I say take start to analyze this if I really just take the fiber product here over a point Given by the trivial bundle O squared Then what I really have to give you is a point of the fuck from 10 curve But recall the points of the fuck from 10 curves. They were really parameterized by until so Let me okay. Let me just call this here H The fiber product so H. So what should it actually parameterize so it parameterizes Points of the fuck from 10 curve, but these were untils up to Frobenius that's sharp plus The modification of the trivial rank two bundles there and so the to determine this modification You really just have to understand what happens in the fiber here And then there's really just somehow given you have this rank two bundle here And then you're modifying this at one place. So it is really just given by a line So plus a point of p1 over the until and so more precisely actually well what parameterizes untils What parameterizes is until this bar qp diamond? model your Frobenius and Then actually it's a p1 over that if you're think about it Some can let me write it as this thing so H maps to this thing and the fibers and Now this thing this is not the fuck from 10 curves is something different But it's actually what parameterizes relevant data. So this thing here is actually we denoted by diff one It's the thing that parameterizes like degree one devices and it's actually this thing that you have to put here So that's a bit confusing that on the fuck from 10 curves the thing that parameterizes points on the fuck from 10 It's not the fuck from 10 curve itself. It's something slightly different, which looks very similar and it's called diff one It doesn't actually make a bunch of a difference for what I said before for example, yeah, so if it's not already over Fp bars and I should base change it. I should really write the one to the idea It's still true that pi one of diff one is the very group Because implicitly I guess because everything here was f over fp bar. I should also base change this here from fp to fp bar And so in this sense it should be And then I mean the fundamental group here This would just be inertia and then you have this extra Frobenius here. You so you get the very group of qp okay, so if you actually fix it point here fix an until here also Then the common fiber product here would just be a p1 over cp Which parameterize the modifications maps to the hacker stack and Then so you can modify the trivial rank to bundle by something in P1 and get a new vector bundle and then there's actually some kind of interesting geometry where So I see two strata and p1 of qp Maps to the line like vector bundle O plus O of one And the complement maps to the complement is called Drinfeld's upper half space omega 2 So it's it's not an algebraic variety or anything, right? This is a complement of this profile I'd set inside here This maps to the vector bundle O of a half and This actually gives you some kind of chart of one to so far We run two or something extremely sticky for us, right? But now you actually get some sensible geometric objects mapping to bun two and then you see that the strata of one too They actually kind some kind of pretty wild geometric objects because suddenly is p1 got stratified into a profile and sentence complement So the geometry of this stack is really rather rather wild somehow Anyway, so This would give you one example of a hacker party to start with a chief here pull it back to this hacker stack and then push forward here and Then you get a sheath on bun two times well this The thing that's still just one point really right? I mean this is not a curve It just has one point like it's just just a field and so it turns out that in this sense If you just apply a hacker operator for one leg So but you don't actually have a choice but to be locally constant this direction because there's anyways only one point And so this inclusion there is actually just an equality and to automatically gets a factorization So this is much easier than in geometric length and so as a curve is a curve And you actually have to prove that something is locally constant here. Your curve is still a point Well, I guess I'm out of time already, so let me stop any question There are perverse sheaths Let me Get down this other board. Yes, maybe one thing I should have said is that like if you Like here we were looking at the simplest kinds of modifications of vector bundles And so you can consider more general modifications of vector bundles. So For example, you get some you can look at the modular space of modifications of the of the trivial G bundle. So this is the the effort in our setting and So this effort in grass manian Well, you can somehow take its CP points or something Because you need to specify it where this modification should happen you some of us need to specify an untold and this I just did and Then this is given by taking the so-called be the wrong points of G and divided by the G be the wrong plus points of G where Be the wrong plus is a traditional purely cot theory name for taking the park from 10 curve and then CP gives you a point on that and you can take the complete local ring at that point Because it's a curve. This should be a complete DVR and it is and Be the wrong is a traction field So this something's also called to be the wrong I find grass mania because it's some a given by this fontan construction of this be the wrong period ring You can also write this as an LG Where LG plus G is some of this thing and LG is kind of this thing and Then there's again a geometric satark equivalence So you can define a perverse chiefs on this FN grass manian at least L plus G equivalent ones and These are equivalent to representations of the dual group and this is actually what the dual group is I Mean when I first learned about Langley and summer I was always totally confused what the dual group is where does it come from? I mean, it's just some ad hoc construction and then okay You can't translate some similar simple conjugates classes from one group to the another and then someone works, but Where does it really come from and it comes from geometric satark here? So When you want to define this heck operators here You somehow you can look at the stack of such modifications of G bundles and large generality But then you want to pull back But then you want to tend to buy something here to make it some Have bounded support and so on and then push down again So you need to tend there with some reasonable thing up here And so the things that you should tend to buy are some of things that only depend on what the modification locally looks like So it should be precisely sheaths on this grass manian and someone to make everything well defined They must be a plus G equivalent and turns out that then the right ones is to consider other perverse ones and they Turn out to be equivalent as a tensor category to the representations of the dual group And so this gives you a construction of the dual group here So you can uproar and doubt this with a makes them to knock in category and get a construction of some group scheme over Over the L in this case and then you can just identify which group scheme it is and you see it is a dual group And so some are also canonically on here acting you have the Galois group So canonically acting on here you also get the Galois group and then it turns out that this is the usual action of gamma on the dual group Except not quite as a cyclotomic twist and so this actually also explains the C group that appears sometimes because Actually the usual action of the dual group is not the right one you should do a cyclotomic twist in it And then this is the one that actually appears in geometry You can usually trivialize it by for example choosing a square root of p but in this case So choosing a square root of the cyclotomic character you can trivialize it And you can also define perverse sheaths on Banji if you wanted to but there's probably not a completely general notion But for the most important things you can define what the perverse sheaths are Is the curve but I mean after all it looks like this is like an absolute park-front-end curve It's not living over any perfecto in case, right? Right Except I mean So the right the question so the difference between Div1 and the park-front-end curve You see Div1 You're modding out by the Frobenius of Qp which doesn't really make any sense But actually when you tilt it, so when you pass this diamond it actually acquires a Frobenius as everything's among characters to be as a Frobenius So you can divide by the Frobenius of Qp. So this kind of guy doesn't actually even live over a spark Qp anymore So and it's also just a diamond. It's not It's not an attic space of a Qp or anything So it's they live in kind of different worlds of one is an attic space of a Qp or It's many many of them because for any kind of C you can build one or any S you can build one The other is just one fixed diamonds that lives in Yeah, in characteristic P only really because you've suddenly divided by a Frobenius of Qp, which is a bit weird The question was why Div1 occurs, right? I mean you need to specify where Yeah, let's actually make this a little more precise. What's actually happening here, right? So I was somehow Giving the impressionistic sketch and then when this track of their objects actually live. So this E and E prime Like what are the S-Void points? So these are vector bundles ranked two in this case over the far front-end curve XS and Now I want to somehow say basically say an isomorphism When you restrict your XS minus one point, so I guess I should someone minus one point beats minus An untold S-Sharp in this case Right because we saw that if you have an untold S-Sharp Then we can actually embed this canonical into wires and then project to XS So what we need to give ourselves in order to say that well It's basically giving an untold of S But what parameterizes untolds of S was it was spark Qp diamond models by definition so we need to We need to take a map to spark Qp diamond and then well because we don't actually need to map to wires But only to XS is actually good enough to have it modular Frobenius Well, so when I say I want to somehow specify the point where this Coconal supported if you actually think about what this actually means You don't actually give a point of the front-end curve because there is even not the front-end curve So this is wouldn't even make sense, but instead You really have this kind of funny Div1 occurs So Farg likes likes to call this a mirror curve And it's called the phone because I mean for any for any D you can classify degree D cut your device So there's also some diff D's, but okay. Thank you