 First of all, it's a great honor for me to be at this conference, so for once I decided to act more responsibly and I've somewhat reduced the scope of this talk, so we might not actually get to what I call fundamental local equivalence. Basically, let me just say in words, the story is like this. So the fundamental local equivalence is a basic local ingredient one needs to formulate quantum languines, but that happens on the Durham side of Riemann-Hilbert and basically due to time constraints I decided to stay in this talk completely on the Betty side of Riemann-Hilbert. So there is an extra story going on maybe on some other occasion. All right, so let me begin. So let me remind you the usual geometric satake. So let's say we start with a curve x and a point and the story will be completely local around this point and just to introduce a notation. So I consider the group scheme geofo x is the group of arcs and the group of loops geofkx and one considers the affine-grasmanian which is the quotient. So let's say everything is taking place over some algebraically closed field. All right, so in the usual geometric satake one considers the following category. So one considers perverse sheaves on the affine-grasmanian that are equivariant with respect to geofo x. It's a geometric analog of the of the spherical Hecker algebra in the theory of Piatty groups. So this category has a natural monoidal structure so in the basic theorem which is known as geometric satake reads that this well as a monoidal category it's equivalent to the category of representations of the Langlund's dual group. All right, so yeah, so I'll say that in a moment. So, you know, there's a bunch of comments I need to add. I just want to make sure which order I'm saying them. I pretty forgot the first one. What's the first comment? Yeah, so first of all well as we can see this category is not just monoidal it's actually symmetric monoidal category. What one can do well in the process of the proof of this theorem one promotes the monoidal structure here to a braided monoidal structure when introduced commutativity constraint that turns out to be symmetric. Okay, now what context is the same? So what do we mean by proper sheaves? I'll comment that on that in a second. So basically this is true in any of the familiar context so one can do a ladic sheaves working over an arbitrary ground field. So in this case this is a ql linear category and we understand g-check as an algebraic group over ql. So it does. So it's also true with fl coefficients or zl coefficients. So let me say all right so another context is one can do so if the ground field is is a characteristic zero one can do d-modules. So in this case if we call the ground field k this is a k linear category and this is also an algebraic group over the same field k or one can work over the ground field being complex numbers and one can consider sheaves in the classical topology and in this case one can take as coefficients in fact any ring any commutative ring. So but for the purpose of this talk we'll take field coefficients so over c. So this is also the is there any finalist condition like traversives are constructively usually and so you maybe want to have taken a part adjust the face of the because it's a union of a variety you want to take a fairly many cells and with both constructivity and the representation also with the old finite generative module over your ground ring so if the ring is not good then you have to know what to say. Yes so let's let's be safe here so for this formulation let's let's take field coefficients just just to be on the safe side so in this case I'll take really constructible perver sheets so they live on finite they're supported on finite dimensional pieces and here I'll take I'll just take finite dimensional representations. Yeah so this theorem is due to many people so kind of if the initial input came from Leustig then I believe it was conjectured by Drinfeld and then it was proved by Ginsberg with coefficients in characteristic zero and Mirkovich Villanin with coefficients in arbitrary ring. All right and okay all right so now so this talk is about generalizing rather deforming this equivalence so this equivalence as stated has the following drawbacks so first of all it does not extend to the derived level so namely instead of perver sheets let's consider the the derived category so my notation for the derived category I don't like to write D I like I write sheaves so and when I when I write sheaves I mean the derived category so I consider the equivalent derived category in the affine-grasmanian but then which part of this condition in a second it is important to allow infinity sometimes yes and I will say that in a moment so we will be in allowing infinite infinite things so let me say okay so at some point I'll have to adopt this convention and I may as well do it now I put no finite I put no finiteness conditions just it's the un unbounded derived category I know but you see if you want any influencing to be a direct limit of things supported on the product of constructiveness like for usual so you have the sheaves which are supported on that on the union of part of closures the fighting when orbit so this but you can never see if like the constant shape for your technique constant shape see on on the graspania is it a ship in your sense we'll I'll address this point in exactly five minutes I'll I will I will talk about this in great detail in just a few minutes yeah so yeah just um yeah we'll I'll introduce what I mean in a moment just notationally so because this is my notation for the derived category whenever I talk about a billion category I'll put a little heart so here this wasn't well it's the heart of the t structure so in this notation this meant an equivalence of a billion categories and so without the heart it would mean the derived category and one may want to compare the equivalent derived category in the fine grass manian with the derived category rev the check and even if your coefficients are characteristic zero these are not at all equivalent so you can explicitly say what this thing is and it's not that it's something else so so that is to say that this is it's a kind of fragile statement it only take it only happens at the at the abelian level so another point is and again this is kind of the main direction of the stock is that as we will see in about half an hour that um so the right hand side admits a quantum deformation so namely there is such a thing as representations of the quantum group and you may try to deform the left hand side but again this equivalence even at the abelian level will completely fail so and so the first thing that I'll do I'll replace this equivalence before any quantum deformation by another one that is better what not want to qualify things by an equivalence that will hold at the derived level and one that will admit a quantum deformation all right so okay specifically to address offers question let me say what I mean by the category of sheaves so we take as an input this the theory of sheaves on algebraic varieties of a finite type so if y is an algebraic variety finite type to it we assign something that called sheaves of y and so basically for now the only functor reality that we need is the following that if you have y1 mapping 2 by 2 what I need is well shriek pullback functor so up a shriek yeah yes so for the particular story that I'll talk about one can make do with without higher categories but it's again it's very fragile so better right away proceed this as as the corresponding high categorical enhancement of the derived category and now so now we are in the in the finite dimensionation and let me say here that I should technically specify what I mean so what I mean is the following in the world of demodules I take the entire unbounded category of of demodules so no finiteness conditions so in the world of constructible sheaves what I do this again it's a technical remark so feel free to ignore it I take the usual constructible category and I'll pass to what's called the incompletion is it just for ships or the derived category the derived category so I have them again in the constructible world I have the higher category well higher categorical enhancement of the constructible derived category of the bounded constructible category the kind of the usual one and then I take it what's called the ind completion is it different from the unbounded unconstructible it is different yes it is different but only was like was was trolling coefficients it's the same right yes but I yes you want ql coefficients so in with let's say if I work with ql coefficients yes I do the following thing I take the constructible category and take its incompletion but in the topological world I'll do the same I'll start with a constructible category as it is yes and then I'll take it in completion so usually you consider complexes with only constructible commodity yes this is what you do yes and for d models you take complexes with with quasi-coherent common no if the demolish I can do either of the following two things for d models what I can do I can do straight away I can take the unbounded derived category with no coherence conditions or or quasi-coherent oh of course quasi-coherence but what they said quasi-coherent the all module homology yes quasi-coherent yeah quasi-coherent yes but the object itself don't have to be quasi-coherent this is the point maybe many cases you can do it yeah all right so so now let me specify what I mean by sheaves on something like the fine grass manian so the fine grass manian can be written as a union of an increasing family well of close of schemes each of which is scheme of finite type let me say so it's an ordered set and if you have this then have this closed embedding and so the derived cat so the category of sheaves on the fine grass manian is by definition well I should write it like this it's the so in other words you should think of an object of the category sheaves on the on the fine grass manian as a compatible family of objects on each of these y alphas and compatible in the sense of shriek pullbacks the shriek pullback to each other so that's the definition again well there are time constraints otherwise I would be able to explain why this is exactly what you want okay so this is okay so let me let me so okay let me make let me add this comment okay okay it's a it's a general useful observation but it's very useful so let me say it like this suppose you have a family of categories indexed by some set so as c alpha c beta and there are functions and let me symbolically denote these functions by l like this and suppose it so happens that these functions actually admit left adjoints and I'll denote them which is exactly what is happening in this case because these are closed embeddings so in this case there is a general lemmon which I learned from luri and it says the following that the limit of the inverse family under these functions is canonical equivalent to the co-limit of the direct family one there's a tiny remark one has to add here namely do you assume that the raw shrink guys are fully transformed? no it's it's completely general so therefore you can think of this resultant category in well either as compatible family under shriek pullbacks or you can think of objects there as coming well if you wish as co-limits of objects that come from individuals individuals c alphas through the low yeah through the push forwards and so and this is like what I said this is like what you said before and this is an explanation why this is exactly what you want you want to think of the sheaves on the entire thing as constructed from sheaves on finite dimensional pieces so this presentation gives that point of view it this happens it doesn't matter for the it doesn't matter for this claim but in this case it happens to be filtered okay so here you are you may work with usual categories and then the usual two limits and two co-limits in the two categories they may also work in your higher setup no so this is this happens in the higher in the higher world not in the does it happen in the world of usual you mean you mean usually meaning triangulated categories no no just a billion of categories without structure this makes sense without putting structure in the category is it true without putting any yes this is true because a abstractly a category is a particular case of infinity category however one has to be careful namely if if the index set is not filtered when you if you if you're if you're taking a co-limit of abstract categories you may of just categories you may end up with a higher category so if the cat if the family of indices is filtered then you will stay within ordinary categories but I still don't see because in the co-limit every object cast from has given one no so it's so one has to be again I'll say in a moment so again this is a technical discussion that okay so what I wanted to say is this that so which world are we in we are in the world of categories that admit all co-limits and this co-limit is taking place in that world so so it's co-limit again taken in the world of categories with all co-limits you can explicitly describe it as follows if these categories are compactly generated but now are they triangulated they have some extra structure what kind of category are you looking are they presentable stable or not they again so that's statement about presentable stable infinity category this is this is a statement about presentable stable presentable cat infinity categories all right okay so precise statement so these functions are necessarily commuted with co-limits because they're the left adjoins and you don't need anything else uh so we are taking we are taking the limits and co-limits in the ambient world of well let me call them dg categories with that contain all co-limits and let me just say one remark but what are what are functions in this category are they functions that commute with co-limits so in this category let me say let me assume for simplicity that we take the functions that commute with all co-limits and in this case i'll so okay i want to simplify things let's let's take that let's assume that let's say that we take functions that commute with co-limits in this case we require that both of these commute with all co-limits although this can be made that can be relaxed yeah so let me just say address opera's question so precisely in this setting suppose these categories are compactly generated like in our case we take some small category and we our c alpha is is obtained as in completion of this sub some category of compact objects so then you can describe this co-limit taken in the world of categories with all co-limits as follows you take the co-limit in the world of small categories and then you take it into completion so in this world and suppose also the cat was in place of the category of indices is filtered in this world every object is really really comes for some particular alpha but in this world i am allowing taking co-limits of those so in other words in when i'm form forming this i'm taking i'm really taking co-limits of objects each of which comes from from c alpha as we do in the in the representation theoretic side all right so that was a useful digression about the nature of categories no it's really useful otherwise we don't know what we're talking about all right so now i want to introduce well the the desired modification of the left hand side so let me first introduce the symbol for it i call it the wittaker category in the fine-gross manian so and so let me say what it will be so and then i'll decipher the definition so it's sheaves on the fine-gross manian in the sense with i just introduced but we're taking sheaves that are equivariant against maximum independent against and on non-degenerate character chi so i'll decipher all i'll decipher what this means in a minute and it will be actually a full subcategory in the initial category so i should add the warning here that usually when you talk about the equivalent category the forgetful the functor of forgetting the equivalents is is not fully faithful but because here we're dealing with a group which is essentially nipotent this forgetful functor will be will be fully faithful so so this definition may look non-obvious for the following reason that well as we just discussed here our objects of our category are essentially of finite dimensional nature so there there are co-limits of of sheaves with finite dimensional support but here we're imposing an equivalence condition with respect to a group such that its orbits are infinite dimensional so we are kind of inherently dealing with objects of an infinite dimensional support so here some care well we need to take some we need to be cautious when giving this definition so i'll give the definition and then the question will be why does this category have any objects and we'll we'll discuss it in a moment so what do you mean by this so first of all because the group n is unipotent you can write this group in scheme as an increasing union of group schemes and i so for example in the in the case of SL2 this will be just the group of loran series and here we'll be taking loran series with a bounded degree of pole so whatever this equivalence condition is this will be again it'll be a full subcategor here and it'll be obtained as the intersection of the subcategories where i impose the equivalence with respect to this increasing family of subgroups so equivalence for the big group is just an intersection well it just imposes equivalence with respect to all of the subgroups so now i'll have to say what each of these categories is so well as i as we discussed here the fine grass manual itself is a union of finding dimensional schemes and for each fixed ni i can assume that all of my y alphas are are preserved by by the action of ni so we'll define this as basically but the same procedures here it will be the inverse limit under shriek pullbacks of sheaves on each of these y alphas with the equivalence condition so and now we are almost in the finite dimensional situation except that these groups are not a finite type but each of this and but so i have this ni acting on y alpha and this action factors through some group which is already an algebraic group of finite type with the kernel being pro-unipotent and so and finally i it's by definition the equivalence condition with respect to this quotient so and i'll say in in the moment well what i mean by this but here we are in a finite dimensional situation so we have we have well finite dimensional algebraic variety and an algebraic group acting on it so now what i mean by equivalence so pardon me so this there is no prime this this is the prime so this was infinite dimensional and i define it in terms of the finite dimensional quotient all right so let's pause for a second and so let me just decide for what this means in the finite dimensional situation so suppose well let me call it n prime suppose n prime is an algebraic group that acts on some y and what do i mean by this chi so chi is a homomorphism of algebraic groups from from to the additive group and so let me assume that n prime is unipotent as it is here so so so now we have to be a little bit more specific about the sheaf theory so suppose that we are either in the world of d modules or we are over a finite field so in the world of d modules on ga there exists what's called the exponential d module and in over a finite field there exists an artenschweier sheaf over ga so let me call them in both cases exponential so it's a particular sheaf on ga so now this equivalence so it's a full subcategory and it consists of all those objects f that have the following property it's not extra structure but property so so let me write it and then i'll write and explain what this notation means where act is the action map of the group on my y so what i want is that when i pull back my sheaf onto the product with respect to the action it splits as a product of what i had along y and the exponential and so the tiny remark is that because n prime was unipotent if this homomorphism exists it's it's unique and moreover all the higher compatibilities whatever they are are satisfied automatically so for a small smoke as far as i remember in the usual theory the last before back anyway the upper shrink is involves a pullback shift in the twist but there is a shift and so the shift does not seem to be because this is like a definition you would do is after star but it's after shrink it depends which chronological degree you put the exponential in no but if the space is infinite dimensional like here but it's no it's finer dimensional we already in the final dimensional situation you don't forget to pull back by kind pardon me you forgot to pull back by kind this x is an g i oh i'm sorry the exponential itself would be you may have to yeah it's i put it in the same chronological degree as the dualizing sheaf all right so this is what you literally do if your sheaf theory is d modules or if you are a finite field but what so but in this talk i want to be as i said on the betty side and i'll actually want to work over complex numbers with sheaves in the classical typology where this will not literally make sense so now i'll explain the remedy how to make sense of this exponential well when you don't have but when you don't have the exponential so it's based on the following trick so i call it the passage from whitaker model to kirilov model although i'm not sure the terminology is very accurate so let's consider the following situation so so suppose you have again the variety y on which just the group g a acts so i'll consider basically a baby model of this and it'll be quite easy to see how to combine this with what i said before to produce an actual definition so in this situation we can't talk about whitaker of y again the definition of mean sheaves on y which are where chi is the identity map from ga to itself so this makes sense but so but now i would and again this definition makes use of the exponential object on ga so but now i would like to replace this definition by something else that does not involve the exponential so and but you need an extra structure on y and that extra structure is the following that instead of just having an action of ga will have an action of the semi-direct product suppose you have this acting on y where gm acts on ga in the standard way so in this case i'll introduce something i'll call kirilov model of y and it's the following it's a full subcategory inside sheaves on y which are gm equivalent and it consists of the following objects that these are those objects such that if i average them against ga so there are two kinds of averaging she can star out use star you get zero well it's another category so there are two forget well let's say you have wittaker of y it sits as a full subcategory inside sheaves of y there's a kirilov of y yes so first of all shriek and star don't matter either or according to what you said because the two pullbacks differ by a comological shift so but now yes gm is not unipotent and when you define that you it's the high tech definition so you need some dissimplification yeah same dissatisfaction yes so we have these two categories but now there is a general lemma that says that there exists a canonical equivalence as abstract categories abstract meaning they're not at all the same as subcategories here that sit very differently but abstractly they are they're equivalent and the two equivalences are are so let me just say the diagram does not commute that's what i'm trying to emphasize there yeah the two entities are equivalent let me say what the functor is in this direction it's the functor you take wittaker sheaf and you you star average it with respect to gm and so as and you as you can see in the kirilov model you never you no longer mention the exponential and so that definition makes sense uh well or an arbitrary ground field and so let me so you pull back to gm cos respect and you pull back by apple star and lower and push over by star yes yes yes star averaging it's the functor right adjoint to the forgetful functor uh functor forgetting the gm equivalence and so by combining these ideas you can see that in our case there is this extra gm basically that comes from the torus you can give the kirilov style category here and so the definition actually makes sense however i'll continue writing wittaker uh but again so you should think of kirilov as the as the definition all right whoa it was a long time 20 more yeah all right so in particular we should take this action but this category is an action by the torus which one as you said you're gonna write it but we should compare this to that no the original category had an action of the torus and then i use this torus action to what which is like here you see yes why had an action of the torus and i could do two things i could take just wittaker and no longer have any action or i can use this gm action to write kirilov just like a philosophical question the wittaker implicitly seems to depend on the choice of the character chi but the kirilov doesn't seem to depend on such a choice yeah so in the kirilov you're saying kirilov does does not seem to depend on this choice yes indeed so suppose you are in this situation then suppose you have two characters that are conjugate but element by an element of gm so it's easy to see that the corresponding wittaker categories are equivalent specifically by means of the action by that element of gm so if you wish in this situation wittaker also explicit does not depend on the choice of chi yeah what's the language at some point the choice of chi should admit it no and that's a remark that i forgot i'll i'll add it right now so as peter is saying is that so my definition seems to depend on the well on the choice of chi so what was the sky so i said it was supposed to be yeah so i didn't specify what chi was so now let me say what chi was okay so it was supposed to be a character to ga and i said it was non-degenerate but which which one do i mean so first of all it's a character therefore it factors through the cometant and if you take the maximum independent modulo it's the cometant it's canonically the product of copies of ga according to vertices of the drunken diagram so so and now okay so now you apply this sum map just some overall vertices and now comes an ugly step well you'll want homomorphism from loran series to ga and well the ugly way to say it you will say okay i'll take the coefficient next to t negative t to the negative one but that involves a choice choice of coordinate and that's not something we want to do so we want to do we want a more canonical procedure so the thing would be canonical if instead of well this is basically loran series so this procedure would be canonical if instead of loran series we had loran one forms then the residue would be canonical and so this is what we want to do and for this we modify so to do this we modify our picture a little bit so we so to have a canonical residue we twist everything so namely as i said so we have a curve x and over curve well implicitly we were considering the constant group scheme g but instead of that we'll twist g by the adjoint action of the torus using a specific t-torcer so which one well how do you get t-torsors you can start with a line bundle and use a co-character so we'll as a low line bundle we choose once and for all a square root of the canonical bundle on the curve and we'll use the co-character two row so this is this is the torsor so two row is homomorphism from gm to my torus so i start with a line bundle which is a torsor for gm i induce and i get a torsor for t so and if you perform this twist then you will see that well unipotent group will also get twisted and you will see that these loran series will be will get replaced by one forms so in this sense the procedure becomes canonical so i choose a square root of the canonical bundle on the curve yes it's a line bundle i.e. gm torsor and i i used two rows homomorphism from gm to the torus i induce the torus i get a t-torcer on the curve so then i have a i have my group scheme g and i use the adjoint action of the torus on my group to to get a non-constant group scheme namely i twist my so if you do this if and if you consider the corresponding version of the fine grass manian then this character becomes canonical because your unipotent gets twisted by this row so basically roots of groups will no longer look like loran series they will look like one forms and or in some of the groups row which which doesn't exist yeah yeah yeah a simple co-weight but it may not exist as a co-weight but two row does and for that reason i need to choose this again how does kai even enter in the definition of the correlative model but does it just not enter in so you're asking so it's that gm that will let's postpone this because all right so let me make a few remarks so we might not even get to the quantum situation but please tell say this story properly so so i gave the definition of whitaker but so you can ask well why is this category non-empty and so let me make an argument that in fact it is kind of empty and then i make and then i explain that it's actually non-empty so if you consider the category and the fine grass manian or any in scheme so this category has a t structure so namely an object is declared to be homologically greater or equal than zero if and only if well remember remember an object here is a compatible family of objects on what we're called y alphas under shriek pullbacks so and we can think of this this compatible family is shriek restrictions of f to the corresponding y alphas so we say that an object is homologically greater equal than zero if and only if all of these restrictions are homologically greater equal than zero and when i talk about the t structure on the particular y alpha i mean the perverse t structure so now you can prove the following general simple emma that every object f in this whitaker category is homologically trivial so it's an epidemic spreading here so so i said this category has a t structure and for that reason for every object it makes sense to take its individual homologies i'm saying that for every object in my whitaker category all of its homologies are zero however the trick here is that so the t structure here is what's called non separated there are objects that non zero objects such that all of whose homologies are zero but the object itself is non zero a typical example of such is that dualizing sheaf so um so it's and i'll say in a moment homologically trivial it means that well we have a t structure for every object and take its individual homologies they will all vanish and so let me say that this is one of this so i learned two main things from jaco blurry one was well to operate with infinity categories but here this is not what's going on another thing that to learn from him is basically how to take care of negative infinity and that actually already appeared in peter's talk in response to offer his question about sheaves being sheaves of hyper sheaves so well before i met jacob i would write this mysterious symbol db of some category and i would i was hoping that this mysterious letter b would basically take care of things that basically i was i kind of had this belief that you writing db is kind of it's some kind of rule of hygiene and if and if you observe this rule if you just write it meticulously you will not get sick and everything will be fine but then i but then so adopted new point of view that so being being hygienic you're trying to get rid of some bacteria but some of these bacteria are really helpful and and these bacteria live at comological negative infinity and that they give rise to a lot of mathematics so if you were to work with bounded derived categories you will never see this but a category okay but now let me give you give an example of this object also answering peter's question perverse on on finite dimensional pieces so let me give an example of an object this is the basic object there so let's consider the orbit of the of the unit point of the fine grass manian by means of the independent group and it's usually called s0 and let's denote by i the local well it's it's also in in scheme and this is in locally closed embedding so the sky gives rise to a map to ga and all right so so now i'll write a scary symbol on the board and i'll try to decipher it so i take the following object in the category of sheaves on s0 i take chi up a shriek of the exponential so what do i mean by this well that's an example of an object which is well if it was not exponential if i took the dualizing shift that would be that dualizing so what do i mean by this well s0 itself is a union of finite dimensional pieces and as we said the category on s0 is this inverse limit when i say dualizing well i was i'm supposed to specify objects on each of the fine dimensional pieces that construct as comprises zero and i take the dualizing and this dualizing tautologically shriek restrict to each other and well the same happens for any shriek pullback so that's what this guy is and so let me just note the basic object delta zero it'll be where i lower shriek is the left adjoint which a priori is partially defined to the functor i up a shriek that is always defined in our category but in this well you have to prove that on this particular object this left adjoint is defined why is it partially defined well we have a functor it may or may not admit left adjoint this left adjoint is defined on some objects but you assume in your general setup that you have those in the c alpha discussion the well the lower shriek but this is not a part of this this is the locally it's a locally closed embedding so let's say in the world of d modules where we have non holonomic guys lower shriek may not be defined on all objects in the better side is always defined and moreover you can do the same produce objects what i call delta lambda where lambda is a dominant co-weight so it's a usual trick well familiar to people notomorphic forms that you can do it so if lambda is a dominant co-weight the sky gives rise to a map from s alpha where this is the orbit that means of the unipotent of well i hope you know what i mean by this it's a point in the loop group that corresponds to co-weight lambda and the t is my notation for the coordinate on the formal disk so so it will be kind of s zero it'll be some comological shift here so so i said that the objects of my category are comologically trivial yet when i view them as objects in the ambient category yet one can prove the following that so you can ask so so you want a homomorphism like this which is equivalent with respect to the given homomorphism chi on n and it exists if and only if lambda is dominant it's kind of usual game in castle menschallek a formula so lemma so please pay attention there exists a t structure on whitaker let me say such that an object f is comologically greater than equal to zero if and only if homo lambda so you can create a t structure on my category which is so to say generated by this object delta lambda so again from the point of view of the initial t structure my all of my objects are trivial yet this what's taken on its own the category that obtained has its own t structure generated by this object's delta lambda so now i'm out of time so we didn't get to any q but let me but at least i can state a theorem so that theorem says this that and this is this theorem is true again in any of the shift erratic context so my desire to state on the betty side had to do with q deformation but we didn't get there the theorem is that there exists an equivalence of categories and that happens at the derived level which is t exact so the category representations has its usual t structure and the whitaker category is it has a t structure defined right here and moreover under this equivalence these objects delta lambda go over well to the usual irreducibles note that lambdas were dominant co weights for g which is the same as dominant weights for g check so kind of the level of parameters that make sense and now in the demodal setup so the usually people work with autonomic or autonomic regular and and there are also big big quasi-query and demos yeah that means that they are coherent which are not aeronomic of course so now in the on the representation side only reducible representation correspond to nice things well i don't know but how can it be that you can allow non yeah so what happens is that in this case the objects of my whitaker category are automatically in the holonomic it happens automatically well basically for the following reason that well on each orbit of n of k well your object all of your objects are because yeah because of the covariance yes the in holonomist is forcing you by the equivalence condition this is this question that i wanted to comment on let me see there's few more just one one that's point number two and there's also point number three okay okay now i already forgot your questions what was it monoidal okay can we so i'm over time can we declare it over and then this these will be questions this is a question period a question period you didn't all right so that's okay monoidal structure so as stated this is just an equivalence of stable infinity categories it's interesting but there are more interesting things to say so namely this is indeed a symmetric monoidal category and you would like to see the the structure here so unlike the usual geometric satake where we had convolution this is not even naturally monoidal category so and here but enters a new world and that of factorization categories so what you so as we saw this category was attached to a point on the curve and there is another the whole new structure there so we allow this well we put we'll allow these categories at a moving point and in fact at multiple moving points and that's what makes this what's called a factorization category now every symmetric monoidal category also gives rise to to a factorization category and so the actual theorem is that we have an equivalence of factorization categories so I should say that what that captures is not the symmetric monoidal structure here but the braided monoidal structure here it actually makes difference at the infinity level so you recover rep g as a braided monoidal category so that's that was that question now luke's question is this so let me juxtapose it with the previous equivalence so here we had this let me write a slightly more complete answer to your question okay so I don't want to work at the abelian level I want so you have this entire what's called spherical category and well this category acts by convolutions on on just sheaves in the fine grass manian preserving any kind of equivalency that you put on the left see this this is really a monoidal category that acts here and in this monoidal structure this monoidal action is in addition to the factorization structure so now let me tell you what it corresponds to on the on the right hand side so as I said this category is not equivalent to rep check it's equivalent to something else so let me tell you what it's equivalent to so this is well quasi-co but you need this intco correction on the following I'm sorry to say dg stack it's point times point over it's a fiber product of a point scheme times point scheme over the Lie algebra divided by the joint action of g check whereas this should be thought of quasi-co of point times g check so because this is derived scheme who's underlying derived stack with underlying classical stack is just point more g check the heart of the t structure is exactly that so that's why originally when it passed through the heart of t structure you see that perverse sheaves are equivalent to rep g check but at the drive level you see the whole thing so we have an action of this and it corresponds to the natural action of this on this so basically you have y times quasi-co herring sheaves on y times y over x always acts on quasi-co herring sheaves on y it's the usual convolution action so this is the pattern that's how things fit all right these were two questions and there is one more question that nobody asked but i wanted to comment on so so and the question is do you really have to go that big to have this equivalence do you really have to do whitaker do you want do you really have to do equivalence with respect to this huge thing n of k the answer is yes and no so if you are at a fixed point you will be able to replace n of k by something much much smaller in fact by many things that are much smaller and i'll tell you by what but this kind of description will not be compatible with factorization so if you want factorization you really need n of k because n of k is the object that factorizes so let me tell you what you can replace n of k by so in fact you have many choices one for each natural number so you can replace them by some subgroup that i'll call i sub n so let me tell you what's the i sub n is in the particular case of s o 2 so n is greater than 1 for s o 2 it looks like this it's a subgroup of matrices a b here it will be t to the minus mc t to the no t to the minus mc t to the 2 mc d where a b c d are in all so you make the positive part larger and larger then grows and the negative part smaller and smaller and each such each of these subgroups also has a homomorphism chi into ga and you can play the same game imposing equivalents and the resulting category will be just will be the same as whittaker so in other words you can have a purely finite dimensional model for this for this whittaker category