 And thanks to all the organizers for invitation. So today I'll talk about a joint work in progress with Roman, physical and co. Pablo, Bozela, Averaz, and Equestral. So as you see from the title, there will be two parts in this talk. So there will be geometric slides which concerning homology of fine springer fibers. And there will be a representation side for representations of quantum groups. So let me just fix some notations. On the geometric side, I will work with the simply connected semi-simple algebraic group defined over C. So this will be G. And then I'll fix Borel subgroup inside G and the maximal torus T inside B. And then, so I'll denote the root datum by, so here is it. So the character lattice for T and co-character lattice and phi is the set of roots and phi check is coroutes. And what I will use very often later is actually the lattice, the co-character for T. So this lattice I will denote it by lambda and then this one's lambda check. Okay. And then we have the well group which will be denoted as W. And a fine well group is the semi-direct product of W with this lattice lambda. And then let me go to the other side. So for representation side, I will work with representations defined over a field K, which is assumed to be algebraic closed. And for this talk, I will mainly concentrate in the case where the characteristic of this field is zero. But actually there will be an analog story for the situation for positive characteristics. I'll make some comments later on this. Okay. And so basically K will be the coefficient when we are taking co-homologies of right is defined over C on the left hand side. And then on representation side, I will consider a work with G check which is the Longlands dual group for G. And then I'll fix a corresponding borrel, Longlands dual borrel subgroup and the maximal torus T check. And then I will denote the corresponding the algebra like this, which are some standard notations. And then so we will be working with quantum groups at the L roots of unity. Okay. So I'll fix L to be an odd integer. And I'll assume that L is prime to three. G has a component, which is G2. So that's the assumptions on L. And then I always denote by Zeta and L roots of a primitive L roots of unity in K or in C. So we'll be studying quantum groups at L roots of unity or just at Zeta. Okay. And then so for this reason, on the left hand side, I will consider the so-called Lth upon while group, which is just the subgroup of Waf consisting of W semi-derived product with L dilated lattice L lambda. And then I would denote the set of simple roots by Sigma and Sigma affine the affine simple roots for this root system. And then I'll write Sigma L for the affine simple roots for WL. Okay. So that's all the notations. Are there any questions? Okay. So then let me start with the geometric part. On the geometric part, we'll be interested in the following varieties. So we'll be considering a fine grass manion as we have seen already many times in this conference. So this is just taking the points of G in the field of Laurent power series and then divided by the G with points in the formal power series. Okay. So I would just denote the likeness. So it's an in scheme. And we have an action of the maximum torus of G on it just by left at multiplication. And also we have a, I'll be considering the C star action by rotation. So by the loop rotation. So this C star action acts by, so if I have A belongs to C star rotation and it will just acts on the variable Z by multiplying by A. Okay. And so as we know, the fixed points of the torus in the fine grass manion is parentized by the co-characters of T. So this was my latest lambda. And then I had this zeta, which is L through sub unity that we have fixed. Okay. And then I view zeta as an element in this loop rotation torus. And therefore it generates an action of a cyclic group of order L on the fine grass manion. I just take the fixed points of that cyclic group. Okay. And then you can do the computation. So the thing is clear that if you just take the zeta point, fixed point in this group, you will get the group, which is G you just raise the power of your variable to the power L. And so this is subgroup. And when you take the fixed points in the affine grass manion, you will get a union of partial affine flag varieties for this group. Okay. So maybe let me explain the notation here. So we had this lattice lambda and on it you have naturally an action of this L, the fine wire group WL. And so the components will be, the connected components of this thing will be indexed by the WL orbits in lambda. And then the thing here is just a partial flag variety which is a quotient of this group. And by a parahoric subgroup whose type is determined by this element first. So our piece and element in lambda and you consider all the simple affine reflections which fix it. And then it gives you a subset inside the set of simple affine roots and this defines you a parahoric subgroup. So this is actually an easy computation. You just look at how this group acting on fixed points and modulo stabilize or and you get this. There are any questions? Maybe I'll give an example. So well, yeah, just some special cases. If our piece is a regular element that is it's not lying on any hyperplanes for affine reflections, then the set will just be empty. And so P, this parahoric subgroup will be the urahoric subgroup for GZL. And in this case, the corresponding component will just be a fine flag variety. And on contrary of our piece zero, then it will be fixed by all the finite simple roots. So this set will be the set of finite simple roots and in this case, the parahoric subgroup will be G double bracket ZL. And you see in this case, the corresponding component is an affine grass mania. So same as before, just we raise the variable Z to the power of L. Okay, okay. So this is the affine grass mania and Zeta fixed points in it. And then we will be interested in affine springer fibers actually in a very particular type of affine springer fibers. So for example, this has been mentioned by Eugene Gorski in his course. So the type we'll consider is we first fix a regular semi-simple element in G. So we may just assume it's lies inside the algebra of the tolerance team. Okay, and then the element we'll consider is Z to the power of L minus one times this regular semi-simple element. So in particular, it's an element in G, in the algebra G double parenthesis Z. And this is irrelevant. And so the affine springer fiber we're considering is actually kind of gamma fixed points in the affine grass mania. So concretely, this means you take the points in the affine grass mania, which is represented by cosets for this group. And you require that if you take the joint action of G inverse on gamma, so G inverse gamma G, you require this element is again an element in the lia algebra of this parahoric subgroup. So this is a definition for affine springer fiber inside the affine grass mania. And again, we'll be considering the zeta point fix in this affine springer fiber. So you see gamma comes from the action it's the derivative for the Taurus action or TK action, so T double parenthesis reaction. So it commutes with the action of zeta. So we can first take zeta fixed points and then take gamma fixed points. So using the decomposition we have to explain just now, this part will just be union of this affine, partial affine flag right, and you're taking gamma fixed point in it. Okay. And I'll just make a remark here to explain why I'm considering this element. So actually you see from here that requiring this condition, a joint G inverse of gamma belongs to this, it's equivalent to saying that I require z power L times S at G minus one belongs to Z times G. So this is the radical of this parabolic, the algebra. So you can check that after taking each of the components, then this gamma fixed point is the same as taking the points G, the cosets P, VARP cosets for inside this partial flag variety and you add this condition which means you ask this element, it belongs to the radical of this parahor, the the algebra of this paracetamol just right for the the algebra. Okay. So this is actually what usually referred to as affine-spotage term varieties. Okay. And here I just don't make the difference of the terminology. And in this term, you can see that up to changing ZL with Z, this is the same as just considering affine-springer fiber for Z times S for this type of elements. Okay. So this will be the, or rather this will be the main object of study for today's talk. And let me give you some properties of this in scheme. So first of all, actually it's an in-scheme, but it's locally a finite dimension. That means it has an infinite menu of irreducible components, but each of the irreducible components has finite dimension. And then you have, so we had an action of the Taurus on the aphongrass manian. And since gamma is an element in T of Z, so it commutes with T, okay. So therefore the T action still, T still acts on this fixed points. And you can check that the T fixed points in this affine-springer fiber is the same as the T points in the aphongrass manian. And this is just a lambda as we have explained. Secondly, so this variety satisfy what is so-called GKM condition, which is, so if you take two fixed points inside the affine-springer fiber, then there is at most one T orbit, one dimensionality orbits connecting them. And thirdly, this springer fiber has admitted a fine-paving. So we know that if you take the orbits for you, the aphongrass manian, and you're just intersecting that affine-paving with this fixed points sub-skim, then you get a fine-paving here, okay. So in particular, from this, you know that the cohomology of this space is even, and in particular it satisfy what is called GKM condition. And so we know that in this situation, we have a combinatorial description of the equivalent cohomology of this space. So more concretely, this combinatorial conditions giving us follows. So we can describe this ring as follows. So it's the same as the following set. So whose elements are just tuples index by lambda, which is fixed points in this scheme. And for each fixed point, you associate an element in the equivalent, t-equivariant cohomology of the point. And then you add in the condition that whenever you have a one-dimensional orbit connecting two points X and Y, you ask these two elements, AX and AY, their difference is divisible by alpha, which is an element here. So this is the same as and so can stop. So alpha is an element here. So this is the condition you put for each one-dimensional orbit. Here, the alpha is actually the character for that one-dimensional TARS. So in other words, Kerner of alpha is the algebra of the stabilizer of it. Again, the co-dimension one TARS. Okay, so we have a nice description of the equivalent cohomology of this affine-springer fiber given here. Okay, are there any questions? Can I ask a question? So for these GRS gammas, is it in general true they satisfy the GKM condition? So they thought GKM have to use the extended course? Wait, or sort of L is the gamma? Yeah, okay, good. Actually, if you don't take the gamma fixed points, then you need to take the T cross rotation torus to get GKM condition. So the GKM condition is not true for the affine-grasp man named itself or the data fixed points. But here, what is remarkable is if you take this affine-springer fiber, then you don't need the rotation torus C star. It will be satisfied for only the finite torus. Okay, and that's in general true, like for gamma equals S times whatever Z to the L minus. Yes, for this type of gamma, I think it is always true. Wait, just hold on. Yeah, I thought, because when L is like at least treated, then you need... No, no, no, yeah, no, no, I don't think. No, no, no, sorry. I think it's only true for actually gamma equals to one. So I'm basically in a gamma equals Z as situation. Okay, okay, cool. Yeah, I just wanted to check. Yeah. Are there other questions? Okay, maybe let me give them an example. So let's consider the case of S of two and I pick, for example, L equals to three, okay? So in this case, lambda will just be two Z. And so if I identify the weight line with Z, then lambda is the real line, so it's two Z. And so if we consider lambda quotient by the upon while group action, then there will be actually two orbits, okay? So it's represented by points in this echo between zero and three. And so there will be a regular orbit corresponding to the point two, okay? And then there will be this single, it's a singular orbit, which is the orbit for zero, okay? And in this case, if you look at, so for our P two as we said, so this is regular, this is really just me, a fun flag variety. And in this case, when we take the gamma fixed points, we get an infinite chain of P one. So meaning that you have fixed points indexed by the orbits of two. So they're in bijection with WL. And then you will have a P one connecting the nearby fixed points. And then in this case, flag bar P zero will be, as we said, this is a fun grass manian. And if you take the gamma fixed point, as I said, so this is corresponding to taking a fine spattation variety. And in this case, you just get a bunch of points. So this is just all the points here. And so the cohomology on, well, the quiverin cohomology of the whole space is the direct sum of the quiverin cohomology for each of the components. And so for, well, it's very easy to compute in this case. So in the first case, it's just a quiverin, t-quiverin cohomology of P one and of this infinite chain of P one and right. And so maybe just one more remark that this description I gave here, when you restrict to each of this orbit, you can identify the fixed points with the quotient of a found while group because it's x transitively on each of these orbits. And so quotient by the corresponding type for this heroric subgroup. And then so the one dimensional orbits, like how to detect when there's a one dimensional orbits between two fixed points, this can be said very explicitly in terms of alcoves. So in the case of S02, you just see that there are one dimensional orbits between nearby alcoves and this can be generalized to arbitrary type. Okay, so this is the example for S02. And then we also mentioned, there are some symmetries on the cohomology of these affine springer fibers. So meaning that there will be left action of the found while group and the right action. So the easiest way to define them is the following. So you consider the restriction of equivalent cohomology to the set of fixed points. Okay, and so this is a subring. And on the right, since the fixed points are isolated, this is just functions on the set of fixed points lambda with coefficient in this ring. And then you can define two actions which where the left action is just, so you have a found while group acting on lambda and you just let it X. So you just, it sends this couple AX to AYX but at the places YX, you twist A by the action of Y on this, okay, on the equivalent cohomology of the point. So here are the lattice parts act trivially and you just have the W action on it. And for the right action, you, for the right action, you just translate the, maybe I should write. For the right action, basically it only acts on the lambda fixed points. By translation, but I'm a bit confused here because usually if we look at the, for each, like if it's on a regular block, then it's just the right action of WL on itself. And here on lambda, I forget how to write it properly. But it's possible to define. Well, up to modifying my follower here, it should be well-defined and the two action commutes. So in any case, we can define left and right of our while group action on this space and you can check that it preserve this sub-brain, okay? So you get two actions here. Actually, there are weights, so you can describe these actions more geometrically. The right action is actually the springer action. So we know, like there, in the finite case, there are while group action acting on homology of springer fibers. And so here is, there is a similar situation. So you have, you can construct an action of the found while group on the homology of this springer fiber. And for the left action, it actually comes from some symmetries of the variety itself. So the lattice part, it comes from the fact that on the springer fiber, as we explained, you have the action of this group. So it has a continuous part, which is T and X tributary, but it also has a lattice part, which is L lambda and it gives an action on the cohomology. And then the W action should come from some monodromy action. Right, so these two actions can be described geometrically. Okay, so these are the symmetries. And then we can consider also the restriction from the cohomology of the ethan grass manian, just the zeta fixed points, to the cohomology of this found springer fiber. And just by restriction, because this is a subskin. And it turns out that it lands inside this invariant parts for the left action of this found well group WL. And you can try to study this space. So one of the theorem we prove in this project is that we have an estimation of the dimension, like a upper bound for the dimension of this space. And this is smaller or equal to h plus one to the power of rank of G, and H is the coaxial number. And we can prove that this is actually an inequality in type A. And by consequence, well, it's part of the argument in the proof shows that this is surjective in type A. And so this equality was also approved in general by Pablo, by Zeta, Averaidz, and even Lucette. So like they used our arguments for this inequality and they prove the other inequality, so which allows them to prove it in general. Okay, and another comment is that, so people are so interested in the so-called elliptic of find springer fiber, where instead of this Z, you take another regular semi-simple topological in your potent element in the find the algebra. And so I just write down an expression of it. So I think I wrote it down for the, every one here should be raised to the L if you want to work with the L form. So just in front of the root vector of height H minus one, you put a Z square and for like negative roots, you put a Z in front of it. Yeah, actually it's not very important, but what is well known is that the cohomology of the elliptic fiber is precisely the right-hand side. And it was also proven that the, I think by union and oblong common union that you have the cohomology of flag right, a fine flag right is maps subjectively onto cohomology of this elliptic fiber. And so you have these two maps are just restrictions and in type A, you can show that it actually factorize and you get an isomorphism of vector spaces between the cohomology of the elliptic fiber and the cohomology of this homogeneous speed fiber taking the WL invariance. And also one can show in type A actually in this W action is trivial and it's the same as the fixed points for the lattice. Okay, so I think this ends the geometric parts, I'll pause for questions. Okay, if there's no questions, I will go on to the quantum group part. So here we go. So let Q just be a formal variable and then we have the quantum group associated with the, as I said, London's dual group G check. And so this is the quantum group defined over the field, rational field K, Francis Q, and generated by EIFI and K lambda and satisfying the usual relations for quantum groups. Okay. And then inside here there are some different integral forms defined over this ring. And the most well-known integral forms are one of them due to the conchini cuts. So this is to just pick the sub algebra over this ring generated by the standard generators EIFI and K lambda. And there's another integral form, which is very often studied is due to lipstick. So this is generated by divided powers of the generators. And so EIFI, so this is divided power. So EIFI divided by quantum and factorial. So EIFI, so yeah, so this is Lucic's quantum group. And actually in between, you can consider some intermediate integral forms. So you can take divided power version on one half of the quantum group and take the other one on the other half. So for example, here I did know that if you mix me is you take the divided powers for the EIs and for the K. But on the negative part for Fs, you just take the usual generators EIFI. And also you can do a similar thing, which I did, you can mix N, which means on the carton part, I just take the usual K lambda without divided powers, but I take divided powers only on the EIs. So you have these four forms of integral forms and in terms of that these forms in the middle have been recently studied, for example, by Dennis Giscari, and they have some very interesting representation theory related to affine the algebra. So you have these four forms of integral forms of affine the algebra. Okay. So anyway, we have these four algebras and we can specialize them to roots of unity. So for any of them, start denote by all the subscripts and if I replace Q by a zeta, I just mean specializing Q to zeta in this ring. Right. And the small quantum group, the small quantum group is, so you have map induced by these inclusions when you specialize to zeta, there are no more inclusions, but you still have maps and the small quantum group is the image of the cast of the continuum quantum group inside the loosely quantum group. So there's another way to describe it. So inside the center because the continuum quantum group, there is a part which is often referred to as the Frobeno center or the L center. So this is generated by the elements EI to the power of L, FI to the power of L and K of L lambda. And as it separates, the same as the coordinate range of the Poisson-Dier group. And one can check that the small quantum group is just, you take the cast of the continuum quantum group and you specialize this part of the center to K, like you quotient by supplementation ideal, meaning that EIL send to zero, FIL send to zero and KL lambda send to one. And so this is a finite dimensional algebra. And so from this perspective, we can also consider some deformation of this algebra. So instead of just specializing to K, you can specialize in two, for example, function on the Taurus. Yeah, completed at one. There's some technical issues that you just ignore it. So this is basically you specialize EIFI to zero, but you keep the KL lambda. Or rather, I should write like one plus the, like I write K as an exponential expansion. And that's the meaning of the, of the competition like one. But let me just ignore this. And also you can define another algebra. So is it a B, which is you, you deform further that you consider, you just specialize the central characters to functions on B minus, which means you just specialize EIL to zero, but you keep on the F's and K's. So these two are some deformation of this small quantum group is it. And so we're mainly interested in this one. And, and here you can identify the completion of these functions on the, on the L Taurus with the completion of the function on the new algebra at zero and, and identified with the homology T equivalent homology of the point. And this ring will be used very often. So denoted by S. Okay. So these are all the algebras. I will be considering. Are there questions? Okay, so then let me tell you what kind of representation. So I'll be interested in. So we'll be interested in the following representation. So recall our lambda was the co-characters for T and the characters for T do. So for the lucid quantum group, I would just interested in finite dimensional modules over over them. So for example, this is the symposar index by, by dominant weights. And for the small quantum group, we'll be considering the so called the L lambda graded use a time modules. So, yeah, so I consider finite dimensional L lambda graded use a time modules. So, yeah, basically in use it, you have the, you have the carton part, but it's this, it's like it's lambda mod L lambda graded. And sorry, so it's, yeah, like you don't have the whole torus action, but you, you only have them mod L. So, so when I consider lambda graded using time modules, I add the torus action back. So this, this category you can think of it as a sort of use it T modules. So, so you're in particular require that the, the use it action and the, and the lattice action is grading is compatible. And you can do the same thing for, for use it to T and use it to pay B. So for this deform that address, you can also define lambda graded modules over over them. Although you need to replace finite dimensional by finite generated over the space range. And so this is mostly for the small and then for this mixed version. I define here for this mixed version. Well, considering their category all. So, so, so you consider again, I'm the graded modules, and you act, you ask the positive parts acts locally unipotent. Okay, as what we do there often for category all. Right, and then for, for all these categories you can study standard objects. For example, here you have baby verma objects and here you have verma modules, you had while modules, and you can, you can study in their relationship under different kinds of functors. So, so here let me explain what kind of relations you have between these categories. So you see by definition that this old mixed end. So, let me recall this you mix and has a, has this cartoon part so it also has, it's a linear over over this ring as here. Okay, so, so this all makes sense as linear category and this use it to model and also as linear category. And actually, the use it to be is just on the category all here is you just specialize as to K. So, and, and use it to mod lambda is a specialization of use it that he learned by specializing this as to K. So you should think of these two categories as deformations of these two categories. And then, and then from this category to to read you Lucy, so let me make a remark on this category so. So by custom lucid equivalence we know that the representation of use it lucid is equivalent as a tensor category to this, a fine parabolic category or just custom lucid category. And so recently there is there is conjecture by gas query and I think it. I'm not certain of the current status but I think it's proven by his students that this only speed is derived equivalent to the affine category. Okay, so, in other words, if you want to extend the usual cash and lucid equivalence to like from this parabolic category to the whole category Oh, this is the correct object to consider. Well, the usual you can also define a category of for lucid quantum group but it turns out that's not the correct object. So that's why I think it's interesting to study this these categories. In any case, so you for for this category you can, you can take you take an object here you can take a maximal quotient, which belongs to, which is interverbal over lucid quantum group. So this gives you a functor gamma, which, yeah, which is a joint to some sort of restriction function. And then, and then from representation of quantum group you can restrict to the to the small quantum. You have functon like this. And finally, so there is also a function connecting umix and omix and and this deformed g1 team modules by first restricting this you mix and to the use it to be I define. And then, and then specialize to use it. But I think that's too technical. Let me just ignore this. This part. So, so we have a commutative diagram of funters between the categories I've introduced. So one. So the first result we prove is a computation of the center of the cat this category here. So this is the deformed use it on what you category. And, and what we say is that it's center is isomorphic as a rain to the equivalent homology of the assigned springer fiber. And before. So this was the object we studied in the first part of the talk. And, and ideas of the proof is the following so we follow kind of GKM description, and you can check that. And as I explained, this is a S me near category. If you extend the scalar to the fraction field of as it actually become semi simple. And it seems it's simple objects are baby Vermont modules, which are indexed by the by London, the fixed points here. So if I have a semi simple object, then, like this category by short amount, your center is just a project. It's just a product of scholars on the simple modules. So therefore, we can then invite the center of this localized category with the co model equivalent homology of the fixed points and also localized to the fraction. And then, after this you can look at the co dimension one situation where you consider a localization for each route you consider a localize localization of S, where you invert all the roots, which are not equal to other positive roots which are not equal to alpha. And then you look at the representation side if you localize to this, this rain, then your category is actually equivalent to a direct sum of of the same kind of categories just rank one group. And there you can do an explicit calculation by writing down all the projectives and maps between them and you can compute the center of that category. And so this, this you can check by hand matches the GKM description. And finally, using some some nice properties of the center you can check it's actually equals the intersection of this co dimension one localization inside the this big localization angle. We have a question in the chapter. Can you say it or I mean, yeah, I should be able to say. Yeah, I do you have a conceptual explanation this proof is somehow reduction to rank one is fine but somehow if you have a purely reason why these two things should be I think it's I think it's better. Yes. So, I think it is something like. So, it's better to explain and in, in terms of like for encounter is p this use it will correspond to this group G one which is the kernel of the Frobenius and and you can you can iterate this procedure and in in in some sense this this module category. So it should correspond to this springer fiber which is the one which is like it is not the most generic one but just the one after Yeah, I don't have a conceptual proof. Maybe I don't require proof but certain explanation. Maybe. Right. Too much proof might be too much of it. Right. It looks a little bit mysterious to me. I agree it's very mysterious, but on the other hand side. So the, the structure of this G1 team one is. Yeah, like, in many aspects it's it's related to this springer fiber. Oh, it's maybe better to say that. So Roman has a, I think with win and Michael McBrain and maybe with Pablo also they. Moral now is to say that this category is causal dual to to microlocal sheeps on this fine springer fiber. And, you know, probably there is a big picture where you can use this. You know, in the finite situation you have causal dual for for cotangent low flag variety and then you can deduce information about a causal dual for springer fibers relating to to this parabolic singular duality. And I think there should be some some kind of similar picture going on, which tells you that the geometry responsible for this, this small quantum group is this, this, this particular fiber. Okay, yeah. Okay. So, right, just, it might be related to some kind of 40 on some type of duality but but I don't know what to say more about it. Okay, so we have, we can prove this isomorphism, and, and if you specialize the scalar. So if you specialize as as to K, and on a co homology side, we know, you know, this call model is pure. And so this is just the usual co homology of the, of the springer fiber. But there's a very complicated issue on the center on that way, the construction of the center is not functorial said, if you specialize here, you only have a map into the center of this known deform the category, but a priori, it's not clear this is, this is So, so we can so a straightforward corollary of the previous theorem is that you have an embedding of the co homology inside the center of the non deformed category. And one of the conjecture, I have it with like we have any and it's like this should be an isomorphism, but for the moment, cannot prove it. And, right, and the remark is that remember we had this decomposition of this a fine springer fiber into into components. And actually this matches well on the representation side on the representation side you have decomposition of this category into blocks. Just by the horizontal central characters. And so, so, so this is the same decomposition. Okay. And another theorem in which is not whose proof is not completely written down at the moment so right there I mean progress is that the center of this omics and the the equivalent co homology of this data fixed point in the underground grass mania. And, and the center of this omics fee is equal to the homology like the non equivalent homology. So, so in this case this specialization should should go well as in the case of usual category all. And then, so we have a commutative diagram, where on the right hand side, you have the, remember you have this restriction map from the co homology of the upon grass mania to this, a fine springer fiber, which lands in the WL fixed parts. And on the left, you can, you can use the relations between these categories to produce a map between the center of this omics be and the center of use it to learn the movies. And the statement so this this map is defined algebraically and and you can check that we can check this is a you have a commutative diagram like this. Yeah, there is a question in the Q&A, maybe you can. There's certain variety. Oh, yes, I think. Yes. I think X alpha is just instead of taking fixed point by T you take the fixed point by this co dimension one course. So I had a co damage for for alpha. Um, yeah, you just define a co dimension one torus by the corner of alpha and then take fixed point by that, I think. Yeah, I think that is, that is X alpha. Okay. Okay, and finally, um, there is. Yeah, finally there's a quantum Frobenius map from this lucid quantum group to the envelope in algebra of G, maybe with completion and it's kind of this you can realize small quantum group that's it's kind of and therefore using the adjoint action of quantum group on on itself, you can produce an action of the group G check on the center of the small quantum group. And so, yeah, so we have two conjectures. So one the first one I have already mentioned will help the center of the non-deformed category matches the co homology of this affine springer fiber. And the second one, we expect that if you take the G invariant part in the center of us data. This is isomorphic to the WL invariance in the co homology of this affine springer fiber. We have some estimation of the dimension of this space. And this space, for example, in type A is the same has the same dimension as the diagonal co invariance. And actually, on this work, this whole work was in was motivated by a conjecture of lecture scan, which conjecture the center which gives this conjecture in type A, this gene G check invariance of the center of a small quantum group has the same dimension of the co invariance. Okay. Yeah, and let me just mention briefly that I explained there, there's two actions of a final world group on the on the co homology and there, you can also construct them on the representation side, where the left action is just by some again by some radical symmetry, you have the L lambda and just acting on shifting the grading and and the final world group action acts by some by some twists. And then the right W action is given is induced by some translation functions. So, on, on using time or lambda you have different kinds of translation functions between blocks and their by drawn actions. And there's a standard way due to vanish time to define some operators on the center and you can check that you get them W L action on the center in that way and these two actions are compatible under this identification with the with metric action. Yeah, and maybe a final remark is that, as I mentioned, for if you take K to be a field of positive coefficient, then a large part of the story is still true. And in this case, the use data module, this category is just what is so called G one team will use. Yeah. But this part is maybe less clear, like this statement is goes the same, but the second part it's more complicated in positive characteristics. Okay, let me end there. Thank you. Any further question. So in positive characteristic, maybe you need a field of characteristic help. This parity localization parity she's and localization to the fixed point. This work of rich and Williamson. So that would relate to maybe the homology of the fixed point to the homology of the whole affine grass mania. Can that be helpful or help understand. I think it's related. So, I think what they study is, is the model for representation of G of the group in characters. So their story is, is like the causal deal between representation of gene positive characteristic and then this, this parity sheets on the on the ground grass mania and, and, and this is a fixed points it's related to the, to the block position. Just to describe this comology of the fixed point locus in characteristic L. Yes, can you recover this description in terms of this localization in terms of the comology of the whole affine grass mania. I mean, you mean localization with respect to this meal L action with with the coefficient in positive characteristics. Yeah, I don't know. I think there's something going on there but I haven't looked this into details. So it turns out the in in positive characteristic. There's more structures on on this, on this range. Yeah. But I don't know what to say for the moment. Any more question. I have a question so you consider the fiber of a gamma and also the whole variety and what happens if you consider more general fiber in the other side. I don't know I. What do you mean by more general fiber. You can take any element from the real. That is to to to exotic question. Yes, I think. I think it should be related to some kinds of W. I do rose. But I don't know what to. I like. You have two versions of. And you have some some some. Some other algebra appearing. For. I didn't understand. So. Can you say it again. So you have to. In the algebra side you have two algebras corresponding to the whole variety and the fiber over gamma. If you consider more more general gamma then you should get more new new algebras. Yes, it should be the center of the new algebra and you explain, but you mentioned double algebra. So you said this new algebra is double algebra. I think it should relate to some kinds of W. If you take the finite analog of this, like, then on the top, it's similar as center of category. Oh, like the variety and the, and the algebra here are sort of causal due to each other. I see. Yeah, so, so I expect if you change gamma, there should be some sort of W address showing up on this side. No, I see, I see. Okay, thank you. Yeah, let's thank. Thank you again. There is a question. Thank you again. Yeah, so, so, so yeah. For this elliptic fiber. Somehow you can, you can think of this split fiber as you can somehow. Can you just read number? My, my memory is correct that you have, you can, you can degenerate this gamma prime to this to this split fiber gamma. So, so it is possible to write down a map from the co homology of this identity fiber to to to this fiber we considered such that this this diagram commute. In general, it's hard to say much about this map. So in general, it's, it's hard to say whether it's subjective or isomorphism. Here we prove it's isomorphism is really using some argument of Carson of them called we can show the image here is contained in this part. So, and then you use this, this on the dimensions, we can show it's isomorphic. But in general, I think that we don't have a theoretical argument, which, which gives this isomorphism. Okay. Now we can really thank her again.