 In nekaj se boš prišlošnja, da imamo otročenje, da imamo otročenje, da se nekaj spremno je posta, da je Okumkov, Aga Nacic in Lokumkov vseh vseh skupov, očasno dregov, časno odpočenje, počkupov, kvantumov, vseh, z equivalentom komoložijom. In zelo vidimo vse najbolj basične vse, ki je rank 1, SL2, in tudi je kratiljno odvrčilo konstrucije. Tako što se možemo vse zelo odličil, tako v svojih dnev, tudi smo zelo vse reprezentacije vzelo, ki je vzelo, tako v enu formulacijo Vorčje frazdej na svetu z vsem svetu p자�u se llama z DNS, dala za dve dnega vsega danetne parametrmi, in z sedm za qantum rup je od inamitega radiялиs na te danetne parametrmi, sva, da je nazirno vshjdil, da je ona pa vsega odpravila sveti afak啊ščenje. In tudi s tem geometričkem interpretacijem, v termi eriptih komoložij, včešte, kaj je zelo, kaj je zelo, na kaj spasje operatori. Zelo je to, da je tukaj, kaj je propostil na Maulijko in Dokunkov, in then Aghanachi Ukov. Včešte, da je tukaj nekaj nekaj, tako je vseč nekaj načinj, načinj pridunj, za kačko kvače, kvače časke, kaj je nekaj načinj vseč, ko je načinj o nullaritvim, načinj resolutim, In je to vseh varajte. In zato ideje je, da je vseh varajte komologijske teori in vseh vseh vseh varajte in občajno občajno občajte. I vseh vseh občajte, da je vseh komutacije in vseh vseh vseh vseh algebri. Vseh vseh kvantumnih grub, da je vseh varajte, način, kaj je in A.D. graf, in jih je reprezentacij, najbolj zelo, kaj je zelo, zelo, zelo, zelo, zelo, zelo, zelo, zelo, zelo, zelo, zelo, zelo, zelo, zelo, zelo, zelo, zelo, zelo, zelo, zelo, zelo, zelo, zelo, zelo, zelo, zelo, zelo, zelo, zelo, zelo, zelo, zelo, zelo, zelo, zelo, zelo, zelo, zelo, zelo, zelo, zelo, zelo, zelo, zelo, zelo, 1, 2, z 1, minus z 2, lambda minus. There is parameter h bar r 1, 3, so 2, 3. So you have this little modification. The usual Young-Baxter equation is equal to the other ordering, 2, 3, z 2 minus z 3, z 2 minus z 3, z 1 minus z 3, lambda minus h bar h 2, and r 2, 3, 1, 2, z 1 minus z 2, lambda. So if you don't have this lambda dependence, this is just the Young-Baxter equation with spectral parameter used by Baxter in integrable models and Young in kind of scattering theory in two dimensions. And now this is this kind of dynamical modification. And so this is an equation which lives in end of v tensor v tensor v, h, if you like. And so for instance, this r 1, 2, so this indices say on which space you are acting, maybe 1, 2, so this is the first. Means that it is r 1, 2, so it is r, sorry, lambda minus h bar mu. Tens of the identity. When you restrict to, so on, if you take a weight space mu. So the dependence on this third variable depends on the vector you are acting on. If the vector has weight mu in the third argument, then you shift by this amount. So this is this equation. And so one definition of the quantum group for SL2 or for SLN is that you have some quadratic relation of the RTT type, which I will not write. And these are the coefficients. And by definition, representation of the quantum group would be some object obejing those quadratic relations. And this object is a difference operator in the variable lambda. It will involve such shifts, but I will not comment much about that. So this is the thing. So this has kind of two roles. One is the coefficient in the quadratic relation, in which it gives one definition of the quantum group. And also this quantum group has evaluation representations. And there is a co-product. And you can, if you switch the ordering of the factor in the tensor product of evaluation representations, the two orderings are related by r matrices obejing these relations. So, right. So now let me say something about these weight functions that were appeared in solutions of this kinesic samologic of different equations. And I will give a construction with one construction of those weight function, which goes back to an old paper of ours. So the question you wrote just above, is it with h after two authors? Here it is two, maybe there are mistakes. There it is three, and it is one. Yeah, but then definition, so you try to define this using the, this is the definition, or? Yes, this is the definition of this notation. So, but it's analogous for the other. I didn't define it for the other orderings, but you always. Is it kind of for, so the v, what kind of h model do you consider, is it some? It's some simple, so it's a direct sum of weight spaces. So v, finite dimension, and, OK. And the notation in this h after two, or if the h is a self factor, which is, it's a mule. So here there is a three, right? Here we are acting on the three, so I fix the weight in the third factor. If you had h2, you would fix the, in the second factor. So this is a 20-year-old with Tarasov and Varchenko, where we describe those weight functions in terms of shuffle products. So maybe unshuffle product. So in that paper we wanted to construct weight functions associated to any kind of evaluation representations. And in particular for evaluation, verma modules. And so for that you, and this also gave a construction of the R matrix for evaluation, verma modules for arbitrarily pairs of weights. And the construction is a functional realization of representations of the elliptic quantum group, if you like. So suppose you have two functions, f of t1, tk prime, I guess, and the function g of t of k double prime variables. And you say k equal to k prime plus k double prime. And then you also have parameters. And I will define a product of those functions, but it will depend on parameters. Parameters that will have z1 up to zn prime associated to the first function, if you like. And then z prime plus 1 up to zn associated to the second function, in some sense. And then you define a shuffle product. And there is also h bar. And you define f star g. And these are symmetric functions. Symmetric holomorphic functions in all variables. Symmetric under permutations. And this will be a function of k variables. You take some of a shuffle, or you can symmetrize over this variable t1 up to tn. You take some of all permutations of the product of t1 tk prime, g of tk prime plus 1, up to tk, time some function phi, which I will define, which depends on the z variables and the parameter t1 tk, I guess. So this is so-called shuffle product. And this function has a definite form. It's phi zh bar t is a product of theta. So theta would be there are three versions, elliptic, trigonometric, and rational. So in the elliptic case, which most generally it is, the odd Jacobi theta function, tl. And then there is a product of theta tl minus za plus h bar. And the product of theta tl tj minus zb. And I should say something about the indices. So here you sum, j will be smaller. It's the first variable, k prime. And l is bigger than k prime. Same here. So here l is bigger than k prime. A is smaller than n prime. So it's the first n by 1. I really want to split into two groups of variables. And here j is smaller, k prime, and b is larger than n prime. So, OK, some formula like this. And so theta is the first Jacobi theta function, the odd Jacobi theta function. And so it degenerates to sine t or t. So this would be the trigonometric and rational case. So this depends on two. So we fix an elliptic curve, which we really write as z plus tau z. This is fixed, and we have these functions here. And then now this has two periods, one and tau. If you let tau go to infinity, you get the trigonometric limit. And if you let 1 to infinity, you get the rational limit. So, now, can I use the third board? So what are the properties of this function? So the idea now is that, OK, there are some elementary properties that, first, that if f and g are holomorphic, then also f star g. It is slightly, so this function has a zero on the lattice. So there is some denominator. But this will be cancelled out because you symmetrize. Symmetrizing will cancel those denominators. So you really get holomorphic functions. And then you have associativity whenever this is defined. So now you have three sets of variables, and the obvious way it will be associative. And then there is some kind of more. A subtle property is that this preserves theta functions, which is vanishing condition. I will define these notions. The functions f and g here, they also depend on the z. At the moment, I fix z. But later we will want to consider this also for variable z. But at the moment it is fixed. There are parameters. But we will vary z z. So I will explain what it is, but let me first say some words about what it is in the limit, in this limit. So in the rational limit, you can replace your functions by polynomials. And so this will map polynomials to polynomials. In the trigonometric limit, you can take trigonometric polynomials. And again it will. And in both cases, there is some sort of vanishing or wheel condition, which is preserved by this product. And we will define it. And in the elliptic case, you need theta functions. But these theta functions require an additional parameter, which is this dynamical parameter. And this is where it enters here. So let me. Are there functions, which just mean rational functions on the corresponding elliptic curve? Or, I mean, what is the space of all theta functions for you? I will define it now, yes. I will define what I mean by theta functions. So let's define it. So I want to define a space theta kn, which depends on the variable z, that one up to z, n, and h bar, and also this dynamical parameter lambda. So this will be functions f t1 up to tk, which are entire functions, holomorphic, and symmetric under permutations. And such that, so theta, yes. So let me say what theta function means first, such that g, so there is some maybe the better ways to write f divided by product of theta of ti minus za over all i and a up to k. And this is a rational function now, obeys some periodicity property. So g of ti plus 1 is g. And g of ti plus tau is exponential of 2 pi i. Lambda is a parameter, which defines the k times h bar. So it's a section of some theta bundle, and there is a parameter lambda appearing there. No, but it is a formula. A elliptic curve is not number time. I mean, tau doesn't appear in the formula. No. Yes, it doesn't. I should write theta. I should write theta, yes, maybe, something like this. So we fix the elliptic curve. I will not. No, but in the second relation, I have an exponential, but I don't see if this exponential contains theta tau. It is a contained here, if you like. And also, if you realize the transformation property for f, then you will have some more complicated multiplier, which includes some theta type multiplier. Also, this depends on tau, right? But the third condition is that f, so maybe this is the first condition, or maybe this is the first. F is independent. I fixed tau, so I look at all the entire function of t1 up to tk with this property. And the property depends on the parameter, like z, and tau, and h bar, right? And then the vanishing condition, so it's not finished yet, so vanishing condition, is that f of t1, tk minus 2, za, za minus h bar is equal to 0. So this is for all a from 1 to n. And so this condition is empty if, by definition, if k is less than 2, in at least 2, variables. So this is the space, and the claim is that if f and g are theta functions, but maybe I should write better this condition. But roughly speaking is that this star product will preserve the property of being a theta functions. But I have to say something about lambda, because, again, these kind of dynamical shifts will occur. And OK, so this I give the definition. So now let us define, take the direct sum of those spaces. So I consider spaces with different number of variables. Theta kn for k from 0 to n. And the same arguments, which I don't write. So this is as an h module with theta kn of weight. So this is c, and in this case, it is weight minus n plus 2k. So you want to be right. So now it's in h module. Now the statement is that the proposition star is a well-defined map of h module from theta n prime, the first group of variables, if you like, z1, zn prime, h bar lambda. So it should write tau, but ah. And here minus h bar h2 tensor theta n double prime, which is capital N minus n prime theta n of z1 up to zn. So again, the meaning of this notation is that you write this as a sum of weight spaces. And on each weight space you have a different shift by the character of h. So this means it preserves the theta function property, and it also preserves the vanishing condition. So I will explain this a little bit. But so in the generic cases, z1 up to zn generic, this will be a functional realization of a tensor product of evaluation representation for SL2. And if you omit the vanishing condition, you will get something which is isomorphic to the tensor product in the generic case, tensor product of verma module, evaluation modules. And the vanishing, or maybe dual verma modules. And the vanishing condition will take the irreducible subspace of the dual verma module. So with the vanishing condition, it will be the tensor product of, in this case, two dimensional representations. Then there are variants if you want to have higher dimensional representations where you have vanishing, different vanishing conditions. But the vanishing conditions are the same in the rational case? Yeah, yes, I think so. Certainly the trigonometric case. But it looks more as a natural solution than a module solution. How do you see the action where else? So you say it's a product of what you see in the action. Yeah, I will comment on that. We will comment on that. At the moment, it's just a map of each module. And it's right. And is it generated at degree 1? Does the product hold the? Yes, yes. This is the next thing. So in the generic case, it is generated by. Just to follow, will it be a later product for the variant elliptic homology? Yeah, so this will be a stable map. It is a stable map. So this will be roughly speaking, this will be elliptic homology. And for this gas manians, cotangent vandalog gas manians, and this will be the elliptic homology of a fixed point set for a subgroup, where you take some subgroup of diagonal matrices of this form. So it's split into two parts, and this will be the stable envelope. Sorry? This time it's not commutative. It's not commutative, no. And this is where the arm matrices will appear. Put down something. So let me just give the example to see this degree 1, which will kind of generate everything. So for n equal to 1, you have that theta 0. So we have two weight spaces. This is weight minus 1. This is c times the function 1, maybe by definition, theta 1. Here you have just one function, which is theta of t minus z plus lambda, something like this. t minus z plus lambda. You have just one variable. And the higher j1 r0 for j bigger than 1, because you cannot satisfy the vanishing condition. So if you don't have the vanishing condition, then you have infindimensional space, and it's kind of one-verbal module. And then the fundamental r matrix for the quantum group is obtained by taking theta 1 z1 h bar lambda minus h bar h2. You can do this construction in two ways. So here it is symmetric under z1 and z2. But you can also do the other ordering theta 1 z1 h bar lambda. And the r matrix is basically this condition here. So this will be the r matrix times the permutation of factors. And so I forget to say that it is an isomorphism. Maybe I should write it here. It's part of the proposition. It is an isomorphism for generic parameters. So you can really invert one of this map and get an r matrix, which will be a rational or a meromorphic function. So you're enjoying using them. So stable envelope is the multiplication in r matrix. Is a combination of the two, right? Yes, yes. So if you like, there are, so, molecular conformity define stable envelope for various chambers. And these are kind of two different chambers. The relation between the two is given by the r matrix. OK. So now, of course, now since you have an isomorphism, you can iterate this and construct the basis of the space of theta functions with vanishing condition. By taking, so, we have basis theta n to 1 zn h bar lambda labeled, maybe we'll do it in each weight space, kn labeled by subset of 1n with k elements, namely, you write omega. So first of all, you say this is c omega 0. This is c omega 1. And so you can now take the products of, so these are the weight function, omega i, which is omega 1, omega i 1, omega i n. And so alpha is in i, if. Sorry? i k. Should be i k. Should be k. I think there are n of them, yes? So you have n. How many do you take in a product? Right, it is, it is, it is, you have the number. It appears there, yes. So alpha in, so, sorry, i alpha is equal to 1 whenever alpha is in i. So these are 0, 1, 0, 1, 1, 1. So these are labeled by subsets of i. You have k variables if, yes. Right. So, so this is a story. So, and so one property of this basis is a sort of triangularity, right? Because you know, we have those factors, which are not visible now, but you have these factors, yeah. You have this kind of factors here, which vanish if you have t equal to z. So it means that you have a kind of triangularity property that omega i. So these are functions now of all sets of variables. Now you can think of z as variables also if you have that t i, right? So now you can specialize, right? So you can, so now you can set t equal to some values of z. So you have k here and n variables here. So you have omega, so this is omega i r functions of all these variables. And if you take zj for j another subset, then you set the variable t1 to be equal to first index of j and so on. So is equal to 0 if j is, if i is smaller than j. So you have sort of triangularity property, which will play a role later, and I'm not describing. And also you have some normalized, so these are the weight functions, and the normalized weight function. You take w i is equal to omega i divided by, well, this is some function I don't want to discuss. Forget about this. Product of theta of tj minus tl plus h bar subar all j different from l. So forget about this. This is some function only depending on lambda and h bar. And so now here you had a holomorphic function. You divide by something, which has zeros. However, so this normalized weight function has the property that it is holomorphic as a function of z. If you do this kind of restrictions, if you substitute for t some variable z, it will be well-defined due to the vanishing condition, in fact. Do you want to solve this whole thing? So this is this basis here with omega 0, the function 1, omega 1 basis for the one-dimensional space. OK, so this is these are these weight functions. And so some slight generalizational fit works for other representations. And in this way, using this trick here, you can construct arm matrices for other representations. Another application of those weight functions was so there is some sort of dual weight functions, which are given by similar construction. And those are coefficients in the eigenvectors for the so-called GELF and ZETLIN algebra. So in the quantum group you have some subalgebra, commutative subalgebra. And you might want to diagonalize. It's given by the determinant and maybe the first matrix element of the L matrix defining the quantum group. You want to diagonalize them. And using weight function you can construct such set of common eigenvectors. OK, but now let me say something about elliptic homology, so equivariant. So in one formulation, so you have now a G, a compact group. And in our case it will be UN, E, an elliptic curve, which will fix, in this case it will be given by this. And now there is a function, so I don't know if you will accept this notation, which contains the same letter in the finite G, C, W complexes to super schemes. So this is the analog of equivariant homology and equivariant case theory, but in the elliptic case it is better not to talk about algebras of functions, because they are known, but the space on which they are defined. So this is really the analog, not of HT. Not of HT, but on spec of HT. HT is a ring, so you can take its spec and it's a scheme. So unit is possibly not affine in this? This is not affine, yes, absolutely. That's the reason it's better to look at schemes. And so this is, I don't know, our source is a paper of 95 of Gimsburg, Capranov Vastro, who defined it axiomatically. They didn't define it, but they wrote a set of axiom that this theory should obey. So the first set of axiom is that it is an extraordinary homology theory, and then you have to say what is the homology of a point, and also there is some axiom relating to the functoriality as a function of G. And so there is a construction also due to Grojnowski. And at least for Toro's axiom, and also for simple, connected groups. Right, so using this axiom, so the construction of Grojnowski, you can, for instance, consider Grasmanyans and find what it is. So this is our point of view. Right, so now, the equivariant homology is a module over the equivariant homology of a point. And so the analog here is that this scheme is a scheme over EG of a point, so EG of x is a scheme. Maybe I should speed up over EG of a point, which is as interpretation in terms of kind of modular space of bundles. But let me write what it is in the cases we like. E of Un is just the elliptical to the n. So it's not a fine. And E for the Toro, ah sorry, this is divided by Sn, which we do know what's by En, and if you take a Toro's, we'll get En. Where the elliptical enters? That's right. It's where it enters. And now maybe I shouldn't. Right, so there are two descriptions. So now let's take x to be the cotangent space, but now we take the Grasmanian of k-plane in n dimensions. And there are two descriptions, one which is in the paper of Ginsburg-Kaprano-Vasco. And the group, so here we take the group, it's called A, is U1 to the k. And you divide by Sn, sorry, n. So this corresponds in terms of quiver varieties, just to have A is G, yes. It's called A. Because G will be Un, so this will be the Toro's. And there will be a different Toro's, which is called T. But so it's following really Maulikov-Konkov here. So if the quiver variety really corresponding to this quiver, or maybe if you have a framing this quiver here. And so here you have k and n. Right, so one way to describe the comology of the Grasmanian is to use the fact that actually the comology generated by churn classes of tautological boundaries. So this is one way, which is already in that paper. And so I think the ea of k and n. First of all, you have the structure map to the point, which is en. And then here you have the characteristic function, right? So this is, so Ginsburg-Kaprano-Vasco define a characteristic map, which generalizes churn classes. So you have u k. So this is a nice way to describe it. So this is really symmetric power. And so it's a pullback diagram. So it's a fiber product of these two objects. And here you have u n. And you have the symmetrization map. And so this is the description. And so it's a fiber product. So this is a fiber product of these times this. And this is, right, this is a churn classes, if you like, of the tautological bundle k dimensional. And then you can also take the quotient of the whole space by the tautological, you get another tautological bundle. It's churn classes here. And this describes how they fit together. So this is one description. And then the second description is by fixed points, by localization. We don't have time to get to the point. So you can, so now you have, so A has fixed point. I mean, the action of A on this grass manian, labeled again by subsets of 1n. Yes, A is u n, u 1 to the n. So this is the cartotoros of u n action, this manian, k. And so you consider just subspaces of dimension k, which are parallel to the coordinate axis with index in i. These are fixed points for the action. And so you have the fixed points, which is a bunch of disjoint union of en, one for each fixed point. And so you have a map from these two, to the equivalent homology of gr. Do you remember that the restriction is injective? So passing to scheme, you have a surjective map here. And there's also a description of how to, so something co-equalizer diagram. You have some description of how the kernel, if you like, is. So for each pair of such i's, so that the intersection is minus 1, is k minus 1. So they differ just in one point. You have some diagonal that you can embed here. So you have just, it's a diagonal. So you look like this, right? So you have, so this is an en. And this is another en of this collection. And they are glued along diagonals xi equal to xj. And the indexes i and j are the one for which this, where they differ. There is some kind of bunch of glued things like that. I think this is an accurate picture for gr. 1, 2. OK, so, right. So now in this Nakajjima variety, you don't consider the grasmanian, but you consider t star of the grasmanian, which is of course homotopy equivalent of the grasmanian. But you want a torus now. So this is why I, so you will take t to be a cross u1, a slightly bigger torus, which now has the right name t. And this is act for the scale of the fibers of the cotangent bundle. And so, right. And then there is a more, a second modification, what you really consider. So et of t star gre kn is just et of gre kn times this additional factor, which comes from this group here. And then you define some extended version, which includes this dynamical parameter, which does have a geometric interpretation in a Ghanachi Gopunkov. But it's called Kehler parameter in that language. So you add another factor of e, the rank of the. So here you have the dynamical. It's in this language it's called Kehler parameter. So, right. So this is the picture one has. And, right, I'm almost over. So, what happens is that this weight function w i and h bar lambda, so there are to be sections of a line bundle of the form tk n times l i on the space in which the equivariant elliptic homologi is embedded. Namely, or maybe the best description is this one. You have e. So you have some bundle on this space here. So here you have the variables t. It's a symmetric function on t, so it is defined here. And here you will have, what do we have? You have the equivariant variable z1 up to zn for the action of the group a. And then you have h bar, which is the equivariant parameter for this action here. And then you have this additional parameter lambda. So it's some sort of theta bundle. But what is important is that it is some kind of fixed bundle times something which depends on i. Sorry. And this comes from the projection onto the base. So it comes from the base. So maybe it is better to define it in this way. Sorry. You have c star. So you have a map, this characteristic map. Maybe it is called c from the equivariant homologi of t star of gr k n. So again, so here you have two maps from the equivariant homologi of the structure map to the homologi of a point. And you also have the characteristic class of the tautological bundle. You take the product, and you pull back. So here you have a section here. This function here is some section of some theta bundle here. You pull it back, you get something which has this form tk n times p star, something which comes from the base from here. And so this is, so now line bundles of the form tk n. So this is some kind of fixed explicitly described bundle times p t star l are called admissible. So a fixed bundle and something which comes from the base. And so now I don't have time to be more detailed. But the claim is that these are matrices and the action of the quantum group map sections of admissible bundles to sections of admissible bundles. So this is the way we can use this. And right. And the stable envelope can also be understood as a map. So maybe I should stop here. So the stable envelope can also be understood in terms of this weight function. So you have a stable envelope from the elliptic homology of t star. Sorry, from the elliptic homology of the fixed point set, which is some vector space which has the same dimension of c2 to the end, the tensor product of evaluation representations to the elliptic homology of t star gr. And you have to understand elliptic homology classes in this sense as sections of certain admissible bundles. So you have a map from sections of admissible bundles to sections of admissible bundles. So the moral of the story is that you have some class of line bundles on the elliptic homology, which are some kind of fixed, distinguished bundle times pullbacks from the base. And everything can be formulated in terms of these bundles. OK, thank you. Other questions? And Okunkov is axiomatic, so they define the armament. It's possible to check that the stable envelope that you define satisfies their axioms? Yes, yes. I mean, this comes from this. Basically, you have this triangularity axiom. And this comes from this shuffle product, because in the shuffle product we have certain factors, which vanish if t. So the axioms are in terms of restriction of w to those fixed point components. And this corresponds to substituting t equal to some values of z. And for that, there is some triangularity property, which is this axiom. So we didn't quite make the connection. But at least the type, we also give an axiomatic definition, which is, I'm sure, it's equivalent to what they write. But we can check it by this triangularity property of weight function. I have a philosophical question. What is possible to use instead of with stuff a matrix belonging to the infinite belonging? So which is not dynamical? Right, yeah, this is an open question. I think one should be able to construct also for the Baxter solution. But I don't know how to do it. So can you say in what generality we can do it now? Yeah, so this we did with Riemani and Varchenko for SL2. And now Tarasov, Riemani and Varchenko have a new paper where they do it for SLN. So that's the present situation. So you can do the Jordan-Puevo crisis, in the case of the Hilbert's Kitchen? Possibly, yes. Implicitely in Aga Načič, Kukumkov, in principally it's for any quiver, right? At least the abstract construction. Here we have an explicitly. Yeah, now I want to put this. That's a question. We can put it. That you should ask Andrei. But I think so, yes. I mean, in principally it's for any quiver. I mean, what they write. I didn't read the whole paper, it is very long, but. I have a question. So you can also, instead of gross manual, you can also consider k-copies, the product with k-copies of a projected space of dimension n. And you can also have a stable envelope in this way. So how can you relate this stable envelope? Envelope to data, the gross manual, kn. I mean, this is the optimization of a stable envelope. Can you do it with weight functions? I'm not sure I know what you're referring to. So you're saying that for product of projected spaces? So if you, in the construction and quiver variety, you question out by the GLK. So if you, instead, you consider the mass motorists in this GLK, and you use this question, you get the product of k-copies of a projected space of dimension n. And you can also have this stable envelope on each copy of this projected space. And you can also have a stable envelope on that. So how is this related to the gross manual? I'm not familiar to that construction, but maybe it corresponds to having non-symmetric weight functions. Non-symmetric under permutation of t1, tk. I don't know. Thank you. Thank you. Other questions? If you take the tangent bundle, but just the cashmoney on itself, you don't have h-bar? You don't have h-bar. All right. So somehow you would like to introduce h-bar in some way, right? But I don't know. I think Gorbunov had some construction where he only considers gross manual, but I forgot. It corresponds to some stretch. Yes. The substance h-bar goes to 0. Right. But it is not exactly his case. OK. So thanks. Thank you.