 Okay, so welcome to part two of the Schubert's Seminar with Professor Andrei Okunkov. Part two is about general flag manifolds. Thanks, thanks. So this is, it will start by somehow completely unrelated to what I said before. So if I have a smooth compact, stably complex manifold M, I can define the genus but if I have a function psi, I can take a product of that function or we'll change the roots of my manifold, integrate over manifold. So this would be some number, which will be additive over disjoint sums and the multiplicative over product. So it's, and in fact factors through the complex Schubertism ring of manifolds. Imagine this must be pretty familiar to people in the seminar. And if I have a compact group action, I can define that the gradient homologian and that will take part in the, this will take someone and I will push that forward. That push forward is the push, integrations push forward to a point. So if in the gradient homologian, this will take valence in gradient homologian point. So in H invariant functions on the Lie algebra. And you can observe that by continuity or many other ways. If you start, of course, if you start with a polynomial, we get a polynomial, but if you start with some other, say analytic function, and you get with analytic function and the, in the region of analyticity, that has to do with, so if you had a region of analyticity in the neighborhood of the Lie algebra of the maximal compact, so you think of functions, one variable is having to do with a group U1. And so if you had some function analytic in some neighborhood of Lie algebra there, you'll get a function analytic in some neighborhood of the Lie algebra compact form in for your a creative homologiable point. Somehow, I hope it's familiar in this context. And it worked just the same in any somehow extraordinary homologous theory, but we're right now, we want to do it in case theory, in gradient typological case theory. And the only difference between the case theory and the homology is that the homology of a case theory of a point, that looks like the group itself modular conjugation. So it looks slightly more non-linear. It's like a torus modular variable as opposed to the current homolog, ordinary homology you get, get of course the Lie algebra modular conjugation. It's like the Lie algebra of the torus modular variable. And all of these objects we will denote by boldface L, boldface lambda with the corresponding subscripts for the reason that this would be, we had some space lambda of parameters before and this lambda will be exactly this, the lambda from the first half of the talk will be exactly the lambda for the second half of the talk. And in particular, this lambda of U1 is just really just the group law. There's some, you need every, there's a group law corresponding to homology theory and that's the group law, the lambda of U1. So it's in other words, concretely in order to claim homology that would be the additive group of complex numbers and in the case theory version, that would be just the non-zero complex numbers with respect to multiplication. So central to our argument is the observation, which is, I don't know, like so many things go back to classics, but in this case, it's not maybe not Langlansk, but rather I.T. Or maybe somebody before I.T. But anyway, certainly stated very clearly by I.T. is that if I have a non-compact manifold, then the genus can sometimes be also defined but the value is in distribution. And that's entirely parallel to the fact that if I have this is maybe a fact that goes back to Gilfant and Harris Chandra is that if I have an infinite dimensional representation, I'd like to define its character and essentially a distribution rather than rather than a function on the group. But I need some kind of test function on the group to get an operator which has a trace and so then I can take the trace and that's how distribution works. And that's exactly the same here. And since it's convenient for us to work in algebraic geometry setting and the particularly convenient weakening of properties which, so, I.T. I.T. conditions are some kind of transversal ellipticity. For us is the particularly convenient weakening of properties is something called geomological properties that already been used by many people in particular. Then HL, then so what is a geomologically proper maybe quotient stack is so if I have a map from the quotient stack to the point and this has a push forward on coherent sheaves namely coherent sheaves, you take the ordinary push forward and together with taking invariance with respect to your group. And it has higher derived things. So higher geomology of that push forward and this map H mod H to X mod H to the point is called geomologically proper if it takes coherent sheaves to complexes of coherent sheaves. So maybe finite complexes of coherent sheaves. I mean, that's, so then you can, and then that's somehow that's what we can work with. And this equivalent general then have very natural extension to some of distributions on smooth geomologically proper stacks. And maybe the nice, maybe you can, throw in some derived stacks. We don't need a general theory or we need this already, you know, the only the only DJs we need for this are just the old good old Cazool, Chevrolet, Alderberg and Czech complexes. So we didn't pursue any kind of general theory but I'm sure it exists. And would be probably interesting to study. And, so, so for orb refills, there's a very interesting study now by John Pardon. And he I think was planning to at least me pursuing complaining to pursue the state, the case of, you know, like general derived stacks not necessarily orb refills. But anyway, I have nothing to say. I'm just saying in my transparency set this probably be interesting. And I guess also at least one person does this sort of thing is John. So maybe more, I don't know. Maybe I'm not just aware of the literature here. So, but for actual application, all we need is just some very simple things. Some just Cazool complex is Chevrolet, Alderberg complex and the Czech complex. That's all we need. And so back to the spectral problem. So we have this identification where I have, you know, this bold face symbols lambda from different parts of the talk. And the way we'll identify is the following. Let Q, this kind of, this is script Q, that is the image of the norm map on the else of your field. So for, in positive characteristic, that is really the subgroup generated by Q. Q is the number of element of your little Q. And that the subgroup of positive real numbers generated by Q, which is why we call it big script Q. But for reals, the image of the norm is everything on the busad. That's the, that's the difference between characteristic zero and characteristic, positive characteristic. And so in this table, we will have, so what would be that lambda? We will denote by old face lambda the dual group. So the homomorphism from this group Q to C star. And so in positive characteristic, that that would be, that would be just C star itself. So it'll be the curve corresponding to the K theory. And in zero characteristic, that group of homomorphism that C and it will correspond to the homology. And correspondingly, we can, for any torus, we can form that group, the corresponding group lambda. And that, that would be just the, that would be just the word, the good old lambda, which we had before. This is where the spectrum lies. And so therefore, therefore since now we examine this table and we see that in terms of old face lambda, which we now define uniformly in terms of group script Q, we at least have a way to topological interpretation of the size and time distribution. And we would also need dependence on the field and the dependence of the field works like this. It's, so, so the completed zeta function it actually satisfies it's really, so completed zeta function. I didn't define what it is, but it actually is, so, the functional equation for completed zeta function relates S and one minus S, just really relate. I mean, just, just equals, so zeta of S equals to zeta of XI of S equals to XI of one minus S. And so in particular, XI has a pole both at one where zeta function has a pole added zero. So XI has two poles. So if we'd like to interpret XI as a sum equivalent genus, then it better be a genus with something two-dimensional. And so because it's means something non-compact in fact, two-dimensional. And the corresponding geometry is you take just, just basically C2, which I think of that is as a quotient of T star C1 divided by C star, C star squared, that sort of take that stack. And we would like, so this is, and it's C star equivalent genus, we would like to be XI of F. You can always, you can always achieve that. So you can always find a genus such that this is true. And so this is how you bring independence on the field. And so now, right. So, right, maybe I'll, yeah, there's a remark which I don't know if you're somehow if you're willing to accept it now or we can take it later. You can, in the positive characteristic case, there's a very nice explanation what the genus is in terms of some sort of derived scheme structure. Maybe I'll come back to this remark later in the talk. And then I'll say so that the spectral decomposition, now here's kind of two main results, the spectral decomposition that we would, that spectral decomposition in this spectral problem, that's a consequence of the two following results. So first says that this distribution, Langvins define some distribution. And we say that distribution is really just the genus of this T star of, you take Langvins dual things and you take this G mod B on both two sides and take T star of that. And since you have division by B Langvins on two sides, this lives over point mod BL on two sides. And in particular, that lives over point mod AL on two sides. And so this is the distribution on two copies of lambda of AL. And the stack is not something, the stack is not anything scary. It's T star and you have a quotient on two sides, which means what does it mean to have a T star of a quotient? Means you have like some kind of causal complex for the moment map. And they also have, when you take the quotient by B on two sides, means there will be first some Chevrolet-Eilenberg complex for the unipotent radical of B. And then they'll take invariance with respect to the torus. So it's not, it's not the, it's maybe, you know, the Q, I don't know, the one of the easiest stacks you can ever study. And like I said, you take the genus, appropriate genus of this object and that would be that this genus is non-compacted, defined as a distribution. And that distribution is the length of distribution. And so, right, so that's first and right. And since this is T star to something, this lives in the world of, you know, T stars. And so in particular, this really depends only on the zeta function and not on the particular genus because so this is, you know, T star to something. And so it's genus gonna be consist of not just, you never have to evaluate your genus of C, you will always need to only evaluate the genus of T star of C. And the genus of T star of C, we decided this is the completed zeta function of our field. So it doesn't really depend on the, it doesn't, their choices of the many possible size that represent the same C here, but that choice, that doesn't depend on the choices you make here. So that's one result. And the second result is that, well, this is T star to, this is basically T star to G mod B, right? For T star G mod B, we have the springer, what's our grad index springer map to the nilpotent elements. And accordingly, we can decompose this space. So this is that space that did I didn't know that, right, I didn't know that the script T this space, this T star to this quotient on two sides. And if you think about what the springer resolution does, then what you're gonna get is the following, you take, first of all, you take the sum overall nilpotent elements up to conjugation. And then for every nilpotent element, you get the two copies of the springer fiber because we take the quotients on both sides, get two copies of the springer fiber. You get the tomes space of the embedding of zero into the slice at your dual the algebra at the corresponding point. It's just, that's just a linear space. It's a tomes space. It's a tomes space of embedding a point into a vector space. And then you mod out by, and then you take the quotient by the centralizer. And that's, this is somehow, this is more or less, this is more or less obvious if you think about it again, in terms of springer resolution. And what I'd like to say in this transparency that this is the group, the maximal compact subgroup in the centralizer arises is just really that there's a maximal compact subgroup of the centralizer of nilpotent element E. And the integration along the springer fibers determine the spectral projection. And the parts about the slice and the unipotent radical of this centralizer, they contribute to the spectral measure. So maybe I'll go make it somehow, for those of you who've maybe keeping score, maybe I'll go like explain what this means in technical terms in a second. Spend a minute on it. And so what does it mean? So how do I know, what does it mean in terms of spectral decomposition? This decomposition of the stack. So, so, so maybe before we go, because we got this detail, then, so, so we wanted to spectral decompose some space, spectral decompose some space is the same as you think of the, of some kind of L2 peering. You would like to spectral decompose that L2 peering. And we said this L2 peering, this is really some equivalent genus of this distribution corresponding to this L2 peering is really some just equivalent genus corresponding to a certain space. And the composition has nothing to do with what the genus is. The composition has to do with just the space, decompose the space. And the genus just go, just the long for the right. So anyway, so we have this decomposition and the first part of the composition is summation of the nilpotent elements. And that's really the summation over nilpotent elements in Langlund's formulas. And the Langlunds would like them, would like to have or in our formula or, so first of all, first of all, there's an orthogonal direct sum over nilpotent elements in the Langlunds dual algebra. So that's the first step. So that's second, when you take the quotient by the centralizer, you can first take the quotient by the nilpotent subgroup and then by the input radical and then by the reductive group. When you take quotient by the reductive group, what do we do? You just integrate, right? Because you need to extract invariance. You need to extract invariance with respect to reductive group. You integrate over maximal compact subgroup. And that really that integration is the mean. You have some L2 peering, you would like to decompose it into some pieces. And in particular, you have some continuous pieces over which you integrate. And that integration is really integration over the maximal compact subgroup of that centralizer. So that's the origin of continuous spectrum. So then in this integration, there is some weight that comes to do, there's some, in your geometric problem, there's some linear factors. So there is the slice and there is a quotient by the unipotent radical. In both cases, it's just some linear object. And so then the corresponding genera give you just a function, just some weight function. And so that's, therefore you get some weight in your spectral decomposition, you get some weight called, in spectral decomposition, it's called spectral measure. And then finally, you need to start, you need to get the function from your original function on your group. Well, you get that comes from, that comes from integration over the Springer fibers. And that's the spectral projectors. Okay, so that sums up my discussion. There's some further directions, which maybe since I'm out of time, maybe I'll postpone the further directions to questions. And then maybe I'll, thank you for your attention now and take questions. Okay, well, thank you very much. Any questions? So there is at least one question is what are the further directions? Okay. Yeah, so there's, we started with many, many restrictions along the lines and more or less all of them can or should be removed and some, you know, we continue. And one way, so maybe I'll, so there's two slides here. I'll try to go through them quickly. So first of all, is that we started with a group which was split and we took the minimal parabola. So, and all the action of, we get that C star, we get the C star of flag manifold and all the kind of the remaining action that's, I mean, there was the remaining action there was action of Q by scaling the cotangent directions. But in general, if you have a generalized time problem, you will have, what you will have to do is you have to take the Langlands in the Douglas-Dul Lee group, you take the double quotient by the parabolic, by the corresponding Durie parabolic and then you take fixed points of inertia subgroup. So that would be still some flag manifolds, a number of copies of flag manifolds, but now with non-trivial action of redinis. So in general, the setup in this paper should be generalized by to the fact that you have you have flag manifolds for a certain group, which is the centralizer of inertia in the Langlands-Dul group. And then that will have an action of Frobenius element and then you should compute in this, do the corresponding equation computation in whatever case you're a homologist, and that would be that, that's one direction in which this was to go. And second direction is very important to go, we took, we took this double quotient on the one side, we took the quotient by the maximal compact subgroup. And it's very important, this called unremifed number theory. So very important to go beyond that. And as you go beyond unremifed situation, you somehow allow more and more ramification. And the first thing you can do, like the first step is you allow maybe instead of key fixed vectors, you take of a horror fixed vectors. And then you get in the subject, which is already, you get directly to kind of issues which already been studied in Schubert calculus. And that's, you know, it's just one example, one paper in which, in which the corresponding ramifications of that theory have been already studied. And so that, then you will have on top of this. So in other words, you will have on top of that stack, you have some Schubert calculus. We compute some, we compute not just some genus of the, of the T star, but rather some genus of the T star and then cupped with some, with some Schubert classes. And the Schubert classes, they, in the Ewa horror fixed case, they already been studied in this paper by, by, well, you can say it's a good paper. I'm sure it's familiar to many of you. And there's, it's a part of some story which is probably also familiar to many of you.