 Thank you for this honor to be invited. So I should say that for me, just like Nick Katz, it's the 50th anniversary of my first visit to IHS, the 68th first time. What happened was that I was in Oxford with Attia and he, before leaving, before I left Oxford, he called Grottenly, my phone has said whether I can stay here for a day or two and he said yes, I had a wonderful discussion and everything, and all together I spent maybe, if I add the total amount of time, maybe two years I spent at IHS, but in different, which played a very big role for me. Now about Offer, I think I first saw him, I think I first saw him, not met him, but saw him maybe 10 years after that when he was, when I went to MIT and he was student at Harvard. And I remember going to, only I remember, I went to lecture by Sir, where he was the only one who understood everything. Yeah, yeah, yeah. Yes. And then several years later, I think we also had some, I think I benefited tremendously from his work on Pervert Chief, which were quite important for me. So, the thing I will talk about today, I have to say that his garbage work will not appear explicitly, but they all, in fact, without his work, this, I couldn't say, not one word, I think, everything I say, it couldn't exist without his work. Okay, so I will talk about some property of reductive groups of algebraic crossfield, which I think it's maybe not so much, but very well known. So, I will always consider K algebraic crossfield, characteristic P can be zero, and G will be connected reductive groups, reductive group. So the classification of such groups was given by Chevalet in 1955, and they are independent. So it's very remarkable between the classifications independent of K, that's the classification. But what I've tried to do, I tried to decompose G into finally many pieces as a partition, which is namely, what I tried to do is following. So let CL of G set of conjugacy classes. And if C is a conjugacy class, then its dimension is known to be between zero and dimension of G mode, maximum torus. And this suggests that there are two important subsets of the set of conjugacy classes. One is set of regular elements, they are all C, so the dimension of C equals, and the other one is center, which are all C where dimension is conjugacy class is zero. And what I try to do, I try to interpolate, I will try to interpolate between those two extremes. So I will define some partition of G in which there is one part and there's another part. So I will need some, so in this partition, each piece will be union of conjugacy classes of fixed dimension. And the most remarkable thing about this partition is that the indexing set for the pieces is independent of a characteristic of a field. So that is the, so now I need some definitions. So I will use, B will be the right of Borel subgroups. W is a VAL group, and then it is known that it can be interpreted as saying that B cross B is union of subsets indexed by the VAL group, which are the orbits of the group G. And then from this you can get a length function on the VAL group, which this equals the dimension of OW minus dimension of B. And then we need a set of simple reflections, these are all S equals one, this generates the VAL group. Now, for us important, so if G is an element of the group, then we want to consider the all Borel subgroups containing G, so this is also called Springer fiber, and then there is a theorem of Sparterstein, which says, I think he was actually my student, but says that BG has all components, well, certain is not empty, but all components have the same dimension. So BG has dimension, we call it BG, so it's a pure dimension, and this number is actually equal to dimension of B minus dimension of conjugastic class of G divided by two. So this was proved by actually by, this is the second equality was proved by Steinberg and classic zero, and by Sparterstein, any characteristic. Okay, now, the set of regular elements has been studied first by constant in a Li-algebra setting and by Steinberg in a group theoretic setting, and Steinberg has given two, so there are two complete different definitions of the set. So one is that this whole conjugate, this a union. I believe that constant was only just in classic zero. Yes, classic zero, yes. Steinberg was in classic zero. Any characteristic, yes, yes. So what can describe the regular elements, it's all element of a group for which this dimension of this RIT BG is zero, that that is because, the same as saying that dimension of the class is as large as possible. And, but there's a completely different definition which is due to Steinberg, so C, conjugastic class, is contained in a regular set. If only if the following is true for some, or any coaxial element, so I will say what it is, of minimal length, W, this, if you take this class, intersect, suppose you fix some B, so C intersected B, W, B is not empty. So Steinberg showed that if these two for one coaxial element or minimal length, then it's true for any other coaxial element. You know, this characterizes the regular. And this was Steinberg, 1965. By the coaxial element, minimal length, this is a product of the single reflections in any order. This is a definition of the word coaxial element or coaxial element of minimal length? Minimal length, of minimal, any conjugate of that, W conjugate. And similarly this central part can be also characterized in two different ways which are similar to this. So one is that all group elements are the dg equals dimension of flag manifold. So a fixed point set in the flag manifold. So it's all elements contained in all Borelza groups. And it's also the case that element g is in the center if g is contained in B unit element times B. But g is not contained in any other, W is not one. So in this description, these regular elements have, you can associate with them in some sense, the conjugacy class of a coaxial element. And here you can associate a conjugacy class of unit element. By the way, the formula I didn't understand. You wanted maybe there is some, maybe it's a small mistake in Spaltenstein's formula because in the rad, in the, you write d, d minus, so first you divide by two, what, that's the dimension of the class, yes? Okay, but then in the regular case, according to your formula, it's dimension of g of t. So you get twice the dimension of the unit for the radical. That's the same as dimension, that's the dimension, not a part of the dimension of flag manifold, is that? This one, so that's. Ah, it's not dimension of b, but okay. Yeah, that's fine. The dimension of, okay, so it's, did you get it? Curly b, yes. Okay, so now I want to try to give, now give a definition of a stratification. So for any element of the group, we look at homology group of this Springer fiber in a top dimension. This is a top dimension because dimension is twice, is the two of g. And then we can map this to homology group. So this with etal homology, homology of the full flag manifold. So there's a natural map, and we define e, g to be the image of this, of this homography, so. What is the criterion for g and g sent? Well, g is in b, g in the center, but it's an earlier, g is in b1, b, and not in b1. Yeah, not in bw, b. But it doesn't seem to be, maybe, for what is your assertion, because you take, it's certainly b as non-central. Okay, well, yeah. All k's, all, ah, all conjugacy class, all, I mean all conjugates of g's, that's fine. Yes. Yes, yes. Okay, so for each element or group, we can associate a subspace of the homology group of the flag manifold. With the Ladi coefficient. And then we say, we define, we say two elements are equivalent if dg equals, so this dimension, this is same, and also eg equals eg prime. So this, obviously, equivalence relation on our reductive group, and the equivalence classes are called strata. And, Of course, you can do it in terms of travel, obviously. Yeah. Yeah, okay, yeah. So it's, yes, and then it's the kind of travel. Yes, but I have to quote some things which are proven in our homologies also, but. So, and it's clear that the strata, each strata is a union of conjugacy classes of fixed dimension because of this condition and because of this result of Spartans time. So strata, each stratum is a union of conjugacy classes of fixed dimension which is given by that formula. So now, actually, this definition is not yet clear that there are finally many strata. So for this, you have to use some properties. So one property is that w is one group but it's well known that it acts naturally on the homology groups of flag manifold because this can be replaced by g-mode t and then on g-mode t, there's a value of action. And there's a claim is that the subspace of the homology group is always w-stable, e of g. So this statement can be deduced. So you can, to prove this, you can assume that g is unipotent and that's an easy reduction. And in unipotent case, this follows from Resato Springer, 1976, assuming p equals zero or p large or myself in 1984 for NEP. So in unipotent case, you have to, and actually at this point, a result of Gabber or Enter in a group of those things. So this w-stable and more than that, in fact, e-g is irreducible as a w-model. And also, e-g appears with multiplicity one in the homology of this plug manifold in the valve representation. And it doesn't appear any lower one. So all these are proved here and here. Of course, it tends to appear higher. What? It tends to appear higher, you know what I'm saying? Higher, yes. Because you get regular representation. Yes, yes, yes, yes. Yes, but not lower. But what you can deduce from this is that this pair of d-g and e-g can be reconstructed from the isomorphism class of this valve representation. So if you have a valve representation asomorphism class, then this d-g is uniquely determined because it's a first degree in which the valve representation occurs. And this subspace also uniquely determined because it's only subspace which carries that representation. So you can view this strata. You can say that you're indexed by a certain set of valve representations. By the way, concerning valve group representations, is it true that they are all defined? Is there some general rationality that they are all over Q? Yeah, they all define the over Q. Valve representation are all defined over Q, yes. There's no news. So strata are in one-to-one correspondence with a certain subset, which I call is a subset of, take all your insertion of valve group, and then it's a subset of this, which are all repetition, which appear in this way. And they are in projection, so strata. So strata are indexed by this subset. So from this description it's clear that they are finite lemony strata. Okay, now we want to give a second, so anyway, these analogous to this and this definition. So now we want to give a second definition, analogous to this and this. So we denote by this symbol all conjugastic class in the valve group. And if you have any conjugastic class in the valve group, you can consider all elements of minimum length. So all W, such that W equals minimum possible. So length achieve minimum value on this conjugastic class. And then for W, in W, we can look at set all elements of the group, such that B and GB, G inverse, is in OW for some B, some Borelsa group. So this is a union of conjugastic classes. But then we also define, for any conjugastic class in valve group, we define G gamma to be G W. The W is element of minimum length. So here is the issue, you have to verify that this definition is correct. Because you have to check that this set is independent of the choice of an element of minimum length. And this one can be, so this definition is correct. You can deduce from the result of Gek and Pfeiffer. So follows from Gek and Pfeiffer. So Gek and Pfeiffer has shown that if you have a conjugastic class in valve group and take two elements of minimum length, then there is some kind of elementary operations which can transform one into the other. And using that is a very, very non-obvious result. But if you know that, then you can check that this set remains the same, independent of W. So this theorem was proved by Gek and Pfeiffer using computer actually, except for a group proved based by computer. But the user of computer was eliminated by her and Nie. So that's a conceptual proof. It's the same her as appeared in Williamson's talk. It's actually all of my students. It was formative. So anyway, so this thing is non-obvious set but it can be verified. Then we denote by delta gamma. So gamma is a conjugastic class in valve group. It's a minimum of dimension of C where C is any conjugastic class of G such that which is contained in G gamma. So this set is a union of conjugastic classes and take all those of minimum dimension. And the minimum dimension is denoted like this. And then you can also define G gamma in a box. Not some, I don't know, better notation. It's a union of all conjugastic classes of minimum dimension. So C contained in CLG, C contained in G gamma, and dimension of C equals delta gamma. Okay, so this, in this way. So this is a constructable set or? It's certainly a constructable set, yes. And then there's a following claim which is actually also not obvious at all. So claim is that if gamma and gamma prime are conjugastic classes in the valve group, then this set G gamma and G gamma prime are either equal or disjoint. And moreover, they cover the entire group. So in this way, you get another stratification, another way to divide the group into pieces, namely all the subsets of this form. And then also claim is that this strat, this differential stratum and this differential stratum is the same. Are they non-empty? Yes, these are all non-empty. Before you had only a subset of here, W. But now you have a quotient of this. So here is strata, so you can see the following. This is strata of G, and then there's a irreducible of W which is a subset of this. And then there's also a bijection with conjugastic class in valve group, but now it's modular, some equivalence relation. And in particular, there's a natural map from conjugastic class in the valve group to irreducible representation of the valve group, whose image is index, will index set of strata. And in a case of type A, this map is a bijection. All other types is not a bijection. And you claim that this is independent of the characteristics? Oh, that's the next claim, yes. Oh, by the way, I should say that the proof of this statement is actually, it is for classical groups. So it's case by case, so classical groups you can check, but in exceptional groups, it relies on use of computer. And in fact, at this point, you have to use representation theory of groups of finite fields. So you have to use everything that is known, the repetition of finite field. So it's not an easy result in some sense, but it's true. So now the, and from this point of view, this regular set appears as G gamma in a box where gamma is the conjugastic class of a coaxial element. And the central part of a group is G gamma in box per gamma's unit element. Okay, so now I want to give another description of this index set, which is, which will make clear that it is independent of characteristic. So the index set of the set of strata. So support G, U is all set of unipotent, well, maybe U, C, L, G, they're all unipotent, set of all unipotent conjugastic classes. So this set is known to be finite, but so actually finite fields are proved by, P equals zero is morose of Jacobson and Maltsev. And actually, if I get this Maltsev, it's something not known. I think some people completely ignore that. So I think it's always attributed to somebody else, to Dinkin or other people, but I check this Maltsev. So this was 1942, 1951, 1944. And then for P, different for P greater than five, Richardson, and then for any P is, my paper, 1960, 1976, which was actually written at IHES. So that's, we have to use this thing. And then the, well, the classification, I will not, so this all the result were classification, which are more precise than finiteness, but I will not give references to that. But what is important is that this set of unipotent classes of G, of course, you can attach to them some, well, you can restrict this map. You can attach them some irreducible version of our group. So there's a map to this subset, which I defined before. So this is a restriction of the map before. But this will be injective, that's also injective. And the fact that it's injective, it also follows from the springer for P0 and P large and for my paper for any P. And then it has an image, which does depend on characteristic. So it's irreducible, so I call irreducible val group index P. So image is this, it depends on characteristic. And now the claim is the following that, so the following claim is irreducible application of the val group with this underlined. So that's the index set for our strata is actually the union of these things. Maybe put here a P prime over all prime numbers, or all prime numbers. So in this description, you see that this set is independent of the characteristic because this side hasn't, here you have all prime numbers. So it's in a path characteristic, so index. And there's another way to look at this, namely, it's also true that if take interpretations of val group in classic zero, it's also contained in this one for any P. And this means that the unipotent conjugastic classes in a group G over complex numbers can be viewed as a subset of the unipotent conjugastic class G over F, so maybe it's P prime here, P prime. So if you have one algebraic closed field of classic P, it always contains as a subset unipotent classes, classic P contains as a subset, and classic zero. And so there is a distinguished way to specialize. Yeah, well, is this because using, because this is in projection with this and this in projection with this, and this is a subset. Obviously, the reason, but is it the case that when you have a group, let's say, over Q, by the way, the classes are parameterized by some covariance as well, they're all defined over Q, I suppose, the unipotent. When you have a split form group, the classes in G, C are defined over the rationales. Yes. Okay, so you suppose you have such a class, you take the, the schematically. That's probably true, but I don't think it has been, I don't think this has been, this most likely is true what you're saying, but the way I know it is by the way. But you don't say that you take the schematically? No, no, no, no. No, no, no, I don't think it could be the case. It could be the case, yes. It was like, but okay, and then you can form the following set, CL of G underlined. You take union of all unipotent conjugacy classes over all for P prime, any prime number. But these sets are not viewed as disjoint. We identify in each, or each P prime, this set is only considered once. So you can do that. So you have this unipotent conjugacy classes in prime two and prime three and five, et cetera. Each of them contains a subset which are these things in classic zero. So this is classic zero. It's always a subset of this. And then you take the union of all those things, but in which this part and this part and this part are identified with each other. So this is the definition of class re-underlined? Yes. Is it joint union? Okay, the union is a whole device. Yes, yes. And actually this is same as this set here. So these two are in bijection. Is the union of, I know when you say that in this union the intersection for different prime numbers are exactly the same. Exactly, exactly thing which come from complex numbers, yes. Yes. And so this set can be also viewed as indexed for the strata. But in fact, in fact there's very little happens here. So for example, in characteristic, if you have classic, some classical group, then this thing for prime two differs from zero, but all the other primes are the same as for classic zero. So this set in this case, in classical group is just you report in classes in characteristic two. In classical but not GL or SR. In GL, everything is classic zero. In classical group is obviously classic two. But if you take E8, then for two and three you have some different, some, this set is not the same. So in E8, this set has 74 elements and these are 71 elements. These are 70 elements and these are 70 elements for all other primes, 70. And if you take the union, you get 75 elements. So 75 elements of each 70 come classic zero and four exist only classic two and one exist only classic three. And for other groups? No, in other groups you can never have two different primes. So either two or three? Yeah, in G2 you can have three and for all other groups you have only two at most. E6 is not union, E7 is two, F4 is only two. So and then you can say the following thing that the following claim. So I said strata or index by something in a point of P but any stratum contains at most one unimportant class and any stratum actually contains, does contain some unimportant class in some characteristic. If you take the stratum and you move it to some, you take the corresponding stratum in some different characteristics, then it will have a unimportant class. Exactly one, some characteristic. Okay, so now I should say that the fact that this, these two sets are the same. It's proof case by case. I think it's a miracle. I don't understand why it is true, but you can check. And also for the dimension and intersection. Yes, yes. You just closed the classification. Yes, yes. Now maybe I have to make some remark, something when I was at IHS the first time, I talked with Glotendieg actually and I thought I proved something using many verification of many cases. And I thought that's not good. But Glotendieg said, oh, that's excellent. That's how the deeper results should be proved like that. With the, so he thought that that was positive actually. So I always remember that Oh, so now there's a second, another miraculous fact. So, so this, the property I said in terms of the elusive repetition of the bar group. So, so if you take the, your group G, it also has a Lagrange's dual. Lagrange's dual has the same bar group as G. And the claim is that this, this set of elusive repetition of bar group is exactly the same. So, so the same, the same for G and, so, so this strata are extremely stable things. They're not only in a plant of characteristic, they also the same for a group and for a dual. What? The static index, the union of the guy, something. Yeah, but this, this set is also, it's also the same for the two. This is, but because of this, since this is, so these bargex are this. But the individual subsets to the P in them. What? The RwP, are they also independent? The RwP is also the same for G and G dual. This WP? The RwP, the union of the guy is characteristic P. Oh, no, no, no. Not the same. No. And is the map from the conjugastic classes to the elusive representations, is it also independent of the guy? Yes, independent, yes. Characteristic and also independent of passing to dual. What is the map for union? The map from conjugastic classes of bar group. So, this map here is independent of P and it's also independent of passing to dual. So, actually one consequence of this is suppose you look at a symplectic group, SP2N over C, and also SO2N plus one over C. And for each of these groups, you make a list of all dimensions of all conjugastic classes. Just write a list of possible dimensions. Then those two lists are identical. So, it's not obvious, I think. So, list of dimensions. So, is the conjugastic classes... Yes, so the particular list of dimensions is independent of the characteristic by the stuff you mentioned before. But why does it follow that the dimensions are the same for G and the dual, because you can recognize, suppose you know that... Yeah. Does it follow from what you said, or just from the... Because for a given irreducible of W, you have to know the first date for which it appears next to date. But yeah, but they're not the same for one group and for a dual, yes. That's obvious, that's the same. Yeah, because it's a... Yes. LAUGHTER Well, it's a... This is a conmodia flag for some coin variant of some war group. So, the... Conmodia rings are... Yeah. Conmodia rings are... Okay, so now I want to... Oh, yes, actually one strange consequence is that the unipotent classes in casting zero... Well, suppose the group is complex numbers, unipotent classes in casting zero, they index some subset of the strata, the old strata which contains some unipotent class. But if you take unipotent class of the dual group, they also can be seen in terms of the original group, because they appear as some strata, they're not strata which contain the unipotent class, some other strata. But unipotent class of dual group are seen in the original group. Some other set of strata. So now I'll give some example in some low rank to see how this stratification looks like. By the way, this stratification, where the world stratification is different... No, not that... Different meaning, but here it's locally closed. What kind of strata are they are? Yeah, so first of all, I think there was some person who proved that they're locally closed, not me, I think it was Karnovali or Jovana Karnovali. You saw that at least in good characteristics, but I think the proof is also true. So strata are locally closed. That's one statement. But a closure of a stratum is not a union of strata. So it's not as good as that. Are they smooth? No, no, no. You've already been GLN or smooth. No, no, sorry, GLN or smooth, but closure is not a union of... And in other types are not smooth. So there is a big stratum, which is the regular one. And this is the case that you can have a natural... In some cases, the stratification is... That is a frontier condition, but still you have a decreasing chain of closed subsets of the differences of the strata. Is that the case here? Like you take the union of strata of dimension, if most something is closed? I think so, yes, yes, yes. This must have been part of this. Yes, part of that. So maybe I'll give an example, some low rank. So GL2, I think is GL2. In this case, there are two strata. So G central and G regular. G is GL3, they are three strata, which is again this one, this one, and everything else. So these are all classes of dimension, dimension zero and six, and dimension four. And if it's simply the group, then it's a little bit more complicated. In this case, there are five strata. So one is all classes of dimension eight. That's a regular set. Then all classes of dimension six. And then dimension four, there are actually two conjugacy classes, which have dimension four. Each one forms a stratum by itself. One class of dimension four. And there's another class of dimension four. And then they're all classes of dimension zero. So these are the five strata. And these four strata, each one, contains a unipotent element over complex numbers. Now I should say, in Cartesian, different from two. If p is different from two, then this unipotent class and this semi-simple class. But if p equals two, then both this and this are unipotent classes. So what happens in classic two, there is an additional unipotent class, which doesn't exist in classic zero. And it forms a stratum. And as I mentioned, the conjugacy classes of the unipotent class of a dual group can be viewed as part of the set of strata. And they are in fact this one, this one, this one, and this one. So what in classic zero? Yes. Different from two. Yes. So that's how you see the unipotent class of a dual group inside the. In characteristic two, the dual group is respect. Yes, and for E8, maybe I just said E8. They are 75 strata. And so 70 contain unipotent class in classic zero. And four of them contain unipotent class only in characteristic two. And one contains unipotent class only in characteristic three. And in characteristic two, sorry, over complex numbers, there's one stratum which has dimension 120, which is union of two conjugacy classes. One is unipotent on a semi-simple. So that can also happen. OK, so I think that's it. Any questions? Limos, a question. GLN is 3220, the same as just the dimension of Jordan? No. Because you kind of, no. Because total number of strata should be number of partitions. And the dimensions, there are too few possible dimensions. Can you say explicitly what it is? Yes, but I think I'll get it wrong if I say it right. But it's written in my paper, so. But I should say for classic zero, for GLN, this composition has been known before. It was, at least in the Lie algebra case, there's a thesis of Dale Peterson, which was at Harvard a long time ago. So he defined the composition of GLN or Lie algebra level into pieces according to partitions. That makes sense for the groups. And it is the same as this one. And it is also the same as there's a notion of sheets. The sheets for any group or any Lie algebra. The sheets form unions of conjugacy classes of fixed dimension, but they are irreducible, but they are not disjoint. So in that sense, they are not so nice. They're not disjoint. Sheets are not disjoint. Can I have two sheets, which intersect? Each sheet is a union of conjugacy. No, no. Each is union of conjugacy of fixed dimension, and they are irreducible. Each is irreducible, but they are not disjoint. What's the relation to your? Well, in GLN, they are exactly the sheets. And other types, each of my things is a union of sheets, but it's not exactly a sheet. The finite union of sheets. Finite union, yes. What's that in singularity? Can you expect something like the finite group singularity? Yeah, I don't know. I don't know. To know GLN, they are non-singular. I think it's known that in other types it's not. But I don't know exactly. GLN, they are non-singular. All the, all those. Yes. We can find this picture again.