it's the 50th anniversary of my first visit to IGS, for the 68th first time. What happened was that I was in Oxford with Atiyah, and he, before leaving, before I left Oxford, he called Groton League, my phone, and 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 IGS, but in different, which played a very big role for me.

Now about, of Gabor, I think I first saw him, not met him, but saw him, maybe 10 years after that when I went to MIT, and he was student at Harvard. And I remember going to, only thing I remember, I went to lecture by Sarah,

where he was the only one who understood everything. Yeah, yeah, yes. And then several years later, I think we also had some, I think I benefited tremendously from his work

called Pervert Chief, which were quite important for me. So, the thing I will talk about today, I have to say that Gabor's work will not appear explicitly, but they all, in fact, without his work,

I couldn't say not one word, I think. Everything I say, it wouldn't exist without his work. Okay, so I will talk about some property of reductive groups of algebraic closed field,

which I think is maybe not so well known. So, I will always consider K algebraic closed field of characteristic P can be zero,

and G will be connected reductive groups, reductive group. So, the classification of such groups was given by Chevalier in 1955, and they are independent, so it's very remarkable, we think they're independent of,

classifications independent of K. That's a classification. But what I've tried to do,

I tried to decompose G into finitely many pieces as a partition, which, 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 mod, maximum torus.

And this suggests that there are two important subsets of the set of conjugacy classes, one is set of regular elements, these 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 tried to do, I tried to interpolate,

I will try to interpolate between those two extremes, but I will define some partition of G in which this one part and this 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 the Wahl group, and then it is known that it can be interpreted as saying that D cross B is union of subsets indexed by the Wahl group, which are the orbits of the group G.

And then, from this, you can get a length function on the Wahl 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 Wahl group. Now, for us, important, so if G is an element of the group, then you want to consider the,

all Borel subgroups containing G. So, this also called Springer fiber. And then, there is a theorem of Spartenstein,

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, call it DG, so it's a pure dimension.

And this number is actually equal to dimension of B, minus dimension of conjugacy class of G, divided by two. So, this was proved by, actually by,

this is, the second equality was proved by Steinberg, in classic zero, and by Spartenstein, any characteristic. Okay, now, the set of regular elements has been studied first by Konstant in Lie algebra setting,

and by Steinberg in group theoretic setting, and Steinberg has given two, so there are two complete different definitions of the set. So, one is that this whole conjugacy, this union,

and one, I believe the constant was just zero. Yes, constant zero, yes, any characteristic, yes, yes.

So, what can describe is regular elements, it's all elemental group, for which this dimension of this variety BG is zero. And that is because, that is 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, conjugacy class,

is contained in a regular set, if only, if the following is true, for some, or any, coxeter element. So, I will say what it is, of minimal length, W.

If you take this class, intersect, suppose you pick some B. So, C intersected BWB is not empty.

So, Steinberg showed that, if it's true for one coxeter element or minimal length, then it's true for any other coxeter element. This characterizes the regular.

And this was Steinberg, 1965. By the way, coxeter element, minimal length, is a product of the single reflections, in any order.

This is a definition of the word coxeter element, or coxeter element of minimal length? Minimal length, or minimal. Any conjugate of that.

Double 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, fixed-point set in a flag manifold. So, all elements contained in all Borel subgroups. 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, if W is not one. So, in this description, these regular elements, you can associate with them, in some sense, the conjugate class of a coxeter element,

and here, you can also associate the conjugate class of unit element. By the way, the formula, I didn't understand. You wrote it, maybe there is some, maybe it's more of a second, it's Paltrowstein's formula,

because in the, you write D, D minus, so first, you divide by two, what, that's the dimension of the class, yeah? Yeah. Okay, but then, in the regular case, according to your formula, it's the dimension of G mod T, so you get twice the dimension of the only potential particle.

That's the same as the, that's the dimension, not the potential, the dimension of flag manifold, is it? This one, so that's. Ah, it's not the dimension of B, but okay. Yeah, the flag manifold. The dimension of, okay, so it's D. Curly B. Curly B, yes. Okay.

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 pi, is two of G, and then we can map this to homology group, so this with et al homology, homology of the full flag manifold, so there's a natural map, and we define EG to be the image of this homomorphism.

What is the criterion for G and G sent? What? G is in B, G is in the center, it's normally G is in B1B and not in B1. Yeah, not in BWB.

But it doesn't seem to be, maybe, so what is your assertion, because you detect, and suddenly B is non-central. Okay, but yeah. All these, all conjugacy class,

all, I mean, all conjugates of G satisfy this. Yes, yes. Okay, so for each element of group,

we can associate the subspace of the homology group of a flag manifold with a larger coefficient. And then we say, we define, we say two elements are equivalent if DG equals, so this dimension is the same,

and also EG equals EG prime. So this, obviously, equivalence relation on the, on our reductive group,

and the equivalence classes are called strata. And the. Of course, you can do it in terms of char groups here.

Oh, yeah, yeah, okay, yeah. So it's this, and then it depends on where. Yes, but I have to quote some things which are proven at our homology, so. So, and it's clear that each stratum

is a union of conjugacy classes of fixed dimension, because of this condition, and because of this result of spartan time. So 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 finitely many strata. So for this, you have to use some properties.

So one property is that W is var group, but it's well known that it acts naturally on the homology groups of flag manifold, because this can be replaced by G mod T,

and then on G mod T, there's a var group 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. To prove this, you can assume that G is unipotent, and that's an easy reduction. And unipotent case, this follows from result of Springer,

1976, assuming P equals zero, or P large, or myself, in 1984, for any P. So unipotent case, you have to, and actually, at this point, result of Gaber, or enter in the proof of those things.

So this W stable. And more than that, in fact, E G is irreducible as a W module. And also, E G appears in multiplicity one,

in the homology of this flag manifold, in this var group representation,

and it doesn't appear in any lower one. So all these are proved here and here.

But it can, of course, it tends to appear higher. What? It tends to appear higher 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 var group representation. So if you have a var group representation as isomorphism class, then this D G is uniquely determined because it's a first degree in which a var group 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 var group representations. By the way, concerning var group representations,

is it true that they are all defined, is there some general rationality that they are all over Q? Yeah, they're all defined over Q. Var group representation, all reputational are defined over Q, yes. There's no, yes. So strata are in one-to-one correspondence

with a certain subset which I call, is a subset of, take all units of variation of var group, and then it's a subset of this, which are all reputational, which appear in this way.

And they are in magic, so strata. So strata are indexed by this subset. So from this description, it's clear that they're fine, actually, many strata. Okay, now we want to give a second, so anyway, this analogous to this and this definition.

So now I want to give a second definition analogous to this and this.

So we denote by this symbol all conjugacy classes in the var group. And if you have any conjugacy class in the var group, you can consider all elements of minimal length.

So all W such that W equals minimum possible. So W length achieve minimum value on this conjugacy 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 more or less a group.

So this is a union of conjugacy classes. But then we also define, for any conjugacy class in var group, we define G gamma to be GW,

where W is element of minimal 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 minimal length. And this one can be, so this definition is correct. You can deduce from a result of Gek and Pfeiffer. So follows from Gek and Pfeiffer.

So Gek and Pfeiffer has shown that if you have a conjugacy class in var group and take two elements of minimal length, then there is some kind of elementary operation which can transform one into the other.

And using that, it's a 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 a computer, actually, in exceptional group, proved based by computer. But using a computer was eliminated by He and Nie. So that's a conceptual proof.

It's the same as appeared in Phil Lamson's talk. Same, same. Which actually, all of my student was former student. So anyway, so this thing is non-obvious set,

but it can be verified. Then we denote by delta gamma. So gamma is a conjugacy class in var group. It's a minimum of dimension of C,

where C is any conjugacy class of G such that, which is contained in G gamma. So this set is a union of conjugacy 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. It's not, well, some, I don't know, better notation. It's a union of all conjugacy 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 constructible set, or? It's certainly a constructible 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 conjugacy classes in the var group, then the 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 the other claim is that this division of stratum and this division of stratum is the same. Are they non-empty? Yes, these are all non-empty, yes.

Before, you had only a subset of E or W. But now you have a quotient of this. So here the strata, so you can see the following. There's a strata. Oh, okay. Of G, and then there's a irreducible of W,

which is a subset of this. And then there's also bijection with conjugacy class in var group, but now it's modulo some equivalence relation. And in particular, there's a natural map

from conjugacy class in the var group to irreducible representation of the var group whose image is index, will index set of strata. And in a case of type A, this map is bijection.

All other types is not a bijection. And you claim that this is independent of the characteristic? 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, classical groups you can check. But except for 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,

representation of finite field. So it's not an easy result in some sense. But not as useful. So now the, and from this point of view,

this regular set appears as G gamma in a box where gamma is the conjugacy class of a cox element. And the central part of a group is G gamma in box where gamma is a unit element.

Okay, so now I want to give another description of this index set, which is, which will make clear that this independent of characteristic.

So the index set of the set of strata. So suppose G, U is all set of unipotent, maybe U, C, L, G, they're all unipotent,

set of all unipotent conjugacy classes. So this set is known to be finite. So actually it was proved, for p equals zero,

it's models of Jacobson and Malzev. Actually, it's Malzev, 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 checked with Malzev. So this was 1942, 1951, 1944. And then for p different, for p greater than five,

Richardson, and then for NEP, my paper, 1960, 1976, which was actually written at IHES, I think. So we have to use this thing. And then the, well, the classification,

I will not, so this also result for classification, which are more precise than finiteness, but I will not give references to that. 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 reduced representation 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 not, 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 NEP. And then it has an image, which does depend on characteristic. So it's irreducible, so I call irreducible var group, index p. So image is this, and it depends on characteristic.

And now, the claim is the following, so the following claim is the irreducible version of the var group, with this underlined,

so that's the index set for our strata, is actually the union of these things, maybe put here p prime, over all prime numbers, all prime numbers.

So in this description, you see that this set is independent of the characteristic, because this side, you have all prime numbers, so it's independent of characteristic, so index.

And there's another way to look at this, namely, ah, it's also true that

you take irreducible versions of var group in characteristic zero, it's also contained in this one, for NEP, and this means that the unipotent

conjugacy classes in a group G over complex numbers can be viewed as a subset of the unipotent conjugacy classes in 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 a subset, unipotent 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 is in projection with this, and this is a subset. Oh, this is the result, but is it the case that when you have the group, let's say, the group over q, by the way, the classes are parameterized by some commentaries as well, they're all defined over q, I suppose, the unipotent.

When you have a split form, the group, the classes in Gc are defined over the rationals. Okay, so suppose you have such a class, you take the, the schema. 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 schematic log? No, no, no, no, no, no, no, no, no, I don't. It would be the case. It could be the case, yes, most likely, but. Okay, and then, 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, for each p prime, you identify, 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 conjugacy zero. So it's conjugacy 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 G underlined? Yes. The disjoint union, okay, union is a perturbation. Yes, yes, and actually this is the 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 the same, which come from complex numbers, yes.

And so this set can be also viewed as index set for the strata. But in fact, there's very little happens here.

So from a characteristic, if you have 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, classical group

is just unipotent classes in characteristic two. In classical, but not GL, always have been. In GL, in GL, everything is classic zero. The classical group is always in characteristic two. But if you take E8, then for two and three you have some different, this set is not the same.

So in E8, this set has 74 elements and this has 71 elements. These are 70 elements and these are 70 elements. For all other primes, it's 70. And if you take the union, you get 75 elements.

So the 75 elements of each 70 comes to classic zero and four exists only in characteristic two and one exists only in characteristic 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.

E7, E6, there's nothing and E7 is two. Therefore, it's only two. So and then you can say the following claim. So I said strata or index by something in a parent of P,

but any stratum contains at most one unipotent class and any stratum actually does contain

some unipotent class in some characteristic. If you take the stratum and you move it to some, you take the corresponding stratum in some different characteristic, then it will have a unipotent class. Exactly one, some characteristic.

Okay, so now I should say that the fact that these two sets are the same,

it's proved case by case. I think it's a miracle. I don't understand why it is true, but you can check. And also, for example, the main intersection is just with the classification? Yes, yes.

Now, maybe I have to make some remark, something when I was at ICS the first time, I talked with Grothendieck actually, and I proved something using many verification of many cases. And I thought that's not good.

I got it and he said, oh, that's excellent. That's how, deeper results should be proved like that. So he thought that was positive, actually. So I always remember that. Oh, so now there's a second, another miraculous fact.

So this, I said in terms of the unisuppression of our group.

So if you take your group G, it also has a Langlands dual. Langlands dual has the same value group as G, and the claim is that this set of the unisupputation of our group is exactly the same.

So it's the same for G. And so these strata are extremely stable things. They're not only in a pant of characteristic, they're also the same for a group and for a dual.

What? It's got this index piece. You are the theologian of the guy, so pick. Yeah, but this set is also the same for the two. This is, well, because of this, since these are, so these budgets are with.

But the individual subsets, there's a P in them. What? The error W P, are they also independent? The error W P is also the same for G and G dual. Which W P? The error W sub P. You define the error. You define the characteristic P.

Oh, no, no, no. Not the same. No. And is the map from the conjugated classes to the reducible representations, is it also independent of the kind? Yes, independent, yes. Characteristic. And also independent of passing to dual.

What is the map from? The map from conjugated classes of bar group. So this map here is independent of P and is also independent of passing to dual.

So actually, one consequence of this is, suppose you look at a symplectic group, S P two N over C, and also S O two N plus one over C. And for each of these groups, you make a list of all dimensions

of all conjugated 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 conjugated 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, O G and the dual, not let's dual, because you can recognize. Suppose you know that, does it follow from what you said or just from the, because for a given irreducible dimension of W,

you have to know the first D for which it appears in H to D. But yeah, but they are not the same for one group and for its dual, yes. That's obvious, that's the same. Yeah, because it's a, yes. Yes.

Yes. Well, this is a, this is a, this is a common flag for this, some coin variant of some bar group. The common rings are, well, what is this? The common rings are, is what? Yes.

Okay, so now I want to, oh yes, actually one strange consequence is that the unipotent classes in classic zero, suppose a group is complex numbers, unipotent class in classic zero, they index some subset of the strata,

all 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. They are set a set of, some other set of strata. So now I'll give some example in some low rank to see how this stratification looks like. So by the way, this stratification where the whole stratification

is different. No, no, no. It's like different meaning, but if it's locally closed, what kind of the strata or what they are? Yeah, so first of all, I think there was some person proved that they are locally closed. Not me, I think it was Carnevale,

Giovanna Carnevale. He showed that at least in good characteristic, but I think the proof is also true with any. 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.

Okay. No, no, no. Already in GLN are not smooth. No, no, sorry. GLN are not smooth, but the closure is not a union of, and in other types are not smooth. So there is a big stratum which is the regular one? Yes. And is it the case that you can have a natural,

in some cases the stratification doesn't satisfy the frontier condition, but still you have a decreasing chain of closed subsets of the differences of the strata. Is it the case here? Like you take the union of strata of dimensionally closed something, is it closed?

I think so, yes, yes, yes, I think so. This must have been part of this. Yes, part of this. 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. If G is GL3, there 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 symplectic 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 form a stratum by itself, one class. Of dimension four. And there's another class of dimension four.

And then there are all classes of dimension zero. So these are the five strata. And this four strata, each one contains a unipotent element over complex numbers.

Now I should say, in characteristic different from two, so if P is different from two, then this is a unipotent class, and this is a semisimple class.

But if P equals two, then both this and this are unipotent classes.

So what happens in classic two, there is 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. It's different from two. Yes.

So that's how you see the unipotent class of dual group inside the... In characteristic two, the dual group is the, of Sb is the... Yes, and for E8, maybe I just,

again for E8, there 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, in complex numbers,

this one stratum, which has dimension 120, which is union of two conjugacy classes, one is unipotent and one is semisimple, so that can also happen.

Okay, so I think that's it. Any other questions?

Limo's naive question. G11 versus zero, the same as just the dimension of the orbit? No. Because you can, no. That would, the 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 before I try, but it's written in my paper, so. But I should say for classic zero,

for GLN, this decomposition 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,

of 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 class 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 union of conjugacy of fixed dimension. Yes. 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. Finite union or not? Finite union, yes. What about singularities? Can you expect something like the finite group singularity equations?

Yeah, I don't know. I don't know. I don't know GLN, they are non-singular, but I think it's known that in other types it's not, but I don't know exactly. GLN, they are non-singular. All those. Yes.

We can thank the speaker again.