Okay, thank you. It's an honor to be here.

The first time I came here was exactly 50 years ago, almost to the day. It's terrific to be here to honor Ofer Gabor from whom I've learned a terrific amount over the years I guess when I first came, the IHS was 10 years old, Ofer was 10 years old.

He may have already known more than I ever would. I don't think I was 10. You were 10? No, but I was not, I was not, ah, I was 10. Okay. No, in thinking about this conference,

there are two people who I really think of as people who should be here, namely Reynaud and Ekedal. Yeah, okay. So let me start. Right, so for a long time I've been interested

in the fact that very simple to write down, which is to say simple for me to remember, local systems have trace functions which are the trace functions of random elements

in various classical groups. So let me give you just one simple example. Let's take n to be a prime number,

and maybe I should say we'll be in some characteristic p if we can stabilize the chalk here. Fp, and we'll have an additive character,

well, to c cross or to ql bar cross or wherever you like, non-trivial. If I have a finite extension, I'll denote by psi sub k,

just psi composed with the trace. So that's a non-trivial additive character there. And the example I wanna put on the board, if n is a prime number and I look at normalization with some gal sum, which is not important right now,

this gal sub k, sum over k, x in k, psi of x to the n plus tx, t is the parameter, quadratic character of x. So here we should be in any characteristic p,

which is on the one hand odd because otherwise quadratic character doesn't make sense and p doesn't divide, and since it's a prime isn't equal to n. Okay, and what's nice about this example,

so here the rank of this local system is the n here, which is fine. And the fact is that if the characteristic p is strictly bigger than one plus twice n,

then the group, the so-called g-geometric, the monodromy group associated to this, the group traces of random elements of which are these sums. This group is, it's usually S-O-N,

if n is in seven, and it's in fact the exceptional group G2 if n is seven. So I'll explain in a minute where this comes from, but the fact that you have this dichotomy

is based actually in a fundamental way on a theorem of offer, namely that, as we'll see in a minute, when p is large compared to the size, we know that the group we're looking at,

even its Lie algebra, operates irreducibly. And Ofer has a theorem about prime dimensional, n being the dimension, prime dimensional representations of semi-simple Lie algebras. We also know by a theorem of Doline that we're talking about a semi-simple group,

and the theorem is that prime dimensional representations, the group in question, if the prime is odd, is either S-O-N or S-L-N, or if n is seven, it could be G2.

And another fact is that whatever p is, the group is either finite or what's written above.

And this condition, p large compared to n, that enters because of a theorem of Feit and Thomson,

which is a kind of wonderful strengthening of one of Jordan's theorems on finite groups.

Jordan's theorem is that if you have a finite subgroup of GLN over the complex numbers, then it has a normal abelian subgroup whose index is bounded by some constant which only depends on n, okay?

And Feit-Thompson theorem says that if G in GL, so this is, so to speak, a different n than that one, if you like, G in GLN-C is finite, and p is greater than one plus two n,

then the PC-Low subgroup of G is a normal subgroup, so it makes sense to say V, and it's abelian.

Now, it's obvious from Jordan's theorem that this will be true if p is huge. Okay, now, the way this is relevant in this sort of game,

so this local system is on the affine line. Now, the fundamental group of the affine line

over an algebraically closed field of characteristic p, this group has no non-trivial prime to p quotients.

So another influence of Ofer is that I wrote no non-trivial prime to p quotients instead of no prime to p quotients. Right, so the way this is relevant is this. Suppose you're in this situation,

you're trying to define if you're finite or not. Okay, so if G were finite, and p is in this large range, then G modulo its PC-Low,

which makes sense because this is a normal subgroup, is prime to p, but it's occurring on A1, so it's trivial, so in other words, G is a p-group, and it's abelian, that's the second part,

but an irreducible representation then of an abelian group is one-dimensional, in other words, rank one, but our rank isn't one. So that's why when p is large, you can rule out the finite case, and you get the conclusion you want.

Do you know about the G arithmetic and G geometric are the same, or? Okay, the answer to that question is yes, if you have the right choice of Gauss sum normalizing factor, then they're equal.

So the answer is yes. Even in the finite case? No, in the finite case, it can matter, and so I'll give an example. Right, so let's take this n to be seven,

so to speak, the G2 case, okay, and let's try p equals three, okay? Now what's going to turn, and for reasons

which I'll explain in the course of the talk, we're going to use the following identity, seven is three cubed plus one divided by three plus one, not false, and this is going to tell us that, as we'll see, G geometric is in fact SU3, three, okay?

And my memory is that G arithmetic could require a quadratic extension to become equal to this, okay?

Let's try another one, let's try p equals 13. Well, then we're going to use the equation, and I'll explain why later, that seven is 13 plus one divided by two, which I'll write as one plus one, okay?

And what's going to come out here is that G geometric is PSL2 over the field of 13 elements, and I don't remember what you have to,

I mean, I don't know the answer to your question off the top of my head for whether it's, when it's G arithmetic or not. So the numbers in your formula for seven are meant to kind of go into the formula for the group somehow of? As we'll see, yes. Okay, so I knew about this Weigth-Thompson theorem

maybe 30 years ago, and I basically regarded it as a kind of red flag,

don't fool around with low characteristic. Let the characteristic be a little bit high compared to the rank, and then you got nice results. And just, I didn't know anything about finite groups, that's still the case, but I just wanted to stay away.

Okay, however, at some point, I actually looked slightly more carefully, which means I actually read the first page carefully, of the Weigth-Thompson paper, and they point out that their result is best possible

because if you take SL2 over FP, this has an irreducible representation, complex representation of dimension P minus one over two.

So it's another one, is it the same one of Weigth-Thompson with the odd order? No, it's a completely different paper. This paper, it's before the odd order paper, and there had been previous theorems of this type

with, maybe P was bigger than N squared, and you had this kind of conclusion. Anyway, so in fact, it has two such, and it has others of dimension,

in fact, P plus one over two, and two of these. So here, of course, two times this plus one is P, so that's the sense in which it can't be improved, and this one is, so to speak, even worse.

So, okay, so on the one hand, I have that. Now, on the other hand, also going back 30 years, maybe 31 years, Van Cubert, who's no longer alive, was coming to my course,

which that year was about Clusterman-Sheeves. It's not important what they are, but I was interested in knowing when Clusterman-Sheeves had healthy monodromy groups,

and Cubert, by an extremely clever analysis, evaluations of Gauss sums, showed that certain local systems had, in fact, finite monodromy groups,

because for these local systems, if you want one of these local systems, the kind I've written down, to have a finite monodromy group, you just, just have to show that its trace function takes algebraic integer values, that there are no denominators.

And so Cubert says that if you look, for instance, so with this normalizing factor, psi of, now this is a different n, x to the n plus t x, or, well, his method gave

also this times a quadratic character, that this would be finite, a finite group, if n is a prime power plus one over two,

so q equals p is gonna give a local system here,

we would have rank n minus one, and with this one, we would have rank n, so if n is q plus one over two, then we'll either have q minus one over two, if we don't have a quadratic character, or we'll have q plus one over two, okay?

And he also proved, by this incredibly clever argument, that if n was q to the d plus one over q plus one, with d odd, which is what you need to be sure that the bottom divides the top, okay,

then this would also give you finite monodromy. And in fact, the technology that he employed can be used to make it, in this case,

slightly better, instead of just transferring with a quadratic character, you can do with any character of order dividing q plus one, not just order two. So, you have all these local systems

that, by these old results of Kubert, have some kind of finite monodromy groups, and I basically didn't think about it anymore for a long time, right? And what changed, how do I get the top one down?

Oh my God, this is terrifying.

So, that's okay, I think I just erased

what I wanted to refer to. Right, so then, I guess it's at this point almost two years ago, I stumbled upon a 2010 paper by Dick Gross.

And in that paper, Dick Gross analyzes what the lean-listic theory does in the case of PGL2 and PU3.

And in both cases, he tells you that by thinking about the lean-listic theory in both of these cases, you make local systems

on P1 minus zero and infinity, whose groups are these groups,

with local monodromy at zero and infinity given, so to speak, explicitly in terms of the group. So, for instance, local monodromy at infinity

will involve a Borel subgroup and its unipod radical. Local monodromy at zero will involve what he calls a Coxeter torus. So, it's completely explicit. And because from thinking about Clusterman

and hypergeometric sheaves, I knew something about local systems like this. So, I saw that if you take these local systems with the known local monodromy and push them out by representations, so there's a slight technical point that you have to understand what happens when you go from PGL down to PSL2.

That's a subgroup of index T. And here, you have to go down to PSU3. But what happens when you push out by representations of these groups and using some rigidity, you can actually prove

that you are getting the geometric monodromy groups to be, in this case, SL or PSL, and in this kind of case, well, SU,

only when D is three, or PSU. But the point is that these numerical formulas about the seven, they're not an accident.

Okay, so this is where I was at a certain point. And then a sort of funny thing happened. Namely, well, two funny things happened.

So, plugging in what Dick Rose had done gave me not the representations of PSL.

So, in this kind of story, where P could be Q, whichever of these dimensions is odd, and since they differ by one, there's only one such, that's a representation of PSL.

And the other one is a faithful representation of SL. You just have to look whether the element minus one, what its trace is to see, okay. And so I knew something about one of these local systems.

I knew it would have PSL too. And the other one should have SL too, but I didn't see how to do this. So, I said, all right, well, these two dimensions differ by one.

So let's try something which looks ridiculous on its face. Let's look at SIM two of the small one, the low dimensional one, and let's look at exterior two of the high dimensional one.

I mean, high and low, they only differ by one. Since they only differ by one, these two local systems at least have the same size as each other. Now, in general, this would be an idiotic thing to do. For instance, if you were talking about

a low dimensional, say a 10 dimensional representation of the symplectic group, and you did this, you'd be getting the Lie algebra. And if you talked about an 11 dimensional representation of the orthogonal group, you'd be getting its Lie algebra. So how could they have anything to do with each other? Nonetheless, it turned out that

in this Q plus one over two game, they were isomorphic. So there are two parts to this statement. One is the statement that in the representation theory of these groups,

when you take SIM two, now, you have to be careful because there are two of each dimension. So the statement is that SIM two of a low one is exterior two of a correctly chosen high one.

So that's one statement. On the other hand, if I'm trying to prove something about my local systems, I need to know that it's also true that my local systems behave in this way. So Ron Evans, a man who can prove any identity about exponential sums that is in fact true,

did the C identity. And because the character table of SL two Q and Q is a power of an odd prime is simple enough,

I could actually see by looking that this was a true statement about the representation theory. Okay, so the next thing that happened was I wrote to Gorelnick, who's a serious expert on finite groups, and I asked him if this was a known thing.

And he said, no. But in fact, he said, the same thing is true, the same thing is true for the symplectic group.

So I have to explain what that means. And for me, it was like amazingly new information. So let me tell you what that was.

So we're going to look at an SL two, and let me write it just as Q to the N.

And Q could be P, but not just SL two of a prime field. Now, of course, if you think of this as automorphism of a two dimensional vector space over this field, it's not hard to see

that you can embed this group into the symplectic group of size two N over F Q. Just thinking about the two dimensional vector space over F Q N as an N dimensional, as a two N dimensional vector space over F Q, and with some obvious symplectic form.

Right, now this group, or PSL, is going to map by the representation theory into some humongous SL Q plus or minus one over two

group like so. And apparently, in the world of simple or nearly simple groups, when you take such a group

and map it into some big SL, by near reducible representation, apparently what's typical is that what you get, the image is already a so-called maximal subgroup. There aren't bigger finite groups in here that contain this image.

But what's special, which came as a complete revelation to me, is that this bigger group has a representation of the same dimension. And in fact, so this group has representations

of dimension Q N plus or minus one over two, two of each flavor. And when you take one of these representations and restrict it to this much smaller group, you're getting the irreducible representation

that we had here. That's somehow, I think, quite remarkable. And for me, it was completely new information. So on the one hand, it made me wonder

about local system or systems to give symplectic groups as monodromy groups. But again, my criterion was, yeah, maybe I should say,

I mentioned Raynaud at the beginning of the talk. Now, of course, Raynaud proved the Abhi-Ankar conjecture that any finite group generated by its PCLOS subgroups will occur as a quotient, a finite quotient of the fundamental group at the affine line over an algebraically closed field of characteristic P.

And any of these groups certainly have that property. So they certainly occur. And therefore, you can write down something that's gonna make them occur. And Abhi-Ankar wrote down lots of things that made them occur. But I wanted simple things that I could remember.

So that was one piece of information. You mentioned somewhere about the image being a maximal sample. Yes. But once you have a representation of the larger symplectic group.

Yes, so this is an example where it's not true. And then, I guess, because of what Dick Rose

had done about SU3, I learned a little about SU n odd, so different n yet, at least three, Q, and Q is an odd prime power.

And its lowest dimensional representations, next to lowest, there's one of dimension, so this n is the d that was over there.

Qn plus one over Q plus one take away one. There's one of these. And their Q of this entire dimension, Qn plus one over Q plus one.

And the, so to speak, naming scheme of these is they're parameterized by multiplicative characters, pi, of order dividing Q plus one. So here, the relevant character is the trivial character,

and here it's the Q non-trivial characters. So, and these were the numbers that Kubert had said, had proven, gave finite groups.

Right, so the obvious conjecture then, was that when you did the local systems

with this kind of n, and characters like this, you were getting the representation theory of SU n, n being odd, like so. And so both, if you like both, hopes are pretty much okay

and that's entirely due to joint work with Tf, which I'll try and explain just the broad basic ideas

of that make this come out okay.

So a difference between the symplectic case and the special unitary case, at least at this point in the exposition,

difference is that in the case of SU, we had our candidate already. On the other hand, we didn't have a candidate for the symplectic case. We thought maybe there was one, but,

so let me go back to the SL2, but let me write it with Q n. And instead of trying to write down,

or writing down candidates for one of these and one of these, I'm gonna write down a reducible local system, which will be the direct sum of one of each. So let me write that down. I would write down, so a suitable normalizing Gauss sum,

the sum over X in K. So instead of writing Q n plus one over two, I'm gonna write Q n plus one and then I'm gonna write, instead of T times X, T times X squared.

Now if I separate out, so to speak, the squareness and the non-squareness, I'm going to get the thing I had before with this divided in half and this divided in half, once alone and once with a quadratic character. So this will be one representation of Q n minus one over two

plus one of dimension Q n plus one over two. So can you say again what you're,

how do you, first of all, homologically, does it correspond to, it corresponds to higher direct images which we're doing from sum? I just want to, if these were divided in half and I just had an X here, I would just think of it as a Fourier transform.

So if you like, this is the Fourier transform of ELPC of X Q n plus one over two, direct sum ELPC of X Q n plus one over two,

tensor quadratic character. Those are my two pieces. But in fact, when you write the thing this way,

you get a different proof from Coobert's proof that this kind of thing will have finite monodromy because there's a paper that came later than Coobert by van der Heer and van der Flut,

which talks about these kind of sums, psi of what they call maybe R of X times X and this R of X is supposed to be maybe what I would call Q linear.

So you see here, here I have X Q n plus X plus T X times X, that's what's inside. So this is my Q linear polynomial and this is my X.

That by a very clever but elementary argument, this kind of local system, well this kind of sum individually with the suitable GAL sum, will have algebraic integer values and therefore that this kind of local system

will have finite monodromy. It's another way of thinking about the same question. Anyway, so at this point, without any underlying conceptual reason,

I said, well, suppose we were to look instead, Q n plus one, let's have a two parameter family,

two plus one plus T X squared. So this is also gonna break up into two pieces.

And so what do we know about this? We know that on the one hand, this is going to land, it's gonna be a sum of two representations of the right dimension and when we put S to zero, we know that we get SL two Q to the n.

And basically the idea is that

we're now going to have a group that's sitting between SL two Q to the n,

R G geometric, because when you specialize you get something smaller. And on a priori grounds, this is gonna end up in a big SL Q n.

And basically, the theorem that Tia proved was that on the one hand, this intermediate group,

and I'll oversimplify this,

but to a first approximation, if you factor this n, then in between, you'll have SL two Q to the n, that'll be in an SP two A of Q to the B,

and that'll be in an SP two N of Q. And in fact, there are actually a few more possibilities, but ballpark, there aren't so many possibilities for what this guy is. And we know that we're talking about the restriction

to this group of a very special representation of the symplectic group, because these special representations already restrict all the way down here to these special representations. And the group theory people know that

if you had a group like this, then when you looked at the square absolute values of traces in these representations, you would be getting powers of Q to the B. And by looking at this local system and choosing S and T not all that cleverly,

you could get a square absolute value equal to Q. So there were some technical things. And in fact, what you end up with is that you'll get SP two N Q from this,

provided that first of all, P doesn't divide this N if you're in characteristic P, and P doesn't divide the power of P that is Q.

So in a kind of overly complicated logs of P of Q. So it's some technical business, but basically that this works. Okay, now the next step is we say,

all right, so here we used the trick that when we put S equal to zero, we got something we knew.

And therefore, whatever we were getting at least contained that. So now what we're going to do is we're going to put P to zero.

So now we know on the one hand that we're inside. So now for this guy, we have a new G geometric when T is zero. We know that we're inside what we had proven before,

which was at this SP two N Q. But now we have something special because when we just take this much of the local system, that's what it means to put P equal to zero. This thing now becomes a direct sum of,

where you have psi of X Q N plus one over Q plus one plus S X times a chi of X where here the order of chi divides Q plus one.

Same, just extracting, breaking off these Q plus first powers. And that's why we need N to be odd so that Q plus one divides Q N plus one. So we'll get this if N is odd.

Okay, right. So now we come to another miracle provided by Tf that if you have a subgroup of the symplectic group such that this Q to the N dimensional special

representation, which was the direct sum of two pieces here, that when you restrict it to your subgroup, it breaks up into Q plus one pieces with the ranks that you're getting.

Then automatically you have this corresponding big representation of the special unitary group, the direct sum of all these pieces. So it's like a miracle. And so my role was to guess some local systems and Tf would tell me, well, if you could just prove

this or that, then by group theory we'd know the rest and I could prove this or that. So thank you very much. Are there any questions or remarks?

So in the last thing, you have also to know that the group is not smaller than the, you said there is a group theoretical result. First of all, there is also the question of the arithmetic and I don't know if the arithmetic comparison of arithmetic and geometric.

Right. Because the values give you things in the arithmetic. Correct. So in the case of the symplectic group, when the theorem applies, so with many things being primed to P, then my memory is that when you,

so these local systems make sense on A1 over the prime field. When you extend scalars in the symplectic case to the Fq, then my memory is that g geometric is already equal to g arithmetic. In the case, in this special unitary thing,

you already need to work over the field of q squared elements just so that you have these characters available and then, at least the a priori argument I could make required a pretty big extension of Fq squared to get g arithmetic down to being g geometric.

And also you have to know that you get exactly the group you expect. You need to know it can't be small. You said that if your representation breaks down. Yes. Then it is, the group is contained in the one you. No, no, no, if I have a subgroup of the symplectic group and the representation breaks up, then that group is?

There is no smaller, ah, you said that there is this SL, no. Which one, which situation are you speaking about? Not this one. Not this one. So when I assume that, to make life simple,

let me assume that both q as a power of p is a prime to p power of p, which sounds ridiculous, and n is also a prime to p, right? Then I know that my monotony group, my symplectic candidate has grouped the full sp,

not one of these, the intermediate things had to be ruled out to get to sp. Yes, okay. Okay, once you have sp, then Tf tells us that if you have a subgroup of the sp, such that this so-called v representation of sp breaks up into pieces of, irreducible pieces

of these sizes, then it's automatically what you want. So that it reduces again by the geometry. Yes, yes. You could just add more parameters, right? You could add more terms there

for different powers of q. Correct. It'd still be symplectic group. So do you know if the covering of the affine n space you get by doing that is related to the Deline-Luv stick variety for sp2n? I can answer that question very quickly, no. But there is a man here who might be able to help you.