 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 it's terrific to be here to honor Ofer Gabber 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 You were 10 Okay No, I In thinking about this conference. There are two people Who I really think of as people who should be here namely Renault and Ekadol and 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 and to be a prime number and Yeah, maybe I should say We'll have will 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 want to put on the board if n is a prime number and I look at Normalization with some galsum, which is not important right now because galsum k the sum over k x in k Psi of x to the n plus t x 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 the rank of this local system is the n here Which is fine and the the fact is that if the characteristic P is strictly bigger than 1 plus twice n then the group So-called g geometric the monogram we 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 isn't seven and it's in fact the exceptional group g2 if n is seven and So I'll explain in a minute where this comes from But the fact that you have this Dicotomy is based and 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 are looking at even it's Lee algebra operates irreducibly it's and Over has a theorem about prime dimensional and being the dimension prime dimensional representations of Semi-simple Lee algebras we also know by a theorem of Deline that we're talking about a semi-simple group and The theorem is that prime dimensional representations the the group in question if the prime is odd on is either s o n or SLN 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 the This condition P large compared to n that enters because of a theorem of Fight and Thompson, 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 fight Thompson theorem says that if g in GL right so So this this is so to speak a different n than that one if you like g and g l and c is finite and P is greater than 1 plus 2 n then the PC low subgroup G is a normal subgroup So it makes sense to say the and it's abelian Now it's it's obvious from 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 and So another influence of offer is that I wrote no non-trivial prime to P quotients instead of no prime to P quotients right, so So the way this is relevant is this Suppose you're in this situation. You're trying to decide if you're fine. I do or not okay, so I'm if g were finite and P is in this large range then I'm G modulo. It's PC low Which makes sense because this is a normal subgroup is prime to P But it's occurring on a one So it's trivial So in other words G is a P group and it's abelian That's the second part and but an irreducible representation then of an abelian group is one-dimensional In other words a rank one, but our rank isn't one Okay, so that's why When P is large you can rule out the finite case and you get The conclusion you want Okay, the answer to that question is yes if if you have the right Choice of galsum normalizing factor then they're equal so the answer is yes No in the finite case it it can matter and so I'll give an example Right so let's take This this end to be seven so so to speak the g2 case Okay, and let's try P equals three Now what's going to turn it we're going and for reasons which I'll explain in the course of the talk We're going to use the following Identity seven is three cube plus one divided by three plus one not false and this is going to tell us that We'll see g geometric is in fact Su three 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? P sl two over the field of 13 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 meant to kind of go into the formula for the group somehow as we'll see yes Okay, so so I know about this like 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. So 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 fight Thompson paper And they point out that their result is best possible because if you take SL 2 over fp This has an irreducible representation Complex representation of dimension p minus 1 over 2 So No, it's a completely different paper This paper it's it's before the out order paper and it's actually it's in 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 has others of Dimension in fact p plus 1 over 2 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 they have that now on the other hand Also going back 30 years, maybe 31 years then Kubert No longer alive was coming to my course Which at that year was about Cluster man sheaves, it's not important what they are, but I was interested in knowing when Cluster man sheaves had healthy monadromi groups and and Kubert By a extremely clever analysis of valuations of galsums Showed that certain local systems Had in fact finite monadromi groups because for these local systems if you want one of these local systems the kind I've written down to have a finite monadromi group you just Just have to show that its trace function takes algebraic integer values that there are no denominators and So Kubert says that if you look for instance, so with this normalizing factor So I of now this is a different n x to the n Plus Tx or Well his method gave also this Times a quadratic character That this would be finite finite group if If n is a prime power plus one over two Okay, so q equals p is Going to 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 monadromi and in fact the 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 these old results of Cubert Have some kind of finite monadromi 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 this is terrifying so I think I just erased what I wanted to refer to Right, so then I guess it's this point almost two years ago. I stumbled upon a Two thousand and ten paper by dick rose and in that paper Dick rose 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 monadromi Zero and infinity given how to speak Explicitly in terms of the group so for instance local monadromi and infinity Will involve a Burrell subgroup in its unipot radical local monadromi and zero will involve what he calls a coxid or torus So it's completely explicit and because From thinking about Cluster men and hyper geometric sheaves. I I knew something about local systems like this So I saw that if you take these local systems with the known local monadromi 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 to you and here you have to go down to PSU 3 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 monadromi groups to be in this case SL or PSL and in this kind of case Well SU It had only when D is 3 or PSU But the point is that this These numerical formulas about 7 they're not an accident. Okay, so This is where I was at a certain point and Then it's 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 in this kind of story Where P could be Q Whichever of these dimensions is odd and since they differed by one there's only 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 to and the other one should have SL to 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 2 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 differed by one since they only differed by one these two Local systems at least have the same size as each other Okay, now in general this would be an idiotic thing to do for instance if 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. Okay, so that there are two parts to the statement One is the statement that In the representation theory of these groups when you take Sim 2 now you have to be careful because there there are two of each dimension so the statement is that sim 2 of a low one is Exterior to 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 okay, so Ron Evans the man who can prove any Identity about exponential sums that is in fact true did the the Psi identity and Because the the character table of SL 2 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 Garelnik whose 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 this emphatic group. No, so I have to explain what that means and For me it was it was like amazingly new information, so Let me tell you what that was so We're going to look at An SL 2 and let me write it just as Q to the end Q could be P But not just SL 2 of a prime field. Okay now of course if you think of this as On a morphisms 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 2n over F Q just thinking about the Two-dimensional vector space over F Q n as an n-dimensional as a 2n-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 1 over 2 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-ducible representation Apparently what's typical is that what you get the image is already so-called maximal subgroup. There aren't bigger finite groups In here that contain this image But what's special and 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 Representations of dimension Qn plus or minus 1 over 2 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 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 Monotv me groups, but again my criterion was Yeah, maybe I should say I mentioned right now at the beginning of the talk now, of course, right now proved The Abiyankar conjecture that any finite group generated by its PC lo subgroups Will occur as A quotient a finite quotient of the fundamental group of 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 going to make them occur and Abiyankar wrote down lots of things that made them occur, but I I wanted simple things that I could remember Okay, so that's that was one piece of information About the image being a maximum Yes, but once you have a representation of the largest effective. Yes, so this is an example where it's not true and then I guess because of What dick rose had done about s u3? I learned a little about S u and odd so different and yet At least 3 q and q is an odd prime power and Its lowest dimensional representations are next to lowest There's one of dimension Q so this n is the d that was over there Qn plus 1 over q plus 1 take away 1 is one of these and they're q of This entire dimension qn plus 1 over q plus 1 and the the so to speak naming scheme of these is They're parameterized By multiplicative characters high of order Dividing q plus 1 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 Kubrick 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 s un and 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 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 differences 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 so let's let me go back the SL to but let me write it with Qn and Instead of trying to write down or writing down candidates For one of these and one of these I'm going to 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 normal normalizing galsum the sum over x in k so So instead of writing qn plus one over tongue in a right qn plus one and then I'm going to 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 the quadratic character. So this will be one representation of qn minus one over two Plus one of Dimension qn plus one over two So can you say again what you're doing? That's about comologically does it correspond to correspond to higher direct image of each degree of sum. I just wanted 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 lpc of x qn plus one over two The rex lpc of x q n plus one over two tensor quadratic character Those are my two pieces and in but in fact when you write the thing this way You get a different proof from Kubbert's proof that this kind of thing will have finite monodromy because there's a paper that came later than Kubbert by on the here and On their fluked Which talks about these kind of some 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 that here I have x qn plus x Plus Tx like 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 Some individually with with the suitable Gauss some will have algebraic integer values and Therefore that this kind of local system will have finite monodromy. So it's another way of thinking about the same question, right? anyway, so at this point without any underlying Conceptual reason I said well suppose we were To look instead qn plus one Let's have a two parameter family two plus one Plus Tx square So this is also going to 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 going to be a sum of two representations of the right dimension and When we put s to zero we know that we get sl2 Q to the n and Basically the idea is that We're now going to have a group that's sitting between sl2 Q to the n our G geometric because we specialize you get something smaller and On our priori grounds, this is going to end up in a big SL Qn and basically the theorem that Tf proved was that on the one hand This this intermediate group and I'll oversimplify this but to a first approximation If you factor this n then in between you'll have sl2 Q to the n that'll be in an sp 2a of q to the b and That'll be in an sp 2n 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 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 Okay, 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 2n q from this Provided that first of all p doesn't divide this end if you're in characteristic p and P doesn't divide the power of p that is q so maybe in a kind of overly complicated logs the p of q so it's some some technical business, but Basically that this works, okay now the next step As we said alright, 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 T 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 what which was at this sp 2n? 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 t equal to zero This thing now becomes a direct sum of Where you have psi of x qn plus one over q plus one plus sx times of chi of x where here the order of chi divides q plus one same Just extracting Breaking up this cute these q plus first powers and that's why we need n to be odd so that q plus one divides qn plus one So we'll get this if n is odd Okay, right so now We come to another miracle provided by T ab That if you have a subgroup of the symplectic group Such that this q to the n special Dimensional special representation, which was the direct sum of two pieces here that when you restrict to here a 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 Yep, would tell me well if you could just prove this or that Then by group three would know the rest and I could prove this or that so thank you very much There are 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 for radical result First of all you are there is also the question of the arithmetic and I don't know if they're Right give you things in the arithmetic correct so the In the case of the symplectic group when when the theorem applies so with many things being primed to pee then My memory is that when you so these local systems make sense over the on a one 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 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 opera or opera or a argument I could make required a pretty big extension of fq squared to get g geometric 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 be small You said that if you have a representation it breaks down. Yes, then it is the group is containing the one works No, no if I have a subgroup of the symplectic group and the representation breaks up Then that group is 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 primed to p All right, then I know that My monotony group in my subplectic candidate has grouped the full sp Not one of these instruments the intermediate things had to be ruled out to get to sp Okay, once you have sp then tf tells us that if you have a subgroup of the sp Such that this relative so-called they representation of sp breaks up Into pieces of irreducible pieces of these sizes, then it's automatically what you want Yes I can answer that question very quickly. No, but there is a man here who might be able to help you