 Thank you very much. I have enormous respect for OFA's work, and it's a great honor to speak here. Also what I will be talking about at the, so basically I want to kind of give an overview about what we know and don't know about the irreducible representations of reductive algebraic groups in characteristic P, so as algebraic representations. And if I haven't completely poorly planned this talk, then at the end I should get to a connection with the decomposition theorem, which is probably the one theorem that I've thought about the most, and this of course OFA had a lot to do with. So the setting is the following. We take G should be a reductive algebraic group over an algebraically closed field K, and the characteristic of K is P, and I'll assume from the outset that it's positive. So we're looking at rep G, which is the category of algebraic representations of G. So if you want these are those representations of this abstract group in which the matrix coefficients are regular functions on G. And so given lambda a dominant weight, we can associate to this some simple highest weight module. So this is a socle of an induced module, and all irreducible representations are of this form. So this I guess is the theorem of Chevalet. And then the question that we want to study is the character of this lambda. So just to give you some idea of what is known about this, so if G is SL2, then this acts on nabla M, which is polynomials in two variables of degree M. And in this case, l of M is the socle of nabla M, and l of zero, l of one, l up to P minus one are simple. So this you can do as an, it's not a difficult exercise. Sorry, these are all simple, and any lm is lm zero tensor, lm one for V. It's a ml, l for Venus. It's actually general theorem in this context, though. Yes, so I'm just trying to give this as an example of a general theorem. So we have these building blocks, finite in many building blocks, where finite depends on P, then we can write. So this is for Venus twist. So pulling back this representation under for Venus. So this kind of SL2 you can do as an exercise. And then, sorry, this is a periodic expansion of M. So M is sum M i, P to the i, zero is less than. And so, and then for SL3, so George, let's explain to me that this was solved by Brayden in 1967. And then SL4, SP4, G2, these were handled by Janssen via the sum formula in the 1970s. So roughly speaking, what Janssen does is he puts a filtration on these nablers and has some formula that gives you some incomplete information about this filtration, but this formula is strong enough to handle these cases. And then SL5, SP6, spin seven. Here there's a handful of missing cases, but not very many. So somehow with algebraic methods, you get somewhere to this range and then beyond that you need something new. And what is, so I just want to give this kind of standard picture. This is this alcove picture. So we consider, so here's our space of dominant weights. And then inside this, we have some arrangement of hyperplanes. So this affine arrangement, so here's minus row. So people have explained to me that you're an expert in the field if you can draw this picture correctly. So I will try, okay? And so here we have this area here is the dominant weights, so. And then for every, for each of these alcoves, we make a choice, so we assume that P is bigger than equal to H, the coccera number. So that says that our weight zero here is not on any hyperplanes. And then we just make an arbitrary choice inside every alcove of a weight. So these are the A's, the alcoves. So the connected components of the complement of this hyperplane arrangement. I should say this is precisely, these walls are precisely the case where the vile character formula, the vile dimension formula is divisible by P. So it's somehow quite a natural picture to draw. So we consider these alcoves. And then for any alcove A, I choose a weight inside that. So if I have an alcove A, I choose a weight. Lambda of A, such that whenever two alcoves are related by a reflection, the corresponding weights are related by a reflection. So there is something with adding rho here that you have some affine vile group and those are fundamental. Yeah, exactly. So this is the... It's kind of a lambda plus two because the zero is not... Exactly, yeah. ...in the interior, so it's somewhat okay. So I take the affine arrangement, I dilate it by P, and I shift it back by minus rho. So that, yeah. And so there's the distance between these hyperplanes is P. But then the chi plus, the domino... Yeah, so I mean... ...the domino, this is chi plus minus rho, no. So, yeah, I should, if I... Yeah, so this picture is slightly inaccurate, namely one off this wall is the dominant... ...dominant weight. So concerning this, I believe that there is a general fact called the vintage principle that tells you that whatever you do, you only have to look at the orbits under the... Exactly. ...all the interesting questions like how... ...develop a model, the composer, and things like that, not the extension, the orbit under... Exactly, yeah. So basically any... So the linkage principle and the translation principle tells us that if we want to answer questions in the representation theory of G, it's enough to answer it for one of these orbits and I'm making some random choice of these orbits. So for example, if we want to understand how some indecomposable module that's corresponding to this weight decomposes, then the only possibilities in its composition series are weights in this orbit. That's what... In the translation, perhaps there are also degenerate cases where you translate to a word? Yes. But somehow these are always simpler than the regular cases. So the regular cases are the hardest. And there's precise ways in which you can say it's simpler when you go to a wall. And so now we have this lusted character formula. 1979, so I'll call this LCF, which is the statement that the... So I'll say the class in the graph in the group, you could also say the character. So I want to rename... So LA is by definition the simple module with this highest weight. Yes, so now I'm indexing everything via alcoves instead of via high-sweat. And it's the statement that this is the sum and there's a sign that I won't go into, m, b, a. And so this is a so-called vial module with character given by vial character formula. This is what we would like to understand if we're interested in this question. And then this is a so-called spherical casualty polynomial. So this is some polynomial that's defined entirely in a combinatorial way starting from this picture. And we take its value at one and this is the conjectured expression. And there's some, so the first thing that one should say is we, so assume, so if alpha check a plus or always less than, so this is a Janssen condition. So we only expect this character formula to hold for simple modules that aren't too high out, but if p is bigger than, yeah, so basically this is enough to know this formula. And so in this example of SL2, you see that there's these p minus one building blocks from which you can get all representations. This is the case here also in general, by Steinberg tensor product here. And so if we know this formula, then we're okay. So MBU is evaluated at one? Yes, the value of a casualty polynomial at one. Any positive root? Sorry, so you just said, so the generalization of this kind of decomposition was written yes to us? Yes. You said that like how should we tell this somehow or some analogous thing? So there's a region here, which is the restricted weights, which I'll actually need in a second. So you're starting after which piece? Hang on a second, one question at a time, please. So this is the restricted weights. So these are those, so I write these in the fundamental weights, all of my digits should be less than p minus one. And it turns out that I can, so for any highest weight, I can do a periodic expansion in terms of the restricted weights. And then if I form the corresponding tensor product, that will be simple. That's Steinberg theorem. So it's enough to know the characters of these guys. And then it turns out that for p not too small, this region actually lies inside this Janssen condition. And so then I'm okay. Yeah, so what's the other question? Yeah, something after which piece? Is this from the left? I'm starting after from this, not me. Yeah, so these will just be non-zero for finitely many b. So finitely many b which are in some sense less than a and same for all two. Exactly. So after all b, because it gives you zeros here. Yes. Plus or minus one, do you get plus? How do you get minus? It's just, it's the num, I mean it's the number of hyperplanes separating b and a. Like minus one to the power of it. But somehow this is, so for the corresponding quantum group, this is a perfect conjecture. There's none of this Janssen condition and it holds always. But this is somewhat problematic for algebraic groups. I mean, even in its formulation it's somewhat difficult to get your head around. And there's a different version which is called Lustig periodic formula which is better, which I'll explain now. Will I ever be able to get that backboard back down? So a basic fact is that if lambda is restricted, so the restricted weights in this example of SL2 precisely this zero up to P minus one, then this L of lambda is simple as a G equals Le G module. And somehow the Lie algebra is much simpler than looking at modules for the Lie algebra is somehow much simpler than looking at modules for the algebraic group. And so it's natural to consider G modules with compatible T action. So T inside G is a maximal torus. So the meaning of this should be clear. So I consider a module with a T action such that when I differentiate the T action I get the same as the T inside here. And these are called... That's the compatibility, like for GK modules then. Exactly. So you have to look at this. It's like GK modules for... It's like, yeah, G... So you have the modules of CAC-CT? Yes, yeah. And so there's a real... I mean, if you're used to talking about characteristic... thinking about characteristic zero, which I guess is less normal in this audience, then... Then, yeah, you should be aware that there's a big difference between... Yeah, but I don't need to tell you this. Between the G... Big G modules and little G modules between algebraic group modules and Lie algebra modules. Okay, so this is the same... Basically, yeah, essentially the same. Yeah. But here there's a big difference. But for simple modules, it's okay. There's essentially no difference. And then... So this is the world of G1T modules. And so here we can... So this is a... I'll write hat for G1T module. So this is a simple high-escape module. And so in the world of representations of G, it's not... So for example, in the representation theory of G, there's the projective modules that kind of pro-objects. And so they're very large and you don't usually work with them. Whereas in the world of G1T modules, you have these nice, finite-dimensional projective covers. So this is a projective cover, as a G1T module. And then the enlisting periodic formula. And this is explicitly non-charactered. This guy? No. So knowing the character of this is basically the same question as knowing the character of this. So very... Yeah, so then we have the enlisting periodic formula. This is something like the reciprocity, like the environmental model, the P over... Exactly. That's exactly what it is. Perfect. Is this enlisting periodic formula, which tells us that the class of P of A hat is DBA at one, delta B hat. So before this was in the original formulation, this is a vial module, so it's got character given by a vial character formula. This guy is called something called a baby verma module. And so this guy, up to shifts in the weight lattice, always has the same character. It's finite-dimensional. So delta B hat is the restricted Li-arger of G. Restricted Li-arger of B. So this is a finite-dimensional kind of standard module. And this is periodic. Periodic cousins. Again, evaluated at one. And so the beauty of the theory, so in some sense, why do we work with G1T modules rather than G modules? The reason is that modules for G, the grading by weight spaces is by something like the character lattice tensored with Z mod PZ. And so it's kind of, and that's very annoying. And so you kind of unwrap this grading so that it becomes a genuine T grading. But now you can tensor by P times an element of the character lattice. And so the representation theory is periodic. So the representation of G1T modules is periodic. And this is the corresponding theory of Casual Nessik polynomials for a periodic situation. So is this finite or infinite sum? This is also a finite sum. So I'll give you some examples. And one should keep in mind Ofer's remark from before, that knowing this, so this remark, is that this lost periodic formula implies a character formula. Those were originally conjectures. Yes, so yes. So I'm just stating, so I'm stating them as formulas at the moment that may or may not hold, and I'll discuss their validity in a second. Yeah, because you already gave previous talks about the function, okay, so I know that, okay. So just some examples. So for SL2, here the picture looks like this. And then the periodic pattern just always, like I'm just giving you the value at one of these periodic polynomials. And so this is just, this is telling you that no matter where you are, your projective has two baby verma module multiplicities in there, like this. And so in SL3, there's two cases. So let's say, for SL2, when you're blowing up so that the baby will be reducible. Yes. What is the P and the baby and how they are, what do you, I didn't understand the blowing part. So you have. So this will be some alcove. Let me just say, okay, some alcove. Yeah, and let's say that zero's inside here. Okay, that's okay, we'll take the basic. By the way, this is just for the basis for any alcove. This is for any alcove. Yes. Okay, but that's. But it's all periodic. It's completely periodic, the situation. And then there is only one alcove left when you're in SL2. So here the projective, so this would be two P minus two. And then the projective would be. And the relation of delta to L is the same as the relation of delta to delta. Yes. So essentially, in order to get right the deltas in terms of the Ls, you transpose this matrix and then to write the Ls in terms of the deltas you take its inverse. So in principle, it's, I mean, this looks like the cleanest answer. It gets very messy when you do this in practice, but I mean, you know, you can do it. So now you see that I'm not an expert. Okay, and so this is representing the structure of certain projective G1T modules. And this is, you can calculate these pictures in some simple algorithmic way, but they're not usually, this is somehow deceptive because they look reasonably simple. So you don't need to go very far before these pictures become very, very complicated. Okay, in the translation, there are several under the translation that you said, there are several equivalent, there are several, so how many classes module translation you have in the case versus three? Two. It's always, it's always what we call, what do you call it? P mod R, the index of connection. No, sorry, sorry, sorry, that's wrong. So the number of cases you have to do is biogroup modulo index of connection. That's N minus one factorial for SLM. Okay, so now I'll talk about the validity of these statements. So one remark is that these two are equivalent and then the status. So one thing that should somehow motivating this is some old conjectures of, basically it was Verma that I think first said that this kind of alcove picture should be behind the representation theory and that it starts looking independent of P in some sense. So we fix a root system and then we can consider primes and the combinatorics is much more complicated below the coxenon number. So here's the coxenon number, egn for SLM. So this is what we call small primes and for many, since the, for a long time, we've known that there's behavior here that's somewhat mysterious and what the hope was was that these formulas would hold for some reasonable bound on H. So for example, maybe for all primes larger than H or for all primes larger than two H or something like that. And then in the 90s, so there's too many names for me to, so I'll just write initials. So this is Cajda Lustig, Kashiwara Tanisaki Lustig, Anderson-Junson Zergl, proved that there is a, there is an N, non-effective, such that this holds for all P bigger than N. But this N was not known in any case. So even, for example, SL5, there's something like one number that Janssen would like to know. And so there's no way you can even check on computer whether this one number is one or two or something. And then in 2008, Phoebig gave an explicit enormous bound. So EG, it's LCF, is true for P bigger than N to the N squared. So for example, 10 to the 100 for SL10. So a very large number. And then, so based on work with a number of different people, so Elias, ah, for SLN, sorry. So based on work with Elias Schuchar-Hert and also some, ah, some number theory with Contorovic and McNamara, there exists, so LCF does not hold for many P, but, so for many P up to an exponential in N for SLN. So basically what we do is construct examples up to some P of the order of C to the N for some C, for some C bigger than one. And so somehow there's this place here that's exponentially far off, well at least exponentially far off, where LCF is valid, but there's this whole world here of medium primes, medium primes where LCF doesn't necessarily hold. What do you mean by for many P? So basically, yeah, I think that we show that it's basically for all P up to up to some exponential bound, yeah. Or, yeah. And how CNN believes that. So C is just, so C is some number bigger than one and independent event. So an example is that if, so just an example of this phenomenon is that if P divides the Nth Fibonacci number, then LCF fails for some LA for SL something like 4N plus 5 in characteristic P. Okay, so it seems extremely interesting that there's certain, so this is a very arithmetic question and that this is somehow happening in the representation theory of SLN is maybe surprising. Well, at least for me. Okay, so I was thinking, what on earth can this be for, but then I realized. Sorry. Oh, I'm sorry. When do I finish quarter to know? Is that right? Okay, so the theorem that I want to state today, so the theorem. So this is a recent work with Simon Rich from Clermont-Ferrand, but it's based on a long project with Acha. Somehow, where all the work is done is in this long project with Acha, Makassumi and Rich. So the statement is that basically this formula holds, so for all P bigger than 2H minus 2, I'll comment on this in a second. This will hold, so it bears a very, it looks almost the same. These are so-called P, these are periodic P-cajolotic polynomials. So I'll explain in more detail what these are in a second, but roughly speaking, cajolotic polynomials are measuring the stalks of intersection cohomology complexes on the flag radium. These are measuring the stalks of some objects called parity sheaves on there. And so the remarks are, so firstly, there's essentially finitely many of these polynomials. So for fixed root system, these PD, BA, RD, BA, for all P bigger than some non-explicit bound. So it's something like if you have a, imagine you have finitely many algebraic varieties, then their integral intersection cohomology will have no torsion above some bound, but you won't necessarily be able to say what that bound is. It's not too big, it's bound for the left. Why not at this one? Yeah, so I mean, we can't re-derive Phoebig's bound. We can't re-derive Phoebig's bound from this, yeah. So this is worse than, so this implies L, C, F, and R, P. There's now about three different proofs of this character formula for large P, but this gives another one. There's no sign in the formula. There's no sign, no. In the previous one, there was. Yes, yeah. So signs are whenever you're expressing symbols in terms of something, and no signs are whenever you're expressing projectives in terms of something. So we conjecture the formula to hold for all P, so it should be completely uniform, and it seems to check out in small examples that we can calculate. But there was a problem with, you said for sufficiently small P, the zero is not in the right place in the diagram, so. But there's some simple modification that seems to work, so with appropriate, with small modification. For all P, you mean without these restrictions at least? Exactly, yeah. So that's a theorem, and then we conjecture that actually you can just cross this out with appropriate modifications for the fact that there won't be regular weights for P bit smaller than the Cox number. And so just to give you some feel, so these P DBA are much harder to calculate than DBA. So you can ask is this any, does it improve the situation at all? Have we just given something, are we just expressing something uncomputable in terms of something else? Uncomputable. I mean, not uncomputable in a formal sense, all these things are computable in a formal sense, but just very difficult. But this should, so experiments suggest. The usual characteristic formulas are given by some inductive procedures that you can calculate in the given term. Yes. And these things, so characteristics which are, let's go to bi geometries, is it still given by some? It's given, it's not, it's not entirely, it's not given entirely in terms of the combinatorics of coccetter group combinatorics. So you need to know something about the root system, but it is computable in the sense that I can type it into my computer and press enter and it gives me an answer in. So I can compute these, these polynomials in many cases, but I can compute them no way, nowhere near as efficient, efficiently as cash-on-seq polynomials. And it's unlikely that we ever can compute them as, is this an answer? No, the question is just, better there is a formula, like that for the cash-on-lustic polynomial, or it involves some, seems to find the algebraic geometry. So there's a formula that involves only linear algebra and combinatorics of coccetter groups, yeah. It doesn't involve some calculation in algebraic geometry that I may or may not be able to do. You expect a quantum analog? Yeah. So somehow the quantum group just, universally you just ignore the P and it seems to be fine. I mean, that's also theorems. No with one. Yes. As Bairman suggests, formula should provide a complete answer in ranks less than or equal to six. So before we use, but at the moment we know A1, A2, A3, B2, and G2, and it's more effective in the sense that we can go from these cases up to rank six, but probably not beyond that. Using the fact that you know this allows you to calculate the character of the else. This is, so I don't understand which sense it is not there, why it is not a problem. I mean, I should say, experiments suggest we should be able to do the calculation of these polynomials in ranks less than or equal to six. Ah, okay. So you don't regard it here? No. At the end of the day, I actually want to know what the character is, and that's what I'm asking about. Is it not clear? So all these experiments suggest that the PDBA should be computable in ranks less than or equal to six. So I mean, another example is, imagine that I ask you for simple highest weight modules over Lie algebra for SL400 or something. We have a formula in terms of characteristic polynomials, but we can never carry out this calculation. The calculation is just too large, and so there was this Atlas project that carried this out for E8, and this was some enormous calculation. What I'm saying is that we should be able to carry out this calculation in ranks less than or equal to six. But that's still not done. So I want to explain what these P polynomials are at least rustly. So if we have F from X to to X a projective morphism of complex varieties, then a special case of the decomposition theorem is that F lower star, so this is RF lower star, and the constant sheaf on X tilde is a semi-simple complex in the sense of perverse sheaves. So it splits as a direct sum of its perverse cohomology groups and each of these perverse cohomology groups is a semi-simple perverse sheaf. The remark is that the fact that we use Q coefficients is essential. So of course X tilde is no single line. I'm sorry, of course. Yes, this is a smooth, yes. So just a simple example which I love is if we take X to be some quadratic cone inside A3, and then we have X tilde to blow up in zero X, then F lower star, so this is isomorphic to the total space of O of minus 2 on P1, and the kind of absolutely essential point in this example is that this zero section here, which is contracted by this map has self-intersection minus 2. So F lower star of the constant sheaf on X tilde is semi-simple. So in this case, it's always a perverse sheaf because this is a semi-small map. But it's semi-simple if and only if the characteristic is not equal to 2. So in characteristic 2, you get some interesting indecomposable. So I'm always talking about the characteristic of the coefficients. So as I said before, the key point is that if F is the zero section and the self-intersection of F is minus 2. So in general, for any P dividing this, you'll have problems. So now we apply this to the flag variety. So let X be the complex. So now I change and consider complex flag variety, and we consider X-X to be a sheabot variety. So X is sheabot variety. Then what are ordinary Kajan-Unsic polynomials? So this is defined to be the intersection column of this guy with Q coefficients. So the stalks are given by Kajan-Unsic polynomials. So now for any reduced expression, we can consider X in the viral group. We can consider this Poz Samuelson resolution. This has a natural multiplication map to G-M-I-B. So at some point I stopped writing subscript Cs, but I hope it's clear. So this is a Poz Samuelson resolution. What is small x? So this is just any element of the viral group. This is Poz Samuelson resolution, which are some very useful combinatorial resolutions of, sorry, this should go to the sheabot variety inside here. And then the decomposition theorem says that Ic of XQ appears as a sum and inside the direct image Q of this. And this leads to a, so if you imagine what's happening here, you have the decomposition theorem tells you that you have this intersection complex support on the open orbit, and then you have stuff supported on smaller orbits, but there you know all the stalks by induction. And so this gives a combinatorial expression for Poz Samuelson polynomials, derived, yeah, I think from the, yeah, always derived. And there's another kind of way of looking at this, which is just consider this whole world of kind of sheabot varieties and their partial resolutions, and imagine that you're allowed to start with constant sheaves on, constant sheaves on their shifts on things that are smooth, and then you're just allowed to push forward and take summands. Then all you'll ever get is intersection commonergy complexes, which is kind of remarkable. And so now you can ask the same question with coefficients of characteristic P. So if you don't want to mention perverse sheaves, you can still characterize this object as, well, you could still attempt to give a definition of it as being the unique summand inside this that's indeed composable and has support on the open locus. By the way, Kazhnyan Lusik, the founder of Poz Samuelson is in 1979, slightly before the composition theory. So what was the original intuition for? So I can use this question to advertise some wonderful notes on George's website, where he gives some notes to his papers. And there's a very, very nice explanation of various things that led to Kazhnyan's polynomials. OK, three minutes. Very good. Five minutes. OK. And then a kind of somewhat surprising fact, which was first noticed by Zogel in certain cases and then generalized by Juton-Mautner and myself, is that there exists so the summand inside such a direct image in any characteristic is well defined up to isomorphism. The indie composable summand is well defined. So this means that if I take two different resolutions, then this direct image sheaf will be very different in principle. But if I just look at the unique indie composable summand that has open support on the Schubert variety, then this will give me a well-defined object up to isomorphism. So in general, Poz Samuelson implicitly aspects is not a direct sum of the perverse sheet. Exactly. And each of those is not semi-simple. And so here you're just using the Kohl-Schmidt. I mean, to know that there is some of the composition in the composable sheaf is defined up to isomorphism. Exactly. This is what you are. Yeah, so I'm very heavily using the Kohl-Schmidt theorem here in this derived category of constructible sheaves. It's well defined up to isomorphism. And this is an example of what we call a parity sheaf. So this depends on the field K. And then we have EX. K is isomorphic to EX. Q in large characteristic. Sorry. I don't know what to say. Oh, is isomorphic to EX. Yeah, you're right. I mean, what I want to say is that ICXLK, if P is for P large, depending on EX. So once we fix EX, there is a P above which there's no torsion in the stocks of or co-stocks of this thing. And once we reduce that mod P, we get this parity sheaf. But this example up here is kind of illustrative of what happens in general. So there'll be some small primes which this agreement does not occur. And then we get this new and genuinely like a very interesting object. And I'll just say as a remark that I find this a very interesting question. So moreover, you can show that if you consider all kind of partial resolutions of Schubert varieties of Bot-Samuelson type and you allow smooth things on. So you allow constant sheaves on smooth things. And then you allow yourself to push forward and take some ends. Then you only get a finite list of objects that are parameterized in the same way as IC sheaves. And I've often thought about what could happen in general, but I seem to get stuck with curves. In general, you're meant for? I mean, is there some class of maps and some class of proper maps that I can fix? Another case where this is true is, for example, all toric maps between toric varieties. So here, this is self-dual by construction. Yes. And the IC is not quite self-dual because of the torsion. Exactly. So this is closer to the question of torsion in IC. But it's a free only. It could be that although IC is not self-dual, it may be the reduction, but we could still be in the composable. Yeah, but it'll never, I guess in this setting, it'll never be this guy if it's not self-dual. Well, obviously, because the self-duality is given by certain map. OK, but then since on the open stratum, you know it. Let me write one more sentence that finishes my talk, and then. So p-cush-under-sync polynomials are defined to be the stalks of I've given by the stalks of these exk. And there's a completely different way of understanding these guys via some diagrammatic algebra, which allows you to compute things with them. But I won't go into that. So thank you very much. Any questions or comments? Just to clarify, we're in the composable. So the intermorphism ring of the intercomposable is a multidimensional algebra. It's not a trivial idea, but please. So my question is, of course, you can have some phenomenon in case of the brightly closed. So the question is whether the intercomposable are the same or if it comes from this one free bar and whether there's a multidimensional ring, that should be going to say it's just OK. OK? It's OK, sir. Those things. Yeah. OK.