 I met, I saw, I never met him actually, but I saw good more only once in my life in a lecture when he was sitting in front and I was sitting far back. But I think over the year, I can say that over the years I at some or other point I read every line of his book on sheaves that for many years this was a very useful source for me. On the other hand, I have learned a lot from French mathematicians, some of them being here. So it's an honor for me to be being invited to speak here. Thank you very much to the organizers. Let me maybe put here the very basic notations. F is the local field, that's the new characteristic P, positive P. Then G is the F points of a connected reductive group over F. Little K is my coefficient field, an arbitrary field of characteristic P. So everything in this talk will be representation theory in characteristic P. And so correspondingly by mod G, I mean the abelian category of smooth G representations in K vector spaces. And then D of G, as you can guess from the title, I will derive something at some point. So D of G will be the unbounded derived category of mod G. So complex is unbounded in both directions of smooth G representations up to quasi-isomorphism. So these are the basic notations. And so let me run you quickly through some background, even though, of course, I mean, this is known to you. Most of this will be known to you, but I also have to set up the notations, further notations. So I is any, for now, I in G, any compact open subgroup. And then we can form x, the compact deduction from I to G of the trivial representation. So of course, all these coefficients in K, I won't mention this again. Which here, of course, is simply free vector space over the corsets. G acting by left translations. And then once we have that, that is a smooth, of course, smooth G representation. And then we can, it's a very big one. And then we can pass to its Hecker algebra, which is just the endomorphism ring. I think I just call it H, which is the endomorphism ring, G endomorphism ring of X, but with the opposite ring structure. So to make X a left, so a right H module. And then, as you know, there is a pair of, you have a pair of joint functors between, so mod H is simply, of course, the category of, let's say, left H modules. And on one side and on the other, we have our category of smooth G representations. And maybe I denote this by T for tensor and this for homology or comology. So if you have a module, we tensor it over H with our module X, which is a right H module, and then G acts on this, why it's actual X. And on the other hand, if we start with a representation, we just take its i-fix vectors and to see the action immediately, we rewrite this by Frobenius' reciprocity as the G homomorphism from X into V. And then, of course, the action, right action on X, it uses a left action on this home space. Now, unfortunately, these functors are very badly behaved. Certainly in characters Dp, for example, they are not exact. And if they are not exact, of course, it's not a particularly clever idea to think we should derive them, say, this way past the derived picture. But this does not mean that we simply derive them. It means more than that. So the first thing we do is we pick a, I mean, this category mod G, I mean, of course, the modules of a ring anyway have projective and injective objects. The category of smooth G representations is known and easy to show that it has enough injective objects. So pick an injective resolution of our X i dot. So these are complex, of course, one-sided to the right, unbounded to the right complex, exact complex apart from X, of course, of injective smooth G representations. So we pick such a resolution and let me write to be clear in mod G. And again form the endomorphism algebra. And again form its endomorphism algebra. But this is now a more complicated object. So we take the endomorphisms. So this is a differential graded algebra. Because by an endomorphism I do not just mean the endomorphisms of this complex, G endomorphisms of this complex, the usual ones, but the ones of every degree. We also allow endomorphisms which raise the degree, which do not go from ij to ij, but from ij to ij plus a certain degree. And this degree is encoded here. So that gives degrading. So this gives a graded object. And then the differential of i on both sides induce a differential between the endomorphisms. And of course the algebra structure is the opposite of composition. So this gives a differential graded algebra. Maybe let me say right away, so maybe I say up to quasi-isomorphism H dot does not depend. Because it seems I have made a choice here, right? I choose the, there's no preferred injective resolution. But if I choose a different one, I get a quasi-isomorphic algebra. It does not depend on the choice of the injective resolution. I get a quasi-isomorphic algebra. And since I, in a moment, I mean, we are only interested in the derived category of this one, then quasi-isomorphic algebras have isomorphic equivalent derived categories. So this is not really a choice. And then secondly, of course the old algebra degree zero comology of this one. Because the usual endomorphisms of an injective resolution up to homotopy correspond exactly to the endomorphisms of the resolved object. And I mean degree zero, somehow the kernel of the differential here, degree zero, these are the usual endomorphisms of this. And then H zero is quotient out by the image of the differential. This kills all the homotopies. Okay, and then there's the following theorem. So now I have to impose an assumption on I. Suppose that I is for P and torsion free. Then, now in this context, we derive the above functors. So on the left side, we have now the derived category of this DGA algebra. And here we still have our DG. And then I can, so the tensor product is right exact. So I have to derive it from the left. And on the other hand, the home is left exact. So I have to derive it to the right. These then are quasi-inverse equivalences of triangulated categories. Okay, so that's the motivation for what I want to talk about today. The assumption as such in this generality, the assumption is not particularly strong because you can always make I a little bit smaller to achieve these two additional properties in any pietic group. On the other hand, since I will, I mean for the following, it is an assumption because I think more or less right now in a moment I will specialize to particular I's which do not in general satisfy this assumption. Let me maybe first, I can still put this here. So this H dot of course is a very complicated object obviously because it's a differential integrating. So to simplify things, we can pass to its homology algebra. So maybe I put this here. So the homology algebra of H dot is the same as the X. So because here we can get rid of this injective resolution, this is just the X algebra of our original big smooth representation in this category. And then again because this was a compact induction by Frobenius reciprocity, this is the same as the homology algebra of our compact open subgroup with coefficients in this representation. So this is already a much more familiar expression. It has one, at first one big disadvantage and most of what I will be saying later is somehow to make this useful thing anyway. I mean here of course the multiplication H dot induces a multiplication on the graded object. So we have here the intrinsic multiplication and here this turns into the opposite of the Yoneda product. The concatenation of extensions but it's not visible. I mean because right axis only once here it's no longer visible, directly visible on the homology of I. So that's what I want to talk about mostly. Certain structural properties of the homology algebra. So the Rachele Olivier. But from now on I make, from now on I is one particular group. From now on I always, also in further sections, always is the pro-P Ivohoi subgroup of G. So I mean the Ivohoi subgroup was occurred already and pro-P means just it's the pro-P Siudo subgroup of the Ivohoi subgroup, which actually is a normal, I mean because of the structure this is a normal subgroup of the Ivohoi subgroup. It's a very simple quotient. And then in this section here also let G be F split. So these are the assumptions for this section. I do not for now, when I will then I will specifically say I make no assumptions about, I mean it's pro-P anyway by definition but I make, for now I make no assumption about torsion freeness. I mean since the, really in some sense the only thing one can explicitly work with is these homology groups. We have to use them to study the multiplication. And so let me introduce a little bit further standard terminology of course. So in G I pick a maximal split torus. In there I have the maximal compact subgroup and in there I have T1, which is the pro-P Siudo here. And what else? So then the corresponding, let's see, it goes the wrong way. So we have of course the extended affine y-group. It's the normalizer modulo T0 but we work for the pro-P we have to work with in larger quotient v tilde which is the normalizer modulo T1. So this is the important group here. And then we have the Brouillatitz decomposition. G is the disjoint union of these double cosets. And so you see this allows us immediately to write our x, right, which was the vector space over the coset g mod i. We can decompose it into xw where xw, let me use this suggestive notation. It's the induction only to this double coset. So it's the vector space over the Iwi modulo i. In particular, it's an i-equivariant. This is i-equivariant. And so the homology of profiler groups commutes with arbitrary direct sum. So it's the direct sum. And let me abbreviate this by h star w where maybe I put this, save some time. So this is just the homology of i coefficients in xw. But what is this? So now we use Shapiro, right? I mean, this is this here. And of course, before Shapiro, we can, I mean, this is a homogeneous space under i. So I can rewrite this as the h star i induction from i in xw to i where iw is the intersection of i and w is conjugate. And now I can use Shapiro, Shapiro's lemma, which says that this is the homology of iw with trivial coefficients. And actually, I mean, this is all very simple. If you go both things at once, then this is just on the coefficients, the evaluation map in w. So this is setting up the technical tool for the following. And so the basic, maybe the first basic result about multiplication is the following, which looks, I mean, I think you will be, in some sense, it reflects something which in the class in the undirected algebra is very well known in its description by generators and relations. Namely, if you multiply two of these pieces here, so hj, different degrees possibly, and w and w tilde, then you end up in a finite direct sum of all u, of course in, maybe I already did this way, iui is contained in the product of these two double cosets, hj and then the degrees add up, of course, u. So that's good, at least, for the beginning. And now let me go through three structural results. So the first one is the anti... I mean, the usual vahori, or propi vahori hj is known to have an anti-involution. And so is the, so has the homology algebra. And the definition looks, the definition is very elementary, maybe at first it doesn't look like it should be anything interesting, but let me do, okay, will I ever get this back down again? Anyway, I have one in this, and actually this is the principle in all three cases. Yeah, right. In all three cases is that when we do something which does not, which kind of looks kind of trivial, but it turns out to be relevant for the multiplication. So here we define this, make this following very elementary definition, define the K-linear automorphism J. So in the, in the, here, in this picture, I-comology picture A of h star x in the following way. So we do this on each piece, h star w, right? I use, we use, where is it? Up there, we use this Shapiro. This is the same as h star Iwk. And then we just conjugate this by w, by w inverse, in fact. And we end up in h star, the conjugate of Iw, I mean you can do this in two seconds, by w inverse is Iw inverse. And then we go back by Shapiro and have h star w inverse. And then the theorem says that J is an anti, anti-automorphism of the commodity algebra. Anti means, maybe I write it down because this is meant in the graded sense now, of course. So J of the product of two elements, there is a sign, right? Minus one to the product of the degrees of the two elements, J beta times J alpha. And the proof, I mean, I will not go in any, any proofs, but I will tell you in the, in the second part, somehow the technical instrument, so that the proof comes from the second section here. So again, let me do something very naive. So the G-equivariant map. I have a map from x, tensor x, back to x. Extremely simple. I have two functions, f, tensor f prime. And I just map them to their point wise product. I view the induction as the elements here as functions with finite supports, point wise product. That's, of course, G-equivariant. And this then gives me, together with the usual cup product, right? The cup product on group combo G goes, I mean, it goes into the new coefficients of the tensor product of the old coefficients and then we can use this map to go on. So it uses what I call here the cup product, h i, h i, i x, tensor h j, i x to h i plus j, i of the tensor product, but I immediately, sorry, I compose it with this map. It goes back to h i, h some degree i of x. Now, so let me make a remark, which seems to underline that this is totally stupid. This thing and this cup product under Shapiro corresponds to the usual cup product on the homology groups of these I w with red. If you have trivial coefficients, you have the usual cup product. That's very easy to see. But it has, of course, the immediate consequence that this cup product is, I mean, right? It's completely component wise. If I have h i, v, cup, h j, w, this is always zero if v is different from w. So why should this be helpful at all? But in some sense, the second main technical statement so in some sense maybe the key technical proposition is the following. And I formulated it somehow vaguely because this is in formulas, this would be complicated. But I give you as a corollary one very nice and simple formula in a special case. OK, so maybe I can put this here. I need some notation. I mean, we do know, let's say we have an alpha in h i, v, and beta in h j, w. Then by the, is it still somewhere? Should be still somewhere. By the first technical proposition here, we know that the product alpha times beta is the sum over this u in the product double coset, certain elements gamma u. And now the technical thing says that the gamma u can explicitly be expressed in terms of the cup product, this cup product, and homological restriction and co-restriction maps. So this is really the tool which allows to compute the product. Let me give you, I find very nice instance of this, in general this is a very complicated formula. But let's suppose that the two elements, L and V, that their lengths add up so that the product element, the length of the product element is the sum of the individual length. Then actually we only have, then the only possible u is actually the product element. We have only one term here. And for this, so then the product alpha beta, so the nice way to write this is the following. Alpha times tau of w, cup tau of V times beta, where this is the standard notation here for the usual hacker operators, tau of something is the characteristic function of this double coset of something in the usual hacker algebra. So this reduces the product here to the cup product and the modules only the multiplication of the degree zero part on everything which is, I mean this is easy to describe. What is true is that if the lengths do not add up then this is still for the highest term in here, one still has this formula, but there are many complicated other terms then. So then let me see is equality. So now I have to assume for the last two points here in this section, I do assume that i is torsion free. The reason for this is that the consequence of this here by Lazar, an old result of Lazar, that all these subgroups, all these open subgroups Iw are Poincare groups of dimension d which is the dimension of g over qp. You can view perhaps g, either you take the dimension as an algebraic group and multiply it by the extension degree or, I mean I should say that this torsion free enforces that the field has to be of characteristic zero at the local field. Or you view it as a periodic league group and take its dimension as a periodic league group. So this means they have homology only up to degree d. So the immediate consequence in particular or homology algebra sits in degree zero to d. So it's zero for star degree bigger than d. And we will use this later. In the highest degree d, all this, the homology of all these groups is one dimensional. Plus the cap product pairing is non-degenerate. I don't, for each of those, I don't write this down. So now we use another map. It's a very simple map. Have g equivalent from x to k. We just, I call it sigma. If you have a function here, I just add up all the values. It has finite support so I can add up all its values. And this induces together with the cap product for the first, our cap product. And then we take this map on the coefficients. And let me, so this is, let me this, you know this for later use, capital sigma h d i little sigma. Ah, actually I don't want to do this in general, right? I mean we want about duality. So I take your d minus i, then I have here h d d d. i with trivial coefficients. And this I identify once and for all. I told you this is one dimensional. Identify this once and for all with k. And then I can use this, I can use this for duality, this induces a duality map. Right? So get k linear x d minus i x linear dual. So I check for linear dual. So it's easy. Let's take for a moment to this, this homology picture. So these are, what is not difficult to see is these are injective, injective with image. It's of course not surjective with image. All what we call finite linear forms. I mean since this is the cup product which was on each IWP is the usual cup product which is non-degenerate. This gives of course immediately the injectivity and it also gives, so finite just means the linear form is finite. If it's non-zero on at most, finitely many of these summands h d minus i v. So this is all on the I picture but then the proposition is the following. So this map, so I'm going back now to the x picture. The above map is an isomorphism of h comma h bimodules. So this is x i, x comma x and this here before the dualizing is x d minus i. If I leave a little space I have to put some decoration here in a moment, x comma x and of course here the dual, linear dual and then let me write simply indicate this finite. So in the moment I only have rewritten these two terms in the x language and I consider the above map. This is of course also a bijective by what we know already so the statement here is that the h bimodule action which comes from the h action on these two terms is reflected here by the h action on these two terms but changed by the anti-involution where j on the left x denotes x into a, x was originally a right h module so I make it now into a left, into a left, no I always mix up and it was a left module, h was acting from the right so I make it into a, what is the usual convention about left and right? I think now it's a left h module via j, via the anti-automorphism. I mean more should be true but we can prove this here the whole, not only the h, the degree zero action on this from the left and from the right it should be something like this should be true for the, I mean the same statement in some sense should be true for the whole algebra structure but this we can't prove yet. But this allows us and I will show you in the last point d I will show you this, the consequence right, I mean this will allow put here d then this is zero here it's the usual Hecker algebra but with some twisted action and since the usual Hecker algebra we know very well and then we also know the twisted action very well we can explore this to get some information about the structure of the top degree piece. So this is the, in this section the last subsection the top homology hd, hd of h. And again here of course if I want to apply this I have to assume that I is torsion free. Maybe I abbreviate this because I want to, I have a still in the last section so maybe I do only the last fact. So let me write down one but maybe I don't have the space here. Let me only write down the cleanest proposition I have to make. So assume that g is semi-simple, simply connected with irreducible root system then. So for the first part I could, some of this is true in more generality but for time reasons let me put this all together. So then the top homology, right we have this map, we have this map to the 2k actually this splits so this is the die sum of hd ik and let me put here trivial plus of course then the kernel of this map sigma where the drift stands, maybe this I put here right away here. This means as h bimod, this is the trivial h, I mean the h structure on here is the trivial one, trivial h bimod but maybe I put here this separately so this here is super singular, it's a huge super singular from the left and from as h module from the left and the right so super singular here is somehow the analog of super cuspital in the characteristic theory maybe. Anyway it is to formulate the last maybe really more structural fact is I need, I mean super singular, there is a certain ideal in the center of the propi hivorik algebra, a specific idea and super singular means that a module is torsion for this, for this, no I should, I think the other J I used for the convolution so super singular means the module is torsion for this ideal and then the last fact is that this super singular, very big super singular module, so then an obvious super singular module, five dimensional one is I just divide h by the ideal which this central ideal generates in h and then the statement is, so I have this finite dimension super singular module and I take its injective hull of h mod jh, again from the left end and from the right. Okay, so far our structural results about the algebra now I want to talk about at the end in the last brief last section, a little bit what I strangely maybe call derived parabolic induction. So let us pick a parabolic subgroup and of course I think it occurs to me I forgot to say this and I introduced the torus, of course there always has to be a relation between the torus, my Ivorori has to lie so to speak in the apartment of the torus and also yet now of course with the p there has to be a relation to the chosen Ivorori subgroup. And then I mean it's a very easy fact that the usual parabolic induction from mod g to mod g is exact, right, so there's nothing to derive, I mean it's on the nose it extends to a functor on the derived category. Sorry, of course, yeah. So what I mean here is of course using the previous theorem how do we describe this on the side of the derived Hecker algebra. So I will now because I have to distinguish right so far my objects had no subscript usually of course when I will right now maybe h dot m or g I mean and so on I indicate by a subscript whether I'm over m or over g. Now let me first reformulate some known thing so define the h which is hg, hm, so usual Ivorori algebraic bimodule which I call x index g comma p and it's just I take the parabolic induction of xm and I take the i g invariance and then maybe I can put this here I have no space then there is this lemma which is just the reformulation of a result of Olivier and Vigneras. The diagram maybe these are two diagrams made into one mod g mod m sorry mod hg and here mod hm m here we have parabolic induction so here we had this pair in each case this pair of a joint functors tg, hg and similarly here tm, hm and here we have just the tensor product by this bimodule and moreover I think this is so far I cannot yet explore this for the derived picture but I think this is something probably very important and moreover this bimodule xg comma p is finally generated projective over I mean only on one side on the hm side over hm. Okay so that's the picture the undefined picture and so let me finish with the derived picture which looks like a kind of a formal consequence so now define right now we need an h dot g h dot m bimodule which I then of course call i dot subscript gp and let's at first let's just imitate what we did above I mean right there was the the invariance taking invariance is the same as home g equivalent home and here from the universal thing so which is here the injective it's injective resolution into here a little leave a little space the parabolic induction of the universal thing over m the injective resolution there so I mean that was actually my first attempt which turned out to be completely wrong so that but also has to resolve this first injectively so this is right I mean the the the parabolic induction does not preserve injectivity so although this is an injective complex it's a parabolic induction is no longer so I take take an injective resolution but the problem lies there let me come back to this later in my last sentence and then maybe I actually yeah then the theorem is that we have diagrams completely analogous to this one so d h dot g g d m h m dot right I mean where the left right let me maybe only repeat it here this where the left derive this was the right derived of the of this one and correspondingly here we here we have the trivial derivation of parabolic induction and we here we have the derived tensor product with this by model this commutative didn't write it there sorry is commutative now as I said this looks as if would be a completely formal consequence but there is a problem there's a huge problem here namely this is so this means passing to an injective resolution as g representation so in mod g I want a bimodule right from the right I want the action of h m dot so it acts on here and then by functoriality it acts on the parabolic induction but if I take now injective some injective resolution in mod g I completely use this lose this action there's no reason why I should preserve it and so that's the huge problem one has to find so one has to take specific so I mean one needs specific or a specific injective resolution functor in the framework in the framework of differential graded categories it has to lift this whole thing I mean the derived categories in some sense a the homotopic category of a differential graded category which takes the all these gradings into account and one has to do this injective resolution business on the level of DG derived categories DG categories and I mean that this is possible is a is a theorem by Schnürer Olaf Schnürer really non-trivial thing so once you have this then the thing becomes a formality but without it it's it would be hopeless okay thank you very much I mean excuse me I probably should know the answer to this but I mean does this thing exist for characteristic zero representations of groups with the regular periodic groups yes but it's nothing additional because there these functors are exact right the the Ivorori propi groups have no comology in characteristic zero so the whole thing is just the usual HEC algebra there's nothing new happening there I mean did you discuss such things in that paper 2007 or is that no I mean I made I made comments yeah I mean the surprising thing is of course there in characteristic zero as you I'm sure you know you do get say for the propi Ivorori you do get a equivalence of categories but not with the full category of smooth representations only those which are generated by its Ivorori in propi Ivorori events whereas here I mean the surprising thing is that under these assumptions of course the higher come on even even right I mean you may have a representation of course in characteristic p every representation has a non-zero Ivorori fix propi Ivorori fix vector but they don't have to generate the representation nevertheless somehow the higher comology sees sees the representation whether it's generated by its Ivorori fix vectors or not this is in the characteristic p case yeah I mean certainly so for completely formal reasons the the right adjoint right once you have because of the equivalence I mean the usual parabolic induction as a rider joint which is left exact and therefore you can derive it so you have a rider joint for this on the derive category and since these are equivalences you have a rider joint here and you can compute it it's the home from this bimodule the left home the problem is with the left adjoint this usual shakifangta is right exact but it's not clear whether one can derive it because for that you would need projective objects in in in the g-re representations which do not exist in characteristic p so I think that's where this I mean probably this fineness property reflects some fineness property of this year which allows us to see that this is also has a left adjoint so you mentioned that the formula gets very complicated when you suppose for example the length of v is one the length of v is one and you have some formula? Yeah but don't ask me to write it down right now I would have to get up because you get then right I mean if you have let's say this is this is s here right but the lengths do not add up then the possible use are I think s w and omega w where omega is of length 0 and you get terms I mean this this one of these terms I think this one is still kind of I'm not sure which one but you have all the terms occur and they are complicated I mean at least so complicated that I don't know them right now without looking up in the paper. Even for v is equal to w is equal to s? Yeah I mean what? No no no you have every omega even in that case. v is equal to w is of length one. Yeah yeah sure and you have you have s squared here and you have omega s here so you still have many terms. No but you think omega is equal to s? No no omega is of length 0 in this case I mean right I mean I was just describing if this is of length one here a reflection what the possible use are and there are there is one which s times w and then there are q minus one omega times times w and so you have them even if w is s as well.