Progress report on derived Hecke
Schneider, Peter
In the modular smooth representation theory of p-adic reductive groups with coefficents in a field of characteristic p (same p!) a close relation between smo...
 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 years, 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 of residue characters, the P, positive P. Then G is the F points of a connected reductive group over F. And 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 complexes 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. So I is any, for now, I in G, any compact open subgroup. And then we can form x, the compact induction 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 cosets. D 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 called 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, here's a pair of a 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 y, y-axis, and on the other hand, if we start with a representation, then we just take its i-fix vectors and to see the action immediately, you rewrite this by Fabianius reciprocity as the G homomorphisms 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, let me say this way, pass to a 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-reps relations 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-repcentrations. So we pick such a resolution and let me write to be clear in mod G and again form the endomorphism algebra. But this is now a more complicated object. So we take the endomorphisms, again, op. So this is a differential graded algebra because by an endomorphism I do not just mean the 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 the grading. We get 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. And let me say, either way, this does not, 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. 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 is just the degree zero comology of this one here. 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. And then there's the following theorem. So now I have to impose a assumption on I suppose that I is pro-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. But 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. Because here we can get rid of this injective resolution this is just the x algebra star 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 Ioneda product, the concatenation of extensions but it's not visible 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. The structural properties of the homology algebra. Joint work with Rachel Olivier. So but from now on I make from now on I is one particular group. So 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 is the pro-p pseudo subgroup of the Ivohoi subgroup which is, which actually is a normal, I mean because of the structure of Piarticli groups 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. And I do not for now, I will not, when I will then I will specifically say it, I make no assumptions about, I mean it's pro-p anyway by definition but I make, for now I make no assumptions about torsion frieness. I mean since the, really in some sense the only thing one can explicitly say is that since the only thing one can explicitly work with is these Comology 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 torsion, then in there I have the maximal compact subgroup and in there I have T1 which is the pro-p pseudo here. And what else, so then the corresponding, let's see, that was the wrong way. So we have of course the extended affine y group, normalizer, modulo T0, but we work with the pro-p we have to work with the larger quotient retilde 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, which was the vector space over the cosets 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. And in particular it's an I-equiviant, this is I-equiviant. And so the comology 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 comology 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's lemma, which says that this is the comology 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 in some sense it reflects something which in the class in the undirect Hickey algebra is very well known in its description by generators and relations. And if you multiply two of these pieces here, so hj, different degrees possibly, and w and w and v tilde, then you end up in a finite direct sum of all u, of course in, maybe I did this way, iu, i contained in the product of these two double cosets, h 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 Iw or p, Iw Hickey algebra is known to have an anti-involution and 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 then maybe 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, 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, here, in this picture, Icology 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, this Iw inverse. Then we go back by Shapiro and have h star w inverse. And the theorem says that J is an anti-automorphism of the Comorji algebra. What I mean is, maybe I write it down, because this is in the graded sense now, of course. So J of the product, 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 proofs, but I will tell you in the second part somehow the technical instrument, so 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, it's 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 of its finite supports, point-wise product. That's of course G-equivariant. And this then gives me, together with the usual cup product, the cup product on group compagny goes into the new coefficients, the tensor product of the old coefficients, and then we can use this map to go on. So it induces what I call here the cup product, h i, h i, i x, tensor h j, i x, 2 h i plus j i of the tensor product, but I immediately, sorry, I compose it with this map, so 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 from the homology groups of these I w with red. If you have trivial coefficients, you have the usual cup product. So that's very easy to see, but it has of course the immediate consequence that it's this cup product, I mean, it's completely component-wise. If I have h i v cup h j w, this is always 0 if v is different from w. So why should this be helpful at all? But in some sense, then the second technical statement, so in some sense maybe the key technical proposition is the following. And I formulate 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. 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 first technical proposition here, we know that the product alpha times beta is the sum over these 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 the homological restriction and co-restriction lapse. So this is really the tool which allows to compute the product. Let me give you a 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 is the sum of the individual length. Then actually we only have 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 nicest 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 Hegger operators, tau of something is the characteristic function of this double coset of something in the usual Hegger algebra. So this reduces the product here to the cup product and the modules, only the multiplication of the degree 0 part on everything, which 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. So this, one still has this formula, but there are many complicated other terms then. So this again is the order on which our process started. So then let me see its quality. 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 is 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 in the local field. Or you view it as a periodically group and take its dimension as a periodically group. So this means they have homology only up to degree d. So the immediate consequence, in particular, our homology algebra sits in degrees 0 to d. So it's 0 for star degree bigger than d. And we will use this later. In highest degree d, the homology of all these groups is one dimensional. Plus, the cap product pairing is non-degenerate for each of those. I don't write this down. So now we use another map. It's a very simple map. I have g, equivalent map 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 in uses together with the cap product. Let me write it out. So we just take our cap product. And then we take this map on the coefficients. And let me, so this is, let me just denote this for later, use capital sigma h d i, little sigma. Actually, I don't want to do this in general. I mean, we want about duality. So I take your d minus i, then I have here h d d i with trivial coefficients. And this I identify once and for all. I told you this is one dimension. 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. So get k linear maps from h i x into h d minus i i x linear dual. I will check for linear dual. So it's easy. Let's stick for a moment to this homology picture. So these are, what is not difficult to see is these are injective with image. Of course, not so objective 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 a linear form is finite. If it's non-zero on at most, finitely many of these summands. 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, h bimodules. So this is xi of x, x. And this here, before the dualizing, is x d minus i of a little space. I have to put some decoration here in the moment, x, 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 made 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, right? And it was a left module, right? H was acting from the right. So I make it into a, what is the usual convention? Left and right. I think now it's a left. Left h module, y at j, where the anti-automorphism. I mean, more should be true, but we can prove this here. This, the whole, not only the h, the degree 0 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 0 here. It's the usual Hacker algebra, but with some twisted action. And since the usual Hacker 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, 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. Let me only write down the cleanest proposition. I have to make R. 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 comology, we have this map to the 2k, actually this split. So this is the direct sum of hd i k. 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-bimodule, this is the trivial h. I mean, the h structure on here is the trivial one, trivial h-bimodule. 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 singularly, there is a certain ideal in the center of the propi hevorary 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 anti-involution. So super singular means the module is torsion for this idea. And then the last fact is that this super singular, very big super singular module. So then an obvious super singular module, the final 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 final dimension super singular module and I take its injective hull of h mod jh, again from the left end and from the right. OK, so far our structural results about the algebra. Now I want to talk about at the end in the last, free-flash section, what I strangely maybe call the Ralph parabolic induction. This is joint. This is Sarah Schwarzke. So let us pick a parabolic subgroup. And of course, I think it occurs to me, I forgot to say this when I introduced the torsion. Of course, there always has to be a relation between the torsion, my evo-hori 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 evo-hori subgroup. And then, I mean, it's a very easy fact that the usual parabolic induction from mod g to mod g is exact. 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. So what I mean here is, of course, how do we, 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 write now, maybe h dot m or g, I mean, and so on. I indicate by a subscript that 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, the proper evo-hori algebra, algebra is bi-module, 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 more space, then there is this lemma, which is just a, we have this here, reformulation over the side 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, mod 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 bi-module. 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. Moreover, this bi-module xg, p is finally generated projective over, I mean, only on one side, on the hm side over hm. OK, so that's the picture, the under derived picture. And so let me finish with the derived picture, which looks like a kind of a formal consequence. So now we need an h dot gh dot m bi-module, which I then, I dot subscript gp. And at first, let's just imitate what we did above. I mean, the invariance is the same as home, gequivriant home, then here, from the universal thing, so which is here, its injective resolution, into here, 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 it also has to resolve this first injectively. So this is right, I mean, 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 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 dh.g dm dhm dot. I mean, these were the left to right. Let me maybe only repeat it here. These were the left derived, this was the right derived of this one. And correspondingly here, here we have the trivial derivation of parabolic induction. And here we have the derived tensor product with this y-module. So this is commutative. I didn't write it there. Sorry, it's commutative. Now, as I said, this looks as if it 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. From the right, I want the action of hm dot. So it acts on here. And then by functoriality, it acts on the parabolic induction. But if I take some injective resolution in mod G, I completely lose this action. There's no reason why I should preserve it. And so that's the huge problem. One has to take specific, so I mean, one needs, specific or a specific injective resolution functor in the framework of differential graded categories. It has to lift this whole thing. I mean, the derived categories in some sense say the homotopic category of a differential graded category, which takes 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 theorem by Schnürer. Olaf Schnürer. Really non-trivial thing. So once you have this, then the thing becomes a formality. But without it, it would be hopeless. OK. Thank you very much. I have a question. I mean, excuse me, I probably shouldn't know the answer to this. But I mean, does this thing exist for characteristic zero representations of groups with a periodic, regular, periodic groups? Yes. But it's nothing additional because there, these functors are exact. Right? The Ivohoi propi groups have no homology 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 comments. Yeah. I mean, the surprising thing is, of course, there in characteristic zero is, I'm sure you know, you do get, say, for the propi Ivohoi, you do get a equivalence of categories, but not with the full category of smooth representations, only those which are generated by its Ivohoi, propi Ivohoi invents. Whereas here, I mean, the surprising thing is that under these assumptions, of course, the higher, even, even, right, I mean, you may have a representation, of course, in characteristic p, every representation has a non-zero Ivohoi, propi Ivohoi fixed vector, but they don't have to generate the representation. Nevertheless, somehow the higher homology sees, sees the representation, whether it's generated by its Ivohoi fixed vectors or not. This is in the characteristic p case. Yeah. I mean, certainly, so for completely formal reasons, the right adjoint, right, once you have, because of the equivalence, I mean, the usual parabolic induction has a right adjoint, which is left exact, and therefore you can derive it. So you have a right adjoint for this on the derive category. And since these are equivalences, you have a right adjoint here and you can compute it. It's the home from this bimodule, the left home. The problem is with the left adjoint, which usually Schaqet functor is right exact, but it's not clear whether one can derive it, because for that you would need projective objects in the g-repsentations which do not exist in characteristic p. So I think that's where this, I mean, probably this finiteness property reflects some finiteness property of this year, which allows us to see that this also has a left adjoint. But this I don't know yet. So you mentioned that the formula gets very complicated when you suppose for example the length of v is 1. The length of v is 1, maybe you have some formula? Yeah, but don't ask me to write it down right now. I would have to look it up. Because you get then, right, I mean, if you have, let's say this is S here. I'm not sure which one, but the lengths do not add up. Then the possible u's are I think S w and omega w, where omega is of length 0. And you get terms, I mean, this, one of these terms I think this one is still kind of, I'm not sure which one. But you have to, 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. If v is equal to w is of length 1. Yeah, yeah, sure. You have S squared here and you have omega S here. So you still have many terms. No, but you take omega is equal to S. If omega is equal to S. So omega is of length 0 in this case. I mean, I was just describing if this is of length 1, a reflection, what the possible u's are. And there is one which S times w and then there are q minus 1 omega times w. And so you have them even if w is S as well. There is no more question. Thank you.
 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 years, 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 of residue characters, the P, positive P.  Then G is the F points of a connected reductive group over F.  And 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 complexes 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.  So I is any, for now, I in G, any compact open subgroup.  And then we can form x, the compact induction 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 cosets.  D 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 called 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, here's a pair of a 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 y, y-axis, and on the other hand, if we start with a representation,  then we just take its i-fix vectors and to see the action immediately, you rewrite this  by Fabianius reciprocity as the G homomorphisms 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, let me say this way, pass to a 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-reps relations 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-repcentrations.  So we pick such a resolution and let me write to be clear in mod G and again form the endomorphism  algebra.  But this is now a more complicated object.  So we take the endomorphisms, again, op.  So this is a differential graded algebra because by an endomorphism I do not just mean the  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 the grading.  We get 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.  And let me say, either way, this does not, 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.  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 is just the degree zero comology of this one  here.  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.  And then there's the following theorem.  So now I have to impose a assumption on I suppose that I is pro-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.  But 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.  Because here we can get rid of this injective resolution this is just the x algebra star  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  Ioneda product, the concatenation of extensions but it's not visible 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.  The structural properties of the homology algebra.  Joint work with Rachel Olivier.  So but from now on I make from now on I is one particular group.  So 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 is the pro-p pseudo  subgroup of the Ivohoi subgroup which is, which actually is a normal, I mean because  of the structure of Piarticli groups 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.  And I do not for now, I will not, when I will then I will specifically say it, I make no  assumptions about, I mean it's pro-p anyway by definition but I make, for now I make no  assumptions about torsion frieness.  I mean since the, really in some sense the only thing one can explicitly say is that  since the only thing one can explicitly work with is these Comology 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 torsion, then in there I have the maximal compact subgroup  and in there I have T1 which is the pro-p pseudo here.  And what else, so then the corresponding, let's see, that was the wrong way.  So we have of course the extended affine y group, normalizer, modulo T0, but we work  with the pro-p we have to work with the larger quotient retilde 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, which was the vector space  over the cosets 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.  And in particular it's an I-equiviant, this is I-equiviant.  And so the comology 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 comology 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's lemma, which says that this is the comology 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 in some sense it reflects something which in the class in the undirect  Hickey algebra is very well known in its description by generators and relations.  And if you multiply two of these pieces here, so hj, different degrees possibly, and w and  w and v tilde, then you end up in a finite direct sum of all u, of course in, maybe I  did this way, iu, i contained in the product of these two double cosets, h 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 Iw or p, Iw Hickey algebra is known to have  an anti-involution and 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 then maybe 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, 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, here, in this picture, Icology 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, this Iw inverse.  Then we go back by Shapiro and have h star w inverse.  And the theorem says that J is an anti-automorphism of the Comorji algebra.  What I mean is, maybe I write it down, because this is in the graded sense now, of course.  So J of the product, 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 proofs, but I will tell you in the second  part somehow the technical instrument, so 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, it's 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 of its finite supports, point-wise  product.  That's of course G-equivariant.  And this then gives me, together with the usual cup product, the cup product on group  compagny goes into the new coefficients, the tensor product of the old coefficients, and  then we can use this map to go on.  So it induces what I call here the cup product, h i, h i, i x, tensor h j, i x, 2 h i plus  j i of the tensor product, but I immediately, sorry, I compose it with this map, so 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  from the homology groups of these I w with red.  If you have trivial coefficients, you have the usual cup product.  So that's very easy to see, but it has of course the immediate consequence that it's  this cup product, I mean, it's completely component-wise.  If I have h i v cup h j w, this is always 0 if v is different from w.  So why should this be helpful at all?  But in some sense, then the second technical statement, so in some sense maybe the key technical  proposition is the following.  And I formulate 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.  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  first technical proposition here, we know that the product alpha times beta is the sum  over these 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 the homological restriction and co-restriction  lapse.  So this is really the tool which allows to compute the product.  Let me give you a 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 is the sum of the individual  length.  Then actually we only have 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 nicest 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 Hegger operators, tau of something is the characteristic function of this double  coset of something in the usual Hegger algebra.  So this reduces the product here to the cup product and the modules, only the multiplication  of the degree 0 part on everything, which 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.  So this, one still has this formula, but there are many complicated other terms then.  So this again is the order on which our process started.  So then let me see its quality.  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  is 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 in the local field.  Or you view it as a periodically group and take its  dimension as a periodically group.  So this means they have homology only up to degree d.  So the immediate consequence, in particular, our homology  algebra sits in degrees 0 to d.  So it's 0 for star degree bigger than d.  And we will use this later.  In highest degree d, the homology of all these groups  is one dimensional.  Plus, the cap product pairing is non-degenerate for each  of those.  I don't write this down.  So now we use another map.  It's a very simple map.  I have g, equivalent map 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 in uses together with the cap product.  Let me write it out.  So we just take our cap product.  And then we take this map on the coefficients.  And let me, so this is, let me just denote this for later,  use capital sigma h d i, little sigma.  Actually, I don't want to do this in general.  I mean, we want about duality.  So I take your d minus i, then I have here h d d i with  trivial coefficients.  And this I identify once and for all.  I told you this is one dimension.  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.  So get k linear maps from h i x into h d minus i i x linear  dual.  I will check for linear dual.  So it's easy.  Let's stick for a moment to this homology picture.  So these are, what is not difficult to see is these  are injective with image.  Of course, not so objective 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 a linear form is finite.  If it's non-zero on at most, finitely many of these summands.  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, h bimodules.  So this is xi of x, x.  And this here, before the dualizing,  is x d minus i of a little space.  I have to put some decoration here in the moment, x, 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 made 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, right?  And it was a left module, right?  H was acting from the right.  So I make it into a, what is the usual convention?  Left and right.  I think now it's a left.  Left h module, y at j, where the anti-automorphism.  I mean, more should be true, but we can prove this here.  This, the whole, not only the h, the degree 0 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 0 here.  It's the usual Hacker algebra, but with some twisted action.  And since the usual Hacker 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, 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.  Let me only write down the cleanest proposition.  I have to make R. 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 comology, we have this map to the 2k,  actually this split.  So this is the direct sum of hd i k.  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-bimodule, this is the trivial h.  I mean, the h structure on here is the trivial one,  trivial h-bimodule.  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 singularly,  there is a certain ideal in the center of the propi  hevorary 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 anti-involution.  So super singular means the module is torsion for this idea.  And then the last fact is that this super singular,  very big super singular module.  So then an obvious super singular module,  the final 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 final dimension  super singular module and I take its injective hull of h  mod jh, again from the left end and from the right.  OK, so far our structural results about the algebra.  Now I want to talk about at the end in the last,  free-flash section, what I strangely maybe call  the Ralph parabolic induction.  This is joint.  This is Sarah Schwarzke.  So let us pick a parabolic subgroup.  And of course, I think it occurs to me,  I forgot to say this when I introduced the torsion.  Of course, there always has to be a relation  between the torsion, my evo-hori 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 evo-hori subgroup.  And then, I mean, it's a very easy fact  that the usual parabolic induction from mod g to mod g  is exact.  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.  So what I mean here is, of course,  how do we, 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 write now, maybe h dot m or g,  I mean, and so on.  I indicate by a subscript that 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, the proper evo-hori algebra, algebra is bi-module,  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 more space,  then there is this lemma, which is just a,  we have this here, reformulation over the side 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,  mod 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 bi-module.  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.  Moreover, this bi-module xg, p is finally generated  projective over, I mean, only on one side, on the hm side  over hm.  OK, so that's the picture, the under derived picture.  And so let me finish with the derived picture, which  looks like a kind of a formal consequence.  So now we need an h dot gh dot m bi-module, which I then,  I dot subscript gp.  And at first, let's just imitate what we did above.  I mean, the invariance is the same as home,  gequivriant home, then here, from the universal thing,  so which is here, its injective resolution,  into here, 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 it also has to resolve this first injectively.  So this is right, I mean, 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 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 dh.g dm dhm dot.  I mean, these were the left to right.  Let me maybe only repeat it here.  These were the left derived, this was the right derived  of this one.  And correspondingly here, here we  have the trivial derivation of parabolic induction.  And here we have the derived tensor product  with this y-module.  So this is commutative.  I didn't write it there.  Sorry, it's commutative.  Now, as I said, this looks as if it  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.  From the right, I want the action of hm dot.  So it acts on here.  And then by functoriality, it acts on the parabolic induction.  But if I take some injective resolution in mod G,  I completely lose this action.  There's no reason why I should preserve it.  And so that's the huge problem.  One has to take specific, so I mean, one needs,  specific or a specific injective resolution functor  in the framework of differential graded categories.  It has to lift this whole thing.  I mean, the derived categories in some sense  say the homotopic category of a differential graded  category, which takes 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 theorem by Schnürer.  Olaf Schnürer.  Really non-trivial thing.  So once you have this, then the thing becomes a formality.  But without it, it would be hopeless.  OK.  Thank you very much.  I have a question.  I mean, excuse me, I probably shouldn't know the answer to this.  But I mean, does this thing exist for characteristic zero  representations of groups with a periodic, regular,  periodic groups?  Yes.  But it's nothing additional because there,  these functors are exact.  Right?  The Ivohoi propi groups have no homology 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 comments.  Yeah.  I mean, the surprising thing is, of course,  there in characteristic zero is, I'm sure you know,  you do get, say, for the propi Ivohoi,  you do get a equivalence of categories,  but not with the full category of smooth representations,  only those which are generated by its Ivohoi,  propi Ivohoi invents.  Whereas here, I mean, the surprising thing is that under  these assumptions, of course, the higher, even,  even, right, I mean, you may have a representation,  of course, in characteristic p, every representation has  a non-zero Ivohoi, propi Ivohoi fixed vector,  but they don't have to generate the representation.  Nevertheless, somehow the higher homology sees,  sees the representation, whether it's generated by its  Ivohoi fixed vectors or not.  This is in the characteristic p case.  Yeah.  I mean, certainly, so for completely formal reasons,  the right adjoint, right, once you have,  because of the equivalence, I mean,  the usual parabolic induction has a right adjoint,  which is left exact, and therefore you can derive it.  So you have a right adjoint for this on the derive category.  And since these are equivalences, you have a right adjoint  here and you can compute it.  It's the home from this bimodule, the left home.  The problem is with the left adjoint,  which usually Schaqet functor is right exact,  but it's not clear whether one can derive it,  because for that you would need projective objects  in the g-repsentations which do not exist in characteristic p.  So I think that's where this, I mean,  probably this finiteness property reflects some finiteness  property of this year, which allows us to see  that this also has a left adjoint.  But this I don't know yet.  So you mentioned that the formula gets very complicated  when you suppose for example the length of v is 1.  The length of v is 1,  maybe you have some formula?  Yeah, but don't ask me to write it down right now.  I would have to look it up.  Because you get then, right, I mean, if you have,  let's say this is S here.  I'm not sure which one, but the lengths do not add up.  Then the possible u's are I think S w and omega w,  where omega is of length 0.  And you get terms, I mean, this,  one of these terms I think this one is still kind of,  I'm not sure which one.  But you have to, 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.  If v is equal to w is of length 1.  Yeah, yeah, sure.  You have S squared here and you have omega S here.  So you still have many terms.  No, but you take omega is equal to S.  If omega is equal to S.  So omega is of length 0 in this case.  I mean, I was just describing if this is of length 1,  a reflection, what the possible u's are.  And there is one which S times w and then there are q minus 1  omega times w.  And so you have them even if w is S as well.  There is no more question.  Thank you.
Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_0_731.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_732_1307.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_1308_4319.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_4320_5471.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_5472_9011.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_9012_9095.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_9096_9515.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_9516_10259.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_10260_11075.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_11076_11987.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_11988_13487.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_13488_13823.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_13824_14531.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_14532_15683.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_15684_16535.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_16536_19403.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_19404_19547.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_19548_21263.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_21264_21815.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_21816_24059.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_24060_24875.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_24876_26567.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_26568_26975.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_26976_27839.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_27840_28007.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_28008_28355.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_28356_28607.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_28608_28847.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_28848_28991.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_28992_29375.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_29376_34235.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_34236_35207.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_35208_35255.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_35256_36047.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_36048_37319.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_37320_38291.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_38292_43847.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_43848_44279.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_44280_44387.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_44388_47111.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_47112_48503.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_48504_48995.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_48996_49643.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_49644_50483.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_50484_51143.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_51144_51203.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_51204_51599.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_51600_51803.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_51804_51923.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_51924_52259.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_52260_52499.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_52500_52967.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_52968_53267.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_53268_54839.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_54840_55103.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_55104_57755.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_57756_59783.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_59784_64559.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_64560_65435.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_65436_65495.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_65496_65543.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_65544_67223.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_67224_69455.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_69456_69767.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_69768_70583.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_70584_70739.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_70740_72275.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_72276_72455.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_72456_73067.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_73068_73811.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_73812_74243.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_74244_77483.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_77484_77555.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_77556_90635.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_90636_90755.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_90756_92687.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_92688_95699.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_95700_96239.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_96240_98291.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_98292_99371.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_99372_99431.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_99432_101243.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_101244_102275.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_102276_102923.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_102924_104063.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_104064_104963.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_104964_105011.png Peter_Schneider_-_Progress_report_on_derived_Hecke_biwbtEdP2r8_105012_105726.png
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 years, 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 of residue characters, the P, positive P. Then G is the F points of a connected reductive group over F. And 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 complexes 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. So I is any, for now, I in G, any compact open subgroup. And then we can form x, the compact induction 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 cosets. D 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 called 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, here's a pair of a 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 y, y-axis, and on the other hand, if we start with a representation, then we just take its i-fix vectors and to see the action immediately, you rewrite this by Fabianius reciprocity as the G homomorphisms 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, let me say this way, pass to a 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-reps relations 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-repcentrations. So we pick such a resolution and let me write to be clear in mod G and again form the endomorphism algebra. But this is now a more complicated object. So we take the endomorphisms, again, op. So this is a differential graded algebra because by an endomorphism I do not just mean the 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 the grading. We get 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. And let me say, either way, this does not, 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. 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 is just the degree zero comology of this one here. 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. And then there's the following theorem. So now I have to impose a assumption on I suppose that I is pro-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. But 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. Because here we can get rid of this injective resolution this is just the x algebra star 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 Ioneda product, the concatenation of extensions but it's not visible 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. The structural properties of the homology algebra. Joint work with Rachel Olivier. So but from now on I make from now on I is one particular group. So 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 is the pro-p pseudo subgroup of the Ivohoi subgroup which is, which actually is a normal, I mean because of the structure of Piarticli groups 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. And I do not for now, I will not, when I will then I will specifically say it, I make no assumptions about, I mean it's pro-p anyway by definition but I make, for now I make no assumptions about torsion frieness. I mean since the, really in some sense the only thing one can explicitly say is that since the only thing one can explicitly work with is these Comology 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 torsion, then in there I have the maximal compact subgroup and in there I have T1 which is the pro-p pseudo here. And what else, so then the corresponding, let's see, that was the wrong way. So we have of course the extended affine y group, normalizer, modulo T0, but we work with the pro-p we have to work with the larger quotient retilde 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, which was the vector space over the cosets 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. And in particular it's an I-equiviant, this is I-equiviant. And so the comology 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 comology 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's lemma, which says that this is the comology 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 in some sense it reflects something which in the class in the undirect Hickey algebra is very well known in its description by generators and relations. And if you multiply two of these pieces here, so hj, different degrees possibly, and w and w and v tilde, then you end up in a finite direct sum of all u, of course in, maybe I did this way, iu, i contained in the product of these two double cosets, h 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 Iw or p, Iw Hickey algebra is known to have an anti-involution and 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 then maybe 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, 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, here, in this picture, Icology 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, this Iw inverse. Then we go back by Shapiro and have h star w inverse. And the theorem says that J is an anti-automorphism of the Comorji algebra. What I mean is, maybe I write it down, because this is in the graded sense now, of course. So J of the product, 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 proofs, but I will tell you in the second part somehow the technical instrument, so 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, it's 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 of its finite supports, point-wise product. That's of course G-equivariant. And this then gives me, together with the usual cup product, the cup product on group compagny goes into the new coefficients, the tensor product of the old coefficients, and then we can use this map to go on. So it induces what I call here the cup product, h i, h i, i x, tensor h j, i x, 2 h i plus j i of the tensor product, but I immediately, sorry, I compose it with this map, so 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 from the homology groups of these I w with red. If you have trivial coefficients, you have the usual cup product. So that's very easy to see, but it has of course the immediate consequence that it's this cup product, I mean, it's completely component-wise. If I have h i v cup h j w, this is always 0 if v is different from w. So why should this be helpful at all? But in some sense, then the second technical statement, so in some sense maybe the key technical proposition is the following. And I formulate 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. 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 first technical proposition here, we know that the product alpha times beta is the sum over these 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 the homological restriction and co-restriction lapse. So this is really the tool which allows to compute the product. Let me give you a 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 is the sum of the individual length. Then actually we only have 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 nicest 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 Hegger operators, tau of something is the characteristic function of this double coset of something in the usual Hegger algebra. So this reduces the product here to the cup product and the modules, only the multiplication of the degree 0 part on everything, which 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. So this, one still has this formula, but there are many complicated other terms then. So this again is the order on which our process started. So then let me see its quality. 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 is 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 in the local field. Or you view it as a periodically group and take its dimension as a periodically group. So this means they have homology only up to degree d. So the immediate consequence, in particular, our homology algebra sits in degrees 0 to d. So it's 0 for star degree bigger than d. And we will use this later. In highest degree d, the homology of all these groups is one dimensional. Plus, the cap product pairing is non-degenerate for each of those. I don't write this down. So now we use another map. It's a very simple map. I have g, equivalent map 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 in uses together with the cap product. Let me write it out. So we just take our cap product. And then we take this map on the coefficients. And let me, so this is, let me just denote this for later, use capital sigma h d i, little sigma. Actually, I don't want to do this in general. I mean, we want about duality. So I take your d minus i, then I have here h d d i with trivial coefficients. And this I identify once and for all. I told you this is one dimension. 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. So get k linear maps from h i x into h d minus i i x linear dual. I will check for linear dual. So it's easy. Let's stick for a moment to this homology picture. So these are, what is not difficult to see is these are injective with image. Of course, not so objective 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 a linear form is finite. If it's non-zero on at most, finitely many of these summands. 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, h bimodules. So this is xi of x, x. And this here, before the dualizing, is x d minus i of a little space. I have to put some decoration here in the moment, x, 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 made 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, right? And it was a left module, right? H was acting from the right. So I make it into a, what is the usual convention? Left and right. I think now it's a left. Left h module, y at j, where the anti-automorphism. I mean, more should be true, but we can prove this here. This, the whole, not only the h, the degree 0 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 0 here. It's the usual Hacker algebra, but with some twisted action. And since the usual Hacker 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, 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. Let me only write down the cleanest proposition. I have to make R. 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 comology, we have this map to the 2k, actually this split. So this is the direct sum of hd i k. 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-bimodule, this is the trivial h. I mean, the h structure on here is the trivial one, trivial h-bimodule. 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 singularly, there is a certain ideal in the center of the propi hevorary 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 anti-involution. So super singular means the module is torsion for this idea. And then the last fact is that this super singular, very big super singular module. So then an obvious super singular module, the final 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 final dimension super singular module and I take its injective hull of h mod jh, again from the left end and from the right. OK, so far our structural results about the algebra. Now I want to talk about at the end in the last, free-flash section, what I strangely maybe call the Ralph parabolic induction. This is joint. This is Sarah Schwarzke. So let us pick a parabolic subgroup. And of course, I think it occurs to me, I forgot to say this when I introduced the torsion. Of course, there always has to be a relation between the torsion, my evo-hori 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 evo-hori subgroup. And then, I mean, it's a very easy fact that the usual parabolic induction from mod g to mod g is exact. 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. So what I mean here is, of course, how do we, 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 write now, maybe h dot m or g, I mean, and so on. I indicate by a subscript that 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, the proper evo-hori algebra, algebra is bi-module, 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 more space, then there is this lemma, which is just a, we have this here, reformulation over the side 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, mod 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 bi-module. 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. Moreover, this bi-module xg, p is finally generated projective over, I mean, only on one side, on the hm side over hm. OK, so that's the picture, the under derived picture. And so let me finish with the derived picture, which looks like a kind of a formal consequence. So now we need an h dot gh dot m bi-module, which I then, I dot subscript gp. And at first, let's just imitate what we did above. I mean, the invariance is the same as home, gequivriant home, then here, from the universal thing, so which is here, its injective resolution, into here, 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 it also has to resolve this first injectively. So this is right, I mean, 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 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 dh.g dm dhm dot. I mean, these were the left to right. Let me maybe only repeat it here. These were the left derived, this was the right derived of this one. And correspondingly here, here we have the trivial derivation of parabolic induction. And here we have the derived tensor product with this y-module. So this is commutative. I didn't write it there. Sorry, it's commutative. Now, as I said, this looks as if it 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. From the right, I want the action of hm dot. So it acts on here. And then by functoriality, it acts on the parabolic induction. But if I take some injective resolution in mod G, I completely lose this action. There's no reason why I should preserve it. And so that's the huge problem. One has to take specific, so I mean, one needs, specific or a specific injective resolution functor in the framework of differential graded categories. It has to lift this whole thing. I mean, the derived categories in some sense say the homotopic category of a differential graded category, which takes 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 theorem by Schnürer. Olaf Schnürer. Really non-trivial thing. So once you have this, then the thing becomes a formality. But without it, it would be hopeless. OK. Thank you very much. I have a question. I mean, excuse me, I probably shouldn't know the answer to this. But I mean, does this thing exist for characteristic zero representations of groups with a periodic, regular, periodic groups? Yes. But it's nothing additional because there, these functors are exact. Right? The Ivohoi propi groups have no homology 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 comments. Yeah. I mean, the surprising thing is, of course, there in characteristic zero is, I'm sure you know, you do get, say, for the propi Ivohoi, you do get a equivalence of categories, but not with the full category of smooth representations, only those which are generated by its Ivohoi, propi Ivohoi invents. Whereas here, I mean, the surprising thing is that under these assumptions, of course, the higher, even, even, right, I mean, you may have a representation, of course, in characteristic p, every representation has a non-zero Ivohoi, propi Ivohoi fixed vector, but they don't have to generate the representation. Nevertheless, somehow the higher homology sees, sees the representation, whether it's generated by its Ivohoi fixed vectors or not. This is in the characteristic p case. Yeah. I mean, certainly, so for completely formal reasons, the right adjoint, right, once you have, because of the equivalence, I mean, the usual parabolic induction has a right adjoint, which is left exact, and therefore you can derive it. So you have a right adjoint for this on the derive category. And since these are equivalences, you have a right adjoint here and you can compute it. It's the home from this bimodule, the left home. The problem is with the left adjoint, which usually Schaqet functor is right exact, but it's not clear whether one can derive it, because for that you would need projective objects in the g-repsentations which do not exist in characteristic p. So I think that's where this, I mean, probably this finiteness property reflects some finiteness property of this year, which allows us to see that this also has a left adjoint. But this I don't know yet. So you mentioned that the formula gets very complicated when you suppose for example the length of v is 1. The length of v is 1, maybe you have some formula? Yeah, but don't ask me to write it down right now. I would have to look it up. Because you get then, right, I mean, if you have, let's say this is S here. I'm not sure which one, but the lengths do not add up. Then the possible u's are I think S w and omega w, where omega is of length 0. And you get terms, I mean, this, one of these terms I think this one is still kind of, I'm not sure which one. But you have to, 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. If v is equal to w is of length 1. Yeah, yeah, sure. You have S squared here and you have omega S here. So you still have many terms. No, but you take omega is equal to S. If omega is equal to S. So omega is of length 0 in this case. I mean, I was just describing if this is of length 1, a reflection, what the possible u's are. And there is one which S times w and then there are q minus 1 omega times w. And so you have them even if w is S as well. There is no more question. Thank you.
the local field of residue characters, the P, be representation theory in characteristic P. And representations of groups with a periodic, characteristic p, every representation has a