 thanks to the organizers, and it's a pleasure to be here. So I would like to talk about some congruence properties of endoclases, the same endoclases that the ones in the previous talk, and the consequences of these congruence properties for the local Jacque-Longlence correspondence. So this is a work joined with Sean Stevens. First, let me fix some notation. So as in the previous talk, F is a non-archimedean, locally compact field of residue characteristic P. H will be the general linear group GLNF for some N, and G will be an inner form of H. So this is a group of the form GLM of D, where D is a central division F-algebra of reduced degree D with MD equal to N. And so I will denote by D of G or DC of G if I want to insist on the fact that I will consider complex representations. D of G is the discrete series of G. So this is the set up to the set of isomorphism classes of essentially a square integrable, smooth, complex representations of the group G. And there is a canonical bijection, the Jacque-Longlence correspondence between DC of G and DC of H, which I will denote like this with a superscript JL. And this correspondence is characterized by a character identity at elliptic regular conjugacy classes in G and H. So what does essentially square integrable representation look like more precisely? Given pi in the discrete series of G, there is a unique integer R dividing M and a unique up to isomorphism, a unique hospital irreducible representation rho of GLM over R of D, such that pi is the unique irreducible quotient of the following parabolically induced representation. Parabolic induction here is normalized in the usual way. So quotient of the parabolic induction to G of rho tensor rho twisted by the unhammified character given by the absolute value of the reduced norm to the power of some integer S of rho, depending on rho tensor and R times. And the last factor is rho twisted by the absolute value of the reduced norm to the power of S of rho times R minus 1. So where S of rho is a positive integer attached to rho, which divides D and is prime to M. So in particular, if G is split, if G is equal to H, D is 1. So this positive integer S of rho is always equal to 1. So you do not see it. So in this, R and rho are uniquely determined by pi. And so in this situation, I will write pi equals delta of rho R. And I will also write S of pi equals S of rho. So this classification of discrete series representation goes back to Bernstein and Zelivinsky in the split case, where G is split and to tally it in the case of an inner form. OK, so now look at the Jacqueline-Glantz transfer pi JL of pi. It is also a discrete series representation for the split group H. And so according to this classification, it can be written uniquely as delta of sigma t for sum t dividing N and sigma, a cospital representation of G L N over t of F. And so we have two pairs. We have rho R on the one hand and the pair sigma t on the other hand. And between the two integers, R and t, we have the following relation. T is equal to R times S of pi. And then the natural problem is what is the relation between rho and sigma compared rho and n sigma? How can one compare rho and sigma? For this, we use Bushnell. So there is a description of cospital representations of G L N and given by compact induction, which is given by Bushnell-Kutzko's theory of types. So for H, for G L N F, this goes back to Bushnell and Kutzko. And the same works for inner forms of G L N F. It generalizes this description of cospital representations given by Bushnell-Kutzko. It has been generalized by Brussus, Grabitz, Schoen-Stevens, and others. In particular, I will not say anything specific about type theory, about this type theoretic description of cospital representations. But in the previous talk, Bushnell introduced the notion of endoclass. And so to any cospital representation of G L N F, one can associate an endoclass. And the same work for inner forms of G L N. To any cospital irreducible representation pi of group G, one can associate an endoclass denoted theta pi. And so I go back to this question. Compare rho and sigma. Rho and sigma are two cospital representations, but for different groups which do not even need to be inner forms of each other. So how can one compare them? So the interest of looking at the endoclass is that the set of all endoclass is the same for all general linear groups G L N and their inner forms. So one can ask the more precise, more specific question. Compare the endoclass of rho with the endoclass of sigma. And the answer to this question is easy to state. The comparison is as simple as possible. Rho and sigma have the same endoclass. So the comparison is easy to state. It is an equality, but the proof of this is more difficult to state due to the very technical definition of endoclasses. And so the proof of this result appeals unexpectedly to the theory of modular representations of G L N and its inner forms, as well as congruence properties of the local jaculine glons correspondence. So in this talk, I would like to show you how to use modular representation theory in order to prove a result about complex representations. So more precisely, the statement whose proof requires modular techniques is the following one, which comes before this one. So let pi and pi prime be two complex discrete series representations of G. Then pi and pi prime have the same endoclass. So I did not say how to associate an endoclass to a discrete series representation. But so if pi is a discrete series representation, you can write it delta of rho r. And then by definition, the endoclass of pi will be, by definition, will be the endoclass of the cuspidal representation rho. So pi and pi prime have the same endoclass if and only if the same is true of their jaculine glons transfer. And so I would like to explain this, this second statement, starting with, as an example, with the finite field case. So in this first part, I will change temporarily the notation and G will be the finite group gln of some finite field, fq of cardinality q. And I will say that two irreducible representations, so irreducible complex representations of G are a rich standard equivalent. We'll write just hc equivalent if they both occur as components of the same parabolically induced representation for some irreducible representation tau of some, where m is some levy subgroup of G. So this is, it is well known that this is an equivalence relation. And this decomposes the set of irreducible representations of G into a rich standard series. So if you prefer, two irreducible representations of G are a rich standard equivalent if and only if they have the same cuspidal support. Okay, so now replace, fix prime number L different from P. So fix prime L different from P, unlike yesterday afternoon. And consider rather than complex representation, consider representations with coefficients in fl bar, given algebraic closure of finite field of characteristic L. So in the setting of mod L representations, you still have a rich standard induction or parabolic induction. And so the notion, the definition of a rich standard equivalence still makes sense. But it is much less obvious. It is not obvious at all that it is an equivalence relation. And it is the, so we have the following non-trivial result, which is due to Richard Dipper and Gordon James, which tells you precisely that a rich standard equivalence on the set of fl bar irreducible representations of G is an equivalence relation. Since you have an equivalence relation, you can, this decomposes, this decomposes the set of mod L irreducible representations of G into, again, a rich standard series. Okay, so now consider a ladic representations of G. So representations with coefficients in ql bar, a fixed algebraic closure of the field of ladic numbers. So consider pi, a ql bar representation of G on ql bar vector space V. So since G is finite, G stabilizes some zl bar lattice l in V. So l is, such an l is not unique, but there is at least one. And if you reduce l, this lattice mod l, you get a finite length fl bar representation. So this is a finite length fl bar representation of G. So this partial representation does depend on the choice of the lattice, but if you consider the semi-simplification of this reduction mod l, it does not depend anymore on the choice of l. So it is independent of l, it only depends on the representation pi you started with. So this denoted rl of pi, which is, by definition, the reduction mod l of pi. Okay, now I can formulate my first definition. Two irreducible complex representations, pi, pi prime of G are said to be one length. If there exists a prime number l different from pi and a field isomorphism i between c and ql bar. So of course it is a field isomorphism, not a topological field isomorphism. Such that, such that, such that if you look at the reduction mod l of pi, not pi, but i star pi. So the pi but with the field of complex numbers replaced by ql bar by using this isomorphism. And if you do the same for pi prime, you look at the reduction mod l of i star pi prime. So the pi and pi prime are said to be one length. If these two semi-simple representations have all their irreducible components in the same Aries-Chandra series. So two remarks. First remark, this definition does not depend on the choice of the field isomorphism. The second remark is that the fact of being one length, it is an equivalence relation, which is, so this is an equivalence, no, sorry, it is not an equivalence relation. But this is, this is weaker than both being in the same Aries-Chandra series of here c. G and being congruent mod l. Being congruent mod l meaning that the reductions mod l are equal or isomorphic. Now we have the following second definition. So being one length is not an equivalence relation. But you can define the, because it is not, it is not transitive in general. But now let us introduce the following definition. So two irreducible representations pi and pi prime. So irreducible complex representations of G are said to be linked if there are a finite family of irreducible representations. So pi 0 is pi, then pi 1, pi 2, etc. Until pi r, the last one, which is pi prime, with pi i complex irreducible. Such that for all i, pi i and pi i plus 1 are one length. And so of course at each step, the prime number that you use may vary. And now this is, now you get an equivalence, you get an equivalence relation. So being linked is transitive, so is an equivalence relation. So an equivalence relation on the set of complex irreducible representations of G. So we have defined an equivalence relation. So it is natural to determine the equivalence classes of this equivalence relation. So what are the equivalence classes? The answer is easy to state, and it is also not difficult to prove. Any two irreducible representations of G are linked. So this is not very exciting. We have defined an equivalence relation. It turns out to be a trivial equivalence relation. Any two irreducible representations of G are linked. But it will start being more interesting once we pass from the finite field case to the periodic case. And this is what I'm going to do now. So second part, now we go back to the periodic case. And so we go back to G equals an inner form of a periodic GLN group. So of course we have normalized a parabolic induction. And instead of a Richandar equivalence, we can define Bernstein equivalence. So two irreducible complex irreducible representations of G are Bernstein equivalence. If pi occurs as a component of some parabolically induced representation from some levy M to G of some irreducible representation 2 of M. And pi prime occurs as a component of not exactly the same parabolically induced representation but you allow tau to be twisted by some unremifed character chi of M. And again, it is well known that this Bernstein equivalence is an equivalence relation. And in other words, it amounts to saying that pi and pi prime have the same, that their hospital support are inertially equivalent. And this way we can decompose here C of G into Bernstein blocks. So now go back to an L-modular setting. So again, fix a prime L different from P and consider FL bar representations of the periodic group G. We still have normalized parabolic induction. So we can extend this definition of Bernstein equivalence. And the theorem which corresponds to the deeper and James theorem in the finite field case is the following one which is due to Veneras when G is split and is extended to the non-split case by Bertow-Mingus and I. So it tells you that Bernstein equivalence on the set of L-modular irreducible representations of G is an equivalence relation. In other words, in the L-modular setting there is a notion of super-cospital support. So be careful because in the modular setting there is a difference between hospital and super-cospital. There is a notion of super-cospital support and, roughly speaking, in other words, this theorem amounts to saying that for representation of an inner form of GLN the super-cospital support is unique, up to, to conjugate is unique. And so this allows one, so to decompose, here FL bar G decomposes into Bernstein blocks. So this looks very much the same as in the complex case, but be careful. This theorem, which holds for GLN and its inner form, so that is the uniqueness of the super-cospital support, or at least up to inertia, in the modular setting it does not hold in general for a general periodic group. Uniqueness of inertia, even up to inertia. Uniqueness of super-cospital support does not hold for L-modular representation of periodic reductive groups in general. And there are explicit counter-examples. And in order to find a counter-example, no need to look for, to look at an exotic exceptional periodic group. It is enough to consider, look at the symplectic group SP8F with L dividing Q squared plus 1, Q being the cardinality of the residue field of F. And for such a symplectic group, there is an example of a hospital representation having two different non-equivalent super-cospital supports. So this already, the counter-example already works, already shows up in the finite field case. It has been observed by Olivier Duda. And it has been lifted from the finite field case to the periodic case by Jean-François Dutt. And as for the finite case, one can define linked complex representations of GLMT. So as for the finite field case, one can define linked representations in the set of complex irreducible representations of G. One has to be slightly more careful when manipulating lattices, but it works. Okay, so we have a notion of being linked for two complex irreducible representations of our periodic group G. And this is an equivalence relation. So same question as in the finite field case, we have an equivalence relation. What are the equivalence classes? So now this is interesting that this equivalence relation is no longer a trivial equivalence relation. And so we have the following theorem. So I will state it for discrete series representation, because this is the case which is interesting when I would talk about the local Jacket-Longlant correspondence later. But there is a more general statement for any irreducible representation, which requires slightly more notation. So suppose pi and pi prime are complex discrete series representations of G. Then pi and pi prime are linked if and only if they have the same underclass. So in other words, the equivalence classes for this equivalence relation, at least when you look at discrete series representation, the equivalence classes are parametrized by underclasses. So there is an easy way. If two representations are one linked, they clearly have the same underclass. And so two linked representations must have the same underclass. The non-obvious and unexpected way is that they converse holds. What does this have to do with our initial problem? So this theorem is relevant to our initial problem because of the following result due to Minger's and I. So let pi and pi prime as usual be complex discrete series representations of G. Then pi prime are linked if and only if the same holds for the Jack-Longlant's transcripts. So of course the first step is to prove this for one linked and by transitivity it propagates to two linked. So this is what I called in the introduction the congruence properties of the Jack-Longlant's, the local Jack-Longlant's correspondence. Now stay speaking, the local Jack-Longlant's correspondence preserves the relation of congruence. Mod any prime number different from P. Ok, so what does it tell you? If you put together this theorem with this one, it tells you that given any discrete series representation of G the endoclass of the Jack-Longlant's transfer of pi only does not really depend on pi, but it only depends on the endoclass of pi. And even better in some sense, in some sense it does not depend on the integer M. So what does that mean? It means that, so first given, so instead of starting with a discrete series representation I start with an endoclass given theta and endoclass and pi. Try any discrete series representation of GLM having endoclass of endoclass theta for any M. Then the endoclass of the Jack-Longlant's transfer of pi only depends on the theta you started with and I denoted theta jl. Second part, so what does it give you? So you get a map which turns out, which one can prove is bijective. A bijective map theta goes to theta jl from the set of endoclasses over F to itself. And so one wants to prove that this bijective map is the identity. And so in order to compute theta jl for any theta, the idea is that it suffices to compute the endoclass of pi jl for a well-chosen pi of endoclass theta for some well-chosen GLM d. Or more precisely for GLM d for some well-chosen M. You want to be able to vary M but not the division algebra. The division algebra remains constant. So well-chosen, what does this mean? I can be more precise, much less than 10 minutes. So well-chosen, what does it mean? So this means first that given an endoclass theta, choose pi to be a hospital representation of GLM d with endoclass theta. And such that the Jacqueline-Gland's transfer of pi is also hospital. And here, so for some M. For the moment I do not specify M. For the moment M is arbitrary. It is completely arbitrary because it has to be large enough so that such a hospital representation pi exists. But if M is large enough, such a pi always exists. So pi hospital such that its Jacqueline-Gland's transfer is also hospital, it is important. So associated with pi, there is a finite un-ramified extension k over f of degree t, t of pi, the torsion number of pi, that is the number of un-ramified characters such that pi twisted by k remains isomorphic to pi. And I assume that I have fixed an embedding of f-algebras of this f-extension k into the simple algebra mm of d and also into mmd of f. Such embeddings are uniquely determined up to conjugacy. Now consider the following diagram. So this is the map I have defined earlier somewhere here. And you also have the set E of k of endoclassies over k. And, of course, you also have Jacqueline-Gland's map. So I will denote it gl sub k and this one gl gl sub f. And there is a vertical map which is subjective which has been defined by Bushnell and Egnard and which is called the restriction map, restriction from k to f. So res k and f is the Bushnell and Egnard restriction map. So from there, following Bushnell and Egnard's analysis of the trace of the hospital representation pi at certain well-chosen elliptic regular conjugacy classes of G, glmd in terms of types, type theory. One can prove that theta jl can be computed as follows. You look at the following diagram there. It can be computed as the restriction of psi, so here this is jlf. The restriction of psi jlk for any endoclass psi above theta. So here you start with the theta here. You have theta, it's transfer theta jl there. You choose any lift psi of theta and theta jl is given by doing this. And now it remains to explain what well-chosen m means. So first, let rho be a hospital representation of endoclass psi of jk, the centralizer of k times in G. Such that its transfer rho jlk is a hospital by definition of endoclass psi jlk of the group hk, which is the centralizer of k times in h. And now Dotto's idea is to use the flexibility we have on the choice of m to choose m large enough. So large enough I think that D divides m should be enough. Such that the centralizers Gk and hk are isomorphic. And if they are isomorphic, then the Jacqueline-Glantz correspondence between these two isomorphic in our forms should be trivial. And if the Jacqueline-Glantz correspondence between these two is trivial, it does nothing at the level of endoclasses. So it is possible to fix an isomorphism to identify these two centralizers such that the Jacqueline-Glantz transfer of rho is equal to rho, which implies at the level of endoclasses that psi jlk is equal to psi. And if you apply this formula, this implies that theta jl is equal to theta. And we are done. Thank you. Is it difficult to say what link means in general for arbitrary deducing representations? Ah, in terms of endoclasses you mean? No, I don't know. I don't know. Yes, so you have to, in order to generalize the theorem I've stated for discrete series representation to any reducible representation, you just need to slightly generalize the notion of endoclass to define a semi-simple endoclass. So, roughly speaking, if you start with an irreducible representation, you look at its cuspidal support and its semi-simple endoclass will be a linear combination of the endoclasses of the cuspidal representations occurring in the cuspidal support with some weight. So, no, it is not difficult to generalize to any reducible representation. Could you please explain a bit more about the counter-example for the non-uniqueness of the cuspidal support? I cannot explain, but I can give more detail. The proof is that you look at tables giving the decomposition matrices of some principle in composable modules. And if you look at the tables carefully, you will see that you have a counter-example. But I can be more precise. So, for sp8q when l divides q squared plus 1, there is a cuspidal irreducible representation by occurring as a component of... So, first, it occurs in the parabolic induction, for this one I call it g. It occurs in the parabolic induction of the trivial character of the split torus fq times the 4 in g. And it also occurs as a component of the following parabolically induced representation, and lg. So, here l is the levy sp4 times gl1 twice. And the representation you induce is theta10, which is a cuspidal representation of sp4. Theta10 times trivial times trivial. And so you see that the levy subgroups you induce from are far from being conjugated. And I can say no more. Some tables have been computed by Olivier Juda, I think. And if you interpret them in terms of what supercuspidal support means, he tells you that. So, being linked makes sense for any other reductive group. Do you know what is their relation with gas? No, because in order to define being linked, we need to know that Aries Chandra equivalence is an equivalence relation in the modular setting. But, yes, it is for a general finite reductive group. It is a natural question. How should the definition of being linked, how should it be modified so that it makes sense? And so that it is interesting.