Congruence properties of endoclasses...
Sécherre, Vincent
Congruence properties of endoclasses and the local Jacquet-Langlands correspondence Consider the local Jacquet-Langlands correspondence between the discrete ...
 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 Jacques Langlance 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-archimedern 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 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. DC of G is the discrete series of G. So this is the set up to the set of isomorphism classes of essentially a square integral smooth complex representations of the group G. And there is a canonical bijection, the Jacques Langlance correspondence between DC of G and DC of H which I will denote like this with a superscript GL. And this correspondence is characterized by a character identity at elliptic regular conjugacy classes in G and H. What does an essentially square integral representation looks like more precisely? So in the 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 row of GL M over R of D such that pi is the unique irreducible quotient of the following parabolically induced representation. So quotient of the parabolic induction to G of row tensor, row twisted by the unhammified character given by the absolute value of the reduced norm to the power of some integer S of row depending on row tensor and R times and the last factor is row twisted by the absolute value of the reduced norm to the power of S of row times R minus 1. So where S of row, so it is a positive integer attached to row 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 row is always equal to 1 so you do not see it. So this gives you, so in this, so R and row are uniquely determined by pi and so in this situation I will write pi equals delta of row R and I will also write S of pi equals S of row. So this classification of discrete series representation goes back to Bernstein and Zalilinsky in the split case where G is split and to Tadich in the case of an inner form. So now look at the Jaqueline-Glans 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 some T dividing N and sigma a cospital representation of GN over T of F. And so we have two pairs, we have row 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. G is equal to R times S of P and then natural problem is what is the relation between row and sigma or compare row and sigma. How can one compare row and sigma? For this we use Bushnell, so there is a description of Caspital representations of GLN and GN given by compact induction which is given by Bushnell-Kutzko's theory of types. So for H, for GLNF this goes back to Bushnell and Kutzko and the same works for inner forms of GLNF, it generalizes this description of Caspital representations given by Bushnell-Kutzko, it has been generalized by Roussous, Grabitz, Schoen-Stevens and that. In particular, I will not say anything specific about type theory, about this type theory, I think description of Caspital representations. In the previous talk Bushnell introduced the notion of endoclass and so to any Caspital representation of GLNF one can associate an endoclass and the same work for inner forms of GLN to any Caspital 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 Caspital representations but for different groups which do not even need to be inner forms of each other. So how can one compare them? The interest of looking at the endoclass is that the endoclasses are the set of all endoclasses is the same for all general linear groups GLN and their inner forms. So one can ask the more precise, more specific question compare the endoclass of Rho and 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 it's the proof of this result appeals unexpectedly to the theory of modular representations of GLN and its inner forms as well as congruence properties of the local Jacqueline-Glerns 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 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 Jacqueline-Glerns transfer. And so I would like to explain 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 complex representations of G are A rich Sandra equivalent we'll write just HT equivalent if they both occur as components of the same parabolically induced representation and yes 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 Sandra series. So if you prefer two irreducible representations of G are A rich Sandra equivalent if and only if they have the same Cuspidal support. So now replace a fixed prime number L different from P. So fix a prime L different from P unlike yesterday afternoon. And consider rather than complex representation consider representations with coefficients in FL bars given algebraic closure of finite field of characteristic L. So in the setting of model representations you still have a rich Sandra induction or parabolic induction. And so the notion the definition of Irish Sandra equivalence still 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 Irish Sandra equivalence on the set of FL bar irreducible representations of G is an equivalence relation. And since you have an equivalence relation you can this decomposes this decomposes the set of mod L irreducible representations of G into again Irish Sandra series. So now consider a latic representations of G. So representations with coefficients in QL bar a fixed algebraic closure of the field of latic numbers. So consider pi a QL bar representation of G on a 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 finite length FL bar representation. So this is a finite length FL bar representation of G. Such a 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 and so it is denoted RL of pi which is by definition the reduction mod L of pi. 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 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 A richand raceries. The two remarks, first remark this definition does not depend on the choice of the field isomorphism. And 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 weaker than both being in the same A richand raceries of here C, G and being congruent mod L. Being congruent mod L meaning that the reductions mod L are equal 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 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 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. So we need two irreducible representations of G are linked. So this is not very exciting. We have defined an equivalence relation that it turns out to be a trivial equivalence relation. So we need 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 am 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 JLN group. So of course we have normalized a parabolic induction. And instead of a Richandar equivalence, we can define Bernstein equivalence. So two irreducible representations of G are linked. And the other irreducible representations are Bernstein equivalent if pi occurs as a component of some parabolically induced representation from some Levy M to G of some irreducible representation 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 the equation 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. We still have normalized parabolic induction. So we can extend this definition of Bernstein equivalence. And the theorem which in the Pellic case, which corresponds to the deeper and James theorem in the finite field case, is the following one, which is due to Veneras in the 1G split and is extended to the non-split case by Bertow-Mingus and Psi. 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-cuspidal support. So be careful because in the modular setting there is a difference between hospital and super-cuspidal. There is a notion of super-cuspidal support and, roughly speaking, in other words, this theorem amounts to saying that for representation of an inner form of GLN, the super-cuspidal support is unique. So this allows one 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-cuspidal support, or at least up to inertia, in the modular setting, it does not hold in general for a general periodic group. So be careful. Uniqueness of inertial, even up to inertia, uniqueness of super-cuspidal 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 at an exotic, exceptional periodic group. It is enough to consider the simplicity group SP8F with L dividing Q squared plus 1, Q being the cardinality of the residue field of F. And for such a simplicity group, there is an example of a capital representation having two different non-equivalent super-cuspidal 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 Duda. Okay. And as for the finite case, one can define linked complex representations of G. 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 the 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. 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 for the, 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 endoclass. 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 endoclasses. So there is an easy way. When, if two representations are one linked, they clearly have the same endoclass. And so two linked representations must have the same endoclass. The non-obvious and unexpected way is that the converse holds. Okay. What does this have to do with our initial problem? So this, this is, this theorem is relevant to our initial problem because of the following result due to Mingas and I. So let pi and pi prime as usual be complex discrete series representations of G. Then pi, pi prime are linked if and only if the same holds for the Jacqueline-Glerns' transfers. So of course, the first step is to prove this for one linked, but then by transitivity, it propagates to, to, to linked. And so this is what I called in the introduction the congruence properties of the Jacqueline-Glerns, the local Jacqueline-Glerns correspondence. How's this speaking? The local Jacqueline-Glerns correspondence preserves the relation of congruence, mod any prime number different from p. Okay. 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 in the class of the Jacqueline-Glerns transfer of pi only does not really depend on pi, but it only depends on the end of class of pi. And even better, in some sense, in some sense, it does not depend on the integer n. So what does that mean? It means that, so first, given, so instead of starting with a discrete series representation, I start with an underclass given theta and underclass and pi any discrete series representation of glmd having underclass of underclass theta for any m. Then the underclass of the Jacqueline-Glerns transfer of pi only depends on the theta you started with. And I denoted theta jl. And 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 underclasses 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 underclass of pi jl for a well-chosen pi of underclass. Theta for some well-chosen glmd. Or more precisely, for glmd 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. Not sure I have 10 minutes. So well-chosen, what does it mean? So this means first that given an underclass theta form, choose pi to be a cuspidal representation of glmd with underclass theta. And such that the jacqueline glmd transfer of pi is also cuspidal. And you see here, so for some m, for the moment I do not specify m. For the moment m is arbitrary, but not completely arbitrary because it has to be large enough so that such a cuspidal representation pi exists. But if m is large enough, such a pi always exists. So pi cuspidal, such that its jacqueline glmd transfer is also cuspidal, it is important. So associated with pi, there is a finite unhammified extension k over f of degree t, t of pi, the torsion number of pi, that is the number of unhammified characters such that pi twisted by kai 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. And 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 endoclases over k. And of course you also have a jacqueline glmd 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 Enyard and which is called the restriction map, restriction from k to f. So res k and f is the Bushnell and Enyard restriction map. So from there, following Bushnell and Enyard's analysis of the trace of the hospital representation pi at a certain well-chosen elliptic regular conjugacy classes of glmd in terms of types, theory, one can prove that theta jl can be computed as follows. You look at the following diagram there, 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 theta jl. So this is jk, the centralizer of k times in g, such that its transfer rho jlk is a hospital of 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 and to choose m 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 Longland's correspondence between these two isomorphic in our forms, should be trivial. And if the Jacqueline Longland's correspondence between these two is trivial, it does nothing at the level of endoclassies. So more precisely, so it is possible to identify these, to fix an isomorphism, to identify these two centralizers such that the Jacqueline Longland's transfer of rho is equal to rho, which implies at the level of endoclassies that psi jlk is equal to psi. And if you apply this formula, if you apply this formula, this implies that theta jl is equal to theta. And we are done. Thank you. Any questions? Is it difficult to say what link means in general? Yeah. General or arbitrary? In terms of endoclassies, you mean? No, 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, how to 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 endoclassies 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 quantum example for the non-uniqueness of the cuspidal support? I cannot explain, but I can give more detail. Because the proof is that you look at tables giving the decomposition matrices of some principle incomposable 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 square plus 1, there is a cuspidal irreducible representation pi 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-storus 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 theta 10, which is a super-cuspidal representation of sp4 theta 10 times trivial times trivial. And so you see that the levy subgroups you induce from are far from being conjugate. And I can say no more. Yeah, so you have, some tables have been computed by Olivier Juda, I think. And if you interpret them in terms of what super-cuspidal support means, it tells you that. So, being linked makes sense for any other group. Do you know what is the relation it has? No, because in order to define that being linked or when linked, we need to know that ari-chandra equivalence is an equivalence relation in the modular setting. But yes, it is a, 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. Okay, then let's thank you.
 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 Jacques Langlance 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-archimedern 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 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.  DC of G is the discrete series of G. So this is the set up to the set of isomorphism classes  of essentially a square integral smooth complex representations of the group G.  And there is a canonical bijection, the Jacques Langlance correspondence between DC of G and  DC of H which I will denote like this with a superscript GL.  And this correspondence is characterized by a character identity at elliptic regular  conjugacy classes in G and H.  What does an essentially square integral representation looks like more precisely?  So in the 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 row of  GL M over R of D such that pi is the unique irreducible quotient of the following parabolically  induced representation.  So quotient of the parabolic induction to G of row tensor, row twisted by the unhammified  character given by the absolute value of the reduced norm to the power of some integer  S of row depending on row tensor and R times and the last factor is row twisted by the  absolute value of the reduced norm to the power of S of row times R minus 1.  So where S of row, so it is a positive integer attached to row 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  row is always equal to 1 so you do not see it.  So this gives you, so in this, so R and row are uniquely determined by pi and so in this  situation I will write pi equals delta of row R and I will also write S of pi equals  S of row.  So this classification of discrete series representation goes back to Bernstein and Zalilinsky in the  split case where G is split and to Tadich in the case of an inner form.  So now look at the Jaqueline-Glans 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 some T dividing N and sigma a cospital representation  of GN over T of F.  And so we have two pairs, we have row 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.  G is equal to R times S of P and then natural problem is what is the relation between row  and sigma or compare row and sigma.  How can one compare row and sigma?  For this we use Bushnell, so there is a description of Caspital representations of GLN and GN  given by compact induction which is given by Bushnell-Kutzko's theory of types.  So for H, for GLNF this goes back to Bushnell and Kutzko and the same works for inner forms  of GLNF, it generalizes this description of Caspital representations given by Bushnell-Kutzko,  it has been generalized by Roussous, Grabitz, Schoen-Stevens and that.  In particular, I will not say anything specific about type theory, about this type theory,  I think description of Caspital representations.  In the previous talk Bushnell introduced the notion of endoclass and so to any Caspital  representation of GLNF one can associate an endoclass and the same work for inner forms  of GLN to any Caspital 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 Caspital  representations but for different groups which do not even need to be inner forms of each  other.  So how can one compare them?  The interest of looking at the endoclass is that the endoclasses are the set of all endoclasses  is the same for all general linear groups GLN and their inner forms.  So one can ask the more precise, more specific question compare the endoclass of Rho and  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 it's the proof of this result appeals unexpectedly to the theory of modular representations  of GLN and its inner forms as well as congruence properties of the local Jacqueline-Glerns  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 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 Jacqueline-Glerns  transfer.  And so I would like to explain 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 complex representations of G are A rich Sandra equivalent  we'll write just HT equivalent if they both occur as components of the same parabolically  induced representation and yes 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 Sandra series.  So if you prefer two irreducible representations of G are A rich Sandra equivalent if and only  if they have the same Cuspidal support.  So now replace a fixed prime number L different from P. So fix a prime L different from P  unlike yesterday afternoon.  And consider rather than complex representation consider representations with coefficients  in FL bars given algebraic closure of finite field of characteristic L.  So in the setting of model representations you still have a rich Sandra induction or parabolic  induction.  And so the notion the definition of Irish Sandra equivalence still 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 Irish Sandra equivalence on the set of  FL bar irreducible representations of G is an equivalence relation.  And since you have an equivalence relation you can this decomposes this decomposes the  set of mod L irreducible representations of G into again Irish Sandra series.  So now consider a latic representations of G. So representations with coefficients in  QL bar a fixed algebraic closure of the field of latic numbers.  So consider pi a QL bar representation of G on a 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 finite  length FL bar representation.  So this is a finite length FL bar representation of G. Such a 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 and so it is denoted RL of pi which is by  definition the reduction mod L of pi.  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 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 A richand raceries.  The two remarks, first remark this definition does not depend on the choice of the field  isomorphism. And 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 weaker than both being in the same A richand raceries of here C, G and being  congruent mod L. Being congruent mod L meaning that the reductions mod L are equal 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 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 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.  So we need two irreducible representations of G are linked. So this is not very exciting.  We have defined an equivalence relation that it turns out to be a trivial equivalence relation.  So we need 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 am 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 JLN group. So  of course we have normalized a parabolic induction. And instead of a Richandar equivalence, we  can define Bernstein equivalence. So two irreducible representations of G are linked.  And the other irreducible representations are Bernstein equivalent if pi occurs as a component  of some parabolically induced representation from some Levy M to G of some irreducible  representation 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  the equation 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. We still have normalized parabolic induction. So we can extend this definition  of Bernstein equivalence. And the theorem which in the Pellic case, which corresponds  to the deeper and James theorem in the finite field case, is the following one, which is  due to Veneras in the 1G split and is extended to the non-split case by Bertow-Mingus and  Psi. 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-cuspidal support. So be careful because in the modular setting there  is a difference between hospital and super-cuspidal. There is a notion of super-cuspidal support  and, roughly speaking, in other words, this theorem amounts to saying that for representation  of an inner form of GLN, the super-cuspidal support is unique. So this allows one 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-cuspidal support, or at least up to inertia, in the  modular setting, it does not hold in general for a general periodic group. So be careful.  Uniqueness of inertial, even up to inertia, uniqueness of super-cuspidal 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 at  an exotic, exceptional periodic group. It is enough to consider the simplicity group  SP8F with L dividing Q squared plus 1, Q being the cardinality of the residue field of F.  And for such a simplicity group, there is an example of a capital representation having  two different non-equivalent super-cuspidal 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  Duda. Okay. And as for the finite case, one can define linked complex representations  of G. 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 the 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. 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  for the, 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 endoclass. 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 endoclasses. So  there is an easy way. When, if two representations are one linked, they clearly have the same  endoclass. And so two linked representations must have the same endoclass. The non-obvious  and unexpected way is that the converse holds. Okay. What does this have to do with our initial  problem? So this, this is, this theorem is relevant to our initial problem because of  the following result due to Mingas and I. So let pi and pi prime as usual be complex discrete  series representations of G. Then pi, pi prime are linked if and only if the same holds for  the Jacqueline-Glerns' transfers. So of course, the first step is to prove this for one linked,  but then by transitivity, it propagates to, to, to linked. And so this is what I called  in the introduction the congruence properties of the Jacqueline-Glerns, the local Jacqueline-Glerns  correspondence. How's this speaking? The local Jacqueline-Glerns correspondence preserves  the relation of congruence, mod any prime number different from p. Okay. 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 in the class of the Jacqueline-Glerns transfer  of pi only does not really depend on pi, but it only depends on the end of class of pi.  And even better, in some sense, in some sense, it does not depend on the integer n. So what  does that mean? It means that, so first, given, so instead of starting with a discrete series  representation, I start with an underclass given theta and underclass and pi any discrete  series representation of glmd having underclass of underclass theta for any m. Then the underclass  of the Jacqueline-Glerns transfer of pi only depends on the theta you started with. And  I denoted theta jl. And 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 underclasses 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 underclass of pi jl for a well-chosen pi of underclass.  Theta for some well-chosen glmd. Or more precisely, for glmd 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. Not sure I have 10 minutes.  So well-chosen, what does it mean? So this means first that given an underclass theta  form, choose pi to be a cuspidal representation of glmd with underclass theta. And such that  the jacqueline glmd transfer of pi is also cuspidal. And you see here, so for some m,  for the moment I do not specify m. For the moment m is arbitrary, but not completely  arbitrary because it has to be large enough so that such a cuspidal representation pi  exists. But if m is large enough, such a pi always exists. So pi cuspidal, such that its  jacqueline glmd transfer is also cuspidal, it is important. So associated with pi, there  is a finite unhammified extension k over f of degree t, t of pi, the torsion number  of pi, that is the number of unhammified characters such that pi twisted by kai 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. And 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 endoclases over k.  And of course you also have a jacqueline glmd 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 Enyard and which is called the restriction map, restriction from k to  f. So res k and f is the Bushnell and Enyard restriction map.  So from there, following Bushnell and Enyard's analysis of the trace of the hospital representation  pi at a certain well-chosen elliptic regular conjugacy classes of glmd in terms of types,  theory, one can prove that theta jl can be computed as follows. You look at the following  diagram there, 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 theta jl.  So this is jk, the centralizer of k times in g, such that its transfer rho jlk is a  hospital of 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 and to choose m  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  Longland's correspondence between these two isomorphic in our forms, should be trivial.  And if the Jacqueline Longland's correspondence between these two is trivial, it does nothing  at the level of endoclassies. So more precisely, so it is possible to identify these, to fix  an isomorphism, to identify these two centralizers such that the Jacqueline Longland's transfer  of rho is equal to rho, which implies at the level of endoclassies that psi jlk is equal  to psi. And if you apply this formula, if you apply this formula, this implies that  theta jl is equal to theta. And we are done. Thank you.  Any questions? Is it difficult to say what link means in general?  Yeah. General or arbitrary?  In terms of endoclassies, you mean? No, 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, how to 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 endoclassies 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 quantum example for the non-uniqueness of the cuspidal support?  I cannot explain, but I can give more detail. Because the proof is that you look at tables  giving the decomposition matrices of some principle incomposable 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 square plus  1, there is a cuspidal irreducible representation pi 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-storus 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 theta 10, which is a super-cuspidal  representation of sp4 theta 10 times trivial times trivial. And so you see that the levy  subgroups you induce from are far from being conjugate. And I can say no more. Yeah, so  you have, some tables have been computed by Olivier Juda, I think. And if you interpret  them in terms of what super-cuspidal support means, it tells you that.  So, being linked makes sense for any other group. Do you know what is the relation it  has? No, because in order to define that being linked or when linked, we need to know that  ari-chandra equivalence is an equivalence relation in the modular setting. But yes,  it is a, 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.  Okay, then let's thank you.
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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 Jacques Langlance 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-archimedern 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 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. DC of G is the discrete series of G. So this is the set up to the set of isomorphism classes of essentially a square integral smooth complex representations of the group G. And there is a canonical bijection, the Jacques Langlance correspondence between DC of G and DC of H which I will denote like this with a superscript GL. And this correspondence is characterized by a character identity at elliptic regular conjugacy classes in G and H. What does an essentially square integral representation looks like more precisely? So in the 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 row of GL M over R of D such that pi is the unique irreducible quotient of the following parabolically induced representation. So quotient of the parabolic induction to G of row tensor, row twisted by the unhammified character given by the absolute value of the reduced norm to the power of some integer S of row depending on row tensor and R times and the last factor is row twisted by the absolute value of the reduced norm to the power of S of row times R minus 1. So where S of row, so it is a positive integer attached to row 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 row is always equal to 1 so you do not see it. So this gives you, so in this, so R and row are uniquely determined by pi and so in this situation I will write pi equals delta of row R and I will also write S of pi equals S of row. So this classification of discrete series representation goes back to Bernstein and Zalilinsky in the split case where G is split and to Tadich in the case of an inner form. So now look at the Jaqueline-Glans 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 some T dividing N and sigma a cospital representation of GN over T of F. And so we have two pairs, we have row 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. G is equal to R times S of P and then natural problem is what is the relation between row and sigma or compare row and sigma. How can one compare row and sigma? For this we use Bushnell, so there is a description of Caspital representations of GLN and GN given by compact induction which is given by Bushnell-Kutzko's theory of types. So for H, for GLNF this goes back to Bushnell and Kutzko and the same works for inner forms of GLNF, it generalizes this description of Caspital representations given by Bushnell-Kutzko, it has been generalized by Roussous, Grabitz, Schoen-Stevens and that. In particular, I will not say anything specific about type theory, about this type theory, I think description of Caspital representations. In the previous talk Bushnell introduced the notion of endoclass and so to any Caspital representation of GLNF one can associate an endoclass and the same work for inner forms of GLN to any Caspital 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 Caspital representations but for different groups which do not even need to be inner forms of each other. So how can one compare them? The interest of looking at the endoclass is that the endoclasses are the set of all endoclasses is the same for all general linear groups GLN and their inner forms. So one can ask the more precise, more specific question compare the endoclass of Rho and 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 it's the proof of this result appeals unexpectedly to the theory of modular representations of GLN and its inner forms as well as congruence properties of the local Jacqueline-Glerns 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 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 Jacqueline-Glerns transfer. And so I would like to explain 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 complex representations of G are A rich Sandra equivalent we'll write just HT equivalent if they both occur as components of the same parabolically induced representation and yes 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 Sandra series. So if you prefer two irreducible representations of G are A rich Sandra equivalent if and only if they have the same Cuspidal support. So now replace a fixed prime number L different from P. So fix a prime L different from P unlike yesterday afternoon. And consider rather than complex representation consider representations with coefficients in FL bars given algebraic closure of finite field of characteristic L. So in the setting of model representations you still have a rich Sandra induction or parabolic induction. And so the notion the definition of Irish Sandra equivalence still 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 Irish Sandra equivalence on the set of FL bar irreducible representations of G is an equivalence relation. And since you have an equivalence relation you can this decomposes this decomposes the set of mod L irreducible representations of G into again Irish Sandra series. So now consider a latic representations of G. So representations with coefficients in QL bar a fixed algebraic closure of the field of latic numbers. So consider pi a QL bar representation of G on a 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 finite length FL bar representation. So this is a finite length FL bar representation of G. Such a 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 and so it is denoted RL of pi which is by definition the reduction mod L of pi. 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 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 A richand raceries. The two remarks, first remark this definition does not depend on the choice of the field isomorphism. And 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 weaker than both being in the same A richand raceries of here C, G and being congruent mod L. Being congruent mod L meaning that the reductions mod L are equal 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 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 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. So we need two irreducible representations of G are linked. So this is not very exciting. We have defined an equivalence relation that it turns out to be a trivial equivalence relation. So we need 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 am 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 JLN group. So of course we have normalized a parabolic induction. And instead of a Richandar equivalence, we can define Bernstein equivalence. So two irreducible representations of G are linked. And the other irreducible representations are Bernstein equivalent if pi occurs as a component of some parabolically induced representation from some Levy M to G of some irreducible representation 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 the equation 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. We still have normalized parabolic induction. So we can extend this definition of Bernstein equivalence. And the theorem which in the Pellic case, which corresponds to the deeper and James theorem in the finite field case, is the following one, which is due to Veneras in the 1G split and is extended to the non-split case by Bertow-Mingus and Psi. 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-cuspidal support. So be careful because in the modular setting there is a difference between hospital and super-cuspidal. There is a notion of super-cuspidal support and, roughly speaking, in other words, this theorem amounts to saying that for representation of an inner form of GLN, the super-cuspidal support is unique. So this allows one 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-cuspidal support, or at least up to inertia, in the modular setting, it does not hold in general for a general periodic group. So be careful. Uniqueness of inertial, even up to inertia, uniqueness of super-cuspidal 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 at an exotic, exceptional periodic group. It is enough to consider the simplicity group SP8F with L dividing Q squared plus 1, Q being the cardinality of the residue field of F. And for such a simplicity group, there is an example of a capital representation having two different non-equivalent super-cuspidal 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 Duda. Okay. And as for the finite case, one can define linked complex representations of G. 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 the 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. 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 for the, 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 endoclass. 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 endoclasses. So there is an easy way. When, if two representations are one linked, they clearly have the same endoclass. And so two linked representations must have the same endoclass. The non-obvious and unexpected way is that the converse holds. Okay. What does this have to do with our initial problem? So this, this is, this theorem is relevant to our initial problem because of the following result due to Mingas and I. So let pi and pi prime as usual be complex discrete series representations of G. Then pi, pi prime are linked if and only if the same holds for the Jacqueline-Glerns' transfers. So of course, the first step is to prove this for one linked, but then by transitivity, it propagates to, to, to linked. And so this is what I called in the introduction the congruence properties of the Jacqueline-Glerns, the local Jacqueline-Glerns correspondence. How's this speaking? The local Jacqueline-Glerns correspondence preserves the relation of congruence, mod any prime number different from p. Okay. 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 in the class of the Jacqueline-Glerns transfer of pi only does not really depend on pi, but it only depends on the end of class of pi. And even better, in some sense, in some sense, it does not depend on the integer n. So what does that mean? It means that, so first, given, so instead of starting with a discrete series representation, I start with an underclass given theta and underclass and pi any discrete series representation of glmd having underclass of underclass theta for any m. Then the underclass of the Jacqueline-Glerns transfer of pi only depends on the theta you started with. And I denoted theta jl. And 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 underclasses 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 underclass of pi jl for a well-chosen pi of underclass. Theta for some well-chosen glmd. Or more precisely, for glmd 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. Not sure I have 10 minutes. So well-chosen, what does it mean? So this means first that given an underclass theta form, choose pi to be a cuspidal representation of glmd with underclass theta. And such that the jacqueline glmd transfer of pi is also cuspidal. And you see here, so for some m, for the moment I do not specify m. For the moment m is arbitrary, but not completely arbitrary because it has to be large enough so that such a cuspidal representation pi exists. But if m is large enough, such a pi always exists. So pi cuspidal, such that its jacqueline glmd transfer is also cuspidal, it is important. So associated with pi, there is a finite unhammified extension k over f of degree t, t of pi, the torsion number of pi, that is the number of unhammified characters such that pi twisted by kai 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. And 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 endoclases over k. And of course you also have a jacqueline glmd 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 Enyard and which is called the restriction map, restriction from k to f. So res k and f is the Bushnell and Enyard restriction map. So from there, following Bushnell and Enyard's analysis of the trace of the hospital representation pi at a certain well-chosen elliptic regular conjugacy classes of glmd in terms of types, theory, one can prove that theta jl can be computed as follows. You look at the following diagram there, 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 theta jl. So this is jk, the centralizer of k times in g, such that its transfer rho jlk is a hospital of 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 and to choose m 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 Longland's correspondence between these two isomorphic in our forms, should be trivial. And if the Jacqueline Longland's correspondence between these two is trivial, it does nothing at the level of endoclassies. So more precisely, so it is possible to identify these, to fix an isomorphism, to identify these two centralizers such that the Jacqueline Longland's transfer of rho is equal to rho, which implies at the level of endoclassies that psi jlk is equal to psi. And if you apply this formula, if you apply this formula, this implies that theta jl is equal to theta. And we are done. Thank you. Any questions? Is it difficult to say what link means in general? Yeah. General or arbitrary? In terms of endoclassies, you mean? No, 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, how to 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 endoclassies 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 quantum example for the non-uniqueness of the cuspidal support? I cannot explain, but I can give more detail. Because the proof is that you look at tables giving the decomposition matrices of some principle incomposable 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 square plus 1, there is a cuspidal irreducible representation pi 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-storus 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 theta 10, which is a super-cuspidal representation of sp4 theta 10 times trivial times trivial. And so you see that the levy subgroups you induce from are far from being conjugate. And I can say no more. Yeah, so you have, some tables have been computed by Olivier Juda, I think. And if you interpret them in terms of what super-cuspidal support means, it tells you that. So, being linked makes sense for any other group. Do you know what is the relation it has? No, because in order to define that being linked or when linked, we need to know that ari-chandra equivalence is an equivalence relation in the modular setting. But yes, it is a, 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. Okay, then let's thank you.
well as congruence properties of the local Jacqueline Longland's correspondence between