 I'd like to thank the organizer for this very nice invitation. So I am very, it's a great pleasure and honor to talk to the ear. So a few memory, personal memory. So I arrived in Paris in 1983 as a student of ENSGF, JF, so for girls. And I have to get registered in some university in Paris. So I went to Paris 7, and I registered as a student of the license and the maitris at the same time, it was the rule of ENS. And I needed to choose some courses. So as a license courses among the list of possible courses, there was one that I took, which was integration by Roger, sorry, and a second course in the level of maitris, which has the funny name of Analyzes III, and also by Godement. And this course was about differential geometry. And both courses were really great, so I learned a lot from them. And so in integration, it was about the big measure. And the theory of measure. And in the second course, I still remember very well Godement on the blackboard. I explained him to us how to define cards. So in French it's cards, cards, cards. And this was very useful for his course. And also for the course on algebraic geometry, I took the semester, the next semester with Le Potit. So and then I passed my aggregation, and the year after, I tried to find an advisor to do my, at that time it was called DEA, Diplomedétude à Profondi, which corresponds now to master degree. And at ENSGF, so Cèvres in Mont-Rouge, there were some mathematicians who had offices there. And among them, there was Marie-France Vigneras. And so she agreed to be my master advisor. And she gave me a paper of Roger Howe on how correspondence for final groups. So it was my first, the first time I met the world of reductive group. So I remember the date, the preside date when she gave me that paper. It was on the 14th of July, 1984. So it was very strange to me that mathematicians are working on the 14th of July. But I was still used to some strange thing already, because Godman was giving his integration course at lunchtime, and the other courses, and that history was at dinner time. And so Marie-France told me that there was a seminar in Jussieu, and that I will need to go at that seminar, but I will understand probably nothing. But the music will be useful for my ear. And she was very strict about it. She said, it is imperative to go. And so on next fall, I went to this seminar, which the title of this conference is taken from. And I remember the third lecture, and it was really great. And the third lecture at the seminar was given by Corinne Blondel, and I was very, very lucky because it was on the paper I have to study as my master degree. And in these days, the seminar was something like a usual kind of seminar. I mean, invited people, visitors or French people to explain her work, but also doing like a running seminar, so studying some difficult subjects which change each year. And that year, it was on the data correspondence over periodic fields. And the seminar was also a good time to learn in a more deeply way one subject. And also, many times, it provides books after the seminar by some of the speakers. And sometimes we're just reading books, and I remember one summer in the beginning when I was reading the Goethe-Mont-Jacques. And I took it with me from my holidays and I was on the beach with Goethe-Mont-Jacques. And because I had to prepare a talk for the seminar on one chapter of it. And so just a few written for you. So this book, which is well known now, it's a reference about data correspondence from the, which was written after the seminar. Another book, by Mughlin and Vaspurgé, which was quoted by Jim Arthur yesterday. And a small story about that book. So as you see, there is an English version. And I met the person who was doing the tradition in English, and Les Lèche-Neps, and at that time, and she told me the following story. So she started first by doing it by automatic way. So she takes the book and just to use some automatic way to translate. And then a new English word show up. And do you, I guess you have no idea what this name, this word. So many adverbs in French, like probablement are translated in English, are probably. And the new word, whichever, was godly. So this is just a story. So it is, but it's good. It's a book which is a paraphrase of a scripture. So godly is good to pure in it. And then some other book from the time of a seminar, especially this one, which is a project started by Michael, and many other French people, not any French writer, but many articles written by French people. OK, just one thing before I go to my talk. June 1987, do you remember what, in that year, there was a special event in Paris. I wonder if someone remembers what it is. Yes, Luminé, you are right. Luminé was part of a special period, period special, which gives also very good books in hysterics. But this year is very important for me because at that time, during the special period, I met for the first time many of you, but in June, it was in Paris. And it was not really included in the special, but it was an event which was closely related, but did run. And in that event, I remember a very nice talk by Hervé Jacquet about, was about one theorem of Valls-Purget, and some of that talk. And so I give you the title of this event because nobody remembers, unfortunately. But for me, it was good because it was the first time I was attended to a conference. So it's probably why I remember. OK, good morning. Ah, yeah. OK. Too long, too long time ago. So now I'm going to my over here. So I will talk about joint work with Ahmed Mosaoui and Martin and Martin Solleveld. And so it's about N-hands, long-distance parameter. So I will take a group G. G will be the F-points of redactives. Connected at a bright group defined over F. As usual, F is a local non-alchymedian film. And the long-distance parameter, or L parameter, is a certain morphism from, so I will take this as a, this group as a version of the Vadorian group in the L group. And for the version of the L group, I will take this one where G, my notation is the following. So GF is a redactive group over C with root datum, dual to that of G. So the situation is well known for all of you. Just a small example. If G is GLN, F, this one is same group. If G is, for instance, if it is SLN, this one will be PGL. And here it is SP. And of course, if the Piardet group is SP to N, you will get SO to N plus 1. If this is SO to N, you will get SO to N over C. And this parameter is supposed to have, I assume to have some good property that the usual one that you can find in Corvallis, in the paper by Boren in Corvallis, for instance. So, what do I want to say? So this is the parameter. And I am especially interested of a notion, I would like to define a notion, first, a notion of hospitality. So as usual, this is, I will talk on objects like this as objects living on the dual side and the automorphic side, which is just the representation side, just the side of representation of G, a smooth representation of G. So, and of course the idea is that this hospitality on this object on the dual side should correspond to the usual notion of hospitality, of hospitality or super-hospitality at the level of representation. And of course we, you know the, it occurs for instance in the talk of Bertrand Le Maire, for instance. You have, it was mentioned, the Bernstein center of the Bernstein decomposition, which is a decomposition of a representation of, a smooth representation of G due to Bernstein as a product of certain, certain full subcategory and these subcategories are indexed, so Bernstein decomposition. So this is the category of smooth representation. You can do it as a product on some element of some category that I will call R as G of G. And this element S are given by, actually I think L will be a levy of G, a levy of a parabolic, of a rational levy of a rational parabolic. And sigma is a irreducible super-hospital, super-hospital representation of L. And this relation is slightly weaker than the conjugate, the G conjugation. We allow to twist by an unamplified character of a levy. So if you believe in local and non-correspondence, which is known to be true in many cases, like for G, equal to GLNF due to the work of Ari Santelor and also never proved by Enyar and thought-proof by Schultz. It's also known for classical groups, thanks to the book by James Archer. And if we add the stabilization, which was conditional to the stabilization of the local transformer, which is now done in this book. So as I said, just a few lectures, but if you come from a number of pages, you will see why there are some things like this. So if you believe in local and non-correspondence, the irreducible representation, I will call this, first this will give you a partition of, as the level of irreducible representation of the set of the smooth dual of G as it will be partitioned, the smooth dual of G in some subset. And I will call this Bernstein series. So this has nothing to do with local and non-correspondence. But now, if you believe in local and non-correspondence, this set should be in bijection in a nice way with certain objects attached to the dual side. And so it means that you should also have a decomposition similar in the dual, on the dual side. And so the idea was to try to define such a partition directly on the dual side without assuming local and non-correspondence. But here, this is just not possible for the reason that to an L parameter, local and non-correspondence should attach an L packet of irreducible representation of G. And in a given L packet, it happens many often that you have simultaneously irreducible super-cospital representation and non-super-cospital representation. Of course, all these non-super-cospital representation need to be discrete. Discrete series because the parameter to have super-cospital in its packet need to be discrete. And that kind of situation already arrived in SP4 or G2 example. So you have a packet of irreducible representation. So it's when G is this group, which contains super-cospital and non-super-cospital. And for instance, we can have a packet which contains, I will say, representation with unipotent reduction. That are with some are super-cospital and some are in the principal series. And so principal series will mean that in the Berthin decomposition, they will occur in the Berthin series attached to a tolus. So the levy L will be a tolus. And the super-cospital occurs in the Berthin series attached to a levy L which is equal to G. So when you look at this and this, it seems that this is such a stupid thing to do because this decomposition is not compatible with local unknowns. In fact, you will see at the end of the talk that it is some course of the composition is in fact compatible. So you cannot define cospitality for an L-parameter. We will turn to the notion of N-amps in French Enrichie, L-parameter. Some people say that the L-parameter is the name of the object and the enhancement will be its... The L-parameter is the last name and the enhancement will be the surname. So in any case, it will be pairs like this where c-phi is an L-parameter and rho is an irreducible representation of some group that I will call s-phi. So here is a definition, so s-phi. So we use the following construction which is due to Jim Arthur. But this construction is compatible also with the one given by Tashokareta. And the service, as you can imagine, this group is the component group of some group and the group has this fancy notation which we'll explain now. So we are on the dual side, so we have the long-long dual of G and this is its adjoint group, so the quotient of the group by its center. And this notation is for the simply connected cover of this one. And we have a long-long parameter. And this group, so this, this one, yeah, this is the definition, it's the centralizer in G-add of the image of phi and the image of phi just means this. And here, this one with one is just the inverse image of this group. So this is taken for the paper by Arthur and a note on the help group. And so we take this group and, of course, the group in general, for GLN it is connected, but in general, no, even no, sorry, in general this group is not connected, as I defined it, and I take, I call S phi its group of component and so the parameter is, and by an irreducible representation of that group. And this kind of situation occurs in a special case, for instance, for representation which, which I admit E-warifix vector, which are a special case of the representation with unipotent reduction. And this occurs in many years ago in an old paper by Lustig on square integrable representations. Okay. So this could be the end of the story, but in fact, the point is for us, the key point is that this group has another interpretation. So I have considered the group, oh, sorry, I need the things to, between the two. Okay, this group, I define, this group, I define it as a centralizer in this group of the anti-image of phi. And then I can also consider not this group, but just the phi group, and it's what I will do now. So I will introduce... This phi is finite. S phi is finite, yes. And I will introduce this group, same notation, the center of Sg of phi of wf. So I just ask that this group only request that the element of the group centralize the image of wf. And so in this group, and also I have this group, and I will also introduce some unipotent element, which is the one you can imagine. So the image by phi, because this, if I do that, I will only consider the restriction of phi to wf, so I need to do something with SL2c part. So I will just consider the image of this element. And this is a unipotent element, and it can be viewed as a unipotent element of g phi. And so we have a group, and this one is not finite in general, and in general it is disconnected, not necessarily connected. And okay, I have this, and sorry, what I call it, g phi, sorry, it's not a unipotent element, it's a unipotent element of a dual group of L of g. And I will consider the group, the group I will call g phi, sorry, is just the component group of this one. And the point is that it is easy to check that, in fact, s phi is... No, no, it's supposed to be right. Sorry, I'm just confusing my notation. This is not the component group. I was right in the beginning, sorry. This is this group, g is an unipotent of a g, which can be viewed as an element of g phi. And the group of component, sorry, it's the group that I call to be more precise, I should call it, i g phi of u phi. So in g phi, I take the group which is... So this is g phi. This is not this, sorry. Excuse me. So we have g phi, so we are in g phi. I can consider the centralizer of... in g phi of u phi. So g phi maybe is possibly disconnected and if you take a unipotent element in it, it could be even more disconnected. And so you take this group and the point is that this group is isomorphic. So it's a consequence of Jacobson, Morozov, and all of them, of course, final groups. So the idea to look at this group, g phi and that group, i phi is due to my student, Ahmed Musawi, and sorry, it was part of his thesis. And the point is that now you are of a complex redactive group, g phi, possibly disconnected and the unipotent element in it and you are looking at irreduci... and rho, which was an irreducible representation of s phi. You can view it as an irreducible representation of that group. So you have a complex league group maybe disconnected, unipotent element in it and a representation of that. And when you see this, you think of springer correspondence or more precisely it's generalized version due to lipstick. Yeah, by reconstruction it was irreducible representation of s phi but when you view s phi as this group you can view it as a representation of that. And so now you can say but you can define a notion of hospitality. This is cuspidol f. Of course we want that phi is discrete. So its image is not contained in a proper levy subgroup of the L group and also a condition on rho and the condition on rho is just that this with the above notation is cuspidol per in this group g phi. So I will not have time to explain what it means. So this notion is due to lipstick in the case when this group is connected. So it is in this paper in Inventionist I think, yeah. This paper about intersection homology complexes. It is the paper in which it defines generalized springer correspondence. And in fact it is possible to look to view the representation rho as some irreducible local system and there is a notion of cuspidol f for local system which involves to look at some perverships on the unipotron variety of the group g phi and on this notion of perversive you have a parabolic induction and Jacky restriction which are just, I mean, of course it's different way of defining them but cuspidality is the same definition as you know the object which are Jacky restriction when you apply the Jacky restriction it is equal to zero if a levy is a proper levy of a group. So it's something similar which that lipstick has done in this setting and this also has a nice property that it also work for L modular representation and this is due to Acha, Anderson, Gito and Rich. Okay, anyway, you have cuspidol pass but since I will not give you the definition I decided to give you at least two examples. So first example is for this group so that you already saw in Vincent Cechard talk so same notation this is a division D is a division algebra so object that you already were familiar with thanks to the Godman-Jacky book so it's a division algebra so the dimension over with center f and so the dimension of D over f is a square and the condition m with our called md so if I impose this condition all this group are in a form of gln over f and for later use I will define so this group are not split and even more they are all non-quasi-split except in the case where D is 1 and in that case we have a quasi-split form which is gln which in fact is split and sorry this is the example now the dual of gln is maybe I should first give you this yes which is well known that all the inner form are the same long-length group which is the long-length groups of the inner it's given by the other quasi-split group in the inner form of the class and of course it's simply connected cover is sln and the group here if you compute it it's a cyclic group and it's n z mod zn now if your parameter has to be discreet so it's the example of hospital and as long as parameter because the fee discreet it has to be the following form so here this is an irreducible representation of the vague group and this one is an irreducible representation of sl2c and we need a name for this dimension we call it d' and then we can check that so it is written in our first paper which is now published in the manuscript which says that this is hospital if and only if of course we take this fee fee as above we need a condition about the objects so the left common multiple of d and n mod d' is equal to n and of course if you look so this is the condition that you get if you just compute hospital pairs and of course if you look at long nose correspondent look at long nose correspondent for g so we attach which is known to be true we attach to your parameter some representation 5p irreducible representation of your group and in general of course even if you take fee discreet long nose parameter this representation is not always supercospital but pi fee is supercospital if and only if the same condition so you can find it for in when f is zero characteristic in the book by Delene Cashdown and Vigneras on the Bernstein center where you also can find the Bernstein center so this means in particular that there is a nice compatibility between the hospitality on the dual side and the hospitality on the group side in this special case a second case that I would like to give to you is so for classical group sp and so so even you can look in the paper by Ahmed which is now in publishing representation theory so another very similar case which is the case of a unitary group given the quasi-split you should make quasi-split and E is a quadratic extension of course of f VL group is just GLN over C some indirect product with a Galois group of E now this is in a slightly nicer way so my group G5 so my group G5 what is this I was following form it's a product of M sigma M sigma E prime some orthogonal group so if you take an arbitrary long lens not M sigma an arbitrary L parameter you will get the group G5 will be a product of that kind of group and times a product of some group of GLN type but because you assume you're considering a hospital L parameter so f has to be discrete then there is no GL part and of course now we compute the group and this group is of course it's a product similar with some similar group in M sigma C of some so my unipotent element I wrote it as a product of unipotent element depending to this decomposition and of course the group is a product of group and SP and so what I need and so U is a product of U sigma and the condition to get cuspidality so in a lipstick paper you also find that if a pair is cuspidal the unipotent element needs to be distinguished which in this case gives you that so the unipotent so U sigma is a unipotent element in this orthogonal group so it corresponds to a partition of M sigma and the partition you have in this situation of cuspidality is the following it has the following form and so this is when sigma is in I plan so I take just I plan and we can present what means this index it means that the corresponding parameter is of orthogonal type or simplistic type but we don't want to enter in this and here it is the following partition 2, 4 2d sigma and so the group are just Z mod 2z Z mod 2z and now so the group I forgot about the orthogonal or simplistic group this group is a product it's a product of Z mod 2z so it is exactly like this so this sigma is the dimension in that way in that way that u sigma is an element of the corresponding orthogonal or simplistic group my notation are not good but anyway I would like to be going too fast so I need to see pair so Z to a minus 1 and this is Z mod 2z so it's a product of copies of copies of Z mod 2z and in SP is similar I would like to have this fancy things I could have just write it Z a but the reason which appears soon I would prefer to write it like this okay yeah and this is but then and my this is the point I will define rho and rho sorry rho is an irreducible representation of group A g phi of u phi so it's a product a tensor product of representation of all these groups and rho is a tensor product of representation that I would call rho sigma for sigma in i prime cup i prime prime and I need to define what are the rho sigma and so rho sigma is defined okay so now it's a non-trivial character of a two-element group in each time I mean it's a rho sigma on the z z i is minus 1 at some power of i but okay at some at some power at some okay to be i or it could be written in more properly and if you do this in a more proper way then you will see so you can explain more precisely what is i depending if I take 2a or 2a minus 1 and the point is that we have the point is we can choose we can compute and we find what we have to put here and then you will get this is the point that epsilon sigma of za minus 2 is always equal is always equal to a minus 1 and if you look at this condition and especially this one these are exactly the condition which are funded by Collette Mughlin in his work about the long-nose parameter for unitary groups and the description of a superacospital and so the condition and you will get that the condition you have on the unipotent element and the condition of this character correspond exactly to the fact that the Jordan set.5 which occurs in Collette and Marco work for instance or in other paper by Collette Mughlin this is as no all and this is the condition that for the character how to be alternated and so it gives a bijection between between cospital pairs and the superacospital representation of quasi-split unitary groups and so the conjecture in generalis is just that these two kind of objects correspond by local long-nose correspondence and then I will pass quickly to the other things so yes oh yes it's a very good question so it's because I'm okay it's it's hidden in the picture but I am in fact considering at the same time all the inner twist of a given group yes and the other inner form of it's why it's not equal to one and in fact yes we can if you want to look to a given group and not to all these inner twist you need to take not all the character of belonging to S5 so first we need to only consider the phi parameter which are relevant for the given group and there is also a notion of relevance for the and we just take the one which are relevant for G and so the yes because this is just using a codevitz we use the codevitz isomorphism and then you know the action of rho on some some group which is containing in S5 and you just take the one which and this group and the action of that group should correspond to the codevitz isomorphism to the this form of inner twist of a group I mean this is okay but it's just not have time just to really go into it but just say yeah just to answer to your question so we have two group we have this group so just to answer to that specific question so this group is just the group which occur if you look at the group which is assumed to be quasi-split then we have this group sorry the center of group and you consider have a connected component of that group and then we have a central X and then S5 is a central extension of some group of that group by S by some group Z so we have this exact short exact sequence and five and one and where this group is just the centralizer of the center sorry of that group mod its intersection with the connected component of the center of the image of phi and then when you have this you look at the restriction of your irreducible character who of S5 to that group and from that group using Godwitz isomorphism you can understand which you can get a character and then there's something to do but it's and this is the same this one so if your group is quasi-split this is enough to consider but if your group is not you need to consider this okay and then okay I will just analyze this and the conjecture is thus there is a bijection between what I will call phi of Lg which is this will be G conjugacy classes of n-n's l-parameter and here I will add cusp to mean that I will only consider the cuspidal 1 and the union of irreducible cuspidal representation of G alpha for all G alpha in such that G alpha is for all the inert all G alpha inert twist of G and if you take here of course in here you can take the one which are G relevant for some G then this one should correspond to here cusp of the same corresponding G if you consider the element here which are G relevant for given G we conjecture that we are in bijection with a super cuspidal irreducible representation of G so as I told you this conjecture is true for quasi-splitonatory group of course for GLN for SLN of the division of the trivlar for classical split classical groups and in that case is due to Ahmed and it's also true for representation with unipotent reduction of any PID group which is split which became split on a unharmified an unharmified extension of F and this is a recent paper in the archive by a student of Eric Obdam Obdam and Martin Sodeveld and of course when when G is simple of adjoint type simple adjoint type this follow follow from realistic classification of representation with unipotent reduction of such kind of group so I guess my time is finished what you say yeah so I would just just two minutes so this is the conjecture then from from that what we can do is we we write on this so then if you take all the for say for 1G the set of Anas-Langlance parameter first it is inside the theorem so Musaoui-Soleveld it's a Gs so we we of course we have this notion of hospitality now we can take as a cuspidal pair so Levy now this one is a cuspidal is an Anas-Langlance parameter for L and the relation is similar as in Bernstein so we have an action of some object which reflect the which correspond by Levy-Langlance correspondence for Chorus 2 of the Anamifile character and we can similarly to Bernstein define this kind of inertial relation on this object so this is the first thing so it's kind of avatar of a Bernstein decomposition but only at the level of irreducible object and the second thing is that this is in bijection with a simple module of a twisted because I do not have time to explain what it is twisted a fine so say Gs I would just say that the root system so when you have a fine algebra there is some root system attached to a final group so it is correspond to the root system of some group some final group and I will not have time to explain what it is and so we have this and the last word of course this decomposition is not the analog on the dual side of a Bernstein decomposition but now if you consider not as I said in the beginning an L packet is not an union of Bernstein component and the Bernstein component is not an union of L packet but from this it's very easy by considering the stable Bernstein center an union of L packet which in this terminal in this setting here it means to considering some union of of this subset and so there is a well defined union of that subset and so we can have a course decomposition of this as some I phi where this one union of this object and so this give a way to link this kind of decomposition not in L packet but in union of L packet and this union of L packet is the one which occurs in the stable Bernstein center so I will stop here Do you have any questions? Yes I have two questions one is just about notation previously the input of phi was the dual group of the complex numbers L, G Now you write G and index is about G check Are you alright? You are perfectly right It's not this It's LL Thank you It's LL It's L It's this And you said you have two questions The other question is is there a natural topology on this? I have no idea I would be very happy to have any insight on this kind of things At the moment it's just set It's just a set of So it's why it's not as nice as a Bernstein decomposition because it's not the decomposition of it's just at the moment you see a visible object but I guess there should be something but it's unclear to me I want to I want to correspond with the modules of the Hecker That's a a bijection, is there more than one way to normalize the bijection? Okay So the definition of a bijection is very complicated and it's completely based on the Lustig construction in the first graded of the Hecker algebra so it passed through that and also the Hecker algebra we get depends on indeterminate and at the moment there are I mean there are too much freedom on the indeterminate I mean we can choose several kind of we can take just one and determinate but if you do that you have the right algebra because in the Piad case we can have different and so but at the moment so it's not the definition is too as to freedom in it but when you specialize for example for classical groups we can for quasi-split symplectic you know for gonadal group we have check using the work of Volcker-Hierman and also the work of Colin Mughlin they correspond to the there is a canonical choice of indeterminate in order to get an algebra which is the right one but I think there is something to do still like I mean to input like L-functions or something like that in the picture which to make it more canonical so in this in this question there is no multiplicity for in a form which appeared all appear with only ones the inner twist appear so it means that sometimes we have two inner forms which correspond to different inner twist so this is the trick the two inner forms can appear can appear, yes