 Merci beaucoup. Je suis la dernière voix et je suis supposé de donner la parole à la Commissaire organisée. Je suis supposé d'être un membre, mais en fait je n'ai pas fait tout. J'ai une meilleure commission, une position pour la Commissaire organisée, au lieu de l'audience. Je vais aussi remercier le Conseil européen pour sa soutien. Je remercie l'EHS, je remercie le directeur Emmanuel Ulmo, et je remercie aussi le staff de l'EHS, Elisabeth Jasseron, le secrétaire, Laurence Boparin, qui est en charge de l'article, et qui a donné la direction de Marvel pour trouver l'Ormai, le staff de l'EHS. La nourriture apparaît en magique, la main-garde est arrêtée, les microphones sont offertées, et les microphones travaillent. Ok, donc nous allons le dire. Ok, maintenant, La Baisse a donné un overview du travail de Gaudemont. Donc je ne vais pas vraiment dupliquer ses efforts, et je vais choisir dans le travail de Gaudemont ce que je pense est important, ou ce qui est important pour moi. Donc, première, Gaudemont a commencé par ce qu'on appelle l'abstract non sens, et cet exemple était apparemment assez motivé. On a pris un groupe G avec le Borel, un groupe A&N. Ensuite, vous avez la représentation de l'L2G, de l'L2G mod N. Et puis pour chaque caractère du split torus A, un caractère module 1, vous avez la représentation d'induce de B de G, induite par Foucaille. Et l'L2G7 est... Cette représentation est directe, continue directe de ces représentations qui, en général, sont irréduciables. Donc, selon Gaudemont, cet exemple très simple était une grande aide en formulant la notion d'une continue de représentation unitaire, et en arrivant à la théorème, qui n'est pas très difficile, mais que toute une représentation unitaire de groupes locales, séprables, qui admettent une représentation comptable, toute représentation unitaire est un nombre continuous d'irréduciables. Maintenant, à Gaudemont, et j'espère pour cette théorème, il a pensé, pour exemple, que par appliquer pour dire L2 de GA GQ G réductif de Q et A, c'est une grande représentation. Bien sûr, on peut décomposer à un nombre irréduciable, mais il espère que ça donnerait information sur la série d'agestants. En fait, il a proposé ça pour moi comme un sujet pour une thesis. Ouf, ouf. Vous devriez aussi mentionner GALFUND Absolument. GALFUND & COMPANY Oui, oui. Ok, donc c'était l'une de la première chose que Gaudemont a regardé. En fait, Gaudemont, n'a pas prouvé cette théorème, mais il a sauté cette théorème sur son travail. En même temps, j'ai trouvé cette théorème abstraite parfois utile. Pour exemple, nous allons dans une situation locale avec G-split et le subgroup de maximum unipotent et le caractère générique et le subgroup de maximum unipotent le caractère générique et le subgroup de maximum unipotent et la position générale. Nous pouvons, bien sûr, essayer de décomposer ça explicitement, mais juste savoir que ça peut être fait est parfois utile. Donc, la théorème abstraite de temps en temps utile. Maintenant, pour aller à quelque chose à concrète plus concrète, dans le quartier seminar Gaudemont a donné une série de paroles et en particulier j'ai donné une parole des formes hospitales mais dans le contexte du subgroup et des formes homomorphiques donc il a introduit dans ce contexte la notion de formes hospitales et il a prouvé que cet espace pour un groupe donné et le fait qu'il soit finalement dimensionnel et pour faire ça, il utilise un très très bon lemma donc l'espace locale mu abondé sur l'espace et nous pouvons considérer l2 de l'excès et de l'espace close et chaque membre est abondé abondé, bien sûr essentiellement abondé Et puis, le lemma donc, la dimension de cet espace est fanatique donc c'est un lemma, qui est un monde très bon et qui peut aussi être utilisé pour prouver la dimensionnelle finale du espace, bien sûr Ok, donc c'était cette théorique sur la forme de casque, c'était un exemple de quelque concrète, vous voyez, une autre constribution, c'était plus ou moins expository, on va retourner au SL2R et peut-être au SL2Z ou dans un groupe concrète. Et nous considérons une forme automatique de SL2R, mais nous considérons une forme, dans le sens de Farid Chandra, c'est-à-dire que la forme est transformée par un caractère de SL2R et c'est une fonction d'organisation de l'opérateur de Casimir et d'une certaine condition de la forme de SL2R. Donc il a eu un cours, probablement l'économie de cet objectif et le cours, les notes du cours sont élevées par Christian Housel et la main chose que j'ai élevée par ceci est que dans la série 4e de la forme de la forme de la série 4e, il s'agit d'une fonction de l'économie, mais la fonction de l'économie classique, donc quelque sorte d'une fonction spéciale. Ils satisfaient la question fonctionnelle en regardant WSNW-SN, mais en tout cas, c'est là où j'ai trouvé l'idée de ma thesis. Par contre, dans son interview avec Laurent Closel, il peut-être dire quelque chose qui n'est pas assez précis, à commencer avec, il n'a pas suggéré ce problème pour moi. Et second, après ma thesis, Langlands m'a demandé de faire un report et Langlands wrote the report, send the letter to Gronmo, and the letter contains something like three or four new ideas, uniqueness of the Whitaker model, the castle man's suggestion of the castle man's shalaka formula, and the machinery of what's called now the Shaidi Langlands method. Langlands Shady method, okay. And it was not at all saying that my thesis was no good, but simply when I had to work much more. Okay, so that's something. And in the same vein, Gaudmo wrote a little note, which was never published, about what could call the Whitaker function for a cover of SL2R, or even for the universal cover. So the analogous Whitaker function for cover of SL2R, and the function are expressed in term of hyper geometric functions. So it's a little note that was never published. Okay, now as Labès told us, Gaudmo pushed very much for the idyllic point of view in the study of automorphic forms, and the point of view of representations. So I won't go over the whole list of papers he published, but there is, of course, the paper reduction theory in idyllic language. So Borrell and Ari Chandra established the reduction theory for real groups with discrete subgroup, arithmetic district subgroup. Gaudmo et Vaye did something similar for the idyllic group, for the reductive group, the discrete subgroup being now the group of rational points. Naturally, his report of Langlance, so he tried to understand the monster paper of Langlance on the Einstein series, and did lecture on it at Bobackey. So he presented some of the idea, and this was presented as the decomposition of L2G mod GQ into a sum or continuous sum of unitary irreducible representation. So that was his formulation of Langlance paper. I remember the lecture at Bobackey, and Gaudmo was jumping up and down, very enthusiastic report. No more elementary. Gaudmo has again, I guess it's before the report of Langlance paper, in a paper called Analyse spectrale des fonctions modulaires, a complete decomposition of the representation of SL2A on L2SL2A modulo SL2Q. And it's a very simple case, but this paper is quite interesting. If you read carefully, you see that he knew about normalizing intertwining operators, and all the things that I thought was important in this paper was the following. In dealing with Einstein series for GL2, you have to consider in a sense a fiber bundle of function transforming this way under the group of triangular matrices. And he used this formula, where he is a Schwarz-Bruyer function, to publish, to produce such sections. And it looks like nothing, but I use this construction over and over and over again. Okay. Now, so all this is public, but there is some secret contribution. First, he had in jadellic context, he knew how to, to analytically continue the Einstein series for the Borel chain split group. It's an argument which is actually standard, we know it for GL2 and we extend, or SL2, and we extend this to GLN by using Hartog's lemma. Another secret contribution is this. Let N be an important group of a Q, say. So we can consider N2 of NA modulo NQ. So it's a compact quotient, so it decomposes into a discrete sum of irreducible representation with finite multiplicity. But in fact, the multiplicity is one. And one can describe explicitly the representation which appear. This was proved first by Calvin Moore by a rather elaborate construction. And Godman found a very simple proof using only the Poisson summation formula on the Lie algebra. And then, the trace formula. So that gives really a very sleek proof, but unfortunately it was never published. Now, Godman was also very interested by the work of Wey, on what we know, we call now the Converse Theorem. So, Wey considers some Dirichlet series. It's twisted by Dirichlet character, sufficiently many Dirichlet characters. He assumes those Dirichlet series twisted at some appropriate analytic property. And then he concluded that sigma n e to the 2i pi nz is an automorphic form. So Godman saw that and gave a lecture, perhaps at the College de France, where he gave an Adélie interpretation. Immediately, he gave a little lecture. And also he wrote a letter to Wey. Wey had a work of a queue. And Godman was producing this similarity over q of i. And in, so it's done again in the style of Wey, not at all like in Jacqueline Glence. But not the less, for me, he used my thesis and the integral representation of the Wittaker function that I obtained in this work. And later, as you know, after the book of Jacqueline Glence was finished, he wrote an introduction to it. And I have not read it, of course. But I understand it's quite useful. But now we come to perhaps the most important contribution of Godman to the theory of modular form is a zeta function of simple algebra. So first, remember, he gives some lecture on test disease. And that's where I learned test disease and the use of Ibel Abel. And then also, what? I cannot hear you. Where did he give his thesis? In Paris, but maybe I think it was at the university, some course. And also, he gave some lectures on the zeta function for a division algebra. He has two lectures in Bourbaki on this subject. And I'm afraid those lectures are not very good, the kind of incorrect. But they go nowhere. But the course he gave was much more focused. And in Jacqueline Glence, we use this, and the converse theorem to go from automorphic form for a quaternion algebra to quaternion automorphic form on GL2. Now I come, really, to the main subject. Zeta, I don't, OK, I write Godman Jacqueline and function. No, it's a subject on which there is a huge, huge literature, either on the local problem, the global problem. They was very interested, Tamagawa. There's really a lot of articles, but some more, they don't get it, don't quite. And one reason is that you cannot obtain what you want without using representation, representation of local group, Adèle, Idèle, infinite dimensional unitary representation, and so forth. Now Godman gave a lot of lectures here and there about this subject. He wrote a letter to Elias Stein, I wrote this letter. And out of them, I wrote the book with Godman, from cover to cover, signed him the notes, and said you have 15 days to make your comments. And of course, I did not expect an answer. So let me just review some of this material. So let's discuss the local theory. And for now, let F be local, but non-archimédiaire. So the notion of representation we have to use is annucible, irreducible representations. And in the sense, Godman failed to get the neural theory because he tried to stay with unitary representations. So let Pi be such a representation. Then here, we have the ingredient. F is a matrix coefficient of Pi. Phi is a Schwarz-Bürer function, the space of n-bien matrices. And you want to consider those zeta integral, not the normalization, s plus n minus 1 over 2. So when the real part of s is sufficiently large, the integral converges. And then it represents a rational function of q to the minus s. q is, of course, the cardinality of the residual field. And in a precise way, zeta integral belongs to Nidil, generated by a certain element, Lspi. And Lspi is a sum of such functions. So Lspi is completely determined by this condition. We normalize it, of course, by demanding that it's an inverse of a polynomial with p0 equal 1. So this describes the behavior of an analytic behavior of those integrals. There is a functional equation of the form I wrote here. fsh is, of course, f of g inverse. And the gamma factor, which appears a functional equation, has this form where epsilon is a monomial. So you want to prove the existence of such an L and such a gamma. Well, you start by the cospital case. Be careful that for N equal 1, any character is to be considered a cospital, because the cospitality condition is empty. So you have to prove this for pi cospital. For N equal 1, it's, of course, state. And for N larger than 1, you can prove it also, essentially using the fact then that the matrix coefficients are compactly supported by the other center. So once you have it for the cospital representation, you are almost free, home free, because you have this little lemma. C4 pi is a component of an induced representation induced by irreducible representation pi 1, pi 2, pi r, for which we know the theorem. Then the theorem is true for any component of the induced representation, any irreducible component. Furthermore, gamma spi is a product of the gamma spi i. For the N factor, it's a little more complicated. L spi, pi being a irreducible component, is not necessarily equal to the product of the L spi i. It belongs to the ideal generated by them. In other words, it's this product times the polynomial. But since any irreducible representation is a component of something induced by supercospital value of the theorem. So it's very simple. But we need, there is a little thing we don't know yet, is to compute or to compute exactly the L factor. This doesn't quite give enough information. So you have to observe the following. Suppose pi is stamped, or suppose pi, let's say, pi square integrable. By that, I mean the central character is unitary, and the coefficient are square integrable modulo center, or, in fact, pi-tamped. Then zeta phi-androïque fs converges for real part of s positive. It's something you have to prove. Goodman proves that for pi square integrable. You may as well do it for pi-tamped, and you need the following. So this function on the group on GLM belongs to the arishandra space, the space of rapidly decreasing function in the sense of arishandra, if real part of s is positive. Therefore, when pi is stamped, this implies the integral converges for real part of s positive, and in particular, then, ls pi is allomorphic for real part of s positive. Il is not quite compactly supported, modulo center. No, no, no, not necessarily. It could be the special representation. And in fact, it's for the special representation that we have to. What is a test function? Phi is a Schwarz function. It's Schwarz on the space of additive matrices. Not on the group. Not on the group. No, the same is true for pi tilde, the contra-agrédient representation. Did I say that pi tilde is a contra-agrédient representation? No, but no, I am telling you. OK. And therefore, if you come to this fraction, a pole of ls pi cannot be a pole of 1 minus s pi tilde, the pole of this joint. This means this fraction is irreducible. And therefore, if you know the gamma factor, when pi is stamped, you know the L factor. OK, so we have solved the problem of pi tampered, ls pi for pi tampered. Now, we have a new ingredient. Suppose we are in the situation of Langland's quotient. This means pi 1, pi 2, pi r are tampered representation, but twisted by appropriate real powers determinant. And suppose, I mean, then we have a Langland's quotient. And if pi is this Langland's quotient, we have to have more information. We have to prove, as this is a separate proof, that ls pi is then equal to exactly the product of ls pi r. So now, we have finished proving our theorem. And we have completely proved, completely determined the l factor. And the only thing, I mean, everything is reduced to compute the epsilon factor for the supercospital, which is, of course, another problem. So this is a local situation for F, not Archimedean, for F real or complex. In the initial note, I discuss, I use only Ari-Chandra module, GK module. Now, it's more convenient to represent for any irreducible GK module by its canonical model, in the sense of Castleman and Wallach. So now, all the representation of this type, those representation are parametrized by representation, semi-sample representation of the vague group, which is not a big thing in this case. So it's more convenient to define ls pi and epsilon ls pi as a l factor of the representation of the vague group. We, yeah, it's just this little vague group. And same thing for the epsilon. And then, you can prove easily. It's not very hard, something similar. Naturally, the statement ls pi times polynomial must be replaced by the following assertion. Allomorphic multiple of ls pi bounded at infinity vertical strip, vertical strip of finite width, and product by any polynomial at the same property. And then, you have a functional equation. And you have to prove, again, that ls pi itself is a sum of such integral. It's a polynomial in S, or in Q, at the top of S? No, no, now we are over R or C. So it's a polynomial in S. I am not sure, because I did not prove it, but I think a function with this property should be equal to such an integral, or perhaps a sum of such integral. OK, so now we have finished the local theory. I forgot to say what happened in Andro-Hemmify's situation, but you can guess what it is. And now I say a word about the global theory. Now we take a pi, which is hospital automorphic. If you wish, you could think of pi as a unitary irreducible representation. And it's not hard to see, I guess, I am over the group of Adèle of a number field F. It's not hard, in any case, to prove that a unitary irreducible representation of such a group is a tensor product and the appropriate sense of irreducible ones. I think Godmore, his note on Jacques Langland, prouve it for an admissible unitary representation, but if you work a little harder, you can prove for any unitary irreducible. OK, so anyway, we have the global integral. Now phi is a function, trans-breath function, on the space of n by n matrices. F is a matrix coefficient of pi, and we want to consider this integral. And of course, what we want to have is that we have this identity, where phi hat is a Fourier transform, additive Fourier transform on the space of n by n matrices. And so a lot of people worked on that. And everybody knew you have to use Poisson's formula, Poisson's summation formula. But somehow you have to get rid of the singular matrices. So people try all kinds of crazy schemes, like using functions which vanishes at singular matrices, but for Fourier transform vanish also at singular matrices. But Godmore saw this is not needed. So you must explain why you can ignore the singular term in the Poisson's summation formula. The term like phi of 0 is because an integral like this on a quotient by the center or something, is automatically 0. Not because of the cuspidality. No, no, it's not necessary. It's simply because this is true for any representation. If it were non 0, your representation would be, which is irreducible, would be the trivial representation. So this kind of thing, we can ignore this. It's 0. Except for n equal to 1. No, no, no, no. If you take a chi, which is not the trivial character. Yes, but 1 is cuspidal in this case. Yeah. And no, 1 is, that's the only one which is not cuspidal. OK. And now you have to explain. How did you get the multiplicative measure? Yeah, it's multiplicative. Now you must explain why you can ignore the non 0 matrices but which are not invertible. It's simply for the following observation. Take the matrices of time m. So it's a single orbit of matrices like this. The group GLN operating by left and right multiplication. And the stabilizer in this action, the stabilizer of this element, contains the unipotent radical of a parabolic subgroup. And then if you do the computation, this shows those terms disappear. You can use the Poisson's summation formula without the singular terms. So what this gives you is this, which is defined by a convergent integral for a part of s sufficiently large. Extend to an entire function of s and we have this functional equation. Combining that with local theory, we have immediately the equivalent assertion that Ls pi is epsilon s pi. So it's a very, very simple theory, really. Two rebarks. When we wrote Jacquet-Langlance, we use rather the AECA method to obtain this functional analytic continuation. And Langlance said, but me, for GLN, this AECA method is not going to work. You can also use something like AECA method and use Fourier coefficient of cosponses. It's a little more complicated. And Arish Chandra, when I finish writing this set, how come you are doing something which applies only to GLN? Well, now a word about... Arish Chandra knew only SL2 or the World General Code. Arish Chandra knew SL2 or the World General Code. That's the big thing. Yeah, that's true. Now a word about the non... What do we do for non-casform? So if Pi is an automorphic representation but not Caspidol, then we did not know it at the time. Pi is a component of an induced representation. This is a globally induced. We are Pi 1 Pi R, a Caspidol automorphic representation but eventually switched by power of the determinant. This is a theorem of Langlance. This means that Lspi is essentially equal to Lspi 1 Lspi R. It's different only at a finite number of places and in particular we also know that Lspi is biomorphic with the same functional equation. So we don't have to do anything more for the other automorphic forms. This is a theory for... for German. It's quite a complete theory. It's not very difficult. Now let me just say a word. Let's see, I have ten minutes. A lot of people are trying to extend Godman Jaquet to other group so that Arish Chandra will be shown to be wrong. And I want just to show you a very modest contribution. I am on GL2 cross GL2 over a number field, now over a local field and I'd like to understand the ranking Selberg, the gamma factor of the ranking Selberg L function, which is defined intrinsically in terms of Medicare function. I do it only for Pi 1, Pi 2 tamper. Now this has a property. If you twist Pi 1 by chi and Pi 2 by chi inverse, you don't change the gamma factor. This means you must replace GL2 cross GL2 by the set, the group of pairs, G1, G2 with the same determinant. And we want to use Planchial formula for this group G. There is a little subtlety that we need. Suppose sigma is an irreducible representation of G contained in Pi 1, Pi 2, transfer product of Pi 1 and Pi 2 restricted to this group G. Of course, Pi 1 and Pi 2 are not unique, so you could have also sigma restricted to sigma content in the restriction of Pi 1 prime, Pi 2 prime. That could happen. But then you can show that you have a relation like this and therefore the gamma factor as I said. Now, using Planchial formulae on the group G, one can show the following. There is a map of the Ari-Chandra space to itself with a property sorry, there is an F here. So E of phi is in the Schwarz space, the Ari-Chandra space and F1, F2, a matrix coefficient of Pi 1, Pi 2 tamper respectively. So, there is such a map. It's well defined and the problem now is to make it explicit. So I'll just give you formula. Now our map from this definition exchange height and life translation. In particular, if you create this transform at 1, you obtain a distribution which is invariant and a conjugation. So assuming this is a function what we have to do is find a class function say theta 1 G1, G2 class function with the following property. So theta must be a certain class function and our goal now is to find a formula for theta. So exactly the same situation as Fourier transform except theta is an unknown class function on this group G and this has to be true for the principal series when Pi 1, Pi 2 is in principal series this puts a restriction of theta and the only reasonable formula for theta is the following D is a common determinant and we consider this integral, Psi is additive character and here we have this integral which is of course an improper integral doesn't converge as long as trace the sum of the trace is not 0 this makes sense you can integrate of a compact set and let the compact set into infinity that the integral has a meaning if the sum of the trace is 0 this blows up but anyway this is the only reasonable formula for theta and highly heuristic computation shows that indeed to the extent this makes sense this transform as a required property and stop here thank you so every is local yeah yeah here is local yeah it's here did you find the same formula ? good all right you said goodbye at the vacuum for Q of i did he write this ? I have some notes but of course he did not never publish it in the decomposition for the important groups you get the rational orbits yeah yeah yeah exactly it's a very neat proof but as usual he did not publish it could you say a few more words about this in connection with other groups well you would like to work this for other group let's say for ranking sale back for JLN well of course you have a similar group but I have no idea what to do I try also to do similarly the triple product so this is really inspired from Laurent Laforgue we had a similar computation for the symmetric square I just follow his method on question let's thank LV again on behalf of the organizers let me thank everybody for coming yeah absolutely