The work of Roger Godement
Jacquet, Hervé
I will discuss the work of Roger Godement. I will present the part of his work which is not accessible to the general public because it consists of letters o...
 So thank you very much. So I'm the last speaker and I'm supposed to be to give thanks to the organizing committee. I'm supposed to be a member. But in fact, I did not do anything. So I am an even better commission position to thanks the organizing committee, the Alpha of the audience, to thank also the European Research Council for its support. Thanks to the IHES. Thanks to its director Emmanuel Ulmo. And thanks also to the Marvello staff of the IHES, Elisabeth Jassrand, secretaire, Laurence Boparin, which in charge of the housing and gave Marvello the direction to find Lornaille. And all the staff, the staff feed us. Food appears like by magic. The blackboard gets erased. Microphone are provided and the microphone work. So let's thank them all. Okay now, Labès gave an overview of the work of Godmore. So I'm not going to really duplicate his efforts and I'm going to select in the work of Godmore what I think is important or what was important to me. So first Godmore started by what we call abstract nonsense. And this example was apparently quite motivating. You take a split reductive group, G with the Borel subgroup A and then you have the representation on L2 of G mod N. And then for every character of the split torus A character of modulus 1, you have the induced representation from B to G induced by Chi. And L2 of G sub n is this representation is direct sum, continuous direct sum of those representations, which in general are irreducible. So according to Godmore, this very simple example was a big help in formulating the notion of continuous sum of unitary representation and arriving at the theorem, which is not very hard, but that any unitary representation of locally compact group, say separable, that is, admits a countable dance subset, any unitary representation is a continuous sum of irreducible ones. Now Godmore, I hope for this theorem, he sought for instance that by applying it to say L2 of GA mod GQ, say G algebraic, reductive algebraic over Q and A is the ideal of Q. This is a big representation, of course we can decompose it into a sum of irreducible, but he hopes that that would give some information on Agestan series. In fact, in fact that he proposes that to me as a subject as a hypothesis. Oof. Oof. Well you should mention also Gadf on this. Yeah, yeah, yeah, absolutely. And the rush of something, Gelfand, Gelfand. Gelfand and company. Well, yeah, yeah. Okay, so that was one of the first things that Godmore looked at. Actually Godmore in his written work did not prove this theorem, but he knew it and his written work converges towards this theorem. Still I find this abstract theorem sometimes useful. For instance, if you go to a local situation, say with G split and the maximum unipotent subgroup and theta a generic character and a maximum unipotent subgroup, theta a character and general position, we could of course try to decompose this explicitly, but just knowing this can be done is sometimes useful. So the abstract non-sense theorem is from time to time useful. Now to go to something concrete, more concrete. In Carton seminar, Godmore gave a series of talks and in particular he gave a talk about cuspid forms, but in the context of the symplectic group and olemorphic forms. So he introduced in that context the notion of cuspid forms and he proved that this space for a given group and a given weight is finite dimensional. And to do that, he used a very nice lemma. So x, a locally compact space, mu, a bounded measure on x, and we can consider the space L2 of x, mu, and the close of space was bounded, of course, essentially bounded. And then the lemma says the dimension of that space is finite. So that's a lemma which is very Goodman-like. And it can be used also to prove the finite dimensionality of the space of cusp form in the general situation. So that was this theorem on cusp form. It was an example of some concrete thing. A third contribution was more or less expository. We go back to SL2R and perhaps SL2Z or another congruence subgroup. And we consider an automorphic form of SL2R, but we consider a form in the sense of Farid Chandra. This means the form is transformed by a character of SL2R and is an eigenfunction of the Casimir operator and some condition of slow growth. So he taught a course, probably the economic one, on this subject. And the course, the notes of the course was written by Christian Usel. And the main thing I return from this, Goodman made those notes available, is that in the Fourier series of such a form appears a wittaker function, but the wittaker function, the classic sub-ganvous special function, they satisfy a functionally equation relating WSNW-SN. And some of those, that's where I find the idea of my thesis. By the way, about my thesis in his interview with Laurent Kloesel, Goodman says something which is not quite accurate. To begin with, he did not suggest this problem to me. And secondly, after my thesis, Langlands was asked to write a report. And so Langlands wrote the report, sends a letter to Goodman, and the letter contains something like three or four new ideas, uniqueness of the wittaker model, the castle man suggestion of the castle man shalaka formula, and the machinery of what's called now the Shady Langlands method. Langlands method. Langlands report. Yeah, Langlands Shady. Langlands Shady method. And it was not at all saying that my thesis was no good, but simply it was not too much raw. Okay, so that's something. And in the same vein, Goodman wrote a little note, which was never published, about what could call the wittaker function for a cover, cover, or even for the cover of SLUR, or even for the universal cover. So the analogous wittaker function for cover of SLUR, and the function are expressed in terms of hyper geometric functions. So it's a little note that was never published. Okay. Now, as Labès told us, Goodman 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 for in idyllic language. So Borel and Ari Chandra have established the Reduction Theory for real groups with discrete subgroup arithmetic district subgroup. And Goodman and Vaill 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 and the Einstein series, and did a lecture on it at Bourbaki. 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 must say, I remember the lecture at Bourbaki, and Goodman was jumping up and down very, very, very enthusiastic report. More elementary. Goodman has again, I guess it's before the report of Langlance paper. In a paper called Analyze spectral 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 the training operators. And another thing that I thought was important in his 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 phi is a Schwarz-Brioche 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 the idyllic context, he knew how to analytically continue the Einstein series for the Morel. Say in the 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 appeared. This was proved first by Calvin Moore by a rather elaborate construction. 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 Na 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 Adelik interpretation. He immediately gave a little lecture and also he wrote a letter to Wey. Wey had work of a Q. And Godman was producing this similarity over Q of i. And so it's done again in this style of Wey, not at all like in Jacqueline Glence. But nonetheless for me he used my thesis and the integral representation of the Wey-Taker 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've 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 the zeta function of simple algebra. So first I remember he gives some lecture on text disease. And that's where I learned text disease and the use of Ibel Abel. And then also what? Was it in Paris or in Paris? 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 zeta function for division algebra. He has two lectures in Boba Key on this subject. And I'm afraid those lectures are not very good. They're kind of incoherent. 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 an automatic form for a quaternion algebra to automatic form on yield 2. Now I come really to the main subject. Zeta, I don't know. OK. I'll write good one. OK. And function. Now it's a subject on which there is a huge, huge literature. I saw the local problem, the global problem. They was very interested. And there's really a lot of articles. But some more they don't get it. Not quite. And one reason is that you cannot obtain what you want without using representation, representation of local group, Adele, Idel, infinite dimensional unitary representation, and so forth. Now Godma 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 Godma 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-archimedes. So the notion of representation we have to use is the unrucible, irreducible representations. And in the sense Godma 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-bion matrices. You want to consider those zeta integral, not the normalization, s plus n minus 1 over 2. So when the real part of F 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, L spi. And L spi is a sum of such functions. So L spi is completely determined by this condition. We normalize it, of course, by demanding that it's the 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 gamma. Well, you start by the cuspidial case. Be careful that for n equal 1, any character is to be considered a cuspidial because the cuspidality condition is empty. So you have to prove this for pi cuspidial. For n equal 1, it's, of course, theta. 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 center. So once you have it for the cuspidial representation, you are almost free, home free, because you have this little lemma. Cos 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 s pi is a product of the gamma s pi i psi. For the n factor, it's a little more complicated. L s pi, pi being an irreducible component, is not necessarily equal to the product of the L s pi i. It belongs to the ideal generated by them in other words. It's this product times the polynomial. Okay. But since any representation, any irreducible representation is a component of something induced by supercuspidial 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 is square integrable module of center. Or, in fact, pi is tampered. Then zeta phi, I do it, 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 tampered, and you need the following. So this function on the group, on GLM, belongs to the Arich-Chandra space, the space of rapidly decreasing function in the sense of Arich-Chandra, if real part of s is positive. Therefore, when pi is tampered, this implies the integral converges for real part of s positive, and in particular, then Ls pi is allomorphic. So real part of s positive. It's not fine, compactly supported. Module of 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, a Schwarz on the space of additive matrices, not on the group. No, the same is true for pi tilde, the contra-gradient representation. Did I say that pi tilde is a contra-gradient representation? No, but no, I'm telling you. Okay, 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 addition. This means this fraction is irreducible, and therefore, if you know the gamma factor, when pi is tampered, you know the L factor. Okay, 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 a Langland's quotient. This means the pi 1, pi 2, pi r are tampered representation, but twisted by appropriate power, real powers determinant. And suppose we are, 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, 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 supercaspital, which is, of course, another problem. So this is a local situation for F, not H-immediate. For F, real or complex. In the initial note, I discuss, I use only Ari-Chandra module, GK module. And now it's more convenient to represent for any irreducible, I mean, to replace any irreducible GK module by its canonical model in the sense of Castleman and Wallach. So now all the representation are 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 s pi as the L factor of the representation of the vague group. 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. At least the statement Ls pi times polynomial must be replaced by the following assertion. The holomorphic multiple of Ls pi bounded at infinity in 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, I suppose. No, no. It's a polynomial in 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. So now we have finished the local theory. I forgot to say what happened in the Andromedic 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 capital automorphic. If you wish, you could think of pi as a unitary irreducible representation. It's not hard to see, I guess, I am over the group of Adel of a number, 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, he is not on Jacques Langland, proves it for an admissible unitary representation. But if you work a little harder, you can prove for any unitary irreducible. So anyway, we have the global integral. Now phi is a function, Schwarz-Breff 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 function for your transform. Additive for your transform. Additive for your 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 your transform vanish also at singular matrices. But Godmore, so this is not needed. So you must explain why you can ignore the singular term in the Poisson's summation formula. Because a term like phi of 0 is because an integral like this on the, well on the 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 nonzero, your representation would be, which is irreducible, would be the trivial representation. So this kind of thing, we can ignore this in 0. Except for n equal to 1. No, no, no, no, if you take a chi, which is not the trivial character. Yes, but one is cuspidality in this case. Yeah. No, one is, that's the only one which is not cuspidality. Okay. And now you have to explain. Yg is a multiplicative armature. Yeah, it's multiplicative. Now you must explain why you can ignore the nonzero 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, say, of matrices like this. The group GLN, called 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 real part of s sufficiently large, extend to an entire function of s, and we have this functional equation. And 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 Jacket Langlands, we used as an echo method to obtain this functional analytic continuation. And Langlands said, but me, for GLN, this echo method is not going to work. It does also, you can also use something like echo method and use for the coefficient of cuss forms. It's a little more complicated. And Ari Chandra, when I finish writing this set, how come you are doing something which applies only to GLN? Well, now a word about. What do we do for non-cuss forms? Yeah, that's true. So if pi is an automorphic representation, but not cuss-pedal, 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, cuss-pedal automorphic representation, but eventually switched by power of the determinant. This is a theorem of Langlands. This means that Ls pi is essentially equal to Ls pi 1 Ls pi r. It's different only at a finite number of places. And in particular, we also know that Ls pi is biomorphic with the same functional equation. So we don't have to do anything more for the other automorphic forms. So this is a theory for GLN. It's quite a complete theory. It's not very difficult. Now let me just say a word. Let's see. I have 10 minutes. A lot of people are trying to extend Godman's jacket to other groups, so that Ari Chandra will be shown to be wrong. And I want just to show you a very modest contribution. I am on GL2, because GL2 over a number field, now over a local field. And I'd like to understand the Rankine-Selberg, the gamma factor of the Rankine-Selberg L function, which is defined intrinsically in terms of Wittaker function. I do it only for pi 1 pi 2 times. 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 by the set, the group of pairs, G1, G2, with the same determinant. And we want to use Planche-Schell formula for this group G. There is a little subtlety thing that we'll need. Suppose sigma is an irreducible representation of G contained in pi 1 pi 2, a 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 contained 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. You can do it little by little. Now, using Planche-Schell formula 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 this Ari-Chandra space, and F1, F2 are matrix coefficient of pi 1 pi 2, time perd respectively. So the research map is 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 evaluate 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 in 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 zero, this makes sense. You can integrate of a compact set, and then let the compact set into infinity, the integral has a meaning. If the sum of the trace is zero, this blows up. But anyway, this is the only reasonable formula for theta, and highly heuristic computation shows that indeed, to extend this makes sense, this transform has a required property. I'll stop here. Thank you. So, every slope of that here. Yeah, yeah, here is local. Yeah. Where is Valsperger? Is here. Did you find the same formula? Good. We said good morning at the very understated for a Q of I. Yeah. Did he write this? I have some notes, but of course he did not, he never published it. And the composition for new total groups, you get the rational orbits as. Yeah, yeah, yeah, exactly. Yes. Yeah, yeah. No, it's a very neat proof, but as usual, we did not publish it. Okay. Could you say a few more words about this in connection with other groups? Well, you would like to work this for other group. For instance, let's say for Rankin's cell back for JLN, for instance. Well, of course, you'll have a similar group, but I have no idea what to do. I try also to do similarly the triple product. So, so this is really inspired from Laurent Lafogue, we had a similar computation for the symmetric square. So I just follow his method. Okay. Well, I guess if there are no more questions, let's thank LV again. And I think that's on behalf of the organizers, let me thank everybody for coming. Yeah, yeah, absolutely. Thank you.
 So thank you very much.  So I'm the last speaker and I'm supposed to be to give thanks to the organizing committee.  I'm supposed to be a member.  But in fact, I did not do anything.  So I am an even better commission position to thanks the organizing committee, the Alpha  of the audience, to thank also the European Research Council for its support.  Thanks to the IHES.  Thanks to its director Emmanuel Ulmo.  And thanks also to the Marvello staff of the IHES, Elisabeth Jassrand, secretaire,  Laurence Boparin, which in charge of the housing and gave Marvello the direction to  find Lornaille.  And all the staff, the staff feed us.  Food appears like by magic.  The blackboard gets erased.  Microphone are provided and the microphone work.  So let's thank them all.  Okay now, Labès gave an overview of the work of Godmore.  So I'm not going to really duplicate his efforts and I'm going to select in the work  of Godmore what I think is important or what was important to me.  So first Godmore started by what we call abstract nonsense.  And this example was apparently quite motivating.  You take a split reductive group, G with the Borel subgroup A and then you have the representation  on L2 of G mod N. And then for every character of the split  torus A character of modulus 1, you have the induced representation from B to G induced  by Chi.  And L2 of G sub n is this representation is direct sum, continuous direct sum of those  representations, which in general are irreducible.  So according to Godmore, this very simple example was a big help in formulating the notion  of continuous sum of unitary representation and arriving at the theorem, which is not  very hard, but that any unitary representation of locally compact group, say separable, that  is, admits a countable dance subset, any unitary representation is a continuous sum of irreducible  ones.  Now Godmore, I hope for this theorem, he sought for instance that by applying it to  say L2 of GA mod GQ, say G algebraic, reductive algebraic over Q and A is the ideal of Q.  This is a big representation, of course we can decompose it into a sum of irreducible,  but he hopes that that would give some information on Agestan series.  In fact, in fact that he proposes that to me as a subject as a hypothesis.  Oof.  Oof.  Well you should mention also Gadf on this.  Yeah, yeah, yeah, absolutely.  And the rush of something, Gelfand, Gelfand.  Gelfand and company.  Well, yeah, yeah.  Okay, so that was one of the first things that Godmore looked at.  Actually Godmore in his written work did not prove this theorem, but he knew it and his  written work converges towards this theorem.  Still I find this abstract theorem sometimes useful.  For instance, if you go to a local situation, say with G split and the maximum unipotent  subgroup and theta a generic character and a maximum unipotent subgroup, theta a character  and general position, we could of course try to decompose this explicitly, but just  knowing this can be done is sometimes useful.  So the abstract non-sense theorem is from time to time useful.  Now to go to something concrete, more concrete.  In Carton seminar, Godmore gave a series of talks and in particular he gave a talk about  cuspid forms, but in the context of the symplectic group and olemorphic forms.  So he introduced in that context the notion of cuspid forms and he proved that this space  for a given group and a given weight is finite dimensional.  And to do that, he used a very nice lemma.  So x, a locally compact space, mu, a bounded measure on x, and we can consider the space  L2 of x, mu, and the close of space was bounded, of course, essentially bounded.  And then the lemma says the dimension of that space is finite.  So that's a lemma which is very Goodman-like.  And it can be used also to prove the finite dimensionality of the space of cusp form in  the general situation.  So that was this theorem on cusp form.  It was an example of some concrete thing.  A third contribution was more or less expository.  We go back to SL2R and perhaps SL2Z or another congruence subgroup.  And we consider an automorphic form of SL2R, but we consider a form in the sense of Farid  Chandra.  This means the form is transformed by a character of SL2R and is an eigenfunction of the Casimir  operator and some condition of slow growth.  So he taught a course, probably the economic one, on this subject.  And the course, the notes of the course was written by Christian Usel.  And the main thing I return from this, Goodman made those notes available, is that in the  Fourier series of such a form appears a wittaker function, but the wittaker function, the classic  sub-ganvous special function, they satisfy a functionally equation relating WSNW-SN.  And some of those, that's where I find the idea of my thesis.  By the way, about my thesis in his interview with Laurent Kloesel, Goodman says something  which is not quite accurate.  To begin with, he did not suggest this problem to me.  And secondly, after my thesis, Langlands was asked to write a report.  And so Langlands wrote the report, sends a letter to Goodman, and the letter contains  something like three or four new ideas, uniqueness of the wittaker model, the castle man suggestion  of the castle man shalaka formula, and the machinery of what's called now the Shady Langlands  method.  Langlands method.  Langlands report.  Yeah, Langlands Shady.  Langlands Shady method.  And it was not at all saying that my thesis was no good, but simply it was not too much  raw.  Okay, so that's something.  And in the same vein, Goodman wrote a little note, which was never published, about what  could call the wittaker function for a cover, cover, or even for the cover of SLUR, or even  for the universal cover.  So the analogous wittaker function for cover of SLUR, and the function are expressed in  terms of hyper geometric functions.  So it's a little note that was never published.  Okay.  Now, as Labès told us, Goodman 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 for in idyllic language.  So Borel and Ari Chandra have established the Reduction Theory for real groups with discrete  subgroup arithmetic district subgroup.  And Goodman and Vaill 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 and the Einstein series, and  did a lecture on it at Bourbaki.  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 must say, I remember the lecture at Bourbaki, and Goodman was jumping up and down very,  very, very enthusiastic report.  More elementary.  Goodman has again, I guess it's before the report of Langlance paper.  In a paper called Analyze spectral 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 the training operators.  And another thing that I thought was important in his 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 phi is a Schwarz-Brioche 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 the idyllic context, he knew how to analytically continue the Einstein  series for the Morel.  Say in the 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 appeared.  This was proved first by Calvin Moore by a rather elaborate construction.  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 Na 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  Adelik interpretation.  He immediately gave a little lecture and also he wrote a letter to Wey.  Wey had work of a Q.  And Godman was producing this similarity over Q of i.  And so it's done again in this style of Wey, not at all like in Jacqueline Glence.  But nonetheless for me he used my thesis and the integral representation of the Wey-Taker  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've 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 the zeta function of simple algebra.  So first I remember he gives some lecture on text disease.  And that's where I learned text disease and the use of Ibel Abel.  And then also what?  Was it in Paris or in Paris?  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 zeta function for division algebra.  He has two lectures in Boba Key on this subject.  And I'm afraid those lectures are not very good.  They're kind of incoherent.  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 an automatic form for a quaternion algebra to automatic  form on yield 2.  Now I come really to the main subject.  Zeta, I don't know.  OK.  I'll write good one.  OK.  And function.  Now it's a subject on which there is a huge, huge literature.  I saw the local problem, the global problem.  They was very interested.  And there's really a lot of articles.  But some more they don't get it.  Not quite.  And one reason is that you cannot obtain what you want without using representation, representation  of local group, Adele, Idel, infinite dimensional unitary representation, and so forth.  Now Godma 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 Godma 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-archimedes.  So the notion of representation we have to use is the unrucible, irreducible representations.  And in the sense Godma 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-bion matrices.  You want to consider those zeta integral, not the normalization, s plus n minus 1 over  2.  So when the real part of F 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, L  spi.  And L spi is a sum of such functions.  So L spi is completely determined by this condition.  We normalize it, of course, by demanding that it's the 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 gamma.  Well, you start by the cuspidial case.  Be careful that for n equal 1, any character is to be considered a cuspidial because the  cuspidality condition is empty.  So you have to prove this for pi cuspidial.  For n equal 1, it's, of course, theta.  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 center.  So once you have it for the cuspidial representation, you are almost free, home free, because you  have this little lemma.  Cos 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 s pi is a product of the gamma s pi i psi.  For the n factor, it's a little more complicated.  L s pi, pi being an irreducible component, is not necessarily equal to the product of  the L s pi i.  It belongs to the ideal generated by them in other words.  It's this product times the polynomial.  Okay.  But since any representation, any irreducible representation is a component of something  induced by supercuspidial 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 is square integrable  module of center.  Or, in fact, pi is tampered.  Then zeta phi, I do it, 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 tampered, and you need the following.  So this function on the group, on GLM, belongs to the Arich-Chandra space, the space of rapidly  decreasing function in the sense of Arich-Chandra, if real part of s is positive.  Therefore, when pi is tampered, this implies the integral converges for real part of s  positive, and in particular, then Ls pi is allomorphic.  So real part of s positive.  It's not fine, compactly supported.  Module of 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, a Schwarz on the space of additive matrices, not on the group.  No, the same is true for pi tilde, the contra-gradient representation.  Did I say that pi tilde is a contra-gradient representation?  No, but no, I'm telling you.  Okay, 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 addition.  This means this fraction is irreducible, and therefore, if you know the gamma factor,  when pi is tampered, you know the L factor.  Okay, 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 a Langland's quotient.  This means the pi 1, pi 2, pi r are tampered representation, but twisted by appropriate  power, real powers determinant.  And suppose we are, 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, 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 supercaspital, which is, of course, another problem.  So this is a local situation for F, not H-immediate.  For F, real or complex.  In the initial note, I discuss, I use only Ari-Chandra module, GK module.  And now it's more convenient to represent for any irreducible, I mean, to replace any  irreducible GK module by its canonical model in the sense of Castleman and Wallach.  So now all the representation are 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 s pi as the L factor of the representation  of the vague group.  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.  At least the statement Ls pi times polynomial must be replaced by the following assertion.  The holomorphic multiple of Ls pi bounded at infinity in 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, I suppose.  No, no.  It's a polynomial in 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.  So now we have finished the local theory.  I forgot to say what happened in the Andromedic 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 capital automorphic.  If you wish, you could think of pi as a unitary irreducible representation.  It's not hard to see, I guess, I am over the group of Adel of a number, 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, he is not on Jacques Langland, proves it for an admissible unitary representation.  But if you work a little harder, you can prove for any unitary irreducible.  So anyway, we have the global integral.  Now phi is a function, Schwarz-Breff 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  function for your transform.  Additive for your transform.  Additive for your 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 your transform vanish also at singular matrices.  But Godmore, so this is not needed.  So you must explain why you can ignore the singular term in the Poisson's summation formula.  Because a term like phi of 0 is because an integral like this on the, well on the 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 nonzero, your representation would be, which is irreducible, would be the trivial  representation.  So this kind of thing, we can ignore this in 0.  Except for n equal to 1.  No, no, no, no, if you take a chi, which is not the trivial character.  Yes, but one is cuspidality in this case.  Yeah.  No, one is, that's the only one which is not cuspidality.  Okay.  And now you have to explain.  Yg is a multiplicative armature.  Yeah, it's multiplicative.  Now you must explain why you can ignore the nonzero 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, say, of matrices like this.  The group GLN, called 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 real part of  s sufficiently large, extend to an entire function of s, and we have this functional  equation.  And 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 Jacket Langlands, we used as an echo method to obtain this  functional analytic continuation.  And Langlands said, but me, for GLN, this echo method is not going to work.  It does also, you can also use something like echo method and use for the coefficient of  cuss forms.  It's a little more complicated.  And Ari Chandra, when I finish writing this set, how come you are doing something which  applies only to GLN?  Well, now a word about.  What do we do for non-cuss forms?  Yeah, that's true.  So if pi is an automorphic representation, but not cuss-pedal, 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, cuss-pedal automorphic representation, but eventually switched by power of the determinant.  This is a theorem of Langlands.  This means that Ls pi is essentially equal to Ls pi 1 Ls pi r.  It's different only at a finite number of places.  And in particular, we also know that Ls pi is biomorphic with the same functional equation.  So we don't have to do anything more for the other automorphic forms.  So this is a theory for GLN.  It's quite a complete theory.  It's not very difficult.  Now let me just say a word.  Let's see.  I have 10 minutes.  A lot of people are trying to extend Godman's jacket to other groups, so that Ari Chandra  will be shown to be wrong.  And I want just to show you a very modest contribution.  I am on GL2, because GL2 over a number field, now over a local field.  And I'd like to understand the Rankine-Selberg, the gamma factor of the Rankine-Selberg L  function, which is defined intrinsically in terms of Wittaker function.  I do it only for pi 1 pi 2 times.  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 by the set, the group of pairs, G1, G2, with the same  determinant.  And we want to use Planche-Schell formula for this group G. There is a little subtlety  thing that we'll need.  Suppose sigma is an irreducible representation of G contained in pi 1 pi 2, a 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  contained 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.  You can do it little by little.  Now, using Planche-Schell formula 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 this Ari-Chandra space, and F1, F2 are matrix coefficient of pi 1 pi  2, time perd respectively.  So the research map is 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 evaluate 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  in 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 zero, this makes sense.  You can integrate of a compact set, and then let the compact set into infinity, the integral  has a meaning.  If the sum of the trace is zero, this blows up.  But anyway, this is the only reasonable formula for theta, and highly heuristic computation  shows that indeed, to extend this makes sense, this transform has a required property.  I'll stop here.  Thank you.  So, every slope of that here.  Yeah, yeah, here is local.  Yeah.  Where is Valsperger?  Is here.  Did you find the same formula?  Good.  We said good morning at the very understated for a Q of I.  Yeah.  Did he write this?  I have some notes, but of course he did not, he never published it.  And the composition for new total groups, you get the rational orbits as.  Yeah, yeah, yeah, exactly.  Yes.  Yeah, yeah.  No, it's a very neat proof, but as usual, we did not publish it.  Okay.  Could you say a few more words about this in connection with other groups?  Well, you would like to work this for other group.  For instance, let's say for Rankin's cell back for JLN, for instance.  Well, of course, you'll have a similar group, but I have no idea what to do.  I try also to do similarly the triple product.  So, so this is really inspired from Laurent Lafogue, we had a similar computation for  the symmetric square.  So I just follow his method.  Okay.  Well, I guess if there are no more questions, let's thank LV again.  And I think that's on behalf of the organizers, let me thank everybody for coming.  Yeah, yeah, absolutely.  Thank you.
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So thank you very much. So I'm the last speaker and I'm supposed to be to give thanks to the organizing committee. I'm supposed to be a member. But in fact, I did not do anything. So I am an even better commission position to thanks the organizing committee, the Alpha of the audience, to thank also the European Research Council for its support. Thanks to the IHES. Thanks to its director Emmanuel Ulmo. And thanks also to the Marvello staff of the IHES, Elisabeth Jassrand, secretaire, Laurence Boparin, which in charge of the housing and gave Marvello the direction to find Lornaille. And all the staff, the staff feed us. Food appears like by magic. The blackboard gets erased. Microphone are provided and the microphone work. So let's thank them all. Okay now, Labès gave an overview of the work of Godmore. So I'm not going to really duplicate his efforts and I'm going to select in the work of Godmore what I think is important or what was important to me. So first Godmore started by what we call abstract nonsense. And this example was apparently quite motivating. You take a split reductive group, G with the Borel subgroup A and then you have the representation on L2 of G mod N. And then for every character of the split torus A character of modulus 1, you have the induced representation from B to G induced by Chi. And L2 of G sub n is this representation is direct sum, continuous direct sum of those representations, which in general are irreducible. So according to Godmore, this very simple example was a big help in formulating the notion of continuous sum of unitary representation and arriving at the theorem, which is not very hard, but that any unitary representation of locally compact group, say separable, that is, admits a countable dance subset, any unitary representation is a continuous sum of irreducible ones. Now Godmore, I hope for this theorem, he sought for instance that by applying it to say L2 of GA mod GQ, say G algebraic, reductive algebraic over Q and A is the ideal of Q. This is a big representation, of course we can decompose it into a sum of irreducible, but he hopes that that would give some information on Agestan series. In fact, in fact that he proposes that to me as a subject as a hypothesis. Oof. Oof. Well you should mention also Gadf on this. Yeah, yeah, yeah, absolutely. And the rush of something, Gelfand, Gelfand. Gelfand and company. Well, yeah, yeah. Okay, so that was one of the first things that Godmore looked at. Actually Godmore in his written work did not prove this theorem, but he knew it and his written work converges towards this theorem. Still I find this abstract theorem sometimes useful. For instance, if you go to a local situation, say with G split and the maximum unipotent subgroup and theta a generic character and a maximum unipotent subgroup, theta a character and general position, we could of course try to decompose this explicitly, but just knowing this can be done is sometimes useful. So the abstract non-sense theorem is from time to time useful. Now to go to something concrete, more concrete. In Carton seminar, Godmore gave a series of talks and in particular he gave a talk about cuspid forms, but in the context of the symplectic group and olemorphic forms. So he introduced in that context the notion of cuspid forms and he proved that this space for a given group and a given weight is finite dimensional. And to do that, he used a very nice lemma. So x, a locally compact space, mu, a bounded measure on x, and we can consider the space L2 of x, mu, and the close of space was bounded, of course, essentially bounded. And then the lemma says the dimension of that space is finite. So that's a lemma which is very Goodman-like. And it can be used also to prove the finite dimensionality of the space of cusp form in the general situation. So that was this theorem on cusp form. It was an example of some concrete thing. A third contribution was more or less expository. We go back to SL2R and perhaps SL2Z or another congruence subgroup. And we consider an automorphic form of SL2R, but we consider a form in the sense of Farid Chandra. This means the form is transformed by a character of SL2R and is an eigenfunction of the Casimir operator and some condition of slow growth. So he taught a course, probably the economic one, on this subject. And the course, the notes of the course was written by Christian Usel. And the main thing I return from this, Goodman made those notes available, is that in the Fourier series of such a form appears a wittaker function, but the wittaker function, the classic sub-ganvous special function, they satisfy a functionally equation relating WSNW-SN. And some of those, that's where I find the idea of my thesis. By the way, about my thesis in his interview with Laurent Kloesel, Goodman says something which is not quite accurate. To begin with, he did not suggest this problem to me. And secondly, after my thesis, Langlands was asked to write a report. And so Langlands wrote the report, sends a letter to Goodman, and the letter contains something like three or four new ideas, uniqueness of the wittaker model, the castle man suggestion of the castle man shalaka formula, and the machinery of what's called now the Shady Langlands method. Langlands method. Langlands report. Yeah, Langlands Shady. Langlands Shady method. And it was not at all saying that my thesis was no good, but simply it was not too much raw. Okay, so that's something. And in the same vein, Goodman wrote a little note, which was never published, about what could call the wittaker function for a cover, cover, or even for the cover of SLUR, or even for the universal cover. So the analogous wittaker function for cover of SLUR, and the function are expressed in terms of hyper geometric functions. So it's a little note that was never published. Okay. Now, as Labès told us, Goodman 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 for in idyllic language. So Borel and Ari Chandra have established the Reduction Theory for real groups with discrete subgroup arithmetic district subgroup. And Goodman and Vaill 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 and the Einstein series, and did a lecture on it at Bourbaki. 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 must say, I remember the lecture at Bourbaki, and Goodman was jumping up and down very, very, very enthusiastic report. More elementary. Goodman has again, I guess it's before the report of Langlance paper. In a paper called Analyze spectral 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 the training operators. And another thing that I thought was important in his 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 phi is a Schwarz-Brioche 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 the idyllic context, he knew how to analytically continue the Einstein series for the Morel. Say in the 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 appeared. This was proved first by Calvin Moore by a rather elaborate construction. 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 Na 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 Adelik interpretation. He immediately gave a little lecture and also he wrote a letter to Wey. Wey had work of a Q. And Godman was producing this similarity over Q of i. And so it's done again in this style of Wey, not at all like in Jacqueline Glence. But nonetheless for me he used my thesis and the integral representation of the Wey-Taker 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've 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 the zeta function of simple algebra. So first I remember he gives some lecture on text disease. And that's where I learned text disease and the use of Ibel Abel. And then also what? Was it in Paris or in Paris? 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 zeta function for division algebra. He has two lectures in Boba Key on this subject. And I'm afraid those lectures are not very good. They're kind of incoherent. 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 an automatic form for a quaternion algebra to automatic form on yield 2. Now I come really to the main subject. Zeta, I don't know. OK. I'll write good one. OK. And function. Now it's a subject on which there is a huge, huge literature. I saw the local problem, the global problem. They was very interested. And there's really a lot of articles. But some more they don't get it. Not quite. And one reason is that you cannot obtain what you want without using representation, representation of local group, Adele, Idel, infinite dimensional unitary representation, and so forth. Now Godma 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 Godma 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-archimedes. So the notion of representation we have to use is the unrucible, irreducible representations. And in the sense Godma 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-bion matrices. You want to consider those zeta integral, not the normalization, s plus n minus 1 over 2. So when the real part of F 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, L spi. And L spi is a sum of such functions. So L spi is completely determined by this condition. We normalize it, of course, by demanding that it's the 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 gamma. Well, you start by the cuspidial case. Be careful that for n equal 1, any character is to be considered a cuspidial because the cuspidality condition is empty. So you have to prove this for pi cuspidial. For n equal 1, it's, of course, theta. 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 center. So once you have it for the cuspidial representation, you are almost free, home free, because you have this little lemma. Cos 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 s pi is a product of the gamma s pi i psi. For the n factor, it's a little more complicated. L s pi, pi being an irreducible component, is not necessarily equal to the product of the L s pi i. It belongs to the ideal generated by them in other words. It's this product times the polynomial. Okay. But since any representation, any irreducible representation is a component of something induced by supercuspidial 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 is square integrable module of center. Or, in fact, pi is tampered. Then zeta phi, I do it, 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 tampered, and you need the following. So this function on the group, on GLM, belongs to the Arich-Chandra space, the space of rapidly decreasing function in the sense of Arich-Chandra, if real part of s is positive. Therefore, when pi is tampered, this implies the integral converges for real part of s positive, and in particular, then Ls pi is allomorphic. So real part of s positive. It's not fine, compactly supported. Module of 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, a Schwarz on the space of additive matrices, not on the group. No, the same is true for pi tilde, the contra-gradient representation. Did I say that pi tilde is a contra-gradient representation? No, but no, I'm telling you. Okay, 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 addition. This means this fraction is irreducible, and therefore, if you know the gamma factor, when pi is tampered, you know the L factor. Okay, 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 a Langland's quotient. This means the pi 1, pi 2, pi r are tampered representation, but twisted by appropriate power, real powers determinant. And suppose we are, 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, 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 supercaspital, which is, of course, another problem. So this is a local situation for F, not H-immediate. For F, real or complex. In the initial note, I discuss, I use only Ari-Chandra module, GK module. And now it's more convenient to represent for any irreducible, I mean, to replace any irreducible GK module by its canonical model in the sense of Castleman and Wallach. So now all the representation are 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 s pi as the L factor of the representation of the vague group. 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. At least the statement Ls pi times polynomial must be replaced by the following assertion. The holomorphic multiple of Ls pi bounded at infinity in 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, I suppose. No, no. It's a polynomial in 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. So now we have finished the local theory. I forgot to say what happened in the Andromedic 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 capital automorphic. If you wish, you could think of pi as a unitary irreducible representation. It's not hard to see, I guess, I am over the group of Adel of a number, 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, he is not on Jacques Langland, proves it for an admissible unitary representation. But if you work a little harder, you can prove for any unitary irreducible. So anyway, we have the global integral. Now phi is a function, Schwarz-Breff 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 function for your transform. Additive for your transform. Additive for your 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 your transform vanish also at singular matrices. But Godmore, so this is not needed. So you must explain why you can ignore the singular term in the Poisson's summation formula. Because a term like phi of 0 is because an integral like this on the, well on the 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 nonzero, your representation would be, which is irreducible, would be the trivial representation. So this kind of thing, we can ignore this in 0. Except for n equal to 1. No, no, no, no, if you take a chi, which is not the trivial character. Yes, but one is cuspidality in this case. Yeah. No, one is, that's the only one which is not cuspidality. Okay. And now you have to explain. Yg is a multiplicative armature. Yeah, it's multiplicative. Now you must explain why you can ignore the nonzero 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, say, of matrices like this. The group GLN, called 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 real part of s sufficiently large, extend to an entire function of s, and we have this functional equation. And 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 Jacket Langlands, we used as an echo method to obtain this functional analytic continuation. And Langlands said, but me, for GLN, this echo method is not going to work. It does also, you can also use something like echo method and use for the coefficient of cuss forms. It's a little more complicated. And Ari Chandra, when I finish writing this set, how come you are doing something which applies only to GLN? Well, now a word about. What do we do for non-cuss forms? Yeah, that's true. So if pi is an automorphic representation, but not cuss-pedal, 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, cuss-pedal automorphic representation, but eventually switched by power of the determinant. This is a theorem of Langlands. This means that Ls pi is essentially equal to Ls pi 1 Ls pi r. It's different only at a finite number of places. And in particular, we also know that Ls pi is biomorphic with the same functional equation. So we don't have to do anything more for the other automorphic forms. So this is a theory for GLN. It's quite a complete theory. It's not very difficult. Now let me just say a word. Let's see. I have 10 minutes. A lot of people are trying to extend Godman's jacket to other groups, so that Ari Chandra will be shown to be wrong. And I want just to show you a very modest contribution. I am on GL2, because GL2 over a number field, now over a local field. And I'd like to understand the Rankine-Selberg, the gamma factor of the Rankine-Selberg L function, which is defined intrinsically in terms of Wittaker function. I do it only for pi 1 pi 2 times. 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 by the set, the group of pairs, G1, G2, with the same determinant. And we want to use Planche-Schell formula for this group G. There is a little subtlety thing that we'll need. Suppose sigma is an irreducible representation of G contained in pi 1 pi 2, a 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 contained 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. You can do it little by little. Now, using Planche-Schell formula 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 this Ari-Chandra space, and F1, F2 are matrix coefficient of pi 1 pi 2, time perd respectively. So the research map is 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 evaluate 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 in 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 zero, this makes sense. You can integrate of a compact set, and then let the compact set into infinity, the integral has a meaning. If the sum of the trace is zero, this blows up. But anyway, this is the only reasonable formula for theta, and highly heuristic computation shows that indeed, to extend this makes sense, this transform has a required property. I'll stop here. Thank you. So, every slope of that here. Yeah, yeah, here is local. Yeah. Where is Valsperger? Is here. Did you find the same formula? Good. We said good morning at the very understated for a Q of I. Yeah. Did he write this? I have some notes, but of course he did not, he never published it. And the composition for new total groups, you get the rational orbits as. Yeah, yeah, yeah, exactly. Yes. Yeah, yeah. No, it's a very neat proof, but as usual, we did not publish it. Okay. Could you say a few more words about this in connection with other groups? Well, you would like to work this for other group. For instance, let's say for Rankin's cell back for JLN, for instance. Well, of course, you'll have a similar group, but I have no idea what to do. I try also to do similarly the triple product. So, so this is really inspired from Laurent Lafogue, we had a similar computation for the symmetric square. So I just follow his method. Okay. Well, I guess if there are no more questions, let's thank LV again. And I think that's on behalf of the organizers, let me thank everybody for coming. Yeah, yeah, absolutely. Thank you.
the work of Godmore what I think is important or contribution of Godman to the theory of modular material. So let's discuss the local theory. And that's the only one which is not cuspidality.