 First, I would like to thank all the organizers for the invitation. It's a great honor for me to participate in this conference dedicated to French school. Personally, I don't have much connection with Professor Gaudemont. When I was an undergraduate, I was educated by American textbooks. I didn't know about Gaudemont until I came to Paris for my master's degree. And because I like to stay in the library, I'm immediately attracted by the textbooks by Professor Gaudemont and together with all his remarks on nuclear weapons and so on, something non-mathematics. I have to admit that all these affairs appeared more interesting than automorphic forms to me. Anyway, and another personal, well, I have to say something related to Gaudemont. OK, so another thing is OK. So when Professor Gaudemont passed away, there is an article called Roger Gaudemont in the Journal of SNF. OK, let's hear. So another news is OK. This article has been translated into Chinese. OK, so it appeared in some Chinese journal which called translations in mathematics, something like that. OK, so yeah. OK, I will just write down the pronunciation of that. Journal called Yiling. Almost immediately after French version. It's also in the last year. I'm the translator, though. This is the only thing I can do in memory of Professor Gaudemont. And I think his remarks on the relation of mathematics to humanity is that society deserves to be heard by the mathematicians in a developing country like China. But unfortunately, according to the website of this journal, the number of downloads so far is one. Maybe people are reading a paper version. I don't know. Anyways, it's not a big problem. So OK. So in the beginning, I thought it's better to speak in French so I prepared my notes in French. So if I switch to English, maybe I will get confused. I will continue in French. And I apologize for my broken French in advance. So OK. Today, I'm going to talk about the stabilization or the pre-stabilization of the TAS formula to help the dress. So I'm going to explain what the word dress means later. So the motivation is the following. So the choice is a reductive group. So it's always supposed to connect to the number F. And we use the TAS formula. So we write on the form. So there is a distribution and a variant. And there are two codes. That is the expression of this distribution. Ring a geometric or orbital, a load, a spectral. It's the distribution of a variant in the TAS formula. The pre-stabilization of the TAS formula is to express this distribution and then the stable distribution. I use the group of pre-parties like this and for the coefficient explicit. And here, it's a ring. And it's all connected by the sum of all the oscillation associated with C. So setting distribution stable is C-print. And here, C-print is always a group of pre-parties. So the existence of this distribution is a part of the program of stabilization. Its definition is quite non-trivial. And once we have this stabilization, we can study a lot of information on the spectral side, on the representation of the model. It's broken. And here, sorry. I haven't explained the F-print function yet. It's the transfer of F. So here, F-print is a test function on the group of pre-print that has a transfer of F. So we talked a lot about this transfer in the previous exhibition. And to start, the idea is to start with the regular part. They have I, elliptic, F. So we simply use the notation of Professor Attur. And we aim to rewrite this distribution, which is a stable distribution on the endoscopic side. So all these elements, gamma, are semi-simple. So the subject today is the same thing. Well, I'm wearing clothes. So we fix N, which is the decor of the clothes in what I'm wearing. So all the hachines, N and N of the unit are finished on F. And we also fix the displacement of mu N in C. C is star. So we replace the points of C by a garment, that is, a central extension of C A by this finished group of mu N. And then to talk about the automotive background, we also fix a sandwich of this garment on C S. And then to replace V, we can lower this central extension to obtain a local garment. And we can ask the question of the analysis of Monique in this case. I want to repeat it because it's a local compact group. And even local profini, in this case, is not immediate. So the question or the haves are a question of nature. Is there a formula for this with this structure for the garment? So yes, it exists. Well, this is a hypothesis. And secondly, stabilization. So very little known about the second question. So in what is on the test function in the formula that has C Thilda? It's the F function. It's basically the equivalent of the mu N like this. Or Z is in mu N. So it's enough to characterize this test function because the other modes of the equations can be treated by a process of regression. So the second question is a question of much more arithmetic than the first one because stabilization uses a lot of archbishopry. So it must be noted that a class of regression is a geometric or archbishopry. So a class of regression is proposed in its article by Bielanski and the linear under the mu N. So this is the regression provided by the central extension bar 4. So it is a note sample. As a facelift on a spec F we take the concept of Zahiski. I don't need this detail. And for the third question they obtained a complete classification in terms of combinational data. You use this table. So it is reasonable to ask the next question. Once we have original arithmetic such as this we show the program of regression to get the data. There are already the conjectures not many of them. So here is a list very incomplete. So there are conjectures by Weisman and also Gann Gaal and more recently we got a geometric bar Gelskrieg and the conjectures. In particular in order to determine we know to define the L group in order to determine the construction of the Lansky line. In fact it can only be of the dual group of the length. So it is F1 and so compared to the reductive group we can ask if there is a stabilization which I already have this question. And we need to propose an endoscopy to determine the Lansky line. Theoda. Weisman did not discuss the question in his article. There is a lot of difficulty because G. Theoda is in fact a group locally compact. There are no stable conjectures so here is the difficulty so the first question is that in the L group there is a ZF which is semi-simple and here there is a ZF which is a group or ZA as long as it is local or global of course. We say that Gamma is good if the stabilizer in the clothing of a light Gamma Theoda goes to the projection and the ZF is the Hesiplok of the stabilizer at the level of G. Why is it important because it is exactly the regular Gamma semi-simple that the ZF the orbital the F the length of the Gamma Theoda on the clothing is non-trivial so these are the orbits that appear in the formula of the RAS for this Theoda the classification is not completely known and if there is a theory of scoping for the Thela of the clothing then this question must have a fairly simple answer for example we can think of the geometization of the family of the RAS so we must compare the base of Stenberg with a good version for the clothing so we have to solve this question so this is the first difficulty of scoping and you have to define the following notions on the you have to define the scoping donate in particular the scoping group associated so in particular these are reductive groups they are not reductive groups so the formula that is stable if it exists it has to deal with reductive groups and then when we look at reductive groups there is a correspondence and there is a bit the C is the scoping group and finally the factor of the sphere the other which is the most complicated and once we have defined all this we can try to prove the fundamental lens and the sphere things like that it is largely unknown but there are already simpler cases firstly it is already non-trivial but it is resolved but it has to be deployed so the proclamation of the long-range for this goal I think it is resolved in the articles of Weismarck and we have obtained a long-range correspondence for the whole of the goal and then SP2 and we consider the clothing but non-trivial so it is a class of clothing very well known these are metapathetic groups here there is already a formalism like that for all these questions it is proposed by Jeffrey Adams Renard and in my thesis and more recently the identity of local characters are demonstrated by Tsaihua Luo in his thesis the complete stabilization is not yet done but we have already had a lot of results and how it works it uses it uses it uses the representation of the equipment like Omega-Sai so it is very useful very powerful but it is more than an analytical tool so it is not possible to generate the third because it is the equipment of SL2 and for 2N so it is an article not published in my thesis by Hiyaga Ikeda so it begins to be done so it is explained it is done by calculations to calculate a rate of cosine or cosine of kubota in particular it also responds to questions so I do not know if it is archived but you can ask Professor Hiyaga or Ikeda so it is the group on the scope here are PCL2 if the rate of the equipment is less than the maximum of SL2 so it is less so if N is less then the equipment looks like SL2 itself so to address this question there is a nature of a series of generalizations the fourth and third at the same time and see if it works so today I make an excuse for the case of SP2 F still local or global I mainly treat the case local because it is more simple and here N is a board that is admitted but as usual I suppose it is coupled with characteristic and CAR and for the synthetic groups I suppose the characteristic is not 2 and we consider a case like that CF or the theoretical points so of course there is a lot of equipment but here we take the one that is not simple that is primitive more precisely these are the equipment built by Masamoto in the direction of the French I suppose the speculation is not the same with the generalization of the systematic and SL2 3D HILAGA IKETA so ok so in the program of the long run for white man there are a lot of issues first the white man is a very well known white man very well known the argument is to determine a bar for the generalization it is to determine a bar N and also a quadratic is the result of X D is FTH or D is the maximum FTH in the general theory we construct the economic way an isogenic to the source TQN which is very explicit once we have an isogenic we can work in the case let's say in the case the case to simplify so we can say that this is a bar first we construct so there is a diagram like this and then a bar so in this case another in this case a white man DQN and in fact not difficult the match of DQN of T and it consists of the point element of a white man of TQN so it gives an approach of classification of good elements but in general there are other good elements that are not in the match it is the source of all difficulties so in another case this isogenic we identify T to the same the power N on BGCT second second word we can write this word quite simply we can write as a direct product of the name or for each word we have a separate extension of the word and then there is another a separate extension of DQN but it is not necessarily a word it can be deployed and in any case we can talk about the name so T is this thing so more precisely there is a restriction of the word but I will not write and all of this would be good, right? yes, of course and also now we know that we can design a copy of T it is obvious the construction is completely cononic depending on this and today there are theorems but it is not there are theorems that classify all the good elements for the theorem it says that the sound of the theorem is the image that is first of all we have of course the image of the reasoning image but it is not enough in general but we can see all the good elements with this sign it is this content in hiragayikeda by explicit calculation and by the same techniques we can also define a notion of stable conduct for this theorem how does it work? so the idea is simple we can enter and then there is a group over there written identified like this over there so z, z, t it is simply s, e, o, t on e, o, i, shop which is a restriction of the law and conjugation stable conjugation for the group Heductive J with respect to is realized by say z, t so it is in the case of s, l, t so in the case of s, l, t well actually there is a problem with the restriction of the law for the whole world but we can prove the result necessary so in the case of s, l, t well conjugation stable and realized by the action of p, t, l, t, f on s, l, t, f and it rises only on an action on clothing so there is a good quantity for stable conjugation but it is not the correct definition you have to add to obtain a reasonable definition so you need a factor of correction and also you need a good definition it is continued in a pre-print archive archive so the result is like this and now the formula is very long I cook here you need to work with tor so we consider a stable conjugation by elements that can be simple but tor maximum but here c is more than p, t, l, t or c, f, p maybe we have to lift it up on clothing it is a stable conjugation and to define we can now write sigma which is a system of sin and we suppose sigma is 3 times c n is equal to c so we were tor dqn dqn dqn by sigma in this way it is possible so it is like dqn-thioda we have elements of the tioda which is another element and if there is no sigma then dqn is simply like this here we can delta the image of dqn-thioda but with sigma we change things like this it is more than tor it is an homogenous space so the result in the preprint it is a cohesion of the adjunct to clarify the element but with dqn-thioda and it is not only about the tioda but about the dqn by the sojourn the sine so it is a cononic and why do we have to add the sine because in the classification of good elements we have to change the image of the sojourn by the sine so here it also appears we use the general definition so we can change it so we can formulate a notion of stability it is a bit weird because it is very different from the reductive stability and we can and we can test this definition we build a package called epilogic to the caressa and then test their stability so it shows that all these notions are always satisfying for me so 10 more minutes so I will talk about the scope now we have a stable conjugation a little bit weird but we can test the definition of the scope so for a long time there was already a dual group defined by Weisman and maybe all John and according to the construction of the dual of C there is only one of N mod it is SO2MC N and so in this case it is the same as the group SP itself and SP2MC N and B we will talk more about the distinct case and I will write twice the same thing because it is a bit different you will not know the details so what is the scope now it is easy to define the definition if N loses so everything is the same as the case of clothing so it is a bit complicated but it is like the case of SP two of them the same definition and if M loses then it is also defined by Adams, Renard and in my thesis so in this case the tones are in projection one with the whole one point and second with the whole there is no symmetry we can not change it and so the group in the scope is two copies the different copies doesn't present in any case it is almost the first group this is simply the first it seems that the case well known of M and K but there is a difference we have defined there is a difference for the third for the third point for the corresponding stop it we have to connect the good elements to the good elements of the under-scopic group here we have the top and the bottom and the group under-scopic defined at the top and let's say we have the TQN the TQN and we can take it according to the general TQN we can take it under-scopic group and in a way a little a little a little compared to the group on the side it is all continued in the general formalism and of course for M and K we have a corresponding stop it by the procedure here it is also a maximum stop we have the over-scopic they see the over-scopic and it gives you a corresponding stable TQN semi-simple there is no change and for N we have to insert with a sign how to define it the minimum for let's say for the part coming from the second component of the under-scopic group it is not very clear sorry in this way we see that this correspondence sees all the elements all the good elements for their clothing to finish we expose if I were to talk about the factor of the transfer so the factors of the transfer if we don't know how to define it in general but you have to check at least the properties I'm good so it's like that good regular semi-simple let's say local so it's not that it's not if it's only the image the image of the TQN that corresponds with the range for the particular correspondence and then it is equivalent to translation by MUN we don't forget the presence of MUN and then it corresponds well to the stable conjugation so that's called the property of Cossick Cologmon but here you have to use the bizarre version of the stable conjugation CAD the show is like that and also in the global case it has a formula of the product a couple of minutes CN is the character so it was finished in the special case of the case and then in the test and in general the case in general but except except for SL2 it's always in HILAGA and KEDA they have the explicit formula but for M to be I can give a fairly easy definition so for for the elements, we can respond to the orbital like this we can take nothing from the image of the solution and then we can simply take nothing except for M now there is a technique of CN so we can choose the image of the solution it's also identified with DQN we can choose the element and then we can use theotatioda and then theotatioda this is theotatioda so it's well defined in the reflection and it's enough to define the transfer factor in the case where theotatioda is equal to the economic rise how to define it so there is a candidate of course we can define with the transfer factor his clothes that is to say the aspect itself by choosing the epangles and things like that so now the question is so to test if it's the good candidate for the owners if there are easy to verify but you also have to verify that we have the transfer and the fundamental which is not yet done but once we have a concrete definition it's easy to see if it works or not but now it's a speculation so I'll stop here thank you question or question often to answer we will be in a situation where we can be in a situation yes yes I think it also works in this case so it's a question what is what is it it's a new launch I think in this case we have to jump on the same case yes on the job there is no fundamental non-standard in this case but in this case it will be more complicated I think