thank you Russians is a very special occasion for me but I just remember that I think the my very first talk in the is was given ten years of ten years ago with 70 birthday of Mo Clemens and a curious name 40 years of alpha of L group right yes okay and so happy birthday Bob and I hope that you keep me his he placed one rose for next eight years to come so well I mean I I don't have much of you know concrete reason to report but I I do get it he may have been working on this project under under endoscopy for a couple of years now and what I'm trying to say you know the it seemed to me that maybe there's some you know some realistic way to approach it right but first I let me you know it's PI up you on no one this I just try to recall some basic thing to put it into context so so so we are interested in these lambdas from Tory attic and conductors so it is about when you have true group H and G reductive group defi over some global fin f given with a home autism's for between the dual groups then you expect as a way to transfer automatic representations so automatic representation of - automatic representation of G so after not quite or if you heard about the packet at them because they Stan so I'm not going to emphasize the important bit because I am pretty sure that you're unconvinced about it let me just mention one of the small but important case from tolerated at least basically known as endoscopic case it's about basically when LH is more left centralizer of some semisimple element LJ twists histone the final product with cumulative work for almost 40 years many people based on the trace formula budesonide by G Martha and also on the transfer and the fundamental lemma it's reverse filter in and too much will contribute something okay so um so the this case is especially important because once I give a compare precisely what Puckett means so we have two state conductors in more precise way and which multiplicity formula and so on but also what has been mean I think the mob has insisted on least it would give you a stable trace formula which should be a basic tool when to you - when you work on the general curve from Tory a letís so around 2000 maybe a little later Langlands proposed a new approach to general authority so he come as he told me beyond endoscopy which is a some kind of new nautical new way of using stable trade formula to with the eye into the oscillating the pocket of automatic automatic spectrum coming from a smaller group in their frontal realities so so in this so new breed of stable choice formula and all this on the spectrum side that should look with something some on you know some of aberrant eagles on the geometric size user button on the others know maybe a limit on the spectrum size should be something like some of a logarithmic derivative of our function so one can attempt to use to look at the points of these I mean the residues of the right hand side to to to have information about all the polar function and therefore to isolate the part of automatic spectrum that come from from H and and there is also another variant that was the same you know the same order of idea is to build trace formula with with instead of D lock just l function incident and so for my two-day talk I'm not I'm not a no concern with this approach try to isolating isolate the autumn of respect to coming from from tolerate is better actually on the use of this formula this is kind of formula make a mmm judge more precise later but kind of new to stable trace formula with the on spectrum side some our function to you know to up as not pro sugar and careful Tory a letís okay so so let me say what about this automatic air functions so it mean it's come from very early in winter in the one of these revolution of papers of lanolin thunder on problem autumn of it forms and he constructed general to morphic our functions now now with histology and with a representation of finite dimension representation at the L group then for every autumn of it forms an order between reputation up G if he boomed I don't write like this L of s by as all the productive all right and so this is convergent for this all apology convergent for when the region party of the complex parameter s is greater than something and we want to prove that this all the product can be metamorphic continued and has satisfied functional equations similar to the remind data functions so the question is the metamorphic continuation and functional equation this is questioned and 1 1 1 K that for all this is known is constant and constant that case is when G is G nln and draw with a standard representation and then all this is known you to know works by first Betim Ogawa and then by God now exactly so if you actually if you admit from toe rarities so the functor heritage would imply this let me call this standard properties of our functions imply the standard properties of automatic our functions because basically by contour it will reduce everything to the case of standard case right every autumn of our function would be standard elf function right and if I not mistaken the the the opposite the eternal truth that is converge theorem if you know everything bottom of your functions then would be able to have authorities just some follow me I'm not sure how much besides we know but I get it is expect that end so I mean in some way build in some loosely speaking you can say that in the front lip quiver to to the properties of L functions so equations okay I'm not saying that digit is able to put two coaches in mocking okay so um so almost at the same time is layer as the endoscopy papers of Langlands there was another approach to general case that for the function equation that was proposed by brother man and cash done so he proposed a way to you know to prove steady property elf function in general so and it just annoys Jessica's generalizations a Godman sake method so so in order to talk about what embankment casually I just you know just recording you very briefly what what Monica is about that big easy to to talk about the butterman cast and set of conjectures so to the standard case so G equals g ln and Rho is instead of representations so this is based on the to let locally so you have G Ln of F V movies no not given in place and the main heroes it is called the short space and here the space of of locally constant functions with comp a support or the space of matrices so I just denoted by M standard a name standard is so G G and n and n standard a displeasure matrices and by n matrices right so in this case you know you can do the zetas integrals so you can use the jetta integrins and fee so for AV by representation of GF v you can do that.i integrals like in like a dead series this so this is very much a very strict Morissette for universities Sophie in the shop space so but f is a metric coefficient a PI and s is complex parameters and look at this function in look at the common denominators and that give you write with your NS by standard and the main ingredient you make the thing works is the the foolish transforms so now because we are looking at function arms on Victor space space of matrices we have the footage and forms fully transforms is these are operators from the shocks play to itself so let me consider standard everywhere and when you look at the the time what you get is the very nice formula just change the full eternal form of fee and then you switch F to F check F Jack of G's is f G minus 1 and 1 minus s and I don't know plus or minus but then minus 1 over 2 plus or minus 1 is the same as P F s and then there is a gamma factors so this come up so the whole point that the gamma factors is doesn't depend it on Fein on F but just on on PI itself and so to this and then the so somehow I mean in some term come out the gamma factors is a million child form of the of the corner of the fully transforms right and and globally you have that the product of overall if by ve so to morphic then you can have a product of gamma s by V is what on places and that can be derived from the poskim summation formula on on the space of matrices so there's a vector space you have the personal summation formula okay so now so what by memcache I'm proposing I cannot have to do the same the same story of a for general case so for every for every G if you roll it should be able to construct a space short space s row depending on row the functional gfv now it's no longer this so of course this is no longer true you don't don't know how to defy it yet okay that should exist this space of functions yeah I'm going to come to this near and you should hear the fully kind of exotic fully transforms and know instead of which I'm from one that space that map is the row soft pay to itself so that you didn't have this curve local functional equation so this gamma is this gamma s PI and it should have this this formula right so that I mean so yes so okay so as Peter point out the main difficulties here is I mean this basic is in general you had here we have used on the some correction is data that is a space of matrices that have this you can add many additions and then we have fully transforms so now I so the point that actually maybe you can you can stream do without it so first of all you need to understand what should we play the space of matrices so first of all days of some V and you some very basic confusion is some of the matrices should be the range above GL n but of course it is not because here you really use the felt that the GL n is G is inside and standard right because if I trust by L for an edge from yalanji that can be extended as locally constant from the composite of the matrices so the sim M is not Leon zebras so in general are going to be M Rho if I'm even just least I mean from the first thing you should see the M row should be M rows will be an Trebek varieties maybe for the first to have normal of Phi and rubric varieties containing G add an open subset such that G times G that's option of G times G on G by left and right translations extend to M I mean look all this integral that is the basic requirement that you you want to action to extend to your varieties and this is actually some you know but I'm surprised that I didn't know it before but this is no it's going to still reductive mono it this is what we add a mono it that was a failure goes developed quite independent of everything else by people on took on the butcher render and VIN burg so so the very optimal rate in one sentence is described combination of theory of reductive group Latin Boren book once I and toric variety and the tae-seok a hormone in a vector space no but that yeah so that is most is in size and bigger things but they are not connect its very own varieties but so we are tamaño it is the a fi normal f ing back properties right with the multiplication structures from on which structures so that the invertible element form an open safari which is reductive group okay well I mean the basic thing that you each want to look think about them mano it is a local representation of finitely many representation of not necessary reduce what can be reducible and then it just take the closure in the spider and morphism that gives you mono eating that is so there's very nice every way I mean surprisingly nice description of reductive knowledge is a follow so if you have G in M right so M is relative mono it so again it's normal affine algebraic varieties with multiplications and G is unit point so that is inverted one point and it's open subset okay and then look at the maximum tourism inside t mg right look with the closures the closures is empty is closed well I assume it is normal so anything maybe not matter now I control Reiter here so let let let's forget about rule let's look at any any deductive Manu it right I try to classify them right and then I look at GA look at the tourists a look at mt closures and this is your toric varieties because it is closure of tourists and toric varieties going by some side right so this is given by some solid conducts slowly convex rational cone or something like this right so some conveyor you don't want have a line inside right because is optimal to be a fine and what I want is this cone so it is NW equivalent cone because have the viable option right so basically you want look at the vector space of characters or cool characters I mean real vector space of T and you want to call you know Sonic invest code that is stable by a group so you're already CD you cannot do it for some a simple group because the vibes just move it around and it just cannot have so many convex code to generate the whole vector space so you have to be reductive not something simple and this kind of GM the center is is a mean it's very essential to be mono it so now come to make a more question you can build so the simplest model you can do is is as follows simplest so we are not even the GM so what what we actually do assume that G is half-gay determinants map to GM and B G Prime and you also add sooner in do inside we have so assume is do split I do not to cable Galois subjected to split things so it is C star and you assume that we have Rho is a representation of G hat in GN via bro and we are assume that the syste app by scalar multiplication so that means the C star here this is a identity right so so now just look at the now we do have the you know when you look at the the space of a vector space of core characters of T everyone with the count of T then you have a call the conscious given by the the weight of this so it's something like this so you have the labels given by the determinant right right and here is the weight of ero and the second level is a weight of simsim square of e Rho and so on right and then you have a core and because we have this positive direction that make your call is Suki convex and then you have your dollars and let this this are counter therefore now Rho is an irreducible and you can want to talk about where 24 irreducible I mean the non-linearity is very interesting but I think confused but probably have to go away from the hilltops Stuckey stuff but so now just stick to irreducible case so they have this mrow this yeah now algebraic varieties M rows contingency right okay I am fine i I erased my the bottom and crash-land proposal but we want that so the M row is going to play the role of the M standard in general so when when I mean when when standard KMOD example Sparrow matrices but the the main I mean the first difficulties in general accept standard case M lossy always singular so this is a always singular and many single I doesn't make sense of a smooth function anymore right so X away from the standard case the model M law is always singular yes the con yes yes and because you know though because it be determinant it and it's not under so these the wave the corn belong to some size space not on the vector subspace okay well it is civility in the end other way but the space this mean the space where the for modifier is moved right so I made once I don't know if we will see it that way so keep in my students erase it but the MT here is very important historic varieties it's not just a device to construct a monoid but you come back to the MT is going to be the space Edison or later just before I write it okay so now I go back to the shock space so in this image when Rho is standard representations then s standard is the space of locally constant functions with compact support in M standard right now in general we don't know we just I mean the basic guest if you do more or less implicit is changing environment crashing papers is it should be function it should be some kind of function with compact support on em roll FV so I mean the the non-compact support of this function become compact emitter the the closure it come Compaq in Siam row so the direction I mean the but it no longer max and talking about smooth function because the spacer is not smooth right so we need but instead of that I need to impose some intelligent conditions so this kind of asymptotic condition on the boundary well may seem little contingent that mean that is this the condition that defied especially local right look on that mean this this should be the space of compact support section of M Rho F V inside some shift I mean in some shift of the shift would encode on this asymptotic condition I don't know where it is but I mean had a presence to exist okay so but what what do you know I mean I do know now is something in this much smaller but it useful ease in this case I mean as for the trace formula into this me that spare by the way tested base test function for a choice formula and E one to have two ingredient let's pair up test function but also one distinguish vectors right like the so in this hair that one distinguish vectors is the characteristic function of M standard oh the Catholic function of the of the integral matrices right but I try to just for me I want to put in a trade show me like this to do is so so by this function is cool is the property that when you look at the it trace on the under my thigh representations that we need to under my thigh local factors right that's why we give you what well until I want to do the same function so this contact let me call it basic functions for bit of basic row of V so the function with the property that trace of yes trace of beta the row on on PI V this representation of local use be the error factors or L at some specific value so in the good much in the word magic KK would be this n minus 1 over 2 so this uses some numbers I don't know it is something like rose again the again the half sum of roots or something of PI V may be a plus minus I have it my Nobin I forget on that side for when PI V is under my thigh and 0 otherwise right so we want this function so this is going to be a environment on the left on the right by the maximum compact right so it will go to clean on the under my fairy representations and you want to the have this properties and this he actually have the nice Pope the system it is the can prove that it is actually give the by the the I see so we start form to let briefly move to the function fin case even I think it is not a serious issue so let F be to be F Q of T and O V to be F Q of T then this place mrow of all become infinite dimension varieties over FQ so you can do the arc space of of M Rho is empty and I claim that this the function that is defy this properties is given by the singularity Razak space listen so this is IC of L M Rho right so this is the result of 20 I would cheer myself and subtle LEDs okay so that is the the first point now the sick the second point is now we seen conduct initially they want to have this s row the whole space of functions and I don't let me just play just put off some some very optimistic conductors is at least there should be a there is a canonical resolution of singularities of em row maybe M row yes so that the Sharks play with the space that now is most encompassing past functions on em yes well it ladies seem to be I mean I it's not just a wish I mean I think they cheat a checksum at least with some small case writing the gn2 and seemed to since we I don't remember but at least the that ki can construct can recall resume singular I'll give this sorry I think I mean I haven't do everything carefully but the resin synthetic for every cement for gn2 for everything yeah I mean now I had to put check together made if if it works into a but nurse I think that the whole point is the in this series I didn't know very well before this butcher and render theory that just not described mono it but it's the common so the found Tory attribute is Molly to map between them just given by some some very simple recipe with corns make on G to call these very simples so you can know on the maps and to grow on this thing together you get a resolution to credit in generally have in this much more messier many many more corn so I get lost with any two-dimensional case but but okay so so now I mean this do I have now let go to see what it is fully transform some of it okay maybe before we move which time for sex I'd like to you know because I won't have to meet left okay so let me get the item about the choice formula so now we have the so we have for every place v we have this new swap space I mean we want to have yes GFP and with one specific vectors beta V beta Rho G right maybe this can builder the Adel across space so that would be you want to check the s is finite set so then the s V V in s the pro GF v enter the limit inductive limit on s right so it just tensor together finitely many places in there take a limit and the limit is defies you have these specific vectors so it just like me concretely just like us for Vanessa put the any function in the short space and away from s we had to put the basic functions so it is complete like in this I mean it's very similar to the usual case basis this is a place of compatible function and this is the cottage facet of the of the integral point and now you can do this of course I have no idea what happened on the Archimedean price but it would be very interesting to work out what's happening and so so the choice formula that was alluded in landlines papers or some people with function and myself is we actually want to constructive an invariant linear functional on this rock space right so that is you see right now expect on site which is one look at the choice of this function on automatic representation that and it just be called this right what do you have you're not spectrum expansion is I mean and I've just discrete sum that should be something like LS of a PI now s is on way you know more s because I put eight to be this special new G Libyan goes after a big rain chance late so it is only Roby come here and PI right and the opener in the and the geometric side is of coitus or burn integral of F the F in this rock space right you've gotten out you know I have as maybe a stable opera integrals maybe a multiplicity and another choice of a product of the inside s a tricep ID every okay so in this way there is no limit process to be done the limit process is with using the with the lower derivative error function here just a linear function on this I mean it had to help me hopefully maybe you need to some some continuations but each other attrition is no longer it it's just the linear form on some big some some other victors place sorry so fighters rightly formally yeah nobody I guess this is the the kind of formula that just mean walk on its some case at least I mean I strongly believe it is I mean the problem is doable in is nothing like Robert Lee riveted other I mean I look at the geometric side more you know I'm better dramatic side but I mean on the geometric side I don't see any analytic problem very serious and they just look very much like like the usual place okay so I think that my time we actually over so I would well if you want but if you if you are not gonna no idea it is better if you are just say one wood is so so environment Kassadin proposal proposal t1 t1 to let assume that this fully turned from is defy right and then the Battlement cuts done popular will be the sum over on G of F now F be some local function global function f da F of gamma you can sum over the same thing of the Fourier transform of F of gamma with some local conditions so that we don't have to worry about the boundaries right you have local condition so this is the kind of nonlinear possum summation formula that by mancation you want to prove I think that it may be more realistic of you to put a mean whitter either just integrate the whole thing over PA ma GA right and in doing so what you get you actually did exactly this I mean up and here you have under on the left hand side you have you have the EDC items in the trade em in this linear form each of the Intel the integration of this sum over the automatic quotient so it have a both dramatic expansion and spectral expansion on the spectrum expansion what you get is some of our function right on the right hand side you get exactly if you're not mistaken the asymmetrical function right so the daddy what you want to prove if I drew function equations and so I mean as you learn you want to prove it by working on the similar forming for non geometric side right so just really want to prove something like this right okay so I mean of course you need to know more about what is f row is to say something but if you know and talk about this afternoon