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3/4 Automorphic forms in higher rank

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3/4 Automorphic forms in higher rank
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Hey a right now so
we were talking about how dimorphic forms and congealed let me briefly recall what we did last time of gene is PNG LNG and we have 3 important subgroups and came and terms we talked about this said the differential operators gene variants on the whole and we had our chances to function which measures the spectral density we talk about Whittaker
functions we did functions are hiding functions of all the differential operators and we had some formula for Whittaker functions and so on so the 1st thing I would like to start with 2 days expunged and while there are several possible free expansions depending on which group you want to free expand on and well 1 of the most common i versions of the expired with respect to any and for expunge is always easier the group is the of and if any false student in a civilian but event is not to them and it's not the BN has only a small abelian part the deal no part of it it's sitting here on the 1st of diagonal and Everything else is a bit harder to treat that's where the 4 expunging looks a bit complicated if you have a masterful he has an additional site Over a low-dimensional group de-mining itself in my mind that modular In the end like this 1 that so and the unit potent upper triangular group and of dimension minus 1 some of the coefficient and 1 of 2 mn minus 1 different from I'm being a bit sloppy with signs here there a plus and minus signs that I'm going to ignore it and 1 up to to him and minus 1 normalized in such a way that these are of absolute value 1 on average but again there should be some the absolute value and perhaps but I them so the MNJ workers with a certain power
and the power Jake times in my jails over it's not important just normalization and Whittaker function the Whittaker function has the index New which is the spectral parameter of the cost for 5 and then the diagonal matrix M them going to ride on the 2nd day a matrix that in the upper left block in mines and block contains the matrix Scott and and the point then and then the matrix why and while I have to bear in mind and what so here this is a major exporter the business product you that this is a single entry that a diagonal matrix press of died matrix these are the foreign corporations the paper and 1 up to him and minus 1 of our great coefficients this is what we call a efficiency and they are related to the set target parameters and to had guiding values In fact will see later that they should polynomials In this attack at parameter all right and you recovered before a coefficient by integrating the form 5 the group so a M 1 up to and then minus 1 divided by the product of the 2 The Power Jade times and minus J over 2 times the Whittaker function this year the product of 5 of you it crimes creative mind let's see 1 you want and so on my list includes minus 1 you and must you take a character of this group here an integrated against protest against the mosque for and His produced for a confession so now I would like to speak about had cooperated and developed a bit of a fake theory on on pg so that people Prime which a prime once and for all and don't country a cooperative's so operators are parametrized "quotation mark sets and so the typical representative visit is a diagonal matrix with people entry and thinks interplay of exponents as for occasional simplicity right capital it's not this group it's just the name for a diagonal matrix the diagonal appealed to power 1 peak 8 I'm so don't always this and that and on we decompose coasted into single process the disjoint union of single "quotation mark inter-regional locals we can associated had cooperated I call it chief of P and or simply to added maps of functions 2 the function that I get by summing or by applying all the schools represented the
argument and this is obviously well defined and again don't mind very because this was a doubly although it's clear that but the Bay Area is to clear the because a matrix With all interest the same and this is also a permutation variant TVA is the same as the signal of false statement by progress and and the at joined the team in the months but in particular but without loss of generality I can scare this in way that will 1st of all I can order so that the smallest of the insist that this so I can make an important in descending order of things and then I can make sure that the last 1 0 simply by scaling so without loss of generality I can assume that anyone can be given to them and so they can assume that his ears I write absolute value for the summer of the AJC and then I find something that's but I think of as the volume of such an Interpol so again if the awarded in descending order then I defined himself to be some J. J the ordering is relevant of course so the first one gets waited just a little the last 1 gets waited alone all right and I defined the bylaw banning W of a just the orbit of AT and divided all signal when source with is and but 1 of the main problems in every hiccup algebra is the multiplication of 2 he cooperate you want to multiply to as you want to write it as a linear combination of unethical operators and that's combinatorial made possible but the very complicated the thing to do multiplication is called but of course we like to do complicated things so did you multiplied to hack operators associated with 2 double process but then you can write it as a combination of "quotation mark with respect to certain matrices that's clear
but the question for which the is this coefficient of the 0 and what is coming to get you compute given and of course is determined but if I give you 2 matrices a and B 2 sets of couples and you can compute a Ltd well you can write down an abstract formula but this holds very little but so only in those the appear whether "quotation mark set down by the Gunnison as contained in government aid gonna be gone and that's something you don't want to compute and then all of this is viewed as the number of pairs and Jake such that gunmen it is gonna Alfred J. B. well for J & B decay of the given coasted representative on his own principal Yukon computer but it's very complicated under bright coming up tourists have worked about this work and there's a lot and there is something called Young tableau the story might give you 2 temples a and but you can write minister can't have it all and then you can find out which of these Walker and this is called the little condition but that is common story a challenge the head to operators of course come with a differential operators because the the operators a gene variants and with themselves and are normal With respect to the usual and act and know the item value of the but must form but that or perhaps the number of people might think you want to emphasize the prime what I said earlier that should polynomials play a role and for those of you we haven't seen short polynomials let me find what they are show polynomial so if you have any trouble the natural numbers including 0 in descending order we define the assure polynomial instead To be a ratio of 2 but determined so it is that is has an entry that the polynomial in any variables take 1 To the Power 8 1 plus and minus 1 up to power in class and minus 1 and then you decrease the powers 1 by 1 and finally makes 1 poet 1 to excellent power this is 1 determinants extend you and you divide this by fundament it's on to the power and money was extend in mind and takes 1 this just 1 1 so this thing is the show polynomial associated with the it's obviously symmetrical normal and some polynomial of rain absolutely and it turns out that the Fourier coefficients that defined up there are just truer polynomials Anderson tactic the aim of Peter the Power became 1 of the people the Polish Karen -minus 1 equal to the market and a bit of space for this show political attached the following and which so OK 1 plus 2 taken minus 1 and then you remove step by step that term and so the last entry is just the 2nd loss in free skate won and the last was so this obviously descending order the use of feisty and Topalov syntactic parameter so in particular the 1 1 1 1 PM so that only the 1st entry appears and everything else is 0 by the way this is not 1 that is 0 the best 1 0 0 0 all of which is that just as you can see from the definition of the south of the Alpha this is what is usually called the Peace Corps
OK 1 way of trying to understand the hiccup algebra it to use the kind of spherical maps the names of the setback by some officers talking that's not that attaches to the Czech operator polynomial sort of a polynomial and it's a ring I and so multiplication becomes very easy because you just multiply 2 polynomials and everybody knows how to multiply 2 polynomials the disadvantage is that it's hard to compute the image it is useful to be normalized but the had cooperators Irene normalize them as follows the children from me the same thing but multiplied by P people to the power of the affair OK and now I can write down the map only the man Omega maps he Taylor To a polynomial With entries rational with only P denominator I need space so that 1 piece and then it's 1 to extend and it's symmetry symmetric polynomial coefficients having only played nominated and as I said the images very hard to compute it depends on the explicit calls representatives so you have to no explicitly the representatives and then do something with them depending on explicit costs representative but it turns out if you if you don't want to know exactly what the images but just roughly what the images that it's very easy to up to a small error term just a short little and the terms of size 1 of peace but this only a polynomial whose coefficients unbounded by 1 of so peak it's large then you can very well approximated the image and this map by the shortly In this notation you can right down the the eigenvalues the idea value 0 for the for the had cooperated the UN normalized G A of the apply them and Omega then you get a polynomial and in this polynomials his substitute and left 1 over to times all well fathers target parameters of the sales so in particular we can guess what we expect how big or how big should love the baby In this normalization well the shore polynomial has mounted coefficients so the expectation this is what you would expect that the size of land that a place and this normalization is roughly I have been normalized this by a factor the topology of a so Peter Paul Myners here and this is a polynomial of degree in some of the green the absolute value of I and I planted Peter and plus 1 over so plus 10 1 2 times absolute value this is roughly the size that we expect To give you an explicit example how we how we can actually work with the SEC operators I would like to show you how to construct an amplified on how to construct on the and my aim is to produce a very explicit amplified the the fact that abstractly and amplified exists can be what can be proved abstractly there's a beautiful treatment in a paper of action together with Leo Silverman they give an abstract arguments about why an amplifier should exist and how you constructed you can learn a a lot from reading this I would like to construct a very explicit and so on see also said the so the whole idea of amplification is based on on Hecker relations so but we need to relations like London peace squared equals 1 plus planned peace this is the prototype of a hacker relations and we want to have something similar in general how can we and ending of obviously there must be some relations and this algebra but I want to have something explicit so let me 1st show you In the language of the attack Tucker maps but what this looks like I have a bit of space he left and the need to Clear Lake we have so let's look at the case and wants to is the goal of operators of interest for us there a cooperator T 1 wonders just the identity and the image his ex-wife sorry this is supposed to be 2 1 0 so that's that's really the people had cooperated and the images X-1 plastics to and then we have to agree to cooperate the key to 0 and it turns out that the image is excellent squared plus thanks to squared plus takes 1 2 times 1 minus 1 of peak provided by Key West and then we have key
1 1 which is simply the identity and I think 1 time 6 2 people here OK and now we want to but cookout and identity of these 3 operators but and we know what we have to look for the sum of these 2 if the square operator and is cooperated so we want to square this 1 and summer these 2 them but there should be an identity and indeed the 2 1 0 PM square miles key to 0 please plus 1 1 equal x 1 x 2 divided by peace with that and that's just a piece of and recalling that our had cooperated the piece a defined here on normalized a bit differently but we would expect that the TP cooperators roughly squared off peace so if you squared and you get P and to get precisely the had relationships on and on now we want to do something said question story which is they Of course not complete with the book and of course not completely multiplicative but they are multiplicative what they are multiplicative if you have an idea form of of of all had cooperated and that's simply the Chinese remainder theorem but if you multiply to heckle operators for different problems than you can just multiply the diagonal matrices 3 1 prime time on the list of under combinatorics but I defined capital Place To be the set of all petitions of and by which I mean the following all improbable anyone up to it In order such that this summer equals so 0 of course must be allowed to some of them can be 0 and then someone's and so on and the has to be and that's what I mean by pies and I consider the practice of polynomials with say complex coefficients although this is completely irrelevant to the variables that a symmetric and of the degree so all symmetric polynomial of ingredient in in variables and it's obvious that the dimension to the continent and I will construct if the relation by looking at various bases of let's consider areas the 1st phase of the obvious all when meals where the is and the buyer orbit of some 18 and runs from so this generator these obviously symmetric polynomials and they have the correct agrees and every polling owners of so this is sort of like a canonical basis for now I would like to look at the 2nd basis we have look for basically I take the show .period as bases it's as elements so I take all a pipe and I looked at the team had cooperated normalized and applied this attack and this gives me a polynomial and I take the collection of these polynomials and this is was essential the polynomials because the image is up to a small area of the show polynomial plus a small but then of course it's also I mean if this is the basis than if I disturb the base of a little but it's still a basis and so I can really writer inequalities but it said to give you an example In the case of a little war we have the heck operator T 4 0 0 1 2 and 3 1 0 0 2 2 0 0 0 2 1 1 0 and was piles for 5 these the partitions but we get and the corresponding 5 show polynomials form the basis of the victims only the most interesting is the 3rd 1 that I'm now going to fine I bye bracket Jr the can topple J 0 0 0 0 and I look at the had cooperated J 0 0 0 0 0 it then this attack they say just the sum of all of absolute
bouts of of the region plus a small OK and I claimed that the following as basis and I will prove that having proved that the current bases which I'm going to have to define his basis will automatically produce the history it was probably the only proof that I'm going to include in this lecture series slightly the following basis for a claim that the generated but the following so I take this attack a map of the following product of cooperative's so I take and then toppled they and it has increased anyone up to here I take the entry age 0 0 0 0 and multiplied together J from 1 too and I claimed that the collection Of the polynomial generate so again let's look at an example just to see what afford what is happening here is very simple just potentially a bit complicated if any of fall the these are the following had cooperated and there is still a T 4 0 0 0 0 than I have 3 0 0 and 1 0 0 this is the 2nd this all the product but then I have t to square up to 2 0 0 0 square this corresponds to the toppled 2 2 0 0 and then I have team 2 0 0 0 and 2 1 0 0 0 square and I have to use 1 0 0 2 4 these are the 5 had whose image I'm going to do and so you see the important thing is that this collection of not contained the identity and so if this is then I can write the identities as a linear combination of these and then I will so what I have to show that this is really about is approach proved to be a partition it and so as I said somewhere out there the image To until the age is this thing I take the product J from 1 end and then some text to the power bill where has absolute value age plus a small error of sites 1 of appeal but I do know this for a moment and just computer what this this thing this polynomial and I want to compute what is this polynomial and then the collection of all these polynomials should be what turns out that this summer a prime prime in the in the pursuit of petitions and use them all all the and vial orbit of prime it's the policy and have some coefficient his pencil agent prime and what is this coefficient With the a prize equals the number of pages CIG non-negative entries such that the rose from 4 columns on murder and also some of the guys it is gage and the summit of G is a pride so if you prescribed in prize the new to and toppled and you construct all matrices which give growth sums and given columns that the vector of growth funds should be the main victim the vector columns and should be the prime factor the other way around it's irrelevant because obviously symmetry so this the coefficient and proving that this is the basis is traveling to showing that the matrix see a a primary fight all possible pairs of petitions is inevitable thanks to show that this matrix is it is stated like this it's a perfect competition problems you construct the matrixes forces here here's the matrix the matrix has dimension piles and each entry is the number of matrices would give role columns show that the matrix
is convertible the market a good competition mathematician and you can perhaps sold it on the spot I'm not going to these problems just the 1st few cases by hand or by mathematical to see if by finding pattern here so showed that the matrix to a prime going into a primary high and is on the way OK so I computed determinants for any puts 1 in equal to any of the 3 and for spies turned out determines what so this cannot be a coincidence so show more strongly show that the tournament equals 1 but so in the case and called for the matrix looks like it's 1 1 1 1 1 1 2 2 3 1 2 3 4 6 1 relief for 712 1 4 6 12 24 and hesitant and obviously it's a matter that has to be some showed that the terminus was when this case just computer terminals 1 on that Mathematica computer but how do you show general and I played for a while I computed eigenvalues doesn't help anything but then I computed the Huesca decomposition turns out to be have a very beautiful decomposition as they transposed when a gain is 1 1 1 1 1 and then something fun so in this case just 1 1 1 1 2 3 1 2 3 and so obviously the determined of things like this cannot be a coincidence typically vis-a-vis the decomposition square roots and this is an integer matrix and this cannot be a coincidence so Is it must happen in general that this matrix but has you left the decomposition with these numbers held something and OK well then I looked it up and I found what is so large Bracken ,comma at Torrey people have known this for a long time it is the so-called Coskun matrix and these other costs that are well known in the theory of short polynomials and so this is well known that this matrix has this decomposition with an integral matrix and once on the diagonal and then everything's OK so as the corollary the success of linear combinations it bounded coefficients how and by the way so all the time I ignored this error 1 of peace and if I really worked with the images of the shore police of the of the operators themselves it's a total mess but it clears up if you work with Shaw polynomials if you let people go to infinity right so you just look at sufficiently large crimes it turns out that this matrix if you define it properly but the without so we with the images of the of the heck operator is still invertible even for people to but if you choose pay large enough that the new Justice Herbert a little bit and then some the insists linear combination with bounded coefficients such that some the coefficients recalled why such but all politicians why product J from 1 2 and the yield a equals "quotation mark equals to 1 1 1 1 1 which is the identity well it's a scale version of the identity of intense and plus 1 over to the other and as a corollary of this is what you typically need for an amplifier for all there existing 1 less than the Lisbon and you amplify has depth and you have to go at most to the nth power but just as in the case and equals to you only need to go up to peace square such that London G then it's bounded from below by what you expect this is the expectation would end you really get P To the power
minus the jury plus and plus 1 or 2 about Jr and you can compute what this page appeal to the college times and my right simply because here you have some Of had operators in the eigenvalues cannot all be small if they equal the identity right at least 1 of them must have a big item value otherwise this can be and questions all right OK so this was a little bit of a crash course on the head theories on PGO N and I want to spend the remaining 10 minutes 1 just briefly mentioning Eizenstat said so far we have been talking about cancer farms most of the time although this discussion is completely independent of whether you forms a cost former not at theory applies in the same way to Eizenstat series but I would like to briefly mentioned as they don't know which 1 and century abstractly they attached to parabolic subgroups all of if you think that was this With the and you was all told you you going to get you to Peter power and I I think you can't you can't get any lower than that in the morning and you all so what about you I think it's it's it's surrender With 1 you know what but there was no way so let me move to Einstein's theory just a very brief discussions so abstractly they attached a pile of parabolic subgroups and I just show you how it works for an equal threat so for any "quotation mark freely that we start by the case and equal to because some of it inductive process but so let me start by the decomposition of the standard about half playing modular until Tuesday at the O 2 space but that is the case in equals to and that the composer into the eyes space plus the hospital's plus the cost of and the constant function should be here as a residue the offensive OK and from this we can go 1 level up there are 3 types of Eisenstein's series corresponding to this decomposition there's minimal parabolic subgroup anymore parabolic which is the that action mentioned at the beginning of his talk this measure the frequency of attendance and In a lecture series and the corresponding items that furious has a spectral parameter you just as as the cost for and the complex variables that and it can be defined by something what they basic that this is the demand for energy and and usually rotation and the power function that I find that I used also to define Whittaker functions so this is a very close analog to the usual Eizenstat series on GAO too it was for a corporations that are essentially device of so the Fourier coefficients of the following formula 1 due to the political then and then take you want you to you free and this is the game with 1 1st for a coefficient and then by the hacker relation to get hold of it just as he said the 3 Langdon's parameters they add up to 0 and it is a generalized device which OK when you get so I am used that
is the power of mentioned it only depends on on the Y coordinator for all of this the matrix you decomposes in event of a decomposition extends Wyandotte only take the wide corridors and I knew of y 0 why J To linear polynomials so it's it's some some powers of J of J & & You this month the fact that perhaps there it's not in the same rights the right right the product right from this L although in and that the exporters are chosen in such a way so the important thing about this power function is that it has the correct guiding values for all the differential operator but so you just choose the powers appropriately to make sure that it's it's adventure but so is the meaning of parabolic subgroup the maximal except and of course in general there are many groups in between but for any readers on the minimum so this is it looks like this here you have a bit of a room you have to buy to block and you can that geometry 4 the EU opposition ,comma you is now cast from Geel to and that's some gunmen and become more pp is parabolic determinant of demands that the power time you of well you would like to write it but this doesn't make sense because demand that it is in jail for me and you want a deal to introduce the 1st project this to the upper left corner with the made with the map pies High said 1 6 2 6 3 6 1 1 1 times wire-to-wire 1 Y 1 1 2 1 thanks 2 1 and white 1 this makes and the foreign corporations I wish after the heck item values of you there where basically convolutions with the with the the identity so for cooperation all the 1 to the value of the 1 times the To the power to attend 1 to the prom and the 3rd time belongs to the constant function up there so you can view viewed in 2 ways either you can view it by just putting the constant function here or you can view it by taking a residue offer minimal parabolic Eisenstein's is so that the general case you can't she what do you think whatever Fourier coefficients In this general case well that's a stupid question there are no for a conference at least known on the agenda for a confession the only free coefficients that exists for this degenerate case are the freak of frequently 1 of the variables of 0 so a M 1 and 2 with M 1 and 2 both being on 0 doesn't exist but the question look at world and I think that's a good point to stop and next time I will do something more analytic and look at the moment of In the 1st letter of the law I yeah well I hate to reference my own papers but but so do you find this a very very pretty prints on the archive by myself and Peter Martin at the end of the year the whole picture businesslike Lakeshore polynomials and and things like that you can find in look by McDonald has correctly spelled no I'm not at all and it's called think it's called whole polynomials I think it's Cambridge University Press thus far but the other ones from right right right it's right of any rights to the ladies called symmetric functions of all polynomials or something like that this is where I learned all these things were worse sure polynomials the fact that he was the grounds very nice about so let's quote this time where are the intensity of the