2/4 Automorphic forms in higher rank

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2/4 Automorphic forms in higher rank
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a role at a time and place and so on not so as I said in in the 1st lecture roughly dividing this course into 3 parts the 1st 2 parts rather short video on this so when 1 and its speed to win and then I'll moved to the 3rd part which is PG 11 and we were in the middle of the discussion of zeal modular faults and Germany yeah so what I would like to discuss briefly at this point is to pick a theory that for the year modular forms I of course only Indiana ran inside case and the ramifications I don't know if this is ever being worked out but it's certainly very complicated so damn is the group S P 2 and that so the analog of the full modular group and take operators parametrized by "quotation mark sets call of the form so we fix a prime p 6 private key gunmen and then a diagonal matrix With entries to power a 1 up to people to power a N N then P power AA-minus a & P 2 Paul are minus a 1 2 in elements and government and we can all of them such that they 1 is greater than 0 and so on and it's in increasing order up to a in and then it continues to increase so a and is bounded by over to an hour's some integer so are as basically did the nitty-gritty of the of the Hickel operator in particular are as 1 then there is only 1 such a procedure that's simply TP and by slide abuse of notation I just write that "quotation mark said that's just 1 1 1 1 1 and then the PPP pick up where there are several heck operators of degree to Of Peace Square that is ,comma the following form forms I have a couple of 1 I have them and then I have a couple of peace in desire for them and then again in a couple of please in mind sigh and Peace Square In particular so this is for 0 list and I listen with possible equality In particular vise 0 but this is just the PPP PPPs of theater the key is 0 2 with the identity and it turns out that the and then clearly I please for 1 less than 5 less than and minus 1 generate they get the people part of the heck Djibril so every nickel operator In please but is a polynomial In these operators they have some nice properties Hermitian with respect to the inner product that we define their commutative and of course by the Chinese remainder theorem everything is multiplicative 2 different crimes everything's nice multiplicative but if I stay with 1 prime then the Hague relations become very complicated take relations are very complicated I this is a very very complicated combinatorial problems to write down a lineage that if you multiply to hear operators and you can write them as a linear combination of other heck operators but the linear combination is very complicated and I don't think so to my knowledge that in general I there is there's snow but closed formula or not even an easy algorithm to compute what the product of 2 here cooperators uses a simple example
teepees square we would like to write this as a linear combination of degree to heckle operators and this is the truth please squared plus P plus 1 TV Peace Square plus Peter the Power three-piece squared .period 1 past the identity but this is the simplest of all nontrivial but Baker relations and they they become completely crazy if you if you want to write down linear combinations for products of cooperators and I would whatever I'm missing headed the University of was only at the idea that it's OK 4 the fall of Norfolk the new modular form there are Manoogian conjectures known just as in the case of gene is 1 of yes yes yeah I have chickens to marriage and conjecture is knowing this this work of myself but as I said that there is no really clear connection between the various attack a parameters and it's clear that for a crawfish and so what does this mean it means that this attack parameters have absolute value 1 under the former this is a list and I will explain what alleged as the animal alright l functions I'm not going to go in any detail but I I will just mention there are 2 types of El functions there is the so-called standard L functions which is of a malfunction of degree 2 in Class 1 and here the analytic continuation and functional equations I and there's the Spindale function all of degree 2 to the power then and here they added again financial equation is no the case enables too but not in general my impression is nevertheless that the Spaniard function is used more often than the standard error function down in a not going to go into into more definitions but certainly in the case any equals to there wasn't degree for the degree 5 those functions all they finally lift because they produced some arithmetically interesting objects the most famous lift is in the case in any pulled to the side of cover the this is the case in equals to and this has been generalized to arbitrary I think even an hour so I added that I will discuss this later so the idea is the following I'm we take a classical Our hollow Wallstreet cast form of genius 1 and wait to came this the space of homomorphic cast forms on the upper half the usual up after of weight to minus 2 and by the theory of half interval modular forms as developed basically by she more than you can associated this half integral weight modular form With the following for expansion it has some coefficients that I normalized to be of absolute value roughly 1 so the exponent of over to 3 over for the U of M Z and this is a how integral form of weight K minus a half of level 4 and it is in accordance plus space which means that M is congruent to either 0 or minus 1 more for so only half of the coefficients Walker didn't
most producers theory tells us that these coefficients but Central twisted Ellabell use of the original function so this is up to normalization injuries only to find up to a scalar multiple but
I can choose a normalization such that the coefficient is the central element you hide the at least for the fundamental discrimination Of yes but yes vote
for none fundamentalist criminals there is also formula but that's a bit more complicated and then I can attach to GE or these coefficients as modular form as follows so I proceed on this blackboard 2 the 2nd had as RealAudio formed capital Air Force that it has a 40 expunged some over matrices to 80 please feel free to do that and that's a zeal modular form of weight case for full of them and the 18th argument explicitly and in certain cases they can be described quite easily if determinant of 2 to to the case over to minus 3 4 climbs the West sees this foray coefficients coefficients determinants of 2 and this holds at least give minus 1 To the Pollack a minus 1 tends to be at 2 key is the fundamental discretion and if not then there is the slightly more complicated formula and that describes the streak of fish I yet so the flurry coefficients are up to normalization of while there basically the Fourier coefficients of this half integral weight modular form so capital this so-called side cool ,comma lives of either G. M this is also
sometimes called the Maastricht CIG and because hands must be investigated as a lot and he called it treats yards shall so special whatever it's like it's untranslatable some but if you if you find this this phrase for large Shiite and refers to or sometimes also called the mast space so the sub space of capital F of this form inside the hall space of Sweden was that means of course only a very small sub space of forms that that lifts but this is sometimes called the masked space inside the space of all the model for what so these these lists are quite interesting but made no mention 2 results there was the chinos .period formula I'm so what you can always do this has nothing to do with lifts would you can always do if you have a Ziegel modular form you can restricted to the diagonal and by this I mean if you have a point that takes plus wise In the zeal about half space of genes too and then you can write his x 1 x 2 x 2 it's 3 plus 5 Y 1 wire to wire to wire frame and you can project this onto the diagonal x 1 x 3 Class Y 1 by Friday and then this lives on 2 copies of the of the the usual upper half plan there is a complex number X 1 plus I Y 1 and there's a complex number X 3 plus I Y 3 so that the restrictions of the diagonal restricts 2 2 copies of the upper half so so you get get a function that lives on on 2 congruence of cautions and what you can do now is you can take the other 2 norm of the year so both of these are of course equipped with an inner product and then you can take deal to norm of the restriction relative to the Al Toon Army of the original function and if you equipped everything with probability measure then it turns out that this ratio is given as follows it's pies square of 15 L 3 half if so now if this is a less so if in Of this over there and it is assumed to be hacker normalized so little heck I informed there is a unique way of doing this putting the 1st
40 coefficient to be 1 of 1 assumes square times 12 overcame minus 1 and then there's some all from normal basis so this an often normal basis all of this case the notice that it has waged to K minus 2 but here we sending over forms of weight came 1 one-half since square 5 times there so this makes good sense meaning this somehow size K and on Lindelof all of these are essentially bonded so you divide by and all of this is also bound so roughly speaking the and the mosque is rather evenly concentrated on the diet it's a very beautiful formula it determined it compares the mosque on the diagonal with the complete Mars and that's given as a mean value of of certain degrees 6 the health functions all is about 1 thing and the other thing is that the function so did this been L function L spin off this list agrees with the original L. function of the past form little if up to some simple factors Zito this Mindscape escape plus 1 times veto of this minor scale plus 2 times spelled death OK and decide to go ,comma lived has been generalized are too arbitrary values of and by Walid was conjectures that projected a thing but but you can among group and was then approved by so this is called the generalization to arbitrary and this was the key which has similar features but it's more complicated for instance in the sense that the 1st sight of goal ,comma 1 can give the Fourier coefficients exactly even fun on fundamental discriminants is a complete disaster in general there are formulas but there at the end of manageable so finally a bit of literature for further reading I'm a very good book and I think 1 of the standard references is spoke Ziege Shimon reform which unfortunately is in Germany but nevertheless I think it's 1 of the standard references there is a shorter book that basically contains a subset of flight textbook by Clinton's listening in English introductory into introductory lectures to Eagle modular forms of something like this there is a very old book by Ziegler something which is called simply the geometry and it was in fact a research paper that appeared in The American Journal of mathematics but it's like 100 pages long and then they decided to make a book out of it so it's it's the exact copy of the original paper but this is the unspoken takes much of the underlying geometry OK company questions the analytical intermission functional operation of this operations in general no no no I think for any calls to him and I'm not sure what the state of the art as with any quotes free time perhaps there is some partial I don't know why but certainly not in general under any other questions I was there was also part of London's nite he found should be for all the Congress of South Miami some of them forms on 1 of the largest owners of the family .period he then yes yes yes sure that so I mean and I'm not going to go into the into the greatest and most most grades detail from so certainly you can do this for Comoran subgroups as well on but I mean this nice formula of Cohen's added to my knowledge exists for the full modular group although of course the general some framework calls and holds in generalities in any case and this certainly is possible for 4 ,comma subgroups that there's nothing special about the modular group except that everything said it is to be held in the world the head of the problems that led to the underlying assets Walt ,comma notoriously there they from it's it's a different combination of of this attack a parameter I mean they are of course defined by our the products and the the on the factor a certain combination of this attack parameters and you can do this in different ways "quotation mark he was forced to those who don't envision as long and overlapping of functions of local the question OK "quotation mark then this ends my short discussion of zeal modular forms and certainly now going to move on to PG but it's so here's the set up all Group and G is PG 11 a foreign I won't mention and in this In this lecture everything is either over our old or over QP but no audience of hasn't U.S. as our decomposition in a cave end of the year pretend matrices K is a P O N gamma for us it's just a cell and that again you can't talk about Congress approves but I'm am not going to do this age this is now the 3rd age it's not the Ziegler behalf claimants notes hyperbolic about half space but it's well a model forging what came the dimension of age is In minus 1 times 10 plus 2 2 over and we choose the following coordinates we choose the Mazama accordance coming from a and k so .period there Little that perhaps it is extends and takes this 1 1 1 1 1 something and why it
is given by 1 Y 1 and so on up to Y minus 1 why 1 OK and W divided Europe OK I so for then greater than to strictly bigger than true but there are no homomorphic forms so everything is a is a must form and so we're we really think of these forms as forms and so we talk about differential operators the it is the algebra of this gene variant differential operator on page and that's the center Of the universal enveloping algebra and this turns out to be a polynomial rings polynomial ring in a minus-1 variables and this can be viewed as polynomial functions on the duel of the Leite Gibraltar Aso 8 is the group German a busily algebra this is the dual vector space and I take all symmetric policy so can think of this as polynomial functions and they should be invariant under the vital this is the so-called shunned rise office and going from here to here so phase the Leo Gerard and it's isomorphic too are to be a minus 1 but we view it as a hyper playing an and when this is the 1st reflection of the fact that we're working with the projective group so we are working with the trace 0 hyperplasia the trend is 0 type of player so a typical element in a house in components and they add up to 0 Tracy reliable clean it up no all they added besides the morphism here since a differential operator 2 well up the map landed the that goes from a star To and for instance the Laplacian the law those 2 I the map that sends you which is new 1 up to new and so this is now an arbitrary element in in a while in a star 2 In cubed minus and over 24 plus some of the square yeah so to reach to each operator In script there is attached a polynomial and this polynomial tells you that if you haven't and I and how to move it forward with these spectral parameters then the item value and the corresponding operator is this .period but it so for me always knew where the notations foreign elements fall in potentially the complex sification of the duel of the algebra and saying modular the buyer groups these are the spectral parameters all Langlands parameters all that out of Norfolk for to each multiform comes with such an Interpol of spectral parameters and if you want to know what is the at the item value of this autumn offered formed with these spectacle parameters and just plug the values into their respective polynomial under the Sizemore it's a case of mean but everything is normalized such that the Roma Nugent conjecture rather Nugent conjecture says that the news real but other people normalize differently for instance in going spoke the unit axis is 1 over and plus ii a star and there some other people other office say that the unitary Exorcist II a star this is a matter of taste of for me the unitary accesses a start although as you know the Roman conjecture is not known not even for any calls to and the best general bounced are imaginary part of the U J is honored by 1 over planned over 2 months hard work plus a half minus 1 away and square 1 and for any votes to better balance unknown so are afforded in equals 3 and 4 but in general this is the best balance and this is due to rule and OK so How many mosques forms are there and how dense are they in some sense and this is
measured by the horrors Chandra C functions there is a general definition for the harsh and receive function for any leading group in terms of root systems I'm not going to go into into detail here but let me just say that our case of the group pg ln the harsh under C function L is given by a constant times the product 1 class of people than J less than carelessly called end congenial blundered J. minus slum decade where London is let no 1 up to London and in stock perhaps in the city I while the defiant on the case well yes that's right except that land from these integration variable later I integrate overland and keep you fix for my favorite automorphic fall but yes they play the same role that energy is the quotient of of government functions that you can simplify and it just looks like 6 times tension hyperbolic of pipe so darned hyperbolic of course only plays the role of excess very small otherwise this is essentially 1 but so geophysics for practical purposes geophysics Texas and while infected absolute value of because of excess negative and also the tension hyperbolic is negative so this is bounded products of 1 plus London share slums OK so unleashed there is some there was some conspiracy between the components of London but if they are sort of generic if you choose a a generic lavender such that all these differences are roughly of the same size up then this is the normal flounder To the power In shows to if you're at the walls of the buyer chamber when some of these are perhaps identical or very close to each other then of course the spectral measures of its smaller so this crops had been Waltz of the budget Of the way Jr as of the density of Moss forms is a bit less but at the walls of the vise chairman and there is a bylaw and this was approved by and let Pete 2009 that this spectral density really measures the density of Moss forms so the number of Custer forms with spectral parameter London plus all old 1 so you take the ball of size 1 about a given parameter land of size land so what what I wrote down here so I mean the court the the precise dating is a little different so that the ball has to be sufficiently large in order to get a lower bound and them then if you if you expand the region than they actually get an asymptotic formula with a with a power-saving evidence so this counts the number of Muslims for instance in the case in equals 2 but you get the usual by law this then holds in general OK so I mentioned earlier I'm using display blackboards rather randomly L so I mentioned earlier for any calls for greater than 2 L there is no discrete series we only have we must forms however we do have lived From Home all 6 forms of further before any equals to we do have lived From homomorphic forms on Geo too so you can take a homomorphic former jail to and for instance take symmetric square and then you get a must on jails OK any questions OK then the next thing I want to discuss the Whittaker functions Whitaker from but it Whittaker functions have 2 arguments while they haven't index and an argument and the index lives on a C-Star modular the buyer groups and the argument lives on H this and the values a complex number OK and they are defined as follows we take the next in a and I write on the 1st off diagnosed as X 1 up to and minus 1 and I don't care about the rest and then I can define a character field of X is just the yield some texture is a character on character on end and Whittaker function satisfies the following properties W New of X that please feel effects times W New said so it
transforms by multiplication From the letter With respect to this character for all of that page and and it satisfies differential equation if I take a deferred differential operator the In scripted then this is London deal the London ideas this map that I fines up there times W of Z football at the end on itself this is what I wanna call of Whittaker functions by no means clear such objects exist and so on but they do exist and you can actually explicitly construct them as follows so indicated these all IT functions of of decide your script construction is as follows you integrate all of function eyes that I'm going to find as a moment I knew you take the long line elements W Long times you times that the of our view the you use the hommage on end and I knew although that depends in fact only on X as sorry depends only on why it's independent of X it's a product that's basically a power function J from 1 to end minus 1 White J To some power L. J of you I'll wear these are linear forms and music and not going to write the balance is a suitable linear polynomials basically the idea is you choose L J in such a way that this poll function I satisfies exactly this differential equations and then the Whittaker function inherits the differential equations from the differential equation 5 you don't have to take the long white by Lallemand you can also take other vise elements here and then you get degenerative Whittaker functions and that come up in the constant terms Of the 4 expunge revise instances so yeah if you take other the violence in the Long Island by the limited degenerate Whittaker functions general Whittaker functions father about violence for other land initially those requirements so it was that I was against a yearly do 1 that a the 48 will yeah well yes I mean probably brilliant can't take trivial by some and 4 jails there is no other but if you take it and I think if you take a nontrivial by them and then it converges OK so I mean OK so you can write this down in principle it's completely explicit it's an integral over the unit potent group of but these Whittaker functions we have to admit they have very poorly understood but as an example the case equals to wall then we can write down explicitly the Whittaker functions of the Whittaker function has an argument here which is a pair of 2 numbers that add up to 0 so it's new minus mu and then traditionally it's just new there is no extra information but nevertheless there is up to scalar multiplication square root of wires you said k I knew of 2 pie why and then it's a matter of taste with further normalize this or not I like to normalize it by multiplying by course and hyperbolic pioneer over to but again that's a matter of taste if you wanna do this or not the best OK function decays exponentially in you so you can compensate for this by multiplying with the course had about work didn't have to do and OK Bessel functions are fairly well understood so this is a Whittaker function that we can handle that in general for and greater than 2 it's very poorly understood the should figure have the final unit because of this isn't the end of the year I forget about and this is so that they because in hyperbolic but if you included or not is a matter of whether you take the so-called completed Whittaker function or not I have to look it up I forget whether this integral produces the completed Whittaker function but they have but we do have some partial knowledge of on general Whittaker functions and what we know is the following so it can be so you can recursively that wouldn't get a degree in Whittaker function by integrating a degree and minus 1 Whittaker function than if you repeat this you can get down to degree to Whittaker functions which abyss of so general Whittaker functions at intervals of a of options so it can be expressed as iterated integral all the Bessel functions of a overcame special which but of course it and it's bigger than you get a whole bunch of individuals and things become very complicated but in principle there is such a formula and that's you to state Eric state I would be leaving the spirit of firm of Emanuel's talk that this is a reflection of the fact that um the Jilin Klostermann some attached to look to the long by an element at least in certain cases can be written as a product of jail to Klosterman comes at the modular appear ways co-prime and I would assume that this is sums of weak form of reflection of this fact that in in certain cases you can reduce the Jilin case to products of G 2
for an eagle 3 on the Big Morrison on for any equals 3 we have a fairly explicit visited I'm not sure I have to look it up on what it is but I think it's more complicated arguments you well perhaps did you you and 1 over you and what it is I don't know exactly I think it's not just the multiplicative convolutions a bit more complicated than that but you can look it up in States paper we know the double Mellin transform all of if we keep gets fixed then in the variables that go into very why I have to entries why wondered why too so the double Mellin transform In why why wanted the power as 1 way to to the power is to BY ONE DAY TWO OF a Y 1 way to is not at it's in fact what you bloom well brother I don't know if you would expect this but certainly is the ratio of gum off so that's fairly simple and then we have a very important formula that's a very strong tool and it's called state's Formula One this is the art comedian Rankin said the theory it's the product of 2 Whittaker functions if you take W alpha of and W beaches of Y determinant of wide to the power and then the correct measure on the group a so in the classical case this is a product of 2 Bessel functions but now you take a product of 2 general Whittaker functions and this is the sort of rank a ratio of grammar but so it is what you would expect sorghum working ,comma is the usual thing I'll be in the 2nd it's best plus ii Alpha for a I hope they get the science writer and Alfred J. minus beach of Boracay hope that's correct where J and K run from 1 basically take the the Langlands parameters and and combined them each with each divided by whatever you expect from rankings Alberta theory Gomaa of enhanced so that's kind of the tale 2 and that in the classical cases and then products so he get kind of symmetric squalor function at 1 gunmen are 1 plus I Alfred J. minus Elephant Cave the government are 1 -minus be time-barred minus beat-up bar case so I don't know if this is readable or not hopefully it's not readable because it's perhaps slightly wrong but and perhaps there is a constant involved but so if you want to get the Constance ride that's also complete nightmare because everybody normalized slightly differently and you can be sure Europe in the and you'll be off by Apollo to pipe OK so what is going on .period Omaha's the usual things can gum up are affairs it is tied to the minors to I can't stand still OK but yeah let me just to let me just write down 1 more formula to wrap this up and then I'm done for 2 days In jail too there is the so-called Kuntoro which Lebedev transform if you integrate the best-looking function of the against the test functions and then integrated back but this time over the index by then you get the original function back and this holds in general and this was recently approved by good friend and control rich so if you start with the test function ever and you integrated against Whittaker function and you call this if head of New but you can recover it from Arafat so if you integrate their hat of you against the Whittaker function and now you integrate with respect to you the new over the spectral measure again said that London would be my integration variable notes whatever Over the away a star and then you get expect culture the business yes yes it is OK that's true but that's that's true but yes I'm talking that both in Qatar which given very explicit formulation of this but certainly you find an abstract version of this what spoken and works well yes OK look fair enough but it on the Net USA yesterday I'm going to conclude the social and it anyway so that sent other and questions but it well that's it for today this is for the last 2 originally from you know how to the amenities and is because of the diversity of of what it was in the in back gate of from receiving the award but the north In general unions the result is just as well I mean you can't be fielded by its I so for jail free if you if you take care so here curious sitting in a star but in and In a ball of radius 1 of size the of roughly 2 to the 3 guidance of mass forms and on the diagonal and so the and then the the flatly not been ejaculate by for yield to a mail size the right so you have te live sitting exactly on the diagonal and so on but in the whole square of size 1 so this is the chief plus
1 2 plus 1 of so in this ball of size 1 you've altogether due to the 3 remorseful so the lifts are in some sense negligible 1st but this