 Okay, so welcome everyone to the Schubert seminar. Before we start, I want to remind you that our next talk is on Monday, October the 10th. So we're getting back to the original schedule. But today we're very happy to have a professor Andrea from Columbia with a special talk on Eisenstein meets Schubert, so please take it away. Thanks, Leonardo. So this is I, yeah, it's really, I'm really grateful for this opportunity to speak and I'm also appreciate you guys meeting at a special time to somehow to avoid conflict with my my teaching here in Berkeley. So, so this is the what we're going to talk about is a subject which is, you know, kind of new to me so I. You know, you'll see my command of the material is somewhat limited so I'll try my best to explain. So, of course in real life Eisenstein and Schubert I don't think they met because they overlap maybe by four years or so but so Eisenstein lived a very, very short influential and tragic life and so by the time he left this world I think Schubert was three or four years old so I don't think they met. Anyway, so, so what is this about so maybe I'll start with discussion of Eisenstein series. What is the, what's the subject is about. So, this is, I, I'll give you the general definitions but then of course I think it's much more informative to much more informative to see what these definitions look like when I specialize them in most important examples. So, while, well, I present general definition we, I think, I think the examples are really the place to learn. And so it in principle there's, if I have. So let's take, there's a will take a reductive group will take today will take a split. We will will be a group over a global field, which is not at both face F blackboard, blackboard for both face F. And then you can form this double portion when of the adelaic points of that group on the one side you mod out by the by the F value points of the group acting that embedded in the adults diagonally. On the other side you mod out by the maximal compact subgroup and for reasons that will be explained below. This is people tend to you denote this double quotient by one. And this is some set and has a natural measure. And so it's natural to study L two of that measure and in particular would like to decompose or many people would like to decompose L two with respect to the action of operators and so on. And we'll see what this thing is. But like I said, maybe the best. So best to turn to examples and so what kind of so what is a global field a global field is either a finite extension of q, or a field of functions over a curve defined over a finite field. And so if my field is q and my group is SL, then this, this double quotient becomes the double quotient when you take real points of SLM and mod them out on the one side by integer by SLM Z. And on the other side by a solid by the maximal compact which is so in our. And geometrically this is this represents a little review in a second this represents is a space of lattices in our N normalized to have co volume one this the volume of fundamental domain is equal to one and taking up to isometry. Where is the, where is the Hickey operators, what are the Hickey operators, Hickey operators in this case involved. First of all, if I have a lattice in our N, I can consider it's ground matrix and it's ground matrix is is an element in this SL and our mod s o and our. That's a, that's a homogenous space and that's negatively curtain in very differential operator on this homogenous space they, they that's part of the Hickey operators to go up last operator on this homogenous space. Some kind of discrete versions of like a finite difference operators. And this come from the fact that if I have function defined the lattices, you can summit over the following finite set you can consider sub lattices. And you're given lattice such that the quotient has a fixed isomorphism type. And that would be, there'll be some finite sum. And if you have a functional lattice you can, you know, summit over that finite set. And that would be, that would be the action of your Hickey operators. So what is somehow concretely, if I have, how would I define a lattice in an international space well I pick a lot of basis of vectors with determined square matrix with determined one so it defines me a basis of vectors, such that the, the parallel pipe the fundamental probably pipe is spanned by those vectors has volume one. And then generate the lattice of covalent one. And then, and then the two basis, two basis, two, you know, two such sets of vectors, generally the same lattice, if, if and only if they differ by a matrix in SLNZ. And so therefore, you know, more the lattices correspond to the points of this quotient. If I take one SLNR by SLNZ. If I have to study lattices up to orthogonal transformation and of course I have to mod out by orthogonal transformations on the other side. So that would be space of lattices up to isometry. And if I, in particular, my pictures here are two dimensional pictures and if I, in this particular two dimensional case that quotient is the, is the familiar quotient of the upper half plane by SLNZ. And this is the figure which you presume all of you seen. And so this is some, some nice, some nice surface with three special kind of three special points to are the lattices that have extra automorphisms, and, and also has this cusp. This is a place where it goes to infinity. And in particular it's non-compact and this is non-compactness is reflected by the fact that if I have a unimodal lattice, the way it can degenerate is it can have one vector to be very short. And so that's, that's, is it the shortest vector gets shorter and shorter, I go into that cusp. And so since I have, I have this Riemannian manifold when this, it's a Riemannian manifold and that's has, it's non-compact, but this has kind of this exponentially thin neck going into the cusp. Then I get, if I study the spectrum of Laplace operator on this manifold, this will be, it will consist of some discrete and also some continuous spectrum. And that's a general the feature. So if in general if I study all of the spaces there, there's a, there's a spectrum, the spectrum of Laplace operator and the Hecchi operator, some mixture of discrete and continuous spectrum. So another kind of global field is, is if I take a curve defined over, so that little k be a finite field, and if I have a curve defined over that finite field, then the functions on this curve would form a global field. So my effort would be functions on some curve defined over a finite field. And then the maximal compact subgroup are these are simply the matrices with value of the group, group elements that this value is in integral adults of the, of that, of my global field. And in this case, this double quotient has also very concrete geometric description. So for, for a number field is some kind of space of lattices and for, for a certain field, these are just k valued points of the stack of G, G, we have a group, we have a stack of G bundles of for that group G over your finite, over your curve C. And, and this this double quotient is really at this are they really the k value points of that stack. And so concretely, if my group is JLN or SLN, so to say JLN, then that stack is just a stack of vector bundles of rank and on my, on my curve. So, you know, K value points would be some countable set. And so, if I have a vector bundles then similarly just like those lattices if I have a function defined a function of a vector bundle. I can sum over over sub bundles such that the such that the there is a morphism type of the torsion sheaf V mod V prime is fixed. And so this gives me this gives me a whole bunch of this, this, these operators give me a whole bunch of invariant finite difference operators, and they're parameterized by first of all by points of my curve by close points of the curve and also by. If I can generally put general group G is parameterized by some double cosets. But so for, for, for JLN that has maybe more concrete description. So those are those are the things that I'd like to study. When I, as I said already before that the space is non compact and concretely say if I think of going back to SLN over over Q. If I have more lattices optometry, then for if I, if, if the radius of rank two, then the way it's going to degenerate is, is some vector the shortest vector will become very short. But if I have some rank and let us then, then the way it's going to degenerate is that if I choose the shortest vector than the, the next short is not in the span of the first one and so on so forth. So some of these will become the distance difference in the norm between some of them would become very, very large. Like for instance could be I can have, you know, two, two vectors which are much, much shorter than the third one, not in their span. And so this way this is, this is, this is, this is to say I'd like to choose in my, my sequence of vectors were all in principle I all have less or equal science some of this less or equal science, I have to switch to the less less science. And then this is equivalent to a date of a parabolic subgroup in SLN. And that's, that's a general pattern. So, in general, if you look at the boundary of that space that is stratified by strata corresponding to two different parabolic in your group. Reflecting this certification Langland's long time ago so Langland's fundamental work on the subject was done some like early 60s has been appeared in in print, maybe sometimes in the 70s, but somehow it's it's very, very classical. And so, so anyway, first step of that work is that is Langland's proofs the decomposition of this L2 and the bundles in terms of parabolic subgroup up to a certain equivalence relation. So this is this is roughly saying, this is, this means that the kind of a function comes, your eigenfunction comes from that part of the boundary so. So, and there are the spectral projections on the corresponding on the corresponding pieces. They have to do with the operators of constant terms and so what that means concretely for SL2 so if I have a, if I have a function on the cell on upper half plane most of the cell to set, then I can, I can integrate it along the, so going back going in my fundamental domain I can integrate it for fixed value of, of the y coordinate integrated from in x from minus over the period and access periodic periodic function x, I can integrate it integrated. If that integral is non zero, then it's some function of white and if that integral is non zero, then the that is that's the part that's the constant charm that corresponds to the part of the function which actually can support on the part of the boundary. So the rest, the rest is, there is the case super exponentially by going to the, into the cusp. And this decomposition there is, there's one piece that corresponds kind of the bulk of the space the group itself that means those are the cusp form and they, they just complicate in the case of a function field they just compactly supported in the case of a the decay very, very fast as you go, as you go to, as you go to infinity. And, and that's, of course, very important that's a very important part of the composition. But in the theory of Eisenstein series this is treated as a black box and the theory of, and the theory of Eisenstein series six to reconstruct the rest of L2 starting from cusp forms on smaller subgroup namely on levy subgroup of the corresponding parabolic. So, so Eisenstein series they take, they take a function on levy subgroup and you can assume it's already trying to understand the world, this picture inductively. So you can already assume that this is a sort of cusp form and second already eigen function of all Hickey operators. And so they take that function as an input and you produce a family of Hickey eigen function on the whole group G. This, this functions are never an L2 we'll see we'll see an example, but their linear span contains the corresponding, the corresponding piece of the L2 decomposition. And, and the problem so so the problems you have so you have a eigen function which are not square normalizable, you would like to know in their linear span, which is which are the things which are which are normalizable and what is the spectrum on this on that part. So that's, you know, this kind of a that's, that's a pretty standard problem like a quantum mechanics when you know something about, if you, if you know you have some differential equation that you know something about its solutions but you understand well the cases which the solution is not square normalizable you would like to know what was the actual spectrum differential operators. Pretty, pretty standard setup and in analysis of operators quantum mechanics, but here, here, here we have a particular setting in which there's particular interest in understanding that spectrum. And, and, and today we will start from the end. So I already said that this is the, the Eisenstein series is some procedure that starts from from a cast form on a smaller sub on the latest subgroup of your original group and produces some interesting function on on the whole group. And then, of course, it's natural to maybe start from the end, which is the farthest, the farthest from cast form so so in this decomposition will start from the minimum, minimal possible verbal except group, namely for Burrell subgroup. And so that is to say we start from the generic but so they, the, the cast form corresponds like the bulk of the space band, and we would like to instead go to the most generic strata of the boundary. And there on the bond there's, there's, you know, the level subgroup there is a torus, and there's not the harmonic analysis on the torus. It's interesting, but maybe not so super interesting so we might as well start with the function one which is already. It's not, you know, it's, it's the general general general character of a torus maybe is only slightly more interesting than function one. So let's just start with the trivial, trivial character. And so then we've denoted corresponding spaces just this L2 of Eisenstein series of the meaning we have the most basic Eisenstein series. This would be, this would be the series that's for us so to Eisenstein actually studied. And so there's some definition what these things are in terms of your self decomposition on your group. I don't know, I mean you there's there's the slides are on my web page and I can maybe put a reference to that in the chat but maybe instead of explaining how this goes in terms of your self decomposition and so forth. Let's just let me. So, you know, if you, if you haven't seen it before maybe, maybe, maybe it's best to to go straight to the example and an example what is it for SL2 q for SL2 q. So I have a space of lattices and two dimensions taken up to up to isometry and how so I need to produce a function of a lattice and and one more variable. And how would I do that I sum over all primitive vectors in my lattice. And, and then I take the norm of the primitive vector and I raise it to the power, the correct power to take is lambda plus one and this one is the is the row for for SL2 and so in general and if you if you paid attention to the previous slide there was a shift by row. And so this is for SL2 is one. And so that's. And so you some you some overall numbers you some overall. So just repeating myself primitive vectors in your lattice, and you take their norm and raise it to some power so it'll be some function of a lattice if this power is this power is you can have them this this series will converge and we'll give you some function of a lattice. And maybe one thing to notice is that what are the primitive vectors module plus minus one those are the same as rational points is rational points of you on. And in general, the, the Eisenstein series for SL2 those are just this would be a sum over P one over your field with some notion and what replaces the norm is is really notion of a height of the corresponding point in P one. So, so this is and right and and and I put one half in front is just because plus minus primitive vector correspond to the same point in P one and so it's, it's we, we're really doing that summation. And then, how this relates to other firms of Eisenstein series you might have seen. Well, sometimes people some not over primitive vectors but all vectors, but the sum over primitive vectors and overall vectors, they differ by just a zeta function because if you have some over all multiples of a given vector you get you get a factor of a zeta function that that make that clear. So, and this is this is this is called the Eisenstein series. And then what's called pseudo Eisenstein series is imagine you have the same sound but now you took a smooth compactly supported functions on positive reels. And instead of something norm, instead of some summing norm to some power you some the value of this function on the norm. So, the object which is now linearly depends on F, and the relation and the relation between what we have on this slide and what we had on the previous slide is just million transform on the group are plus the multiplicative group of positive real numbers. Those are the powers are the characters. The powers are the characters of that group and then generally we have a function of a group you want to do somewhat decomposed into the characters there's there's a general free transform which for a multiplicative group of positive real numbers is usually called million transform. The reason to introduce the reason one introduces the series is because this function is visibly so if F is a compactly supported function that this is visibly not on your group, whether the previous function like I said is never an L2 and will back will see in a second that is never an L2. But so this is, this is a slightly technical point but this is if you'd like to generate, you'd like to actually generate this pen and L2 of Eisenstein series this is, that's how you do it. So what would that be for SLN? Well, for SLN is, you can anticipate what's going to be replacement of a primitive vector. So there was really you take, you took a rational subspace and you choose the generator of the corresponding of the corresponding of you had a lattice you take a rational subspace you take the generator of the group of the intersection so for SLN, you take a flag of rational subspaces. You have this induced lattices, and you look at their volumes, rather, you know, like the determinant of the intersection of your lattice with the corresponding subspace. And that would be, this would be a height zeta function for now flag manifold of your, of your, this, this, this is a, so you measure a height of rational point using some line bundle of course for flag manifold there are many line bundles. And if you record the height for, you know, it's a function of a line bundle, you get, you get some expression like this and this is such general its expressions like this are called height zeta functions in such a big arithmetic geometry but in concretely this is what it is. So you measure, you take these flags of subspaces and you measure in some sense, like, you know, some some norm of that flag. And then, and then if you have a function field, what is it the function field. You have a, so an analog, an analog of allows us for a function field to the G bundle of your curve. And so if I have a G bundle of your curve, then you also have a G mod B bundle the associated G mod B bundle, namely just the quotient of your principle bundle by the by the subgroup B. And then the analog of, of, of this, I mean, this line look of this kind of rational points of that of the G mod B are the you you count the sections of that G mod B bundle over your curve, according to their degree. And so then shifted by row. And so that's, that's really a generous that this is a super familiar object to anybody who's done some kind of grown within theory. So you just really go and count sections of some vibrational curves and some geometry weighted by the degree. The degree here is a is a vector. So you get a weighting by some monomial. And so that's, that's, so that's the kind of object with Eisenstein series is for in this, in this, you know, in this example Eisenstein series is this generating function. So it's a function, so it's a function over G bundle for every G bundle. And what we would like to do with the G mod bundle is just count all sections according to their degree. So that's seems like a reasonable thing to do. And, and I further like lands proves general fact that Eisenstein series is Eisenstein series were all were were defined. So all the series I consider there's some series has some radio, some, some region of convergence, like, you know, maybe going here, this was convergent for for lambda from the real part of lambda sufficiently large. And that Eisenstein slow language proves so that there is a analytic continuation of functional equation that this thing satisfies some email. Go back to the transparency. So there's analytic in general. So there's the region of convergence and some kind of positive veil chamber. And then analytic continuation to other parts of the space of lambdas. This is, this is like the action of w. And in fact, and in fact, if you analytically continue to some other part then you get some constant that depends on lambda. The interesting constant the kind of important for what what's happening. And, like I said, all of this series are eigen functions of, of Hickey operators and their, their eigen values are always w invariant. And therefore, and moreover, so and moreover, they're, the eigen values are doubly invariant and lambda in, if you have a field of characteristics zero. And in fact, they're dependent q to the lambda only if you have a field over a finite field with q elements. And so in particular, they're, they're q to the lambda is a function which stoop I over log q periodic in lambda. So the Eisenstein spectrum therefore abstractly a subset in this bold face lambda this bold face lambda will be important. And that were the bold face lambda it's either a lead lead algebra of the Langlunds dual torus, or the Langlunds dual torus itself. And, and for now we all we need to know about Langlunds dual torus is the torus dual to the maximum torus of your braille subgroups so that's, that's not something nothing further is required. So, and so what does what does the spectrum looks for SL two against me we had we had this expression. And with this expression one can just kind of eyeball this expression and see that. So, we would like to understand some sort of, you know, what does the expression looks like in the case of my fundamental domain. And by, like I said by some kind of eyeballing of this expression one can see the falling fact that there is. It has the following asymptotic form. In as so I said, so why is the imaginary part of that is the, you know, the coordinate going into the case that it has this asymptotic form, which, which if you're, I know it's not a quantum mechanics seminar it's a Schubert seminar but if you're if you, if those of you somehow interested in differential operators you would recognize that there's some kind of like in comic and reflected wave. If you'd like and so. In any way so this is has this expression and we have to pay attention to both the coefficient the sort of reflection coefficient which is, which contains in itself some slightly modified version of dream and zeta function this is called completed zeta function. And also to the exponents so, so the exponents say that from the exponents you see that this is never an L2 because the, the measure of integration and the dx divide over y square, and you can never. If you square this expression, there'll always be at least one term which is not integrable with respect to that measure. And, however, if I look at it more closely, then, then I realize that if my lambda is purely imaginary, then, then this is borderline L2 that the integral model to norm diverges but just only only borderline diverges so so it makes sense that will contribute to the discrete spectrum and in fact, the discrete spectrum will contribute the expression lambda mean expression from going from minus infinity to minus one fourth. And as this famous one forth, but, but another feature is that this, like I said this size some kind of version of these is completed version of dream and zeta function, and in particular has a pole at lambda equals one. And so if I pick the residue of that, that pole, then I get a constant function. And a constant function suit needs an eigenfunction of Laplace operator and moreover it's core integral, which is actually an L2 so it's not constant function is not an Eisenstein series but it's a residue in the pole of an Eisenstein series so that's, that's the kind of. That's, and that's, and that's the complete description of the Eisenstein spectrum for assault too. And there's the following. We're in an obvious way to state this result. Some are also due to England's and that's in this, you know, later, later editions by Arthur and many other brilliant mathematicians. So this is, this is this is that you can say that like, like, like, result like this well let's think of a matrix lambda minus lambda. So they think of lambda the eigenvalue is as being an L matrix like lambda minus lambda. And let's consider the equation that it's commented with some unknown matrix is twice time E. This at once applies that is a new pot matrix and so there's a particular solution in the characteristic of that new pot of metrics for any lead algebra there's a finite set of semi simple elements called characteristics of new pot elements. And so, and then the above description of the spectrum you can phrase as saying that this met. Some long days in the spectrum. This happens if and only if this lambda minus lambda equals the characteristic modular the maximal compact subgroup in the centralizer of the bottom down. So it's clear if you have a new potent element, then characteristic is defined only in the solution of this age commutator equals to ease only defined up to up to centralizer of E. And what you do the ambiguity that you're allowed to take is in the lead algebra of the centralizer of the maximal compact in the central. So what how does this work in this example. So if he is non zero important, then this characteristic is one minus one. It's, it's centralizer important is unipot. So it has no, it's maximal compact is trivial, it's the algebra is zero, and we conclude that lambda equals to one. Then other possibilities we have the zero new button, and then this characteristic is also zero. However, it's the centralizer is everything the maximal compact in the centralizer as you to. And this for we conclude that lambda has to be purely matched. So that's, so this is, this is a general principle. There's an vision by Langlans this the general principle has to do with the fact that this this lambda, which is the space of all possible lambda modular doubly. These are the semi simple conjugacy classes and either the lead algebra or the lead group. It's the lead algebra of the Langlans doubly group or, or the on the group itself. And the way to state the spectrum is to consider better. It's better. So important elements are the same as SL to triples and so let's let's consider a homomorphism from SL to the lead algebra of the Langlans double group. So the centralizer of that homomorphism sees up by, and then of course we denote by h in e we denote the images of the standard h and e. And so the, the, the main theory will prove was that mean not the mean, the theory will prove them David and the first installment of our, of our paper is the following it's it's really. It's really so. Langlans himself did this computations for, for all groups of Frank to in his, in his fundamental work on the subject. And the computations for G two were kind of amazing computations just just unbelievable how he was able to do it for G two it's pretty amazing. Some people did it for many different groups and, and so I guess we were the first to a do it for a eight and be do it uniformly for all groups but, but, but of course yeah I shouldn't mean I shouldn't really. I mean I should really hide the fact that this is, this is, this is some sort of a may mean this is, this is a great piece of mathematics largely not due to us at all I mean it's by Langlans and many, many people we have this. We have this, you know, point of view which is, you know, I, I hope people will appreciate but it's not the little the contribution of others. And the answer is that this the spectrum is always the image you take the union overall overall conjugacy class of cell to homomorphisms and you take. Well, in the additive case you do have the characteristic plus the lead algebra of the maximal compact, or in the group in the positive characteristics you take q to the characteristic over to plus the actual compact, plus the actual and, and not only we determine the spectrum will actually give an explicit formula for the spectrum decomposition so. So, and the first step, so I'm a little behind the schedule here but it's okay. So, as the first step is we, is the same as Langlans so we, we are first step is just the same as Langlans it says. It says so Langlans what he did, the reason I discussed, the reason I discussed the, the notion of studies in Stein series is because, but technically the proof goes through, through analysis of the formula for the L to pairing between the L to pairing is in Stein series like to, you'd like to decompose the space you might as well to decompose decompose the L to pairing. And so then you have that, or better to say, you can think of a just like the L to pairing have a distribution two copies of your, of your lambda salam the, or my the both days lambda is where the, is where the lambdas take place. And two copies of that I have a distribution that, that tells me what is a viable corresponding to the Eisenstein series, and what is the real tool in a product. And, and the spectral decomposition is really decomposition of that distribution to pieces that spectral pieces. Langlans gives a certain current contour integral formula for that distribution and, and his approach and approach of all the old people I mentioned on the previous transparency was to do residue calculus in that formula, which is, which is that part we should purely avoid. Anyway, so this brings us to a point where I was supposed to make a five minute break so maybe we make that five minute break now and, and you tell me when it ends. Sounds good. Maybe, yeah, five minutes break. I'm going to stop the recording in case there are any questions. Please go ahead.