Yeah, so as I said in the first lecture, I'm roughly dividing this course into three parts.

The first two parts are rather short. They are on SON1 and SP2n. And then I'll move to the third part, which is PGLN. And we were in the middle of the discussion of Ziegel modular forms. And yeah, so what I would like to discuss briefly at this point is Hecke theory for

Ziegel modular forms. Of course, only in the unramified case. The ramified case, I don't know if this has ever been worked out, but it is certainly

very complicated. So gamma is the group SP2nz. So the analog of the full modular group. And Hecke operators are parametrized by double cosets of the form.

So we fix a prime P, fix a prime P. Gamma and then a diagonal matrix with entries P

to power a1 up to P to power an, and then P to the power r minus an, P to power r minus

a1, 2n elements and gamma. And we can order them such that a1 is greater than 0 and so on, and it's in increasing

order up to an, and then it continues to increase. So an is bounded by r over 2. And r is some integer. So r is basically the degree of the Hecke operator.

In particular, if r is 1, then there is only one such Hecke operator. That's simply TP. And by slight abuse of notation, I just write the double coset. That's just 11111 and then PPPPP.

There are several Hecke operators of degree 2. Ti of P squared is of the following form.

I have a couple of 1s, i of them. And then I have a couple of Ps, n minus i of them. And then again, a couple of Ps, n minus i and P squared.

In particular, so this is for 0 less than i less than n with possible equality. In particular, if i is 0, then this is just PPPPPPP, so this is the identity.

So T0 of P squared is the identity. And it turns out that TP and then TIP for 1 less than i less than n minus 1 generate

the P part of the Hecke algebra. So every Hecke operator in P is a polynomial in these operators.

They have some nice properties. At least there are Hermitian with respect to the inner product that we defined.

They are commutative. And of course by the Chinese remainder theorem, everything is multiplicative. So if I take two different primes, then everything is nicely multiplicative. But if I stay with one prime, then the Hecke relations become very complicated.

Hecke relations are very complicated. This is a very, very complicated combinatorial problem to write down a linear.

So if you multiply two Hecke operators, you can write them as a linear combination of other Hecke operators. But the linear combination is very complicated. And I don't think, so to my knowledge, in general, there is no closed formula or not

even an easy algorithm to compute what the product of two Hecke operators is. Here is a simple example, Tp squared. We would like to write this as a linear combination of degree two Hecke operators. And this is T2 p squared plus p plus 1 T1 p squared plus p to the power 3 p squared

p1 times the identity.

But this is the simplest of all non-trivial Hecke relations. And they become completely crazy. If you want to write down linear combinations for products of Hecke operators.

What am I missing? You don't have to write a double coset for chi of a square. Oh, yeah, yeah, yeah, yeah, yeah, OK.

For holomorphic Ziegel modular forms, the Ramanujan conjecture is known.

Just as in the case of genus 1, yes, yes.

Genus 2 Ramanujan conjecture is known. This is work of Weisauer. But as I said, there is no really clear connection between the Satake parameters and

the Fourier coefficients. And so what does this mean? It means that the Satake parameters have absolute value 1 unless the form f is a lift.

And I will explain what a lift is in a moment.

All right, L functions. I'm not going to go into any detail. I will just mention there are two types of L functions. There is the so-called standard L function, which is an L function of degree 2n plus 1.

And here the analytic continuation and functional equation are known.

And there is the spinor L function of degree 2 to the power n. And here the analytic and functional equation is known in the case n equals 2, but not in general.

My impression is nevertheless that the spinor L function is used more often than the standard L function. Yeah, in any case, I'm not going to go into more definitions. But certainly in the case n equals 2, there is a degree 4 and a degree 5 L function.

OK, finally, lifts, because they produce some arithmetically interesting objects. The most famous lift is in the case in n equals 2, the Saito Kurokawa lift.

This is the case n equals 2. And this has been generalized to arbitrary, I think even n or so, that I will discuss this later. So the idea is the following.

We take a classical holomorphic cusp form of genus 1 and weight 2k minus 2. So this is the space of holomorphic cusp forms on the upper half plane, the usual upper half plane of weight 2k minus 2.

And by the theory of half integral modular forms as developed basically by Shimura, you can associate to this half integral weight modular form with the following Fourier expansion. It has some coefficients that I normalize to be of absolute value roughly 1.

So the exponent is k over 2 minus 3 over 4 e of mz. And this is a half integral weight form of weight k minus a half of level 4.

And it is in conants plus space, which means that m is congruent to either 0 or minus 1 mod 4. So only half of the coefficients occur.

And then Walz-Bergé's theorem tells us that these coefficients are central twisted l values of the original function f. So this is up to normalization.

I mean, g is only defined up to a scalar multiple. But I can choose a normalization such that the coefficient is the central l value, chi d, at least for d, a fundamental discriminant.

Yes. Yeah. For non-fundamental discriminants, there is also a formula, but that's a bit more complicated.

And then I can attach to g or these coefficients a Ziegler modular form as follows. So I proceed on this blackboard.

To this, I can attach a Ziegler modular form, capital F of z. It has a Fourier expansion. Sum over matrices T, At, E of trace Tz. And that's a Ziegler modular form of weight k for sp4z.

And the At's are given explicitly. And in certain cases, they can be described quite easily. It's determinant of 2t to the k over 2 minus 3 over 4 times c, where c is this Fourier

coefficient, determinant of 2t. And this holds at least if minus 1 to the power k minus 1 times det 2t is a fundamental discriminant. And if not, then there is a slightly more complicated formula that describes these Fourier

coefficients. Yeah. So the Fourier coefficients are up to normalization. Well, they're basically the Fourier coefficients of this half integral weight modular form.

So capital F is the so-called Saito Kurokawa lift of either f or g.

This is also sometimes called the Mass-Spitzialchar, because Hans Mass investigated this a lot, and he called it Spitzialchar, so special whatever, I guess it's untranslatable. But if you find this phrase, Spitzialchar, and it refers to.

Or sometimes it's also called the mass space. So the subspace of capital F of this form inside the whole space of Siegel. I mean, it's, of course, only a very small subspace of forms that are lifts. But this is sometimes called the mass space inside the space of all Siegel modular forms.

OK. So these lifts are quite interesting. Let me mention two results. There is Iccino's period formula. So what you can always do, this has nothing to do with lifts.

What you can always do if you have a Siegel modular form, you can restrict it to the diagonal. And by this I mean, if you have a point z x plus i y in the Siegel upper half space of genus 2, then you can write this as x1, x2, x2, x3 plus i y1, y2, y2, y3.

And you can project this onto the diagonal x1, x3 plus i y1, y3. And then this lives on two copies of the usual upper half plane.

There is a complex number x1 plus i y1, and there is a complex number x3 plus i y3. So the restriction to the diagonal restricts to two copies of the upper half plane. So you get a function that lives on two congruence

quotients. And what you can do now is, you can take the L2 norm of the,

so both of these are of course equipped with an inner product. And you can take the L2 norm of the restriction relative to the L2 norm of the original function. And if you equip everything with probability measures,

then it turns out that this ratio is given as follows. It's pi squared over 15 L3 half f. So now f is a lift. So f is a lift of this f over there. And f is assumed to be Hecke normalized.

So little f is a Hecke eigenform. There is a unique way of doing this, putting the first Fourier coefficient to be 1. 1 sim square f times 12 over k minus 1.

And then a sum over an orthonormal basis. So this is an orthonormal basis of sk. So notice that f has weight 2k minus 2. But here we are summing over forms of weight k.

1 half sim square phi times f. So this makes good sense. I mean this sum has size k and on Lindelof all of these are essentially bounded.

So you divide by k and all of this is also bounded. So roughly speaking the mass is rather evenly concentrated on the diagonal. But it's a very beautiful formula. It compares the mass on the diagonal with the complete mass.

And it's given as a mean value of certain degree 6 L-functions. OK, so that's one thing. And the other thing is that the L-function, so the the spinner L-function, L-spin of

this lift f agrees with the original L-function of the cusp form little f up to some simple factors. Zeta of s minus k plus 1 times zeta of s minus k plus 2 times L f s.

OK, and this side to Kurokawa lift has been generalized to arbitrary values of n.

Well, it was conjectured, I think, by Duke and Imamoglu. And it was then proved by Ikeda. So this is called the Ikeda lift. Generalization to arbitrary n.

This is the Ikeda lift, which has similar features. But it's more complicated, for instance, in the sense that for Saito Kurokawa, one can give the Fourier coefficients exactly, even for non-fundamental discriminants. This is a complete disaster in general. There are formulas, but they are not manageable.

OK, so finally, a bit of literature for further reading. A very good book, and I think one of the standard references is Freitag's book

Siegel-Chamodelformen, which unfortunately is in German. Nevertheless, I think it's one of the standard references. There is a shorter book that basically contains a subset of Freitag's book by Klingen. This is in English, introductory lectures to Siegel modular forms or something like this.

There's a very old book by Siegel, which is called Symplectic Geometry. 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 the exact copy of the original paper. This, Siegel's book contains much of the underlying geometry. OK, any questions? Is the analytic continuation functional equation known for the spinner L functions in general?

No, no. I think for n equals 2. And I'm not sure what the state of the art is with n equals 3. Perhaps there is some partial, I don't know. But certainly not in general.

OK, any other questions? There are a lot of works of Kaufman and Zagir about Shumor Elite from the Congrens subgroup. And we start not the gun market forms on model groups, but on Congrens subgroups. And then leave them for half waveforms.

Give the analogous here. Yes, sure. So I'm not going to go into the greatest, most greatest detail. So certainly you can do this for Congrens subgroups as well. But I mean, this nice formula of Cohen-Zagir, to my knowledge,

exists for the full modular group. Although, of course, the general framework holds in generality. In any case, this certainly is possible for Congrens subgroups. There's nothing special about the modular group, except that everything's simpler.

For these two L functions, I mean, how should one think about the fact that it's two different ones? Are they meant to reflect different underlying aspects of the form? Well, combinatorially, it's a different combination of the Satake parameters.

I mean, they are, of course, defined by Euler products. And the Euler factor is a certain combination of the Satake parameters. And you can do this in different ways. Well, I mean, at least for SP4, there's two fundamental representations of SP4. And these are the Langlands L functions attached to these two fundamental representations.

So I'll prove that they are quite independent. OK. Well, then this ends my short discussion of Ziegal modular forms. And so I'm now going to move on to PGLN.

OK. So here's the setup. Our group G is PGLN of R. I won't mention adelts in this lecture. Everything is either over R or over QP.

But no adelts. OK. It has an Eva-Zawack decomposition, NaK. N are the unipotent matrices. K is PON.

Gamma for us is just SLNZ. Again, you can talk about congruent subgroups, but I'm not going to do this. H. This is now a third H. It's not the Ziegal upper half plane. It's not hyperbolic upper half space.

But it's a model for G mod K. The dimension of H is n minus 1 times n plus 2 over 2. And we choose the following coordinates.

We choose the Eva-Zawack coordinates coming from A and K. So a point Z, little z, perhaps, is x times y. And x is 11111 something.

And y is given by 1, y1, and so on, up to y n minus 1, y1.

OK. And w is the Weyl group. OK.

So for n greater than 2, strictly bigger than 2, there are no holomorphic forms. So everything is a mass form. And so we really think of these forms as mass forms.

And so we talk about differential operators. D is the algebra of G-invariant differential operators on H.

And that's the center of the universal enveloping algebra.

And this turns out to be a polynomial ring in n minus 1 variables.

And this can be viewed as polynomial functions on the dual of the Lie algebra of A. So A is the group. German A is the Lie algebra.

This is the dual vector space. And I take all symmetric powers. So I can think of this as polynomial functions. And they should be invariant under the Weyl group. This is the so-called Howey-Chandra isomorphism, going from here to here.

So A is the Lie algebra. And it's isomorphic to r to the n minus 1. But we view it as a hyperplane in Rn. I mean, this is a reflection of the fact that we are working with the projective group.

So we are working with the trace 0 hyperplane. Trace 0 hyperplane. So a typical element in A has n components, and they add up to 0. Trace 0 hyperplane in Rn.

OK, and this isomorphism here sends a differential operator to, well, a map lambda d that goes from A star to C.

And for instance, the Laplacian goes to the map that sends mu, which is mu 1 up to mu n.

So this is now an arbitrary element in it. Well, in A star to n cubed minus n over 24 plus sum of the squares.

To each operator in script D, there is attached a polynomial. And this polynomial tells you that if you have an eigenform, an automorphic form with these spectral parameters, then the eigenvalue under the corresponding operator is this polynomial.

OK, so for me, always, mu is the notation for an element, well, and potentially the complexification of the dual of the Lie algebra,

say, modulo the Weyl group. These are the spectral parameters, or Langlands parameters, of an automorphic form.

So each automorphic form comes with such an n-tuple of spectral parameters. And if you want to know what is the eigenvalue of this automorphic form with these spectral parameters, then you just plug the values into the respective polynomial under this isomorphism.

OK, so for me, everything is normalized such that the Ramanujan conjecture says that

the mu's are real. Other people normalize differently. For instance, in Goldfeld's book, the unitary axis is

1 over n plus i a star. And some other people, other authors, say that the unitary axis is i a star.

This is a matter of taste. For me, the unitary axis is a star.

OK, as you know, the Ramanujan conjecture is not known, not even for n equals 2. And the best general bounds are imaginary part of mu j is bounded by 1 over 2 plus a

half minus 1 over n squared plus 1. And for n equals 2, better bounds are known. So are for n equals 3 and 4. But in general, these are the best bounds. And this is due to Luo, Rodnik, and Sarnak.

OK, so how many mass forms are there? How dense are they, in some sense?

And this is measured by the Harris-Chandler C function. There is a general definition for the Harris-Chandler C function for any Lie group in terms of root systems.

I'm not going to go into detail here. Let me just say that in our case of the group PGLN, the Harris-Chandler C function is given by a constant times the product 1 less or

equal than j, less than k, less or equal than n, g of lambda j minus lambda k, where lambda is lambda 1 up to lambda n in A star, perhaps A star C.

Well, let's define it only on A star.

Well, yes, that's right, except that lambda for me is an integration variable. Later, I integrate over lambda, and I keep mu fixed for my favorite automorphic form. But yes, they play the same role. And g is a quotient of gamma functions that you can simplify.

And it just looks like x times tangent hyperbolic of pi x. So the tangent hyperbolic, of course, only plays a role if x is very small. Otherwise, this is essentially 1. So g of x, for practical purposes, g of x is x.

Well, in fact, it's absolute value of x. Because if x is negative, then also the tangent hyperbolic is negative. So this is bounded by product of 1 plus lambda j minus lambda k.

OK, so unless there is some conspiracy between the components of lambda, if they are sort of generic, if you choose a generic lambda,

such that all these differences are roughly of the same size, then this is the norm of lambda to the power n choose 2. If you're at the walls of the Weyl chamber, when some of these are perhaps identical or very close to each other,

then, of course, the spectral measure is a bit smaller. So this drops at the walls of the Weyl chamber, of the Weyl chambers.

So the density of mass forms is a bit less at the walls of the Weyl chamber. And there is a Weyl law. And this was proved by Muller and Lapid, 2009, that this spectral density really measures the density of mass forms.

So the number of cusp forms with spectral parameter lambda plus O of 1.

So you take a ball of size 1 about a given parameter lambda is of size c lambda. So what I wrote down here.

So, I mean, the precise statement is a little different. So the ball has to be sufficiently large in order to get a lower bound. And then if you expand the region, then they actually get an asymptotic formula with a power saving error term.

So this counts the number of mass forms. For instance, in the case n equals 2, you get the usual Weyl law. And this then holds in general. OK, so I mentioned earlier,

so I'm using these blackboards rather randomly. So I mentioned earlier, for n greater than 2, there is no discrete series.

We only have mass forms. However, we do have lifts from holomorphic forms for n equals 2.

We do have lifts from holomorphic forms on GL2. So you can take a holomorphic form on GL2 and, for instance, take the symmetric square

and then you get a mass form on GL3. OK, any questions? OK, then next thing I want to discuss are Whitaker functions.

Whitaker functions.

OK, Whitaker functions have two arguments. Well, they have an index and an argument. And the index lives on AC star modulo the Weyl group. And the argument lives on h, this h here.

And the value is a complex number. OK, and they are defined as follows. We take an x in A, and I write down the first off-diagonal

as x1 up to xn minus 1. And I don't care about the rest.

And then I can define a character. Theta of x is just e of some xj. This is a character on n, character on n.

And then a Whitaker function satisfies the following properties. W mu of xz is theta of x times w mu of z. So it transforms by multiplication from the left with respect to this character for all z in h,

x in n. And it satisfies a differential equation.

If I take a differential operator d in script D, then this is lambda d of mu. So lambda d is this map that I defined up there times w mu of z for all d in D.

OK, so this is what I want to call a Whitaker function. It's by no means clear that such objects exist and so on.

But they do exist, and you can actually explicitly construct them as follows. So in particular, these are all eigenfunctions of this algebra script D. Construction is as follows.

You integrate over n a function i that I'm going to define in a moment, i mu. You take the long Weil element, w long, times u times z theta bar of u du.

u is the Haar measure on n. And i mu of z depends, in fact, only on x. Sorry, it depends only on y. It's independent of x.

It's a product. It's basically a power function. j from 1 to n minus 1, yj to some power lj of mu, where these are linear forms in the mu's that I'm not going to write down.

So lj is a suitable linear polynomial. Basically, the idea is so you choose lj in such a way that this power function i mu satisfies exactly this differential equation.

And then the Whittaker function inherits the differential equation from the differential equation of i. You don't have to take the long Weil element. You can also take other Weil elements here, and then you get degenerate Whittaker functions

that come up in the constant terms of the Fourier expansion of Eisenstein series. So yeah, if you take other Weil elements and the long Weil element, you get degenerate Whittaker functions. Degenerate Whittaker functions for other Weil elements.

For other Weil elements, will the integral be convergent? Sorry, what's that? For other Weil elements, other than the long element, the integral will be convergent?

The what? The integral will be convergent? Ah. For GL2, there are two. Well, yes. I mean, probably you can't take the trivial Weil element. And for GL2, there is no other Weil element.

But I think if you take a non-trivial Weil element, then it converges. OK, so you can write this down. In principle, it's completely explicit. It's an integral of the unipotent group. But these Whittaker functions, we have to admit they are very poorly understood.

OK, as an example, the case n equals 2. Well, then we can write down explicitly the Whittaker function.

So the Whittaker function has an argument here, which is a pair of two numbers that add up to 0. So it's mu minus mu. And then traditionally, it's just mu. There is no extra information. But nevertheless, z is up to scalar multiplication, square root of y,

e of x, k i mu of 2 pi y. And then it's a matter of taste whether you further normalize this or not. I like to normalize it by multiplying by cosine hyperbolic pi mu over 2.

But again, that's a matter of taste if you want to do this or not. The Bessel-K function decays exponentially in mu. So you can compensate for this by multiplying with the cosine hyperbolic. Or you don't have to do this. And OK, so Bessel functions are fairly well understood.

So this is a Whittaker function that we can handle. But in general, for n greater than 2, it's very poorly understood. So if you take the construction you gave for n equals 2,

do you get this cosine hyperbolic? You should just literally take this integral. I forget. So the cosine hyperbolic, if you include it 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 or not.

OK, but we do have some partial knowledge on general Whittaker functions. And what we know is the following. So you can recursively get a degree n Whittaker function by integrating a degree n minus 1 Whittaker function. Then if you repeat this, you can get down to degree 2 Whittaker functions,

which are Bessel functions. So general Whittaker functions are integrals over Bessel functions. So it can be expressed as iterated integrals over Bessel functions, over k Bessel functions.

But of course, if n gets big, then you get a whole bunch of integrals and things become very complicated. But in principle, there is such a formula, and that's u to state, Eric's state.

I would believe in the spirit of Emmanuel's talk that this is a reflection of the fact that the GLN closed diamond sum attached to the long Weyl element, at least in certain cases, can be written as a product of GL2 closed diamond sums

if the moduli are pairwise co-prime. And I would assume that this is some weak form of reflection of this fact, that in certain cases, you can reduce the GLN case to products of GL2.

For n equals 3, a bit more is known. So for n equals 3, we have a fairly explicit... The integration is a complicated combination? I'm not sure. I have to look it up. I think it's more complicated.

You get arguments u. Well, perhaps you get u and 1 over u. I don't know exactly. I think it's not just a multiplicative convolution. It's a bit more complicated than that. But you can look it up in State's paper. We know the double-Melling transform.

If we keep x fixed, then in the variable y, I have two entries, y1 and y2. So the double-Melling transform in y, y1 to the power s1,

y2 to the power s2, dy1, dy2 over y1, y2 is known. And it's, in fact, what you... Well, perhaps I don't know if you would expect this, but certainly it's a ratio of gamma factors. 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. This is the Archimedean Rankin-Selberg theory.

It's the product of two Whittaker functions. If you take w alpha of y and w beta of y bar, determinant of y to the power s, and then the correct measure on the group A.

So in the classical case, this is a product of two Bessel functions. But now you take a product of two general Whittaker functions. And this is the sort of Rankin-Selberg ratio of gamma factors. So it is what you would expect.

So gamma, okay, gamma r is the usual thing. I'll define it in a second. It's s plus i alpha... I hope I get the signs right.

Alpha j minus beta bar k. I hope that's correct. Where j and k run from one to n. So basically you take the Langlands parameters and combine them each with each. Divided by whatever you expect from Rankin-Selberg theory,

gamma r of n s. So that's kind of zeta of two in the classical case. So here you get a kind of symmetric square l function at one.

Gamma r one plus i alpha j minus alpha k. Gamma r one minus i beta bar j minus beta bar k. So I don't know if this is readable or not.

Hopefully it's not readable because it's perhaps slightly wrong. And perhaps there is a constant involved. So if you want to get the constants right that's also a complete nightmare because everybody normalizes slightly differently and you can be sure you're in the end you'll be off by a power two pi. Okay so what is gamma r? Gamma r is the usual thing.

Gamma r of s is pi to the minus s over two times gamma of s over two.

Okay let me just write down one more formula to wrap this up and then I'm done for today. In gl2 there is the so-called Kontorovich-Lebedev transform.

If you integrate the Bessel K function against the test function and then integrate it back but this time over the index then you get the original function back. And this holds in general.

And this was recently proved by Goldfeld and Kontorovich. So if you start with a test function f and you integrate it against a Whittaker function and you call this f hat of mu then you can recover f from f hat.

So if you integrate f hat of mu against the Whittaker function

and now you integrate with respect to mu d mu over the spectral measure. Okay I said that lambda would be my integration variable now it's mu whatever. Over a star then you get f back.

So this is the Whittaker functional theorem right? Yes, yes, yes. I mean it isn't that was proved maybe by Walmart. Okay that's true, that's true but yes.

Goldfeld and Kontorovich give a very explicit formulation of this but certainly you find an abstract version of this in Wallach's book and in Wallach's work. Yes okay, okay fair enough.

Didn't you say yesterday I'm going to conclude the session? OK, anyway, so that's it. Are there any questions? OK, well, that's it for today. This is Peter already asked for questions,

but if you just want to take questions only from the organizers, then ask the questions now. And if that's OK, yeah. Regarding the leaves, this may be a stupid question. How many percent, what is the percentage you can get in the leaf, which you can get from a lower order?

In general, you mean? In a special case, for instance. Well, I mean, if you look at Weil's law, so for GL3, if you take, so here you're sitting in A star,

and in a ball of radius one of size T, you have roughly T to the three guys of mass forms, and on the diagonal, so the Galba-Jaki lifts,

by Weil's law for GL2, they are of size T, right? So you have T lifts sitting exactly on the diagonal, but in the whole square of size one, so this is T, T plus one, T, T plus one. So in this ball of size one, you have altogether

T to the three mass forms. So the lifts are in some sense negligible.