In this post, I will answer the second question posed in Part I of this story by providing an example of the phenomenon explained in that question. Recall that the question was roughly whether there was a direct summand of the space of automorphic forms (on some group) and a subquotient of that direct summand whose cohomologies were genuinely different. I will construct here examples of such direct summands for the group whose archimedean component has cohomology which is one dimensional and concentrated in degree , and which has a subquotient whose archimedean component has cohomology which is also one dimensional but concentrated in degree , such that these cohomology spaces are not related by a connecting homomorphism.

We will use the machinery developed in the previous two posts to do this. The key to being able to apply that machinery will be a result of Henry Kim in his paper where he determines the residual spectrum of .

## Preliminaries on the group

We take to be the group over of matrices such that , where denotes transpose of and

.

Then is defined to be the group over of matrices such for some depending on . The map is a character, called the similitude character of the group, and is often written . The similitude adds a component to the center; the group has finite center , while the center of is a one dimensional .

We consider the Siegel parabolic subgroup, which we will denote , in what follows. This is the subgroup of of matrices of the form

.

There are, of course, relations among these asterisks. For example, up to similitude, the top left block is the transpose-inverse of the bottom right block.

The Siegel parabolic is a maximal parabolic subgroup. The Levi of , which we’ll write as , is isomorphic to . The unipotent radical, , is three dimensional.

We will also need to consider the other maximal parabolic subgroup, called the Klingen parabolic. It will be denoted with Levi decomposition . It consists of matrices of the form

.

Then and is again three dimensional.

But I really don’t think about this group matricially, at least not in this context; I added the matrix descriptions for psychological comfort only. Instead I will think about and and using the root lattice. Recall that is the split group of type (or , if you like). The group is just a central extension of and therefore has the same root lattice. So there are two simple roots, and the square of the length of one is twice that of the other. Let’s call the long root and the short one. Then the root lattice looks as in the following hand-drawn picture:

The positive cone is shown below, outlined in blue, as well as the dominant cone, shaded in green:

The Siegel parabolic is the short root parabolic, meaning that contains the root groups corresponding . And is the long root parabolic, so that contains the root groups corresponding to . The unipotent radicals of either of these parabolics just consist of the root groups for the positive roots not already contained in the Levis. Below is a picture in the root lattice which shows all of this data.

Below we will be interested in the structure of the Lie algebra of as a representation of via the adjoint action. This can essentially be read off the root lattice as well. The central component of will of course act trivially. As a representation of the remaining component, however, breaks up into a sum of two representations of , corresponding to the two root strings in in the direction of . The character of the maximal torus extends to a character of this and is given by the determinant. In essence, it only sees the center of , because it is perpendicular to . The representation of which is given by the root string in of length two (consisting of and ) is up to central twist given by the two dimensional standard representation of . Actually, it is exactly given by which one can see by the placement of this root string relative to and the determinant character . All of this summarized in the following picture:

## Siegel parabolic Eisenstein series

Now I am going to construct the summand of the space of automorphic forms on that we will examine. It will consist of Eisenstein series induced from the Siegel parabolic and derivatives thereof. Recall that these spaces were discussed in Part II of this story.

So we need to start with a cuspidal automorphic representation of the Levi . This will consist therefore of the data of a cuspidal automorphic representation of and a character of . We take the character on to be the trivial character . For the component, let’s start with a classical cuspidal holomorphic eigenform with trivial nebentypus and (even) weight . Take to be the unitary cuspidal automorphic representation associated with . We make the following assumption on .

**Assumption.** We assume which, under our normalization, is to say that .

Let . Then, in the notation of Part II, . If is the irreducible representation of of highest weight , and if is the class of the pair , then the space we’ll be interested in is , again all in the notation of Part II.

Let’s make this more explicit. The modulus character of is given by the sum of the roots in which, by above, is . The root gives the determinant, so a twist by is the same as a twist by . Then the space is the space of all Eisenstein series coming from the induction

along with their derivatives at . We may denote such an Eisenstein series by or, if we wish to vary the power of the modulus character, by , for a section of the above induction. Here, by “section” we mean that is varying with in such a way that its restriction to a given maximal compact subgroup is independent of . Some would be more precise and call this a “flat section” of the induction. By the Iwasawa decomposition, any element of the above induction at a given can be interpolated into a unique flat section.

Now note that we do not take any residues of these Eisenstein series when constructing ; indeed these Eisenstein series have no poles. The reason for this will be explained now, and has to do with our assumption on the vanishing of the -function of at its central point.

It is well known that the singularities of Eisenstein series have the same locations as those of their constant terms. Since an Eisenstein series as above is induced from a cuspidal automorphic representation on a maximal parabolic, all the constant terms vanish except those along parabolics conjugate to . As for the constant term along , the Langlands–Shahidi method readily gives an answer in terms of -functions and intertwining operators.

Recall that the constant term along of is defined as

for . When the evaluation of the section at is suitably interpreted as a complex-valued function on , the constant term is given by

.

Here, is the unique nonidentity Weyl group element that preserves the set of roots in (so by the picture of in the root lattice above, is the simple reflection across the line perpendicular to ) and is an intertwining operator, given for in a certain half plane by

where it is holomorphic in , and given by meromorphic continuation elsewhere. The resulting section is a section of the induction at , not at . The reason for this is that acts by a minus sign on , the complexified Lie algebra of the center of the -component of , and the complex -plane is identified with , where this is playing the role that it did in Parts II and III of this story. (However, note that is slightly different here, in that I’m now excluding the Lie algebra of the center from this definition of $\mathfrak{a}$. This is just because I have fixed the trivial character on the center already, and would rather not complicate notations with it.) The intertwining operator itself is an operator which is -equivariant (i.e., intertwining) between the induction of and that of , where acts on the space of by , where is an adelic point of the -component of the Levi , and now is being viewed as the nontrivial element in the Weyl group of that . But .

In any case, the summand in the constant term above decomposes into a product of local expressions. Indeed, writing as a restricted tensor product over all places, the induction of to then decomposes as well as

,

where the restriction on the tensor product means that at almost all unramified places , a pure tensor must have as its component a fixed choice of spherical vector . Here the are also varying in a flat section. Then taking to be decomposable, the intertwining operator breaks up into a product of local intertwining operators which are given by

.

If is a finite place at which , then the sections and are related by a quotient of local -functions as follows.

An -function for is determined by the data of the Satake parameters for and a representation of the dual group of the -component of . When dualizing the root datum for itself, which is self-dual, the long and short roots switch, making Levi the Klingen parabolic a natural incarnation of the dual group of . Now acts on the Lie algebra of the unipotent radical , and this Lie algebra has a two step filtration whose associated graded is , where as representations of the -component of , and ; we saw this already in the previous section. Then the results of Gindikin–Karpelevich and Langlands–Shahidi dictate that

,

when is unramified. Here the multiples of come from the fact that corresponds to .

Then these results can be upgraded to something global, at least at the unramified places. The final result is this. Let be the set of finite places where is not spherical. Then

.

Here, denotes a partial -function.

To get back to the poles of our Eisenstein series, we first note that because is unitary and has trivial central character. Therefore this factor has a pole at . Luckily, the zero of at , which we assumed existed above, cancels this pole! Since the factors of the denominator of the quotient of -functions above are in the region of convergence (since is tempered by Deligne) we know that it does not vanish. Thus, the unramified factor of the constant term has no pole.

Then to say whether or not there are choices of for which has a pole at , the issue becomes whether or not the local intertwining operators at or at can have poles. This is what H. Kim needed to examine in his paper mentioned at the beginning of this post. In particular, he proves that the only Siegel parabolic Eisenstein series which have poles are the ones coming from eigenforms whose -functions do not vanish at their central points, and the only possible poles with are at . Therefore we are safe at because of our assumption. The values of our Eisenstein series at and are related by a functional equation. Therefore, we are also safe to the left of the line , i.e., none of our Eisenstein series have poles! This finally explains why we do not need to worry about residual Eisenstein series, and the methods of Part III will apply to compute the cohomology of .

Before we do this, let me quickly make a technical remark. Strictly speaking, Kim’s paper proves the result we need for , not for . To get the result for , it suffices to note that and are the same as varieties. Therefore, The Eisenstein series for both groups are sums over the same set of -points, and the values of the Eisenstein series are determined by those of the Eisenstein series. So it was really enough to study the poles in the case.

## Cohomology

Now we compute the cohomology of using the results of the previous post. Fix a maximal compact subgroup of . Note that the derived group of is , so we will be considering the -cohomology of .

This will not be a difficult computation, given the results of the previous post. The point will be to determine the length of a particular Weyl element. In fact, we need to find the element of the Weyl group of such that the following conditions, whose meanings we will recall, hold:

,

Here is what this all means. The set is the set of elements of the Weyl group with for all positive roots in . The only such root is , so this first condition is that has . Since are the only short positive roots, this means must be or .

In the next condition, is the highest weight of , which in our case was . The weight is, of course, half the sum of the positive roots, so . Thus . Also, is the center of the component of , and since is in , is trivial. And is the determinant, so generates the character group of . Finally, is the differential of the central character of restricted to . The central character of is just as discussed above, so is just . So the second condition says

.

Since , the only way this can happen is if moves to , in which case it will move to , and so , which will cancel with .

Finally, for the last condition, we note that is the space orthogonal to , the complexified Lie algebra of , and hence characters of are orthogonal to those of ; They are in the direction of and generated by . The character is the infinitesimal character of the archimedean component of . Since is a holomorphic discrete series of weight , this character is supported on the orbit of times the simple root of under the Weyl group of . In other words, it is supported on . Thus needs to equal . Though this is actually automatic for any satisfying the second condition, so we may ignore this third condition. But we needed to check anyway that is could be satisfied, otherwise there would be no and our cohomology computation would yield the zero group.

Now the Weyl group of is the dihedral group of order . The first condition above restricts us to four elements: The inverses of two that send to itself, and the inverses of the two that send to . These are the inverses of and , and of and , where denotes the reflection through the line perpendicular to a root . Thus

.

We easily compute that and , while . So the second condition that forces .

The length of is, of course, equal to , so now we can use the results of the previous post to compute the cohomology

.

it is equal to

.

Here is the multiplicity of in . We have

.

The holomorphic discrete series of weight on has one-dimensional -cohomology in (middle) degree , when tensored with the highest weight representation of weight . Thus the above cohomology picks out a one dimensional space from and leaves us with the following -module:

.

The cohomology is concentrated only in this degree.

If we were to instead compute the cohomology of the induction space, call it , of , we would get

by the results of the previous post, because is one dimensional in degrees and . This cohomology also vanishes outside degrees and .

## An answer to our question, finally!

We just saw that the induction above has the same cohomology in degrees and that has in just degree . We will now explain why it has a quotient whose archimedean component is irreducible, and whose cohomology is concentrated in degree . This will be pretty quick, actually, now that we set up all the theory.

Now it turns out that the induction

has a subrepresentation which is discrete series. (Note that now the occurs in the exponent with a sign.) I’m not exactly sure who to attribute this fact to. I know how one could go about proving it, and my advisor seems to know this well, but unfortunately I can’t point to a reference in the literature. If anyone knows, please comment below.

In any case, if we choose our section to have lying in this subrepresentation, then examining our formula for the constant term above, we see that , that is, the intertwining operator kills . This is because the intertwining operator at infinity kills tempered representations, and of course discrete series are such. Thus the Eisenstein series with in this discrete series subrepresentation are themselves discrete series at infinity, since they must transform as -modules like their constant terms.

Now, on the other hand, the space of Eisenstein series evaluated at is dual to the space just considered, because is self dual. And thus the considerations above give us a quotient which, at infinity, is the representation dual to the discrete series just considered. So it is still discrete series (just of a different Harish-Chandra parameter) and hence occurs in middle degree cohomology, degree and degree only. This finally gives the example discussed in the first post in this series.

I should remark, in connection with the computations of the cohomology of induced representations, that the space of Eisenstein series that we constructed, but without derivatives, may not be isomorphic to the induction we considered at the end of the previous section. This is because the summand of the constant term may not vanish if the section maps nontrivially to the nontempered Langlands quotient of the induction. An exception to this is if the -function of vanishes at its central value to order more than . In that case the space of Eisenstein series is isomorphic to an induction space, and the degree cohomology sees the discrete series quotient just constructed, and the degree cohomology sees the Eisenstein cohomology. So I’ll end this series of posts with one more question, to which I do not know the answer now, even though I expect it is affirmative.

**Question. **Let be the space of Eisenstein series constructed as above, but with no derivatives. If vanishes to order exactly at , then do we still have an isomorphism

?