The Skinner–Urban method

At the beginning of this month I defended my PhD thesis, and the amount of work (and pages!) that went into my thesis is the reason this blog hasn’t been updated since last May. For anyone interested, here is a link to it on Columbia’s academic commons, but let me give a brief synopsis of it here.

The main result of my thesis concerns the Bloch–Kato Selmer group of the symmetric cube of the Galois representation attached to a cuspidal eigenform of level $1$ ; under some standard conjectures I construct nontrivial elements in such Selmer groups.

In a little more detail, let $F$ be a cuspidal eigenform of level $1$ and weight $k$ , and $\rho_F$ its Galois representation as constructed by Deligne. We also have the unitary cuspidal automorphic representation $\pi_F$ attached to $F$ , and it turns out that since $F$ has level $1$ , the $L$ -function

$L(s,\pi_F,\mathrm{Sym}^3)$

always vanishes at the central point $s=1/2$ (in fact its functional equation has sign $-1$ ). The Bloch–Kato conjecture then predicts (after unraveling normalizations involving Tate/determinant twists) that the Bloch–Kato Selmer group

$H_f^1(\mathbb{Q},(\mathrm{Sym}^3\rho_F)^\vee(\tfrac{3k}{2}-1))$

is nontrivial, and I prove this assuming a version of the global Langlands correspondence for cohomological automorphic representations for the exceptional group $G_2$ , as well as specific cases of the Arthur conjectures. Let me describe the general method through which the argument goes now.

The Skinner–Urban method constructs nontrivial elements in the Bloch–Kato Selmer groups of certain Galois representations and was pioneered in this paper and this paper of Skinner and Urban. I usually like to summarize it using the following diagram, whose notation I will explain below.

Here we are starting with a nice automorphic representation $\pi$ of a reductive $\mathbb{Q}$ -group $M$ . When made precise, this adjective “nice” with which I qualified $\pi$ should imply that $\pi$ has attached to it a $p$ -adic Galois representation $\rho_\pi$ which itself is nice in an appropriate sense (i.e., it is geometric in the sense of Fontaine–Mazur); this is the content of the arrow labelled “Langlands” in the diagram. If we assume for simplicity that $M$ is split, then $\rho_\pi$ will be a continuous representation of the absolute Galois group $G_{\mathbb{Q}}$ of $\mathbb{Q}$ into the group $M^\vee(\overline{\mathbb{Q}}_p)$ of $\overline{\mathbb{Q}}_p$ -points of the dual reductive group $M^\vee$ to $M$ .

Now let $R:M^\vee\to GL_n$ be a finite dimensional algebraic representation of $M^\vee$ . Then we can consider the composition $R\circ \rho_\pi$ , which is a nice representation $G_{\mathbb{Q}}\to GL_n(\overline{\mathbb{Q}}_p)$ . It has attached to it an $L$ -function $L(s,R\circ \rho_\pi)$ which should satisfy a functional equation. Let us assume things are normalized so that the center of the functional equation for $L(s,R\circ \rho_\pi)$ is located at $s=0$ . Then the purpose of the Skinner–Urban method is to prove implications of the form

$L(0,R\circ\rho_\pi)=0\quad\Longrightarrow\quad H_f^1(\mathbb{Q},(R\circ\rho_\pi)^\vee(1))\ne 0,$

which are predicted by the Bloch–Kato conjecture.

The way the method does this is by traversing the top arch of the diagram. The first thing to do is to embed $M$ inside a bigger reductive group $G$ as the Levi of some parabolic $P$ . (Realistically, one usually embeds $M$ times some $\mathbb{G}_m$ factors in $G$ as a Levi, but these factors cause no trouble and I will ignore them.)

On the right side of the diagram, to the parabolic $P$ will correspond a parabolic $P^\vee$ in the dual group $G^\vee$ of $G$ . This parabolic $P^\vee$ is just the one whose Levi contains the coroots corresponding to the roots contained in $M$ . Its Levi will indeed be the dual group $M^\vee$ to $M$ .

Now let us traverse the arch. The first step is to ascend the arrow labelled “Eisenstein.” This means we must construct a functorial lift of $\pi$ to an automorphic representation $\Pi$ of $G$ . This must be done by some process which resembles parabolic induction, usually with Eisenstein series (though it is sometimes better to use certain CAP representations).

Now the functorial lift $\Pi$ must be constructed in anticipation of the next step, which is to $p$ -adically deform $\Pi$ in a generically cuspidal family of automorphic representations $\mathcal{E}$ . One can usually construct such a family if one knows that $\Pi$ satisfies certain properties locally at the archimedean place. For example, often knowing that $\Pi$ is a certain type of discrete series at $\infty$ is enough. This family $\mathcal{E}$ will vary in weight and is usually constructed as a piece of an eigenvariety (whence the nomenclature). It should be parametrized by, say, an affinoid rigid analytic space over $\mathbb{Q}_p$ .

The next step is to move from the left, automorphic side to the right, Galois side along the top of the diagram. What this means is that, assuming that we have an appropriate Langlands correspondence for $G$ , we must construct a rigid analytic family $\rho_{\mathcal{E}}$ of Galois representations which interpolates those attached to the classical, cuspidal members of $\mathcal{E}$ . Ideally, having such a family of Galois representations would mean is that $\rho_{\mathcal{E}}$ is a representation $G_{\mathbb{Q}}\to G^\vee(\mathrm{Frac}(\mathcal{A}))$ which is continuous in a certain sense, where $\mathcal{A}$ is the affinoid ring of analytic functions on $\mathcal{E}$ . In reality, we cannot expect the construction of $\rho_{\mathcal{E}}$ to be so direct, but roughly whatever object we get in its place should have properties which resemble the ones just listed.

The final step in this process is to descend the arrow labelled “Ribet.” This involves constructing a particular lattice $\rho_{\mathcal{L}}$ in $\rho_{\mathcal{E}}$ (again in an appropriate sense of the word “lattice”) and specializing it at the point in $\mathcal{E}$ corresponding to $\Pi$ . If done correctly, one will not quite reobtain the representation $\rho_\pi$ , but instead one will get a representation $\rho_{\overline{\mathcal{L}}}:G_{\mathbb{Q}}\to P^\vee(\overline{\mathbb{Q}}_p)$ which does not factor through $M^\vee(\overline{\mathbb{Q}}_p)$ . The failure to factor through the Levi $M^\vee(\overline{\mathbb{Q}}_p)$ should be measured by a nontrivial cocycle $\sigma$ which gives a Galois cohomology class in $H^1(\mathbb{Q},(R\circ\rho_\pi)^\vee(1))$ . One must then show that this class satisfies the appropriate local conditions, putting it in the correct Bloch–Kato Selmer group.

The one thing left that I should explain is where the representation $R$ shows up in this process. In order to carry out the Skinner–Urban method, certain pieces of numerology must be satisfied by the objects at play, and among them is the following. The Levi $M^\vee$ of $P^\vee$ will act on the unipotent radical $N^\vee$ of $P^\vee$ by the adjoint action. The Jordan–Hölder constituents of $N^\vee$ will therefore be representations of $M^\vee$ , and we must require (among other things) that the last constituent be isomorphic to $R$ . (One then sees $R$ on the automorphic side, for example, in the constant terms of Eisenstein series constructed from $\pi$ by the Langlands–Shahidi method.)

In my next post (or maybe posts, depending on how long they get) I want to describe how the first step of this method (that is, the construction of the functorial lift $\Pi$ ) goes in the particular case of my thesis, and at the same time, correct an error I made in my previous post. In short, the error is that there is actually no Eisenstein class in degree $2$ for $GSp_4$ as constructed in that post, contrary to what I thought (and also contrary to what my advisor says in an example in his eigenvarieties paper!). I’ll try to explain where this error comes from and how its resolution affects my work and the aforementioned example in my advisor’s paper.

Eisenstein series and a cohomological phenomenon (Part IV)

[Edit, March 2022: I made some mistakes in the writing of this post. I’ve decided that, instead of rewriting and reposting them, I’m going to leave them as they are but insert some “edits,” like this one, where there are mistakes.]

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 $\mathrm{GSp}_4$ whose archimedean component has cohomology which is one dimensional and concentrated in degree $2$ , and which has a subquotient whose archimedean component has cohomology which is also one dimensional but concentrated in degree $3$ , 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 $\mathrm{Sp}_4$ .

Preliminaries on the group

We take $\mathrm{Sp}_4$ to be the group over $\mathbb{Q}$ of matrices $g\in\mathrm{GL}_4$ such that $gJ\,^tg=J$ , where $\,^tg$ denotes transpose of $g$ and

$J=\begin{pmatrix} 0&0&1&0\\ 0&0&0&1\\ -1&0&0&0\\ 0&-1&0&0 \end{pmatrix}$ .

Then $\mathrm{GSp}_4$ is defined to be the group over $\mathbb{Q}$ of matrices $g\in\mathrm{GL}_4$ such $gJ\,^tg=\nu_g J$ for some $\nu_g\ne 0$ depending on $g$ . The map $g\mapsto \nu_g$ is a character, called the similitude character of the group, and is often written $\nu(g)$ . The similitude adds a component to the center; the group $\mathrm{Sp}_4$ has finite center $\{\pm 1\}$ , while the center of $\mathrm{GSp}_4$ is a one dimensional $\mathrm{GL}_1$ .

We consider the Siegel parabolic subgroup, which we will denote $P$ , in what follows. This is the subgroup of $\mathrm{GSp}_4$ of matrices of the form

$\begin{pmatrix} *&*&*&*\\ *&*&*&*\\ 0&0&*&*\\ 0&0&*&* \end{pmatrix}$ .

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 $P$ is a maximal parabolic subgroup. The Levi of $P$ , which we’ll write as $M$ , is isomorphic to $\mathrm{GL}_2\times \mathrm{GL}_1$ . The unipotent radical, $N$ , is three dimensional.

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

$\begin{pmatrix} *&0&*&*\\ *&*&*&*\\ *&0&*&*\\ 0&0&0&* \end{pmatrix}$ .

Then $M_Q\cong\mathrm{GL}_2\times\mathrm{GL}_1$ and $N_Q$ 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 $\mathrm{GSp}_4$ and $P$ and $Q$ using the root lattice. Recall that $\mathrm{Sp}_4$ is the split group of type $\textrm{C}_2$ (or $\textrm{B}_2$ , if you like). The group $\mathrm{GSp}_4$ is just a central extension of $\mathrm{Sp}_4$ 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 $\alpha$ the long root and $\beta$ 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 $P$ is the short root parabolic, meaning that $M$ contains the root groups corresponding $\pm\beta$ . And $Q$ is the long root parabolic, so that $M_Q$ contains the root groups corresponding to $\pm\alpha$ . 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 $N_Q$ as a representation of $M_Q$ via the adjoint action. This can essentially be read off the root lattice as well. The central $\mathrm{GL}_1$ component of $M_Q$ will of course act trivially. As a representation of the remaining $\mathrm{GL}_2$ component, however, $\mathrm{Lie}(N_Q)$ breaks up into a sum of two representations of $\mathrm{GL}_2$ , corresponding to the two root strings in $N_Q$ in the direction of $\alpha$ . The character $\alpha+2\beta$ of the maximal torus extends to a character of this $\mathrm{GL}_2$ and is given by the determinant. In essence, it only sees the center of $M_Q$ , because it is perpendicular to $\alpha$ . The representation of $\mathrm{GL}_2$ which is given by the root string in $N_Q$ of length two (consisting of $\beta$ and $\alpha+\beta$ ) is up to central twist given by the two dimensional standard representation $\mathrm{Std}$ of $\mathrm{GL}_2$ . Actually, it is exactly given by $\mathrm{Std}$ which one can see by the placement of this root string relative to $0$ and the determinant character $\alpha+2\beta$ . 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 $\mathrm{GSp}_4$ 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 $M$ . This will consist therefore of the data of a cuspidal automorphic representation of $\mathrm{GL}_2$ and a character of $\mathrm{GL}_1$ . We take the character on $\mathrm{GL}_1$ to be the trivial character $1$ . For the $\mathrm{GL}_2$ component, let’s start with a classical cuspidal holomorphic eigenform $f$ with trivial nebentypus and (even) weight $k\geq 4$ . Take $\pi$ to be the unitary cuspidal automorphic representation associated with $f$ . We make the following assumption on $f$ .

Assumption. We assume $L(\pi,1/2)=0$ which, under our normalization, is to say that $L(f,k/2)=0$ .

Let $\pi_{-}=\pi\otimes\vert\det\vert^{-1/2}$ . Then, in the notation of Part II, $\tilde\pi_-=\pi$ . If $V$ is the irreducible representation of $\mathrm{GSp}_4$ of highest weight $\frac{k-4}{2}(\alpha+\beta)$ , and if $\varphi$ is the class of the pair $(P,\pi_-\boxtimes 1)$ , then the space we’ll be interested in is $\mathcal{A}_{V,\varphi}$ , again all in the notation of Part II.

Let’s make this more explicit. The modulus character $\delta_P$ of $P$ is given by the sum of the roots in $N$ which, by above, is $3\alpha+3\beta$ . The root $\alpha+\beta$ gives the determinant, so a twist by $\vert\det\vert^{1/2}$ is the same as a twist by $\delta_P^{1/6}$ . Then the space $\mathcal{A}_{V,\varphi}$ is the space of all Eisenstein series coming from the induction

$\mathrm{Ind}_P^{\mathrm{GSp}_4}((\pi\otimes\delta_P^{s+1/2})\boxtimes 1),\qquad s=-1/6$

along with their derivatives at $s=-1/6$ . We may denote such an Eisenstein series by $E(-1/6,\phi)$ or, if we wish to vary the power of the modulus character, by $E(s,\phi)$ , for $\phi$ a section of the above induction. Here, by “section” we mean that $\phi$ is varying with $s$ in such a way that its restriction to a given maximal compact subgroup is independent of $s$ . 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 $s$ can be interpolated into a unique flat section.

Now note that we do not take any residues of these Eisenstein series when constructing $\mathcal{A}_{V,\varphi}$ ; 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 $L$ -function of $f$ 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 $E(s,\phi)$ as above is induced from a cuspidal automorphic representation on a maximal parabolic, all the constant terms vanish except those along parabolics conjugate to $P$ . As for the constant term along $P$ , the Langlands–Shahidi method readily gives an answer in terms of $L$ -functions and intertwining operators.

Recall that the constant term along $P$ of $E(s,\phi)$ is defined as

$E_P(s,\phi)(g)=\int_{N(\mathbb{Q})\backslash N(\mathbb{A})}E(s,\phi)(ng)\,dn$

for $g\in\mathrm{GSp}_4(\mathbb{A})$ . When the evaluation $\phi_s$ of the section $\phi$ at $s$ is suitably interpreted as a complex-valued function on $\mathrm{GSp}_4(\mathbb{A})$ , the constant term is given by

$E_P(s,\phi)=\phi_s+M(w,\phi_s)$ .

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

$M(w,\phi_s)(g)=\int_{N(\mathbb{A})}\phi_s(nwg)\,dn$

where it is holomorphic in $s$ , and given by meromorphic continuation elsewhere. The resulting section is a section of the induction at $-s$ , not at $s$ . The reason for this is that $w$ acts by a minus sign on $\mathfrak{a}$ , the complexified Lie algebra of the center of the $\mathrm{GL}_2$ -component of $M$ , and the complex $s$ -plane is identified with $\mathfrak{a}^\vee$ , where this $\mathfrak{a}^\vee$ is playing the role that it did in Parts II and III of this story. (However, note that $\mathfrak{a}$ 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 $\mathrm{GSp}_4(\mathbb{A})$ -equivariant (i.e., intertwining) between the induction of $\pi$ and that of $\pi^w$ , where $\pi^w(g)$ acts on the space of $\pi$ by $\pi^w(m)=\pi(wmw)$ , where $m$ is an adelic point of the $\mathrm{GL}_2$ -component of the Levi $M$ , and now $w$ is being viewed as the nontrivial element in the Weyl group of that $\mathrm{GL}_2$ . But $\pi^w\cong\pi^\vee\cong\pi$ .

In any case, the summand $M(w,\phi)(-s)$ in the constant term above decomposes into a product of local expressions. Indeed, writing $\pi=\bigotimes^{'}\pi_v$ as a restricted tensor product over all places, the induction of $\pi$ to $\mathrm{GSp}_4$ then decomposes as well as

$\mathrm{Ind}_{P(\mathbb{A})}^{\mathrm{GSp}_4(\mathbb{A})}((\pi\otimes\delta_{P(\mathbb{A})}^{s+1/2})\boxtimes 1)=\bigotimes^{'}\mathrm{Ind}_{P(\mathbb{Q}_v)}^{\mathrm{GSp}_4(\mathbb{Q}_v)}((\pi_v\otimes\delta_{P(\mathbb{Q}_v)}^{s+1/2})\boxtimes 1)$ ,

where the restriction on the tensor product means that at almost all unramified places $v$ , a pure tensor must have as its $v$ component a fixed choice of spherical vector $\phi_{v,s}^{\mathrm{sph}}$ . Here the $\phi_{v,s}^{\mathrm{sph}}$ are also varying in a flat section. Then taking $\phi_s=\bigotimes^{'}\phi_{v,s}$ to be decomposable, the intertwining operator breaks up into a product of local intertwining operators which are given by

$M_v(w,\phi_{v,s})(g)=\int_{N(\mathbb{Q_v})}\phi_{s}(nwg)\,dn$ .

If $v$ is a finite place at which $\phi_{v,s}=\phi_{v,s}^{\mathrm{sph}}$ , then the sections $M_v(w,\phi_{v,s})$ and $\phi_{v,-s}=\phi_{v,-s}^{\mathrm{sph}}$ are related by a quotient of local $L$ -functions as follows.

An $L$ -function for $\pi_v$ is determined by the data of the Satake parameters for $\pi_v$ and a representation of the dual group of the $\mathrm{GL}_2$ -component of $M$ . When dualizing the root datum for $\mathrm{GSp}_4$ itself, which is self-dual, the long and short roots switch, making Levi $M_Q$ the Klingen parabolic a natural incarnation of the dual group of $M$ . Now $M_Q$ acts on the Lie algebra of the unipotent radical $N_Q$ , and this Lie algebra has a two step filtration whose associated graded is $R_1\oplus R_2$ , where as representations of the $\mathrm{GL}_2$ -component of $M_Q$ , $R_1=\mathrm{Std}$ and $R_2=\det$ ; we saw this already in the previous section. Then the results of Gindikin–Karpelevich and Langlands–Shahidi dictate that

$M_v(w,\phi_{v,s}^{\mathrm{sph}})=\frac{L(3s,\pi_v,\mathrm{Std})L(6s,\pi_v,\det)}{L(3s+1,\pi_v,\mathrm{Std})L(6s+1,\pi_v,\det)}\phi_{v,-s}^{\mathrm{sph}}$ ,

when $v$ is unramified. Here the multiples of $3$ come from the fact that $3s$ corresponds to $s(2\rho_P)\in\mathfrak{a}^\vee$ .

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

$E_P(s,\phi)=\phi_{s}+M_\infty(w,\phi_{\infty,s})\otimes\bigotimes_{v\in S}M_v(w,\phi_{v,s})\otimes\frac{L^S(3s,\pi,\mathrm{Std})L^S(6s,\pi,\det)}{L^S(3s+1,\pi,\mathrm{Std})L^S(6s+1,\pi,\det)}\bigotimes_{v\notin S}^{'}\phi_{v,-s}^{\mathrm{sph}}$ .

Here, $L^S$ denotes a partial $L$ -function.

To get back to the poles of our Eisenstein series, we first note that $L^S(6s,\pi,\det)=\zeta^S(6s)$ because $\pi$ is unitary and has trivial central character. Therefore this factor has a pole at $s=1/6$ . Luckily, the zero of $L^S(3s,\pi,\mathrm{Std})$ at $s=1/6$ , which we assumed existed above, cancels this pole! Since the factors of the denominator of the quotient of $L$ -functions above are in the region of convergence (since $\pi_v$ 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 $\phi$ for which $E(s,\phi)$ has a pole at $s=1/6$ , the issue becomes whether or not the local intertwining operators at $v\in S$ or at $\infty$ 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 $f$ whose $L$ -functions do not vanish at their central points, and the only possible poles with $\mathrm{Re}(s)\geq 0$ are at $s=1/6$ . Therefore we are safe at $s=1/6$ because of our assumption. The values of our Eisenstein series at $s=1/6$ and $s=-1/6$ are related by a functional equation. Therefore, we are also safe to the left of the line $\mathrm{Re}(s)$ , i.e., none of our Eisenstein series $E(-1/6,\phi)$ 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 $\mathcal{A}_{V,\varphi}$ .

Before we do this, let me quickly make a technical remark. Strictly speaking, Kim’s paper proves the result we need for $\mathrm{Sp}_4$ , not for $\mathrm{GSp}_4$ . To get the result for $\mathrm{GSp}_4$ , it suffices to note that $P\backslash\mathrm{GSp}_4$ and $(P\cap\mathrm{Sp}_4)\backslash\mathrm{Sp}_4$ are the same as varieties. Therefore, The Eisenstein series for both groups are sums over the same set of $\mathbb{Q}$ -points, and the values of the $\mathrm{GSp}_4$ Eisenstein series are determined by those of the $\mathrm{Sp}_4$ Eisenstein series. So it was really enough to study the poles in the $\mathrm{Sp}_4$ case.

Cohomology

Now we compute the cohomology of $\mathcal{A}_{V,\varphi}$ using the results of the previous post. Fix a maximal compact subgroup $K_\infty$ of $\mathrm{GSp}_4(\mathbb{R})$ . Note that the derived group of $\mathrm{GSp}_4$ is $\mathrm{Sp}_4$ , so we will be considering the $(\mathfrak{sp}_4,K_\infty)$ -cohomology of $\mathcal{A}_{V,\varphi}\otimes V$ .

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 $w_0$ of the Weyl group of $\mathrm{GSp}_4$ such that the following conditions, whose meanings we will recall, hold:

$w_0\in W^P,\qquad w_0(\Lambda+\rho)|_A+\chi_{\pi_-}=0,\qquad -w_0(\Lambda+\rho)|_{\mathfrak{h}\cap\mathfrak{m}_0}\textrm{ is in }\chi_{\mathrm{inf},\pi}$ ,

Here is what this all means. The set $W^P$ is the set of elements $w$ of the Weyl group with $w^{-1}\gamma>0$ for all positive roots $\gamma$ in $M$ . The only such root $\gamma$ is $\beta$ , so this first condition is that $w_0$ has $w_0^{-1}\beta>0$ . Since $\beta,\alpha+\beta$ are the only short positive roots, this means $w_0^{-1}\beta$ must be $\beta$ or $\alpha+\beta$ .

In the next condition, $\Lambda$ is the highest weight of $V$ , which in our case was $\frac{k-4}{2}(\alpha+\beta)$ . The weight $\rho$ is, of course, half the sum of the positive roots, so $\rho=\frac{1}{2}(3\alpha+4\beta)$ . Thus $\Lambda+\rho=\frac{k-1}{2}(\alpha+\beta)+\frac{1}{2}\beta$ . Also, $A$ is the center of the $\mathrm{GL}_2$ component of $M$ , and since $\beta$ is in $M$ , $\beta|_A$ is trivial. And $(\alpha+\beta)|_M$ is the determinant, so $\frac{1}{2}(\alpha+\beta)$ generates the character group of $A$ . Finally, $\chi_{\pi_-}$ is the differential of the central character of $\pi_-$ restricted to $A$ . The central character of $\pi_-$ is just $\vert\det\vert^{-1/2}$ as discussed above, so $\chi_{\pi_-}$ is just $-\frac{1}{2}(\alpha+\beta)|_A$ . So the second condition says

$w_0(\frac{k-1}{2}(\alpha+\beta)+\frac{1}{2}\beta)|_A-\frac{1}{2}(\alpha+\beta)|_A=0$ .

Since $k\geq 4$ , the only way this can happen is if $w_0$ moves $\beta$ to $\alpha+\beta$ , in which case it will move $\alpha+\beta$ to $\pm\beta$ , and so $w_0(\frac{k-1}{2}(\alpha+\beta)+\frac{1}{2}\beta)|_A=(\pm\frac{k-1}{2}\beta+\frac{1}{2}(\alpha+\beta))|_A=\frac{1}{2}(\alpha+\beta)|_A$ , which will cancel with $\chi_{\pi_-}$ .

Finally, for the last condition, we note that $\mathfrak{h}\cap\mathfrak{m}_0$ is the space orthogonal to $\mathfrak{a}$ , the complexified Lie algebra of $A$ , and hence characters of $\mathfrak{h}\cap\mathfrak{m}_0$ are orthogonal to those of $A$ ; They are in the direction of $\beta$ and generated by $\frac{1}{2}\beta$ . The character $\chi_{\pi}$ is the infinitesimal character of the archimedean component $\pi_\infty$ of $\pi$ . Since $\pi_\infty$ is a holomorphic discrete series of weight $k$ , this character is supported on the orbit of $(k-1)$ times the simple root of $M$ under the Weyl group of $M$ . In other words, it is supported on $\pm\frac{k-1}{2}\beta$ . Thus $w_0(\frac{k-1}{2}(\alpha+\beta)+\frac{1}{2}\beta)$ needs to equal $\pm\frac{k-1}{2}\beta\pm\frac{1}{2}(\alpha+\beta)$ . Though this is actually automatic for any $w_0$ 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 $w_0$ and our cohomology computation would yield the zero group.

Now the Weyl group $W$ of $\mathrm{GSp}_4$ is the dihedral group of order $8$ . The first condition above restricts us to four elements: The inverses of two that send $\beta$ to itself, and the inverses of the two that send $\beta$ to $\alpha+\beta$ . These are the inverses of $1$ and $w_\alpha w_\beta w_\alpha$ , and of $w_\alpha$ and $w_\beta w_\alpha$ , where $w_\gamma$ denotes the reflection through the line perpendicular to a root $\gamma$ . Thus

$W^P=\{1,w_\alpha,w_\alpha w_\beta,w_\alpha w_\beta w_\alpha\}$ .

We easily compute that $(w_\alpha w_\beta)(\beta)=-(\alpha+\beta)$ and $(w_\alpha w_\beta w_\alpha)(\beta)=\beta$ , while $w_\alpha \beta=\alpha+\beta$ . So the second condition that $w_0\beta=\alpha+\beta$ forces $w_0=w_\alpha$ .

The length $\ell$ of $w_\alpha$ is, of course, equal to $1$ , so now we can use the results of the previous post to compute the cohomology

$H^{r+\ell}(\mathfrak{sp}_4,K_\infty;\mathcal{A}_{V,\varphi}\otimes V)$ .

it is equal to

$\mathrm{Ind}_{M(\mathbb{A}_f)}^{\mathrm{GSp}_4(\mathbb{A}_f)}H^r(\mathfrak{sl}_2,\mathrm{O}(2);\pi\otimes E_{w_\alpha(\Lambda+\rho)-\rho})^{m(\pi)}$ .

Here $m(\pi)=1$ is the multiplicity of $\pi$ in $L^2$ . We have

$w_\alpha(\Lambda+\rho)-\rho=(\frac{k-1}{2}\beta+\frac{1}{2}(\alpha+\beta))-\frac{3}{2}(\alpha+\beta)-\frac{1}{2}\beta=\frac{k-2}{2}\beta-(\alpha+\beta)$ .

The holomorphic discrete series of weight $k$ on $\mathrm{GL}_2(\mathbb{R})$ has one-dimensional $(\mathfrak{sl}_2,\mathrm{O}(2))$ -cohomology in (middle) degree $1$ , when tensored with the highest weight representation of weight $k-2$ . Thus the above cohomology picks out a one dimensional space from $\pi_\infty\otimes E_{w_\alpha(\Lambda+\rho)-\rho}$ and leaves us with the following $\mathrm{GSp}_4(\mathbb{A}_f)$ -module:

$H^{2}(\mathfrak{sp}_4,K_\infty;\mathcal{A}_{V,\varphi}\otimes V)=\mathrm{Ind}_{M(\mathbb{A}_f)}^{\mathrm{GSp}_4(\mathbb{A}_f)}(\pi_f\otimes\delta_P^{1/2-1/6})$ .

The cohomology is concentrated only in this degree.

If we were to instead compute the cohomology of the induction space, call it $\sigma$ , of $\pi_-\boxtimes 1$ , we would get

$H^{2}(\mathfrak{sp}_4,K_\infty,\sigma)=H^{3}(\mathfrak{sp}_4,K_\infty,\sigma)=\mathrm{Ind}_{M(\mathbb{A}_f)}^{\mathrm{GSp}_4(\mathbb{A}_f)}((\pi_f\otimes\delta_P^{1/2-1/6})\boxtimes 1)$

by the results of the previous post, because $\bigwedge^*\mathfrak{a}^\vee$ is one dimensional in degrees $0$ and $1$ . This cohomology also vanishes outside degrees $2$ and $3$ .

An answer to our question, finally!

We just saw that the induction above has the same cohomology in degrees $2$ and $3$ that $\mathcal{A}_{V,\varphi}$ has in just degree $2$ . We will now explain why it has a quotient whose archimedean component is irreducible, and whose cohomology is concentrated in degree $3$ . This will be pretty quick, actually, now that we set up all the theory.

Now it turns out that the induction

$\mathrm{Ind}_{P(\mathbb{R})}^{\mathrm{GSp_4}(\mathbb{R})}((\pi_\infty\otimes\delta_{P(\mathbb{R})}^{1/2+1/6})\boxtimes 1)$

has a subrepresentation which is discrete series. (Note that now the $1/6$ 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 $\phi=\bigotimes^{'}\phi_v$ to have $\phi_{\infty,1/6}$ lying in this subrepresentation, then examining our formula for the constant term above, we see that $E_P(1/6,\phi)=\phi_{1/6}$ , that is, the intertwining operator kills $\phi_{1/6}$ . This is because the intertwining operator at infinity kills tempered representations, and of course discrete series are such. Thus the Eisenstein series with $\phi_\infty$ in this discrete series subrepresentation are themselves discrete series at infinity, since they must transform as $(\mathfrak{sp}_4,K_\infty)$ -modules like their constant terms.

Now, on the other hand, the space of Eisenstein series evaluated at $-1/6$ [Edit, March 2022: Here is the mistake! There are issues with evaluating the Eisenstein series constructed above at $s=-1/6$ ; the constant term picks up a pole coming from a zero from the $L$ -function in its denominator. Consequently, the corresponding induced representation at this point is not a subrepresentation of the space of Eisenstein series.] is dual to the space just considered, because $\pi$ 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 $3$ and degree $3$ 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 $M(w,\phi_{1/6})$ of the constant term may not vanish if the section $\phi_{\infty,1/6}$ maps nontrivially to the nontempered Langlands quotient of the induction. An exception to this is if the $L$ -function of $f$ vanishes at its central value to order more than $1$ . In that case the space of Eisenstein series is isomorphic to an induction space, and the degree $3$ cohomology sees the discrete series quotient just constructed, and the degree $2$ cohomology sees the Eisenstein cohomology. So I’ll end this series of posts with one more question [Edit, March 2022: In view of the mistake above, feel free to ignore this question!] to which I do not know the answer now, even though I expect it is affirmative.

Question. Let $\Pi$ be the space of Eisenstein series $E(-1/6,\phi)$ constructed as above, but with no derivatives. If $L(s,f)$ vanishes to order exactly $1$ at $s=k/2$ , then do we still have an isomorphism

$H^*(\mathfrak{sp}_4,K_\infty;\Pi)\cong H^*(\mathfrak{sp}_4,K_\infty;\mathrm{Ind}_P^{\mathrm{GSp}_4}((\pi\otimes\delta_P^{1/2-1/6})\boxtimes 1))$ ?

Eisenstein series and a cohomological phenomenon (Part III)

In the last post, I explained how the space of automorphic forms decomposes as a direct sum of a space of cusp forms and spaces consisting of Eisenstein series and their residues and derivatives. I’ll recall what this decomposition looked like in a moment, but as in the previous post, I’ll first set the notation that will be in play throughout this post.

Notation for the group:

$G$ is a reductive algebraic group over $\mathbb{Q}$ .
$\mathfrak{g}$ is the complexified Lie algebra of $G$ .
$\mathfrak{g}_0$ is the complexified Lie algebra of the derived group of $G$ .
$K$ is a maximal compact subgroup of $G(\mathbb{A})$ .
$K_\infty$ is a maximal compact subgroup of $G(\mathbb{R})$ .
$V$ is a finite dimensional irreducible complex representation of $G$ .

Notation for parabolic subgroups:

$P$ is a parabolic $\mathbb{Q}$ -subgroup of $G$ .
$M$ is the Levi subgroup of $P$ .
$N$ is the unipotent radical of $P$ .
$A$ is the maximal split torus in the center of $M$ .
$M_0$ is the subgroup of $M$ such that $P=M_0AN$ is the Langlands decomposition of $P$ .
$\mathfrak{p},\mathfrak{m},\mathfrak{m}_0,\mathfrak{a},\mathfrak{n}$ are the relevant complex Lie algebras.

In the last post, we defined, for a cuspidal automorphic representation $\pi$ , an induction space $W_{\tilde\pi}$ , and given $f\in W_{\tilde\pi}$ and $\lambda\in\mathfrak{a}^\vee$ , we defined an Eisenstein series $E(\lambda,f)$ . Let $\chi_\pi\in\mathfrak{a}^\vee$ be the differential of the central character of $\pi$ at infinity. Let $\varphi$ be the class of the pair $(P,\pi)$ as described previously.

The decomposition from the last post took the form

$\mathcal{A}_V=\bigoplus_{\textrm{classes }[P]}\bigoplus_{\varphi\in\Phi_{V,[P]}}\mathcal{A}_{V,\varphi}$ .

Here $\mathcal{A}_{V,\varphi}$ was the space of Eisenstein series, their derivatives, and their residues, coming from $\pi$ at the point $\chi_\pi\in\mathfrak{a}^\vee$ . Recall that $W_{\tilde\pi}$ is isomorphic to a parabolic induction. To be precise, we have

$W_{\tilde\pi}\cong\mathrm{Ind}_{M(\mathbb{A}_f)\times(\mathfrak{g}_0\cap\mathfrak{p},K_\infty\cap P(\mathbb{R}))}^{G(\mathbb{A}_f)\times(\mathfrak{g},K_\infty)}(\tilde\pi)^{m(\pi)}$

where $m(\pi)$ is the cuspidal multiplicity of $\pi$ .

Now we need to make an assumption that will be in force throughout this post.

Assumption. For every $f\in W_{\tilde\pi}$ , the Eisenstein series $E(\lambda,f)$ is regular at $\lambda=\chi_\pi$

Under this assumption, there is a nice description of the space $A_{V,\varphi}$ . Let $\mathrm{Sym}(\mathfrak{a}^\vee)$ be the symmetric algebra on the $\mathbb{C}$ -vector space $\mathfrak{a}$ . We can view $\mathrm{Sym}(\mathfrak{a}^\vee)$ as a space of differential operators on $\mathfrak{a}^\vee$ in the standard way. Then we can define a map $W_{\tilde{\pi}}\otimes\mathrm{Sym}(\mathfrak{a}^\vee)\to\mathcal{A}_{V,\varphi}$ by

$f\otimes D\mapsto D_{\lambda=\chi_{\pi}}E(\lambda,f)$ .

This map is surjective by definition under our assumption. Though we note that in order to get it to be a map of $G(\mathbb{A}_f)\times(\mathfrak{g},K_\infty)$ -modules, we have to give the space $W_{\tilde{\pi}}\otimes\mathrm{Sym}(\mathfrak{a}^\vee)$ a specific $G(\mathbb{A}_f)\times(\mathfrak{g},K_\infty)$ -module structure explained in Li–Schwermer, page 155. And one has to be more careful than they let on; the space $\mathrm{Sym}(\mathfrak{a}^\vee)$ must be viewed as a space of distributions given by differential operators supported at the point $\rho_P+\chi_\pi\in\mathfrak{a}^\vee$ (Li and Schwermer do not emphasize that they’re supported at a point). Then Theorem 14 of the Franke paper mentioned above proves that the map just defined is an isomorphism under this regularity hypothesis. See Franke–Schwermer‘s proof of their Theorem 1.4 for an explanation of why this follows (in particular their remarks a the end of p. 775). In other words, unless there are residual Eisenstein series, there are no relations amongst the Eisenstein series and their derivatives in the space $\mathcal{A}_{V,\varphi}$ .

Cohomology of Eisenstein series

So it suffices to compute the cohomology of $W_{\tilde{\pi}}\otimes\mathrm{Sym}(\mathfrak{a}^\vee)$ , which Franke does in Section 7.4 of his paper. Let’s recall how this is done, because it’s a nice computation. We’ll actually follow the explanation in the textbook of Borel–Wallach of their Theorem III.3.3, which is very similar but gives more information.

Recall that $V$ is our finite dimensional representation of $G$ . We have first

$H^i(\mathfrak{g}_0,K_\infty;W_{\tilde\pi}\otimes\mathrm{Sym}(\mathfrak{a}^\vee)\otimes V)$

is isomorphic to $m(\pi)$ copies of

$H^i(\mathfrak{g}_0,K_\infty;\mathrm{Ind}_{M(\mathbb{A}_f)\times(\mathfrak{g}_0\cap\mathfrak{p},K_\infty\cap P(\mathbb{R}))}^{G(\mathbb{A}_f)\times(\mathfrak{g},K_\infty)}(\tilde\pi\otimes\mathrm{Sym}(\mathfrak{a}^\vee)\otimes V)$

because one can interchange an induction like this with a tensor product by a finite dimensional representation, and $\mathrm{Sym}(\mathfrak{a}^\vee)$ is a colimit of such. Then we can pull out the induction over the finite adelic groups, and Frobenius reciprocity allows us to manipulate the archimedean induction and shows us that the above space is isomorphic to

$\mathrm{Ind}_{M(\mathbb{A}_f)}^{G(\mathbb{A}_f)}H^i(\mathfrak{g}_0\cap\mathfrak{p},K_\infty\cap P(\mathbb{R});\tilde\pi\otimes\mathrm{Sym}(\mathfrak{a}^\vee)\otimes V).$

Now $\mathfrak{p}\cap\mathfrak{g}_0$ decomposes as $\mathfrak{m}\oplus\mathfrak{n}$ , where $\mathfrak{m}$ and $\mathfrak{n}$ are the complexified Lie algebras of $M_0$ and $N$ , respectively. We use a spectral sequence to get rid of the $\mathfrak{n}$ , as follows. There’s an $E_2$ spectral sequence with

$E_2^{p,q}=H^p(\mathfrak{m},K_\infty\cap P(\mathbb{R});H^q(\mathfrak{n},\tilde\pi\otimes\mathrm{Sym}(\mathfrak{a}^\vee)\otimes V))$

converging to our cohomology space above with $i=p+q$ . By the Künneth formula, the inner cohomology group is given by

$H^q(\mathfrak{n},V)\otimes\tilde\pi\otimes\mathrm{Sym}(\mathfrak{a}^\vee)$

because $\mathfrak{n}$ acts trivially on $\tilde\pi$ and $\mathrm{Sym}(\mathfrak{a}^\vee)$ . So to continue, we need to know how to compute the spaces $H^q(\mathfrak{n},V)$ . This is given by the Kostant decomposition, which we recall. Fix a Borel $B$ in $P$ , and let $\mathfrak{h}\subset\mathfrak{g}$ be the corresponding Cartan subalgebra. Then fixing $B$ also fixes a positive system on the roots of $\mathfrak{h}$ in $\mathfrak{g}$ . Let $W^P$ be those elements $w$ of the Weyl group of $\mathfrak{h}$ in $\mathfrak{g}$ for which $w^{-1}\alpha>0$ for every positive root $\alpha$ occuring in $\mathfrak{m}$ . If $\Lambda$ denotes the highest weight of the representation $V$ , then

$H^q(\mathfrak{n},V)\cong\bigoplus_{w\in W^P,\,\,\ell(w)=q}E_{w(\Lambda+\rho)-\rho}$

where $\ell(w)$ is the length of $w$ , $E_\nu$ is the representation of $M$ with highest weight $\nu$ , and $\rho$ is half the sum of the positive roots. Thus we now have

$E_2^{p,q}=\bigoplus_{w\in W^P,\,\,\ell(w)=q}H^p(\mathfrak{m},K_\infty\cap P(\mathbb{R});\tilde\pi\otimes\mathrm{Sym}(\mathfrak{a}^\vee)\otimes E_{w(\Lambda+\rho)-\rho})$ .

We want to use the Künneth formula on this, using that $\mathfrak{m}=\mathfrak{m}_0\oplus\mathfrak{a}$ . Note that for a weight $\nu$ , $E_{\nu}$ is the (exterior) tensor product of the $\mathfrak{m}_0$ -module obtained by restricting $E_\nu$ to $\mathfrak{m}_0$ , with the character $\mathbb{C}_{\nu|_{A}}$ of $\mathfrak{a}$ . Furthermore, $\mathrm{Sym}(\mathfrak{a}^\vee)$ has trivial $\mathfrak{m}$ -action. Thus by Künneth,

$H^p(\mathfrak{m},K_\infty\cap P(\mathbb{R});\tilde\pi\otimes\mathrm{Sym}(\mathfrak{a}^\vee)\otimes E_{w(\Lambda+\rho)-\rho})$

$=\bigoplus_{j+k=p}H^j(\mathfrak{m}_0,K_\infty\cap P(\mathbb{R});\tilde\pi\otimes E_{w(\Lambda+\rho)-\rho})\otimes H^k(\mathfrak{a},\mathrm{Sym}(\mathfrak{a}^\vee)\otimes\mathbb{C}_{(w(\Lambda+\rho)-\rho)|_A})$ .

Now Franke notes that $H^k(\mathfrak{a},\mathrm{Sym}(\mathfrak{a}^\vee)\otimes\mathbb{C}_{\nu})$ is one dimensional and concentrated in degree zero if $\nu=-\rho_P-\chi_\pi$ , and is otherwise zero. Thus we must have

$(w(\Lambda+\rho)-\rho)|_A+\rho_p+\chi_\pi=0$

for this not to vanish. But $\rho|_A=\rho_P$ , so this just says

$w(\Lambda+\rho)|_A+\chi_\pi=0$

and thus we get

$E_2^{p,q}=\bigoplus_{w\in W^P,\,\,\ell(w)=q,\,\,w(\Lambda+\rho)|_A+\chi_\pi=0}H^p(\mathfrak{m}_0,K_\infty\cap P(\mathbb{R});\tilde\pi\otimes E_{w(\Lambda+\rho)-\rho})$ .

The infinitesimal character of $\tilde\pi$ must match that of $E_{w(\Lambda+\rho)-\rho}^\vee$ , which is the orbit of $-w(\Lambda+\rho)|_{\mathfrak{h}\cap\mathfrak{m}_0}$ under the Weyl group of $M$ . Since $\Lambda+\rho$ is dominant and regular and $w\in W^P$ and $W^P$ is a set of coset representatives for the Weyl group of $G$ modulo that of $M$ , this condition pins down this orbit uniquely; that is, the orbit of $-w_1(\Lambda+\rho)|_{\mathfrak{h}\cap\mathfrak{m}_0}$ and $-w_2(\Lambda+\rho)|_{\mathfrak{h}\cap\mathfrak{m}_0}$ cannot coincide for $w_1,w_2\in W^P$ distinct. Thus, since we know as well that $w(\Lambda+\rho)|_A=-\chi_\pi$ , the dominance and regularity of $\Lambda+\rho$ shows that we can determine $w$ uniquely. If it exists, it is the unique element of the Weyl group of $G$ such that

$w\in W^P,\qquad w(\Lambda+\rho)|_A+\chi_\pi=0,\qquad -w(\Lambda+\rho)|_{\mathfrak{h}\cap\mathfrak{m}_0}\textrm{ is in }\chi_{\mathrm{inf},\tilde\pi}$ ,

where $\chi_{\mathrm{inf},\tilde\pi}$ is the infinitesimal character of $\tilde\pi$ . Assume such a $w$ exists and let $\ell=\ell(w)$ for this $w$ . Then we just showed that $E_2^{p,q}=0$ unless $q=\ell$ , and that

$E_2^{p,\ell}=H^p(\mathfrak{m}_0,K_\infty\cap P(\mathbb{R});\tilde\pi\otimes E_{w(\Lambda+\rho)-\rho})$ .

Thus the spectral sequence degenerates immediately, and thus we finally get

$H^i(\mathfrak{g}_0,K_\infty;W_{\tilde\pi}\otimes\mathrm{Sym}(\mathfrak{a}^\vee)\otimes V)=\mathrm{Ind}_{M(\mathbb{A}_f)}^{G(\mathbb{A}_f)}H^i(\mathfrak{m}_0,K_\infty\cap P(\mathbb{R});\tilde\pi\otimes E_{w(\Lambda+\rho)-\rho})^{m(\pi)}$ !

Cohomology of the induction

Now we compute the cohomology of the induced representation.

$\mathrm{Ind}_{M(\mathbb{A}_f)\times(\mathfrak{g}_0\cap\mathfrak{p},K_\infty\cap P(\mathbb{R}))}^{G(\mathbb{A}_f)\times(\mathfrak{g},K_\infty)}(\tilde\pi\otimes\mathbb{C}_{\rho_P+\chi_\pi})$ .

In a previous version of this post, I claimed that this sits as a subspace of $\mathcal{A}_{V,\varphi}$ under our assumption, and that it is the space of Eisenstein series $E(\chi_\pi,f)$ with no derivatives. I want to take this moment to say that this may not be correct. I will explain what goes wrong in an example at the end of the next post.

In any case, the cohomology of this induction can be computed in much the same way as that of $\mathcal{A}_{V,\varphi}$ . This is actually what Borel and Wallach do in their book. It is

$H^i(\mathfrak{g}_0,K_\infty;W_{\tilde\pi}\otimes\mathbb{C}_{\rho_P+\chi_\pi}\otimes V)$ .

One proceeds as above, and eventually reaches a step where instead of computing $H^*(\mathfrak{a},\mathrm{Sym}(\mathfrak{a}^\vee)\otimes\mathbb{C}_{\nu})$ , one instead needs to compute $H^*(\mathfrak{a},\mathbb{C}_{\rho_P+\chi_\pi}\otimes\mathbb{C}_{\nu})$ . But this is easy; it is trivial unless $\nu=-\rho_P-\chi_\pi$ , in which case it is the exterior algebra on the vector space $\mathfrak{a}^\vee$ . Substituting this back into the computation above, one gets

$H^i(\mathfrak{g}_0,K_\infty;W_{\tilde\pi}\otimes\mathbb{C}_{\rho_P+\chi_\pi}\otimes V)$

$=\bigoplus_{i=j+k}H^j(\mathfrak{m}_0,K_\infty\cap P(\mathbb{R});\tilde\pi\otimes E_{w(\Lambda+\rho)-\rho})^{m(\pi)}\otimes\bigwedge^k\mathfrak{a}^\vee$ ,

where $w$ is as above. Notice that the result of the previous computation is one of the factors in the tensor product on the right hand side.

If this cohomology is nontrivial, the only time it ever coincides with the cohomology of $\mathcal{A}_{V,\varphi}$ is when $P=G$ so that $A$ is trivial. Otherwise the cohomology of this induced representation is bigger. Even when $A$ is one dimensional, the cohomology space just above is the direct sum of two cohomology spaces, namely the cohomology of $\mathcal{A}_{V,\varphi}$ , and the same thing again but shifted up in degree by one.

This discrepancy is essentially what we will use in the next post to answer the question we posed in the first post: We will construct a space of regular Eisenstein series on $\mathrm{GSp}_4$ and embed it in a space that also includes its derivatives, and we will find a quotient of the space of Eisenstein series without derivatives which has cohomology in a different degree than the space of these Eisenstein series and their derivatives. Strictly speaking, however, it will only work this way if we impose strong conditions on $\pi$ . But still we’ll be able to answer our question next post.

Eisenstein series and a cohomological phenomenon (Part II)

I’m going to continue now with my story about Eisenstein cohomology. This post will be about the Franke–Schwermer decomposition of the space of automorphic forms. Let me warn the reader that post will be technical and heavy on notation. If you are not too invested in the details of these things, it is probably worthwhile to skip to the theorem at the end of this section to see the shape of this decomposition, and just read the paragraphs after it. For the convenience of the reader, I am also going to give a glossary of notation that will be used in this post, as well as notation that will be introduced later. Let me do this now.

Notation for the group:

$G$ is a reductive algebraic group over $\mathbb{Q}$ .
$\mathfrak{g}$ is the complexified Lie algebra of $G$ .
$\mathfrak{g}_0$ is the complexified Lie algebra of the derived group of $G$ .
$K$ is a maximal compact subgroup of $G(\mathbb{A})$ .
$K_\infty$ is a maximal compact subgroup of $G(\mathbb{R})$ .
$V$ is a finite dimensional irreducible complex representation of $G$ .

Notation for parabolic subgroups:

$P$ is a parabolic $\mathbb{Q}$ -subgroup of $G$ .
$M$ is the Levi subgroup of $P$ .
$N$ is the unipotent radical of $P$ .
$A$ is the maximal split torus in the center of $M$ .
$M_0$ is the subgroup of $M$ such that $P=M_0AN$ is the Langlands decomposition of $P$ .
$\mathfrak{p},\mathfrak{m},\mathfrak{m}_0,\mathfrak{a},\mathfrak{n}$ are the relevant complex Lie algebras.

Notation to be introduced in more detail below:

$pi$ is a cuspidal automorphic representation of $M$ .
$\chi_\pi$ is the archimedean central character of $pi$ (viewed either as a character of $A(\mathbb{R})$ or of $\mathfrak{a}$ ).
$\tilde\pi$ is the unitary normalization of $\pi$ .
$W_{\tilde\pi}$ is a parabolic induction space constructed from $\tilde\pi$ .
$m(\pi)$ is the cuspidal multiplicity of $\pi$ .
$E(\lambda,f)$ is an Eisenstein series.
$\Phi_{V,[P]}$ is a set of equivalence classes of pairs $(P,\pi)$ .
$\mathcal{A}_V$ is the space of automorphic forms on $G$ which are killed by a power of the annihilator of $V^\vee$ in the center of the universal enveloping algebra of $\mathfrak{g}$ .
$\mathcal{A}_{V,[P]}$ is the subspace of $\mathcal{A}_V$ of forms which are negligible along parabolics not associate to $P$ .
$\mathcal{A}_{V,\varphi}$ is, for $\varphi\in\Phi_{V,[P]}$ , the space of all possible residues and derivatives of Eisenstein series $E(\chi_\pi,f)$ for $f\in W_{\tilde\pi}$ .

A decomposition of the space of automorphic forms

In the paper I mentioned in the previous post, Franke was able to break up the space of automorphic forms on $G$ according to the parabolic subgroups of $G$ . This decomposition was further refined in a paper of Franke–Schwermer, and I will describe this refinement briefly, after I define the Eisenstein series. See Chapter 1 of the paper of Li–Schwermer for a nice exposition of this.

Now let $P$ be a parabolic subgroup of $G$ defined over $\mathbb{Q}$ . Let $M$ be the Levi of $P$ and let $N$ be the unipotent radical. Write $A$ for the maximal split torus in the center of $M$ . Let $\pi$ now be a cuspidal automorphic representation of $M$ with archimedean central character $\chi_\pi$ , and renormalize $\pi$ to get a representation $\tilde{\pi}$ which is unitary.

Next we define $W_{\tilde{\pi}}$ be the space of smooth, $K$ -finite, complex valued functions $f$ on $M(\mathbb{Q})N(\mathbb{A})A(\mathbb{R})^\circ\backslash G(\mathbb{A})$ such that for any $g\in G(\mathbb{A})$ , the function on $M(\mathbb{A})$ given by $m\mapsto f(mg)$ belongs to the space $L_{\mathrm{cusp}}^2(M(\mathbb{Q})A(\mathbb{R})^\circ\backslash M(\mathbb{A}),1)[\tilde{\pi}]$ , where the brackets denote isotypic component and the $1$ denotes trivial central character.

What is this space? Well, one should view it as a parabolic induction. In fact, it is true that

$W_{\tilde{\pi}}\cong\mathrm{Ind}_{P(\mathbb{A})}^{G(\mathbb{A})}(\tilde\pi)^{m(\pi)}$

where the induction is a (non-normalized) smooth, $K$ -finite induction, and $m(\pi)$ is the multiplicity of $\tilde\pi$ in $L_{\mathrm{cusp}}^2(M(\mathbb{Q})\backslash M(\mathbb{A}),1)$ .

Now, taking $\mathfrak{a}$ to be the complexified Lie algebra of $A$ and $\mathfrak{a}^\vee$ to be its dual, if given an $f\in W_{\tilde{\pi}}$ and $\lambda\in\mathfrak{a}^\vee$ , one defines the Eisenstein series

$E(\lambda,f)(g)=\sum_{\gamma\in P(\mathbb{Q})\backslash G(\mathbb{Q})}e^{\langle H_P(\gamma g),\lambda+\rho_P\rangle}f(\gamma g)$ ,

where $\rho_P$ is the character of $\mathfrak{a}$ induced from the modulus character of $P$ (given by half the sum of the positive roots in $N$ , if we had fixed a Borel contained in $P$ ) and where $H_P$ is the usual height function on $G$ defined by $P$ . In the case when $P$ is a maximal parabolic, for instance, one has

$e^{\langle H_P(pk),\rho_P\rangle}=\delta_P^{1/2}(p),\qquad p\in P,\,\,k\in K$ ,

where $\delta_P$ is the usual modulus character on $P$ .

Now the series defining $E(\lambda,f)(g)$ only converges for $\lambda$ sufficiently deep inside a positive cone in $\mathfrak{a}^\vee$ , but it defines a holomorphic function of $\lambda$ there, and continues meromorphically to all of $\mathfrak{a}^\vee$ . The Eisenstein series $E(\lambda,f)(g)$ is an automorphic form, and as $f$ varies, these give an automorphic realization of (a central twist of) the parabolic induction of $\pi$ .

So given $P$ and $\pi$ as above, we defined an Eisenstein series. But different choices $P$ ‘s and $\pi$ ‘s may define the same Eisenstein series. So Franke and Schwermer define an equivalence relation on pairs $(P,\pi)$ , consisting of a parabolic and a cuspidal automorphic representation on its Levi, so that they give the Eisenstein series. They furthermore restrict attention only to those $\pi$ whose infinitesimal characters match that of $V^\vee$ (where $V$ is, as above, a finite dimensional irreducible representation of $G$ ). See the Section 1.3 of the paper of Li–Schwermer linked above for a precise description of this equivalence relation. If $(P,\pi)$ and $(P',\pi')$ are equivalent, then $P$ and $P'$ are associate. We denote by $[P]$ the associate class of parabolics represented by $P$ . Let us also denote the set of equivalence classes of the pairs $(P,\pi)$ as above, with the afformentioned restriction on central characters, by $\Phi_{V,[P]}$ .

Let $\mathcal{J}$ be the annihilator of $V^\vee$ in the center of the universal enveloping algebra, and write $\mathcal{A}_V$ for the space of automorphic forms which are annihilated by a power of $\mathcal{J}$ . (These are the forms which can possibly contribute to the cohomology of $G$ with coefficients in $V$ .) Given $[P]$ , we let $\mathcal{A}_{V,[P]}$ be the space of forms $f$ in $\mathcal{A}_V$ whose constant terms along parabolics $Q$ are orthogonal to the space of functions on the Levi of $Q$ (these forms are said to be “negligible” along $Q$ ) for any $Q$ not in $[P]$ . This space includes any Eisenstein series coming from a cuspidal automorphic representation of the levi of $P$ .

For $\varphi=(P,\pi)\in\Phi_{V,[P]}$ , let $\chi_\pi\in\mathfrak{a}^\vee$ also denote the differential of the real component of the central character of $\pi$ . Define $\mathcal{A}_{V,\varphi}$ to be the space spanned by all possible residues and derivatives at $\lambda=\chi_\pi$ of Eisenstein series $E(\lambda,f)$ as $f$ ranges through $W_{\tilde{\pi}}$ . Then we can finally state the decomposition theorem of Franke–Schwermer:

Theorem. We have a direct sum decomposition

$\mathcal{A}_V=\bigoplus_{\textrm{classes }[P]}\bigoplus_{\varphi\in\Phi_{V,[P]}}\mathcal{A}_{V,\varphi}$ .

It is worth noting that the space $\mathcal{A}_{V,[G]}$ is the space of cusp forms on $G$ satisfying the archimedean infinitesimal central character condition explained above. So this breaks the space of automorphic forms (with that condition) into a direct sum of a space of cusp forms with spaces that consist of all residues and derivatives of Eisenstein series which are induced from a given cuspidal automorphic representation of a Levi subgroup.

This decomposition therefore gives a decomposition of cohomology of $G$ as follows:

$\qquad H^*(\mathfrak{g}_0,K_\infty;\mathcal{A}_G\otimes V)$

$=H^*(\mathfrak{g}_0,K_\infty;L_{\mathrm{cusp}}^2(G(\mathbb{Q})\backslash G(\mathbb{A}))\otimes V)\oplus\bigoplus_{[P]}\bigoplus_{\varphi}H^*(\mathfrak{g}_0,K_\infty;\mathcal{A}_{V,\varphi}\otimes V)$ .

The leading summand of this decomposition is called the cuspidal cohomology of $G$ , and the rest is the Eisenstein cohomology. In the next post I will compute a given summand of the Eisenstein cohomology under the favorable hypothesis that there are no residual Eisenstein series.

Eisenstein series and a cohomological phenomenon (Part I)

[Edit, March 2022: I made some mistakes in the writing of this post and its three sequels. I’ve decided that, instead of rewriting and reposting them, I’m going to leave them as they are but insert some “edits,” like this one, where there are mistakes.]

This is the first in a series of posts which will be about a phenomenon I noticed and thought was interesting while studying the Eisenstein cohomology of arithmetic groups. To explain this phenomenon, however, we need to do some setup.

Setup and the main question(s)

Let’s start with some notation. Let $G$ be a reductive algebraic group over $\mathbb{Q}$ . Consider the adelic quotient space $G(\mathbb{Q})\backslash G(\mathbb{A})$ and the space $C^\infty(G(\mathbb{Q})\backslash G(\mathbb{A}))$ of smooth functions on it. We view this space as a $(\mathfrak{g},K_\infty)\times G(\mathbb{A}_f)$ -module, where $\mathfrak{g}$ is the complexified Lie algebra of $G$ , $K_\infty$ is the maximal compact subgroup of $G(\mathbb{R})$ , and $\mathbb{A}_f$ denotes the finite adeles of $\mathbb{Q}$ . Let $\mathfrak{g}_0$ be the complexified Lie algebra of the derived group of $G$ (so remove the center from $\mathfrak{g}_0$ ). Then we have the following theorem, which was originally a conjecture of Borel.

Theorem. Let $V$ be a finite dimensional irreducible complex representation of $G(\mathbb{C})$ . Let $\mathcal{A}_G$ denote the $(\mathfrak{g},K_\infty)\times G(\mathbb{A}_f)$ -module of automorphic forms for $G$ . Then the inclusion

$\mathcal{A}_G\hookrightarrow C^\infty(G(\mathbb{Q})\backslash G(\mathbb{A}))$

induces an isomorphism on the level of $(\mathfrak{g},K_\infty)$ -cohomology when tensored with $V$ . That is

$H^*(\mathfrak{g},K_\infty;\mathcal{A}_G\otimes V)\cong H^*(\mathfrak{g},K_\infty;C^\infty(G(\mathbb{Q})\backslash G(\mathbb{A}))\otimes V).$

This was first proved by Franke in his paper published in 1998.

However, finding exactly which automorphic representations actually contribute a nonzero subspace to this cohomology can prove to be quite difficult. In fact, one can even ask the following basic question.

Question 1. Let $\pi\hookrightarrow\mathcal{A}_G$ be an (irreducible) automorphic representation which is a subspace of the space of automorphic forms on $G$ . Is the map

$H^*(\mathfrak{g},K_\infty; \pi)\to H^*(\mathfrak{g},K_\infty;\mathcal{A}_G)$

injective?

In other words, does an automorphic representation with nonvanishing cohomology necessarily contribute its cohomology to that of $G$ ?

I do not know the answer to this question [Edit, March 2022: Actually I do, and the answer is “no.”] but I do know the answer to a related question (Question 2 below). But before I get to that, let me explain why the answer to Question 1 is “yes” if the representation $\pi$ is cuspidal.

It’s not too hard, actually. In fact, as we will see in the next post, the space of cusp forms splits off as a direct summand of the space of automorphic forms. This summand itself decomposes as a direct sum, equal to the smooth, $K$ -finite vectors ( $K$ is a maximal compact subgroup of $G(\mathbb{A})$ ) in the cuspidal spectrum $L_{\mathrm{cusp}}^2(G(\mathbb{Q})\backslash G(\mathbb{A}))$ , which has

$L_{\mathrm{cusp}}^2(G(\mathbb{Q})\backslash G(\mathbb{A}))\cong\bigoplus_\pi V_\pi^{m(\pi)}$ .

Here, the sum is a Hilbert direct sum over all irreducible subrepresentations $(\pi,V_\pi)$ of $L_{\mathrm{disc}}^2(G(\mathbb{Q})\backslash G(\mathbb{A}))$ , and $m(\pi)$ is the multiplicity of $\pi$ in this space. This Hilbert direct sum becomes a usual direct sum when we replace this space by the subspace of smooth, $K$ -finite vectors. But $(\mathfrak{g},K_\infty)$ -cohomology only sees the smooth, $K$ -finite vectors, so the result follows.

It is maybe worth remarking that we have a decomposition as a Hilbert direct sum of the discrete spectrum

$L_{\mathrm{disc}}^2(G(\mathbb{Q})\backslash G(\mathbb{A}))\cong\bigoplus_\pi V_\pi^{m_\pi}$

In fact,

$L_{\mathrm{disc}}^2(G(\mathbb{Q})\backslash G(\mathbb{A}))=L_{\mathrm{cusp}}^2(G(\mathbb{Q})\backslash G(\mathbb{A}))\oplus L_{\mathrm{res}}^2(G(\mathbb{Q})\backslash G(\mathbb{A}))$ ,

so this argument would seem to show that the answer to the above question is “yes” for residual Eisenstein series as well. However, this does not follow immediately, and I am not sure whether this result is true for residual Eisenstein series. [Edit, March 2022: Actually this already fails for the simplest example of residual Eisenstein series on $GL_2$ .] What’s the issue in general? Well some automorphic representations, namely those coming from Eisenstein series, do not appear as direct summands of $\mathcal{A}_G$ , but rather appear in nontrivial extensions of themselves with other automorphic representations. Residual Eisenstein series themselves can appear as subrepresentations of spaces consisting of Eisenstein series, their derivatives, and their residues, and these spaces do not themselves decompose as direct sums.

In any case, I would like to ask (and eventually answer) the following question, alluded to above.

Question 2. Let $\sigma$ be a direct summand (as a $G(\mathbb{A}_f)\times(\mathfrak{g},K_\infty)$ -module) of the space $\mathcal{A}_G$ of automorphic forms on $G$ . Let $\pi$ be an irreducible subquotient of $\sigma$ . Is it possible for $H^i(\mathfrak{g},K_\infty;\pi)$ to be bigger than $H^i(\mathfrak{g},K_\infty;\sigma)$ ?

The answer, as you may have expected, is “yes, this is possible.” I will give an example in a later post on $\mathrm{GSp}_4$ of a $\pi$ and a $\sigma$ such that the cohomology of $\pi$ occurs only in (middle) degree $3$ , while the cohomology of $\sigma$ occurs only in degree $2$ . But that’s not surprising, really; maybe there’s some chain of long exact sequences which relates the cohomology of $\pi$ and that of $\sigma$ , and maybe there’s a connecting homomorphism in one of them that raises the degree from $2$ to $3$ . But actually, and this is what I find interesting, this is NOT the case. In fact, I’ll show there’s a subspace $\sigma_0\hookrightarrow\sigma$ with $\pi$ as a quotient, which has cohomology in both degrees $2$ and $3$ . This means that the cohomology of this intermediary space $\sigma_0$ sees separately both the cohomology of $\pi$ and of $\sigma$ , even though the cohomology of $\mathcal{A}_G$ only sees that of $\sigma$ . [Edit, March 2022: Actually I can’t construct such a $\sigma_0$ . I’ll make some edits to part IV of this post explaining what goes wrong.]

But before I give this example, in the next couple of posts I will need to give some relevant background on Eisenstein cohomology and the decomposition of the space of automorphic forms due to Franke and Franke–Schwermer. Stay tuned!

A Post on Primes

Hi all, I thought I should finally start my own math blog. Why add another math blog to the pile? Well for fun, and because I’ve finally been working in this field long enough to have sufficiently many small insights to fill out a blog.

Probably most of the posts here will be about number theory, especially things related to Iwasawa theory and the $p$ -adic theory of automorphic forms, whence the name.

But for this first post I want to do something more elementary. Today I’ll talk about some proofs of the fact that there are infinitely many prime numbers, and connect them in a way that I haven’t seen elsewhere in the literature.

Furstenberg’s proof

It is fairly well known that in the middle of the twentieth century, Furstenburg found a proof that there are infinitely many primes using elementary point set topology. But it is not known by everyone, so let me recall it here.

What Furstenberg does is give $\mathbb{Z}$ the smallest topology where the arithmetic progressions are open. The observation to make here is that every nonempty open subset of $\mathbb{Z}$ under this topology is infinite.

Now assume for sake of contradiction that there are only finitely many primes. For a prime $p$ , let $U_p$ be the open set consisting of numbers which are coprime to $p$ ; this is indeed open as it is a union of arithmetic progressions, more specifically it is the union of the nonzero residue classes modulo $p$ . Then the set

$\bigcap_{p\textrm{ prime}}U_p$

is open, being a finite intersection of opens, but is easily computed to be the set $\{\pm 1\}$ . Contradiction.

After stripping away the topology, this is essentially Euclid’s method of proof. But this way of viewing things lends itself well to being recast as follows.

A recasting of Furstenberg’s proof

Now I will put this proof in an “adelic” setting. Let $\widehat{\mathbb{Z}}$ be the profinite completion of the integers. So

$\widehat{\mathbb{Z}}=\varprojlim_{n\geq 1}\mathbb{Z}/n\mathbb{Z}$ .

The group $\widehat{\mathbb{Z}}$ is endowed here with its standard profinite topology. The subset $\mathbb{Z}$ is then dense in $\widehat{\mathbb{Z}}$ .

Now by the Chinese remainder theorem,

$\widehat{\mathbb{Z}}\cong\prod_{p\textrm{ prime}}\mathbb{Z}_p$

as topological groups, where $\mathbb{Z}_p$ is the topological group (actually ring) of $p$ -adic integers.

Assume that there are only finitely many primes now, so that the product above is finite. We recall that the group of units $\mathbb{Z}_p^\times$ is open in $\mathbb{Z}_p$ , being a union of cosets of the ideal $p\mathbb{Z}_p$ . So under this assumption, the subset

$U=\prod_{p\textrm{ prime}}\mathbb{Z}_p^\times$

is open in $\widehat{\mathbb{Z}}$ . The contradiction comes when we consider $\mathbb{Z}\cap U$ : On the one hand, this equals $\mathbb{Z}^\times=\{\pm 1\}$ ; but on the other hand, this is the intersection of an infinite dense subset with an open subset, and therefore must be infinite. This is absurd, and so we are done.

The thing to note here is that the topology on $\mathbb{Z}$ from Furstenberg’s proof is exactly the subspace topology that $\mathbb{Z}$ inherits from $\widehat{\mathbb{Z}}$ . Indeed this is what inspired me to think of this version of the proof. But this recasting is nice because it can be related to yet another proof of the infinitude of primes. This can be done as follows.

Relation to Euler’s proof

In the early 1700’s, Euler gave the following proof that there are infinitely many primes. Define the zeta function

$\zeta(s)=\sum_{n=1}^\infty \frac{1}{n^s}$ ,

where $s>1$ is a real variable. Then this infinite sum converges absolutely, and even decomposes as a product

$\zeta(s)=\prod_{p\textrm{ prime}}\frac{1}{1-p^{-s}}$ .

Why? Expand the terms on the right hand side as geometric series, multiply them, and observe that every integer factors uniquely into primes.

Now assuming for sake of contradiction that there are only finitely many primes, the product above can be evaluated at $s=1$ and gives

$\sum_{n=1}^\infty \frac{1}{n}=\prod_{p\textrm{ prime}}\frac{1}{1-p^{-1}}$

with the product on the right being finite. But the sum on the left diverges; indeed, we have

$1+(\frac{1}{2}+\frac{1}{3})+(\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7})+\dotsb> 1+(\frac{1}{2}+\frac{1}{2})+(\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4})+\dotsb=1+1+1+\dotsb$ .

This is the contradiction sought.

How does this relate to the adelic proof above? Well first let’s put a Haar measure $\mu$ on the compact group $\widehat{\mathbb{Z}}$ which gives it measure $1$ , and similarly give $\mathbb{Z}_p$ such a measure $\mu_p$ . Then the measure $\mu$ is the product of the measures $\mu_p$ .

Above, we got our contradiction using the set $U$ , which was the product over all $p$ of $\mathbb{Z}_p^\times$ . But the measure of this set is easily seen to be

$\mu(U)=\prod_{p\textrm{ prime}}\frac{p-1}{p}=\prod_{p\textrm{ prime}}(1-p^{-1})=\zeta(1)^{-1}$

because each $\mathbb{Z}_p^\times$ is the union of the $(p-1)$ non-identity cosets of $p\mathbb{Z}_p$ , and $p\mathbb{Z}_p$ has measure $\frac{1}{p}$ .

Unfortunately I do not see a way to turn this around and use abstract measure theory to prove the divergence of $\zeta(1)$ . This seems to be a fact which is somehow strictly stronger than the infinitude of primes.