# Automorphic Hamiltonians, Epstein zeta functions, and Kronecker limit
formulas

Adrienne Sands  School of Mathematics, University of Minnesota, Minneapolis,
MN 55455 MIT Lincoln Laboratory, 244 Wood St, Lexington, MA 02421
<EMAIL_ADDRESS>This manuscript is dedicated to my advisor, Paul
Garrett.

###### Abstract.

First, we recount a history of how certain methods using natural self-adjoint
operators have, thus far, failed to prove the Riemann Hypothesis. In Section
2, we set the analytical context necessary to have genuine proofs in later
sections, rather than attractive heuristics. In Section 3, we recall the
utility of designed pseudo-Laplacians by reproving meromorphic continuation of
certain Eisenstein series and proving a spacing result for zeros of
$\zeta_{k}(s)$ for $k$ a complex quadratic field with negative determinant, as
in [BG20] and [Gar18]. In Section 4, we construct an automorphic Hamiltonian
which has purely discrete spectrum on
$L^{2}\left(SL_{r}(\operatorname{\mathbb{Z}})\backslash
SL_{r}(\operatorname{\mathbb{R}})/SO(r,\operatorname{\mathbb{R}})\right)$,
identify its ground state, and show how it can characterize a nuclear Fréchet
automorphic Schwartz space.

###### 2022 Mathematics Subject Classification:

Number Theory

This material is based upon work supported by the National Science Foundation
Graduate Research Fellowship Program under Grant #39202.

††copyright: ©2021: Adrienne Sands

##  1\. Introduction

In 1972, Montgomery and Dyson heuristically related the spacing of zeros of
the Riemann zeta function to the distribution of eigenvalues of random
Hermitian matrices used in nuclear physics. The eigenstates of an important
class of quantum mechanical operators satisfy this distribution, suggesting a
mysterious connection between mathematical physics and number theory. Some
time in the 1910s, Pólya and Hilbert independently speculated, in effect, that
one might prove the Riemann Hypothesis if the zeros $s$ of $\zeta(s)$
corresponded to eigenvalues $\lambda_{s}=s(s-1)$ of a self-adjoint operator.
While it is certainly possible to construct an operator with eigenvalues
parametrized by zeros, self-adjointness is apparently unverifiable by
available machinery [BC95].

Efforts to prove that eigenvalues of certain self-adjoint operators correspond
to zeros $s$ of $\zeta(s)$ have been dramatic. In his masters thesis [Ha77],
H. Haas attempted to numerically compute eigenvalues $\lambda_{s}=s(s-1)$ of
the invariant Laplace-Beltrami operator (see Section 2.2) on
$SL_{2}(\operatorname{\mathbb{Z}})\backslash\operatorname{\mathfrak{H}}$. H.
Stark and D. Hejhal happened upon his list of $s$-values via A. Terras and
recognized zeros of $\zeta(s)$ and the Dirichlet $L$-function
$L(s,\chi_{-3})$, a striking discovery which suggested that the Pólya-Hilbert
dream might be realized with a natural operator. Hejhal immediately began
efforts to replicate Haas’s results using more careful numerical methods but
found that exactly these zeros were missing. Naturally, he wondered how flawed
numerical procedures could produce garbage output with such number theoretic
significance. He determined that Haas had misapplied Henrici’s collocation
method of numerically solving differential equations [FHM67], inadvertently
allowing non-smoothness of eigenfunctions at the corners $\omega=e^{2\pi i/3}$
of the usual fundamental domain for $SL_{2}(\operatorname{\mathbb{Z}})$, and
had actually solved an inhomogeneous equation
$(\Delta-\lambda_{s})u_{s}=\delta_{\omega}^{\text{afc}}$, with
$\delta_{\omega}^{\text{afc}}=\sum_{\gamma\in\Gamma}\delta^{\mathfrak{H}}_{\omega}\circ\gamma$
the automorphic Dirac delta at the third and sixth roots of unity [Hej81].
Since the “eigenfunctions” corresponding to zeros of $\zeta(s)$ were solutions
to an inhomogeneous equation (rather than $(\Delta-\lambda_{s})u=0)$, the
special values in Haas’s list were not necessarily genuine eigenvalues of a
self-adjoint operator, and no inferences (such as the Riemann Hypothesis)
could be made from the special $\lambda_{s}$’s.

Nonetheless, certain properties of automorphic Green’s functions (see Section
3.2) had already been investigated in [Els73], [Neu73], [Fay77], [Nie73], and
others. For example, the constant term of $u_{s}$ is

$c_{P}u_{s}(iy)=\int_{0}^{1}u_{s}(x+iy)dx=\frac{y^{1-s}E_{s}(\omega)}{2s-1},\hskip
14.22636pt\text{for }y\geq 1,$

and it had long been known that

$E_{s}(\omega)=\left(\frac{\sqrt{3}}{2}\right)^{s/2}\cdot\frac{\zeta(s)L(s,\chi_{-3})}{\zeta(2s)}.$

Although these connections are provocative, there is no obvious connection to
the (genuine) eigenvalues of a self-adjoint operator.

In the early 1980s, mathematical physicist Y. Colin de Verdière speculated on
a way to promote these numbers to genuine eigenvalues by considering variants
of $\Delta$ with additional true eigenfunctions [CdV82]. In particular, he
considered Friedrichs’ self-adjoint extension $\tilde{\Delta}_{\delta}$ of a
restriction of $\Delta$ (see Section 2.8) to automorphic test functions in the
kernel of $\delta_{\omega}^{\text{afc}}$. These so-called pseudo-Laplacians
allow certain non-smooth functions as genuine eigenfunctions, so the special
values in Haas’ list might be (true) eigenvalues of a self-adjoint operator.
However, a technical hazard obstructs such a construction: the Friedrichs
extension for second-order elliptic operators limits the non-smoothness of
eigenfunctions to a $+1$-index Sobolev space, but solutions to
$(\Delta-\lambda_{s})u=\delta_{\omega}^{\text{afc}}$ are in a $1-\varepsilon$
Sobolev space. In his 1983 paper, Colin de Verdière suggested one might
overcome this issue by projecting solutions to the orthogonal complement of
the discrete spectrum of
$L^{2}(SL_{2}(\operatorname{\mathbb{Z}})\backslash\operatorname{\mathfrak{H}})$.

Almost forty years later, [BG20] validated and refined the observations of
Hejhal and Colin de Verdière. After projecting, not exactly to the orthogonal
complement of the discrete spectrum, but to the larger non-cuspidal spectrum,
all eigenvalues $\lambda_{s}$, if any at all, of the pseudo-Laplacian
$\tilde{\Delta}_{\theta}$ would correspond to zeros of $\zeta(s)$ and
$L(s,\chi_{-3})$; however, assuming Montgomery’s 1972 pair correlation
conjecture about the spacing of zeros, at most 94% of the zeros of $\zeta(s)$
can appear in this basic construction. Since the alleged remaining 6% probably
aren’t special, it is reasonable to believe that no zeros appear in the
simplest implementation of Colin de Verdière’s idea.

Central, but not unique, to the work of Bombieri-Garrett is a sufficient
analytical viewpoint to give genuine proofs of attractive heuristics, for
example, from physics. After all, physicists [Dir28], [Dir30], [Tho35], [BP35]
used solvable models such as $(-\Delta-\lambda)u=\delta$ to successfully
predict experimental outcomes since at least the 1930s, often rewriting the
corresponding differential equation as a perturbation of $-\Delta$ by a
singular potential $\delta$ (see Section 2.9). Apparently, these differential
equations were only understood rigorously after [BF61].

The spectral theory of unbounded self-adjoint operators on global automorphic
Sobolev spaces (see Section 2.6) provides a reasonable context to discuss
partial differential equations in automorphic forms, from solvable models to
eigenvalue equations of quantum mechanical Hamiltonians, perturbations of
$-\Delta$ by smooth confining potentials $q$ (see Section 2.9). After
establishing an appropriate analytical context in Section 2 and expounding on
aforementioned results in Section 3, we construct an automorphic Hamiltonian
$-\Delta+q$ whose self-adjoint Friedrichs extension has purely discrete
spectrum on $SL_{r}(\operatorname{\mathbb{Z}})\backslash
SL_{r}(\operatorname{\mathbb{R}})/SO(r,\operatorname{\mathbb{R}})$ and
characterizes an automorphic Schwartz space as a nuclear Fréchet space.

##  2\. Background

In this chapter, we recall classical results. In Section 2.1, we exhibit a
single (adelic) Siegel set which covers automorphic quotients $G_{k}\backslash
G_{\mathbb{A}}$ and define a suitable notion of height on such quotients to
simplify computations. In Section 2.2, to facilitate construction of an
automorphic Hamiltonian $-\Delta+q$, we recall a coordinate-free
characterization of the invariant Laplacian $\Delta$ on quotients
$\Gamma\backslash G/K$. In Sections 2.4 and 2.5, we recall analytical
properties of degenerate Eisenstein series attached to certain maximal proper
parabolic subgroups of $SL_{r}(\operatorname{\mathbb{R}})$, which motivate our
choice of confining potential $q$. In Section 2.6, we characterize the global
automorphic Sobolev spaces of functions on which our Hamiltonian is defined.
In Section 2.7, we recall Schwartz’ kernel theorem and offer a convenient
notion of a nuclear Fréchet automorphic Schwartz space. In Section 2.8, we
recall Friedrichs’ construction of self-adjoint extensions of densely-defined
semi-bounded symmetric operators on Hilbert spaces. In Section 2.9, we briefly
recount the perturbation theory of linear operators, placing our Hamiltonians
in a larger context.

###  2.1. Reduction Theory

To simplify analysis in later sections, we exhibit a single (adelic) Siegel
set that covers the quotient $SL_{r}(\operatorname{\mathbb{Z}})\backslash
SL_{r}(\operatorname{\mathbb{R}})/SO_{r}(\operatorname{\mathbb{R}})$ and
define a suitable notion of height. We largely follow the discussion in
[Gar18]*Section 3, which follows Godement’s Bourbaki talk on reduction theory
[God64], and refer the reader to the classic texts [Bor66b] and [Spr94]. For
convenience to the reader, we include some foundational material about the
adele group $G_{\operatorname{\mathbb{A}}}=GL_{r}(\operatorname{\mathbb{A}})$.

####  2.1.1. The adele group
$G_{\operatorname{\mathbb{A}}}=GL_{r}(\operatorname{\mathbb{A}})$

Let $k$ be an arbitrary number field with integers $\mathfrak{o}$, completions
$k_{v}$, and local rings of integers $\mathfrak{o}_{v}$ at non-archimedean
places. Let $G_{v}=GL_{r}(k_{v})$ with center $Z_{v}$. For non-archimedean
$v$, let $K_{v}=GL_{r}(\mathfrak{o}_{v})$. For real $v$, let $K_{v}$ be the
standard orthogonal group $O_{r}(\operatorname{\mathbb{R}})=\\{g\in
GL_{r}(\operatorname{\mathbb{R}}):g^{\top}g=1_{r}\\}$, and for complex $v$,
let $K_{v}$ be the standard unitary group $U_{r}=\\{g\in
GL_{r}(\operatorname{\mathbb{C}}):g^{*}g=1_{r}\\}$.

Let $\ell$ be the number of non-isomorphic archimedean completions of $k$.
That is, $\ell=\ell_{1}+\ell_{2}$, where $\ell_{1}$ is the number of real
completions, $\ell_{2}$ is the number of complex completions (not counting
conjugates), and $[k:\operatorname{\mathbb{Q}}]=\ell_{1}+2\ell_{2}$. Let
$Z^{+}$ be the positive real scalar matrices diagonally imbedded across all
archimedean places $v$ by the map

$\delta:t\mapsto(...,t^{1/\ell},...)\hskip 19.91692pt(\text{for }t>0)$

The map $\delta$ gives a section of the idele norm map

$|\cdot|:t\mapsto\prod_{v}|t_{v}|_{v}.$

That is, $|\delta(t)|=t$.

The group $P_{v}$ of $v$-adic points of a standard parabolic $P=P^{F}$ is the
stabilizer of $F_{v}\subset k_{v}^{r}$ in $G_{v}$ with the same shape as $P$.
Similarly, $N_{v}=N^{P}_{v}$ and $M_{v}=M^{P}_{v}$ are the $v$-adic points of
the unipotent radical $N^{P}$ and the standard Levi component $M^{P}$ of a
standard parabolic $P$, respectively. As expected, we have Iwasawa
decompositions $G_{v}=P_{v}\cdot K_{v}$ for a standard minimal parabolic $P$
and Cartan decompositions $G_{v}=K_{v}M_{v}K_{v}$ for $M$ the standard Levi
component of the minimal parabolic (for proofs, see [Gar18]*Claims 3.2.1 and
3.2.2, for example). The Cartan decompositions follow from the spectral
theorem for symmetric or Hermitian operators at archimedean places and from
the structure theorem for finitely-generated modules at finite places.

The corresponding adele group is
$G_{\operatorname{\mathbb{A}}}=GL_{r}(\operatorname{\mathbb{A}})$, $r$-by-$r$
matrices with entries in $\operatorname{\mathbb{A}}$ and determinant in the
ideles $\operatorname{\mathbb{J}}$. This group is also a colimit of products

$G_{S}=\prod_{v\in S}G_{v}\times\prod_{v\notin S}K_{v},$

where finite sets $S$ of places $v$ are ordered by containment. Similarly,
$P_{\operatorname{\mathbb{A}}}$, $M^{P}_{\operatorname{\mathbb{A}}}$,
$N^{P}_{\operatorname{\mathbb{A}}}$, and $Z_{\operatorname{\mathbb{A}}}$ are
the adelic forms of those groups. We also let
$K_{\operatorname{\mathbb{A}}}=\prod_{v}K_{v}\subset
G_{\operatorname{\mathbb{A}}}$.

####  2.1.2. Reduction theory for $G_{k}\backslash
G_{\operatorname{\mathbb{A}}}$

For $k$ an arbitrary number field and $G_{k}=GL_{r}(k)$, we identify a
suitable notion of height $\eta$ on $G_{k}\backslash
G_{\operatorname{\mathbb{A}}}$ and recall that a single adelic Siegel set
$\mathfrak{S}_{t,C}$ covers the quotient $G_{k}\backslash
G_{\operatorname{\mathbb{A}}}$.

Let $P_{k}$, $M^{P}_{k}$, and $N^{P}_{k}$ be the corresponding subgroups of
$G_{k}$ with entries in $k$. We omit a proof of discreteness of $G_{k}$ in
$G_{\operatorname{\mathbb{A}}}$, since the proof easily generalizes from the
proof for $GL_{2}$ (see [Gar18]*Claim 2.2.1, for example). Let

$G^{1}=\\{g\in G_{\operatorname{\mathbb{A}}}:|\det g|=1\\}$

such that $G_{\operatorname{\mathbb{A}}}=Z^{+}\times G^{1}$. By the product
formula, $\prod_{v\leq\infty}|t|_{v}=1$ for $t\in k^{\times}$, we observe that
$G_{k}\subset G^{1}$, and, in particular, $G_{k}$ is still discrete in
$Z^{+}\backslash G_{\operatorname{\mathbb{A}}}\approx G^{1}$.

In contrast to the $GL_{2}$, the notion of a single numerical height for
$GL_{r}$ is initially replaced by the family of $(r-1)$ standard positive
simple roots

$\alpha_{i}\begin{pmatrix}m_{1}\\\ &\ddots&\\\
&&m_{r}\end{pmatrix}=\frac{m_{i}}{m_{i+1}}\hskip 19.91692pt(1\leq i<r),$

which are characters on the Levi component $M^{\text{min}}$ of the standard
minimal parabolic $P^{\text{min}}=P^{1,...,1}$. These roots make sense on
$M_{k}$, $M_{v}$, and $M_{\operatorname{\mathbb{A}}}$ and take values in
$k^{\times}$, $k_{v}^{\times}$, and $\operatorname{\mathbb{J}}$, respectively.

For reasons that will become apparent, we need a notion of height on
$\operatorname{\mathbb{A}}^{r}$ to identify a suitable notion of height on
$G_{k}\backslash G_{\operatorname{\mathbb{A}}}$. Let
$G_{\operatorname{\mathbb{A}}}$ act on the right on
$\operatorname{\mathbb{A}}^{r}$ by matrix multiplication. For real places $v$
of $k$, the local height function on $k_{v}^{r}$ is
$h_{v}(x)=\sqrt{x_{1}^{2}+...+x_{r}^{2}}$. For complex $v$, take
$h_{v}(x)=|x_{1}|_{\operatorname{\mathbb{C}}}+...+|x_{r}|_{\operatorname{\mathbb{C}}}$,
with
$|z|_{\operatorname{\mathbb{C}}}=|N_{\operatorname{\mathbb{R}}}^{\operatorname{\mathbb{C}}}z|_{\operatorname{\mathbb{R}}}$
to avoid disturbing the product formula. For non-archimedean $v$, let
$h_{v}(x)=\sup_{i}|x_{i}|_{v}$. By design, the isometry groups of the height
functions $h_{v}$ are the compact subgroups $K_{v}$. That is, the $K_{v}$
preserve heights $h_{v}$.

A primitive vector $x\in\operatorname{\mathbb{A}}^{r}$ is of the form
$x=x_{o}\cdot g$ for $g\in G_{\operatorname{\mathbb{A}}}$ and $x_{o}\in
k^{r}-\\{0\\}$. For $x=(x_{1},...,x_{r})\in k^{r}$ and almost all non-
archimedean places $v$, the components $x_{i}$ are in $\mathfrak{o}_{v}$ and
have local greatest common divisor 1; furthermore, elements of the adele group
$g\in G_{\operatorname{\mathbb{A}}}$ are in $K_{v}$ almost everywhere, so this
is not changed by multiplication by $g$. Thus, primitive vectors
$x=(x_{1},...,x_{r})$ have the property that the components $x_{i}$ are
locally integrable and have local greatest common divisor 1 almost everywhere.
Since almost all finite primes have local height 1, it makes sense to define a
global height function on primitive vectors as the product of local height
functions: $h(x)=\prod_{v}h_{v}(x_{v})$. As recalled in [Gar18]*Claim 3.3.2,
for example, this global height function $h(x)$ on primitive vectors
$x\in\operatorname{\mathbb{A}}^{r}$ has several convenient properties which
facilitate the reduction theory of $G_{k}\backslash
G_{\operatorname{\mathbb{A}}}$:

  1. (1)

For fixed $g\in
G_{\operatorname{\mathbb{A}}}=GL_{r}(\operatorname{\mathbb{A}})$ and fixed
$c>0$,

$\text{card}\left(k^{\times}\backslash\\{x\in k^{r}-\\{0\\}:h(x\cdot
g)<c\\}\right)<\infty,$

  2. (2)

For compact $C\subset G_{\operatorname{\mathbb{A}}}$, there are positive
implied constants such that for all $g\in C$ and primitive
$x\in\operatorname{\mathbb{A}}^{r}$,

$h(x)\ll_{C}h(x\cdot g)\ll_{C}h(x).$

  3. (3)

For $t\in k^{\times}$, via the product formula $\prod_{v\leq\infty}|t|_{v}=1$
for $t\in k^{\times}$,

$h(t\cdot x)=\prod_{v\leq\infty}h_{v}(t\cdot
x)=\prod_{v\leq\infty}|t|_{v}\cdot
h_{v}(x)=\prod_{v\leq\infty}|t|_{v}\cdot\prod_{v\leq\infty}h_{v}(x)=1\cdot
h(x)$

Finally, let

$\eta_{v}(g_{v})=|\det g_{v}|_{v}\cdot h_{v}(e_{r}\cdot g_{v})^{-r}\hskip
28.45274pt(\text{for }g_{v}\in G_{v},\\{e_{i}\\}\text{ the standard basis for
}k^{r})$

and $\eta(g)=\prod_{v}\eta_{v}(g_{v})$ for $g=(g_{v})_{v}\in
G_{\operatorname{\mathbb{A}}}$. In what follows, this is a suitable notion of
height on $G_{k}\backslash G_{\operatorname{\mathbb{A}}}$. We recall several
analytic properties:

######  Lemma 2.1.

The height function $\eta(g)=\prod_{v}|\det g_{v}|_{v}\cdot h_{v}(e_{r}\cdot
g_{v})^{-r}$ is right $K_{\operatorname{\mathbb{A}}}$-invariant, left
$P_{k}^{r-1,1}$-invariant, and $Z_{\operatorname{\mathbb{A}}}$-invariant.

###### Proof.

The isometry groups of the height functions $h_{v}$ are the compact subgroups
$K_{v}$, so $\eta(g)$ is right $K_{\operatorname{\mathbb{A}}}$-invariant. By
the product formula, for $p=\begin{pmatrix}a&b\\\ 0&d\end{pmatrix}\in
P^{r-1,1}_{k}\subset G^{1}$,

$\displaystyle\eta(p\cdot g)$ $\displaystyle=|\det pg|\cdot h(e_{r}\cdot
pg)^{-r}=|\det pg|\cdot h(de_{r}\cdot g)^{-r}$ $\displaystyle=|\det
p|\cdot|\det g|\cdot|d|^{-r}\cdot h(e_{r}\cdot g)^{-r}=1\cdot|\det g|\cdot
h(e_{r}\cdot g)^{-r}=\eta(g).$

For $z=\begin{pmatrix}t&&&\\\ &&\ddots&\\\ &&&t\end{pmatrix}\in
Z_{\operatorname{\mathbb{A}}}$

$\displaystyle\eta(z\cdot g)$ $\displaystyle=|\det zg|\cdot h\left(e_{r}\cdot
zg\right)^{-r}=|\det zg|\cdot h(te_{r}\cdot g)^{-r}$ $\displaystyle=|\det
z|\cdot|\det g|\cdot|t|^{-r}\cdot h(e_{r}\cdot
g)^{-r}=|t|^{r}\cdot|t|^{-r}\cdot\eta(g)=\eta(g)$

∎

######  Lemma 2.2 (For example, [Gar18]*Corollary 3.3.3).

For any $g\in G_{\operatorname{\mathbb{A}}}$, there are finitely-many
$\gamma\in P_{k}^{r-1,1}\backslash G_{k}$ such that

$\eta(\gamma\cdot g)>\eta(g).$

Thus, the supremum $\sup_{\gamma}\eta(\gamma\cdot g)$ is attained and is
finite.

###### Proof.

This lemma hinges on the natural bijection $P^{r-1,1}_{k}\backslash
G_{k}\longleftrightarrow k^{\times}\backslash(k^{r}-\\{0\\})$ given by

$P^{r-1,1}_{k}\cdot\begin{pmatrix}*&...&*\\\ \vdots&\ddots&\vdots\\\
c_{1}&...&c_{r}\end{pmatrix}\mapsto k^{\times}\cdot(c_{1},...,c_{r}).$

for any invertible matrix with bottom row $(c_{1},...,c_{r})$. Indeed, $G_{k}$
is transitive on non-zero vectors, and $P^{r-1,1}_{k}$ is the stabilizer,
acting on the right, of the line (minus a point) $k^{\times}\cdot e_{r}$. By
property 1 of the global height function $h$, there are finitely-many $x\in
k^{\times}\backslash(k^{r}-\\{0\\})$ such that $h(x\cdot g)<c$ (equivalently,
such that $h(x\cdot g)^{-1}>c^{-1})$. Then, the bijection above and
$G_{k}$-invariance of $|\det g|$ yields our assertion. ∎

Finally, we exhibit a single (adelic) Siegel set covering $G_{k}\backslash
G_{\operatorname{\mathbb{A}}}$. A standard Siegel set aligned to a (standard)
parabolic $P$ is of the form

$\mathfrak{S}^{P}=\mathfrak{S}^{P}_{t,C}=\\{g=nmk:n\in C,m\in
M_{\operatorname{\mathbb{A}}},k\in K_{\operatorname{\mathbb{A}}},\text{ and
}|\alpha_{i}(m)|\geq t,\text{ for }1\leq i<r\\}$

for $t\in(0,\infty)$, $C\subset N^{P}_{\operatorname{\mathbb{A}}}$ compact,
and $|\cdot|$ the idele norm. For $P=P^{min}$ the minimal parabolic, we have
the following:

######  Theorem 2.3 (For example, [Gar18]*Theorem 3.3.1).

Given a number field $k$, there is some $t>0$ and compact $C\subset
N^{P}_{\operatorname{\mathbb{A}}}$ such that
$G_{k}\cdot\mathfrak{S}^{P}_{t,C}=G_{\operatorname{\mathbb{A}}}$.

###### Proof.

We include a proof to illustrate the role of the $P^{r-1,1}$ parabolic (see
Section 2.4) in the reduction theory of $G_{k}\backslash G_{\mathbb{A}}$. The
proof proceeds by induction on $r$, reducing to the case $r=2$ treated in many
sources, including [Gar18]*Section 2.2.

Given $g\in G_{\operatorname{\mathbb{A}}}$ and by Lemma 2.2,
$h(x^{\prime}\cdot g)$ is minimized at some $x\in k^{r}-\\{0\\}$ with
$h(x\cdot g)>0$. Let $\gamma_{o}\in G_{k}$ such that $x=e_{r}\gamma_{o}$.
Since $h(x\cdot g)=h(e_{r}\gamma_{o}\cdot g)$ is minimal, $\eta(\gamma_{o}g)$
is maximal among all values $\eta(\gamma\cdot\gamma_{o}g)$ for $\gamma\in
G_{k}$. By Iwasawa, there is $\theta\in K$ such that $q=\gamma_{o}g\theta\in
P_{\operatorname{\mathbb{A}}}^{r-1,1}$. Then $h(\gamma_{o}g\theta)=|q_{r,r}|$
where $q_{i,j}$ is the $ij^{th}$ entry of $q$, and $\eta(q)$ is maximal among
all values $\eta(\gamma\cdot g)$ for $\gamma\in G_{k}$.

Let $H\subset M_{\operatorname{\mathbb{A}}}^{r-1,1}$ be the subgroup of
$G_{\operatorname{\mathbb{A}}}$ fixing $e_{r}$ and stabilizing the subspace
spanned by $e_{1},...,e_{r-1}$, so $H\approx
GL_{r-1}(\operatorname{\mathbb{A}})$. By induction on $r$, beginning at $r=2$
as treated in many sources including [Gar18]*Section 2.2, by acting on
$q=\lambda_{o}g\theta$ on the left by $H_{k}$ and on the right by
$H_{\operatorname{\mathbb{A}}}\cap K_{\operatorname{\mathbb{A}}}$, we can
suppose that $q\in P_{\operatorname{\mathbb{A}}}^{min}$ and
$|q_{i,i}/q_{i+1,i+1}|\geq t$ for $i<r-1$, without altering $\eta(q)$.
Certainly, since

$h(e_{r}\cdot q)\leq h(x^{\prime}\cdot q)\hskip 14.22636pt\text{for all
}x^{\prime}\in k^{r}-\\{0\\},$

we have $h(e_{r}\cdot q)\leq h(x^{\prime}\cdot q)$ for the subset of vectors
of the form $x^{\prime}=(0,...,0,x_{r-1},x_{r})$. Thus, the lower right 2-by-2
block $q^{\prime}$ of $q$ is reduced as an element of $GL_{2}(\mathbb{A})$.
This reduces to the $r=2$ case treated in [Gar18]*Theorem 2.2.7, yielding

$|q_{r-1,r-1}|/|q_{rr}|\geq t,\hskip 14.22636pt\text{for sufficiently small
}t$

and proving the theorem. ∎

###  2.2. The invariant Laplacian

We characterize the invariant Laplacian descended from Casimir, prove its
invariance on automorphic quotients $\Gamma\backslash G/K$, and recall facts
about its eigenfunctions. To maintain some generality in our discussion and,
incidentally, certify invariance, it is useful to characterize the invariant
Laplacian on certain automorphic quotients without choosing coordinates.

Let $G$ be a semisimple Lie group with maximal compact subgroup $K$ and
discrete subgroup $\Gamma$. The universal enveloping algebra $U\mathfrak{g}$
is a canonical quotient of the universal associative algebra $A\mathfrak{g}$,
also known as the tensor algebra $\large\otimes^{\bullet}\mathfrak{g}$. The
associative algebra $U\mathfrak{g}$ is universal in the sense that any linear
map $\varphi:\mathfrak{g}\to B$ to an associative algebra $B$ which respects
the Lie bracket

$\varphi([x,y])=\varphi(x)\varphi(y)-\varphi(y)\varphi(x)\hskip
14.22636pt\text{(for }x,y\in\mathfrak{g})$

induces a unique associative algebra homomorphism $\Phi:U\mathfrak{g}\to B$.
That is, the following diagram commutes:

${U\mathfrak{g}}$${\mathfrak{g}}$${B.}$$\scriptstyle{\exists!\hskip
2.84526pt\Phi}$$\scriptstyle{i}$$\scriptstyle{\forall\hskip 2.84526pt\varphi}$

The Casimir element $\Omega$ is the simplest non-trivial $G$-invariant element
in $U\mathfrak{g}$ and gives a two-sided $G$-invariant differential operator
on $G$ which descends to the invariant Laplacian on quotients $G/K$ and
$\Gamma\backslash G/K$. Casimir can be characterized as the image of the
$G$-invariant element
$\text{id}_{\mathfrak{g}}\in\operatorname{End}_{\operatorname{\mathbb{C}}}(\mathfrak{g})$
under the map $\zeta$ given by

Coordinate-free characterization of Casimir

${\operatorname{End}_{\operatorname{\mathbb{C}}}(\mathfrak{g})}$${\mathfrak{g}\otimes\mathfrak{g}^{*}}$${\mathfrak{g}\otimes\mathfrak{g}}$${A\mathfrak{g}}$${U\mathfrak{g}}$${\text{id}_{\mathfrak{g}}}$${\zeta(\text{id}_{\mathfrak{g}})=\Omega.}$$\scriptstyle{\approx}$$\scriptstyle{\zeta}$$\scriptstyle{\approx\hskip
2.84526ptvia\hskip
2.84526pt\langle\rangle}$$\scriptstyle{incl}$$\scriptstyle{quot}$

The first map in the composition is the inverse of the canonical isomorphism
$V\otimes V^{*}\to\operatorname{End}V$ for finite-dimensional vector spaces.
The second map is induced from $\langle-,v\rangle\mapsto v:V^{*}\to V$ for
$\langle-,-\rangle$ a $G$-equivariant, non-degenerate bilinear form on
$\mathfrak{g}$, such as the trace form $\langle x,y\rangle=\text{tr}(xy)$ on
$\mathfrak{so}_{r},\mathfrak{sl}_{r}$, and $\mathfrak{gl}_{r}$. The third map
is an inclusion, and the fourth is a quotient map.

The above prescription tells us how to express the Casimir element
$\Omega=\zeta(\text{id}_{\mathfrak{g}})$ in terms of any basis
$x_{1},...,x_{r}$ of $\mathfrak{g}$: For $\lambda_{1},...,\lambda_{r}$ the
corresponding dual basis of the dual $\mathfrak{g}^{*}$, characterized by
$\lambda_{i}(x_{j})=\delta_{ij}$, and $x_{1}^{*},...,x_{r}^{*}$ the dual basis
for $\mathfrak{g}$ in terms of $\langle,\rangle$, characterized by $\langle
x_{i},x_{j}^{*}\rangle=\delta_{ij}$,

${\operatorname{End}_{\operatorname{\mathbb{C}}}(\mathfrak{g})}$${\mathfrak{g}\otimes\mathfrak{g}^{*}}$${\mathfrak{g}\otimes\mathfrak{g}}$${A\mathfrak{g}}$${U\mathfrak{g}}$${\text{id}_{\mathfrak{g}}}$${\sum_{i}x_{i}\otimes\lambda_{i}}$${\sum_{i}x_{i}\otimes
x_{i}^{*}}$${\sum_{i}x_{i}\otimes
x_{i}^{*}}$${\sum_{i}x_{i}x_{i}^{*}=\Omega.}$$\scriptstyle{\approx}$$\scriptstyle{\zeta}$$\scriptstyle{\approx\hskip
2.84526ptvia\hskip
2.84526pt\langle\rangle}$$\scriptstyle{incl}$$\scriptstyle{quot}$

That is, we have an intrinsic description of the Casimir element which does
not depend on a choice of basis $x_{1},...,x_{r}$. Furthermore, we can show
that $\Omega=\zeta(\text{id}_{\mathfrak{g}})$ is $G$-invariant in
$U\mathfrak{g}$ without showing by change-of-basis that the defined object is
independent of choice of basis:

######  Theorem 2.4.

The Casimir operator $\Omega=\zeta(\text{id}_{\mathfrak{g}})$ is a
$G$-invariant element of $U\mathfrak{g}$.

###### Proof.

Since the action of $G$ respects all maps in the composition, $\zeta$ is
$G$-equivariant. Furthermore, $\text{id}_{\mathfrak{g}}$ commutes with any
endomorphism of $\mathfrak{g}$, so $\Omega=\zeta(\text{id}_{\mathfrak{g}})$ is
certainly $G$-invariant element of $U\mathfrak{g}$:

$g\hskip 2.84526pt\zeta(\text{id}_{\mathfrak{g}})\hskip
2.84526ptg^{-1}=\zeta(g\hskip 2.84526pt\text{id}_{\mathfrak{g}}\hskip
2.84526ptg^{-1})=\zeta(g\hskip 2.84526ptg^{-1}\hskip
2.84526pt\text{id}_{\mathfrak{g}})=\zeta(\text{id}_{\mathfrak{g}}).$

∎

Incidentally, this intrinsic description renders trivial the proof of
invariance of Casimir on automorphic quotients:

######  Theorem 2.5.

For $G$ a semisimple Lie group with maximal compact $K$ and discrete subgroup
$\Gamma$, the Casimir operator $\Omega=\zeta(\text{id}_{\mathfrak{g}})$
descends to an invariant differential operator on $\Gamma\backslash G/K$.

###### Proof.

On one hand, the (right differentiation) action of $x\in\mathfrak{g}$ on
functions $f$ on $G$ by

$(x\cdot f)(g)=\frac{d}{dt}\bigg{|}_{t=0}f(g\cdot e^{tx})$

is left $G$-invariant (and therefore left $\Gamma$-invariant) by
associativity. Certainly, this extends to the left $G$-invariant action of
$U\mathfrak{g}$. On the other hand, if the differential operator comes from
the $\text{Ad }G$-invariant elements of $U\mathfrak{g}$ given by

$(U\mathfrak{g})^{G}=\\{\alpha\in U\mathfrak{g}:g\alpha g^{-1}=\alpha,\text{
for all }g\in G\\},$

then the action is right $G$-invariant (and therefore right $K$-invariant) by
the commutation property. By Theorem 2.4, $\Omega\in(U\mathfrak{g})^{G}$ and,
thus, descends to the quotient $\Gamma\backslash G/K$. ∎

###  2.3. Eigenvalues of integral operators on eigenfunctions of Casimir

As in the proof of a pre-trace formula in, for example, [Gar18]*Section 12.1,
we recall that the eigenvalues of integral operators on Eisenstein series or
strong-sense cuspforms are essentially determined by their eigenvalues for the
Laplacian. This follows from the non-trivial fact that a suitable
representation $V_{f}$ generated by right translates of a
$\Delta$-eigenfunction $f$ is isomorphic to a subquotient of the principal
series representation $I_{s}$. That subtler point is an instance of a
subquotient theorem [HC54], which was eventually strengthened to the
subrepresentation theorem [Cas78/80],[CM82]. Eigenvalues depend only on the
isomorphism class of $V_{f}$, so they can be computed on the (much simpler)
subquotient of the principal series representation.

We follow the discussion in [Gar18]*Chapter 12. Consider an integral operator
attached to a compactly supported measure $\eta$ on the group $G$ acting on a
quasi-complete, locally convex topological vector space $V$ of automorphic
forms. For a continuous action $G\times V\to V$, the action of $\eta$ is

$\eta\cdot v=\int_{G}g\cdot v\hskip 2.84526ptd\eta(g),\hskip
14.22636pt\text{for }v\in V,$

as a Gelfand-Pettis integral.

Quasi-complete, locally convex topological vector spaces $V,W$ with continuous
actions $G\times V\to V$ and $G\times W\to W$ are representations of $G$, and
continuous $\operatorname{\mathbb{C}}$-linear maps $T:V\to W$ respecting the
action of $G$ are $G$-homomorphisms. By properties of Gelfand-Pettis
integrals, $G$-homomorphisms commute with the action of integral operators
attached to compactly supported measures:

$T(\eta\cdot v)=T\left(\int_{G}g\cdot v\hskip
2.84526ptd\eta(g)\right)=\int_{G}T(g\cdot v)\hskip
2.84526ptd\eta(g)=\int_{G}g\cdot T(v)\hskip 2.84526ptd\eta(g)=\eta\cdot T(v).$

In particular, $G$-homomorphisms preserve the eigenvalues and eigenvectors of
integral operators.

For $v$ in a (quasi-complete, locally-convex) $G$-representation space $V$,
the subrepresentation generated by $v$ is the topological closure of the span
of finite linear combinations of images $g\cdot v$ of $v$ by $g\in G$. By
properties of Gelfand-Pettis integrals (as shown in [Gar18]*Claim 12.1.3, for
example), a strong-sense cuspform or Eisenstein series $f$ is the unique
$K$-invariant vector in the subrepresentation $V_{f}$ it generates under right
translation, up to a constant. For any left and right $K$-invariant compactly
supported measure $\eta$, the integral operator action

$(\eta\cdot f)(x)=\int_{G}g\cdot f(x)\hskip
2.84526ptd\eta(g)=\int_{G}f(xg)\hskip 2.84526ptd\eta(g)$

produces another right $K$-invariant vector in the subrepresentation space
$V_{f}$; thus $\eta\cdot f$ is a scalar multiple of $f$. Let
$\chi_{f}(\eta)\in\operatorname{\mathbb{C}}$ denote the eigenvalue:

$\eta\cdot f=\chi_{f}(\eta)\cdot f.$

Since $G$-homomorphisms preserve the eigenvalues of integral operators, the
scalar $\chi_{f}(\eta)$ can be computed in any image of $V_{f}$. For
$\Omega\cdot f=r\ell(r-\ell)s(s-1)\cdot f$, the representation generated by
$f$ is isomorphic to an unramified principal series

$I_{s}=\left\\{\varphi\in C^{\infty}(G):\varphi\left(\begin{pmatrix}a&*\\\
0&d\end{pmatrix}\cdot g\right)=\frac{|\det a|^{(r-\ell)s}}{|\det d|^{\ell
s}}\cdot\varphi(g)\right\\},\hskip 14.22636pt(\text{with
}s\in\operatorname{\mathbb{C}})$

under right translation by $G$. The Iwasawa decomposition
$G=P^{\ell,r-\ell}\cdot K$ shows that the space of $K$-fixed vectors is one-
dimensional. By an explicit form of the proof of the Harish-Chandra
isomorphism (see [HC51] and [Gar17b], for example), we can compute eigenvalues
of $\Omega$ on Eisenstein series or strong-sense level-one/spherical cuspforms
by computing eigenvalues on $I_{s}$.

###  2.4. Degenerate Eisenstein series attached to maximal proper parabolics

In this section, we define and recall analytical properties of degenerate
Eisenstein series $E_{s,\varphi}^{P}$ attached to maximal proper parabolic
subgroups $P=P^{\ell,r-\ell}\subset SL_{r}(\operatorname{\mathbb{R}})$. For
this classical geometric algebra, we primarily follow the discussion in
[Gar18]*Sections 3.1, 3.11.

#### Parabolic subgroups of $SL_{r}(\operatorname{\mathbb{R}})$

Let $k$ be a number field. A flag $F$ in $k^{r}$ is a nested sequence of
$k$-subspaces

$V_{1}\subset...\subset V_{m}\subset k^{r}$

The corresponding parabolic subgroup $P=P^{F}$ is the stabilizer of the flag
$F$. The maximal proper parabolic subgroups are stabilizers $P^{V}$ of flags
consisting of single proper subspaces $V\subset k^{r}$.

With $e_{1},e_{2},...,e_{r}$ the standard basis for $k^{r}$, identify
$k^{d}=ke_{1}+...+ke_{d}$. By transitivity of $G$ on ordered bases of $k^{r}$,
every orbit in the action of $G$ on flags has a unique representative among
the standard flags, namely for some ordered partition
$d_{1}+d_{2}+...+d_{m}=r$ with $0<d_{j}\in\operatorname{\mathbb{Z}}$, the
corresponding standard flag is

$F^{d_{1},...,d_{m}}=\left(k^{d_{1}}\subset k^{d_{1}+d_{2}}\subset....\subset
k^{d_{1}+d_{2}+...+d_{m}}\right).$

The standard proper parabolic subgroup $P^{d_{1},...,d_{m}}$ of $G$ is the
stabilizer of the flag $F^{d_{1},...,d_{m}}$. The standard maximal proper
parabolics are the block-upper triangular matrices

$P^{\ell,r-\ell}=\left\\{\begin{pmatrix}a&b\\\ 0&d\end{pmatrix}:a\in
SL_{\ell}(k),b=\ell\times(r-\ell),d\in SL_{r-\ell}(k)\right\\},$

where the off-diagonal blocks are sized to fit. The unique associate parabolic
of $P=P^{\ell,r-\ell}$ is the subgroup $P^{r-\ell,\ell}$, defined as expected.
We observe that $P^{\ell,r-\ell}$ is self-associate when $r=2\ell$.

The unipotent radical $N^{P}$ of a parabolic $P$ stabilizing a flag
$F=(V_{1}\subset...\subset V_{m})$ is the subgroup that fixes the quotients
$V_{m}/V_{m-1}$ pointwise. For the standard maximal proper parabolic
$P^{\ell,r-\ell}$, the unipotent radical is

$N^{\ell,r-\ell}=\left\\{\begin{pmatrix}1_{\ell}&*\\\
0&1_{r-\ell}\end{pmatrix}\right\\}.$

The standard Levi-Malcev component $M^{P}$ is the subgroup of $P$ with all the
blocks above the diagonal 0. For the standard maximal proper parabolic
$P^{\ell,r-\ell}$, the Levi component is

$M^{\ell,r-\ell}=\left\\{\begin{pmatrix}a&0\\\ 0&d\end{pmatrix}\right\\}.$

The proof of the Iwasawa decomposition $G=P^{\ell,r-\ell}\cdot K$ is an
iteration of the $r=2$ case, and the Levi-Malcev decomposition
$P^{\ell,r-\ell}=N^{\ell,r-\ell}\cdot M^{\ell,r-\ell}$ is simply an expression
of the behavior of matrix multiplication in block decompositions.

#### Degenerate Eisenstein series

Degenerate Eisenstein series play a subtler role in Plancherel or the spectral
decomposition of $L^{2}(\Gamma\backslash G/K)$ than do cuspidal-data
Eisenstein series; however, as was known to Selberg and appeared in [Lan76],
they do appear as $(r-2)$-fold residues of a minimal-parabolic Eisenstein
series. However, as multi-residues of Eisenstein series are not necessary for
the current discussion, we adapt [Gar18]*Section 3.11 and [Gar09].

Let $G=SL_{r}(\operatorname{\mathbb{R}})$, $P=P^{\ell,r-\ell}$ a maximal
proper parabolic subgroup, $K=SO(r,\operatorname{\mathbb{R}})$ the maximal
compact, $\Gamma=SL_{r}(\operatorname{\mathbb{Z}})$, and

$\varphi^{P}_{s}(nmk)=\frac{|\det a|^{(r-\ell)s}}{|\det d|^{\ell s}},\hskip
14.22636ptm=\begin{pmatrix}a&0\\\ 0&d\end{pmatrix}.$

Note that $\varphi_{s}^{P}$ is the unique spherical vector in a (degenerate)
unramified principal series representation of $G$, and is, therefore, left
$N_{\operatorname{\mathbb{A}}}^{P}$ invariant. Choosing a convenient basis for
$\mathfrak{g}$, we can compute the $\Omega$-eigenvalue explicitly:

######  Lemma 2.6.

For $\Omega$ the Casimir operator on $G$,

$\Omega\cdot\varphi^{P}_{s}=r\ell(r-\ell)(s^{2}-s)\cdot\varphi^{P}_{s}.$

###### Proof.

Let $h_{i}$ be the diagonal matrix with 1 at the $i^{th}$ position and 0’s
elsewhere; furthermore, for $i<j$, let $x_{ij}$ be the matrix with a 1 at the
$ij^{th}$ location and 0’s elsewhere, and for $i>j$, let $y_{ij}$ be the
matrix with a 1 at the $ij^{th}$ location and 0’s elsewhere. The
$h_{i},x_{ij}$, and $y_{ij}$, form an orthogonal basis $\\{x_{k}\\}$ for
$\mathfrak{g}$ with $h_{i}^{*}=h_{i}$, $x_{ij}^{*}=y_{ji}$,
$y_{ij}^{*}=x_{ji}$, and $[x_{ij},y_{ji}]=-h_{i}+h_{j}$. Thus, we have

$\displaystyle\Omega$ $\displaystyle=\sum_{k}x_{k}x^{*}_{k}=\sum_{1\leq i\leq
r}h_{i}^{2}+\sum_{i<j}x_{ij}y_{ji}+\sum_{i>j}y_{ij}x_{ji}=\sum_{1\leq i\leq
r}h_{i}^{2}+\sum_{i<j}x_{ij}y_{ji}+y_{ji}x_{ij}$ $\displaystyle=\sum_{1\leq
i\leq r}h_{i}^{2}+\sum_{i<j}2x_{ij}y_{ji}+[x_{ij},y_{ji}]=\sum_{1\leq i\leq
r}h_{i}^{2}+\sum_{i<j}2x_{ij}y_{ji}-h_{i}+h_{j}$

Since $x_{ij}\in\mathfrak{n}$ and acts first (after conjugating) in the middle
term $\sum_{i<j}2x_{ij}y_{ij}$, this term acts by 0 on left $N^{P}$ invariant
functions, such as $\varphi^{P}_{s}$. Thus,

$\Omega\cdot\varphi^{P}_{s}=\sum_{i}h_{i}^{2}\cdot\varphi^{P}_{s}+\sum_{i<j}(-h_{i}+h_{j})\cdot\varphi^{P}_{s}.$

Although, as usual, $h\in\mathfrak{g}$ acts on the right, Casimir commutes
with conjugation by $G$, so we can let the associated differential operators
$h_{i}$ act on the left. Thus, by direct calculation,

$\displaystyle h_{i}\cdot\varphi^{P}_{s}(g)$
$\displaystyle=\begin{cases}s(r-\ell)\cdot\varphi_{s}^{P}(g)&1\leq
i\leq\ell\\\ -s\ell\cdot\varphi^{P}_{s}(g)&\ell<i\leq r,\end{cases}$

such that,

$\displaystyle\sum_{i}h_{i}^{2}\cdot\varphi^{P}_{s}$
$\displaystyle=\sum_{1\leq
i\leq\ell}h_{i}^{2}\cdot\varphi^{P}_{s}+\sum_{\ell<i\leq
r}h_{i}^{2}\cdot\varphi^{P}_{s}$
$\displaystyle=[\ell\left(s(r-\ell)\right)^{2}+(r-\ell)(-s\ell)^{2}]\cdot\varphi_{s}^{P}=s^{2}\ell(r-\ell)r\cdot\varphi_{s}^{P}$
$\displaystyle\sum_{i<j}(-h_{i}+h_{j})\cdot\varphi^{P}_{s}$
$\displaystyle=\sum_{1\leq
i<j\leq\ell}0\cdot\varphi^{P}_{s}+\sum_{\ell<i<j\leq
r}0\cdot\varphi^{P}_{s}+\sum_{1\leq i\leq\ell<j\leq
r}(-h_{i}+h_{j})\cdot\varphi^{P}_{s}$ $\displaystyle=-(r-\ell)\sum_{1\leq
i\leq\ell}h_{i}\cdot\varphi^{P}_{s}+\ell\sum_{\ell<j\leq
r}h_{j}\cdot\varphi^{P}_{s}$ $\displaystyle=[-(r-\ell)\ell
s(r-\ell)+\ell(r-\ell)(-s\ell)]\cdot\varphi_{s}^{P}=-s\ell(r-\ell)r\cdot\varphi_{s}^{P}.$

Summing these identities yields our claim. ∎

The degenerate Eisenstein series associated to $P$ and corresponding to
$\varphi^{P}_{s}$ is

$E^{P}_{s,\varphi}(g)=\sum_{{\gamma\in
P\cap\Gamma\backslash\Gamma}}\varphi^{P}_{s}(\gamma\cdot g).$

We recall basic analytic properties which will be useful in the sequel:

######  Theorem 2.7.

The degenerate Eisenstein series $E^{P}_{s,\varphi}(g)$ converges absolutely
and uniformly on compacta for $\text{Re}(s)>{1}$.

###### Proof.

For example, this follows from Godement’s Criterion (see [Bor66a]), as in the
proof of [Gar18]*Claim 3.11.1. ∎

Of course, since $\varphi^{P}_{s}$ is right $K$-invariant and winding up to
form the corresponding Eisensteins series is a $G$-homomorphism, the
degenerate Eisenstein series is also right $K$-invariant in the region of
convergence. Furthermore, the meromorphic continuation of the degenerate
Eisenstein series is right $K$-invariant by the identity principle.

######  Theorem 2.8.

The degenerate Eisenstein series $\displaystyle E^{P}_{s,\varphi}$ is an
eigenvector of Casimir $\Omega$ with eigenvalue
$\lambda_{s}=r\ell(r-\ell)s(s-1)$. In particular, the eigenvalue is invariant
under $s\to 1-s$.

###### Proof.

Granting convergence in the $C^{\infty}$ topology, in $\text{Re}(s)>1$, and
using the fact that $\Omega$ commutes with translations by $\Gamma$, letting
$\Omega\cdot\varphi^{P}_{s}=\lambda_{s}\cdot\varphi^{P}_{s}$ (as in Lemma
2.6),

$\displaystyle\Omega\cdot E^{P}_{s,\varphi}$
$\displaystyle=\Omega\sum_{{\gamma\in
P\cap\Gamma\backslash\Gamma}}\varphi^{P}_{s}(\gamma\cdot g)=\sum_{{\gamma\in
P\cap\Gamma\backslash\Gamma}}(\Omega\cdot\varphi^{P}_{s})(\gamma\cdot g)$
$\displaystyle=\lambda_{s}\sum_{{\gamma\in
P\cap\Gamma\backslash\Gamma}}\varphi^{P}_{s}(\gamma\cdot g)=\lambda_{s}\cdot
E^{P}_{s,\varphi}.$

Finally, the eigenvalue property holds for the meromorphic continuation of
$E^{P}_{s,\varphi}$ by the identity principle. ∎

###  2.5. Epstein zeta functions

For $Q$ a real, positive-definite $r\times r$ matrix, the corresponding
Epstein zeta function is

$Z_{r}(Q,s)=\sum_{0\neq v\in\operatorname{\mathbb{Z}}^{r}}(v\hskip
2.84526ptQ\hskip 2.84526ptv^{\top})^{-s},\hskip
14.22636pt\text{Re}(s)>\frac{r}{2}.$

[Eps03] introduced this function as a more general function satisfying a
functional equation similar to the one satisfied by the Riemann zeta function.
With $Q=gg^{\top}$ for $g\in SL_{r}(\operatorname{\mathbb{R}})$, we obtain a
family of functions on $SL_{r}(\operatorname{\mathbb{R}})$, parametrized by
$s\in\operatorname{\mathbb{C}}$. To clarify the normalization (such that the
first pole is at $s=1$) and make certain analytical assertions precise,
recall:

######  Lemma 2.9.

For $g\in SL_{r}(\operatorname{\mathbb{R}})$, the Epstein zeta function
$Z_{r}(gg^{\top},s)$ is essentially a degenerate Eisenstein series. More
precisely,

$Z_{r}\left(gg^{\top},\frac{rs}{2}\right)=2\zeta\left({rs}\right)E^{P}_{s,\varphi}(g),$

for $\zeta(s)$ Riemann’s zeta function and $E^{P}_{s,\varphi}(g)$ a degenerate
Eisenstein series associated to the $(r-1,1)$-parabolic subgroup of
$SL_{r}(\operatorname{\mathbb{R}})$.

###### Proof.

We derive a re-normalization of [Gol15]*Equation 10.7.4. Since
$v(gg^{\top})v^{\top}=|v\cdot g|^{2}$ is homogeneous of degree 2, we can
rewrite this Epstein zeta function as

$Z_{r}\left(gg^{\top},\frac{rs}{2}\right)=\sum_{0\neq
v\in\operatorname{\mathbb{Z}}^{r}}{|v\cdot
g|^{-2(rs/2)}}=\zeta(rs)\sum_{\text{ prim.
}v\in\operatorname{\mathbb{Z}}^{r}}{|v\cdot g|^{-rs}},$

where $\zeta(rs)$ appears by pulling out the gcd’s of $v$.

Let $\Gamma=SL_{r}(\operatorname{\mathbb{Z}})$, $P=P^{r-1,1}$ the maximal
proper parabolic subgroup

$P=\left\\{\begin{pmatrix}(r-1)\times(r-1)&*\\\ 0&1\times
1\end{pmatrix}\right\\}$

of $SL_{r}(\operatorname{\mathbb{R}})$, and $\Lambda$ the set of primitive
vectors in $\operatorname{\mathbb{Z}}^{r}$. Notice $\Gamma$ acts transitively
on $\Lambda$. By the orbit-stabilizer theorem, $\\{\pm 1\\}\backslash\Lambda$
is in bijection with the coset space $\Gamma_{x}\backslash\Gamma$, for
$\Gamma_{x}$ the isotropy group of the $\operatorname{\mathbb{Z}}$-module
generated by $x\in\Lambda$. Since $P\cap\Gamma$ stabilizes the lattice
generated by the standard basis vector $e_{r}=(0,...,0,1)\in\Lambda$,
$P\cap\Gamma$ is the isotropy group $\Gamma_{e_{r}}$. Thus, $\\{\pm
1\\}\backslash\Lambda$ is in bijection with the coset space
$P\cap\Gamma\backslash\Gamma$ by

$\Psi:\\{\pm
1\\}\cdot(c_{1},...,c_{r})\mapsto(P\cap\Gamma)\cdot\begin{pmatrix}*&...&*\\\
\vdots&\ddots&\vdots\\\ c_{1}&...&c_{r}\end{pmatrix}.$

Let $\varphi_{s}^{P}$ be the data for the degenerate Eisenstein series as in
2.4. By the Iwasawa decomposition $G=N^{P}\cdot M^{P}\cdot K$, for any
$g=\begin{pmatrix}1_{r-1}&*\\\ 0&1\end{pmatrix}\begin{pmatrix}a&0\\\
0&d\end{pmatrix}\cdot k\in SL_{r}(\operatorname{\mathbb{R}})$,

$\displaystyle\varphi^{P}_{s}(g)=\varphi^{P}_{s}(nmk)=|d|^{-rs}=|e_{r}\cdot
g|^{-rs}.$

Similarly, for $v\in\\{\pm 1\\}\backslash\Lambda$,

$\varphi^{P}_{s}\left(\Psi(v)g\right)=|e_{r}\cdot\Psi(v)g|^{-rs}=|v\cdot
g|^{-rs}.$

Thus, we conclude that

$\displaystyle Z_{r}\left(gg^{\top},\frac{rs}{2}\right)$
$\displaystyle=\zeta(rs)\sum_{v\in\Lambda}{|v\cdot
g|^{-rs}}=2\zeta(rs)\sum_{v\in\\{\pm 1\\}\backslash\Lambda}{|v\cdot g|^{-rs}}$
$\displaystyle=2\zeta(rs)\sum_{\Psi(v)\in
P\cap\Gamma\backslash\Gamma}\varphi_{s}^{P}(\Psi(v)g)=2\zeta(rs)E^{P}_{s,\varphi}(g),$

yielding our claim. ∎

We recall the proof of meromorphic continuation and functional equation, as in
many sources such as [Gol15]*Section 10.7, of the degenerate Eisenstein series
$E^{P}_{s,\varphi}$ by Poisson summation à la Riemann’s proof for $\zeta(s)$
(see [Rie59] and [Gar15]). Certainly, this proof yields meromorphic
continuation and functional equation of the Epstein zeta function
$Z_{r}(gg^{\top},s)$, as in [Eps03] and [Ter73].

######  Theorem 2.10.

The degenerate Eisenstein series $E^{P}_{s,\varphi}$ associated to the
$(r-1,1)$-parabolic subgroup of $SL_{r}(\operatorname{\mathbb{R}})$
meromorphically continues to $\operatorname{\mathbb{C}}$ with a simple pole at
$\displaystyle s=1$ with residue =
$\displaystyle\frac{\pi^{r/2}}{{2}\Gamma\left(r/2\right)\zeta(r)}$.
Furthermore, $E^{P}_{s,\varphi}$ satisfies the functional equation

$\pi^{-\frac{rs}{2}}\Gamma\left(\frac{rs}{2}\right)\zeta(rs)E^{P}_{s,\varphi}(g)=\pi^{-\frac{r}{2}(1-s)}\Gamma\left(\frac{r(1-s)}{2}\right)\zeta\left(r-rs\right){E^{P}_{1-s,\varphi}\left((g^{\top})^{-1}\right)}.$

###### Proof.

As usual, let $v\in\mathbb{Z}^{r}$ act on $SL_{r}(\mathbb{R})$ by matrix
multiplication, and consider the Gaussian

$\varphi(v)=e^{-\pi|v|^{2}}.$

For $g\in SL_{r}(\mathbb{R})$, let

$\Theta(g)=\sum_{v\in\mathbb{Z}^{r}}\varphi(v\cdot
g)=\sum_{v\in\mathbb{Z}^{r}}e^{-\pi|v\cdot g|^{2}}$

and (for reasons which will become apparent) consider the integral
representation

$\mathcal{I}(g)=\int_{0}^{\infty}t^{2s}(\Theta(tg)-1)\frac{dt}{t}=\int_{0}^{\infty}t^{2s}\left(\sum_{0\neq
v\in\operatorname{\mathbb{Z}}^{r}}e^{-\pi|v\cdot tg|^{2}}\right)\frac{dt}{t}$

where $t$ acts on $SL_{r}(\mathbb{R})$ by scalar multiplication. Since
$\Theta$ decays rapidly, we can integrate $\mathcal{I}$ termwise. By the
change of variables $t\mapsto t/(\sqrt{\pi}|v\cdot g|)$ and Lemma 2.9,

$\displaystyle\mathcal{I}(g)$ $\displaystyle=\sum_{0\neq
v\in\mathbb{Z}^{r}}\int_{0}^{\infty}t^{2s}e^{-\pi|v\cdot
tg|^{2}}\frac{dt}{t}=\sum_{0\neq v\in\mathbb{Z}^{r}}(\sqrt{\pi}|v\cdot
g|)^{-2s}\int_{0}^{\infty}t^{2s}e^{-t^{2}}\frac{dt}{t}$
$\displaystyle=\frac{1}{2}\pi^{-s}\sum_{0\neq v\in\mathbb{Z}^{r}}|v\cdot
g|^{-2s}\int_{0}^{\infty}u^{s}e^{-u}\frac{du}{u}=\frac{1}{2}\pi^{-s}\Gamma(s)Z_{r}\left(gg^{\top},s\right)$
$\displaystyle=\pi^{-s}\Gamma(s)\zeta(2s)E^{P}_{2s/r,\varphi}(g).$

We will use this integral representation to prove the meromorphic continuation
of $E^{P}_{s,\varphi}$ as in Riemann’s corresponding argument for $\zeta(s)$.
First, break the integral into two parts at $t=1$. By elementary estimates,
the integral from 1 to $\infty$ is uniformly and absolutely convergent. Thus

$\int_{1}^{\infty}t^{2s}(\Theta(tg)-1)\frac{dt}{t}=\text{entire in }s.$

To prove meromorphy of the integral from 0 to 1, we use Poisson summation on
the kernel of $\mathcal{I}$. By direct calculation,

${\text{Fourier transform of }(v\to\varphi(v\cdot
tg))=v\to{t^{-r}}\text{det}(g)^{-1}\varphi\left(v\cdot\frac{(g^{\top})^{-1}}{t}\right),}$

such that Poisson summation asserts

${\Theta(tg)={t^{-r}}\text{det}(g)^{-1}\Theta\left(\frac{(g^{\top})^{-1}}{t}\right)}.$

For now, assume Re($s)>\frac{r}{2}$. By the Poisson summation assertion,
change of variables $t\mapsto 1/t$, and explicit evaluation of some elementary
integrals,

$\displaystyle\int_{0}^{1}t^{2s}(\Theta(tg)-1)\frac{dt}{t}$
$\displaystyle=\int_{0}^{1}t^{2s}\left({t^{-r}}\text{det}(g)^{-1}[\Theta\left(\frac{(g^{\top})^{-1}}{t}\right)-1]+{t^{-r}}\text{det}(g)^{-1}-1\right)\frac{dt}{t}$
$\displaystyle=\int_{1}^{\infty}t^{-2s}\left({t^{r}}\text{det}(g)^{-1}[\Theta\left(t(g^{\top})^{-1}\right)-1]+{t^{r}}\text{det}(g)^{-1}-1\right)\frac{dt}{t}$
$\displaystyle=\text{det}(g)^{-1}\int_{1}^{\infty}{t^{r-2s}}\left(\Theta\left(t(g^{\top})^{-1}\right)-1\right)\frac{dt}{t}+{\frac{\text{det}(g)^{-1}}{2s-r}}-\frac{1}{2s}$
$\displaystyle=\text{entire in
}s+{\frac{\text{det}(g)^{-1}}{2s-r}}-\frac{1}{2s},$

since the integral from 1 to $\infty$ converges nicely by elementary
estimates. Thus, for $g\in SL_{r}(\operatorname{\mathbb{R}})$,

$\displaystyle\pi^{-s}\Gamma(s)\zeta(2s)E^{P}_{2s/r,\varphi}(g)$
$\displaystyle=\int_{1}^{\infty}t^{2s}(\Theta(tg)-1)\frac{dt}{t}+\int_{0}^{1}t^{2s}(\Theta(tg)-1)\frac{dt}{t}$
$\displaystyle=\int_{1}^{\infty}t^{2s}(\Theta(tg)-1)\frac{dt}{t}+\int_{1}^{\infty}{t^{r-2s}}\left(\Theta\left(t(g^{\top})^{-1}\right)-1\right)\frac{dt}{t}$
$\displaystyle\hskip 28.45274pt+{\frac{1}{2s-r}}-\frac{1}{2s}$
$\displaystyle=\text{entire in }s+{\frac{1}{2s-r}}-\frac{1}{2s}.$

The rational expressions have meromorphic continuations, so the right-hand
side of the expression yields a meromorphic continuation of
$E^{P}_{s,\varphi}$. The right-hand side of the expression is also visibly
invariant under $s\to(\frac{r}{2}-s)$, so $E^{P}_{2s/r,\varphi}$ satisfies the
functional equation

$\pi^{-s}\Gamma(s)\zeta(2s)E^{P}_{2s/r,\varphi}(g)=\pi^{-(\frac{r}{2}-s)}\Gamma\left(\frac{r}{2}-s\right)\zeta(r-2s){E^{P}_{1-2s/r,\varphi}\left((g^{\top})^{-1}\right)}.$

Renormalizing the complex parameter in the degenerate Eisenstein series,

$\pi^{-\frac{rs}{2}}\Gamma\left(\frac{rs}{2}\right)\zeta(rs)E^{P}_{s,\varphi}(g)=\pi^{-\frac{r}{2}(1-s)}\Gamma\left(\frac{r(1-s)}{2}\right)\zeta\left(r-rs\right){E^{P}_{1-s,\varphi}\left((g^{\top})^{-1}\right)}.$

Finally, we see that the only poles of
$\pi^{-s}\Gamma(s)\zeta(2s)E^{P}_{2s/r,\varphi}$ are at $\displaystyle s=0$
and $r/2$ with constant residues $-\frac{1}{2},\frac{1}{2}$. Using standard
facts about $\Gamma(s)$ and $\zeta(s)$, we recover assertions for
$E^{P}_{s,\varphi}$ itself. At $s=0$, $\Gamma(s)$ has a simple pole of residue
$1$ and (the analytic continuation of) $\zeta(s)$ takes value $-\frac{1}{2}$,
so $E^{P}_{2s/r,\varphi}$ and $E^{P}_{s,\varphi}$ are holomorphic at $s=0$. At
$s=r/2$, $\pi^{-s}\Gamma(s)$ is holomorphic. Since the infinite product for
$\zeta(2s)$ converges for $\text{Re}(s)>1/2$, $\zeta(2s)$ is nonzero in this
region. Thus $E^{P}_{2s/r,\varphi}$ has a simple pole at $s=r/2$ with constant
residue
$\displaystyle\frac{\pi^{r/2}}{2\Gamma\left(\frac{r}{2}\right)\zeta(r)}$ and
poles at $\rho/2$ for all non-trivial zeros $\rho$ of $\zeta(s)$ in
$0<\text{Re}(s)<\frac{1}{2}$. Renormalizing the complex parameter in the
degenerate Eisenstein series, $E^{P}_{s,\varphi}$ has a simple pole at $s=1$
with constant residue
$\displaystyle\frac{\pi^{r/2}}{2\Gamma\left(\frac{r}{2}\right)\zeta(r)}$ and
poles at $\rho/r$ for all non-trivial zeros $\rho$ of $\zeta(s)$ in
$0<\text{Re}(s)<\frac{1}{2}$.

∎

By Lemma 2.9,

######  Corollary 2.10.1.

For $g\in SL_{r}(\operatorname{\mathbb{R}})$, the Epstein zeta function
$Z_{r}(gg^{\top},s)$ meromorphically continues to $\mathbb{C}$ with a simple
pole at $s=r/2$ with constant residue
$\displaystyle\frac{\pi^{r/2}}{\Gamma\left(r/2\right)}$. Furthermore,
$Z_{r}(gg^{\top},s)$ satisfies the functional equation

$\pi^{-s}\Gamma(s)Z_{r}(gg^{\top},s)=\pi^{-\left(\frac{r}{2}-s\right)}\Gamma\left(\frac{r}{2}-s\right){Z_{r}\left((gg^{\top})^{-1},\frac{r}{2}-s\right).}$

###  2.6. Global automorphic Sobolev spaces

Global automorphic Sobolev spaces provide a reasonable context to discuss
solutions to partial differential equations in automorphic forms. We briefly
recall results here and refer the interested reader to [DeC21] and
[Gar18]*Chapter 12.

Let $\Delta$ be the invariant Laplacian on $\Gamma\backslash G/K$ descended
from Casimir (see Section 2.2). For $\Xi$ a locally compact, Hausdorff,
$\sigma$-compact topological space parametrizing cuspforms, Eisenstein series,
and their residues, and $d\xi$ a positive regular Borel measure on $\Xi$, we
have $L^{2}$-expansions

$f=\int_{\Xi}\langle f,\Phi_{\xi}\rangle\cdot\Phi_{\xi}\hskip 2.84526ptd\xi$

and Plancherel

$\left|f\right|^{2}_{L^{2}}=\int_{\Xi}\left|\langle
f,\Phi_{\xi}\rangle\right|^{2}d\xi,$

at first for test functions $C_{c}^{\infty}(\Gamma\backslash G/K)$, then
extended to $L^{2}(\Gamma\backslash G/K)$ by continuity.

Many of the $\Delta$-eigenfunctions $\Phi_{\xi}$ appearing in the
$L^{2}$-decomposition are not in $L^{2}(\Gamma\backslash G/K)$, but they are
all smooth, and we can arrange that $\xi\to\Phi_{\xi}$ is a continuous
$C^{\infty}(\Gamma\backslash G/K)$-valued function on $\Xi$. Since integration
of elements of $C^{\infty}(\Gamma\backslash G/K)$ against a fixed test
function $f\in C_{c}^{\infty}(\Gamma\backslash G/K)$ is a continuous linear
functional on $C^{\infty}(\Gamma\backslash G/K)$,

$\xi\mapsto\langle f,\Phi_{\xi}\rangle\hskip 14.22636pt(\text{for fixed }f\in
C_{c}^{\infty}(\Gamma\backslash G/K))$

is a continuous $\mathbb{C}$-valued function on $\Xi$ and thus has pointwise
values. The implied integrals in the spectral expansion do not converge for
all $f$ in $L^{2}$ (and $L^{2}$-expansions do not reliably converge
pointwise), but Plancherel asserts that the literal integrals
$f\to(\xi\to\langle f,\Phi_{\xi}\rangle)$ on test functions extend to an
isometry $\mathcal{F}:L^{2}(\Gamma\backslash G/K)\to L^{2}(\Xi)$ and that the
spectral coefficients $\langle f,\Phi_{\xi}\rangle$ for $f$ in $L^{2}$ are
extensions by continuity of the literal integrals for test functions.

On the other hand, Plancherel does not assert anything about pointwise values
of cuspforms or Eisenstein series, or about residues of Eisenstein series.
Even in the simplest cases, as was known to Hecke and Maass in the early 20th
century, certain linear combinations of pointwise values of $E_{s}$ are
essentially $\zeta_{k}(s)/\zeta_{k}(2s)$ for complex quadratic extensions $k$
of $\mathbb{Q}$, and sharp pointwise bounds on the critical line would imply
Lindelöf [Gar09].

In a suitable global automorphic Sobolev space and as described precisely in
what follows, $\Delta$ differentiates spectral expansions of test functions
term-wise; then, integration by parts gives

$\Delta f=\int_{\Xi}\langle\Delta f,\Phi_{\xi}\rangle\Phi_{\xi}\hskip
2.84526ptd\xi=\int_{\Xi}\lambda_{\xi}\cdot\langle
f,\Phi_{\xi}\rangle\Phi_{\xi}\hskip 2.84526ptd\xi\hskip 28.45274pt(\text{for
}f\in C_{c}^{\infty}(\Gamma\backslash G/K)).$

For each non-negative integer $k$, we define an inner product $\langle
f,g\rangle_{\operatorname{\mathfrak{B}}^{k}}$ on
$C_{c}^{\infty}(\Gamma\backslash G/K)$ by

$\langle
f,g\rangle_{\operatorname{\mathfrak{B}}^{k}}=\langle(1-\Delta)^{k}f,g\rangle=\int_{\Gamma\backslash
G/K}(1-\Delta)^{k}f\cdot\bar{g},$

and the $k$-th global automorphic Sobolev space
$\operatorname{\mathfrak{B}}^{k}$ is the completion of
$C_{c}^{\infty}(\Gamma\backslash G/K)$ with respect to the norm
$|f|_{\operatorname{\mathfrak{B}}^{k}}=\langle
f,f\rangle^{1/2}_{\operatorname{\mathfrak{B}}^{k}}$.

Since $P(-\Delta)$ is non-negative on test functions for any polynomial $P$
with non-negative real coefficients,
$|f|_{\operatorname{\mathfrak{B}}^{k+1}}\geq|f|_{\operatorname{\mathfrak{B}}^{k}}$
for test functions, and there is a canonical continuous injection
$\operatorname{\mathfrak{B}}^{k+1}\to\operatorname{\mathfrak{B}}^{k}$ with
dense image [Gar18]*Claim 12.3.9. By design,
$\Delta:C_{c}^{\infty}(\Gamma\backslash G/K)\to
C_{c}^{\infty}(\Gamma\backslash G/K)$ is continuous when the source is given
the $\operatorname{\mathfrak{B}}^{k+2}$ topology and the target is given the
$\operatorname{\mathfrak{B}}^{k}$ topology, so $\Delta$ extends by continuity
to a continuous linear map

$\Delta:\operatorname{\mathfrak{B}}^{k+2}(\Gamma\backslash
G/K)\to\operatorname{\mathfrak{B}}^{k}(\Gamma\backslash G/K)\hskip
14.22636pt(\text{for }0\leq k\in\mathbb{Z}).$

We characterize negative-index Sobolev spaces as the Hilbert space duals
$\operatorname{\mathfrak{B}}^{-k}=(\operatorname{\mathfrak{B}}^{k})^{*}$ and
identify $\operatorname{\mathfrak{B}}^{0}=L^{2}(\Gamma\backslash G/K)$ with
its dual via the Riesz-Fréchet map $\Lambda:f\to\langle-,f\rangle$ composed
with pointwise conjugation $c:f\to\overline{f}$. Since the continuous
inclusions
$i:\operatorname{\mathfrak{B}}^{k+1}\hookrightarrow\operatorname{\mathfrak{B}}^{k}$
have dense image, their adjoints
$i^{*}:\operatorname{\mathfrak{B}}^{-k}\hookrightarrow\operatorname{\mathfrak{B}}^{-k-1}$
are also continuous injections with dense image. Thus, we have a chain of
continuous linear maps between Hilbert spaces:

${...}$${\operatorname{\mathfrak{B}}^{2}}$${\operatorname{\mathfrak{B}}^{1}}$${\operatorname{\mathfrak{B}}^{0}}$${(\operatorname{\mathfrak{B}}^{0})^{*}}$${\operatorname{\mathfrak{B}}^{-1}}$${\operatorname{\mathfrak{B}}^{-2}}$${...}$$\scriptstyle{i}$$\scriptstyle{i}$$\scriptstyle{i}$$\scriptstyle{\Lambda\circ
c}$$\scriptstyle{\approx}$$\scriptstyle{i^{*}}$$\scriptstyle{i^{*}}$$\scriptstyle{i^{*}}$

Continuous $L^{2}$-differentiation for positive-index Sobolev spaces gives an
extension by continuity from $\Delta:C_{c}^{\infty}(\Gamma\backslash G/K)\to
C_{c}^{\infty}(\Gamma\backslash G/K)$, equipped with the
$\operatorname{\mathfrak{B}}^{-k}$ and $\operatorname{\mathfrak{B}}^{-k-2}$
topologies, to a continuous map
$\Delta:\operatorname{\mathfrak{B}}^{-k}\to\operatorname{\mathfrak{B}}^{-k-2}$
on negative-index Sobolev spaces, with the extension to
$\Delta:\operatorname{\mathfrak{B}}^{1}\to\operatorname{\mathfrak{B}}^{-1}$
characterized by

$((1-\Delta)f)(F)=\langle
f,\bar{F}\rangle_{\operatorname{\mathfrak{B}}^{1}},\hskip 14.22636pt\text{for
}f,F\in\operatorname{\mathfrak{B}}^{1}.$

To bookend the Sobolev tower, consider

$\operatorname{\mathfrak{B}}^{\infty}=\lim_{0\leq
k\in\mathbb{Z}}\operatorname{\mathfrak{B}}^{2k},\hskip
14.22636pt\operatorname{\mathfrak{B}}^{-\infty}=\text{colim}_{\hskip
2.84526pt0\leq k\in\mathbb{Z}}\operatorname{\mathfrak{B}}^{-2k}.$

Since every continuous linear functional on a limit of Banach spaces factors
through some limitand when the image of the limit is dense in the limitands
(see [Gar17a]), $\operatorname{\mathfrak{B}}^{-\infty}$ is the dual of
$\operatorname{\mathfrak{B}}^{\infty}$. By characterization of projective
limits, there is a unique, continuous map
$\Delta:\operatorname{\mathfrak{B}}^{\infty}\to\operatorname{\mathfrak{B}}^{\infty}$
induced from the family of compatible maps
$\operatorname{\mathfrak{B}}^{\infty}\hookrightarrow\operatorname{\mathfrak{B}}^{k+2}\to\operatorname{\mathfrak{B}}^{k}$.
Similarly, there is a unique, continuous map
$\Delta:\operatorname{\mathfrak{B}}^{-\infty}\to\operatorname{\mathfrak{B}}^{-\infty}$
which is compatible with the family of inclusions
$\operatorname{\mathfrak{B}}^{-k}\to\operatorname{\mathfrak{B}}^{-k-2}\hookrightarrow\operatorname{\mathfrak{B}}^{-\infty}$,
by characterization of the colimit.

For $0\leq k\in\operatorname{\mathbb{Z}}$ and
$\mathcal{F}:L^{2}(\Gamma\backslash G/K)\to L^{2}(\Xi)$ the spectral
transform, certainly $\mathcal{F}\operatorname{\mathfrak{B}}^{k}$ is contained
in $V^{k}$, where

${V^{s}=\\{\text{measurable }v\text{ on }\Xi:(1-\lambda_{\xi})^{s/2}\cdot v\in
L^{2}(\Xi)\\}},\hskip 14.22636pt\text{for }s\in\operatorname{\mathbb{R}}.$

For any $s\in\operatorname{\mathbb{R}}$, we can endow $V^{s}$ with a Hilbert-
space structure from the norm

$|v|^{2}_{V^{s}}=\int_{\Xi}(1-\lambda_{\xi})^{s}|v(\xi)|^{2}\hskip
2.84526ptd\xi.$

As one might expect, there is a continuous inclusion $V^{s}\to V^{t}$ with
dense image for $s>t$, and the space $V^{-s}$ is naturally the Hilbert space
dual $(V^{s})^{*}$ of $V^{s}$, with Hermitian pairing given by the
asymmetrical extension of the Hermitian pairing $V^{0}\times
V^{0}\to\operatorname{\mathbb{C}}$ by $v\times w\to\langle
v,w\rangle_{V^{0}}$:

$\langle v,w\rangle_{V^{s}\times V^{-s}}=\int_{\Xi}v(\xi)\hskip
2.84526pt\overline{w(\xi)}\hskip 2.84526ptd\xi.$

The spectral transform $\mathcal{F}:C_{c}^{\infty}(\Gamma\backslash G/K)\to
L^{2}(\Xi)$ induces Hilbert space isomorphisms
$\mathcal{F}:\operatorname{\mathfrak{B}}^{2k}\to V^{2k}$ for all non-negative
integers $k$. The case $k=0$ is the Plancherel theorem. Let
$M_{\xi}(v)=(1-\lambda_{\xi})\cdot v$ be the multiplication operator. We
observe that $\mathcal{F}$ intertwines the operators
$1-\Delta:\operatorname{\mathfrak{B}}^{2k}\to\operatorname{\mathfrak{B}}^{2k-2}$
and $M_{\xi}:V^{2k}\to V^{2k-2}$ for Sobolev spaces of positive index;
furthermore, dualization gives the same result on negative-index spaces. Thus,
the following diagram commutes for $1\leq k\in\operatorname{\mathbb{Z}}$:

${\operatorname{\mathfrak{B}}^{2k}}$${\operatorname{\mathfrak{B}}^{2k-2}}$${V^{2k}}$${V^{2k-2}.}$$\scriptstyle{1-\Delta}$$\scriptstyle{\approx}$$\scriptstyle{\approx}$$\scriptstyle{\mathcal{F}}$$\scriptstyle{\approx}$$\scriptstyle{\mathcal{F}}$$\scriptstyle{\approx}$$\scriptstyle{M_{\xi}}$

This allows us to define an isomorphism
$\mathcal{F}:\operatorname{\mathfrak{B}}^{-2k}\to V^{-2k}$ as the adjoint to
the isomorphism $\mathcal{F}^{-1}:V^{2k}\to\operatorname{\mathfrak{B}}^{2k}.$

The odd-index case is slightly complicated since
$\operatorname{\mathfrak{B}}^{2k+1}$ is not naturally isomorphic to
$\operatorname{\mathfrak{B}}^{0}$, but there is no issue since
$\mathcal{F}:\operatorname{\mathfrak{B}}^{1}\to V^{1}$ is an isometry to its
(dense) image and $\operatorname{\mathfrak{B}}^{1}$ is complete. The
$\operatorname{\mathfrak{B}}^{1}\to\operatorname{\mathfrak{B}}^{-1}$ case
follows since $\mathcal{F}^{*}\circ M_{\xi}\circ\mathcal{F}=1-\Delta$ on test
functions, by the characterization of adjoints and de-symmetrized Plancherel.
As in the even-index case, by design,
$(1-\Delta):\operatorname{\mathfrak{B}}^{2k+1}\to\operatorname{\mathfrak{B}}^{2k-1}$
and $M_{\xi}:V^{2k+1}\to V^{2k-1}$ are isomorphisms, intertwined by the
Hilbert space isomorphism $\mathcal{F}:\operatorname{\mathfrak{B}}^{2k+1}\to
V^{2k+1}$ induced from the spectral transform on test functions. Furthermore,
we can define an isomorphism
$\mathcal{F}:\operatorname{\mathfrak{B}}^{-1-2k}\to V^{-1-2k}$ as the adjoint
to the isomorphism
$\mathcal{F}^{-1}:V^{2k+1}\to\operatorname{\mathfrak{B}}^{2k+1}.$

Thus, for $k\in\operatorname{\mathbb{Z}}$, the Hilbert space $V^{k}$ and
$M_{\xi}$ are spectral-side mirrors of $\operatorname{\mathfrak{B}}^{k}$ and
$1-\Delta$; furthermore, the spectral side allows us to define
$\operatorname{\mathfrak{B}}^{s}$ for any $s\in\operatorname{\mathbb{R}}$ as
completion of test functions with respect to the norm
$|f|_{\operatorname{\mathfrak{B}}^{s}}=|\mathcal{F}f|_{V^{s}}$. Sobolev
imbedding makes explicit the classical regularity of functions in automorphic
Sobolev spaces:

######  Theorem 2.11 (Sobolev imbedding for automorphic functions).

For $r=\dim_{\mathbb{R}}\Gamma\backslash G/K$ and $s>\frac{r}{2}$,
$\operatorname{\mathfrak{B}}^{s}\subset C^{o}(\Gamma\backslash G/K)$.
Furthermore, for $f\in C_{c}^{\infty}(\Gamma\backslash G/K)$ and compact
$C\subset\Gamma\backslash G/K$, we have $\sup_{z_{o}\in
C}|f(z_{o})|\ll_{C,s}|f|_{\operatorname{\mathfrak{B}}^{s}}$ and

$\lim_{k}\int_{\xi:|\lambda_{\xi}|\leq\ell}\langle
f,\Phi_{\xi}\rangle\cdot\Phi_{\xi}\hskip 2.84526ptd\xi=f\hskip
14.22636pt(\text{in }C^{o}).$

###### Proof.

The proofs in [Gar18]*Claim 12.3.19, Theorem 12.4.6 use a precursor to the
trace formulas in [Sel54], [Sel56], [Hej76], and [Iwa02], namely the pre-trace
formula

$\int_{\xi:|\lambda_{\xi}|\leq T^{2}}|\Phi_{\xi}(z_{o})|^{2}\ll_{C}T^{r}\hskip
14.22636pt(\text{as }T\to\infty),$

to give a uniform bound on
$\int_{\Xi}|\Phi_{\xi}(z_{o})|^{2}\cdot(1-\lambda_{\xi})^{-\frac{r}{2}-\varepsilon}$,
for $z_{o}$ in compact $C\subset\Gamma\backslash G/K$:

$\displaystyle\int_{\Xi}|\Phi_{\xi}(z_{o})|^{2}\cdot(1-\lambda_{\xi})^{-\frac{r}{2}-\varepsilon}=\sum_{\ell=1}^{\infty}\int_{\xi:\ell-1\leq|\lambda_{\xi}|<\ell}|\Phi_{\xi}(z_{o})|^{2}\cdot(1-\lambda_{\xi})^{-\frac{r}{2}-\varepsilon}$
$\displaystyle\ll_{C}\sum_{\ell=1}^{\infty}\int_{\xi:|\lambda_{\xi}|<\ell}|\Phi_{\xi}(z_{o})|^{2}\cdot\left((1+\ell)^{-\frac{r}{2}-\varepsilon}-(1+(1+\ell))^{-\frac{r}{2}-\varepsilon}\right)$
$\displaystyle\ll_{C}\sum_{\ell=1}^{\infty}\int_{\xi:|\lambda_{\xi}|<\ell}|\Phi_{\xi}(z_{o})|^{2}\cdot(1+\ell)^{-\frac{r}{2}-\varepsilon-1}\ll_{C}\sum_{\ell=1}^{\infty}\ell^{\frac{r}{2}}\cdot(1+\ell)^{-\frac{r}{2}-\varepsilon-1}<\infty.$

Since $\\{\xi\in\Xi:|\lambda_{\xi}|\leq\ell\\}$ is compact and
$\xi\to\Phi_{\xi}$ is a continuous $C^{\infty}(\Gamma\backslash G/K)$-valued
function on $\Xi$, the integrals

$f_{\ell}=\int_{\xi:|\lambda_{\xi}|\leq\ell}\langle
f,\Phi_{\xi}\rangle\cdot\Phi_{\xi}(z)\hskip 2.84526ptd\xi,\hskip
14.22636pt\text{for }f\in C_{c}^{\infty}(\Gamma\backslash G/K)$

exist as $C^{\infty}(\Gamma\backslash G/K)$-valued Gelfand-Pettis integrals,
so are certainly continuous. By the spectral characterization of
$\operatorname{\mathfrak{B}}^{s}$, $f_{\ell}\to f$ in the
$\operatorname{\mathfrak{B}}^{s}$ topology.

For $s>\frac{r}{2}$, $f\in\operatorname{\mathfrak{B}}^{s}$, and $z_{o}\in
C\subset\Gamma\backslash G/K$ compact,

$\displaystyle\left|\int_{\Xi}\mathcal{F}f(\xi)\cdot\Phi(z_{o})\hskip
2.84526ptd\xi\right|$
$\displaystyle=\left|\int_{\Xi}\mathcal{F}f(\xi)(1-\lambda_{\xi})^{s/2}\cdot(1-\lambda_{\xi})^{-s/2}\Phi(z_{o})\hskip
2.84526ptd\xi\right|$
$\displaystyle\leq\left(\int_{\Xi}|\mathcal{F}f(\xi)|^{2}(1-\lambda_{\xi})^{s}\hskip
2.84526ptd\xi\right)^{1/2}\cdot\left(\int_{\Xi}|\Phi(z_{o})|^{2}\cdot(1-\lambda_{\xi})^{-s}\hskip
2.84526ptd\xi\right)^{1/2}$
$\displaystyle=|f|_{\operatorname{\mathfrak{B}}^{s}}\hskip
2.84526pt\cdot\left(\int_{\Xi}|\Phi(z_{o})|^{2}\cdot(1-\lambda_{\xi})^{-s}\hskip
2.84526ptd\xi\right)^{1/2}\ll_{C,s}|f|_{\operatorname{\mathfrak{B}}^{s}}$

by Cauchy-Schwarz-Bunyakowsky and the uniform bound above. That is, the
$C^{o}$-norm (sup norm on compact) of the function
$z_{o}\to\int_{\Xi}\mathcal{F}f(\xi)\cdot\Phi(z_{o})\hskip 2.84526ptd\xi$
exists and is dominated by $|f|_{\operatorname{\mathfrak{B}}^{s}}$. Since
$f_{\ell}\to f$ in $\operatorname{\mathfrak{B}}^{s}$ for a dense subset of
$\operatorname{\mathfrak{B}}^{s}$ and $\operatorname{\mathfrak{B}}^{s}$
convergence implies $C^{o}$ convergence,
$\operatorname{\mathfrak{B}}^{s}\subset C^{o}$ and, for
$f\in\operatorname{\mathfrak{B}}^{s}$, $f^{\ell}\to f$ in $C^{o}$ (that is,
pointwise and uniformly on compact). ∎

###  2.7. Schwartz’ Kernel Theorem, nuclearity

In Section 4, and alternatively to [HC66] and [Cas89], we construct a Schwartz
space on $\Gamma\backslash G/K$ as a projective limit of Hilbert spaces with
Hilbert-Schmidt transition maps. This limit is nuclear Fréchet, so a kernel
theorem follows immediately for general categorical reasons. Following
[Gar20], we recall foundational results.

For $A,B,C$ all $k$-vector spaces without topologies and $\otimes_{k}$ the
usual tensor product of vector spaces, the Cartan-Eilenberg adjunction is

$\text{Hom}_{k}(A,\text{Hom}_{k}(B,C))\approx\text{Hom}_{k}(A\otimes_{k}B,C)\hskip
14.22636pt(\text{by }\varphi(a\otimes b)\to\varphi(a)(b)),$

and the special case $C=k$ becomes

$\text{Hom}_{k}(A,B^{*})\approx(A\otimes_{k}B)^{*}.$

For topological vector spaces $V$ and $W$, a genuine (categorical) tensor
product is a topological vector space $X$ and a continuous linear map
$j:V\times W\to X$ such that for every continuous bilinear map $K:V\times W\to
Y$ to a topological vector space $Y$, there is a unique continuous linear map
$k:X\to Y$ fitting into the commutative diagram

${X}$${V\times W}$${Y.}$$\scriptstyle{\exists!\hskip
2.84526ptk}$$\scriptstyle{j}$$\scriptstyle{K}$

By an easy corollary of Baire category (see e.g. [Gar20]*Claim 15.1),
separately continuous maps on Hilbert spaces are also jointly continuous, so
the lack of specification is justified.

When there is a genuine tensor product of topological vector spaces $V$ and
$W$, Cartan-Eilenberg yields a Schwartz kernel theorem. However, there is no
(categorical) tensor product in the category of Hilbert spaces and continuous
linear maps [Gar10]. While it is possible to put an inner product on the
algebraic tensor product $V\otimes_{alg}W$, and the completion
$V\otimes_{HS}W$ with respect to the associated norm is a Hilbert space (the
space of Hilbert-Schmidt operators), this Hilbert space does not have the
universal property in the categorical characterization of the tensor product,
since there are (jointly) continuous $\beta:V\times W\to X$ to Hilbert spaces
$X$ which do not factor through any continuous linear map $B:V\otimes_{HS}W\to
X$. That is, not all continuous operators are Hilbert-Schmidt.

For example, let $V=\ell^{2}$ with orthonormal basis $\\{v_{i}\\}$, $W=V^{*}$
with the corresponding dual basis $\lambda_{i}(v)=\langle v,v_{i}\rangle$, and
$X=\mathbb{C}$. Since $\beta(v,\lambda)=\lambda(v)$ is (jointly) continuous,
we might expect there is some continuous linear map
$B:V\otimes_{HS}V^{*}\to\mathbb{C}$ such that for all coefficients $c_{ij}$

$B\left(\sum_{ij}c_{ij}v_{i}\otimes_{HS}\lambda_{j}\right)=\sum_{ij}c_{ij}\beta(v_{i},\lambda_{j})=\sum_{i}c_{ii}.$

However, $\sum_{i}\frac{1}{i}v_{i}\otimes_{HS}\lambda_{i}$ is in
$V\otimes_{HS}V^{*}$, but the necessary value of $B$ is impossible since
$\sum_{i}\frac{1}{i}$ diverges.

Nuclear spaces have genuine tensor products [Gar20]*Theorem 7.5, and countable
projective (directed) limits of Hilbert spaces with Hilbert-Schmidt transition
maps are the simplest types of nuclear spaces [Gro55]: Let $V,W,V_{1},W_{1}$
be Hilbert spaces with Hilbert-Schmidt maps $S:V_{1}\to V$ and $T:W_{1}\to W$.
For any (jointly) continuous $\beta:V\times W\to X$ there is a unique
continuous (and Hilbert-Schmidt) map $B:V_{1}\otimes_{HS}W_{1}\to X$ giving a
commutative diagram

${V_{1}\otimes_{HS}W_{1}}$${V\otimes_{HS}W}$${V_{1}\times W_{1}}$${V\times
W}$${X.}$$\scriptstyle{B}$$\scriptstyle{S\times T}$$\scriptstyle{\beta}$

Let $v_{i}$, $w_{j}$ be orthonormal bases for $V$ and $W$, respectively. By
the continuity of $\beta$,

$\sum_{ij}\beta(Sv_{i},Tw_{j})\leq
C\sum_{ij}|Sv_{i}|^{2}\cdot|Tw_{j}|^{2}=C\cdot|S|^{2}_{HS}\cdot|T|^{2}_{HS}<\infty.$

A continuous map $B:V_{1}\otimes_{HS}W_{1}\to X$ fitting into the diagram
would have

$B\left(\sum_{ij}c_{ij}v_{i}\otimes
w_{j}\right)=\sum_{ij}c_{ij}\beta(Sv_{i},Tw_{j}),$

and since

$\sum_{ij}|c_{ij}\beta(Sv_{i},Tw_{j})|^{2}\leq\sum_{ij}|c_{ij}|^{2}\cdot\sum_{ij}|\beta(Sv_{i},Tw_{j})|^{2}\leq\sum_{ij}|c_{ij}|^{2}\cdot\left(C\cdot|S|^{2}_{HS}\cdot|T|^{2}_{HS}\right)$

by Cauchy-Schwarz-Bunyakowsky, $B$ is Hilbert-Schmidt (and exists).

The simplest examples of nuclear spaces are the projective limits
$H^{\infty}(\mathbb{T}^{m})$ of non-negative index Sobolev spaces on products
of circles

$H^{s}(\operatorname{\mathbb{T}}^{m})=\left\\{\sum_{\xi\in\operatorname{\mathbb{Z}}^{m}}c_{\xi}e^{i\xi\cdot
x}\in
L^{2}(\operatorname{\mathbb{T}}^{m}):|c_{\xi}|^{2}\dot{(}1+|\xi|^{2})^{s}<\infty\right\\}\hskip
14.22636pt(\text{for }s\geq 0),$

originally defined as completions of smooth functions on
$\operatorname{\mathbb{T}}^{m}$. The Hilbert-Schmidt property is proven by a
Rellich lemma, and the tensor product

$H^{\infty}(\operatorname{\mathbb{T}}^{m})\otimes
H^{\infty}(\operatorname{\mathbb{T}}^{n})\approx
H^{\infty}(\operatorname{\mathbb{T}}^{m+n})$

is constructed as

$\lim_{s}H^{s}(\operatorname{\mathbb{T}}^{m})\otimes\lim_{s}H^{s}(\operatorname{\mathbb{T}}^{n})=\lim_{s}\bigg{(}H^{s}(\operatorname{\mathbb{T}}^{m})\otimes_{HS}H^{s}(\operatorname{\mathbb{T}}^{n})\bigg{)}.$

Then, Cartan Eilenberg yields the simplest example of Schwartz’ Kernel
Theorem: there is an isomorphism

$\text{Hom}^{o}(H^{\infty}(\operatorname{\mathbb{T}}^{m}),H^{-\infty}(\operatorname{\mathbb{T}}^{n}))\approx
H^{-\infty}(\operatorname{\mathbb{T}}^{m+n})$

where $\text{Hom}^{o}$ is the space of continuous linear maps and
$H^{-\infty}$ is equipped with the weak-dual topology. The isomorphism is
induced by

$(f\to(F\to\Phi(f\otimes F)))\leftarrow\Phi,$

and the distribution $\Phi\in H^{-\infty}(\operatorname{\mathbb{T}}^{m+n})$
producing a given continuous map $H^{\infty}(\operatorname{\mathbb{T}}^{m})\to
H^{-\infty}(\operatorname{\mathbb{T}}^{n})$ is the Schwartz kernel of the map.

###  2.8. Friedrichs’ extensions of restrictions of semi-bounded operators

Crucial to [LP76]*pp. 204-206, [CdV82], [BG20], and Section 4 is Friedrichs’
self-adjoint extension of semi-bounded operators on Hilbert spaces [Fri34].

For $T$ a symmetric, semi-bounded operator with domain $D$ dense in some
Hilbert space $V$, Friedrichs’ extension
$\tilde{T}:\tilde{D}=\tilde{T}^{-1}V\to V$ is characterized by

$\langle(1+T)v,(1+\tilde{T})^{-1}w\rangle=\langle v,w\rangle\hskip
14.22636pt\text{(for }v\in D,w\in V).$

Unlike the other self-adjoint extensions of symmetric operators classified by
Stone [Sto32] and vonNeumann [vN30], Friedrichs’ extension has certain nice
properties: for $V^{1}$ the completion of $D$ with respect to $\langle
v,w\rangle_{1}=\langle(1+T)v,w\rangle$, the resolvent $(1+\tilde{T})^{-1}:V\to
V^{1}$ is continuous when $V^{1}$ has the finer topology and compact if the
inclusion $V^{1}\to V$ is compact with the finer topology on $V^{1}$.

As in [BG20] and [Gar18]*Section 11.2, we can re-characterize Friedrichs’
extension to facilitate finer analysis in terms of distributions and even
meromorphically continue certain Eisenstein series beyond the critical line
(see Section 3.4):

Assume the Hilbert space $V$ has a complex conjugation map $c$ that commutes
with $T$ such that the composition of the Riesz-Fréchet map is a complex-
linear isomorphism between $V$ and its dual $V^{*}$, and let $j:V^{1}\to V$ be
the continuous injection induced from the identity $D\to D$ with adjoint
$j^{*}:V^{*}\to(V^{1})^{*}=V^{-1}$. The continuous linear map
$T^{\\#}=j^{*}\circ c\circ j:V^{1}\to V^{-1}$ given by

$T^{\\#}(x)(y)=\langle x,\bar{y}\rangle_{1}\hskip 28.45274pt(\text{for }x,y\in
V^{1})$

is a topological isomorphism by Riesz-Fréchet, and we may use this map to more
explicitly identify the domain $\tilde{D}=\tilde{T}^{-1}V$ of the Friedrichs
extension $\tilde{T}$. Notice that $T^{\\#}$ is an extension of $j^{*}\circ
c\circ\tilde{T}$: for $x=\tilde{T}^{-1}x^{\prime}\in\tilde{D}$ with
$x^{\prime}\in V$ and any $y\in V^{1}$,

$(T^{\\#}x)(y)=\langle x,\bar{y}\rangle_{1}=((j^{*}\circ
c)x^{\prime})(y)=((j^{*}\circ c\circ\tilde{T})x)(y),$

and for $x\in V^{1}$ such that $T^{\\#}x=(j^{*}\circ c)y$ for some $y\in V$,
and all $z\in V^{1}$,

$\langle z,\bar{x}\rangle_{1}=(T^{\\#}x)(z)=((j^{*}\circ
c)y)(z)=(cy)(jz)=\langle jz,\bar{y}\rangle=\langle
z,\tilde{T}^{-1}\bar{y}\rangle_{1}.$

That is, the domain of $\tilde{T}$ is

$\tilde{D}:=\tilde{T}^{-1}V=\\{x\in V^{1}:T^{\\#}x\in(j^{*}\circ c)V\\}.$

To legitimize the discussion in later sections, we discuss Friedrichs’
extension of certain restrictions of $T$, following [BG20]. Let $\Theta\subset
D$ be a $T$-subspace which is stable under conjugation,
$V_{\Theta}=\Theta^{\perp}$ the orthogonal complement to $\Theta$ in $V$, and
$D_{\Theta}=D\cap V_{\Theta}$. Certainly $D_{\Theta}\subset V^{1}\cap
V_{\Theta}$, so the $V^{1}$-closure of $D_{\Theta}$ is a subset of $V^{1}\cap
V_{\Theta}$. However, $D_{\Theta}$ is not necessarily $V^{1}$-dense in
$V^{1}\cap V_{\Theta}$. As in the cases of interest, we assume $D_{\Theta}$ is
$V^{1}$-dense in $V^{1}\cap V_{\Theta}$. Under this assumption, the
restriction $T_{\Theta}$ of $T$ to $D_{\Theta}$ is densely-defined and
symmetric on $V_{\Theta}$ with Friedrichs extension
$\tilde{T}_{\Theta}:\tilde{D}_{\Theta}\to V_{\Theta}$ and further extension

$T_{\Theta}^{\\#}:V^{1}\cap V_{\Theta}\to(V^{1}\cap V_{\Theta})^{*},$

characterized by

$(T_{\Theta}^{\\#}x)(y)=\langle x,\bar{y}\rangle_{1}\hskip
28.45274pt(\text{for }x,y\in V^{1}\cap V_{\Theta}).$

For $i_{\Theta}:V^{1}\cap V_{\Theta}\to V^{1}$ and $j_{\Theta}:V^{1}\cap
V_{\Theta}\to V_{\Theta}$ the inclusions with adjoints $i^{*}_{\Theta}$ and
$j_{\Theta}^{*}$, we have the following diagram:

${\tilde{T}\subset
T^{\\#}:}$${V^{1}}$${V}$${V^{-1}}$${\tilde{T}_{\Theta}\subset
T^{\\#}_{\Theta}:}$${V^{1}\cap V_{\Theta}}$${V_{\Theta}}$${(V^{1}\cap
V_{\Theta})^{*},}$$\scriptstyle{j}$$\scriptstyle{j^{*}\circ\hskip
2.84526ptc}$$\scriptstyle{i^{*}_{\Theta}}$$\scriptstyle{j_{\Theta}}$$\scriptstyle{i_{\Theta}}$$\scriptstyle{j_{\Theta}^{*}\hskip
2.84526pt\circ\hskip 2.84526ptc}$$\scriptstyle{i}$

which has a useful compatibility $T_{\Theta}^{\\#}=i_{\Theta}^{*}\circ
T^{\\#}\circ i_{\Theta}$, since

$((i^{*}_{\Theta}\circ T^{\\#}\circ i_{\Theta})x)(y)=(T^{\\#}\circ
i_{\Theta}x)(i_{\Theta}y)=\langle
i_{\Theta}x,i_{\Theta}\bar{y}\rangle_{1}=\langle
x,\bar{y}\rangle_{1}=(T^{\\#}_{\Theta}x)(y)$

for $x,y\in V^{1}\cap V^{\Theta}$. By injectivity of the inclusion
$i:V_{\Theta}\to V$,

$\tilde{D}_{\Theta}=\\{x\in V^{1}\cap
V_{\Theta}:T^{\\#}_{\Theta}x\in(j_{\Theta}^{*}\circ c)V_{\Theta}\\}=\\{x\in
V^{1}\cap V_{\Theta}:T^{\\#}_{\Theta}x\in(i^{*}_{\Theta}\circ j^{*}\circ
c)V\\}.$

Furthermore, $T_{\Theta}^{\\#}x=(i^{*}_{\Theta}\circ j^{*}\circ c)y$ for $y\in
V$ if and only if $(T^{\\#}\circ i_{\Theta})x=(j^{*}\circ c)y+\theta$ for some
$\theta\in\ker i^{*}_{\Theta}$, the $V^{-1}$ closure of $(j^{*}\circ
c)\Theta$.

###  2.9. Perturbations of linear operators

In his introduction to [Kat66], Kato credits Rayleigh [Ray94] and Schrödinger
[Sch28] with the creation of perturbation theory of linear operators, the
study of physical systems which deviate slightly from a simpler ideal system
for which a (more) complete solution is known. Although such operators had
been used by physicists to successfully predict physical phenomena (see e.g.
[Dir28], [Dir30], [Tho35], [BP35]), analytical considerations such as
convergence and self-adjointness were not rigorously considered at least until
a series of papers by Rellich, Friedrichs, Kato, and many others decades
later. Self-adjointness of Hamiltonians $-\Delta+q$ was originally established
by the Kato-Rellich theorem of 1939 and the Kato, Lions, Lax-Milgram, Nelson
(KLMN) theorem. [BF61], [Kat72], [Sim73], [Sim73b] and many others greatly
broadened the class of potentials for which the corresponding Schrödinger
operator was essentially self-adjoint.

At the time, and currently, solvable models such as
$(\Delta-\lambda)u=\delta(u)\cdot\delta$ were casually rewritten by physicists
as $((-\Delta+\delta)-\lambda)u=0$, viewing $-\Delta+\delta$ as a perturbation
of $-\Delta$ by a singular potential. Indeed, as is now well understood,
$(\Delta-\lambda)u=\delta(u)\cdot\delta$ can be rearranged to
$(\Delta-\delta\otimes\delta-\lambda)u=0$.

Complementing the theory of singular potentials, [Kat51] established a fairly
complete theory for Hamiltonians $S=-\Delta+q$, with $q$ in one of four
classes $L^{2}+L^{\infty}$,
$L^{2}+(L^{\infty})_{\varepsilon}=\\{f:f=x+g_{\varepsilon}:x\in
L^{2},\left\lVert g_{\varepsilon}\right\rVert_{\infty}<\varepsilon\\}$,
$L^{2}$, and $L^{2}\cap L^{1}$. [Sim71] extended this theory beyond $L^{2}$
potentials to match physical phenomema by defining the sum of operators
$-\Delta+q$ as a quadratic form, as suggested by Nelson and Faris, instead of
as the sum of the Laplacian and a multiplication operator.

##  3\. Applications of modern analysis of automorphic forms

In Section 3.1, we show how to use automorphic spectral expansions to solve
differential equations on the Sobolev spaces defined in Section 2.6. Section
3.2 demonstrates the utility of this viewpoint by computing the simplest
automorphic Green’s function. In Section 3.3, we recall the seemingly
paradoxical Lax-Phillips discretization of the continuous spectrum of $\Delta$
and demonstrate how to identify the “exotic” eigenfunctions essential to
reprove meromorphic continuation of certain Eisenstein series in [CdV82] and
[Gar18], recounted in Section 3.4. Most provocatively, in Section 3.5, we
demonstrate how the natural self-adjoint operators of Colin de Verdière, Lax-
Phillips, and Bombieri-Garrett have some implications for the spacing of zeros
of $\zeta_{k}(s)$ on the critical line, for
$k=\operatorname{\mathbb{Q}}(\sqrt{d})$ quadratic fields with $d<0$.

###  3.1. Solving automorphic differential equations by division

We are now equipped to solve certain differential equations in automorphic
forms.

Given $f\in\operatorname{\mathfrak{B}}^{-\infty}(\Gamma\backslash G/K)$ and
$\lambda\in\operatorname{\mathbb{C}}$, we want to solve

$(\Delta-\lambda)u=f$

for $u\in\operatorname{\mathfrak{B}}^{-\infty}(\Gamma\backslash G/K)$, when
possible. Applying the spectral transform $\mathcal{F}$ to both sides yields

$\mathcal{F}f=\mathcal{F}(\Delta-\lambda)u=(\lambda_{\xi}-\lambda)\mathcal{F}u,$

where both $\mathcal{F}f$ and $\mathcal{F}u$ are in weighted $L^{2}$ spaces on
the spectral parameter space $\Xi$. For complex $\lambda\notin(-\infty,0]$,
the function $\lambda_{\xi}-\lambda$ is bounded away from 0 on $\Xi$, and we
can divide to obtain

$\mathcal{F}u=\frac{\mathcal{F}f}{\lambda_{\xi}-\lambda},$

such that

$u=\int_{\Xi}\frac{\mathcal{F}f(\xi)}{\lambda_{\xi}-\lambda}\cdot\Phi_{\xi}\hskip
2.84526ptd\xi\hskip 28.45274pt(\text{convergent in
}\operatorname{\mathfrak{B}}^{-\infty})$

is a solution. In fact, this solution is unique in
$\operatorname{\mathfrak{B}}^{-\infty}$, since any solution $v$ to the
homogeneous equation would have spectral expansion
$v=\int_{\Xi}\mathcal{F}v(\xi)\cdot\Phi_{\xi}\hskip 2.84526ptd\xi$ for
$\mathcal{F}v=0$ almost everywhere.

For $\lambda\in(-\infty,0]$, solutions to $(\Delta-\lambda)u=f$ are
expressible in terms of the eigenvalues $\lambda_{\xi}$ and eigenfunctions
$\Phi_{\xi}$ of $\Delta$. Consider the simplest case
$G=SL_{2}(\operatorname{\mathbb{R}})$, where
$f\in\operatorname{\mathfrak{B}}^{-\infty}(\operatorname{\Gamma\backslash
G/K})$ has spectral expansion

$f=\sum_{\text{cfm }F}\langle f,F\rangle\cdot F+\frac{\langle f,1\rangle\cdot
1}{\langle 1,1\rangle}+\frac{1}{4\pi i}\int_{(\frac{1}{2})}\langle
f,E_{s}\rangle\cdot E_{s}\hskip 2.84526ptds\hskip 28.45274pt(\text{convergent
in }\operatorname{\mathfrak{B}}^{-\infty}),$

where the indicated pairings and integrals are extensions by continuity of the
literal pairings and integrals. Denote $\lambda_{s}=s(s-1)$. For
$\text{Re}(w)>\frac{1}{2}$, and $w\neq 1$, the equation
$(\Delta-\lambda_{w})u=f$ has solution

$u=\sum_{\text{cfm }F}\frac{\langle f,F\rangle\cdot
F}{\lambda_{s_{F}}-\lambda_{w}}+\frac{\langle f,1\rangle\cdot
1}{(\lambda_{1}-\lambda_{w})\cdot\langle 1,1\rangle}+\frac{1}{4\pi
i}\int_{(\frac{1}{2})}\frac{\langle f,E_{s}\rangle\cdot
E_{s}}{\lambda_{s}-\lambda_{w}}ds,$

converging in at least $\operatorname{\mathfrak{B}}^{-\infty}$. For
$f\in\operatorname{\mathfrak{B}}^{r}$, the spectral characterization shows
that $u\in\operatorname{\mathfrak{B}}^{r+2}$ with the spectral expansion
converging in the same Sobolev space.

To meromorphically continue the solution to $(\Delta-\lambda_{w})u=f$ as a
function of $w$, consider $u=u_{w}$ as a holomorphic or meromorphic function-
valued function of $w$. By the spectral characterization of
$\operatorname{\mathfrak{B}}^{r}$ and $\operatorname{\mathfrak{B}}^{r+2}$, the
cuspidal component

$w\to u_{w}^{\text{cusp}}=\sum_{\text{cfm }F}\frac{\langle f,F\rangle\cdot
F}{\lambda_{s_{F}}-\lambda_{w}}$

is visibly a meromorphic $\operatorname{\mathfrak{B}}^{r+2}$-valued function
of $w\in\operatorname{\mathbb{C}}$, with poles at most at $w=s_{F}$, discrete
points in $\operatorname{\mathbb{C}}$ corresponding to the eigenvalues of
cuspforms. The constant component is similar.

The continuous spectrum component

$w\to u_{w}^{\text{cts}}=\frac{1}{4\pi i}\int_{(\frac{1}{2})}\frac{\langle
f,E_{s}\rangle\cdot E_{s}}{\lambda_{s}-\lambda_{w}}ds$

does not generally meromorphically continue as an
$\operatorname{\mathfrak{B}}^{r+2}$-valued function: for
$\lambda_{w}\leq-1/4$, if $(\Delta-\lambda_{w})u=f$ has a solution
$u\in\operatorname{\mathfrak{B}}^{-\infty}$, then $\langle f,E_{w}\rangle=0$
in the strong sense that $\langle f,E_{s}\rangle/(\lambda_{s}-\lambda_{w})$ is
locally integrable near $s=w$. However, we will show that for $X$ a quasi-
complete, locally convex topological vector space containing
$\operatorname{\mathfrak{B}}^{r+2}$ and Eisenstein series, the continuous
spectrum component has a meromorphic continuation to
$\operatorname{\mathbb{C}}$ as an $X$-valued function of $w$, with functional
equation

$u_{w}^{\text{cts}}=u_{1-w}^{\text{cts}}-\frac{\langle f,E_{w}\rangle\cdot
E_{w}}{2w-1}.$

The following topological vector space contains both suitable global
automorphic Sobolev spaces and Eisenstein series. Let $\mathcal{E}$ be the
Fréchet space $C^{\infty}(\Gamma\backslash G/K)$ and $\mathcal{D}$ the space
of test functions $C^{\infty}_{c}(\Gamma\backslash G/K)$. The quotient $X$ of
$\operatorname{\mathfrak{B}}^{r}\oplus\hskip 2.84526pt\mathcal{E}$ by the
closure of the anti-diagonal copy
$\mathcal{D}^{-\Delta}=\\{(\varphi,-\varphi):\varphi\in\mathcal{D}\\}$ of
$\mathcal{D}$ is a topological vector space fitting into the pushout diagram

${\mathcal{D}}$${\operatorname{\mathfrak{B}}^{r}}$${\mathcal{E}}$${X}$incinc

and is unique up to unique isomorphism. Furthermore, the seminorms

$\nu_{\mu}(f)=\sup_{w\in K}\mu(f(w))\hskip 14.22636pt(\text{for
}K\subset\Omega\text{ compact})$

give $X$ a quasi-complete, locally-convex topology.

We now prove the meromorphic continuation of the continuous component
$u^{\text{cts}}_{w}$. At first for $\text{Re}(w)>\frac{1}{2}$,

$\displaystyle u_{w}^{\text{cts}}$ $\displaystyle=\frac{1}{4\pi
i}\int_{(\frac{1}{2})}\frac{\langle f,E_{s}\rangle\cdot E_{s}-\langle
f,E_{w}\rangle\cdot E_{w}}{\lambda_{s}-\lambda_{w}}ds+\frac{1}{4\pi
i}\int_{(\frac{1}{2})}\frac{\langle f,E_{w}\rangle\cdot
E_{w}}{\lambda_{s}-\lambda_{w}}ds$ $\displaystyle=\frac{1}{4\pi
i}\int_{(\frac{1}{2})}\frac{\langle f,E_{s}\rangle\cdot E_{s}-\langle
f,E_{w}\rangle\cdot E_{w}}{\lambda_{s}-\lambda_{w}}ds-\frac{\langle
f,E_{w}\rangle\cdot E_{w}}{2(2w-1)}.$

The integral looks better behaved near $s=w$, but the appearance is misleading
since it is not a literal integral. With $t=\text{Im}(s)$, we can split the
remaining integral at some height $T$:

$\displaystyle\int_{(\frac{1}{2})}\frac{\langle f,E_{s}\rangle\cdot
E_{s}-\langle f,E_{w}\rangle\cdot E_{w}}{\lambda_{s}-\lambda_{w}}\hskip
2.84526ptds$ $\displaystyle=\int_{|t|>T}\frac{\langle f,E_{s}\rangle\cdot
E_{s}-\langle f,E_{w}\rangle\cdot E_{w}}{\lambda_{s}-\lambda_{w}}\hskip
2.84526ptds$ $\displaystyle+\int_{|t|\leq T}\frac{\langle f,E_{s}\rangle\cdot
E_{s}-\langle f,E_{w}\rangle\cdot E_{w}}{\lambda_{s}-\lambda_{w}}\hskip
2.84526ptds.$

The integral over $|t|>T$ is

$\int_{|t|>T}\frac{\langle f,E_{s}\rangle\cdot
E_{s}}{\lambda_{s}-\lambda_{w}}ds-\langle f,E_{w}\rangle
E_{w}\cdot\int_{|t|>T}\frac{ds}{\lambda_{s}-\lambda_{w}}.$

While the first term is $\operatorname{\mathfrak{B}}^{r+2}$-valued, the second
term takes values $X$, unless $\langle f,E_{w}\rangle$ vanishes. The integrand
of

$\int_{|t|\leq T}\frac{\langle f,E_{s}\rangle\cdot E_{s}-\langle
f,E_{w}\rangle\cdot E_{w}}{\lambda_{s}-\lambda_{w}}ds$

is a compactly supported, holomorphic $X$-valued function of the complex
variables $s,w$ away from the diagonal $s=w$. By inspection of the vector-
valued power-series expansion of the integral [Gar18]*Claim 15.8.1, there is
also cancellation on the diagonal, so the integrand is a holomorphic
$X$-valued function of $s,w$. For any complex-analytic $X$-valued function
$f(s,w)$ on some domain
$\Omega_{1}\times\Omega_{2}\subset\operatorname{\mathbb{C}}^{2}$, the function
$s\to(w\to f(s,w))$ is a holomorphic $\text{Hol}(\Omega_{2},X)$-valued
function on $\Omega_{1}$. Thus for $\Omega$ an appropriate bounded open set
containing the set where $|t|\leq T$, the integrand over $|t|\leq T$ is a
compactly-supported, continuous $\text{Hol}(\Omega,X)$-valued function of $s$,
and has a Gelfand-Pettis integral in $\text{Hol}(\Omega,X)$. That is, $w\to
u_{w}^{\text{cts}}$ has a meromorphic continuation as an $X$-valued function
of $w$.

Having proven its meromorphic continuation, we may use the integral expression
of $u_{w}^{\text{cts}}$ to derive its functional equation. At first for
$\text{Re}(w)<\frac{1}{2}$,

$\displaystyle u_{w}^{\text{cts}}$ $\displaystyle=\frac{1}{4\pi
i}\int_{(\frac{1}{2})}\frac{\langle f,E_{s}\rangle\cdot E_{s}-\langle
f,E_{w}\rangle\cdot E_{w}}{\lambda_{s}-\lambda_{w}}ds-\frac{\langle
f,E_{w}\rangle\cdot E_{w}}{2(2w-1)}$ $\displaystyle=\frac{1}{4\pi
i}\int_{(\frac{1}{2})}\frac{\langle f,E_{s}\rangle\cdot
E_{s}}{\lambda_{s}-\lambda_{w}}ds-\frac{\langle f,E_{w}\rangle\cdot
E_{w}}{(2w-1)}=u_{1-w}^{\text{cts}}-\frac{\langle f,E_{w}\rangle\cdot
E_{w}}{(2w-1)},$

since $\text{Re}(1-w)>\frac{1}{2}$ when $\text{Re}(w)<\frac{1}{2}$. This
identity extends to all $w\in\operatorname{\mathbb{C}}$ away from poles by the
identity principle.

###  3.2. Computing the simplest automorphic Green’s Function

To illustrate the usefulness of the previous discussion, we compute the
automorphic Green’s function on the simplest quotient. [Hub55] and [Sel54] had
independently considered such matters in the context of lattice-point problems
in hyperbolic spaces. [Neu73] considers the convergence and meromorphic
continuation of an automorphic Green’s function formed by winding up free-
space Green’s functions. Complications or failure of spectral expressions to
converge pointwise can be avoided by considering convergence in suitable
global automorphic Sobolev spaces, as in [Gar18]*Example 12.6, [DeC12],
[DeC16], [DeC21] and Section 2.6.

Since $\operatorname{\mathfrak{B}}^{s}\subset C^{o}$ for any $s>\frac{r}{2}$,
$f\to f(z)=\delta_{z}(f)$ is a continuous linear functional on
$\operatorname{\mathfrak{B}}^{s}$. That is,
$\delta_{z}\in\operatorname{\mathfrak{B}}^{-s}$ with spectral expansion

$\delta_{z}=\int_{\Xi}\overline{\Phi_{\xi}}(z)\cdot\Phi_{\xi}d\xi\hskip
14.22636pt(\text{convergent in }\operatorname{\mathfrak{B}}^{-s}).$

Thus, for $G=SL_{2}(\operatorname{\mathbb{R}})$, $\text{Re}(w)>\frac{1}{2}$
and $w\neq 1$, the equation $(\Delta-\lambda_{w})u=\delta_{z}$ has a solution
$u=u_{w}$ given by

$u=\sum_{\text{cfm }F}\frac{\overline{F(z)}\cdot
F}{\lambda_{s_{F}}-\lambda_{w}}+\frac{1}{(\lambda_{1}-\lambda_{w})\cdot\langle
1,1\rangle}+\frac{1}{4\pi i}\int_{(\frac{1}{2})}\frac{E_{1-s}(z)\cdot
E_{s}}{\lambda_{s}-\lambda_{w}}ds\hskip 14.22636pt(\text{convergent in
}\operatorname{\mathfrak{B}}^{1-\varepsilon})$

for any $\varepsilon>0$. Furthermore, $w\to u_{w}$ has a meromorphic
continuation in a topological vector space $X$, which includes
$\operatorname{\mathfrak{B}}^{1-\varepsilon}$ and Eisenstein series.

We may also compute the constant term of this Green’s function in a more
direct style than [Nie73] and [Fay77]:

######  Theorem 3.1.

For $\text{Re}(w)>\frac{1}{2}$, $w\notin(\frac{1}{2},1]$, and
$a\geq\text{Im}(z)$, the solution $u_{w}=u_{w,z}$ in
$\operatorname{\mathfrak{B}}^{1-\varepsilon}$ of the equation
$(\Delta-\lambda_{w})u_{w}=\delta_{z}$ has constant term

$c_{P}u_{w}(ia)=\int_{0}^{1}u_{w}(x+ia)dx=a^{1-w}\cdot\frac{E_{w}(z)}{1-2w},$

where $z=x+iy$ on $G/K\approx\mathfrak{H}$.

###### Proof.

We recall the proof of [Gar18]*Theorem 12.6.1. The orbits of
$(N\cap\Gamma)\backslash N$ are compact and codimension 1, so the distribution

$\eta_{a}f=c_{P}f(ia)=\int_{0}^{1}f(x+ia)dx$

which evaluates the constant term at height $a$ is in
$\operatorname{\mathfrak{B}}^{-\frac{1}{2}-\varepsilon}$ for every
$\varepsilon>0$. Since $u_{w}\in\operatorname{\mathfrak{B}}^{1-\varepsilon}$
for every $\varepsilon>0$, $\eta_{a}$ gives a continuous linear functional on
a Sobolev space containing $u_{w}$, and by the extended asymmetrical form of
Plancherel,

$\displaystyle\eta_{a}(u_{w})$
$\displaystyle=\int_{\Xi}\mathcal{F}\eta_{a}\cdot\overline{\mathcal{F}u_{w}}=\frac{\eta_{a}(1)\cdot\delta_{z}(1)}{(\lambda_{1}-\lambda_{w})\langle
1,1\rangle}+\frac{1}{4\pi
i}\int_{(\frac{1}{2})}\frac{\eta_{a}E_{1-s}\cdot\delta_{z}E_{s}}{\lambda_{s}-\lambda_{w}}ds$
$\displaystyle=\frac{1}{(\lambda_{1}-\lambda_{w})\langle
1,1\rangle}+\frac{1}{4\pi
i}\int_{(\frac{1}{2})}\frac{(a^{1-s}+c_{1-s}a^{s})\cdot
E_{s}(z)}{\lambda_{s}-\lambda_{w}}ds.$

By the functional equation for the Eisenstein series and the change of
variables $s$ to $1-s$, the integral of $c_{1-s}a^{s}\cdot
E_{s}(z)/(\lambda_{s}-\lambda_{w})$ produces a copy of the integral of
$a^{1-s}\cdot E_{s}(z)/(\lambda_{s}-\lambda_{w})$ such that

$\eta_{a}(u_{w})=\frac{1}{(\lambda_{1}-\lambda_{w})\langle
1,1\rangle}+\frac{1}{2\pi i}\int_{(\frac{1}{2})}\frac{a^{1-s}\cdot
E_{s}(z)}{\lambda_{s}-\lambda_{w}}ds.$

From the theory of the constant term [Gar18]*Theorem 8.1.1, the Eisenstein
series $E_{s}(z)$ is asymptotically dominated by its constant term
$y^{s}+c_{s}y^{1-s}$. For $a\geq y$ and by elementary estimates, the contour
$\text{Re}(s)=\frac{1}{2}$ can be pushed indefinitely to the right, picking up
residues at $s=1$ (due to the pole of $E_{s}$) and at $s=w$ (due to the
denominator). The constant-function term exactly cancels the residue at $s=1$
and since $\lambda_{s}-\lambda_{w}=(s-w)(s+w-1)$,

$\eta_{a}(u_{w})=-\text{Res}_{s=w}a^{1-s}\cdot\frac{E_{s}(z)}{\lambda_{s}-\lambda_{w}}=-a^{1-w}\cdot\frac{E_{w}(z)}{w-1+w}.$

This expression gives the meromorphic continuation of $w\to\eta_{a}(u_{w})$,
which is the constant term of the meromorphic continuation of $u_{w}$ to a
larger topological vector space $X$, assuming the topology on $X$ is fine
enough such that $\eta_{a}$ is a continuous linear functional on it. ∎

###  3.3. Partial discretization of the continuous spectrum

Friedrichs’ extension of $\Delta$ restricted to test functions in the Lax-
Phillips space of $L^{2}$ pseudo-cuspforms has purely discrete spectrum,
consisting of genuine cuspforms and infinitely-many “exotic” eigenfunctions
which are not eigenfunctions of $\Delta$ [LP76]*pp. 204-206. The seemingly
paradoxical discretization of the continuous spectrum of $\Delta$ is essential
to the proof of meromorphic continuation of the simplest Eisenstein series in
[CdV82], and [Gar18]*Section 11.11 generalizes the method to prove meromorphic
continuation of Eisenstein series for maximal proper parabolic subgroups
$P^{r_{1},r_{2}}\subset GL_{r_{1}+r_{2}}(\operatorname{\mathbb{R}})$, as an
alternate proof to [MW89]*Appendix and Theorem 2.5 in Section 2.4.

The notion of pseudo-cuspforms is simplest for unicuspidal quotients
$\Gamma\backslash G/K$, namely

$L^{2}_{a}(\Gamma\backslash G/K)=\\{f\in L^{2}(\Gamma\backslash
G/K):c_{P}f(g)=0\text{ for }\eta(g)\geq a\\},$

for $P$ any standard parabolic and $\eta(n_{x}m_{y}k)$ a suitable notion of
height. Vanishing of the constant term $c_{P}f$ above height $a$ is equivalent
to orthogonality $\langle f,\Psi_{\varphi}\rangle_{L^{2}}=0$ to pseudo-
Eisenstein series $\Psi_{\varphi}$ with test-function data $\varphi\in
C_{c}^{\infty}(0,\infty)$ supported on $[a,\infty)$.

While density of $D_{a}=C_{c}^{\infty}(\Gamma\backslash G/K)\cap
L^{2}_{a}(\Gamma\backslash G/K)$ in $L^{2}_{a}$ is not obvious, $D_{a}$ is
provably dense in $L^{2}_{a}(\Gamma\backslash G/K)$ for $a\gg 1$ [Gar18]*Lemma
10.3.1. The proof approximates $f\in L^{2}_{a}(\Gamma\backslash G/K)$ by test
functions, then uses the $a\gg 1$ condition to consider well-behaved smooth
cut-offs of the constant term near height $a$, with the width of the cut-off
region shrinking to 0. The averaged action images $\psi_{n}\cdot f$ are
smooth, but the averaging smears the support of the constant term, depending
on the support of $\psi_{n}$. The condition $a\gg 1$ ensures that the standard
Siegel set $\mathfrak{S}_{a}$ has the property that
$\mathfrak{S}_{a}\cap\gamma\mathfrak{S}_{a}\neq 0$ implies
$\gamma\in\Gamma\cap P$ such that the natural smooth cutting-off of the
constant term near the given height interacts with the constant-term vanishing
in a controlled manner. Such $a\gg 1$ exists by reduction theory
[Gar18]*Section 1.5. While ensuring the required density, the $a\gg 1$
condition means that the proof mechanism does not immediately yield
discreteness of genuine cuspforms $L^{2}_{o}$. However, it is possible to
salvage with a little work [Gar18]*Theorem 7.1.1.

The restriction $\Delta_{a}$ of $\Delta$ to $D_{a}$ is symmetric, semi-
bounded, and densely defined, so it has a self-adjoint Friedrichs extension
$\tilde{\Delta}_{a}$. Let $\operatorname{\mathfrak{B}}_{a}^{1}$ be the
completion of $D_{a}$ with respect to the Sobolev-like norm

$|f|^{2}_{\operatorname{\mathfrak{B}}_{a}^{1}}=\langle(1-\Delta)f,f\rangle_{L^{2}}\hskip
14.22636pt(\text{for }f\in D_{a}).$

Since the inclusion
$L^{2}_{a}\hookrightarrow\operatorname{\mathfrak{B}}_{a}^{1}$ is compact when
$\operatorname{\mathfrak{B}}_{a}^{1}$ has the finer topology [LP76]*pp.
204-206, [Gar18]*Sections 10.7-10.8, $\tilde{\Delta}_{a}$ has compact
resolvent on $L^{2}_{a}(\Gamma\backslash G/K)$ as the composition of a
continuous map and compact inclusion. Proof of compactness of the inclusion
relies on tail estimates [LP76]*pp. 204-206, [Gar18]*Sections 10.4-10.6. In
the simplest case, the crucial estimate is as follows. Given $\varepsilon>0$,
a cut-off $c\geq a$ can be made sufficiently large so that the image of the
$\operatorname{\mathfrak{B}}_{a}^{1}$-unit ball $B$ lies in a single
$\varepsilon$-ball in $L^{2}(\operatorname{\Gamma\backslash G/K})$. That is,
for $f\in\operatorname{\mathfrak{B}}_{a}^{1}$,

$\lim_{c\to\infty}\int_{y>c}|f(z)|^{2}\frac{dxdy}{y^{2}}\to 0\hskip
14.22636pt(\text{uniformly for }|f|_{\operatorname{\mathfrak{B}}^{1}}\leq 1).$

The seeming paradox is that $L^{2}_{a}(\Gamma\backslash G/K)$ contains the
space of genuine $L^{2}$-cuspforms and an infinite-dimensional space of
pseudo-Eisenstein series. For example, for $a^{\prime}<a$ with $a^{\prime}$
still large enough such that
$\gamma\mathfrak{S}_{a^{\prime}}\cap\mathfrak{S}_{a^{\prime}}\neq\phi$ implies
$\gamma\in\Gamma\cap P$ and for $\varphi\in C_{c}^{\infty}(0,\infty)$
supported on $[a^{\prime},a]$, the pseudo-Eisenstein series
$\Psi_{\varphi}(g)=\sum_{\gamma\in\Gamma}\varphi(\gamma g)$ is identically 0
in the region $\eta(g)>a$. By the standard spectral decomposition, these
pseudo-Eisenstein series are integrals of Eisenstein series, so some part of
the continuous spectrum of $\Delta$ becomes discrete for $\tilde{\Delta}_{a}$.

Identification of exotic eigenfunctions is also non-trivial. In the simplest
case $\operatorname{\Gamma\backslash G/K}$, the truncated Eisenstein series
$\wedge^{a}E_{s}$ is an eigenfunction for $\tilde{\Delta}_{a}$ if and only if
$c_{P}E_{w}(x+ia)=0$, and the spectral characterization of the global
automorphic Sobolev spaces $\operatorname{\mathfrak{B}}^{s}$ shows that all
eigenvalues $\lambda_{w}<-\frac{1}{4}$ arise this way. That is, all
eigenfunctions for $\lambda_{w}<-\frac{1}{4}$ are truncated Eisenstein series
[LP76]*pp. 204-206, [BG20]. These truncated Eisenstein series are in $L^{2}$
by the theory of the constant term [Gar18]*Claim 1.11.3, but are not smooth,
and therefore not eigenfunctions of the elliptic differential operator
$\Delta$.

We illustrate an identification method in the simplest case, following [CdV82]
with detailed explanation in [Gar18]. Take $a\gg 1$ and use the coordinates
$z=x+iy\in\mathfrak{H}$. First we show
$(\tilde{\Delta}_{a}-\lambda_{w})\wedge^{a}E_{w}=0$. That is, we will show
$(\Delta-\lambda_{w})\wedge^{a}E_{w}=c\cdot\eta_{a}$ for some constant
$c=c(w,a)$. Away from $y=a$, $(\Delta-\lambda_{w})\wedge^{a}E_{w}(z)=0$
locally. In $y\gg 1$, $\Delta$ annihilates all Fourier components of $E_{w}$
but the constant term, and in the lower part of a sufficiently large Siegel
set, the operator also annihilates the constant term. To compute near $y=a$,
let $H$ be the Heaviside function $H(y)=0$ for $y<0$ and $H(y)=1$ for $y>0$.
Near $y=a$ (and with all equalities and derivatives in
$\operatorname{\mathfrak{B}}^{-1}$ sense),

$\displaystyle(\Delta-\lambda_{w})\wedge^{a}E_{w}(z)$
$\displaystyle=(\Delta-\lambda_{w})\left(H(a-y)\cdot(y^{w}+c_{w}y^{1-w})\right)$
$\displaystyle=(y^{2}\frac{\partial^{2}}{\partial
y^{2}}-w(w-1))\left(H(a-y)\cdot(y^{w}+c_{w}y^{1-w})\right)$
$\displaystyle=y^{2}\left(\delta^{\prime}_{a}\cdot(y^{w}+c_{w}y^{1-w})-2\delta_{a}\cdot(wy^{w-1}+(1-w)c_{w}y^{-w})\right).$

For $y^{w}+c_{w}y^{1-w}=0$, the term with $\delta^{\prime}_{a}$ vanishes, and
for $y=a$, the rest simplifies to

$-2a\delta_{a}\cdot(wa^{w}+(1-w)c_{w}a^{1-w})=-2\delta_{a}\cdot(2w-1)a^{w+1}.$

On functions of $y$ independent of $x$, this is
$2(2w-1)a^{w+1}\cdot\eta_{a}\in\operatorname{\mathfrak{B}}^{-1}$. If
$a^{w}+c_{w}a^{1-w}\neq 0$, the term with $\delta_{a}^{\prime}$ remains, and
the expression is not inside $\operatorname{\mathfrak{B}}^{-1}$ nor in the
domain of $\tilde{\Delta}_{a}$. That is, $\wedge^{a}E_{w}$ is an eigenfunction
only if $a^{w}+c_{w}a^{1-w}=0$.

For $G=SL_{r}(\operatorname{\mathbb{R}})$, the notion of pseudo-cuspform is
more complicated. Let $A$ be the standard maximal torus of diagonal real
matrices

$A=\left\\{\begin{pmatrix}a_{1}&\\\ &\ddots\\\
&&a_{r}\end{pmatrix}:a_{i}\in\operatorname{\mathbb{R}}\right\\},$

$A^{+}$ the subgroup of positive real diagonal matrices, and
$\Phi=\\{\alpha_{i}(a)=\frac{a_{i}}{a_{i+1}}:i=1,...,r-1\\}$ a choice of
positive simple roots. By reduction theory [Gar18]*Section 3.3, there is a
sufficently large Siegel set $\mathfrak{S}=\mathfrak{S}_{t_{o}}=\\{nmk:n\in
N^{\text{min}},m\in A^{+},k\in K,\alpha(m)\geq t_{o}\text{ for all
}\alpha\in\Phi\\}$ associated to the standard minimal parabolic
$P^{\text{min}}$ such that $\Gamma\cdot\mathfrak{S}=G$. For real $a\gg 1$, and
$Y_{a}\subset\mathfrak{S}$ given by

$Y_{a}=\\{nmk\in\mathfrak{S}:\alpha(m)\geq a\text{ for some
}\alpha\in\Phi\\},$

the relevant space of pseudo-cuspforms is $V_{a}$, the $L^{2}$-closure of

$D_{a}=\\{f\in C_{c}^{\infty}(\Gamma\backslash G/K):\text{ for }g\in
Y_{a},c_{P}f(g)=0,\text{ for all standard parabolics }P\\}.$

This space is certainly a subset of

$L^{2}_{a}=\\{f\in L^{2}(\Gamma\backslash G/K):\text{ for }g\in
Y_{a},c_{P}f(g)=0,\text{ for all standard parabolics }P\\},$

but density of $D_{a}$ in the bigger space $L^{2}_{a}$ is not necessary for
proving meromorphic continuation of cuspidal-data Eisenstein series.

As in the simplest case, we can prove that for sufficiently high cut-off
heights $\eta$, there must be infinitely-many eigenfunctions for
$\tilde{\Delta}_{a}$ that were not eigenfunctions for $\Delta$. For example,
let $P=P^{r,r}\subset SL_{2r}$ be a self-associate maximal proper parabolic,
$f$ a cusp form on the Levi component $M=M^{P}$, and $E_{s,f}$ the
corresponding cuspidal-data Eisenstein series with constant term
$c_{P}E_{s,f}$. For $A^{P}$ the center of $M$, $M^{1}$ the subgroup of $M$
consisting of matrices in $r$-by-$r$ blocks $\begin{pmatrix}a&0\\\
0&d\end{pmatrix}$ with $\det a=1=\det d$, and $s\in\operatorname{\mathbb{C}}$
such that $c_{P}E_{s,f}(mm_{1})=0$ for all $m_{1}\in M^{1}$ and $m\in A^{P}$
with $\alpha_{r}(m)=a$, the truncation $\wedge^{a}E_{s,f}$ is a
$\tilde{\Delta}_{a}$ eigenfunction in $V_{a}$ [Gar18]*Claim 11.11.1.

###  3.4. Meromorphic continuation of Eisenstein series

Here we recall an argument illustrating the utility of singular potentials.

Although Selberg had already proven meromorphic continuation of Eisenstein
series on rank-one quotients in [Sel56] (see also [Roe56a],[Roe56b]) and
Langlands [Lan76] had already treated the general case, Colin de Verdière
[CdV82] demonstrated how one can use extensions of restrictions of the
Laplacian to give an alternate proof of meromorphic continuation of the
simplest Eisenstein series (see also [LP76]*pp. 204-206, [Mu96]), with
analytic continuation of $\zeta(s)$, as in [Lan76], [Lan71], and [Sha78], as a
corollary. Furthermore, the method generalizes to prove meromorphic
continuation of cuspidal-data Eisenstein series. For a general result, see
[MW95]*Appendix, and for maximal proper parabolics in
$GL_{r}(\operatorname{\mathbb{R}})$, see [Gar18]*Sections 11.7-11.13 or
Theorem 2.5 in Section 2.

In short, compactness of the inclusion map
$\operatorname{\mathfrak{B}}^{1}_{a}\hookrightarrow L^{2}_{a}$ for $a\gg 1$
yields compactness of the resolvent $(1-\tilde{\Delta}_{a})^{-1}$ on
$L^{2}_{a}$ and meromorphy of $a\to(1-\tilde{\Delta}_{a})^{-1}$ as an
operator-valued function taking values in the space of operators
$T:L^{2}_{a}\to L^{2}_{a}$. Eisenstein series differ from truncated Eisenstein
series by elementary functions, yielding meromorphic continuation.

First, we recall the argument for meromorphic continuation up to the critical
line of the Eisenstein series on $\Gamma\backslash
G/K=SL_{2}(\operatorname{\mathbb{Z}})\backslash\mathfrak{H}$, already known to
Roelcke, et alia: For $\tilde{\Delta}$ the Friedrichs extension of the
restriction of $\Delta$ to $D=C_{c}^{\infty}(\Gamma\backslash G/K)$ with
domain $\tilde{D}$ and $\operatorname{\mathfrak{B}}^{k}$ the index $k$ global
automorphic Sobolev space, we have the inclusions
$\operatorname{\mathfrak{B}}^{2}\subset\tilde{D}\subset\operatorname{\mathfrak{B}}^{1}$.
As usual, take $a\gg 1$. The quotient $\Gamma\backslash G/K$ is the union of a
compact part $X_{\text{cpt}}$ and a geometrically simpler, non-compact part

$X_{\infty}=\Gamma_{\infty}\backslash\\{x+iy\in\mathfrak{H}:y\geq
a\\}\approx(\operatorname{\mathbb{Z}}\backslash\operatorname{\mathbb{R}})\times[a,\infty).$

Let $a<a^{\prime}<a^{\prime\prime}$, and define a smooth cut-off function on
$G/K\approx\mathfrak{H}$ given by

$\tau(g)=\tau(x+iy)=\begin{cases}1&y>a^{\prime}\\\
0&y<a^{\prime\prime}.\end{cases}$

We may define a pseudo-Eisenstein series $h_{s}$ by winding up the smoothly
cut-off function $\tau(g)\cdot\text{Im}(g)^{s}$:

$h_{s}(g)=\sum_{\gamma\in\Gamma_{\infty}\backslash\Gamma}\tau(\gamma
g)\cdot\text{Im}(\gamma g)^{s}.$

Since $\tau$ is supported on $y\geq a^{\prime\prime}$, there is at most one
non-vanishing summand in the expression for $h_{s}$ for any $g\in G/K$, and
convergence is not an issue. Thus, the function-valued function $s\to h_{s}$
is entire.

Next, consider

$\tilde{E}_{s}=h_{s}-(\tilde{\Delta}-\lambda_{s})^{-1}(\Delta-\lambda_{s})h_{s}.$

For all $s\in\operatorname{\mathbb{C}}$, the smooth function
$(\Delta-\lambda_{s})h_{s}$ is supported on the image of the compact set
$a\leq y\leq a^{\prime}$ in $\Gamma\backslash G/K$ and is therefore in
$C_{c}^{\infty}(\Gamma\backslash G/K)$. For $\lambda_{s}\notin(-\infty,0]$,
that is for $s\in\operatorname{\mathbb{C}}$ such that
$\text{Re}(s)>\frac{1}{2}$ and $s\notin(\frac{1}{2},1]$, the resolvent
$(\tilde{\Delta}-\lambda_{s})^{-1}:L^{2}\to\operatorname{\mathfrak{B}}^{1}$
exists as an everywhere-defined, continuous operator. That is,
$s\to(\tilde{\Delta}-\lambda_{s})^{-1}$ is a holomorphic operator-valued
function on
$S_{\rho}=\\{s\in\operatorname{\mathbb{C}}:\text{Re}(s)>\frac{1}{2},s\notin\left(\frac{1}{2},1\right]\\}$
such that $s\to\tilde{E}_{s}-h_{s}$ is a holomorphic
$\operatorname{\mathfrak{B}}^{1}$-valued function on $S_{\rho}$. However,
$s\to\tilde{E}_{s}-h_{s}$ is non-trivial. Since $h_{s}\notin L^{2}$ and
$(\tilde{\Delta}-\lambda_{s})^{-1}(\Delta-\lambda_{s})h_{s}\in\operatorname{\mathfrak{B}}^{1}\subset
L^{2}$ for $s\in S_{\rho}$, their difference $\tilde{E}_{s}$ cannot vanish on
$S_{\rho}$.

We claim $s\to\tilde{E}_{s}$ is a meromorphic
$\operatorname{\mathfrak{B}}^{1}$-valued function on $S_{\rho}$ and gives a
meromorphic continuation of $E_{s}$ to
$\text{Re}(s)>\frac{1}{2},s\notin(\frac{1}{2},1]$, and it suffices to prove
that $u=\tilde{E}_{s}-h_{s}$ is the unique element of $\tilde{D}$ such that

$(\tilde{\Delta}-\lambda_{s})u=-(\Delta-\lambda_{s})h_{s}=:-H_{s}.$

Uniqueness follows from Friedrichs’ construction, since
$(\tilde{\Delta}-\lambda_{s}):\tilde{D}\to L^{2}(\Gamma\backslash G/K)$ is a
bijection for $s\in S_{\rho}$. Furthermore, $\tilde{E}_{s}-h_{s}$ is in
$C^{\infty}(\Gamma\backslash G/K)\cap L^{2}(\Gamma\backslash G/K)$ with

$\displaystyle\Delta(\tilde{E}_{s}-h_{s})$
$\displaystyle=(\Delta-\lambda_{s})(\tilde{E}_{s}-h_{s})+\lambda_{s}(\tilde{E}_{s}-h_{s})$
$\displaystyle=(\Delta-\lambda_{s})(\tilde{\Delta}-\lambda_{s})^{-1}(\Delta-\lambda_{s})h_{s}+\lambda_{s}(\tilde{E}_{s}-h_{s})$
$\displaystyle=(\Delta-\lambda_{s})h_{s}+\lambda_{s}(\tilde{E}_{s}-h_{s})\in
L^{2},$

since $(\tilde{\Delta}-\lambda_{s})^{-1}=(\Delta-\lambda_{s})^{-1}$ on
$S_{\rho}$ and $(\Delta-\lambda_{s})h_{s}\in C_{c}^{\infty}(\Gamma\backslash
G/K)$, the domain of $\Delta$. Thus,
$\tilde{E}_{s}-h_{s}\in\operatorname{\mathfrak{B}}^{2}\subset\tilde{D}$, and
the following computation is legitimate:

$\displaystyle(\tilde{\Delta}-\lambda_{s})(\tilde{E}_{s}-h_{s})=(\tilde{\Delta}-\lambda_{s})\left(h_{s}-(\tilde{\Delta}-\lambda_{s})^{-1}H_{s}-h_{s}\right)=-H_{s}.$

We also have

$(\tilde{\Delta}-\lambda_{s})(E_{s}-h_{s})=-(\Delta-\lambda_{s})h_{s}=-H_{s},$

since $E_{s}$ is a $\Delta$-eigenfunction with eigenvalue $\lambda_{s}$. By
uniqueness, $\tilde{E}_{s}-h_{s}$ = $E_{s}-h_{s}$ for $s\in S_{\rho}$, so
$\tilde{E}_{s}$ gives an analytic continuation of $s\to E_{s}$ to $S_{\rho}$
as an $h_{s}+\operatorname{\mathfrak{B}}^{1}$-valued function.

Furthermore, Friedrichs’ construction gives a bound for the $L^{2}$ norm of
$E_{s}-h_{s}$. Since $(\Delta-\lambda_{s})h_{s}$ has support in the image of
$a\leq y\leq a^{\prime}$,

$|(\Delta-\lambda_{s})h_{s}|^{2}_{2}\leq\int_{0}^{1}\int_{a}^{a^{\prime}}(|\delta
h_{s}|+|\lambda_{s}h_{s}|)^{2}\hskip
2.84526pt\frac{dxdy}{y^{2}}\ll_{a,a^{\prime}}|\lambda_{s}|^{2}.$

Denote $\lambda_{s}=c+di$. For $s\in S_{\rho}$ and $v\in L^{2}$ arbitrary,
since $\tilde{\Delta}$ is negative-definite and self-adjoint,

$|(\tilde{\Delta}-\lambda_{s})v|_{2}^{2}=|(\tilde{\Delta}-c)v|_{2}^{2}-i\cdot
d\langle(\tilde{\Delta}-c)v,v\rangle_{2}+i\cdot d\langle
v,(\tilde{\Delta}-c)\rangle_{2}+d^{2}|v|^{2}{\geq}d^{2}|v|^{2}.$

Thus, for $s\in S_{\rho}$, the operator norm of the resolvent is estimated by

$\left\lVert(\tilde{\Delta}-\lambda_{s})^{-1}\right\rVert^{2}\leq\frac{1}{d^{2}}=\frac{1}{2(\text{Re}(s)-\frac{1}{2})\cdot\text{Im}(s)}\hskip
14.22636pt(\text{for Re}(s)>\frac{1}{2},\text{Im}(s)\neq 0),$

such that

$\displaystyle|E_{s}-h_{s}|^{2}_{2}\leq\left\lVert(\tilde{\Delta}-\lambda_{s})^{-1}\right\rVert^{2}\cdot|(\Delta-\lambda_{s})h_{s}|^{2}_{2}\ll_{a,a^{\prime}}\frac{|\lambda_{s}|}{(\text{Re}(s)-\frac{1}{2})\cdot\text{Im}(s)}\hskip
2.84526pt.$

The simplest Eisenstein series has constant term
$c_{P}E_{s}=y^{s}+c_{s}y^{1-s}$, for $\displaystyle
c_{s}=\frac{\xi(2s-1)}{\xi(2s)}$ and $\xi(s)=\pi^{-s/2}\Gamma(s/2)\zeta(s)$
the completed zeta-function, so analytic continuation of $E_{s}$ to
$\text{Re}(s)>\frac{1}{2}$ analytically continues $c_{s}$ to the same region,
yielding the analytic continuation of $\zeta(s)$ to $\text{Re}(s)>0$ off the
interval $[0,1]$.

The re-characterization of pseudo-Laplacians [Gar18]*Section 11.3 yields
meromorphic continuation of Eisenstein series beyond the critical line and
without the spectral decomposition of $L^{2}$ (which uses this meromorphic
continuation). Referring to the notation of the last section, take
$V=L^{2}(\Gamma\backslash G/K)$, $c$ pointwise conjugation, $D$ the space of
automorphic test functions, $T=1-\Delta\big{|}_{D}$, $a\gg 1$ large enough
such that $D_{a}=D\cap L^{2}_{a}$ is dense in the Lax-Phillips space of
pseudo-cuspforms $L^{2}_{a}$, and $\Theta$ the space of pseudo-Eisenstein
series $\Phi_{\varphi}$ with data $\varphi\in C_{c}^{\infty}(a,\infty)$.
Notice $V_{\Theta}=\Theta^{\perp}=L^{2}_{a}$, $V^{1}$ is the global
automorphic Sobolev space $\operatorname{\mathfrak{B}}^{1}$ with Hilbert space
dual $\operatorname{\mathfrak{B}}^{-1}$, and the relevant diagram becomes

${\tilde{\Delta}\subset\Delta:}$${\operatorname{\mathfrak{B}}^{1}}$${L^{2}}$${\operatorname{\mathfrak{B}}^{-1}}$${\tilde{\Delta}_{a}\subset\Delta_{a}^{\\#}:}$${\operatorname{\mathfrak{B}}^{1}\cap
L^{2}_{a}}$${L^{2}_{a}}$${(\operatorname{\mathfrak{B}}^{1}\cap
L^{2}_{a})^{*}.}$$\scriptstyle{j}$$\scriptstyle{j^{*}\circ\hskip
2.84526ptc}$$\scriptstyle{i^{*}_{\Theta}}$$\scriptstyle{j_{\Theta}}$$\scriptstyle{i_{\Theta}}$$\scriptstyle{j_{\Theta}^{*}\hskip
2.84526pt\circ\hskip 2.84526ptc}$$\scriptstyle{i}$

By spectral theory on multi-tori (and without the spectral decomposition of
$\operatorname{\mathfrak{B}}^{1}$ or $\operatorname{\mathfrak{B}}^{-1}$), the
evaluation of the constant term at height $a$ is in the
$\operatorname{\mathfrak{B}}^{-1}$ closure of $\Theta$ for $a\gg 1$
[Gar18]*Section 11.3. Furthermore, $D_{a}$ is
$\operatorname{\mathfrak{B}}^{1}$-dense in
$\operatorname{\mathfrak{B}}^{1}\cap L^{2}_{a}(\Gamma\backslash G/K)$
[Gar18]*Section 11.4. Thus, $\tilde{T}_{\Theta}u=f$ for $f\in V_{\Theta}$ if
and only if $u\in V^{1}\cap V_{\Theta}$ and $(T^{\\#}\circ
i_{\Theta})u=(j^{*}\circ c\circ i)f+\theta$ for some $\theta\in\ker
i^{*}_{\Theta}$. That is, $\tilde{\Delta}_{a}u=f$ for $f\in
L^{2}_{a}(\Gamma\backslash G/K)$ if and only if
$u\in\operatorname{\mathfrak{B}}^{1}\cap L^{2}_{a}(\Gamma\backslash G/K)$ and
$\Delta u=f+c\cdot\eta_{a}$ for some constant $c$.

The proof of further meromorphic continuation starts as above, with
$\tilde{\Delta}_{a}$ in place of $\tilde{\Delta}$. Let $h_{s}$ be the pseudo-
Eisenstein series formed from the smooth cut-off $\tau\cdot y^{s}$ of $y^{s}$
and consider

$\tilde{E}_{a,s}=h_{s}-(\tilde{\Delta}_{a}-\lambda_{s})^{-1}(\Delta-\lambda_{s})h_{s}.$

Since $(\Delta-\lambda_{s})h_{s}$ is compactly supported for all
$s\in\operatorname{\mathbb{C}}$, it is in $L^{2}_{a}$ for $a\gg 1$ large
enough. For $\lambda_{s}\notin(-\infty,0]$,
$(\tilde{\Delta}_{a}-\lambda_{s})^{-1}:L^{2}_{a}\to\tilde{D}_{a}$ is a
bijection of $L^{2}_{a}$ to the domain of $\tilde{\Delta}_{a}$, so
$u=\tilde{E}_{s,a}-h_{s}$ is the unique element of the domain of
$\tilde{\Delta}_{a}$ satisfying

$(\tilde{\Delta}_{a}-\lambda_{s})u=-(\Delta-\lambda_{s})h_{s}.$

The map $s\to h_{s}$ is entire, so meromorphy of the resolvent
$s\to(\tilde{\Delta}_{a}-\lambda_{s})^{-1}$ for $s\in S_{\rho}$ yields the
meromorphy of $s\to(\tilde{E}_{a,s}-h_{s})$ as a
$(\operatorname{\mathfrak{B}}^{1}\cap L^{2}_{a})$-valued function in the same
region.

Next we identify $c_{P}\tilde{E}_{a,s}$. Since
$(\tilde{\Delta}_{a}-\lambda_{s})^{-1}$ maps $(\Delta-\lambda_{s})h_{s}$ to
$L^{2}_{a}$, the constant term of $\tilde{E}_{a,s}$ is the constant term of
$h_{s}$ above $y=a$, namely $y^{s}$. More generally,

$-(\Delta-\lambda_{s})h_{s}=(\tilde{\Delta}_{a}-\lambda_{s})(\tilde{E}_{s,a}-h_{s})=(\Delta-\lambda_{s})(\tilde{E}_{a,s}-h_{s})+C_{s}\cdot\eta_{a}$

as distributions for some constant $C_{s}$. Rearranging yields a particularly
useful identity:

$(\Delta-\lambda_{s})\tilde{E}_{a,s}=-C_{s}\cdot\eta_{a}\hskip
14.22636pt(\text{as distributions}).$

Since $\Delta$ is $G$-invariant, it commutes with evaluation of the constant
term, so the distribution $(\Delta-\lambda_{s})c_{P}\tilde{E}_{a,s}$ vanishes
away from $y=a$. On $0<y<a$, the distributional differential equation

$(y^{2}\frac{\partial^{2}}{\partial y^{2}}-\lambda_{s})u=0$

has solutions exactly of the form $A_{s}y^{s}+B^{s}y^{1-s}$ for constants
$A_{s},B_{s}$, so $c_{P}\tilde{E}_{a,s}$ must be of this form in $0<y<a$.
Since $s\to\tilde{E}_{a,s}$ is meromorphic, so are $A_{s},B_{s}$. In summary,

$c_{P}\tilde{E}_{a,s}=\begin{cases}A_{s}y^{s}+B_{s}y^{1-s}&(0<y<a)\\\
y^{s}&(y>a).\end{cases}$

Furthermore, $c_{P}\tilde{E}_{a,s}$ continuous at $y=a$. By construction,
$h_{s}$ is smooth, and
$(\tilde{\Delta}_{a}-\lambda_{s})^{-1}f\in\operatorname{\mathfrak{B}}^{1}$ for
all $f\in L_{a}^{2}(\Gamma\backslash G/K)$. Thus $\tilde{E}_{a,s}$ is locally
in $\operatorname{\mathfrak{B}}^{1}$ in the sense that
$\psi\cdot\tilde{E}_{a,s}$ is in $\operatorname{\mathfrak{B}}^{1}$ for any
smooth cut-off $\psi\in C_{c}^{\infty}(\Gamma\backslash G/K)$. In fact, we can
choose a cutoff such that
$c_{P}(\psi\cdot\tilde{E}_{a,s})\in\operatorname{\mathfrak{B}}^{1}\cap
C^{o}(\Gamma\backslash G/K)$ by Sobolev imbedding on
$\operatorname{\mathbb{R}}$ [Gar18]*Section 11.5.

Meromorphic continuation follows from using another pseudo-Eisenstein series.
Let $\chi_{[a,\infty)}$ be the characteristic function of $[a,\infty)$ and

$\beta_{a,s}=\chi_{[a,\infty)}(y)\cdot(A_{s}y^{s}+B_{s}y^{1-s}-y^{s}).$

The support of $\beta_{a,s}$ is inside the set where $y\geq a$, so the
corresponding pseudo-Eisenstein series
$\Phi_{a,s}(g)=\sum_{\Gamma_{\infty}\backslash\Gamma}\beta_{a,s}(\gamma g)$
has as at most one non-zero summand for each $g\in G/K$ and therefore
converges for all $s\in\operatorname{\mathbb{C}}$. By [Gar18]*Section 11.5,
$A_{s}\cdot E_{s}=\tilde{E}_{a,s}+\Phi_{a,s}=B_{s}\cdot E_{1-s}$ for
$A_{s},B_{s}$ not identically 0. Thus, $s\to E_{s}$ meromorphically continues
as a $(h_{s}+\operatorname{\mathfrak{B}}^{1})$-valued function with functional
equation $A_{s}\cdot E_{s}=B_{s}\cdot E_{1-s}$. Certainly, this functional
equation reduces to the standard functional equation $E_{s}=c_{s}E_{1-s}$, and
one can prove that $s\to E_{s}$ takes values in the space of smooth functions
of moderate growth, by the theory of vector-valued integrals and Gelfand-
Pettis corollaries [Gar18]*Section 11.5.

###  3.5. Spacing of zeros of zeta

As appears in [BG20], exotic eigenfunction expansions also have implications
for the spacing of zeros of $\zeta_{k}(s)$ on the critical line, for
$k=\mathbb{Q}(\sqrt{d})$ quadratic fields with $d<0$.

For $a\gg_{d}1$, let $\eta_{a}$ be evaluation of the constant term at height
$a$ and $\theta$ (the restriction of) the sum of Dirac deltas at the Heegner
points attached to $k$. For $u_{w}$ the meromorphically continued solution of
$(\Delta-\lambda_{w})u_{w}=\theta$ and $v_{w}$ that of
$(\Delta-\lambda_{w})v_{w}=\eta_{a}$, a linear combination $xu_{w}+yv_{w}$ is
in $\ker\theta\cap\ker\eta_{a}$ if

$\begin{cases}0=\theta(xu_{w}+yv_{w})=\theta(u_{w})x+\theta(v_{w})y\\\
0=\eta_{a}(xu_{w}+yv_{w})=\eta_{a}(u_{w})x+\eta_{a}(v_{w})y\end{cases}$

which has non-trivial solution if and only if

$\theta(u_{w})\cdot\eta_{a}(v_{w})-\eta_{a}(u_{w})\cdot\theta(v_{w})=0.$

From the spectral expansions and pairings, by residues,

$\eta_{a}(v_{w})=\frac{a^{w}+c_{w}a^{1-w}}{1-2w}\cdot a^{1-w}.$

Since $\theta E_{w}$ is essentially $\displaystyle\zeta_{k}(w)/\zeta(2w)$,

$\eta_{a}(u_{w})=\theta(v_{w})=\frac{\theta(E_{w})a^{1-w}}{1-2w}=\frac{\zeta_{k}(w)a^{1-w}}{\zeta(2w)(1-2w)}.$

Any exotic eigenfunction of the Friedrichs extension $\tilde{T}_{a,\theta}$ of
$\Delta$ restricted to $\ker\theta\cap\ker\eta_{a}$ is in
$\ker\theta\cap\ker\eta_{a}$, so the identities above imply that
$\lambda_{w}<-1/4$ is a $\tilde{T}_{a,\theta}$-eigenvalue if and only if

$\theta(u_{w})\cdot\frac{a^{w}+c_{w}a^{1-w}}{1-2w}\cdot
a^{1-w}-\left(\frac{\zeta_{k}(w)a^{1-w}}{\zeta(2w)(1-2w)}\right)^{2}=0.$

Next consider the Friedrichs extension $\tilde{T}_{\geq a}$ of $\Delta$
restricted to the space of pseudo-Eisenstein series with test function data
and constant term vanishing at heights $y\geq a$. By a slight adaptation of
the proof for $\tilde{\Delta}_{a}$, the operator $\tilde{T}_{\geq a}$ has
purely discrete spectrum on the $L^{2}$-closure of $D_{\geq a}$, and for
$\\{f_{n}\\}$ an orthogonal basis of $\tilde{T}_{\geq a}$-eigenfunctions, each
$f_{n}\in L^{2}_{a}$ has eigenvalue $\lambda_{s_{n}}$ satisfying
$a^{s_{n}}+c_{s_{n}}a^{1-s_{n}}=0$.

Let $j:\operatorname{\mathfrak{B}}^{1}\cap
L^{2}_{a}\to\operatorname{\mathfrak{B}}^{1}$ be the inclusion map,
$j^{*}:\operatorname{\mathfrak{B}}^{-1}\to(\operatorname{\mathfrak{B}}^{1}\cap
L^{2}_{a})^{*}$ its adjoint, and $\tilde{T}^{\\#}_{\geq
a}:\operatorname{\mathfrak{B}}^{1}\cap
L^{2}_{a}\to(\operatorname{\mathfrak{B}}^{1}\cap L^{2}_{a})^{*}$ the extension
of $\tilde{T}_{\geq a}$ given by

$\tilde{T}^{\\#}_{\geq a}(u)(v)=\langle
u,\bar{v}\rangle_{\operatorname{\mathfrak{B}}^{1}}.$

Since $\theta$ is (the restriction of) a compactly supported real measure,
$\theta\in\operatorname{\mathfrak{B}}^{-1}$. There is a spectral expansion
convergent in $(\operatorname{\mathfrak{B}}^{1}\cap
L^{2}_{a})^{*}=j^{*}\operatorname{\mathfrak{B}}^{-1}$, a quotient of
$\operatorname{\mathfrak{B}}^{-1}$:

$j^{*}\theta=\sum_{n}(j^{*}\theta)(f_{n})\cdot
f_{n}=\sum_{n}\theta(jf_{n})\cdot f_{n}=\sum_{n}\theta(f_{n})\cdot f_{n}\hskip
14.22636pt(\text{convergent in }j^{*}\operatorname{\mathfrak{B}}^{-1}).$

If $(\Delta-\lambda_{w})u_{w}=\theta$ and
$u_{w}\in\operatorname{\mathfrak{B}}^{1}\cap L^{2}_{a}$, then certainly
$(\tilde{T}^{\\#}_{\geq a}-\lambda_{w})u_{w}=j^{*}\theta$. Noting that $j$
identifies $u_{w}$ with its image, this yields

$j^{*}\theta=(j^{*}\circ(\Delta-\lambda_{w})\circ
j)u_{w}=(\tilde{T}^{\\#}_{\geq a}-\lambda_{w})u_{w}.$

The latter equation can be solved by division, producing a spectral expansion

$u_{w}=\sum_{n}\frac{\theta(f_{n})}{\lambda_{s_{n}}-\lambda_{w}}\cdot
f_{n}\hskip 14.22636pt(\text{convergent in
}\operatorname{\mathfrak{B}}^{1}\cap
L^{2}_{a}\subset\operatorname{\mathfrak{B}}^{1}).$

By de-symmetrized Plancherel, the condition $\theta(u_{w})=0$ is

$0=\theta(u_{w})=\sum_{n}\frac{\theta(f_{n})^{2}}{\lambda_{s_{n}}-\lambda_{w}}\hskip
2.84526pt.$

By the intermediate value theorem, there is exactly one solution to the latter
equation between successive parameters $s_{n}$ with $\theta(f_{n})=0$, so the
equations $(\Delta-\lambda_{w})u=\theta$ and $\theta u=0$ have a solution $u$
for at most one $w$ in each interval
$\text{Im}(s_{n})\leq\text{Im}(w)\leq\text{Im}(s_{n+1})$.

For $\log\log t$ large, the spacing of the spectral parameters $s_{n}$ such
that $a^{s_{n}}+c_{s_{n}}a^{1-s_{n}}=0$ is essentially regular. Namely,
$\psi(t)=\arg\xi(1+2it)$ satisfies [Tit86]

$\psi(t)=t\log t+O\left(\frac{t\log t}{\log\log t}\right)\hskip
14.22636pt\text{and}\hskip 14.22636pt\psi^{\prime}(t)=\log t+O\left(\frac{\log
t}{\log\log t}\right).$

Thus, given $\varepsilon>0$, there is a $t_{o}$ sufficiently large such that
for $w_{1},w_{2}\in\\{\frac{1}{2}+it\in\operatorname{\mathbb{C}}:t\geq
t_{o}\\}$ such that $\lambda_{w_{1}}$, $\lambda_{w_{2}}$ are eigenvalues of
the Friedrichs extension $\tilde{T}_{\theta}$ of $\Delta$ restricted to test
functions in the kernel of $\theta$,

$\left|\text{Im}(w_{1})-\text{Im}(w_{2})\right|\geq(1-\varepsilon)\cdot\frac{\pi}{\log
t}.$

Finally, let

$J(w)=\frac{h_{d}^{2}}{-\lambda_{w}\cdot\langle 1,1\rangle}+\frac{1}{4\pi
i}\int_{(1/2)}\left|\frac{\zeta_{k}(s)}{\zeta(2s)}\right|^{2}-\left|\frac{\zeta_{k}(w)}{\zeta(2w)}\right|^{2}\frac{ds}{\lambda_{s}-\lambda_{w}},$

with $w=\frac{1}{2}+i\tau$. After re-arranging, the condition that
$\lambda_{w}$ be an eigenvalue of $\tilde{T}_{a,\theta}$ (or $\tilde{T}_{\geq
a,\theta}$) becomes

$\cos(\tau\log a+\psi(\tau))\cdot J(w)=\sin(\tau\log
a+\psi(\tau))\cdot\frac{\theta E_{1-w}\cdot\theta E_{w}}{2\tau}.$

Between any two consecutive zeros of $\cos(\tau\log a+\psi(\tau))$, there is a
unique $\tau$ such that $\lambda_{\frac{1}{2}+i\tau}$ is an eigenvalue of
$\tilde{T}_{a,\theta}$ [Tit86]. That is, half the on-line zeros $w_{o}$ of
$J(w)$, if any, repel upward the on-line zeros $s_{0}$ of $\zeta_{k}(s)$, in
the sense that, given $\varepsilon>0$, there is a sufficiently large $t_{o}$
such that above a zero $w_{0}$ of $J(w)$ with $\text{Im}(w_{0})\geq t_{o}$,
the next on-line zero $s_{0}$ of $\zeta_{k}(s)$ must satisfy

$\left|\text{Im}(s_{0})-\text{Im}(w_{0})\right|\geq(1/2-\varepsilon)\cdot\text{average
spacing}\geq(1/2-\varepsilon)\cdot\frac{\pi}{\log t}.$

##  4\. Automorphic Hamiltonians

Building on [Dir28], [Fri34], [Gro53], [LP76]*pp. 204-206, [CdV82], [BG20],
and others, we construct a Hamiltonian which discretizes
$L^{2}(\Gamma\backslash G/K)$ and will be shown to characterize a nuclear
Fréchet automorphic Schwartz space.

###  4.1. Invariant Dirac operators

Given an $r$-dimensional vector space $V$ over a field $K$ equipped with a
quadratic form $q:V\to K$, the coefficients of a (suitable) Dirac operator are
in the Clifford algebra. The Clifford algebra is a unital associative algebra
$C\ell(V,q)$ and a linear map $i:V\to C\ell(V,q)$ characterized by the
following universal property: given any unital associative algebra $A$ over
$K$ and a linear map $j:V\to A$ such that for all $v\in V$,

$j(v)^{2}=-q(v)1_{A},$

there is a unique algebra homomorphism $J:C\ell(V,q)\to A$ such that the
following diagram commutes:

${C\ell(V,q)}$${V}$${A.}$$\scriptstyle{\exists!\hskip
2.84526ptJ}$$\scriptstyle{i}$$\scriptstyle{\forall\hskip 2.84526ptj}$

Additionally, as proposed in [Che54] and [Bou59], $C\ell(V,q)$ can be
constructed as the quotient of the tensor algebra $T(V)$ over $V$ by the two-
sided ideal generated by elements of the form $v\otimes v+q(v)\cdot 1$.

To motivate examples, it is useful to note that the relation
$j(v)^{2}=-q(v)1_{A}$ in $A$ imposes orthogonality relations which describe
its structure in coordinates. Namely, for $\\{e_{i}\\}$ an orthonormal basis
of $V$ embedded in $C\ell(V)$, the relations are

$e_{i}^{2}=-1,\hskip 14.22636pte_{i}e_{j}=-e_{i}e_{j}\hskip 8.53581pt(i\neq
j).$

For example, with $q(v)=\left\lVert v\right\rVert$ the usual Euclidean metric,

$\displaystyle
C\ell(\operatorname{\mathbb{R}},q)\approx\operatorname{\mathbb{C}},\hskip
14.22636ptC\ell(\operatorname{\mathbb{R}}^{2},q)\approx\mathbb{H},$

for $\mathbb{H}$ the Hamiltonian quaternions. Alternatively, since all
positive-definite quadratic forms on an $r$-dimensional complex vector space
are equivalent to $q(z_{1},...,z_{r})=z_{1}^{2}+...+z_{r}^{2}$,

$\displaystyle
C\ell(\operatorname{\mathbb{C}},q)\approx\operatorname{\mathbb{C}}\oplus\operatorname{\mathbb{C}},\hskip
14.22636ptC\ell(\operatorname{\mathbb{C}}^{2},q)\approx\operatorname{\mathbb{C}}\otimes_{\operatorname{\mathbb{R}}}\mathbb{H},$

where $\operatorname{\mathbb{C}}\otimes_{\operatorname{\mathbb{R}}}\mathbb{H}$
are the biquaternions. In what follows, we assume the standard positive-
definite quadratic form and denote $C\ell(V,q)$ as $C\ell(V)$.

Next, we recall properties of invariant Dirac operators on Lie groups $G$. As
in Section 2.2, we offer a coordinate-free characterization which facilitates
the proof of invariance.

For $\mathfrak{g}={\mathfrak{p}}\oplus\mathfrak{k}$ a Cartan decomposition and
$\rho$ the $K$-equivariant map given below, the invariant Dirac operator is
$\mathbb{D}=\rho(\text{id}_{\mathfrak{p}})$.

Coordinate-free characterization of the invariant Dirac operator

${\operatorname{End}_{\operatorname{\mathbb{R}}}(\mathfrak{p})}$${\mathfrak{p}\otimes\mathfrak{p}^{*}}$${\mathfrak{p}\otimes\mathfrak{p}}$${C\ell(\mathfrak{p})\otimes
U\mathfrak{g}}$${\text{id}_{\mathfrak{p}}}$${\rho(\text{id}_{\mathfrak{p}})=\mathbb{D}.}$$\scriptstyle{\approx}$$\scriptstyle{\rho}$$\scriptstyle{\approx\hskip
2.84526ptvia\hskip 2.84526pt\langle\rangle}$$\scriptstyle{incl}$

This coordinate-free characterization allows us to express the invariant Dirac
operator $\mathbb{D}=\rho(\text{id}_{\mathfrak{p}})$ in terms of any
orthogonal basis $e_{1},...,e_{r}$ of $\mathfrak{p}$ embedded in
$C\ell(\mathfrak{p})$: For $\lambda_{1},...,\lambda_{r}$ the corresponding
dual basis of the dual $\mathfrak{p}^{*}$, which is characterized by
$\lambda_{i}(e_{j})=\delta_{ij}$, and $X_{1},...,X_{r}$ the dual basis for
$\mathfrak{p}$ in terms of $\langle-,-\rangle$, characterized by $\langle
e_{i},X_{j}\rangle=\delta_{ij}$, we have

${\operatorname{End}_{\operatorname{\mathbb{R}}}(\mathfrak{p})}$${\mathfrak{p}\otimes\mathfrak{p}^{*}}$${\mathfrak{p}\otimes\mathfrak{p}}$${C\ell(\mathfrak{p})\otimes
U\mathfrak{g}}$${\text{id}_{\mathfrak{p}}}$${\sum_{i}e_{i}\otimes\lambda_{i}}$${\sum_{i}e_{i}\otimes
X_{i}}$${\sum_{i}e_{i}\otimes
X_{i}=\mathbb{D}.}$$\scriptstyle{\approx}$$\scriptstyle{\rho}$$\scriptstyle{\approx\hskip
2.84526ptvia\hskip 2.84526pt\langle\rangle}$$\scriptstyle{incl}$

Since $\rho$ is $K$-equivariant and $\text{id}_{\mathfrak{p}}$ is
$K$-invariant, $\operatorname{\mathbb{D}}=\rho(\text{Id}_{\mathfrak{p}})$ is
$K$-invariant and descends to the quotient $G/K$. Furthermore, right
differentiation by $\mathfrak{g}$ commutes with left multiplication by $G$ by
associativity, so $\operatorname{\mathbb{D}}$ descends to the further quotient
$\Gamma\backslash G/K$ for $\Gamma\subseteq G$ discrete.

Furthermore, $\operatorname{\mathbb{D}}$ has the expected property:

######  Theorem 4.1.

For $\operatorname{\mathbb{D}}$ the invariant Dirac operator,
$\operatorname{\mathbb{D}}^{2}=-\Delta$ on $G/K$.

###### Proof.

The coordinate-wise orthogonality relations of the Clifford algebra
$\operatorname{C\ell}(\mathfrak{p})$ yield