# Bounds for theta sums in higher rank I111Research supported by EPSRC grant
EP/S024948/1

Jens Marklof and Matthew Welsh

(School of Mathematics, University of Bristol, Bristol BS8 1UG, U.K.
22 March 2022)

###### Abstract

Theta sums are finite exponential sums with a quadratic form in the
oscillatory phase. This paper establishes new upper bounds for theta sums in
the case of smooth and box truncations. This generalises a classic 1977 result
of Fiedler, Jurkat and Körner for one-variable theta sums and, in the multi-
variable case, improves previous estimates obtained by Cosentino and Flaminio
in 2015. Key steps in our approach are the automorphic representation of theta
functions and their growth in the cusps of the underlying homogeneous space.

###### Contents

  1. 1 Introduction
  2. 2 Heisenberg, symplectic, and Jacobi groups
    1. 2.1 Iwasawa decomposition and Haar measure
    2. 2.2 Parabolic subgroups
    3. 2.3 Schrödinger and Segal-Shale-Weil representations
  3. 3 The subgroups $\Gamma$ and $\tilde{\Gamma}$
  4. 4 Theta functions and asymptotics
  5. 5 Proof of the main theorems
    1. 5.1 Heights and volumes
    2. 5.2 Proof of theorem 1.1
    3. 5.3 Proof of theorem 1.2

##  1 Introduction

Consider the exponential sum

$\theta_{f}(M,X,\bm{x},\bm{y})=\sum_{\bm{m}\in\mathbb{Z}^{n}}f(M^{-1}(\bm{m}+\bm{x}))\;\mathrm{e}\left(\tfrac{1}{2}\,\bm{m}X\prescript{t}{}{\\!\bm{m}}+\bm{m}\prescript{t}{}{\\!\bm{y}}\right),$
(1.1)

where $f:\mathbb{R}^{n}\to\mathbb{R}$ is a rapidly decaying cut-off function,
$M\in\mathbb{R}_{>0}$, $X$ a real symmetric $n\times n$ matrix, and
$\bm{x},\bm{y}\in\mathbb{R}^{n}$ (represented as row vectors). We also use the
shorthand $\mathrm{e}(z)=\mathrm{e}^{2\pi\mathrm{i}z}$.

We refer to $\theta_{f}$ as a theta sum. If, for example
$f(\bm{x})=\exp(-\pi\bm{x}P\prescript{t}{}{\\!\bm{x}})$ for some positive
definite matrix $P$, we obtain the classical Siegel theta series

$\theta_{f}(M,X,\bm{0},\bm{y})=\sum_{\bm{m}\in\mathbb{Z}^{n}}\mathrm{e}\left(\tfrac{1}{2}\,\bm{m}Z\prescript{t}{}{\\!\bm{m}}+\bm{m}\prescript{t}{}{\\!\bm{y}}\right),$
(1.2)

with $Z=X+\mathrm{i}Y$ and $Y=M^{-2}P$. If, on the other hand,
$f=\chi_{\mathcal{B}}$ is the characteristic function of a bounded set
$\mathcal{B}\subset\mathbb{R}^{n}$ we have the finite sum

$\theta_{f}(M,X,\bm{x},\bm{y})=\sum_{\bm{m}\in\mathbb{Z}^{n}\cap(M\mathcal{B}-\bm{x})}\mathrm{e}\left(\tfrac{1}{2}\,\bm{m}X\prescript{t}{}{\\!\bm{m}}+\bm{m}\prescript{t}{}{\\!\bm{y}}\right).$
(1.3)

In this case we will also use the notation $\theta_{f}=\theta_{\mathcal{B}}$.

The following theorem, which is our first main result, gives an upper bound on
the values of theta sums in the limit of large $M$, when the truncation
function is in the class of complex-valued Schwartz functions
$\mathcal{S}(\mathbb{R}^{n})$ .

######  Theorem 1.1.

Fix $f\in\mathcal{S}(\mathbb{R}^{n})$ and let $\psi:[0,\infty)\to[1,\infty)$
be an increasing function such that series

$\sum_{k\geq 0}\psi(k)^{-(2n+2)}$ (1.4)

converges. Then there exists a subset
$\mathcal{X}(\psi)\subset\mathbb{R}^{n\times n}_{\mathrm{sym}}$ of full
Lebesgue measure such that for $M\geq 1$, $X\in\mathcal{X}(\psi)$,
$\bm{x},\bm{y}\in\mathbb{R}^{n}$ we have

$\theta_{f}(M,X,\bm{x},\bm{y})=O_{f,X}\big{(}M^{\frac{n}{2}}\psi(\log
M)\big{)}$ (1.5)

The implied constants in (1.5) are independent of $M$, $\bm{x}$ and $\bm{y}$.

This theorem follows from a geometric representation of $\theta_{f}$ as an
automorphic function and an application of a dynamical Borel-Cantelli lemma
for flows on homogeneous spaces, theorem 1.7 in [8]. The special case of
theorem 1.1 for general smooth theta sums in one variable was considered in
[14].

Upper bounds for smooth multi-variable theta sums, with an additional linear
average in $X$, have played an important role in understanding the value
distribution of quadratic forms, see for example the work of Götze [6],
Buterus, Götze, Hille and Margulis [1] and the first named author [12, 13].

The second main result of this paper deals with the subtler case when $f$ is
the characteristic function of a rectangular box $\mathcal{B}$. In this
setting Cosentino and Flaminio [3] established the bound

$\theta_{\mathcal{B}}(M,X,\bm{0},\bm{y})=O_{X,\epsilon}\big{(}M^{\frac{n}{2}}(\log
M)^{n+\frac{1}{2n+2}+\epsilon}\big{)}$ (1.6)

for the unit cube $\mathcal{B}=[0,1]^{n}$, any $\epsilon>0$ and almost every
$X$. The following theorem improves on this by a factor of $(\log M)^{n}$ and
produces a uniform bound for rectangular boxes of the form
$\mathcal{B}=(0,b_{1})\times\cdots\times(0,b_{n})$, with
$b_{i}\in\mathbb{R}_{>0}$ ranging over compacta.

######  Theorem 1.2.

Fix a compact subset $\mathcal{K}\subset\mathbb{R}_{>0}^{n}$, and choose
$\psi$ as in theorem 1.1. Then there exists a subset
$\mathcal{X}(\psi)\subset\mathbb{R}^{n\times n}_{\mathrm{sym}}$ of full
Lebesgue measure such that

$\theta_{\mathcal{B}}(M,X,\bm{x},\bm{y})=O_{X}\big{(}M^{\frac{n}{2}}\psi(\log
M)\big{)}$ (1.7)

for all $M\geq 1$, $\bm{b}=(b_{1},\ldots,b_{n})\in\mathcal{K}$,
$X\in\mathcal{X}(\psi)$, $\bm{x},\bm{y}\in\mathbb{R}^{n}$. The implied
constants are independent of $M$, $\bm{b}$, $\bm{x}$ and $\bm{y}$.

To compare this with the bound obtained in [3], note that
$\psi(t)=t^{\frac{1}{2n+2}+\epsilon}$ satisfies (1.4) and thus resulting bound
(1.7) indeed improves (1.6) by a factor of $(\log M)^{n}$. The paper [3] also
established the stronger bound

$\theta_{\mathcal{B}}(M,X,\bm{x},\bm{y})=O_{X}\big{(}M^{\frac{n}{2}}\big{)}$
(1.8)

for “bounded-type” $X$ that are badly approximable by rationals (these form a
set of measure zero), and weaker bounds for $X$ that satisfy more relaxed
Diophantine conditions. These same bounds can also be obtained from our
techniques, but with no further improvements.

In the case $n=1$ our estimate (1.7) matches the optimal results found by
Fiedler, Jurkat and Körner [5]. For $n>1$, obtaining the lower bounds in these
papers (which follow from the harder part of the Borel-Cantelli lemma) is more
subtle, and we hope to develop an approach to this elsewhere.

The bounds for the theta sum in (1.2) and (1.5) are uniform in the shift
$\bm{x}$ and the linear phase $\bm{y}$. In forthcoming work [15] we will
consider improved bounds valid for almost all
$(X,\bm{x},\bm{y})\subset\mathbb{R}^{n\times
n}_{\mathrm{sym}}\times\mathbb{R}^{n}\times\mathbb{R}^{n}$, generalising the
results for $n=1$ found in [4].

This paper is organised as follows. We begin in section 2 by recalling some
basic facts about the Heisenberg group and symplectic group
$\mathrm{Sp}(n,\mathbb{R})$ as well as their semi-direct product, the Jacobi
group, including the Iwasawa decomposition, Haar measure, and parabolic
subgroups. We then review the Schrödinger and Segal-Shale-Weil representations
of the Heisenberg and symplectic group, respectively. Following the method of
[11], these representations are used to define theta functions in section 4.

The theta functions satisfy an automorphy condition on a certain, morally-
speaking discrete subgroup of the Jacobi group. This subgroup is discussed in
section 3. Its projection to the symplectic group is just the integral
symplectic group $\mathrm{Sp}(n,\mathbb{Z})$. The bulk of section 3 concerns a
fundamental domain (a slight modification of Siegel’s classic fundamental
domain [18] based on the work of [7]) and its properties.

In section 4 we define the theta functions and state their automorphy
properties before analysing their asymptotic behaviour. While for the proof of
theorems 1.1 and 1.2 we only need the upper bound contained in corollary 4.5,
the full asymptotics contained in theorem 4.4 may be of independent interest.
The proof of theorem 4.4 combines the properties of the fundamental domain
constructed in section 3 and basic estimates for sums over integers together
with the Langlands decompositions of the maximal parabolic subgroups of the
symplectic group.

We prove theorems 1.1 and 1.2 in section 5. Apart from the upper bound in
corollary 4.5, our method relies on upper bounds for the measure of rapidly
diverging orbits under a particular one-parameter diagonal action in the
symplectic group as well as (for theorem 1.2) a resolution of the singular
cutoff function in (1.3) using an $n$-parameter diagonal action. The estimates
for the first part are largely based on the easy part of the proof of theorem
1.7 in [8], which is also a main input into the method in [3]. The
complications arising from the $n$-parameter flow however prevent a
straightforward application of this theorem, so we instead proceed more
directly with a self-contained proof.

##  2 Heisenberg, symplectic, and Jacobi groups

We define the ($2n+1$)-dimensional Heisenberg group $H$ to be the set
$\mathbb{R}^{n}\times\mathbb{R}^{n}\times\mathbb{R}$ with multiplication given
by

$(\bm{x}_{1},\bm{y}_{1},t_{1})(\bm{x}_{2},\bm{y}_{2},t_{2})=\big{(}\bm{x}_{1}+\bm{x}_{2},\bm{y}_{1}+\bm{y}_{2},t_{1}+t_{2}+\tfrac{1}{2}(\bm{y}_{1}\prescript{t}{}{\\!\bm{x}_{2}}-\bm{x}_{1}\prescript{t}{}{\\!\bm{y}_{2}})\big{)}.$
(2.1)

The rank $n$ symplectic group $G=\mathrm{Sp}(n,\mathbb{R})$ is defined by

$G=\\{g\in\mathrm{GL}(2n,\mathbb{R}):gJ_{0}\prescript{t}{}{\\!g}=J_{0}\\}$
(2.2)

where

$J_{0}=\begin{pmatrix}0&-I\\\ I&0\end{pmatrix}$ (2.3)

with $I$ the $n\times n$ identity matrix. We have the alternative
characterization

$G=\left\\{\begin{pmatrix}A&B\\\
C&D\end{pmatrix}:A\prescript{t}{}{\\!B}=B\prescript{t}{}{\\!A},\
C\prescript{t}{}{\\!D}=D\prescript{t}{}{\\!C},\
A\prescript{t}{}{\\!D}-B\prescript{t}{}{\\!C}=I\right\\}.$ (2.4)

The group $G$ acts by on $H$ via

$(\bm{x},\bm{y},t)^{g}=(\bm{x}A+\bm{y}C,\bm{x}B+\bm{y}D,t)$ (2.5)

where

$g=\begin{pmatrix}A&B\\\ C&D\end{pmatrix}.$ (2.6)

Since $g$ preserves the symplectic form $J_{0}$ used to define the
multiplication (2.1), this action is by automorphisms, i.e.
$(h_{1}h_{2})^{g}=h_{1}^{g}h_{2}^{g}$. We define the semi-direct product group
$H\rtimes G$, called the Jacobi group, to be the set of all $(h,g)$, $h\in G$
and $g\in G$, with multiplication given by

$(h_{1},g_{1})(h_{2},g_{2})=(h_{1}h_{2}^{g_{1}^{-1}},g_{1}g_{2}).$ (2.7)

###  2.1 Iwasawa decomposition and Haar measure

The intersection $K=G\cap\mathrm{O}(2n)$ is a maximal compact subgroup of $G$
and

$Q\mapsto k(Q)=\begin{pmatrix}\mathrm{Re}(Q)&-\mathrm{Im}(Q)\\\
\mathrm{Im}(Q)&\mathrm{Re}(Q)\end{pmatrix}$ (2.8)

defines an isomorphism from the unitary group $\mathrm{U}(n)$ to $K$. The
Iwasawa decomposition of $G$ with respect to $K$ implies that any $g\in G$ can
be written uniquely as

$g=\begin{pmatrix}A&B\\\ C&D\end{pmatrix}=\begin{pmatrix}I&X\\\
0&I\end{pmatrix}\begin{pmatrix}Y^{\frac{1}{2}}&0\\\
0&\prescript{t}{}{Y}^{-\frac{1}{2}}\end{pmatrix}k(Q)$ (2.9)

where $X$ and $Y$ are symmetric, $Y$ is positive definite, and
$Q\in\mathrm{U}(n)$. Here we have chosen $Y^{\frac{1}{2}}$ to by upper-
triangular with positive diagonal entries, and we often further decompose
$Y=UV\prescript{t}{}{U}$ with $U$ upper-triangular unipotent and $V$ positive
diagonal. We also note that $Y^{-\frac{1}{2}}$ is always interpreted as
$(Y^{\frac{1}{2}})^{-1}$, not $(Y^{-1})^{\frac{1}{2}}$. We make frequent use
of the following expressions for the $X$, $Y$, and $Q$ coordinates,

$\displaystyle Y$
$\displaystyle=(C\prescript{t}{}{\\!C}+D\prescript{t}{}{\\!D})^{-1}$
$\displaystyle X$
$\displaystyle=(A\prescript{t}{}{\\!C}+B\prescript{t}{}{\\!D})(C\prescript{t}{}{\\!C}+D\prescript{t}{}{\\!D})^{-1}$
$\displaystyle Q$
$\displaystyle=(C\prescript{t}{}{\\!C}+D\prescript{t}{}{\\!D})^{-\frac{1}{2}}(D+iC),$
(2.10)

where as before
$(C\prescript{t}{}{\\!C}+D\prescript{t}{}{\\!D})^{\frac{1}{2}}$ is chosen to
be upper-triangular with positive diagonal entries.

The Haar measure on $G$ can be easily expressed in terms of the Iwasawa
decomposition. For

$g=\begin{pmatrix}I&X\\\ 0&I\end{pmatrix}\begin{pmatrix}UV^{\frac{1}{2}}&0\\\
0&\prescript{t}{}{U}^{-1}V^{-\frac{1}{2}}\end{pmatrix}k(Q),$ (2.11)

the Haar measure $\mu$ on $G$ is given by

$\differential\mu(g)=\left(\prod_{1\leq i\leq j\leq n}\differential
x_{ij}\right)\left(\prod_{1\leq i<j\leq n}\differential
u_{ij}\right)\left(\prod_{1\leq j\leq n}v_{jj}^{-n+j-2}\differential
v_{jj}\right)\differential Q.$ (2.12)

Here $\differential Q$ denotes the Haar measure on $\mathrm{U}(n)$ and
$\differential x_{ij}$, $\differential u_{ij}$, $\differential v_{jj}$ are
respectively the Lebesgue measures on the entries of $X$, $U$, $V$.

We note that if $g=\begin{pmatrix}A&B\\\ C&D\end{pmatrix}$ with $D$
invertible, then we can write

$g=\begin{pmatrix}I&BD^{-1}\\\
0&I\end{pmatrix}\begin{pmatrix}\prescript{t}{}{\\!D}^{-1}&0\\\
0&D\end{pmatrix}\begin{pmatrix}I&0\\\ D^{-1}C&I\end{pmatrix}.$ (2.13)

Therefore the set of $g\in G$ having the form

$g=\begin{pmatrix}I&X\\\ 0&I\end{pmatrix}\begin{pmatrix}A&0\\\
0&\prescript{t}{}{\\!A}^{-1}\end{pmatrix}\begin{pmatrix}I&0\\\
T&I\end{pmatrix}$ (2.14)

for $X$, $T$ symmetric and $A\in\mathrm{GL}(n,\mathbb{R})$ is open and dense
in $G$. We claim that in these coordinates we have, up to multiplication by a
positive constant,

$\differential\mu(g)=(\det A)^{-2n-1}\left(\prod_{i\leq j}\differential
x_{ij}\right)\left(\prod_{i,j}\differential a_{ij}\right)\left(\prod_{i\leq
j}\differential t_{ij}\right).$ (2.15)

where $\differential x_{ij}$, $\differential a_{ij}$, $\differential t_{ij}$
are the Lebesgue measure on the entries of $X$, $A$, $T$.

To verify (2.15) up to a positive constant it suffices to check that the right
side is invariant under left multiplication by generators of $G$. The
invariance under matrices $\begin{pmatrix}I&X_{1}\\\ 0&I\end{pmatrix}$ with
$X_{1}$ symmetric is obvious, and the invariance under matrices
$\begin{pmatrix}A_{1}&0\\\ 0&\prescript{t}{}{\\!A}_{1}^{-1}\end{pmatrix}$
follows from

$\begin{pmatrix}A_{1}&0\\\
0&\prescript{t}{}{\\!A}_{1}^{-1}\end{pmatrix}\begin{pmatrix}I&X\\\
0&I\end{pmatrix}\begin{pmatrix}A&0\\\
0&\prescript{t}{}{\\!A}^{-1}\end{pmatrix}\\\
=\begin{pmatrix}I&A_{1}X\prescript{t}{}{\\!A}_{1}\\\
0&I\end{pmatrix}\begin{pmatrix}A_{1}A&0\\\
0&\prescript{t}{}{\\!A}_{1}^{-1}\prescript{t}{}{\\!A}^{-1}\end{pmatrix}$
(2.16)

and that the replacements $X\leftarrow
A_{1}^{-1}X\prescript{t}{}{\\!A}_{1}^{-1}$, $A\leftarrow A_{1}^{-1}A$ change

$\prod_{i\leq j}\differential x_{ij}\leftarrow(\det A_{1})^{-n-1}\prod_{i\leq
j}\differential x_{ij},\quad\prod_{i,j}\differential a_{ij}\leftarrow(\det
A_{1})^{-n}\prod_{i,j}\differential a_{ij}.$ (2.17)

To verify the invariance under $\begin{pmatrix}0&-I\\\ I&0\end{pmatrix}$ we
may restrict further to the set of $g$ of the form (2.14) with $X$ invertible,
as this is still an open, dense set. We then have

$\begin{pmatrix}0&-I\\\ I&0\end{pmatrix}\begin{pmatrix}I&X\\\
0&I\end{pmatrix}\begin{pmatrix}A&0\\\
0&\prescript{t}{}{\\!A}^{-1}\end{pmatrix}\begin{pmatrix}I&0\\\
T&I\end{pmatrix}\\\ =\begin{pmatrix}I&-X^{-1}\\\
0&I\end{pmatrix}\begin{pmatrix}X^{-1}A&0\\\
0&\prescript{t}{}{\\!X}\prescript{t}{}{\\!A}^{-1}\end{pmatrix}\begin{pmatrix}I&0\\\
T+\prescript{t}{}{\\!A}X^{-1}A&I\end{pmatrix}.$ (2.18)

The invariance then follows from the fact that the replacement $X\leftarrow
X^{-1}$ changes

$\prod_{i\leq j}\differential x_{ij}\leftarrow(\det X)^{-n-1}\prod_{i\leq
j}\differential x_{ij}.$ (2.19)

###  2.2 Parabolic subgroups

We recall that conjugacy classes of parabolic subgroups of $G$ are in
bijection with subsets of the $n$ positive simple roots, see for example
section 4.5.3 of [19]. Here we make the choice of positive simple roots
$\alpha_{1},\dots,\alpha_{n}$ where, for $1\leq l<n$,

$\alpha_{l}\begin{pmatrix}A&0\\\ 0&A^{-1}\end{pmatrix}=a_{l}a_{l+1}^{-1}$
(2.20)

and

$\alpha_{n}\begin{pmatrix}A&0\\\ 0&A^{-1}\end{pmatrix}=a_{n}^{2}.$ (2.21)

Here

$A=\begin{pmatrix}a_{1}&\cdots&0\\\ \vdots&\ddots&\vdots\\\
0&\cdots&a_{n}\end{pmatrix}$ (2.22)

is positive diagonal. See for example section 5.1 of [19].

The parabolic corresponding to a subset
$L\subset\\{\alpha_{1},\dots,\alpha_{n}\\}$ is given by

$P_{L}=N\bigcap_{\alpha\in L}Z(\mathrm{ker}(\alpha))$ (2.23)

where $Z(\mathrm{ker}(\alpha))$ is the centraliser in $G$ of the kernel of the
root $\alpha$ and

$N=\bigg{\\{}\begin{pmatrix}U&X\prescript{t}{}{U}^{-1}\\\
0&\prescript{t}{}{U}^{-1}\end{pmatrix}:\\\ U\ \mathrm{upper\ triangular\
unipotent},\ X\ \mathrm{symmetric}\bigg{\\}}.$ (2.24)

The maximal parabolic subgroups correspond to subsets $L$ of size $n-1$ and we
denote them by $P_{l}$, $1\leq l\leq n$ corresponding to root $\alpha_{l}$ not
in the set $L$. For $1\leq l<n$, we write an arbitrary element of $P_{l}$ as

$\begin{pmatrix}I&R_{l}&T_{l}-S_{l}\prescript{t}{}{\\!R}_{l}&S_{l}\\\
0&I&\prescript{t}{}{\\!S}_{l}&0\\\ 0&0&I&0\\\
0&0&-\prescript{t}{}{\\!R}_{l}&I\end{pmatrix}\begin{pmatrix}a_{l}I&0&0&0\\\
0&I&0&0\\\ 0&0&a_{l}^{-1}I&0\\\
0&0&0&I\end{pmatrix}\begin{pmatrix}U_{l}&0&0&0\\\ 0&A_{l}&0&B_{l}\\\
0&0&\prescript{t}{}{U}_{l}^{-1}&0\\\ 0&C_{l}&0&D_{l}\end{pmatrix}$ (2.25)

where $R_{l}$ and $S_{l}$ are $l\times(n-l)$ matrices, $T_{l}$ is $l\times l$
symmetric, $a_{l}>0$, $U_{l}\in\mathrm{GL}(l,\mathbb{R})$ with $\det U_{l}=\pm
1$, and $g_{l}=\begin{pmatrix}A_{l}&B_{l}\\\
C_{l}&D_{l}\end{pmatrix}\in\mathrm{Sp}(n-l,\mathbb{R})$. For $l=n$, we write
an arbitrary element of $P_{n}$ as

$\begin{pmatrix}I&T_{n}\\\ 0&I\end{pmatrix}\begin{pmatrix}a_{n}I&0\\\
0&a_{n}^{-1}I\end{pmatrix}\begin{pmatrix}U_{n}&0\\\
0&\prescript{t}{}{\\!U_{n}}^{-1}\end{pmatrix}$ (2.26)

where $T_{n}$ is $n\times n$ symmetric, $a_{n}>0$, and
$U_{n}\in\mathrm{GL}(n,\mathbb{R})$ with $\det U_{n}=\pm 1$. The
factorizations (2.25), (2.26) are in fact the Langlands decompositions of
$P_{l}$, $P_{n}$, which write an arbitrary element of the parabolic subgroup
as a product of elements of a nilpotent subgroup, a diagonal subgroup, and a
semi-simple subgroup. For general considersations regarding the Langlands
decomposition, see section 7.7 of [10]. The author’s lecture notes [16]
contain explicit calculations for the symplectic group $G$ along these lines.

###  2.3 Schrödinger and Segal-Shale-Weil representations

The Schrödinger representation $W$ of $H$ acts on $L^{2}(\mathbb{R}^{n})$ by
the unitary transformations

$W(\bm{x},\bm{y},t)f(\bm{x}_{0})=\mathrm{e}(-t+\tfrac{1}{2}\bm{x}\prescript{t}{}{\\!\bm{y}}+\bm{x}_{0}\prescript{t}{}{\bm{y}})f(\bm{x}_{0}+\bm{x}).$
(2.27)

We remark that this definition of the Schrödinger representation differs
slightly from the conventional one; they are of course unitarily equivalent.

Given $g\in G$, we obtain another representation $W^{g}$ of $H$ by
$W^{g}(h)=W(h^{g})$. By the Stone-von Neumann theorem, there exists unitary
operators $R(g)$ on $L^{2}(\mathbb{R}^{n})$ such that

$W^{g}=R(g)^{-1}WR(g).$ (2.28)

The relation (2.28) actually defines $R(g)$ up to a scalar multiple.
Regardless of the choice of this scalar (which we make below), we have

$R(g_{1}g_{2})=\rho(g_{1},g_{2})R(g_{1})R(g_{2})$ (2.29)

for a nontrivial, unitary cocycle $\rho:G\times G\to\mathbb{C}$. Thus $R$
defines a projective representation of $G$, which is called the Segal-Shale-
Weil representation. The projective representation $R$ can be extended to a
true representation of the metaplectic group – the simply connected double
cover of $G$, but we do not make use of this construction.

The following proposition gives expressions for $R(g)$ for certain $g$ and on
a dense subset of $L^{2}(\mathbb{R}^{n})$. In particular the proposition makes
precise the choice of scalar multiple in our definition of $R$.

######  Proposition 2.1.

Let $f\in\mathcal{S}(\mathbb{R}^{n})$. Then for

$g=\begin{pmatrix}A&B\\\ 0&\prescript{t}{}{\\!A}^{-1}\end{pmatrix},$ (2.30)

we have

$R(g)f(\bm{x})=|\det
A|^{\frac{1}{2}}\mathrm{e}\left(\frac{1}{2}\bm{x}A\prescript{t}{}{\\!B}\prescript{t}{}{\\!\bm{x}}\right)f(\bm{x}A),$
(2.31)

and for

$g=J_{l}=\begin{pmatrix}I&0&0&0\\\ 0&0&0&-I\\\ 0&0&I&0\\\
0&I&0&0\end{pmatrix},$ (2.32)

with square blocks of size $l$, $n-l$, $l$, and $n-l$ along the diagonal,
$0\leq l\leq n$, we have

$R(g)f(\bm{x})=\int_{\mathbb{R}^{n-l}}f(\bm{x}^{(1)},\bm{y}^{(2)})\mathrm{e}(-\bm{x}^{(2)}\prescript{t}{}{\\!\bm{y}}^{(2)})\differential\bm{y}^{(2)}$
(2.33)

with $\bm{x}=\begin{pmatrix}\bm{x}^{(1)}&\bm{x}^{(2)}\end{pmatrix}$. Moreover,
for $g$ of the form (2.30) and any $g^{\prime}\in G$, we have
$\rho(g,g^{\prime})=\rho(g^{\prime},g)=1$.

This proposition is a summary of various calculations found in [11]. The
forthcoming lecture notes by the authors [16] will give self-contained proofs.

We remark that together with the Bruhat decomposition

$G=\bigcup_{0\leq l\leq n}P_{n}J_{l}P_{n}$ (2.34)

where $P_{n}$ is the maximal parabolic subgroup (2.26), proposition 2.1 allows
one to compute $R(g)$ for any $g\in G$. For example, for

$g=\begin{pmatrix}A&B\\\ C&D\end{pmatrix}=\begin{pmatrix}I&AC^{-1}\\\
0&I\end{pmatrix}\begin{pmatrix}0&-I\\\ I&0\end{pmatrix}\begin{pmatrix}C&D\\\
0&\prescript{t}{}{C}^{-1}\end{pmatrix}$ (2.35)

with $C$ invertible, we have

$R(g)f(\bm{x})=|\det
C|^{-\frac{1}{2}}\mathrm{e}\left(\frac{1}{2}\bm{x}AC^{-1}\prescript{t}{}{\\!\bm{x}}\right)\\\
\int_{\mathbb{R}^{n}}f(\bm{y})\mathrm{e}\left(\frac{1}{2}\bm{y}C^{-1}D\prescript{t}{}{\\!\bm{y}}-\bm{x}\prescript{t}{}{\\!C}^{-1}\prescript{t}{}{\\!\bm{y}}\right)\differential\bm{y}.$
(2.36)

##  3 The subgroups $\Gamma$ and $\tilde{\Gamma}$

We denote by $\Gamma$ the discrete subgroup
$\Gamma=\mathrm{Sp}(n,\mathbb{Z})\subset G$. For

$\gamma=\begin{pmatrix}A&B\\\ C&D\end{pmatrix}\in\Gamma,$ (3.1)

we set $h_{\gamma}=(\bm{r},\bm{s},0)\in H$ where the entries or $\bm{r}$ are
$0$ or $\frac{1}{2}$ depending on whether the corresponding diagonal entry of
$C\prescript{t}{}{D}$ is even or odd, and the entries of $\bm{s}$ are $0$ or
$\frac{1}{2}$ depending on whether the corresponding diagonal entry of
$A\prescript{t}{}{B}$ is even or odd. We now define the group
$\tilde{\Gamma}\subset H\rtimes G$ by

$\tilde{\Gamma}=\\{((\bm{m},\bm{n},t)h_{\gamma},\gamma)\in H\rtimes
G:\gamma\in\Gamma,\bm{m}\in\mathbb{Z}^{n},\bm{n}\in\mathbb{Z}^{n},t\in\mathbb{R}\\}.$
(3.2)

We note that this is a subgroup of $G$ because, modulo left multiplication by
elements $(\bm{m},\bm{n},t)\in H$ with $\bm{m},\bm{n}\in\mathbb{Z}^{n}$ and
$t\in\mathbb{R}$, we have
$h_{\gamma_{1}\gamma_{2}}=h_{\gamma_{1}}h_{\gamma_{2}}^{\gamma_{1}^{-1}}$ for
any $\gamma_{1},\gamma_{2}\in\Gamma$. Indeed, we have that $2\bm{r}$ and
$2\bm{s}$, where $h_{\gamma_{1}\gamma_{2}}=(\bm{r},\bm{s},0)$, has the same
parity as the diagonal entries of

$(C_{1}A_{2}+D_{1}C_{2})\prescript{t}{}{(C_{1}B_{2}+D_{1}D_{2})}$ (3.3)

and

$(A_{1}A_{2}+B_{1}D_{2})\prescript{t}{}{(A_{1}B_{2}+B_{1}D_{2})},$ (3.4)

which, in view of $A_{2}\prescript{t}{}{D}_{2}-B_{2}\prescript{t}{}{C_{2}}=I$,
have the same parity as the diagonal entries of

$C_{1}A_{2}\prescript{t}{}{B}_{2}\prescript{t}{}{C}_{1}+D_{1}C_{2}\prescript{t}{}{D}_{2}\prescript{t}{}{D}_{1}+C_{1}\prescript{t}{}{D}_{1}$
(3.5)

and

$A_{1}A_{2}\prescript{t}{}{B}_{2}\prescript{t}{}{A}_{1}+B_{1}C_{2}\prescript{t}{}{D}_{2}\prescript{t}{}{B}_{1}+A_{1}\prescript{t}{}{B}_{1}.$
(3.6)

On the other hand, we have

$h_{\gamma_{1}}h_{\gamma_{2}}^{\gamma_{1}^{-1}}=\left(\bm{r}_{1}+\bm{r}_{2}\prescript{t}{}{D}_{1}-\bm{s}_{2}\prescript{t}{}{C}_{1},\bm{s}_{1}-\bm{r}_{2}\prescript{t}{}{B}_{1}+\bm{s}_{2}\prescript{t}{}{A}_{1},*\right),$
(3.7)

where $\gamma_{\gamma_{j}}=(\bm{r}_{j},\bm{s}_{j},0)$. The entries of two
times the vectors on the right of (3.7) have the same parity as the diagonal
entries of (3.5) and (3.6), as claimed.

We say that a closed set $\mathcal{D}\subset G$ is a fundamental domain for
$\Gamma\backslash G$ if

  1. 1.

for all $g\in G$ there exists $\gamma\in\Gamma$ such that $\gamma
g\in\mathcal{D}$ and

  2. 2.

if for $g\in\mathcal{D}$ there is a non-identity $\gamma\in\Gamma$ such that
$\gamma g\in\mathcal{D}$, then $g$ is contained in the boundary of
$\mathcal{D}$.

Following Siegel [18], we define $\mathcal{D}$ to be the set of all

$g=\begin{pmatrix}I&X\\\ 0&I\end{pmatrix}\begin{pmatrix}Y^{\frac{1}{2}}&0\\\
0&\prescript{t}{}{Y}^{-\frac{1}{2}}\end{pmatrix}k(Q)\in G$ (3.8)

such that

  1. 1.

$|\det(C(X+\mathrm{i}Y)+D)|\geq 1$ for all $\begin{pmatrix}A&B\\\
C&D\end{pmatrix}\in\Gamma$,

  2. 2.

$Y\in\mathcal{D}^{\prime}$, a fundamental domain for the action of
$\mathrm{GL}(n,\mathbb{Z})$ on $n\times n$ positive symmetric matrices, and

  3. 3.

$|x_{ij}|\leq\frac{1}{2}$, where $x_{ij}$ are the entries of $X$.

We note that since (2.1) implies that

$Y(\gamma
g)=\prescript{t}{}{\\!(C(X-\mathrm{i}Y)+D)}^{-1}Y(C(X+\mathrm{i}Y)+D)^{-1},$
(3.9)

the first condition implies that for $g\in\mathcal{D}$,
$|\det(Y(g))|\geq|\det(Y(\gamma g))|$ for all $\gamma\in\Gamma$. We also note
that Siegel chooses $\mathcal{D}^{\prime}$ to be the set of positive definite
symmetric $Y$ such that $Y^{-1}$ is in Minkowski’s classical fundamental
domain. However, here we choose $\mathcal{D}^{\prime}$ to be the set of $Y$
such that $Y^{-1}$ is in Grenier’s fundamental domain, see [7] and [19].

Following [7] and [19], we define
$\mathcal{D}^{\prime}=\mathcal{D}^{\prime}_{n}$ recursively as follows. We set
$\mathcal{D}_{1}^{\prime}=\\{y>0\\}$ and

$\mathcal{D}^{\prime}_{2}=\bigg{\\{}\begin{pmatrix}1&r_{1}\\\
0&1\end{pmatrix}\begin{pmatrix}v_{1}&0\\\
0&v_{2}\end{pmatrix}\begin{pmatrix}1&0\\\ r_{1}&1\end{pmatrix}:\\\
r_{1}^{2}+\frac{v_{1}}{v_{2}}\geq 1,\ 0\leq u\leq\frac{1}{2},\
v_{1},v_{2}>0\bigg{\\}},$ (3.10)

the standard fundamental domain for $\mathrm{GL}(2,\mathbb{Z})$. For $n>2$ we
define $\mathcal{D}_{n}^{\prime}$ to be the set of

$Y=\begin{pmatrix}1&\bm{r}_{1}\\\ 0&I\end{pmatrix}\begin{pmatrix}v_{1}&0\\\
0&Y_{1}\end{pmatrix}\begin{pmatrix}1&0\\\
\prescript{t}{}{\\!\bm{r}_{1}}&I\end{pmatrix}$ (3.11)

such that

  1. 1.

$v_{1}=v_{1}(Y)\geq v_{1}(AY\prescript{t}{}{\\!A})$ for all
$A\in\mathrm{GL}(n,\mathbb{Z})$,

  2. 2.

$Y_{1}(Y)\in\mathcal{D}_{n-1}^{\prime}$, and

  3. 3.

$|r_{j}|\leq\frac{1}{2}$ and $0\leq r_{1}\leq\frac{1}{2}$, where $r_{j}$ are
the entries of $\bm{r}_{1}$.

This is proven to be a fundamental domain in [7] and [19]. In general, the
motivation for using this fundamental domain is the box-shaped cusp, but here
the primary advantage is its recursive definition, which we make frequent use
of below. We remark that one can construct a fundamental domain for
$\Gamma\backslash G$ with a box-shaped cusp by maximising $v_{1}$ over all of
$\Gamma$, not just $\mathrm{GL}(n,\mathbb{Z})$. This approach is utilised in
the second paper in this series [15]. However we do not need this feature
here, and in fact maximising the determinant in the fundamental domain as we
have done is useful in what follows, see the proofs of lemmas 5.2 and 5.2.

The following proposition records some useful properties of $\mathcal{D}$.

######  Proposition 3.1.

Let $g\in\mathcal{D}$ and write

$g=\begin{pmatrix}I&X\\\ 0&I\end{pmatrix}\begin{pmatrix}Y^{\frac{1}{2}}&0\\\
0&Y^{-\frac{1}{2}}\end{pmatrix}k(Q),\quad Y=\begin{pmatrix}1&\bm{r}_{1}\\\
0&I\end{pmatrix}\begin{pmatrix}v_{1}&0\\\
0&Y_{1}\end{pmatrix}\begin{pmatrix}1&0\\\
\prescript{t}{}{\\!\bm{r}_{1}}&I\end{pmatrix},$ (3.12)

and also

$Y=UV\prescript{t}{}{U},\quad V=\begin{pmatrix}v_{1}&\cdots&0\\\
\vdots&\ddots&\vdots\\\ 0&\cdots&v_{n}\end{pmatrix}.$ (3.13)

Then we have

  1. 1.

$v_{n}\geq\frac{\sqrt{3}}{2}$ and $v_{j}\geq\frac{3}{4}v_{j+1}$ for $1\leq
j\leq n-1$ and

  2. 2.

for all
$\bm{x}=\begin{pmatrix}x^{(1)}&\bm{x}^{(2)}\end{pmatrix}\in\mathbb{R}^{n}$,

$\bm{x}Y\prescript{t}{}{\\!\bm{x}}\asymp_{n}v_{1}(x^{(1)})^{2}+\bm{x}^{(2)}Y_{1}\prescript{t}{}{\\!\bm{x}}^{(2)}.$
(3.14)

###### Proof.

We apply $|\det(C(X+\mathrm{i}Y)+D)|^{2}\geq 1$ for

$C=\begin{pmatrix}0&\cdots&0&0\\\ \vdots&\ddots&\vdots&\vdots\\\
0&\cdots&0&0\\\ 0&\cdots&0&1\end{pmatrix},\quad
D=\begin{pmatrix}1&\cdots&0&0\\\ \vdots&\ddots&\vdots&\vdots\\\
0&\cdots&1&0\\\ 0&\cdots&0&0\end{pmatrix},$ (3.15)

which may be completed to $\begin{pmatrix}A&B\\\ C&D\end{pmatrix}\in\Gamma$
with $A=D$ and $B=C$. We have

$|\det(C(X+\mathrm{i}Y)+D)|^{2}=x_{nn}^{2}+v_{n}^{2},$ (3.16)

where $x_{nn}$ is the $(n,n)$ entry of $X$. Since the entries of $X$ are at
most a half in absolute value, $v_{n}^{2}\geq 1-x_{nn}^{2}\geq\frac{3}{4}$ as
required.

We have

$Y=\begin{pmatrix}1&\bm{r}_{1}\\\ 0&I\end{pmatrix}\begin{pmatrix}v_{1}&0\\\
0&Y_{1}\end{pmatrix}\begin{pmatrix}1&0\\\
-\bm{r}_{1}&I\end{pmatrix}\in\mathcal{D}^{\prime}$ (3.17)

and we note that to demonstrate $v_{j}\geq\frac{3}{4}v_{j+1}$, it suffices to
consider $j=1$ by the inductive construction of $\mathcal{D}^{\prime}$. We
apply the minimality of $v_{1}^{-1}$ for an element
$\gamma\in\mathrm{GL}(n,\mathbb{Z})$ having first row
$\begin{pmatrix}0&1&0&\cdots&0\end{pmatrix}$. We find that

$v_{1}^{-1}\leq v_{1}^{-1}r^{2}+v_{2}^{-1},$ (3.18)

where $r$ is the first entry of $\bm{r}_{1}$. Since $|r|\leq\frac{1}{2}$, it
follows that $v_{1}\geq\frac{3}{4}v_{2}$.

To demonstrate the second part of the proposition, we let
$\bm{y}_{1},\dots,\bm{y}_{n}$ denote the rows of

$Y^{\frac{1}{2}}=\begin{pmatrix}1&\bm{r}_{1}\\\
0&I\end{pmatrix}\begin{pmatrix}v_{1}^{\frac{1}{2}}&0\\\
0&Y_{1}^{\frac{1}{2}}\end{pmatrix}.$ (3.19)

Setting $\bm{y}=x_{2}\bm{y}_{2}+\cdots+x_{n}\bm{y}_{n}$, where the $x_{j}$ are
the entries of $\bm{x}$, our aim is to prove that for some constants
$0<c_{1}<1<c_{2}$ depending only on $n$,

$c_{1}\left(||\bm{y}_{1}||^{2}x_{1}^{2}+||\bm{y}||^{2}\right)\leq||x_{1}\bm{y}_{1}+\bm{y}||^{2}\leq
c_{2}\left(||\bm{y}_{1}||^{2}x_{1}^{2}+||\bm{y}||^{2}\right),$ (3.20)

from which the lower bound in (3.14) follows as $||\bm{y}_{1}||^{2}\geq
v_{1}$. The upper bound in (3.14) follows from $v_{1}\gg||\bm{y}_{1}||^{2}$,
which is verified below, see (3.26). Expanding the expression in the middle of
(3.20), we find that it is enough to show that

$2|x_{1}\bm{y}_{1}\prescript{t}{}{\\!\bm{y}}|\leq(1-c_{1})\left(||\bm{y}_{1}||^{2}x_{1}^{2}+||\bm{y}||^{2}\right),$
(3.21)

and

$2|x_{1}\bm{y}_{1}\prescript{t}{}{\\!\bm{y}}|\leq(c_{2}-1)\left(||\bm{y}_{1}||^{2}x_{1}^{2}+||\bm{y}||^{2}\right).$
(3.22)

The upper bound (3.22) is trivial if $c_{2}=2$, and the upper bound (3.21)
would follow from

$|\bm{y}_{1}\prescript{t}{}{\\!\bm{y}}|\leq(1-c_{1})||\bm{y}_{1}||\;||\bm{y}||.$
(3.23)

We let $0<\phi_{1}<\pi$ denote the angle between $\bm{y}_{1}$ and $\bm{y}$ and
$0<\phi_{2}<\frac{\pi}{2}$ denote the angle between $\bm{y}_{1}$ and the
hyperplane $\mathrm{span}(\bm{y}_{2},\dots,\bm{y}_{n})$. We have
$\phi_{2}\leq\mathrm{min}(\phi_{1},\pi-\phi_{1})$, and so
$|\cos\phi_{1}|\leq|\cos\phi_{2}|$. We bound $\cos\phi_{2}$ away from $1$ by
bounding $\sin\phi_{2}$ away from $0$.

We have

$|\sin\phi_{2}|=\frac{||\bm{y}_{1}\wedge\cdots\wedge\bm{y}_{n}||}{||\bm{y}_{1}||\;||\bm{y}_{2}\wedge\cdots\wedge\bm{y}_{n}||}=\frac{v_{1}^{\frac{1}{2}}}{||\bm{y}_{1}||},$
(3.24)

so it suffices to show that $v_{1}^{\frac{1}{2}}\gg||\bm{y}_{1}||$. Here
$\wedge$ denotes the usual wedge product on $\mathbb{R}^{n}$ and the norm on
$\bigwedge^{k}\mathbb{R}^{n}$ is given by

$||\bm{a}_{1}\wedge\cdots\wedge\bm{a}_{k}||^{2}=\det\begin{pmatrix}\bm{a}_{1}\\\
\vdots\\\
\bm{a}_{k}\end{pmatrix}\begin{pmatrix}\prescript{t}{}{\bm{a}}_{1}&\cdots&\prescript{t}{}{\bm{a}}_{k}\end{pmatrix}.$
(3.25)

Using the inductive construction of $\mathcal{D}^{\prime}$ and the fact that
the entries of $\bm{r}_{1}(Y),\bm{r}_{1}(Y_{1}),\dots$ are at most
$\frac{1}{2}$ in absolute value, we observe that $U$ has entries bounded by a
constant depending only on $n$. We find that

$||\bm{y}_{1}||^{2}\ll v_{1}+\cdots+v_{n}\ll v_{1}$ (3.26)

with the implied constant depending on $n$. ∎

##  4 Theta functions and asymptotics

Following [11], for $f\in\mathcal{S}(\mathbb{R}^{n})$ we define the theta
function $\Theta_{f}:H\rtimes G\mapsto\mathbb{C}$ by

$\Theta_{f}(h,g)=\sum_{\bm{m}\in\mathbb{Z}^{n}}(W(h)R(g)f)(\bm{m}).$ (4.1)

Setting $h=(\bm{x},\bm{y},t)$,

$g=\begin{pmatrix}I&X\\\ 0&I\end{pmatrix}\begin{pmatrix}Y^{\frac{1}{2}}&0\\\
0&\prescript{t}{}{Y}^{-\frac{1}{2}}\end{pmatrix}k(Q),$ (4.2)

and $f_{Q}=R(k(Q))f$, we have from (2.27) and (2.31) that

$\Theta_{f}(h,g)=(\det
Y)^{\frac{1}{4}}\mathrm{e}\left(-t+\frac{1}{2}\bm{x}\prescript{t}{}{\\!\bm{y}}\right)\\\
\sum_{\bm{m}\in\mathbb{Z}^{n}}f_{Q}\left((\bm{m}+\bm{x})Y^{\frac{1}{2}}\right)\mathrm{e}\left(\frac{1}{2}(\bm{m}+\bm{x})X\prescript{t}{}{\\!(\bm{m}+\bm{x})}+\bm{m}\prescript{t}{}{\\!\bm{y}}\right).$
(4.3)

Thus for $f(\bm{x})=\exp(-\pi\bm{x}\prescript{t}{}{\\!\bm{x}})$, $Q=I$, and
$h=(0,0,0)$, we recover $(\det Y)^{\frac{1}{4}}$ times the classical Siegel
theta series that is holomorphic in $Z=X+iY$.222 In fact, one can show that
$f(\bm{x})=\exp(-\pi\bm{x}\prescript{t}{}{\\!\bm{x}})$ is a simultaneous
eigenfunction of all the $R(k(Q))$, $Q\in\mathrm{U}(n)$, see [11]. This
together with theorem 4.1 establishes the automorphy of the holomorphic theta
function.

The following theorem establishes the automorphy of $\Theta_{f}$ under
$\tilde{\Gamma}$, which we recall is defined at the beginning of section 3.

######  Theorem 4.1.

For all $(uh_{\gamma},\gamma)\in\tilde{\Gamma}$ and $(h,g)\in H\rtimes G$,
there is a complex number $\varepsilon(\gamma)$ of modulus $1$ such that

$\Theta_{f}\left((uh_{\gamma},\gamma)(h,g)\right)=\varepsilon(\gamma)\rho(\gamma,g)\mathrm{e}\left(-t+\frac{1}{2}\bm{m}\prescript{t}{}{\\!\bm{n}}\right)\Theta_{f}(h,g),$
(4.4)

where $u=(\bm{m},\bm{n},t)$.

This theorem is proved in [11] but with $\tilde{\Gamma}$ replaced by a finite
index subgroup. The automorphy under the full group $\tilde{\Gamma}$ is proved
in [17], however only for the special function
$f(\bm{x})=\exp(-\pi\bm{x}\prescript{t}{}{\\!\bm{x}})$. In [11] it is shown
that this function is an eigenfunction for all of the operators $R(k(Q))$,
$Q\in\mathrm{U}(n)$. Moreover, it can be seen from the theory built there that
the automorphy for any Schwartz function follows from that for
$\exp(-\pi\bm{x}\prescript{t}{}{\\!\bm{x}})$. A self-contained proof along the
lines of [11] is presented in forthcoming notes by the authors [16]. We also
remark that $\varepsilon(\gamma)$ can be expressed as a kind of Gauss sum as
shown in [11] and the author’s notes, but we do not make use of this here.

We recall that for $Q\in\mathrm{U}(n)$ and $f\in L^{2}(\mathbb{R}^{n})$, we
let $f_{Q}=R(k(Q))f$. The following lemma states that if $f$ is a Schwartz
function, then the $f_{Q}$ are “uniformly Schwartz.”

######  Lemma 4.2.

Let $f\in\mathcal{S}(\mathbb{R}^{n})$. Then for all $A>0$ and multi-indices
$\alpha\geq 0$, there exist constants $c_{f}(\alpha,A)$ such that for all
$Q\in\mathrm{U}(n)$,

$\left|\left(\frac{\partial}{\partial\bm{x}}\right)^{\alpha}f_{Q}(\bm{x})\right|\leq
c_{f}(\alpha,A)(1+||\bm{x}||)^{-A}.$ (4.5)

###### Proof.

Since $f$ is Schwartz, so are the Fourier transforms of $f$ with respect to
any subset of the variables. For a subset $S\subset\\{1,\dots,n\\}$, multi-
index $\alpha\geq 0$, and $A>0$, we let $c_{f}^{S}(\alpha,A)$ be constants
such that

$\left|\left(\frac{\partial}{\partial\bm{x}}\right)^{\alpha}f^{S}(\bm{x})\right|\leq
c_{f}^{S}(\alpha,A)(1+||\bm{x}||)^{-A}$ (4.6)

where $f^{S}$ is the Fourier transform of $f$ in the variables having indices
in $S$.

We now consider $f_{Q}$ for $Q\in\mathrm{U}(n)$ diagonal with the first $n-l$
entries $1$ and the last $l$ entries $\mathrm{e}^{\mathrm{i}\phi_{j}}$ with
$0<\phi_{j}<\pi$. We let $S\subset\\{1,\dots,n\\}$ be the set of indices $j$,
$n-l<j\leq n$, such that
$\phi_{j}\in\left(0,\frac{\pi}{4}\right)\cup\left(\frac{3\pi}{4},\pi\right)$
and we write $Q=Q^{\prime}Q_{S}$ where $Q_{S}$ is diagonal with $(j,j)$ entry
$\mathrm{i}$ if $j\in S$ and $1$ if $j\not\in S$. We have

$f_{Q}=\rho(k(Q^{\prime}),k(Q_{S}))R(k(Q^{\prime}))R(k(Q_{S}))f,$ (4.7)

and we recall that $|\rho(k(Q^{\prime}),k(Q_{S}))|=1$.

We write

$k(Q^{\prime})=\begin{pmatrix}I&0&0&0\\\ 0&D&0&-C\\\ 0&0&I&0\\\
0&C&0&D\end{pmatrix},$ (4.8)

with $C$, $D$ diagonal, the entries of $C$ being $\cos\phi_{j}$ or
$\sin\phi_{j}$ depending on whether $j\in S$ or not, the entries of $D$ being
$-\sin\phi_{j}$ or $\cos\phi_{j}$ depending on whether $j\in S$ or not. We
note that the entries of $C$ are at least $\frac{1}{\sqrt{2}}$.

Writing

$k(Q^{\prime})=\begin{pmatrix}I&0&0&0\\\ 0&I&0&C^{-1}D\\\ 0&0&I&0\\\
0&0&0&I\end{pmatrix}\begin{pmatrix}I&0&0&0\\\ 0&0&0&-I\\\ 0&0&I&0\\\
0&I&0&0\end{pmatrix}\begin{pmatrix}I&0&0&0\\\ 0&C&0&D\\\ 0&0&I&0\\\
0&0&0&C^{-1}\end{pmatrix},$ (4.9)

using proposition 2.1, and noting that $R(k(Q_{S}))=f^{S}$, we compute

$f_{Q}(\bm{x}^{(1)},\bm{x}^{(2)})=\rho(k(Q^{\prime}),k(Q_{S}))|\det
C|^{-\frac{1}{2}}\int_{\mathbb{R}^{l}}f^{S}(\bm{x}^{(1)},\bm{y}^{(2)})\\\
\mathrm{e}\left(\frac{1}{2}\bm{x}^{(2)}C^{-1}D\prescript{t}{}{\\!\bm{x}}^{(2)}-\bm{x}^{(2)}C^{-1}\prescript{t}{}{\\!\bm{y}}^{(2)}+\frac{1}{2}\bm{y}^{(2)}C^{-1}D\prescript{t}{}{\\!\bm{y}}^{(2)}\right)\differential\bm{y}^{(2)}.$
(4.10)

Now as the entries of $C$ are between $\frac{1}{\sqrt{2}}$ and $1$, and the
entries of $D$ are at most $\frac{1}{\sqrt{2}}$ in absolute value, integration
by parts and (4.6) shows that

$\left|\left(\frac{\partial}{\partial\bm{x}}\right)^{\alpha}f_{Q}(\bm{x})\right|\ll(1+||\bm{x}||)^{-A},$
(4.11)

with implied constant depending on $f$, $S$, $\alpha$, and $A$.

We observe that for real orthogonal $Q_{1}$,
$R(k(Q_{1}))f(\bm{x})=f(\bm{x}Q_{1})$, so (4.11) implies

$\left|\left(\frac{\partial}{\partial\bm{x}}\right)^{\alpha}f_{Q_{1}QQ_{2}}\bm{x})\right|\ll(1+||\bm{x}||)^{-A},$
(4.12)

for any orthogonal $Q$. It now suffices to show that any unitary matrix
$Q_{0}$ can be written as $Q_{1}QQ_{2}$ with $Q$ having the special form above
and $Q_{1}$, $Q_{2}$ real orthogonal. Writing
$Q_{0}\prescript{t}{}{\\!Q}_{0}=X+iY$ with $X$, $Y$ real and symmetric, we
note that since $Q_{0}\prescript{t}{}{\\!Q}_{0}$ is unitary,
$X^{2}+Y^{2}+i(XY-YX)$ is the identity. It follows that $X$, $Y$ commute, and
thus can be simultaneously diagonalized by an orthogonal matrix $Q_{1}$. We
have $Q_{0}\prescript{t}{}{\\!Q}_{0}=Q_{1}Q^{2}\prescript{t}{}{\\!Q}_{1}$ with
$Q$ diagonal, and so $Q_{2}=Q^{-1}Q_{1}^{-1}Q_{0}$ is orthogonal. Finally, we
may permute the diagonal entries of $Q$ and change their signs so that the
special form above holds. ∎

We now turn to analysing the behaviour of the theta function $\Theta_{f}$, $f$
a Schwartz function, in the cusp of $\tilde{\Gamma}\backslash H\rtimes G$. We
repeatedly use the easy bounds recorded in the following lemma.

######  Lemma 4.3.

For real numbers $A>\frac{1}{2}$, $|x|\leq\frac{1}{2}$ and $v,y>0$, we have

$\sum_{\begin{subarray}{c}m\in\mathbb{Z}\\\ m\neq
0\end{subarray}}(v(m+x)^{2}+y)^{-A}\ll_{A}v^{-\frac{1}{2}}(v+y)^{-A+\frac{1}{2}}.$
(4.13)

and, if in addition $v\leq ay$ with $a>0$,

$\sum_{m\in\mathbb{Z}}(v(m+x)^{2}+y)^{-A}\ll_{a,A}v^{-\frac{1}{2}}y^{-A}$
(4.14)

for $A>\frac{1}{2}$.

###### Proof.

We have

$\sum_{\begin{subarray}{c}m\in\mathbb{Z}\\\ m\neq
0\end{subarray}}(v(m+x)^{2}+y)^{-A}\leq\sum_{0<|m|\leq\sqrt{\frac{y}{v}}}y^{-A}+\sum_{|m|>\sqrt{\frac{y}{v}}}v^{-A}|m+x|^{-2A}.$
(4.15)

The first sum here is $0$ if $y<v$, otherwise it is at most
$v^{-\frac{1}{2}}y^{-A+\frac{1}{2}}$. The second sum is at most

$2^{2A}v^{-A}\sum_{|m|>\sqrt{\frac{y}{v}}}|m|^{-2A},$ (4.16)

which is $\ll_{A}v^{-A}$ if $y<v$ and
$\ll_{A}v^{-\frac{1}{2}}y^{-A+\frac{1}{2}}$ otherwise. The estimate (4.13) now
follows as $v^{-A}$, respectively $v^{-\frac{1}{2}}y^{-A+\frac{1}{2}}$, is
$\ll v^{-\frac{1}{2}}(v+y)^{-A+\frac{1}{2}}$ if $y<v$, respectively $y\geq v$.

Turning to (4.14), we have

$\sum_{m\in\mathbb{Z}}(v(m+x)^{2}+y)^{-A}\leq\sum_{|m|\leq\sqrt{\frac{ay}{v}}}y^{-A}+\sum_{|m|>\sqrt{\frac{ay}{v}}}v^{-A}|m+x|^{-2A}.$
(4.17)

The first sum here is $\ll_{a}v^{-\frac{1}{2}}y^{-A+\frac{1}{2}}$, while the
second sum is at most

$2^{2A}v^{-A}\sum_{|m|>\sqrt{\frac{ay}{v}}}|m|^{-2A}\ll_{a,A}v^{-\frac{1}{2}}y^{-A+\frac{1}{2}},$
(4.18)

so (4.14) follows immediately. ∎

The following theorem, while a little complicated, gives an asymptotic formula
for $\Theta_{f}(h,g)$ as $g\to\infty$ inside the fundamental domain
$\mathcal{D}$. We describe the relevant neighbourhoods of $\infty$ using the
Langlands decomposition (2.25) of the parabolic subgroups $P_{l}$ with $1\leq
l<n$, see (4.19). The semi-simple part of the Langlands decomposition of this
parabolic is a copy of $\mathrm{Sp}(n-l,\mathbb{R})$, and our asymptotic
formula for $\Theta_{f}$ has a theta function associated to
$\mathrm{Sp}(n-l,\mathbb{R})$ for a main term, see (4.4).

######  Theorem 4.4.

Let $f\in\mathcal{S}(\mathbb{R}^{n})$, $g\in\mathcal{D}$, and
$h=(\bm{x},\bm{y},t)\in H$ with the entries of $\bm{x}$ and $\bm{y}$ all at
most $\frac{1}{2}$ in absolute value. For $1\leq l\leq n$ we write

$g=\begin{pmatrix}I&0&T_{l}&S_{l}\\\ 0&I&\prescript{t}{}{\\!S}_{l}&0\\\
0&0&I&0\\\ 0&0&0&I\end{pmatrix}\begin{pmatrix}U_{l}&R_{l}&0&0\\\ 0&I&0&0\\\
0&0&\prescript{t}{}{\\!U_{l}}^{-1}&0\\\
0&0&-\prescript{t}{}{\\!R_{l}}\prescript{t}{}{\\!U_{l}}^{-1}&I\end{pmatrix}\\\
\begin{pmatrix}I&0&0&0\\\ 0&I&0&X_{l}\\\ 0&0&I&0\\\
0&0&0&I\end{pmatrix}\begin{pmatrix}V_{l}^{\frac{1}{2}}&0&0&0\\\
0&Y_{l}^{\frac{1}{2}}&0&0\\\ 0&0&V_{l}^{\frac{1}{2}}&0\\\
0&0&0&\prescript{t}{}{\\!Y_{l}}^{-\frac{1}{2}}\end{pmatrix}k(Q)$ (4.19)

where $R_{l}$, $S_{l}$ are $l\times(n-l)$ matrices, $T_{l}$ is $l\times l$
symmetric, $U_{l}$ is $l\times l$ upper-triangular unipotent, $X_{l}$ is
$(n-l)\times(n-l)$ symmetric, $V_{l}$ is $l\times l$ positive diagonal,
$Y_{l}$ is $(n-l)\times(n-l)$ positive definite symmetric, and
$Q\in\mathrm{U}(n)$.

We have

$\displaystyle\Theta_{f}(h,g)=$ $\displaystyle(\det V_{l})^{\frac{1}{4}}(\det
Y_{l})^{\frac{1}{4}}\mathrm{e}\left(-t+\frac{1}{2}\bm{x}_{l}\prescript{t}{}{\\!\bm{y}_{l}}\right)$
$\displaystyle\sum_{\bm{m}^{(2)}\in\mathbb{Z}^{n-l}}f_{Q}\left(\bm{x}_{l}^{(1)}V_{l}^{\frac{1}{2}},(\bm{m}^{(2)}+\bm{x}_{l}^{(2)})Y_{l}^{\frac{1}{2}}\right)$
$\displaystyle\qquad\qquad\qquad\mathrm{e}\left(\frac{1}{2}(\bm{m}^{(2)}+\bm{x}_{l}^{(2)})X_{l}\prescript{t}{}{\\!(\bm{m}^{(2)}+\bm{x}_{l}^{(2)})}+\bm{m}^{(2)}\prescript{t}{}{\\!\bm{y}_{l}}^{(2)}\right)$
$\displaystyle+O_{A,f}\left((\det
V_{l})^{\frac{1}{4}}(v_{l}+\bm{x}V\prescript{t}{}{\\!\bm{x}})^{-A}\right),$
(4.20)

where

$V=\begin{pmatrix}v_{1}&&\\\ &\ddots&\\\ &&v_{n}\end{pmatrix},$ (4.21)

and, with
$\begin{pmatrix}\bm{x}&\bm{y}\end{pmatrix}=\begin{pmatrix}\bm{x}^{(1)}&\bm{x}^{(2)}&\bm{y}^{(1)}&\bm{y}^{(2)}\end{pmatrix}$,

$\begin{pmatrix}\bm{x}_{l}^{(1)}&\bm{x}_{l}^{(2)}&\bm{y}_{l}^{(1)}&\bm{y}_{l}^{(2)}\end{pmatrix}\\\
=\begin{pmatrix}\bm{x}^{(1)}&\bm{x}^{(2)}&\bm{y}^{(1)}&\bm{y}^{(2)}\end{pmatrix}\begin{pmatrix}I&0&T_{l}&S_{l}\\\
0&I&\prescript{t}{}{\\!S}_{l}&0\\\ 0&0&I&0\\\
0&0&0&I\end{pmatrix}\begin{pmatrix}U_{l}&R_{l}&0&0\\\ 0&I&0&0\\\
0&0&\prescript{t}{}{\\!U_{l}}^{-1}&0\\\
0&0&-\prescript{t}{}{\\!R_{l}}\prescript{t}{}{\\!U_{l}}^{-1}&I\end{pmatrix}.$
(4.22)

###### Proof.

Comparing the expressions

$g=\begin{pmatrix}I&X\\\ 0&I\end{pmatrix}\begin{pmatrix}Y^{\frac{1}{2}}&0\\\
0&\prescript{t}{}{Y}^{-\frac{1}{2}}\end{pmatrix}k(Q)$ (4.23)

and

$g=\begin{pmatrix}I&0&T_{l}&S_{l}\\\ 0&I&\prescript{t}{}{\\!S}_{l}&0\\\
0&0&I&0\\\ 0&0&0&I\end{pmatrix}\begin{pmatrix}U_{l}&R_{l}&0&0\\\ 0&I&0&0\\\
0&0&\prescript{t}{}{\\!U_{l}}^{-1}&0\\\
0&0&-\prescript{t}{}{\\!R_{l}}\prescript{t}{}{\\!U_{l}}^{-1}&I\end{pmatrix}\\\
\begin{pmatrix}I&0&0&0\\\ 0&I&0&X_{l}\\\ 0&0&I&0\\\
0&0&0&I\end{pmatrix}\begin{pmatrix}V_{l}^{\frac{1}{2}}&0&0&0\\\
0&Y_{l}^{\frac{1}{2}}&0&0\\\ 0&0&V_{l}^{\frac{1}{2}}&0\\\
0&0&0&\prescript{t}{}{\\!Y_{l}}^{-\frac{1}{2}}\end{pmatrix}k(Q),$ (4.24)

we find that

$X=\begin{pmatrix}T_{l}+R_{l}X_{l}\prescript{t}{}{\\!R}_{l}&S_{l}+R_{l}X_{l}\\\
\prescript{t}{}{\\!S_{l}}+X_{l}\prescript{t}{}{\\!R}_{l}&X_{l}\end{pmatrix}$
(4.25)

and

$Y^{\frac{1}{2}}=\begin{pmatrix}U_{l}V_{l}^{\frac{1}{2}}&R_{l}Y_{l}^{\frac{1}{2}}\\\
0&Y_{l}^{\frac{1}{2}}\end{pmatrix}.$ (4.26)

Recalling from (4.3) that

$\Theta_{f}(h,g)=(\det
Y)^{\frac{1}{4}}\mathrm{e}\left(-t+\frac{1}{2}\bm{x}\prescript{t}{}{\\!\bm{y}}\right)\\\
\sum_{\bm{m}\in\mathbb{Z}^{n}}f_{Q}\left((\bm{m}+\bm{x})Y^{\frac{1}{2}}\right)\mathrm{e}\left(\frac{1}{2}(\bm{m}+\bm{x})X\prescript{t}{}{\\!(\bm{m}+\bm{x})}+\bm{m}\prescript{t}{}{\\!\bm{y}}\right),$
(4.27)

we express each term of the sum as

$\displaystyle
f_{Q}\left((\bm{m}+\bm{x})Y^{\frac{1}{2}}\right)\mathrm{e}\left(\frac{1}{2}\bm{x}\prescript{t}{}{\\!\bm{y}}+\frac{1}{2}(\bm{m}+\bm{x})X\prescript{t}{}{\\!(\bm{m}+\bm{x})}+\bm{m}\prescript{t}{}{\\!\bm{y}}\right)$
$\displaystyle=f_{Q}\left((\bm{m}^{(1)}U_{l}+\bm{x}_{l}^{(1)})V_{l}^{\frac{1}{2}},(\bm{m}^{(1)}R_{j}+\bm{m}^{(2)}+\bm{x}_{l}^{(2)})Y_{l}^{\frac{1}{2}}\right)$
$\displaystyle\qquad\mathrm{e}\bigg{(}\frac{1}{2}\bm{m}^{(1)}\left(T_{l}+R_{l}X_{l}\prescript{t}{}{\\!R_{l}}\right)\prescript{t}{}{\\!\bm{m}}^{(1)}$
$\displaystyle\qquad\qquad+\bm{m}^{(1)}\prescript{t}{}{\\!\big{(}\bm{y}^{(1)}+\bm{x}^{(1)}(T_{l}+R_{l}X_{l}\prescript{t}{}{\\!R_{l}})+(\bm{m}^{(2)}+\bm{x}^{(2)})(\prescript{t}{}{\\!S_{l}}+X_{l}\prescript{t}{}{\\!R_{l}})\big{)}}\bigg{)}$
$\displaystyle\qquad\mathrm{e}\bigg{(}\frac{1}{2}\bm{x}_{l}^{(1)}\prescript{t}{}{\\!\bm{y}_{l}}^{(1)}+\frac{1}{2}\bm{x}_{l}^{(2)}\prescript{t}{}{\\!\bm{y}_{l}}^{(2)}$
$\displaystyle\qquad\qquad+\frac{1}{2}(\bm{m}^{(2)}+\bm{x}^{(2)})X_{l}\prescript{t}{}{\\!(\bm{m}^{(2)}+\bm{x}^{(2)})}+\bm{m}^{(2)}\prescript{t}{}{\\!\bm{y}_{l}}^{(2)}\bigg{)},$
(4.28)

where $\bm{m}=\begin{pmatrix}\bm{m}^{(1)}&\bm{m}^{(2)}\end{pmatrix}$, and
$\begin{pmatrix}\bm{x}_{l}&\bm{y}_{l}\end{pmatrix}=\begin{pmatrix}\bm{x}_{l}^{(1)}&\bm{x}_{l}^{(2)}&\bm{y}_{l}^{(1)}&\bm{y}_{l}^{(2)}\end{pmatrix}$
is given by (4.22). We observe from (4) that the main term in (4.4) is the sum
over those $\bm{m}=\begin{pmatrix}\bm{m}^{(1)}&\bm{m}^{(2)}\end{pmatrix}$ with
$\bm{m}^{(1)}=0$.

To bound the contribution of the terms with $\bm{m}^{(1)}\neq 0$, we proceed
by induction on $l$, making use of the recursive definition of the fundamental
domain $\mathcal{D}^{\prime}_{n}$ containing $Y$. For $l=1$, the contribution
of $m^{(1)}\neq 0$ is, by lemma 4.2,

$\displaystyle\ll(\det
Y)^{\frac{1}{4}}\sum_{\begin{subarray}{c}m^{(1)}\in\mathbb{Z}\\\ m^{(1)}\neq
0\end{subarray}}\sum_{\bm{m}^{(2)}\in\mathbb{Z}^{n-1}}\left(1+(\bm{m}+\bm{x})Y\prescript{t}{}{\\!(\bm{m}+\bm{x})}\right)^{-A}$
$\displaystyle\quad\ll v_{1}^{\frac{1}{4}}(\det
Y_{1})^{\frac{1}{4}}\sum_{\begin{subarray}{c}m^{(1)}\in\mathbb{Z}\\\
m^{(1)}\neq 0\end{subarray}}\sum_{\bm{m}^{(2)}\in\mathbb{Z}^{n-1}}$
$\displaystyle\qquad\left(1+v_{1}(m^{(1)}+x^{(1)})^{2}+(\bm{m}^{(2)}+\bm{x}^{(2)})Y_{1}\prescript{t}{}{\\!(\bm{m}^{(2)}+\bm{x}^{(2)})}\right)^{-A}$
(4.29)

by proposition 3.1. Applying (4.13) with $v=v_{1}$, $x=x^{(1)}$,
$y=1+(\bm{m}^{(2)}+\bm{x}^{(2)})Y_{1}\prescript{t}{}{\\!(\bm{m}^{(2)}+\bm{x}^{(2)})}$,
and renaming $\bm{m}^{(2)}=\bm{m}_{1}$, $\bm{x}^{(2)}=\bm{x}_{1}$ (not to be
confused with (4.22)), this is

$\displaystyle\ll v_{1}^{-\frac{1}{4}}(\det
Y_{1})^{\frac{1}{4}}\sum_{\bm{m}_{1}\in\mathbb{Z}^{n-1}}\left(v_{1}+(\bm{m}_{1}+\bm{x}_{1})Y_{1}\prescript{t}{}{\\!(\bm{m}_{1}+\bm{x}_{1})}\right)^{-A}$
$\displaystyle\quad\ll v_{1}^{-\frac{1}{4}}v_{2}^{\frac{1}{4}}(\det
Y_{2})^{\frac{1}{4}}\sum_{m_{1}^{(1)}\in\mathbb{Z}}\sum_{\bm{m}_{1}^{(2)}\in\mathbb{Z}^{n-2}}$
$\displaystyle\qquad\left(v_{1}+v_{2}(m_{1}^{(1)}+x_{1}^{(1)})^{2}+(\bm{m}_{1}^{(2)}+\bm{x}_{1}^{(2)})Y_{1}\prescript{t}{}{\\!(\bm{m}_{1}^{(2)}+\bm{x}_{1}^{(2)})}\right)^{-A}$
(4.30)

by proposition 3.1, recalling that $Y_{1}\in\mathcal{D}^{\prime}_{n-1}$.
Applying (4.14) with $v=v_{2}$, $x=x_{1}^{(1)}$,
$y=v_{1}+(\bm{m}_{1}^{(2)}+\bm{x}_{1}^{(2)})Y_{1}\prescript{t}{}{\\!(\bm{m}_{1}^{(2)}+\bm{x}_{1}^{(2)})}\gg
v_{2}$, this is

$\ll v_{1}^{-\frac{1}{4}}v_{2}^{-\frac{1}{4}}(\det
Y_{2})^{\frac{1}{4}}\sum_{\bm{m}_{1}^{(2)}\in\mathbb{Z}^{n-2}}\left(v_{1}+(\bm{m}_{1}^{(2)}+\bm{x}_{1}^{(2)})Y_{2}\prescript{t}{}{\\!(\bm{m}_{1}^{(2)}+\bm{x}_{1}^{(2)})}\right)^{-A}.$
(4.31)

Continuing in this way, we eventually obtain the bound

$\ll v_{1}^{-\frac{1}{4}}\cdots v_{n}^{-\frac{1}{4}}(v_{1})^{-A}\ll
v_{1}^{\frac{1}{4}}(v_{1}+\bm{x}V\prescript{t}{}{\\!\bm{x}})^{-A},$ (4.32)

thus establishing (4.4) for $l=1$.

For $l>1$, we see by induction, lemma 4.2, and proposition 3.1 that we need to
bound

$(\det Y)^{\frac{1}{4}}\sum_{\begin{subarray}{c}\bm{m}^{(1)}=(0\
m)\in\mathbb{Z}^{l}\\\ m\neq
0\end{subarray}}\sum_{\bm{m}^{(2)}\in\mathbb{Z}^{n-l}}\\\
\left((\bm{m}^{(1)}+\bm{x}^{(1)})V_{l}\prescript{t}{}{\\!(\bm{m}^{(1)}+\bm{x}^{(1)})}+(\bm{m}^{(2)}+\bm{x}^{(2)})Y_{l}\prescript{t}{}{\\!(\bm{m}^{(2)}+\bm{x}^{(2)})}\right)^{-A}.$
(4.33)

Applying (4.13) with $v=v_{l}$, $x$ the last entry of $\bm{x}^{(1)}$,
$y=\bm{x}^{(1)}V_{l}\prescript{t}{}{\\!\bm{x}^{(1)}}-v_{l}x^{2}$, and renaming
$\bm{m}^{(2)}=\bm{m}_{1}$, $\bm{x}^{(2)}=\bm{x}_{1}$, this is

$\displaystyle\ll v_{l}^{-\frac{1}{2}}(\det V_{l})^{\frac{1}{4}}(\det
Y_{l})^{\frac{1}{4}}$
$\displaystyle\quad\sum_{\bm{m}_{1}\in\mathbb{Z}^{n-l}}\left(v_{l}+\bm{x}^{(1)}V_{l}\prescript{t}{}{\\!\bm{x}^{(1)}}+(\bm{x}_{1}+\bm{m}_{1})Y_{l}\prescript{t}{}{\\!(\bm{x}_{1}+\bm{m}_{1})}\right)^{-A}$
$\displaystyle\ll v_{l}^{-\frac{1}{2}}(\det V_{l})^{\frac{1}{4}}(\det
Y_{l})^{\frac{1}{4}}$
$\displaystyle\quad\sum_{m_{1}^{(1)}\in\mathbb{Z}}\sum_{\bm{m}_{1}^{(2)}\in\mathbb{Z}^{n-l-1}}\Big{(}v_{l}+\bm{x}^{(1)}V_{l}\prescript{t}{}{\\!\bm{x}^{(1)}}+$
$\displaystyle\qquad
v_{l+1}(m_{1}^{(1)}+x_{1}^{(1)})^{2}+(\bm{m}_{1}^{(2)}+\bm{x}_{1}^{(2)})Y_{l+1}\prescript{t}{}{\\!(\bm{m}_{1}^{(2)}+\bm{x}_{1}^{(2)})}\Big{)}^{-A}$
(4.34)

by proposition 3.1 and $Y_{l}\in\mathcal{D}_{n-l}^{\prime}$. Applying (4.14)
repeatedly as we did in the $l=1$ case, we obtain the bound

$\ll(\det V_{l})^{\frac{1}{4}}v_{l}^{-\frac{1}{2}}v_{l+1}^{-\frac{1}{4}}\cdots
v_{n}^{-\frac{1}{4}}(v_{l}+\bm{x}^{(1)}V_{l}\prescript{t}{}{\\!\bm{x}^{(1)}})^{-A}\\\
\ll(\det V_{l})^{\frac{1}{4}}(v_{l}+\bm{x}V\prescript{t}{}{\\!\bm{x}})^{-A}$
(4.35)

as required. ∎

Since $v_{n}\geq\frac{\sqrt{3}}{2}$ for $g\in\mathcal{D}$ by proposition 3.1,
we obtain the following corollary.

######  Corollary 4.5.

For a Schwartz function $f\in\mathcal{S}(\mathbb{R}^{n})$, $g\in\mathcal{D}$,
and $h=(\bm{x},\bm{y},t)\in H$ with the entries of $\bm{x}$ and $\bm{y}$ at
most $\frac{1}{2}$ in absolute value, we have

$\Theta_{f}(h,g)\ll_{f}(\det
V)^{\frac{1}{4}}(1+\bm{x}V\prescript{t}{}{\\!\bm{x}})^{-A}$ (4.36)

where

$g=\begin{pmatrix}I&X\\\ 0&I\end{pmatrix}\begin{pmatrix}Y^{\frac{1}{2}}&0\\\
0&\prescript{t}{}{Y}^{-\frac{1}{2}}\end{pmatrix}k(Q)$ (4.37)

with $Y=UV\prescript{t}{}{U}$ as usual.

##  5 Proof of the main theorems

Having the bounds from corollary 4.5, we now proceed to the proof of theorems
1.1 and 1.2. In the smooth setting of theorem 1.1, we need to construct a
distance-like (DL) function that captures the bounds in corollary 4.5. This
will enable us to directly apply theorem 1.7 in [8] modulo a standard argument
that allows us to pass from a full measure set in $\Gamma g\in\Gamma\backslash
G$ to a full measure set on the unstable foliation parametrized by
$X\in\mathbb{R}^{n\times n}_{\mathrm{sym}}$. The proof of theorem 1.2 is more
involved, and requires modifications of the method in [8] to enable a
resolution of the singular cutoff function in (1.3). To this end we need to
uniformly manage many points in $\Gamma\backslash G$.

We note that theorem 1.7 in [8] is also a main input in the method of [3].

###  5.1 Heights and volumes

We define the height function $D:\Gamma\backslash G\to\mathbb{R}_{>0}$ by

$D\left(\Gamma g\right)=\max_{\gamma\in\Gamma}\det V(\gamma g)=\det
V(\gamma_{0}g)$ (5.1)

where $\gamma_{0}$ is such that $\gamma_{0}g\in\mathcal{D}$ and we write

$g=\begin{pmatrix}U&X\prescript{t}{}{U}^{-1}\\\
0&\prescript{t}{}{U}^{-1}\end{pmatrix}\begin{pmatrix}V^{\frac{1}{2}}&0\\\
0&V^{-\frac{1}{2}}\end{pmatrix}k(Q)$ (5.2)

with $V=V(g)$ positive diagonal as usual. We remark that from corollary 4.5
and the automorphy of $\Theta_{f}$, theorem 4.1, we have $\Theta_{f}(h,g)\ll
D(\Gamma g)^{\frac{1}{4}}$ for all $(h,g)\in H\rtimes G$ with the implied
constant depending only on $f$. We also remark that the logarithm of $D$ is a
distance-like function in the sense of [8], see also [3].

We begin by estimating the measure of the set on which $D$ is large, thus
verifying one of the required properties for the logarithm of $D$ to be a
$\frac{n+1}{2}$-DL function, see [8]. This estimate is also found in [3] and
the relevant change of variables in [9].

######  Lemma 5.1.

Let $\mu$ be Haar measure on $G$ and $R>0$. We have

$\mu(\\{\Gamma g\in\Gamma\backslash G:D(\Gamma g)\geq R\\})\ll
R^{-\frac{n+1}{2}}$ (5.3)

with the implied constant depending only on $n$.

###### Proof.

We recall that $g\in\mathcal{D}$ is written as

$g=\begin{pmatrix}U&X\prescript{t}{}{U}^{-1}\\\
0&\prescript{t}{}{U}^{-1}\end{pmatrix}\begin{pmatrix}V^{\frac{1}{2}}&0\\\
0&V^{-\frac{1}{2}}\end{pmatrix}k(Q)$ (5.4)

for $U$ upper-triangular unipotent, $X$ symmetric, $Q\in\mathrm{U}(n)$, and

$V=V(g)=\begin{pmatrix}v_{1}&\cdots&0\\\ \vdots&\ddots&\vdots\\\
0&\cdots&v_{n}\end{pmatrix}$ (5.5)

positive diagonal. The Haar measure $\mu$ is then Lebesgue measure with
respect to the entries of $X$ and the off-diagonal entries of $U$,
$\mathrm{U}(n)$-Haar measure on $Q$, and the measure given by

$v_{1}^{-n-1}v_{2}^{-n}\cdots v_{n}^{-2}\differential v_{1}\differential
v_{2}\cdots\differential v_{n}$ (5.6)

on $V$.

From the construction of $\mathcal{D}$, it is clear that the entries of $U$
and $X$ are constrained to a compact region. Since $\mathrm{U}(n)$ is also
compact, we have by proposition 3.1 that

$\mu(\\{g\in\mathcal{D}_{n}:\det V(g)\geq R\\})\\\
\ll\underset{\begin{subarray}{c}v_{j}\geq\frac{3}{4}v_{j+1}\\\ v_{1}\cdots
v_{n}\geq R\end{subarray}}{\int\cdots\int}v_{1}^{-n-1}v_{2}^{-n}\cdots
v_{n}^{-2}\differential v_{1}\differential v_{2}\cdots\differential v_{n}.$
(5.7)

Changing variables $v_{j}=\exp(u_{j})$, the integral in (5.7) is

$\underset{\begin{subarray}{c}u_{j}-u_{j+1}\geq\log\frac{3}{4}\\\
u_{1}+\cdots+u_{n}\geq\log
R\end{subarray}}{\int\cdots\int}\exp(-nu_{1}-(n-1)u_{2}-\cdots-
u_{n})\differential u_{1}\differential u_{2}\cdots\differential u_{n}.$ (5.8)

We now make the linear change of variables $s_{j}=u_{j}-u_{j+1}$ for $j<n$ and
$s_{n}=u_{1}+\cdots+u_{n}$. This transformation has determinant $n$ and its
inverse is given by

$u_{j}=-\frac{1}{n}\sum_{1\leq i<j}is_{i}+\frac{1}{n}\sum_{j\leq
i<n}(n-i)s_{i}+\frac{1}{n}s_{n}.$ (5.9)

We find that the exponent in (5.8) is then

$-\sum_{1\leq j\leq n}(n-j+1)u_{j}=-\frac{n+1}{2}s_{n}-\sum_{1\leq
j<n}\frac{j(n-j)}{2}s_{j}.$ (5.10)

As $\frac{j(n-j)}{2}>0$ for $j<n$, the bound (5.3) follows. ∎

We now control the change in the height function $D$ under a geodesic flow by
a fixed distance. This estimate should be compared to the requirement in [8]
that distance-like functions be uniformly continuous.

######  Lemma 5.2.

For $\Gamma g\in\Gamma\backslash G$ and $|s|\leq 1$, we have

$D\bigg{(}\Gamma g\begin{pmatrix}\mathrm{e}^{s}I&0\\\
0&\mathrm{e}^{-s}I\end{pmatrix}\bigg{)}\asymp D(\Gamma g)$ (5.11)

with implied constants depending only on $n$.

###### Proof.

For arbitrary $g\in G$ and $|s|\leq 1$, we set

$g_{s}=g\begin{pmatrix}\mathrm{e}^{s}I&0\\\ 0&\mathrm{e}^{-s}I\end{pmatrix}.$
(5.12)

We first claim that

$\det V(g_{s})\asymp\det V(g)$ (5.13)

for all $g\in G$.

As usual we have

$g=\begin{pmatrix}I&X\\\ 0&I\end{pmatrix}\begin{pmatrix}Y^{\frac{1}{2}}&0\\\
0&\prescript{t}{}{Y}^{-\frac{1}{2}}\end{pmatrix}k(Q),$ (5.14)

and we note that $\det Y=\det V(g)$. Writing

$Q=R+\mathrm{i}S,$ (5.15)

we have

$g_{s}=\begin{pmatrix}I&X\\\
0&I\end{pmatrix}\begin{pmatrix}\mathrm{e}^{s}Y^{\frac{1}{2}}R&-\mathrm{e}^{-s}Y^{\frac{1}{2}}S\\\
\mathrm{e}^{s}\prescript{t}{}{Y}^{-\frac{1}{2}}S&\mathrm{e}^{-s}\prescript{t}{}{Y}^{-\frac{1}{2}}R\end{pmatrix},$
(5.16)

so in view of (2.1)

$Y(g_{s})=Y^{\frac{1}{2}}\left(\mathrm{e}^{2s}S\prescript{t}{}{\\!S}+\mathrm{e}^{-2s}R\prescript{t}{}{\\!R}\right)^{-1}\prescript{t}{}{Y}^{\frac{1}{2}}.$
(5.17)

The ratio of the right to the left side of (5.13) is then

$\det(\mathrm{e}^{2s}S\prescript{t}{}{\\!S}+\mathrm{e}^{-2s}R\prescript{t}{}{\\!R}).$
(5.18)

Using the diagonalization argument from the proof of lemma 4.2, we can
multiply by orthogonal matrices to make $R$ and $S$ diagonal with entries
$\cos\phi_{j}$ and $\sin\phi_{j}$. The determinant (5.18) is then

$\prod_{1\leq j\leq
n}(\mathrm{e}^{2s}\sin^{2}\phi_{j}+\mathrm{e}^{-2s}\cos^{2}\phi_{j}).$ (5.19)

Since $|s|\leq 1$, this is clearly bounded from above by a constant depending
on $n$, and since $\sin^{2}\phi_{j}$ and $\cos^{2}\phi_{j}$ cannot both be
less than $\frac{1}{2}$, it is also bounded away from $0$. This establishes
(5.13).

Now we have

$D(\Gamma g_{s})=\max_{\gamma\in\Gamma}\det V(\gamma g_{s})=\det
V(\gamma_{0}g_{s})$ (5.20)

for some $\gamma_{0}\in\Gamma$. By (5.13) we have $D(\Gamma g_{s})\ll\det
V(\gamma_{0}g)\leq D(\Gamma g)$. The same reasoning with $g_{s}$ replaced by
$g$ leads to the reverse inequality, establishing (5.11). ∎

The following lemma is similar to lemma 5.2 in that we control the change in
$D$ under a particular action. Here the action is more general, however we
only need to consider small neighbourhoods in $\Gamma\backslash G$.

######  Lemma 5.3.

There exists a constant $\epsilon_{n}>0$ depending only on $n$ such that for
all $\Gamma g\in\Gamma\backslash G$, $A\in\mathrm{GL}(n,\mathbb{R})$
satisfying $||A-I||\leq\epsilon_{n}$, and symmetric $T$ satisfying
$||T||\leq\epsilon_{n}$, we have

$D\bigg{(}\Gamma g\begin{pmatrix}A&0\\\
0&\prescript{t}{}{\\!A}^{-1}\end{pmatrix}\begin{pmatrix}I&0\\\
T&I\end{pmatrix}\bigg{)}\asymp D(\Gamma g).$ (5.21)

###### Proof.

As in the proof of lemma 5.2, it suffices to show that for all $g\in G$,

$\det V(gg_{A}g_{T})\asymp\det V(g),$ (5.22)

where

$g_{A}=\begin{pmatrix}A&0\\\ 0&\prescript{t}{}{\\!A}^{-1}\end{pmatrix},\quad
g_{T}=g\begin{pmatrix}I&0\\\ T&I\end{pmatrix}.$ (5.23)

We write

$g=\begin{pmatrix}I&X\\\ 0&I\end{pmatrix}\begin{pmatrix}Y^{\frac{1}{2}}&0\\\
0&\prescript{t}{}{Y}^{-\frac{1}{2}}\end{pmatrix}\begin{pmatrix}R&-S\\\
S&R\end{pmatrix},$ (5.24)

where $R+\mathrm{i}S\in\mathrm{U}(n)$. We compute

$Y(gg_{A}g_{T})^{-1}\\\
=\prescript{t}{}{Y}^{-\frac{1}{2}}\left((SA+R\prescript{t}{}{\\!A}^{-1}T)\prescript{t}{}{\\!(SA+R\prescript{t}{}{\\!A}^{-1}T)}+R\prescript{t}{}{\\!A}^{-1}A^{-1}\prescript{t}{}{\\!R}\right)Y^{-\frac{1}{2}},$
(5.25)

so the ratio of the left and right sides of (5.22) is

$\displaystyle\det((SA+R\prescript{t}{}{\\!A}^{-1}T)\prescript{t}{}{\\!(SA+R\prescript{t}{}{\\!A}^{-1}T)}+R\prescript{t}{}{\\!A}^{-1}A^{-1}\prescript{t}{}{\\!R})$
$\displaystyle\quad=$.

Recalling that $R\prescript{t}{}{\\!R}+S\prescript{t}{}{\\!S}=I$, we have

$SA\prescript{t}{}{\\!A}\prescript{t}{}{\\!S}+R\prescript{t}{}{\\!A}^{-1}A^{-1}\prescript{t}{}{\\!R}=I+O(\epsilon_{n})$
(5.26)

if $||A-I||\leq\epsilon_{n}$. It follows that $\epsilon_{n}$ can be made
sufficiently small so that the symmetric matrix

$-I+SA\prescript{t}{}{\\!A}\prescript{t}{}{\\!S}+R\prescript{t}{}{\\!A}^{-1}A^{-1}\prescript{t}{}{\\!R}\\\
+R\prescript{t}{}{\\!A}^{-1}T\prescript{t}{}{\\!A}\prescript{t}{}{\\!S}+SATA^{-1}\prescript{t}{}{\\!R}+R\prescript{t}{}{\\!A}^{-1}T^{2}A^{-1}\prescript{t}{}{\\!R}$
(5.27)

has all eigenvalues less than $\frac{1}{n}$, say, in absolute value, and
(5.22) follows. ∎

###  5.2 Proof of theorem 1.1

In this section we sketch a proof of theorem 1.1, appealing to the method in
[8]. A complete proof of theorem 1.1 can be obtained from the proof of theorem
1.2 in the following section by only considering the $\bm{j}=(0,\ldots,0)$
term in the dyadic expansion, for a general Schwartz function $f$ rather than
the compactly supported function $f_{n}$ considered.

We have

$\theta_{f}(M,X,\bm{x},\bm{y})=M^{\frac{n}{2}}\Theta_{f}(h,g_{M,X})$ (5.28)

where $h=(\bm{x},\bm{y},0)$ and

$g_{M,X}=\begin{pmatrix}I&X\\\ 0&I\end{pmatrix}\begin{pmatrix}M^{-1}I&0\\\
0&MI\end{pmatrix}.$ (5.29)

By corollary 4.5 we have

$\theta_{f}(M,X,\bm{x},\bm{y})\ll_{f}M^{\frac{n}{2}}D(\Gamma
g_{M,X})^{\frac{1}{4}}.$ (5.30)

Now for the proof of the easy part of theorem 1.7 in [8], we observe that one
only needs the upper bound in lemma 5.1 instead of the matching lower bound in
the definition of $\frac{n+1}{2}$-DL functions. In addition, one does not need
that the function $\log D(\Gamma g)$ be uniformly continuous; lemma 5.2
suffices. We therefore have that

$D\left(\Gamma g\begin{pmatrix}M^{-1}I&0\\\
0&MI\end{pmatrix}\right)^{\frac{1}{4}}\ll_{g}\psi(\log M)$ (5.31)

for almost all $\Gamma g\in\Gamma\backslash G$ as long as

$\sum_{k\geq 0}\psi(k)^{-2n-2}<\infty$ (5.32)

with $\psi:[0,\infty)\to[1,\infty)$ increasing.

To finish our proof sketch, we consider the set of $X\in\mathbb{R}^{n\times
n}_{\mathrm{sym}}$ such that there exist $A\in\mathrm{GL}(n,\mathbb{R})$ with
$||A-I||\leq\epsilon_{n}$ and $T\in\mathbb{R}^{n\times n}_{\mathrm{sym}}$ with
$||T||\leq\epsilon_{n}$ so that

$\Gamma g=\Gamma\begin{pmatrix}I&X\\\ 0&I\end{pmatrix}\begin{pmatrix}A&0\\\
0&\prescript{t}{}{A}^{-1}\end{pmatrix}\begin{pmatrix}I&0\\\ T&I\end{pmatrix}$
(5.33)

satisfies the bound (5.31). From the Haar measure calculation (2.15) we see
that this set of $X$ has full measure, and from lemma 5.3 and (5.31) we see
that

$D(\Gamma g_{M,X})^{\frac{1}{4}}\ll_{X}\psi(\log M)$ (5.34)

for all $X$ in this set. Theorem 1.1 then follows.

###  5.3 Proof of theorem 1.2

We record the following lemma that dyadically decomposes the indicator
function of the open interval $(0,1)$, noting that both the singularities at
$0$ and $1$ need to be resolved.

######  Lemma 5.4.

There exists a smooth, compactly supported function
$f_{1}:(0,1)\to\mathbb{R}_{\geq 0}$ such that

$\chi_{1}(x)=\sum_{j\geq
0}\left(f_{1}\left(2^{j}x\right)+f_{1}\left(2^{j}(1-x)\right)\right),$ (5.35)

where $\chi_{1}$ is the indicator function of the open unit interval $(0,1)$.

We note that [2] has an explicit construction of a function $f_{1}$ satisfying
(5.35) that is however only twice differentiable. As we make no effort here to
determine constants in our estimates, we sacrifice explicitness for
smoothness.

###### Proof.

We let $f_{0}$ be a non-negative, smooth function such that $f_{0}(x)=0$ for
$x\leq 0$, $f_{0}(x)=1$ for $x\geq 1$, and

$f_{0}(x)+f_{0}(1-x)=1$ (5.36)

for all $0\leq x\leq 1$. We then define the smooth function $f_{1}$ on the
interval by

$f_{1}(x)=\begin{cases}0&\mathrm{if\ }x\leq\frac{1}{6}\\\
f_{0}(6x-1)&\mathrm{if\ }\frac{1}{6}\leq x\leq\frac{1}{3}\\\
f_{0}(2-3x)&\mathrm{if\ }\frac{1}{3}\leq x\leq\frac{2}{3}\\\ 0&\mathrm{if\
}\frac{2}{3}\leq x.\end{cases}$ (5.37)

Let us now consider the expression

$\sum_{j\geq 0}\left(f_{1}(2^{j}x)+f_{1}(2^{j}(1-x))\right),$ (5.38)

which is clearly $0$ if $x\not\in(0,1)$. If $0<x\leq\frac{1}{3}$ then
$f_{1}(2^{j}(1-x))=0$ for all $j\geq 0$, and $f_{1}(2^{j}x)$ is nonzero for
exactly two values of $j\geq 0$, say $j_{0}$ and $j_{0}+1$. We have

$f_{1}(2^{j_{0}}x)+f_{1}(2^{j_{0}+1}x)=f_{0}(2^{j_{0}}6x-1)+f_{0}(2-2^{j_{0}}6x)=1$
(5.39)

by (5.36). We similarly find that (5.38) is $1$ for $\frac{2}{3}\leq x<1$.
When $\frac{1}{3}\leq x\leq\frac{2}{3}$, only the $j=0$ term in (5.38) is
nonzero. We have

$f_{1}(x)+f_{1}(1-x)=f_{0}(2-3x)+f_{0}(3x-1)=1$ (5.40)

by the condition (5.36). ∎

For a subset $S\subset\\{1,\dots,n\\}$ and
$\bm{j}=(j_{1},\dots,j_{n})\in\mathbb{Z}^{n}$ with $j_{i}\geq 0$, we define

$g_{\bm{j},S}=\begin{pmatrix}A_{\bm{j}}E_{S}&0\\\
0&A_{\bm{j}}^{-1}E_{S}\end{pmatrix}\in G,$ (5.41)

where $E_{S}$ is diagonal with $(i,i)$ entry $-1$ if $i\in S$, $+1$ if
$i\not\in S$ and

$A_{\bm{j}}=\begin{pmatrix}2^{j_{1}}&\cdots&0\\\ \vdots&\ddots&\vdots\\\
0&\cdots&2^{j_{n}}\end{pmatrix}.$ (5.42)

We also set

$h_{S}=(\bm{x}_{S},0,0)\in H$ (5.43)

where $\bm{x}_{S}$ has $i$th entry $-1$ if $i\in S$ and $0$ if $i\not\in S$.

We observe that from lemma 5.4 we have

$\chi(\bm{x})=\sum_{\bm{j}\geq
0}\sum_{S}f_{n}\left((\bm{x}+\bm{x}_{S})E_{S}A_{\bm{j}}\right),$ (5.44)

where $\chi$ is the indicator function of the open unit cube $(0,1)^{n}$,

$f_{n}(x_{1},\dots,x_{n})=\prod_{1\leq j\leq n}f_{1}(x_{j}),$ (5.45)

and the sums are over $\bm{j}\in\mathbb{Z}^{n}$ with nonnegative entries and
all subsets $S$ of $\\{1,\dots,n\\}$. The characteristic function of the
rectangular box $\mathcal{B}=(0,b_{1})\times\cdots\times(0,b_{n})$ is
therefore

$\chi_{\mathcal{B}}(\bm{x})=\chi(\bm{x}B^{-1})=\sum_{\bm{j}\geq
0}\sum_{S}f_{n}\left((\bm{x}B^{-1}+\bm{x}_{S})E_{S}A_{\bm{j}}\right),$ (5.46)

where $B$ is the diagonal matrix with coefficients $b_{1},\ldots,b_{n}$.

Recalling the Schrödinger representation $W$ and the Segal-Shale-Weil
representation $R$, we have

$\chi(\bm{x})=\sum_{\bm{j}\geq
0}\sum_{S}2^{-\frac{1}{2}(j_{1}+\cdots+j_{n})}\left(W(h_{S})R(g_{\bm{j},S})f_{n}\right)(\bm{x})$
(5.47)

and

$\begin{split}\chi_{\mathcal{B}}(\bm{x})&=(\det
B)^{\frac{1}{2}}(R(\begin{pmatrix}B^{-1}&0\\\
0&B\end{pmatrix})\chi)(\bm{x})\\\ &=(\det B)^{\frac{1}{2}}\sum_{\bm{j}\geq
0}\sum_{S}2^{-\frac{1}{2}(j_{1}+\cdots+j_{n})}\left(R(\begin{pmatrix}B^{-1}&0\\\
0&B\end{pmatrix})W(h_{S})R(g_{\bm{j},S})f_{n}\right)(\bm{x}).\end{split}$
(5.48)

We note that for $(h,g)\in H\rtimes G$,

$W(h)R(g)W(h_{S})R(g_{\bm{j},S})=W(hh_{S}^{g^{-1}})R(gg_{\bm{j},S}),$ (5.49)

and so, as we are interested in the theta sums (1.3) with the sharp cutoff
$\chi$, it is natural to consider the expression

$\tilde{\Theta}_{\chi}(h,g)=\sum_{\bm{j}\geq
0}\sum_{S}2^{-\frac{1}{2}(j_{1}+\cdots+j_{n})}\Theta_{f_{n}}\left(hh_{S}^{g^{-1}},gg_{\bm{j},S}\right).$
(5.50)

The convergence of this expression for almost every $g\in G$ is a corollary of
lemma 5.5 below.

Motivated by bounding (5.50) via corollary 4.5, for $C>0$ and
$\psi:[0,\infty)\to[1,\infty)$ an increasing function, we define
$\mathcal{G}_{\bm{j}}(\psi,C)$ to be the set of $\Gamma g\in\Gamma\backslash
G$ such that

$D\bigg{(}\Gamma gg_{\bm{j},S}\begin{pmatrix}\mathrm{e}^{-s}I&0\\\
0&\mathrm{e}^{s}I\end{pmatrix}\bigg{)}\leq C^{4}\psi(s)^{4}$ (5.51)

for all $S\subset\\{1,\cdots,n\\}$, and $s\geq 1$.

######  Lemma 5.5.

Suppose $\psi$ satisfies

$\sum_{k\geq 0}\psi(k)^{-(2n+2)}\leq C_{\psi}$ (5.52)

for some $C_{\psi}>0$. Then

$\mu\left(\Gamma\backslash G-\mathcal{G}_{\bm{j}}(\psi,C)\right)\ll
C_{\psi}C^{-(2n+2)}$ (5.53)

with the implied constant depending only on $n$.

###### Proof.

Suppose $\Gamma g\not\in\mathcal{G}_{\bm{j}}(\psi,C)$, so

$D\bigg{(}\Gamma gg_{\bm{j},S}\begin{pmatrix}\mathrm{e}^{-s}I&0\\\
0&\mathrm{e}^{s}I\end{pmatrix}\bigg{)}\geq C^{4}\psi(s)^{4}$ (5.54)

for some $S\subset\\{1,\dots,n\\}$ and $s\geq 1$. Applying lemma 5.2 and the
fact that $\psi$ is increasing, we find that there is an integer $k\geq 0$
such that

$D\bigg{(}\Gamma gg_{\bm{j},S}\begin{pmatrix}\mathrm{e}^{-k}I&0\\\
0&\mathrm{e}^{k}I\end{pmatrix}\bigg{)}\gg C^{4}\psi(k)^{4}.$ (5.55)

Applying lemma 5.1 together with the fact that right multiplication is volume
preserving, we have that the volume of the set of $\Gamma g\in\Gamma\backslash
G$ satisfying (5.55) for a particular $S$ and $k$ is

$\ll C^{-(2n+2)}\psi(k)^{-(2n+2)}.$ (5.56)

We obtain the required estimate by bounding the volume of the union of these
sets over $S$ and $k$ by the sum of (5.56) over the relevant ranges. ∎

We now have all the ingredients for the proof of theorem 1.2.

###### Proof of theorem 1.2.

From (5.44) we express $\theta_{\mathcal{B}}(M,X,\bm{x},\bm{y})$ as

$\sum_{S\subset\\{1,\dots,n\\}}\sum_{\bm{j}\geq
0}f_{n}\left(\frac{1}{M}(\bm{m}+\bm{x}+M\bm{x}_{S}B)B^{-1}E_{S}A_{\bm{j}}\right)\\\
\mathrm{e}\left(\frac{1}{2}(\bm{m}+\bm{x})X\prescript{t}{}{\\!(\bm{m}+\bm{x})}+\bm{m}\prescript{t}{}{\\!\bm{y}}\right).$
(5.57)

Using (5.47), (5.49) we break the inner sum of (5.57) as

$M^{\frac{n}{2}}(\det B)^{\frac{1}{2}}\sum_{\begin{subarray}{c}\bm{j}\geq 0\\\
2^{j_{i}}b_{j_{i}}^{-1}\leq
M\end{subarray}}2^{-\frac{1}{2}(j_{1}+\cdots+j_{n})}\Theta_{f_{n}}\left(hh_{S}^{g_{MB,X}^{-1}},g_{MB,X}g_{\bm{j},S}\right)\\\
+\sum_{\begin{subarray}{c}\bm{j}\geq 0\\\
\max_{i}2^{j_{i}}b_{j_{i}}^{-1}>M\end{subarray}}f_{n}\left(\frac{1}{M}(\bm{m}+\bm{x}+M\bm{x}_{S}B)B^{-1}E_{S}A_{\bm{j}}\right)\mathrm{e}\left(\frac{1}{2}(\bm{m}+\bm{x})X\prescript{t}{}{\\!(\bm{m}+\bm{x})}+\bm{m}\prescript{t}{}{\\!\bm{y}}\right),$
(5.58)

where $h=(\bm{x},\bm{y},0)$ and

$g_{B,X}=\begin{pmatrix}I&X\\\ 0&I\end{pmatrix}\begin{pmatrix}B^{-1}&0\\\
0&B\end{pmatrix}.$ (5.59)

We first consider the second line of (5.58). Suppose that
$L\subset\\{1,\dots,n\\}$ is not empty and that $2^{j_{l}}>b_{j_{l}}M$ for all
$l\in L$. Then the compact support of $f_{1}$ implies that the sum over
$\bm{m}^{(L)}$, the vector of entries of $\bm{m}$ with index in $L$, has a
bounded number of terms. We write

$(\bm{m}+\bm{x})X\prescript{t}{}{\\!(\bm{m}+\bm{x})}=(\bm{m}^{(L)}+\bm{x}^{(L)})X^{(L,L)}\prescript{t}{}{\\!(\bm{m}^{(L)}+\bm{x}^{(L)})}\\\
+2(\bm{m}^{(L)}+\bm{x}^{(L)})X^{(L,L^{\prime})}\prescript{t}{}{\\!(\bm{m}^{(L^{\prime})}+\bm{x}^{(L^{\prime})})}+(\bm{m}^{(L^{\prime})}+\bm{x}^{(L^{\prime})})X^{(L^{\prime},L^{\prime})}\prescript{t}{}{\\!(\bm{m}^{(L^{\prime})}+\bm{x}^{(L^{\prime})})},$
(5.60)

where $L^{\prime}$ is the complement of $L$, and $X^{(L_{1},L_{2})}$ is the
matrix of entries of $X$ with row and column indices in $L_{1}$ and $L_{2}$
respectively. We have that
$f_{n}\left(\frac{1}{M}(\bm{m}+\bm{x}+M\bm{x}_{S}B)B^{-1}E_{S}A_{\bm{j}}\right)$
factors as

$f_{\\#L}\left(\frac{1}{M}(\bm{m}^{(L)}+\bm{x}^{(L)}+M\bm{x}_{S}^{(L)})(B^{(L,L)})^{-1}E_{S}^{(L,L)}A_{\bm{j}}^{(L,L)}\right)\\\
\times
f_{\\#L^{\prime}}\left(\frac{1}{M}(\bm{m}^{(L^{\prime})}+\bm{x}^{(L^{\prime})}+M\bm{x}_{S}^{(L^{\prime})})(B^{(L^{\prime},L^{\prime})})^{-1}E_{S}^{(L^{\prime},L^{\prime})}A_{\bm{j}}^{(L^{\prime},L^{\prime})}\right),$
(5.61)

and so, by inclusion-exclusion and the boundedness of $f_{\\#L}$, the terms
$\bm{j}$ of (5.57) with $\bm{j}_{l}>b_{j_{i}}M$ for some $i$ is at most a
constant (depending only on $n$) times

$\sum_{\begin{subarray}{c}L\subset\\{1,\dots,n\\}\\\
L\neq\emptyset\end{subarray}}\sum_{S\subset
L}\sum_{\bm{m}^{(L)}}\big{|}\theta_{\mathcal{B}^{(L^{\prime})}}(M,X^{L^{\prime},L^{\prime}},\bm{x}^{(L^{\prime})},\bm{y}^{(L^{\prime})}+(\bm{m}^{(L)}+\bm{x}^{(L)})X^{(L,L^{\prime})})\big{|},$
(5.62)

where the sum over $\bm{m}^{(L)}$ has a bounded number of terms,
$\mathcal{B}^{(L^{\prime})}$ is the edge of $\mathcal{B}$ associatated to
$L^{\prime}$, and we have used the decomposition (5.44) to express
$\theta_{\mathcal{B}^{(L^{\prime})}}(M,X^{L^{\prime},L^{\prime}},\bm{x}^{(L^{\prime})},\bm{y}^{(L^{\prime})}+(\bm{m}^{(L)}+\bm{x}^{(L)})X^{(L,L^{\prime})})$
as

$\sum_{S^{\prime}\subset
L^{\prime}}\sum_{\bm{j}_{L^{\prime}}}\sum_{\bm{m}_{L^{\prime}}}f_{\\#L^{\prime}}\left(\frac{1}{M}(\bm{m}^{(L^{\prime})}+\bm{x}^{(L^{\prime})}+M\bm{x}_{S}^{(L^{\prime})})(B^{(L^{\prime},L^{\prime})})^{-1}E_{S}^{(L^{\prime},L^{\prime})}A_{\bm{j}}^{(L^{\prime},L^{\prime})}\right)\\\
\times\mathrm{e}\left(\tfrac{1}{2}(\bm{m}^{(L^{\prime})}+\bm{x}^{(L^{\prime})})X^{(L^{\prime},L^{\prime})}\prescript{t}{}{\\!(\bm{m}^{(L^{\prime})}+\bm{x}^{(L^{\prime})})}+\bm{m}^{(L^{\prime})}\prescript{t}{}{(\bm{y}^{(L^{\prime})}+(\bm{m}^{(L)}+\bm{x}^{(L)})X^{(L,L^{\prime})})}\right).$
(5.63)

When $L=\\{1,\dots,n\\}$ or $n=1$, the corresponding part of (5.62) is clearly
bounded. Proceeding by induction on $n>1$, for any other $L$, there are full
measure subsets $\mathcal{X}^{(n-\\#L)}$ such that if
$X^{(L^{\prime},L^{\prime})}\in\mathcal{X}^{(n-\\#L)}$, the corresponding part
of (5.62) is $\ll M^{\frac{n-\\#L}{2}+\epsilon}$ for any $\epsilon>0$. It
follows that (5.62) is $\ll M^{\frac{n}{2}}$ assuming that $X$ is such that
$X^{(L^{\prime},L^{\prime})}\in\mathcal{X}^{(n-\\#L)}$ for all nonempty
$L\subset\\{1,\dots,n\\}$.

We now consider the part of (5.58) with $\bm{j}$ such that
$2^{j_{i}}b_{j_{i}}^{-1}\leq M$. We set $\mathcal{X}_{\bm{j}}(\psi,C)$ to be
the set of $X\in\mathbb{Z}^{n\times
n}_{\mathrm{sym}}\backslash\mathbb{R}^{n\times n}_{\mathrm{sym}}$ such that
there exist $A\in\mathrm{GL}(n,\mathbb{R})$ and $T\in\mathbb{R}^{n\times
n}_{\mathrm{sym}}$ satisfying
$\sup_{B\in\mathcal{K}}||A_{\bm{j}}^{-1}BAA_{\bm{j}}-I||\leq\epsilon_{n}$,
$||T||\leq\epsilon_{n}$, and

$\Gamma g=\Gamma\begin{pmatrix}I&X\\\ 0&I\end{pmatrix}\begin{pmatrix}A&0\\\
0&\prescript{t}{}{\\!A}^{-1}\end{pmatrix}\begin{pmatrix}I&0\\\
T&I\end{pmatrix}\in\mathcal{G}_{\bm{j}}(\psi,C).$ (5.64)

Here $\mathcal{K}$ is the compact subset in theorem 1.2 identified with the
compact subset of diagonal matrices $B$ in $\mathrm{GL}(n,\mathbb{R})$ in the
obvious way. We then set

$\mathcal{X}(\psi)=\left(\mathbb{Z}^{n\times
n}_{\mathrm{sym}}+\bigcup_{C>0}\bigcap_{\bm{j}\geq
0}\mathcal{X}_{\bm{j}}(\psi,C2^{\frac{1}{4}(j_{1}+\cdots+j_{n})})\right)\\\
\cap\bigcap_{L\subset\\{1,\dots,n\\}}\\{X\in\mathbb{R}^{n\times
n}_{\mathrm{sym}}:X^{(L^{\prime},L^{\prime})}\in\mathcal{X}^{(n-\\#L)}\\}\subset\mathbb{R}^{n\times
n}_{\mathrm{sym}}.$ (5.65)

We now verify that with $\psi$ satisfying the conditions of theorems 1.1, 1.2,
$\mathcal{X}(\psi)$ has full measure, noting (again by induction on $n$) that
it is enough to show that

$\bigcup_{C>0}\bigcap_{\bm{j}\geq
0}\mathcal{X}_{\bm{j}}(\psi,C2^{\frac{1}{4}(j_{1}+\cdots+j_{n})})$ (5.66)

has full measure in $\mathbb{Z}^{n\times
n}_{\mathrm{sym}}\backslash\mathbb{R}^{n\times n}_{\mathrm{sym}}$. First we
suppose that the Lebesgue measure of the complement of
$\mathcal{X}_{\bm{j}}(\psi,C)$ is greater than some $\epsilon>0$. Then, using
the expression (2.15) for the Haar measure on $G$, we have

$\mu\left(\Gamma\backslash G-\mathcal{G}_{\bm{j}}(\psi,C)\right)\gg\epsilon,$
(5.67)

with implied constant depending only on $n$ and $\mathcal{K}$. From lemma 5.5
it follows that

$\mathrm{meas}\left(\mathbb{Z}^{n\times
n}_{\mathrm{sym}}\backslash\mathbb{R}^{n\times
n}_{\mathrm{sym}}-\mathcal{X}_{\bm{j},}(\psi,C)\right)\ll C_{\psi}C^{-2n-2},$
(5.68)

and we find that

$\mathrm{meas}\left(\mathbb{Z}^{n\times
n}_{\mathrm{sym}}\backslash\mathbb{R}^{n\times
n}_{\mathrm{sym}}-\bigcup_{C>0}\bigcap_{\bm{j}\geq
0}\mathcal{X}_{\bm{j}}(\psi,C2^{\frac{1}{4}(j_{1}+\cdots+j_{n})})\right)\\\
\ll\lim_{C\to\infty}\sum_{\bm{j}\geq
0}C_{\psi}C^{-2n-2}2^{-\frac{n+1}{2}(j_{1}+\cdots+j_{n})}=0$ (5.69)

as required.

Now let us suppose that $X\in\mathcal{X}(\psi)$, so in particular the coset
$\mathbb{Z}^{n\times n}_{\mathrm{sym}}+X$ is in
$\mathcal{X}_{\bm{j}}(\psi,C2^{\frac{1}{4}(j_{1}+\cdots+j_{n})})$ for some
$C>0$ (independent of $\bm{j}$) and all $\bm{j}\geq 0$. We have from corollary
4.5 and the definition of the height function $D$ that

$\ll
M^{\frac{n}{2}}\sum_{S\subset\\{1,\dots,n\\}}\sum_{\begin{subarray}{c}\bm{j}\geq
0\\\ 2^{j_{i}}b_{j_{i}}^{-1}\leq
M\end{subarray}}2^{-\frac{1}{2}(j_{1}+\cdots+j_{n})}D(\Gamma
g_{MB,X}g_{\bm{j},S})^{\frac{1}{4}}$ (5.70)

bounds the first line of (5.58). Now for all $\bm{j}\geq 0$ there is a
$g\in\mathcal{G}_{\bm{j}}(\psi,C2^{\frac{1}{4}(j_{1}+\cdots+j_{n})})$ having
the form

$g=\begin{pmatrix}I&X\\\ 0&I\end{pmatrix}\begin{pmatrix}A&0\\\
0&\prescript{t}{}{\\!A}^{-1}\end{pmatrix}\begin{pmatrix}I&0\\\
T&I\end{pmatrix}$ (5.71)

with $||A_{\bm{j}}^{-1}BAA_{\bm{j}}-I||\leq\epsilon_{n}$ and
$||T||\leq\epsilon_{n}$. We have

$gg_{\bm{j},S}\begin{pmatrix}\frac{1}{M}I&0\\\ 0&MI\end{pmatrix}\\\
=g_{MB,X}g_{\bm{j},S}\begin{pmatrix}E_{S}A_{\bm{j}}^{-1}BAA_{\bm{j}}E_{S}&0\\\
0&E_{S}A_{\bm{j}}B^{-1}\prescript{t}{}{\\!A}^{-1}A_{\bm{j}}^{-1}E_{S}\end{pmatrix}\begin{pmatrix}I&0\\\
\frac{1}{M^{2}}A_{\bm{j}}TA_{\bm{j}}&I\end{pmatrix},$ (5.72)

and so lemma 5.3 implies

$D(g_{MB,X}g_{\bm{j},S})\asymp
D\left(gg_{\bm{j},S}\begin{pmatrix}\frac{1}{M}I&0\\\
0&MI\end{pmatrix}\right)\ll C2^{\frac{1}{4}(j_{1}+\cdots+j_{n})}\psi(\log M)$
(5.73)

since $g\in\mathcal{G}_{\bm{j}}(\psi,C2^{\frac{1}{4}(j_{1}+\cdots+j_{n})})$
and $2^{j_{i}}\leq Mb_{i}$ gives

$\frac{1}{M^{2}}||A_{\bm{j}}TA_{\bm{j}}||\leq||T||\leq\epsilon_{n}.$ (5.74)

It follows that (5.70) is bounded by

$\ll CM^{\frac{n}{2}}\psi(\log M)\sum_{\bm{j}\geq
0}2^{-\frac{1}{4}(j_{1}+\cdots+j_{n})}\ll CM^{\frac{n}{2}}\psi(\log M),$
(5.75)

and theorem 1.2 follows. ∎

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