# The Weyl law for congruence subgroups and arbitrary $K_{\infty}$-types

Werner Müller  Universität Bonn
Mathematisches Institut
Endenicher Allee 60
D – 53115 Bonn, Germany<EMAIL_ADDRESS>
###### Abstract.

Let $G$ be a reductive algebraic group over ${\mathbb{Q}}$ and $\Gamma\subset
G({\mathbb{Q}})$ an arithmetic subgroup. Let $K_{\infty}\subset
G({\mathbb{R}})$ be a maximal compact subgroup. We study the asymptotic
behavior of the counting functions of the cuspidal and residual spectrum,
respectively, of the regular representation of $G({\mathbb{R}})$ in
$L^{2}(\Gamma\backslash G({\mathbb{R}}))$ of a fixed $K_{\infty}$-type
$\sigma$. A conjecture, which is due to Sarnak, states that the counting
function of the cuspidal spectrum of type $\sigma$ satisfies Weyl’s law and
the residual spectrum is of lower order growth. Using the Arthur trace formula
we reduce the conjecture to a problem about $L$-functions occurring in the
constant terms of Eisenstein series. If $G$ satisfies property (L), introduced
by Finis and Lapid, we establish the conjecture. This includes classical
groups over a number field.

###### Key words and phrases:

Weyl law, locally symmetric spaces

###### 1991 Mathematics Subject Classification:

Primary: 11F72, Secondary: 58J35

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Arithmetic manifolds
  4. 4 Eisenstein series and intertwining operators
  5. 5 Normalizing factors
  6. 6 Logarithmic derivatives of local intertwining operators
  7. 7 The residual spectrum
  8. 8 The spectral side of the trace formula
  9. 9 Geometric side of the trace formula
  10. 10 Proof of the main theorem

##  1\. Introduction

Let $G$ be a connected semisimple algebraic group over ${\mathbb{Q}}$ and
$\Gamma\subset G({\mathbb{Q}})$ an arithmetic subgroup, which we assume to be
torsion free. A basic problem in the theory of automorphic forms is the study
of the spectral resolution of the regular representation $R_{\Gamma}$ of
$G({\mathbb{R}})$ in $L^{2}(\Gamma\backslash G({\mathbb{R}}))$. Of particular
importance is the determination of the structure of the discrete spectrum. Let
$L^{2}_{\mbox{\k dis}}(\Gamma\backslash G({\mathbb{R}}))$ be the discrete part
of $L^{2}(\Gamma\backslash G({\mathbb{R}}))$, i.e., the closure of the span of
all irreducible subrepresentations of $R_{\Gamma}$. Denote by
$R_{\Gamma,\mbox{\k dis}}$ the corresponding restriction of $R_{\Gamma}$.
Denote by $\Pi_{\mbox{\k dis}}(G({\mathbb{R}}))$ the set of isomorphism
classes of irreducible unitary representations of $G({\mathbb{R}})$, which
occur in $R_{\Gamma}$. By definition we have

(1.1) $R_{\Gamma,\mbox{\k dis}}=\widehat{\bigoplus}_{\pi\in\Pi_{\mbox{\k
dis}}(G({\mathbb{R}}))}m_{\Gamma}(\pi)\pi,$

where

$m_{\Gamma}(\pi)=\dim\operatorname{Hom}_{G({\mathbb{R}})}(\pi,R_{\Gamma})=\dim\operatorname{Hom}_{G({\mathbb{R}})}(\pi,R_{\Gamma,\mbox{\k
dis}})$

is the multiplicity with which $\pi$ occurs in $R_{\Gamma}$. Apart from
special cases, as for example discrete series representations, one cannot hope
to describe the multiplicity function $m_{\Gamma}$ on $\Pi(G({\mathbb{R}}))$
explicitly. Therefor it is feasible to study asymptotic questions such as the
limit multiplicity problem [FLM2] and the Weyl law, which is the subject of
this article.

To begin with we recall that the discrete spectrum decomposes into the
cuspidal and the residual spectrum. Let ${\bf K}_{\infty}$ be a maximal
compact subgroup of $G({\mathbb{R}})$. Let
${\mathcal{Z}}({\mathfrak{g}}_{\mathbb{C}})$ be the center of the universal
enveloping algebra of the complexification of the Lie algebra ${\mathfrak{g}}$
of $G({\mathbb{R}})$. Recall that a cusp form for $\Gamma$ is a smooth and
right ${\bf K}_{\infty}$-finite function $\phi\colon\Gamma\backslash
G({\mathbb{R}})\to{\mathbb{C}}$ which is a simultaneous eigenfunction of
${\mathcal{Z}}({\mathfrak{g}}_{\mathbb{C}})$ and which satisfies

(1.2) $\int_{\Gamma\cap N_{P}({\mathbb{R}})\backslash
N_{P}({\mathbb{R}})}\phi(nx)dn=0$

for all unipotent radicals $N_{P}$ of proper rational parabolic subgroups $P$
of $G$. By Langlands’ theory of Eisenstein series [La1], cusp forms are the
building blocks of the spectral resolution. We note that each cusp form
$\phi\in C^{\infty}(\Gamma\backslash G({\mathbb{R}}))$ is rapidly decreasing
on $\Gamma\backslash G({\mathbb{R}})$ and hence square integrable. Let
$L^{2}_{\mbox{\k cus}}(\Gamma\backslash G({\mathbb{R}}))$ be the closure of
the linear span of all cusp forms. The restriction of the regular
representation $R_{\Gamma}$ to $L^{2}_{\mbox{\k cus}}(\Gamma\backslash
G({\mathbb{R}}))$ decomposes discretely and $L^{2}_{\mbox{\k
cus}}(\Gamma\backslash G({\mathbb{R}}))$ is a subspace of $L^{2}_{\mbox{\k
dis}}(\Gamma\backslash G({\mathbb{R}}))$. Denote by $L^{2}_{\mbox{\k
res}}(\Gamma\backslash G({\mathbb{R}}))$ the orthogonal complement of
$L^{2}_{\mbox{\k cus}}(\Gamma\backslash G({\mathbb{R}}))$ in $L^{2}_{\mbox{\k
dis}}(\Gamma\backslash G({\mathbb{R}}))$. This is the residual subspace. Let
$(\sigma,V_{\sigma})$ be an irreducible unitary representation of ${\bf
K}_{\infty}$. Set

(1.3) $L^{2}(\Gamma\backslash G({\mathbb{R}}),\sigma):=(L^{2}(\Gamma\backslash
G({\mathbb{R}}))\otimes V_{\sigma})^{{\bf K}_{\infty}}.$

Define the subspaces $L^{2}_{\mbox{\k dis}}(\Gamma\backslash
G({\mathbb{R}}),\sigma)$, $L^{2}_{\mbox{\k cus}}(\Gamma\backslash
G({\mathbb{R}}),\sigma)$ and $L^{2}_{\mbox{\k res}}(\Gamma\backslash
G({\mathbb{R}}),\sigma)$ in a similar way. Then $L^{2}_{\mbox{\k
cus}}(\Gamma\backslash G({\mathbb{R}}),\sigma)$ is the space of cusp forms
with fixed ${\bf K}_{\infty}$-type $\sigma$. Let
$\widetilde{X}:=G({\mathbb{R}})/{\bf K}_{\infty}$ be the Riemannian symmetric
space associated to $G({\mathbb{R}})$ and $X=\Gamma\backslash\widetilde{X}$
the corresponding locally symmetric space. Since we assume that $\Gamma$ is
torsion free, $X$ is a manifold. Let $E_{\sigma}\to\Gamma\backslash X$ be the
locally homogeneous vector bundle associated to $\sigma$ and let
$L^{2}(X,E_{\sigma})$ be the space of square integrable sections of
$E_{\sigma}$. There is a canonical isomorphism

(1.4) $L^{2}(\Gamma\backslash G({\mathbb{R}}),\sigma)\cong
L^{2}(X,E_{\sigma}).$

Let $\Omega_{G({\mathbb{R}})}\in{\mathcal{Z}}({\mathfrak{g}}_{\mathbb{C}})$ be
the Casimir element of $G({\mathbb{R}})$. Then
$-\Omega_{G({\mathbb{R}})}\otimes\operatorname{Id}$ induces a self-adjoint
operator $\Delta_{\sigma}$ in the Hilbert space $L^{2}(\Gamma\backslash
G({\mathbb{R}}),\sigma)$ which is bounded from below. With respect to the
isomorphism (1.4) we have

(1.5)
$\Delta_{\sigma}=(\nabla^{\sigma})^{\ast}\nabla^{\sigma}-\lambda_{\sigma}\operatorname{Id},$

where $\nabla^{\sigma}$ is the canonical invariant connection in $E_{\sigma}$
and $\lambda_{\sigma}$ denotes the Casimir eigenvalue of $\sigma$. In
particular, if $\sigma_{0}$ is the trivial representation, then
$L^{2}(\Gamma\backslash G({\mathbb{R}}),\sigma_{0})\cong L^{2}(X)$ and
$\Delta_{\sigma_{0}}$ equals the Laplacian $\Delta$ on $X$.

The restriction of $\Delta_{\sigma}$ to the subspace $L^{2}_{\mbox{\k
dis}}(\Gamma\backslash G({\mathbb{R}}),\sigma)$ has pure point spectrum
consisting of eigenvalues $\lambda_{0}(\sigma)<\lambda_{1}(\sigma)<\cdots$ of
finite multiplicities. Let ${\mathcal{E}}(\lambda_{i}(\sigma))$ be the
eigenspace corresponding to $\lambda_{i}(\sigma)$. Then we define the
eigenvalue counting function $N_{\Gamma,\mbox{\k dis}}(\lambda,\sigma)$,
$\lambda\geq 0$, by

(1.6) $N_{\Gamma,\mbox{\k
dis}}(\lambda,\sigma)=\sum_{\lambda_{i}(\sigma)\leq\lambda}\dim{\mathcal{E}}(\lambda_{i}(\sigma)).$

The counting functions $N_{\Gamma,\mbox{\k cus}}(\lambda,\sigma)$ and
$N_{\Gamma,\mbox{\k res}}(\lambda,\sigma)$ of the cuspidal and residual
spectrum are defined by considering the restriction of $\Delta_{\sigma}$ to
the cuspidal and residual subspace, respectively. The main goal is to
determine the asymptotic behavior of the counting functions as
$\lambda\to\infty$. If $X$ is compact, the Weyl law holds. Recall that for a
compact Riemannian manifold $X$ of dimension $n$, the Weyl law states that the
number $N_{X}(\lambda)$ of eigenvalues $\lambda_{i}\leq\lambda$, counted with
multiplicity, of the Laplace operator $\Delta$ of $X$ satisfies

(1.7)
$N_{X}(\lambda)=\frac{\operatorname{vol}(X)}{(4\pi)^{n}\Gamma(\frac{n}{2}+1)}\lambda^{n/2}+o(\lambda^{(n/2})$

as $\lambda\to\infty$. A standard method to prove (1.7) is the heat equation
method. Using the wave equation, one gets a more precise version with an
estimation of the remainder term:

(1.8)
$N_{X}(\lambda)=\frac{\operatorname{vol}(X)}{(4\pi)^{n}\Gamma(\frac{n}{2}+1)}\lambda^{n/2}+O(\lambda^{(n-1)/2})$

as $\lambda\to\infty$. This is due to Avakumovic [Av] and Hörmander. Without
further assumptions on the Riemannian manifold, the remainder term is optimal
[Av]. More generally, one can consider the Bochner-Laplace operator
$\Delta_{E}$ for a Hermitian vector bundle $E\to X$ with Hermitian connection.
There is a similar formula (1.7) for the eigenvalue counting function
$N_{X}(\lambda,E)$ of $\Delta_{E}$. The only difference is the rank of $E$
which appears on the right hand side in the leading coefficient.

For non-uniform lattices $\Gamma$ the self-adjoint operator $\Delta_{\sigma}$
has a large continuous spectrum so that alomost all eigenvalues of
$\Delta_{\sigma}$ will be embedded in the continuous spectrum which makes it
very difficult to study them. A number of results are known for the spherical
cuspidal spectrum. The first results concerning the growth of the cuspidal
spectrum are due to Selberg [Se1]. He proved that for every congruence
subgroup $\Gamma\subset\operatorname{SL}(2,{\mathbb{Z}})$, the counting
function of the cuspidal spectrum satisfies Weyl’s law, i.e., one has

(1.9)
$N_{\Gamma}(\lambda)=\frac{\operatorname{vol}(\Gamma\backslash{\mathbb{H}}^{2})}{4\pi}\lambda^{2}-\frac{2m}{\pi}\lambda\log\lambda+O(\lambda),$

as $\lambda\to\infty$. This shows that for congruence subgroups eigenvalues
exist in abundance. On the other hand, based on their work on the dissolution
of cusp forms under deformation of lattices, Phillips and Sarnak [Sa2]
conjectured that except for the Teichmüller space of the once punctured torus,
the point spectrum of the Laplacian on $\Gamma\backslash{\mathbb{H}}^{2}$ for
a generic non-uniform lattice $\Gamma$ in $\operatorname{SL}(2,{\mathbb{R}})$
is finite and is contained in $[0,1/4)$. In the more general context of
manifolds with cusps Colin de Verdère [CV] has shown that under a generic
compactly supported conformal deformation of the metric of a non-compact
hyperbolic surface of finite area all eigenvalues $\lambda\geq 1/4$ are
dissolved.

If $\operatorname{rank}(G)>1$, the situation is very different. By the results
of Margulis, we have rigidity of irreducible lattices and irreducible lattices
are arithmetic. One expects that arithmetic groups have a large discrete
spectrum. The following conjecture is due to Sarnak [Sa1].

######  Conjecture 1.1.

Let $\Gamma\subset G({\mathbb{Q}})$ be a congruence subgroup. Then for every
$\nu\in\Pi({\bf K}_{\infty})$, $N_{\Gamma,\mbox{\k cus}}(\lambda;\nu)$
satisfies Weyl’s law and $N_{\Gamma,\mbox{\k res}}(\lambda;\nu)$ is of lower
order growth.

There are some general results concerning the conjecture. Let $G$ be a
connected real semisimple Lie group, $K$ a maximal compact subgroup of $G$,
and $\Gamma\subset G$ a torsion free lattice. Let $n=\dim G/K$. Donnelly [Do,
Theorem 9.1] has established the following upper bound for the cuspidal
spectrum

(1.10) $\limsup_{\lambda\to\infty}\frac{N_{\Gamma,\mbox{\k
cus}}(\lambda;\nu)}{\lambda^{n/2}}\leq\frac{\dim(\nu)\operatorname{vol}(\Gamma\backslash
G/K)}{(4\pi)^{n/2}\Gamma(\frac{n}{2}+1)},$

which holds for every $\nu\in\Pi(K)$. Concerning the residual spectrum, it was
proved in [Mu1, Theorem 0.1] that for a general lattice one has

(1.11) $N_{\Gamma,\mbox{\k res}}(\lambda;\nu)\ll 1+\lambda^{2n}.$

However, this is not the optimal bound that one expects. In general, one would
expect that the residual spectrum is of order $O(\lambda^{n/2})$ and for
arithmetic groups of order $O(\lambda^{(n-1)/2})$ as $\lambda\to\infty$.

Conjecture (1.1) has been verified in a number of cases. Most of the results
are obtained for the spherical spectrum. The first result in higher rank is
due to S.D. Miller [Mil] who established the Weyl law for spherical cusp forms
for $\Gamma=\operatorname{SL}(3,{\mathbb{Z}})$. The author [Mu3] proved it for
a principal congruence subgroup
$\Gamma\subset\operatorname{SL}(n,{\mathbb{Z}})$. The method of proof follows
Selberg’s approach and uses the trace formula. Then Lindenstrauss and
Venkatesh [LV] proved the Weyl law for spherical cusps forms in great
generality, namely for congruence subgroups $\Gamma\subset\bf
G({\mathbb{R}})$, where $\bf G$ is a split adjoint semisimple group over
${\mathbb{Q}}$. The method is different. It uses Hecke operators to eliminate
the contribution of Eisenstein series. For congruence subgroups of
$\operatorname{SL}(n,{\mathbb{Z}})$, E. Lapid and W. Müller [LM] established
the Weyl law for the cuspidal spectrum with an estimation of the remainder
term. The order of the remainder term is
$O(\lambda^{(d-1)/2}(\log\lambda)^{\max(n,3)})$ where
$d=\dim\operatorname{SL}(n,{\mathbb{R}})/\operatorname{SO}(n)$. The method is
also based on the Arthur trace formula as in [Mu3]. However, the argument is
simplified and strengthened, which corresponds to the use of the wave equation
in the derivation of the Weyl law for a compact Riemannian manifold. Recently
T. Finis and E. Lapid [FL2] estimated the remainder term for the cuspidal
spectrum of a locally symmetric space $X=\Gamma\backslash{\bf
G}({\mathbb{R}})/{\bf K}_{\infty}$, where $\bf G$ is a simply connected,
simple Chevalley group and $\Gamma$ a congruence subgroup of $\bf
G({\mathbb{Z}})$. The method also uses Hecke operators as in [LV], but in a
slightly different way. The estimation they obtain is $O(\lambda^{d-\delta})$,
where $d=\dim X$ and $\delta>0$ some constant which is not further specified.
In [FM], T. Finis and J. Matz included Hecke operators. They studied the
asymptotic behavior of the traces of Hecke operators for the spherical
discrete spectrum.

For the non-spherical case, the Weyl law was proved in [Mu3] for a principal
congruence subgroup of $\operatorname{SL}(n,{\mathbb{Z}})$. Recently, A. Maiti
[Ma] has generalized the approach of Lindenstrauss and Venkatesh [LV] to
establish the Weyl law for cusp forms and arbitrary $K_{\infty}$-types. As in
[LV], the method works for a semi-simple, split, adjoint linear algebraic
group over ${\mathbb{Q}}$. It provides no results for the residual spectrum.

Concerning the residual spectrum, there is the general upper bound (1.11),
which, however, is not the expected optimal one. For
$\operatorname{rank}(G)=1$, the residual spectrum is known to be finite. For
$\operatorname{GL}(n)$ the residual spectrum has been determined by Mœglin and
Waldspurger [MW]. This has been used in [Mu3, Proposition 3.6] to prove that
in this case the residual spectrum is of lower order growth.,

The main goal of the present paper is to prove Conjecture (1.1) for a certain
class of reductive groups including classical groups over a number field. We
use the Arthur trace formula to reduce the proof of the conjecture to a
problem about automorphic $L$-functions occurring in the constant terms of
Eisenstein series. This problem can be dealt with if the reductive group $G$
satisfies property (L), which was introduced by Finis and Lapid in [FL1,
Definition 3.4]. Let $G$ be a reductive group over ${\mathbb{Q}}$. As usual,
let $G({\mathbb{R}})^{1}$ denote the intersection of the kernels of the
homorphisms $|\chi|\colon G({\mathbb{R}})\to{\mathbb{R}}^{>0}$, where $\chi$
ranges over the ${\mathbb{Q}}$-rational characters of $G$. Then our main
result is the following theorem. vvv

######  Theorem 1.2.

Let $G_{0}$ be a connected reductive algebraic group over a number field $F$
which satisfies property (L). Let
$G=\operatorname{Res}_{F/{\mathbb{Q}}}(G_{0}/F)$. Let ${\bf K}_{\infty}\subset
G({\mathbb{R}})^{1}$ be a maximal compact subgroup and let $n=\dim
G({\mathbb{R}})^{1}/{\bf K}_{\infty}$. Let $\Gamma\subset G({\mathbb{Q}})$ be
a torsion free congruence subgroup. Then for every $\nu\in\Pi({\bf
K}_{\infty})$ we have

(1.12) $N_{\Gamma,\mbox{\k
cus}}(\lambda;\nu)\sim\frac{\dim(\nu)vol(\Gamma\backslash
G({\mathbb{R}})^{1}/{\bf
K}_{\infty})}{(4\pi)^{n/2}\Gamma(\frac{n}{2}+1)}\lambda^{n/2},\quad\lambda\to\infty.$

and

(1.13) $N_{\Gamma,\mbox{\k res}}(\lambda;\nu)\ll
1+\lambda^{(n-1)/2},\quad\lambda>0.$

Thus in order to establish the Weyl law and the estimation of the residual
spectrum for every ${\bf K}_{\infty}$-type, we are reduced to the verification
that $G_{0}$ satisfies property (L). For $\operatorname{GL}(n)$ the relevant
$L$-functions are the Rankin-Selberg $L$-functions, which are known to satisfy
the pertinent properties. Using Arthur’s work on functoriality from classical
groups to $\operatorname{GL}(n)$, T. Finis and E. Lapid [FL1, Theorem 3.11]
proved that quasi-split classical groups over a number field $F$ satisfy
property (L). Moreover, they also proved that inner forms of
$\operatorname{GL}(n)$ and the exceptional group $G_{2}$ over a number field
$F$ satisfy property (L). In fact, one expects that property (L) holds for all
reductive groups. Currently, we only know [FL1, Theorem 3.11]. Together with
Theorem 1.2 this leads to the following corollary.

######  Corollary 1.3.

Let $F$ be a number field and let $G_{0}$ be one of the following groups over
$F$:

  1. (1)

$\operatorname{GL}(n)$ and its inner forms.

  2. (2)

Quasi-split classical groups.

  3. (3)

The exceptional group $G_{2}$.

Let ${G}=\operatorname{Res}_{F/{\mathbb{Q}}}({G_{0}}/F)$. Let $\Gamma\subset
G({\mathbb{Q}})$ be a congruence subgroup and $\nu\in\Pi({\bf K}_{\infty})$.
Then (1.12) and (1.13) hold.

Our approach to prove Theorem 1.2 is a generalization of the heat equation
method to the non-compact setting. The basic tool is the Arthur trace formula.
This requires to pass to the adelic setting. We will work with reductive
groups over a number field $F$. However, for the rest of the introduction we
will assume that $F={\mathbb{Q}}$. So let $G$ be a connected reductive group
defined over ${\mathbb{Q}}$. Let ${\mathbb{A}}$ be the ring of adeles of
${\mathbb{Q}}$. Let $G({\mathbb{A}})^{1}:=\cap_{\chi}\ker|\chi|$, where $\chi$
runs over the rational characters of $G$. Denote by $T_{G}$ the split
component of the center of $G$ and let $A_{G}$ be the component of the
identity of $T_{G}({\mathbb{R}})$. Then

$G({\mathbb{A}})=A_{G}\times G({\mathbb{A}})^{1}.$

We replace $\Gamma\backslash G({\mathbb{R}})^{1}$ by the adelic quotient
$A_{G}G({\mathbb{Q}})\backslash
G({\mathbb{A}})/K_{f}=G({\mathbb{Q}})\backslash G({\mathbb{A}})^{1}/K_{f}$,
where $K_{f}\subset G({\mathbb{A}}_{f})$ is an open compact subgroup. Let
$\Pi(G({\mathbb{A}}))$ (resp. $\Pi(G({\mathbb{A}})^{1})$) be the set of
equivalence classes of irreducible unitary representations of
$G({\mathbb{A}})$ (resp. $G({\mathbb{A}})^{1}$). We identify a representation
of $G({\mathbb{A}})^{1}$ with a representation of $G({\mathbb{A}})$, which is
trivial on $A_{G}$. Let $L^{2}_{\mbox{\k dis}}(A_{G}G({\mathbb{Q}})\backslash
G({\mathbb{A}}))$ be the closure of the span of all irreducible
subrepresentations of the regular representation $R$ of $G({\mathbb{A}})$ in
$L^{2}(A_{G}G({\mathbb{Q}})\backslash G({\mathbb{A}}))$. Denote by
$\Pi_{\mbox{\k dis}}(G({\mathbb{A}}))$ the subspace of all
$\pi\in\Pi(G({\mathbb{A}}))$ which are equivalent to a subrepresentation of
the regular representation of $G({\mathbb{A}})$ in
$L^{2}(A_{G}G({\mathbb{Q}})\backslash G({\mathbb{A}}))$. Note that this is a
countable set. Denote by $R_{\mbox{\k dis}}$ the restriction of $R$ to
$L^{2}_{\mbox{\k dis}}(A_{G}G({\mathbb{Q}})\backslash G({\mathbb{A}}))$. Then

(1.14) $R_{\mbox{\k dis}}\cong\widehat{\bigoplus}_{\pi\in\Pi_{\mbox{\k
dis}}(G({\mathbb{A}}))}m(\pi)\pi,$

where

(1.15) $m(\pi)=\dim\operatorname{Hom}(\pi,L^{2}(A_{G}G({\mathbb{Q}})\backslash
G({\mathbb{A}}))$

is the multiplicity with which $\pi$ occurs in
$L^{2}(A_{G}G({\mathbb{Q}})\backslash G({\mathbb{A}}))$. Any
$\pi\in\Pi(G({\mathbb{A}}))$ can be written as
$\pi=\pi_{\infty}\otimes\pi_{f}$, where $\pi_{\infty}$ and $\pi_{f}$ are
irreducible unitary representations of $G({\mathbb{R}})$ and
$G({\mathbb{A}}_{f})$, respectively. Let ${\mathcal{H}}_{\pi_{\infty}}$ and
${\mathcal{H}}_{\pi_{f}}$ denote the Hilbert space of the representation
$\pi_{\infty}$ and $\pi_{f}$, respectively. Let $K_{f}\subset
G({\mathbb{A}}_{f})$ be an open compact subgroup. Denote by
${\mathcal{H}}_{\pi_{f}}^{K_{f}}$ the subspace of $K_{f}$-invariant vectors in
${\mathcal{H}}_{\pi_{f}}$. Let $G({\mathbb{R}})^{1}=G({\mathbb{A}})^{1}\cap
G({\mathbb{R}})$. Given $\pi\in\Pi(G({\mathbb{A}}))$, denote by
$\lambda_{\pi_{\infty}}$ the Casimir eigenvalue of the restriction of
$\pi_{\infty}$ to $G({\mathbb{R}})^{1}$. Let $\nu\in\Pi({\bf K}_{\infty})$.
Then we define the adelic counting function of the discrete spectrum by

(1.16) $N_{\mbox{\k
dis}}^{K_{f},\nu}(\lambda):=\sum_{\begin{subarray}{c}\pi\in\Pi_{\mbox{\k
dis}}(G({\mathbb{A}}))\\\
-\lambda_{\pi_{\infty}}\leq\lambda\end{subarray}}m(\pi)\dim({\mathcal{H}}_{\pi_{f}}^{K_{f}})\dim({\mathcal{H}}_{\pi_{\infty}}\otimes
V_{\nu})^{K_{\infty}}.$

In the same way we define the counting functions $N_{\mbox{\k
cus}}^{K_{f},\nu}(\lambda)$ and $N_{\mbox{\k res}}^{K_{f},\nu}(\lambda)$ of
the cuspidal and residual spectrum, respectively. The adelic version of
Theorem 1.2 is then

######  Theorem 1.4.

Let $G_{0}$ be a connected reductive algebraic group over a number field $F$.
Assume that $G_{0}$ satisfies property (L). Let
$G=\operatorname{Res}_{F/{\mathbb{Q}}}(G_{0})$ be the group that is obtained
from $G_{0}$ by restriction of scalars. Let ${\bf K}_{\infty}$ be a maximal
compact subgroup of $G({\mathbb{R}})^{1}$. Let
$d:=\dim(G({\mathbb{R}})^{1}/{\bf K}_{\infty})$. Let $K_{f}\subset
G({\mathbb{A}}_{f})$ be an open compact subgroup and let $\nu\in\Pi({\bf
K}_{\infty})$. Then we have

(1.17) $N_{\mbox{\k
cus}}^{K_{f},\nu}(\lambda)=\frac{\dim(\nu)\operatorname{vol}(A_{G}G({\mathbb{Q}})\backslash
G({\mathbb{A}})/K_{f})}{(4\pi)^{d/2}\Gamma(\frac{d}{2}+1)}\lambda^{d/2}+o(\lambda^{d/2}),\quad\lambda\to\infty,$

and

(1.18) $N_{\mbox{\k
res}}^{K_{f},\nu}(\lambda)\ll(1+\lambda^{(n-1)/2}),\quad\lambda>0.$

To deduce Theorem 1.2 from the adelic version, we recall that there exist
finitely many congruence subgroups $\Gamma_{i}\subset G({\mathbb{Q}})$,
$i=1,...,m$, such that

(1.19) $A_{G}G({\mathbb{Q}})\backslash
G({\mathbb{A}})/K_{f}=\bigsqcup_{i=1}^{m}\Gamma_{i}\backslash
G({\mathbb{R}})^{1}.$

(see sect. 3). Denote by $N_{\Gamma_{i},\mbox{\k cus}}(\lambda,\nu)$ the
counting function for the cuspidal spectrum $L^{2}(\Gamma_{i}\backslash
G({\mathbb{R}})^{1})\otimes V_{\nu})^{K_{\infty}}$. Then it follows that

(1.20) $N_{\mbox{\k
cus}}^{K_{f},\nu}(\lambda)=\sum_{i=1}^{m}N_{\Gamma_{i},\mbox{\k
cus}}(\lambda,\nu).$

This is used to derive Theorem 1.2 from Theorem 1.4.

To prove Theorem 1.4 we start with the estimation of the residual counting
function, which is needed to establish the Weyl law. For this purpose we use
Langlands’ description of the residual spectrum in terms of iterated residues
of Eisenstein series [La1, Ch. 7], [MW, V.3.13]. Using the Maass-Selberg
relations, the problem is finally reduced to the estimation of the number of
real poles of the normalizing factors of intertwining operators, which appear
in the constant terms of Eisenstein series. To obtain the appropriate bounds,
we need that $G$ satisfies property (L) which was introduced by Finis and
Lapid [FL1, Definition 3.4]. In this way we get (1.18).

To prove the Weyl law, we use the Arthur trace formula. We will work with
groups over a number field $F$. However, in order to explain the method we
will simply assume that $F={\mathbb{Q}}$. We proceed as in [Mu3]. We choose
test functions $\phi^{\nu}_{t}\in C_{c}^{\infty}(G({\mathbb{A}})^{1})$, $t>0$,
which at the infinite place are obtained from the heat kernel $H_{t}^{\nu}$ of
the Bochner-Laplace operator $\widetilde{\Delta}_{\nu}$ on the symmetric space
$\widetilde{X}=G({\mathbb{R}})^{1}/{\bf K}_{\infty}$ and which at the finite
places is given by the normalized characteristic function of $K_{f}$ (see
(8.11) for the precise definition). Then we insert $\phi_{t}^{\nu}$ into the
spectral side $J_{\text{spec}}$ of the trace formula and study the asymptotic
behavior of $J_{\text{spec}}(\phi_{t}^{\nu})$ as $t\to 0$. The spectral side
is a sum of distributions $J_{\text{spec},M}$ associated to conjugacy classes
of Levi subgroups $M$ of $G$. For $M=G$ we have

(1.21) $J_{\text{spec},G}(\phi^{\nu}_{t})=\sum_{\pi\in\Pi_{\mbox{\k
dis}}(G({\mathbb{A}}))}m(\pi)e^{t\lambda_{\pi_{\infty}}}\dim({\mathcal{H}}_{\pi_{f}}^{K_{f}})\dim({\mathcal{H}}_{\pi_{\infty}}\otimes
V_{\nu})^{K_{\infty}},$

which is the contribution of the discrete spectrum to the spectral side. For
$M\neq G$, the main ingredients of the distributions $J_{\text{spec},M}$ are
logarithmic derivatives of intertwining operators. The intertwining operators
can be normalized by certain meromorphic functions. Then the logarithmic
derivatives of the intertwining operators are expressed in terms of
logarithmic derivatives of the normalizing factors and logarithmic derivatives
of the local normalized intertwining operators. In fact, we only need to
control integrals of logarithmic derivatives which simplifies the problem. To
deal with the integrals of logarithmic derivatives of the normalizing factors,
we use property (TWN+) [FL1, Definition 3.3]. By [FL1, Proposition 3.8],
property (TWN+) is a consequence of property (L) [FL1, Definition 3.4], which
we assume to be satisfied by $G$. To deal with the local intertwining
operators, we follow essentially the approach used in [FLM2]. The final result
is Theorem 8.5, which states that if $G$ satisfies property (L), then

(1.22)
$J_{\text{spec}}(\phi_{t}^{\nu})=J_{\text{spec},G}(\phi_{t}^{\nu})+O(t^{-(d-1)/2})$

as $t\to 0$.

Next we come to the geometric side $J_{\mbox{\k geom}}(\phi_{t}^{\nu})$. Its
asymptotic behavior as $t\to 0$ has been determined in [MM2, Theorem 1.1]. We
will briefly recall the main steps of the proof and determine the leading
coefficient. By the trace formula we have
$J_{\text{spec}}(\phi_{t}^{\nu})=J_{\mbox{\k geom}}(\phi_{t}^{\nu})$, which
together with (1.21) and (1.22) leads to

(1.23) $\begin{split}\sum_{\pi\in\Pi_{\mbox{\k
dis}}(G({\mathbb{A}}))}m(\pi)\dim({\mathcal{H}}_{\pi_{f}}^{K_{f}})\dim&\left({\mathcal{H}}_{\pi_{\infty}}\otimes
V_{\nu}\right)^{K_{\infty}}\,e^{t\lambda_{\pi_{\infty}}}\\\
&=\frac{\dim(\nu)\,\operatorname{vol}(X(K_{f}))}{(4\pi)^{d/2}}t^{-d/2}+O(t^{-(d-1)/2})\end{split}$

as $t\to 0$. Applying Karamata’s theorem, we obtain the adelic Weyl law
(1.17).

##  2\. Preliminaries

We will mostly use the notation of [FLM1]. Let $G$ be a reductive algebraic
group defined over a number field $F$. We fix a minimal parabolic subgroup
$P_{0}$ of $G$ defined over $F$ and a Levi decomposition $P_{0}=M_{0}U_{0}$,
both defined over $F$. Let $T_{0}$ be the $F$-split component of the center of
$M_{0}$. Let ${\mathcal{F}}$ be the set of parabolic subgroups of $G$ which
contain $M_{0}$ and are defined over $F$. Let ${\mathcal{L}}$ be the set of
subgroups of $G$ which contain $M_{0}$ and are Levi components of groups in
${\mathcal{F}}$. For any $P\in{\mathcal{F}}$ we write

$P=M_{P}N_{P},$

where $N_{P}$ is the unipotent radical of $P$ and $M_{P}$ belongs to
${\mathcal{L}}$.

Let $M\in{\mathcal{L}}$. Denote by $T_{M}$ the $F$-split component of the
center of $M$. Put $T_{P}=T_{M_{P}}$. With our previous notation, we have
$T_{0}=T_{M_{0}}$. Let $L\in{\mathcal{L}}$ and assume that $L$ contains $M$.
Then $L$ is a reductive group defined over $F$ and $M$ is a Levi subgroup of
$L$. We shall denote the set of Levi subgroups of $L$ which contain $M$ by
${\mathcal{L}}^{L}(M)$. We also write ${\mathcal{F}}^{L}(M)$ for the set of
parabolic subgroups of $L$, defined over $F$, which contain $M$, and
${\mathcal{P}}^{L}(M)$ for the set of groups in ${\mathcal{F}}^{L}(M)$ for
which $M$ is a Levi component. Each of these three sets is finite. If $L=G$,
we shall usually denote these sets by ${\mathcal{L}}(M)$, ${\mathcal{F}}(M)$
and ${\mathcal{P}}(M)$.

Let $W_{0}=N_{G(F)}(T_{0})/M_{0}$ be the Weyl group of $(G,T_{0})$, where
$N_{G(F)}(H)$ denotes the normalizer of $H$ in $G(F)$. For any $s\in W_{0}$ we
choose a representative $w_{s}\in G(F)$. Note that $W_{0}$ acts on
${\mathcal{L}}$ by $sM=w_{s}Mw_{s}^{-1}$. For $M\in{\mathcal{L}}$ let
$W(M)=N_{G(F)}(M)/M$, which can be identified with a subgroup of $W_{0}$.

Let $X(M)_{F}$ be the group of characters of $M$ which are defined over $F$.
Put

(2.1) ${\mathfrak{a}}_{M}:=\operatorname{Hom}(X(M)_{F},{\mathbb{R}}).$

This is a real vector space whose dimension equals that of $T_{M}$. Its dual
space is

${\mathfrak{a}}_{M}^{\ast}=X(M)_{F}\otimes{\mathbb{R}}.$

We shall write,

(2.2)
${\mathfrak{a}}_{P}={\mathfrak{a}}_{M_{P}}\quad\text{and}\quad{\mathfrak{a}}_{0}={\mathfrak{a}}_{M_{0}}.$

For any $L\in{\mathcal{L}}(M)$ we identify ${\mathfrak{a}}_{L}^{\ast}$ with a
subspace of ${\mathfrak{a}}_{M}^{\ast}$. We denote by ${\mathfrak{a}}_{M}^{L}$
the annihilator of ${\mathfrak{a}}_{L}^{\ast}$ in ${\mathfrak{a}}_{M}$. Then
$r=\dim{\mathfrak{a}}_{0}^{G}$ is the semisimple rank of $G$. We set

(2.3)
${\mathcal{L}}_{1}(M)=\\{L\in{\mathcal{L}}(M):\dim\mathfrak{a}_{M}^{L}=1\\}$

and

(2.4)
${\mathcal{F}}_{1}(M)=\bigcup_{L\in{\mathcal{L}}_{1}(M)}{\mathcal{P}}(L).$

Let $\Sigma_{P}\subset{\mathfrak{a}}_{P}^{\ast}$ be the set of reduced roots
of $T_{P}$ on the Lie algebra ${\mathfrak{n}}_{P}$ of $N_{P}$. Let
$\Delta_{P}$ be the subset of simple roots of $P$, which is a basis for
$({\mathfrak{a}}_{P}^{G})^{\ast}$. Denote by $\Sigma_{M}$ the set of reduced
roots of $T_{M}$ on the Lie algebra of $G$. For any $\alpha\in\Sigma_{M}$ we
denote by $\alpha^{\vee}\in\mathfrak{a}_{M}$ the corresponding co–root. Let
$P_{1}$ and $P_{2}$ be parabolic subgroups with $P_{1}\subset P_{2}$. Then
${\mathfrak{a}}_{P_{2}}^{\ast}$ is embedded into
${\mathfrak{a}}_{P_{1}}^{\ast}$, while ${\mathfrak{a}}_{P_{2}}$ is a natural
quotient vector space of ${\mathfrak{a}}_{P_{1}}$. The group $M_{P_{2}}\cap
P_{1}$ is a parabolic subgroup of $M_{P_{2}}$. Let $\Delta_{P_{1}}^{P_{2}}$
denote the set of simple roots of $(M_{P_{2}}\cap P_{1},T_{P_{1}})$. It is a
subset of $\Delta_{P_{1}}$. For a parabolic subgroup $P$ with $P_{0}\subset P$
we write $\Delta_{0}^{P}:=\Delta_{P_{0}}^{P}$.

Let ${\mathbb{A}}$ be the ring of adeles of $F$, ${\mathbb{A}}_{f}$ the ring
of finite adeles and $F_{\infty}=F\otimes_{\mathbb{Q}}{\mathbb{R}}$. We fix a
maximal compact subgroup ${\bf K}=\prod_{v}{\bf K}_{v}={\bf
K}_{\infty}\cdot{\bf K}_{f}$ of $G({\mathbb{A}})=G(F_{\infty})\cdot
G({\mathbb{A}}_{f})$. We assume that the maximal compact subgroup ${\bf
K}\subset G({\mathbb{A}})$ is admissible with respect to $M_{0}$ [Ar6, $\S$
1]. Let $H_{M}:M({\mathbb{A}})\rightarrow\mathfrak{a}_{M}$ be the homomorphism
given by

(2.5)
$e^{\left\langle\chi,H_{M}(m)\right\rangle}=\lvert\chi(m)\rvert_{\mathbb{A}}=\prod_{v}\lvert\chi(m_{v})\rvert_{v}$

for any $\chi\in X(M)_{F}$ and denote by $M({\mathbb{A}})^{1}\subset
M({\mathbb{A}})$ the kernel of $H_{M}$.

Let $G_{1}=\operatorname{Res}_{F/{\mathbb{Q}}}(G)$ be the group over
${\mathbb{Q}}$ obtained from $G$ by restriction of scalars [We]. Similar for
any $M\in{\mathcal{L}}$ let $M_{1}:=\operatorname{Res}_{F/{\mathbb{Q}}}(M)$.
Let $T_{M_{1}}$ be the ${\mathbb{Q}}$-split component of the center of
$M_{1}$. For $M\in{\mathcal{L}}$ let $A_{M}$ denote the connected component of
the identity of $T_{M_{1}}({\mathbb{R}})$, which is viewed as a subgroup of
$T_{M}({\mathbb{A}}_{F})$ via the diagonal embedding of ${\mathbb{R}}$ into
$F_{\infty}$. Note that it follows from the properties of the restriction of
scalars that $M_{1}({\mathbb{A}}_{\mathbb{Q}})^{1}\cong
M({\mathbb{A}}_{F})^{1}$. Thus we have

$M({\mathbb{A}}_{F})=A_{M}\times M({\mathbb{A}}_{F})^{1}.$

Let $L^{2}_{\operatorname{disc}}(A_{M}M(F)\backslash M({\mathbb{A}}))$ be the
discrete part of $L^{2}(A_{M}M(F)\backslash M({\mathbb{A}}))$, i.e., the
closure of the sum of all irreducible subrepresentations of the regular
representation of $M({\mathbb{A}})$. We denote by
$\Pi_{\operatorname{disc}}(M({\mathbb{A}}))$ the countable set of equivalence
classes of irreducible unitary representations of $M({\mathbb{A}})$ which
occur in the decomposition of the discrete subspace
$L^{2}_{\operatorname{disc}}(A_{M}M(F)\backslash M({\mathbb{A}}))$ into
irreducible representations. Let $L^{2}_{\mbox{\k cus}}(A_{M}M(F)\backslash
M({\mathbb{A}}))$ be the subspace of cusp forms. Denote by $\Pi_{\mbox{\k
cus}}(M({\mathbb{A}}))$ the set of equivalence classes of irreducible unitary
representations of $M({\mathbb{A}})$ which occur in the decomposition of the
space of cusp forms $L^{2}_{\mbox{\k cus}}(A_{M}M(F)\backslash
M({\mathbb{A}}))$ into irreducible representations.

Let ${\mathfrak{g}}$ and ${\mathfrak{k}}$ denote the Lie algebras of
$G(F_{\infty})$ and ${\bf K}_{\infty}$, respectively. Let $\theta$ be the
Cartan involution of $G(F_{\infty})$ with respect to ${\bf K}_{\infty}$. It
induces a Cartan decomposition $\mathfrak{g}=\mathfrak{p}\oplus\mathfrak{k}$.
We fix an invariant bi-linear form $B$ on $\mathfrak{g}$ which is positive
definite on $\mathfrak{p}$ and negative definite on $\mathfrak{k}$. This
choice defines a Casimir operator $\Omega$ on $G(F_{\infty})$, and we denote
the Casimir eigenvalue of any $\pi\in\Pi(G(F_{\infty}))$ by $\lambda_{\pi}$.
Similarly, we obtain a Casimir operator $\Omega_{{\bf K}_{\infty}}$ on ${\bf
K}_{\infty}$ and write $\lambda_{\tau}$ for the Casimir eigenvalue of a
representation $\tau\in\Pi({\bf K}_{\infty})$ (cf. [BG, $\S$ 2.3]). The form
$B$ induces a Euclidean scalar product $(X,Y)=-B(X,\theta(Y))$ on
$\mathfrak{g}$ and all its subspaces. For $\tau\in\Pi({\bf K}_{\infty})$ we
define $\lVert\tau\rVert$ as in [CD, $\S$ 2.2]. Note that the restriction of
the scalar product $(\cdot,\cdot)$ on ${\mathfrak{g}}$ to ${\mathfrak{a}}_{0}$
gives ${\mathfrak{a}}_{0}$ the structure of a Euclidean space. In particular,
this fixes Haar measures on the spaces ${\mathfrak{a}}_{M}^{L}$ and their
duals $({\mathfrak{a}}_{M}^{L})^{\ast}$. We follow Arthur in the corresponding
normalization of Haar measures on the groups $M({\mathbb{A}})$ ([Ar1, $\S$
1]).

Let $H$ be a topological group. We will denote by $\Pi(H)$ the set of
equivalence classes of irreducible unitary representations of $H$.

Next we introduce the space ${\mathcal{C}}(G({\mathbb{A}})^{1})$ of Schwartz
functions. For any compact open subgroup $K_{f}$ of $G({\mathbb{A}}_{f})$ the
space $G({\mathbb{A}})^{1}/K_{f}$ is the countable disjoint union of copies of
$G(F_{\infty})^{1}=G(F_{\infty})\cap G({\mathbb{A}})^{1}$ and therefore, it is
a differentiable manifold. Any element
$X\in\mathcal{U}({\mathfrak{g}}^{1}_{\infty})$ of the universal enveloping
algebra of the Lie algebra ${\mathfrak{g}}_{\infty}^{1}$ of
$G(F_{\infty})^{1}$ defines a left invariant differential operator $f\mapsto
f\ast X$ on $G({\mathbb{A}})^{1}/K_{f}$. Let
${\mathcal{C}}(G({\mathbb{A}})^{1};K_{f})$ be the space of smooth right
$K_{f}$-invariant functions on $G({\mathbb{A}})^{1}$ which belong, together
with all their derivatives, to $L^{1}(G({\mathbb{A}})^{1})$. The space
${\mathcal{C}}(G({\mathbb{A}})^{1};K_{f})$ becomes a Fréchet space under the
seminorms

$\|f\ast X\|_{L^{1}(G({\mathbb{A}})^{1})},\quad
X\in\mathcal{U}({\mathfrak{g}}^{1}_{\infty}).$

Denote by ${\mathcal{C}}(G({\mathbb{A}})^{1})$ the union of the spaces
${\mathcal{C}}(G({\mathbb{A}})^{1};K_{f})$ as $K_{f}$ varies over the compact
open subgroups of $G({\mathbb{A}}_{f})$ and endow
${\mathcal{C}}(G({\mathbb{A}})^{1})$ with the inductive limit topology.

##  3\. Arithmetic manifolds

In this section we introduce the adelic description of the locally symmetric
spaces we will work with. We also explain the relation to the usual set up.

Let $G$ be a reductive algebraic group over a number field $F$. Fix a faithful
$F$-rational representation $\rho\colon G\to\operatorname{GL}(V)$ and an
${{\mathcal{O}}}_{F}$-lattice $\Lambda$ in the representation space $V$ such
that the stabilizer of
$\hat{\Lambda}:=\hat{\mathcal{O}}_{F}\otimes\Lambda\subset{\mathbb{A}}_{f}\otimes
V$ in $G({\mathbb{A}}_{f})$ is the group ${\bf K}_{f}$. Since the maximal
compact subgroups of $\operatorname{GL}({\mathbb{A}}_{f}\otimes V)$ are
precisely the stabilizers of lattices, it is easy to see that such a lattice
exists. For any non-zero ideal ${\mathfrak{n}}$ of ${\mathcal{O}}_{F}$, let

${\bf K}({\mathfrak{n}})={\bf K}_{G}({\mathfrak{n}})=\\{g\in
G({\mathbb{A}}_{f})\colon\rho(g)v\equiv
v\quad(\hskip-10.0pt\mod\;{\mathfrak{n}}\hat{\Lambda}),\;v\in\hat{\Lambda}\\}$

be the principal congruence subgroup of level ${\mathfrak{n}}$. Note that
${\bf K}({\mathfrak{n}})$ is a factorizable normal subgroup of ${\bf K}_{f}$.
Moreover, the groups ${\bf K}({\mathfrak{n}})$ form a neighborhood base of the
identity element in $G({\mathbb{A}}_{f})$, i.e., every compact open subgroup
$K_{f}\subset G({\mathbb{A}}_{f})$ contains a $K({\mathfrak{n}})$ for some
ideal ${\mathfrak{n}}$. We denote by
$N({\mathfrak{n}}):=[{\mathfrak{o}}_{F}\colon{\mathfrak{n}}]$ the ideal norm
of ${\mathfrak{n}}$.

A subgroup $\Gamma\subset G(F)$ is a congruence subgroup if it contains a a
finite-index subgroup of the form $\Gamma({\mathfrak{n}}):=G(F)\cap{\bf
K}({\mathfrak{n}})$ for some ideal ${\mathfrak{n}}$. This definition of a
congruence subgroup is independent of the choice of a faithful representation,
i.e., it is intrinsic to the $F$-group $G$. Let $K_{f}\subset
G({\mathbb{A}}_{f})$ be a compact open subgroup. Then there exists an ideal
${\mathfrak{n}}$ of ${\mathcal{O}}_{F}$ such that ${\bf
K}({\mathfrak{n}})\subset K_{f}$. Let $\Gamma_{K_{f}}:=G(F)\cap K_{f}$. Then
$\Gamma({\mathfrak{n}})\subset\Gamma_{K_{f}}$ is a finite index subgroup. Thus
$\Gamma_{K_{f}}$ is a congruence subgroup of $G(F)$.

By [Bo1, $\S$ 5.6] the double coset space $G(F)\backslash
G({\mathbb{A}})/G(F_{\infty})K_{f}$ is finite. Let $x_{1}=1,x_{2},\dots,x_{l}$
be a set of representatives in $G({\mathbb{A}}_{f})$ of the double cosets.
Then the groups

(3.1) $\Gamma_{i}:=\left(G(F_{\infty})\times x_{i}K_{f}x_{i}^{-1}\right)\cap
G(F),\quad 1\leq i\leq l,$

are arithmetic subgroups of $G(F_{\infty})$ and the action of $G(F_{\infty})$
on the space of double cosets $A_{G}G(F)\backslash G({\mathbb{A}})/K_{f}$
induces the following decomposition into $G(F_{\infty})$-orbits:

(3.2) $A_{G}G(F)\backslash
G({\mathbb{A}})/K_{f}\cong\bigsqcup_{i=1}^{l}\left(\Gamma_{i}\backslash
G(F_{\infty})^{1}\right),$

where $G(F_{\infty})^{1}=G(F_{\infty})/A_{G}$. Given a function $f$ on
$G({\mathbb{A}})$, let $f_{i}$ be the function on $G(F_{\infty})$ which is
defined by $g\mapsto f(x_{i}\cdot g)$, $g\in G(F_{\infty})$. Then the map
$f\mapsto(f_{i})_{i=1}^{l}$ yields an isomorphism of $G(F_{\infty})$-modules

(3.3) $L^{2}(A_{G}G(F)\backslash
G({\mathbb{A}}))^{K_{f}}\cong\bigoplus_{i=1}^{l}L^{2}(\Gamma_{i}\backslash
G(F_{\infty})^{1})$

[BJ, 4.3]. We note that, in general, $l>1$. However, if $G$ is semisimple,
simply connected, and without any $F$-simple factors $H$ for which
$H(F_{\infty})$ is compact, then by strong approximation we have

$G(F)\backslash G({\mathbb{A}})/K_{f}\cong\Gamma\backslash G(F_{\infty}),$

where $\Gamma=(G(F_{\infty})\times K_{f})\cap G(F)$. In particular this is the
case for $G=\operatorname{SL}(n)$. Since (3.3) is an isomorphism of
$G(F_{\infty})$-modules, it holds also for the discrete spectrum, i.e., we
have an isomorphism of $G(F_{\infty})$-modules

(3.4) $L^{2}_{\mbox{\k dis}}(A_{G}G(F)\backslash
G({\mathbb{A}}))^{K_{f}}\cong\bigoplus_{i=1}^{l}L^{2}_{\mbox{\k
dis}}(\Gamma_{i}\backslash G(F_{\infty})^{1}).$

Let $P\in{\mathcal{L}}$ with Levi decomposition $P=M\ltimes N$. Let $f\in
L^{2}(A_{G}G(F)\backslash G({\mathbb{A}}))$ correspond to
$(f_{i})_{i=1}^{l}\in\oplus_{i=1}^{l}L^{2}(\Gamma_{i}\backslash
G(F_{\infty})^{1})$ as above. As explained in [BJ, Sect. 4.4], $f$ is a cusp
function if and only if each $f_{i}$, $i=1,...,l$, is a cusp function. Hence
(3.3) induces an isomorphism

(3.5) $L^{2}_{\mbox{\k cus}}(A_{G}G(F)\backslash
G({\mathbb{A}}))^{K_{f}}\cong\bigoplus_{i=1}^{l}L^{2}_{\mbox{\k
cus}}(\Gamma_{i}\backslash G(F_{\infty})^{1})$

of $G(F_{\infty})$-modules. Since $L^{2}_{\mbox{\k res}}(\cdot)$ is the
orthogonal complement of $L^{2}_{\mbox{\k cus}}(\cdot)$ in $L^{2}(\cdot)$, it
follows from (3.4) and (3.5), that we also have an isomorphism of
$G(F_{\infty})$-modules

(3.6) $L^{2}_{\mbox{\k res}}(A_{G}G(F)\backslash
G({\mathbb{A}}))^{K_{f}}\cong\bigoplus_{i=1}^{l}L^{2}_{\mbox{\k
res}}(\Gamma_{i}\backslash G(F_{\infty})^{1}).$

Given $\tau\in\Pi(G(F_{\infty}))$, let $m(\tau)$ be the multiplicity with
which the representation $\tau$ occurs in $L^{2}_{\mbox{\k
dis}}(A_{G}G(F)\backslash G({\mathbb{A}}))^{K_{f}}$. Then

(3.7) $m(\tau)=\sum_{\begin{subarray}{c}\pi\in\Pi_{\mbox{\k
dis}}(G({\mathbb{A}}))\\\
\tau=\pi_{\infty}\end{subarray}}m(\pi)\dim({\mathcal{H}}_{\pi^{\prime}_{f}}^{K_{f}}),$

where $\pi=\pi_{\infty}\otimes\pi_{f}$. Similarly, let $m_{\Gamma_{i}}(\tau)$
be the multiplicity with which $\tau$ occurs in $L^{2}_{\mbox{\k
dis}}(\Gamma_{i}\backslash G(F_{\infty})^{1})$. Since (3.4) is an isomorphism
of $G(F_{\infty})^{1}$-modules, it follows that

(3.8) $\sum_{\begin{subarray}{c}\pi\in\Pi_{\mbox{\k dis}}(G({\mathbb{A}}))\\\
\pi_{\infty}=\tau\end{subarray}}m(\pi)\dim({\mathcal{H}}_{\pi^{\prime}_{f}}^{K_{f}})=\sum_{j=1}^{l}m_{\Gamma_{j}}(\tau).$

Let ${\bf K}_{\infty}\subset G(F_{\infty})^{1}$ be a maximal compact subgroup.
Let

(3.9) $\widetilde{X}:=G(F_{\infty})^{1}/{\bf K}_{\infty}$

be the associated global Riemannian symmetric space. Given an open compact
subgroup $K_{f}\subset G({\mathbb{A}}_{f})$, we define the arithmetic manifold
$X(K_{f})$ by

(3.10) $X(K_{f}):=G(F)\backslash(\widetilde{X}\times
G({\mathbb{A}}_{f})/K_{f}).$

By (3.2) we have

(3.11)
$X(K_{f})=\bigsqcup_{i=1}^{l}\left(\Gamma_{i}\backslash\widetilde{X}\right),$

where each component $\Gamma_{i}\backslash\widetilde{X}$ is a locally
symmetric space. We will assume that $K_{f}$ is neat. Then $X(K_{f})$ is a
locally symmetric manifold of finite volume.

Let $\nu\in\Pi({\bf K}_{\infty})$. Let $\widetilde{E}_{\nu}\to\widetilde{X}$
be the homogeneous vector bundle associated to $\nu$. Denote by
$C^{\infty}(\widetilde{X},\widetilde{E}_{\nu})$ the space of smooth sections
of $\widetilde{E}_{\nu}$. Let

(3.12)
$\displaystyle\begin{split}C^{\infty}(G(F_{\infty})^{1},\nu):=\\{f:G(F_{\infty})^{1}\rightarrow
V_{\nu}\colon f\in C^{\infty},\>f(gk)=&\nu(k^{-1})f(g),\\\ &\forall g\in
G(F_{\infty})^{1},\,\forall k\in{\bf K}_{\infty}\\}.\end{split}$

Let $L^{2}(G(F_{\infty})^{1},\nu)$ be the corresponding $L^{2}$-space. There
is a canonical isomorphism

(3.13) $\widetilde{A}\colon C^{\infty}(\widetilde{X},\widetilde{E}_{\nu})\cong
C^{\infty}(G(F_{\infty})^{1},\nu),$

(see [Mia, p. 4]). $\widetilde{A}$ extends to an isometry of the corresponding
$L^{2}$-spaces.

Over each component of $X(K_{f})$, $\widetilde{E}_{\sigma}$ induces a locally
homogeneous Hermitian vector bundle
$E_{i,\sigma}\to\Gamma_{i}\backslash\widetilde{X}$. Let

$E_{\sigma}:=\bigsqcup_{i=1}^{l}E_{i,\sigma}.$

Then $E_{\sigma}$ is a vector bundle over $X(K_{f})$ which is locally
homogeneous. Let $L^{2}(X(K_{f}),E_{\sigma})$ be the space of square
integrable sections of $E_{\sigma}$.

##  4\. Eisenstein series and intertwining operators

In this section we recall some basic facts about Eisenstein series and
intertwining operators, which are the main ingredients of the spectral side of
the Arthur trace formula.

Let $M\in{\mathcal{L}}$ and $P\in{\mathcal{P}}(M)$ with $P=M\ltimes N_{P}$.
Recall that we denote by $\Sigma_{P}\subset{\mathfrak{a}}_{P}^{*}$ the set of
reduced roots of $T_{M}$ on the Lie algebra $\mathfrak{n}_{P}$ of $N_{P}$. Let
$\Delta_{P}$ be the subset of simple roots of $P$, which is a basis for
$({\mathfrak{a}}_{P}^{G})^{*}$. Write ${\mathfrak{a}}_{P,+}^{*}$ for the
closure of the Weyl chamber of $P$, i.e.

$\mathfrak{a}_{P,+}^{*}=\\{\lambda\in\mathfrak{a}_{M}^{*}:\left\langle\lambda,\alpha^{\vee}\right\rangle\geq
0\text{ for all
}\alpha\in\Sigma_{P}\\}=\\{\lambda\in\mathfrak{a}_{M}^{*}:\left\langle\lambda,\alpha^{\vee}\right\rangle\geq
0\text{ for all }\alpha\in\Delta_{P}\\}.$

Denote by $\delta_{P}$ the modulus function of $P({\mathbb{A}})$. Let
$\bar{\mathcal{A}}^{2}(P)$ be the Hilbert space completion of

$\\{\phi\in C^{\infty}(M(F)U_{P}({\mathbb{A}})\backslash
G({\mathbb{A}})):\delta_{P}^{-\frac{1}{2}}\phi(\cdot x)\in
L^{2}_{\operatorname{disc}}(A_{M}M(F)\backslash M({\mathbb{A}})),\ \forall
x\in G({\mathbb{A}})\\}$

with respect to the inner product

$(\phi_{1},\phi_{2})=\int_{A_{M}M(F)N_{P}({\mathbb{A}})\backslash
G({\mathbb{A}})}\phi_{1}(g)\overline{\phi_{2}(g)}\ dg.$

Let $\alpha\in\Sigma_{M}$. We say that two parabolic subgroups
$P,Q\in{\mathcal{P}}(M)$ are _adjacent_ along $\alpha$, and write
$P|^{\alpha}Q$, if $\Sigma_{P}\cap-\Sigma_{Q}=\\{\alpha\\}$. Alternatively,
$P$ and $Q$ are adjacent if the group $\langle P,Q\rangle$ generated by $P$
and $Q$ belongs to ${\mathcal{F}}_{1}(M)$ (see (2.4) for its definition). Any
$R\in{\mathcal{F}}_{1}({\mathcal{M}})$ is of the form $\langle P,Q\rangle$,
where $P,Q$ are the elements of ${\mathcal{P}}(M)$ contained in $R$. We have
$P|^{\alpha}Q$ with
$\alpha^{\vee}\in\Sigma_{P}^{\vee}\cap{\mathfrak{a}}^{R}_{M}$. Interchanging
$P$ and $Q$ changes $\alpha$ to $-\alpha$.

For any $P\in{\mathcal{P}}(M)$ let $H_{P}\colon
G({\mathbb{A}})\rightarrow{\mathfrak{a}}_{P}$ be the extension of $H_{M}$ to a
left $N_{P}({\mathbb{A}})$-and right ${\bf K}$-invariant map. Denote by
${\mathcal{A}}^{2}(P)$ the dense subspace of $\bar{\mathcal{A}}^{2}(P)$
consisting of its ${\bf K}$\- and $\mathfrak{Z}$-finite vectors, where
$\mathfrak{Z}$ is the center of the universal enveloping algebra of
$\mathfrak{g}\otimes{\mathbb{C}}$. That is, ${\mathcal{A}}^{2}(P)$ is the
space of automorphic forms $\phi$ on $N_{P}({\mathbb{A}})M(F)\backslash
G({\mathbb{A}})$ such that $\delta_{P}^{-\frac{1}{2}}\phi(\cdot k)$ is a
square-integrable automorphic form on $A_{M}M(F)\backslash M({\mathbb{A}})$
for all $k\in{\bf K}$. Let $\rho(P,\lambda)$,
$\lambda\in{\mathfrak{a}}_{M,{\mathbb{C}}}^{*}$, be the induced representation
of $G({\mathbb{A}})$ on $\bar{\mathcal{A}}^{2}(P)$ given by

$(\rho(P,\lambda,y)\phi)(x)=\phi(xy)e^{\left\langle\lambda,H_{P}(xy)-H_{P}(x)\right\rangle}.$

It is isomorphic to the induced representation

$\operatorname{Ind}_{P({\mathbb{A}})}^{G({\mathbb{A}})}\left(L^{2}_{\operatorname{disc}}(A_{M}M(F)\backslash
M({\mathbb{A}}))\otimes
e^{\left\langle\lambda,H_{M}(\cdot)\right\rangle}\right).$

For $\phi\in{\mathcal{A}}^{2}(P)$ and
$\lambda\in{\mathfrak{a}}^{\ast}_{P,{\mathbb{C}}}$, the associated Eisenstein
series is defined by

(4.1) $E(g,\phi,\lambda):=\sum_{\gamma\in P(F)\backslash G(F)}\phi(\gamma
g)e^{(\lambda+\rho_{p})(H_{P}(\gamma g))}.$

The series converges absolutely and locally uniformly in $g$ and $\lambda$ for
$\operatorname{Re}(\lambda)$ sufficiently regular in the positive Weyl chamber
of ${\mathfrak{a}}_{P}^{\ast}$ ([MW, II.1.5]. By Langlands [La1] the
Eisenstein series can be continued analytically to a meromorphic function of
$\lambda\in{\mathfrak{a}}^{\ast}_{P,{\mathbb{C}}}$. Its singularities lie
along hypersurfaces defined by root equations.

Let $M,M_{1}\in{\mathcal{L}}$. Let
$W({\mathfrak{a}}_{M},{\mathfrak{a}}_{M_{1}})$ be the set of isomorphisms from
${\mathfrak{a}}_{M}$ onto ${\mathfrak{a}}_{M_{1}}$ obtained by restricting
elements in $W_{0}$, the Weyl group of $(G,T_{0})$, to ${\mathfrak{a}}_{M}$.
Each $s\in W({\mathfrak{a}}_{M},{\mathfrak{a}}_{M_{1}})$ has a representative
$w_{s}$ in $G(F)$. Given $s\in W({\mathfrak{a}}_{M},{\mathfrak{a}}_{M_{1}})$,
$P\in{\mathcal{P}}(M)$ and $P_{1}\in{\mathcal{P}}(M_{1})$, let

(4.2)
$M_{P_{1}|P}(s,\lambda):{\mathcal{A}}^{2}(P)\to{\mathcal{A}}^{2}(P_{1}),\quad\lambda\in{\mathfrak{a}}_{M,{\mathbb{C}}}^{*},$

be the standard _intertwining operator_ [Ar9, $\S$ 1], which is the
meromorphic continuation in $\lambda$ of the integral

(4.3) $\begin{split}[M_{P_{1}|P}&(s,\lambda)\phi](x)\\\
&=\int_{N_{P_{1}}({\mathbb{A}})\cap
w_{s}N_{P}({\mathbb{A}})w_{s}^{-1}\backslash
N_{P_{1}}({\mathbb{A}})}\phi(w_{s}^{-1}nx)e^{(\lambda+\rho_{P})(H_{P}(w_{s}^{-1}nx))}e^{-(s\lambda+\rho_{P_{1}})(H_{P_{1}}(x))}\,dn,\end{split}$

for $\phi\in{\mathcal{A}}^{2}(P)$, $x\in G({\mathbb{A}})$. Let $M=M_{1}$. Then
for $Q,P\in{\mathcal{P}}(M)$ and $1\in W({\mathfrak{a}}_{M})$ the identity
element, we put

(4.4) $M_{Q|P}(\lambda):=M_{Q|P}(1,\lambda).$

Recall that $L^{2}_{\mbox{\k dis}}(A_{M}M(F)\backslash M({\mathbb{A}}))$
decomposes as the completed direct sum of its $\pi$-isotopic components for
$\pi\in\Pi_{\mbox{\k dis}}(M({\mathbb{A}}))$. We have a corresponding
decomposition of $\bar{\mathcal{A}}^{2}(P)$ as a direct sum of Hilbert spaces
$\hat{\oplus}_{\pi\in\Pi_{\mbox{\k
dis}}(M({\mathbb{A}}))}\bar{\mathcal{A}}^{2}_{\pi}(P)$ and the corresponding
algebraic sum decomposition

(4.5) ${\mathcal{A}}^{2}(P)=\bigoplus_{\pi\in\Pi_{\mbox{\k
dis}}(M({\mathbb{A}}))}{\mathcal{A}}^{2}_{\pi}(P).$

We further decompose ${\mathcal{A}}^{2}_{\pi}(P)$ according to the action of
${\bf K}_{\infty}$ into isotypic subspaces

(4.6) ${\mathcal{A}}^{2}_{\pi}(P)=\bigoplus_{\nu\in\Pi({\bf
K}_{\infty})}{\mathcal{A}}^{2}_{\pi}(P)^{\nu}.$

Furthermore, for an open compact subgroup $K_{f}\subset G({\mathbb{A}}_{f})$
let ${\mathcal{A}}^{2}_{\pi}(P)^{K_{f}}$ be the subspace of $K_{f}$-invariant
functions in ${\mathcal{A}}^{2}_{\pi}(P)$ and for $\nu\in\Pi({\bf
K}_{\infty})$ we let ${\mathcal{A}}^{2}_{\pi}(P)^{K_{f},\nu}$ be the
$\nu$-isotypic subspace of ${\mathcal{A}}^{2}_{\pi}(P)^{K_{f}}$.

Given $\pi\in\Pi_{\mbox{\k dis}}(M({\mathbb{A}}))$, let
$(\operatorname{Ind}_{P({\mathbb{A}})}^{G({\mathbb{A}})}(\pi),{\mathcal{H}}_{P}(\pi))$
be the induced representation. Let ${\mathcal{H}}^{0}_{P}(\pi)$ be the
subspace of ${\mathcal{H}}_{P}(\pi)$, consisting of all
$\phi\in{\mathcal{H}}_{P}(\pi)$ which are right ${\bf K}$-finite and right
${\mathcal{Z}}({\mathfrak{g}}_{\mathbb{C}})$-finite. There is a canonical
isomorphism of $G({\mathbb{A}}_{f})\times(\mathfrak{g}_{{\mathbb{C}}},{\bf
K}_{\infty})$-modules

(4.7) $j_{P}:\operatorname{Hom}(\pi,L^{2}(A_{M}M(F)\backslash
M({\mathbb{A}})))\otimes{\mathcal{H}}_{P}^{0}(\pi)\rightarrow{\mathcal{A}}^{2}_{\pi}(P).$

If we fix a unitary structure on $\pi$ and endow
$\operatorname{Hom}(\pi,L^{2}(A_{M}M(F)\backslash M({\mathbb{A}})))$ with the
inner product $(A,B)=B^{\ast}A$ (which is a scalar operator on the space of
$\pi$), the isomorphism $j_{P}$ becomes an isometry. Let

(4.8) $M_{Q|P}(\pi,\lambda):=M_{Q|P}(\lambda)|_{{\mathcal{A}}^{2}_{\pi}(P)}$

be the restriction of $M_{Q|P}(\lambda)$ to the subspace
${\mathcal{A}}^{2}_{\pi}(P)$. Suppose that $P|^{\alpha}Q$. The operator
$M_{Q|P}(\pi,z):=M_{Q|P}(\pi,z\varpi)$, where
$\varpi\in{\mathfrak{a}}_{M}^{\star}$ is such that
$\langle\varpi,\alpha^{\vee}\rangle=1$, admits a normalization by a global
factor $n_{\alpha}(\pi,z)$ which is a meromorphic function in
$z\in{\mathbb{C}}$. We may write

(4.9) $M_{Q|P}(\pi,z)\circ j_{P}=n_{\alpha}(\pi,z)\cdot
j_{Q}\circ(\operatorname{Id}\otimes R_{Q|P}(\pi,z))$

where $R_{Q|P}(\pi,z)=\otimes_{v}R_{Q|P}(\pi_{v},z)$ is the product of the
locally defined normalized intertwining operators and $\pi=\otimes_{v}\pi_{v}$
[Ar9, $\S$ 6], (cf. [Mu2, (2.17)]). In many cases, the normalizing factors can
be expressed in terms automorphic $L$-functions [Sh1], [Sh2].

For any $P,Q\in{\mathcal{P}}(M)$ there exists a sequence of parabolic
subgroups $P_{0},...,P_{k}$ and roots $\alpha_{1},...,\alpha_{k}\in\Sigma_{M}$
such that $P=P_{0}$, $Q=P_{k}$, and $P_{i-1}|^{\alpha_{i}}P_{i}$ for
$i=1,...,k$. By the product rule for intertwining operators we have

(4.10) $M_{Q|P}(\pi,\lambda)=M_{P_{k}|P_{k-1}}(\pi,\lambda)\circ
M_{P_{k-1}|P_{k-2}}(\pi,\lambda)\circ\cdots\circ
M_{P_{1}|P_{0}}(\pi,\lambda).$

Thus the study of the operators $M_{Q|P}(\pi,\lambda)$ is reduced to the case
where $Q,P\in{\mathcal{P}}(M)$ are adjacent along some root
$\alpha\in\Sigma_{M}$. Let

(4.11)
$n_{Q|P}(\pi,\lambda):=\prod_{\alpha\in\Sigma_{P}\cap\Sigma_{\bar{Q}}}n_{\alpha}(\pi,\lambda(\alpha^{\vee}))$

The product is a meromorphic function of
$\lambda\in{\mathfrak{a}}_{M,{\mathbb{C}}}^{\ast}$. Then
$M_{Q|P}(\pi,\lambda)$ is normalized by $n_{Q|P}(\pi,\lambda)$, i.e.,

(4.12) $M_{Q|P}(\pi,\lambda)\circ j_{P}=n_{Q|P}(\pi,\lambda)\cdot
j_{Q}\circ(\operatorname{Id}\otimes R_{Q|P}(\pi,\lambda)).$

Recall that $\pi=\otimes_{v}\pi_{v}$, where $\pi_{v}\in\Pi(M(F_{v}))$. If
${\mathcal{H}}_{P}(\pi_{v})$ is the Hilbert space of the induced
representation $\operatorname{Ind}_{P(F_{v})}^{G(F_{v})}(\pi_{v})$, then one
has

${\mathcal{H}}_{P}(\pi)\cong\bigotimes_{v}{\mathcal{H}}_{P}(\pi_{v})$

and, with respect to this isomorphism, it follows that $R_{Q|P}(\pi,\lambda)$
is the product of the corresponding local normalized intertwining operators

(4.13) $R_{Q|P}(\pi,\lambda)=\otimes_{v}R_{Q|P}(\pi_{v},\lambda)$

[Ar4], [Ar9, $\S$ 6], [Mu2, $\S$ 2].

##  5\. Normalizing factors

In this section we consider the global normalizing factors of intertwining
operators. The goal is to estimate the number of singular hyperplanes of
normalizing factors which intersect a given compact set. The normalizing
factors can be expressed in terms of $L$-functions. To begin with we recall
some basic facts about $L$-functions. As above, we assume that $G$ is a
reductive group over a number field $F$. Recall that
$A_{G}=T_{G_{1}}({\mathbb{R}})^{0}$, where $T_{G_{1}}$ is the
${\mathbb{Q}}$-split part of the connected component of the center of
$G_{1}=\operatorname{Res}_{F/{\mathbb{Q}}}(G)$, viewed as a subgroup of
$T_{G_{1}}({\mathbb{A}}_{\mathbb{Q}})$ and hence of $G({\mathbb{A}}_{F})$.

Recall that we denote by $\Pi_{\mbox{\k dis}}(G({\mathbb{A}}))$ the set of
equivalence classes of automorphic representations of $G({\mathbb{A}})$ which
occur in the discrete spectrum of $L^{2}(A_{G}G(F)\backslash
G({\mathbb{A}}))$. For any $\pi=\otimes_{v}\pi_{v}\in\Pi_{\mbox{\k
dis}}(G({\mathbb{A}}))$ let $S(\pi)$ be the finite set of places of $F$
containing all archimedean places and such that for each finite place $v\in
S(\pi)$ at least one of the following conditions holds:

  1. (1)

$v$ is archimedean.

  2. (2)

$F/{\mathbb{Q}}$ is ramified at $v$.

  3. (3)

$G$ is ramified at $v$, i.e., either $G$ is not quasi-split over $F_{v}$ or
$G$ does not split over an unramified extension of $F_{v}$.

  4. (4)

For every hyperspecial maximal compact subgroup $K_{v}$ of $G(F_{v})$,
$\pi_{v}$ does not have a nonzero vector which is invariant under $K_{v}$.

Let $S_{\infty}$ denote the set of archimedean places of $F$ and let
$S_{f}(\pi)$ denote the set of non-archimedean places in $S(\pi)$. Thus
$S(\pi)=S_{\infty}\cup S_{f}(\pi)$. For any $v\in S_{f}(\pi)$ let $q_{v}$
denote the order of the residue field of $F_{v}$. Let
$S_{{\mathbb{Q}},f}(\pi)$ be the set of rational primes which lie below the
primes in $S_{f}(\pi)$. Also set $S_{\mathbb{Q}}(\pi):=\\{\infty\\}\cup
S_{{\mathbb{Q}},f}(\pi\\}$.

Let $W_{F}$ be the Weil group of $F$ and let ${}^{L}G$ be the Langlands
$L$-group of $G$ [Bo2]. Let $r\colon^{L}G\to\operatorname{GL}(N,{\mathbb{C}})$
be a continuous and $W_{F}$-semisimple $N$-dimensional complex representation
of ${}^{L}G$. For any $\pi\in\Pi_{\mbox{\k dis}}(G({\mathbb{A}}))$ and any
place $v$ of $F$ with $v\not\in S(\pi)$ let $t_{\pi_{v}}\in^{L}G$ be the
Hecke-Frobenius parameter of $\pi_{v}$. Then the local $L$-function
$L_{v}(s,\pi,r)$ is defined by

(5.1)
$L_{v}(s,\pi,r):=\operatorname{det}\left(\operatorname{Id}-r(t_{\pi_{v}})q_{v}^{-s}\right)^{-1}.$

Since $\pi$ is unitary, the $|r(t_{\pi_{v}})|$ are bounded by $q_{v}^{c}$,
where $c$ depends only on $G$ and $r$, [Bo2], [La2]. Therefore, for $S\supset
S(\pi)$ the partial $L$-function

(5.2) $L^{S}(s,\pi,r):=\prod_{v\not\in S}L_{v}(s,\pi,r)$

converges absolutely and uniformly on compact subsets of
$\operatorname{Re}(s)>c+1$. One of the goals of the Langlands program is to
show that each of these $L$-functions admits a meromorphic extension to the
entire complex plane and satisfies a functional equation. This is far from
being proved. In [FL1, Definition 2.1], Finis and Lapid formulated a precise
version of the expected functional equation. According to this definition,
$(G,r)$ has property (FE), if for any $\pi\in\Pi_{\mbox{\k
dis}}(G({\mathbb{A}}_{F})$ the partial $L$-function $L^{S(\pi)}(s,\pi,r)$
admits a meromorphic continuation to ${\mathbb{C}}$ with a functional equation
of the form

(5.3) $L^{S(\pi)}(s,\pi,r)=\left(\prod_{p\in
S_{\mathbb{Q}}(\pi)}\gamma_{p}(s,\pi,r))\right)L^{S(\pi)}(1-s,\pi,r^{\vee}),$

where for each $p\in S_{{\mathbb{Q}},f}(\pi)$,
$\gamma_{p}(s,\pi,r)=R_{p}(p^{-s})$ for some rational function $R_{p}$ and

(5.4)
$\gamma_{\infty}(s,\pi,r)=C_{\infty}\prod_{i=1}^{m}\frac{\Gamma_{\mathbb{R}}(1-s+\alpha_{i}^{\vee})}{\Gamma_{\mathbb{R}}(s+\alpha_{i})}$

for certain parameters
$\alpha_{1},...,\alpha_{m},\alpha_{1}^{\vee},...,\alpha_{m}^{\vee}\in{\mathbb{C}}$
and a constant $C_{\infty}$. By [FL1, Lemma 2.2] the parameters
$\alpha_{1}^{\vee},...,\alpha_{m}^{\vee}$ are determined by
$\alpha_{1},...,\alpha_{m}$. Moreover, the integer $m$ is uniquely determined.
The parameters $\alpha_{1},...,\alpha_{m}$ are said to be reduced, if
$\alpha_{i}+\alpha_{j}$ is not a negative odd integer for any $1\leq i,j\leq
m$. By [FL1, Lemma 2.2] one may choose the parameters
$\alpha_{1},...,\alpha_{m}$ to be reduced. Assuming that this is satisfied,
Finis and Lapid introduce the reduced $L$-factor at the Archimedean place by

(5.5) $L_{\infty}^{\mbox{\k
red}}(s,\pi,r):=\prod_{i=1}^{m}\Gamma_{\mathbb{R}}(s+\alpha_{i}).$

Now let $p\in S_{{\mathbb{Q}},f}(\pi)$. Then by [FL1, (2.7)],
$\gamma_{p}(s,\pi,r)$ can be written in a unique way as

(5.6)
$\gamma_{p}(s,\pi,r)=c_{p}p^{\left(\frac{1}{2}-s\right){\mathfrak{e}}_{p}(\pi,r)}P_{p}(p^{-s})/\bar{P}_{p}(p^{s-1}),$

where $c_{p}\in{\mathbb{C}}^{\ast}$,
${\mathfrak{e}}_{p}(\pi,r)\in{\mathbb{Z}}$, and $P_{p}$ is a polynomial with
$P_{p}(0)=1$ such that no zeros $\alpha$ and $\beta$ of $P_{p}$ satisfy
$\alpha\bar{\beta}=p^{-1}$. Then Finis and Lapid define the reduced $L$-factor
at $p$ by

(5.7) $L_{p}^{\mbox{\k red}}(s,\pi,r):=P_{p}(p^{-s})^{-1},\quad p\in
S_{{\mathbb{Q}},f}(\pi),$

and introduce the reduced completed $L$-function by

(5.8) $L^{\mbox{\k red}}(s,\pi,r):=\bigl{(}\prod_{p\in
S_{\mathbb{Q}}(\pi)}L_{p}^{\mbox{\k
red}}(s,\pi,r)\bigr{)}L^{S(\pi)}(s,\pi,r).$

There is also a corresponding reduced epsilon factor $\epsilon^{\mbox{\k
red}}(s,\pi,r)$, which is defined by

(5.9) $\epsilon^{\mbox{\k red}}(s,\pi,r)=c_{\infty}\prod_{p\in
S_{{\mathbb{Q}},f}(\pi)}c_{p}p^{\left(\frac{1}{2}-s\right){\mathfrak{e}}_{p}(\pi,r)}={\mathfrak{n}}(\pi,r)^{\frac{1}{2}-s}\prod_{p\in
S_{\mathbb{Q}}(\pi)}c_{p},$

where

(5.10) ${\mathfrak{n}}(\pi,r)=\prod_{p\in
S_{{\mathbb{Q}},f}(\pi)}p^{{\mathfrak{e}}_{p}(\pi,r)}.$

Then the functional equation (5.3) becomes

(5.11) $L^{\mbox{\k red}}(s,\pi,r)=\epsilon^{\mbox{\k
red}}(s,\pi,r)L^{\mbox{\k red}}(1-s,\pi,r^{\vee}).$

In [FL1, Definition 2.4] a stronger version of property (FE) is introduced.
The pair $(G,r)$ is said to satisfy property (FE+), if it satisfies (FE) and
in addition some uniformity conditions for $\gamma_{\infty}$ and $P_{p}$ are
fulfilled. For the precise statement see [FL1, Definition 2.4].

The normalizing factors are described in [FL1, Sect. 3]. To recall the
description, we need to introduce some notation. Let $M\in{\mathcal{L}}$ and
$\alpha\in\Sigma_{M}$. Let $\tilde{M}_{\alpha}$ be the Levi subgroup of $M$ of
co-rank one, defined in [FL1, p. 254], together with the map $p^{\mbox{\k
sc}}\colon\tilde{M}^{\mbox{\k sc}}_{\alpha}\to\tilde{M}_{\alpha}$, which is
also defined in [FL1, p. 254]. Furthermore, let $U_{\alpha}$ be the unipotent
subgroup of $G$ corresponding to $\alpha$. Thus the eigenvalues of $T_{M}$
acting on the Lie algebra of $U_{\alpha}$ are positive integer multiples of
$\alpha$. The adjoint action of ${}^{L}M$ on
$\operatorname{Lie}(^{L}U_{\alpha})$ factors through the composed homomorphism
${}^{L}M\to^{L}\tilde{M}_{\alpha}$. The contragredient of the adjoint
representation of ${}^{L}\tilde{M}_{\alpha}$ on
$\operatorname{Lie}(^{L}U_{\alpha})$ is decomposed as $\oplus_{j=1}^{l}r_{j}$
into irreducible representations $r_{j}$.

By T. Finis and E. Lapid [FL1, Definition 3.4], $G$ satisfies property (L), if
for any standard Levi subgroup $M$, any $\alpha\in\Sigma_{M}$, and any
irreducible constituent $r=r_{j}$ as above, the pair $(\tilde{M}_{\alpha},r)$
satisfies properties (FE+) [FL1, Definition 2.4] and the conductor condition
(CC) [FL1, Definition 2.9].

Assume that $G$ satisfies property (L). Then one can describe the normalizing
factors in terms of $L$-functions. Let $\pi\in\Pi_{\mbox{\k
dis}}(M({\mathbb{A}}))$ and let $n_{\alpha}(\pi,s)$ be the normalizing factor
as in (4.9). First note that $n_{\alpha}(\pi,s)$ satisfies the functional
equation

(5.12) $n_{\alpha}(\pi,s)\overline{n_{\alpha}(\pi,-\bar{s})}=1.$

Next recall that for $\operatorname{Re}(s)\gg 0$, $n_{\alpha}(\pi,s)$
factorizes as

(5.13) $n_{\alpha}(\pi,s)=\prod_{v}n_{\alpha,v}(\pi_{v},s).$

By [FL1, Lemma 2.13] there exist $\sigma\in\Pi_{\mbox{\k
dis}}(\tilde{M}_{\alpha}({\mathbb{A}}))$ and a character $\chi$ of
$\tilde{M}_{\alpha}({\mathbb{A}})$, which is trivial on
$\tilde{M}_{\alpha}(F)p^{\mbox{\k sc}}(\tilde{M}_{\alpha}^{\mbox{\k sc}})$,
such that $\sigma\chi$ is a subrepresentation of
$\pi|_{\tilde{M}_{\alpha}({\mathbb{A}})}$. Let

(5.14) $m(\sigma,s):=\prod_{j=1}^{l}\frac{\epsilon^{\mbox{\k
red}}(1,\sigma,r_{j})L^{\mbox{\k red}}(js,\sigma,r_{j})}{L^{\mbox{\k
red}}(js+1,\sigma,r_{j})}.$

Then at the end of the proof of Proposition 3.8 in [FL1, p. 259] it has been
shown that there exists $C>0$ such that

(5.15) $n_{\alpha}(\pi,s)=C\cdot m(\sigma,s)\cdot\prod_{j=1}^{l}\prod_{p\in
S_{\mathbb{Q}}(\sigma)}\frac{L_{p}^{\mbox{\k
red}}(js+1,\sigma,r_{j})}{L_{p}^{\mbox{\k red}}(js,\sigma,r_{j})}\prod_{v\in
S(\sigma)}n_{\alpha,v}(\pi,s).$

We use this formula to estimate the number of poles of $n_{\alpha}(\pi,s)$.
First we consider the two finite products. For this purpose we need to
estimate the cardinality of $S(\sigma)$ and $S_{\mathbb{Q}}(\sigma)$. Recall
that the level of $\sigma$ is defined as
$\operatorname{level}(\sigma)=N({\mathfrak{n}})$, where ${\mathfrak{n}}$ is
the largest ideal of ${\mathcal{O}}_{F}$ such that $\sigma^{{\bf
K}({\mathfrak{n}})\cap\tilde{M}_{\alpha}}\neq 0$. We obviously have

$|S(\sigma)|,|S_{\mathbb{Q}}(\sigma)|\leq
C(1+\log\operatorname{level}(\sigma))$

for some constant $C>0$ which is independent of $\sigma$. Furthermore, recall
that by [FL1, Sec. 2.3], $\operatorname{level}(\pi;p^{\mbox{\k sc}})$ is
defined as $\operatorname{level}(\pi;p^{\mbox{\k sc}})=N({\mathfrak{n}})$,
where ${\mathfrak{n}}$ is the largest ideal in ${\mathcal{O}}_{F}$ such that
$\pi^{{\bf K}({\mathfrak{n}})\cap p^{\mbox{\k sc}}(\tilde{M}^{\mbox{\k
sc}}_{\alpha})}\neq 0$.

By [FL1, Lemma 2.13] there exists $N_{1}\in{\mathbb{N}}$, which depends only
on $p^{\mbox{\k sc}}$ and $G$, such that for every $\pi\in\Pi_{\mbox{\k
dis}}(M({\mathbb{A}}))$ the corresponding representation
$\sigma\in\Pi_{\mbox{\k dis}}(\tilde{M}_{\alpha}({\mathbb{A}}))$ is such that
$\operatorname{level}(\sigma)$ divides
$N_{1}\operatorname{level}(\pi;p^{\mbox{\k sc}})$. Thus there exists $C>0$
such that

(5.16) $|S_{{\mathbb{Q}}}(\sigma)|,|S(\sigma)|\leq
C\log\operatorname{level}(\pi;p^{\mbox{\k sc}})$

for all $\pi$ and $\sigma$ which are related as above.

Now consider the last product on the right hand side of (5.15). Let $v\in
S_{f}(\sigma)$. By [FL1, p. 255, (1)], $n_{\alpha,v}(\pi,s)$ is a rational
function in $X=q_{v}^{-s}$, whose degree is bounded in terms of $G$ only and
which is regular and non-zero at $X=0$. Hence the poles of
$n_{\alpha,v}(\pi,s)$ form a finite union of arithmetic progressions with
imaginary difference and the number of progressions is bounded by a constant
that depends only on $G$.

If $v\in S_{\infty}(\sigma)$, then by [FL1, p. 255, (2)] we have

(5.17)
$n_{\alpha,v}(\pi,s)=c_{v}\prod_{i=1}^{N_{v}}\frac{\Gamma_{\mathbb{R}}(j_{i}s+\alpha_{i})}{\Gamma_{\mathbb{R}}(j_{i}s+\alpha_{i}+1)},$

where $c_{v}\neq 0$, $\alpha_{1},...,\alpha_{N_{v}}\in{\mathbb{C}}$ and the
integers $N_{v}\geq 1$ and $j_{i}\geq 1$, $i=1,...,N_{v}$ are bounded in terms
of $G$ only. Recall that $\Gamma(z)$ has no zeros and the poles are simple and
occur at the negative integers. Let $R>0$. It follow that the number of poles
of $n_{\alpha,v}(\pi,s)$ in a fixed half-strip $|\operatorname{Im}(s)|\leq R$,
$\operatorname{Re}(s)\geq-R$ is bounded by a constant independent of $\pi$.
Thus by (5.16) it follows that for every $R>0$ there exists $C_{1}>0$, which
is independent of $\pi$, such that the number of poles in the half-strip
$|\operatorname{Im}(s)|\leq R$, $\operatorname{Re}(s)\geq-R$, counted with
their order, of the last product is bounded by
$C_{1}\log\operatorname{level}(\pi;p^{\mbox{\k sc}})$.

Next we deal with the product over $S_{\mathbb{Q}}(\sigma)$, Let $p\in
S_{{\mathbb{Q}},f}(\sigma)$. By (5.7), there exist a polynomial
$P_{p}(x;\sigma,r_{j})$ such that $L_{p}^{\mbox{\k
red}}(s,\sigma,r_{j})=P_{p}(p^{-s};\sigma,r_{j})^{-1}$. By definition,
$(G,r_{j})$ satisfies property (FE+) [FL1, Definition 2.4]. By (2) of this
definition, the degree of $P_{p}(x;\sigma,r_{j})$ is bounded in terms of
$(G,r_{j})$ only. Thus the poles of $L_{p}(s,\sigma,r_{j})$ form a finite
union of arithmetic progressions with imaginary difference and the number of
progressions is bounded by a constant that depends only on $(G,r_{j})$. Hence
for every $R>0$ there exists $C>0$, which depends only on $(G,r_{j})$, such
that the number of poles of $L_{p}(s,\sigma,r_{j})$ in the strip
$|\operatorname{Im}(s)|\leq R$ is bounded by $C$. For $p=\infty$ we use (5.5).
By [FL1, Definition 2.4, (3)], there exists $\beta\in{\mathbb{R}}$ which
depends only on $(G,r_{j})$ such that the reduced parameters $\alpha_{i}$
satisfy $\operatorname{Re}(\alpha_{i})\geq-\beta$, $i=1,...,l$, and
$\gamma_{\infty}(s,\sigma,r_{j})$ has no zeros in
$\operatorname{Re}(s)>\beta$. So it follows as above, that the number of poles
of $L_{p}^{\mbox{\k red}}(js+1,\sigma,r_{j})/L_{p}^{\mbox{\k
red}}(js,\sigma,r_{j})$ in the half-strip $|\operatorname{Im}(s)|\leq R$,
$\operatorname{Re}(s)\geq-R$, counted with their order, is bounded by a
constant independent of $\pi$. Using (5.16) it follows that for each $R>0$
there exists $C_{2}>0$ such that the number of poles of the product over
$S_{{\mathbb{Q}}}(\sigma)$, counted with their order, in the half-strip
$|\operatorname{Im}(s)|\leq R$, $\operatorname{Re}(s)\geq-R$, is bounded by
$C_{2}\log\operatorname{level}(\pi;p^{\mbox{\k sc}})$.

So it remains to consider $m(\sigma,s)$. Let $r:=r_{j}$ for some $j$ and let

(5.18) $\Lambda(s,\sigma,r):={\mathfrak{n}}(\sigma,r)^{s/2}L^{\mbox{\k
red}}(s,\sigma,r),$

where ${\mathfrak{n}}(\sigma,r)$ is defined by (5.10) Then, using functional
equation (5.11) and the definition of the epsilon factor by (5.9), it follows
that $\Lambda(s,\sigma,r)$ satisfies

(5.19) $\Lambda(s,\sigma,r)=\epsilon^{\mbox{\k
red}}(\frac{1}{2},\sigma,r)\overline{\Lambda(1-\bar{s},\sigma,r)}.$

By (5.9) and (5.18) we get

(5.20)
$\frac{\Lambda(s,\sigma,r)}{\overline{\Lambda(-\bar{s},\sigma,r)}}=\frac{\epsilon^{\mbox{\k
red}}(1,\sigma,r)L^{\mbox{\k red}}(s,\sigma,r)}{L^{\mbox{\k
red}}(s+1,\sigma,r)}.$

Thus by the definition (5.14) it follows that

(5.21)
$m(\sigma,s)=\prod_{j=1}^{l}\frac{\Lambda(js,\sigma,r_{j})}{\overline{\Lambda(-j\bar{s},\sigma,r_{j})}}.$

As explained in the proof of [FL1, Proposition 2.6], $\Lambda(s,\sigma,r)$ is
the quotient of two holomorphic functions of order one. Therefore
$\Lambda(s,\sigma,r)$ admits a Hadamard factorization

(5.22) $\Lambda(s,\sigma,r)=e^{a+bs}s^{n(0)}\prod_{\rho\neq
0}\bigl{[}(1-s/\rho)e^{s/\rho}\bigr{]}^{n(\rho)},$

where $a,b\in{\mathbb{C}}$, the product ranges over the zeros and poles of
$\Lambda(s)$ different from 0, and $n(\rho)$ is the order of the function
$\Lambda(s)$ at $s=\rho$. In the poof of [FL1, Proposition 2.6] it was shown
that the conditions of property (FE+), [FL1, Definition 2.4], together with
the functional equation (5.19) imply that there exists $A\geq 1$, depending
only on $G$ and $r$, such that all zeros and poles of $\Lambda(s,\sigma,r)$
lie in the strip $1-A\leq\operatorname{Re}(s)\leq A$. Moreover, by [FL1,
(2.12)] we have for $T\geq 0$

(5.23)
$\sum_{\rho\colon|\operatorname{Im}(\rho)-T|<2}|n(\rho)|\ll\log{\mathfrak{n}}(\sigma,r)+\sum_{p\in
S_{{\mathbb{Q}},f}(\sigma)}\log
p+\log{\mathfrak{c}}_{\infty}(\sigma,r)+\log(1+|T|)+1,$

where ${\mathfrak{c}}_{\infty}(\pi,r)$ is the archimedean conductor defined by
[FL1, (2.6)] and ${\mathfrak{n}}(\sigma,r)$ the finite conductor (5.10). Since
we assume that $G$ satisfies property (L), $(\tilde{M}_{\alpha},r_{j})$
satisfies property (CC). Let $\Lambda(\pi_{\infty};p^{\mbox{\k sc}})$ be
defined by [FL1, (2.18)]. Then by [FL1, (2.14), (2.15)] and [FL1, Lemma 2.13]
we obtain

(5.24)
$\sum_{\rho\colon|\operatorname{Im}(\rho)-T|<2}|n(\rho)|\ll\log(|T|+2)+\log\operatorname{level}(\pi;p^{\mbox{\k
sc}})+\log\Lambda(\pi_{\infty};p^{\mbox{\k sc}}).$

We combine (5.24) with the results above concerning the other factors
occurring in (5.15). Note that by the functional equation (5.12), the poles of
$n_{\alpha}(\pi,s)$ are contained in a strip $|\operatorname{Re}(s)|\leq C$
for some $C>0$. We can summarize our results as follows. Denote by
$\Sigma_{\alpha}(\pi)$ the poles of $n_{\alpha}(\pi,s)$. Given
$\rho\in\Sigma_{\alpha}(\pi)$, denote by $n(\rho)$ its order. Then combined
with the results above concerning the other factors occurring in (5.15), we
obtain the following proposition.

######  Proposition 5.1.

Assume that $G$ satisfies property (L). Let $M\in{\mathcal{L}}$,
$\pi\in\Pi_{\mbox{\k dis}}(M({\mathbb{A}}))$, and $\alpha\in\Sigma_{M}$. Let
$\Sigma_{\alpha}(\pi)$ be the set of poles of $n_{\alpha}(\pi,s)$ and for any
$\rho\in\Sigma_{\alpha}(\pi)$ denote by $n(\rho)$ the order of the pole
$\rho$. Then for every $R>0$ there exist $C>0$ such that

(5.25)
$\sum_{\rho\in\Sigma_{\alpha}(\pi),|\operatorname{Im}(\rho)|<R}|n(\rho)|\leq
C(1+\log\operatorname{level}(\pi;p^{\mbox{\k
sc}})+\log\Lambda(\pi_{\infty};p^{\mbox{\k sc}})).$

Let $\pi\in\Pi_{\mbox{\k dis}}(M({\mathbb{A}}))$. Let $W_{P}(\pi_{\infty})$ be
the set of minimal $K_{\infty}$-types of
$\operatorname{Ind}_{P({\mathbb{R}})}^{G({\mathbb{R}})}(\pi_{\infty})$. Then
$W_{P}(\pi_{\infty})$ is a non empty finite subset of $\Pi(K_{\infty})$. Let
$\lambda_{\pi_{\infty}}$ be the Casimir eigenvalue of $\pi_{\infty}$ and for
each $\tau\in\Pi(K_{\infty})$, let $\lambda_{\tau}$ be the Casimir eigenvalue
of $\tau$. Put

(5.26) $\Lambda_{\pi_{\infty}}:=\min_{\tau\in
W_{P}(\pi_{\infty})}\sqrt{\lambda_{\pi_{\infty}}^{2}+\lambda_{\tau}^{2}}.$

Then by [FLM2, (10)] one has

(5.27) $\Lambda(\pi_{\infty};p^{\mbox{\k
sc}})\ll_{G}1+\Lambda^{2}_{\pi_{\infty}}.$

Let $K_{f}$ be an open compact subgroup of $G({\mathbb{A}}_{f})$. Put

(5.28)
$\Pi(M({\mathbb{A}});K_{f}):=\\{\pi\in\Pi(M({\mathbb{A}})\colon\pi_{f}^{K_{f}\cap
M({\mathbb{A}}_{f})}\neq 0\\}.$

Furthermore, given $\nu\in\Pi({\bf K}_{\infty})$, let

(5.29)
$\Pi(M({\mathbb{A}});K_{f},\nu)=\\{\pi\in\Pi(M({\mathbb{A}});K_{f})\colon[\operatorname{Ind}_{P({\mathbb{R}})}^{G({\mathbb{R}})}(\pi_{\infty})|_{K_{\infty}}\colon\nu]>0\\}$

and put

(5.30) $\Pi_{\mbox{\k dis}}(M({\mathbb{A}});K_{f},\nu)=\Pi_{\mbox{\k
dis}}(M({\mathbb{A}}))\cap\Pi(M({\mathbb{A}});K_{f},\nu).$

Now recall the definition of $\operatorname{level}(\pi;p^{\mbox{\k sc}})$
[FL1, Sec. 2.3]. It follows that there exists $C>0$ such that all for
$\pi\in\Pi(M({\mathbb{A}});K_{f})$ we have
$\operatorname{level}(\pi;p^{\mbox{\k sc}})\leq C$. Furthermore, by (5.26)
there exists $C_{1}>0$ such that for all
$\pi\in\Pi(M({\mathbb{A}});K_{f},\nu)$ one has $\Lambda_{\pi_{\infty}}^{2}\leq
C_{1}(1+\lambda_{\pi_{\infty}}^{2})$, and it follows from (5.27) that in this
case $\Lambda(\pi_{\infty};p^{\mbox{\k
sc}})\ll_{G}(1+\lambda_{\pi_{\infty}}^{2})$. In this way we get

######  Corollary 5.2.

Assume that $G$ satisfies property (L). Let $K_{f}$ be an open compact
subgroup of $G({\mathbb{A}}_{f})$ and $\nu\in\Pi(K_{\infty})$. Let
$M\in{\mathcal{L}}$ and $\alpha\in\Sigma_{M}$. Let the notation be as above.
For every $R>0$ there exist $C>0$ such that

(5.31)
$\sum_{\rho\in\Sigma_{\alpha}(\pi),|\operatorname{Im}(\rho)|<R}|n(\rho)|\leq
C(1+\log(1+\lambda_{\pi_{\infty}}^{2}))$

for all $\pi\in\Pi_{\mbox{\k dis}}(M({\mathbb{A}});K_{f},\nu)$.

##  6\. Logarithmic derivatives of local intertwining operators

In this section we prove some auxiliary results for local intertwining
operators. To begin with we recall some facts concerning local intertwining
operators and normalizing factors. Let $M\in{\mathcal{L}}$ and
$P,Q\in{\mathcal{P}}(M)$. Let $v$ be a place of $F$. If $v$ is finite, let
$K_{v}$ be an open compact subgroup of $G(F_{v})$ and if $v\in S_{\infty}$,
let $K_{v}$ be a maximal compact subgroup of $G(F_{v})$. Let
$\pi_{v}\in\Pi(M(F_{v}))$. Given
$\lambda\in{\mathfrak{a}}^{\ast}_{M,{\mathbb{C}}}$, let
$(I_{P}^{G}(\pi_{v},\lambda),{\mathcal{H}}_{P}(\pi_{v}))$ denote the induced
representation. Let
${\mathcal{H}}_{P}^{0}(\pi_{v})\subset{\mathcal{H}}_{P}(\pi_{v})$ be the
subspace of $K_{v}$-finite functions. Let

$J_{Q|P}(\pi_{v},\lambda)\colon{\mathcal{H}}_{P}^{0}(\pi_{v})\to{\mathcal{H}}_{Q}^{0}(\pi_{v})$

be the local intertwining operator between the induced representations
$I_{P}^{G}(\pi_{v},\lambda)$ and $I_{Q}^{G}(\pi_{v},\lambda)$ [Sh1]. It is
proved in [Ar4], [CLL, Lecture 15] that there exist scalar valued meromorphic
functions $r_{Q|P}(\pi_{v},\lambda)$ of
$\lambda\in{\mathfrak{a}}_{P,{\mathbb{C}}}^{\ast}$ such that the normalized
intertwining operators

(6.1)
$R_{Q|P}(\pi_{v},\lambda)=r_{Q|P}(\pi_{v},\lambda)^{-1}J_{Q|P}(\pi_{v},\lambda)$

satisfy the conditions $(R_{1})-(R_{8})$ of Theorem 2.1 of [Ar4]. We recall
some facts about the local normalizing factors. First assume that $v$ is a
finite valuation of $F$ with $q_{v}\in{\mathbb{N}}$ the cardinality of the
residue field of $F_{v}$. Furthermore assume that
$\dim({\mathfrak{a}}_{M}/{\mathfrak{a}}_{G})=1$ and $\pi_{v}$ is square
integrable. Let $P\in{\mathcal{P}}(M)$ and let $\alpha$ be the unique simple
root of $(P,T_{M})$. Then Langlands [CLL, Lecture 15] has shown that there
exists a rational function $V_{P}(\pi_{v},z)$ of one variable such that

(6.2)
$r_{\overline{P}|P}(\pi_{v},\lambda)=V_{P}(\pi_{v},q_{v}^{-\lambda(\widetilde{\alpha})}),$

where $\widetilde{\alpha}\in{\mathfrak{a}}_{M}$ is uniquely determined by
$\alpha$. For the construction of $V_{P}$ see also [Mu2, Sect. 3]. In this
reference, only the case ${\mathbb{Q}}_{v}$ has been discussed. However, the
case $F_{v}$ can be dealt with in exactly the same way. We need the following
lemma.

######  Lemma 6.1.

Let $M\in{\mathcal{L}}$ be such that
$\dim({\mathfrak{a}}_{M}/{\mathfrak{a}}_{G})=1$. There exists $C>0$ such that
for all $P\in{\mathcal{P}}(M)$ and all $\pi\in\Pi(M(F_{v}))$ the number of
zeros of the rational function $V_{P}(\pi,z)$ is less than or equal to $C$.

For the proof see [MM2, Lemma 10.1]. Again, the proof has been carried out for
${\mathbb{Q}}_{v}$. It extends to $F_{v}$ without any changes.

The main goal of this section is to estimate the logarithmic derivatives of
the normalized intertwining operators $R_{Q|P}(\pi,\lambda)$. For
$G=\operatorname{GL}(n)$ such estimates were derived in [MS, Proposition 0.2].
The proof depends on a weak version of the Ramanujan conjecture, which is not
available in general. Therefore we will establish only an integrated version
of it, which however, is sufficient for our purpose. For $\pi\in\Pi_{\mbox{\k
dis}}(M({\mathbb{A}}))$ denote by ${\mathcal{H}}_{P}(\pi)$ the Hilbert space
of the induced representation $I_{P}^{G}(\pi,\lambda)$. Furthermore, for an
open compact subgroup $K_{f}\subset G({\mathbb{A}}_{f})$ and $\nu\in\Pi({\bf
K}_{\infty})$, denote by ${\mathcal{H}}_{P}(\pi)^{K_{f}}$ the subspace of
vectors, which are invariant under $K_{f}$ and let
${\mathcal{H}}_{P}(\pi)^{K_{f},\nu}$ denote the $\nu$-isotypical subspace of
${\mathcal{H}}_{P}(\pi)^{K_{f}}$. Let $P,Q\in{\mathcal{P}}(M)$ be adjacent
parabolic subgroups. Then $R_{Q|P}(\pi,\lambda)$ depends on a single variable
$s\in{\mathbb{C}}$ and we will write

$R^{\prime}_{Q|P}(\pi,s_{0}):=\frac{d}{ds}R_{Q|P}(\pi,s)\big{|}_{s=s_{0}}$

for any regular $s_{0}\in{\mathbb{C}}$.

######  Proposition 6.2.

Let $M\in{\mathcal{L}}$, and let $P,Q\in{\mathcal{P}}(M)$ be adjacent
parabolic subgroups. Let $K_{f}\subset G({\mathbb{A}}_{f})$ be an open compact
subgroup and let $\nu\in\Pi({\bf K}_{\infty})$. Then there exists $C>0$ such
that

(6.3)
$\int_{{\mathbb{R}}}\Big{\|}R_{Q|P}(\pi,iu)^{-1}R^{\prime}_{Q|P}(\pi,iu)\big{|}_{{\mathcal{H}}_{P}(\pi)^{K_{f},\nu}}\Big{\|}e^{-tu^{2}}du\leq
Ct^{-1/2}$

for all $0<t\leq 1$ and $\pi\in\Pi_{\mbox{\k dis}}(M({\mathbb{A}}))$ with
${\mathcal{H}}_{P}(\pi)^{K_{f},\nu}\neq 0$.

###### Proof.

We may assume that $K_{f}$ is factorisable, i.e., $K_{f}=\prod_{v}K_{v}$. Let
$S$ be the finite set of finite places such that $K_{v}$ is not hyperspecial.
Since $P$ and $Q$ are adjacent, by standard properties of normalized
intertwining operators [Ar4, Theorem 2.1] we may assume that $P$ is a maximal
parabolic subgroup and $Q=\overline{P}$, the opposite parabolic subgroup to
$P$. By [Ar4, Theorem 2.1, (R8)], $R_{\overline{P}|P}(\pi_{v},s)^{K_{v}}$ is
independent of $s$ if $v$ is finite and $v\notin S$. Thus we have

(6.4)
$\begin{split}R_{\overline{P}|P}(\pi,s)^{-1}R^{\prime}_{\overline{P}|P}(\pi,s)\big{|}_{H_{P}(\pi)^{K_{f},\nu}}&=R_{\overline{P}|P}(\pi_{\infty},s)^{-1}R^{\prime}_{\overline{P}|P}(\pi_{\infty},s)\big{|}_{{\mathcal{H}}_{P}(\pi_{\infty})^{\nu}}\\\
&+\sum_{v\in
S}R_{\overline{P}|P}(\pi_{v},s)^{-1}R^{\prime}_{\overline{P}|P}(\pi_{v},s)\big{|}_{{\mathcal{H}}_{P}(\pi_{v})^{K_{v}}}\end{split}$

This reduces our problem to the operators at the local places. We distinguish
between the archimedean and the non-archimedean case.

Case 1: $v<\infty$. Define
$A_{v}\colon{\mathbb{C}}\to\operatorname{End}({\mathcal{H}}_{P}(\pi_{v})^{K_{v}})$
by

$A_{v}(q_{v}^{-s}):=R_{\overline{P}|P}(\pi_{v},s)\big{|}_{{\mathcal{H}}_{P}(\pi_{v})^{K_{v}}}.$

This is a meromorphic function with values in the space of endomorphisms of a
finite dimensional vector space. It has the following properties. By the
unitarity of $R_{\overline{P}|P}(\pi_{v},iu)$, $u\in{\mathbb{R}}$, it follows
that $A_{v}(z)$ is holomorphic for $z\in S^{1}$ and satisfies
$\|A_{v}(z)\|\leq 1$, $|z|=1$. By [Ar4, Theorem 2.1], the matrix coefficients
of $A_{v}(z)$ are rational functions. Recall that the operators
$R_{\overline{P}|P}(\pi_{v},iu)$ are unitary. As in [FLM2, (14)] we get

(6.5)
$\begin{split}\int_{{\mathbb{R}}}\Big{\|}&R_{\overline{P}|P}(\pi_{v},iu)^{-1}R^{\prime}_{\overline{P}|P}(\pi_{v},iu)\big{|}_{{\mathcal{H}}_{P}(\pi_{v})^{K_{v}}}\Big{\|}e^{-tu^{2}}du=\int_{\mathbb{R}}\left\|R^{\prime}_{\overline{P}|P}(\pi_{v},iu)\Big{|}_{{\mathcal{H}}_{P}(\pi_{v})^{K_{v}}}\right\|e^{-tu^{2}}du\\\
&\leq 2\sum_{n=0}^{\infty}\exp\left(-t\frac{4\pi^{2}n^{2}}{(\log
q_{v})^{2}}\right)\int_{0}^{\frac{2\pi}{\log
q_{v}}}\left\|R^{\prime}_{\overline{P}|P}(\pi_{v},iu)\Big{|}_{{\mathcal{H}}_{P}(\pi_{v})^{K_{v}}}\right\|du\\\
&\leq 2\left(1+\int_{0}^{\infty}\exp\left(-t\frac{4\pi^{2}x^{2}}{(\log
q_{v})^{2}}\right)dx\right)\int_{0}^{\frac{2\pi}{\log
q_{v}}}\left\|R^{\prime}_{\overline{P}|P}(\pi_{v},iu)\Big{|}_{{\mathcal{H}}_{P}(\pi_{v})^{K_{v}}}\right\|du\\\
&=\left(2+\frac{\log q_{v}}{\pi}\cdot
t^{-1/2}\right)\int_{S^{1}}\left\|A_{v}^{\prime}(z)\right\||dz|.\end{split}$

As explained above, $A_{v}$ satisfies the assumptions of [FLM2, Corollary
5.18]. Denote by $z_{1},...,z_{m}\in{\mathbb{C}}\setminus S^{1}$ be the poles
of $A_{v}(z)$. Then $(z-z_{1})\cdots(z-z_{m})A_{v}(z)$ is a polynomial of
degree $n$ with coefficients in
$\operatorname{End}({\mathcal{H}}_{P}(\pi_{v})^{K_{v}})$ and by [FLM2,
Corollary 5.18] we get

(6.6)
$\|A_{v}^{\prime}(z)\|\leq\max\left(\max(n-m,0)+\sum_{j\colon|z_{j}|>1}\frac{|z_{j}|^{2}-1}{|z_{j}-z|^{2}},\;\sum_{j\colon|z_{j}|<1}\frac{1-|z_{j}|^{2}}{|z_{j}-z|^{2}}\right),\quad
z\in S^{1}.$

Now observe that

$\frac{1}{2\pi}\int_{S^{1}}\frac{1-|z_{0}|^{2}}{|z-z_{0}|^{2}}|dz|=1.$

$z_{0}\in{\mathbb{C}}$, $|z_{0}|<1$. This follows from the fact that the
integrant is the Poisson kernel and so the integral is the unique harmonic
function on the unit disc which is equal to 1 on the boundary. This is the
constant function 1. Hence by (6.6) we get

(6.7) $\int_{S^{1}}\|A^{\prime}_{v}(z)\|\;|dz|\leq 2\pi\max(m,n).$

Next we estimate $m$ and $n$. First consider $m$. Let
$J_{\overline{P}|P}(\pi_{v},s)$ be the usual intertwining operator so that

$R_{\overline{P}|P}(\pi_{v},s)=r_{\overline{P}|P}(\pi_{v},s)^{-1}J_{\overline{P}|P}(\pi_{v},s),$

where $r_{\overline{P}|P}(\pi_{v},s)$ is the normalizing factor [Ar4]. By
[Sh1, Theorem 2.2.2] there exists a polynomial $p(z)$ with $p(0)=1$ whose
degree is bounded independently of $\pi_{v}$, such that
$p(q_{v}^{-s})J_{\overline{P}|P}(\pi_{v},s)$ is holomorphic on ${\mathbb{C}}$.
To deal with the normalizing factor we use (6.2) together with Lemma 6.1 to
count the number of poles of $r_{\overline{P}|P}(\pi_{v},s)^{-1}$. This leads
to a bound for $m$ which depends only on $G$. To estimate $n$ we fix an open
compact subgroup $K_{v}$ of $G(F_{v})$. Our goal is now to estimate the order
at $\infty$ of any matrix coefficient of $R_{\overline{P}|P}(\pi_{v},s)$
regarded as a function of $z=q_{v}^{-s}$. Write $\pi_{v}$ as Langlands
quotient $\pi_{v}=J_{R}^{M}(\delta_{v},\mu)$ where $R$ is a parabolic subgroup
of $M$, $\delta_{v}$ a square integrable representation of $M_{R}(F_{v})$ and
$\mu\in({\mathfrak{a}}^{\ast}_{R}/{\mathfrak{a}}^{\ast}_{M})_{\mathbb{C}}$
with $\operatorname{Re}(\mu)$ in the chamber attached to $R$. Then by [Ar4, p.
30] we have

$R_{\overline{P}|P}(\pi_{v},s)=R_{\overline{P}(R)|P(R)}(\delta_{v},s+\mu)$

with respect to the identifications described in [Ar4, p. 30]. Here $s$ is
identified with a point in
$({\mathfrak{a}}_{R}^{\ast}/{\mathfrak{a}}_{G}^{\ast})_{\mathbb{C}}$ with
respect to the canonical embedding
${\mathfrak{a}}_{M}^{\ast}\subset{\mathfrak{a}}_{G}^{\ast}$. Using again the
factorization of normalized intertwining operators we reduce the problem to
the case of a square-integrable representation $\delta_{v}$. Moreover
$\delta_{v}$ has to satisfy
$[I_{P}^{G}(\delta_{v},s)\big{|}_{K_{v}}\colon\mathbf{1}]\geq 1$. By [Si,
Lemma 1] we have

(6.8) $[I_{P}^{G}(\delta_{v},s)\big{|}_{K_{v}}\colon\mathbf{1}]\geq
1\Leftrightarrow[\delta_{v}\big{|}_{K_{v}\cap M(F_{v})}\colon\mathbf{1}]\geq
1$

Let $\Pi_{2}(M(F_{v}))$ be the space of square-integrable representations of
$M(F_{v})$. This space has a manifold structure [HC1], [Si]. By [HC1, Theorem
10] the set of square-integrable representations $\Pi_{2}(M(F_{v}),K_{v})$ of
$M(F_{v})$ with $[\delta_{v}\big{|}_{K_{v}\cap M(F_{v})}\colon\mathbf{1}]\geq
1$ is a compact subset of $\Pi_{2}(M(F_{v}))$. Under the canonical action of
$i{\mathfrak{a}}_{M}$, the set $\Pi_{2}(M(F_{v}),K_{v})$ decomposes into a
finite number of orbits. For $\mu\in i{\mathfrak{a}}_{M}$ and
$\delta_{v}\in\Pi_{2}(M(F_{v}),K_{v})$, let
$(\delta_{v})_{\mu}\in\Pi_{2}(M(F_{v}),K_{v})$ be the result of the canonical
action. Then it follows that

$R_{\overline{P}|P}((\delta_{v})_{\mu},\lambda)=R_{\overline{P}|P}(\delta_{v},\lambda+\mu).$

In this way our problem is finally reduced to the consideration of the matrix
coefficients of $R_{\overline{P}|P}(\pi_{v},s)\big{|}_{K_{v}}$ for a finite
number of representations $\pi_{v}$. This implies that $n$ is bounded by a
constant which is independent of $\pi_{v}$. Together with (6.7) it follows
that for each finite place $v$ of $F$ and each open compact subgroup $K_{v}$
of $G(F_{v})$ there exists $C_{v}>0$ such that

(6.9)
$\int_{{\mathbb{R}}}\Big{\|}R_{\overline{P}|P}(\pi_{v},iu)^{-1}R^{\prime}_{\overline{P}|P}(\pi_{v},iu)\big{|}_{{\mathcal{H}}_{P}(\pi_{v})^{K_{v}}}\Big{\|}e^{-tu^{2}}du\leq
C_{v}t^{-1/2}$

for all $0<t\leq 1$ and $\pi_{v}\in\Pi(M(F_{v}))$ with
$I_{P}^{G}(\pi_{v})\big{|}_{{\mathcal{H}}_{P}(\pi_{v})^{K_{v}}}\neq 0$.

Case 2: $v=\infty$. To begin with we need a modification of [FLM2, Lemma
5.19].

######  Lemma 6.3.

Let $z_{j}\in{\mathbb{C}}\setminus i{\mathbb{R}}$, $j=1,...,m$, and let
$b(z)=(z-z_{1})\cdots(z-z_{m})$. Suppose that
$A\colon{\mathbb{C}}\setminus\\{z_{1},...,z_{m}\\}\to V$ is such that
$\|A(z)\|\leq 1$ for all $z\in i{\mathbb{R}}$ and $b(z)A(z)$ is a polynomial
in $z\in{\mathbb{C}}$ (necessarily of degree $\leq m$) with coefficients in
$V$. Then

$\int_{{\mathbb{R}}}\|A^{\prime}(iu)\|e^{-tu^{2}}du\leq 2\pi m$

for all $0\leq t$.

###### Proof.

For any $w\in{\mathbb{C}}$, let $\phi_{w}(z):=\frac{z+\bar{w}}{z-w}$, and set

$\phi_{>}(z)=\sum_{j:\operatorname{Re}(z_{j})>0}\phi_{z_{j}}(z),\quad\phi_{<}(z)=\sum_{j:\operatorname{Re}(z_{j})<0}\phi_{z_{j}}(z).$

Applying [BE, Theorem 4], it follows that

(6.10)
$\|A^{\prime}(z)\|\leq\max\\{|\phi^{\prime}_{>}(z)|,|\phi^{\prime}_{<}(z)|\\}\leq|\phi^{\prime}_{>}(z)|+|\phi^{\prime}_{<}(z)|,\quad
z\in i{\mathbb{R}}.$

Now observe that for $w=x+iy\in{\mathbb{C}}\setminus i{\mathbb{R}}$, one has
$|\phi_{w}^{\prime}(z)|=\frac{2|x|}{|z-w|^{2}}=\frac{2|x|}{x^{2}+(u-y)^{2}}$
for $z=iu$, $u\in{\mathbb{R}}$. So we get

$\begin{split}\int_{{\mathbb{R}}}|\phi^{\prime}_{w}(iu)|e^{-tu^{2}}du&\leq\int_{\mathbb{R}}\frac{2|x|}{x^{2}+(u-y)^{2}}du=\frac{2}{|x|}\int_{\mathbb{R}}\frac{1}{1+(\frac{u}{|x|}-\frac{y}{|x|})^{2}}du\\\
&=2\int_{\mathbb{R}}\frac{1}{1+(u-\frac{y}{|x|})^{2}}du=2\int_{\mathbb{R}}\frac{du}{1+u^{2}}=2\pi.\end{split}$

Together with (6.10) the lemma follows. ∎

As above let $M\in{\mathcal{L}}$ with
$\dim({\mathfrak{a}}_{M}/{\mathfrak{a}}_{G})=1$ and $P\in{\mathcal{P}}(M)$.
Let $\pi_{\infty}\in\Pi(M(F_{\infty}))$ and $\nu\in\Pi({\bf K}_{\infty})$. As
explained in [MS, Appendix], there exist $w_{1},...,w_{r}\in{\mathbb{C}}$ and
$m\in{\mathbb{N}}$ such that the poles of
$R_{\overline{P}|P}(\pi_{\infty},s)\big{|}_{{\mathcal{H}}_{P}(\pi_{\infty})^{\nu}}$
are contained in $\cup_{j=1}^{r}\\{w_{j}-k\colon k=1,...,m\\}$. Moreover, by
[MS, Proposition A.2] there exists $c>0$ which depends only on $G$, such that

(6.11) $r\leq c,\quad m\leq c(1+\|\nu\|).$

Let $A\colon{\mathbb{C}}\to{\mathcal{H}}_{P}(\pi_{\infty})^{\nu}$ be defined
by

$A(z):=R_{\overline{P}|P}(\pi_{\infty},z)\big{|}_{{\mathcal{H}}_{P}(\pi_{\infty})^{\nu}}$

and let $b(z)=\prod_{j=1}^{r}\prod_{k=1}^{m}(z-w_{j}+k)$. Then it follows from
($R_{6}$) of [Ar4, Theorem 2.1] that $b(z)A(z)$ is a polynomial function.
Moreover, by unitarity of $R_{\overline{P}|P}(\pi_{\infty},it)$,
$t\in{\mathbb{R}}$, we have $\|A(it)\|=1$. Thus $A(z)$ satisfies the
assumptions of [FLM2, Lemma 5.19]. Thus by Lemma 6.3 and (6.11) we get

(6.12)
$\begin{split}\int_{\mathbb{R}}\Big{\|}R_{\overline{P}|P}(\pi_{\infty},iu)^{-1}R_{\overline{P}|P}^{\prime}(\pi_{\infty},iu)\big{|}_{{\mathcal{H}}_{P}(\pi_{\infty})^{\nu}}&\Big{\|}e^{-tu^{2}}du=\int_{\mathbb{R}}\|A^{\prime}(iu)\|e^{-tu^{2}}du\\\
&\leq 2\pi r\cdot m\leq 2\pi c^{2}(1+\|\nu\|).\end{split}$

Combining (6.4), (6.9) and (6.12), the proposition follows.

######  Remark 6.4.

For $G=\operatorname{GL}_{n}$ it is proved in [MS, Proposition 0.2] that the
corresponding bounds hold for the derivatives of the local intertwining
operators itself. This follows from a weak version of the Ramanujan
conjecture, which implies that the poles of the local intertwining operators
are uniformly bounded away from the imaginary axis. For the integrated
derivatives the distance of the poles from the imaginary axis does not matter.

∎

##  7\. The residual spectrum

The goal of this section is to estimate the growth of the counting function of
the residual spectrum. To this end we recall the construction of the residual
spectrum. By Langlands [La1, Ch. 7], [MW, V.3.13], $L^{2}_{\mbox{\k
res}}(G(F)\backslash G({\mathbb{A}})^{1})$ is spanned by iterated residues of
cuspidal Eisenstein series. Let us briefly recall this construction.

Let $P=M\ltimes N$ be a $F$-rational parabolic subgroup of $G$. If
$\alpha\in\Sigma_{P}$, denote by $\alpha^{\vee}$ the co-root associated to
$\alpha$. Given $\alpha\in\Sigma_{P}$ and $c\in{\mathbb{R}}$, we set

$H(\alpha,c):=\\{\Lambda\in{\mathfrak{a}}^{\ast}_{\mathbb{C}}\colon\Lambda(\alpha^{\vee})=c\\}.$

An affine subspace ${\mathcal{H}}\subset{\mathfrak{a}}^{\ast}_{\mathbb{C}}$ is
called admissible, if ${\mathcal{H}}$ is the intersection of such hyperplanes.
Suppose that ${\mathcal{H}}_{1}\supset{\mathcal{H}}_{2}$ are two admissible
affine subspaces of ${\mathfrak{a}}^{\ast}_{\mathbb{C}}$ and
${\mathcal{H}}_{2}$ is of co-dimension one in ${\mathcal{H}}_{1}$. Let
$\Phi(\Lambda)$ be a meromorphic function on ${\mathcal{H}}_{1}$ whose
singularities lie along hyperplanes which are admissible as subspaces of
${\mathfrak{a}}^{\ast}_{\mathbb{C}}$. Choose a real unit vector $\Lambda_{0}$
in ${\mathcal{H}}_{1}$ which is normal to ${\mathcal{H}}_{2}$. Let $\delta>0$
be such that $\Phi(\Lambda+z\Lambda_{0})$ has no singularities in the
punctured disc $0<|z|<2\delta$. Then we can define a meromorphic function
$\operatorname{Res}_{{\mathcal{H}}_{2}}\Phi$ on ${\mathcal{H}}_{2}$ by

$\operatorname{Res}_{{\mathcal{H}}_{2}}\Phi(\Lambda):=\frac{\delta}{2\pi
i}\int_{0}^{1}\Phi(\Lambda+\delta e^{2\pi i\vartheta}\Lambda_{0})d(e^{2\pi
i\vartheta}),$

The singularities of $\operatorname{Res}_{{\mathcal{H}}_{2}}\Phi$ lie on the
intersections with ${\mathcal{H}}_{2}$ of the singular hyperplanes of $\Phi$
different from ${\mathcal{H}}_{2}$. Now consider a complete flag

${\mathfrak{a}}^{\ast}_{\mathbb{C}}={\mathcal{H}}_{p}\supset{\mathcal{H}}_{p-1}\supset\cdots\supset{\mathcal{H}}_{1}\supset{\mathcal{H}}_{0}=\\{\Lambda_{0}\\}$

of affine admissible subspaces of ${\mathfrak{a}}^{\ast}_{\mathbb{C}}$ and let
$\Lambda_{i}\in{\mathcal{H}}_{i}$ be a real unit vector which is normal to
${\mathcal{H}}_{i-1}$, $i=1,\dots,p$. We call
${\mathcal{F}}=\\{{\mathcal{H}}_{i},\Lambda_{i}\\}$ an admissible flag. Let
$\Phi$ be a meromorphic function on ${\mathfrak{a}}^{\ast}_{\mathbb{C}}$ whose
singularities lie along admissible hyperplanes of
${\mathfrak{a}}^{\ast}_{\mathbb{C}}$. Then we define $\Phi_{i}$ inductively by

$\Phi_{p}=\Phi,\;\Phi_{i}=\operatorname{Res}_{{\mathcal{H}}_{i}}\Phi_{i+1},\quad
i=0,\dots,p-1.$

Set

$\operatorname{Res}_{{\mathcal{F}}}\Phi:=\Phi_{0}.$

This is the iterated residue of $\Phi$ at $\Lambda_{0}$.

Now let ${\mathcal{A}}^{2}_{\mbox{\k cus}}(P)$ the subspace of functions
$\phi\in{\mathcal{A}}^{2}(P)$ such that for almost all $x\in G({\mathbb{A}})$,
the function $\phi_{x}(m):=\phi(mx)$ on $M(F)\backslash M({\mathbb{A}})^{1}$
lies in the space $L^{2}_{\mbox{\k cus}}(M(F)\backslash M({\mathbb{A}})^{1})$.
For $\pi\in\Pi_{\mbox{\k cus}}(M({\mathbb{A}})^{1})$ let
${\mathcal{A}}^{2}_{\mbox{\k cus},\pi}(P)$ be the subspace of functions
$\phi\in{\mathcal{A}}^{2}_{\mbox{\k cus}}(P)$ such that each of the functions
$\phi_{x}$ lies in the subspace $L^{2}_{\mbox{\k cus},\pi}(M(F)\backslash
M(A)^{1})$ ( isotypical subspace). Let $\phi\in{\mathcal{A}}^{2}_{\mbox{\k
cus}}(P)$. As shown by Langlands [La1, §7], the singularities of the
Eisenstein series $E(\phi,\lambda)$ lie along hyperplanes of
${\mathfrak{a}}^{\ast}_{\mathbb{C}}$ which are defined by equations of the
form $\Lambda(\alpha^{\vee})=w$, $w\in{\mathbb{C}}$, $\alpha\in\Sigma_{P}$.
Let $H(\alpha_{i},c_{i})$, $i=0,\dots,p-1$, be a set of singular hyperplanes
of $E(\phi,\lambda)$ with $\cap_{i}H(\alpha_{i},c_{i})=\\{\Lambda_{0}\\}$. Set
${\mathcal{H}}_{i}:=\cap_{j\geq i}H(\alpha_{j},c_{j})$, $i=0,\dots,p-1$, and
${\mathcal{H}}_{P}={\mathfrak{a}}^{\ast}_{\mathbb{C}}$. Choose real unit
vectors $\Lambda_{i}\in{\mathcal{H}}_{i}$ normal to ${\mathcal{H}}_{i-1}$.
Then ${\mathcal{F}}:=\\{{\mathcal{H}}_{i},\Lambda_{i}\\}$ is an admissible
flag. Furthermore, let $\varphi\in C_{c}^{\infty}({\mathfrak{a}})$ and let
$\hat{\varphi}(\Lambda)$ be its Fourier transform. It is holomorphic on
${\mathfrak{a}}^{\ast}_{\mathbb{C}}$. Put

$\psi:=\operatorname{Res}_{{\mathcal{F}}}\bigl{[}E(\phi,\Lambda)\hat{\varphi}(\Lambda)\bigr{]}.$

Note that $\psi$ depends only on the derivatives of $\hat{\varphi}$ at
$\Lambda_{0}$. Let ${\mathcal{C}}({\mathfrak{a}}^{\ast})$ be the positive cone
in ${\mathfrak{a}}^{\ast}$ spanned by the simple roots of $(P,A)$. If
$\Lambda_{0}\in{\mathcal{C}}({\mathfrak{a}}^{\ast})$ then $\psi$ is square
integrable. Let $\Omega_{M}\in{\mathcal{Z}}({\mathfrak{m}}_{\mathbb{C}})$ be
the Casimir element of $M(F_{\infty})$. Assume that $\Omega_{M}\phi=\mu\phi$.
Then it follows that

(7.1) $\Omega\psi=(\|\Lambda_{0}\|^{2}-\|\rho_{P}\|^{2}+\mu)\psi.$

[HC2, p. 29]. Moreover $\|\Lambda_{0}\|\leq\|\rho_{P}\|$. As shown by
Langlands [La1, Theorem 7.2], [MW, V.3.13], $L^{2}_{\mbox{\k
res}}(G(F)\backslash G({\mathbb{A}})^{1})$ is spanned by all such $\psi$,
where $P$ runs over the standard parabolic subgroups of $G$, $\pi$ over
$\Pi_{\mbox{\k cus}}(M_{P}({\mathbb{A}})^{1})$ and $\phi$ over a basis of
${\mathcal{A}}^{2}_{\mbox{\k cus},\pi}(P)$. For all details concerning the
description of the discrete and residual spectrum see [MW, Theorem V.3.13, p.
221], and [MW, Corollary, p. 224]. Moreover, the question of positivity is
dealt with by [MW, Corollary VI.1.6 (d), p. 255].

Let $K_{f}$ be an open compact subgroup of $G({\mathbb{A}}_{f})$. Let
$L^{2}_{\mbox{\k res}}(A_{G}G(F)\backslash G({\mathbb{A}}))^{K_{f}}$ be the
subspace of $K_{f}$-invariant functions of $L^{2}_{\mbox{\k
res}}(A_{G}G(F)\backslash G({\mathbb{A}}))$. Moreover, for $\nu\in\Pi({\bf
K}_{\infty})$ let $L^{2}_{\mbox{\k res}}(A_{G}G(F)\backslash
G({\mathbb{A}}))^{K_{f},\nu}$ be the $\nu$-isotypical subspace of
$L^{2}_{\mbox{\k res}}(A_{G}G(F)\backslash G({\mathbb{A}}))^{K_{f}}$. Then
$L^{2}_{\mbox{\k res}}(A_{G}G(F)\backslash G({\mathbb{A}}))^{K_{f},\nu}$ is
spanned by residues as above, where for a given pair $(P,\pi)$, $\phi$ runs
over a basis of ${\mathcal{A}}^{2}_{\mbox{\k cus},\pi}(P)^{K_{f},\sigma}$.
Recall that ${\mathcal{A}}^{2}_{\mbox{\k cus},\pi}(P)^{K_{f},\sigma}$ is
finite dimensional. So the estimation of the counting function of the residual
spectrum is reduced to the following problems:

  1. (1)

Estimation of $\dim{\mathcal{A}}^{2}_{\mbox{\k cus},\pi}(P)^{K_{f},\sigma}$ in
terms of $\pi$, $K_{f}$, and $\sigma$.

  2. (2)

For a given cuspidal Eisenstein series $E(\phi,\Lambda)$,
$\phi\in{\mathcal{A}}^{2}_{\mbox{\k cus},\pi}(P)$, we need to estimate the
number of its singular hyperplanes, counted to multiplicity, which are real
and intersect a given compact set containing the origin.

We start with (1). Let $\pi\in\Pi_{\mbox{\k cus}}(M({\mathbb{A}}))$,
$\pi=\pi_{\infty}\otimes\pi_{f}$. Let ${\mathcal{H}}_{P}(\pi_{\infty})$ (resp.
${\mathcal{H}}_{P}(\pi_{f})$) be the Hilbert space of the induced
representation
$\operatorname{Ind}_{P(F_{\infty})}^{G(F_{\infty})}(\pi_{\infty})$ (resp.
$\operatorname{Ind}_{P({\mathbb{A}}_{f})}^{G({\mathbb{A}}_{f})}$). Denote by
${\mathcal{H}}_{P}(\pi_{\infty})_{\sigma}$ the $\sigma$-isotypical subspace of
${\mathcal{H}}_{P}(\pi_{\infty})$ and by ${\mathcal{H}}_{P}(\pi_{f})^{K_{f}}$
the subspace of $K_{f}$-invariant vectors of ${\mathcal{H}}_{P}(\pi_{f})$. Let
$m(\pi)$ denote the multiplicity with which $\pi\in\Pi_{\mbox{\k
cus}}(M({\mathbb{A}}))$ occurs in $L^{2}_{\mbox{\k cus}}(A_{G}M(F)\backslash
M({\mathbb{A}}))$. Then by (4.7) we obtain

(7.2)
$\dim{\mathcal{A}}^{2}_{\pi}(P)^{K_{f},\sigma}=m(\pi)\dim({\mathcal{H}}_{P}(\pi_{f})^{K_{f}})\dim({\mathcal{H}}_{P}(\pi_{\infty})_{\sigma}).$

Using Frobenius reciprocity [Kn, p. 208] we get

$[\operatorname{Ind}_{P(F_{\infty})}^{G(F_{\infty})}(\pi_{\infty})|_{{\bf
K}_{\infty}}\colon\sigma]=\sum_{\tau\in\Pi({\bf
K}_{M,\infty})}[\pi_{\infty}|_{{\bf
K}_{M,\infty}}\colon\tau]\cdot[\sigma|_{{\bf K}_{M,\infty}}\colon\tau].$

For $\tau\in\Pi({\bf K}_{M,\infty})$ let ${\mathcal{H}}_{\pi_{\infty}}(\tau)$
denote the $\tau$-isotypical subspace of ${\mathcal{H}}_{\pi_{\infty}}$. Then
we obtain

(7.3)
$\dim({\mathcal{H}}_{P}(\pi_{\infty})_{\sigma})\leq\dim(\sigma)\sum_{\tau\in\Pi({\bf
K}_{M,\infty})}\dim({\mathcal{H}}_{\pi_{\infty}}(\tau))\cdot[\sigma|_{{\bf
K}_{M,\infty}}\colon\tau].$

Next we consider $\pi_{f}=\otimes_{v<\infty}\pi_{v}$. Replacing $K_{f}$ by a
subgroup of finite index, if necessary, we can assume that
$K_{f}=\prod_{v<\infty}K_{v}$. For any $v<\infty$, denote by
${\mathcal{H}}_{P}(\pi_{v})$ the Hilbert space of the induced representation
$\operatorname{Ind}_{P(F_{v})}^{G(F_{v})}(\pi_{v})$. Let
${\mathcal{H}}_{P}(\pi_{v})^{K_{v}}$ be the subspace of $K_{v}$-invariant
vectors. Then $\dim{\mathcal{H}}_{P}(\pi_{v})^{K_{v}}=1$ for almost all $v$
and

${\mathcal{H}}_{P}(\pi_{f})^{K_{f}}\cong\bigotimes_{v<\infty}{\mathcal{H}}_{P}(\pi_{v})^{K_{v}}.$
(7.4)
$\begin{split}\operatorname{Ind}^{G(F_{v})}_{P(F_{v})}(\pi_{v})^{K_{v}}&=\left(\operatorname{Ind}_{P({\mathcal{O}}_{v})}^{G({\mathcal{O}}_{v})}(\pi_{v})\right)^{K_{v}}\\\
&\hookrightarrow\bigoplus_{G({\mathcal{O}}_{v})/K_{v}}\operatorname{Ind}^{K_{v}}_{K_{v}\cap
P}(\pi_{v})^{K_{v}}\\\
&\cong\bigoplus_{G({\mathcal{O}}_{v})/K_{v}}\pi_{v}^{K_{v}\cap P}.\end{split}$

Let $K_{M,f}=K_{f}\cap M({\mathbb{A}}_{f})$. For $\sigma\in\Pi(K_{\infty})$
let

${\mathcal{F}}_{M}(\sigma):=\\{\tau\in\Pi(K_{M,\infty})\colon[\sigma|_{K_{M,\infty}}\colon\tau]>0\\}.$

Combining (7.2) - (7.4), it follows that there exists $C>0$, which depends on
$\sigma$, such that

(7.5) $\dim{\mathcal{A}}^{2}_{\pi}(P)^{K_{f},\sigma}\leq
Cm(\pi)\dim({\mathcal{H}}_{\pi_{f}}^{K_{M,f}})\sum_{\tau\in{\mathcal{F}}(\sigma)}\dim({\mathcal{H}}_{\pi_{\infty}}(\tau)).$

In order to deal with (2), we use the inner product formula for truncated
cuspidal Eisenstein series proved by Langlands [La3, Sect. 9], [Ar2, Lemma
4.2]. We recall the formula. Let $T\in{\mathfrak{a}}_{0}^{+}$ be sufficiently
regular. For $\phi\in{\mathcal{A}}^{2}_{\mbox{\k cus}}(P)$ let
$\Lambda^{T}E(g,\phi,\lambda)$ be the truncated Eisenstein series [Ar2, Sect.
1]. Let $\phi\in{\mathcal{A}}_{\mbox{\k cus}}^{2}(P)$ and
$\phi^{\prime}\in{\mathcal{A}}^{2}_{\mbox{\k cus}}(P^{\prime})$. Then we have
the following inner product formula

(7.6) $\begin{split}&\int_{G(F)\backslash
G({\mathbb{A}})^{1}}\Lambda^{T}E(g,\phi,\lambda)\overline{\Lambda^{T}E(g,\phi^{\prime},\lambda^{\prime})}dg\\\
&=\sum_{Q}\sum_{s}\sum_{s^{\prime}}\operatorname{vol}({\mathfrak{a}}^{G}_{Q}/{\mathbb{Z}}(\Delta^{\vee}_{Q}))\frac{e^{(s\lambda+s^{\prime}\lambda^{\prime})(T)}}{\prod_{\alpha\in\Delta_{Q}}(s\lambda+s^{\prime}\bar{\lambda^{\prime}})(\alpha^{\vee})}(M_{Q|P}(s,\lambda)\phi,M_{Q|P^{\prime}}(s^{\prime},\lambda^{\prime})\phi^{\prime}),\end{split}$

where $Q$ runs over all standard parabolic subgroups, $s\in
W({\mathfrak{a}}_{P},{\mathfrak{a}}_{Q})$, and $s^{\prime}\in
W({\mathfrak{a}}_{P^{\prime}},{\mathfrak{a}}_{Q})$, as meromorphic functions
of $\lambda\in{\mathfrak{a}}^{\ast}_{P,{\mathbb{C}}}$ and
$\lambda^{\prime}\in{\mathfrak{a}}^{\ast}_{P^{\prime},{\mathbb{C}}}$ [Ar5,
Prop. 15.3], [Ar2, Lemma 4.2]. It follows from the inner product formula that
in order to settle (2), it suffices to estimate the corresponding number of
singular hyperplanes of the intertwining operators
$M_{Q|P}(s,\lambda)|_{{\mathcal{A}}^{2}_{\mbox{\k cus},\pi}(P)}$ for
$\pi\in\Pi_{\mbox{\k cus}}(M({\mathbb{A}}))$. To deal with this problem, we
reduce it to the case of $M_{Q|P}(\lambda)|_{{\mathcal{A}}^{2}_{\mbox{\k
cus},\pi}(P)}$, $Q,P\in{\mathcal{P}}(M)$, $\pi\in\Pi_{\mbox{\k
cus}}(M({\mathbb{A}}))$. Let $M,M_{1}\in{\mathcal{L}}$ and let
$P\in{\mathcal{P}}(M)$, $P_{1}\in{\mathcal{P}}(M_{1})$. Suppose that $P$ and
$P_{1}$ are associated and let $t\in
W({\mathfrak{a}}_{M},{\mathfrak{a}}_{M_{1}})$. Let $w_{t}\in G(F)$ be a
representative of $t$. Then $M_{1}:=w_{t}Mw_{t}^{-1}$ and
$tP=w_{t}Pw_{t}^{-1}$ is a parabolic subgroup which belongs to
${\mathcal{P}}(M_{1})$. The restriction of $t$ to
${\mathfrak{a}}_{M}\subset{\mathfrak{a}}_{0}$ defines an element in
$W({\mathfrak{a}}_{M},{\mathfrak{a}}_{tM})$. Let

$t\colon{\mathcal{A}}^{2}(P)\to{\mathcal{A}}^{2}(tP)$

be the linear operator defined by $(t\phi)(x)=\phi(w_{t}^{-1}x)$, $x\in
G({\mathbb{A}})$. By [Ar3, Lemma 1.1] there exists
$T_{0}\in{\mathfrak{a}}_{0}$ such that

$H_{P_{0}}(w_{t}^{-1})=T_{0}-t^{-1}T_{0}.$

Then by [Ar9, (1.5)] one has

(7.7)
$M_{P_{1}|P}(s,\lambda)t^{-1}=M_{P_{1}|tP}(st^{-1},t\lambda)e^{(\lambda+\rho_{P})(T_{0}-t^{-1}T_{0})}$

for $s\in W({\mathfrak{a}}_{M},{\mathfrak{a}}_{M_{1}})$. Thus as far as the
singular hyperplanes of $M_{P_{1}|P}(s,\lambda)$ are concerned, we can assume
that $P_{1}$ and $P$ have the same Levi component $M$, that is
$P,P_{1}\in{\mathcal{P}}(M)$. Let $t\in W({\mathfrak{a}}_{M})$. By the
functional equation [Ar9, (1.2)] we have

(7.8) $M_{P_{1}|P}(t,\lambda)=M_{P_{1}|tP}(1,t\lambda)M_{tP|P}(t,\lambda).$

Using (7.7) with $s=t$, we get

$M_{tP|P}(t,\lambda)=M_{tP|tP}(1,t\lambda)te^{(\lambda+\rho_{P})(T_{0}-t^{-1}T_{0})}.$

Since $M_{tP|tP}(1,\lambda)=\operatorname{Id}$, it follows that

(7.9)
$M_{P_{1}|P}(t,\lambda)=M_{P_{1}|tP}(1,t\lambda)te^{(\lambda+\rho_{P})(T_{0}-t^{-1}T_{0})}.$

Thus we are reduced to the consideration of the singular hyperplanes of the
intertwining operators $M_{Q|P}(\lambda)=M_{Q|P}(1,\lambda)$. Given
$\pi\in\Pi_{\mbox{\k cus}}(M_{P}({\mathbb{A}}))$, an open compact subgroup
$K_{f}\subset G({\mathbb{A}}_{f})$ and $\sigma\in\Pi({\bf K}_{\infty})$, we
need to estimate the singular hyperplanes, counted to multiplicity, of
$M_{Q|P}(\pi,\lambda)^{K_{f},\sigma}$, which are real and intersect a fixed
compact set. By (4.12) the problem is reduces to the consideration of the
normalizing factors $n_{Q|P}(\pi,\lambda)$ and the normalized intertwining
operators $R_{Q|P}(\pi,\lambda)$ restricted to ${\mathcal{A}}^{2}_{\mbox{\k
cus},\pi}(P)^{K_{f},\sigma}$.

To begin with we consider the normalized intertwining operators (4.13). Let
$v$ be a place of $F$. For $\pi_{v}\in\Pi(G(F_{v}))$ let

$J_{Q|P}(\pi_{v},\lambda)\colon{\mathcal{H}}_{P}^{0}(\pi_{v})\to{\mathcal{H}}_{Q}^{0}(\pi_{v}),\quad\lambda\in{\mathfrak{a}}_{Q,{\mathbb{C}}}^{\ast},$

be the local intertwining operator and

(7.10)
$R_{Q|P}(\pi_{v},\lambda):=n_{Q|P}(\pi_{v},\lambda)^{-1}J_{Q|P}(\pi_{v},\lambda).$

the local normalized intertwining operator. The operators
$R_{Q|P}(\pi_{v},\lambda)$ satisfy properties $(R_{1}),...,(R_{8})$ of [Ar4,
Theorem 2.1]. Let $\pi=\otimes_{v}\pi_{v}\in\Pi_{\mbox{\k
dis}}(G({\mathbb{A}}))$. There exists a finite set of places $S(\pi)$ of $F$,
containing the Archimedean places, such that for all $v\not\in S$, $G/F_{v}$
and $\pi_{v}$ are unramified. For $v\not\in S(\pi)$, let ${\bf K}_{v}$ be
hyperspecial and assume that $\phi_{v}\in{\mathcal{H}}_{P}(\pi_{v})^{{\bf
K}_{v}}$. Then by $(R_{8})$ we have

$R_{Q|P}(\pi_{v},\lambda)\phi_{v}=\phi_{v},\quad s\not\in S(\pi).$

Hence the product (4.13) runs only over $v\in S(\pi)$ and therefore, it is
well defined for all $\lambda\in{\mathfrak{a}}^{\ast}_{M,{\mathbb{C}}}$. So it
suffices to consider the local intertwining operators
$R_{Q|P}(\pi_{v},\lambda)$. Let $P_{0},...,P_{k}\in{\mathcal{P}}(M)$ and
$\alpha_{1},...,\alpha_{k}\in\Sigma_{M}$ such that $P=P_{0}$, $Q=P_{k}$ and
$P_{i-1}\big{|}^{\alpha_{i}}P_{i}$ for $i=1,...,k$. By [Ar4, ($R_{2}$)] we get

(7.11) $R_{Q|P}(\pi_{v},\lambda)=R_{P_{k}|P_{k-1}}(\pi_{v},\lambda)\circ
R_{R_{k-1}|P_{k-2}}(\pi_{v},\lambda)\circ\cdots\circ
R_{P_{1}|P_{0}}(\pi_{v},\lambda).$

Hence we can assume that $Q,P\in{\mathcal{P}}(M)$ are adjacent along some root
$\alpha\in\Sigma_{M}$. Then $R_{Q|P}(\pi_{v},\lambda)$ depends only on
$\lambda(\alpha^{\vee})$. First consider the case $v<\infty$. Let $q_{v}$ be
the order of the residue field of $F_{v}$. By $(R_{6})$,
$R_{Q|P}(\pi_{v},\lambda)$ is a rational function of
$q_{v}^{-\lambda(\alpha^{\vee})}$. Furthermore, by [Sh1, Theorem 2.2.2] there
exists a polynomial $Q_{v}(\pi_{v},s)$ with $Q_{v}(\pi_{v},0)=1$, such that

$Q_{v}(\pi_{v},q_{v}^{-\lambda(\alpha^{\vee})})J_{Q|P}(\pi_{v},\lambda)$

is a holomorphic and non-zero function of
$\lambda\in{\mathfrak{a}}_{M,{\mathbb{C}}}^{\ast}$. Moreover, the degree of
the polynomial $Q_{v}$ is independent of $\pi_{v}\in\Pi(M(F_{v}))$ By (7.10)
it follows that

$n_{Q|P}(\pi_{v},\lambda)Q_{v}(\pi_{v},q_{v}^{-\lambda(\alpha^{\vee})})R_{Q|P}(\pi_{v},\lambda)$

is holomorphic on ${\mathfrak{a}}_{M,{\mathbb{C}}}^{\ast}$. The normalizing
factors $n_{Q|P}(\pi_{v},\lambda)$ satisfy properties similar to the
corresponding properties $(R_{1}),...,(R_{8})$ satisfied by the local
intertwining operators [Ar4, Theorem 2.1]. In particular, there exists a
rational function $n_{\alpha}(\pi_{v},s)$ such that

$n_{Q|P}(\pi_{v},\lambda)=n_{\alpha}(\pi_{v},q_{v}^{-\lambda(\alpha^{\vee})}).$

Now observe that for every $R>0$ and $z\in{\mathbb{C}}$ the number of
solutions of $q_{v}^{-s}=z$ in the disc $|s|\leq R$ is bounded by
$1+(2\pi)^{-1}\log(q_{v})R$. Hence it suffices to estimate the number of zeros
of $n_{\alpha}(\pi_{v},s)$ and $Q_{v}(\pi_{v},s)$, respectively. As mentioned
above, the degree of $Q_{v}(\pi,s)$ is bounded independently of
$\pi\in\Pi(M(F_{v}))$. The rational function $n_{\alpha}(\pi_{v},s)$ has been
described in [Mu2, (3.6)]. It follows from [Mu2, Lemma 3.1] and [Mu2, (3.6)]
that there exists $C>0$ such that for all $M\in{\mathcal{L}}(M_{0})$ and all
square integrable $\pi\in\Pi(M(F_{v}))$ the number of poles and zeros of
$n_{\alpha}(\pi,s)$ is less than $C$. Now let $\pi$ be tempered. It is known
that $\pi$ is an irreducible constituent of an induced representation
$\operatorname{Ind}_{R}^{M}(\sigma)$, where $M_{R}$ is an admissible Levi
subgroup of $M$ and $\sigma\in\Pi(M_{R}(F_{v}))$ is square integrable modulo
$A_{R}$. Then by [Ar4, (2.2)] we are reduced to the square integrable case. In
general, $\pi$ is a Langlands quotient of an induced representation
$\operatorname{Ind}_{R}^{M}(\sigma,\mu)$, where $M_{R}$ is an admissible Levi
subgroup of $M$, $\sigma\in\Pi_{\mbox{\k temp}}(M_{R}(F_{v}))$, and $\mu$ is
point in the chamber of ${\mathfrak{a}}_{R}^{\ast}/{\mathfrak{a}}_{M}^{\ast}$.
Now we use [Ar4, (2.3)] to reduce to the tempered case, which proves that
there exists $C>0$ such that the number of poles and zeros of
$n_{\alpha}(\pi,s)$ is less than $C$ for all $\pi\in\Pi(M(F_{v}))$.

The case $v\in S_{\infty}$ has been already treated in section 6. See (6.11)
and the text above (6.11).

Now we can summarize our results. Using (4.11), we obtain the following
proposition.

######  Proposition 7.1.

Let $M\in{\mathcal{L}}(M_{0})$. Let $K_{f}\subset G({\mathbb{A}}_{f})$ be an
open compact subgroup and $\sigma\in\Pi({\bf K}_{\infty})$. For every $R>0$
there exists $C>0$ such that for all $Q,P\in{\mathcal{P}}(M)$ and
$\pi\in\Pi(M({\mathbb{A}}))$ the number of singular hyperplanes of
$R_{Q|P}(\pi,\lambda)\big{|}_{{\mathcal{A}}^{2}_{\pi}(P)^{K_{f},\sigma}}$,
which intersects the ball of radius $R$ in
${\mathfrak{a}}_{M,{\mathbb{C}}}^{\ast}$, is bounded by $C$.

Next we consider the global normalizing factors. By (4.11),
$n_{Q|P}(\pi,\lambda)$ is the product of the normalizing factors
$n_{\alpha}(\pi,\lambda(\alpha^{\vee}))$,
$\alpha\in\Sigma_{P}\cap\Sigma_{\bar{Q}}$. Thus our problem is reduced to the
estimation of the number of real poles, counted to multiplicity, of the
meromorphic function $n_{\alpha}(\pi,s)$. Let
$\Sigma_{\alpha}^{\mathbb{R}}(\pi)$ be the set of real poles of
$n_{\alpha}(\pi,s)$. Let $\pi\in\Pi_{\mbox{\k
dis}}(M({\mathbb{A}});K_{f},\sigma)$. Then it follows from Corollary 5.2 that
there exists $C>0$

(7.12) $\sum_{\rho\in\Sigma_{\alpha}^{\mathbb{R}}(\pi)}n(\rho)\leq
C(1+\log(1+\lambda^{2}_{\pi_{\infty}})).$

Now we can summarize our results as follows. For arbitrary
$Q,P\in{\mathcal{P}}(M)$, the global normalizing factor $n_{Q|P}(\pi,\lambda)$
is the product of the functions $n_{\alpha}(\pi,\lambda(\alpha^{\vee}))$,
$\alpha\in\Sigma_{P}\cap\Sigma_{\bar{Q}}$ (4.11). Note that
$\\#\Sigma_{P}\leq\dim{\mathfrak{n}}_{P}$. Let
$d_{P}:=\dim{\mathfrak{n}}_{P}$. Then it follows from (7.12) and Proposition
(7.1) that there exists $C>0$ such for every $\pi\in\Pi_{\mbox{\k
cus}}(M({\mathbb{A}});K_{f},\sigma)$ and every $\phi\in{\mathcal{A}}_{\mbox{\k
cus},\pi}^{2}(P)$ the number of singular hyperplanes of the Eisenstein series
$E(\phi,\lambda)$, counted with multiplicity, which are real and intersect the
ball
$\\{\lambda\in{\mathfrak{a}}_{M,{\mathbb{C}}}^{\ast}\colon\|\lambda\|\leq\|\rho_{P}\|\\}$
is bounded by

(7.13) $C(1+\log(1+\lambda^{2}_{\pi_{\infty}}))^{d_{P}},$

Now recall from the beginning of this section that the residual spectrum is
spanned by iterated residues of Eisenstein series $E(\phi,\lambda)$ with
respect to complete flags of affine admissible subspaces of
${\mathfrak{a}}_{M,{\mathbb{C}}}^{\ast}$. For $\lambda\geq 0$ let

$\Pi_{\mbox{\k cus}}(M({\mathbb{A}});\lambda):=\\{\pi\in\Pi_{\mbox{\k
cus}}(M({\mathbb{A}}))\colon-\lambda_{\pi_{\infty}}\leq\lambda\\}.$

We note that there exists $C\in{\mathbb{R}}$ such that
$C\leq-\lambda_{\pi_{\infty}}$ for all $\pi\in\Pi_{\mbox{\k
dis}}(M({\mathbb{A}}))$. Using (7.1) and (7.5), it follows that there exist
$C_{0},C_{1}>0$ such that

(7.14) $\begin{split}N_{\mbox{\k res}}^{K_{f},\sigma}(\lambda)\leq
C_{1}\sum_{P\supset P_{0}}&(1+\log(1+\lambda^{2}))^{d_{P}}\\\
&\cdot\sum_{\tau\in{\mathcal{F}}_{M_{P}}(\sigma)}\sum_{\pi\in\Pi_{\mbox{\k
cus}}(M_{P}({\mathbb{A}});\lambda+C_{0})}m(\pi)\dim({\mathcal{H}}_{\pi_{f}}^{K_{M_{P},f}})\dim({\mathcal{H}}_{\pi_{\infty}}(\tau)).\end{split}$

For a given $P$ let $M=M_{P}$. As in (3.2) there exist finitely many lattices
$\Gamma_{M,i}\subset M(F_{\infty})$, $i=1,...,k$, such that

(7.15) $A_{M}M(F)\backslash
M({\mathbb{A}})/K_{M,f}\cong\bigsqcup_{i=1}^{k}(\Gamma_{M,i}\backslash
M(F_{\infty})^{1}).$

Thus we get an isomorphism of $M(F_{\infty})$-modules

(7.16) $L^{2}_{\mbox{\k cus}}(A_{M}M(F)\backslash
M({\mathbb{A}}))^{K_{M,f}}\cong\bigoplus_{i=1}^{k}L^{2}_{\mbox{\k
cus}}(\Gamma_{M,i}\backslash M(F_{\infty})^{1}).$

And hence for $\tau\in\Pi(K_{M,\infty})$ we get

(7.17) $L^{2}_{\mbox{\k cus}}(A_{M}M(F)\backslash
M({\mathbb{A}}))^{K_{M,f},\tau}\cong\bigoplus_{i=1}^{k}L^{2}_{\mbox{\k
cus}}(\Gamma_{M,i}\backslash(M(F_{\infty})^{1}\otimes
V_{\tau})^{K_{M,\infty}}.$

Let $\widetilde{X}_{M}=A_{M}\backslash M(F_{\infty})/K_{M,\infty}$. Let
$E_{\tau,i}\to\Gamma_{M,i}\backslash\widetilde{X}_{M}$ be the locally
homogeneous vector bundle associated to $\tau$. Let $N_{\mbox{\k
cus}}^{\Gamma_{M,i}}(\lambda;\tau)$ be the eigenvalues counting function for
the Casimir operator acting in $L^{2}_{\mbox{\k
cus}}(\Gamma_{M,i}\backslash\widetilde{X}_{M},E_{\tau,i})$. It follows from
(7.17) that

(7.18) $\sum_{\pi\in\Pi_{\mbox{\k
cus}}(M({\mathbb{A}});\lambda)}m(\pi)\dim({\mathcal{H}}^{K_{M,f}}_{\pi_{f}})\dim({\mathcal{H}}_{\pi_{\infty}}(\tau))=\sum_{i=1}^{k}N_{\mbox{\k
cus}}^{\Gamma_{M,i}}(\lambda;\tau).$

Let $m_{M}:=\dim\widetilde{X}_{M}$. Then by [Do, Theorem 9.1] we get

(7.19) $N_{\mbox{\k cus}}^{\Gamma_{M,i}}(\lambda;\tau)\ll
1+\lambda^{m_{M}/2}.$

Thus by (7.18) it follows that there exists $C>0$ such that

(7.20) $\sum_{\tau\in{\mathcal{F}}_{M}(\sigma)}\sum_{\pi\in\Pi_{\mbox{\k
cus}}(M({\mathbb{A}});\lambda+C_{0})}m(\pi)\dim({\mathcal{H}}^{K_{M,f}}_{\pi_{f}})\dim({\mathcal{H}}_{\pi_{\infty}}(\tau))\leq
C(1+\lambda^{m_{M}/2}).$

Let $n:=\dim\widetilde{X}$ and $m_{0}(G):=\max\\{m_{M_{P}}\colon P\neq G\\}$.
Note that $d_{P}\leq r$ for all $P$. Then by (7.14) and (7.20) we obtain

(7.21) $N_{\mbox{\k res}}^{K_{f},\sigma}(\lambda)\leq
C(1+\log(1+\lambda^{2}))^{n}(1+\lambda^{m_{0}(G)/2}).$

Now observe that $\widetilde{X}\cong\widetilde{X}_{M}\times A_{M}\times
N_{P}(F_{\infty})$ [Bo3, Sect. 4.2, (3)]. Hence $m_{M}\leq n-2$, and therefore
$m_{0}(G)\leq n-2$. Thus we get

(7.22) $N_{\mbox{\k res}}^{K_{f},\sigma}(\lambda)\leq
C(1+\lambda^{(n-1)/2}),\quad\lambda\geq 0,$

where $C>0$ depends on $K_{f}$ and $\sigma$. This completes the proof of the
second statement of Theorem 1.4.

Next we wish to extend this result to any Levi subgroup
$L\in{\mathcal{L}}(M)$. Recall that for any pair of elements
$Q\in{\mathcal{P}}(L)$ and $R\in{\mathcal{P}}^{L}(M)$ there exists a unique
$P\in{\mathcal{P}}(M)$ such that $P\subset Q$ and $P\cap L=R$. Then $P$ is
denoted by $Q(R)$. Let $R,R^{\prime}\in{\mathcal{P}}^{L}(M)$,
$\pi\in\Pi_{\mbox{\k dis}}(M({\mathbb{A}}))$, and $Q\in{\mathcal{P}}(L)$. Then
for any $k\in{\bf K}$ and $\phi\in{\mathcal{A}}^{2}_{\pi}(Q(R))$, the function
$\phi_{k}$ on $M({\mathbb{A}})$, which is defined by $\phi_{k}(m):=\phi(mk)$,
$m\in M({\mathbb{A}})$, belongs to ${\mathcal{A}}^{2}_{\pi}(R)$, and one has

(7.23)
$(M_{Q(R^{\prime})|Q(R)}(\pi,\lambda)\phi)_{k}=M_{R^{\prime}|R}(\pi,\lambda)\phi_{k})$

[Ar9, (1.3)]. Furthermore, the normalizing factors satisfy

(7.24) $n_{Q(R^{\prime})|Q(R)}(\pi,\lambda)=n_{R^{\prime}|R}(\pi,\lambda)$

[Ar4, Sect. 2]. Thus the considerations above continue to hold for each
standard Levi subgroup $L$ of $G$. For $M\in{\mathcal{L}}(M_{0})$ and
$\sigma\in\Pi({\bf K}_{\infty})$ let $\sigma_{M}:=\sigma|_{{\bf
K}_{\infty}\cap M(F_{\infty})}$. Denote by $N_{M,\mbox{\k
res}}^{K_{M,f},\sigma_{M}}(\lambda)$ the counting function of the residual
spectrum for $M$ with respect to $(K_{M,f},\sigma_{M})$. Let
$m_{0}(M):=\max\\{m_{M_{R}}\colon R\in{\mathcal{P}}^{M},\;R\neq M\\}$. Then,
summarizing our results, we get

(7.25) $N_{M,\mbox{\k res}}^{K_{M,f},\sigma_{M}}(\lambda)\leq
C(1+\log(1+\lambda^{2}))^{r_{M}}\lambda^{m_{0}(M)/2}$

for $\lambda\geq 0$. As above it follows that
$m_{0}(M)\leq\dim\widetilde{X}_{M}-2$, and we obtain the following
proposition.

######  Proposition 7.2.

Assume that $G$ satisfies condition (L). Let $K_{f}$ be an open compact
subgroup of $G({\mathbb{A}}_{f})$ and $\sigma\in\Pi({\bf K}_{\infty})$. Let
$M\in{\mathcal{L}}$ and $m_{M}:=\dim\widetilde{X}_{M}$. Then there exists
$C>0$ such that

(7.26) $N_{M,\mbox{\k res}}^{K_{M,f},\sigma_{M}}(\lambda)\leq
C(1+\lambda^{(m_{M}-1)/2})$

for $\lambda\geq 0$.

##  8\. The spectral side of the trace formula

In this section we apply the spectral side of the (non-invariant) trace
formula of Arthur [Ar1], [Ar2], to the heat kernel. The goal is to prove that
the leading term of the asymptotic expansion as $t\to 0$ is given by the trace
of the heat operator, restricted to the point spectrum.

To begin with we briefly recall the structure of the spectral side. Let
$L\supset M$ be Levi subgroups in ${\mathcal{L}}$, $P\in\mathcal{P}(M)$, and
let $m=\dim\mathfrak{a}_{L}^{G}$ be the co-rank of $L$ in $G$. Denote by
$\mathfrak{B}_{P,L}$ the set of $m$-tuples
$\underline{\beta}=(\beta_{1}^{\vee},\dots,\beta_{m}^{\vee})$ of elements of
$\Sigma_{P}^{\vee}$ whose projections to ${\mathfrak{a}}_{L}$ form a basis for
${\mathfrak{a}}_{L}^{G}$. For any
$\underline{\beta}=(\beta_{1}^{\vee},\dots,\beta_{m}^{\vee})\in\mathfrak{B}_{P,L}$
let $\operatorname{vol}(\underline{\beta})$ be the co-volume in
${\mathfrak{a}}_{L}^{G}$ of the lattice spanned by $\underline{\beta}$ and let

$\displaystyle\Xi_{L}(\underline{\beta})$
$\displaystyle=\\{(Q_{1},\dots,Q_{m})\in{\mathcal{F}}_{1}(M)^{m}:\ \
\beta_{i}^{\vee}\in{\mathfrak{a}}_{M}^{Q_{i}},\,i=1,\dots,m\\}$
$\displaystyle=\\{\langle P_{1},P_{1}^{\prime}\rangle,\dots,\langle
P_{m},P_{m}^{\prime}\rangle):\ \
P_{i}|^{\beta_{i}}P_{i}^{\prime},\,i=1,\dots,m\\}.$

Given $Q,P\in{\mathcal{P}}(M)$, let
$M_{Q|P}(\lambda)\colon{\mathcal{A}}^{2}(P)\to{\mathcal{A}}^{2}(Q)$,
$\lambda\in{\mathfrak{a}}^{\ast}_{M,{\mathbb{C}}}$, be the intertwining
operator defined by (4.4).

For any smooth function $f$ on ${\mathfrak{a}}_{M}^{*}$ and
$\mu\in{\mathfrak{a}}_{M}^{*}$ denote by $D_{\mu}f$ the directional derivative
of $f$ along $\mu\in{\mathfrak{a}}_{M}^{*}$. For a pair $P_{1}|^{\alpha}P_{2}$
of adjacent parabolic subgroups in ${\mathcal{P}}(M)$ write

(8.1)
$\delta_{P_{1}|P_{2}}(\lambda)=M_{P_{2}|P_{1}}(\lambda)D_{\varpi}M_{P_{1}|P_{2}}(\lambda):{\mathcal{A}}^{2}(P_{1})\rightarrow{\mathcal{A}}^{2}(P_{2}),$

where $\varpi\in{\mathfrak{a}}_{M}^{*}$ is such that
$\left\langle\varpi,\alpha^{\vee}\right\rangle=1$. 111Note that this
definition differs slightly from the definition of $\delta_{P_{1}|P_{2}}$ in
[FLM1]. Equivalently, writing
$M_{P_{1}|P_{2}}(\lambda)=\Phi(\left\langle\lambda,\alpha^{\vee}\right\rangle)$
for a meromorphic function $\Phi$ of a single complex variable, we have

$\delta_{P_{1}|P_{2}}(\lambda)=\Phi(\left\langle\lambda,\alpha^{\vee}\right\rangle)^{-1}\Phi^{\prime}(\left\langle\lambda,\alpha^{\vee}\right\rangle).$

Recall that for $P,Q\in{\mathcal{P}}(M)$, $\langle P,Q\rangle$ denotes the
group generated by $P$ and $Q$. For any $m$-tuple
$\mathcal{X}=(Q_{1},\dots,Q_{m})\in\Xi_{L}(\underline{\beta})$ with
$Q_{i}=\langle P_{i},P_{i}^{\prime}\rangle$,
$P_{i}|^{\beta_{i}}P_{i}^{\prime}$, denote by
$\Delta_{\mathcal{X}}(P,\lambda)$ the expression

(8.2)
$\frac{\operatorname{vol}(\underline{\beta})}{m!}M_{P_{1}^{\prime}|P}(\lambda)^{-1}\delta_{P_{1}|P_{1}^{\prime}}(\lambda)M_{P_{1}^{\prime}|P_{2}^{\prime}}(\lambda)\cdots\delta_{P_{m-1}|P_{m-1}^{\prime}}(\lambda)M_{P_{m-1}^{\prime}|P_{m}^{\prime}}(\lambda)\delta_{P_{m}|P_{m}^{\prime}}(\lambda)M_{P_{m}^{\prime}|P}(\lambda).$

Recall the (purely combinatorial) map
$\mathcal{X}_{L}:\mathfrak{B}_{P,L}\to{\mathcal{F}}_{1}(M)^{m}$ with the
property that
$\mathcal{X}_{L}(\underline{\beta})\in\Xi_{L}(\underline{\beta})$ for all
$\underline{\beta}\in\mathfrak{B}_{P,L}$ as defined in [FLM1, pp.
179–180].222The map $\mathcal{X}_{L}$ depends in fact on the additional choice
of a vector $\underline{\mu}\in(\mathfrak{a}^{*}_{M})^{m}$ which does not lie
in an explicit finite set of hyperplanes. For our purposes, the precise
definition of $\mathcal{X}_{L}$ is immaterial.

For any $s\in W(M)$ let $L_{s}$ be the smallest Levi subgroup in
${\mathcal{L}}(M)$ containing $w_{s}$. We recall that
$\mathfrak{a}_{L_{s}}=\\{H\in\mathfrak{a}_{M}\mid sH=H\\}$. Set

$\iota_{s}=\lvert\operatorname{det}(s-1)_{\mathfrak{a}^{L_{s}}_{M}}\rvert^{-1}.$

For $P\in{\mathcal{F}}(M_{0})$ and $s\in W(M_{P})$ let
$M(P,s):{\mathcal{A}}^{2}(P)\to{\mathcal{A}}^{2}(P)$ be as in [Ar3, p. 1309].
$M(P,s)$ is a unitary operator which commutes with the operators
$\rho(P,\lambda,h)$ for $\lambda\in{\mathrm{i}}\mathfrak{a}_{L_{s}}^{*}$.
Finally, we can state the refined spectral expansion.

######  Theorem 8.1 ([FLM1]).

For any $h\in C_{c}^{\infty}(G({\mathbb{A}})^{1})$ the spectral side of
Arthur’s trace formula is given by

(8.3) $J_{\text{spec}}(h)=\sum_{[M]}J_{\text{spec},M}(h),$

$M$ ranging over the conjugacy classes of Levi subgroups of $G$ (represented
by members of $\mathcal{L}$), where

(8.4) $J_{\text{spec},M}(h)=\frac{1}{\lvert W(M)\rvert}\sum_{s\in
W(M)}\iota_{s}\sum_{\underline{\beta}\in\mathfrak{B}_{P,L_{s}}}\int_{{\mathrm{i}}(\mathfrak{a}^{G}_{L_{s}})^{*}}\operatorname{tr}(\Delta_{\mathcal{X}_{L_{s}}(\underline{\beta})}(P,\lambda)M(P,s)\rho(P,\lambda,h))\
d\lambda$

with $P\in\mathcal{P}(M)$ arbitrary. The operators are of trace class and the
integrals are absolutely convergent with respect to the trace norm and define
distributions on ${\mathcal{C}}(G({\mathbb{A}})^{1})$.

Now we apply the trace formula to the heat kernel. We recall its definition.
For details see [MM1, $\S$ 3]. Recall that the underlying symmetric space is

$\widetilde{X}=G(F_{\infty})^{1}/{\bf K}_{\infty},$

where $G(F_{\infty})^{1}=G({\mathbb{A}})^{1}\cap G(F_{\infty})$. Note that
$G(F_{\infty})^{1}$ is semisimple and

(8.5) $G(F_{\infty})=G(F_{\infty})^{1}\cdot A_{G}.$

Given $\nu\in\Pi({\bf K}_{\infty})$, let $\widetilde{E}_{\nu}\to\widetilde{X}$
be the associated homogeneous vector bundle. Let
$\widetilde{\Delta}_{\nu}=(\widetilde{\nabla}^{\nu})^{\ast}\widetilde{\nabla}^{\nu}$
be the Bochner-Laplace operator acting in the space
$C^{\infty}(\widetilde{X},\widetilde{E}_{\nu})$ of smooth sections of
$\widetilde{E}_{\nu}$. This is a $G(F_{\infty})^{1}$-invariant second order
elliptic differential operator. Since $\widetilde{X}$ is complete,
$\widetilde{\Delta}_{\nu}$, regarded as operator in
$L^{2}(\widetilde{X},\widetilde{E}_{\nu})$ with domain the smooth compactly
supported sections, is essentially self-adjoint [LaM, p. 155]. Its self-
adjoint extension will also be denoted by $\widetilde{\Delta}_{\nu}$. Let
$\Omega\in{\mathcal{Z}}({\mathfrak{g}}_{\mathbb{C}})$ and $\Omega_{{\bf
K}_{\infty}}\in{\mathcal{Z}}({\mathfrak{k}})$ be the Casimir operators of
${\mathfrak{g}}$ and ${\mathfrak{k}}$, respectively, where the latter is
defined with respect to the restriction of the normalized Killing form of
$\mathfrak{g}$ to $\mathfrak{k}$. Then with respect to the isomorphism (3.13)
we have

(8.6) $\displaystyle\widetilde{\Delta}_{\nu}=-R(\Omega)+\nu(\Omega_{{\bf
K}_{\infty}}),$

where $R$ denotes the right regular representation of $G(F_{\infty})$ in
$C^{\infty}(G(F_{\infty}),\nu)$ (see [Mia, Proposition 1.1]).

Let $e^{-t\widetilde{\Delta}_{\nu}}$, $t>0$, be the heat semigroup generated
by $\widetilde{\Delta}_{\nu}$. It commutes with the action of
$G(F_{\infty})^{1}$. With respect to the isomorphism (3.13) we may regard
$e^{-t\widetilde{\Delta}_{\nu}}$ as a bounded operator in
$L^{2}(G(F_{\infty})^{1},\nu)$, which commutes with the action of
$G(F_{\infty})^{1}$. Hence it is a convolution operator, i.e., there exists a
smooth map

(8.7) $H_{t}^{\nu}\colon G(F_{\infty})^{1}\to\operatorname{End}(V_{\nu})$

such that

$(e^{-t\widetilde{\Delta}_{\nu}}\phi)(g)=\int_{G(F_{\infty})^{1}}H_{t}^{\nu}(g^{-1}g^{\prime})(\phi(g^{\prime}))\;dg^{\prime},\quad\phi\in
L^{2}(G(F_{\infty})^{1},\nu).$

The kernel $H_{t}^{\nu}$ satisfies

(8.8)
$\displaystyle{H}^{\nu}_{t}(k^{-1}gk^{\prime})=\nu(k)^{-1}\circ{H}^{\nu}_{t}(g)\circ\nu(k^{\prime}),\>\forall
k,k^{\prime}\in{\bf K}_{\infty},\forall g\in G(F_{\infty})^{1}.$

Moreover, proceeding as in the proof of [BM, Proposition 2.4] it follows that
$H^{\nu}_{t}$ belongs to
$({\mathscr{C}}^{q}(G(F_{\infty})^{1})\otimes\operatorname{End}(V_{\nu}))^{K_{\infty}\times
K_{\infty}}$ for all $q>0$, where ${\mathscr{C}}^{q}(G(F_{\infty})^{1})$ is
Harish-Chandra’s Schwartz space of $L^{q}$-integrable rapidly decreasing
functions on $G(F_{\infty})^{1}$. Put

(8.9) $h_{t}^{\nu}(g):=\operatorname{tr}H_{t}^{\nu}(g),\quad g\in
G(F_{\infty})^{1}.$

Then $h_{t}^{\nu}\in{\mathscr{C}}^{q}(G(F_{\infty})^{1})$ for all $q>0$. We
extend $h_{t}^{\nu}$ to a function on $G(F_{\infty})$ by

$h_{t}^{\nu}(ag)=h_{t}^{\nu}(g),\quad g\in G(F_{\infty})^{1},\;a\in A_{G}.$

Let ${\bf 1}_{K_{f}}\colon G({\mathbb{A}}_{f})\to{\mathbb{C}}$ be the
characteristic function of $K_{f}$. Put

(8.10) $\chi_{K_{f}}:=\frac{{\bf 1}_{K_{f}}}{\operatorname{vol}(K_{f})}$

and

(8.11) $\phi_{t}^{\nu}(g):=h_{t}^{\nu}(g_{\infty})\chi_{K_{f}}(g_{f})$

for $g=g_{\infty}\cdot g_{f}\in G({\mathbb{A}})=G(F_{\infty})\cdot
G({\mathbb{A}}_{f})$. Now observe that all derivatives of $\phi_{t}^{\nu}$
belong to $L^{1}(G({\mathbb{A}})^{1})$. Thus $\phi_{t}^{\nu}$ belongs to
${\mathcal{C}}(G({\mathbb{A}});K_{f})$ (see section 2 for its definition). By
Theorem 8.1, $J_{\text{spec}}$ is a distribution on
${\mathcal{C}}(G({\mathbb{A}});K_{f})$. Thus we can insert $\phi_{t}^{\nu}$
into the trace formula and by Theorem 8.1 we get

(8.12)
$J_{\text{spec}}(\phi_{t}^{\nu})=\sum_{[M]}J_{\text{spec},M}(\phi_{t}^{\nu}),$

where the sum ranges over the conjugacy classes of Levi subgroups of $G$ and
$J_{\text{spec},M}(\phi_{t}^{\tau,p})$ is given by (8.4). To analyze these
terms, we proceed as in [MM1, Section 13]. Recall that the operator
$\Delta_{\mathcal{X}}(P,\lambda)$, which appears in the formula (8.4), is
defined by (8.2). Its definition involves the intertwining operators
$M_{Q|P}(\lambda)$. If we replace $M_{Q|P}(\lambda)$ by its restriction
$M_{Q|P}(\pi,\lambda)$ to ${\mathcal{A}}^{2}_{\pi}(P)$, we obtain the
restriction $\Delta_{\mathcal{X}}(P,\pi,\lambda)$ of
$\Delta_{\mathcal{X}}(P,\lambda)$ to ${\mathcal{A}}^{2}_{\pi}(P)$. Similarly,
let $\rho_{\pi}(P,\lambda)$ be the induced representation in
$\bar{{\mathcal{A}}}^{2}_{\pi}(P)$. Fix $s\in W(M)$ and
$\beta\in\mathfrak{B}_{P,L_{s}}$. Then for the integral on the right of (8.4)
with $h=\phi_{t}^{\nu}$ we get

(8.13) $\sum_{\pi\in\Pi_{\mbox{\k
dis}}(M({\mathbb{A}}))}\int_{i({\mathfrak{a}}^{G}_{L_{s}})^{*}}\operatorname{Tr}\left(\Delta_{\mathcal{X}_{L_{s}}(\underline{\beta})}(P,\pi,\lambda)M(P,\pi,s)\rho_{\pi}(P,\lambda,\phi^{\nu}_{t})\right)\;d\lambda.$

In order to deal with the integrand, we need the following result. Let $\pi$
be a unitary admissible representation of $G(F_{\infty})$. Let
$A\colon{\mathcal{H}}_{\pi}\to{\mathcal{H}}_{\pi}$ be a bounded operator which
is an intertwining operator for $\pi|_{{\bf K}_{\infty}}$. Then
$A\circ\pi(h_{t}^{\nu})$ is a finite rank operator. Define an operator
$\tilde{A}$ on ${\mathcal{H}}_{\pi}\otimes V_{\nu}$ by
$\tilde{A}:=A\otimes\operatorname{Id}$. Then by [MM1, (9.13)] we have

(8.14)
$\operatorname{Tr}(A\circ\pi(h_{t}^{\nu}))=e^{t(\pi(\Omega)-\nu(\Omega_{{\bf
K}_{\infty}}))}\operatorname{Tr}\left(\tilde{A}|_{({\mathcal{H}}_{\pi}\otimes
V_{\nu})^{{\bf K}_{\infty}}}\right).$

We will apply this to the induced representation $\rho_{\pi}(P,\lambda)$. Let
$m(\pi)$ denote the multiplicity with which $\pi$ occurs in the regular
representation of $M({\mathbb{A}})$ in $L^{2}_{\mbox{\k
dis}}(A_{M}M(F)\backslash M({\mathbb{A}}))$. Then

(8.15)
$\rho_{\pi}(P,\lambda)\cong\oplus_{i=1}^{m(\pi)}\operatorname{Ind}_{P({\mathbb{A}})}^{G({\mathbb{A}})}(\pi,\lambda).$

Fix positive restricted roots of ${\mathfrak{a}}_{P}$ and let
$\rho_{{\mathfrak{a}}_{P}}$ denote the corresponding half-sum of these roots.
For $\xi\in\Pi(M(F_{\infty}))$ and $\lambda\in{\mathfrak{a}}^{\ast}_{P}$ let

$\pi_{\xi,\lambda}:=\operatorname{Ind}_{P(F_{\infty})}^{G(F_{\infty})}(\xi\otimes
e^{i\lambda})$

be the unitary induced representation. Let $\xi(\Omega_{M})$ be the Casimir
eigenvalue of $\xi$. Define a constant $c(\xi)$ by

(8.16)
$c(\xi):=-\langle\rho_{{\mathfrak{a}}_{P}},\rho_{{\mathfrak{a}}_{P}}\rangle+\xi(\Omega_{M}).$

Then for $\lambda\in{\mathfrak{a}}^{\ast}_{P}$ one has

(8.17) $\pi_{\xi,\lambda}(\Omega)=-\|\lambda\|^{2}+c(\xi)$

(see [Kn, Theorem 8.22]). Denote by
$\widetilde{\Delta}_{\mathcal{X}_{L_{s}}(\underline{\beta})}(P,\pi,\nu,\lambda)$
resp. $\widetilde{M}(P,\pi,\nu,s)$ the extensions of
$\widetilde{\Delta}_{\mathcal{X}_{L_{s}}(\underline{\beta})}(P,\pi,\nu,\lambda)$
resp. $M(P,\pi,\nu,s)$ to operators on ${\mathcal{A}}^{2}_{\pi}(P)\otimes
V_{\nu}$ as above. Using the definition of $\phi^{\nu}_{t}$, (8.15) and
(8.14), it follows that (8.13) is equal to

(8.18) $\begin{split}\sum_{\pi\in\Pi_{\mbox{\k
dis}}(M({\mathbb{A}}))}&e^{tc(\pi_{\infty})}\\\
&\cdot\int_{i({\mathfrak{a}}^{G}_{L_{s}})^{*}}e^{-t\|\lambda\|^{2}}\operatorname{Tr}\left(\bigl{(}\widetilde{\Delta}_{\mathcal{X}_{L_{s}}(\underline{\beta})}(P,\pi,\nu,\lambda)\widetilde{M}(P,\pi,\nu,s)\bigr{)}\big{|}_{({\mathcal{A}}^{2}_{\pi}(P)^{K_{f}}\otimes
V_{\nu})^{{\bf K}_{\infty}}}\right)\;d\lambda.\end{split}$

Since $M(P,\pi,s)$ is unitary, (8.18) can be estimated by

(8.19) $\begin{split}\sum_{\pi\in\Pi_{\mbox{\k