# Weyl’s law for arbitrary Archimedean type

Ayan Maiti1 1Department of Mathematics, Purdue University<EMAIL_ADDRESS>
###### Abstract.

We generalize the work of Lindenstrauss and Venkatesh establishing Weyl’s Law
for cusp forms from the spherical spectrum to arbitrary Archimedean type.
Weyl’s law for the spherical spectrum gives an asymptotic formula for the
number of cusp forms that are bi-$K_{\infty}$ invariant in terms of eigenvalue
$T$ of the Laplacian. We prove an analogous asymptotic holds for cusp forms
with Archimedean type $\tau$, where the main term is multiplied by
$\dim{\tau}$. While in the spherical case the surjectivity of the Satake Map
was used, in the more general case that is not available and we use Arthur’s
Paley-Wiener theorem and multipliers.

### Mathematics Subject Classification

Primary $11F72$, $22E55$, Secondary $32N15$.

##  1\. Introduction

The purpose of this article is to prove Weyl’s law for the cuspidal
automorphic forms, generalizing a result of Lindenstrauss and Venkatesh [LV]
from spherical case (bi-$K_{\infty}$ invariant) to the cuspforms of arbitrary
$K_{\infty}$-type.
Let $G$ be a semisimple linear algebraic group which is split and adjoint over
$\mathbb{Q}$. Let $G(\mathbb{R})$ be the $\mathbb{R}$-points of $G$. Let
$\Gamma\subset G(\mathbb{R})$ be an arithmetic subgroup, which we assume to be
torsion-free for simplicity. Let $K_{\infty}$ be a maximal compact subgroup of
$G(\mathbb{R})$. Let $L^{2}(\Gamma\backslash G(\mathbb{R}))$ be the space of
square integrable $\Gamma$ invariant functions on $G(\mathbb{R})$. Let
$\mathcal{Z}(\mathfrak{g}_{\mathbb{C}})$ be the center of universal enveloping
algebra of the complexification of the Lie algebra $\mathfrak{g}$ of
$G(\mathbb{R})$. A cuspform for $\Gamma$ is a smooth and $K_{\infty}$-finite
complex-valued functions $f$, which is a simultaneous eigenfunction of
$\mathcal{Z}(\mathfrak{g}_{\mathbb{C}})$ and which satisfies

$\int\limits_{\Gamma\bigcap N_{P}(\mathbb{R})\backslash
N_{P}(\mathbb{R})}f(nx)dn=0,$

for all unipotent radicals $N_{P}$ of proper rational parabolic subgroups $P$
of $G$ [La]. It can be shown that cusp forms are square- integrable. Let
$L_{\text{cusp}}^{2}(\Gamma\backslash G(\mathbb{R}))$ be the closure of the
linear span of all cusp forms. Let $R$ be the right regular representation of
$G(\mathbb{R})$ on $L^{2}(\Gamma\backslash G(\mathbb{R}))$. Suppose
$(\tau,V_{\tau})$ denotes an irreducible finite dimensional representation of
the maximal compact $K_{\infty}$. We let

$(L^{2}(\Gamma\backslash G(\mathbb{R}))\otimes V_{\tau})^{K_{\infty}}$

be the space of homogeneous vector bundle on the Riemannian symmetric space
$G(\mathbb{R})/K_{\infty}$. These are space of functions that satisfies the
following condition:

$f(gk)=\tau(k^{-1})f(g).$

Let $\Omega_{G(\mathbb{R})}$ be the casimir operator in
$\mathcal{Z}(\mathfrak{g}_{\mathbb{C}})$, the center of the universal
enveloping algebra. Then $-\Omega_{G(\mathbb{R})}\otimes Id$ induces a self
adjoint operator $\Delta_{\tau}$ whose restriction to

$L_{\text{cusp}}^{2}(\Gamma\backslash
G(\mathbb{R}),\tau):=(L_{\text{cusp}}^{2}(\Gamma\backslash
G(\mathbb{R}))\otimes V_{\tau})^{K_{\infty}}$

has pure point spectrum with finite multiplicities. Let us denote them as

$0\leq\nu_{1}(\tau)<\nu_{2}(\tau)\cdots$

with finite multiplicities. Suppose $\mathcal{E}(\nu_{i}(\tau))$ denotes the
respective eigenspace corresponding to the eigenvalue $\nu_{i}$. We define the
eigenvalue counting function as

$N_{\text{cusp}}^{\Gamma}(\nu,\tau)=\sum_{\nu_{i}(\tau)\leq\nu}\text{dim}(\mathcal{E}(\nu_{i}(\tau))).$

Let $\mathbb{H}$ be the upper half plane and let $\Gamma$ be a congruence
subgroup of $SL(2,\mathbb{Z})$. Let $\Delta$ be the hyperbolic laplacian on
$\mathbb{H}$. Using the notations above let $N_{\text{cusp}}^{\Gamma}(\nu)$ be
the eigenfunction counting function for eigenvalues upto $\nu$. Selberg [Se],
using his celebrated trace formula for the group $SL(2,\mathbb{R})$, proved
the following version of Weyl’s law:

$\lim_{\nu\to\infty}N_{\text{cusp}}^{\Gamma}(\nu)\sim\text{Vol}(\Gamma\backslash\mathbb{H})\frac{\nu}{4\pi}.$

If $d=\text{dim}G(\mathbb{R})/K_{\infty}$, then it has been conjectured by
Sarnak [Sa] that for $d>1$ and for an irreducible lattice $\Gamma$:

$\limsup_{\nu\to\infty}\frac{N_{\text{cusp}}^{\Gamma}(\nu)}{\nu^{d/2}}\sim\frac{\text{vol}(\Gamma\backslash
G)}{(4\pi)^{d/2}\Gamma(d/2+1)},$

where $\Gamma(n)$ denotes the Gamma function.
A similar conjecture was made by Müller [Mü1] for
$N_{\text{disc}}^{\Gamma}(\nu,\tau)$, the counting function for the discrete
spectrum of the Laplace operator $\Delta_{\tau}$. This conjecture states that
for any arithmetic subgroup and any $K_{\infty}$-type $\tau$ we have:

$\limsup_{\nu\to\infty}\frac{N_{\text{cusp}}^{\Gamma}(\nu,\tau)}{\nu^{d/2}}\sim\frac{\text{vol}(\Gamma\backslash
G)\text{dim}(\tau)}{(4\pi)^{d/2}\Gamma(d/2+1)}.$

Up to now this conjecture has been proved for the following cases: for the
congruence subgroups of $SO(n,1)$ by Reznikov [Rez], congruence subgroups of
$\text{Res}_{F/\mathbb{Q}}SL_{2}$, where $F$ is totally real field, by Efrat
[Ef], for $\Gamma=SL_{3}(\mathbb{Z})$ by Stephen D. Miller [Mi], and for
torsion free arithmetic subgroups of $SL_{n}(\mathbb{R})$ by Müller [Mü2].
Labesse and Müller [LM] proved a weak version of Weyl’s law for almost simply
connected, simply connected, semisimple algebraic groups. To explain their
method we introduce the following notations: Let $G(\mathbb{A})$ be the group
of adelic points of algebraic group $G$ defined over $\mathbb{Q}$. Let
$K=K_{\infty}\times K_{f}$ be an open compact subgroup of $G(\mathbb{A})$.
Then by Strong Approximation Theorem we have that for
$\Gamma=G(\mathbb{Q})\cap K_{f}$:

$L_{\text{cusp}}^{2}(G(\mathbb{Q})\backslash G(\mathbb{A})/K_{f})\simeq
L_{{\text{cusp}}}^{2}(\Gamma\backslash G(\mathbb{R})).$

To understand the spectral side of the Arthur-Selberg trace formula we need to
give a representation theoretic point of view of the eigenvalue counting
function $N_{\text{cusp}}^{\Gamma}(\nu,\tau)$. Let
$\Pi_{\text{cusp}}(G(\mathbb{A}))$ be the unitary irreducible cuspidal
subrepresentations of the regular representation of $G(\mathbb{A})$ on
$L_{\text{cusp}}^{2}(G(\mathbb{Q})\backslash G(\mathbb{A})/K_{f})$. Let
$\Pi_{\text{cusp}}(G(\mathbb{R}))$ be the subrepresentations of the regular
representations of $G(\mathbb{R})$ acting on

$L_{\text{cusp}}^{2}(G(\mathbb{Q})\backslash G(\mathbb{A})/K_{f}).$

Any element $\pi\in\Pi_{\text{cusp}}(G(\mathbb{A}))$ can be written as
$\pi=\pi_{\infty}\otimes\pi_{f}$, where
$\pi_{\infty}\in\Pi_{\text{cusp}}(G(\mathbb{R}))$. Let
$H_{\pi_{\infty}}(\tau)$ be the $\tau$-isotypical subspace of
$(\pi_{\infty},H_{\pi_{\infty}})$. Let $H_{\pi_{f}}^{K_{f}}$ be the subspace
of $K_{f}$-fixed vectors in $(\pi_{f},H_{\pi_{f}})$. Let $m(\pi_{\infty})$,
resp $m(\pi)$ denote the multiplicity with which $\pi_{\infty}$, resp $\pi$
occurs as a subrepresentation of $G(\mathbb{R})$, resp $G(\mathbb{A})$ in the
discrete subspace $L_{\text{cusp}}^{2}(G(\mathbb{Q})\backslash
G(\mathbb{A})/K_{f})$. Then we have the following :

$m(\pi_{\infty})=\sum_{\pi^{{}^{\prime}}\in\Pi_{\text{cusp}}(G(\mathbb{A}))}m(\pi^{{}^{\prime}})\text{dim}H_{\pi_{f}}^{K_{f}}$

for all $\pi^{{}^{\prime}}$ such that
$\pi_{\infty}^{{}^{\prime}}=\pi_{\infty}$. Suppose $\nu_{\pi}$ denotes the
Casimir eigenvalue of $\pi_{\infty}$. Then we take the sub-collection
$\Pi_{\text{cusp}}(G(\mathbb{A}))_{\nu}$ such that $|\nu_{\pi}|\leq\nu$.
Similarly we define $\Pi_{\text{cusp}}(G(\mathbb{R}))_{\nu}$. Then we have :

$\sum_{\pi_{\infty}\in\Pi_{\text{cusp}}(G(\mathbb{R}))_{\nu}}m(\pi_{\infty})\text{dim}(\text{Hom}_{K_{\infty}}(H_{\pi_{\infty}}(\tau),V_{\tau}))=N_{\text{cusp}}^{\Gamma}(\nu,\tau).$

The usual idea of proving the asymptotic formula for the counting functions is
to apply the Arthur-Selberg trace formula for a family of test functions on
$G(\mathbb{A})$ whose Archimedean part arise from the integral kernel function
of the integral operator $e^{-t\Delta_{\tau}}$, for $0\leq t<1$, and the non-
Archimedean parts are idempotents $e_{K_{f}}$. In the spectral side the terms
corresponding to the innerproduct of Eisenstein series will contribute
trivially when $t\to 0$ as shown by Müeller [Mü2] in the case of
$SL_{n}(\mathbb{R})$. But the calculation is delicate for arbitrary groups. In
this regard it will be useful to find test functions such that convolution
operators with respect to them have purely cuspidal image.
Let $S$ be a finite set of non-archimedean places. Then, following the simple
trace formula introduced by Flicker and Kazdhan [FK], Labesse and Müller [LM]
considered the test functions decomposed as $f=f_{\infty}\otimes f_{S}\otimes
e_{K_{f}^{S}}$, where $f_{S}$ are the pseudo-coefficients of Steinberg
representation of $G(\mathbb{Q}_{S})$ acting on
$L_{\text{cusp}}^{2}(G(\mathbb{Q})\backslash G(\mathbb{A})/K_{f})$. Hence the
image of the right regular representation with respect to the above test
function projects into the subspace
$L_{\text{cusp}}^{2}(G(\mathbb{Q})\backslash G(\mathbb{A})/K_{f},S)$,
generated by the vectors of automorphic representations which are Steinberg at
places in $S$. Define the eigenvalue counting function
$N_{\text{cusp}}^{\Gamma}(\nu,\tau,S)$ with respect to $S$ in
$L_{\text{cusp}}^{2}(G(\mathbb{Q})\backslash G(\mathbb{A})/K_{f},S)$. Using
this idea they were able to show that:

$\limsup_{\nu\to\infty}\frac{N_{\text{cusp}}^{\Gamma}(\nu,\tau,S)}{\nu^{d/2}}=\frac{C_{S}(\Gamma)\text{vol}(\Gamma\backslash
G)\text{dim}(\tau)}{(4\pi)^{d/2}\Gamma(d/2+1)}.$

But the non-triviality of the constant $C_{S}(\Gamma)$ would depend on the
choice of the compact set $K_{f}$, as $\Gamma=G(\mathbb{Q})\cap K_{f}$, where
$K_{p}$ for $p\in S$ lies inside the minimal parahoric compact subgroup.
To get the full Weyl’s law for spherical cuspforms on semisimple Algebraic
group of split and adjoint type, Lindenstrauss and Venkatesh [LV] were able to
find a collection of test functions from the spherical Hecke algebra
$C_{c}^{\infty}(G_{S}//K_{S})$ that has purely cuspidal image, where $S$ is a
set of places containing the Archimedean places, and $K_{S}=K_{\infty}\times
K_{f}$, where $K_{\infty}$ is a maximal compact subgroup of $G(\mathbb{R})$
and $K_{f}$ is a hyperspecial maximal compact subgroup of
$G_{S\backslash\infty}$. They use the Satake Isomorphism for spherical Hecke
algebra to prove the existence of such functions. Also the arithmetic subgroup
is chosen as a congruence subgroup of $G(\mathbb{Z}[S^{-1}])$, so that the
projection of $\Gamma$ on the finite number of inequivalent conjugacy classes
of parabolic subgroups of $G_{S}$ have large center. Hence there are
constraints on the spectral parameters of Eisenstein series at different
places. They used the Paley-Wiener Theorem for the spherical functions to
choose the family of test functions of the form $\phi_{n}\star f_{t}$, for
$n\in\mathbb{Z},0\leq t<1$, where $\phi_{n}$’s are the family of test
functions constructed to get the purely cuspidal image. Now instead of using
Arthur’s trace formula they use a partial trace formula introduced by Miller
[Mi] to get the lower bound with the same constant as Donnelly’s upperbound
[Do].
Jack Buttcane in [BU] proved the Weyl’s Law for the case $G=GL(3,\mathbb{R})$,
using Kuznetsov trace formula.

Our theorem, which is a generalization of the result of Lindenstrauss and
Venkatesh [LV], to the case of cuspforms of arbitrary $K_{\infty}$-type is the
following:

######  Theorem.

Let $G$ be a semi-simple, split, adjoint linear algebraic group over
$\mathbb{Q}$. Let $G_{\infty}=G(\mathbb{R})$ be the real points of $G$, and
let $\Gamma$ be a torsion free arithmetic subgroup of $G$. Suppose
$d=\text{dim}(G_{\infty}/K_{\infty})$. Let $(\tau,V_{\tau})$ be an irreducible
finite-dimensional representation of $K_{\infty}$. Let
$N_{\text{cusp}}^{\Gamma}(T,\tau)$ be the counting function of the cuspidal
eigenfunctions of $\Delta_{\tau}$ with eigenvalue $\leq T$. Then we have the
following asymptotic formula:

$\displaystyle\frac{N_{\text{cusp}}^{\Gamma}(T,\tau)}{T^{d/2}}\sim\frac{\text{dim}(\tau)\text{vol}(\Gamma\backslash
G_{\infty})}{(4\pi)^{d/2}\Gamma(d/2+1)},\quad\text{as}\quad T\to\infty.$ (1)

We follow the same methodology as in Lindenstrauss-Venkatesh [LV] to prove the
Weyl’s law (without the remainder term) for an arbitrary irreducible
$K_{\infty}$-type. But as the Abel-Satake map is not surjective in this case,
we can not use the same construction to get the non-trivial test function
whose image under convolution lies inside the cuspidal space. Hence, we use
the Arthur’s Paley-Wiener theorem for Archimedean [Ar3] and non-Archimedean
[Ar4] cases. We note that we are only concerned with the main term of the
Weyl’s law. To get an estimation of the error terms as derived by Müller, one
has to use the full trace formula and estimates on the constant terms of the
Eisenstein series.
This draft is organized as follows: in section 2 we discuss the necessary
preliminaries of Harmonic Analysis of reductive groups over real and $p$-adic
fields. In section 3 we prove some estimates of Plancherel Measure necessary
to prove the asymptotic formula of the Main term of the Weyl’s law, section 4
we prove a condition on test functions so that their convolution image is
purely cuspidal, section 5 and 6 we provide the analysis to derive Theorem
mentioned above.

I would like to thank my advisor Professor Mahdi Asgari for suggesting this
problem, and Professors Roberto Camporesi, Jayce Getz, Werner Müller, Sug Woo
Shin and Roger Zierau for their help and useful inputs at various stages of
this work.

##  2\. Preliminaries

In this section we will recall some basic facts of Harmonic Analysis.(see
[LV]).

###  2.1. Parabolic Subgroups

Let $G$ be a semisimple split adjoint linear algebraic group over
$\mathbb{Q}$. Let $S$ be a finite set of places containing $\infty$. We fix a
minimal parabolic, i.e. a Borel subgroup, $P_{0}\supset A_{0}$, where $A_{0}$
is a maximal $\mathbb{Q}$-spilt torus. Suppose $N_{0}=R_{u}(P_{0})$ is the
unipotent radical of $P_{0}$. We have the Levi decomposition
$P_{0}=M_{0}N_{0}$. Let $P$ be a parabolic subgroup containing $P_{0}$ with a
Levi decomposition $P=M_{P}N_{P}$. Moreover we let $A_{P}=$ Split part of
$Z(M_{P})$, where $Z$ denotes the center.
Let $F=\mathbb{Q}_{p}$ or $\mathbb{R}$. Fix a maximal split torus $A_{0}$ in
$G(F)$. We denote by $W=W(G,A_{0})$ the Weyl group of $G(F)$ with respect to
$A_{0}$. Let $\Phi=\Phi(G,A_{0})$ be the set of roots. Fix a minimal parabolic
subgroup $B$ containing $A_{0}$. The choice of $B$ determines the set of
simple roots $\Pi$ and the set of positive roots $\Phi^{+}\subset\phi$. If
$\alpha\in\Phi^{+}$, we write $\alpha>0$.
Let $P=MN\subset G(F)$ be a standard parabolic subgroup of $G(F)$. We denote
by $\Pi_{M}\subset\Phi$ the corresponding set of simple roots. Let $A_{M}$ be
the split component of the center of $M$, $X(M)_{F}$ the group of $F$-rational
characters of $M$. If $\Pi_{M}=\Theta$, we also use $A_{\Theta}$ to denote
$A_{M}$ . Hence, $A_{\emptyset}=A$ and $A_{\Pi}=A_{G}$.
The restriction homomorphism $X(M)_{F}\mapsto X(A_{M})_{F}$ is injective and
has a finite cokernel. Therefore, we have a canonical linear isomorphism:

$\mathfrak{a}_{M}^{*}=X(M)_{F}\otimes_{\mathbb{Z}}\mathbb{R}\cong
X(A_{M})_{F}\otimes_{\mathbb{Z}}\mathbb{R}.$

If $L$ is a standard parabolic subgroups such that $L\subset M$, then

$A_{M}\subset A_{L}\subset L\subset M.$

The restriction $X(M)_{F}\mapsto X(L)_{F}$ induces an injective map and its
restriction induces a linear injection
$i_{M}^{L}:\mathfrak{a}_{M}^{*}\mapsto\mathfrak{a}_{L}^{*}$. The restriction
$X(A_{L})_{F}\mapsto X(A_{M})_{F}$ induces a linear surjection
$r_{M}^{L}:\mathfrak{a}_{L}^{*}\mapsto\mathfrak{a}_{M}^{*}$. Let
$(\mathfrak{a}_{M}^{L})^{*}$ be the kernel of the restriction $r_{M}^{L}$.
Then

$\mathfrak{a}_{L}^{*}=i_{M}^{L}(\mathfrak{a}_{M}^{*})\oplus(\mathfrak{a}_{M}^{L})^{*}.$

There is a homomorphism $H_{M}:M\mapsto\mathfrak{a}_{M}=Hom(X(M),\mathbb{R})$
such that:

$\left|\nu(m)\right|_{F}=\begin{cases}q^{(\nu,H_{M}(m))},&\text{if
}F=\mathbb{Q}_{p}\\\ e^{(\nu,H_{M}(m))},&\text{if }F=\mathbb{R}\end{cases}$

, for all $m\in M$ and $\nu\in X(M)_{F}$.
We set $G_{S}=G(\mathbb{Q}_{S})$, $A_{0,S}=A_{0}(\mathbb{Q}_{S})$,
$M_{0,S}=M_{0}(\mathbb{Q}_{S})$ and $N_{0,S}=N_{0}(\mathbb{Q}_{S})$. We denote
a parabolic subgroup over $\mathbb{Q}_{S}$ as
$P_{S}=M(\mathbb{Q}_{S})N(\mathbb{Q}_{S})$ with its corresponding Levi
decomposition. We can think of this parabolic as a direct product of parabolic
subgroups of a product of groups. Let $G_{\infty}=G(\mathbb{R})$. We have an
Iwasawa decomposition $G_{\infty}=N_{\infty}A_{\infty}^{o}K_{\infty}$, where
$K_{\infty}$ is a maximal compact subgroup of $G(\mathbb{R})$.
Let $K_{S}=K_{\infty}\prod\limits_{p<\infty}G(\mathbb{Z}_{p})$, where
$G(\mathbb{Z}_{p})$ is a maximal compact subgroup of $G(\mathbb{Q}_{p})$ for
all prime $p$. We will assume that $S$ has the property that, for each finite
$p\in S$ and for each parabolic $P(\mathbb{Q}_{p})$ containing
$A_{0}(\mathbb{Q}_{p})$, $K_{p}\bigcap M_{P}(\mathbb{Q}_{p})$ is the
stabilizer in $M_{P}(\mathbb{Q}_{p})$ of a special vertex in the building of
$M_{P}(\mathbb{Q}_{p})$; and moreover this vertex belongs to the apartment
associated to the maximal torus $A_{0}(\mathbb{Q}_{p})$. This condition is
satisfied for almost all finite $p$. Moreover, $K_{\infty}\bigcap
M_{P}(\mathbb{R})$ is a maximal compact subgroup of $M_{P}(\mathbb{R})$, and
$K_{S}\bigcap M(\mathbb{Q}_{S})$ is a maximal compact subgroup of
$M(\mathbb{Q}_{S})$.
The map $N(\mathbb{Q}_{S})\times M(\mathbb{Q}_{S})\times K_{S}\mapsto G_{S}$
is surjective (Iwasawa). We equip each $G(\mathbb{Q}_{p})$, for $p$ finite,
with the Haar measure which assigns $G(\mathbb{Z}_{p})$ the mass 1. We equip
$K_{\infty}$ with the Haar measure of mass 1, and then choose the Haar measure
on $G_{\infty}$ which is compatible with the Riemannian metric defined on the
Riemannian symmetric space $G_{\infty}/K_{\infty}$. Let $\Phi^{+}$ be the
system of positive roots of $A_{0,S}$ with respect to $N_{0,S}$ and let
$\Delta\subset\Phi^{+}$ be the set of simple roots. Let $\delta_{S}$ be the
square root of the modulus character of $A_{0,S}$.
The following lemma which is due to Harish-chandra going to describe the
correspondence between parabolic subgroups of $G(\mathbb{Q}_{p})$ contained in
some parabolic subgroup $Q(\mathbb{Q}_{p})$ and the parabolic subgroups of
$M_{Q}(\mathbb{Q}_{p})$ for all $p\in S$.

######  Lemma 1.

There is an one to one correspondence between Parabolic subgroup
$P(\mathbb{Q}_{p})$ of $G(\mathbb{Q}_{p})$ which are contained in
$Q(\mathbb{Q}_{p})$, and parabolic subgroups ${}^{*}P(\mathbb{Q}_{p})$ of
$M_{Q}(\mathbb{Q}_{p})$. The correspondence are as follows: If
$Q(\mathbb{Q}_{p})=M_{Q}(\mathbb{Q}_{p})N_{Q}(\mathbb{Q}_{p})$ and
$P(\mathbb{Q}_{p})=M_{P}(\mathbb{Q}_{p})N_{P}(\mathbb{Q}_{p})$ are the
corresponding Levi decompositions, then the Levi decomposition of
${}^{*}P_{Q}=P(\mathbb{Q}_{p})\cap
M_{Q}(\mathbb{Q}_{p})=M_{P}(\mathbb{Q}_{p})N_{Q}^{P}(\mathbb{Q}_{p})$, where
$A_{P}(\mathbb{Q}_{p})=A_{Q}^{P}(\mathbb{Q}_{p})A_{Q}(\mathbb{Q}_{p})$,
$N_{P}(\mathbb{Q}_{p})=N_{Q}^{P}(\mathbb{Q}_{p})N_{Q}(\mathbb{Q}_{p})$.

###  2.2. Congruence Subgroup

We choose a congruence subgroup $\Gamma\subset G(\mathbb{Z}[S^{-1}])$, which
is torsion free. The number of $\Gamma$\- orbits of proper
$\mathbb{Q}$-parabolic subgroups is finite. Let us denote their representative
as $\\{P_{1},P_{2},...,P_{r}\\}$. We conjugate them by appropriate elements of
$G(\mathbb{Q})$ so that the $P_{i}(\mathbb{Q}_{S})$ contain the minimal
parabolic subgroup $M_{0,S}A_{0,S}N_{0,S}$. We denote them as
$Q_{i,S}=M_{i,S}A_{i,S}N_{i,S}$, and their corresponding conjugating elements
as $\delta_{i}\in G(\mathbb{Q})$ (i.e.
$\delta_{i}P_{i}\delta_{i}^{-1}=Q_{i}$). Let $M_{i,S}=M_{Q_{i},S}$,
$N_{i,S}=N_{Q_{i},S}$ and $A_{i,S}=A_{Q_{i},S}$. Moreover we put
$\Gamma_{i}=\delta_{i}\Gamma\delta_{i}^{-1}$, $\Gamma_{N_{i,S}}=\Gamma_{i}\cap
N_{i,S}$ and $\Gamma_{A_{i,S}}=\Gamma_{i}\cap A_{i,S}$.

Let $X^{*}(M(\mathbb{Q}_{S}))_{\mathbb{Q}_{S}}$ be the set of $\mathbb{Q}_{S}$
characters of $M(\mathbb{Q}_{S})$, the Levi subgroup of $P_{S}$. The dual of
this space, which can be identified with the Lie algebra of the maximal split
part of the center of $M(\mathbb{Q}_{S})$ is

$\mathfrak{a}_{M(\mathbb{Q}_{S})}=\text{Hom}(X^{*}(M(\mathbb{Q}_{S}))_{\mathbb{Q}_{S}},\mathbb{R})$

For $\nu_{S}\in X^{*}(M(\mathbb{Q}_{S}))_{\mathbb{Q}_{S}}$, we have the
following Harish-Chandra homomorphism $H_{M(\mathbb{Q}_{S})}$:

$e^{\langle H_{M(\mathbb{Q}_{S})}(m),\nu_{S}\rangle}=\prod_{p\in
S}\lvert\nu_{p}(m_{p})\rvert_{p}.$

Let $\omega_{S}$ be the irreducible unitary square integrable admissible
representation of $M(\mathbb{Q}_{S})$ which is trivial on $A(\mathbb{Q}_{S})$.
We define the set of equivalence irreducible classes of $\omega_{S}$ as
$\mathcal{E}_{2}(M(\mathbb{Q}_{S}))$. For $\nu_{S}\in
X^{*}(M(\mathbb{Q}_{S}))_{\mathbb{Q}_{S}}\otimes\mathbb{C}=\mathfrak{a}_{M(\mathbb{Q}_{S}),\mathbb{C}}^{*}$,
we can define the following induced representation on $G_{S}$ with parameters
$(\omega_{S},\nu_{S})$:

$\displaystyle\text{Ind}(\omega_{S},\nu_{S})=\prod_{p\in
S}\text{Ind}(\omega_{p},\nu_{p}).$

###  2.3. Test Functions

Let $(\tau,V_{\tau})$ be an irreducible $K_{\infty}$-type, i.e. an irreducible
finite dimensional representation of $K_{\infty}$. Suppose $d_{\tau}$ and
$\chi_{\tau}$ denote the dimension and the character of the above
representation respectively. Let $C_{c}^{\infty}(G(\mathbb{R}),\tau,\tau)$ be
the following space of functions

$\bigg{\\{}\phi_{\infty}:G(\mathbb{R})\rightarrow\text{End}(V_{\tau}),\phi_{\infty}(k_{1}gk_{2})=\tau(k_{2}^{-1})\phi_{\infty}(g)\tau(k_{1}^{-1})\bigg{\\}}.$

A function $\Phi_{\infty}\in C_{c}^{\infty}(G(\mathbb{R}))$ is called
bi-$K_{\infty}$-finite, if the following condition is satisfied:

$\Phi_{\infty}(x)=\int\limits_{K_{\infty}}\int\limits_{K_{\infty}}d_{\tau}\chi_{\tau}(k)\Phi_{\infty}(k^{-1}xk^{\prime})d_{\tau}\chi_{\tau}({k^{\prime}}^{-1})dk^{\prime}dk.$

A function $\Phi_{\infty}$ is called $K_{\infty}$-central if
$\Phi_{\infty}(kxk^{-1})=\Phi_{\infty}(x)$ for all $k\in K_{\infty}$ and for
all $x\in G(\mathbb{R})$. We denote the convolution algebra of
bi-$K_{\infty}$-finite and $K_{\infty}$-central functions as
$C_{c}^{\infty}(G(\mathbb{R}))_{K_{\infty}}^{K_{\infty}}$. This convolution
algebra is isomorphic to the $\text{End}(V_{\tau})$-valued algebra defined
above via the following isomorphism :

$\displaystyle C_{c}^{\infty}(G(\mathbb{R}),\tau,\tau)\cong
C_{c}^{\infty}(G(\mathbb{R}))_{K_{\infty}}^{K_{\infty}}$
$\displaystyle\phi_{\infty}\mapsto\Phi_{\infty}=d_{\tau}\text{Tr}\phi_{\infty}$
$\displaystyle\int\limits_{K_{\infty}}\Phi_{\infty}(gk)\tau(k)dk=\phi_{\infty}(g)\mapsfrom\Phi_{\infty}(g)$

At the non-Archimedean places of $S^{\prime}=S\backslash\infty$ we define the
Hecke algebra as the space of compactly supported, locally constant functions.
We denote this space as $C_{c}^{\infty}(G(\mathbb{Q}_{S^{\prime}}))$. We
define the co-center of this Hecke algebra as the following quotient:

$\bar{\mathcal{H}}(G(\mathbb{Q}_{S^{\prime}})):=\frac{C_{c}^{\infty}(G(\mathbb{Q}_{S^{\prime}}))}{[C_{c}^{\infty}(G(\mathbb{Q}_{S^{\prime}})),C_{c}^{\infty}(G(\mathbb{Q}_{S^{\prime}}))]}.$

Moreover we choose the functions from this space which are
bi-${K^{\prime}}_{S^{\prime}}$ invariant, where ${K^{\prime}}_{S^{\prime}}$ is
an arbitrary compact subgroup of the maximal compact subgroup
$K_{S^{\prime}}$. We denote this subspace as
$\bar{\mathcal{H}}({K^{\prime}}_{S^{\prime}}\backslash
G_{S^{\prime}}/{K^{\prime}}_{S^{\prime}})$. Let
$\Phi_{S^{\prime}}\in\bar{\mathcal{H}}({K^{\prime}}_{S^{\prime}}\backslash
G_{S^{\prime}}/{K^{\prime}}_{S^{\prime}})$. Hence we can combine the
End$(V_{\tau})$-valued function $\phi_{\infty}$ at the Archimedean place with
$\Phi_{S^{\prime}}$ to obtain an endomorphism valued test function on $G_{S}$
and denote the set containing these functions as:

$C_{c}^{\infty}(G(\mathbb{R}),\tau,\tau)\otimes\bar{\mathcal{H}}({K^{\prime}}_{S^{\prime}}\backslash
G_{S^{\prime}}/{K^{\prime}}_{S^{\prime}}).$

We would denote the scalar valued counterpart of the above space as

$C_{c}^{\infty}(G(\mathbb{R}))_{K_{\infty}}^{K_{\infty}}\otimes\bar{\mathcal{H}}({K^{\prime}}_{S^{\prime}}\backslash
G_{S^{\prime}}/{K^{\prime}}_{S^{\prime}}).$

Let $L^{2}(\Gamma\backslash G_{S},V_{\tau})$ be the following set:

$\bigg{\\{}f:\Gamma\backslash G_{S}\mapsto
V_{\tau}:f(gk_{\infty})=\tau(k_{\infty})^{-1}f(g),(f_{1},f_{2})=\int_{\Gamma\backslash
G_{S}}\langle f_{1}(x),f_{2}(x)\rangle_{V_{\tau}}dx\bigg{\\}}.$

Here the inner product makes sense as $\text{Vol}(\Gamma\backslash
G_{S})<\infty$. Also as $\Gamma$ is chosen to be a torsion free congruence
subgroup, $\Gamma\backslash G(\mathbb{R})$ is a manifold. Elements of
$C_{c}^{\infty}(G(\mathbb{R}),\tau,\tau)\otimes\mathcal{C}({K^{\prime}}_{S^{\prime}}\backslash
G_{S^{\prime}}/{K^{\prime}}_{S^{\prime}})$ acts on this space via convolution.
Let $R$ be the right regular representation of $G(\mathbb{R})$ on
$L^{2}(\Gamma\backslash G(\mathbb{R}))$. Let $\Omega_{G(\mathbb{R})}$ be the
Casimir operator in $\mathcal{Z}(\mathfrak{g}_{\mathbb{C}})$, the center of
the universal enveloping algebra. Then $-\Omega_{G(\mathbb{R})}\otimes Id$
induces a self adjoint operator $\Delta_{\tau}$ whose restriction to

$L_{\text{cusp}}^{2}(\Gamma\backslash
G(\mathbb{R}),\tau):=(L_{\text{cusp}}^{2}(\Gamma\backslash
G(\mathbb{R}))\otimes V_{\tau})^{K_{\infty}}$

has pure point spectrum with finite multiplicities. Let us denote them as

$0<\lambda_{1}(\tau)<\lambda_{2}(\tau)...$

with finite multiplicities. Suppose $\mathcal{E}(\lambda_{i}(\tau))$ denotes
the eigenspace corresponding to the eigenvalue $\lambda_{i}(\tau)$. We have
the counting function as

$N_{\text{cusp}}^{\Gamma}(T,\tau)=\sum_{\lambda_{i}(\tau)\leq\sqrt{T}}\text{dim}(\mathcal{E}(\lambda_{i}(\tau))).$

We can redefine the above counting function by representation theoretic means
in the following way: Let $\Pi_{\text{cusp}}(G(\mathbb{Q}_{S}))$ be the set of
unitary irreducible cuspidal subrepresentations of the regular representation
of $G(\mathbb{Q}_{S})$ on $L_{\text{cusp}}^{2}(\Gamma\backslash
G(\mathbb{Q}_{S})/K_{S^{\prime}}^{^{\prime}})$. Let
$\Pi_{\text{cusp}}(G(\mathbb{R}))$ be the set of subrepresentations of the
regular representations of $G(\mathbb{R})$ acting on
$L_{\text{cusp}}^{2}(\Gamma\backslash
G(\mathbb{Q}_{S})/K_{S^{\prime}}^{^{\prime}})$. Any element
$\pi\in\Pi_{\text{cusp}}(G(\mathbb{Q}_{S}))$ can be written as
$\pi=\pi_{\infty}\otimes\pi_{S\backslash\infty}$, where
$\pi_{\infty}\in\Pi_{\text{cusp}}(G(\mathbb{R}))$. Let
$H_{\pi_{\infty}}(\tau)$ be the $\tau$-isotypical subspace of
$(\pi_{\infty},H_{\pi_{\infty}})$. Let
$H_{\pi_{S\backslash\infty}}^{K^{\prime}}$ be the subspace of
$K_{S^{\prime}}^{^{\prime}}$-fixed vectors in
$(\pi_{S\backslash\infty},H_{\pi_{S\backslash\infty}})$. Let
$m(\pi_{\infty})$, resp $m(\pi)$, be the multiplicity with which
$\pi_{\infty}$, resp $\pi$, occurs as a subrepresentation of $G(\mathbb{R})$,
resp $G(\mathbb{Q}_{S})$, on the cuspidal subspace
$L_{\text{cusp}}^{2}(\Gamma\backslash
G(\mathbb{Q}_{S})/K_{S^{\prime}}^{^{\prime}})$. Then we have

$m(\pi_{\infty})=\sum_{\pi^{{}^{\prime}}\in\Pi_{\text{cusp}}(G(\mathbb{Q}_{S}))}m(\pi^{{}^{\prime}})\text{dim}H_{\pi_{S\backslash\infty}}^{K^{\prime}}$

for all $\pi^{{}^{\prime}}$ such that
$\pi_{\infty}^{{}^{\prime}}=\pi_{\infty}$. Suppose $\nu_{\pi}$ denotes the
Casimir eigenvalue of $\pi_{\infty}$. Then we take the subcollection
$\Pi_{\text{cusp}}(G(\mathbb{Q}_{S}))_{T}$ whose elements satisfies
$|\nu_{\pi}|^{2}\leq T$. Similarly we define
$\Pi_{\text{cusp}}(G(\mathbb{R}))_{T}$. Then we have

$\sum_{\pi_{\infty}\in\Pi_{\text{cusp}}(G(\mathbb{R}))_{T}}m(\pi_{\infty})\text{dim}\text{Hom}_{K_{\infty}}(H_{\pi_{\infty}}(\tau),V_{\tau})=N_{\text{cusp}}^{\Gamma}(T,\tau).$

###  2.4. Fourier Transform

We now define the scalar valued Fourier transform of functions on
$C_{c}^{\infty}(G(\mathbb{R}))_{K_{\infty}}^{K_{\infty}}$. Let
$P_{\infty}=M_{\infty}^{1}A_{\infty}N_{\infty}$ be the Langlands decomposition
of a standard cuspidal parabolic subgroup of $G(\mathbb{R})$. Choose
$\omega_{\infty}\in\mathcal{E}_{2}(M_{\infty}^{1})$. Suppose
$\theta_{\omega_{\infty}}$ denotes its character. Let $d_{\omega_{\infty}}$ be
the formal degree of $\omega_{\infty}$. Let $\tau$ be the double
representation of $K_{\infty}$ on $L^{2}(K_{\infty}\times K_{\infty})$
obtained from $(\tau,V_{\tau})$. Let $\tau_{M_{\infty}}$ be the restriction of
$\tau$ to $K_{\infty}\bigcap M_{\infty}$. We let
$L_{\omega}^{2}(M_{\infty},\tau_{M_{\infty}})$ be the set of
$\tau_{M_{\infty}}$-spherical functions on $L^{2}(M_{\infty})\otimes
L^{2}(K_{\infty}\times K_{\infty})$. The norm in this space is defined as:

$\lVert\psi\rVert^{2}=\int_{M_{\infty}}\int_{K_{\infty}\times
K_{\infty}}\|\psi(k_{1}:m:k_{2})\|^{2}dk_{1}dk_{2}dm.$

It can be made into a Hilbert algebra with the multiplication via

$(\psi_{1}\psi_{2})(k_{1}:m:k_{2})=\int_{M_{\infty}}\int_{K_{\infty}}\psi_{1}(k_{1}:\tilde{m}:k^{-1})\psi_{2}(k:\tilde{m}^{-1}m:k_{2})dkd\tilde{m}.$

The Fourier transform of functions $\Phi_{\infty}\in
C_{c}^{\infty}(G(\mathbb{R}))$ is defined as in [Ar2]:

$\Phi_{\infty}\mapsto\widehat{\Phi_{\infty}}(\omega_{\infty},\nu_{\infty})\in
L_{\omega}^{2}(M_{\infty},\tau_{M_{\infty}}),$

where the formula of $\widehat{\Phi_{\infty}}(\omega_{\infty},\nu_{\infty})$
is

$\widehat{\Phi_{\infty}}(\omega_{\infty},\nu_{\infty})(k_{1}:m:k_{2})=d_{\omega_{\infty}}\int\limits_{M_{\infty}}\int\limits_{A_{\infty}}\int\limits_{N_{\infty}}\Phi_{\infty}(k_{1}nam\widetilde{m}k_{2})\theta_{\omega_{\infty}}(\widetilde{m}^{-1})e^{(-\nu_{\infty}+\rho_{\infty})\ln(a)}dndad\widetilde{m}.$

Next We define the operator valued Fourier transform of $\Phi_{\infty}\in
C_{c}^{\infty}(G(\mathbb{R}))_{K_{\infty}}^{K_{\infty}}$. Let
$\pi_{\infty}\in\widehat{G(\mathbb{R})}(\tau)$, the unitary irreducible
representation of $G(\mathbb{R})$ that contains $\tau$ upon restriction to
$K_{\infty}$. Then $\Phi_{\infty}\mapsto\pi_{\infty}(\Phi_{\infty})$ defines
the operator valued Fourier transform on the space of endomorphisms of finite
dimensional vector space. From Harish-Chandra’s sub-representation theorem we
know that $\pi_{\infty}$ is isomorphic to an irreducible subrepresentation of
a induced representation from an cuspidal parabolic $P_{\infty}$ with
parameters $(\omega_{\infty},\nu_{\infty})$. Let us denote the induced
representation as $\text{Ind}(\omega_{\infty},\nu_{\infty})$. Then we have the
following relation [Ar2]:

$d_{\omega_{\infty}}\text{Tr}(\text{Ind}(\omega_{\infty},\nu_{\infty})\Phi_{\infty})=\int\limits_{K_{\infty}}\widehat{\Phi_{\infty}}(\omega_{\infty},\nu_{\infty})(k^{-1}:1:k)dk.$

Let $m_{\pi_{\infty}}(\tau)$ be the multiplicity with which $\tau$ appears in
the decomposition of $\pi_{\infty}$ restricted to $K_{\infty}$. Suppose
$\pi_{\infty,\tau}(\Phi_{\infty})$ is the restriction of $\pi_{\infty}$ on
$\mathcal{H}_{\pi_{\infty}}(\tau)$. Then we can define the spherical Fourier
transform [Cmp1]
$\mathcal{F}(\Phi_{\infty})(\pi_{\infty})\in\text{End}(\mathbb{C}^{m_{\pi_{\infty}}(\tau)})$
as follows:

$\pi_{\infty,\tau}(\Phi_{\infty})=\mathbbm{1}_{\tau}\otimes\mathcal{F}(\Phi_{\infty})(\pi_{\infty}).$

Let $\widetilde{\Phi_{\infty}}(x)=\overline{\Phi_{\infty}(x^{-1})}$. Then
$\pi_{\infty}(\widetilde{\Phi_{\infty}})=\pi_{\infty}(\Phi_{\infty})^{*}$, the
conjugate transpose of $\pi_{\infty}(\Phi_{\infty})$. Moreover
$\text{Tr}\pi_{\infty}(\Phi_{\infty}\star\widetilde{\Phi_{\infty}})=\left|\left|\pi_{\infty}(\Phi_{\infty})\right|\right|_{\text{HS}}^{2}$.
Let $\mu_{\infty}(\omega_{\infty},\nu_{\infty})$ be the Harish-chandra $\mu$
function corresponding to induced parameters $(\omega_{\infty},\nu_{\infty})$.
Let $\mathcal{P}$ be the set of associated classes of parabolic subgroups. The
Plancherel inversion of $\Phi_{\infty}$ has the following formula:

$\Phi_{\infty}\star\widetilde{\Phi_{\infty}}(e)=\sum\limits_{\mathcal{P}}n(\mathcal{P})^{-1}\sum\limits_{P\in\mathcal{P}}\sum\limits_{\mathcal{E}_{2}(M_{\infty})}d_{\omega}(\frac{1}{2\pi
i})^{q}\int\limits_{i\mathfrak{a}_{\infty}^{*}}\left|\left|\text{Ind}(\omega_{\infty},\nu)(\Phi_{\infty})\right|\right|_{\text{HS}}^{2}\mu_{\infty}(\omega_{\infty},\nu)d\nu.$

Similarly, at the non-Archimedean place $p$ if we assume the induced
parameters are $(\omega_{p}\otimes\nu_{p})$, then the the Plancherel Measure
$\mu_{p}(\omega_{p})$ is defined over a connected compact manifold
$\mathcal{O}_{2}(M(\mathbb{Q}_{p}))$ for each
$\omega_{p}\in\mathcal{E}_{2}(M(\mathbb{Q}_{p}))$. We denote by
$d_{\omega_{p}}$ the Euclidean measure of the connected compact manifold. Then
we have the following Placnherel inversion formula for
$\Phi_{p}\in\mathcal{C}(K^{\prime}_{p}\backslash
G(\mathbb{Q}_{p})/K^{\prime}_{p})$:

$\Phi_{p}\star\widetilde{\Phi_{p}}(e)=\sum\limits_{\mathcal{P}}n(\mathcal{P})^{-1}\sum\limits_{P\in\mathcal{P}}\sum\limits_{\mathcal{E}_{2}(M_{p})}d_{\omega_{p}}\int\limits_{\mathcal{O}_{2}(M(\mathbb{Q}_{p}))}\left|\left|\text{Ind}(\omega_{p})(\Phi_{p})\right|\right|_{\text{HS}}^{2}\mu_{p}(\omega_{p})d\omega_{p}.$

##  3\. Plancherel Measure and estimates

In this section we are going to review the explicit formula of Plancherel
measure in the case of Reductive Lie group (Real and $p-$adic) and their
various estimates which is one of the essential part to prove the main term of
the Weyl’s law. The references for this section are [HC1] and [HC2].

###  3.1. Real case

We now review some necessary formulas in the Real case.

####  3.1.1. product formula

For this section let us fix a parabolic subgroup $(P,A)$ of $G(\mathbb{R})$,
with the corresponding Langlands Decomposition $P=M^{1}AN$. Let
$\omega_{\infty}\in\mathcal{E}_{2}(M^{1})$, be a square integrable unitary
irreducible class of representation of $M^{1}$. Let
$\mu(\omega_{\infty},\nu_{\infty})$ for
$\omega_{\infty}\in\mathcal{E}_{2}(M^{1})$ and
$\nu_{\infty}\in\mathfrak{a}_{\mathbb{C}}^{*}$ be the Harish-Chandra $\mu$
function of the pair $(G(\mathbb{R}),P)$. Denote by $\Sigma$, the set of all
roots of $(P,A)$. A root $\alpha\in\Sigma$ is called reduced if
$k\alpha\notin\Sigma$ for $0\leq k<1/2$. Let $\Phi$ be a set of all reduced
roots. For any $\alpha\in\Phi$ put

$\displaystyle\mathfrak{n}_{\alpha}=\bigoplus_{k\alpha:{k\geq
1}}\mathfrak{n}(\alpha),$ (2)

where $\mathfrak{n}(\alpha)=\\{x\in\mathfrak{g}:[H,X]=\alpha(H)X,\forall
H\in\mathfrak{a}\\}$. Let $N_{\alpha}$ be the analytic subgroup corresponding
to $\mathfrak{n}_{\alpha}$.
Let $\sigma_{\alpha}$ be the hyper-plane given by $\alpha=0$ in
$\mathfrak{a}$. Let $Z_{\alpha}$ be the centralizer of $\sigma_{\alpha}$. We
put $M_{\alpha}=\prescript{0}{}{Z_{\alpha}}$, $A_{\alpha}=M_{\alpha}\cap A$
and $\theta(N_{\alpha})=\bar{N_{\alpha}}$. Then we can define the following
parabolic subgroups with their corresponding Langlands decompositions

$\prescript{*}{}{P_{\alpha}}=M^{1}A_{\alpha}N_{\alpha},\quad\bar{\prescript{*}{}{P_{\alpha}}}=M^{1}A_{\alpha}\bar{N_{\alpha}},\quad
P=M^{1}AN.$

We can now define the product formula for the Harish-chandra $\mu$ function.
Suppose $\mu(\omega_{\infty},\nu_{\infty}^{\alpha})$ denotes the corresponding
$\mu(\omega_{\infty},\nu_{\infty})$ for the group
$(M_{\alpha},\prescript{*}{}{P_{\alpha}})$. Here
$\nu_{\infty}\in\mathfrak{a}_{\mathbb{C}}^{*}$ and $\nu_{\infty}^{\alpha}$
denotes the restriction of $\nu_{\infty}$ to $A_{\alpha}$, and
$\omega\in\mathcal{E}_{2}(M^{1})$. Then for a suitable constant $C_{G}$
depending on $G$ we have the following product formula [HC2, Theorem 12, p.
145]

$\displaystyle\mu(\omega_{\infty},\nu_{\infty})=C_{G}\prod_{\alpha\in\Phi}\mu(\omega_{\infty},\nu_{\infty}^{\alpha}).$
(3)

Here we have that prk $M_{\alpha}=0$, prk$\prescript{*}{}{P_{\alpha}}=1$.

####  3.1.2. Explicit formula

When prk $G=0$, and prk $P=1$ we have the following two possibilities

  * •

$\mathcal{E}_{2}(G)\neq\emptyset$

  * •

$\mathcal{E}_{2}(G)=\emptyset$

Here the first condition is equivalent to Rank $G=$Rank $K$. We write down the
formula for the $\mu$ function in each case.

  * •

[HC1, Theorem 1, section 24] We consider the second condition first. Let us
introduce some notations. Let $Q$ be the set of positive roots of
$(\mathfrak{g},\mathfrak{h})$, where $\mathfrak{h}$ is a $\theta$-stable
Cartan subalgebra of $\mathfrak{g}$. Now $Q$ is the union of three disjoint
parts $Q_{I},Q_{R},Q_{C}$, set of imaginary, real and complex roots
respectively. Let $H_{\alpha}$ be the unique element in $\mathfrak{h}$ such
that $(H_{\alpha},H)=\alpha(H)$, for all $H\in\mathfrak{h}$, where $(,)$
denotes the Killing form. With respect to Cartan involution we have the
decomposition $\mathfrak{g}=\mathfrak{k}\bigoplus\mathfrak{p}$. Moreover we
have $\mathfrak{h}=\mathfrak{h}_{I}\oplus\mathfrak{h}_{R}$, and a fixed
parabolic subgroup with Langlands decomposition
$P=M^{1}e^{\mathfrak{h}_{R}}N.$ We put

$\widetilde{w_{I}}=\prod_{\alpha\in
Q_{I}}H_{\alpha},\quad\widetilde{w_{R}}=\prod_{\alpha\in
Q_{R}}H_{\alpha},\quad\widetilde{w_{+}}=\prod_{\alpha\in Q_{C}}H_{\alpha}.$

Suppose $\mathfrak{h}_{I}$ is a cartan subalgebra of $\mathfrak{k}$. Then the
second condition is satisfied.
From the theorem of Harish-Chandra [HC1, section 23,Theorem 1] we know that
$\omega_{\infty}\in\mathcal{E}_{2}(M^{1})$ corresponds to an element in orbit
of $H^{*^{\prime}}_{I}$ under the action of $W(M^{1}/H_{I})$, where
$H^{*^{\prime}}_{I}$ is a subspace of $H^{*}_{I}$, which is the Cartan
subgroup of $M^{1}$ in $K\cap M^{1}$(we call this element $\lambda$ which lies
in the lie algebra of $H_{I}^{*^{\prime}}$). Let $a^{*}$ be the element in
$H^{*^{\prime}}_{I}$ that corresponds to $\omega_{\infty}$. Put
$\lambda=\lambda(a^{*})\in i\mathfrak{h}_{I}^{*^{\prime}}$. Then we have the
following formula of the $\mu$ function:

$\displaystyle\mu(\omega_{\infty},\nu_{\infty})=C\widetilde{w_{+}}(\lambda+\nu_{\infty})=C\prod_{\alpha\in
Q_{C}}\left|(\lambda+\nu_{\infty},\alpha)\right|.$ (4)

Here the constant $C$ depends on $G(\mathbb{R})$ and $M^{1}$. In this case one
can see that if $Q_{R}$ is empty [Wal, 2.3.5, p. 58], then dim $N=|Q_{C}|$. So
$\mu$ is a polynomial in $\nu_{\infty}$ of degree dim $N$. Hence as
$t\to\infty$ we have

$\mu(\omega_{\infty},t\nu_{\infty})\sim C_{\nu_{\infty}}t^{\text{dim}(N)},$

for a non zero positive constant $C_{\nu_{\infty}}$.

Therefore when $G=M_{\alpha}$, $N=N_{\alpha}$,
$e^{\mathfrak{h}_{\mathbf{R}}}=A_{\alpha}$ we have,

$\mu(\omega_{\infty},t\nu_{\infty}^{\alpha})\sim
C_{\nu^{\alpha}_{\infty}}t^{\text{dim}(N_{\alpha})}\quad\text{as}\quad
t\to\infty.$

  * •

[HC1, Sec. 36, p. 190] Next we consider the case of rank $G$= rank $K$. Let
$\mathfrak{h}$ be a $\theta$-stable Cartan subalgebra of $\mathfrak{g}$ with
$\mathfrak{h}=\mathfrak{h}_{I}\oplus\mathfrak{h}_{R}$, dim
$(e^{\mathfrak{h}_{\mathbf{R}}})=1$, and a fixed parabolic subgroup with
Langlands decomposition $P=M^{1}e^{\mathfrak{h}_{R}}N$. So dim $N=1+|Q_{C}|.$

Let $H_{I}$ be the analytic subgroup corresponding to $\mathfrak{h}_{I}$. Let
$Q$ be the set of positive roots of $(\mathfrak{g},\mathfrak{h})$ which is the
disjoint union of imaginary$(Q_{I})$, real$(Q_{R})$ and complex
roots$(Q_{C})$. Let us denote by $\alpha$ the unique root in $Q_{R}$. For
$a^{*}\in H^{*}_{I}$, let us define:

$\mu_{0}(a^{*},\nu_{\infty}):=d(a^{*})^{-1}\text{Tr}\left(\frac{\pi
i\nu_{\infty}^{\alpha}\sinh{\pi i\nu_{\infty}^{\alpha}}}{\cosh{\pi
i\nu_{\infty}^{\alpha}}-\frac{(-1)^{\rho_{\alpha}}}{2}(\sigma_{a^{*}}(\gamma)+\sigma_{a^{*}}(\gamma^{-1}))}\right).$

Here $d(a^{*})$ is the degree,
$\nu_{\infty}^{\alpha}=2\frac{(\nu_{\infty},\alpha)}{(\alpha,\alpha)}$,
$\rho_{\alpha}=2\frac{(\rho,\alpha)}{(\alpha,\alpha)}\in\mathbb{Z}$, where
$\rho$ is the half of the sum of the positive roots of
$(\mathfrak{g},\mathfrak{h})$), $\sigma_{\alpha}$ is the irreducible
representation of $H_{I}$ whose character is $a^{*}$ and $\gamma$ is a fixed
element in $H_{I}$. So the above expression has the form:

$u(z)=\frac{z\sinh{\pi z}}{\cosh{\pi z}+k},$

for a fixed real number $k\geq-1$.

As $\omega_{\infty}\in\mathcal{E}_{2}(M^{1})$ corresponds to an element
$a^{*}\in H^{*^{\prime}}_{I}$, which we denote by
$\lambda=\lambda(a^{*})\in\mathfrak{h}_{I}^{*^{\prime}}$ as the corresponding
element in the Lie algebra of $H_{I}$.
As before we define:

$\widetilde{{w}_{+}}(\omega_{\infty}:\nu_{\infty})=\prod_{\alpha\in
Q_{C}}\left|(\lambda+\nu_{\infty},\alpha)\right|.$

Moreover, we define

$\mu_{0}(\omega_{\infty},\nu_{\infty})=\frac{1}{|W(M^{1}/H_{I})|}\sum_{W(M^{1}/H_{I})}\mu_{0}(sa^{*},\nu_{\infty}).$

With these notations in mind we can write the Harish-Chandra $\mu$-function as
follows

$\displaystyle\mu(\omega_{\infty},\nu_{\infty})=C\lvert\alpha\rvert\mu_{0}(\omega_{\infty},\nu_{\infty})\widetilde{{w}_{+}}(\omega_{\infty}:\nu_{\infty}).$
(5)

We can show that $u(tz)\sim C_{z}t$, as $t\to\infty$, when $z\in\mathbb{R}$
and $C_{z}>0$. On the other hand
$\widetilde{{w}_{+}}(\omega_{\infty}:\nu_{\infty})$ is a polynomial in
$\nu_{\infty}$ with degree $\text{dim}(N)-1$. Therefore we obtain the same
asymptotic expression

$\mu(\omega_{\infty},t\nu_{\infty})\sim
C_{\nu}t^{\text{dim}N}\quad\text{as}\quad t\to\infty.$

With the previous notation(i.e. when $G=M_{\alpha}$, $N=N_{\alpha}$,
$e^{\mathfrak{h}_{\mathbf{R}}}=A_{\alpha}$) we have,
$\mu(\omega_{\infty},t\nu_{\infty}^{\alpha})\sim
C_{\nu^{\alpha}}t^{\text{dim}(N_{\alpha})}$ as $t\to\infty$.

####  3.1.3. Asymptotic estimate

Because we have the polar decomposition $G=KP$, where $P=MAN$ is the Langlands
decomposition, we have

$\text{dim}(G/K)=\text{dim}(\frac{M^{1}}{K\cap
M^{1}})+\text{dim}(A)+\text{dim}(N).$

Therefore in the case where $P=P_{0}$ is the minimal parabolic we have

$\text{dim}(G/K)=\text{dim}(A_{0})+\text{dim}(N_{0})=\text{dim}(A_{0})+\sum_{\alpha\in\Phi}\text{dim}(N_{0,\alpha}),$

and we have the following estimate of the density of the Plancherel measure:

$\displaystyle\int_{\nu_{\infty}\in
i\mathfrak{a}_{0,\mathbb{R}}^{*}:(\nu_{\infty},\nu_{\infty})\leq
t^{2}}\mu(\omega_{\infty},\nu_{\infty})d\nu\sim
C_{G}C_{\nu_{\infty}}t^{\text{dim}(G/K)},\quad\text{as}\quad t\to\infty.$ (6)

Now for the pair $(M_{\alpha},A_{\alpha})$ we have polynomial bound. Here we
invoke [HC1, Thm 1, Sec. 25]. The theorem states that $\mu$ can be extended to
the whole complex plane meromorphically. Moreover there exists $C,r\geq 0$
such that:

$\lvert\mu(\omega_{\infty},\nu_{\infty})\rvert\leq
C(1+\lVert\text{Im}(\nu_{\infty})\rVert)^{r}.$

Our job is to find an explicit value of $r$ in the inequality mentioned in
Harish-Chandra’s paper.

The case when $\mathcal{E}_{2}(M_{\alpha})=\emptyset$, we have that
$\mu(\omega_{\infty},\nu_{\infty}^{\alpha})$ is a polynomial in $\nu_{\infty}$
of degree $\text{dim}(N_{\alpha})$. So in this scenario we can take
$r=\text{dim}(N_{\alpha})$. And similarly for the other case we can arrive at
the same estimate, as $\sup_{z\in\mathbb{C}}\lvert u(z)\rvert\leq(1+\lvert
z\rvert)$, whenever $z$ is real. And the other part, namely
$\widetilde{{w}_{+}}(\omega_{\infty}:\nu_{\infty})$, is a polynomial in
$\nu_{\infty}$ of degree $\text{dim}(N_{\alpha})-1$.
Hence combining with the product formula mentioned above we conclude that for
some $C^{\prime}>0$

$\displaystyle\mu(\omega_{\infty},\nu_{\infty})\leq
C^{\prime}(1+|\lvert\nu_{\infty}\rvert|)^{\text{dim}(N)}.$ (7)

Remark: Important point to note here the constant $C^{\prime}$ does not really
matter in terms of finding out the main term of the Weyl’s law. Only constant
that could matter is $C_{G}$. [LV, Sec. 6.3]).

###  3.2. The $p-$adic case

The comment that we are going to make is due to [HC3, P. 355]. The Plancherel
measure in this case is defined on $\mathcal{E}_{2}(M_{p})$, as evident from
the formulas written above. For this case this set is compact,(can be denoted
as $\sqcup_{\omega_{p}\in\mathcal{E}_{2}(M_{p})}\mathcal{O}_{\omega_{p}}$)
hence the asymptotic estimate will not change if we are to consider the
plancherel inversion formula for the group $G_{S}$.
Hence combining the above two subsection we arrive at the estimate that

$\displaystyle\int_{\nu\in
i\mathfrak{a}_{0,\infty}^{*}\times\mathcal{E}_{2}(M_{p}):(\nu_{\infty},\nu_{\infty})\leq
t^{2}}\mu(\omega,\nu)d\nu
d\omega_{p}\sim\alpha(G)t^{\text{dim}(G_{\infty}/K_{\infty})}.$ (8)

##  4\. Condition for purely cuspidal image

In this section we provide the necessary condition on the space of scalar
valued test functions so that the image of convolution operator on scalar
valued $K_{\infty}$-finite automorphic forms only consists of cuspidal
$K_{\infty}$-finite automorphic forms . We closely follow [LV, Prop. 3, second
proof].
First we need some preparation. We recall a couple of lemmas due to Harish-
Chandra regarding vanishing condition of Schwartz functions.
Let us recall some of the notations mentioned already in the preliminaries.
Let $Q=M_{Q}N_{Q}$ be a standard parabolic subgroup of $G(\mathbb{R})$ and
$G(\mathbb{Q}_{p})$. Let $\mathcal{C}(G(\mathbb{R}),\tau)$ be the Harish-
Chandra Schwartz space of vector valued function which are $\tau$-spherical.
These are functions from $G(\mathbb{R})$ to $V_{\tau}\subset
L^{2}(K_{\infty}\times K_{\infty})$, where $V_{\tau}$ is viewed as a double
representation of $\tau$. The action of $\tau$ could be defined as

$\tau(k)\phi(k_{1}:g:k_{2})\tau(k^{\prime})=\phi(k_{1}k:g:k^{\prime}k_{2}).$

Let $\mathcal{C}_{\text{cusp}}(M_{Q},\tau_{M_{Q}})$ be the space of functions
which are cuspidal, $\tau_{M_{Q}\cap K_{\infty}}$-spherical,
$\mathfrak{Z}(M_{Q})$-finite and $A_{Q}$-invariant. The $L^{2}$-completion of
this space is generated by the square integrable matrix coefficients of
finitely many classes of isomorphic unitary irreducible discrete series
representations of $M_{Q}$ which are $A_{Q}$-invariant. For more information
about this space see [Ar3, Chp. I,Sec. 2]. Define:

$\phi_{\infty}^{Q}(la)=\int_{N_{Q}}\phi_{\infty}(nla)dn,\quad\phi_{\infty}\in\mathcal{C}(G(\mathbb{R}),\tau),$

for all $la\in M_{Q}^{1}A_{Q}$. Moreover we write $\phi_{\infty}^{Q}\sim 0$ if
the following holds:

$\int_{M_{Q}/A_{Q}}\Big{(}f(l),\phi_{\infty}^{Q}(la)\Big{)}dl=0,\quad\forall
f\in\mathcal{C}_{\text{cusp}}(M_{Q},\tau_{M_{Q}^{1}})$

for all $a\in A_{Q}$, where $(,)$ denotes the inner product in the vector
space $V_{\tau}$ (which is viewed as a double representation space of the
action of $\tau$ on $L^{2}(K_{\infty}\times K_{\infty})$). Then by [HC3, vol
IV, p. 149] we have the following.

######  Lemma 2 (Archimedean case).

Let $\phi_{\infty}$ be an element in $\mathcal{C}(G(\mathbb{R}),\tau)$ such
that $\phi_{\infty}^{Q}\sim 0$ for all parabolic subgroups
$Q(\mathbb{R})\subset G(\mathbb{R})$. Then $\phi_{\infty}=0$.

We now modify the above conditions for scalar valued bi-$K_{\infty}$-finite
functions, by using the ideas mentioned in [HC3, vol IV, p. 175]. We denote
the corresponding scalar valued function as $\Phi_{\infty}$. Hence again we
write:
we will write $\Phi_{\infty}^{Q}\sim 0$ if

$\int_{M_{Q}^{1}}f(l)\Phi_{\infty}^{Q}(la)dl=0,\quad\forall a\in A_{Q}.$

The above integral is a function defined on $A_{Q}$, the split part of the
center of $M_{Q}$. Hence if $\Phi_{\infty}$ is a compactly supported smooth
function on $G(\mathbb{R})$, then the integral above is also a compactly
supported function defined on $A_{Q}$. We recall some characterization of
parabolic subgroups of standard Levi subgroups due to Harish-chandra

######  Lemma 3.

Let $p\in S$. There is an one to one correspondence between parabolic subgroup
$P(\mathbb{Q}_{p})$ of $G(\mathbb{Q}_{p})$ which are contained in
$Q(\mathbb{Q}_{p})$, and parabolic subgroups ${}^{*}P(\mathbb{Q}_{p})$ of
$M_{Q}(\mathbb{Q}_{p})$. The correspondence are as follows: If
$Q(\mathbb{Q}_{p})=M_{Q}(\mathbb{Q}_{p})N_{Q}(\mathbb{Q}_{p})$ and
$P(\mathbb{Q}_{p})=M_{P}(\mathbb{Q}_{p})N_{P}(\mathbb{Q}_{p})$ are the
corresponding Levi decompositions, then ${}^{*}P_{Q}=P(\mathbb{Q}_{p})\cap
M_{Q}(\mathbb{Q}_{p})=M_{P}(\mathbb{Q}_{p})N_{Q}^{P}(\mathbb{Q}_{p})$ is the
corresponding Levi decomposition, where
$A_{P}(\mathbb{Q}_{p})=A_{Q}^{P}(\mathbb{Q}_{p})A_{Q}(\mathbb{Q}_{p})$,
$N_{P}(\mathbb{Q}_{p})=N_{Q}^{P}(\mathbb{Q}_{p})N_{Q}(\mathbb{Q}_{p})$.

######  Lemma 4.

Let $\Phi\in
C_{c}^{\infty}(G(\mathbb{R}))_{K_{\infty}}^{K_{\infty}}\otimes\bar{\mathcal{H}}(G(K_{S^{{}^{\prime}}}^{{}^{\prime}}\backslash\mathbb{Q}_{S^{\prime}}/K_{S^{{}^{\prime}}}^{{}^{\prime}}))$.
Assume that:

$\displaystyle\text{Ind}(\omega_{S},\nu_{S})(\Phi)=0,$ (9)

for all $i$, for all parabolic $P_{S}\subset Q_{i,S}$ whose Archimedean part
is the chosen minimal parabolic subgroup, for all $\omega_{S}$, equivalence
classes of unitary irreducible representation of $M(\mathbb{Q}_{S})$ whose
Archimedean component is discrete series and non-Archimedean components are
cuspidal representations, such that
$\omega_{\infty}\subset\tau|_{K_{\infty}\cap M_{\infty}}$, for all
$\nu_{S}\in\mathfrak{a}_{P_{0,S},\mathbb{C}}^{*}$ satisfying
$\nu_{S}\lvert_{\Gamma_{A_{i,S}}}=1$, then $\Phi$ satisfies the following
equation for all $i$, for all $k_{1},k_{2}\in K_{S}$ and for all $m\in
M_{i,S}$:

$\displaystyle\sum_{\Gamma_{A_{i,S}}}\int_{N_{i,S}}\Phi(k_{1}n\gamma
mk_{2})dn=0.$ (10)

###### Proof.

The equation (9) implies

$\int\limits_{M(\mathbb{Q}_{S})}\int\limits_{N(\mathbb{Q}_{S})}\int\limits_{{K_{S}}}\Phi(k_{1}mnk_{2})\omega_{S}(m^{-1})e^{(-\nu_{S}+\rho_{S})H_{P_{S}}(m^{-1})}dk_{2}dndm=0,$

for all $k_{1}\in K_{S}$, for all $P(\mathbb{Q}_{S})=P_{S}\subset
Q_{i,S}=Q_{i}(\mathbb{Q}_{S})$, and for all
$\nu_{S}\in\mathfrak{a}_{P_{0,S},\mathbb{C}}^{*}$ such that
$\nu_{S}|_{\Gamma_{A_{i,S}}}=1$. (we will suppress the iteration i in our
discussion that follows) Hence we can drop the integration on $K_{S}$ to get:

$\int\limits_{M(\mathbb{Q}_{S})}\int\limits_{N(\mathbb{Q}_{S})}\Phi(k_{1}mnk_{2})\omega_{S}(m^{-1})e^{(-\nu_{S}+\rho_{S})H_{P_{S}}(m^{-1})}dndm=0.$

Breaking the group $N(\mathbb{Q}_{S})$ as product of
$N_{Q}^{P}(\mathbb{Q}_{S})$ and $N_{Q}(\mathbb{Q}_{S})$ we have the following:

$\int\limits_{M(\mathbb{Q}_{S})}\int\limits_{N_{Q}^{P}(\mathbb{Q}_{S})}\int\limits_{N_{Q}(\mathbb{Q}_{S})}\Phi(k_{1}mn^{\prime}n^{\prime\prime}k_{2})\omega_{S}(m^{-1})e^{(-\nu_{S}+\rho_{S})H_{P_{S}}(m)}dn^{\prime\prime}dn^{\prime}dm=0.$
(11)

As
${}^{*}P_{Q}(\mathbb{Q}_{S})=M(\mathbb{Q}_{S})N_{Q}^{P}(\mathbb{Q}_{S})\subset
M_{Q,S}$, is a parabolic subgroup of $M_{Q,S}$, we have $\Gamma_{A,S}\subset
M(\mathbb{Q}_{S})$ and $\Gamma_{A,S}$ centralizes $M(\mathbb{Q}_{S})$. Hence
(11) implies

$\displaystyle\int\limits_{\frac{M(\mathbb{Q}_{S})}{\Gamma_{A,S}}}\int\limits_{N_{Q}^{P}(\mathbb{Q}_{S})}\int\limits_{N_{Q}(\mathbb{Q}_{S})}\sum\limits_{\Gamma_{A,S}}\Phi(k_{1}m_{1}\gamma
n^{\prime}n^{\prime\prime}k_{2})\omega_{S}(m_{1}^{-1})$ (12) $\displaystyle
e^{(-\nu_{S}+\rho_{S})H_{P_{S}}(m_{1})}dn^{\prime\prime}dn^{\prime}dm_{1}=0.$

We will apply Fubini’s theorem to change the order of the integral on
$N_{Q}(\mathbb{Q}_{S})$ and sum on $\Gamma_{A,S}$ (as $\Phi$ is compactly
supported the integral above is convergent). Moreover we can break down the
set $\frac{M(\mathbb{Q}_{S})}{\Gamma_{A,S}}$ into product of
$\frac{M(\mathbb{Q}_{S})}{A(\mathbb{Q}_{S})}$ and
$\frac{A(\mathbb{Q}_{S})}{\Gamma_{A,S}}$. Hence we can think of the above
integral as the Fourier transform of the following integral:

$\int\limits_{\frac{M(\mathbb{Q}_{S})}{A(\mathbb{Q}_{S})}}\int\limits_{N_{Q}^{P}(\mathbb{Q}_{S})}\int\limits_{N_{Q}(\mathbb{Q}_{S})}\sum\limits_{\Gamma_{A,S}}\Phi(k_{1}m_{1}\gamma
n^{\prime}n^{\prime\prime}k_{2})f_{S}(m_{1})dn^{\prime\prime}dn^{\prime}dm_{1},$

where $f_{S}$ is the product of coefficient of discrete series (at the
Archimedean place) and cuspidal representations (at the non-Archimedean
places) of $M(\mathbb{Q}_{S})$. Therefore, using the injectivity of Fourier
transform on functions defined on $M_{Q,S}$ which are compactly supported
modulo the central direction of $M_{Q,S}$ (which holds for non-Archimedean
case by combining [B, Th. 25], [BDK] and [CH, Sec. 5.7]), eq. 12 implies

$\displaystyle\sum_{\Gamma_{A_{i,S}}}\int_{N_{i,S}}\Phi(k_{1}n\gamma
mk_{2})dn=0.$ (13)

∎

######  Lemma 5.

If $\Phi\in
C_{c}^{\infty}(G(\mathbb{R}))_{K_{\infty}}^{K_{\infty}}\otimes\bar{\mathcal{H}}(G(K_{S^{{}^{\prime}}}^{{}^{\prime}}\backslash\mathbb{Q}_{S^{\prime}}/K_{S^{{}^{\prime}}}^{{}^{\prime}}))$
satisfies the (10) then $\Phi$ maps the elements in
$L_{\text{loc}}^{1}(\Gamma\backslash G_{S})$ to
$L_{\text{cusp}}^{2}(\Gamma\backslash G_{S})$ as a convolution operator.

###### Proof.

We fix $1\leq i\leq r$ and let $\Psi\in L^{2}(\Gamma\backslash
G_{S})_{K_{\infty}}$ and $\Psi_{i}(g)=\Psi(\delta_{i}g)$. Then it is easy to
see that $\Psi_{i}\in L^{2}(\Gamma_{i}\backslash G_{S})_{K_{\infty}}$. To get
the purely cuspidal image we need the following:
For $\Phi=\Phi_{\infty}\Phi_{S\backslash\infty}$, where $\Phi_{\infty}\in
C_{c}^{\infty}(G(\mathbb{R}))_{K_{\infty}}^{K_{\infty}}$,
$\Phi_{S\backslash\infty}\in\bar{\mathcal{H}}(G(K_{S^{{}^{\prime}}}^{{}^{\prime}}\backslash\mathbb{Q}_{S^{\prime}}/K_{S^{{}^{\prime}}}^{{}^{\prime}}))$
and $\Psi_{i}\in L^{2}(\Gamma_{i}\backslash G_{S})_{K_{\infty}}$,

$\int_{\Gamma_{N_{i,S}}\backslash N_{i,S}}\Psi_{i}\star\Phi(nx)dn=0.$

This is equivalent to:

$\int_{\Gamma_{N_{i,S}}\backslash
N_{i,S}}\int_{G_{S}}\Phi(y^{-1}nx)\Psi_{i}(y)dydn=0.$

Now we break down the integral on $G_{S}$ as an integral on
$\Gamma_{i}\backslash G_{S}$ and a discrete sum on $\Gamma_{i}$ to get the
equivalent relation:

$\int\limits_{\Gamma_{N_{i,S}}\backslash
N_{i,S}}\sum_{\Gamma_{i}}\int_{\Gamma_{i}\backslash
G_{S}}\Phi(y^{-1}\gamma^{-1}nx)\Psi(y)dydn=0,$

for all $x,y$. This is equivalent to:

$\int\limits_{\Gamma_{N_{i,S}}\backslash
N_{i,S}}\sum_{\Gamma_{i}}\Phi(y^{-1}\gamma^{-1}nx)dn=0.$

After swapping the integral and the sum (as the sum over $\Gamma_{i}$ is
locally finite, and $\Phi$ is compactly supported, we can use the Monotone
Convergence Theorem or Dominated Convergence Theorem) and rearranging the
domains, we get a sum over $(\Gamma_{i}\bigcap N_{i,S})\backslash\Gamma_{i}$
and an integral on $N_{i,S}$ Therefore, we arrive at the equivalent condition:

$\displaystyle\sum_{\Gamma_{N_{i,S}}\backslash\Gamma_{i}}\int_{N_{i,S}}\Phi(y^{-1}\gamma
nx)dn=0.$ (14)

We replace $x$ and $y$ with their respective Iwasawa decompositions, i.e.
$x=m_{1}n_{1}k_{1}$ and $y=m_{2}n_{2}k_{2}$. Moreover, we can write the sum
over $\frac{\Gamma}{\Gamma\bigcap N_{i,S}}$ as sum over ${\Gamma\bigcap
A_{i,S}}=\Gamma_{A_{i,S}}$ cosets. Therefore as $M_{i,S}$ centralizes
$A_{i,S}$, we can write $k_{2}^{-1}n_{2}^{-1}m_{2}^{-1}\gamma
nm_{1}n_{1}k_{1}=k_{2}^{-1}n_{2}^{-1}\gamma m_{2}^{-1}nm_{1}n_{1}k_{1}$. As
$M_{i,S}$ normalizes $N_{i,S}$, with a modular factor we have

$k_{2}^{-1}n_{2}^{-1}\gamma
m_{2}^{-1}nm_{1}n_{1}k_{1}=k_{2}^{-1}n_{2}^{-1}n^{\prime}\gamma
m_{2}^{-1}m_{1}n_{1}k_{1}.$

But as the modular factor, a scalar, only depends on $m_{2}$, and $\gamma$
belongs to a discrete subgroup, we can ignore that above. Therefore we can
finally write the argument of $\Phi$ as
$k_{2}^{-1}n_{2}^{-1}n^{\prime}n^{\prime\prime}\gamma
m_{2}^{-1}m_{1}k_{1}=k_{2}n\gamma mk_{2}$. Hence, we can rewrite the condition
in (14) as follows: For all $k_{1},k_{2}\in K_{S}$ and for all $m\in M_{i,S}$
[LV, (4.6)]

$\displaystyle\sum_{\Gamma_{A_{i,S}}}\int_{N_{i,S}}\Phi(k_{1}n\gamma
mk_{2})dn=0.$ (15)

Consequently (10) is the sufficient condition. ∎

In the next step we need to find a non-zero combined test functions on
$G_{S}$, which is bi-$K_{\infty}$-finite,$K_{\infty}$-central and compactly
supported at the Archimedean place and a function from the Hecke algebra at
the non-Archimedean places that satisfies the above conditions. For parabolic
subgroups $P_{S}$, whose Archimedean component is the chosen standard minimal
parabolic subgroup, (9) would become as follows: For all
$\nu_{S}\in\mathfrak{a}_{P_{S}}^{*}\supset\mathfrak{a}_{Q_{i,S}}^{*}$ such
that whenever for all i, $\nu_{S}|_{\Gamma_{A_{i,S}}}=1$, we have

$\displaystyle\text{Ind}_{P_{S}}^{G_{S}}(\omega_{S},\nu_{S})(\Phi)=0,$ (16)

for all $\omega_{S}$ discrete series representation of $M_{P,S}$ such that
$\tau|_{M_{0,\infty}}\supset\omega_{\infty}$. By the description of arithmetic
tori [PR, Thm. 5.12], $\nu_{S}$ should have the property that
$\nu_{p}=\nu_{q}$, for all $p,q\in S$. Let $\mathfrak{Z}(G(\mathbb{Q}_{p}))$
be the ring of regular functions defined on union of Benrstein components
$\Omega(G(\mathbb{Q}_{p}))$ ([MT, 2.3.1]). Combining for all $p\in
S\backslash\infty$ we define $\mathfrak{Z}(\mathbb{Q}_{S\backslash\infty})$ to
be the set of Fourier transform of Bernstein center for
$G(\mathbb{Q}_{S\backslash\infty})$. Let $z_{i}$ be the elements in the
Bernstein center for each $Q_{i}(\mathbb{Q}_{S\backslash\infty})$ [MT, 2.2.1].
Let $\hat{z_{i}}\in\mathfrak{Z}(G(\mathbb{Q}_{p}))$. We can form the test
function

$\Phi=\Phi_{\infty}\prod_{i}(z_{i}\star\mathbbm{1}_{K^{\prime}}).$

We have to find a regular function $R$ defined on
$\Omega(G(\mathbb{Q}_{S\backslash\infty}))$ such that
$R(\nu_{S\backslash\infty})=0$, whenever $\nu_{p}=\nu_{q}$, for all $p,q\in S$
and $\nu_{p}\in\mathfrak{a}_{P_{S}}^{*}$. We can find a polynomial that
satisfies this property for every $P_{S}\subset Q_{i,S}$. Hence, we could
apply the Arthur’s Paley-Wiener Theorem at the Archimedean place and matrix
Paley-Wiener Theorem for Hecke alegebra by Bernstein [B, Thm. 25] at the non-
archimedean place to construct a non-zero test function $\Phi$.

##  5\. Partial trace formula

In this section we write the partial trace formula. We choose a test function
whose Archimedean component is a $\tau-$spherical function belonging to the
convolution algebra $C_{c}^{\infty}(G(\mathbb{R}),\tau,\tau)$, satisfying the
identity

$\phi(k_{1}gk_{2})=\tau(k_{2})^{-1}\phi(g)\tau(k_{1})^{-1}$

and a scalar valued function at the non-archimedean places from
$\bar{\mathcal{H}}(G(K_{S^{{}^{\prime}}}^{{}^{\prime}}\backslash\mathbb{Q}_{S^{\prime}}/K_{S^{{}^{\prime}}}^{{}^{\prime}}))$,
the Hecke algebra. Suppose it also satisfies the condition of cuspidality
described in the previous section. It acts on $\Gamma-$invariant $L^{2}$
eigensection ($K_{\infty}$-finite,$K_{S^{\prime}}^{{}^{\prime}}$-fixed)
$e_{\lambda}(x)$ (orthonormal with respect to the inner product mentioned in
the introduction) of the Casimir operator(defined for sections of vector
bundles), with the eigenvalue parameter defined as $\lambda$. We define the
convolution action as follows:

$\displaystyle e_{\lambda}\star\phi(x)$
$\displaystyle=\int_{G_{S}}\phi(y^{-1}x)e_{\lambda}(y)dy$
$\displaystyle=\int_{\Gamma\backslash
G_{S}}\sum_{\gamma^{-1}\in\Gamma}\phi(y^{-1}\gamma^{-1}x)e_{\lambda}(y)dy$
$\displaystyle=\int_{\Gamma\backslash G_{S}}K(x,y)e_{\lambda}(y)dy$
$\displaystyle=\int_{\Gamma\backslash G_{S}/K_{\infty}}K(x,y)e_{\lambda}(y)dy$

In the last equation we have used the fact that $K(x,y)e_{\lambda}(y)$ is
$K_{\infty}-$invariant on $y$. Then the spectral expansion of $K(x,y)$ can be
written as follows:

$K(x,y)=\sum_{\lambda,\mu}(e_{\lambda}\star\phi,e_{\mu})e_{\mu}(x)\otimes
e_{\lambda}(y)^{*},$

where $e_{\mu}(y)^{*}$ denotes the dual vector which acts on $f(y)$ through
the pairing $\langle f(y),e_{\mu}(y)\rangle_{V_{\tau}}$ on the fiber
$(E_{\tau})_{y}$ [Dui, (7.3)]. If we let $x=y$, then the spectral side will
have the following form:

$K(x,x)=\sum_{\lambda,\mu}(e_{\lambda}\star\phi,e_{\mu})e_{\mu}(x)\otimes
e_{\lambda}(x)^{*}.$

Therefore, taking the trace on both sides we get

$\text{Tr}K(x,x)=\sum_{\lambda}(e_{\lambda}\star\phi,e_{\lambda})e_{\lambda}(x)\otimes
e_{\lambda}(x)^{*}.$

Consider a compact subset $\Omega\subset\Gamma\backslash G_{S}$, whose measure
is arbitrarily close to Vol$(\Gamma\backslash G_{S})$. We take the pre-image
of $\Omega$ in $G_{S}$ and call it $\tilde{\Omega}$. Unwinding the sum on the
left hand side we get

$TrK(x,x)=Tr\phi(e)+\sum_{\gamma\in Z}\text{Tr}\phi(x^{-1}\gamma x),$

The set $Z$ will have the following form :

$Z=\Big{(}\Gamma\backslash\\{e\\}\Big{)}\bigcup\Big{(}xgx^{-1}:x\in\tilde{\Omega},\text{x
lies in support of Tr}(\phi)\Big{)}.$

The cardinality of $Z$ would be finite, and would depend only on
$\tilde{\Omega}$ and the support of Tr$(\phi)$.
Integrating both sides over $\tilde{\Omega}$, we obtain the following:

$\int_{\tilde{\Omega}}TrK(x,x)dx\leq\int_{\Gamma\backslash
G_{S}/K_{\infty}}TrK(x,x)dx=\sum_{\lambda}(e_{\lambda}\star\phi,e_{\lambda})$

To make sure we have a self adjoint convolution operator we need
$\phi(x)=\overline{\phi(x^{-1})^{T}}$. To achieve the self-adjointness we
replace $\phi$ with $\phi\star\widetilde{\phi}$, where
$\widetilde{\phi}(x)=\overline{\phi(x^{-1})^{T}}$. Hence, the right hand side
of the above inequality becomes
$\sum_{\lambda}(e_{\lambda}\star\phi,e_{\lambda}\star\phi)$.
Now by a theorem of Gelfand, Graev and Piatetski-Shapiro [Bmp, Prop. 3.2.3]
which states that the convolution operator on the scalar-valued automorphic
forms is a compact operator. Therefore we obtain:
$\sum_{\lambda}(e_{\lambda}\star\phi,e_{\lambda}\star\phi)<\infty$. Hence, we
have

$\displaystyle\text{Tr}(\phi\star\widetilde{\phi}(e))\text{Vol}(\Omega)+\sum_{\gamma\in
Z}\int_{\Omega}\text{Tr}(\phi\star\tilde{\phi})(x^{-1}\gamma
x)\leq\sum\limits_{\lambda}(e_{\lambda}\star\phi,e_{\lambda}\star\phi).$ (17)

We now give a representation theoretic interpretation of
$\sum_{\lambda}(e_{\lambda}\star\phi,e_{\lambda}\star\phi)$. Let
$\pi_{\infty}\in\Pi_{\text{cusp}}(G(\mathbb{R}),\tau)$ be the Archimedean part
of irreducible unitary representation $\pi_{S}$ which appears as a
subrepresentation of right regular representation of $G(\mathbb{Q}_{S})$ on
$L^{2}_{\text{cusp}}(\Gamma\backslash G(\mathbb{Q}_{S}),\tau)$ with
multiplicities $m(\pi_{\infty})$, and let $H_{\pi_{\infty}}$ be the
corresponding Hilbert space. Let $H_{\pi_{\infty}}(\tau)$ be the
$\tau-$isotypic subspace. Then using [BM, Thm. 3.3] we have:

$\displaystyle\sum_{\lambda}(e_{\lambda}\star\phi,e_{\lambda}\star\phi)=\sum\limits_{\Pi_{\text{cusp}}(G(\mathbb{R}),\tau)}m(\pi_{\infty})\bigg{(}\sum\limits_{i=1}^{m}(e_{i}\star\phi,e_{i}\star\phi)\bigg{)},$
(18)

where
$m=\text{dim}(\text{Hom}_{K_{\infty}}(\mathcal{H}_{\pi_{\infty}}(\tau),V_{\tau}))$.

##  6\. An approximation lemma

In this section we find a family of test functions

$H_{S,t}=H_{\infty,t}\cdot\mathbbm{1}_{K^{\prime}},$

for $0\leq t<1$ that satisfy certain approximations. For the rest of the
section and beyond we will write $S\backslash\infty=S^{\prime}$. Here
$K^{\prime}=K_{S\backslash\infty}^{{}^{\prime}}$. We prove a slight
generalization of [LV, Lemmma 2] below.
Let $\mathfrak{h}=i\mathfrak{h}_{K_{\infty}}+\mathfrak{a}_{0,\infty}$ be the
Cartan subalgebra of $\mathfrak{U(\mathfrak{g_{\infty}})}$. Let
$\mathfrak{h}_{\mathbb{C}}=\mathfrak{h}\otimes\mathbb{C}$ be the
complexification of the Cartan subalgebra. Let $\mathfrak{h}_{\mathbb{C}}^{*}$
be the dual of the Cartan subalgebra. We fix $0<\epsilon<1$. By [LV,
(5.12)-(5.15)] we know there exist a non-empty open set of Schwarz functions
$\psi$ defined on cylinders

$\\{\lambda_{\infty}\in\mathfrak{h}_{\mathbb{C}}^{*}:\lvert\text{Re}(\lambda_{\infty})\rvert\leq
a\\}$

that satisfy the following conditions:

  * •

$0\leq\psi(\lambda_{\infty})<1,\quad\text{when}\quad\lVert\lambda_{\infty}\rVert\leq
1,\lambda_{\infty}\in\mathfrak{h}_{K_{\infty}}^{*}+i\mathfrak{a}_{0,\infty}^{*}$.

  * •

$\int\limits_{i\mathfrak{a}_{0,\infty}^{*}}\left|\psi(\nu_{K_{\infty}}+\nu_{\infty})-\chi(\nu_{K_{\infty}}+\nu_{\infty})\right|(1+\lVert\nu_{\infty}\rVert)^{\text{dim}(N_{0})}d\nu_{\infty}\leq\epsilon$,
for fixed $\nu_{K_{\infty}}\in\mathfrak{h}_{K_{\infty},\mathbb{C}}^{*}$ such
that $\text{Re}(\nu_{K_{\infty}})$ is bounded.

  * •

$\sup\limits_{\lVert\lambda_{\infty}\rVert>1}(1+\lVert\lambda_{\infty}\rVert)^{d+1}\left|\psi(\lambda_{\infty})\right|\leq\epsilon$.

Here $\chi(\nu_{K_{\infty}}+\nu_{\infty})$ denotes the characteristic function
of the sphere $\lVert\nu_{K_{\infty}}+\nu_{\infty}\rVert\leq 1$. Without loss
of generality we may assume that $\psi$ can be extended to a holomorphic
function on $\mathfrak{h}_{\mathbb{C}}^{*}$, as the Fourier transform of
$\psi$ has compact support. Let $\mathbb{O}_{\mathbb{C}}$ be the orthogonal
group of $\mathfrak{h}_{\mathbb{C}}^{*}$ with respect to the inner product
$\langle,\rangle$. This inner product is induced from the Killing form on
$\mathfrak{h}_{\mathbb{C}}$. By averaging we can make $\psi$ as
$\mathbb{O}_{\mathbb{C}}-$ invariant function. Therefore
$\psi(\lambda_{\infty})$ depends only on
$\langle\lambda_{\infty},\lambda_{\infty}\rangle$. Let $d_{\omega_{\infty}}$
denote the degree of equivalent classes of square integrable irreducible
representations $\omega_{\infty}$ of $M_{0,\infty}$. Using Frobenius
Reciprocity we see that for the case of minimal parabolic $M_{0,\infty}$, we
have

$[\text{Ind}(\omega_{\infty},\nu_{\infty})|_{K_{\infty}}:\tau]=[\tau|_{M_{0,\infty}}:\omega_{\infty}].$

Therefore,

$\sum\limits_{\omega_{\infty}\in\mathcal{E}_{2}(M_{0,\infty})}d_{\omega_{\infty}}[\tau|_{M_{0,\infty}}:\omega_{\infty}]=\sum\limits_{\omega_{\infty}\in\mathcal{E}_{2}(M_{0,\infty})}d_{\omega_{\infty}}[\text{Ind}(\omega_{\infty},\nu_{\infty})|_{K_{\infty}}:\tau]=d_{\tau}.$

Put $m_{\omega_{\infty}}=[\tau|_{M_{0,\infty}}:\omega_{\infty}]$. Let
$C_{c}^{\infty}(\mathfrak{h})$ be the set of compactly supported smooth
functions (i.e. set of multipliers for the convolution algebra
$C_{c}^{\infty}(G(\mathbb{R}))_{K_{\infty}}^{K_{\infty}}$). Then by the
Euclidean Paley-Wiener Theorem there exists $\zeta\in
C_{c}^{\infty}(\mathfrak{h})$ such that its Laplace-Fourier transform
$\hat{\zeta}(\lambda_{\infty})$ satisfies:

$\hat{\zeta}(\lambda_{\infty})=\psi(\lambda_{\infty}),\quad\forall\lambda_{\infty}\in\mathfrak{h}_{\mathbb{C}}^{*}.$

There exists a function $H_{\infty}^{\sharp}\in
C_{c}^{\infty}(G(\mathbb{R}))_{K_{\infty}}^{K_{\infty}}$ such that
$\text{Ind}(\omega_{\infty},\nu_{\infty})(H_{\infty}^{\sharp})=\text{Ind}(\omega_{\infty},\nu_{\infty})(d_{\tau}\chi_{\tau})$
[GV, Lemma 1.3.2]. Now using the Arthur’s theorem on multiplier we can choose
a family of functions $H_{\infty,t,\zeta}^{\sharp}\in
C_{c}^{\infty}(G(\mathbb{R}))_{K_{\infty}}^{K_{\infty}}$, such that their
operator valued Fourier transforms are

$\text{Ind}(\omega_{\infty},\nu_{\infty})(H_{\infty,t,\zeta})=\hat{\zeta}(\nu_{\omega_{\infty}}+t\nu_{\infty})\text{Ind}(\omega_{\infty},\nu_{\infty})(d_{\tau}\chi_{\tau}),$

for $0<t\leq 1$. Let
$H_{S,t,\zeta}^{\sharp}=H_{\infty,t,\zeta}^{\sharp}\cdot\mathbbm{1}_{K^{\prime}}$,
for $K^{\prime}$ an arbitrarily chosen open compact subgroup of
$G_{S^{\prime}}$. Then

$\displaystyle\left|\left|\widehat{H_{\infty,t,\zeta}^{\sharp}}(\omega_{\infty},\nu_{\infty})\right|\right|^{2}=\int\limits_{K_{\infty}}\left|\widehat{H_{\infty,t,\zeta}^{\sharp}}(\omega,\nu)(1:1:k)\right|^{2}dk=d_{\omega}\left|\left|\text{Ind}(\omega_{\infty},\nu_{\infty})(H_{\infty,t,\zeta}^{\sharp})\right|\right|_{\text{HS}}^{2}.$

Therefore, from the above choice of Schwartz function we have

$\left|\left|\widehat{H_{\infty,t,\zeta}^{\sharp}}(\omega_{\infty},\nu_{\infty})\right|\right|^{2}=d_{\tau}d_{\omega}m_{\omega_{\infty}}\lVert\psi(\nu_{\omega_{\infty}}+t\nu_{\infty})\rVert^{2}.$

The following estimate will be instrumental in proving the main estimate in
Weyl’s law. Let $0<\epsilon<1$ and choose $H_{\infty,t,\zeta}^{\sharp}$ that
depending on $\epsilon$.

######  Lemma 6.

There exists $C_{1}>0$ such that for sufficiently small $0<t\leq 1$ and for
the minimal Parabolic $P_{0,\infty}=M_{0,\infty}A_{0,\infty}N_{0,\infty}$ we
have

$\displaystyle\left|t^{d}\sum_{\omega\in\mathcal{E}_{2}(M_{0,\infty})}\int_{i\mathfrak{a}_{0,\infty}^{*}}\left|\left|\widehat{H_{\infty,t,\zeta}^{\sharp}}(\omega_{\infty},\nu_{\infty})\right|\right|^{2}\mu(\omega_{\infty},\nu_{\infty})d\nu-
d_{\tau}^{2}\alpha(G_{\infty})\right|\leq C_{1}\epsilon.$

###### Proof.

Recall the Plancherel inversion formula at the real place

$\displaystyle
f\star\tilde{f}(1)=\sum_{\mathcal{P}}n(\mathcal{P})^{-1}\sum_{P\in\mathcal{P}}\sum_{\omega\in\mathcal{E}_{2}(M_{\infty})}(\frac{1}{2\pi
i})^{q}\int_{i\mathfrak{a}_{M_{\infty}}^{*}}\left|\left|\widehat{f}(\omega,\nu)\right|\right|^{2}\mu(\omega,\nu)d\nu.$
(19)

Here, $\mathcal{P}$ denotes the associated classes pf parabolic subgroups. The
integer $q$ denotes the dimension of respective
$i\mathfrak{a}_{M_{\infty}}^{*}$. We are interested on the summand that
corresponds to the minimal parabolic. For the sum and integral involving the
minimal parabolic subgroup
$P_{0,\infty}=M_{0,\infty}A_{0,\infty}N_{0,\infty}$, and
$\text{dim}(i\mathfrak{a}_{0,\infty}^{*})=r$ we have

$\limsup_{t\to
0}\left|t^{d}\sum_{\omega_{\infty}\in\mathcal{E}_{2}(M_{0,\infty})}(\frac{1}{2\pi
i})^{r}\int_{i\mathfrak{a}_{0,\infty}^{*}}\lVert\widehat{{H_{\infty,t,\zeta}^{\sharp}}}(\omega_{\infty},\nu_{\infty})\rVert^{2}\mu(\omega_{\infty},\nu_{\infty})d\nu_{\infty}-d_{\tau}^{2}\alpha(G_{\infty})\right|$
$=\limsup_{t\to
0}\left|t^{d}\sum_{\omega_{\infty}\in\mathcal{E}_{2}(M_{0,\infty})}(\frac{1}{2\pi
i})^{r}\int_{i\mathfrak{a}_{0,\infty}^{*}}\lVert\widehat{{H_{\infty,t,\zeta}^{\sharp}}}(\omega_{\infty},\nu_{\infty})\rVert^{2}\mu(\omega_{\infty},\nu_{\infty})d\nu_{\infty}-d_{\tau}^{2}\alpha(G_{\infty})\right|$
$\leq\left|\limsup_{t\to
0}t^{d}\sum_{\omega_{\infty}\in\mathcal{E}_{2}(M_{0,\infty})}(\frac{1}{2\pi
i})^{r}\int_{i\mathfrak{a}_{0,\infty}^{*}}d_{\tau}d_{\omega}m_{\omega_{\infty}}\left|\lVert\psi(\nu_{\omega_{\infty}}+t\nu_{\infty})\rVert^{2}-\chi(\nu_{\omega_{\infty}}+t\nu_{\infty})\right|\mu(\omega_{\infty},\nu_{\infty})d\nu_{\infty}\right|$
$\leq\mathopen{\left\lvert\vphantom{{\limsup_{t\to
0}t^{d}\sum_{\omega_{\infty}\in\mathcal{E}_{2}(M_{0,\infty})}(\frac{1}{2\pi
i})^{r}\int_{i\mathfrak{a}_{0,\infty}^{*}}d_{\tau}d_{\omega}m_{\omega_{\infty}}\left|\lVert\psi(\nu_{\omega_{\infty}}+\nu_{\infty})\rVert^{2}-\chi(\nu_{\omega_{\infty}}+\nu_{\infty})\right|\mu(\omega_{\infty},t^{-1}\nu_{\infty})d(t^{-1}\nu_{\infty})}}\kern-1.2pt\right.}\limsup_{t\to
0}t^{d}\sum_{\omega_{\infty}\in\mathcal{E}_{2}(M_{0,\infty})}(\frac{1}{2\pi
i})^{r}\int_{i\mathfrak{a}_{0,\infty}^{*}}d_{\tau}d_{\omega}m_{\omega_{\infty}}\left|\lVert\psi(\nu_{\omega_{\infty}}+\nu_{\infty})\rVert^{2}-\chi(\nu_{\omega_{\infty}}+\nu_{\infty})\right|\mu(\omega_{\infty},t^{-1}\nu_{\infty})d(t^{-1}\nu_{\infty})\mathclose{\left.\kern-1.2pt\vphantom{{\limsup_{t\to
0}t^{d}\sum_{\omega_{\infty}\in\mathcal{E}_{2}(M_{0,\infty})}(\frac{1}{2\pi
i})^{r}\int_{i\mathfrak{a}_{0,\infty}^{*}}d_{\tau}d_{\omega}m_{\omega_{\infty}}\left|\lVert\psi(\nu_{\omega_{\infty}}+\nu_{\infty})\rVert^{2}-\chi(\nu_{\omega_{\infty}}+\nu_{\infty})\right|\mu(\omega_{\infty},t^{-1}\nu_{\infty})d(t^{-1}\nu_{\infty})}}\right\rvert}$
$\leq\left|\sum_{\omega_{\infty}\in\mathcal{E}_{2}(M_{0,\infty})}(\frac{1}{2\pi
i})^{r}\int_{i\mathfrak{a}_{0,\infty}^{*}}d_{\tau}d_{\omega}m_{\omega_{\infty}}\left|\lVert\psi(\nu_{\omega_{\infty}}+\nu_{\infty})\rVert^{2}-\chi(\nu_{\omega_{\infty}}+\nu_{\infty})\right|\Big{(}1+\lVert\nu_{\infty}\rVert\Big{)}^{\text{dim}(N_{0})}d(\nu_{\infty})\right|$
$\leq C_{1}\epsilon.$

In the last step we use the second condition of $\psi$ defined in the
beginning of this section.

∎

##  7\. Plancherel Inversion and test functions

In this section we describe the choice of test functions. We start by
recalling a result of Camporesi, which identifies the endomorphism valued
convolution algebra with scaler valued functions.

######  Proposition.

[Cmp1, Prop 2.1] Let $\tau$ be the irreducible $K_{\infty}$-type as before of
dimension $d_{\tau}$. then the endomorphism valued convolution algebra
isomorphic to scalar valued bi-$K_{\infty}$-finite, $K_{\infty}$-central
function space. The anti-isomorphism is given by the following map

$\displaystyle f\mapsto F=d_{\tau}\text{Tr}(f).$ (20)

Moreover it satisfies the following relations:

$d_{\tau}\text{Tr}(f_{1}\star f_{2})=d_{\tau}\text{Tr}(f_{2})\star
d_{\tau}\text{Tr}(f_{1}),$ $d_{\tau}\chi_{\tau}\star F=F=F\star
d_{\tau}\chi_{\tau}.$

###  7.1. Test Functions

The following steps will describe our test functions. We closely follow [LV,
Lemma 2,3].
⚫ We choose a function
$\Phi_{S}^{\sharp}=\Phi_{\infty}^{\sharp}\Phi_{S^{\prime}}$, where
$\Phi_{\infty}^{\sharp}\in
C_{c}^{\infty}(G(\mathbb{R}))_{K_{\infty}}^{K_{\infty}}$ and
$\Phi_{S^{\prime}}\in\bar{\mathcal{H}}(G(K_{S^{{}^{\prime}}}^{{}^{\prime}}\backslash\mathbb{Q}_{S^{\prime}}/K_{S^{{}^{\prime}}}^{{}^{\prime}}))$,
so that the cuspidality condition holds true, i.e. $\Phi_{S}^{\sharp}$ satisfy
$(8)$. By the isomorphism in $(20)$ we have a function $\phi_{\infty}\in
C_{c}^{\infty}(G(\mathbb{R}),\tau,\tau)$ so that
$d_{\tau}\text{Tr}\phi_{\infty}=\Phi_{\infty}^{\sharp}$. Let $\phi_{S}$ be the
product of $\phi_{\infty}$ by
$\Phi_{S^{\prime}}\in\bar{\mathcal{H}}(G(K_{S^{{}^{\prime}}}^{{}^{\prime}}\backslash\mathbb{Q}_{S^{\prime}}/K_{S^{{}^{\prime}}}^{{}^{\prime}}))$.
⚫ Next we choose a family of functions

$h_{\infty,t,\zeta}\in
C_{c}^{\infty}(G(\mathbb{R}),\tau,\tau)\quad\text{for}\quad 0<t\leq 1.$

From the properties mentioned in the previous section, we can choose an entire
Schwartz function $\widehat{H_{\infty,\zeta}}(\omega,\nu)$ that satisfies the
Lemma 6. We form a family $h_{\infty,t,\zeta}$ for $0<t\leq 1$, so that
$d_{\tau}\text{Tr}h_{\infty,t,\zeta}=H_{\infty,t,\zeta}^{\sharp}$. We multiply
$h_{\infty,t,\zeta}$ with $\mathbbm{1}_{K^{\prime}}$, and call this function
$h_{S,t,\zeta}$.
⚫ Finally, choose a sequence $\Phi_{n,S}^{\sharp}$ that satisfies the
condition of cuspidality. Let $Z_{\infty}\in C_{c}^{\infty}(\mathfrak{h})$ be
an element in the set of Archimedean multipliers. Then by Euclidean Paley-
Wiener Theorem, $\hat{Z}_{\infty}$ is bounded on the set
$\\{\lambda_{\infty}\in\mathfrak{h}_{\mathbb{C}}^{*}:\text{Im}(\lambda_{\infty})=0\\}$.
If we choose $\Phi_{S}$ so that it satisfies the condition of cuspidality,
then the fourier transform of elements of the Bernstein center at the non-
Archimedean places is bounded on the set of unitary unramified characters.
Suppose the bound is $B>0$. Following [LV, p. 245-246] we construct such a
sequence. Let

$P_{n}(Z_{S})=1-\Big{(}1-\frac{(Z_{S})^{2}}{B^{2}}\Big{)}^{n}.$

Let $\Phi_{S}^{\sharp}=Z_{S}\star f_{S}$, where
$\lVert\pi_{\infty}(f_{\infty})\rVert_{\text{HS}}^{2}=1$ and
$f_{S\backslash\infty}=\mathbbm{1}_{K^{\prime}}$. As the multipliers on the
space of bi-$K_{\infty}-$finite compactly supported smooth functions on
$G(\mathbb{R})$ and multipliers on space of locally constant functions on
$G(\mathbb{Q}_{S^{\prime}})$ are equipped with convolution, (thought of as a
multiplication) we can define a sequence
$\Phi_{n,S}^{\sharp}=P_{n}(Z_{S})\star f_{S}\star\widetilde{f_{S}}$. $P_{n}$
will satisfy the following properties

  * •

$P_{n}(0)=0$

  * •

$\text{Ind}(\omega_{S},\nu_{S})(\Phi_{n,S}^{\sharp})=P_{n}(\hat{Z}_{S})(\text{Ind}(\omega_{S},\nu_{S})(f_{S})\text{Ind}(\omega_{S},\nu_{S})(f_{S})^{*}).$
Notice that here multiplication of Fourier transforms are defined as [Ar1,
part II,pp. 1.1 ]. Therefore $\Phi_{n,S}^{\sharp}$ will satisfy (9).

Let $\phi_{n,S}$ be the endomorphism valued test functions corresponding to
$\Phi_{n,S}^{\sharp}$ as $\phi_{n,S}$. We apply the partial trace formula on
the family of test functions $\phi_{n,S}\star h_{S,t,\zeta}$. Write
$\phi_{n,S,t,\zeta}=\phi_{n,S}\star h_{S,t,\zeta}$. We obtain

$\displaystyle
d_{\tau}\text{Tr}((\phi_{n,S,t,\zeta}\star\widetilde{\phi_{n,S,t,\zeta}})(e))\text{Vol}(\Omega)+d_{\tau}\sum_{\gamma\in
Z}\int_{\Omega}\text{Tr}(\phi_{n,S,t,\zeta}\star\widetilde{\phi_{n,t,S}})(x^{-1}\gamma
x)$ $\displaystyle\leq
d_{\tau}\sum\limits_{\lambda}(e_{\lambda}\star\phi_{n,S,t,\zeta}\star\widetilde{\phi_{n,S,t,\zeta}},e_{\nu})$
$\displaystyle=d_{\tau}\sum\limits_{\lambda}\sum\limits_{-\nu_{\pi_{\infty}}=\lambda}m(\pi)\text{Tr}(\pi(\text{Tr}(\phi_{n,,S,t,\zeta}\star\widetilde{\phi_{n,S,t,\zeta}}))).$
(21)

###  7.2. Plancherel Inversion

We now recall the Plancherel Theorem at Archimedean place as in [Ar1, part II,
(2.1)]

$\displaystyle\widetilde{\Phi_{n,\infty,t,\zeta}^{\sharp}}\star\Phi_{n,\infty,t,\zeta}^{\sharp}(e)\widetilde{\Phi_{n,S^{\prime}}}\star\Phi_{n,S^{\prime}}(e)=\sum_{\mathcal{P}\in\text{Cl}(G_{\infty})}n(\mathcal{P})^{-1}\sum_{P\in\mathcal{P}}\sum_{\omega_{\infty}\in\mathcal{E}_{2}(M_{\infty})}(\frac{1}{2\pi
i})^{q}$
$\displaystyle\int_{i\mathfrak{a_{\infty}^{*}}}\left|\left|\text{Ind}(\omega_{\infty},\nu_{\infty})(\Phi_{n,\infty,t,\zeta}^{\sharp})\right|\right|_{\text{HS}}^{2}\mu(\omega_{\infty},\nu_{\infty})d\nu_{\infty}\widetilde{\Phi_{n,S^{\prime}}}\star\Phi_{n,S^{\prime}}(e).$

We have the following convergences as $n\to\infty$

$\text{Ind}(\omega_{S^{\prime}})(\Phi_{n,S^{\prime}})\to\text{Ind}(\omega_{S^{\prime}})(\mathbbm{1}_{K^{\prime}}),\quad\text{and}\quad\lVert\text{Ind}(\omega_{\infty},\nu_{\infty})(\Phi_{n,\infty}^{\sharp})\rVert\to
1.$

Now if we take the limit inside the norm (due to continuity) and inside the
Fourier transformation (due to isometry)[Ar2, p. 4719] the above integrand
converges to

$\left|\left|\widehat{H_{\infty,t,\zeta}^{\sharp}}(\omega_{\infty},\nu_{\infty})\right|\right|^{2}\mu(\omega_{\infty},\nu_{\infty}).$

Therefore, we see that integrand corresponding to the minimal parabolic
summand in the Plancherel Formula can be divided into two sets $X=\\{\nu\in
i\mathfrak{a}_{0,\infty}^{*}:P_{n}(\hat{Z}_{S})\leq\epsilon\\}$ and its
complement $X^{c}$. As we take $\lim{\epsilon\to 0}$, the set $X$ will have
measure 0, and on $X^{c}$ the integrand will become
$\lVert\widehat{H_{\infty,t,\zeta}^{\sharp}}(\omega_{\infty},\nu_{\infty})\rVert^{2}\mu(\omega_{\infty},\nu_{\infty})$.
From the discussion above, it is clear that the following estimate will be
enough for us to arrive at the main term as $\limsup{t\to 0}$.

######  Lemma 7.

For all $n$, there exists $C_{1}>0$ such that

$\displaystyle\mathopen{\left\lvert\vphantom{{t^{d}\sum_{\mathcal{P}\in\text{Cl}(G_{\infty})}n(\mathcal{P})^{-1}\sum_{P\in\mathcal{P}}\sum_{\omega_{\infty}\in\mathcal{E}_{2}(M_{\infty})}(\frac{1}{2\pi
i})^{q}\int\limits_{i\mathfrak{a}_{\infty}^{*}}\left|\left|\widehat{{H_{\infty,t,\zeta}^{\sharp}}}(\omega_{\infty},\nu_{\infty})\right|\right|^{2}\mu(\omega_{\infty},\nu_{\infty})d\nu_{\infty}\widetilde{\Phi_{n,S^{\prime}}}\star\Phi_{n,S^{\prime}}(e)-d_{\tau}^{2}\alpha(G)}}\kern-1.2pt\right.}t^{d}\sum_{\mathcal{P}\in\text{Cl}(G_{\infty})}n(\mathcal{P})^{-1}\sum_{P\in\mathcal{P}}\sum_{\omega_{\infty}\in\mathcal{E}_{2}(M_{\infty})}(\frac{1}{2\pi
i})^{q}\int\limits_{i\mathfrak{a}_{\infty}^{*}}\left|\left|\widehat{{H_{\infty,t,\zeta}^{\sharp}}}(\omega_{\infty},\nu_{\infty})\right|\right|^{2}\mu(\omega_{\infty},\nu_{\infty})d\nu_{\infty}$
$\displaystyle\widetilde{\Phi_{n,S^{\prime}}}\star\Phi_{n,S^{\prime}}(e)-d_{\tau}^{2}\alpha(G)\mathclose{\left.\kern-1.2pt\vphantom{{t^{d}\sum_{\mathcal{P}\in\text{Cl}(G_{\infty})}n(\mathcal{P})^{-1}\sum_{P\in\mathcal{P}}\sum_{\omega_{\infty}\in\mathcal{E}_{2}(M_{\infty})}(\frac{1}{2\pi
i})^{q}\int\limits_{i\mathfrak{a}_{\infty}^{*}}\left|\left|\widehat{{H_{\infty,t,\zeta}^{\sharp}}}(\omega_{\infty},\nu_{\infty})\right|\right|^{2}\mu(\omega_{\infty},\nu_{\infty})d\nu_{\infty}\widetilde{\Phi_{n,S^{\prime}}}\star\Phi_{n,S^{\prime}}(e)-d_{\tau}^{2}\alpha(G)}}\right\rvert}\leq
C_{1}\epsilon.$

###### Proof.

We can ignore the terms related to the $p-$adic Plancherel formula as the
tempered parameters in this case are finite disjoint unions of compact
orbifolds, hence those terms will be automatically bounded. Therefore, we only
concentrate on the Archimedean part. We have

$\left|t^{d}\sum_{\mathcal{P}\in\text{Cl}(G_{\infty})}n(\mathcal{P})^{-1}\sum_{P\in\mathcal{P}}\sum_{\omega_{\infty}\in\mathcal{E}_{2}(M_{\infty})}(\frac{1}{2\pi
i})^{q}\int_{i\mathfrak{a}_{\infty}^{*}}\left|\left|\widehat{{H_{\infty,t,\zeta}^{\sharp}}}(\omega_{\infty},\nu_{\infty})\right|\right|^{2}\mu(\omega_{\infty},\nu_{\infty})d\nu_{\infty}-d_{\tau}^{2}\alpha(G)\right|$
$=\left|t^{d}\times(\text{non-minimal
terms})+t^{d}\sum_{\omega_{\infty}\in\mathcal{E}_{2}(M_{0,\infty})}(\frac{1}{2\pi
i})^{r}\int_{i\mathfrak{a}_{0,\infty}^{*}}\left|\left|\widehat{{H_{\infty,t,\zeta}^{\sharp}}}(\omega_{\infty},\nu_{\infty})\right|\right|^{2}\mu(\omega_{\infty},\nu_{\infty})d\nu_{\infty}-d_{\tau}^{2}\alpha(G)\right|$

The Plancherel density corresponding to the non-minimal parabolic subgroups
will have the following asymptotic estimate. For some integer $l<d$, we have

$\int_{i\mathfrak{a}_{P,\infty}^{*}}\mu(\omega_{\infty},t^{-1}\nu_{\infty})d(t^{-1}\nu_{\infty})\sim
t^{-l}\quad\text{as}\quad t\to 0.$

Therefore, the non-minimal terms will tend to 0 as $t\to 0$. And the
approximation for the other term was dealt with in Lemma 6. ∎

Therefore, We see from (21) the main term corresponding to the trivial
conjugacy class is asymptotic to

$d_{\tau}^{2}\alpha(G)\text{Vol}(\Omega)t^{-d}\quad\text{as}\quad t\to 0.$

##  8\. Bounds for the non-trivial classes

To get the estimates for non-trivial conjugacy classes on the geometric side
we write the Fourier inversion formula of Harish-chandra with respect to
Eisenstein integrals. To this end we use the formula $(1.1)$ in [Ar3, Chap.
III Sec. 1]. It gives

$\displaystyle H_{\infty,t}(x)|_{(1,1)}$
$\displaystyle=\sum\limits_{\mathcal{P}}\lvert{\mathcal{P}\rvert}^{-1}\sum_{P\in\mathcal{P}}{\lvert
W(\mathfrak{a}_{P})\rvert}^{-1}\int\limits_{i\mathfrak{a}_{\infty}^{*}}E_{P}(x_{\infty},\mu_{P}(\nu_{\infty})\widehat{H_{\infty,P}}(t\nu_{\infty}),\nu_{\infty})_{(1:1)}d\nu_{\infty},$

where $\mathcal{P}$ denotes an associated class of parabolic subgroups, the
function $H_{\infty,t}$ lies in $C_{c}^{\infty}(G(\mathbb{R}),\tau)$. Note
that $C_{c}^{\infty}(G(\mathbb{R}),\tau)$ is isomorphic to
$C_{c}^{\infty}(G(\mathbb{R}))_{K_{\infty}}$ via the relation:

$H_{\infty,t}(x)|_{k_{1},k_{2}}=H_{\infty,t}(k_{1}xk_{2}).$

We concentrate on the part of the above series and integral corresponding to
the minimal parabolic. We obtain the following inequality for the summand
corresponding to minimal parabolic $P_{0,\infty}$:

$\displaystyle\int\limits_{i\mathfrak{a}_{0,\infty}^{*}}\lvert
E_{P_{0}}(x_{\infty},\mu_{P_{0}}(\nu_{\infty})\widehat{H_{\infty,P_{0}}}(t\nu_{\infty}),\nu_{\infty})|_{(1:1)}\rvert
d\nu_{\infty}$
$\displaystyle\leq\int\limits_{i\mathfrak{a}_{0,\infty}^{*}}\int\limits_{K_{\infty}}\lvert\widehat{H_{\infty,P_{0}}}(t\nu)(1:m(kx):1)\rvert\mu_{P_{0}}(\nu)\lvert
e^{(\nu+\rho)H(kx_{\infty})}\rvert dkd\nu$ (22)

The right hand side of the above inequality is bounded by

$\int\limits_{i\mathfrak{a}_{0,\infty}^{*}}\Bigg{(}\lVert\widehat{H_{\infty,P_{0}}}(t\nu)\rVert\mu_{P_{0}}(\nu)\int\limits_{K_{\infty}}\lvert
e^{(\nu+\rho)H(kx)}\rvert dk\Bigg{)}d\nu.$

Moreover, we have $\int_{K_{\infty}}\lvert e^{(\nu+\rho)H(kx)}\rvert
dkd\nu\leq C_{1}(1+\lVert\nu\rVert)^{-1/2}$ when $x\notin K_{\infty}$ and lies
in a fixed compact set[LV, Lemma 3]. Now making a change of variable
$\nu_{\infty}\to\frac{\nu_{\infty}}{t}$, using the Paley-Wiener bound of
$\lVert\widehat{H_{\infty,P}}(\nu_{\infty})\rVert$, and using the bound of the
Plancherel Measure from (7) from section 3, we obtain the following
inequality:

$\displaystyle\int\limits_{i\mathfrak{a}_{0,\infty}^{*}}\lvert
E_{P}(x_{\infty},\mu_{P}(\nu)\widehat{H_{\infty,P}}(t\nu)(1:1),\nu)\rvert
d\nu\leq C_{2}t^{-d+1/2}.$ (23)

Hence, we have the bound

$\displaystyle\left|H_{\infty,t}(x)|_{(1,1)}\right|\leq C_{2}t^{-d+1/2}.$ (24)

Now we apply the above bound for the Archimedean part of the integrand
corresponding to the non-trivial conjugacy class of $\gamma\in\Gamma$ in (21).
Notice that similar to [LV, (6.3)] we can assume that the support of
$d_{\tau}\text{Tr}(\phi_{n,S,t,\zeta}\star\widetilde{\phi_{n,S,t,\zeta}})$
that lies inside $K_{\infty}$ will have measure zero when projected onto
$G(\mathbb{R})$. Moreover,

$d_{\tau}\text{Tr}(\phi_{n,\infty,t,\zeta}\star\widetilde{\phi_{n,\infty,t,\zeta}})=\Phi_{n,\infty,t,\zeta}^{\sharp}\star\widetilde{\Phi_{n,\infty,t,\zeta}^{\sharp}}.$

This is a bi-$K_{\infty}$-finite function. Hence we can apply the above
discussion. Arguing as in [LV, pg. 243] for the lower bound of Weyl’s law, we
can see that the terms corresponding to a non-trivial conjugacy class in (21)
is bounded by $c|Z|t^{-d+1/2}$. Notice that as $\Gamma$ injects into $G_{S}$
diagonally, the cardinality of $Z$ is finite. Moreover the $L^{1}$ norm of
$\Phi_{n,S^{\prime}}\mathbbm{1}_{K^{\prime}}$ are bounded by a constant for
all $n$. Therefore it follows from (21) that

$\displaystyle\left|d_{\tau}\text{Tr}((\widetilde{\phi_{n,S,t,\zeta}}\star\phi_{n,S,t,\zeta})(e))\text{Vol}(\Omega)+d_{\tau}\sum_{\gamma\in
Z}\int_{\Omega}\text{Tr}(\widetilde{\phi_{n,S,t,\zeta}}\star\phi_{n,S,t,\zeta})(x^{-1}\gamma
x)\right|<\sum_{\lambda}C_{\phi_{n,S,t,\zeta}}$

or,

$\displaystyle\left|d_{\tau}\text{Tr}((\widetilde{\phi_{n,S,t,\zeta}}\star\phi_{n,S,t,\zeta})(e))\text{Vol}(\Omega)\right|-c|Z|t^{-d+1/2}<\sum_{\lambda}C_{\phi_{n,S,t,\zeta}}$

or,

$\displaystyle\left|t^{d}\widetilde{\Phi_{n,S,t,\zeta}^{\sharp}}\star\Phi_{n,S,t,\zeta}^{\sharp}(e)\text{Vol}(\Omega)\right|-c|Z|t^{1/2}<t^{d}\sum_{\lambda}C_{\phi_{n,S,t,\zeta}}.$

Here,
$C_{\phi_{n,S,t,\zeta}}=(e_{\lambda}\star\phi_{n,S,t,\zeta},e_{\lambda}\star\phi_{n,S,t,\zeta})$.
From (21) and [BM, Lemma 3.3] we have that

$\displaystyle\sum\limits_{\lambda}(e_{\lambda}\star\phi_{n,S,t,\zeta},e_{\lambda}\star\phi_{n,S,t,\zeta})=\sum\limits_{\Pi_{\text{cusp}}(G(\mathbb{Q}_{S}),\tau)}m(\pi)\left|\left|\pi(\Phi_{n,S,t,\zeta}^{\sharp})\right|\right|_{\text{HS}}^{2}.$
(25)

The multiplicities of $\pi_{\infty}$ can be written as:

$\displaystyle
m(\pi_{\infty})=\sum\limits_{\Pi_{\text{cusp}}(G(\mathbb{Q}_{S}),\tau)}^{{}^{\prime}}m(\pi^{^{\prime}})\text{dim}\mathcal{H}_{\pi^{^{\prime}}}^{K_{S^{\prime}}^{{}^{\prime}}},$
(26)

where the sum is over $\pi^{^{\prime}}$ whose Archimedean component is
$\pi_{\infty}$. If we let $n\to\infty$, we have that
$\pi(\Phi_{n,t,S,\zeta}^{\sharp})\to\text{dim}\mathcal{H}_{\pi^{^{\prime}}}^{K_{S^{\prime}}^{{}^{\prime}}}\pi_{\infty}(H_{\infty,t,\zeta}^{\sharp})$.
Hence rewriting the $(25)$ we get

$\displaystyle\sum\limits_{\lambda}(e_{\lambda}\star\phi_{n,S,t,\zeta},e_{\lambda}\star\phi_{n,S,t,\zeta})=\sum\limits_{\Pi_{\text{cusp}}(G(\mathbb{R}),\tau)}m(\pi_{\infty})\left|\left|\pi_{\infty}\Bigg{(}H_{\infty,t,\zeta}^{\sharp}\Bigg{)}\right|\right|_{\text{HS}}^{2}.$
(27)

The sum on the right hand side of (27), could be divided into two parts,
$\lVert\lambda_{\pi_{\infty}}\rVert^{2}\leq t^{-2}$ and
$\lVert\lambda_{\pi_{\infty}}\rVert^{2}\geq t^{-2}$. Here
$\lambda_{\pi_{\infty}}$ denotes the infinitesimal character of
$\pi_{\infty}$. The representations $\pi_{\infty}$ is a subrepresentations of
a non-unitary principle series representation with parameters
$\omega_{\infty}\otimes\nu_{\infty}\otimes 1$. The infinitesimal character of
$\lambda_{\pi_{\infty}}$ can be written as
$\nu_{\omega_{\infty}}+\nu_{\infty}$, where $\nu_{\omega_{\infty}}$ is the
infinitesimal character of $\omega_{\infty}$ [Kn, Prop. 8.22]. Let
$d_{\omega_{\infty}}$ be the formal degree of $\omega_{\infty}$. We have the
following inequality of Hilbert-Schmidt norm in terms of Fourier transform
$\widehat{H_{\infty,t,\zeta}^{\sharp}}(\omega_{\infty},\nu_{\infty})$:

$\displaystyle\left|\left|\widehat{H_{\infty,t,\zeta}^{\sharp}}(\omega_{\infty},\nu_{\infty})\right|\right|^{2}\geq
d_{\omega_{\infty}}\left|\left|\pi_{\infty}\Bigg{(}H_{\infty,t,\zeta}^{\sharp}\Bigg{)}\right|\right|_{\text{HS}}^{2}.$

Using the choice of the Schwartz function the right hand side of (27) is
bounded by

$\displaystyle\sum\limits_{\lVert\nu_{\omega_{\infty}}+\nu_{\infty}\rVert\leq
t^{-2}}d_{\tau}m(\pi_{\infty})\text{dim}(\text{Hom}(\mathcal{H}_{\pi_{\infty}}(\tau),V_{\tau}))\left|\psi(t\nu_{\infty})\right|^{2}$
(28)

Hence, using our earlier notations, we have

$\displaystyle\sum\limits_{\lVert\nu_{\omega_{\infty}}+\nu_{\infty}\rVert\leq
t^{-2}}d_{\tau}m(\pi_{\infty})\text{dim}(\text{Hom}_{K_{\infty}}(\mathcal{H}_{\pi_{\infty}}(\tau),V_{\tau}))\left|\psi(t\nu_{\infty})\right|^{2}$
$\displaystyle\leq d_{\tau}N_{\text{cusp}}^{\Gamma}(t^{-2},\tau)$

##  9\. Main Theorem

In this last section we put all our earlier result together to prove pur main
asymptotic formula. Suppose $\Delta_{\tau}$ is the self-adjoint Casimir
operator acting on $L_{\text{cusp}}^{2}(\Gamma\backslash G_{S},\tau)$ with
pure point spectrum
$0<\nu_{0}(\tau)\leq\nu_{1}(\tau)\leq\nu_{2}(\tau)...\to\infty$ Let
$\mathcal{E}(\nu_{i}(\tau))$ denote the space of eigenvectors with eigenvalue
$\nu_{i}(\tau)$. Define

$N_{\text{cusp}}^{\Gamma}(T^{2},\tau)=\sum_{\nu_{i}(\tau)\leq
T^{2}}\text{dim}\mathcal{E}(\nu_{i}(\tau)).$

Let $M$ be a Riemannian Manifold. Suppose $C(M)$ denotes the product of volume
of $M$, the volume of the Euclidean unit ball in $\mathbb{R}^{\text{dim}(M)}$
and $(2\pi)^{-\text{dim}(M)}$. Collecting all the results in the previous
section, we prove the following:

######  Theorem.

Let $G$ be a semi-simple, connected, algebraic group over $\mathbb{Q}_{S}$.
Assume that $G$ is also split and adjoint type. Let $\Gamma\subset
G(\mathbb{Z}[S^{-1}])$ be a congruence subgroup with no torsion element. Let
$X_{\infty}=G_{\infty}/K_{\infty}$ and $d=\text{dim}_{\mathbb{R}}X_{\infty}$.
Let $\tau$ be an irreducible representation of $K_{\infty}$ of dimension
$d_{\tau}$. Then there exists a constant $C(\Gamma\backslash X_{\infty})>0$,
such that

$N_{\text{cusp}}^{\Gamma}(T^{2},\tau)\sim d_{\tau}C(\Gamma\backslash
X_{\infty})T^{d}\quad\text{as}\quad T\to\infty.$

###### Proof.

We make a change of variable $t=\frac{1}{T}$, and prove the asymptotic as
$\limsup_{t\to 0}$. Let us apply the partial trace formula in $(17)$ with
$\phi$ being the test function
$\phi_{n,S,t,\zeta}\star\widetilde{\phi_{n,S,t,\zeta}}$ in Section 8. Taking
the limit as $n\to\infty$ and using $(21)$ the inequality becomes

$\displaystyle\text{Tr}((h_{S,t,\zeta}\star\widetilde{h_{S,t,\zeta}})(e))\text{Vol}(\Omega)+\sum_{\gamma\in
Z}\int_{\Omega}\text{Tr}(h_{S,t,\zeta}\star\widetilde{h_{S,t,\zeta}})(x^{-1}\gamma
x)\leq N_{\text{cusp}}^{\Gamma}(t^{-2},\tau).$ (29)

Now from Section 8 and Lemma 7 we can conclude that the term corresponding to
the identity class will be asymptotic to
$d_{\tau}\alpha(G)t^{-d}\text{Vol}(\Gamma\backslash G_{S})$ as $\limsup_{t\to
0}$ and as $\lim_{n\to\infty}$. And from $(24)$ section 9 we can show that as
we take $\limsup_{t\to 0}$ the terms corresponding to non-identity class will
converge to 0. This is done exactly as in the proof of the lower bound in the
Weyl law, [LV, page 243]. There exist $\Gamma_{\infty,i}$, for finitely many
$i$ such that

$\Gamma\backslash G_{S}/K_{S}=\bigcup\limits_{i}\Gamma_{\infty,i}\backslash
G(\mathbb{R})/K_{\infty}.$

For each $i$, let $N_{i}^{\Gamma}(T,\tau)$ be the eigenvalue counting function
for the space $\Gamma_{\infty,i}\backslash G(\mathbb{R})/K_{\infty}$. That
this same asymptotic term along with the constant
$C(\Gamma_{\infty,i}\backslash X_{\infty})$ is an upper bound for the right
hand side has been proved by in greater generality by Donnelly [Do]. To prove,

$\alpha(G)\text{Vol}(\Gamma\backslash
G_{S})=\sum\limits_{i}C(\Gamma_{\infty,i}\backslash X_{\infty}),$

we argue as in [LV, Sec.6.3]. Therefore it establishes the asymptotic formula
in the statement of the theorem.

∎

### Declaration

The author declares that there is no conflict of interest. Any dataset that
may have been generated during the current study is available from the author.

## References

  * [Ar1] J. Arthur, Harmonic Analysis of the Schwartz space on a Reductive Lie group I and II, Preprint.
”http://www.math.toronto.edu/arthur/pdf/20160115093856221.pdf”

  * [Ar2] J. Arthur, A theorem on the Schwartz space of a reductive Lie group, Proc. Nat. Acad. Sci. USA 72, No. 12 (1975), 4718-4719.

  * [Ar3] J. Arthur, A Paley-Wiener theorem for real reductive groups, Acta Mathematica 150 (1983), 1-89
”http://www.math.toronto.edu/arthur/pdf/15.pdf”

  * [Ar4] J. Arthur, The Trace Paley Wiener Theorem for Schwartz Functions, Contemporary Mathematics 177 (1994), 171-180.

  * [B] J. Bernstein. Draft of : Representations of p-adic groups, Fall 1992. Lectures by Joseph Bernstein, Written by Karl E. Rumelhart

  * [BDK] J. Bernstein, P. Deligne, D. Kazhdan, Trace Paley-Wiener theorem for reductive p-adic groups, J. d’Analyse Math. 47 (1986), 180–192.

  * [BM] D. Barbasch, H. Moscovici, L2-Index and the Selberg Trace Formula” by in Journal Of Functional Analysis 53 (1983), 151-201.

  * [BU] J. Buttcane, Kuznetsov, Petersson and Weyl on GL(3), II: The generalized principal series forms, Mathematische Annalen volume 380, pages231–266 (2021)

  * [Bmp] D. Bump, Automorphic forms and Representations, Cambridge studies in Advanced Mathematics 55.

  * [Cmp1] R. Camporesi, The spherical transform for homogeneous vector bundles over Riemannian symmetric spaces, Journal of Lie Theory 7 (1997) 29–60.

  * [Cmp2] R. Camporesi, The Helgason Fourier Transform For Homogeneous Vector Bundles Over Riemannian Symmetric Spaces, Pacific journal of mathematics 179, No. 2, 1997.

  * [CH] D. Ciubotaru, X. He, Cocenters of p-adic Groups, III: Elliptic and Rigid Cocenters, Peking Mathematical Journal 4(2) 2017.

  * [DKV] J. Duistermaat, J. Kolk, V. Varadarajan Functions, flows and oscillatory integrals on flag manifolds and conjugacy classes in real semisimple lie groups, Compositio Mathematica, tome 49, no.3 (1983) 309-398.

  * [Dui] J. Duistermaat, The heat Kernel Lefschetz fixed point formula for the spin-C dirac operator, Modern Birkhäuser Classics.

  * [Do] H. Donnelly, On the cuspidal spectrum for finite volume symmetric spaces, J. Differential Geom. 17 (1982), 239–253.

  * [Ef] I. Efrat, The Selberg Trace Formula for $PSL_{2}(\mathbb{R})$, Mem. Amer. Math. Soc. 65, no. 359 (1987), A. M. S., Providence, RI.

  * [FK] Y. Flicker, D. Kazhdan, A simple trace formula, J. Analyse Math. 50 (1988), 189–200.

  * [GV] R. Gangolli, V.S. Varadarajan Harmonic Analysis of Spherical Functions on Real Reductive Groups, Springer-Verlag Berlin Heidelberg New York London Paris Tokyo.

  * [HC1] Harish-Chandra, Harmonic Analysis on Real Reductive Groups III. The Maass-Selberg Relations and the Plancherel Formula, Annals of Mathematics, Second Series, Vol. 104, No. 1 (Jul., 1976), 117-201

  * [HC2] Harish-Chandra, ”On the theory of the Eisenstein integral” in Conference on Harmonic Analysis (Univ. Maryland, College Park), Lecture Notes in Mathematics 266 (1971), Berlin, New York: Springer-Verlag, 123–149.

  * [HC3] Harish-Chandra, Collected Papers, Springer, Berlin 1984.

  * [Kn] A. Knapp, Representation theory of Semisimple groups, an Overview based on examples,Princeton University Press, Princeton, New jersey, 1986.

  * [La] R. Langlands, On the Functional Equations Satisfied by Eisenstein Series, Lecture Notes in Math. 544, Springer-Verlag, New York, 1976.

  * [LM] J.-P. Labesse, W. Müller, Weak Weyl’s law for congruence subgroups, math.RT/0404037.

  * [LM1] E. Lapid, W. Müller, Spectral asymptotics for arithmetic quotients of $SL(n,\mathbb{R})/SO(n,\mathbb{R})$,Duke Mathematical Journal 149 (2009).

  * [LV] E. Lindenstrauss, A. Venkatesh, Existence and Weyl’s law for spherical cusp forms,GAFA, Geom. func. anal. 17 (2007) 220-251.

  * [Rez] A. Reznikov, Eisenstein matrix and existence of cusp forms in rank one symmetric spaces, Geom. Funct. Anal. 3 (1993), 79–105.

  * [Mi] S. Miller, On the existence and temperedness of cusp forms for $SL(3,\mathbb{Z})$. J. Reine Angew. Math. 533 (2001), 127–169.

  * [Mü1] W. Müller, Eigenvalue estimates for locally symmetric spaces of finite volume, in Partial Differential Equations (Holzhau, 1988) 179–196, Teubner-Texte Math. 112, Leipzig, 1989.

  * [Mü2] W. Müller, Weyl’s law for the cuspidal spectrum of $SL(n)$, Annals of Mathematics, 165 (2007), 275–333.

  * [MT] A. Moy, M. Tadic The Bernstein Center In Terms Of Invariant Locally Integrable Functions, Representation Theory An Electronic Journal of the American Mathematical Society 6, (November 19, 2002), 313–329.

  * [PR] V. Platonov, A. Rapinchuk, Rachel Rowen, Algebraic Groups and Number Theory, Academic Press, Dec 7, 1993

  * [Sa] P. Sarnak, On cusp forms, in The Selberg Trace Formula and Related Topics (Brunswick, Maine, 1984) (D. Hejhal et al., eds.), Contemp. Math. 53, 393–407.

  * [Se] A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. (N.S.) 20 (1956), 47–87.

  * [Wal] N. Wallach, Real Reductive Groups I, Volume 132,1st Edition, Academy Press, 1988

  * [W] G. Warner, Harmonic Analysis on Semi-Simple Lie Groups II, Springer-Verlag Berlin Heidelberg New York 1972.