# Von Neumann Dimensions and Trace Formulas I: Limit Multiplicities

Jun Yang

###### Abstract.

Given a connected semisimple Lie group $G$ and an arithmetic subgroup
$\Gamma$, it is well-known that each irreducible representation $\pi$ of $G$
occurs in the discrete spectrum $L^{2}_{\text{disc}}(\Gamma\backslash G)$ of
$L^{2}(\Gamma\backslash G)$ with at most a finite multiplicity
$m_{\Gamma}(\pi)$. While $m_{\Gamma}(\pi)$ is unknown in general, we are
interested in its limit as $\Gamma$ is taken to be in a tower of lattices
$\Gamma_{1}\supset\Gamma_{2}\supset\dots$. For a bounded measurable subset $X$
of the unitary dual $\widehat{G}$, we let $m_{\Gamma_{n}}(X)$ be the sum of
the multiplicity $m_{\Gamma_{n}}(\pi)$ over all $\pi$ in $X$. Let $H_{X}$ be
the direct integral of the irreducible representations in $X$ with respect to
the Plancherel measure of $G$, which is also a module over the group von
Neumann algebra $\mathcal{L}\Gamma_{n}$. We prove:

$\lim\limits_{n\to\infty}\cfrac{m_{\Gamma_{n}}(X)}{\dim_{\mathcal{L}\Gamma_{n}}H_{X}}=1$,

for any bounded subset $X$ of $\widehat{G}$, when i) $\Gamma_{n}$’s are
cocompact, or, ii) $G=\operatorname{SL}(n,\mathbb{R})$ and $\\{\Gamma_{n}\\}$
are principal congruence subgroups.

††AMS 2010 Mathematics Subject Classification: 46L10, 20G05, 20G35. ††✉ Jun
Yang<EMAIL_ADDRESS>Harvard University, Cambridge, MA, USA††This work
was supported in part by the ARO Grant W911NF-19-1-0302 and the ARO MURI Grant
W911NF-20-1-0082.

###### Contents

  1. 1 Introduction: an example on $\operatorname{SL}(2,\mathbb{R})$
  2. 2 The trace formulas and dominant terms
    1. 2.1 The Selberg trace formula
    2. 2.2 The Arthur trace formula
    3. 2.3 The dominant term on the geometric side
  3. 3 The multiplicities problem
    1. 3.1 The multiplicities in $L^{2}(\Gamma\backslash G)$
    2. 3.2 Sauvageot’s density theorems
  4. 4 The von Neumann dimensions of direct integrals
    1. 4.1 The group von Neumann algebra and the trace
    2. 4.2 A theorem of von Neumann dimension
    3. 4.3 The proof of the main theorem

##  1\. Introduction: an example on $\operatorname{SL}(2,\mathbb{R})$

In this section, we introduce a multiplicity problem of square-integrable
irreducible representations of $G=\operatorname{SL}(2,\mathbb{R})$ on
$L^{2}_{\text{cusp}}(\Gamma\backslash G)$ for some arithmetic subgroups
$\Gamma$. It is one of the motivations of this article.

We first $\Gamma=\operatorname{SL}(2,\mathbb{Z})$ and $\Gamma(N)$ be the
principal congruence subgroup of level $n$ defined by

$\Gamma(N)=\Bigl{\\{}\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\operatorname{SL}(2,\mathbb{Z}):a,d\equiv
1\pmod{N},b,c\equiv 0\pmod{N}\Bigl{\\}}$.

Consider the right quasi-regular representation of $G$ on
$L^{2}(\Gamma(N)\backslash G)$ given by $(R(g)\phi)(x)=\phi(xg)$ for $\phi\in
L^{2}(\Gamma(N)\backslash G)$, $g\in G$. It is well known (see [28]) to be
reducible and can be decomposed as

$L^{2}(\Gamma(N)\backslash G)=L^{2}_{\text{cusp}}(\Gamma(N)\backslash G)\oplus
L^{2}_{\text{cont}}(\Gamma(N)\backslash G)\oplus\mathbb{C}$.

Here $L^{2}_{\text{cusp}}(\Gamma(N)\backslash G)$ is the cuspidal part, which
is a direct sum of irreducible representations with finite multiplicities,
i.e.,

$L^{2}_{\text{cusp}}(\Gamma(N)\backslash G)=\sum m_{\Gamma(N)}(\pi)\cdot\pi$,
$m_{\Gamma(N)}(\pi)<\infty$ for each $\pi$,

and $L^{2}_{\text{cont}}(\Gamma(N)\backslash G)$ is a direct integral of
irreducible representations given by the Eisenstein series.

The multiplicities $m_{\Gamma(N)}(\pi)$ are still unknown in general, except
for some special families of irreducible representations including the
discrete series of $\operatorname{SL}(2,\mathbb{R})$ (see [26] for an
introduction of discrete series). Let $S_{k}(\Gamma)$ be the space of cusp
forms of weight $k$ for a Fuchsian group $\Gamma$. We have the following
result (see [22] Theorem 2.10).

######  Lemma 1.1.

For the discrete series $\pi_{k}$, we have $m_{\Gamma(N)}(\pi_{k})=\dim
S_{k}(\Gamma(N))$.

By applying the dimension formulas of cusp forms (see [14] Chapter 3.9), we
obtain

(1)
$m_{\Gamma(N)}(\pi_{k})=(\frac{k-1}{24}-\frac{1}{4N})N^{3}\prod_{p|N}(1-\frac{1}{p^{2}})$

for all $N>2$.

On the other hand, let $H_{k}$ be the underlying Hilbert space of the discrete
series $\pi_{k}$. As $H_{k}$ is a module over the group $\Gamma(N)$, we can
further prove that it is also a module over the group von Neumann algebra
$\mathcal{L}(\Gamma(N))$ (see Section 4.1 for the definition). Hence $H_{k}$
has a von Neumann dimension $\dim_{\mathcal{L}(\Gamma(N))}H_{k}$ over
$\mathcal{L}(\Gamma(N))$. Indeed, if a discrete group $\Gamma$ is ICC
(infinite conjugacy class, see also 4.1), this dimension totally determines
the equivalence class of $\mathcal{L}(\Gamma)$-module, i.e.,
$\dim_{\mathcal{L}(\Gamma)}H_{1}=\dim_{\mathcal{L}(\Gamma)}H_{2}$ if and only
if $H_{1},H_{2}$ are isomorphic as $\mathcal{L}(\Gamma)$-module.

We consider a lattice $\Gamma$ in a Lie group $G$. Suppose $(\pi,H)$ is a
discrete series representation of $G$ and let $d(\pi)$ be the formal dimension
of $\pi$ (see [33] Chapter 16). We have

######  Lemma 1.2 (Goodman-la Harpe-Jones[23]).

$\dim_{\mathcal{L}(\Gamma)}H=\operatorname{vol}(\Gamma\backslash G)\cdot
d(\pi)$

By Example 3.3.4 in [23], we know
$\dim_{\mathcal{L}(\operatorname{PSL}(2,\mathbb{Z}))}H_{k}=\frac{k-1}{12}$. As
$\operatorname{SL}(2,\mathbb{Z})=(\mathbb{Z}/2\mathbb{Z})\rtimes\operatorname{PSL}(2,\mathbb{Z})$,
we have
$\dim_{\mathcal{L}(\operatorname{SL}(2,\mathbb{Z}))}H_{k}=\frac{k-1}{24}$.
Since $[\Gamma\colon\Gamma(n)]=N^{3}\prod_{p|N}(1-\frac{1}{p^{2}})$, we can
conclude

(2)
$\dim_{\mathcal{L}(\Gamma(N))}H_{k}=\frac{k-1}{24}N^{3}\prod_{p|N}(1-\frac{1}{p^{2}}).$

Thus we obtain:

######  Corollary 1.3.

For a discrete series $(\pi_{k},H_{k})$ of $\operatorname{SL}(2,\mathbb{R})$,
we have

$\lim_{N\to\infty}\cfrac{m_{\Gamma(N)}(\pi_{k})}{\dim_{\mathcal{L}(\Gamma(N))}H_{k}}=1$.

Proof: Comparing Equations 1 and 2, we obtain
$\frac{m_{\Gamma(N)}(\pi_{k})}{\dim_{\mathcal{L}(\Gamma(N))}H_{k}}=\frac{k-1-6/N}{k-1}$
and then take the limit.

While the explicit multiplicities of most irreducible representations are
still unknown, the limit multiplicities have been studied since 1970s. In the
case of towers of uniform lattices, DeGeorge and Wallach got the first results
for discrete series of Lie groups [11] and later for bounded sets of
irreducible representations in rank one group [12]. Delorme [13] finally
solved the problem for bounded sets of irreducible representations in all Lie
groups. See also [1] for a recent approach.

For the non-uniform lattices (or most arithmetic subgroups), Savin [36]
obtained the results on discrete series in his thesis at first, which is based
on the work by Rohlfs and Speh [34]. Then Deitmar and Hoffmann proved the
results on certain towers of arithmetic subgroups in rank one group. Recently,
Finis and Lapid solved the case of congruence subgroups in
$\operatorname{SL}(n,\mathbb{R})$ [20, 17], which are based on their study of
the spectral side of Arthur’s trace formulas [18, 16].

The goal of this paper is to extend Corollary 1.3 to some general settings. In
the rest of this paper, we generalize this result mainly in the following
aspects:

  1. (1)

from a single discrete series representation to any bounded subset of the
unitary dual $\widehat{G}$ of $G$;

  2. (2)

from $\operatorname{SL}(2,\mathbb{R})$ to the towers of uniform lattices in an
arbitrary semisimple Lie group,

  3. (3)

from $\operatorname{SL}(2,\mathbb{R})$ to $\operatorname{SL}(n,\mathbb{R})$
with its the principal congruence subgroups.

Finally, we are able to prove:

######  Theorem 1.4 (The Main Theorem).

Let $G$ be a semisimple simply-connected Lie group. Let $X$ be a bounded
subset of the unitary dual of $G$ and $H_{X}$ be the direct integral of the
irreducible representations of $G$ in $X$. We have:

$\lim\limits_{n\to\infty}\cfrac{m_{\Gamma_{n}}(X)}{\dim_{\mathcal{L}\Gamma_{n}}H_{X}}=1$,

when i) $\Gamma_{n}$’s are cocompact, or ii)
$G=\operatorname{SL}(n,\mathbb{R})$ and $\\{\Gamma_{n}\\}$ are principal
congruence subgroups.

##  2\. The trace formulas and dominant terms

We have a brief review of the Arthur-Selberg trace formulas and give the
dominant terms in these formulas. We mainly follow [3, 4, 19].

Let $\mathbf{G}$ be a reductive group over $\mathbb{Q}$. The group
$G(\mathbb{A})$ acts naturally on $L^{2}(G(\mathbb{Q})\backslash
G(\mathbb{A}))$ by

$R(g)\phi(x)=\phi(xg)$

for $\phi\in L^{2}(G(\mathbb{Q})\backslash G(\mathbb{A}))$ and $g\in
G(\mathbb{A}))$. Let $C^{\infty}_{\text{c}}{(G(\mathbb{A}))}$ be the complex
algebra of smooth, compactly supported function on $G(\mathbb{A}))$. Given
$f\in C^{\infty}_{\text{c}}{(G(\mathbb{A}))}$, we may define

$(R(f)\phi)(x)=\int_{G(\mathbb{A}))}f(g)R(g)\phi(x)dg=\int_{G(\mathbb{A}))}f(g)\phi(xg)dg$.

If we define the kernel

$K(x,y)=K_{f}(x,y)\colon=\sum\limits_{\gamma\in G(\mathbb{Q})}f(x^{-1}\gamma
y)$,

we have $(R(f)\phi)(x)=\int_{G(\mathbb{Q})\backslash
G(\mathbb{A})}K(x,y)\phi(y)dy$.

###  2.1. The Selberg trace formula

We first assume $\mathbf{G}$ is anisotropic and hence the quotient space
$G(\mathbb{Q})\backslash G(\mathbb{A})$ is compact. Let $\mathcal{O}$ be the
set of conjugacy classes in $G(\mathbb{Q})$ and $o\in\mathcal{O}$ be a
conjugacy class. We may define

$K_{o}(x,y)=\sum\limits_{\gamma\in o}f(x^{-1}\gamma y)$

and obtain $K(x,y)=\sum\limits_{o\in\mathcal{O}}K_{o}(x,y)$. On the other
hand, the representation $R$ decomposes into a direct sum of irreducible
representations with finite multiplicities, i.e.,
$L^{2}(G(\mathbb{Q})\backslash
G(\mathbb{A}))=\oplus_{\chi\in\mathcal{X}}L^{2}(G(\mathbb{Q})\backslash
G(\mathbb{A}))_{\chi}$. Here $L^{2}(G(\mathbb{Q})\backslash
G(\mathbb{A}))_{\chi}=m(\chi)\cdot\chi$, which is $m(\chi)$ copies of the
irreducible representation $\chi$. Assume $\mathcal{B}_{\chi}$ is a
orthonormal basis of $L^{2}(G(\mathbb{Q})\backslash G(\mathbb{A}))_{\chi}$.
Then

$K_{\chi}(x,y)=K_{f,\chi}(x,y)\colon=\sum_{\phi\in\mathcal{B}_{\chi}}(R(f))\phi(x)\cdot\overline{\phi(y)}$

converges. Now we let

  1. (1)

$k_{\chi}(x,f)=K_{\chi}(x,x)$ and $J_{\chi}(f)=\int_{G(\mathbb{Q})\backslash
G(\mathbb{A})}k_{\chi}(x,f)dx$,

  2. (2)

$k_{o}(x,f)=K_{o}(x,x)$ and $J_{o}(f)=\int_{G(\mathbb{Q})\backslash
G(\mathbb{A})}k_{o}(x,f)dx$.

If we let ${\gamma}$ be a representatives of $o\in\mathcal{O}$ and
$H_{\gamma}=\\{h\in H|h\gamma h^{-1}=\gamma\\}$ for a group $H$ containing
$\gamma$, we get

$J_{o}(f)=\operatorname{vol}(G(\mathbb{Q})_{\gamma}\backslash
G(\mathbb{A})_{\gamma})\int_{G(\mathbb{A})_{\gamma}\backslash
G(\mathbb{A})}f(x^{-1}\gamma x)dx$.

######  Theorem 2.1.

Assuming $G(\mathbb{Q})\backslash G(\mathbb{A})$ is compact, we have

(3) $\displaystyle\operatorname{tr}R(f)=\sum\limits_{o\in
O}J_{o}(f)=\sum\limits_{\chi\in\mathcal{X}}J_{\chi}(f)$

for any $f\in C^{\infty}_{c}{(G(\mathbb{A}))}$.

For the classical setting, we start with a real Lie group $G$ and a lattice
$\Gamma\subset G$. Consider the representation $R_{\Gamma}$ of $G$ on
$L^{2}(\Gamma\backslash G)$ given by $(R_{\Gamma}(g)\phi)(x)=\phi(xg)$ for
$x,g\in G$. Let $C^{\infty}_{c}(G)$ be the space of smooth function on $G$
with compact support. For $f\in C^{\infty}_{c}(G)$ and a representation
$(\pi,H)$ of $G$, we let

$\pi(f)v=\int_{G}f(g)\pi(g)vdg$.

If $\pi$ is irreducible, $\pi(f)$ is a trace class operator and we let
$\theta_{\pi}(f)=\operatorname{tr}\pi(f)$. Note for the representation
$R_{\Gamma}$, we have
$(R_{\Gamma}(f)\phi)(x)=\int_{G}f(g)R_{\Gamma}(g)\phi(x)dg=\int_{G}f(g)\phi(xg)dg$.

It is known that $\Gamma\backslash G$ is compact if and only if the reductive
part of $\mathbf{G}$ is anisotropic (see [32] Theorem 4.12). In this case,
$L^{2}(\Gamma\backslash G)$ can be decomposed into a direct sum of irreducible
representations of $G$ with each of finite multiplicity, i.e.,

$L^{2}(\Gamma\backslash G)=\oplus m_{\Gamma}(\pi)\cdot\pi$

with $m_{\Gamma}(\pi)=\dim\operatorname{Hom}_{G}(\pi,L^{2}(\Gamma\backslash
G))<\infty$ for each $\pi$.

By taking the test function in Theorem 2.1 to be $f\otimes 1_{K}$ for a
maximal compact subgroup $K$ of $G(\mathbb{A}^{\text{fin}})$ with $f\in
C^{\infty}_{c}(G)$ (see Section 3.1), we get the following result for the
lattice $\Gamma$ in the real Lie group $G$.

######  Corollary 2.2 (The Selberg trace formula).

If $\Gamma\backslash G$ is compact, $R_{\Gamma}(f)$ is of trace class and

(4)
$trR_{\Gamma}(f)=\sum_{\pi\in\widehat{G}}m_{\Gamma}(\pi)\theta_{\pi}(f)=\sum_{\gamma\in[\Gamma]}\operatorname{vol}(\Gamma_{\gamma}\backslash
G_{\gamma})\int_{\Gamma_{\gamma}\backslash G}f(x^{-1}\gamma x)dx$

###  2.2. The Arthur trace formula

We assume $\mathbf{G}$ is not necessarily anisotropic and
$G(\mathbb{Q})\backslash G(\mathbb{A})$ may not be compact. Assume $B$ is a
Borel subgroup defined over $\mathbb{Q}$, $M_{0}$ is a Levi factor of $B$
defined over $\mathbb{Q}$, $P$ is a standard parabolic subgroup defined over
$\mathbb{Q}$ (i.e., $P_{0}=B\subset P$), $N_{P}=R_{u}(P)$ (the unipotent
radical of $P$), $M_{P}$ is the unique Levi component of $P$ such that
$M_{0}\subset M_{P}$.

We also assume $A_{P}$ is the split component of the center of $M_{P}$ and
$Z=A_{G}$, $\Delta_{0}=\Delta_{B}$ is a base for a root system. We will mostly
use the notations of [3, 4, 6] and [21] as follows:

  * •

$\mathfrak{a}_{P}=\operatorname{Hom}(X(M_{P})_{\mathbb{Q}},\mathbb{R})$ where
$X(M_{P})_{\mathbb{Q}}$ is the $\mathbb{Q}$-characters of $M_{P}$,
$\mathfrak{a}_{P}^{*}=X(M_{P})_{\mathbb{Q}}\otimes\mathbb{R}$ and
$\mathfrak{a}_{P}^{+}=\\{H\in\mathfrak{a}_{P}|\alpha(H)>0,\forall\alpha\in\Delta_{P}\\}$.

  * •

$\gamma=\gamma_{s}\gamma_{u}$, which is the decomposition such that
$\gamma_{s}$ is semisimple and $\gamma_{u}$ is unipotent.

  * •

$\mathcal{O}$ is the set of $G(\mathbb{Q})$-semisimple conjugacy class of
$G(\mathbb{Q})$ ($\gamma\cong\beta$ if $\gamma_{s}$ and $\beta_{s}$ are
$G(\mathbb{Q})$-conjugate).

  * •

$o\in\mathcal{O}$ is a conjugacy class in $G(\mathbb{Q})$.

  * •

$\mathcal{X}$ is the set of equivalence classes of pairs $(M,\rho)$, where $M$
is a Levi subgroup of $G$ and $\rho\in\widehat{M(\mathbb{A})^{1}}$
($(M,\rho)\sim(M^{\prime},\rho^{\prime})$ if there is an
$s\in\Omega(\mathfrak{a},\mathfrak{a}^{\prime})$ such that the representation
$(s\rho)(m^{\prime})=\rho(w_{s}^{-1}mw_{s})$ is unitarily equivalent to
$\rho^{\prime}$).

  * •

For a pair of parabolic groups $P_{1}\subset P_{2}$, $\Delta^{P_{2}}_{P_{1}}$
is the set of simple roots of $(M_{P_{2}}\cap P_{1},A_{P_{1}})$ and
$\hat{\Delta}^{P_{2}}_{P_{1}}=\\{\varpi_{\alpha}|\alpha\in\Delta^{P_{2}}_{P_{1}}\\}$,
i.e, the dual basis for $\Delta^{P_{2}}_{P_{1}}$.

  * •

$\hat{\tau}_{P}$ is the characteristic function on $\mathfrak{a}_{0}$ of
$\\{H\in\mathfrak{a}_{0}|\varpi(H)>0,\varpi\in\hat{\Delta}^{G}_{P}\\}$.

  * •

For $m=\prod_{v}m_{v}\in M(\mathbb{A})$, let $H_{M}(m)\in\mathfrak{a}_{p}$
given by

$e^{\langle
H_{M}(m),\chi\rangle}=|\chi(m)|=\prod\limits_{v}|\chi(m_{v})|_{v}$,
$\forall\chi\in X(M)_{\mathbb{Q}}$.

  * •

$x=nmak\in G(\mathbb{A})$ with $n\in G(\mathbb{A}),m\in M(\mathbb{A})^{1},a\in
A(\mathbb{R})^{0}$ and $k\in K$.

  * •

$H(x)=H_{M}(ma)=H_{M}(a)\in\mathfrak{a}_{p}$.

Let $T\in\mathfrak{a}_{0}^{+}$ be suitably regular, i.e., $\alpha(T)$ is
sufficiently large for all $\alpha\in\Delta_{0}$. For a parabolic subgroup
$P$, there are kernels $K_{P,o}=\sum\limits_{\gamma\in M(\mathbb{Q})\cap
o}\int_{N(\mathbb{A})}f(x^{-1}\gamma ny)dn$ and $K_{P,\chi}$ (see [3] p.923
and p.935 for the precise definitions). Then Arthur is able to define the
truncated kernels and distributions $J^{T}_{o},J^{T}_{\chi}$ as follows:

  1. (1)

$k^{T}_{o}(x,f)=\sum\limits_{P}(-1)\dim(A_{P}/Z)\sum\limits_{\delta\in
P(\mathbb{Q})\backslash G(\mathbb{Q})}K_{P,o}(\delta x,\delta
x)\cdot\hat{\tau}_{p}(H(\delta x)-T)$.

  2. (2)

$k^{T}_{\chi}(x,f)=\sum\limits_{P}(-1)\dim(A_{P}/Z)\sum\limits_{\delta\in
P(\mathbb{Q})\backslash G(\mathbb{Q})}K_{P,\chi}(\delta x,\delta
x)\cdot\hat{\tau}_{p}(H(\delta x)-T)$.

  3. (3)

$J^{T}_{o}(f)=\int_{G(\mathbb{Q})\backslash
G(\mathbb{A})^{1}}k^{T}_{o}(x,f)dx$.

  4. (4)

$J^{T}_{\chi}(f)=\int_{G(\mathbb{Q})\backslash
G(\mathbb{A})^{1}}k^{T}_{\chi}(x,f)dx$.

Let $\mathcal{X}(G)=\\{(M,\rho)\in\mathcal{X}|M=G\\}$. We reach a coarse trace
formula, which is firstly given in [4] Chapter 5.

######  Theorem 2.3 (The Arthur trace formula).

For any $f\in C^{\infty}_{c}(G(\mathbb{A})^{1})$ and any suitably regular
$T\in\mathfrak{a}_{0}^{+}$, we have

(5)
$\displaystyle\sum\limits_{o\in\mathcal{O}}J^{T}_{o}(f)=\sum\limits_{\chi\in\mathcal{X}}J^{T}_{\chi}(f)$

Moreover, the trace formula of $R(f)$ is given by

$\operatorname{tr}R_{\text{cusp}}(f)=\sum\limits_{o\in\mathcal{O}}J^{T}_{o}(f)-\sum\limits_{\chi\in\mathcal{X}\backslash\mathcal{X}(G)}J^{T}_{\chi}(f)$.

###  2.3. The dominant term on the geometric side

Consider the adelic case at first. Let $F$ be a number field and
$V,V_{\infty}$ and $V_{f}$ be the set of places, Archimedean and non-
Archimedean places of $F$ respectively. Let $\mathbb{A}$ be adele ring of $F$
and $A_{\text{fin}}\subset\mathbb{A}$ be restricted product over the finite
places.

Suppose $S\subset V$ is a finite set containing $V_{\infty}$. Let
$F_{S}=\prod_{v\in S}F_{s}$ and $\mathbb{A}^{S}=\prod^{\prime}_{v\in
V\backslash S}F_{s}$ so that $\mathbb{A}=F_{S}\times\mathbb{A}^{S}$. We define

  1. (1)

$G(F_{S})^{1}=\underset{\chi\in\operatorname{Hom}(G(F_{S}),F^{\times})}{\bigcap}\\{\ker|\chi|\colon
G(F_{S})\to\mathbb{R}_{+}\\}$,

  2. (2)

$G(\mathbb{A})^{1}=\underset{\chi\in\operatorname{Hom}(G(\mathbb{A}),F^{\times})}{\bigcap}\\{\ker|\chi|\colon
G(\mathbb{A})\to\mathbb{R}_{+}\\}$,

where $|\cdot|$ is the product of valuations on $F_{S}$ and $\mathbb{A}$
respectively.

We will consider the representation of $G(F_{S})$ on $L^{2}(G(F)\backslash
G(\mathbb{A})^{1}/K)$ for an open compact subgroup $K$ of $G(\mathbb{A}^{S})$.
In particular, it will reduce to the representation of $G(F_{\infty})$ on
$L^{2}(\Gamma_{K}\backslash F_{\infty})$ if we take $S=\\{\infty\\}$ and
$\Gamma_{K}=G(F)\cap K$.

Let $J(f)$ be the distribution defined by Equation 3 or 5 in Section 2 for
$f\in C^{\infty}_{c}(G(F_{S}))$, which also depends on $G(F)\backslash
G(\mathbb{A})^{1}$ is compact or not. The goal of this subsection is to prove

$\lim\limits_{n\to\infty}\cfrac{\operatorname{vol}(G(F)\backslash
G(\mathbb{A})^{1})f(1)}{J(f\otimes 1_{K_{n}})}=1$

for certain towers of open compact subgroups $\\{K_{n}\\}_{n\geq 1}$ of
$G(\mathbb{A}^{S})$.

Let us assume $\Gamma$ is a uniform lattice in the semisimple Lie group $G$.
We add a subscript such as $R_{\Gamma}$ and $J_{\Gamma}$ for the
representation of $G$ on $L^{2}(\Gamma\backslash G)$ and the corresponding
trace formulas as an emphasis on the lattice $\Gamma$. Since $\Gamma\backslash
G$ is compact, $J_{\Gamma}(f)$ is the trace $\operatorname{tr}R_{\Gamma}(f)$
and we obtain

$J_{\Gamma}(f)=\operatorname{tr}R_{\Gamma}(f)=\sum_{\pi\in\widehat{G}}J_{\pi,\Gamma}(f)=\sum_{o\in\mathcal{O}}J_{o,\Gamma}(f)$.

Let $J_{\\{1\\},\Gamma}(f)=\operatorname{vol}(\Gamma\backslash G)f(1)$, the
contribution of the identity to the geometric side of the trace formula.

We take a tower of uniform lattices $\\{\Gamma_{n}\\}_{n\geq 1}$, such that
$\Gamma_{n}\trianglelefteq\Gamma_{1}$, $[\Gamma_{1}:\Gamma_{n}]<\infty$ and
$\cap_{n\geq 1}\Gamma_{n}=\\{1\\}$.

######  Proposition 2.4.

With the assumption of uniform lattice $\\{\Gamma_{n}\\}$ above, we have

$\lim\limits_{n\to\infty}\cfrac{J_{\\{1\\},\Gamma_{n}}(f)}{J_{\Gamma_{n}}(f)}=1$.

Proof: Following the Equation (2) in [10], we obtain

$\operatorname{tr}R_{\Gamma_{n}}(\phi)=J_{\\{1\\},\Gamma_{n}}(\phi)+\sum_{\gamma\neq
1}s_{n}(\gamma)\operatorname{vol}(\Gamma_{j}\backslash
G)\operatorname{vol}(\Gamma_{\gamma}\backslash
G_{\gamma})\int_{\Gamma_{\gamma}\backslash G}\phi(x^{-1}\gamma x)dx$,

where $0\leq s_{n}(\gamma)\leq\operatorname{vol}(\Gamma\backslash G)^{-1}$. As
$\cap_{n\geq 1}\Gamma_{n}=\\{1\\}$, $\lim_{n\to\infty}s_{n}(\gamma)$ for all
$\gamma\neq 1$. By [10] Theorem 2, we have
$\operatorname{vol}(\Gamma_{n}\backslash
G)^{-1}\cdot\lim_{n\to\infty}\operatorname{tr}R_{\Gamma_{n}}(\phi)=\phi(1)$.
Hence
$\lim\limits_{n\to\infty}\frac{J_{\\{1\\},\Gamma_{n}}(\phi)}{J_{\Gamma_{n}}(\phi)}=\lim\limits_{n\to\infty}\frac{J_{\\{1\\},\Gamma_{n}}(\phi)}{\operatorname{tr}R_{\Gamma_{n}}(\phi)}=1$.

Now we let $G$ be a reductive group over a number field $F$. Let
$K=K_{\infty}K_{\text{fin}}$ be a maximal compact subgroup of
$G(\mathbb{A})=G(\mathbb{A}_{F})$. By fixing a faithful $F$-rational
representation $\rho\colon G(F)\to\operatorname{GL}(m,F)$ for some $m>0$, we
let $\Lambda\subset F^{m}$ be an $\mathcal{O}_{F}$-lattice such that the
stablilizer of $\widehat{\Lambda}=\mathcal{O}_{F}\otimes_{F}\Lambda$ in
$G(A_{\text{fin}})$ is $K_{\text{fin}}$.

For a non-trivial ideal $I$ of $\mathcal{O}_{F}$, we let

$K(I)=\\{g\in G(A_{\text{fin}})|\rho(g)v\equiv
v\pmod{I\cdot\widehat{\Lambda}},v\in\widehat{\Lambda}\\}$

be the principal congruence subgroup of level $I$. We also denote the ideal
norm of $I$ by $N(I)=[\mathcal{O}_{F}\colon I]$. Consider a descending tower
of ideals $I_{1}\supsetneq I_{2}\supsetneq I_{3}\supsetneq\cdots$ such that
each $I_{k}$ is prime to (the prime ideals in) $S$. We obtain the
corresponding tower of principal congruence subgroups:

$K_{1}\supsetneq K_{2}\supsetneq K_{3}\supsetneq\cdots$,

where $K_{n}=K(I_{n})$. By factoring into prime ideals, the family
$\\{I_{n}\\}_{n\geq 1}$ satisfies either one of the following properties:

  1. (1)

there exists a prime ideal $\mathfrak{p}$ such that each $\mathfrak{p}^{k}$ is
eventually contained in the tower, i.e., for any $k\geq 1$, there is $N_{k}>0$
such that $\mathfrak{p}^{k}\subset I_{n}$ for all $n\geq n_{k}$, or,

  2. (2)

there exists infinitely many prime ideals $\\{\mathfrak{p}_{k}\\}_{k\geq 1}$
such that for each $k$, there exist $M_{k}>0$ such that
$\mathfrak{p}_{k}\subset I_{n}$ for all $n\geq M_{k}$.

In either of these two cases, we have

######  Lemma 2.5.

$\cap_{n\geq 1}I_{n}=\\{0\\}$ and $\cap_{n\geq 1}K_{n}=\\{1\\}$.

Recall the equivalence class of unipotent elements in $G(F)$, which is the
element $\gamma=\gamma_{s}\gamma_{u}$ with the semisimple component $r_{s}=1$
(see [5] p.1240). Let

$J_{\text{unip}}^{T}(f)$, $f\in C^{\infty}_{c}(G(\mathbb{A})^{1})$.

be the contribution of this equivalence class on the geometric side of the
trace formula 5. We will consider the function of the form $f=h_{S}\otimes
1_{K_{n}}$ with $h_{S}\in C^{\infty}_{c}(G(F_{S})^{1})$.

######  Lemma 2.6.

For $h_{S}\in C^{\infty}_{c}(G(F_{S})^{1})$,
$\lim\limits_{n\to\infty}J(h_{S}\otimes
1_{K_{n}})=\lim\limits_{n\to\infty}J_{\text{unip}}(h_{S}\otimes 1_{K_{n}})$.

Proof: Let $D_{h}=\text{supp}(h_{S})\subset G(F_{S})^{1}$ be the compact
support of $h_{S}$. Then $\text{supp}(h_{S}\otimes 1_{K_{n})})=D_{h}K_{n}$ is
compact and hence it intersects finitely many semisimple-conjugate class
$o\in\mathcal{O}$.

Consider the trace formula and Equation 5, only the classes $o$’s (and its
$G(\mathbb{A})$-conjugations) which intersect infinitely many $D_{h}K_{n}$
contributes a non-trivial $J_{o}(h_{S}\otimes 1_{K_{n}})$ to the limit
$\lim\limits_{n\to\infty}J(h_{S}\otimes 1_{K_{n}})$.

Suppose the $G(\mathbb{A})$ conjugacy classes of elements in $o$ intersects
$D_{h}K_{n}$ for infinitely many $n$, i.e., $\\{g\gamma g^{-1}|g\in
G(\mathbb{A}),\gamma\in o\\}\cap D_{h}K_{n}\neq\emptyset$ for infinitely many
$n$. Take some $\gamma\in o$. By fixing a faithful $F$-representation
$\rho\colon G(F)\to\operatorname{GL}(m)$, we let $p(x)\in F[x]$ be the
characteristic polynomial of $\rho(\gamma)-1$ (a $m$-by-$m$ matrix over $F$).
Suppose $p(x)=x^{m}+a_{m-1}x^{m-1}+\cdots+a_{0}$ with all $a_{i}\in F$. By
Lemma 2.5, we know $a_{i}$ belongs to infinitely many $I_{n}$, or,
equivalently $a_{i}=0$. Hence $p(x)=x^{m}$ and $\gamma$ is unipotent.

The unipotent contribution $J_{\text{unip}}(h_{S}\otimes 1_{K_{n}})$ can be
further reduced to the the one from the identity as follows. We let $I_{S}$ be
a product of prime ideals in at the places of $S$ and
$K_{S-S_{\infty}}(I_{S})$ be the $S-S_{\infty}$ component of the compact group
$K(I_{S})$.

We also let $C^{\infty}_{\Omega}(G(F_{S})^{1})$ be the set of smooth functions
with compact support contained in a compact subset $\Omega$ of $G(F_{S})^{1}$.
For each $k\geq 0$, we let $\mathcal{B}_{k}$ be the $k$-th component of the
universal enveloping algebra $\mathcal{U}(\mathfrak{g}_{\mathbb{C}})$, where
$\mathfrak{g}_{\mathbb{C}}$ is the complexified Lie algebra of the Lie group
$G(F_{\infty})$. We set $\|h\|_{k}=\sum_{X\in\mathcal{B}_{k}}\|X\circ
h\|_{L^{1}(G(\mathbb{A})^{1}}$ for $h\in C^{\infty}_{\Omega}(G(F_{S})^{1})$.

The following result is a special case of Proposition 3.1 in [20], whose proof
is mainly based on Theorem 3.1 and Theorem 4.2 in [5].

######  Proposition 2.7 (Finis-Lapid-Müller).

There exists an integer $k\geq 0$ such that for any compact subset $\Omega$ of
$G(F_{S})^{1}$, we have a constant $C_{\Omega}>0$ and

$|J_{\text{unip}}(h_{S}\otimes 1_{K_{n}})-\operatorname{vol}(G(F)\backslash
G(\mathbb{A})^{1})h_{S}(1)|\leq C_{\Omega}\frac{(1+\log
N(I_{S}I))^{d_{0}}}{N(I)}\|h_{S}\|_{k}$

for any bi-$K_{S-S_{\infty}}(I_{S})$-invariant function $h_{S}\in
C^{\infty}_{\Omega}(G(F_{S})^{1})$.

Then, combining Lemma 2.6 we can obtain:

######  Corollary 2.8.

For $h_{S}\in C^{\infty}_{c}(G(F_{S})^{1})$, we have

$\lim\limits_{n\to\infty}\cfrac{\operatorname{vol}(G(F)\backslash
G(\mathbb{A})^{1})h_{S}(1)}{J(h_{S}\otimes 1_{K^{S}(n)})}=1$.

##  3\. The multiplicities problem

This section is devoted to the multiplicity of bounded subsets of the unitary
dual, instead of a single irreducible representation.

###  3.1. The multiplicities in $L^{2}(\Gamma\backslash G)$

Let $G=\mathbf{G}(\mathbb{R})^{0}$, the connected component of the real group
obtained from an almost simple group $\mathbf{G}$ over $\mathbb{Q}$. By fixing
a faithful $\mathbb{Q}$-embedding $\rho:\mathbf{G}\to GL_{n}$, we have an
arithmetic group $\Gamma$ commensurable with
$G\cap\operatorname{GL}_{n}(\mathbb{Z})$. Let $\widehat{G}$ be the unitary
dual of $G$ and $\widehat{G}_{\text{temp}}\subset\widehat{G}$ be the tempered
dual. Let us consider the following two cases.

1\. $\Gamma\backslash G$ is compact. As introduced in Section 2.1,
$L^{2}(\Gamma\backslash G)$ can be decomposed into a direct sum of irreducible
representations of $G$ with each of finite multiplicity, i.e.,

$L^{2}(\Gamma\backslash G)=\oplus m_{\Gamma}(\pi)\cdot\pi$

with
$m_{\Gamma}(\pi)\colon=\dim\operatorname{Hom}_{G}(\pi,L^{2}(\Gamma\backslash
G))<\infty$ for each $\pi\in\widehat{G}$.

2\. $\Gamma\backslash G$ is not compact. If $G$ is semisimple, we have
$\Gamma\backslash G$ is of finite (Haar) measure (see [31] Theorem 4.13). The
regular representation has both discrete and continuous spectra:
$L^{2}(\Gamma\backslash G)=L^{2}_{\text{disc}}(\Gamma\backslash G)\oplus
L^{2}_{\text{disc}}(\Gamma\backslash G)$. The discrete spectrum can be written
as the direct sum of cuspidal and residue subspaces:
$L^{2}_{\text{disc}}(\Gamma\backslash G)=L^{2}_{\text{cusp}}(\Gamma\backslash
G)\oplus L^{2}_{\text{res}}(\Gamma\backslash G)$. which can be decomposed
further into a direct sum of irreducible representations with finite
multiplicities, i.e.,

$L^{2}_{\text{disc}}(\Gamma\backslash G)=\oplus m_{\Gamma}(\pi)\cdot\pi$

with
$m_{\Gamma}(\pi)\colon=\dim\operatorname{Hom}_{G}(\pi,L^{2}_{\text{disc}}(\Gamma\backslash
G))=\dim\operatorname{Hom}_{G}(\pi,L^{2}(\Gamma\backslash G))$ is finite for
each $\pi\in\widehat{G}$.

We say $X\subset\widehat{G}$ is bounded if it is relatively compact under the
Fell topology.

######  Definition 3.1 (The multiplicity for $X\subset\widehat{G}$).

For a bounded $X\subset\widehat{G}$, we define the multiplicity of $X$ to be
the sum of the multiplicities of the irreducible representations in $X$, i.e.,

$m_{\Gamma}(X)\colon=\sum_{\pi\in X}m_{\Gamma}(\pi)$.

Borel and Garland proved the finiteness of $m_{\Gamma}(X)$ by considering the
spectrum of a certain Laplacian (see [9] Theorem 3, Theorem 4.6 and also [24]
Theorem 1.1.3).

######  Theorem 3.1 (Borel-Garland).

Let $G=\mathbf{G}(\mathbb{R})^{0}$ for a connected semisimple group
$\mathbf{G}$ over $\mathbb{Q}$ and $X\subset\widehat{G}$ is bounded. We have
$m_{\Gamma}(X)<\infty$.

For a subset $X\subset\widehat{G(F_{S})^{1}}$, we call it bounded if it is
relatively compact under the Fell topology (see [35]).

######  Definition 3.2 (The multiplicity for $\widehat{G(F_{S})^{1}}$).

Suppose $K$ is a compact open subgroup of $G(\mathbb{A}^{S})$. Let $\sigma$ be
an irreducible representation of $G(F_{S})^{1}$ and
$X\subset\widehat{G(F_{S})^{1}}$ be a bounded subset.

  1. (1)

The multiplicity of $\sigma$ with respect to $K$ is defined as

$m_{K}(\sigma)\colon=\dim\operatorname{Hom}_{G(F_{S})^{1}}(\sigma,L^{2}(G(\mathbb{Q})\backslash
G(\mathbb{A})^{1}/K))$.

  2. (2)

The multiplicity of $X$ with respect to $K$ is defined as

$m_{K}(X)\colon=\sum_{\sigma\in X}m_{K}(\sigma)$.

For an irreducible representation $\pi$ of $G(\mathbb{A})^{1}$, we write
$\pi=\pi_{S}\otimes\pi^{S}$, where $\pi_{S}$ and $\pi^{S}$ denote the
components of the representations of $G(F_{S})^{1}$ and $G(\mathbb{A}^{S})$
respectively. As shown in Theorem 3.1, $m_{K}(X)$ is finite and hence well-
defined. If we treat $L^{2}(G(\mathbb{Q})\backslash G(\mathbb{A})^{1}/K)$ as
the subspace of $K$-right invariant functions in
$L^{2}(G(\mathbb{Q})\backslash G(\mathbb{A})^{1}))$, we have

$m_{K}(\sigma)=\sum_{\pi\in\widehat{G(\mathbb{A})^{1}},\pi_{S}=\sigma}\dim\operatorname{Hom}_{G(\mathbb{A})^{1}}(\pi,L^{2}(G(\mathbb{Q})\backslash
G(\mathbb{A})^{1}))\dim(\pi^{S})^{K}$.

If we take $S=V_{\infty}$ and $\mathbf{G}$ is semisimple, simply connected,
and without any $F$-simple factors $H$ such that $H(F_{\infty})$ is compact
and $K$ is an open compact subgroup of $G(\mathbb{A}_{\text{fin}})$, we know
$\Gamma_{K}=G(\mathbb{F})\cap K$ is a lattice in the seimisimple Lie group
$G(F_{\infty})$.

######  Lemma 3.2.

With the assumption above, we have $m_{\Gamma_{K}}(\pi)=m_{K}(\pi)$ for any
$\pi\in\widehat{G(F_{\infty})^{1}}$ and $m_{\Gamma_{K}}(X)=m_{K}(X)$ for any
bounded $X\subset\widehat{G(F_{\infty})^{1}}$

Proof: It follows the fact $G(\mathbb{Q})\backslash G(\mathbb{A})/K$ can be
identified with $\Gamma_{K}\backslash G(F_{\infty})$, which leads to a
$G(F_{\infty})$-isomorphism $L^{2}(\Gamma_{K}\backslash G(F_{\infty}))\cong
L^{2}(G(\mathbb{Q})\backslash G(\mathbb{A})^{1}/K)$ (see [27] Chapter 6 and
[32] Chapter 7.4.).

For a finite set $S$ and a function $\phi$ on $\widehat{G(F_{S})^{1}}$, we
define

$m_{K}(\phi)\colon=\int_{\widehat{G(F_{S})^{1}}}\phi(\pi)dm_{K}(\pi)$

as its integral with respect to the measure given by multiplicities above. If
$1_{X}$ is the characteristic function of $X$, i.e., $1_{X}(\pi)=1$ if $\pi\in
X$ and $0$ otherwise, $m_{K}(1_{X})=m_{K}(X)$.

For $f\in C^{\infty}_{\text{c}}(G(F_{S})^{1})$, we let
$\widehat{f}(\pi)=\operatorname{tr}\pi(f)$, the distribution character of
$\pi$. Let $R_{\text{disc}}$ denote the action of $G(\mathbb{A})$ on the
discrete subspace $L^{2}(G(\mathbb{Q})\backslash G(\mathbb{A})^{1})$.

######  Proposition 3.3.

For $f\in C^{\infty}_{\text{c}}(G(F_{S})^{1})$, we have

$\operatorname{tr}R_{\text{disc}}(f\otimes\frac{1_{K}}{\operatorname{vol}(K)})=m_{K}(\hat{f})$.

Proof: Observe for the component $\pi^{S}$ of representation of
$G(\mathbb{A}^{S})$, we have

$\displaystyle\operatorname{tr}\pi^{S}(1_{K})$
$\displaystyle=\int_{G(\mathbb{A}^{S})}1_{K}(x)\pi^{S}(x^{-1})d\mu^{S}(x)$
$\displaystyle=\int_{K}\pi^{S}(x^{-1})d\mu^{S}(x)=\operatorname{vol}(K)\dim(\pi^{S})^{K},$

where we apply the fact that $\int_{K}\sigma(x)d\mu^{S}(x)=0$ for any non-
trivial irreducible representation $\sigma$ of $K$. Hence we obtain

$\displaystyle\operatorname{tr}R_{\text{disc}}(f\otimes\frac{1_{K}}{\operatorname{vol}(K)})$
$\displaystyle=\frac{1}{\operatorname{vol}(K)}\sum_{\pi\in\widehat{G(\mathbb{A})^{1}}}m(\pi)\operatorname{tr}\pi(f\otimes
1_{K})$
$\displaystyle=\frac{1}{\operatorname{vol}(K)}\sum_{\pi\in\widehat{G(\mathbb{A})^{1}}}m(\pi)\operatorname{tr}\pi_{S}(f)\operatorname{tr}\pi^{S}(1_{K})$
$\displaystyle=\frac{1}{\operatorname{vol}(K)}\sum_{\pi\in\widehat{G(\mathbb{A})^{1}}}m(\pi)\operatorname{tr}\pi_{S}(f)\operatorname{vol}(K)\dim(\pi^{S})^{K}$
$\displaystyle=\sum_{\sigma\in\widehat{G(F_{S})^{1}}}m_{K}(\sigma)\operatorname{tr}\sigma(f)=m_{K}(\widehat{f}).$

We also give the following result which connects the trace formulas for adelic
groups and Lie groups.

######  Corollary 3.4.

Let $\Gamma_{K}=G(F)\cap K$ with an open compact subgroup $K$ of
$G(\mathbb{A}_{\rm{fin}})$. We have

$\operatorname{tr}R_{\text{disc}}(f\otimes\frac{1_{K}}{\operatorname{vol}(K)})=\operatorname{tr}R_{\Gamma_{K}}(f)$.

for all $f\in C^{\infty}_{c}(G(F_{\infty})^{1})$.

Proof: It follows the fact $m_{K}(\widehat{f})=m_{\Gamma_{K}}(\widehat{f})$ in
Lemma 3.2, $m_{\Gamma_{K}}(\widehat{f})=\operatorname{tr}R_{\Gamma_{K}}(f)$
and Proposition 3.3.

###  3.2. Sauvageot’s density theorems

We have a brief review of the results in [35]. See also [37] for an
alternative approach and corrections.

For an open compact subgroup $K$ of $G(\mathbb{A}^{S})$, we define a measure
on $\widehat{G(F_{S})^{1}}$ by

$\nu_{K}(X)\colon=\frac{\operatorname{vol}(K)}{\operatorname{vol}(G(\mathbb{Q})\backslash
G(\mathbb{A})^{1})}m_{K}(X)$

for any bounded subset $X$ of $\widehat{G(F_{S})^{1}}$ and $m_{K}$ is the
multipilicity defined in Chapter 3.1.

Let $K_{1}\supsetneq K_{2}\supsetneq\cdots$ be a sequence of open compact
subgroups of $G(\mathbb{A}^{S})$. Given a bounded subset $X$ of
$\widehat{G(F_{S})^{1}}$ and $C\geq 0$, we write

$\lim\limits_{n\to\infty}\nu_{K}(X)=C$,

if for any $\varepsilon>0$, there exists $N=N(\varepsilon)>0$ such that
$|\nu_{K_{n}}(X)-C|<\varepsilon$ for all $n\geq N$.

Let $\mathcal{H}(G(F_{S})^{1})$ be the complex algebra of smooth, compactly-
supported, bi-$K_{S}$-finite functions on $G(F_{S})^{1}$.

######  Lemma 3.5 ([35] Corollaire 6.2).

For $\varepsilon>0$ and any bounded
$X\subset\widehat{G(F_{S})^{1}}\backslash\widehat{G(F_{S})^{1}}_{\text{temp}}$,
there is $\Psi\in\mathcal{H}(G(F_{S})^{1})$ such that

$\widehat{\Psi}|_{\widehat{G(F_{S})^{1}}}\geq 0$,
$\nu(\widehat{\Psi})<\varepsilon$ and $\widehat{\Psi}|_{X}\geq 1$.

Given a function $f$ defined on $\widehat{G(F_{S})^{1}}_{\text{temp}}$, we
also denote by $f$ the function on $\widehat{G(F_{S})^{1}}$, which is extended
by $0$ on the untempered part.

######  Lemma 3.6 ([35] Theoreme 7.3(b)).

For $\varepsilon>0$ and any $\nu$-integrable function $f$ on
$\widehat{G(F_{S})^{1}}_{\text{temp}}$, there exists
$\phi,\psi\in\mathcal{H}(G(F_{S})^{1})$ such that

$|f(\pi)-\widehat{\phi}(\pi)|\leq\widehat{\psi}(\pi)$ and
$\nu(\widehat{\psi})<\varepsilon$.

Here we obtain one of the main results in [35] and we also provide a proof for
completeness.

######  Theorem 3.7 (Sauvageot).

Suppose $\lim\limits_{n\to\infty}\nu_{K_{n}}(\widehat{\phi})=\phi(1)$ for all
$\phi\in\mathcal{H}(G(F_{S})^{1})$. We have

$\lim\limits_{n\to\infty}\nu_{K_{n}}(X)=\nu(X)$

for all bounded subset $X$ of $\widehat{G(F_{S})^{1}}$.

Proof: First, we show the contribution from the untempered part is negligible
in the limit. For a bounded subset $X_{0}$ of
$\widehat{G(F_{S})^{1}}\backslash\widehat{G(F_{S})^{1}}_{\text{temp}}$ and
$\varepsilon>0$, we let $\Psi\in\mathcal{H}(G(F_{S})^{1})$ satisfies Lemma 3.5
with respect to $X$. We have
$\nu_{K_{n}}(X)\leq\nu_{K_{n}}(\widehat{\Psi})\leq|\nu_{K_{n}}(\widehat{\Psi})-\psi(1)|+\psi(1)<2\varepsilon$
for all $n\geq N_{1}$ with some $N_{1}\geq 0$.

For the tempered part, we fix a bounded subset $X_{1}$ of
$\widehat{G(F_{S})^{1}}_{\text{temp}}$ with the same $\varepsilon$ above. Let
$\phi,\psi\in\mathcal{H}(G(F_{S})^{1})$ satisfy Lemma 3.6 with respect to the
function $f=1_{X_{1}}$ on $\widehat{G(F_{S})^{1}}_{\text{temp}}$ and
$\varepsilon$. By assumption, we have
$|\nu_{K_{n}}(\widehat{\phi})-\phi(1)|<\varepsilon$ and
$|\nu_{K_{n}}(\widehat{\psi})-\psi(1)|<\varepsilon$ for all $n\geq N_{2}$ with
some $N_{2}\geq 0$. Hence, for $n\geq N_{2}$, we obtain

$\displaystyle|\nu_{K_{n}}(X_{1})-\nu(X_{1})|$
$\displaystyle\leq|\nu_{K_{n}}(X_{1})-\nu_{K_{n}}(\widehat{\phi})|+|\nu_{K_{n}}(\widehat{\phi})-\phi(1)|+|\phi(1)-\nu(X)|$
$\displaystyle\leq|\nu_{K_{n}}(\widehat{\phi})-\phi(1)|+\nu_{K_{n}}(\widehat{\psi})+\psi(1)$
$\displaystyle\leq|\nu_{K_{n}}(\widehat{\phi})-\phi(1)|+|+|\nu_{K_{n}}(\widehat{\psi})-\psi(1)|+2\psi(1)<4\varepsilon.$

Hence, for the bounded set $X$ of $\widehat{G(F_{S})^{1}}$, let $X=X_{0}\sqcup
X_{1}$ be the decomposition into its untempered and tempered parts. We have

$\displaystyle|\nu_{K_{n}}(X)-\nu(X)|$
$\displaystyle=|\nu_{K_{n}}(X)-\nu(X_{1})|=|\nu_{K_{n}}(X_{1})-\nu(X)|+\nu_{K_{n}}(X_{0})$
$\displaystyle\leq 4\varepsilon+2\varepsilon=6\varepsilon$

for all $N\geq\max\\{N_{1},N_{2}\\}$.

##  4\. The von Neumann dimensions of direct integrals

###  4.1. The group von Neumann algebra and the trace

Let $\Gamma$ be a countable group with the counting measure. Let
$\\{\delta_{\gamma}\\}_{\gamma\in\Gamma}$ be the usual orthonormal basis of
$l^{2}(\Gamma)$. We also let $\lambda$ and $\rho$ be the left and right
regular representations of $\Gamma$ on $l^{2}(\Gamma)$ respectively. For all
$\gamma,\gamma^{\prime}\in\Gamma$, we have
$\lambda(\gamma^{\prime})\delta_{\gamma}=\delta_{\gamma^{\prime}\gamma}$ and
$\rho(\gamma^{\prime})\delta_{\gamma}=\delta_{\gamma\gamma^{\prime-1}}$. Let
$\mathcal{L}(\Gamma)$ be the strong operator closure of the complex linear
span of $\lambda(\gamma)$’s (or equivalently, $\rho(\gamma)$’s). This is the
group von Neumann algebra of $\Gamma$. There is a canonical faithful normal
tracial state $\tau_{\Gamma}$, or simply $\tau$, on $\mathcal{L}(\Gamma)$,
which is given by

$\tau(x)=\langle x\delta_{e},\delta_{e}\rangle_{l^{2}(\Gamma)}$,
$x\in\mathcal{L}(\Gamma)$.

Hence $\mathcal{L}(\Gamma)$ is a finite von Neumann algebra (which must be of
type I or $\text{II}_{1}$).

More generally, for a tracial von Neumann algebra $M$ with the trace $\tau$,
we consider the GNS representation of $M$ on the Hilbert space constructed
from the completion of $M$ with respect to the inner product $\langle
x,y\rangle_{\tau}=\tau(xy^{*})$. The underlying space will be denoted by
$L^{2}(M,\tau)$, or simply $L^{2}(M)$.

Consider a normal unital representation $\pi\colon M\to B(H)$ with both $M$
and $H$ separable. There exists an isometry $u\colon H\to L^{2}(M)\otimes
l^{2}(\mathbb{N})$, which commutes with the actions of $M$:

$u\circ\pi(x)=(\lambda(x)\otimes\operatorname{id}_{l^{2}(\mathbb{N})})\circ
u$, $\forall x\in M$,

where $\lambda\colon M\mapsto L^{2}(M)$ denotes the left action. Then
$p=uu^{*}$ is a projection in $B(L^{2}(M)\otimes l^{2}(\mathbb{N}))$ such that
$H\cong p(L^{2}(M)\otimes l^{2}(\mathbb{N}))$. We have the following result
(see [2] Proposition 8.2.3).

######  Proposition 4.1.

The correspondence $H\mapsto p$ above defines a bijection between the set of
equivalence classes of left $M$-modules and the set of equivalence classes of
projections in $(M^{\prime}\cap B(L^{2}(M)))\otimes B(l^{2}(\mathbb{N}))$.

The von Neumann dimension of the $M$-module $H$ are defined to be
$(\tau\otimes\operatorname{Tr})(p)$ and denoted by $\dim_{M}(H)$, which takes
its value in $[0,\infty]$. We have:

  1. (1)

$\dim_{M}(\oplus_{i}H_{i})=\sum_{i}\dim_{M}(H_{i})$.

  2. (2)

$\dim_{M}(L^{2}(M))=1$.

Note $\dim_{M}(H)$ depends on the trace $\tau$. If $M$ is a finite factor,
i.e., $Z(M)\cong\mathbb{C}$, there is a unique normal tracial state (see [25,
29]) and we further have:

  1. (3)

$\dim_{M}(H)=\dim_{M}(H^{\prime})$ if and only if $H$ and $H^{\prime}$ are
isomorphic as $M$-modules (provided $M$ is a factor).

When $M$ is not a factor, there is a $Z(M)$-valued trace which determines the
isomorphism class of an $M$-module (see [8]).

In the following sections, we will consider the group von Neumann algebra
$\mathcal{L}(\Gamma)$ with the canonical trace $tr(x)=\langle
x\delta_{e},\delta_{e}\rangle$. Hence the von Neumann dimension of
$\mathcal{L}(\Gamma)$ is the one uniquely determined by this trace. Note a
discrete group $\Gamma$ is called an infinite conjugacy class (ICC) group if
every nontrivial conjugacy class $C_{\gamma}=\\{g\gamma g^{-1}|g\in\Gamma\\}$,
$\gamma\neq e$, is infinite. It is well-known that $\mathcal{L}(\Gamma)$ is a
$\rm{II}_{1}$ factor if and only if $\Gamma$ is a nontrivial ICC group.

Now we consider the case that $\Gamma$ is a discrete subgroup of a locally
compact unimodular type I group $G$. Let $\mu$ be a Haar measure of $G$. A
measurable set $D\subset G$ is called a fundamental domain for $\Gamma$ if $D$
satisfies $\mu(G\backslash\cup_{\gamma\in\Gamma}\gamma D)=0$ and
$\mu(\gamma_{1}D\cap\gamma_{2}D)=0$ if $\gamma_{1}\neq\gamma_{2}$ in $\Gamma$.
In this section, we always assume $\Gamma$ is a lattice, i.e.,
$\mu(D)<\infty$. The measure $\mu(D)$ is called covolume of $\Gamma$ and will
be denoted by $\operatorname{covol}(\Gamma)$. Note the covolume depends on the
Haar measure $\mu$ (see Remark 4.3).

There is a natural isomorphism $L^{2}(G)\cong l^{2}(\Gamma)\otimes
L^{2}(D,\mu)$ given by

$\phi\mapsto\sum_{\gamma\in\Gamma}\delta_{\gamma}\otimes\phi_{\gamma}$ with
$\phi_{\gamma}(z)=\phi(\gamma\cdot z)$,

where $z\in D$ and $\gamma\in\Gamma$. The restriction representation
$\lambda_{G}|_{\Gamma}$ of $\Gamma$ is the tensor product of
$\lambda_{\Gamma}$ on $l^{2}(\Gamma)$ and the identity operator
$\operatorname{id}$ on $L^{2}(D,\mu)$. Hence we obtain the von Neumann algebra
$\lambda_{G}(\Gamma)^{\prime\prime}\cong\mathcal{L}(\Gamma)\otimes\mathbb{C}=\mathcal{L}(\Gamma)$,
which will be denoted by $M$ throughout this section. Please note
$L^{2}(M)=l^{2}(\Gamma)$.

###  4.2. A theorem of von Neumann dimension

Suppose $X$ is a measurable subset of $\widehat{G}$ with the Plancherel
measure $\nu(X)<\infty$. Define

$H_{X}=\int_{X}^{\oplus}H_{\pi}d\nu(\pi)$,

which is the direct integral of the spaces $H_{\pi}$ with $\pi\in X$. It is a
module over $G$, its lattice $\Gamma$, and also the group von Neumann algebra
$\mathcal{L}(\Gamma)$.

We state a result on the von Neumann dimension of direct integrals. One may
refer to [38] Section 4 for the proof.

######  Theorem 4.2.

Let $G$ be a locally compact unimodular type I group with Haar measure $\mu$.
Let $\nu$ be the Plancherel measure on the unitary dual $\widehat{G}$ of $G$.
Suppose $\Gamma$ is a lattice in $G$ and $\mathcal{L}(\Gamma)$ is the group
von Neumann algebra of $\Gamma$. Let $X\subset\widehat{G}$ such that
$\nu(X)<\infty$ and $H_{X}=\int_{X}^{\oplus}H_{\pi}d\nu(\pi)$. We have

$\dim_{\mathcal{L}(\Gamma)}(H_{X})=\operatorname{covol}(\Gamma)\cdot\nu(X)$.

######  Remark 4.3.

  1. (1)

If $\mu^{\prime}=k\cdot\mu$ is another Haar measure on $G$ for some $k>0$, the
covolumes are related by
$\operatorname{covol}^{\prime}(\Gamma)=\mu^{\prime}(G/\Gamma)=k^{\prime}\cdot\mu(G/\Gamma)=k\cdot\operatorname{covol}(\Gamma)$.
But the induced Plancherel measure $\nu^{\prime}=k^{-1}\cdot\nu$ and the
dependencies cancel out in the formula above.

  2. (2)

There is a relevant approach by H. Peterson and A. Valette [31]. They study
the von Neumann dimension over locally compact groups. The group von Neumann
algebra is equipped with a semifinite tracial weight instead of a tracial
state for a discrete group. It is motivated by the study of $L^{2}$-Betti
number of locally compact groups [30].

If $\pi$ is an atom in $\widehat{G}$, i.e., $\nu(\\{\pi\\})>0$, the
irreducible representation $\pi$ is a discrete series and $\nu(\\{\pi\\})$ is
just the formal dimension of $\pi$ [15, 33]. Under this assumption, if $G$ is
a real Lie group that has discrete series and $\Gamma$ is an ICC group, the
theorem reduces to the special case of a single representation (see [23]
Theorem 3.3.2)

$\dim_{\mathcal{L}(\Gamma)}(H_{\pi})=\operatorname{covol}(\Gamma)\cdot
d_{\pi}$.

This is motivated by the geometric construction of discrete series of Lie
groups by M. Atiyah and W. Schmid [7].

###  4.3. The proof of the main theorem

We will prove the main theorem. We first give the proof for a tower of uniform
lattices.

######  Theorem 4.4.

[a tower of uniform lattices] Let
$\Gamma_{1}\supsetneq\Gamma_{2}\supsetneq\cdots$ be a normal tower of
cocompact lattice in a semisimple real Lie group $G$ such that $\cap_{n\geq
1}\Gamma_{n}=\\{1\\}$. For any bounded subset $X$ of $\widehat{G}$, we have

$\lim\limits_{n\to\infty}\cfrac{m(X,\Gamma_{n})}{\dim_{\mathcal{L}\Gamma_{n}}H_{X}}=1$.

Proof: Recall that $m_{\Gamma_{K}}(X)=\operatorname{vol}(\Gamma_{K}\backslash
G(F_{\infty}))\nu_{K}(X)$ by definition and
$\dim_{\mathcal{L}\Gamma_{n}}H_{X}=\operatorname{vol}(\Gamma_{K}\backslash
G(F_{\infty}))\nu(X)$ by Theorem 4.2. We need to show
$\lim\limits_{n\to\infty}\nu_{K_{n}}(X)=\nu(X)$, which reduces to
$\lim\limits_{n\to\infty}\nu_{K_{n}}(\widehat{\phi})=\phi(1)$ for all $\phi\in
C^{\infty}_{c}(G(F_{\infty})^{1})$ by Theorem 3.7.

From Proposition 3.3, we know

$\displaystyle\operatorname{tr}R_{\text{disc}}(\phi\otimes\frac{1_{K}}{\operatorname{vol}(K)})$
$\displaystyle=m_{K}(\widehat{\phi})=\operatorname{vol}(G(\mathbb{Q})\backslash
G(\mathbb{A})^{1})/K)\cdot\nu_{K}(\widehat{\phi}),$

which is to say $\operatorname{tr}R_{\text{disc}}(\phi\otimes
1_{K})=\operatorname{vol}(G(\mathbb{Q})\backslash
G(\mathbb{A})^{1}))\cdot\nu_{K}(\widehat{\phi})$. By Proposition 2.4, we have
$\lim\limits_{n\to\infty}\operatorname{tr}R_{\text{disc}}(\phi\otimes
1_{K_{n}})=\operatorname{vol}(G(\mathbb{Q})\backslash
G(\mathbb{A})^{1}))\cdot\phi(1)$. Hence
$\lim\limits_{n\to\infty}\nu_{K_{n}}(\widehat{\phi})=\phi(1)$.

For the non-uniform case, the distribution $J(f)$ in Equation 5 will no longer
be the trace of $R_{\text{disc}}(f)$, which leads to the main task for most
arithmetic subgroups. Fortunately, Finis-Lapid-Müller proved the following
result on the limit of the spectral side of Equation 5 (see [20] Corollary
7.8).

######  Theorem 4.5.

[Finis-Lapid-Müller] Suppose $G=\operatorname{SL}(n)$. Let $\\{I_{n}\\}$ be a
family of descending integral ideals in $\mathcal{O}_{F}$ prime to $S$ and
$K_{n}=K(I_{n})$ be the compact subgroups of $G(\mathbb{A}^{S})$ given by
$I_{n}$. We have

$\lim_{n\to\infty}J(h_{S}\otimes
1_{K_{n}})=\lim_{n\to\infty}\operatorname{tr}R_{\text{disc}}(h_{S}\otimes
1_{K_{n}})$

for any $h_{S}\in C^{\infty}_{c}(G(F_{S})^{1})$.

Then we are able to prove:

######  Corollary 4.6.

[principal congruence subgroups in $\operatorname{SL}(n,\mathbb{R})$] Let
$\Gamma_{1}\supsetneq\Gamma_{2}\supsetneq\cdots$ be a tower of principal
congruence subgroups in $G=\operatorname{SL}(n,\mathbb{R})$. For any bounded
subset $X$ of $\widehat{G}$, we have

$\lim\limits_{n\to\infty}\cfrac{m(X,\Gamma_{n})}{\dim_{\mathcal{L}\Gamma_{n}}H_{X}}=1$.

Proof: As shown in Theorem 4.4, it suffices to prove
$\lim\limits_{n\to\infty}\nu_{K_{n}}(\widehat{\phi})=\phi(1)$ for all $\phi\in
C^{\infty}_{c}(G(F_{\infty})^{1})$. By Proposition 3.3 and Theorem 4.5, we
know

$\lim\limits_{n\to\infty}\operatorname{vol}(G(\mathbb{Q})\backslash
G(\mathbb{A})^{1}))\cdot\nu_{K_{n}}(\widehat{\phi})=\lim\limits_{n\to\infty}\operatorname{tr}R_{\text{disc}}(\phi\otimes
1_{K_{n}})=\lim\limits_{n\to\infty}J(\phi\otimes 1_{K_{n}})$.

As $\lim\limits_{n\to\infty}\frac{\operatorname{vol}(G(F)\backslash
G(\mathbb{A})^{1})\phi(1)}{J(\phi\otimes 1_{K_{n}})}=1$ in Corollary 2.8, we
obtain $\lim\limits_{n\to\infty}\nu_{K_{n}}(\widehat{\phi})=\phi(1)$.

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