# Plancherel Measures of Reductive Adelic Groups and von Neumann Dimensions

Jun Yang

###### Abstract

Given a number field $F$ and a reductive group $G$ over $F$, the unitary dual
$\widehat{G(\mathbb{A}_{F})}$ of the adelic group $G(\mathbb{A}_{F})$ and the
Placherel measure $\nu_{G(\mathbb{A}_{F})}$ on it can be determined by the
Plancherel measure of its local groups $G(F_{v})$. Given a subset
$X\subset\widehat{G(\mathbb{A}_{F})}$ of finite Plancherel measure, let
$H_{X}$ be the direct integral of the irreducible representations in $X$.
Besides a $G(\mathbb{A}_{F})$-module and a $G(F)$-module, $H_{X}$ is also a
module over the group von Neumann algebra $\mathcal{L}(G(F))$, hence there is
a canonical dimension $\dim_{\mathcal{L}(G(F))}H_{X}\in[0,\infty)$. It is
proved that the Plancherel measure of $G(\mathbb{A}_{F})$ coincides with the
dimension over the algebra $\mathcal{L}(G(F))$:

$\dim_{\mathcal{L}(G(F))}H_{X}=\nu_{G(\mathbb{A}_{F})}(X)$,

if $G$ is semisimple, simply connected and $G(\mathbb{A}_{F})$ is equipped
with the Tamagawa measure.

††2010 Mathematics Subject Classification. 20G05, 22D25.††✉ Jun Yang
<EMAIL_ADDRESS>University, Cambridge, MA 02138, 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
  2. 2 The Plancherel measure of an adelic group
  3. 3 The von Neumann dimensions of direct integrals
  4. 4 A product formula
  5. 5 The dimension and the adelic Plancherel measure

##  1 Introduction

Let $F$ be a number field and $G$ be a reductive algebraic group over $F$. The
unitary irreducible representations of the adelic group $G(\mathbb{A}_{F})$
are of great interest as the automorphic representation side of the global
Langlands correspondence. It is then natural to consider the Plancherel
measure on the equivalence classes of these representations, i.e., the unitary
dual $\widehat{G(\mathbb{A}_{F})}$. It can be shown that the dual space
$\widehat{G(\mathbb{A}_{F})}$ with the Plancherel measure
$\nu_{G(\mathbb{A}_{F})}$ is the restricted product of its local dual
$\widehat{G(F_{v})}$ with respect to the unramified representations (see
Theorem 2.6).

Meanwhile, the study of the von Neumann algebra $\mathcal{L}(\Gamma)$ of a
countable discrete group $\Gamma$ is one of the most challenging topics in
operator algebras. Very little is known of how $\mathcal{L}(\Gamma)$ depends
on the group $\Gamma$. Especially, a well known conjecture asks whether
$\mathcal{L}(\Gamma_{1})\cong\mathcal{L}(\Gamma_{2})$ implies
$\Gamma_{1}\cong\Gamma_{2}$ provided $\Gamma_{1},\Gamma_{2}$ are free groups
or lattices in a real Lie group [14]. In this paper, we consider the group von
Neumann algebra $\mathcal{L}(G(F))$ of the countable group $G(F)$ defined over
the number field $F$. The modules over $\mathcal{L}(G(F))$ can be described by
a canonical dimension function $\dim_{\mathcal{L}(G(F))}$, which determines
the isomorphic class of an $\mathcal{L}(G(F))$-module if $G(F)$ is an infinite
conjugacy class (ICC) group (or equivalently, $\mathcal{L}(G(F))$ is a factor
of type $\text{II}_{1}$).

We show the coincidence of the Plancherel measures $\nu_{G(\mathbb{A}_{F})}$
of the adelic group $G(\mathbb{A}_{F})$ and the dimensions over the von
Neumann algebra $\mathcal{L}(G(F))$ of the group $G(F)$. More precisely, let
$X$ be a subset of $\widehat{G(\mathbb{A}_{F})}$ with a finite Plancherel
measure, i.e., $\nu_{G(\mathbb{A}_{F})}(X)<\infty$. Denoting the underlying
Hilbert space of an irreducible representation $\pi$ by $H_{\pi}$, we let
$H_{X}$ be the direct integral of the spaces $H_{\pi}$’s such that the
isomorphism class $[\pi]\in X$, which is
$H_{X}=\int_{X}^{\oplus}H_{\pi}d\nu_{G(\mathbb{A}_{F})}([\pi])$. The following
result (Theorem 5.2) is proved :

######  Theorem.

Let $G$ be a simply connected semisimple algebraic group defined over a number
field $F$. Then, for a subset $X$ of $\widehat{G(\mathbb{A}_{F})}$ with finite
Plancherel measure, we have

$\dim_{\mathcal{L}(G(F))}H_{X}=\nu_{G(\mathbb{A}_{F})}(X)$.

In another word, one can tell the Placherel measure of the adelic group
$G(\mathbb{A}_{F})$ just by treating the representations as modules over the
von Neumann algebra $\mathcal{L}(G(F))$ of the discrete subgroup $G(F)$. In
Section 2, we give an explicit description of the Plancherel measure of an
adelic group. In Section 3, we construct the Hilbert space from the direct
integral of irreducible representations of $G(\mathbb{A}_{F})$ and discuss its
von Neumann over $\mathcal{L}(G(F))$. In Section 4, we reach a product formula
involving the covolume of a lattice. An example is also provided. In Section
5, we prove the theorem above.

Acknowledgements The author would like to thank V. Jones for encouraging me to
work on these problems. The author is grateful for the detailed comments from
D. Bisch, F. Radulescu, and many fruitful conversations with A. Jaffe, B.
Mazur, D. Vogan, and W. Schmid.

##  2 The Plancherel measure of an adelic group

Let $G$ be a locally compact Hausdorff group with a Haar measure $\mu$. Let
$\widehat{G}$ be the unitary dual of $G$, i.e., the set of equivalence classes
$[\pi]$ of irreducible unitary representations $(\pi,H_{\pi})$ of $G$. For
convenience, we will not distinguish a representation $\pi$ and its
equivalence class $[\pi]$ In the following sections, we will let $\pi$ denote
its equivalence class $[\pi]$ or also a representative in it.

We first give a brief review of type I groups following [8, 10]. A unitary
representation $(\pi,H)$ of $G$ is called primary if the von Neumann algebra
$\mathcal{A}(\pi)\subset B(H)$ generated by $\\{\pi(g)|g\in G\\}$ is a factor,
i.e., its center $Z(A(\pi))$ are scalar multiples of the identity. If $\pi$ is
a direct sum of irreducible representations, then $\pi$ is primary if and only
if all these irreducible summands are unitary equivalent.

A group $G$ is called type I if every primary representation of $G$ is a
direct sum of copies of some irreducible representation. That is to say,
$\mathcal{A}(\pi)$ is a type I factor for every primary representation $\pi$.
It is known that compact groups, connected semisimple Lie groups, connected
real algebraic groups, reductive $p$-adic groups, and also the adelic group
$G(\mathbb{A})$ of a connected reductive group $G$ defined over a number field
are type I (see [4, 7, 10, 15]).

Let us consider the regular representations of a type I locally compact group
$G$ and the Fourier transform. Let $\rho,\lambda$ be the right and left
regular representation of $G$ on $L^{2}(G)$:

$(\rho(g)f)(x)=R_{g}f(x)=f(xg)$, $(\lambda(g)f)(x)=L_{g}f(x)=f(g^{-1}x)$

for $f\in L^{2}(G)$. This gives us the two sided regular representation $\tau$
of $G\times G$ on $L^{2}(G)$:
$(\tau(g,h)f)(x)=R_{g}L_{h}f(x)=L_{h}R_{g}f(x)=f(h^{-1}xg)$. If $f\in
L^{1}(G)$, the Fourier transform of $f$ is defined to be the measurable field
of operators over $\widehat{G}$ given by

$\hat{f}(\pi)=\int_{G}f(x)\pi(x^{-1})d\mu(x)$,

with a representative of a equivalence class $[\pi]\in\widehat{G}$. We have
the convolution $\widehat{f_{1}*f_{2}}=\widehat{f_{1}}\cdot\widehat{f_{2}}$
[10, 16]. Now we let

$\mathcal{J}^{1}=L^{1}(G)\cap L^{2}(G)$,
$\mathcal{J}^{2}=\operatorname{span}\\{f*g|f,g\in\mathcal{J}^{1}\\}$.

For a unitary representation $(\pi,H_{\pi})$ of $G$, we denote its
contragradient by $(\overline{\pi},H_{\overline{\pi}})$. We let
$\operatorname{Tr}$ denote the usual trace defined on the trace-class
operators.

######  Theorem 2.1 (The Plancherel Theorem).

Suppose $G$ is a second-countable, unimodular, type I group. There is a
measure $\nu$ on $\widehat{G}$, uniquely determined once the Haar measure
$\mu$ on $G$ is fixed, with the following properties.

  1. 1.

The Fourier transform $f\mapsto\hat{f}$ maps $\mathcal{J}^{1}$ into
$\int_{G}^{\oplus}H_{\pi}\otimes H_{\overline{\pi}}d\nu(\pi)$ and it extends
to a unitary map from $L^{2}(G)$ onto $\int_{G}^{\oplus}H_{\pi}\otimes
H_{\overline{\pi}}d\nu(\pi)$ that intertwines the two-sided regular
representation $\tau$ with
$\int_{G}^{\oplus}{\pi}\otimes{\overline{\pi}}d\nu(\pi)$.

  2. 2.

For $f_{1},f_{2}\in\mathcal{J}^{1}$, one has the Parseval formula

$\int_{G}f_{1}(x)\overline{f_{2}(x)}d\mu(x)=\int_{\widehat{G}}\operatorname{Tr}[\hat{f_{1}}(\pi)\hat{f_{2}}(\pi)^{*}]d\nu(\pi)$.

  3. 3.

For $f\in\mathcal{J}^{2}$, one has the Fourier inversion formula

$f(x)=\int_{\widehat{G}}\operatorname{Tr}[{\pi}(x)^{*}\hat{f}(\pi)]d\nu(\pi)$,
$x\in G$.

Indeed, the Plancherel measure can be determined by either the second or third
property described above. This theorem also implies the decomposition of right
and left regular representation of $G$:

$\rho\cong\int_{\widehat{G}}^{\oplus}\pi\otimes{\operatorname{id}_{\overline{H_{\pi}}}}~{}d\nu(\pi)$,
$\lambda\cong\int_{\widehat{G}}^{\oplus}{\operatorname{id}_{H_{\pi}}}\otimes\overline{\pi}~{}d\nu(\pi)$.

The measure $\nu$ above is called the Plancherel measure on $\widehat{G}$.
Note if $G$ is abelian, the Plancherel measure is a Haar measure; if $G$ is
compact, the Plancherel measure is given by $\nu(E)=\sum_{\pi\in
E}\dim_{\mathbb{C}}{H_{\pi}}$ for $E\subset\widehat{G}$. The support of $\nu$
is not always all of $\widehat{G}$. Indeed, ${\rm supp}(\nu)=\widehat{G}$ if
and only if $G$ is amenable (see [8]).

######  Remark 2.2.

For a Hilbert space $H$, let $H^{*}$ be its dual space, which is isomorphic to
the conjugate space $\overline{H}$. We may identify the tensor Hilbert space
$H\otimes H^{*}$ with the Hilbert-Schmidt operators on $H$, denoted by
$B(H)_{\rm HS}$, which is equipped with the inner product $\langle
x,y\rangle_{B(H)_{\rm HS}}=\operatorname{Tr}(xy^{*})$. Taking an orthonormal
basis $\\{e_{i}\\}_{i\geq 1}$ of $H$, there is an isometric isomorphism
$\Psi\colon B(H)_{\rm HS}\to H\otimes H^{*}$ given as

$\Psi(T)=\sum_{i,j}\langle Te_{j},e_{i}\rangle\cdot e_{i}\otimes e_{j}^{*}$,
$T\in B(H)_{\rm HS}$,

such that $\operatorname{Tr}(TT^{*})=\|\Psi(T)\|_{H\otimes H^{*}}^{2}$.

Now let $F$ be a number field and $G$ be a reductive linear algebraic group
defined over $F$. We let $V$ ($V_{f}$ and $V_{\infty}$, respectively) denote
the set of equivalence classes of places (finite places and infinite places,
respectively) of $F$ and $F_{v}$ denotes the local fields at $v\in V$.

Let $G_{v}=G(F_{v})$ and $G(\mathbb{A}_{F})=\prod_{v\in
V_{\infty}}G(F_{v})\times\prod^{\prime}_{v\in V_{f}}G(F_{v})$, the group over
the adele $\mathbb{A}_{F}$ of $F$. For each $v\in V_{f}$, let $K_{v}$ be a
special open maximal compact subgroup of $G_{v}$. Indeed, for almost all $v$,
$K_{v}$ can be taken to be $G(\mathcal{O}_{v})$, the integral points of
$G_{v}$. For each finite place $v\in V_{f}$, we fix a Haar measure $\mu_{v}$
on the local group $G_{v}=G(F_{v})$. We may assume it is normalized in the
sense that $\mu_{v}(K_{v})=1$. Let $\nu_{v}$ be the Planacherel measure on
$\widehat{G_{v}}$ determined by $\mu_{v}$.

Denote an irreducible unitary representation of $G_{v}$ by
$(\pi_{v},H_{\pi_{v}})$, which will also stand for the isomorphism class in
the unitary dual $\widehat{G_{v}}$. Hence we have
$L^{2}(G_{v})=\int_{\widehat{G_{v}}}H_{\pi_{v}}\otimes
H_{\pi_{v}}^{*}d\nu_{v}(\pi_{v})$ by the Plancherel Theorem 2.1. Consider the
characteristic function $\chi_{v}$ of $K_{v}$, i.e.,

$\chi_{v}(x)=\begin{cases}1,&\text{if}\ x\in K_{v};\\\
0,&\text{otherwise}.\end{cases}$

Observe $\|\chi_{v}\|_{L^{2}(G_{v})}=\mu_{v}(K_{v})=1$.

Now we take an irreducible representation $(\pi_{v},H_{v})$ from
$\widehat{G_{v}}$ and consider the Fourier transform $\widehat{\chi_{v}}$ of
$\chi_{v}$ at $\pi_{v}$, which is given by

$\widehat{\chi_{v}}(\pi_{v})=\int_{x\in
G_{v}}\chi_{v}(x)\pi_{v}(x^{-1})d\mu_{v}(x)$.

Please note $\widehat{\chi_{v}}(\pi_{v})$ is a Hilbert-Schmidt operator, or
equivalently, a vector in $H_{\pi_{v}}\otimes H_{\overline{\pi_{v}}}$ almost
everywhere on $\widehat{G_{v}}$ (see 2.2).

######  Lemma 2.3.

If $\widehat{\chi_{v}}(\pi_{v})$ is nonzero, we have
$\|\widehat{\chi_{v}}(\pi_{v})\|_{H_{\pi_{v}}\otimes
H_{\overline{\pi_{v}}}}=1$.

Proof: Since the group $K_{v}$ is compact, the restriction $\pi_{v}|_{K_{v}}$
can be written as a direct sum of irreducible subrepresentations of $K_{v}$,
i.e., $\pi_{v}|_{K_{v}}=\bigoplus_{i\in I}\rho_{i}$. Then the Fourier
transform of the characteristic function $\chi_{v}$ can be obtained as
follows:

$\displaystyle\widehat{\chi_{v}}(\pi_{v})$
$\displaystyle=\int_{G_{v}}\chi_{v}(x)\pi_{v}(x^{-1})d\mu_{v}(x)=\int_{K_{v}}\pi_{v}(x^{-1})d\mu_{v}(x)$
$\displaystyle=\bigoplus_{i\in I}\int_{x\in
K_{v}}\rho_{i}(x^{-1})d\mu_{v}(x).$

If $\rho_{i}$ is a nontrivial irreducible representation, we have
$\int_{K_{v}}\rho(x^{-1})d\mu_{v}(x)=0$. But for the trivial representation
${\rm 1}_{K_{v}}$, $\int_{K_{v}}{\rm 1}_{K_{v}}(x^{-1})d\mu_{v}(x)=1$.
Therefore $\widehat{\chi_{v}}(\pi_{v})$ is the multiplicity of ${\rm
1}_{K_{v}}$ in $\pi_{v}$.

It is well-known that an irreducible unitary representation of $G(F_{v})$
contains ${\rm 1}_{K_{v}}$ at most once (which are the unramified ones, see
[9] Theorem 2). Hence $\widehat{\chi_{v}}(\pi_{v})$ is $0$ or a projection on
$B(H_{\pi_{v}})$ with rank $1$, or equivalently, the zero vector or a unit
vector in the tensor space $H_{\pi_{v}}\otimes H_{\overline{\pi_{v}}}$.

Let $X_{v}\subset\widehat{G_{v}}$ be the subset of unramified representations
of $G_{v}$.

######  Corollary 2.4.

We have $\nu_{v}(X_{v})=1$, the Plancherel measure of unramified
representations is $1$.

Proof: It follows the fact that

$1=\|\chi_{v}\|_{L^{2}(G_{v})}^{2}=\int_{\widehat{G_{v}}}\|\widehat{\chi_{v}}(\pi_{v})\|^{2}d\nu_{v}(\pi_{v})$,

where $\|\widehat{\chi_{v}}(\pi_{v})\|=1$ for unramified representations and
$\|\widehat{\chi_{v}}(\pi_{v})\|=0$ otherwise.

######  Remark 2.5 (The restricted product of measure spaces [5]).

Let $\\{(X_{i},\mu_{i}):i\in I\\}$ be a set of measure spaces with subsets
$Y_{i}\subset X_{i}$ such that $\mu_{i}(Y_{i})=1$ for all but finitely many
$i\in I$. Write $(X_{S},\nu_{S})=(\prod_{i\in S}X_{i})\times(\prod_{i\notin
F}Y_{i})$ for a finite subset $S\subset I$. Note for $S^{\prime}\subset S$,
there is a natural embedding of measure spaces $X_{S^{\prime}}\subset X_{S}$.
The restricted product of $\\{(X_{i},\mu_{i}):i\in I\\}$ with respect to
$\\{Y_{i}:i\in I\\}$ is defined by

$\prod^{\prime}_{i\in
I}(X_{i},Y_{i},\mu_{i})=\varinjlim_{S}X_{S}=\\{(x_{i})_{i\in I}\in\prod
X_{i}|x_{i}\in Y_{i}\text{~{}for~{}almost~{}all~{}}i\\}$,

which will also be denoted simply by $(X,\nu)$. A subset $Z\subset X$ is
measurable if and only if $Z\cap X_{S}$ is measurable for all finite $S$ and
its measure is given by $\mu(S)=\sup_{S}\nu_{S}(Z\cap X_{S})$.

Now we take the restricted product of the measures $\\{\mu_{v}\\}_{v\in V}$ as
a Haar measure on the adelic group $G(\mathbb{A})$, denoted by
$\mu_{G_{\mathbb{A}}}$, or simply $\mu$. Let $\nu_{G(\mathbb{A})}$ be the
Plancherel measure of $\widehat{G_{\mathbb{A}}}$ determined by $\mu$.

######  Theorem 2.6.

$(\widehat{G_{\mathbb{A}}},\nu_{G(\mathbb{A})})=\prod^{\prime}_{v\in
V}(\widehat{G_{v}},X_{v},\nu_{v})$.

Proof: We have
$L^{2}(G_{\mathbb{A}},\mu_{\mathbb{A}})=\bigotimes^{\prime}_{v\in
V}(L^{2}(G_{v}),\chi_{v})$ by [5] Corollary 2.3, where the restriced product
is taken with respect to the unit vectors $\widehat{\chi_{v}}$. This can be
rewritten as

$L^{2}(G_{\mathbb{A}})=\bigotimes^{\prime}_{v}\left(\int_{\widehat{G_{v}}}H_{\pi_{v}}\otimes
H_{\overline{\pi_{v}}}d\nu_{v}(\pi_{v})\right)$.

As shown in [9], an irreducible representation $\pi$ of $G(\mathbb{A})$ can be
written as $\pi\cong\bigotimes\pi_{v}$, where each $\pi_{v}$ is an irreducible
unitary representation of $G_{v}$. Almost all $\pi_{v}$ are unramified, i.e.,
$\pi_{v}\in X_{v}$ for all but finitely many $v$.

Take a finite set $S\subset V$ and consider the function $h_{S}=(\otimes_{v\in
S}~{}h_{v})\otimes(\otimes_{v\in V\backslash S}~{}\chi_{v})$ with
$h_{v}\in\mathcal{J}^{2}_{G_{v}}$. For any $x=(x_{v})_{v\in V}\in
G_{\mathbb{A}}$, we let $S_{x}=\\{v\in V|x_{v}\notin K_{v}\\}$, the finite set
of the places $v$ whose local coordinate $x_{v}$ is outside of the compact
group $K_{v}$. Then the Fourier inversion formula of $h$ at $x$ is given by

$\displaystyle h((x_{v})_{v\in V})=$
$\displaystyle\int_{\widehat{G_{\mathbb{A}}}}\operatorname{Tr}(\widehat{h}(\pi))\pi((x_{v})_{v\in
V})^{*}d\nu_{\mathbb{A}}(\pi)$ (1) $\displaystyle=$
$\displaystyle\int_{\widehat{G_{\mathbb{A}}}}\left(\prod_{v\in
S}\operatorname{Tr}(\widehat{h_{v}}(\pi_{v}))\pi_{v}(x_{v})^{*})\cdot\prod_{v\in
V\backslash
S}\operatorname{Tr}(\widehat{\chi_{K_{v}}}(\pi_{v})\pi_{v}(x_{v})^{*})\right)d\nu_{\mathbb{A}}(\otimes\pi_{v})$
$\displaystyle=$
$\displaystyle\int_{\widehat{G_{\mathbb{A}}}}\left(\prod_{v\in
S}\operatorname{Tr}(\widehat{h_{v}}(\pi_{v}))\pi_{v}(x_{v})^{*})\cdot\prod_{v\in
S_{x}\backslash
S}\operatorname{Tr}(\widehat{\chi_{K_{v}}}(\pi_{v})\pi_{v}(x_{v})^{*})\right)\cdot$
$\displaystyle~{}~{}~{}~{}~{}~{}\left(\prod_{v\in V\backslash(S\cup
S_{x})}\delta_{X_{v}}(\pi_{v})\right)d\nu_{\mathbb{A}}(\otimes^{\prime}\pi_{v})$

Observing that for $x_{v}\in K_{v}$,
$\widehat{\chi_{v}}(\pi_{v})\circ\pi_{v}(x_{v})=\int_{K_{v}}\pi_{v}(y^{-1}x_{v})d\mu_{v}(y)$
depends only on $\pi_{v}$, we have $\delta_{X_{v}}(\pi_{v})=1$ or $0$ depends
on whether $\pi_{v}$ is unramified or not. On the other hand, we have

$\displaystyle h((x_{v})_{v\in V})=$ $\displaystyle\prod_{v\in
S}h_{v}(x_{v})\cdot\prod_{v\in V\backslash S}\chi_{K_{v}}(x_{v})$ (2)
$\displaystyle=$ $\displaystyle\prod_{v\in
S}\int_{\widehat{G_{v}}}\operatorname{Tr}(\widehat{h_{v}}(\pi_{v})\pi_{v}(x_{v})^{*})d\nu_{v}(\pi_{v})\cdot\prod_{v\in
V\backslash
S}\int_{\widehat{G_{v}}}\operatorname{Tr}(\widehat{\chi_{K_{v}}}(\pi_{v})\pi_{v}(x_{v})^{*})d\nu_{v}(\pi_{v})$
$\displaystyle=$ $\displaystyle\prod_{v\in
S}\int_{\widehat{G_{v}}}\operatorname{Tr}(\widehat{h_{v}}(\pi_{v})\pi_{v}(x_{v})^{*})d\nu_{v}(\pi_{v})\cdot\prod_{v\in
S_{x}\backslash
S}\int_{\widehat{G_{v}}}\operatorname{Tr}(\widehat{\chi_{K_{v}}}(\pi_{v})\pi_{v}(x_{v})^{*})d\nu_{v}(\pi_{v})$
$\displaystyle~{}~{}~{}~{}~{}~{}\cdot\prod_{v\in V\backslash(S\cup
S_{x})}\int_{\widehat{G_{v}}}\delta_{X_{v}}(\pi_{v})d\nu_{v}(\pi_{v}),$

where the last product is $1$ by Corollary 2.4.

Observe the Plancherel measure is uniquely determined by the Fourier inversion
formula. The functions of the same form as $h$ above span a dense subset of
the $L^{2}$-space. Comparing the equation (1) with (2), we conclude
$(\widehat{G_{\mathbb{A}}},\nu_{\mathbb{A}})=\prod^{\prime}(\widehat{G_{v}},X_{v},\nu_{v})$.

Note an irreducible unitary representation $\pi=\otimes\pi_{v}$ is a discrete
series representation (an irreducible direct summand in the $L^{2}$-space) if
and only if $\nu_{\mathbb{A}}(\\{\pi\\})>0$. But $\nu_{v}(\pi_{v})=0$ for
almost all $v$. Hence $\nu_{\mathbb{A}}(\pi)=\prod^{\prime}\nu_{p}(\pi_{v})=0$
and we obtain the following known result:

######  Corollary 2.7.

$G(\mathbb{A})$ has no discrete series representations.

##  3 The von Neumann dimensions of direct integrals

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 (hence it is 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 [1] Chapter 8).

######  Proposition 3.1.

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 [13,
21]) 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 [3]).

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 4.2).

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

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 $\pi\in X$. Suppose
$\\{e_{k}(\pi)\\}_{k\geq 1}$ is an orthonormal basis of its underlying Hilbert
space $H_{\pi}$. We have the following natural $G$-equivariant isometric
isomorphism from $H_{X}$ to a subspace of $L^{2}(G)$:

$\displaystyle H_{X}{}{}{}{}{}{}{}{}{}$
$\displaystyle\cong~{}~{}~{}\int_{X}^{\oplus}H_{\pi}\otimes
e_{1}(\pi)^{*}d\nu(\pi)$ (3) $\displaystyle
v=\int_{X}^{\oplus}v(\pi)d\nu(\pi){}{}$
$\displaystyle\mapsto~{}~{}~{}\int_{X}v(\pi)\otimes e_{1}(\pi)d\nu(\pi),$

Therefore we will not distinguish these two spaces and denote them both by
$H_{X}$.

Consider the projection $P_{X}\colon L^{2}(G)\to H_{X}$ defined on a dense
subspace of $L^{2}(G)$ as follows:

$\int_{\widehat{G}}^{\oplus}\left(\sum\limits_{i,j\geq
1}a_{i,j}(\pi)e_{j}(\pi)\otimes
e_{i}(\pi)^{*}\right)d\nu(\pi)\mapsto\int_{\widehat{X}}^{\oplus}\left(\sum\limits_{j\geq
1}a_{1,j}(\pi)e_{j}(\pi)\otimes e_{1}(\pi)^{*}\right)d\nu(\pi)$ ,

where all but finite $a_{i,j}(\pi)\in\mathbb{C}$ are zero for each $\pi$.
Observe that $P_{X}$ commutes with the right regular representation $\rho$,
i.e., $\rho\circ P_{X}=P_{X}\circ\rho$. We have a unitary representation of
$G$ on $H_{X}$ and denote it by $\rho_{X}$.

Given two vectors $v=\int_{X}^{\oplus}v(\pi)d\nu(\pi)$ and
$w=\int_{X}^{\oplus}w(\pi)d\nu(\pi)$ in $H_{X}$ with $v(\pi),w(\pi)\in
H_{\pi}$, we have $v(\pi)\otimes w(\pi)^{*}\in H_{\pi}\otimes H_{\pi}^{*}$.
The corresponding Hilbert-Schmidt operator will also be denoted by
$v(\pi)\otimes w(\pi)^{*}$ (see Remark 2.2). We define a function on $G$ by

$C_{v,w}(g)=\langle\rho_{X}(g^{-1})v,w\rangle_{H_{X}}=\int_{X}\operatorname{Tr}(\pi(g)^{*}v(\pi)\otimes
w(\pi)^{*})d\nu(\pi)$.

######  Lemma 3.2.

For $v,w\in H_{X}$, we have $C_{v,w}\in L^{2}(G)$. Moreover,

$\langle
C_{v_{1},w_{1}},C_{v_{2},w_{2}}\rangle_{L^{2}(G)}=\int_{X}\operatorname{Tr}(v_{1}(\pi)\otimes
w_{1}(\pi)^{*}\cdot w_{2}(\pi)\otimes v_{2}(\pi)^{*})d\nu(\pi)$,

for any vectors $v_{1},v_{2},w_{1},w_{2}\in H_{X}$.

Proof: Since $v_{i}(\pi)\otimes w_{i}(\pi)^{*}$ is Hilbert-Schmidt,
$v_{1}(\pi)\otimes w_{1}(\pi)^{*}\cdot w_{2}(\pi)\otimes v_{2}(\pi)^{*}$ is a
trace-class operator. The integral is bounded by the norm of $v,w$ in $H_{X}$
and hence well-defined. As $v(\pi)=w(\pi)=0$ for $\pi\in\widehat{G}-X$, we
have

$C_{v,w}(g)=\int_{X}\operatorname{Tr}(\pi(g)^{*}v(\pi)\otimes
w(\pi)^{*})d\nu(\pi)=\int_{\widehat{G}}\operatorname{Tr}(\pi(g)^{*}v(\pi)\otimes
w(\pi)^{*})d\nu(\pi)$.

By Theorem 2.1, we conclude that $v\otimes w^{*}=\int_{X}v(\pi)\otimes
w(\pi)^{*}d\nu(\pi)=\widehat{C_{v,w}}$, the Fourier transform of the function
$C_{v,w}$ on $G$. Thus we obtain

$\displaystyle\langle C_{v_{1},w_{1}},C_{v_{2},w_{2}}\rangle_{L^{2}(G)}$
$\displaystyle=\int_{\widehat{G}}\operatorname{Tr}\left(\widehat{C_{v_{1},w_{1}}}(\pi)\widehat{C_{v_{1},w_{1}}}(\pi)^{*}\right)d\nu(\pi)$
$\displaystyle=\int_{X}\operatorname{Tr}\left(\widehat{C_{v_{1},w_{1}}}(\pi)\widehat{C_{v_{1},w_{1}}}(\pi)^{*}\right)d\nu(\pi)$
$\displaystyle=\int_{\widehat{X}}\operatorname{Tr}\left(v_{1}(\pi)\otimes
w_{1}(\pi)^{*}\cdot w_{2}(\pi)\otimes v_{2}(\pi)^{*}\right)d\nu(\pi)$

Then it is easy to get $C_{v,w}\in L^{2}(G)$.

Let $\\{d_{k}\\}_{k\geq 1}$ be an orthonormal basis of $L^{2}(D)$. Note the
restricted representation $\rho_{X}|_{\Gamma}$ makes the Hilbert space $H_{X}$
an $M$-module, whose von Neumann dimension can be obtained as follows.

######  Lemma 3.3.

With the assumption above, we have

$\dim_{M}(H_{X})=\sum_{k\geq 1}\|Pd_{k}\|_{H_{X}}^{2}$.

Proof: Let $u$ be the inclusion $H_{X}\to L^{2}(G)$. We have $u^{*}u={\rm
id}_{H_{X}}$ and $uu^{*}=P_{X}$. Note $L^{2}(G)\cong L^{2}(M)\otimes
L^{2}(D,dg)$, where $L^{2}(M)$ is the standard $M$-module and $L^{2}(D,dg)$ is
regarded as a trivial $M$-module. Thus, by definition (see Proposition 3.1),
we know

$\dim_{M}(H_{X})=\operatorname{Tr}_{M^{\prime}\cap B(L^{2}(G))}(P)$,

where $M^{\prime}\cap B(L^{2}(G))=\\{T\in B(L^{2}(G))|Tx=xT,~{}\forall x\in
M\\}$, the commutant of $M$ on $L^{2}(G)$. On the right-hand side,

$\operatorname{Tr}_{M^{\prime}\cap
B(L^{2}(G))}=\operatorname{tr}_{M^{\prime}\cap
B(L^{2}(M))}\otimes\operatorname{Tr}_{B(L^{2}(D))}$

is the natural trace on $M^{\prime}$.

The commutant $M^{\prime}$ is generated by the finite sums of the form

$x=\sum_{\gamma\in\Gamma}\rho_{\gamma}\otimes a_{\gamma}$,

where $\rho_{\gamma}=J\lambda(\gamma)J\in M^{\prime}\cap L^{2}(M)$ (here
$J\colon L^{2}(M)\to L^{2}(M)$ is the conjugate linear isometry extended from
$x\mapsto x^{*}$) and $a_{\gamma}$ is a finite rank operator in $B(L^{2}(D))$.

Let $d_{m}^{*}\otimes d_{n}$ denotes the operator $\xi\mapsto\langle
d_{m},\xi\rangle\cdot d_{n}$ on $L^{2}(D)$. Then each $a_{\gamma}$ can be
written as $a_{\gamma}=\sum_{m,n\geq 1}a_{\gamma,m,n}d_{m}^{*}\otimes d_{n}$
with $a_{\gamma,m,n}\in\mathbb{C}$ and all but finite terms of $a_{m,n}$ are
trivial. Thus we obtain

$\operatorname{Tr}_{M^{\prime}}(\rho_{\gamma}\otimes
a_{\gamma})=\operatorname{tr}_{M}(\lambda_{\gamma})\sum_{m\geq
1}a_{\gamma,m,m}=\delta_{e}(\lambda)\operatorname{Tr}_{L^{2}(D)}(a_{\gamma})$.

This is equivalent to say

$\operatorname{Tr}_{M^{\prime}}(x)=\operatorname{Tr}_{L^{2}(D)}(a_{e})$.

Let $Q$ be the projection of $L^{2}(G)$ onto
$L^{2}(D)\cong\mathbb{C}\delta_{e}\otimes L^{2}(D)$. Then
$\operatorname{Tr}_{L^{2}(D)}=\operatorname{Tr}_{L^{2}(G)}(QyQ)$ for $y\in
B(L^{2}(D))$. We have

$\operatorname{Tr}_{M^{\prime}}(x)=\operatorname{Tr}_{L^{2}(D)}(a_{e})=\operatorname{Tr}_{L^{2}(G)}(Qa_{e}Q)=\operatorname{Tr}_{L^{2}(G)}(QxQ)$
(4)

As $P$ is a strong limit of elements that have the same form as $x$ above and
the traces are normal, formula (4) holds for $x=P$ and we obtain

$\displaystyle\dim_{M}(H_{X})$
$\displaystyle=\operatorname{Tr}_{M^{\prime}}(P)=\operatorname{Tr}_{L^{2}(G,dg)}(QPQ)$
$\displaystyle=\sum\nolimits_{k\geq 1}\langle
QPQd_{k},d_{k}\rangle_{L^{2}(G)}=\sum\nolimits_{k\geq 1}\langle
Qd_{k},PQd_{k}\rangle_{L^{2}(G)}$ $\displaystyle=\sum\nolimits_{k\geq
1}\langle d_{k},Pd_{k}\rangle_{L^{2}(G)}=\sum\nolimits_{k\geq 1}\langle
Pd_{k},Pd_{k}\rangle_{L^{2}(G)}$ $\displaystyle=\sum\nolimits_{k\geq 1}\langle
Pd_{k},Pd_{k}\rangle_{H_{X}}=\sum\nolimits_{n\geq 1}\|Pd_{k}\|_{H_{X}}^{2}$

##  4 A product formula

######  Theorem 4.1.

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

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

Proof: We take a vector $\eta=\int_{X}^{\oplus}\eta(\pi)d\nu(\pi)$ in $H_{X}$
such that $\|\eta(\pi)\|_{H_{\pi}}^{2}=\frac{1}{\nu(X)}$ almost everywhere in
$X$. Then $\eta$ is a unit vector in $H_{X}$ and also in $L^{2}(G)$.

Observe $\\{\delta_{\gamma}\otimes d_{k}\\}_{\gamma\in\Gamma,n\geq 1}$ is an
orthogonal basis of $L^{2}(G,\mu)$ via the isomorphism $L^{2}(G)\cong
l^{2}(\Gamma)\otimes L^{2}(D,\mu)$. Note $\\{d_{k}\\}_{k\geq 1}$ can be
regarded as functions in $L^{2}(G)$ with support in $D$. We have
$\delta_{\gamma}\otimes d_{k}=\rho_{G}(\gamma)d_{k}$ for $k\geq 1$ and
$\gamma\in\Gamma$. Hence we have

$1=\|\rho_{X}(g)\eta\|_{H_{X}}^{2}=\|\rho(g)\eta\|_{L^{2}(G)}^{2}=\sum_{\gamma\in\Gamma,k\geq
1}|\langle\rho(g)\eta,\rho(\lambda)d_{k}\rangle_{L^{2}(G)}|^{2}$.

Consequently, we obtain:

$\displaystyle{\rm covol}(\Gamma)$
$\displaystyle=\int_{D}1d\mu(g)=\int_{D}\sum_{\gamma\in\Gamma,k\geq
1}|\langle\rho(\lambda)^{*}\rho(g)\eta,d_{k}\rangle|^{2}d\mu(g)$
$\displaystyle=\sum_{k\geq 1}\int_{G}|\langle
P\rho(g)\eta,d_{k}\rangle_{L^{2}(G)}|^{2}d\mu(g)=\sum_{k\geq
1}\int_{G}|\langle\rho(g)\eta,Pd_{k}\rangle_{H_{X}}|^{2}d\mu(g)$
$\displaystyle=\sum_{k\geq
1}\int_{G}|\langle\rho(g)^{*}Pd_{k},\eta\rangle|^{2}d\mu(g)=\sum_{k\geq
1}\langle C_{Pd_{k},\eta},C_{Pd_{k},\eta}\rangle_{L^{2}(G)}$
$\displaystyle=\sum_{k\geq
1}\int_{X}\operatorname{Tr}((Pd_{n})(\pi)\otimes\eta(\pi)^{*}\cdot(\eta(\pi)\otimes(Pd_{k})(\pi)^{*})d\nu(\pi)$
$\displaystyle=\sum_{k\geq
1}\int_{X}\langle(Pd_{n})(\pi)\otimes\eta(\pi)^{*},(Pd_{n})(\pi)\otimes\eta(\pi)^{*}\rangle_{H_{\pi}\otimes
H_{\pi}^{*}}d\nu(\pi)$ $\displaystyle=\sum_{k\geq
1}\int_{X}\|\eta(\pi)\|_{H_{\pi}}^{2}\cdot\|(Pd_{k})(\pi)\|_{H_{\pi}}^{2}d\nu(\pi)$
$\displaystyle=\frac{1}{\nu(X)}\sum_{n\geq 1}\|Pd_{k}\|_{H_{X}}^{2},$

which is $\dim_{M}(H_{X})\cdot\nu(X)^{-1}$ by Lemma 3.3. Hence we get
$\dim_{M}(H_{X})=\operatorname{covol}(\Gamma)\cdot\nu(X)$.

######  Remark 4.2.

  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.

The conditions of $G$ above are satisfied for all connected semisimple Lie
groups and connected reductive algebraic groups over local fields and adelic
rings (see Section 2).

If $\pi$ is an atom in $\widehat{G}$, i.e., $\nu(\\{\pi\\})>0$, we can show
$\pi$ is a discrete series representation and $\nu(\\{\pi\\})$ is just the
formal dimension of $\pi$ [8, 24]. 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 [12] Theorem
3.3.2)

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

This is motivated by the study of discrete series of Lie groups by M. Atiyah
and W. Schmid [2]. (See also [25] for a discussion of the $p$-adic
$\operatorname{GL}(2)$ together with its cuspidal and Steinberg
representations.)

It is known that a Lie group $G$ has discrete series if and only if
$\operatorname{rank}G=\operatorname{rank}K$ for a maximal compact subgroup
$K$. Thus it excludes a large family of Lie groups, e.g,
$\operatorname{SL}_{n}(\mathbb{R})$ ($n\geq 2$),
$\operatorname{SO}(n,\mathbb{C})$, $\operatorname{GL}_{n}(\mathbb{C})$. A
natural question is what is the analog for other Lie groups or other
irreducible representations that are not discrete series. This is one of the
motivations for this result.

Before closing this section, I would like to introduce an example of
$\operatorname{SL}(2,\mathbb{R})$ as an application of Theorem 4.1.

######  Example 4.3.

Consider $G=\operatorname{SL}(2,\mathbb{R})$ and
$\mathcal{H}=\\{x+iy|x,y\in\mathbb{R},y>0\\}$, the Poincaré upper-half plane
with a $\operatorname{SL}(2,\mathbb{R})$-invariant measure $y^{-2}dxdy$.
Observe that $\operatorname{PSL}(2,\mathbb{R})/\\{\operatorname{SO}(2)/\\{\pm
1\\}\\}\cong\mathcal{H}$. We fix a Haar measure on
$\operatorname{SO}(2)/\\{\pm 1\\}$ with total measure $1$ and then take the
Haar measure $\mu$ on $\operatorname{PSL}(2,\mathbb{R})$ to be the product
measure on $\mathcal{H}\times\\{\operatorname{SO}(2)/\\{\pm 1\\}\\}$. Let
$\nu$ be the Plancherel measure given by $\mu$.

The classification of irreducible unitary representations of
$\operatorname{SL}(2,\mathbb{R})$ is well-known (see [11]). However, the
support of the Plancherel measure $\nu$ on the unitary dual
$\widehat{\operatorname{SL}(2,\mathbb{R})}$ contains only the following two
families of irreducible representations (see [10, 17]).

  1. 1.

Discrete series representations: $\\{\pi_{n}^{\pm}|n\geq 2\\}$. The underlying
space of $\pi_{n}^{+}$ is given as

$H_{n}^{+}=\\{f\colon\mathcal{H}\to\mathbb{C}\text{
holomorphic}~{}|\int_{\mathcal{H}}|f(x+iy)|^{2}y^{n-2}dxdy<\infty\\}$.

For $g=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\operatorname{SL}(2,\mathbb{R})$ and $f\in H_{n}^{+}$, the
action is given by
$(\pi_{k}^{+}(g)f)(z)=(bz+d)^{-k}f\left(\frac{az+b}{cz+d}\right)$. The
representation $\pi_{n}^{-}$ is defined to be the contragradient of
$\pi_{n}^{+}$. We have
$\nu(\\{\pi_{n}^{+}\\})=\nu(\\{\pi_{n}^{-}\\})=\frac{n-1}{4\pi}$ for $n\geq
2$.

  2. 2.

Principal series representations: $\\{\pi^{\pm}_{it}|t>0\\}$. The spaces
$\\{H_{it}^{\pm}\\}_{t>0}$ for them are all $L^{2}(\mathbb{R})$ with distinct
actions given by

$\pi^{+}_{it}\begin{pmatrix}a&b\\\
c&d\end{pmatrix}f(x)=|bx+d|^{-1-it}f\left(\frac{ax+c}{bx+d}\right)$,

$\pi^{-}_{it}\begin{pmatrix}a&b\\\ c&d\end{pmatrix}f(x)={\rm
sgn}(bx+d)|bx+d|^{-1-it}f\left(\frac{ax+c}{bx+d}\right)$.

Let $P=\\{g=\begin{pmatrix}a&b\\\ 0&a^{-1}\end{pmatrix}|a,b\in\mathbb{R},a\neq
0\\}$ with characters given by $\chi_{t}^{\pm}\begin{pmatrix}a&b\\\
0&a^{-1}\end{pmatrix}=\varepsilon^{\pm}(a)|a|^{it}$, where
$\varepsilon^{+}(a)=1$ and $\varepsilon^{-}(a)=\rm{sgn}(a)$. One can show
$\pi^{\pm}_{it}=\operatorname{Ind}_{P}^{G}(\chi_{t}^{\pm})$. We have
$d\nu(\pi_{t}^{+})=\frac{t}{8\pi}\tanh{\frac{\pi t}{2}}dt$ and
$d\nu(\pi_{t}^{-})=\frac{t}{8\pi}\coth{\frac{\pi t}{2}}dt$.

For the lattice $\operatorname{PSL}(2,\mathbb{Z})$, we have
$\operatorname{covol}(\Gamma)=\frac{\pi}{3}$. By Theorem 4.1, we list the
dimensions of some modules over
$\mathcal{L}(\operatorname{PSL}(2,\mathbb{Z}))$ (which is a factor of type
${\rm II}_{1}$).

  1. 1.

(Discrete part) Let $X=\\{\pi_{n_{1}}^{+},\dots,\pi_{n_{k}}^{+}\\}$. We have
$H_{X}=\oplus_{1\leq i\leq k}\pi^{+}_{n_{i}}$. Hence

$\dim_{\mathcal{L}(\operatorname{PSL}(2,\mathbb{Z}))}H_{X}=\frac{\sum_{i=1}^{k}n_{i}}{12}-\frac{k}{12}$.

Note the dimension above remains the same if we replace any $\pi_{n_{i}}^{+}$
by its dual $\pi_{n_{i}}^{-}$.

  2. 2.

(Continuous part) Let $Y=\\{\pi^{+}_{it}|t\in[a,b]\\}$ for some $0<a<b$. We
have $H_{Y}=\int_{[a,b]}^{\oplus}H_{\pi_{it}^{+}}d\nu(\pi_{it}^{+})$. Hence

$\dim_{\mathcal{L}(\operatorname{PSL}(2,\mathbb{Z}))}H_{Y}=\frac{1}{24}\int_{a}^{b}t\tanh{\frac{\pi
t}{2}}dt$.

##  5 The dimension and the adelic Plancherel measure

Let $G$ be a reductive group defined over a number field $F$. Assume $G$ is of
dimension $d$ over $F$ so that we are able to take a nonzero left-invariant
$d$-form $\omega$ on $G$. For each infinite place $v\in V_{\infty}$, $\omega$
determines a left-invariant different form $\omega_{v}$ on the smooth manifold
$G(F_{v})$. This in turn gives a Haar measure $\mu_{v}$ of $G(F_{v})$. For
each finite place $v\in V_{f}$, let $p_{v}\in\mathbb{Z}$ be the prime number
lying under $v$. Let $\mu_{v}$ be the Haar measure on $G(F_{v})$, which is
uniquely determined by the corresponding $d$-form $\omega_{v}$ on the
$p_{v}$-adic analytic manifold $G(F_{v})$. The Tamagawa measure, denoted by
$\mu_{G(\mathbb{A}_{F})}$, or simply $\mu$, is defined to be the restricted
product of the measures $\\{\mu_{v}\\}_{v\in V}$ on the adelic group
$G(\mathbb{A}_{F})=\prod_{v\in V_{\infty}}G(F_{v})\times\prod^{\prime}_{v\in
V_{f}}G(F_{v})$, which is independent of the choice of $\omega$ [23].

The volume of $G(\mathbb{A}_{F})/G(F)$ (if it exists) is called the Tamagawa
number of $G$ and denoted by $\tau_{F}(G)$, or simply $\tau(G)$. In fact, for
any finite extension of number fields $L/F$, we can prove
$\tau_{L}(G)=\tau_{F}(G)$ [22]. It suffices to consider the case
$F=\mathbb{Q}$. The following result was conjectured by A. Weil and proved by
R. Langlands [20], K. Lai [19], R. Kottwitz [18] and V. Chernousov [6].

######  Theorem 5.1.

If $G$ is a simply connected semisimple group, then $\tau(G)=1$.

By fixing the Tamagama measure $\mu$ on $G(\mathbb{A}_{F})$ as a Haar measure
and the corresponding Plancherel measure $\nu$ on
$\widehat{G(\mathbb{A}_{F})}$, we have the following result.

######  Theorem 5.2.

Let $G$ be a simply connected semisimple algebraic group over a number field
$F$. Let $X\subset\widehat{G(\mathbb{A}_{F})}$ such that $\nu(X)<\infty$ and
$H_{X}=\int_{\pi\in X}^{\oplus}H_{\pi}d\nu_{G(\mathbb{A}_{F})}(\pi)$. We have

$\dim_{\mathcal{L}(G(F))}H_{X}=\nu_{G(\mathbb{A}_{F})}(X)$.

Proof: By Theorem 4.1, we have

$\dim_{\mathcal{L}(G(F))}H_{X}=\operatorname{covol}(G(F))\cdot\nu_{G(\mathbb{A}_{F})}(X)$.

Then it follows Theorem 5.1.

Here we give a criterion for the case when $\mathcal{L}(G(F))$ is a type
$\text{II}_{1}$ factor, which is related to the local groups at some infinite
places.

######  Proposition 5.3.

If there is an infinite place $v$ such that $G(F_{v})$ is center-free and
connected, $G(F)$ is an ICC group and hence $\mathcal{L}(G(F))$ is a
$\rm{II}_{1}$ factor.

Proof: First, we observe $G(F)$ is dense in each local group $G(F_{v})$ if the
place $v$ is archimedean ($v\in V_{\infty}$).

Take $\gamma\in G(F)$ and consider its conjugacy class $C_{\gamma}=\\{g\gamma
g^{-1}|g\in G(F)\\}$ in $G(F)$. The map $\gamma\to C_{\gamma}$ defined on
$G(F)$ can be extended to the group $G(F_{v})$ by continuity. If $C_{\gamma}$
is finite, $C_{\gamma}=\overline{C_{\gamma}}=\\{g\gamma g^{-1}|g\in
G(F_{v})\\}$, which can be denoted as $Y=\\{\gamma,g_{1}\gamma
g_{1}^{-1},\dots,g_{N}\gamma g_{N}^{-1}\\}$.

Let $G(F_{v})$ act on the set $Y$ by conjugation. The stabilizer of $\gamma$
is $Z_{\gamma}(G(F_{v}))=\\{g\in G(F_{v})|g\gamma g^{-1}=\gamma\\}$. Note
$N+1=|Y|=|G(F_{v})x|=|G(F_{v})/Z_{\gamma}(G(F_{v}))|$. That is to say,
$Z_{\gamma}(G(F_{v}))$ is a closed subgroup of $G(F_{v})$ of finite index.
Therefore $Z_{\gamma}(G(F_{v}))=G(F_{v})$ by the connectedness. As $G(F_{v})$
is center-free, $\gamma=e$ and $G(F)$ is an ICC group.

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