# Global Jacquet-Langlands Correspondence of Plancherel measures

Jun Yang

###### Abstract

We prove the global Jacquet-Langlands correspondence of $\operatorname{GL}(n)$
preserves each local Plancherel density if the Tamagawa measures are taken as
the Haar measures.

AMS 2010 Mathematics Subject Classification: 46L10, 20G05, 20G35. This work
was supported in part by the ARO Grant W911NF-19-1-0302 and the ARO MURI Grant
W911NF-20-1-0082.

##  1 Introduction

The irreducible representations of an algebraic group $G$ over a local field
$F_{v}$ are highly related to the irreducible representations of its inner
form, which is an $F_{v}$-group such that $G(\overline{F_{v}})\cong
H(\overline{F_{v}})$. For $G=\operatorname{GL}(n)$ and non-archimedean
$F_{v}$, the local Jacquet-Langlands correspondence gives a well-defined
pairing between the set of equivalence classes of square-integrable
representations of $\operatorname{GL}(n,F_{v})$ and the set of equivalence
classes of square-integrable representations of $\operatorname{GL}(r,D)$ for
some division algebra $D$ over $F_{v}$ such that $\dim_{F_{v}}D=(n/r)^{2}$
(see [10]).

Assuming $\pi$ and $\pi^{\prime}$ are matched under such a correspondence, it
is also known that their formal degrees are proportional, i.e.,
$\frac{\deg(\pi^{\prime})}{\deg(\pi)}$ is a constant that does not depend on
either $\pi$ or $\pi^{\prime}$ (see [21, 9]) (but depends on the Haar
measures). This result has been generalized to tempered representations with
their Plancherel measures in [2]. The key technique for the proof is the
comparison of the trace formulas of certain functions on the two different
groups (see [1, Chapter 2]).

One may expect such a matching of formal degrees or Plancherel measures for
the global Jacquet-Langlands correspondence, which establishes the paring of
(cuspidal) automorphic representations of the adelic groups. There are mainly
three key features to point out:

  1. 1.

There is only one canonical Haar measure, the Tamagawa measure on the adelic
group while the Haar measures on the local groups can be chosen with any
normalizations.

  2. 2.

The automorphic representations are unramified almost everywhere so they
cannot be square-integrable. We must compare the Plancherel measures instead
of formal degrees.

  3. 3.

For a local group over an archimedean field such as
$\operatorname{GL}(n,\mathbb{R})$, it has no square-integrable representations
(unless $n=1,2$) while the non-archimedean ones always have Steinberg
representations for us to compare their formal degrees.

By considering all these, we take local and global Tamagawa measures, consider
the Plancherel densities $\mathop{}\\!\mathrm{d}\nu_{v}$ with respect to the
canonical Harish-Chandra measures on the tempered dual of each local group
$G(F_{v})$ and finally obtain the following result.

######  Theorem.

Let $D$ be a division algebra over a number field $F$. Suppose
$\pi=\otimes\pi_{v}$ and $\pi^{\prime}=\otimes\pi^{\prime}_{v}$ are
automorphic representations of $\operatorname{GL}(n,\mathbb{A})$ and
$\operatorname{GL}(r,D_{\mathbb{A}})$, which are matched by the global
Jacquet-Langlands correspondence. If both $\pi,\pi^{\prime}$ are tempered
111Noting the generalized Ramanujan conjecture of $\operatorname{GL}(n)$ says
all the cuspidal representations of $\operatorname{GL}(n,\mathbb{A})$ are
tempered (see [22]), it is reasonable to assume both $\pi,\pi^{\prime}$ are
tempered., then $\pi_{v}$ and $\pi^{\prime}_{v}$ have the same Plancherel
density, i.e.,

$\mathop{}\\!\mathrm{d}\nu_{v}(\pi_{v})=\mathop{}\\!\mathrm{d}\nu^{\prime}_{v}(\pi^{\prime}_{v})$

for each place $v$.

Let us outline the proof of this result as well as the article. We review some
basic facts in Section 2, where the $\operatorname{GL}(2)$-case is stated as a
special case. We then discuss the local correspondences for the archimedean
and non-archimedean places in Section 3 and Section 4. In Section 5, we put
the local results together to reach the main result on the global
correspondence.

##  2 Preliminaries

We review some basic facts about division algebras over number fields and
Jacquet-Langlands correspondence of $\operatorname{GL}(n)$ with some necessary
notations included. An example on $\operatorname{GL}(2)$ is provided as a
special case of our main result.

###  2.1 Division algebras over number fields

Let $F$ be a number field and $\mathcal{O}_{F}$ be the ring of integers of
$F$. Let $V$ be the set of places of $F$, $V_{\infty}$ be the set of infinite
places of $F$ and $V_{f}$ be the set of finite places of $F$. For a place $v$
of $F$, we let $F_{v}$ be the completion of $F$ with respect to $v$. For $v\in
V_{f}$, we denote by $P_{v}$ the prime ideal in $\mathcal{O}_{F}$ associated
with $v$. Let $\mathcal{O}_{v}$ be the ring of integers of $F_{v}$,
$\mathfrak{m}_{v}$ be its maximal ideal and
$k(v)=\mathcal{O}_{v}/\mathfrak{m}_{v}\cong\mathcal{O}_{F}/P_{v}$ be its
residue field. Let $q_{v}$ be the cardinality of $k(v)$.

Let $A$ be a finite-dimensional simple algebra over $F$. By the Wedderburn-
Artin theorem, $A\cong\operatorname{Mat}(n,D)$ for some division algebra $D$
over $F$ such that the center $Z(D)\cong F$. We know $\dim_{F}D=d^{2}$ for
some positive integer $d$.

For each $v\in V$, let $A_{v}=A\otimes_{F}F_{v}$. We know that there is a
division algebra $D_{v}$ 222Here $D_{v}$ is not the algebra
$D\otimes_{F}F_{v}$ unless $n=1$.  over $F_{v}$ such that

$A_{v}\cong\operatorname{Mat}(n_{v},D_{v})$

for some $n_{v}\geq 1$ with $n_{v}$ divided by $n$. It is known that
$\dim_{F_{v}}D_{v}=d_{v}^{2}$ for some $d_{v}\geq 1$ which satisfies
$d_{v}n_{v}=dn$. We say $A$ (or $A^{\times}$) splits at $v$ if $d_{v}=1$ and
$n_{v}=nd$, in which case $A_{v}\cong\operatorname{GL}(nd,F_{v})$. We let

  * •

$S$ = the set of places at which $A$ (or $A^{\times}$) does not split.

It is known that $S$ is a finite subset of $V$. For $v\in V_{f}$, the
following notations will be used throughout this paper.

  * •

$O(D_{v}),O(A_{v})$ = the maximal order of $D_{v}$ and $A_{v}$, respectively.

  * •

$J(D_{v}),J(A_{v})$ = the maximal ideal of $O(D_{v})$ and $O(A_{v})$,
respectively.

Assume $K=F$ or $F_{v}$ for some $v\in V$. In the rest of this section, we
denote the $K$-algebra $A$ or $A_{v}$ simply by $A$. For $a\in A$, let
$\lambda_{a}\in\operatorname{Hom}_{K}(A,A)$ be the left multiplication by $a$.
We denote by $\operatorname{ch}_{A/K}(a;X)$ be the characteristic polynomial
of $\lambda$cqns write $\operatorname{ch}_{A/K}(a;X)=\sum_{0\leq i\leq
m}c_{m}(a)X^{m}$ with each $c_{i}(a)\in K$ and $c_{m}(a)=1$. We follow [19,
§2] for these definitions and notations.

  1. 1.

The trace $\operatorname{Tr}_{A/K}\colon A\to K$ is defined to be
$\operatorname{Tr}_{A/K}(a)=-c_{m-1}(a)$;

  2. 2.

The norm $\operatorname{Nm}_{A/K}\colon A\to K$ is defined to be
$\operatorname{Nm}_{A/K}(a)=(-1)^{m}c_{0}$.

It is known that there exists a finite extension $K_{1}$ of $K$ such that
$A\otimes_{K}K_{1}\cong\operatorname{Mat}(m,K_{1})$. The reduced
characteristic polynomial $\operatorname{rdch}_{A/K}(a;X)$ is defined to be
the characteristic polynomial of $1\otimes a$ under this isomorphism. It is
known that $\operatorname{rdch}(a;X)\in K[X]$ for each $a\in A$ and
$\operatorname{rdch}_{A/K}(a;X)^{m}=\operatorname{ch}_{A/K}(a;X)$.

  1. 3.

The reduced trace $\operatorname{tr}\colon A\to K$ is defined by
$\operatorname{tr}(a)=\operatorname{Tr}(a\otimes
1)=\frac{\operatorname{Tr}(a)}{m}$.

  2. 4.

The reduced norm $\operatorname{nr}\colon A\to K$ is defined by
$\operatorname{nr}(a)=\operatorname{Nm}(a\otimes
1)=\operatorname{Nm}(a)^{1/m}$.

Let $R$ be a noetherian domain and $K$ be the fraction field of $R$. Let $B$
be a semisimple algebra over $K$ and $I\subset B$ be an $R$-lattice. Suppose
$\dim_{K}B=m$. For $\alpha_{1},\cdots,\alpha_{m}\in B$, we let
$d(\alpha_{1},\cdots,\alpha_{m})=\det([\operatorname{tr}(\alpha_{i}\alpha_{j})]_{i,j})$.

  1. 5.

the discriminant $\operatorname{disc}_{R}(I)$ is the $R$-submodule of $F$
generated by the set

$\\{d(\alpha_{1},\cdots,\alpha_{m})\\}$.

It can be shown that if $I$ has a free $R$-basis $e_{1},\cdots,e_{m}$, then
$\operatorname{disc}_{R}(I)=\det([\operatorname{tr}(\alpha_{i}\alpha_{j})]_{i,j})\cdot
R$, which is a principal ideal (see [19, §10]).

  2. 6.

the absolute norm $\operatorname{N}(\mathfrak{a})$ of a fractional ideal
$\mathfrak{a}$ of $R$ is defined to be
$\operatorname{N}(\mathfrak{a})=[R\colon\mathfrak{a}]$.

  3. 7.

the norm on $A_{v}$ is defined to be
$\|x\|_{A_{v}}=\|\operatorname{Nm}_{A_{v}/F_{v}}(x)\|_{F_{v}}$.

###  2.2 Jacquet-Langlands correspondence of $\operatorname{GL}(n)$

Let $G=\operatorname{GL}(nd,F)$ and
$G^{\prime}=A^{\times}=\operatorname{GL}(n,D)$. For each $v\in V$, we let
$G_{v}=A^{\times}_{v}=\operatorname{GL}(nd,F_{v})$ and
$G^{\prime}_{v}=A^{\prime\times}_{v}=\operatorname{GL}(nr_{v},D_{v})$. Let
$\mathbb{A}$ be the adele ring of $F$. The following notations will be used
throughout this paper.

  * •

$G=\operatorname{GL}(nd,F)$, $G_{v}=\operatorname{GL}(nd,F_{v})$ and
$G_{\mathbb{A}}=\operatorname{GL}(nd,\mathbb{A})$

  * •

$G^{\prime}=GL(n,D)$, $G^{\prime}_{v}=\operatorname{GL}(nr_{v},D_{v})$ and
$G^{\prime}_{\mathbb{A}}=\operatorname{GL}(n,D(\mathbb{A}))$.

  * •

$Z(\mathbb{A})=$ the center of $G_{\mathbb{A}}$ (and also of
$G^{\prime}_{\mathbb{A}}$).

  * •

$\Pi_{0}(G),\Pi_{0}(G^{\prime})$ = the set of equivalence classes of
irreducible cuspidal subrepresentations of $L^{2}(G(F)Z(\mathbb{A})\backslash
G(\mathbb{A}))$ and $L^{2}(G^{\prime}(F)Z(\mathbb{A})\backslash
G^{\prime}(\mathbb{A}))$ respectively.

  * •

$\Pi(H),\Pi_{\operatorname{temp}}(H),\Pi_{2}(H)$ 333If the center of a group
$H$ is not compact, there will be no square-integrable representation and we
usually take the irreducible representations which are square-integrable
irreducible modulo the center of $H$ = the set of equivalence class of
unitary, tempered and square-integrable irreducible representations of a group
$H$ respectively.

  * •

$\mathcal{H}(G_{v})$ = the Hecke algebra of the group $G_{v}$ (smooth or
locally constant compactly supported complex functions on $G_{v}$ according
$F_{v}$ is archimdean or not, respectively). $\mathcal{H}(G^{\prime}_{v})$ is
defined similarly.

If $H$ is a linear algebraic group over a local field $K$, we let
$\theta_{\pi}$ be the character of $\pi$ defined on the set $H^{s}$ of regular
semisimple 444An element $g\in H$ is called regular semisimple if the
characteristic polynomial of $g$ has distinct roots over an algebraic closure
of $K$ group elements such that
$\operatorname{tr}\pi(f)=\int_{G^{s}}\theta_{\pi}(g)f(g)dg$. For $g\in G_{v}$
and $g^{\prime}\in G^{\prime}_{v}$, we say $g$ corresponds to $g^{\prime}$,
denoted by $g\leftrightarrow g^{\prime}$, if they have the same characteristic
polynomial.

######  Theorem 2.1.

  1. 1.

For each $v\in V$, there exists a unique injective map

$\operatorname{JL}_{v}\colon\Pi(G_{v})\to\Pi(G^{\prime}_{v})$

such that for each $\pi_{v}\in\Pi(G_{v})$ we have
$\theta_{\pi}(g)=(-1)^{nd-n}\theta_{\operatorname{JL}_{v}(\pi_{v})}(g^{\prime})$
for all $g\in G_{v}$ and $g^{\prime}\in G^{\prime}_{v}$ such that
$g^{\prime}\leftrightarrow g$.

  2. 2.

There exists a unique map

$\operatorname{JL}\colon\Pi_{0}(G^{\prime})\to\Pi_{0}(G)$

such that if $\pi=\otimes_{v\in V}\pi_{v}=\operatorname{JL}(\pi^{\prime})$ for
some $\pi^{\prime}=\otimes_{v\in V}\in\Pi_{0}(G^{\prime})$, then

    1. (a)

$\operatorname{JL}_{v}(\pi_{v})=\pi^{\prime}_{v}$ if $v\in S$;

    2. (b)

$\pi_{v}\cong\pi^{\prime}_{v}$ if $v\notin S$.

This theorem is proved for $\operatorname{GL}(2)$ by H.Jacquet and R.Langlands
in [14] and extended to $\operatorname{GL}(n)$ by I. Badulecu and D. Renard in
[4, 6] for number fields. The function field case is proved by I. Badulescu
and P. Roche in [7].

###  2.3 An example on $\operatorname{GL}(2)$

In this section, we restrict ourselves to a quaternion algebra $D$ over the
number field $F$. We follow [17, 24] for harmonic analysis on it. We take the
Haar measures on the groups $D^{\times}(F_{v})$ and
$\operatorname{GL}(2,F_{v})$ as follows.

  1. 1.

If $v$ is archimedean, we let

$\mathop{}\\!\mathrm{d}\mu_{v}(x)=\frac{|\det(\operatorname{tr}(e_{i}e_{j}))|^{1/2}}{\|x\|_{D_{v}}}\cdot\prod\mathop{}\\!\mathrm{d}x_{i}$,

where $\\{e_{i}\\}$ is an $\mathbb{R}$-basis of $D_{v}$ and $x=\sum
x_{i}e_{i}\in D_{v}$.

  2. 2.

If $v$ is non-archimedean, we choose an additive Haar measure
$\mathop{}\\!\mathrm{d}x_{v}$ on $D_{v}$ such that the volume of a maximal
order is $1$. Assume $v$ lies above a prime number $p$. We define the
multiplicative Haar measure

$\mathop{}\\!\mathrm{d}\mu_{v}(x)=\frac{\|\det(\operatorname{tr}(e_{i}e_{j}))\|_{\mathbb{Q}_{p}}^{1/2}}{(1-q_{v}^{-1})\|x\|_{D_{v}}}\mathop{}\\!\mathrm{d}x_{v}$,

where $\\{e_{i}\\}$ is a $\mathbb{Z}_{p}$-basis of the maximal order.

We can show these Haar measures are constructed from the self-dual additive
measures with respect to the canonical characters on $D_{v}$ (see [17, Lemma
7.5.6]). Hence these $\mu_{v}$ are called the local Tamagawa measures of
$D^{\times}$. Moreover, the product $\mu=\prod_{v\in V}\mu_{v}$ is the
Tamagawa measure on $D^{\times}(\mathbb{A})$ (see [17, §7.6]).

Let $\deg(\pi)$ be the formal degree of an irreducible square-integrable
representation $\pi$. For $v\in S$, we let

$\operatorname{JL}_{v}\colon\Pi_{\operatorname{disc}}(\operatorname{GL}(2,F_{v}))\to\Pi(D^{\times}(F_{v}))$,

be the local Jacquet-Langlands correspondence of $\operatorname{GL}(2)$. We
take these local Tamagawa measures $\mu_{v}$’s on $\operatorname{GL}(2,F_{v})$
and $D^{\times}(F_{v})$ respectively. We take the canonical Harish-Chandra
measure $\mathop{}\\!\mathrm{d}\omega_{v}$ on
$\Pi_{\operatorname{temp}}(\operatorname{GL}(2,F_{v}))$ and
$\Pi_{\operatorname{temp}}(D^{\times}(F_{v}))$ respectively (see Section 3.2
and 4.2). Let $\deg$ be the formal degree determined by $\mu_{v}$ (where
$\mathop{}\\!\mathrm{d}\omega_{v}$ is the counting measure). Let
$\mathop{}\\!\mathrm{d}\nu_{v},\mathop{}\\!\mathrm{d}\nu^{\prime}_{v}$ be the
density of the local Plancherel measure and formal degree determined by
$\mu_{v}$ with respect to $\mathop{}\\!\mathrm{d}\omega$. The following
results are known.

  1. 1.

If $v\in S\cap V_{f}$, we have $\deg(\pi)=\deg(\operatorname{JL}_{v}(\pi))$ by
[14, §15].

  2. 2.

If $v\in S\cap V_{\infty}$, we have
$\deg(\pi)=\deg(\operatorname{JL}_{v}(\pi))$ by Lemma 3.1.

  3. 3.

If $v\notin S$, we have $\pi\cong\operatorname{JL}_{v}(\pi)$ as
$\operatorname{GL}(2,F_{v})\cong D^{\times}(F_{v})$ and their Plancherel
densities are then automatically identical.

Note the formal degree is the Plancherel measure of an atom (or a square-
integrable representation) in the tempered dual. The local Jacquet-Langlands
correspondence of tempered representations reduces to the square-integrable
representations for $\operatorname{GL}(2)$ if $v\in S$. From the discussion
above, we can conclude:

######  Proposition 2.2.

Suppose $\pi=\otimes_{v\in
V}\pi_{v}\in\Pi_{0}(\operatorname{GL}(2,\mathbb{A}))$,
$\pi^{\prime}=\otimes_{v\in
V}\pi^{\prime}_{v}\in\Pi_{0}(D^{\times}(\mathbb{A}))$ such that
$\operatorname{JL}(\pi^{\prime})=\pi$ and both $\pi,\pi^{\prime}$ are
tempered, we have

$\mathop{}\\!\mathrm{d}\nu_{v}(\pi_{v})=\mathop{}\\!\mathrm{d}\nu^{\prime}_{v}(\pi^{\prime}_{v})$

for each place $v\in V$.

Note the assumption on the temperedness of $\pi$ is Ramanujan conjecture.
There are cupidal representations of
$\operatorname{GL}(2,\mathbb{A}_{\mathbb{Q}})$ constructed from cusp forms of
congruence subgroups of $\operatorname{SL}(2,\mathbb{Z})$, which satisfy this
condition (see [11, Proposition 5.21]).

This result is generalized to $\operatorname{GL}(n)$ and tempered
representations in Section 5.

##  3 The local correspondence of archimedean $\operatorname{GL}(n)$

We prove the local Jacquet-Langlands correspondence of archimedean
$\operatorname{GL}(n)$ preserve the Plancherel measures. We will focus on the
real case instead of the complex one where all division algebra split as the
field is algebraically closed.

###  3.1 The local Tamagawa measures and formal degrees

We discuss the central simple finite dimensional algebras over $\mathbb{R}$.
Let $A$ be a finite-dimensional simple algebra over $\mathbb{R}$. It is known
that $A\cong\operatorname{Mat}(n,\mathbb{R})$ or
$A\cong\operatorname{Mat}(n,\mathbb{H})$ where $\mathbb{H}$ is the Hamilton
quaternion algebra. Suppose $\dim_{\mathbb{R}}A=m$ and we choose an
$\mathbb{R}$-basis $e_{1},\cdots,e_{m}$ for $A$. We define the (additive) Haar
measure $\mu_{\mathbb{R}}^{+}$ on $B$ to be given as

$\mathop{}\\!\mathrm{d}x=|d(e_{1},\cdots,e_{m})|^{1/2}\mathop{}\\!\mathrm{d}x_{1}\cdots\mathop{}\\!\mathrm{d}x_{m}$,

where $x=\sum_{1\leq i\leq m}x_{i}e_{i}$, $d(\cdots)$ is the discriminant of a
basis and $\mathop{}\\!\mathrm{d}x_{i}$ is the Lebesgue measure of
$\mathbb{R}$. Then the multiplicative Haar measure on $A^{\times}$ is defined
as

$\mathop{}\\!\mathrm{d}\mu_{\mathbb{R}}(x)=\frac{\mathop{}\\!\mathrm{d}\mu_{\mathbb{R}}^{+}(x)}{\|x\|_{D}}$,

where $\|x\|_{A}=|\operatorname{Nm}_{A/\mathbb{R}}(x)|$ for $x\in A^{\times}$.
We call it the local Tamagawa measure of $A^{\times}$ and will be taken as the
Haar measures on $A^{\times}$.

Let us consider the square-integrable irreducible representations of the
quaternion algebras over $\mathbb{R}$, which are $4$-dimensional simple
algebras that are isomorphic to either $\operatorname{Mat}(2,\mathbb{R})$ or
$\mathbb{H}$.

  1. 1.

For $D=\operatorname{Mat}(2,\mathbb{R})$, we know
$D^{\times}=\operatorname{GL}(2,\mathbb{R})=\mathbb{R}^{\times}_{>0}\cdot\operatorname{SL}^{\pm}(2,\mathbb{R})$.
Let $H_{n}^{+}$ and $H_{n}^{-}$ be the holomorphic and anti-holomorphic
discrete series representations of $\operatorname{SL}(2,\mathbb{R})$
respectively (see [15]). The square-integrable irreducible representations of
$\operatorname{GL}(2,\mathbb{R})$ are given by

$H_{n}(\omega):=\omega\otimes\operatorname{Ind}_{\operatorname{SL}(2,\mathbb{R})}^{\operatorname{SL}^{\pm}(2,\mathbb{R})}H_{n}^{+}=\omega\otimes(H_{n}^{+}\oplus
H_{n}^{-})$, $n\geq 1$,

where $\omega$ is a character of $\mathbb{R}^{\times}_{>0}$.

  2. 2.

For $D=\mathbb{H}$, we know
$D^{\times}=\mathbb{H}^{\times}=\mathbb{R}^{\times}_{>0}\cdot\mathbb{H}^{1}$,
where
$\mathbb{H}^{1}=\\{x\in\mathbb{H}|\operatorname{Nm}(x)=1\\}\cong\operatorname{SU}(2)$.
Thus the irreducible representations of $\mathbb{H}^{\otimes}$ are given by

$V_{n}(\omega)=\omega\otimes\mathbb{C}^{n}$,

where $\omega$ is a character of $\mathbb{R}^{\times}_{>0}$ and
$\mathbb{C}^{n}$ is the unique $n$-dimensional irreducible representation of
$\operatorname{SU}(2)$.

Recall the local Jacquet-Langlands correspondence for
$\operatorname{GL}(2,\mathbb{R})$ (see [16, §1]):

$\operatorname{JL}_{\mathbb{R}}\colon\Pi_{2}(\operatorname{GL}(2,\mathbb{R}))\to\Pi(\mathbb{H}^{\times})$
given by $\operatorname{JL}_{\mathbb{R}}(H_{n}(\omega))=V_{n}(\omega)$.

The measure $\mu_{\mathbb{R}}$ above will be taken as Haar measures on
$\operatorname{GL}(2,\mathbb{R})$ and $\mathbb{H}^{\times}$. Observe
$\mathbb{H}^{\times}$ is compact modulo its center $\mathbb{R}$. We identify
$\Pi(\mathbb{H}^{\times})$ and $\Pi_{2}(\mathbb{H}^{\times})$. Thus the formal
degree of $\pi\in\Pi(\mathbb{H}^{\times})$ is given by
$\frac{\dim_{\mathbb{C}}\pi}{\mu_{\mathbb{R}}(\mathbb{H}^{\times}/\mathbb{R}^{*})}$,
where we treat $\mathbb{H}^{\times}/\mathbb{R}^{*}$ as a fundamental domain of
$\mathbb{H}^{\times}$ with respect to the action of $\mathbb{R}^{*}$.

######  Lemma 3.1.

$\operatorname{JL}_{\mathbb{R}}$ preserves the formal degrees. In particular,
the formal degrees of $H_{n}(\omega)$ and $V_{n}(\omega)$ are
$\frac{n}{2\pi^{2}}$.

Proof: Observe
$\operatorname{PSL}(2,\mathbb{R})=\mathcal{H}\times\operatorname{PSO}(2)$,
where $\mathcal{H}=\\{x+iy|x\in\mathbb{R},y\in\mathbb{R}_{>0}\\}$ is the upper
half-plane. If we take $y^{-2}dxdy$ for the measure of $\mathcal{H}$ and
normalized measure for $\operatorname{PSO}(2)$, the formal degree of
$H^{+}_{n}$ is $\frac{n}{4\pi}$ by [12, §3.3.d]. Thus, for the Haar measure
$\mu_{\mathbb{R}}$ above, $\operatorname{PSO}(2)$ is of total mass $\pi$ and
the formal degree of $H_{n}^{+}$ is $\frac{n}{4\pi^{2}}$. Hence the formal
degree of $H_{n}(\omega)$ is $\frac{n}{2\pi^{2}}$.

For $\mathbb{H}^{\times}$, we know the
$\mu_{\mathbb{R}}(\mathbb{H}^{1})=4\pi^{2}$ (see [25, Lemma 29.5.9]). Observe
$\mathbb{H}^{\times}/\mathbb{R}^{*}=\operatorname{SU}(2)/\\{\pm I\\}$, we have
$\mu_{\mathbb{R}}(\mathbb{H}^{\times}/\mathbb{R}^{*})=2\pi^{2}$ where
$\mathbb{H}^{\times}/\mathbb{R}^{*}\subset\mathbb{H}^{\times}$ is a
fundamental domain for the $\mathbb{R}^{*}$-action. Thus the formal degree of
$V_{n}(\omega)$ is
$\frac{\dim_{\mathbb{C}}{V_{n}(\omega)}}{\mu_{\mathbb{R}}(\mathbb{H}^{\times}/\mathbb{R}^{*})}=\frac{n}{2\pi^{2}}$.

###  3.2 The correspondences of archimedean Plancherel measures

We first review the tempered irreducible representations of
$\operatorname{GL}(n,\mathbb{R})$ and $\operatorname{GL}(n,\mathbb{H})$.

Let $G$ be a real reductive group. Let $P_{0}$ be the minimal parabolic
subgroup of $G$ and $P_{0}={}^{0}M_{P_{0}}A_{0}N_{0}$ be its Langlands
decomposition (see [27, §2.2.7]). Given a parabolic pair $P={}^{0}{M}_{P}AN$,
a unitary representation $\omega$ of $M$ and $\nu\in\mathfrak{a}^{*}$ (where
$\mathfrak{a}$ is the Lie algebra of $A$), we let

$\operatorname{Ind}_{P}^{G}(\omega,i\nu)=\operatorname{Ind}_{(^{0}{M}_{P}AN)}^{G}(\omega\otimes\exp(i\nu)\otimes
1)$

be the induced representation of $G$. The tempered dual of $G$ can be
described by the Harish-Chandra’s Plancherel formula:

$f(g)=\sum\limits_{(P,A)\succ(P_{0},A_{0})}C_{A}\sum\limits_{\omega\in\Pi_{2}(^{0}{M}_{P})}d(\omega)\cdot\int_{\mathfrak{a}^{*}}\theta_{\operatorname{Ind}_{P}^{G}(\omega,i\nu)}(R(g)f)\mu(\omega,i\nu)d\nu,$
(1)

where $f$ is in the space of Schwartz functions and $\nu$ is a measure on
$\mathfrak{a}^{*}$ (see [28, §13.4]).

The tempered dual of $\operatorname{GL}(n,\mathbb{R})$ can be described as
follows. Observe $\operatorname{GL}(n,\mathbb{R})$ has square-integrable
irreducible representations (modulo the center) if and only if $n=1$ or $2$.
Hence the Levi factor $M={}^{0}{M}_{P}A$ we actually need in Equation 2 for
$\operatorname{GL}(n,\mathbb{R})$ is the products of
$\operatorname{GL}(2,\mathbb{R})$ and $\operatorname{GL}(2,\mathbb{R})$ Hence
we take $\sigma=(n_{1},\cdots,n_{s})$ be a partition of $n$ into $1$’s and
$2$’s. For each $\operatorname{GL}(n_{i},\mathbb{R})$, we take a square-
integrable irreducible representation (modulo the center) which can be
completely classified as follows:

  * •

For $\operatorname{GL}(1,\mathbb{R})=\mathbb{R}^{\times}$, let
$\chi_{t,+}(g)=|g|_{\mathbb{R}}^{t}$ and
$\chi_{t,-}(g)=\operatorname{sgn}(g)\otimes|g|_{\mathbb{R}}^{t}$ for
$t\in\mathbb{C}$.

  * •

For $\operatorname{GL}(2,\mathbb{R})$, let
$H_{k}(\omega)=\omega\otimes\operatorname{Ind}_{\operatorname{SL}(2,\mathbb{R})}^{\operatorname{SL}^{\pm}(2,\mathbb{R})}H_{k}^{+}=\omega\otimes(H_{k}^{+}\oplus
H_{k}^{-})$ for $k\geq 1$.

Hence each tempered irreducible representation $\pi$ is of the form
$\operatorname{Ind}_{P}^{G}(\pi_{1}\otimes\cdots\otimes\pi_{s})$ with the Levi
factor $M_{P}\cong\prod_{1\leq i\leq s}\operatorname{GL}(n_{i},\mathbb{R})$
and each $\pi_{i}$ taken from one of representations of
$\operatorname{GL}(n_{i},\mathbb{R})$ above.

For the tempered dual of $\operatorname{GL}(n,\mathbb{H})$, we know that it
has square-integrable irreducible representations (modulo the center) if and
only if $n=1$ 555The maximal compact subgroup of
$\operatorname{GL}(n,\mathbb{H})$ is $\operatorname{Sp}(n)$. It is a maximal
Cartan subgroup, or equivalently
$\operatorname{rank}\operatorname{Sp}(n)=\operatorname{rank}\operatorname{GL}(n,\mathbb{H})$
($=n$) if and only if $n=1$.. They come from the following ones:

  * •

$V_{k}(\omega)=\omega\otimes\mathbb{C}^{k}$, where $\mathbb{C}^{k}$ is the
unique $k$-dimensional irreducible representation of
$\mathbb{H}^{1}=\\{x\in\mathbb{H}|\operatorname{Nm}(x)=1\\}\cong\operatorname{SU}(2)$
and $\omega$ is a character of $\mathbb{R}_{>0}^{\times}$.

Hence the induced representations of the form
$\operatorname{Ind}^{G}_{\prod_{1\leq i\leq
n}\mathbb{H}^{\times}}(V_{k_{1}}(\omega_{1})\otimes\cdots\otimes
V_{k_{n}}(\omega_{n}))$ give the tempered dual of
$\operatorname{GL}(n,\mathbb{H})$.

The local Jacquet-Langlands correspondence of
$\operatorname{GL}(n,\mathbb{R})$ is given explicitly in [6, §13] as a map on
the Grothendieck groups of unitary representations:

$\operatorname{JL}_{\mathbb{R}}\colon\mathbb{Z}\cdot\Pi({\operatorname{GL}(2n,\mathbb{R})})\to\mathbb{Z}\cdot\Pi(\operatorname{GL}(n,\mathbb{H}))$.

The restriction of the correspondence on the tempered representations can be
described as follows.

  1. 1.

If each factor of the Levi subgroup $M$ is $\operatorname{GL}(2,\mathbb{R})$
(the partition $\sigma=(2,\cdots,2)$), then

$\operatorname{JL}_{\mathbb{R}}(\operatorname{Ind}(H_{k_{1}}(\omega_{1})\otimes\cdots\otimes\operatorname{Ind}(H_{k_{n}}(\omega_{n})))=\operatorname{Ind}(\operatorname{Ind}(V_{k_{1}}(\omega_{1})\otimes\cdots\otimes\operatorname{Ind}(V_{k_{n}}(\omega_{n})))$

for $k_{i}\geq 1$ and each $\omega$ is a character of
$\mathbb{R}_{>0}^{\times}$.

  2. 2.

If there exists a factor $\operatorname{GL}(1,\mathbb{R})$ of the Levi
subgroup $M$ (the partition $\sigma$ contains at least one $1$), then

$\operatorname{JL}_{\mathbb{R}}(\operatorname{Ind}(\pi_{1}\otimes\cdots\otimes\pi_{s}))=0$

for any representations $\pi_{1}\otimes\cdots\otimes\pi_{s}$ of $M$. Note $0$
stands as the zero element in the Grothendieck group but not the trivial
representation.

By Equation 1, we identify $\mathfrak{a}^{*}$ with $\mathbb{R}^{n}$ and take
the Lebesgue measure as the Harish-Chandra canonical measure on the tempered
dual (see[§7] [13]), which will be denoted as
$\mathop{}\\!\mathrm{d}\omega_{\mathbb{R}}$. We let $\pi,\pi^{\prime}$ denote
tempered irreducible representation of $\operatorname{GL}(2n,\mathbb{R})$ and
$\operatorname{GL}(n,\mathbb{H})$ respectively.

######  Proposition 3.2.

Let $\pi,\pi^{\prime}$ be nontrivial tempered irreducible representations of
$\operatorname{GL}(2n,\mathbb{R})$ and $\operatorname{GL}(n,\mathbb{H})$ such
that $\operatorname{JL}_{\mathbb{R}}(\pi)=\pi^{\prime}$. We have

$\mathop{}\\!\mathrm{d}\nu_{v}(\pi)=\mathop{}\\!\mathrm{d}\nu^{\prime}_{v}(\pi^{\prime})$,

if the Haar measures are taken to be the local Tamagawa measures above.

Proof: It suffices to compare each term of Equation 1 for
$G=\operatorname{GL}(2n,\mathbb{R})$ and
$G^{\prime}=\operatorname{GL}(n,\mathbb{H})$. Note the domain
$\mathfrak{a}^{*}$ is identical for both $G=\operatorname{GL}(2n,\mathbb{R})$
and $G^{\prime}=\operatorname{GL}(n,\mathbb{H})$ as the Levi factors
$\prod_{i=1}^{n}\operatorname{GL}(2,\mathbb{R})$ and
$\prod_{i=1}^{n}\mathbb{H}^{\times}$ has the same $A$-factor
$(\mathbb{R}^{\times}_{>0})^{n}$.

The term $\deg(\omega)$ is the formal degree of a square-integrable
representation of $\prod_{1\leq i\leq n}\operatorname{GL}(2,\mathbb{R})$ and
$\prod_{1\leq i\leq n}\operatorname{GL}(1,\mathbb{H})$ respectively. Thus, by
Lemma 3.1 and the explicit $\operatorname{JL}_{\mathbb{R}}$ above, we have

$\deg(\prod_{1\leq i\leq n}\pi_{i})=\prod_{1\leq i\leq
n}\deg(\pi_{i})=\prod_{1\leq i\leq
n}\deg(\operatorname{JL}_{\mathbb{R}}(\pi_{i}))=\deg(\operatorname{JL}_{\mathbb{R}}(\prod_{1\leq
i\leq n}\pi_{i}))$

if each $\pi\\_i\in\Pi_{2}(\operatorname{GL}(2,\mathbb{R}))$.

Note the factor $C_{A}=|W(A)\gamma_{A}c_{A}|^{-1}$ (see [28, §13.4.1]). Here
$W(A)$ is the Weyl group of $G$ (or $G^{\prime}$) with respect to
$A\cong(\mathbb{R}_{>0}^{\times})^{n}$, which are identical for
$G,G^{\prime}$. The term $\gamma_{A}$ is defined on [28, pp 163,231], which is
identical for each parabolic group that has such an $A$ factor. It then
follows [28, Proposition 12.5.3] that the $\gamma_{A}$ term is identical. The
term $c_{A}$ is defined in [28, §13.3.2], which is uniquely determined by $A$.

For $\mu(\omega,i\nu)$, we adapt the proof of [1, §2 Lemma 2.1]. Fix the Levi
subgroup $M=\prod_{i=1}^{n}\operatorname{GL}(2,\mathbb{R})$ of $G$. Consider a
function $f\in\mathcal{H}(G)$ (the Hecke algebra of $G$) such that ${\rm
supp}(\theta_{f})\subset\\{\operatorname{Ind}_{P}^{G}(\pi)|\pi\in\Pi_{2}(M),M\subset
P\\}$. Thus Equation 1 for such an $f$ can be rewritten as

$f(1)=\gamma_{M}\cdot\int_{\Pi_{2}(M)}d(\pi)\nu_{M}(\pi)f_{M}(\pi)d\pi,$ (2)

where we combine the representations of ${}^{0}M_{P}$ and $A$ (or
$\mathfrak{a}$, equivalently) into $\Pi(M)=\Pi(^{0}M_{P}A)$ and write
$f_{M}(\pi)=\operatorname{tr}(\operatorname{Ind}_{P}^{G}(\pi)(f))$.

Let $M^{\prime}=\prod_{1\leq i\leq n}\mathbb{H}^{\times}$ and let
$\operatorname{JL}_{\mathbb{R}}\colon\Pi_{2}(M)\to\Pi_{2}(M^{\prime})$, the
correspondence obtained from Lemma 3.1. There exists a function $f^{\prime}$
on $\Pi_{\operatorname{temp}}(G^{\prime})$ such that

  1. 1.

$f^{\prime}=\theta_{h}$ for some $h\in\mathcal{H}(G^{\prime})$;

  2. 2.

${\rm
supp}(f^{\prime})\subset\\{\operatorname{Ind}_{P^{\prime}}^{G^{\prime}}(\pi^{\prime})|\pi^{\prime}\in\operatorname{JL}_{\mathbb{R}}(\Pi_{2}(M)),M^{\prime}\subset
P^{\prime}\\}$.

It is known that $f(1)=c\cdot h(1)$ for some $c\in\\{1,-1\\}$ and
$f^{\prime}(\operatorname{Ind}_{P^{\prime}}^{G^{\prime}}(\pi^{\prime}))=c^{-1}\cdot
f_{M}(\pi)$ if $\operatorname{JL}_{\mathbb{R}}\colon\pi\to\pi^{\prime}$. The
Plancherel formula of $h$ gives us

$f(1)=\gamma_{M^{\prime}}\cdot\int_{\Pi_{2}(M)}d(\operatorname{JL}_{\mathbb{R}}(\pi))\nu_{M^{\prime}}(\operatorname{JL}_{\mathbb{R}}(\pi))f_{M}(\pi)d\pi.$
(3)

Note the terms $\gamma_{M}$ and $\gamma_{M^{\prime}}$ are identical under our
choice of Haar measures. By the trace Paley-Weiner theorem for real Lie groups
(see [8, Theorem 1]), we compare Equation 2 with Equation 3 and obtain

$d(\pi^{\prime})\cdot\mu_{M^{\prime}}(\pi^{\prime})=d(\pi)\cdot\mu_{M}(\pi)$,

if $\pi\in\Pi_{\operatorname{disc}}(M)$ and
$\operatorname{JL}_{\mathbb{R}}(\pi)=\pi^{\prime}$. Thus
$\mu_{M^{\prime}}(\pi^{\prime})=\mu_{M}(\pi)$ as the formal degrees are shown
to be identical above. This completes the proof.

##  4 The local correspondence of non-archimedean $\operatorname{GL}(n)$

We fix a non-archimedean place $v\in V_{f}$ throughout this section. Consider
the $F_{v}$-algebra $A_{v}\cong\operatorname{Mat}(n_{v},D_{v})$ and its
multiplcative group $A_{v}^{\times}\cong\operatorname{GL}(n_{v},D_{v})$. We
will define Haar measures in Section 4.1 and prove the local Jacquet-Langlands
correspondence of the Plancherel densities on the tempered dual of
$\operatorname{GL}(n_{v},D_{v})$ in Section 4.2.

###  4.1 The local Tamagawa measures and maximal orders

Let us consider the valuation map on $D_{v}$ given as
$w_{v}=v\circ\operatorname{Nm}_{D_{v}/F_{v}}\colon D_{v}\to\mathbb{Z}$. Let
$O(D_{v})$ be the maximal $\mathcal{O}_{v}$-order of $D_{v}$ and $J(D_{v})$ be
the maximal ideal of $O(D_{v})$. It is known that (see [19, Theorem 12.8 and
13.2])

$O(D_{v})=\\{x\in D_{v}|w_{v}(x)\geq 0\\}$ and $J(D_{v})=\\{x\in
D_{v}|w_{v}(x)>0\\}$,

which are both unique up to isomorphism.

For the $F_{v}$-algebra $A_{v}\cong\operatorname{Mat}(n_{v},D_{v})$, we let
$O(A_{v})$ be its maximal $\mathcal{O}_{v}$-order and $J(A_{v})$ be the
maximal two-sided ideal of $O(A_{v})$. It is known that (see [19, Theorem
17.3])

$O(A_{v})=\operatorname{Mat}(n_{v},O(D_{v}))$ and $J(A_{v})=J(D_{v})O(A_{v})$.

The additive Haar measure $\mu^{+}_{v}$ on $A_{v}$ is defined to be the one
normalized in the sense that its maximal order has volume $1$, i.e.,
$\mu^{+}_{v}(O(A_{v})))=1$. The multiplicative Haar measure $\mu^{\times}_{v}$
is defined by

$\mathop{}\\!\mathrm{d}\mu^{\times}_{v}(x)=(1-1/q_{v})^{-1}\frac{\mathop{}\\!\mathrm{d}\mu^{+}_{v}(x)}{\|x\|_{A_{v}}}$,

where $\|x\|_{A_{v}}$ is the norm on $A_{v}$ defined as
$\|x\|_{A_{v}}=\|\operatorname{Nm}_{A_{v}/F_{v}}(x)\|_{F_{v}}$.

######  Lemma 4.1.

$\mu^{\times}_{v}(O^{\times}(A_{v}))=(1-1/q_{v})^{-1}\cdot\prod_{i=1}^{n_{v}}(1-(q_{v}^{d_{v}})^{i})$.

Proof: By [19, Theorem 13.2], $O(D_{v})/J(D_{v})$ is a skewfield over $F_{v}$.
It is known that $\dim_{F_{v}}O(D_{v})/J(D_{v})=d_{v}$ (see [18, Theorem
14.3]). Consider the short exact sequence

$1\to
1+J(D_{v})\operatorname{Mat}(n_{v},O_{v})\to\operatorname{GL}(n_{v},O(D_{v})))\to\operatorname{GL}(n_{v},O(D_{v})/J(D_{v}))\to
1$.

As $\|x\|_{A_{v}}=1$ for $x\in\operatorname{GL}(n_{v},O(D_{v})))$, we know
that

$\displaystyle\mu^{\times}_{v}(O(A_{v}))$
$\displaystyle=(1-1/q_{v})^{-1}\cdot\mu^{+}_{v}(1+J_{v}\cdot\operatorname{Mat}(n_{v},O_{v}))\cdot|\operatorname{GL}(n_{v},O(D_{v})/J(D_{v}))|$
$\displaystyle=(1-1/q_{v})^{-1}\cdot
q_{v}^{-d_{v}n_{v}^{2}}\cdot\prod_{i=0}^{nr_{v}-1}((q_{v}^{d_{v}})^{n_{v}}-(q_{v}^{d_{v}})^{i})$
$\displaystyle=(1-1/q_{v})^{-1}\cdot\prod_{i=1}^{n_{v}}(1-(q_{v}^{d_{v}})^{i}).$

In particular, for the split case when $r_{v}=d$, $d_{v}=1$ and
$O(A_{v})^{\times}\cong\operatorname{GL}(nd,\mathcal{O}_{v})$, we have

$\mu^{\times}_{v}(\operatorname{GL}(nd,\mathcal{O}_{v}))=(1-1/q_{v})^{-1}\prod_{i=1}^{nd}(1-q_{v}^{-i})$.

Now we define the local Tamagawa measure on
$A_{v}^{\times}\cong\operatorname{GL}(n_{v},D_{v})$ as

$\mu_{v}=\operatorname{N}(\operatorname{disc}_{\mathbb{Z}_{p}}(O(A_{v})))^{-1/2}\cdot\mu^{\times}_{v}$.

It is self-dual to the standard unitary character $\psi_{A_{v}}\colon A_{v}\to
F_{v}$ constructed as follows (see [25, Proposition 29.7.7]):

  1. 1.

For $x\in\mathbb{Q}_{p}$, we let $\langle
x\rangle_{\mathbb{Q}_{p}}\in\mathbb{Q}$ such that $0\leq\langle
x\rangle_{\mathbb{Q}_{p}}<1$ and $x-\langle
x\rangle_{\mathbb{Q}_{p}}\in\mathbb{Z}_{p}$.

  2. 2.

For $x\in F_{v}$ with $F_{v}/\mathbb{Q}_{p}$ a finite extension, we let
$\langle
x\rangle_{F_{v}}=\langle\operatorname{Tr}_{F_{v}/\mathbb{Q}_{p}}(x)\rangle_{\mathbb{Q}_{p}}$
and $\psi_{F_{v}}(x)=\exp(2\pi i\langle x\rangle_{F_{v}})$

  3. 3.

For $x\in A_{v}$, we let $\psi_{A_{v}}(x)=\psi_{F_{v}}(\operatorname{tr}(x))$.

Observe $\operatorname{disc}_{\mathbb{Z}_{p}}(\mathcal{O}_{v})$ is the local
discriminant $d(F_{v})$ if $v$ is above $p$.

######  Lemma 4.2.

$\operatorname{N}(\operatorname{disc}_{\mathbb{Z}_{p}}(O(A_{v})))=d(F_{v})^{n^{2}d^{2}}q_{v}^{d_{v}(d_{v}-1)n_{v}^{2}}$

Proof: Observe $O(A_{v})$ is a lattice in $A_{v}$ of free rank
$n_{v}^{2}d_{v}^{2}=n^{2}d^{2}$ and
$O(A_{v})=\operatorname{Mat}(n_{v},O(D_{v}))$. We have

$\displaystyle\operatorname{N}(\operatorname{disc}_{\mathbb{Z}_{p}}(O(A_{v})))$
$\displaystyle=\operatorname{N}(\operatorname{disc}_{\mathbb{Z}_{p}}(\mathcal{O}_{v}))^{n^{2}d^{2}}\operatorname{N}(\operatorname{disc}_{\mathcal{O}_{v}}(O(A_{v})))$
$\displaystyle=\operatorname{N}(\operatorname{disc}_{\mathbb{Z}_{p}}(\mathcal{O}_{v}))^{n^{2}d^{2}}\cdot\operatorname{N}(\operatorname{disc}_{\mathcal{O}_{v}}(O(D_{v})))^{n_{v}^{2}}.$

By [19, Theorem 14.9],
$\operatorname{disc}_{\mathcal{O}_{v}}(O(D_{v}))=P_{v}^{d_{v}(d_{v}-1)}$. Thus
$\operatorname{N}(\operatorname{disc}_{\mathcal{O}_{v}}(O(D_{v})))=q_{v}^{d_{v}(d_{v}-1)}$
and
$\operatorname{N}(\operatorname{disc}_{\mathbb{Z}_{p}}(O(A_{v})))=\operatorname{N}(\operatorname{disc}_{\mathbb{Z}_{p}}(\mathcal{O}_{v}))^{n^{2}d^{2}}q_{v}^{d_{v}(d_{v}-1)n_{v}^{2}}$.

Combining Lemma 4.1 and Lemma 4.2, we obtain the volume of $O^{\times}(A_{v})$
under the local Tamagawa measure.

######  Corollary 4.3.

$\mu_{v}(O^{\times}(A_{v}))=d(F_{v})^{-\frac{1}{2}n^{2}d^{2}}q_{v}^{-\frac{1}{2}d_{v}(d_{v}-1)n_{v}^{2}}(1-1/q_{v})^{-1}\prod_{i=1}^{n_{v}}(1-(q_{v}^{d_{v}})^{i})$.

###  4.2 The correspondence of non-archimedean Plancherel measures

In this section, we simply write $G$ for $G_{v}=\operatorname{GL}(n,F_{v})$
and $G^{\prime}$ for $G^{\prime}_{v}=\operatorname{GL}(n_{v},D_{v})$. We
discuss the tempered dual and the canonical Harish-Chandra measure on them as
described in [26].

Let $M$ be a Levi subgroup and $\operatorname{Im}X(M)$ be the group of unitary
unramified characters of $M$. Denote by $\Pi_{2}(M)$ the set of equivalence
classes of the square-integrable irreducible representations (modulo the
center) of $M$. The group $\operatorname{Im}X(M)$ has a natural action on
$\Pi_{2}(M)$ given as $\chi\circ\pi=\chi\pi$ for
$\pi\in\Pi_{2}(M),\chi\in\operatorname{Im}X(M)$. Let $P(G)$ be the set of
semi-standard parabolic subgroups of $G$. For $P\in P(G)$, we let $P=MU$ be
the Levi decomposition. We define

$\Theta=\\{(\mathcal{O},P)|P=MU\in P(G),\mathcal{O}\subset\Pi_{2}(M)\text{ is
an }\operatorname{Im}X(M)\text{-orbit}\\}$.

Assuming $W$ be the Weyl group of $G_{v}$ and $W_{M}$ be the Wey group of $M$,
we let $W(G|M)=\\{s\in W|s\cdot M=M\\}/W_{M}$ and
$\operatorname{Stab}(\mathcal{O},M)=\\{s\in
W(G|M)|s\mathcal{O}=\mathcal{O}\\}$.

For $\omega\in\Pi_{2}(M)$, we let $\operatorname{Ind}^{G}_{P}(\omega)$ be the
normalized induced representation from $\omega$. Thus we get a map
$(\mathcal{O},P=MU)\to\Pi_{\operatorname{temp}}(G)$ given by
$\omega\mapsto\operatorname{Ind}^{G}_{P}(\omega)$. This gives a bijection

$\bigsqcup\limits_{(\mathcal{O},P=MU)\in\Theta}(\mathcal{O},P=MU)/\operatorname{Stab}(\mathcal{O},M)\to\operatorname{Irr}^{\rm
t}(G)$.

Thus $\Pi_{\operatorname{temp}}(G)$ can be identified with the left-hand side
as a disjoint union of countably many compact orbifolds.

Let $A_{M}$ be the center of $M$. We assign a Haar measure $\mu_{0}$ to the
group $\operatorname{Im}X(A_{M})$ of unramified unitary characters on $A_{M}$
such that its total mass is $1$. Consider the map
$f\colon\operatorname{Im}X(M)\to\mathcal{O}$ given by
$\chi\mapsto\omega\otimes\chi$ and the restriction map
$h\colon\operatorname{Im}X(M)\to\operatorname{Im}X(A_{m})$. Then we defined a
measure $\mathop{}\\!\mathrm{d}\omega_{v}$ on $\mathcal{O}$ such that
$f^{*}\omega=h^{*}\mu_{0}$ on $\operatorname{Im}X(M)$.

Let $E\subset\mathcal{O}$ be a fundamental set of the action of
$\operatorname{Stab}(\mathcal{O},M)$. We define
$\mathop{}\\!\mathrm{d}\omega_{v}$ to be the measure on $E$ as follows: for
any $\operatorname{Stab}(\mathcal{O},M)$-invariant integrable function $F$ on
$\mathcal{O}$,
$\int_{\mathcal{O}}F\mathop{}\\!\mathrm{d}\omega_{v}=|\operatorname{Stab}(\mathcal{O},M)|\cdot\int_{E}F\mathop{}\\!\mathrm{d}\omega_{v}$.

For $f\in\mathcal{H}(G)$, the orbital integral of $f$ is a function on the set
regular semi-simple elements of $G$ given by
$\Phi(f;g)=\int_{G/Z_{G}(g)}f(xgx^{-1})dx$ for $g\in G^{s}$, where $Z_{G}(g)$
is the centralizer of $g$ in $G$. The orbit integral for $G^{\prime}$ can be
defined similarly. Recall that $f^{\prime}\in\mathcal{H}(G)$ and
$f\in\mathcal{H}(G^{\prime})$ are corresponded, denoted by $f\leftrightarrow
f^{\prime}$ if

  1. 1.

if $g\leftrightarrow g^{\prime}$ for $g\in G,g^{\prime}\in G^{\prime}$, then
$\Phi(f;g)=(-1)^{n_{v}-n}\cdot\Phi(f^{\prime};g^{\prime})$;

  2. 2.

if for $g\in G$ there does not exist $g^{\prime}\in G^{\prime}$ such that
$g\leftrightarrow g^{\prime}$, then $\Phi(f;g)=0$.

For $f^{\prime}\in\mathcal{H}(G^{\prime})$, there exists $f\in\mathcal{H}(G)$
such that $f\leftrightarrow f^{\prime}$ and if ${\rm supp}(f)\subset G^{s}$,
there exists $f^{\prime}\in\mathcal{H}(G^{\prime})$ such that
$f\leftrightarrow f^{\prime}$ (see [5, §2.5]).

Let $\operatorname{St}_{G},\operatorname{St}_{G}^{\prime}$ be the Steinberg
representations of $G,G^{\prime}$ respectively. The following result is known
and we add a short proof for completeness.

######  Lemma 4.4.

If $f\leftrightarrow f^{\prime}$, we have
$\frac{f(1)}{\deg(\operatorname{St}_{G})}=\frac{f^{\prime}(1)}{\deg(\operatorname{St}_{G^{\prime}})}$.

Proof: For $g\in G,g^{\prime}\in G^{\prime}$, their conjugacy class
$C_{G}(g),C_{G^{\prime}}(g^{\prime})$ are closed in $G,G^{\prime}$
respectively. For a unipotent conjugacy class $O$, we let $\Lambda_{O}(f)$ be
the orbital integral over $O$, i.e., $\Lambda_{O}(f)=\int_{O}f(xgx^{-1})dx$.
By [23, Theorem 2.1.1], if $g$ is sufficiently close to $1$, we have

$\Phi(f,g)=\sum_{O{\rm~{}unipotent}}\Gamma_{O}^{G}(g)\cdot\Lambda_{O}(f)$ (4)

for some function $\Gamma^{G}_{O}\colon G^{s}\to\mathbb{R}$ that is identical
on each unipotent $O$. Moreover, $\Lambda_{1}(f)=f(1)$ and all these also
holds for $G^{\prime}$. By [20], we know
$\Gamma_{1}^{G}=\frac{(-1)^{n}}{\deg(\operatorname{St}_{G})}$ and
$\Gamma_{1}^{G^{\prime}}=\frac{(-1)^{n_{v}}}{\deg(\operatorname{St}_{G^{\prime}})}$.
It then follows Equation 4 of $f,f^{\prime}$ at $g=1,g^{\prime}=1$
respectively.

Please note while $f(1),f^{\prime}(1)$ and
$\deg(\operatorname{St}_{G}),\deg(\operatorname{St}_{G}^{\prime})$ depend on
the choice of the Haar measure on $G,G^{\prime}$, these dependences cancel out
in the equation of the two quotients.

We take the Haar measures $\mu_{v}$ defined in Section 4.1 on
$\operatorname{GL}(nd,F_{v}),\operatorname{GL}(n_{v},D_{v})$ and let
$\nu_{v},\nu_{v}^{\prime}$ be the Plancherel measure on
$\Pi_{\operatorname{temp}}(\operatorname{GL}(nd,F_{v})),\Pi_{\operatorname{temp}}(\operatorname{GL}(n_{v},D_{v}))$
respectively. We denote by
$\mathop{}\\!\mathrm{d}\nu_{v},\mathop{}\\!\mathrm{d}\nu^{\prime}_{v}$ the
Plancherel densities with respect to the canonical Harish-Chandra measure
$\mathop{}\\!\mathrm{d}\omega_{v}$.

Note the local Jacquet-Langlands correspondence $\operatorname{JL}_{v}$ gives
a homeomorphism from a subset
$Y\subset\Pi_{\operatorname{temp}}(\operatorname{GL}(nd,F_{v}))$ to
$\Pi_{\operatorname{temp}}(\operatorname{GL}(n_{v},D_{v}))$ (see [2, §7 pp.
63]).

######  Proposition 4.5.

If $\operatorname{JL}_{v}(\pi)=\pi^{\prime}$ with $\pi,\pi^{\prime}$ tempered
and nontrivial, we have

$\mathop{}\\!\mathrm{d}\nu_{v}(\pi)=\mathop{}\\!\mathrm{d}\nu^{\prime}_{v}(\pi^{\prime})$,

if the Haar measures are taken to be the local Tamagawa measures $\mu_{v}$.

Proof: Assuming $f\leftrightarrow f^{\prime}$ and $\pi^{\prime}$ is an
irreducible representation of $G^{\prime}$ induced from some Levi subgroup
$M^{\prime}=\prod\operatorname{GL}(m_{i},D_{v})$ with $\sum m_{i}=n_{v}$, we
know

$\theta_{\pi^{\prime}}(f^{\prime})=\theta_{\operatorname{JL}_{v}(\pi^{\prime})}(f)$
(5)

by [3, Proposition 3.6]. It is also proved that $\theta_{\pi}(f)=0$ if $\pi$
is not an induced representation comes from a Levi subgroup
$M=\prod\operatorname{GL}(n_{i},F_{v})$ such that all $n_{i}$ are divided by
$d$ (see [3, §3]).

We compare the Plancherel formulas of $G,G^{\prime}$ at $1$. By Lemma 4.4, we
obtain

$\frac{\int_{Y}\theta_{\pi}(f)\mathop{}\\!\mathrm{d}\nu(\pi)}{\deg(\operatorname{St}_{G})}=\frac{\int_{\Pi_{\operatorname{temp}}(G^{\prime})}\theta_{\pi^{\prime}}(f^{\prime})\mathop{}\\!\mathrm{d}\nu^{\prime}(\pi^{\prime})}{\deg(\operatorname{St}_{G^{\prime}})}.$
(6)

By [3, Lemma 3.4], the characters $\theta_{\pi}(f^{*})$ spans the space of
compactly supported functions on $\Pi_{\operatorname{temp}}(G^{\prime})$ (here
$f^{*}(g)=f(g^{-1})$). By Equation 5 and 6, the measures
$\frac{\mathop{}\\!\mathrm{d}\nu|_{Y}}{\deg(\operatorname{St}_{G})}$ and
$\frac{\mathop{}\\!\mathrm{d}\nu^{\prime}}{\deg(\operatorname{St}_{G}^{\prime})}$
coincides on a dense subset of integrable function on
$\Pi_{\operatorname{temp}}(G^{\prime})\cong
Y\subset\Pi_{\operatorname{temp}}(G)$. Thus
$\frac{\mathop{}\\!\mathrm{d}\nu|_{Y}}{\deg(\operatorname{St}_{G})}=\frac{\mathop{}\\!\mathrm{d}\nu^{\prime}}{\deg(\operatorname{St}_{G}^{\prime})}$
and it then suffices to discuss
$\deg(\operatorname{St}_{G}),\deg(\operatorname{St}_{G}^{\prime})$.

If the Haar measures of $\operatorname{GL}(n,F_{v})$ and
$\operatorname{GL}(n_{v},D_{v})$ are normalized such that the maximal compact
subgroups $\operatorname{GL}(n,\mathcal{O}_{v})$ and
$\operatorname{GL}(n_{v},O(D_{v}))$ are of total mass $1$, we have
$\deg(\operatorname{St}_{G})=\frac{1}{n}\prod_{i=1}^{n-1}(q^{i}-1),\deg(\operatorname{St}_{G}^{\prime})=\frac{1}{n}\prod_{i=1}^{n_{v}-1}(q^{d_{v}i}-1)$
(see [9, §2]). We now take the Haar measures $\mu_{v}$ above. By Corollary
4.3, the volumes of these maximal compact subgroups are given as follows.

  1. 1.

If $A_{v}\cong\operatorname{Mat}(n_{v},D_{v})$ and
$\dim_{F_{v}}D_{v}=d_{v}^{2}$, we have
$O^{\times}(A_{v})\cong\operatorname{GL}(n_{v},O(D_{v}))$ and

$\mu_{v}(\operatorname{GL}(n_{v},O(D_{v})))=d(F_{v})^{-\frac{1}{2}n^{2}d^{2}}q_{v}^{-\frac{1}{2}d_{v}(d_{v}-1)n_{v}^{2}}(1-1/q_{v})^{-1}\prod_{i=1}^{n_{v}}(1-(q_{v}^{d_{v}})^{i})$;

  2. 2.

If $A_{v}\cong\operatorname{Mat}(n,F_{v})$, we have
$O^{\times}(A_{v})\cong\operatorname{GL}(n,\mathcal{O}_{v})$ and

$\mu_{v}(\operatorname{GL}(n,\mathcal{O}_{v}))=d(F_{v})^{-\frac{1}{2}n^{2}d^{2}}(1-1/q_{v})^{-1}\prod_{i=1}^{n}(1-(q_{v}^{d_{v}})^{i})$.

Thus we get
$\frac{\mu_{v}(\operatorname{GL}(n_{v},O(D_{v})))}{\mu_{v}(\operatorname{GL}(n,\mathcal{O}_{v}))}=q_{v}^{-\frac{1}{2}d_{v}(d_{v}-1)n_{v}^{2}}\prod_{i=1,d_{v}\nmid
i}^{nd}(1-q^{-i})^{-1}$. The quotient of the two Plancherel densities is then
renormalized as

$\displaystyle\frac{\mathop{}\\!\mathrm{d}\nu_{v}^{\prime}(\pi_{v}^{\prime})}{\mathop{}\\!\mathrm{d}\nu_{v}(\pi_{v})}$
$\displaystyle=\prod_{i=1,d_{v}\nmid
i}^{nd}(q_{v}^{i}-1)^{-1}\cdot\prod_{i=1,d_{v}\nmid i}^{nd}(1-q^{-i})\cdot
q_{v}^{\frac{1}{2}d_{v}(d_{v}-1)n_{v}^{2}}$
$\displaystyle=\prod_{i=1,d_{v}\nmid
i}^{nd}\frac{1-q_{v}^{-i}}{q_{v}^{i}-1}\cdot
q_{v}^{\frac{1}{2}d_{v}(d_{v}-1)n_{v}^{2}}=\prod_{i=1,d_{v}\nmid
i}^{nd}q_{v}^{-i}\cdot q_{v}^{\frac{1}{2}d_{v}(d_{v}-1)n_{v}^{2}}$
$\displaystyle=\frac{\prod_{j=1}^{n_{v}}q_{v}^{d_{v}\cdot
j}}{\prod_{i=1}^{nd}q_{v}^{i}}\cdot
q_{v}^{\frac{1}{2}d_{v}(d_{v}-1)n_{v}^{2}}=q_{v}^{\frac{1}{2}d_{v}n_{v}(n_{v}+1)}\cdot
q_{v}^{-\frac{1}{2}n_{v}d_{v}(n_{v}d_{v}+1)}\cdot
q_{v}^{\frac{1}{2}d_{v}(d_{v}-1)n_{v}^{2}}$
$\displaystyle=q_{v}^{\frac{1}{2}n_{v}d_{v}(n_{v}+1+n_{v}d_{v}-n_{v}-n_{v}d_{v}-1)}=1.$

##  5 The global correspondence of Plancherel measures

We take the local Tamagawa measure $\mu_{v}$ as the Haar measure on the local
groups $\operatorname{GL}(nd,F_{v})$ and $\operatorname{GL}(n_{v},D_{v})$ over
$F_{v}$ respectively, which are defined in Section 3.1 and Section 4.1. It is
known that $\prod_{v\in V}\mu_{v}$ is the Tamagawa measure on the adelic
groups $\operatorname{GL}(nd,\mathbb{A})$ and
$\operatorname{GL}(n,D(\mathbb{A}))$ respectively (see [25, §29.8]).

For each $v$, we let $\mathop{}\\!\mathrm{d}\nu_{v}$ be the Plancherel density
determined by $\mu_{v}$ with respect to the Harish-Chandra measure
$\mathop{}\\!\mathrm{d}\omega_{v}$ on the local tempered dual
$\Pi_{\operatorname{temp}}(\operatorname{GL}(n_{v},D_{v}))$ and
$\Pi_{\operatorname{temp}}(\operatorname{GL}(n_{v},D_{v}))$ respectively.

######  Theorem 5.1.

Suppose $\pi=\otimes_{v\in
V}\pi_{v}\in\Pi_{0}(\operatorname{GL}(nd,\mathbb{A}))$,
$\pi^{\prime}=\otimes_{v\in
V}\pi^{\prime}_{v}\in\Pi_{0}(\operatorname{GL}(n,D(\mathbb{A}))$ such that
$\operatorname{JL}(\pi^{\prime})=\pi$ and both $\pi,\pi^{\prime}$ are
tempered, we have

$\mathop{}\\!\mathrm{d}\nu_{v}(\pi_{v})=\mathop{}\\!\mathrm{d}\nu^{\prime}_{v}(\pi^{\prime}_{v})$

for each place $v\in V$.

Proof: Note $A\otimes_{F}F_{v}\cong\operatorname{Mat}(nd,F_{v})$ if and only
if $v\notin S$. For $v\notin S$, we have
$\operatorname{GL}(n,D_{v})\cong\operatorname{GL}(nd,F_{v})$,
$\pi_{v}\cong\pi^{\prime}_{v}$ and
$\mathop{}\\!\mathrm{d}\nu_{v}(\pi_{v})=\mathop{}\\!\mathrm{d}\nu^{\prime}_{v}(\pi^{\prime}_{v})$.
For $v\in S$, it follows Proposition 3.2 and Proposition 4.5.

######  Remark 5.2.

For an adelic group $G(\mathbb{A})$, a Plancherel measurable set in its
tempered dual $\Pi_{\operatorname{temp}}(G(\mathbb{A}))$ is always a union of
the product form $\left(\prod_{v\in T}\pi_{v}\right)\cdot\left(\prod_{v\in
V\backslash T}\Pi_{\rm urmf}(G(F_{v}))\right)$, where $\Pi_{\rm urmf}$ is the
set of equivalence class of unramified irreducible representations. Thus the
Plancherel density of a single representation $\pi=\prod_{v\in V}\pi_{v}$ with
respect to the product $\prod_{v\in v}\mathop{}\\!\mathrm{d}\omega_{v}$ (of
the canonical Harish-Chandra measures) is not convergent in general and not
well-defined.

## References

  * [1] J. Arthur and L. Clozel.  Simple algebras, base change, and the advanced theory of the trace formula, volume 120 of Annals of Mathematics Studies.  Princeton University Press, Princeton, NJ, 1989.
  * [2] A.-M. Aubert and R. Plymen.  Plancherel measure for ${\rm GL}(n,F)$ and ${\rm GL}(m,D)$: explicit formulas and Bernstein decomposition.  J. Number Theory, 112(1):26–66, 2005.
  * [3] A. I. Badulescu.  Un résultat de transfert et un résultat d’intégrabilité locale des caractères en caractéristique non nulle.  J. Reine Angew. Math., 565:101–124, 2003.
  * [4] A. I. Badulescu.  Global Jacquet-Langlands correspondence, multiplicity one and classification of automorphic representations.  Invent. Math., 172(2):383–438, 2008.  With an appendix by Neven Grbac.
  * [5] A. I. Badulescu.  The trace formula and the proof of the global Jacquet-Langlands correspondence.  In Relative aspects in representation theory, Langlands functoriality and automorphic forms, volume 2221 of Lecture Notes in Math., pages 197–250. Springer, Cham, 2018.
  * [6] A. I. Badulescu and D. Renard.  Unitary dual of ${\rm GL}(n)$ at Archimedean places and global Jacquet-Langlands correspondence.  Compos. Math., 146(5):1115–1164, 2010.
  * [7] A. I. Badulescu and P. Roche.  Global Jacquet-Langlands correspondence for division algebras in characteristic $p$.  Int. Math. Res. Not. IMRN, (7):2172–2206, 2017.
  * [8] L. Clozel and P. Delorme.  Le théorème de Paley-Wiener invariant pour les groupes de Lie réductifs.  Invent. Math., 77(3):427–453, 1984.
  * [9] L. Corwin, A. Moy, and P. J. Sally, Jr.  Degrees and formal degrees for division algebras and ${\rm GL}_{n}$ over a $p$-adic field.  Pacific J. Math., 141(1):21–45, 1990.
  * [10] P. Deligne, D. Kazhdan, and M.-F. Vignéras.  Représentations des algèbres centrales simples $p$-adiques.  In Representations of reductive groups over a local field, Travaux en Cours, pages 33–117. Hermann, Paris, 1984.
  * [11] S. S. Gelbart.  Automorphic forms on adèle groups.  Annals of Mathematics Studies, No. 83. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1975.
  * [12] F. M. Goodman, P. de la Harpe, and V. F. R. Jones.  Coxeter graphs and towers of algebras, volume 14 of Mathematical Sciences Research Institute Publications.  Springer-Verlag, New York, 1989.
  * [13] Harish-Chandra.  Harmonic analysis on real reductive groups. I. The theory of the constant term.  J. Functional Analysis, 19:104–204, 1975.
  * [14] H. Jacquet and R. P. Langlands.  Automorphic forms on ${\rm GL}(2)$.  Lecture Notes in Mathematics, Vol. 114. Springer-Verlag, Berlin-New York, 1970.
  * [15] A. W. Knapp.  Representations of ${\rm GL}_{2}({\bf R})$ and ${\rm GL}_{2}({\bf C})$.  In Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, Proc. Sympos. Pure Math., XXXIII, pages 87–91. Amer. Math. Soc., Providence, R.I., 1979.
  * [16] A. W. Knapp and J. D. Rogawski.  Applications of the trace formula.  In Representation theory and automorphic forms (Edinburgh, 1996), volume 61 of Proc. Sympos. Pure Math., pages 413–431. Amer. Math. Soc., Providence, RI, 1997.
  * [17] C. Maclachlan and A. W. Reid.  The arithmetic of hyperbolic 3-manifolds, volume 219 of Graduate Texts in Mathematics.  Springer-Verlag, New York, 2003.
  * [18] F. Radulescu.  Free group factors and Hecke operators.  In The varied landscape of operator theory, volume 17 of Theta Ser. Adv. Math., pages 241–257. Theta, Bucharest, 2014.
  * [19] I. Reiner.  Maximal orders, volume 28 of London Mathematical Society Monographs. New Series.  The Clarendon Press, Oxford University Press, Oxford, 2003.  Corrected reprint of the 1975 original, With a foreword by M. J. Taylor.
  * [20] J. D. Rogawski.  An application of the building to orbital integrals.  Compositio Math., 42(3):417–423, 1980/81.
  * [21] J. D. Rogawski.  Representations of ${\rm GL}(n)$ and division algebras over a $p$-adic field.  Duke Math. J., 50(1):161–196, 1983.
  * [22] P. Sarnak.  Notes on the generalized Ramanujan conjectures.  In Harmonic analysis, the trace formula, and Shimura varieties, volume 4 of Clay Math. Proc., pages 659–685. Amer. Math. Soc., Providence, RI, 2005.
  * [23] J. A. Shalika.  A theorem on semi-simple ${\cal P}$-adic groups.  Ann. of Math. (2), 95:226–242, 1972.
  * [24] M.-F. Vignéras.  Arithmétique des algèbres de quaternions, volume 800 of Lecture Notes in Mathematics.  Springer, Berlin, 1980.
  * [25] J. Voight.  Quaternion algebras, volume 288 of Graduate Texts in Mathematics.  Springer, Cham, [2021] ©2021.
  * [26] J.-L. Waldspurger.  La formule de Plancherel pour les groupes $p$-adiques (d’après Harish-Chandra).  J. Inst. Math. Jussieu, 2(2):235–333, 2003.
  * [27] N. R. Wallach.  Real reductive groups. I, volume 132 of Pure and Applied Mathematics.  Academic Press, Inc., Boston, MA, 1988.
  * [28] N. R. Wallach.  Real reductive groups. II, volume 132-II of Pure and Applied Mathematics.  Academic Press, Inc., Boston, MA, 1992.

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