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bibarxiv

A note on the Sauvageot density principle

Yugo Takanashi Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-Ku, Tokyo 153-8914, Japan [email protected]

Abstract.

In this short note, we address a gap in the proof of Sauvageot’s paper [Sau97] pointed out in [NelsonVenkatesh2021-orbitmethod] and provide a complete proof of its main theorem.

1. Introduction

The Sauvageot density principle for reductive groups over local fields was proved in [Sau97] and it has been applied to prove well known results; see, for example, [Shi12], [FLM15] and [MikrosBergeron2017]. It was noted in [Shi12]*Appendix A that there are some minor errors in the proof of this result. More recently, in [NelsonVenkatesh2021-orbitmethod]*p.159, footnote, it was pointed out that there is a gap in the proof. In this short note, we fill the gap in the proof pointed out in [NelsonVenkatesh2021-orbitmethod]. We also fix some notational and logical errors in the proof, pointed out in [Shi12].

Roughly speaking, to prove the density principle, we need to approximate a function on the unitary dual using the Fourier transform of compactly supported functions. The proof given in [Sau97] utilizes only the functions that factor through the space of Bernstein components. However, the map from the unitary dual to the union of Bernstein components has non-trivial fibers, and the functions used by Sauvageot cannot separate the points within a single fiber. We will explain why this method by Sauvageot actually fails, see Remark 6.1. This appears to be the gap identified in [NelsonVenkatesh2021-orbitmethod]*p.159, footnote.

The main idea to fill the gap is to utilize a module version of the Stone-Weierstrass theorem, see Lemma 6.2, and the measure theoretic analog of this result, see Theorem 6.11. This enables us to distinguish points within each fiber of the map, allowing us to recover Sauvageot’s original result.

In the last part of this paper, we will present two applications of Sauvegeot’s density principle. One is the automorphic Plancherel density theorem by Shin. We will give a somewhat extended version of his result. The other one is an application of the Weyl law by Finis-Matz and Eikemeier, which produces spherical cusp forms of a quasi-split group with prescribed local properties.

1.1. Notation and assumptions

Let $F$ be a local field or global field of characteristic 0. Let $\mathbb{G}$ be a connected reductive group over $F$ . We denote by $G$ the set of $F$ -valued points of $\mathbb{G}$ . Unless otherwise stated, we will identify $\mathbb{G}$ and $G=\mathbb{G}(F)$ . Also, we will only consider the sets of $F$ -valued points of $F$ -subgroups of $\mathbb{G}$ and we referring to them simply as subgroups.

We use the following standard notation.

•

Let $A_{G}$ denote the maximal split torus of the center $Z_{G}$ of $G$ .
•

Let $\mathcal{P}^{G}(M)$ denote the set of parabolic subgroups of $G$ with the Levi factor $M$ .
•

Let $\mathcal{X}(G)$ denote the group of unramified characters of $G$ . Let $\mathcal{X}_{\mathrm{unit}}(G)$ denote the group of unitary unramified characters of $G$ .
•

Let $X^{*}(M)$ be the group of rational characters of $M$ over $F$ . We set $\mathfrak{a}^{*}_{M}=X^{*}(M)\otimes_{\mathbb{Z}}\mathbb{R}$ and $\mathfrak{a}^{*}_{M,\mathbb{C}}=X^{*}(M)\otimes_{\mathbb{Z}}\mathbb{C}$ . We have the natural map

$\displaystyle\mathfrak{a}^{*}_{M,\mathbb{C}}\to\mathcal{X}(M)\colon\lambda\mapsto\chi_{\lambda}.$
•

Let $W(G,M)$ denote the relative Weyl group for $M$ .

If $F=\mathbb{R}$ , let $\operatorname{Lie}(G)$ or $\mathfrak{g}$ denote the Lie algebra of $G$ . We always fix a maximal compact subgroup of $G$ and we denote it by $K$ . By a representation of a real reductive group, we mean an admissible $(\mathfrak{g}\otimes_{\mathbb{R}}\mathbb{C},K)$ over $\mathbb{C}$ . If we consider a topological representation, we take the Casselman-Wallach globalization. If $F$ is a $p$ -adic field, by a representation of $p$ -adic groups, we mean a smooth admissible representation over $\mathbb{C}$ .

1.2. Acknowledgment

I became aware of Sauvageot’s paper while collaborating with Satoshi Wakatsuki. I would like to thank Satoshi Wakatsuki for his encouragement and for careful reading. I would like to thank Sug Woo Shin for his encouragement, careful reading and helpful comments which greatly improved the readability of this article. Some parts of this paper are based on his work [Shi12]. I would like to thank Masaki Natori and Mao Hoshino for their careful reading and for pointing out some mistakes in definition 6.6 and Lemma 6.7. I would like to thank Yoichi Mieda for his careful reading. This work is supported by JSPS Grant-in-Aid for JSPS Fellows 23KJ0403.

2. The space of infinitesimal characters

2.1. Definition

We first recall the elementary facts about the space of infinitesimal characters, following [BernsteinDeligneKazhdan1986]*2.

Let $F,G$ as in section 1.1.

The notion of infinitesimal characters for reductive groups over $\mathbb{R}$ is well known; they correspond to the characters of the commutative algebra $Z(U(\mathfrak{g}_{\mathbb{C}}))$ . Let $\Theta(G)$ denote the set of infinitesimal characters of $G$ .

Definition 2.1.

We assume that $F$ is a $p$ -adic field. We call a pair $(M,\sigma)$ , consisting of a Levi subgroup $M$ of $G$ and a supercuspidal representation $\sigma$ of $M$ a cuspidal datum of $G$ . We refer to a $G$ -conjugacy class of a cuspidal datum of $G$ as an infinitesimal character of $G$ . We write $\Theta(G)$ for the set of infinitesimal characters. We write $[M,\sigma]_{G}$ for the class of $(M,\sigma)$ .

The following subrepresentation theorem is well known.

Theorem 2.2 ([Renard2010-p-adiques]*VI 5.4, Théorème).

We assume that $F$ is a $p$ -adic field. Let $\pi$ be an irreducible representation of $G$ .

(1)

There exists a cuspidal datum $(M,\sigma)$ and a parabolic subgroup $P\in\mathcal{P}^{G}(M)$ such that $\pi\subset i^{G}_{P}(\sigma)$ .
(2)

Let $(M,\sigma)$ and $(M^{\prime},\sigma^{\prime})$ be two cuspidal data. We assume that there exist parabolic subgroups $P\in\mathcal{P}^{G}(M)$ and $P^{\prime}\in\mathcal{P}^{G}(M)$ such that the parabolic inductions $i^{G}_{P}(\sigma)$ and $i^{G}_{P^{\prime}}(\sigma^{\prime})$ have a common irreducible component. Then, the infinitesimal characters $[M,\sigma]_{G}$ and $[M^{\prime},\sigma^{\prime}]_{G}$ are equal. Also, the converse holds true.
(3)

In the case of $(2)$ , the semisimplifications of $i^{G}_{P}(\sigma)$ and $i^{G}_{P^{\prime}}(\sigma^{\prime})$ are equal.

Construction 2.3.

We assume that $F$ is a $p$ -adic field. Let $(M,\sigma)$ be a cuspidal datum of $G$ . We call the image of the map

(2.1)

\displaystyle\mathcal{X}(M)\to\Theta(G)\colon\chi\mapsto[M,\sigma\otimes\chi]_{G}

a connected component of $\Theta(G)$ containing $[M,\sigma]_{G}$ . We denote it by $C=C_{[M,\sigma]}$ . We denote the stabilizer of $\sigma$ in $\mathcal{X}(M)$ by $\mathcal{X}(M)_{\sigma}$ . Also, we denote the stabilizer of $C=C_{[M,\sigma]}$ in $W(G,M)$ by $W(G,M)_{C}$ .

Through the above map, we can identify the set $C_{[M,\sigma]}$ with the quotient of the complex algebraic torus $\mathcal{X}(M)/\mathcal{X}(M)_{\sigma}$ by the finite group $W(G,M)_{C}$ . Thus, we endow the set $C=C_{[M,\sigma]}$ with the structure of affine algebraic variety $(\mathcal{X}(M)/\mathcal{X}(M)_{\sigma})//W(G,M)_{C}$ . We also endow the set $\Theta(G)$ with the structure of a countable disjoint union of complex algebraic varieties as the countable disjoint union of connected components. Note that we can identify the connected component $C$ with the image of the map

(2.2)

\displaystyle\mathfrak{a}^{*}_{M,\mathbb{C}}\to\Theta(G)\colon\lambda\mapsto[M,\sigma\otimes\chi_{\lambda}]_{G}.

This definition can be used to define the similar notion in the archimedean case.

2.2. A ring of functions and hermitian involutions

Definition 2.4.

We assume that $F$ is a $p$ -adic field. Let $\mathcal{A}(G)$ denote the ring of regular functions on $\Theta(G)$ supported on finite connected components of $\Theta(G)$ . We assume that $F=\mathbb{R}$ . Let $\theta$ be a Cartan involution of $G(\mathbb{R})$ . In this case, let $\mathfrak{h}\subset\operatorname{Lie}(G)$ be the Lie algebra of a $\theta$ -stable maximally split maximal torus $H\subset G$ . Then, the subalgebra decomposes as $\mathfrak{a}_{0}\oplus\mathfrak{t}_{0}$ with respect to the Cartan involution. The space $\mathfrak{a}_{0}$ (resp. $\mathfrak{t}_{0}$ ) is the $-1$ -eigenspace (resp. $+1$ -eigenspace) with respect to the action of $\theta$ . We set $\mathfrak{h}_{\mathbb{R}}=\mathfrak{a}_{0}\oplus i\mathfrak{t}_{0}$ . Then, we have the Fourier-Laplace transform

\displaystyle C_{c}^{\infty}(\mathfrak{h}_{\mathbb{R}})^{W(G(\mathbb{C}),H(\mathbb{C}))}\to\mathcal{PW}(\mathfrak{h}_{\mathbb{R}}\otimes_{\mathbb{R}}\mathbb{C})^{W(G(\mathbb{C}),H(\mathbb{C}))},

from the space of compactly supported functions to the space of Paley-Wiener functions. We denote by $\mathcal{A}(G)$ the ring of functions on the right hand side.

Definition 2.5.

We assume that $F$ is a $p$ -adic field. We define the anti-holomorphic involution on the set $\Theta(G)$ by $[M,\sigma]_{G}\mapsto[M,\sigma^{h}]_{G}$ , where $\sigma^{h}=\overline{\sigma}^{\vee}$ . We denote the image by $[M,\sigma]_{G}^{h}$ . Let $\Theta(G)_{\mathrm{herm}}$ denote the set of fixed points of this involution. We assume that $F=\mathbb{R}$ . The linear bijection $\operatorname{Lie}(G)\to\operatorname{Lie}(G)\colon X\mapsto-X$ extends $\mathbb{C}$ -semilinearly to the semilinear isomorphism $Z(U(\operatorname{Lie}(G)_{\mathbb{C}}))\to Z(U(\operatorname{Lie}(G)_{\mathbb{C}}))$ . We denote this map by $z\mapsto z^{h}$ . This involution induces the anti-holomorphic involution $(\cdot)^{h}\colon\Theta(G)\to\Theta(G)$ .

Lemma 2.6.

We denote the pullback of an element $a\in\mathcal{A}(G)$ by the map $h$ by $a^{h}$ . Then, the map $a\mapsto\overline{a^{h}}$ induces the $\mathbb{C}$ -linear involution of the algebra $\mathcal{A}(G)$ .

3. The space of induced representations

We introduce the space of induced representations from essentially discrete series representations, which is similar to the space of infinitesimal characters.

Definition 3.1.

We call a pair $(M,\sigma)$ which consists of a Levi subgroup $M$ of $G$ and an essentially discrete series representation $\sigma$ of $M$ a discrete pair of $G$ . We refer to a $G$ -conjugacy class of a discrete pair of $G$ as a discrete datum of $G$ . We denote the class of $(M,\sigma)$ by $\{M,\sigma\}_{G}$ and the set of discrete data of $G$ by $\Theta_{\mathrm{disc}}(G)$ .

Lemma 3.2.

Let $(M,\sigma)$ and $(M^{\prime},\sigma^{\prime})$ be discrete data of $G$ and $P\in\mathcal{P}^{G}(M)$ and $P^{\prime}\in\mathcal{P}^{G}(M^{\prime})$ . Then, the semisimplifications $i_{P}^{G}(\sigma)$ and $i_{P^{\prime}}^{G}(\sigma^{\prime})$ are equal if and only if the discrete data $\{M,\sigma\}_{G}$ and $\{M^{\prime},\sigma^{\prime}\}_{G}$ are equal.

Proof.

First, by the character formula for parabolic induction [Lipsman1971]*Theorem 9.1 and [vanDijk1972]*5.3, the character of $i_{P}^{G}(\sigma)$ is independent from the choice of $P$ . Hence, we assume the real part of the central character $\operatorname{Re}(\chi_{\sigma})$ of $\sigma$ is positive with respect to $P$ and $P$ is standard. Here, the positivity means that $\operatorname{Re}(\chi_{\sigma})$ is in the closure of the cone defined by the fundamental weight associated to $P$ (that is the relative Weyl chamber associated to $P$ ).

Now, we take the Levi subgroup $M_{1}$ which contains $M$ such that $\operatorname{Re}(\chi_{\sigma})\in\mathfrak{a}_{M_{1}}^{*}$ and regular in $\mathfrak{a}_{M_{1}}^{*}$ . Then, we obtain the standard parabolic subgroup $P_{1}$ of $G$ with Levi subgroup $M_{1}$ and $\operatorname{Re}(\chi_{\sigma})$ is positive with respect to $P_{1}$ . Also, we write the representation $\sigma$ as $\sigma_{u}\otimes\operatorname{Re}(\chi_{\sigma})$ , where $\sigma_{u}$ is a unitary discrete series representation. In this situation, we have $i_{P}^{G}(\sigma)=i_{P_{1}}^{G}(i^{M_{1}}_{P\cap M_{1}}(\sigma_{u})\otimes\operatorname{Re}(\chi_{\sigma}))$ . By decomposing $i^{M_{1}}_{P\cap M_{1}}(\sigma_{u})$ to the irreducible tempered representations and taking their semisimplifications, we obtain the sum of standard modules. By the remark in the proof of [Clo86]*Proposition 2.1 and [Hecht1979]*Proposition 6.2, the standard representations form a finite basis of the Grothendieck group of representations $\pi^{\prime}$ of $G$ with $\inf_{G}(\pi^{\prime})=\inf_{G}(i^{G}_{P}(\sigma))$ .

If we apply this procedure to the representation $i_{P^{\prime}}^{G}(\sigma^{\prime})$ to obtain $M_{1}^{\prime}$ , $P_{1}^{\prime}$ and so on. By assumption, we see that $P_{1}=P_{1}^{\prime}$ , $M_{1}=M_{1}^{\prime}$ , $\operatorname{Re}(\chi_{\sigma})=\operatorname{Re}(\chi_{\sigma^{\prime}})$ and $(M,\sigma_{u})$ and $(M^{\prime},\sigma^{\prime}_{u})$ are conjugate by an element $w\in M_{1}(F)$ , i.e. $\sigma^{\prime}_{u}=w(\sigma_{u})$ and $M^{\prime}=w(M)$ . As the character $\operatorname{Re}(\chi_{\sigma})$ is in $\mathfrak{a}_{M_{1}}^{*}$ , the element $w$ acts this element trivially and we have $w(M,\sigma)=(M^{\prime},\sigma^{\prime})$ . Hence the result. ∎

Remark 3.3.

We identify $\{M,\sigma\}_{G}$ with the semisimplification $i^{G}_{P}(\sigma)_{ss}$ of the induced representation $i^{G}_{P}(\sigma)$ . Using Lemma 3.2 and 2.3, we can endow the set $\Theta_{\mathrm{disc}}(G)$ with the countable union of complex algebraic varieties and we can also define the notion of connected component. We omit the details.

Construction 3.4.

We define the map

\displaystyle\mathrm{inf}_{G}\colon\Theta_{\mathrm{disc}}(G)\to\Theta(G)

defined as follows.

Let $\{M,\sigma\}_{G}$ be a discrete pair of $G$ .

•

If $F=\mathbb{R}$ , the map is just taking the infinitesimal character of $i^{G}_{P}(\sigma)$ for any $P\in\mathcal{P}^{G}(M)$ .
•

If $F$ is a $p$ -adic field, we can embed $\sigma$ in $i^{M}_{Q}(\tau)$ for some cuspidal datum $(L,\tau)\in\Theta(M)$ and $Q\in\mathcal{P}^{M}(L)$ . Then, we set $\inf_{G}(\{M,\sigma\}_{G})=[L,\tau]_{G}$ . This map is well-defined by Theorem 2.2 and Lemma 3.2.

The following lemma is obvious.

Lemma 3.5.

The map

\displaystyle\mathrm{inf}_{G}\colon\Theta_{\mathrm{disc}}(G)\to\Theta(G)

taking the infinitesimal characters of representations is continuous and proper. Also, each fiber is a finite set.

Proof.

See [NelsonVenkatesh2021-orbitmethod]*p.157, line 1 for the first statement. We prove the second statement. Note that the set of $G$ -conjugacy classes of Levi subgroups of $G$ is finite. Thus, the second statement follows from [Wallach1988realreductiveI]*Theorem 5.5.6 if $F=\mathbb{R}$ , and [Renard2010-p-adiques]*VI 6.2, Lemme if $F$ is a $p$ -adic field and the fact that the parabolic induction of any irreducible representation is of finite length. ∎

Definition 3.6.

Let $\Theta_{\mathrm{disc}}(G)_{\mathrm{temp}}$ denote the closed subset of $\Theta_{\mathrm{disc}}(G)$ which consists of classes $\{M,\sigma\}_{G}$ with $\sigma$ unitary discrete series. This set also endows with the structure of the countable union of algebraic varieties over $\mathbb{R}$ , as the countable union of quotients of compact tori or $\mathbb{R}$ -vector spaces by finite groups.

Remark 3.7.

We can identify the element $\{M,\sigma\}_{G}\in\Theta_{\mathrm{disc}}(G)_{\mathrm{temp}}$ with the semisimple representation $i^{G}_{P}(\sigma)$ . As the spaces $\Theta_{\mathrm{disc}}(G),\Theta(G)$ and $\Theta_{\mathrm{disc}}(G)_{\mathrm{temp}}$ are countable unions of quotients of locally Euclidean spaces by finite groups, we have the notion of smooth functions, holomorphic functions on these spaces and so on.

The following is well known, for example, see [Wal03]*Proposition III 4.1 and [Harish-Chandra1972-Eisensteinintegral]*Lemma 12.

Lemma 3.8.

Let $\{M,\sigma\}_{G}$ and $\{M^{\prime},\sigma^{\prime}\}_{G}$ be elements of $\Theta_{\mathrm{disc}}(G)_{\mathrm{temp}}$ . Then, if the parabolic inductions $i^{G}_{P}(\sigma)$ and $i^{G}_{P^{\prime}}(\sigma^{\prime})$ have a common irreducible constituent, then we have the equality $\{M,\sigma\}_{G}=\{M^{\prime},\sigma^{\prime}\}_{G}$ .

4. On the generic irreducibility theorem

As noted in [Shi12], the proof of the generic irreducibility theorem [Sau97]*Théorème 3.2 appears to contain an error. Konno [Konno2003]*Theorem 4.2 provided a remedy. However, a stronger hypothesis than [Sau97]*Théorème 3.2 was used in [Konno2003]*p.403, l.24. We will show that this strengthening is unnecessary by proving the following lemma. Furthermore, the proof of [Sau97]*Corollaire 3.3 seems to require the original version of the theorem.

Lemma 4.1.

Let $V$ be a real vector space. Let $S$ be a finite subset in $V$ . Let $T^{\vee}$ be a finite subset in the linear dual $V^{\vee}$ of $V$ . If no element of $T^{\vee}$ vanishes at an element in $S$ , then there exist $v\in\mathrm{span}_{\mathbb{R}}(S)$ such that $\lambda(v)\neq 0$ for all $\lambda\in T^{\vee}$ .

Proof.

Let $W=\mathrm{span}_{\mathbb{R}}(S)$ . Then, every element of $T^{\vee}$ has non-zero restriction on $W$ . Using the standard hyperplane avoidance technique, we can take $v\in W$ as in the statement. ∎

Let $P=MN$ be a parabolic subgroup of $G$ with a Levi decomposition $P=MN$ . Let $\Phi(G,A_{M})$ be the set of weights of the action of $A_{M}$ on $\mathrm{Lie}(G)$ .

Theorem 4.2.

For every irreducible representation $\sigma$ of $M$ , there exists a neighborhood $\mathcal{U}$ of $0\in(\mathfrak{a}_{M})^{*}_{\mathbb{C}}$ such that $i_{P}^{G}(\sigma\otimes\chi_{\lambda})$ is irreducible for any element $\lambda\in\mathcal{U}$ such that $\alpha^{\vee}(\lambda)\neq 0$ , $\alpha\in\Phi(G,A_{M})$ .

Proof.

If $F=\mathbb{R}$ , this follows from [Renard2024]*Theorem 1.1. We assume that $F$ is a $p$ -adic field. Then, the proof is the same as that in [Konno2003]*Theorem 4.2, if we combine the argument at [Konno2003]*p.403, l.24 with the previous lemma. ∎

5. The unitary dual and the Plancherel measure

We will review some definitions and results on the unitary dual and the Plancherel measures for reductive groups over local fields.

Definition 5.1.

We write $\operatorname{Irr}_{\mathrm{unit}}(G)$ for the unitary dual of $G$ . This space is naturally endowed with the Fell topology. Let $\operatorname{Irr}_{\mathrm{temp}}(G)$ denote the closed subset of $\operatorname{Irr}_{\mathrm{unit}}(G)$ consisting of tempered representations.

We fix a Haar measure on $G$ throughout. Also, the space $\Theta_{\mathrm{disc}}(G)$ is a countable union of quotients of locally Euclidean spaces by finite groups. Thus, we have the quotient measure on $\Theta_{\mathrm{disc}}(G)$ , induced from the locally Euclidean measures. This measure naturally restricts to the subspace $\Theta_{\mathrm{disc}}(G)_{\mathrm{temp}}$ . Let $d\pi$ denote this measure, where $\pi=i^{G}_{P}(\sigma)\in\Theta_{\mathrm{disc}}(G)_{\mathrm{temp}}$ .

Theorem 5.2 ([Wal03]Théorème VIII 1.1, [Wallach1992RRGII]Theorem 13.4).

There exists a positive smooth function $\mu^{G}(\pi)$ of moderate growth on $\Theta_{\mathrm{disc}}(G)_{\mathrm{temp}}$ such that we have

\displaystyle f(1)=\int_{\pi\in\Theta_{\mathrm{disc}}(G)_{\mathrm{temp}}}\operatorname{tr}\pi(f)\mu^{G}(\pi)\,\mathrm{d}\pi

for any smooth function $f\in C_{c}^{\infty}(G)$ .

Definition 5.3.

We call the measure $\mu^{G}(\pi)d\pi$ the Plancherel measure of $G$ and we denote it by $\mu^{G}$ . Let $\widehat{f}$ denote the function $\pi\mapsto\operatorname{tr}\pi(f)$ on $\operatorname{Irr}_{\mathrm{unit}}(G)$ or $\Theta_{\mathrm{disc}}(G)$ . We refer to it as the Fourier transform of $f\in C_{c}^{\infty}(G)$ . We denote the set of Fourier transforms for $K$ -finite elements in $C_{c}^{\infty}(G)$ by $\mathcal{FT}(G)$ .

Definition 5.4.

Let $f\in C_{c}^{\infty}(G)$ be a smooth function. We set

\displaystyle f^{*}(g)=\overline{f(g^{-1})}.

Lemma 5.5.

Let $\pi$ be a unitary admissible representation of $G$ of finite length. Let $f\in C_{c}^{\infty}(G)$ be a smooth $K$ -finite function. Then, we have

\displaystyle\operatorname{tr}\pi(f^{*})=\overline{\operatorname{tr}\pi(f)}.

Proof.

We fix an orthonormal basis $\mathcal{B}$ of $V$ . Then, we have

	$\displaystyle\operatorname{tr}\pi(f^{*})$	$\displaystyle=\sum_{v\in\mathcal{B}}(\pi(f^{*})v,v)$
		$\displaystyle=\sum_{v\in\mathcal{B}}\int_{G}\overline{f(g^{-1})}(\pi(g)v,v)\,\mathrm{d}g$
		$\displaystyle=\sum_{v\in\mathcal{B}}\int_{G}\overline{f(g)(\pi(g)v,v)}\,\mathrm{d}g$
		$\displaystyle=\overline{\operatorname{tr}\pi(f)}.$

Note that by admissibility, the above sums are finite sums. Hence the result. ∎

Lemma 5.6.

The subset of $\Theta_{\mathrm{disc}}(G)_{\mathrm{temp}}$ which consists of parabolic inductions of regular discrete series representations in the sense of [Sau97]*p.172, line 30 is a dense open subset in $\Theta_{\mathrm{disc}}(G)_{\mathrm{temp}}$ which is conull with respect to $d\pi$ . Thus, we can identify $\Theta_{\mathrm{disc}}(G)_{\mathrm{temp}}$ with $\operatorname{Irr}_{\mathrm{temp}}(G)$ up to a null set with respect to $d\pi$ .

On this subset, we have an equality of induced topologies from the spaces $\Theta_{\mathrm{disc}}(G)_{\mathrm{temp}}$ and $\operatorname{Irr}_{\mathrm{temp}}(G)$ .

Proof.

The first statement follows from Theorem 4.2. The second statement follows from the argument in [Beuzart-Plessis2021-Plancherel-GLnE-GLnF]*p.245, line 33. ∎

Definition 5.7.

By the previous lemma, the measure $\mu^{G}$ defines a Borel measure on $\operatorname{Irr}_{\mathrm{temp}}(G)$ and also on $\operatorname{Irr}_{\mathrm{unit}}(G)$ . The measures are also denoted by $\mu^{G}$ .

6. A variant of the Stone-Weierstrass theorem

In this section, we prove the main approximation theorem Theorem 6.11 which will be used to complete the proof of the Sauvageot density principle.

Before proving the approximation theorem, we recall the gap pointed out in [NelsonVenkatesh2021-orbitmethod]*p.159, footnote.

Remark 6.1.

In [Sau97]*p.181, line 4, for any function $f$ as [Sau97]*p.180, (1) and any positive number $\epsilon>0$ , it is stated that there exist functions $g,h\in\mathcal{A}(G)$ such that

	$\displaystyle\lvert f-g\circ\mathrm{inf}_{G}\rvert$	$\displaystyle\leq h\circ\mathrm{inf}_{G}$
	$\displaystyle\mu^{G}(h\circ\mathrm{inf}_{G})$	$\displaystyle<\epsilon.$

We assume that $G$ has a pair of discrete series representations $\pi_{1}\neq\pi_{2}$ such that $\mathrm{inf}_{G}(\pi_{1})=\inf_{G}(\pi_{2})$ . Such a pair exists if $G=G_{2}$ , see [GS23theta]*Proposition 3.2, (ii). We take the characteristic function of $\{\pi_{1}\}$ as $f$ and this satisfies the condition $(1)$ of [Sau97]*p.180. If there exist functions $g,h\in\mathcal{A}(G)$ such that

	$\displaystyle\lvert f-g\circ\mathrm{inf}_{G}\rvert$	$\displaystyle\leq h\circ\mathrm{inf}_{G}$
	$\displaystyle\mu^{G}(h\circ\mathrm{inf}_{G})$	$\displaystyle<\epsilon,$

then, we have

	$\displaystyle\lvert 1-g\circ\mathrm{inf}_{G}(\pi_{1})\rvert$	$\displaystyle\leq h\circ\mathrm{inf}_{G}(\pi_{1}),$
	$\displaystyle\lvert g\circ\mathrm{inf}_{G}(\pi_{2})\rvert$	$\displaystyle\leq h\circ\mathrm{inf}_{G}(\pi_{2}).$

This implies that we have $h\circ\inf_{G}(\pi_{1})\geq\frac{1}{2}$ . Finally, we obtain $\mu^{G}(h\circ\mu^{G})\geq\frac{1}{2}d(\pi_{1})$ , where $d(\pi_{1})$ is the formal degree of $\pi_{1}$ and this gives a contradiction if $\epsilon<\frac{1}{2}d(\pi_{1})$ .

Thus, to approximate the functions in [Sau97]*p.180, (1) in the previous sense, we have to utilize more functions.

6.1. Approximation on locally compact Hausdorff spaces

Let $X,Y$ be locally compact Hausdorff spaces and $p\colon X\to Y$ be a proper map. Let $X^{\infty}=X\coprod\{\infty_{X}\}$ denote the one-point compactification of $X$ . Then, we can extend $p$ to a continuous map $p^{\infty}\colon X^{\infty}\to Y^{\infty}$ . We have the identification of the function spaces:

C_{0}(X)\overset{\cong}{\to}\{f\in C(X^{\infty})\,|\,f(\infty_{X})=0\}\colon f\mapsto f

by the extension $f(\infty_{X})=0$ . This isomorphism preserves the norms $||\cdot||_{\infty}$ . The space $C_{0}(X)$ has the norm, so it defines a metric on this space. Also we can take the distance

d_{X}(f,S)=\inf_{g\in S}{d_{X}(f,g)}

for any element $f\in C_{0}(X)$ and subset $S\subset C_{0}(X)$ .

We have the Stone-Weierstrass theorem for subalgebras of $C_{0}(Y)$ , for example, see [Rud91]*p.122. We utilize the following module version of the Stone-Weierstrass theorem, which is a special case of theorems proved by Nachbin and Prolla. We give a proof of the result to make this paper as self-contained as possible.

Lemma 6.2 (c.f. [Nachbin1965]p.54, Theorem 1, [Prolla1994]Theorem 9).

Let $A$ be a dense $\mathbb{C}$ -subalgebra of $C_{0}(Y)$ . Let $p^{*}A$ be a $\mathbb{C}$ -subalgebra of $C_{0}(X)$ which consists of the pullback of elements in $A$ by $p$ . Let $M$ be a subspace of $C_{0}(X)$ which is a $p^{*}A$ -module with respect to pointwise multiplication. Then, for any $f\in C_{0}(X)$ we have

d_{X}(f,M)=\sup_{y\in Y}{d_{p^{-1}(y)}(f|p^{-1}(y),M|p^{-1}(y))},

where $M|p^{-1}(y)$ denotes the set of functions obtained by restriction of elements in $M$ to the compact set $p^{-1}(y)$ and the distance in the right hand side is taken in $C(p^{-1}(y))$ .

Proof.

By one-point compactification, we can assume that $X$ and $Y$ are compact and the space $A$ and $M$ contain constant functions. Also, by replacing $Y$ by the image of $X$ , we can assume that $p$ is surjective.

We obviously have the inequality

d_{X}(f,M)\geq\sup_{y\in Y}{d_{p^{-1}(y)}(f|p^{-1}(y),M|p^{-1}(y))},

so we will prove the converse. Let $D$ denote the right hand side of this inequality and take $\epsilon>0$ . It suffices to show that left hand side is less than $D+\epsilon$ .

By the definition of supremum, for each $y\in Y$ , there exists an element $g_{y}\in M$ such that

d_{p^{-1}(y)}(f|p^{-1}(y),g_{y}|p^{-1}(y))<D+\epsilon.

By a simple argument using $p$ is a closed map, we can take an open neighborhood $y\in V_{y}$ such that on $U_{y}=p^{-1}(V_{y})$ we have

d_{U_{y}}(f|U_{y},g_{y}|U_{y})<D+\epsilon.

Since the open subsets $V_{y}$ cover the space $Y$ , we can take a finite open subcover $\{V_{i}=V_{y_{i}}\,|\,i=1,2,...,n\}$ . We can also take a partition of unity $\rho_{i}$ associated with this cover.

Set $g_{i}=g_{y_{i}}$ and

g=\sum_{i}p^{*}(\rho_{i})g_{i}\in C_{0}(X).

Then, we have

	$\displaystyle\lvert f(x)-g(x)\rvert$	$\displaystyle=\lvert\sum_{i}\rho_{i}(p(x))(f(x)-g_{i}(x))\rvert$
		$\displaystyle\leq\sum_{i}\rho_{i}(p(x))\lvert f(x)-g_{i}(x)\rvert$
		$\displaystyle<(\sum_{i}\rho_{i})(p(x))(D+\epsilon)$
		$\displaystyle=D+\epsilon.$

By usual Stone–Weierstrass’ theorem, we can approximate each $\rho_{i}$ by an element $\sigma_{i}$ of $A$ . As $M$ is a $p^{*}A$ -module by multiplication, the function

h=\sum_{i}p^{*}(\sigma_{i})g_{i}

belongs to $M$ and approximates $g$ . Hence the result. ∎

Remark 6.3.

The results by Nachbin and Prolla can be applied to more general settings, see [Nachbin1965]*p.54, Theorem 1 and [Prolla1994]*Definition 2, Theorem 9.

6.2. Approximation on measure spaces

Let $X,Y$ be locally compact Hausdorff spaces and $p\colon X\to Y$ be a proper map. Let $\mu_{X}$ be a Radon measure on $X$ and we denote the pushforward measure $p_{*}\mu_{X}$ by $\mu_{Y}$ . Then, $\mu_{Y}$ is also a Radon measure on $Y$ .

Definition 6.4.

Let $(Z,\mu)$ be a topological space with a positive Borel measure. A function $\phi\colon Z\to\mathbb{C}$ is said to be $\mu$ -Riemann integrable i it is bounded, compactly supported and its set of discontinuous points is $\mu$ -null set. We denote the set of $\mu$ -Riemann integrable functions by $\mathcal{RI}(Z,\mu)$ . We assume that $Z$ is a locally compact Hausdorff space and $\mu$ is a positive Radon measure on $Z$ . Let $\mathcal{L}^{1}(Z,\mu)$ denote the set of $\mu$ -integrable functions on $Z$ .

Example 6.5.

The characteristic function $\mathbf{1}_{U}$ of an open set $U$ is $\mu$ -Riemann integrable if and only if $U$ is relatively compact and its boundary $\partial U$ is a $\mu$ -null set.

Definition 6.6.

Let $M$ be a subspace of $\mathcal{L}^{1}(X,\mu_{X})$ such that any element $m\in M$ is $\mu_{X}$ -integrable and stable under pointwise complex conjugation. Let $\tau_{M}$ denote the linear topology on $\mathcal{L}^{1}(X,\mu_{X})$ whose fundamental system of neighborhoods of $0$ is generated by

U_{\epsilon}=\{h\in\mathcal{L}^{1}(X,\mu_{X})\,|\,\\ \text{There exists $m\in M$ such that }\lvert h\rvert\leq m,\mu_{X}(m)<\epsilon\},

for $\epsilon>0$ .

Lemma 6.7.

Let $\overline{M}^{\tau_{M}}$ denote the closure of $M$ with respect to $\tau_{M}$ . Then, we have $\tau_{M}=\tau_{\overline{M}^{\tau_{M}}}$ .

Proof.

As $M$ is stable under complex conjugation, it suffices to show the lemma after replacing $\mathcal{L}^{1}(X,\mu_{X})$ with the subspace $\mathcal{L}^{1}(X,\mu_{X},\mathbb{R})$ of real valued functions and $M$ with the subspace $M_{\mathbb{R}}$ of real valued functions in $M$ . If we consider the topology defined by $\overline{M}^{\tau_{M}}$ , we write the open subset $U_{\epsilon}$ as $U^{\prime}_{\epsilon}$ , i.e.

U^{\prime}_{\epsilon}=\{h\in\mathcal{L}^{1}(X,\mu_{X})\,|\,\\ \text{There exists $m^{\prime}\in\overline{M}^{\tau_{M}}$ such that }\lvert h\rvert\leq m^{\prime},\mu_{X}(m^{\prime})<\epsilon\}.

Then, it is trivial that we have $U_{\epsilon}\subset U^{\prime}_{\epsilon}$ . We claim that if $\epsilon^{\prime}<\epsilon$ , then we have $U^{\prime}_{\epsilon^{\prime}}\subset U_{\epsilon}$ . We assume that a function $h\in\mathcal{L}^{1}(X,\mu_{X})$ is in $U^{\prime}_{\epsilon^{\prime}}$ . Then, we can find $m^{\prime}\in\overline{M}^{\tau_{M}}$ and we have

	$\displaystyle\lvert h\rvert\leq m^{\prime},$
	$\displaystyle\mu_{X}(m^{\prime})<\epsilon^{\prime}.$

We set $\delta=\epsilon-\epsilon^{\prime}$ . Then, we have $\delta>0$ . As the function $m^{\prime}$ is in $\overline{M}^{\tau_{M}}$ , there exist functions $n_{1}\in M$ and $n_{2}\in M$ such that $\lvert m^{\prime}-n_{1}\rvert\leq n_{2}$ and $\mu_{X}(n_{2})<\frac{1}{2}\delta$ . Then, we have

\displaystyle\lvert h\rvert\leq n_{1}+n_{2}

and

\displaystyle\mu_{X}(n_{1}+n_{2})<\epsilon^{\prime}+2\times\frac{1}{2}\delta=\epsilon.

Thus, we have $U_{\epsilon^{\prime}}^{\prime}\subset U_{\epsilon}$ . ∎

Corollary 6.8.

We have $\overline{\overline{M}^{\tau_{M}}}^{\mathcal{T}_{\overline{M}^{\tau_{M}}}}=\overline{M}^{\tau_{M}}$ .

The following lemma is [Sau97]*Lemme 2.3.

Lemma 6.9.

Let $A\subset C_{0}(Y)$ be a dense $\mathbb{C}$ -subalgebra. Let $M\subset\mathcal{L}^{1}(X,\mu_{X})$ be a $p^{*}A$ -module with respect to pointwise multiplication and stable under complex conjugation, such that any element $m\in M$ is $\mu_{X}$ -integrable. We assume that $M$ also satisfies the following condition: For any function $m\in M$ , there exists a function $n\in M$ such that $\lvert m\rvert\leq n$ .

Then, for any function $m\in M$ and $c\in C_{0}(Y)$ , the function $(p^{*}c)m$ is in $\overline{M}^{\tau_{M}}$ .

Proof.

We take a function $n\in M$ as in the hypothesis of the lemma. There exists a function $a\in A$ such that $\lvert c-a\rvert\leq\epsilon$ with $\epsilon$ sufficiently small with respect to $\mu_{X}(n)$ . Then, we have

\displaystyle\lvert(p^{*}c)m-(p^{*}a)m\rvert\leq\epsilon n.

Hence the result. ∎

Remark 6.10.

Note that the proof of Lemma 6.9 and Corollary 6.8 does not use the Hausdorff property of $X$ .

Theorem 6.11.

Let $A$ be a dense $\mathbb{C}$ -subalgebra of $C_{0}(Y)$ . Let $M\subset C_{0}(X)\cap\mathcal{L}^{1}(X,\mu_{X})$ be a $p^{*}A$ -module with respect to pointwise multiplication. We assume the following conditions:

(1)

$M$ is stable under pointwise complex conjugation.
(2)

The restriction of the elements of $M$ to $p^{-1}(y)$ is dense in $C(p^{-1}(y))$ for every $y\in Y$ .
(3)

For any function $m\in M$ , there exists a function $n\in M$ such that $\lvert m\rvert\leq n$ .

Then, we have $\mathcal{RI}(X,\mu_{X})\subset\overline{M}^{\tau_{M}}$ .

Proof.

It suffices to show that we have $\mathcal{RI}(X,\mu_{X})\subset\overline{\overline{M}^{\tau_{M}}}^{\mathcal{T}_{\overline{M}^{\tau_{M}}}}$ by Corollary 6.8. We will write down this condition. Take any function $\phi\in\mathcal{RI}(X,\mu_{X})$ and any positive number $\epsilon>0$ . We want to show that there exist functions $m^{\prime}_{1},m^{\prime}_{2}\in\overline{M}^{\tau_{M}}$ such that we have

	$\displaystyle\lvert\phi-m^{\prime}_{1}\rvert$	$\displaystyle\leq m^{\prime}_{2}$
	$\displaystyle\mu_{X}(m^{\prime}_{2})$	$\displaystyle<\epsilon.$

We fix $\epsilon>0$ . By assumption, the support of $\phi$ is compact. Thus, we can take a positive function $c\in C_{c}(Y)$ such that on the support of $\phi$ , we have $p^{*}c=1$ . By the assumptions $(2),(3)$ for $M$ , it is easy to find a positive function $m_{c}\in M$ such that we have $p^{*}c\leq m_{c}$ on $X$ . Indeed, we claim that for any point $x\in X$ , we can find $m_{x}\in M$ such that $m_{x}(x)>0$ . This claim clearly proves the existence of the function $m_{c}$ . By the assumption $(2)$ , we can take a function $n_{x}\in M$ such that we have $n_{x}(x)\neq 0$ . Then, the assumption $(3)$ implies that there exists $m_{x}\in M$ such that we have $\lvert n_{x}\rvert\leq m_{x}$ . We have $m_{x}(x)\geq\lvert n_{x}(x)\rvert>0$ and this proves the claim.

We take a positive number $\epsilon_{1}>0$ such that

\displaystyle\epsilon_{1}<\frac{1}{3||m_{c}||_{\infty}+\mu_{X}(m_{c})}\epsilon.

By [Bou04]*IV 12 Lemma 5, we obtain functions $c_{1},c_{2}\in C_{c}(X)$ such that

	$\displaystyle\lvert\phi-c_{1}\rvert$	$\displaystyle\leq c_{2}$
	$\displaystyle\mu_{X}(c_{2})$	$\displaystyle\leq\epsilon_{1}.$

We apply Lemma 6.2 to obtain functions $m_{1},m_{2}\in M$ such that

	$\displaystyle\lvert c_{1}-m_{1}\rvert$	$\displaystyle<\epsilon_{1}$
	$\displaystyle\lvert c_{2}-m_{2}\rvert$	$\displaystyle<\epsilon_{1}.$

At this stage, we have

	$\displaystyle\lvert\phi-(p^{*}c)m_{1}\rvert$	$\displaystyle\leq(p^{}c)\lvert\phi-c_{1}\rvert+(p^{}c)\lvert c_{1}-m_{1}\rvert$
		$\displaystyle\leq(p^{}c)m_{2}+(p^{}c)\lvert c_{1}-m_{1}\rvert+(p^{*}c)\lvert c_{2}-m_{2}\rvert$
		$\displaystyle\leq(p^{*}c)m_{2}+\epsilon_{1}m_{c}+\epsilon_{1}m_{c}$
		$\displaystyle=(p^{*}c)m_{2}+2\epsilon_{1}m_{c}.$

Here, we have

	$\displaystyle\mu_{X}((p^{*}c)m_{2})$	$\displaystyle\leq\mu_{X}((p^{}c)\lvert m_{2}-c_{2}\rvert)+\mu_{X}((p^{}c)c_{2})$
		$\displaystyle\leq\epsilon_{1}\mu_{X}(p^{*}c)+\epsilon_{1}\|\|c\|\|_{\infty}.$

Thus, we have

\displaystyle\mu_{X}((p^{*}c)m_{2}+2\epsilon_{1}m_{c})\leq(3||m_{c}||_{\infty}+\mu_{X}(m_{c}))\epsilon_{1}<\epsilon

and $(p^{*}c)m_{1},(p^{*}c)m_{2}+2\epsilon_{1}m_{c}\in\overline{M}^{\tau_{M}}$ by Lemma 6.9. Hence the result. ∎

Example 6.12.

We take

•

the space $\Theta_{\mathrm{disc}}(G)_{\mathrm{temp}}$ as $X$ ,
•

the space $\Theta(G)_{\mathrm{herm}}$ as $Y$ ,
•

the map $\inf_{G}$ as $p$ ,
•

the Radon measure $\mu^{G}$ as $\mu_{X}$ ,
•

the algebra $\mathcal{A}(G)$ as $A$ and
•

the module $\mathcal{FT}(G)$ as $M$ , see definition 5.3.

The space $\mathcal{FT}(G)$ is an $\mathcal{A}(G)$ -module by the Paley-Wiener theorem [ClozelDelorme1990]*Théorème 1 if $F=\mathbb{R}$ and [Renard2010-p-adiques]*VI.10.3 Théorème if $F$ is a $p$ -adic field. Then, Lemmas in [Sau97]*Lemme 3.1, 3.4, 3.5, Lemma 2.6, Lemma 3.5, Lemma 3.8, Lemma 5.5, and the linear independence of irreducible characters imply that this sextuple satisfies the condition of Theorem 6.11. Also, the similar result holds for any reductive group schemes over a product of local fields.

Definition 6.13.

Let $\operatorname{Irr}_{\mathrm{unit}}(G)_{\mathrm{good}}$ denote the subset of $\operatorname{Irr}_{\mathrm{unit}}(G)$ which consists of the parabolic inductions associated to a discrete pair $(M,\sigma)$ with $\sigma$ unitary and regular in the sense of [Sau97]*p.172, line 30. Let $\operatorname{Irr}_{\mathrm{unit}}(G)_{\mathrm{bad}}$ denote the complement in $\operatorname{Irr}_{\mathrm{unit}}(G)$ of $\operatorname{Irr}_{\mathrm{unit}}(G)_{\mathrm{good}}$ .

Remark 6.14.

We have $\mu^{G}(\operatorname{Irr}_{\mathrm{unit}}(G)_{\mathrm{bad}})=0$ by Theorem 4.2.

Corollary 6.15.

Let $G$ be a reductive group scheme over a product of local fields of characteristic $0$ . Let $\phi$ be a function in $\mathcal{RI}(\operatorname{Irr}_{\mathrm{temp}}(G),\mu^{G})$ and $\epsilon$ be a positive real number. Then, there exist functions $h_{1},h_{2}\in C_{c}^{\infty}(G)$ such that we have

	$\displaystyle\lvert\phi-\widehat{h_{1}}\rvert$	$\displaystyle\leq\widehat{h_{2}},$
	$\displaystyle\mu^{G}(\widehat{h_{2}})$	$\displaystyle<\epsilon,$

on the set $\operatorname{Irr}_{\mathrm{unit}}(G)_{\mathrm{good}}$ .

Proof.

We take a function $\phi\in\mathcal{RI}(\operatorname{Irr}_{\mathrm{temp}}(G),\mu^{G})$ . Then, by Theorem 6.11 and Example 6.12, we can find functions $h_{1},h_{2}\in C_{c}^{\infty}(G)$ such that we have

	$\displaystyle\lvert\phi-\widehat{h_{1}}\rvert$	$\displaystyle\leq\widehat{h_{2}},$
	$\displaystyle\mu^{G}(\widehat{h_{2}})$	$\displaystyle<\epsilon,$

on the subset ${\operatorname{Irr}_{\mathrm{unit}}(G)}_{\mathrm{good}}$ in $\operatorname{Irr}_{\mathrm{unit}}(G)$ , through the identification explained in Lemma 5.6. Hence the result. ∎

7. The Sauvageot density principle

In this section, we will prove the following result.

Proposition 7.1 ([Sau97]*Théorème 5.4).

Let $G$ be a reductive group schemes over a product of local fields. Let $C$ be a compact subset of $\Theta(G)$ and let $\epsilon>0$ be a positive real number. Then, there exists a function $h\in C_{c}^{\infty}(G)$ which satisfies the following conditions.

(1)

We have $\widehat{h}\geq 0$ on $\operatorname{Irr}_{\mathrm{unit}}(G)$ .
(2)

We have $\mu^{G}(\widehat{h})\leq\epsilon$ .
(3)

For any $\pi\in\operatorname{Irr}_{\mathrm{unit}}(G)_{\mathrm{bad}}$ with $\inf_{G}(\pi)\in C$ , we have

$\displaystyle\widehat{h}(\pi)\geq 1.$

Before proving this theorem, we first show that this theorem and Theorem 6.11 imply the following Sauvageot density principle.

Theorem 7.2 (Sauvageot’s density principle, [Sau97]*Théorème 7.3).

Let $\phi$ be a bounded, Borel measurable, and compactly supported function on $\operatorname{Irr}_{\mathrm{temp}}(G)\subset\operatorname{Irr}_{\mathrm{unit}}(G)$ such that the set of discontinuity of $\phi$ is a $\mu^{G}$ -null set. Let $\epsilon>0$ be a positive real number. Then, there exist functions $h_{1},h_{2}\in C_{c}^{\infty}(G)$ such that we have

	$\displaystyle\lvert\phi-\widehat{h_{1}}\rvert$	$\displaystyle\leq\widehat{h_{2}}$
	$\displaystyle\mu^{G}(\widehat{h_{2}})$	$\displaystyle<\epsilon$

on $\operatorname{Irr}_{\mathrm{unit}}(G)$ .

Proof that Proposition 7.1 implies Theorem 7.2.

We take a function $c\in C_{c}(\Theta(G))$ such that $c^{\prime}=\inf_{G}^{*}c$ is equal to $1$ on the support of $\phi$ . We take a positive number $\epsilon_{1}$ such that

\displaystyle\epsilon_{1}<\frac{1}{2(||c||_{\infty}+1)}\epsilon.

We take the functions $h^{\prime}_{1},h^{\prime}_{2}\in C_{c}^{\infty}(G)$ satisfying the condition of Corollary 6.15 for $\phi$ and $\epsilon_{1}$ . Then, we have

	$\displaystyle\lvert\phi-\widehat{h^{\prime}_{1}}\rvert$	$\displaystyle\leq\widehat{h^{\prime}_{2}}$
	$\displaystyle\mu^{G}(\widehat{h^{\prime}_{2}})$	$\displaystyle<\epsilon_{1}$

on the complement of $\operatorname{Irr}_{\mathrm{unit}}(G)_{\mathrm{bad}}$ . Then, we have

\displaystyle\lvert\phi-c^{\prime}\widehat{h^{\prime}_{1}}\rvert\leq c^{\prime}\widehat{h^{\prime}_{2}}.

By Proposition 7.1, we can take an element $h^{\prime}_{3}\in C_{c}^{\infty}(G)$ such that we have

	$\displaystyle\lvert\phi-c^{\prime}\widehat{h^{\prime}_{1}}\rvert+c^{\prime}\widehat{h^{\prime}_{2}}$	$\displaystyle\leq\widehat{h^{\prime}_{3}}$
	$\displaystyle\mu^{G}(\widehat{h^{\prime}_{3}})$	$\displaystyle<\epsilon_{1}$

on the set $\operatorname{Irr}_{\mathrm{unit}}(G)_{\mathrm{bad}}$ . Then, we have

	$\displaystyle\lvert\phi-c^{\prime}\widehat{h^{\prime}_{1}}\rvert$	$\displaystyle\leq c^{\prime}\widehat{h^{\prime}_{2}}+\widehat{h^{\prime}_{3}}$
	$\displaystyle\mu^{G}(c^{\prime}\widehat{h^{\prime}_{2}}+\widehat{h^{\prime}_{3}})$	$\displaystyle<(\|\|c\|\|_{\infty}+1)\epsilon_{1}<\frac{1}{2}\epsilon.$

By applying Corollary 6.8, Lemma 6.9 and Remark 6.10, we can find functions $h_{1},h_{2}\in C_{c}^{\infty}(G)$ such that we have

	$\displaystyle\lvert\phi-\widehat{h_{1}}\rvert$	$\displaystyle\leq\widehat{h_{2}}$
	$\displaystyle\mu^{G}(\widehat{h_{2}})$	$\displaystyle<\epsilon.$

Hence the result. ∎

Lemma 7.3.

[Sau97]*Corollaire 6.2 Proposition 7.1 for reductive groups over local fields implies the general case of Proposition 7.1.

Thus, we return to the setting where $G$ is a reductive group over a local field $F$ .

Lemma 7.4 (c.f. [Sau97]*Lemme 2.6).

Let $V$ be a finite dimensional $\mathbb{R}$ -vector space and let $\Gamma$ be a discrete subgroup of $V$ . We set $X=V/\Gamma$ . For any compact set $C$ of $X$ and a family of open neighborhood $U_{x}$ for each point $x\in C$ , there exists a finite covering $\{W_{i}\}_{i\in I}$ of $C$ by relatively compact open subsets $W_{i}$ which satisfy the following property.

(1)

For any $i\in I$ , there exists a point $x\in C$ such that $\overline{W_{i}}\subset V_{x}$ .
(2)

Each point of $\overline{W_{i}}$ is contained in at most $2^{\dim X}$ elements of $\{\overline{W_{j}}\}_{j\in I}$ .

Proof.

We may assume that $V=\mathbb{R}^{n}$ and $\Gamma=\mathbb{Z}^{m}$ for $m\leq n$ . Let $\{p_{i}\}_{i=1,...,n}$ be the standard coordinate functions on $\mathbb{R}^{n}$ . We set $\mathcal{U}=\{U_{x}\}_{x\in K}$ . Let $K_{\epsilon}$ denote the closed $\epsilon$ -neighborhood of $K$ . Then, as $K$ is compact, there exists an $\epsilon>0$ such that

\displaystyle K\subset K^{\prime}=K_{\epsilon}\subset\bigcup\mathcal{U}.

We consider the subgroup $\frac{1}{2^{k}}\mathbb{Z}^{n}/\mathbb{Z}^{m}$ for sufficiently large $k$ . This defines a family of boxes in $X$ with width $\frac{1}{2^{k}}$ , where a box in $X$ with width $r$ is the projection of the sets in $\mathbb{R}^{n}$ of the form

\displaystyle\{y\in X\,|\,\lvert p_{i}(y)-p_{i}(c)\rvert<r/2,i=1,2,...,n\}

for some $c\in X$ and $r>0$ . For each element $B$ in the family above, we take a box $B^{\prime}$ with width $\frac{11}{10}\frac{1}{2^{k}}$ , roughly speaking, the thickened box $B^{\prime}$ obtained from $B$ . We will denote the family of the thickened boxes by $\mathcal{B}_{k}$ .

We set $\mathcal{B}_{k}(K)=\{B\in\mathcal{B}_{k}\,|\,\overline{B}\cap K\neq\emptyset\}$ . This set is obviously finite, as the set $K$ is compact. By enlarging $k$ , we may assume that

\displaystyle K\subset\bigcup_{B\in\mathcal{B}_{k}(K)}B\subset\bigcup_{B\in\mathcal{B}_{k}(K)}\overline{B}\subset K^{\prime}\subset\bigcup\mathcal{U}.

As $K^{\prime}$ is compact, the set $\bigcup_{B\subset\mathcal{B}_{k}(K)}\overline{B}$ is compact in $X$ . Thus, applying the argument using the Lebesgue number for this set and the covering induced by $\mathcal{U}$ , we divide each box in $\mathcal{B}_{k}(K)$ into $2^{n}$ boxes several times to obtain a new set of boxes $\mathcal{B}$ in $X$ , such that for any element $B\in\mathcal{B}$ , there exists an element $U\in\mathcal{U}$ such that $\overline{B}\subset U$ . We take the set $\mathcal{B}$ as the set $\{W_{i}\}_{i}$ in the statement. Then, the first property is satisfied by definition. By noting that the projection $V\twoheadrightarrow V/\Gamma$ is a local isomorphism and counting locally, it is easy to see that the second property holds true for this set. ∎

We recall the notion of vertical bands with compact real part, which are used in the proof.

Definition 7.5.

First, assume that $F=\mathbb{R}$ . In this case, by a vertical band with compact real part in $\mathfrak{h}^{*}\otimes_{\mathbb{R}}\mathbb{C}$ , we mean a subset $B$ of $\mathfrak{h}^{*}\otimes_{\mathbb{R}}\mathbb{C}$ with compact real part with respect to $\mathfrak{h}^{*}_{\mathbb{R}}$ . We call the projection of a vertical band with compact real part in $\mathfrak{h}^{*}\otimes_{\mathbb{R}}\mathbb{C}$ onto $\Theta(G)$ , a vertical band with compact real part in $\Theta(G)$ .

We assume that $F$ is a $p$ -adic field. By a vertical band with compact real part in $\mathfrak{a}^{*}_{M,\mathbb{C}}$ , we mean a subset of $\mathfrak{a}^{*}_{M,\mathbb{C}}$ with compact real part with respect to $X^{*}(M)\otimes_{\mathbb{Z}}\mathbb{R}$ . We call the finite union of images of vertical bands with compact real part in $\mathfrak{a}^{*}_{M,\mathbb{C}}$ under the map $(2.2)$ , a vertical band with compact real part in $\Theta(G)$ .

The proof of Proposition 7.1.

The proof we give is essentially the same as that in [Sau97]*Théorème 5.4, but we fix minor notational errors pointed out in [Shi12]*Appendix A.

We take arbitrary positive numbers $\epsilon_{1},\epsilon_{2},\epsilon_{3}$ . Also, we fix a compact neighborhood $C_{1}$ of $C$ .

By [Sau97]*Lemme 5.1, for any point $\theta\in C$ , we can find an open neighborhood $V_{\theta,\epsilon_{1}}\subset C_{1}$ of $\theta$ and a function $h_{\theta,\epsilon_{1}}\in C_{c}^{\infty}(G)$ , such that the following condition holds: For any irreducible representation $\pi$ of $G$ with $\inf_{G}(\pi)\in V_{\theta,\epsilon_{1}}$ , we have

	$\displaystyle 0\leq$	$\displaystyle\widehat{h_{\theta,\epsilon_{1}}}(\pi)\leq\epsilon_{1},\quad\pi\in\operatorname{Irr}_{\mathrm{unit}}(G)_{\mathrm{good}},$
	$\displaystyle 1\leq$	$\displaystyle\widehat{h_{\theta,\epsilon_{1}}}(\pi)\leq c_{G},\quad\pi\in\operatorname{Irr}_{\mathrm{unit}}(G)_{\mathrm{bad}},$

where $c_{G}$ is a positive constant depending only on $G$ .

Then, by Lemma 7.4, there exists a subset $\{\theta_{i}\}_{i=1,2,...,N(\epsilon_{1})}$ of $C$ and an open covering $\{W_{i}\}_{i=1,...,N(\epsilon_{1})}$ of $C$ such that

	$\displaystyle\sum_{i=1,...,N(\epsilon_{1})}\mathbf{1}_{\overline{W_{i}}}$	$\displaystyle\leq c^{\prime}_{G}\mathbf{1}_{C_{1}}$
	$\displaystyle\overline{W_{i}}$	$\displaystyle\subset V_{\theta_{i},\epsilon_{1}},$

where $\mathbf{1}_{Z}$ denotes the characteristic function of $Z$ and $c^{\prime}_{G}$ is a positive constant which only depends on $G$ .

We set

\displaystyle M(\epsilon_{1})=\sup_{i=1,...,N(\epsilon_{1}),\pi\in\operatorname{Irr}_{\mathrm{unit}}(G)}\lvert\widehat{h_{\theta_{i},\epsilon_{1}}}(\pi)\rvert.

By [Sau97]*Lemme 3.4, there exists a vertical band $B$ with compact real part in $\Theta(G)$ containing $C_{1}$ such that the functions $\widehat{\phi_{\theta_{i},\epsilon_{1}}}(\pi)$ are zero if $\pi\in\operatorname{Irr}_{\mathrm{unit}}(G)$ and $\inf_{G}(\pi)\not\in B$ . We will choose an open neighborhood $W^{\prime}_{i}$ of $\overline{W_{i}}$ in $V_{\theta,\epsilon_{1}}$ such that we have

\displaystyle{\mathrm{inf}_{G}}_{*}(\mu^{G})(\bigcup_{i=1,...,N(\epsilon_{1})}(W_{i}^{\prime}\setminus\overline{W_{i}})\,)\leq\epsilon_{2}.

This follows from the outer regularity of the measure ${\mathrm{inf}_{G}}_{*}(\mu^{G})$ , see [Rud87]*Theorem 2.18.

We choose a function $a_{i}\in\mathcal{A}(G)$ by applying [Sau97]*Lemme 5.2. Then, the function $a_{i}$ is real on $\Theta(G)_{\mathrm{herm}}$ and we have

	$\displaystyle 0\leq a_{i}(\theta)$	$\displaystyle\leq 1+\epsilon_{3},\quad\theta\in B\cap\Theta(G)_{\mathrm{herm}},$
	$\displaystyle a_{i}(\theta)$	$\displaystyle\geq 1,\quad\theta\in\overline{W_{i}}\cap\Theta(G)_{\mathrm{herm}},$
	$\displaystyle a_{i}(\theta)$	$\displaystyle\leq\epsilon_{3},\quad\theta\in(B\setminus W^{\prime}_{i})\cap\Theta(G)_{\mathrm{herm}}.$

We consider the function

\displaystyle\sum_{i=1,...,N(\epsilon_{1})}(\mathrm{inf}_{G}^{*}a_{i})\widehat{h_{\theta_{i},\epsilon_{1}}}.

First, this function is of the form $\widehat{h_{1}}\in\mathcal{F}(G)$ . We consider the value of this function on each set:

•

On the set $\inf_{G}^{-1}(\bigcup_{i}\overline{W_{i}})\cap\operatorname{Irr}_{\mathrm{unit}}(G)_{\mathrm{good}}$ , we have

$\displaystyle\lvert\widehat{h_{1}}\rvert\leq(1+\epsilon_{3})\epsilon_{1}c^{\prime}_{G}(\mathbf{1}_{C_{1}}\circ\mathrm{inf}_{G}).$

•

On the set $\inf_{G}^{-1}(\bigcup_{i}\overline{W_{i}})\cap\operatorname{Irr}_{\mathrm{unit}}(G)_{\mathrm{bad}}$ , we have

\displaystyle\lvert\widehat{h_{1}}\rvert\leq(1+\epsilon_{3})c_{G}c^{\prime}_{G}(\mathbf{1}_{C_{1}}\circ\mathrm{inf}_{G})(\mathbf{1}_{\operatorname{Irr}_{\mathrm{unit}}(G)_{\mathrm{bad}}}).

•

On the set $\inf_{G}^{-1}(\bigcup_{i}(W^{\prime}_{i}\setminus\overline{W_{i}}))\cap\operatorname{Irr}_{\mathrm{unit}}(G)$ , we have

\displaystyle\lvert\widehat{h_{1}}\rvert\leq(1+\epsilon_{3})M(\epsilon_{1})N(\epsilon_{1})(\mathbf{1}_{\bigcup_{i=1,...,N(\epsilon_{1})}(W_{i}^{\prime}\setminus\overline{W_{i}})}\circ\mathrm{inf}_{G}).

•

Otherwise, we have

$\displaystyle\lvert\widehat{h_{1}}\rvert\leq\epsilon_{3}\sum_{i}\lvert\widehat{h_{\theta_{i},\epsilon_{1}}}\rvert.$

We take a positive function $\widehat{h_{2}}$ with $\widehat{h_{2}}\geq 1$ on $C_{1}$ by applying [Sau97]*Lemme 5.3. We set

\displaystyle\widehat{h^{\prime}}=\widehat{h_{1}}+\epsilon_{3}M(\epsilon_{1})N(\epsilon_{1})\widehat{h_{2}}.

By considering whether $\pi\in\inf_{G}^{-1}(V_{\theta_{i},\epsilon_{1}})$ or not, it is easy to see that we have $\widehat{h^{\prime}}\geq 0$ on $C_{1}$ . Similarly, on the set $\inf_{G}^{-1}(C_{1})\cap\operatorname{Irr}_{\mathrm{unit}}(G)_{\mathrm{bad}}$ , we have $\widehat{h^{\prime}}\geq 1$ . By the estimate above, if we choose sufficiently small numbers $\epsilon_{1},\epsilon_{2},\epsilon_{3}$ , we have

\displaystyle\mu^{G}(\widehat{h^{\prime}})<\epsilon/2.

By the argument starting from [Sau97]*p.178, line 12, we can replace this function $h^{\prime}$ to obtain a new function $h\in C_{c}^{\infty}(G)$ which satisfies the statement of our proposition. As this procedure is explained completely in [Sau97], we do not repeat the argument. ∎

8. Applications

8.1. Some lemmas

Let $\mathbb{G}$ be a reductive group scheme over a product $A$ of local fields of characteristic $0$ . We set $G=\mathbb{G}(A)$ . Let $\mu^{G}$ denote the Plancherel measure of $G$ . Let $\{\mu^{G}_{n}\}_{n\geq 1}$ be a sequence of positive Borel measures on $\operatorname{Irr}_{\mathrm{unit}}(G)$ .

Lemma 8.1 ([Sau97]*Proposition 1.3).

Let $\mu^{G}_{n}$ be a sequence of positive Borel measures on $\operatorname{Irr}_{\mathrm{unit}}(G)$ . We assume that for any $h\in C_{c}^{\infty}(G)$ , we have

\displaystyle\lim_{n\to\infty}\mu^{G}_{n}(\widehat{h})=\mu^{G}(\widehat{h}).

Then, it follows that

\displaystyle\lim_{n\to\infty}\mu^{G}_{n}(\phi)=\mu^{G}(\phi)

for any $\phi\in\mathcal{RI}(\operatorname{Irr}_{\mathrm{temp}}(G),\mu^{G})$ .

Proof.

Let $\epsilon$ be any positive real number. By Theorem 7.2, we can find functions $h_{1},h_{2}\in C_{c}^{\infty}(G)$ such that

	$\displaystyle\lvert\phi-\widehat{h_{1}}\rvert\leq\widehat{h_{2}},$
	$\displaystyle\mu^{G}(\widehat{h_{2}})<\epsilon.$

We have

\lvert\mu^{G}_{n}(\phi)-\mu^{G}(\phi)\rvert\leq\lvert\mu^{G}_{n}(\phi)-\mu^{G}_{n}(\widehat{h_{1}})\rvert+\lvert\mu^{G}(\phi)-\mu^{G}(\widehat{h_{1}})\rvert\\ +\lvert\mu^{G}_{n}(\widehat{h_{1}})-\mu^{G}(\widehat{h_{1}})\rvert.

By the assumption on $h_{1},h_{2}$ , we have

	$\displaystyle\lvert\mu^{G}_{n}(\phi)-\mu^{G}_{n}(\widehat{h_{1}})\rvert\leq\mu^{G}_{n}(h_{2}),$
	$\displaystyle\lvert\mu^{G}(\phi)-\mu^{G}(\widehat{h_{1}})\rvert\leq\mu^{G}(h_{2}).$

By substituting these inequalities into the previous inequality and considering the limit superior, we obtain

\displaystyle\limsup_{n\to\infty}\lvert\mu^{G}_{n}(\phi)-\mu^{G}(\phi)\rvert\leq 2\mu^{G}(\widehat{h_{2}})<2\epsilon.

As we can take $\epsilon>0$ arbitrarily, we obtain

\lim_{n\to\infty}\mu^{G}_{n}(\phi)=\mu^{G}(\phi).\qed

We also give a lemma to globalize local characters. Let $\Lambda=\Lambda_{1}\coprod\Lambda_{2}$ be a set such that $\Lambda_{2}$ is finite. For $\lambda\in\Lambda_{1}$ , let $\mathcal{P}_{\lambda}=(G_{\lambda},K_{\lambda})$ be a pair consisting of a locally compact Hausdorff group $G_{\lambda}$ and its open compact subgroup $K_{\lambda}$ . For $\lambda\in\Lambda_{2}$ , let $\mathcal{P}_{\lambda}=G_{\lambda}$ be a locally compact Hausdorff space. Let $G$ be a restricted product $\prod_{\lambda\in\Lambda_{1}}^{\prime}(G_{\lambda},K_{\lambda})\times\prod_{\lambda\in\Lambda_{2}}G_{\lambda}$ of the family $\mathcal{P}_{\lambda}$ . Then, $G$ is locally compact. Let $\Gamma$ be a discrete subgroup of $G$ such that the map $\Gamma\hookrightarrow G\twoheadrightarrow G_{\lambda_{0}}$ is injective for an element $\lambda_{0}\in\Lambda$ .

Lemma 8.2.

Let $S$ be a finite subset of $\Lambda_{1}$ which does not contain $\lambda_{0}$ . Let $\chi_{S}$ be a continuous character of $K_{S}=\prod_{\lambda\in S}K_{\lambda}$ . Then, there exists a character $\chi$ of $G/\Gamma$ , such that we have

	$\displaystyle\chi_{\restriction_{K_{S}}}$	$\displaystyle=\chi_{S},$
	$\displaystyle\chi_{\prod_{\Lambda_{1}\setminus(S\cup\{\lambda_{0}\})}K_{\lambda}}$	$\displaystyle=\mathbf{1}_{\prod_{\Lambda_{1}\setminus(S\cup\{\lambda_{0}\})}K_{\lambda}}.$

Proof.

By the hypothesis on $\lambda_{0}$ , we have

\displaystyle\Gamma\cap\prod_{\lambda\neq\lambda_{0}}K_{\lambda}=\{1\}.

Also, the group $\prod_{\lambda\neq\lambda_{0}}K_{\lambda}$ is compact, the image of the map

\displaystyle\prod_{\lambda\neq\lambda_{0}}K_{\lambda}\to G/\Gamma.

is compact and closed. Thus, the subgroup $\Gamma\prod_{\lambda\neq\lambda_{0}}K_{\lambda}/\Gamma$ of $G/\Gamma$ is closed and isomorphic to $\prod_{\lambda\neq\lambda_{0}}K_{\lambda}$ . By the Pontryagin duality, there exists a unitary extension $\chi$ of the character $\chi_{S}\otimes\mathbf{1}_{\prod_{\Lambda_{1}\setminus(S\cup\{\lambda_{0}\})}K_{\lambda}}$ to $G/\Gamma$ . ∎

Corollary 8.3.

Let $T$ be a torus over a number field $k$ , and let $S$ be a finite set of finite places of $k$ . Suppose that $\chi_{S}$ be a character of the maximal compact subgroup $T(k_{S})_{c}$ of $T(k_{S})$ . Then, there exists a unitary character $\chi^{\prime}$ of $T(\mathbb{A}_{k})/T(k)$ which extends $\chi_{S}$ on $T(k_{S})_{c}$ and unramified outside $S$ .

We assume that $S$ is a singleton $\{u_{0}\}$ and $A_{T_{u_{0}}}=A_{T}\times_{k}k_{u_{0}}$ . Let $\chi_{u_{0}}$ be a unitary character of $T(k_{u_{0}})$ . Then, there exists a unitary character $\chi$ of $T(\mathbb{A}_{k})/T(k)$ which extends $\chi_{u_{0}}$ on $T(k_{u_{0}})$ and unramified outside $\{u_{0}\}$ .

Proof.

The first statement follows from the previous lemma by taking the set of places of $k$ as $\Lambda$ , the set of finite places of $k$ as $\Lambda_{1}$ , the set of infinite places of $k$ as $\Lambda_{2}$ and an infinite place $k$ as $\lambda_{0}$ . We now consider the second statement. By applying the first statement to obtain a global character $\chi^{\prime}$ which extends ${\chi_{u_{0}}}_{\restriction{T(k_{u_{0}})_{c}}}$ . Considering the twist by $\chi^{\prime}$ , we may assume that ${\chi_{u_{0}}}_{\restriction{T(k_{u_{0}})_{c}}}=1$ . Note that the group $T(k_{u_{0}})_{c}$ is the intersection of the kernels of unramified characters of $T(k_{u_{0}})$ . Thus, the character $\chi_{u_{0}}$ is unramified. By the assumption $A_{T_{u_{0}}}=A_{T}\times_{k}k_{u_{0}}$ , we have the injection

\displaystyle T(k_{u_{0}})/T(k_{u_{0}})_{c}\hookrightarrow T(\mathbb{A}_{k})/T(\mathbb{A}_{k})^{1},

and the unramified character $\chi_{u_{0}}$ can be extended to an automorphic character $\chi$ of $T(\mathbb{A}_{k})$ by the Pontryagin duality. Hence the result. ∎

8.2. Shin’s automorphic density theorem

Let $k$ be a number field with a real place $u_{0}$ . Thus, we have $k_{u_{0}}\cong\mathbb{R}$ . Let $G$ be a connected reductive group over $k$ . We assume that the center $Z_{G}$ of $G$ is an induced torus over $k$ . For a finite set $S$ of places of $k$ , we denote the group $\prod_{v\in S}G(k_{v})$ by $G_{S}$ . Similarly, we set $G^{S}=\prod_{v\not\in S}^{\prime}G(F_{v})$ .

Let $\lambda$ be an automorphic unitary character of $Z_{G}(\mathbb{A}_{k})$ . As we have $Z_{G}(\mathbb{A}_{k})=(Z_{G})_{S}\times Z_{G}^{S}$ , we can decompose this central character as $\lambda=\lambda_{S}\boxtimes\lambda^{S}$ . By [MoeglinWaldspurger2016-Stablisation2]*VI 2.8, we have the invariant trace formula with the fixed central character $\lambda$ .

Definition 8.4.

Let $\operatorname{Irr}_{\mathrm{unit}}(G(F_{S}),\lambda_{S})$ be a set of unitary representations of $G(F_{S})$ with the central character $\lambda_{S}$ . This set is also endowed with the restriction of the Fell topology on $\operatorname{Irr}_{\mathrm{unit}}(G(F_{S}))$ . A Borel subset $U_{S}$ of $\operatorname{Irr}_{\mathrm{unit}}(G_{S},\lambda_{S})$ is said to be $\mu^{G_{S}}$ -regular if the measure $\mu^{G_{S}}(\partial U_{S})$ of the boundary is equal to zero.

Theorem 8.5 (c.f. [Shi12]*Theorem 4.11).

Let $k$ be a number field with a real place $u_{0}$ . Let $G$ be a connected reductive group over $k$ . Let $\lambda$ be an automorphic unitary character of $Z_{G}(\mathbb{A}_{k})$ . Let $S$ be a finite set of places of $k$ containing all the infinite places and assume that $G$ and $\lambda$ is unramified outside $S$ . We set $S^{\prime}=S\setminus\{u_{0}\}$ .

We assume the following.

•

The center $Z_{G}$ of $G$ is induced.
•

The group $G(k_{u_{0}})$ has a discrete series representation with character $\lambda_{u_{0}}$ and also we have $A_{G}\times_{k}k_{u_{0}}=A_{G_{k_{u_{0}}}}$ .
•

We are given a $\mu^{G_{S^{\prime}}}$ -regular relatively quasi-compact set $U_{S^{\prime}}\subset\operatorname{Irr}_{\mathrm{unit}}(G(k_{S^{\prime}}),\lambda_{S^{\prime}})$ such that $\mu^{G_{S^{\prime}}}(U_{S^{\prime}})>0$ .

Then, there exists a cuspidal automorphic representation $\Pi$ of $G(\mathbb{A}_{k})$ with the central character $\lambda$ such that

•

we have $\Pi_{S^{\prime}}\in U_{S^{\prime}}$ ,
•

the representation $\Pi_{v}$ is unramified for $v\not\in S$ , and
•

the representation $\Pi_{u_{0}}$ is a discrete series representation of $G(k_{u_{0}})$ and the infinitesimal character of $\Pi_{u_{0}}$ is sufficiently regular.

Proof.

The proof is essentially the same as [Shi12]*Theorem 4.11. We use the invariant trace formula for the fixed central character $\lambda$ constructed in [MoeglinWaldspurger2016-Stablisation2]*VI 2.8, instead of the usual Arthur trace formula.

We first recall some definitions which are similar to those in [Shi12]. We set $G^{\natural}=G/Z_{G}$ . Let $\Pi_{\mathrm{cusp}}(G(\mathbb{A}_{k}),\lambda)$ denote the set of cuspidal automorphic representations of $G(\mathbb{A}_{k})$ with the central character $\lambda$ and $m_{\mathrm{cusp}}(\pi)$ be the multiplicity of $\pi$ in the space of cuspidal automorphic forms on $G(\mathbb{A}_{k})$ . Let $\{\xi_{n}\}_{n\geq 1}$ be a sequence of the finite dimensional algebraic representation of $G(k_{u_{0}}\times_{\mathbb{R}}\mathbb{C})$ with the central character $\lambda_{u_{0}}$ on $Z_{G}(k_{u_{0}})$ , such that $\lim_{n\to\infty}\xi_{n}=\infty$ in the sense of [Shi12]*Definition 3.5. We define the functional $\mu^{\mathrm{cusp}}_{\phi^{S^{\prime}},\xi_{n},\lambda}$ on $C_{c}^{\infty}(G(F_{S^{\prime}}),\lambda_{S^{\prime}}^{-1})$ as

\mu^{\mathrm{cusp}}_{\phi^{S^{\prime}},\xi_{n},\lambda}(f)\\ =\frac{(-1)^{q(G_{u_{0}})}}{\dim(\xi_{n})\operatorname{vol}(G^{\natural}(k)\backslash G^{\natural}(\mathbb{A}_{k}))}\sum_{\pi\in\Pi_{\mathrm{cusp}}(G(\mathbb{A}_{k}),\lambda)}m_{\mathrm{cusp}}(\pi)\operatorname{tr}\pi(f\otimes\phi^{S}\otimes\phi_{\xi_{n}}).

Here, the function $\phi_{\xi_{n}}\in C_{c}^{\infty}(G(k_{u_{0}}),\lambda_{u_{0}}^{-1})$ is the Euler-Poincare function associated with $\xi_{n}$ and $\phi^{S}$ is the unit of the ring of unramified functions in $C_{c}^{\infty}(G(\mathbb{A}^{S}_{k}),(\lambda^{S})^{-1})$ . Let $\mathcal{FT}(G(F_{S^{\prime}}),\lambda_{S^{\prime}}^{-1})$ denote the space of Fourier transforms of functions in $C_{c}^{\infty}(G(k_{S^{\prime}}),\lambda_{S^{\prime}}^{-1})$ . The last expression is equal to

\frac{\operatorname{vol}(K^{S}Z_{G}(\mathbb{A}_{k})/Z_{G}(\mathbb{A}_{k}))}{\dim(\xi_{n})\operatorname{vol}(G^{\natural}(k)\backslash G^{\natural}(\mathbb{A}_{k}))}\\ \times\sum_{\pi\in\Pi_{\mathrm{cusp}}(G(\mathbb{A}_{k}),\lambda),\pi^{K^{S}}\neq 0,\text{$\xi_{n}$-cohomological}}m_{\mathrm{cusp}}(\pi)\operatorname{tr}\pi_{S^{\prime}}(f)

if $n$ is large.

The functional $\mu^{\mathrm{cusp}}_{\phi^{S^{\prime}},\xi_{n},\lambda}$ factors through the map $C_{c}^{\infty}(G(k_{S^{\prime}}),\lambda_{S^{\prime}}^{-1})\to\mathcal{FT}(G(F_{S^{\prime}}),\lambda_{S^{\prime}}^{-1})$ taking the Fourier transform. We denote the resulting functional on $\mathcal{FT}(G(F_{S^{\prime}}),\lambda_{S^{\prime}}^{-1})$ by $\widehat{\mu}^{\mathrm{cusp}}_{\phi_{n}^{S^{\prime}},\xi,\lambda}$ . Applying [Shi12]*Lemma 4.9 and $\phi_{\xi_{n}}(1)=\dim(\xi_{n})$ , we have

\displaystyle\lim_{n\to\infty}\widehat{\mu}^{\mathrm{cusp}}_{\phi^{S^{\prime}},\xi_{n},\lambda}(\widehat{f})=f(1)=\mu^{S^{\prime}}(\widehat{f}),

for any $f\in C_{c}^{\infty}(G(F_{S^{\prime}}),\lambda_{S^{\prime}}^{-1})$ . The assumption $A_{G}\times_{k}k_{u_{0}}=A_{G_{u_{0}}}$ is used here to apply the result [GKM1997]*Theorem 5.2. An analog of the Sauvageot density principle for the fixed central character $\lambda^{-1}_{S^{\prime}}$ proved in [Binder2019]*Proposition 3.3.2 by using Theorem 7.2. Hence, the computation above and an analog of Lemma 8.1 implies the result. ∎

Remark 8.6.

Instead of assuming that the center $Z_{G}$ is induced, we can prove a similar result by assuming that $G_{v}$ has an anisotropic center over $k_{v}$ for $v\in S$ , by considering the Poisson summation formula and finiteness of the class number for $Z_{G}$ .

8.3. Globalization to spherical cusp forms

Let $k$ be a number field. We set $G=\mathrm{PGL}_{n}(\mathbb{A}_{k})$ . Let $S_{\infty}$ be the set of infinite places of $k$ . Let $A_{0}$ be a maximal split torus of $G$ over $k$ . We denote the group $W(G,A_{0})$ by $W_{\mathbb{R}}$ . Note that the space of spherical tempered representations $\operatorname{Irr}_{\mathrm{temp},\mathrm{sph}}(G_{S_{\infty}})$ is an open and closed subset of $\operatorname{Irr}_{\mathrm{temp}}(G_{S_{\infty}})$ . This space is identified with $i\mathfrak{a}^{*}_{0}/W_{\mathbb{R}}$ . Thus, for any subset $\Omega$ and positive number $t\in\mathbb{R}_{>0}$ , we have the dilatation $t\Omega$ of $\Omega$ .

Theorem 8.7.

For any finite set of finite places $S$ , $\mu^{G_{S}}$ -regular relatively quasi-compact set $U_{S}$ of $\operatorname{Irr}_{\mathrm{unit}}(G_{S})$ with positive measure and any open subset $U_{\infty}$ of $\operatorname{Irr}_{\mathrm{temp},\mathrm{sph}}(G_{S_{\infty}})$ with positive measure, there exists a cuspidal automorphic representation of $\mathrm{PGL}_{n}(\mathbb{A}_{k})$ such that

•

we have $\Pi_{S}\in U_{S}$ ,
•

$\pi_{v}$ is unramified for $v\not\in S$ , and
•

$\Pi_{S_{\infty}}$ is spherical and the representation is in the set $tU_{\infty}$ for some $t\geq 1$ .

Proof.

This follows from [Eikemeier2022]*Theorem 1.2 and Theorem 7.2 and we will give the details below. The first result is an extension of the result [FinisMatz2021]*Theorem 1.1 to quasi-split groups, see the footnote of [FinisMatz2021]*p.1038.

Now, we give the details. Let $T_{0}$ be maximal split torus of $G$ and $\mathfrak{t}_{0}$ be its Lie algebra. Then, the infinitesimal characters of spherical representations of $G_{S_{\infty}}$ are parametrized by the quotient $(\mathfrak{t}^{*}_{0}\otimes_{\mathbb{R}}\mathbb{C})/W_{\mathbb{R}}$ . Those for unitary spherical representations are in the subset $i\mathfrak{t}^{*}_{0}/W_{\mathbb{R}}$ . For any bounded set $B\subset(\mathfrak{t}^{*}_{0}\otimes_{\mathbb{R}}\mathbb{C})/W_{\mathbb{R}}$ and for any smooth function $\tau\in C_{c}^{\infty}(G(\mathbb{A}_{k,\mathrm{fin}}))$ , we set

\displaystyle m(B,\tau)=\sum_{\lambda\in B}\sum_{\pi\in\Pi_{\mathrm{disc}}(G(\mathbb{A}_{k})),\pi_{\infty}^{K_{\infty}}\neq 0,W_{\mathbb{R}}\lambda_{\pi_{\infty}}=W_{\mathbb{R}}\lambda}m_{\mathrm{disc}}(\pi)\operatorname{tr}\pi_{\mathrm{fin}}(\tau).

This counts the contribution of spherical discrete automorphic representations with the infinitesimal character in the $W_{\mathbb{R}}$ -orbit of $B$ . Let $\Omega\subset i\mathfrak{t}^{*}_{0}$ be a bounded open subset with the rectifiable boundary in the sense of [FinisMatz2021]*p.1094, line 15. By shrinking $\Omega$ , we can always achieve this condition. For $t\geq 1$ , let $\Lambda_{\Omega}(t)$ be the volume of $t\Omega$ with respect to the Plancherel measure, see [Eikemeier2022]*p.14, line 25 for an explicit description. Then, by [Eikemeier2022]*Theorem 1.2, we have

\displaystyle\lim_{t\to\infty}\frac{1}{\Lambda_{\Omega}(t)}m(t\Omega,\tau)=\tau(1)=\mu^{G_{S}}(\widehat{\tau_{S}}).

We take $\tau=\tau_{S}\otimes\bigotimes_{v\not\in S}\mathbf{1}_{K_{v}}$ . Note that if we write $m(B,\tau_{S})$ for $m(B,\tau)$ , then, we have

	$\displaystyle m(B,\tau_{S})$	$\displaystyle=\sum_{\lambda\in B}\sum_{\pi\in\Pi_{\mathrm{disc}}(G(\mathbb{A}_{k})),\pi^{K^{S}K_{\infty}}\neq 0,W_{\mathbb{R}}\lambda_{\pi_{\infty}}=W_{\mathbb{R}}\lambda}m_{\mathrm{disc}}(\pi)\operatorname{tr}\pi_{S}(\tau_{S})$
		$\displaystyle=\sum_{\lambda\in B}\sum_{\pi\in\Pi_{\mathrm{disc}}(G(\mathbb{A}_{k})),\pi^{K^{S}K_{\infty}}\neq 0,W_{\mathbb{R}}\lambda_{\pi_{\infty}}=W_{\mathbb{R}}\lambda}m_{\mathrm{disc}}(\pi)\widehat{\tau_{S}}(\pi_{S}).$

Thus, if for any $F\in\mathcal{RI}(\operatorname{Irr}_{\mathrm{unit}}(G_{S}),\mu^{G_{S}})$ we set

\displaystyle\mu^{G_{S}}_{\mathrm{disc},t}(F)=\frac{1}{\Lambda_{\Omega}(t)}\sum_{\lambda\in B}\sum_{\pi\in\Pi_{\mathrm{disc}}(G(\mathbb{A}_{k})),\pi^{K^{S}K_{\infty}}\neq 0,W_{\mathbb{R}}\lambda_{\pi_{\infty}}=W_{\mathbb{R}}\lambda}m_{\mathrm{disc}}(\pi)F(\pi_{S}),

then, we have

\displaystyle\lim_{t\to\infty}\mu^{G_{S}}_{\mathrm{disc},t}(\widehat{\tau_{S}})=\mu^{G_{S}}(\widehat{\tau_{S}}).

Then, Lemma 8.1 implies the result. Note that we can replace $m_{\mathrm{disc}}(\pi)$ with $m_{\mathrm{cusp}}(\pi)$ , see [FLM15]*Remark 1.1. ∎

Remark 8.8.

The argument of this type can be applied to more general (not necessarily adjoint) quasi-split reductive groups, see [Eikemeier2022]*Theorem 1.2.

A note on the Sauvageot density principle

Abstract.

1. Introduction

1.1. Notation and assumptions

1.2. Acknowledgment

2. The space of infinitesimal characters

2.1. Definition

Definition 2.1.

Theorem 2.2 ([Renard2010-p-adiques]*VI 5.4, Théorème).

Construction 2.3.

2.2. A ring of functions and hermitian involutions

Definition 2.4.

Definition 2.5.

Lemma 2.6.

3. The space of induced representations

Definition 3.1.

Lemma 3.2.

Proof.

Remark 3.3.

Construction 3.4.

Lemma 3.5.

Proof.

Definition 3.6.

Remark 3.7.

Lemma 3.8.

4. On the generic irreducibility theorem

Lemma 4.1.

Proof.

Theorem 4.2.

Proof.

5. The unitary dual and the Plancherel measure

Definition 5.1.

Theorem 5.2 ([Wal03]*Théorème VIII 1.1, [Wallach1992RRGII]*Theorem 13.4).

Definition 5.3.

Definition 5.4.

Lemma 5.5.

Proof.

Lemma 5.6.

Proof.

Definition 5.7.

6. A variant of the Stone-Weierstrass theorem

Remark 6.1.

6.1. Approximation on locally compact Hausdorff spaces

Lemma 6.2 (c.f. [Nachbin1965]*p.54, Theorem 1, [Prolla1994]*Theorem 9).

Proof.

Remark 6.3.

6.2. Approximation on measure spaces

Definition 6.4.

Example 6.5.

Definition 6.6.

Lemma 6.7.

Proof.

Corollary 6.8.

Lemma 6.9.

Proof.

Remark 6.10.

Theorem 6.11.

Proof.

Example 6.12.

Definition 6.13.

Remark 6.14.

Corollary 6.15.

Proof.

7. The Sauvageot density principle

Proposition 7.1 ([Sau97]*Théorème 5.4).

Theorem 7.2 (Sauvageot’s density principle, [Sau97]*Théorème 7.3).

Proof that Proposition 7.1 implies Theorem 7.2.

Lemma 7.3.

Lemma 7.4 (c.f. [Sau97]*Lemme 2.6).

Proof.

Definition 7.5.

The proof of Proposition 7.1.

8. Applications

8.1. Some lemmas

Lemma 8.1 ([Sau97]*Proposition 1.3).

Proof.

Lemma 8.2.

Proof.

Corollary 8.3.

Proof.

Theorem 5.2 ([Wal03]Théorème VIII 1.1, [Wallach1992RRGII]Theorem 13.4).

Lemma 6.2 (c.f. [Nachbin1965]p.54, Theorem 1, [Prolla1994]Theorem 9).