2.1 Algebraic groups

Let ${\mathcal{O}}$ denote the ring of integers of $F$. Let $\mathbb{A}$ and $\mathbb{A}_{f}$ be the adeles and finite adeles of $F$.

Let $G$ and $\unicode[STIX]{x1D703}$ be as in Theorems 1.1 and 1.2. Let $H$ again denote the identity component of the group of fixed points of $\unicode[STIX]{x1D703}$. We let $T\subset G$ and $T_{H}\subset H$ be maximal tori defined over $F$ with $T_{H}\subset T$. We fix an $F$-embedding $\unicode[STIX]{x1D70C}:G\rightarrow \text{SL}_{d}$. Let $Z$ be the centre of $G$.

Consider $\text{SL}_{d}$ as a group scheme over ${\mathcal{O}}$ in the standard way, and take the schematic closures of $T_{H}$, $T$, $H$, and $G$ in $\text{SL}_{d}/{\mathcal{O}}$. These closures are group schemes over ${\mathcal{O}}$, and all the inclusions between them over $F$ extend to closed embeddings over ${\mathcal{O}}$. By [Reference TitsTit79, 3.9], we may choose $D\in \mathbb{N}$ such that, over ${\mathcal{O}}[1/D]$, all fibers of these group schemes are smooth, connected and reductive.

Let $X^{\ast }(T)$ and $X_{\ast }(T)$ denote the group of characters and cocharacters of $T\times _{F}\overline{F}$. Let $\unicode[STIX]{x1D6E5}$ be the set of roots of $T$ in $G$, and let $\unicode[STIX]{x1D6E5}^{+}$ be a choice of positive roots. Let $W$ be the Weyl group of $(G,T)$ over $\overline{F}$. We define

$$\begin{eqnarray}X_{\ast }^{+}(T)=\{\unicode[STIX]{x1D707}\in X_{\ast }(T):\langle \unicode[STIX]{x1D707},\unicode[STIX]{x1D6FC}\rangle \geqslant 0,\unicode[STIX]{x1D6FC}\in \unicode[STIX]{x1D6E5}^{+}\}.\end{eqnarray}$$

Similarly, we may define $\unicode[STIX]{x1D6E5}_{H},\unicode[STIX]{x1D6E5}_{H}^{+}$, $W_{H}$ and $X_{\ast }^{+}(T_{H})$. Letting $\unicode[STIX]{x1D70C}$ and $\unicode[STIX]{x1D70C}_{H}$ denote, as usual, the half-sum of positive roots for $G$ and $H$, we introduce the functions

$$\begin{eqnarray}\displaystyle \Vert \unicode[STIX]{x1D707}\Vert ^{\ast } & = & \displaystyle \underset{w\in W}{\max }\langle w\unicode[STIX]{x1D707},\unicode[STIX]{x1D70C}\rangle ,\nonumber\\ \displaystyle \Vert \unicode[STIX]{x1D707}\Vert _{H}^{\ast } & = & \displaystyle \underset{w\in W_{H}}{\max }\langle w\unicode[STIX]{x1D707},\unicode[STIX]{x1D70C}_{H}\rangle \nonumber\end{eqnarray}$$

on $X_{\ast }(T)$ and $X_{\ast }(T_{H})$, respectively. Our assumption that $G$ is semisimple implies that $\Vert \!\unicode[STIX]{x1D707}\!\Vert ^{\ast }$ is a norm; the condition that $\Vert \unicode[STIX]{x1D707}\Vert ^{\ast }=\Vert -\unicode[STIX]{x1D707}\Vert ^{\ast }$ follows from the fact that $\unicode[STIX]{x1D70C}$ and $-\unicode[STIX]{x1D70C}$ lie in the same Weyl orbit. While $\Vert \unicode[STIX]{x1D707}\Vert _{H}^{\ast }$ also has the property that $\Vert \unicode[STIX]{x1D707}\Vert _{H}^{\ast }=\Vert -\unicode[STIX]{x1D707}\Vert _{H}^{\ast }$, it is trivial on cocharacters arising from the centre of $H$.

Definition 2.1. We say that $G$ is $H$-large if there exists a non-zero $\unicode[STIX]{x1D707}\in X_{\ast }(T_{H})$ such that

(3)$$\begin{eqnarray}2\Vert \unicode[STIX]{x1D707}\Vert _{H}^{\ast }\geqslant \Vert \unicode[STIX]{x1D707}\Vert ^{\ast }.\end{eqnarray}$$

Note that this definition is exactly the opposite of the definition of $H$ being small in $G$ given in [Reference MarshallMar14, Definition 8.1]. By [Reference MarshallMar14, Lemma 8.2], the definition of $H$-large is independent of the choice of tori $T_{H}\subset T$.

2.2 Local fields

If $v$ is a place of $F$, we denote the completion by $F_{v}$. If $v$ is finite, we denote the ring of integers, uniformizer and cardinality of the residue field by ${\mathcal{O}}_{v}$, $\unicode[STIX]{x1D71B}_{v}$ and $q_{v}$, respectively. If $v\nmid D\infty$, we have the following consequences of our assumptions on $D$ and $\unicode[STIX]{x1D70C}$ above.

1. (i) We have $G({\mathcal{O}}_{v})=\unicode[STIX]{x1D70C}^{-1}(\unicode[STIX]{x1D70C}(G_{v})\cap \text{SL}_{d}({\mathcal{O}}_{v}))$ and $H({\mathcal{O}}_{v})=\unicode[STIX]{x1D70C}^{-1}(\unicode[STIX]{x1D70C}(H_{v})\cap \text{SL}_{d}({\mathcal{O}}_{v}))$, so that $G({\mathcal{O}}_{v})\cap H_{v}=H({\mathcal{O}}_{v})$.

2. (ii) $G({\mathcal{O}}_{v})$ and $H({\mathcal{O}}_{v})$ are hyperspecial maximal compact subgroups of $G_{v}$ and $H_{v}$, respectively.

3. (iii) If $T$ (and hence $T_{H}$) split at $v$, the subgroups $G({\mathcal{O}}_{v})$ and $H({\mathcal{O}}_{v})$ correspond to points in the Bruhat–Tits buildings of $G_{v}$ and $H_{v}$ that lie in the apartments of $T$ and $T_{H}$, respectively.

Property (i) is a consequence of the inclusions $H\subset G\subset \text{SL}_{d}$ extending to closed embeddings over ${\mathcal{O}}_{v}$. Property (ii) is [Reference TitsTit79, 3.8] or [Reference Bruhat and TitsBT84, 5.1.40]. Property (iii) is [Reference Bruhat and TitsBT84, 5.1.33], once we use [Reference Bruhat and TitsBT84, 5.1.40] to show that the ${\mathcal{O}}_{v}$ group scheme $T/{\mathcal{O}}_{v}$ is the standard one associated with $T_{v}$. We let ${\mathcal{P}}$ be the set of finite places of $F$ that do not divide $D$ and at which $T$ splits. If $v\in {\mathcal{P}}$, our assumptions imply that $G_{v}$ has a Cartan decomposition

$$\begin{eqnarray}G_{v}=\coprod _{\unicode[STIX]{x1D707}\in X_{\ast }^{+}(T)}G({\mathcal{O}}_{v})\unicode[STIX]{x1D707}(\unicode[STIX]{x1D71B}_{v})G({\mathcal{O}}_{v})\end{eqnarray}$$

with respect to $G({\mathcal{O}}_{v})$ and $T$.

2.3 Metrics

For any place $v$ of $F$ and $g\in G(F_{v})$ let $\Vert g\Vert _{v}$ denote the maximum of the $v$-adic norms of the matrix entries of $\unicode[STIX]{x1D70C}(g)$. For $g\in G(\mathbb{A}_{f})$, let $\Vert g\Vert _{f}=\prod _{v\nmid \infty }\Vert g_{v}\Vert _{v}$. Fix a left-invariant Riemannian metric on $G(F_{v_{0}})$. Let $d(\cdot ,\cdot )$ be the associated distance function. We define $d(x,y)=\infty$ when $x$ and $y$ are in different connected components of $G(F_{v_{0}})$ with the topology of a real manifold.

2.4 Compact subgroups

We choose compact subgroups $K_{v}$ of $G_{v}$ for all $v$ such that:

• ∙ $K_{v_{0}}=H_{v_{0}}$;

• ∙ $K_{v}=G_{v}$ for all other real places;

• ∙ $\unicode[STIX]{x1D70C}(K_{v})\subset \text{SL}_{d}({\mathcal{O}}_{v})$ for all finite $v$;

• ∙ $K_{v}=G({\mathcal{O}}_{v})$ for finite places $v\nmid D$; and

• ∙ $K=\prod _{v\nmid \infty }K_{v}$ is open in $G(\mathbb{A}_{f})$.

By the remarks in the proof of Lemma 2.2, $K_{v_{0}}$ is a maximal compact connected subgroup of $G_{v_{0}}$. We shall suppose that $K_{v}$ for $v|D$ is sufficiently small to ensure that the finite group $Z(F)\cap K$ is reduced to $\{e\}$.

2.5 Measure normalizations

For any place $v$ of $F$, let $\unicode[STIX]{x1D707}_{G,v}^{\text{can}}$ be the canonical measure on $G(F_{v})$ as defined by Gross in [Reference GrossGro97, § 11]; we recall this construction in § 6.1. Then for all finite places $v\nmid D$ we have $\unicode[STIX]{x1D707}_{G,v}^{\text{can}}(K_{v})=1$. We may then form the product measure $\unicode[STIX]{x1D707}_{G}^{\text{can}}=\prod _{v}\unicode[STIX]{x1D707}_{G,v}^{\text{can}}$ on $G(\mathbb{A})$. All convolutions (local and global) on $G$ will be defined with respect to these measures. If $f\in C_{c}^{\infty }(G(\mathbb{A}))$, we define the operator $\unicode[STIX]{x1D70B}(f)$ on $L^{2}(G(F)\backslash G(\mathbb{A}))$ by

$$\begin{eqnarray}[\unicode[STIX]{x1D70B}(f)\unicode[STIX]{x1D719}](x)=\int _{G(\mathbb{A})}\unicode[STIX]{x1D719}(xg)f(g)\,d\unicode[STIX]{x1D707}_{G}^{\text{can}}(g).\end{eqnarray}$$

If $f\in C_{c}^{\infty }(G(\mathbb{A}))$, we define $f^{\ast }$ by $f^{\ast }(g)=\overline{f}(g^{-1})$, so that $\unicode[STIX]{x1D70B}(f)$ and $\unicode[STIX]{x1D70B}(f^{\ast })$ are adjoints.

The choice of canonical measure for $G$ is imposed by the use of the Arthur–Selberg trace formula in § 6; indeed, we want a uniform way of normalizing measures on the collection of connected reductive groups appearing as centralizers. We can afford to be more casual with measure normalizations for $H$, in light of our treatment of the geometric side of the relative trace formula in § 5. For finite places $v$ we choose Haar measures $dh_{v}$ on $H(F_{v})$ so that $H(F_{v})\cap K_{v}$ is assigned measure 1. Because $H_{v}$ is compact for archimedean $v$, at these places we choose Haar measures $dh_{v}$ so that $H_{v}$ has volume 1. We set $dh=\prod _{v}dh_{v}$.

2.6 Hecke algebras

If $S$ is any finite set of finite places prime to $D$, let ${\mathcal{H}}_{S}$ be the convolution algebra of functions on $G(F_{S})$ that are compactly supported and bi-invariant under $K_{S}$. Similarly, ${\mathcal{H}}_{f}$ is the convolution algebra of functions on $G(\mathbb{A}_{f}^{D})$ that are compactly supported and bi-invariant under $K^{D}$. We identify ${\mathcal{H}}_{S}$ and ${\mathcal{H}}_{f}$ with subalgebras of $C_{c}^{\infty }(G(\mathbb{A}_{f}))$ in the natural way. If $G_{\infty }^{0}$ denotes the connected component of the identity in $G_{\infty }$ in the real topology, we define ${\mathcal{H}}_{\infty }$ to be the subspace of $C_{c}^{\infty }(G_{\infty }^{0})$ consisting of functions that are bi-invariant under $K_{\infty }$, and define ${\mathcal{H}}={\mathcal{H}}_{\infty }\otimes {\mathcal{H}}_{f}$.

To control for the degree of Hecke operators in our later estimates, it will be convenient to work with the truncated Hecke algebras defined in [Reference Shin and TemplierST16]. Recall from § 2.1 the Weyl-invariant norm $\Vert \cdot \Vert ^{\ast }$ on $X_{\ast }(T)$. Let $v$ be a finite place not dividing $D$. Because $G_{v}$ is unramified, there is a maximal $F_{v}$-split torus $A_{v}$ in $G_{v}$ such that $K_{v}$ corresponds to a point in the apartment of $A_{v}$. We may conjugate $A_{v}$ inside $T$ over $\overline{F}_{v}$, and obtain a norm $\Vert \cdot \Vert _{v}$ on $X_{\ast }(A_{v})$. We then define

$$\begin{eqnarray}{\mathcal{H}}_{v}^{{\leqslant}\unicode[STIX]{x1D705}}=\text{span}_{\mathbb{ C}}\,\{\mathbf{1}_{K_{v}\unicode[STIX]{x1D707}(\unicode[STIX]{x1D71B}_{v})K_{v}}:\unicode[STIX]{x1D707}\in X_{\ast }(A_{v}),\,\Vert \unicode[STIX]{x1D707}\Vert _{v}\leqslant \unicode[STIX]{x1D705}\};\end{eqnarray}$$

note that here and elsewhere, $\mathbf{1}_{A}$ denotes the characteristic function of the set $A$. If $S$ is any finite set of finite places not dividing $D$, we set ${\mathcal{H}}_{S}^{{\leqslant}\unicode[STIX]{x1D705}}=\bigotimes _{v\in S}{\mathcal{H}}_{v}^{{\leqslant}\unicode[STIX]{x1D705}}$.

We note that ${\mathcal{H}}_{v}^{{\leqslant}\unicode[STIX]{x1D705}}$ is independent of the choices made in its definition. First, the norm $\Vert \cdot \Vert _{v}$ is independent of the conjugation of $A_{v}$ inside $T$. To see this, we note that $\Vert \unicode[STIX]{x1D706}\Vert ^{\ast }$ can also be defined by taking $\mathbb{G}_{m}\overset{\unicode[STIX]{x1D706}}{\longrightarrow }T\overset{\text{Ad}}{\longrightarrow }\text{GL}(\mathfrak{g})$, decomposing the representation of $\mathbb{G}_{m}$ on $\mathfrak{g}$ into characters in $X^{\ast }(\mathbb{G}_{m})\simeq \mathbb{Z}$, and taking half the sum of the positive terms. This lets us extend $\Vert \cdot \Vert ^{\ast }$ to all of $X_{\ast }(G)$ in a conjugation-invariant way, and $\Vert \cdot \Vert _{v}$ is the restriction of this to $X_{\ast }(A_{v})$. Secondly, if $A_{v}$ and $A_{v}^{\prime }$ are two tori whose apartments contain the point corresponding to $K_{v}$, then by [Reference TitsTit79, § 2.2.1] they are conjugate by an element of $K_{v}$, and this conjugation respects the norms on $X_{\ast }(A_{v})$ and $X_{\ast }(A_{v}^{\prime })$. This implies that ${\mathcal{H}}_{v}^{{\leqslant}\unicode[STIX]{x1D705}}$ is independent of $A_{v}$.

2.7 A lemma on double cosets

We shall need bounds on the volume of the support of Hecke operators on $G_{v}$ and $H_{v}$ at places where these groups are split. Such bounds are proved by Macdonald [Reference MacdonaldMac71] for split simply connected semisimple groups, but we shall need them in the general split reductive case, and derive them now from a result of Gross [Reference GrossGro96].

In this subsection we let $F$ be a $p$-adic field with integer ring ${\mathcal{O}}$, uniformizer $\unicode[STIX]{x1D71B}$, and residue field $k$, and let $q=\#k$. We let $G$ be a split connected reductive group over $F$. Let $T\subset G$ be a maximal split torus, and $K\subset G$ a hyperspecial maximal compact subgroup corresponding to a point in the apartment of $T$. Let $\Vert \cdot \Vert ^{\ast }$ be the function on $X_{\ast }(T)$ as in § 2.1.

Proof. Let $P_{\unicode[STIX]{x1D706}}$ be the parabolic subgroup associated with $\unicode[STIX]{x1D706}$, so that

$$\begin{eqnarray}\text{Lie}(P_{\unicode[STIX]{x1D706}})=\text{Lie}(T)+\bigoplus _{\langle \unicode[STIX]{x1D6FC},\unicode[STIX]{x1D706}\rangle \geqslant 0}\text{Lie}(G)_{\unicode[STIX]{x1D6FC}}.\end{eqnarray}$$

By Proposition 7.4 of [Reference GrossGro96], we have $\#K\unicode[STIX]{x1D706}(\unicode[STIX]{x1D71B})K/K=q^{2\Vert \unicode[STIX]{x1D706}\Vert ^{\ast }-\dim (G/P_{\unicode[STIX]{x1D706}})}\#(G/P_{\unicode[STIX]{x1D706}})(k)$. Now

$$\begin{eqnarray}\#(G/P_{\unicode[STIX]{x1D706}})(k)=\#G(k)/\#P_{\unicode[STIX]{x1D706}}(k)\quad \text{and}\quad \#P_{\unicode[STIX]{x1D706}}(k)=\#M_{\unicode[STIX]{x1D706}}(k)q^{\dim U_{\unicode[STIX]{x1D706}}},\end{eqnarray}$$

where $P_{\unicode[STIX]{x1D706}}=M_{\unicode[STIX]{x1D706}}U_{\unicode[STIX]{x1D706}}$ is the standard Levi decomposition, and the first equality follows from [Reference BorelBor91, Corollary 16.5]. It follows that

$$\begin{eqnarray}q^{-\text{dim}(G/P_{\unicode[STIX]{x1D706}})}\#(G/P_{\unicode[STIX]{x1D706}})(k)=q^{\dim M_{\unicode[STIX]{x1D706}}-\dim G}\#G(k)/\#M_{\unicode[STIX]{x1D706}}(k).\end{eqnarray}$$

The formula of Steinberg [Reference SteinbergSte68] for the number of points of a reductive group over a finite field implies that $\#G(k)/q^{\dim G}=f(q^{-1})$ and $\#M_{\unicode[STIX]{x1D706}}(k)/q^{\dim M_{\unicode[STIX]{x1D706}}}=f_{\unicode[STIX]{x1D706}}(q^{-1})$, where $f$ and $f_{\unicode[STIX]{x1D706}}$ are polynomials with constant term 1 that depend only on the root data of $G$ and $M_{\unicode[STIX]{x1D706}}$, respectively. This completes the proof.◻

2.8 Lie algebras

Let $\mathfrak{g}$ be the real Lie algebra of $G(F_{\infty })$, and let $\mathfrak{g}=\mathfrak{k}+\mathfrak{p}$ be the Cartan decomposition associated with $K_{\infty }$. Let $\mathfrak{a}\subset \mathfrak{p}$ be a maximal abelian subalgebra. We let $\unicode[STIX]{x1D6E5}_{\mathbb{R}}$ be the roots of $\mathfrak{a}$ in $\mathfrak{g}$, and let $\unicode[STIX]{x1D6E5}_{\mathbb{R}}^{+}$ be a choice of positive roots. We let $W_{\mathbb{R}}$ be the Weyl group of $\unicode[STIX]{x1D6E5}_{\mathbb{R}}$. For $\unicode[STIX]{x1D6FC}\in \unicode[STIX]{x1D6E5}_{\mathbb{R}}$, we let $m(\unicode[STIX]{x1D6FC})$ denote the dimension of the corresponding root space. We denote the Killing form on $\mathfrak{g}$ and $\mathfrak{g}^{\ast }$ by $\langle \cdot ,\cdot \rangle$, and let $\Vert \cdot \Vert$ be the associated norm on $\mathfrak{a}$ and $\mathfrak{a}^{\ast }$.

Let $c(\unicode[STIX]{x1D709})$ be Harish-Chandra’s $c$-function; see [Reference HelgasonHel08, ch. II, § 3.3]. Then it is known [Reference Duistermaat, Kolk and VaradarajanDKV79, (3.44)] that the upper bound

(4)$$\begin{eqnarray}|c(\unicode[STIX]{x1D709})|^{-2}\ll \unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D709})\end{eqnarray}$$

holds for all $\unicode[STIX]{x1D709}\in \mathfrak{a}^{\ast }$, where we have put

$$\begin{eqnarray}\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D709})=\mathop{\prod }_{\unicode[STIX]{x1D6FC}\in \unicode[STIX]{x1D6E5}_{\mathbb{R}}^{+}}(1+|\langle \unicode[STIX]{x1D6FC},\unicode[STIX]{x1D709}\rangle |)^{m(\unicode[STIX]{x1D6FC})}.\end{eqnarray}$$

2.9 Proof of Proposition 1.3

We now recall Borel’s theorem on the existence of rationally defined Cartan involutions on real semisimple Lie algebras. This is the central ingredient in the proof of Proposition 1.3.

The corresponding statement for simply connected semisimple groups, Proposition 1.3, can be deduced from the Lie algebra version, as follows.

Let $G^{\prime }$ be as in Proposition 1.3 and let $\mathfrak{g}^{\prime }/\mathbb{R}$ be its Lie algebra. Let $F$ be a totally real number field and $v_{0}$ a real place of $F$. Applying Proposition 2.4 to $\mathfrak{g}^{\prime }$ with this choice of $F$ and $v_{0}$, we obtain a corresponding $\mathfrak{g}/F$ and involution $\unicode[STIX]{x1D703}$. Let $\text{Int}(\mathfrak{g})$ be the identity component of $\text{Aut}(\mathfrak{g})$, and let $G$ be the simply connected cover of $\text{Int}(\mathfrak{g})$. Then $G$ has Lie algebra $\mathfrak{g}$, and $\unicode[STIX]{x1D703}$ induces an involution of $G$ which we also denote by $\unicode[STIX]{x1D703}$ (as the derivative at the identity of the former is equal to the latter). Because $G^{\prime }$ and $G_{v_{0}}$ are connected, simply connected semisimple groups with isomorphic Lie algebras, they are isomorphic. If $v\neq v_{0}$ is real, then $\mathfrak{g}_{v}$ is of compact type and $G(F_{v})$ is compact.

By Proposition 2.4, $\unicode[STIX]{x1D703}$ induces a Cartan involution of $\mathfrak{g}_{v_{0}}$. Because $G$ is simply connected, $G_{v_{0}}$ is connected in the real topology by the Cartan connectedness theorem [Reference Onishchik and VinbergOV94, ch. 4, §2.2, Theorem 2.2], and it follows that $\unicode[STIX]{x1D703}$ also induces a Cartan involution of $G_{v_{0}}$ (in the sense that the fixed point set $G_{v_{0}}^{\unicode[STIX]{x1D703}}$ is a maximal compact subgroup of $G_{v_{0}}$). This completes the proof of Proposition 1.3.

2.10 Congruence subgroups and adelic quotients

Recall from § 2.1 the integer $D$, and from § 2.4 the compact open subgroups $K_{v}$ for finite places $v$ and their product $K=\prod _{v\nmid \infty }K_{v}$ inside $G(\mathbb{A}_{f})$. We now define congruence subgroups of $K_{v}$ and $K$ to be used throughout the paper.

Fix an integer $N\geqslant 1$ prime to $D$. For $v\nmid D\infty$ we write $K_{v}(N)$ for the level $N$ principal congruence subgroup, given by $K_{v}(N)=\unicode[STIX]{x1D70C}^{-1}(\unicode[STIX]{x1D70C}(G_{v})\cap \text{SL}_{d}({\mathcal{O}}_{v})(N))$. Here $\text{SL}_{d}({\mathcal{O}}_{v})(N)$ denotes the principal congruence subgroup of $\text{SL}_{d}({\mathcal{O}}_{v})$. Let $S$ be a finite set of finite places prime to $D$. Then we put

$$\begin{eqnarray}K(N)=\mathop{\prod }_{v\nmid D\infty }K_{v}(N)\mathop{\prod }_{v\mid D}K_{v}\quad \text{and}\quad K^{S}(N)=\mathop{\prod }_{\substack{ v\nmid D\infty \\ v\notin S}}K_{v}(N)\mathop{\prod }_{v\mid D}K_{v}.\end{eqnarray}$$

For every finite $v$ we define $K_{H,v}=H_{v}\cap K_{v}$. Letting

$$\begin{eqnarray}K_{H}=\mathop{\prod }_{v\nmid \infty }K_{H,v}\quad \text{and}\quad K_{H}^{S}=\mathop{\prod }_{v\notin S\cup \infty }K_{H,v},\end{eqnarray}$$

then the congruence manifold of interest to us is

$$\begin{eqnarray}Y_{N}=G(F)\backslash G(\mathbb{A})/K(N)K_{H}K_{\infty }.\end{eqnarray}$$

Let $\text{Vol}_{N}$ be the volume assigned by $\prod _{v\nmid \infty }\unicode[STIX]{x1D707}_{G,v}^{\text{can}}$ to the compact open subgroup $K(N)K_{H}$.

Finally, we write

$$\begin{eqnarray}[G]=G(F)\backslash G(\mathbb{A})\quad \text{and}\quad [H]=H(F)\backslash H(\mathbb{A})\end{eqnarray}$$

for the automorphic spaces associated with $G$ and $H$. We shall identify $L^{2}(Y_{N})$ with the functions in $L^{2}([G])$ fixed under $K(N)K_{H}K_{\infty }$.

2.11 Hecke–Maass forms

Let ${\mathcal{D}}$ be the algebra of differential operators on $G_{\infty }^{0}/K_{\infty }$ that are invariant under the left action of $G_{\infty }^{0}$. Note that if we define $K_{v_{0}}^{+}$ to be the group of fixed points of $\unicode[STIX]{x1D703}$ on $G_{v_{0}}$, and let $K_{\infty }^{+}$ be the group obtained by replacing $K_{v_{0}}$ with $K_{v_{0}}^{+}$ in the definition of $K_{\infty }$, then $K_{\infty }^{+}$ is a maximal compact subgroup of $G_{\infty }$, we have $G_{\infty }/K_{\infty }^{+}\simeq G_{\infty }^{0}/K_{\infty }$, and the elements of ${\mathcal{D}}$ are also invariant under the larger group $G_{\infty }$. It follows that ${\mathcal{D}}$ descends to an algebra of operators on $Y_{N}$ in a natural way.

We define a Hecke–Maass form to be a function on $Y_{N}$ that is an eigenfunction of the ring ${\mathcal{D}}$ on $Y_{N}$ and the Hecke algebra ${\mathcal{H}}_{f}$ (and hence of ${\mathcal{H}}$). If $\unicode[STIX]{x1D713}$ is a Hecke–Maass form and $\unicode[STIX]{x1D714}\in {\mathcal{H}}$, then, using the notation in § 2.5, we define $\widehat{\unicode[STIX]{x1D714}}(\unicode[STIX]{x1D713})$ by the equation $\unicode[STIX]{x1D70B}(\unicode[STIX]{x1D714})\unicode[STIX]{x1D713}=\widehat{\unicode[STIX]{x1D714}}(\unicode[STIX]{x1D713})\unicode[STIX]{x1D713}$. We define the spectral parameter of $\unicode[STIX]{x1D713}$ to be the unique $\unicode[STIX]{x1D709}\in \mathfrak{a}_{\mathbb{C}}^{\ast }/W_{\mathbb{R}}$ such that $\unicode[STIX]{x1D713}$ has the same eigenvalues under the action of ${\mathcal{D}}$ as the associated spherical function $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D709}}$. The Laplace eigenvalue of $\unicode[STIX]{x1D713}$ is given by $(\unicode[STIX]{x1D6E5}+C_{1}(G)+\langle \unicode[STIX]{x1D709},\unicode[STIX]{x1D709}\rangle )\unicode[STIX]{x1D713}=0$ for some $C_{1}(G)\in \mathbb{R}$.

3.1 The trace formula

Let $N$ be a positive integer prime to $D$. Let $D^{\prime }$ be a sufficiently large multiple of $D$, depending only on $G$, to be chosen in § 6.4. Let $T$ be a finite set of finite places prime to $D^{\prime }$ and $N$. Let $\unicode[STIX]{x1D709}\in \mathfrak{a}^{\ast }$. We will use the trace formula to count Hecke–Maass forms on $Y_{N}$ with spectral parameter near $\unicode[STIX]{x1D709}$, weighted by a Hecke operator at $T$. To do this, we shall consider test functions of the form $\unicode[STIX]{x1D719}=\mathbf{1}_{N}^{T}\otimes k_{T}\otimes k_{\unicode[STIX]{x1D709}}$, for $k_{T}\in {\mathcal{H}}_{T}$ and $k_{\unicode[STIX]{x1D709}}\in {\mathcal{H}}_{\infty }$, where we have put

(5)$$\begin{eqnarray}\mathbf{1}_{N}^{T}=\mathbf{1}_{K^{T}(N)K_{H}^{T}}.\end{eqnarray}$$

We introduce the following condition on $k_{\unicode[STIX]{x1D709}}$.

$(\mathsf{P}_{\!\!\infty })$:

$k_{\unicode[STIX]{x1D709}}$ is supported in $\{g\in G(F_{\infty }):d(g,H_{v_{0}})<1\}$ and satisfies

$$\begin{eqnarray}k_{\unicode[STIX]{x1D709}}(g)\ll \unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D709})(1+\Vert \unicode[STIX]{x1D709}\Vert d(g,H_{v_{0}}))^{-1/2}.\end{eqnarray}$$

If $U$ is any finite set of finite places, we define $q_{U}=\prod _{v\in U}q_{v}$. Our asymptotic trace formula is given in the following theorem.

Theorem 3.1. There are constants $A,B,\unicode[STIX]{x1D6FF},\unicode[STIX]{x1D702}>0$ such that the following statement holds. Let $N$, $T$ and $\unicode[STIX]{x1D709}$ be as above, and let $\unicode[STIX]{x1D705}\geqslant 0$. For any $k_{T}\in {\mathcal{H}}_{T}^{{\leqslant}\unicode[STIX]{x1D705}}$ and any $k_{\unicode[STIX]{x1D709}}\in {\mathcal{H}}_{\infty }$ satisfying $(\mathsf{P}_{\infty })$ we have

$$\begin{eqnarray}\text{Vol}_{N}\mathop{\sum }_{i\geqslant 0}\widehat{k_{T}\otimes k_{\unicode[STIX]{x1D709}}}(\unicode[STIX]{x1D713}_{i})=\unicode[STIX]{x1D707}_{G}^{\text{can}}([G])k_{T}(1)k_{\unicode[STIX]{x1D709}}(1)+O(N^{-\unicode[STIX]{x1D6FF}}q_{T}^{A\unicode[STIX]{x1D705}+B}\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D709})(1+\Vert \unicode[STIX]{x1D709}\Vert )^{-\unicode[STIX]{x1D702}}\Vert k_{T}\Vert _{\infty }),\end{eqnarray}$$

where $\{\unicode[STIX]{x1D713}_{i}\}$ runs over an orthonormal basis of Hecke–Maass forms for $L^{2}(Y_{N})$. The implied constant depends only on $G$, $\unicode[STIX]{x1D703}$ and $K$.

We note that the exponents $A$, $B$ and $\unicode[STIX]{x1D6FF}$ are ineffective, and this is the only source of ineffectiveness of the exponent in Theorems 1.1 and 1.2. As discussed in § 1.1, this comes from using theorems of Finis and Lapid and of Cluckers, Gordon and Halupczok to bound orbital integrals.

3.2 The relative trace formula

For $\unicode[STIX]{x1D713}\in C^{\infty }(G(F)\backslash G(\mathbb{A}))$ we consider the $H$-automorphic period

$$\begin{eqnarray}{\mathcal{P}}_{H}(\unicode[STIX]{x1D713})=\int _{H(F)\backslash H(\mathbb{A})}\unicode[STIX]{x1D713}(h)\,dh.\end{eqnarray}$$

Our relative trace formula will give the average size of these periods over an orthonormal basis $\{\unicode[STIX]{x1D713}_{i}\}$ of Hecke–Maass forms for $L^{2}(Y_{N})$.

Let $N$ be a positive integer prime to $D$, $S$ a finite set of finite places prime to $D$ and $N$, and $\unicode[STIX]{x1D709}\in \mathfrak{a}^{\ast }$. We now consider test functions of the form $\unicode[STIX]{x1D719}=\mathbf{1}_{N}^{S}\,\otimes \,k_{S}\,\otimes \,k_{\unicode[STIX]{x1D709}}$, for $k_{S}\in {\mathcal{H}}_{S}$ and $k_{\unicode[STIX]{x1D709}}\in {\mathcal{H}}_{\infty }$. As well as condition $(\mathsf{P}_{\infty })$, we shall use the following condition on $k_{S}$.

$(\mathsf{P}_{S})$:

There is $R>1$ such that $\Vert g\Vert _{S}\leqslant R$ for all $g\in \text{supp}(k_{S})$.

The main term of our formula is expressed in terms of the averaging map

$$\begin{eqnarray}\unicode[STIX]{x1D6F1}_{H}:L^{1}(G(F_{S}))\longrightarrow L^{1}(G(F_{S})/H(F_{S}))\end{eqnarray}$$

given by

$$\begin{eqnarray}\unicode[STIX]{x1D6F1}_{H}k_{S}(g)=\int _{H(F_{S})}k_{S}(gh)\,dh.\end{eqnarray}$$

With this notation in hand, we may state our asymptotic relative trace formula.

Proposition 3.2. There is $A>0$ such that the following statement holds. Let $N$, $S$ and $\unicode[STIX]{x1D709}$ be as above. Let $k_{S}\in {\mathcal{H}}_{S}$ satisfy $(\mathsf{P}_{S})$ and $k_{\unicode[STIX]{x1D709}}\in {\mathcal{H}}_{\infty }$ satisfy $(\mathsf{P}_{\!\!\infty })$. Then

$$\begin{eqnarray}\displaystyle & & \displaystyle \text{Vol}_{N}\mathop{\sum }_{i\geqslant 0}\widehat{k_{S}\,\otimes \,k_{\unicode[STIX]{x1D709}}}(\unicode[STIX]{x1D713}_{i})|{\mathcal{P}}_{H}(\unicode[STIX]{x1D713}_{i})|^{2}\nonumber\\ \displaystyle & & \displaystyle \quad =\text{vol}([H])\unicode[STIX]{x1D6F1}_{H}k_{S}(1)k_{\unicode[STIX]{x1D709}}(1)+O\big(\#(\text{supp}\,k_{S}/K_{S})\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D709})(1+\Vert \unicode[STIX]{x1D709}\Vert )^{-1/4}N^{-1/4}R^{A}\Vert k_{S}\Vert _{\infty }\big),\nonumber\end{eqnarray}$$

where $\{\unicode[STIX]{x1D713}_{i}\}$ runs over an orthonormal basis of Hecke–Maass forms for $L^{2}(Y_{N})$.

3.3 Proof of Theorems 1.1 and 1.2

Recall from § 2.2 that ${\mathcal{P}}$ denotes the set of finite places of $F$ that do not divide $D$ and at which $T$ splits. For $v\in {\mathcal{P}}$ and $\unicode[STIX]{x1D707}\in X_{\ast }(T)$, we define $\unicode[STIX]{x1D70F}(v,\unicode[STIX]{x1D707})\in {\mathcal{H}}_{v}$ to be the function supported on $K_{v}\unicode[STIX]{x1D707}(\unicode[STIX]{x1D71B}_{v})K_{v}$ and taking the value $q_{v}^{-\Vert \unicode[STIX]{x1D707}\Vert ^{\ast }}$ there. Let $P$ be a positive integer and put

(6)$$\begin{eqnarray}S=\{v\in {\mathcal{P}}:P/2\leqslant q_{v}<P,\;v\nmid N,\;v\nmid D^{\prime }\}.\end{eqnarray}$$

Choose any non-zero $\unicode[STIX]{x1D708}\in X_{\ast }(T)$, and let

$$\begin{eqnarray}\unicode[STIX]{x1D714}_{S}=\mathop{\sum }_{v\in S}\unicode[STIX]{x1D70F}(v,\unicode[STIX]{x1D708}),\end{eqnarray}$$

where, as in § 2.6, we are identifying ${\mathcal{H}}_{v}$ with a subalgebra of ${\mathcal{H}}_{S}$ in the natural way.

In § 4 we will deduce the following weighted bounds from our two trace formulae.

Proposition 3.3. For any $\unicode[STIX]{x1D708}$ and $P$, we have

(7)$$\begin{eqnarray}\text{Vol}_{N}\mathop{\sum }_{\Vert \Re \unicode[STIX]{x1D709}_{i}-\unicode[STIX]{x1D709}\Vert \leqslant 1}|\widehat{\unicode[STIX]{x1D714}_{S}}(\unicode[STIX]{x1D713}_{i})|^{2}\ll P\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D709})(1+P^{A}N^{-\unicode[STIX]{x1D6FF}}(1+\Vert \unicode[STIX]{x1D709}\Vert )^{-\unicode[STIX]{x1D6FF}}).\end{eqnarray}$$

In addition, if $\unicode[STIX]{x1D708}\in X_{\ast }(T_{H})$ satisfies the $H$-large inequality (3), then there is $Q>0$ depending on $\unicode[STIX]{x1D708}$ such that

(8)$$\begin{eqnarray}\text{Vol}_{N}\mathop{\sum }_{\Vert \Re \unicode[STIX]{x1D709}_{i}-\unicode[STIX]{x1D709}\Vert \leqslant Q}|\widehat{\unicode[STIX]{x1D714}_{S}}(\unicode[STIX]{x1D713}_{i})|^{2}|{\mathcal{P}}_{H}(\unicode[STIX]{x1D713}_{i})|^{2}\gg _{\unicode[STIX]{x1D716}}P^{2-\unicode[STIX]{x1D716}}\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D709})(1+O(P^{A}N^{-\unicode[STIX]{x1D6FF}}(1+\Vert \unicode[STIX]{x1D709}\Vert )^{-\unicode[STIX]{x1D6FF}})).\end{eqnarray}$$

The exponent $A$ and the implied constants depend on the choice of $\unicode[STIX]{x1D708}$.

Theorems 1.1 and 1.2 follow easily from Proposition 3.3, as we now explain. Choose $0\neq \unicode[STIX]{x1D708}\in X_{\ast }(T_{H})$ satisfying (3), and choose $P$ to be a small power of $(1+\Vert \unicode[STIX]{x1D709}\Vert )N$. We then find $\unicode[STIX]{x1D6FF}>0$ such that for $(1+\Vert \unicode[STIX]{x1D709}\Vert )N$ sufficiently large,

(9)$$\begin{eqnarray}\text{Vol}_{N}\mathop{\sum }_{\Vert \Re \unicode[STIX]{x1D709}_{i}-\unicode[STIX]{x1D709}\Vert \leqslant Q}|\widehat{\unicode[STIX]{x1D714}_{S}}(\unicode[STIX]{x1D713}_{i})|^{2}\ll \unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D709})(1+\Vert \unicode[STIX]{x1D709}\Vert )^{\unicode[STIX]{x1D6FF}}N^{\unicode[STIX]{x1D6FF}}\end{eqnarray}$$

and

(10)$$\begin{eqnarray}\text{Vol}_{N}\mathop{\sum }_{\Vert \Re \unicode[STIX]{x1D709}_{i}-\unicode[STIX]{x1D709}\Vert \leqslant Q}|\widehat{\unicode[STIX]{x1D714}_{S}}(\unicode[STIX]{x1D713}_{i})|^{2}|{\mathcal{P}}_{H}(\unicode[STIX]{x1D713}_{i})|^{2}\gg _{\unicode[STIX]{x1D716}}\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D709})(1+\Vert \unicode[STIX]{x1D709}\Vert )^{2\unicode[STIX]{x1D6FF}-\unicode[STIX]{x1D716}}N^{2\unicode[STIX]{x1D6FF}-\unicode[STIX]{x1D716}}.\end{eqnarray}$$

Comparing (9) and (10) we find that there exist $\unicode[STIX]{x1D6FF}>0$ and $Q>1$ such that, for every $\unicode[STIX]{x1D709}\in \mathfrak{a}^{\ast }$ and every $N$ with $(1+\Vert \unicode[STIX]{x1D709}\Vert )N$ large, there is a Hecke–Maass form $\unicode[STIX]{x1D713}_{i}$ on $Y_{N}$ with spectral parameter $\Vert \unicode[STIX]{x1D709}_{i}-\unicode[STIX]{x1D709}\Vert \leqslant Q$ and satisfying $|{\mathcal{P}}_{H}(\unicode[STIX]{x1D713}_{i})|\gg (1+\Vert \unicode[STIX]{x1D709}_{i}\Vert )^{\unicode[STIX]{x1D6FF}}N^{\unicode[STIX]{x1D6FF}}$. This implies the same lower bound on $\Vert \unicode[STIX]{x1D713}_{i}\Vert _{\infty }$, and in particular proves Theorem 1.1.

It remains to refine this power growth to obtain the stated lower bounds of Theorem 1.2. For this, we will make a special choice of orthonormal basis of Hecke–Maass forms for $L^{2}(Y_{N})$. First recall that we have a Hilbert direct sum decomposition

$$\begin{eqnarray}L^{2}(Y_{N})=\bigoplus _{\unicode[STIX]{x1D70B}}m(\unicode[STIX]{x1D70B})\unicode[STIX]{x1D70B}^{K(N)K_{H}K_{\infty }},\end{eqnarray}$$

the sum ranging over irreducible representations of $G_{\infty }^{0}\times G(\mathbb{A}_{f})$ in $L^{2}(G(F)\backslash G(\mathbb{A}))$ satisfying $\unicode[STIX]{x1D70B}^{K(N)K_{H}K_{\infty }}\neq 0$, each occurring with multiplicity $m(\unicode[STIX]{x1D70B})$. Each $\unicode[STIX]{x1D70B}$ gives rise to a spectral datum $c_{\unicode[STIX]{x1D70B}}$ (as defined in the paragraph preceding Theorem 1.2), and this assignment $\unicode[STIX]{x1D70B}\mapsto c_{\unicode[STIX]{x1D70B}}$ is finite-to-one. Given a spectral datum $c$ for $(G,N)$, the space $V(N,c)$ is the direct sum of $m(\unicode[STIX]{x1D70B})\unicode[STIX]{x1D70B}^{K(N)K_{H}K_{\infty }}$ over all $\unicode[STIX]{x1D70B}$ in the fiber over $c$.

Let $\{c_{i}\}$ be an enumeration of the spectral data, and for each $c_{i}$ write $V_{i}=V(N,c_{i})$ and $\unicode[STIX]{x1D709}_{i}=\unicode[STIX]{x1D709}(c_{i})$. The automorphic period ${\mathcal{P}}_{H}$ defines a linear functional on each finite-dimensional vector space $V_{i}$, and its kernel is of codimension at most 1. If the kernel is of codimension 1 we let $\unicode[STIX]{x1D719}_{i}\in V_{i}$ be a unit normal to it, and otherwise choose $\unicode[STIX]{x1D719}_{i}\in V_{i}$ to be an arbitrary unit vector. We may complete this set of vectors to an orthonormal basis of Hecke–Maass forms for $L^{2}(Y_{N})$, to which we apply (9) and (10). Since ${\mathcal{H}}$ acts as a scalar on $V_{i}$ we may write the left-hand side of (9) as

$$\begin{eqnarray}\text{Vol}_{N}\mathop{\sum }_{\substack{ c_{i} \\ \Vert \Re \unicode[STIX]{x1D709}_{i}-\unicode[STIX]{x1D709}\Vert \leqslant Q}}|\widehat{\unicode[STIX]{x1D714}_{S}}(\unicode[STIX]{x1D719}_{i})|^{2}\dim V_{i},\end{eqnarray}$$

and the left-hand side of (10) as

$$\begin{eqnarray}\text{Vol}_{N}\mathop{\sum }_{\substack{ c_{i} \\ \Vert \Re \unicode[STIX]{x1D709}_{i}-\unicode[STIX]{x1D709}\Vert \leqslant Q}}|\widehat{\unicode[STIX]{x1D714}_{S}}(\unicode[STIX]{x1D719}_{i})|^{2}|{\mathcal{P}}_{H}(\unicode[STIX]{x1D719}_{i})|^{2}.\end{eqnarray}$$

We obtain Theorem 1.2 by comparing the right-hand sides of (9) and (10) as before. ◻

4.1 The archimedean test function

Let $G_{\infty }^{0}$ denote the connected component of the identity in $G_{\infty }$ in the real topology. If $\unicode[STIX]{x1D707}\in \mathfrak{a}_{\mathbb{C}}^{\ast }$, we define $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D707}}$ to be the corresponding spherical function on $G_{\infty }^{0}$. If $k_{\infty }\in C_{c}^{\infty }(G_{\infty }^{0})$, we define its Harish-Chandra transform by

$$\begin{eqnarray}\widehat{k}_{\infty }(\unicode[STIX]{x1D707})=\int _{G_{\infty }^{0}}k_{\infty }(g)\unicode[STIX]{x1D711}_{-\unicode[STIX]{x1D707}}(g)\,d\unicode[STIX]{x1D707}_{G,\infty }^{\text{can}}(g).\end{eqnarray}$$

Note that this is related to our earlier notation as follows: if $\unicode[STIX]{x1D713}$ is a Maass form with spectral parameter $\unicode[STIX]{x1D707}$ and $k_{\infty }$ is bi-$K_{\infty }$-invariant, then $\widehat{k}_{\infty }(\unicode[STIX]{x1D713})=\widehat{k}_{\infty }(-\unicode[STIX]{x1D707})$.

We shall choose $k_{\unicode[STIX]{x1D709}}$ so that its Harish-Chandra transform concentrates around $-\unicode[STIX]{x1D709}\in \mathfrak{a}^{\ast }$. For this we first take a function $h_{0}\in C^{\infty }(\mathfrak{a}^{\ast })$ of Paley–Wiener type that is real, non-negative and satisfies $h_{0}(0)=1$. We implicitly extend $h_{0}$ to a function on $\mathfrak{a}_{\mathbb{C}}^{\ast }$. Let

$$\begin{eqnarray}h_{\unicode[STIX]{x1D709}}^{0}(\unicode[STIX]{x1D708})=\mathop{\sum }_{w\in W_{\mathbb{R}}}h_{0}(w\unicode[STIX]{x1D708}-\unicode[STIX]{x1D709}),\end{eqnarray}$$

and let $k_{\unicode[STIX]{x1D709}}^{0}$ be the bi-$K_{\infty }$-invariant function on $G_{\infty }^{0}$ satisfying $\widehat{k}_{\unicode[STIX]{x1D709}}^{0}(-\unicode[STIX]{x1D707})=h_{\unicode[STIX]{x1D709}}^{0}(\unicode[STIX]{x1D707})$. We define $k_{\unicode[STIX]{x1D709}}=k_{\unicode[STIX]{x1D709}}^{0}\ast k_{\unicode[STIX]{x1D709}}^{0}$ and $h_{\unicode[STIX]{x1D709}}=(h_{\unicode[STIX]{x1D709}}^{0})^{2}$. We have $\widehat{k}_{\unicode[STIX]{x1D709}}(-\unicode[STIX]{x1D707})=h_{\unicode[STIX]{x1D709}}(\unicode[STIX]{x1D707})$, so that if $\unicode[STIX]{x1D713}$ has spectral parameter $\unicode[STIX]{x1D707}$ as before then $\widehat{k}_{\unicode[STIX]{x1D709}}(\unicode[STIX]{x1D713})=h_{\unicode[STIX]{x1D709}}(\unicode[STIX]{x1D707})$. We define a unitary spectral parameter to be a spectral parameter of a spherical unitary representation of $G_{\infty }^{0}$. We have $\Vert \Im \unicode[STIX]{x1D707}\Vert \leqslant \Vert \unicode[STIX]{x1D70C}\Vert$ for any unitary spectral parameter $\unicode[STIX]{x1D707}$ [Reference HelgasonHel00, ch. IV, Theorem 8.1]. If $\unicode[STIX]{x1D707}\in \mathfrak{a}_{\mathbb{C}}^{\ast }$ is unitary, Lemma 4.1 implies that $h_{\unicode[STIX]{x1D709}}(\unicode[STIX]{x1D707})=h_{\unicode[STIX]{x1D709}}^{0}(\unicode[STIX]{x1D707})h_{\unicode[STIX]{x1D709}}^{0}(\overline{\unicode[STIX]{x1D707}})=|h_{\unicode[STIX]{x1D709}}^{0}(\unicode[STIX]{x1D707})|^{2}\geqslant 0$.

Proof. Note that $k_{\unicode[STIX]{x1D709}}$ is of compact support independent of $\unicode[STIX]{x1D709}$ from the Paley–Wiener theorem of [Reference GangolliGan71]; we may thus take $h_{0}$ so that the support of $k_{\unicode[STIX]{x1D709}}$ lies in $\{g\in G(F_{\infty }):d(g,H_{v_{0}})<1\}$ for all $\unicode[STIX]{x1D709}$.

We have the inversion formula

(11)$$\begin{eqnarray}k_{\unicode[STIX]{x1D709}}(g)=\frac{1}{|W_{\mathbb{R}}|}\int _{\mathfrak{a}^{\ast }}\widehat{k}_{\unicode[STIX]{x1D709}}(\unicode[STIX]{x1D707})\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D707}}(g)|c(\unicode[STIX]{x1D707})|^{-2}\,d\unicode[STIX]{x1D707}.\end{eqnarray}$$

We can now quote either [Reference Blomer and PohlBP16, Theorem 2] or [Reference Matz and TemplierMT15, Proposition 7.2] and apply our conditions on $G_{\infty }$, to find

(12)$$\begin{eqnarray}\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D707}}(x)\,\ll _{C}\,(1+\Vert \unicode[STIX]{x1D707}\Vert d(x,K_{\infty }))^{-1/2},\end{eqnarray}$$

for $\unicode[STIX]{x1D707}\in \mathfrak{a}^{\ast }$ and $x$ in a compact set $C\subset G_{\infty }$. In the inversion formula (11), we apply (4) to bound the Plancherel density function, (12) to bound the spherical function, and recall the concentration of $\widehat{k}_{\unicode[STIX]{x1D709}}$ around $-\unicode[STIX]{x1D709}$, to obtain

$$\begin{eqnarray}k_{\unicode[STIX]{x1D709}}(x)\ll \unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D709})(1+\Vert \unicode[STIX]{x1D709}\Vert d(x,K_{\infty }))^{-1/2},\end{eqnarray}$$

whence the claim. ◻

The following lemma will be helpful when truncating spectral sums. We let $B(\unicode[STIX]{x1D707})\subset \mathfrak{a}^{\ast }$ denote the unit ball around $\unicode[STIX]{x1D707}\in \mathfrak{a}^{\ast }$.

Proof. Let $0<\unicode[STIX]{x1D6FF}<1$. Let $b\in C_{0}^{\infty }(\mathfrak{a})$ be non-negative, supported in the $\unicode[STIX]{x1D6FF}$-ball around $0$, and satisfy $\int b=1$. We further assume that $b=b_{0}\ast b_{0}$ for some even real-valued $b_{0}$, which ensures that $\widehat{b}(\unicode[STIX]{x1D708})$ is non-negative for $\unicode[STIX]{x1D708}\in \mathfrak{a}^{\ast }$. We will show that $h_{0}=\widehat{b}$ suffices in our construction of $h_{\unicode[STIX]{x1D707}}$.

We let $C>0$ be a constant depending only on $G$ that may vary from line to line. We start by showing that $\Re \widehat{b}(\unicode[STIX]{x1D708})\geqslant -C\unicode[STIX]{x1D6FF}$ for all $\unicode[STIX]{x1D708}\in \mathfrak{a}_{\mathbb{C}}^{\ast }$ with $\Vert \Im \unicode[STIX]{x1D708}\Vert \leqslant \Vert \unicode[STIX]{x1D70C}\Vert$. We have

$$\begin{eqnarray}\displaystyle \widehat{b}(\unicode[STIX]{x1D708}) & = & \displaystyle \int _{\mathfrak{a}}b(H)e^{-i\unicode[STIX]{x1D708}(H)}\,dH\nonumber\\ \displaystyle & = & \displaystyle \int _{\mathfrak{a}}b(H)[e^{-i\Re \unicode[STIX]{x1D708}(H)}+(e^{-i\unicode[STIX]{x1D708}(H)}-e^{-i\Re \unicode[STIX]{x1D708}(H)})]\,dH\nonumber\\ \displaystyle & = & \displaystyle \widehat{b}(\Re \unicode[STIX]{x1D708})+\int _{\mathfrak{a}}b(H)e^{-i\Re \unicode[STIX]{x1D708}(H)}(e^{\Im \unicode[STIX]{x1D708}(H)}-1)\,dH.\nonumber\end{eqnarray}$$

As $\Vert \Im \unicode[STIX]{x1D708}\Vert \leqslant \Vert \unicode[STIX]{x1D70C}\Vert$, we have $|e^{\Im \unicode[STIX]{x1D708}(H)}-1|\leqslant C\unicode[STIX]{x1D6FF}$ for all $H\in \text{supp}(b)$, so that

$$\begin{eqnarray}\Re \widehat{b}(\unicode[STIX]{x1D708})\geqslant \widehat{b}(\Re \unicode[STIX]{x1D708})-C\unicode[STIX]{x1D6FF}\geqslant -C\unicode[STIX]{x1D6FF}\end{eqnarray}$$

as required.

We now take $h_{0}=\widehat{b}$, and construct $h_{\unicode[STIX]{x1D707}}^{0}$ and $h_{\unicode[STIX]{x1D707}}$ as before. If we choose $\unicode[STIX]{x1D6FF}$ small, we will have $\Re h_{0}(\unicode[STIX]{x1D708})\geqslant 1/2$ for all $\unicode[STIX]{x1D708}$ with $\Vert \Re \unicode[STIX]{x1D708}\Vert \leqslant 1$, $\Vert \Im \unicode[STIX]{x1D708}\Vert \leqslant \Vert \unicode[STIX]{x1D70C}\Vert$. Therefore if $\unicode[STIX]{x1D706}\in \mathfrak{a}_{\mathbb{C}}^{\ast }$ is unitary with $\Re \unicode[STIX]{x1D706}\in B(\unicode[STIX]{x1D707})$, the bound $\Vert \Im \unicode[STIX]{x1D706}\Vert \leqslant \Vert \unicode[STIX]{x1D70C}\Vert$ implies that

$$\begin{eqnarray}\displaystyle \Re h_{\unicode[STIX]{x1D707}}^{0}(\unicode[STIX]{x1D706}) & = & \displaystyle \mathop{\sum }_{w\in W_{\mathbb{R}}}\Re h_{0}(w\unicode[STIX]{x1D706}-\unicode[STIX]{x1D707})\nonumber\\ \displaystyle & {\geqslant} & \displaystyle \Re h_{0}(\unicode[STIX]{x1D706}-\unicode[STIX]{x1D707})-C\unicode[STIX]{x1D6FF}\geqslant 1/4.\nonumber\end{eqnarray}$$

This gives $h_{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D706})=|h_{\unicode[STIX]{x1D707}}^{0}(\unicode[STIX]{x1D706})|^{2}\geqslant 1/16$ as required.◻

4.2 The $S$-adic test function

We choose the test function at places in $S$ to be $k_{S}=\unicode[STIX]{x1D714}_{S}\unicode[STIX]{x1D714}_{S}^{\ast }\in {\mathcal{H}}_{S}$, where $\unicode[STIX]{x1D714}_{S}$ is as in § 3.3.

Proof. (a) Because supports add under convolution, it suffices to show this for $g\in \text{supp}(\unicode[STIX]{x1D70F}(v,\unicode[STIX]{x1D708}))$ (or $\text{supp}(\unicode[STIX]{x1D70F}(v,-\unicode[STIX]{x1D708}))$, which is similar and we omit) for some $v\in S$. Our assumption that $\unicode[STIX]{x1D70C}(K_{w})\subset \text{SL}_{d}({\mathcal{O}}_{w})$ for all finite $w$ means that we only need to estimate $\Vert \unicode[STIX]{x1D70C}(g)\Vert _{v}$. Consider $\unicode[STIX]{x1D70C}$ as a representation of $G$, and let $\unicode[STIX]{x1D6FA}\subset X^{\ast }(T)$ be the multiset of weights of this representation. It follows that $\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D708}(\unicode[STIX]{x1D71B}_{v}))$ is semisimple with eigenvalues $\{\unicode[STIX]{x1D71B}_{v}^{\langle \unicode[STIX]{x1D714},\unicode[STIX]{x1D708}\rangle }:\unicode[STIX]{x1D714}\in \unicode[STIX]{x1D6FA}\}$. By [Reference Shin and TemplierST16, Lemma 2.17], there is $x\in \text{GL}_{d}({\mathcal{O}}_{v})$ such that $x\unicode[STIX]{x1D70C}(T_{v})x^{-1}$ is diagonal, and so if we define $A=\max \{-\langle \unicode[STIX]{x1D714},\unicode[STIX]{x1D708}\rangle :\unicode[STIX]{x1D714}\in \unicode[STIX]{x1D6FA}\}$ then we have $\Vert \unicode[STIX]{x1D70C}(\unicode[STIX]{x1D708}(\unicode[STIX]{x1D71B}_{v}))\Vert _{v}=\Vert x\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D708}(\unicode[STIX]{x1D71B}_{v}))x^{-1}\Vert _{v}=q_{v}^{A}<P^{A}$. Because $\unicode[STIX]{x1D70C}(G({\mathcal{O}}_{v}))\subset \text{SL}_{d}({\mathcal{O}}_{v})$, the same holds for $g\in \text{supp}(\unicode[STIX]{x1D70F}(v,\unicode[STIX]{x1D708}))$.

(b) We note that

(13)$$\begin{eqnarray}k_{S}=\mathop{\sum }_{v\in S}\unicode[STIX]{x1D70F}(v,\unicode[STIX]{x1D708})\unicode[STIX]{x1D70F}(v,\unicode[STIX]{x1D708})^{\ast }+\mathop{\sum }_{\substack{ v,w\in S \\ v\neq w}}\unicode[STIX]{x1D70F}(v,\unicode[STIX]{x1D708})\unicode[STIX]{x1D70F}(w,\unicode[STIX]{x1D708})^{\ast }.\end{eqnarray}$$

The bound is clear for the second sum, because the terms satisfy $\Vert \unicode[STIX]{x1D70F}(v,\unicode[STIX]{x1D708})\unicode[STIX]{x1D70F}(w,\unicode[STIX]{x1D708})^{\ast }\Vert _{\infty }\leqslant 1$ and the supports of the terms are disjoint. For the first sum, we have

$$\begin{eqnarray}\Vert \unicode[STIX]{x1D70F}(v,\unicode[STIX]{x1D708})\unicode[STIX]{x1D70F}(v,\unicode[STIX]{x1D708})^{\ast }\Vert _{\infty }\leqslant \Vert \unicode[STIX]{x1D70F}(v,\unicode[STIX]{x1D708})\Vert _{2}^{2},\end{eqnarray}$$

and $\Vert \unicode[STIX]{x1D70F}(v,\unicode[STIX]{x1D708})\Vert _{2}\ll 1$ follows from Lemma 2.3.

(c) We may write the first sum in (13) as a linear combination of $\unicode[STIX]{x1D70F}(v,\unicode[STIX]{x1D707})$ with $\unicode[STIX]{x1D707}$ lying in a finite set depending on $\unicode[STIX]{x1D708}$. The bound now follows from Lemma 2.3. The second sum may be treated similarly.◻

4.3 Proof of Proposition 3.3

We are now in a position to prove Proposition 3.3. We first note that $k_{\unicode[STIX]{x1D709}}(1)\asymp \unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D709})$, where the upper and lower bounds follow from property ($\mathsf{P}_{\infty }$) and [Reference Duistermaat, Kolk and VaradarajanDKV79, (3.44a)], respectively. Moreover, $\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D709})$ is bounded above (and below) by a power of $(1+\Vert \unicode[STIX]{x1D709}\Vert )$.

We now apply Proposition 3.2 with test functions $k_{S}k_{\unicode[STIX]{x1D709}}$. Moreover, for every pair $v,w\in S$ we put $T=\{v,w\}$ and apply Theorem 3.1 with test function $k_{T}k_{\unicode[STIX]{x1D709}}$ where $k_{T}=\unicode[STIX]{x1D70F}(v,\unicode[STIX]{x1D708})\unicode[STIX]{x1D70F}(w,\unicode[STIX]{x1D708})^{\ast }$; we then sum over such pairs $v,w$. As a result, we deduce the existence of constants $A>2$ (depending on $\unicode[STIX]{x1D708}$) and $\unicode[STIX]{x1D6FF}>0$ such that

(14)$$\begin{eqnarray}\text{Vol}_{N}\mathop{\sum }_{i\geqslant 0}|\widehat{\unicode[STIX]{x1D714}_{S}}(\unicode[STIX]{x1D713}_{i})|^{2}h_{\unicode[STIX]{x1D709}}(\unicode[STIX]{x1D709}_{i})\asymp \unicode[STIX]{x1D714}_{S}\unicode[STIX]{x1D714}_{S}^{\ast }(1)\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D709})+O(P^{A}N^{-\unicode[STIX]{x1D6FF}}\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D709})(1+\Vert \unicode[STIX]{x1D709}\Vert )^{-\unicode[STIX]{x1D6FF}})\end{eqnarray}$$

and

(15)$$\begin{eqnarray}\text{Vol}_{N}\mathop{\sum }_{i\geqslant 0}|\widehat{\unicode[STIX]{x1D714}_{S}}(\unicode[STIX]{x1D713}_{i})|^{2}|{\mathcal{P}}_{H}(\unicode[STIX]{x1D713}_{i})|^{2}h_{\unicode[STIX]{x1D709}}(\unicode[STIX]{x1D709}_{i})\asymp \unicode[STIX]{x1D6F1}_{H}\unicode[STIX]{x1D714}_{S}\unicode[STIX]{x1D714}_{S}^{\ast }(1)\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D709})+O(P^{A}N^{-\unicode[STIX]{x1D6FF}}\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D709})(1+\Vert \unicode[STIX]{x1D709}\Vert )^{-\unicode[STIX]{x1D6FF}}).\end{eqnarray}$$

The error term in (14) was obtained by observing that $\unicode[STIX]{x1D70F}(v,\unicode[STIX]{x1D708})\unicode[STIX]{x1D70F}(w,\unicode[STIX]{x1D708})^{\ast }\in {\mathcal{H}}_{T}^{{\leqslant}\unicode[STIX]{x1D705}}$, for some $\unicode[STIX]{x1D705}$ depending only on $\unicode[STIX]{x1D708}$, and $q_{T}=q_{v}q_{w}<P^{2}$, so that $q_{T}^{A\unicode[STIX]{x1D705}+B}$ is bounded by a power of $P$; we must also insert the $L^{\infty }$ norm estimate for $k_{T}$ coming from the proof of Lemma 4.4. The error term in (15) was obtained by taking $B$ as in Lemma 4.4 and setting $R=P^{B}$ in condition $(\mathsf{P}_{S})$, and inserting the $L^{\infty }$ norm estimate of Lemma 4.4. It remains then to explicate the size of $\unicode[STIX]{x1D714}_{S}\unicode[STIX]{x1D714}_{S}^{\ast }(1)$ and $\unicode[STIX]{x1D6F1}_{H}\unicode[STIX]{x1D714}_{S}\unicode[STIX]{x1D714}_{S}^{\ast }(1)$ (the latter under our extra condition on $\unicode[STIX]{x1D708}$), and truncate the spectral sums. We continue to use the convention that the values of the exponents $A,\unicode[STIX]{x1D6FF}>0$ can vary from line to line.

We begin with (14). Lemma 4.4 (b) gives $\unicode[STIX]{x1D714}_{S}\unicode[STIX]{x1D714}_{S}^{\ast }(1)\ll P$. By positivity we may truncate the spectral sum to obtain the upper bound

$$\begin{eqnarray}\text{Vol}_{N}\mathop{\sum }_{\Vert \Re \unicode[STIX]{x1D709}_{i}-\unicode[STIX]{x1D709}\Vert \leqslant 1}|\widehat{\unicode[STIX]{x1D714}_{S}}(\unicode[STIX]{x1D713}_{i})|^{2}h_{\unicode[STIX]{x1D709}}(\unicode[STIX]{x1D709}_{i})\ll P\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D709})(1+P^{A}N^{-\unicode[STIX]{x1D6FF}}(1+\Vert \unicode[STIX]{x1D709}\Vert )^{-\unicode[STIX]{x1D6FF}}).\end{eqnarray}$$

Combined with Lemma 4.3, this implies (7).

We next examine (15). We begin by observing that if $\unicode[STIX]{x1D707}\in \mathfrak{a}^{\ast }$, then we may again truncate the spectral sum and apply Lemma 4.3 to obtain

(16)$$\begin{eqnarray}\text{Vol}_{N}\mathop{\sum }_{\Vert \Re \unicode[STIX]{x1D709}_{i}-\unicode[STIX]{x1D707}\Vert \leqslant 1}|\widehat{\unicode[STIX]{x1D714}_{S}}(\unicode[STIX]{x1D713}_{i})|^{2}|{\mathcal{P}}_{H}(\unicode[STIX]{x1D713}_{i})|^{2}\ll \unicode[STIX]{x1D6F1}_{H}\unicode[STIX]{x1D714}_{S}\unicode[STIX]{x1D714}_{S}^{\ast }(1)\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D707})+P^{A}\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D707})(1+\Vert \unicode[STIX]{x1D707}\Vert )^{-\unicode[STIX]{x1D6FF}}N^{-\unicode[STIX]{x1D6FF}}.\end{eqnarray}$$

We next use this to truncate the spectral sum about $\unicode[STIX]{x1D709}$.

Lemma 4.5. For any $Q>1$ the left-hand side of (15) can be written as

$$\begin{eqnarray}\displaystyle & & \displaystyle \text{Vol}_{N}\mathop{\sum }_{\Vert \Re \unicode[STIX]{x1D709}_{i}-\unicode[STIX]{x1D709}\Vert \leqslant Q}|\widehat{\unicode[STIX]{x1D714}_{S}}(\unicode[STIX]{x1D713}_{i})|^{2}|{\mathcal{P}}_{H}(\unicode[STIX]{x1D713}_{i})|^{2}h_{\unicode[STIX]{x1D709}}(\unicode[STIX]{x1D709}_{i})\nonumber\\ \displaystyle & & \displaystyle \quad +\,O_{M}(\unicode[STIX]{x1D6F1}_{H}\unicode[STIX]{x1D714}_{S}\unicode[STIX]{x1D714}_{S}^{\ast }(1)\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D709})Q^{-M})+O(P^{A}N^{-\unicode[STIX]{x1D6FF}}\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D709})(1+\Vert \unicode[STIX]{x1D709}\Vert )^{-\unicode[STIX]{x1D6FF}}).\nonumber\end{eqnarray}$$

Proof. We must show that the sum

$$\begin{eqnarray}\mathfrak{S}=\text{Vol}_{N}\mathop{\sum }_{\Vert \Re \unicode[STIX]{x1D709}_{i}-\unicode[STIX]{x1D709}\Vert >Q}|\widehat{\unicode[STIX]{x1D714}_{S}}(\unicode[STIX]{x1D713}_{i})|^{2}|{\mathcal{P}}_{H}(\unicode[STIX]{x1D713}_{i})|^{2}h_{\unicode[STIX]{x1D709}}(\unicode[STIX]{x1D709}_{i})\end{eqnarray}$$

satisfies

$$\begin{eqnarray}\mathfrak{S}=O_{M}(\unicode[STIX]{x1D6F1}_{H}\unicode[STIX]{x1D714}_{S}\unicode[STIX]{x1D714}_{S}^{\ast }(1)\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D709})Q^{-M})+O(P^{A}N^{-\unicode[STIX]{x1D6FF}}\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D709})(1+\Vert \unicode[STIX]{x1D709}\Vert )^{-\unicode[STIX]{x1D6FF}}).\end{eqnarray}$$

Break the region in the positive chamber $\mathfrak{a}_{+}^{\ast }$ defined by $\Vert \unicode[STIX]{x1D707}-\unicode[STIX]{x1D709}\Vert >Q$ into an overlapping union of unit balls $B(\unicode[STIX]{x1D707}_{n})$ centred at points $\unicode[STIX]{x1D707}_{n}\in \mathfrak{a}^{\ast }$. Because unitary spectral parameters $\unicode[STIX]{x1D707}$ satisfy $\Vert \Im \unicode[STIX]{x1D707}\Vert \leqslant \Vert \unicode[STIX]{x1D70C}\Vert$, they also satisfy the rapid decay estimate $h_{\unicode[STIX]{x1D709}}(\unicode[STIX]{x1D707})\,\ll _{M}\,\Vert \Re \unicode[STIX]{x1D707}-\unicode[STIX]{x1D709}\Vert ^{-M}$. Applying this on each ball gives

$$\begin{eqnarray}\mathop{\sum }_{\Re \unicode[STIX]{x1D709}_{i}\in B(\unicode[STIX]{x1D707}_{n})}|\widehat{\unicode[STIX]{x1D714}_{S}}(\unicode[STIX]{x1D713}_{i})|^{2}|{\mathcal{P}}_{H}(\unicode[STIX]{x1D713}_{i})|^{2}h_{\unicode[STIX]{x1D709}}(\unicode[STIX]{x1D709}_{i})\,\ll _{M}\,\Vert \unicode[STIX]{x1D707}_{n}-\unicode[STIX]{x1D709}\Vert ^{-M}\mathop{\sum }_{\Re \unicode[STIX]{x1D709}_{i}\in B(\unicode[STIX]{x1D707}_{n})}|\widehat{\unicode[STIX]{x1D714}_{S}}(\unicode[STIX]{x1D713}_{i})|^{2}|{\mathcal{P}}_{H}(\unicode[STIX]{x1D713}_{i})|^{2}.\end{eqnarray}$$

Applying (16) and summing over $n$, we obtain

$$\begin{eqnarray}\mathfrak{S}\,\ll _{M}\,\unicode[STIX]{x1D6F1}_{H}\unicode[STIX]{x1D714}_{S}\unicode[STIX]{x1D714}_{S}^{\ast }(1)\mathop{\sum }_{n}\Vert \unicode[STIX]{x1D707}_{n}-\unicode[STIX]{x1D709}\Vert ^{-M}\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D707}_{n})+P^{A}N^{-\unicode[STIX]{x1D6FF}}\mathop{\sum }_{n}\Vert \unicode[STIX]{x1D707}_{n}-\unicode[STIX]{x1D709}\Vert ^{-M}(1+\Vert \unicode[STIX]{x1D707}_{n}\Vert )^{-\unicode[STIX]{x1D6FF}}\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D707}_{n}).\end{eqnarray}$$

From $\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D707}_{n})\ll \Vert \unicode[STIX]{x1D707}_{n}-\unicode[STIX]{x1D709}\Vert ^{k}\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D709})$, where $k$ is the number of roots of $G$ counted with multiplicity, we may simplify this to

$$\begin{eqnarray}\mathfrak{S}\,\ll _{M}\,\unicode[STIX]{x1D6F1}_{H}\unicode[STIX]{x1D714}_{S}\unicode[STIX]{x1D714}_{S}^{\ast }(1)\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D709})\mathop{\sum }_{n}\Vert \unicode[STIX]{x1D707}_{n}-\unicode[STIX]{x1D709}\Vert ^{-M}+\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D709})P^{A}N^{-\unicode[STIX]{x1D6FF}}\mathop{\sum }_{n}\Vert \unicode[STIX]{x1D707}_{n}-\unicode[STIX]{x1D709}\Vert ^{-M}(1+\Vert \unicode[STIX]{x1D707}_{n}\Vert )^{-\unicode[STIX]{x1D6FF}}.\end{eqnarray}$$

The first sum is $\ll _{M}\,Q^{-M}$. We bound the second sum by breaking it into $\Vert \unicode[STIX]{x1D707}_{n}\Vert \leqslant \Vert \unicode[STIX]{x1D709}\Vert /2$ and the complement. The first sum is $\ll (1+\Vert \unicode[STIX]{x1D709}\Vert )^{-M}$, and the second is $\ll Q^{-M}(1+\Vert \unicode[STIX]{x1D709}\Vert )^{-\unicode[STIX]{x1D6FF}}$. Both of these are dominated by $(1+\Vert \unicode[STIX]{x1D709}\Vert )^{-\unicode[STIX]{x1D6FF}}$, which finally shows that

$$\begin{eqnarray}\mathfrak{S}\,\ll _{M}\,\unicode[STIX]{x1D6F1}_{H}\unicode[STIX]{x1D714}_{S}\unicode[STIX]{x1D714}_{S}^{\ast }(1)\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D709})Q^{-M}+P^{A}N^{-\unicode[STIX]{x1D6FF}}\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D709})(1+\Vert \unicode[STIX]{x1D709}\Vert )^{-\unicode[STIX]{x1D6FF}},\end{eqnarray}$$

as desired. ◻

We combine Lemma 4.5 (taking any $M>0$ and large enough $Q$) and (15) to obtain

$$\begin{eqnarray}\text{Vol}_{N}\mathop{\sum }_{\Vert \Re \unicode[STIX]{x1D709}_{i}-\unicode[STIX]{x1D709}\Vert \leqslant Q}|\widehat{\unicode[STIX]{x1D714}_{S}}(\unicode[STIX]{x1D713}_{i})|^{2}|{\mathcal{P}}_{H}(\unicode[STIX]{x1D713}_{i})|^{2}h_{\unicode[STIX]{x1D709}}(\unicode[STIX]{x1D709}_{i})\asymp \unicode[STIX]{x1D6F1}_{H}\unicode[STIX]{x1D714}_{S}\unicode[STIX]{x1D714}_{S}^{\ast }(1)\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D709})+O(P^{A}N^{-\unicode[STIX]{x1D6FF}}\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D709})(1+\Vert \unicode[STIX]{x1D709}\Vert )^{-\unicode[STIX]{x1D6FF}}).\end{eqnarray}$$

We now make use of the critical assumptions that $G$ is $H$-large, and $\unicode[STIX]{x1D708}\in X_{\ast }(T_{H})$ satisfies (3), to bound the right-hand side from below.

Proof. Note that for any $v\in {\mathcal{P}}$ and $\unicode[STIX]{x1D708}\in X_{\ast }(T_{H})$ we have

$$\begin{eqnarray}q_{v}^{\Vert \unicode[STIX]{x1D708}\Vert ^{\ast }}\int _{H_{v}}\unicode[STIX]{x1D70F}(v,\unicode[STIX]{x1D708})(x)\,dx=\text{vol}(H_{v}\cap K_{v}\unicode[STIX]{x1D708}(\unicode[STIX]{x1D71B}_{v})K_{v})\geqslant \text{vol}(K_{H,v}\unicode[STIX]{x1D708}(\unicode[STIX]{x1D71B}_{v})K_{H,v})\gg q_{v}^{2\Vert \unicode[STIX]{x1D708}\Vert _{H}^{\ast }},\end{eqnarray}$$

where we have used our assumption that $T$, and hence $T_{H}$, is split at $v$, and applied the lower bound from Lemma 2.3. If $\unicode[STIX]{x1D708}\in X_{\ast }(T_{H})$ satisfies (3), then so does $-\unicode[STIX]{x1D708}$, and applying the above bound with $\pm \unicode[STIX]{x1D708}$, we get $\int _{H(F_{S})}\unicode[STIX]{x1D70F}(v,\unicode[STIX]{x1D708})\unicode[STIX]{x1D70F}(w,\unicode[STIX]{x1D708})^{\ast }\gg 1$ if $v\neq w$. Summing over $v,w\in S$ yields the claim.◻

We deduce from the above lemma that

$$\begin{eqnarray}\text{Vol}_{N}\mathop{\sum }_{\Vert \Re \unicode[STIX]{x1D709}_{i}-\unicode[STIX]{x1D709}\Vert \leqslant Q}|\widehat{\unicode[STIX]{x1D714}_{S}}(\unicode[STIX]{x1D713}_{i})|^{2}|{\mathcal{P}}_{H}(\unicode[STIX]{x1D713}_{i})|^{2}h_{\unicode[STIX]{x1D709}}(\unicode[STIX]{x1D709}_{i})\gg _{\unicode[STIX]{x1D716}}P^{2-\unicode[STIX]{x1D716}}\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D709})(1+O(P^{A}N^{-\unicode[STIX]{x1D6FF}}(1+\Vert \unicode[STIX]{x1D709}\Vert )^{-\unicode[STIX]{x1D6FF}})).\end{eqnarray}$$

The bound $h_{\unicode[STIX]{x1D709}}(\unicode[STIX]{x1D709}_{i})\ll 1$ completes the proof of (8).

5.1 Bounding the off-diagonal contributions

In this section we establish Corollary 5.2, which was used in the proof of Proposition 3.2 above. It is based on the following Diophantine lemma which shows, roughly speaking, that any $\unicode[STIX]{x1D6FE}\in G(F)-H(F)$ cannot be too close to $H_{v}$ for various $v$. We shall use the notation introduced in § 2.3.

Corollary 5.2. There is $A>0$ such that the following statement holds. Let $N$, $S$ and $\unicode[STIX]{x1D709}$ be as in § 3.2. Let $k_{S}\in {\mathcal{H}}_{S}$ satisfy $(\mathsf{P}_{S})$ and $k_{\unicode[STIX]{x1D709}}\in {\mathcal{H}}_{\infty }$ satisfy $(\mathsf{P}_{\!\!\infty })$. Put $\unicode[STIX]{x1D719}=\mathbf{1}_{N}^{S}\,\otimes \,k_{S}\,\otimes \,k_{\unicode[STIX]{x1D709}}$. Then, for all $\unicode[STIX]{x1D6FE}\in G(F)-H(F)$ and all $x,y\in H(\mathbb{A})$, we have

$$\begin{eqnarray}\unicode[STIX]{x1D719}(x^{-1}\unicode[STIX]{x1D6FE}y)\ll \unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D709})(1+\Vert \unicode[STIX]{x1D709}\Vert )^{-1/4}N^{-1/4}R^{A}\Vert k_{S}\Vert _{\infty }.\end{eqnarray}$$

Proof. Let $\unicode[STIX]{x1D6FA}_{H}\subset H(\mathbb{A})$ be a compact set containing a fundamental domain for $[H]$. We assume that $\unicode[STIX]{x1D6FA}_{H}=\unicode[STIX]{x1D6FA}_{H,D\infty }\times \prod _{v\nmid D\infty }K_{H,v}$ after possibly enlarging $D$. Because $G(F)-H(F)$ is bi-invariant under $H(F)$, we may assume that $x,y\in \unicode[STIX]{x1D6FA}_{H}$.

We may also clearly assume that $\unicode[STIX]{x1D719}(x^{-1}\unicode[STIX]{x1D6FE}y)\neq 0$. It then follows from Property $(\mathsf{P}_{S})$ that $\Vert x^{-1}\unicode[STIX]{x1D6FE}y\Vert _{S}\leqslant R$; in fact we have $\Vert x^{-1}\unicode[STIX]{x1D6FE}y\Vert _{f}\leqslant R$, using the condition $x^{-1}\unicode[STIX]{x1D6FE}y\in K^{S}(N)K_{H}^{S}\subset K^{S}$. When combined with $x,y\in \unicode[STIX]{x1D6FA}_{H}$, this gives $\Vert \unicode[STIX]{x1D6FE}\Vert _{f}\ll R$. We may now apply part (i) of Lemma 5.1, to find that $d(\unicode[STIX]{x1D6FE},H_{v_{0}})\gg R^{-A}$. All together, since $x,y\in H_{v_{0}}$, we deduce that $d(x^{-1}\unicode[STIX]{x1D6FE}y,H_{v_{0}})\gg R^{-A}$. Similarly, from $k_{\unicode[STIX]{x1D709}}(x^{-1}\unicode[STIX]{x1D6FE}y)\neq 0$ and Property $(\mathsf{P}_{\!\!\infty })$ it follows that $d(x^{-1}\unicode[STIX]{x1D6FE}y,H_{v_{0}})<1$, and hence $d(\unicode[STIX]{x1D6FE},H_{v_{0}})\ll 1$.

Suppose that $1+\Vert \unicode[STIX]{x1D709}\Vert \geqslant N$. We have $(k_{S}\,\otimes \,k_{\unicode[STIX]{x1D709}})(x^{-1}\unicode[STIX]{x1D6FE}y)\leqslant \Vert k_{S}\Vert _{\infty }k_{\unicode[STIX]{x1D709}}(x^{-1}\unicode[STIX]{x1D6FE}y)$. We then combine $(\mathsf{P}_{\!\!\infty })$ with $d(x^{-1}\unicode[STIX]{x1D6FE}y,H_{v_{0}})\gg R^{-A}$ to get

$$\begin{eqnarray}\displaystyle k_{\unicode[STIX]{x1D709}}(x^{-1}\unicode[STIX]{x1D6FE}y) & \ll & \displaystyle \unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D709})(1+\Vert \unicode[STIX]{x1D709}\Vert d(x^{-1}\unicode[STIX]{x1D6FE}y,H_{v_{0}}))^{-1/2}\nonumber\\ \displaystyle & \ll & \displaystyle \unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D709})(1+\Vert \unicode[STIX]{x1D709}\Vert R^{-A})^{-1/2}\nonumber\\ \displaystyle & \ll & \displaystyle \unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D709})(1+\Vert \unicode[STIX]{x1D709}\Vert )^{-1/2}R^{A/2}\nonumber\\ \displaystyle & \ll & \displaystyle \unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D709})(1+\Vert \unicode[STIX]{x1D709}\Vert )^{-1/4}N^{-1/4}R^{A/2},\nonumber\end{eqnarray}$$

which completes the proof in this case.

Now suppose that $1+\Vert \unicode[STIX]{x1D709}\Vert <N$. Because $\Vert \unicode[STIX]{x1D6FE}\Vert _{f}\ll R$, part (ii) of Lemma 5.1 implies that there are $A,C>0$ such that if $N>CR^{A}$, then there is a place $v|N$ for which $\unicode[STIX]{x1D6FE}_{v}\notin K(N)_{v}K_{H,v}$. Because $x,y\in \unicode[STIX]{x1D6FA}_{H}$, we have $x,y\in K_{H,v}$, and so $x^{-1}\unicode[STIX]{x1D6FE}y\notin K(N)_{v}K_{H,v}$. It follows that if $N>CR^{A}$, then $\unicode[STIX]{x1D719}(x^{-1}\unicode[STIX]{x1D6FE}y)=0$. We may rephrase this as saying that

$$\begin{eqnarray}\unicode[STIX]{x1D719}(x^{-1}\unicode[STIX]{x1D6FE}y)\leqslant \Vert k_{S}k_{\unicode[STIX]{x1D709}}\Vert _{\infty }N^{-1}CR^{A}\ll \Vert k_{S}\Vert _{\infty }\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D709})N^{-1}R^{A},\end{eqnarray}$$

and the bound $N^{-1}\leqslant N^{-1/2}(1+\Vert \unicode[STIX]{x1D709}\Vert )^{-1/2}$ completes the proof.◻

6.1 Canonical and Tamagawa measures

If $G$ is a general connected reductive group over $F$, Gross [Reference GrossGro97, (1.5)] attaches to $G$ an Artin–Tate motive

$$\begin{eqnarray}M_{G}=\bigoplus _{d\geqslant 1}M_{G,d}(1-d)\end{eqnarray}$$

with coefficients in $\mathbb{Q}$. Here $M_{G,d}$ is an Artin motive and $(1-d)$ denotes the Tate twist. We let $\unicode[STIX]{x1D716}(M_{G})$ be the $\unicode[STIX]{x1D716}$-factor of this motive, which is given by

$$\begin{eqnarray}\unicode[STIX]{x1D716}(M_{G})=|d_{F}|^{\dim G/2}\mathop{\prod }_{d\geqslant 1}{\mathcal{N}}_{F/\mathbb{Q}}(\mathfrak{f}(M_{G,d}))^{d-1/2},\end{eqnarray}$$

where $\mathfrak{f}(M_{G,d})$ denotes the conductor of the Artin motive $M_{G,d}$ (see [Reference GrossGro97, (9.8)]). We let $L(M_{G_{v}}^{\vee }(1))$ denote the $L$-function of the local motive $M_{G_{v}}^{\vee }(1)$, and $L(M_{G}^{\vee }(1))$ and $\unicode[STIX]{x1D6EC}(M_{G}^{\vee }(1))$ denote the finite and completed $L$-functions of $M_{G}^{\vee }(1)$. Then $L(M_{G_{v}}^{\vee }(1))$ is a positive real number, and $L(M_{G}^{\vee }(1))$ and $\unicode[STIX]{x1D6EC}(M_{G}^{\vee }(1))$ are finite if $Z(G)$ does not contain an $F$-split torus (see [Reference GrossGro97, Proposition 9.4]). From now on we shall assume that $G$ satisfies this condition.

In [Reference GrossGro97, § 11] Gross defines a canonical measure $|\unicode[STIX]{x1D714}_{G_{v}}|$ on $G_{v}$ at any place of $F$. We define $\unicode[STIX]{x1D707}_{G,v}^{\text{can}}=L(M_{G_{v}}^{\vee }(1))\cdot |\unicode[STIX]{x1D714}_{G_{v}}|$ as in [Reference GrossGro97]. When $v$ is finite and $G$ is unramified at $v$, $\unicode[STIX]{x1D707}_{G,v}^{\text{can}}$ assigns volume 1 to a hyperspecial subgroup of $G(F_{v})$, and so we can define the measure $\unicode[STIX]{x1D707}_{G}^{\text{can}}=\prod _{v}\unicode[STIX]{x1D707}_{G,v}^{\text{can}}$ on $G(\mathbb{A})$.

Now let $\unicode[STIX]{x1D714}$ be a non-zero differential form of top degree on $G$ defined over $F$. For each $v$, one may associate with $\unicode[STIX]{x1D714}$ a Haar measure $|\unicode[STIX]{x1D714}|_{v}$ on $G(F_{v})$. For almost all $v$, $L(M_{G_{v}}^{\vee }(1))\cdot |\unicode[STIX]{x1D714}|_{v}$ assigns volume 1 to a hyperspecial subgroup of $G(F_{v})$. Let $\unicode[STIX]{x1D707}_{G}^{\text{Tam}}$ be the Tamagawa measure on $G(\mathbb{A})$, which is defined by

$$\begin{eqnarray}\unicode[STIX]{x1D707}_{G}^{\text{Tam}}=\unicode[STIX]{x1D6EC}(M_{G}^{\vee }(1))^{-1}|d_{F}|^{-\text{dim}\,G/2}\bigotimes _{v}L(M_{G_{v}}^{\vee }(1))|\unicode[STIX]{x1D714}|_{v}\end{eqnarray}$$

(see [Reference GrossGro97, (10.2)]) and satisfies [Reference KottwitzKot88, p. 629]

$$\begin{eqnarray}\unicode[STIX]{x1D707}_{G}^{\text{Tam}}(G(F)\backslash G(\mathbb{A}))=|\unicode[STIX]{x1D70B}_{0}(Z(\widehat{G})^{\text{Gal}(\overline{F}/F)})||\text{ker}^{1}(F,Z(\widehat{G}))|^{-1}.\end{eqnarray}$$

The comparison between $\unicode[STIX]{x1D707}_{G}^{\text{can}}$ and $\unicode[STIX]{x1D707}_{G}^{\text{Tam}}$ is given by [Reference GrossGro97, Theorem 11.5]

(18)$$\begin{eqnarray}\frac{\unicode[STIX]{x1D707}_{G}^{\text{can}}}{\unicode[STIX]{x1D707}_{G}^{\text{Tam}}}=\unicode[STIX]{x1D716}(M_{G})\unicode[STIX]{x1D6EC}(M_{G}^{\vee }(1)).\end{eqnarray}$$

6.2 The trace formula

We now assume that $G$ is anisotropic. The trace formula is a distributional identity

$$\begin{eqnarray}I_{\text{spec}}(\unicode[STIX]{x1D719},\unicode[STIX]{x1D707}_{G}^{\text{can}})=I_{\text{geom}}(\unicode[STIX]{x1D719},\unicode[STIX]{x1D707}_{G}^{\text{can}}),\end{eqnarray}$$

for $\unicode[STIX]{x1D719}\in C_{c}^{\infty }(G(\mathbb{A}))$. More precisely,

$$\begin{eqnarray}I_{\text{spec}}(\unicode[STIX]{x1D719},\unicode[STIX]{x1D707}_{G}^{\text{can}})=\mathop{\sum }_{\unicode[STIX]{x1D70B}}m(\unicode[STIX]{x1D70B})\text{tr}(\unicode[STIX]{x1D70B}(\unicode[STIX]{x1D719})),\end{eqnarray}$$

where $\unicode[STIX]{x1D70B}$ runs over all irreducible subrepresentations of $L^{2}(G(F)\backslash G(\mathbb{A}))$ occurring with multiplicity $m(\unicode[STIX]{x1D70B})$, and

$$\begin{eqnarray}I_{\text{geom}}(\unicode[STIX]{x1D719},\unicode[STIX]{x1D707}_{G}^{\text{can}})=\mathop{\sum }_{\{\unicode[STIX]{x1D6FE}\}}\frac{\unicode[STIX]{x1D707}_{I_{\unicode[STIX]{x1D6FE}}}^{\text{can}}(I_{\unicode[STIX]{x1D6FE}}(F)\backslash I_{\unicode[STIX]{x1D6FE}}(\mathbb{A}))}{|G_{\unicode[STIX]{x1D6FE}}(F):I_{\unicode[STIX]{x1D6FE}}(F)|}O_{\unicode[STIX]{x1D6FE}}(\unicode[STIX]{x1D719}),\end{eqnarray}$$

where $\{\unicode[STIX]{x1D6FE}\}$ runs over all $G(F)$-conjugacy classes, $G_{\unicode[STIX]{x1D6FE}}$ is the centralizer of $\unicode[STIX]{x1D6FE}$ in $G$, $I_{\unicode[STIX]{x1D6FE}}$ is the connected component of $G_{\unicode[STIX]{x1D6FE}}$, and

$$\begin{eqnarray}O_{\unicode[STIX]{x1D6FE}}(\unicode[STIX]{x1D719})=\int _{I_{\unicode[STIX]{x1D6FE}}(\mathbb{A})\backslash G(\mathbb{A})}\unicode[STIX]{x1D719}(x^{-1}\unicode[STIX]{x1D6FE}x)\,d\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FE}}(x).\end{eqnarray}$$

(See, for example, [Reference KottwitzKot86, § 9.1] for this formulation of the geometric side.) The measure $\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FE}}$ above denotes the quotient measure $d\unicode[STIX]{x1D707}_{G}^{\text{can}}/d\unicode[STIX]{x1D707}_{I_{\unicode[STIX]{x1D6FE}}}^{\text{can}}$.

We shall bound the terms in $I_{\text{geom}}(\unicode[STIX]{x1D719},\unicode[STIX]{x1D707}_{G}^{\text{can}})$ using the Weyl discriminant. For any $v$ and $\unicode[STIX]{x1D6FE}\in G_{v}$, this is defined by

$$\begin{eqnarray}D_{v}(\unicode[STIX]{x1D6FE})=|\!\det (1-\text{Ad}(\unicode[STIX]{x1D6FE})_{|\mathfrak{g}_{v}/\mathfrak{g}_{v,\unicode[STIX]{x1D6FE}}})|_{v},\end{eqnarray}$$

where $\mathfrak{g}_{v,\unicode[STIX]{x1D6FE}}$ denotes the centralizer of $\unicode[STIX]{x1D6FE}$ in $\mathfrak{g}_{v}$. If $S$ is any set of places and $\unicode[STIX]{x1D6FE}\in G(F)$, we define $D_{S}(\unicode[STIX]{x1D6FE})=\prod _{v\in S}D_{v}(\unicode[STIX]{x1D6FE})$ and $D^{S}(\unicode[STIX]{x1D6FE})=\prod _{v\notin S}D_{v}(\unicode[STIX]{x1D6FE})$.

6.3 Bounding volumes

We again let $G$ denote a group satisfying the conditions of Theorem 1.1. Throughout the rest of this section, $A,B$ and $C$ will denote sufficiently large positive constants that may vary from line to line, and will never depend on a choice of place of $F$.

In preparation for the following result, we introduce some additional notation. Given $\unicode[STIX]{x1D705}\geqslant 0$ and a finite set of finite places $T$ away from $D$, we write $U_{T}^{{\leqslant}\unicode[STIX]{x1D705}}$ for the open compact subset $\text{supp}\,{\mathcal{H}}_{T}^{{\leqslant}\unicode[STIX]{x1D705}}$. Furthermore, we denote by ${\mathcal{C}}_{T}^{{\leqslant}\unicode[STIX]{x1D705}}$ the set of $G(F)$-conjugacy classes of elements in $G(F)-Z(F)$ whose $G(\mathbb{A})$-conjugacy classes have non-empty intersection with $K^{T}\cdot U_{T}^{{\leqslant}\unicode[STIX]{x1D705}}\cdot U_{\infty }$, where $U_{\infty }=\{g\in G(F_{\infty }):d(g,H_{v_{0}})<1\}$.

Proposition 6.1. There exist $A,B>0$ such that for any $\unicode[STIX]{x1D705}\geqslant 0$, any finite set of finite places $T$ away from $D$, and any $\{\unicode[STIX]{x1D6FE}\}\in {\mathcal{C}}_{T}^{{\leqslant}\unicode[STIX]{x1D705}}$, we have

$$\begin{eqnarray}\unicode[STIX]{x1D707}_{I_{\unicode[STIX]{x1D6FE}}}^{\text{can}}(I_{\unicode[STIX]{x1D6FE}}(F)\backslash I_{\unicode[STIX]{x1D6FE}}(\mathbb{A}))\ll q_{T}^{A\unicode[STIX]{x1D705}+B}.\end{eqnarray}$$

The implied constant depends only on $G$.

Proof. Let $S_{D}$ denote the set of places dividing $D$. Put $S_{\unicode[STIX]{x1D6FE}}=\{v\notin S_{D}\cup \infty \cup T:D_{v}(\unicode[STIX]{x1D6FE})\neq 1\}$. We begin by noting that for any $\{\unicode[STIX]{x1D6FE}\}\in {\mathcal{C}}_{T}^{{\leqslant}\unicode[STIX]{x1D705}}$ we have

(19)$$\begin{eqnarray}\unicode[STIX]{x1D707}_{I_{\unicode[STIX]{x1D6FE}}}^{\text{can}}(I_{\unicode[STIX]{x1D6FE}}(F)\backslash I_{\unicode[STIX]{x1D6FE}}(\mathbb{A}))\ll q_{S_{\unicode[STIX]{x1D6FE}}}^{B}q_{T}^{B},\end{eqnarray}$$

where the implied constant depends only on $G$. Indeed, from the proof of [Reference Shin and TemplierST16, Corollary 6.16] we have

$$\begin{eqnarray}\unicode[STIX]{x1D716}(M_{I_{\unicode[STIX]{x1D6FE}}})L(M_{I_{\unicode[STIX]{x1D6FE}}}^{\vee }(1))\ll \mathop{\prod }_{v\in \text{Ram}(I_{\unicode[STIX]{x1D6FE}})}q_{v}^{B}\ll q_{S_{\unicode[STIX]{x1D6FE}}}^{B}q_{T}^{B},\end{eqnarray}$$

where $\text{Ram}(I_{\unicode[STIX]{x1D6FE}})$ is the set of finite places where $I_{\unicode[STIX]{x1D6FE}}$ is ramified. Note that the last bound follows from the inclusion $\text{Ram}(I_{\unicode[STIX]{x1D6FE}})\subset S_{D}\cup S_{\unicode[STIX]{x1D6FE}}\cup T$, which follows from [Reference KottwitzKot86, Proposition 7.1]. Moreover, from the definition of the local archimedean factors in [Reference GrossGro97, (7.1) and (7.2)], combined with [Reference Shin and TemplierST16, Proposition 6.3], we find that $L_{\infty }(M_{I_{\unicode[STIX]{x1D6FE}}}^{\vee }(1))\ll 1$, the implied constant depending only on $G$. Finally, Corollary 8.12 and Lemma 8.13 of [Reference Shin and TemplierST16] imply $\unicode[STIX]{x1D707}_{I_{\unicode[STIX]{x1D6FE}}}^{\text{Tam}}(I_{\unicode[STIX]{x1D6FE}}(F)\backslash I_{\unicode[STIX]{x1D6FE}}(\mathbb{A}))\ll 1$. By combining these estimates with (18) we obtain (19).

Now for $\{\unicode[STIX]{x1D6FE}\}\in {\mathcal{C}}_{T}^{{\leqslant}\unicode[STIX]{x1D705}}$ we have

$$\begin{eqnarray}D_{v}(\unicode[STIX]{x1D6FE})\leqslant \left\{\begin{array}{@{}ll@{}}q_{v}^{A\unicode[STIX]{x1D705}+B}\quad & \text{ for }v\in T,\\ C\quad & \text{ for }v\mid \infty ,\\ 1\quad & \text{ for }v\notin T\cup \infty .\end{array}\right.\end{eqnarray}$$

From this and the product formula we deduce that

(20)$$\begin{eqnarray}1=\mathop{\prod }_{v\in T}D_{v}(\unicode[STIX]{x1D6FE})\mathop{\prod }_{v\in S_{D}\cup \infty }D_{v}(\unicode[STIX]{x1D6FE})\mathop{\prod }_{v\in S_{\unicode[STIX]{x1D6FE}}}D_{v}(\unicode[STIX]{x1D6FE})\ll q_{T}^{A\unicode[STIX]{x1D705}+B}q_{S_{\unicode[STIX]{x1D6FE}}}^{-1},\end{eqnarray}$$

since $D_{v}(\unicode[STIX]{x1D6FE})\leqslant q_{v}^{-1}$ for every $v\in S_{\unicode[STIX]{x1D6FE}}$. Inserting this into (19) gives the proposition.◻

6.4 Bounding adelic orbital integrals

We shall prove Theorem 3.1 by inserting test functions of the form $\unicode[STIX]{x1D719}=\mathbf{1}_{N}^{T}\,\otimes \,k_{T}\,\otimes \,k_{\unicode[STIX]{x1D709}}$ into the trace formula, where $k_{T}\in {\mathcal{H}}_{T}$, $k_{\unicode[STIX]{x1D709}}\in {\mathcal{H}}_{\infty }$, and $\mathbf{1}_{N}^{T}$ was defined in (5). We now estimate the orbital integrals of these functions.

Proposition 6.2. There are constants $A,B,\unicode[STIX]{x1D6FF},\unicode[STIX]{x1D702}>0$ such that the following statement holds. Let $T$ be a finite set of finite places away from $N$ and $D^{\prime }$. Let $\unicode[STIX]{x1D709}\in \mathfrak{a}^{\ast }$ and $\unicode[STIX]{x1D705}\geqslant 0$. Then, for $k_{T}\in {\mathcal{H}}_{T}^{{\leqslant}\unicode[STIX]{x1D705}}$, any $k_{\unicode[STIX]{x1D709}}\in {\mathcal{H}}_{\infty }$ satisfying $(\mathsf{P}_{\infty })$, and any $\unicode[STIX]{x1D6FE}\in G(F)-Z(F)$, we have

$$\begin{eqnarray}O_{\unicode[STIX]{x1D6FE}}(\mathbf{1}_{N}^{T}\,\otimes \,k_{T}\,\otimes \,k_{\unicode[STIX]{x1D709}})\ll N^{-\unicode[STIX]{x1D6FF}}\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D709})(1+\Vert \unicode[STIX]{x1D709}\Vert )^{-\unicode[STIX]{x1D702}}q_{T}^{A\unicode[STIX]{x1D705}+B}\Vert k_{T}\Vert _{\infty },\end{eqnarray}$$

where the implied constant depends only on $G$, $\unicode[STIX]{x1D703}$ and $K$.

Proof. We may write $k_{T}$ as a linear combination of at most $q_{T}^{A\unicode[STIX]{x1D705}+B}$ terms of the form $\bigotimes _{v\in T}\,1_{K_{v}\unicode[STIX]{x1D707}_{v}(\unicode[STIX]{x1D71B}_{v})K_{v}}\in {\mathcal{H}}_{T}^{{\leqslant}\unicode[STIX]{x1D705}}$, whose sup norms are all bounded by $\Vert k_{T}\Vert _{\infty }$. We may therefore assume that $k_{T}$ is a multiple of $\bigotimes _{v\in T}\,1_{K_{v}\unicode[STIX]{x1D707}_{v}(\unicode[STIX]{x1D71B}_{v})K_{v}}$. This assumption implies that the orbital integral factorizes as $\prod _{v}O_{\unicode[STIX]{x1D6FE}}(\unicode[STIX]{x1D719}_{v})$, where for any $\unicode[STIX]{x1D6FE}_{v}\in G(F_{v})$ we have

$$\begin{eqnarray}O_{\unicode[STIX]{x1D6FE}_{v}}(\unicode[STIX]{x1D719}_{v})=\int _{I_{\unicode[STIX]{x1D6FE}_{v}}(F_{v})\backslash G(F_{v})}\unicode[STIX]{x1D719}_{v}(x_{v}^{-1}\unicode[STIX]{x1D6FE}_{v}x_{v})\,d\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FE},v}(x_{v})\end{eqnarray}$$

and $\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FE},v}=\unicode[STIX]{x1D707}_{G,v}^{\text{can}}/\unicode[STIX]{x1D707}_{I_{\unicode[STIX]{x1D6FE}_{v}},v}^{\text{can}}$. It therefore suffices to work place by place.

In [Reference Shin and TemplierST16, Theorem 14.1] it is shown that, if we choose $D^{\prime }$ to be a sufficiently large (ineffective) multiple of $D$ depending only on $G$, then

(21)$$\begin{eqnarray}|O_{\unicode[STIX]{x1D6FE}}(k_{T})|\leqslant q_{T}^{A\unicode[STIX]{x1D705}+B}D_{T}(\unicode[STIX]{x1D6FE})^{-1/2}\Vert k_{T}\Vert _{\infty }.\end{eqnarray}$$

We may prove the following bound for the integral at infinity using the results of § 7.

Lemma 6.3. For any $0<\unicode[STIX]{x1D702}<1/2$, we have the bound

$$\begin{eqnarray}O_{\unicode[STIX]{x1D6FE}}(k_{\unicode[STIX]{x1D709}})\ll \unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D709})(1+\Vert \unicode[STIX]{x1D709}\Vert )^{-\unicode[STIX]{x1D702}}D_{\infty }(\unicode[STIX]{x1D6FE})^{-3/4}.\end{eqnarray}$$

Proof. Let $G_{\text{cpt}}$ and $G_{\text{cpt},v_{0}}$ be the groups associated with $G_{\infty }$ and $G_{v_{0}}$ by Definition 7.2. We begin by showing that, as a consequence of Proposition 7.3, the following statement holds. Let $0<\unicode[STIX]{x1D702}<1/2$, and let $k_{\unicode[STIX]{x1D709}}\in {\mathcal{H}}_{\infty }$ satisfy property $(\mathsf{P}_{\infty })$. Then

(22)$$\begin{eqnarray}O_{\unicode[STIX]{x1D6FE}}(k_{\unicode[STIX]{x1D709}})\ll \unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D709})(1+\Vert \unicode[STIX]{x1D709}\Vert )^{-\unicode[STIX]{x1D702}}D_{\infty }(\unicode[STIX]{x1D6FE})^{-3/4}\end{eqnarray}$$

for every semisimple $\unicode[STIX]{x1D6FE}\in G_{\infty }-G_{\text{cpt}}$.

To see how (22) follows from Proposition 7.3, first note that for any non-negative $f\in C_{c}^{\infty }(G_{\infty }^{0})$ such that $f=1$ on the support of $k_{\unicode[STIX]{x1D709}}$, we have, using property $(\mathsf{P}_{\infty })$,

$$\begin{eqnarray}\displaystyle k_{\unicode[STIX]{x1D709}}(g) & \ll & \displaystyle \unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D709})(1+\Vert \unicode[STIX]{x1D709}\Vert d(g,H_{v_{0}}))^{-1/2}f(g)\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D709})(1+\Vert \unicode[STIX]{x1D709}\Vert d(g,H_{v_{0}}))^{-\unicode[STIX]{x1D702}}f(g)\nonumber\\ \displaystyle & \ll & \displaystyle \unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D709})(1+\Vert \unicode[STIX]{x1D709}\Vert )^{-\unicode[STIX]{x1D702}}\,d(g,H_{v_{0}})^{-\unicode[STIX]{x1D702}}f(g).\nonumber\end{eqnarray}$$

Thus

$$\begin{eqnarray}O_{\unicode[STIX]{x1D6FE}}(k_{\unicode[STIX]{x1D709}})\ll \unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D709})(1+\Vert \unicode[STIX]{x1D709}\Vert )^{-\unicode[STIX]{x1D702}}O_{\unicode[STIX]{x1D6FE}}(fd(\cdot ,H_{v_{0}})^{-\unicode[STIX]{x1D702}}),\end{eqnarray}$$

to which we may apply (27). Indeed, since $Z(G)$ is finite, the function $\Vert X(g)\Vert _{0}$ used there satisfies $\Vert X(g)\Vert _{0}\ll d(g,H_{v_{0}})$.

The proof of Lemma 6.3 then follows from (22) once we have verified that a non-central element $\unicode[STIX]{x1D6FE}\in G(F)$ cannot lie in $G_{\text{cpt}}$. Because $G_{\infty }=\prod _{v|\infty }G_{v}$, we have $G_{\text{cpt}}=G_{\text{cpt},v_{0}}\times \prod _{v\neq v_{0}}G_{v}$, and so it suffices to verify that $\unicode[STIX]{x1D6FE}_{v_{0}}\notin G_{\text{cpt},v_{0}}$.

In the case at hand, $G_{\text{cpt},v_{0}}$ is a normal subgroup of $G_{v_{0}}$ which, as $G$ is semisimple, is compact. If we let $H^{+}$ be the fixed point set of $\unicode[STIX]{x1D703}$ in $G$ (we write $H^{+}$ to distinguish it from its identity component $H$), then $H_{v_{0}}^{+}$ is a maximal compact subgroup of $G_{v_{0}}$, and so we have $G_{\text{cpt},v_{0}}\subset gH_{v_{0}}^{+}g^{-1}$ for all $g\in G_{v_{0}}$. Thus, if $\unicode[STIX]{x1D6FE}_{v_{0}}\in G_{\text{cpt},v_{0}}$, we have $\unicode[STIX]{x1D6FE}\in gH^{+}(F)g^{-1}$ for all $g\in G(F)$. The group

$$\begin{eqnarray}\mathop{\bigcap }_{g\in G(F)}gH^{+}g^{-1}\end{eqnarray}$$

is a proper $F$-subgroup of $G$; it is normal, since its normalizer contains the Zariski-dense set $G(F)$. Our assumption that $G$ is $F$-almost simple implies that it must be contained in $Z(G)$. We therefore have $\unicode[STIX]{x1D6FE}\in Z(F)$, a contradiction.◻

It remains then to address the size of the orbital integral at finite places away from $T$. This is provided by the following result. We retain the set-up from the statement of Proposition 6.2; in particular, we recall the notation $\mathbf{1}_{N}^{T}$ from (5).

Lemma 6.4. There are constants $\unicode[STIX]{x1D6FF},A,B,C>0$ such that for $\{\unicode[STIX]{x1D6FE}\}\in {\mathcal{C}}_{T}^{{\leqslant}\unicode[STIX]{x1D705}}$ we have

(23)$$\begin{eqnarray}O_{\unicode[STIX]{x1D6FE}}(\mathbf{1}_{N}^{T})\ll N^{-\unicode[STIX]{x1D6FF}}q_{T}^{A\unicode[STIX]{x1D705}+B}D^{T\infty }(\unicode[STIX]{x1D6FE})^{-C}.\end{eqnarray}$$

Proof. Recall the sets $S_{D}$ and $S_{\unicode[STIX]{x1D6FE}}$ from the proof of Proposition 6.1. Let $S_{N}$ denote the set of places dividing $N$.

• ∙ If $v\notin S_{D}\cup S_{N}\cup S_{\unicode[STIX]{x1D6FE}}\cup T\cup \infty$, then $K_{v}(N)K_{H_{v}}=K_{v}$ and we have $O_{\unicode[STIX]{x1D6FE}}(\mathbf{1}_{K_{v}})=1$; see, for example, [Reference KottwitzKot86, Corollary 7.3].

• ∙ If $v\in S_{D}$, then $K_{v}(N)K_{H_{v}}=K_{v}K_{H_{v}}$ and a general bound of Kottwitz [Reference Shin and TemplierST16, Theorem 13.1] establishes that $O_{\unicode[STIX]{x1D6FE}}(\mathbf{1}_{K_{v}K_{H_{v}}})\,\ll _{v}\,D_{v}(\unicode[STIX]{x1D6FE})^{-1/2}$.

• ∙ If $v\in S_{N}\cup S_{\unicode[STIX]{x1D6FE}}$ we argue as follows.

We begin by estimating the orbital integrals at places $v\in S_{N}$, for which we will use as a critical input the work of Finis and Lapid [Reference Finis and LapidFL18]. As the setting of [Reference Finis and LapidFL18] is that of $\mathbb{Q}$-groups, we shall need to restrict scalars from $F$ to $\mathbb{Q}$ to properly invoke their results. We thus let $p$ denote the rational prime over which $v$ lies, and we note that all places lying over $p$ belong to $S_{N}$. We set $K_{p}=\prod _{v\mid p}K_{v}$, and $K_{p}(N)K_{H,p}=\prod _{v\mid p}K_{v}(N)K_{H,v}$. Factorize $N=\prod _{p|N}p^{n_{p}}$ and put $N_{p}=p^{n_{p}}$.

Let $\unicode[STIX]{x1D707}_{G,p}^{\text{can}}$ (respectively, $\unicode[STIX]{x1D707}_{I_{\unicode[STIX]{x1D6FE}},p}^{\text{can}}$) be the product measure on $G_{p}=\prod _{v\mid p}G_{v}$ (respectively, $I_{\unicode[STIX]{x1D6FE},p}=\prod _{v\mid p}I_{\unicode[STIX]{x1D6FE},v}$), and let $\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FE},p}$ be the quotient measure. Let $\text{Ad}:I_{\unicode[STIX]{x1D6FE},p}\backslash G_{p}\rightarrow G_{p}$ be the map $g\mapsto g^{-1}\unicode[STIX]{x1D6FE}g$, which is a proper map as $\unicode[STIX]{x1D6FE}$ is semisimple. By definition,

$$\begin{eqnarray}O_{\unicode[STIX]{x1D6FE}}(\mathbf{1}_{K_{p}(N)K_{H_{p}}})=\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FE},p}(\text{Ad}^{-1}(K_{p}(N)K_{H_{p}})).\end{eqnarray}$$

As $\text{Ad}^{-1}(K_{p})$ is compact and right-invariant under $K_{p}$, we may break it up into a finite number of orbits $\coprod \overline{g}_{i}K_{p}$, where $g_{i}\in G_{p}$ and $\overline{g}_{i}$ denotes its image in $I_{\unicode[STIX]{x1D6FE},p}\backslash G_{p}$. This gives

$$\begin{eqnarray}O_{\unicode[STIX]{x1D6FE}}(\mathbf{1}_{K_{p}(N)K_{H_{p}}})=\mathop{\sum }_{i}\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FE},p}(\overline{g}_{i}K_{p}\cap \text{Ad}^{-1}(K_{p}(N)K_{H_{p}})).\end{eqnarray}$$

Let $g\in G_{p}$ be such that $x:=g^{-1}\unicode[STIX]{x1D6FE}g\in K_{p}$. The fibers of the reduction map $gK_{p}\rightarrow \overline{g}K_{p}$ are the left orbits of $I_{\unicode[STIX]{x1D6FE},p}\cap gK_{p}g^{-1}$, so for any $K_{p}^{\prime }\subset K_{p}$ we have

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FE},p}(\overline{g}K_{p}\cap \text{Ad}^{-1}(K_{p}^{\prime })) & = & \displaystyle \frac{\unicode[STIX]{x1D707}_{G,p}^{\text{can}}(h\in gK_{p}:h^{-1}\unicode[STIX]{x1D6FE}h\in K_{p}^{\prime })}{\unicode[STIX]{x1D707}_{I_{\unicode[STIX]{x1D6FE}},p}^{\text{can}}(I_{\unicode[STIX]{x1D6FE},p}\cap gK_{p}g^{-1})}\nonumber\\ \displaystyle & = & \displaystyle \frac{\unicode[STIX]{x1D707}_{G,p}^{\text{can}}(k\in K_{p}:k^{-1}xk\in K_{p}^{\prime })}{\unicode[STIX]{x1D707}_{I_{\unicode[STIX]{x1D6FE}},p}^{\text{can}}(I_{\unicode[STIX]{x1D6FE},p}\cap gK_{p}g^{-1})}.\nonumber\end{eqnarray}$$

Taking the quotient of these with $K_{p}^{\prime }$ equal to $K_{p}$ and $K_{p}(N)K_{H_{p}}$, and using the fact that $\unicode[STIX]{x1D707}_{G,p}^{\text{can}}(K_{p})=1$ for all $v\notin S_{D}\cup \infty$, gives

$$\begin{eqnarray}\frac{\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FE},p}(\overline{g}K_{p}\cap \text{Ad}^{-1}(K_{p}(N)K_{H_{p}}))}{\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FE},p}(\overline{g}K_{p})}=\unicode[STIX]{x1D707}_{G,p}^{\text{can}}(k\in K_{p}:k^{-1}xk\in K_{p}(N)K_{H_{p}}).\end{eqnarray}$$

One can deduce from [Reference Finis and LapidFL18, Propositions 5.10 and 5.11] (see § 6.5 below) that there are constants $\unicode[STIX]{x1D716},\unicode[STIX]{x1D6FF}>0$ (independent of $p$) such that if $x\in K_{p}$ is conjugate to an element of $G(F)-Z(F)$ and satisfies $D_{p}(x)>N_{p}^{-\unicode[STIX]{x1D716}}$ then

(24)$$\begin{eqnarray}\unicode[STIX]{x1D707}_{G,p}^{\text{can}}(k\in K_{p}:k^{-1}xk\in K_{p}(N)K_{H,p})\ll N_{p}^{-\unicode[STIX]{x1D6FF}}.\end{eqnarray}$$

(In particular, the implied constant is independent of $p$.) On the other hand, $\sum _{i}\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FE},p}(\overline{g}_{i}K_{p})=\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FE},p}(\text{Ad}^{-1}(K_{p}))=O_{\unicode[STIX]{x1D6FE}}(\mathbf{1}_{K_{p}})$. Noting that $D_{p}(\unicode[STIX]{x1D6FE})=D_{p}(g_{i}^{-1}\unicode[STIX]{x1D6FE}g_{i})$ for all $i$, we deduce that for $D_{p}(\unicode[STIX]{x1D6FE})>N_{p}^{-\unicode[STIX]{x1D716}}$ we have

(25)$$\begin{eqnarray}O_{\unicode[STIX]{x1D6FE}}(\mathbf{1}_{K_{p}(N)K_{H_{p}}})\ll N_{p}^{-\unicode[STIX]{x1D6FF}}\mathop{\sum }_{i}\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FE},p}(\overline{g}_{i}K_{p})=N_{p}^{-\unicode[STIX]{x1D6FF}}O_{\unicode[STIX]{x1D6FE}}(\mathbf{1}_{K_{p}}).\end{eqnarray}$$

In the remaining range $D_{p}(\unicode[STIX]{x1D6FE})\leqslant N_{p}^{-\unicode[STIX]{x1D716}}$, we have

$$\begin{eqnarray}O_{\unicode[STIX]{x1D6FE}}(\mathbf{1}_{K_{p}(N)K_{H_{p}}})\leqslant O_{\unicode[STIX]{x1D6FE}}(\mathbf{1}_{K_{p}})\leqslant N_{p}^{-\unicode[STIX]{x1D6FF}}D_{p}(\unicode[STIX]{x1D6FE})^{-\unicode[STIX]{x1D6FF}/\unicode[STIX]{x1D716}}O_{\unicode[STIX]{x1D6FE}}(\mathbf{1}_{K_{p}}).\end{eqnarray}$$

Since $D_{v}(\unicode[STIX]{x1D6FE})\leqslant 1$ for all $v\notin T\cup \infty$, we may combine these as

$$\begin{eqnarray}\mathop{\prod }_{v\in S_{N}}O_{\unicode[STIX]{x1D6FE}}(\mathbf{1}_{K_{v}(N)K_{H_{v}}})\ll N^{-\unicode[STIX]{x1D6FF}}D_{S_{N}}(\unicode[STIX]{x1D6FE})^{-C}\mathop{\prod }_{v\in S_{N}}O_{\unicode[STIX]{x1D6FE}}(\mathbf{1}_{K_{v}}).\end{eqnarray}$$

Note that we may shrink $\unicode[STIX]{x1D6FF}$ to absorb the implied constant in (25) for $p$ sufficiently large, so that the implied constant above only depends on $G$ and not the number of factors of $N$.

We now return to the product of orbital integrals over all $v\in S_{N}\cup S_{\unicode[STIX]{x1D6FE}}$. Recalling that $O_{\unicode[STIX]{x1D6FE}}(\mathbf{1}_{K_{v}})=1$ for $v\notin S_{\unicode[STIX]{x1D6FE}}$, we have just shown that

$$\begin{eqnarray}\displaystyle \mathop{\prod }_{v\in S_{N}}O_{\unicode[STIX]{x1D6FE}}(\mathbf{1}_{K_{v}(N)K_{H_{v}}})\mathop{\prod }_{\substack{ v\in S_{\unicode[STIX]{x1D6FE}} \\ v\notin S_{N}}}O_{\unicode[STIX]{x1D6FE}}(\mathbf{1}_{K_{v}}) & \ll & \displaystyle N^{-\unicode[STIX]{x1D6FF}}D_{S_{N}}(\unicode[STIX]{x1D6FE})^{-C}\mathop{\prod }_{v\in S_{N}}O_{\unicode[STIX]{x1D6FE}}(\mathbf{1}_{K_{v}})\mathop{\prod }_{\substack{ v\in S_{\unicode[STIX]{x1D6FE}} \\ v\notin S_{N}}}O_{\unicode[STIX]{x1D6FE}}(\mathbf{1}_{K_{v}})\nonumber\\ \displaystyle & = & \displaystyle N^{-\unicode[STIX]{x1D6FF}}D_{S_{N}}(\unicode[STIX]{x1D6FE})^{-C}\mathop{\prod }_{v\in S_{\unicode[STIX]{x1D6FE}}}O_{\unicode[STIX]{x1D6FE}}(\mathbf{1}_{K_{v}}).\nonumber\end{eqnarray}$$

For $v\in S_{\unicode[STIX]{x1D6FE}}$, we proceed as follows. If $v\nmid D^{\prime }$ we again apply [Reference Shin and TemplierST16, Theorem 14.1] to get $O_{\unicode[STIX]{x1D6FE}}(\mathbf{1}_{K_{v}})\leqslant q_{v}^{B}D_{v}(\unicode[STIX]{x1D6FE})^{-1/2}$, and if $v\mid D^{\prime }$ we apply [Reference Shin and TemplierST16, Theorem 13.1] to get $O_{\unicode[STIX]{x1D6FE}}(\mathbf{1}_{K_{v}})\,\ll _{v}\,D_{v}(\unicode[STIX]{x1D6FE})^{-1/2}$. Combining these gives

$$\begin{eqnarray}\mathop{\prod }_{v\in S_{\unicode[STIX]{x1D6FE}}}O_{\unicode[STIX]{x1D6FE}}(\mathbf{1}_{K_{v}})\leqslant q_{S_{\unicode[STIX]{x1D6FE}}}^{B}D_{S_{\unicode[STIX]{x1D6FE}}}(\unicode[STIX]{x1D6FE})^{-C}.\end{eqnarray}$$

Since $\{\unicode[STIX]{x1D6FE}\}\in {\mathcal{C}}_{T}^{{\leqslant}\unicode[STIX]{x1D705}}$ we may invoke (20) to obtain

$$\begin{eqnarray}\mathop{\prod }_{v\in S_{\unicode[STIX]{x1D6FE}}}O_{\unicode[STIX]{x1D6FE}}(\mathbf{1}_{K_{v}})\leqslant q_{S_{\unicode[STIX]{x1D6FE}}}^{B}D_{S_{\unicode[STIX]{x1D6FE}}}(\unicode[STIX]{x1D6FE})^{-C}\ll q_{T}^{A\unicode[STIX]{x1D705}+B}D_{S_{\unicode[STIX]{x1D6FE}}}(\unicode[STIX]{x1D6FE})^{-C}.\end{eqnarray}$$

Putting these estimates together completes the proof of Lemma 6.4. ◻

Remark 6.5. As the authors point out in [Reference Shin and TemplierST16, Remark 7.4], the bound [Reference Shin and TemplierST16, Theorem 14.1] is uniform in the place $v\nmid D^{\prime }$ whereas the bound [Reference Shin and TemplierST16, Theorem 13.1] of Kottwitz applies to $v\mid D^{\prime }$ but is not uniform in $v$. As we allow the implied constant in Proposition 6.2 to depend on the group, this non-uniformity is not an issue.

Taken together (and using the product rule for the product of Weyl discriminants), the estimates in (21), Lemma 6.3 and Lemma 6.4 imply Proposition 6.2.◻