2.1 Brief structure theory of reductive groups

Let $T_0$ denote a fixed maximal F-split torus of G. Let $M_0$ denote the centralizer of $T_0$ in G, which is a minimal Levi subgroup. Let $\mathcal {L}$ denote the finite set of Levi subgroups defined over F containing $M_0$ . Given $M\in \mathcal {L}$ , let $\mathcal {L}(M) \subset \mathcal {L}$ denote the (finite) set of Levi subgroups containing M and $\mathcal {P}(M)$ denote the (finite) set of parabolic subgroups defined over F with Levi subgroup M. We also fix a minimal parabolic subgroup $P_0 \supset M_0$ .

Let $T_M$ denote the split part of the identity component of the centre of M. We denote ${W(M) := N_{G}(M)/M}$ , where $N_G(M)$ denotes the normalizer of M in G. We denote the Weyl group of $(G,T_0)$ by $W_0 := W(M_0)$ . We can identify $W(M)$ as a subgroup of $W_0$ .

Let $\mathfrak {a}_M^*$ denote the $\mathbb {R}$ -vector space spanned by the lattice $X^*(M)$ of F-rational characters of M, and let $\mathfrak {a}_{M,\mathbb {C}}^* := \mathfrak {a}_M^*\otimes _{\mathbb {R}} \mathbb {C}$ . For $M_1 \subset M_2$ , it holds that $\mathfrak {a}_{M_2}^* \subset \mathfrak {a}_{M_1}^*$ with a well-defined complement $\mathfrak {a}_{M_1}^* = \mathfrak {a}_{M_2}^*\oplus (\mathfrak {a}_{M_1}^{M_2})^*$ . Let $\mathfrak {a}_M$ denote the dual space of $\mathfrak {a}_M^*$ , and we fix a pairing $\langle ,\rangle $ between $\mathfrak {a}_M$ and $\mathfrak {a}_M^*$ . For $P\in \mathcal {P}(M)$ , we write $\mathfrak {a}_P =\mathfrak {a}_M$ .

We choose a $W_0$ -invariant inner product on $\mathfrak {a}_0:= \mathfrak {a}_{M_0}$ , which also defines measures on $\mathfrak {a}_M$ and $\mathfrak {a}_M^*$ for all $M\in \mathcal {L}$ . We normalize all measures as in [Reference Arthur6, Section 7], similar to [Reference Finis, Lapid and Müller21].

For $P\in \mathcal {P}(M)$ , we write $M_P=M$ . Let $N_P$ denote the unipotent radical of P. Let $\Sigma _P \subset (\mathfrak {a}_{P}^{G})^*$ denote the set of roots of $T_M$ on the Lie algebra of $N_P$ . Let $\Delta _P$ denote the subset of simple roots of $\Sigma _P$ . Let $\Sigma _M := \cup _{P\in \mathcal {P}(M)}\Sigma _P$ . Given two parabolics $P,Q\in \mathcal {P}(M)$ , we say that P and Q are adjacent along $\alpha \in \Sigma _M$ and write $P|^{\alpha } Q$ if $\Sigma _P = \Sigma _Q \setminus \{-\alpha \} \cup \{\alpha \}$ .

Let $A_0$ denote the identity component of $T_0(\mathbb {R})$ . We let $A_M := A_0 \cap T_M(\mathbb {R})$ . Let $H_M\colon M(\mathbb {A})\to \mathfrak {a}_M$ denote the natural homomorphism defined by

$$\begin{align*}e^{\langle \chi, H_M(m) \rangle} := |\chi(m)|_{\mathbb{A}} = \prod_v |\chi(m_v)|_v \end{align*}$$

for any $\chi \in X^*(M)$ . Here, $|\cdot |_{\mathbb {A}}$ denotes the homomorphism from $\mathbb {A}^\times \to \mathbb {R}^+$ given by $|\cdot |_{\mathbb {A}}:=\prod _v|\cdot |_v$ , where $|\cdot |_v$ denotes the usual absolute value on $F_v$ . Let $M(\mathbb {A})^1 \subset M(\mathbb {A})$ denote the kernel of $H_M$ . It holds that $M(\mathbb {A}) = A_M \times M(\mathbb {A})^1$ .

We fix a maximal compact subgroup $K_\infty $ of $G(F_\infty )$ and a maximal open-compact subgroup $K_f$ of $G(\mathbb {A}_f)$ . Finally, we fix the maximal compact subgroup $K_{\mathbb {A}} := K_\infty K_f$ of $G(\mathbb {A})$ and assume that it is admissible relative to $M_0$ , so that for every $P\in \mathcal {P}(M_0)$ , an Iwasawa decomposition $G(\mathbb {A}) = P(\mathbb {A})K_{\mathbb {A}}$ holds.

Given $M\in \mathcal {M}$ and $P\in \mathcal {P}(M)$ , let $H_P\colon G(\mathbb {A}) \to \mathfrak {a}_P$ denote the extension of $H_M$ to a left $N_P(\mathbb {A})$ and right $K_{\mathbb {A}}$ -invariant map.

3.1 Automorphic forms

Given $M\in \mathcal {L}$ , let

$$\begin{align*}L^2_{\mathrm{disc}}(M(F) \backslash M(\mathbb{A})^1) \cong L^2_{\mathrm{disc}}(M(F)A_M \backslash M(\mathbb{A})) \end{align*}$$

denote the discrete part of the spectrum of $L^2(M(F)A_M\backslash M(\mathbb {A}))$ , that is the Hilbert sum of all irreducible subrepresentations of $M(\mathbb {A})$ on the space.

Let $\Pi _2(M)$ denoteFootnote ⁴ the countable set of equivalence classes of irreducible unitary representation of $M(\mathbb {A})$ appearing in $L^2_{\mathrm {disc}}(M(F) \backslash M(\mathbb {A})^1)$ . Each element $\pi \in \Pi _2(M)$ is an abstract unitary representation of $M(\mathbb {A})$ . By abuse of notations, we consider each $\pi \in \Pi _2(M)$ also as a representation of $M(\mathbb {A})^1$ . By Flath’s tensor product theorem (see [Reference Flath23]), each $\pi \in \Pi _2(M)$ can be considered as a tensor product of local representations $\pi = \otimes _v \pi _v$ , where $\pi _v$ is a representation of $M(F_v)$ . We denote the finite part of the representation $\otimes _{v<\infty }\pi _v$ by $\pi _f$ , and the infinite part of the representation $\otimes _{v\mid \infty }\pi _v$ by $\pi _\infty $ .

Let $P\in \mathcal {P}(M)$ and $\mathcal {A}^2(P)$ denote the set of automorphic forms $\varphi $ on $N_P(\mathbb {A})M(F)\backslash G(\mathbb {A})$ that are square-integrable, that is $\varphi $ satisfies

$$\begin{align*}\delta_P^{-1/2}\varphi(\cdot k) \in L^2_{\mathrm{disc}}(A_MM(F) \backslash M(\mathbb{A})),\quad \forall k\in K_{\mathbb{A}}, \end{align*}$$

where $\delta _P$ is the modular character attached to P (in particular, $\varphi (amk)=\delta _P^{1/2}(a)\varphi (mk)$ for $a\in A_M$ , $m\in M(\mathbb {A})^1$ and $k\in K_{\mathbb {A}}$ ). Then $\mathcal {A}^2(P)$ is a pre-Hilbert space, with an inner product

$$\begin{align*}\langle\varphi_1,\varphi_2\rangle_{\mathcal{A}^2(P)} := \intop_{A_M M(F) N_P(\mathbb{A}) \backslash G(\mathbb{A})} \varphi_1(g) \overline{\varphi_2(g)} \,\mathrm{d} g. \end{align*}$$

Sometimes, we shorthand $\|\cdot \|_{\mathcal {A}^2(G)}$ as $\|\cdot \|_2$ . Let $\overline {\mathcal {A}^2(P)}$ denote the Hilbert space completion of $\mathcal {A}^2(P)$ . For $\pi \in \Pi _2(M)$ , we will also denote $\mathcal {A}^2_\pi (P)\subset \mathcal {A}^2(P)$ to be the subspace of $\varphi $ , such that $\delta _P^{-1/2}\varphi (\cdot k)$ lies in the $\pi $ -isotypic subspace of $L^2_{\mathrm {disc},\pi }(A_M M(F) \backslash M(\mathbb {A}))\subset L^2_{\mathrm {disc}}(A_M M(F) \backslash M(\mathbb {A}))$ for every $k\in K_{\mathbb {A}}$ .

Given $\lambda \in i(\mathfrak {a}_P^G)^*$ , let $\rho (P,\lambda ,\cdot )$ denote the unitary representation of $G(\mathbb {A})$ on $\overline {\mathcal {A}^2(P)}$ , given by

$$\begin{align*}\rho(P,\lambda,y)(\varphi)(x) := \varphi(xy)e^{\langle \lambda, H_P(xy)-H_P(x) \rangle},\quad y\in G(\mathbb{A}).\end{align*}$$

The representation $\rho (P,\lambda ,\cdot )$ is isomorphic to

$$\begin{align*}\mathrm{Ind}_{P(\mathbb{A})}^{G(\mathbb{A})}\left(L^2_{\mathrm{disc}}(A_M M(F) \backslash M(\mathbb{A}))\otimes e^{\langle \lambda, H_M(\cdot) \rangle}\right) \end{align*}$$

(see [Reference Finis, Lapid and Müller22, Section 4.2]). We remark that the action of every compact subgroup of $G(\mathbb {A})$ on $\mathcal {A}^2(P)$ does not depend on $\lambda $ .

Recall that a cuspidal datum $\chi $ is a $G(F)$ -conjugacy class of pairs $(L,\sigma )$ , where L is a Levi subgroup of G defined over F and $\sigma $ is a cuspidal representation of $L(\mathbb {A})$  (see [Reference Arthur6, Section 12] for a detailed discussion). We have a coarse spectral decomposition

$$\begin{align*}L^2_{\mathrm{disc}}(A_M M(F) \backslash M(\mathbb{A})) = \widehat{\bigoplus}_{\chi}L^2_{\mathrm{disc},\chi}(A_M M(F) \backslash M(\mathbb{A}))\end{align*}$$

(see [Reference Arthur6, (12.5)]). Here and elsewhere, $\widehat {\oplus }$ denotes a Hilbert space direct sum. We have a similar coarse spectral decomposition of $\mathcal {A}^2(P)$ (see [Reference Arthur6, p.66–67]). Let $\mathcal {A}^2_\chi (P)$ denote the $\chi $ -part of this spectral decomposition. Finally, we also write

$$ \begin{align*} L^2_{\mathrm{disc},\chi,\pi}(A_M M(F) \backslash M(\mathbb{A})) := L^2_{\mathrm{disc},\chi}(A_M M(F) \backslash M(\mathbb{A})) \cap L^2_{\mathrm{disc},\pi}(A_M M(F) \backslash M(\mathbb{A})) \end{align*} $$

and $\mathcal {A}^2_{\chi ,\pi }(P) := \mathcal {A}^2_{\chi }(P) \cap \mathcal {A}^2_{\pi }(P)$ .

3.2 Some discussion of the coarse spectral expansion

Since this topic is less standard, for the convenience of the reader, we add here a discussion of the coarse spectral decomposition.

We start with the group $\mathrm {GL}_n$ , in which case, the coarse spectral decomposition is well understood. Thanks to the multiplicity one theorem, each cuspidal representation $\sigma $ of a Levi subgroup L appears with multiplicity one, that is

$$\begin{align*}\dim \mathrm{Hom}_{L(\mathbb{A})}(\sigma, L^2(A_L L(F) \backslash L(\mathbb{A}))) \le 1. \end{align*}$$

Next, let us describe some of the results of [Reference Mœglin and Waldspurger44]. First, we recall that given a cuspidal representation $\sigma $ of $\mathrm {GL}_a(\mathbb {A})$ and $b>0$ , one may construct a Speh representation $\operatorname {Speh}(\sigma ,b)$ of $\mathrm {GL}_{ab}(\mathbb {A})$ , which is the unique irreducible quotient of the induction of $\sigma ^{\otimes b}$ from the Levi subgroup $\mathrm {GL}_a(\mathbb {A})^b$ and twisted by the character $\mathrm {GL}_a(\mathbb {A})^b\ni (g_1,g_2,\dots ,g_b)\mapsto |\det (g_1)|^{(b-1)/2}|\det (g_2)|^{(b-3)/2}\dots |\det (g_b)|^{(1-b)/2}$ . It holds that $\operatorname {Speh}(\sigma ,b) \in \Pi _2(\mathrm {GL}_{ab})$ , and each representation $\pi \in \Pi _2(\mathrm {GL}_n)$ is isomorphic to a unique Speh representation $\operatorname {Speh}(\sigma ,b)$ for some $b\mid n$ and cuspidal representation $\sigma $ of $\mathrm {GL}_{n/b}(\mathbb {A})$ . Note that the $b=1$ case corresponds to the cuspidal representations in $\Pi _2(\mathrm {GL}_n)$ .

Now, if M is a Levi subgroup of $\mathrm {GL}_n$ of a parabolic P, for each $\pi \in \Pi _2(M)$ , there is a unique cuspidal datum $\chi $ , such that $\mathcal {A}^2_{\chi ,\pi }(P)\ne \{0\}$ , and the multiplicity of $\pi $ in the space $L^2_{\mathrm {disc}}(A_MM(F) \backslash M(\mathbb {A}))$ is equal to $1$ . More precisely, if $M \cong \mathrm {GL}_{n_1}\times \dots \times \mathrm {GL}_{n_k}$ is the standard Levi subgroup, and $\pi = \pi _1 \otimes \dots \otimes \pi _k$ , then one can write (uniquely) $n_i = a_i b_i$ , find a cuspidal representation $\sigma _i \in \Pi _2(\mathrm {GL}_{a_i})$ and write $\pi _i = \operatorname {Speh}(\sigma _i,b_i)$ . In this case, let $L:=\mathrm {GL}_{a_1}^{b_1}\times \dots \times \mathrm {GL}_{a_k}^{b_k}$ and $\sigma := \sigma _1^{\otimes b_1}\otimes \dots \otimes \sigma _k^{\otimes b_k} \in \Pi _2(L)$ be cuspidal. If $\chi $ is the cuspidal datum corresponding to $(L,\sigma )$ , then $\mathcal {A}^2_{\chi ,\pi }(P)\ne \{0\}$ .

The procedure can be reversed, but one should be a bit careful about the fact that cuspidal datum is only defined up to equivalence. In general, given a Levi L and $\sigma \in \Pi _2(L)$ cuspidal, let $\chi :=(L,\sigma )$ be the cuspidal datum and P be a standard parabolic containing L as a Levi subgroup. Then there are at most $n_P:= |\sum _{P'\sim P} W(\mathfrak {a}_{P},\mathfrak {a}_{P'})|\le |W_0|=n!$ other pairs $(L',\sigma ')$ of Levi subgroups $L'\supset M_0$ and cuspidal representations $\sigma '\in \Pi _2(L')$ in $\chi $ (see [Reference Arthur6, Section 12]). Given such a pair $(L,\sigma )$ and a Levi subgroup $L\subset M$ , we get at most one residual representation in $L^2_{\mathrm {disc},\chi }(A_M M(F) \backslash M(\mathbb {A}))$ . All in all, there are at most $n!$ pairs of standard parabolics P and $\pi \in \Pi _2(M_P)$ , such that $\mathcal {A}^2_{\chi ,\pi }(P)\ne \{0\}$ .

For groups that are different than $\mathrm {GL}_n$ , the situation becomes much more involved. An interesting case is the group $G=G_2$ . By [Reference Mœglin and Waldspurger45, Appendix III], for $\chi = (M_0,\mathrm {triv})$ (the torus and the trivial representation of it), there are infinitely many different $\pi \in \Pi _2(G)$ , such that

$$\begin{align*}L^2_{\mathrm{disc},\chi,\pi}(A_G G(F) \backslash G(\mathbb{A}))\ne \{0\} \end{align*}$$

(see also [Reference Kim35, Reference Žampera55] for complete results about the residual spectrum in this case). Moreover, by [Reference Gan, Gurevich and Jiang24], for some $\pi \in \Pi _2(G)$ , there are two different cuspidal data $\chi _1,\chi _2$ , such that

$$\begin{align*}L^2_{\mathrm{disc},\chi_i,\pi}(A_G G(F) \backslash G(\mathbb{A}))\ne \{0\}, \end{align*}$$

with $\chi _1$ associated with the subgroup G and $\chi _2$ associated with a proper parabolic subgroup. Finally, the multiplicity of $\pi $ in $L^2_{\mathrm {disc},\chi _1,\pi }(A_G G(F) \backslash G(\mathbb {A}))$ can be arbitrarily large.

3.3 Eisenstein series

Given $\varphi \in \mathcal {A}^2(P)$ and $\lambda \in (\mathfrak {a}_{P,\mathbb {C}}^G)^*$ , we define a corresponding Eisenstein series, as

$$\begin{align*}\mathrm{Eis}(\varphi,\lambda)(g) := \sum_{\gamma \in P(F) \backslash G(F)} \varphi(\gamma g) e^{\langle \lambda, H_P(\gamma g)\rangle }. \end{align*}$$

The above converges absolutely if $\lambda $ is sufficiently dominantFootnote ⁵ . One can then, via Langlands’ deep work (see, e.g. [Reference Langlands36]), meromorphically continue $\mathrm {Eis}(\varphi ,\lambda )$ to all $\lambda \in (\mathfrak {a}_{P,\mathbb {C}}^G)^*$ .

It holds that the Eisenstein series intertwines the representation $\rho (P,\lambda ,\cdot )$ . Namely, it holds that

$$\begin{align*}\mathrm{Eis}(\varphi,\lambda)(gy) = \mathrm{Eis}(\rho(P,\lambda,y)\varphi,\lambda)(g), \end{align*}$$

for $y\in G(\mathbb {A})$ .

3.4 Truncation

Recall the minimal parabolic subgroup $P_0 \in \mathcal {P}(M_0)$ . Given $T\in \mathfrak {a}_{M_0}^G$ , let

$$\begin{align*}d(T):= \min_{\alpha \in \Delta_{P_0}}\{\alpha(T)\}. \end{align*}$$

Given T with $d(T)$ sufficiently large, Arthur (following Langlands; see [Reference Arthur6, Section 13]) defined certain truncation operator $\Lambda ^T$ , which acts on locally integrable functions on $G(F) \backslash G(\mathbb {A})^1$ . We do not need the exact definition of $\Lambda ^T$ but just the following properties of it.

Let f be a locally integrable function on $G(F)\backslash G(\mathbb {A})^1$ .

1. 1. $\Lambda ^Tf$ is rapidly decaying. In particular, if $\lambda \in (\mathfrak {a}_{P,\mathbb {C}}^G)^*$ is away from any pole of $\mathrm {Eis}(\varphi ,\lambda )$ , then $\Lambda ^T\mathrm {Eis}(\varphi ,\lambda )$ is square-integrable.

2. 2. Given any compact $\Omega \subset G(F)\backslash G(\mathbb {A})^1$ , there is a $C(\Omega )>0$ , such that for every $T\in \mathfrak {a}_{M_0}^G$ with $d(T)>C(\Omega )$ , it holds that $\Lambda ^Tf\mid _\Omega =f\mid _\Omega $ (see [Reference Lapid40, Lemma 6.2]).

3. 3. When $\alpha (T)\le \alpha (T')$ for every $\alpha \in \Delta _{P_0}$ , it holds that $\Lambda ^{T'}\Lambda ^{T}=\Lambda ^T$ and $\|\Lambda ^{T'}f\|_2\ge \|\Lambda ^T f\|_2$ (see [Reference Finis and Lapid20, Section 3.3]).

3.5 Intertwining operators

We follow [Reference Finis, Lapid and Müller22, Section 4.2] to define intertwining operators and to record their relevant properties.

For $P,Q \in \mathcal {P}(M)$ and $\lambda \in \mathfrak {a}^*_{M,\mathbb {C}}$ , we define the standard intertwining operator

$$\begin{align*}M_{Q|P}(\lambda)\colon \mathcal{A}^2(P) \to \mathcal{A}^2(Q) \end{align*}$$

by

(3.1) $$ \begin{align} (M_{Q|P}(\lambda)\varphi)(x) := \intop_{N_Q(\mathbb{A})\cap N_P(\mathbb{A}) \backslash N_Q(\mathbb{A})}\varphi(nx) e^{\langle \lambda, H_P(nx) - H_Q(x)\rangle} \,\mathrm{d} n. \end{align} $$

The above integral converges absolutely for sufficiently dominant $\lambda $ . It can then be meromorphically continued for all $\lambda \in \mathfrak {a}^*_{M,\mathbb {C}}$ . It can be checked that $M_{Q\mid P}(\lambda )^{-1}=M_{P\mid Q}(\lambda )$ and for $\lambda \in i(\mathfrak {a}_M^G)^*$ the operator $M_{Q\mid P}(\lambda )$ is unitary.

For each $\pi \in \Pi _2(M)$ , we denote the restriction of $M_{Q\mid P}(\lambda )$ on $\mathcal {A}^2_\pi (P)$ by $M_{Q\mid P}(\pi ,\lambda ):\mathcal {A}^2_\pi (P)\to \mathcal {A}^2_\pi (Q)$ . On the other hand, we can define an intertwiner from

$$\begin{align*}M^{\mathrm{I}}_{Q\mid P}(\lambda,\pi):\mathrm{Ind}_{P(\mathbb{A})}^{G(\mathbb{A})}\pi\to\mathrm{Ind}_{Q(\mathbb{A})}^{G(\mathbb{A})}\pi\end{align*}$$

that can be densely defined on K-finite and $Z\left (\mathrm {Lie}(G(F_\infty ))\otimes _{\mathbb {R}}\mathbb {C}\right )$ -finite subspace of $\mathrm {Ind}_{P(\mathbb {A})}^{G(\mathbb {A})}\pi $ as in (3.1) (first, for sufficiently dominant $\lambda $ , then by meromorphic continuation). Then we have a unique canonical isomorphism of $G(\mathbb {A}_f)\times \left (\mathrm {Lie}(G(F_\infty ))\otimes _{\mathbb {R}}\mathbb {C},K_\infty \right )$ -modules (see [Reference Finis, Lapid and Müller22, Section 4.2])

$$\begin{align*}j_P\,\colon\, \mathrm{Hom}_{M(\mathbb{A})}\left(\pi,L^2(A_MM(F) \backslash M(\mathbb{A}))\right)\otimes \mathrm{Ind}_{P(\mathbb{A})}^{G(\mathbb{A})}\pi \to \mathcal{A}_\pi^2(P) \end{align*}$$

(similarly, $j_Q$ ) that can be characterized by

$$\begin{align*}j_Q\circ\left[\mathrm{Id}\otimes M^{\mathrm{I}}_{Q\mid P}(\lambda,\pi)\right]=M_{Q\mid P}(\lambda,\pi)\circ j_P\end{align*}$$

(see [Reference Müller48, (2.3)] and discussion around it).

Endowing $\mathrm {Hom}_{M(\mathbb {A})}\left (\pi ,L^2(A_MM(F) \backslash M(\mathbb {A}))\right )$ with the inner product $\langle f_1,f_2\rangle = f_2^*f_1$ , where $*$ denotes adjoint (notice that by Schur’s lemma, $f_2^* f_1$ is a scalar), we note that $j_P$ is an isometry.

Now, in the rest of this subsection, let $P|^\alpha Q$ for some $\alpha \in \Sigma _M$ . In this case, $M_{Q\mid P}(\lambda )$ depends only upon $\langle \lambda ,\alpha ^\vee \rangle $ . Let $\varpi \in \mathfrak {a}_M^*$ be such that $\langle \varpi ,\alpha ^\vee \rangle =1$ . We have

(3.2) $$ \begin{align} M_{Q|P}(s\varpi)\mid_{\mathcal{A}^2_\pi(P)} \circ j_P = n_\alpha(\pi,s) \cdot j_Q \circ \left(\mathrm{Id}\otimes R_{Q|P}(\pi,s)\right), \end{align} $$

for any $s\in \mathbb {C}$ avoiding poles of the operators on both sides of above. Here, $n_\alpha (\pi ,s)$ , a global normalizing factor, is a meromorphic function in s. Moreover, if $s\in i\mathbb {R}$ , then $|n_\alpha (\pi ,s)|=1$ . The operator

$$\begin{align*}R_{Q|P}(\pi,s):\mathrm{Ind}_{P(\mathbb{A})}^{G(\mathbb{A})}\pi\to \mathrm{Ind}_{Q(\mathbb{A})}^{G(\mathbb{A})}\pi\end{align*}$$

is a factorizable normalized intertwining operator, so that

$$\begin{align*}R_{Q|P}(\pi,s)=\prod_v R_{Q|P}(\pi_v,s),\quad\pi = \otimes_v \pi_v.\end{align*}$$

Here, for almost all v. Moreover, if $s\in i\mathbb {R}$ , then $R_{Q\mid P}(\pi ,s)$ is unitary (see [Reference Arthur6, Theorem 21.4] for details).

3.6 Langlands inner product formula

For $Q\in \mathcal {P}(M)$ , we let

$$\begin{align*}\theta_Q(\Lambda) := v_{\Delta_Q}^{-1}\prod_{\beta \in \Delta_Q}\langle \Lambda,\beta^\vee \rangle,\quad \Lambda\in \mathfrak{a}_{M_0,\mathbb{C},}^* \end{align*}$$

where $v_{\Delta _Q}$ is the covolume of the lattice in $\mathfrak {a}_Q^G$ spanned by $\{\beta ^\vee \mid \beta \in \Delta _Q\}$ .

For $T\in \mathfrak {a}_{M_0}^G$ , we also define $Y_Q(T)$ as the projection to $\mathfrak {a}_M^G$ of $t^{-1}(T-T_0)+T_0$ , where $t \in W_0$ is an element, such that $P_0 \subset tQ $ , and $T_0 \in a_M^G$ is the element defined by Arthur in [Reference Arthur2, Lemma 1.1] (also see the discussion around [Reference Arthur6, (9.4)]).

Let $\chi = (M,\pi )$ be a cuspidal datum and $\varphi ,\varphi ' \in \mathcal {A}^2_{\chi ,\pi }(P)$ , namely, $\varphi $ and $\varphi '$ are cusp forms. In this case, Langlands, generalizing the classical Maass–Selberg relations, obtained a formula for the inner product of two truncated Eisenstein series attached to $\varphi $ and $\varphi '$ (see [Reference Arthur6, Proposition 15.3]). The formula involves sums over various Weyl elements and parabolic subgroups of quantities involving (twisted) intertwining operators, $\theta _Q$ , and $Y_Q$ . There is also an approximate version of the formula, due to Arthur in [Reference Arthur5], when $\varphi $ or $\varphi '$ are not cuspidal. However, we do not need any of these formulæ here, but only a special case that we describe below.

For any two parabolic subgroups $P,Q\supset P_0$ , let $W(\mathfrak {a}_P,\mathfrak {a}_Q)$ be the set of distinct linear isomorphisms from $\mathfrak {a}_P\subset \mathfrak {a}_{P_0}$ to $\mathfrak {a}_Q\subset \mathfrak {a}_{P_0}$ , induced by elements of $W_0$ . For any $Q\in \mathcal {P}(M)$ and $P_0\subset P$ , there is a natural action of $W(\mathfrak {a}_P,\mathfrak {a}_Q)$ on $\Pi _2(M)$ . Following [Reference Arthur6, Section 15], we call $\chi $ unramified if, for every $(M,\pi )\in \chi $ , the stabilizer of $\pi $ in $W(\mathfrak {a}_P,\mathfrak {a}_P)$ is trivial. We warn again that this notion should not be confused with the unramifiedness of representations of reductive groups over local fields.

In the unramified case, we record the Langlands inner product formula.

Lemma 3.1. Let $\chi :=(M,\pi )$ be any unramified cuspidal datum and $\varphi ,\varphi '\in \mathcal {A}_{\chi ,\pi }^2(P)$ . Then we have

$$ \begin{align*} \langle \Lambda^T \mathrm{Eis}(\varphi,\lambda), \Lambda^T \mathrm{Eis}(\varphi',\lambda) \rangle_{\mathcal{A}^2(G)} = \lim_{\Lambda \to 0} \sum_{Q\in \mathcal{P}(M)} e^{\langle \Lambda, Y_Q(T) \rangle}\frac{\langle M_{Q|P}(\lambda+\Lambda)\varphi,M_{Q|P}(\lambda)\varphi'\rangle_{\mathcal{A}^2(Q)}}{\theta_Q(\Lambda)}. \end{align*} $$

4.1 Parameters associated with a representation

Let $M\in \mathcal {L}$ . The goal of this subsection is to associate various parameters measuring the complexity of a certain representation $\pi \in \Pi _2(M)$ .

4.1.1 Non-Archimedean complexity

First, following [Reference Finis, Lapid and Müller22, Section 5.1], for any open-compact subgroup $K\subset K_f$ , we recall the notion $\mathrm {level}(K)$ . Fix $\iota :G\to \mathrm {GL}(V)$ a faithful F-rational representation and an $\mathfrak {o}$ -lattice $\Lambda \subset V$ so that the stabilizer of $\hat {\mathfrak {o}}\otimes \Lambda \subset \mathbb {A}_f\otimes V$ in $G(\mathbb {A}_f)$ is $K_f$ . Then, for any ideal $\mathfrak {q}\subset \mathfrak {o}$ , we define the principal congruence subgroup of level $\mathfrak {q}$ by

$$\begin{align*}K({\mathfrak{q}}):=\{k \in K_f\mid \iota(k)v \equiv v \quad\mod(\mathfrak{q}(\hat{\mathfrak{o}}\otimes\Lambda)), v\in \hat{\mathfrak{o}}\otimes\Lambda\}.\end{align*}$$

Now, we define $\mathrm {level}(K)$ to be the index $[\mathfrak {o}:\mathfrak {q}_K]$ , where $\mathfrak {q}_K\subset \mathfrak {o}$ is the largest ideal $\mathfrak {q}$ , such that $K\supset K({\mathfrak {q}})$ . On the other hand, given a representation $\pi _f$ of $G(\mathbb {A}_f)$ , we define $\mathrm {level}(\pi _f)$ as the minimum over $\mathrm {level}(K)$ , such that $\pi _f$ has a nontrivial K-invariant vector.Footnote ⁶

Given a representation $\pi _f$ of $M(\mathbb {A}_f)$ and $\alpha \in \Sigma _M$ , we recall the quantity $\mathrm {level}_M(\pi _f,\hat {M}_\alpha ^+)$ , as defined in [Reference Finis, Lapid and Müller22, Sections 5.1–5.2], which measures the complexity of $\pi _f$ . In this paper, we do not need to know $\mathrm {level}_M(\pi _f,\hat {M}_\alpha ^+)$ explicitly, but only an upper bound that we describe below.

Lemma 4.1. Let $K\subset K_f$ be an open-compact subgroup. Assume that $\pi \in \Pi _2(M)$ is such that $\mathcal {A}^2_\pi (P)^{K} \neq \{0\}$ for some $P\in \mathcal {P}(M)$ , that is the parabolic induction of $\pi _f$ to $G(\mathbb {A}_f)$ has a K-invariant vector. Then we have

$$\begin{align*}\mathrm{level}_M(\pi_f,\hat{M}_\alpha^+)\le \mathrm{level}(K). \end{align*}$$

Proof. It follows from [Reference Finis, Lapid and Müller22, Lemma 5.12] that

$$\begin{align*}\mathrm{level}_M(\pi_f,\hat{M}_\alpha^+)\le \mathrm{level}(K;G_M^+),\end{align*}$$

where $G_M^+\subset G$ is a certain subgroup and $\mathrm {level}(K;G_M^+)$ denotes the relative level of K in respect to $G_M^+$ as defined in [Reference Finis, Lapid and Müller22, Section 5.1]. The lemma follows from the fact that $\mathrm {level}(K;G_M^+)\le \mathrm {level}(K)$ (see [Reference Finis, Lapid and Müller22, Section 5.1]).

4.1.2 Archimedean complexity

Let $\pi _\infty $ be a smooth representation of $M(F_\infty )$ . Let $\alpha \in \Sigma _M$ . We recall the quantity $\Lambda _M(\pi _\infty ;\hat {M}_\alpha )$ , as defined in [Reference Finis, Lapid and Müller22, Sections 5.1–5.2], which measures the complexity of $\pi _\infty $ as a representation of $M(F_\infty )$ . In this paper, we do not need to know $\Lambda _M(\pi _\infty ;\hat {M}_\alpha )$ explicitly, but only an upper bound that we describe below.

Let $\nu (\pi _\infty )$ be the eigenvalue for $\pi _\infty $ of the Casimir operator of $M(F_\infty )$ . Similarly, given $\tau \in \widehat {K_\infty }$ , let $\nu (\tau )$ be the Casimir eigenvalue of $\tau $ .

Lemma 4.2. Let $\tau \in \widehat {K_\infty }$ . Assume that $\pi \in \Pi _2(M)$ is such that $\mathcal {A}_\pi ^2(P)^{\tau }\neq \{0\}$ for some $P\in \mathcal {P}(M)$ , that is, the parabolic induction of $\pi _\infty $ to $G(F_\infty )$ has a nonzero vector whose $K_\infty $ -type is $\tau $ . Then we have

$$\begin{align*}\Lambda_M(\pi_\infty;\hat{M}_\alpha)\ll 1+\nu(\pi_\infty)^2+\nu(\tau)^2.\end{align*}$$

Proof. It follows from [Reference Finis, Lapid and Müller22, (10)–(11)] that

$$\begin{align*}\Lambda_M(\pi_\infty;\hat{M}_\alpha) \ll \nu(\pi_\infty)^2+\min_{\tau^\prime}\nu(\tau^\prime)^{2},\end{align*}$$

where $\tau '\in \widehat {K_\infty }$ runs over the lowest $K_\infty $ -types of $\mathrm {Ind}_{P(F_\infty )}^{G(F_\infty )}\pi _\infty $ . We conclude by noting that the lowest $K_\infty $ -types have the smallest Casimir eigenvalues (equivalently, the smallest infinitesimal characters).

4.2 The spherical function

In the rest of this section, we only work over the Archimedean places, that is $F_\infty $ (see [Reference Duistermaat, Kolk and Varadarajan15, Section 3] for references to the material below).

Let $G_\infty := A_G \backslash G(F_\infty )$ and $K_\infty $ be a maximal compact subgroup in $G_\infty $ . Choose a maximal $\mathbb {R}$ -split torus $T_\infty \subset G_\infty $ , which we may assume to contain $T_0(F_\infty )$ and contained in $M_0(F_\infty )$ . Let ${W_\infty := N_{G_\infty }(T_\infty )/T_\infty }$ be the Weyl group of $G_\infty $ relative to $T_\infty $ . Let $T_\infty ^0$ be the connected component of the identity of $T_\infty $ and $\mathfrak {a}_\infty $ be the Lie algebra of $T_\infty ^0$ .

We remark that for any Levi $M\supset M_0$ , the embedding $T_\infty \to M(F_\infty )$ defines a projection $\mathfrak {a}_\infty \twoheadrightarrow \mathfrak {a}_{M}^G$ and an embedding $(\mathfrak {a}_{M}^G)^* \hookrightarrow \mathfrak {a}_\infty ^*$ (see [Reference Arthur6, pp.118]).

We let $M_{0,\infty }$ be the Levi subgroup, which is the normalizer of $T_\infty $ in $G_\infty $ . Let $B_\infty $ be a Borel subgroup containing $M_{0,\infty }$ as a Levi subgroup and also containing $P_0(F_\infty )$ . Let $N_\infty $ be the unipotent radical of $B_\infty $ . We have the Iwasawa decomposition $G_\infty = B_\infty K_\infty = N_\infty T_\infty ^0 K_\infty $ .

Using the exponential map from $\mathfrak {a}_\infty $ to $T_\infty ^0$ , each element $\mu \in \mathfrak {a}_{\infty ,\mathbb {C}}^*:=(\mathfrak {a}_\infty \otimes _{\mathbb {R}}\mathbb {C})^*$ defines a character $\chi _\mu $ of $T_\infty $ , which is trivial on $T_\infty /T_\infty ^0$ . It extends, via $N_\infty $ -invariance, to a character of $B_\infty $ . Let $\delta _\infty \colon B_\infty \to \mathbb {R}_{>0}$ be the modular character of $B_\infty $ , which is attached to the character $\chi _{2\rho _\infty }$ , where $2\rho _\infty $ is the sum of all the roots of the action of $T_\infty $ on the Lie algebra of $N_\infty $ , weighted by the dimensions of the corresponding root spaces.

Given $g \in G_\infty $ , we let $a(g)$ be its $T_\infty ^0$ part according to the Iwasawa decomposition $G_\infty = N_\infty T_\infty ^0 K_\infty $ . We define the spherical function $\eta _\mu \colon G_\infty \to \mathbb {C}$ corresponding to $\mu \in \mathfrak {a}_{\infty ,\mathbb {C}}^*$ by

$$\begin{align*}\eta_\mu(g) := \intop_{K_\infty}\chi_{\mu+\rho_\infty}(a(kg))\,\mathrm{d} k. \end{align*}$$

Thus, $\eta _\mu $ is bi $K_\infty $ -invariant. Here, $\,\mathrm {d} k$ denotes the probability Haar measure on $K_\infty $ .

4.3 Spherical representation

We call an irreducible admissible representation $(\pi ,V)$ of $G_\infty $ spherical if $\pi $ has a nonzero $K_\infty $ -invariant vector. We can construct all irreducible admissible spherical representations of $G_\infty $ from the unitarily induced principal series representations. Let $\mu \in \mathfrak {a}_{\infty ,\mathbb {C}}^*$ and $\mathrm {Ind}_{B_\infty }^{G_\infty }\chi _\mu $ denotes the normalized parabolic induction of $\chi _\mu $ from $B_\infty $ to $G_\infty $ . It is an admissible representation and has a unique irreducible spherical subquotient. Conversely, for any irreducible spherical representation $(\pi ,V)$ , we can find a (unique up to $W_\infty $ -action) $\mu _\pi \in \mathfrak {a}_{\infty ,\mathbb {C}}^*$ , such that $\pi $ appears as the unique spherical subquotient of $\mathrm {Ind}_{B_\infty }^{G_\infty }\chi _{\mu _\pi }$ . In this case, we call $\mu _\pi $ the Langlands parameter of $\pi $ . When $\pi $ is also unitary, it holds that $\|\Re (\mu _\pi )\| \le \|\rho _\infty \|$ and $1+\|\mu _\pi \|^2\asymp 1+ \nu (\pi )$ (more precisely, $\nu (\pi ) = \|\rho _\infty \|^2- \|\Re (\mu _\pi )\|^2 + \|\Im (\mu _\pi )\|^2$ ) (see [Reference Duistermaat, Kolk and Varadarajan15, (3.17)]).

If $\pi $ is spherical, then the $K_\infty $ -invariant vector of $\pi $ is unique up to multiplication by a scalar. Let be the characteristic function on $K_\infty $ . Then is well-defined and acts as a projection onto the $K_\infty $ -invariant subspace of V. Let $0\neq v\in \pi $ be a $K_\infty $ -invariant vector. It follows from the definition of the spherical function that .

Let $M_\infty $ be any Levi subgroup of $G_\infty $ , having $T_\infty $ as a maximal split torus. We can associate to each irreducible admissible spherical (containing a nonzero $K_\infty \cap M_\infty $ -invariant vector) representation $\pi $ of $M_\infty $ a Langlands parameter $\mu _\pi \in \mathfrak {a}_{\infty ,\mathbb {C}}^*$ , defined up to $W_{M,\infty }=N_{M_\infty }(T_\infty )/T_\infty \subset W_\infty $ . Moreover, if $Q_\infty $ is a parabolic subgroup of $G_\infty $ with $M_\infty $ as its Levi subgroup, $\mathrm {Ind}_{Q_\infty }^{G_\infty }\pi $ has a unique spherical subrepresentation. On the other hand, if $\mathrm {Ind}_{Q_\infty }^{G_\infty }\pi $ has a $K_\infty $ -invariant vector, then $\pi $ necessarily has a $K_\infty \cap M_\infty $ -invariant vector. In any case, by the transitivity of induction, $\mu _\pi $ is also the Langlands parameter of the spherical subquotient of $\mathrm {Ind}_{Q_\infty }^{G_\infty }\pi $ (however, the new parameter is defined up to $W_\infty $ ).

4.4 The spherical transform

We call the convolution algebra on $C_c^\infty (K_\infty \backslash G_\infty / K_\infty )$ the spherical Hecke algebra of $G_\infty $ . For ${h \in C_c^\infty (K_\infty \backslash G_\infty / K_\infty )}$ , we let $\tilde {h}\colon \mathfrak {a}_{\infty ,\mathbb {C}}^* \to \mathbb {C}$ be the spherical transform of h, defined by

$$\begin{align*}\tilde{h}(\mu) := \intop_{G_\infty } h(g) \eta_\mu(g) \,\mathrm{d} g. \end{align*}$$

If $\pi $ is spherical and $0\neq v\in \pi $ is a $K_\infty $ -invariant vector, then it holds that

$$\begin{align*}\pi(h)v = \intop_{G_\infty} h(g) \pi(g) v \,\mathrm{d} g = \tilde{h}(\mu_\pi)v. \end{align*}$$

We define the Abel–Satake transform (also known as Harish-Chandra transform) to be the map $\mathcal {S}:C_c(K_\infty \backslash G_\infty / K_\infty )\to C_c(T_\infty )$ defined by

$$\begin{align*}h\mapsto \mathcal{S} h : a\mapsto \delta_\infty(a)^{1/2} \intop_{N_\infty}h(an)\,\mathrm{d} n, \end{align*}$$

where $\,\mathrm {d} n$ is a Haar measure on $N_\infty $ . Since $\mathcal {S} h$ is left- $K_\infty \cap T_\infty $ -invariant, it is actually a map on $T_\infty ^0$ . We have the exponential map $\exp \colon \mathfrak {a}_\infty \to T_\infty ^0$ , which gives an identification of $\mathfrak {a}_\infty $ with $T_\infty ^0$ . So we may as well realize $\mathcal {S} h \in C_c(\mathfrak {a}_\infty )$ after precomposing with the $\exp $ map. It holds that $\mathcal {S} f$ is $W_\infty $ -invariant. In fact, Gangolli showed that

$$\begin{align*}\mathcal{S}\colon C_c^ \infty(K_\infty \backslash G_\infty / K_\infty) \to C_c^\infty(\mathfrak{a}_\infty)^{W_\infty}\end{align*}$$

is an isomorphism of topological algebras (see [Reference Duistermaat, Kolk and Varadarajan15, eq.(3.21)] (also see [Reference Gangolli25]). Let us denote the Fourier–Laplace transform $C_c(\mathfrak {a}_\infty )\to C(\mathfrak {a}_{\infty ,\mathbb {C}}^*)$ by the map

$$\begin{align*}h\mapsto \hat{h}:\mu\mapsto \intop_{\mathfrak{a}_\infty} h(\alpha) e^{\langle\mu, \alpha\rangle}\,\mathrm{d}\alpha,\end{align*}$$

Harish-Chandra showed that the spherical transform $\tilde {h} = \widehat {\mathcal {S}(h)}$ . Here, $\,\mathrm {d}\alpha $ denotes the Lebesgue measure inherited from the $\mathbb {R}$ -vector space structure of $\mathfrak {a}_\infty $ .

4.5 The Paley–Wiener theorem

Recall the Abel–Satake transform $\mathcal {S}$ above. Gangolli (see [Reference Duistermaat, Kolk and Varadarajan15, eq.(3.22)]) proved that if $h\in C_c^\infty (\mathfrak {a}_\infty )^{W_\infty }$ is such that $\mathrm {supp} (h) \subset \{\alpha \in \mathfrak {a}_\infty \mid \|\alpha \|\le b\}$ , then

(4.1) $$ \begin{align} \mathrm{supp} (\mathcal{S}^{-1} h) \subset K_\infty \{\exp \alpha\mid\alpha \in \mathfrak{a}_\infty, \|\alpha\|\le b\}K_\infty. \end{align} $$

We also record the classical Paley–Wiener theorem, which implies that for $h\in C_c^\infty (\mathfrak {a}_\infty )^{W_\infty }$ with support in $\{\alpha \in \mathfrak {a}_\infty \mid \|\alpha \|\le b\}$ , we have

(4.2) $$ \begin{align} |\hat{h}(\mu)|\ll_{N,h} \exp(b\|\Re (\mu) \|)(1+\|\mu\|)^{-N} \end{align} $$

for every $N\ge 0$ .

5.1 Property (TWN+)

In [Reference Finis and Lapid18, Definition 3.3]Footnote ⁷ , the authors defined the (TWN+) property, strong version, of a reductive group. Property (TWN+) is a global property that concerns the size of the logarithmic derivative of $n_\alpha (\pi ,s)$ . Below, we record an implication of property (TWN+) which we will use in this article.

Lemma 5.1. Let G satisfy property (TWN+). Let $M\in \mathcal {L}$ be proper, $\pi \in \Pi _2(M)$ and $\alpha \in \Sigma _M$ . Let $K\subset K_f$ be open-compact and $\tau \in \widehat {K_\infty }$ so that $\mathcal {A}^2_\pi (P)^{\tau ,K}\neq \{0\}$ for some $P\in \mathcal {P}(M)$ . Then it holds that

$$\begin{align*}\intop_{t_0}^{t_0+1} \left|\frac{n^{\prime}_\alpha(\pi,it)}{n_\alpha(\pi,it)}\right| \,\mathrm{d} t \ll \log \left(1+|t_0|+\nu(\pi_\infty)+\nu(\tau)+\mathrm{level}(K)\right), \end{align*}$$

for any $t_0\in \mathbb {R}$ .

Finally, we keep a record that property (TWN+) is not vacuous.

5.2 Property (BD)

In [Reference Finis and Lapid19, Definitions 1–2], the authors defined the (BD) property of a reductive group. Property (BD) is a local property that concerns the size of the logarithmic derivative of $R_{Q\mid P}(\pi _v,s)$ . Below, we record an implication of property (BD) which we will use in this article.

Lemma 5.3. Let G satisfy property (BD) at every place. Let $M\in \mathcal {L}$ be proper, $\pi \in \Pi _2(M)$ and $P,Q\in \mathcal {P}(M)$ be adjacent. Let $K\subset K_f$ be open-compact and $\tau \in \widehat {K_\infty }$ so that $\mathcal {A}^2_\pi (P)^{\tau ,K}\neq \{0\}$ . Then it holds that

$$\begin{align*}\intop_{t_0}^{t_0+1} \,\left\|R_{Q|P}(\pi,it)^{-1} R_{Q|P}^\prime(\pi,it)\vert_{\left(\mathrm{Ind}_{P(\mathbb{A})}^{G(\mathbb{A})}\pi\right)^{\tau,K}}\right\|_{\mathrm{op}} \,\mathrm{d} t \ll \log \left(1+\nu(\tau)+\mathrm{level}(K)\right), \end{align*}$$

for any $t_0\in \mathbb {R}$ .

To prove Lemma 5.3, we use a different formulation of property (BD), namely, [Reference Finis, Lapid and Müller22, Definition 5.9]. This formulation is implied by the formulation in [Reference Finis and Lapid19, Definitions 1–2] (see [Reference Finis and Lapid19, Remark 3]).

Proof. We closely follow the proof of [Reference Finis, Lapid and Müller22, Proposition 5.16]. As $R_{Q\mid P}(\pi ,s)=\prod _v R_{Q\mid P}(\pi _v,s)$ and as $R_{Q\mid P}(\pi _v,s)\vert _{\left (\mathrm {Ind}_{P(F_v)}^{G(F_v)}\pi _v\right )^{K_v}}$ is independent of s for almost all v (see [Reference Finis, Lapid and Müller22, Remark 5.13]), it is enough to show that

$$\begin{align*}\intop_{t_0}^{t_0+1} \,\left\|R_{Q|P}(\pi_\infty,it)^{-1} R_{Q|P}^\prime(\pi_\infty,it)\vert_{\left(\mathrm{Ind}_{P(F_\infty)}^{G(F_\infty)}\pi_\infty\right)^{\tau}}\right\|_{\mathrm{op}} \,\mathrm{d} t \ll \log \left(1+\nu(\tau)\right),\end{align*}$$

and for each non-Archimedean v

$$\begin{align*}\intop_{t_0}^{t_0+1} \,\kern1pt\left\|R_{Q|P}(\pi_v,it)^{-1} R_{Q|P}^\prime(\pi_v,it)\vert_{\left(\mathrm{Ind}_{P(F_v)}^{G(F_v)}\pi_v\right)^{K_v}}\right\|_{\mathrm{op}} \,\mathrm{d} t \ll \log \left(1+\mathrm{level}(K_v)\right),\end{align*}$$

where $K=\prod _{v<\infty }K_v$ and $\mathrm {level}(K_v)$ is the v-adic factor of $\mathrm {level}(K)$ and the implied constant is uniform in v (i.e. uniform in the order $q_v$ of the corresponding residue fields).

To prove the non-Archimedean part, we only modify [Reference Finis, Lapid and Müller22, (14)]. Keeping the same notations (in particular, denoting $R_{Q\mid P}(\pi _v,s)\vert _{\left (\mathrm {Ind}_{P(F_v)}^{G(F_v)}\pi _v\right )^{K_v}} = A_v(q_v^{-s})$ ) and assumptions as in [Reference Finis, Lapid and Müller22], we bound

$$ \begin{align*} \intop_{t_0}^{t_0+1} \,\kern1pt&\left\|R_{Q|P}(\pi_v,it)^{-1} R_{Q|P}^\prime(\pi_v,it)\vert_{\left(\mathrm{Ind}_{P(F_v)}^{G(F_v)}\pi_v\right)^{K_v}}\right\|_{\mathrm{op}} \,\mathrm{d} t\\ &=\log q_v\intop_{t_0}^{t_0+1} \|A^{\prime}_v(q_v^{-it})\|_{\mathrm{op}} \,\mathrm{d} t \ll \log q_v\intop_{S^1}\|A^{\prime}_v(z)\|_{\mathrm{op}}|\,\mathrm{d} z|, \end{align*} $$

where the last estimate follows by changing variable $q_v^{-it}\mapsto z$ . In the first equality above, the factor $\log q_v$ comes from

$$ \begin{align*}R^{\prime}_{Q\mid P}(\pi_v,s) = -\log q_v A^{\prime}_v(q_v^{-s}).\end{align*} $$

In the second estimate above, we have changed variable

$$ \begin{align*}q_v^{-it}\mapsto z,\quad \frac{\,\mathrm{d} z}{z} = -i \log q_v\,\mathrm{d} t.\end{align*} $$

On the other hand, as t moves along $[t_0,t_0+1]\subset \mathbb {R}$ , the variable z winds $S^1$ asymptotically $\log q_v$ times, uniformly in $t_0$ . From here, we follow the discussion after [Reference Finis, Lapid and Müller22, (14)] and conclude.

To prove the Archimedean part, we first modify [Reference Finis, Lapid and Müller22, Lemma 5.19] and its proof. Keeping the same notations and assumptions as in [Reference Finis, Lapid and Müller22], we claim that

$$\begin{align*}\intop_{t_0}^{t_0+1}\|A'(it)\|_{\mathrm{op}}\,\mathrm{d} t \ll \sum_{1\le j\le m:\,u_j\neq 0} \min \left(1,\frac{1}{|u_j|}\right).\end{align*}$$

Here, $A(s)=A_\infty (s)=R_{Q\mid P}(\pi _\infty ,s)\vert _{\left (\mathrm {Ind}_{P(F_\infty )}^{G(F_\infty )}\pi _\infty \right )^{\tau }}$ . Following the proof of [Reference Finis, Lapid and Müller22, Lemma 5.19], we see that the above claim follows from the following claim

$$\begin{align*}\intop_{t_0}^{t_0+1} |\phi_w'(it)| \,\mathrm{d} t \ll \min \left( 1, 1/|u|\right), \end{align*}$$

where $w=u+iv$ with $u,v\in \mathbb {R}$ ,

$$\begin{align*}\phi_w(z):=\frac{z+\overline{w}}{z-w},\quad |\phi_w'(it)| = \frac{2|u|}{u^2+(t-v)^2}.\end{align*}$$

If $u\neq 0$ , we clearly have

$$\begin{align*}\intop_{t_0}^{t_0+1}|\phi^{\prime}_w(it)|\,\mathrm{d} t \ll \intop_{t_0}^{t_0+1}\frac{1}{|u|}\,\mathrm{d} t \ll \frac{1}{|u|}.\end{align*}$$

We also have

$$\begin{align*}\intop_{t_0}^{t_0+1}|\phi^{\prime}_w(it)|\,\mathrm{d} t \le \intop_{\mathbb{R}}\frac{2|u|}{u^2+(t-v)^2}\,\mathrm{d} t =\left.2\arctan\left(\frac{t-v}{|u|}\right)\right\vert{}^{\infty}_{-\infty} = 2\pi.\end{align*}$$

Thus, the claim follows.

Now, it follows from the discussion and the proof in [Reference Finis, Lapid and Müller22, pp.620–621] that

$$\begin{align*}\intop_{t_0}^{t_0+1} \,\left\|R_{Q|P}(\pi_\infty,it)^{-1} R_{Q|P}^\prime(\pi_\infty,it)\vert_{\left(\mathrm{Ind}_{P(F_\infty)}^{G(F_\infty)}\pi_\infty\right)^{\tau}}\right\|_{\mathrm{op}} \,\mathrm{d} t \ll \sum_{j=1}^r\sum_{k=1}^m\min\left(1,\frac{1}{|u_j-ck|}\right),\end{align*}$$

where $|u_j|,r\ll _G 1$ ,

$$\begin{align*}m\ll 1+\|\tau\|\, \asymp 1+\nu(\tau)^2,\end{align*}$$

and $c>0$ depends only on M. We conclude the proof in the Archimedean case by noting that the double sum above is bounded by $\ll _{G,M}\log m$ .

Finally, we keep a record that the property (BD) is not vacuous.

Fix $T\in \mathfrak {a}_{M_0}^G$ . Let $M\in \mathcal {L}$ , and fix $P\in \mathcal {P}(M)$ . For $Q\in \mathcal {P}(M)$ and $\lambda ,\Lambda \in i(\mathfrak {a}_M^G)^*$ , we define an operator $\mathcal {M}_Q^T(\lambda ,\Lambda ,P)$ acting on $\mathcal {A}^2(P)$ by

(5.1) $$ \begin{align} \mathcal{M}_Q^T(\lambda,\Lambda,P) := e^{\langle \Lambda,Y_Q(T) \rangle }M_{Q|P}(\lambda)^{-1}M_{Q|P}(\lambda+\Lambda). \end{align} $$

Similarly, for $L\in \mathcal {L}(M)$ , $Q\in \mathcal {P}(L)$ and $\lambda ,\Lambda \in i(\mathfrak {a}_L^G)^* \subset i(\mathfrak {a}_M^G)^*$ , we define

$$\begin{align*}\mathcal{M}_Q^T(\lambda,\Lambda,P) := \mathcal{M}_{Q_1}^T(\lambda,\Lambda,P),\end{align*}$$

for arbitrary $Q_1\subset Q$ , $Q_1 \in \mathcal {P}(M)$ . The choice of $Q_1$ does not matter here. Here, in the left-hand side, $\lambda ,\Lambda $ are realized as restrictions on $\mathfrak {a}_L$ (see [Reference Arthur6, pp.133]).

Following definitions in [Reference Arthur6, Section 21], the set $\{ \mathcal {M}_Q^T(\lambda ,\Lambda ,P) \}_{Q\in \mathcal {P}(L)}$ is an example of a ‘ $(G,L)$ -family’ (see [Reference Arthur6, Section 17] for a general discussion on $(G,M)$ -family). For $L\in \mathcal {L}(M)$ and $\lambda \in i(\mathfrak {a}_L^G)^*$ , we define the operator $\mathcal {M}^T_L(\lambda ,P)$ as the value at $\Lambda =0$ of

$$\begin{align*}\sum_{Q \in \mathcal{P}(L)}\frac{\mathcal{M}_Q^T(\lambda,\Lambda,P)}{\theta_Q(\Lambda)}. \end{align*}$$

Below, we prove the main estimate on the operator norm of $\mathcal {M}^T_L(\lambda ,P)$ .

Proposition 5.5. Assume that G satisfies properties (TWN+) and (BD). Fix $M\in \mathcal {L}$ , $P\in \mathcal {P}(M)$ and $L\in \mathcal {L}(M)$ . Let $\tau \in \widehat {K_\infty }$ , $K\subset K_f$ be an open-compact and $\pi \in \Pi _2(M)$ . Then, for every $\lambda _0 \in i(\mathfrak {a}_L^G)^*$ ,

$$\begin{align*}\intop_{\substack{\lambda\in i(\mathfrak{a}_L^G)^* \\ \|\lambda-\lambda_0\|\le 1}}\left\|\mathcal{M}^T_L(\lambda,P)|_{\mathcal{A}^2_\pi(P)^{\tau,K}} \right\|_{\mathrm{op}} \,\mathrm{d} \lambda \ll \left(\|T\|\,\log \left(1+\nu(\tau)+\nu(\pi_\infty)+\|\lambda_0\|+\mathrm{level}(K)\right)\right)^{\dim \mathfrak{a}_L^G}, \end{align*}$$

where $T\in \mathfrak {a}_{M_0}^G$ with sufficiently large $d(T)$ .

Before proving Proposition 5.5, we give a proof of Theorem 2.

Proof of Theorem 2.

Using Lemma 3.1 and the fact that $M_{Q\mid P}(\lambda )$ unitary for $\lambda \in i(\mathfrak {a}_M^G)^*$ , and recalling Equation (5.1), we can write

$$\begin{align*}\|\Lambda^T \mathrm{Eis}(\varphi,\lambda)\|_2^2 = \langle \mathcal{M}^T_M(\lambda,P) \varphi, \varphi \rangle_{\mathcal{A}^2(P)}. \end{align*}$$

We conclude using Proposition 5.5 and that $\varphi $ is a unit.

The rest of this section is devoted to the proof of Proposition 5.5.

Let $P|^\alpha Q$ for some $\alpha \in \Sigma _M$ . Let $\varpi \in \mathfrak {a}_M^*$ be such that $\langle \varpi ,\alpha ^\vee \rangle =1$ . We define

(5.2) $$ \begin{align} \delta_{Q|P}(\lambda) := M_{Q|P}^{-1}(\lambda)D_{\varpi}M_{Q|P}(\lambda) \end{align} $$

as in [Reference Finis, Lapid and Müller22, Section 4.3] as an operator from $\mathcal {A}^2(P)$ to $\mathcal {A}^2(P)$ . Here $D_{\varpi }M_{Q|P}(\lambda )$ is the derivative of $M_{Q|P}(\lambda )$ in the direction $\varpi $ . The definition is independent of the choice of $\varpi $ ; see [Reference Finis, Lapid and Müller22, pp.608].

Lemma 5.6. Keep the notations as above and assumptions as in Proposition 5.5. We have

$$\begin{align*}\intop_{t_0}^{t_0+1} \,\left\| \delta_{Q|P}(it\varpi)|_{\mathcal{A}^2_{\pi}(P)^{\tau,K}} \right\|_{\mathrm{op}} \,\mathrm{d} t \ll \log\left(1+\nu(\tau)+\nu(\pi_\infty)+|t_0|+\mathrm{level}(K)\right), \end{align*}$$

for any $t_0\in \mathbb {R}$ .

Proof. We take the inverse on both sides of (3.2) and compose it with the derivative of (3.2) and obtain

$$\begin{align*}j_P^{-1}\circ \delta_{Q|P}(s\varpi) \circ j_P = \frac{n^{\prime}_\alpha(\pi,s)}{n_\alpha(\pi,s)}\mathrm{Id}+R_{Q|P}^{-1}(\pi,s)R_{Q|P}^\prime(\pi,s). \end{align*}$$

Recall that $j_P$ can be chosen to be isometric, that is $\|j_P\|=1$ . We conclude the proof by applying Lemmas 5.1 and 5.3.

Given $L \in \mathcal {L}(M)$ , we let $\mathfrak {B}_{P,L}$ be the set of tuples $\underline {\beta } = (\beta _1^\vee ,...,\beta _m^\vee )$ of $\Sigma _P^\vee $ with $m:=\dim \mathfrak {a}^G_L$ , such that their projection to $\mathfrak {a}_L^G$ form a basis of $\mathfrak {a}_L^G$ . We denote $\mathrm {vol}(\underline {\beta })$ to be the covolume of the lattice generated by the projection of $\beta _1^\vee ,...,\beta _m^\vee $ in $\mathfrak {a}_L^G$ . Let

$$\begin{align*}\Xi_L(\underline{\beta}):=\{(P_1,...,P_m,P_1^\prime,...,P_m^\prime) \in \mathcal{P}(M)^{2m} \mid P_i,P_i^\prime \in \mathcal{P}(M), P_i|^{\beta_i}P_i^\prime\}. \end{align*}$$

In [Reference Finis, Lapid and Müller21, pp.179–180], (also see [Reference Finis, Lapid and Müller22, pp.608–609]) the authors define a mapFootnote ⁹

$$\begin{align*}\mathfrak{B}_{P,L}\ni\underline{\beta}\mapsto \xi(\underline{\beta})\in\Xi_L(\underline{\beta}).\end{align*}$$

For $P|^\beta P'$ , we find $\alpha _{P,P^\prime }\in \mathfrak {a}_M^*$ , such that

$$\begin{align*}Y_{P}(T)-Y_{P'}(T) = \langle \alpha_{P,P^\prime},T-T_0 \rangle \beta^\vee. \end{align*}$$

Such an $\alpha _{P,P'}$ exists (see [Reference Finis, Lapid and Müller21, pp.186]). Finally, for $\xi \in \Xi _L(\underline {\beta })$ , we define the operator

(5.3) $$ \begin{align} \Delta_{\xi}^T(P,\lambda) := \frac{\mathrm{vol}(\underline{\beta})}{m!}M_{P\mid P_1^{\prime}}(\lambda) \bigg(\delta_{P_1|P_1^{\prime}}(\lambda)&+\langle\alpha_{P_1,P_1^{\prime}},T-T_0\rangle\mathrm{Id}\bigg) M_{P_1^{\prime}|P_2^{\prime}}(\lambda)\nonumber\\ &\quad\dots \bigg(\delta_{P_m|P_m^{\prime}}(\lambda)+\langle{\alpha}_{P_m,P_m^{\prime}},T-T_0\rangle\mathrm{Id}\bigg)M_{P_m^{\prime}|P}(\lambda). \end{align} $$

Note that the definition of $\Delta _\xi ^T(P,\lambda )$ , here, looks a little different from [Reference Finis, Lapid and Müller21, pp.186], although they are the same. This is because the definition of $\delta _{P_i\mid P_i'}$ in [Reference Finis, Lapid and Müller21, pp.179] is different than ours (also compare with the definition of $\Delta ^T_\xi (P,\lambda )$ in [Reference Finis, Lapid and Müller22, pp.609]).

Following, Proposition 5.7 expresses the operator $\mathcal {M}^T_L$ in terms of $\Delta ^T_\xi $ . A proof of this is essentially contained in [Reference Finis, Lapid and Müller21, Theorem 4].

Proposition 5.7. Keep the same notations as in Proposition 5.5. We have

$$\begin{align*}\mathcal{M}_L^T(P,\lambda) = \sum_{\underline{\beta}\in \mathfrak{B}_{P,L}} \Delta_{\xi(\underline{\beta})}^T(P,\lambda) \end{align*}$$

as operators on $\mathcal {A}^2_{\pi }(P)^{\tau ,K}$ .

Proof. We closely follow the proof of [Reference Finis, Lapid and Müller21, Theorem 4]. The Taylor expansion $\mathcal {A}_Q$ of

$$\begin{align*}M_{Q\mid P}(\lambda)^{-1}M_{Q\mid P}(\lambda+\Lambda),\quad \text{at }\Lambda=0\end{align*}$$

form a compatible family (in the sense of [Reference Finis, Lapid and Müller21]), when restricted to the finite-dimensional subspace $\mathcal {A}_\pi ^2(P)^{\tau ,K}$ . This follows from the argument in [Reference Finis, Lapid and Müller21, pp.185] for $s=1$ . On the other hand, the Taylor series $c_Q$ of $e^{\langle \Lambda ,Y_Q(T)\rangle }$ at $\Lambda =0$ form a scalar-valued compatible family. Thus, following the argument in [Reference Finis, Lapid and Müller21, pp.186], we conclude that $c_Q\mathcal {A}_Q$ form a compatible family when restricted to the finite-dimensional subspace $\mathcal {A}_\pi ^2(P)^{\tau ,K}$ . Hence, as in [Reference Finis, Lapid and Müller21], we compute the limit of (5.1) at $\Lambda =0$ using [Reference Finis and Lapid17, Theorem 8.1], which concludes the proof.

Finally, we can prove Proposition 5.5.

Proof of Proposition 5.5.

Using Proposition 5.7, we see that it is enough to bound for each $\beta \in \mathfrak {B}_{P,L}$ ,

$$\begin{align*}\intop_{\substack{{\lambda\in i(\mathfrak{a}_L^G)^*} \\ \|\lambda-\lambda_0\|\le 1}}\left\| \Delta_{\xi(\underline{\beta})}^T(P,\lambda)|_{\mathcal{A}^2_\pi(P)^{\tau,K}} \right\|_{\mathrm{op}} \,\mathrm{d} \lambda \ll \left(\|T\|\,\log \left(1+\nu(\tau)+\nu(\pi_\infty)+\|\lambda_0\|+\mathrm{level}(K)\right)\right)^{m}. \end{align*}$$

Let $\vartheta _i\in (\mathfrak {a}_L^G)^*$ be the dual basis to $\beta _j^\vee $ . Using this basis, the above integral can be majorized by

$$\begin{align*}\ll \intop_{\substack{{\lambda=i\sum_{j=1}^mt_j\vartheta_j}\\{t_j\in\mathbb{R},\,|t_j-t_{0,j}| \le C}}}\left\|\Delta_{\xi(\underline{\beta})}^T(P,\lambda)|_{\mathcal{A}^2_\pi(P)^{\tau,K}} \right\|_{\mathrm{op}} \,\mathrm{d} \lambda,\end{align*}$$

where $it_{0,j}:=\langle \lambda _0,\beta _j^\vee \rangle $ and C is a constant, depending only on the root system. Now we recall (5.3) and use the fact that $M_{Q|P}(\lambda )$ is unitary for purely imaginary $\lambda $ to see that the above integral can be bounded by

$$\begin{align*}\prod_{j=1}^m\intop_{|t_j - t_{0,j}| \le C} \left(\left\|\delta_{P_j|P_j^\prime}(i\vartheta_jt_j)|_{\mathcal{A}^2_\pi(P^{\prime}_j)^{\tau,K}}\right\|_{\mathrm{op}}+\|T\|\right) \,\mathrm{d} t_j. \end{align*}$$

We conclude the proof by dividing the above interval in the j’th integral in $O(1)$ number of length one subinterval centring the points of size $O(t_{0,j})$ and applying Lemma 5.6 for the integral over each subinterval.

7.1 Choice of test function

We freely use the notations of Section 4.2. Recall that $\mathfrak {a}_\infty $ is the Lie algebra of a maximal $\mathbb {R}$ -split torus of $G(F_\infty )$ . Let $\delta>0$ be a sufficiently small constant. We choose a fixed $L^1$ -normalized nonnegative test function $f_{0,\delta }$ on $\mathfrak {a}_\infty $ supported on the ball of radius $\delta /2$ around $0$ . Let $f_{\delta }:=f_{0,\delta }*f_{0,\delta }^*$ , where for any $f\in C_c(\mathfrak {a}_\infty )$ , we define $f^*(\alpha ) := \overline {f(-\alpha )}$ . It holds that $\widehat {f^*}(\mu ) = \overline {\hat {f}(-\overline {\mu })}$ . Thus, $\widehat {f_{\delta }}$ is nonnegative on $i\mathfrak {a}_\infty ^*$ . Note that $f_{\delta }$ is also nonnegative, $L^1$ -normalized and supported on the ball of radius $\delta $ around $0$ .

We fix

(7.1) $$ \begin{align} C_0:=2\|\rho_\infty\|+1. \end{align} $$

Now, we make $\delta>0$ small enough so that

(7.2) $$ \begin{align} \Re(\widehat{f_\delta}(\mu))\ge 1/2,\quad\text{ for }\|\mu\|\le C_0. \end{align} $$

This follows from continuity and the fact that $\widehat {f_\delta }(0)=1$ .

Given $\mu _0 \in i\mathfrak {a}_\infty ^*$ , we define $f_{\mu _0} \in C_c^\infty (\mathfrak {a}_\infty )^{W_\infty }$ by

$$\begin{align*}f_{\mu_0}(\alpha) := \sum_{w \in W_\infty} f_\delta(w\alpha)e^{-\langle \mu_0,w\alpha \rangle}, \end{align*}$$

so that

$$\begin{align*}\widehat{f_{\mu_0}}(\mu)=\sum_{w\in W_\infty}\widehat{f_\delta}(w\mu-\mu_0),\quad \mu\in\mathfrak{a}^*_{\infty,\mathbb{C}}.\end{align*}$$

Finally, we define

$$\begin{align*}h_{\mu_0} := \mathcal{S}^{-1} (f_{\mu_0}^* *f_{\mu_0})\in C_c^\infty(K_\infty\backslash G(F_\infty)/K_\infty).\end{align*}$$

Thus, the spherical transform of $h_{\mu _0}$ is

$$\begin{align*}\tilde{h}_{\mu_0}(\mu)=\widehat{f_{\mu_0}}(\mu)\overline{\widehat{f_{\mu_0}}(-\overline{\mu})}\end{align*}$$

(see Section 4.4). Then $h_{\mu _0}$ satisfies the following properties:

1. 1. The support of $h_{\mu _0}$ is bounded independently of $\mu _0$ , which follows from (4.1).

2. 2. $\tilde {h}_{\mu _0}(\mu ) \ge 0$ , if $\mu = -w\overline {\mu }$ for some $w\in W_\infty $ ; in particular, if $\mu \in i\mathfrak {a}_\infty ^*$ , or more generally, if $\mu $ is the Langlands parameter of a unitary representation. This is immediate from $W_\infty $ -invariance of $\widehat {f_{\mu _0}}$ .

3. 3. There is a constant b depending only on the support of $f_\delta $ , such that for every $N>0$ , it holds that

   $$\begin{align*}|\tilde{h}_{\mu_0}(\mu)| \ll_{N} \sum_{w \in W_\infty}\exp(b\|\Re(\mu)\|)(1+\|w\mu-\mu_0\|)^{-N}, \end{align*}$$
   which follows from (4.2).

4. 4. For $C_0$ as in (7.1), we have that if $\|\mu -\mu _0\| \le C_0$ , then $|\tilde {h}_{\mu _0}(\mu )| \ge 1/10$ , perhaps after making $\delta>0$ smaller.

To see the last claim, it suffices to show that

$$\begin{align*}\Re\left(\widehat{f_{\mu_0}}(\mu)\right) \ge 1/3,\quad\text{ for }\|\mu-\mu_0\|=\|-\overline{\mu}-\mu_0\|<C_0.\end{align*}$$

Working similarly to the proof of [Reference Finis, Lapid and Müller21, Lemma 3.2], below, we prove the above sufficient formulation. We see that

$$ \begin{align*} \widehat{f_\delta}(\mu) &=\intop_{\mathfrak{a}_\infty}f_\delta(\alpha)\left(e^{i\langle\Im(\mu),\alpha\rangle}-e^{i\langle\Im(\mu),\alpha\rangle}+e^{\langle\mu,\alpha\rangle}\right)\,\mathrm{d}\alpha\\ &=\widehat{f_\delta}\left(i\Im(\mu)\right) + \intop_{\mathfrak{a}_\infty}f_\delta(\alpha)e^{i\langle\Im(\mu),\alpha\rangle}\left(e^{\langle\Re(\mu),\alpha\rangle}-1\right)\,\mathrm{d}\alpha. \end{align*} $$

As $\|\Re (\mu )\|\le C_0$ , we have $|e^{\langle \Re (\mu ),\alpha \rangle }-1|<C_1\delta $ for $\alpha $ lying in a $\delta $ -radius ball around the origin and for certain $C_1$ depending on $C_0$ . Consequently, we have

$$\begin{align*}\Re(\widehat{f_\delta}(\mu)) \ge - C_1\delta,\end{align*}$$

as $\widehat {f_\delta }\left (i\Im (\mu )\right )\ge 0$ . Thus, upon applying (7.2), and the fact that

$$\begin{align*}\|\Re(w\mu-\mu_0)\|= \|\Re(\mu)\|=\|\Re(\mu-\mu_0)\|\le \|\mu-\mu_0\|\le C_0,\end{align*}$$

we obtain

$$ \begin{align*} \Re(\widehat{f_{\mu_0}}(\mu)) =\Re(\widehat{f_\delta}(\mu-\mu_0))+\sum_{w\neq 1}\Re(\widehat{f_\delta}(w\mu-\mu_0)) \ge 1/2-C_2 \delta, \end{align*} $$

for some $C_2$ depending on $C_0$ and the group. We conclude by making $\delta $ sufficiently small.

7.2 Langlands parameters

Let $\chi :=(M_1,\pi _1)$ be a cuspidal datum, determined by a Levi subgroup $M_1$ and a representation $\pi _1 \in \Pi _2(M_1)$ . Let $P_1 \supset P_0$ be such that $M_1 = M_{P_1}$ . We assume that $\pi _{1,\infty }$ is spherical, that is, has a nonzero $K_\infty \cap M_1(F_\infty )$ -invariant vector, so we may associate to it a Langlands parameter $\mu _{\pi _1}$ .

We first notice that $\chi $ is actually an equivalence class, whose size by [Reference Arthur6, Section 12], is at most

$$\begin{align*}n_{P_1} = \sum_{P_0 \subset P'\sim P_1} |W(\mathfrak{a}_{P_1},\mathfrak{a}_{P'})|. \end{align*}$$

We let $(M_1,\pi _1),...,(M_k,\pi _k)$ be the elements in this equivalence class, with $k\le n_{P_1} \le n_{P_0}$ , and $M_i = M_{P_i}$ for $P_i \supset P_0$ , all associated with $P_1$ . Notice that the Langlands parameter $\mu _{\pi _i}$ is in the $W_\infty $ -orbit of $\mu _{\pi _1}$ , so $\|\mu _{\pi _i}\|=\|\mu _{\pi _1}\|$ .

Now, let $P\supset P_0$ and $\pi \in \Pi _2(M_P)$ so that $\mathcal {A}_{\chi ,\pi }^2(P)^{K_\infty K}\neq \{0\}$ . It follows from the Langlands construction of the residual spectrum that $\pi $ is defined using iterated residues from an Eisenstein series of one of the $\pi _i$ , $1\le i\le k$ , for some $P_i \sim P$ . Let $\rho _\pi \in (\mathfrak {a}_{M_i}^M)^* \subset (\mathfrak {a}_{M_0}^G)^* \subset (\mathfrak {a}_\infty )^*$ be the point where the residual representation is defined by iterated residues (see, e.g. [Reference Arthur5, Section 2]). Then $\mu _\pi = \mu _{\pi _i}+\rho _\pi $ . The crucial point here is that $\rho _\pi $ is real and universally bounded (in fact, $\|\rho _\pi \|\le \|\rho _\infty \|$ , where $\rho _\infty $ is as in Section 4.2). Therefore, $\|\mu _{\pi }-\mu _{\pi _i}\|\le \|\rho _\infty \|$ and

(7.3) $$ \begin{align} 1+\|\mu_{\pi}\|\asymp 1+\|\mu_{\pi_i}\|. \end{align} $$

The above properties of $\rho _\pi $ follow from Langlands’ construction of the residual spectrum (see e.g. [Reference Müller47, pp.1152, Proof of Lemma 4.3]).

Next, the Langlands parameter of $\mathrm {Ind}_{P(F_\infty )}^{G(F_\infty )}\left (\pi _\infty \otimes e^{\langle \lambda , H_{M_P}(\cdot ) \rangle }\right )$ for $\lambda \in (\mathfrak {a}_{P,\mathbb {C}}^G)^*$ is $\mu _{\pi ,\lambda }:=\mu _\pi +\lambda $ .Footnote ¹⁰ If $\lambda \in i(\mathfrak {a}_P^G)^*$ , then this induction is unitary. Correspondingly, it holds that

• • $\|\Re (\mu _{\pi ,\lambda })\|\le \|\rho _{\infty }\|$ , and

• • there is a $w\in W_\infty $ so that $-\overline {\mu _{\pi ,\lambda }}=w\mu _{\pi ,\lambda }$ (as a matter of fact, we may choose $w \in N_{M_P(F_\infty )}(T_\infty )/T_\infty \subset W_\infty $ )

(see, e.g. [Reference Duistermaat, Kolk and Varadarajan15, Proposition 3.4].

We now prove Theorem 3.

Proof of Theorem 3.

We use property (3) in Section 3.4 to assume that T in Theorem 3 is such that $d(T)$ is sufficiently large (in terms of G and F) but fixed.

We assume that the cuspidal datum $\chi :=(M,\pi _0)$ with the equivalence class $(M_1,\pi _1),\dots ,(M_k,\pi _k)$ as above. Recall that $\|\mu _{\pi _0}\|\asymp \|\mu _{\pi _i}\|$ .

Let $i\mathfrak {a}_\infty \ni \mu _j := i \Im (\mu _{\pi _j})+\lambda _0$ for $j=1,\dots ,k$ . We construct $h_{\mu _j}$ as in Section 7.1, with the choice of $C_0$ given in (7.1). Now let $h_j \in C_c^\infty (G(\mathbb {A})^1)$ be of the form

Note that Item 1 implies that support of $h_j$ has bounded measure.

Clearly, for any $P\supset P_0$ and $\lambda \in i(\mathfrak {a}_{P}^G)^*$ , the operator $\rho (P,\lambda ,h_j)$ projects on $\mathcal {A}^2(P)^{K_\infty K}$ . Moreover, for any $\pi \in \Pi _2(M_P)$ and $\varphi \in \mathcal {A}^2_{\chi ,\pi }(P)^{K_\infty K}$ , we have

$$\begin{align*}\rho(P,\lambda,h)\varphi = \tilde{h}_{\mu_j}(\mu_{\pi,\lambda})\varphi. \end{align*}$$

Recall from the discussion about the Langlands parameters above that there exists $w\in W_\infty $ so that $w\mu _{\pi ,\lambda }=-\overline {\mu _{\pi ,\lambda }}$ . Thus, using Item 2, we obtain that

$$\begin{align*}\tilde{h}_{\mu_j}(\mu_{\pi,\lambda})\ge 0.\end{align*}$$

On the other hand, if $\|\lambda -\lambda _0\|\le 1$ , and $P\supset P_0$ and $\pi \in \Pi _2(M_P)$ are such that $\mathcal {A}_{\chi ,\pi }^2(P)\neq \{0\}$ , then by the discussion above, there is $1\le j\le k$ , such that

$$ \begin{align*} \|\mu_{\pi,\lambda}-\mu_j\|\le \|\Re(\mu_{\pi_j})\| + \|\rho_\pi\|+ \|\lambda-\lambda_0\| \le 2\|\rho_\infty\|+1 =C_0. \end{align*} $$

Hence, applying Item 4 it holds that

$$\begin{align*}\tilde{h}_{\mu_j}(\mu_{\pi,\lambda}) \ge 1/10. \end{align*}$$

Now, we fix $T\in \mathfrak {a}_{M_0}^G$ with sufficiently large $d(T)$ and apply Proposition 6.1 to write

$$\begin{align*}J_\chi^T(h_j) = \sum_{P\supset P_0}\frac{1}{n_P}\sum_{\pi\in\Pi_2(M_P)}\sum_{\varphi\in\mathcal{B}_{\chi,\pi}(P)^{K_\infty K}}\intop_{i(\mathfrak{a}_P^G)^*}\tilde{h}_{\mu_j}(\mu_{\pi,\lambda})\|\Lambda^T\mathrm{Eis}(\varphi,\lambda)\|^2_2\,\mathrm{d}\lambda.\end{align*}$$

Here, $\mathcal {B}_{\chi ,\pi }(P)^{K_\infty K}$ is an orthonormal basis of $\mathcal {A}_{\chi ,\pi }^2(P)^{K_\infty K}$ . Using the above properties of $\tilde {h}_{\mu _j}$ , we obtain that

$$\begin{align*}\sum_j J_\chi^T(h_j)\gg \sum_{P\supset P_0}\sum_{\pi\in\Pi_2(M_P)}\sum_{\varphi\in\mathcal{B}_{\chi,\pi}(P)^{K_\infty K}}\intop_{\substack{{\lambda\in i(\mathfrak{a}_P^G)^*} \\ \|\lambda-\lambda_0\|\le 1}}\|\Lambda^T\mathrm{Eis}(\varphi,\lambda)\|^2_2\,\mathrm{d}\lambda.\end{align*}$$

Note that the right-hand side above equals to the left-hand side of the estimate in Theorem 3.

Hence, it remains to bound $J_\chi ^T(h_j)$ for $j=1,\dots ,k$ . As the proof is the same for all j without loss of generality, we will bound for $j=1$ and write $h=h_1$ .

Using Proposition 6.2, we see that

$$\begin{align*}J_\chi^T(h)\ll \sum_{P\supset P_0}\max_{L\in\mathcal{L}(M_P)}\max_{w\in W^L(M_P)}\sum_{\pi\in\Pi_2(M_P)}\intop_{i(\mathfrak{a}_{L}^G)^*} \mathrm{tr}\left(\mathcal{M}_{L}^T(P,\lambda)M(P,w) \rho(P,\lambda,h)|_{\mathcal{A}_{\chi,\pi}^2(P)}\right)\,\mathrm{d}\lambda. \end{align*}$$

Recall that $\rho (P,\lambda ,h)\mid _{\mathcal {A}_{\chi ,\pi }^2(P)}$ projects on $\mathcal {A}_{\chi ,\pi }^2(P)^{K_\infty K}$ and acts there by the scalar $\tilde {h}_{\mu _1}(\mu _\pi +\lambda )$ . Noting that $M(P,w)$ is unitary and commutes with $\rho (P,\lambda ,h)$ , we get that the integral on the right-hand side above is bounded by

$$\begin{align*}\dim\,\left(\mathcal{A}_{\chi,\pi}^2(P)^{K_\infty K}\right)\intop_{i(\mathfrak{a}_{L}^G)^*}\tilde{h}_{\mu_1}(\mu_\pi+\lambda) \left\|\mathcal{M}_{L}^T(P,\lambda)|_{\mathcal{A}_{\chi,\pi}^2(P)^{K_\infty K}}\right\|_{\mathrm{op}}\,\mathrm{d} \lambda. \end{align*}$$

Applying Item 3, we can bound the inner integral in the above display by

(7.4) $$ \begin{align} \ll_N\max_{w\in W_\infty}\intop_{i(\mathfrak{a}_L^G)^*}\left(1+\|w\lambda+wi\Im(\mu_\pi)-\lambda_0-i\Im(\mu_{\pi_1})\|\right)^{-N}\left\|\mathcal{M}_{L}^T(P,\lambda)|_{\mathcal{A}_{\chi,\pi}^2(P)^{K_\infty K}}\right\|_{\mathrm{op}} \,\mathrm{d} \lambda. \end{align} $$

For each $w\in W_\infty $ , we define

$$\begin{align*}\lambda_1:=\lambda_{1,w,\pi,\pi_1}:=w^{-1}\left(\lambda_0-i\Im(w\mu_\pi-\mu_{\pi_1})\right)\in i(\mathfrak{a}_\infty)^*.\end{align*}$$

Using (7.3), we obtain

$$\begin{align*}\|\lambda_1\|\ll \|\lambda_0\|+\|\mu_\pi\|+\|\mu_{\pi_1}\|\,\asymp \|\lambda_0\|+\sqrt{\nu(\pi_{1,\infty})}.\end{align*}$$

Thus, for each w, we bound the integral on the right-hand side of (7.4) by

(7.5) $$ \begin{align} \intop_{i(\mathfrak{a}_L^G)^*}\left(1+\|\lambda-\lambda_1\|\right)^{-N}&\left\|\mathcal{M}_{L}^T(P,\lambda)|_{\mathcal{A}_{\chi,\pi}^2(P)^{K_\infty K}}\right\|_{\mathrm{op}} \,\mathrm{d} \lambda\nonumber\\ &\qquad\le\sum_{m=1}^\infty\frac{1}{m^N}\intop_{{\substack{{\lambda\in i(\mathfrak{a}_L^G)^*} \\ m-1\le\|\lambda-\lambda_1\|\le m}}}\left\|\mathcal{M}_{L}^T(P,\lambda)|_{\mathcal{A}_{\chi,\pi}^2(P)^{K_\infty K}}\right\|_{\mathrm{op}} \,\mathrm{d} \lambda. \end{align} $$

We can find $\{\eta _l\}_{l=1}^{n}\subset i(\mathfrak {a}_L^G)^*$ with $\|\eta _l\|\le m$ and $n\ll m^{\dim \mathfrak {a}_L^G}$ , such that

$$\begin{align*}\{\lambda:\|\lambda-\lambda_1\| \le m\}\subset \cup_{l=1}^n\{\lambda:\|\lambda-\lambda_1-\eta_l\|\le 1\}.\end{align*}$$

Thus, the integral on the right-hand side of (7.5) is bounded by

$$\begin{align*}\sum_{l=1}^n\intop_{{\substack{{\lambda\in i(\mathfrak{a}_L^G)^*} \\ \|\lambda-\lambda_1-\eta_l\|\le 1}}}\left\|\mathcal{M}_{L}^T(P,\lambda)|_{\mathcal{A}_{\chi,\pi}^2(P)^{K_\infty K}}\right\|_{\mathrm{op}} \,\mathrm{d} \lambda.\end{align*}$$

We apply Proposition 5.5 to estimate the above integral and obtain that

$$ \begin{align*} \intop_{i(\mathfrak{a}_{L}^G)^*}\tilde{h}_{\mu_1}(\mu_\pi+\lambda) &\left\|\mathcal{M}_{L}^T(P,\lambda)|_{\mathcal{A}_{\chi,\pi}^2(P)^{K_\infty K}}\right\|_{\mathrm{op}} \,\mathrm{d} \lambda\\ &\qquad\ll_{N}\max_{w\in W_\infty}\sum_{m=1}^\infty\frac{1}{m^N}\sum_{l=1}^n\left(\|T\|\,\log\left(1+\|\lambda_1+\eta_l\|+\nu(\pi_\infty)+\mathrm{level}(K)\right)\right)^{\dim\mathfrak{a}_L^G.} \end{align*} $$

Using the bounds of $\lambda _1$ , $\eta _l$ , and n, the right-hand side above is bounded by

$$ \begin{align*} \ll_{N}\sum_{m=1}^\infty\frac{1}{m^{N-\dim\mathfrak{a}_L^G}}&\left(\|T\|\,\log\left(1+m+\|\lambda_0\|+\nu(\pi_{1,\infty})+\mathrm{level}(K)\right)\right)^{\dim\mathfrak{a}_L^G}\\ &\ll\left(\|T\|\,\log\left(1+\|\lambda_0\|+\nu(\pi_{1,\infty})+\mathrm{level}(K)\right)\right)^{\dim\mathfrak{a}_L^G}, \end{align*} $$

by making N large enough. Thus, we obtain

$$ \begin{align*} J_\chi^T(h) \ll \sum_{P\supset P_0}\max_{L\in\mathcal{L}(M_P)}\sum_{\pi\in\Pi_2(M_P)}\dim\ &\left(\mathcal{A}_{\chi,\pi}^2(P)^{K_\infty K}\right)\\ &\qquad\times\left(\|T\|\,\log\left(1+\|\lambda_0\|+\nu(\pi_{1,\infty})+\mathrm{level}(K)\right)\right)^{\dim\mathfrak{a}_L^G}. \end{align*} $$

We conclude noting that $\max _{L\in \mathcal {L}(M_P)}\dim \mathfrak {a}_L^G=\dim \mathfrak {a}_M^G$ .

We now prove Corollary 1.6 and Theorem 1.

Proof of Theorem 1.

We will prove the slightly stronger statement given in Remark 1.2.

The classification of the residual spectrum for $\mathrm {GL}_n$ due to Mœglin and Waldspurger [Reference Mœglin and Waldspurger45] (see Section 3.2) implies that for every $\chi $ and $P\supset P_0$ it holds that

$$\begin{align*}\#\{\pi\in\Pi_2(M_P)\mid \mathcal{A}^2_{\chi,\pi}(P)\neq \{0\}\}\ll_n 1.\end{align*}$$

The classification also implies that the multiplicity of $\pi $ in $L^2_{\mathrm {disc},\chi }(A_{M_P}M_P(F) \backslash M(\mathbb {A}))$ is at most $1$ . So the representation $\rho (P,0,\cdot )$ on $\mathcal {A}^2_{\chi ,\pi }(P)$ is isomorphic to $\mathrm {Ind}_{P(\mathbb {A})}^{G(\mathbb {A})} \pi $ , and, in particular

$$\begin{align*}\mathcal{A}^2_{\chi,\pi}(P)^{K_\infty K} = (\mathrm{Ind}_{P(\mathbb{A})}^{G(\mathbb{A})} \pi)^{K_\infty K.} \end{align*}$$

On the other hand, we may write $\pi = \otimes _v^\prime \pi _v$ and obtain

$$\begin{align*}(\mathrm{Ind}_{P(\mathbb{A})}^{G(\mathbb{A})} \pi)^{K_\infty K} = \bigotimes{}^\prime(\mathrm{Ind}_{P(F_v)}^{G(F_v)} \pi_v)^{K_v},\end{align*}$$

where $K_\infty K = \prod _v K_v$ .

Now using uniform admissibility due to Bernstein [Reference Bernšteĭn10], which says that

$$\begin{align*}\dim(\mathrm{Ind}_{P(F_v)}^{G(F_v)} \pi_v)^{K_v}\ll_{K_v} 1, \end{align*}$$

we deduce that

$$\begin{align*}F(\chi;\mathrm{triv},K) = \sum_{P\supset P_0}\sum_{\pi\in\Pi_2(M_P)}\dim \left(\mathcal{A}^2_{\chi,\pi}(P)^{K_\infty K}\right) \ll_{n,K} 1. \end{align*}$$

Let $\chi _0$ be a cuspidal datum and $\pi _0\in \Pi _2(M_P)$ , such that $\varphi _0\in \mathcal {A}^2_{\chi _0,\pi _0}(P)^{K_\infty K}$ . Then we have $\nu (\varphi _0)\asymp \nu (\pi _{0,\infty })$ . Now the result follows from Corollary 1.6 along with Propositions 5.2 and 5.4.