2.1 Basic notations

We use adèlic language. Let $r\ge 3$ . For any ring R by $G(R)$ , we denote the set of points $\mathrm {GL}_r(R)/R^\times $ . In this paper, R will denote the adèles $\mathbb {A}$ over $\mathbb {Q}$ or the local fields $\mathbb {R},\mathbb {Q}_p$ or rational numbers $\mathbb {Q}$ or the local ring $\mathbb {Z}_p$ . We drop the ring R from the notation $G(R)$ if the ring is clear from the context.

Let N be the maximal unipotent subgroup of G consisting of upper triangular matrices. For $q\in \mathbb {A}^{r-1}$ , we define a character of $N(\mathbb {A})$ by

$$ \begin{align*}\psi_q(n(x))=\psi_0\left(\sum_{i=1}^{r-1}q_ix_{i,i+1}\right),\quad n(x):=(x_{i,j})_{i,j},\end{align*} $$

where $\psi _0$ is an additive character of $\mathbb {Q}\backslash \mathbb {A}$ . We abbreviate $\psi _{(1,\dots ,1)}$ by $\psi $ . We call $\psi _q$ nondegenerate if $q_i\neq 0$ for $1\le i\le r-1$ ; otherwise, we call $\psi _q$ degenerate.

Let A be the set of diagonal matrices in G, which we identify with $\begin {pmatrix}A_{r-1}&\\&1\end {pmatrix}$ , where $A_{r-1}$ is the set of diagonal matrices in $\mathrm {GL}(r-1)$ . We parametrise elements of $A_{r-1}$ as $a(y):=\mathrm {diag}(y_1\dots y_{r-1},\dots , y_{r-1})$ Let $K:=\prod _{p\le \infty }K_p$ be the standard maximal compact in $G(\mathbb {A})$ , where $K_p:=G(\mathbb {Z}_p)$ for $p<\infty $ and $K_\infty :=\mathrm {PO}_r(\mathbb {R})$ .

For any factorisable function f on $G(\mathbb {A})$ by $f_p$ , we denote the pth component of f, which is a function on $G(\mathbb {Q}_p)$ .

2.2 Domains and measures

We fix Haar measures on G and its subgroups, and a G-invariant measure of $N\backslash G$ . If the subgroup is compact, then we normalise the Haar measure to be a probability measure. Let $\delta $ denote the modular character on A. It is defined by

$$ \begin{align*}\delta(a(y)) := \prod_{j=1}^{r-1} |y_j|^{j(r-1-j)}\end{align*} $$

and is trivially extended to $NA$ .

To integrate over $N(\mathbb {R})\backslash G(\mathbb {R})$ , we use two different types of coordinates according to efficiency. The first one is Bruhat (with respect to the standard maximal parabolic) coordinates. First, note that the set of elements of the form $\begin {pmatrix}h&b^t\\c&0\end {pmatrix}$ with $h\in \mathrm {GL}_{r-1}(\mathbb {R})$ and row vectors $b,c\in \mathbb {R}^{r-1}$ has zero measure in $\mathrm {GL}_r(\mathbb {R})$ with respect to its Haar measure. Thus while integrating over $G(\mathbb {R})$ , we integrate over the points of the form

$$ \begin{align*}\begin{pmatrix}h&b^t\\&1\end{pmatrix}\begin{pmatrix}\mathrm{I}_{r-1}&\\c&1\end{pmatrix},\quad h\in \mathrm{GL}_{r-1}(\mathbb{R})\text{ and } b,c\in\mathbb{R}^{r-1} \text{ row vectors}.\end{align*} $$

Similarly, the set of points of the form

$$ \begin{align*}g = \begin{pmatrix}h&\\&1\end{pmatrix}\begin{pmatrix}\mathrm{I}_{r-1}&\\c&1\end{pmatrix},\quad h\in N_{r-1}(\mathbb{R})\backslash\mathrm{GL}_{r-1}(\mathbb{R})\text{ and } c\in\mathbb{R}^{r-1}\text{ row vector}\end{align*} $$

has full measure in $N(\mathbb {R})\backslash G(\mathbb {R})$ . We use these coordinates to integrate over $N(\mathbb {R})\backslash G(\mathbb {R})$ using the invariant measure

$$ \begin{align*}dg=\frac{dh}{|\det(h)|}dc,\end{align*} $$

where $dc$ denotes the Lebesgue measure and $dh$ is the $\mathrm {GL}_{r-1}(\mathbb {R})$ -invariant Haar measure on $N_{r-1}(\mathbb {R})\backslash \mathrm {GL}_{r-1}(\mathbb {R})$ . Description of the above invariant measure follows from [Reference Knapp23, eq. (5.14)] and discussion above that. However, there is a more direct way to see this. Let $\phi \in C_c(G(\mathbb {R}))$ be measurable. Then it follows from [Reference Goldfeld14, Proposition 1.4.3] that

$$ \begin{align*}\int_{G(\mathbb{R})} \phi(g) dg = \int_{\mathbb{R}^{n^2-1}}\phi\left[\begin{pmatrix}h&b^t\\c&1\end{pmatrix}\right]\frac{\prod_{i,j}d_L h_{ij}\prod_id_Lb_i\prod_i d_Lc_i}{\left|\det\left[\begin{pmatrix}h&b^t\\c&1\end{pmatrix}\right]\right|^n},\end{align*} $$

where $d_Lx$ denotes the Lebesgue measure on $\mathbb {R}$ . Noting that

$$ \begin{align*}\begin{pmatrix}h&b^t\\c&1\end{pmatrix}=\begin{pmatrix}\mathrm{I}_{r-1}&b^t\\&1\end{pmatrix}\begin{pmatrix}h-b^tc\\&1\end{pmatrix}\begin{pmatrix}\mathrm{I}_{r-1}&\\c&1\end{pmatrix},\end{align*} $$

we can write

$$ \begin{align*}\int_{G(\mathbb{R})} \phi(g) dg = \int_{\mathbb{R}^{n^2-1}}\phi\left[\begin{pmatrix}\mathrm{I}_{r-1}&b^t\\&1\end{pmatrix}\begin{pmatrix}h-b^tc\\&1\end{pmatrix}\begin{pmatrix}\mathrm{I}_{r-1}&\\c&1\end{pmatrix}\right]\frac{\prod_{i,j}d_L h_{ij}\prod_id_Lb_i\prod_id_Lc_i}{|\det(h-b^tc)|^n}.\end{align*} $$

Fixing $b,c$ and changing variables $h_{ij}\mapsto h_{ij}+(b^tc)_{ij}$ , we can write the above as

$$ \begin{align*}\int_{G(\mathbb{R})} \phi(g) dg = \int_{\mathbb{R}^{n^2-1}}\phi\left[\begin{pmatrix}\mathrm{I}_{r-1}&b^t\\&1\end{pmatrix}\begin{pmatrix}h\\&1\end{pmatrix}\begin{pmatrix}\mathrm{I}_{r-1}&\\c&1\end{pmatrix}\right]\frac{\prod_{i,j}d_L h_{ij}\prod_id_Lb_i\prod_id_Lc_i}{|\det(h)|^n}.\end{align*} $$

Noting that $\frac {\prod _{i,j}d_Lh_{ij}}{|\det (h)|^{n-1}}=dh$ and taking the $N(\mathbb {R})$ -quotient on the left, we deduce the invariant measure on $N(\mathbb {R})\backslash G(\mathbb {R})$ .

On the other hand, when we integrate on $N_{r-1}(\mathbb {R})\backslash \mathrm {GL}_{r-1}(\mathbb {R})$ , we use Iwasawa coordinates. We write

$$ \begin{align*}N_{r-1}(\mathbb{R})\backslash\mathrm{GL}_{r-1}(\mathbb{R})\ni h=a(y)k,\quad a(y)\in A_{r-1}, k\in K_{r-1},\end{align*} $$

where $K_{r-1}$ is the standard maximal compact in $\mathrm {GL}_{r-1}$ , with the measure

$$ \begin{align*}dh=\frac{\prod_id^\times y_i}{\delta(a(y))}dk,\end{align*} $$

where $dk$ is the probability Haar measure on $K_{r-1}$ .

Let $\mathbb {X}:=G(\mathbb {Q})\backslash G(\mathbb {A})$ . We fix a fundamental domain $\mathbb {X}$ in $G(\mathbb {A})$ of the form

$$ \begin{align*}D\times K_f,\quad D\subseteq G(\mathbb{R}), K_f:=\prod_{p<\infty}K_p\end{align*} $$

that is contained in a Siegel domain of the form $\mathbb {S}\times K_f$ , where

(2.1) $$ \begin{align} \mathbb{S}:=\{G(\mathbb{R})\ni g=n(x)\begin{pmatrix}a(y)&\\&1\end{pmatrix}k\mid |x_{i,j}|<1,|y_i|> y_0, k\in K_\infty\}, \end{align} $$

where $y_0>0$ is an explicit constant dependent only on the group. The above follows from strong approximation for $\mathrm {GL}(n)$ and [Reference Goldfeld14, §1.3].

We equip $\mathbb {X}$ with the $G(\mathbb {A})$ -invariant probability measure that in Iwasawa coordinates is given by

$$ \begin{align*}\mathbb{X}\ni g=n(x)\begin{pmatrix}a(y)&\\&1\end{pmatrix}k,\quad dg=\prod_{j,k}dx_{j,k}\frac{\prod_id^\times y_i}{\delta(a(y))|\det(a(y))|}dk,\end{align*} $$

where $n(x)\in N(\mathbb {R})$ and $dx_{i,j}$ is the usual Lebesgue measure. Note that $\delta \begin {pmatrix}a(y)&\\&1\end {pmatrix}= \delta (a(y))|\det (a(y))|$ .

2.3 Automorphic representations

We briefly describe the classes of local and global representations that are relevant in this paper. We refer to [Reference Mœglin and Waldspurger28], [Reference Cogdell and Piatetski-Shapiro12, §5] for details.

Let $\hat {\mathbb {X}}$ be the isomorphism class of irreducible unitary automorphic representations that are unramified at all finite places and appear in the spectral decomposition of $L^2(\mathbb {X})$ . Similarly, by $\hat {\mathbb {X}}_{\mathrm {gen}}$ , we denote the subclass of generic representations in $\hat {\mathbb {X}}$ : that is, the class of representations that have (unique) Whittaker models.

We first mention the Langlands description for $\hat {\mathbb {X}}_{\mathrm {gen}}$ . We take a partition $r=r_1+\dots +r_k$ . Let $\pi _j$ be a unitary cuspidal automorphic representation for $\mathrm {GL}_{r_j}(\mathbb {Q})$ (if $r_j=1$ , we take $\pi _j$ to be a unitary Hecke character). Consider the normalised parabolic induction $\Pi $ from the Levi $\mathrm {GL}(r_1)\times \dots \times \mathrm {GL}(r_k)$ to G of the tensor product $\pi _1\otimes \dots \otimes \pi _k$ . There exists a unique irreducible constituent of $\Pi $ , which we denote by the isobaric sum $\pi _1\boxplus \dots \boxplus \pi _k$ . Then the Langlands classification says that every element in $\hat {\mathbb {X}}_{\mathrm {gen}}$ is isomorphic to an isobaric sum $\boxplus _{j=1}^{k'}\pi ^{\prime }_j$ for some partition $r=\sum _{j=1}^{k'}r^{\prime }_j$ and some cuspidal representation $\pi ^{\prime }_j$ of $\mathrm {GL}(r^{\prime }_j)$ .

We recall from [Reference Mœglin and Waldspurger28] that we call $\pi '\in \hat {\mathbb {X}}$ a discrete series if $\pi '$ appears discretely in the spectral decomposition of $L^2(\mathbb {X})$ . The elements of $\pi '$ are square-integrable automorphic forms for $G(\mathbb {Q})$ . Mœglin–Waldspurger classified the discrete series for $G(\mathbb {Q})$ via the iterated residues of generic automorphic forms; see [Reference Mœglin and Waldspurger28]. The Langlands description of $\hat {\mathbb {X}}$ says that every element in $\hat {\mathbb {X}}$ is isomorphic to an isobaric sum $\boxplus _{j=1}^{k'}\pi ^{\prime }_j$ for some partition $r=\sum _{j=1}^{k'}r^{\prime }_j$ and some discrete series representation $\pi ^{\prime }_j$ of $\mathrm {GL}(r^{\prime }_j)$ .

We fix an automorphic Plancherel measure $d\mu _{\mathrm {aut}}$ on $\hat {\mathbb {X}}$ compatible with the invariant probability measure on $\mathbb {X}$ . If $\pi $ is a discrete series, then $d\mu _{\mathrm {aut}}(\pi )$ is absolutely continuous to the counting measure at $\pi $ . On the other hand, if $\pi $ is an Eisenstein series induced from a twisted discrete series $\pi '|.|^\lambda $ , where $\pi '$ is a discrete series on a Levi subgroup M and $\lambda $ lies in the purely imaginary dual of the Cartan subalgebra of M, then $d\mu _{\mathrm {aut}}(\pi )$ is absolute continuous to the product of the counting measure at $\pi '$ and the Lebesgue measure at $\lambda $ .

For any $\pi \in \hat {\mathbb {X}}$ , we denote the pth component of $\pi $ by $\pi _p$ for $p\le \infty $ . The generalised Ramanujan conjecture predicts that if $\pi $ is cuspidal, then $\pi _p$ is tempered for all $p\le \infty $ . In this paper, we assume that certain cuspidal representations are $\vartheta $ -tempered at the archimedean place, whose definition we recall below.

First we describe the Langlands description of the generic representations of $G(\mathbb {R})$ . Let $r'\in \{1,2\}$ and $\sigma $ be an essentially square integrable (square integrable mod centre) representation of $\mathrm {GL}_{r'}(\mathbb {R})$ . That is, if $r'=2$ , then $\sigma $ is a discrete series of $\mathrm {GL}_2(\mathbb {R})$ ; and if $r'=1$ , then $\sigma $ is a unitary character of $\mathrm {GL}_1(\mathbb {R})$ . By the Langlands classification, we know that any generic unitary irreducible representation $\xi $ of $G(\mathbb {R})$ is isomorphic to a normalised parabolic induction of

$$ \begin{align*}\sigma_1|\det|^{s_1}\otimes\dots\otimes\sigma_k|\det|^{s_k},\end{align*} $$

from a Levi subgroup attached to a partition of $r=\sum _{j=1}^k r_j$ with $r_j\in \{1,2\}$ , where $\sigma _j$ is an essentially square integrable representation of $\mathrm {GL}(r_j)$ and $s_j\in \mathbb {C}$ with $\sum _{j=1}^k r_js_j=0$ and $\Re (s_1)\ge \dots \ge \Re (s_k)$ .

Let $\vartheta \geq 0$ . We say that $\xi $ is $\vartheta $ -tempered if all such $s_i$ have real parts in $[-\vartheta ,\vartheta ]$ . By [Reference Müller and Speh26], if $\pi $ is cuspidal, then $\pi _\infty $ is $\vartheta $ -tempered with $\vartheta = 1/2 - 1/(1+r^2)$ .

We denote the analytic conductor of $\pi $ by $C(\pi )$ . Note that as $\pi \in \hat {\mathbb {X}}$ is unramified at all the finite places, we have $C(\pi )=C(\pi _\infty )$ . If $\{\mu _i\}\in \mathbb {C}^r$ are the Langlands parameters of $\pi _\infty $ , then we define (see [Reference Iwaniec and Sarnak15]) $C(\pi _\infty ):=\prod _{i=1}^r(1+|\mu _i|)$ .

2.4 Maximal Eisenstein series

Let P be the standard parabolic subgroup in G attached to the $r=(r-1)+1$ partition. We choose a generalised principal series vector

$$ \begin{align*} f_s\in \mathcal{I}_{r-1,1}(s):=\mathrm{Ind}_{P(\mathbb{A})}^{G(\mathbb{A})}|\det|^{s}\boxplus |.|^{-(r-1)s},\quad s\in \mathbb{C}, \end{align*} $$

by

(2.2) $$ \begin{align} f_s(g)=f_{s,\Phi}(g):=\int_{\mathbb{A}^\times}\Phi(te_rg)|\det(tg)|^sd^\times t, \end{align} $$

where $\Phi \in \mathcal {S}(\mathbb {A}^r)$ is a Schwartz–Bruhat, factorisable function and $e_r=(0,\dots ,0,1)\in \mathbb {A}^r$ . The integral in equation (2.2) converges for $\Re (s)>1/r$ and then can be extended meromorphically to the whole complex plane. By $\hat {\Phi }$ , we denote Fourier transform of $\Phi $ that is defined by

$$ \begin{align*}\hat{\Phi}(x):=\int_{\mathbb{A}^r}\Phi(u)\psi_0(x_1u_1+\dots++x_ru_r)du.\end{align*} $$

We abbreviate $f_{s,\hat {\Phi }}$ as $\hat {f}_s$ . We record the following transformation property of $f_s$ , which can be seen from equation (2.2):

(2.3) $$ \begin{align} f_s\left[\begin{pmatrix}h&\\&1\end{pmatrix}g\right]=|\det(h)|^sf_s(g),\quad h\in \mathrm{GL}_{r-1}(\mathbb{A}),g\in G(\mathbb{A}). \end{align} $$

Finally, we define the maximal Eisenstein series associated to $f_s$ by

$$ \begin{align*} \mathrm{Eis}(f_{s})(g):=\sum_{\gamma\in P(\mathbb{Q})\backslash G(\mathbb{Q})}f_s(\gamma g). \end{align*} $$

The above definition is valid for s in a right half plane and then can be extended to all of $\mathbb {C}$ by meromorphic continuation. From [Reference Jacquet and Shalika21, §4], [Reference Cogdell11, §2.3.1], we know that for $f_{s,\Phi }\in \mathcal {I}_{r-1,1}(s)$ with some $\Phi \in \mathcal {S}(\mathbb {A}^r)$ , the maximal parabolic Eisenstein series $\mathrm {Eis}(f_{s,\Phi })$ has at most simple poles at $s=0$ and $s=1$ . The residues at these poles are independent of g.

Let $\tilde {P}$ be the maximal parabolic subgroup in G attached to the partition $r=1+(r-1)$ (the associate parabolic to P). We can similarly construct an associated Eisenstein series from a vector $\tilde {f}_s\in \mathcal {I}_{1,r-1}(s)$ defined analogously. All of the properties of an Eisenstein series associated to P hold analogously for the same associated to $\tilde {P}$ .

2.5 Genericity and Kirillov model

We briefly review the Whittaker and Kirillov models of a generic representation of G over a local field; see [Reference Jacquet16] for details. In this subsection, we only work locally, without mentioning the underlying local field. Fix a nondegenerate additive character $\psi $ of $N<G$ . Consider the space of Whittaker functions on G by

$$ \begin{align*}\mathcal{W}(G):=\left\lbrace W\in C^\infty(G)\middle| \begin{aligned} & W(ng)=\psi(n)W(g),n\in N, g\in G;\\& W \text{ grows at most polynomially in }g \end{aligned}\right\rbrace,\end{align*} $$

on which G acts by right translation.

We call an irreducible representation $\pi $ of G generic if there exists a G-equivariant embedding $\pi \hookrightarrow \mathcal {W}(G)$ . For generic $\pi $ , we identify $\pi $ with its image in $\mathcal {W}(G)$ , which we call the Whittaker model of $\pi $ under this embedding.

It is known (for example, see [Reference Jacquet16] for the case of an archimedean local field) from the theory of the Kirillov model that if $\pi $ is an irreducible generic representation of $\mathrm {PGL}(r)$ , then

$$ \begin{align*}\pi\ni W\mapsto \left\lbrace\mathrm{GL}(r-1)\ni g\mapsto W\left[\begin{pmatrix}g&\\&1\end{pmatrix}\right]\right\rbrace\end{align*} $$

is injective and the space of the restricted Whittaker functions in the right-hand side, which is called the Kirillov model, is isomorphic to $\pi $ as well. It is also known that the space $C_c^\infty (N_{r-1}(\mathbb {R})\backslash \mathrm {GL}_{r-1}(\mathbb {R}),\psi )$ is contained in $\pi $ under this realisation; see [Reference Jacquet16, Proposition 5].

If $\pi $ is also unitary, then we can give a unitary structure on its Whittaker model by the inner product

$$ \begin{align*}\langle W_1,W_2\rangle :=\int_{N_{r-1}(\mathbb{R})\backslash\mathrm{GL}_{r-1}(\mathbb{R})}W_1\left[\begin{pmatrix}h&\\&1\end{pmatrix}\right]\overline{W_2\left[\begin{pmatrix}h&\\&1\end{pmatrix}\right]}dh;\end{align*} $$

that is, we have $\langle W_1(.g),W_2(.g)\rangle =\langle W_1,W_2\rangle $ for $g\in G$ .

2.6 Zeta integrals

We review the theory of the $\mathrm {GL}(r)\times \mathrm {GL}(r)$ Rankin–Selberg integral. We refer to [Reference Cogdell11] for details. We choose $\phi _0\in \pi _0$ with a factorisable Whittaker function $W_{0}$ and a maximal Eisenstein series $\mathrm {Eis}(f_s)$ attached to some vector $f_s$ in the generalised Principal series $\mathcal {I}_{r-1,1}(s)$ , as defined in Section 2.4. Let $\pi \in \hat {\mathbb {X}}$ and $\phi \in \pi $ be any automorphic form. One defines the global Rankin–Selberg integral [Reference Cogdell11, §2.3.2] of $\phi _0$ and $\phi $ by

$$ \begin{align*} \Psi(f_s,\phi_0,\bar{\phi}):=\int_{G(\mathbb{Q})\backslash G(\mathbb{A})}\phi_0(g)\overline{\phi(g)}\mathrm{Eis}(f_s)(g)dg, \end{align*} $$

where $s\in \mathbb {C}$ is such that $\mathrm {Eis}(f_s)$ is regular. As $\phi _0$ is cuspidal, the above integral converges absolutely. For s in a right half plane performing a standard unfolding-folding, one gets

$$ \begin{align*}\Psi(f_s,\phi_0,\bar{\phi})=\int_{P(\mathbb{Q})\backslash G(\mathbb{A})}\phi_0(g)\overline{\phi(g)}f_s(g)dg.\end{align*} $$

The above integral representation of $\Psi (f_s,\phi _0,\bar {\phi })$ has a meromorphic continuation to all $s\in \mathbb {C}$ . It is known that if $\pi $ and $\pi _0$ are cuspidal, then the only possible poles of $\Psi $ are simple and can occur at $\Re (s)=0,1$ .

We may choose $f_s$ to be factorisable, which can be done by choosing $\Phi \in \mathcal {S}(\mathbb {A}^r)$ , as in Section 2.4, to be factorisable. Furthermore, if we assume that $\pi $ is generic and $\phi \in \pi $ has a factorisable Whittaker function $W_\phi $ , then for all $s\in \mathbb {C}$ , the global zeta integral is Eulerian – that is, factors in local zeta integrals

$$ \begin{align*} \Psi(f_s,\phi,\bar{\phi})=\Psi_\infty(f_{s,\infty},W_{0,\infty},\overline{W_{\phi,\infty}})\prod_{p<\infty}\Psi_p(f_{s,p},W_{0,p},\overline{W_{\phi,p}}), \end{align*} $$

where the local zeta integral $\Psi _\infty $ is defined by

$$ \begin{align*} \Psi_\infty(f_{s,\infty},W_{0,\infty}, \overline{W_{\phi,\infty}}):&=\int_{N(\mathbb{R})\backslash G(\mathbb{R})}W_{0,\infty}(g)\overline{W_{\phi,\infty}(g)}f_{s,\infty}(g) dg\\ &=\int_{N(\mathbb{R})\backslash \mathrm{GL}_r(\mathbb{R})}W_{0,\infty}(g)\overline{W_{\phi,\infty}(g)}\Phi_{\infty}(e_rg)|\det(g)|^sdg, \end{align*} $$

for s being in some right half plane and then can be meromorphically continued to the whole complex plane. Similarly, the nonarchimedean zeta integral $\Psi _p$ is defined by replacing $\infty $ with p and $\mathbb {R}$ with $\mathbb {Q}_p$ . It is known that if $\pi _{0,p}$ and $\pi _p$ are unitary and $\vartheta _0$ and $\vartheta $ tempered, respectively, then the above integral representation of $\Psi _p$ is valid for $\Re (s)\ge 1/2$ if $\vartheta +\vartheta _0 < 1/2$ and $p\le \infty $ (this can be seen in the archimedean case from the bounds of the Whittaker functions in Lemma 7.2).

We record the local functional equation satisfied by $\Psi _\infty $ . From [Reference Cogdell11, Theorem 3.2], we have

$$ \begin{align*} \int_{N(\mathbb{R})\backslash \mathrm{GL}_r(\mathbb{R})} &\tilde{W}_{0,\infty}(g){\tilde{W}_{\phi,\infty}(g)}\hat{\Phi}_{\infty}(e_rg)|\det(g)|^{1-s}dg\\&\quad=\gamma_\infty(s,\pi_{0,\infty}\otimes{\pi}_\infty,\psi)\int_{N(\mathbb{R})\backslash \mathrm{GL}_r(\mathbb{R})}W_{0,\infty}(g){W_{\phi,\infty}(g)}\Phi_{\infty}(e_rg)|\det(g)|^sdg. \end{align*} $$

Here $\tilde {W}$ denotes the contragredient Whittaker function of W defined by $\tilde {W}(g):=W(wg^{-t})$ , where w is the long Weyl element in $G(\mathbb {R})$ and $\gamma _\infty (.,.,\psi )$ denotes the local archimedean $\gamma $ -factor. As the additive character $\psi $ is fixed throughout the paper, we drop $\psi $ from the notation of $\gamma _\infty $ . Folding the above integrals over $\mathbb {R}^\times $ , we can also rewrite the local functional equation as

(2.4) $$ \begin{align} \Psi_\infty(\hat{f}_{1-s,\infty}, \tilde{W}_{0,\infty},\overline{\tilde{W}_{\infty}})=\gamma_\infty(s,\pi_{0,\infty}\otimes\bar{\pi}_{\infty})\Psi_\infty(f_{s,\infty},W_{0,\infty},\overline{W_{\infty}}), \end{align} $$

for any $W_\infty \in \pi _\infty $ , and f is related to $\Phi $ according to equation (2.2). From the definition of the $\gamma $ -factors (see [Reference Cogdell11, p. 120]), one can check that $|\gamma _\infty (1/2,\Pi )|=1$ if $\Pi $ is unitary.

2.7 Plancherel formula

We refer to [Reference Michel and Venkatesh27, §2.2] for a more detailed discussion of the Plancherel formula.

Recall the automorphic Plancherel measure $d\mu _{\mathrm {aut}}$ on $\hat {\mathbb {X}}$ from Section 2.3. Let $\phi _1,\phi _2\in C^\infty (\mathbb {X})$ with rapid decay at all cusps. We record a Plancherel formula (i.e., a spectral decomposition) of the inner product between $\phi _1$ and $\phi _2$ ,

(2.5) $$ \begin{align} \langle \phi_1,\phi_2\rangle = \int_{\hat{\mathbb{X}}}\sum_{\phi\in\mathcal{B}(\pi)}\langle \phi_1,\phi\rangle\langle \phi,\phi_2\rangle d\mu_{\mathrm{aut}}(\pi), \end{align} $$

where $\mathcal {B}(\pi )$ is an orthonormal basis of $\pi $ and

$$ \begin{align*}\langle f_1,f_2\rangle:=\int_{\mathbb{X}}f_1(g)\overline{f_2(g)}dg.\end{align*} $$

The identity equation (2.5) is independent of choice of $\mathcal {B}(\pi )$ .

Rapid decay properties of $\phi _i$ imply that all the inner products on the right-hand side of equation (2.5) converge. One can show by the trace class property in $L^2(\mathbb {X})$ of some inverse Laplacian that the right-hand side of equation (2.5) converges absolutely.

2.8 Spectral weights

Let $\pi ,\pi _0\in \hat {\mathbb {X}}_{\mathrm {gen}}$ with $W_{0,\infty }\in \pi _{0,\infty }$ and $f_s\in \mathcal {I}_{r-1,1}(s)$ . We define the spectral weight

(2.6) $$ \begin{align} J({f_{s,\infty}W_{0,\infty}},\pi_\infty):=\sum_{W_\infty\in\mathcal{B}(\pi_\infty)}|\Psi_\infty(f_{s,\infty},W_{0,\infty}, \overline{W_\infty})|^2. \end{align} $$

Here $\mathcal {B}(\pi _\infty )$ is an orthonormal basis of $\pi _\infty $ . The sum in the right-hand side of equation (2.6) is absolutely convergent and is independent of choice of $\mathcal {B}(\pi _\infty )$ ; see [Reference Baruch and Mao5, Appendix 4].

The definition of J involves only the archimedean components of the representations and functions. In fact, one can define J for any irreducible generic unitary representation $\sigma $ of $G(\mathbb {R})$ and $\beta \in C^\infty (N(\mathbb {R})\backslash G(\mathbb {R}),\psi _\infty )$ with sufficient decay at infinity, by

$$ \begin{align*} J(\beta,\sigma):=\sum_{W\in\mathcal{B}(\sigma)}\left|\int_{N(\mathbb{R})\backslash G(\mathbb{R})}\beta(g)\overline{W(g)}dg\right|^2. \end{align*} $$

Then, using the Whittaker–Plancherel formula (see [Reference Wallach33, Chapter 15]), one can obtain that

(2.7) $$ \begin{align} \int_{\widehat{G(\mathbb{R})}}J(\beta,\sigma)d\mu_{\mathrm{loc}}(\sigma)=\int_{N(\mathbb{R})\backslash G(\mathbb{R})}|\beta(g)|^2dg=:\|\beta\|^2_{L^2(N(\mathbb{R})\backslash G(\mathbb{R}))}. \end{align} $$

Here $\widehat {G(\mathbb {R})}$ is the tempered unitary dual of $G(\mathbb {R})$ equipped with the local Plancherel measure $d\mu _{\mathrm {loc}}$ compatible with the chosen Haar measure on $G(\mathbb {R})$ (see [Reference Brumley and Milićević6, §4.13.2]).

2.9 Analytic newvectors

Analytic newvectors are certain approximate archimedean analogues of the classical nonarchimedean newvectors pioneered by Casselman [Reference Casselmann8] and Jacquet–Piatetski-Shapiro–Shalika [Reference Jacquet, Piatetski-Shapiro and Shalika20]. Let $K_0(p^N)\subset \mathrm {PGL}_r(\mathbb {Z}_p)$ be the subgroup of matrices whose last rows are congruent to $(0,\dots ,0,*)\mod p^N$ . Let $\sigma $ be a generic irreducible representation of $\mathrm {PGL}_r(\mathbb {Q}_p)$ , and let $N_0$ be the minimal nonnegative integer such that $\sigma $ contains a nonzero vector v that is invariant by $K_0(p^{N_0})$ . Let $C(\sigma )$ be the conductor of $\sigma $ , which can be defined in terms of the local gamma factor attached to $\sigma $ . Then the main theorem of [Reference Casselmann8, Reference Jacquet, Piatetski-Shapiro and Shalika20] states that the real number $p^{N_0}$ is equal to $C(\sigma )$ . One calls such a v a newvector of $\sigma $ . In [Reference Jacquet, Piatetski-Shapiro and Shalika20], the authors call newvectors essential vectors; and in some literature, the authors call them newforms.

In [Reference Jana and Nelson19] the authors produce an approximate analogue of this theorem at the archimedean place. Let $X>1$ be tending to infinity and $\tau>0$ be sufficiently small but fixed. We define an approximate congruence subgroup $K_0(X,\tau )\subseteq \mathrm {PGL}_r(\mathbb {R})$ , which is an archimedean analogue of the subgroup $K_0(p^N)$ , in the following way: it is the image in $\mathrm {PGL}_r(\mathbb {R})$ of

(2.8) $$ \begin{align} \left\{\begin{pmatrix} a&b\\c&d \end{pmatrix}\in\mathrm{GL}_{r}(\mathbb{R})\middle| \begin{aligned} & a \in \mathrm{GL}_{r-1}(\mathbb{R}),\quad |a - 1_{r-1}| < \tau, \quad |b|<\tau, \\ & d \in \mathrm{GL}_1(\mathbb{R}),\quad |c|<\frac{\tau}{X}, \quad |d-1|<\tau \end{aligned} \right\}. \end{align} $$

Here, $|.|$ denotes an arbitrary fixed norm on the corresponding spaces of matrices. Fix $0\le \vartheta <1/2$ . Then in [Reference Jana and Nelson19, Theorem 1], the authors show that for all $\epsilon>0$ , there is a $\tau>0$ such that for all generic irreducible unitary $\vartheta $ -tempered representation $\pi $ of $\mathrm {PGL}_r(\mathbb {R})$ , there is a unit vector $v\in \pi $ such that

$$ \begin{align*}\|\pi(g)v-v\|_\pi<\epsilon\quad \text{ for all } g\in K_0(C(\pi),\tau),\end{align*} $$

where $C(\pi )$ is the analytic conductor of $\pi $ . We call such a vector v an analytic newvector of $\pi $ .

The authors also prove that [Reference Jana and Nelson19, Theorem 7] any unit vector v in the Kirillov model of $\pi $ that can be given by a function in $C_c^\infty (N_{r-1}(\mathbb {R})\backslash \mathrm {GL}_{r-1}(\mathbb {R}),\psi _\infty )^{\mathrm {O}_{r-1}(\mathbb {R})}$ is a newvector. Moreover, v can be chosen in a way such that if W is the image of v in the corresponding Whittaker model, then also

$$ \begin{align*}|W(g)-W(1)|<\epsilon\end{align*} $$

for all $g\in K_0(C(\pi ),\tau )$ and $W(1)\asymp 1$ .

2.10 Main theorem

Theorem 2.1. Let $r\ge 3$ and X be tending to infinity. Let $\pi _0$ be a fixed cuspidal representation in $\hat {\mathbb {X}}$ such that $\pi _{0,\infty }$ is $\vartheta _0$ -tempered for some $0\le \vartheta _0< 1/(r^2+1)$ . We define a weight function

$$ \begin{align*}J_X:\hat{\mathbb{X}}_{\mathrm{gen}}\to\mathbb{R}_{\ge 0},\end{align*} $$

as in equation (4.5), which satisfies the following properties:

• ○ $J_X(\pi )$ only depends on the archimedean component of $\pi $ (with an abuse of notation, we write $J_X(\pi )=J_X(\pi _\infty )$ ).

• ○ If $\pi _\infty $ is $\vartheta $ -tempered such that $\vartheta +\vartheta _0<1/2$ and $C(\pi _\infty )<X$ , then $J_X(\pi _\infty )\gg _{\pi _0} 1$ .

• ○ $\int _{\widehat {G(\mathbb {R})}}J_X(\pi _\infty )d\mu _{\mathrm {loc}}(\pi _\infty )=X^{r-1}$ .

And finally, we have

$$ \begin{align*}\int_{\hat{\mathbb{X}}_{\mathrm{gen}}}\frac{|L(1/2,\tilde{\pi}\otimes \pi_0)|^2}{\ell(\pi)}{J_X(\pi)}d\mu_{\mathrm{aut}}(\pi)= X^{r-1}\left(r\frac{\zeta(r/2)^2}{\zeta(r)}L(1,\pi_0,\mathrm{Ad})\log X + O_{\pi_0}(1)\right),\end{align*} $$

where $\tilde {\pi }$ is the contragredient of $\pi $ . Here $\ell (\pi )$ is defined as in equation (4.3) and only depends on the nonarchimedean data of $\pi $ .

If $\pi $ is cuspidal, then $\ell (\pi )\asymp L(1,\pi ,\mathrm {Ad})$ with an absolute implied constant and thus $\ell (\pi ) \ll _\epsilon C(\pi )^\epsilon $ , which follows from [Reference Li25].

We note that if $\pi \in \hat {\mathbb {X}}_{\mathrm {gen}}$ , then $\pi _\infty $ is $\vartheta $ -tempered for $\vartheta <1/2-1/(r^2+1)$ , which is a result in [Reference Müller and Speh26]. Thus the $\vartheta _0$ -temperedness assumption of $\pi _{0,\infty }$ in Theorem 2.1 implies that $J_X(\pi )\gg 1$ for all $\pi \in \hat {\mathbb {X}}_{\mathrm {gen}}$ with $C(\pi )<X$ . Moreover, the family

$$ \begin{align*}\mathcal{F}_X^{\,\mathrm{gen}}:=\{\text{generic automorphic representations }\pi\text{ of }\mathrm{PGL}_r(\mathbb{Z})\text{ with }C(\pi)<X\}\end{align*} $$

has $\ell (\pi )^{-1}$ -weighted cardinality $\asymp X^{r-1}$ . This is essentially contained in the proof of [Reference Jana and Nelson19, Theorem 9]. Hence, $J_X$ can be realised as a smoothened characteristic function of the $\ell (\pi )^{-1}$ -weighted family $\mathcal {F}_X^{\,\mathrm {gen}}$ .

Consequently, we have an immediate corollary of Theorem 2.1.

Corollary 2.2. Let $\pi _0$ be as in Theorem 2.1. Then

$$ \begin{align*}\sum_{\substack{{C(\pi)<X}\\{\hat{\mathbb{X}}\ni\pi\mathrm{ cuspidal}}}}\frac{|L(1/2,\pi\otimes \pi_0)|^2}{L(1,\pi,\mathrm{Ad})}\ll_{\pi_0} X^{r-1}\log X\end{align*} $$

as X tends to infinity.

This is the sharpest possible (Lindelöf on average) second-moment estimate of the cuspidal Rankin–Selberg central L-values.

4.1 Choices of the local components

We start by choosing various vectors and auxiliary test functions. Let $\pi _0\in \hat {\mathbb {X}}_{\mathrm {gen}}$ be the fixed cuspidal representation as in Theorem 2.1. Let $\phi _0\in \pi _0$ with Whittaker function $W_0=\bigotimes _{p\le \infty }W_{0,p}$ , such that

$$ \begin{align*}W_{0,p}\text{ are unramified for }p<\infty\text{ with } W_{0,p}(1)=1.\\[-15pt]\end{align*} $$

Here and elsewhere in the paper, the index set $\{p\le \infty \}$ denotes the set of places $\{\infty \}\cup \{p\text { prime in }\mathbb {Z}\}$ .

We choose $W_{0,\infty }\in \pi _{0,\infty }$ so that

$$ \begin{align*}\|W_{0,\infty}\|_{\pi_{0,\infty}}=1,\text{ and }W_{0,\infty}\left[\begin{pmatrix}.&\\&1\end{pmatrix}\right]\in C_c^\infty(N_{r-1}(\mathbb{R})\backslash\mathrm{GL}_{r-1}(\mathbb{R}),\psi_\infty)^{\mathrm{O}_{r-1}(\mathbb{R})},\\[-15pt]\end{align*} $$

whose existence is guaranteed by the theory of Kirillov model. We choose $\mathcal {S}(\mathbb {A}^r)\ni \Phi =\bigotimes _{p\le \infty }\Phi _p$ with

$$ \begin{align*}\Phi_p:=\mathrm{char}({\mathbb{Z}_p^r})=\hat{\Phi}_p\text{ for }p<\infty,\\[-15pt]\end{align*} $$

and for $\tau>0$ sufficiently small but fixed

$$ \begin{align*}\Phi_\infty\in C_c^\infty(B_\tau(0,\dots,0,1)),\\[-15pt]\end{align*} $$

such that $\Phi _\infty $ is nonnegative and has values sufficiently concentrated near $1$ . Here $B_\tau $ denotes the ball of radius $\tau $ . Thus $\Phi _\infty $ can be thought as a smoothened characteristic function of $B_\tau (0,\dots ,0,1)$ .

Let $f_s:=f_{s,\Phi }\in \mathcal {I}_{r-1,1}(s)$ be associated to $\Phi $ according to equation (2.2). The choice of $\Phi _\infty $ ensures that there exist sufficiently small $\tau _1,\tau _2>0$ (depending on $\tau $ and $\Phi _\infty $ ) such that $f_{1/2,\infty }\left [\begin {pmatrix}\mathrm {I}_{r-1}&\\c&1\end {pmatrix}\right ]$ is supported on $|c|\le \tau _1$ and has values $\asymp 1$ for $|c|\le \tau _2$ . This implies that

$$ \begin{align*}\int_{\mathbb{R}^{r-1}}\left|f_{1/2,\infty}\right|^2\left[\begin{pmatrix}\mathrm{I}_{r-1}&\\c&1\end{pmatrix}\right]dc\asymp 1,\\[-15pt]\end{align*} $$

with an absolute implied constant. We renormalise $\Phi _\infty $ by an absolute constant so that the above integral is $1$ .

4.2 Computation of the spectral side

Let $\mathrm {Eis}(f_s):=\mathrm {Eis}(f_{s,\Phi })$ be the maximal Eisenstein series attached to $f_{s,\Phi }\in \mathcal {I}_{r-1,1}(s)$ , which is defined in Section 4.1. Let $X>1$ be a large number tending to infinity and

$$ \begin{align*}A(\mathbb{A})\ni x:=(x_p)_p,\quad x_\infty:=\mathrm{diag}(X,\dots,X,1)\in A(\mathbb{R})\text{ and }x_p=1 \text{ for all } p<\infty.\\[-15pt]\end{align*} $$

Our point of departure is the following period, which we write in two different ways:

(4.1) $$ \begin{align} \int_{\mathbb{X}}|\phi_0(g)|^2|\mathrm{Eis}(f_s)(gx)|^2dg=\langle \phi_0 \mathrm{Eis}(f_s)(.x),\phi_0\mathrm{Eis}(f_s)(.x)\rangle.\\[-15pt]\nonumber \end{align} $$

We use the Parseval relation in equation (2.5) and the notations in Section 2.6 to write the right-hand side of equation (4.1) as

(4.2) $$ \begin{align} \int_{\hat{\mathbb{X}}}\sum_{\phi\in\mathcal{B}(\pi)}|\Psi(f_s(.x),\phi_0,\bar{\phi})|^2d\mu_{\mathrm{aut}}(\pi),\\[-15pt]\nonumber \end{align} $$

where $\mathcal {B}(\pi )$ is an orthonormal basis of $\pi $ .

Proof. For $\Re (s)$ sufficiently large, we have (see Section 2.6)

$$ \begin{align*}\Psi(f_s,\phi_0,\bar{\phi})=\int_{P(\mathbb{Q})\backslash G(\mathbb{A})}\phi_0(g)\overline{\phi(g)}f_s(g)dg.\end{align*} $$

We follow the computation of [Reference Cogdell11, p.104–105]. We use the Fourier expansion

$$ \begin{align*}\phi_0(g)=\sum_{\gamma\in N(\mathbb{Q})\backslash P(\mathbb{Q})}W_0(\gamma g)\end{align*} $$

and the left $P(\mathbb {Q})$ -invariance of $f_s$ and unfold over $P(\mathbb {Q})$ to get

$$ \begin{align*}\Psi(f_s,\phi_0,\bar{\phi})=\int_{N(\mathbb{Q})\backslash G(\mathbb{A})}W_0(g)\overline{\phi(g)}f_s(g)dg.\end{align*} $$

We fold the last integral over $N(\mathbb {A})$ and use the left N-equivariance of $W_0$ and the left N-invariance of $f_s$ to obtain

$$ \begin{align*}\Psi(f_s,\phi_0,\bar{\phi})=\int_{N(\mathbb{A})\backslash G(\mathbb{A})}W_0(g)f_s(g)\int_{N(\mathbb{Q})\backslash N(\mathbb{A})}\overline{\phi(ng)}\psi(n)dn dg.\end{align*} $$

By definition, the inner integral vanishes as $\phi $ is nongeneric. Finally, by analytic continuation of $\Psi $ , we extend the result for all $s\in \mathbb {C}$ .

Thus Lemma 4.1 allows us to reduce the integral in equation (4.2) only over $\hat {\mathbb {X}}_{\mathrm {gen}}$ . Once we restrict to $\pi \in \hat {\mathbb {X}}_{\mathrm {gen}}$ , we can use the Eulerian property of the zeta integral $\Psi $ as in Section 2.6. If $\phi \in \pi $ with $\|\phi \|_\pi =1$ and Whittaker function $W_\phi =\bigotimes _{p\le \infty }W_{p}$ such that $W_{p}$ is unramified and $W_{p}(1)=1$ for $p<\infty $ , then by Schur’s lemma, we have

(4.3) $$ \begin{align} \|\phi\|^2_\pi=\ell(\pi)\|W_\infty\|^2_{\pi_\infty}, \end{align} $$

where $\ell (\pi )$ only depends on the nonarchimedean data of $\pi $ . A standard Rankin–Selberg computationFootnote ² yields that $\ell (\pi )\asymp L(1,\pi ,\mathrm {Ad})$ for a cuspidal $\pi $ . In fact, in our case, $\ell (\pi )$ is equal to $L(1,\pi ,\mathrm {Ad})$ up to a positive constant dependent only on n.

Another standard computation [Reference Cogdell11, Theorem 3.3] shows that

(4.4) $$ \begin{align} \Psi_p(f_{s,p},W_{0,p},\overline{W_p})=L_p(s,\pi_0\otimes\bar{\pi}),\quad p<\infty, \end{align} $$

where $f_s$ is as chosen in Section 4.1 and $L_p(s,.)$ denotes the unramified p-adic Euler factor of $L(s,.)$ . Thus by meromorphic continuation, we have

$$ \begin{align*} \Psi(f_s,\phi_0,\bar{\phi})={L(s,\pi_0\otimes\bar{\pi})}\Psi_\infty(f_{s,\infty},W_{0,\infty},\overline{W_\infty}) \end{align*} $$

for all $s\in \mathbb {C}$ whenever both sides of the above are defined. Using the equation above, and recalling the harmonic weight in equation (4.3) and the spectral weight in equation (2.6), we obtain that equation (4.2) is equal to

$$ \begin{align*} \int_{\hat{\mathbb{X}}_{\mathrm{gen}}}\frac{|L(s,\pi_0\otimes\bar{\pi})|^2}{\ell(\pi)}J(f_{s,\infty}(.x_\infty) W_{0,\infty},\pi_\infty)d\mu_{\mathrm{aut}}(\pi). \end{align*} $$

We appeal to the holomorphicity of the zeta integrals to specify $s=1/2$ and define a normalised spectral weight $J_X(\pi _\infty )$ by

(4.5) $$ \begin{align} J_X(\pi):=J_X(\pi_\infty):=X^{r-1}J(f_{1/2,\infty}(.x_\infty) W_{0,\infty},\pi_\infty). \end{align} $$

Thus we write the main equation of our proof:

(4.6) $$ \begin{align} \int_{\hat{\mathbb{X}}_{\mathrm{gen}}}\frac{|L(1/2,\pi_0\otimes\bar{\pi})|^2}{\ell(\pi)}J_X(\pi)d\mu_{\mathrm{aut}}(\pi)=X^{r-1}\langle |\phi_0|^2, |\mathrm{Eis}(f_{1/2})(.x)|^2\rangle. \end{align} $$

4.3 Computation of the period side

Again recall the choices of the local factors in Section 4.1. We write $f_{s}=\bigotimes _{p\le \infty }f_{s,p}$ ; then for $k_p\in K_p$ ,

$$ \begin{align*} f_{s,p}(k_p)=\int_{\mathbb{Q}_p^\times}\Phi_p(te_rk_p)|\det(tk_p)|^sd^\times t. \end{align*} $$

Here we fix Haar measures $d^\times t$ on $\mathbb {Q}_p^\times $ and (respectively, $dt$ on $\mathbb {Q}_p$ ) such that $\mathrm {vol}(\mathbb {Z}_p^\times )=1$ (respectively, $\mathrm {vol}(\mathbb {Z}_p)=1$ ).

First, we record that $f_{s,p}$ is an unramified vector in $\mathcal {I}_{r-1,1}(s)_p$ and $\tilde {f}_{s,p}$ is an unramified vector in $\mathcal {I}_{1,r-1}(1-s)_p$ , which is a generalised principal series attached to the opposite parabolic of P. Note that $te_rk_p\in \mathbb {Z}_p^r$ if and only if $t\in \mathbb {Z}_p$ . Thus

$$ \begin{align*}f_{s,p}(k_p)=\sum_{m=0}^\infty p^{-mrs}=(1-p^{-rs})^{-1},\end{align*} $$

for $\Re (s)>0$ . Similarly, using equation (3.5), we have

$$ \begin{align*}\tilde{f}_{s,p}(k_p)=f_{1-s,\hat{\Phi}_p}(wk_p^{-t})=(1-p^{-r(1-s)})^{-1}\end{align*} $$

for $\Re (s)<1$ . Thus for g in the fundamental domain of $\mathbb {X}$ , we can write

(4.7) $$ \begin{align} f_s(g)=\zeta(rs)f_{s,\infty}(g_\infty) \end{align} $$

and

(4.8) $$ \begin{align} \tilde{f}_{s}(g)=\zeta(r-rs)\tilde{f}_{s,\infty}(g_\infty), \end{align} $$

for all $s\in \mathbb {C}$ , which can be achieved by meromorphic continuation.

Let $\Re (s)$ be sufficiently small. From Proposition 3.3, we can see that among the constant terms of $\mathrm {Eis}(f_{1/2+s})(g)$ , the terms that do not lie in $L^2(\mathbb {X})$ are $f_{1/2+s}(g)$ and $\tilde {f}_{1/2+s}(g)$ . Similarly, one checks that the constant terms of

$$ \begin{align*}\overline{\mathrm{Eis}(f_{1/2})}\mathrm{Eis}(f_{1/2+s})-\overline{f_{1/2}}f_{1/2+s}-\overline{\tilde{f}_{1/2}}\tilde{f}_{1/2+s}\end{align*} $$

are integrable in $L^2(\mathbb {X})$ . Inspired by this, we define a regularised Eisenstein series of the form

(4.9) $$ \begin{align} \tilde{E}_s:=\overline{\mathrm{Eis}(f_{1/2})}\mathrm{Eis}(f_{1/2+s})-\mathrm{Eis}(\overline{f_{1/2}}f_{1/2+s})-\mathrm{Eis}(\overline{\tilde{f}_{1/2}}\tilde{f}_{1/2+s}). \end{align} $$

Proof of Theorem 2.1

Recall equations (4.6) and (4.9). We write the inner product on the right-hand side of equation (4.6) as

$$ \begin{align*}\lim_{s\to 0}\langle |\phi_0|^2,\tilde{E}_s(.x)\rangle+\lim_{s\to 0}\left[\langle |\phi_0|^2, \mathrm{Eis}(\overline{f_{1/2}}f_{1/2+s})(.x)\rangle+\langle |\phi_0|^2,\mathrm{Eis}(\overline{\tilde{f}_{1/2}}\tilde{f}_{1/2+s})(.x)\rangle\right].\end{align*} $$

The second term is the degenerate term as in equation (5.1). From equation (5.4), Proposition 5.2 and Lemma 5.1, we obtain that the second term above is

$$ \begin{align*}rL(1,\pi_0,\mathrm{Ad})\frac{\zeta(r/2)^2}{\zeta(r)}\log X + O_{\pi_0}(1).\end{align*} $$

On the other hand, we write the first term above, which is the regularised term, as

$$ \begin{align*}\lim_{s\to 0}\int_{\mathbb{X}}|\phi_0|^2(gx^{-1})\overline{\tilde{E}_s(g)}dg,\end{align*} $$

and bound this by

$$ \begin{align*}\|\phi_0\|^2_{L^\infty(\mathbb{X})}\int_{\mathbb{X}}|\tilde{E}_s(g)|dg.\end{align*} $$

From Proposition 6.1, we know that the last integral is convergent for s being sufficiently small, and $\tilde {E}_s$ is holomorphic in a sufficiently small neighbourhood of $s=0$ . Thus, using Cauchy’s residue theorem, we can write the above limit as

$$ \begin{align*}\int_{|s|=\epsilon}\frac{1}{s}\int_{\mathbb{X}}|\phi_0|^2(gx^{-1})\overline{\tilde{E}_s(g)}dg\frac{ds}{2\pi i}\end{align*} $$

for some arbitrary small but fixed $\epsilon>0$ . Applying Proposition 6.1 once again, we confirm the above integral is $O_{\phi _0,\epsilon }(1)$ .

Now nonnegativity and the first property of the spectral weight $J_X(\pi )$ follow from the definition in equation (4.5). The second property follows from Proposition 7.1. Finally, the third property follows from equation (2.7) and Lemma 5.1.