Test Vectors for Nonarchimedean Godement-Jacquet Zeta Integrals

TL;DR

This paper constructs explicit matrix coefficients and Schwartz functions for nonarchimedean GL(n) representations, ensuring the Godement-Jacquet zeta integral equals the associated L-function, thus providing concrete tools for understanding these integrals.

Contribution

It explicitly identifies choices of data that realize the Godement-Jacquet zeta integral as the L-function for nonarchimedean GL(n) representations.

Findings

Existence of explicit matrix coefficient and Schwartz function choices

Zeta integral equals the L-function for these choices

Provides concrete realization of the Godement-Jacquet integral

Abstract

Given an induced representation of Langlands type $(\pi,V)$ of $\mathrm{GL}_n(F)$ with $F$ nonarchimedean, we show that there exist explicit choices of matrix coefficient $\beta$ and Schwartz-Bruhat function $\Phi$ for which the Godement-Jacquet zeta integral $Z(s,\beta,\Phi)$ attains the $L$-function $L(s,\pi)$.

Equations

$$Z(s,\beta ,\Phi )\cdot ={\int }_{{\mathrm{GL}}_n(F)}\beta (g)\Phi (g){\left|\det g\right|}^{s+\frac{n-1}{2}}dg,$$

$$Z(s,\beta ,\Phi )=L(s,\pi ).$$

$$\Phi(x)=\begin{dcases*}1&if $x\in\operatorname{Mat}_{n\times n}(\mathcal{O})$,\\ 0&otherwise.\end{dcases*}$$

$$\Phi(x)\mathrel{\hbox to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\begin{dcases*}\frac{\omega_{\pi}^{-1}\left(x_{n,n}\right)}{\operatorname*{vol}(K_{0}(\mathfrak{p}^{c(\pi)}))}&if $x\in\operatorname{Mat}_{n\times n}(\mathcal{O})$ with $x_{n,1},\ldots,x_{n,n-1}\in\mathfrak{p}^{c(\pi)}$ and $x_{n,n}\in\mathcal{O}^{\times}$,\\ 0&otherwise,\end{dcases*}$$

$$Z(s,\beta ,\Phi )=L(s,\pi ).$$

$$\pi =\text{vbox}\text{leavevmode}\text{resizebox}0.0pt⊞{\text{displaylimits}}_{j=1}^r{\pi }_j\cdot ={\mathrm{Ind}}_{\mathrm{P}(F)}^{{\mathrm{GL}}_n(F)}\text{vbox}\text{leavevmode}\text{resizebox}0.0pt⊠{\text{displaylimits}}_{j=1}^r{\pi }_j.$$

$$\Lambda \left(\pi (u)\cdot v\right)={\psi }_n(u)\Lambda (v)$$

$$f^{\circ }(uag)=f^{\circ }(g){\delta }_n^{1/2}(a)\prod _{i=1}^n|a_i{|}^{t_i}$$

$$f^{\circ }(1_n)=\prod _{i=1}^{n-1}\prod _{j=i+1}^n{\zeta }_F(1+t_i-t_j),{\zeta }_F(s)\cdot =\frac{1}{1-q^{-s}}.$$

$$W^{\circ }(g)\cdot ={\int }_{{\mathrm{N}}_n(F)}{f}^{\circ }(w_{n}ug)\overline{{\psi }_n}(u)du,$$

$$K_0\left({\mathfrak{p}}^m\right)\cdot =\left\{k\in K:k_{n,1},\dots ,k_{n,n-1}\in {\mathfrak{p}}^m\right\}.$$

$$V_{\pi }^K\cdot =\left\{v\in V_{\pi }:\pi (k)\cdot v=v\text{ for all }k\in K\right\}$$

$$V_{\pi }^{K_0({\mathfrak{p}}^m)}\cdot =\left\{v\in V_{\pi }:\pi (k)\cdot v={\omega }_{\pi }(k_{n,n})v\text{ for all }k\in K_0({\mathfrak{p}}^m)\right\}$$

$${\Pi }^m(v)$$

$${\xi }^m(k)$$

$${\Pi }^{c(\pi )}(v)=⟨v,\widetilde{v^{\circ }}⟩v^{\circ },$$

$$\Psi \left(s,W^{\circ },W^{\prime \circ }\right)\cdot =\underset{{\mathrm{N}}_{n-1}(F)\backslash {\mathrm{GL}}_{n-1}(F)}{\int }{W}^{\circ }\left(\begin{array}{cc}g& 0\\ 0& 1\end{array}\right)W^{\prime \circ }(g){\left|\det g\right|}^{s-\frac{1}{2}}dg$$

$$\Psi \left(s,W^{\circ },W^{\prime \circ },{\Phi }^{\circ }\right)\cdot =\underset{{\mathrm{N}}_n(F)\backslash {\mathrm{GL}}_n(F)}{\int }{W}^{\circ }(g)W^{\prime \circ }(g)\Phi (e_{n}g){\left|\det g\right|}^{s}dg$$

$$\Phi^{\circ}(x_{1},\ldots,x_{n})\mathrel{\hbox to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\begin{dcases*}\frac{\omega_{\pi}^{-1}(x_{n})}{\operatorname*{vol}(K_{0}(\mathfrak{p}^{c(\pi)}))}&if $c(\pi)>0$, $x_{1},\ldots,x_{n-1}\in\mathfrak{p}^{c(\pi)}$, and $x_{n}\in\mathcal{O}^{\times}$,\\ 1&if $c(\pi)=0$ and $x_{1},\ldots,x_{n}\in\mathcal{O}$,\\ 0&otherwise.\end{dcases*}$$

$$\begin{gathered} W^{\prime \circ }(g)={\left|\det g\right|}^{t_1^{\prime }+\frac{n-1}{2}}{\int }_{{\mathrm{GL}}_{n-1}(F)}{W}_0^{\prime \circ }(h){\left|\det h\right|}^{-t_1^{\prime }-\frac{n}{2}} \\ \times \underset{{\mathrm{Mat}}_{(n-1)\times 1}(F)}{\int }{\Phi }^{\prime }\left(h^{-1}\left(\begin{array}{cc}{1}_{n-1}& v\end{array}\right)g\right)\psi (e_{n-1}v)dvdh, \end{gathered}$$

$$\Phi^{\prime}(x)\mathrel{\hbox to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\begin{dcases*}1&if $x\in\operatorname{Mat}_{(n-1)\times n}(\mathcal{O})$,\\ 0&otherwise.\end{dcases*}$$

$$f^{\prime \circ }(1_n)$$

$$f^{\prime \circ }(uag)$$

$$f^{\prime \circ }(gk)$$

$$f^{\prime \circ }(g)\cdot ={\left|\det g\right|}^{t_1^{\prime }+\frac{n-1}{2}}{\int }_{{\mathrm{GL}}_{n-1}(F)}{\Phi }^{\prime }\left(h^{-1}\left(\begin{array}{cc}0& 1_{n-1}\end{array}\right)g\right)f_0^{\prime \circ }(h){\left|\det h\right|}^{-t_1^{\prime }-\frac{n}{2}}dh.$$

$$f_0^{\prime \circ }(1_{n-1})$$

$$f_0^{\prime \circ }(u^{\prime }{a}^{\prime }h)$$

$$f_0^{\prime \circ }(h{k}^{\prime })$$

$$W^{\prime \circ }(g)\cdot ={\int }_{{\mathrm{N}}_n(F)}{f}^{\prime \circ }(w_{n}ug){\psi }_n(u)du,$$

$$f^{\prime \circ }(1_n)=f_0^{\prime \circ }(1_{n-1}){\int }_{{\mathrm{N}}_{n-1}(F)}{\int }_{{\mathrm{A}}_{n-1}(F)}{\Phi }^{\prime }\left(\begin{array}{cc}0& a^{\prime }{u}^{\prime }\end{array}\right)\prod _{i=2}^n|a_i^{\prime }{|}^{-t_i^{\prime }}{\delta }_{n-1}^{1/2}(a^{\prime }){\left|\det a^{\prime }\right|}^{t_1^{\prime }+\frac{n}{2}}{d}^{\times }{a}^{\prime }d{u}^{\prime },$$

$$f_0^{\prime \circ }(1_{n-1})\prod _{j=2}^{n-1}\prod _{\ell =j+1}^n{\int }_{\mathcal{O}}d{u}_{j,\ell }^{\prime }\prod _{i=2}^n{\int }_{\mathcal{O}\setminus \{0\}}|a_i^{\prime }{|}^{1+t_1^{\prime }-t_i^{\prime }}{d}^{\times }{a}_i^{\prime }.$$

$$\underset{{\mathrm{N}}_{n-1}(F)\backslash {\mathrm{GL}}_{n-1}(F)}{\int }{W}_0^{\prime \circ }(h){\left|\det h\right|}^{s-\frac{1}{2}}{\int }_{{\mathrm{GL}}_n(F)}{W}^{\circ }\left(\left(\begin{array}{cc}h& 0\\ 0& 1\end{array}\right)g\right)\Phi (g){\left|\det g\right|}^{s+t_1^{\prime }+\frac{n-1}{2}}dgdh,$$

$$\Phi (g)={\int }_K{\xi }^{c(\pi )}(k)\Phi (k^{-1}g)dk,$$

Full text

Research supported by the European Research Council grant agreement 670239.

1. Introduction

Let $F$ be a nonarchimedean local field with ring of integers $\mathcal{O}$, maximal ideal $\mathfrak{p}$, and uniformiser $\varpi$, so that $\varpi\mathcal{O}=\mathfrak{p}$ and $\mathcal{O}/\mathfrak{p}\cong\mathbb{F}_{q}$ for some prime power $q$. We normalise the absolute value $|\cdot|$ on $F$ such that $|\varpi|=q^{-1}$.

Let $(\pi,V_{\pi})$ be a generic irreducible admissible smooth representation of $\mathrm{GL}_{n}(F)$, where $F$ is a nonarchimedean local field. Given a matrix coefficient $\beta(g)=\langle\pi(g)\cdot v_{1},\widetilde{v_{2}}\rangle$ of $\pi$, where $v_{1}\in V_{\pi}$ and $v_{2}\in V_{\widetilde{\pi}}$, and given a Schwartz–Bruhat function $\Phi\in\mathscr{S}(\operatorname{Mat}_{n\times n}(F))$, we define the Godement–Jacquet zeta integral [GJ72, Jac79]

[TABLE]

which is absolutely convergent for $\Re(s)$ sufficiently large. The test vector problem for Godement–Jacquet zeta integrals is the following.

Test Vector Problem**.**

Given a generic irreducible admissible smooth representation $(\pi,V_{\pi})$ of $\mathrm{GL}_{n}(F)$, determine the existence of $K$-finite vectors $v_{1}\in V_{\pi}$, $\widetilde{v_{2}}\in V_{\widetilde{\pi}}$, and a Schwartz–Bruhat function $\Phi\in\mathscr{S}(\operatorname{Mat}_{n\times n}(F))$ such that

[TABLE]

The archimedean analogue of this problem has been resolved for $F=\mathbb{C}$ by Ishii [Ish19] and for $F=\mathbb{R}$ by Lin [Lin18]111The author has been unable to verify certain aspects of [Lin18]. In particular, the functions constructed in [Lin18, (6.5) and (6.7)] are defined only on the maximal compact subgroup $K=\mathrm{O}(n)$ of $\mathrm{GL}_{n}(\mathbb{R})$. For these functions to be elements of certain induced representations of $\mathrm{GL}_{n}(\mathbb{R})$, they must transform under the action of diagonal matrices $a=\operatorname{diag}(a_{1},\ldots,a_{n})\in\mathrm{A}_{n}(\mathbb{R})$ in a specified manner, and this action does not seem to be compatible with the definitions [Lin18, (6.5) and (6.7)] when $k\in K$ is taken to be a diagonal orthogonal matrix.. For nonarchimedean $F$, the spherical case is resolved in [GJ72, Lemma 6.10]: one takes $v_{1}$ and $v_{2}$ to be spherical vectors and

[TABLE]

We solve the ramified case of this problem.

Theorem 1.2**.**

Let $(\pi,V_{\pi})$ be a generic irreducible admissible smooth representation of $\mathrm{GL}_{n}(F)$ of conductor exponent $c(\pi)>0$. Let $\beta(g)$ denote the matrix coefficient $\langle\pi(g)\cdot v^{\circ},\widetilde{v^{\circ}}\rangle$, where $v^{\circ}\in V_{\pi}$ is the newform of $\pi$ normalised such that $\beta(1_{n})=1$. Define the Schwartz–Bruhat function $\Phi\in\mathscr{S}(\operatorname{Mat}_{n\times n}(F))$ by

[TABLE]

where $\omega_{\pi}$ denotes the central character of $\pi$ and the congruence subgroup $K_{0}(\mathfrak{p}^{c(\pi)})$ is as in (3.1). Then for $\Re(s)$ sufficiently large,

[TABLE]

2. Induced Representations of Langlands Type

Rather than working with generic irreducible admissible smooth representations, we will work in the more general setting of induced representations of Langlands type; see [CP-S17, Section 1.5] for further details.

Given representations $\pi_{1},\ldots,\pi_{r}$ of $\mathrm{GL}_{n_{1}}(F),\ldots,\mathrm{GL}_{n_{r}}(F)$, where $n_{1}+\cdots+n_{r}=n$, we form the representation $\pi_{1}\boxtimes\cdots\boxtimes\pi_{r}$ of $\mathrm{M}_{\mathrm{P}}(F)$, where $\boxtimes$ denotes the outer tensor product and $\mathrm{M}_{\mathrm{P}}(F)$ denote the block-diagonal Levi subgroup of the standard parabolic subgroup $\mathrm{P}(F)=\mathrm{P}_{(n_{1},\ldots,n_{r})}(F)$ of $\mathrm{GL}_{n}(F)$. We then extend this representation trivially to a representation of $\mathrm{P}(F)$. By normalised parabolic induction, we obtain an induced representation $\pi$ of $\mathrm{GL}_{n}(F)$,

[TABLE]

When $\pi_{1},\ldots,\pi_{r}$ are irreducible and essentially square-integrable, $\pi_{1}\boxplus\cdots\boxplus\pi_{r}$ is said to be an induced representation of Whittaker type; such a representation is admissible and smooth. Moreover, if each $\pi_{j}$ is of the form $\sigma_{j}\left|\det\right|^{t_{j}}$, where $\sigma_{j}$ is irreducible, unitary, and square-integrable, and $\Re(t_{1})\geq\cdots\geq\Re(t_{r})$, then $\pi$ is said to be an induced representation of Langlands type. Every irreducible admissible smooth representation $\pi$ of $\mathrm{GL}_{n}(F)$ is isomorphic to the unique irreducible quotient of some induced representation of Langlands type. If $\pi$ is also generic, then it is isomorphic to some (necessarily irreducible) induced representation of Langlands type.

An induced representation of Langlands type $(\pi,V_{\pi})$ is isomorphic to its Whittaker model $\mathcal{W}(\pi,\psi)$, the image of $V_{\pi}$ under the map $v\mapsto\Lambda(\pi(\cdot)\cdot v)$, where $\Lambda:V_{\pi}\to\mathbb{C}$ is the unique (up to scalar multiplication) nontrivial Whittaker functional associated to an additive character $\psi$ of $F$. This is a continuous linear functional that satisfies

[TABLE]

for all $v\in V_{\pi}$ and $u\in\mathrm{N}_{n}(F)$, where $\mathrm{N}_{n}(F)$ denotes the unipotent radical of the standard minimal parabolic subgroup and \psi_{n}(u)\mathrel{\hbox to0.0pt{\raisebox{1.29167pt}{\cdot}\hss}\raisebox{-1.29167pt}{\cdot}}=\psi(u_{1,2}+u_{2,3}+\cdots+u_{n-1,n}).

An induced representation of Langlands type $\pi$ is said to be spherical if it has a $K$-fixed vector, where K\mathrel{\hbox to0.0pt{\raisebox{1.29167pt}{\cdot}\hss}\raisebox{-1.29167pt}{\cdot}}=\mathrm{GL}_{n}(\mathcal{O}). Such a spherical representation $\pi$ must be a principal series representation of the form $|\cdot|^{t_{1}}\boxplus\cdots\boxplus|\cdot|^{t_{n}}$; furthermore, the subspace of $K$-fixed vectors must be one-dimensional. This $K$-fixed vector, unique up to scalar multiplication, is called the spherical vector of $\pi$. In the induced model of $\pi$, the normalised spherical vector is the unique smooth right $K$-invariant function $f^{\circ}:\mathrm{GL}_{n}(F)\to\mathbb{C}$ satisfying

[TABLE]

for all $u\in\mathrm{N}_{n}(F)$, $a=\operatorname{diag}(a_{1},\ldots,a_{n})\in\mathrm{A}_{n}(F)\cong F^{n}$, the subgroup of diagonal matrices, and $g\in\mathrm{GL}_{n}(F)$, where \delta_{n}(a)\mathrel{\hbox to0.0pt{\raisebox{1.29167pt}{\cdot}\hss}\raisebox{-1.29167pt}{\cdot}}=\prod_{i=1}^{n}|a_{i}|^{n-2i+1} denotes the modulus character of the standard minimal parabolic subgroup, and normalised such that

[TABLE]

The normalised spherical Whittaker function $W^{\circ}$ in the Whittaker model $\mathcal{W}(\pi,\psi)$ is given by the analytic continuation of the Jacquet integral

[TABLE]

where $w_{n}=\operatorname{antidiag}(1,\ldots,1)$ is the long Weyl element. The Jacquet integral is absolutely convergent if $\Re(t_{1})>\cdots>\Re(t_{n})$ [JS83, Section 3] and extends holomorphically as a function of the complex variables $t_{1},\ldots,t_{n}$ [CS80]. The Haar measure on $\mathrm{N}_{n}(F)$ is $du=\prod_{j=1}^{n-1}\prod_{\ell=j+1}^{n}du_{j,\ell}$, where for $u_{j,\ell}\in F$, $du_{j,\ell}$ is the additive Haar measure on $F$ normalised to give $\mathcal{O}$ volume $1$. With this normalisation of Haar measures and with $\psi$ an unramified additive character of $F$, the normalised spherical vector $W^{\circ}\in\mathcal{W}(\pi,\psi)$ satisfies $W^{\circ}(1_{n})=1$.

3. The Newform

For each nonnegative integer $m$, we define the congruence subgroup $K_{0}(\mathfrak{p}^{m})$ of $K$ by

[TABLE]

Theorem 3.2** ([JP-SS81, Théorème (5)]).**

Let $(\pi,V_{\pi})$ be an induced representation of Langlands type of $\mathrm{GL}_{n}(F)$. Then either $\pi$ is spherical, so that

[TABLE]

is one-dimensional, or $\pi$ is ramified, in which case $V_{\pi}^{K}$ is empty and there exists a minimal positive integer $m=c(\pi)$ for which the vector subspace

[TABLE]

is nontrivial; moreover, $V_{\pi}^{K_{0}(\mathfrak{p}^{c(\pi)})}$ is one-dimensional.

Definition 3.3**.**

The vector $v^{\circ}\in V_{\pi}^{K_{0}(\mathfrak{p}^{c(\pi)})}$, unique up to scalar multiplication, is called the newform of $\pi$. The nonnegative integer $c(\pi)$ is called the conductor exponent of $\pi$, where we set $c(\pi)=0$ if $\pi$ is spherical.

For each $m$, we may view $V_{\pi}^{K_{0}(\mathfrak{p}^{m})}$ as the image of the projection map $\Pi^{m}:V_{\pi}\to V_{\pi}$ given by

[TABLE]

Here $dk$ is the Haar measure on the compact group $K$ normalised to give $K$ volume $1$. In particular, for any $v\in V_{\pi}$, we have that

[TABLE]

where $v^{\circ}\in V_{\pi}^{K_{0}(\mathfrak{p}^{c(\pi)})}$ and $\widetilde{v^{\circ}}\in V_{\widetilde{\pi}}^{K_{0}(\mathfrak{p}^{c(\pi)})}$ are normalised such that $\langle v^{\circ},\widetilde{v^{\circ}}\rangle=1$.

We write $W^{\circ}$ for the newform in the Whittaker model $\mathcal{W}(\pi,\psi)$ normalised such that $W^{\circ}(1_{n})=1$, where $\psi$ is an unramified additive character; we also normalise $v^{\circ}\in V_{\pi}$ and the Whittaker functional $\Lambda$ such that $\Lambda(v^{\circ})=W^{\circ}(1_{n})=1$. Note that if $\pi$ is spherical, then the newform in the Whittaker model is precisely the normalised spherical Whittaker function.

A key property of $W^{\circ}$ is the fact that it is a test vector for certain Rankin–Selberg integrals.

Theorem 3.7** (Jacquet–Piatetski-Shapiro–Shalika [JP-SS81, Théorème (4)], Jacquet [Jac12], Matringe [Mat13, Corollary 3.3]).**

Let $\pi$ be an induced representation of Langlands type, and let $W^{\circ}\in\mathcal{W}(\pi,\psi)$ denote the newform in the Whittaker model. Then for any spherical representation of Langlands type $\pi^{\prime}$ of $\mathrm{GL}_{n-1}(F)$ with normalised spherical Whittaker function $W^{\prime\circ}\in\mathcal{W}\left(\pi^{\prime},\overline{\psi}\right)$, the $\mathrm{GL}_{n}\times\mathrm{GL}_{n-1}$ Rankin–Selberg integral

[TABLE]

is equal to the Rankin–Selberg $L$-function $L(s,\pi\times\pi^{\prime})$.

Here the Haar measure on $\mathrm{GL}_{n}(F)$ is that induced from the Iwasawa decomposition $\mathrm{GL}_{n}(F)=\mathrm{N}_{n}(F)\mathrm{A}_{n}(F)K$, namely $dg=du\,\delta_{n}^{-1}(a)\,d^{\times}a\,dk$, where $d^{\times}a=\prod_{i=1}^{n}d^{\times}a_{i}$ with the multiplicative Haar measure on $F^{\times}$ given by $d^{\times}a_{i}=\zeta_{F}(1)|a_{i}|^{-1}\,da_{i}$.

Theorem 3.9** (Kim [Kim10, Theorem 2.1.1]).**

Let $\pi$ be an induced representation of Langlands type, and let $W^{\circ}\in\mathcal{W}(\pi,\psi)$ denote the newform in the Whittaker model. Then for any spherical representation of Langlands type $\pi^{\prime}$ of $\mathrm{GL}_{n}(F)$ with normalised spherical Whittaker function $W^{\prime\circ}\in\mathcal{W}\left(\pi^{\prime},\overline{\psi}\right)$, the $\mathrm{GL}_{n}\times\mathrm{GL}_{n}$ Rankin–Selberg integral

[TABLE]

is equal to the Rankin–Selberg $L$-function $L(s,\pi\times\pi^{\prime})$, where e_{n}\mathrel{\hbox to0.0pt{\raisebox{1.29167pt}{\cdot}\hss}\raisebox{-1.29167pt}{\cdot}}=(0,\ldots,0,1)\in\operatorname{Mat}_{1\times n}(F) and $\Phi^{\circ}\in\mathscr{S}(\operatorname{Mat}_{1\times n}(F))$ is given by

[TABLE]

4. A Propagation Formula

We now present a propagation formula for spherical Whittaker functions. This is a recursive formula for a $\mathrm{GL}_{n}(F)$ Whittaker function in terms of a $\mathrm{GL}_{n-1}(F)$ Whittaker function.

Lemma 4.1**.**

Let $\pi^{\prime}=|\cdot|^{t_{1}^{\prime}}\boxplus\cdots\boxplus|\cdot|^{t_{n}^{\prime}}$ be a spherical representation of Langlands type of $\mathrm{GL}_{n}(F)$. Then the normalised spherical Whittaker function $W^{\prime\circ}\in\mathcal{W}(\pi^{\prime},\overline{\psi})$ satisfies

[TABLE]

where $W_{0}^{\prime\circ}\in\mathcal{W}(\pi_{0}^{\prime},\overline{\psi})$ is the normalised spherical Whittaker function of the spherical representation of Langlands type \pi_{0}^{\prime}\mathrel{\hbox to0.0pt{\raisebox{1.29167pt}{\cdot}\hss}\raisebox{-1.29167pt}{\cdot}}=|\cdot|^{t_{2}^{\prime}}\boxplus\cdots\boxplus|\cdot|^{t_{n}^{\prime}} of $\mathrm{GL}_{n-1}(F)$ and $\Phi^{\prime}\in\mathscr{S}(\operatorname{Mat}_{(n-1)\times n}(F))$ is the Schwartz–Bruhat function

[TABLE]

Proof.

Let $f^{\prime\circ}$ be the normalised spherical vector in the induced model of $\pi^{\prime}$, so that

[TABLE]

for all $u\in\mathrm{N}_{n}(F)$, $a=\operatorname{diag}(a_{1},\ldots,a_{n})\in\mathrm{A}_{n}(F)$, $g\in\mathrm{GL}_{n}(F)$, and $k\in K$. We claim that $f^{\prime\circ}$ is also given by the Godement section

[TABLE]

Here $f_{0}^{\prime\circ}$ is the normalised spherical vector in the induced model of $\pi_{0}^{\prime}$, so that

[TABLE]

for all $u^{\prime}\in\mathrm{N}_{n-1}(F)$, $a^{\prime}=\operatorname{diag}(a_{2}^{\prime},\ldots,a_{n}^{\prime})\in\mathrm{A}_{n-1}(F)$, $h\in\mathrm{GL}_{n-1}(F)$, and $k^{\prime}\in\mathrm{GL}_{n-1}(\mathcal{O})$. We then insert the identity (4.6) into the Jacquet integral

[TABLE]

write $w_{n}=\left(\begin{smallmatrix}0&1\\ w_{n-1}&0\end{smallmatrix}\right)$ and $u=\left(\begin{smallmatrix}u^{\prime}&0\\ 0&1\end{smallmatrix}\right)\left(\begin{smallmatrix}1_{n-1}&v\\ 0&1\end{smallmatrix}\right)$ for $u^{\prime}\in\mathrm{N}_{n-1}(F)$ and $v\in\operatorname{Mat}_{(n-1)\times 1}(F)$, and make the change of variables $h\mapsto w_{n-1}u^{\prime}h$ to obtain the identity (4.2).

So it remains to show that $f^{\prime\circ}$ is indeed given by (4.6). We first show that this is an element of the induced model of $\pi^{\prime}$, just as in [Jac09, Proposition 7.1]. We replace $g$ with $\left(\begin{smallmatrix}1&v\\ 0&u^{\prime}\end{smallmatrix}\right)\left(\begin{smallmatrix}a_{1}&0\\ 0&a^{\prime}\end{smallmatrix}\right)g$, where $v\in\operatorname{Mat}_{1\times(n-1)}(F)$, $u^{\prime}\in\mathrm{N}_{n-1}(F)$, $a_{1}\in F^{\times}$, and $a^{\prime}\in\mathrm{A}_{n-1}(F)$. Upon making the change of variables $h\mapsto u^{\prime}a^{\prime}h$ and using (4.8), we see that (4.4) is satisfied. Next, we check that $f^{\prime\circ}$ given by (4.6) satisfies (4.5), which follows easily from the fact that $\Phi^{\prime}(xk)=\Phi^{\prime}(x)$ for all $x\in\operatorname{Mat}_{(n-1)\times n}(F)$ and $k\in K$. Finally, we confirm the normalisation (4.3). To see this, we use the Iwasawa decomposition $h=u^{\prime}a^{\prime}k^{\prime}$ in (4.6), in which case the Haar measure is $dh=\delta_{n-1}^{-1}(a^{\prime})\,du^{\prime}\,d^{\times}a^{\prime}\,dk^{\prime}$. The integral over $\mathrm{GL}_{n-1}(\mathcal{O})\ni k^{\prime}$ is trivial. We then make the change of variables $u^{\prime}\mapsto{u^{\prime}}^{-1}$, $a^{\prime}\mapsto{a^{\prime}}^{-1}$, so that

[TABLE]

recalling (4.8). Writing $du^{\prime}=\prod_{j=2}^{n-1}\prod_{\ell=j+1}^{n}du_{j,\ell}^{\prime}$ and $d^{\times}a^{\prime}=\prod_{i=2}^{n}d^{\times}a_{i}^{\prime}$ and making the change of variables $u_{j,\ell}^{\prime}\mapsto{a_{j}^{\prime}}^{-1}u_{j,\ell}^{\prime}$, this becomes

[TABLE]

The integral over $\mathcal{O}\ni u_{j,\ell}^{\prime}$ is $1$, while the integral over $\mathcal{O}\setminus\{0\}\ni a_{i}^{\prime}$ is $\zeta_{F}(1+t_{1}^{\prime}-t_{i}^{\prime})$. Recalling the normalisation (4.7) of $f_{0}^{\prime\circ}(1_{n-1})$, we see that (4.3) is indeed satisfied. ∎

5. Proof of Theorem 1.2

Proof of Theorem 1.2.

Let $\pi$ be a ramified induced representation of Langlands type of $\mathrm{GL}_{n}(F)$, so that $c(\pi)>0$, and let $\pi^{\prime}=|\cdot|^{t_{1}^{\prime}}\boxplus\cdots\boxplus|\cdot|^{t_{n}^{\prime}}$ be an arbitrary spherical representation of Langlands type of $\mathrm{GL}_{n}(F)$. We insert the identity (4.2) for the normalised spherical Whittaker function $W^{\prime\circ}\in\mathcal{W}(\pi^{\prime},\overline{\psi})$ into the $\mathrm{GL}_{n}\times\mathrm{GL}_{n}$ Rankin–Selberg integral (3.10). Just as in [Jac09, Equation (8.1)], we fold the integration over $\mathrm{N}_{n-1}(F)\backslash\mathrm{N}_{n}(F)\cong\operatorname{Mat}_{(n-1)\times 1}(F)\ni v$ and make the change of variables $g\mapsto\left(\begin{smallmatrix}h&0\\ 0&1\end{smallmatrix}\right)g$. In this way, we find that $\Psi(s,W^{\circ},W^{\prime\circ},\Phi^{\circ})$ is equal to

[TABLE]

with \Phi(x)\mathrel{\hbox to0.0pt{\raisebox{1.29167pt}{\cdot}\hss}\raisebox{-1.29167pt}{\cdot}}=\Phi^{\circ}(e_{n}x)\Phi^{\prime}\left(\begin{pmatrix}1_{n-1}&0\end{pmatrix}x\right) as in (1.3).

We claim that

[TABLE]

with $\xi^{c(\pi)}$ as in (3.5). Indeed, $\xi^{c(\pi)}(k)$ vanishes unless $k\in K_{0}(\mathfrak{p}^{c(\pi)})$, in which case $\Phi(k^{-1}g)$ vanishes unless $g\in\operatorname{Mat}_{n\times n}(\mathcal{O})$ with $g_{n,1},\ldots,g_{n,n-1}\in\mathfrak{p}^{c(\pi)}$ and $g_{n,n}\in\mathcal{O}^{\times}$. Then as $k^{-1}\in K_{0}(\mathfrak{p}^{c(\pi)})$, it is easily checked that

[TABLE]

using the fact that $e_{n}k^{-1}g\prescript{t}{}{e_{n}}-e_{n}k^{-1}\prescript{t}{}{e_{n}}g_{n,n}\in\mathfrak{p}^{c(\pi)}$, $e_{n}k^{-1}\prescript{t}{}{e_{n}}k_{n,n}-1\in\mathfrak{p}^{c(\pi)}$, and $c(\omega_{\pi})\leq c(\pi)$. Thus (5.2) follows.

We insert (5.2) into (5.1) and make the change of variables $g\mapsto kg$, so that the integral over $K\ni k$ is

[TABLE]

We note that

[TABLE]

where \beta(g)\mathrel{\hbox to0.0pt{\raisebox{1.29167pt}{\cdot}\hss}\raisebox{-1.29167pt}{\cdot}}=\langle\pi(g)\cdot v^{\circ},\widetilde{v^{\circ}}\rangle, recalling (3.4) and (3.6), so that

[TABLE]

Combining (5.1) with (5.2) and (5.3), we find that

[TABLE]

recalling the definitions (1.1) of the Godement–Jacquet zeta integral and (3.8) of the $\mathrm{GL}_{n}\times\mathrm{GL}_{n-1}$ Rankin–Selberg integral. From Theorems 3.9 and 3.7,

[TABLE]

Moreover, [JP-SS83, (9.5) Theorem] implies that

[TABLE]

Since $L(s,\pi\times\pi_{0}^{\prime})$ is not uniformly zero, we conclude that

[TABLE]

Acknowledgements

The author would like to thank the anonymous referee for many helpful suggestions and corrections.