# Product of Rankin-Selberg convolutions and a new proof of Jacquet’s local
converse conjecture

Pan Yan  Department of Mathematics, The University of Arizona, Tucson, AZ
85721, USA<EMAIL_ADDRESS>and  Qing Zhang  School of Mathematics and
Statistics, Huazhong University of Science and Technology, Wuhan, 430074,
China<EMAIL_ADDRESS>Dedicated to our advisor Professor Jim Cogdell on
the occasion of his 70th birthday.

###### Abstract.

In this article, we construct a family of integrals which represent the
product of Rankin-Selberg $L$-functions of
${\mathrm{GL}}_{l}\times{\mathrm{GL}}_{m}$ and of
${\mathrm{GL}}_{l}\times{\mathrm{GL}}_{n}$ when $m+n<l$. When $n=0$, these
integrals are those defined by Jacquet–Piatetski-Shapiro–Shalika up to a
shift. In this sense, these new integrals generalize Jacquet–Piatetski-
Shapiro–Shalika’s Rankin-Selberg convolution integrals. We study basic
properties of these integrals. In particular, we define local gamma factors
using this new family of integrals. As an application, we obtain a new proof
of Jacquet’s local converse conjecture using these new integrals.

###### Key words and phrases:

Rankin-Selberg convolution, $L$-functions, gamma factors, local converse
theorem

###### 2010 Mathematics Subject Classification:

11F70, 22E50

The first named author is partially supported by an AMS-Simons Travel Grant.
The second named author is partially supported by NSFC grant 12371010.

###### Contents

  1. 1 Introduction
  2. 2 The global integrals
  3. 3 The local integrals
  4. 4 A local converse theorem
  5. 5 Proof of the local converse theorem

##  1\. Introduction

Let $F$ be a global field and let ${\mathbb{A}}$ be its ring of adeles. The
Rankin-Selberg integral theory of $L$-functions for
${\mathrm{GL}}_{n}({\mathbb{A}})\times{\mathrm{GL}}_{m}({\mathbb{A}})$
developed by Jacquet, Piatetski-Shapiro and Shalika in a series of papers, see
[JPSS, Jacquet-Shalika-EulerI, Jacquet-Shalika-EulerII] for example, and also
the notes by Cogdell [Cogdell:Fields, Cogdell:IAS] for a survey and more
references, play important roles in the study of $L$-functions and automorphic
representations. One of the most important applications of Rankin-Selberg
integrals is the proof of converse theorems, see [JPSS-GL3I, JPSS-GL3II] for
the converse theorems for ${\mathrm{GL}}(3)$ and [CPS:converseI,
CPS:converseII] for the converse theorems for general ${\mathrm{GL}}_{n}$. The
converse theorems have become one important tool in establishing the Langlands
functorial lifting, see [CKPSS-functorialclassical, CKPSS-functorial, CPSS-
functoriality-quasisplit, Kim-Shahidi, Kim-functorial, CFK2018] for a few
examples.

The first goal of this article is to construct a family of Rankin-Selberg
integrals which generalize those of Jacquet, Piatetski-Shapiro and Shalika.
More precisely, we present a family of Rankin-Selberg type integrals which, up
to a shift, represents the product Rankin-Selberg $L$-function

$L(s_{1},\pi\times\tau_{1})L(s_{2},\widetilde{\pi}\times\widetilde{\tau}_{2}),$

where $\pi$ is an irreducible cuspidal automorphic representation of
${\mathrm{GL}}_{l}({\mathbb{A}})$, $\tau_{1}$ (resp. $\tau_{2}$) is an
irreducible cuspidal representation of ${\mathrm{GL}}_{m}({\mathbb{A}})$
(resp. ${\mathrm{GL}}_{n}({\mathbb{A}})$). Here $l$ is a positive integer and
$m,n$ are non-negative integers with $m+n<l$. If $n=0$, our integrals
degenerate to those defined by Jacquet–Piatetski-Shapiro–Shalika (JPSS for
abbreviation). In this sense, our integrals indeed generalize the JPSS Rankin-
Selberg convolution integrals.

To give more details, we introduce some notations. For an integer $j$ with
$0\leq j\leq l-m-n-1$, we set $k=l-m-n-1-j$ and consider the embedding
$\iota_{j}:{\mathrm{GL}}_{m+n}\to{\mathrm{GL}}_{l}$ given by

$\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\mapsto\begin{pmatrix}I_{j}&&&&\\\
&a&&b&\\\ &&1&&\\\ &c&&d&\\\ &&&&I_{k}\end{pmatrix}$

for $a\in{\mathrm{Mat}}_{m\times m},b\in{\mathrm{Mat}}_{m\times
n},c\in{\mathrm{Mat}}_{n\times m},d\in{\mathrm{Mat}}_{n\times n}.$ Given an
irreducible cuspidal automorphic representation $\pi$ (resp.
$\tau_{1},\tau_{2}$) of ${\mathrm{GL}}_{l}({\mathbb{A}})$ (resp.
${\mathrm{GL}}_{m}({\mathbb{A}}),{\mathrm{GL}}_{n}({\mathbb{A}})$), we
consider the integral

$I_{j}(\phi,f_{{\bf{s}}})=\int_{{\mathrm{GL}}_{m+n}(F)\backslash{\mathrm{GL}}_{m+n}({\mathbb{A}})}\phi^{\psi}_{Y_{j}}(\iota_{j}(h))E(h,f_{{\bf{s}}})dh.$

Here $\phi\in\pi$ is a cusp form, $\phi^{\psi}_{Y_{j}}$ is a certain Fourier
coefficient of $\phi$ along certain subgroup $Y_{j}\subset{\mathrm{GL}}_{l}$.
Moreover, ${\bf{s}}=(s_{1},s_{2})$ is a pair of complex numbers and
$E(h,f_{{\bf{s}}})$ is the standard Eisenstein series on
${\mathrm{GL}}_{m+n}({\mathbb{A}})$ associated with a section $f_{{\bf{s}}}$
in the representation induced from
$\tau_{1}||^{s_{1}-1/2}\otimes\tau_{2}||^{-s_{2}+1/2}$ on the standard Levi
subgroup of ${\mathrm{GL}}_{m+n}$ with partition $(m,n)$. See §2 for the
unexplained notations.

######  Theorem 1.1.

The integral $I_{j}(\phi,f_{{\bf{s}}})$ converges absolutely and uniformly in
vertical strips for each variable $s_{1},s_{2}$ in ${\bf{s}}=(s_{1},s_{2})$,
away from the poles of Eisenstein series. The integral is Eulerian, and for
decomposing data, for any given ${\bf{s}}$, up to a holomorphic function, the
integral is equal to

$\frac{L^{S}(s_{1}+\frac{k-j}{2},\pi\times\tau_{1})L^{S}(s_{2}-\frac{k-j}{2},\widetilde{\pi}\times\widetilde{\tau}_{2})}{L^{S}(s_{1}+s_{2},\tau_{1}\times\widetilde{\tau}_{2})},$

where $\widetilde{\pi}$ (resp. $\widetilde{\tau}_{2}$) is the contragredient
representation of $\pi$ (resp. $\tau_{2}$), and
$L^{S}(s_{1}+\frac{k-j}{2},\pi\times\tau_{1})$ denotes the partial Rankin-
Selberg L-function of $\pi\times\tau_{1}$. Here $S$ is a finite set of places
which contains all infinite places and outside $S$, $\pi,\tau_{1}$ and
$\tau_{2}$ are unramified.

Theorem 1.1 is proved in §2 and §3. In addition, we also prove the existence
of local gamma factors. More precisely, let $\Psi(W,f_{{\bf{s}}};j)$ be the
local zeta integral in the unfolding of $I_{j}(\phi,f_{{\bf{s}}})$. Here $W$
is a Whittaker function of a local representation $\pi_{v}$ and $f_{{\bf{s}}}$
is a section in the local induced representation by abuse of notation. Then we
prove that there exists a local gamma factor
$\Gamma({\bf{s}},\pi,(\tau_{1},\tau_{2}),\psi)$ such that

$\Psi(W,M(f_{{\bf{s}}});0)=\Gamma({\bf{s}},\pi,(\tau_{1},\tau_{2}),\psi)\Psi(W,f_{{\bf{s}}};0).$

Here $M$ denotes an intertwining operator. See §3 for more details. Here we
remark that if $n=0$, the local zeta integral $\Psi(W,f_{{\bf{s}}};j)$ is
exactly the JPSS local zeta integral and when $l=2r+1$ and $m=n$, then the
local zeta integral $\Psi(W,f_{{\bf{s}}};r-m)$ is the local zeta integral of
${\mathrm{U}}_{2r+1,E/F}\times{\mathrm{Res}}_{E/F}({\mathrm{GL}}_{n})$ at
split places as considered in [BAS:Uodd]. In the above definition of local
gamma factors, we only used the integral when $j=0$. Although we don’t address
it here, it should not be too hard to consider a similar local functional
equation for general $j$ so that it will degenerate to the JPSS local
functional equation for general $j$. Moreover, as it is suggested by the
unramified calculation, we expect that

(1.1)
$\Gamma({\bf{s}},\pi,(\tau_{1},\tau_{2}),\psi)=\frac{\gamma(s_{1}+(k-j)/2,\pi\times\tau_{1},\psi)\gamma(s_{2}+(j-k)/2,\widetilde{\pi}\times\widetilde{\tau}_{2},\psi)}{\gamma(s_{1}+s_{2},\tau_{1}\times\widetilde{\tau}_{2})}.$

Here the gamma factors on the right side are those defined by JPSS or by
Shahidi [Shahidi-on-certain-L-functions, Shahidi-gamma]. The proof of this
expected property is standard and it will be addressed in our sequel paper
[Yan-Zhang].

As we mentioned above, one important application of the JPSS Rankin-Selberg
integrals is the proof of the converse theorems given by Cogdell and
Piatetski-Shapiro in [CPS:converseI, CPS:converseII], which roughly says that
for an admissible irreducible representation $\pi$ of
${\mathrm{GL}}_{l}({\mathbb{A}})$, if $L(s,\pi\times\tau)$ is “nice” (see
[CPS:converseI]*page 165 for the definition) for all irreducible cuspidal
automorphic representation $\tau$ of ${\mathrm{GL}}_{m}({\mathbb{A}})$ with
$1\leq m\leq l-2$, then $\pi$ is cuspidal automorphic. In applications to the
functoriality problems, it is desirable to reduce the number of twists used in
the converse theorem. In this direction, one important open question is the
following

######  Conjecture 1.2 (Jacquet’s global converse conjecture, see
[CPS:converseII]*§8, Conjecture 1).

Let $\pi=\otimes^{\prime}_{v}\pi_{v}$ be an irreducible admissible generic
representation of ${\mathrm{GL}}_{l}({\mathbb{A}})$ such that its central
character is trivial on $F^{\times}$ and its $L$-function $L(s,\pi)$ is
convergent in some half plane. If $L(s,\pi\times\tau)$ is nice for all
irreducible cuspidal automorphic representation $\tau$ of
${\mathrm{GL}}_{m}({\mathbb{A}})$ with $1\leq m\leq[l/2]$, then $\pi$ is
cuspidal automorphic.

After many years of the original proof given in [CPS:converseI,
CPS:converseII], it seems very hard to use the original JPSS integral to
attack the above conjecture. We expect that our new family of integrals might
be useful in the above problem. In fact, assuming the expected property of the
gamma factors (1.1), the condition that $L(s,\pi\times\tau)$ is nice for all
irreducible cuspidal automorphic representation $\tau$ of ${\mathrm{GL}}_{m}$
with $1\leq m\leq[l/2]$ will give us an extra new family of equalities of
integrals besides those JPSS integrals.

Although we don’t know how to attack the above Jacquet’s global converse
conjecture at this moment, in this paper, assuming (1.1), we illustrate the
above idea by giving a new proof of the following

######  Conjecture 1.3 (Jacquet’s local converse conjecture).

Let $F$ be a non-archimedean local field and let $\pi_{1},\pi_{2}$ be two
supercuspidal representations of ${\mathrm{GL}}_{l}(F)$ with the same central
character. If
$\gamma(s,\pi_{1}\times\tau,\psi)=\gamma(s,\pi_{2}\times\tau,\psi)$ for all
irreducible generic representation $\tau$ of ${\mathrm{GL}}_{m}(F)$ with
$1\leq m\leq[l/2]$, then $\pi_{1}\cong\pi_{2}$.

As proved in [JNS], one can drop the supercuspidal condition in the above
conjecture. In fact, what we proved is the following

######  Theorem 1.4 (Theorem 4.1).

Let $F$ be a non-archimedean local field and let $\pi_{1},\pi_{2}$ be two
irreducible supercuspidal representations of ${\mathrm{GL}}_{l}(F)$ with the
same central character. If
$\Gamma({\bf{s}},\pi_{1}\times(\tau_{1},\tau_{2}),\psi)=\Gamma({\bf{s}},\pi_{2}\times(\tau_{1},\tau_{2}),\psi)$
for all irreducible generic representations $\tau_{1}$ (resp. $\tau_{2}$) of
${\mathrm{GL}}_{m}(F)$ (resp. ${\mathrm{GL}}_{n}(F)$) with $0\leq
n\leq[l/2],0\leq m\leq[l/2]$, then $\pi_{1}\cong\pi_{2}$.

Local converse theorems for ${\mathrm{GL}}_{l}$ using twists up to $l-1$ and
$l-2$ have been proved in [Henniart, CPS:converseII, chen]. The Jacquet’s
local converse conjecture has been proved in [chai] and [Jacquet-Liu]
independently. Our new contribution here is to use the new family of
integrals. A proof of Jacquet’s local converse conjecture along this method
was promised in [Sp(2r)]*§8.2 and in [U(2r+1)]*Introduction, where it was
(incorrectly) believed that the integrals of
${\mathrm{U}}_{l,E/F}\times{\mathrm{Res}}_{E/F}({\mathrm{GL}}_{m})$ at split
places for a quadratic extension $E/F$ as developed in [BAS:Uodd] were enough.
As explained above, these integrals are just our new family of integrals when
$m=n$. It turns out that we need to use the whole new family of integrals.
Proof of Theorem 1.4 uses partial Bessel functions developed in [CST] and is
indeed similar as outlined in [Sp(2r)]*§8.2 and in [U(2r+1)]*Introduction.
Similar methods have been successfully used in proving local converse theorems
of other classical groups over $p$-adic fields and $G_{2}$ over finite fields,
see [Sp(2r), U(2r+1), Liu-Zhang:classical, Liu-Zhang:G2, Jo, HL:p-adic,
SO(4)]. See [Liu-Zhang:G2] for more references on local converse problems.

In this paper, we only considered the integrals which represents the product
of Rankin-Selberg $L$-functions ${\mathrm{GL}}_{l}\times{\mathrm{GL}}_{m}$ and
${\mathrm{GL}}_{l}\times{\mathrm{GL}}_{n}$ when $m+n<l$. It is natural to ask
if similar construction is generalizable to the case when $m+n\geq l$. We will
address this question in future work.

The paper is organized as follows. In §2, we introduce the global integrals
and discuss the absolute convergence, functional equation, and the unfolding
computations of the global integrals. §3 is devoted to the local theory of the
integrals. We prove the existence of a local gamma factor
$\Gamma({\bf{s}},\pi,(\tau_{1},\tau_{2}),\psi)$, and carry out the local
unramified computation for the local integrals when all data are unramified.
In §4, we restate Theorem 1.4 and prepare some necessary tools for the proof.
In particular, we recall the notions of partial Bessel functions and a result
from [CST]. Theorem 1.4 is proved in §5. Actually, we prove a slightly more
general result (see Theorem 5.1).

To conclude the introduction, we introduce some notations which will be used
throughout the paper. For a positive integer $k$, let $I_{k}$ be the identity
$k\times k$ matrix. Let $B_{k}=T_{k}N_{k}\subset{\mathrm{GL}}_{k}$ the
standard upper triangular Borel subgroup, with $T_{k}$ the group of diagonal
matrices and $N_{k}$ the upper triangular unipotent subgroup. Let
$\overline{N}_{k}$ be the opposite of $N_{k}$, i.e., $\overline{N}_{k}$ is the
lower triangular unipotent subgroup of ${\mathrm{GL}}_{k}$. For positive
integers $m,n$, let ${\mathrm{Mat}}_{m\times n}$ be the set of $m\times n$
matrices. We consider the following subgroups of ${\mathrm{GL}}_{m+n}$ given
by

$M_{m,n}=\left\\{{\begin{pmatrix}g_{1}&\\\
&g_{2}\end{pmatrix},g_{1}\in{\mathrm{GL}}_{m},g_{2}\in{\mathrm{GL}}_{n}}\right\\},N_{m,n}=\left\\{{\begin{pmatrix}I_{m}&X\\\
&I_{n}\end{pmatrix},X\in{\mathrm{Mat}}_{m\times n}}\right\\},$

and $P_{m,n}=M_{m,n}N_{m,n}$. Denote $w_{m,n}=\begin{pmatrix}&I_{m}\\\
I_{n}&\end{pmatrix}$.

## Acknowledgement

We thank our advisor Jim Cogdell for his guidance and support over the years.
It is our pleasure to dedicate this paper to him on the occasion of his 70th
birthday. Some ideas of this paper grew out from the second named author
thesis work under the direction of Professor Cogdell and we would like to
thank him for a lot of fruitful communications related to this project. We
would like to thank Professor Terrence Tao, who answered the second named
author a question on MathOverFlow about Littlewood-Richardson coefficients and
also generously allowed us to reproduce his answer in our paper, see §3.4. The
second named author thanks the support of a start-up funding of Huazhong
University of Science and Technology.

##  2\. The global integrals

In this section, let $F$ be a global field and ${\mathbb{A}}$ be its ring of
adeles.

###  2.1. Eisenstein series

Notice that the modulus character of $P_{m,n}$ is given by

$\delta_{P_{m,n}}({\mathrm{diag}}(a_{1},a_{2}))=|\det(a_{1})|^{n}|\det(a_{2})|^{-m},\quad
a_{1}\in{\mathrm{GL}}_{m},a_{2}\in{\mathrm{GL}}_{n}.$

Let $\tau_{1}$ (resp. $\tau_{2}$) be an irreducible automorphic cuspidal
representation of ${\mathrm{GL}}_{m}({\mathbb{A}})$ (resp.
${\mathrm{GL}}_{n}({\mathbb{A}})$), we write
${\boldsymbol{\tau}}=(\tau_{1},\tau_{2})$. Given a pair of complex numbers
${\bf{s}}:=(s_{1},s_{2})$, we consider the normalized induced representation

${\mathrm{I}}({\bf{s}},{\boldsymbol{\tau}}):={\mathrm{Ind}}_{P_{m,n}({\mathbb{A}})}^{{\mathrm{GL}}_{m+n}({\mathbb{A}})}(\tau_{1}|\det|^{s_{1}-\frac{1}{2}}\otimes\tau_{2}|\det|^{-s_{2}+\frac{1}{2}}).$

Concretely, we associate with each
$u\in{\mathrm{I}}({\bf{s}},{\boldsymbol{\tau}})$ the function
$f_{\bf{s}}(h)=(u(h))(1),h\in{\mathrm{GL}}_{m+n}({\mathbb{A}})$. Thus the
space ${\mathrm{I}}({\bf{s}},{\boldsymbol{\tau}})$ consists of all functions
$f_{\bf{s}}:{\mathrm{GL}}_{m+n}({\mathbb{A}})\to{\mathbb{C}}$ satisfying

$f_{{\bf{s}}}({\mathrm{diag}}(a,b)uh)=|\det(a)|^{s_{1}+\frac{n-1}{2}}|\det(b)|^{-s_{2}+\frac{1-m}{2}}\varphi_{h}(a,b),$

where,
$a\in{\mathrm{GL}}_{m}({\mathbb{A}}),b\in{\mathrm{GL}}_{n}({\mathbb{A}}),u\in
N_{m,n}({\mathbb{A}}),h\in{\mathrm{GL}}_{m+n}({\mathbb{A}})$ and for a fixed
$h$, the function $(a,b)\mapsto\varphi_{h}(a,b)$ is a cusp form in the space
of $\tau=\tau_{1}\boxtimes\tau_{2}$ of the group
$M_{m,n}({\mathbb{A}})={\mathrm{GL}}_{m}({\mathbb{A}})\times{\mathrm{GL}}_{n}({\mathbb{A}})$.

Denote $\widehat{\bf{s}}:=(s_{2},s_{1}),1-\widehat{\bf{s}}:=(1-s_{2},1-s_{1})$
and $\widehat{\boldsymbol{\tau}}:=(\tau_{2},\tau_{1})$. There is a standard
intertwining operator

$M_{w_{m,n}}:{\mathrm{I}}({\bf{s}},{\boldsymbol{\tau}})\to{\mathrm{I}}(1-\widehat{\bf{s}},\widehat{\boldsymbol{\tau}})$

defined by

$M_{w_{m,n}}f_{{\bf{s}}}(g)=\int_{N_{n,m}({\mathbb{A}})}f_{{\bf{s}}}\left(w_{m,n}ug\right)du.$

Notice that ${\mathrm{I}}(1-\widehat{\bf{s}},\widehat{\boldsymbol{\tau}})$ is
the induced representation

${\mathrm{Ind}}_{P_{n,m}({\mathbb{A}})}^{{\mathrm{GL}}_{m+n}({\mathbb{A}})}(\tau_{2}|\det|^{(1-s_{2})-\frac{1}{2}}\otimes\tau_{1}|\det|^{-(1-s_{1})+\frac{1}{2}}),$

which consists of all functions $f_{1-\widehat{\bf{s}}}$ satisfying

$f_{1-\widehat{\bf{s}}}({\mathrm{diag}}(a,b)uh)=|\det(a)|^{1-s_{2}+\frac{m-1}{2}}|\det(b)|^{-(1-s_{1})-\frac{n-1}{2}}\varphi_{h}(a,b).$

In the above equation, ${\mathrm{diag}}(a,b)\in M_{n,m}({\mathbb{A}}),u\in
N_{n,m}({\mathbb{A}}),h\in{\mathrm{GL}}_{m+n}({\mathbb{A}}),$ and for a fixed
$h$, the function $(a,b)\mapsto\varphi_{h}(a,b)$ is a cusp form in the space
of $\widehat{\tau}:=\tau_{2}\otimes\tau_{1}$ of the group
$M_{n,m}({\mathbb{A}}).$

Given $f_{{\bf{s}}}\in{\mathrm{I}}({\bf{s}},{\boldsymbol{\tau}})$, we consider
the Eisenstein series

$E(h,f_{{\bf{s}}})=\sum_{\gamma\in
P_{m,n}(F)\backslash{\mathrm{GL}}_{m+n}(F)}f_{\bf{s}}(\gamma h).$

Similarly, we can also consider the Eisenstein series

$E(h,f_{1-\widehat{\bf{s}}})=\sum_{\gamma\in
P_{n,m}(F)\backslash{\mathrm{GL}}_{m+n}(F)}f_{1-\widehat{\bf{s}}}(\gamma h),$

for
$f_{1-\widehat{\bf{s}}}\in{\mathrm{I}}(1-\widehat{\bf{s}},\widehat{\boldsymbol{\tau}}).$

###  2.2. Global integrals

Fix a positive integer $l$. Let $m,n$ be non-negative integers such that
$l>m+n$. For a non-negative integer $j$ with $0\leq j\leq l-m-n-1$, we set
$k=l-m-n-1-j\geq 0$ and consider the embedding

$\iota_{j,m,n}:{\mathrm{GL}}_{m+n}\to{\mathrm{GL}}_{l}$ $\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\mapsto\begin{pmatrix}I_{j}&&&&\\\ &a&&b&\\\ &&1&&\\\
&c&&d&\\\ &&&&I_{k}\end{pmatrix}$

for $a\in{\mathrm{Mat}}_{m\times m},b\in{\mathrm{Mat}}_{m\times
n},c\in{\mathrm{Mat}}_{n\times m},d\in{\mathrm{Mat}}_{n\times n}.$ we also
consider $s_{j,m,n}\in{\mathrm{GL}}_{l}$ defined by

$s_{j,m,n}=\begin{pmatrix}0&I_{m}&0&0&0\\\ 0&0&0&I_{n}&0\\\ I_{j}&0&0&0&0\\\
0&0&1&0&0\\\ 0&0&0&0&I_{k}\end{pmatrix}.$

Then the embedding $\iota_{j,m,n}:{\mathrm{GL}}_{m+n}\to{\mathrm{GL}}_{l}$ can
be written as

$\iota_{j,m,n}(h)=(s_{j,m,n})^{-1}\begin{pmatrix}h&\\\
&I_{j+1+k}\end{pmatrix}s_{j,m,n},\quad h\in{\mathrm{GL}}_{m+n}.$

Next, we consider the subgroup $Y_{j,m,n}$ of ${\mathrm{GL}}_{l}$ defined by

$Y_{j,m,n}=\left\\{{\begin{pmatrix}u&*&*\\\ &I_{m+n+1}&*\\\
&&v\end{pmatrix},u\in N_{j},v\in N_{k}}\right\\}.$

To ease the notation, if $m,n$ are understood, we usually drop $m,n$ from the
subscripts from the above notations. For example, we may write $Y_{j,m,n}$ as
$Y_{j}$. We now define a character $\psi_{j}$ on $Y_{j}(F)\backslash
Y_{j}({\mathbb{A}})$ by

$\psi_{j}(y)=\psi\left(\sum_{i=1}^{j-1}y_{i,i+1}+\sum_{i=j+m+n+2}^{l-1}y_{i,i+1}+y_{j-1,j+m+1}+y_{j+m+1,j+m+n+2}\right),$

for $y=(y_{p,q})_{1\leq p,q\leq l}\in Y_{j}({\mathbb{A}}).$

######  Lemma 2.1.

For $h\in{\mathrm{GL}}_{m+n}({\mathbb{A}})$, $y\in Y_{j}({\mathbb{A}})$, we
have

  1. (1)

$\iota_{j}(h)^{-1}y\iota_{j}(h)\in Y_{j}$, and

  2. (2)

$\psi_{j}(\iota_{j}(h)^{-1}y\iota_{j}(h))=\psi_{j}(y).$

###### Proof.

This follows from a simple matrix calculation. ∎

Let $\pi$ be an irreducible cuspidal automorphic representation of
${\mathrm{GL}}_{l}({\mathbb{A}})$ and for $\phi\in V_{\pi}$, we consider the
following Fourier coefficient of $\phi$ along $Y_{j}$:

$\phi_{Y_{j},\psi_{j}}(h)=\int_{Y_{j}(F)\backslash
Y_{j}({\mathbb{A}})}\phi(y\iota_{j}(h))\psi_{j}^{-1}(y)dy,\quad
h\in{\mathrm{GL}}_{m+n}({\mathbb{A}}).$

By Lemma 2.1, $\phi_{Y,\psi}$ is left ${\mathrm{GL}}_{m+n}(F)$-invariant. Thus
for $f_{\bf{s}}\in{\mathrm{I}}({\bf{s}},{\boldsymbol{\tau}})$, we can consider
the integral

$I_{j}(\phi,f_{\bf{s}}):=\int_{{\mathrm{GL}}_{m+n}(F)\backslash{\mathrm{GL}}_{m+n}({\mathbb{A}})}\phi_{Y_{j},\psi_{j}}(h)E(h,f_{\bf{s}})dh.$

Similarly, we can also consider $I_{j}(\phi,M_{w_{m,n}}(f_{{\bf{s}}}))$.

######  Proposition 2.2.

The integral $I_{j}(\phi,f_{{\bf{s}}})$ converges absolutely and uniformly in
vertical strips in ${\mathbb{C}}$ for each variable $s_{1},s_{2}$ in
${\bf{s}}=(s_{1},s_{2})$, away from the poles of the Eisenstein series.
Moreover, away from the poles of $E(h,f_{\bf{s}})$ and
$E(h,M_{w_{m,n}}(f_{\bf{s}}))$, we have

$I_{j}(\phi,f_{{\bf{s}}})=I_{j}(\phi,M_{w_{m,n}}(f_{{\bf{s}}})).$

###### Proof.

The second statement follows from the functional equation of the Eisenstein
series. For the first statement, it is sufficient to show that
$\phi_{Y_{j},\psi_{j}}$ is rapidly decreasing. The proof is similar to other
situations appeared elsewhere, see [BAS:Uodd]*Lemma 2.1 for one example. We
provide some details below following the same argument as in [BAS:Uodd]*Lemma
2.1.

Let $\Omega$ be a compact subset of $B_{m+n}({\mathbb{A}})$. Let $c$ be a real
number with $0<c<1$, and we define a set $A_{c}$ as follows. We embed the
positive real numbers diagonally in the archimedean part of
${\mathbb{A}}^{\times}$, and $1$ at the finite part of
${\mathbb{A}}^{\times}$. Denote the image of this embedding by
$\mathbb{R}_{+,\Delta}$. Then $A_{c}$ is the set of all
${\mathrm{diag}}(t_{1},\dots,t_{m+n})$, such that
$t_{i}\in\mathbb{R}_{+,\Delta}$ and $t_{1}\geq ct_{2}\geq
c^{2}t_{3}\geq\dots\geq c^{m+n-1}t_{m+n}\geq c^{m+n}$. Then
$\mathcal{S}=\Omega A_{c}K_{{\mathrm{GL}}_{m+n}({\mathbb{A}})}$ is a Siegel
domain for ${\mathrm{GL}}_{m+n}({\mathbb{A}})$. Similarly, let
$\mathcal{S}^{\prime}=\Omega^{\prime}A_{c}^{\prime}K_{{\mathrm{GL}}_{l}({\mathbb{A}})}$
be a Siegel domain for ${\mathrm{GL}}_{l}({\mathbb{A}})$, where
$\iota_{j}(\Omega)\subset\Omega^{\prime}$ is a compact subset of
$B_{l}({\mathbb{A}})$ and $A^{\prime}_{c}$ is similarly defined. We take $c$
small enough and $\Omega,\Omega^{\prime}$ large enough, so that
${\mathrm{GL}}_{l}({\mathbb{A}})={\mathrm{GL}}_{l}(F)\mathcal{S}^{\prime}$,
and ${\mathrm{GL}}_{m+n}({\mathbb{A}})={\mathrm{GL}}_{m+n}(F)\mathcal{S}$. Now
let $h=\omega ak\in\mathcal{S}$, where $\omega\in\Omega$,
$a={\mathrm{diag}}(t_{1},\dots,t_{m+n})\in A_{c}$, and $k\in
K_{{\mathrm{GL}}_{m+n}({\mathbb{A}})}$. Associated to $a$, we define

$b={\mathrm{diag}}(c^{j}t_{1},c^{j-1}t_{1},\dots,ct_{1},I_{m},t_{m},I_{n},c^{-1}t_{m+n},c^{-2}t_{m+n},\dots,c^{-k}t_{m+n}).$

Then $b\iota_{j}(a)\in A_{c}^{\prime}$. Let
$\Omega_{b}^{\prime}=\Omega^{\prime}\cup\Omega^{\prime}\cdot b^{-1}$. For
fixed $a\in A_{c}$, $\Omega_{b}^{\prime}$ is a compact subset of
$B_{l}({\mathbb{A}})$ which contains $\Omega^{\prime}$. Let
$\mathcal{S}^{\prime}_{b}=\Omega^{\prime}_{b}A_{c}^{\prime}K_{{\mathrm{GL}}_{l}({\mathbb{A}})}$.
This is a Siegel domain for ${\mathrm{GL}}_{l}({\mathbb{A}})$, which contains
$\mathcal{S}^{\prime}$. Thus, $h=(\omega
b^{-1})(ba)k\in\mathcal{S}_{b}^{\prime}$. We fix a compact subset
$Y_{j,0}\subset Y_{j}({\mathbb{A}})$ such that
$Y_{j}({\mathbb{A}})=Y_{j}(F)Y_{j,0}$. We may assume that
$Y_{j,0}\subset\Omega^{\prime}$. Then we have

(2.1) $|\phi_{Y_{j},\psi_{j}}(h)|\leq\int_{Y_{j,0}}|\phi(y\iota_{j}(\omega
b^{-1}(ba)k))|dy.$

Let $N>0$ be given. Since $\phi$ is rapidly decreasing in
$\mathcal{S}^{\prime}$, there exists a constant $c_{0}$ such that for all
$\omega^{\prime}\in\Omega^{\prime}$, $a^{\prime}\in A_{c}^{\prime}$, and
$k^{\prime}\in K_{{\mathrm{GL}}_{l}({\mathbb{A}})}$, we have

(2.2) $|\phi(\omega^{\prime}a^{\prime}k^{\prime})|\leq
c_{0}\|a^{\prime}\|^{-N}.$

Here, $\|\cdot\|$ is the norm on ${\mathrm{GL}}_{l}({\mathbb{A}})$ defined by

$\|g\|=\prod_{v}\|g_{v}\|_{v}$

where $g\in{\mathrm{GL}}_{l}({\mathbb{A}})$, $v$ runs over all places of $F$,
and $\|g_{v}\|_{v}$ is the local norm on ${\mathrm{GL}}_{l}(F_{v})$ defined by

$\|g_{v}\|_{v}=\max\\{|(g_{v})_{i,j}|_{v},|(g_{v}^{-1})_{i,j}|_{v}:1\leq
i,j\leq l\\}.$

When passing from the Siegel domain $\mathcal{S}^{\prime}$ to the Siegel
domain $\mathcal{S}^{\prime}_{b}$, the constant $c_{0}$ in (2.2) can be
replaced by $c_{0}\|b^{-1}\|^{N_{0}}=c_{0}\|b\|^{N_{0}}$, for some positive
number $N_{0}$, which does not depend on $b$ (see [MoeglinWaldspurger, Sec.
I.2.10, I.2.11]). Thus, in the integrand in (2.1), we have

$|\phi(y\iota_{j}(\omega b^{-1}(ba)k))|\leq
c_{0}\|b\|^{N_{0}}\|b\iota_{j}(a)\|^{-N}.$

Notice that

$\begin{split}\|b\|=&\max\\{c^{j}t_{1},c^{j-1}t_{1},\dots,ct_{1},t_{m},c^{-1}t_{m+n},c^{-2}t_{m+n},\dots,c^{-k}t_{m+n},\
\\\ &\
c^{-j}t_{1}^{-1},c^{-j+1}t_{1}^{-1},\dots,c^{-1}t_{1}^{-1},t_{m}^{-1},ct_{m+n}^{-1},c^{2}t_{m+n}^{-1},\dots,c^{k}t_{m+n}^{-1}\\}\\\
=&\max\\{ct_{1},c^{-j}t_{1}^{-1},t_{m},t_{m}^{-1},c^{-k}t_{m+n},ct_{m+n}^{-1}\\}\\\
\leq&c^{\max\\{1,-j,-k\\}}\|a\|\end{split}$

and

$\begin{split}\|b\iota_{j}(a)\|=&\max\\{c^{j}t_{1},c^{j-1}t_{1},\dots,ct_{1},t_{1},t_{2},\dots,t_{m+n},c^{-1}t_{m+n},c^{-2}t_{m+n}\dots,c^{-k}t_{m+n},\
\\\ &\
c^{-j}t_{1}^{-1},c^{-j+1}t_{1}^{-1},\dots,c^{-1}t_{1}^{-1},t_{1}^{-1},\dots,t_{m+n}^{-1},ct_{m+n}^{-1},c^{2}t_{m+n}^{-1},\dots,c^{k}t_{m+n}^{-1}\\}\\\
\geq&\max\\{t_{1},t_{2},\dots,t_{m+n},t_{1}^{-1},t_{2}^{-1},\dots,t_{m+n}^{-1}\\}\\\
=&\|a\|.\end{split}$

We conclude that

(2.3) $|\phi(y\iota_{j}(\omega ak))|\leq c_{1}\|a\|^{N_{0}-N}$

where $c_{1}$ is a positive constant, depending on $c$ and $c_{0}$. Since
$Y_{j,0}$ is compact, we combine (2.2) and (2.3) to conclude that
$\phi_{Y_{j},\psi_{j}}$ is rapidly decreasing in $\mathcal{S}$. This completes
the proof. ∎

###  2.3. Unfolding of the global integral $I_{j}(\phi,f_{{\bf{s}}})$

For integers $m,n\geq 0$, denote

$Z_{m,n}=\left\\{\begin{pmatrix}I_{m}&0&z\\\ &1&0\\\
&&I_{n}\end{pmatrix}:z\in{\mathrm{Mat}}_{m\times
n}\right\\}\subset{\mathrm{GL}}_{m+n+1}.$

For a cusp form $\phi$ on
${\mathrm{GL}}_{m+n+1}(F)\backslash{\mathrm{GL}}_{m+n+1}({\mathbb{A}})$, we
define its constant term along $Z_{m,n}$ by

$\phi_{Z_{m,n}}(g)=\int_{Z_{m,n}(F)\backslash
Z_{m,n}({\mathbb{A}})}\phi\left(zg\right)dz.$

We have the following expansion of $\phi_{Z_{m,n}}$.

######  Lemma 2.3.

For $\phi\in{\mathcal{A}}_{0}({\mathrm{GL}}_{m+n+1})$, the space of cusp forms
on ${\mathrm{GL}}_{m+n+1}(F)\backslash{\mathrm{GL}}_{m+n+1}({\mathbb{A}})$, we
have

$\displaystyle\phi_{Z_{m,n}}(g)=\sum_{\begin{subarray}{c}\gamma_{1}\in
N_{m}(F)\backslash{\mathrm{GL}}_{m}(F),\\\ \gamma_{2}\in
N_{n}(F)\backslash{\mathrm{GL}}_{n}(F)\end{subarray}}W_{\phi}^{\psi}\left(\begin{pmatrix}\gamma_{1}&&\\\
&1&\\\ &&\gamma_{2}\end{pmatrix}g\right),$

where $W_{\phi}^{\psi}$ is the $\psi$-Whittaker function of $\phi$.

Note that when $n=0$, the above expansion is just the usual Fourier expansion
of cusp forms, due to Piatetski-Shapiro [PS:Fourier] and Shalika
[Shalika:multiplicityone]. On the other hand, the above version expansion is
an easy consequence of the result of Piatetski-Shapiro and Shalika. We give a
sketch of the proof below.

###### Proof.

Let

$Q_{m}=\left\\{{\begin{pmatrix}g_{1}&x\\\
&1\end{pmatrix}:g_{1}\in{\mathrm{GL}}_{m},x\in{\mathrm{Mat}}_{m\times
1}}\right\\}$

be the usual mirabolic subgroup of ${\mathrm{GL}}_{m+1}$. We consider the
function $\phi_{1}$ on $Q_{m}(F)\backslash Q_{m}({\mathbb{A}})$ defined by

$\phi_{1}\left(\begin{pmatrix}g_{1}&x\\\
&1\end{pmatrix}\right)=\phi_{Z_{m,n}}\left(\begin{pmatrix}g_{1}&x&\\\ &1&\\\
&&I_{n}\end{pmatrix}g\right).$

Then $\phi_{1}$ is a cuspidal automorphic form on $Q_{m}(F)\backslash
Q_{m}({\mathbb{A}})$ in the sense that for any parabolic subgroup $P=MU$ of
$Q_{m}$ with unipotent subgroup $U$, we have

$\int_{U(F)\backslash U({\mathbb{A}})}\phi_{1}(uq)du=0,\quad\forall q\in
Q_{m}({\mathbb{A}}).$

This can be checked easily using cuspidality of $\phi$, see
[Cogdell:IAS]*Lemma 2.2 for a similar situation. Thus by the Fourier expansion
for $\phi_{1}$ we get that

$\phi_{1}(I_{m+1})=\sum_{\gamma_{1}\in
N_{m}(F)\backslash{\mathrm{GL}}_{m}(F)}W_{\phi_{1}}^{\psi}\left(\begin{pmatrix}\gamma_{1}&\\\
&1\end{pmatrix}\right),$

where $W_{\phi_{1}}^{\psi}$ is the standard $\psi$-Whittaker function of
$\phi_{1}$. Plugging in the definitions, we get that

$\displaystyle\phi_{Z_{m,n}}(g)$ $\displaystyle=\sum_{\gamma_{1}\in
N_{m}(F)\backslash{\mathrm{GL}}_{m}(F)}\int\phi\left(\begin{pmatrix}u&x&z\\\
&1&\\\ &&I_{n}\end{pmatrix}\begin{pmatrix}\gamma_{1}&&\\\ &1&\\\
&&I_{n}\end{pmatrix}g\right)\psi^{-1}(u)\psi^{-1}(x_{m})dudxdz,$

where $u=(u_{ij})\in N_{m}({\mathbb{A}})$,
$\psi^{-1}(u)=\psi^{-1}(\sum_{i}u_{i,i+1})$ and $x_{m}$ is the last component
of $x$. Similarly, we consider the mirabolic subgroup $Q_{n}^{\prime}$ of
${\mathrm{GL}}_{n+1}$ of the form

$Q_{n}^{\prime}=\left\\{{\begin{pmatrix}1&y\\\
0&g_{2}\end{pmatrix},y\in{\mathrm{Mat}}_{1\times
n},g_{2}\in{\mathrm{GL}}_{n}}\right\\}.$

For fixed $\gamma_{1}$ and $g$, we consider the function $\phi_{2}$ on
$Q_{n}^{\prime}(F)\backslash Q_{n}^{\prime}({\mathbb{A}})$ defined by

$\displaystyle\phi_{2}\left(\begin{pmatrix}1&y\\\ 0&g_{2}\end{pmatrix}\right)$
$\displaystyle=\int\phi\left(\begin{pmatrix}u&x&z\\\ &1&y\\\
&&g_{2}\end{pmatrix}\begin{pmatrix}\gamma_{1}&&\\\ &1&\\\
&&I_{n}\end{pmatrix}g\right)\psi^{-1}(u)\psi^{-1}(x_{m})dudxdz.$

Again, $\phi_{2}$ is a cusp form on $Q_{n}^{\prime}(F)\backslash
Q_{n}^{\prime}({\mathbb{A}})$. By a slightly variant form of the Fourier
expansion, see for example [CPS:converseII]*§1, Proposition, we have

$\phi_{2}(I_{n+1})=\sum_{\gamma_{2}\in
N_{n}(F)\backslash{\mathrm{GL}}_{n}(F)}W_{\phi_{2}}^{\psi}\left(\begin{pmatrix}1&\\\
&\gamma_{2}\end{pmatrix}\right).$

Note that

$\displaystyle W_{\phi_{2}}^{\psi}\left(\begin{pmatrix}1&\\\
&\gamma_{2}\end{pmatrix}\right)$
$\displaystyle=\int\phi\left(\begin{pmatrix}u&x&z\\\ &1&y\\\
&&v\end{pmatrix}\begin{pmatrix}\gamma_{1}&&\\\ &1&\\\
&&\gamma_{2}\end{pmatrix}g\right)\psi^{-1}(uv)\psi^{-1}(x_{m}+y_{1})dxdydudv$
$\displaystyle=W_{\phi}^{\psi}\left(\begin{pmatrix}\gamma_{1}&&\\\ &1&\\\
&&\gamma_{2}\end{pmatrix}g\right),$

where $y_{1}$ in the first integral is the first component of $y$. The result
follows. ∎

######  Theorem 2.4.

The integral $I_{j}(\phi,f_{{\bf{s}}})$ is Eulerian. More precisely, in the
region of absolute convergence, we have

$\displaystyle
I_{j}(\phi,f_{{\bf{s}}})=\int_{N_{m+n}({\mathbb{A}})\backslash{\mathrm{GL}}_{m+n}({\mathbb{A}})}\int_{\overline{U}^{j,m,n}({\mathbb{A}})}W_{\phi}^{\psi}\left(\overline{u}\eta_{j}\iota_{j}(h)\right)\xi_{f_{\bf{s}}}^{\psi^{-1}}(h)d\overline{u}dh$

where

$\displaystyle\overline{U}^{j,m,n}$
$\displaystyle=\left\\{{\overline{u}(x,y)=\begin{pmatrix}I_{m}&&&&\\\
x&I_{j}&&&\\\ &&1&&\\\ &&&I_{k}&\\\
&&&y&I_{n}\end{pmatrix}:\begin{subarray}{c}x\in{\mathrm{Mat}}_{j\times m}\\\
y\in{\mathrm{Mat}}_{n\times k}\end{subarray}}\right\\},$
$\displaystyle\eta_{j}$
$\displaystyle:=\eta_{j,m,n}=\begin{pmatrix}&I_{m}&&&\\\ I_{j}&&&&\\\ &&1&&\\\
&&&&I_{k}\\\ &&&I_{n}&\end{pmatrix},$
$\displaystyle\xi_{f_{\bf{s}}}^{\psi^{-1}}(h)$
$\displaystyle=\int_{N_{m}(F)\backslash N_{m}({\mathbb{A}})\times
N_{n}(F)\backslash
N_{n}({\mathbb{A}})}f_{{\bf{s}}}\left(\begin{pmatrix}u_{1}&\\\
&u_{2}\end{pmatrix}h\right)\psi(u_{1})\psi(u_{2})du_{1}du_{2}.$

###### Proof.

For simplicity, in the following, we write $G(F)\backslash G({\mathbb{A}})$ as
$[G]$ for an algebraic group $G$ over $F$. Plugging in the definition of
Eisenstein series, we have

(2.4) $\displaystyle I_{j}(\phi,f_{s})$
$\displaystyle=\int_{P_{m,n}(F)\backslash{\mathrm{GL}}_{m+n}({\mathbb{A}})}\phi_{Y_{j},\psi_{j}}(h)f_{\bf{s}}(h)dh$
$\displaystyle=\int_{M_{m,n}(F)N_{m,n}({\mathbb{A}})\backslash{\mathrm{GL}}_{m+n}({\mathbb{A}})}\int_{[N_{m,n}]}\phi_{Y_{j},\psi_{j}}(uh)duf_{\bf{s}}(h)dh$
$\displaystyle=\int_{M_{m,n}(F)N_{m,n}({\mathbb{A}})\backslash{\mathrm{GL}}_{m+n}({\mathbb{A}})}\phi_{Y_{j},\psi_{j},N_{m,n}}(h)f_{\bf{s}}(h)dh,$

where

(2.5) $\displaystyle\phi_{Y_{j},\psi_{j},N_{m,n}}(h)$
$\displaystyle=\int_{[N_{m,n}]}\phi_{Y_{j},\psi_{j}}(uh)du$
$\displaystyle=\int_{[Y_{j}]\times[N_{m,n}]}\phi(y\iota_{j}(uh))\psi_{j}^{-1}(y)dudy.$

For

(2.6) $\displaystyle y=\begin{pmatrix}v_{1}&x_{1}&x_{2}&x_{3}&z\\\
&I_{m}&&&y_{3}\\\ &&1&&y_{2}\\\ &&&I_{n}&y_{1}\\\ &&&&v_{2}\end{pmatrix}\in
Y_{j}({\mathbb{A}}),\quad u=\begin{pmatrix}I_{m}&t\\\ &I_{n}\end{pmatrix}\in
N_{m,n}({\mathbb{A}}),$

we have

(2.7)
$\displaystyle\eta_{j}y\iota_{j}(u)\eta_{j}^{-1}=\begin{pmatrix}I_{m}&0&0&y_{3}&t\\\
x_{1}&v_{1}&x_{2}&z&x_{3}\\\ 0&0&1&y_{2}&0\\\ 0&0&0&v_{2}&0\\\
0&0&0&y_{1}&I_{n}\end{pmatrix},$

where
$v_{1}\in[N_{j}],v_{2}\in[N_{k}],(x_{1},x_{2},x_{3})\in[{\mathrm{Mat}}_{j\times(m+n+1)}],z\in[{\mathrm{Mat}}_{j\times
k}],(y_{3},y_{2},y_{1})^{t}\in[{\mathrm{Mat}}_{(m+n+1)\times
k}],t\in[{\mathrm{Mat}}_{m\times n}]$. Since $\phi$ is left
${\mathrm{GL}}_{l}(F)$-invariant and $\eta_{j,m,n}\in{\mathrm{GL}}_{l}(F)$, we
have

(2.8)
$\phi_{Y_{j},\psi_{j},N_{m,n}}(h)=\int_{[Y_{j}]\times[\iota_{j}(N_{m,n})]}\phi\left(\begin{pmatrix}I_{m}&0&0&y_{3}&t\\\
x_{1}&v_{1}&x_{2}&z&x_{3}\\\ 0&0&1&y_{2}&0\\\ 0&0&0&v_{2}&0\\\
0&0&0&y_{1}&I_{n}\end{pmatrix}\eta_{j}\iota_{j}(h)\right)\psi_{j}^{-1}(y)dydu.$

Write

$Z=\begin{pmatrix}y_{3}&t\\\
z&x_{3}\end{pmatrix}\in{\mathrm{Mat}}_{(m+j)\times(n+k)}({\mathbb{A}}).$

In the right side integral of (2.8), there is an inner integral

$\displaystyle\int_{[{\mathrm{Mat}}_{(m+j)\times(n+k)}]}\phi\left(\begin{pmatrix}I_{m+j}&&Z\\\
&1&\\\ &&I_{n+k}\end{pmatrix}g\right)dZ,$

which is

(2.9) $\displaystyle\sum_{\begin{subarray}{c}\gamma_{1}\in
N_{m+j}(F)\backslash{\mathrm{GL}}_{m+j}(F)\\\ \gamma_{2}\in
N_{n+k}(F)\backslash{\mathrm{GL}}_{n+k}(F)\end{subarray}}W_{\phi}^{\psi}\left(\begin{pmatrix}\gamma_{1}&&\\\
&1&\\\ &&\gamma_{2}\end{pmatrix}g\right)$

by Lemma 2.3. Plugging (2.9) into (2.8), we get

(2.10) $\displaystyle\phi_{Y_{j},\psi_{j},N_{m,n}}(h)$
$\displaystyle=\sum_{\gamma_{1},\gamma_{2}}\int
W_{\phi}^{\psi}\left(\begin{pmatrix}\gamma_{1}&&\\\ &1&\\\
&&\gamma_{2}\end{pmatrix}\begin{pmatrix}I_{m}&0&0&0&0\\\
x_{1}&v_{1}&x_{2}&0&0\\\ 0&0&1&y_{2}&0\\\ 0&0&0&v_{2}&0\\\
0&0&0&y_{1}&I_{n}\end{pmatrix}\eta_{j}\iota_{j}(h)\right)\psi_{j}^{-1}(y)dy.$

To simplify the above integral (2.10), we consider its inner integral with
respect to $x_{2}=[x^{1},\dots,x^{j}]\in[{\mathrm{Mat}}_{j\times 1}]$ first,
which is

$\displaystyle\int_{(F\backslash{\mathbb{A}})^{j}}W_{\phi}^{\psi}\left(\begin{pmatrix}\gamma_{1}&&\\\
&1&\\\ &&\gamma_{2}\end{pmatrix}\begin{pmatrix}I_{m}&0&0&0&0\\\
0&I_{j}&x_{2}&0&0\\\ 0&0&1&0&0\\\ 0&0&0&I_{k}&0\\\
0&0&0&0&I_{n}\end{pmatrix}\begin{pmatrix}I_{m}&0&0&0&0\\\ x_{1}&v_{1}&0&0&0\\\
0&0&1&y_{2}&0\\\ 0&0&0&v_{2}&0\\\
0&0&0&y_{1}&I_{n}\end{pmatrix}\eta_{j}\iota_{j}(h)\right)\psi^{-1}(x^{j})dx_{2}.$

Write $\gamma_{1}=(\gamma_{pq})_{1\leq p,q\leq m+j}$, then we have

$\gamma_{1}\begin{pmatrix}0\\\ x_{2}\end{pmatrix}=\begin{pmatrix}*\\\ *\\\
\vdots\\\
\gamma_{m+j,m+1}x^{1}+\gamma_{m+j,m+2}x^{2}+\dots+\gamma_{m+j,m+j}x^{j}\end{pmatrix}.$

Thus we get

$\displaystyle W_{\phi}^{\psi}\left(\begin{pmatrix}\gamma_{1}&&\\\ &1&\\\
&&\gamma_{2}\end{pmatrix}\begin{pmatrix}I_{m}&0&0&0&0\\\ 0&I_{j}&x_{2}&0&0\\\
0&0&1&0&0\\\ 0&0&0&I_{k}&0\\\ 0&0&0&0&I_{n}\end{pmatrix}g\right)=$
$\displaystyle\psi(\gamma_{m+j,m+1}x^{1}+\dots+\gamma_{m+j,m+j}x^{j})$
$\displaystyle\quad\cdot W_{\phi}^{\psi}\left(\begin{pmatrix}\gamma_{1}&&\\\
&1&\\\ &&\gamma_{2}\end{pmatrix}g\right),$

with

$g=\begin{pmatrix}I_{m}&0&0&0&0\\\ x_{1}&v_{1}&0&0&0\\\ 0&0&1&y_{2}&0\\\
0&0&0&v_{2}&0\\\ 0&0&0&y_{1}&I_{n}\end{pmatrix}\eta_{j}\iota_{j}(h).$

Thus the inner integral of (2.10) with respect to $x_{2}$ is

$\displaystyle\int_{(F\backslash{\mathbb{A}})^{j}}\psi(\gamma_{m+j,m+1}x^{1}+\dots+(\gamma_{m+j,m+j}-1)x^{j})dx^{1}\dots
dx^{j}W_{\phi}^{\psi}\left(\begin{pmatrix}\gamma_{1}&&\\\ &1&\\\
&&\gamma_{2}\end{pmatrix}g\right).$

The above integral over $x^{1},\dots,x^{j}$ is $1$ if
$\gamma_{m+j,m+1}=\dots=\gamma_{m+j,m+j-1}=0$ and $\gamma_{m+j,m+j}=1$, and is
zero otherwise. Note that if $\gamma_{m+j,m+1}=\dots=\gamma_{m+j,m+j-1}=0$, as
an element of the coset $N_{m+j}(F)\backslash{\mathrm{GL}}_{m+j}(F)$, we can
write

$\gamma_{1}=\begin{pmatrix}\gamma_{1}^{\prime}&\\\
&1\end{pmatrix}\begin{pmatrix}I_{m}&&\\\ &I_{j-1}&\\\ \xi&&1\end{pmatrix},$

with $\gamma_{1}^{\prime}\in
N_{m+j}(F)\backslash{\mathrm{GL}}_{m+j}(F),\xi\in{\mathrm{Mat}}_{1\times
m}(F).$ By changing the summation notation, integral (2.10) becomes

(2.11) $\displaystyle\phi_{Y_{j},\psi_{j},N_{m,n}}(h)=$
$\displaystyle\sum_{\begin{subarray}{c}\gamma_{1}\in
N_{m+j-1}(F)\backslash{\mathrm{GL}}_{m+j-1}(F)\\\ \gamma_{2}\in
N_{n+k}(F)\backslash{\mathrm{GL}}_{n+k}(F)\end{subarray}}\sum_{\xi\in F^{m}}$
$\displaystyle\int W_{\phi}^{\psi}\left(\begin{pmatrix}\gamma_{1}&&\\\
&I_{2}&\\\ &&\gamma_{2}\end{pmatrix}\begin{pmatrix}I_{m}&&&&\\\ &I_{j-1}&&&\\\
\xi&&1&&\\\ &&&1&\\\ &&&&I_{n+k}\end{pmatrix}\begin{pmatrix}I_{m}&0&0&0&0\\\
x_{1}&v_{1}&0&0&0\\\ 0&0&1&y_{2}&0\\\ 0&0&0&v_{2}&0\\\
0&0&0&y_{1}&I_{n}\end{pmatrix}\eta_{j}\iota_{j}(h)\right)$
$\displaystyle\quad\cdot\psi^{-1}(v_{1})\psi^{-1}(v_{2})\psi^{-1}(y^{1})dx_{1}dy_{1}dy_{2}dv_{1}dv_{2}.$

Here $y^{1}$ is the first component of the $y_{2}\in{\mathrm{Mat}}_{1\times
k}.$ In (2.11), the summation over $\xi$ could be absorbed into the integral
over the last row of $x_{1}$. Thus we get

(2.12) $\displaystyle\phi_{Y_{j},\psi_{j},N_{m,n}}(h)=$
$\displaystyle\sum_{\begin{subarray}{c}\gamma_{1}\in
N_{m+j-1}(F)\backslash{\mathrm{GL}}_{m+j-1}(F)\\\ \gamma_{2}\in
N_{n+k}(F)\backslash{\mathrm{GL}}_{n+k}(F)\end{subarray}}\int_{(F\backslash{\mathbb{A}})^{*}}\int_{{\mathbb{A}}^{m}}$
$\displaystyle\int W_{\phi}^{\psi}\left(\begin{pmatrix}\gamma_{1}&&\\\
&I_{2}&\\\ &&\gamma_{2}\end{pmatrix}\begin{pmatrix}I_{m}&0&0&0&0&0\\\
x_{1}^{\prime}&v_{1}^{\prime}&p&0&0&0\\\ (x_{j1},\dots,x_{jm})&0&1&0&0&0\\\
0&0&0&1&y_{2}&0\\\ 0&0&0&0&v_{2}&0\\\
0&0&0&0&y_{1}&I_{n}\end{pmatrix}\eta_{j}\iota_{j}(h)\right)$
$\displaystyle\quad\cdot\psi^{-1}(v_{1})\psi^{-1}(v_{2})\psi^{-1}(y^{1})(\prod_{t=1}^{m}dx_{jt})dx_{1}^{\prime}dy_{1}dy_{2}dv_{1}dv_{2},$

where we wrote $x_{1}=\begin{pmatrix}x_{1}^{\prime}\\\
(x_{j1},\dots,x_{jm})\end{pmatrix}$, $v_{1}=\begin{pmatrix}v_{1}^{\prime}&p\\\
&1\end{pmatrix}$ with $p\in[{\mathrm{Mat}}_{(j-1)\times 1}]$, and $*$ in
$(F\backslash{\mathbb{A}})^{*}$ denotes the number of variables other than the
part in $(x_{j1},\dots,x_{jm})$. We next compute the inner integral over the
$p$-part, which is similar as above. Note that
$\psi(v_{1})=\psi(v_{1}^{\prime})\psi(p^{j-1})$, where
$p=(p^{1},\dots,p^{j-1})^{t}$. For $\gamma_{1}\in{\mathrm{GL}}_{m+j-1}(F)$,
and $p=(p^{1},\dots,p^{j-1})^{t}$ we have

$\gamma_{1}\begin{pmatrix}0_{m\times 1}\\\ p\end{pmatrix}=\begin{pmatrix}*\\\
\vdots\\\ *\\\
\gamma_{m+j-1,m+1}p^{1}+\dots\gamma_{m+j-1,m+j-1}p^{j-1}\end{pmatrix}.$

Thus the inner integral over $p$ in (2.12) is

$\displaystyle\int_{(F\backslash{\mathbb{A}})^{j-1}}\psi(\gamma_{m+j-1,m+1}p^{1}+\dots+(\gamma_{m+j-1,m+j-1}-1)p^{j-1})\prod_{t}dp^{t}W_{\phi}^{\psi}\left(\begin{pmatrix}\gamma_{1}&&\\\
&1&\\\ &&\gamma_{2}\end{pmatrix}g\right),$

for certain appropriate $g$ which should be self-evident from the context. The
above integral is 1 if $\gamma_{m+j-1,m+1}=\dots=\gamma_{m+j-1,m+j-2}=0$ and
$\gamma_{m+j-1,m+j-1}=1$, and is zero otherwise. In this case, we can write
that

$\gamma_{1}=\begin{pmatrix}\gamma_{1}^{\prime}&\\\
&1\end{pmatrix}\begin{pmatrix}I_{m}&&\\\ &I_{j-2}&\\\ \xi&&1\end{pmatrix}$

as an element in the coset $N_{m+j-1}(F)\backslash{\mathrm{GL}}_{m+j-1}(F)$,
where $\gamma_{1}^{\prime}\in
N_{m+j-2}(F)\backslash{\mathrm{GL}}_{m+j-2}(F),\xi\in F^{m}$. Similarly as
above, by absorbing the summation over $\xi$, we get that

$\displaystyle\phi_{Y_{j},\psi_{j},N_{m,n}}(h)=$
$\displaystyle\sum_{\begin{subarray}{c}\gamma_{1}\in
N_{m+j-2}(F)\backslash{\mathrm{GL}}_{m+j-2}(F)\\\ \gamma_{2}\in
N_{n+k}(F)\backslash{\mathrm{GL}}_{n+k}(F)\end{subarray}}\int_{(F\backslash{\mathbb{A}})^{*}}\int_{{\mathbb{A}}^{2m}}$
$\displaystyle\int W_{\phi}^{\psi}\left(\begin{pmatrix}\gamma_{1}&&\\\
&I_{3}&\\\ &&\gamma_{2}\end{pmatrix}\begin{pmatrix}I_{m}&0&0&0&0&0&0\\\
x_{1}^{\prime\prime}&v_{1}^{\prime\prime}&p^{\prime}&0&0&0&0\\\
(x_{j-1,1},\dots,x_{j-1,m})&0&1&0&0&0&0\\\
(x_{j1},\dots,x_{jm})&0&0&1&0&0&0\\\ 0&0&0&0&1&y_{2}&0\\\ 0&0&0&0&0&v_{2}&0\\\
0&0&0&0&0&y_{1}&I_{n}\end{pmatrix}\eta_{j}\iota_{j}(h)\right)$
$\displaystyle\quad\cdot\psi^{-1}(v_{1}^{\prime})\psi^{-1}(v_{2})\psi^{-1}(y^{1})(\prod_{i=j-1}^{j}\prod_{t=1}^{m}dx_{it})dx_{1}^{\prime\prime}dy_{1}dy_{2}dv_{1}^{\prime}dv_{2},$

where $v_{1}^{\prime}=\begin{pmatrix}v_{1}^{\prime\prime}&p^{\prime}\\\
&1\end{pmatrix}$. An induction argument shows that

$\displaystyle\phi_{Y_{j},\psi_{j},N_{m,n}}(h)=$
$\displaystyle\sum_{\begin{subarray}{c}\gamma_{1}\in
N_{m}(F)\backslash{\mathrm{GL}}_{m}(F)\\\ \gamma_{2}\in
N_{n+k}(F)\backslash{\mathrm{GL}}_{n+k}(F)\end{subarray}}\int_{(F\backslash{\mathbb{A}})^{*}}\int_{{\mathrm{Mat}}_{j\times
m}({\mathbb{A}})}$ $\displaystyle
W_{\phi}^{\psi}\left(\begin{pmatrix}\gamma_{1}&&\\\ &I_{j+1}&\\\
&&\gamma_{2}\end{pmatrix}\begin{pmatrix}I_{m}&&&&&\\\ x&I_{j}&&&&\\\ &&1&&&\\\
&&&1&y_{2}&\\\ &&&&v_{2}&\\\
&&&&y_{1}&I_{n}\end{pmatrix}\eta_{m,n}j(h)\right)$
$\displaystyle\cdot\psi^{-1}(v_{2})\psi^{-1}(y^{1})dxdy_{1}dy_{2}dv_{2}.$

The integral over $y_{1},y_{2},v_{2}$ can be done similarly and we have

$\displaystyle\phi_{Y_{j},\psi_{j},N_{m,n}}(h)=$
$\displaystyle\sum_{\begin{subarray}{c}\gamma_{1}\in
N_{m}(F)\backslash{\mathrm{GL}}_{m}(F)\\\ \gamma_{2}\in
N_{n}(F)\backslash{\mathrm{GL}}_{n}(F)\end{subarray}}\int_{{\mathrm{Mat}}_{j\times
m}({\mathbb{A}})}\int_{{\mathrm{Mat}}_{n\times k}({\mathbb{A}})}$
$\displaystyle W_{\phi}^{\psi}\left(\begin{pmatrix}\gamma_{1}&&\\\
&I_{j+k+1}&\\\ &&\gamma_{2}\end{pmatrix}\begin{pmatrix}I_{m}&&&&\\\
x&I_{j}&&&\\\ &&1&&\\\ &&&I_{k}&\\\
&&&y&I_{n}\end{pmatrix}\eta_{j}\iota_{j}(h)\right)dydx$ $\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}\gamma_{1}\in
N_{m}(F)\backslash{\mathrm{GL}}_{m}(F)\\\ \gamma_{2}\in
N_{n}(F)\backslash{\mathrm{GL}}_{n}(F)\end{subarray}}\int_{{\mathrm{Mat}}_{j\times
m}({\mathbb{A}})}\int_{{\mathrm{Mat}}_{n\times k}({\mathbb{A}})}$
$\displaystyle W_{\phi}^{\psi}\left(\begin{pmatrix}I_{m}&&&&\\\ x&I_{j}&&&\\\
&&1&&\\\ &&&I_{k}&\\\
&&&y&I_{n}\end{pmatrix}\eta_{j}\iota_{j}\left(\begin{pmatrix}\gamma_{1}&\\\
&\gamma_{2}\end{pmatrix}h\right)\right)dydx$

We now plug the above formula into (2.4) to get

(2.13)
$\begin{split}I_{j}(\phi,f_{{\bf{s}}})=&\int_{M_{m,n}(F)N_{m,n}({\mathbb{A}})\backslash{\mathrm{GL}}_{m+n}({\mathbb{A}})}\phi_{Y_{j},\psi_{j},N_{m,n}}(h)f_{{\bf{s}}}(h)dh\\\
=&\int_{(N_{m}(F)\times
N_{n}(F))N_{m,n}({\mathbb{A}})\backslash{\mathrm{GL}}_{m+n}({\mathbb{A}})}\int_{\overline{U}^{j,m,n}({\mathbb{A}})}W_{\phi}^{\psi}\left(\overline{u}\eta_{j}\iota_{j}(h)\right)f_{{\bf{s}}}(h)d\overline{u}dh.\end{split}$

In order to justify this step, we need to show that the double integral in the
second line of (2.13) converges absolutely. This will be done in Subsection
2.5. From (2.13), we obtain

$\displaystyle I_{j}(\phi,f_{{\bf{s}}})=$
$\displaystyle\int_{N_{m+n}({\mathbb{A}})\backslash{\mathrm{GL}}_{m+n}({\mathbb{A}})}\int_{\overline{U}^{j,m,n}({\mathbb{A}})}\int_{N_{m}(F)\backslash
N_{m}({\mathbb{A}})}\int_{N_{n}(F)\backslash
N_{n}({\mathbb{A}})}W_{\phi}^{\psi}\left(\overline{u}\eta_{j}\iota_{j}\left(\begin{pmatrix}u_{1}&\\\
&u_{2}\end{pmatrix}h\right)\right)$ $\displaystyle\cdot
f_{\bf{s}}\left(\begin{pmatrix}u_{1}&\\\
&u_{2}\end{pmatrix}h\right)du_{2}du_{1}dydxdh$ $\displaystyle=$
$\displaystyle\int_{N_{m+n}({\mathbb{A}})\backslash{\mathrm{GL}}_{m+n}({\mathbb{A}})}\int_{\overline{U}^{j,m,n}({\mathbb{A}})}\int_{N_{m}(F)\backslash
N_{m}({\mathbb{A}})}\int_{N_{n}(F)\backslash
N_{n}({\mathbb{A}})}W_{\phi}^{\psi}\left(\overline{u}\eta_{j}\iota_{j}\left(\begin{pmatrix}u_{1}&\\\
&u_{2}\end{pmatrix}h\right)\right)$ $\displaystyle\cdot
f_{\bf{s}}\left(\begin{pmatrix}u_{1}&\\\
&u_{2}\end{pmatrix}h\right)du_{2}du_{1}dydxdh$ $\displaystyle=$
$\displaystyle\int_{N_{m+n}({\mathbb{A}})\backslash{\mathrm{GL}}_{m+n}({\mathbb{A}})}\int_{\overline{U}^{j,m,n}({\mathbb{A}})}W_{\phi}^{\psi}\left(\overline{u}\eta_{j}\iota_{j}(h)\right)\xi_{f_{\bf{s}}}^{\psi^{-1}}(h)d\overline{u}dh.$

The result follows. ∎

###  2.4. Unfolding of $I_{j}(\phi,M_{w_{m,n}}(f_{\bf{s}}))$

######  Theorem 2.5.

The integral $I_{j}(\phi,M_{w_{m,n}}(f_{\bf{s}}))$ is Eulerian. More
precisely, in the region of absolute convergence, we have

$\displaystyle
I_{j}(\phi,\widetilde{f}_{{\bf{s}}})=\int_{N_{n+m}({\mathbb{A}})\backslash{\mathrm{GL}}_{n+m}({\mathbb{A}})}\int_{\overline{V}^{j,m,n}}W_{\phi}^{\psi}\left(\overline{u}\gamma_{n,m}\begin{pmatrix}h&\\\
&I_{l-m-n}\end{pmatrix}s_{j,m,n}\right)\xi_{\widetilde{f}_{\bf{s}}}^{\psi^{-1}}(h)d\overline{u}dh,$

where

$\displaystyle\widetilde{f}_{\bf{s}}$ $\displaystyle=M_{w_{m,n}}(f_{\bf{s}}),$
$\displaystyle\overline{V}^{j,m,n}$
$\displaystyle=\left\\{{\begin{pmatrix}I_{n}&0&0&0&0\\\ x&I_{j}&0&0&0\\\
&&1&0&0\\\ &&&I_{k}&0\\\ &&&y&I_{m}\end{pmatrix}:x\in{\mathrm{Mat}}_{j\times
n},y\in{\mathrm{Mat}}_{m\times k}}\right\\}=\overline{U}^{j,n,m},$
$\displaystyle\gamma_{n,m}$ $\displaystyle=\begin{pmatrix}I_{n}&&\\\
&&I_{l-m-n}\\\ &I_{m}&\end{pmatrix}$
$\displaystyle\xi_{\widetilde{f}_{\bf{s}}}^{\psi^{-1}}(h)$
$\displaystyle=\int_{N_{n}(F)\backslash N_{n}({\mathbb{A}})\times
N_{m}(F)\backslash
N_{m}({\mathbb{A}})}\widetilde{f}_{{\bf{s}}}\left(\begin{pmatrix}u_{1}&\\\
&u_{2}\end{pmatrix}h\right)\psi(u_{1})\psi(u_{2})du_{1}du_{2}.$

Notice that

$\eta_{j,m,n}s_{j,m,n}^{-1}=\begin{pmatrix}I_{m}&&\\\ &&I_{l-m-n}\\\
&I_{n}&\end{pmatrix}=\gamma_{m,n}.$

The proof is similar to the proof of Theorem 2.4. We give some details for
completeness.

###### Proof.

In the following, we assume that $m\geq n$. If $n\leq m$, the matrix
calculation performed below is a little bit different, but other parts of the
proof go through and the result is the same. Note that
$\widetilde{f}_{\bf{s}}\in{\mathrm{I}}(1-\widehat{\bf{s}},\widehat{\boldsymbol{\tau}})$
is left invariant under $N_{n,m}({\mathbb{A}})$. Thus we have

(2.14)
$\begin{split}I_{j}(\phi,\widetilde{f}_{\bf{s}})&=\int_{P_{n,m}(F)\backslash{\mathrm{GL}}_{n+m}({\mathbb{A}})}\phi_{Y_{j},\psi_{j}}(h)\widetilde{f}_{{\bf{s}}}(h)dh\\\
&=\int_{M_{n,m}(F)N_{n,m}({\mathbb{A}})\backslash{\mathrm{GL}}_{n+m}({\mathbb{A}})}\int_{N_{n,m}(F)\backslash
N_{n,m}({\mathbb{A}})}\phi_{Y_{j},\psi_{j}}(uh)\widetilde{f}_{\bf{s}}(h)dudh\\\
&=\int_{M_{n,m}(F)N_{n,m}({\mathbb{A}})\backslash{\mathrm{GL}}_{n+m}({\mathbb{A}})}\phi_{Y_{j},\psi_{j},N_{n,m}}(h)\widetilde{f}_{\bf{s}}(h)dh,\end{split}$

where

$\displaystyle\phi_{Y_{j},\psi_{j},N_{n,m}}(h):$
$\displaystyle=\int_{N_{n,m}(F)\backslash
N_{n,m}({\mathbb{A}})}\phi_{Y_{j},\psi_{j}}(uh)du$
$\displaystyle=\int_{N_{n,m}(F)\backslash
N_{n,m}({\mathbb{A}})}\int_{Y_{j}(F)\backslash
Y_{j}({\mathbb{A}})}\phi(y\iota_{j}(u)\iota_{j}(h))\psi_{j}^{-1}(y)dydu.$

Since $\phi$ is left ${\mathrm{GL}}_{l}(F)$-invariant, we have

$\phi(y\iota_{j}(u)\iota_{j}(h))=\phi\left(\gamma_{n,m}s_{j,m,n}y\iota_{j}(u)s_{j,m,n}^{-1}\gamma_{n,m}^{-1}\gamma_{n,m}\begin{pmatrix}h&\\\
&I_{l-m-n}\end{pmatrix}s_{j,m,n}\right).$

Write

$\displaystyle y=\begin{pmatrix}v_{1}&x_{1}&x_{1}^{\prime}&x_{2}&x_{3}&z\\\
&I_{n}&&&&y_{3}\\\ &&I_{m-n}&&&y_{3}^{\prime}\\\ &&&1&&y_{2}\\\
&&&&I_{n}&y^{\prime}_{1}\\\ &&&&&v_{2}\end{pmatrix}\in Y({\mathbb{A}}),\quad
u=\begin{pmatrix}I_{n}&t_{1}&t_{2}\\\ &I_{m-n}&\\\ &&I_{n}\end{pmatrix}\in
N_{n,m}({\mathbb{A}}),$

with $v_{1}\in N_{j}({\mathbb{A}}),v_{2}\in N_{k}({\mathbb{A}})$ and other
variables in appropriate matrices spaces. A matrix calculation shows that

$\gamma_{n,m}s_{j,m,n}y\iota_{j}(u)s_{j,m,n}^{-1}\gamma_{n,m}^{-1}=\begin{pmatrix}I_{n}&0&0&y_{3}&t_{1}&t_{2}\\\
x_{1}&v_{1}&x_{2}&z&x_{1}^{\prime}+x_{1}t_{1}&x_{3}+x_{1}t_{2}\\\
&&1&y_{2}&0&0\\\ &&&v_{2}&0&0\\\ &&&y_{3}^{\prime}&I_{m-n}&0\\\
&&&y^{\prime}_{1}&0&I_{n}\end{pmatrix}.$

Thus we get

$\displaystyle\phi_{Y_{j},\psi_{j},N_{n,m}}(h)$
$\displaystyle=\int_{[Y_{j}]\times[N_{n,m}]}\phi\left(\begin{pmatrix}I_{n}&0&0&y_{3}&t_{1}&t_{2}\\\
x_{1}&v_{1}&x_{2}&z&x_{1}^{\prime}&x_{3}\\\ &&1&y_{2}&0&0\\\ &&&v_{2}&0&0\\\
&&&y_{3}^{\prime}&I_{m-n}&0\\\
&&&y^{\prime}_{1}&0&I_{n}\end{pmatrix}\gamma_{n,m}\begin{pmatrix}h&\\\
&I_{l-m-n}\end{pmatrix}s_{j}\right)\psi_{j}^{-1}(y)dydu.$

Denote

$Z=\begin{pmatrix}y_{3}&t_{1}&t_{2}\\\
z&x_{1}^{\prime}&x_{3}\end{pmatrix}\in[{\mathrm{Mat}}_{(n+j)\times(m+k)}].$

Then inside the integral $\phi_{Y_{j},\psi_{j},N_{n,m}}(h)$, there is an inner
integral

$\int_{[{\mathrm{Mat}}_{n+j}\times(m+k)]}\phi\left(\begin{pmatrix}I_{n+j}&&Z\\\
&1&\\\ &&I_{m+k}\end{pmatrix}g\right)dZ,$

which, by Lemma 2.3, equals to

$\displaystyle\sum_{\begin{subarray}{c}\gamma_{1}\in
N_{n+j}(F)\backslash{\mathrm{GL}}_{n+j}(F)\\\ \gamma_{2}\in
N_{m+k}(F)\backslash{\mathrm{GL}}_{m+k}(F)\end{subarray}}W_{\phi}^{\psi}\left(\begin{pmatrix}\gamma_{1}&&\\\
&1&\\\ &&\gamma_{2}\end{pmatrix}g\right).$

Thus we get

$\displaystyle\phi_{Y_{j},\psi_{j},N_{n,m}}(h)=\sum_{\begin{subarray}{c}\gamma_{1}\in
N_{n+j}(F)\backslash{\mathrm{GL}}_{n+j}(F)\\\ \gamma_{2}\in
N_{m+k}(F)\backslash{\mathrm{GL}}_{m+k}(F)\end{subarray}}\int
W_{\phi}^{\psi}\left(\begin{pmatrix}\gamma_{1}&&\\\ &1&\\\
&&\gamma_{2}\end{pmatrix}\begin{pmatrix}I_{n}&0&0&0&0\\\
x_{1}&v_{1}&x_{2}&0&0\\\ &&1&y_{2}&0\\\ &&&v_{2}&0\\\ &&&y_{1}&I_{m}\\\
\end{pmatrix}\gamma_{n,m}hs_{j}\right)$

where $y_{1}=\begin{pmatrix}y_{3}^{\prime}\\\
y_{1}^{\prime}\end{pmatrix}\in[{\mathrm{Mat}}_{m\times k}],$ and
$h=\begin{pmatrix}h&\\\ &I_{l-m-n}\end{pmatrix}$. Note that the above formula
is similar to (2.10). By the same method as in the proof of Theorem 2.4, we
get that

$\displaystyle\phi_{Y,\psi,N^{\prime}}(h)$
$\displaystyle=\sum_{\begin{subarray}{c}\gamma_{1}\in
N_{n}(F)\backslash{\mathrm{GL}}_{n}(F)\\\ \gamma_{2}\in
N_{m}(F)\backslash{\mathrm{GL}}_{m}(F)\end{subarray}}\int_{\overline{V}^{j,m,n}({\mathbb{A}})}W_{\phi}^{\psi}\left(\begin{pmatrix}\gamma_{1}&&\\\
&I_{l-m-n}&\\\
&&\gamma_{2}\end{pmatrix}\overline{v}\gamma_{n,m}hs_{j}\right)d\overline{v}$
$\displaystyle=\sum_{\begin{subarray}{c}\gamma_{1}\in
N_{n}(F)\backslash{\mathrm{GL}}_{n}(F)\\\ \gamma_{2}\in
N_{m}(F)\backslash{\mathrm{GL}}_{m}(F)\end{subarray}}\int_{\overline{V}^{j,m,n}({\mathbb{A}})}W_{\phi}^{\psi}\left(\overline{v}\gamma_{n,m}\begin{pmatrix}\gamma_{1}&&\\\
&\gamma_{2}&\\\ &&I_{l-m-n}\end{pmatrix}hs_{j}\right)d\overline{v}.$

Plugging the above equation into (2.14), we get that

$\displaystyle I(\phi,\widetilde{f}_{\bf{s}})$
$\displaystyle=\int_{\\{(N_{n}(F)\times
N_{m}(F))N_{n,m}({\mathbb{A}})\\}\backslash{\mathrm{GL}}_{n+m}({\mathbb{A}})}\int_{\overline{V}^{m,n}({\mathbb{A}})}W_{\phi}^{\psi}(\bar{v}\gamma_{n,m}hs_{j})\widetilde{f}_{\bf{s}}(h)d\overline{v}dh$
$\displaystyle=\int_{N_{n+m}({\mathbb{A}})\backslash{\mathrm{GL}}_{n+m}({\mathbb{A}})}\int_{\overline{V}^{m,n}({\mathbb{A}})}W_{\phi}^{\psi}(\overline{v}\gamma_{n,m}hs_{j})$
$\displaystyle\quad\cdot\int_{[N_{n}]\times[N_{m}]}\widetilde{f}_{s}\left(\begin{pmatrix}u_{1}&\\\
&u_{2}\end{pmatrix}\right)\psi(u_{1})\psi(u_{2})du_{1}du_{2}d\overline{v}dh$
$\displaystyle=\int_{N_{n+m}({\mathbb{A}})\backslash{\mathrm{GL}}_{n+m}({\mathbb{A}})}\int_{\overline{V}^{m,n}({\mathbb{A}})}W_{\phi}^{\psi}(\overline{v}\gamma_{n,m}hs_{m,n})\xi^{\psi^{-1}}_{\widetilde{f}_{\bf{s}}}(h)d\overline{v}dh.$

The result follows. ∎

###  2.5. Convergence and justifications

In this subsection, we prove the convergence of the double integral in (2.13),
for ${\mathrm{Re}}(s_{1})\gg 0,{\mathrm{Re}}(s_{2})\gg 0$. This is standard
and similar to many other situations like [BAS:Uodd]. Using the Iwasawa
decomposition, and the fact that $(N_{m}(F)\times
N_{n}(F))\backslash(N_{m}({\mathbb{A}})\times N_{n}({\mathbb{A}}))$ is
compact, the convergence of the double integral in (2.13), for
${\mathrm{Re}}(s_{1})\gg 0,{\mathrm{Re}}(s_{2})\gg 0$, quickly reduces to the
convergence of

(2.15)
$\int_{T_{m+n}({\mathbb{A}})}\|t\|^{N_{0}}|\det(a)|^{{\mathrm{Re}}(s_{1})+c_{1}}|\det(b)|^{-{\mathrm{Re}}(s_{2})+c_{2}}\int_{\overline{U}^{j,m,n}({\mathbb{A}})}|W_{\phi}^{\psi}\left({\bf{t}}_{m,n}(a,b)\overline{u}\right)|d\overline{u}dt$

where $t={\mathrm{diag}}(a,b)$ with $a\in T_{m}({\mathbb{A}}),b\in
T_{n}({\mathbb{A}})$ and
${\bf{t}}_{m,n}(a,b)=\eta_{j}\iota_{j}(t)\eta_{j}^{-1}={\mathrm{diag}}(a,I_{l-m-n},b)$.
Here, $N_{0}$, $c_{1}$ and $c_{2}$ are fixed given positive numbers. Note that
the integration over $K_{{\mathrm{GL}}_{l}({\mathbb{A}})}$ is dropped, using a
similar reasoning as in [BAS:Uodd, Remark 4.7] (in conjunction with Lemma 2.7
and (2.21)). We may assume that the Whittaker function $W_{\phi}^{\psi}$
decomposes as $\prod_{v}W_{v}$, where $W_{v}$ is a local $\psi_{v}$-Whittaker
function of $\pi_{v}$, such that outside of a finite set $S$ of places
(including the archimedean ones), $\pi_{v}$ is unramified, and
$W_{v}=W_{v}^{0}$ is the normalized unramified $\psi_{v}$-Whittaker function
of $\pi_{v}$ whose value at the identity is equal to 1. We assume that for $v$
outside of $S$, $\psi_{v}$ is unramified. It suffices to prove, for
${\mathrm{Re}}(s_{1})\gg 0,{\mathrm{Re}}(s_{2})\gg 0$, that we have

(2.16)
$\prod_{v}\int_{T_{m+n}(F_{v})}\|t\|^{N_{0}}_{v}|\det(a)|_{v}^{{\mathrm{Re}}(s_{1})+c_{1}}|\det(b)|_{v}^{-{\mathrm{Re}}(s_{2})+c_{2}}\int_{\overline{U}^{j,m,n}(F_{v})}|W_{v}\left({\bf{t}}_{m,n}(a,b)\overline{u}\right)|d\overline{u}dt<\infty.$

######  Lemma 2.6.

Let $v$ be a finite place of $F$. For fixed
$W_{v}\in{\mathcal{W}}(\pi_{v},\psi_{v})$, and $t={\mathrm{diag}}(a,b)$ with
$a\in T_{m}(F_{v}),b\in T_{n}(F_{v})$, the function

$\overline{u}\mapsto
W_{v}({\bf{t}}_{m,n}(a,b)\overline{u}),\quad\overline{u}\in\overline{U}^{j,m,n}(F_{v})$

has compact support in $\overline{U}^{j,m,n}(F_{v})$. If $W_{v}=W_{v}^{0}$,
then this support is in $\overline{U}^{j,m,n}({\mathcal{O}}_{v})$.

###### Proof.

The proof is a standard “root killing” argument and it is similar to the proof
of [BAS:Uodd]*Lemma 4.1. We omit the details. ∎

Suppose $v$ is finite. By Lemma 2.6, for the local integral of (2.16) at $v$,
it suffices to show

(2.17)
$\int_{T_{m+n}(F_{v})}\|t\|^{N_{0}}_{v}|\det(a)|_{v}^{{\mathrm{Re}}(s_{1})+c_{1}}|\det(b)|_{v}^{-{\mathrm{Re}}(s_{2})+c_{2}}|W_{v}\left({\bf{t}}_{m,n}(a,b)\right)|dt<\infty$

for ${\mathrm{Re}}(s_{i})\gg 0$. Now we recall gauge estimates on Whittaker
functions in [JPSS-GL3I, Section 2]. A gauge on ${\mathrm{GL}}_{l}(F_{v})$ is
a function $\xi$ on ${\mathrm{GL}}_{l}(F_{v})$ which is invariant on the left
under $N_{l}(F_{v})$, on the right under
${\mathrm{GL}}_{l}({\mathcal{O}}_{v})$, and which on $T_{l}(F_{v})$ has the
form

(2.18) $\xi(t)=|t_{1}t_{2}\cdots
t_{l-1}|_{v}^{-c}\Phi(t_{1},t_{2},\cdots,t_{l-1})$

for

$t={\mathrm{diag}}(t_{1}t_{2}\cdots t_{l},t_{2}\cdots
t_{l},\cdots,t_{l-1}t_{l},t_{l})\in T_{l}(F_{v}),$

where $c\geq 0$ is a real number and $\Phi\geq 0$ is a Schwartz-Bruhat
function on $F_{v}^{l-1}$. In particular, $\xi$ is invariant under the center
of ${\mathrm{GL}}_{l}(F_{v})$. Write $a\in T_{m}(F)$ and $b\in T_{n}(F)$ as

$\begin{split}a&={\mathrm{diag}}(a_{1}\cdots a_{m},a_{2}\cdots
a_{m},\dots,a_{m-1}a_{m},a_{m}),\\\
b&={\mathrm{diag}}(b_{1}^{-1},b_{1}^{-1}b_{2}^{-1},\dots,b_{1}^{-1}b_{2}^{-1}\cdots
b_{n}^{-1}),\end{split}$

with $a_{i}\in F^{\times},b_{j}\in F^{\times}$. Then

$\begin{split}|\det(a)|_{v}&=|a_{1}a_{2}^{2}\cdots
a_{m-1}^{m-1}a_{m}^{m}|_{v},\\\ |\det(b)|_{v}&=|b_{1}^{n}b_{2}^{n-1}\cdots
b_{n-1}^{2}b_{n}|_{v}^{-1},\end{split}$

and

$\displaystyle{\bf{t}}_{m,n}$ $\displaystyle({\mathrm{diag}}(a,b))=$
$\displaystyle{\mathrm{diag}}(a_{1}\cdots a_{m},a_{2}\cdots
a_{m},\dots,a_{m-1}a_{m},a_{m},1,1,\dots,1,b_{1}^{-1},b_{1}^{-1}b_{2}^{-1},\dots,b_{1}^{-1}b_{2}^{-1}\cdots
b_{n}^{-1}).$

Then for a gauge $\xi$ on ${\mathrm{GL}}_{l}(F_{v})$, it follows from (2.18)
that there is some real number $c\geq 0$ and a Schwartz-Bruhat function $\Phi$
on $F_{v}^{l-1}$ such that

(2.19) $\xi({\bf{t}}_{m,n}({\mathrm{diag}}(a,b)))=|a_{1}\cdots
a_{m}b_{1}\cdots
b_{n}|_{v}^{-c}\Phi(a_{1},a_{2},\dots,a_{m},1,\dots,1,b_{1},b_{2},\dots,b_{n}).$

Write $|\omega_{\pi}|_{v}=\alpha^{c_{0}}$, where $\alpha$ is a non-negative
real-valued function on $F_{v}^{\times}$ and $c_{0}$ is a real number. By
[JPSS-GL3I, Proposition 2.3.6], for any Whittaker function
$W_{v}\in\mathcal{W}(\pi_{v},\psi_{v})$, there is a gauge $\xi$ such that

(2.20) $|W_{v}\otimes\alpha^{-c_{0}/l}|\leq\xi.$

Then (2.17) follows from (2.19) and the estimate (2.20). This proves that the
product in (2.16) over finite places is convergent.

Now we turn to the archimedean places. Let $v$ be an archimedean place, so
$F_{v}$ is either $\mathbb{R}$ or $\mathbb{C}$. We recall the notion of gauge
[JacquetShalika-archimedean] in this setting, which is slightly different from
the non-archimedean case. Let $\chi$ be a sum of positive characters of
$T_{l}(F_{v})$ trivial on the center of ${\mathrm{GL}}_{l}(F_{v})$. An
homogeneous gauge on ${\mathrm{GL}}_{l}(F_{v})$ is a function $\xi$ on
${\mathrm{GL}}_{l}(F_{v})$ of the form

$\xi(ntk)=\chi(t)\Phi(t_{1},t_{2},\cdots,t_{l-1}),$

where $n\in N_{l}(F_{v})$, $t={\mathrm{diag}}(t_{1},\cdots,t_{l})\in
T_{l}(F_{v})$, $k$ is in the maximal compact subgroup $K_{l}$ of
${\mathrm{GL}}_{l}(F_{v})$, and $\Phi>0$ is a rapidly decreasing function in
$l-1$ variables. Here, $\Phi$ being rapidly decreasing means that, for every
set of integers $N_{i}$, $1\leq i\leq l-1$, there is a constant $C>0$ such
that

$\Phi(t_{1},t_{2},\cdots,t_{l-1})\leq C\prod_{i}(1+|t_{i}|_{v}^{2})^{-N_{i}}.$

We have the following estimate.

######  Lemma 2.7.

Let $v$ be an archimedean place. Let $\xi$ be an homogeneous gauge on
${\mathrm{GL}}_{l}(F_{v})$. Let $a\in T_{m}(F_{v})$ and $b\in T_{n}(F_{v})$,
with Iwasawa decompositions

$\begin{split}a=n_{1}{\mathrm{diag}}(t_{1},\cdots,t_{m})k_{1},\quad
b=n_{2}{\mathrm{diag}}(t_{l-n+1},\cdots,t_{l})k_{2},\end{split}$

where $t_{1},\cdots,t_{m},t_{l-n+1},\cdots,t_{l}$ are positive real numbers.
Set $t_{m+1}=t_{m+2}=\cdots=t_{l-n}=1$. Given positive integers
$M_{1},\cdots,M_{j},N_{1},\cdots,N_{n}$, $L_{1},\cdots,L_{l-1}$, there exists
a positive constant $C>0$ such that

$\begin{split}&\xi\left(\begin{pmatrix}a&&&&\\\ &I_{j}&&&\\\ &&1&&\\\
&&&I_{k}&\\\ &&&&b\end{pmatrix}\begin{pmatrix}I_{m}&&&&\\\ x&I_{j}&&&\\\
&&1&&\\\ &&&I_{k}&\\\ &&&y&I_{n}\end{pmatrix}\right)\\\
\leq&C\prod_{i=1}^{j}(1+||x_{i}||^{2})^{-M_{i}}\prod_{i=1}^{n}(1+||y_{i}||^{2})^{-N_{i}}\prod_{i=1}^{l-1}\left(1+\left|\frac{t_{i}}{t_{i+1}}\right|^{2}\right)^{-L_{i}}\chi(t_{1},t_{2},\cdots,t_{l}).\end{split}$

Here, $\chi$ is a fixed sum of positive characters of $T_{l}(F_{v})$.

###### Proof.

The proof is similar to that of [JacquetShalika-archimedean, Lemma 5.2]. See
also [BAS:Uodd, Lemma 4.6]. We omit the details. ∎

By [JacquetShalika-archimedean, Proposition 2.1], for any Whittaker function
$W_{v}\in\mathcal{W}(\pi_{v},\psi_{v})$, there is a gauge $\xi$ on
${\mathrm{GL}}_{l}(F_{v})$ such that

(2.21) $|W_{v}(g)|\leq\xi(g),\quad g\in{\mathrm{GL}}_{l}(F_{v}).$

Combining (2.21) with Lemma 2.7, we conclude that the archimedean part of the
product in (2.16) is convergent. This complete the proof of the convergence of
the double integral in (2.13).

##  3\. The local integrals

In this section, let $F$ be a local field. Let $\psi$ be a nontrivial additive
character of $F$. We still fix a positive integer $l$ and non-negative
integers $m,n$ such that $m+n\leq l-1$. For $0\leq j\leq l-m-n-1$, we set
$k=l-m-n-1-j$.

###  3.1. Definition of the local zeta integrals

Let $\pi$ be an irreducible generic representation of ${\mathrm{GL}}_{l}(F)$
and let ${\mathcal{W}}(\pi,\psi)$ be its Whittaker model. Let
$(\tau_{1},V_{\tau_{1}})$ (resp. $(\tau_{2},V_{\tau_{2}})$) be an irreducible
generic representation of ${\mathrm{GL}}_{m}(F)$ (resp.
${\mathrm{GL}}_{n}(F)$). As in the last section, we write
${\boldsymbol{\tau}}=(\tau_{1},\tau_{2})$ and
$\widehat{\boldsymbol{\tau}}=(\tau_{2},\tau_{1})$. Let
${\bf{s}}=(s_{1},s_{2})$ be a pair of complex numbers. Then we can consider
the induced representation

${\mathrm{I}}({\bf{s}},{\boldsymbol{\tau}}):={\mathrm{Ind}}_{P_{m,n}(F)}^{{\mathrm{GL}}_{m+n}(F)}(\tau_{1}||^{s_{1}-\frac{1}{2}}\boxtimes\tau_{2}||^{-s_{2}+1/2}).$

We fix $\psi^{-1}$-Whittaker functionals $\lambda_{i}$ of $\tau_{i}$. Recall
that a section $f_{{\bf{s}}}\in{\mathrm{I}}({\bf{s}},{\boldsymbol{\tau}})$ is
a function $f_{\bf{s}}:{\mathrm{GL}}_{m+n}(F)\to V_{\tau_{1}}\boxtimes
V_{\tau_{2}}$ satisfying certain quasi-invariance properties. We consider the
${\mathbb{C}}$-valued function

$\xi_{f_{\bf{s}}}:{\mathrm{GL}}_{m+n}(F)\times{\mathrm{GL}}_{m}(F)\times{\mathrm{GL}}_{n}(F)\to{\mathbb{C}}$

defined by

$\xi_{f_{\bf{s}}}(h,a_{1},a_{2})=\lambda_{1}\boxtimes\lambda_{2}(\tau_{1}(a_{1})\boxtimes\tau_{2}(a_{2})(f_{\bf{s}}(h))).$

Set
${\mathcal{W}}({\bf{s}},{\boldsymbol{\tau}},\psi^{-1})=\left\\{{\xi_{f_{\bf{s}}}:f_{\bf{s}}\in{\mathrm{I}}({\bf{s}},{\boldsymbol{\tau}})}\right\\}$.
Note that an element $\xi_{\bf{s}}$ satisfies

$\displaystyle\xi_{{\bf{s}}}\left(\begin{pmatrix}b_{1}&\\\
&b_{2}\end{pmatrix}uh,a_{1},a_{2}\right)=|\det(b_{1})|^{s_{1}+\frac{n-1}{2}}|\det(b_{2})|^{-s_{2}-\frac{m-1}{2}}\xi_{{\bf{s}}}(h,a_{1}b_{1},a_{2}b_{2}),$

for
$a_{1},b_{1}\in{\mathrm{GL}}_{m}(F),a_{2},b_{2}\in{\mathrm{GL}}_{n}(F),u\in
N_{m,n}(F),h\in{\mathrm{GL}}_{m+n}(F)$. In particular

$\displaystyle\xi_{\bf{s}}\left(\begin{pmatrix}u_{1}&\\\
&u_{2}\end{pmatrix}uh,I_{m},I_{n}\right)$
$\displaystyle=\psi^{-1}(u_{1})\psi^{-1}(u_{2})\xi_{s}(h,I_{m},I_{n}),$

for $u_{1}\in N_{m}(F),u_{2}\in N_{n}(F),u\in
N_{m,n}(F),h\in{\mathrm{GL}}_{m+n}(F).$ We usually write
$\xi_{{\bf{s}}}(h,I_{m},I_{n})$ as $\xi_{{\bf{s}}}(h)$ for simplicity.

Similarly, we can consider the space
${\mathcal{W}}(1-\widehat{\bf{s}},\widehat{\boldsymbol{\tau}},\psi^{-1})=\left\\{{\xi_{f_{1-\widehat{\bf{s}}}}:f_{1-\widehat{\bf{s}}}\in{\mathrm{I}}(1-\widehat{\bf{s}},\widehat{\boldsymbol{\tau}})}\right\\}$.
Note that the intertwining operator on the induced representations gives an
intertwining operator

$M_{w_{m,n}}:{\mathcal{W}}({\bf{s}},{\boldsymbol{\tau}},\psi^{-1})\to{\mathcal{W}}(1-\widehat{\bf{s}},\widehat{\boldsymbol{\tau}},\psi^{-1})$

defined by

$M_{w_{m,n}}(\xi_{\bf{s}})(h,a_{1},a_{2})=\int_{N_{n,m}(F)}\xi_{\bf{s}}(w_{m,n}uh,a_{2},a_{1})du,$

where $a_{1}\in{\mathrm{GL}}_{n}(F),a_{2}\in{\mathrm{GL}}_{m}(F)$.

For $W\in{\mathcal{W}}(\pi,\psi)$,
$\xi_{\bf{s}}\in{\mathcal{W}}({\bf{s}},{\boldsymbol{\tau}},\psi^{-1})$, and
for $j$ with $0\leq j\leq l-m-n-1$, we consider the local zeta integrals

(3.1)
$\Psi(W,\xi_{\bf{s}};j):=\int_{N_{m+n}(F)\backslash{\mathrm{GL}}_{m+n}(F)}\int_{\overline{U}^{j,m,n}(F)}W\left(\overline{u}\gamma_{m,n}\begin{pmatrix}h&\\\
&I_{l-m-n}\end{pmatrix}\right)\xi_{\bf{s}}(h)d\overline{u}dh,$

where we recall that

$\displaystyle\overline{U}^{j,m,n}$
$\displaystyle=\left\\{{\overline{u}(x,y)=\begin{pmatrix}I_{m}&&&&\\\
x&I_{j}&&&\\\ &&1&&\\\ &&&I_{k}&\\\
&&&y&I_{n}\end{pmatrix}:\begin{subarray}{c}x\in{\mathrm{Mat}}_{j\times m}\\\
y\in{\mathrm{Mat}}_{n\times k}\end{subarray}}\right\\},$
$\displaystyle\gamma_{m,n}$
$\displaystyle=\eta_{j,m,n}s_{j,m,n}^{-1}=\begin{pmatrix}I_{m}&&\\\
&&I_{l-m-n}\\\ &I_{n}&\end{pmatrix}.$

Here we remark that the natural numbers $m,n$ appeared in the local zeta
integral (3.1) are determined by the section $\xi_{{\bf{s}}}$, which is an
element of
${\mathrm{Ind}}_{P_{m,n}(F)}^{{\mathrm{GL}}_{m+n}(F)}(\tau_{1}||^{s_{1}-1/2}\otimes\tau_{2}||^{-s_{2}+1/2})$.
In particular, if we take
$\widetilde{\xi}_{1-\widehat{\bf{s}}}\in{\mathcal{W}}(1-\widehat{\bf{s}},\widehat{\boldsymbol{\tau}},\psi^{-1})$,
we should have

(3.2)
$\Psi(W,\widetilde{\xi}_{1-\widehat{\bf{s}}};j)=\int_{N_{m+n}(F)\backslash{\mathrm{GL}}_{m+n}(F)}\int_{\overline{U}^{j,n,m}(F)}W\left(\overline{u}\gamma_{n,m}\begin{pmatrix}h&\\\
&I_{l-m-n}\end{pmatrix}\right)\widetilde{\xi}_{1-\widehat{\bf{s}}}(h)d\overline{u}dh.$

######  Remark 3.1.

In this remark, we assume that $F$ is a global field. If
$\phi=\otimes\phi_{v}$ is a cusp form on ${\mathrm{GL}}_{l}({\mathbb{A}})$ and
$f_{{\bf{s}}}=\otimes
f_{{\bf{s}},v}\in{\mathrm{I}}({\bf{s}},{\boldsymbol{\tau}})$ is a pure tensor
of a global section, then Theorem 2.4 and Theorem 2.5 imply that

$I_{j}(\phi,f_{\bf{s}})=\prod_{v}\Psi(\rho(s_{j,m,n})W_{v},\xi_{f_{{\bf{s}},v}};j),\quad
I_{j}(\phi,\widetilde{f}_{\bf{s}})=\prod_{v}\Psi(\rho(s_{j,m,n})W_{v},\xi_{\widetilde{f}_{{\bf{s}},v}};j).$

Here $\rho$ denotes the right translation.

######  Remark 3.2.

In this remark, we consider the degenerate case when $m>0$ and $n=0$. In this
case, ${\boldsymbol{\tau}}=\tau_{1}$ is just a representation of
${\mathrm{GL}}_{m}(F)$, and ${\bf{s}}=s$ is a single complex number. Moreover,
an element $\xi_{\bf{s}}$ has the form
$\xi_{\bf{s}}(h)=W^{\prime}(h)|\det(h)|^{s-1/2}$ and we have
$M_{w_{m,0}}(\xi_{\bf{s}})=\xi_{{\bf{s}}}.$ Thus

$\displaystyle\Psi(W,\xi_{{\bf{s}}};j)=\int_{N_{m}(F)\backslash{\mathrm{GL}}_{m}(F)}\int_{{\mathrm{Mat}}_{j\times
m}(F)}$ $\displaystyle W\left(\begin{pmatrix}I_{m}&&\\\ x&I_{j}&\\\
&&I_{l-m-j}\end{pmatrix}\begin{pmatrix}h&\\\ &I_{l-m}\end{pmatrix}\right)$
$\displaystyle\cdot W^{\prime}(h)|\det(h)|^{s-1/2}dxdh,$

and

$\displaystyle\Psi(W,M_{w_{m,0}}(\xi_{\bf{s}});j)=\int_{N_{m}(F)\backslash{\mathrm{GL}}_{m}(F)}\int_{{\mathrm{Mat}}_{m\times
k}(F)}$ $\displaystyle W\left(\begin{pmatrix}I_{j+1}&&\\\ &I_{k}&\\\
&y&I_{m}\end{pmatrix}\begin{pmatrix}&I_{l-m}\\\
I_{m}&\end{pmatrix}\begin{pmatrix}h&\\\ &I_{l-m}\end{pmatrix}\right)$
$\displaystyle\cdot W^{\prime}(h)|\det(h)|^{s-1/2}dydh.$

Here we notice that $\gamma_{m,0}=I_{l}$ while
$\gamma_{0,m}=\begin{pmatrix}&I_{l-m}\\\ I_{m}&\end{pmatrix}.$ A simple change
of variable shows that

$\displaystyle\Psi(W,\xi_{\bf{s}};j)=\int_{N_{m}(F)\backslash{\mathrm{GL}}_{m}(F)}\int_{{\mathrm{Mat}}_{j\times
m}(F)}$ $\displaystyle W\left(\begin{pmatrix}h&&\\\ x&I_{j}&\\\
&&I_{l-m-j}\end{pmatrix}\right)W^{\prime}(h)|\det(h)|^{s-1/2-j}dxdh.$

One can compare the above integral with that defined by Jacquet–Piatetski-
Shapiro–Shalika in [JPSS] and observe that

(3.3)
$\Psi(W,\xi_{{\bf{s}}};j)=\Psi^{\mathrm{JPSS}}(s-j+\frac{l-m-1}{2},W,W^{\prime};j),$

where $\Psi^{\mathrm{JPSS}}$ denotes the integral defined in [JPSS]*p.387. On
the other hand, for $W\in{\mathcal{W}}(\pi,\psi)$, we denote
$\widetilde{W}(g)=W(J_{l}{}^{t}\\!g^{-1})$, which represents a Whittaker
function of the contragredient representation $\widetilde{\pi}$ of $\pi$. It
is easy to check that

$\displaystyle\Psi(W,M_{w_{m,0}}(\xi_{\bf{s}});j)=\int_{N_{m}(F)\backslash{\mathrm{GL}}_{m}(F)}\int_{{\mathrm{Mat}}_{k\times
m}(F)}$ $\displaystyle\widetilde{W}\left(\begin{pmatrix}h&&\\\ y&I_{k}&\\\
&&I_{j+1}\end{pmatrix}\begin{pmatrix}I_{m}&\\\ &J_{l-m}\end{pmatrix}\right)$
$\displaystyle\widetilde{W}^{\prime}(h)|\det(h)|^{-s+1/2-k}dydh.$

Thus we get

(3.4)
$\Psi(W,M_{w_{m,0}}(\xi_{{\bf{s}}});j)=\Psi^{\mathrm{JPSS}}\left(1-s-k+\frac{l-m-1}{2},\rho\left(\begin{pmatrix}I_{m}&\\\
&J_{l-m}\end{pmatrix}\right)\widetilde{W},\widetilde{W}^{\prime};k\right).$

######  Remark 3.3.

If $l=2r+1$ and $m=n$ with $1\leq m\leq r$, then the integral
$\Psi(W,\xi_{{\bf{s}}};r-m)$ is the local zeta integral of
${\mathrm{U}}_{E/F}(2r+1)\times{\mathrm{Res}}_{E/F}({\mathrm{GL}}_{m})$ at
split places as in [BAS:Uodd], where $E/F$ is a quadratic extension of global
fields.

######  Proposition 3.4.

The local zeta integrals $\Psi(W,\xi_{{\bf{s}}};j)$ are absolutely convergent
for ${\mathrm{Re}}(s_{i})\gg 0$ for $i=1,2$. Over nonarchimedean local fields,
there exist $W$ and $\xi_{\bf{s}}$, such that the integral is absolutely
convergent and equals 1, for all ${\bf{s}}$. Over archimedean fields, for any
${\bf{s}}$, there are choices of data $(W^{i},\xi_{{\bf{s}}}^{i})$ such that
$\sum_{i}\Psi(W^{i},\xi_{{\bf{s}}}^{i};j)$ is holomorphic and nonzero in a
neighborhood of ${\bf{s}}$.

###### Proof.

For $n=0$, this was already proved in [JPSS] over nonarchimedean local fields
and in [JacquetShalika-archimedean] over archimedean fields. Very similar
statements can be found in many other places in the literature, for example,
[Soudry1993], [Soudry1995], [GinzburgRallisSoudry1998], [BAS:Uodd], and
[CFK2022]. We provide some details here for completeness.

First, we consider the case where $F$ is nonarchimedean. By the Iwasawa
decomposition, and the fact that smooth vectors are finite under the maximal
compact subgroup, we get that $\Psi(W,\xi_{{\bf{s}}};j)$ is a finite sum of
integrals of the form

$\int_{T_{m+n}(F)}\int_{\overline{U}^{m,n}(F)}W^{\prime}({\bf{t}}_{m,n}(a,b)\overline{u})d\overline{u}W_{\tau_{1}}(a)W_{\tau_{2}}(b)|\det(a)|^{s_{1}+\frac{n-1}{2}-j}|\det(b)|^{-s_{2}-\frac{m-1}{2}+k}\delta_{B_{m+n}}(t)^{-1}dt$

where $W^{\prime}\in\mathcal{W}(\pi,\psi)$,
$W_{\tau_{1}}\in\mathcal{W}(\tau_{1},\psi^{-1})$,
$W_{\tau_{2}}\in\mathcal{W}(\tau_{2},\psi^{-1})$, $t={\mathrm{diag}}(a,b)$
with $a\in T_{m}(F),b\in T_{n}(F)$ and
${\bf{t}}_{m,n}(a,b)={\mathrm{diag}}(a,I_{l-m-n},b)$. Here the term
$|\det(a)|^{-j}|\det(b)|^{k}$ comes from conjugating ${\bf{t}}_{m,n}(a,b)$ to
the left of $\overline{u}$ and making a change of variables on $\overline{u}$.
By Lemma 2.6, the last integral is a finite sum of integrals of the form

(3.5)
$\int_{T_{m+n}(F)}W^{\prime}({\bf{t}}_{m,n}(a,b))W_{\tau_{1}}(a)W_{\tau_{2}}(b)|\det(a)|^{s_{1}+\frac{n-1}{2}-j}|\det(b)|^{-s_{2}-\frac{m-1}{2}+k}\delta_{B_{m+n}}(t)^{-1}dt.$

Now we recall the asymptotic expansion of Whittaker functions [JPSS, Section
2.5]. There is a finite set $X_{l}$ of functions on $T_{l}(F)$ such that for
every $W\in\mathcal{W}(\pi,\psi)$ we have

$W(t)=\sum_{\chi\in
X_{l}}\omega_{\pi}(t_{l})\phi_{\chi}(t_{1},t_{2},\cdots,t_{l-1})\chi(t)$

where $t={\mathrm{diag}}(t_{1}t_{2}\cdots t_{l},t_{2}\cdots
t_{l},\cdots,t_{l-1}t_{l},t_{l})\in T_{l}(F)$ and
$\phi_{\chi}\in\mathcal{S}(F^{l-1})$. Then for every
$W\in\mathcal{W}(\pi,\psi)$, we have

(3.6) $|W(t)|\leq\sum_{\eta\in
Y_{l}}\phi_{\eta}(t_{1},t_{2},\cdots,t_{l-1})\eta(t)$

where $\phi_{\eta}\in\mathcal{S}(F^{l-1})$ is non-negative and $\eta$ varies
in another finite set $Y_{l}$ of finite functions on $T_{l}(F)$. Applying the
majorization (3.6) to $W^{\prime}$ (and the analogous ones for $W_{\tau_{1}}$
and $W_{\tau_{2}}$), we obtain the absolute convergence of the integral (3.5)
for ${\mathrm{Re}}(s_{i})\gg 0$ for $i=1,2$. Hence $\Psi(W,\xi_{{\bf{s}}};j)$
is absolutely convergent for ${\mathrm{Re}}(s_{i})\gg 0$ for $i=1,2$.

We continue to assume that $F$ is nonarchimedean. Since
$N_{m+n}(F)T_{m+n}(F)\overline{N}_{m+n}(F)$ is an open dense subset of
${\mathrm{GL}}_{m+n}(F)$ whose complement has Haar measure zero, we may
rewrite $\Psi(W,\xi_{{\bf{s}}};j)$ as

(3.7)
$\begin{split}\int_{T_{m+n}(F)}\int_{\overline{N}_{m+n}(F)}\int_{\overline{U}^{j,m,n}(F)}&W\left(\overline{u}\gamma_{m,n}\begin{pmatrix}t\overline{v}&\\\
&I_{l-m-n}\end{pmatrix}\right)\xi_{\bf{s}}(\overline{v},a,b)\\\
&|\det(a)|^{s_{1}+\frac{n-1}{2}}|\det(b)|^{-s_{2}-\frac{m-1}{2}}\delta_{B_{m+n}}(t)^{-1}d\overline{u}d\overline{v}dt,\end{split}$

where $t={\mathrm{diag}}(a,b)$ with $a\in T_{m}(F)$, $b\in T_{n}(F)$. Similar
to [Soudry1993, Proposition 6.1], we choose $\xi_{\bf{s}}$ to have support in
$B_{m+n}(F)\cdot\mathcal{V}_{1}$, where $\mathcal{V}_{1}$ is a small open
compact subgroup of ${\mathrm{GL}}_{m+n}(F)$, and such that
$\xi_{\bf{s}}(u,b_{1},b_{2})=W_{\tau_{1}}(b_{1})W_{\tau_{2}}(b_{2})$ for
$u\in\mathcal{V}_{1}$, $b_{1}\in T_{m}(F),b_{2}\in T_{n}(F)$. Here,
$W_{\tau_{i}}\in\mathcal{W}(\tau_{i},\psi^{-1})$ for $i=1,2$. We choose
$\mathcal{V}_{1}$ so small that $W$ is fixed by
$\pi({\mathrm{diag}}(\overline{v},I_{l-m-n}))$ for
$\overline{v}\in\mathcal{V}_{1}$. Thus, $\Psi(W,\xi_{{\bf{s}}};j)$ is equal to

$\begin{split}\mathrm{vol}(\mathcal{V}_{1}\cap\overline{N}_{m+n}(F))\cdot\int_{T_{m+n}(F)}\int_{\overline{U}^{j,m,n}(F)}&W\left(\overline{u}\gamma_{m,n}\begin{pmatrix}t&\\\
&I_{l-m-n}\end{pmatrix}\right)W_{\tau_{1}}(a)W_{\tau_{2}}(b)\\\
&|\det(a)|^{s_{1}+\frac{n-1}{2}}|\det(b)|^{-s_{2}-\frac{m-1}{2}}\delta_{B_{m+n}}(t)^{-1}d\overline{u}dt.\end{split}$

We conjugate ${\mathrm{diag}}(t,I_{l-m-n})$ to the left of $\overline{u}$ and
make a change of variable in $\overline{u}$ to get

$\begin{split}\mathrm{vol}(\mathcal{V}_{1}\cap\overline{N}_{m+n}(F))\cdot\int_{T_{m+n}(F)}\int_{\overline{U}^{j,m,n}(F)}&\rho(\gamma_{m,n})W\left(\begin{pmatrix}a&&\\\
&I_{l-m-n}&\\\
&&b\end{pmatrix}\overline{u}\right)W_{\tau_{1}}(a)W_{\tau_{2}}(b)\\\
&|\det(a)|^{s_{1}+\frac{n-1}{2}-j}|\det(b)|^{-s_{2}-\frac{m-1}{2}+k}\delta_{B_{m+n}}(t)^{-1}d\overline{u}dt.\end{split}$

Now we choose $W$, $W_{\tau_{1}}$ and $W_{\tau_{2}}$ such that the function

$(a,b,\overline{u})\mapsto\rho(\gamma_{m,n})W\left(\begin{pmatrix}a&&\\\
&I_{l-m-n}&\\\
&&b\end{pmatrix}\overline{u}\right)W_{\tau_{1}}(a)W_{\tau_{2}}(b)$

is the characteristic function of a small neighborhood of
$(I_{m},I_{n},I_{l})$. Thus the integral can be made constant.

Now we assume $F$ is archimedean. Similar to [Soudry1993, Lemma 5.2], there is
a positive integer $A_{0}$, such that for any $\xi_{{\bf{s}}}$, there is a
constant $c_{{\bf{s}}}>0$, such that

$|\xi_{{\bf{s}}}({\mathrm{diag}}(a,b)k)|\leq
c_{\bf{s}}|\det(a)|^{{\mathrm{Re}}(s_{1})+\frac{n-1}{2}}|\det(b)|^{-{\mathrm{Re}}(s_{2})-\frac{m-1}{2}}\|{\mathrm{diag}}(a,b)\|^{A_{0}},$

where $a\in T_{m}(F),b\in T_{n}(F)$, and $k$ is in the maximal compact
subgroup $K_{l}$ of ${\mathrm{GL}}_{l}(F)$. We then use the Iwasawa
decomposition, (2.21) and Lemma 2.7 to conclude the absolute convergence of
$\Psi(W,\xi_{{\bf{s}}};j)$.

Now we prove the non-vanishing of the integrals when $F$ is archimedean. Write
$\Psi(W,\xi_{{\bf{s}}};j)$ in the form (3.7). Choose $\xi_{{\bf{s}}}$ to have
support in $P_{m,n}(F)\cdot\overline{N}_{m+n}(F)$, and assume

$\displaystyle\xi_{{\bf{s}}}\left(\begin{pmatrix}b_{1}&\\\
&b_{2}\end{pmatrix}u\overline{v},a_{1},a_{2}\right)=|\det(b_{1})|^{s_{1}+\frac{n-1}{2}}|\det(b_{2})|^{-s_{2}-\frac{m-1}{2}}\varphi_{1}(\overline{v})W_{\tau_{1}}(a_{1}b_{1})W_{\tau_{2}}(a_{2}b_{2}),$

for
$a_{1},b_{1}\in{\mathrm{GL}}_{m}(F),a_{2},b_{2}\in{\mathrm{GL}}_{n}(F),u\in
N_{m,n}(F),\overline{v}\in\overline{N}_{m+n}(F)$,
$W_{\tau_{i}}\in\mathcal{W}(\tau_{i},\psi^{-1})$ for $i=1,2$, and
$\varphi_{1}\in C_{c}^{\infty}(\overline{N}_{m+n}(F))$. With this choice,
$\Psi(W,\xi_{{\bf{s}}};j)$ is equal to an integral of the form

(3.8)
$\begin{split}\int_{T_{m+n}(F)}\int_{\overline{N}_{m+n}(F)}\int_{\overline{U}^{j,m,n}(F)}&W\left(\overline{u}\gamma_{m,n}\begin{pmatrix}t\overline{v}&\\\
&I_{l-m-n}\end{pmatrix}\right)\varphi_{1}(\overline{v})W_{\tau_{1}}(a)W_{\tau_{2}}(b)\\\
&|\det(a)|^{s_{1}+\frac{n-1}{2}}|\det(b)|^{-s_{2}-\frac{m-1}{2}}\delta_{B_{m+n}}(t)^{-1}d\overline{u}d\overline{v}dt.\end{split}$

We consider the $d\overline{v}$ integration first. By the Dixmier-Malliavin
Theorem [DixmierMalliavin1978], a linear combination of the $d\overline{v}$
integrals represents a general element of $\mathcal{W}(\pi,\psi)$. Thus, a
suitable linear combination of integrals of the form (3.8) gives an integral
of the form

$\begin{split}\int_{T_{m+n}(F)}\int_{\overline{U}^{j,m,n}(F)}&W\left(\overline{u}\gamma_{m,n}\begin{pmatrix}t&\\\
&I_{l-m-n}\end{pmatrix}\right)W_{\tau_{1}}(a)W_{\tau_{2}}(b)\\\
&|\det(a)|^{s_{1}+\frac{n-1}{2}}|\det(b)|^{-s_{2}-\frac{m-1}{2}}\delta_{B_{m+n}}(t)^{-1}d\overline{u}dt.\end{split}$

We conjugate ${\mathrm{diag}}(t,I_{l-m-n})$ to the left of $\overline{u}$ to
get

$\begin{split}\int_{T_{m+n}(F)}\int_{\overline{U}^{j,m,n}(F)}&\rho(\gamma_{m,n})W\left(\begin{pmatrix}a&&\\\
&I_{l-m-n}&\\\
&&b\end{pmatrix}\overline{u}\right)W_{\tau_{1}}(a)W_{\tau_{2}}(b)\\\
&|\det(a)|^{s_{1}+\frac{n-1}{2}-j}|\det(b)|^{-s_{2}-\frac{m-1}{2}+k}\delta_{B_{m+n}}(t)^{-1}d\overline{u}dt.\end{split}$

Now we choose $W$ so that
$\rho(\gamma_{m,n})W(t\overline{u})=\rho(\gamma_{m,n})W(t)\varphi_{2}(\overline{u})$
for $t\in B_{l}(F)$, $\overline{u}\in U^{j,m,n}(F)$ and $\varphi_{2}\in
C_{c}^{\infty}(U^{j,m,n}(F))$. Then the above integral becomes

$\begin{split}\int_{\overline{U}^{j,m,n}(F)}\varphi_{2}(\overline{u})d\overline{u}\cdot\int_{T_{m+n}(F)}&\rho(\gamma_{m,n})W\left(\begin{pmatrix}a&&\\\
&I_{l-m-n}&\\\ &&b\end{pmatrix}\right)W_{\tau_{1}}(a)W_{\tau_{2}}(b)\\\
&|\det(a)|^{s_{1}+\frac{n-1}{2}-j}|\det(b)|^{-s_{2}-\frac{m-1}{2}+k}\delta_{B_{m+n}}(t)^{-1}dt.\end{split}$

The $d\overline{u}$ integral is a nonzero constant for appropriate
$\varphi_{2}$. For appropriate $W,W_{\tau_{1}},W_{\tau_{2}}$, the $dt$
integral is holomorphic and nonzero in a neighborhood of any given ${\bf{s}}$.
This proves that there is a linear combination of the local integrals
$\Psi(W,\xi_{{\bf{s}}};j)$ which is holomorphic and nonzero in a neighborhood
of any given ${\bf{s}}$. ∎

###  3.2. Local functional equations

######  Proposition 3.5.

There exists a meromorphic function
$\Gamma({\bf{s}},\pi\times{\boldsymbol{\tau}},\psi)$ such that

$\Psi(W,M_{w_{m,n}}(\xi_{{\bf{s}}});0)=\Gamma({\bf{s}},\pi\times{\boldsymbol{\tau}},\psi)\Psi(W,\xi_{{\bf{s}}};0),$

for any $W\in{\mathcal{W}}(\pi,\psi)$ and
$\xi_{{\bf{s}}}\in{\mathcal{W}}({\bf{s}},{\boldsymbol{\tau}},\psi^{-1})$.

###### Proof.

Recall that

$Y_{0,m,n}=\left\\{{\begin{pmatrix}I_{m+n+1}&v^{\prime}\\\
&v\end{pmatrix}:v^{\prime}\in{\mathrm{Mat}}_{(m+n+1)\times(l-m-n-1)},v\in
N_{l-m-n-1}}\right\\},$

and we have defined a character $\psi_{0}$ on $Y_{0,m,n}(F)$. One can check
that

(3.9)
$\displaystyle\begin{split}\Psi(\rho(y)W,\xi_{{\bf{s}}};0)&=\psi_{0}(y)\Psi(W,\xi_{{\bf{s}}};0),\quad\forall
y\in Y_{0,m,n}(F),\\\ \Psi(\rho\left(\begin{pmatrix}h&\\\
&I_{l-m-n}\end{pmatrix}\right)W,\rho(h)\xi_{{\bf{s}}};0)&=\Psi(W,\xi_{{\bf{s}}};0),\quad\forall
h\in{\mathrm{GL}}_{m+n}(F).\end{split}$

Denote by

$H=\left\\{{\begin{pmatrix}h&&*\\\ &1&*\\\ &&v\end{pmatrix},v\in
N_{l-m-n-1}}\right\\}={\mathrm{GL}}_{m+n}\ltimes Y_{0,m,n}.$

One can define a representation $\nu_{{\bf{s}}}$ of $H(F)$ by
$\nu_{{\bf{s}}}|_{{\mathrm{GL}}_{m+n}(F)}={\mathrm{I}}({\bf{s}},{\boldsymbol{\tau}})$
and $\nu_{{\bf{s}}}|_{Y_{0,m,n}(F)}=\psi_{0}$. Then (3.9) implies that the
bilinear form $(W,\xi_{{\bf{s}}})\mapsto\Psi(W,\xi_{{\bf{s}}})$ defines an
element in

${\mathrm{Hom}}_{H(F)}(\pi\otimes\nu_{{\bf{s}}},1).$

Similarly, $\Psi(W,M_{w_{m,n}}(\xi_{{\bf{s}}}))$ satisfies the same quasi-
invariance property (3.9) and thus

$(W,\xi_{{\bf{s}}})\mapsto\Psi(W,M_{w_{m,n}}(\xi_{{\bf{s}}}))$

also defines an element in
${\mathrm{Hom}}_{H(F)}(\pi\otimes\nu_{{\bf{s}}},1)$. By the main result of
[Chen-Sun] (or by [JPSS]*Proposition 2.11 if $F$ is non-archimedean), we have
$\dim_{{\mathbb{C}}}{\mathrm{Hom}}_{H(F)}(\pi\otimes\nu_{{\bf{s}}},1)\leq 1$
excluding a discrete set of ${\bf{s}}$. This shows the existence of the gamma
factor. ∎

######  Remark 3.6.

If $m>0$ and $n=0$, by Remark 3.2, we can see that

$\Gamma({\bf{s}},\pi\times{\boldsymbol{\tau}},\psi)=\omega_{\tau_{1}}(-1)^{l-1}\gamma^{\mathrm{JPSS}}(s+\frac{l-m-1}{2},\pi\times\tau_{1},\psi),$

where ${\boldsymbol{\tau}}=\tau_{1}$ and $\gamma^{\mathrm{JPSS}}$ is the
Jacquet–Piatetski-Shapiro–Shalika local gamma factor as defined in
[JPSS]*Theorem 2.7.

######  Remark 3.7.

To get a local functional equation of $\Psi(W,\xi_{{\bf{s}}};j)$ for general
$j$ with $0\leq j\leq l-m-n-1$, one should be able to prove the following
analogue of the main result of [Chen-Sun]. Recall that we have defined

$Y_{j,m,n}=\left\\{{\begin{pmatrix}u&*&*\\\ &I_{m+n+1}&*\\\
&&v\end{pmatrix}:u\in N_{j}(F),v\in N_{k}(F)}\right\\}$

and a character $\psi_{j}$ on $Y_{j,m,n}$. Consider

$H_{j}=\left\\{{\begin{pmatrix}u&*&*&*&*\\\ &a&&b&*\\\ &&1&&*\\\ &c&&d&*\\\
&&&&v\end{pmatrix},u\in N_{j},v\in N_{k},\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in{\mathrm{GL}}_{m+n}(F)}\right\\}={\mathrm{GL}}_{m+n}\ltimes
Y_{j,m,n}.$

For a representation $\sigma$ of ${\mathrm{GL}}_{m+n}(F)$, we can define a
representation $\nu_{\psi}$ of $H_{j}(F)$ such that
$\nu|_{{\mathrm{GL}}_{m+n}}=\sigma$ and
$\nu_{\psi}|_{Y_{j,m,n}}=\psi_{j,m,n}$. Then for an irreducible smooth
representation $\pi$ of ${\mathrm{GL}}_{l}(F)$, one should have

$\dim{\mathrm{Hom}}_{H_{j}}(\pi,\nu_{\psi})\leq 1.$

If $n=0$ or $m=0$, this was the main result of [Chen-Sun]. If $m=n$, this is
the uniqueness of Bessel model of ${\mathrm{GL}}_{l}(F)$, which was proved in
[GGP] using the multiplicity one result of [AGRS]. On the other hand, when
$n=0$, Jacquet–Piatetski-Shapiro–Shalika (see [JPSS-GL3I]*Theorem 4.5 and
[JPSS]*Theorem 2.7) directly showed that

(3.10)
$\Psi(W,M_{w_{m,n}}(\xi_{{\bf{s}}});j)=\Gamma((s_{1}-j,s_{2}+j),\pi\times(\tau_{1},\tau_{2}),\psi)\Psi(W,\xi_{{\bf{s}}};j),$

for any $W\in{\mathcal{W}}(\pi,\psi)$ and
$\xi_{{\bf{s}}}\in{\mathcal{W}}({\bf{s}},(\tau_{1},\tau_{2}),\psi^{-1})$ and
for any $j$ with $0\leq j\leq l-m-n-1$. We expect this is true in general even
we have trouble to extend the proof of [JPSS] to our case.

###  3.3. Unramified calculation

In this subsection, let $F$ be a non-archimedean local field with ring of
integers ${\mathcal{O}}$. Let $\varpi\in{\mathcal{O}}$ be a fixed uniformizer
and $q=|{\mathcal{O}}/(\varpi)|$. Our goal in this subsection is to compute
the local zeta integral (3.1) when everything is unramified. In particular, we
assume that $\pi$ is unramified with Satake parameters
$\mathbf{\alpha}={\mathrm{diag}}(\alpha_{1},\dots,\alpha_{l})\in{\mathrm{GL}}_{l}({\mathbb{C}})$
and $\tau_{1}$ (resp. $\widetilde{\tau}_{2}$) is unramified with Satake
parameters
$\mathbf{\beta}^{1}={\mathrm{diag}}(\beta_{1}^{1},\dots,\beta_{m}^{1})\in{\mathrm{GL}}_{m}({\mathbb{C}})$
(resp.
$\mathbf{\beta}^{2}={\mathrm{diag}}(\beta_{1}^{2},\dots,\beta_{n}^{2})\in{\mathrm{GL}}_{n}({\mathbb{C}})$).
Moreover, we assume that $W\in{\mathcal{W}}(\pi,\psi)$ is the Whittaker
function normalized by $W(I_{l})=1$, $\xi_{\bf{s}}$ is the Whittker function
associated with the normalized spherical section
$f_{\bf{s}}\in{\mathrm{I}}({\bf{s}},{\boldsymbol{\tau}})$. By Iwasawa
decomposition ${\mathrm{GL}}_{m+n}(F)=N_{m+n}(F)T_{m+n}(F)K_{m+n}$, where
$K_{m+n}={\mathrm{GL}}_{m+n}({\mathcal{O}})$, we have

$\displaystyle\Psi(W,\xi_{\bf{s}};j)$
$\displaystyle=\int_{T_{m+n}(F)}\int_{\overline{U}^{j,m,n}(F)}W(\overline{u}\gamma_{m,n}{\mathrm{diag}}(t,I_{l-m-n}))\xi_{\bf{s}}(t)\delta_{B_{m+n}}(t)^{-1}d\overline{u}dt$
$\displaystyle=\int_{T_{m+n}(F)}\int_{\overline{U}^{m,n}(F)}W({\bf{t}}_{m,n}(a,b)\overline{u})\xi_{\bf{s}}(t)|\det(a)|^{-j}|\det(b)|^{k}\delta_{B_{m+n}}(t)^{-1}d\overline{u}dt$
$\displaystyle=\int_{T_{m+n}(F)}\int_{\overline{U}^{m,n}(F)}W({\bf{t}}_{m,n}(a,b)\overline{u})W_{\tau_{1}}(a)W_{\tau_{2}}(b)$
$\displaystyle\quad\cdot|\det(a)|^{s_{1}+\frac{n-1}{2}-j}|\det(b)|^{-s_{2}-\frac{m-1}{2}+k}\delta_{B_{m+n}}(t)^{-1}d\overline{u}dt$

where $t={\mathrm{diag}}(a,b)$ with $a\in T_{m}(F),b\in T_{n}(F)$ and
${\bf{t}}_{m,n}(a,b)={\mathrm{diag}}(a,I_{l-m-n},b)$. Here the term
$|\det(a)|^{-j}|\det(b)|^{k}$ comes from a modulus character when we change
variables on $\overline{u}$ and the term $\delta_{B_{m+n}}(t)^{-1}$ comes from
the corresponding Haar measure when we use the Iwasawa decomposition. By Lemma
2.6, we have

(3.11) $\displaystyle\Psi(W,\xi_{\bf{s}};j)=$
$\displaystyle\int_{T_{m+n}(F)}W({\bf{t}}_{m,n}(a,b))W_{\tau_{1}}(a)W_{\tau_{2}}(b)$
$\displaystyle\quad\cdot|\det(a)|^{s_{1}+\frac{n-1}{2}-j}|\det(b)|^{-s_{2}-\frac{m-1}{2}+k}\delta_{B_{m+n}}^{-1}\left(\begin{pmatrix}a&\\\
&b\end{pmatrix}\right)dadb$ $\displaystyle=$
$\displaystyle\int_{T_{m+n}(F)}W({\bf{t}}_{m,n}(a,b^{*}))W_{\tau_{1}}(a)W_{\tau_{2}}(b^{*})$
$\displaystyle\quad\cdot|\det(a)|^{s_{1}+\frac{n-1}{2}-j}|\det(b)|^{s_{2}+\frac{m-1}{2}-k}\delta_{B_{m+n}}^{-1}\left(\begin{pmatrix}a&\\\
&b^{*}\end{pmatrix}\right)dadb$

where $b^{*}=J_{n}^{t}\\!b^{-1}J_{n}^{-1}$, with $J_{n}=\begin{pmatrix}&&1\\\
&\iddots&\\\ 1&&\end{pmatrix}$. Note that the function $b\mapsto
W_{\tau_{2}}(b^{*})$ is just the normalized Whittaker function of
$\widetilde{\tau}_{2}$, namely,
$W_{\tau_{2}}(b^{*})=W_{\widetilde{\tau}_{2}}(b).$ We use the following
notations following [Jacquet-Shalika-EulerI]. For $m$-tuple
$\mathbf{x}=(x_{1},\dots,x_{m})$, we write
$\varpi^{\bf{x}}=(\varpi^{x_{1}},\dots,\varpi^{x_{m}})$. Then (3.11) can be
written as

(3.12) $\displaystyle\Psi(W,\xi_{{\bf{s}}};j)=$
$\displaystyle\sum_{{\bf{x}},{\bf{y}}}W(\varpi^{({\bf{x}},0,{\bf{y}}^{*})})W_{\tau_{1}}(\varpi^{{\bf{x}}})W_{\widetilde{\tau}_{2}}(\varpi^{\bf{y}})\delta_{B_{m+n}}^{-1}(\varpi^{({\bf{x}},{\bf{y}}^{*})})$
$\displaystyle\cdot|\det(\varpi^{\bf{x}})|^{s_{1}+\frac{n-1}{2}-j}|\det(\varpi^{{\bf{y}}})|^{s_{1}+\frac{m-1}{2}-k}.$

Here ${\bf{x}}$ (resp. ${\bf{y}}$) runs over all $m$-tuples (resp. $n$-tuples)
of all integers, ${\bf{y}}^{*}=(-y_{n},\dots,-y_{1})$ for
${\bf{y}}=(y_{1},\dots,y_{n})$ and $({\bf{x}},0,{\bf{y}}^{*})$ denotes the
$l$-tuple $(x_{1},\dots,x_{m},0,\dots,0,-y_{n},\dots,-y_{1})$ with $l-m-n$
zeros in the middle. Denote $T^{+}(m)$ the $m$-tuples of integers
${\bf{x}}=(x_{1},\dots,x_{m})$ such that $x_{1}\geq x_{2}\geq\dots\geq
x_{m}\geq 0$. Similarly, we define $T^{+}(n)$. By the Shintani-Casselman-
Shalika formula [Shintani, CasselmanShalika], we have
$W(\varpi^{({\bf{x}},0,{\bf{y}}^{*})})=0$ unless ${\bf{x}}\in T^{+}(m)$ and
${\bf{y}}\in T^{+}(n)$. If ${\bf{x}}\in T^{+}(m)$ and ${\bf{y}}\in T^{+}(n)$,
we have
$W(\varpi^{({\bf{x}},0,{\bf{y}}^{*})})=\delta_{B_{l}}^{1/2}(\varpi^{({\bf{x}},0,{\bf{y}}^{*})})S_{({\bf{x}},0,{\bf{y}}^{*})}(\alpha)$.
Here $S_{({\bf{x}},0,{\bf{y}}^{*})}$ denotes the Schur polynomial associated
with $({\bf{x}},0,{\bf{y}}^{*})$ (see [Fulton-Harris]*Appendix A), or more
explicitly,

$S_{({\bf{x}},0,{\bf{y}}^{*})}(\alpha)=\prod_{1\leq i<j\leq
l}(\alpha_{i}-\alpha_{j})^{-1}\cdot\det\begin{pmatrix}\alpha_{1}^{x_{1}+l-1}&\dots&\alpha_{l}^{x_{1}+l-1}\\\
\vdots&&\vdots\\\ \alpha_{1}^{-y_{1}}&\dots&\alpha_{l}^{-y_{1}}\end{pmatrix}.$

Moreover, for ${\bf{x}}\in T^{+}(m),{\bf{y}}\in T^{+}(n)$, we have

$W_{\tau_{1}}(\varpi^{{\bf{x}}})=\delta_{B_{m}}^{1/2}(\varpi^{{\bf{x}}})S_{{\bf{x}}}(\beta^{1}),\quad
W_{\widetilde{\tau}_{2}}(\varpi^{{\bf{y}}})=\delta_{B_{n}}^{1/2}(\varpi^{{\bf{y}}})S_{{\bf{y}}}(\beta^{2}).$

We can check that

$\displaystyle\delta_{B_{m}}(\varpi^{{\bf{x}}})$
$\displaystyle=|\varpi^{x_{1}}|^{m-1}|\varpi^{x_{2}}|^{m-3}\dots|\varpi^{x_{m}}|^{-m+1},$
$\displaystyle\delta_{B_{n}}(\varpi^{{\bf{y}}})$
$\displaystyle=|\varpi^{y_{1}}|^{n-1}|\varpi^{y_{2}}|^{n-3}\dots|\varpi^{y_{n}}|^{-n+1},$
$\displaystyle\delta_{B_{m+n}}(\varpi^{({\bf{x}},{\bf{y}}^{*})})$
$\displaystyle=|\varpi^{x_{1}}|^{m-1}|\varpi^{x_{2}}|^{m-3}\dots|\varpi^{x_{m}}|^{-m+1}|\varpi^{y_{n}}|^{-n+1}\dots|\varpi^{y_{1}}|^{n-1}|\det(\varpi^{{\bf{x}}})|^{n}|\det(\varpi^{{\bf{y}}})|^{m},$
$\displaystyle\delta_{B_{l}}(\varpi^{({\bf{x}},0,{\bf{y}}^{*})})$
$\displaystyle=|\varpi^{x_{1}}|^{m-1}\dots|\varpi^{x_{m}}|^{1-m}|\varpi^{y_{n}}|^{1-n}\dots|\varpi^{y_{1}}|^{n-1}|\det(\varpi^{{\bf{x}}})|^{l-m}|\det(\varpi^{{\bf{y}}})|^{l-n}.$

Combining the above formulas, (3.12) becomes

(3.13) $\displaystyle\Psi(W,\xi_{{\bf{s}}};j)$
$\displaystyle=\sum_{\begin{subarray}{c}{\bf{x}}\in T^{+}(m)\\\ {\bf{y}}\in
T^{+}(n)\end{subarray}}S_{({\bf{x}},0,{\bf{y}}^{*})}(\alpha)S_{{\bf{x}}}(\beta^{1})S_{{\bf{y}}}(\beta^{2})|\det(\varpi^{{\bf{x}}})|^{s_{1}+\frac{k-j}{2}}|\det(\varpi^{{\bf{y}}})|^{s_{2}+\frac{j-k}{2}}.$

######  Proposition 3.8.

We keep the notations as above. Then

$\Psi(W,\xi_{\bf{s}};j)=\frac{L(s_{1}+\frac{k-j}{2},\pi\times\tau_{1})L(s_{2}-\frac{k-j}{2},\widetilde{\pi}\times\widetilde{\tau}_{2})}{L(s_{1}+s_{2},\tau_{1}\times\widetilde{\tau}_{2})}.$

Recall that $k=l-m-n-1-j$ and thus $k-j=l-m-n-1-2j.$

If $n=0$, the above formula is the unramified calculation of the
Jacquet–Piatetski-Shapiro–Shalika integral, see [Jacquet-Shalika-
EulerI]*Proposition 2.4 and also [Cogdell:Fields, Cogdell:IAS]. If
$l=2r+1,m=n$ and $j=r-m=k$, the above unramified calculation is done in [GPS]
(when $r=1$), in [Tamir] (for general $r$ when $m=n=r$) with slightly
different normalization, and in [BAS:Uodd] (when $m=n<r$), where this was the
unramified calculation of $L$-functions for
${\mathrm{U}}_{2r+1,E/F}\times{\mathrm{Res}}_{E/F}({\mathrm{GL}}_{r})$ at
split places for a quadratic extension $E/F$.

###### Proof.

Without loss of generality, we assume that $m\geq n$. Write
$T_{1}=q^{-(s_{1}+\frac{k-j}{2})},T_{2}=q^{-(s_{2}+\frac{j-k}{2})}$. For an
$m$-tuple ${\bf{x}}=(x_{1},\dots,x_{m})$, denote
$|{\bf{x}}|=\sum_{i=1}^{m}x_{i}$. An $m$-tuple ${\bf{x}}\in T^{+}(m)$ can be
identified with a partition of $|{\bf{x}}|$ and can be represented by an Young
diagram, see [Fulton-Harris]*§4 for example. We can then write (3.13) as

(3.14) $\Psi(W,\xi_{{\bf{s}}};j)=\sum_{\begin{subarray}{c}{\bf{x}}\in
T^{+}(m)\\\ {\bf{y}}\in
T^{+}(n)\end{subarray}}S_{({\bf{x}},0,{\bf{y}}^{*})}(\alpha)S_{{\bf{x}}}(\beta^{1})S_{{\bf{y}}}(\beta^{2})T_{1}^{|{\bf{x}}|}T_{2}^{|{\bf{y}}|}.$

On the other hand, we have

$L(s_{1}+s_{2},\tau_{1}\times\widetilde{\tau}_{2})=\det(I-\beta^{1}\otimes\beta^{2}T_{1}T_{2})^{-1}=\sum_{e\geq
0}{\mathrm{Tr}}({\mathrm{Sym}}^{e}(\beta^{1}\otimes\beta^{2}))(T_{1}T_{2})^{e}.$

Thus we get that

(3.15)
$\displaystyle\begin{split}L(s_{1}+s_{2},\tau_{1}\times\widetilde{\tau}_{2})\Psi(W,\xi_{{\bf{s}}};j)=\sum_{\begin{subarray}{c}{\bf{x}}\in
T^{+}(m),{\bf{y}}\in T^{+}(n),e\geq
0\end{subarray}}&S_{({\bf{x}},0,{\bf{y}}^{*})}(\alpha)S_{{\bf{x}}}(\beta^{1})S_{{\bf{y}}}(\beta^{2})\\\
&\cdot{\mathrm{Tr}}({\mathrm{Sym}}^{e}(\beta^{1}\otimes\beta^{2}))T_{1}^{|{\bf{x}}|+e}T_{2}^{|{\bf{y}}|+e}.\end{split}$

Since

$L(s_{1}+\frac{k-j}{2},\pi\times\tau_{1})=\sum_{c\geq
0}{\mathrm{Tr}}({\mathrm{Sym}}^{c}(\alpha\otimes\beta^{1}))T_{1}^{c},$

and

$L(s_{2}+\frac{j-k}{2},\widetilde{\pi}\times\widetilde{\tau}_{2})=\sum_{d\geq
0}{\mathrm{Tr}}({\mathrm{Sym}}^{d}(\widetilde{\alpha}\otimes\beta^{2}))T_{2}^{d},$

where $\widetilde{\alpha}={\mathrm{diag}}(a_{1}^{-1},\dots,a_{l}^{-1})$ is the
Satake parameter for $\widetilde{\pi},$ we get that

(3.16) $\displaystyle
L(s_{1}+\frac{k-j}{2},\pi\times\tau_{1})L(s_{2}+\frac{j-k}{2},\widetilde{\pi}\times\widetilde{\tau}_{2})=\sum_{c\geq
0,d\geq
0}{\mathrm{Tr}}({\mathrm{Sym}}^{c}(\alpha\otimes\beta^{1})){\mathrm{Tr}}({\mathrm{Sym}}^{d}(\widetilde{\alpha}\otimes\beta^{2}))T_{1}^{c}T_{2}^{d}.$

Comparing (3.15) and (3.16), in order to prove Proposition 3.8, it suffices to
show

(3.17)
$\displaystyle\begin{split}{\mathrm{Tr}}({\mathrm{Sym}}^{c}(\alpha\otimes\beta^{1})){\mathrm{Tr}}({\mathrm{Sym}}^{d}(\widetilde{\alpha}\otimes\beta^{2}))=\sum_{e\geq
0}\sum_{\begin{subarray}{c}{\bf{x}}\in T^{+}(m),{\bf{y}}\in T^{+}(n),e\geq
0\\\
|{\bf{x}}|=c-e,|{\bf{y}}|=d-e\end{subarray}}&S_{({\bf{x}},0,{\bf{y}}^{*})}(\alpha)S_{{\bf{x}}}(\beta^{1})S_{{\bf{y}}}(\beta^{2})\\\
&\cdot{\mathrm{Tr}}({\mathrm{Sym}}^{e}(\beta^{1}\otimes\beta^{2})).\end{split}$

By [Jacquet-Shalika-EulerI]*Proposition 2.4, we have

${\mathrm{Tr}}({\mathrm{Sym}}^{e}(\beta^{1}\otimes\beta^{2}))=\sum_{{\bf{z}}\in
T^{+}(n),|{\bf{z}}|=e}S_{({\bf{z}},0_{m-n})}(\beta^{1})S_{{\bf{z}}}(\beta^{2}).$

Here ${\bf{z}}=(z_{1},\dots,z_{n})$ can be identified with a partition of
$e=|{\bf{z}}|$ with at most $n$-parts (since $m\geq n$ by our assumption) and
$S_{{\bf{z}}}$ (resp. $S_{({\bf{z}},0_{m-n})}$) is the Schur polynomial
defined by ${\bf{z}}$ with $n$ (resp. $m$) variables. Similarly,

$\displaystyle{\mathrm{Tr}}({\mathrm{Sym}}^{c}(\alpha\otimes\beta^{1}))$
$\displaystyle=\sum_{\mathbf{u}\in
T^{+}(m),|{\bf{u}}|=c}S_{({\bf{u}},0_{l-m})}(\alpha)S_{{\bf{u}}}(\beta^{1}),$
$\displaystyle{\mathrm{Tr}}({\mathrm{Sym}}^{d}(\widetilde{\alpha}\otimes\beta^{2}))$
$\displaystyle=\sum_{\mathbf{v}\in
T^{+}(n),|{\bf{v}}|=d}S_{({\bf{v}},0_{l-n})}(\widetilde{\alpha})S_{{\bf{u}}}(\beta^{2}).$

A simple matrix calculation shows that

$S_{({\bf{v}},0_{l-n})}(\widetilde{\alpha})=S_{(0_{l-n},{\bf{v}}^{*})}(\alpha).$

See also [Fulton-Harris]*Exercise 15.50 for a representation theoretic
explanation of this formula. Thus the left hand side of (3.17) becomes

$\displaystyle LHS=\sum_{{\bf{u}}\in T^{+}(m),|{\bf{u}}|=c}\sum_{{\bf{v}}\in
T^{+}(n),|{\bf{v}}|=d}S_{({\bf{u}},0_{l-m})}(\alpha)S_{(0_{l-n},{\bf{v}}^{*})}(\alpha)S_{{\bf{u}}}(\beta^{1})S_{{\bf{v}}}(\beta^{2}),$

while the right side of (3.17) becomes

$\displaystyle RHS=\sum_{{\bf{z}}\in
T^{+}(n)}\sum_{\begin{subarray}{c}{\bf{x}}\in T^{+}(m),{\bf{y}}\in
T^{+}(n),e\geq 0\\\
|{\bf{x}}|=c-|{\bf{z}}|,|{\bf{y}}|=d-|{\bf{z}}|\end{subarray}}$ $\displaystyle
S_{({\bf{x}},0,{\bf{y}}^{*})}(\alpha)S_{{\bf{x}}}(\beta^{1})S_{({\bf{z}},0)}(\beta^{1})S_{{\bf{y}}}(\beta^{2})S_{{\bf{z}}}(\beta^{2})$

By Littlewood-Richardson rule, see [Fulton-Harris]*(A.8) or [Mac]*§I.9, we
have

$\displaystyle S_{{\bf{x}}}(\beta^{1})S_{{\bf{z}}}(\beta^{1})=$
$\displaystyle\sum_{{\bf{u}}\in
T^{+}(m),|{\bf{u}}|=c}c_{{\bf{x}},{\bf{z}}}^{\bf{u}}S_{{\bf{u}}}(\beta^{1}),$
$\displaystyle S_{{\bf{y}}}(\beta^{2})S_{{\bf{z}}}(\beta^{2})=$
$\displaystyle\sum_{{\bf{v}}\in
T^{+}(n),|{\bf{v}}|=d}c_{{\bf{y}},{\bf{z}}}^{\bf{v}}S_{{\bf{v}}}(\beta^{2}),$

where in the first equation, $({\bf{z}},0_{m-n})$ is identified with
$|{\bf{z}}|$ as a partition of $e=|{\bf{z}}|$ with at most $n$ parts, and
$c_{{\bf{x}},{\bf{z}}}^{\bf{u}},c_{{\bf{y}},{\bf{z}}}^{\bf{u}}$ are the
Littlewood-Richardson coefficients as defined in [Fulton-Harris]*page 454 or
[Mac]*§I.9.2. Thus

$\displaystyle RHS=\sum_{{\bf{u}}\in T^{+}(m),|{\bf{u}}|=c}\sum_{{\bf{v}}\in
T^{+}(n),|{\bf{v}}|=d}\sum_{\begin{subarray}{c}{\bf{x}}\in
T^{+}(m),{\bf{y}},{\bf{z}}\in T^{+}(n)\\\
|{\bf{x}}|+|{\bf{z}}|=c,|{\bf{y}}|+|{\bf{z}}|=d\end{subarray}}c_{{\bf{x}},{\bf{z}}}^{\bf{u}}c_{{\bf{y}},{\bf{z}}}^{\bf{v}}S_{({\bf{x}},0,{\bf{y}}^{*})}(\alpha)S_{{\bf{u}}}(\beta^{1})S_{{\bf{v}}}(\beta^{2}).$

Thus in order to prove (3.17) and hence Proposition 3.8, it suffices to prove
that for any ${\bf{u}}\in T^{+}(m),{\bf{v}}\in T^{+}(n)$, one has

(3.18) $\displaystyle
S_{({\bf{u}},0_{l-m})}(\alpha)S_{(0_{l-n},{\bf{v}}^{*})}(\alpha)=\sum_{\begin{subarray}{c}{\bf{x}}\in
T^{+}(m),{\bf{y}},{\bf{z}}\in T^{+}(n)\\\
|{\bf{x}}|+|{\bf{z}}|=c,|{\bf{y}}|+|{\bf{z}}|=d\end{subarray}}c_{{\bf{x}},{\bf{z}}}^{\bf{u}}c_{{\bf{y}},{\bf{z}}}^{\bf{v}}S_{({\bf{x}},0,{\bf{y}}^{*})}(\alpha).$

For ${\bf{v}}=(v_{1},\dots,v_{n})\in T^{+}(n)$, we write
$\widetilde{\bf{v}}=(v_{1},\dots,v_{1},v_{1}-v_{n},\dots,v_{1}-v_{2},0)\in
T^{+}(l)$. Then
$S_{(0_{l-n},{\bf{v}}^{*})}(\alpha)=S_{{\bf{v}}}(\alpha)D_{-v_{1}}(\alpha)$,
where $D_{-v_{1}}(\alpha)=\det^{-v_{1}}(\alpha)$ following the notation of
[Fulton-Harris]*§15.5. Thus using Littlewood-Richardson rule again, we have

$\displaystyle
S_{({\bf{u}},0_{l-m})}(\alpha)S_{(0_{l-n},{\bf{v}}^{*})}(\alpha)$
$\displaystyle=D_{-v_{1}}(\alpha)\sum_{\lambda\in
T^{+}(l),|\lambda|=|\widetilde{v}|+|{\bf{u}}|}c_{\widetilde{\bf{v}},{\bf{u}}}^{\lambda}S_{\lambda}(\alpha).$

Write $\lambda=(\lambda_{1},\dots,\lambda_{l})$. By the definition of
Littlewood-Richardson coefficients, if
$c_{\widetilde{\bf{v}},{\bf{u}}}^{\lambda}\neq 0$, we must have
$\lambda_{m+1}=\dots=\lambda_{l-n-1}=v_{1}$, which means that
$S_{\lambda}\cdot D_{-v_{1}}=S_{(\lambda_{1}-v_{1},\dots,\lambda_{l}-v_{1})}$
must be of the form $S_{({\bf{x}},0_{l-m-n},{\bf{y}}^{*})}$ for ${\bf{x}}\in
T^{+}(m)$ and ${\bf{y}}\in T^{+}(n)$. Thus we get

$\displaystyle
S_{({\bf{u}},0_{l-m})}(\alpha)S_{(0_{l-n},{\bf{v}}^{*})}(\alpha)=\sum_{{\bf{x}}\in
T^{+}(m),{\bf{y}}\in
T^{+}(n)}c_{\widetilde{\bf{v}},{\bf{u}}}^{\lambda}S_{({\bf{x}},0,{\bf{y}}^{*})},$

where
$\lambda=(\lambda_{1},\dots,\lambda_{l})=({\bf{x}},0,{\bf{y}}^{*})+(v_{1},\dots,v_{1}).$
Note that $|{\bf{u}}|-|{\bf{v}}|=|{\bf{x}}|-|{\bf{y}}|$. Thus in order to
prove (3.18), it suffices to show that for any fixed ${\bf{u}},{\bf{x}}\in
T^{+}(m)$ and ${\bf{v}},{\bf{y}}\in T^{+}(n)$ with
$|{\bf{u}}|-|{\bf{x}}|=|{\bf{v}}|-|{\bf{y}}|$,

(3.19) $\displaystyle
c_{\widetilde{\bf{v}},{\bf{u}}}^{\lambda}=\sum_{{\bf{z}}\in
T^{+}(n)}c_{{\bf{x}},{\bf{z}}}^{{\bf{u}}}c_{{\bf{y}},{\bf{z}}}^{{\bf{v}}},$

where
$\lambda=(\lambda_{1},\dots,\lambda_{l})=({\bf{x}},0,{\bf{y}}^{*})+(v_{1},\dots,v_{1}).$
The formula (3.19) was proved by Professor T. Tao in a MathOverflow answer
[Tao] using the hive model for Littlewood-Richardson coefficients introduced
in [Knutson-Tao]. A proof of (3.19) based on Tao’s MathOverflow answer [Tao]
will be reproduced in §3.4 after we introduce some necessary notations and
tools. ∎

######  Remark 3.9.

Here we give an example of (3.18). We take $l=4,m=2,n=1$ and
${\bf{u}}=(2,1),{\bf{v}}=(2)$. One can check that there are 3 choices of
${\bf{z}}$, which are ${\bf{z}}=(0),{\bf{z}}=(1),{\bf{z}}=(2)$, and
correspondingly, there are 3 choices of ${\bf{y}}$ given by
${\bf{y}}=(2),{\bf{y}}=(1),{\bf{y}}=(0)$. When ${\bf{z}}=(0)$, we must have
${\bf{z}}=(2,1)$ and when ${\bf{z}}=(2)$, we must have ${\bf{x}}=(1)=(1,0)$.
But when ${\bf{z}}=(1)$, there are two choices of ${\bf{x}}$, which are
${\bf{x}}=(1,1)$ or ${\bf{x}}=(2)=(2,0)$. One can check that in each case,
$c_{{\bf{x}},{\bf{z}}}^{{\bf{u}}}c_{{\bf{y}},{\bf{z}}}^{{\bf{v}}}=1$. Thus
(3.18) becomes

$S_{(2,1,0,0)}\cdot
S_{(0,0,0,-2)}=S_{(2,1,0,-2)}+S_{(1,0,0,0)}+S_{(1,1,0,-1)}+S_{(2,0,0,-1)},$

which could be checked directly using Littlewood-Richardson rule by noting
that $S_{(0,0,0,-2)}=S_{(2,2,2,0)}\cdot D_{-2}$, where $D_{-2}=\det^{-2}$.

###  3.4. Proof of Tao’s formula (3.19)

An integral n-hive is an array of integers $a_{ij}$ for $0\leq i,j,i+j\leq n$
placed in the vertices of triangles of the following shape

Figure 1. hive

which satisfies all of the following rhombus inequalities: for each rhombus of
the following types

the sum of the two integers at the obtuse vertices must be greater than or
equal to the sum of the two integers at the acute vertices.

######  Theorem 3.10 (Knutson-Tao, [Knutson-Tao]).

Let
${\bf{x}}=(x_{1},\dots,x_{n}),{\bf{y}}=(y_{1},\dots,y_{n}),{\bf{z}}=(z_{1},\dots,z_{n})$
be partitions with $|{\bf{z}}|=|{\bf{x}}|+|{\bf{y}}|$, then
$c_{{\bf{x}},{\bf{y}}}^{{\bf{z}}}$ is the number of $n$-hives with boundary
labels

Here the arrow and the number $x_{i}$ (resp. $y_{j},z_{k}$) on the arrow
indicates that the numbers increase by $x_{i}$ (resp. $y_{j},z_{k}$) along the
direction indicated by the arrow. One can normalize the above $n$-hive by
assign any integer to any fixed vertex.

We note that different normalization will give the same number of hives. The
above theorem is proved in [Knutson-Tao]. See also the appendix of [Buch] for
a different proof given by W. Fulton.

######  Remark 3.11.

We give a simple example which also appeared in [Buch]. We have
$c_{(2,1),(2,1)}^{(3,2,1)}=2$, which can be computed in the following way.
There are exactly two $3$-hives with boundary conditions given below,

$\begin{matrix}&&&3&&&\\\ &&&&&\\\ &&3&&5&&\\\ &&&&&\\\ &2&&x&&6&\\\ &&&&&&\\\
0&&3&&5&&6\end{matrix},$

which are given by $x=4,5$.

We temporarily call the following object an anti-n-hive: an array of integers
placed in the vertices of triangles of the shape as Figure 1 which satisfies
the “reverse” rhombus inequalities: for each rhombus below

the sum of the two integers at the obtuse vertices must be less than or equal
to the sum of the two integers at the acute vertices.

For any $n$-hive, if we switch the sign of the number at each vertices, we
will get an anti-$n$-hive. Note that, this process will change the boundary
conditions, which gives us the following direct corollary.

######  Corollary 3.12.

Let
${\bf{x}}=(x_{1},\dots,x_{n}),{\bf{y}}=(y_{1},\dots,y_{n}),{\bf{z}}=(z_{1},\dots,z_{n})$
be partitions with $|{\bf{z}}|=|{\bf{x}}|+|{\bf{y}}|$, then
$c_{{\bf{x}},{\bf{y}}}^{{\bf{z}}}$ is the number of anti-$n$-hives with
boundary labels

Here the arrow and the number $x_{i}$ (resp. $y_{j},z_{k}$) on the arrow
indicates that the numbers increase by $x_{i}$ (resp. $y_{j},z_{k}$) along the
direction indicated by the arrow. One can normalize the above $n$-hive by
assign any integer to any fixed vertex.

Now we can prove Tao’s formula (3.19), which we restate it below.

######  Proposition 3.13.

Let $l,m,n$ be non-negative integers with $l\leq m+n+1$ and $m\geq n$. Given
${\bf{x}},{\bf{u}}\in T^{+}(m),{\bf{y}},{\bf{v}}\in T^{+}(n)$ with
$|{\bf{u}}|-|{\bf{x}}|=|{\bf{v}}|-|{\bf{y}}|\geq 0$, then

$c_{\widetilde{\bf{v}},{\bf{u}}}^{\lambda}=\sum_{{\bf{z}}\in
T^{+}(n)}c_{{\bf{x}},{\bf{z}}}^{{\bf{u}}}c_{{\bf{y}},{\bf{z}}}^{{\bf{v}}}.$

Here
${\bf{u}}=(u_{1},\dots,u_{m}),{\bf{x}}=(x_{1},\dots,x_{m}),{\bf{y}}=(y_{1},\dots,y_{n}),{\bf{v}}=(v_{1},\dots,v_{n})$,
${\bf{y}}^{*}=(-y_{n},\dots,-y_{2},-y_{1})$,
$\widetilde{\bf{v}}=(0_{l-n},{\bf{v}}^{*})+(v_{1},\dots,v_{1})=(v_{1},\dots,v_{1},v_{1}-v_{n},\dots,v_{1}-v_{2},0)\in
T^{+}(l)$, and
$\lambda=({\bf{x}},0_{l-m-n},{\bf{y}}^{*})+(v_{1},\dots,v_{1})\in T^{+}(l)$.
Moreover, ${\bf{u}}$ in $c_{\widetilde{\bf{v}},{\bf{u}}}^{\lambda}$ is viewed
as an element in $T^{+}(l)$ in the obvious way, namely,
${\bf{u}}=({\bf{u}},0_{l-n})$.

###### Proof.

By Theorem 3.10 and Corollary 3.12, one can see that
$c_{\widetilde{\bf{v}},{\bf{u}}}^{\lambda}$ is the number of anti-$l$-hives
with boundary conditions indicated below,

where $v_{1}$ in the left side boundary and bottom boundary means
$(v_{1},\dots,v_{1})\in T^{+}(l)$. Here the two interior line segments are not
important here. For each hive above, we assume that its vertex integers are
given by $(a_{ij})_{0\leq i,j,i+j\leq l}$ placed as in Figure 1. Then
$(a_{ij}-(i+j)v_{1})_{0\leq i,j,i+j\leq l}$ is also an anti-$l$-hive which has
the boundary conditions as indicated in the following Figure 2. We also
normalized the anti-$l$-hive so that the top vertex has value $0$.

Figure 2. boundary condition for anti-hives which represents
$c_{\widetilde{\bf{v}},{\bf{u}}}^{\lambda}$

Thus $c_{\widetilde{\bf{v}},{\bf{u}}}^{\lambda}$ is the number of
anti-$l$-hives with boundary conditions as in Figure 2. Using the reverse
rhombus inequality, we can check that an anti-$l$-hive as above must vanish
completely in the quadrilateral $ABEF$ (including each sides) in Figure 3.
Moreover, inside the trapezoid $BCDE$, the values of the hive on each
horizontal line are the same. In particular, this means that there exists a
${\bf{z}}\in T^{+}(n)$ such that the boundary condition on $CB$ and $DE$ are
both given by ${\bf{z}}^{*}$.

Figure 3.

Thus such a hive is uniquely determined by its values in the anti-hives $BGC$
and $FDH$, with the indicated boundary conditions as in Figure 3. Conversely,
given anti-hives $BGC$ and $FDH$ with boundary conditions as in Figure 3, we
get an anti-hive with the boundary condition as in Figure 2 using a reverse
process. Finally, note that the number of anti-hives $BGC$ is
$c_{{\bf{y}},{\bf{z}}}^{{\bf{v}}}$ and the number of anti-hives $FDH$ is
$c_{{\bf{x}},{\bf{z}}}^{{\bf{u}}}$. Thus we get

$c_{\widetilde{\bf{v}},{\bf{u}}}^{\lambda}=\sum_{{\bf{z}}\in
T^{+}(n)}c_{{\bf{x}},{\bf{z}}}^{{\bf{u}}}c_{{\bf{y}},{\bf{z}}}^{{\bf{v}}}.$

This concludes the proof. ∎

##  4\. A local converse theorem

In the rest of this paper, we assume that $F$ is a non-archimedean local
field. Let ${\mathcal{O}}$ be the ring of integers of $F$, ${\mathfrak{p}}$ be
the maximal ideal of ${\mathcal{O}}$ and let $\varpi\in{\mathfrak{p}}$ be a
fixed uniformizer. The purpose of the rest of this paper is to prove the
following

######  Theorem 4.1.

Let $l$ be a positive integer and let $\pi_{1},\pi_{2}$ be two irreducible
supercuspidal representations of ${\mathrm{GL}}_{l}(F)$ with the same central
character. If
$\Gamma({\bf{s}},\pi_{1}\times(\tau_{1},\tau_{2}),\psi)=\Gamma({\bf{s}},\pi_{2}\times(\tau_{1},\tau_{2}),\psi)$
for all irreducible generic representations $\tau_{1}$ (resp. $\tau_{2}$) of
${\mathrm{GL}}_{m}(F)$ (resp. ${\mathrm{GL}}_{n}(F)$) with $0\leq
n\leq[l/2],0\leq m\leq[l/2]$, then $\pi_{1}\cong\pi_{2}$.

######  Remark 4.2.

If $l=2r$ is even and $m=n=r$, we have not defined the gamma factor
$\Gamma({\bf{s}},\pi\times(\tau_{1},\tau_{2}),\psi)$ yet, because our local
zeta integral (3.1) and hence our local gamma factor defined from that in
Proposition 3.5 require $m+n<l$. In the case if $l=2r,m=n=l$, the
corresponding local gamma factor used in Theorem 4.1 is the one defined from
the local zeta integral of unitary group
${\mathrm{U}}_{E/F}(2r)\times{\mathrm{Res}}_{E/F}({\mathrm{GL}}_{r})$ at a
split place, see [BAS:Uodd] and [Morimoto]. Actually, the properties of this
gamma factor is well studied. In particular, it has been shown that it is the
product of Jacquet–Piatetski-Shapiro–Shalika local gamma factors after
normalization, see [Morimoto]. We will review its definition in §4.1.

######  Remark 4.3.

Note that if $m=n=0$, then condition
$\Gamma({\bf{s}},\pi_{1}\times(\tau_{1},\tau_{2}),\psi)=\Gamma({\bf{s}},\pi_{2}\times(\tau_{1},\tau_{2}),\psi)$
is empty. If $m>0$ and $n=0$, the corresponding gamma factor
$\Gamma({\bf{s}},\pi_{1}\times(\tau_{1},\tau_{2}),\psi)$ is exactly a
Jacquet–Piatetski-Shapiro–Shalika local gamma factor up to a shift, see Remark
3.6.

Here we recall the Jacquet’s local converse conjecture

######  Conjecture 4.4.

Let $\pi_{1},\pi_{2}$ be two irreducible generic representations of
${\mathrm{GL}}_{l}(F)$ with the same central character. If
$\gamma^{\mathrm{JPSS}}(s,\pi_{1}\times\tau,\psi)=\gamma^{\mathrm{JPSS}}(s,\pi_{2}\times\tau,\psi)$
for any irreducible generic representation $\tau$ of ${\mathrm{GL}}_{m}(F)$
with $1\leq m\leq[l/2]$, then $\pi_{1}\cong\pi_{2}$.

One can assume that $\pi_{1},\pi_{2}$ are supercuspidal and remove the central
character restriction after the work of [JNS]. The above conjecture was proved
in [chai] and [Jacquet-Liu] independently. In the next remark, we will explain
that our Theorem 4.1 indeed gives a new proof of Conjecture 4.4 modulo a
standard fact on our gamma factors which will be given in a sequel paper [Yan-
Zhang].

######  Remark 4.5.

We denote by ${\mathcal{C}}(0)$ the condition that $\pi_{1},\pi_{2}$ have the
same central character, which is always assumed, and for $t\geq 1$, we denote
${\mathcal{C}}(t):={\mathcal{C}}(t;\pi_{1},\pi_{2})$ the following condition
for $\pi_{1},\pi_{2}:$

$\Gamma({\bf{s}},\pi_{1}\times(\tau_{1},\tau_{2}),\psi)=\Gamma({\bf{s}},\pi_{2}\times(\tau_{1},\tau_{2}),\psi),$

for any irreducible generic representation $\tau_{1}$ (resp. $\tau_{2}$) of
${\mathrm{GL}}_{m}(F)$ (resp. ${\mathrm{GL}}_{n}(F)$) with $0\leq m,n\leq t$.
To compare our result with Jacquet’s local converse conjecture, we also denote
by ${\mathcal{C}}^{\prime}(t):={\mathcal{C}}^{\prime}(t;\pi_{1},\pi_{2})$ the
condition:
$\gamma^{\mathrm{JPSS}}(s,\pi_{1}\times\tau,\psi)=\gamma^{\mathrm{JPSS}}(s,\pi_{2}\times\tau,\psi)$
for any irreducible generic representation $\tau$ of ${\mathrm{GL}}_{m}(F)$
with $1\leq m\leq t$. The condition ${\mathcal{C}}(t)$ is stronger than
${\mathcal{C}}^{\prime}(t)$ by Remark 3.6 and Remark 4.3. Thus it seems that
the result of Theorem 4.1 is weaker than the Jacquet’s local converse
conjecture as proved in [chai, Jacquet-Liu]. However, in a sequel paper [Yan-
Zhang], we will show that $\Gamma({\bf{s}},\pi\times(\tau_{1},\tau_{2}),\psi)$
is the product
$\gamma^{\mathrm{JPSS}}(s_{1}+\frac{l-m-n-1}{2},\pi\times\tau_{1},\psi)\gamma^{\mathrm{JPSS}}(s_{2}-\frac{l-m-n-1}{2},\widetilde{\pi}\times\widetilde{\tau}_{2},\psi)$
up to a normalizing factor which only depends on $\tau_{1},\tau_{2}$. Note
that
$\gamma^{\mathrm{JPSS}}(1-s,\widetilde{\pi}\times\widetilde{\tau},\psi)\gamma^{\mathrm{JPSS}}(s,\pi\times\tau,\psi)=1$.
Thus the condition ${\mathcal{C}}(t)$ is in fact equivalent to
${\mathcal{C}}^{\prime}(t)$. So our proof of Theorem 4.1 gives a new proof of
Jacquet’s local converse conjecture.

The proof of Theorem 4.1 will be given in the next section. In the rest of
this section, we introduce some necessary tools which will be used in the
proof of Theorem 4.1.

###  4.1. On the gamma factors for
${\mathrm{GL}}_{2r}\times({\mathrm{GL}}_{r},{\mathrm{GL}}_{r})$

Recall that if $m+n\leq l-1$, for generic representation $\pi$ of
${\mathrm{GL}}_{l}(F)$, $\tau_{1}$ (resp. $\tau_{2}$) of
${\mathrm{GL}}_{m}(F)$ (resp. ${\mathrm{GL}}_{n}(F)$), our local gamma factor
$\Gamma({\bf{s}},\pi\times(\tau_{1},\tau_{2}),\psi)$ is defined by the local
functional equation

$\Psi(W,M_{w_{m,n}}(\xi_{{\bf{s}}});0)=\Gamma({\bf{s}},\pi\times(\tau_{1},\tau_{2}),\psi)\Psi(W,\xi_{{\bf{s}}};0),$

for all $W\in{\mathcal{W}}(\pi,\psi)$ and
$\xi_{{\bf{s}}}\in{\mathcal{W}}({\bf{s}},(\tau_{1},\tau_{2}),\psi^{-1}).$ See
Proposition 3.5. For $W\in{\mathcal{W}}(\pi,\psi)$, we have
$\rho(\gamma_{m,n}^{-1})W$ is also an element in ${\mathcal{W}}(\pi,\psi)$.
Thus we have

(4.1)
$\displaystyle\Psi(\rho(\gamma_{m,n}^{-1})W,M_{w_{m,n}}(\xi_{{\bf{s}}});0)=\Gamma({\bf{s}},\pi\times(\tau_{1},\tau_{2}),\psi)\Psi(\rho(\gamma_{m,n}^{-1})W,\xi_{{\bf{s}}};0),$

for all
$\xi_{\bf{s}}\in{\mathcal{W}}({\bf{s}},(\tau_{1},\tau_{2}),\psi^{-1})$. Here
$\rho$ denotes the right translation and $\gamma_{m,n}$ is the element in
${\mathrm{GL}}_{l}$ as defined after (3.1). The local functional equation
(4.1) is the one we will use to prove our local converse theorem.

As explained in Remark 4.2, we also need the local gamma factors for
$\Gamma({\bf{s}},\pi\times(\tau_{1},\tau_{2}),\psi)$ when $l=2r$ and $m=n=r$,
which is not covered in our previous sections. This local gamma factor has
been defined in [BAS:Uodd] and studied in [Morimoto]. We recall the definition
now.

We first endowed $F^{2r}\oplus F^{2r}$ a symplectic structure
$\langle{~{},~{}}\rangle$ defined by

$\langle{(u_{1},u_{2}),(v_{1},v_{2})}\rangle=2(u_{1}J_{2r}v_{2}^{t}-v_{2}J_{2r}u_{2}^{t}),$

where $u_{i},v_{i}\in F^{2r}$ are viewed as row vectors. For a nontrivial
additive character $\psi$ of $F$ and for a character $\mu$ of $F^{\times}$, we
can consider the Weil representation $\omega_{\psi^{-1},\mu,\mu^{-1}}$ of
${\mathrm{GL}}_{2r}(F)$, see [Morimoto]*§2.2. Note that we used a little bit
different normalization. The Weil representation
$\omega_{\psi^{-1},\mu,\mu^{-1}}$ can be realized on the space
${\mathcal{S}}(F^{r}\times F^{r})$, the Bruhat-Schwatz functions on $F^{2r}$.
This is the Schrödinger model of the Weil representation. For example, we have
the well-known formula

$\left(\omega_{\psi^{-1}.\mu,\mu^{-1}}\left(\begin{pmatrix}I_{r}&X\\\
&I_{r}\end{pmatrix}\right)\phi\right)(x,y)=\psi(xXJ_{r}y^{t})\phi(x,y),X\in{\mathrm{Mat}}_{r\times
r}(F).$

In the following, we assume that $\mu$ is understood and omit it from the
notation.

Now let $\pi$ be an irreducible generic representation of
${\mathrm{GL}}_{2r}(F)$, $(\tau_{1},\tau_{2})$ be a pair of irreducible
generic representations of ${\mathrm{GL}}_{r}(F)$ and ${\bf{s}}=(s_{1},s_{2})$
be a pair of complex numbers. For
$W\in{\mathcal{W}}(\pi,\psi),\xi_{\bf{s}}\in{\mathcal{W}}({\bf{s}},(\tau_{1},\tau_{2},),\psi^{-1})$,
and $\phi\in{\mathcal{S}}(F^{2r})$, we consider the local zeta integral

$\Psi(W,\xi_{{\bf{s}}},\phi)=\int_{N_{2r}(F)\backslash{\mathrm{GL}}_{2r}(F)}W(g)\xi_{{\bf{s}}}(g)(\omega_{\psi^{-1}}(g)\phi)(e_{r},e_{r})dg,$

where $e_{r}\in F^{r}$ is the vector $(0,0,\dots,0,1)$. There exists a
meromorphic function $\Gamma({\bf{s}},\pi\times(\tau_{1},\tau_{2}),\mu,\psi)$
such that

(4.2)
$\Psi(W,M_{w_{r,r}}\xi_{{\bf{s}}},\phi)=\Gamma({\bf{s}},\pi\times(\tau_{1},\tau_{2}),\mu,\psi)\Psi(W,\xi_{{\bf{s}}},\phi)$

for any
$W\in{\mathcal{W}}(\pi,\psi),\xi_{{\bf{s}}}\in{\mathcal{W}}({\bf{s}},(\tau_{1},\tau_{2}),\psi^{-1})$
and $\phi\in{\mathcal{S}}(F^{r}\times F^{r}).$ Note that, in [BAS:Uodd] and
[Morimoto], there is only a single complex variable involved in the local zeta
integral and local gamma factor. Here we still use two variables case.

###  4.2. Howe vectors

Our strategy of the proof of Theorem 4.1 is along the lines of that given in
[Sp(2r)] and [U(2r+1)]. One basic tool for us is the partial Bessel functions
associated with Howe vectors as developed in [Baruch]. Here we recall the
basic construction. Let $\psi$ be a fixed unramified additive character of $F$
and we also view $\psi$ as a character of the maximal unipotent subgroup
$N_{l}\subset{\mathrm{GL}}_{l}(F)$ in the usual way. For an integer $i\geq 0$,
we consider the open compact subgroup
$K_{{\mathrm{GL}}_{l}}^{i}:=I_{l}+{\mathrm{Mat}}_{l\times
l}({\mathfrak{p}}^{i})$ of ${\mathrm{GL}}_{l}(F)$. Consider the character
$\theta_{i}$ of $K_{{\mathrm{GL}}_{l}}^{i}$ defined by

$\theta_{i}(k)=\psi(\varpi^{-2i}(\sum_{s=1}^{l-1}k_{s,s+1})),\quad
k=(k_{st})_{1\leq s,t\leq l}\in K_{{\mathrm{GL}}_{l}}^{i}.$

One can check that $\theta_{i}$ is indeed a character of
$K_{{\mathrm{GL}}_{l}}^{i}$. Consider the element

$d_{i}={\mathrm{diag}}(\varpi^{-i(l-1)},\varpi^{-i(l-3)},\dots,\varpi^{i(l-3)},\varpi^{i(l-1)}),$

and $H_{l}^{i}=d_{i}K^{i}d_{i}^{-1}$, which is still an open compact subgroup
of ${\mathrm{GL}}_{l}(F)$. One sees that $H_{l}^{i}$ has the form

$H_{l}^{i}=\begin{pmatrix}1+{\mathfrak{p}}^{i}&{\mathfrak{p}}^{-i}&{\mathfrak{p}}^{-3i}&\dots\\\
{\mathfrak{p}}^{3i}&1+{\mathfrak{p}}^{i}&{\mathfrak{p}}^{-i}&\dots\\\
{\mathfrak{p}}^{5i}&{\mathfrak{p}}^{3i}&1+{\mathfrak{p}}^{i}&\dots\\\
\dots&\dots&\dots&\dots\end{pmatrix}.$

We consider the character $\psi_{i}$ of $H_{l}^{i}$ defined by

$\psi_{i}(h):=\theta_{i}(d_{i}^{-1}hd_{i}),\quad h\in H_{l}^{i}.$

For a subgroup $U\subset{\mathrm{GL}}_{l}(F)$, we denote $U^{i}:=U\cap
H_{l}^{i}$. For example, $N_{l}^{i}$ denotes $N_{l}\cap H_{l}^{i}$. We also
usually drop $l$ from the notation if $l$ is understood. It is easy to see
that $\psi_{i}|_{N_{l}^{i}}=\psi|_{N_{l}^{i}}$.

Let $(\pi,V)$ be an irreducible generic representation of
${\mathrm{GL}}_{l}(F)$ and for $v\in V$, we consider

$v_{i}=\frac{1}{{\mathrm{vol}}(N_{l}^{i})}\int_{N_{l}^{i}}\psi_{i}^{-1}(u)\pi(u)vdu.$

If $W\in{\mathcal{W}}(\pi,\psi)$ is the Whittaker function associated with
$v$, then we denote $W_{i}=W_{v_{i}}$. Note that

$W_{i}(u_{1}gu_{2})=\psi(u_{1})\psi_{i}(u_{2})W_{i}(g),\quad\forall
g\in{\mathrm{GL}}_{l}(F),u_{1}\in N_{l},u_{2}\in N_{l}^{i}.$

Actually, that exists a positive integer $C(v)>0$, such that $W_{i}$ satisfies
the additional quasi-invariance property

(4.3) $W_{i}(ugh)=\psi(u)\psi_{i}(h)W_{i}(g)$

for all $u\in N_{l},g\in{\mathrm{GL}}_{l}(F),h\in H_{l}^{i}$ if $i>C(v)$, see
[Baruch]*Lemma 3.2. According the proof [Baruch]*Lemma 3.2, one can take
$C(v)$ to be the integer such that $v$ is fixed by $\pi(K^{C(v)}_{l})$.

Let $\omega$ be a character of $F^{\times}$ and we consider the space
$C_{c}^{\infty}({\mathrm{GL}}_{l}(F),\omega)$ consisting of smooth function
$f$ on $G$ such that $f$ is compactly supported modulo $Z_{l}$, the center of
${\mathrm{GL}}_{l}(F)$, and $f(zg)=\omega(z)f(g)$. If $\pi$ is supercuspidal,
let ${\mathcal{M}}(\pi)$ be the space of matrix coefficients of $\pi$. Then
${\mathcal{M}}(\pi)\subset C_{c}^{\infty}({\mathrm{GL}}_{l}(F),\omega_{\pi})$.
For $f\in{\mathcal{M}}(\pi)$, following [CST]*page 2089, we consider the
function

$W^{f}(g)=\int_{N_{l}}\psi^{-1}(u)f(ug)du.$

Note that the integral is convergent by assumption and defines an element in
${\mathcal{W}}(\pi,\psi)$. Moreover, for an appropriate choice of $f$, we can
assume that $W^{f}(I_{l})=1$. See [CST]*page 2089-2090. Thus we can consider
$W^{f}_{i}$. We also use the notation

${\mathcal{B}}_{i}(g,f)=W^{f}_{i}(g),\quad g\in{\mathrm{GL}}_{l}(F).$

###  4.3. Weyl elements which support Bessel functions

Let $\Delta=\Delta({\mathrm{GL}}_{l})$ be the set of simple roots of
${\mathrm{GL}}_{l}(F)$. Then $\Delta=\\{\alpha_{k}:1\leq k\leq l-1\\}$, where

$\alpha_{k}({\mathrm{diag}}(t_{1},\dots,t_{l}))=t_{k}/t_{k+1},\quad{\mathrm{diag}}(t_{1},\dots,t_{l})\in
T_{l}(F).$

Let ${\bf{W}}={\bf{W}}({\mathrm{GL}}_{l})$ be the Weyl group of
${\mathrm{GL}}_{l}(F)$. We sometimes identify ${\bf{W}}$ with the permutation
matrix in ${\mathrm{GL}}_{l}(F)$. Denote by $e$ the identiy element in
${\bf{W}}$, which is represented by $I_{l}\in{\mathrm{GL}}_{l}(F)$. For
$w\in{\bf{W}}$, denote $C(w)=BwB$, where $B=B_{l}$ is the upper triangular
subgroup of ${\mathrm{GL}}_{l}(F)$. There is a Bruhat order on ${\bf{W}}$,
which is recalled as follows. Given $w_{1},w_{2}\in{\bf{W}}$, then $w_{1}\leq
w_{2}$ (or $w_{2}\geq w_{1}$) if and only if
$C(w_{1})\subset\overline{C(w_{2})}$. For $w\in{\bf{W}}$, we denote
$\Omega_{w}=\coprod_{w^{\prime}\geq w}C(w^{\prime})$. Then $C(w)$ is closed in
$\Omega_{w}$ and $\Omega_{w}$ is open in $G$.

Let
${\mathrm{B}}({\mathrm{GL}}_{l})=\left\\{{w\in{\bf{W}}({\mathrm{GL}}_{l}):\alpha\in\Delta,w\alpha>0\implies
w\alpha\in\Delta}\right\\}$, which is the set of Weyl elements that can
support partial Bessel functions.

Let $w_{0}=J_{l}\in{\mathrm{GL}}_{l}(F)$, which represents the longest Weyl
element of ${\mathrm{GL}}_{l}(F)$. It is well-known that $w\in{\mathrm{B}}(G)$
if and only if $w_{0}w$ is the longest Weyl element of the Levi subgroup of a
standard parabolic subgroup of ${\mathrm{GL}}_{l}(F)$. For
$w\in{\mathrm{B}}(G)$, let $P_{w}=M_{w}N_{w}$ be the corresponding parabolic
subgroup such that $w_{0}w=w_{0}^{M_{w}}$, where $M_{w}$ is the Levi subgroup
of $P_{w}$ and $w_{0}^{M_{w}}$ is the longest Weyl element of $M_{w}$. Let
$\theta_{w}$ be the subset of $\Delta$ which consists all simple roots in
$M_{w}$. Then we have the relation

$\theta_{w}=\left\\{{\alpha\in\Delta|w\alpha>0}\right\\}\subset\Delta.$

The assignment $w\mapsto\theta_{w}$ is a bijection between ${\mathrm{B}}(G)$
and subsets of $\Delta$. Moreover, it is known that the assignment
$w\mapsto\theta_{w}$ is order-reversing, i.e., $w^{\prime}\leq w$ if and only
if $\theta_{w}\subset\theta_{w^{\prime}}$, see [CPSS05, Proposition 2.11]. For
example, we have $\theta_{w_{0}}=\emptyset$ and $\theta_{e}=\Delta$.

Given a subset $\theta\subset\Delta$, we will write the corresponding Weyl
element in ${\mathrm{B}}({\mathrm{GL}}_{l})$ by $w_{\theta}$. For an integer
$k$ with $1\leq k\leq l-1$, denote

$\overline{w}_{k}=\begin{pmatrix}&I_{l-k}\\\ I_{k}&\end{pmatrix}.$

######  Lemma 4.6.

For every $k$ with $1\leq k\leq l-1$, we have
$\overline{w}_{k}=w_{\Delta-\left\\{{\alpha_{k}}\right\\}}.$

###### Proof.

We have

$w_{0}\overline{w}_{k}=\begin{pmatrix}J_{k}&\\\ &J_{l-k}\end{pmatrix},$

which is the longest Weyl element of the Levi subgroup

$M_{\overline{w}_{k}}=\left\\{{\begin{pmatrix}a&\\\
&b\end{pmatrix}:a\in{\mathrm{GL}}_{k}(F),b\in{\mathrm{GL}}_{l-k}(F)}\right\\}.$

The set of simple roots in $M_{\overline{w}_{k}}$ is
$\Delta-\left\\{{\alpha_{k}}\right\\}$. Thus we have
$\overline{w}_{k}\in{\mathrm{B}}({\mathrm{GL}}_{l})$ and
$\theta_{\overline{w}_{k}}=\Delta-\left\\{{\alpha_{k}}\right\\}$. ∎

Denote

$\widetilde{w}_{n,m}=\begin{pmatrix}&&I_{n}\\\ &I_{l-m-n}&\\\
I_{m}&&\end{pmatrix}.$

######  Lemma 4.7.

For positive integers $m,n$ with $1\leq m+n\leq l-1$, we have
$\theta_{\widetilde{w}_{n,m}}=\Delta-\left\\{{\alpha_{m},\alpha_{l-n}}\right\\}$.

###### Proof.

We have

$w_{0}\widetilde{w}_{n,m}=\begin{pmatrix}J_{m}&&\\\ &J_{l-m-n}&\\\
&&J_{n}\end{pmatrix},$

which is the longest Weyl element in the Levi subgroup

$M_{\widetilde{w}_{n,m}}=\begin{pmatrix}a&&\\\ &b&\\\
&&c\end{pmatrix},a\in{\mathrm{GL}}_{n},b\in{\mathrm{GL}}_{l-m-n},c\in{\mathrm{GL}}_{m}.$

Thus
$\theta_{\widetilde{w}_{n,m}}=\Delta-\left\\{{\alpha_{m},\alpha_{l-n}}\right\\}.$
∎

Given $w,w^{\prime}\in{\mathrm{B}}({\mathrm{GL}}_{l})$ with $w>w^{\prime}$,
define (following Jacquet [J])

$d_{B}(w,w^{\prime})=\max\left\\{{m|\textrm{ there exist }w^{\prime}_{i}\in
B(G)\textrm{ with
}w=w^{\prime}_{m}>w^{\prime}_{m-1}>\dots>w^{\prime}_{0}=w^{\prime}}\right\\}.$

The number $d_{B}(w,w^{\prime})$ is called the Bessel distance of
$w,w^{\prime}$. By [CPSS05, Proposition 2.1] and Lemma 4.6, the set of
elements in ${\mathrm{B}}(G)$ which has Bessel distance 1 with the element
$e\in{\mathrm{B}}(G)$ are $\left\\{{\overline{w}_{k},1\leq k\leq
l-1}\right\\}$, i.e.,

(4.4) $\left\\{{w|d_{B}(w,e)=1}\right\\}=\left\\{{\overline{w}_{k}|1\leq k\leq
l-1}\right\\}.$

For $w,w^{\prime}\in{\bf{W}}$ with $w<w^{\prime}$, we denote by
$[w,w^{\prime}]$ the closed Bruhat interval
$\left\\{{w^{\prime\prime}\in{\bf{W}}({\mathrm{GL}}_{l})|w\leq
w^{\prime\prime}\leq w^{\prime}}\right\\}$.

###  4.4. Cogdell-Shahidi-Tsai’s theory on partial Bessel functions

In this subsection, we review certain basic properties of partial Bessel
functions developed by Cogdell-Shahidi-Tsai recently in [CST].

For $w\in{\mathrm{B}}({\mathrm{GL}}_{l})$, we denote

(4.5) $A_{w}=\left\\{{a\in T_{l}(F)|\alpha(a)=1\textrm{ for all
}\alpha\in\theta_{w}}\right\\}.$

The set $A_{w}$ is in fact the center of $M_{w}$.

######  Theorem 4.8 (Cogdell-Shahidi-Tsai).

Let $\omega$ be a character of $F^{\times}$.

  1. (1)

Let $w\in{\bf{W}}$, $m>0$ and $f\in C_{c}^{\infty}(\Omega_{w},\omega)$.
Suppose ${\mathcal{B}}_{i}(wa,f)=0$ for all $a\in A_{w}$. Then there exists
$f_{0}\in C_{c}^{\infty}(\Omega_{w}-C(w),\omega)$, such that for sufficiently
large $i$ depending only on $f$, we have
${\mathcal{B}}_{i}(g,f)={\mathcal{B}}_{i}(g,f_{0})$ for all
$g\in{\mathrm{GL}}_{l}(F)$.

  2. (2)

Let $w\in{\mathrm{B}}({\mathrm{GL}}_{l})$. Let $\Omega_{w,0}$ and
$\Omega_{w,1}$ be $N_{l}\times N_{l}$ and $T_{l}$-invariant open sets of
$\Omega_{w}$ such that $\Omega_{w,0}\subset\Omega_{w,1}$ and
$\Omega_{w,1}-\Omega_{w,0}$ is a union of Bruhat cells $C(w^{\prime})$ such
that $w^{\prime}$ does not support a Bessel function, i.e.,
$w^{\prime}\notin{\mathrm{B}}({\mathrm{GL}}_{l})$. Then for any $f_{1}\in
C_{c}^{\infty}(\Omega_{w,1},\omega)$ there exists $f_{0}\in
C_{c}^{\infty}(\Omega_{w,0},\omega)$ such that for all sufficiently large $i$
depending only on $f_{1}$, we have
${\mathcal{B}}_{i}(g,f_{0})={\mathcal{B}}_{i}(g,f_{1}),$ for all
$g\in{\mathrm{GL}}_{l}(F)$.

###### Proof.

Part (1) is [CST, Lemma 5.13] and part (2) is [CST, Lemma 5.14]. ∎

######  Corollary 4.9.

Let $f_{1},f_{2}\in C_{c}^{\infty}({\mathrm{GL}}_{l}(F),\omega)$ with
$W^{f_{1}}(I_{l})=W^{f_{2}}(I_{l})=1$. Then there exist functions
$f_{\overline{w}_{k}}\in C_{c}^{\infty}(\Omega_{\overline{w}_{k}},\omega)$ for
all $k$ with $1\leq k\leq l-1$ such that for sufficiently large $i$ (depending
only on $f_{1},f_{2}$) we have

${\mathcal{B}}_{i}(g,f_{1})-{\mathcal{B}}_{i}(g,f_{2})=\sum_{k=1}^{l-1}{\mathcal{B}}_{i}(g,f_{\overline{w}_{k}}),\quad\forall
g\in G.$

This is essentially [CST, Proposition 5.3], see [CST]*page 2115 for a similar
identity. Almost identical proofs in similar situations are given in
[Sp(2r)]*Corollary 4.7 and [U(2r+1)]*Corollary 2.7. We omit the proof here and
just remark that each term in the expansion of the right side comes from the
Weyl elements which has Bessel distance 1 from the trivial Weyl element
$e\in{\bf{W}}({\mathrm{GL}}_{l})$, namely the elements in the set (4.4).

###  4.5. Construction of certain sections of induced representations

Let $m,n$ be two positive integers and $\tau_{1}$ (resp. $\tau_{2}$) be an
irreducible generic representation of ${\mathrm{GL}}_{m}(F)$ (resp.
${\mathrm{GL}}_{n}(F)$) and let ${\bf{s}}=(s_{1},s_{2})$. Consider

$N_{m,n}=\left\\{{u_{m,n}(x)=\begin{pmatrix}I_{m}&x\\\
&I_{n}\end{pmatrix},x\in{\mathrm{Mat}}_{m\times
n}}\right\\},\overline{N}_{m,n}=\left\\{{\overline{u}_{m,n}(x):=\begin{pmatrix}I_{m}&\\\
x&I_{n}\end{pmatrix},x\in{\mathrm{Mat}}_{n\times m}}\right\\},$

and

$\overline{N}_{m,n}^{k}=\left\\{{\overline{u}_{m,n}(x)\left|\begin{pmatrix}I_{m}&&\\\
&I_{l-m-n}&\\\ x&&I_{n}\end{pmatrix}\in H_{l}^{k}\right.}\right\\}.$

Here we identify $N_{m,n}$ etc. with its $F$-rational points and recall that
$H_{l}^{k}$ is defined in Section 4.2.

Let $D$ be a compact open subset of $N_{m,n}$. For $x\in D$ and a positive
integer $i$, we consider the set

$S(x,k)=\left\\{{\overline{y}\in\overline{N}_{m,n}:\overline{y}x\in
P_{m,n}\overline{N}_{m,n}^{k}}\right\\}.$

######  Lemma 4.10.

  1. (1)

For any positive integer $c$, there exists a positive integer
$k_{1}=k_{1}(D,c)$ such that for all $k\geq k_{1},x\in D,\overline{y}\in
S(x,k)$, we can write

$\overline{y}x=u_{m,n}(x_{1}){\mathrm{diag}}(a,b)\overline{u}_{m,n}(y_{1}),$

with $a\in K^{c}_{{\mathrm{GL}}_{m}},b\in K_{{\mathrm{GL}}_{n}}^{c}$. Here
$u_{m,n}(x_{1})\in
N_{m,n},\overline{u}_{m,n}(y_{1})\in\overline{N}_{m,n}^{k}$. We recall that
$K^{c}_{{\mathrm{GL}}_{m}}=I_{m}+{\mathrm{Mat}}_{m\times
m}({\mathfrak{p}}^{c})$.

  2. (2)

There exists an integer $k_{2}=k_{2}(D)$ such that
$S(x,k)=\overline{N}_{m,n}^{k}$ for all $x\in D$ and $k\geq k_{2}$.

###### Proof.

This is an analogue of [Baruch]*Lemma 4.1, [Sp(2r)]*Lemma 5.1 and the proof is
also similar. We provide a sketch below. For $x\in D$ and $\overline{y}\in
S(x,k)$, we assume that
$\overline{y}x=u_{m,n}(x_{1}){\mathrm{diag}}(a,b)\overline{u}_{m,n}(y_{1})$
for some
$a\in{\mathrm{GL}}_{m}(F),b\in{\mathrm{GL}}_{n}(F),x_{1}\in{\mathrm{Mat}}_{m\times
n},y_{1}\in{\mathrm{Mat}}_{n\times m}$ with
$\overline{u}_{m,n}(y_{1})\in\overline{N}_{m,n}^{k}$. By abuse of notation, we
also write $\overline{y}=\overline{u}_{m,n}(y),x=u_{m,n}(x)$. Then from the
equation

$\overline{y}^{-1}u_{m,n}(x_{1}){\mathrm{diag}}(a,b)=x\overline{u}_{m,n}(-y_{1}),$

we get

(4.6) $\begin{pmatrix}a&x_{1}b\\\
-ya&(I_{n}-yx_{1})b\end{pmatrix}=\begin{pmatrix}I_{m}-xy_{1}&x\\\
-y_{1}&I_{n}\end{pmatrix}.$

We can solve that $a=I_{m}-xy_{1}$ and $b=I_{n}+y_{1}a^{-1}x$. Since when
$x\in D$, the entries of $x$ are bounded, and the entries of $y_{1}$ go to
zero as $k\to\infty$, we can take $k$ large enough such that
$a=I_{m}-xy_{1}\in K_{{\mathrm{GL}}_{m}}^{c}$ and $b=I_{n}+y_{1}a^{-1}x\in
K_{{\mathrm{GL}}_{n}}^{c}.$ This proves (1).

By (4.6), we have
$y=y_{1}a^{-1}=y_{1}(I_{m}-xy_{1})^{-1}=y_{1}(I_{m}+xy_{1}+(xy_{1})^{2}+\dots)$.
Again, since each entry of $x$ is bounded, we may take $k$ large such that the
entries of $y_{1}(xy_{1})^{t}$ are so large so that
$\overline{u}_{m,n}(y_{1}(xy_{1})^{t})\in\overline{N}_{m,n}^{k}$ for $t\geq
0$. This shows that for $k$ large, we have
$\overline{u}_{m,n}(y)\in\overline{N}_{m,n}^{k}$ and thus
$S(x,k)\subset\overline{N}_{m,n}^{k}$ since
$\overline{y}=\overline{u}_{m,n}(y)$ is arbitrarily chosen. See [Sp(2r)]*Lemma
5.1 for a similar and more detailed argument.

Take $x\in D$, we need to show $\overline{N}_{m,n}^{k}\subset S(x,k)$ for $k$
large. As above, we write $x=u_{m,n}(x)$ by abuse of notation. We first assume
that $k$ is so large such that if
$\overline{u}_{m,n}(y)\in\overline{N}_{m,n}^{k}$, then $I_{n}+yx$ is
invertible and $I_{n}-x(I_{n}+yx)^{-1}y$ is also invertible. This can be done
because $x$ has bounded entries and $y$ has small entries if
$\overline{u}_{m,n}(y)\in\overline{N}_{m,n}^{k}$ when $k$ large. Then we have

$\overline{u}_{m,n}(y)u_{m,n}(x)=u_{m,n}(x_{1}){\mathrm{diag}}(a,b)\overline{u}_{m,n}(y_{1}),$

with $b=I_{n}+yx,a=I_{n}-b^{-1}y,x_{1}=xb^{-1}$ and $y_{1}=(I_{n}+yx)^{-1}y$.
In particular, $\overline{u}_{m,n}(y)u_{m,n}(x)\in P_{m,n}\overline{N}_{m,n}$.
To show $\overline{u}_{m,n}(y)\in S(x,k)$ for $k$ large, it suffices to show
that one can choose $k$ large so that the above
$\overline{u}_{m,n}(y_{1})\in\overline{N}_{m,n}^{k}$. Notice that
$y_{1}=(I_{n}+yx)^{-1}y$ with bounded entries in $x$ and small entries in $y$,
the argument is the same the above step. We are done. ∎

Given $v_{j}\in V_{\tau_{j}}$, the space of $\tau_{j}$, for $j=1,2$, we
consider the following $\tau_{1}\boxtimes\tau_{2}$-valued function on
${\mathrm{GL}}_{m+n}(F)$.