# Petersson Inner Products and Whittaker–Fourier Periods on Even Special
Orthogonal and Symplectic Groups

Yeongseong Jo  Department of Mathematics Education, Ewha Womans University,
Seoul 03760, Republic of Korea
jo.59<EMAIL_ADDRESS>Dedicated to James Cogdell
with admiration

###### Abstract.

In this article, we would like to formulate a relation between the square norm
of Whittaker–Fourier coefficients on even special orthogonal and symplectic
groups and Petersson inner products along with the critical value of
$L$-functions up to constants. We follow the path of Lapid and Mao to reduce
it to the conjectural local identity. Our strategy is based on the work of
Ginzburg–Rallis–Soudry on automorphic descent. We present the analogue result
for odd special orthogonal groups, which is conditional on unfolding Whittaker
functions of descents.

###### Key words and phrases:

Petersson inner products, Whittaker–Fourier coefficients, Automorphic
descents, Automorphic forms and representations.

###### 2020 Mathematics Subject Classification:

Primary 11F67; Secondary 11F70, 22E50

##  1\. Introduction

Let ${\rm GL}_{n}(\mathbb{A})$ be a general linear group over a number field
$F$ and $\mathbb{A}$ the ring of adeles of $F$. We let $\pi$ denote an
irreducible cuspidal representation. We let $B$ denote a Borel subgroup of
${\rm GL}_{n}$, $N$ the unipotent radical of $B$ and fix a non-degenerate
character $\psi_{N}$ of $N(\mathbb{A})$, trivial on $N(F)$. For a cusp form
$\varphi$ of ${\rm GL}_{n}(F)\backslash{\rm GL}_{n}(\mathbb{A})$, we consider
the Whittaker–Fourier coefficient

$\mathcal{W}^{\psi_{N}}(\varphi):=\int_{N(F)\backslash
N(\mathbb{A})}\varphi(u)\psi^{-1}_{N}(u)\,du.$

If $\pi$ is an irreducible cuspidal representation, then non-zero
Whittaker–Fourier coefficient $\mathcal{W}^{\psi_{N}}(\varphi)$ gives a
realization of $\pi$ in the space of Whittaker functions on ${\rm
GL}_{n}(\mathbb{A})$. By local multiplicity one, the space of Whittaker
functions depends only on $\pi$ as an abstract representation and $\pi$ is
decomposable in the sense that

$\pi\cong\otimes_{v}\mathbb{W}^{\psi_{N,v}}(\pi_{v})$

where $\mathbb{W}^{\psi_{N,v}}(\pi_{v})$ is the local Whittaker model of
$\pi_{v}$. Upon further assuming that $\mathcal{W}^{\psi_{N}}(\varphi)$ is
decomposable, we have a factorization of Whittaker functions

$\mathcal{W}^{\psi_{N}}(\varphi)=\prod_{v}W_{v},\quad
W_{v}\in\mathbb{W}^{\psi_{N,v}}(\pi_{v}).$

It therefore provides a useful tool for understanding $\pi$, both
computationally and conceptually.

The construction of Whittaker functions has had numerous significant impact in
global integrals of Rankin-Selberg type. The prototype integral representation
is the original construction for ${\rm GL}_{2}\times{\rm GL}_{2}$ by R. Rankin
and A. Selberg in late 1930. Their analogues for higher rank has been
developed and studied in the pioneering work by Jacquet, Piatetski–Shapiro,
and Shalika [JS81a, JS81b, JPSS83] with some parts completed only in recent
years by Jacquet [Jac09]. The Rankin-Selberg integral has been investigated
extensively in the literature and are the subject matter in modern number
theory of automorphic representations and their $L$-functions. Among other
things, the Rankin–Selberg integral expresses the Petersson inner product in
terms of a canonical inner product on the local Whittaker model of $\pi_{v}$
under the decomposition $\pi=\otimes_{v}\pi_{v}$ [JS81a, §4]. The global
factor which shows up in this expression is the residue at $s=1$ of the
adjoint $L$-function $L(s,\pi,{\rm Ad})$ of $\pi$ (or alternatively,
$L(s,\pi\times\pi^{\vee})$) where $\pi^{\vee}$ is the contragredient of $\pi$.

We try to make the statement more precise, and to generalize it to other
groups. Let ${\rm GL}_{n}(\mathbb{A})^{1}=\\{g\in{\rm
GL}_{n}(\mathbb{A})\;|\;|\det g|=1\\}$. It is know from [LM15, §4] that ${\rm
vol}({\rm GL}_{n}(F)\backslash{\rm GL}_{n}(\mathbb{A})^{1})=1$. We let
$\zeta^{S}_{F}(s)=\prod_{v\notin S}\zeta_{F_{v}}(s)$, $\Re s>1$ be the partial
Dedekind zeta function. Similar notation applies to the partial adjoint
$L$-function $L^{S}(s,\pi,{\rm Ad})$. For a cusp form $\varphi$ and another
cusp form $\varphi^{\vee}$ on ${\rm GL}_{n}(F)\backslash{\rm
GL}_{n}(\mathbb{A})$, we define the Petersson inner product by

$\langle\varphi,\varphi^{\vee}\rangle_{{\rm GL}_{n}(F)\backslash{\rm
GL}_{n}(\mathbb{A})^{1}}:=\int_{{\rm GL}_{n}(F)\backslash{\rm
GL}_{n}(\mathbb{A})^{1}}\varphi(g)\varphi^{\vee}(g)\,dg,$

while on $\mathbb{W}^{\psi_{N,v}}(\pi_{v})$ an invariant inner product is
given by

$J_{v}\left(1,W_{v},\check{W}_{v}\right):=\int_{({\rm GL}_{n-1}(F_{v})\cap
N(F_{v}))\backslash{\rm GL}_{n-1}(F_{v})}W_{v}\begin{pmatrix}g&\\\
&1\end{pmatrix}\check{W}_{v}\begin{pmatrix}g&\\\ &1\end{pmatrix}\,dg$

for non-archimedean places $v$ of $F$ [Jo23, Proposition 3.4] and archimedean
places $v$ of $F$ [HJ24, Remark 5.8]. It is natural to compare the local
invariant inner product with the Petersson inner product. The relation has
been established in the pioneering work of Jacquet and Shalika [JS81a, §4];

(1.1) $\langle\varphi,\varphi^{\vee}\rangle_{{\rm GL}_{n}(F)\backslash{\rm
GL}_{n}(\mathbb{A})^{1}}=\frac{L^{S}(1,\pi,{\rm
Ad})}{\prod_{j=1}^{n}\zeta_{F}^{S}(j)}\prod_{v\in
S}J_{v}\left(1,W_{v},\check{W}_{v}\right).$

(We ask readers to refer to [Zha14]*Proposition 3.1 [LM15]*p.477, Lemma 4.4).
Recently the above product formula can be further refined by the product of
so-called naïve Rankin–Selberg $L$-factors over all places at $s=1$, upon
taking the global newform [HJ24, Definition 8.1] and then specializing the
value of the complex variable $s$ in the period integral [HJ24, Theorem 8.4].

As a consequence of the formula of Jacquet and Shalika (1.1), we take the
$\psi_{N}$-th Fourier coefficient of a matrix coefficient of $\pi$ and turn
our attention to associating it with the product of Whittaker functions. The
integral

$\int_{N(\mathbb{A})}\langle\pi(u)\varphi,\varphi^{\vee}\rangle_{{\rm
GL}_{n}(F)\backslash{\rm GL}_{n}(\mathbb{A})^{1}}\psi_{N}^{-1}(u)\,du$

does not converge. In fact, even the local integrals

$\int_{N(F_{v})}\langle\pi_{v}(u_{v})\varphi_{v},\varphi_{v}^{\vee}\rangle_{v}\psi^{-1}_{N}(u_{v})\,du_{v}$

(where $\varphi_{v}$ and $\varphi_{v}^{\vee}$ are now taken from the space of
$\pi_{v}$ and $\pi^{\vee}_{v}$ respectively and
$\langle\cdot,\cdot\rangle_{v}$ is the canonical pairing) do not converge
unless $\pi_{v}$ is square-integrable. Given a finite set of places $S$, it is
possible to regularize the integral in such a way that

$\int^{\rm st}_{N(F_{S})}\langle\pi(u)\varphi,\varphi^{\vee}\rangle_{{\rm
GL}_{n}(F)\backslash{\rm GL}_{n}(\mathbb{A})^{1}}\psi^{-1}_{N}(u)\,du$

just as in [LM15]. In particular, if $S$ consists solely of non-archimedean
places, then

$\int^{\rm st}_{N(F_{S})}\langle\pi(u)\varphi,\varphi^{\vee}\rangle_{{\rm
GL}_{n}(F)\backslash{\rm
GL}_{n}(\mathbb{A})^{1}}\psi^{-1}_{N}(u)\,du=\int_{N^{\ast}}\langle\pi(u)\varphi,\varphi^{\vee}\rangle_{{\rm
GL}_{n}(F)\backslash{\rm GL}_{n}(\mathbb{A})^{1}}\psi^{-1}_{N}(u)\,du$

for any sufficiently large compact open subgroup $N^{\ast}$ of $N(F_{S})$. The
definition of the regularized integral is different in the archimedean case
(cf. [Mor24, §4.3]), but in this paper we do not use regularized integrals
over archimedean fields directly. From the viewpoint of local multiplicity
one, there exist constants $c_{\pi_{v}}$ depending on $\pi_{v}$ such that

(1.2)
$\mathcal{W}^{\psi_{N}}(\varphi)\mathcal{W}^{\psi^{-1}_{N}}(\varphi^{\vee})=\left(\prod_{v\in
S}c^{-1}_{\pi_{v}}\right)\frac{\prod_{j=1}^{n}\zeta_{F}^{S}(j)}{L^{S}(1,\pi\times\pi^{\vee})}\int^{\rm
st}_{N(F_{S})}\langle\pi(u)\varphi,\varphi^{\vee}\rangle_{{\rm
GL}_{n}(F)\backslash{\rm GL}_{n}(\mathbb{A})^{1}}\psi^{-1}_{N}(u)\;du$

for all $\varphi=\otimes_{v}\varphi_{v}\in\pi$,
$\varphi^{\vee}=\otimes_{v}\varphi^{\vee}_{v}\in\pi^{\vee}$, and all $S$
sufficiently large [LM15, (1.2)]. Implicit here is the existence and non-
vanishing of
$\left.\dfrac{\prod_{j=1}^{n}\zeta_{F}^{S}(js)}{L^{S}(s,\pi\times\pi^{\vee})}\right|_{s=1}$.
Lapid and Mao [LM15, Theorem 4.1, Lemma 4.4] utilize the theory of
Rankin–Selbeg integrals to show that $c_{\pi_{v}}=1$ over all places $v$ for
irreducible cuspidal representation of ${\rm GL}_{n}$, while a similar result
was proved independently in the work of Sakellaridis and Venkatesh [SV17,
§6.3, §18.3]. We mention in this context a slightly modified formulation
presented in [Liu16, §2.2].

It is desirable to extend this relation (1.2) based on the analog of the
equality (1.1) to symplectic and special orthogonal groups. The purpose of the
paper is to compute Whittaker–Fourier coefficients on symplectic and special
orthogonal groups to reduce the global conjectural identity to a local
corresponding identity.

  * •

In Theorem 3.14, we establish the explicit formula relating the product of
Whittaker–Fourier coefficients of cusp forms on adelic quotients of split and
quasi-split even special orthogonal groups to the Petersson inner product. The
formulation relies on the validity of Working Hypothesis 3.5 and is built upon
Proposition 3.13.

  * •

In Theorem 4.11, we establish the explicit formula relating the product of
Whittaker–Fourier coefficients of cusp forms on adelic quotients of symplectic
groups to the Petersson inner product. The formulation relies on the validity
of Working Hypothesis 4.2 and is built upon Proposition 4.10.

In Theorem 5.14, we also study the formula relating the square norm of
Whittaker–Fourier coefficients on adelic quotients of split odd special
orthogonal groups to the product of and special value of automorphic partial
$L$-functions under Working Hypothesis 5.5. Whereas the formulation in odd
special orthogonal cases is built upon Proposition 5.13 as before, it relies
on the ongoing work of Lapid and Mao (Theorem 5.8), which concerns with
unfolding Whittaker functions of the global descent generated by Gelfand–Graev
coefficients. It is noteworthy that the bulk of Section 5 is intended for
gathering more or less expository groundwork toward a local conjecture
identity, rather than concentrating on proving precise statements.

In this note, we follow the path of the adaptation of [LM16] to study
Whittaker–Fourier coefficients for unitary groups and shed a light on the
product of Whittaker–Fourier coefficients (1.2). The key ingredient is the
theory of automorphic descent developed by Ginzburg–Rallis–Soudry [GRS99,
GRS11]. We address that this approach is different from that of Waldspurger
[Wal81] who uses the Shimura correspondence as well as from the relative trace
formula approach [Zha14]. At the moment, the descent theory is more thoroughly
understood in the case of metaplectic groups than in the case of special
orthogonal and symplectic groups. In addition there are some expected
properties of the descent, specifically irreducibility of global descents
(Working Hypothesis 3.5 and 4.2) in the even special orthogonal and symplectic
case just as assumed in [LM16], and irreducibility of global descents (Working
Hypothesis 5.5) and Theorem 5.8 of Lapid–Mao in odd special orthogonal case.
Although it is likely that the similar method in metaplectic case work, we do
not take up the issue of bridging theses gaps here. This is because the
ultimate goal is to settle down the local identity, once the reduction to the
local identity from the global identity is achieved in the current project. In
this regard, we will take for granted the expected properties of the descent
for special orthogonal and symplectic groups. Therefore our results are
conditional.

In the case of classical groups of different rank cases, it is one of the most
important topic in modern number theory to explore these types of deep
relations between certain period integrals of automorphic forms and critical
value of $L$-functions. In 1990’s, Gross and Prasad [GP92, GGP12] predicted a
relationship between the non-vanishing of Gross-Prasad period on special
orthogonal groups of co-rank $1$ and the non-vanishing of certain
$L$-functions. Recently the influential work of Ichino and Ikeda [II10] has
refined the global Gan–Gross-Prasad conjecture [GGP12] to a conjectural
formula which computes the norm of the period integral in terms of the central
$L$-value in the case of orthogonal group of co-rank $1$. The construction of
period integrals for cuspidal representations of unitary groups ${\rm
U}_{n}\times{\rm U}_{m}$ depends on the parity of $n-m$, so one has to treat
the case $n-m\neq 0$ even (Fourier–Jacobi cases) and $n-m$ odd (Bessel cases)
separately. For unitary groups ${\rm U}_{n+1}\times{\rm U}_{n}$ and $n=1,2$,
the similar work was later accomplished by R. N. Harris [Har14]. W. Zhang
[Zha14] has recently made a remarkable progress toward refined formula for
${\rm U}_{n+1}\times{\rm U}_{n}$ taking the relative trace formula approach
initiated by Jacquet [Jac89]. The reader may consult to [BLX24, Liu16] for
incomplete list of Jacobi and Bessel cases on unitary groups.

The case of the metaplectic group $\widetilde{\rm SL}_{2}(=\widetilde{\rm
Sp}_{2})$-the two-fold cover of the special linear group ${\rm SL}_{2}$ goes
back to the seminal work of Waldspurger on the Fourier coefficients of half-
integral weight modular forms [Wal81]. His result has been revisited in a
flurry of work since then (see [LM15, LM17, Qiu19] and the references
therein). Later Lapid and Mao [LM17] generalize the relation between the
Whittaker–Fourier coefficients and the Petersson inner product to arbitrary
metaplectic groups $\widetilde{\rm Sp}_{2n}$. While these groups are not
algebraic, they behave in many ways like algebraic groups. More importantly,
the descent method applies to metaplectic groups and give rise to irreducible
representations. The formula relating the Whittaker–Fourier coefficients to
the Peterson inner product for quasi-split unitary groups is carried out by
Lapid and Mao [LM15] but under the assumption of irreducibility of global
descents.

As seen in (1.2), for all places $v$ of $F$, the local constant $c_{\pi_{v}}$
depending on $\pi_{v}$ is given as a proportionality constant of two local
Whittaker periods. Afterwords, Lapid and Mao [LM17-2] determine a value for
the $c_{\pi_{v}}$, which is provided with a root number
$c_{\pi_{v}}=\varepsilon(\frac{1}{2},\pi_{v},\psi_{v})$ in the case of
metaplectic groups though they only consider $p$-adic places $v$. Very
recently, Morimoto [Mor24] resolves what is known as Lapid–Mao conjecture,
which specifies the value $c_{\pi_{v}}$ as the central character
$\omega_{\pi}$ of $\pi$ evaluated at $\tau$ satisfying
$-\tau=\mathfrak{c}(\tau)$ for all places $v$ in the case of unitary groups.
Here $\mathfrak{c}$ is a Galois involution over a quadratic extension of
$F_{v}$. It turns out that this result is preceded by his earlier work on even
unitary groups at non-split finite places [Mor22]. The main significance of
this task is that the presumptive local identity is expected to be equivalent
to the formal degree conjecture of Hiraga–Ichino–Ikeda [HII08] in the case of
generic discrete series representations. Ichino–Lapid–Mao [ILM17] use this
method to establish the formal degree conjecture for the case of odd special
orthogonal groups and metaplectic groups. For unitary group cases, Morimoto
[Mor22, Mor24] generalizes the technique to work out the formal degree
conjecture. It is perhaps fair to raise a natural question as to analyzing the
exact value of $c_{\pi_{v}}$ in the context of special orthogonal and
symplectic groups. As was pointed out briefly, we follow the robust argument
proposed by Lapid and Mao to attack the explicit identity in author’s ongoing
collaborations. However we think that our current content keeps our exposition
a reasonable length and we hope to do so in the future.

Let us describe the organization of this paper in more detail. We set up basic
notations in Section 2. In Section 3 through Section 5, we introduce and study
the descent construction, both locally and globally. We then explore analytic
properties of Rakin–Selberg–Shimura type integral and compute it for
unramified data. After piecing together ingredients of the descent
constructions and Euler products from Petersson inner products, we present our
main result which allows us to reduce to a local statement. Section 3 is
devoted to the case for both quasi-split and split even orthogonal groups and
we dedicate Section 4 to the case for symplectic groups. In Section 5, we
present results in the case of odd special orthogonal groups, which parallel
the analogue of previous cases.

##  2\. Notation and Preliminaries

###  2.1. Local settings

Let $F$ be a local field of character zero. By abuse of notation, we will use
the same letter for an algebraic group over $F$ and its group of $F$-points.
We denote by ${\rm Irr}\,Q$ the set of equivalence classes of smooth complex
irreducible representations of the group of $F$-points of an algebraic group
$Q$ over $F$. We also write $\delta_{Q}$ for the modulus function of $Q$. If
$Q$ is quasi-split and $\psi_{N_{Q}}$ is a non-degenerate character of a
maximal unipotent subgroup $N_{Q}$, we denote by ${\rm Irr}_{{\rm
gen},\psi_{N_{Q}}}Q$ the subset of representations that are generic with
respect to $\psi_{N_{Q}}$. We suppress $\psi_{N_{Q}}$ from the notation and
write it as ${\rm Irr}_{\rm gen}Q$, if it is clear from the context or is
irrelevant. We denote by $\boxplus$ the normalized parabolic induction for the
general linear group, which we call the isobaric sum. Specifically, we let
${\rm P}_{(n_{1},\dotsm,n_{r})}(F)$ be the standard upper parabolic subgroup
of ${\rm GL}_{n}(F)$ with $n_{1}+\dotsm+n_{r}=n$. Given representations
$\pi_{i}$ of ${\rm GL}_{n_{i}}(F)$ we form an induced representation $\pi$ of
${\rm GL}_{n}(F)$ by normalized parabolic induction;

$\pi:=\mathop{\vphantom{\bigoplus}\mathchoice{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}}\displaylimits_{i=1}^{r}\pi_{i}={\rm
Ind}^{{\rm GL}_{n}(F)}_{{\rm
P}_{(n_{1},\dotsm,n_{r})}(F)}(\pi_{1}\otimes\dotsm\otimes\pi_{r}).$

We caution that the modular character $\delta_{Q}$ in Lapid and Mao [LM16,
LM17] is really $\delta^{-1}_{Q}$ in our notation. Our convention is
consistent with that in Kaplan [Kap15].

Let $F^{n}$ be a $n$-dimensional space of column vectors. We let
$\\{^{t}\xi_{i}\,|\,1\leq i\leq n\\}$ be the standard column basis for
$F^{n}$. We let $1_{n}$ denote the $n\times n$ identity matrix and $0_{m,l}$
the $m\times l$ zero matrix. Let

$w_{n}=\begin{pmatrix}\vspace{-2ex}&&1\\\ &\iddots&\vspace{-2ex}\\\
1&&\end{pmatrix}$

be the $n\times n$ matrix with ones on non-principal diagonal and zeros
elsewhere. Let ${\rm Mat}_{l,m}$ be the vector space of $l\times m$ matrices.
For $x\in{\rm Mat}_{l,m}$, let $\check{x}$ be the twisted transpose matrix in
${\rm Mat}_{m,l}$ given by $\check{x}=w_{m}\prescript{t}{}{x}w_{l}$. Let
$g\mapsto g^{\ast}$ be the outer automorphism of ${\rm GL}_{n}$ given by
$g^{\ast}=w^{-1}_{n}\prescript{t}{}{g^{-1}}w_{n}$. Let $\tau\in F^{\times}$
and we let $G^{\prime}$ denote one of the quasi-split group ${\rm
SO}_{2n+2,\tau}$, the split group ${\rm Sp}_{2n}$ or the split group ${\rm
SO}_{2n+1}$. Let $\mathbb{M}^{\prime}={\rm GL}_{n}$. We let
$Z_{n}=N^{\prime}_{\mathbb{M}^{\prime}}$ denote the group of unipotent upper
triangular matrices of ${\rm GL}_{n}$. Let $\mathfrak{t}={\rm
diag}(1,-1,\dotsm,(-1)^{n-1})\in{\rm GL}_{n}$. Let $K_{{\rm GL}_{n}(F)}$ be
the standard maximal compact subgroup of ${\rm GL}_{n}(F)$. Let
$\widetilde{\rm GL}_{n}$ be the two-fold metaplectic cover of ${\rm GL}_{n}$.
We write element of $\widetilde{\rm GL}_{n}$ as pairs $(g,\varepsilon)$,
$g\in{\rm GL}_{n}$, $\varepsilon=\pm 1$, where the multiplication rule is
given by Banks–Levy–Sepanski’s cocycle (Refer to, for example, [Tak14, §1.1]
for detailed conventions.) When $g\in{\rm GL}_{n}$, we write
$\tilde{g}=(g,1)\in\widetilde{\rm GL}_{n}$. For each subgroup $H\subseteq{\rm
GL}_{n}$, we denote the preimage of $H$ via the canonical projection
$\widetilde{\rm GL}_{n}\rightarrow{\rm GL}_{n}$ by $\tilde{H}$ when the base
field is clear from the context. We let $\delta^{\prime}_{n,2n}$ and
$\delta^{\prime\prime}_{n,2n}$ be $n\times 2n$ matrices such that

$\delta^{\prime}_{n,2n}=\begin{pmatrix}1&\vspace{-.2cm}&&&\\\
&0\;1&\vspace{-.2cm}&&\\\ &&0\;1&\vspace{-.2cm}&\\\
&&&\ddots&\vspace{-.2cm}\\\
&&&&1\;0\end{pmatrix}\quad\text{and}\quad\delta^{\prime\prime}_{n,2n}=\begin{pmatrix}0\;1&\vspace{-.2cm}&&&\\\
&0\;1&\vspace{-.2cm}&&\\\ \ &&\ddots&\vspace{-.2cm}&\\\ &&&1&\vspace{-.2cm}\\\
&&&&0\;1\end{pmatrix}.$

Let $\epsilon_{i,j}\in{\rm Mat}_{l,m}$ be the matrix one at the $(i,j)-$entry
and zero elsewhere.

###  2.2. Global settings

Let $F$ be a number field and $\mathbb{A}$ its ring of adeles. Most of the
previous notation has an obvious meaning in the global context. Some changes
in the global context are as follows. Let $S$ be a finite set of places of
$F$. Let $F_{S}=\prod_{v\in S}F_{v}$ and $\mathcal{O}_{S}$ be the ring of
$S$-integers of $F$. Let $|\cdot|$ denote the standard ideal norm of
$\mathbb{A}^{\times}$. We put
$\mathbb{A}^{1}=\\{a\in\mathbb{A}^{\times}\,|\,|a|=1\\}$ and ${\rm
GL}_{n}(\mathbb{A})^{1}=\\{g\in{\rm GL}_{n}(\mathbb{A})\;|\;\det
g\in\mathbb{A}^{1}\\}$. The global double cover $\widetilde{\rm
GL}_{n}(\mathbb{A})$ of ${\rm GL}_{n}(\mathbb{A})$ is compatible with the
local double covers at each place. (See, for example, [Tak14, §1.2] for
complete regards.)

We write ${\rm Cusp}\,Q$ for the set of equivalence class of irreducible
cuspidal automorphic representations of $Q(\mathbb{A})$. For any automorphic
forms $\varphi$, $\varphi^{\vee}$ on $G^{\prime}(F)\backslash
G^{\prime}(\mathbb{A})$, let
$\langle\varphi,\varphi^{\vee}\rangle_{G^{\prime}}$ be the canonical inner
product on $G^{\prime}(F)\backslash G^{\prime}(\mathbb{A})$ defined by

(2.1) $\langle\varphi,\varphi^{\vee}\rangle_{G^{\prime}}:={\rm
vol}(G^{\prime}(F)\backslash
G^{\prime}(\mathbb{A}))^{-1}\int_{G^{\prime}(F)\backslash
G^{\prime}(\mathbb{A})}\varphi(g)\varphi^{\vee}(g)\,dg,$

provided that the integral converges. Let $N^{\prime}$ be a maximal unipotent
subgroup of $G^{\prime}$ and $\psi_{N^{\prime}}$ a fixed non-degenerate
character of $N^{\prime}(\mathbb{A})$, trivial on $N^{\prime}(F)$. For an
automorphic form $\varphi$ on $G^{\prime}(\mathbb{A})$, we define the
Whittaker–Fourier coefficient $\mathcal{W}^{\psi_{N^{\prime}}}(g,\varphi)$ by

$\mathcal{W}^{\psi_{N^{\prime}}}(g,\varphi):={(\rm
vol}(N^{\prime}(F)\backslash
N^{\prime}(\mathbb{A})))^{-1}\int_{N^{\prime}(F)\backslash
N^{\prime}(\mathbb{A})}\varphi(ng)\psi^{-1}_{N^{\prime}}(n)\,dn,\quad g\in
G^{\prime}(\mathbb{A}).$

We abbreviate $\mathcal{W}^{\psi_{N^{\prime}}}(e,\varphi)$ to
$\mathcal{W}^{\psi_{N^{\prime}}}(\varphi)$. Let $\mathcal{S}(\mathbb{A}^{n})$
be the space of Schwartz-Bruhat functions on $\mathbb{A}^{n}$. Given a finite
set of places $S$ of $F$, we defined in [LM15] a regularized integral

$\int^{\rm st}_{N^{\prime}(F_{S})}f(u)\,du$

for a suitable class of smooth functions $f$ on $N^{\prime}(F_{S})$. If $S$
consists only of non-archimedean places then

$\int^{\rm st}_{N^{\prime}(F_{S})}f(u)\,du=\int_{N^{\prime}_{1}}f(u)\,du$

for any sufficiently large compact open subgroup $N^{\prime}_{1}$ of
$N^{\prime}(F_{S})$, whereas an ad-hoc definition is given in the archimedean
case.

##  3\. The Case for Even Special Orthogonal Groups

###  3.1. Groups, Embeddings, and Weyl Elements

Let $G={\rm SO}_{4n+3}$ be the even split special orthogonal group defined by

$G={\rm SO}_{4n+3}=\\{g\in{\rm
GL}_{4n+3}\,|\,\prescript{t}{}{g}w_{4n+3}g=w_{4n+3}\\}.$

The group $G$ acts on the space $F^{4n+3}$ with the standard basis
$e_{1},\dotsm,e_{2n+1},e_{0},e_{-1-2n},\dotsm,e_{-1}$. We denote by
$\langle\cdot,\cdot\rangle$ the symmetric form defined on the space $F^{4n+3}$
by $\langle u,v\rangle=\prescript{t}{}{u}w_{4n+3}v$ for any $u,v\in F^{4n+3}$.
Let $\tau\in F^{\times}$ and we define $G^{\prime}={\rm
SO}_{2n+2,\tau}\subseteq G$ consisting of elements fixing
$e_{1},\dotsm,e_{n},e_{-n},\dotsm,e_{-1}$ and
$e_{2n+1}+\frac{\tau}{2}e_{-1-2n}$. Let $\mathcal{V}$ be the orthogonal
complement of the space generated by elements
$e_{1},\dotsm,e_{n},e_{-n},\dotsm,e_{-1}$ and
$e_{2n+1}+\frac{\tau}{2}e_{-1-2n}$.

When $\tau=\beta^{2}$ is a square, we use

$\mathcal{B}_{G^{\prime}}=\\{e_{n+1},\dotsm,e_{2n},e_{0}+\frac{1}{\beta}e_{2n+1}-\frac{\beta}{2}e_{-1-2n},\frac{1}{2}e_{0}-\frac{1}{2\beta}e_{2n+1}+\frac{\beta}{4}e_{-1-2n},e_{-2n},\dotsm,e_{-n-1}\\}.$

[GRS11, §9.3] as an ordered basis for $\mathcal{V}$. The group $G^{\prime}$ is
given by

$G^{\prime}={\rm SO}_{2n+2,\tau}=\\{g\in{\rm
GL}_{2n+2}\,|\,\prescript{t}{}{g}w_{2n+2}g=w_{2n+2}\\}.$

The group $G^{\prime}$, viewed as an algebraic group over $F$, is called split
if $\tau\in F^{2}$.

When $\tau$ is not a square, we take
$\mathcal{B}_{G^{\prime}}=\\{e_{n+1},\dotsm,e_{2n},e_{0},e_{2n+1}-\frac{\tau}{2}e_{-1-2n},e_{-2n},\dotsm,e_{-n-1}\\}$
[GRS11, §9.3] as an ordered basis for $\mathcal{V}$. The group $G^{\prime}$ is
given by

$G^{\prime}={\rm SO}_{2n+2,\tau}=\\{g\in{\rm
GL}_{2n+2}\,|\,\prescript{t}{}{g}\begin{pmatrix}[r]&&\vspace{-.1cm}&w_{n}\\\
&1&0&\\\ \vspace{-.1cm}&0&-\tau&\\\
w_{n}&&&\end{pmatrix}g=\begin{pmatrix}[r]&&\vspace{-.1cm}&w_{n}\\\ &1&0&\\\
\vspace{-.1cm}&0&-\tau&\\\ w_{n}&&&\end{pmatrix}\\}.$

The group $G^{\prime}$, viewed as an algebraic group $F$, is called quasi-
split if it is non-split over $F$ and split over a quadratic extension of
$F(\sqrt{\tau})$.

Let $\mathcal{B}^{\prime}_{G^{\prime}}$ be the basis of
$\mathcal{V}\oplus\langle e_{2n+1}+\frac{\tau}{2}e_{-1-2n}\rangle$ obtained by
adding $e_{2n+1}+\frac{\tau}{2}e_{-1-2n}$ to $\mathcal{B}_{G^{\prime}}$ as the
$n+2^{th}$ vector. In coordinates, the embedding $\eta:{\rm
SO}_{2n+2,\tau}\rightarrow{\rm SO}_{4n+3}$ is given by

$\eta:\begin{pmatrix}A_{1}&A_{2}&A_{3}&A_{4}\\\ A_{5}&A_{6}&A_{7}&A_{8}\\\
A_{9}&A_{10}&A_{11}&A_{12}\\\
A_{13}&A_{14}&A_{15}&A_{16}\end{pmatrix}\mapsto{\rm
diag}(1_{n},\mathcal{M}\begin{pmatrix}A_{1}&A_{2}&&A_{3}&A_{4}\\\
A_{5}&A_{6}&\vspace{-.1cm}&A_{7}&A_{8}\\\ \vspace{-.1cm}&&1&&\\\
A_{9}&A_{10}&&A_{11}&A_{12}\\\
A_{13}&A_{14}&&A_{15}&A_{16}\end{pmatrix}\mathcal{M}^{-1},1_{n}),$

where $A_{1}$ and $A_{16}$ are $n\times n$ matrices, $A_{6}$ and $A_{11}$ are
$1\times 1$ matrices, and

$\mathcal{M}={\rm
diag}(1_{n+1},\begin{pmatrix}\;\;\;\frac{1}{\beta}&1&-\frac{1}{2\beta}\\\
\;\;\;1&0&\;\;\;\frac{1}{2}\\\
-\frac{\beta}{2}&\frac{\beta^{2}}{2}&\;\;\;\frac{\beta}{4}\end{pmatrix},1_{n+1}),\quad\text{split
${\rm SO}_{2n+2,\tau}$}$

and

$\mathcal{M}={\rm diag}(1_{n+1},\begin{pmatrix}[r]0&1&1\\\ 1&0&0\\\
0&\frac{\tau}{2}&-\frac{\tau}{2}\\\ \end{pmatrix},1_{n+1}),\quad\text{quasi-
split ${\rm SO}_{2n+2,\tau}$}$

are the change of coordinate matrices from the standard basis to
$\mathcal{B}^{\prime}_{G^{\prime}}$. For example [GRS11, (9.10)], if ${\rm
SO}_{2n+2,\tau}$ is split,

$g=\begin{pmatrix}1_{n}&x_{1}&x_{2}&y\\\ &1&0&-\check{x}_{2}\\\
&&1&-\check{x}_{1}\\\ &&&1_{n}\end{pmatrix},\quad\eta(g)={\rm
diag}(1_{n},\begin{pmatrix}1_{n}&(x_{4},x_{3},-\frac{2}{\tau}x_{4})&y\\\
&1_{3}&\begin{pmatrix}\frac{2}{\tau}\check{x}_{4}\\\ -\check{x}_{3}\\\
-\check{x}_{4}\end{pmatrix}\\\ &&1_{n}\end{pmatrix},1_{n})\in{\rm SO}_{4n+3}$

with $x_{3}$ and $x_{4}$ certain linear combinations of $x_{1}$ and $x_{2}$,
and if ${\rm SO}_{2n+2,\tau}$ is quasi-split, a standard unipotent subgroup
$g$ in ${\rm SO}_{2n+2,\tau}$ and the image $\eta(g)$ in ${\rm SO}_{4n+3}$
with respect to the standard basis $F^{4n+3}$ are

$g=\begin{pmatrix}1_{n}&x_{1}&2x_{2}&y\\\ &1&0&-2\check{x}_{1}\\\
&&1&\frac{2}{\tau}\check{x}_{2}\\\ &&&1_{n}\end{pmatrix},\quad\eta(g)={\rm
diag}(1_{n},\begin{pmatrix}1_{n}&(x_{2},x_{1},-\frac{2}{\tau}x_{2})&y\\\
&1_{3}&\begin{pmatrix}\frac{2}{\tau}\check{x}_{2}\\\ -\check{x}_{1}\\\
-\check{x}_{2}\end{pmatrix}\\\ &&1_{n}\end{pmatrix},1_{n})\in{\rm SO}_{4n+3}.$

Let $P^{\prime}=M^{\prime}\ltimes U^{\prime}$ be the Seigel parabolic subgroup
of $G^{\prime}$ with the Levi part
$M^{\prime}=\varrho^{\prime}(\mathbb{M}^{\prime})$ and the Seigel unipotent
subgroup $U^{\prime}$, where $\varrho^{\prime}:m\mapsto{\rm
diag}(1_{n},m,1_{3},m^{\ast},1_{n})$. Let us set

$\mathfrak{a}_{n}=\\{x\in{\rm Mat}_{n,n}\,|\,\check{x}=-x\\}.$

Then $U^{\prime}=U^{\prime}_{1}\ltimes U^{\prime}_{0}$ with

$U^{\prime}_{0}=\\{{\rm diag}(1_{n},\begin{pmatrix}1_{n}&&u\\\ &1_{3}&\\\
&&1_{n}\end{pmatrix},1_{n})\,|\,u\in\mathfrak{a}_{n}\\}$

and

$U^{\prime}_{1}=\\{{\rm
diag}(1_{n},\begin{pmatrix}1_{n}&(v_{2},v_{1},-\frac{2}{\tau}v_{2})&\\\
&1_{3}&\begin{pmatrix}\frac{2}{\tau}\check{v}_{2}\\\ -\check{v}_{1}\\\
-\check{v}_{2}\end{pmatrix}\\\ &&1_{n}\end{pmatrix},1_{n})\,|\,v_{1},v_{2}\in
F^{n}\\}.$

Let $N^{\prime}$ (resp. $N$) be the standard maximal unipotent subgroup of
$G^{\prime}$ (resp. $G$). Then $N^{\prime}=N^{\prime}_{M^{\prime}}\ltimes
U^{\prime}$ with
$N^{\prime}_{M^{\prime}}=\varrho^{\prime}(N^{\prime}_{\mathbb{M}^{\prime}})$.
Let $V$ be the unipotent radical in $G$ of the standard parabolic subgroup
with Levi ${\rm GL}^{n}_{1}\times{\rm SO}_{2n+3}$:

$V=\\{\begin{pmatrix}v&u&\;\;\;y\\\ &1_{2n+3}&-\check{u}\\\
&&\;\;\;v^{\ast}\end{pmatrix}\in{\rm SO}_{4n+3}\,|\,v\in Z_{n},u\in{\rm
Mat}_{n,2n+3}\\}.$

Let $P^{\diamond}=M^{\diamond}\ltimes U^{\diamond}$ (resp. $P=M\ltimes U$) be
the standard Seigel parabolic subgroup of ${\rm SO}_{2n+3}$ (resp. $G$) with
the Levi part $M^{\diamond}$ (resp. $M$) isomorphic to ${\rm GL}_{n+1}$ (rep.
${\rm GL}_{2n+1}$). We let $\mathbb{M}={\rm GL}_{2n+1}$ and use the
isomorphism $\varrho(h)={\rm diag}(h,1,h^{\ast})$ to identify $\mathbb{M}$
with $M$. Let

$w^{\prime}_{U^{\diamond}}={\rm
diag}(1_{n},\begin{pmatrix}&\vspace{-.1cm}&1_{2n+1}\\\ &1&\vspace{-.1cm}\\\
1_{2n+1}&&\end{pmatrix},1_{n})$

represent the longest $M^{\diamond}$-reduced Weyl element of ${\rm
SO}_{2n+3}$. Let $w_{U}=\begin{pmatrix}&\vspace{-.1cm}&1_{2n+1}\\\
\vspace{-.1cm}&1&\\\ 1_{2n+1}&&\end{pmatrix}$ represent the longest
$M$-reduced Weyl element of ${\rm SO}_{4n+3}$. We write [GRS11, (4.14),(4.15)]

$\gamma=w_{U}(w^{\prime}_{U^{\diamond}})^{-1}=\begin{pmatrix}&1_{n+1}&&&\vspace{-.15cm}\\\
&&\vspace{-.15cm}&&1_{n}\\\ \vspace{-.15cm}&&1&&\\\
1_{n}&&&\vspace{-.15cm}&\\\ &&&1_{n+1}&\end{pmatrix}.$

Let us denote

$V_{\gamma}=V\cap\gamma^{-1}N\gamma=\\{\begin{pmatrix}v&\vspace{-.1cm}&&u&\\\
&1_{n+1}&\vspace{-.1cm}&&-\check{u}\\\ &&1&\vspace{-.1cm}&\\\
&&&1_{n+1}&\vspace{-.15cm}\\\ &&&&v^{\ast}\end{pmatrix}\,|\,v\in
Z_{n},u\in{\rm Mat}_{n,n+1}\\}.$

Let $N_{M}$ (resp. $N_{\mathbb{M}}$) be the standard maximal unipotent
subgroup of $M$ (resp. $\mathbb{M}$). Then $N=N_{M}\ltimes U$ with
$N_{M}=\varrho(N_{\mathbb{M}})$. Let $K=K_{{\rm GL}_{4n+3}(F)}\cap G$ and
$K^{\prime}=K_{{\rm GL}_{2n+2}(F)}\cap G^{\prime}$. Then $K$ (resp.
$K^{\prime}$) is a maximal compact subgroup of G (resp. $G^{\prime}$). Let
$E:=F(\sqrt{\tau})$ be the discriminant field over $F$. Let $\chi_{\tau}$ be
the quadratic character associated to $E/F$ by the class field theory.

###  3.2. Additive Characters

We fix a non-trivial additive character $\psi$ of $F$ and a non-degenerate
character $\psi_{N_{\mathbb{M}}}$ of $N_{\mathbb{M}}$. Let $\psi_{\circ}$ be
non-trivial characters of $F$ given by

$\psi_{\circ}(x)=\psi_{N_{\mathbb{M}}}(1_{2n+1}+x\epsilon_{n+1,n+2})$

for $x\in F$. As explained in [LM17, Remark 6.4] (cf. [LM16, §4.1, §6.1]), the
statements in the sequel will not depend on the choice of
$\psi_{N_{\mathbb{M}}}$. For convenience, we set

(3.1) $\psi_{N_{\mathbb{M}}}(u)=\psi(u_{1,2}+\dotsm+u_{2n,2n+1}).$

Thus $\psi_{\circ}=\psi$. The character $\psi$ will determine characters of
several other unipotent groups. Let $\psi_{N_{M}}$ be the non-degenerate
character of $N_{M}$ given by
$\psi_{N_{M}}(\varrho(u))=\psi_{N_{\mathbb{M}}}(u)$. We denote by $\psi_{N}$
the degenerate character on $N$ given by $\psi_{N}(uv)=\psi_{N_{M}}(u)$ for
any $u\in N_{M}$ and $v\in U$. Let $\psi_{N^{\prime}_{M^{\prime}}}$ be the
non-degenerate character of $N^{\prime}_{M^{\prime}}$ such that
$\psi_{N^{\prime}_{M^{\prime}}}(\varrho(u))=\psi_{N^{\prime}_{\mathbb{M^{\prime}}}}(u)$.
With the choice of $\psi_{N_{\mathbb{M}}}$, we have

$\psi_{N^{\prime}_{\mathbb{M}^{\prime}}}(u^{\prime})=\psi(u^{\prime}_{1,2}+\dotsm+u^{\prime}_{n-1,n}).$

Let $\psi_{U^{\prime}}$ be the character on $U^{\prime}$ given by
$\psi_{U^{\prime}}(u)=\psi(u_{2n,2n+2})$. Then the non-degenerate character
$\psi_{N^{\prime}}$ is given by

$\psi_{N^{\prime}}(uv)=\psi_{N^{\prime}_{M^{\prime}}}(u)\psi_{U^{\prime}}(v)=\psi(u_{n+1,n+2}+\dotsm+u_{2n-1,2n}+v_{2n,2n+2})$

with $u\in N^{\prime}_{M^{\prime}}$ and $v\in U^{\prime}$. Our choice is
consistent with [GRS11, (9.11)] as we have specified $\lambda=1$ and $\mu=0$.
An element in $V$ can be written as $vu$ where $u$ fixes $e_{1},\dotsm,e_{n}$,
$v$ fixes $e_{n+1},e_{n+2},\dotsm,e_{-n-1}$ and we set [GRS11, (3.48)]

$\psi_{V}(vu)=\psi^{-1}(v_{1,2}+\dotsm+v_{n-1,n})\psi^{-1}(v_{n,n+1}+\frac{\tau}{2}v_{n,n+3}).$

###  3.3. Whittaker Models and the Intertwining Operator

Let $\pi$ be an irreducible generic representation of $\mathbb{M}$ with its
Whittaker model $\mathbb{W}^{\psi_{N_{\mathbb{M}}}}(\pi)$ with respect to the
character $\psi_{N_{\mathbb{M}}}$. Similarly we use the notation
$\mathbb{W}^{\psi^{-1}_{N_{\mathbb{M}}}}$, $\mathbb{W}^{\psi_{N_{M}}}$,
$\mathbb{W}^{\psi^{-1}_{N_{M}}}$.

Given $g\in G$, we define $\nu(g)$ by $\nu(u\varrho(m)k)=|\det m|$ for any
$u\in U$, $m\in\mathbb{M}$, and $k\in K$ via the Iwasawa decomposition. For
any $f\in C^{\infty}(G)$ and $s\in\mathbb{C}$, define
$f_{s}(g)=f(g)\nu(g)^{s}$, $g\in G$. Let ${\rm
Ind}(\mathbb{W}^{\psi_{N_{M}}}(\pi))$ be the space of $G$-smooth left
$U$-invariant function $W:G\rightarrow\mathbb{C}$ such that for all $g\in G$,
the function $m\mapsto\delta^{\frac{1}{2}}_{P}(m)W(mg)$ on $M$ belongs to
$\mathbb{W}^{\psi_{N_{M}}}(\pi)$. Similarly we define ${\rm
Ind}(\mathbb{W}^{\psi^{-1}_{N_{M}}}(\pi))$, ${\rm
Ind}(\mathbb{W}^{\psi_{N_{\mathbb{M}}}}(\pi))$, ${\rm
Ind}(\mathbb{W}^{\psi^{-1}_{N_{\mathbb{M}}}}(\pi))$. For any $s\in\mathbb{C}$,
we have a representation $I(s,g)$ on the space ${\rm
Ind}(s,\mathbb{W}^{\psi_{N_{M}}}(\pi))$ given by
$(I(s,g)W)_{s})(x)=W_{s}(xg)$, $x,g\in G$.

We define the intertwining operator [Kap15, §3.2];

$M(s,\pi):{\rm Ind}(s,\mathbb{W}^{\psi_{N_{M}}}(\pi))\rightarrow{\rm
Ind}(-s,\mathbb{W}^{\psi_{N_{M}}}(\pi^{\vee}))$

by (the analytic continuation of)

$M(s,\pi)W(g)=\nu(g)^{s}\int_{U}W_{s}(\varrho(\mathfrak{t})w_{U}ug)\,du,$

where $\mathfrak{t}\in{\rm GL}_{2n+1}$ is introduced in order to preserve the
character $\psi_{N_{M}}$.

In case where $F$ is $p$-adic and $\pi$ and $\psi$ are unramified, and there
exist (necessarily unique) $K$-fixed elements $W^{\circ}\in{\rm
Ind}(\mathbb{W}^{\psi_{N_{M}}}(\pi))$ and $\check{W}^{\circ}\in{\rm
Ind}(\mathbb{W}^{\psi_{N_{M}}}(\pi^{\vee}))$ such that
$W^{\circ}(e)=\check{W}^{\circ}(e)=1$, then we have [Sha88, §2] (assuming
${\rm vol}(U\cap K)=1$)

(3.2) $M(s,\pi)W^{\circ}=\frac{L(2s,\pi,{\rm Sym}^{2})}{L(2s+1,\pi,{\rm
Sym}^{2})}\check{W}^{\circ},$

The following result is an analogue of [LM16, Proposition 2.1] and [LM17,
Proposition 4.1].

######  Proposition 3.1.

Suppose that $\pi\in{\rm Irr}_{\rm gen}\,\mathbb{M}$ such that
$\pi\cong\pi^{\vee}$. Then $M(s,\pi)$ is holomorphic at $s=\frac{1}{2}$.

###### Proof.

We write $\pi=\sigma_{1}\boxplus\dotsm\boxplus\sigma_{k}$ where
$\sigma_{1},\dotsm,\sigma_{k}$ are essentially square-integrable. By
decomposing $M(s,\pi)$ into a product of co-rank one intertwining operators,
it is enough to show that the following intertwining operators

  1. $(\mathrm{i})$

$\sigma_{i}|\cdot|^{s}\boxplus\sigma_{j}^{\vee}|\cdot|^{-s}\rightarrow\sigma_{j}^{\vee}|\cdot|^{-s}\boxplus\sigma_{i}|\cdot|^{s},\quad
i,j=1,2,\dotsm,k,i\neq j$,

  2. $(\mathrm{ii})$

$M(s,\sigma_{i}),\quad i=1,2,\dotsm,k$,

are holomorphic at $s=\frac{1}{2}$. The holomorphy of $(\mathrm{i})$ is a
consequence of [LM17, Lemma 4.2] and the irreducibility of
$\sigma_{i}\boxplus\sigma^{\vee}_{j}$. If $F$ is an archimedean local field,
the holomorphy of $(\mathrm{ii})$ follows from [LM17, Lemma 4.3] and the
irreducibility of $\sigma_{i}\boxplus\sigma_{i}^{\vee}$. Under the condition
of [LM17, Lemma 4.3], we have the stronger conclusion from the very recent
paper [Luo24, §2.4 Main Theorem 1] that $L(2s,\sigma_{i},{\rm
Sym}^{2})^{-1}M(s,\sigma_{i})$ is holomorphic for the $p$-adic case. ∎

###  3.4. Period and Distinction

####  3.4.1. Global period

Let $\vartheta$ be the space of the global exceptional representation $\Theta$
of $\widetilde{\rm GL}_{n}(\mathbb{A})$, which is the Langlands quotient of a
certain induced representation of $\widetilde{\rm GL}_{n}(\mathbb{A})$
inherited from the metaplectic preimage $\tilde{B}_{n}(\mathbb{A})$ of the
Borel subgroup $B_{n}(\mathbb{A})$. The best reference for the construction of
this representation is [Tak14, §2.1]. Let $F$ be a number field. We say that
$\pi\in{\rm Cusp}\,\mathbb{M}$ is of orthogonal symplectic type if

$\int_{{\rm GL}_{n}(F)\backslash{\rm
GL}_{n}(\mathbb{A})^{1}}\varphi(g)\Theta(g)\overline{\Theta^{\prime}}(g)\,dg\neq
0$

for some $\varphi$ in the space of $\pi$, and $\Theta,\Theta^{\prime}$ in the
space of $\vartheta$. The following characterization is due to Bump–Ginzburg
[BG92], Takeda [Tak14], and Kaplan–Yamana [KY17, Theorem A].

######  Proposition 3.2.

Let $\pi\in{\rm Cusp}\,\mathbb{M}$. Then $\pi$ is of orthogonal symplectic
type if and only if $L^{S}(s,\pi,{\rm Sym}^{2})$ has a pole at $s=1$.

We let ${\rm OScusp}_{k}\,\mathbb{M}$ be the set of automorphic
representations $\pi$ of the form $\pi_{1}\boxplus\dotsm\boxplus\pi_{k}$,
where $\pi_{i}\in{\rm Cusp}\,{\rm GL}_{n_{i}}(F)$, $i=1,\dotsm,k$ are pairwise
inequivalent (that is, $\pi_{i}\not\cong\pi_{j}$ if $i\neq j$) and of
orthogonal symplectic type with $n_{1}+\dotsm+n_{k}=2n+1$ and $n_{i}>1$ for
all $i$. In particular $\pi^{\vee}_{i}\cong\pi_{i}$ and the central character
$\omega_{\pi}$ of $\pi$ is trivial on $(\mathbb{A}^{\times})^{2}$. We let
${\rm OScusp}\,\mathbb{M}=\cup_{k}{\rm OScusp}_{k}\,\mathbb{M}$.

####  3.4.2. Local distinction

Let $F$ be a local field. If $F$ is $p$-adic, we say that $\pi\in{\rm
Irr}\,{\rm GL}_{n}(F)$ is of orthogonal symplectic type if there is an
exceptional representation $\theta$ of $\widetilde{\rm GL}_{n}(F)$ (defined as
in Bump–Ginzburg [BG92] and Takeda [Tak14]) such that $\pi$ admits a non-
trivial continuous functionals $L$ on the space
$\pi\times\theta\times\theta^{\vee}$. In particular, the central character
$\omega_{\pi}$ of $\pi$ is trivial on $(F^{\times})^{2}$ (cf. [Kap16,
Corollary 4.3]).

Let $P_{\circ}\subseteq\mathbb{M}$ be the standard parabolic subgroup in
$\mathbb{M}$ whose Levi subgroup is isomorphic to $\prod_{i=1}^{k}{\rm
GL}_{n_{i}}$ with $n_{1}+\dotsm+n_{k}=2n+1$ and $n_{i}>1$ for all $i$. If
$F=\mathbb{R}$ or $\mathbb{C}$, we let $\pi={\rm
Ind}^{\mathbb{M}}_{P_{\circ}}\otimes_{i=1}^{k}\pi_{i}$ be a self-dual
representation of $\mathbb{M}$ (i.e., $\pi\cong\pi^{\vee}$), where $\pi_{i}$,
$i=1,\dotsm,k$ are pairwise inequivalent (i.e., $\pi_{i}\not\cong\pi_{j}$ if
$i\neq j$) and each $\pi_{i}$ owns a non-trivial continuous functionals $L$ on
the space $\pi\times\theta\times\theta^{\vee}$ with $n_{1}+\dotsm+n_{k}=2n+1$
and $n_{i}>1$ for all $i$. Such a representation $\pi={\rm
Ind}^{\mathbb{M}}_{P_{\circ}}\otimes_{i=1}^{k}\pi_{i}$ is called a
representation of orthogonal symplectic type.

For $F$ a local field, we write ${\rm Irr}_{\rm os}\,{\rm GL}_{n}(F)$ for the
set of irreducible representations of orthogonal symplectic type. Clearly, if
$\pi$ is of orthogonal symplectic type in the global setting, then all its
local components $\pi_{v}$ are irreducible and of orthogonal symplectic type
as well.

######  Lemma 3.3.

Assume that $F$ is a $p$-adic field.

  1. $(\mathrm{1})$

([Kap17, Corollary 4.19]) Suppose that $\pi\in{\rm Irr}_{\rm os}\,{\rm
GL}_{n}(F)$. Then $\pi^{\vee}\cong\pi$.

  2. $(\mathrm{2})$

([Kap17, Theorem A-(1)]) Suppose that $\pi\in{\rm Irr}\,{\rm GL}_{n}(F)$ is
square integrable. Then $\pi\in{\rm Irr}_{\rm os}\,{\rm GL}_{n}(F)$ if and
only if $L(s,\pi,{\rm Sym}^{2})$ has a pole at $s=0$.

  3. $(\mathrm{3})$

([Kap16, Theorem 3]) Suppose that $\pi_{i}\in{\rm Irr}_{\rm os}\,{\rm
GL}_{n_{i}}(F)$, $i=1,2$ and the parabolic induction $\pi_{1}\boxplus\pi_{2}$
is irreducible. Then $\pi_{1}\boxplus\pi_{2}\in{\rm Irr}_{\rm os}\,{\rm
GL}_{n_{1}+n_{2}}(F)$.

For the sake of completeness, we recall the following classification theorem,
due to Kaplan, of the set ${\rm Irr}_{\rm gen,os}\,{\rm GL}_{n}(F)$ of generic
representations of orthogonal symplectic types.

######  Proposition 3.4.

([Kap17, Theorem A-(2)]) Assume that $F$ is a $p$-adic field. Then the set
${\rm Irr}_{\rm gen,os}\,{\rm GL}_{n}(F)$ consists of the irreducible
representations of the form

$\pi=(\sigma_{1}\boxplus\sigma^{\vee}_{1})\boxplus\dotsm\boxplus(\sigma_{k}\boxplus\sigma^{\vee}_{k})\boxplus\tau_{1}\boxplus\dotsm\boxplus\tau_{\ell}$

where $\sigma_{1},\dotsm,\sigma_{k}$ are essentially square-integrable and
$\tau_{1},\dotsm,\tau_{\ell}$ are square-integrable of orthogonal symplectic
type (i.e., $L(0,\tau_{i},{\rm Sym}^{2})=\infty$ for all $i=1,\dotsm,\ell$).

###  3.5. Descent map

####  3.5.1. Global descent

We now go to the global setting. For any automorphic form $\varphi$ on
$G(\mathbb{A})$, let ${\rm GG}(\varphi)$ be the Gelfand–Graev coefficient (a
function on $G^{\prime}(F)\backslash G^{\prime}(\mathbb{A})$)

${\rm GG}(\varphi)(g)=\int_{V(F)\backslash
V(\mathbb{A})}\varphi(vg)\psi_{V}^{-1}(v)\,dv,\quad g\in
G^{\prime}(\mathbb{A}).$

The automorphic representation $\pi$ of $\mathbb{M}(\mathbb{A})$ in ${\rm
OScusp}\,\mathbb{M}$ is realized on the space of Eisenstein series induced
from $\pi_{1}\otimes\dotsm\otimes\pi_{k}$. We view $\pi$ as a representation
of $M(\mathbb{A})$ via $\varrho$. Let $\mathcal{A}(\pi)$ be the space of
functions $\varphi:M(F)U(\mathbb{A})\backslash
G(\mathbb{A})\rightarrow\mathbb{C}$ such that
$m\mapsto\delta^{\frac{1}{2}}_{P}(m)\varphi(mg)$, $m\in M(\mathbb{A})$ belongs
to the space of $\pi$ for all $g\in G(\mathbb{A})$. Let
$\varphi_{s}(g)=\nu(g)^{s}\varphi(g)$. We define Eisenstein series

$\mathcal{E}(s,\varphi)=\sum_{\gamma\in P(F)\backslash G(F)}\varphi_{s}(\gamma
g),\quad\varphi\in\mathcal{A}(\pi).$

The space of $\pi$ is invariant under the conjugation by $w_{U}$. Then we
define intertwining operator
$M(s,\pi):\mathcal{A}(\pi)\rightarrow\mathcal{A}(\pi)$ given by

$M(s,\pi)\varphi(g)=\nu(g)^{s}\int_{U(\mathbb{A})}\varphi_{s}(w_{U}ug)\,du.$

Both $\mathcal{E}(s,\varphi)$ and $M(s,\pi)$ converge absolutely for ${\rm
Re}(s)\gg 0$ and extend meromorphically to $\mathbb{C}$. It follows from
[GRS11, Theorem 2.1] that both $\mathcal{E}(s,\varphi)$ and $M(s,\pi)$ have a
pole of order $k$ at $s=\frac{1}{2}$. Let

$\mathcal{E}_{-k}\varphi=\lim_{s\rightarrow\frac{1}{2}}\left(s-\frac{1}{2}\right)^{k}\mathcal{E}(s,\varphi)$

and

$M_{-k}=\lim_{s\rightarrow\frac{1}{2}}\left(s-\frac{1}{2}\right)^{k}M(s,\pi)$

be the leading coefficient in the Laurent series expansion at $s=\frac{1}{2}$.
The constant term

$\mathcal{E}^{U}_{-k}\varphi(g)=\int_{U(F)\backslash
U(\mathbb{A})}\mathcal{E}_{-k}\varphi(ug)\,du$

of $\mathcal{E}_{-k}\varphi$ along $U$ is given by

$\mathcal{E}^{U}_{-k}\varphi(g)=(M_{-k}(\varphi)(g))_{-\frac{1}{2}}.$

The global descent of $\pi$ (with respect to $\psi_{N_{\mathbb{M}}}$) is the
space $\mathcal{D}_{\psi}(\pi)$ generated by ${\rm
GG}(\mathcal{E}_{-k}\varphi)$ with $\varphi\in\mathcal{A}(\pi)$. It is known
from [GRS11, Theorem 3.1] that $\mathcal{D}_{\psi}(\pi)$ is multiplicity free.
By [GRS11, Theorem 9.6], $\mathcal{D}_{\psi}(\pi)$ is a non-trivial cuspidal
(and globally generic) automorphic representation of $G^{\prime}$. We expect
it to be irreducible. This result will be the analogue of [GJS12, Theorem 2.3]
in metaplectic group case and [Mor18, Theorem 1.1] in even unitary group case.
It is likely that the same methods of [GJS12, Theorem 2.3] and [Mor18, Theorem
1.1] no doubt work in the case at hand. Unfortunately, to the best of our
knowledge this has not been carried out in the literature. Since this is
beyond the scope of the current paper we simply make it a working hypothesis.

######  Working Hypothesis 3.5.

When $\pi\in{\rm OScusp}\,\mathbb{M}$, $\mathcal{D}_{\psi}(\pi)$ is
irreducible.

The image of the weak lift is then a consequence of [GRS11, Theorem 11.2]. In
particular, [GRS11, Theorem 9.6 (2)] describes the image for unramified
places.

######  Proposition 3.6.

With the above Working Hypothesis 3.5,
$\pi\mapsto\sigma=\mathcal{D}_{\psi}(\pi)$ defines a bijection between ${\rm
OScusp}\,\mathbb{M}$ and ${\rm Cusp}_{\psi_{N^{\prime}}}\,G^{\prime}$, the set
of $\psi_{N^{\prime}}$-generic cuspidal automorphic representation of
$G^{\prime}$. Moreover $\sigma^{\vee}$ weekly lifts to
$\pi\boxplus\omega_{\pi}\chi_{\tau}$.

####  3.5.2. Local descent

We go back to the local setup. Suppose now that $F$ is local fields. Let
$\pi\in{\rm Irr}_{\rm gen}\,M$. For any $W\in{\rm
Ind}(\mathbb{W}^{\psi_{N_{M}}}(\pi))$ and $s\in\mathbb{C}$, we define a
function on $G^{\prime}$:

$A^{\psi}(s,g,W)=\int_{V_{\gamma}\backslash V}W_{s}((-1)^{n}\gamma
vg)\psi_{V}^{-1}(v)\,dv,\quad g\in G^{\prime}.$

The basic properties of $A^{\psi}$ are summarized in the following
proposition. Its proof is identical to [LM16, Lemmata 4.5 and 4.9] and [LM17,
Lemmata 5.1 and 8.1], and will be omitted.

######  Proposition 3.7.

Suppose that $\pi\in{\rm Irr}_{\rm gen}\,M$. Let $W\in{\rm
Ind}(\mathbb{W}^{\psi_{N_{M}}}(\pi))$ and $s\in\mathbb{C}$. Then

  1. $(\mathrm{1})$

The integral $A^{\psi}(s,g,W)$ is well-defined and absolutely convergent
uniformly for $s\in\mathbb{C}$ and $g\in G^{\prime}$ in compact sets.
Therefore $A^{\psi}(s,g,W)$ is entire in $s$ and smooth in $g$. In the non-
archimedean case the integrand is compactly supported on $V_{\gamma}\backslash
V$.

  2. $(\mathrm{2})$

For any $W\in{\rm Ind}(\mathbb{W}^{\psi_{N_{M}}}(\pi))$ and $s\in\mathbb{C}$,
the function $g\mapsto A^{\psi}(s,g,W)$ is smooth and
$(N^{\prime},\psi_{N^{\prime}})$-equivariant.

  3. $(\mathrm{3})$

For any $g,x\in G^{\prime}$, we have

(3.3) $A^{\psi}(s,g,I(s,x)W)=A^{\psi}(s,gx,W).$

  4. $(\mathrm{4})$

Suppose that $\pi$ is unramified, $\psi$ is unramified, and $W^{\circ}\in{\rm
Ind}(\mathbb{W}^{\psi_{N_{M}}}(\pi))$ is the standard unramified vector. Then
$A^{\psi}(s,e,W^{\circ})\equiv 1$ (assuming ${\rm vol}(V\cap K)={\rm
vol}(V_{\gamma}\cap K)=1$).

Let $\pi\in{\rm Irr}_{\rm gen}\,\mathbb{M}$, regarded also as a representation
of $M$ via $\varrho$. By the same argument as in [GRS99, Theorem in §1.3] in
conjunction with [LM17, Remark 4.13], for any non-zero subrepresentation
$\pi^{\prime}$ of ${\rm Ind}(\mathbb{W}^{\psi_{N_{M}}}(\pi))$ there exists
$W^{\prime}\in\pi^{\prime}$ such that $A^{\psi}(s,\cdot,W^{\prime})\not\equiv
0$. Upon twisting $\pi$, we deduce that for any given $s_{0}\in\mathbb{C}$,
there exists $W^{\prime}\in\pi^{\prime}$ such that
$A^{\psi}(s_{0},\cdot,W^{\prime})\not\equiv 0$.

Assume now that $\pi\in{\rm Irr}_{\rm gen,os}\,\mathbb{M}$. By Proposition
3.1, $M(s,\pi)$ is holomorphic at $s=\frac{1}{2}$. The local descent of $\pi$
is the space $\mathcal{D}_{\psi}(\pi)$ of Whittaker functions on $G^{\prime}$
generated by
$A^{\psi}\left(-\frac{1}{2},\cdot,M\left(\frac{1}{2},\pi\right)W\right)$ with
$W\in{\rm Ind}(\mathbb{W}^{\psi_{N_{M}}}(\pi))$. By the above observation,
$\mathcal{D}_{\psi}(\pi)\neq 0$.

Let $\pi^{\prime}$ be the image of ${\rm Ind}(\mathbb{W}^{\psi_{N_{M}}}(\pi))$
under $M\left(\frac{1}{2},\pi\right)$. In virtue of (3.3), the space
$\mathcal{D}_{\psi}(\pi)$ is canonically a quotient of the $G^{\prime}$-module
$J_{V}(\pi^{\prime})$ of the $V$-coinvariant of $\pi^{\prime}$. In this
perspective, we view $J_{V}(\pi^{\prime})$ as the abstract descent, whereas
$\mathcal{D}_{\psi}(\pi)$ as the explicit descent.

####  3.5.3. Whittaker function of descent

We let $\hat{\psi}_{N}$ be a degenerate character on $N$ that is trivial on
$U$ and for $u\in N_{\mathbb{M}}$

$\hat{\psi}_{N}(\varrho(u)):=\psi(u_{1,2}+\dotsm+u_{n,n+1}-u_{n+1,n+2}-\dotsm-
u_{2n,2n+1}).$

to be consistent with the additive character in [GRS11, (8.14)]. Thus
$\psi_{\circ}(x)=\psi(x)$. By [LM17, Remark 6.4] (cf. [LM16, §4.1, §6.1]), we
will verify Theorem 3.8 for a specific choice of $\psi_{N_{\mathbb{M}}}$ given
by

$\psi_{N_{\mathbb{M}}}(u)=\psi(u_{1,2}+\dotsm+u_{n,n+1}-\frac{\tau}{2}u_{n+1,n+2}-u_{n+2,n+3}-\dotsm-
u_{2n,2n+1}).$

Henceforth $\psi_{\circ}(x)=\psi(x)$. Let

$\delta=\left(\begin{array}[]{cc|c|cc}\delta^{\prime}_{n,2n}&0&0&0&0_{n\times
2n}\\\ 0&1&0&0&0\\\ 0_{n\times 2n}&0&0&0&\delta^{\prime}_{n,2n}\\\ \hline\cr
0&0&1&0&0\\\ \hline\cr\delta^{\prime\prime}_{n,2n}&0&0&0&0_{n\times 2n}\\\
0&0&0&1&0\\\ 0_{n\times 2n}&0&0&0&\delta^{\prime\prime}_{n\times
2n}\end{array}\right).$

We let $\kappa=\varrho(\kappa^{\prime})$, where $\kappa^{\prime}$ is the Weyl
element of $\mathbb{M}$ such that $\kappa^{\prime}_{2i,i}$, $i=1,\dotsm,n$,
$\kappa^{\prime}_{2i-1,n+i}=1$, $i=1,\dotsm,n+1$, and all other entires are
zero. Let $\epsilon={\rm diag}(A,\dotsm,A,1_{3},A^{\ast},\dotsm A^{\ast})$,
where $A=\begin{pmatrix}[r]\frac{1}{2}&1\\\
-\frac{\tau}{4}&\frac{\tau}{2}\end{pmatrix}$ (cf. [GRS11, (9.57)]). Here $A$
is repeated $n$ times. Writing

$A=\begin{pmatrix}1&\frac{2}{\tau}\\\ &1\end{pmatrix}\begin{pmatrix}1&\\\
&\frac{\tau}{2}\end{pmatrix}\begin{pmatrix}[r]1&\\\
-\frac{1}{2}&1\end{pmatrix},$

we obtain $\epsilon=\epsilon_{U}\epsilon_{T}\epsilon_{\overline{U}}$, where

$\begin{split}&\epsilon_{U}={\rm diag}(\begin{pmatrix}1&\frac{2}{\tau}\\\
&1\end{pmatrix},\dotsm,\begin{pmatrix}1&\frac{2}{\tau}\\\
&1\end{pmatrix},1_{3},\begin{pmatrix}[r]1&-\frac{2}{\tau}\\\
&1\end{pmatrix},\dotsm,\begin{pmatrix}[r]1&-\frac{2}{\tau}\\\
&1\end{pmatrix}),\\\ &\epsilon_{\overline{U}}={\rm
diag}(\begin{pmatrix}[r]1&\\\
-\frac{1}{2}&1\end{pmatrix},\dotsm,\begin{pmatrix}[r]1&\\\
-\frac{1}{2}&1\end{pmatrix},1_{3},\begin{pmatrix}1&\\\
\frac{1}{2}&1\end{pmatrix},\dotsm,\begin{pmatrix}1&\\\
\frac{1}{2}&1\end{pmatrix}),\\\ \end{split}$

and

$\epsilon_{T}=\left(\begin{pmatrix}1&\\\
&\frac{\tau}{2}\end{pmatrix},\dotsm\begin{pmatrix}1&\\\
&\frac{\tau}{2}\end{pmatrix},1_{3},\begin{pmatrix}\frac{2}{\tau}&\\\
&1\end{pmatrix},\dotsm\begin{pmatrix}\frac{2}{\tau}&\\\
&1\end{pmatrix}\right).$

Let $X$ be the subspace of $\mathfrak{a}_{2n+1}$ consisting of the strictly
upper triangular matrices. We regard $\mathbb{A}^{n}$ as subspace of
$\mathbb{A}^{2n+1}$ by the embedding

${}^{t}(a_{1},\dotsm,a_{n})\in\mathbb{A}^{n}\mapsto
a:={{}^{t}(}a_{1},\dotsm,a_{n},0,\dotsm,0),\quad(\text{\rm namely $a_{i}=0$
for $i>n$}).$

We let

$\overline{\ell}(a,x)=\left(\begin{array}[]{c|c|c}1_{2n+1}&&\\\
\hline\cr-\check{a}&1&\\\ \hline\cr x&a&1_{2n+1}\end{array}\right)$

for $x\in\mathfrak{a}_{2n+1}$ and
$a\in\mathbb{A}^{n}\subseteq\mathbb{A}^{2n+1}$. Let $Y$ be the subgroup of
$N_{M}$ consisting of the matrices of the form
$\varrho\left(\begin{pmatrix}1_{n}&y\\\ &1_{n+1}\end{pmatrix}\right)$ where
$y$ is lower triangular, (namely $y_{i,j}=0$ if $j>i$).

######  Theorem 3.8.

$($Reformulation of [GRS11, Theorem 9.6, part (1)]$)$ Let $\pi\in{\rm
OScusp}\,\mathbb{M}$ and $\varphi\in\mathcal{A}(\pi)$. Then for any $g\in
G^{\prime}$ we have

$\mathcal{W}^{\psi_{N^{\prime}}}(g,{\rm
GG}(\mathcal{E}_{-k}\varphi))=\int_{V_{\gamma}(\mathbb{A})\backslash
V(\mathbb{A})}\mathcal{W}^{\psi_{N}}((-1)^{n}\gamma
vg,\mathcal{E}_{-k}\varphi)\psi^{-1}_{V}(v)\,dv$

where the integral is absolutely convergent.

###### Proof.

We use Tamagawa measure in the proof. It is enough to prove the required
identity for $g=e$. The expression for $\mathcal{W}^{\psi_{N^{\prime}}}(e,{\rm
GG}(\mathcal{E}_{-k}\varphi))$ in [GRS11, Theorem 9.6, part (1)] is (with
$\alpha=\tau$ and $\lambda=1$ in their notation):

$\int_{Y(\mathbb{A})}\left(\int_{\mathbb{A}^{n}}\left(\int_{X(\mathbb{A})}\mathcal{W}^{\hat{\psi}_{N}}((-1)^{n}\overline{\ell}(a,x)\delta\epsilon\kappa
y,\mathcal{E}_{-k}\varphi)\,dx\right)da\right)dy.$

We note from [GRS11, §8.2, (8.15)] that
$\delta\epsilon_{U}\delta^{-1}=\begin{pmatrix}1_{2n+1}&\vspace{-.1cm}&-\epsilon_{\delta}\\\
&1&\vspace{-.1cm}\\\ &&1_{2n+1}\end{pmatrix}$, where $\epsilon_{\delta}={\rm
diag}(-\frac{2}{\tau}\cdot 1_{n},0,\frac{2}{\tau}\cdot 1_{n})$. For any $x\in
X$ and $a\in\mathbb{A}^{n}\subseteq\mathbb{A}^{2n+1}$, we proceed to write

$\left(\begin{array}[]{c|c|c}1_{2n+1}&&\\\ \hline\cr-\check{a}&1&\\\ \hline\cr
x&a&1_{2n+1}\end{array}\right)\left(\begin{array}[]{c|c|c}1_{2n+1}&&-\epsilon_{\delta}\\\
\hline\cr&1&\\\ \hline\cr&&1_{2n+1}\end{array}\right)\\\
=\left(\begin{array}[]{c|c|c}(1_{2n+1}-x\epsilon_{\delta})^{\ast}&\epsilon_{\delta}(1_{2n+1}-x\epsilon_{\delta})^{-1}a&-\epsilon_{\delta}\\\
\hline\cr&1&\check{a}\epsilon_{\delta}\\\
\hline\cr&&1_{2n+1}-x\epsilon_{\delta}\end{array}\right)\left(\begin{array}[]{c|c|c}1_{2n+1}&&\\\
\hline\cr-\check{a}^{\prime}&1&\\\ \hline\cr
x^{\prime}&a^{\prime}&1_{2n+1}\end{array}\right),$

where
$x^{\prime}=(1_{2n+1}-x\epsilon_{\delta})^{-1}x=x+x\epsilon_{\delta}x+\dotsm+(x\epsilon_{\delta})^{2n}x$
and $a^{\prime}=(1_{2n+1}-x\epsilon_{\delta})^{-1}a$. (Note that
$(1_{2n+1}-x\epsilon_{\delta})^{\ast}=(1_{2n+1}-\epsilon_{\delta}x)^{-1}$.)
After changing the variables $x\mapsto(1_{2n+1}-x\epsilon_{\delta})x$ and
$a\mapsto(1_{2n+1}-x\epsilon_{\delta})a$, we get

$\int_{Y(\mathbb{A})}\left(\int_{\mathbb{A}^{n}}\left(\int_{X(\mathbb{A})}\hat{\psi}_{N}(\varrho((1_{2n+1}-x\epsilon_{\delta})^{\ast}))\mathcal{W}^{\hat{\psi}_{N}}((-1)^{n}\overline{\ell}(a,x)\delta\epsilon_{T}\epsilon_{\overline{U}}\kappa
y,\mathcal{E}_{-k}\varphi)\,dx\right)da\right)dy.$

Since $x_{2n+1-k,2n+2-k}=-x_{k,k+1}$, we have

$\hat{\psi}_{N}(\varrho((1_{2n+1}-x\epsilon_{\delta})^{\ast}))=\hat{\psi}_{N}(\varrho((1_{2n+1}-\epsilon_{\delta}x)^{-1}))=\psi^{-1}\left(\frac{2}{\tau}x_{n,n+1}\right)$

from which it follows that

$\int_{Y(\mathbb{A})}\left(\int_{\mathbb{A}^{n}}\left(\int_{X(\mathbb{A})}\psi^{-1}\left(\frac{2}{\tau}x_{n,n+1}\right)\mathcal{W}^{\hat{\psi}_{N}}((-1)^{n}\overline{\ell}(a,x)\delta\epsilon_{T}\epsilon_{\overline{U}}\kappa
y,\mathcal{E}_{-k}\varphi)\,dx\right)da\right)dy.$

Furthermore, $\delta\epsilon_{T}\delta^{-1}={\rm
diag}(1_{n+1},\frac{2}{\tau}1_{n},1,\frac{\tau}{2}1_{n},1_{n+1})$. This
element conjugates $\hat{\psi}_{N}$ to $\psi_{N}$ and
$\psi(\frac{2}{\tau}x_{n,n+1})$ to $\psi(x_{n,n+1})$, so by conjugating we
obtain

$\int_{Y(\mathbb{A})}\left(\int_{\mathbb{A}^{n}}\left(\int_{X(\mathbb{A})}\psi^{-1}(x_{n,n+1})\mathcal{W}^{\psi_{N}}((-1)^{n}\overline{\ell}(a,x)\delta\epsilon_{\overline{U}}\kappa
y,\mathcal{E}_{-k}\varphi)\,dx\right)da\right)dy.$

With the direct computation just as in [LM16, Proof of Proposition 8.2]), we
find $\kappa^{-1}\epsilon_{\overline{U}}\kappa\in Y$ and
$\delta\kappa=\gamma$. Upon making change of variables
$y\mapsto(\kappa^{-1}\epsilon_{\overline{U}}\kappa)^{-1}y$, we obtain that the
above equals

$\int_{Y(\mathbb{A})}\left(\int_{\mathbb{A}^{n}}\left(\int_{X(\mathbb{A})}\psi^{-1}(x_{n,n+1})\mathcal{W}^{\psi_{N}}((-1)^{n}\overline{\ell}(a,x)\delta\gamma
y,\mathcal{E}_{-k}\varphi)\,dx\right)da\right)dy.$

Note that the group $\gamma^{-1}\overline{\ell}(\mathbb{A}^{n},X)\gamma$ is
equal to

$\\{\begin{pmatrix}1_{n}&(x_{1},x^{\prime}_{1})&x^{\prime\prime}_{1}&\vspace{-.07cm}&x_{2}\\\
&1_{n+1}&&\vspace{-.07cm}&\\\ &&1&&\check{x}^{\prime\prime}_{1}\\\
&&&1_{n+1}&\begin{pmatrix}\check{x}^{\prime}_{1}\\\
\check{x}_{1}\end{pmatrix}\\\
&&&\vspace{-.07cm}&1_{n}\end{pmatrix}\,|\,x_{1}\in{\rm Mat}_{n,n}\;\text{\rm
strictly upper
triangular},x_{2}\in\mathfrak{a}_{n},x^{\prime}_{1},x^{\prime\prime}_{1}\in\mathbb{A}^{n}\\},$

so that $(\gamma^{-1}\overline{\ell}(\mathbb{A}^{n},X)\gamma Y)\rtimes
V_{\gamma}=V$. In conclusion, we arrive at

$\int_{V_{\gamma}(\mathbb{A})\backslash
V(\mathbb{A})}\mathcal{W}^{\psi_{N}}((-1)^{n}\gamma
v,\mathcal{E}_{-k}\varphi)\psi^{-1}_{V}(v)\,dv$

provided that it converges, since
$\psi_{V}(\gamma^{-1}\overline{\ell}(a,x)\gamma)=\psi(x_{n,n+1})$ for $x\in
X(\mathbb{A})$ and $a\in\mathbb{A}^{n}$. ∎

###  3.6. Local Rankin–Selberg–Shimura integrals

We are now going back to our choice of $\psi_{N_{\mathbb{M}}}$ specified in
(3.1). Now let $F$ be either $p$-adic or archimedean, $\pi\in{\rm Irr}_{\rm
gen}\,M$ and $\sigma\in{\rm Irr}_{{\rm
gen},\psi^{-1}_{N^{\prime}}}\,G^{\prime}$ with Whittaker model
$\mathbb{W}^{\psi^{-1}_{N^{\prime}}}(\sigma)$. For any
$W^{\prime}\in\mathbb{W}^{\psi^{-1}_{N^{\prime}}}(\sigma)$ and $W\in{\rm
Ind}(\mathbb{W}^{\psi_{N_{M}}}(\pi))$, we define the local
Rankin–Selberg–Shimura type integral

$J(s,W^{\prime},W):=\int_{N^{\prime}\backslash
G^{\prime}}W^{\prime}(g)A^{\psi}(s,g,W)\,dg.$

The analytic properties of this integral were extensively investigated by
Kaplan [Kap10, Kap12, Kap15].

######  Proposition 3.9.

Suppose that $\pi\in{\rm Irr}_{\rm gen}\,M$ and $\sigma\in{\rm Irr}_{{\rm
gen},\psi^{-1}_{N^{\prime}}}\,G^{\prime}$. Let
$W^{\prime}\in\mathbb{W}^{\psi^{-1}_{N^{\prime}}}(\sigma)$ and $W\in{\rm
Ind}(\mathbb{W}^{\psi_{N_{M}}}(\pi))$. Then

  1. $(\mathrm{1})$

$J(s,W^{\prime},W)$ converges in some right-half plane (depending only on
$\pi$ and $\sigma$) and admits a meromorphic continuation in $s$.

  2. $(\mathrm{2})$

For any $s\in\mathbb{C}$, we can choose $W^{\prime}$ and $W$ such that
$J(s,W^{\prime},W)\neq 0$.

  3. $(\mathrm{3})$

If $\pi$, $\sigma$, and $\psi$ are unramified, $W^{\circ}\in{\rm
Ind}(\mathbb{W}^{\psi_{N_{M}}}(\pi))$ is the standard unramified vector and
${W^{\prime}}^{\circ}$ is $K^{\prime}$-invariant with
${W^{\prime}}^{\circ}(e)=1$, then

$J(s,{W^{\prime}}^{\circ},W^{\circ})=\frac{{\rm vol}(K^{\prime})}{{\rm
vol}(N^{\prime}\cap
K^{\prime})}\frac{L(s+\frac{1}{2},\sigma\times\pi)}{L(2s+1,\pi,{\rm
Sym}^{2})}$

assuming the Haar measure on $V$ and $V_{\gamma}$ are normalized so that ${\rm
vol}(V\cap K)$ and ${\rm vol}(V_{\gamma}\cap K)$ are all $1$.

  4. $(\mathrm{4})$

$J(s,W^{\prime},W)$ satisfies local functional equations:

$J(-s,W^{\prime},M(s,\pi)W)=c(s,\sigma,\pi,\tau)\frac{\gamma(s+\frac{1}{2},\sigma\times\pi,\psi)}{\gamma(2s+1,\pi,{\rm
Sym}^{2},\psi)}J(s,W^{\prime},W)$

where
$c(s,\sigma,\pi,\tau)=(\omega_{\sigma}(-1)|\tau|^{s}(-1,\tau),\gamma_{\psi}(\tau))^{2n+1}\omega_{\pi}((-1)^{n+1}\tau)$
and $\gamma(s,\pi,{\rm Sym}^{2},\psi)$ is the gamma factor defined by Shahidi
(pertaining to the symmetric square representation).

It is noteworthy that $\displaystyle{\rm
vol}(K^{\prime})=(\prod_{j=1}^{n}\zeta_{F}(2j)L(n+1,\chi_{\tau}))^{-1}.$

###  3.7. Factorization of Gelfand–Graev periods

We define for $\varphi\in\mathcal{A}(\pi)$ an automorphic form on
$G(\mathbb{A})$;

$A^{\psi}(s,g,\varphi):=\int_{V_{\gamma}(\mathbb{A})\backslash
V(\mathbb{A})}\mathcal{W}^{\psi_{N_{M}}}((-1)^{n}\gamma
vg,\varphi_{s})\psi^{-1}_{V}(v)\,dv,\quad g\in G^{\prime}.$

The global descent $\mathcal{D_{\psi}}(\pi)$ and the local descent
$\mathcal{D}_{\psi}(\pi_{v})$ are related in the following way:

######  Proposition 3.10.

Suppose that $\varphi\in\mathcal{A}(\pi)$ is a factorizable vector and
$\mathcal{W}^{\psi_{N_{M}}}(\cdot,\varphi)=\prod_{v}W_{v}$ with $W_{v}\in{\rm
Ind}(\mathbb{W}^{\psi_{N_{M}}}(\pi_{v}))$. Then for any sufficiently large
finite set of places $S$, we have

(3.4) $\mathcal{W}^{\psi_{N^{\prime}}}(g,{\rm
GG}(\mathcal{E}_{-k}\varphi))=\frac{m^{S}_{-k,1}(\pi)}{2^{k}}\prod_{v\in
S}A^{\psi}\left(-\frac{1}{2},g_{v},M\left(\frac{1}{2},\pi_{v}\right)W_{v}\right),$

where

$m^{S}_{-k,1}(\pi)=\frac{\lim_{s\rightarrow 1}(s-1)^{k}L^{S}(s,\pi,{\rm
Sym}^{2})}{L^{S}(2,\pi,{\rm Sym}^{2})}.$

###### Proof.

We note that

$\mathcal{W}^{\psi_{N}}(\mathcal{E}_{-k}\varphi)=\mathcal{W}^{\psi_{N_{M}}}(\mathcal{E}^{U}_{-k}\varphi)=\mathcal{W}^{\psi_{N_{M}}}((M_{-k}(\varphi))_{-\frac{1}{2}}).$

With this in hand, we rewrite Theorem 3.8 in terms of the global transform
$A^{\psi}$ as

(3.5) $\mathcal{W}^{\psi_{N^{\prime}}}(g,{\rm
GG}(\mathcal{E}_{-k}\varphi))=A^{\psi}\left(-\frac{1}{2},g,M_{-k}(\varphi)\right).$

We can identify $\mathcal{A}(\pi)$ with ${\rm
Ind}^{G(\mathbb{A})}_{P(\mathbb{A})}\,\pi=\otimes_{v}{\rm
Ind}^{G(F_{v})}_{P(F_{v})}\,\pi_{v}$. For ${\rm Re}(s)\gg 0$, we have

$\mathcal{W}^{\psi_{N_{M}}}(g,M(s,\pi)\varphi)=\int_{N_{M}(F)\backslash
N_{M}(\mathbb{A})}M(s,\pi)\varphi(ng)\psi^{-1}_{N_{M}}(n)\,dn\\\
=\int_{N_{M}(F)\backslash
N_{M}(\mathbb{A})}\int_{U(\mathbb{A})}\varphi(\varrho(\mathfrak{t})w_{U}ung)\psi^{-1}_{N_{M}}(n)\,du\,dn=\int_{U(\mathbb{A})}\mathcal{W}^{\psi_{N_{M}}}(\varrho(\mathfrak{t})w_{U}ug,\varphi)\,du$

from which it follows that for $S$ large enough

$\mathcal{W}^{\psi_{N_{M}}}(M(s,\pi)\varphi)\\\ =m^{S}(s,\pi)\left(\prod_{v\in
S}M(s,\pi_{v})W_{v}\right)\prod_{v\notin S}W_{v},$

as meromorphic functions in $s\in\mathbb{C}$. Here

$m^{S}(s,\pi)=\frac{L^{S}(2s,\pi,{\rm Sym}^{2})}{L^{S}(2s+1,\pi,{\rm
Sym}^{2})}.$

Taking the limit as $s\rightarrow\frac{1}{2}$, we get for $S$ sufficiently
large,

(3.6) $\mathcal{W}^{\psi_{N_{M}}}(M_{-k}\varphi)=m^{S}_{-k}(\pi)\prod_{v\in
S}M\left(\frac{1}{2},\pi_{v}\right)W_{v}\prod_{v\not\in S}W_{v}$

where

$m^{S}_{-k}(\pi)=\lim_{s\rightarrow\frac{1}{2}}\left(s-\frac{1}{2}\right)^{k}m^{S}(s,\pi)=2^{-k}\frac{\lim_{s\rightarrow\frac{1}{2}}L^{S}(2s,\pi,{\rm
Sym}^{2})}{L(2,\pi,{\rm Sym}^{2})}\\\ =2^{-k}\frac{\lim_{s\rightarrow
1}(s-1)^{k}L^{S}(s,\pi,{\rm Sym}^{2})}{L(2,\pi,{\rm
Sym}^{2})}=\frac{m^{S}_{-k,1}(\pi)}{2^{k}}.$

For any factorizable $\varphi\in\mathcal{A}(\pi)$, the desired conclusion
follows from (3.5) and (3.6). ∎

We unfold Petersson inner products of automorphic forms against the
Gelfand–Graev coefficient of Eisenstein series [GRS11, Theorem 10.3, (10.4)]
and then perform unramified computations [GRS11, (10.60)] to find the
following Euler product.

######  Proposition 3.11.

Let $\pi\in{\rm OScusp}_{k}\,\mathbb{M}$ and
$\sigma=\mathcal{D}_{\psi^{-1}}(\pi)$. Assume our Working Hypothesis 3.5. Let
$\varphi^{\prime}\in\sigma$ be an irreducible $\psi^{-1}_{N^{\prime}}$-generic
cuspidal automorphic representation of $G^{\prime}$ and
$\varphi\in\mathcal{A}(\pi)$. Suppose that
$\mathcal{W}^{\psi^{-1}_{N^{\prime}}}(\cdot,\varphi^{\prime})=\prod_{v}W^{\prime}_{v}$
and $\mathcal{W}^{\psi_{N_{M}}}(\cdot,\varphi)=\prod_{v}W_{v}$. Then for any
sufficiently large finite set of places $S$, we have

(3.7) $\langle\varphi^{\prime},{\rm
GG}(\mathcal{E}(s,\varphi))\rangle_{G^{\prime}}=\frac{1}{2}\cdot\frac{{\rm
vol}(N^{\prime}(\mathcal{O}_{S})\backslash
N^{\prime}(F_{S}))}{\prod_{j=1}^{n}\zeta^{S}_{F}(2j)L^{S}(n+1,\chi_{\tau})}\cdot\frac{L^{S}(s+\frac{1}{2},\sigma\times\pi)}{L^{S}(2s+1,\pi,{\rm
Sym}^{2})}\prod_{v\in S}J(s,W^{\prime}_{v},W_{v}),$

where on the right-hand side we take the unnormalized Tamagawa measure on
$G^{\prime}(F_{S})$, $N^{\prime}(F_{S})$, $V(F_{S})$ and $V_{\gamma}(F_{S})$
(which are independent of the choice of gauge forms when $S$ is sufficiently
large.)

###### Proof.

It is worthwhile to mention that the volume of $G^{\prime}(F)\backslash
G^{\prime}(\mathbb{A})$, appearing on the right-hand side of (2.1), is $2$
equal to the Tamagawa number when we use the Tamagawa measure as in [LM16,
LM17]. We apply [GRS11, Theorem 10.3, (10.4)] in order to unfold

(3.8) $\langle\varphi^{\prime},{\rm
GG}(\mathcal{E}(s,\varphi))\rangle_{G^{\prime}}=\frac{1}{2}\cdot{\rm
vol}(N^{\prime}(F)\backslash
N^{\prime}(\mathbb{A}))\int_{N^{\prime}(\mathbb{A})\backslash
G^{\prime}(\mathbb{A})}\mathcal{W}^{\psi^{-1}_{N^{\prime}}}(g,\varphi^{\prime})A^{\psi}(s,g,\varphi)\,dg.$

Similar to [LM16, Propositions 5.7 and 8.4], our desired result is immediate
from (3.8) in conjunction with the unramified computation [Kap10, Kap15] (cf.
[GRS11, (10.60)]);

(3.9) $J(s,{W^{\prime}_{v}}^{\circ},W^{\circ}_{v})=\frac{{\rm
vol}(K^{\prime})}{{\rm vol}(N^{\prime}(F_{v})\cap
K^{\prime})}\frac{L(s+\frac{1}{2},\sigma_{v}\times\pi_{v})}{L(2s+1,\pi_{v},{\rm
Sym}^{2})}$

with $K^{\prime}=K_{{\rm GL}_{2n+2}(F_{v})}\cap G^{\prime}(F_{v})$. Since we
use the Tamagawa measure, the factor $\Delta^{S}_{{\rm
SO}_{2n+2,\tau}}(1)^{-1}$ shows up, where $\Delta^{S}_{{\rm
SO}_{2n+2,\tau}}(s)=L(M^{\vee}(s))$ is the partial $L$-function of the dual
motive $M^{\vee}$ of the motive $M$ associated to ${\rm SO}_{2n+2,\tau}$ by
Gross [Gro97]. According to [Xue17]*Notation and Convention [II10]

$\Delta^{S}_{{\rm
SO}_{2n+2,\tau}}(1)=\prod_{j=1}^{n}\zeta^{S}_{F}(2j)L^{S}(n+1,\chi_{\tau})$

for any finite set of places $S$. ∎

###  3.8. Petersson inner products and Fourier coefficients

Let $F$ be a local field. We let $\pi\in{\rm Irr}_{\rm gen,os}\,\mathbb{M}$.
We say that $\pi$ is good if the following conditions are satisfied for all
$\psi$;

  1. $(\mathrm{i})$

$\mathcal{D}_{\psi}(\pi)$ is irreducible.

  2. $(\mathrm{ii})$

$J(s,W^{\prime},W)$ is holomorphic at $s=\frac{1}{2}$ for any
$W^{\prime}\in\mathcal{D}_{\psi^{-1}}(\pi)$ and $W\in{\rm
Ind}(\mathbb{W}^{\psi_{N_{M}}}(\pi))$.

  3. $(\mathrm{iii})$

For any $W^{\prime}\in\mathcal{D}_{\psi^{-1}}(\pi)$,
$J\left(\frac{1}{2},W^{\prime},W\right)$ factors through the map
$W\mapsto\left(A^{\psi}\left(-\frac{1}{2},\cdot,M\left(\frac{1}{2},\pi\right)W\right)\right).$

We know from (3.3) that for $x\in G^{\prime}$ we have

$J(s,\sigma(x)W^{\prime},I(s,x)W)=J(s,W^{\prime},W),$

where $\sigma=\mathcal{D}_{\psi^{-1}}(\pi)$. Therefore if $\pi$ is good, there
is a non-degenerate $G^{\prime}$-invariant pairing $[\cdot,\cdot]$ on
$\mathcal{D}_{\psi^{-1}}(\pi)\times\mathcal{D}_{\psi}(\pi)$ such that

$J\left(\frac{1}{2},W^{\prime},W\right)=\left[W^{\prime},A^{\psi}\left(-\frac{1}{2},\cdot,M\left(\frac{1}{2},\pi\right)W\right)\right]$

for any $W^{\prime}\in\mathcal{D}_{\psi^{-1}}(\pi)$ and $W\in{\rm
Ind}(\mathbb{W}^{\psi_{N_{M}}}(\pi))$. By [LM15, §2], when $\pi$ is good,
there exists $c_{\pi}$ such that

(3.10) $\int^{\rm
st}_{N^{\prime}}J\left(\frac{1}{2},\sigma(u)W^{\prime},W\right)\psi_{N^{\prime}}(u)\,du=c_{\pi}W^{\prime}(e)A^{\psi}\left(-\frac{1}{2},e,M\left(\frac{1}{2},\pi\right)W\right).$

######  Proposition 3.12.

Suppose that $F$ is $p$-adic, $\pi\in{\rm Irr}_{\rm gen,os}\,\mathbb{M}$, and
$\psi$ are unramified. Then $\sigma=\mathcal{D}_{\psi^{-1}}(\pi)$ is
irreducible and unramified. Let $W^{\circ}\in{\rm
Ind}(\mathbb{W}^{\psi_{N_{M}}}(\pi))$ be the standard unramified vector and
let ${W^{\prime}}^{\circ}$ be $K^{\prime}$-invariant with
${W^{\prime}}^{\circ}(e)=1$. Then holds with $c_{\pi}=1$.

###### Proof.

The first part is proved exactly as in the proof of [LM17, Lemma 5.4] relying
on [GRS11, Theorem 5.6] instead of [GRS11, Theorem6.4]. Assuming ${\rm
vol}(U\cap K)=1$ as in (3.2), we know that

${\rm vol}(K^{\prime})={\rm vol}(U\cap
K)(\zeta^{S}_{F}(2j)L^{S}(n+1,\chi_{\tau}))^{-1}=(\zeta^{S}_{F}(2j)L^{S}(n+1,\chi_{\tau}))^{-1}.$

We combine it and [LM15, Proposition 2.14] to see that

$\int^{\rm
st}_{N^{\prime}}J\left(\frac{1}{2},\sigma(u){W^{\prime}}^{\circ},W^{\circ}\right)\psi^{-1}_{N^{\prime}}(u)\,du=\frac{{\rm
vol}(N^{\prime}\cap K^{\prime})}{{\rm
vol}(K^{\prime})}\frac{J\left(\frac{1}{2},{W^{\prime}}^{\circ},W^{\circ}\right)}{L(1,\sigma,{\rm
Ad})}.$

It follows from the description of $\sigma$ in this case [GRS11, Theorem 5.6]
together with [GGP12, §7] that

$L(1,\sigma,{\rm
Ad})=L(1,\pi\boxplus\omega_{\pi}\chi_{\tau},\wedge^{2})=L(1,\pi,\wedge^{2})L(1,\pi\times\omega_{\pi}\chi_{\tau})$

and
$L(s,\sigma\times\pi)=L(s,\pi\times\pi)L(s,\omega_{\pi}\chi_{\tau}\times\pi)$.
We recall that $\pi=\pi^{\vee}$ in our case. Hence (3.2) aligned with of (3.9)
implies that the above is equal to

$\frac{1}{L(1,\pi,\wedge^{2})L(1,\pi\times\omega_{\pi}\chi_{\tau})}\frac{L(1,\pi\times\pi)L(1,\omega_{\pi}\chi_{\tau}\times\pi)}{L(2,\pi,{\rm
Sym}^{2})}\\\ =\frac{L(1,\pi,{\rm Sym}^{2})}{L(2,\pi,{\rm
Sym}^{2})}=M\left(\frac{1}{2},\pi\right)W^{\circ}(e)={W^{\prime}}^{\circ}(e)A^{\psi}\left(-\frac{1}{2},e,M\left(\frac{1}{2},\pi\right)W^{\circ}\right)$

that we seek for. ∎

At first glance, $c_{\pi}$ implicitly depends on the choice of Haar measures
on $G^{\prime}$ (the former used in the definition of $J(s,W^{\prime},W)$) and
$U$ (the latter used in the definition of the intertwining operator), but not
on any other groups (refer to [LM17, Remark 5.2] for more details). However
the vector spaces underlying the Lie algebras of $G^{\prime}$ and $U$ are both
canonically isomorphic as vector spaces to

$\\{X\in{\rm Mat}_{2n+1,2n+1}(F)\,|\,{X^{t}}\,w_{2n+1}=-w_{2n+1}X\\}.$

We can thereby identify the gauge forms on $G^{\prime}$ and $U$ and so
$c_{\pi}$ does not depend on any choice if we use the unnormalized Tamagawa
measure on $G^{\prime}$ and $U$ with respect to the same gauge form. We will
say that the Haar measures on $G^{\prime}$ and $U$ are compatible in this
case.

######  Proposition 3.13.

Let $\pi\in{\rm OScusp}_{k}\,\mathbb{M}$ and
$\sigma=\mathcal{D}_{\psi^{-1}}(\pi)$. Assume our Working Hypothesis 3.5. Let
$\varphi^{\prime}\in\sigma$ be an irreducible $\psi^{-1}_{N^{\prime}}$-generic
cuspidal automorphic representation of $G^{\prime}$ and
$\varphi\in\mathcal{A}(\pi)$. Suppose that
$\mathcal{W}^{\psi^{-1}_{N^{\prime}}}(\cdot,\varphi^{\prime})=\prod_{v}W^{\prime}_{v}$
and $\mathcal{W}^{\psi_{N_{M}}}(\cdot,\varphi)=\prod_{v}W_{v}$. Then for all
$v$ $\pi_{v}$ is good. Moreover for any sufficiently large finite set of
places $S$, we have

(3.11) $\langle\varphi^{\prime},{\rm
GG}(\mathcal{E}_{-k}\varphi)\rangle_{G^{\prime}}=\frac{1}{2}\cdot\frac{{\rm
vol}(N^{\prime}(\mathcal{O}_{S})\backslash
N^{\prime}(F_{S}))}{\prod_{j=1}^{n}\zeta^{S}_{F}(2j)L^{S}(n+1,\chi_{\tau})}\cdot
n^{S}_{-k,1}\prod_{v\in S}J\left(\frac{1}{2},W^{\prime}_{v},W_{v}\right),$

where

$n^{S}_{-k,1}=\frac{\lim_{s\rightarrow
1}(s-1)^{k}L^{S}(s,\pi\times\pi)L^{S}(s,\pi\times\omega_{\pi}\chi_{\tau})}{L^{S}(2,\pi,{\rm
Sym}^{2})}.$

###### Proof.

We confirm that $\pi_{v}$ is good for all $v$. To this end, it follows from
(3.4) together with the irreducibility of the global descent that
$\mathcal{D}_{\psi^{-1}}(\pi_{v})$ is irreducible for all $v$.

When $\pi\in{\rm OScusp}_{k}\,\mathbb{M}$ and
$\sigma=\mathcal{D}_{\psi^{-1}}(\pi)$, we obtain
$L^{S}(s,\sigma\times\pi)=L^{S}(s,\pi\times\pi)L^{S}(s,\omega_{\pi}\chi_{\tau}\times\pi)$,
which has a pole of order $k$ at $s=\frac{1}{2}$. On the one hand,
$J(s,W^{\prime}_{v},W_{v})$ is non-vanishing at $s=\frac{1}{2}$ for suitable
$W^{\prime}_{v}$ and $W_{v}$. Consequently the right-hand side of (3.7) (when
$\varphi^{\prime}\in\sigma$) has a pole of order at least $k$ at
$s=\frac{1}{2}$ for proper $\varphi^{\prime}$ and $\varphi$. On the other
hand, the left-hand side of (3.7) has a pole of order at most $k$ at
$s=\frac{1}{2}$ because this is true for $\mathcal{E}(s,\varphi)$ and
$\varphi^{\prime}$ is rapidly decreasing. Multiplying (3.7) by
$(s-\frac{1}{2})^{k}$ and then taking the limit as $s\rightarrow\frac{1}{2}$,
we conclude that $J(s,W^{\prime}_{v},W_{v})$ is holomorphic at $s=\frac{1}{2}$
for all $v$, and for $\varphi^{\prime}\in\sigma$ and
$\varphi\in\mathcal{A}(\pi)$ we get

(3.12) $\langle\varphi^{\prime},{\rm
GG}(\mathcal{E}_{-k}\varphi)\rangle_{G^{\prime}}=\frac{1}{2}\cdot\frac{{\rm
vol}(N^{\prime}(\mathcal{O}_{S})\backslash
N^{\prime}(F_{S}))}{\prod_{j=1}^{n}\zeta^{S}_{F}(2j)L^{S}(n+1,\chi_{\tau})}\cdot
n^{S}_{-k,1}\prod_{v\in S}J\left(\frac{1}{2},W^{\prime}_{v},W_{v}\right).$

We fix a place $v_{0}$. We assume that $W_{v_{0}}$ is such that
$A^{\psi}\left(-\frac{1}{2},\cdot,M\left(\frac{1}{2},\pi_{v_{0}}\right)W_{v_{0}}\right)\equiv
0$. Then by (3.4), $\mathcal{W}^{\psi_{N^{\prime}}}(\cdot,{\rm
GG}(\mathcal{E}_{-k}\varphi))\equiv 0$ and therefore ${\rm
GG}(\mathcal{E}_{-k}\varphi)\equiv 0$ benefited from the irreducibility and
genericity of the descent. We conclude from (3.12) that
$J(\frac{1}{2},W^{\prime}_{v_{0}},W_{v_{0}})=0$, which amounts to saying that
$\pi_{v_{0}}$ is good. ∎

While we expect that any irreducible generic representation of orthogonal
symplectic type $\pi_{v}$ is good, the proof that we give under the assumption
that $\pi_{v}$ is the local component of $\pi\in{\rm OScusp}_{k}\,\mathbb{M}$
is valid at the very least for the case of our interest, which pertains to the
global period integrals. Our proof here requires a local and global argument.
It behoves us to highlight the fact that the global approach is overkill and
indirect; one may directly prove the desired result by using a local
functional equation [LM17, Proposition 5.3]. For the sake of brevity, we only
include this indirect mean in keeping with the spirit of [LM16, Propositions
5.7 and 8.4].

######  Theorem 3.14.

Let $\pi\in{\rm OScusp}_{k}\,\mathbb{M}$ and $\sigma=\mathcal{D}_{\psi}(\pi)$.
Assume our Working Hypothesis 3.5. Let $S$ be a finite set of places including
all the archimedean places such that $\pi$ and $\psi$ are all unramified
outside $S$. Then for any $\varphi\in\sigma$ and
$\varphi^{\vee}\in\sigma^{\vee}$ which are fixed under $K^{\prime}_{v}$ for
all $v\notin S$, we have

(3.13)
$\quad\quad\mathcal{W}^{\psi_{N^{\prime}}}(\varphi)\mathcal{W}^{\psi^{-1}_{N^{\prime}}}(\varphi^{\vee})=2^{1-k}\left(\prod_{v\in
S}c^{-1}_{\pi_{v}}\right)\frac{\prod_{j=1}^{n}\zeta^{S}_{F}(2j)L^{S}(n+1,\chi_{\tau})}{L^{S}(1,\pi,\wedge^{2})L^{S}(1,\pi\times\omega_{\pi}\chi_{\tau})}\\\
\times({\rm vol}(N^{\prime}(\mathcal{O}_{S})\backslash
N^{\prime}(F_{S})))^{-1}\int^{\rm
st}_{N^{\prime}(F_{S})}\langle\sigma(u)\varphi,\varphi^{\vee}\rangle_{G^{\prime}}\psi^{-1}_{N^{\prime}}(u)\;du.\quad\quad$

###### Proof.

Since the descent of $\pi$ is the space
$\mathcal{D}_{\psi_{N_{\mathbb{M}}}}(\pi)$ generated by ${\rm
GG}(\mathcal{E}_{-k}\varphi^{\flat})$ for
$\varphi^{\flat}\in\mathcal{A}(\pi)$, we initially deal with this for the case
$\varphi^{\vee}={\rm GG}(\mathcal{E}_{-k}\varphi^{\sharp})$ with
$\varphi^{\sharp}\in\mathcal{A}(\pi)$, and this identity (3.13) then extends
via linearity to all $\varphi^{\vee}$. Appealing to Proposition 3.10, we see
that

$\mathcal{W}^{\psi_{N^{\prime}}}(\varphi)\mathcal{W}^{\psi^{-1}_{N^{\prime}}}({\rm
GG}(\mathcal{E}_{-k}\varphi^{\sharp}))=\mathcal{W}^{\psi_{N^{\prime}}}(\varphi)\frac{m^{S}_{-k,1}(\pi)}{2^{k}}\prod_{v\in
S}A^{\psi^{-1}}\left(-\frac{1}{2},e,M\left(\frac{1}{2},\pi_{v}\right)W^{\sharp}_{v}\right).$

Relying on the fact that $\pi_{v}$ is good in Proposition 3.13, we use
Proposition 3.12 to insert the identity (3.10) for all $v$ with
$c_{\pi_{v}}=1$ for almost all $v$. In this way, we write

(3.14)
$\mathcal{W}^{\psi_{N^{\prime}}}(\varphi)\mathcal{W}^{\psi^{-1}_{N^{\prime}}}({\rm
GG}(\mathcal{E}_{-k}\varphi^{\sharp}))=\frac{m^{S}_{-k,1}(\pi)}{2^{k}}\left(\prod_{v\in
S}c^{-1}_{\pi_{v}}\int^{\rm
st}_{N^{\prime}(F_{v})}J\left(\frac{1}{2},\sigma_{v}(u_{v})W_{v},W^{\sharp}_{v}\right)\psi^{-1}_{N^{\prime}(F_{v})}(u_{v})\,du_{v}\right).$

Integrating (3.11) over $N^{\prime}(F_{v})$ for $v\in S$, a product of
regularized integrals becomes

(3.15) $\prod_{v\in S}\int^{\rm
st}_{N^{\prime}(F_{v})}J\left(\frac{1}{2},\sigma_{v}(u_{v})W_{v},W^{\sharp}_{v}\right)\psi^{-1}_{N^{\prime}(F_{v})}(u_{v})\,du_{v}\\\
=\frac{2\prod_{j=1}^{n}\zeta^{S}_{F}(2j)L^{S}(n+1,\chi_{\tau})}{{\rm
vol}(N^{\prime}(\mathcal{O}_{S})\backslash N^{\prime}(F_{S}))\cdot
n^{S}_{-k,1}}\int^{\rm st}_{N^{\prime}(F_{S})}\langle\sigma(u)\varphi,{\rm
GG}(\mathcal{E}_{-k}\varphi^{\sharp}\rangle_{G^{\prime}}\psi^{-1}_{N^{\prime}}(u)\;du.$

The factorization
$L^{S}(s,\pi\times\pi)=L^{S}(s,\pi\times\pi^{\vee})=L^{S}(s,\pi,{\rm
Sym}^{2})L^{S}(s,\pi,\wedge^{2})$ permits us to simplify the ratio

(3.16)
$m^{S}_{-k,1}/n^{S}_{-k,1}=1/(L^{S}(1,\pi,\wedge^{2})L^{S}(1,\pi\times\omega_{\pi}\chi_{\tau}))$

whose partial $L$-function occurring in the denominator is nothing but the
adjoint $L$-function $L^{S}(s,\sigma,{\rm Ad})$ for $\sigma$. The desired
conclusion follows from (3.14), (3.15), and (3.16). ∎

##  4\. The Case for Symplectic Groups

###  4.1. Groups, Embeddings, and Weyl Elements

Let $G$ be the symplectic group defined by

$G={\rm Sp}_{4n}=\\{g\in{\rm
GL}_{4n}\,|\,\prescript{t}{}{g}J_{4n}g=J_{4n}\\},\quad\text{where}\quad
J_{4n}=\begin{pmatrix}&w_{2n}\\\ -w_{2n}&\end{pmatrix},$

and $\prescript{t}{}{g}$ is the transpose of $g$. The group $G$ acts on a $4n$
dimensional space with the standard basis
$e_{1},\dotsm,e_{2n},e_{-2n},e_{-2n+1},\dotsm,e_{-1}$. We let

$G^{\prime}={\rm Sp}_{2n}=\\{g\in{\rm
GL}_{2n}\,|\,\prescript{t}{}{g}J_{2n}g=J_{2n}\\},\quad\text{where}\quad
J_{2n}=\begin{pmatrix}&w_{n}\\\ -w_{n}&\end{pmatrix}.$

The group $G^{\prime}$ is embedded as a subgroup of $G$ via
$g\mapsto\eta(g):={\rm diag}(1_{n},g,1_{n})$, and thus it is the subgroup of
$G$ consisting of elements fixing $e_{1},\dotsm,e_{n}$ and
$e_{-n},\dotsm,e_{-1}$. Let $P^{\prime}=M^{\prime}\ltimes U^{\prime}$ be the
Seigel parabolic subgroup of $G^{\prime}$ with the Levi part
$M^{\prime}=\varrho^{\prime}(\mathbb{M}^{\prime})$ and the Seigel unipotent
subgroup $U^{\prime}$, where $\varrho^{\prime}:m\mapsto{\rm
diag}(1_{n},m,m^{\ast},1_{n})$. We set

$\mathfrak{s}_{n}=\\{x\in{\rm Mat}_{n,n}\,|\,\check{x}=x\\}.$

Then

$U^{\prime}=\\{{\rm diag}(1_{n},\begin{pmatrix}1_{n}&u\\\
&1_{n}\end{pmatrix},1_{n})\,|\,u\in\mathfrak{s}_{n}\\}.$

Let $N^{\prime}$ be the standard maximal unipotent subgroup of $G^{\prime}$.
Then $N^{\prime}=N^{\prime}_{M^{\prime}}\ltimes U^{\prime}$ with
$N^{\prime}_{M^{\prime}}=\varrho^{\prime}(N^{\prime}_{\mathbb{M}^{\prime}})$.
Let $V$ be the unipotent radical of the standard parabolic subgroup of ${\rm
Sp}_{4n}$, whose Levi part is isomorphic to ${\rm GL}^{n}_{1}\times{\rm
Sp}_{2n}$:

$V=\\{\begin{pmatrix}v&u&y\\\ &1_{2n}&\check{u}\\\
&&v^{\ast}\end{pmatrix}\in{\rm Sp}_{4n}\,|\,v\in Z_{n},u\in{\rm
Mat}_{n,2n}\\}.$

Let $P=M\ltimes U$ be the Seigel parabolic subgroup of $G$ with $U$ the
standard maximal unipotent subgroup and Levi part $M=\varrho(\mathbb{M})$,
where $\mathbb{M}={\rm GL}_{2n}$ and $\varrho:h\mapsto{\rm diag}(h,h^{\ast})$.
Let

$w^{\prime}_{U^{\prime}}={\rm diag}(1_{n},\begin{pmatrix}&1_{n}\\\
-1_{n}&\end{pmatrix},1_{n})\in G^{\prime}$

represent the longest $M^{\prime}$-reduced Weyl element of $G^{\prime}$. Let
$w_{U}=\begin{pmatrix}&1_{2n}\\\ -1_{2n}&\end{pmatrix}\in G$ represent the
longest $M$-reduced Weyl element of $G$. We write [GRS11, (4.14),(4.15)]

$\gamma=w_{U}({w^{\prime}_{U^{\prime}}})^{-1}=\begin{pmatrix}&1_{n}&\vspace{-.1cm}&\\\
\vspace{-.1cm}&&&1_{n}\\\ -1_{n}&&\vspace{-.1cm}&\\\ &&1_{n}&\end{pmatrix}.$

Let us denote

$V_{\gamma}=V\cap\gamma^{-1}N\gamma=\\{\begin{pmatrix}v&\vspace{-.1cm}&u&\\\
&1_{n}&\vspace{-.1cm}&\check{u}\\\ &&1_{n}&\vspace{-.1cm}\\\
&&&v^{\ast}\end{pmatrix}\,|\,v\in Z_{n},u\in{\rm Mat}_{n,n}\\}.$

Let $N_{M}$ (resp. $N_{\mathbb{M}}$) be the standard maximal unipotent
subgroup of $M$ (resp. $\mathbb{M}$). Let $K=K_{{\rm GL}_{4n}}\cap{\rm
Sp}_{4n}$. Then $K$ is a maximal compact subgroup of ${\rm Sp}_{4n}$;
similarly for $K^{\prime}$.

Let $\tilde{G}^{\prime}=\widetilde{\rm Sp}_{n}$ be the metaplectic group which
is the two-fold cover of $G^{\prime}$. We write elements of
$\tilde{G}^{\prime}$ as pairs $(g,\varepsilon)$, $g\in G$, $\varepsilon=\pm 1$
where the multiplication rule is given by Rao’s cocycle [Rao93, §5] (see also
[Szp13, §1.3]). When $g\in G^{\prime}$, we write
$\tilde{g}=(g,1)\in\tilde{G}^{\prime}$. (Of course, $g\mapsto\tilde{g}$ is not
a group homomorphism). For each subgroup $H$ of $G$, let $\tilde{H}$ be the
inverse image of $H$ under the canonical projection $\tilde{G}\rightarrow G$.
We let $\tilde{M}$ the double cover of $M$ inside $G$ and we will identify $U$
with a subgroup $\tilde{U}$ via $u\mapsto\tilde{u}$. For $g\in G^{\prime}$,
let $\tilde{\eta}(g)=(\eta(g),1)\in\widetilde{\rm Sp}_{4n}$. When $F$ is
$p$-adic, and $p\neq 2$, we view $K^{\prime}$ as a subgroup of
$\tilde{G}^{\prime}$.

###  4.2. Additive Characters

We fix a non-trivial additive character $\psi$ of $F$ and a non-degenerate
character $\psi_{N_{\mathbb{M}}}$ of $N_{\mathbb{M}}$. Let $\psi_{\circ}$ be
the non-trivial character of $F$ given by

$\psi_{\circ}(x)=\psi_{N_{\mathbb{M}}}(1_{2n}+x\epsilon_{n,n+1}).$

As illustrated in [LM17, Remark 6.4] (cf. [LM16, §4.1, §6.1]), the results
stated below are independent of the choice of $\psi_{N_{\mathbb{M}}}$. For
convenience, we set

(4.1) $\psi_{N_{\mathbb{M}}}(u)=\psi(u_{1,2}+\dotsm+u_{2n-1,2n}).$

Thus $\psi_{\circ}=\psi$. We define the characters of various unipotent
groups. We denote by $\psi_{N}$ the degenerate character on $N$ given by
$\psi_{N}(uv)=\psi_{N_{M}}(u)$ for any $u\in N_{M}$ and $v\in U$. Let
$\psi_{N_{M}}$ be the non-degenerate character of $N_{M}$ given by
$\psi_{N_{M}}(\varrho(u))=\psi_{N_{\mathbb{M}}}(u)$. Let
$\psi_{N^{\prime}_{M^{\prime}}}$ be the non-degenerate character of
$N^{\prime}_{M^{\prime}}$ such that
$\psi_{N^{\prime}_{M^{\prime}}}(\varrho(u))=\psi_{N^{\prime}_{\mathbb{M}^{\prime}}}(u)$.
With the choice of $\psi_{N_{\mathbb{M}}}$, we have

$\psi_{N^{\prime}_{\mathbb{M}^{\prime}}}(u^{\prime})=\psi(u^{\prime}_{1,2}+\dotsm+u^{\prime}_{n-1,n}).$

Let $\psi_{U^{\prime}}$ be the character on $U^{\prime}$ given by
$\psi_{U^{\prime}}(u)=\psi(\frac{1}{2}u_{2n,2n+1})^{-1}$. Then the non-
degenerate character $\psi_{N^{\prime}}$ [LM17, §2.2] is given by

$\psi_{N^{\prime}}(uv)=\psi_{N^{\prime}_{M^{\prime}}}(u)\psi_{U^{\prime}}(v)=\psi(u_{n+1,n+2}+\dotsm+u_{2n-1,2n}-\frac{1}{2}v_{2n,2n+1})$

with $u\in N^{\prime}_{M^{\prime}}$ and $v\in U^{\prime}$. (Cf. we have
specified $\lambda=-1/2$ in their notation [GRS11, (9.9)]). An element in $V$
can be written as $vu$ where $u$ fixes $e_{1},\dotsm,e_{n}$, $v$ fixes
$e_{n+1},e_{n+2},\dotsm,e_{-n-1}$ and we set [GRS11, (3.60)]

$\psi_{V}(vu)=\psi^{-1}(v_{1,2}+\dotsm+v_{n-1,n}).$

###  4.3. Weil Representation

Let $W$ be a $2n$-dimensional space over $F$, equipped with a non-degenerate,
symplectic form $\langle\cdot,\cdot\rangle$. We let $\mathcal{H}_{W}=W\oplus
F$ be the Heisenberg group attached to $W$ with the multiplication

$(x,t)\cdot(y,z)=\left(x+y,t+z+\frac{1}{2}\langle x,y\rangle\right).$

We write

$W=W_{+}\oplus W_{-}$

as a direct sum of two maximal isotropic subspaces of $W$, which are dual
under $\langle\cdot,\cdot\rangle$. Then the group ${\rm Sp}(W)$ acts on the
right on $W$. We write a prototype element of ${\rm Sp}(W)$ as
$\begin{pmatrix}A&B\\\ C&D\end{pmatrix}$ where $A\in{\rm Hom}(W_{+},W_{+})$,
$B\in{\rm Hom}(W_{+},W_{-})$, $C\in{\rm Hom}(W_{-},W_{+})$, and $D\in{\rm
Hom(W_{-},W_{-})}$. Let $\gamma_{\psi}$ be Weil’s factor given by
$\gamma_{\psi}(a)=\gamma(\psi_{a})/\gamma(\psi)$, where $\gamma(\psi)$ is
Weil’s index, which is an eighth root of unity, and $\psi_{a}=\psi(a\cdot)$.
We denote by $\gamma_{\psi}$ in [GRS11, (1.4)] the Weil index denoted by
$\gamma_{\psi_{1/2}}$ in our notation. Let $\widetilde{\rm Sp}(W)$ be the
metaplectic two-fold cover of ${\rm Sp}(W)$ with respect to the Rao cocycle
determined by the splitting [Rao93, §5] (see also [Szp13, §1.3]). We consider
the Weil representation $\omega_{\psi}$ of the group
$\mathcal{H}_{W}\rtimes\widetilde{\rm Sp}(W)$ on the Schwartz–Bruhat space
$\mathcal{S}(W_{+})$. Explicitly for any $\Phi\in\mathcal{S}(W_{+})$ and $X\in
W_{+}$, the action of $\mathcal{H}_{W}$ is given by

(4.2) $\displaystyle\omega_{\psi}(a,0)\Phi(X)$ $\displaystyle=\Phi(X+a),\quad
a\in W_{+},$ (4.3) $\displaystyle\omega_{\psi}(b,0)\Phi(X)$
$\displaystyle=\psi(\langle X,b\rangle)\Phi(X),\quad b\in W_{-},$ (4.4)
$\displaystyle\omega_{\psi}(0,t)\Phi(X)$ $\displaystyle=\psi(t)\Phi(X),\quad
t\in F,$

while the action of $\widetilde{\rm Sp}(W)$ is given by

$\begin{split}\omega_{\psi}\left(\begin{pmatrix}g&\\\
&g^{\ast}\end{pmatrix},\varepsilon\right)\Phi(X)&=\varepsilon\gamma_{\psi}(\det
g)|\det g|^{\frac{1}{2}}\Phi(Xg),\quad g\in{\rm GL}(W_{+}),\\\
\omega_{\psi}\left(\begin{pmatrix}1_{n}&B\\\
&1_{n}\end{pmatrix},\varepsilon\right)\Phi(X)&=\varepsilon\psi\left(\frac{1}{2}\langle
X,XB\rangle\right)\Phi(X),\quad B\in{\rm Hom}(W_{+},W_{-})\text{\;\;self-
dual}.\end{split}$

We now take $W=F^{2n}$ with the standard symplectic form

$\langle(x_{1},\dotsm,x_{2n}),(y_{1},\dotsm,y_{2n})\rangle=\sum_{i=1}^{n}x_{i}y_{2n+1-i}-\sum_{i=1}^{n}y_{i}x_{2n+1-i}$

and the standard polarization $W_{+}=\\{x_{1},\dotsm,x_{n},0,\dotsm,0\\}$,
$W_{-}=\\{(0,\dotsm,0,y_{1},\dotsm,y_{n})\\}$, with the standard basis
$e_{1},\dotsm,e_{n},e_{-n},\dotsm,e_{-1}$. For $X=(x_{1},\dotsm,x_{n})$ and
$Y=(y_{1},\dotsm,y_{n})\in F^{n}$, we define

$\langle X,Y\rangle^{\prime}=\langle X,Y\begin{pmatrix}&1_{n}\\\
-1_{n}&\end{pmatrix}\rangle=x_{1}y_{n}+\dotsm+x_{n}y_{1}.$

For $\Phi$ in the Schwartz–Bruhat space $\mathcal{S}(F^{n})$, we define the
Fourier transform

$\hat{\Phi}(X)=\int_{F^{n}}\Phi(Y)\psi\left(\langle
X,Y\rangle^{\prime}\right)\,dY,$

where we take the self-dual Haar measure on $F$ with respect to $\psi$ and the
product measure on $F^{n}$. Then, reapplied on $\mathcal{S}(F^{n})$, the Weil
representation satisfies

$\begin{split}\omega_{\psi}(\widetilde{\rho^{\prime}(g)})\Phi(X)&=\beta_{\psi}(\rho^{\prime}(g))|\det
g|^{\frac{1}{2}}\Phi(Xg),\quad g\in\mathbb{M}^{\prime},\\\
\omega_{\psi}(\widetilde{w^{\prime}_{U^{\prime}}})\Phi(X)&=\beta_{\psi}(w^{\prime}_{U^{\prime}})\hat{\Phi}(X),\\\
\omega_{\psi}\left(\widetilde{\begin{pmatrix}1_{n}&B\\\
&1_{n}\end{pmatrix}}\right)\Phi(X)&=\psi\left(\frac{1}{2}\langle
X,XB\rangle^{\prime}\right)\Phi(X),\quad B\in\mathfrak{s}_{n},\end{split}$

where $\beta_{\psi}$ is a certain eight root of unity, which satisfies
$\beta_{\psi}(\cdot)^{-1}=\beta_{\psi^{-1}}(\cdot)$.

Let $V_{0}\subset V$ be the unipotent radical of the standard parabolic
subgroup of $G$ with Levi ${\rm GL}^{n-1}_{1}\times{\rm Sp}_{2n+2}$. Then the
map

$v\mapsto
v_{\mathcal{H}_{F^{2n}}}:=\left((v_{n,n+j})_{j=1,2,\dotsm,2n},\frac{1}{2}v_{n,3n+1}\right)$

gives an isomorphism from $V/V_{0}$ to a Heisenberg group
$\mathcal{H}_{F^{2n}}$. Then we may regard $\omega_{\psi}$ as a representation
of $V/V_{0}\rtimes\widetilde{\rm Sp}(F^{2n})$. Starting with
$\psi_{N_{\mathbb{M}}}$, and with $\psi_{\circ}$ and $\psi_{V}$ as above, we
extend $\omega_{\psi_{0}}$ to a representation
$\omega_{\psi_{N_{\mathbb{M}}}}$ of $V\rtimes\tilde{G}^{\prime}$ by setting

(4.5)
$\omega_{\psi_{N_{\mathbb{M}}}}(v\tilde{g})\Phi=\psi_{V}(v)\omega_{\psi_{\circ}}(v_{\mathcal{H}_{F^{2n}}})(\omega_{\psi_{\circ}}(\tilde{g})\Phi),\quad
v\in V,g\in G^{\prime}.$

Our $\omega_{\psi_{\circ}}$ corresponds to the definition given in [GRS11,
(1.4)] with the conductor $\psi_{\circ}$ there replaced by
$\psi_{\circ}(\frac{1}{2}x)$. We find it more convenient to use this
convention and our convention follows from those of Kaplan [Kap15, §3.3.1,
Remark 4.4] and of Lapid and Mao [LM16, LM17].

###  4.4. Whittaker Models and the Intertwining Operator

Let $\pi$ be an irreducible generic representation of $\mathbb{M}$ and let
$\Pi$ denote the genuine representation $\pi\times\gamma_{\psi}$ of the double
cover $\tilde{M}$ of $M$ defined by
$(\pi\times\gamma_{\psi})(\varrho(g),\varepsilon)=\varepsilon\gamma_{\psi}(\det
g)\pi(g)$ for $(g,\varepsilon)\in\mathbb{M}\times\\{\pm 1\\}$. We denote by
${\rm Irr}_{{\rm gen},\psi_{N_{M}}}\,\tilde{M}$ the class of irreducible
genuine generic representations of $\tilde{M}$ with respect to $\psi_{N_{M}}$.
The analogous notation applies to ${\rm Irr}_{{\rm
gen},\psi_{N_{\mathbb{M}}}}\,\tilde{\mathbb{M}}$. We use
$\mathbb{W}^{\psi_{N_{\mathbb{M}}}}(\pi)$ to denote the (uniquely determined)
Whittaker space of $\pi$ with respect to the character
$\psi_{N_{\mathbb{M}}}$. Similarly we use the notation
$\mathbb{W}^{\psi^{-1}_{N_{\mathbb{M}}}}(\pi)$,
$\mathbb{W}^{\psi_{N_{M}}}(\pi)$, $\mathbb{W}^{\psi^{-1}_{N_{M}}}(\pi)$,
$\mathbb{W}^{\psi_{N_{M}}}(\Pi)$, $\mathbb{W}^{\psi^{-1}_{N_{M}}}(\Pi)$,
$\mathbb{W}^{\psi_{N_{\mathbb{M}}}}(\Pi)$,
$\mathbb{W}^{\psi^{-1}_{N_{\mathbb{M}}}}(\Pi)$.

Using the Iwasawa decomposition, we extend the character
$\varrho(g)\mapsto|\det g|$, $g\in\mathbb{M}$ to a right $K$ left
$U$-invariant $\nu(g)$ function on $G$. For any genuine function $f\in
C^{\infty}(\tilde{G})$ and $s\in\mathbb{C}$ define
$f_{s}(\tilde{g})=f(\tilde{g})\nu(g)^{s}$, $g\in G$. Let ${\rm
Ind}(\mathbb{W}^{\psi_{N_{M}}}(\Pi))$ be the space of $\tilde{G}$-smooth,
genuine, left $\tilde{U}$-invariant function
$W:\tilde{G}\rightarrow\mathbb{C}$ such that for all $g\in G$, the function
$\tilde{m}\mapsto\delta^{\frac{1}{2}}_{P}(m)W(\tilde{m}\tilde{g})$ on
$\tilde{M}$ belongs to $\mathbb{W}^{\psi_{N_{M}}}(\Pi)$. Similarly we define
${\rm Ind}(\mathbb{W}^{\psi^{-1}_{N_{M}}}(\Pi))$, ${\rm
Ind}(\mathbb{W}^{\psi_{N_{M}}}(\pi))$, ${\rm
Ind}(\mathbb{W}^{\psi^{-1}_{N_{M}}}(\pi))$, ${\rm
Ind}(\mathbb{W}^{\psi_{N_{\mathbb{M}}}}(\pi))$, ${\rm
Ind}(\mathbb{W}^{\psi^{-1}_{N_{\mathbb{M}}}}(\pi))$. For any $s\in\mathbb{C}$,
we define the intertwining operator [Kap15, §3.3]

$M(s,\Pi):{\rm Ind}(s,\mathbb{W}^{\psi_{N_{M}}}(\Pi))\rightarrow{\rm
Ind}(-s,\mathbb{W}^{\psi_{N_{M}}}(\Pi^{\vee}))$

by (the analytic continuation of)

$M(s,\Pi)\tilde{W}(\tilde{g})=\nu(g)^{s}\int_{U}\tilde{W}_{s}(\tilde{\varrho}(\mathfrak{t})\tilde{w}_{U}\tilde{u}\tilde{g})\,du$

where $\mathfrak{t}\in{\rm GL}_{2n}$ is introduced in order to preserve the
character $\psi_{N_{M}}$ and
$\widetilde{\varrho}(m)=(\varrho(m),1)\in\tilde{G}$.

In the case when $F$ is $p$-adic with $p$ odd, [Szp13, Lemma 4.2], and there
exist (necessarily unique) $\tilde{K}$-fixed elements
$\tilde{W}^{\circ}\in{\rm Ind}(\mathbb{W}^{\psi_{N_{M}}}(\Pi))$ and
$\tilde{W}^{{\vee}^{\circ}}\in{\rm
Ind}(\mathbb{W}^{\psi_{N_{M}}}(\Pi^{\vee}))$ such that

(4.6) $M(s,\Pi)\tilde{W}^{\circ}={\rm vol}(U\cap K)\frac{L(2s,\pi,{\rm
Sym}^{2})}{L(2s+1,\pi,{\rm Sym}^{2})}\tilde{W}^{{\vee}^{\circ}}.$

The following result is an analogue of [LM16, Proposition 2.1] and [LM17,
Proposition 4.1].

######  Proposition 4.1.

Let $\Pi$ be a genuine representation of $\tilde{\mathbb{M}}$ such that
$\Pi\cong\Pi^{\vee}$. Then $M(s,\Pi)$ is holomorphic at $s=\frac{1}{2}$.

###### Proof.

We let $\Pi$ be the irreducible generic quotient of a representation
parabolically induced from
$(\sigma_{1}\times\gamma_{\psi})\otimes\dotsm\otimes(\sigma_{k}\times\gamma_{\psi})$
where $\sigma_{1},\dotsm,\sigma_{k}$ are essentially square-integrable. It can
be seen in the course of the proof of [Szp13, Lemma 3.3] that $M(s,\Pi)$ is
the restriction of $M(s,\pi)$ where
$\pi=\sigma_{1}\boxplus\dotsm\boxplus\sigma_{k}$. We decompose $M(s,\pi)$ as
the composition of co-rank one intertwining operators. Just as in Proposition
3.1, the problem boils down to the holomorphy of the following intertwining
operators

  1. $(\mathrm{i})$

$\sigma_{i}|\cdot|^{s}\boxplus\sigma_{j}^{\vee}|\cdot|^{-s}\rightarrow\sigma_{j}^{\vee}|\cdot|^{-s}\boxplus\sigma_{i}|\cdot|^{s},\quad,i,j=1,\dotsm,k,i\neq
j$,

  2. $(\mathrm{ii})$

$M(s,\sigma_{i}),\quad i=1,2,\dotsm,k$,

at $s=1/2$. Then we can derive from [LM17, Lemma 4.2] and the irreducibility
of $\sigma_{i}\boxplus\sigma_{j}^{\vee}$ that the intertwining operator
$(\mathrm{i})$ is holomorphic at $s=1/2$. The holomorphy of $(\mathrm{ii})$ is
an immediate consequence of [LM17, Lemma 4.3] and the irreducibility of
$\sigma_{i}\boxplus\sigma_{i}^{\vee}$. ∎

###  4.5. Descent map

####  4.5.1. Global Descent

Suppose that $F$ is a number field. Let
$\tilde{G}^{\prime}(\mathbb{A}):=\widetilde{\rm Sp}_{2n}(\mathbb{A})$ be the
metaplectic two-fold cover of $G^{\prime}(\mathbb{A})$ which splits over
$G^{\prime}(F)$. The global double cover $\tilde{G}^{\prime}(\mathbb{A})$ of
$G^{\prime}(\mathbb{A})$ is compatible with the local double covers at each
place. (See, for example, [JS07, §2.2] for conventions.) Let
$P_{\circ}\subseteq\mathbb{M}$ be the standard parabolic subgroup in
$\mathbb{M}$ whose Levi subgroup is isomorphic to $\prod_{i=1}^{k}{\rm
GL}_{n_{i}}$ with $n_{1}+\dotsm+n_{k}=2n$ and $n_{i}>1$ for all $i$. For
$\pi\in{\rm OScusp}_{k}\,\mathbb{M}$, we associate it to the genuine
representation $\Pi={\rm
Ind}^{\tilde{\mathbb{M}}}_{\tilde{P}_{\circ}}\,\widetilde{\pi}$, where
$\tilde{\pi}$ acts as (cf. [GRS11, (2.11)])

(4.7) $\tilde{\pi}((m,\varepsilon))=\prod_{1\leq i<j\leq k}(\det m_{i},\det
m_{j})\prod_{i=1}^{k}\gamma_{\psi}(\det
m_{i})\otimes_{i=1}^{k}\pi_{i}(m_{i}),$

for $m={\rm diag}(m_{1},\dotsm,m_{k})\in\mathbb{M}$. We let ${\rm
Scusp}_{k}\,\tilde{\mathbb{M}}$ be the set of representations of $\Pi$
constructed as above and let ${\rm Scusp}\,\tilde{\mathbb{M}}=\cup_{k}{\rm
Scusp}_{k}\,\tilde{\mathbb{M}}$. Any genuine representation $\Pi$ in ${\rm
Scusp}\,\tilde{\mathbb{M}}$ is called a representation of symplectic type.

For $\Sigma\in{\rm Scusp}\,\tilde{\mathbb{M}}$, we view it as a representation
$\Pi$ of $\tilde{M}$ via
$\Sigma((\varrho(m),\varepsilon)):=\Pi((m,\varepsilon))$. Let
$\mathcal{A}(\Pi)$ be the space of functions
$\tilde{\varphi}:\tilde{M}(F)\tilde{U}(\mathbb{A})\backslash\tilde{G}(\mathbb{A})\rightarrow\mathbb{C}$
such that
$\tilde{m}\mapsto\delta_{P}^{\frac{1}{2}}(m)\varphi(\tilde{m}\tilde{g})$,
$m\in M(\mathbb{A})$ belongs to the space of $\Pi$ for all
$\tilde{g}\in\tilde{G}(\mathbb{A})$. For $\Phi\in\mathcal{S}(\mathbb{A}^{n})$
we define the theta function

$\Theta^{\Phi}_{\psi_{N_{\mathbb{M}}}}(v\tilde{g})=\sum_{\xi\in
F^{n}}\omega_{\psi_{N_{\mathbb{M}}}}(v\tilde{g})\Phi(\xi),\quad v\in
V(\mathbb{A}),g\in G^{\prime}(\mathbb{A}).$

let ${\rm FJ}_{\psi_{N_{\mathbb{M}}}}$ be the Fourier–Jacobi coefficient (a
function on $G^{\prime}(F)\backslash G^{\prime}(\mathbb{A})$)

${\rm
FJ}_{\psi_{N_{\mathbb{M}}}}(\tilde{\varphi},\Phi)(g)=\int_{V(F)\backslash
V(\mathbb{A})}\tilde{\varphi}(\tilde{v}\tilde{g})\Theta^{\Phi}_{\psi^{-1}_{N_{\mathbb{M}}}}(v\tilde{g})\,dv,\quad
g\in G^{\prime}(\mathbb{A}).$

Let $\tilde{\varphi}_{s}(g)=\nu(g)^{s}\tilde{\varphi}(\tilde{g})$. We define
Eisenstein series

$\mathcal{E}(s,\tilde{\varphi})=\sum_{\gamma\in P(F)\backslash
G(F)}\varphi_{s}(\tilde{\gamma}\tilde{g}),\quad\tilde{\varphi}\in\mathcal{A}(\Pi).$

The space of $\Pi$ is invariant under the conjugation by $\tilde{w}_{U}$. Then
we define intertwining operator
$M(s,\Pi):\mathcal{A}(\Pi)\rightarrow\mathcal{A}(\Pi)$ given by

$M(s,\Pi)\tilde{\varphi}(\tilde{g})=\nu(g)^{s}\int_{U(\mathbb{A})}\tilde{\varphi}_{s}(\tilde{w}_{U}\tilde{u}\tilde{g})\,du.$

Both $\mathcal{E}(s,\tilde{\varphi})$ and $M(s,\Pi)$ converge absolutely for
${\rm Re}(s)\gg 0$ and extend meromorphically to $\mathbb{C}$. It follows from
[GRS11, Theorem 2.1] that both $\mathcal{E}(s,\tilde{\varphi})$ and $M(s,\Pi)$
have a pole of order $k$ at $s=\frac{1}{2}$. Let

$\mathcal{E}_{-k}\tilde{\varphi}=\lim_{s\rightarrow\frac{1}{2}}\left(s-\frac{1}{2}\right)^{k}\mathcal{E}(s,\tilde{\varphi})$

and

$M_{-k}=\lim_{s\rightarrow\frac{1}{2}}\left(s-\frac{1}{2}\right)^{k}M(s,\Pi)$

be the leading coefficient in the Laurent series expansion at $s=\frac{1}{2}$.
The constant term

$\mathcal{E}^{U}_{-k}\tilde{\varphi}(\tilde{g})=\int_{U(F)\backslash
U(\mathbb{A})}\mathcal{E}_{-k}\tilde{\varphi}(\tilde{u}\tilde{g})\,du$

of $\mathcal{E}_{-k}\tilde{\varphi}$ along $U$ is given by

$\mathcal{E}^{U}_{-k}\tilde{\varphi}(\tilde{g})=(M_{-k}(\tilde{\varphi})(\tilde{g}))_{-\frac{1}{2}}.$

The descent of $\Pi$ (with respect to $\psi_{N_{\mathbb{M}}}$) is the space
$\mathcal{D}_{\psi}(\Pi)$ generated by ${\rm
FJ}_{\psi_{N_{\mathbb{M}}}}(\mathcal{E}_{-k}\tilde{\varphi},\Phi)$ with
$\tilde{\varphi}\in\mathcal{A}(\Pi)$ and $\Phi\in\mathcal{S}(\mathbb{A}^{n})$.
It is known from [GRS11, Theorem 3.1] that $\mathcal{D}_{\psi}(\Pi)$ is
multiplicity free. By [GRS11, Theorem 9.7], $\mathcal{D}_{\psi}(\Pi)$ is a
non-trivial cuspidal (and globally generic) automorphic representation of
$G^{\prime}$. We expect it to be irreducible. This result will be the analogue
of [GJS12, Theorem 2.3] in metaplectic group case and [Mor18, Theorem 1.1] in
even unitary group case. It is likely that the method of [GJS12, Theorem 2.3]
and [Mor18, Theorem 1.1] is extended to the case at hand.

######  Working Hypothesis 4.2.

When $\Pi\in{\rm Scusp}\,\tilde{\mathbb{M}}$, $\mathcal{D}_{\psi}(\Pi)$ is
irreducible.

The image of the weak lift is then a consequence of [GRS11, Theorem 11.2]. In
particular, [GRS11, Theorem 9.7 (2)] describe the image of the lift for
unramified places.

######  Proposition 4.3.

With the above Working Hypothesis 4.2, the map $\Pi={\rm
Ind}^{\tilde{\mathbb{M}}}_{\tilde{P}_{\circ}}\tilde{\pi}\mapsto\sigma=\mathcal{D}_{\psi}(\Pi)$
defines a bijection between ${\rm Scusp}\,\tilde{\mathbb{M}}$ and ${\rm
Cusp}_{\psi_{N^{\prime}}}\,G^{\prime}$, the set of $\psi_{N^{\prime}}$-generic
irreducible cuspidal automorphic representations of $G^{\prime}$. Moreover
$\sigma^{\vee}$ weekly lifts to $\pi\boxplus\omega_{\pi}$.

####  4.5.2. Local descent

We go back to the local setup. Suppose now that $F$ is local fields. Let
$\Pi\in{\rm Irr}_{{\rm gen}}\,\tilde{M}$. For any $\tilde{W}\in{\rm
Ind}(\mathbb{W}^{\psi_{N_{M}}}(\Pi))$, $\Phi\in\mathcal{S}(F^{n})$, and
$s\in\mathbb{C}$, we define a function on $G^{\prime}$:

$A^{\psi}(s,g,\tilde{W},\Phi)=\int_{V_{\gamma}\backslash
V}\tilde{W}_{s}(\tilde{\gamma}\tilde{v}\tilde{\eta}(g))\omega_{\psi_{N_{\mathbb{M}}}}(v\tilde{g})\Phi(\xi_{n})\,dv,\quad
g\in G^{\prime}.$

The basic properties of $A^{\psi}$ are summarized in the following
proposition. Its proof is identical to [LM16, Lemmata 4.5 and 4.9] and [LM17,
Lemmata 5.1 and 8.1], and will be omitted.

######  Proposition 4.4.

Suppose that $\Pi\in{\rm Irr}_{{\rm gen}}\,\tilde{M}$. Let $\tilde{W}\in{\rm
Ind}(\mathbb{W}^{\psi_{N_{M}}}(\Pi))$, $\Phi\in\mathcal{S}(F^{n})$, and
$s\in\mathbb{C}$. Then

  1. $(\mathrm{1})$

The integral $A^{\psi}(s,g,\tilde{W},\Phi)$ is well defined and absolutely
convergent uniformly for $s\in\mathbb{C}$ and $g\in G^{\prime}$ in compact
sets. Thus $A^{\psi}(s,g,\tilde{W},\Phi)$ is entire in $s$ and smooth in $g$.
In the non-archimedean case the integrand is compactly supported on
$V_{\gamma}\backslash V$.

  2. $(\mathrm{2})$

For any $\tilde{W}\in{\rm Ind}(\mathbb{W}^{\psi_{N_{M}}}(\Pi))$ and
$s\in\mathbb{C}$, the function $g\mapsto A^{\psi}(s,g,\tilde{W},\Phi)$ is
smooth and $(N^{\prime},\psi_{N^{\prime}})$-equivariant.

  3. $(\mathrm{3})$

For any $g,x\in G^{\prime}$ and $v\in V$, we have

(4.8)
$A^{\psi}(s,g,I(s,\tilde{v}\tilde{\eta}(x))\tilde{W},\omega_{\psi^{-1}_{N_{\mathbb{M}}}}(\tilde{v}\tilde{x})\Phi)=A^{\psi}(s,gx,\tilde{W},\Phi).$

  4. $(\mathrm{4})$

Suppose that $p\neq 2$, $\Pi$ is unramified, $\psi$ is unramified,
$\tilde{W}^{\circ}\in{\rm Ind}(\mathbb{W}^{\psi_{N_{M}}}(\Pi))$ is the
standard unramified vector, and $\Phi_{0}={\bf 1}_{\mathcal{O}^{n}}$. Then
$A^{\psi}(s,e,\tilde{W}^{\circ},\Phi_{0})\equiv 1$ (assuming ${\rm vol}(V\cap
K)={\rm vol}(V_{\gamma}\cap K)=1$).

Let $\pi_{i},i=1,\dotsm,k$ be pairwise inequivalent (that is,
$\pi_{i}\not\cong\pi_{j}$ if $i\neq j$) and of orthogonal symplectic type with
$n_{1}+\dotsm+n_{k}=2n$ with $n_{i}>1$ for all $i$. Let $\pi={\rm
Ind}^{\mathbb{M}}_{P_{\circ}}\otimes\pi_{i}$ of $\mathbb{M}$. We relate it to
the genuine representation $\Pi={\rm
Ind}^{\tilde{\mathbb{M}}}_{\tilde{P_{\circ}}}\tilde{\pi}$, where $\tilde{\pi}$
is defined in (4.7). Such a genuine representation $\Pi$ is called a
representation of symplectic type.

For $F$ a local field, we write ${\rm Irr}_{\rm sym}\tilde{\rm GL}_{n}(F)$ for
the set of irreducible genuine representations of symplectic type. It is clear
that if a genuine representation $\Pi$ is of symplectic type in the global
setting, then all its local components $\Pi_{v}$ are irreducible and of
symplectic type as well.

Let $\Pi\in{\rm Irr}_{\rm gen}\,\tilde{\mathbb{M}}$, considered also as a
representation of $\tilde{M}$ via $\varrho$. By the similar argument as in
[GRS99, Theorem in §1.3] together with [LM17, Remark 4.13], for any non-zero
subrepresentation $\Pi^{\prime}$ of ${\rm
Ind}(\mathbb{W}^{\psi_{N_{M}}}(\Pi))$, there exist
$\tilde{W}^{\prime}\in\Pi^{\prime}$ and $\Phi\in\mathcal{S}(F^{n})$ such that
$A^{\psi}(s,\cdot,\tilde{W}^{\prime},\Phi)\not\equiv 0$.

We assume that $\Pi\in{\rm Irr}_{{\rm gen,sym}}\tilde{\mathbb{M}}$. By
Proposition 4.1, $M(s,\Pi)$ is holomorphic at $s=\frac{1}{2}$. The local
descent of $\pi$ is the space $\mathcal{D}_{\psi}(\Pi)$ of Whittaker functions
on $G^{\prime}$ generated by
$A^{\psi}(-\frac{1}{2},\cdot,M(\frac{1}{2},\Pi)\tilde{W},\Phi)$ with
$\tilde{W}\in{\rm Ind}(\mathbb{W}^{\psi_{N_{M}}}(\Pi))$ and
$\Phi\in\mathcal{S}(F^{n})$. By the above observation,
$\mathcal{D}_{\psi}(\Pi)\neq 0$.

Let $\Pi^{\prime}$ be the image of ${\rm Ind}(\mathbb{W}^{\psi_{N_{M}}}(\Pi))$
under $M(\frac{1}{2},\Pi)$. Thanks to (4.8), the space
$\mathcal{D}_{\psi}(\Pi)$ is canonically a quotient of the $G^{\prime}$-module
$J_{V}(\Pi)$ of the $V$-coinvariant of $\Pi^{\prime}$. In this regard, we view
$J_{V}(\Pi^{\prime})$ as the abstract descent, whereas
$\mathcal{D}_{\psi}(\Pi)$ as the explicit descent.

####  4.5.3. Whittaker function of descent

We let $\hat{\psi}_{N}$ be a degenerate character on $N$ that is trivial on
$U$ and for $u\in N_{\mathbb{M}}$

$\hat{\psi}(\varrho(u)):=\psi(u_{1,2}+\dotsm+u_{n,n+1}-u_{n+1,n+2}-\dotsm-
u_{2n-1,2n}).$

to be consistent with the additive character in [GRS11, (8.14)]. Thus
$\psi_{\circ}(x)=\psi(x)$. By [LM17, Remark 6.4] (cf. [LM16, §4.1, §6.1]), we
confirm Theorem 4.5 for a specific choice of $\psi_{N_{\mathbb{M}}}$ given by

$\psi_{N_{\mathbb{M}}}(u)=\psi(u_{1,2}+\dotsm+u_{n-1,n}+2u_{n,n+1}-u_{n+1,n+2}-\dotsm-
u_{2n-1,2n}).$

just as in [GRS11, Theorem 10.3]. Henceforth $\psi_{\circ}(x)=\psi(2x)$. Let

$\delta=\left(\begin{array}[]{c|c}\delta^{\prime}_{n,2n}&0_{n\times 2n}\\\
0_{n\times 2n}&\delta^{\prime}_{n,2n}\\\
\hline\cr\delta^{\prime\prime}_{n,2n}&0_{n\times 2n}\\\ 0_{n\times
2n}&\delta^{\prime\prime}_{n,2n}\end{array}\right).$

We let $\kappa=\varrho(\kappa^{\prime})$, where $\kappa^{\prime}$ is the Weyl
element of $\mathbb{M}$ such that
$\kappa^{\prime}_{2i,i}=\kappa^{\prime}_{2i-1,n+i}=1$, $i=1,\dotsm,n$, and all
other entries are zero. Let $\epsilon={\rm
diag}(A,\dotsm,A,A^{\ast},\dotsm,A^{\ast})$, where $A=\begin{pmatrix}[r]1&1\\\
-1&1\end{pmatrix}$. Here $A$ is repeated $n$ times (cf. [GRS11, Proof of
Proposition 9.5]). Writing

$A=\begin{pmatrix}1&1\\\ &1\end{pmatrix}\begin{pmatrix}2&\\\
&1\end{pmatrix}\begin{pmatrix}[r]1&\\\ -1&1\end{pmatrix}$

we get $\epsilon=\epsilon_{U}\epsilon_{T}\epsilon_{\overline{U}}$, where

$\epsilon_{U}={\rm diag}(\begin{pmatrix}1&1\\\
&1\end{pmatrix},\dotsm,\begin{pmatrix}1&1\\\
&1\end{pmatrix},\begin{pmatrix}[r]1&-1\\\
&1\end{pmatrix},\dotsm,\begin{pmatrix}[r]1&-1\\\ &1\end{pmatrix}),$
$\epsilon_{\overline{U}}={\rm diag}(\begin{pmatrix}[r]1&\\\
-1&1\end{pmatrix},\dotsm,\begin{pmatrix}[r]1&\\\
-1&1\end{pmatrix},\begin{pmatrix}1&\\\
1&1\end{pmatrix},\dotsm,\begin{pmatrix}1&\\\ 1&1\end{pmatrix}),$

and

$\epsilon_{T}={\rm diag}(\begin{pmatrix}2&\\\
&1\end{pmatrix},\dotsm,\begin{pmatrix}2&\\\
&1\end{pmatrix},\begin{pmatrix}1&\\\
&\frac{1}{2}\end{pmatrix},\dotsm,\begin{pmatrix}1&\\\
&\frac{1}{2}\end{pmatrix}).$

Let $X$ be the subspace of $\mathfrak{s}_{2n}$ consisting of the strictly
upper triangular matrices. We let

$\overline{\ell}(x)=\begin{pmatrix}1_{2n}&\\\ x&1_{2n}\end{pmatrix}$

for $x\in\mathfrak{s}_{2n}$. Let $Y$ be the subgroup of $N_{M}$ consisting of
the matrices of the form $\varrho(\begin{pmatrix}1_{n}&y\\\
&1_{n}\end{pmatrix})$ where $y$ is lower triangular, (namely $y_{i,j}=0$ if
$j>i$). Let $Y^{\prime}$ be the subgroup of $Y$ consisting of the matrices of
the form $\varrho(\begin{pmatrix}1_{n}&x\\\ &1_{n}\end{pmatrix})$ where $x$ is
lower triangular with last row $0$ and
$a^{+}=\varrho(\begin{pmatrix}1_{n}&z\\\ &1_{n}\end{pmatrix})$ where the last
row of $z$ is $a$ and all other rows are zero.

######  Theorem 4.5.

$($Reformulation of [GRS11, Theorem 9.7, part (1)]$)$ Let $\Pi\in{\rm
Scusp}\,\mathbb{M}$ and $\tilde{\varphi}$. Then for any $g\in G^{\prime}$ we
have

$\mathcal{W}^{\psi_{N^{\prime}}}(g,{\rm
FJ}_{\psi_{N_{\mathbb{M}}}}(\mathcal{E}_{-k}\tilde{\varphi},\Phi))=\int_{V_{\gamma}(\mathbb{A})\backslash
V(\mathbb{A})}\mathcal{W}^{\psi_{N}}(\tilde{\gamma}\tilde{v}\tilde{\eta}(g),\mathcal{E}_{-k}\tilde{\varphi})\omega_{\psi^{-1}_{N_{\mathbb{M}}}}(v\tilde{g})\Phi(\xi_{n})\,dv,$

where the integral is absolutely convergent.

###### Proof.

We use Tamagawa measure in the proof. It is enough to prove the required
identity for $g=e$. The expression for $\mathcal{W}^{\psi_{N^{\prime}}}(e,{\rm
FJ}_{\psi_{N_{\mathbb{M}}}}(\mathcal{E}_{-k}\tilde{\varphi},\Phi)$ in [GRS11,
Theorem 9.7, part (1)] is (with $\lambda=-1$ in their notation):

(4.9)
$\int_{\mathbb{A}^{n}}\left(\int_{Y^{\prime}(\mathbb{A})}\left(\int_{X(\mathbb{A})}\mathcal{W}^{\hat{\psi}_{N}}(\tilde{\overline{\ell}}(x)\tilde{\delta}\tilde{\epsilon}\tilde{\kappa}\tilde{y}^{\prime}\tilde{a}^{+},\mathcal{E}_{-k}\tilde{\varphi})\Phi(a)\,dx\right)\,dy^{\prime}\right)\,da.$

Combining $\tilde{y}^{\prime}$ and $\tilde{a}^{+}$, it is evident from (4.2)
and (4.5) that we can rewrite the above expression as

$\int_{Y(\mathbb{A})}\left(\int_{X(\mathbb{A})}\mathcal{W}^{\hat{\psi}_{N}}(\tilde{\overline{\ell}}(x)\tilde{\delta}\tilde{\epsilon}\tilde{\kappa}\tilde{y},\mathcal{E}_{-k}\tilde{\varphi})\omega_{\psi^{-1}_{N_{\mathbb{M}}}}(y)\Phi(0)\,dx\right)\,dy.$

We note from [GRS11, §8.2, (8.15)] that
$\delta\epsilon_{U}\delta^{-1}=\begin{pmatrix}[r]1_{2n}&-1_{2n}\\\
&1_{2n}\end{pmatrix}$. For any $x\in X(\mathbb{A})$, we proceed to write

$\begin{pmatrix}1_{2n}&\\\
x&1_{2n}\end{pmatrix}\begin{pmatrix}[r]1_{2n}&-1_{2n}\\\
&1_{2n}\end{pmatrix}=\begin{pmatrix}(1_{2n}-x)^{-1}&-1_{2n}\\\
&1_{2n}-x\end{pmatrix}\begin{pmatrix}1_{2n}&\\\
x^{\prime}&1_{2n}\end{pmatrix},$

where $x^{\prime}=(1_{2n}-x)^{-1}x=x+x^{2}+\dotsm+x^{2n-1}$. (Note that
$(1_{2n}-x)^{-1}=(1_{2n}-x)^{\ast}$.) After a change of variable
$x\mapsto(1_{2n}-x)x$ we get

$\int_{Y(\mathbb{A})}\left(\int_{X(\mathbb{A})}\psi(x_{n,n+1})\mathcal{W}^{\hat{\psi}_{N}}(\tilde{\overline{\ell}}(x)\tilde{\delta}\tilde{\epsilon}_{T}\tilde{\epsilon}_{\overline{U}}\tilde{\kappa}\tilde{y},\mathcal{E}_{-k}\tilde{\varphi})\omega_{\psi^{-1}_{N_{\mathbb{M}}}}(y)\Phi(0)\,dx\right)\,dy$

because the relation $x_{k,k+1}=x_{2n-k,2n+1-k}$ yields that
$\hat{\psi}_{N}(\varrho((1_{2n}-x)^{-1}))=\psi(x^{\prime}_{n,n+1})=\psi(x_{n,n+1})$.

In addition, $\delta\epsilon_{T}\delta^{-1}={\rm diag}(2\cdot
1_{n},1_{n},1_{n},\frac{1}{2}\cdot 1_{n})$. This element corresponds
$\hat{\psi}_{N}$ to $\psi_{N}$, and stabilizes $\psi(x_{n,n+1})$, so by
conjugating we obtain

$\int_{Y(\mathbb{A})}\left(\int_{X(\mathbb{A})}\psi(x_{n,n+1})\mathcal{W}^{\psi_{N}}(\tilde{\overline{\ell}}(x)\tilde{\delta}\tilde{\epsilon}_{\overline{U}}\tilde{\kappa}\tilde{y},\mathcal{E}_{-k}\tilde{\varphi})\omega_{\psi^{-1}_{N_{\mathbb{M}}}}(y)\Phi(0)\,dx\right)\,dy$

Note that
$\kappa^{-1}\epsilon_{\overline{U}}\kappa=\varrho(\begin{pmatrix}[r]1_{n}&-1_{n}\\\
&1_{n}\end{pmatrix})$ and $\gamma=\delta\kappa$. Upon changing variables
$y\mapsto(\kappa^{-1}\epsilon_{\overline{U}}\kappa)^{-1}y$, we get by (4.2)
and (4.5) that the above equals

$\int_{Y(\mathbb{A})}\left(\int_{X}\psi(x_{n,n+1})\mathcal{W}^{\psi_{N}}(,\tilde{\overline{\ell}}(x)\tilde{\gamma}\tilde{y},\mathcal{E}_{-k}\tilde{\varphi})\omega_{\psi^{-1}_{N_{\mathbb{M}}}}(y)\Phi(\xi_{n})\,dx\right)\,dy.$

We note that

$\gamma^{-1}\overline{\ell}(X)\gamma=\\{\begin{pmatrix}1_{n}&x_{1}&\vspace{-.07cm}&x_{2}\\\
&1_{n}&&\vspace{-.07cm}\\\ \vspace{-.07cm}&&1_{n}&\check{x}_{1}\\\
&&&1_{n}\end{pmatrix}\,|\,x_{1}\in{\rm Mat}_{n,n}\;\text{\rm strictly upper
triangular},x_{2}\in\mathfrak{s}_{n}\\},$

so that $(\gamma^{-1}\overline{\ell}(X)\gamma Y)\rtimes V_{\gamma}=V$. In
conclusion, we end up at

(4.10) $\int_{V_{\gamma}(\mathbb{A})\backslash
V(\mathbb{A})}\mathcal{W}^{\psi_{N}}(\tilde{\gamma}\tilde{v},\mathcal{E}_{-k}\tilde{\varphi})\omega_{\psi^{-1}_{N_{\mathbb{M}}}}(v)\Phi(\xi_{n})\,dx$

provided that it converges, since by (4.4) and (4.5)
$\omega_{\psi^{-1}_{N_{\mathbb{M}}}}(\gamma^{-1}\overline{\ell}(x)\gamma)$
acts by the scalar $\psi(x_{n,n+1})$ for $x\in X(\mathbb{A})$. Apparently
(4.9) and (4.10) are equal up to an appropriate sign $\mathfrak{e}\in\\{\pm
1\\}$. We can actually determine the precise constant $\mathfrak{e}=1$ by
using the algorithm for computing the cocycle. But since this computation is
extremely tedious, though not so deep, we do not include detailed calculation
of $\mathfrak{e}$. ∎

###  4.6. Local Rankin–Selberg–Shimura integrals

We are now going back to our choice of $\psi_{N_{\mathbb{M}}}$ specified in
(4.1). Now let $F$ be either $p$-adic or archimedean, $\Pi\in{\rm Irr}_{\rm
gen}\,\tilde{M}$ and $\sigma\in{\rm Irr}_{{\rm
gen},\psi^{-1}_{N^{\prime}}}\,G^{\prime}$ with Whittaker model
$\mathbb{W}^{\psi^{-1}_{N^{\prime}}}(\sigma)$. For any
$W^{\prime}\in\mathbb{W}^{\psi^{-1}_{N^{\prime}}}(\sigma)$, $\tilde{W}\in{\rm
Ind}(\mathbb{W}^{\psi_{N_{M}}}(\Pi))$, and $\Phi\in\mathcal{S}(F^{n})$, we
define the local Rankin–Selberg–Shimura type integral

$J(s,W^{\prime},\tilde{W},\Phi):=\int_{N^{\prime}\backslash
G^{\prime}}W^{\prime}(g)A^{\psi}(s,g,\tilde{W},\Phi)\,dg.$

The analytic properties of this integral were studied by
Ginzburg–Rallis–Soudry [GRS98] and subsequently established by Kaplan [Kap15].

######  Proposition 4.6.

Suppose that $\Pi\in{\rm Irr}_{{\rm gen}}\,\tilde{M}$ and $\sigma\in{\rm
Irr}_{{\rm gen},\psi^{-1}_{N^{\prime}}}\,G^{\prime}$. Let
$W^{\prime}\in\mathbb{W}^{\psi^{-1}_{N^{\prime}}}(\sigma)$, $\tilde{W}\in{\rm
Ind}(\mathbb{W}^{\psi_{N_{M}}}(\Pi))$, and $\Phi\in\mathcal{S}(F^{n})$. Then

  1. $(\mathrm{1})$

$J(s,W^{\prime},\tilde{W},\Phi)$ converges in some right-half plane (depending
only on $\pi$ and $\sigma$) and admits a meromorphic continuation in $s$.

  2. $(\mathrm{2})$

For any $s\in\mathbb{C}$, we can choose $W^{\prime}$, $\tilde{W}$, and $\Phi$
such that $J(s,W^{\prime},\tilde{W},\Phi)\neq 0$.

  3. $(\mathrm{3})$

If $\Pi$, $\sigma$, and $\psi$ are unramified, $\tilde{W}^{\circ}\in{\rm
Ind}(\mathbb{W}^{\psi_{N_{M}}}(\Pi))$ is the standard unramified vector,
$\Phi_{0}={\bf 1}_{\mathcal{O}^{n}}$, and ${W^{\prime}}^{\circ}$ is
$K^{\prime}$-invariant with ${W^{\prime}}^{\circ}(e)=1$, then

$J(s,{W^{\prime}}^{\circ},\tilde{W}^{\circ},\Phi_{0})=\frac{{\rm
vol}(K^{\prime})}{{\rm vol}(N^{\prime}\cap
K^{\prime})}\frac{L(s+\frac{1}{2},\sigma\times\pi)}{L(2s+1,\pi,{\rm
Sym}^{2})}$

assuming the Haar measure on $V$ and $V_{\gamma}$ are normalized so that ${\rm
vol}(V\cap K)$ and ${\rm vol}(V_{\gamma}\cap K)$ are all $1$.

  4. $(\mathrm{4})$

$J(s,W^{\prime},\tilde{W},\Phi)$ satisfies local functional equations:

$J(-s,W^{\prime},M(s,\Pi)\tilde{W},\Phi)=\omega^{n}_{\pi}(-1)\frac{\gamma(s+\frac{1}{2},\sigma\times\pi,\psi)}{C^{\widetilde{\rm
Sp}_{2n}}_{\psi}(s+\frac{1}{2},\pi,w_{U})}J(s,W^{\prime},\tilde{W},\Phi)$

where $C^{\widetilde{\rm Sp}_{2n}}_{\psi}(s+\frac{1}{2},\pi,w_{U})$ is the
Langlands-Shahidi local coefficient for the metaplectic group defined by
Szpruch [Szp13].

It is worthwhile to mention that $\displaystyle{\rm
vol}(K^{\prime})=(\prod_{j=1}^{n}\zeta_{F}(2i))^{-1}.$

###  4.7. Factorization of Fourier–Jacobi periods

We define for $\tilde{\varphi}\in\mathcal{A}(\Pi)$ an automorphic form on
$G(\mathbb{A})$ by

$A^{\psi}(s,g,\tilde{\varphi},\Phi):=\int_{V_{\gamma}(\mathbb{A})\backslash
V(\mathbb{A})}\mathcal{W}^{\psi_{N}}(\tilde{\gamma}\tilde{v}\tilde{\eta}(g),\tilde{\varphi}_{s})\omega_{\psi^{-1}_{N_{\mathbb{M}}}}(v\tilde{g})\Phi(\xi_{n})\,dv.$

The global descent $\mathcal{D}_{\psi}(\Pi)$ and the local descent
$\mathcal{D}_{\psi}(\Pi_{v})$ are related in the following way:

######  Proposition 4.7.

Let $\Phi=\otimes_{v}\Phi_{v}\in\mathcal{S}(\mathbb{A}^{n})$ and
$\tilde{\varphi}\in\mathcal{A}(\Pi)$ be factorizable vectors. Suppose that
$\mathcal{W}^{\psi_{N_{M}}}(\cdot,\tilde{\varphi})=\prod_{v}\tilde{W}_{v}$
with $\tilde{W}_{v}\in{\rm Ind}(\mathbb{W}^{\psi_{N_{M}}}(\Pi_{v}))$. Then for
any sufficiently large finite set of places $S$, we have

(4.11) $\mathcal{W}^{\psi_{N^{\prime}}}(g,{\rm
FJ}_{\psi_{N_{\mathbb{M}}}}(\mathcal{E}_{-k}\tilde{\varphi},\Phi))=\frac{m^{S}_{-k,1}(\pi)}{2^{k}}\cdot\prod_{v\in
S}A^{\psi}\left(-\frac{1}{2},g_{v},M\left(\frac{1}{2},\Pi_{v}\right)\tilde{W}_{v},\Phi_{v}\right),$

where

$m^{S}_{-k,1}(\pi)=\frac{\lim_{s\rightarrow 1}(s-1)^{k}L^{S}(s,\pi,{\rm
Sym}^{2})}{L^{S}(2,\pi,{\rm Sym}^{2})}.$

###### Proof.

We note that

$\mathcal{W}^{\psi_{N}}(\mathcal{E}_{-k}\tilde{\varphi})=\mathcal{W}^{\psi_{N_{M}}}(\mathcal{E}^{U}_{-k}\tilde{\varphi})=\mathcal{W}^{\psi_{N_{M}}}(M_{-k}(\tilde{\varphi})_{-\frac{1}{2}}).$

With this in hand, we rewrite Theorem 4.5 in terms of the global transforms
$A^{\psi}$ as

(4.12) $\mathcal{W}^{\psi_{N^{\prime}}}(g,{\rm
FJ}_{\psi_{N_{\mathbb{M}}}}(\mathcal{E}_{-k}\tilde{\varphi},\Phi))=A^{\psi}\left(-\frac{1}{2},g,M_{-k}(\tilde{\varphi}),\Phi\right).$

We can identify $\mathcal{A}(\Pi)$ with ${\rm
Ind}^{\tilde{G}(\mathbb{A})}_{\tilde{P}(\mathbb{A})}\,\Pi=\otimes_{v}{\rm
Ind}^{\tilde{G}(F_{v})}_{\tilde{P}(F_{v})}\,\Pi_{v}$. For ${\rm Re}(s)\gg 0$,
we have

$\mathcal{W}^{\psi_{N_{M}}}(\tilde{g},M(s,\Pi)\tilde{\varphi})=\int_{N_{M}(F)\backslash
N_{M}(\mathbb{A})}M(s,\Pi)\varphi(\tilde{n}\tilde{g})\psi^{-1}_{N_{M}}(n)\,dn\\\
=\int_{N_{M}(F)\backslash
N_{M}(\mathbb{A})}\int_{U(\mathbb{A})}\tilde{\varphi}(\tilde{\varrho}(\mathfrak{t})\tilde{w}_{U}\tilde{u}\tilde{n}\tilde{g})\psi^{-1}_{N_{M}}(n)\,du\,dn=\int_{U(\mathbb{A})}\mathcal{W}^{\psi_{N_{M}}}(\tilde{\rho}(\mathfrak{t})\tilde{w}_{U}\tilde{u}\tilde{g},\tilde{\varphi})\,du$

from which it follows that for $S$ sufficiently large

$\mathcal{W}^{\psi_{N_{M}}}(\tilde{g},M(s,\Pi)\tilde{\varphi})=m^{S}(s,\pi)\left(\prod_{v\in
S}M(s,\Pi_{v})\tilde{W}_{v}\right)\prod_{v\notin S}\tilde{W}_{v},$

as meromorphic functions in $s\in\mathbb{C}$. Here

$m^{S}(s,\pi)=\frac{L^{S}(2s,\pi,{\rm Sym}^{2})}{L^{S}(2s+1,\pi,{\rm
Sym}^{2})}.$

Taking the limit as $s\rightarrow\frac{1}{2}$, we get for $S$ large enough,

(4.13)
$\mathcal{W}^{\psi_{N_{M}}}(M_{-k}\tilde{\varphi})=m^{S}_{-k}(\pi)\prod_{v\in
S}M\left(\frac{1}{2},\Pi_{v}\right)\tilde{W}_{v}\prod_{v\notin
S}\tilde{W}_{v},$

where

$m^{S}_{-k}(\pi)=\lim_{s\rightarrow\frac{1}{2}}\left(s-\frac{1}{2}\right)^{k}m^{S}(s,\pi)=2^{-k}\frac{\lim_{s\rightarrow
1}^{k}L^{S}(s,\pi,{\rm Sym}^{2})}{L(2,\pi,{\rm
Sym}^{2})}=\frac{m^{S}_{-k,1}(\pi)}{2^{k}}.$

For any factorizable $\tilde{\varphi}\in\mathcal{A}(\Pi)$, the desired
conclusion follows from (4.12) and (4.13) ∎

We unfold the Petersson inner product of automorphic forms against the
Fourier–Jacobi coefficient of Eisenstein series [GRS11, Theorem 10.4, (10.6)]
and then perform unramified computations [GRS11, (10.62)] to deduce the
following Euler product.

######  Proposition 4.8.

Let $\Pi\in{\rm Scusp}_{k}\,\tilde{\mathbb{M}}$ and
$\sigma=\mathcal{D}_{\psi^{-1}}(\Pi)$. Let $\varphi^{\prime}\in\sigma$ be an
irreducible $\psi^{-1}_{N^{\prime}}$-generic cuspidal representation of
$G^{\prime}$ and $\tilde{\varphi}\in\mathcal{A}(\Pi)$. Suppose that
$\mathcal{W}^{\psi^{-1}_{N^{\prime}}}(\cdot,\varphi^{\prime})=\prod_{v}W^{\prime}_{v}$,
$\mathcal{W}^{\psi_{N_{M}}}(\cdot,\tilde{\varphi})=\prod_{v}\tilde{W}_{v}$,
$\Phi=\otimes_{v}\Phi_{v}$. Then for any sufficiently large finite set of
places $S$, we have

(4.14) $\langle\varphi^{\prime},{\rm
FJ}_{\psi_{N_{\mathbb{M}}}}(\mathcal{E}(s,\tilde{\varphi}),\Phi)\rangle_{G^{\prime}}=\frac{{\rm
vol}(N^{\prime}(\mathcal{O}_{S})\backslash
N^{\prime}(F_{S}))}{\prod_{j=1}^{n}\zeta^{S}_{F}(2j)}\cdot\frac{L^{S}(s+\frac{1}{2},\sigma\times\pi)}{L^{S}(2s+1,\pi,{\rm
Sym}^{2})}\prod_{v\in S}J(s,W^{\prime}_{v},\tilde{W}_{v},\Phi_{v}),$

where on the right-hand side we take the unnormalized Tamagawa measure on
$G^{\prime}(F_{S})$, $N^{\prime}(F_{S})$, $V(F_{S})$ and $V_{\gamma}(F_{S})$
(which are independent of the choice of gauge forms )

###### Proof.

It is worthwhile to give our attention to the fact that the volume of
$G^{\prime}(F)\backslash G^{\prime}(\mathbb{A})$, appearing on the right-hand
side of (2.1), is 1 equal to the Tamagawa number when we use the Tamagawa
measure as in [LM16, LM17]. We apply [GRS11, Theorem 10.4, (10.5)] in order to
unfold

(4.15) $\langle\varphi^{\prime},{\rm
FJ}_{\psi_{N_{\mathbb{M}}}}(\mathcal{E}(s,\tilde{\varphi}),\Phi)\rangle_{G^{\prime}}={\rm
vol}(N^{\prime}(F)\backslash
N^{\prime}(\mathbb{A}))\int_{N^{\prime}(\mathbb{A})\backslash
G^{\prime}(\mathbb{A})}\mathcal{W}^{\psi^{-1}_{N^{\prime}}}(g,\varphi^{\prime})A^{\psi}(s,g,\tilde{\varphi},\Phi)\,dg.$

Similar to [LM16, Propositions 5.7 and 8.4], our desired result is immediate
from (4.15) in conjunction with the unramified computation [GRS98] (cf.
[GRS11, (10.62)]):

(4.16)
$J(s,{W^{\prime}_{v}}^{\circ},\tilde{W}_{v}^{\circ},\Phi_{0})=\frac{{\rm
vol}(K^{\prime})}{{\rm vol}(N^{\prime}(F_{v})\cap
K^{\prime})}\frac{L(s+\frac{1}{2},\sigma_{v}\times\pi_{v})}{L(2s+1,\pi_{v},{\rm
Sym}^{2})}$

with $K^{\prime}=K_{{\rm GL}_{2n}(F_{v})}\cap G^{\prime}(F_{v})$. Since we use
the Tamagawa measure, the factor $\Delta^{S}_{{\rm Sp}_{2n}}(1)^{-1}$ pops up,
where $\Delta^{S}_{{\rm Sp}_{2n}}(1)=L(M^{\vee}(s))$ is the partial
$L$-function of the dual motive $M^{\vee}$ of the motive $M$ associated to
$\Delta^{S}_{{\rm Sp}_{2n}}(1)$ by Gross [Gro97]. Thanks to [Xue17, Notation
and Convention],

$\Delta^{S}_{{\rm Sp}_{2n}}(1)=\prod_{j=1}^{n}\zeta^{S}_{F}(2i)$

for any finite set of places $S$. ∎

###  4.8. Petersson inner products and Fourier coefficients

Let $F$ denote a local field and let $\Pi$ be a genuine representation of
$\tilde{\mathbb{M}}(F)$ of symplectic type. We say that $\Pi$ is good if the
following condition are satisfied for all $\psi$:

  1. $(\mathrm{1})$

$\mathcal{D}_{\psi}(\Pi)$ is irreducible as a representation of $G^{\prime}$.

  2. $(\mathrm{2})$

$J(s,W^{\prime},\tilde{W},\Phi)$ is holomorphic at $s=\frac{1}{2}$ for any
$W^{\prime}\in\mathcal{D}_{\psi^{-1}}(\Pi)$, $\tilde{W}\in{\rm
Ind}^{\psi_{N_{M}}}(\Pi)$, and $\Phi\in\mathcal{S}(F^{n})$.

  3. $(\mathrm{3})$

For any $W^{\prime}\in\mathcal{D}_{\psi^{-1}}(\Pi)$,
$J(\frac{1}{2},W^{\prime},\tilde{W},\Phi)$ factors through the map
$\tilde{W}\mapsto(A^{\psi}(-\frac{1}{2}),\cdot,M(\frac{1}{2},\Pi)\tilde{W},\Phi)$.

We know from (4.8) that for $x\in G^{\prime}$ and $v\in V$ we have

$J(s,\sigma(x)W^{\prime},I(s,\tilde{v}\tilde{\eta}(x))\tilde{W},\omega_{\psi^{-1}_{N_{\mathbb{M}}}}(v\tilde{x})\Phi)=J(s,W^{\prime},\tilde{W},\Phi)$

where $\sigma=\mathcal{D}_{\psi^{-1}}(\Pi)$. Henceforth if $\Pi$ is good,
there is a non-degenerate $G^{\prime}$-invariant pairing $[\cdot,\cdot]$ on
$\mathcal{D}_{\psi^{-1}}(\Pi)\times\mathcal{D}_{\psi}(\Pi)$ such that

$J\left(\frac{1}{2},W^{\prime},\tilde{W},\Phi\right)=\left[W^{\prime},A^{\psi}\left(-\frac{1}{2},\cdot,M\left(\frac{1}{2},\Pi\right)\tilde{W},\Phi\right)\right]$

for any $W^{\prime}\in\mathcal{D}_{\psi^{-1}}(\Pi)$, $\tilde{W}\in{\rm
Ind}(\mathcal{W}^{\psi_{N_{M}}}(\Pi))$, and $\Phi\in\mathcal{S}(F^{n})$.

(4.17) $\int^{\rm
st}_{N^{\prime}}J\left(\frac{1}{2},\sigma(u)W^{\prime},\tilde{W},\Phi\right)\psi_{N^{\prime}}(u)\,du=c_{\Pi}W^{\prime}(e)A^{\psi}\left(-\frac{1}{2},e,M\left(\frac{1}{2},\Pi\right)\tilde{W},\Phi\right).$

######  Proposition 4.9.

Suppose that $F$ is $p$-adic with $p\neq 2$, $\Pi$ is of symplectic type and
unramified and $\psi$ is unramified. Then
$\sigma=\mathcal{D}_{\psi^{-1}}(\Pi)$ is irreducible and unramifield. Let
$\tilde{W}^{\circ}\in{\rm Ind}(\mathbb{W}^{\psi_{N_{M}}}(\Pi))$ be the
standard unramified $\tilde{K}$-vector and let ${W^{\prime}}^{\circ}$ be
$K^{\prime}$-invariant with $W^{\prime}(e)=1$. Then holds with $c_{\Pi}=1$.

###### Proof.

The first part of the proof is similar to the one of [LM17, Lemma 5.4], so we
omit it. Assuming ${\rm vol}(U\cap K)=1$ as in (4.6), we deduce

${\rm vol}(K^{\prime})={\rm vol}(U\cap
K)(\prod_{j=1}^{n}\zeta^{S}_{F}(2j))^{-1}=(\prod_{j=1}^{n}\zeta^{S}_{F}(2j))^{-1}.$

In virtue of it along with [LM15, Proposition 2.14], we find that

$\int^{\rm
st}_{N^{\prime}}J\left(\frac{1}{2},\sigma(u){W^{\prime}}^{\circ},\tilde{W}^{\circ},\Phi_{0}\right)\psi^{-1}_{N^{\prime}}(u)\,du=\frac{{\rm
vol}(N^{\prime}\cap K^{\prime})}{{\rm
vol}(K^{\prime})}\frac{J(\frac{1}{2},{W^{\prime}}^{\circ},W^{\circ},\Phi_{0})}{L(1,\sigma,{\rm
Ad})}$

It is verified from the description of $\sigma$ in this case [GRS11, Theorem
6.4] together with [GGP12, §7] that

$L(1,\sigma,{\rm
Ad})=L(1,\pi\boxplus\omega_{\pi},\wedge^{2})=L(1,\pi,\wedge^{2})L(1,\pi\times\omega_{\pi})$

and $L(s,\sigma\times\pi)=L(s,\pi\times\pi)L(s,\omega_{\pi}\times\pi)$. We
recall that $\pi=\pi^{\vee}$ in our case. Thus (4.6) aligned with of (4.16)
implies that the above is equal to

$\frac{1}{L(1,\pi,\wedge^{2})L(1,\pi\times\omega_{\pi})}\frac{L(1,\pi\times\pi)L(1,\omega_{\pi}\times\pi)}{L(2,\pi,{\rm
Sym}^{2})}=\frac{L(1,\pi,{\rm Sym}^{2})}{L(2,\pi,{\rm Sym}^{2})}\\\
=M\left(\frac{1}{2},\Pi\right)\tilde{W}^{\circ}(e)={W^{\prime}}^{\circ}(e)A^{\psi}\left(-\frac{1}{2},e,M\left(\frac{1}{2},\Pi\right)\tilde{W}^{\circ},\Phi_{0}\right)$

that we look for. ∎

At first glance, $c_{\Pi}$ implicitly depends on the choice of Haar measure on
$G^{\prime}$ (the former exploited in the definition of
$J(s,W^{\prime},\tilde{W},\Phi)$) and $U$ (the latter used in the definition
of the intertwining operator), but not on any other groups (refer to [LM17,
Remark 5.2] for further details). However the vector spaces underlying the Lie
algebras of $G^{\prime}$ and $U$ are both canonically isomorphic as vectors
spaces to

$\\{X\in{\rm Mat}_{2n,2n}(F)\,|\,{X^{t}}w_{2n}=w_{2n}X\\}.$

We can henceforth identify the gauge forms on $G^{\prime}$ and $U$. With this
in hand, $c_{\Pi}$ does not depend on any choice provided that we use the
unnormalized Tamagawa measure on $G^{\prime}$ and $U$ with respect to the same
gauge form. We will say that the Haar measures on $G^{\prime}$ and $U$ are
compatible in this case.

######  Proposition 4.10.

Let $\Pi\in{\rm Scusp}_{k}\,\widetilde{\mathbb{M}}$ and
$\sigma=\mathcal{D}_{\psi^{-1}}(\Pi)$. Assume our Working Hypothesis 4.2. Let
$\varphi^{\prime}\in\sigma$ be an irreducible $\psi^{-1}_{N^{\prime}}$-generic
cuspidal automorphic representation of $G^{\prime}$ and
$\tilde{\varphi}\in\mathcal{A}(\Pi)$. Suppose that
$\mathcal{W}^{\psi^{-1}_{N^{\prime}}}(\cdot,\varphi^{\prime})=\prod_{v}W^{\prime}_{v}$,
$\mathcal{W}^{\psi_{N_{M}}}(\cdot,\tilde{\varphi})=\prod_{v}\tilde{W}_{v}$ ,
$\Phi=\otimes_{v}\Phi_{v}$. Then for all $v$, $\Pi_{v}$ is good. Moreover for
any sufficiently large finite set of places $S$, we have

(4.18) $\langle\varphi^{\prime},{\rm
FJ}_{\psi_{N_{\mathbb{M}}}}(\mathcal{E}_{-k}\tilde{\varphi},\Phi)\rangle_{G^{\prime}}=\frac{{\rm
vol}(N^{\prime}(\mathcal{O}_{S})\backslash
N^{\prime}(F_{S}))}{\prod_{j=1}^{n}\zeta^{S}_{F}(2j)}\cdot
n^{S}_{-k,1}\prod_{v\in
S}J\left(\frac{1}{2},W^{\prime}_{v},\tilde{W}_{v},\Phi_{v}\right),$

where

$n^{S}_{-k,1}=\frac{\lim_{s\rightarrow
1}L^{S}(s,\pi\times\pi)L^{S}(s,\pi\times\omega_{\pi})}{L^{S}(2,\pi,{\rm
Sym}^{2})}.$

###### Proof.

We confirm that $\Pi_{v}$ is good for all $v$. To this end, we draw the
conclusion from (4.11) together with the irreducibility of the global descent
that $\mathcal{D}_{\psi}(\Pi_{v})$ is irreducible for all $v$.

When $\Pi\in{\rm Scusp}_{k}\,\tilde{\mathbb{M}}$ and
$\sigma\in\mathcal{D}_{\psi^{-1}}(\Pi)$, we obtain
$L^{S}(s,\sigma\times\pi)=L^{S}(s,\pi\times\pi)L^{S}(s,\omega_{\pi}\times\pi)$,
which has a pole of order $k$ at $s=\frac{1}{2}$. On the one hand,
$J(s,W^{\prime}_{v},\tilde{W}_{v},\Phi_{v})$ is non-vanishing for suitable
$W^{\prime}_{v}$, $\tilde{W}_{v}$, and $\Phi_{v}$, from which it follows that
the right-hand side of (4.14) (when $\varphi^{\prime}\in\sigma$) has a pole of
order at most $k$ at $s=\frac{1}{2}$ for appropriate $\varphi^{\prime}$,
$\tilde{\varphi}$, and $\Phi$. On the other hand, the left-hand side of (4.14)
has a pole of order at most $k$ because this is true for
$\mathcal{E}(s,\tilde{\varphi})$ and $\varphi^{\prime}$ is rapidly decreasing.
Multiplying (4.14) by $(s-\frac{1}{2})^{k}$ and then taking the limit as
$s\rightarrow\frac{1}{2}$, we conclude that
$J(s,W^{\prime}_{v},\tilde{W}_{v},\Phi_{v})$ is holomorphic at $s=\frac{1}{2}$
for all $v$, and for $\varphi^{\prime}\in\sigma$,
$\tilde{\varphi}\in\mathcal{A}(\Pi)$, and $\Phi\in\mathcal{S}(\mathbb{A}^{n})$
we get

(4.19) $\langle\varphi^{\prime},{\rm
FJ}_{\psi_{N_{\mathbb{M}}}}(\mathcal{E}_{-k}\tilde{\varphi},\Phi)\rangle_{G^{\prime}}=\frac{{\rm
vol}(N^{\prime}(\mathcal{O}_{S})\backslash
N^{\prime}(F_{S}))}{\prod_{j=1}^{n}\zeta^{S}_{F}(2j)}\cdot
n^{S}_{-k,1}\prod_{v\in
S}J\left(\frac{1}{2},W^{\prime}_{v},\tilde{W}_{v},\Phi_{v}\right).$

We fix a place $v_{0}$. We assume that $\tilde{W}_{v_{0}}$ and $\Phi_{v_{0}}$
are such that
$A^{\psi}\left(-\frac{1}{2},\cdot,M\left(\frac{1}{2},\Pi_{v_{0}}\right)\tilde{W}_{v_{0}},\Phi_{v_{0}}\right)\equiv
0$. Thereafter by (4.11),
$\tilde{\mathcal{W}}^{\psi_{\tilde{N}^{\prime}}}(\cdot,{\rm
FJ}_{\psi_{N_{\mathbb{M}}}}(\mathcal{E}_{-k}\tilde{\varphi},\Phi))\equiv 0$.
With this in hand, the irreducibility and genericity of the descent ensure
that ${\rm
FJ}_{\psi_{N_{\mathbb{M}}}}(\mathcal{E}_{-k}\tilde{\varphi},\Phi)\equiv 0$. We
deduce from (4.19) that
$J\left(\frac{1}{2},W^{\prime}_{v_{0}},\tilde{W}_{v_{0}},\Phi_{v_{0}}\right)=0$.
Putting all together, $\Pi_{v_{0}}$ is good. ∎

While we expect that any irreducible generic genuine representation of
symplectic type $\Pi_{v}$ is good, the proof that we give under the assumption
that $\Pi_{v}$ is the local component of $\Pi\in{\rm
Scusp}_{k}\,\tilde{\mathbb{M}}$ is valid at the very least for the case of our
interest which pertains to the global period integrals. Our proof here
requires a local to global argument. Whereas the global approach is overkill
and indirect, one may directly prove the desired result by using a local
functional equation [LM17, Proposition 5.3]. For the sake of brevity, we only
include this indirect mean in keeping with the spirit of [LM16, Propositions
5.7 and 8.4].

######  Theorem 4.11.

Let $\Pi\in{\rm Scusp}_{k}\,\tilde{\mathbb{M}}$ and
$\sigma=\mathcal{D}_{\psi}(\Pi)$. Assume our Working Hypothesis 4.2. Let $S$
be a finite set of places including all the archimedean places and even places
such that $\Pi$ and $\psi$ are all unramified outside $S$. Then for any
$\varphi\in\sigma$ and $\varphi^{\vee}\in\sigma^{\vee}$ which are fixed under
$K^{\prime}_{v}$ for all $v\notin S$, we have

(4.20)
$\quad\mathcal{W}^{\psi_{N^{\prime}}}(\varphi)\mathcal{W}^{\psi^{-1}_{N^{\prime}}}(\varphi^{\vee})=2^{-k}\left(\prod_{v\in
S}c^{-1}_{\Pi_{v}}\right)\frac{\prod_{j=1}^{n}\zeta^{S}_{F}(2j)}{L^{S}(1,\pi,\wedge^{2})L^{S}(1,\pi\times\omega_{\pi})}\\\
\times({\rm vol}(N^{\prime}(\mathcal{O}_{S})\backslash
N^{\prime}(F_{S})))^{-1}\int^{\rm
st}_{N^{\prime}(F_{S})}\langle\sigma(u)\varphi,\varphi^{\vee}\rangle_{G^{\prime}}\psi^{-1}_{N^{\prime}}(u)\,du.\quad$

###### Proof.

Since the descent of $\Pi$ is generated by ${\rm
FJ}_{\psi_{N_{\mathbb{M}}}}(\mathcal{E}_{-k}\tilde{\varphi}^{\flat},\Phi)$ for
$\tilde{\varphi}^{\flat}\in\mathcal{A}(\Pi)$ and
$\Phi\in\mathcal{S}(\mathbb{A}^{n})$, we initially take up this for the case
$\varphi^{\vee}={\rm
FJ}_{\psi_{N_{\mathbb{M}}}}(\mathcal{E}_{-k}\tilde{\varphi}^{\sharp},\Phi)$
with $\tilde{\varphi}^{\sharp}\in\mathcal{A}(\Pi)$, and this identity (4.20)
then extends via linearity to all $\varphi^{\vee}$. In view of Proposition
4.7, we find that

$\mathcal{W}^{\psi_{N^{\prime}}}(\varphi)\mathcal{W}^{\psi^{-1}_{N^{\prime}}}({\rm
FJ}_{\psi_{N_{\mathbb{M}}}}(\mathcal{E}_{-k}\tilde{\varphi}^{\sharp},\Phi))=\mathcal{W}^{\psi_{N^{\prime}}}(\varphi)\frac{m^{S}_{-k,1}(\pi)}{2^{k}}\prod_{v\in