# Test Vectors for Archimedean Period Integrals

Peter Humphries and Yeongseong Jo  Department of Mathematics, University of
Virginia, Charlottesville, VA 22904, USA<EMAIL_ADDRESS>https://sites.google.com/view/peterhumphries/ Department of Mathematics and
Statistics, The University of Maine, Orono, ME 04469, USA
<EMAIL_ADDRESS>
###### Abstract.

We study period integrals involving Whittaker functions associated to generic
irreducible Casselman–Wallach representations of $\mathrm{GL}_{n}(F)$, where
$F$ is an archimedean local field. Via the archimedean theory of newforms for
$\mathrm{GL}_{n}$ developed by the first author, we prove that newforms are
weak test vectors for several period integrals, including the
$\mathrm{GL}_{n}\times\mathrm{GL}_{n}$ Rankin–Selberg integral, the Flicker
integral, and the Bump–Friedberg integral. By taking special values of these
period integrals, we deduce that newforms are weak test vectors for
Rankin–Selberg periods, Flicker–Rallis periods, and Friedberg–Jacquet periods.
These results parallel analogous results in the nonarchimedean setting proven
by the second author, which use the nonarchimedean theory of newforms for
$\mathrm{GL}_{n}$ developed by Jacquet, Piatetski-Shapiro, and Shalika. By
combining these archimedean and nonarchimedean results, we prove the existence
of weak test vectors for certain global period integrals of automorphic forms.

###### Key words and phrases:

Archimedean newform theory, archimedean Rankin–Selberg integral, local and
global period integrals, test vectors

###### 2020 Mathematics Subject Classification:

11F70 (primary); 11F67, 22E45, 22E50 (secondary)

##  1\. Introduction

###  1.1. Test Vectors for $\mathrm{GL}_{n}\times\mathrm{GL}_{n}$
Rankin–Selberg Integrals

A period integral of automorphic forms over a number field $F$ is said to be
_Eulerian_ if it factorises as a product of period integrals over local
fields. A quintessential example is the $\mathrm{GL}_{n}\times\mathrm{GL}_{n}$
Rankin–Selberg integral

$\int\limits_{\mathrm{Z}_{n}(\mathbb{A}_{F})\mathrm{GL}_{n}(F)\backslash\mathrm{GL}_{n}(\mathbb{A}_{F})}\varphi_{\pi}(g)\varphi_{\sigma}(g)E(g,s;\Phi,\omega_{\pi}\omega_{\sigma})\,dg$

involving two automorphic forms $\varphi_{\pi}$ and $\varphi_{\sigma}$ lying
in two automorphic representations $\pi$ and $\sigma$ of
$\mathrm{GL}_{n}(\mathbb{A}_{F})$ as well as a distinguished Eisenstein series
$E(g,s;\Phi,\omega_{\pi}\omega_{\sigma})$ associated to a Schwartz–Bruhat
function $\Phi\in\mathscr{S}(\mathbb{A}_{F}^{n})$ and a product
$\omega_{\pi}\omega_{\sigma}$ of central characters. If $\varphi_{\pi}$,
$\varphi_{\sigma}$, and $\Phi$ are pure tensors, then by unfolding, this
global period integral factorises as a product over all places $v$ of $F$ of
local $\mathrm{GL}_{n}\times\mathrm{GL}_{n}$ Rankin–Selberg integrals

$\Psi(s,W_{\pi_{v}},W_{\sigma_{v}},\Phi_{v})\coloneqq\int\limits_{\mathrm{N}_{n}(F_{v})\backslash\mathrm{GL}_{n}(F_{v})}W_{\pi_{v}}(g_{v})W_{\sigma_{v}}(g_{v})\Phi_{v}(e_{n}g_{v})\left|\det
g_{v}\right|_{v}^{s}\,dg_{v},$

where $W_{\pi_{v}}$ and $W_{\sigma_{v}}$ are the local Whittaker functions
associated to $\varphi_{\pi}$ and $\varphi_{\sigma}$.

The local period integrals $\Psi(s,W_{\pi_{v}},W_{\sigma_{v}},\Phi_{v})$
represent the local $\mathrm{GL}_{n}\times\mathrm{GL}_{n}$ Rankin–Selberg
$L$-function $L(s,\pi_{v}\times\sigma_{v})$, where $\pi_{v}$ and $\sigma_{v}$
are the generic irreducible admissible smooth representations of
$\mathrm{GL}_{n}(F_{v})$ occurring in the tensor product decompositions of
$\pi$ and $\sigma$. More precisely, if $v$ is a nonarchimedean place with
residue field of order $q$, then the quotient
$\Psi(s,W_{\pi_{v}},W_{\sigma_{v}},\Phi_{v})/L(s,\pi_{v}\times\sigma_{v})$ is
a polynomial in $q^{s}$ and $q^{-s}$, and in particular is entire. If $v$ is
an archimedean place, then the quotient
$\Psi(s,W_{\pi_{v}},W_{\sigma_{v}},\Phi_{v})/L(s,\pi_{v}\times\sigma_{v})$ is
entire and of finite order in vertical strips.

While the quotient
$\Psi(s,W_{\pi_{v}},W_{\sigma_{v}},\Phi_{v})/L(s,\pi_{v}\times\sigma_{v})$ is
always entire regardless of the choice of Whittaker functions $W_{\pi_{v}}$
and $W_{\sigma_{v}}$ and Schwartz–Bruhat function $\Phi_{v}$, for many
applications, one requires something stronger, namely that for particular
choices of $W_{\pi_{v}}$, $W_{\sigma_{v}}$, and $\Phi_{v}$, this quotient be
nicely behaved — in particular, nonvanishing apart from a prescribed
collection of values of $s\in\mathbb{C}$. When the representations $\pi_{v}$
and $\sigma_{v}$ are both unramified, there exists an explicit Schwartz–Bruhat
function $\Phi_{v}$ for which this quotient is _exactly_ equal to $1$ when
$W_{\pi_{v}}$ and $W_{\sigma_{v}}$ are chosen to be _spherical_ Whittaker
functions. This motivates the following problem.

######  Strong Test Vector Problem.

Given generic irreducible admissible smooth representations $\pi_{v}$ and
$\sigma_{v}$ of $\mathrm{GL}_{n}(F_{v})$, determine the existence of Whittaker
functions $W_{\pi_{v}}$ and $W_{\sigma_{v}}$ in the Whittaker models of
$\pi_{v}$ and $\sigma_{v}$ and a Schwartz–Bruhat function
$\Phi_{v}\in\mathscr{S}(F_{v}^{n})$ for which

$\Psi(s,W_{\pi_{v}},W_{\sigma_{v}},\Phi_{v})=L(s,\pi_{v}\times\sigma_{v}).$

We call such a triple $(W_{\pi_{v}},W_{\sigma_{v}},\Phi_{v})$ a _strong test
vector_ for the $\mathrm{GL}_{n}\times\mathrm{GL}_{n}$ Rankin–Selberg
integral. In full generality, this problem remains unresolved, though some
special cases have been settled:

  * •

When $n=2$ and $F_{v}$ is nonarchimedean for several families of
representations $\pi_{v}$ and $\sigma_{v}$ by Kim [Kim10, Chapters 4 and 5];

  * •

When $n=2$ and $F_{v}$ is archimedean by Jacquet [Jac72, Theorem 7.2], S-W.
Zhang [Zha01, Proposition 2.5.2], Miyazaki [Miy18, Theorem 6.1] and Hirano,
Ishii, and Miyazaki [HIM21, Appendix A];

  * •

When at least one of the two representations $\pi_{v}$ and $\sigma_{v}$ is
unramified by Kim [Kim10, Theorem 2.1.1] for $F_{v}$ nonarchimedean and by the
first author [Hum20a, Theorem 4.18] for $F_{v}$ archimedean;

  * •

When both representations are principal series of prescribed forms and $F_{v}$
is archimedean by Ishii and Miyazaki [IM20, Theorem 2.9].

We focus on a weaker yet more tractable problem, where we are satisfied with
finding a triple $(W_{\pi_{v}},W_{\sigma_{v}},\Phi_{v})$ for which
$\Psi(s,W_{\pi_{v}},W_{\sigma_{v}},\Phi_{v})$ is an explicit polynomial
multiple of $L(s,\pi_{v}\times\sigma_{v})$. Associated to $\pi_{v}$ and
$\sigma_{v}$ are _unramified_ representations $\pi_{v,\mathrm{ur}}$ and
$\sigma_{v,\mathrm{ur}}$ whose standard $L$-functions are such that
$L(s,\pi_{v,\mathrm{ur}})=L(s,\pi_{v})$ and
$L(s,\sigma_{v,\mathrm{ur}})=L(s,\sigma_{v})$. We then define the _naïve
Rankin–Selberg $L$-function_ associated to $\pi_{v}$ and $\sigma_{v}$ to be
$L(s,\pi_{v,\mathrm{ur}}\times\sigma_{v,\mathrm{ur}})$, which we show is an
explicit polynomial multiple of $L(s,\pi_{v}\times\sigma_{v})$.

######  Weak Test Vector Problem.

Given generic irreducible admissible smooth representations $\pi_{v}$ and
$\sigma_{v}$ of $\mathrm{GL}_{n}(F_{v})$, determine the existence of Whittaker
functions $W_{\pi_{v}}$ and $W_{\sigma_{v}}$ in the Whittaker models of
$\pi_{v}$ and $\sigma_{v}$ and a Schwartz–Bruhat function
$\Phi_{v}\in\mathscr{S}(F_{v}^{n})$ for which

$\Psi(s,W_{\pi_{v}},W_{\sigma_{v}},\Phi_{v})=L(s,\pi_{v,\mathrm{ur}}\times\sigma_{v,\mathrm{ur}}).$

We call such a triple $(W_{\pi_{v}},W_{\sigma_{v}},\Phi_{v})$ a _weak test
vector_ for the $\mathrm{GL}_{n}\times\mathrm{GL}_{n}$ Rankin–Selberg
integral. The second author [Jo21, Theorem 1.1 (i)] resolved this problem when
$F_{v}$ is nonarchimedean via the theory of nonarchimedean newforms due to
Jacquet, Piatetski-Shapiro, and Shalika [JP-SS81]. We resolve this problem
when $F_{v}$ is archimedean via the theory of archimedean newforms introduced
by the first author [Hum20a].

######  Theorem 1.1.

Let $F_{v}$ be an archimedean local field and let $\pi_{v}$ and $\sigma_{v}$
be generic irreducible admissible smooth representations of
$\mathrm{GL}_{n}(F_{v})$. Then there exist Whittaker functions $W_{\pi_{v}}$
and $W_{\sigma_{v}}$ in the Whittaker models of $\pi_{v}$ and $\sigma_{v}$ and
a Schwartz–Bruhat function $\Phi_{v}\in\mathscr{S}(F_{v}^{n})$ for which

$\Psi(s,W_{\pi_{v}},W_{\sigma_{v}},\Phi_{v})=L(s,\pi_{v,\mathrm{ur}}\times\sigma_{v,\mathrm{ur}}).$

A more precise statement of this result is given in Theorem 5.6.

###  1.2. Test Vectors for Archimedean Period Integrals

Theorem 1.1 is but one of many results that we prove pertaining to weak test
vectors for archimedean period integrals. There are many local period
integrals involving integrals of Whittaker functions that represent
distinguished $L$-functions. As well as for the
$\mathrm{GL}_{n}\times\mathrm{GL}_{n}$ Rankin–Selberg integral, we resolve the
analogue of the weak test vector problem for the following:

  * •

The modified $\mathrm{GL}_{n}\times\mathrm{GL}_{n}$ Rankin–Selberg integral
introduced by Sakellaridis given by (5.15). This represents
$L(s_{2},\pi_{v}\times\sigma_{v})L(s_{1}-\frac{n-1}{2},\sigma_{v})$, the
product of the $\mathrm{GL}_{n}\times\mathrm{GL}_{n}$ Rankin–Selberg
$L$-function and the standard $L$-function. In Theorem 5.14, we prove the
existence of a weak test vector $(W_{\pi_{v}},W_{\sigma_{v}},\Phi_{v})$ for
which this integral is equal to
$L(s_{2},\pi_{v,\mathrm{ur}}\times\sigma_{v,\mathrm{ur}})L(s_{1}-\frac{n-1}{2},\sigma_{v})$.

  * •

The modified $\mathrm{GL}_{n}\times\mathrm{GL}_{n-1}$ Rankin–Selberg integral
introduced by Sakellaridis given by (5.17). This represents
$L(s_{2}+\frac{1}{2},\pi_{v}\times\sigma_{v})L(s_{1}-\frac{n-2}{2},\sigma_{v})$,
the product of the $\mathrm{GL}_{n}\times\mathrm{GL}_{n-1}$ Rankin–Selberg
$L$-function and the standard $L$-function. In Theorem 5.16, we prove that
when $\sigma_{v}$ is unramified, there exists of a strong test vector
$(W_{\pi_{v}},W_{\sigma_{v}},\Phi_{v})$ for which this integral is equal to
$L(s_{2}+\frac{1}{2},\pi_{v}\times\sigma_{v})L(s_{1}-\frac{n-1}{2},\sigma_{v})$.

  * •

The Flicker integral given by (6.11). This represents the Asai $L$-function
$L(s,\pi_{v},\mathrm{As})$. In Theorem 6.10, we prove the existence of a weak
test vector $(W_{\pi_{v}},\Phi_{v})$ for which this integral is equal to
$L(s,\pi_{v,\mathrm{ur}},\mathrm{As})$.

  * •

The Bump–Friedberg integral given by (7.13). This represents
$L(s_{1},\pi_{v})L(s_{2},\pi_{v},\wedge^{2})$, the product of the standard
$L$-function and the exterior square $L$-function. In Theorem 7.12, we prove
the existence of a weak test vector $(W_{\pi_{v}},\Phi_{v})$ for which this
integral is equal to
$L(s_{1},\pi_{v})L(s_{2},\pi_{v,\mathrm{ur}},\wedge^{2})$.

The period integrals listed above either involve a single complex variable $s$
or a pair of complex variables $s_{1},s_{2}$. In certain situations, these
complex variables can be specialised to special values, at which point these
period integrals are linear functionals for particular representations. We
list below the periods of interest that arise from specialising these complex
variables.

  * •

The $\mathrm{GL}_{n}\times\mathrm{GL}_{n}$ Rankin–Selberg period given by
(5.11). When $\pi_{v}$ and $\sigma_{v}$ are unitary, we show in Theorem 5.10
the existence of a test vector $(W_{\pi_{v}},W_{\sigma_{v}})$ for which this
period is equal to the special $L$-value
$L(1,\pi_{v,\mathrm{ur}}\times\sigma_{v,\mathrm{ur}})/L(n,\omega_{\pi_{v,\mathrm{ur}}}\omega_{\sigma_{v,\mathrm{ur}}})$,
where $\omega_{\pi_{v,\mathrm{ur}}}$ and $\omega_{\sigma_{v,\mathrm{ur}}}$
denote the central characters of $\pi_{v,\mathrm{ur}}$ and
$\sigma_{v,\mathrm{ur}}$.

  * •

The Flicker–Rallis period given by (6.15). When $\pi_{v}$ is unitary, we show
in Theorem 6.14 the existence of a test vector $W_{\pi_{v}}$ for which this
period is equal to the special $L$-value
$L(1,\pi_{v,\mathrm{ur}},\mathrm{As})/L(n,\omega_{\pi_{v,\mathrm{ur}}}|_{\mathbb{R}^{\times}})$.

  * •

The Friedberg–Jacquet period given by (7.17). When $\pi_{v}$ is unitary, we
show in Theorem 7.16 the existence of a test vector $W_{\pi_{v}}$ for which
this period is equal to the special $L$-value
$L(\frac{1}{2},\pi_{v})L(1,\pi_{v,\mathrm{ur}},\wedge^{2})/L(\frac{n}{2},\omega_{\pi_{v,\mathrm{ur}}})$.

###  1.3. Comparisons between Archimedean and Nonarchimedean Results

Our results on test vectors for archimedean period integrals parallel
analogous results of the second author [Jo21] in the nonarchimedean setting.
The strategy of proof in both settings is similar: via the Iwasawa
decomposition and the fact that Whittaker newforms transform on the right
under the maximal compact subgroup $K_{v}$ of $\mathrm{GL}_{n}(F_{v})$ in a
prescribed manner, these integrals can be reduced to integrals over a diagonal
torus $\mathrm{A}_{n}(F_{v})$.

When $F_{v}$ is nonarchimedean, the behaviour of the Whittaker newform when
restricted to the diagonal torus has a closed form via the
Shintani–Casselman–Shalika formula [CS80, Shi76] together with work of
Matringe [Mat13] and Miyauchi [Miy14]. This allows one to _directly_ evaluate
the period integrals of interest.

When $F_{v}$ is archimedean, on the other hand, no such closed form exists for
the Whittaker newform when restricted to the diagonal torus. Instead, there
exists a _propagation formula_ due to the first author [Hum20a, Lemmata 9.8
and 9.17] that gives a recursive formula for this Whittaker newform in terms
of a Whittaker newform associated to a representation of
$\mathrm{GL}_{n-1}(F_{v})$; for unramified representations, such an identity
is due independently to Gerasimov, Lebedev, and Oblezin [GLO08, Proposition
4.1] and to Ishii and Stade [IS13, Proposition 2.1]. In this way, the period
integrals of interest can be evaluated via _induction_. This approach has been
previously used to successfully solve the strong test vector problem for the
unramified $\mathrm{GL}_{n}\times\mathrm{GL}_{n}$ Rankin–Selberg integral
[Sta02, Theorem 1.1] and the unramified Bump–Friedberg integral [Sta01,
Theorem 3.3]. For the modified $\mathrm{GL}_{n}\times\mathrm{GL}_{n}$ and
$\mathrm{GL}_{n}\times\mathrm{GL}_{n-1}$ Rankin–Selberg integrals introduced
by Sakellaridis and for the Flicker integral, our results are new even for the
unramified case.

Known results for weak test vectors for nonarchimedean period integrals
encompass more than just the analogues of the archimedean results listed in
Section 1.2. Weak test vectors have been determined for
$\mathrm{GL}_{n}\times\mathrm{GL}_{m}$ Rankin–Selberg integrals with $n>m$
[BKL20, Theorem 1.1], for Jacquet–Shalika integrals [MY13, Theorem 1.1], and
for Bump–Ginzburg integrals [Jo21, Theorem 6.3]. In turn, these give weak test
vectors for $\mathrm{GL}_{n}\times\mathrm{GL}_{m}$ Rankin–Selberg periods,
Jacquet–Shalika periods, and Bump–Ginzburg periods.

In the archimedean setting, on the other hand, we do _not_ expect to be able
to find such weak test vectors for these various period integrals, with the
exception of the $\mathrm{GL}_{n}\times\mathrm{GL}_{n-1}$ Rankin–Selberg
integral, if we enforce the condition that these test vectors be right
$K_{v}$-finite. Indeed, even if all representations are unramified and all
Whittaker functions are spherical, it is widely believed that these period
integrals are _not_ a nonzero polynomial multiple of the $L$-function
associated to such a period integral [Bum89, §2.6]. A prototypical example of
this phenomenon is the identity for the unramified
$\mathrm{GL}_{n}\times\mathrm{GL}_{n-2}$ archimedean Rankin–Selberg integral
due to Ishii and Stade [IS13, Theorem 3.2], namely

$\int\limits_{\mathrm{N}_{n-2}(\mathbb{R})\backslash\mathrm{GL}_{n-2}(\mathbb{R})}W_{\pi_{\infty}}\begin{pmatrix}g_{\infty}&0\\\
0&1_{2}\end{pmatrix}W_{\sigma_{\infty}}(g_{\infty})\left|\det
g_{\infty}\right|_{\mathbb{R}}^{s-1}\,dg_{\infty}\\\
=L(s,\pi_{\infty}\times\sigma_{\infty})\frac{1}{4\pi
i}\int_{\sigma-i\infty}^{\sigma+i\infty}\frac{L\left(w,\widetilde{\pi_{\infty}}\right)}{L(s+w,\sigma_{\infty})}\,dw.$

###  1.4. Global Applications

In conjunction with the nonarchimedean results on test vectors in [Jo21], our
archimedean results have global applications. Each local period integral has a
global counterpart, and we are able to show the existence of weak test vectors
for these global period integrals. More precisely, we resolve the weak test
vector problem for the following:

  * •

The global $\mathrm{GL}_{n}\times\mathrm{GL}_{n}$ Rankin–Selberg integral
given by (8.5) in Theorem 8.4.

  * •

The global modified $\mathrm{GL}_{n}\times\mathrm{GL}_{n}$ Rankin–Selberg
integral introduced by Sakellaridis given by (8.11) in Theorem 8.10.

  * •

The global modified $\mathrm{GL}_{n}\times\mathrm{GL}_{n-1}$ Rankin–Selberg
integral introduced by Sakellaridis given by (8.13) in Theorem 8.12.

  * •

The global Flicker integral given by (8.15) in Theorem 8.14.

  * •

The global Bump–Friedberg integral given by (8.20) and (8.21) in Theorem 8.19.

By specialising the value of the complex variables $s$ or $s_{1},s_{2}$ in
these period integrals, our results give weak test vectors for certain
periods:

  * •

The global $\mathrm{GL}_{n}\times\mathrm{GL}_{n}$ Rankin–Selberg period given
by (8.8) in Theorem 8.7.

  * •

The global Flicker–Rallis period given by (8.18) in Theorem 8.17.

  * •

The global Friedberg–Jacquet period given by (8.27) in Theorem 8.26.

##  2\. Preliminaries

From here on, we work over an archimedean local field. Since all that follows
is local, we drop the usage of the subscript $v$. We list below some standard
facts on the representation theory of $\mathrm{GL}_{n}(F)$ with $F$
archimedean. Much of this is well-known and appears elsewhere in the
literature; see in particular [Hum20a, §2].

###  2.1. Groups and Measures

####  2.1.1. Local Fields

Let $F$ be an archimedean local field, so that $F$ is either $\mathbb{R}$ or
$\mathbb{C}$. We denote by $|\cdot|_{F}$ the absolute value on $F$, which we
normalise such that

$|x|_{F}=\begin{dcases*}\max\\{x,-x\\}&if $F=\mathbb{R}$,\\\ x\overline{x}&if
$F=\mathbb{C}$.\end{dcases*}$

When the local field is clear from the context, we omit the subscript $F$ in
our notation and simply write $|\cdot|=|\cdot|_{F}$. We also let
$\|\cdot\|\coloneqq|\cdot|^{1/2}_{\mathbb{C}}$ denote the standard module on
$\mathbb{C}$.

We fix a nontrivial additive character $\psi_{F}=\psi$ of $F$. For
$F=\mathbb{R}$, we choose $\psi(x)\coloneqq\exp(2\pi ix)$, while for
$F=\mathbb{C}$, we choose $\psi(x)\coloneqq\exp(2\pi i(x+\overline{x}))$; in
Section 6, we will also work with a slightly different nontrivial additive
character $\psi_{\mathbb{C}/\mathbb{R}}$ of $\mathbb{C}$ that is trivial when
restricted to $\mathbb{R}$. We normalise the Haar measure $dx$ on $F$ so that
it is self-dual with respect to $\psi$. For $F=\mathbb{R}$, $dx$ is simply the
Lebesgue measure, whereas for $F=\mathbb{C}$, $dx$ is twice the Lebesgue
measure. The multiplicative Haar measure $d^{\times}x$ on $F^{\times}$ is
$\zeta_{F}(1)|x|^{-1}dx$, where

$\zeta_{F}(s)\coloneqq\begin{dcases*}\pi^{-\frac{s}{2}}\Gamma\left(\dfrac{s}{2}\right)&if
$F=\mathbb{R}$,\\\ 2(2\pi)^{-s}\Gamma(s)&if $F=\mathbb{C}$.\end{dcases*}$

####  2.1.2. Groups and Haar Measures

We write $1_{n}$ to denote the $n\times n$ identity matrix. Let
$\\{e_{i}.:1\leq i\leq n\\}$ be the standard row basis of $F^{n}$. Let
$\mathrm{P}(F)=\mathrm{P}_{(n_{1},\ldots,n_{r})}(F)$ denote the standard upper
parabolic subgroup of $\mathrm{GL}_{n}(F)$ of type
$(n_{1},n_{2},\ldots,n_{r})$ with $n=n_{1}+n_{2}+\cdots+n_{r}$. This has the
Levi decomposition
$\mathrm{P}(F)=\mathrm{N}_{\mathrm{P}}(F)\mathrm{M}_{\mathrm{P}}(F)$, where
the block-diagonal Levi subgroup $\mathrm{M}_{\mathrm{P}}(F)$ is isomorphic to
$\mathrm{GL}_{n_{1}}(F)\times\cdots\times\mathrm{GL}_{n_{r}}(F)$, while the
unipotent radical $\mathrm{N}_{\mathrm{P}}(F)$ of $\mathrm{P}(F)$ consists of
upper triangular matrices with block-diagonal entries
$(1_{n_{1}},\ldots,1_{n_{r}})$. When $\mathrm{P}(F)$ is the standard Borel
(and minimal parabolic) subgroup
$\mathrm{P}_{(1,\ldots,1)}(F)\eqqcolon\mathrm{B}_{n}(F)$ of upper triangular
matrices, we write $\mathrm{N}_{\mathrm{P}}(F)\eqqcolon\mathrm{N}_{n}(F)\cong
F^{n(n-1)/2}$, the subgroup of unipotent upper triangular matrices, and
$\mathrm{M}_{\mathrm{P}}(F)\eqqcolon\mathrm{A}_{n}(F)\cong(F^{\times})^{n}$,
the subgroup of diagonal matrices. We write $\mathrm{Z}_{n}(F)\cong
F^{\times}$ to denote to the centre consisting of scalar matrices. We define
$\mathrm{P}_{n}(F)$, the mirabolic subgroup of $\mathrm{GL}_{n}(F)$, given by

$\mathrm{P}_{n}(F)=\left\\{\begin{pmatrix}h&\prescript{t}{}{x}\\\
0&1\end{pmatrix}:h\in\mathrm{GL}_{n-1}(F),\ x\in F^{n-1}\right\\}.$

The modulus character of a parabolic subgroup
$\mathrm{P}(F)=\mathrm{P}_{(n_{1},\ldots,n_{r})}(F)$ is

$\delta_{\mathrm{P}(F)}(m)=\prod_{j=1}^{r}\left|\det
m_{j}\right|^{n-2(n_{1}+n_{2}+\cdots+n_{j-1})-n_{j}}$

for any
$m=\operatorname{blockdiag}(m_{1},\ldots,m_{r})\in\mathrm{M}_{\mathrm{P}}(F)$.
We let $K_{n}$ denote the maximal compact subgroup of $\mathrm{GL}_{n}(F)$,
unique up to conjugacy, so that

$K_{n}=\begin{dcases*}\mathrm{O}(n)&if $F=\mathbb{R}$,\\\ \mathrm{U}(n)&if
$F=\mathbb{C}$.\end{dcases*}$

We have the Iwasawa decomposition
$\mathrm{GL}_{n}(F)=\mathrm{N}_{n}(F)\mathrm{A}_{n}(F)K_{n}$. We normalise the
Haar measure $dg$ on $\mathrm{GL}_{n}(F)\ni g$ via the Iwasawa decomposition
$g=uak$ with $u\in\mathrm{N}_{n}(F)$, $a\in\mathrm{A}_{n}(F)$, and $k\in
K_{n}$, so that

$dg=\delta^{-1}_{\mathrm{B}_{n}(F)}(a)\,du\,d^{\times}a\,dk.$

Here $du=\prod_{1\leq i<j\leq n}du_{i,j}$, where $du_{i,j}$ is the Haar
measure on $F$, while $d^{\times}a=\prod_{j=1}^{n}d^{\times}a_{j}$, where
$d^{\times}a_{j}$ is the Haar measure on $F^{\times}$, and $dk$ is the
probability Haar measure on $K_{n}$. The modulus character of
$\mathrm{B}_{n}(F)$ is simply
$\delta_{\mathrm{B}_{n}(F)}(a)=\prod_{j=1}^{n}|a_{j}|^{n-2j+1}$.

The Iwasawa decomposition also gives rise to the Haar measure on
$\mathrm{N}_{n}(F)\backslash\mathrm{GL}_{n}(F)$ via
$dg=\delta_{B_{n}(F)}^{-1}(a)\,d^{\times}a\,dk$ for $g=ak$ with
$a\in\mathrm{A}_{n}(F)$ and $k\in K_{n}$. There is an alternate expression for
this Haar measure via the Iwasawa decomposition
$g=(z1_{n})\begin{pmatrix}h&0\\\ 0&1\end{pmatrix}k$ associated to the maximal
parabolic subgroup $\mathrm{P}_{(n-1,1)}(F)$ of $\mathrm{GL}_{n}(F)$ of type
$(n-1,1)$. Here $z\in F^{\times}$,
$h\in\mathrm{N}_{n-1}(F)\backslash\mathrm{GL}_{n-1}(F)$, and $k\in K_{n}$; the
Haar measure on $\mathrm{N}_{n}(F)\backslash\mathrm{GL}_{n}(F)$ becomes

$dg=\left|\det h\right|^{-1}\,d^{\times}z\,dh\,dk,$

where $d^{\times}z$ denotes the multiplicative Haar measure on $F^{\times}$
and $dh$ denotes the Haar measure on
$\mathrm{N}_{n-1}(F)\backslash\mathrm{GL}_{n-1}(F)$.

###  2.2. Representations of $\mathrm{GL}_{n}(F)$

####  2.2.1. Isobaric Sums

Given representations $(\pi_{1},V_{\pi_{1}}),\ldots,(\pi_{r},V_{\pi_{r}})$ of
$\mathrm{GL}_{n_{1}}(F),\ldots,\mathrm{GL}_{n_{r}}(F)$ with
$n=n_{1}+\cdots+n_{r}$, we form the representation
$\mathop{\vphantom{\bigotimes}\mathchoice{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxtimes$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxtimes$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxtimes$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxtimes$}}}}}\displaylimits_{j=1}^{r}\pi_{j}$
of $\mathrm{M}_{\mathrm{P}}(F)$, where $\boxtimes$ denote the outer tensor
product. We then extend this representation trivially to a representation of
$\mathrm{P}(F)$. We obtain a normalised parabolically induced representation
$(\pi,V_{\pi})$ of $\mathrm{GL}_{n}(F)$ by

$\pi\coloneqq\operatorname{Ind}^{\mathrm{GL}_{n}(F)}_{\mathrm{P}(F)}\mathop{\vphantom{\bigotimes}\mathchoice{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxtimes$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxtimes$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxtimes$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxtimes$}}}}}\displaylimits_{j=1}^{r}\pi_{j},$

where $V_{\pi}$ denote the space of smooth functions $f:\mathrm{GL}_{n}(F)\to
V_{\pi_{1}}\otimes\cdots\otimes V_{\pi_{r}}$, upon which $\pi$ on $V_{\pi}$
via right translation, namely, $(\pi(h)\cdot f)(g)\coloneqq f(gh)$, that
satisfy

$f(umg)=\delta^{1/2}_{\mathrm{P}(F)}(m)\bigotimes_{j=1}^{r}\pi_{j}(m_{j})\cdot
f(g),$

for any $u\in\mathrm{N}_{\mathrm{P}}(F)$,
$m=\operatorname{blockdiag}(m_{1},\ldots,m_{r})\in\mathrm{M}_{\mathrm{P}}(F)$,
and $g\in\mathrm{GL}_{n}$. The induced representation $\pi$ is called the
_isobaric sum_ of $\pi_{1},\ldots,\pi_{r}$, which we denote by

$\pi\coloneqq\mathop{\vphantom{\bigoplus}\mathchoice{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}}\displaylimits_{j=1}^{r}\pi_{j}.$

####  2.2.2. Essential Square-Integrable Representations

Essential square-integrable representations of $\mathrm{GL}_{n}(\mathbb{C})$
exist only for $n=1$. An essentially square-integrable representation of
$\mathrm{GL}_{1}(\mathbb{R})=\mathbb{R}^{\times}$ must be a character of the
form $\pi(x)=e^{i\kappa\arg(x)}|x|^{t}_{\mathbb{C}}$ for some
$\kappa\in\mathbb{Z}$ and $t\in\mathbb{C}$, where $e^{i\arg(x)}\coloneqq
x/\|x\|$.

Essential square-integrable representations of $\mathrm{GL}_{n}(\mathbb{R})$
exist only for $n\in\\{1,2\\}$. An essentially square-integrable
representation of $\mathrm{GL}_{1}(\mathbb{R})=\mathbb{R}^{\times}$ must be a
character of the form
$\pi(x)=\operatorname{sgn}(x)^{\kappa}|x|^{t}_{\mathbb{R}}$ for some
$\kappa\in\\{0,1\\}$ and $t\in\mathbb{C}$, where
$\operatorname{sgn}(x)\coloneqq x/\|x\|$. We view
$\mathrm{GL}_{1}(\mathbb{C})$ as a subgroup of $\mathrm{GL}_{2}(\mathbb{R})$
via the identification $a+ib\mapsto\begin{pmatrix}a&b\\\ -b&a\end{pmatrix}$.
For $\kappa\neq 0$, the essential discrete series representation of weight
$\|\kappa\|+1$,

$D_{\|\kappa\|+1}\otimes\left|\det\right|^{t}_{\mathbb{R}}\coloneqq\operatorname{Ind}^{\mathrm{GL}_{2}(\mathbb{R})}_{\mathrm{GL}_{1}(\mathbb{C})}e^{i\kappa\arg}|\cdot|^{t}_{\mathbb{C}}\cong\operatorname{Ind}^{\mathrm{GL}_{2}(\mathbb{R})}_{\mathrm{GL}_{1}(\mathbb{C})}e^{-i\kappa\arg}|\cdot|^{t}_{\mathbb{C}}$

is essentially square-integrable. Every essentially square-integrable
representation of $\mathrm{GL}_{2}(\mathbb{R})$ is of the form
$\pi=D_{\kappa}\otimes\left|\det\right|^{t}_{\mathbb{R}}$ for some integer
$\kappa\geq 2$ and $t\in\mathbb{C}$.

####  2.2.3. Induced Representations of Whittaker and Langlands Types

A representation $\pi$ of $\mathrm{GL}_{n}(F)$ is said to be an _induced
representation of Whittaker type_ if it is the isobaric sum of
$\pi_{1},\pi_{2},\ldots,\pi_{r}$ and each $\pi_{j}$ is essentially square-
integrable. Such a representation is an admissible smooth Fréchet
representation of moderate growth and of finite length. Induced
representations of Whittaker type are Casselman–Wallach representations
[Cas89, Wal92], which is to say admissible smooth Fréchet representations of
moderate growth and finite length. In addition, if each $\pi_{j}$ is of the
form $\sigma_{j}\otimes\left|\det\right|^{t_{j}}$, where $\sigma_{j}$ is
irreducible, unitary, square-integrable, and
$\Re(t_{1})\geq\Re(t_{2})\geq\cdots\geq\Re(t_{r})$, then $\pi$ is said to be
an _induced representation of Langlands type_.

Induced representations of Whittaker and Langlands type need not be
irreducible. Nonetheless, every generic irreducible Casselman–Wallach
representation of $\mathrm{GL}_{n}(F)$ is isomorphic to some (necessarily
irreducible) induced representation of Langlands type. For this reason, we
will work more generally with induced representations of Langlands type, since
this encompasses generic irreducible Casselman–Wallach representations.

A _spherical induced representation of Whittaker type_ of $\mathrm{GL}_{n}(F)$
is a representation of the form
$\pi=\mathop{\vphantom{\bigoplus}\mathchoice{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}}\displaylimits_{j=1}^{n}\pi_{j}$
with each $\pi_{j}$ an unramified character of $F^{\times}$, namely a
character of the form $\pi_{j}=|\cdot|^{t_{j}}$ with $t_{j}\in\mathbb{C}$.
Such a representation has a $K_{n}$-fixed vector, which is unique up to scalar
multiplication; this is called the _spherical vector_.

####  2.2.4. The Whittaker Model

Let $(\pi,V_{\pi})$ be an induced representation of Whittaker type. We let
$\psi_{n}$ denote an additive character of $\mathrm{N}_{n}(F)$ defined by

$\psi_{n}(u)\coloneqq\psi\left(\sum_{j=1}^{n-1}u_{j,j+1}\right)$

for all $u\in\mathrm{N}_{n}(F)$. A _Whittaker functional_
$\Lambda:V_{\pi}\to\mathbb{C}$ is a continuous linear functional that
satisfies

$\Lambda(\pi(u)\cdot v)=\psi_{n}(u)\Lambda(v)$

for all $v\in V_{\pi}$ and $u\in\mathrm{N}_{n}(F)$. If $\pi$ is additionally
irreducible, then the space
${\operatorname{Hom}}_{\mathrm{N}_{n}(F)}(\pi,\psi)$ of Whittaker functionals
of $\pi$ is at most one-dimensional. If the space is one-dimensional, it
admits a unique functional up to scalar multiplication, and the representation
$\pi$ is said to be _generic_.

Let $\mathcal{W}(\pi,\psi)$ denote the _Whittaker model_ of $\pi$, which is
the image of $V_{\pi}$ under the map $v\mapsto\Lambda(\pi(\cdot)\cdot v)$. The
Whittaker model $\mathcal{W}(\pi,\psi)$ consists of Whittaker functions
$W:\mathrm{GL}_{n}(F)\to\mathbb{C}$ of the form
$W(g)\coloneqq\Lambda(\pi(g)\cdot v)$. Every induced representation of
Langlands type is generic and isomorphic to its unique Whittaker model
$\mathcal{W}(\pi,\psi)$. Although an induced representation of Whittaker type
$\pi$ affords a one-dimensional space of Whittaker functionals, the map
$v\mapsto\Lambda(\pi(\cdot)\cdot v)$ need not to be injective, so the
Whittaker model may only be a model of a quotient of $\pi$.

Let
$\pi=\mathop{\vphantom{\bigoplus}\mathchoice{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}}\displaylimits_{j=1}^{r}\pi_{j}$
be an induced representation of Whittaker type of $\mathrm{GL}_{n}(F)$, so
that each $\pi_{j}$ is of the form $e^{i\kappa_{j}\arg}|\cdot|^{t_{j}}$ if
$F=\mathbb{C}$ or either $\operatorname{sgn}^{\kappa_{j}}|\cdot|^{t_{j}}$ or
$D_{\kappa_{j}}\otimes\left|\det\right|^{t_{j}}$ if $F=\mathbb{R}$. The
induced model of $\pi$ consists of smooth functions
$f:\mathrm{GL}_{n}(F)\to\bigotimes_{j=1}^{r}V_{\pi_{j}}$. Each $V_{\pi_{j}}$
may itself be identified with a space of smooth functions from
$\mathrm{GL}_{n_{j}}(F)$ to $\mathbb{C}$ [Hum20a, Lemma 8.1]. Evaluating such
a function at the identity $1_{n_{j}}$ for each $j\in\\{1,\ldots,r\\}$, we may
thereby view an element $f$ of the induced model $V_{\pi}$ of $\pi$ as a
smooth function from $\mathrm{GL}_{n}(F)$ to $\mathbb{C}$ [Hum20a, Corollary
8.2].

Given $f\in V_{\pi}$, we define the _Jacquet integral_

$W_{f}(g)\coloneqq\int_{\mathrm{N}_{n}(F)}f(w_{n}ug)\overline{\psi_{n}}(u)\,du,$

where $w_{n}\coloneqq\operatorname{antidiag}(1,\ldots,1)$ denotes the long
Weyl element in $\mathrm{GL}_{n}(F)$. This integral converges absolutely if
$\Re(t_{1})>\Re(t_{2})>\cdots>\Re(t_{r})$ and defines a Whittaker function
$W_{f}\in\mathcal{W}(\pi,\psi)$; that is, as a function of $f\in V_{\pi}$,
$\Lambda(f)\coloneqq W_{f}(1_{n})$ defines a Whittaker functional, which is
therefore unique up to scalar multiplication. Moreover, the Jacquet integral
provides a Whittaker functional for _all_ induced representations of Whittaker
type, and not just those for which $\Re(t_{1})>\Re(t_{2})>\cdots>\Re(t_{r})$,
via analytic continuation in the sense of Wallach [Wal92].

####  2.2.5. $L$-Functions

We put

$d_{F}\coloneqq[F:\mathbb{R}]=\begin{dcases*}1&if $F=\mathbb{R}$,\\\ 2&if
$F=\mathbb{C}$.\end{dcases*}$

The integral representation of the zeta function $\zeta_{F}(s)$ is

$\zeta_{F}(s)=\int_{F^{\times}}\exp(-d_{F}\pi\|x\|^{2})|x|^{s}\,d^{\times}x,$

which converges absolutely for $\Re(s)>0$ and extends meromorphically to the
entire complex plane. In particular, if $\omega$ is an unramified character of
$F^{\times}$, so that $\omega=|\cdot|^{t}$ for some $t\in\mathbb{C}$, then for
$\Re(s)>-\Re(t)$, we have that

(2.1)
$L(s,\omega)=\int_{F^{\times}}\omega(x)\exp(-d_{F}\pi\|x\|^{2})|x|^{s}\,d^{\times}x.$

Given an induced representation of Whittaker type
$\pi=\mathop{\vphantom{\bigoplus}\mathchoice{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}}\displaylimits_{j=1}^{r}\pi_{j}$
of $\mathrm{GL}_{n}(F)$, the local Langlands correspondence, as explicated by
Knapp [Kna94], gives that the standard $L$-function of $\pi$ [GJ72] is

$L(s,\pi)=\prod_{j=1}^{r}L(s,\pi_{j}),$

where the $L$-function of the essentially square-integrable representation
$\pi_{j}$ is

$L(s,\pi_{j})=\begin{dcases*}\zeta_{\mathbb{C}}\left(s+t_{j}+\frac{\|\kappa_{j}\|}{2}\right)&if
$F=\mathbb{C}$ and
$\pi_{j}=e^{i\kappa_{j}\arg}|\cdot|_{\mathbb{C}}^{t_{j}}$,\\\
\zeta_{\mathbb{R}}(s+t_{j}+\kappa_{j})&if $F=\mathbb{R}$ and
$\pi_{j}=\operatorname{sgn}^{\kappa_{j}}|\cdot|_{\mathbb{R}}^{t_{j}}$,\\\
\zeta_{\mathbb{R}}\left(s+t_{j}+\frac{\kappa_{j}-1}{2}\right)\zeta_{\mathbb{R}}\left(s+t_{j}+\frac{\kappa_{j}+1}{2}\right)&if
$F=\mathbb{R}$ and
$\pi_{j}=D_{\kappa_{j}}\otimes\left|\det\right|_{\mathbb{R}}^{t_{j}}$.\end{dcases*}$

##  3\. Archimedean Newform Theory

We now survey the theory of newforms for induced representations of Whittaker
type of $\mathrm{GL}_{n}(F)$ introduced by the first author [Hum20a]. While
newforms over nonarchimedean fields are usually defined in terms of vectors
fixed under certain congruence subgroups, newforms over archimedean fields are
instead defined in terms of vectors lying in distinguished $K_{n}$-types. We
first recall some properties of representations of $K_{n}$, as well as
properties of distinguished models of such representations.

###  3.1. Representation Theory of $K_{n}$

The equivalence classes of finite-dimensional irreducible representations of
the orthogonal group

$\mathrm{O}(n)\coloneqq\\{k\in\operatorname{Mat}_{n\times
n}(\mathbb{R}):k\prescript{t}{}{k}=1_{n}\\}$

are parametrised by the set of highest weights, which may be identified with
$n$-tuples of nonnegative integers of the form

$\mu=\\{\mu_{1},\ldots,\mu_{m},\underbrace{\eta,\ldots,\eta}_{\text{$n-2m$
times}},\underbrace{0,\ldots,0}_{\text{$m$ times}}\\}\in\mathbb{Z}^{n},$

where $m\in\\{0,\ldots,\lfloor\frac{n}{2}\rfloor\\}$,
$\mu_{1}\geq\cdots\geq\mu_{m}\geq 1$, and $\eta\in\\{0,1\\}$.

Similarly, the equivalence class of finite-dimensional irreducible
representations of the unitary group

$\mathrm{U}(n)\coloneqq\\{k\in\operatorname{Mat}_{n\times
n}(\mathbb{C}):k\prescript{t}{}{\overline{k}}=1_{n}\\}$

are parametrised by the set of highest weights, which may be identified with
$n$-tuples of integers $\mu=(\mu_{1},\ldots,\mu_{n})\in\mathbb{Z}^{n}$ that
are nonincreasing, so that $\mu_{1}\geq\cdots\geq\mu_{n}$.

In both settings, to each $\tau\in\widehat{K_{n}}$, the set of equivalence
class of irreducible representations of $K_{n}$, one can associate a
nonnegative integer $\deg\tau$ called the _Howe degree_ of $\tau$ [How89]. The
Howe degree of an irreducible representation $\tau$ of highest weight $\mu$ is

$\deg\tau\coloneqq\sum_{j=1}^{n}\|\mu_{j}\|.$

When $n=1$, these irreducible representations are simply characters.
Characters $\chi$ of $K_{1}$ are of the form

$\chi(x)=\begin{dcases*}\operatorname{sgn}(x)^{\kappa}&if
$K_{1}=\mathrm{O}(1)$,\\\ e^{i\kappa\arg(x)}&if
$K_{1}=\mathrm{U}(1)$,\end{dcases*}$

where $\kappa\in\\{0,1\\}$ if $K_{1}=\mathrm{O}(1)\cong\mathbb{Z}/2\mathbb{Z}$
and $\kappa\in\mathbb{Z}$ if
$K_{1}=\mathrm{U}(1)\cong\mathbb{R}/\mathbb{Z}\cong S^{1}$. In either case,
the conductor exponent of $\chi$ is $c(\chi)\coloneqq\|\kappa\|$.

###  3.2. Spaces of Homogeneous Harmonic Polynomials

Let $\chi$ be a character of $\mathrm{O}(1)$. We let $m$ be a nonnegative
integer for which $m\geq c(\chi)$ and $m\equiv c(\chi)\pmod{2}$. Let
$\mathcal{P}_{\chi,m}(\mathbb{R}^{n})$ denote the space of degree $m$
homogeneous polynomials with central character $\chi$, and let
$\mathcal{H}_{\chi,m}(\mathbb{R}^{n})$ denote the subspace of
$\mathcal{P}_{\chi,m}(\mathbb{R}^{n})$ of harmonic homogeneous polynomials;
$\mathcal{H}_{\chi,m}(\mathbb{R}^{n})$ is a model for the irreducible
representation of $\mathrm{O}(n)$ with central character $\chi$ and highest
weight $(m,0,\ldots,0)$, which we denote by $\tau_{\chi,m}$.

Similarly, let $\chi$ be a character of $\mathrm{U}(1)$. We let $m$ be a
nonnegative integer for which $m\geq c(\chi)$ and $m\equiv c(\chi)\pmod{2}$.
Let $\mathcal{P}_{\chi,m}(\mathbb{C}^{n})$ denote the space of degree $m$
homogeneous polynomials with central character $\chi$, and let
$\mathcal{H}_{\chi,m}(\mathbb{C}^{n})$ denote the subspace of
$\mathcal{P}_{\chi,m}(\mathbb{C}^{n})$ of harmonic homogeneous polynomials;
$\mathcal{H}_{\chi,m}(\mathbb{C}^{n})$ is a model for the irreducible
representation of $\mathrm{U}(n)$ with central character $\chi$ and highest
weight $(m_{1},0,\ldots,0,-m_{2})$, which we also denote by $\tau_{\chi,m}$.
The nonnegative integers $m_{1},m_{2}$ are such that $m_{1}+m_{2}=m$ and
$m_{1}-m_{2}=\ell$, where $\chi=e^{i\ell\arg}$, so that
$c(\chi)=\max\\{\ell,-\ell\\}$, and elements of
$\mathcal{P}_{\chi,m}(\mathbb{C}^{n})$ have bidegree $(m_{1},m_{2})$.

In both cases, every irreducible representation of $K_{n}$ whose restriction
to $K_{n-1}$ contains the trivial representation is of the form
$\tau_{\chi,m}$ for some character $\chi$ of $K_{1}$ and some integer $m\geq
c(\chi)$; moreover, the Howe degree $\deg\tau_{\chi,m}$ of $\tau_{\chi,m}$ is
simply the nonnegative integer $m$. The action of the group $K_{n}\ni k$ on
the space $\mathcal{P}_{\chi,m}(F^{n})\ni P$ is via right translation, namely
$(\tau_{\chi,m}(k)\cdot P)(x)\coloneqq P(xk)$. The homogeneity of a polynomial
$P$ in $\mathcal{P}_{\chi,m}(F^{n})$ means that

(3.1) $P(\lambda
x)=\chi\left(\frac{\lambda}{\|\lambda\|}\right)\|\lambda\|^{m}P(x)$

for all $x\in F^{n}$ and $\lambda\in F^{\times}$, while polynomials in
$\mathcal{H}_{\chi,m}(F^{n})$ are additionally annihilated by the Laplacian

$\Delta=\begin{dcases*}\sum_{j=1}^{n}\frac{\partial^{2}}{\partial
x^{2}_{j}}&if $F=\mathbb{R}$,\\\ 4\sum_{i=1}^{n}\frac{\partial^{2}}{\partial
x_{j}\partial\overline{x_{j}}}&if $F=\mathbb{C}$.\end{dcases*}$

We record the following key properties of homogeneous polynomials in
$\mathcal{P}_{\chi,m}(F^{n})$.

######  Proposition 3.2 (Cf. [Hum20a, Lemmata 7.1 and 7.7]).

There exists a unique $K_{n-1}$-invariant polynomial $P_{\chi,m}^{\circ}$ in
$\mathcal{H}_{\chi,m}(F^{n})$ satisfying $P_{\chi,m}^{\circ}(e_{n})=1$, where
the group $K_{n-1}$ is embedded in $K_{n}$ via
$k^{\prime}\mapsto\begin{pmatrix}k^{\prime}&0\\\ 0&1\end{pmatrix}$. In
particular, for all $k\in K_{n}$, we have that

(3.3)
$P_{{\chi},m}^{\circ}(e_{n}k)=\overline{P_{\chi,m}^{\circ}}(e_{n}k^{-1}).$

We define an inner product on $\mathcal{P}_{\chi,m}(F^{n})\ni P,Q$ via

$\langle P,Q\rangle\coloneqq\int_{K_{n}}P(e_{n}k)\overline{Q}(e_{n}k)\,dk.$

Our first utilisation of $P^{\circ}_{{\chi},m}$ is the following, which is
known as the addition theorem for $\mathcal{H}_{\chi,m}(F^{n})$.

######  Proposition 3.4 (Cf. [AH12, Theorem 2.9]).

Let $\\{Q_{\ell}\\}$ be an orthonormal basis of $\mathcal{H}_{\chi,m}(F^{n})$.
Then for any $x\in\mathbb{R}^{n}$ and $k\in K_{n}$, we have that

$\sum_{\ell=1}^{\dim\tau_{\chi,m}}Q_{\ell}(x)\overline{Q_{\ell}}(e_{n}k)=\dim\tau_{\chi,m}P_{{\chi},m}^{\circ}(xk^{-1}).$

We make crucial use of the fact that for all $P\in\mathcal{H}_{\chi,m}(F^{n})$
and $k\in K_{n}$, $P(e_{n}k)$ is equal to a matrix coefficient of
$\tau_{\chi,m}$. This can be thought of as an explicit form of Schur
orthogonality.

######  Proposition 3.5 (Cf. [Hum20a, Lemmata 7.5 and 7.12]).

The reproducing kernel for $\mathcal{H}_{\chi,m}(F^{n})$ is the homogeneous
harmonic polynomial $(\dim\tau_{\chi,m})P^{\circ}_{\chi,m}(x)$, while the
reproducing kernel for $\mathcal{P}_{\chi,\ell}(F^{n})$ is the homogeneous
polynomial

(3.6) $\sum_{\begin{subarray}{c}m=c(\chi)\\\ m\equiv
c(\chi)\pmod{2}\end{subarray}}^{\ell}(x\prescript{t}{}{\overline{x}})^{\frac{\ell-m}{2}}(\dim\tau_{\chi,m})P^{\circ}_{\chi,m}(x),$

so that for all $k\in K_{n}$,

(3.7) $\displaystyle P(e_{n}k)$
$\displaystyle=\int_{K_{n}}P(e_{n}k^{\prime}k)(\dim\tau_{\chi,m})\overline{P_{\chi,m}^{\circ}}(e_{n}k^{\prime})\,dk^{\prime}$
for all $P\in\mathcal{H}_{\chi,m}(F^{n})$, (3.8) $\displaystyle P(e_{n}k)$
$\displaystyle=\int_{K_{n}}P(e_{n}k^{\prime}k)\sum_{\begin{subarray}{c}m=c(\chi)\\\
m\equiv
c(\chi)\pmod{2}\end{subarray}}^{\ell}(\dim\tau_{\chi,m})\overline{P_{\chi,m}^{\circ}}(e_{n}k^{\prime})\,dk^{\prime}$
for all $P\in\mathcal{P}_{\chi,\ell}(F^{n})$.

###### Proof.

The first assertion is justified in [Hum20a, Lemmata 7.5 and 7.12]. The second
assertion follows from the first upon recalling that every homogeneous
polynomial $P\in\mathcal{P}_{\chi,\ell}(F^{n})$ of degree $\ell$ and central
character $\chi$ admits a decomposition of the form

$P(x)=\sum_{\begin{subarray}{c}m=c(\chi)\\\ m\equiv
c(\chi)\pmod{2}\end{subarray}}^{\ell}\left(x\prescript{t}{}{\overline{x}}\right)^{\frac{\ell-m}{2}}P_{m}(x)$

for some harmonic homogeneous polynomials
$P_{m}\in\mathcal{H}_{\chi,m}(F^{n})$ [Rud08, Theorem 12.1.3]. ∎

###  3.3. Archimedean Newform Theory

Let $(\pi,V_{\pi})$ be an induced representation of Whittaker type of
$\mathrm{GL}_{n}(F)$. Since $\pi$ is admissible,
$\operatorname{Hom}_{K_{n}}(\tau,\pi|_{K_{n}})$ is finite-dimensional for any
irreducible representation $\tau$ of $K_{n}$. We say that $\tau$ is a _$K_{n}$
-type of $\pi$_ if $\operatorname{Hom}_{K_{n}}(\tau,\pi|_{K_{n}})$ is
nontrivial, and we call $\dim\operatorname{Hom}_{K_{n}}(\tau,\pi|_{K_{n}})$
the _multiplicity_ of $\tau$ in $\pi$. The fundamental result proven in
[Hum20a] is the existence of a distinguished $K_{n}$-type of $\pi$ that occurs
with multiplicity one.

######  Theorem 3.9 ([Hum20a, Theorem 4.7]).

Let $(\pi,V_{\pi})$ be an induced representation of Whittaker type of
$\mathrm{GL}_{n}(F)$. Among the $K_{n}$-types $\tau_{\chi,m}$ of $\pi$ whose
restriction to $K_{n-1}$ contains the trivial representation, there exists a
unique such $K_{n}$-type of minimal Howe degree $m$. Furthermore, this
$K_{n}$-type $\tau_{\chi,m}$ occurs with multiplicity one, and the subspace of
$V_{\pi}$ of $\tau_{\chi,m}$-isotypic $K_{n-1}$-invariant vectors is one-
dimensional.

######  Remark 3.10.

By considering the restrictions of $\pi$ and $\tau_{\chi,m}$ to the centre of
$K_{n}$, we observe that
$\operatorname{Hom}_{K_{n}}(\tau_{\chi,m},\pi|_{K_{n}})$ is trivial if the
central character $\chi$ of $\tau_{\chi,m}$ is not equal to

$\chi_{\pi}\coloneqq\omega_{\pi}|_{K_{1}},$

the restriction of the central character $\omega_{\pi}$ of $\pi$ to $K_{1}$.

######  Definition 3.11 ([Hum20a, Definition 4.8]).

Let $(\pi,V_{\pi})$ be an induced representation of Whittaker type of
$\mathrm{GL}_{n}(F)$. We define the _newform $K_{n}$-type_
$\tau_{\chi_{\pi},c(\pi)}$ to be the $K_{n}$-type of minimal Howe degree
$m=c(\pi)$ whose restriction to $K_{n-1}$ contains the trivial representation.
We define the _conductor exponent_ $c(\pi)$ of $\pi$ to be the Howe degree of
the newform $K_{n}$-type. The nonzero $\tau_{\chi_{\pi},c(\pi)}$-isotypic
$K_{n-1}$-invariant vector $v^{\circ}\in V_{\pi}$, unique up to scalar
multiplication, is called the _newform_ of $\pi$.

######  Remark 3.12.

As proven by the first author in [Hum20b], this definition of the conductor
exponent and the newform, as well as the existence of the newform
$K_{n}$-type, is consistent with the _nonarchimedean_ definition of the
conductor exponent and the newform first introduced by Jacquet, Piatetski-
Shapiro, and Shalika [JP-SS81].

When $V_{\pi}$ is the induced model of $\pi$, we may normalise the newform
$v^{\circ}\in V_{\pi}$ as in [Hum20a, Corollary 8.17 and Definition 9.2],
which gives us a _canonically normalised newform_. When $\pi$ is an induced
representation of Langlands type, the Whittaker model $\mathcal{W}(\pi,\psi)$
is a model of $\pi$, which is given by the analytic continuation of the
Jacquet integral of the induced model. The image of the canonically normalised
newform under this map is called the _Whittaker newform_ and denoted by
$W_{\pi}^{\circ}$ [Hum20a, §9.1]. When $\pi$ is unramified, the Whittaker
newform $W_{\pi}^{\circ}$ is simply the spherical Whittaker function.

The following lemma may be thought as the archimedean analogue of [Hum20b,
Theorem 4.16].

######  Lemma 3.13 (Cf. [Hum20a, Theorem 4.17]).

For all $g\in\mathrm{GL}_{n}(F)$ and $k^{\prime}\in K_{n-1}$, the Whittaker
newform $W_{\pi}^{\circ}\in\mathcal{W}(\pi,\psi)$ of an induced representation
of Langlands type $(\pi,V_{\pi})$ of $\mathrm{GL}_{n}(F)$ satisfies

(3.14)
$\displaystyle\dim\tau_{\chi_{\pi},c(\pi)}\int_{K_{n}}W_{\pi}^{\circ}(gk)P_{\chi_{\pi},c(\pi)}^{\circ}(e_{n}k^{-1})\,dk$
$\displaystyle=W_{\pi}^{\circ}(g),$ (3.15) $\displaystyle
W^{\circ}_{\pi}\left(g\begin{pmatrix}k^{\prime}&0\\\ 0&1\end{pmatrix}\right)$
$\displaystyle=W^{\circ}_{\pi}(g).$

###### Proof.

The identity (3.15) is simply the fact that $W_{\pi}^{\circ}$ is right
$K_{n-1}$-invariant. Since $W_{\pi}^{\circ}$ is additionally
$\tau_{\chi_{\pi},c(\pi)}$-isotypic, it satisfies

$\dim\tau_{\chi_{\pi},c(\pi)}\int_{K_{n}}W_{\pi}^{\circ}(gk)\operatorname{Tr}\tau_{\chi_{\pi},c(\pi)}(k^{-1})\,dk=W_{\pi}^{\circ}(g)$

for all $g\in\mathrm{GL}_{n}(F)$. Upon replacing $g$ with
$g\begin{pmatrix}k^{\prime}&0\\\ 0&1\end{pmatrix}$, integrating over
$K_{n-1}\ni k^{\prime}$, and making the change of variables $k^{\prime}\mapsto
k^{\prime-1}$ and $k\mapsto\begin{pmatrix}k^{\prime}&0\\\ 0&1\end{pmatrix}k$,
we deduce that

$\dim\tau_{\chi_{\pi},c(\pi)}\int_{K_{n}}W_{\pi}^{\circ}(gk)\int_{K_{n-1}}\operatorname{Tr}\tau_{\chi_{\pi},c(\pi)}\left(k^{-1}\begin{pmatrix}k^{\prime}&0\\\
0&1\end{pmatrix}\right)\,dk^{\prime}\,dk=W_{\pi}^{\circ}(g).$

From Proposition 3.2 and (3.7), the inner integral is equal to

$\frac{\left\langle\tau_{\chi_{\pi},c(\pi)}(k^{-1})\cdot
P_{\chi_{\pi},c(\pi)}^{\circ},P_{\chi_{\pi},c(\pi)}^{\circ}\right\rangle}{\left\langle
P_{\chi_{\pi},c(\pi)}^{\circ},P_{\chi_{\pi},c(\pi)}^{\circ}\right\rangle}=P_{\chi_{\pi},c(\pi)}^{\circ}(e_{n}k^{-1}).\qed$

##  4\. Langlands Parameters

We now associate an induced representation of Langlands type $\pi$ of
$\mathrm{GL}_{n}(F)$ to a distinguished spherical induced representation of
Langlands type $\pi_{\mathrm{ur}}$ of $\mathrm{GL}_{n}(F)$ defined in terms of
the Langlands parameters of $\pi$.

######  Definition 4.1.

The Langlands parameters associated to an induced representation of Whittaker
type
$\pi=\mathop{\vphantom{\bigoplus}\mathchoice{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}}\displaylimits_{j=1}^{r}\pi_{j}$
of $\mathrm{GL}_{n}(F)$ are the $n$-tuple of complex numbers
$(\alpha_{\pi,1},\ldots,\alpha_{\pi,n})$ given by

$\alpha_{\pi,\ell}\coloneqq\begin{dcases*}t_{j}+\frac{\|\kappa_{j}\|}{2}&if
$\ell=n_{1}+\cdots+n_{j}$ and
$\pi_{j}=e^{i\kappa_{j}\arg}|\cdot|_{\mathbb{C}}^{t_{j}}$,\\\
t_{j}+\kappa_{j}&if $\ell=n_{1}+\cdots+n_{j}$ and
$\pi_{j}=\operatorname{sgn}^{\kappa_{j}}|\cdot|_{\mathbb{R}}^{t_{j}}$,\\\
t_{j}+\frac{\kappa_{j}+1}{2}&if $\ell=n_{1}+\cdots+n_{j}-1$ and
$\pi_{j}=D_{\kappa_{j}}\otimes\left|\det\right|_{\mathbb{R}}^{t_{j}}$,\\\
t_{j}+\frac{\kappa_{j}-1}{2}&if $\ell=n_{1}+\cdots+n_{j}$ and
$\pi_{j}=D_{\kappa_{j}}\otimes\left|\det\right|_{\mathbb{R}}^{t_{j}}$.\end{dcases*}$

######  Proposition 4.2.

Given an induced representation of Whittaker type $\pi$ of
$\mathrm{GL}_{n}(F)$, there exists a spherical induced representation of
Langlands type $\pi_{\mathrm{ur}}$ of $\mathrm{GL}_{n}(F)$ for which
$L(s,\pi)=L(s,\pi_{\mathrm{ur}})$.

###### Proof.

There exists a permutation $\sigma$ for which the Langlands parameters of
$\pi$ are such that
$\Re(\alpha_{\pi,\sigma(1)})\geq\cdots\geq\Re(\alpha_{\pi,\sigma(n)})$. Let
$\pi_{\mathrm{ur}}\coloneqq\mathop{\vphantom{\bigoplus}\mathchoice{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}}\displaylimits_{j=1}^{n}|\cdot|^{\alpha_{\pi,\sigma(j)}}$.
This isobaric sum of unramified characters is a spherical induced
representation of Langlands type of $\mathrm{GL}_{n}(F)$ that satisfies

$\displaystyle L(s,\pi_{\mathrm{ur}})$
$\displaystyle=\prod_{j=1}^{n}\zeta_{F}\left(s+\alpha_{\pi,\sigma(j)}\right)$
$\displaystyle=\prod_{j=1}^{r}\begin{dcases*}\zeta_{\mathbb{C}}\left(s+t_{j}+\frac{\|\kappa_{j}\|}{2}\right)&if
$\pi_{j}=e^{i\kappa_{j}\arg}|\cdot|_{\mathbb{C}}^{t_{j}}$,\\\
\zeta_{\mathbb{R}}(s+t_{j}+\kappa_{j})&if
$\pi_{j}=\operatorname{sgn}^{\kappa_{j}}|\cdot|_{\mathbb{R}}^{t_{j}}$,\\\
\zeta_{\mathbb{R}}\left(s+t_{j}+\frac{\kappa_{j}-1}{2}\right)\zeta_{\mathbb{R}}\left(s+t_{j}+\frac{\kappa_{j}+1}{2}\right)&if
$\pi_{j}=D_{\kappa_{j}}\otimes\left|\det\right|_{\mathbb{R}}^{t_{j}}$,\end{dcases*}$
$\displaystyle=\prod_{j=1}^{r}L(s,\pi_{j})$ $\displaystyle=L(s,\pi).\qed$

######  Remark 4.3.

Proposition 4.2 ensures the existence of such a spherical representation
$\pi_{\mathrm{ur}}$ of $\mathrm{GL}_{n}(F)$, but does not guarantee the
_uniqueness_. Indeed, if there are at least two Langlands parameters whose
real parts are equal but whose imaginary parts are not, then there exists more
than one permutation $\sigma$ for which the condition
$\Re(\alpha_{\pi,\sigma(1)})\geq\cdots\geq\Re(\alpha_{\pi,\sigma(n)})$ is met,
and hence more than one spherical induced representation of Langlands type
having the same $L$-function as $\pi$. Nonetheless, the spherical Whittaker
function $W_{\pi_{\mathrm{ur}}}^{\circ}\in\mathcal{W}(\pi_{\mathrm{ur}},\psi)$
_is_ uniquely determined, since if two spherical induced representations of
Langlands type have the same $L$-function, then their spherical Whittaker
functions are equal due to the Whittaker–Plancherel theorem (see [Hum20a,
Lemma 10.5] and Lemma 4.6).

######  Remark 4.4.

A result analogous to Proposition 4.2 also holds in the nonarchimedean
setting. A notable difference in this setting is that if $\pi$ is ramified,
then $\pi_{\mathrm{ur}}$ is a spherical representation of $\mathrm{GL}_{m}(F)$
with $m<n$.

The central character $\omega_{\pi_{\mathrm{ur}}}$ of $\pi_{\mathrm{ur}}$ is
closely related to the central character $\omega_{\pi}$ of $\pi$.

######  Lemma 4.5.

Let $\pi$ be an induced representation of Langlands type of
$\mathrm{GL}_{n}(F)$. Then for all $z\in F^{\times}$,

$\omega_{\pi}(z)\overline{\chi_{\pi}}\left(\frac{z}{\|z\|}\right)\|z\|^{c(\pi)}=\omega_{\pi_{\mathrm{ur}}}(z).$

###### Proof.

We write
$\pi=\mathop{\vphantom{\bigoplus}\mathchoice{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}}\displaylimits_{j=1}^{r}\pi_{j}$,
so that the central character of $\pi$ is
$\omega_{\pi}=\prod_{j=1}^{r}\omega_{\pi_{j}}$, where the central character
$\omega_{\pi_{j}}$ of $\pi_{j}$ is

$\omega_{\pi_{j}}=\begin{dcases*}e^{i\kappa_{j}\arg}|\cdot|_{\mathbb{C}}^{t_{j}}&if
$\pi_{j}=e^{i\kappa_{j}\arg}|\cdot|_{\mathbb{C}}^{t_{j}}$,\\\
\operatorname{sgn}^{\kappa_{j}}|\cdot|_{\mathbb{R}}^{t_{j}}&if
$\pi_{j}=\operatorname{sgn}^{\kappa_{j}}|\cdot|_{\mathbb{R}}^{t_{j}}$,\\\
\operatorname{sgn}^{\kappa_{j}\pmod{2}}|\cdot|_{\mathbb{R}}^{t_{j}}&if
$\pi_{j}=D_{\kappa_{j}}\otimes\left|\det\right|_{\mathbb{R}}^{t_{j}}$.\end{dcases*}$

Thus for $F=\mathbb{C}$,
$\omega_{\pi}=e^{i\kappa_{\pi}\arg}|\cdot|_{\mathbb{C}}^{t_{\pi}}$ where
$\kappa_{\pi}\coloneqq\sum_{j=1}^{n}\kappa_{j}$ and
$t_{\pi}\coloneqq\sum_{j=1}^{n}t_{j}$, while for $F=\mathbb{R}$, we have that
$\omega_{\pi}=\operatorname{sgn}^{\kappa_{\pi}}|\cdot|_{\mathbb{R}}^{t_{\pi}}$,
where $\kappa_{\pi}\in\\{0,1\\}$ is such that
$\kappa_{\pi}\equiv\sum_{j=1}^{r}\kappa_{j}\pmod{2}$ and
$t_{\pi}\coloneqq\sum_{j=1}^{r}t_{j}$. In particular,

$\chi_{\pi}\left(\frac{z}{\|z\|}\right)=\begin{dcases*}e^{i\kappa_{\pi}\arg(z)}&if
$F=\mathbb{C}$,\\\ \operatorname{sgn}(z)^{\kappa_{\pi}}&if
$F=\mathbb{R}$.\end{dcases*}$

From [Hum20a, Theorem 4.15 and Section 5.2.1], we have that
$c(\pi)=\sum_{j=1}^{r}\|\kappa_{j}\|$, so that

$\|z\|^{c(\pi)}=\begin{dcases*}|z|_{\mathbb{C}}^{(\|\kappa_{1}\|+\cdots+\|\kappa_{r}\|)/2}&if
$F=\mathbb{C}$,\\\ |z|_{\mathbb{R}}^{\kappa_{1}+\cdots+\kappa_{r}}&if
$F=\mathbb{R}$.\end{dcases*}$

From this, we see that

$\omega_{\pi}(z)\overline{\chi_{\pi}}\left(\frac{z}{\|z\|}\right)\|z\|^{c(\pi)}=|z|^{(t_{1}+\cdots+t_{r})+(\|\kappa_{1}\|+\cdots+\|\kappa_{r}\|)/d_{F}}.$

On the other hand, the central character of $\pi_{\mathrm{ur}}$ is
$\omega_{\pi_{\mathrm{ur}}}=|\cdot|^{t_{\pi_{\mathrm{ur}}}}$ with
$t_{\pi_{\mathrm{ur}}}\coloneqq\sum_{j=1}^{r}(t_{j}+\|\kappa_{j}\|/d_{F})$.
Thus

$\omega_{\pi_{\mathrm{ur}}}(z)=|z|^{(t_{1}+\cdots+t_{r})+(\|\kappa_{1}\|+\cdots+\|\kappa_{r}\|)/d_{F}}.\qed$

We introduce the following by-now well-known lemma. Over nonarchimedean
fields, this lemma plays an important role in the proofs of the stability of
$\gamma$-factors [JS85, Lemma 3.2] and of the local converse theorem [Zha18].
Our case is the archimedean analogue (cf. [Hum20a, Lemma 10.5]) of a result of
Jacquet, Piateski-Shapiro, and Shalika [JP-SS81, Lemme (3.5)].

######  Lemma 4.6.

Let $\pi$ be an induced representation of Langlands type of
$\mathrm{GL}_{n}(F)$. Then

$W_{\pi}^{\circ}\begin{pmatrix}g&0\\\
0&1\end{pmatrix}=W_{\pi_{\mathrm{ur}}}^{\circ}\begin{pmatrix}g&0\\\
0&1\end{pmatrix}\quad\text{for all $g\in\mathrm{GL}_{n-1}(F)$.}$

###### Proof.

We claim that for every spherical induced representation of Langlands type
$\sigma$ of $\mathrm{GL}_{n-1}(F)$ with spherical Whittaker function
$W_{\sigma}^{\circ}\in\mathcal{W}(\sigma,\overline{\psi})$,

$\int\limits_{\mathrm{N}_{n-1}(F)\backslash\mathrm{GL}_{n-1}(F)}\left(W_{\pi}^{\circ}\begin{pmatrix}g&0\\\
0&1\end{pmatrix}-W_{\pi_{\mathrm{ur}}}^{\circ}\begin{pmatrix}g&0\\\
0&1\end{pmatrix}\right)W_{\sigma}^{\circ}(g)\left|\det
g\right|^{s-\frac{1}{2}}\,dg=0$

for $\Re(s)$ sufficiently large; by the Whittaker–Plancherel theorem (see
[Hum20a, Lemma 10.5]), this implies the desired equality. Indeed, we have by
[Hum20a, Theorem 4.17] that

$\displaystyle\int\limits_{\mathrm{N}_{n-1}(F)\backslash\mathrm{GL}_{n-1}(F)}W_{\pi}^{\circ}\begin{pmatrix}g&0\\\
0&1\end{pmatrix}W_{\sigma}^{\circ}(g)\left|\det g\right|^{s-\frac{1}{2}}\,dg$
$\displaystyle=L(s,\pi\times\sigma),$
$\displaystyle\int\limits_{\mathrm{N}_{n-1}(F)\backslash\mathrm{GL}_{n-1}(F)}W_{\pi_{\mathrm{ur}}}^{\circ}\begin{pmatrix}g&0\\\
0&1\end{pmatrix}W_{\sigma}^{\circ}(g)\left|\det g\right|^{s-\frac{1}{2}}\,dg$
$\displaystyle=L(s,\pi_{\mathrm{ur}}\times\sigma),$

and these are equal as we can write
$\sigma=\mathop{\vphantom{\bigoplus}\mathchoice{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}}\displaylimits_{j=1}^{n-1}|\cdot|^{t_{j}}$
for some $t_{j}\in\mathbb{C}$, and then

$L(s,\pi\times\sigma)=\prod_{j=1}^{n-1}L(s+t_{j},\pi)=\prod_{j=1}^{n-1}L(s+t_{j},\pi_{\mathrm{ur}})=L(s,\pi_{\mathrm{ur}}\times\sigma).\qed$

##  5\. Rankin–Selberg Integrals

Given induced representations of Whittaker type $\pi$ and $\sigma$ of
$\mathrm{GL}_{n}(F)$, Whittaker functions $W_{\pi}\in\mathcal{W}(\pi,\psi)$
and $W_{\sigma}\in\mathcal{W}(\sigma,\overline{\psi})$, and a Schwartz–Bruhat
function $\Phi\in\mathscr{S}(F^{n})$, the
$\mathrm{GL}_{n}\times\mathrm{GL}_{n}$ Rankin–Selberg integral is defined by

$\Psi(s,W_{\pi},W_{\sigma},\Phi)\coloneqq\int\limits_{\mathrm{N}_{n}(F)\backslash\mathrm{GL}_{n}(F)}W_{\pi}(g)W_{\sigma}(g)\Phi(e_{n}g)\left|\det
g\right|^{s}\,dg.$

The local Rankin–Selberg $L$-function $L(s,\pi\times\sigma)$ is defined via
the local Langlands correspondence as delineated in [Kna94]. This integral
converges absolutely for $\Re(s)$ sufficiently large and extends
meromorphically to the entire complex plane. Jacquet [Jac09, Theorem 2.3] has
shown that $\Psi(s,W_{\pi},W_{\sigma},\Phi)$ is a holomorphic multiple of
$L(s,\pi\times\sigma)$ and that the quotient

$\frac{\Psi(s,W_{\pi},W_{\sigma},\Phi)}{L(s,\pi\times\sigma)}$

is of finite order in vertical strips.

###  5.1. Test Vectors for $\mathrm{GL}_{n}\times\mathrm{GL}_{n}$
Rankin–Selberg Integrals and Periods

For $x\in\operatorname{Mat}_{n\times m}(F)$, we define the Schwartz–Bruhat
function $\Phi_{\mathrm{ur}}\in\mathscr{S}(\operatorname{Mat}_{n\times m}(F))$
by

(5.1)
$\Phi_{\mathrm{ur}}(x)\coloneqq\exp(-d_{F}\pi\operatorname{Tr}(x\prescript{t}{}{\overline{x}}))=\begin{dcases*}\exp(-\pi\operatorname{Tr}(x\prescript{t}{}{x}))&if
$F=\mathbb{R}$,\\\
\exp(-2\pi\operatorname{Tr}(x\prescript{t}{}{\overline{x}})&if
$F=\mathbb{C}$.\end{dcases*}$

It is readily seen that
$\Phi_{\mathrm{ur}}(xk)=\Phi_{\mathrm{ur}}(k^{\prime}x)=\Phi_{\mathrm{ur}}(x)$
for all $k\in K_{m}$ and $k^{\prime}\in K_{n}$; that is, $\Phi_{\mathrm{ur}}$
is right $K_{m}$-invariant and left $K_{n}$-invariant.

Our goal is to evaluate the $\mathrm{GL}_{n}\times\mathrm{GL}_{n}$
Rankin–Selberg integral when $\pi$ and $\sigma$ may be ramified. Our first
step is to reduce this integral to an integral over
$\mathrm{N}_{n-1}(F)\backslash\mathrm{GL}_{n-1}(F)$.

######  Proposition 5.2.

Let $\pi$ and $\sigma$ be induced representations of Langlands type of
$\mathrm{GL}_{n}(F)$ with Whittaker newforms
$W_{\pi}^{\circ}\in\mathcal{W}(\pi,\psi)$ and
$W_{\sigma}^{\circ}\in\mathcal{W}(\sigma,\overline{\psi})$. Let
$\Phi\in\mathscr{S}(F^{n})$ be the right $K_{n}$-finite Schwartz–Bruhat
function of the form $\Phi(x)=P(x)\exp(-d_{F}\pi
x\prescript{t}{}{\overline{x}})$, where the distinguished homogeneous
polynomial
$P\in\mathcal{P}_{\overline{\chi_{\pi}\chi_{\sigma}},c(\pi)+c(\sigma)}(F^{n})$
is given by

(5.3) $P(x)\coloneqq\sum_{\begin{subarray}{c}m=c(\chi_{\pi}\chi_{\sigma})\\\
m\equiv
c(\chi_{\pi}\chi_{\sigma})\pmod{2}\end{subarray}}^{c(\pi)+c(\sigma)}(x\prescript{t}{}{\overline{x}})^{\frac{c(\pi)+c(\sigma)-m}{2}}(\dim\tau_{\chi_{\pi}\chi_{\sigma},m})\overline{P_{\chi_{\pi}\chi_{\sigma},m}^{\circ}}(x).$

Then the $\mathrm{GL}_{n}\times\mathrm{GL}_{n}$ Rankin–Selberg integral
$\Psi(s,W_{\pi}^{\circ},W_{\sigma}^{\circ},\Phi)$ is equal to

(5.4)
$L(ns,\omega_{\pi_{\mathrm{ur}}}\omega_{\sigma_{\mathrm{ur}}})\int\limits_{\mathrm{N}_{n-1}(F)\backslash\mathrm{GL}_{n-1}(F)}W_{\pi}^{\circ}\begin{pmatrix}h&0\\\
0&1\end{pmatrix}W_{\sigma}^{\circ}\begin{pmatrix}h&0\\\
0&1\end{pmatrix}\left|\det h\right|^{s-1}\,dh.$

###### Proof.

The absolute convergence of the ensuing integral can be justified from [Jac09,
Proposition 3.3 and Lemma 3.5]. By the Iwasawa decomposition
$g=(z1_{n})\begin{pmatrix}h&0\\\ 0&1\end{pmatrix}k$ for
$\mathrm{N}_{n}(F)\backslash\mathrm{GL}_{n}(F)$, the
$\mathrm{GL}_{n}\times\mathrm{GL}_{n}$ Rankin–Selberg integral
$\Psi(s,W_{\pi}^{\circ},W_{\sigma}^{\circ},\Phi)$ is equal to

$\int_{F^{\times}}\omega_{\pi}\omega_{\sigma}(z)|z|^{ns}\int\limits_{\mathrm{N}_{n-1}(F)\backslash\mathrm{GL}_{n-1}(F)}\left|\det
h\right|^{s-1}\\\ \times\int_{K_{n}}W_{\pi}^{\circ}\left(\begin{pmatrix}h&0\\\
0&1\end{pmatrix}k\right)W_{\sigma}^{\circ}\left(\begin{pmatrix}h&0\\\
0&1\end{pmatrix}k\right)\Phi(ze_{n}k)\,dk\,dh\,d^{\times}z.$

We now insert the identities (3.14) for both $W_{\pi}^{\circ}(g)$ and
$W_{\sigma}^{\circ}(g)$ with $g$ replaced by $\begin{pmatrix}h&0\\\
0&1\end{pmatrix}k$ and the variables of integration being $k_{1}\in K_{n}$ for
$W_{\pi}^{\circ}$ and $k_{2}\in K_{n}$ for $W_{\sigma}^{\circ}$. We then
interchange the order of integration and make the change of variables
$k\mapsto k^{-1}$, $k_{1}\mapsto k^{-1}k_{1}$, and $k_{2}\mapsto k^{-1}k_{2}$,
arriving at

$\int_{F^{\times}}\omega_{\pi}\omega_{\sigma}(z)|z|^{ns}\int\limits_{\mathrm{N}_{n-1}(F)\backslash\mathrm{GL}_{n-1}(F)}\left|\det
h\right|^{s-1}\int_{K_{n}}W_{\pi}^{\circ}\left(\begin{pmatrix}h&0\\\
0&1\end{pmatrix}k_{1}\right)\int_{K_{n}}W_{\sigma}^{\circ}\left(\begin{pmatrix}h&0\\\
0&1\end{pmatrix}k_{2}\right)\\\
\times\dim\tau_{\chi_{\pi},c(\pi)}\dim\tau_{\chi_{\sigma},c(\sigma)}\int_{K_{n}}P_{\chi_{\pi},c(\pi)}^{\circ}(e_{n}k_{1}^{-1}k)P_{\chi_{\sigma},c(\sigma)}^{\circ}(e_{n}k_{2}^{-1}k)\Phi(ze_{n}k^{-1})\,dk\,dk_{2}\,dk_{1}\,dh\,d^{\times}z.$

By the addition theorem, Proposition 3.4, the last line is

$\sum_{\ell_{1}=1}^{\dim\tau_{\chi_{\pi},c(\pi)}}\sum_{\ell_{2}=1}^{\dim\tau_{\chi_{\sigma},c(\sigma)}}Q_{\ell_{1}}(e_{n}k_{1}^{-1})Q_{\ell_{2}}^{\prime}(e_{n}k_{2}^{-1})\int_{K_{n}}\overline{Q_{\ell_{1}}}(e_{n}k)\overline{Q_{\ell_{2}}^{\prime}}(e_{n}k)\Phi(ze_{n}k^{-1})\,dk,$

where $\\{Q_{\ell_{1}}\\}$ and $\\{Q_{\ell_{2}}^{\prime}\\}$ are orthonormal
bases of $\mathcal{H}_{\chi_{\pi},c(\pi)}(F^{n})$ and
$\mathcal{H}_{\chi_{\sigma},c(\sigma)}(F^{n})$.

To proceed further, we observe that for $z\in F^{\times}$ and $k\in K_{n}$,we
have that

$\Phi(ze_{n}k^{-1})=\overline{\chi_{\pi}\chi_{\sigma}}\left(\frac{z}{\|z\|}\right)\|z\|^{c(\pi)+c(\sigma)}e^{-d_{F}\pi\|z\|^{2}}\sum_{\begin{subarray}{c}m=c(\chi_{\pi}\chi_{\sigma})\\\
m\equiv
c(\chi_{\pi}\chi_{\sigma})\pmod{2}\end{subarray}}^{c(\pi)+c(\sigma)}(\dim\tau_{\chi_{\pi}\chi_{\sigma},m})P_{\chi_{\pi}\chi_{\sigma},m}^{\circ}(e_{n}k)$

by the definition of the Schwartz–Bruhat function $\Phi\in\mathscr{S}(F^{n})$,
the homogeneity of $P_{\chi_{\pi}\chi_{\sigma},m}^{\circ}$ as in (3.1), and
the identity (3.3). By (3.6), the sum over $m$ is the reproducing kernel for
$\mathcal{P}_{\overline{\chi_{\pi}\chi_{\sigma}},c(\pi)+c(\sigma)}(F^{n})$,
and so the integral over $K_{n}\ni k$ is
$\overline{Q_{\ell_{1}}}(e_{n})\overline{Q_{\ell_{2}}^{\prime}}(e_{n})$ by
(3.8). Using the addition theorem, Proposition 3.4, in _reverse_ and then
using (3.14) to evaluate the integrals over $K_{n}\ni k_{1}$ and $K_{n}\ni
k_{2}$, we find that $\Psi(s,W_{\pi}^{\circ},W_{\sigma}^{\circ},\Phi)$ is
equal to

$\int_{F^{\times}}\omega_{\pi}\omega_{\sigma}(z)\overline{\chi_{\pi}\chi_{\sigma}}\left(\frac{z}{\|z\|}\right)\|z\|^{c(\pi)+c(\sigma)}|z|^{ns}e^{-d_{F}\pi\|z\|^{2}}\,d^{\times}z\\\
\times\int\limits_{\mathrm{N}_{n-1}(F)\backslash\mathrm{GL}_{n-1}(F)}W_{\pi}^{\circ}\begin{pmatrix}h&0\\\
0&1\end{pmatrix}W_{\sigma}^{\circ}\begin{pmatrix}h&0\\\
0&1\end{pmatrix}\left|\det h\right|^{s-1}\,dh.$

It remains to recall Lemma 4.5, which, by (2.1), shows that the integral over
$F^{\times}\ni z$ is
$L(ns,\omega_{\pi_{\mathrm{ur}}}\omega_{\sigma_{\mathrm{ur}}})$. ∎

######  Remark 5.5.

The same proof remains valid in the nonarchimedean setting using the theory of
the newform $K_{n}$-type and $p$-adic spherical harmonics [Hum20b].

With this in hand, we are now able to evaluate this Rankin–Selberg integral by
reducing to the spherical case.

######  Theorem 5.6.

With the notation and hypotheses of Proposition 5.2, the
$\mathrm{GL}_{n}\times\mathrm{GL}_{n}$ Rankin–Selberg integral

$\Psi(s,W_{\pi}^{\circ},W_{\sigma}^{\circ},\Phi)$

is equal to $L(s,\pi_{\mathrm{ur}}\times\sigma_{\mathrm{ur}})$.

###### Proof.

According to Lemma 4.6 and Proposition 5.2, we have the identity

$\Psi(s,W_{\pi}^{\circ},W_{\sigma}^{\circ},\Phi)=L(ns,\omega_{\pi_{\mathrm{ur}}}\omega_{\sigma_{\mathrm{ur}}})\int\limits_{\mathrm{N}_{n-1}(F)\backslash\mathrm{GL}_{n-1}(F)}W_{\pi_{\mathrm{ur}}}^{\circ}\begin{pmatrix}h&0\\\
0&1\end{pmatrix}W_{\sigma_{\mathrm{ur}}}^{\circ}\begin{pmatrix}h&0\\\
0&1\end{pmatrix}\left|\det h\right|^{s-1}\,dh.$

On the other hand, with $\Phi_{\mathrm{ur}}\in\mathscr{S}(F^{n})$ given by
(5.1), the same calculation shows that the right-hand side above is equal to
$\Psi(s,W_{\pi_{\mathrm{ur}}}^{\circ},W_{\sigma_{\mathrm{ur}}}^{\circ},\Phi_{\mathrm{ur}})$.
Stade’s formula [Sta02, Theorem 1.1] then gives the identity

$\Psi(s,W_{\pi_{\mathrm{ur}}}^{\circ},W_{\sigma_{\mathrm{ur}}}^{\circ},\Phi_{\mathrm{ur}})=L(s,\pi_{\mathrm{ur}}\times\sigma_{\mathrm{ur}}).\qed$

######  Remark 5.7.

Stade’s formula is only proved for $F=\mathbb{R}$ in [Sta02], though from
[Sta95, Proposition 2.1], the same method can be used to prove this formula
for $F=\mathbb{C}$. Different proofs of Stade’s formula, valid for
$F\in\\{\mathbb{R},\mathbb{C}\\}$, were given in more general settings by the
first author [Hum20a, Theorem 4.18] and independently by Ishii and Miyazaki
[IM20, Theorem 2.9]; both proofs are based on the work of Jacquet [Jac09].

As previously highlighted, the triple
$(W_{\pi}^{\circ},W_{\sigma}^{\circ},\Phi)$ is a weak test vector, rather than
a strong test vector, for the $\mathrm{GL}_{n}\times\mathrm{GL}_{n}$
Rankin–Selberg integral, since in general the naïve Rankin–Selberg
$L$-function $L(s,\pi_{\mathrm{ur}}\times\sigma_{\mathrm{ur}})$ is not equal
to $L(s,\pi\times\sigma)$. Nonetheless, these two $L$-functions are closely
related.

######  Proposition 5.8.

Given induced representations of Whittaker type $\pi$ of $\mathrm{GL}_{n}(F)$
and $\sigma$ of $\mathrm{GL}_{m}(F)$, there exists a polynomial $p(s)$ for
which

$L(s,\pi_{\mathrm{ur}}\times\sigma_{\mathrm{ur}})=p(s)L(s,\pi\times\sigma).$

###### Proof.

For
$\pi=\mathop{\vphantom{\bigoplus}\mathchoice{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}}\displaylimits_{j=1}^{r}\pi_{j}$
and
$\sigma=\mathop{\vphantom{\bigoplus}\mathchoice{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}}\displaylimits_{\ell=1}^{\rho}\sigma_{\ell}$,
with each $\pi_{j}$ and $\sigma_{\ell}$ essentially square-integrable, we have
that

$L(s,\pi\times\sigma)=\prod_{j=1}^{r}\prod_{\ell=1}^{\rho}L(s,\pi_{j}\times\sigma_{\ell})$

via the local Langlands correspondence [Kna94]. Thus it suffices to consider
the case where $\pi$ and $\sigma$ are both essentially square-integrable.

Suppose first that $F=\mathbb{C}$, so that
$\pi=e^{i\kappa\arg}|\cdot|_{\mathbb{C}}^{t}$, and
$\sigma=e^{i\lambda\arg}|\cdot|_{\mathbb{C}}^{u}$. Then since
$\zeta_{\mathbb{C}}(s)\coloneqq 2(2\pi)^{-s}\Gamma(s)$, and recalling the fact
that $\Gamma(s+1)=s\Gamma(s)$, we have that

$\frac{L(s,\pi_{\mathrm{ur}}\times\sigma_{\mathrm{ur}})}{L(s,\pi\times\sigma)}=\frac{\zeta_{\mathbb{C}}\left(s+t+u+\frac{\|\kappa\|+\|\lambda\|}{2}\right)}{\zeta_{\mathbb{C}}\left(s+t+u+\frac{\|\kappa+\lambda\|}{2}\right)}\\\
=\begin{dcases*}1&if
$\operatorname{sgn}(\kappa)=\operatorname{sgn}(\lambda)$,\\\
(2\pi)^{-\min\\{\|\kappa\|,\|\lambda\|\\}}\prod_{m=0}^{\min\\{\|\kappa\|,\|\lambda\|\\}-1}\left(s+t+u+\frac{\|\kappa+\lambda\|}{2}+m\right)&if
$\operatorname{sgn}(\kappa)\neq\operatorname{sgn}(\lambda)$.\end{dcases*}$

Next suppose that $F=\mathbb{R}$ and that
$\pi=\operatorname{sgn}^{\kappa}|\cdot|_{\mathbb{R}}^{t}$, and
$\sigma=\operatorname{sgn}^{\lambda}|\cdot|_{\mathbb{R}}^{u}$. Then since
$\zeta_{\mathbb{R}}(s)\coloneqq\pi^{-s/2}\Gamma(s/2)$, we have that

$\frac{L(s,\pi_{\mathrm{ur}}\times\sigma_{\mathrm{ur}})}{L(s,\pi\times\sigma)}=\frac{\zeta_{\mathbb{R}}\left(s+t+u+\kappa+\lambda\right)}{\zeta_{\mathbb{R}}\left(s+t+u+\|\kappa-\lambda\|\right)}=\begin{dcases*}1&if
$(\kappa,\lambda)\neq(1,1)$,\\\ \pi^{-1}\left(\frac{s+t+u}{2}\right)&if
$(\kappa,\lambda)=(1,1)$.\end{dcases*}$

If $F=\mathbb{R}$, $\pi=D_{\kappa}\otimes\left|\det\right|_{\mathbb{R}}^{t}$,
and $\sigma=\operatorname{sgn}^{\lambda}|\cdot|_{\mathbb{R}}^{u}$, then

$\displaystyle\frac{L(s,\pi_{\mathrm{ur}}\times\sigma_{\mathrm{ur}})}{L(s,\pi\times\sigma)}$
$\displaystyle=\frac{\zeta_{\mathbb{R}}\left(s+t+u+\frac{\kappa-1}{2}+\lambda\right)\zeta_{\mathbb{R}}\left(s+t+u+\frac{\kappa+1}{2}+\lambda\right)}{\zeta_{\mathbb{R}}\left(s+t+u+\frac{\kappa-1}{2}\right)\zeta_{\mathbb{R}}\left(s+t+u+\frac{\kappa+1}{2}\right)}$
$\displaystyle=\begin{dcases*}1&if $\lambda=0$,\\\
\pi^{-1}\left(\frac{s+t+u+\frac{\kappa-1}{2}}{2}\right)&if
$\lambda=1$.\end{dcases*}$

An analogous identity holds if
$\pi=\operatorname{sgn}^{\kappa}|\cdot|_{\mathbb{R}}^{t}$, and
$\sigma=D_{\lambda}\otimes\left|\det\right|_{\mathbb{R}}^{u}$.

Finally, if $F=\mathbb{R}$,
$\pi=D_{\kappa}\otimes\left|\det\right|_{\mathbb{R}}^{t}$, and
$\sigma=D_{\lambda}\otimes\left|\det\right|_{\mathbb{R}}^{u}$, then

$\displaystyle\frac{L(s,\pi_{\mathrm{ur}}\times\sigma_{\mathrm{ur}})}{L(s,\pi\times\sigma)}$
$\displaystyle=\frac{\zeta_{\mathbb{R}}\left(s+t+u+\frac{\kappa+\lambda}{2}-1\right)\zeta_{\mathbb{R}}\left(s+t+u+\frac{\kappa+\lambda}{2}\right)}{\zeta_{\mathbb{R}}\left(s+t+u+\frac{\kappa+\lambda}{2}-1\right)\zeta_{\mathbb{R}}\left(s+t+u+\frac{\kappa+\lambda}{2}\right)}$
$\displaystyle\hskip
56.9055pt\times\frac{\zeta_{\mathbb{R}}\left(s+t+u+\frac{\kappa+\lambda}{2}\right)\zeta_{\mathbb{R}}\left(s+t+u+\frac{\kappa+\lambda}{2}+1\right)}{\zeta_{\mathbb{R}}\left(s+t+u+\frac{\|\kappa-\lambda\|}{2}\right)\zeta_{\mathbb{R}}\left(s+t+u+\frac{\|\kappa-\lambda\|}{2}+1\right)}$
$\displaystyle=\pi^{-\min\\{\kappa,\lambda\\}}\prod_{m=0}^{\left\lceil\frac{1}{2}\min\\{\kappa,\lambda\\}\right\rceil-1}\left(\frac{s+t+u+\frac{\|\kappa-\lambda\|}{2}}{2}+m\right)$
$\displaystyle\hskip
56.9055pt\times\prod_{m=0}^{\left\lfloor\frac{1}{2}\min\\{\kappa,\lambda\\}\right\rfloor-1}\left(\frac{s+t+u+\frac{\|\kappa-\lambda\|}{2}+1}{2}+m\right).\qed$

When $s=1$, the integral appearing in (5.4) is known as the _Rankin–Selberg
period_. In general, this integral need not converge at $s=1$. When $\pi$ and
$\sigma$ are _unitary_ , however, convergence is guaranteed by the following
lemma. We omit the proof of this lemma, since it is standard [JS81a,
Proposition 3.17]; it relies upon bounds for Whittaker functions by a gauge,
namely [JS90, §4 Propositions 2 and 3].

######  Lemma 5.9.

Let $\pi$ and $\sigma$ be unitary generic irreducible Casselman–Wallach
representations of $\mathrm{GL}_{n}(F)$. For any
$W_{\pi}\in\mathcal{W}(\pi,\psi)$ and
$W_{\sigma}\in\mathcal{W}(\sigma,\overline{\psi})$, the integral

$\int\limits_{\mathrm{N}_{n-1}(F)\backslash\mathrm{GL}_{n-1}(F)}W_{\pi}\begin{pmatrix}h&0\\\
0&1\end{pmatrix}W_{\sigma}\begin{pmatrix}h&0\\\ 0&1\end{pmatrix}\left|\det
h\right|^{s-1}\,dh$

converges absolutely for $\Re(s)\geq 1$.

Archimedean components of cuspidal automorphic representations are unitary
generic irreducible Casselman–Wallach representations twisted by a (possibly
nonunitary) unramified character. In this regard, our unitary assumption is
sufficient for potential global applications therein (cf. Section 8). For
$\sigma=\widetilde{\pi}$, the Rankin–Selberg period (in a slightly modified
form as the inner product) appears in the work of Feigon, Lapid, and Offen
[FLO, Appendix A.1], Gelbart, Jacquet, and Rogawski [GJR01, Lemma 3.3], and W.
Zhang [Zha14, (3.2)]), which are all based on the pioneering result of Jacquet
and Shalika [JS81a, Proposition 3.17]. Notably, Venkatesh [Ven06, §7] has
evaluated the Rankin–Selberg period for nonarchimedean $F$ when
$\sigma=\widetilde{\pi}$ and both Whittaker functions are newforms, so that
this is simply the square of the $L^{2}$-norm of the newform. We prove an
archimedean analogue.

######  Theorem 5.10.

Let $\pi$ and $\sigma$ be unitary generic irreducible Casselman–Wallach
representations of $\mathrm{GL}_{n}(F)$ with Whittaker newforms
$W_{\pi}^{\circ}\in\mathcal{W}(\pi,\psi)$ and
$W_{\sigma}^{\circ}\in\mathcal{W}(\sigma,\overline{\psi})$. Then the
$\mathrm{GL}_{n}\times\mathrm{GL}_{n}$ Rankin–Selberg period

(5.11)
$\int\limits_{\mathrm{N}_{n-1}(F)\backslash\mathrm{GL}_{n-1}(F)}W_{\pi}^{\circ}\begin{pmatrix}h&0\\\
0&1\end{pmatrix}W_{\sigma}^{\circ}\begin{pmatrix}h&0\\\ 0&1\end{pmatrix}\,dh$

is equal to

$\frac{L(1,\pi_{\mathrm{ur}}\times\sigma_{\mathrm{ur}})}{L(n,\omega_{\pi_{\mathrm{ur}}}\omega_{\sigma_{\mathrm{ur}}})}.$

###### Proof.

Having Lemma 5.9 in mind, we take $s=1$ in Proposition 5.2 and invoke Theorem
5.6. ∎

We say that $\pi\otimes\sigma$ is _$\mathrm{GL}_{n}(F)$ -distinguished_ if
$\operatorname{Hom}_{\mathrm{GL}_{n}(F)}(\pi\otimes\sigma,\mathbf{1})$ is
nontrivial. If $\pi$ and $\sigma$ are irreducible, this condition amounts to
saying that $\sigma\simeq\widetilde{\pi}$.

######  Remark 5.12 ([Bar03, §10]).

We define a $\mathrm{P}_{n}(F)$-invariant bilinear form
$\beta:\mathcal{W}(\pi,\psi)\times\mathcal{W}(\sigma,\overline{\psi})\to\mathbb{C}$
by

(5.13)
$\beta(W_{\pi},W_{\sigma})\coloneqq\int\limits_{\mathrm{N}_{n-1}(F)\backslash\mathrm{GL}_{n-1}(F)}W_{\pi}\begin{pmatrix}h&0\\\
0&1\end{pmatrix}W_{\sigma}\begin{pmatrix}h&0\\\ 0&1\end{pmatrix}\,dh.$

It can be deduced from Theorem 5.10 that $\beta$ is a nontrivial bilinear
form. Furthermore, Baruch [Bar03] has shown that any
$\mathrm{P}_{n}(F)$-invariant pairing is $\mathrm{GL}_{n}(F)$-invariant; the
proof is purely local, whereas a local-to-global approach can be extracted
from [Zha14, Proposition 3.1]. Thus $\beta$ gives rise to a nontrivial
$\mathrm{GL}_{n}(F)$-invariant bilinear form on
$\mathcal{W}(\pi,\psi)\times\mathcal{W}(\sigma,\overline{\psi})$.

###  5.2. Modified Rankin–Selberg Integrals by Sakellaridis

Sakellaridis [Sak12, §5] has introduced new types of
$\mathrm{GL}_{n}\times\mathrm{GL}_{n}$ and
$\mathrm{GL}_{n}\times\mathrm{GL}_{n-1}$ Rankin–Selberg integrals that extend
the classical theory of Rankin–Selberg integrals due to Jacquet, Piatetski-
Shapiro, and Shalika [JP-SS83]. We solve the weak test vector problems for
these integrals. For these modified Rankin–Selberg integrals, we let
$G^{\operatorname{diag}}$ denote the image of $G$ under the diagonal embedding
$\mathrm{GL}_{n}\hookrightarrow\mathrm{GL}_{n}\times\mathrm{GL}_{n}$ given by
$g\mapsto(g,g)$ with $G$ a subgroup of $\mathrm{GL}_{n}(F)$.

######  Theorem 5.14.

Let $\pi$ and $\sigma$ be induced representations of Langlands type of
$\mathrm{GL}_{n}(F)$ with Whittaker newforms
$W^{\circ}_{\pi}\in\mathcal{W}(\pi,\psi)$ and
$W^{\circ}_{\sigma}\in\mathcal{W}(\sigma,\overline{\psi})$. Let
$\Phi\in\mathscr{S}(F^{n})$ be the right $K_{n}$-finite Schwartz–Bruhat
function of the form $\Phi(x)=P(x)\exp(-d_{F}\pi
x\prescript{t}{}{\overline{x}})$, where the distinguished homogeneous
polynomial
$P\in\mathcal{P}_{\overline{\chi_{\pi}\chi_{\sigma}},c(\pi)+c(\sigma)}(F^{n})$
is given by (5.3), and let
$\Phi^{\ast}\in\mathscr{S}(\operatorname{Mat}_{n\times n}(F))$ be the
bi-$K_{n}$-finite Schwartz–Bruhat function of the form
$\Phi^{\ast}(x)=P^{\ast}(e_{n}x)\exp(-d_{F}\pi\operatorname{Tr}(x\prescript{t}{}{\overline{x}}))$,
where the distinguished homogeneous polynomial
$P^{\ast}\in\mathcal{H}_{\overline{\chi_{\sigma}},c(\sigma)}(F^{n})$ is given
by

$P^{\ast}(x)\coloneqq(\dim\tau_{\chi_{\sigma},c(\sigma)})\overline{P_{\chi_{\sigma},c(\sigma)}^{\circ}}(x).$

Then for $\Re(s_{1})$ and $\Re(s_{2})$ sufficiently large, the modified
$\mathrm{GL}_{n}(F)\times\mathrm{GL}_{n}(F)$ Rankin–Selberg integral by
Sakellaridis

(5.15)
$\int\limits_{\mathrm{N}^{\operatorname{diag}}_{n}(F)\backslash\mathrm{GL}_{n}(F)\times\mathrm{GL}_{n}(F)}W^{\circ}_{\pi}(g_{1})W^{\circ}_{\sigma}(g_{2})\Phi^{\ast}(g^{-1}_{1}g_{2})\Phi(e_{n}g_{1})\left|\frac{\det
g_{2}}{\det g_{1}}\right|^{s_{1}}\left|\det
g_{1}\right|^{s_{2}}\,dg_{2}\,dg_{1}$

is equal to

$L\left(s_{2},\pi_{\mathrm{ur}}\times\sigma_{\mathrm{ur}}\right)L\left(s_{1}-\frac{n-1}{2},\sigma\right).$

###### Proof.

We make the change of variables $g_{2}\mapsto g_{1}g_{2}$. The ensuing
integral becomes

$\int\limits_{\mathrm{N}_{n}(F)\backslash\mathrm{GL}_{n}(F)}W^{\circ}_{\pi}(g_{1})\Phi(e_{n}g_{1})\left|\det
g_{1}\right|^{s_{2}}\int_{\mathrm{GL}_{n}(F)}W^{\circ}_{\sigma}(g_{1}g_{2})\Phi^{\ast}(g_{2})\left|\det
g_{2}\right|^{s_{1}}\,dg_{2}\,dg_{1}.$

The absolute convergence of the above double integral follows from [Jac09,
Lemma 3.2 (ii), Proposition 3.3, and Lemma 3.5]. Applying [Hum20a, Lemma 9.6]
to the innermost integral yields

$L\left(s_{1}-\frac{n-1}{2},\sigma\right)\int\limits_{\mathrm{N}_{n}(F)\backslash\mathrm{GL}_{n}(F)}W^{\circ}_{\pi}(g_{1})W^{\circ}_{\sigma}(g_{1})\Phi(e_{n}g_{1})\left|\det
g_{1}\right|^{s_{2}}\,dg_{1}.$

The latter is the $\mathrm{GL}_{n}\times\mathrm{GL}_{n}$ Rankin–Selberg
integral, which is $L(s_{2},\pi_{\mathrm{ur}}\times\sigma_{\mathrm{ur}})$ by
Theorem 5.6. ∎

######  Theorem 5.16.

Let $\pi$ be an induced representation of Langlands type of
$\mathrm{GL}_{n}(F)$ with Whittaker newform
$W^{\circ}_{\pi}\in\mathcal{W}(\pi,\psi)$ and let $\sigma$ be a spherical
induced representation of Langlands type of $\mathrm{GL}_{n-1}(F)$ with
spherical Whittaker function
$W^{\circ}_{\sigma}\in\mathcal{W}(\sigma,\overline{\psi})$. Then for
$\Re(s_{1})$ and $\Re(s_{2})$ sufficiently large, the modified
$\mathrm{GL}_{n}(F)\times\mathrm{GL}_{n-1}(F)$ Rankin–Selberg integral by
Sakellaridis

(5.17)
$\int\limits_{\mathrm{N}^{\operatorname{diag}}_{n-1}(F)\backslash\mathrm{GL}_{n-1}(F)\times\mathrm{GL}_{n-1}(F)}W^{\circ}_{\pi}\begin{pmatrix}g_{1}&0\\\
0&1\end{pmatrix}W^{\circ}_{\sigma}(g_{2})\Phi_{\mathrm{ur}}\left(g_{1}^{-1}g_{2}\right)\left|\frac{\det
g_{2}}{\det g_{1}}\right|^{s_{1}}\left|\det
g_{1}\right|^{s_{2}}\,dg_{2}\,dg_{1}$

is equal to

$L\left(s_{2}+\frac{1}{2},\pi\times\sigma\right)L\left(s_{1}-\frac{n-2}{2},\sigma\right)$

where the Schwartz–Bruhat function
$\Phi_{\mathrm{ur}}\in\mathscr{S}(\operatorname{Mat}_{(n-1)\times(n-1)}(F))$
is given by (5.1).

###### Proof.

We make the change of variables $g_{2}\mapsto g_{1}g_{2}$ in order to obtain

$\int\limits_{\mathrm{N}_{n-1}(F)\backslash\mathrm{GL}_{n-1}(F)}W^{\circ}_{\pi}\begin{pmatrix}g_{1}&0\\\
0&1\end{pmatrix}\left|\det
g_{1}\right|^{s_{2}}\int_{\mathrm{GL}_{n-1}(F)}W^{\circ}_{\sigma}(g_{1}g_{2})\Phi_{\mathrm{ur}}(g_{2})\left|\det
g_{2}\right|^{s_{1}}\,dg_{2}\,dg_{1}.$

The absolute convergence of the above double integral can be shown by [Jac09,
Lemma 3.2 (ii) and Section 8.3]. Then [Hum20a, Lemma 9.6] implies that this is
equal to

$L\left(s_{1}-\frac{n-2}{2},\sigma\right)\int\limits_{\mathrm{N}_{n-1}(F)\backslash\mathrm{GL}_{n-1}(F)}W^{\circ}_{\pi}\begin{pmatrix}g_{1}&0\\\
0&1\end{pmatrix}W^{\circ}_{\sigma}(g_{1})\left|\det
g_{1}\right|^{s_{2}}\,dg_{1}.$

From [Hum20a, Theorem 4.17], the latter integral is simply
$L(s_{2}+1/2,\pi\times\sigma)$. ∎

We take this occasion to complete weak test vector problems for modified
Rankin–Selberg integrals over a nonarchimedean local field $F$. The missing
ingredient is a convolution section identity for the Whittaker newform, namely
the nonarchimedean analogue of [Hum20a, Lemma 9.6]. The identity can be viewed
as a generalization of an identity of Godement–Jacquet [GJ72, Lemma 6.10] (cf.
[Sak12, (5-2)]).

######  Lemma 5.18.

Let $F$ be a nonarchimedean local field and let $\pi$ be an induced
representation of Langlands type of $\mathrm{GL}_{n}(F)$ with Whittaker
newform $W^{\circ}_{\pi}\in\mathcal{W}(\pi,\psi)$. Then for all
$h\in\mathrm{GL}_{n}(F)$ and for $\Re(s)$ sufficiently large, we have that

$\int_{\mathrm{GL}_{n}(F)}W^{\circ}_{\pi}(hg)\Phi^{\ast}(g)\left|\det
g\right|^{s+\frac{n-1}{2}}\,dg=L(s,\pi)W^{\circ}_{\pi}(h),$

where $\Phi^{\ast}\in\mathscr{S}(\operatorname{Mat}_{n\times n}(F))$ is the
bi-$K_{n}$-finite Schwartz–Bruhat function

(5.19) $\Phi^{\ast}(x)\\\ \coloneqq\begin{dcases*}1&if $c(\pi)=0$,
$x\in\operatorname{Mat}_{n\times n}(\mathcal{O})$, and
$x_{n,1},\ldots,x_{n,n}\in\mathcal{O}$,\\\
\frac{\omega_{\pi}^{-1}(x_{n,n})}{\operatorname{vol}(K_{0}(\mathfrak{p}^{c(\pi)}))}&if
$c(\pi)>0$, $x\in\operatorname{Mat}_{n\times n}(\mathcal{O})$,
$x_{n,1},\ldots,x_{n,n-1}\in\mathfrak{p}^{c(\pi)}$, and
$x_{n,n}\in\mathcal{O}^{\times}$,\\\ 0&otherwise.\end{dcases*}$

Here $\mathcal{O}$ denotes the ring of integers of $F$, $\mathfrak{p}$ denotes
the maximal ideal of $\mathcal{O}$, $c(\pi)$ denotes the conductor exponent of
$\pi$, and for $m>0$, $K_{0}(\mathfrak{p}^{m})$ denotes the congruence
subgroup

$K_{0}(\mathfrak{p}^{m})\coloneqq\\{k\in\mathrm{GL}_{n}(\mathcal{O}):k_{n,1},\ldots,k_{n,n-1}\in\mathfrak{p}^{m}\\},$

which has volume $q^{-(m-1)(n-1)}(q-1)/(q^{n}-1)$, where
$q\coloneqq\\#\mathcal{O}/\mathfrak{p}$.

###### Proof.

Let
$\sigma=\mathop{\vphantom{\bigoplus}\mathchoice{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}}\displaylimits_{j=1}^{n}|\cdot|^{t_{j}}$
and
$\sigma_{0}\coloneqq\mathop{\vphantom{\bigoplus}\mathchoice{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}}\displaylimits_{j=2}^{n}|\cdot|^{t_{j}}$
be induced representations of Langlands type of $\mathrm{GL}_{n}(F)$ and
$\mathrm{GL}_{n-1}(F)$ with spherical Whittaker functions
$W_{\sigma}^{\circ}\in\mathcal{W}(\sigma,\overline{\psi})$ and
$W_{\sigma_{0}}^{\circ}\in\mathcal{W}(\sigma_{0},\overline{\psi})$. By [Kim10,
Theorem 2.1.1], the $\mathrm{GL}_{n}\times\mathrm{GL}_{n}$ Rankin–Selberg
integral

$\Psi(s,W_{\pi}^{\circ},W_{\sigma}^{\circ},\Phi^{\circ})\coloneqq\int\limits_{\mathrm{N}_{n}(F)\backslash\mathrm{GL}_{n}(F)}W_{\pi}^{\circ}(g)W_{\sigma}^{\circ}(g)\Phi^{\circ}(e_{n}g)\left|\det
g\right|^{s}\,dg$

is equal to

$L(s,\pi\times\sigma)=L(s+t_{1},\pi)L(s,\pi\times\sigma_{0}),$

where $\Phi^{\circ}\in\mathscr{S}(F^{n})$ is the Schwartz–Bruhat function

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

On the other hand, we may insert into the
$\mathrm{GL}_{n}\times\mathrm{GL}_{n}$ Rankin–Selberg integral the propagation
formula [Hum21, Lemma 4.1]

$W_{\sigma}^{\circ}(g)=\left|\det
g\right|^{t_{1}+\frac{n-1}{2}}\int_{\mathrm{GL}_{n-1}(F)}W_{\sigma_{0}}^{\circ}(h)\left|\det
h\right|^{-t_{1}-\frac{n}{2}}\\\
\times\int\limits_{\operatorname{Mat}_{(n-1)\times
1}(F)}\Phi^{\prime}\left(h^{-1}\begin{pmatrix}1_{n-1}&v\end{pmatrix}g\right)\psi(e_{n-1}v)\,dv\,dh,$

where $\Phi^{\prime}\in\mathscr{S}(\operatorname{Mat}_{(n-1)\times n}(F))$ is
the Schwartz–Bruhat function

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

unfold the integration, and make the change of variables
$g\mapsto\begin{pmatrix}h&0\\\ 0&1\end{pmatrix}g$ in order to see that

$\Psi(s,W_{\pi}^{\circ},W_{\sigma}^{\circ},\Phi^{\circ})\\\
=\int\limits_{\mathrm{N}_{n-1}(F)\backslash\mathrm{GL}_{n-1}(F)}W_{\sigma_{0}}^{\circ}(h)\left|\det
h\right|^{s-\frac{1}{2}}\int_{\mathrm{GL}_{n}(F)}W_{\pi}^{\circ}\left(\begin{pmatrix}h&0\\\
0&1\end{pmatrix}g\right)\Phi^{\ast}(g)\left|\det
g\right|^{s+t_{1}+\frac{n-1}{2}}\,dg\,dh$

since
$\Phi^{\ast}(g)=\Phi^{\prime}(\begin{pmatrix}1_{n-1}&0\end{pmatrix}g)\Phi^{\circ}(e_{n}g)$.
So letting $W_{\pi}^{\prime}\in\mathcal{W}(\pi,\psi)$ be given by

$W_{\pi}^{\prime}(h)\coloneqq\frac{1}{L(w,\pi)}\int_{\mathrm{GL}_{n}(F)}W_{\pi}^{\circ}(hg)\Phi^{\ast}(g)\left|\det
g\right|^{w+\frac{n-1}{2}}\,dg$

with $\Re(w)$ sufficiently large, we see that for every induced representation
of Langlands type $\sigma_{0}$ of $\mathrm{GL}_{n-1}(F)$ with spherical
Whittaker function
$W_{\sigma_{0}}^{\circ}\in\mathcal{W}(\sigma_{0},\overline{\psi})$,

$\int\limits_{\mathrm{N}_{n-1}(F)\backslash
GL_{n-1}(F)}\left(W_{\pi}^{\prime}\begin{pmatrix}h&0\\\
0&1\end{pmatrix}-W_{\pi}^{\circ}\begin{pmatrix}h&0\\\
0&1\end{pmatrix}\right)W_{\sigma_{0}}^{\circ}(h)\left|\det
h\right|^{s-\frac{1}{2}}\,dh=0$

for $\Re(s)$ sufficiently large due to [JP-SS81, Théorème (4)]. Since
$W_{\pi}^{\prime}$ is right $\mathrm{GL}_{n-1}(\mathcal{O})$-invariant, as we
may make the change of variables $g\mapsto\begin{pmatrix}k^{\prime-1}&0\\\
0&1\end{pmatrix}g$ and use the fact that $\Phi$ is left
$\mathrm{GL}_{n-1}(\mathcal{O})$-invariant, we therefore have that
$W_{\pi}^{\prime}\begin{pmatrix}h&0\\\
0&1\end{pmatrix}=W_{\pi}^{\circ}\begin{pmatrix}h&0\\\ 0&1\end{pmatrix}$ for
all $h\in\mathrm{GL}_{n-1}(F)$ by [JP-SS81, Lemme (3.5)]. Invoking the
uniqueness of the Kirillov model of $\pi$, we deduce that
$W_{\pi}^{\prime}(g)=W_{\pi}^{\circ}(g)$ for all $g\in\mathrm{GL}_{n}(F)$. ∎

We present following consequences of the convolution section identity in Lemma
5.18, which follow in the exact same manner as in the archimedean setting.

######  Corollary 5.21.

Let $F$ be a nonarchimedean local field, and let $\pi$ and $\sigma$ be induced
representations of Langlands type of $\mathrm{GL}_{n}(F)$ with Whittaker
newforms $W_{\pi}^{\circ}\in\mathcal{W}(\pi,\psi)$ and
$W_{\sigma}^{\circ}\in\mathcal{W}(\sigma,\overline{\psi})$. Let
$\Phi^{\ast}\in\mathscr{S}(\operatorname{Mat}_{n\times n}(F))$ and
$\Phi^{\circ}\in\mathscr{S}(F^{n})$ be the Schwartz–Bruhat functions given by
(5.19) and (5.20). Then for $\Re(s_{1})$ and $\Re(s_{2})$ sufficiently large,
the modified $\mathrm{GL}_{n}(F)\times\mathrm{GL}_{n}(F)$ Rankin–Selberg
integral by Sakellaridis

$\int\limits_{\mathrm{N}^{\operatorname{diag}}_{n}(F)\backslash\mathrm{GL}_{n}(F)\times\mathrm{GL}_{n}(F)}W^{\circ}_{\pi}(g_{1})W^{\circ}_{\sigma}(g_{2})\Phi^{\ast}(g^{-1}_{1}g_{2})\Phi^{\circ}(e_{n}g_{1})\left|\frac{\det
g_{2}}{\det g_{1}}\right|^{s_{1}}\left|\det
g_{1}\right|^{s_{2}}\,dg_{2}\,dg_{1}$

is equal to

$L\left(s_{2},\pi_{\mathrm{ur}}\times\sigma_{\mathrm{ur}}\right)L\left(s_{1}-\frac{n-1}{2},\sigma\right).$

######  Corollary 5.22.

Let $F$ be a nonarchimedean local field, let $\pi$ be an induced
representation of Langlands type of $\mathrm{GL}_{n}(F)$ with Whittaker
newform $W^{\circ}_{\pi}\in\mathcal{W}(\pi,\psi)$, and let $\sigma$ be a
spherical induced representation of Langlands type of $\mathrm{GL}_{n-1}(F)$
with spherical Whittaker function
$W^{\circ}_{\sigma}\in\mathcal{W}(\sigma,\overline{\psi})$. Then for
$\Re(s_{1})$ and $\Re(s_{2})$ sufficiently large, the modified
$\mathrm{GL}_{n}(F)\times\mathrm{GL}_{n-1}(F)$ Rankin–Selberg integral by
Sakellaridis

$\int\limits_{\mathrm{N}^{\operatorname{diag}}_{n-1}(F)\backslash\mathrm{GL}_{n-1}(F)\times\mathrm{GL}_{n-1}(F)}W^{\circ}_{\pi}\begin{pmatrix}g_{1}&0\\\
0&1\end{pmatrix}W^{\circ}_{\sigma}(g_{2})\Phi_{\mathrm{ur}}\left(g_{1}^{-1}g_{2}\right)\left|\frac{\det
g_{2}}{\det g_{1}}\right|^{s_{1}}\left|\det
g_{1}\right|^{s_{2}}\,dg_{2}\,dg_{1}$

is equal to

$L\left(s_{2}+\frac{1}{2},\pi\times\sigma\right)L\left(s_{1}-\frac{n-2}{2},\sigma\right)$

where the Schwartz–Bruhat function
$\Phi_{\mathrm{ur}}\in\mathscr{S}(\operatorname{Mat}_{(n-1)\times(n-1)}(F))$
is given by

$\Phi_{\mathrm{ur}}(x)\coloneqq\begin{dcases*}1&if
$x\in\operatorname{Mat}_{(n-1)\times(n-1)}(\mathcal{O})$,\\\
0&otherwise.\end{dcases*}$

For a pair of spherical induced representations of Langlands type over
nonarchimedean local fields, such formulæ are due to Sakellaridis [Sak12, §5].

##  6\. Flicker Integrals

We define an additive character on $\mathbb{C}$ by
$\psi_{\mathbb{C}/\mathbb{R}}(z)\coloneqq e^{-2\pi(z-\overline{z})}$, so that
$\psi_{\mathbb{C}/\mathbb{R}}(x+iy)=e^{-4\pi iy}$ for $x,y\in\mathbb{R}$; this
additive character is trivial when restricted to $\mathbb{R}$. We let

$\psi_{\mathbb{C}/\mathbb{R},n}(u)\coloneqq\psi_{\mathbb{C}/\mathbb{R}}\left(\sum_{j=1}^{n-1}u_{j,j+1}\right)$

denote the corresponding character of $\mathrm{N}_{n}(\mathbb{C})\ni u$. Given
an induced representation of Whittaker type $\pi$ of
$\mathrm{GL}_{n}(\mathbb{C})$, a Whittaker function
$W_{\pi}\in\mathcal{W}(\pi,\psi_{\mathbb{C}/\mathbb{R}})$, and a
Schwartz–Bruhat function $\Phi\in\mathscr{S}(\mathbb{R}^{n})$, the
$\mathrm{GL}_{n}$ Flicker integral [Fli93, FZ95] is defined by

$\Psi(s,W_{\pi},\Phi)\coloneqq\int\limits_{\mathrm{N}_{n}(\mathbb{R})\backslash\mathrm{GL}_{n}(\mathbb{R})}W_{\pi}(g)\Phi(e_{n}g)\left|\det
g\right|_{\mathbb{R}}^{s}\,dg.$

Once more, this integral converges absolutely for $\Re(s)$ sufficiently and
extends meromorphically to the entire complex plane. The local Asai
$L$-function $L(s,\pi,\mathrm{As})$ is defined via the local Langlands
correspondence as described accurately in [B-P21, Section 3.2].
Beuzart–Plessis [B-P21, Theorem 3.5] has shown that $\Psi(s,W_{\pi},\Phi)$ is
a holomorphic multiple of $L(s,\pi,\mathrm{As})$ and that the quotient

$\frac{\Psi(s,W_{\pi},\Phi)}{L(s,\pi,\mathrm{As})}$

is of finite order in vertical strips.

###  6.1. The Spherical Calculation

We recall the Iwasawa decomposition

$\mathrm{GL}_{n}(\mathbb{C})=\mathrm{N}_{n}(\mathbb{C})\mathrm{A}_{n}(\mathbb{C})\mathrm{U}(n).$

Since every element of $\mathrm{A}_{n}(\mathbb{C})$ may be written as the
product of an element of $\mathrm{A}_{n}(\mathbb{R})$ and of
$\mathrm{A}_{n}(\mathbb{C})\cap\mathrm{U}(n)$, we also have the Iwasawa
decomposition

$\mathrm{GL}_{n}(\mathbb{C})=\mathrm{N}_{n}(\mathbb{C})\mathrm{A}_{n}(\mathbb{R})\mathrm{U}(n).$

We write $g=uak$, where $g\in\mathrm{GL}_{n}(\mathbb{C})$,
$u\in\mathrm{N}_{n}(\mathbb{C})$,
$a=\operatorname{diag}(a_{1},\ldots,a_{n})\in\mathrm{A}_{n}(\mathbb{R})$, and
$k\in\mathrm{U}(n)$. The Haar measure on $\mathrm{GL}_{n}(\mathbb{C})$ becomes

$dg=2^{n}\delta_{\mathrm{B}_{n}(\mathbb{R})}^{-2}(a)\,dk\,d^{\times}a\,du.$

Here the additional factor of $2^{n}$ arises from the fact that the diagonal
torus is chosen to be $\mathrm{A}_{n}(\mathbb{R})$ in place of
$\mathrm{A}_{n}(\mathbb{C})$, recalling that the Haar measure on $\mathbb{C}$
is twice the Lebesgue measure, while for $a\in\mathrm{A}_{n}(\mathbb{R})$, we
observe that
$\delta_{\mathrm{B}_{n}(\mathbb{R})}^{-2}(a)=\delta_{\mathrm{B}_{n}(\mathbb{C})}^{-1}(a)$.

We first require the following propagation formula.

######  Lemma 6.1.

Let
$\pi=\mathop{\vphantom{\bigoplus}\mathchoice{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}}\displaylimits_{j=1}^{n}|\cdot|_{\mathbb{C}}^{t_{j}}$
and
$\pi_{0}\coloneqq\mathop{\vphantom{\bigoplus}\mathchoice{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}}\displaylimits_{j=2}^{n}|\cdot|_{\mathbb{C}}^{t_{j}}$
be spherical induced representations of Langlands type of
$\mathrm{GL}_{n}(\mathbb{C})$ and $\mathrm{GL}_{n-1}(\mathbb{C})$ respectively
with spherical Whittaker functions
$W_{\pi}^{\circ}\in\mathcal{W}(\pi,\psi_{\mathbb{C}/\mathbb{R}})$ and
$W_{\pi_{0}}^{\circ}\in\mathcal{W}(\pi_{0},\psi_{\mathbb{C}/\mathbb{R}})$.
Then for
$a=\operatorname{diag}(a_{1},\ldots,a_{n})\in\mathrm{A}_{n}(\mathbb{R})$, we
have that

(6.2) $W_{\pi}^{\circ}(a)=2^{n-1}\left|\det
a\right|_{\mathbb{R}}^{2t_{1}}\delta_{\mathrm{B}_{n}(\mathbb{R})}(a)\int_{\mathrm{A}_{n-1}(\mathbb{R})}W_{\pi_{0}}^{\circ}(a^{\prime})\prod_{j=1}^{n-1}\exp\left(-2\pi
a_{j}^{\prime 2}a_{j+1}^{-2}\right)\\\ \times\exp\left(-2\pi
a_{j}^{\prime-2}a_{j}^{2}\right)\left|\det
a^{\prime}\right|_{\mathbb{R}}^{-2t_{1}}\delta_{\mathrm{B}_{n-1}(\mathbb{R})}^{-1}(a^{\prime})\,d^{\times}a^{\prime}.$

###### Proof.

From [Hum20a, Lemma 9.14], we have the identity

$W_{\pi_{0}}^{\circ}\begin{pmatrix}g&0\\\ 0&a_{n}\end{pmatrix}=\left|\det
g\right|_{\mathbb{C}}^{t_{1}+\frac{n-1}{2}}|a_{n}|_{\mathbb{C}}^{t_{1}-\frac{n-1}{2}}\\\
\times\int_{\mathrm{GL}_{n-1}(\mathbb{C})}W_{\pi_{0}}^{\circ}(h)\Phi_{1}(h^{-1}g)\Phi_{2}(a_{n}^{-1}e_{n-1}h)\left|\det
h\right|_{\mathbb{C}}^{-t_{1}-\frac{n}{2}+1}\,dh$

for $g\in\mathrm{GL}_{n}(\mathbb{C})$ and $a_{n}\in\mathbb{C}^{\times}$, where
$\Phi_{1}\in\mathscr{S}(\operatorname{Mat}_{(n-1)\times(n-1)}(\mathbb{C}))$
and $\Phi_{2}\in\mathscr{S}(\operatorname{Mat}_{1\times(n-1)}(\mathbb{C}))$
are given by

$\Phi_{1}(x_{1})\coloneqq\exp(-2\pi\operatorname{Tr}(x_{1}\prescript{t}{}{\overline{x_{1}}})),\qquad\Phi_{2}(x_{2})\coloneqq\exp(-2\pi
x_{2}\prescript{t}{}{\overline{x_{2}}}).$

We employ the Iwasawa decomposition in order to write
$h=u^{\prime}a^{\prime}k^{\prime}$, where
$u^{\prime}\in\mathrm{N}_{n-1}(\mathbb{C})$,
$a^{\prime}=\operatorname{diag}(a_{1}^{\prime},\ldots,a_{n-1}^{\prime})\in\mathrm{A}_{n-1}(\mathbb{R})$,
and $k^{\prime}\in\mathrm{U}(n-1)$. As
$W_{\pi_{0}}^{\circ}(u^{\prime}a^{\prime}k^{\prime})=\psi_{\mathbb{C}/\mathbb{R},n-1}(u^{\prime})W_{\pi_{0}}^{\circ}(a^{\prime})$,
the integral over $\mathrm{U}(n-1)\ni k^{\prime}$ is trivial.

Now we specify
$g=\operatorname{diag}(a_{1},\ldots,a_{n-1})\in\mathrm{A}_{n-1}(\mathbb{R})$
and $a_{n}\in\mathbb{R}$, so that $\Phi_{2}(a_{n}^{-1}e_{n-1}h)=\exp(-2\pi
a_{n-1}^{\prime 2}a_{n}^{-2})$. We make the change of variables
$u^{\prime}\mapsto u^{\prime-1}$, then $u_{i,j}^{\prime}\mapsto
a_{i}^{\prime}a_{j}^{-1}u_{i,j}^{\prime}$, and finally evaluate the integrals
over $\mathbb{C}\ni u_{i,j}^{\prime}$; they are equal to $1$ if $j\neq i+1$
and to $\exp(-2\pi a_{i}^{\prime 2}a_{i+1}^{-2})$ if $j=i+1$. This gives the
desired identity. ∎

We also require a convolution section identity.

######  Lemma 6.3.

Let
$\pi=\mathop{\vphantom{\bigoplus}\mathchoice{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}}\displaylimits_{j=1}^{n}|\cdot|_{\mathbb{C}}^{t_{j}}$
be a spherical induced representation of Langlands type of
$\mathrm{GL}_{n}(\mathbb{C})$ with spherical Whittaker functions
$W_{\pi}^{\circ}\in\mathcal{W}(\pi,\psi_{\mathbb{C}/\mathbb{R}})$. Then for
$a^{\prime}=\operatorname{diag}(a_{1}^{\prime},\ldots,a_{n}^{\prime})\in\mathrm{A}_{n}(\mathbb{R})$,
we have that for $\Re(s)$ sufficiently large,

(6.4) $L(s,\pi)W_{\pi}^{\circ}(a^{\prime})\\\
=2^{n}\int_{\mathrm{A}_{n}(\mathbb{R})}W_{\pi}^{\circ}(a^{\prime}a)\prod_{j=1}^{n}\exp(-2\pi
a_{j}^{2})\prod_{j=1}^{n-1}\exp\left(-2\pi a_{j}^{\prime
2}a_{j+1}^{\prime-2}a_{j+1}^{-2}\right)\left|\det
a\right|_{\mathbb{R}}^{2s}\delta_{\mathrm{B}_{n}(\mathbb{R})}^{-1}(a)\,d^{\times}a.$

###### Proof.

From [Hum20a, Lemma 9.6], we have the identity

$L(s,\pi)W_{\pi}^{\circ}(a^{\prime})=\int_{\mathrm{GL}_{n-1}(\mathbb{C})}W_{\pi}^{\circ}(a^{\prime}h)\Phi_{\mathrm{ur}}(h)\left|\det
h\right|_{\mathbb{C}}^{s+\frac{n-1}{2}}\,dh,$

with $\Phi_{\mathrm{ur}}\in\mathscr{S}(\operatorname{Mat}_{n\times
n}(\mathbb{C}))$ given by (5.1). We employ the Iwasawa decomposition in order
to write $h=uak$, where $u\in\mathrm{N}_{n}(\mathbb{C})$,
$a=\operatorname{diag}(a_{1},\ldots,a_{n})\in\mathrm{A}_{n}(\mathbb{R})$, and
$k\in\mathrm{U}(n)$. As
$W_{\pi}^{\circ}(a^{\prime}uak)=\psi_{\mathbb{C}/\mathbb{R},n-1}(a^{\prime}ua^{\prime-1})W_{\pi}^{\circ}(a^{\prime}a)$,
the integral over $\mathrm{U}(n)\ni k$ is trivial. We make the change of
variables $u_{i,j}\mapsto a_{j}^{-1}u_{i,j}$, and finally evaluate the
integrals over $\mathbb{C}\ni u_{i,j}$; they are equal to $1$ if $j\neq i+1$
and to $\exp(-2\pi a_{i}^{\prime 2}a_{i+1}^{\prime-2}a_{i+1}^{-2})$ if
$j=i+1$. We arrive at the identity (6.4). ∎

We are now able to prove that when $\pi$ is a spherical induced representation
of Langlands type of $\mathrm{GL}_{n}(\mathbb{C})$, the spherical Whittaker
function $W_{\pi}^{\circ}\in\mathcal{W}(\pi,\psi_{\mathbb{C}/\mathbb{R}})$ is
a strong test vector for the Flicker integral.

######  Theorem 6.5.

Let
$\pi=\mathop{\vphantom{\bigoplus}\mathchoice{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}}\displaylimits_{j=1}^{n}|\cdot|_{\mathbb{C}}^{t_{j}}$
be a spherical induced representation of Langlands type of
$\mathrm{GL}_{n}(\mathbb{C})$ with spherical Whittaker function
$W_{\pi}^{\circ}\in\mathcal{W}(\pi,\psi_{\mathbb{C}/\mathbb{R}})$. Then with
$\Phi_{\mathrm{ur}}\in\mathscr{S}(\mathbb{R}^{n})$ given by (5.1), the Flicker
integral $\Psi(s,W_{\pi}^{\circ},\Phi_{\mathrm{ur}})$ is equal to

$L(s,\pi,\mathrm{As})\coloneqq\prod_{j=1}^{n}\zeta_{\mathbb{R}}(s+2t_{j})\prod_{1\leq
j<\ell\leq n}\zeta_{\mathbb{C}}(s+t_{j}+t_{\ell}).$

###### Proof.

We prove this by induction. The base case $n=1$ is trivially true since the
$\mathrm{GL}_{1}$ Flicker integral is simply

$\int_{\mathbb{R}^{\times}}\exp(-\pi
x^{2})|x|_{\mathbb{R}}^{s+2t_{1}}\,d^{\times}x=\zeta_{\mathbb{R}}(s+2t_{1})=L(s,\pi,\mathrm{As}).$

For the general case, we shall show that if
$\pi=\mathop{\vphantom{\bigoplus}\mathchoice{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}}\displaylimits_{j=1}^{n}|\cdot|_{\mathbb{C}}^{t_{j}}$
and
$\pi_{0}\coloneqq\mathop{\vphantom{\bigoplus}\mathchoice{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}}\displaylimits_{j=2}^{n}|\cdot|_{\mathbb{C}}^{t_{j}}$,
then the $\mathrm{GL}_{n}$ Flicker integral satisfies the identity

(6.6)
$\Psi(s,W_{\pi}^{\circ},\Phi_{\mathrm{ur}})=\zeta_{\mathbb{R}}(s+2t_{1})L(s+t_{1},\pi_{0})I(s,W_{\pi_{0}}^{\circ},\Phi_{\mathrm{ur}}),$

where the $\mathrm{GL}_{n-1}$ Flicker integral on the right-hand side involves
the spherical Whittaker function
$W_{\pi_{0}}^{\circ}\in\mathcal{W}(\pi_{0},\psi_{\mathbb{C}/\mathbb{R}})$ and
the Schwartz–Bruhat function
$\Phi_{\mathrm{ur}}\in\mathscr{S}(\mathbb{R}^{n-1})$ given by (5.1). Since

$L(s+t_{1},\pi_{0})=\prod_{\ell=2}^{n}\zeta_{\mathbb{C}}(s+t_{1}+t_{\ell}),$

this implies the result by the induction hypothesis.

We first note that the $\Psi(s,W_{\pi}^{\circ},\Phi_{\mathrm{ur}})$ is equal
to

$\int_{\mathrm{A}_{n}(\mathbb{R})}W_{\pi}^{\circ}(a)\exp(-\pi
a_{n}^{2})\left|\det
a\right|_{\mathbb{R}}^{s}\delta_{\mathrm{B}_{n}(\mathbb{R})}^{-1}(a)\,d^{\times}a$

via the Iwasawa decomposition $g=ak$, since the integral over
$\mathrm{O}(n)\ni k$ is trivial. We insert the propagation formula (6.2) and
relabel
$a=\operatorname{diag}(a_{1},\ldots,a_{n})\in\mathrm{A}_{n}(\mathbb{R})$ as
$\begin{pmatrix}y&0\\\ 0&a\end{pmatrix}$, where now $y\in\mathbb{R}^{\times}$
and
$a=\operatorname{diag}(a_{1},\ldots,a_{n-1})\in\mathrm{A}_{n-1}(\mathbb{R})$.
The Flicker integral $\Psi(s,W_{\pi}^{\circ},\Phi_{\mathrm{ur}})$ becomes

$2^{n-1}\int_{\mathrm{A}_{n-1}(\mathbb{R})}\exp(-\pi a_{n-1}^{2})\left|\det
a\right|_{\mathbb{R}}^{s+2t_{1}}\int_{\mathrm{A}_{n-1}(\mathbb{R})}W_{\pi_{0}}^{\circ}(a^{\prime})\prod_{j=1}^{n-2}\exp\left(-2\pi
a_{j+1}^{\prime-2}a_{j}^{2}\right)\\\ \times\prod_{j=1}^{n-1}\exp\left(-2\pi
a_{j}^{\prime 2}a_{j}^{-2}\right)\left|\det
a^{\prime}\right|_{\mathbb{R}}^{-2t_{1}}\delta_{\mathrm{B}_{n-1}(\mathbb{R})}^{-1}(a^{\prime})\int_{\mathbb{R}^{\times}}\exp(-2\pi
a_{1}^{\prime-2}y^{2})|y|_{\mathbb{R}}^{s+2t_{1}}\,d^{\times}y\,d^{\times}a^{\prime}\,d^{\times}a.$

We make the change of variables $y\mapsto 2^{-1/2}a_{1}^{\prime}y$, then the
change of variables $a_{j}^{\prime}\mapsto 2^{-1/2}a_{j}^{\prime}a_{j}$ and
$a_{j}\mapsto 2^{1/2}a_{j}^{\prime-1}a_{j+1}^{\prime}a_{j}$, and finally
$a_{n-1}\mapsto 2^{1/2}a_{n-1}$. We arrive at the identity

$\int_{\mathbb{R}^{\times}}\exp(-\pi
y^{2})|y|_{\mathbb{R}}^{s+2t_{1}}\,d^{\times}y\int_{\mathrm{A}_{n-1}(\mathbb{R})}\exp(-\pi
a_{n-1}^{\prime 2})\left|\det
a^{\prime}\right|_{\mathbb{R}}^{s}\delta_{\mathrm{B}_{n-1}(\mathbb{R})}^{-1}(a^{\prime})\\\
\times
2^{n-1}\int_{\mathrm{A}_{n-1}(\mathbb{R})}W_{\pi_{0}}^{\circ}(a^{\prime}a)\prod_{j=1}^{n-1}\exp(-2\pi
a_{j}^{2})\prod_{j=1}^{n-2}\exp\left(-2\pi a_{j}^{\prime
2}a_{j+1}^{\prime-2}a_{j+1}^{-2}\right)\\\ \times\left|\det
a\right|_{\mathbb{R}}^{2s+2t_{1}}\delta_{\mathrm{B}_{n-1}(\mathbb{R})}^{-1}(a)\,d^{\times}a\,d^{\times}a^{\prime}.$

The integral over $\mathbb{R}^{\times}\ni y$ is
$\zeta_{\mathbb{R}}(s+2t_{1})$, while the last two lines are
$L(s+t_{1},\pi_{0})W_{\pi_{0}}^{\circ}(a^{\prime})$ by the convolution section
identity (6.4). So this is

$\zeta_{\mathbb{R}}(s+2t_{1})L(s+t_{1},\pi_{0})\int_{\mathrm{A}_{n-1}(\mathbb{R})}W_{\pi_{0}}^{\circ}(a^{\prime})\exp(-\pi
a_{n-1}^{\prime 2})\left|\det
a^{\prime}\right|_{\mathbb{R}}^{s}\delta_{\mathrm{B}_{n-1}(\mathbb{R})}^{-1}(a^{\prime})\,d^{\times}a^{\prime},$

which is precisely (6.6) by the Iwasawa decomposition
$g=a^{\prime}k^{\prime}$, since the integral over $\mathrm{O}(n-1)\ni
k^{\prime}$ is trivial. ∎

###  6.2. Test Vectors for Flicker Integrals and Flicker–Rallis Periods

We proceed to the more general case where $\pi$ may be ramified. Our first
step is to reduce the Flicker integral to an integral over
$\mathrm{N}_{n-1}(\mathbb{R})\backslash\mathrm{GL}_{n-1}(\mathbb{R})$.

######  Proposition 6.7.

Let $\pi$ be an induced representation of Langlands type of
$\mathrm{GL}_{n}(\mathbb{C})$ with Whittaker newform
$W_{\pi}^{\circ}\in\mathcal{W}(\pi,\psi_{\mathbb{C}/\mathbb{R}})$. Let
$\Phi\in\mathscr{S}(\mathbb{R}^{n})$ be the right $\mathrm{O}(n)$-finite
Schwartz–Bruhat function of the form $\Phi(x)=P(x)\exp(-\pi
x\prescript{t}{}{x})$, where the distinguished homogeneous polynomial
$P\in\mathcal{P}_{\chi_{\pi}|_{\mathrm{O}(1)},c(\pi)}(\mathbb{R}^{n})$ is
given by

$P(x)\coloneqq\sum_{\begin{subarray}{c}m=c(\chi_{\pi}|_{\mathrm{O}(1)})\\\
m\equiv
c(\chi_{\pi}|_{\mathrm{O}(1)})\pmod{2}\end{subarray}}^{c(\pi)}\left(x\prescript{t}{}{x}\right)^{\frac{c(\pi)-m}{2}}(\dim\tau_{\chi_{\pi}|_{\mathrm{O}(1)},m})\overline{P_{\chi_{\pi}|_{\mathrm{O}(1)},m}^{\circ}}(x).$

Then the Flicker integral $\Psi(s,W_{\pi}^{\circ},\Phi)$ is equal to

(6.8)
$L\left(ns,\omega_{\pi_{\mathrm{ur}}}|_{\mathbb{R}^{\times}}\right)\int\limits_{\mathrm{N}_{n-1}(\mathbb{R})\backslash\mathrm{GL}_{n-1}(\mathbb{R})}W_{\pi}^{\circ}\begin{pmatrix}g&0\\\
0&1\end{pmatrix}\left|\det g\right|_{\mathbb{R}}^{s-1}\,dg.$

The proof of this proceeds along similar lines to that of Proposition 5.2.

###### Proof.

We use the Iwasawa decomposition $g=(z1_{n})\begin{pmatrix}h&0\\\
0&1\end{pmatrix}k$ for
$\mathrm{N}_{n}(\mathbb{R})\backslash\mathrm{GL}_{n}(\mathbb{R})$ to see that
the Flicker integral $\Psi(s,W_{\pi}^{\circ},\Phi)$ is equal to

$\int_{\mathbb{R}^{\times}}\omega_{\pi}(z)|z|_{\mathbb{R}}^{ns}\int\limits_{\mathrm{N}_{n-1}(\mathbb{R})\backslash\mathrm{GL}_{n-1}(\mathbb{R})}\left|\det
h\right|_{\mathbb{R}}^{s-1}\int_{\mathrm{O}(n)}W_{\pi}^{\circ}\left(\begin{pmatrix}h&0\\\
0&1\end{pmatrix}k\right)\Phi(ze_{n}k)\,dk\,dh\,d^{\times}z.$

We insert the identity (3.14) for $W_{\pi}^{\circ}(g)$ with $g$ replaced by
$\begin{pmatrix}h&0\\\ 0&1\end{pmatrix}k$ and the variable of integration
being $k^{\prime}\in\mathrm{U}(n)$, then interchange the order of integration
and make the change of variables $k\mapsto k^{-1}$ and $k^{\prime}\mapsto
kk^{\prime}$. We end up with

$\int_{\mathbb{R}^{\times}}\omega_{\pi}(z)|z|_{\mathbb{R}}^{ns}\int\limits_{\mathrm{N}_{n-1}(\mathbb{R})\backslash\mathrm{GL}_{n-1}(\mathbb{R})}\left|\det
h\right|_{\mathbb{R}}^{s-1}\int_{\mathrm{U}(n)}W_{\pi}^{\circ}\left(\begin{pmatrix}h&0\\\
0&1\end{pmatrix}k^{\prime}\right)\\\
\times\dim\tau_{\chi_{\pi},c(\pi)}\int_{\mathrm{O}(n)}P_{\chi_{\pi},c(\pi)}^{\circ}(e_{n}k^{\prime-1}k^{-1})\Phi(ze_{n}k^{-1})\,dk\,dk^{\prime}\,dh\,d^{\times}z.$

By the addition theorem, Proposition 3.4, the last line is

$\sum_{\ell=1}^{\dim\tau_{\chi_{\pi},c(\pi)}}Q_{\ell}(e_{n}k^{\prime-1})\int_{\mathrm{O}(n)}\overline{Q_{\ell}}(e_{n}k)\Phi(ze_{n}k^{-1})\,dk,$

where $\\{Q_{\ell}\\}$ is an orthonormal basis of
$\mathcal{H}_{\chi_{\pi},c(\pi)}(\mathbb{C}^{n})$. By the homogeneity of
$Q_{\ell}$, (3.1), we observe that the restrictions of these polynomials to
$\mathbb{R}^{n}$ are elements of
$\mathcal{P}_{\chi_{\pi}|_{\mathrm{O}(1)},c(\pi)}(\mathbb{R}^{n})$.

We now use the fact that

$\Phi(ze_{n}k^{-1})=\overline{\chi_{\pi}}\left(\frac{z}{\|z\|}\right)\|z\|^{c(\pi)}e^{-\pi
z^{2}}\sum_{\begin{subarray}{c}m=c(\chi_{\pi}|_{\mathrm{O}(1)})\\\ m\equiv
c(\chi_{\pi}|_{\mathrm{O}(1)})\pmod{2}\end{subarray}}^{c(\pi)}(\dim\tau_{\chi_{\pi}|_{\mathrm{O}(1)},m})P_{\chi_{\pi}|_{\mathrm{O}(1)},m}^{\circ}(e_{n}k)$

for $z\in\mathbb{R}^{\times}$ and $k\in\mathrm{O}(n)$ by the homogeneity of
$P_{\chi_{\pi}|_{\mathrm{O}(1)},m}^{\circ}$ as in (3.1) coupled with the
identity (3.3). By (3.6), the sum over $m$ is the reproducing kernel of
$\mathcal{P}_{\chi_{\pi}|_{\mathrm{O}(1)},c(\pi)}(\mathbb{R}^{n})$, and so the
integral over $\mathrm{O}(n)\ni k$ is $\overline{Q_{\ell}}(e_{n})$ by (3.8).
Using the addition theorem, Proposition 3.4, in _reverse_ and then using
(3.14) to evalute the integral over $\mathrm{U}(n)\ni k^{\prime}$, we find
that $I(s,W_{\pi}^{\circ},\Phi)$ is equal to

$\int_{\mathbb{R}^{\times}}\omega_{\pi}(z)\overline{\chi_{\pi}}\left(\frac{z}{\|z\|}\right)\|z\|^{c(\pi)}|z|_{\mathbb{R}}^{ns}e^{-\pi
z^{2}}\,d^{\times}z\int\limits_{\mathrm{N}_{n-1}(\mathbb{R})\backslash\mathrm{GL}_{n-1}(\mathbb{R})}W_{\pi}^{\circ}\begin{pmatrix}h&0\\\
0&1\end{pmatrix}\left|\det h\right|_{\mathbb{R}}^{s-1}\,dh.$

It remains to recall Lemma 4.5, which, by (2.1), shows that the integral over
$\mathbb{R}^{\times}\ni z$ is
$L(ns,\omega_{\pi_{\mathrm{ur}}}|_{\mathbb{R}^{\times}})$. ∎

######  Remark 6.9.

Just as for Proposition 5.2, the same proof remains valid in the
nonarchimedean setting.

Finally, we use Theorem 6.5 to show that when $\pi$ is an induced
representation of Langlands type of $\mathrm{GL}_{n}(\mathbb{C})$, the newform
$W_{\pi}^{\circ}\in\mathcal{W}(\pi,\psi_{\mathbb{C}/\mathbb{R}})$ is a weak
test vector for the Flicker integral. The proof proceeds by reducing the
problem to the spherical case.

######  Theorem 6.10.

With the notation and hypotheses of Proposition 6.7, the Flicker integral

(6.11)
$\Psi(s,W_{\pi}^{\circ},\Phi)\coloneqq\int\limits_{\mathrm{N}_{n}(\mathbb{R})\backslash\mathrm{GL}_{n}(\mathbb{R})}W_{\pi}^{\circ}(g)\Phi(e_{n}g)\left|\det
g\right|_{\mathbb{R}}^{s}\,dg$

is equal to $L(s,\pi_{\mathrm{ur}},\mathrm{As})$.

###### Proof.

By Lemma 4.6 and Proposition 6.7, we have that

$\Psi(s,W_{\pi}^{\circ},\Phi)=L\left(ns,\omega_{\pi_{\mathrm{ur}}}|_{\mathbb{R}^{\times}}\right)\int\limits_{\mathrm{N}_{n-1}(\mathbb{R})\backslash\mathrm{GL}_{n-1}(\mathbb{R})}W_{\pi_{\mathrm{ur}}}^{\circ}\begin{pmatrix}h&0\\\
0&1\end{pmatrix}\left|\det h\right|_{\mathbb{R}}^{s-1}\,dh.$

On the other hand, the same calculation shows that the right-hand side is
equal to $\Psi(s,W_{\pi_{\mathrm{ur}}}^{\circ},\Phi_{\mathrm{ur}})$ with
$\Phi_{\mathrm{ur}}\in\mathscr{S}(\mathbb{R}^{n})$ given by (5.1). It remains
to invoke Theorem 6.5. ∎

In general, the naïve Asai $L$-function $L(s,\pi_{\mathrm{ur}},\mathrm{As})$
is not equal to $L(s,\pi,\mathrm{As})$. Nonetheless, these two $L$-functions
are closely related.

######  Proposition 6.12.

Given an induced representation of Whittaker type $\pi$ of
$\mathrm{GL}_{n}(\mathbb{C})$, there exists a polynomial $p(s)$ for which

$L(s,\pi_{\mathrm{ur}},\mathrm{As})=p(s)L(s,\pi,\mathrm{As}).$

###### Proof.

For
$\pi=\mathop{\vphantom{\bigoplus}\mathchoice{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}}\displaylimits_{j=1}^{n}\pi_{j}$,
we have that

$L(s,\pi,\mathrm{As})=\prod_{j=1}^{n}L(s,\pi_{j},\mathrm{As})\prod_{1\leq
j<\ell\leq n}L(s,\pi_{j}\times\pi_{\ell})$

via the local Langlands correspondence (cf. [B-P21, Lemma 3.2.1]). Recalling
Proposition 5.8, it thereby suffices to consider the case where $\pi$ is
essentially square-integrable.

For $\pi$ essentially square-integrable, so that
$\pi=e^{i\kappa\arg}|\cdot|_{\mathbb{C}}^{t}$ and
$\pi_{\mathrm{ur}}=|\cdot|_{\mathbb{C}}^{t+\|\kappa\|/2}$, the fact that
$\zeta_{\mathbb{R}}(s)\coloneqq\pi^{-s/2}\Gamma(s/2)$ and that
$\Gamma(s+1)=s\Gamma(s)$ means that for $\kappa^{\prime}\in\\{0,1\\}$
satisfying $\kappa\equiv\kappa^{\prime}\pmod{2}$,

$\frac{L(s,\pi_{\mathrm{ur}},\mathrm{As})}{L(s,\pi,\mathrm{As})}=\frac{\zeta_{\mathbb{R}}\left(s+2t+\|\kappa\|\right)}{\zeta_{\mathbb{R}}\left(s+2t+\kappa^{\prime}\right)}=\pi^{-\frac{\|\kappa\|-\kappa^{\prime}}{2}}\prod_{m=0}^{\frac{1}{2}(\|\kappa\|-\kappa^{\prime})-1}\left(\frac{s+2t+\kappa^{\prime}}{2}+m\right).\qed$

When $s=1$, the integral appearing in (6.8) is known as the _Flicker–Rallis
period_. In general, this integral need not converge at $s=1$. When $\pi$ is
_unitary_ , however, convergence is guaranteed by the following lemma, whose
proof we omit as it is a standard application [Kem15b]*Lemma 7.1;
[FZ95]*Proposition 1 of bounds for Whittaker functions by a gauge [JS90, §4
Propositions 2 and 3].

######  Lemma 6.13.

Let $\pi$ be a unitary generic irreducible Casselman–Wallach representation of
$\mathrm{GL}_{n}(\mathbb{C})$. For any
$W_{\pi}\in\mathcal{W}(\pi,\psi_{\mathbb{C}/\mathbb{R}})$, the integral

$\int\limits_{\mathrm{N}_{n-1}(\mathbb{R})\backslash\mathrm{GL}_{n-1}(\mathbb{R})}W_{\pi}\begin{pmatrix}h&0\\\
0&1\end{pmatrix}\left|\det h\right|_{\mathbb{R}}^{s-1}\,dh$

converges absolutely for $\Re(s)\geq 1$.

We explicitly evaluate the Flicker–Rallis period at the Whittaker newform. The
nonvanishing of this integral (in a slightly modified form) is implicitly
described in work of Gelbart, Jacquet, and Rogawski [GJR01, §2], Kemarsky
[Kem15a]*§1; [Kem15b]*§8, and W. Zhang [Zha14, (3.14) and (3.21)].

######  Theorem 6.14.

Let $\pi$ be a unitary generic irreducible Casselman–Wallach representation of
$\mathrm{GL}_{n}(\mathbb{C})$ with Whittaker newform
$W_{\pi}^{\circ}\in\mathcal{W}(\pi,\psi_{\mathbb{C}/\mathbb{R}})$. Then the
Flicker–Rallis period

(6.15)
$\int\limits_{\mathrm{N}_{n-1}(\mathbb{R})\backslash\mathrm{GL}_{n-1}(\mathbb{R})}W_{\pi}^{\circ}\begin{pmatrix}h&0\\\
0&1\end{pmatrix}\,dh$

is equal to

$\frac{L(1,\pi_{\mathrm{ur}},\mathrm{As})}{L\left(n,\omega_{\pi_{\mathrm{ur}}}|_{\mathbb{R}^{\times}}\right)}.$

###### Proof.

With Lemma 6.13 in hand, we take $s=1$ in Proposition 6.7 and combine this
with Theorem 6.10. ∎

######  Remark 6.16.

The nonarchimedean analogue of this result has been resolved by
Anandavardhanan and Matringe [AM17, Theorem 1.1].

We say that $\pi$ is _$\mathrm{GL}_{n}(\mathbb{R})$ -distinguished_ if
$\operatorname{Hom}_{\mathrm{GL}_{n}(\mathbb{R})}(\pi,\mathbf{1})$ is
nontrivial.

######  Remark 6.17 ([Kem15b, §6]).

We define the $\mathrm{P}_{n}(\mathbb{R})$-invariant linear functional
$\vartheta^{\flat}:\mathcal{W}(\pi,\psi_{\mathbb{C}/\mathbb{R}})\to\mathbb{C}$
by

(6.18)
$\vartheta^{\flat}(W_{\pi})\coloneqq\int\limits_{\mathrm{N}_{n-1}(\mathbb{R})\backslash\mathrm{GL}_{n-1}(\mathbb{R})}W_{\pi}\begin{pmatrix}h&0\\\
0&1\end{pmatrix}\,dh.$

Owing to Theorem 6.14, the linear functional $\vartheta^{\flat}$ is
nontrivial. In addition, Kemarsky [Kem15a, Theorem 1.1] shows that a
$\mathrm{P}_{n}(\mathbb{R})$-invariant linear functional extends to a
$\mathrm{GL}_{n}(\mathbb{R})$-invariant linear functional in a purely local
manner. Alternatively, a local-to-global method may be adapted as in [GJR01,
p. 185–186] (cf. [Zha14, Proposition 3.2]). Hence $\vartheta^{\flat}$ gives
rise to a nontrivial $\mathrm{GL}_{n}(\mathbb{R})$-invariant linear functional
on the Whittaker model $\mathcal{W}(\pi,\psi_{\mathbb{C}/\mathbb{R}})$.

##  7\. Bump–Friedberg Integrals

For $n=2m$, we define the embedding
$J:\mathrm{GL}_{m}(F)\times\mathrm{GL}_{m}(F)\to\mathrm{GL}_{n}(F)$ by

$J(g,g^{\prime})_{k,\ell}\coloneqq\begin{dcases*}g_{i,j}&if $k=2i-1$ and
$\ell=2j-1$,\\\ g^{\prime}_{i,j}&if $k=2i$ and $l=2j$,\\\
0&otherwise.\end{dcases*}$

We denote by
$\mathrm{M}_{m,m}(F)\cong\mathrm{GL}_{m}(F)\times\mathrm{GL}_{m}(F)$ the
standard Levi subgroup of $\mathrm{GL}_{n}(F)$ associated with the partition
$(m,m)$. Let

$w_{m,m}\coloneqq\begin{pmatrix}1&2&\cdots&m&|&m+1&m+2&\cdots&2m\\\
1&3&\cdots&2m-1&|&2&4&\cdots&2m\end{pmatrix}.$

We then set

$\displaystyle\mathrm{H}_{m,m}(F)$ $\displaystyle\coloneqq
w_{m,m}\mathrm{M}_{m,m}(F)w^{-1}_{m,m}$
$\displaystyle=\left\\{J(g,g^{\prime})=w_{m,m}\operatorname{diag}(g,g^{\prime})w^{-1}_{m,m}:\operatorname{diag}(g,g^{\prime})\in\mathrm{M}_{m,m}(F)\right\\}.$

Similarly, for $n=2m+1$, we define the embedding
$J:\mathrm{GL}_{m+1}(F)\times\mathrm{GL}_{m}(F)\to\mathrm{GL}_{n}(F)$ by

$J(g,g^{\prime})_{k,\ell}\coloneqq\begin{dcases*}g_{i,j}&if $k=2i-1$ and
$\ell=2j-1$,\\\ g^{\prime}_{i,j}&if $k=2i$ and $\ell=2j$,\\\
0&otherwise.\end{dcases*}$

We denote by
$\mathrm{M}_{m+1,m}(F)\cong\mathrm{GL}_{m+1}(F)\times\mathrm{GL}_{m}(F)$ the
standard Levi subgroup associated to the partition $(m+1,m)$ of $2m+1$. Let
$w_{m+1,m}\coloneqq w_{m+1,m+1}|_{\mathrm{GL}_{2m+1}(F)}$, so that

$w_{m+1,m}\coloneqq\begin{pmatrix}1&2&\cdots&m+1&|&m+2&m+3&\cdots&2m&2m+1\\\
1&3&\cdots&2m+1&|&2&4&\cdots&2m-2&2m\end{pmatrix},$

and then set

$\displaystyle\mathrm{H}_{m+1,m}(F)$ $\displaystyle\coloneqq
w_{m+1,m}\mathrm{M}_{m+1,m}(F)w^{-1}_{m+1,m}$
$\displaystyle=\left\\{J(g,g^{\prime})=w_{m+1,m}\operatorname{diag}(g,g^{\prime})w^{-1}_{m+1,m}:\operatorname{diag}(g,g^{\prime})\in\mathrm{M}_{m+1,m}(F)\right\\}.$

To make the above description much more transparent, we provide prototypical
elements in the cases of $\mathrm{H}_{2,2}(F)$ and $\mathrm{H}_{3,2}(F)$.

######  Example 7.1.

The group $\mathrm{H}_{2,2}(F)$ consists of invertible matrices of the form

$J\left(\begin{pmatrix}g_{1,1}&g_{1,2}\\\
g_{2,1}&g_{2,2}\end{pmatrix},\begin{pmatrix}g^{\prime}_{1,1}&g^{\prime}_{1,2}\\\
g^{\prime}_{2,1}&g^{\prime}_{2,2}\end{pmatrix}\right)=\begin{pmatrix}g_{1,1}&0&g_{1,2}&0\\\
0&g^{\prime}_{1,1}&0&g^{\prime}_{1,2}\\\ g_{2,1}&0&g_{2,2}&0\\\
0&g^{\prime}_{2,1}&0&g^{\prime}_{2,2}\end{pmatrix}.$

The group $\mathrm{H}_{3,2}(F)$ consists of invertible matrices of the form

$J\left(\begin{pmatrix}g_{1,1}&g_{1,2}&g_{1,3}\\\ g_{2,1}&g_{2,2}&g_{2,3}\\\
g_{3,1}&g_{3,2}&g_{3,3}\end{pmatrix},\begin{pmatrix}g^{\prime}_{1,1}&g^{\prime}_{1,2}\\\
g^{\prime}_{2,1}&g^{\prime}_{2,2}\end{pmatrix}\right)=\begin{pmatrix}g_{1,1}&0&g_{1,2}&0&g_{1,3}\\\
0&g^{\prime}_{1,1}&0&g^{\prime}_{1,2}&0\\\ g_{2,1}&0&g_{2,2}&0&g_{2,3}\\\
0&g^{\prime}_{2,1}&0&g^{\prime}_{2,2}&0\\\
g_{3,1}&0&g_{3,2}&0&g_{3,3}\end{pmatrix}.$

Given an induced representation of Whittaker type $\pi$ of
$\mathrm{GL}_{n}(F)$, a Whittaker function $W_{\pi}\in\mathcal{W}(\pi,\psi)$,
and a Schwartz–Bruhat function $\Phi\in\mathscr{S}(F^{m})$, where
$m\coloneqq\lfloor\frac{n}{2}\rfloor$, the Bump–Friedberg integral [BF90] is
given by

$B(s_{1},s_{2},W_{\pi},\Phi)\\\
\coloneqq\begin{dcases*}\int\limits_{\mathrm{N}_{m}(F)\backslash\mathrm{GL}_{m}(F)}\int\limits_{\mathrm{N}_{m}(F)\backslash\mathrm{GL}_{m}(F)}W_{\pi}(J(g,g^{\prime}))\Phi(e_{m}g^{\prime})\left|\det
g\right|^{s_{1}-\frac{1}{2}}\left|\det
g^{\prime}\right|^{s_{2}-s_{1}+\frac{1}{2}}\,dg\,dg^{\prime}&\\\ & for
$n=2m$,\\\
\int\limits_{\mathrm{N}_{m}(F)\backslash\mathrm{GL}_{m}(F)}\int\limits_{\mathrm{N}_{m+1}(F)\backslash\mathrm{GL}_{m+1}(F)}W_{\pi}(J(g,g^{\prime}))\Phi(e_{m+1}g)\left|\det
g\right|^{s_{1}}\left|\det
g^{\prime}\right|^{s_{2}-s_{1}}\,dg\,dg^{\prime}&\\\ & for
$n=2m+1$.\end{dcases*}$

Alternatively, one can write this as an integral over
$(\mathrm{N}_{n}(F)\cap\mathrm{H}_{m,m}(F))\backslash\mathrm{H}_{m,m}(F)$ for
$n=2m$ and as an integral over
$(\mathrm{N}_{n}(F)\cap\mathrm{H}_{m+1,m}(F))\backslash\mathrm{H}_{m+1,m}(F)$
for $n=2m+1$ (cf. [Mat15, Mat17]). The Bump–Friedberg integrals converges
absolutely for $\Re(s)$ sufficiently large and extends meromorphically to the
entire complex plane. The local exterior square $L$-function
$L(s,\pi,\wedge^{2})$ is defined via the local Langlands correspondence as
illustrated in [Kna94] and [B-P21, Section 3.2]. We take the local
Bump–Friedberg $L$-function $L(s_{1},s_{2},\pi,\mathrm{BF})$ to be
$L(s_{1},\pi)L(s_{2},\pi,\wedge^{2})$. For our purposes, it will often be
convenient to write $B(s,W_{\pi},\Phi)$ in place of $B(s,2s,W_{\pi},\Phi)$ and
$L(s,\pi,\mathrm{BF})$ in place of $L(s,\pi)L(2s,\pi,\wedge^{2})$ when $n=2m$
is even.

###  7.1. The Spherical Calculation

We summarise a propagation formula for $\mathrm{GL}_{n}(F)$ Whittaker
functions in terms of $\mathrm{GL}_{n-1}(F)$ Whittaker functions and a
convolution section identity for radial parts. We omit the proofs, since they
are essentially identical to the corresponding proofs of Lemmata 6.1 and 6.3.

######  Lemma 7.2 (Cf. Lemma 6.1).

Let
$\pi=\mathop{\vphantom{\bigoplus}\mathchoice{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}}\displaylimits_{j=1}^{n}|\cdot|^{t_{j}}$
and
$\pi_{0}\coloneqq\mathop{\vphantom{\bigoplus}\mathchoice{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}}\displaylimits_{j=2}^{n}|\cdot|^{t_{j}}$
be the spherical induced representations of Langlands type of
$\mathrm{GL}_{n}(F)$ and $\mathrm{GL}_{n-1}(F)$ respectively with spherical
Whittaker functions $W^{\circ}_{\pi}\in\mathcal{W}(\pi,\psi)$ and
$W^{\circ}_{\pi_{0}}\in\mathcal{W}(\pi_{0},\psi)$. Then for
$a=\operatorname{diag}(a_{1},\ldots,a_{n})\in\mathrm{A}_{n}(F)$, we have

(7.3) $W^{\circ}_{\pi}(a)=\left|\det
a\right|^{t_{1}}\delta^{1/2}_{\mathrm{B}_{n}(F)}(a)\int_{\mathrm{A}_{n-1}(F)}W^{\circ}_{\pi_{0}}(a^{\prime})\prod_{j=1}^{n-1}\exp(-d_{F}\pi\|a^{\prime}_{j}a^{-1}_{j+1}\|^{2})\\\
\times\exp(-d_{F}\pi\|a^{\prime-1}_{j}a_{j}\|^{2})\left|\det
a^{\prime}\right|^{-t_{1}}\delta^{-1/2}_{\mathrm{B}_{n-1}(F)}(a^{\prime})\,d^{\times}a^{\prime}.$

######  Lemma 7.4 (Cf. Lemma 6.3).

Let
$\pi=\mathop{\vphantom{\bigoplus}\mathchoice{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}}\displaylimits_{j=1}^{n}|\cdot|^{t_{j}}$
be a spherical induced representation of Langlands type of
$\mathrm{GL}_{n}(F)$ with spherical Whittaker function
$W^{\circ}_{\pi}\in\mathcal{W}(\pi,\psi)$. Then for
$a=\operatorname{diag}(a_{1},\ldots,a_{n})\in\mathrm{A}_{n}(F)$, we have

$L(s,\pi)W^{\circ}_{\pi}(a^{\prime})=\int_{\mathrm{A}_{n}(F)}W^{\circ}_{\pi}(a^{\prime}a)\prod_{j=1}^{n}\exp(-d_{F}\pi\|a_{j}\|^{2})\prod_{j=1}^{n-1}\exp(-d_{F}\pi\|a^{\prime}_{j}a^{\prime-1}_{j+1}a^{-1}_{j+1}\|^{2})\\\
\times\left|\det
a\right|^{s}_{F}\delta^{-1/2}_{\mathrm{B}_{n}(F)}(a)\,d^{\times}a.$

We will use these identities after reducing Bump–Friedberg integrals to
integrals over a torus.

######  Proposition 7.5.

Let
$\pi=\mathop{\vphantom{\bigoplus}\mathchoice{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}}\displaylimits_{j=1}^{n}|\cdot|^{t_{j}}$
be a spherical induced representation of Langlands type of
$\mathrm{GL}_{n}(F)$ with spherical Whittaker function
$W^{\circ}_{\pi}\in\mathcal{W}(\pi,\psi)$, and for
$m=\lfloor\frac{n}{2}\rfloor$, let $\Phi_{\mathrm{ur}}\in\mathscr{S}(F^{m})$
be given by (5.1).

  1. _(i)_

For $n=2m$, the Bump–Friedberg integral
$B(s_{1},s_{2},W_{\pi}^{\circ},\Phi_{\mathrm{ur}})$ is equal to

$\int_{\mathrm{A}_{n}(F)}W^{\circ}_{\pi}(b)\exp(-d_{F}\pi\|b_{n}\|^{2})\left|b_{1}b_{3}\cdots
b_{n-1}\right|^{s_{1}}\left|b_{2}b_{4}\cdots
b_{n-2}\right|^{s_{2}-s_{1}}\delta^{-\frac{1}{2}}_{\mathrm{B}_{n}(F)}(b)\,d^{\times}b.$

  2. _(ii)_

For $n=2m+1$, the Bump–Friedberg integral
$B(s_{1},s_{2},W_{\pi}^{\circ},\Phi_{\mathrm{ur}})$ is equal to

$\int_{\mathrm{A}_{n}(F)}W^{\circ}_{\pi}(b)\exp(-d_{F}\pi\|b_{n}\|^{2})\left|b_{1}b_{3}\cdots
b_{n-2}\right|^{s_{1}}\left|b_{2}b_{4}\cdots
b_{n-1}\right|^{s_{2}-s_{1}}\delta^{-\frac{1}{2}}_{\mathrm{B}_{n}(F)}(b)\,d^{\times}b.$

###### Proof.

We prove this only for $n=2m$; the case $n=2m+1$ follows analogously.
Exploiting the Iwasawa decomposition $g=ak$ and
$g^{\prime}=a^{\prime}k^{\prime}$ for
$\mathrm{N}_{m}(F)\backslash\mathrm{GL}_{m}(F)$, the Bump–Friedberg integral
$B(s_{1},s_{2},W_{\pi}^{\circ},\Phi_{\mathrm{ur}})$ is equal to

$\int_{\mathrm{A}_{m}(F)}\int_{\mathrm{A}_{m}(F)}W^{\circ}_{\pi}(J(a,a^{\prime}))\exp(-d_{F}\pi\|a^{\prime}_{m}\|^{2})\left|\det
a\right|^{s_{1}-\frac{1}{2}}\left|\det
a^{\prime}\right|^{s_{2}-s_{1}+\frac{1}{2}}\\\
\times\delta^{-1}_{\mathrm{B}_{m}(F)}(a)\delta^{-1}_{\mathrm{B}_{m}(F)}(a^{\prime})\,d^{\times}a\,d^{\times}a^{\prime}.$

Since $\left|\det a\right|^{-\frac{1}{2}}\left|\det
a^{\prime}\right|^{\frac{1}{2}}\delta^{-1}_{\mathrm{B}_{m}(F)}(a)\delta^{-1}_{\mathrm{B}_{m}(F)}(a^{\prime})=\delta^{-\frac{1}{2}}_{\mathrm{B}_{n}(F)}(J(a,a^{\prime}))$,
this is equal to

$\int_{\mathrm{A}_{m}(F)}\int_{\mathrm{A}_{m}(F)}W^{\circ}_{\pi}(J(a,a^{\prime}))\exp(-d_{F}\pi\|a^{\prime}_{m}\|^{2})\left|\det
a\right|^{s_{1}}\left|\det
a^{\prime}\right|^{s_{2}-s_{1}}\delta^{-\frac{1}{2}}_{\mathrm{B}_{n}(F)}(J(a,a^{\prime}))\,d^{\times}a\,d^{\times}a^{\prime}.$

The result now follows by writing
$a=\operatorname{diag}(b_{1},b_{3},\ldots,b_{n-1})$ and
$a^{\prime}=\operatorname{diag}(b_{2},b_{4},\ldots,b_{n})$ and letting
$b\coloneqq(b_{1},\ldots,b_{n})\in\mathrm{A}_{n}(F)$. ∎

We are now able to prove that when $\pi$ is a spherical induced representation
of Langlands type of $\mathrm{GL}_{n}(F)$, the spherical Whittaker function
$W_{\pi}^{\circ}\in\mathcal{W}(\pi,\psi)$ is a strong test vector for the
Bump–Friedberg integral.

######  Theorem 7.6.

Let
$\pi=\mathop{\vphantom{\bigoplus}\mathchoice{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}}\displaylimits_{j=1}^{n}|\cdot|^{t_{j}}$
be a spherical induced representation of Langlands type of
$\mathrm{GL}_{n}(F)$ with spherical Whittaker function
$W^{\circ}_{\pi}\in\mathcal{W}(\pi,\psi)$, and for
$m=\lfloor\frac{n}{2}\rfloor$, let $\Phi_{\mathrm{ur}}\in\mathscr{S}(F^{m})$
be given by (5.1). Then the Bump–Friedberg integral
$B(s_{1},s_{2},W_{\pi}^{\circ},\Phi_{\mathrm{ur}})$ is equal to

$L(s_{1},s_{2},\pi,\mathrm{BF})\coloneqq
L(s_{1},\pi)L(s_{2},\pi,\wedge^{2})=\prod_{\ell=1}^{n}\zeta_{F}(s_{1}+t_{\ell})\prod_{1\leq
j<k\leq n}\zeta_{F}(s_{2}+t_{j}+t_{k}).$

###### Proof.

We prove this by induction. For the base case $n=2$, the Bump–Friedberg
integral is

$\int_{F^{\times}}\int_{F^{\times}}W_{\pi}^{\circ}\begin{pmatrix}a_{1}&0\\\
0&a_{2}\end{pmatrix}\exp(-d_{F}\pi\|a_{2}\|^{2})|a_{1}|^{s_{1}-\frac{1}{2}}|a_{2}|^{s_{2}-s_{1}+\frac{1}{2}}\,d^{\times}a_{1}\,d^{\times}a_{2}.$

From (7.3), this is equal to

$\int_{F^{\times}}\int_{F^{\times}}\exp(-d_{F}\pi\|a_{2}\|^{2})|a_{1}|^{s_{1}+t_{1}}|a_{2}|^{s_{2}-s_{1}+t_{1}}\\\
\times\int_{F^{\times}}\exp(-d_{F}\pi\|a^{\prime}a_{2}^{-1}\|^{2})\exp(-d_{F}\pi\|a^{\prime-1}a_{1}\|^{2})|a^{\prime}|^{t_{2}-t_{1}}\,d^{\times}a^{\prime}\,d^{\times}a_{1}\,d^{\times}a_{2}.$

We interchange the order of integration and make the change of variables
$a_{1}\mapsto a^{\prime}a_{1}$ and $a^{\prime}\mapsto a^{\prime}a_{2}$,
yielding

$\int_{F^{\times}}\exp(-d_{F}\pi\|a_{1}\|^{2})|a_{1}|^{s_{1}+t_{1}}\,d^{\times}a_{1}\int_{F^{\times}}\exp(-d_{F}\pi\|a_{2}\|^{2})|a_{2}|^{s_{2}+t_{1}+t_{2}}\,d^{\times}a_{2}\\\
\times\int_{F^{\times}}\exp(-d_{F}\pi\|a^{\prime}\|^{2})|a^{\prime}|^{s_{1}+t_{2}}\,d^{\times}a^{\prime},$

which is precisely $L(s_{1},s_{2},\pi,\mathrm{BF})$.

Now we proceed to the induction step. We suppose that the desired identity
holds for $n=2m-1$ and we prove this for $n=2m$. We insert the recursive
formula for $W_{\pi}^{\circ}(a)$ from Lemma 7.2 into the expression given in
Proposition 7.5 __(i)__ for the Bump–Friedberg integral
$B(s_{1},s_{2},W_{\pi}^{\circ},\Phi_{\mathrm{ur}})$. We then relabel
$b\in\mathrm{A}_{n}(F)$ with $\operatorname{diag}(y,b)$, where now $y\in
F^{\times}$ and
$b=\operatorname{diag}(b_{1},b_{2},\ldots,b_{n-1})\in\mathrm{A}_{n-1}(F)$. The
modulus character $\delta^{-1/2}_{\mathrm{B}_{n}(F)}(b)$ is cancelled out and
we arrive at

$\int_{\mathrm{A}_{n-1}(F)}\exp(-d_{F}\pi\|b_{n-1}\|^{2})\left|b_{2}b_{4}\cdots
b_{n-2}\right|^{s_{1}+t_{1}}\left|b_{1}b_{3}\cdots
b_{n-1}\right|^{s_{2}-s_{1}+t_{1}}\\\
\times\int_{\mathrm{A}_{n-1}(F)}W^{\circ}_{\pi_{0}}(b^{\prime})\prod_{j=1}^{n-1}\exp(-d_{F}\pi\|b^{\prime}_{j}b^{-1}_{j}\|^{2})\prod_{j=1}^{n-2}\exp(-d_{F}\pi\|b^{\prime-1}_{j+1}b_{j}\|^{2})\left|\det
b^{\prime}\right|^{-t_{1}}\delta^{-\frac{1}{2}}_{\mathrm{B}_{n-1}(F)}(b^{\prime})\\\
\times\int_{F^{\times}}\exp(-d_{F}\pi\|b^{\prime-1}_{1}y\|^{2})|y|^{s_{1}+t_{1}}\,d^{\times}y\,d^{\times}b^{\prime}\,d^{\times}b.$

We interchange the order of integration and perform the change of variables
$y\mapsto b^{\prime}_{1}y$ and $b_{j}\mapsto b^{\prime}_{j+1}b_{j}$ for $1\leq
j\leq n-2$. The Bump–Friedberg integral
$B(s_{1},s_{2},W_{\pi}^{\circ},\Phi_{\mathrm{ur}})$ becomes

$\int_{\mathrm{A}_{n-1}(F)}\prod_{j=1}^{n-1}\exp(-d_{F}\pi\|b_{j}\|^{2})\left|b_{2}b_{4}\cdots
b_{n-2}\right|^{s_{1}+t_{1}}\left|b_{1}b_{3}\cdots
b_{n-1}\right|^{s_{2}-s_{1}+t_{1}}\\\
\times\int_{\mathrm{A}_{n-1}(F)}W^{\circ}_{\pi_{0}}(b^{\prime})\prod_{j=1}^{n-2}\exp(-d_{F}\pi\|b^{\prime}_{j}b^{\prime-1}_{j+1}b^{-1}_{j}\|^{2})\exp(-d_{F}\pi\|b^{\prime}_{n-1}b^{-1}_{n-1}\|^{2})\delta^{-\frac{1}{2}}_{\mathrm{B}_{n-1}(F)}(b^{\prime})\\\
\times\left|b^{\prime}_{1}b^{\prime}_{3}\cdots
b^{\prime}_{n-1}\right|^{s_{1}}\left|b^{\prime}_{2}b^{\prime}_{4}\cdots
b^{\prime}_{n-2}\right|^{s_{2}-s_{1}}\int_{F^{\times}}\exp(-d_{F}\pi\|y\|^{2})|y|^{s_{1}+t_{1}}\,d^{\times}y\,d^{\times}b^{\prime}\,d^{\times}b.$

The integral over $F^{\times}\ni y$ is simply $\zeta_{F}(s_{1}+t_{1})$. We
make the change of variables $b^{\prime}_{j}\mapsto b^{\prime}_{j}b_{j}$ and
then interchange the order of the integration. This leads us to

$\zeta_{F}(s_{1}+t_{1})\int_{\mathrm{A}_{n-1}(F)}\exp(-d_{F}\pi\|b^{\prime}_{n-1}\|^{2})\left|b^{\prime}_{1}b^{\prime}_{3}\cdots
b^{\prime}_{n-1}\right|^{s_{1}}\left|b^{\prime}_{2}b^{\prime}_{4}\cdots