2.1.1. Local fields and absolute values

Let F be a local field, and denote by $|\cdot |_F$ the absolute value on F: for nonarchimedean F with ring of integers $\mathcal {O}$ , maximal ideal $\mathfrak {p}$ and uniformiser $\varpi $ , so that $\varpi \mathcal {O} = \mathfrak {p}$ and $\mathcal {O} / \mathfrak {p} \cong \mathbb {F}_q$ for some finite field of order q, this absolute value is normalised such that $|\varpi |_F = q^{-1}$ , whereas for archimedean F, this is normalised such that

$$\begin{align*}|x|_F = \begin{cases} \max\{x,-x\} & \text{if }F = \mathbb{R}, \\ x \overline{x} & \text{if }F = \mathbb{C}. \end{cases}\end{align*}$$

When the local field is clear from context, we write $|\cdot |$ in place of $|\cdot |_F$ . We also let denote the standard modulus on $\mathbb {C}$ .

2.1.2. Haar measures on F and $F^{\times }$

We let $dx$ denote the Haar measure on F normalised such that it is self-dual with respect to a fixed nontrivial additive character $\psi = \psi _F$ of F. For $F = \mathbb {R}$ , we choose , so that $dx$ is the Lebesgue measure; for $F = \mathbb {C}$ , we choose , so that $dx$ is twice the Lebesgue measure; finally, we choose $\psi $ to be unramified when F is nonarchimedean, in which case $dx$ gives $\mathcal {O}$ volume $1$ . The multiplicative Haar measure $d^{\times } x$ for $F^{\times }$ is defined to be $\zeta _F(1) |x|^{-1} \, dx$ , where

2.1.3. Subgroups of $\operatorname {\mathrm {GL}}_n(F)$ and the Iwasawa decomposition

For each r-tuple of positive integers $(n_1,\ldots ,n_r) \in \mathbb {N}^r$ for which $n_1 + \cdots + n_r = n$ , let $\mathrm {P}(F) = \mathrm {P}_{(n_1,\ldots ,n_r)}(F)$ denote the associated standard upper parabolic subgroup of $\operatorname {\mathrm {GL}}_n(F)$ containing the standard Borel subgroup of upper triangular matrices. 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 $\operatorname {\mathrm {GL}}_{n_1}(F) \times \cdots \times \operatorname {\mathrm {GL}}_{n_r}(F)$ , while the unipotent radical $\mathrm {N}_{\mathrm {P}}(F)$ of $\mathrm {P}(F)$ consists of upper triangular matrices with block-diagonal entries $(1_{n_1},\ldots ,1_{n_r})$ ; here we have written $1_n$ to denote the $n \times n$ identity matrix. When $\mathrm {P}(F)$ is the standard Borel (and minimal parabolic) subgroup $\mathrm {P}_{(1,\ldots ,1)}(F)$ , we write , the subgroup of unipotent upper triangular matrices, and , the subgroup of diagonal matrices.

The maximal compact subgroup $K_n$ of $\operatorname {\mathrm {GL}}_n(F)$ , unique up to conjugacy, is

$$\begin{align*}K_n = \begin{cases} \operatorname{\mathrm{O}}(n) & \text{if }F = \mathbb{R}, \\ \operatorname{\mathrm{U}}(n) & \text{if }F = \mathbb{C}, \\ \operatorname{\mathrm{GL}}_n(\mathcal{O}) & \text{if }F\text{ is nonarchimedean}. \end{cases}\end{align*}$$

When the context is clear, we write K in place of $K_n$ . Given a standard parabolic subgroup $\mathrm {P}(F)$ , we have the Iwasawa decomposition $\operatorname {\mathrm {GL}}_n(F) = \mathrm {P}(F) K_n = \mathrm {N}_{\mathrm {P}}(F) \mathrm {M}_{\mathrm {P}}(F) K_n$ . Note that the Iwasawa decomposition is not unique since $\mathrm {M}_{\mathrm {P}}(F)$ intersects $K_n$ nontrivially.

2.1.4. Haar measures on $\operatorname {\mathrm {GL}}_n(F)$ and its subgroups

We normalise the Haar measure $dg$ on $\operatorname {\mathrm {GL}}_n(F) \ni g$ via the Iwasawa decomposition $g = uak$ for the standard Borel subgroup, so that $dg = \delta _n^{-1}(a) \, du \, d^{\times } a \, dk$ . Here, $du = \prod _{j = 1}^{n - 1} \prod _{\ell = j + 1}^{n} du_{j,\ell }$ for $u \in \mathrm {N}_n(F)$ with upper triangular entries $u_{j,\ell } \in F$ , $d^{\times } a = \prod _{j = 1}^{n} d^{\times } a_j$ for $a \in \mathrm {A}_n(F)$ with diagonal entries $a_j \in F^{\times }$ , $\delta _n(a) = \prod _{j = 1}^{n} |a_j|^{n - 2j + 1}$ denotes the modulus character of the Borel subgroup, and $dk$ is the Haar measure on the compact group $K_n \ni k$ normalised to give $K_n$ volume $1$ (so that when $F = \mathbb {R}$ and $n = 1$ , in which case $K_1 = \operatorname {\mathrm {O}}(1) = \{\pm 1\} \cong \mathbb {Z}/2\mathbb {Z}$ , this is just half the counting measure).

More generally, given a standard parabolic subgroup $\mathrm {P}(F) = \mathrm {N}_{\mathrm {P}}(F) \mathrm {M}_{\mathrm {P}}(F)$ of $\operatorname {\mathrm {GL}}_n(F)$ , the Haar measure $dg$ on $\operatorname {\mathrm {GL}}_n(F) \ni g$ is given by $dg = \delta _{\mathrm {P}}^{-1}(m) \, du \, d^{\times } m \, dk$ with respect to the Iwasawa decomposition $g = umk$ , where for $m = \operatorname {\mathrm {blockdiag}}(m_1,\ldots ,m_r)$ , the modulus character is

$$\begin{align*}\delta_{\mathrm{P}}(m) = \prod_{j = 1}^{r} \left|\det m_j\right|{}^{n - 2(n_1 + \cdots + n_{j - 1}) - n_j}\end{align*}$$

and $d^{\times } m = \prod _{j = 1}^{r} dm_j$ with $dm_j$ the Haar measure on $\operatorname {\mathrm {GL}}_{n_j}(F) \ni m_j$ normalised via the Iwasawa decomposition for the standard Borel subgroup of $\operatorname {\mathrm {GL}}_{n_j}(F)$ .

2.2.1. Isobaric sums

Given representations $(\pi _1,V_{\pi _1}), \ldots , (\pi _r,V_{\pi _r})$ of $\operatorname {\mathrm {GL}}_{n_1}(F), \ldots , \operatorname {\mathrm {GL}}_{n_r}(F)$ , where F is a local field and $n_1 + \cdots + n_r = n$ , we form the representation $\pi _1 \boxtimes \cdots \boxtimes \pi _r$ of $\mathrm {M}_{\mathrm {P}}(F)$ , where $\boxtimes $ denotes the outer tensor product and $\mathrm {M}_{\mathrm {P}}(F)$ denotes the block-diagonal Levi subgroup of the standard (upper) parabolic subgroup $\mathrm {P}(F) = \mathrm {P}_{(n_1,\ldots ,n_r)}(F)$ of $\operatorname {\mathrm {GL}}_n(F)$ . We then extend this representation trivially to a representation of $\mathrm {P}(F)$ . By normalised parabolic induction, we obtain an induced representation $(\pi ,V_{\pi })$ of $\operatorname {\mathrm {GL}}_n(F)$ ,

where $V_{\pi }$ denotes the space of smooth functions $f : \operatorname {\mathrm {GL}}_n(F) \to V_{\pi _1} \otimes \cdots \otimes V_{\pi _r}$ that satisfy

$$\begin{align*}f(umg) = \delta_{\mathrm{P}}^{1/2}(m) \pi_1(m_1) \otimes \cdots \otimes \pi_r(m_r) \cdot f(g),\end{align*}$$

for any $u \in \mathrm {N}_{\mathrm {P}}(F)$ , $m = \operatorname {\mathrm {blockdiag}}(m_1,\ldots ,m_r) \in \mathrm {M}_{\mathrm {P}}(F)$ , and $g \in \operatorname {\mathrm {GL}}_n(F)$ , and the action of $\pi $ on $V_{\pi }$ is by right translation – namely, . We call $\pi $ the isobaric sum of $\pi _1,\ldots ,\pi _r$ , which we denote by

2.2.2. Induced representations of Whittaker and Langlands types

A representation $\pi $ of $\operatorname {\mathrm {GL}}_n(F)$ is said to be an induced representation of Whittaker type if it is the isobaric sum of $\pi _1,\ldots ,\pi _r$ and each $\pi _j$ is irreducible and essentially square-integrable. Such a representation is admissible and smooth; moreover, if F is archimedean, then it is a Fréchet representation of moderate growth and of finite length. The contragredient of an induced representation of Whittaker type $\pi = \pi _1 \boxplus \cdots \boxplus \pi _r$ is $\widetilde {\pi } = \widetilde {\pi _1} \boxplus \cdots \boxplus \widetilde {\pi _r}$ , which is again an induced representation of Whittaker type. If each $\pi _j$ is additionally of the form $\sigma _j \otimes \left |\det \right |{}^{t_j}$ , where $\sigma _j$ is irreducible, unitary and square-integrable, and $\Re (t_1) \geq \cdots \geq \Re (t_r)$ , then $\pi $ is said to be an induced representation of Langlands type.

2.2.3. Whittaker models

A Whittaker functional $\Lambda : V_{\pi } \to \mathbb {C}$ of an admissible smooth representation $(\pi ,V_{\pi })$ of $\operatorname {\mathrm {GL}}_n(F)$ is a continuous linear functional that satisfies

$$\begin{align*}\Lambda(\pi(u) \cdot v) = \psi_n(u) \Lambda(v)\end{align*}$$

for all $v \in V_{\pi }$ and $u \in \mathrm {N}_n(F)$ ; here,

If $\pi $ is additionally irreducible, then the space of Whittaker functionals of $\pi $ is at most one-dimensional. If this space is indeed one-dimensional, so that there exists a unique such functional up to scalar multiplication, then $\pi $ is said to be generic. Every induced representation of Langlands type $(\pi ,V_{\pi })$ admits a nontrivial Whittaker functional $\Lambda $ ; moreover, $\pi $ is isomorphic to its unique Whittaker model $\mathcal {W}(\pi ,\psi )$ , which is the image of $V_{\pi }$ under the map $v \mapsto \Lambda (\pi (\cdot ) \cdot v)$ , so that $\mathcal {W}(\pi ,\psi )$ consists of Whittaker functions on $\operatorname {\mathrm {GL}}_n(F)$ of the form . An induced representation of Whittaker type $\pi $ also has a one-dimensional space of Whittaker functionals, but the map $v \mapsto \Lambda (\pi (\cdot ) \cdot v)$ need not be injective, so that the Whittaker model may only be a model of a quotient of  $\pi $ .

2.2.4. Irreducible representations

For nonarchimedean F, every irreducible admissible smooth representation $\pi $ of $\operatorname {\mathrm {GL}}_n(F)$ is isomorphic to the unique irreducible quotient of some induced representation of Langlands type. If $\pi $ is also generic, then it is isomorphic to some (necessarily irreducible) induced representation of Langlands type [Reference Casselman and ShahidiCS98].

For archimedean F, we recall that a Casselman–Wallach representation of $\operatorname {\mathrm {GL}}_n(F)$ is an admissible smooth Fréchet representation of moderate growth and of finite length [Reference WallachWal92; Reference Bernstein and KrötzBK14]; in particular, induced representations of Whittaker type are Casselman–Wallach representations. Every irreducible Casselman–Wallach representation $\pi $ of $\operatorname {\mathrm {GL}}_n(F)$ is isomorphic to the unique irreducible quotient of some induced representation of Langlands type. Again, if $\pi $ is additionally generic, then it is isomorphic to some (necessarily irreducible) induced representation of Langlands type.

2.2.5. Spherical Representations

An induced representation of Whittaker type $\pi $ is said to be spherical if it has a K-fixed vector. Such a spherical representation $\pi $ must then be a principal series representation of the form $|\cdot |^{t_1} \boxplus \cdots \boxplus |\cdot |^{t_n}$ ; furthermore, the subspace of K-fixed vectors must be one-dimensional. This K-fixed vector, which is unique up to scalar multiplication, is called the spherical vector of $\pi $ . For a spherical representation of Langlands type $\pi $ , the spherical Whittaker function $W^{\circ }$ in the Whittaker model $\mathcal {W}(\pi ,\psi )$ is given by the Jacquet integral

where is the long Weyl element and $f^{\circ }$ is the canonically normalised spherical vector in the induced model: the unique smooth function ${f^{\circ } : \operatorname {\mathrm {GL}}_n(F) \to \mathbb {C}}$ satisfying

$$\begin{align*}f^{\circ}(1_n) = \prod_{j = 1}^{n - 1} \prod_{\ell = j + 1}^{n} \zeta_F(1 + t_j - t_{\ell}),\!\!\qquad f^{\circ}(uag) = f^{\circ}(g) \delta_n^{1/2}(a) \prod_{j = 1}^{n} |a_j|^{t_j}, \qquad\!\! f^{\circ}(gk) = f^{\circ}(g)\end{align*}$$

for all $u \in \mathrm {N}_n(F)$ , $a = \operatorname {\mathrm {diag}}(a_1,\ldots ,a_n) \in \mathrm {A}_n(F)$ , $g \in \operatorname {\mathrm {GL}}_n(F)$ , and $k \in K$ . The Jacquet integral of $f^{\circ }$ converges absolutely when $\Re (t_1)> \cdots > \Re (t_n)$ and extends holomorphically to all of $\mathbb {C}^n \ni (t_1,\ldots ,t_n)$ (cf. Section 9.1). For nonarchimedean F, the normalisation of $W^{\circ }$ is such that $W^{\circ }(1_n) = 1$ .

2.3.1. Rankin–Selberg integrals

We recall the definition of Rankin–Selberg integrals over a local field F; see [Reference Cogdell, Cogdell, Kim and MurtyCog04] for further details. Given induced representations of Whittaker type $\pi $ of $\operatorname {\mathrm {GL}}_n(F)$ and $\pi '$ of $\operatorname {\mathrm {GL}}_m(F)$ with $m \leq n$ , we take Whittaker functions $W \in \mathcal {W}(\pi ,\psi )$ and $W' \in \mathcal {W}\left (\pi ',\overline {\psi }\right )$ and form the local $\operatorname {\mathrm {GL}}_n \times \operatorname {\mathrm {GL}}_m$ Rankin–Selberg integral defined by

(2.1)

(2.2)

where $\Phi \in \mathscr {S}(\operatorname {\mathrm {Mat}}_{1 \times n}(F))$ is a Schwartz–Bruhat function and . These integrals converge absolutely for $\Re (s)$ sufficiently large and extend meromorphically to the entire complex plane via the local functional equation.

2.3.2. Godement–Jacquet zeta integrals

Following [Reference Godement and JacquetGJ72; Reference JacquetJac79], we define Godement–Jacquet zeta integrals over a local field F. Given an induced representation of Whittaker type $(\pi ,V_{\pi })$ of $\operatorname {\mathrm {GL}}_n(F)$ , we take $v_1 \in V_{\pi }$ , $\widetilde {v_2} \in V_{\widetilde {\pi }}$ , with associated matrix coefficient of $\pi $ , and form the local Godement–Jacquet zeta integral defined by

(2.3)

where $\Phi \in \mathscr {S}(\operatorname {\mathrm {Mat}}_{n \times n}(F))$ is a Schwartz–Bruhat function. This integral converges absolutely for $\Re (s)$ sufficiently large and extends meromorphically to the entire complex plane via the local functional equation.

2.4.1. Rankin–Selberg L-functions and standard L-functions

For nonarchimedean F, the Rankin–Selberg L-function $L(s,\pi \times \pi ')$ is the generator of the $\mathbb {C}[q^s,q^{-s}]$ -fractional ideal of $\mathbb {C}(q^{-s})$ generated by the family of Rankin–Selberg integrals $\Psi (s,W,W')$ (or $\Psi (s,W,W',\Phi )$ if $m = n$ ) with $W \in \mathcal {W}(\pi ,\psi )$ and $W' \in \mathcal {W}\left (\pi ',\overline {\psi }\right )$ (and $\Phi \in \mathscr {S}(\operatorname {\mathrm {Mat}}_{1 \times n}(F))$ if $m = n$ ) with $L(s,\pi \times \pi ')$ normalised to be of the form $P(q^{-s})^{-1}$ for some $P(q^{-s}) \in \mathbb {C}[q^{-s}]$ whose constant term is $1$ . For archimedean F, the Rankin–Selberg L-function $L(s,\pi \times \pi ')$ is defined via the local Langlands correspondence as explicated in [Reference Knapp, Jannsen, Kleiman and SerreKna94].

Similarly, for nonarchimedean F, the standard L-function $L(s,\pi )$ is the generator of the $\mathbb {C}[q^s,q^{-s}]$ -fractional ideal of $\mathbb {C}(q^{-s})$ generated by the family of Godement–Jacquet zeta integrals $Z(s,\beta ,\Phi )$ with $v_1 \in V_{\pi }$ , $\widetilde {v_2} \in V_{\widetilde {\pi }}$ , and $\Phi \in \mathscr {S}(\operatorname {\mathrm {Mat}}_{n \times n}(F))$ , with $L(s,\pi )$ normalised to be of the form $P(q^{-s})^{-1}$ for some $P(q^{-s}) \in \mathbb {C}[q^{-s}]$ whose constant term is  $1$ . For archimedean F, the standard L-function $L(s,\pi )$ is defined via the local Langlands correspondence as explicated in [Reference Knapp, Jannsen, Kleiman and SerreKna94].

In both settings, upon decomposing $\pi $ and $\pi '$ as isobaric sums

(2.4)

we have the identities

(2.5) $$ \begin{align} L(s,\pi) = \prod_{j = 1}^{r} L(s,\pi_j), \qquad L(s,\pi \times \pi') = \prod_{j = 1}^{r} \prod_{j' = 1}^{r'} L(s,\pi_j \times \pi_{j'}^{\prime}). \end{align} $$

Moreover, Rankin–Selberg L-functions involving twists by a character are related to standard L-functions via the identity

(2.6) $$ \begin{align} L(s,\pi \times |\cdot|^t) = L(s + t,\pi). \end{align} $$

2.4.2. L-functions for representations of $\operatorname {\mathrm {GL}}_n(\mathbb {C})$

Essentially square-integrable representations of $\operatorname {\mathrm {GL}}_n(\mathbb {C})$ exist only for $n = 1$ , in which case the representation must be a character of the form $\pi (z) = \chi ^{\kappa }(z) |z|_{\mathbb {C}}^t$ for some $\kappa \in \mathbb {Z}$ and $t \in \mathbb {C}$ , where $z \in \operatorname {\mathrm {GL}}_1(\mathbb {C}) = \mathbb {C}^{\times }$ and $\chi $ is the canonical character

The L-function of $\pi $ is

(2.7) $$ \begin{align} L(s,\pi) = \zeta_{\mathbb{C}}\left(s + t + \frac{\|\kappa\|}{2}\right), \end{align} $$

where

(2.8)

This integral representation of $\zeta _{\mathbb {C}}(s)$ converges absolutely for $\Re (s)> 0$ and extends meromorphically to the entire complex plane with poles at nonpositive integers. The contragredient of $\pi $ is $\widetilde {\pi } = \chi ^{-\kappa } |\cdot |_{\mathbb {C}}^{-t}$ , so that

(2.9) $$ \begin{align} L(s,\widetilde{\pi}) = \zeta_{\mathbb{C}}\left(s - t + \frac{\|\kappa\|}{2}\right). \end{align} $$

2.4.3. L-functions for representations of $\operatorname {\mathrm {GL}}_n(\mathbb {R})$

Essentially square-integrable representations of $\operatorname {\mathrm {GL}}_n(\mathbb {R})$ exist only for $n \in \{1,2\}$ . An essentially square-integrable representation of $\operatorname {\mathrm {GL}}_1(\mathbb {R})$ must be a character of the form $\pi (x) = \chi ^{\kappa }(x) |x|_{\mathbb {R}}^t$ for some $\kappa \in \{0,1\}$ and $t \in \mathbb {C}$ , where $x \in \operatorname {\mathrm {GL}}_1(\mathbb {R}) = \mathbb {R}^{\times }$ and $\chi $ is the canonical character

The L-function of $\pi $ is

(2.10) $$ \begin{align} L(s,\pi) = \zeta_{\mathbb{R}}(s + t + \kappa), \end{align} $$

where

(2.11)

This integral representation of $\zeta _{\mathbb {R}}(s)$ converges absolutely for $\Re (s)> 0$ and extends meromorphically to the entire complex plane with poles at nonpositive even integers. The contragredient is $\widetilde {\pi } = \chi ^{\kappa } |\cdot |_{\mathbb {R}}^{-t}$ , so that

(2.12) $$ \begin{align} L(s,\widetilde{\pi}) = \zeta_{\mathbb{R}}(s - t + \kappa). \end{align} $$

For $n = 2$ , we note that

$$\begin{align*}|\cdot|_{\mathbb{R}}^t \boxplus \chi |\cdot|_{\mathbb{R}}^t \cong \operatorname{\mathrm{Ind}}_{\operatorname{\mathrm{GL}}_1(\mathbb{C})}^{\operatorname{\mathrm{GL}}_2(\mathbb{R})} |\cdot|_{\mathbb{C}}^t,\end{align*}$$

where $\operatorname {\mathrm {GL}}_1(\mathbb {C})$ is viewed as a subgroup of $\operatorname {\mathrm {GL}}_2(\mathbb {R})$ via the identification . For $\kappa \neq 0$ , the essentially discrete series representation of weight $\|\kappa \| + 1$ ,

is essentially square-integrable, and every essentially square-integrable representation of $\operatorname {\mathrm {GL}}_2(\mathbb {R})$ is of the form $\pi = D_{\kappa } \otimes \left |\det \right |{}_{\mathbb {R}}^t$ for some integer $\kappa \geq 2$ and $t \in \mathbb {C}$ . The L-function of $\pi $ is

(2.13) $$ \begin{align} L(s,\pi) = \zeta_{\mathbb{C}}\left(s + t + \frac{\kappa - 1}{2}\right) = \zeta_{\mathbb{R}}\left(s + t + \frac{\kappa - 1}{2}\right) \zeta_{\mathbb{R}}\left(s + t + \frac{\kappa + 1}{2}\right). \end{align} $$

The contragredient of $\pi $ is $\widetilde {\pi } = D_{\kappa } \otimes \left |\det \right |{}_{\mathbb {R}}^{-t}$ , so that

(2.14) $$ \begin{align} L(s,\widetilde{\pi}) = \zeta_{\mathbb{C}}\left(s - t + \frac{\kappa - 1}{2}\right) = \zeta_{\mathbb{R}}\left(s - t + \frac{\kappa - 1}{2}\right) \zeta_{\mathbb{R}}\left(s - t + \frac{\kappa + 1}{2}\right). \end{align} $$

2.4.4. Epsilon factors

To any induced representations of Whittaker type $\pi $ of $\operatorname {\mathrm {GL}}_n(F)$ and $\pi '$ of $\operatorname {\mathrm {GL}}_m(F)$ and any nontrivial additive character $\psi $ of F, one can associate the epsilon factor $\varepsilon (s,\pi ,\psi )$ of $\pi $ , $\varepsilon (s,\pi ',\psi )$ of $\pi '$ , and $\varepsilon (s,\pi \times \pi ',\psi )$ of $\pi \times \pi '$ , which arise via the local functional equations for the Godement–Jacquet zeta integral and $\operatorname {\mathrm {GL}}_n \times \operatorname {\mathrm {GL}}_m$ Rankin–Selberg integral, respectively. In particular, the local functional equation for the Godement–Jacquet zeta integral is

(2.15) $$ \begin{align} \frac{Z(1 - s,\widetilde{\beta},\widehat{\Phi})}{L(1 - s,\widetilde{\pi})} = \varepsilon(s,\pi,\psi) \frac{Z(s,\beta,\Phi)}{L(s,\pi)}, \end{align} $$

where with denoting the transpose of g, while the Fourier transform $\widehat {\Phi }$ is

Note that this normalisation of the Fourier transform differs from that in [Reference Godement and JacquetGJ72] and that the epsilon factor is dependent on this normalisation. For the local functional equation for the $\operatorname {\mathrm {GL}}_n \times \operatorname {\mathrm {GL}}_m$ Rankin–Selberg integral, see, for example, [Reference Jacquet, Ginzburg, Lapid and SoudryJac09, Theorem 2.1].

Upon decomposing $\pi $ and $\pi '$ as isobaric sums as in (2.4), we have the identities

(2.16) $$ \begin{align} \varepsilon(s,\pi,\psi) = \prod_{j = 1}^{r} \varepsilon(s,\pi_j,\psi), \qquad \varepsilon(s,\pi \times \pi',\psi) = \prod_{j = 1}^{r} \prod_{j' = 1}^{r'} \varepsilon(s,\pi_j \times \pi_{j'}^{\prime},\psi) \end{align} $$

by [Reference Jacquet, Piatetski-Shapiro and ShalikaJP-SS83, Theorem (3.1), Proposition (8.4), Proposition (9.4)] for nonarchimedean F and via the local Langlands correspondence for archimedean F. When $\pi '$ is the trivial representation of $\operatorname {\mathrm {GL}}_1(F)$ , we have the equality $\varepsilon (s,\pi ,\psi ) = \varepsilon (s,\pi \times \pi ',\psi )$ between the epsilon factor of $\pi $ defined via the Godement–Jacquet zeta integral and the epsilon factor of $\pi \times \pi '$ defined via the $\operatorname {\mathrm {GL}}_n \times \operatorname {\mathrm {GL}}_1$ Rankin–Selberg integral.

For nonarchimedean F, the epsilon factors $\varepsilon (s,\pi ,\psi )$ and $\varepsilon (s,\pi \times \pi ',\psi )$ are units in $\mathbb {C}[q^s,q^{-s}]$ of the form

(2.17) $$ \begin{align} \varepsilon(s,\pi,\psi) = \varepsilon\left(\frac{1}{2},\pi,\psi\right) q^{-c(\pi) \left(s - \frac{1}{2}\right)}, \qquad \varepsilon(s,\pi \times \pi',\psi) = \varepsilon\left(\frac{1}{2},\pi \times \pi',\psi\right) q^{-c(\pi \times \pi') \left(s - \frac{1}{2}\right)} \end{align} $$

for some nonnegative integers $c(\pi )$ and $c(\pi \times \pi ')$ .

For archimedean F, we have that

(2.18) $$ \begin{align} \varepsilon(s,\pi,\psi) = \begin{cases} i^{-\kappa} &\text{if }F = \mathbb{R}\text{ and }\pi = \chi^{\kappa} |\cdot|_{\mathbb{R}}^t\text{ for }\kappa \in \{0,1\}, \\ i^{-\kappa} &\text{if }F = \mathbb{R}\text{ and }\pi = D_{\kappa} \otimes \left|\det\right|{}_{\mathbb{R}}^t\text{ for }\kappa \geq 2, \\ i^{-\|\kappa\|} &\text{if }F = \mathbb{C}\text{ and }\pi = \chi^{\kappa} |\cdot|_{\mathbb{C}}^t\text{ for }\kappa \in \mathbb{Z}, \end{cases} \end{align} $$

which, via (2.15), may be used to determine $\varepsilon (s,\pi ,\psi )$ and $\varepsilon (s,\pi \times \pi ',\psi )$ for any induced representations of Whittaker type $\pi $ and $\pi '$ , not just for essentially square-integrable representations.

4.2.1. Highest weight theory for $\operatorname {\mathrm {U}}(n)$

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

are parametrised by the set of highest weights, which we may identify with n-tuples of integers that are nonincreasing:

We denote by $\tau _{\mu }$ an irreducible representation of $\operatorname {\mathrm {U}}(n)$ with highest weight $\mu \in \Lambda _n$ . The Howe degree of $\tau _{\mu }$ is given by

(4.2) $$ \begin{align} \deg \tau_{\mu} = \sum_{j = 1}^{n} \|\mu_j\|. \end{align} $$

Remark 4.3. There is another natural invariant that one associates to an irreducible representation $\tau _{\mu }$ – namely, its Vogan norm $\|\tau _{\mu }\|_{\mathrm {V}}$ [Reference VoganVog81], which is

$$\begin{align*}\|\tau_{\mu}\|_{\mathrm{V}}^2 = \sum_{j = 1}^{n} \left(\mu_j + n + 1 - 2j\right)^2.\end{align*}$$

4.2.2. Highest weight theory for $\operatorname {\mathrm {O}}(n)$

The equivalence classes of finite-dimensional irreducible representations $\tau _{\mu }$ of the orthogonal group

are parametrised by an n-tuple of integers $\mu $ , which we again call highest weights (though unlike $\operatorname {\mathrm {U}}(n)$ , the compact Lie group $\operatorname {\mathrm {O}}(n)$ is not connected). These highest weights $\mu $ are precisely those for which the highest weight vector of the irreducible representation of $\operatorname {\mathrm {U}}(n)$ with highest weight $\mu $ generates $\tau _{\mu }$ when restricted to $\operatorname {\mathrm {O}}(n)$ , and are of the form

$$\begin{align*}\mu = (\mu_1,\ldots,\mu_m,\underbrace{\eta,\ldots,\eta}_{n - 2m \text{ times}},\underbrace{0,\ldots,0}_{m \text{ times}}) \in \mathbb{N}_0^n,\end{align*}$$

where $m \in \{0,\ldots ,\lfloor \frac {n}{2}\rfloor \}$ , $\mu _1 \geq \cdots \geq \mu _m \geq 1$ , $\eta \in \{0,1\}$ and . We again let $\Lambda _n$ denote the set of highest weights; note, in particular, that $\Lambda _1 = \{0,1\}$ . We denote by $\tau _{\mu }$ an irreducible representation of $\operatorname {\mathrm {O}}(n)$ with highest weight $\mu \in \Lambda _n$ . The Howe degree of  $\tau _{\mu }$  is

(4.4) $$ \begin{align} \deg \tau_{\mu} = \sum_{j = 1}^{n} \mu_j. \end{align} $$

Remark 4.5. The Vogan norm $\|\tau _{\mu }\|_{\mathrm {V}}$ of $\tau _{\mu }$ is

$$\begin{align*}\|\tau_{\mu}\|_{\mathrm{V}}^2 = \sum_{j = 1}^{m} \left(\mu_j + n - 2j\right)^2 + \sum_{j = m + 1}^{\left\lfloor \frac{n}{2} \right\rfloor} (n - 2j)^2.\end{align*}$$

5.2.1. The conductor exponent for essentially square-integrable representations

For $\pi = \chi ^{\kappa } |\cdot |^t$ , where $\kappa \in \{0,1\}$ for $F = \mathbb {R}$ and $\kappa \in \mathbb {Z}$ for $F = \mathbb {C}$ , the conductor exponent is $c(\pi ) = \|\kappa \|$ . For $\pi = D_{\kappa } \otimes \left |\det \right |{}^t$ , where $\kappa \geq 2$ , the conductor exponent is $c(\pi ) = \kappa $ . Via Theorem 4.15, this allows one to calculate the conductor exponent of any induced representation of Whittaker type.

5.2.2. $\operatorname {\mathrm {GL}}_2$ Examples

Let us consider the classical setting of automorphic forms on the upper half-plane.

Remark 5.4. The newform K-type of the archimedean component of the automorphic representation associated to an odd Maaß form of weight zero is not its minimal K-type: the newform K-type is $\tau _{(2,0)}$ , whereas the minimal K-type is the determinant representation $\tau _{(1,1)}$ . Classically, this manifests itself via the fact that an odd Maaß form f of weight zero is not a test vector for the Rankin–Selberg integral

$$\begin{align*}\int_{0}^{\infty} f(iy) y^{s - \frac{1}{2}} \, \frac{dy}{y}.\end{align*}$$

Instead, one must use Maaß raising and lowering operators on such a Maaß form to obtain a test vector.

5.2.3. A non-application: the analytic conductor and analytic newvector

We have introduced in Section 4.3 the notion of the conductor exponent of a generic irreducible Casselman–Wallach representation $\pi $ as a measure of the extent of ramification. This can also be thought of as quantifying the (representation-theoretic) complexity of $\pi $ .

There is another well-known quantification of the complexity of $\pi $ : the analytic conductor. First introduced by Iwaniec and Sarnak [Reference Iwaniec, Sarnak, Alon, Bourgain, Connes, Gromov and MilmanIwSa00, (31)], this is defined on essentially square-integrable representations $\pi $ by

and extended to arbitrary induced representations of Whittaker type $\pi = \pi _1 \boxplus \cdots \boxplus \pi _r$ via multiplicativity:

One can relate this to the asymptotic behaviour of the local $\gamma $ -factor $\gamma (s,\pi ,\psi )$ as s tends to $1/2$ ; see, in particular, [Reference Jana and NelsonJN19, Lemma 3.1 (1)] and also [Reference Conrey, Farmer, Keating, Rubinstein and SnaithCFKRS05, Section 3.1] (the latter also alludes to how this similarly encompasses the conductor at the nonarchimedean places).

These two quantifications are entirely distinct. This can be seen most clearly for spherical representations $\pi = |\cdot |^{it_f} \boxplus |\cdot |^{-it_f}$ of $\operatorname {\mathrm {GL}}_2(\mathbb {R})$ occurring as the archimedean component of an automorphic representation associated to an even Maaß form f of weight zero. As discussed in Example 5.3, the conductor exponent of such a representation $\pi $ is $c(\pi ) = 0$ ; however, the analytic conductor is $\mathfrak {q}(\pi ) = (1 + \|t_f\|)^2$ . This distinction boils down to the fact that the analytic conductor $\mathfrak {q}(\pi )$ is better thought of as being a measure of the size of the L-function $L(s,\pi )$ rather than a measure of the complexity of the representation itself.

Similarly, the newform is unrelated to the analytic newvectors introduced by Jana and Nelson in [Reference Jana and NelsonJN19]. Analytic newvectors are a different archimedean analogue of newforms; they are a family of vectors in a generic irreducible unitary Casselman–Wallach representation that are almost invariant under certain subsets (but not subgroups) of $\operatorname {\mathrm {GL}}_n(F)$ that are archimedean analogues of the congruence subgroups $K_1(\mathfrak {p}^m)$ . Notably, analytic newvectors are not necessarily $K_n$ -finite, though they are $K_{n - 1}$ -invariant; in particular, they lie in the subspace spanned by the newform and all oldforms. Note that Jana and Nelson call the newform an ‘algebraic newvector’.

5.2.4. Global Eulerian integrals

We mention a global application of the resolution of the test vector problem for Rankin–Selberg integrals. In order to do so, we require some discussion of global automorphic forms.

Let $\psi _{\mathbb {A}_{\mathbb {Q}}}$ denote the standard additive character of $\mathbb {A}_{\mathbb {Q}}$ that is unramified at every place of $\mathbb {Q}$ . Given a global number field F, define the additive character of $\mathbb {A}_F$ . The conductor of $\psi _{\mathbb {A}_F}$ is the inverse different $\mathfrak {d}^{-1}$ of F; furthermore, there is a finite idèle $d \in \mathbb {A}_F^{\times }$ representing $\mathfrak {d}$ such that $\psi _{\mathbb {A}_F} = \bigotimes _v \psi _v^{d_v}$ , where the additive character $\psi _v^{d_v}$ of $F_v$ of conductor $\mathfrak {d}_v^{-1}$ is defined by with $\psi _v$ an unramified additive character of $F_v$ as in Section 2.1.2.

Let $(\pi ,V_{\pi })$ be a cuspidal automorphic representation of $\operatorname {\mathrm {GL}}_n(\mathbb {A}_F)$ with $n \geq 2$ , where $V_{\pi }$ is a space of automorphic forms on $\operatorname {\mathrm {GL}}_n(\mathbb {A}_F)$ . Let $W_{\varphi } \in \mathcal {W}(\pi ,\psi _{\mathbb {A}_F})$ denote the Whittaker function of $\varphi \in V_{\pi }$ , so that

where $du$ denotes the Tamagawa measure. If $W_{\varphi }$ is a pure tensor, we may write $W_{\varphi } = \prod _v W_{\varphi ,v}$ with $W_{\varphi ,v} \in \mathcal {W}(\pi _v,\psi _v^{d_v})$ , where the generic irreducible admissible smooth representations $\pi _v$ are the local components of the automorphic representation $\pi = \bigotimes _v \pi _v$ . We define the global newform $\varphi ^{\circ } \in V_{\pi }$ to be such that at each place v of F, we have that with $W_v \in \mathcal {W}(\pi _v,\psi _v)$ the (local) Whittaker newform as in Theorems 3.14 and 4.17.

Proposition 5.5. Let F be a global number field of absolute discriminant $D_{F/\mathbb {Q}}$ , and let $(\pi ,V_{\pi })$ be a cuspidal automorphic representation of $\operatorname {\mathrm {GL}}_n(\mathbb {A}_F)$ with $n \geq 2$ . Then the global newform $\varphi ^{\circ } \in V_{\pi }$ is such that for all cuspidal automorphic representations $(\pi ',V_{\pi '})$ of $\operatorname {\mathrm {GL}}_{n - 1}(\mathbb {A}_F)$ that are everywhere unramified with global newforms $\varphi ^{\prime \circ } \in V_{\pi '}$ and for $\Re (s)$ sufficiently large, the global $\operatorname {\mathrm {GL}}_n \times \operatorname {\mathrm {GL}}_{n - 1}$ Rankin–Selberg integral

with $dg$ the Tamagawa measure is equal, up to multiplication by a constant dependent only on F, to the product of $\omega _{\pi '}^{-1}(d) D_{F/\mathbb {Q}}^{\frac {n(n - 1)s}{2}}$ , where $\omega _{\pi '}$ denotes the central character of $\pi '$ , and of the global completed Rankin–Selberg L-function .

Proof. By unfolding,

$$ \begin{align*} I(s,\varphi^{\circ},\varphi^{\prime\circ}) & = \int\limits_{\mathrm{N}_{n - 1}(\mathbb{A}_F) \backslash \operatorname{\mathrm{GL}}_{n - 1}(\mathbb{A}_F)} W_{\varphi^{\circ}}\begin{pmatrix} g & 0 \\ 0 & 1 \end{pmatrix} W_{\varphi^{\prime\circ}}(g) \left|\det g\right|{}_{\mathbb{A}_F}^{s - \frac{1}{2}} \, dg \\ & = c_{F/\mathbb{Q}} \prod_v \Psi(s,W_{\varphi^{\circ},v},W_{\varphi^{\prime\circ},v}), \end{align*} $$

where the local Rankin–Selberg integrals $\Psi (s,W_{\varphi ^{\circ },v},W_{\varphi ^{\prime \circ },v})$ are as in (2.1), and ${c_{F/\mathbb {Q}}> 0}$ is an absolute constant dependent only on F that arises from the compatibility of the global Tamagawa measure on $\mathrm {N}_{n - 1}(\mathbb {A}_F) \backslash \operatorname {\mathrm {GL}}_{n - 1}(\mathbb {A}_F)$ compared to the local Haar measure on $\mathrm {N}_{n - 1}(F_v) \backslash \operatorname {\mathrm {GL}}_{n - 1}(F_v)$ as normalised in Section 2.1. Upon making the change of variables $g_v^{\prime } \mapsto \operatorname {\mathrm {diag}}(d_v^{1 - n},\ldots ,d_v^{-1}) g_v^{\prime }$ and using the fact that $W_v^{\prime }(\operatorname {\mathrm {diag}}(d_v^{-1},\ldots ,d_v^{-1}) g_v^{\prime }) = \omega _{\pi _v^{\prime }}^{-1}(d_v) W_v^{\prime }(g_v^{\prime })$ , we see that

$$\begin{align*}\Psi(s,W_{\varphi^{\circ},v},W_{\varphi^{\prime\circ},v}) = \omega_{\pi_v^{\prime}}^{-1}(d_v) |d_v|_v^{-\frac{n(n - 1) s}{2} + \frac{n(n - 1)(2n - 1)}{12}} \Psi(s,W_v,W_v^{\prime}).\end{align*}$$

The result now follows from Theorems 3.14 and 4.17.

A similar result holds for global $\operatorname {\mathrm {GL}}_n \times \operatorname {\mathrm {GL}}_n$ Rankin–Selberg integrals via Theorems 3.18 and 4.18. These involve an Eisenstein series

$$\begin{align*}E(g,s;\Phi,\eta) = \left|\det g\right|{}_{\mathbb{A}_F}^s \int_{F^{\times} \backslash \mathbb{A}_F^{\times}} \Theta_{\Phi}^{\prime}(a,g) \eta(a) |a|_{\mathbb{A}_F}^{ns} \, d^{\times} a,\end{align*}$$

where $\eta : F^{\times } \backslash \mathbb {A}_F^{\times } \to \mathbb {C}^{\times }$ is a Hecke character, $\Phi \in \mathscr {S}(\operatorname {\mathrm {Mat}}_{1 \times n}(\mathbb {A}_F))$ is a Schwartz–Bruhat function, and

Proposition 5.7. Let F be a number field of absolute discriminant $D_{F/\mathbb {Q}}$ , and let $(\pi ,V_{\pi })$ and $(\pi ',V_{\pi '})$ be automorphic representations of $\operatorname {\mathrm {GL}}_n(\mathbb {A}_F)$ with central characters $\omega _{\pi }$ and $\omega _{\pi '}$ . Suppose that at least one of $\pi $ and $\pi '$ is cuspidal and that $\pi $ and $\pi '$ have disjoint ramification. Then there exists a right K-finite Schwartz–Bruhat function $\Phi \in \mathscr {S}(\operatorname {\mathrm {Mat}}_{1 \times n}(\mathbb {A}_F))$ such that for $\Re (s)$ sufficiently large, the global $\operatorname {\mathrm {GL}}_n \times \operatorname {\mathrm {GL}}_n$ Rankin–Selberg integral

with $dg$ the Tamagawa measure is equal, up to multiplication by a constant dependent only on F, to the product of $D_{F/\mathbb {Q}}^{\frac {n(n - 1)s}{2}}$ and of the global completed Rankin–Selberg L-function .

We also may show the existence of a test vector for the global Godement–Jacquet zeta integral via Theorems 3.20 and 4.23.

Proposition 5.9. Let F be a number field, and let $(\pi ,V_{\pi })$ be an automorphic representation of $\operatorname {\mathrm {GL}}_n(\mathbb {A}_F)$ . Then there exists a matrix coefficient and a bi-K-finite Schwartz–Bruhat function $\Phi \in \mathscr {S}(\operatorname {\mathrm {Mat}}_{n \times n}(\mathbb {A}_F))$ such that for $\Re (s)$ sufficiently large, the global Godement–Jacquet zeta integral

with $dg$ the Tamagawa measure is equal, up to multiplication by a constant dependent only on F, to the global completed standard L-function .

A similar result also holds for the Piatetski-Shapiro–Rallis integral [Reference Piatetski-Shapiro and RallisP-SR87], since this doubling integral is equal to the Godement–Jacquet zeta integral [Reference Piatetski-Shapiro and RallisP-SR87, Proposition 3.2].

Remark 5.10. In fact, we show something slightly stronger – namely, that

$$\begin{align*}\int_{\operatorname{\mathrm{GL}}_n(\mathbb{A}_F)} \varphi^{\circ}(hg) \Phi(g) \left|\det g\right|{}_{\mathbb{A}_F}^{s + \frac{n - 1}{2}} \, dg\end{align*}$$

is equal, up to multiplication by a constant dependent only on F, to $\Lambda (s,\pi ) \varphi ^{\circ }(h)$ for all $h \in \operatorname {\mathrm {GL}}_n(\mathbb {A}_F)$ .

5.3.1. Nonarchimedean strategies

As discussed in Section 3, there are several approaches towards developing nonarchimedean newform theory for $\operatorname {\mathrm {GL}}_n$ . We briefly examine the challenges in transporting each of these methods to the archimedean setting.

Matringe [Reference MatringeMat13] uses the nonarchimedean theory of Bernstein–Zelevinsky derivatives to explicitly construct the Whittaker newform. The archimedean theory of Bernstein–Zelevinsky derivatives is less well-developed (though see [Reference Aizenbud, Gourevitch and SahiAGS15] and [Reference ChaiCha15] for two different approaches), and it does not seem straightforward to transport Matringe’s proof to this setting.

The method of Jacquet [Reference JacquetJac12] does not use Bernstein–Zelevinsky derivatives; the proof, however, is nonconstructive and only shows the existence of a $K_{n - 1}$ -invariant test vector for $\operatorname {\mathrm {GL}}_n \times \operatorname {\mathrm {GL}}_{n - 1}$ Rankin–Selberg integrals. In the archimedean setting, it seems difficult to describe the newform K-type when one only knows of the existence of such a test vector.

Finally, the method of Miyauchi [Reference MiyauchiMiy14] assumes the existence of the newform in $V_{\pi }^{K_1(\mathfrak {p}^{c(\pi )})}$ and uses the action of certain Hecke operators to derive a recursive relation between certain values of the newform in the Whittaker model; using this, one can then show that the newform in the Whittaker model is a test vector for Rankin–Selberg integrals. The archimedean analogue of this is to assume the existence of the newform in the newform K-type and use the action of certain differential operators (arising from elements of the centre of the universal enveloping algebra) to derive systems of partial differential equations satisfied by the newform in the Whittaker model. This is essentially the approach undertaken in [Reference Hirano and OdaHO09] and [Reference Ishii and OdaIO14] (where the Whittaker function studied lies in the minimal K-type, not the newform K-type); already for $\operatorname {\mathrm {GL}}_3(\mathbb {C})$ , however, this leads to immense combinatorial difficulties in solving these systems of partial differential equations.

5.3.2. The archimedean strategy

We take a different path. To prove the existence of the newform and newform K-type, we use Frobenius reciprocity to reduce the problem to branching rules on the associated maximal compact subgroups. Here, we benefit from the fact that, unlike in the nonarchimedean setting, irreducible representations of K and explicit branching rules are well understood, and the induced representations of Whittaker type are particularly easy to describe, since essentially square-integrable representations do not exist for $\operatorname {\mathrm {GL}}_n(F)$ with $n \geq 3$ . This approach also determines the dimension of spaces of oldforms and yields the additivity and inductivity of the conductor exponent. All of this is proven in Section 6.

To study the newform in more detail, with an eye towards formulæ for the Whittaker newform that are beneficial for evaluating $\operatorname {\mathrm {GL}}_n \times \operatorname {\mathrm {GL}}_{n - 1}$ and $\operatorname {\mathrm {GL}}_n \times \operatorname {\mathrm {GL}}_n$ Rankin–Selberg integrals, we require additional knowledge, given in Section 7, of a particular model of the newform K-type – namely, a space of homogeneous harmonic polynomials. Using this, we explicitly construct the newform in the induced model of $\pi $ in Section 8.

We give three different constructions of the newform in the induced model: via the Iwasawa decomposition, via convolution sections and via Godement sections. Each construction has its advantages and disadvantages. The construction via the Iwasawa decomposition is straightforward but lacks a direct relation to Whittaker functions. The construction via convolution sections, following work of Jacquet [Reference Jacquet, Hida, Ramakrishnan and ShahidiJac04], gives a recursive formula for the newform in terms of a convolution of the newform itself with an explicit standard Schwartz function; this gives an immediate resolution of the test vector problem for archimedean Godement–Jacquet integrals. Finally, following Jacquet [Reference Jacquet, Ginzburg, Lapid and SoudryJac09], the newform is shown to be given as an element of a Godement section, which gives a recursive formula for the newform in terms of an integral of a distinguished newform of a representation of $\operatorname {\mathrm {GL}}_{n - 1}(F)$ against a particular standard Schwartz function; this is limited to certain induced representations of Whittaker type, but is invaluable as an inductive step.

With these formulæ in hand, we then express the newform in the Whittaker model via the Jacquet integral in Section 9. The usage of convolution sections and Godement sections gives us recursive formulæ for the Whittaker newform. The latter, in particular, gives what we call a propagation formula: this is a recursive formula for $\operatorname {\mathrm {GL}}_n(F)$ Whittaker functions in terms of $\operatorname {\mathrm {GL}}_{n - 1}(F)$ Whittaker functions.

Our expression for the newform via convolution sections gives a quick resolution of the test vector problem for the Godement–Jacquet zeta integral. Our strategy for resolving the test vector problems for $\operatorname {\mathrm {GL}}_n \times \operatorname {\mathrm {GL}}_n$ and $\operatorname {\mathrm {GL}}_n \times \operatorname {\mathrm {GL}}_{n - 1}$ Rankin–Selberg integrals follows an approach pioneered by Jacquet [Reference Jacquet, Ginzburg, Lapid and SoudryJac09] (which is also followed in [Reference Ishii and MiyazakiIM22]). We employ a double induction argument presented in Section 10. This type of argument is due to Jacquet [Reference Jacquet, Ginzburg, Lapid and SoudryJac09] (who, in turn, attributes this strategy to Shalika): it expresses the $\operatorname {\mathrm {GL}}_n \times \operatorname {\mathrm {GL}}_n$ Rankin–Selberg integral as the product of a $\operatorname {\mathrm {GL}}_n \times \operatorname {\mathrm {GL}}_{n - 1}$ Rankin–Selberg integral and a $\operatorname {\mathrm {GL}}_n$ Godement–Jacquet zeta integral, and similarly expresses the $\operatorname {\mathrm {GL}}_n \times \operatorname {\mathrm {GL}}_{n - 1}$ Rankin–Selberg integral as a product of a $\operatorname {\mathrm {GL}}_{n - 1} \times \operatorname {\mathrm {GL}}_{n - 1}$ Rankin–Selberg integral and a $\operatorname {\mathrm {GL}}_{n - 1}$ Godement–Jacquet zeta integral. (In fact, we find a slightly more direct approach via convolution sections that masks the presence of Godement–Jacquet zeta integrals.)

5.3.3. Additional remarks on the proofs

We emphasise that the proofs of Theorems 4.17 and 4.18, given in Section 10, are independent of the proofs in [Reference Gerasimov, Lebedev and OblezinGLO08, Reference Ishii and StadeIsSt13, Reference StadeSta01, Reference StadeSta02] of the unramified cases. Although our proofs are somewhat involved when ramification is present, they are particularly simple for spherical representations. In particular, these give proofs of Stade’s formulæ – namely, the unramified cases of Theorems 4.17 and 4.18. These reproofs of Stade’s formulæ also follow from the proofs of [Reference Ishii and MiyazakiIM22, Theorems 2.5 and 2.9]; they are essentially implicit in the work of Jacquet [Reference Jacquet, Ginzburg, Lapid and SoudryJac09].

Notably, we do not explicitly make use of the action of the universal enveloping algebra of the complexified Lie algebra of $\operatorname {\mathrm {GL}}_n(F)$ as differential operators on Whittaker functions, nor do we require any calculations involving Mellin transforms. In this regard, our construction of Whittaker functions is entirely distinct to that of much previous work on archimedean Whittaker functions [Reference Hirano, Ishii, Miyazaki, Hamahata, Ichikawa, Murase and SuganoHIM12, Reference Hirano, Ishii and MiyazakiHIM16; Reference Hirano and OdaHO09, Reference Ishii and OdaIO14, Reference Ishii and StadeIsSt13, Reference PopaPop08, Reference StadeSta90, Reference StadeSta95, Reference StadeSta01, Reference StadeSta02]. In particular, our proofs of Theorems 4.17 and 4.18 demonstrate that the Jacquet integral is adequate for the direct computation of archimedean Rankin–Selberg integrals, contrary to an assertion of Ishii and Oda [Reference Ishii and OdaIO14, p. 1288], provided one couples this with the usage of convolution sections and Godement sections.