2.1 Cohomological representations

Let $\mathbb {K}$ be an archimedean local field. Thus, it is a topological field that is topologically isomorphic to $\mathbb {R}$ or $\mathbb {C}$. Its complexification

\[ \mathbb{K}\otimes_{\mathbb{R}}\mathbb{C}=\prod_{\iota\in \mathcal{E}_\mathbb{K}} \mathbb{C}, \]

where $\mathcal {E}_\mathbb {K}$ denotes the set of all continuous field embeddings $\iota : \mathbb {K}\rightarrow \mathbb {C}$. Note that $\mathcal {E}_\mathbb {R}$ consists of the inclusion map, and $\mathcal {E}_\mathbb {C}$ consists of the identity map and the complex conjugation.

Fix an integer $n\geqslant 1$, and fix a weight

\[ \mu^\iota=(\mu_1^\iota\geqslant \mu_2^\iota\geqslant \cdots \geqslant \mu_n^\iota)\in \mathbb{Z}^n \]

for every $\iota \in \mathcal {E}_\mathbb {K}$. Write $\mu :=\{\mu ^\iota \}_{\iota \in \mathcal {E}_\mathbb {K}}$, and denote by $F_{\mu }$ the irreducible algebraic representation of ${\mathrm {GL}}_n(\mathbb {K}\otimes _\mathbb {R} \mathbb {C})=\prod _{\iota \in \mathcal {E}_\mathbb {K}} {\mathrm {GL}}_n(\mathbb {C})$ of highest weight $\mu$. Recall that all algebraic representations of algebraic groups are assumed to be finite-dimensional.

We say that $\mu$ is pure if

\[ \mu_{1}^\iota+\mu_{n}^{\bar \iota}=\mu_{2}^{\iota}+\mu_{n-1}^{\bar \iota}=\cdots=\mu_{n}^{\iota}+\mu_{1}^{\bar \iota}, \]

for every $\iota \in \mathcal {E}_\mathbb {K}$, where $\bar \iota$ denotes the composition of $\iota$ with the complex conjugation. We suppose that $\mu$ is pure. Denote by $\Omega (\mu )$ the set of isomorphism classes of irreducible Casselman–Wallach representations $\pi _\mu$ of ${\mathrm {GL}}_{n}(\mathbb {K})$ such that:

• – $\pi _\mu$ is generic, and essentially unitarizable in the sense that $\pi _\mu \otimes \chi '$ is unitarizable for some character $\chi '$ of ${\mathrm {GL}}_n(\mathbb {K})$; and

• – the total continuous cohomology

  \[ \operatorname{H}_{\mathrm{ct}}^*({\mathrm{GL}}_n(\mathbb{K})^0; F_\mu^\vee\otimes \pi_\mu)\neq \{0\}. \]

We remark that no such $\pi _\mu$ exists when $\mu$ is not pure (see [Reference ClozelClo90, Lemma 4.9]).

By [Reference ClozelClo90, § 3],

(6)\begin{equation} \#(\Omega(\mu))=\begin{cases} 2, & \hbox{if } \mathbb{K}\cong {\mathbb{R}} \hbox{ and } n \hbox{ is odd;}\\ 1, & \hbox{otherwise.} \end{cases} \end{equation}

Write $\operatorname {sgn}_{\mathbb {K}^\times }:\mathbb {K}^\times \rightarrow \mathbb {C}^\times$ for the quadratic character that is nontrivial if and only if $\mathbb {K}\cong \mathbb {R}$, and define the sign character

\[ \operatorname{sgn}:=\operatorname{sgn}_{\mathbb{K}^\times}\circ\det \]

of a general linear group ${\mathrm {GL}}_n(\mathbb {K})$. Then in the first case of (6) the two members of $\Omega (\mu )$ are twists of each other by the sign character, and in the second case of (6) the only representation in $\Omega (\mu )$ is isomorphic to its own twist by the sign character. Recall that by [Reference ClozelClo90, Lemma 3.14],

\[ \operatorname{H}_{\mathrm{ct}}^i({\mathrm{GL}}_n(\mathbb{K})^0; F_\mu^\vee\otimes \pi_\mu)=\{0\},\quad \textrm{if $i< b_{n,\mathbb{K}}$,} \]

and

\[ \operatorname{H}_{\mathrm{ct}}^{b_{n,\mathbb{K}}}({\mathrm{GL}}_n(\mathbb{K})^0; F_\mu^\vee\otimes \pi_\mu) \cong \begin{cases} 1_{\mathbb{K}^\times}\oplus \operatorname{sgn}_{\mathbb{K}^\times}, & \hbox{if } \mathbb{K}\cong {\mathbb{R}} \hbox{ and } n \hbox{ is even,}\\ \varepsilon_{\pi_\mu}, & \hbox{otherwise,} \end{cases} \]

as representations of $\pi _0(\mathbb {K}^\times )$, where $1_{\mathbb {K}^\times }$ denotes the trivial character of $\mathbb {K}^\times$, and $\varepsilon _{\pi _\mu }$ denotes the central character of $F_\mu ^\vee \otimes \pi _\mu$. Here and henceforth we make the identification

\[ \pi_0({\mathrm{GL}}_n(\mathbb{K})) = \pi_0(\mathbb{K}^\times) \quad (\pi_0 \textrm{ indicates the set of connected components}) \]

through the determinant map ${\mathrm {GL}}_n(\mathbb {K})\to \mathbb {K}^\times$. Note that $\varepsilon _{\pi _\mu }$ is equal to either $1_{\mathbb {K}^\times }$ or $\operatorname {sgn}_{\mathbb {K}^\times }$ for $\mathbb {K}\cong {\mathbb {R}}$ and $n$ odd, and is trivial otherwise.

For every commutative ring $R$, let $\mathrm {B}_n(R)$ be the subgroup of ${\mathrm {GL}}_n(R)$ consisting of all the upper triangular matrices, and let $\mathrm {N}_n(R)$ be the subgroup of matrices in $\mathrm B_n(R)$ whose diagonal entries are $1$. Likewise let $\bar {\mathrm {B}}_n(R)$ be the subgroup of ${\mathrm {GL}}_n(R)$ consisting of all the lower triangular matrices, and let $\bar {\mathrm {N}}_n(R)$ be the subgroup of matrices in $\bar {\mathrm {B}}_n(R)$ whose diagonal entries are $1$. Let $\mathrm {T}_n(R)$ be the subgroup of diagonal matrices in ${\mathrm {GL}}_n(R)$.

Note that the invariant spaces $(F_\mu )^{\mathrm {N}_n(\mathbb {K}\otimes _\mathbb {R} \mathbb {C})}$ and $(F_\mu ^\vee )^{\bar {\mathrm {N}}_n(\mathbb {K}\otimes _\mathbb {R} \mathbb {C})}$ are one-dimensional. We shall fix a generator $v_\mu \in (F_\mu )^{\mathrm {N}_n(\mathbb {K}\otimes _\mathbb {R} \mathbb {C})}$ and a generator $v_\mu ^\vee \in (F_\mu ^\vee )^{\bar {\mathrm {N}}_n(\mathbb {K}\otimes _\mathbb {R} \mathbb {C})}$ such that their pairing

(7)\begin{equation} \langle v_\mu, v_\mu^\vee\rangle =1. \end{equation}

To be more concrete, by the Borel–Weil–Bott theorem [Reference BottBot57] we can realize $F_\mu$ as the algebraic induction

(8)\begin{equation} F_\mu= {}^{\rm alg} {\mathrm{Ind}}^{{\mathrm{GL}}_n(\mathbb{K}\otimes_\mathbb{R} \mathbb{C})}_{\bar{\mathrm{B}}_n(\mathbb{K}\otimes_\mathbb{R} \mathbb{C})}\chi_\mu, \end{equation}

which consists of all algebraic functions $f: {\mathrm {GL}}_n(\mathbb {K}\otimes _\mathbb {R} \mathbb {C})\to \mathbb {C}$ such that

\[ f(\bar bg)=\chi_\mu(\bar b)f(g) \quad \textrm{for all $\bar b\in \bar{\mathrm{B}}_n(\mathbb{K}\otimes_\mathbb{R} \mathbb{C})$ and $g\in {\mathrm{GL}}_n(\mathbb{K}\otimes_\mathbb{R} \mathbb{C})$}. \]

Here $\chi _\mu = \otimes _{\iota \in {{\mathcal {E}}_\mathbb {K}}}\chi _{\mu ^{\iota }}$ denotes the algebraic character of $\mathrm {T}_n(\mathbb {K}\otimes _\mathbb {R} \mathbb {C})$ corresponding to the weight $\mu \in (\mathbb {Z}^n)^{{\mathcal {E}}_\mathbb {K}}$, to be viewed as an algebraic character of $\bar {\mathrm {B}}_n(\mathbb {K}\otimes _\mathbb {R} \mathbb {C})$ as usual. Then we realize $v_\mu$ as the ${\mathrm {N}}_n(\mathbb {K}\otimes _\mathbb {R} \mathbb {C})$-invariant algebraic function $f$ in $F_\mu$ such that

\[ f(1_n)=1\quad (1_n\textrm{ denotes the identity element of ${\mathrm{GL}}_n(\mathbb{K}\otimes_\mathbb{R} \mathbb{C})$}). \]

Similarly, we realize $F_\mu ^\vee$ as the algebraic induction

\[ F_\mu^\vee= {}^{\rm alg} {\mathrm{Ind}}^{{\mathrm{GL}}_n(\mathbb{K}\otimes_\mathbb{R} \mathbb{C})}_{{\mathrm{B}}_n(\mathbb{K}\otimes_\mathbb{R} \mathbb{C})}\chi_{-\mu}, \]

and realize $v_\mu ^\vee$ as the ${\bar {\mathrm {N}}}_n(\mathbb {K}\otimes _\mathbb {R} \mathbb {C})$-invariant algebraic function $f^\vee$ in $F_\mu ^\vee$ such that $f^\vee (1_n)=1$. The invariant pairing $\langle \,,\,\rangle : F_\mu \times F_\mu ^\vee \rightarrow \mathbb {C}$ is determined by the equality (7). Note that as a linear functional on $F_\mu ^\vee$, $v_\mu$ equals the evaluation map at $1_n$. Similarly, $v_\mu ^\vee$ equals the evaluation map at $1_n$ as a linear functional on $F_\mu$.

Fix a unitary character

(9)\begin{equation} \psi_\mathbb{R}: \mathbb{R}\rightarrow \mathbb{C}^\times,\quad x\mapsto e^{2\pi \mathrm{i}x}, \end{equation}

which induces a unitary character

(10)\begin{equation} \psi_\mathbb{K}: \mathbb{K}\rightarrow \mathbb{C}^\times, \quad x\mapsto \psi_\mathbb{R}\bigg(\sum_{\iota\in \mathcal{E}_\mathbb{K}} \iota(x)\bigg). \end{equation}

This further induces a unitary character

(11)\begin{equation} \psi_{n,\mathbb{K}}: \mathrm{N}_n(\mathbb{K})\rightarrow \mathbb{C}^\times, \quad [x_{i,j}]_{1\leqslant i,j\leqslant n} \mapsto \psi_\mathbb{K} \bigg((-1)^n\cdot \sum_{i=1}^{n-1} x_{i,i+1}\bigg). \end{equation}

By abuse of notation, we will still use $\psi _{n,\mathbb {K}}$ to denote the space ${\mathbb {C}}$ carrying the representation of $\mathrm {N}_n(\mathbb {K})$ corresponding to the character $\psi _{n,\mathbb {K}}$. Similar notation will be freely used for other characters. Let $\pi _\mu \in \Omega (\mu )$. Recall that the space ${\mathrm {Hom}}_{ \mathrm {N}_n(\mathbb {K})}(\pi _\mu, \psi _{n,\mathbb {K}})$ is one-dimensional. Fix a generator

(12)\begin{equation} \lambda_\mu\in {\mathrm{Hom}}_{ \mathrm{N}_n(\mathbb{K})}(\pi_\mu, \psi_{n,\mathbb{K}}), \end{equation}

to be called the Whittaker functional on $\pi _\mu$.

Write $0_{n,\mathbb {K}}$ for the zero element of $(\mathbb {Z}^n)^{\mathcal {E}_\mathbb {K}}$. Then $F_{0_{n,\mathbb {K}}}$ is the trivial representation. Specifying the above argument to the case when $\mu =0_{n,\mathbb {K}}$, we take a representation $\pi _{0_{n,\mathbb {K}}}\in \Omega (0_{n,\mathbb {K}})$, together with the Whittaker functional $\lambda _{0_{n,\mathbb {K}}}\in {\mathrm {Hom}}_{ \mathrm {N}_n(\mathbb {K})}(\pi _{0_{n,\mathbb {K}}}, \psi _{n,\mathbb {K}})\setminus \{0\}$.

Throughout this article, we assume that the representation $\pi _{0_{n,\mathbb {K}}} \in \Omega (0_{n,\mathbb {K}})$ is chosen such that $\pi _{0_{n,\mathbb {K}}}$ and $F_\mu ^\vee \otimes \pi _\mu$ have the same central character, to be denoted by $\varepsilon _{n,\mathbb {K}}$.

2.2 Explicit translations

We will prove the following result in this subsection.

Proposition 2.1 There is a unique element $\jmath _\mu \in {\mathrm {Hom}}_{{\mathrm {GL}}_n(\mathbb {K})}(\pi _{0_{n,\mathbb {K}}}, F_\mu ^\vee \otimes \pi _\mu )$ such that the following diagram commutes.

Moreover, $\jmath _\mu$ induces a linear isomorphism

\[ \jmath_\mu : \operatorname{H}_{\mathrm{ct}}^i({\mathrm{GL}}_n(\mathbb{K})^0; \pi_{0_{n,\mathbb{K}}})\xrightarrow{\sim} \operatorname{H}_{\mathrm{ct}}^i({\mathrm{GL}}_n(\mathbb{K})^0; F_\mu^\vee\otimes \pi_\mu) \]

of representations of $\pi _0(\mathbb {K}^\times )$ for each $i\in \mathbb {Z}$.

It is known from the Vogan–Zuckerman theory of cohomological representations (see Proposition 1.2 and § 5 of [Reference Vogan and ZuckermanVZ84]) that

(13)\begin{equation} \dim {\mathrm{Hom}}_{{\mathrm{GL}}_n(\mathbb{K})}(\pi_{0_{n,\mathbb{K}}}, F_\mu^\vee\otimes \pi_\mu)=1. \end{equation}

We first recall the realization of $\pi _\mu$ and introduce a certain principal series representation $I_\mu$ of ${\mathrm {GL}}_n(\mathbb {K})$. Define a character

\[ \rho_n:=\otimes^n_{i=1}\lvert\, \cdot \,\rvert_\mathbb{K}^{(n+1)/2-i} \quad (\lvert\,\cdot\,\rvert_\mathbb{K} \textrm{ denotes the normalized absolute value}) \]

of $\mathrm {T}_n(\mathbb {K})$. For $\iota \in {\mathcal {E}}_\mathbb {K}$, define the half-integers

(14)\begin{equation} \tilde{\mu}_i^\iota : = \mu_i^\iota + \frac{n+1}{2}-i, \quad i =1, \ldots, n. \end{equation}

Then $\{(\tilde {\mu }^\iota _1,\tilde \mu ^\iota _2,\ldots, \tilde \mu ^\iota _n)\}_{\iota \in {\mathcal {E}}_\mathbb {K}}$ is the infinitesimal character of the algebraic representation $F_\mu$.

For $\mathbb {K} \cong \mathbb {R}$, $a, b\in {\mathbb {C}}$ with $a-b\in {\mathbb {Z}}\setminus \{0\}$, denote by $D_{a, b}$ the essentially square-integrable irreducible Casselman–Wallach representation of ${\mathrm {GL}}_2(\mathbb {K})$ with infinitesimal character $(a,b)$. Note that such a representation is unique up to isomorphism. In this article we use the notation $\widehat {\otimes }$ to denote the completed projective tensor product of locally convex topological vector spaces (see [Reference TrèvesTrè67, Definition 43.5]).

If $n$ is even, then

\[ \pi_\mu\cong {\mathrm{Ind}}^{{\mathrm{GL}}_n(\mathbb{K})}_{\bar{{\mathrm{P}}}_n(\mathbb{K})}\big(D_{\tilde{\mu}^\iota_1, \tilde{\mu}^\iota_n}\widehat{\otimes} \cdots \widehat{\otimes} D_{\tilde{\mu}^\iota_{{n}/{2}}, \tilde{\mu}^\iota_{n/2+1}}\big) \quad (\textrm{normalized smooth induction}), \]

where $\bar {{\mathrm {P}}}_n$ is the lower triangular parabolic subgroup of type $(2,\ldots, 2)$. More precisely, the above normalized smooth induction consists of all smooth maps $f: {\mathrm {GL}}_n(\mathbb {K}) \to D_{\tilde {\mu }^\iota _1, \tilde {\mu }^\iota _n}\widehat {\otimes } \cdots \widehat {\otimes } D_{\tilde {\mu }^\iota _{n/{2}}, \tilde {\mu }^\iota _{{n}/{2}+1}}$ such that

(15)\begin{equation} f(\bar p g) = \delta_{\bar{\mathrm{P}}_n}^{1/2}(\bar p)\cdot (\bar p\cdot(f(g)))\quad \textrm{for all $\bar p\in \bar{\mathrm{P}}_n$ and $g\in {\mathrm{GL}}_n(\mathbb{K})$,} \end{equation}

where $\delta _{\bar {\mathrm {P}}_n}$ denotes the modular character of $\bar {\mathrm {P}}_n$. Thereafter ${\mathrm {Ind}}$ always denotes the normalized smooth induction which is similarly defined as above.

If $n$ is odd, then

\[ \pi_\mu\cong {\mathrm{Ind}}^{{\mathrm{GL}}_n(\mathbb{K})}_{\bar{{\mathrm{P}}}_n(\mathbb{K})}\big(D_{\tilde{\mu}^\iota_1, \tilde{\mu}^\iota_n}\widehat{\otimes} \cdots \widehat{\otimes} D_{\tilde{\mu}^\iota_{(n-1)/2}, \tilde{\mu}^\iota_{(n+3)/2}}\otimes (\cdot)^{\tilde{\mu}^\iota_{(n+1)/2}}\varepsilon_{n,\mathbb{K}} \big), \]

where $\bar {{\mathrm {P}}}_n$ is the lower triangular parabolic subgroup of type $(2,\ldots, 2, 1)$, and we recall that $\varepsilon _{n,\mathbb {K}}=1_{\mathbb {K}^\times }$ or $\operatorname {sgn}_{\mathbb {K}^\times }$ is the common central character of $F_\mu ^\vee \otimes \pi _\mu$ and $\pi _{0_{n,\mathbb {K}}}$.

For $\mathbb {K}\cong \mathbb {C}$, we have that

\[ \pi_\mu\cong {\mathrm{Ind}}^{{\mathrm{GL}}_n(\mathbb{K})}_{\bar{\mathrm{B}}_n(\mathbb{K})} \big(\iota^{\tilde{\mu}^\iota_1} \bar{\iota}^{\tilde{\mu}^{\bar{\iota}}_n}\otimes\cdots\otimes \iota^{\tilde{\mu}^\iota_n} \bar{\iota}^{\tilde{\mu}^{\bar{\iota}}_1}\big), \]

where for $a, b\in {\mathbb {C}}$ with $a-b\in {\mathbb {Z}}$, $\iota ^a \bar {\iota }^b$ denotes the character

\[ \iota^a \bar{\iota}^b: \mathbb{K}^\times \to {\mathbb{C}}^\times, \quad z\mapsto \iota(z)^{a-b} (\iota(z)\bar{\iota}(z))^b. \]

In both the real and complex cases, we define the principal series representation

\[ I_\mu := {\mathrm{Ind}}^{{\mathrm{GL}}_n(\mathbb{K})}_{\bar{\mathrm{B}}_n(\mathbb{K})} (\chi_\mu \cdot \rho_n\cdot ( \varepsilon_{n,\mathbb{K}}\circ \det)), \]

so that $I_\mu$ and $\pi _\mu$ have the same central character.

Proof. By [Reference JacquetJac09, Lemma 2.5], $I_\mu ^\vee$ has a unique irreducible subrepresentation, which is also the unique generic irreducible subquotient. This implies the first statement of the lemma.

For $\mathbb {K}\cong {\mathbb {R}}$, by the well-known realization of essentially square-integrable representations of ${\mathrm {GL}}_2({\mathbb {R}})$ as quotients of principal series representations, $\pi _\mu$ is a quotient of

\[ {\mathrm{Ind}}^{{\mathrm{GL}}_n(\mathbb{K})}_{\bar{\mathrm{B}}_n(\mathbb{K})} (w(\chi_\mu\cdot \rho_n)\cdot( \varepsilon_{n,\mathbb{K}}\circ \det)) \]

for a certain $w\in W_n$. Here $W_n$ is the subgroup of permutation matrices in $\textrm {GL}_n({\mathbb {Z}})$ which is identified with the Weyl group and acts on $\mathrm {T}_n(\mathbb {K})$ by conjugation, and thus it acts on the set of characters of $\mathrm {T}_n(\mathbb {K})$. The above representation and $I_\mu$ have the same irreducible constituents. Hence, $\pi _\mu$ is isomorphic to the unique generic irreducible subquotient of $I_\mu$, which is in fact a quotient as we mentioned at the beginning of the proof.

For $\mathbb {K}\cong {\mathbb {C}}$, the lemma follows easily from [Reference Jacquet and LanglandsJL70, Theorem 6.2] (for the case of ${\mathrm {GL}}_2({\mathbb {C}})$) and parabolic induction in stages.

The group ${\mathrm {N}}_n(\mathbb {K})$ is equipped with the Haar measure

(16)\begin{equation} \operatorname{d}\! u:=\prod_{1\leqslant i< j\leqslant n} \,\operatorname{d}\!u_{i,j}, \quad u=[u_{i,j}]_{1\leqslant i,j\leqslant n}\in {\mathrm{N}}_n(\mathbb{K}), \end{equation}

where $\operatorname {d}\!u_{i,j}$ is the self-dual Haar measure on $\mathbb {K}$ with respect to $\psi _\mathbb {K}$. By [Reference WallachWal92, Theorem 15.4.1],

(17)\begin{equation} \dim {\mathrm{Hom}}_{\mathrm{N}_n(\mathbb{K})}(I_\mu, \psi_{n,\mathbb{K}}) =1 \end{equation}

and there is a unique $\lambda _\mu '\in {\mathrm {Hom}}_{\mathrm {N}_n(\mathbb {K})}(I_\mu, \psi _{n,\mathbb {K}})$ such that

(18)\begin{equation} \lambda_\mu'(f)=\int_{\mathrm{N}_n(\mathbb{K})} f(u)\overline{\psi_{n,\mathbb{K}}}(u)\operatorname{d}\! u \end{equation}

for all $f\in I_\mu$ such that $f|_{\mathrm {N}_n(\mathbb {K})}\in {\mathcal {S}}(\mathrm {N}_n(\mathbb {K}))$. Here and henceforth, for a Nash manifold $X$, denote by ${\mathcal {S}}(X)$ the space of Schwartz functions on $X$ (see [Reference Du ClouxDC91, Reference Aizenbud and GourevitchAG08]).

As usual, an element $u\otimes f\in F_\mu ^\vee \otimes I_\mu$ is identified with the function

\[ {\mathrm{GL}}_n(\mathbb{K})\rightarrow F_\mu^\vee, \quad g\mapsto f(g) \cdot u. \]

Then $F_\mu ^\vee \otimes I_\mu$ is identified with the space of $F_\mu ^\vee$-valued smooth functions $\varphi$ on ${\mathrm {GL}}_n(\mathbb {K})$ satisfying that

\[ \varphi(bx)=\big(((\varepsilon_{n,\mathbb{K}}\circ \det)\cdot \chi_\mu)(b)\big)\cdot \varphi(x),\quad \textrm{for all }b\in \bar{\mathrm{B}}_n(\mathbb{K}),\ x\in {\mathrm{GL}}_n(\mathbb{K}), \]

on which ${\mathrm {GL}}_n(\mathbb {K})$ acts by

\[ (g . \varphi)(x):=g. (\varphi(xg)),\quad \textrm{where } \, g, \, x \in {\mathrm{GL}}_n(\mathbb{K}). \]

Define a ${\mathrm {GL}}_n(\mathbb {K})$-homomorphism

(19)\begin{equation} \begin{array}{rcl} \imath_\mu:I_{0_{n,\mathbb{K}}} & \rightarrow & F_\mu^\vee \otimes I_\mu,\\ f & \mapsto & \big(g\mapsto f(g)\cdot (g^{-1}. v_\mu^\vee)\big). \end{array} \end{equation}

Lemma 2.3 The map $\imath _\mu$ satisfies that

(20)\begin{equation} (v_\mu \otimes \lambda_\mu' ) \circ \imath_\mu = \lambda_{0_{n,\mathbb{K}}}'. \end{equation}

Proof. Recall that $v_\mu \in F_\mu$ is ${\mathrm {N}}_n(\mathbb {K}\otimes _{\mathbb {R}} \mathbb {C})$-invariant, and $\langle v_\mu, v_\mu ^\vee \rangle =1$. For $f\in I_{0_n,\mathbb {K}}$ with $f|_{\mathrm {N}_n(\mathbb {K})}\in {\mathcal {S}}(\mathrm {N}_n(\mathbb {K}))$, we have that

\begin{align*} \big((v_\mu \otimes \lambda_\mu' ) \circ \imath_\mu\big)(f) & = \int_{\mathrm{N}_n(\mathbb{K})} \langle v_\mu, \imath_\mu(f)(u)\rangle \overline{\psi_{n,\mathbb{K}}}(u)\operatorname{d}\! u \\ & = \int_{\mathrm{N}_n(\mathbb{K})} f(u)\overline{\psi_{n,\mathbb{K}}}(u)\operatorname{d}\! u \\ & = \lambda_{0_{n,\mathbb{K}}}'(f). \end{align*}

This proves (20), in view of [Reference WallachWal92, Theorem 15.4.1].

By Lemma 2.2 and (17), there is a unique $p_\mu \in {\mathrm {Hom}}_{{\mathrm {GL}}_n(\mathbb {K})}(I_\mu, \pi _\mu )$ such that

(21)\begin{equation} \lambda_\mu\circ p_\mu=\lambda_\mu'. \end{equation}

Let $J_\mu :={\mathrm {Ker}}(p_\mu )$, which is the largest subrepresentation of $I_\mu$ such that

\[ \textrm{Dim} \, J_\mu < \textrm{Dim}\, I_\mu. \]

Here and below, $\textrm {Dim}$ indicates the Gelfand–Kirillov dimension of a Casselman–Wallach representation of ${\mathrm {GL}}_n(\mathbb {K})$. Likewise, we have $J_{0_{n,\mathbb {K}}}:={\mathrm {Ker}}(p_{0_{n,\mathbb {K}}})\subset I_{0_{n,\mathbb {K}}}$.

Lemma 2.4 It holds that

\[ \imath_\mu(J_{0_{n,\mathbb{K}}})\subset F_\mu^\vee \otimes J_\mu. \]

Proof. It suffices to show that

\[ \tilde{\imath}_\mu(F_\mu \otimes J_{0_{n, \mathbb{K}}} )\subset J_\mu, \]

where $\tilde {\imath }_\mu \in {\mathrm {Hom}}_{{\mathrm {GL}}_n(\mathbb {K})}(F_\mu \otimes I_{0_{n, \mathbb {K}}}, I_\mu )$ is the linear map induced by $\iota _\mu$. This follows from the fact that (see [Reference VoganVog78, Lemma 2.2])

\[ \textrm{Dim} \, ( F_\mu \otimes J_{0_{n, \mathbb{K}}} ) = \textrm{Dim} \, J_{0_{n, \mathbb{K}}}. \]

□

By Lemma 2.4, there is a unique $\jmath _\mu \in {\mathrm {Hom}}_{{\mathrm {GL}}_n(\mathbb {K})}(\pi _{0_{n,\mathbb {K}}}, F_\mu ^\vee \otimes \pi _\mu )$ such that the following diagram commutes.

(22)

By (20) and (21),

\begin{align*} (v_\mu\otimes \lambda_\mu)\circ \jmath_\mu\circ p_{0_{n,\mathbb{K}}} & = (v_\mu\otimes \lambda_\mu)\circ ( {\rm id}_{F_\mu^\vee}\otimes p_\mu) \circ \imath_\mu \\ & = (v_\mu \otimes \lambda_\mu')\circ \imath_\mu \\ & = \lambda'_{0_{n,\mathbb{K}}} \\ & = \lambda_{0_{n, \mathbb{K}}}\circ p_{0_{n, \mathbb{K}}}, \end{align*}

which implies that

\[ (v_\mu\otimes \lambda_\mu)\circ \jmath_\mu = \lambda_{0_{n, \mathbb{K}}}. \]

This proves the existence part of Proposition 2.1. The uniqueness follows from (13). The last statement of the proposition follows from [Reference Vogan and ZuckermanVZ84, § 5].

3.1 Some cohomology spaces

For simplicity, write

\[ \operatorname{H}_\mu:= \operatorname{H}_{\mathrm{ct}}^{b_{n,\mathbb{K}}}({\mathrm{GL}}_n(\mathbb{K})^0; F_\mu^\vee\otimes \pi_\mu), \]

which is of dimension 1 or 2. As in Proposition 2.1, we have a linear isomorphism

\[ \jmath_\mu :\operatorname{H}_{0_{n,\mathbb{K}}} \xrightarrow{\sim} \operatorname{H}_\mu \]

of representations of $\pi _0(\mathbb {K}^\times )$.

Fix a maximal compact subgroup

(23)\begin{equation} K_{n,\mathbb{K}} :=\begin{cases} {\mathrm{O}}(n), & {\rm if\ }\mathbb{K}\cong{\mathbb{R}}; \\[-1.5pt] {\mathrm{U}}(n), & {\rm if \ }\mathbb{K}\cong{\mathbb{C}} \end{cases} \end{equation}

of ${\mathrm {GL}}_n(\mathbb {K})$. The determinant homomorphism yields identifications

\[ \pi_0({\mathrm{GL}}_n(\mathbb{K}))=\pi_0(K_{n,\mathbb{K}})=\pi_0(\mathbb{K}^\times). \]

We use the corresponding lowercase Gothic letter to denote the Lie algebra of a Lie group. For example, the Lie algebra of $K_{n,\mathbb {K}}$ will be denoted by $\frak {k}_{n,\mathbb {K}}$. Put

\[ d_{n,\mathbb{K}}:=b_{n+1,\mathbb{K}}+b_{n,\mathbb{K}}=\dim_\mathbb{R} (\mathfrak{g}\mathfrak{l}_n(\mathbb{K})/\mathfrak{k}_{n,\mathbb{K}})= \begin{cases} \dfrac{n(n+1)}{2}, & \hbox{if } \mathbb{K}\cong \mathbb{R};\\[5pt] n^2, & \hbox{if } \mathbb{K}\cong \mathbb{C}. \end{cases} \]

Define a one-dimensional real vector space

\[ \omega_{n,\mathbb{K}}(\mathbb{R}):=\wedge^{d_{n,\mathbb{K}}} (\mathfrak{g}\mathfrak{l}_n(\mathbb{K})/\mathfrak{k}_{n,\mathbb{K}}). \]

Put

\[ \omega_{n,\mathbb{K}}:=\omega_{n,\mathbb{K}}(\mathbb{R})\otimes_\mathbb{R} \mathbb{C}, \]

which is naturally a representation of $\pi _0(\mathbb {K}^\times )$ that is isomorphic to $\operatorname {sgn}_{\mathbb {K}^\times }^{n-1}$. Here and henceforth, we also view $1_{\mathbb {K}^\times }$ and $\operatorname {sgn}_{\mathbb {K}^\times }$ as representations of $\pi _0(\mathbb {K}^\times )$. Then there is an identification

(24)\begin{equation} \operatorname{H}_{\mathrm{ct}}^{d_{n,\mathbb{K}}}({\mathrm{GL}}_n(\mathbb{K})^0; \omega_{n,\mathbb{K}})=1_{\mathbb{K}^\times} \end{equation}

of representations of $\pi _0(\mathbb {K}^\times )$.

Write $\omega _{n,\mathbb {K}}^+$ and $\omega _{n,\mathbb {K}}^-$ for the two connected components of $\omega _{n,\mathbb {K}}(\mathbb {R})\setminus \{0\}$, which are viewed as left invariant orientations on ${\mathrm {GL}}_n(\mathbb {K})/K_{n,\mathbb {K}}^0$. The complex orientation space of $\omega _{n,\mathbb {K}}(\mathbb {R})$ is defined to be the one-dimensional space

(25)\begin{equation} \mathfrak{O}_{n,\mathbb{K}}:=\frac{\mathbb{C}\cdot \omega_{n,\mathbb{K}}^+ \oplus \mathbb{C}\cdot \omega_{n,\mathbb{K}}^- }{\{a( \omega_{n,\mathbb{K}}^+ + \omega_{n,\mathbb{K}}^-)\, : \, a\in \mathbb{C}\}}. \end{equation}

Then $\pi _0(\mathbb {K}^\times )=\pi _0(K_{n,\mathbb {K}})$ acts on $\mathfrak {O}_{n,\mathbb {K}}$ by $\operatorname {sgn}_{\mathbb {K}^\times }^{n-1}$, through the right translation on ${\mathrm {GL}}_n(\mathbb {K})/K_{n,\mathbb {K}}^0$. We identify $\omega _{n,\mathbb {K}}^*\otimes \mathfrak {O}_{n,\mathbb {K}}$ with the space of invariant measures on ${\mathrm {GL}}_n(\mathbb {K})/ K_{n,\mathbb {K}}^0$ in the obvious way. Here and as usual, a superscript $*$ over a vector space indicates the dual space. Denote by $\frak {M}_{n,\mathbb {K}}$ the one-dimensional space of invariant measures on ${\mathrm {GL}}_n(\mathbb {K})$. By push-forward of measures through the map ${\mathrm {GL}}_n(\mathbb {K})\to {\mathrm {GL}}_n(\mathbb {K})/ K_{n,\mathbb {K}}^0$, we have an identification

\[ \frak{M}_{n,\mathbb{K}} = \omega_{n,\mathbb{K}}^*\otimes \mathfrak{O}_{n,\mathbb{K}}. \]

In view of this and (24), we have that

\[ \operatorname{H}_{\mathrm{ct}}^{d_{n,\mathbb{K}}}({\mathrm{GL}}_n(\mathbb{K})^0; \mathfrak{M}_{n,\mathbb{K}}^*) \otimes \frak{O}_{n,\mathbb{K}}= {\mathbb{C}}. \]

3.2 Archimedean modular symbols and archimedean period relations

Now we assume that $n\geqslant 2$. Let $\nu \in (\mathbb {Z}^{n-1})^{\mathcal {E}_\mathbb {K}}$ be a highest weight and assume that it is pure. Then, as before, we have representations $F_\nu$ and $\pi _\nu$ of ${\mathrm {GL}}_{n-1}(\mathbb {K}\otimes _{\mathbb {R}}{\mathbb {C}})$ and ${\mathrm {GL}}_{n-1}(\mathbb {K})$, respectively, an element $v_\nu \in F_\nu$, an element $v_\nu ^\vee \in F_\nu ^\vee$, and a Whittaker functional $\lambda _\nu$ on $\pi _\nu$. The representation $\pi _{0_{n-1,\mathbb {K}}}$ is determined by $\pi _\nu$ as before, and we have a linear isomorphism

\[ \jmath_\nu :\operatorname{H}_{0_{n-1,\mathbb{K}}} \xrightarrow{\sim} \operatorname{H}_\nu \]

of representations of $\pi _0(\mathbb {K}^\times )$.

As usual, we have an embedding

(26)\begin{equation} \imath: {\mathrm{GL}}_{n-1}(R)\hookrightarrow {\mathrm{GL}}_n(R),\quad g\mapsto \left[ \begin{array}{@{}cc@{}} g & 0 \\ 0 & 1 \\ \end{array} \right], \end{equation}

where $R$ is an arbitrary commutative ring. We also view ${\mathrm {GL}}_{n-1}(R)$ as a subgroup of ${\mathrm {GL}}_n(R)\times {\mathrm {GL}}_{n-1}(R)$ via the diagonal embedding

(27)\begin{equation} {\mathrm{GL}}_{n-1}(R) \hookrightarrow {\mathrm{GL}}_n(R)\times {\mathrm{GL}}_{n-1}(R), \quad g\mapsto \bigg(\begin{bmatrix} g & 0 \\ 0 & 1 \end{bmatrix}, g \bigg). \end{equation}

For all $k,l\in {\mathbb {N}}$, denote by $R^{k\times l}$ the set of $k\times l$ matrices with entries in $R$.

Recall the pure weight $\mu \in (\mathbb {Z}^n)^{{\mathcal {E}}_\mathbb {K}}$ from § 2.1. Put $\xi :=(\mu,\nu )$. Write $F_\xi :=F_\mu \otimes F_\nu$. Assume that $\xi$ is balanced in the sense that there is an integer $j$ such that

(28)\begin{equation} {\mathrm{Hom}}_{{\mathrm{GL}}_{n-1}(\mathbb{K}\otimes_\mathbb{R} \mathbb{C})}(F_\xi^\vee, \otimes_{\iota\in \mathcal{E}_\mathbb{K}}{\det}^j) \neq 0. \end{equation}

For each $k\in {\mathbb {N}}$, write

\[ w_k:=\left[ \begin{array}{@{}cccc@{}} 0 & \cdots & 0 & 1 \\ 0 & \cdots & 1 & 0 \\ & \cdots & \cdots & \\ 1 & 0 & \cdots & 0 \\ \end{array} \right]\in {\mathrm{GL}}_k({\mathbb{Z}}). \]

Following [Reference Li, Liu, Su and SunLLSS23], define a series $\{z_k\in {\mathrm {GL}}_k({\mathbb {Z}})\}_{k\in {\mathbb {N}}}$ of matrices inductively by

\[ z_0:=\varnothing\ (\textrm{the unique element of ${\mathrm{GL}}_0({\mathbb{Z}})$}), \quad z_1:=[1], \]

and

(29)\begin{equation} z_k := \left[ \begin{array}{@{}cc@{}} w_{k-1} & 0 \\ 0 & 1 \\ \end{array} \right] \left[ \begin{array}{@{}cc@{}} {}^t z_{k-2}^{-1} & 0 \\ 0 & 1_2 \\ \end{array} \right] \left[ \begin{array}{@{}cc@{}} {}^tz_{k-1} w_{k-1} z_{k-1} & {}^t e_{k-1}\\ 0 & 1 \\ \end{array} \right], \quad \textrm{for all $k\geqslant~2$.} \end{equation}

Here, and as usual, a left superscript $t$ over a matrix indicates the transpose, $1_2$ stands for the $2\times 2$ identity matrix, and $e_{k-1}:=[0,\ldots, 0,1]\in {\mathbb {Z}}^{1\times (k-1)}$.

Let $j\in \mathbb {Z}$ be as in (28). The following proposition follows from the fact that

\[ \big(\bar{\mathrm{B}}_n(\mathbb{C}) \times \bar{\mathrm{B}}_{n-1}(\mathbb{C})\big) \cdot (z_n, z_{n-1}) \cdot {\mathrm{GL}}_{n-1}({\mathbb{C}}) \]

is Zariski open in ${\mathrm {GL}}_n(\mathbb {C}) \times {\mathrm {GL}}_{n-1}(\mathbb {C})$ (see [Reference Li, Liu, Su and SunLLSS23, Lemma 1.1]).

Proposition 3.1 There is a unique element

\[ \phi_{\xi,j}\in{\mathrm{Hom}}_{{\mathrm{GL}}_{n-1}(\mathbb{K}\otimes_\mathbb{R} \mathbb{C})}(F_\xi^\vee , \otimes_{\iota\in \mathcal{E}_\mathbb{K}}{\det}^j) \]

such that

\[ \phi_{\xi,j}((z_n^{-1} . v_\mu^\vee)\otimes (z_{n-1}^{-1}. v_\nu^\vee))=1. \]

Fix a quadratic character $\chi _\mathbb {K}$ of $\mathbb {K}^\times$. Define characters

(30)\begin{equation} \chi_{\mathbb{K},t}:=\chi_\mathbb{K}\cdot \lvert\,\cdot\,\rvert_\mathbb{K}^t,\quad \chi_{\mathbb{K}}^{(j)}:=\chi_{\mathbb{K}}\cdot \operatorname{sgn}_{\mathbb{K}^\times}^j\quad (t\in \mathbb{C}), \end{equation}

and, more generally,

\[ \chi_{\mathbb{K},t}^{(j)}:=\chi_{\mathbb{K}}\cdot\lvert\,\cdot\,\rvert_\mathbb{K}^t\cdot \operatorname{sgn}_{\mathbb{K}^\times}^j, \]

of the group $\mathbb {K}^\times$. When no confusion arises, for every commutative ring $R$, every character of $R^\times$ is identified with a character of ${\mathrm {GL}}_{n-1}(R)$ via the pullback through the determinant homomorphism. In particular, $\chi _{\mathbb {K},t}^{(j)}$ is also viewed as a character of ${\mathrm {GL}}_{n-1}(\mathbb {K})$.

Put

\[ \pi_\xi:= \pi_\mu\widehat{\otimes} \pi_\nu. \]

We have the normalized Rankin–Selberg integral (see [Reference JacquetJac09])

\begin{align*} \operatorname{Z}^\circ_\xi (\cdot, s, \chi_\mathbb{K})&\in {\mathrm{Hom}}_{{\mathrm{GL}}_{n-1}(\mathbb{K})}\big(\pi_\xi \otimes \frak{M}_{n-1,\mathbb{K}}, \, \chi_{\mathbb{K}, -s+{1}/{2}}\big)\\ &={\mathrm{Hom}}_{{\mathrm{GL}}_{n-1}(\mathbb{K})}\big( \pi_\xi, \chi_{\mathbb{K}, -s+{1}/{2}}\otimes \mathfrak{M}^*_{n-1,\mathbb{K}}\big), \end{align*}

such that

(31)\begin{align} \operatorname{Z}^\circ_\xi( f\otimes f' \otimes m, s, \chi_\mathbb{K}) & := \frac{1}{\operatorname{L}(s, \pi_\mu\times\pi_\nu)} \nonumber\\[4pt] & \quad \cdot \int_{\mathrm{N}_{n-1}(\mathbb{K})\backslash {\mathrm{GL}}_{n-1}(\mathbb{K})} \lambda_\mu\bigg(\begin{bmatrix} g & 0 \\ 0 & 1 \end{bmatrix}\cdot f\bigg) \lambda_{\nu} (g\cdot f') \nonumber\\[4pt] &\quad \cdot \chi_{\mathbb{K}}(\det g) \cdot \lvert\det g\rvert_\mathbb{K}^{s-{1}/{2}} \,{\rm d}\bar{m} (g) \end{align}

for all $f\in \pi _\mu$, $f' \in \pi _\nu$, $m \in \frak {M}_{n-1,\mathbb {K}}$, and $s\in {\mathbb {C}}$ with the real part ${\rm Re}(s)$ sufficiently large (it extends to all $s\in {\mathbb {C}}$ by holomorphic continuation). Here and henceforth, $\bar {m}$ is the quotient measure on ${\mathrm {N}}_{n-1}(\mathbb {K}) \backslash {\mathrm {GL}}_{n-1}(\mathbb {K})$ induced by $m$. Recall that a Haar measure on $\mathrm {N}_{n-1}(\mathbb {K})$ has been fixed as in (16).

Put

\[ {\mathrm{H}}_{\chi_\mathbb{K}^{(j)}} := {\mathrm{H}}_{\rm ct}^0( {\mathrm{GL}}_{n-1}(\mathbb{K})^0; \chi_\mathbb{K}^{(j)}). \]

Let $\phi _{\xi,j}$ be as in Proposition 3.1. Then we have a ${\mathrm {GL}}_{n-1}(\mathbb {K})$-equivariant continuous linear map

(32)\begin{align} \phi_{\xi,j}\otimes \operatorname{Z}_{\xi}^\circ(\cdot, \tfrac{1}{2}+j, \chi_\mathbb{K}): F_\xi^\vee\otimes \pi_\xi &\rightarrow \ (\otimes_{\iota \in \mathcal{E}_\mathbb{K}}{\det}^{j})\otimes (\chi_{\mathbb{K}, -j}\otimes \mathfrak{M}^*_{n-1,\mathbb{K}}) \nonumber\\ &= \chi_\mathbb{K}^{(j)} \otimes \mathfrak{M}^*_{n-1,\mathbb{K}}. \end{align}

By restriction of cohomology, this induces a linear map

\begin{align*} &\wp_{\xi, \chi_\mathbb{K}, j}: \operatorname{H}_\mu\otimes \operatorname{H}_\nu\otimes \operatorname{H}_{\chi_\mathbb{K}^{(j)}} \otimes \mathfrak{O}_{n-1,\mathbb{K}} \\ & \quad= \operatorname{H}_{\mathrm{ct}}^{d_{n-1,\mathbb{K}}}({\mathrm{GL}}_n(\mathbb{K})^0\times {\mathrm{GL}}_{n-1}(\mathbb{K})^0; F_\xi^\vee\otimes \pi_\xi\otimes \chi_\mathbb{K}^{(j)}) \otimes \frak{O}_{n-1,\mathbb{K}}\\ &\quad \rightarrow \operatorname{H}_{\mathrm{ct}}^{d_{n-1,\mathbb{K}}}({\mathrm{GL}}_{n-1}(\mathbb{K})^0; \mathfrak{M}^*_{n-1,\mathbb{K}})\otimes \mathfrak{O}_{n-1,\mathbb{K}} = {\mathbb{C}}. \end{align*}

We call this map the archimedean modular symbol, which is nonzero by the non-vanishing hypothesis that is proved in [Reference SunSun17].

Specifying the above argument to the case when $\xi =\xi _0:=(0_{n, \mathbb {K}}, 0_{n-1, \mathbb {K}})$ and $j=0$, we get a linear map (with $\chi _\mathbb {K}$ replaced by $\chi _\mathbb {K}^{(j)}$)

(33)\begin{equation} \wp_{\xi_0,\chi_\mathbb{K}^{(j)}, 0} : \operatorname{H}_{0_{n,\mathbb{K}}}\otimes \operatorname{H}_{0_{n-1,\mathbb{K}}}\otimes \operatorname{H}_{\chi_\mathbb{K}^{(j)}}\otimes \mathfrak{O}_{n-1,\mathbb{K}} \to {\mathbb{C}}. \end{equation}

The archimedean period relation is the following theorem.

Theorem 3.2 Let the notation and assumptions be as above. Let

(34)\begin{equation} \Omega_{\mu,\nu, j}':= {\mathrm{i}}^{ j (n(n-1)/2) [\mathbb{K}\, :\, {\mathbb{R}}]}\cdot c'_\mu \cdot c_\nu \cdot \varepsilon_{\mu,\nu}, \end{equation}

where

\begin{align*} & c'_\mu := \prod^{n-1}_{i=1} ((-1)^n \mathrm{i})^{(n-i)\sum_{\iota\in {\mathcal{E}}_\mathbb{K}}\mu^{\iota}_i}, \\ & c_\nu: = \prod^{n-1}_{i=1} ((-1)^n \mathrm{i})^{(n-i)\sum_{\iota \in {\mathcal{E}}_\mathbb{K}}\nu^{\iota}_i}, \quad and\\ & \varepsilon_{\mu,\nu}:=\prod_{i>k, \, i+k\leqslant n}(-1)^{\sum_{\iota\in {\mathcal{E}}_\mathbb{K}}(\mu_i^\iota+\nu_k^\iota)}. \end{align*}

Then the following diagram commutes.

4.1 Reduction to principal series representations

Recall that in § 2.2 we have defined a principal series representation $I_\mu$ with a Whittaker functional $\lambda _\mu ' \in {\mathrm {Hom}}_{\mathrm {N}_n(\mathbb {K})}(I_\mu, \psi _{n, \mathbb {K}})$, and a unique $p_\mu \in {\mathrm {Hom}}_{{\mathrm {GL}}_{n}(\mathbb {K})}(I_\mu, \pi _\mu )$ such that

\[ \lambda_\mu \circ p_\mu =\lambda_\mu'. \]

We have also defined $\imath _\mu \in {\mathrm {Hom}}_{{\mathrm {GL}}_n(\mathbb {K})}(I_{0_{n,\mathbb {K}}}, F_\mu ^\vee \otimes I_\mu )$ such that

\[ (v_\mu \otimes \lambda_\mu' ) \circ \imath_\mu = \lambda_{0_{n,\mathbb{K}}}' \]

and that the diagram (22) commutes. We have similar data for $\nu$. Put

\[ I_\xi := I_\mu\widehat \otimes I_\nu, \quad p_\xi:=p_\mu \otimes p_\nu, \quad \imath_\xi:=\imath_\mu \otimes \imath_\nu. \]

Define the normalized Rankin–Selberg integral

\begin{align*} \operatorname{Z}_\xi^{\diamond}(\cdot, s, \chi_\mathbb{K}) & = \frac{1}{\operatorname{L}(s, \pi_\mu\times\pi_\nu)} \operatorname{Z}_\xi(\cdot, s, \chi_\mathbb{K}) \\ & \in {\mathrm{Hom}}_{{\mathrm{GL}}_{n-1}(\mathbb{K})}\big(I_\xi \otimes \frak{M}_{n-1,\mathbb{K}}, \, \chi_{\mathbb{K}, -s+{1}/{2}}\big) \end{align*}

as the composition

\[ I_\xi \otimes \frak{M}_{n-1,\mathbb{K}} \xrightarrow{p_\xi\otimes{\rm id}} \pi_\xi \otimes \frak{M}_{n-1,\mathbb{K}} \xrightarrow{ \operatorname{Z}_\xi^{\circ}(\cdot, s, \chi_\mathbb{K})} \chi_{\mathbb{K}, -s+{1}/{2}}. \]

Then

\begin{align*} \operatorname{Z}^\diamond_{\xi}( f\otimes f' \otimes m, s, \chi_\mathbb{K}) &= \frac{1}{\operatorname{L}(s, \pi_\mu\times\pi_\nu)} \\ & \quad \cdot \int_{\mathrm{N}_{n-1}(\mathbb{K})\backslash {\mathrm{GL}}_{n-1}(\mathbb{K})} \lambda'_\mu\bigg(\begin{bmatrix} g & 0 \\ 0 & 1 \end{bmatrix}.f\bigg) \lambda'_{\nu} (g. f') \\ &\quad \cdot \chi_{\mathbb{K}}(\det g) \cdot \lvert\det g\rvert_\mathbb{K}^{s-{1}/{2}} \operatorname{d}\! \bar{m} (g), \end{align*}

for $f\in I_\mu$, $f' \in I_\nu$, $m \in \frak {M}_{n-1,\mathbb {K}}$, and $s\in \mathbb {C}$ with ${\rm Re}(s)$ sufficiently large.

In view of all the above, by the multiplicity one theorem [Reference Aizenbud and GourevitchAG09, Reference Sun and ZhuSZ12], there exists a unique entire function $\Xi _{\mu, \nu, j}(s)$ such that the following diagram commutes for all $s\in {\mathbb {C}}$.

(36)

In the rest of this section we compute the function $\Xi _{\mu, \nu, j}(s)$ and show that it is a nonzero constant whose inverse is equal to $\Omega '_{\mu, \nu, j}$ given by (34). The main ingredient of the computation is [Reference Li, Liu, Su and SunLLSS23].

4.2 Integral over the open orbit

For the convenience of the reader, we describe the main result of [Reference Li, Liu, Su and SunLLSS23]. Write $\widehat {\mathbb {K}^\times }$ for the set of all (unitary or not) characters of $\mathbb {K}^\times$. Let $\varrho =(\varrho _1,\ldots, \varrho _n) \in (\widehat {\mathbb {K}^\times })^n$, viewed as a character of $\bar {\mathrm {B}}_n(\mathbb {K})$ as usual, and let

\[ I(\varrho) :={\mathrm{Ind}}^{{\mathrm{GL}}_n(\mathbb{K})}_{\bar{\rm B}_n(\mathbb{K})} \varrho \]

be the corresponding principal series representation of ${\mathrm {GL}}_n(\mathbb {K})$. Similarly let $\varrho ' = (\varrho _1',\ldots, \varrho _{n-1}')\in (\widehat {\mathbb {K}^\times })^{n-1}$ and let $I(\varrho ')$ be the corresponding principal series representation of ${\mathrm {GL}}_{n-1}(\mathbb {K})$. We have a meromorphic family of unnormalized Rankin–Selberg integrals

\[ \operatorname{Z}(\cdot, s, \chi_\mathbb{K}) \in {\mathrm{Hom}}_{{\mathrm{GL}}_{n-1}(\mathbb{K})}\big(I(\varrho)\widehat{\otimes} I(\varrho') \otimes \frak{M}_{n-1,\mathbb{K}}, \, \chi_{\mathbb{K}, -s+{1}/{2}}\big) \]

such that

\begin{align*} & \operatorname{Z}( f\otimes f' \otimes m, s, \chi_\mathbb{K}) \\ &\quad = \int_{\mathrm{N}_{n-1}(\mathbb{K})\backslash {\mathrm{GL}}_{n-1}(\mathbb{K})} \lambda'_\varrho\bigg(\begin{bmatrix} g & 0 \\ 0 & 1 \end{bmatrix}.f\bigg) \lambda'_{\varrho'} (g. f') \cdot \chi_{\mathbb{K}}(\det g) \cdot \lvert\det g\rvert_\mathbb{K}^{s-{1}/{2}} \operatorname{d}\! \bar{m} (g) \end{align*}

for all $f\in I(\varrho )$, $f' \in I(\varrho ')$, $m \in \frak {M}_{n-1,\mathbb {K}}$, and $s\in \mathbb {C}$ with ${\rm Re}(s)$ sufficiently large, where $\lambda '_\varrho \in {\mathrm {Hom}}_{{\mathrm {N}}_n(\mathbb {K})}(I(\varrho ), \psi _{n,\mathbb {K}})$ and $\lambda '_{\varrho '}\in {\mathrm {Hom}}_{{\mathrm {N}}_{n-1}(\mathbb {K})}(I(\varrho '), \psi _{n-1,\mathbb {K}})$ are defined in the way similar to (18).

Let

(37)\begin{equation} z:=(z_n, z_{n-1})\in {\mathrm{GL}}_n(\mathbb{Z})\times {\mathrm{GL}}_{n-1}({\mathbb{Z}}), \end{equation}

where $z_n\in {\mathrm {GL}}_n(\mathbb {Z})$ is defined inductively in (29). The right action of ${\mathrm {GL}}_{n-1}(\mathbb {K})$ on the flag variety $(\bar {\mathrm {B}}_n(\mathbb {K}) \times \bar {\mathrm {B}}_{n-1}(\mathbb {K}) )\backslash ({\mathrm {GL}}_n(\mathbb {K}) \times {\mathrm {GL}}_{n-1}(\mathbb {K}))$ has a unique open orbit

(38)\begin{equation} \big(\big(\bar{\mathrm{B}}_n(\mathbb{K})\times \bar{\mathrm{B}}_{n-1}(\mathbb{K})\big) z\big) \cdot {\mathrm{GL}}_{n-1}(\mathbb{K}). \end{equation}

Note that

\[ I(\varrho)\widehat{\otimes} I(\varrho') = {\mathrm{Ind}}^{{\mathrm{GL}}_n(\mathbb{K})\times {\mathrm{GL}}_{n-1}(\mathbb{K})}_{\bar{\rm B}_n(\mathbb{K})\times \bar{\rm B}_{n-1}(\mathbb{K})}\varrho\otimes \varrho'. \]

Following [Reference Li, Liu, Su and SunLLSS23], we first formally define

\[ \Lambda(\cdot, s, \chi_\mathbb{K} )\in {\mathrm{Hom}}_{{\mathrm{GL}}_{n-1}(\mathbb{K})}\big(I(\varrho) \widehat \otimes I(\varrho') \otimes \frak{M}_{n-1,\mathbb{K}}, \, \chi_{\mathbb{K}, -s+{1}/{2}}\big) \]

as the integral over the above open orbit, that is,

(39)\begin{align} & \Lambda( \phi \otimes m, s, \chi_\mathbb{K}) \nonumber\\ &\quad := \int_{ {\mathrm{GL}}_{n-1}(\mathbb{K})}\phi \bigg(z_n \left[ \begin{array}{@{}cc@{}} g & 0 \\ 0 & 1 \\ \end{array} \right], z_{n-1} g\bigg)\cdot \chi_\mathbb{K}(\det g) \cdot \lvert\det g\rvert^{s-{1}/{2}}_{\mathbb{K}}\operatorname{d}\! m(g) \end{align}

for $\phi \in I(\varrho ) \widehat \otimes I(\varrho ')$ and $m\in \frak {M}_{n-1,\mathbb {K}}$.

Define

\[ \operatorname{sgn}(\varrho, \varrho', \chi_\mathbb{K}):=\prod_{i>k, \, i+k\leqslant n} (\varrho_i \cdot \varrho'_k \cdot\chi_\mathbb{K})(-1), \]

and a meromorphic function

\[ \gamma_{\psi_\mathbb{K}^{(n)}}(s, \varrho, \varrho', \chi_\mathbb{K}):=\prod_{i+k\leqslant n} \gamma(s, \varrho_i \cdot \varrho'_k \cdot \chi_\mathbb{K}, \psi_\mathbb{K}^{(n)}), \]

where $\psi _\mathbb {K}^{(n)}$ is the additive character $\mathbb {K}\to {\mathbb {C}}^\times$, $x\mapsto \psi _\mathbb {K}((-1)^n x)$,

\[ \gamma(s, \omega, \psi_\mathbb{K}^{(n)}) = \varepsilon(s,\omega, \psi_\mathbb{K}^{(n)}) \cdot \frac{\operatorname{L}(1-s, \omega^{-1})}{\operatorname{L}(s, \omega)} \]

is the local gamma factor of a character $\omega \in \widehat {\mathbb {K}^\times }$, and $\varepsilon (s, \omega, \psi _\mathbb {K}^{(n)})$ is the local epsilon factor, defined following [Reference TateTat79, Reference JacquetJac79, Reference KudlaKud03]. For convenience also define

\[ \varepsilon_{\psi_\mathbb{K}^{(n)}}(s, \varrho, \varrho', \chi_\mathbb{K}):=\prod_{i+k\leqslant n} \varepsilon(s, \varrho_i \cdot \varrho'_k\cdot \chi_\mathbb{K}, \psi_\mathbb{K}^{(n)}). \]

Finally, define a meromorphic function

\[ \Gamma_{\psi_\mathbb{K}^{(n)}}(s, \varrho, \varrho', \chi_\mathbb{K}):=\operatorname{sgn}(\varrho, \varrho', \chi_\mathbb{K}) \cdot \gamma_{\psi_\mathbb{K}^{(n)}}(s, \varrho, \varrho', \chi_\mathbb{K}). \]

For a character $\omega \in \widehat {\mathbb {K}^\times }$, denote by ${\rm ex}(\omega )$ the real number such that

\[ \lvert\omega\rvert = \lvert\,\cdot\,\rvert_\mathbb{K}^{{\rm ex}(\omega)}. \]

Consider the complex manifold

\[ \mathcal{M}:={\mathbb{C}}\times (\widehat{\mathbb{K}^\times})^n \times (\widehat{\mathbb{K}^\times})^{n-1} \]

and its nonempty open subset

\[ \Omega:=\bigg\{ (s, \varrho, \varrho') \in \mathcal{M} \left|\, \begin{aligned} & \mathrm{ex}(\varrho_i)+\mathrm{ex}(\varrho'_k)+{\mathrm{Re}}(s)<1 \textrm{ whenever } i+k\leqslant n,\\ & \mathrm{ex}(\varrho_i)+\mathrm{ex}(\varrho'_k)+{\mathrm{Re}}(s)>0 \textrm{ whenever } i+k> n \end{aligned}\right.\!\bigg\}. \]

Theorem 4.2 [Reference Li, Liu, Su and SunLLSS23, Theorem 1.6(b)] Assume that $(s,\varrho,\varrho ')\in \Omega$. Then the integral (39) converges absolutely, and

(40)\begin{equation} \Lambda ( \phi \otimes m, s, \chi_\mathbb{K}) = \Gamma_{\psi_\mathbb{K}^{(n)}}(s, \varrho, \varrho', \chi_\mathbb{K}) \cdot \operatorname{Z}( \phi \otimes m, s, \chi_\mathbb{K}). \end{equation}

We remark that the right-hand side of (40) is holomorphic as a function of the variable $s\in \Omega _{\varrho, \varrho '}:=\{s\in \mathbb {C}\, : \, (s, \varrho, \varrho ')\in \Omega \}$ (see [Reference Li, Liu, Su and SunLLSS23, Remark 1.7]).

Let $(I(\varrho ) \widehat \otimes I(\varrho '))^\sharp \subset I(\varrho ) \widehat \otimes I(\varrho ')$ be the subspace of $\phi \in I(\varrho ) \widehat \otimes I(\varrho ')$ such that

\[ \phi |_{z\cdot {\mathrm{GL}}_{n-1}(\mathbb{K})}\in \mathcal{S}(z\cdot {\mathrm{GL}}_{n-1}(\mathbb{K})). \]

Then for every $\varrho \in (\widehat {\mathbb {K}^\times })^n$, $\varrho '\in (\widehat {\mathbb {K}^\times })^{n-1}$ and $\phi \in (I(\varrho )\widehat \otimes I(\varrho '))^\sharp$, the integral (39) converges absolutely and is an entire function of $s\in {\mathbb {C}}$. We deduce the following consequence of Theorem 4.2.

Proof. Let $\mathcal {C}$ be the connected component of $(\widehat {\mathbb {K}^\times })^n$ containing $\varrho$, and let $\mathcal {C}'$ be the connected component of $(\widehat {\mathbb {K}^\times })^{n-1}$ containing $\varrho '$. Write $K_\mathbb {K}:=K_{n,\mathbb {K}}\times K_{n-1,\mathbb {K}}$. Define

\[ C^\infty_{\mathcal{C}, \mathcal{C}'}(K_\mathbb{K}):=\bigg\{ f \in C^\infty(K_\mathbb{K}) \left| \begin{array}{@{}l@{}} f(b\cdot k) = ( \varrho \otimes \varrho')(b) \cdot f(k), \\ \textrm{for all } b\in K_\mathbb{K}\cap (\bar{\mathrm{B}}_n(\mathbb{K}) \times \bar{\mathrm{B}}_{n-1}(\mathbb{K})), \ k\in K_\mathbb{K} \end{array}\right.\!\bigg\}, \]

which only depends on $\mathcal {C}$ and $\mathcal {C}'$, not on the particular choices of $\varrho$ and $\varrho '$.

Consider the natural map

\[ K_\mathbb{K} \to (\bar{\mathrm{B}}_n(\mathbb{K}) \times \bar{\mathrm{B}}_{n-1}(\mathbb{K}) )\backslash ({\mathrm{GL}}_n(\mathbb{K}) \times {\mathrm{GL}}_{n-1}(\mathbb{K})), \]

which is surjective by the Iwasawa decomposition. Let $K_\mathbb {K}^\sharp \subset K_\mathbb {K}$ be the preimage of the open orbit (38) under the above map. Fix $f \in C^\infty _{\mathcal {C}, \mathcal {C}'}(K_\mathbb {K})$ such that $f|_{K_\mathbb {K}^\sharp }\in {\mathcal {S}}(K_\mathbb {K}^\sharp )$. Then there is a unique

\[ \phi_{\varrho,\varrho'}:=\phi_{f,\varrho,\varrho'}\in (I(\varrho)\widehat\otimes I(\varrho'))^\sharp \]

such that

\[ \phi_{\varrho, \varrho'}|_{K_\mathbb{K}} = f. \]

Let ${\mathcal {M}}^\circ :={\mathbb {C}}\times \mathcal {C}\times \mathcal {C}'$, which is a connected component of ${\mathcal {M}}$. When $(\varrho, \varrho ')$ varies in $\mathcal {C}\times \mathcal {C}'$, the integral $\Lambda (\phi _{\varrho,\varrho '}\otimes m, s, \chi _\mathbb {K})$ is clearly holomorphic on ${\mathcal {M}}^\circ$. By [Reference JacquetJac09, § 8.1], we also have that

\[ \Gamma_{\psi_\mathbb{K}^{(n)}}(s, \varrho, \varrho', \chi_\mathbb{K}) \cdot \operatorname{Z}( \phi_{\varrho,\varrho'} \otimes m, s, \chi_\mathbb{K}) \]

is meromorphic on ${\mathcal {M}}^{\circ }$. Since the equality

\[ \Lambda(\phi_{\varrho,\varrho'}\otimes m, s, \chi_\mathbb{K}) = \Gamma_{\psi_\mathbb{K}^{(n)}} (s, \varrho, \varrho', \chi_\mathbb{K}) \cdot \operatorname{Z}( \phi_{\varrho,\varrho'} \otimes m, s, \chi_\mathbb{K}) \]

holds on $\Omega \cap {\mathcal {M}}^\circ$, which is nonempty and open, it holds over all ${\mathcal {M}}^\circ$ by the uniqueness of meromorphic continuation. The corollary then follows by noting that every $\phi \in (I(\varrho )\widehat \otimes I(\varrho '))^\sharp$ equals $\phi _{f,\varrho, \varrho '}$ for some $f \in C^\infty _{\mathcal {C}, \mathcal {C}'}(K_\mathbb {K})$ such that $f|_{K_\mathbb {K}^\sharp }\in {\mathcal {S}}(K_\mathbb {K}^\sharp )$.

4.3 A commutative diagram

We now specify the above discussion to the principal series representations $I_\mu$ and $I_\nu$. Define $\varrho ^\mu = (\varrho ^\mu _1, \ldots, \varrho ^\mu _n)\in (\widehat {\mathbb {K}^\times })^n$, where

\[ \varrho^\mu_i := \varepsilon_{n,\mathbb{K}} \lvert\,\cdot\,\rvert_\mathbb{K}^{(n+1)/2-i} \prod_{\iota \in {\mathcal{E}}_\mathbb{K}} \iota^{\mu_i^{\iota}}\in \widehat{\mathbb{K}^\times}, \quad i =1,\ldots, n, \]

so that $I_\mu = I(\varrho ^\mu )$ in the above notation. Likewise we define $\varrho ^\nu = (\varrho ^\nu _1,\ldots, \varrho ^\nu _{n-1})\in (\widehat {\mathbb {K}^\times })^{n-1}$ so that $I_\nu = I(\varrho ^\nu )$, and put $I_\xi ^\sharp : = (I_\mu \widehat \otimes I_\nu )^\sharp$. Similar to $\varepsilon _{n,\mathbb {K}}$, one defines $\varepsilon _{n-1,\mathbb {K}}$ as the common central character of $F_\nu ^\vee \otimes \pi _\nu$ and $\pi _{0_{n-1,\mathbb {K}}}$.

The integral (39) defines a nonzero linear functional

\[ \Lambda_\xi (\cdot, s, \chi_\mathbb{K})\in {\mathrm{Hom}}_{{\mathrm{GL}}_{n-1}(\mathbb{K})}\big(I_\xi^\sharp, \, \chi_{\mathbb{K}, -s+{1}/{2}} \otimes \frak{M}_{n-1,\mathbb{K}}^* \big). \]

By Corollary 4.3,

(41)\begin{align} \Lambda_\xi (\cdot, s, \chi_\mathbb{K}) & = \Gamma_{\psi_\mathbb{K}^{(n)}} (s, \varrho^\mu, \varrho^\nu, \chi_\mathbb{K}) \cdot \operatorname{Z}_\xi(\cdot, s, \chi_\mathbb{K}) \nonumber\\ & = \Gamma_{\psi_\mathbb{K}^{(n)}} (s, \varrho^\mu, \varrho^\nu, \chi_\mathbb{K}) \cdot \operatorname{L}(s, \pi_\mu\times \pi_\nu) \cdot \operatorname{Z}^\diamond_\xi(\cdot, s, \chi_\mathbb{K}) \end{align}

holds on $I_\xi ^\sharp \otimes \frak {M}_{n-1,\mathbb {K}}$. Recall that $\imath _\xi = \imath _\mu \otimes \imath _\nu \in {\mathrm {Hom}}_{{\mathrm {GL}}_{n-1}(\mathbb {K})}(I_{\xi _0}, F_{\xi }^\vee \otimes I_\xi )$. It is clear that

\[ \imath_\xi(I_{\xi_0}^\sharp)\subset F_\xi^\vee\otimes I_\xi^\sharp. \]

Proposition 4.4 The following diagram commutes.

Proof. Recall from Proposition 3.1 that $\phi _{\xi,j}\in {\mathrm {Hom}}_{{\mathrm {GL}}_{n-1}(\mathbb {K}\otimes _\mathbb {R} \mathbb {C})}(F_\xi ^\vee, \otimes _{\iota \in \mathcal {E}_\mathbb {K}}{\det }^j)$ and

\[ \phi_{\xi, j}(z^{-1}. v_\xi^\vee)=1, \]

where $v_\xi ^\vee :=v_\mu ^\vee \otimes v_\nu ^\vee$. For $\phi \in I_{\xi _0}$ and $g\in {\mathrm {GL}}_{n-1}(\mathbb {K})\subset {\mathrm {GL}}_n(\mathbb {K})\times {\mathrm {GL}}_{n-1}(\mathbb {K})$ (see (27) for the inclusion), we have that

\begin{align*} \phi_{\xi, j} ( \imath_\xi(\phi) (zg)) &= \phi_{\xi, j}\big(\phi(zg)\cdot (g^{-1}z^{-1}. v_\xi^\vee)\big) \quad (\textrm{see} \;{(19)}) \\ &= \phi(zg)\cdot \phi_{\xi, j}(g^{-1}z^{-1}. v_\xi^\vee) \\ &= \phi(zg) \cdot (\otimes_{\iota\in \mathcal{E}_\mathbb{K}}{\det}^{-j})(g). \end{align*}

Assume that $\phi \in I_{\xi _0}^\sharp$. By (39), we have that

\begin{align*} & \big( \phi_{\xi,j}\otimes \Lambda_\xi(\cdot, s+j, \chi_\mathbb{K})\big)(\imath_{\xi}(\phi)\otimes m) \\[4pt] &\quad= \int_{ {\mathrm{GL}}_{n-1}(\mathbb{K})} \phi_{\xi, j} ( \imath_\xi(\phi) (zg)) \cdot \chi_\mathbb{K}(\det g) \cdot \lvert\det g\rvert^{s+j-{1}/{2}}_{\mathbb{K}}\operatorname{d}\! m(g) \\[4pt] &\quad= \int_{ {\mathrm{GL}}_{n-1}(\mathbb{K})} \phi(zg) \cdot (\otimes_{\iota\in \mathcal{E}_\mathbb{K}}{\det}^{-j})(g) \cdot \chi_\mathbb{K}(\det g) \cdot \lvert\det g\rvert^{s+j-{1}/{2}}_{\mathbb{K}}\operatorname{d}\! m(g) \\[4pt] &\quad= \int_{ {\mathrm{GL}}_{n-1}(\mathbb{K})} \phi(zg) \cdot \chi_\mathbb{K}^{(j)}(\det g) \cdot \lvert\det g\rvert^{s-{1}/{2}}_{\mathbb{K}}\operatorname{d}\! m(g) \\[4pt] &\quad= \Lambda_{\xi_0}(\phi\otimes m, s, \chi_\mathbb{K}^{(j)}), \end{align*}

where $m\in \frak {M}_{n-1,\mathbb {K}}$. This proves the proposition.

Corollary 4.5 Let the notation be as above. Then

(42)\begin{equation} \Xi_{\mu, \nu, j}(s) \cdot \frac{\Gamma_{\psi_\mathbb{K}^{(n)}} (s+j, \varrho^\mu, \varrho^\nu, \chi_\mathbb{K})}{\Gamma_{\psi_\mathbb{K}^{(n)}} (s, \varrho^{0_{n,\mathbb{K}}}, \varrho^{0_{n-1,\mathbb{K}}}, \chi_\mathbb{K}^{(j)})} \cdot \frac{\operatorname{L}(s+j, \pi_\mu\times \pi_\nu)}{\operatorname{L}(s, \pi_{0_{n,\mathbb{K}}}\times \pi_{0_{n-1,\mathbb{K}}})}=1 \end{equation}

as meromorphic functions of the variable $s\in {\mathbb {C}}$.

4.4 Archimedean local factors

To finish the proof, it remains to evaluate the function $\Xi _{\mu, \nu, j}(s)^{-1}$ given by (42) and show that it is equal to the constant $\Omega '_{\mu, \nu, j}$ given by (34) as in Theorem 3.2. To this end, we first recall some standard facts about archimedean local L-factors and epsilon factors that we need, following [Reference KnappKna94]. Let

\[ \Gamma_\mathbb{K}(s)=\begin{cases} \pi^{-s/2}\Gamma(s/2), & \textrm{if }\mathbb{K}\cong{\mathbb{R}}, \\[4pt] 2(2\pi)^{-s}\Gamma(s), & \textrm{if }\mathbb{K}\cong{\mathbb{C}}, \end{cases} \]

where $\Gamma (s)$ is the standard gamma function. Recall the Legendre duplication formula

(43)\begin{equation} \Gamma_{\mathbb{C}}(s) = \Gamma_{\mathbb{R}}(s) \Gamma_{\mathbb{R}}(s+1). \end{equation}

Recall the additive character $\psi _\mathbb {K}$ that is defined in (10) by using $\psi _{\mathbb {R}}$.

If $\mathbb {K}\cong {\mathbb {R}}$, then the following hold true.

• – For all $t\in {\mathbb {C}}$ and $\delta \in \{0,1\}$,

  \[ \operatorname{L}(s, \lvert\,\cdot\,\rvert_\mathbb{K}^t \operatorname{sgn}_{\mathbb{K}^\times}^\delta)= \Gamma_{\mathbb{R}}(s+t +\delta), \]
  and
  \[ \varepsilon(s, \lvert\,\cdot\,\rvert_\mathbb{K}^t \operatorname{sgn}_{\mathbb{K}^\times}^\delta, \psi_\mathbb{K}^{(n)})=((-1)^n {\rm i})^\delta. \]

• – For all $a, b\in {\mathbb {C}}$ with $a-b\in {\mathbb {Z}}\setminus \{0\}$, and $t$, $\delta$ as above,

  \[ \operatorname{L}(s, D_{a, b} \times \lvert\,\cdot\,\rvert_\mathbb{K}^t \operatorname{sgn}_{\mathbb{K}^\times}^\delta)= \Gamma_{\mathbb{C}}\big(s+t+\max\{a,b\}\big). \]
  Here and henceforth, for any $a', b'\in {\mathbb {C}}$ with $a'-b'\in {\mathbb {Z}}$,
  \[ \max\{a', b'\}: = \begin{cases} a', & \textrm{if }a'-b' \geqslant 0; \\ b', & \textrm{otherwise}. \end{cases} \]

• – For all $a, b, a', b'\in {\mathbb {C}}$ with $a-b, a'-b'\in {\mathbb {Z}}\setminus \{0\}$,

  \begin{align*} \operatorname{L}(s, D_{a, b} \times D_{a', b'}) &= \Gamma_{\mathbb{C}}\big(s+\max\{a+a', b+b'\}\big) \cdot \Gamma_{\mathbb{C}}\big(s+ \max\{a+b', b+a'\}\big). \end{align*}

If $\mathbb {K}\cong {\mathbb {C}}$, then for all $a, b\in {\mathbb {C}}$ with $a-b\in {\mathbb {Z}}$,

\[ \operatorname{L}(s, \iota^a \bar{\iota}^b )= \Gamma_{\mathbb{C}}\big(s+\max\{a, b\}\big) \]

and

\[ \varepsilon(s, \iota^a \bar{\iota}^b, \psi_\mathbb{K}^{(n)}) = ((-1)^n \mathrm{i})^{\lvert a-b\rvert}. \]

By the well-known branching rule for ${\mathrm {GL}}_n({\mathbb {C}})$, we have that $j\in {\mathbb {Z}}$ is a balanced place for $\xi$ if and only if

\[ -\mu^{\iota}_n \geqslant \nu^{\iota}_1+ j \geqslant - \mu^{\iota}_{n-1} \geqslant \nu^{\iota}_2+j \geqslant \cdots \geqslant \nu^{\iota}_{n-1}+j \geqslant - \mu^{\iota}_1 \]

for every $\iota \in {\mathcal {E}}_\mathbb {K}$. Equivalently, by [Reference RaghuramRag16, Corollary 2.35], $j\in \mathbb {Z}$ is a balanced place for $\xi$ if and only if

\[ m_{\mu, \nu}^-\leqslant j\leqslant m_{\mu, \nu}^+, \]

where

\begin{align*} & m_{\mu, \nu}^- := \max\{-\mu_{n-i}^{\iota}-\nu_i^{\iota}: 1\leqslant i\leqslant n-1, \iota \in {\mathcal{E}}_\mathbb{K}\}, \\ & m_{\mu, \nu}^+ := \min\{-\mu_{n+1-i}^{\iota}-\nu_i^{\iota}: 1\leqslant i\leqslant n-1, \iota \in {\mathcal{E}}_\mathbb{K}\}. \end{align*}

Recall the half-integers $\tilde {\mu }^\iota _i$ ($i=1,\ldots, n$) given by (14). The following result can be easily checked by using the above result (see the proof of [Reference RaghuramRag16, Lemma 2.24]).

Lemma 4.6 Assume that $\xi =(\mu,\nu )$ is balanced. Then

\[ \tilde{\mu}^\iota_i + \tilde{\nu}^\iota_k - \tilde{\mu}^{\bar{\iota}}_{n+1-i} - \tilde{\nu}^{\bar{\iota}}_{n-k} \]

is positive if $i+k\leqslant n$, and is negative otherwise, for every $\iota \in {\mathcal {E}}_\mathbb {K}$.

Now we establish the following result, which thereby finishes the proof of Theorem 4.1.

Proof. We use the notation of Theorem 3.2. It is clear that

\[ \frac{\operatorname{sgn}(\varrho^\mu, \varrho^\nu, \chi_\mathbb{K})}{\operatorname{sgn}(\varrho^{0_{n,\mathbb{K}}}, \varrho^{0_{n-1,\mathbb{K}}}, \chi_\mathbb{K}^{(j)})} = \prod_{i>k, \, i+k \leqslant n}(-1)^{\sum_{\iota\in {\mathcal{E}}_\mathbb{K}}(\mu_i^\iota+\nu_k^\iota+j)} =(-1)^{ j (n^2(n-1)/2)[\mathbb{K}: \, {\mathbb{R}}]}\cdot \varepsilon_{\mu, \nu}. \]

To evaluate the contribution from the local gamma and $\operatorname {L}$-factors, we consider the real and complex cases separately.

(i) Assume that $\mathbb {K}\cong {\mathbb {R}}$. Then ${\mathcal {E}}_\mathbb {K}=\{\iota \}$. Using Lemma 4.6, it is easy to check that

\[ \operatorname{L}(s, \pi_\mu\times \pi_\nu) = \prod_{i+k\leqslant n} \Gamma_{\mathbb{C}}(s + \tilde{\mu}^\iota_i + \tilde{\nu}^\iota_k). \]

For a character $\omega \in \widehat {\mathbb {K}^\times }$, write $\delta (\omega )\in \{0, 1\}$ such that $\omega (-1) = (-1)^{\delta (\omega )}$. Then

(44)\begin{equation} \varepsilon(s, \omega, \psi_\mathbb{K}^{(n)})= ((-1)^n \mathrm{i})^{\delta(\omega)}, \end{equation}

and it is clear that

(45)\begin{equation} j+ \mu^\iota_i + \nu^\iota_k - \delta(\varrho^\mu_i \varrho^\nu_k \chi_\mathbb{K}) + \delta(\varrho^{0_{n,\mathbb{K}}}_i \varrho^{0_{n-1,\mathbb{K}}}_k \chi_\mathbb{K}^{(j)}) \in 2{\mathbb{Z}}. \end{equation}

We have that

\[ \frac{\operatorname{L}(1-s, (\varrho^\mu_i \varrho^\nu_k\chi_\mathbb{K})^{-1})}{\operatorname{L}(s, \varrho^\mu_i\varrho^\nu_k \chi_\mathbb{K})} = \frac{\Gamma_{\mathbb{R}}(1-s-\tilde{\mu}^{\iota}_i - \tilde{\nu}^\iota_k+\delta(\varrho^\mu_i \varrho^\nu_k \chi_\mathbb{K}))}{\Gamma_{\mathbb{R}}(s+\tilde{\mu}^\iota_i + \tilde{\nu}^{\iota}_k+\delta(\varrho^\mu_i \varrho^\nu_k \chi_\mathbb{K}))}. \]

It follows from (43) that

(46)\begin{align} & \bigg( \prod_{i+k\leqslant n} \frac{\operatorname{L}(1-s, (\varrho^\mu_i \varrho^\nu_k \chi_\mathbb{K} )^{-1})}{\operatorname{L}(s, \varrho^\mu_i\varrho^\nu_k\chi_\mathbb{K})} \bigg) \cdot \operatorname{L}(s, \pi_\mu\times \pi_\nu)\nonumber\\ &\quad = \prod_{i+k\leqslant n}\big(\Gamma_{\mathbb{R}}(s+\tilde{\mu}^\iota_i + \tilde{\nu}^\iota_k+1-\delta(\varrho^\mu_i \varrho^\nu_k \chi_\mathbb{K})) \cdot\Gamma_{\mathbb{R}}(1-s- \tilde{\mu}^\iota_i - \tilde{\nu}^\iota_k+ \delta(\varrho^\mu_i \varrho^\nu_k \chi_\mathbb{K}))\big). \end{align}

By (44), (45), (46) and the formula

\[ \Gamma_{\mathbb{R}}(s+ \ell )\cdot \Gamma_{\mathbb{R}}(2-s- \ell ) = {\rm i}^\ell \cdot \Gamma_{\mathbb{R}}(s)\cdot \Gamma_{\mathbb{R}}(2-s),\quad \ell \in 2 {\mathbb{Z}}, \]

we find that

\begin{align*} & \frac{\gamma_{\psi_\mathbb{K}^{(n)}} (s+j, \varrho^\mu, \varrho^\nu, \chi_\mathbb{K})}{\gamma_{\psi_\mathbb{K}^{(n)}} (s, \varrho^{0_{n,\mathbb{K}}}, \varrho^{0_{n-1,\mathbb{K}}}, \chi_\mathbb{K}^{(j)})} \cdot \frac{\operatorname{L}(s+j, \pi_\mu\times \pi_\nu)}{\operatorname{L}(s, \pi_{0_{n,\mathbb{K}}}\times \pi_{0_{n-1,\mathbb{K}}})} \nonumber\\ &\quad = \prod_{i+k\leqslant n} \bigg(\frac{ \varepsilon(s+j, \varrho^\mu \varrho^\nu \chi_\mathbb{K}, \psi_\mathbb{K}^{(n)})}{\varepsilon (s, \varrho^{0_{n,\mathbb{K}}} \varrho^{0_{n-1,\mathbb{K}}} \chi_\mathbb{K}^{(j)}, \psi_\mathbb{K}^{(n)})} \cdot {\rm i}^{j+ \mu_i^\iota+\nu^\iota_k - \delta(\varrho^\mu \varrho^\nu \chi_\mathbb{K}) + \delta(\varrho^{0_{n,\mathbb{K}}} \varrho^{0_{n-1,\mathbb{K}}} \chi_\mathbb{K}^{(j)})} \bigg) \\ &\quad = \prod_{i+k\leqslant n} ((-1)^n \mathrm{i})^{j+\mu^\iota_i + \nu^\iota_k} \\ &\quad = (-1)^{j {n^2(n-1)}/{2}}\cdot \mathrm{i}^{j{n(n-1)}/{2}} \cdot c_\mu' \cdot c_\nu. \end{align*}

(ii) Assume that $\mathbb {K}\cong {\mathbb {C}}$. Then $\chi _\mathbb {K}^{(j)}$ is trivial, which will be omitted from the notation for convenience. Using Lemma 4.6 again, we find that

\[ \operatorname{L}(s, \pi_\mu\times \pi_\nu) = \prod_{i+k\leqslant n, \, \iota\in {\mathcal{E}}_\mathbb{K}} \Gamma_{\mathbb{C}}(s + \tilde{\mu}^\iota_i +\tilde{\nu}^\iota_k). \]

We have that

\[ \frac{\operatorname{L}(1-s, (\varrho^\mu_i \varrho^\nu_k)^{-1})}{\operatorname{L}(s, \varrho^\mu_i\varrho^\nu_k)} = \frac{\Gamma_{\mathbb{C}}(1-s-\min_{\iota \in{\mathcal{E}}_\mathbb{K}}\{\tilde{\mu}^{\iota}_i + \tilde{\nu}^\iota_k\})}{\Gamma_{\mathbb{C}}(s+\max_{\iota\in {\mathcal{E}}_\mathbb{K}}\{\tilde{\mu}^\iota_i + \tilde{\nu}^{\iota}_k\})}. \]

It follows that

(47)\begin{align} & \bigg(\prod_{i+k\leqslant n} \frac{\operatorname{L}(1-s, (\varrho^\mu_i \varrho^\nu_k)^{-1})}{\operatorname{L}(s, \varrho^\mu_i\varrho^\nu_k)} \bigg) \cdot \operatorname{L}(s, \pi_\mu\times \pi_\nu)\nonumber\\ &\quad = \prod_{i+k\leqslant n}\Big ( \Gamma_{\mathbb{C}} \Big(s+\min_{\iota\in{\mathcal{E}}_\mathbb{K}}\{\tilde{\mu}^\iota_i + \tilde{\nu}^\iota_k\}\Big) \cdot\Gamma_{\mathbb{C}}\bigg(1-s-\min_{\iota\in{\mathcal{E}}_\mathbb{K}}\{\tilde{\mu}^\iota_i + \tilde{\nu}^\iota_k\}\Big)\Big). \end{align}

Using (47) and the formula

\[ \Gamma_{\mathbb{C}}(s+ \ell)\cdot\Gamma_{\mathbb{C}}(1-s-\ell) = (-1)^\ell \cdot \Gamma_{\mathbb{C}}(s)\cdot \Gamma_{\mathbb{C}}(1-s),\quad \ell \in {\mathbb{Z}}, \]

we find that

(48)\begin{align} & \frac{\gamma_{\psi_\mathbb{K}^{(n)}} (s+j, \varrho^\mu, \varrho^\nu)}{\gamma_{\psi_\mathbb{K}^{(n)}} (s, \varrho^{0_{n,\mathbb{K}}}, \varrho^{0_{n-1,\mathbb{K}}})} \cdot \frac{\operatorname{L}(s+j, \pi_\mu\times \pi_\nu)}{\operatorname{L}(s, \pi_{0_{n,\mathbb{K}}}\times \pi_{0_{n-1,\mathbb{K}}})} \nonumber\\ &\quad = \prod_{i+k\leqslant n} \big( \varepsilon(s+j, \varrho^\mu_i \varrho^\nu_k, \psi_\mathbb{K}^{(n)}) \cdot (-1)^{j+\min_{\iota\in{\mathcal{E}}_\mathbb{K}} \{ \mu^\iota_i + \nu^\iota_k\}}\big). \end{align}

We have the local epsilon factor

\[ \varepsilon(s+j, \varrho^\mu_i \varrho^\nu_k, \psi_\mathbb{K}^{(n)}) = ((-1)^n \mathrm{i})^{\max_{\iota\in{\mathcal{E}}_\mathbb{K}}\{\mu^\iota_i + \nu^\iota_k \} - \min_{\iota\in {\mathcal{E}}_\mathbb{K}} \{ \mu^{ \iota}_i + \nu^{ \iota}_k\} }. \]

Hence, (48) is equal to

\[ \prod_{i+k \leqslant n, \, \iota\in {\mathcal{E}}_\mathbb{K}} ((-1)^n \mathrm{i})^{j + \mu^\iota_i + \nu^\iota_k } = (-1)^{jn^2(n-1)}\cdot \mathrm{i}^{j n(n-1)}\cdot c'_\mu \cdot c_\nu. \]

This finishes the proof of the proposition.

5.1 Preliminaries

Denote ${\mathrm {Ind}}^{{\mathrm {GL}}_n(\mathbb {K})}_{{\rm N}_n(\mathbb {K})} \psi _{n,\mathbb {K}}$ the smooth induction which consists of all functions $f: {\mathrm {GL}}_n(\mathbb {K})\to {\mathbb {C}}$ such that:

• – $f$ is right invariant under some open compact subgroup of ${\mathrm {GL}}_n(\mathbb {K})$;

• – $f(ug)= \psi _{n,\mathbb {K}}(u)f(g)$ for all $u\in {\rm N}_n(\mathbb {K})$ and $g\in {\mathrm {GL}}_n(\mathbb {K})$.

This is a smooth representation of ${\mathrm {GL}}_n(\mathbb {K})$ under the right translation.

Let $p$ be the residue characteristic of $\mathbb {K}$, and $\mu _{p^\infty }\subset {\mathbb {C}}^\times$ be the subgroup of $p$th power roots of unity. Recall the cyclotomic character

\[ {\rm Aut}(\mathbb{Q}(\mu_{p^\infty})/\mathbb{Q})\to {\mathbb{Z}}_p^\times, \quad \sigma\mapsto t_{\sigma, p} \]

defined by requiring that

(49)\begin{equation} \sigma( \zeta )=\zeta^{t_{\sigma, p}} \quad \textrm{for all } \zeta \in \mu_{p^\infty}. \end{equation}

Write $\sigma \mapsto t_{\sigma, \mathbb {K}}$ for the composition

\[ {\mathrm{Aut}}({\mathbb{C}}/\mathbb{Q}) \xrightarrow{\rm restriction} {\mathrm{Aut}}(\mathbb{Q}(\mu^\infty_p)/\mathbb{Q}) \xrightarrow{\sigma\mapsto t_{\sigma, p}}{\mathbb{Z}}_p^\times \subset \mathbb{K}^\times. \]

Following [Reference HarderHar83, pp. 79–80] and [Reference MahnkopfMah05, p. 594], define

(50)\begin{equation} \mathbf{t}_{n, \sigma, \mathbb{K}} := {\rm diag}(t^{-(n-1)}_{\sigma, \mathbb{K}}, \ldots, t_{\sigma, \mathbb{K}}^{-1}, 1) \in {\mathrm{GL}}_n(\mathbb{K}), \end{equation}

and define an action of ${\mathrm {Aut}}({\mathbb {C}})$ on ${\mathrm {Ind}}^{{\mathrm {GL}}_n(\mathbb {K})}_{{\rm N}_n(\mathbb {K})} \psi _{n,\mathbb {K}}$ by

(51)\begin{equation} {}^\sigma \! f(g):=\sigma\big( f(\mathbf{t}_{n, \sigma,\mathbb{K}} \cdot g) \big), \end{equation}

where $\sigma \in {\mathrm {Aut}}({\mathbb {C}})$, $f\in {\mathrm {Ind}}^{{\mathrm {GL}}_n(\mathbb {K})}_{{\rm N}_n(\mathbb {K})}\psi _{n,\mathbb {K}}$, and $g\in {\mathrm {GL}}_n(\mathbb {K})$.

Let $\Pi _\mathbb {K}$ be a generic irreducible smooth representations of ${\mathrm {GL}}_n(\mathbb {K})$, with a fixed Whittaker functional

\[ \lambda_\mathbb{K} \in {\mathrm{Hom}}_{\mathrm{N}_n(\mathbb{K})}(\Pi_\mathbb{K}, \psi_{n, \mathbb{K}})\setminus\{0\}. \]

Using $\lambda _\mathbb {K}$, we realize $\Pi _\mathbb {K}$ as a subrepresentation of ${\mathrm {Ind}}^{{\mathrm {GL}}_n(\mathbb {K})}_{{\rm N}_n(\mathbb {K})} \psi _{n,\mathbb {K}}$ by

(52)\begin{equation} \Pi_\mathbb{K} \to {\mathrm{Ind}}^{{\mathrm{GL}}_n(\mathbb{K})}_{{\rm N}_n(\mathbb{K})} \psi_{n,\mathbb{K}},\quad u\mapsto \big(g\mapsto \lambda_\mathbb{K}(g.u)\big). \end{equation}

Put

\[ {}^\sigma\Pi_\mathbb{K}: = \sigma(\Pi_\mathbb{K}) \subset {\mathrm{Ind}}^{{\mathrm{GL}}_n(\mathbb{K})}_{{\rm N}_n(\mathbb{K})}\psi_{n,\mathbb{K}}, \]

which is also a generic irreducible smooth representations of ${\mathrm {GL}}_n(\mathbb {K})$ with a fixed Whittaker functional (the evaluation map at the identity matrix).

Let $\chi _\mathbb {K}: \mathbb {K}^\times \to {\mathbb {C}}^\times$ be a character. Let $\frak {c}(\chi _\mathbb {K})$ and $\frak {c}(\psi _\mathbb {K})$ be the conductors of $\chi _\mathbb {K}$ and $\psi _\mathbb {K}$, which are ideal and fractional ideal of ${\mathcal {O}}_\mathbb {K}$, respectively. Here ${\mathcal {O}}_\mathbb {K}$ denotes the ring of integers of $\mathbb {K}$. Fix $y_\mathbb {K} \in \mathbb {K}^\times$ such that

\[ \frak{c}(\psi_\mathbb{K})= y_\mathbb{K} \cdot \frak{c}(\chi_\mathbb{K}). \]

The local Gauss sum is defined by

(53)\begin{equation} {\mathcal{G}}(\chi_\mathbb{K}):={\mathcal{G}}(\chi_\mathbb{K}, \psi_\mathbb{K}, y_\mathbb{K}): = \int_{{\mathcal{O}}_\mathbb{K}^\times} \chi_\mathbb{K}(x)^{-1} \cdot \psi_\mathbb{K}(y_\mathbb{K} x)\operatorname{d}\! x, \end{equation}

where $\operatorname {d}\! x$ is the normalized Haar measure so that ${\mathcal {O}}_\mathbb {K}^\times$ has total volume 1. Note that ${\mathcal {G}}(\chi _\mathbb {K})=1$ when $\frak {c}(\chi _\mathbb {K})=\frak {c}(\psi _\mathbb {K})={\mathcal {O}}_\mathbb {K}$. For every $\sigma \in {\mathrm {Aut}}(\mathbb {C})$, it is easily checked that $\frak {c}({}^\sigma \chi _\mathbb {K})=\frak {c}(\chi _\mathbb {K})$, and

(54)\begin{equation} {\mathcal{G}}(\chi_\mathbb{K}, \psi_\mathbb{K}, y_\mathbb{K})={}^\sigma \chi_\mathbb{K}(t_{\sigma, \mathbb{K}})\cdot {\mathcal{G}}({}^\sigma\chi_\mathbb{K}, \psi_\mathbb{K}, y_\mathbb{K}). \end{equation}

5.2 Non-archimedean period relation

Suppose that $n\geqslant 2$, and $\Sigma _\mathbb {K}$ is a generic irreducible smooth representation of ${\mathrm {GL}}_{n-1}(\mathbb {K})$ with a fixed Whittaker functional

\[ \lambda'_\mathbb{K} \in {\mathrm{Hom}}_{\mathrm{N}_{n-1}(\mathbb{K})}(\Sigma_\mathbb{K}, \psi_{n-1, \mathbb{K}})\setminus \{0\}. \]

As before, we use $\lambda _\mathbb {K}'$ to realize $\Sigma _\mathbb {K}$ as a subrepresentation of ${\mathrm {Ind}}^{{\mathrm {GL}}_{n-1}(\mathbb {K})}_{{\rm N}_{n-1}(\mathbb {K})}\psi _{n-1,\mathbb {K}}$, and we have a subrepresentation ${}^\sigma \Sigma _\mathbb {K} \subset {\mathrm {Ind}}^{{\mathrm {GL}}_{n-1}(\mathbb {K})}_{{\rm N}_{n-1}(\mathbb {K})}\psi _{n-1,\mathbb {K}}$ for every $\sigma \in {\mathrm {Aut}}(\mathbb {C})$.

As in the archimedean case, denote by $\frak {M}_{n-1,\mathbb {K}}$ the one-dimensional space of invariant measures on ${\mathrm {GL}}_{n-1}(\mathbb {K})$. Fix the Haar measure on $\mathrm {N}_{n-1}(\mathbb {K})$ to be the product of self-dual Haar measures on $\mathbb {K}$ with respect to $\psi _\mathbb {K}$, as in (16). Then each $m\in \frak {M}_{n-1,\mathbb {K}}$ induces a quotient measure $\bar {m}$ on $\mathrm {N}_{n-1}(\mathbb {K})\backslash {\mathrm {GL}}_{n-1}(\mathbb {K})$.

Let $\chi _{\Sigma _\mathbb {K}}$ denote the central character of $\Sigma _\mathbb {K}$. For every $\sigma \in {\mathrm {Aut}}(\mathbb {C})$, it is clear that ${}^\sigma (\chi _{\Sigma _\mathbb {K}}) = \chi _{{}^\sigma \Sigma _\mathbb {K}}$. Similar to (54), we also have that

(55)\begin{equation} {\mathcal{G}}(\chi_{\Sigma_\mathbb{K}}, \psi_\mathbb{K}, y'_\mathbb{K})={}^\sigma \chi_\mathbb{K}(t_{\sigma, \mathbb{K}})\cdot {\mathcal{G}}(\chi_{{}^\sigma\Sigma_\mathbb{K}}, \psi_\mathbb{K}, y'_\mathbb{K}), \end{equation}

where $y'_\mathbb {K}\in \mathbb {K}^\times$ satisfies that $\frak {c}(\psi _\mathbb {K})= y'_\mathbb {K} \cdot \frak {c}(\chi _{\Sigma _\mathbb {K}})$.

We call an invariant measure $m$ on ${\mathrm {GL}}_{n-1}(\mathbb {K})$ rational if $m(K)\in \mathbb {Q}$ for every open compact subgroup $K$ of ${\mathrm {GL}}_{n-1}(\mathbb {K})$. All rational measures on ${\mathrm {GL}}_{n-1}(\mathbb {K})$ form a rational structure of $\frak {M}_{n-1,\mathbb {K}}$. By using this rational structure, we get a $\sigma$-linear isomorphism $\sigma : \frak {M}_{n-1,\mathbb {K}}\rightarrow \frak {M}_{n-1,\mathbb {K}}$. By taking the tensor product of the above $\sigma$-linear isomorphism with the $\sigma$-linear isomorphisms as defined in (51), we get a $\sigma$-linear isomorphism

(56)\begin{equation} \sigma: \Pi_\mathbb{K} \otimes \Sigma_\mathbb{K} \otimes \chi_{\mathbb{K}, s-{1}/{2}}\otimes \frak{M}_{n-1,\mathbb{K}}\rightarrow {}^\sigma \Pi_\mathbb{K} \otimes {}^\sigma\Sigma_\mathbb{K} \otimes {}^\sigma(\chi_{\mathbb{K}, s-{1}/{2}})\otimes \frak{M}_{n-1,\mathbb{K}}, \end{equation}

where $s\in \mathbb {C}$.

Similar to (31), we have the normalized Rankin–Selberg integrals

\[ \operatorname{Z}^\circ(\cdot, s, \chi_\mathbb{K})\in {\mathrm{Hom}}_{{\mathrm{GL}}_{n-1}(\mathbb{K})}(\Pi_\mathbb{K} \otimes \Sigma_\mathbb{K} \otimes \chi_{\mathbb{K}, s-{1}/{2}} \otimes \frak{M}_{n-1,\mathbb{K}}, {\mathbb{C}}) \]

and

\[ \operatorname{Z}^\circ(\cdot, s, {}^\sigma \chi_\mathbb{K})\in {\mathrm{Hom}}_{{\mathrm{GL}}_{n-1}(\mathbb{K})}( {}^\sigma \Pi_\mathbb{K} \otimes {}^\sigma \Sigma_\mathbb{K} \otimes ({}^\sigma \chi_{\mathbb{K}})_{s-{1}/{2}} \otimes \frak{M}_{n-1,\mathbb{K}}, {\mathbb{C}}), \]

where $({}^\sigma \chi _{\mathbb {K}})_{s-{1}/{2}} :={}^\sigma \chi _{\mathbb {K}}\cdot \lvert \,\cdot \,\rvert _\mathbb {K}^{s-{1}/{2}}$, which equals ${}^\sigma (\chi _{\mathbb {K}, s-{1}/{2}})$ when $s\in \frac {1}{2}+{\mathbb {Z}}$.

Following the idea of Harder [Reference HarderHar83, § III] (for $n=2$), Mahnkopf [Reference MahnkopfMah05, § 3.4] and Raghuram [Reference RaghuramRag10, § 3.3], we formulate the non-archimedean period relation as in the following proposition.

Proposition 5.1 For all $s_0\in \frac {1}{2}+{\mathbb {Z}}$ and $\sigma \in {\mathrm {Aut}}({\mathbb {C}})$, the following diagram commutes.

Proof. Note that

\[ \operatorname{L}(s, \Pi_\mathbb{K} \times \Sigma_\mathbb{K} \times \chi_\mathbb{K}) = P(q^{{1}/{2}-s})^{-1} \]

for a polynomial $P(X)\in {\mathbb {C}}[X]$. For $\sigma \in {\mathrm {Aut}}({\mathbb {C}})$, denote by ${}^\sigma P(X)\in {\mathbb {C}}[X]$ the polynomial obtained by applying $\sigma$ to the coefficients of the polynomial $P(X)$. Following the proof of [Reference ClozelClo90, Lemma 4.6], and by noting that the local Rankin–Selberg $\operatorname {L}$-function does not depend on $\psi _\mathbb {K}$, it is easy to show that

\[ \operatorname{L}(s, {}^\sigma \Pi_\mathbb{K} \times {}^\sigma\Sigma_\mathbb{K} \times {}^\sigma\chi_\mathbb{K})= {}^\sigma P(q^{{1}/{2}-s})^{-1}. \]

Specifying $s$ to $s_0\in \frac {1}{2}+{\mathbb {Z}}$, we obtain that

(57)\begin{equation} \operatorname{L}(s_0, {}^\sigma\Pi_\mathbb{K} \times {}^\sigma\Sigma_\mathbb{K} \times {}^\sigma\chi_\mathbb{K}) = \sigma(\operatorname{L}(s_0, \Pi_\mathbb{K}\times \Sigma_\mathbb{K} \times \chi_\mathbb{K})). \end{equation}

For $f\in \Pi _\mathbb {K}$, $f'\in \Sigma _\mathbb {K}$, $m\in \frak {M}_{n-1,\mathbb {K}}$, and $s_0\in \frac {1}{2}+{\mathbb {Z}}$ large enough, by (51) and (57) we have that

\begin{align*} & \operatorname{Z}^\circ({}^\sigma\!f \otimes {}^\sigma\!f' \otimes {}^\sigma m, s_0, {}^\sigma\chi_\mathbb{K}))\\ &\quad = \frac{1}{\operatorname{L}(s_0, {}^\sigma\Pi_\mathbb{K}\times {}^\sigma \Sigma_\mathbb{K} \times {}^\sigma\chi_\mathbb{K})} \\ &\qquad \cdot \int_{{\mathrm{N}}_{n-1}(\mathbb{K})\backslash{\mathrm{GL}}_{n-1}(\mathbb{K})} {}^\sigma\!f \bigg(\! \begin{bmatrix} g & 0 \\ 0 & 1 \end{bmatrix}\!\bigg) \cdot {}^\sigma\!f' (g) \cdot {}^\sigma\chi_\mathbb{K} (\det g) \cdot \lvert\det g\rvert_{\mathbb{K}}^{s_0-{1}/{2}} \operatorname{d}\! \overline{{}^\sigma m}(g) \\ &\quad = \frac{1}{\sigma(\operatorname{L}(s_0, \Pi_\mathbb{K}\times \Sigma_\mathbb{K} \times \chi_\mathbb{K}))} \int_{{\mathrm{N}}_{n-1}(\mathbb{K})\backslash {\mathrm{GL}}_{n-1}(\mathbb{K})} \sigma\bigg(f \bigg( \mathbf{t}_{n, \sigma, \mathbb{K}} \begin{bmatrix} g & 0 \\ 0 & 1\end{bmatrix}\!\bigg)\bigg) \\ &\qquad \cdot \sigma\big( f' (\mathbf{t}_{n-1, \sigma, \mathbb{K}} \cdot g) \big) \cdot {}^\sigma\chi_\mathbb{K} (\det g) \cdot \lvert\det g\rvert_{\mathbb{K}}^{s_0-{1}/{2}} \operatorname{d}\!\overline{{}^\sigma m}(g)\\ &\quad = \sigma\bigg( \frac{1}{\operatorname{L}(s_0, \Pi_\mathbb{K}\times \Sigma_\mathbb{K} \times \chi_\mathbb{K})} \cdot \int_{{\mathrm{N}}_{n-1}(\mathbb{K})\backslash {\mathrm{GL}}_{n-1}(\mathbb{K})} f \bigg( \!\begin{bmatrix} t_{\sigma,\mathbb{K}}^{-1}\mathbf{t}_{n-1, \sigma, \mathbb{K}}\, g & 0 \\ 0 & 1\end{bmatrix}\!\bigg) \\ &\qquad \cdot f' (\mathbf{t}_{n-1, \sigma, \mathbb{K}} \cdot g) \chi_\mathbb{K} (\det g) \cdot \lvert\det g\rvert_{\mathbb{K}}^{s_0-{1}/{2}} \operatorname{d}\!\overline{ m}(g)\bigg)\\ &\quad = {}^\sigma \chi_{\Sigma_\mathbb{K}}(t_{\sigma, \mathbb{K}})\cdot {}^\sigma \chi_\mathbb{K}(t_{\sigma, \mathbb{K}})^ {{n(n-1)}/{2}}\cdot \sigma( \operatorname{Z}^\circ(f \otimes f' \otimes m, s_0,\chi_\mathbb{K})). \end{align*}

By [Reference Jacquet, Piatetskii-Shapiro and ShalikaJPSS83, Theorem 2.7] the map $s\mapsto \operatorname {Z}^\circ (f\otimes f' \otimes m, s, \chi _\mathbb {K})$ is an element of the ring ${\mathbb {C}}[q^{s-{1}/{2}}, q^{{1}/{2}-s}]$. Therefore, the above equality holds for all $s_0\in \frac {1}{2}+{\mathbb {Z}}$. Hence, by (54) and (55), the diagram in the proposition is commutative.

6.1 Canonical generators of the cohomology spaces

Put $K_{n,\infty } :=\prod _{v|\infty } K_{n, \mathrm {k}_v}$ ($n\geqslant 1$), where $K_{n, \mathrm {k}_v}$ is the standard maximal compact subgroup of ${\mathrm {GL}}_n(\mathrm {k}_v)$ as in (23) for an archimedean place $v$ of $\mathrm {k}$. Define a one-dimensional real vector space

\[ \omega_{n,\infty}({\mathbb{R}}):=\wedge^{d_{n,\infty}} (\frak{gl}_{n}(\mathrm{k}_\infty) / \frak{k}_{n,\infty}), \]

where

\[ d_{n,\infty}:= \sum_{v|\infty} d_{n, \mathrm{k}_v} = \dim_{\mathbb{R}} (\frak{gl}_{n}(\mathrm{k}_\infty) / \frak{k}_{n,\infty}). \]

Put

\[ \omega_{n,\infty}:=\omega_{n,\infty}({\mathbb{R}})\otimes_{\mathbb{R}} {\mathbb{C}}. \]

Similar to (25) in the archimedean case, denote by $\frak {O}_{n,\infty }$ the complex orientation space of $\omega _{n,\infty }$, and put

(58)\begin{equation} \widetilde{\frak{O}}_{n,\infty}:= \frak{O}_{n-1,\infty}\otimes \cdots\otimes \frak{O}_{1,\infty}\otimes \frak{O}_{0,\infty}. \end{equation}

By convention, we set $\widetilde {\frak {O}}_{0,\infty }:=\frak {O}_{0,\infty }:={\mathbb {C}}$. For any $m\geqslant 0$, we identify $\frak {O}_{m,\infty }\otimes \frak {O}_{m,\infty }$ with ${\mathbb {C}}$ in the obvious way. Then we have that

(59)\begin{equation} \widetilde{\frak{O}}_{n,\infty} \otimes \widetilde{\frak{O}}_{n-1,\infty} = \frak{O}_{n-1,\infty}. \end{equation}

Let $\mu =\{\mu ^\iota \}_{\iota \in {\mathcal {E}}_\mathrm {k}} \in ({\mathbb {Z}}^n)^{{\mathcal {E}}_\mathrm {k}}$ be a highest weight that is pure as in the introduction. For every archimedean place $v$ of $\mathrm {k}$, view ${\mathcal {E}}_{\mathrm {k}_v}$ as a subset of ${\mathcal {E}}_{\mathrm {k}}$ in the obvious way, and set

(60)\begin{equation} \mu_v:=\{\mu^{\iota}\}_{\iota\in {\mathcal{E}}_{\mathrm{k}_v}} \in ({\mathbb{Z}}^n)^{{\mathcal{E}}_{\mathrm{k}_v}}. \end{equation}

Put

\[ \Omega(\mu) := \left\{ \widehat \otimes_{v|\infty} \pi_{\mu_v}: \pi_{\mu_v}\in \Omega(\mu_v)\right\} \]

and

\[ {\mathcal{H}}_{\mu} :=\bigoplus_{\pi_{\mu}\in \Omega(\mu)} {\mathcal{H}}(\pi_\mu), \]

where

\[ {\mathcal{H}}(\pi_\mu):= {\mathrm{H}}_\mathrm{ct}^{b_{n,\infty}}({\mathbb{R}}^\times_+\backslash {\mathrm{GL}}_n(\mathrm{k}_\infty)^0; F_\mu^\vee\otimes \pi_{\mu})\otimes \widetilde{\frak{O}}_{n,\infty}. \]

Here $b_{n,\infty } = \sum _{v|\infty } b_{n, \mathrm {k}_v}$ is as in the introduction, and ${\mathbb {R}}^\times _+$ is identified with a central subgroup of ${\mathrm {GL}}_n(\mathrm {k}_\infty )$ via the diagonal embedding.

Recall from the introduction that $F_\mu$ is an irreducible algebraic representation of ${\mathrm {GL}}_n(\mathrm {k}\otimes _\mathbb {Q} \mathbb {C})$ of highest weight $\mu$. It has a decomposition

\[ F_\mu=\otimes_{v|\infty} F_{\mu_v}. \]

For every archimedean place $v$ of $\mathrm {k}$, we have fixed a generator $v_{\mu _v}\in (F_{\mu _v})^{\mathrm {N}_n(\mathrm {k}_v\otimes _{\mathbb {R}} {\mathbb {C}})}$. This yields a generator

\[ v_{\mu}:=\otimes v_{\mu_v}\in (F_{\mu})^{\mathrm{N}_n(\mathrm{k}\otimes_\mathbb{Q} {\mathbb{C}})}. \]

We remark that the representation $F_\mu$ is unique up to isomorphism, and the pair $(F_\mu,v_\mu )$ is more rigid in the sense that it is unique up to a unique isomorphism. Also recall from (12) the Whittaker functional $\lambda _{\mu _v}$ on $\pi _{\mu _v}\in \Omega (\mu _v)$. By tensor product, this induces the Whittaker functional $\lambda _\mu$ on every $\pi _\mu \in \Omega (\mu )$.

Let $\varepsilon \in \widehat {\pi _0(\mathrm {k}_\infty ^\times )}$. Denote the $\varepsilon$-isotypic component of ${\mathcal {H}}_\mu$ by ${\mathcal {H}}_\mu [\varepsilon ]$ (similar notation will be used without further explanation). Then ${\mathcal {H}}_\mu [\varepsilon ]$ is one-dimensional. In what follows, we will define a canonical generator $\kappa _{\mu, \varepsilon }$ of ${\mathcal {H}}_\mu [\varepsilon ]$, which is determined by the pairs $(F_\mu,v_\mu )$ and $(\pi _\mu, \lambda _\mu )$. Here we suppose that $\pi _\mu$ is the unique representation in $\Omega (\mu )$ such that $\mathcal {H}(\pi _\mu )[\varepsilon ]\neq \{0\}$.

We first consider the case that $\mu =0_{n,\infty }$, the zero weight. For $n=1$, we naturally identify ${\mathcal {H}}_{0_{1,\infty }}[\varepsilon ]$ with ${\mathbb {C}}$, and put $\kappa _{0_{1,\infty },\varepsilon }:=1$ under this identification.

For $n\geqslant 2$, fix

\[ \pi_{0_{n,\infty}} = \widehat \otimes_{v|\infty} \pi_{0_{n,\mathrm{k}_v}}\in \Omega(0_{n,\infty})\quad {\rm and}\quad \pi_{0_{n-1,\infty}}= \widehat \otimes_{v|\infty} \pi_{0_{n-1, \mathrm{k}_v}}\in \Omega(0_{n-1,\infty}), \]

with fixed Whittaker functionals

\[ \lambda_{0_{n, \infty}}\in {\mathrm{Hom}}_{ \mathrm{N}_n(\mathrm{k}_\infty)}(\pi_{0_{n,\infty}}, \psi_{n,\infty})\setminus\{0\} \]

and

\[ \lambda_{0_{n-1, \infty}}\in {\mathrm{Hom}}_{ \mathrm{N}_{n-1}(\mathrm{k}_\infty)}(\pi_{0_{n-1,\infty}}, \psi_{n-1,\infty})\setminus\{0\}. \]

Denote by $\frak {M}_{n-1, \infty }$ the one-dimensional space of invariant measures on ${\mathrm {GL}}_{n-1}(\mathrm {k}_\infty )$. Similar to the local case at each archimedean place, we have an identification

\[ \frak{M}_{n-1,\infty} = \omega_{n-1, \infty}^* \otimes \frak{O}_{n-1,\infty} \]

by push-forward of measures. Similar to (31), we have the normalized Rankin–Selberg integral

\[ \operatorname{Z}^\circ(\cdot, s ) \in {\mathrm{Hom}}_{{\mathrm{GL}}_{n-1}(\mathrm{k}_\infty)} (\pi_{0_{n,\infty}}\widehat{\otimes}\pi_{0_{n-1,\infty}}\otimes \lvert\det\rvert_{\mathrm{k}_\infty}^{s-{1}/{2}}\otimes \frak{M}_{n-1,\infty}, {\mathbb{C}}). \]

In view of (59), we define a map ${\mathcal {P}}_{\infty, 0}$ to be the composition of

\begin{align*} &{\mathcal{P}}_{\infty, 0}: {\mathcal{H}}(\pi_{0_{n,\infty}}) \otimes {\mathcal{H}}(\pi_{0_{n-1,\infty}}) \\ &\quad \rightarrow {\mathrm{H}}_\mathrm{ct}^{d_{n-1,\infty}}( {\mathrm{GL}}_{n-1}(\mathrm{k}_\infty)^0; \pi_{0_{n,\infty}} \widehat\otimes \pi_{0_{n-1,\infty}}) \otimes \frak{O}_{n-1,\infty} \\ &\quad \rightarrow {\mathrm{H}}_\mathrm{ct}^{d_{n-1,\infty}}({\mathrm{GL}}_{n-1}(\mathrm{k}_\infty)^0; \frak{M}_{n-1,\infty}^*)\otimes \frak{O}_{n-1,\infty}={\mathbb{C}}, \end{align*}

where the first arrow is the restriction of cohomology composed with the cup product, and the last arrow is the map induced by the linear functional

\[ \operatorname{Z}^\circ(\cdot, \tfrac{1}{2}) : \pi_{0_{n,\infty}} \widehat\otimes \pi_{0_{n-1,\infty}} \rightarrow \frak{M}_{n-1,\infty}^*. \]

By the non-vanishing hypothesis that is proved in [Reference SunSun17], these modular symbols for all $\pi _{0_{n,\infty }}\in \Omega (0_{n,\infty })$ and $\pi _{0_{n-1,\infty }}\in \Omega (0_{n-1,\infty })$ give the following non-degenerate pairings for all $\varepsilon \in \widehat {\pi _0(\mathrm {k}_\infty ^\times )}$, still denoted by

\[ {\mathcal{P}}_{\infty, 0}: {\mathcal{H}}_{0_{n,\infty}}[\varepsilon]\times {\mathcal{H}}_{0_{n-1,\infty}}[\varepsilon] \to {\mathbb{C}}. \]

We inductively define $\kappa _{0_{n,\infty }, \varepsilon }$ by requiring that

\[ {\mathcal{P}}_{\infty, 0}(\kappa_{0_{n,\infty}, \varepsilon}, \kappa_{0_{n-1,\infty}, \varepsilon})=1. \]

In general, we define

\[ \kappa_{\mu, \varepsilon} := \jmath_\mu (\kappa_{0_{n,\infty},\varepsilon}), \]

where

\[ \jmath_\mu: {\mathcal{H}}_{0_{n,\infty}}[\varepsilon] \to {\mathcal{H}}_\mu[\varepsilon] \]

is the isomorphism induced by the local ones in Proposition 2.1.

6.2 Some actions of ${\mathrm {Aut}}(\mathbb {C})$

Recall the additive character $\psi _{\mathbb {R}}$ from (9). Denote by ${\mathbb {A}}_\mathbb {Q}$ the adele ring of $\mathbb {Q}$. Fix a nontrivial additive character of $\mathrm {k}\backslash {\mathbb {A}}$ as the composition of

(61)\begin{equation} \psi: \mathrm{k}\backslash {\mathbb{A}} \xrightarrow{{\rm Tr}_{\mathrm{k}/\mathbb{Q}}} \mathbb{Q}\backslash {\mathbb{A}}_\mathbb{Q} \to \mathbb{Q}\backslash {\mathbb{A}}_\mathbb{Q} / \widehat{\mathbb{Z}} = {\mathbb{R}}/{\mathbb{Z}} \xrightarrow{\psi_{\mathbb{R}}} {\mathbb{C}}^\times, \end{equation}

where ${\rm Tr}_{\mathrm {k}/\mathbb {Q}}$ is the trace map, and $\widehat {\mathbb {Z}}$ is the profinite completion of ${\mathbb {Z}}$. Write $\psi = \otimes _v\psi _v$, where $\psi _v$ is a character of $\mathrm {k}_v$ for each place $v$ of $\mathrm {k}$. By using $\psi$, we define the character $\psi _{n}$ of $\mathrm {N}_n(\mathbb {A})$ as in (11) ($n\geqslant 1$). Then we have a decomposition $\psi _n = \psi _{n, f}\otimes \psi _{n,\infty }$, where $\psi _{n, f}$ and $\psi _{n,\infty }$ are characters of ${\rm N}_n({\mathbb {A}}_f)$ and ${\rm N}_n(\mathrm {k}_\infty )$, respectively. Here ${\mathbb {A}}_f$ denotes the finite adele ring of $\mathrm {k}$ so that ${\mathbb {A}}={\mathbb {A}}_f\times \mathrm {k}_\infty$.

For every $\sigma \in {\mathrm {Aut}}(\mathbb {C})$, put

\[ \mathbf{t}_{n, \sigma} := (\mathbf{t}_{n,\sigma, \mathrm{k}_v})_{v\nmid \infty}\in {\mathrm{GL}}_n({\mathbb{A}}_f) \qquad (\textrm{see}\; (\mbox{50})), \]

and define an action of ${\mathrm {Aut}}({\mathbb {C}})$ on ${\mathrm {Ind}}^{{\mathrm {GL}}_n({\mathbb {A}}_f)}_{{\rm N}_n({\mathbb {A}}_f)}\psi _{n,f}$ (the smooth induction) by

(62)\begin{equation} {}^\sigma\!f(g):=\sigma\big( f(\mathbf{t}_{ n, \sigma} \cdot g)\big), \end{equation}

where $f\in {\mathrm {Ind}}^{{\mathrm {GL}}_n({\mathbb {A}}_f)}_{{\rm N}_n({\mathbb {A}}_f)} \psi _{n,f}$ and $g\in {\mathrm {GL}}_n(\mathbb {A}_f)$.

Let $\Pi _f$ be a generic irreducible smooth representation of ${\mathrm {GL}}_n({\mathbb {A}}_f)$, with a fixed Whittaker functional

\[ \lambda_f\in {\mathrm{Hom}}_{{\rm N}_n({\mathbb{A}}_f)}(\Pi_f, \psi_{n,f})\setminus\{0\}. \]

As before, we use $\lambda _f$ to realize $\Pi _f$ as a subrepresentation of ${\mathrm {Ind}}^{{\mathrm {GL}}_n({\mathbb {A}}_f)}_{{\rm N}_n({\mathbb {A}}_f)}\psi _{n,f}$, namely the Whittaker model of $\Pi _f$. The rationality field of $\Pi _f$, denoted by $\mathbb {Q}(\Pi _f)$, is the fixed field of the group of field automorphisms $\sigma \in {\rm Aut}({\mathbb {C}})$ such that ${}^\sigma \Pi _f :=\sigma (\Pi _f)= \Pi _f$.

Let $\Pi$ be an irreducible subrepresentation of $\mathcal {A}^\infty ({\mathrm {GL}}_n(\mathrm {k})\backslash {\mathrm {GL}}_n({\mathbb {A}}))$. If $\Pi$ is cuspidal, then the exponent of $\Pi$ is defined to be the real number ${\rm ex}(\Pi )$ such that $\Pi \otimes \lvert \det \rvert _{\mathbb {A}}^{-{\rm ex}(\Pi )}$ is unitarizable, where $\lvert \, \cdot \, \rvert _{\mathbb {A}}$ is the normalized absolute value on ${\mathbb {A}}$.

We say that $\Pi$ is tamely isobaric if

(63)\begin{equation} \Pi \cong {\mathrm{Ind}}^{{\mathrm{GL}}_n({\mathbb{A}})}_{P({\mathbb{A}})} (\Pi_1 \widehat\otimes_{\mathrm{i}}\cdots \widehat\otimes_{\mathrm{i}} \Pi_r)\quad({\rm cf.}~(\mbox{15})), \end{equation}

for a standard parabolic subgroup $P$ of ${\mathrm {GL}}_n$ with Levi subgroup $M_P= {\mathrm {GL}}_{n_1}\times \cdots \times {\mathrm {GL}}_{n_r}$, and irreducible cuspidal subrepresentations $\Pi _i$ of $\mathcal {A}^\infty ({\mathrm {GL}}_n(\mathrm {k})\backslash {\mathrm {GL}}_{n_i}({\mathbb {A}}))$, $i=1,\ldots, r$, that have the same exponent. Here $\widehat \otimes _{\rm i}$ denotes the completed inductive tensor product (see [Reference TrèvesTrè67, Definition 43.5]). We can view the right-hand side of (63) as a space of smooth automorphic forms by using the Eisenstein series (see [Reference LanglandsLan79, Proposition 2] and [Reference Grobner and LinGL21, § 1.4.3] for more details).

Suppose that $\Pi _f$ and $\Pi _\infty$ are the finite and infinite part of $\Pi$, respectively, so that $\Pi = \Pi _f \otimes \Pi _\infty$. Now we assume that $\Pi$ is tamely isobaric and regular algebraic (in the sense of [Reference ClozelClo90] such that (1) holds). By the proof of [Reference GrobnerGro18, Lemma 1.2] (see also [Reference Grobner and LinGL21, § 1.4.3]), for every $\sigma \in {\mathrm {Aut}}({\mathbb {C}})$, ${}^\sigma \Pi _f :=\sigma (\Pi _f)$ given by (62) is the finite part of a unique irreducible subrepresentation ${}^\sigma \Pi$ of ${\mathcal {A}}^\infty ({\mathrm {GL}}_n(\mathrm {k})\backslash {\mathrm {GL}}_n({\mathbb {A}}))$. Moreover, ${}^\sigma \Pi$ is also tamely isobaric and regular algebraic.

Remark 6.1 More precisely, the above assertion holds when $\Pi$ is cuspidal and regular algebraic by [Reference ClozelClo90, Theorem 3.13]. In general, if $\Pi$ is tamely isobaric as in (63) and is regular algebraic, then

\[ \Xi_1\widehat \otimes_{\mathrm{i}}\cdots\widehat \otimes_{\mathrm{i}} \Xi_r\\ :=(\Pi_1\widehat\otimes_{\mathrm{i}}\cdots\widehat \otimes_{\mathrm{i}}\Pi_r)\otimes\rho_P \]

is a regular algebraic irreducible cuspidal subrepresentation of ${\mathcal {A}}^\infty (M_P(\mathrm {k})\backslash M_P({\mathbb {A}}))$, where $\rho _P :=\delta _P^{{1}/{2}}$ is the square root of the modular character $\delta _P$ of $P(\mathbb {A})$. Then we have that

\[ {}^\sigma\Pi \cong {\mathrm{Ind}}^{{\mathrm{GL}}_n({\mathbb{A}})}_{P({\mathbb{A}})} \big({}^\sigma \Xi_1\widehat \otimes_{\mathrm i}\cdots \widehat \otimes_{\mathrm{i}} {}^\sigma\Xi_r\big)\otimes \rho_P^{-1}. \]

Recall that the rationality field of $\Pi$ is defined to be $\mathbb {Q}(\Pi ):=\mathbb {Q}(\Pi _f)$. Let ${\mathrm {Aut}}({\mathbb {C}}/\mathbb {Q}(\Pi ))$ act on $\Pi _f$ by (62). It is known that $\mathbb {Q}(\Pi )$ is a number field and $(\Pi _f)^{{\mathrm {Aut}}({\mathbb {C}}/\mathbb {Q}(\Pi ))}$ is a $\mathbb {Q}(\Pi )$-rational structure of $\Pi _f$ (see [Reference Raghuram and ShahidiRS08b, Lemma 3.2]).

As in the introduction, suppose that $F_\mu$ is an irreducible algebraic representation of ${\mathrm {GL}}_n(\mathrm {k}\otimes _{\mathbb {Q}}\mathbb {C})$ whose highest weight $\mu =\{\mu ^\iota \}_{\iota \in {\mathcal {E}}_\mathrm {k}}\in ({\mathbb {Z}}^n)^{{\mathcal {E}}_\mathrm {k}}$ is pure of weight $w_\mu \in {\mathbb {Z}}$ so that

\[ \mu^\iota_1+\mu^{\bar{\iota}}_n = \cdots = \mu^\iota_n + \mu^{\bar{\iota}}_1= w_\mu\quad \textrm{for all }\iota\in {\mathcal{E}}_\mathrm{k}. \]

Similar to the local case (8), we realize $F_\mu$ as the algebraic induction

\[ F_\mu = {}^{\rm alg}{\mathrm{Ind}}^{{\mathrm{GL}}_n(\mathrm{k} \otimes_\mathbb{Q} {\mathbb{C}})}_{\bar{{\mathrm{B}}}_n(\mathrm{k}\otimes_\mathbb{Q}{\mathbb{C}})}\chi_\mu, \]

and realize $v_\mu \in F_\mu$ as the $\mathrm {N}_n(\mathrm {k}\otimes _{\mathbb {Q}} \mathbb {C})$-invariant function that has value $1$ at the identity matrix. Then the generator $v_\mu ^\vee \in (F_\mu ^\vee )^{\bar {\mathrm {N}}_n(\mathrm {k}\otimes _{\mathbb {Q}} \mathbb {C})}$ is identified with the evaluation map at the identity matrix. Similarly, $F_\mu ^\vee$ is realized as the algebraic induction

\[ F_\mu^\vee = {}^{\rm alg}{\mathrm{Ind}}^{{\mathrm{GL}}_n(\mathrm{k} \otimes_\mathbb{Q} {\mathbb{C}})}_{{{\mathrm{B}}}_n(\mathrm{k}\otimes_\mathbb{Q}{\mathbb{C}})}\chi_{-\mu}. \]

For every $\sigma \in {\mathrm {Aut}}({\mathbb {C}})$, write

\[ {}^\sigma\!\mu = \{ \mu^{\sigma^{-1} \circ \iota}\}_{\iota \in {\mathcal{E}}_\mathrm{k}}\!. \]

As a consequence of the purity lemma [Reference ClozelClo90, Lemma 4.9], $\mu$ necessarily satisfies the condition that (see [Reference GrobnerGro18, Lemma 1.3])

\[ \mu^{\sigma\circ\bar{\iota}} = \mu^{\overline{\sigma\circ\iota}}\quad \textrm{for all $\sigma\in {\mathrm{Aut}}({\mathbb{C}})$ and $\iota\in {\mathcal{E}}_\mathrm{k}$}. \]

Therefore, ${}^\sigma \!\mu$ is also pure of weight $w_\mu$. The rationality field $\mathbb {Q}(F_\mu )$ is defined to be the fixed field of the group of field automorphisms $\sigma \in {\mathrm {Aut}}({\mathbb {C}})$ such that ${}^\sigma \!\mu = \mu$.

Let ${\mathrm {Aut}}({\mathbb {C}})$ act on the space of algebraic functions on ${\mathrm {GL}}_n(\mathrm {k} \otimes _\mathbb {Q}{\mathbb {C}})$ by

(64)\begin{equation} ({}^\sigma\!f)(x):= \sigma( f(\sigma^{-1}x))\quad (x\in {\mathrm{GL}}_n(\mathrm{k}\otimes_\mathbb{Q}{\mathbb{C}})), \end{equation}

where ${\mathrm {Aut}}({\mathbb {C}})$ acts on ${\mathrm {GL}}_n(\mathrm {k} \otimes _\mathbb {Q}{\mathbb {C}})$ through its action on the second factor of $\mathrm {k}\otimes _\mathbb {Q} {\mathbb {C}}$. Then

\[ \sigma(F_\mu)= F_{ {}^\sigma\!\mu} \quad \textrm{and} \quad \sigma(F_\mu^\vee)= F_{ {}^\sigma\!\mu}^\vee. \]

Define

(65)\begin{equation} {\mathcal{X}}_{n}:=({\mathbb{R}}^\times_+\cdot {\mathrm{GL}}_{n}(\mathrm{k}))\backslash {\mathrm{GL}}_{n}({\mathbb{A}})/ K_{n,\infty}^0. \end{equation}

For every open compact subgroup $K_f$ of ${\mathrm {GL}}_n({\mathbb {A}}_f)$, the finite-dimensional representation $F_\mu ^\vee$ defines a sheaf on ${\mathcal {X}}_n/K_f$, which is still denoted by $F_\mu ^\vee$. Let ${\mathrm {H}}^{b_{n,\infty }}({\mathcal {X}}_n/ K_f, F_\mu ^\vee )$ be the sheaf cohomology group, and define

(66)\begin{align} {\mathcal{H}}^{b_{n,\infty}}({\mathcal{X}}_n, F_\mu^\vee)& := {\mathrm{H}}^{b_{n,\infty}}({\mathcal{X}}_n, F_\mu^\vee) \otimes \widetilde{\frak{O}}_{n,\infty} \nonumber\\ &:= \varinjlim_{K_f}{\mathrm{H}}^{b_{n,\infty}}({\mathcal{X}}_n/ K_f, F_\mu^\vee)\otimes \widetilde{\frak{O}}_{n,\infty}, \end{align}

where $K_f$ runs over the directed system of open compact subgroups of ${\mathrm {GL}}_n({\mathbb {A}}_f)$.

Note that $\widetilde {\frak {O}}_{n,\infty }$ has a natural $\mathbb {Q}$-structure. For every $\sigma \in {\mathrm {Aut}}({\mathbb {C}})$, the map (64) induces a $\sigma$-linear map

(67)\begin{equation} \sigma: {\mathcal{H}}^{b_{n,\infty}}({\mathcal{X}}_n, F_\mu^\vee) \to {\mathcal{H}}^{b_{n,\infty}}({\mathcal{X}}_n, F_{{}^\sigma\!\mu}^\vee). \end{equation}

Put

\[ {\mathrm{GL}}_n({\mathbb{A}})^\natural:= {\mathrm{GL}}_n({\mathbb{A}}_f)\times \pi_0(\mathrm{k}_\infty^\times). \]

Then both the domain and codomain of the map (67) are naturally smooth representations of ${\mathrm {GL}}_n({\mathbb {A}})^\natural$, and the map (67) is ${\mathrm {GL}}_n({\mathbb {A}})^\natural$-equivariant.

6.3 Definition of the Whittaker periods

As above, $\Pi$ is an irreducible subrepresentation of ${\mathcal {A}}^\infty ({\mathrm {GL}}_n(\mathrm {k})\backslash {\mathrm {GL}}_n(\mathbb {A}))$ which is assumed to be tamely isobaric and regular algebraic. Fix the Haar measure on $\mathrm {N}_n({\mathbb {A}})$ to be the product of self-dual Haar measures on ${\mathbb {A}}$ with respect to $\psi$, as in (16). Then we have a nonzero continuous linear functional

(68)\begin{equation} \lambda \in {\mathrm{Hom}}_{\mathrm{N}_n({\mathbb{A}})}(\Pi, \psi_n),\quad \varphi\mapsto \int_{\mathrm{N}_n(\mathrm{k})\backslash\mathrm{N}_n({\mathbb{A}})} \varphi(u) \cdot \overline{\psi_n}(u) \operatorname{d}\! u. \end{equation}

By the uniqueness of Whittaker models, we have a factorization

(69)\begin{equation} \lambda = \lambda_f \otimes \lambda_\infty, \end{equation}

where

\[ \lambda_f\in {\mathrm{Hom}}_{\mathrm{N}_n({\mathbb{A}}_f)}(\Pi_f, \psi_{n,f}) \]

as before, and

\[ \lambda_\infty\in {\mathrm{Hom}}_{\mathrm{N}_n(\mathrm{k}_\infty)}(\Pi_\infty, \psi_{n,\infty}). \]

More generally, for every $\sigma \in {\mathrm {Aut}}(\mathbb {C})$, let ${}^\sigma \lambda \in {\mathrm {Hom}}_{\mathrm {N}_n({\mathbb {A}})}({}^\sigma \Pi, \psi _n)$ be the Whittaker functional defined by the integrals as in (68). Similar to (69), we also have factorizations

\[ {}^\sigma \Pi= {}^\sigma\Pi_f \otimes {}^\sigma \Pi_\infty\quad\textrm{and}\quad {}^\sigma\lambda = {}^\sigma \lambda_f \otimes {}^\sigma \lambda_\infty. \]

Recall that ${}^\sigma \Pi _f :=\sigma (\Pi _f)$ is realized as a space of Whittaker functions so that ${}^\sigma \lambda _f$ is realized as the evaluation map at the identity matrix.

Suppose that $F_\mu$ is the coefficient system of $\Pi$ as in the introduction. Then $F_{{}^\sigma \!\mu }$ is the coefficient system of ${}^\sigma \Pi$ (cf.  [Reference ClozelClo90, Theorem 3.13] and [Reference GrobnerGro18, Corollary 1.4]) so that

\[ {\mathcal{H}}({}^\sigma\Pi_\infty) :={\mathrm{H}}_\mathrm{ct}^{b_{n,\infty}}({\mathbb{R}}^\times_+\backslash {\mathrm{GL}}_n(\mathrm{k}_\infty)^0; F_{ {}^\sigma\!\mu}^\vee\otimes {}^\sigma\Pi_\infty)\otimes \widetilde{\frak{O}}_{n,\infty}\neq \{0\}. \]

Consequently, $\mathbb {Q}(F_\mu )\subset \mathbb {Q}(\Pi )$. Put

\[ {\mathcal{H}}({}^\sigma\Pi) :={\mathrm{H}}_\mathrm{ct}^{b_{n,\infty}}({\mathbb{R}}^\times_+\backslash {\mathrm{GL}}_n(\mathrm{k}_\infty)^0; F_{ {}^\sigma\!\mu}^\vee\otimes {}^\sigma\Pi)\otimes \widetilde{\frak{O}}_{n,\infty}. \]

Then we have the canonical isomorphism

\[ \iota_{\rm can}: {}^\sigma\Pi_f\otimes {\mathcal{H}}({}^\sigma\Pi_\infty) \to {\mathcal{H}}({}^\sigma\Pi). \]

Following [Reference ClozelClo90, Lemma 3.15] and [Reference GrobnerGro18, Proposition 1.6], we have a ${\mathrm {GL}}_n({\mathbb {A}})^\natural$-equivariant embedding

(70)\begin{equation} \iota_{\Pi}: \Pi_f\otimes {\mathcal{H}}(\Pi_\infty) = {\mathcal{H}}(\Pi) \hookrightarrow {\mathcal{H}}^{b_{n,\infty}}({\mathcal{X}}_n, F_\mu^\vee). \end{equation}

Let $\varepsilon \in \widehat { \pi _0(\mathrm {k}_\infty ^\times )}$ be the character $\varepsilon _{\Pi _\infty }\cdot \operatorname {sgn}_\infty ^{(n-1)(n-2)/{2}}$ when $n$ is odd, and be arbitrary when $n$ is even. Then we have a ${\mathrm {GL}}_n({\mathbb {A}})^\natural$-equivariant linear embedding

\[ \iota_{\Pi, \varepsilon}: \Pi_f\otimes {\mathcal{H}}(\Pi_\infty)[\varepsilon] = {\mathcal{H}}(\Pi)[\varepsilon] \hookrightarrow {\mathcal{H}}^{b_{n,\infty}}({\mathcal{X}}_n, F_\mu^\vee). \]

Proposition 6.2 Let the notation and assumptions be as above. Then

\[ \dim_{{\mathrm{GL}}_n({\mathbb{A}})^\natural}({\mathcal{H}}(\Pi)[\varepsilon], {\mathcal{H}}^{b_{n,\infty}}({\mathcal{X}}_n, F_\mu^\vee))=1, \]

and for every $\sigma \in {\mathrm {Aut}}(\mathbb {C})$ the map (67) induces the following commutative diagram.

Moreover, under the action given by the left vertical arrow of the above diagram, $({\mathcal {H}}(\Pi )[\varepsilon ])^{{\mathrm {Aut}}({\mathbb {C}}/\mathbb {Q}(\Pi ))}$ is a $\mathbb {Q}(\Pi )$-rational structure of ${\mathcal {H}}(\Pi )[\varepsilon ]$.

We equip ${\mathcal {H}}(\Pi )[\varepsilon ]$ with the action of ${\mathrm {Aut}}({\mathbb {C}}/\mathbb {Q}(\Pi ))$ given by Proposition 6.2. Write

\[ \Pi^\natural:=\Pi_f\otimes \varepsilon \cong {\mathcal{H}}(\Pi)[\varepsilon], \]

where $\varepsilon \in \widehat { \pi _0(\mathrm {k}_\infty ^\times )}$ is identified with ${\mathbb {C}}$ as a vector space. We equip $\Pi ^\natural$ with the action of ${\mathrm {Aut}}({\mathbb {C}}/\mathbb {Q}(\Pi ))$ given by its action on $\Pi _f$ as in (62) and its natural action on $\mathbb {C}$.

Lemma 6.4 There exists a generator

\[ \omega_{\Pi^\natural}\in {\mathrm{Hom}}_{{\mathrm{GL}}_n({\mathbb{A}})^\natural}(\Pi^\natural, {\mathcal{H}}(\Pi)[\varepsilon]) \]

that is ${\mathrm {Aut}}({\mathbb {C}}/\mathbb {Q}(\Pi ))$-equivariant. Moreover, such a generator is unique up to multiplication by scalar in $\mathbb {Q}(\Pi )^\times$.

Proof. Recall that $(\Pi ^\natural )^{{\mathrm {Aut}}({\mathbb {C}}/\mathbb {Q}(\Pi ))}$ is a $\mathbb {Q}(\Pi )$-rational structure of $\Pi ^\natural$ (see [Reference Raghuram and ShahidiRS08b, Lemma 3.2]), and $({\mathcal {H}}(\Pi )[\varepsilon ])^{{\mathrm {Aut}}({\mathbb {C}}/\mathbb {Q}(\Pi ))}$ is a $\mathbb {Q}(\Pi )$-rational structure of ${\mathcal {H}}(\Pi )[\varepsilon ]$ (Proposition 6.2). Let $\overline {\mathbb {Q}}$ denote the field of algebraic numbers in $\mathbb {C}$. By the multiplicity one property of new vectors, the $\overline {\mathbb {Q}}$-rational structure of $\Pi _f$ is unique up to homotheties (see the proof of [Reference ClozelClo90, Theorem 3.13], and [Reference WaldspurgerWal85, Chapter I]). It follows that $\Pi ^\natural$ and ${\mathcal {H}}(\Pi )[\varepsilon ]$ are isomorphic over $\overline {\mathbb {Q}}$. Since $(\Pi ^\natural )^{{\mathrm {Aut}}({\mathbb {C}}/\overline {\mathbb {Q}})}$ is irreducible (as a smooth representation of ${\mathrm {GL}}_n({\mathbb {A}})^\natural$ over $\overline {\mathbb {Q}}$), ${\mathrm {Aut}}(\overline {\mathbb {Q}}/\mathbb {Q}(\Pi ))$ acts continuously on the one-dimensional $\overline {\mathbb {Q}}$-vector space (with the discrete topology)

\[ {\mathrm{Hom}}_{{\mathrm{GL}}_n({\mathbb{A}})^\natural}\big((\Pi^\natural)^{{\mathrm{Aut}}({\mathbb{C}}/\overline{\mathbb{Q}})}, ({\mathcal{H}}(\Pi)[\varepsilon])^{{\mathrm{Aut}}({\mathbb{C}}/\overline{\mathbb{Q}})}\big). \]

This implies the existence of $\omega _{\Pi ^\natural }$ by [Reference SpringerSpr98, Proposition 11.1.6]. The uniqueness is obvious.

Fix $\omega _{\Pi ^\natural }$ as in Lemma 6.4. For $\sigma \in {\mathrm {Aut}}({\mathbb {C}})$, put

\[ {}^\sigma \Pi^{\natural}:={}^\sigma\Pi_f\otimes \varepsilon. \]

The $\sigma$-linear isomorphisms $\sigma : \Pi ^{\natural }\to {}^\sigma \Pi ^{\natural }$ and

\[ \sigma: {\mathcal{H}}(\Pi)[\varepsilon] \to {\mathcal{H}}({}^\sigma\Pi)[\varepsilon] \quad (\mbox{see Proposition 6.2}) \]

induce a $\sigma$-linear isomorphism

\[ \sigma: {\mathrm{Hom}}_{{\mathrm{GL}}_n({\mathbb{A}})^\natural}(\Pi^\natural, {\mathcal{H}}(\Pi)[\varepsilon]) \to {\mathrm{Hom}}_{{\mathrm{GL}}_n({\mathbb{A}})^\natural}({}^\sigma \Pi^{\natural}, {\mathcal{H}}({}^\sigma\Pi)[\varepsilon]). \]

Using this isomorphism, we define

\[ \omega_{{}^\sigma\Pi^{\natural}}:=\sigma(\omega_{\Pi^\natural})\in {\mathrm{Hom}}_{{\mathrm{GL}}_n({\mathbb{A}})^\natural}({}^\sigma\Pi^{\natural}, {\mathcal{H}}({}^\sigma\Pi)[\varepsilon]). \]

Unraveling definitions, we have the following commutative diagram.

(71)

Recall form § 6.1 that the pairs $(F_\mu,v_\mu )$ and $(\Pi _\infty, \lambda _\infty )$ determine a generator $\kappa _{\mu, \varepsilon }$ of ${\mathcal {H}}(\Pi _\infty )[\varepsilon ]={\mathcal {H}}_\mu [\varepsilon ]$. More generally, for every $\sigma \in {\mathrm {Aut}}(\mathbb {C})$, the pairs $(F_{{}^\sigma \!\mu },v_{{}^\sigma \!\mu })$ and $({}^\sigma \Pi _\infty, {}^\sigma \lambda _\infty )$ determine a generator $\kappa _{{}^\sigma \!\mu, \varepsilon }$ of ${\mathcal {H}}({}^\sigma \Pi _\infty )[\varepsilon ]={\mathcal {H}}_{{}^\sigma \!\mu }[\varepsilon ]$.

Definition 6.5 For every $\sigma \in {\mathrm {Aut}}(\mathbb {C})$, the Whittaker period $\Omega _\varepsilon ({}^\sigma \Pi )\in {\mathbb {C}}^\times$ is the unique scalar such that the following diagram commutes.

(72)

Up to scalar multiplication by $\mathbb {Q}(\Pi )^\times$, the Whittaker periods defined above are independent of the choice of the generator $\omega _{\Pi ^\natural }$. More precisely, we have the following lemma.

Lemma 6.6 Let $c\in \mathbb {Q}(\Pi )^\times$ so that $\omega _{\Pi ^\natural }' := c\cdot \omega _{\Pi ^\natural }\in {\mathrm {Hom}}_{{\mathrm {GL}}_n({\mathbb {A}})^\natural }(\Pi ^\natural, {\mathcal {H}}(\Pi )[\varepsilon ])$ is another generator that is ${\mathrm {Aut}}({\mathbb {C}}/\mathbb {Q}(\Pi ))$-equivariant, which defines a corresponding family of Whittaker periods $\{\Omega '_\varepsilon ({}^\sigma \Pi )\}_{\sigma \in {\mathrm {Aut}}({\mathbb {C}})}$. Then for all $\sigma \in {\mathrm {Aut}}({\mathbb {C}})$,

\[ \sigma\bigg(\frac{\Omega'_\varepsilon(\Pi)}{\Omega_\varepsilon(\Pi)}\bigg) = \frac{\Omega'_\varepsilon({}^\sigma\Pi)}{\Omega_\varepsilon({}^\sigma\Pi)}= \sigma(c^{-1}). \]

For every $\sigma \in {\mathrm {Aut}}({\mathbb {C}})$, we define a $\sigma$-linear map

(73)\begin{equation} \sigma: {\mathcal{H}}(\Pi_\infty)[\varepsilon] \to {\mathcal{H}}({}^\sigma \Pi_\infty)[\varepsilon] \end{equation}

such that

\[ \sigma(\kappa_{\mu,\varepsilon}) = \kappa_{{}^\sigma\!\mu,\varepsilon}. \]

Proposition 6.7 For all $\sigma \in {\mathrm {Aut}}({\mathbb {C}})$, the diagram

(74)

commutes, where the left vertical arrow is the $\sigma$-linear map induced by the map $\sigma : \Pi _f\to {}^\sigma \Pi _f$ and the map (73).

7.1 Rankin–Selberg integrals

In this subsection, let $\Pi$ be an irreducible cuspidal subrepresentation of ${\mathcal {A}}^\infty ({\mathrm {GL}}_n(\mathrm {k})\backslash {\mathrm {GL}}_n({\mathbb {A}}))$ ($n\geqslant 2$), and let $\Sigma$ be an irreducible tamely isobaric subrepresentation of ${\mathcal {A}}^\infty ({\mathrm {GL}}_{n-1}(\mathrm {k})\backslash {\mathrm {GL}}_{n-1}({\mathbb {A}}))$.

As in (68), we have Whittaker functionals

\[ \lambda \in {\mathrm{Hom}}_{\mathrm{N}_n({\mathbb{A}})}(\Pi, \psi_n)\quad \textrm{and}\quad \lambda' \in {\mathrm{Hom}}_{\mathrm{N}_{n-1}({\mathbb{A}})}(\Sigma, \psi_{n-1}) \]

defined by integrals, and as in (69), we have decompositions

\[ \lambda = \lambda_f \otimes \lambda_\infty\quad \textrm{and}\quad \lambda' = \lambda'_f \otimes \lambda'_\infty, \]

with

\[ \lambda_f\in {\mathrm{Hom}}_{\mathrm{N}_n({\mathbb{A}}_f)}(\Pi_f, \psi_{n,f}), \quad \lambda_\infty \in {\mathrm{Hom}}_{\mathrm{N}_n(\mathrm{k}_\infty)}(\Pi_\infty, \psi_{n,\infty}), \quad \Pi=\Pi_f\otimes \Pi_\infty, \]

and

\[ \lambda'_f\in {\mathrm{Hom}}_{\mathrm{N}_{n-1}({\mathbb{A}}_f)}(\Sigma_f, \psi_{n-1,f}), \quad \lambda'_\infty \in {\mathrm{Hom}}_{\mathrm{N}_{n-1}(\mathrm{k}_\infty)}(\Sigma_\infty, \psi_{n-1,\infty}), \quad \Sigma=\Sigma_f\otimes \Sigma_\infty. \]

Let $\chi : \mathrm {k}^\times \backslash {\mathbb {A}}^\times \to {\mathbb {C}}^\times$ be a Hecke character. Similar to (30), for each $t\in \mathbb {C}$ define a character

(75)\begin{equation} \chi_t:= \chi \cdot \lvert\,\cdot\,\rvert_{\mathbb{A}}^t: \mathbb{A}^\times \rightarrow \mathbb{C}^\times. \end{equation}

As usual, write

\[ \chi=\otimes_{v} \chi_v=\chi_f \otimes \chi_\infty \quad \textrm{and}\quad \chi_t = \chi_{f, t} \otimes \chi_{\infty, t}. \]

Denote by $\frak {M}_{n-1}$ and $\frak {M}_{n-1,f}$ the one-dimensional spaces of invariant measures on ${\mathrm {GL}}_{n-1}({\mathbb {A}})$ and ${\mathrm {GL}}_{n-1}({\mathbb {A}}_f)$, respectively, so that

\[ \frak{M}_{n-1} = \frak{M}_{n-1, f}\otimes \frak{M}_{n-1,\infty}. \]

Similar to (31), we have the finite part of the normalized Rankin–Selberg integral

\[ \operatorname{Z}^\circ(\cdot, s, \chi_f)\in {\mathrm{Hom}}_{{\mathrm{GL}}_{n-1}({\mathbb{A}}_f)}(\Pi_f\otimes \Sigma_f \otimes \chi_{f, s-{1}/{2}}\otimes \frak{M}_{n-1,f}, {\mathbb{C}}), \]

and the normalized Rankin–Selberg integral at infinity

\[ \operatorname{Z}^\circ(\cdot, s, \chi_\infty) \in {\mathrm{Hom}}_{{\mathrm{GL}}_{n-1}(\mathrm{k}_\infty)} (\Pi_\infty\widehat{\otimes}\Sigma_\infty\otimes \chi_{\infty, s-{1}/{2}}\otimes \frak{M}_{n-1,\infty}, {\mathbb{C}}). \]

Define the global Rankin–Selberg period integral

\[ \operatorname{Z}(\cdot, s, \chi) : \, \Pi \widehat{\otimes} \Sigma\otimes \chi_{s-{1}/{2}} \otimes \frak{M}_{n-1} \rightarrow {\mathbb{C}} \]

by

\[ \operatorname{Z}(\varphi\otimes \varphi'\otimes 1\otimes m):=\int_{ {\mathrm{GL}}_{n-1}(\mathrm{k})\backslash {\mathrm{GL}}_{n-1}({\mathbb{A}})} \varphi\bigg(\begin{bmatrix} g & 0 \\ 0 & 1\end{bmatrix}\bigg) \cdot \varphi'(g) \cdot \chi(g)\cdot \lvert\det g\rvert_{\mathbb{A}}^{s-{1}/{2}} \operatorname{d}\! \bar{m}(g), \]

where $\varphi \in \Pi$, $\varphi '\in \Sigma$, $m\in \frak {M}_{n-1}$ and $\bar {m}$ is the quotient measure of $m$. Here

\[ \Pi\widehat\otimes \Sigma:= (\Pi_\infty\widehat{\otimes}\Sigma_\infty)\otimes (\Pi_f\otimes \Sigma_f). \]

The following proposition reformulates the Euler factorization of Rankin–Selberg period integrals established in [Reference Jacquet and ShalikaJS81b, p. 796, (7)].

Proposition 7.1 For $\Pi$, $\Sigma$, $\chi$ as above, and all $s\in {\mathbb {C}}$, the following diagram commutes.

7.2 Modular symbols and modular symbols at infinity

From now on, further assume that $\Pi$ and $\Sigma$ are regular algebraic with balanced coefficient systems $F_\mu$ and $F_\nu$, respectively, and assume that $\chi$ has finite order.

Similar to (66) we have the following space given by sheaf cohomology with compact support:

(76)\begin{align} {\mathcal{H}}_c^{b_{n,\infty}}({\mathcal{X}}_n, F_\mu^\vee)& := {\mathrm{H}}_c^{b_{n,\infty}}({\mathcal{X}}_n, F_\mu^\vee) \otimes \widetilde{\frak{O}}_{n,\infty} \nonumber\\ &:= \varinjlim_{K_f}{\mathrm{H}}_c^{b_{n,\infty}}({\mathcal{X}}_n/ K_f, F_\mu^\vee)\otimes \widetilde{\frak{O}}_{n,\infty}, \end{align}

where $K_f$ runs over the directed system of open compact subgroups of ${\mathrm {GL}}_n({\mathbb {A}}_f)$. This is also a smooth representation of ${\mathrm {GL}}_n(\mathbb {A})^\natural$ and we have a natural ${\mathrm {GL}}_n(\mathbb {A})^\natural$-equivariant linear map

\[ \iota_\mu: {\mathcal{H}}_c^{b_{n,\infty}}({\mathcal{X}}_n, F_\mu^\vee)\rightarrow {\mathcal{H}}^{b_{n,\infty}}({\mathcal{X}}_n, F_\mu^\vee). \]

Since $\Pi$ is cuspidal, there is a natural embedding

\[ \iota_\Pi': {\mathcal{H}}(\Pi) \hookrightarrow {\mathcal{H}}^{b_{n,\infty}}_c({\mathcal{X}}_n, F_\mu^\vee) \]

such that $\iota _\mu \circ \iota _\Pi '= \iota _\Pi$ (see [Reference ClozelClo90, Lemma 3.15]).

Put

\[ \widetilde{{\mathcal{X}}}_{n-1}:= {\mathrm{GL}}_{n-1}(\mathrm{k})\backslash {\mathrm{GL}}_{n-1}({\mathbb{A}})/ K_{n-1,\infty}^0. \]

The embedding $\imath : {\mathrm {GL}}_{n-1}({\mathbb {A}}) \hookrightarrow {\mathrm {GL}}_n({\mathbb {A}})$ given by (26) induces a proper map, still denoted by

\[ \imath: \widetilde{{\mathcal{X}}}_{n-1} \to {\mathcal{X}}_n, \]

which induces a map

\[ \imath^*: {\mathcal{H}}^{b_{n,\infty}}_c({\mathcal{X}}_n, F_\mu^\vee) \to {\mathcal{H}}^{b_{n,\infty}}_c(\widetilde{{\mathcal{X}}}_{n-1}, F_\mu^\vee). \]

The natural map $\wp : \widetilde {{\mathcal {X}}}_{n-1}\to {\mathcal {X}}_{n-1}$ induces a map

\[ \wp^*: {\mathcal{H}}^{b_{n-1,\infty}}({\mathcal{X}}_{n-1}, F_\nu^\vee) \to {\mathcal{H}}^{b_{n-1,\infty}}(\widetilde{{\mathcal{X}}}_{n-1}, F_\nu^\vee). \]

Since $\xi :=(\mu, \nu )$ is assumed to be balanced, by [Reference RaghuramRag16, Theorem 2.21] (see also [Reference Kasten and SchmidtKS13, Theorem 2.3] and [Reference Grobner and HarrisGH16, Lemma 4.7]) we have that

\[ \{j\in {\mathbb{Z}} \, :\, j\textrm{ is balanced for }\xi\} =\big\{j\in {\mathbb{Z}} \, : \, \tfrac{1}{2}+j \textrm{ is a critical place of }\Pi\times\Sigma\big\}. \]

Recall that a half-integer $\frac {1}{2}+j$ is a critical place of $\Pi \times \Sigma$ if it is a pole of neither $\operatorname {L}(s, \Pi _\infty \times \Sigma _\infty )$ nor $\operatorname {L}(1-s, \Pi _\infty ^\vee \times \Sigma _\infty ^\vee )$.

Let $j$ be a balanced place for $\xi$. Define an algebraic character

\[ \delta_{ j}:=\otimes_{\iota\in {\mathcal{E}}_\mathrm{k}} {\det}^j \]

of ${\mathrm {GL}}_{n-1}(\mathrm {k} \otimes _\mathbb {Q}{\mathbb {C}})$. Put

\[ \operatorname{H}(\chi_j) : = {\mathrm{H}}^0_{\rm ct}({\mathbb{R}}^\times_+\backslash {\mathrm{GL}}_{n-1}(\mathrm{k}_\infty)^0; \delta_{j}^\vee\otimes \chi_{ j})\quad(\textrm{see}\; (\mbox{75}) \textrm{for the definition of}\; {\chi_j}). \]

Then we have a natural injective map

\[ \iota_j: \operatorname{H}(\chi_j) \hookrightarrow {\mathrm{H}}^0({\mathcal{X}}_{n-1}, \delta_{ j}^\vee). \]

With the notation as before, we have the generators

\[ v^\vee_\mu := \otimes_{\iota\in {\mathcal{E}}_\mathrm{k}} v_{\mu^\iota}^\vee\in (F_\mu^\vee)^{\bar{\mathrm{N}}_n(\mathrm{k}\otimes_\mathbb{Q} \mathbb{C})}\quad \textrm{and}\quad v_\nu^\vee := \otimes_{\iota\in {\mathcal{E}}_\mathrm{k}} v_{\nu^\iota}^\vee\in (F_\nu^\vee)^{\bar{\mathrm{N}}_n(\mathrm{k}\otimes_\mathbb{Q} \mathbb{C})}. \]

Put $F_\xi := F_\mu \otimes F_\nu$ and $v_\xi ^\vee : = v_\mu ^\vee \otimes v_\nu ^\vee$. Recall from (37) the element

\[ z=(z_n, z_{n-1})\in {\mathrm{GL}}_n(\mathbb{Z})\times {\mathrm{GL}}_{n-1}(\mathbb{Z})\subset {\mathrm{GL}}_n(\mathrm{k})\times {\mathrm{GL}}_{n-1}(\mathrm{k}). \]

By Proposition 3.1, we have a unique element

\[ \phi_{\xi, j}\in {\mathrm{Hom}}_{{\mathrm{GL}}_{n-1}(\mathrm{k} \otimes_{{\mathbb{R}}}{\mathbb{C}})}(F_\xi^\vee \otimes \delta_{ j}^\vee, \mathbb{C}) \]

such that $\phi _{\xi, j} \big ( z.v_\xi ^\vee \otimes 1 \big )=1$. Then $\phi _{\xi, j}$ induces a linear map

\[ \phi_{\xi, j}: {\mathrm{H}}^{d_{n-1,\infty}}_c(\widetilde{{\mathcal{X}}}_{n-1}, F_\xi^\vee)\otimes {\mathrm{H}}^0(\widetilde{{\mathcal{X}}}_{n-1}, \delta_{ j}^\vee) \to {\mathrm{H}}^{d_{n-1,\infty}}_c(\widetilde{{\mathcal{X}}}_{n-1}, {\mathbb{C}}). \]

Put

\[ \frak{M}_{n-1}^\natural: = \frak{M}_{n-1, f}\otimes \frak{O}_{n-1,\infty}. \]

Note that $\widetilde {{\mathcal {X}}}_{n-1}/K_f$ is an orientable manifold when $K_f$ is a sufficiently small open compact subgroup of ${\mathrm {GL}}_{n-1}({\mathbb {A}}_f)$, and pairing with the fundamental class yields a linear map

\[ \int_{\widetilde{{\mathcal{X}}}_{n-1}}: {\mathrm{H}}^{d_{n-1,\infty}}_c (\widetilde{{\mathcal{X}}}_{n-1}, {\mathbb{C}}) \otimes \frak{M}_{n-1}^\natural \to {\mathbb{C}}. \]

See [Reference MahnkopfMah05, § 5.1] for more explanations.

In view of (59), define the modular symbol ${\mathcal {P}}_{j}$ to be the composition of

(77)\begin{align} & {\mathcal{P}}_{j}: {\mathcal{H}}(\Pi)\otimes {\mathcal{H}}(\Sigma) \otimes \operatorname{H}(\chi_j) \otimes \frak{M}_{n-1, f} \nonumber\\ &\quad\xrightarrow{\iota'_\Pi\otimes\iota_\Sigma\otimes\iota_j\otimes {\rm id}} {\mathcal{H}}^{b_{n,\infty}}_c({\mathcal{X}}_n, F_\mu^\vee)\otimes {\mathcal{H}}^{b_{n-1,\infty}}({\mathcal{X}}_{n-1}, F_\nu^\vee) \otimes {\mathrm{H}}^0({\mathcal{X}}_{n-1}, \delta_{ j}^\vee) \otimes \frak{M}_{n-1, f} \nonumber\\ &\quad\xrightarrow{\imath^*\otimes \wp^* \otimes \wp^*\otimes {\rm id}} {\mathcal{H}}^{b_{n,\infty}}_c(\widetilde{{\mathcal{X}}}_{n-1}, F_\mu^\vee)\otimes {\mathcal{H}}^{b_{n-1,\infty}}(\widetilde{{\mathcal{X}}}_{n-1}, F_\nu^\vee) \otimes {\mathrm{H}}^0(\widetilde{{\mathcal{X}}}_{n-1}, \delta_{ j}^\vee) \otimes \frak{M}_{n-1, f} \nonumber\\ &\quad\xrightarrow{\cup\otimes {\rm id}} {\mathrm{H}}^{d_{n-1,\infty}}_c(\widetilde{{\mathcal{X}}}_{n-1}, F_\xi^\vee \otimes \delta_{ j}^\vee)\otimes \frak{O}_{n-1,\infty} \otimes \frak{M}_{n-1, f} \nonumber\\ &\quad\xrightarrow{\phi_{\xi, j}\otimes {\rm id}} {\mathrm{H}}^{d_{n-1,\infty}}_c (\widetilde{{\mathcal{X}}}_{n-1}, {\mathbb{C}}) \otimes \frak{M}_{n-1, f}^\natural \nonumber\\ &\quad\xrightarrow{\int_{\widetilde{{\mathcal{X}}}_{n-1}}} {\mathbb{C}}. \end{align}

Recall that we have the normalized Rankin–Selberg integral at infinity

\begin{align*} \operatorname{Z}^\circ(\cdot, s, \chi_\infty) & \in {\mathrm{Hom}}_{{\mathrm{GL}}_{n-1}(\mathrm{k}_\infty)} (\Pi_\infty\widehat{\otimes}\Sigma_\infty\otimes \chi_{\infty, s-{1}/{2}}\otimes \frak{M}_{n-1,\infty}, {\mathbb{C}}) \\ & = {\mathrm{Hom}}_{{\mathrm{GL}}_{n-1}(\mathrm{k}_\infty)} ( \Pi_\infty\widehat{\otimes}\Sigma_\infty\otimes \chi_{\infty, s-{1}/{2}}, \frak{M}_{n-1,\infty}^*), \end{align*}

where, as before, $*$ stands for the dual space. Put

\begin{align*} \operatorname{H}(\chi_{\infty, j}) : & = {\mathrm{H}}^0_{\rm ct}({\mathbb{R}}^\times_+\backslash {\mathrm{GL}}_{n-1}(\mathrm{k}_\infty)^0; \delta_{j}^\vee\otimes \chi_{\infty, j}) \\ & = {\mathrm{H}}^0_{\rm ct}({\mathbb{R}}^\times_+\backslash {\mathrm{GL}}_{n-1}(\mathrm{k}_\infty)^0; \chi_\infty\cdot\operatorname{sgn}_\infty^j ), \end{align*}

where $\operatorname {sgn}_\infty$ is given as in the introduction.

Analogous to the archimedean modular symbol defined in § 3.2, we define the modular symbol at infinity, which is denoted by ${\mathcal {P}}_{\infty,j}$, to be the composition of

\begin{align*} & {\mathcal{P}}_{\infty,j}: {\mathcal{H}}(\Pi_\infty) \otimes {\mathcal{H}}(\Sigma_\infty) \otimes {\mathrm{H}}(\chi_{\infty, j}) \\ &\quad \rightarrow {\mathrm{H}}_\mathrm{ct}^{d_{n-1,\infty}}( {\mathrm{GL}}_{n-1}(\mathrm{k}_\infty)^0; (\Pi_\infty\widehat{\otimes} \Sigma_\infty\otimes \chi_{\infty, j}) \otimes (F_\xi^\vee\otimes \delta_{ j}^\vee)) \otimes \frak{O}_{n-1,\infty} \\ &\quad \rightarrow {\mathrm{H}}_\mathrm{ct}^{d_{n-1,\infty}}({\mathrm{GL}}_{n-1}(\mathrm{k}_\infty)^0; \frak{M}_{n-1,\infty}^*)\otimes \frak{O}_{n-1,\infty}={\mathbb{C}}, \end{align*}

where the first arrow is the restriction of cohomology composed with the cup product, and the last arrow is the map induced by the linear functional

\[ \operatorname{Z}^\circ(\cdot, \tfrac{1}{2}+j, \chi_\infty) \otimes \phi_{\xi,j}:(\Pi_\infty\widehat\otimes \Sigma_\infty\otimes \chi_{\infty, j})\otimes (F_\xi^\vee \otimes \delta_{ j}^\vee) \rightarrow \frak{M}_{n-1,\infty}^*. \]

Proposition 7.2 (Cf. [Reference MahnkopfMah05, (5.3)] and [Reference JanuszewskiJan19, § 4.6]) Let the notation and assumptions be as above. Then the diagram

commutes, where the left vertical arrow $\iota _{\rm can}$ is the natural isomorphism.

Proof. Define $\frak {q}_{n,\infty } := \big ( \frak {gl}_{n}(\mathrm {k}_\infty )/ ({\mathbb {R}} \oplus \frak {k}_{n,\infty })\big )\otimes _{\mathbb {R}} {\mathbb {C}}$. We have a map

\[ ( \wedge^{b_{n,\infty}}\frak{q}_{n,\infty})^* \otimes (\wedge^{b_{n-1,\infty}}\frak{q}_{n-1,\infty})^* \to \omega_{n-1,\infty}^* = \wedge^{d_{n-1,\infty}}\big((\frak{gl}_{n-1}(\mathrm{k}_\infty)/\frak{k}_{n-1,\infty})\otimes_{\mathbb{R}}{\mathbb{C}}\big)^* \]

induced by restriction. By the identification of continuous cohomology and relative Lie algebra cohomology [Reference Hochschild and MostowHM62, Theorem 6.1], as well as the explicit determination of the relative Lie algebra cohomology [Reference WallachWal88, Proposition 9.4.3], we have that

\[ {\mathcal{H}}(\Pi_\infty) =\big((\wedge^{b_{n,\infty}}\frak{q}_{n,\infty})^* \otimes \Pi_\infty \otimes F_\mu^\vee\big)^{K_{n,\infty}^0} \otimes \widetilde{\frak{O}}_{n,\infty} \]

and

\[ {\mathcal{H}}(\Sigma_\infty) =\big((\wedge^{b_{n-1,\infty}}\frak{q}_{n-1,\infty})^* \otimes \Sigma_\infty \otimes F_\nu^\vee\big)^{K_{n-1,\infty}^0}\otimes \widetilde{\frak{O}}_{n-1,\infty}. \]

By definition of ${\mathcal {P}}_{\infty, j}$, the top horizontal arrow of the diagram is identified with the composition of

\begin{align*} & \Pi_f\otimes \Sigma_f\otimes \chi_{f, j} \otimes \frak{M}_{n-1,f} \otimes \big((\wedge^{b_{n,\infty}}\frak{q}_{n,\infty})^* \otimes \Pi_\infty \otimes F_\mu^\vee\big)^{K_{n,\infty}^0}\otimes \widetilde{\frak{O}}_{n,\infty} \\[4pt] &\qquad \otimes \big((\wedge^{b_{n-1,\infty}}\frak{q}_{n-1,\infty})^* \otimes \Sigma_\infty \otimes F_\nu^\vee\big)^{K_{n-1,\infty}^0} \otimes \widetilde{\frak{O}}_{n-1,\infty} \otimes \delta_{j}^\vee \otimes \chi_{\infty, j} \\[4pt] &\quad\xrightarrow{\textrm{restriction}} \Pi_f\otimes\Sigma_f \otimes \chi_{f, j}\otimes \frak{M}_{n-1, f} \otimes \omega_{n-1,\infty}^*\otimes \Pi_\infty \widehat{\otimes} \Sigma_\infty \otimes F_\xi^\vee \otimes \frak{O}_{n-1,\infty} \otimes\delta_{ j}^\vee \otimes \chi_{\infty, j}\\[4pt] &\quad= (\Pi_f\otimes\Sigma_f \otimes \chi_{f, j}\otimes \frak{M}_{n-1,f}) \otimes ( \Pi_\infty \widehat{\otimes} \Sigma_\infty\otimes \chi_{\infty, j} \otimes \frak{M}_{n-1,\infty})\otimes (F_\xi^\vee\otimes \delta_{j}^\vee) \\[4pt] &\quad\rightarrow {\mathbb{C}}, \end{align*}

where the last map is given by

\[ \operatorname{Z}^\circ(\cdot, \tfrac{1}{2}+j, \chi_f)\otimes \operatorname{Z}^\circ(\cdot, \tfrac{1}{2}+j, \chi_\infty) \otimes \phi_{\xi,j}. \]

Using fast decreasing differential forms as in [Reference BorelBor81, § 5.6], the bottom arrow of the diagram is identified with the composition of

\begin{align*} &\big((\wedge^{b_{n,\infty}}\frak{q}_{n,\infty} )^* \otimes \Pi \otimes F_\mu^\vee\big)^{K_{n,\infty}^0} \otimes \widetilde{\frak{O}}_{n,\infty} \\[4pt] &\qquad \otimes \big((\wedge^{b_{n-1,\infty}}\frak{q}_{n-1,\infty} )^* \otimes \Sigma \otimes F_\nu^\vee\big)^{K_{n-1,\infty}^0} \otimes \widetilde{\frak{O}}_{n-1,\infty} \otimes \delta_{ j}^\vee \otimes \chi_j \otimes \frak{M}_{n-1, f}\\[4pt] &\quad\xrightarrow{\textrm{restriction}} \omega_{n-1,\infty}^*\otimes \Pi \widehat{\otimes} \Sigma \otimes F_\xi^\vee \otimes \frak{O}_{n-1,\infty}\otimes \delta_j^\vee\otimes \chi_j \otimes \frak{M}_{n-1,f } \\[4pt] &\quad= ( \Pi \widehat{\otimes} \Sigma \otimes \chi_j \otimes \frak{M}_{n-1}) \otimes (F_\xi^\vee \otimes \delta_{ j}^\vee) \\[4pt] &\quad\xrightarrow{\operatorname{Z}(\cdot, \frac{1}{2}+j, \chi)\otimes \phi_{\xi, j}} {\mathbb{C}}. \end{align*}

The proposition then follows from Proposition 7.1.