2.1.1 Quaternion algebras

Let $X(E : F)$ denote the set of $F$-isomorphism classes of central simple algebras over $F$ containing $E$. Then we recall that the map $\varepsilon \mapsto D_{\varepsilon }$ gives a bijection between $F^\times / \mathrm {N}_{E / F}(E^\times )$ and $X(E : F)$ (see [Reference Furusawa and ShalikaFS03, Lemma 1.3]) where

(2.1.1) \begin{equation} D_\varepsilon := \bigg\{\!\begin{pmatrix} a & \varepsilon b\\ b^\sigma & a^\sigma \end{pmatrix} : a, b \in E \bigg\}\quad\text{for $\varepsilon\in F^\times$}. \end{equation}

Here we regard $E$ as a subalgebra of $D_\varepsilon$ by

\[ E \ni a \mapsto \begin{pmatrix}a & 0\\ 0 & a^\sigma \end{pmatrix} \in D_\varepsilon. \]

We also note that $D_\varepsilon \simeq \mathrm {Mat}_{2 \times 2}(F)$ when $\varepsilon \in \mathrm {N}_{E/ F}(E^\times )$. The canonical involution $D_\varepsilon \ni x\mapsto \bar {x}\in D_\varepsilon$ is given by

\[ \bar{x}=\begin{pmatrix} a^\sigma & -\varepsilon b\\ -b^\sigma & a\end{pmatrix}\quad\text{for $x=\begin{pmatrix}a & \varepsilon b\\ b^\sigma & a^\sigma\end{pmatrix}$}. \]

We denote the reduced trace of $D$ by $\mathrm {tr}_D$.

2.1.2 Orthogonal groups

For a non-negative integer $n$, a symmetric matrix $S_n\in \mathrm {Mat}_{(2n+2) \times (2n+2)}(F)$ is defined inductively by

(2.1.2)\begin{equation} S_0: = \begin{pmatrix}2 & 0\\ 0 & -2d \end{pmatrix}\quad \text{and} \quad S_n: = \begin{pmatrix}0 & 0 & 1\\ 0 & S_{n-1} & 0\\ 1 & 0 & 0\end{pmatrix}\quad \text{for $n\ge 1$}. \end{equation}

We recall that $E=F(\eta )$ where $\eta ^2=d$. Then we denote the corresponding orthogonal group, the special orthogonal group and the similitude orthogonal group by

(2.1.3)\begin{equation} \mathrm{O}(S_n)=\mathrm{O}_{n+2,n},\quad \mathrm{SO}(S_n)= \mathrm{SO}_{n+2,n}\quad \text{and}\quad \mathrm{GO}(S_n)=\mathrm{GO}_{n+2,n}, \end{equation}

respectively. Let $\mathrm {GSO}_{n+2,n}$ denote the identity component of $\mathrm {GO}_{n+2,n}$. Thus,

(2.1.4)\begin{equation} \mathrm{GSO}_{n+2,n}(F) = \{g \in \mathrm{GO}_{n+2,n}(F): \det (g) = \lambda(g)^{n+1}\}, \end{equation}

where

(2.1.5)\begin{equation} \mathrm{GO}_{n+2,n}(F) = \{g \in \mathrm{GL}_{2n+2}(F) :{}^{t}g\, S_{n} \,g = \lambda(g)S_n,\ \lambda(g) \in F^\times\}. \end{equation}

For a positive integer $n$, we denote by $J_{2n}$ the $2n\times 2n$ symmetric matrix with ones on the non-principal diagonal and zeros elsewhere, i.e.

(2.1.6)\begin{equation} J_2=\begin{pmatrix}0 & 1\\1 & 0\end{pmatrix}\quad\text{and}\quad J_{2(n+1)}=\begin{pmatrix}0 & 0 & 1\\0 & J_{2n} & 0\\1 & 0 & 0\end{pmatrix}\quad\text{for $n\ge 1$}. \end{equation}

Then the similitude orthogonal group $\mathrm {GO}_{n,n}$ is defined by

(2.1.7)\begin{equation} \mathrm{GO}_{n,n}(F):=\{g \in\mathrm{GL}_{2n}(F): {}^{t}g\, J_{2n}\, g = \lambda(g) J_{2n}, \lambda(g)\in F^\times\} \end{equation}

and we denote by $\mathrm {GSO}_{n,n}$ its identity component, which is given by

(2.1.8)\begin{equation} \mathrm{GSO}_{n,n}(F) = \{g \in \mathrm{GO}_{n,n}(F) : \det (g) = \lambda(g)^{n}\}. \end{equation}

2.1.3 Quaternionic unitary groups

Let $D$ be a quaternion algebra over $F$ containing $E$. Recall that $G_D$ denotes the similitude quaternionic unitary group of degree $2$ defined by (1.4.2).

We define a similitude quaternionic unitary group $\mathrm {GU}_{3,D}$ of degree $3$ by

(2.1.9)\begin{equation} \mathrm{GU}_{3,D}(F):= \{g \in \mathrm{GL}_3(D) : {}^t \bar{g} \,{\bf J}_\eta\, g = \lambda(g) {\bf J}_\eta,\ \lambda(g)\in F^\times\}, \end{equation}

where we define a skew-hermitian matrix ${\bf J}_\eta$ by

(2.1.10)\begin{equation} {\bf J}_\eta := \begin{pmatrix} 0 & 0 & \eta\\0 & \eta & 0\\ \eta & 0 & 0\end{pmatrix}. \end{equation}

Here $\bar {A}=(\bar {a}_{ij})$ for $A=(a_{ij})\in \mathrm {Mat}_{m \times n}(D)$. Let us denote by $\mathrm {GSU}_{3,D}$ the identity component of $\mathrm {GU}_{3,D}$. Then unlike the orthogonal case, as noted in [Reference Mœglin, Vigneras and WaldspurgerMVW87, p. 21–22], we have

\[ \mathrm{GSU}_{3,D}(F) = \mathrm{GU}_{3,D}(F) \]

and

\[ \text{$\mathrm{GSU}_{3,D}(F_v) = \mathrm{GU}_{3,D}(F_v)$ when $D \otimes_F F_v$ is not split.} \]

Moreover, when $D\otimes _F F_v$ is split at a place $v$ of $F$, we have

(2.1.11)\begin{equation} \mathrm{GU}_{3, D}(F_v)\simeq \begin{cases} \mathrm{GO}_{4,2}(F_v) & \text{if $E\otimes F_v$ is a quadratic extension of $F_v$};\\ \mathrm{GO}_{3,3}(F_v) & \text{if $E\otimes F_v\simeq F_v\oplus F_v$.} \end{cases} \end{equation}

We also define $\mathrm {GU}_{1,D}$ by

(2.1.12)\begin{equation} \mathrm{GU}_{1,D}(F):= \{\alpha\in D^\times :\bar{\alpha} \eta \alpha = \lambda(\alpha) \eta,\ \lambda(\alpha)\in F^\times\} \end{equation}

and denote its identity component by $\mathrm {GSU}_{1,D}$. Then we note that

(2.1.13)\begin{align} \mathrm{GSU}_{1,D}(F)&=\{\alpha\in D^\times:\bar{\alpha} \eta \alpha = \mathrm{n}_D(\alpha) \eta\} \nonumber\\ &=\{x\in D^\times\mid x\eta=\eta x\}=T_\eta , \end{align}

where $T_\eta$ is defined by (1.4.6) with $\xi =\eta$ and $\mathrm {n}_D$ denotes the reduced norm of $D$.

2.1.4 Unitary groups

Suppose that $D=D_\varepsilon$ defined by (2.1.1). Then we define $\mathrm {GU}_{4,\varepsilon }$ a similitude unitary group of degree $4$ by

(2.1.14)\begin{equation} \mathrm{GU}_{4,\varepsilon}(F):=\{g \in \mathrm{GL}_4(E) : {}^{t}g^\sigma \mathcal{J}_\varepsilon g = \lambda(g) \mathcal{J}_\varepsilon,\ \lambda(g)\in F^\times\}, \end{equation}

where we define a hermitian matrix $\mathcal {J}_\varepsilon$ by

\[ \mathcal{J}_\varepsilon := \begin{pmatrix} 0 & 0 & 0 & 1\\ 0 & -1 & 0 & 0\\ 0 & 0 & \varepsilon & 0\\ 1 & 0 & 0 & 0\end{pmatrix}. \]

Here $A^\sigma =(a_{ij}^\sigma )$ for $A=(a_{ij})\in \mathrm {Mat}_{m \times n}(E)$. Then we have

(2.1.15)\begin{equation} \mathrm{GU}_{4,\varepsilon}\simeq \begin{cases} \mathrm{GU}_{2,2}, & \text{when $D$ is split, i.e. $\varepsilon\in \mathrm{N}_{E/ F}(E^\times)$};\\ \mathrm{GU}_{3,1}, & \text{when $D$ is non-split, i.e. $\varepsilon\notin \mathrm{N}_{E/ F}(E^\times)$}. \end{cases} \end{equation}

We also define $\mathrm {GU}_{2,\varepsilon }$ a similitude unitary group of degree $2$ by

(2.1.16)\begin{equation} \mathrm{GU}_{2,\varepsilon}(F) := \{ g \in \mathrm{GL}_{2}(E) : {}^{t}g^\sigma J_\varepsilon g =\lambda(g) J_\varepsilon,\ \lambda(g) \in F^\times\}\quad \text{where}\ J_\varepsilon=\begin{pmatrix}-1 & 0\\0 & \varepsilon \end{pmatrix}. \end{equation}

2.2.1 $\mathrm {PGSU}_{3,D}\simeq \mathrm {PGU}_{4,\varepsilon }$

Suppose that $D=D_\varepsilon$. Then we may naturally realize $\mathrm {GSU}_{3,D}(F)$ as a subgroup of $\mathrm {GL}_6(E)$. We note that

\[ \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0\\ 0 & -\varepsilon & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & -\varepsilon & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & -\varepsilon \end{pmatrix}\,{}^{t}\bar{g}\, \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0\\ 0 & -\varepsilon & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & -\varepsilon & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & -\varepsilon \end{pmatrix}^{-1}= {}^{t}g^\sigma \]

and

\[ \begin{pmatrix} 0 & 1 & 0 & 0 & 0 & 0\\ -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 & -1 & 0 \end{pmatrix}\,{}^{t}\bar{g}\, \begin{pmatrix} 0 & 1 & 0 & 0 & 0 & 0\\ -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 & -1 & 0 \end{pmatrix}^{-1}= {}^{t}g. \]

Thus, in this realization, we have

(2.2.1)\begin{equation} \mathrm{GSU}_{3,D}(F) = \{ g \in \mathrm{GSO}_{3,3}(E) : {}^{t} g^\sigma\, \mathcal{J}_\varepsilon^\circ \,g =\lambda(g)\mathcal{J}_{\varepsilon}^\circ,\ \lambda(g)\in F^\times \}, \end{equation}

where

\[ \mathcal{J}_{\varepsilon}^\circ =- \begin{pmatrix} 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & \varepsilon\\ 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & \varepsilon & 0 & 0\\ 1 & 0 & 0 & 0 & 0 & 0\\ 0 & \varepsilon & 0 & 0 & 0 & 0 \end{pmatrix}. \]

Here we recall that

(2.2.2)\begin{equation} \mathrm{GSO}_{3,3}(E) \simeq \mathrm{GL}_4(E) \times \mathrm{GL}_1(E) / \{ (z, z^{-2} ) :z \in E^\times \}. \end{equation}

In fact, the isomorphism (2.2.2) is realized as follows. Let us take the standard basis

\[ b_1 ={}^{t}(1,0,0,0),\quad b_2 ={}^{t}(0,1,0,0),\quad b_3 ={}^{t}(0,0,1,0),\quad b_4 ={}^{t}(0,0,0,1), \]

of $E^4$. Then we may consider $V:=\wedge ^2 E^4$ as an orthogonal space over $E$ with a quadratic form $(\,{,}\,)_V$ defined by

\[ v_1\wedge v_2=(v_1 ,v_2)_V \cdot b_1 \wedge b_2 \wedge b_3 \wedge b_4 \]

for $v_1, v_2 \in V$. As a basis of $V$ over $E$, we take $\{ \varepsilon _i: 1\le i\le 6 \}$ given by

\[ \varepsilon_1 =b_1 \wedge b_2,\quad \varepsilon_2 =b_1 \wedge b_3,\quad \varepsilon_3 = b_1 \wedge b_4,\quad \varepsilon_4 = b_2 \wedge b_3,\quad \varepsilon_5 = b_4 \wedge b_2,\quad \varepsilon_6 = b_3 \wedge b_4. \]

Let the group $\mathrm {GL}_4(E) \times \mathrm {GL}_1(E)$ act on $V$ by $(g, a)(w_1 \wedge w_2) = a \cdot (gw_1 \wedge gw_2)$ where $w_1,w_2\in E^4$. This action defines a homomorphism

(2.2.3)\begin{equation} \mathrm{GL}_4(E) \times \mathrm{GL}_1(E)\to \mathrm{GSO}_{3,3}(E), \end{equation}

where we take $\{\varepsilon _i: 1\le i\le 6\}$ as a basis of $V$ and the homomorphism (2.2.3) induces the isomorphism (2.2.2). By a direct computation we observe that $(-\mathcal {J}_\varepsilon, 1)$ is mapped to $\mathcal {J}_\varepsilon ^\circ$ under (2.2.3) and the restriction of the homomorphism (2.2.3) gives a homomorphism

(2.2.4)\begin{equation} \mathrm{GU}_{4,\varepsilon}(F)\to \mathrm{GSU}_{3,D}(F). \end{equation}

Then it is easily seen that the isomorphism

(2.2.5)\begin{equation} \Phi_D : \mathrm{PGU}_{4,\varepsilon}(F)\overset{\sim}{\to} \mathrm{PGSU}_{3,D}(F) \end{equation}

is induced.

2.2.2 $\mathrm {PGU}_{2,2} \simeq \mathrm {PGSO}_{4,2}$

When $\varepsilon \in \mathrm {N}_{E / F}(E^\times )$, the quaternion algebra $D=D_\varepsilon$ is split and the isomorphism (2.2.5) gives an isomorphism $\mathrm {PGU}_{2,2}\simeq \mathrm {PGSO}_{4,2}$. We recall the concrete realization of this isomorphism. First we define $\mathrm {GU}_{2,2}$ by

\begin{align*} \mathrm{GU}_{2,2} &:= \{g\in\mathrm{GL}_4(E): {}^tg^\sigma\, J_4\,g =\lambda(g)J_4, \lambda(g)\in F^\times\}, \quad \text{where $J_4=\begin{pmatrix} 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0\\ 0 & 1 & 0 & 0\\ 1 & 0 & 0 & 0 \end{pmatrix}$} \end{align*}

as (2.1.6). Let

\[ {\mathcal{V}}: = \left\{B((x_i)_{1\le i\le 6}):= \left(\begin{matrix} 0 & \eta x_1 & x_3 + \eta x_4 & x_2\\ -\eta x_1 & 0 & x_5 & -x_3 +\eta x_4\\ -x_3 -\eta x_4 & -x_5 & 0 & \eta^{-1} x_6\\ -x_2 & x_3 -\eta x_4 & -\eta^{-1} x_6 & 0\\ \end{matrix}\right): x_i \in F (1\le i\le6)\right\}. \]

We define $\Psi : {\mathcal {V}} \rightarrow F$ by

\[ \Psi (B): = {\rm Tr} \bigg(B\, \begin{pmatrix} 0 & 1_2\\ 1_2 & 0\end{pmatrix} \,{}^{t}B^\sigma\, \begin{pmatrix}0 & 1_2\\ 1_2 & 0\end{pmatrix}\bigg). \]

Then we have

\[ \Psi(B((x_i)_{1\le i\le 6}))=-4\{x_1x_6+x_2x_5-(x_3^2-dx_4^2)\}. \]

Let $\mathrm {GSU}_{2,2}$ denote the identity component of $\mathrm {GU}_{2,2}$, i.e.

\[ \mathrm{GSU}_{2,2} = \{ g \in \mathrm{GU}_{2,2}: \det(g) = \lambda(g)^2 \}. \]

We let $\mathrm {GSU}_{2,2}$ act on $\mathcal {V}$ by

\[ \mathrm{GSU}_{2,2}\times \mathcal{V}\ni (g,B)\mapsto (wg w)B(w\,{}^tg w)\in\mathcal{V}, \quad\text{where $w=\begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0\end{pmatrix}$.} \]

Then this action induces a homomorphism $\phi : \mathrm {GSU}_{2,2} \rightarrow \mathrm {GO}({\mathcal {V}})$. We note that

\[ \lambda(\phi(g)) = \det(g)\quad \text{for $g\in \mathrm{GSU}_{2,2}$} \]

and this implies that the image of $\phi$ is contained in $\mathrm {GSO}(\mathcal {V})$. As a basis of ${\mathcal {V}}$, we may take

\begin{gather*} f_1=\begin{pmatrix} 0 & \eta & 0 & 0\\ -\eta & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ \end{pmatrix},\quad f_2=\begin{pmatrix} 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ -1 & 0 & 0 & 0\\ \end{pmatrix},\quad f_3=\begin{pmatrix} 0 & 0 & 1 & 0\\ 0 & 0 & 0 & -1\\ -1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ \end{pmatrix},\\ f_4= \begin{pmatrix} 0 & 0 & \eta & 0\\ 0 & 0 & 0 & \eta\\ -\eta & 0 & 0 & 0\\ 0 & -\eta & 0 & 0\\ \end{pmatrix},\quad f_5=\begin{pmatrix} 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0\\ \end{pmatrix},\quad f_6=\begin{pmatrix} 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & \eta^{-1}\\ 0 & 0 & -\eta^{-1} & 0\\ \end{pmatrix}. \end{gather*}

With respect to this basis, we may regard $\phi$ as a homomorphism from $\mathrm {GSU}_{2,2}$ to $\mathrm {GO}_{4,2}$, where the group $\mathrm {GO}_{4,2}$ is given by (2.1.5) for $n=2$. Let us consider $\mathrm {GSU}_{2,2}\rtimes E^\times$ where the action of $\alpha \in E^\times$ on $g\in \mathrm {GSU}_{2,2}$ is given by

\[ \alpha \cdot g = \begin{pmatrix} \alpha & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & (\alpha^{\sigma})^{-1} \end{pmatrix}\,g \,\begin{pmatrix} \alpha & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & (\alpha^{\sigma})^{-1} \end{pmatrix}^{-1}. \]

Then as in [Reference MorimotoMor14a, p. 32–34], $\phi$ may be extended to $\mathrm {GSU}_{2,2}\rtimes E^\times$ and we have a homomorphism $\mathrm {GSU}_{2,2}\rtimes E^\times \to \mathrm {PGSO}_{4,2}$ which induces the isomorphism

(2.2.6)\begin{equation} \Phi: \mathrm{PGU}_{2,2} \overset{\sim}{\to} \mathrm{PGSO}_{4,2}. \end{equation}

2.3.1 Bessel periods on $G=\mathrm {GSp}_2$

Though we already introduced Bessel periods on $G_D$ in general as (1.4.11), we would like to describe them concretely in the case of $G$ here for our explicit pull-back computations in the next section.

Let $P$ be the Siegel parabolic subgroup of $G$ with the Levi decomposition $P = MN$ where

\[ M(F) = \bigg\{\!\begin{pmatrix} g & 0\\ 0 & \lambda \cdot {}^tg^{-1} \end{pmatrix}: \begin{aligned} & g \in \mathrm{GL}_2(F),\\ & \lambda \in F^\times \end{aligned}\bigg\},\quad N(F) = \bigg\{\!\begin{pmatrix} 1 & X\\ 0 & 1\end{pmatrix} : X \in \mathrm{Sym}_2(F)\bigg\}.\label{d:N} \]

Here $\mathrm {Sym}_n(F)$ denotes the set of $n$ by $n$ symmetric matrices with entries in $F$ for a positive integer $n$. For $S \in \mathrm {Sym}_2(F)$, let us define a character $\psi _{S}$ of $N(\mathbb {A})$ by

\[ \psi_{S} \begin{pmatrix} 1 & X\\ 0 & 1 \end{pmatrix} = \psi[\mathrm{tr}(SX)]. \]

For $S\in \mathrm {Sym}_2(F)$ such that $\det S\ne 0$, let

\[ T_S := \{ g \in\mathrm{GL}_2 : {}^{t}gSg = \det(g)S\}.\label{T_S} \]

We identify $T_S$ with the subgroup of $G$ given by

\[ \bigg\{\!\begin{pmatrix} g & 0\\ 0 & \det (g) \cdot {}^{t}g^{-1} \end{pmatrix} : g \in T_S\bigg\}. \]

Definition 2.1 Let us take $S\in \mathrm {Sym}_2(F)$ such that $T_S(F)$ is isomorphic to $E^\times$. Let $\pi$ be an irreducible cuspidal automorphic representation of $G(\mathbb {A})$ whose central character is trivial and $V_\pi$ its space of automorphic forms. Fix an $F$-isomorphism $T_S(F)\simeq E^\times$. Let $\Lambda$ be a character of $\mathbb {A}_E^\times / E^\times$ such that $\Lambda \mid _{\mathbb {A}^\times }$ is trivial. We regard $\Lambda$ as a character of $T_S(\mathbb {A}) / \mathbb {A}^\times \, T_S(F)$.

Then for $\varphi \in V_\pi$, we define $B_{S,\Lambda,\psi }(\varphi )$, the $(S,\Lambda,\psi )$-Bessel period of $\varphi$ by

(2.3.1)\begin{equation} B_{S, \Lambda,\psi}(\varphi) = \int_{\mathbb{A}^\times \,T_{S}(F) \backslash T_{S}(\mathbb{A})} \int_{N(F) \backslash N(\mathbb{A})} \varphi(uh) \Lambda^{-1}(h) \psi_{S}^{-1}(u) \, du\, dh. \end{equation}

We say that $\pi$ has the $(S,\Lambda,\psi )$-Bessel period when $B_{S, \Lambda,\psi }\not \equiv 0$ on $V_\pi$. Then we also say that $\pi$ has the $(E,\Lambda )$-Bessel period as in Definition 1.2.

2.3.2 Bessel periods on $\mathrm {GSU}_{3,D}$

Let us introduce Bessel periods on the group $\mathrm {GSU}_{3,D}$ defined in § 2.1.3. Let $P_{3,D}$ be a maximal parabolic subgroup of $\mathrm {GSU}_{3,D}$ with the Levi decomposition $P_{3,D}=M_{3,D}N_{3,D}$ where

\[ M_{3,D} = \left\{\begin{pmatrix} g & 0 & 0\\ 0 & h & 0\\ 0 & 0 & g \end{pmatrix}:\begin{aligned} & g\in D^\times,\\ & h \in T_\eta,\\ & \mathrm{n}_D(g)=\mathrm{n}_D(h) \end{aligned}\right\},\quad N_{3,D} = \left\{\begin{pmatrix} 1 & A^\prime & B\\ 0 & 1 & A\\ 0 & 0 & 1 \end{pmatrix} \in \mathrm{GSU}_{3,D}\right\}.\label{d:N3,D} \]

As for $T_\eta$, we recall (2.1.13) and $T_\eta \simeq E^\times$. For $X \in D^\times$, we define a character $\psi _{X, D}$ of $N_{3, D}(\mathbb {A})$ by

\[ \psi_{X, D} \begin{pmatrix} 1 & A^\prime & B\\ 0 & 1 & A\\ 0 & 0 & 1 \end{pmatrix} = \psi [\mathrm{tr}_D(XA)]. \]

Then the identity component of the stabilizer of $\psi _{X, D}$ in $M_{3,D}$ is

\[ M_{X, D} = \left\{\!\begin{pmatrix} h^{X} & 0 & 0\\ 0 & h & 0\\ 0 & 0 & h^{X} \end{pmatrix}: h \in T_\eta \right\},\quad \text{where}\ h^X = XhX^{-1}.\label{M_X D} \]

We identify $M_X$ with $T_\eta$ by

(2.3.2)\begin{equation} M_{X, D}\ni \begin{pmatrix} h^{X} & 0 & 0\\ 0 & h & 0\\ 0 & 0 & h^{X}\end{pmatrix} \mapsto h\in T_\eta \end{equation}

and we fix an $F$-isomorphism $T_\eta \simeq E^\times$.

Definition 2.2 Let $\sigma _D$ be an irreducible cuspidal automorphic representation of $\mathrm {GSU}_{3,D}(\mathbb {A})$ and $V_{\sigma _D}$ its space of automorphic forms. Let $\chi$ be a character of $\mathbb {A}_E^\times / E^\times$ and we regard $\chi$ as a character of $M_{X,D}(\mathbb {A}) / M_{X,D}(F)$. Suppose that $\chi |_{\mathbb {A}^\times } = \omega _{\sigma _D}$, the central character of $\sigma _D$.

Then for $\varphi \in V_{\sigma _D}$, we define $\mathcal {B}^D_{X, \chi, \psi } (\varphi )$, the $(X,\chi, \psi )$-Bessel period of $\varphi$ by

(2.3.3) \begin{align} \mathcal{B}^D_{X, \chi, \psi} (\varphi) &= \int_{\mathbb{A}^\times M_{X,D}(F) \backslash M_{X,D}(\mathbb{A})} \int_{N_{3,D}(F) \backslash N_{3,D}(\mathbb{A})} \varphi(uh) \chi (h)^{-1} \psi_{X, D}(u)^{-1} \, du \, dh. \end{align}

2.3.3 Bessel periods on $\mathrm {GU}_{4,\varepsilon }$

In light of the accidental isomorphism (2.2.5), Bessel periods on the group $\mathrm {GU}_{4,\varepsilon }$ is defined as follows.

Let $P_{4,\varepsilon }$ be a maximal parabolic subgroup of $\mathrm {GU}_{4,\varepsilon }$ with the Levi decomposition $M_{4,\varepsilon }N_{4,\varepsilon }$ where

\begin{align*} M_{4,\varepsilon}(F) &= \left\{\!\begin{pmatrix} a & 0 & 0\\ 0 & g & 0\\ 0 & 0 & \lambda(g) (a^\sigma)^{-1} \end{pmatrix}: a \in E^\times, g \in \mathrm{GU}_{2,\varepsilon}(F) \right\},\\ N_{4,\varepsilon}(F) &= \left\{\!\begin{pmatrix} 1 & A & B\\ 0 & 1_2 & A^\prime\\ 0 & 0 & 1 \end{pmatrix} \in \mathrm{GU}_{4,\varepsilon}(F) \right\}. \end{align*}

Let us take an anisotropic vector $e\in E^4$ of the form ${}^{t}(0, \ast, \ast, 0)$. Then we define a character $\chi _e$ of $N_{4,\varepsilon }(\mathbb {A})$ by

\[ \chi_e (u)= \psi( (ue, b_1)_\varepsilon),\quad \text{where $(x,y)_\varepsilon={}^tx^\sigma J_{\varepsilon}y$}. \]

Here we recall that $J_\varepsilon$ is as given in (2.1.16) and $b_1={}^t(1,0,0,0)$. Let $D_e$ denote the subgroup of $M_{4,\varepsilon }$ given by

\[ D_{e} := \left\{\!\begin{pmatrix}1 & 0 & 0\\ 0 & h & 0\\ 0 & 0 & 1\end{pmatrix}: h \in \mathrm{U}_{2,\varepsilon},\ h e = e \right\}. \]

Then the group $D_e(\mathbb {A})$ stabilizes the character $\chi _e$ by conjugation. We note that

\[ D_e(F) \simeq \mathrm{U}_1(F):= \{ a \in E^\times : \bar{a}a =1 \}. \]

Hence, for a character $\Lambda$ of $\mathbb {A}_E^\times$ which is trivial on $\mathbb {A}^\times$, we may regard $\Lambda$ as a character of $D_e(\mathbb {A})$ by $d \mapsto \Lambda (\det d)$. Then we define a character $\chi _{e, \Lambda }$ of $R_e(\mathbb {A})$ where $R_e:=D_eN_{4,\varepsilon }$ by

(2.3.4)\begin{equation} \chi_{e, \Lambda}(ts) := \Lambda(t) \chi_e(s)\quad \text{for}\ t \in D_e(\mathbb{A}),\enspace s \in N_{4,\varepsilon}(\mathbb{A}). \end{equation}

Definition 2.3 For a cusp form $\varphi$ on $\mathrm {GU}_{4,\varepsilon }(\mathbb {A}_F)$ with a trivial central character, we define $B_{e, \Lambda,\psi }(\varphi )$, the $(e, \Lambda,\psi )$-Bessel period of $\varphi$, by

(2.3.5)\begin{equation} B_{e, \Lambda,\psi}(\varphi) = \int_{D_e(F) \backslash D_e(\mathbb{A}_F)} \int_{N_{4,\varepsilon}(F) \backslash N_{4,\varepsilon}(\mathbb{A}_F)} \chi_{e, \Lambda}(ts)^{-1}\, \varphi(ts) \, ds \, dt. \end{equation}

2.3.4 Bessel periods on $\mathrm {GSO}_{4,2}$ and $\mathrm {GSO}_{3,3}$

By combining the accidental isomorphisms (2.2.5) and (2.2.6) in the split case, we shall define Bessel periods on $\mathrm {GSO}_{4,2}$ and $\mathrm {GSO}_{3,3}$ as the following.

Let $P_{4,2}$ denote a maximal parabolic subgroup of $\mathrm {GSO}_{4,2}$ with the Levi decomposition $P_{4,2}= M_{4,2}N_{4,2}$ where

\[ M_{4,2} = \left\{\!\begin{pmatrix} g & 0 & 0\\0 & h & 0\\0 & 0 & g^\ast\cdot \det h\end{pmatrix} : \begin{aligned} & g \in \mathrm{GL}_2,\\ & h \in \mathrm{GSO}_{2 ,0} \end{aligned}\right\},\quad N_{4,2}= \left\{\!\begin{pmatrix} 1_2 & A^\prime & B\\ 0 & 1_2 & A\\ 0 & 0 & 1_2 \end{pmatrix} \in \mathrm{GSO}_{4,2}\right\}.\label{d:N4,2} \]

Here

\[ g^\ast = \begin{pmatrix}0 & 1\\1 & 0\end{pmatrix} {}^{t}g^{-1} \begin{pmatrix}0 & 1\\1 & 0\end{pmatrix}\quad\text{for}\ g\in\mathrm{GL}_2. \]

Then for $X\in \mathrm {Mat}_{2\times 2}(F)$, we define a character $\psi _X$ of $N_{4,2}(\mathbb {A})$ by

\[ \psi_{X} \begin{pmatrix}1_2 & A^\prime & B\\ 0 & 1_2 & A\\0 & 0 & 1_2 \end{pmatrix}= \psi [\mathrm{tr}(XA)]. \]

Suppose that $\det X \ne 0$ and let

\[ M_{X} := \left\{\!\begin{pmatrix} (\det h) \cdot (h^X)^\ast & 0 & 0\\ 0 & h & 0\\ 0 & 0 & h^X \end{pmatrix} : h \in \mathrm{GSO}_{2,0} \right\},\label{M_X} \]

where $h^X = XhX^{-1}$ . Then $M_X(\mathbb {A})$ stabilizes the character $\psi _X$ and $M_X$ is isomorphic to $\mathrm {GSO}_{2,0}$. We fix an isomorphism $\mathrm {GSO}_{2,0}(F)\simeq E^\times$ and we regard a character of $\mathbb {A}_E^\times$ as a character of $M_X(\mathbb {A})$.

Definition 2.4 Let $\sigma$ be an irreducible cuspidal automorphic representation of $\mathrm {GSO}_{4,2}(\mathbb {A})$ with its space of automorphic forms $V_\sigma$ and the central character $\omega _\sigma$. For a character $\chi$ of $\mathbb {A}_E^\times$ such that $\chi |_{\mathbb {A}^\times } = \omega _{\sigma }$, we define $\mathcal {B}_{X,\chi, \psi }(\varphi )$, the $(X, \chi, \psi )$-Bessel period of $\varphi \in V_\sigma$ by

(2.3.6)\begin{equation} \mathcal{B}_{X, \chi, \psi}(\varphi) = \int_{N_{4,2}(F) \backslash N_{4,2}(\mathbb{A})} \int_{M_{X}(F) \mathbb{A}^\times \backslash M_{X}(\mathbb{A})} \varphi(u h) \chi(h)^{-1} \psi_{X}(u)^{-1} \, du \, dh. \end{equation}

When $d \in (F^\times )^2$, we know that $\mathrm {GSO}(S_2) \simeq \mathrm {GSO}_{3,3}$. Hence, as above, for a cusp form $\varphi$ on $\mathrm {GSO}_{3,3}$ with central character $\omega$ and characters $\Lambda _1, \Lambda _2$ of $\mathbb {A}^\times / F^\times$ such that $\Lambda _1 \Lambda _2=\omega$, we define $(X, \Lambda _1, \Lambda _2, \psi )$-Bessel period by

\[ \mathcal{B}_{X, \Lambda, \psi}(\varphi) = \int_{N_{4,2}(F) \backslash N_{4,2}(\mathbb{A})} \int_{M_{X}(F) \mathbb{A}^\times \backslash M_{X}(\mathbb{A})} \varphi(u h) \chi_{\Lambda_1, \Lambda_2}(h)^{-1} \psi_{X}(u)^{-1} \, du \, dh. \]

Here, since $M_{4,2} \simeq \mathrm {GL}_2 \times \mathrm {GSO}_{1,1}$ and $\mathrm {GSO}_{1,1}(F) = \big \{\big (\begin{smallmatrix} a & \\ & b\end{smallmatrix}\big ) : a, b \in F^\times \big \}$, we define a character $\chi _{\Lambda _1, \Lambda _2}$ of $\mathrm {GSO}_{1,1}(\mathbb {A})$ by

\[ \chi_{\Lambda_1, \Lambda_2} \begin{pmatrix} a & \\ & b\end{pmatrix} =\Lambda_1(a) \Lambda_2(b). \]

When $\omega$ is trivial, we have $\Lambda _2 = \Lambda _1^{-1}$. In this case, we simply call $(X, \Lambda _1, \Lambda _1^{-1}, \psi )$-Bessel period as $(X, \Lambda _1, \psi )$-Bessel period and simply write $\chi _{\Lambda _1, \Lambda _1^{-1}} = \Lambda _1$.

3.1.1 Symplectic-orthogonal theta correspondence with similitudes

Let $X$ (respectively, $Y$) be a finite-dimensional vector space over $F$ equipped with a non-degenerate alternating (respectively, symmetric) bilinear form. Assume that $\dim _F Y$ is even. We denote their similitude groups by $\mathrm {GSp}(X)$ and $\mathrm {GO}(Y)$, and, their isometry groups by $\mathrm {Sp}(X)$ and $\mathrm {O}(Y)$, respectively. We denote the identity component of $\mathrm {GO}(Y)$ and $\mathrm {O}(Y)$ by $\mathrm {GSO}(Y)$ and $\mathrm {SO}(Y)$, respectively. We let $\mathrm {GSp}(X)$ (respectively, $\mathrm {GO}(Y)$) act on $X$ from right (respectively, left). The space $Z = X \otimes Y$ has a natural non-degenerate alternating form $\langle \,{,} \, \rangle$, and we have an embedding $\mathrm {Sp}(X) \times \mathrm {O}(Y) \rightarrow \mathrm {Sp}(Z)$ defined by

(3.1.1)\begin{equation} (x \otimes y)(g, h) = x g \otimes h^{-1}y,\quad \text{for } x \in X,\enspace y \in Y,\enspace h \in \mathrm{O}(Y),\enspace g \in \mathrm{Sp}(X). \end{equation}

Fix a polarization $Z = Z_{+} \oplus Z_{-}$. Let us denote by $(\omega _{\psi }, \mathcal {S}(Z_{+}(\mathbb {A})))$ the Schrödinger model of the Weil representation of $\widetilde {\mathrm {Sp}}(Z)$ corresponding to this polarization with the Schwartz–Bruhat space $\mathcal {S}(Z_{+})$ on $Z_{+}$. We denote a typical element of $\mathrm {Sp}(Z)$ by

\[ \begin{pmatrix} A & B\\ C & D\\ \end{pmatrix},\quad\text{where}\ \begin{cases} A \in {\rm Hom}(Z_{+} , Z_{+}), & B \in {\rm Hom}(Z_{+} , Z_{-}),\\ C \in {\rm Hom}(Z_{-} , Z_{+}), & D \in {\rm Hom}(Z_{-} , Z_{-}). \end{cases} \]

Then the action of $\omega _{\psi }$ on $\phi \in \mathcal {S}(Z_+)$ is given by the following formulas:

(3.1.2)\begin{equation} \omega_{\psi}\bigg(\begin{pmatrix} A & B\\ 0 & ^{t}A^{-1}\\ \end{pmatrix},\varepsilon\bigg)\phi(z_{+})= \varepsilon \frac{\gamma _{\psi }(1)}{\gamma _{\psi }({\rm det}A)} |{\rm det}(A)|^{1/2} \psi \bigg(\frac{1}{2} \langle z_{+}A ,z_{+}B \rangle\bigg)\phi(z_{+}A), \end{equation}

(3.1.3)\begin{equation} \omega_{\psi}\bigg(\begin{pmatrix} 0 & I\\ -I & 0 \end{pmatrix},\varepsilon\bigg)\phi(z_{+}) =\varepsilon (\gamma _{\psi }(1))^{-\dim Z_{+}} \int _{Z_{+}} \psi\bigg(\bigg\langle z^{\prime}, z \begin{pmatrix} 0 & I\\ -I & 0\\ \end{pmatrix}\bigg\rangle \bigg)\phi(z^{\prime}) \, d z^{\prime}, \end{equation}

where $\gamma _{\psi }(t)$ is a certain eighth root of unity called the Weil factor. Moreover, since the embedding given by (3.1.1) splits in the metaplectic group $\mathrm {Mp}(Z)$, we obtain the Weil representation of $\mathrm {Sp}(X, \mathbb {A}) \times \mathrm {O}(Y, \mathbb {A})$ by restriction. We also denote this representation by $\omega _\psi$.

We have a natural homomorphism

\[ i : \mathrm{GSp}(X) \times \mathrm{GO}(Y) \rightarrow \mathrm{GSp}(Z) \]

given by the action (3.1.1). Then we note that $\lambda (i(g,h)) = \lambda (g)\lambda (h)^{-1}$. Let

\[ R:=\{(g,h) \in \mathrm{GSp}(X)\times \mathrm{GO}(Y) \,|\, \lambda(g) = \lambda(h)\}\supset \mathrm{Sp}(X) \times \mathrm{O}(Y). \]

We may define an extension of the Weil representation of $\mathrm {Sp}(X, \mathbb {A}) \times \mathrm {O}(Y, \mathbb {A})$ to $R(\mathbb {A})$ as follows. Let $X= X_+ \oplus X_-$ be a polarization of $X$ and use the polarization $Z_{\pm } = X_{\pm } \otimes Y$ of $Z$ to realize the Weil representation $\omega _\psi$. Then we note that

\[ \omega_{\psi}(1, h)\phi(z) = \phi(i (h)^{-1} z)\quad \text{for $h\in\mathrm{O}(\mathbb{A})$ and $\phi\in \mathcal{S}(Z_+(\mathbb{A}))$}. \]

Thus, we define an action $L$ of $\mathrm {GO}(Y,\mathbb {A})$ on $\mathcal {S}(Z_+(\mathbb {A}))$ by

\[ L(h)\phi(z)=|\lambda(h)|^{-({1}/{8}) \dim~X \cdot \dim~Y}\phi(i (h)^{-1}z). \]

Then we may extend the Weil representation $\omega _\psi$ of $\mathrm {Sp}(X, \mathbb {A}) \times \mathrm {O}(Y, \mathbb {A})$ to $R(\mathbb {A})$ by

\[ \omega_{\psi}(g, h) \phi = \omega_\psi(g_1, 1) L(h)\phi\quad \text{for $\phi\in\mathcal{S}(Z_+(\mathbb{A}))$ and $(g,h)\in R(\mathbb{A})$}, \]

where

\[ g_1 = g \begin{pmatrix}\lambda(g)^{-1} & 0\\ 0 & 1 \end{pmatrix} \in \mathrm{Sp}(X,\mathbb{A}). \]

In general, for any polarization $Z= Z_+^\prime \oplus Z_-^\prime$, there exists an $\mathrm {Sp}(X, \mathbb {A}) \times \mathrm {O}(Y)(\mathbb {A})$-isomorphism $p : \mathcal {S}(Z_+(\mathbb {A})) \to \mathcal {S}(Z_+^\prime (\mathbb {A}))$ given by an integral transform (see Ichino and Prasanna [Reference Ichino and PrasannaIP21, Lemma 3.3]). Let us denote the realization of the Weil representation of ${\mathrm {Sp}(X, \mathbb {A}) \times \mathrm {O}(Y)(\mathbb {A})}$ on $\mathcal {S}(Z_+^\prime (\mathbb {A}))$ by $\omega _\psi ^\prime$. Then we may extend $\omega _\psi ^\prime$ to $R(\mathbb {A})$ by

\[ \omega_\psi^\prime(g,h)=p\circ \omega_\psi(g,h)\circ p^{-1}\quad \text{for $(g, h) \in R(\mathbb{A})$}. \]

For $\phi \in \mathcal {S}(Z_{+}(\mathbb {A}))$, we define the theta kernel $\theta ^\phi$ by

\[ \theta_{\psi}^{\phi}(g , h) = \theta^{\phi}(g , h) := \sum_{z_{+} \in Z_{+}(F)} \omega_{\psi} (g,h)\phi(z_{+}) \quad \text{for $(g, h) \in R(\mathbb{A})$}. \]

Let

(3.1.4)\begin{equation} \mathrm{GSp}(X, \mathbb{A})^{+}= \{g \in \mathrm{GSp}(X, \mathbb{A}) \mid \, \lambda(g) = \lambda(h) \text{ for some } h \in \mathrm{GO}(Y, \mathbb{A})\} \end{equation}

and $\mathrm {GSp}(X, F)^{+} = \mathrm {GSp}(X, \mathbb {A})^{+} \cap \mathrm {GSp}(X, F)$.

As in [Reference Harris and KudlaHK92, § 5.1], for a cusp form $f$ on $\mathrm {GSp}(X, \mathbb {A})^{+}$, we define its theta lift to $\mathrm {GO}(Y, \mathbb {A})$ by

\[ \Theta_\psi^{X, Y} (f, \phi)(h) = \Theta(f, \phi)(h) := \int_{\mathrm{Sp}(X, F) \backslash \mathrm{Sp}(X, \mathbb{A})} \theta ^{\phi}(g_1g , h)f(g_1 g) \, dg_1 \]

for $h \in \mathrm {GO}(Y, \mathbb {A})$, where $g \in \mathrm {GSp}(X, \mathbb {A})^+$ is chosen so that $\lambda (g) = \lambda (h)$. It defines an automorphic form on $\mathrm {GO}(Y, \mathbb {A})$. For a cuspidal automorphic representation $(\pi _+, V_{\pi _+})$ of $\mathrm {GSp}(X, \mathbb {A})^+$, we denote by $\Theta _\psi (\pi _+)$ the theta lift of $\pi _+$ to $\mathrm {GO}(Y, \mathbb {A})$. Namely,

\[ \Theta_\psi^{X, Y}(\pi_+) = \Theta_\psi(\pi_+): = \{\Theta (f, \phi) : f \in V_{\pi_+}, \phi \in \mathcal{S}(Z_+(\mathbb{A}))\}. \]

Furthermore, for an irreducible cuspidal automorphic representation $(\pi, V_\pi )$ of $\mathrm {GSp}(X, \mathbb {A})$, we define

\[ \Theta_\psi(\pi): = \Theta_\psi(\pi|_{\mathrm{GSp}(X, \mathbb{A})^+}), \]

where $\pi |_{\mathrm {GSp}(X, \mathbb {A})^+}$ denotes the automorphic representation of $\mathrm {GSp}(X, \mathbb {A})^+$ with its space of automorphic forms $\{\varphi |_{\mathrm {GSp}(X, \mathbb {A})^+} : \varphi \in V_\pi \}$.

As for the opposite direction, for a cusp form $f^\prime$ on $\mathrm {GO}(Y, \mathbb {A})$, we define its theta lift $\Theta (f^\prime, \phi )$ to $\mathrm {GSp}(X, \mathbb {A})^+$ by

\[ \Theta (f^\prime, \phi)(g): = \int _{\mathrm{O}(Y, F) \backslash \mathrm{O}(Y, \mathbb{A})} \theta ^{\phi}(g , h_1h)f(h_1 h) \, dh_1\quad \text{for $g\in\mathrm{GSp}(X,\mathbb{A})^+$}, \]

where $h \in \mathrm {GO}(Y,\mathbb {A})$ is chosen so that $\lambda (g) = \lambda (h)$. For an irreducible cuspidal automorphic representation $(\sigma, V_\sigma )$ of $\mathrm {GO}(Y, \mathbb {A})$, we define the theta lift $\Theta _\psi (\sigma )$ of $\sigma$ to $\mathrm {GSp}(X, \mathbb {A})^+$ by

\[ \Theta_\psi(\sigma):=\{\Theta (f^\prime, \phi): f^\prime\in V_\sigma,\ \phi \in \mathcal{S}(Z_+(\mathbb{A}))\}. \]

Moreover, we extend $\theta (f^\prime, \phi )$ to an automorphic form on $\mathrm {GSp}(X, \mathbb {A})$ by the natural embedding

\[ \mathrm{GSp}(X, F)^+ \backslash \mathrm{GSp}(X, \mathbb{A})^+ \rightarrow \mathrm{GSp}(X, F) \backslash \mathrm{GSp}(X, \mathbb{A}) \]

and extension by zero. Then we define the theta lift $\Theta _\psi (\sigma )$ of $\sigma$ to $\mathrm {GSp}(X, \mathbb {A})$ as the $\mathrm {GSp}(X,\mathbb {A})$ representation generated by such $\theta (f^\prime,\phi )$ for $f^\prime \in V_\sigma$ and $\phi \in \mathcal {S}(Z_+(\mathbb {A}))$.

For some $X$ and $Y$, theta correspondence for the dual pair $(\mathrm {GSp}(X)^+, \mathrm {GO}(Y))$ gives theta correspondence between $\mathrm {GSp}(X)^+$ and $\mathrm {GSO}(Y)$ by the restriction of representations of $\mathrm {GO}(Y)$ to $\mathrm {GSO}(Y)$. Indeed, when $\dim X=4$ and $\dim Y = 6$, we may consider theta correspondence for the pair $(\mathrm {GSp}(X)^+, \mathrm {GSO}(Y))$. In Gan and Takeda [Reference Gan and TakedaGT10, Reference Gan and TakedaGT11b], they study the case when $\mathrm {GSO}(Y) \simeq \mathrm {GSO_{3,3}}$ or $\mathrm {GSO}_{5,1}$, and, in [Reference MorimotoMor14a], the case when $\mathrm {GSO}(Y) \simeq \mathrm {GSO_{4,2}}$ is studied. In these cases, for a cusp form $f$ on $\mathrm {GSp}(X, \mathbb {A})^{+}$, we denote by $\theta (f, \phi )$ the restriction of $\Theta (f, \phi )$ to $\mathrm {GSO}(Y, \mathbb {A})$. Moreover, for a cuspidal automorphic representation $(\pi _+, V_{\pi _+})$ of $\mathrm {GSp}(X, \mathbb {A})^+$, we define the theta lift $\theta _\psi (\pi _+)$ of $\pi _+$ to $\mathrm {GSO}(Y, \mathbb {A})$ by

\[ \theta_\psi^{X, Y}(\pi_+) = \theta_\psi(\pi_+) := \{\theta (f, \phi) : f \in V_{\pi_+},\, \phi \in \mathcal{S}(Z_+(\mathbb{A}))\}. \]

Similarly, for a cusp form $f^\prime$ on $\mathrm {GSO}(Y, \mathbb {A})$, we define its theta lift $\theta (f^\prime, \phi )$ to $\mathrm {GSp}(X, \mathbb {A})^+$ by

\[ \theta (f^\prime, \phi)(g): = \int _{\mathrm{SO}(Y, F) \backslash \mathrm{SO}(Y, \mathbb{A})} \theta^{\phi}(g , h_1h)f(h_1 h) \, dh_1\quad \text{for $g\in\mathrm{GSp}(X,\mathbb{A})^+$}, \]

where $h \in \mathrm {GSO}(Y,\mathbb {A})$ is chosen so that $\lambda (g) = \lambda (h)$. We extend it to an automorphic from on $\mathrm {GSp}(X, \mathbb {A})$ as above. For a cuspidal automorphic representation $(\sigma, V_\sigma )$ of $\mathrm {GSO}(Y, \mathbb {A})$, we define the theta lift $\theta _\psi (\sigma )$ of $\sigma$ to $\mathrm {GSp}(X, \mathbb {A})^+$ by

\[ \theta_\psi(\sigma):= \{\theta (f^\prime, \phi): f^\prime\in V_\sigma,\ \phi \in \mathcal{S}(Z_+(\mathbb{A}))\}. \]

3.1.2 Pull-back of the global Bessel periods for the dual pairs $(\mathrm {GSp}_2,\mathrm {GSO}_{4,2})$ and $(\mathrm {GSp}_2,\mathrm {GSO}_{3,3})$

Our goal here is to prove the pull-back formula (3.1.6).

First we introduce the set-up. Let $X$ be the space of four-dimensional row vectors over $F$ equipped with the symplectic form

\[ \langle w_1,w_2\rangle=w_1\begin{pmatrix}0 & 1_2\\-1_2 & 0\end{pmatrix}\,{}^tw_2. \]

Let us take the standard basis of $X$ and name the basis vectors as

(3.1.5)\begin{equation} x_1 = (1, 0, 0, 0),\quad x_2 = (0, 1, 0, 0),\quad x_{-1} = (0, 0, 1, 0),\quad x_{-2} = (0, 0, 0, 1). \end{equation}

Then the matrix representation of $\mathrm {GSp}(X)$ with respect to the standard basis is $G=\mathrm {GSp}_2$ defined by (1.4.3). We let $G$ act on $X$ from the right.

Let $Y$ be the space of six-dimensional column vectors over $F$ equipped with the non-degenerate symmetric bilinear form

\[ (v_1,v_2)={}^tv_1 S_2 v_2, \]

where the symmetric matrix $S_2$ is given by (2.1.2). Let us take the standard basis of $Y$ and name the basis vectors as

\begin{align*} y_{-2} &= {}^{t}(1, 0, 0, 0, 0, 0),\quad y_{-1} = {}^{t}(0, 1, 0, 0, 0, 0),\\ e_1&= {}^{t}(0, 0, 1, 0, 0, 0),\quad e_2 = {}^{t}(0, 0, 0, 1, 0, 0),\\ y_{1} &= {}^{t}(0, 0, 0, 0, 1, 0),\quad y_{2} = {}^{t}(0, 0, 0, 0, 0, 1). \end{align*}

We note that $( y_i , y_j )=\delta _{ij}, ( e_1 , e_1)=2$ and $( e_2 , e_2) =-2d$. Since $d \in F^\times \setminus (F^\times )^2$, with respect to the standard basis, the matrix representations of $\mathrm {GO}(Y)$ and $\mathrm {GSO}(Y)$ are $\mathrm {GO}_{4,2}$ defined by (2.1.5) and $\mathrm {GSO}_{4,2}$ defined by (2.1.4), respectively. In this section, we also study the theta correspondence for the dual pair $(\mathrm {GSp}(X), \mathrm {GSO}_{3,3})$, for which, we may use the above matrix representation with $d \in (F^\times )^2$. Hence, in the remaining of this section, we study theta correspondence for $(\mathrm {GSp}(X), \mathrm {GSO}(Y))$ for an arbitrary $d \in F^\times$.

We shall denote $\mathrm {GSp}(X, \mathbb {A})^+$ by $G(\mathbb {A})^+$ and also $\mathrm {GSp}(X, F)^+$ by $G(F)^+$. We note that when $d \in (F^\times )^2$, $\mathrm {GSp}(X)^+ = \mathrm {GSp}(X)$.

Let $Z=X\otimes Y$ and we take a polarization $Z=Z_+\oplus Z_-$ as follows. First we take $X=X_+\oplus X_-$ where

\[ X_{+} = F \cdot x_1 + F \cdot x_2 \quad \text{and} \quad X_{-}= F \cdot x_{-1} + F \cdot x_{-2} \]

as the polarization of $X$. Then we decompose $Y$ as $Y=Y_+\oplus Y_0\oplus Y_-$ where

\[ Y_{+} =F \cdot y_1+ F \cdot y_2,\quad Y_0 = F \cdot e_1 + F \cdot e_2\quad \text{and}\quad Y_{-} =F \cdot y_{-1} + F \cdot y_{-2}. \]

Then let

\[ Z_{\pm} = (X \otimes Y_{\pm})\oplus(X_{\pm} \otimes Y_0), \]

where the double sign corresponds. To simplify the notation, we sometimes write $z_+\in Z_+$ as $z_+=(a_1,a_2;b_1,b_2)$ when

\[ z_{+} = a_1 \otimes y_1 + a_2 \otimes y_2 + b_1 \otimes e_1 + b_2 \otimes e_2\in Z_+,\quad \text{where $ a_i \in X,\ b_i \in X_{+}\, (i=1,2)$}. \]

Let us compute the pull-back of $(X, \chi, \psi )$-Bessel periods on $\mathrm {GSO}(Y)$ defined by (2.3.6) with respect to the theta lift from $G$.

Proposition 3.1 Let $(\pi,V_\pi )$ be an irreducible cuspidal automorphic representation of $G(\mathbb {A})$ whose central character is $\omega _\pi$ and $\chi$ a character of $\mathbb {A}_E^\times$ such that $\chi \mid _{\mathbb {A}^\times }=\omega _\pi ^{-1}$. Let $X\in \mathrm {Mat}_{2\times 2}(F)$ such that $\det X\ne 0$.

Then for $f \in V_\pi$ and $\phi \in \mathcal {S}(Z_+(\mathbb {A}))$, we have

(3.1.6)\begin{equation} \mathcal{B}_{X, \chi, \psi}(\theta(f: \phi)) = \int_{N(\mathbb{A}) \backslash G^1(\mathbb{A})} B_{S_X, \chi^{-1},\psi}(\pi(g)f)(\omega_\psi(g, 1) \phi)(v_X) \, dg, \end{equation}

where $B_{S_X, \chi ^{-1},\psi }$ is the $(S_X,\chi ^{-1},\psi )$-Bessel period on $G$ defined by (2.3.1).

Here, for $X= \big (\begin{smallmatrix}x_{11} & x_{12}\\x_{21} & x_{22} \end{smallmatrix}\big )$, we define a vector $v_X\in Z_+$ by

(3.1.7)\begin{equation} v_X := \bigg(x_{-2}, x_{-1};\frac{x_{21}}{2}x_1+\frac{x_{11}}{2} x_2, -\frac{x_{22}}{2d}x_1-\frac{x_{12}}{2d} x_2\bigg) \end{equation}

and a $2$-by-$2$ symmetric matrix $S_X$ by

(3.1.8)\begin{equation} S_X: = \frac{1}{4d}{}^{t}(J_2 \,{}^{t}X J_2)S_0 ( J_2\,{}^{t}X J_2). \end{equation}

We regard $\chi$ as a character of $\mathrm {GSO}(S_X)(\mathbb {A})$ by

(3.1.9)\begin{equation} \mathrm{GSO}(S_X) \ni k \mapsto \chi((J_2{}^{t}XJ_2) k (J_2{}^{t}XJ_2)^{-1})\in\mathbb{C}^\times. \end{equation}

In particular, the $(S_X, \chi ^{-1},\psi )$-Bessel period does not vanish on $V_\pi$ if and only if the $(X,\chi, \psi )$-Bessel period does not vanish on $\theta _\psi (\pi )$.

Proof. We compute the $(X,\chi, \psi )$-Bessel period defined by (2.3.6) in stages. We consider subgroups of $N_{4,2}$ given by

(3.1.10)\begin{align} N_0(F) &= \left\{u_0(x) :=\left.\begin{pmatrix} 1 & -{}^{t}X_0 S_1 & 0\\ 0 & 1_4 & X_0\\ 0 & 0 & 1 \end{pmatrix}\right| X_0 = \begin{pmatrix} x\\0\\0\\0\end{pmatrix}\right\}; \end{align}

(3.1.11)\begin{align} N_1(F) &= \left\{u_1(s_1, t_1) :=\left.\begin{pmatrix} 1 & -{}^{t}X_1 S_1 & -\dfrac{1}{2}{}^{t}X_1 S_1 X_1\\ 0 & 1_4 & X_1\\ 0 & 0 & 1 \end{pmatrix}\right| X_1 = \begin{pmatrix} 0\\s_1\\t_1\\0 \end{pmatrix}\right\}; \end{align}

(3.1.12)\begin{align} N_2(F) &= \left\{u_2(s_2, t_2) := \left.\begin{pmatrix} 1 & 0 & 0 & 0 & 0\\ 0 & 1 & -{}^{t}X_2 S_0 & -\dfrac{1}{2}{}^{t}X_2 S_0 X_2 & 0\\ 0 & 0 & 1_2 & X_2 & 0\\ 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 1 \end{pmatrix}\right| X_2 = \begin{pmatrix} s_2\\ t_2 \end{pmatrix}\right\}, \end{align}

where $S_0$ and $S_1$ are given by (2.1.2). Then we have

\[ N_0 \lhd N_0 N_1 \lhd N_0 N_1 N_2 =N_{4.2}. \]

Thus, we may write

(3.1.13)\begin{align} \mathcal{B}_{X, \chi, \psi}(\theta(f: \phi)) &= \int_{\mathbb{A}^\times M_{X}(F) \backslash M_X(\mathbb{A})} \int_{(F \backslash \mathbb{A}_{F})^2} \int_{(F \backslash \mathbb{A}_{F})^2} \int_{F \backslash \mathbb{A}_{F}} \theta(f, \phi)(u_0(x) u_1(s_1, t_1) u_2(s_2, t_2) h) \nonumber\\ &\quad \times \psi(x_{21}s_1+x_{22}t_1+x_{11}s_2+x_{12}t_2)^{-1}\chi (h)^{-1} \, {dx} \,ds_1 \,dt_1 \,ds_2 \,dt_2 \,dh. \end{align}

For $h \in \mathrm {GSO}(Y, \mathbb {A})$, let us define

\[ W_0(\theta(f: \phi))(h):= \int_{F \backslash \mathbb{A}_F} \theta(f, \phi)(u_0(x)h)\,{dx}. \]

From the definition of the theta lift, we have

(3.1.14)\begin{align} & W_0(\theta(f, \phi))(h) \nonumber\\ &\quad =\int_{F \backslash \mathbb{A}_F} \int_{G^1(F) \backslash G^1(\mathbb{A}_F)} \sum_{a_i \in X, b_i \in X_{+}} (\omega_\psi(g_1 \lambda_s(\nu(h)), u_0(x) h) \phi)(a_1, a_2; b_1, b_2) \nonumber\\ &\qquad \times f(g_1 \lambda_s(\lambda(h)))\, dg_1 \, {dx}. \end{align}

Here, for $a \in \mathbb {A}^\times$, we let

\[ \lambda_s(a) = \begin{pmatrix} 1_2 & 0\\0 & a \cdot 1_2 \end{pmatrix}. \]

Since $Z_{-}(1, u_0(x))= Z_{-}$ and we have

\[ z_{+}(1, u_0(x)) = z_{+} + (x \cdot a_1 \otimes y_{-2} -x \cdot a_2 \otimes y_{-1}), \]

we observe that

(3.1.15)\begin{align} (\omega_\psi(1, u_0(x)) \phi)(z_+) &= \psi \big(\tfrac{1}{2} \langle z_+, x \cdot a_1 \otimes y_{-2} -x \cdot a_2 \otimes y_{-1} \rangle \big) \phi(z_+) \nonumber\\ &= \psi (-x \langle a_1, a_2 \rangle) \phi(z_+). \end{align}

Thus, in the summation of the right-hand side of (3.1.14), only $a_i$ such that $\langle a_1, a_2 \rangle = 0$ contributes to the integral $W_0(\theta (f, \phi ))$, and we obtain

\begin{align*} W_0(\theta(f, \phi))(h) &= \int_{G^1(F) \backslash G^1(\mathbb{A}_F)}\\ &\quad \times\sum_{\substack{a_i \in X, \langle a_1, a_2 \rangle =0,\\ b_i \in X_{+}}} (\omega_\psi(g_1 \lambda_s(\lambda(h)), h) \phi) (a_1, a_2; b_1, b_2) f(g_1 \lambda_s(\lambda(h)))\,dg_1. \end{align*}

Since the space spanned by $a_1$ and $a_2$ is isotropic, there exists $\gamma \in G^1(F)$ such that $a_1 \gamma ^{-1}, a_2 \gamma ^{-1} \in X_{-}$. Let us define an equivalence relation $\sim$ on $(X_{-})^{2}$ by

\[ (a_1 , a_2) \sim (a_1^\prime , a_2^\prime) \underset{\text{def.}}{\Longleftrightarrow} \text{there exists $\gamma\in G^1(F)$ such that $a_i^\prime = a_i \gamma$ for $i=1,2$}. \]

Let us denote by $\mathcal {X}_{-}$ the set of equivalence classes $(X_{-})^2 / \sim$ and by $\overline {(a_1, a_2)}$ the equivalence class containing $(a_1,a_2)\in (X_{-})^2$. Then we may write $W_0 (\theta (f, \phi )) (h)$ as

\begin{align*} & \int_{G^1(F) \backslash G^1(\mathbb{A}_F)} \sum_{\overline{(a_1 , a_2)} \in \mathcal{X}_{-}}\ \sum_{\gamma \in V(a_1 , a_2) \backslash G^1(F)}\ \sum_{b_i \in X_{+}} (\omega_\psi (g_1 \lambda_s(\lambda(h)) , h) \phi)(a_1\gamma, a_2 \gamma; b_1, b_2)\\ &\quad \times f(g_1 \lambda_s(\lambda(h))) \, dg_1. \end{align*}

Here

\[ V(a_1, a_2) = \{g \in G^1(F) \mid \text{$a_ig =a_i$ for $i=1,2$}\}. \]

Lemma 3.1 For any $g \in G(\mathbb {A})^+$ and $h \in \mathrm {GSO}(Y, \mathbb {A})$ such that $\lambda (g) = \lambda (h)$,

\[ \sum_{b_i \in X_{+}}(\omega_\psi (g, h) \phi)(a_1 \gamma, a_2 \gamma, b_1, b_2) =\sum_{b_i \in X_{+}}(\omega_\psi (\gamma g, h) \phi)(a_1 , a_2 , b_1, b_2). \]

Further, by an argument similar to the one for $W_0 (\theta (f, \phi )) (h)$, we shall prove the following lemma.

Lemma 3.2 For any $g \in G(\mathbb {A})^+$ and $h \in \mathrm {GSO}(Y, \mathbb {A})$ such that $\lambda (g) = \lambda (h)$,

\begin{align*} & \int_{(F \backslash \mathbb{A}_F)^{2}} \psi^{-1}(x_{21}s_1+x_{22}t_1) (\omega_\psi (g , u_{1}(s_1, t_1) h) \phi)(a_1 , a_2 , b_1, b_2)\,ds_1\,dt_1\\ &\quad = \left\{\begin{array}{@{}ll} (\omega_\psi (g , h)\phi)(a_1 , a_2 , b_1, b_2) & \text{if $ \langle a_2, b_1 \rangle =-\dfrac{x_{21}}{2}$ and $\langle a_2, b_2 \rangle= \dfrac{x_{22}}{2d}$};\\ & \\ 0 & \text{otherwise}\\ \end{array}\right. \end{align*}

and

\begin{align*} & \int_{(F \backslash \mathbb{A}_F)^{2}} \psi^{-1}(x_{11} s_2+x_{12}t_2) (\omega_\psi(g , u_{2}(s_2, t_2) h) \phi)(a_1 , a_2 , b_1, b_2)\, ds_2\, dt_2\\ &\quad =\left\{\begin{array}{@{}ll} (\omega_\psi (g , h)\phi)(a_1 , a_2 , b_1, b_2) & \text{if $ \langle a_1 , b_1 \rangle =-\dfrac{x_{11}}{2}$ and $\langle a_1, b_2 \rangle =\dfrac{x_{12}}{2d}$};\\ & \\ 0 & \text{otherwise}.\\ \end{array}\right. \end{align*}

Proof. Since $Z_-\, (1, u_1(s_1, t_1)) = Z_-$ and we have

\begin{align*} z_+(1, u_1(s_1, t_1)) &= z_{+} +2s_1 (b_1 \otimes y_{-2}) - 2dt_1(b_2 \otimes y_{-2})\\ &\quad +(-s_1^2+2dt_1^2)a_2 \otimes y_{-2}-s_1a_2 \otimes e_1-t_1 a_2 \otimes e_2, \end{align*}

we obtain

\begin{align*} & (\omega_\psi(1, u_1(s_1, t_1)) \phi)(z_+) = \psi \big(\tfrac{1}{2} (2s_1\langle a_2, b_1 \rangle -2dt_1 \langle a_2, b_2 \rangle)\big)\\ &\qquad \times\psi \big(\tfrac{1}{2} ((-s_1^2+2dt_1^2) \langle a_2, a_2 \rangle -2s_1 \langle b_1, a_2 \rangle +2dt_1 \langle b_2, a_2 \rangle)\big)\phi(z_+)\\ &\quad =\psi (2s_1\langle a_2, b_1 \rangle -2dt_1 \langle a_2, b_2 \rangle)\phi(z_+). \end{align*}

Then the first assertion readily follows.

Similarly, since $Z_- \,(1, u_2(s_2, t_2)) = Z_-$ and we have

\[ z_+\,(1, u_2(s_2, t_2)) = z_+ + a_1 \otimes ((s_2^2-dt_2^2)y_{-1}-s_2 e_1-t_2e_2 ) + 2s_2 b_1 \otimes y_{-1} -2dt_2 b_2 \otimes y_{-1}, \]

we obtain

\begin{align*} & \omega(1, u_2(s_2, t_2)) \phi(z_+)= \psi \big(\tfrac{1}{2} (-2s_2 \langle b_1, a_1 \rangle +2dt_2 \langle b_2, a_1 \rangle)\big)\\ &\qquad \times\psi \big(\tfrac{1}{2}(2s_2 \langle a_1, b_1 \rangle -2dt_2 \langle a_1, b_2 \rangle)\big)\phi(z_+)\\ &\quad =\psi (2s_2 \langle a_1, b_1 \rangle -2dt_2 \langle a_1, b_2 \rangle) \phi(z_+) \end{align*}

and the second assertion follows.

Lemma 3.2 implies that

\begin{align*} \mathcal{B}_{X, \chi, \psi}(\theta(f: \phi)) &= \int_{\mathbb{A}^\times M_{X}(F) \backslash M_X(\mathbb{A})} \int_{G^1(F) \backslash G^1(\mathbb{A}_F)} \chi (h)^{-1}\\ &\quad \times\sum _{ \overline{(a_1 , a_2)} \in \mathcal{X}_{-}} \sum_{\gamma \in V(a_1 , a_2) \backslash G^1(F)} \sum_{\substack{ b_i \in X_{+}, \langle a_i, b_1 \rangle = {{x_{i1}}/{2}},\\ \langle a_i, b_2 \rangle =- ({x_{i2}}/{2d}) }}\\ &\quad \times(\omega_\psi (\gamma g_1 \lambda_s(\lambda(h)) , h) \phi) (a_1, a_2, b_1, b_2) f(g_1 \lambda_s(\lambda(h))) \, dg_1\,dh. \end{align*}

We note that $a_1$ and $a_2$ are linearly independent from the conditions on $a_i$ and $\det (X) \ne 0$. Since $a_i \in X_-$ and $\dim X_- =2$, we may take $(a_1,a_2)=(x_{-2},x_{-1})$ as a representative. Then we should have

\[ b_1 =\frac{x_{21}}{2}x_1+\frac{x_{11}}{2} x_2,\quad b_2= -\frac{x_{22}}{2d}x_1-\frac{x_{12}}{2d} x_2. \]

Hence, we get

(3.1.16)\begin{align} \mathcal{B}_{X, \chi, \psi}(\theta(f: \phi)) &= \int_{\mathbb{A}^\times M_{X}(F) \backslash M_X(\mathbb{A})} \int_{G^1(F) \backslash G^1(\mathbb{A}_F)}\chi(h)^{-1} \nonumber\\ &\quad \times \sum_{\gamma \in N(F) \backslash G^1(F)} (\omega_\psi (\gamma g_1 \lambda_s(\lambda(h)), h) \phi)(v_X) f(g_1 \lambda_s(\lambda(h))) \, dg_1\,dh \nonumber\\ &= \int_{N(\mathbb{A}) \backslash G^1(\mathbb{A}_F)} \int_{\mathbb{A}^\times M_{X}(F) \backslash M_X(\mathbb{A})} \int_{N(F) \backslash N(\mathbb{A})} \nonumber\\ &\quad \times\chi(h)^{-1}\omega (v g_1 \lambda_s(\lambda(h)), h) \phi(v_X) f(v g_1 \lambda_s(\lambda(h))) \, dv\,dg_1\,dh , \end{align}

where we put $v_X = (x_{-2}, x_{-1}; ( {x_{21}}/{2})x_1+{{x_{11}}/{2}} x_2, -({x_{22}}/{2d})x_1-({x_{12}}/{2d}) x_2)$.

For $u=\big (\begin{smallmatrix}1_2 & A\\0 & 1_2\end{smallmatrix}\big )$ where $A= \big (\begin{smallmatrix}a & b\\ b & c\end{smallmatrix}\big ) \in \mathrm {Sym}^2$, we have

\begin{align*} & \bigg( x_{-2} \otimes y_1 + x_{-1} \otimes y_2+ \bigg(\frac{x_{21}}{2}x_1+\frac{x_{11}}{2} x_2 \bigg) \otimes e_1 + \bigg( -\frac{x_{22}}{2d}x_1-\frac{x_{12}}{2d} x_2 \bigg)\otimes e_2 \bigg)(u,1) \\ &\quad = x_{-2} \otimes y_1 + x_{-1} \otimes y_2+ \bigg(\frac{x_{21}}{2}(x_1 +a x_{-1}+b x_{-2} )+ \frac{x_{11}}{2} (x_2+bx_{-1} +c x_{-2})\bigg) \otimes e_1 \\ &\qquad +\bigg(-\frac{x_{22}}{2d}(x_1 +a x_{-1}+b x_{-2} )- \frac{x_{12}}{2d} (x_2+bx_{-1} +c x_{-2})\bigg)\otimes e_2. \end{align*}

Hence, when we put

\begin{align*} S_X &= \frac{1}{4d}{}^{t}(J_2{}^{t}XJ_2)S_0 (J_2{}^{t}XJ_2)\\ &= \frac{1}{2d} \begin{pmatrix} x_{22}^2-dx_{21}^2 & x_{22}x_{12}-dx_{21}x_{11} \\ x_{22}x_{12}-dx_{21}x_{11} & x_{12}^2-dx_{11}^2 \end{pmatrix}\in \mathrm{Sym}^2(F), \end{align*}

for $u\in N(\mathbb {A})$, we have

\[ (\omega_\psi(ug\lambda_s(\lambda(h)), h) \phi)(v_X) = \psi_{S_X} (u)^{-1} \omega_\psi(g\lambda_s(\lambda(h)), h) \phi(v_X). \]

Therefore, we get

\begin{align*} & \int_{N(\mathbb{A}) \backslash G^1(\mathbb{A}_F)} \int_{\mathbb{A}^\times M_{X}(F) \backslash M_X(\mathbb{A})} \int_{N(F) \backslash N(\mathbb{A})} \chi(h)^{-1} \\ &\qquad \times (\omega_\psi (g_1 \lambda_s(\lambda(h)), h) \phi)(v_X) f(u g_1 \lambda_s(\lambda(h))) \psi_{S_X}(u)^{-1} \, du \, dh\, dg_1 \\ &\quad =\int_{N(\mathbb{A}) \backslash G^1(\mathbb{A}_F)} \int_{\mathbb{A}^\times M_{X}(F) \backslash M_X(\mathbb{A})} \int_{N(F) \backslash N(\mathbb{A})} \chi (h)^{-1} \\ &\qquad \times\omega_\psi (\lambda_s(\lambda(h)) g_1, h) \phi(v_X) |\lambda(h)|^3 f(u \lambda_s(\lambda(h)) g_1)\psi_{S_X}(u)^{-1}\, du \, dh\, dg_1. \end{align*}

By a direct computation, we see that

\[ (\omega_\psi (\lambda_s(\lambda(h)) g_1, h) \phi)(v_X) =|\lambda(h)|^{-3} (\omega_\psi (h_0 \lambda_s(\lambda(h)) g_1, 1) \phi)(v_X) \]

when we let

\[ h = \begin{pmatrix} (\det h) (h^X)^\ast & 0 & 0\\ 0 & h & 0\\ 0 & 0 & h^X \end{pmatrix},\quad h_0 = \begin{pmatrix} ({}^{t}XJ_2)^{-1} {}^{t}h ({}^{t}XJ_2) & 0\\ 0 & (J_2X) h^{-1} (J_2X)^{-1} \end{pmatrix}. \]

For $g \in \mathrm {GSO}(S_0)$, we have ${}^{t}g = wgw$ and we may write

\[ h_0 = \begin{pmatrix} (J_2{}^{t}XJ_2)^{-1} {}^{t}h (J_2{}^{t}XJ_2) & 0\\ 0 & {}^{t}((J_2{}^{t}XJ_2)^{-1} {}^{t}h (J_2{}^{t}XJ_2))^{-1} \end{pmatrix}. \]

Since we have

\[ \mathrm{GSO}(S_X) = (J_2{}^{t}XJ_2)^{-1} \mathrm{GSO}(S_0) (J_2{}^{t}XJ_2), \]

we get

(3.1.17)\begin{align} & \int_{N(\mathbb{A}) \backslash G^1(\mathbb{A}_F)} \int_{\mathbb{A}^\times T_{S_X}(F) \backslash T_{S_X}(\mathbb{A})} \int_{N(F) \backslash N(\mathbb{A})} \chi(h) \nonumber\\ &\qquad \times(\omega_\psi (g_1, 1) \phi)(v_X) f(u h g_1) \psi_{S_X}(u)^{-1} \, du \, dh\, dg_1 \nonumber\\ &\quad =\int_{N(\mathbb{A}) \backslash G^1(\mathbb{A}_F)} B_{S_X, \chi^{-1}}(\pi(g_1)f) (\omega_\psi (g_1, 1) \phi)(v_X) \,dg_1 , \end{align}

where we regard $\chi$ as a character of $\mathrm {GSO}(S_X)(\mathbb {A})$ by (3.1.9).

Finally, the last statement concerning the equivalence of the non-vanishing conditions on the $(S_X,\chi ^{-1},\psi )$-Bessel period and the $(X,\chi )$-Bessel period follows from the pull-back formula (3.1.6) by an argument similar to that in the proof of Proposition 2 in [Reference Furusawa and MorimotoFM17].

3.2.1 Theta correspondence for quaternionic dual pair with similitudes

Let $D$ be a quaternion division algebra over $F$. Let $X_D$ (respectively, $Y_D$) be a right (respectively, left) $D$-vector space of finite rank equipped with a non-degenerate hermitian bilinear form $(\,{,}\,)_{X_D}$ (respectively, non-degenerate skew-hermitian bilinear form $\langle \, {,} \, \rangle _{Y_D}$). Hence $(\,{,}\,)_{X_D}$ and $\langle \, {,} \,\rangle _{Y_D}$ are $D$-valued $F$-bilinear form on $X_D$ and $Y_D$ satisfying

\begin{gather*} \overline{(x ,x^\prime)_{X_D}} = (x^\prime,x)_{X_D},\quad (xa,x^\prime b)_{X_D} = \bar{a}(x,x^\prime)_{X_D} b, \\ \overline{\langle y,y^\prime \rangle_{Y_D}} = -\langle y^\prime,y\rangle_{Y_D}, \quad \langle ay,y^\prime b \rangle_{Y_D} = a \langle y,y^\prime \rangle_{Y_D} \bar{b}, \end{gather*}

for $x, x^\prime \in X_D$, $y, y^\prime \in Y_D$ and $a,b\in D$. We denote the isometry group of $X_D$ and $Y_D$ by $\mathrm {U}(X_D)$ and $\mathrm {U}(Y_D)$, respectively. Then the space $Z_D = X_D \otimes _D Y_D$ is regarded as a symplectic space over $F$ with the non-degenerate alternating form $\langle \,{,} \, \rangle$ defined by

(3.2.1)\begin{equation} \langle x_1 \otimes y_1, x_2 \otimes y_2 \rangle = \mathrm{tr}_D ((x_1, x_2)_{X_D} \overline{\langle y_1, y_2 \rangle_{Y_D}}) \in F \end{equation}

and we have a homomorphism $\mathrm {U}(X_D) \times \mathrm {U}(Y_D) \rightarrow \mathrm {Sp}(Z_D)$ defined by

(3.2.2)\begin{equation} (x \otimes y)(g, h) = x g \otimes h^{-1}y\quad \text{for $x \in X, y \in Y$, $h \in \mathrm{U}(Y_D)$ and $g \in \mathrm{U}(X_D)$.} \end{equation}

As in the case when $D \simeq \mathrm {Mat}_{2 \times 2}$, this mapping splits in the metaplectic group $\mathrm {Mp}(Z_D)$. Hence we have the Weil representation $\omega _\psi$ of $\mathrm {U}(X_D, \mathbb {A}) \times \mathrm {U}(Y_D, \mathbb {A})$ by restriction.

From now on, we suppose that the rank of $X_D$ is $2k$ and $X_D$ is maximally split, in the sense that its maximal isotropic subspace has rank $k$.

Let us denote by $\mathrm {GU}(X_D)$ (respectively, $\mathrm {GU}(Y_D)$) the similitude unitary group of $X_D$ (respectively, $Y_D$) with the similitude character $\lambda _D$ (respectively, $\nu _D$). In addition, we denote the identity component of $\mathrm {GU}(Y_D)$ by $\mathrm {GSU}(Y_D)$. Then the action (3.2.2) extends to a homomorphism