2.1 Notation and convention

In this paper, for a group G of Type $A_n$ (resp. $B_n$ , $C_n$ , $D_n$ , $G_2$ , $E_6$ , $E_7$ ), we use $std_G$ to denote the n-dimensional (resp. $2n+1$ -dimensional, $2n$ -dimensional, $2n$ -dimensional, 7-dimensional, 27-dimensional, 56-dimensional) standard representation of G. We use ${\mathrm {Spin}}_{2n}$ (resp. ${\mathrm {Spin}}_{2n+1}$ ) to denote the Spin representation of the reductive group of Type $D_n$ (resp. $B_n$ ) and we use ${\mathrm {HSpin}}_{2n}$ to denote the Half-Spin representation of reductive group with Type $D_n$ . We use $Sym^n$ (resp. $\wedge ^n$ ) to denote the n-th symmetric power (resp. exterior power) of a reductive group of Type A. We use $\wedge _{0}^{3}$ to denote the third fundamental representation of a reductive group of Type $C_3$ . Lastly, for a representation $\rho $ of G, we use $\rho ^\vee $ to denote the dual representation and $T(\rho )$ to denote $\rho \oplus \rho ^\vee $ .

In this paper, we always use l to denote the similitude character of a similitude group. If we have two similitude group $GH_1$ and $GH_2$ , we let

$$ \begin{align*}G(H_1\times H_2)=\{(h_1,h_2)\in GH_1\times GH_2|\;l(h_1)=l(h_2)\},\end{align*} $$

$$ \begin{align*}S(GH_1\times GH_2)=\{(h_1,h_2)\in GH_1\times GH_2|\;l(h_1)l(h_2)=1\}.\end{align*} $$

Similarly we can also define $G(H_1\times \cdots \times H_n)$ and $S(GH_1\times \cdots \times GH_n)$ . For example,

$$ \begin{align*}S({\mathrm{GL}}_{2}^{3})=S({\mathrm{GL}}_2\times {\mathrm{GL}}_2\times {\mathrm{GL}}_2)=\{(h_1,h_2,h_3)\in {\mathrm{GL}}_{2}^{3}|\;\det(h_1h_2h_3)=1\}.\end{align*} $$

All the nilpotent orbits considered in this paper are principal in a Levi subgroup (this is also the case in [Reference Ben-Zvi, Sakellaridis and Venkatesh1]). As a result, we will use the Levi subgroup or just the root type of the Levi subgroup to denote the nilpotent orbit, i.e. we will use a Levi subgroup L of G to denote the nilpotent orbit of ${\mathfrak {g}}$ that is principal in L (the zero nilpotent orbit is still denoted by $1$ ). For a split reductive group G, we will use $T_G$ to denote a maximal split torus of G (a minimal Levi subgroup).

For a BZSV quadruple $\hat {\Delta }=(\hat {G},\hat {G},\hat {\rho },1)$ , there are many other quadruples that is essentially equal to $\hat {\Delta }$ up to some central isogeny. To be specific, one can take any group $\hat {H}$ of the same root Type as $\hat {G}$ such that the representation $\hat {\rho }$ can also be defined on $\hat {H}$ . Then one can choose any group $\hat {G}'$ containing $\hat {H}$ such that $\hat {G}'=\hat {H}Z_{\hat {G}'}$ . The quadruple $(\hat {G}',\hat {H},\hat {\rho }, 1)$ is essentially equal to $\hat {\Delta }$ up to some central isogeny. For example, both $({\mathrm {PGL}}_{2}^3,{\mathrm {PGL}}_2,0,1)$ and $({\mathrm {GL}}_{2}^{3},{\mathrm {GL}}_2,0,1)$ can be viewed as trilinear ${\mathrm {GL}}_2$ -model. The dual quadruple of them are $({\mathrm {SL}}_{2}^3,{\mathrm {SL}}_{2}^3,\hat {\rho },1)$ and $({\mathrm {GL}}_{2}^3,S({\mathrm {GL}}_{2}^3),\hat {\rho },1)$ where $\hat {\rho }$ is the tensor product of the standard representations of ${\mathrm {SL}}_{2}\times {\mathrm {SL}}_2\times {\mathrm {SL}}_2$ and $S({\mathrm {GL}}_{2}\times {\mathrm {GL}}_2\times {\mathrm {GL}}_2)$ respectively, and they are equal to each other up to some central isogeny. While there are various choices of dual quadruples pairs $(\Delta ,\hat \Delta )$ associated to $\hat \rho $ due to the isogeny issue, in this paper, for each representation $\hat {\rho }$ in [Reference Knop30], we will only write down one quadruple $\Delta =(G,H,\rho _H,\iota )$ whose dual quadruple $\hat \Delta $ is $(\hat {G},\widehat {G/Z_\Delta },\hat {\rho },1)$ where $Z_{\Delta }=Z_G\cap ker(\rho _H)$ .

2.2 Theta correspondence for classical groups

In this paper we will frequently use theta correspondence for classical groups. We will briefly review it in this subsection. We start with the theta correspondence for the general linear group. Let $n\geq m\geq 1$ and $G=H_1\times H_2={\mathrm {GL}}_n\times {\mathrm {GL}}_m$ . We use V to denote the underlying vector space of the representation $\rho =std_{{\mathrm {GL}}_n}\otimes std_{{\mathrm {GL}}_m}$ of G. For $\varphi \in {\mathcal {S}}(V({\mathbb A}))$ , we define the theta function

$$ \begin{align*}\Theta_{\psi}^{\varphi}(g)=\sum_{X\in V(k)}\rho(g)\varphi(X),\;g\in G({\mathbb A})\end{align*} $$

which is an automorphic function on $G({\mathbb A})=H_1({\mathbb A})\times H_2({\mathbb A})$ . Let $\pi $ be a cuspidal automorphic representation of $H_2({\mathbb A})$ . For $\phi \in L^{2}(H_2(k){\backslash } H_2({\mathbb A}))_{\pi }$ , the integral

$$ \begin{align*}\int_{H_2(k){\backslash} H_2({\mathbb A})} \Theta_{\psi}^{\varphi}(h_1,h_2)\phi(h_2)dh_2\end{align*} $$

gives an automorphic function on $H_1({\mathbb A})$ which will be denoted by $\Theta (\phi )$ .

Theorem 2.2 [Reference Li34].

We have

$$ \begin{align*}\{\Theta(\phi)|\;\phi\in L^{2}(H_2(k){\backslash} H_2({\mathbb A}))_{\pi}\}=\{E(\phi',1)|\; \phi'\in L^{2}(H_2(k){\backslash} H_2({\mathbb A}))_{\pi}\}\end{align*} $$

where $E(\phi ',1)$ is the Eisenstein series on $H_1({\mathbb A})={\mathrm {GL}}_n({\mathbb A})$ induced from $\phi '$ and the identity function on ${\mathrm {GL}}_{n-m}({\mathbb A})$ that belongs to the induced representation $Ind_{P}^{H_1}((\pi \otimes 1)\delta _{P}^{1/2})$ (here P is a parabolic subgroup of $H_1={\mathrm {GL}}_n$ of type $(m,n-m)$ ). Moreover, for $\phi _1,\phi _2\in L^{2}(H_2(k){\backslash } H_2({\mathbb A}))_{\pi }$ , we have the Rallis inner product formula

$$ \begin{align*}& \int_{H_2(k){\backslash} H_2({\mathbb A})/Z_{H_2}({\mathbb A})} \int_{H_1(k){\backslash} H_1({\mathbb A})} \int_{H_1(k){\backslash} H_1({\mathbb A})} \Theta_{\psi}^{\varphi}(h_1,h_2) E(\phi_1,1)(h_1) \\&\qquad\quad \overline{\Theta_{\psi}^{\varphi}(h_1',h_2)E(\phi_2,1)(h_1')} dh_1dh_1' dh_2\end{align*} $$

$$ \begin{align*} \qquad\qquad``=\text{''} L(\frac{n-m+1}{2},\pi)\cdot \int_{H_1(k){\backslash} H_1({\mathbb A})/Z_{H_1}({\mathbb A})} E(\phi_1,1)(h_1)\overline{E(\phi_2,1)(h_1)}dh_2.\end{align*} $$

Here the integrals on both sides require regularization.

For the unramified computation, we also need the local theta correspondence for unramified representation. Let F be a p-adic local field that is a local place of k. We use $\phi _{\rho }(h_1,h_2)$ to denote the local spherical matrix coefficient of the Weil representation with $\phi _{\rho }(1,1)=1$ . Let $\pi $ be a tempered unramified representation of $H_2(F)$ , $\phi _\pi $ (resp. $\phi _{\pi ,1}$ ) be the unramified matrix coefficient of $\pi $ (resp. $Ind_{{\mathrm {GL}}_m\times {\mathrm {GL}}_{n-m}}^{{\mathrm {GL}}_n}(\pi \otimes 1)$ ) with $\phi _\pi (1)=\phi _{\pi ,1}(1)=1$ .

Theorem 2.4 [Reference Li34].

With the notation above, we have

$$ \begin{align*}\int_{H_2(F)} \phi_{\rho}(h_1,h_2) \phi_\pi(h_2) dh_2=L\left(\frac{n-m+1}{2},\pi\right)\cdot \phi_{\pi,1}(h_1).\end{align*} $$

Next we study the theta correspondence between ${\mathrm {SO}}_{2n}$ and ${\mathrm {Sp}}_{2m}$ with $n> m\geq 1$ . Let $G=H_1\times H_2={\mathrm {SO}}_{2n}\times {\mathrm {Sp}}_{2m}$ and we use V to denote the underlying vector space of the representation $\rho =std_{{\mathrm {SO}}_{2n}}\otimes std_{{\mathrm {Sp}}_{2m}}$ of G. Let Y be a maximal isotropic subspace of V, we can define $\Theta _{\psi }^{\varphi }(g)$ an automorphic function on $G({\mathbb A})$ as in the introduction, for any Schwartz function $\varphi $ on Y.

Let $\Pi $ be a cuspidal tempered global Arthur packet of $H_2({\mathbb A})={\mathrm {Sp}}_{2m}({\mathbb A})$ and let $\Pi '$ be its lifting to $H_1({\mathbb A})={\mathrm {SO}}_{2n}({\mathbb A})$ under the map ${\mathrm {SO}}_{2m+1}({\mathbb C})\times {\mathrm {SL}}_2({\mathbb C})\rightarrow {\mathrm {SO}}_{2n}({\mathbb A})$ whose restriction to ${\mathrm {SL}}_2$ is the principal embedding from ${\mathrm {SL}}_2$ to ${\mathrm {SO}}_{2n-2m-1}$ (if $n>m+1$ then $\Pi '$ is a nontempered Arthur L-packet)Footnote ⁷ . For $\phi \in L^{2}(H_2(k){\backslash } H_2({\mathbb A}))_{\pi }$ , the integral

$$ \begin{align*}\int_{H_2(k){\backslash} H_2({\mathbb A})} \Theta_{\psi}^{\varphi}(h_1,h_2)\phi(h_2) dh_2\end{align*} $$

gives an automorphic function on $H_1({\mathbb A})={\mathrm {SO}}_{2n}({\mathbb A})$ which will be denoted by $\Theta (\phi )$ . Then the following theorem holds.

Theorem 2.5 [Reference Kudla and Rallis32, Reference Yamana49, Reference Gan, Qiu and Takeda20].

With the notation above, the representation

$$ \begin{align*}\{\Theta(\phi)|\;\phi\in L^{2}({\mathrm{Sp}}_{2m}(k){\backslash} {\mathrm{Sp}}_{2m}({\mathbb A}))_{\Pi}\}\end{align*} $$

of $\ {\mathrm {SO}}_{2n}({\mathbb A})$ is a direct sum of some distinct irreducible representations belonging to the Arthur L-packet $\Pi '$ of $H_1({\mathbb A})={\mathrm {SO}}_{2n}({\mathbb A})$ . Moreover, for $\phi _1,\phi _2\in \Pi '$ , we have the Rallis inner product formula

$$ \begin{align*}\int_{H_2(k){\backslash} H_2({\mathbb A})} \int_{H_1(k){\backslash} H_1({\mathbb A})} \int_{H_1(k){\backslash} H_1({\mathbb A})} \Theta_{\psi}^{\varphi}(h_1,h_2) \phi_1(h_1) \overline{\Theta_{\psi}^{\varphi}(h_1',h_2)\phi_2(h_1')}dh_1dh_1' dh_2\end{align*} $$

$$ \begin{align*}``=\text{''} L(n-m,\Pi)\cdot \int_{H_1(k){\backslash} H_1({\mathbb A})} \phi_1(h_1)\phi_2(h_1)dh_1.\end{align*} $$

For the unramified computation, we also need the local theta correspondence for unramified representation. Let F be a p-adic local field that is a local place of k. We use $\phi _{\rho }(h_1,h_2)$ to denote the local spherical matrix coefficient of the Weil representation with $\phi _{\rho }(1,1)=1$ . Let $\pi $ be a tempered unramified representation of $H_2(F)$ and $\pi '$ be its lifting to $H_1(F)$ (which is also unramified). Let $\phi _\pi $ (resp. $\phi _{\pi '}$ ) be the unramified matrix coefficient of $\pi $ (resp. $\pi '$ ) with $\phi _\pi (1)=\phi _{\pi '}(1)=1$ .

Theorem 2.6 [Reference Li34].

With the notation above, we have

$$ \begin{align*}\int_{H_2(F)} \phi_{\rho}(h_1,h_2) \phi_\pi(h_2) dh_2=L(n-m,\pi)\cdot \phi_{\pi'}(h_1).\end{align*} $$

The theta correspondence between ${\mathrm {SO}}_{2m}$ and $ {\mathrm {Sp}}_{2n}$ (resp. ${\mathrm {GSO}}_{2n}$ and ${\mathrm {GSp}}_{2m}$ , ${\mathrm {GSO}}_{2m}$ and ${\mathrm {GSp}}_{2n}$ ) is similar and we will skip it here.

2.3 Anomaly-free symplectic representations

In this subsection we will recall the definition of anomaly-free symplectic representations. Let G be a split reductive group, $T\subset G$ be a maximal split torus and $\rho :G\rightarrow {\mathrm {Sp}}(V)$ is a symplectic representation of G. Since T is split, the restriction of the representation $(\rho , V)$ to T can be decomposed into a direct sum of two representations that are dual to each other, i.e.

$$ \begin{align*}(\rho|_T,V)\simeq (\rho_0,W)\oplus (\rho_{0}^\vee,W).\end{align*} $$

2.4 A conjecture of the duality under induction

We recall the notion from the introduction. Let $\Delta =(G,H,\rho _H,\iota )$ be a BZSV quadruple. Let L be the centralizer of $\{\iota (diag(t,t^{-1}))|\;t\in {\mathrm {GL}}_1\}$ in G. Then L is a Levi subgroup of G with $H\subset L$ . We have defined

$$ \begin{align*}\Delta_{red}=(L,H,\rho_{H,\iota},1)\end{align*} $$

where the representation $\rho _{H,\iota }$ was defined in (2). It is clear that $\Delta $ is reductive if and only if $\Delta =\Delta _{red}$ .

In Section 4.2.2 of [Reference Ben-Zvi, Sakellaridis and Venkatesh1], Ben-Zvi–Sakellaridis–Venkatesh made a conjecture about the relation between the dual of $\Delta $ and $\Delta _{red}$ . To state their conjecture, we first need a definition.

Definition 2.9. Let L be a Levi subgroup of G and $\rho $ be an irreducible representation of L with the highest weight $\varpi _L$ . There exists a Weyl element w of G such that $w\varpi _L$ is a dominant weight of G Footnote ⁸ . We define $(\rho )_{L}^{G}$ to be the irreducible representation of G whose highest weight is $w\varpi _L$ . In general, if $\rho =\oplus _i\rho _i$ is a finite-dimensional representation of L with $\rho _i$ irreducible, we define

$$ \begin{align*}(\rho)_{L}^{G}=\oplus_i(\rho_i)_{L}^{G}.\end{align*} $$

Now we are ready to state the conjecture.

Conjecture 2.10 (Ben-Zvi–Sakellaridis–Venkatesh, [Reference Ben-Zvi, Sakellaridis and Venkatesh1]).

With the notation above, if the dual of $\Delta _{red}$ is given by $\hat {\Delta }_{red}=(\hat {L}, \hat {H}_L', \rho ',\hat {\iota }')$ , then the dual of $\Delta $ is given by

$$ \begin{align*}(\hat{G},\hat{H}', (\rho')_{\hat{H}_L'}^{\hat{H}'},\hat{\iota}' )\end{align*} $$

where $\hat {H}'$ is generated by $\hat {H}_L'$ and $\{Im(\iota _\alpha )|\;\alpha \in \Delta _{\hat {G}}-\Delta _{\hat {L}}\}$ Footnote ⁹ . Here $\Delta _{\hat {G}}$ (resp. $\Delta _{\hat {L}}$ ) is the set of simple roots of $\hat {G}$ (resp. $\hat {L}$ ) and $\iota _\alpha :{\mathrm {SL}}_2\rightarrow \hat {G}$ is the embedding associated to $\alpha $ .

Assuming the above conjecture, it would suffice to develop an algorithm for the duality in the reductive case, i.e. when $\Delta =(G,H,\rho _H,1)$ (where $\iota =1$ ). Within this setting, two particularly important special cases arise:

• • the case when $\rho _H=0$ (i.e. the spherical variety case);

• • the case when $H=G$ (i.e. the symplectic vector space case).

As mentioned in the introduction, Ben-Zvi, Sakellaridis, and Venkatesh provided an algorithm in Section 4 of [Reference Ben-Zvi, Sakellaridis and Venkatesh1] for computing the dual Hamiltonian space in the spherical variety case. In this paper, we develop an algorithm for the symplectic vector space case (our approach will be explained in the next subsection).

Ultimately, our goal is to integrate these two algorithms to establish a general procedure for all reductive cases. If successful, this could be combined with Conjecture 2.10 to produce a general algorithm for computing the duality of any anomaly-free hyperspherical Hamiltonian space.

2.5 General strategy

Let $\hat {\Delta }=(\hat {G},\hat {G},\hat {\rho },1)$ be a quadruple such that $\hat {\rho }$ is an anomaly-free symplectic representation of $\hat {G}$ , and it appears in Tables 1, 2, 11, 12, 22 of [Reference Knop30]. Our goal is to write down a dual quadruple (up to isogeny) $\Delta =(G,H,\rho _H,\iota )$ .

The data in Knop’s tables of [Reference Knop30], besides $(\hat {G},\hat {\rho })$ , also contains the following two items: a Levi subgroup $\hat {L}$ of $\hat {G}$ and a Weyl group $\hat {W}_V$ written in the form of $W_{\hat {H}}$ where $\hat {H}$ is the root type (e.g. $A_n,B_n,C_n$ , etc). (In [Reference Knop30] the notations are $L,G,W_V$ in place of $\hat {L},\hat {G},\hat {W}_V$ respectively.) Our key observation is that two data $(H,\iota )$ of the dual quadruple $\Delta =(G,H,\rho _H,\iota )$ are given by the following properties.

Property 2.11 provides an algorithm to compute H and $\iota $ in the dual quadruple $\Delta =(G,H,\rho _H,\iota )$ . It remains to determine the symplectic representation $\rho _H$ . At this moment we do not have a systematic way to construct $\rho _H$ . Instead, we propose a $\rho _H$ in an ad hoc way and then provide evidence for the duality between $\Delta =(G,H,\rho _H,\iota )$ and $(\hat {G},\widehat {G/Z_\Delta },\hat {\rho },1)$ .

We provide two strong pieces of evidence for the duality. The first one is evidence for the period integral conjecture (Conjecture 1.5), i.e., Theorem 1.13 and 1.15. The second evidence is for nonreductive models. For those models, we will show that the duality is compatible with Conjecture 2.10, i.e. Theorem 1.18.

In the sections that follow, we will go through Knop’s list of representations $\hat \rho $ (i.e. Tables 1, 2, 11, 12, and 22 in [Reference Knop30]). For each $\hat {\rho }$ we write down a quadruple $(G,H,\rho _H,\iota )$ . When the quadruple is not reductive, we will also write down $\Delta _{red}$ which is dual to another representation $(\hat L,\hat \rho _L)$ in Knop’s list and verify that Theorem 1.18 holds.

For cases in Table 21, 22, 23 and 24, we give references where either the local relative character is calculated in the unramified places, or a Rankin-Selberg integral provides the local unramified Rankin-Selberg factor, thus verifying Theorem 1.13. We will also verify Theorem 1.15 for the global periods associated to the dual side $\hat \Delta $ for cases in Table 21, 23 and 25. In Section 8, we summarize our findings in Table 21-26.

Lastly, in Section 9 we will deal with Table S of [Reference Knop30]. We will describe an algorithm to glue to dual Hamiltonian spaces for representations in Table S of [Reference Knop30] and then we will prove Theorem 1.21.

3.1 The reductive case

In this subsection we consider the reductive cases, i.e., the ones in Table 1. The nilpotent orbit $\iota $ is trivial for all these cases so we will ignore it.

For (1.1) with $p=2m$ (resp. $p=2m+2$ ), the associated quadruple $\Delta $ is

(5) $$ \begin{align} (G,H,\rho_H)=({\mathrm{SO}}_{2m+1}\times {\mathrm{SO}}_{2m},{\mathrm{SO}}_{2m}, 0) \end{align} $$

(6) $$ \begin{align} (\text{resp.} (G,H,\rho_H)=({\mathrm{SO}}_{2m+1}\times {\mathrm{SO}}_{2m+2},{\mathrm{SO}}_{2m+1}, 0)) \end{align} $$

which is just the reductive Gross-Prasad model. The unramified computations in [Reference Ichino and Ikeda27] prove Theorem 1.13 in these two cases. For the dual side, Theorem 2.5 applied to the theta correspondence between ${\mathrm {SO}}_{2m}\times {\mathrm {Sp}}_{2m}$ (resp. ${\mathrm {SO}}_{2m+2}\times {\mathrm {Sp}}_{2m}$ ) implies Conjecture 1.5(2) and this proves Theorem 1.15.

For (1.3) with $m=2$ , the associated quadruple $\Delta $ is

$$ \begin{align*}(G,H,\rho_H)=({\mathrm{GSp}}_6\times {\mathrm{GSp}}_4,G({\mathrm{Sp}}_4\times {\mathrm{Sp}}_2),0)\end{align*} $$

which is the model $({\mathrm {GSp}}_6\times {\mathrm {GSp}}_4,G({\mathrm {Sp}}_4\times {\mathrm {Sp}}_2))$ studied in [Reference Wan and Zhang46]. The unramified computations in [Reference Wan and Zhang46] prove Theorem 1.13 in this case.

For (1.3) with $m=3$ , the associated quadruple $\Delta $ is

$$ \begin{align*}(G,H,\rho_H)=({\mathrm{GSp}}_6\times {\mathrm{GSpin}}_{7},S({\mathrm{GSp}}_6\times {\mathrm{GSpin}}_{7}),std_{{\mathrm{Sp}}_6}\otimes {\mathrm{Spin}}_7).\end{align*} $$

For (1.3) with $m=4$ , the associated quadruple $\Delta $ is

(7) $$ \begin{align} (G,H,\rho_H)=({\mathrm{GSp}}_6\times {\mathrm{GSpin}}_{9},S({\mathrm{GSp}}_6\times {\mathrm{GSpin}}_{8}),std_{{\mathrm{Sp}}_6}\otimes {\mathrm{HSpin}}_8). \end{align} $$

Theorem 1.13 and 1.15 for two cases can be established by the same argument as Model (11.11) of [Reference Knop30] (see (23) and (22) of Section 5.1) together with the triality of $D_4$ .

For (1.6), it is clear that the generic stabilizer of $\hat {\rho }$ in $\hat {G}$ is not connected, hence it does not belong to the current framework of BZSV duality. However, for this specific case, by the work of [Reference Ginzburg, Jiang and Rallis22], we expect there is an associated quadruple of the form $({\mathrm {GL}}_2,{\mathrm {GL}}_2,\rho _H,1)$ where $\rho _H$ is no longer an anomaly free symplectic representation, but rather we understand that $\rho _H$ corresponds to the theta series on $H={\mathrm {GL}}_2$ defined via the cubic covering of ${\mathrm {GL}}_2$ as in [Reference Ginzburg, Jiang and Rallis22]. There is a covering group involved in the integral since the generic stabilizer is not connected. In [Reference Ginzburg, Jiang and Rallis22] it is established that the nonvanishing of ${\mathcal {P}}_{H,\iota ,\rho _H}(\phi )$ is equivalent to the nonvanishing of $L(1/2,\Pi ,\hat \rho )$ . We expect further that Conjecture 1.5(1) holds in this case.

By the discussion above, the strongly tempered quadruple associated to Table 1 (without the row corresponding to (1.6)) is given as follows. Note that $\iota $ is trivial for all these cases.

Table 3 Dual quadruples of Table 1.

3.2 The nonreductive case

In this subsection we consider the nonreductive cases, i.e., the ones in Table 2.

For (1.1) with $p=2n<2m$ , the associated quadruple $\Delta $ isFootnote ¹⁰

(8) $$ \begin{align} ({\mathrm{SO}}_{2m+1}\times {\mathrm{SO}}_{2n},{\mathrm{SO}}_{2n},0,({\mathrm{GL}}_1)^{n}\times {\mathrm{SO}}_{2m-2n+1}\times T_{{\mathrm{SO}}_{2n}}) \end{align} $$

and it is the Gross-Prasad period for ${\mathrm {SO}}_{2m+1}\times {\mathrm {SO}}_{2n}$ . For (1.1) with $p=2n>2m+2$ , the associated quadruple $\Delta $ is

$$ \begin{align*}({\mathrm{SO}}_{2m+1}\times {\mathrm{SO}}_{2n},{\mathrm{SO}}_{2m+1},0,T_{{\mathrm{SO}}_{2m+1}}\times ({\mathrm{GL}}_1)^{m}\times {\mathrm{SO}}_{2n-2m})\end{align*} $$

and it is still the Gross-Prasad period for ${\mathrm {SO}}_{2m+1}\times {\mathrm {SO}}_{2n}$ . In these two cases $\Delta _{red}$ are given by (5), (6). It is clear that Theorem 1.18 holds in these two cases. The unramified computation in [Reference Ichino and Ikeda27] proves Theorem 1.13 for these two cases. Theorem 2.5 applied to the theta correspondence between ${\mathrm {SO}}_{2n}\times {\mathrm {Sp}}_{2m}$ implies Conjecture 1.5(2) and proves Theorem 1.15 for these two cases.

For (1.3) when $m=1$ , the associated quadruple $\Delta $ is

(9) $$ \begin{align} ({\mathrm{GSp}}_6\times {\mathrm{GL}}_2,{\mathrm{GL}}_2,0,({\mathrm{GL}}_3\times {\mathrm{GL}}_1)\times T_{{\mathrm{GL}}_2}) \end{align} $$

and it is the model $({\mathrm {GSp}}_6\times {\mathrm {GL}}_2,{\mathrm {GL}}_2\ltimes U)$ studied in [Reference Wan and Zhang46]. In this case $\Delta _{red}=(({\mathrm {GL}}_2)^3,{\mathrm {GL}}_2,0,1)$ (which a special case of (6) with $m=1$ ). It is clear that Theorem 1.18 holds in this case and the unramified computation in [Reference Wan and Zhang46] proves Theorem 1.13 in this case.

For (1.3) when $m>4$ , the associated quadruple $\Delta $ is

$$ \begin{align*}({\mathrm{GSpin}}_{2m+1}\times {\mathrm{GSp}}_6,S({\mathrm{GSpin}}_8\times {\mathrm{GSp}}_6),std_{{\mathrm{Sp}}_6}\otimes {\mathrm{HSpin}}_8, L)\end{align*} $$

where L is the Levi subgroup whose projection to ${\mathrm {GSpin}}_{2m+1}$ (resp. ${\mathrm {GSp}}_6$ ) is of the form $({\mathrm {GL}}_1)^4\times {\mathrm {GSpin}}_{2m-7}$ (resp. the maximal torus). The nilpotent orbit induces a Bessel period for the unipotent radical of the parabolic subgroup $P=MU$ with $M=({\mathrm {GL}}_1)^{m-4}\times {\mathrm {GSpin}}_9\times {\mathrm {GSp}}_6$ whose stabilizer is ${\mathrm {GSpin}}_8\times {\mathrm {GSp}}_6$ and we can naturally embed H into the stabilizer. In this case $\Delta _{red}$ is given by (7) and it is clear that Theorem 1.18 holds. Theorem 1.13 and 1.15 for this model can be established by the same argument as (27) in Section 5.2 together with the triality of $D_4$ .

For (1.4), the associated quadruple $\Delta $ is

(10) $$ \begin{align} ({\mathrm{GSp}}_8\times {\mathrm{GL}}_2,G({\mathrm{SL}}_2\times {\mathrm{SL}}_2),0,{\mathrm{GL}}_3\times {\mathrm{GL}}_1\times {\mathrm{GL}}_1\times T_{{\mathrm{GL}}_2}). \end{align} $$

The nilpotent orbit induces a Bessel period for the unipotent radical of the parabolic subgroup $P=MU$ with $M={\mathrm {GL}}_2\times {\mathrm {GSp}}_4\times {\mathrm {GL}}_2$ whose stabilizer is $G({\mathrm {SL}}_2\times {\mathrm {SL}}_2)\times {\mathrm {GL}}_2$ . We embeds H into the stabilizer so that the induced embedding from H into M is given by the natural embeddings of H into ${\mathrm {GSp}}_4$ and into ${\mathrm {GL}}_2\times {\mathrm {GL}}_2$ . In this case $\Delta _{red}=({\mathrm {GSp}}_4\times {\mathrm {GL}}_2\times {\mathrm {GL}}_2,G({\mathrm {SL}}_2\times {\mathrm {SL}}_2),0,1)$ which is essentially the Gross-Prasad model for ${\mathrm {SO}}_5\times {\mathrm {SO}}_4$ . If we replace the cusp form on ${\mathrm {GL}}_2$ by an Eisenstein series, we recover the Rankin-Selberg integrals in [Reference Bump and Ginzburg7]. It is clear that Theorem 1.18 holds in this case and the result in [Reference Bump and Ginzburg7] proves Theorem 1.13 in this case.

For (1.5) when $n=11$ , the associated quadruple $\Delta $ is

(11) $$ \begin{align} ({\mathrm{GSp}}_{10},{\mathrm{GL}}_2,0,{\mathrm{GL}}_5\times {\mathrm{GL}}_1) \end{align} $$

and it is the model $({\mathrm {GSp}}_{10},{\mathrm {GL}}_2\ltimes U)$ studied in [Reference Wan and Zhang46]. In this case $\Delta _{red}=(({\mathrm {GL}}_2)^3,{\mathrm {GL}}_2,0,1)$ . It is clear that Theorem 1.18 holds in this case and the unramified computation in [Reference Wan and Zhang46] proves Theorem 1.13 in this case.

For (1.5) when $n=12$ , the associated quadruple $\Delta $ is

$$ \begin{align*}({\mathrm{GSO}}_{12},{\mathrm{GL}}_2,0,{\mathrm{GL}}_6\times {\mathrm{GL}}_1)\end{align*} $$

and it is the model $({\mathrm {GSO}}_{12},{\mathrm {GL}}_2\ltimes U)$ studied in [Reference Wan and Zhang46]. In this case $\Delta _{red}=(({\mathrm {GL}}_2)^3,{\mathrm {GL}}_2,0,1)$ . It is clear that Theorem 1.18 holds in this case and the unramified computation in [Reference Wan and Zhang46] proves Theorem 1.13 in this case.

For (1.5) when $n=13$ , the associated quadruple $\Delta $ is

$$ \begin{align*}({\mathrm{GSp}}_{12},{\mathrm{GSp}}_4,0,{\mathrm{GL}}_3\times {\mathrm{GL}}_3\times {\mathrm{GL}}_1).\end{align*} $$

The nilpotent orbit induces a Bessel period for the unipotent radical of the parabolic subgroup $P=MU$ with $M={\mathrm {GL}}_4\times {\mathrm {GSp}}_4$ whose stabilizer is $H={\mathrm {GSp}}_4$ . In this case $\Delta _{red}=({\mathrm {GSp}}_4\times {\mathrm {GL}}_4,{\mathrm {GSp}}_4,0,1)$ which is essentially the Gross-Prasad model for ${\mathrm {SO}}_6\times {\mathrm {SO}}_5$ . It is clear that Theorem 1.18 holds in this case. In this case the unramified computation can be done in a similar way as [Reference Wan and Zhang46], which will give Theorem 1.13. Specifically, following the approach outlined in Sections 2.4–2.5 of [Reference Wan and Zhang46], one need only verify equations (2.21) and (2.23), as well as Lemma 2.32 of loc. cit., for the present model. Equations (2.21) and (2.23) can be established via straightforward matrix computation, while Lemma 2.32 can be verified by direct calculation. The argument is similar to the models studied in [Reference Wan and Zhang46].

For (1.7), the associated quadruple $\Delta $ is

(12) $$ \begin{align} ({\mathrm{GL}}_6,{\mathrm{GL}}_2,0,{\mathrm{GL}}_3\times {\mathrm{GL}}_3) \end{align} $$

and it is the Ginzburg-Rallis model $({\mathrm {GL}}_6,{\mathrm {GL}}_2\ltimes U)$ studied in [Reference Wan and Zhang46]. In this case $\Delta _{red}=(({\mathrm {GL}}_2)^3,{\mathrm {GL}}_2,0,1)$ . It is clear that Theorem 1.18 holds in this case and the unramified computation in [Reference Wan and Zhang46] proves Theorem 1.13 in this case.

For (1.11), the associated quadruple $\Delta $ is

$$ \begin{align*}(E_{7},{\mathrm{PGL}}_2,0,GE_6)\end{align*} $$

and it is the model $(E_7,{\mathrm {PGL}}_2\ltimes U)$ studied in [Reference Wan and Zhang46]. In this case $\Delta _{red}=(({\mathrm {PGL}}_2)^3,{\mathrm {PGL}}_2,0,1)$ . It is clear that Theorem 1.18 holds in this case and the unramified computation in [Reference Wan and Zhang46] proves Theorem 1.13 in this case.

By the discussion above, the strongly tempered quadruple associated to Table 2 is given as follows. Here for $\iota $ , we only list the root type of the Levi subgroup L of G such that $\iota $ is principal in L.

Table 4 Dual quadruples of Table 2.

4.1 The reductive case

In this subsection we consider the reductive cases, i.e., the ones in Table 5. The nilpotent orbit $\iota $ is trivial for all these cases so we will ignore it.

For (2.1) with $m=n$ , the associated quadruple $\Delta $ is given by

(13) $$ \begin{align} (G,H,\rho_H)=({\mathrm{GL}}_n\times {\mathrm{GL}}_n,{\mathrm{GL}}_n,T(std_{{\mathrm{GL}}_n})). \end{align} $$

For (2.1) with $m=n+1$ and (2.4) with $n=2$ , the associated quadruple $\Delta $ is given by

(14) $$ \begin{align} (G,H,\rho_H)=({\mathrm{GL}}_{n+1}\times {\mathrm{GL}}_n,{\mathrm{GL}}_n,0). \end{align} $$

The period integrals in these two cases are exactly the Rankin-Selberg integral for ${\mathrm {GL}}_n\times {\mathrm {GL}}_n$ and ${\mathrm {GL}}_{n+1}\times {\mathrm {GL}}_n$ in [Reference Jacquet, Piatetskii-Shapiro and Shalika28]. The result in loc. cit. proves Conjecture 1.5(1). The unramified computation in [Reference Harris26] and [Reference Xue48] imply Theorem 1.13. For the dual side, Theorem 2.2 applied to the theta correspondence for ${\mathrm {GL}}_n\times {\mathrm {GL}}_{n+1}$ and ${\mathrm {GL}}_n\times {\mathrm {GL}}_n$ imply Conjecture 1.5(2) and this proves Theorem 1.15.

For (2.3), the generic stabilizer of $\hat {\rho }$ in $\hat {G}$ is not connected, hence it does not belong to the current framework of the BZSV duality. However, for this specific case, by the Rankin-Selberg integral in [Reference Bump and Ginzburg8, Reference Patterson and Piatetski-Shapiro37, Reference Takeda42], we know that the dual integral should be the one in [Reference Bump and Ginzburg8, Reference Patterson and Piatetski-Shapiro37, Reference Takeda42]. As the generic stabilizer is not connected, there are covering groups involved in the integral.

For (2.6) with $m=n=2$ , the associated quadruple $\Delta $ is given by

(15) $$ \begin{align} (G,H,\rho_H)=({\mathrm{GSp}}_4\times {\mathrm{GL}}_2, G({\mathrm{SL}}_2\times {\mathrm{SL}}_2), T(std_{{\mathrm{GL}}_2,2})) \end{align} $$

where the embedding of H into G is given by the canonical embedding from ${\mathrm {GSpin}}_4=G({\mathrm {SL}}_2\times {\mathrm {SL}}_2)$ into ${\mathrm {GSpin}}_5={\mathrm {GSp}}_4$ and the projection of $G({\mathrm {SL}}_2\times {\mathrm {SL}}_2)$ into ${\mathrm {GL}}_2$ via the first ${\mathrm {GL}}_2$ -copy. The representation $\rho _H$ is the standard representation of the second ${\mathrm {GL}}_2$ -copy of H. This integral is essentially the Gross-Prasad model for ${\mathrm {SO}}_5\times {\mathrm {SO}}_4$ except we replace the cusp form on one ${\mathrm {GL}}_2$ -copy by the theta series. The unramified computation in [Reference Ichino and Ikeda27] proves Theorem 1.13 in this case. For the dual side, Conjecture 1.5(2) would follow from Theorem 2.2 applied to the theta correspondence of ${\mathrm {GL}}_2\times {\mathrm {GL}}_4$ and Gan-Gross-Prasad conjecture (Conjecture 9.11 of [Reference Gan, Gross and Prasad14]) for nontempered Arthur packet for the pair $({\mathrm {GL}}_4\times {\mathrm {GSp}}_4,{\mathrm {GSp}}_4)$ which is essentially the Gross-Prasad period for ${\mathrm {SO}}_6\times {\mathrm {SO}}_5$ . This proves Theorem 1.15.

For (2.6) with $m=2, n=3$ , the associated quadruple $\Delta $ is given by

$$ \begin{align*}(G,H,\rho_H)=({\mathrm{GSp}}_4\times {\mathrm{GL}}_3, {\mathrm{GSp}}_4\times {\mathrm{GL}}_3, T(std_{{\mathrm{GSp}}_4}\otimes std_{{\mathrm{GL}}_3})).\end{align*} $$

By the theta correspondence for ${\mathrm {GL}}_3\times {\mathrm {GL}}_4$ (note that the theta function constructed from $T(std_{{\mathrm {GSp}}_4}\otimes std_{{\mathrm {GL}}_3})$ is the restriction of the theta function from $T(std_{{\mathrm {GL}}_4}\otimes std_{{\mathrm {GL}}_3})$ ), the integral over ${\mathrm {GL}}_3$ of a cusp form on ${\mathrm {GL}}_3$ with the theta series associated to $\rho _H$ produces an Eisenstein series of ${\mathrm {GL}}_4$ induced from the cusp form on ${\mathrm {GL}}_3$ and the trivial character of ${\mathrm {GL}}_1$ . Then the integral over ${\mathrm {GSp}}_4$ is just the period integral for the pair $({\mathrm {GL}}_4\times {\mathrm {GSp}}_4,{\mathrm {GSp}}_4)$ which is essentially the Gross-Prasad period for ${\mathrm {SO}}_6\times {\mathrm {SO}}_5$ . The unramified computation in [Reference Ichino and Ikeda27] and Theorem 2.4 applied to theta correspondence for ${\mathrm {GL}}_3\times {\mathrm {GL}}_4$ proves Theorem 1.13 in this case. For the dual side, Conjecture 1.5(2) follows from Theorem 2.2 applied to the theta correspondence of ${\mathrm {GL}}_4\times {\mathrm {GL}}_3$ and the global period integral conjecture for the pair $({\mathrm {GL}}_4\times {\mathrm {GSp}}_4,{\mathrm {GSp}}_4)$ (which is essentially the Gross-Prasad period for ${\mathrm {SO}}_6\times {\mathrm {SO}}_5$ ) in [Reference Gan, Gross and Prasad13]. This proves Theorem 1.15.

For (2.6) with $m=2, n=4$ , the associated quadruple $\Delta $ is

(16) $$ \begin{align} (G,H,\rho_H)=({\mathrm{GSp}}_4\times {\mathrm{GL}}_4, S({\mathrm{GSp}}_4\times {\mathrm{GL}}_4),std_{{\mathrm{Sp}}_4}\otimes \wedge^2\oplus T(std_{{\mathrm{GL}}_4})). \end{align} $$

By the theta correspondence for ${\mathrm {GSp}}_4\times {\mathrm {GSO}}_6$ , the integral over ${\mathrm {Sp}}_4$ of a cusp form on ${\mathrm {GSp}}_4$ with the theta series associated to $\rho _H$ produces an automorphic form of ${\mathrm {GL}}_4$ . Then the integral over ${\mathrm {GL}}_4$ is just the Rankin-Selberg integral of ${\mathrm {GL}}_4\times {\mathrm {GL}}_4$ as in [Reference Jacquet, Piatetskii-Shapiro and Shalika28]Footnote ¹¹ . The Rankin-Selberg integral in [Reference Jacquet, Piatetskii-Shapiro and Shalika28], the unramified computation in [Reference Xue48], and Theorems 2.2 and 2.4 applied to theta correspondence for ${\mathrm {GSp}}_4\times {\mathrm {GSO}}_6$ proves Conjecture 1.5(1) and Theorem 1.13 in this case. For the dual side, Conjecture 1.5(2) follows from Theorem 2.2 applied to the theta correspondence of ${\mathrm {GL}}_4\times {\mathrm {GL}}_4$ and the global period integral conjecture for the pair $({\mathrm {GL}}_4\times {\mathrm {GSp}}_4,{\mathrm {GSp}}_4)$ (which is essentially the Gross-Prasad period for ${\mathrm {SO}}_6\times {\mathrm {SO}}_5$ ) in [Reference Gan, Gross and Prasad13]. This proves Theorem 1.15. This is a very interesting case because both $\Delta $ and $\hat {\Delta }$ are strongly tempered and they are not equal to each other.

For (2.6) with $m=2, n=5$ , the associated quadruple $\Delta $ is

(17) $$ \begin{align} (G,H,\rho_H)= ({\mathrm{GSp}}_4\times {\mathrm{GL}}_5, S({\mathrm{GSp}}_4\times {\mathrm{GL}}_4),std_{{\mathrm{Sp}}_4}\otimes \wedge^2). \end{align} $$

By the theta correspondence for ${\mathrm {GSp}}_4\times {\mathrm {GSO}}_6$ , the integral over ${\mathrm {Sp}}_4$ of a cusp form on ${\mathrm {GSp}}_4$ with the theta series associated to $\rho _H$ produces an automorphic form of ${\mathrm {GL}}_4$ . Then the integral over ${\mathrm {GL}}_4$ is just the Rankin-Selberg integral of ${\mathrm {GL}}_5\times {\mathrm {GL}}_4$ . The Rankin-Selberg integral in [Reference Jacquet, Piatetskii-Shapiro and Shalika28], the unramified computation in [Reference Harris26], and Theorems 2.5 and 2.6 applied to theta correspondence ${\mathrm {GSp}}_4\times {\mathrm {GSO}}_6$ proves Conjecture 1.5(1) and Theorem 1.13 in this case. For the dual side, Conjecture 1.5(2) follows from Theorem 2.2 applied to the theta correspondence of ${\mathrm {GL}}_4\times {\mathrm {GL}}_5$ and the global period integral conjecture for the pair $({\mathrm {GL}}_4\times {\mathrm {GSp}}_4,{\mathrm {GSp}}_4)$ (which is essentially the Gross-Prasad period for ${\mathrm {SO}}_6\times {\mathrm {SO}}_5$ ) in [Reference Gan, Gross and Prasad13]. This proves Theorem 1.15.

For (2.6) with $m=n=3$ , the associated quadruple $\Delta $ is given by

(18) $$ \begin{align} (G,H,\rho_H)= ({\mathrm{GSpin}}_7\times {\mathrm{GL}}_3,{\mathrm{GSpin}}_6\times {\mathrm{GL}}_3,T({\mathrm{HSpin}}_6\otimes std_{{\mathrm{GL}}_3})). \end{align} $$

By the theta correspondence for ${\mathrm {GL}}_3\times {\mathrm {GL}}_4$ (note that ${\mathrm {GSpin}}_6$ is essentially ${\mathrm {GL}}_4$ up to some central isogeny which won’t affect the unramified computation) the integral over ${\mathrm {GL}}_3$ of a cusp form on ${\mathrm {GL}}_3$ with the theta series associated to $\rho _H$ produces an Eisenstein series of ${\mathrm {GSpin}}_6$ induced from the cusp form on ${\mathrm {GL}}_3$ and the trivial character of ${\mathrm {GL}}_1$ . Then the integral over ${\mathrm {GSpin}}_6$ is just the period integral for the Gross-Prasad model of ${\mathrm {GSpin}}_7\times {\mathrm {GSpin}}_6$ . The unramified computation in [Reference Ichino and Ikeda27] and Theorem 2.4 applied to theta correspondence for ${\mathrm {GL}}_3\times {\mathrm {GL}}_4$ proves Theorem 1.13 in this case. For the dual side, Conjecture 1.5(2) follows from Theorem 2.5 applied to the theta correspondence of ${\mathrm {GSp}}_6\times {\mathrm {GSO}}_6$ and the Rankin-Selberg integral of ${\mathrm {GL}}_4\times {\mathrm {GL}}_3$ . This proves Theorem 1.15.

By the discussion above, the strongly tempered quadruple associated to Table 5 is given as follows. Note that $\iota $ is trivial for all these cases.

Table 7 Dual quadruples of Table 5.

4.2 The nonreductive case

For (2.1) with $m>n+1$ and (2.4) with $n>2$ , the associated quadruple $\Delta $ is given by

$$ \begin{align*}(G,H,\rho_H,\iota)=({\mathrm{GL}}_m\times {\mathrm{GL}}_n,{\mathrm{GL}}_n,0, ({\mathrm{GL}}_1)^{n}\times {\mathrm{GL}}_{m-n}\times T_{{\mathrm{GL}}_n} ).\end{align*} $$

When $m-n$ is odd (resp. even), the nilpotent orbit induces a Bessel period (resp. Fourier-Jacobi period) for the unipotent radical of the parabolic subgroup $P=MU$ with $M=({\mathrm {GL}}_1)^{m-n-1}\times {\mathrm {GL}}_{n+1}\times {\mathrm {GL}}_n$ (resp. $M=({\mathrm {GL}}_1)^{m-n}\times {\mathrm {GL}}_{n}\times {\mathrm {GL}}_n$ ) whose stabilizer in M is ${\mathrm {GL}}_n\times {\mathrm {GL}}_n$ . We can diagonally embed H into the stabilizer. This is the Gan-Gross-Prasad model for the general linear group defined in Section 13 of [Reference Gan, Gross and Prasad13]. In this case $\Delta _{red}$ is given by the quadruple (14) (resp. (13)). It is clear that Theorem 1.18 holds in this case. The period integral in this case is closely related to the Rankin-Selberg integral in [Reference Jacquet, Piatetskii-Shapiro and Shalika28]. However the difference is not negligible and we do not claim Theorem 1.13 for this case. For the dual side, Conjecture 1.5(2) follows from Theorem 2.2 applied to the theta correspondence for ${\mathrm {GL}}_n\times {\mathrm {GL}}_m$ . This proves Theorem 1.15.

For (2.2) with $n=2m$ , the associated quadruple $\Delta $ is given by

$$ \begin{align*}({\mathrm{GL}}_{2m},{\mathrm{GL}}_m,T(std_{{\mathrm{GL}}_m}),({\mathrm{GL}}_2)^m).\end{align*} $$

The nilpotent orbit induces a Bessel period for the unipotent radical of the parabolic subgroup $P=MU$ with $M={\mathrm {GL}}_m\times {\mathrm {GL}}_m$ whose stabilizer in M is $H={\mathrm {GL}}_m$ . In this case $\Delta _{red}$ is given by (13). It is clear that Theorem 1.18 holds in this case. The period integral in this case is exactly the Rankin-Selberg integral in [Reference Jacquet and Shalika29]. The result in loc. cit. proves Conjecture 1.5(1) and Theorem 1.13.

For (2.2) with $n=2m+1$ , the associated quadruple $\Delta $ is given by

$$ \begin{align*}({\mathrm{GL}}_{2m+1},{\mathrm{GL}}_m,0,({\mathrm{GL}}_2)^m\times {\mathrm{GL}}_1).\end{align*} $$

The nilpotent orbit induces a Fourier-Jacobi period for the unipotent radical of the parabolic subgroup $P=MU$ with $M={\mathrm {GL}}_m\times {\mathrm {GL}}_1\times {\mathrm {GL}}_m$ whose stabilizer in M is ${\mathrm {GL}}_n\times {\mathrm {GL}}_1$ . We can naturally embed H into the stabilizer. In this case $\Delta _{red}$ is given by (13). It is clear that Theorem 1.18 holds in this case. The period integral in this case is exactly the Rankin-Selberg integral in [Reference Jacquet and Shalika29]. The result in loc. cit. proves Conjecture 1.5(1) and Theorem 1.13.

For (2.5), the associated quadruple $\Delta $ is given by

$$ \begin{align*}({\mathrm{SO}}_{2m+1},{\mathrm{SO}}_2,0,{\mathrm{SO}}_{2m-1}\times {\mathrm{GL}}_1).\end{align*} $$

It is the Gross-Prasad model of ${\mathrm {SO}}_{2m+1}\times {\mathrm {SO}}_2$ and $\Delta _{red}$ is given by (5) when $m=1$ . It is clear that Theorem 1.18 holds in this case. The unramified computation in [Reference Ichino and Ikeda27] proves Theorem 1.13. For the dual side, Conjecture 1.5(2) follows from Theorem 2.5 applied to the theta correspondence for ${\mathrm {Sp}}_{2m}\times {\mathrm {SO}}_2$ and this proves Theorem 1.15.

For (2.6) with $m>2, n=2$ , the associated quadruple $\Delta $ is given by

$$ \begin{align*}(G,H,\rho_H,\iota)=({\mathrm{GSpin}}_{2m+1}\times {\mathrm{GL}}_2, G({\mathrm{SL}}_2\times {\mathrm{SL}}_2), T(std_{{\mathrm{GL}}_2}),({\mathrm{GL}}_1)^{2}\times {\mathrm{GSpin}}_{2m-3}\times T_{{\mathrm{GL}}_2,2}).\end{align*} $$

The nilpotent orbit $\iota $ induces a Bessel period on the unipotent radical of the parabolic subgroup $P=MU$ with $M={\mathrm {GSpin}}_{5}\times ({\mathrm {GL}}_1)^{m-2}\times {\mathrm {GL}}_2$ whose stabilizer in M is ${\mathrm {GSpin}}_4\times {\mathrm {GL}}_2$ . We then embeds $H=G({\mathrm {SL}}_2\times {\mathrm {SL}}_2)$ into ${\mathrm {GSpin}}_4\times {\mathrm {GL}}_2$ via the identity map on ${\mathrm {GSpin}}_4$ and the projection of $G({\mathrm {SL}}_2\times {\mathrm {SL}}_2)$ into ${\mathrm {GL}}_2$ via the first ${\mathrm {GL}}_2$ -copy. The representation $\rho _H$ is the standard representation of the second ${\mathrm {GL}}_2$ -copy of H. This integral is essentially the Gross-Prasad model for ${\mathrm {GSpin}}_{2m+1}\times {\mathrm {GSpin}}_4$ except we replace the cusp form on one ${\mathrm {GL}}_2$ -copy by theta series. In this case $\Delta _{red}$ is given by (15). It is clear that Theorem 1.18 holds in this case. The unramified computation in [Reference Ichino and Ikeda27] proves Theorem 1.13. For the dual side, Conjecture 1.5(2) follows from Theorem 2.5 applied to the theta correspondence for ${\mathrm {GSp}}_{2n}\times {\mathrm {GSO}}_4$ and the Rankin-Selberg integral of ${\mathrm {GL}}_2\times {\mathrm {GL}}_1$ . This proves Theorem 1.15.

For (2.6) with $m=2, n>5$ , the associated quadruple $\Delta $ is

$$ \begin{align*}({\mathrm{GSp}}_4\times {\mathrm{GL}}_n, S({\mathrm{GSp}}_4\times {\mathrm{GL}}_4),std_{{\mathrm{Sp}}_4}\otimes \wedge^2,T_{{\mathrm{GSp}}_4}\times ({\mathrm{GL}}_1)^{4}\times {\mathrm{GL}}_{n-4}).\end{align*} $$

When n is odd (resp. even), the nilpotent orbit induces a Bessel period (resp. Fourier-Jacobi period) for the unipotent radical of the parabolic subgroup $P=MU$ with $M={\mathrm {GSp}}_4\times {\mathrm {GL}}_5\times ({\mathrm {GL}}_1)^5$ (resp. $M={\mathrm {GSp}}_4\times {\mathrm {GL}}_4\times ({\mathrm {GL}}_1)^4$ ) whose stabilizer in M is ${\mathrm {GSp}}_4\times {\mathrm {GL}}_4$ . We can naturally embed H into the stabilizer. In this case $\Delta _{red}$ is given by (17) (resp. (16)). It is clear that Theorem 1.18 holds in this case. For the dual side, Conjecture 1.5(2) would follow from Theorem 2.2 applied to the theta correspondence of ${\mathrm {GL}}_n\times {\mathrm {GL}}_4$ and the global period integral conjecture for the pair $({\mathrm {GL}}_4\times {\mathrm {GSp}}_4,{\mathrm {GSp}}_4)$ (which is essentially the Gross-Prasad period for ${\mathrm {SO}}_6\times {\mathrm {SO}}_5$ ) in [Reference Gan, Gross and Prasad13]. This proves Theorem 1.15.

For (2.6) with $m>3,n=3$ , the associated quadruple $\Delta $ is given by

$$ \begin{align*}({\mathrm{GSpin}}_{2m+1}\times {\mathrm{GL}}_3,{\mathrm{GSpin}}_6\times {\mathrm{GL}}_3,T({\mathrm{HSpin}}_6\otimes std_{{\mathrm{GL}}_3}),({\mathrm{GL}}_1)^{3}\times {\mathrm{GSpin}}_{2m-5}\times T_{{\mathrm{GL}}_3}).\end{align*} $$

The nilpotent orbit $\iota $ induces a Bessel period on the unipotent radical of the parabolic subgroup $P=MU$ with $M={\mathrm {GSpin}}_{7}\times ({\mathrm {GL}}_1)^{m-3}\times {\mathrm {GL}}_3$ whose stabilizer in M is $H={\mathrm {GSpin}}_6\times {\mathrm {GL}}_3$ . In this case $\Delta _{red}$ is given by (18). It is clear that Theorem 1.18 holds in this case. The unramified computation in [Reference Ichino and Ikeda27] and Theorem 2.4 applied to theta correspondence for ${\mathrm {GL}}_4\times {\mathrm {GL}}_3$ proves Theorem 1.13. For the dual side, Conjecture 1.5(2) follows from Theorem 2.5 applied to the theta correspondence of ${\mathrm {GSp}}_{2n}\times {\mathrm {GSO}}_6$ and the Rankin-Selberg period for ${\mathrm {GL}}_4\times {\mathrm {GL}}_3$ . This proves Theorem 1.15.

For (2.7) with $m=2k$ , the associated quadruple $\Delta $ is

$$ \begin{align*}({\mathrm{GSpin}}_{2k},{\mathrm{GSpin}}_3,T({\mathrm{Spin}}_3),{\mathrm{GL}}_1\times {\mathrm{GSpin}}_{2k-2}).\end{align*} $$

This is essentially the Gross-Prasad model for ${\mathrm {GSpin}}_{2k}\times {\mathrm {GSpin}}_3$ except we replace the cusp form on ${\mathrm {GSpin}}_3$ by a theta series. In this case $\Delta _{red}$ is given by (13) when $n=2$ . It is clear that Theorem 1.18 holds in this case. The unramified computation in [Reference Ichino and Ikeda27] proves Theorem 1.13.

For (2.7) with $m=2k+1$ , the generic stabilizer of $\hat {\rho }$ in $\hat {G}$ is not connected, hence it does not belong to the current framework of the BZSV duality. However, for this specific case, by the Rankin-Selberg integral in [Reference Ginzburg, Rallis and Soudry24], we know that the dual integral should be the one in [Reference Ginzburg, Rallis and Soudry24]. As the generic stabilizer is not connected, there are covering groups involved in the integral.

For (2.8) with $n=7$ , the associated quadruple $\Delta $ is given by

$$ \begin{align*}({\mathrm{GSp}}_6,{\mathrm{GL}}_2,T(std_{{\mathrm{GL}}_2}),{\mathrm{GL}}_3\times {\mathrm{GL}}_1).\end{align*} $$

This is essentially the same as the quadruple (9) except we replace the cusp form on ${\mathrm {GL}}_2$ by theta series. The period integral in this case is exactly the Rankin-Selberg integral in [Reference Bump and Ginzburg6] and $\Delta _{red}$ is given by (13) when $m=2$ . It is clear that Theorem 1.18 holds in this case. The unramified computation in [Reference Wan and Zhang46] proves Theorem 1.13.

For (2.8) with $n=9$ , the associated quadruple $\Delta $ is

(19) $$ \begin{align} ({\mathrm{GSp}}_8,G({\mathrm{SL}}_2\times {\mathrm{SL}}_2),T(std_{{\mathrm{GL}}_2,2}),{\mathrm{GL}}_3\times {\mathrm{GL}}_1\times {\mathrm{GL}}_1). \end{align} $$

where $std_{{\mathrm {GL}}_2,2}$ is the standard representation of the second ${\mathrm {GL}}_2$ -copy. This is essentially the same as the quadruple (10) except we replace the cusp form on ${\mathrm {GL}}_2$ by theta series and the period integral in this case is exactly the Rankin-Selberg integral in [Reference Bump and Ginzburg7]. In this case $\Delta _{red}$ is given by (15). It is clear that Theorem 1.18 holds in this case. The result in [Reference Bump and Ginzburg7] proves Theorem 1.13.

For (2.8) with $n=10$ , the associated quadruple $\Delta $ is

$$ \begin{align*}({\mathrm{PGSO}}_{10},{\mathrm{GL}}_2,0,{\mathrm{GL}}_4\times {\mathrm{GL}}_1).\end{align*} $$

The nilpotent orbit $\iota $ induces a Fourier-Jacobi period on the unipotent radical of the parabolic subgroup $P=MU$ with $M={\mathrm {GL}}_2\times {\mathrm {GL}}_2\times {\mathrm {SO}}_2$ whose stabilizer in M is $H={\mathrm {GL}}_2$ (here the embedding is given by $h\mapsto (h,h,diag(\det (h),1))$ ). In this case $\Delta _{red}$ is given by (13) when $n=2$ . It is clear that Theorem 1.18 holds in this case. This integral is very close to the Rankin-Selberg integral in [Reference Ginzburg17], though we again do not claim Theorem 1.13 in this case.

For (2.9), the generic stabilizer of $\hat {\rho }$ in $\hat {G}$ is not connected, hence it does not belong to the current framework of the BZSV duality. However, for this specific case, by the Rankin-Selberg integral in [Reference Ginzburg16], we know that the dual integral should be the one in [Reference Ginzburg16]. As the generic stabilizer is not connected, there are covering groups involved in the integral.

For (2.10), the associated quadruple $\Delta $ is

$$ \begin{align*}(GE_6, {\mathrm{GL}}_3,T(std_{{\mathrm{GL}}_3}),D_4).\end{align*} $$

In this case $\Delta _{red}$ is given by (13) when $n=3$ . The period integral associated to it is exactly the Rankin-Selberg integral in [Reference Ginzburg15]. It is clear that Theorem 1.18 holds in this case. The result in [Reference Ginzburg15] proves Theorem 1.13.

By the discussion above, the strongly tempered quadruple associated to Table 6 is given as follows. Here for $\iota $ , we only list the root type of the Levi subgroup L of G such that $\iota $ is principal in L.

Table 8 Dual quadruples of Table 6.

5.1 The reductive case

In this subsection we consider the reductive cases, i.e., the ones in Table 9. The nilpotent orbit $\iota $ is trivial for all these cases so we will ignore it.

For (11.7), the associated quadruple $\Delta $ is

(20) $$ \begin{align} ({\mathrm{GSp}}_4\times {\mathrm{GSpin}}_8\times {\mathrm{GL}}_2,S({\mathrm{GSpin}}_8\times G({\mathrm{Sp}}_4\times {\mathrm{SL}}_2)), std_{{\mathrm{Sp}}_4}\otimes std_{{\mathrm{Spin}}_8}\oplus {\mathrm{HSpin}}_8\otimes std_{{\mathrm{SL}}_2}). \end{align} $$

Note that when we take Borel Eisenstein series on ${\mathrm {GSp}}_4$ and ${\mathrm {GL}}_2$ , this period integral recovers the Rankin-Selberg integral in [Reference Ginzburg and Hundley18]. This is because the theta lifting of a Borel Eisenstein series on ${\mathrm {GSp}}_4$ to ${\mathrm {GSO}}_8$ is the two-variable Eisenstein series on ${\mathrm {GSO}}_8$ appeared in the Rankin-Selberg integral of [Reference Ginzburg and Hundley18] and the theta lifting of a Borel Eisenstein series on ${\mathrm {GL}}_2$ to ${\mathrm {GSO}}_8$ is the one-variable Eisenstein series on ${\mathrm {GSO}}_8$ appeared in the Rankin-Selberg integral of [Reference Ginzburg and Hundley18] up to a triality automorphism on ${\mathrm {GSO}}_8$ (we need the triality automorphism here becuase we are using the half Spin representation instead of the standard representation in $\rho _H$ ). The result in loc. cit. proves Theorem 1.13 in this case. This quadruple is self-dual.

For (11.9), the associated quadruple $\Delta $ is given by

$$ \begin{align*}({\mathrm{GSp}}_6\times {\mathrm{GSO}}_4,S(G({\mathrm{Sp}}_4\times {\mathrm{SL}}_2)\times {\mathrm{GSO}}_4), std_{{\mathrm{Sp}}_4}\times std_{{\mathrm{SO}}_4}).\end{align*} $$

By the theta correspondence for ${\mathrm {GSO}}_4\times {\mathrm {GSp}}_4$ , the integral over ${\mathrm {SO}}_4$ of a cusp form on ${\mathrm {GSO}}_4$ with the theta series associated to $\rho _H$ produces an automorphic form on ${\mathrm {GSp}}_4$ . Then the integral over $G({\mathrm {Sp}}_4\times {\mathrm {SL}}_2)$ is just the period integral for the pair $({\mathrm {GSp}}_6\times {\mathrm {GSp}}_4,G({\mathrm {Sp}}_4\times {\mathrm {Sp}}_2))$ in [Reference Wan and Zhang46]. The unramified computation in [Reference Wan and Zhang46] and Theorem 2.6 applied to theta correspondence for ${\mathrm {GSO}}_4\times {\mathrm {GSp}}_4$ proves Theorem 1.13 in this case.

For (11.10), the associated quadruple $\Delta $ is given by

(21) $$ \begin{align} ({\mathrm{GL}}_4\times {\mathrm{GSO}}_4,S({\mathrm{GSp}}_4\times {\mathrm{GSO}}_4),std_{{\mathrm{SO}}_4}\times std_{{\mathrm{Sp}}_4}). \end{align} $$

By the theta correspondence for ${\mathrm {GSO}}_4\times {\mathrm {GSp}}_4$ , the integral over ${\mathrm {SO}}_4$ of a cusp form on ${\mathrm {GSO}}_4$ with the theta series associated to $\rho _H$ produces an automorphic form on ${\mathrm {GSp}}_4$ . Then the integral over ${\mathrm {GSp}}_4$ is just the period integral for the pair $({\mathrm {GL}}_4\times {\mathrm {GSp}}_4,{\mathrm {GSp}}_4)$ which is essentially the Gross-Prasad model for ${\mathrm {SO}}_6\times {\mathrm {SO}}_5$ . The unramified computation in [Reference Ichino and Ikeda27] and Theorem 2.6 applied to theta correspondence for ${\mathrm {GSO}}_4\times {\mathrm {GSp}}_4$ proves Theorem 1.13 in this case. For the dual side, Conjecture 1.5(2) would follow from the theta correspondence for ${\mathrm {SO}}_{6}\times {\mathrm {Sp}}_{4}$ (here we view ${\mathrm {SL}}_2\times {\mathrm {SL}}_2$ as a subgroup of ${\mathrm {Sp}}_4$ ) and the global period integral conjecture for the Gross-Prasad model ${\mathrm {SO}}_{5}\times {\mathrm {SO}}_{4}$ in [Reference Gan, Gross and Prasad13]. This proves Theorem 1.15.

For (11.11) when $p=2m+1$ , the associated quadruple $\Delta $ is given by

(22) $$ \begin{align} ({\mathrm{SO}}_{2m+1}\times {\mathrm{Sp}}_{2m},{\mathrm{SO}}_{2m+1}\times {\mathrm{Sp}}_{2m}, std_{{\mathrm{SO}}_{2m+1}}\otimes std_{{\mathrm{Sp}}_{2m}}\oplus std_{{\mathrm{Sp}}_{2m}}). \end{align} $$

By the theta correspondence for ${\mathrm {SO}}_{2m+2}\times {\mathrm {Sp}}_{2m}$ , the integral over ${\mathrm {Sp}}_{2m}$ of a cusp form on ${\mathrm {Sp}}_{2m}$ with the theta series associated to $\rho _H$ produces an automorphic form on ${\mathrm {SO}}_{2m+2}$ . Then the integral over ${\mathrm {SO}}_{2m+1}$ is just the period integral for the Gross-Prasad period for ${\mathrm {SO}}_{2m+2}\times {\mathrm {SO}}_{2m+1}$ . The unramified computation in [Reference Ichino and Ikeda27] and Theorem 2.6 applied to theta correspondence for ${\mathrm {SO}}_{2m+2}\times {\mathrm {Sp}}_{2m}$ proves Theorem 1.13 in this case. This quadruple is self-dual and it is clear that Conjecture 1.5 would follow from the theta correspondence for ${\mathrm {SO}}_{2m+2}\times {\mathrm {Sp}}_{2m}$ and the global period integral conjecture for the Gross-Prasad model of ${\mathrm {SO}}_{2m+2}\times {\mathrm {SO}}_{2m+1}$ in [Reference Gan, Gross and Prasad13]. This proves Theorem 1.15.

For (11.11) when $p=2m-1$ , the associated quadruple $\Delta $ is given by

(23) $$ \begin{align} ({\mathrm{SO}}_{2m+1}\times {\mathrm{Sp}}_{2m-2},{\mathrm{SO}}_{2m}\times {\mathrm{Sp}}_{2m-2}, std_{{\mathrm{SO}}_{2m}}\otimes std_{{\mathrm{Sp}}_{2m-2}}). \end{align} $$

By the theta correspondence for ${\mathrm {SO}}_{2m}\times {\mathrm {Sp}}_{2m-2}$ , the integral over ${\mathrm {Sp}}_{2m-2}$ of a cusp form on ${\mathrm {Sp}}_{2m}$ with the theta series associated to $\rho _H$ produces an automorphic form on ${\mathrm {SO}}_{2m}$ . Then the integral over ${\mathrm {SO}}_{2m}$ is just the Gross-Prasad period for ${\mathrm {SO}}_{2m+1}\times {\mathrm {SO}}_{2m}$ . The unramified computation in [Reference Ichino and Ikeda27] and Theorem 2.6 applied to theta correspondence for ${\mathrm {SO}}_{2m}\times {\mathrm {Sp}}_{2m-2}$ proves Theorem 1.13 in this case. For the dual side, Conjecture 1.5(2) would follow from the theta correspondence for ${\mathrm {SO}}_{2m}\times {\mathrm {Sp}}_{2m-2}$ and the global period integral conjecture for the Gross-Prasad model ${\mathrm {SO}}_{2m}\times {\mathrm {SO}}_{2m+1}$ in [Reference Gan, Gross and Prasad13]. This proves Theorem 1.15.

By the discussion above, the strongly tempered quadruple associated to Table 9 is given as follows (note that $\iota $ is trivial for all these cases) where

$$ \begin{align*}\ast=({\mathrm{GSp}}_4\times {\mathrm{GSpin}}_8\times {\mathrm{GL}}_2,S({\mathrm{GSpin}}_8\times G({\mathrm{Sp}}_4\times {\mathrm{SL}}_2)), std_{{\mathrm{Sp}}_4}\otimes std_{{\mathrm{Spin}}_8}\oplus {\mathrm{HSpin}}_8\otimes std_{{\mathrm{SL}}_2})\end{align*} $$

Table 11 Dual quadruples of Table 9.

5.2 The nonreductive case

For (11.1) when $n=2k$ , the associated quadruple $\Delta $ is

$$ \begin{align*}({\mathrm{GSpin}}_{2k}\times {\mathrm{GSO}}_4,S({\mathrm{GSp}}_4\times {\mathrm{GSO}}_4),std_{{\mathrm{SO}}_4}\times std_{{\mathrm{Sp}}_4}, {\mathrm{GSpin}}_{2k-4}\times ({\mathrm{GL}}_1)^2\times T_{{\mathrm{GSO}}_4}).\end{align*} $$

The nilpotent orbit $\iota $ induces a Bessel period on the unipotent radical of the parabolic subgroup $P=MU$ with $M={\mathrm {GSpin}}_{6}\times ({\mathrm {GL}}_1)^{k-3}\times {\mathrm {GSO}}_4$ whose stabilizer in M is ${\mathrm {GSpin}}_5\times {\mathrm {GSO}}_4$ . We can embed H into the stabilizer as in (21) and $\Delta _{red}$ is given by (21). It is clear that Theorem 1.18 holds in this case. The unramified computation in [Reference Ichino and Ikeda27] and Theorem 2.6 applied to theta correspondence for ${\mathrm {GSO}}_4\times {\mathrm {GSp}}_4$ proves Theorem 1.13 in this case. For the dual side, Conjecture 1.5(2) would follow from the theta correspondence for ${\mathrm {SO}}_{2k}\times {\mathrm {Sp}}_{4}$ (here we view ${\mathrm {SL}}_2\times {\mathrm {SL}}_2$ as a subgroup of ${\mathrm {Sp}}_4$ ) and the global period integral conjecture for the Gross-Prasad model ${\mathrm {SO}}_{5}\times {\mathrm {SO}}_{4}$ in [Reference Gan, Gross and Prasad13]. This proves Theorem 1.15.

For (11.2), the associated quadruple $\Delta $ is

$$ \begin{align*}({\mathrm{GSO}}_{12},S({\mathrm{GSp}}_4\times {\mathrm{GSO}}_4),0, {\mathrm{GL}}_2\times {\mathrm{GL}}_2\times ({\mathrm{GL}}_1)^3).\end{align*} $$

The nilpotent orbit $\iota $ induces a Fourier-Jacobi period on the unipotent radical of the parabolic subgroup $P=MU$ with $M={\mathrm {GL}}_4\times {\mathrm {GSO}}_4$ whose stabilizer in M is H. In this case $\Delta _{red}$ is given by (21). It is clear that Theorem 1.18 holds in this case.

For (11.3), we first introduce a reductive quadruple which belongs to Table S of [Reference Knop30]. Let $G=({\mathrm {GL}}_2)^5$ and $H=S({\mathrm {GL}}_2\times {\mathrm {GL}}_2\times {\mathrm {GL}}_2)$ where the embedding $H\rightarrow G$ is given by mapping the first ${\mathrm {GL}}_2$ -copy into the first ${\mathrm {GL}}_2$ -copy, and mapping the second (resp. third) ${\mathrm {GL}}_2$ -copy diagonally into the second and third (resp. fourth and fifth) ${\mathrm {GL}}_2$ -copy. Let $\rho _H=std_{{\mathrm {GL}}_2}\otimes std_{{\mathrm {GL}}_2}\otimes std_{{\mathrm {GL}}_2}$ be the triple product representation and $\iota $ be trivial. The quadruple

(24) $$ \begin{align} \Delta_0=(G,H,\rho_H,\iota)=(({\mathrm{GL}}_2)^5,S({\mathrm{GL}}_2\times {\mathrm{GL}}_2\times {\mathrm{GL}}_2),std_{{\mathrm{GL}}_2}\otimes std_{{\mathrm{GL}}_2}\otimes std_{{\mathrm{GL}}_2},1) \end{align} $$

will be used to explain several models in this paper. This quadruple comes from Table S of [Reference Knop30], it is obtained by combining two copies of Model (S.3) with $n=4$ . We claim the dual quadruple is given by

$$ \begin{align*}\hat{\Delta}_0=(\hat{G},\widehat{G/Z_{\Delta}},\hat{\rho},1),\;\hat{\rho}=std_{{\mathrm{GL}}_2,1}\otimes std_{{\mathrm{GL}}_2,2}\otimes std_{{\mathrm{GL}}_2,3}\oplus std_{{\mathrm{GL}}_2,1}\otimes std_{{\mathrm{GL}}_2,4}\otimes std_{{\mathrm{GL}}_2,5}\end{align*} $$

where $std_{{\mathrm {GL}}_2,i}$ represents the standard representation of the i-th ${\mathrm {GL}}_2$ -copy. To justify the duality, we will prove Theorem 1.13 and Theorem 1.15 for this case.

We start with Theorem 1.13. By the theta correspondence for ${\mathrm {GSp}}_2\times {\mathrm {GSO}}_4$ , the integral of a cusp form on the first ${\mathrm {GL}}_2$ -copy with the theta series produces cusp forms on the other two ${\mathrm {GL}}_2$ -copies of H. Then the period integral over the remaining two copies of ${\mathrm {GL}}_2$ are just the period for two trilinear ${\mathrm {GL}}_2$ -models (i.e., the first, second, third ${\mathrm {GL}}_2$ -copies and the first, fourth, fifth ${\mathrm {GL}}_2$ -copies). Then Theorem 1.13 follows from the unramified computation in [Reference Ichino and Ikeda27]. In fact, in this case, Conjecture 1.5(1) follows from the result in [Reference Harris and Kudla25] (for the trilinear ${\mathrm {GL}}_2$ -model) and Theorem 2.6 applied to theta correspondence for ${\mathrm {GSp}}_2\times {\mathrm {GSO}}_4$ . For the dual side, Conjecture 1.5(2) in this case is also a direct consequence of the result in [Reference Harris and Kudla25] (for the trilinear ${\mathrm {GL}}_2$ -model) and Theorem 2.5 applied to theta correspondence for ${\mathrm {GSp}}_2\times {\mathrm {GSO}}_4$ . This proves Theorem 1.15. Later in Section 9, we will use a similar argument to prove Theorem 1.21 for most of the cases.

For (11.3) the associated quadruple $\Delta $ is

(25) $$ \begin{align} ({\mathrm{GSO}}_{12}\times {\mathrm{PGL}}_2,S({\mathrm{GL}}_2\times {\mathrm{GSO}}_4),0,{\mathrm{GL}}_4\times ({\mathrm{GL}}_1)^3\times T_{{\mathrm{PGL}}_2}). \end{align} $$

The nilpotent orbit $\iota $ induces a Fourier-Jacobi period on the unipotent radical of the parabolic subgroup $P=MU$ with $M={\mathrm {GL}}_2\times {\mathrm {GL}}_2\times {\mathrm {GSO}}_4\times {\mathrm {PGL}}_2$ whose stabilizer in M is $S({\mathrm {GL}}_2\times {\mathrm {GSO}}_4)\times {\mathrm {GL}}_2$ . We can embed H into the stabilizer by mapping the ${\mathrm {GL}}_2$ -copy of H into the ${\mathrm {GL}}_2$ -copy of the stabilizer and by mapping ${\mathrm {GSO}}_4={\mathrm {GL}}_2\times {\mathrm {GL}}_2/{\mathrm {GL}}_{1}^{diag}$ into ${\mathrm {GSO}}_4\times {\mathrm {PGL}}_2$ via the idenity map on ${\mathrm {GSO}}_4$ and the projection map ${\mathrm {GSO}}_4={\mathrm {GL}}_2\times {\mathrm {GL}}_2/{\mathrm {GL}}_{1}^{diag}\rightarrow {\mathrm {PGL}}_2$ via the firts ${\mathrm {GL}}_2$ -copy of ${\mathrm {GSO}}_4$ . It is clear that the induced embedding from H into M is the same as (24). In this case $\Delta _{red}$ is given by (24). It is clear that Theorem 1.18 holds in this case.

For (11.4), the associated quadruple $\Delta $ is

$$ \begin{align*}({\mathrm{GSp}}_4\times {\mathrm{GSpin}}_{12}, S({\mathrm{GSpin}}_8\times G({\mathrm{Sp}}_4\times {\mathrm{SL}}_2)), std_{{\mathrm{Sp}}_4}\otimes std_{{\mathrm{Spin}}_8}, T_{{\mathrm{GSp}}_4}\times {\mathrm{GL}}_2\times ({\mathrm{GL}}_1)^5).\end{align*} $$

The nilpotent orbit $\iota $ induces a Fourier-Jacobi period on the unipotent radical of the parabolic subgroup $P=MU$ with $M={\mathrm {GSp}}_4\times {\mathrm {GL}}_2\times {\mathrm {GSpin}}_8$ whose stabilizer in M is ${\mathrm {GSp}}_4\times S({\mathrm {GL}}_2\times {\mathrm {GSpin}}_8)$ and we can naturally embed H into the stabilizer. In this case $\Delta _{red}$ is given by (20). It is clear that Theorem 1.18 holds in this case.

For (11.6), the associated quadruple $\Delta $ is

(26) $$ \begin{align} ({\mathrm{GSO}}_{8}\times {\mathrm{GSO}}_4,S({\mathrm{GL}}_2\times {\mathrm{GSO}}_4),0,{\mathrm{GL}}_2\times ({\mathrm{GL}}_1)^3\times T_{{\mathrm{GSO}}_4}). \end{align} $$

The nilpotent orbit $\iota $ induces a Fourier-Jacobi period on the unipotent radical of the parabolic subgroup $P=MU$ with $M={\mathrm {GSO}}_4\times {\mathrm {GL}}_2\times {\mathrm {GSO}}_4$ whose stabilizer in M is $S({\mathrm {GSO}}_4\times {\mathrm {GL}}_2)\times {\mathrm {GSO}}_4$ . We can embed H into the stabilizer by making the ${\mathrm {GL}}_2$ -copy of H into the ${\mathrm {GL}}_2$ -copy of the stabilizer and by mapping the ${\mathrm {GSO}}_4$ -copy of H diagonally into the two ${\mathrm {GSO}}_4$ -copies of the stabilizer. It is clear that the induced embedding from H into M is the same as (24). In this case $\Delta _{red}$ is given by (24). It is clear that Theorem 1.18 holds in this case.

For (11.11) when $p=2k+1>2m+1$ , the associated quadruple $\Delta $ is

$$ \begin{align*}({\mathrm{SO}}_{2m+1}\times {\mathrm{Sp}}_{2k},{\mathrm{SO}}_{2m+1}\times {\mathrm{Sp}}_{2m},std_{{\mathrm{SO}}_{2m+1}}\otimes std_{{\mathrm{Sp}}_{2m}}, T_{{\mathrm{SO}}_{2m+1}}\times {\mathrm{Sp}}_{2k-2m}\times ({\mathrm{GL}}_1)^m ).\end{align*} $$

The nilpotent orbit $\iota $ induces a Fourier-Jacobi period on the unipotent radical of the parabolic subgroup $P=MU$ with $M={\mathrm {Sp}}_{2m}\times ({\mathrm {GL}}_1)^{k-m}\times {\mathrm {SO}}_{2m+1}$ whose stabilizer in M is H. In this case $\Delta _{red}$ is given by (22). It is clear that Theorem 1.18 holds in this case. For the dual side, Conjecture 1.5(2) would follow from Theorem 2.5 applied to the theta correspondence for ${\mathrm {Sp}}_{2m}\times {\mathrm {SO}}_{2k+2}$ and the Gan-Gross-Prasad conjecture (Conjecture 9.11 of [Reference Gan, Gross and Prasad14]) for nontempered Arthur packet of the Gross-Prasad model of ${\mathrm {SO}}_{2k+2}\times {\mathrm {SO}}_{2k+1}$ . This proves Theorem 1.15.

For (11.11) when $p=2n-1<2m-1$ , the associated quadruple $\Delta $ is

(27) $$ \begin{align} ({\mathrm{SO}}_{2m+1}\times {\mathrm{Sp}}_{2n-2},{\mathrm{SO}}_{2n}\times {\mathrm{Sp}}_{2n-2}, std_{{\mathrm{SO}}_{2n}}\otimes std_{{\mathrm{Sp}}_{2n-2}},{\mathrm{SO}}_{2m-2n+1}\times ({\mathrm{GL}}_1)^n\times T_{{\mathrm{Sp}}_{2n-2}}). \end{align} $$

In this case $\Delta _{red}$ is given by (23). It is clear that Theorem 1.18 holds in this case. By the theta correspondence for ${\mathrm {SO}}_{2n}\times {\mathrm {Sp}}_{2n-2}$ , the integral over ${\mathrm {Sp}}_{2n-2}$ of a cusp form on ${\mathrm {Sp}}_{2n}$ with the theta series associated to $\rho _H$ produces an automorphic form on ${\mathrm {SO}}_{2n}$ . Then the integral over ${\mathrm {SO}}_{2n}$ is just the Gross-Prasad period for ${\mathrm {SO}}_{2m+1}\times {\mathrm {SO}}_{2n}$ . The unramified computation in [Reference Ichino and Ikeda27] and Theorem 2.6 applied to theta correspondence for ${\mathrm {SO}}_{2n}\times {\mathrm {Sp}}_{2n-2}$ proves Theorem 1.13 in this case. For the dual side, Conjecture 1.5(2) would follow from the theta correspondence for ${\mathrm {Sp}}_{2m}\times {\mathrm {SO}}_{2n}$ and the global period integral conjecture for the Gross-Prasad period of ${\mathrm {SO}}_{2n}\times {\mathrm {SO}}_{2n-1}$ in [Reference Gan, Gross and Prasad13]. This proves Theorem 1.15.

For (11.12), the associated quadruple $\Delta $ is

$$ \begin{align*}({\mathrm{GSpin}}_7, {\mathrm{GL}}_2, S({\mathrm{GL}}_2\times {\mathrm{GL}}_2), std_{{\mathrm{GL}}_2}, {\mathrm{GL}}_2\times ({\mathrm{GL}}_1)^2).\end{align*} $$

The nilpotent orbit $\iota $ induces a Fourier-Jacobi period on the unipotent radical of the parabolic subgroup $P=MU$ with $M={\mathrm {GSpin}}_3\times {\mathrm {GL}}_2$ whose stabilizer in M is H. The representation $\rho _H$ is the standard representation on the first ${\mathrm {GL}}_2$ -copy. In this case $\Delta _{red}$ is given by (22) when $m=1$ . It is clear that Theorem 1.18 holds in this case.

By the discussion above, the strongly tempered quadruple associated to Table 10 is given as follows. Here for $\iota $ , we only list the root type of the Levi subgroup L of G such that $\iota $ is principal in L and $\ast =({\mathrm {GSpin}}_4\times {\mathrm {GSpin}}_{12}, S({\mathrm {GSpin}}_8\times G({\mathrm {Sp}}_4\times {\mathrm {SL}}_2)), std_{{\mathrm {Sp}}_4}\otimes std_{{\mathrm {Spin}}_8}).$

Table 12 Dual quadruples of Table 10.

6.1 The reductive case

In this subsection we consider the reductive cases, i.e., the ones in Table 13. The nilpotent orbit $\iota $ is trivial for all these cases so we will ignore it.

For (12.5), the associated quadruple $\Delta $ is

(28) $$ \begin{align} ({\mathrm{GL}}_6\times {\mathrm{GL}}_2,{\mathrm{GL}}_2\times S({\mathrm{GL}}_4\times {\mathrm{GL}}_2),\wedge^2\otimes std_{{\mathrm{GL}}_2}). \end{align} $$

Here the first ${\mathrm {GL}}_2$ -copy maps into the ${\mathrm {GL}}_6$ part of G, while the second ${\mathrm {GL}}_2$ -copy maps into the ${\mathrm {GL}}_2$ part of G, the representation $\wedge ^2\otimes std_{{\mathrm {GL}}_2}$ stands for the tensor product of the exterior square representation of ${\mathrm {GL}}_4$ with the standard representation of the second ${\mathrm {GL}}_2$ -copy. An unramified computation similar to [Reference Ichino and Ikeda27] and [Reference Wan and Zhang46] can confirm the duality in this case.

During the peer review process, an anonymous referee provided a beautiful argument connecting the period of this quadruple to the exterior cube L-function. To be specific, let $\Pi =\pi \otimes \tau $ be a cuspidal representation on ${\mathrm {GL}}_6\times {\mathrm {GL}}_2$ . After integrating $\tau $ against the theta series associated to the representation $\wedge ^2\otimes std_{{\mathrm {GL}}_2}$ , we get the Speh representation $Speh(\tau ,2)$ on ${\mathrm {GL}}_4$ (this is the theta correspondence between ${\mathrm {Sp}}_2$ and ${\mathrm {SO}}_6$ up to isogeny). Then the global period integral becomes

$$ \begin{align*}\int_{[{\mathrm{GL}}_2\times {\mathrm{GL}}_4]/{\mathrm{GL}}_1({\mathbb A})} \varphi(diag(g_1,g_2))E_\tau(g_2) dg_1dg_2 ,\;\varphi\in \pi,\;E_\tau\in Speh(\tau,2).\end{align*} $$

We add a s-variable in the integral and define

$$ \begin{align*}I(\varphi,E_\tau,s)&=\int_{[{\mathrm{GL}}_4\times {\mathrm{GL}}_2]/{\mathrm{GL}}_1({\mathbb A})} |\det(g_1)/\det(g_2)^2|^s \varphi(diag(g_1,g_2))E_\tau(g_1) dg_1dg_2 ,\\&\qquad\varphi\in \pi,\;E_\tau\in Speh(\tau,2)\end{align*} $$

Lemma 6.1.

$$ \begin{align*}I(\varphi,E_\tau,s)=\int_{H"({\mathbb A}){\backslash} ({\mathrm{GL}}_4({\mathbb A})\times {\mathrm{GL}}_2({\mathbb A}))}|\det(g_1)/\det(g_2)^2|^s \varphi_{GR}(g)E_{\tau,\psi}(g_1) d(g_1,g_2)\end{align*} $$

where

$$ \begin{align*}\varphi_{GR}(g)=\int_{[{\mathrm{GL}}_2]/{\mathrm{GL}}_1({\mathbb A})}\int_{[Mat_{2\times 2}]^3} \varphi(\begin{pmatrix}I_2 & Z & X \\ 0 & I_2 & Y \\ 0 & 0 & I_2 \end{pmatrix}diag(h,h,h)g)\psi(tr(Z+Y))\ dXdYdZdh\end{align*} $$

is the period integral of $\varphi $ with respect to the Ginzburg-Rallis model (12) and

$$ \begin{align*}E_{\tau,\psi}(g_1)&=\int_{[Mat_{2\times 2}]}E_\tau(\begin{pmatrix} I_2 & U \\ 0 & I_2\end{pmatrix} g_1)\psi(-U)\ dU;\\H"&=\{\begin{pmatrix}h& X\\ 0 & h \end{pmatrix}\times (h)|\; h\in {\mathrm{GL}}_2,X\in Mat_{2\times 2}\}\subset {\mathrm{GL}}_4\times {\mathrm{GL}}_2.\end{align*} $$

Proof. We expand the function $\varphi (diag(g_1,g_2))$ along the unipotent subgroup $\begin {pmatrix}I_4 & X \\ 0 & I_2\end {pmatrix}$ :

$$ \begin{align*}\varphi(\begin{pmatrix}g_1 & 0 \\ 0 & g_2 \end{pmatrix})=\sum_{\xi} \int_{[Mat_{4\times 2}]}\varphi(\begin{pmatrix}I_4 & X \\ 0 & I_2\end{pmatrix}\begin{pmatrix}g_1 & 0 \\ 0 & g_2 \end{pmatrix})\xi(X)dX\end{align*} $$

where $\xi $ runs over characters of $[Mat_{4\times 2}]$ . Under the adjoint action of ${\mathrm {GL}}_4(k)\times {\mathrm {GL}}_2(k)$ , the set of characters $\xi $ have three orbits, which correspond to elements in $Mat_{2\times 4}(k)$ of rank $0,1,2$ respectively. Using the cuspidal condition of $\Pi $ it is easy to see that the contribution of rank $0$ and rank $1$ terms to $I(\varphi ,E_\tau ,s)$ is zero. So we only need to consider the rank $2$ terms, which can be written as

$$ \begin{align*}\varphi(\begin{pmatrix}g_1 & 0 \\ 0 & g_2 \end{pmatrix})=\sum_{g\in H'(k){\backslash} ({\mathrm{GL}}_4(k)\times {\mathrm{GL}}_2(k))} \int_{[Mat_{2\times 2}]^2}\varphi(g^{-1}\begin{pmatrix}I_2 & 0 & X \\ 0 & I_2 & Y \\ 0 & 0 & I_2 \end{pmatrix}g\begin{pmatrix}g_1 & 0 \\ 0 & g_2 \end{pmatrix})\psi(tr(Y))dXdY\end{align*} $$

with

$$ \begin{align*}H'=\{\begin{pmatrix}h_1& X\\ 0 & h_2 \end{pmatrix}\times (h_2)|\; h_i\in {\mathrm{GL}}_2,X\in Mat_{2\times 2}\}\subset {\mathrm{GL}}_4\times {\mathrm{GL}}_2,\end{align*} $$

and we embed $H'$ into ${\mathrm {GL}}_6$ with $(a,b)\mapsto diag(a,b)$ . Unwinding, we have

$$ \begin{align*}I(\varphi,E_\tau,s)=\int_{H'({\mathbb A}){\backslash} ({\mathrm{GL}}_4({\mathbb A})\times {\mathrm{GL}}_2({\mathbb A}))}J(g_1,g_2)|\det(g_1)/\det(g_2)^2|^s \ d(g_1,g_2);\end{align*} $$

with

$$ \begin{align*}J(g_1,g_2)&= \int_{[H']/{\mathrm{GL}}_1({\mathbb A})}\int_{[Mat_{2\times 2}]^2} \varphi(\begin{pmatrix}I_2 & 0 & X \\ 0 & I_2 & Y \\ 0 & 0 & I_2 \end{pmatrix}\begin{pmatrix}g_1'g_1 & 0\\ 0 & g_2'g_2 \end{pmatrix})\\& \quad \times E_\tau(g_1'g_1) |\det(g_1')/\det(g_2')^2|^s \psi(tr(Y))dXdYd(g_1',g_2'). \end{align*} $$

Further expanding with $\eta $ running through all characters of $[Mat_{2\times 2}]$ :

$$ \begin{align*}J(g_1,g_2)&=\sum_\eta \int_{[H']/{\mathrm{GL}}_1({\mathbb A})}\int_{[Mat_{2\times 2}]^3} \varphi(\begin{pmatrix}I_2 & Z & X \\ 0 & I_2 & Y \\ 0 & 0 & I_2 \end{pmatrix}\begin{pmatrix}g_1'g_1 & 0\\ 0 & g_2'g_2 \end{pmatrix})\\& \quad \times E_\tau(g_1'g_1) |\det(g_1')/\det(g_2')^2|^s\eta(Z)\psi(tr(Y))\ dZ dXdYd(g_1',g_2').\end{align*} $$

Under the adjoint action of ${\mathrm {GL}}_2(k)\times {\mathrm {GL}}_2(k)$ , the set of characters $\eta $ have three orbits, which correspond to elements in $Mat_{2\times 2}(k)$ of rank $0,1,2$ respectively. Using the cuspidal condition of $\Pi $ it is easy to see that the contribution of rank $0$ and rank $1$ terms to $J(g_1,g_2)$ is zero. The rank 2 characters $\eta $ are the orbit of $H'/H"$ acting on the character $Z\mapsto \psi (tr(Z)) $ . Thus

$$ \begin{align*}J(g_1,g_2)&=\sum_{H'(k)/H"(k)} \int_{[H']/{\mathrm{GL}}_1({\mathbb A})}\int_{[Mat_{2\times 2}]^3} \varphi(h\begin{pmatrix}I_2 & Z & X \\ 0 & I_2 & Y \\ 0 & 0 & I_2 \end{pmatrix}\begin{pmatrix}g_1'g_1 & 0\\ 0 & g_2'g_2 \end{pmatrix})\\& \quad \times E_\tau(hg_1'g_1) |\det(g_1')/\det(g_2')^2|^s\psi(tr(Y+Z))\ dZ dXdYd(g_1',g_2'),\end{align*} $$

which unwinds to

$$ \begin{align*}&\int_{H"({\mathbb A})\backslash H'({\mathbb A})} \int_{[H"]/{\mathrm{GL}}_1({\mathbb A})}\int_{[Mat_{2\times 2}]^3} \varphi(h\begin{pmatrix}I_2 & Z & X \\ 0 & I_2 & Y \\ 0 & 0 & I_2 \end{pmatrix}\begin{pmatrix}g_1'g_1 & 0\\ 0 & g_2'g_2 \end{pmatrix})\\& \quad \times E_\tau(hg_1'g_1) |\det(g_1')/\det(g_2')^2|^s \psi(tr(Y+Z))\ dZ dXdY dhd(g_1',g_2').\end{align*} $$

Write an element in $H"$ as $(\begin {pmatrix}h & U \\ 0 & h \end {pmatrix},h)$ and change variable $Z\mapsto Z-U$ , the above becomes:

$$ \begin{align*}&\int_{H"({\mathbb A})\backslash H'({\mathbb A})} \int_{[{\mathrm{GL}}_2]/{\mathrm{GL}}_1({\mathbb A})}\int_{[Mat_{2\times 2}]^4} \varphi(\mathrm{diag}(h,h,h)\begin{pmatrix}I_2 & Z & X \\ 0 & I_2 & Y \\ 0 & 0 & I_2 \end{pmatrix}\begin{pmatrix}g_1'g_1 & 0\\ 0 & g_2'g_2 \end{pmatrix})\times\\& \quad \times E_\tau(\begin{pmatrix}h & U \\ 0 & h \end{pmatrix}g_1'g_1) |\det(g_1')/\det(g_2')^2|^s\psi(tr(Y+Z-U))\ dU\ dZ dXdY\ dh d(g_1',g_2').\end{align*} $$

Now

$$\begin{align*}\int_{[Mat_{2\times 2}]}E_\tau(\begin{pmatrix}h & U \\ 0 & h \end{pmatrix}\cdot) \psi(tr(-U))\ dU=E_{\tau,\psi}(\mathrm{ diag}(h,h)\cdot)\end{align*}$$

By Theorem 13 and Claim 8 of [Reference Cai, Friedberg, Ginzburg and Kaplan9], $E_{\tau ,\psi }$ is a Shalika model of $Speh(\tau ,2)$ , namely $E_{\tau ,\psi }(\mathrm {diag}(h,h)\cdot )=E_{\tau ,\psi }(\cdot )$ . Therefore

$$ \begin{align*}J(g_1,g_2)&=\int_{H"({\mathbb A})\backslash H'({\mathbb A})}\int_{[{\mathrm{GL}}_2]/{\mathrm{GL}}_1({\mathbb A})}\int_{[Mat_{2\times 2}]^4} \varphi(\mathrm{diag}(h,h,h)\begin{pmatrix}I_2 & Z & X \\ 0 & I_2 & Y \\ 0 & 0 & I_2 \end{pmatrix}\begin{pmatrix}g_1'g_1 & 0\\ 0 & g_2'g_2 \end{pmatrix})\\& \quad \times E_{\tau,\psi}(g_1'g_1) |\det(g_1')/\det(g_2')^2|^s\psi(tr(Y+Z))\ dZ dXdY\ dh d(g_1',g_2'),\end{align*} $$

or simply

$$ \begin{align*}J(g_1,g_2)=\int_{H"({\mathbb A})\backslash H'({\mathbb A})}\varphi_{GR}(\begin{pmatrix}g_1'g_1 & 0\\ 0 & g_2'g_2 \end{pmatrix}) E_{\tau,\psi}(g_1'g_1)|\det(g_1')/\det(g_2')^2|^s\ d(g_1',g_2').\end{align*} $$

The lemma follows from further combine the integration over $H"({\mathbb A})\backslash H'({\mathbb A})'$ and $ H'({\mathbb A}){\backslash } ({\mathrm {GL}}_4({\mathbb A})\times {\mathrm {GL}}_2({\mathbb A})) $ .

Since the Ginzburg-Rallis model yields the exterior cube L-function, we expect the Eulerian integral above to coincide with the L-function of the tensor product of the standard representations of ${\mathrm {GL}}_6$ and ${\mathrm {GL}}_2$ which would explain the duality for the quadruple (28). We thank the referee for this beautiful argument. As the unramified calculation of this Eulerian integral will be computed in the PhD thesis of a student of the third author, we will not reproduce it here.

For (12.7) with $m=1$ , the associated quadruple $\Delta $ is

$$ \begin{align*}({\mathrm{GL}}_4\times{\mathrm{GL}}_2,{\mathrm{GL}}_2\times {\mathrm{GL}}_2,0).\end{align*} $$

This is the model $({\mathrm {GL}}_4\times {\mathrm {GL}}_2,{\mathrm {GL}}_2\times {\mathrm {GL}}_2)$ studied in [Reference Wan and Zhang46] and the unramified computation in [Reference Wan and Zhang46] proves Theorem 1.13 in this case. For the dual side, Conjecture 1.5(2) would follow from Theorem 2.5 applied to the theta correspondence of ${\mathrm {GSp}}_{2}\times {\mathrm {GSO}}_{6}$ and Gan-Gross-Prasad conjecture (Conjecture 9.11 of [Reference Gan, Gross and Prasad14]) for nontempered Arthur packet of the Rankin-Selberg integral of ${\mathrm {GL}}_4\times {\mathrm {GL}}_4$ . This proves Theorem 1.15.

For (12.7) with $m=2$ , the associated quadruple $\Delta $ is

$$ \begin{align*}({\mathrm{GL}}_4\times {\mathrm{GSp}}_4, {\mathrm{GL}}_4\times {\mathrm{GSp}}_4, T(std_{{\mathrm{GL}}_4}\otimes std_{{\mathrm{GSp}}_4})).\end{align*} $$

Observe that this is the dual to the quadruple in (16), thus both Theorems 1.13 and 1.15 have been proved there.

For (12.7) with $m=3$ , the associated quadruple $\Delta $ is

(29) $$ \begin{align} ({\mathrm{GSpin}}_7\times {\mathrm{GSpin}}_6,{\mathrm{GSpin}}_6\times {\mathrm{GSpin}}_6,T({\mathrm{HSpin}}_6\otimes {\mathrm{HSpin}}_6)). \end{align} $$

By the theta correspondence for ${\mathrm {GL}}_4\times {\mathrm {GL}}_4$ , the integral over the second ${\mathrm {GSpin}}_6$ -copy of a cusp form on ${\mathrm {GSpin}}_6$ with the theta series associated to $\rho _H$ produces the same cusp form with an extra central value of the Spin L-function. Then the integral over the other copy of ${\mathrm {GSpin}}_6$ is just the period integral for the Gross-Prasad model ${\mathrm {GSpin}}_7\times {\mathrm {GSpin}}_6$ . The unramified computation in [Reference Ichino and Ikeda27] and Theorem 2.4 applied to theta correspondence for ${\mathrm {GL}}_4\times {\mathrm {GL}}_4$ proves Theorem 1.13 in this case. For the dual side, Conjecture 1.5(2) follows from the theta correspondence for ${\mathrm {GSp}}_6\times {\mathrm {GSO}}_6$ and the Rankin-Selberg integral of ${\mathrm {GL}}_4\times {\mathrm {GL}}_4$ . This proves Theorem 1.15.

For (12.8), the associated quadruple $\Delta $ is

(30) $$ \begin{align} ({\mathrm{GL}}_2\times {\mathrm{GL}}_4\times {\mathrm{GL}}_2,S({\mathrm{GL}}_2\times {\mathrm{GL}}_4)\times {\mathrm{GL}}_2, std_{{\mathrm{GL}}_2}\otimes \wedge^2\oplus T(std_{{\mathrm{GL}}_4}\times std_{{\mathrm{GL}}_2})). \end{align} $$

Note that when we put Eisenstein series on both ${\mathrm {GL}}_2$ copies, this period integral recovers the Rankin-Selberg integral in [Reference Pollack and Shah38]. The result in [Reference Pollack and Shah38] proves Theorem 1.13 in this case. This quadruple is self-dual.

By the discussion above, the strongly tempered quadruple associated to Table 13 is given as follows ( $\iota $ is trivial for all these cases) where

$$ \begin{align*}\ast=({\mathrm{GL}}_2\times {\mathrm{GL}}_4\times {\mathrm{GL}}_2,S({\mathrm{GL}}_2\times {\mathrm{GL}}_4)\times {\mathrm{GL}}_2, std_{{\mathrm{GL}}_2}\otimes \wedge^2\oplus T(std_{{\mathrm{GL}}_4}\times std_{{\mathrm{GL}}_2})).\end{align*} $$

Table 15 Dual quadruples of Table 13.

6.2 The nonreductive case

For (12.1), we first introduce a reductive quadruple which belongs to Table S of [Reference Knop30]. Let $G=({\mathrm {GL}}_2)^4$ and $H=S({\mathrm {GL}}_2\times {\mathrm {GL}}_2\times {\mathrm {GL}}_2)$ where the embedding $H\rightarrow G$ is given by mapping the first two ${\mathrm {GL}}_2$ -copies into the first two ${\mathrm {GL}}_2$ -copy, and mapping the last ${\mathrm {GL}}_2$ -copy diagonally into the third and fourth ${\mathrm {GL}}_2$ -copy. Let $\rho _H=std_{{\mathrm {GL}}_2}\otimes std_{{\mathrm {GL}}_2}\otimes std_{{\mathrm {GL}}_2}\oplus T(std_{{\mathrm {GL}}_2,2})$ where $std_{{\mathrm {GL}}_2,i}$ represents the standard representation of the i-th ${\mathrm {GL}}_2$ -copy and $\iota $ be trivial. This quadruple

(31) $$ \begin{align} \Delta_0=(G,H,\rho_H,\iota)=(({\mathrm{GL}}_2)^4,S({\mathrm{GL}}_2\times {\mathrm{GL}}_2\times {\mathrm{GL}}_2),std_{{\mathrm{GL}}_2}\otimes std_{{\mathrm{GL}}_2}\otimes std_{{\mathrm{GL}}_2}\oplus T(std_{{\mathrm{GL}}_2,2}),1) \end{align} $$

is almost the same as (24) except we replace the cusp form on one ${\mathrm {GL}}_2$ -copy by theta series. It is obtained by combining Model (S.3) and (S.11) in Table S of [Reference Knop30] with $n=4$ and $m=2$ . We claim the dual quadruple is given by

$$ \begin{align*}\hat{\Delta}_0=(\hat{G},\widehat{G/Z_{\Delta}},\hat{\rho},1),\;\hat{\rho}=T(std_{{\mathrm{GL}}_2,1}\otimes std_{{\mathrm{GL}}_2,2})\oplus std_{{\mathrm{GL}}_2,1}\otimes std_{{\mathrm{GL}}_2,3}\otimes std_{{\mathrm{GL}}_2,4}.\end{align*} $$

We can use the same argument as in (24) to prove Theorem 1.13 and Theorem 1.15 for this case.

For (12.1), the associated quadruple $\Delta $ is

$$ \begin{align*}({\mathrm{GSO}}_{12},S({\mathrm{GL}}_2\times {\mathrm{GSO}}_4),T(std_{{\mathrm{GL}}_2}),{\mathrm{GL}}_4\times ({\mathrm{GL}}_1)^3).\end{align*} $$

The attached period integral is the same as model in (25) except we replace the cusp form on ${\mathrm {GL}}_2$ by theta series. In this case $\Delta _{red}$ is given by (31) and it is clear that Theorem 1.18 holds in this case.

For (12.2), the associated quadruple $\Delta $ is

$$ \begin{align*}({\mathrm{GSpin}}_{10}\times {\mathrm{GL}}_2, S({\mathrm{GL}}_2\times {\mathrm{GSpin}}_6)\times {\mathrm{GL}}_2, T({\mathrm{HSpin}}_6\otimes std_{{\mathrm{GL}}_2}),{\mathrm{GL}}_2\times ({\mathrm{GL}}_1)^{4}\times T_{{\mathrm{GL}}_2})\end{align*} $$

The nilpotent orbit $\iota $ induces a Fourier-Jacobi period on the unipotent radical of the parabolic subgroup $P=MU$ with $M={\mathrm {GL}}_2\times {\mathrm {GSpin}}_6\times {\mathrm {GL}}_2$ whose stabilizer in M is H. In this case $\Delta _{red}$ is given by (30). It is clear that Theorem 1.18 holds in this case.

For (12.3), the associated quadruple $\Delta $ is

$$ \begin{align*}({\mathrm{GSO}}_{8}\times {\mathrm{GL}}_2,S({\mathrm{GL}}_2\times {\mathrm{GSO}}_4),T(std_{{\mathrm{GL}}_2}),{\mathrm{GL}}_2\times ({\mathrm{GL}}_1)^3\times T_{{\mathrm{GL}}_2}).\end{align*} $$

The attached period integral is the same as the model (26) except we replace the cusp form on one ${\mathrm {GL}}_2$ -copy by theta series. In this case $\Delta _{red}$ is given by (31) and it is clear that Theorem 1.18 holds in this case.

For (12.6), we first introduce a reductive quadruple from Table S of [Reference Knop30] (it is obtained by combining Model (S.10) and Model (S.3) with $n=4$ )

(32) $$ \begin{align} (G,H,\rho_H,\iota)=({\mathrm{GL}}_2\times {\mathrm{GL}}_2\times {\mathrm{GL}}_2,{\mathrm{GL}}_2\times {\mathrm{GL}}_2,T(std_{{\mathrm{GL}}_2}\otimes std_{{\mathrm{GL}}_2}),1) \end{align} $$

where H embeds into G by mapping the first ${\mathrm {GL}}_2$ -copy into the first ${\mathrm {GL}}_2$ -copy and mapping the second ${\mathrm {GL}}_2$ -copy diagonally into the second and third ${\mathrm {GL}}_2$ -copy. We claim the dual quadruple is given by

$$ \begin{align*}(\hat{G},\widehat{G/Z_{\Delta}},\hat{\rho},1),\;\hat{\rho}=T(std_{{\mathrm{GL}}_2,1})\oplus std_{{\mathrm{GL}}_2,1}\otimes std_{{\mathrm{GL}}_2,2}\otimes std_{{\mathrm{GL}}_2,3}\end{align*} $$

where $std_{{\mathrm {GL}}_2,i}$ is the standard representation of the i-th ${\mathrm {GL}}_2$ -copy. To justify the duality, we will prove Theorem 1.13 and Theorem 1.15 for this case.