iy}\mathop{}\\!\mathrm{d}y}{(y-2i)^{m}}$ $\displaystyle=(-i\epsilon)^{m}(2\pi
i)^{m}\frac{(m-1)!}{(m-1)!\pi^{m}}e^{-4\pi}=(2\epsilon)^{m}e^{-4\pi};$

hence $\lambda=(2\epsilon)^{m}.$

By the equivariance properties of $\phi_{2},$

$M[\phi_{2}]\in\operatorname{Ind}_{\overline{B}(\mathbb{R})}^{\operatorname{PGL}_{2}(\mathbb{R})}|\cdot|^{1/2}$

is a section of weight two for $\operatorname{SO}(2)$. Thus it is determined
by:

(81)
$M[\phi_{2}](1)=\int\phi_{2}(x,y,0,0)\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}y=\frac{1}{\pi}.$

Now, our local zeta integral is given by

$(2\epsilon)^{m}\cdot\left(\int_{N\backslash\operatorname{PGL}_{2}(F_{v})}W^{-}_{m+2,\psi}(g_{1})[\phi_{2}](g_{1})W^{+}_{m,\psi^{-1}}(g_{1})\mathop{}\\!\mathrm{d}g_{1}\right)\cdot\left(\int_{N\backslash\operatorname{PGL}_{2}(F_{v})}W^{\epsilon}_{m,\psi}(g_{2})W^{-\epsilon}_{m,\psi^{-1}}(g_{2})\varphi^{0}(g_{2})\mathop{}\\!\mathrm{d}g_{2}\right).$

Since both integrands are right $\operatorname{SO}(2)$-invariant, and since
the Haar measure on $\operatorname{PGL}_{2}(\mathbb{R})$ is given by

$\mathop{}\\!\mathrm{d}g=\frac{\mathop{}\\!\mathrm{d}a\mathop{}\\!\mathrm{d}t\mathop{}\\!\mathrm{d}\theta}{\pi
t^{2}},\;\;\;g=\begin{pmatrix}1&a\\\ 0&1\end{pmatrix}\begin{pmatrix}t&0\\\
0&1\end{pmatrix}\begin{pmatrix}\cos\theta&\sin\theta\\\
-\sin\theta&\cos\theta\end{pmatrix},\;\;t\in\mathbb{R}^{\times},\theta\in[0,\pi),$

we obtain

(82) $\frac{(2\epsilon)^{m}}{\pi}\left(\int_{0}^{\infty}t^{m}e^{-4\pi
t}\mathop{}\\!\mathrm{d}t\right)\cdot\left(\int_{0}^{\infty}t^{m-1}e^{-4\pi
t}\mathop{}\\!\mathrm{d}t\right)=\frac{\epsilon^{m}\cdot
m!(m-1)!}{\pi^{2m+2}2^{3m+2}},$

as claimed.

To show that

$\mathcal{Z}_{v}\left(\phi_{1}^{\prime},\phi_{2}^{\prime},W^{-}_{m+2,\psi}\otimes
W^{\epsilon}_{m,\psi},W^{-\epsilon}_{m,\psi}\right)=0,$

we may ignore scalar factors. The inner integral

(83)
$I^{\prime}(g_{1},g_{2})=\int_{\operatorname{SL}_{2}(F_{v})}\omega(h_{1}h_{c},\bm{p}_{Z}(g_{1},g_{2}))\widehat{\phi}^{\prime}_{1}(1,0,0,-1)W^{-\epsilon}_{m,\psi}(h_{1}h_{c})\mathop{}\\!\mathrm{d}h_{1},\;\;c=\operatorname{det}(g_{1})\operatorname{det}(g_{2})$

must be of the form

(84)
$I^{\prime}(g_{1},g_{2})=W^{\prime}(g_{1})W^{-\epsilon}_{m,\psi^{-1}}(g_{2}),$

where $W^{\prime}$ is a weight $m+2$ vector in the $\psi^{-1}$-Whittaker model
of the discrete series representation of $\operatorname{GL}_{2}(\mathbb{R})$
of weight $m$. To calculate $W^{\prime}$, let

(85) $X\coloneqq\begin{pmatrix}-i&1\\\ 1&i\end{pmatrix}\in\mathfrak{gl}_{2}$

be the $\operatorname{SO}(2)$-raising operator. Then, up to scalar,

(86) $W^{\prime}=X\cdot W_{m,\psi^{-1}}^{+}=\begin{pmatrix}-i&0\\\
0&i\end{pmatrix}W^{+}_{m,\psi^{-1}}+2\begin{pmatrix}0&1\\\
0&0\end{pmatrix}\cdot W_{m,\psi^{-1}}^{+}-imW_{m,\psi^{-1}}^{+}.$

In particular, $W^{\prime}\begin{pmatrix}t^{1/2}&0\\\
0&t^{-1/2}\end{pmatrix}=0$ and

(87) $W^{\prime}\begin{pmatrix}-t^{1/2}&0\\\
0&t^{-1/2}\end{pmatrix}=-2imt^{m/2}e^{-2\pi t}+8\pi it^{m/2+1}e^{-2\pi t}$

(up to scalar) for $t>0$. To complete the proof, we will show the integral

($\ast$)
$\int_{N\backslash\operatorname{PGL}_{2}(F_{v})}W^{\prime}(g_{1})[\phi_{2}](g_{1})W^{+}_{m,\psi^{-1}}(g_{1})\mathop{}\\!\mathrm{d}g_{1}$

vanishes. Since $[\phi_{2}]$ is a section of weight zero, ($\ast$) is
proportional to

$\displaystyle(\ast\ast)$ $\displaystyle=\int_{0}^{\infty}(-2imt^{m/2}+8\pi
it^{m/2+1})\cdot t\cdot t^{m/2}e^{-4\pi
t}\frac{\mathop{}\\!\mathrm{d}t}{t^{2}}$
$\displaystyle=-2im\int_{0}^{\infty}t^{m-1}e^{-4\pi
t}\mathop{}\\!\mathrm{d}t+8\pi i\int_{0}^{\infty}e^{-4\pi
t}t^{m}\mathop{}\\!\mathrm{d}t$
$\displaystyle=-2im\frac{(m-1)!}{(4\pi)^{m}}+8\pi i\frac{m!}{(4\pi)^{m+1}}$
$\displaystyle=0.$

∎

###  7.2. Cohomological span of special cycle

#### 7.2.1.

Let
$H=\operatorname{GL}_{2}\times_{\mathbb{G}_{m}}\operatorname{GL}_{2}\subset\operatorname{GSp}_{4}$,
viewed as an algebraic group over $F$. Then $\bm{H}$ possesses a Shimura
datum, and we have a natural embedding of pro-algebraic varieties

$\iota:S(\bm{H})\hookrightarrow S(\operatorname{\mathbf{GSp}}_{4})\times
S({\operatorname{\mathbf{GL}}_{2}}),$

induced from the map on the level of groups:
$(h_{1},h_{2})\mapsto((h_{1},h_{2}),h_{1})$. For all weights $\bm{m}$ as in
(5.3.1) above, abbreviate by ${\mathcal{W}}_{\bm{m}}$ the local system
$\mathcal{V}_{(\bm{m}-2,0)}^{\vee}\boxtimes\mathcal{V}_{\bm{m}-2}$ on
$S(\operatorname{\mathbf{GSp}}_{4})\times S(\operatorname{\mathbf{GL}}_{2})$.
Note that the constant local system $\underline{\mathbb{Q}(\bm{m})}$ on
$S(\bm{H})$ is a direct factor with multiplicity one of the pullback
$\iota^{\ast}({\mathcal{W}}_{\bm{m}})$, and in particular, we have a composite
map (well-defined up to a scalar):

(88) $H^{4d}_{c}(S(\operatorname{\mathbf{GSp}}_{4})\times
S(\operatorname{\mathbf{GL}}_{2}),{\mathcal{W}}^{\vee}_{\bm{m}})\to
H^{4d}_{c}(S(\bm{H}),\iota^{\ast}({\mathcal{W}}^{\vee}_{\bm{m}}))\to
H^{4d}_{c}(S({\bm{H}}),\mathbb{Q}(\bm{m})).$

######  Definition 7.2.2.

The cycle class $[\mathcal{Z}]\in
H^{4d}(S(\operatorname{\mathbf{GSp}}_{4})\times
S(\operatorname{\mathbf{GL}}_{2}),{\mathcal{W}}_{\bm{m}})(2d)$ is the image of
the fundamental class of $S(\bm{H})$ under the map

$H^{0}(S(\bm{H}),\mathbb{Q}(\bm{m}))\to
H^{2d}(S(\operatorname{\mathbf{GSp}}_{4})\times
S(\operatorname{\mathbf{GL}}_{2}),{\mathcal{W}}_{\bm{m}})(2d)$

induced by the dual of (88). We write

$[\mathcal{Z}]_{\ast}:H^{3d}_{c}(S(\operatorname{\mathbf{GSp}}_{4}),\mathcal{V}_{(\bm{m}-2,0)})(d)\to
H^{d}(S(\operatorname{\mathbf{GL}}_{2}),\mathcal{V}_{\bm{m}-2})$

for the induced map.

#### 7.2.3.

Let $\pi$ be an automorphic cuspidal representation of
$\operatorname{GL}_{2}(\mathbb{A}_{F})$ of weight $\bm{m}$ whose central
character has infinity type $\omega_{\bm{m}}$. If $\pi$ is defined over $E$,
recall that the trace map induces a perfect pairing:

$\langle\cdot,\cdot\rangle:H^{d}_{c}(S(\operatorname{\mathbf{GL}}_{2}),\mathcal{V}_{\bm{m}-2,E})[\pi_{f}]\times
H^{d}(S(\operatorname{\mathbf{GL}}_{2}),\mathcal{V}^{\vee}_{\bm{m}-2,E})[\pi_{f}^{\vee}]\to
H^{2d}_{c}(S(\operatorname{\mathbf{GL}}_{2}),E)\to E(d).$

######  Proposition 7.2.4.

Let $\pi$ be as above, and let $\pi_{1},\pi_{2}$ be as in (5.3.1), with
$\Pi=\Pi_{S}(\pi_{1},\pi_{2})$, for some $S=S_{f}\sqcup S_{\infty}$ such that
$|S|$ is even.

  1. (1)

For choices of signs $\bm{\epsilon},\bm{\epsilon}^{\prime}$, let
$\sigma_{\bm{\epsilon},\bm{\epsilon}^{\prime}}:\bm{\tau}^{\bm{\epsilon}}_{\bm{m},S_{\infty}}\to\mathbb{C}$
be the projection onto the weight
$(-\bm{\epsilon}^{\prime}\bm{m},0)$-component (hence $\sigma$ is trivial
unless $S_{\infty}=\emptyset$ and $\bm{\epsilon}=\bm{\epsilon}^{\prime}$).
Then the following diagram commutes up to a nonzero scalar:

${\left(\Pi\otimes\bm{\tau}_{\bm{m},S_{\infty}}^{\bm{\epsilon}}\right)^{\bm{K}_{2}}\otimes\left(\pi^{\vee}\otimes\bm{\chi}_{-\bm{\epsilon}^{\prime}\bm{m}}^{\vee}\right)^{\bm{K}_{1}}}$${\Pi\otimes\pi}$${H^{\ast}_{(2)}(S(\operatorname{\mathbf{GSp}}_{4}),\mathcal{V}_{(\bm{m}-2,0),\mathbb{C}})[\Pi_{S_{f}}]\otimes
H^{\ast}_{(2)}(S(\operatorname{\mathbf{GL}}_{2}),\mathcal{V}^{\vee}_{\bm{m}-2,\mathbb{C}})[\pi_{f}^{\vee}]}$${H_{c}^{3d}(S(\operatorname{\mathbf{GSp}}_{4}),\mathcal{V}_{(\bm{m}-2,0),\mathbb{C}})[\Pi_{S_{f}}]\otimes
H_{c}^{d}(S(\operatorname{\mathbf{GL}}_{2}),\mathcal{V}_{\bm{m}-2,\mathbb{C}}^{\vee})[\pi_{f}]}$${H^{d}(S(\operatorname{\mathbf{GL}}_{2}),\mathcal{V}_{\bm{m}-2,\mathbb{C}})\otimes
H_{c}^{d}(S(\operatorname{\mathbf{GL}}_{2}),\mathcal{V}_{\bm{m}-2,\mathbb{C}}^{\vee})}$${\mathbb{C}}$${\mathbb{C}}$$\scriptstyle{\sigma_{\bm{\epsilon},\bm{\epsilon}^{\prime}}\otimes\operatorname{id}}$$\scriptstyle{\operatorname{cl}_{S}^{\bm{\epsilon}}\otimes\operatorname{cl}_{\bm{\epsilon}^{\prime}}^{\prime}}$$\scriptstyle{\mathcal{P}_{S}}$$\scriptstyle{\sim}$$\scriptstyle{{[\mathcal{Z}]}_{\ast}\otimes\operatorname{id}}$$\scriptstyle{\langle\cdot,\cdot\rangle}$

  2. (2)

Suppose $S_{\infty}=\emptyset$ and $\bm{\epsilon}=\bm{\epsilon}$. After fixing
isomorphisms

$\Pi_{f}\simeq\left(\Pi\otimes\bm{\tau}^{\bm{\epsilon}}_{\bm{m},\emptyset}\right)^{\bm{K}_{2}},\;\;\pi_{f}^{\vee}\simeq\left(\pi^{\vee}\otimes\bm{\chi}_{-\bm{\epsilon}\bm{m}}^{\vee}\right)^{\bm{K}_{1}},$

the composites with the map from (1):

$\Pi_{f}\otimes\pi_{f}^{\vee}\xrightarrow{\;\sim\;}\left(\Pi\otimes\bm{\tau}^{\bm{\epsilon}}_{\bm{m},\emptyset}\right)^{\bm{K}_{2}}\otimes\left(\pi^{\vee}\otimes\bm{\chi}_{-\bm{\epsilon}\bm{m}}^{\vee}\right)^{\bm{K}_{1}}\longrightarrow\mathbb{C}$

are independent of $\bm{\epsilon}$ up to a nonzero scalar.

###### Proof.

Let $\ell:V_{(\bm{m}-2,0)}\otimes V_{\bm{m}-2}^{\vee}\to\mathbb{Q}(\bm{m})$ be
an $H(F)$-invariant projection. There exists a basis $\mathbbm{1}_{\bm{H}}$ of
$\wedge^{4d}\mathfrak{p}_{H}$ such that

$\int_{S(\bm{H})}\omega=\int_{[Z_{H}\backslash
H]}h^{\ast}\omega(\mathbbm{1}_{\bm{H}})\mathop{}\\!\mathrm{d}h$

for all top-degree forms $\omega$ on $S(\bm{H})$. Then by definition,

$\langle[\mathcal{Z}]_{\ast}\alpha,\beta\rangle=\int_{[Z_{H}\backslash
H]}\ell\left[h^{\ast}\iota^{\ast}(\alpha\wedge\beta)(\mathbbm{1}_{\bm{H}})\right]\mathop{}\\!\mathrm{d}h.$

The composite map:

(89)
$\begin{split}\bm{\tau}_{\bm{m},S_{\infty}}^{\bm{\epsilon}}\otimes\bm{\chi}^{\vee}_{-\bm{\epsilon}^{\prime}\bm{m}}\to\wedge^{\bm{p}(\bm{\epsilon},S_{\infty}),\bm{q}(\bm{\epsilon},S_{\infty})}\mathfrak{p}_{\operatorname{GSp}_{4}}^{\ast}\otimes\wedge^{1-\bm{p}(\bm{\epsilon}^{\prime}),1-\bm{q}(\bm{\epsilon}^{\prime})}\mathfrak{p}_{\operatorname{GL}_{2}}^{\ast}\otimes
V_{(\bm{m}-2,0)}\otimes V_{\bm{m}-2}\\\
\xrightarrow{\iota^{\ast}\otimes\ell}\wedge^{\bm{p}(\bm{\epsilon},S_{\infty})+1-\bm{p}(\bm{\epsilon}^{\prime}),\bm{q}(\bm{\epsilon},S_{\infty})+1-\bm{q}(\bm{\epsilon}^{\prime})}\mathfrak{p}_{H}\xrightarrow{\mathbbm{1}_{\bm{H}}}\mathbb{C}\end{split}$

is a map of $U(1)^{d}\times U(1)^{d}$-modules, where the action on
$\bm{\chi}_{-\bm{\epsilon}^{\prime}\bm{m}}$,
$\wedge^{\bm{p}(\bm{\epsilon}^{\prime}),\bm{q}(\bm{\epsilon}^{\prime})}\mathfrak{p}_{\operatorname{GL}_{2}}$,
and $V_{\bm{m}-2}$ is through projection to the first factor. In particular,
(89) is trivial unless $S_{\infty}=\emptyset$ and
$\bm{\epsilon}=\bm{\epsilon}^{\prime}$, in which case it is proportional to
the projection onto the weight $(-\bm{\epsilon}\bm{m},0)$-component of
$\bm{\tau}_{\bm{m},\emptyset}^{\bm{\epsilon}}$; moreover, a direct calculation
shows it is nonzero, and (1) follows.

For (2), let

$g_{\bm{\epsilon}}=\begin{pmatrix}\epsilon_{v}&&&\\\ &\epsilon_{v}&&\\\
&&1&\\\ &&&1\end{pmatrix}_{v|\infty}\in G(F\otimes\mathbb{R})\subset
G(\mathbb{A}_{F}),\;\;\;\;g^{\prime}_{\bm{\epsilon}}=\begin{pmatrix}\epsilon_{v}&\\\
&1\end{pmatrix}_{v|\infty}\in G^{\prime}(F\otimes\mathbb{R})\subset
G^{\prime}(\mathbb{A}_{F}).$

We have an obvious commutative diagram

${\left(\Pi\otimes\bm{\tau}^{\bm{\epsilon}}_{\bm{m},\emptyset}\right)^{\bm{K}_{2}}\otimes\left(\pi^{\vee}\otimes\bm{\chi}_{-\bm{\epsilon}\bm{m}}^{\vee}\right)^{\bm{K}_{1}}}$${\Pi\otimes\pi^{\vee}}$${\left(\Pi\otimes\bm{\tau}^{\bm{+}}_{\bm{m},\emptyset}\right)^{\bm{K}_{2}}\otimes\left(\pi^{\vee}\otimes\bm{\chi}^{\vee}_{-\bm{m}}\right)^{\bm{K}_{1}}}$${\Pi\otimes\pi^{\vee}}$$\scriptstyle{\sigma_{\bm{\epsilon},-\bm{\epsilon}}\otimes\operatorname{id}}$$\scriptstyle{\sigma_{\bm{+},\bm{-}}\otimes\operatorname{id}}$

in which the vertical arrows are translation by
$(g_{\bm{\epsilon}},g^{\prime}_{\bm{\epsilon}})$ and $\pm$ stands for the
constant sign $(\pm)_{v}|\infty$. However, since
$(g_{\bm{\epsilon}},g^{\prime}_{\bm{\epsilon}})$ lies in
$H(\mathbb{A}_{F})\subset G(\mathbb{A}_{F})\times G^{\prime}(\mathbb{A}_{F}),$
this translation has no effect on the period integral $\mathcal{P}_{S}$, and
(2) follows. ∎

######  Theorem 7.2.5.

Let $\pi_{1},\pi_{2},\pi$ be cuspidal automorphic representations of
$G^{\prime}(\mathbb{A})$ of weights $\bm{m}+2$, $\bm{m}$, and $\bm{m}$,
respectively, where $\bm{m}=(m_{v})_{v|\infty}$ for positive integers $m_{v}$.
Assume that the central characters of $\pi_{1}$ and $\pi_{2}$ agree, and that
the the central characters of $\pi_{1},$ $\pi_{2}$, and $\pi$ all have
infinity type $\omega_{\bm{m}}$. Let $\Pi_{S_{f}}$ be as in (5.3.1) for set
$S_{f}$ of finite places of $F$. Then, for any coefficient field $E\supset
F^{c}$ such that $\Pi,\pi_{i},$ and $\pi$ are defined over $E$, the induced
map

$[\mathcal{Z}]_{\ast}:H^{3d}_{!}(S(\operatorname{\mathbf{GSp}}_{4}),\mathcal{V}_{(\bm{m}-2,0)})(d)[\Pi_{S_{f}}]\to
H^{d}_{!}(S(\operatorname{\mathbf{GL}}_{2}),\mathcal{V}_{\bm{m}-2})[\pi_{f}]$

is trivial unless $\pi=\pi_{2}$ and $S_{f}=\emptyset$. In the latter case,
$[\mathcal{Z}]_{\ast}$ takes the form:

$\Pi_{\emptyset}\otimes
H_{!}^{3d}(S(\operatorname{\mathbf{GSp}}_{4}),\mathcal{V}_{(\bm{m}-2,0)})_{\Pi_{\emptyset}}(d)\xrightarrow{\ell\otimes
s}\pi_{2,f}\otimes
H_{!}^{d}(S(\operatorname{\mathbf{GL}}_{2}),\mathcal{V}_{\bm{m}-2})_{\pi_{2,f}},$

where $s$ is an surjection and $\ell$ is a nontrivial $E$-linear map.

###### Proof.

Without loss of generality, suppose $E=\mathbb{C}$. By Proposition 7.2.4,
Theorem 6.2.2, and Theorem 6.5.2, we immediately reduce to the case
$S_{f}=\emptyset$ and $\pi=\pi_{2}$. In this case, write
$\Pi=\Pi_{\emptyset}(\pi_{1},\pi_{2})$. Under the decomposition

$H^{3d}_{!}(S(\operatorname{\mathbf{GSp}}_{4}),\mathcal{V}_{(\bm{m}-2,0),\mathbb{C}})[\Pi_{f}]=\bigoplus_{S_{\infty}}\bigoplus_{\bm{\epsilon}}H^{\bm{p}(\bm{\epsilon},S_{\infty}),\bm{q}(\bm{\epsilon},S_{\infty})}_{(2)}(S(\operatorname{\mathbf{GSp}}_{4}),\mathcal{V}_{(\bm{m}-2,0),\mathbb{C}})[\Pi_{f}]$

provided by 5.3.3 and 5.3.5, Proposition 7.2.4(1) implies that
$[\mathcal{Z}]_{\ast}$ is trivial on components with
$S_{\infty}\neq\emptyset$, and maps
$H^{\bm{p}(\bm{\epsilon},\emptyset),\bm{q}(\bm{\epsilon},\emptyset)}_{(2)}(S(\operatorname{\mathbf{GSp}}_{4}),\mathcal{V}_{(\bm{n}-2,0),\mathbb{C}})(d)[\Pi_{f}]$
to
$H^{\bm{p}(\bm{\epsilon}),\bm{q}(\bm{\epsilon})}_{(2)}(S(\operatorname{\mathbf{GL}}_{2}),\mathcal{V}_{\bm{m}-2,\mathbb{C}})[\pi_{2,f}]$.
Moreover, by Proposition 7.2.4(2) and Proposition 3.5.3,
$[\mathcal{Z}]_{\ast}$ is a pure tensor $\ell\otimes s$, and $s$ is surjective
provided it is nontrivial. Thus, for any single choice of $\bm{\epsilon}$, it
suffices to show that

(90)
$\begin{split}H^{\bm{p}(\bm{\epsilon},\emptyset),\bm{q}(\bm{\epsilon},\emptyset)}_{(2)}(S(\operatorname{\mathbf{GSp}}_{4}),\mathcal{V}_{(\bm{m}-2,0),\mathbb{C}})[\Pi_{f}]\otimes
H_{(2)}^{\bm{1}-p(\bm{\epsilon}),\bm{1}-q(\bm{\epsilon})}(S(\operatorname{\mathbf{GL}}_{2}),\mathcal{V}_{\bm{m}-2,\mathbb{C}})[\pi_{2,f}^{\vee}]\xrightarrow{\;\;\langle[\mathcal{Z}]_{\ast}\cdot,\cdot\rangle\;\;}\mathbb{C}\end{split}$

is nontrivial.

Indeed, let

$\bm{\varphi}^{\bm{\epsilon}}_{\infty}=\otimes_{v|\infty}\varphi^{\epsilon_{v}}_{m_{v}}\in\mathcal{S}_{F\otimes\mathbb{R}}(\langle
e_{2},e_{4}\rangle\otimes
V)\otimes\bm{\tau}_{\bm{m},\emptyset}^{\bm{\epsilon}}\otimes\bm{\chi}_{-\bm{\epsilon}}^{\vee},$

where $\varphi^{\epsilon_{v}}_{m_{v}}$ is the vector-valued Schwartz function
of Proposition 7.1.7. Also let

(91) $\theta_{\bm{\epsilon}}:\mathcal{S}_{\mathbb{A}_{F,f}}(\langle
e_{2},e_{4}\rangle\otimes
V)\twoheadrightarrow\left(\Pi\otimes\bm{\tau}^{\bm{\epsilon}}_{\bm{m},\emptyset}\right)^{\bm{K}_{2}}$

be the $\mathbb{C}$-linear map

$\phi_{f}\mapsto\theta_{\phi_{f}\otimes\varphi^{\bm{\epsilon}}_{\infty}}(f_{1}\otimes
f_{2}),$

where $f_{1}\in\pi_{1}$ and $f_{2}\in\pi_{2}$ are nonzero newforms of weights
$-(\bm{m}+2)$ and $\bm{\epsilon}\bm{m},$ respectively.

Now Proposition 7.2.4 and Proposition 7.1.8 imply that the composite map

(92) $\mathcal{S}_{\mathbb{A}_{F,f}}(\langle e_{2},e_{4}\rangle\otimes
V)\otimes(\pi_{2}^{\vee}\otimes\bm{\chi}^{\vee}_{-\bm{\epsilon}\bm{m}})^{\bm{K}_{1}}\xrightarrow{\theta_{\bm{\epsilon}}\otimes\operatorname{id}}\left(\Pi\otimes\bm{\tau}_{\bm{m},\emptyset}^{\bm{\epsilon}}\right)^{\bm{K}_{2}}\otimes(\pi_{2}^{\vee}\otimes\bm{\chi}^{\vee}_{-\bm{\epsilon}\bm{m}})^{\bm{K}_{1}}\xrightarrow{\langle[\mathcal{Z}]_{\ast}\circ\operatorname{cl}_{\emptyset}^{\bm{\epsilon}},\operatorname{cl}^{\prime}_{\bm{\epsilon}}\rangle}\mathbb{C}$

is given by

(93)
$\phi_{f}\otimes\beta\mapsto\mathcal{P}_{\emptyset}(\theta_{\phi_{f}\otimes\overline{\varphi}_{\infty}^{\bm{\epsilon}}}(f_{1}\otimes
f_{2}),\beta)$

up to a nonzero scalar, where

$\overline{\varphi}_{\infty}^{\bm{\epsilon}}=\otimes_{v}\overline{\varphi}_{m_{v}}^{\epsilon_{v}}$

for $\overline{\varphi}_{m_{v}}^{\epsilon_{v}}\in\mathcal{S}_{F_{v}}(\langle
e_{2},e_{4}\rangle\otimes V)$ as in Proposition 7.1.8. Suppose given
factorizable test data

$\phi_{i,f}=\otimes_{v\nmid\infty}\phi_{i,v}\in\mathcal{S}_{\mathbb{A}_{F,f}}(V),\;\;i=1,2$

and

$\beta=\otimes_{v}\beta_{v}\in(\pi_{2}^{\vee}\otimes\bm{\chi}^{\vee}_{-\bm{\epsilon}\bm{m}})^{\bm{K}_{1}};$

we also fix decompositions of the global Whittaker functionals as in Theorem
6.5.2. Then, from Theorem 6.5.2 and Proposition 7.1.10, we conclude the
formula

(94)
$\langle[\mathcal{Z}]_{\ast}\circ\operatorname{cl}_{\emptyset}^{\bm{\epsilon}}(\theta_{\bm{\epsilon}}(\phi_{1,f}\otimes\phi_{2,f})),\operatorname{cl}_{\bm{\epsilon}}^{\prime}(\beta)\rangle\doteq\frac{L^{S}(1,\pi_{1}\times\pi_{2}^{\vee})L^{S}(1,\operatorname{Ad}\pi_{2})}{\zeta_{F}^{S}(2)}\prod_{\begin{subarray}{c}v\in
S\\\
v\nmid\infty\end{subarray}}\frac{\mathcal{Z}_{v}(\phi_{1,v},\phi_{2,v},f_{1,v}\otimes
f_{2,v},\beta_{v})}{1-q_{v}^{-1}}\cdot\prod_{v|\infty}\frac{W_{\beta_{v}}(1)}{e^{-2\pi}},$

where $\doteq$ denotes equality up to a nonzero constant. The nonvanishing of
the right-hand side, which follows from Theorem 6.5.2, concludes the proof. ∎

######  Remark 7.2.6.

In fact, since the formula (94) is uniform in $\bm{\epsilon}$, Proposition
7.2.4(2) was not strictly necessary for the proof of the theorem.

##  8\. Non-tempered theta lifts on $\operatorname{GSp}_{6}$

###  8.1. Arthur parameters

#### 8.1.1.

Let $G$ be a split reductive group over $F$, and recall that a local Arthur
parameter for $G$ is a $\widehat{G}$-conjugacy class of homomorphisms

$\psi_{v}:WD_{F_{v}}\times\operatorname{SL}_{2}(\mathbb{C})\to\widehat{G},$

such that the restriction to $WD_{F_{v}}$ is a bounded Langlands parameter. If
$v$ is non-archimedean and $\psi_{v}$ is unramified, then it determines a
local Langlands parameter $\varphi_{v}$ by the rule

(95) $\varphi_{v}(w)=\psi_{v}\left(w,\begin{pmatrix}|w|^{1/2}_{v}&0\\\
0&|w|^{-1/2}_{v}\end{pmatrix}\right).$

An unramified representation of $G(F_{v})$ is said to have Arthur parameter
$\psi_{v}$ if its Langlands parameter is determined by the rule (95).

#### 8.1.2.

In [4], Arthur defines discrete global parameters for $\operatorname{Sp}_{2n}$
to be formal (unordered) sums

(96) $\oplus\pi_{i}[d_{i}],$

where:

  * •

$\pi_{i}$ are cuspidal automorphic representations of
$\operatorname{PGL}_{n_{i}}$;

  * •

$d_{i}\geq 0$ are integers such that $\sum n_{i}d_{i}=2n+1$;

  * •

$\pi_{i}$ is conjugate symplectic if $d_{i}$ is even, and conjugate orthogonal
if $d_{i}$ is odd;

  * •

the pairs $(\pi_{i},d_{i})$ are distinct.

The integers $d_{i}$ are to be interpreted as the dimensions of irreducible
representations of $\operatorname{SL}_{2}(\mathbb{C})$. Moreover, a global
Arthur parameter $\psi$ induces a local Arthur parameter $\psi_{v}$ for each
place $v$ of $F$ via the local Langlands classification for
$\operatorname{GL}_{m}$. The reason we must take this rather circuitous
definition of global Arthur parameters is the lack of a global Langlands
group.

#### 8.1.3.

In [4], Arthur classified discrete automorphic representations of
$\operatorname{Sp}_{2n}$ by constructing local and global packets for these
parameters, denoted $\Pi_{\psi_{v}}$ and $\Pi_{\psi}$, respectively. An
automorphic representation
$\pi\subset\mathcal{A}_{(2)}(\operatorname{Sp}_{2n}(\mathbb{A}_{F}))$ belongs
to $\Pi_{\psi}$ if and only if, for almost all places $v$, $\pi_{v}$ is the
spherical representation determined by $\varphi_{v}$. Although a full
endoscopic classification for $\operatorname{GSp}_{2n}$ is not available, one
can obtain partial results using the pullback along the natural map

$\iota:\operatorname{Sp}_{2n}\to\operatorname{GSp}_{2n}.$

Indeed, for our purposes the following rather weak result is sufficient.

######  Proposition 8.1.4.

Let
$\pi_{1},\pi_{2}\subset\mathcal{A}_{(2)}(\operatorname{GSp}_{2n}(\mathbb{A}_{F}))$
be nearly equivalent discrete automorphic representations of
$\operatorname{GSp}_{2n}(\mathbb{A}_{F})$, and suppose that some irreducible
constituent of $\iota^{\ast}\pi_{1}$ has parameter $\psi$. Then for all places
$v$ of $F$, any irreducible constituent of the admissible
$\operatorname{Sp}_{2n}(F_{v})$-module $\iota^{\ast}\pi_{2}$ belongs to the
Arthur packet $\Pi_{\psi_{v}}$.

###### Proof.

Let $\pi_{0,v}$ be an irreducible constituent of $\iota^{\ast}\pi_{2,v}$.
There exists a vector $f\in\pi_{2}$ generating an irreducible
$\operatorname{Sp}_{2n}(\mathbb{A}_{F}))$-module
$\pi_{0}\subset\mathcal{A}_{(2)}(\mathbb{A}_{F})$ whose component at $v$ is
$\pi_{0,v}$. By construction, $\pi_{0}$ has Arthur parameter $\psi$, and so
the result follows from [4]. ∎

###  8.2. Theta lift from $\operatorname{GSO}(V_{B})$ to
$\operatorname{GSp}_{6}$

#### 8.2.1.

For the remainder of this section, fix a _non-split_ quaternion algebra $B$
over $F$, and let $\pi$ be a tempered automorphic representation of
$PB^{\times}$. We consider the representation $\pi\boxtimes\mathbbm{1}$ of
$\operatorname{GSO}(V_{B})\simeq B^{\times}\times B^{\times}/\mathbb{G}_{m}$
and its theta lift $\Theta(\pi\boxtimes\mathbbm{1})$ to
$\operatorname{GSp}_{6}(\mathbb{A}_{F})$; this is well-defined because $V_{B}$
is anisotropic, and descends to $\operatorname{PGSp}_{6}(\mathbb{A}_{F})$
because the similitude theta lift preserves central characters. Although this
is not strictly necessary for our argument, it follows from [9] that
$\Theta(\pi\boxtimes\mathbbm{1})\neq 0$ for any such $\pi$. However,
$\Theta(\pi\boxtimes\mathbbm{1})$ need not be irreducible.

######  Proposition 8.2.2.

The theta lift $\Theta(\pi\boxtimes\mathbbm{1})$ lies in the $L^{2}$ subspace
$\mathcal{A}_{(2)}(\operatorname{PGSp}_{6}(\mathbb{A}_{F}))$.

By the usual criterion for square-integrability [23, Lemma I.4.11], we must
check that, for each standard parabolic subgroup $P=MN$ of
$\operatorname{GSp}_{6},$ the characters of $Z(M)$ appearing in the cuspidal
component of the normalized Jacquet module

$\Theta(\pi\boxtimes\mathbbm{1})_{N}\otimes\delta_{P}^{-1/2}$

all lie in the interior of the cone spanned by the _negatives_ of the
characters appearing in the action of $Z(M)$ on $N$. Since
$\pi\boxtimes\mathbbm{1}$ is not a theta lift from
$\operatorname{GSp}_{2}=\operatorname{GL}_{2},$ [27, Theorem I.1.1] implies
that the Jacquet modules are given by:

(97)
$\Theta(\pi\boxtimes\mathbbm{1})_{N}=\begin{cases}|\cdot|^{2}\Theta^{\prime}(\pi\boxtimes\mathbbm{1}),&M=\operatorname{GSp}_{4}\times\operatorname{GL}_{1}\\\
0,&\text{ otherwise}.\end{cases}$

Here $\Theta^{\prime}(\pi\boxtimes\mathbbm{1})$ denotes the theta lift to
$\operatorname{GSp}_{4},$ and $|\cdot|$ is the norm character of
$\operatorname{GL}_{1}.$ On the other hand, the action of $Z(M)$ on $N$ is
through positive powers of $|\cdot|$, and $\delta_{P}=|\cdot|^{6}$; thus the
criterion for square-integrability is satisfied.

######  Proposition 8.2.3.

Suppose $\Pi$ is an irreducible constituent of
$\Theta(\pi\boxtimes\mathbbm{1})$.

  1. (1)

For all non-archimedean $v$, if $\pi_{v}$ is unramified with local Langlands
parameter $\phi_{v}$, then $\Pi_{v}$ is unramified with an Arthur parameter
$\psi_{v}$ such that the composite

$W_{F_{v}}\xrightarrow{\psi_{v}}\widehat{\operatorname{PGSp}_{6}}=\operatorname{Spin}_{7}\xrightarrow{r_{\text{spin}}}\operatorname{GL}_{8}$

is given by

$\phi_{v}\otimes S_{2}\oplus S_{3}\oplus S_{1},$

where $S_{i}$ is the $i$-dimensional irreducible representation of
$\operatorname{SL}_{2}$.

  2. (2)

Any irreducible constituent of $\iota^{\ast}\Pi$ has global Arthur parameter

$\operatorname{JL}(\pi)[2]\oplus\mathbbm{1}[3].$

###### Proof.

For (1), Propositions 8.2.2 and 4.2.3 imply that $\Pi_{v}$ is an irreducible
constituent of $\Theta_{v}(\pi_{v}\boxtimes\mathbbm{1})$ for all $v$. Since
$\pi_{v}$ is tempered at all unramified $v$,
$\operatorname{Ind}_{\operatorname{GSO}(V)(F_{v})}^{\operatorname{GO}(V)(F_{v})}\pi_{v}\boxtimes\mathbbm{1}$
is irreducible. Adopting the notation of (4.3.2) with
$G=\operatorname{GSO}(V)$ and $H=\operatorname{GSp}_{6}$,
$\pi_{v}\boxtimes\mathbbm{1}$ is the spherical representation $\pi_{\chi}$ for
the unramified character of $T_{G}(F_{v})$ defined by
$(-\log_{q}\alpha_{v}-1/2,1/2-\log_{q}\alpha_{v},\log_{q}\alpha_{v})$, where
$\left\\{\alpha_{v},\alpha_{v}^{-1}\right\\}$ are the Sataka parameters of
$\pi_{v}$. Then Proposition 4.3.3 implies that
$\Theta_{v}(\pi_{v}\boxtimes\mathbbm{1})$ is the irreducible representation
$\sigma_{\mu}$ with
$\mu(\alpha_{v})=(-\log_{q}\alpha_{v}-1/2,1/2-\log_{q}\alpha_{v},-1,1/2+\log_{q}\alpha_{v})$.
Recall that any $\mu$ determines an unramified Langlands parameter for $H$:
the characters
$x_{i},\lambda\in\operatorname{Hom}_{F_{v}}(T_{G},\mathbb{G}_{m})$ correspond
to cocharacters in
$\operatorname{Hom}_{\mathbb{C}}(\mathbb{G}_{m},\widehat{T}_{G})$, and any
unramified character $\mu=(\beta_{1},\cdots,\beta_{3},t)$ may be viewed as the
element

$\lambda(q^{-t})\prod_{i}x_{i}(q^{-\beta_{i}})\in\widehat{T}_{H}(\mathbb{C}).$

Then the Langlands parameter of $\sigma_{\mu}$ is the conjugacy class of the
unramified map

$\phi_{\mu}:W_{F_{v}}\to\widehat{T}_{H}(\mathbb{C})\hookrightarrow\widehat{H}(\mathbb{C})$

such that $\phi(\operatorname{Frob}_{v})$ is the element corresponding to
$\mu$. Now, the eigenvalues of $r_{\text{spin}}\circ\phi_{\mu}$ on
$\mathbb{C}^{8}$ are given by $q^{-t}\prod_{i\in S}q^{-\beta_{i}}$ as $S$
ranges over subsets of $\left\\{1,2,3\right\\}$. Hence in our case, the eight
Frobenius eigenvalues are (with multiplicity) $q^{\pm 1/2}\alpha_{v}^{\pm 1}$,
$q,$ $q^{-1}$, 1, and 1. These Frobenius eigenvalues correspond to the
unramified Arthur parameter $\phi_{v}\otimes S_{2}\oplus S_{3}\oplus S_{1}$
according to (95).

The second claim follows from the endoscopic classification for
$\operatorname{Sp}_{6}$, since the local Arthur parameter for any irreducible
constituent of $\iota^{\ast}\Pi_{v}$ is the composition of $\psi_{v}$ with the
projection $\operatorname{Spin}_{7}\to\operatorname{SO}_{7}$ (cf. [26]). ∎

###  8.3. Archimedean Arthur packet and cohomology

#### 8.3.1.

In order to construct Hodge classes from $\Theta(\pi\boxtimes\mathbbm{1})$, it
is essential to understand the structure of the cohomological local Arthur
packet $\Pi_{\psi_{v}}$, where $\psi$ is a global parameter for
$\operatorname{Sp}_{6}$ of the kind in Proposition 8.2.3(2) and $v$ is
archimedean. We now recall the construction of $\Pi_{\psi_{v}}$ due to Adams
and Johnson [2]. (This construction agrees with [4] by [1, 3].)

#### 8.3.2.

We first fix an anisotropic maximal torus $T$ of $G=\operatorname{Sp}_{6}$.
However, we depart from our usual coordinates for $G$, and choose a _complex_
basis of the $2n$-dimensional symplectic space with respect to which

(98) $\mathfrak{g}_{\mathbb{C}}=\left\\{\begin{pmatrix}A&B\\\
C&-A^{T}\end{pmatrix}\bigg{|}A,B,C\in\mathfrak{gl}_{3,\mathbb{C}},\;B=B^{T},\;C=C^{T}\right\\},$
(99) $T_{\mathbb{C}}=\left\\{\begin{pmatrix}\lambda_{1}&&&&&\\\
&\lambda_{2}&&&&\\\ &&\lambda_{3}&&&\\\ &&&\lambda_{1}^{-1}&&\\\
&&&&\lambda_{2}^{-1}&\\\
&&&&&\lambda_{3}^{-1}\end{pmatrix}\right\\}\simeq(\mathbb{C}^{\times})^{3}\subset
G_{\mathbb{C}},$

and complex conjugation acts by

(100) $\begin{pmatrix}A&B\\\
C&-A^{T}\end{pmatrix}\mapsto\begin{pmatrix}-\overline{A}^{T}&\overline{C}\\\
\overline{B}&\overline{A}.\end{pmatrix}$

We choose the Borel subgroup $B$ of $G_{\mathbb{C}}$ given by

(101) $B=\left\\{\begin{pmatrix}A&B\\\ 0&A^{-T}\end{pmatrix}\in
G_{\mathbb{C}}\bigg{|}A\text{ is upper triangular}\right\\}.$

Let $\theta$ be the Cartan involution of $G$ that acts on
$\mathfrak{g}_{\mathbb{C}}$ by

(102) $\begin{pmatrix}A&B\\\
C&-A^{T}\end{pmatrix}\mapsto\begin{pmatrix}A&-B\\\ -C&-A^{T}\end{pmatrix}.$

The corresponding maximal compact subgroup of $G$ is $K=U(3)$. The absolute
Weyl group of $G$ is

(103) $W(G,T)=\left\\{\pm 1\right\\}^{3}\rtimes S_{3},$

where $S_{3}$ acts on $T$ by permuting $(\lambda_{1},\lambda_{2},\lambda_{3})$
and $\left\\{\pm 1\right\\}^{3}$ acts by

$(e_{1},e_{2},e_{3})\cdot(\lambda_{1},\lambda_{2},\lambda_{3})=(\lambda_{1}^{e_{1}},\lambda_{2}^{e_{2}},\lambda_{3}^{e_{3}}).$

The relative Weyl group $W_{\mathbb{R}}(G,T)$ of $G$ may be identified with
$S_{3}.$

#### 8.3.3.

Let $L\subset G$ be the unique $\mathbb{R}$-subgroup isomorphic to $U(2)\times
SU(1,1)$ such that

(104) $L_{\mathbb{C}}=\left\\{\left(\begin{array}[]{ccc|ccc}&&&&&\\\
\lx@intercol\hfil\smash{\raisebox{6.0pt}{$A$}}\hfil\lx@intercol&&&&\\\
&&a&&&b\\\ \hline\cr\\\\[-12.0pt] &&&&&\\\
&&&\lx@intercol\hfil\smash{\raisebox{6.0pt}{$A^{-T}$}}\hfil\lx@intercol&\\\
&&c&&&d\end{array}\right)\Bigg{|}A\in\operatorname{GL}_{2},\;\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\operatorname{SL}_{2}\right\\}\subset G_{\mathbb{C}}.$

Then $L_{\mathbb{C}}$ is the Levi factor of a $\theta$-stable standard
parabolic $Q\subset G_{\mathbb{C}},$ which does not descend to $\mathbb{R}$.
The absolute Weyl group $W(L,T)$ is given by

(105) $W(L,T)=\left\\{(1,1,\pm
1)\right\\}\times\left\\{(1),(12)\right\\}\subset W(G,T).$

Note that the local Arthur parameter factors as

(106)
$\psi_{v}:W_{F_{v}}\times\operatorname{SL}_{2}(\mathbb{C})\to{}^{L}L\to{}^{L}G.$

If $\operatorname{JL}(\pi_{v})$ is the discrete series representation of
$\operatorname{GL}_{2}(\mathbb{R})$ of weight $2k\geq 6,$ then, in the
notation of [2],

(107)
$\Pi_{\psi_{v}}=\left\\{A_{\mathfrak{q}_{w}}(w^{-1}\cdot(k-3,k-3,0))\,:\,w\in
S\right\\},$

where $(k-3,k-3,0)$ is viewed as a character of $T$,

(108) $S=W(L,T)\backslash W(G,T)/W_{\mathbb{R}}(G,T),$

and, for $w\in S$, $Q_{w}$ is the $\theta$-stable parabolic subgroup
$w^{-1}Qw$ of $G_{\mathbb{C}}$.

#### 8.3.4.

Choose representatives

(109) $w_{0}=1,\;\;w_{1}=(-1,1,1),\;\;w_{2}=(-1,-1,1)$

of $S$ in $W(G,T)$, and label the elements of $\Pi_{\psi_{v}}$ as

(110)
$\Pi_{\psi_{v},i}=A_{\mathfrak{q}_{w_{i}}}(w_{i}^{-1}\cdot(k-3,k-3,0)),\;\;0\leq
i\leq 2.$

######  Proposition 8.3.5.

For any complex, irreducible algebraic representation $V$ of $G$,

$H^{\ast}(\mathfrak{g},K;\Pi_{\psi_{v},i}\otimes V)=0$

unless $V$ is the highest weight representation $V_{(k-3,k-3,0)}.$ If
$V=V_{(k-3,k-3,0)}$, then the dimensions of the nonzero cohomology groups of
the representations $\Pi_{\psi_{v},i}$ are computed as follows.

$\dim_{\mathbb{C}}H^{5,0}(\mathfrak{g},K;\Pi_{\psi_{v},0}\otimes
V)=\dim_{\mathbb{C}}H^{6,1}(\mathfrak{g},K;\Pi_{\psi_{v},0}\otimes V)=1$
$\displaystyle\dim_{\mathbb{C}}H^{2,2}(\mathfrak{g},K;\Pi_{\psi_{v},1}\otimes
V)$
$\displaystyle=\dim_{\mathbb{C}}H^{4,4}(\mathfrak{g},K;\Pi_{\psi_{v},1}\otimes
V)=1$
$\displaystyle\dim_{\mathbb{C}}H^{3,3}(\mathfrak{g},K;\Pi_{\psi_{v},1}\otimes
V)$ $\displaystyle=2$
$\dim_{\mathbb{C}}H^{0,5}(\mathfrak{g},K;\Pi_{\psi_{v},2}\otimes
V)=\dim_{\mathbb{C}}H^{1,6}(\mathfrak{g},K;\Pi_{\psi_{v},2}\otimes V)=1$

###### Proof.

Write $Q_{w_{i}}=L_{i}U_{i}.$ The proposition follows from [33, Proposition
6.19], together with the calculations:

$\displaystyle\mathfrak{u}_{0}=\left(\begin{array}[]{ccc|ccc}&&\ast&\ast&\ast&\ast\\\ \lx@intercol\hfil\smash{\raisebox{6.0pt}{$I$}}\hfil\lx@intercol&\ast&\ast&\ast&\ast\\\ 0&0&1&\ast&\ast&0\\\ \hline\cr\\\\[-12.0pt] &&&&&0\\\ &&&\lx@intercol\hfil\smash{\raisebox{6.0pt}{$I$}}\hfil\lx@intercol&0\\\ &&&\ast&\ast&1\end{array}\right),\;\;\;$ $\displaystyle W_{\mathbb{R}}(L_{0},T)=\left\\{1,(12)\right\\},$ $\displaystyle\mathfrak{u}_{1}=\left(\begin{array}[]{ccc | ccc}1&&&&&\\\ \ast&1&\ast&&\ast&\ast\\\ \ast&&1&&\ast&\\\ \hline\cr\\\\[-12.0pt] \ast&&\ast&&\ast&\\\ &&&&1&\\\ \ast&&&&\ast&1\end{array}\right),\;\;\;$ $\displaystyle W_{\mathbb{R}}(L_{1},T)=\left\\{1\right\\},$ $\displaystyle\mathfrak{u}_{2}=\left(\begin{array}[]{ccc | ccc}&&0&&&\\\ \lx@intercol\hfil\smash{\raisebox{6.0pt}{$I$}}\hfil\lx@intercol&0&&&\\\ \ast&\ast&1&&&\\\ \hline\cr\\\\[-12.0pt] \ast&\ast&\ast&&&\ast\\\ \ast&\ast&\ast&\lx@intercol\hfil\smash{\raisebox{6.0pt}{$I$}}\hfil\lx@intercol&\ast\\\ \ast&\ast&0&0&0&1\end{array}\right),\;\;\;$ $\displaystyle W_{\mathbb{R}}(L_{2},T)=\left\\{1,(12)\right\\}.$

∎

#### 8.3.6.

If $V=V_{(k-3,k-3,0)}$, then $V$ has a $K$-stable Hodge decomposition of
weight $m=2k-6$ defined by

(111) $V^{p,q}=\left\\{v\in V:z\cdot v=z^{n-2p}v,\;\forall z\in Z_{K}\simeq
U(1)\right\\},\;p+q=m.$

The decomposition (111) induces a refined decomposition of the Lie algebra
cohomology, cf. [37]:

$H^{p,q}(\mathfrak{g},K;\Pi_{\psi_{v},i}\otimes
V)=\bigoplus_{\begin{subarray}{c}r+s=m\\\ r,s\geq
0\end{subarray}}H^{(p,q);(r,s)}(\mathfrak{g},K;\Pi_{\psi_{v},i}\otimes V).$

For later use, we now calculate this decomposition.

######  Proposition 8.3.7.

Let $V=V_{(k-3,k-3,0)}$. The dimensions of the nonzero refined components of
the cohomology groups of the representations $\Pi_{\psi_{v},i}$ are computed
as follows.

$\displaystyle\dim_{\mathbb{C}}H^{(5,0);(2k-6,0)}(\mathfrak{g},K;\Pi_{\psi_{v},0}\otimes
V)$
$\displaystyle=\dim_{\mathbb{C}}H^{(6,1);(2k-6,0)}(\mathfrak{g},K;\Pi_{\psi_{v},0}\otimes
V=1$
$\displaystyle\dim_{\mathbb{C}}H^{(2,2);(k-3,k-3)}(\mathfrak{g},K;\Pi_{\psi_{v},1}\otimes
V)$
$\displaystyle=\dim_{\mathbb{C}}H^{(4,4);(k-3,k-3)}(\mathfrak{g},K;\Pi_{\psi_{v},1}\otimes
V)=1$
$\displaystyle\dim_{\mathbb{C}}H^{(3,3);(k-3,k-3)}(\mathfrak{g},K;\Pi_{\psi_{v},1}\otimes
V)$ $\displaystyle=2$
$\displaystyle\dim_{\mathbb{C}}H^{(0,5);(0,2k-6)}(\mathfrak{g},K;\Pi_{\psi_{v},2}\otimes
V)$
$\displaystyle=\dim_{\mathbb{C}}H^{(1,6);(0,2k-6)}(\mathfrak{g},K;\Pi_{\psi_{v},2}\otimes
V)=1$

###### Proof.

The proof is similar to [14, Proposition 11.4]. Consider the set of coset
representatives for $W_{\mathbb{R}}(G,T)\backslash W(G,T)$ given by

(112) $W_{0}\coloneqq\left\\{w\in
W(G,T):w^{-1}(\Delta_{c}^{+})\subset\Delta^{+}\right\\},$

where $\Delta^{+}$ is the set of positive roots with respect to the Borel
subgroup (101) and $\Delta^{+}_{c}\subset\Delta^{+}$ is the subset of compact
roots (i.e. the positive roots of $K$). In the notation of [37], let $\mu$ and
$\lambda$ be the highest characters of $Z_{K}$ appearing in the action of $K$
on $\mathfrak{g}_{\mathbb{C}}$ and $V$, respectively; identifying characters
of $Z_{K}$ with $\mathbb{Z}$, $\mu=2$ and $\lambda=2k-6=m$ (the weight of
$V$). Also let $\Lambda=(k-3,k-3,0)$, considered as a character of $T$. By
[37, §5], if

$H^{(p,q);(n-p,m+p-n)}(\mathfrak{g},K;\Pi_{\psi_{v},i}\otimes V)\neq 0$

for any $i$, then there exists $w\in W_{0}$ with $\ell(w)=p,$ such that the
$K$-representation of highest weight $w(\Lambda+\rho)-\rho$ has central
character $\lambda-n\mu=2k-2n-6.$ Here $\rho=(3,2,1)$ is the half sum of
positive roots for $\mathfrak{g}_{\mathbb{C}}$. Thus each $w\in W_{0}$ defines
exactly one choice of $p$ and $n$ such that a nonzero contribution of bidegree
$(p,q);(n-p,m+p-n)$ is possible. These choices are summarized in the following
table.

$\begin{array}[]{c|c |c | c | c}w\cdot(a,b,c)&p=\ell(w)&w(\Lambda+\rho)-\rho&n&\text{possible types}\\\ \hline\cr(a,b,c)&0&(k-3,k-3,0)&0&(0,q);(0,2k-6)\\\ (a,b,-c)&1&(k-3,k-3,-2)&1&(1,q);(0,2k-6)\\\ (a,c,-b)&2&(k-3,-1,-k)&k-1&(2,q);(k-3,k-3)\\\ (b,c,-a)&3&(k-4,-1,-k-1)&k&(3,q);(k-3,k-3)\\\ (a,-c,-b)&3&(k-3,-3,-k)&k&(3,q);(k-3,k-3)\\\ (b,-c,-a)&4&(k-4,-3,-k-1)&k+1&(4,q);(k-3,k-3)\\\ (c,-b,-a)&5&(-2,-k-1,-k-1)&2k-1&(5,q);(2k-6,0)\\\ (-c,-b,-a)&6&(-4,-k-1,-k-1)&2k&(6,q);(2k-6,0)\end{array}$

Comparing with Proposition 8.3.5 completes the proof. ∎

###  8.4. Contributions to the cohomology of Shimura varieties

#### 8.4.1.

Consider the Shimura variety for $\operatorname{\mathbf{GSp}}_{6}$ as in §3.4.
Following the notation of (3.1.1), we obtain a local system
$\mathcal{V}_{(\bm{m}-2,\bm{m}-2,0)}$ of $\mathbb{Q}(\bm{m})$-vector spaces.
Let $\sigma_{m_{v}}$ be the unique irreducible representation of $K_{3,v}$
with trivial central character and whose restriction to $U(3)$ has highest
weight $(m_{v}+1,0,-m_{v}-1)$, and let $\bm{\sigma}_{\bm{m}}$ be the
representation $\otimes_{v|\infty}\sigma_{m_{v}}$ of $\bm{K}_{3}.$ One
calculates that

(113)
$\dim\operatorname{Hom}_{\bm{K}_{3}}\left(\bm{\sigma}_{\bm{m}},V_{(\bm{m}-2,\bm{m}-2,0),\mathbb{C}}\otimes\wedge^{\bm{2},\bm{2}}\mathfrak{p}^{\ast}_{\operatorname{GSp}_{6}}\right)=1,$

where $(\bm{2},\bm{2})$ is the constant plectic Hodge type. Thus we have, from
(31), a class map

(114)
$\left(\mathcal{A}_{(2)}(\operatorname{GSp}_{6}(\mathbb{A}_{F}))\otimes\bm{\sigma}_{\bm{m}}\right)^{\bm{K}_{3}}\to
H^{\bm{2},\bm{2}}_{(2)}(S(\operatorname{\mathbf{GSp}}_{6}),{\mathcal{V}}_{(\bm{m}-2,\bm{m}-2,0),\mathbb{C}}).$

#### 8.4.2.

We now choose a totally indefinite, non-split quaternion algebra $B$ over $F$.
Let $\pi$ be an auxiliary automorphic representation of
$PB^{\times}(\mathbb{A}_{F})$ of weight $2\bm{m}+2=(2m_{v}+2)_{v|\infty}$.

######  Lemma 8.4.3.

If $\widetilde{\Pi}$ is a discrete automorphic representation of
$\operatorname{GSp}_{6}(\mathbb{A}_{F})$ which is nearly equivalent to a
constituent of the theta lift $\iota^{\ast}\Theta(\pi\boxtimes\mathbbm{1})$,
then we have:

  1. (1)

View $\mathcal{V}_{(\bm{m}-2,\bm{m}-2,0),\mathbb{C}}$ as a variation of
structures of weight 0. Then the $L^{2}$-cohomology

$H_{(2)}^{4d}(S(\operatorname{\mathbf{GSp}}_{6}),{\mathcal{V}}_{(\bm{m}-2,\bm{m}-2,0),\mathbb{C}})[\widetilde{\Pi}_{f}]$

is purely of Hodge type $(2d,2d)$ for the (non-plectic) Hodge structure on
$L^{2}$ cohomology defined in [37].

  2. (2)

The Galois group $\operatorname{Gal}(\overline{\mathbb{Q}}/F^{c})$ acts on the
intersection cohomology

$IH^{4d}(S(\operatorname{\mathbf{GSp}}_{6})_{\overline{\mathbb{Q}}},{\mathcal{V}}_{(\bm{m}-2,\bm{m}-2,0),\overline{\mathbb{Q}}_{\ell}})[\widetilde{\Pi}_{f}]$

via $\chi^{-2d}$, where $\chi$ is the $\ell$-adic cyclotomic character.

###### Proof.

For (1), it follows from Propositions 8.1.4, 8.2.3, and 8.3.5 that

$H_{(2)}^{4d}(S(\operatorname{\mathbf{GSp}}_{6}),{\mathcal{V}}_{(\bm{m}-2,\bm{m}-2,0),\mathbb{C}})[\widetilde{\Pi}_{f}]\simeq\widetilde{\Pi}_{f}\otimes\bigotimes_{v|\infty}H^{2,2}({\mathfrak{\operatorname{GSp}}_{6,\mathbb{R}}},K_{3};\pi_{v}^{\text{sm}}\otimes
V_{(m_{v}-2,m_{v}-2,0),\mathbb{C}}).$

Thus it follows from Proposition 8.3.7 and from [37] that the cohomology is
pure of Hodge type $(m,m)$ in Zucker’s normalization, where $m=\sum m_{v}$.
However, since $V_{\bm{m},\mathbb{C}}$ has trivial central character and hence
total weight 0 in the algebraic normalization, we must twist by $(2d-m,2d-m)$,
which shows the claim. For (2), choose a compact open subgroup $K=\prod
K_{v}\subset\operatorname{GSp}_{6}(\mathbb{A}_{F,f})$. It suffices to show
that $\operatorname{Frob}_{p}$ acts trivially on

$H\coloneqq
IH^{4d}(S_{K}(\operatorname{\mathbf{GSp}}_{6})_{\overline{\mathbb{Q}}},{\mathcal{V}}_{(\bm{m}-2,\bm{m}-2,0),\overline{\mathbb{Q}}_{\ell}})[\widetilde{\Pi}_{f}]$

for almost all $p$ such that $p$ splits completely in $F^{c}$ and $K_{p}$ is
hyperspecial. Assume without loss of generality that $\widetilde{\Pi}_{p}$ is
unramified with local Langlands parameter

$\phi_{p}:W_{\mathbb{Q}_{p}}\to{}^{L}\operatorname{\mathbf{GSp}}_{6}=\operatorname{GSpin}_{7}(\mathbb{C})^{d}\times
W_{\mathbb{Q}_{p}},$

and consider the $8d$-dimensional representation defined by the composite:

(115)
$W_{\mathbb{Q}_{p}}\xrightarrow{\phi_{p}}\operatorname{GSpin}_{7}(\mathbb{C})^{d}\xrightarrow{r_{\text{spin}}^{\otimes
d}}\operatorname{GL}_{8d}(\mathbb{C}).$

After picking an isomorphism $\mathbb{C}\cong\overline{\mathbb{Q}}_{\ell}$,
(115) defines an $8d$-dimensional $\ell$-adic unramified local Galois
representation $V_{p}$. By [20, §2], since $p$ splits in $F(\bm{m})$, the
action of the geometric Frobenius $\operatorname{Frob}_{p}^{-1}$ on $H$
satisfies the characteristic polynomial of $p^{3}\operatorname{Frob}_{p}^{-1}$
on $V_{p}$. Now by Proposition 8.2.3, for almost every such $p$ the
representation $V_{p}$ is given by

$\bigotimes_{v|p}\left(\overline{\mathbb{Q}}_{\ell}(-1)\oplus\overline{\mathbb{Q}}_{\ell}^{2}\oplus\overline{\mathbb{Q}}_{\ell}(1)\oplus\rho_{\pi}|_{F_{v}}\oplus\rho_{\pi}|_{F_{v}}(1)\right),$

where $\rho_{\pi}$ is the $2$-dimensional $\ell$-adic Galois representation
associated to $\pi$, which we normalize to have weight one. (Recall that
$\rho_{\pi}$ is pure since $\pi$ is discrete series at a finite place of $F$
[25].) On the other hand, it is known that $H$ is pure of weight $4d$;
comparing with the weights in $V_{p}$, it follows that
$\operatorname{Frob}_{p}$ acts as $p^{-2d}$ on $H$. ∎

##  9\. Triple product periods

###  9.1. The vector-valued period problem

#### 9.1.1.

Let $\bm{m}$, $\pi_{1},$ $\pi_{2},$ $\bm{\epsilon}$,
$\bm{\tau}^{\bm{\epsilon}}_{\bm{m}}$, and $\bm{\sigma}_{\bm{m}}$ be as in
(5.3.1) and (8.4.1), and let $B$ be a _non-split_ totally indefinite
quaternion algebra over $F$, ramified at a set $S$ of places of $F$ at which
$\pi_{i}$ are both discrete series.

#### 9.1.2.

For auxiliary automorphic representations $\pi$ of
$PB(\mathbb{A}_{F})^{\times}$ of weight $2\bm{m}+2,$ we will consider triple
product period integrals of $\Theta(\pi\boxtimes\mathbbm{1})$ along the
subgroup

(116)
$\widetilde{H}\coloneqq(\operatorname{GSp}_{4}\times_{\mathbb{G}_{m}}\operatorname{GL}_{2})\subset\operatorname{GSp}_{6}.$

The maximal compact-modulo-center subgroup of
$\widetilde{H}(F\otimes\mathbb{R})$ is

(117)
$(\bm{K}_{2}\times\bm{K}_{1})_{0}\coloneqq(\bm{K}_{2}\times\bm{K}_{1})\cap\widetilde{H}(F\otimes\mathbb{R}).$

To define the vector-valued period integral, note that (by the classical
branching law for unitary groups), the space

(118)
$\operatorname{Hom}_{(\bm{K}_{2}\times\bm{K}_{1})_{0}}(\bm{\sigma}_{\bm{m}}\otimes\bm{\tau}^{\bm{\epsilon}}_{\bm{m}}\otimes\bm{\chi}^{\vee}_{-\bm{\epsilon}\bm{m}},\mathbb{C})$

is one-dimensional, say with generator $\ell$.

We then define, for any auxiliary representation $\pi$ of
$PB(\mathbb{A}_{F})^{\times}$ of weight $2\bm{m}+2,$ the triple product
period:

(119)
$\begin{split}\widetilde{\mathcal{P}}_{S,\pi_{1},\pi_{2},\pi}^{\bm{\epsilon}}(\alpha,\beta,\gamma)&=\int_{[\widetilde{H}]}\ell(\alpha(g,g^{\prime})\otimes\beta(g)\otimes\gamma(g^{\prime}))\mathop{}\\!\mathrm{d}(g,g^{\prime})\neq
0,\\\
&\alpha\in(\Theta(\pi\boxtimes\mathbbm{1})\otimes\bm{\sigma}_{\bm{m}})^{\bm{K}_{3}},\;\;\beta\in(\Pi_{S}(\pi_{1}\otimes\pi_{2})\otimes\bm{\tau}^{\bm{\epsilon}}_{\bm{m}})^{\bm{K}_{2}},\;\;\gamma\in(\pi_{2}^{\vee}\otimes\bm{\chi}^{\vee}_{-\bm{\epsilon}\bm{m}})^{\bm{K}_{1}}.\end{split}$

Since we will not give a precise formula for
$\widetilde{\mathcal{P}}^{\bm{\epsilon}}_{S,\pi_{1},\pi_{2},\pi}$ and are only
interested in its non-vanishing, we ignore the problem of normalization. The
non-vanishing of (119), for a good choice of $\pi$, is the key input to the
non-vanishing of the Hodge classes we construct in the next section.

#### 9.1.3.

The strategy for calculating (119) is to use the seesaw diagram:

${\operatorname{GSO}(V_{B})\times_{\mathbb{G}_{m}}\operatorname{GSO}(V_{B})}$${\operatorname{GSp}_{6}}$${\operatorname{GSO}(V_{B})}$${\widetilde{H}}$

There are two main inputs to the non-vanishing of our period integral (for a
good choice of $\pi$): the first is a vector-valued version of the usual
global seesaw identity, and the second is a non-vanishing result for the
vector-valued theta lifts along the “other” diagonal in the seesaw diagram,
i.e. from $\operatorname{GSp}_{4}$ and $\operatorname{GL}_{2}$ to
$\operatorname{GSO}(V_{B})$.

###  9.2. Vector-valued seesaw identity

#### 9.2.1.

Continuing with the notation from (7.1.1), let $m\geq 2$ be an integer and let
$\sigma_{m}$ be the unique representation of $K_{3}$ of trivial central
character and whose restriction to $U(3)$ has highest weight $(m+1,0,-m-1)$.
Let

$\widetilde{\varphi}_{m}\in(S_{\leq
2m+2}^{0}(3)\otimes\sigma_{m}\otimes\chi_{-(2m+2),0})^{(K_{3}\times L)_{0}}$

be a generator, which makes sense by Proposition 7.1.3. If $(K_{2}\times
K_{1})_{0}$ is the intersection of $K_{3}$ with
$\operatorname{GSp}_{4,\mathbb{R}}\times_{\mathbb{G}_{m}}\operatorname{GL}_{2,\mathbb{R}}$
inside $\operatorname{GSp}_{6,\mathbb{R}}$, then we have, for any
$\epsilon=\pm 1,$

(120) $\dim\operatorname{Hom}_{(K_{2}\times
K_{1})_{0}}(\sigma_{m},\tau_{m}^{\epsilon}\otimes\chi_{-\epsilon
m}^{\vee})=1;$

let $\ell_{\epsilon}$ denote a generator. Also let $(K_{1}\times K_{2}\times
L)_{0}=(K_{1}\times K_{2})_{0}\times L\cap(K_{3}\times L)_{0}.$

######  Proposition 9.2.2.

The Schwartz function

$\ell_{\epsilon}(\widetilde{\varphi}_{m})\in\left(\mathcal{S}_{\mathbb{R}}(\langle
e_{2},e_{4},e_{6}\rangle\otimes
V)\otimes\tau_{m}^{\epsilon}\otimes\chi_{-\epsilon
m}^{\vee}\otimes\chi_{2m+2,0}\right)^{(K_{1}\times K_{2}\times L)_{0}}$

is a nonzero scalar multiple of the tensor product

$\varphi_{m}^{\epsilon}\otimes\phi_{m}^{\epsilon}\in\left(\left(\mathcal{S}_{\mathbb{R}}(\langle
e_{2},e_{4}\rangle\otimes
V)\otimes\tau_{m}^{\epsilon}\otimes\chi^{\vee}_{m+2,-\epsilon
m}\right)^{(K_{2}\times L)_{0}}\otimes\left(\mathcal{S}_{\mathbb{R}}(\langle
e_{6}\rangle\otimes V)\otimes\chi^{\vee}_{-\epsilon m}\otimes\chi_{-m,\epsilon
m}\right)^{(K_{1}\times L)_{0}}\right)\bigg{|}_{(K_{1}\times K_{2}\times
L)_{0}}.$

###### Proof.

Assume $\epsilon=+$; the other case is similar. Let $\phi\in S_{\leq
2m+2}^{0}(3)$ be the contraction of
$\varphi_{m}^{\epsilon}\otimes\phi_{m}^{\epsilon}$ with any nonzero vector.
Then $\phi$ generates the irreducible $U(2)\times U(1)$-representation of
highest weight $(1,-m-1,m)$, and it suffices to show that $U(3)\cdot\phi$ is
the irreducible representation of highest weight $(m+1,0,-m-1)$. Without loss
of generality, assume $U(3)\cdot\phi$ is irreducible, say with highest weight
$(a,b,c)$. It follows (using the branching law for unitary groups) that

(121) $\begin{split}a\geq 1\geq b\geq-m-1\geq c,\\\ a+b+c=0.\end{split}$

On the other hand, considering Proposition 7.1.3(2), we have

(122) $|a|+|b|+|c|\leq 2m+2.$

The combination of (121) and (122) force $(a,b,c)=(m+1,0,-m-1)$. ∎

#### 9.2.3.

We now return to the global situation. Choose an isomorphism
$V_{B}\otimes_{F}\mathbb{R}\simeq V\otimes_{F}\mathbb{R}$, which induces an
isomorphism
$\operatorname{GSO}(V_{B})(F\otimes\mathbb{R})\simeq\operatorname{GSO}(V)(F\otimes\mathbb{R})$.
Then let
$\bm{L}=\prod_{v|\infty}L\subset\operatorname{GSO}(V_{B})(F\otimes\mathbb{R})$,
and similarly for $(\bm{K}_{n}\times\bm{L})_{0}$, etc. We fix vector-valued
Schwartz functions as follows:

(123)
$\begin{split}\bm{\varphi}^{-\bm{\epsilon}}_{\bm{m}}=\otimes_{v|\infty}\varphi^{-\epsilon_{v}}_{m_{v}}&\in\left(\mathcal{S}_{F\otimes\mathbb{R}}(\langle
e_{2},e_{4}\rangle\otimes
V)\otimes\bm{\tau}_{\bm{m}}^{-\bm{\epsilon}}\otimes\bm{\chi}^{\vee}_{(\bm{m}+2),\bm{\epsilon}\bm{m}}\right)^{(\bm{K}_{2}\times\bm{L})_{0}}\text{
\;\; (Proposition \ref{prop: one-dimensional harmonic})}\\\
&\simeq\left(\mathcal{S}_{F\otimes\mathbb{R}}(\langle
e_{2},e_{4}\rangle\otimes
V_{B})\otimes\bm{(}\tau_{\bm{m}}^{\bm{\epsilon}})^{\vee}\otimes\bm{\chi}_{-(\bm{m}+2),-\bm{\epsilon}\bm{m}}\right)^{(\bm{K}_{2}\times\bm{L})_{0}}\\\
\bm{\phi}^{-\bm{\epsilon}}_{\bm{m}}=\otimes_{v|\infty}\phi_{m_{v}}^{-\epsilon_{v}}&\in\left(\mathcal{S}_{F\otimes\mathbb{R}}(\langle
e_{6}\rangle\otimes
V)\otimes\bm{\chi}_{-\bm{\epsilon}\bm{m}}\otimes\bm{\chi}^{\vee}_{\bm{m},-\bm{\epsilon}\bm{m}}\right)^{(\bm{K}_{1}\times\bm{L})_{0}}\text{
\;\; (Proposition \ref{prop: S02 covariance})}\\\
&\simeq\left(\mathcal{S}_{F\otimes\mathbb{R}}(\langle e_{6}\rangle\otimes
V_{B})\otimes\bm{\chi}_{-\bm{\epsilon}\bm{m}}\otimes\bm{\chi}^{\vee}_{\bm{m},-\bm{\epsilon}\bm{m}}\right)^{(\bm{K}_{1}\times\bm{L})_{0}}\\\
\widetilde{\bm{\varphi}}_{\bm{m}}=\otimes_{v|\infty}\widetilde{\varphi}_{m_{v}}&\in\left(\mathcal{S}_{\mathbb{R}}(\langle
e_{2},e_{4},e_{6}\rangle
V)\otimes\bm{\sigma}_{\bm{m}}\otimes\bm{\chi}_{-(2\bm{m}+2),0}\right)^{(\bm{K}_{2}\times\bm{L})_{0}}\\\
&\simeq\left(\mathcal{S}_{\mathbb{R}}(\langle e_{2},e_{4},e_{6}\rangle\otimes
V_{B})\otimes\bm{\sigma}_{\bm{m}}\otimes\bm{\chi}_{-(2\bm{m}+2),0}\right)^{(\bm{K}_{2}\times\bm{L})_{0}}\end{split}$

######  Proposition 9.2.4.

Let $\ell$ be as above. For all

$\alpha\in\left(\mathcal{A}_{0}(\operatorname{PGSO}(V_{B})(\mathbb{A}_{F}))\otimes\bm{\chi}_{2\bm{m}+2,0}\right)^{\bm{L}},\;\;\beta\in\left(\Pi_{S}(\pi_{1},\pi_{2})\otimes\bm{\tau}_{\bm{m}}^{\bm{\epsilon}}\right)^{\bm{K}_{2}},\;\;\gamma\in\left(\pi_{2}^{\vee}\otimes\bm{\chi}_{-\bm{\epsilon}\bm{m}}^{\vee}\right)^{\bm{K}_{1}},$
$\phi_{1,f}\in\mathcal{S}_{\mathbb{A}_{F,f}}(\langle e_{2},e_{4}\rangle\otimes
V_{B}),\;\;\phi_{2,f}\in\mathcal{S}_{\mathbb{A}_{F,f}}(\langle
e_{6}\rangle\otimes V_{B}),$

and up to a nonzero scalar depending on the normalizations, we have the
identity:

(124)
$\begin{split}\int_{[Z_{\widetilde{H}}\backslash\widetilde{H}]}\ell\left(\theta_{\phi_{1,f}\otimes\phi_{2,f}\otimes\widetilde{\bm{\varphi}}_{\bm{m}}}(\alpha)(g,g^{\prime})\otimes\beta(g)\otimes\gamma(g^{\prime})\right)\mathop{}\\!\mathrm{d}(g,g^{\prime})=\\\
\int_{[\operatorname{PGSO}(V_{B})]}\alpha(g)\theta_{\phi_{1,f}\otimes\bm{\varphi}_{\bm{m}}^{-\bm{\epsilon}}}(\beta)(g)\theta_{\phi_{2,f}\otimes\bm{\phi}_{\bm{m}}^{-\bm{\epsilon}}}(\gamma)(g)\mathop{}\\!\mathrm{d}g.\end{split}$

###### Proof.

This is formal from Proposition 9.2.2 and the usual seesaw identity, i.e.
exchanging the order of integration. ∎

###  9.3. Proof of the non-vanishing result

######  Proposition 9.3.1.

Let $\pi_{i}^{B}$ be the Jacquet-Langlands transfers of $\pi_{i}$ to
$B(\mathbb{A}_{F})^{\times}$.

  1. (1)

The map

$\theta_{\bm{\varphi}^{-\bm{\epsilon}}_{\bm{m}}}:\mathcal{S}_{\mathbb{A}_{F,f}}(\langle
e_{2},e_{4}\rangle\otimes
V_{B})\otimes(\Pi_{S}(\pi_{1},\pi_{2})\otimes\bm{\tau}^{\bm{\epsilon}})^{\bm{K}_{2}}\to\left(\mathcal{A}(\operatorname{GSO}(V_{B}))(\mathbb{A}_{F})\otimes\bm{\chi}_{-(\bm{m}+2),-\bm{\epsilon}\bm{m}}\right)^{\bm{L}},$

defined by

$(\phi,\alpha)\mapsto\theta_{\phi\otimes\bm{\varphi}^{-\bm{\epsilon}}_{\bm{m}}}(\alpha),$

has image containing
$\left((\pi_{1}^{B}\boxtimes\pi_{2}^{B})\otimes\bm{\chi}_{-(\bm{m}+2),-\bm{\epsilon}\bm{m}}\right)^{\bm{L}}.$

  2. (2)

The map

$\theta_{\bm{\phi}^{-\bm{\epsilon}}_{\bm{m}}}:\mathcal{S}_{\mathbb{A}_{F,f}}(\langle
e_{6}\rangle\otimes
V_{B})\otimes\left(\pi_{2}^{\vee}\otimes\bm{\chi}_{-\bm{\epsilon}\bm{m}}^{\vee}\right)^{\bm{K}_{1}}\to\left(\mathcal{A}(\operatorname{GSO}(V_{B}))(\mathbb{A}_{F})\otimes\bm{\chi}^{\vee}_{\bm{m},-\bm{\epsilon}\bm{m}}\right)^{\bm{L}},$

defined by

$(\phi,\alpha)\mapsto\theta_{\phi\otimes\bm{\phi}_{\bm{m}}^{-\bm{\epsilon}}}(\alpha),$

has image containing
$\left(((\pi_{2}^{B})^{\vee}\boxtimes(\pi_{2}^{B})^{\vee})\otimes\bm{\chi}^{\vee}_{\bm{m},-\bm{\epsilon}\bm{m}}\right)^{\bm{L}}.$

###### Proof.

In the general setup of §4, suppose $\Theta_{V,W}(\pi)=\Pi$ for cuspidal
automorphic representations $\pi$ of $G(V)(\mathbb{A}_{F})$ and $\Pi$ of
$H(W)(\mathbb{A}_{F})$. Then by definition we have a surjective composite

(125) $\mathcal{S}_{\mathbb{A}_{F}}(W_{2}\otimes
V)\xrightarrow{\theta}\mathcal{A}(R_{0}(\mathbb{A}_{F}))\twoheadrightarrow\Pi\otimes\pi^{\vee}.$

Now, the theta kernel satisfies

$\theta(\phi)(g,h)=\overline{\theta(\overline{\phi})}\left(\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}g\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix},h\right)$

(cf. [28]), so we deduce that
$\overline{\Pi\otimes\pi^{\vee}}=\Pi^{\vee}\otimes\pi$ also appears in the
spectrum of the theta kernel. (Recall that the central characters of $\Pi$ and
$\pi$ must agree since the central character of the Weil representation is
trivial.) In particular $\Theta_{W,V}(\Pi)$ contains the nonzero irreducible
constituent $\pi$. For (1), take $W=W_{4},$ $V=V_{B},$ and
$\Pi=\Pi_{S}(\pi_{1},\pi_{2})$. As in the proof of Proposition 4.2.3, the
global theta lift gives rise to a nontrivial map:

(126) $\mathcal{S}_{\mathbb{A}_{F}}(\langle e_{2},e_{4}\rangle\otimes
V_{B})\twoheadrightarrow\Pi_{S}(\pi_{1},\pi_{2})^{\vee}\otimes\Theta_{W_{4},V_{B}}(\Pi_{S}(\pi_{1},\pi_{2})).$

(Since $\operatorname{GSO}(V_{B})$ is anisotropic, all theta lifts are square-
integrable.) The map (126) is a restricted tensor product of local maps. To
prove the proposition, it suffices to show that, for all $v|\infty$ and for
some vector $0\neq\ell\in\tau_{m_{v}}^{\epsilon_{v}}$, the contraction
$\ell(\varphi_{m_{v}}^{-\epsilon_{v}})$ has nontrivial image under the local
component

(127) $\mathcal{S}_{F_{v}}(\langle e_{2},e_{4}\rangle\otimes
V_{B})\twoheadrightarrow\Pi^{+}(\pi_{1,v},\pi_{2,v})^{\vee}\otimes\left(\pi_{1,v}^{B}\boxtimes\pi_{2,v}^{B}\right)$

of (126). Now, by local Howe duality for the disconnected similitude group
$\operatorname{GO}(V_{B})$, the local theta lift
$\Theta_{W_{4},V_{B}}(\Pi^{+}(\pi_{1,v},\pi_{2,v})$ is irreducible when viewed
as a representation of $\operatorname{GO}(V_{B})(F_{v})$ [13, 28]. Hence

$\Theta_{W_{4},V_{B}}(\Pi^{+}(\pi_{1,v},\pi_{2,v})=\pi_{1,v}^{B}\boxtimes\pi_{2,v}^{B}\oplus\pi_{2,v}^{B}\boxtimes\pi_{1,v}^{B},$

and the map (127) factors as

$\mathcal{S}_{F_{v}}(\langle e_{2},e_{4}\rangle\otimes
V_{B})\twoheadrightarrow\Pi^{+}(\pi_{1,v},\pi_{2,v})^{\vee}\otimes\left(\pi_{1,v}^{B}\boxtimes\pi_{2,v}^{B}\oplus\pi_{2,v}^{B}\boxtimes\pi_{1,v}^{B}\right)\twoheadrightarrow\Pi^{+}(\pi_{1,v},\pi_{2,v})^{\vee}\otimes\left(\pi_{1,v}^{B}\boxtimes\pi_{2,v}^{B}\right).$

We now note that $\ell(\varphi_{m_{v}}^{\epsilon_{v}})$ is a harmonic in the
sense of [13] by Proposition 7.1.3, and generates a $U(2)$-type that appears
in $\Pi^{+}(\pi_{1,v},\pi_{2,v})^{\vee}.$ It follows from [13] that its image
under

$\mathcal{S}_{F_{v}}(\langle e_{2},e_{4}\rangle\otimes
V_{B})\twoheadrightarrow\Pi^{+}(\pi_{1,v},\pi_{2,v})^{\vee}\otimes\left(\pi_{1,v}^{B}\boxtimes\pi_{2,v}^{B}\oplus\pi_{2,v}^{B}\boxtimes\pi_{1,v}^{B}\right)$

is nontrivial. However, $\ell(\varphi_{m_{v}}^{\epsilon_{v}})$ generates the
$L$-type $\chi_{-(m_{v}+2),\epsilon_{v}m_{v}}^{\vee}$, which does not appear
in $\pi_{2,v}^{B}\boxtimes\pi_{1,v}^{B}$. It follows that
$\ell(\varphi_{m_{v}}^{\epsilon_{v}})$ has nonzero image under (127). This
proves (1). The proof of (2) is analogous. ∎

Finally we come to the main result of this section:

######  Lemma 9.3.2.

There exists an automorphic representation $\pi$ of
$PB(\mathbb{A}_{F})^{\times}$ of weight $2\bm{m}+2$, along with vectors

$\alpha\in(\Theta(\pi\boxtimes\mathbbm{1})\otimes\bm{\sigma}_{\bm{m}})^{\bm{K}_{3}},\;\;\beta\in(\Pi_{S}(\pi_{1}\otimes\pi_{2})\otimes\bm{\tau}^{\bm{\epsilon}}_{\bm{m}})^{\bm{K}_{2}},\;\;\gamma\in(\pi_{2}^{\vee}\otimes\bm{\chi}^{\vee}_{-\bm{\epsilon}\bm{m}})^{\bm{K}_{1}},$

such that:

$\widetilde{\mathcal{P}}_{S,\pi_{1},\pi_{2},\pi}^{\bm{\epsilon}}(\alpha,\beta,\gamma)=\int_{[\widetilde{H}]}\ell(\alpha(g,g^{\prime})\otimes\beta(g)\otimes\gamma(g^{\prime}))\mathop{}\\!\mathrm{d}(g,g^{\prime})\neq
0.$

###### Proof.

First, fix newforms

$f_{1}\in\pi_{1}^{B},\;\;f^{\bm{\epsilon}}_{2}\in\pi_{2}^{B},\;\;f_{2}^{\vee}\in(\pi_{2}^{B})^{\vee},\;\;(f_{2}^{\bm{\epsilon}})^{\vee}\in(\pi_{2}^{B})^{\vee}$

of weights $\bm{m}+2,$ $\bm{\epsilon}\bm{m}$, $\bm{m}$, and
$-\bm{\epsilon}\bm{m}$, respectively. Then Proposition 9.3.1 implies that we
may choose vectors

$\beta\in\left(\Pi_{S}(\pi_{1},\pi_{2})\otimes\bm{\tau}^{\bm{\epsilon}}_{\bm{m}}\right)^{\bm{K}_{2}},\;\;\gamma\in(\pi_{2}^{\vee}\otimes\bm{\chi}^{\vee}_{-\bm{\epsilon}\bm{m}})^{\bm{K}_{1}}$

and Schwartz functions

$\phi_{1,f}\in\mathcal{S}_{\mathbb{A}_{F,f}}(\langle e_{2},e_{4}\rangle\otimes
V_{B}),\;\;\phi_{2,f}\in\mathcal{S}_{\mathbb{A}_{F,f}}(\langle
e_{6}\rangle\otimes V_{B})$

such that:

(128)
$\theta_{\phi_{1,f}\otimes\bm{\varphi}^{-\bm{\epsilon}}_{\bm{m}}}=f_{1}\otimes
f_{2}^{\bm{\epsilon}};\;\;\;\theta_{\phi_{2,f}\otimes\bm{\varphi}^{-\bm{\epsilon}}_{\bm{m}}}(\gamma)=f_{2}^{\vee}\otimes(f_{2}^{\bm{\epsilon}})^{\vee}.$

Now, the automorphic function $g\mapsto f_{1}(g)\cdot f_{2}^{\vee}(g)$
corresponds to a Hilbert modular form on $B^{\times}$ of weight $2\bm{m}+2$
and trivial central character. We may therefore choose some automorphic
representation $\pi$ of $PB(\mathbb{A}_{F})^{\times}$ of weight $2\bm{m}+2,$
with a vector $\alpha_{0}$ of weight $-(2\bm{m}+2)$, such that

$\int_{[PB^{\times}]}\alpha_{0}(g)f_{1}(g)f_{2}^{\vee}(g)\mathop{}\\!\mathrm{d}g\neq
0.$

Now, we turn $\alpha_{0}$ into an automorphic form $\alpha$ on
$\operatorname{PGSO}(V_{B})(\mathbb{A}_{F})$ by setting
$\alpha(\bm{p}_{Z}(g_{1},g_{2}))=\alpha_{0}(g_{1})$. It is clear that $\alpha$
is a vector in
$\left((\pi\boxtimes\mathbbm{1})\otimes\bm{\chi}_{2\bm{m}+2,0}\right)^{\bm{L}}$.
Then Proposition 9.2.4 allows us to compute:

$\widetilde{\mathcal{P}}_{S,\pi_{1},\pi_{2},\pi}(\theta_{\phi_{1,f}\otimes\phi_{2,f}\otimes\widetilde{\bm{\varphi}}_{\bm{m}}}(\alpha),\beta,\gamma)=\left(\int_{[PB^{\times}]}\alpha_{0}(g)f_{1}(g)f_{2}^{\vee}(g)\mathop{}\\!\mathrm{d}g\right)\cdot\left(\int_{[PB^{\times}]}f_{2}^{\bm{\epsilon}}(g)(f_{2}^{\bm{\epsilon}})^{\vee}(g)\mathop{}\\!\mathrm{d}g\right)\neq
0.$

∎

##  10\. Proof of main result: Hodge classes in the non-generic case

###  10.1. Construction

#### 10.1.1.

Consider the inclusions of Shimura varieties:

(129)
$S(\operatorname{\mathbf{GSp}}_{6})\xleftarrow{\iota_{1}}S(\widetilde{\bm{H}})\xrightarrow{\iota_{2}}S(\operatorname{\mathbf{GSp}}_{4})\times
S(\operatorname{\mathbf{GL}}_{2}),$

where

$\widetilde{H}\coloneqq\operatorname{GSp}_{4}\times_{\mathbb{G}_{m}}\operatorname{GL}_{2}\subset\operatorname{GSp}_{6}.$

Note that $\iota_{1}^{\ast}\mathcal{V}_{(\bm{m}-2,\bm{m}-2,0)}$ contains
$\iota_{2}^{\ast}\mathcal{W}_{\bm{m}}$ as a direct factor with multiplicity
one. Since $\iota_{2}$ is an open and closed embedding at sufficiently small
level, one obtains from (129) a map

(130)
$H^{i}(S(\operatorname{\mathbf{GSp}}_{6}),{\mathcal{V}}_{(\bm{m}-2,\bm{m}-2,0)})\to
H^{i}(S(\operatorname{\mathbf{GSp}}_{4})\times
S(\operatorname{\mathbf{GL}}_{2}),\mathcal{W}_{\bm{m}}).$

The Hodge classes we construct will be the images of classes on
$S(\operatorname{\mathbf{GSp}}_{6})$ under the map (130).

#### 10.1.2.

Let $\pi_{1},\pi_{2},$ and $\Pi_{S_{f}}$ be as in (5.3.1), where $|S_{f}|\geq
2$ is even. We let $B$ be the unique quaternion algebra over $F$ which is
ramified at $S_{f}$ and split at all archimedean places. For any finite set
$\Sigma\supset S_{f}$ of places of $F$, including all infinite ones, we
consider the unramified Hecke algebra with $\mathbb{Q}$-coefficients:

(131)
$\widetilde{\mathbb{T}}^{\Sigma}=\otimes_{v\not\in\Sigma}\mathcal{H}(\operatorname{GSp}_{6}(F_{v}),\operatorname{GSp}_{6}(\mathcal{O}_{v})).$

For an auxiliary automorphic representation $\pi$ of $PB^{\times}$ which is
tempered, unramified outside of $\Sigma$, and of weight $2\bm{m}+2,$ the Hecke
action on $\Theta(\pi\boxtimes\mathbbm{1})$ defines a maximal ideal
$I^{\Sigma}\subset\widetilde{\mathbb{T}}^{\Sigma}$.

######  Definition 10.1.3.

Fix $\pi$ and $\Sigma$ as above, a sufficiently small compact open subgroup
$K=\prod K_{v}\subset\operatorname{GSp}_{6}(\mathbb{A}_{F_{f}})$ such that
$K_{v}=\operatorname{GSp}_{6}(\mathcal{O}_{v})$ for $v\not\in\Sigma$, and a
coefficient field $E\supset\mathbb{Q}(\bm{m})$ over which $\Pi_{S_{f}}$ is
defined. Then we define

$\operatorname{Hdg}_{E}(\pi,K,\Sigma)\subset
H^{4d}(S(\operatorname{\mathbf{GSp}}_{4})\times
S(\operatorname{\mathbf{GL}}_{2}),\mathcal{W}_{\bm{m},E})(2d)\left[\Pi^{\vee}_{S_{f}}\boxtimes\pi_{2}\right]$

to be the image of the Tate twist of the composite map

$\begin{split}IH^{4d}(S_{K}(\operatorname{\mathbf{GSp}}_{6})^{\ast},\mathcal{V}_{(\bm{m}-2,\bm{m}-2,0),E})[I^{\Sigma}]\to
H^{4d}(S(\operatorname{\mathbf{GSp}}_{6}),\mathcal{V}_{(\bm{m}-2,\bm{m}-2,0),E})\xrightarrow{(\ref{EQ:
back from big Shimura righty
cohomology})}H^{4d}(S(\operatorname{\mathbf{GSp}}_{4})\times
S(\operatorname{\mathbf{GL}}_{2}),\mathcal{W}_{\bm{m},E})\\\
\twoheadrightarrow H_{!}^{4d}(S(\operatorname{\mathbf{GSp}}_{4})\times
S(\operatorname{\mathbf{GL}}_{2}),\mathcal{W}_{\bm{m},E})\left[\Pi_{S_{f}}^{\vee}\boxtimes\pi_{2}\right].\end{split}$

######  Lemma 10.1.4.

Any $\xi\in\operatorname{Hdg}_{E}(\pi,K,\Sigma)$ is a Hodge class of weight
$(0,0)$. Moreover, the image of $\xi$ in

$H_{\rm{\acute{e}t},!}^{4d}\left((S(\operatorname{\mathbf{GSp}}_{4})\times
S(\operatorname{\mathbf{GL}}_{2})_{\overline{\mathbb{Q}}},\mathcal{W}_{\bm{m},E_{\lambda}}\right)(2d)\left[\Pi_{S_{f}}^{\vee}\boxtimes\pi_{2,f}\right]$

is $\operatorname{Gal}(\overline{\mathbb{Q}}/F^{c})$-invariant, where
$\lambda$ is any finite prime of $E$.

###### Proof.

Consider the following commutative diagram:

${H_{(2)}^{4d}(S_{K}(\operatorname{\mathbf{GSp}}_{6}),\mathcal{V}_{(\bm{m}-2,\bm{m}-2,0),\mathbb{C}})(2d)[I^{\Sigma}]}$${H^{4d}(S_{K}(\operatorname{\mathbf{GSp}}_{6}),\mathcal{V}_{(\bm{m}-2,\bm{m}-2,0),\mathbb{C}})(2d)}$${H_{(2)}^{4d}(S(\operatorname{\mathbf{GSp}}_{4})\times
S(\operatorname{\mathbf{GL}}_{2}),\mathcal{W}_{\bm{m},\mathbb{C}})(2d)\left[\Pi_{S_{f}}^{\vee}\boxtimes\pi_{2}\right]}$${H_{!}^{4d}(S(\operatorname{\mathbf{GSp}}_{4})\times
S(\operatorname{\mathbf{GL}}_{2}),\mathcal{W}_{\bm{m},\mathbb{C}})(2d)\left[\Pi^{\vee}_{S_{f}}\boxtimes\pi_{2}\right].}$$\scriptstyle{\sim}$

The left and bottom arrows in this diagram are maps of pure Hodge structures,
and by construction
$\operatorname{Hdg}_{E}(\pi,K,\Sigma)\otimes_{E}\mathbb{C}$ is the image of
the composite from the top left to the bottom right. We wish to show that
every element of
$H_{(2)}^{4d}(S_{K}(\operatorname{\mathbf{GSp}}_{6}),\mathcal{V}_{(\bm{m}-2,\bm{m}-2,0),\mathbb{C}})(2d)[I^{\Sigma}]$
has weight $(0,0)$. If $\widetilde{\Pi}$ is an automorphic representation of
$\operatorname{GSp}_{6}(\mathbb{A}_{F})$ such that
$\widetilde{\Pi}^{K}_{f}\neq 0$ is annihilated by $I^{\Sigma}$, then there
exists a Galois element
$\sigma\in\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ such that
$\widetilde{\Pi}$ is nearly equivalent to a constituent of
$\Theta(\pi\boxtimes\mathbbm{1})^{\sigma}$. Since the spherical theta
correspondence is defined over $\mathbb{Q}$, it follows that $\widetilde{\Pi}$
is nearly equivalent to a constituent of
$\Theta(\pi^{\sigma}\boxtimes\mathbbm{1})$. Thus, by Lemma 8.4.3,
$\operatorname{Hdg}_{E}(\pi,K,\Sigma)$ consists of hodge classes of weight
$(0,0)$.

The Galois invariance is similar, replacing $\mathbb{C}$ with
$\overline{\mathbb{Q}}_{\ell}$ and $L^{2}$ cohomology with absolute
intersection cohomology in the diagram above. ∎

###  10.2. Nonvanishing

#### 10.2.1.

To test the non-degeneracy of the subspace
$\operatorname{Hdg}_{(}\pi,K,\Sigma)$, we will use the following proposition.

######  Proposition 10.2.2.

Let $\Pi=\Pi_{S},$ where $S=S_{f}\sqcup S_{\infty}$, and choose an auxiliary
$\pi$ as above. Suppose given

$\alpha\in\left(\Theta(\pi\boxtimes\mathbbm{1})\otimes\bm{\sigma}_{\bm{m}}\right)^{{\bm{K}}_{3}}.$

  1. (1)

Fix choices of signs $\bm{\epsilon}=\left\\{\epsilon_{v}\right\\}_{v|\infty}$
and $\bm{\epsilon}^{\prime}=\left\\{\epsilon_{v}\right\\}_{v|\infty}$. Then:

$\langle\iota_{2,\ast}\circ\iota^{\ast}_{1}(\operatorname{cl}(\alpha)),\operatorname{cl}_{S_{\infty}}^{\bm{\epsilon}}(\beta)\boxtimes\operatorname{cl}^{\prime}_{\bm{\epsilon}^{\prime}}(\gamma)=\begin{cases}\widetilde{\mathcal{P}}_{S,\pi_{1},\pi_{2},\pi}(\alpha,\beta,\gamma),&\text{if
}S_{\infty}=\emptyset\text{ and }\bm{\epsilon}=\bm{\epsilon}^{\prime};\\\
0,&\text{otherwise.}\end{cases}$

  2. (2)

After choosing isomorphisms

$\Pi_{S_{f}}\simeq\left(\Pi\otimes\bm{\tau}_{\bm{m}}^{\bm{\epsilon}}\right)^{\bm{K}_{3}},\;\;\pi_{2,f}^{\vee}\simeq(\pi_{2}^{\vee}\otimes\bm{\chi}_{-\bm{\epsilon}\bm{m}})^{\bm{K}_{1}},$

the composite maps

$\Pi_{S_{f}}\otimes\pi^{\vee}_{2,f}\xrightarrow{\operatorname{cl}^{\bm{\epsilon}}\otimes\operatorname{cl}^{-\bm{\epsilon}}}H^{4d}_{(2)}(S(\operatorname{\mathbf{GSp}}_{4})\times
S(\operatorname{\mathbf{GL}}_{2}),\mathcal{W}_{\bm{m},\mathbb{C}})\xrightarrow{\langle\iota_{2,\ast}\iota_{1}^{\ast}\zeta,\cdot\rangle}\mathbb{C}$

are independent of $\bm{\epsilon}$ up to a scalar.

###### Proof.

The proof is essentially identical to Proposition 7.2.4. The only new
ingredient is the calculation of the
$(\bm{K}_{2}\times\bm{K}_{1})_{0}$-equivariant composite:

(132)
$\begin{split}\bm{\sigma}_{\bm{m}}\otimes\bm{\tau}_{\bm{m},S_{\infty}}^{\bm{\epsilon}}\otimes\bm{\chi}_{-\bm{\epsilon}^{\prime}\bm{m}}^{\vee}\to\wedge^{\bm{2},\bm{2}}\mathfrak{p}^{\ast}_{\operatorname{GSp}_{6}}\otimes
V_{(\bm{m}-2,\bm{m}-2,0),\mathbb{C}}\otimes\wedge^{\bm{p}(\bm{\epsilon},S_{\infty}),\bm{q}(\bm{\epsilon},S_{\infty})}\mathfrak{p}^{\ast}_{\operatorname{GSp}_{4}}\otimes
V_{(\bm{m}-2,0),\mathbb{C}}\\\
\otimes\wedge^{1-\bm{p}(\bm{\epsilon}^{\prime}),1-\bm{q}(\bm{\epsilon}^{\prime})}\mathfrak{p}^{\ast}_{\operatorname{GL}_{2}}\otimes
V_{\bm{m}-2,\mathbb{C}}^{\vee}\to\wedge^{3+\bm{p}(\bm{\epsilon},S_{\infty})-\bm{p}(\bm{\epsilon}^{\prime}),3+\bm{q}(\bm{\epsilon},S_{\infty})-\bm{q}(\bm{\epsilon}^{\prime})}\mathfrak{p}^{\ast}_{\widetilde{H}}\xrightarrow{\mathbbm{1}_{\widetilde{H}}}\mathbb{C},\end{split}$

which is automatically trivial unless $S_{\infty}=\emptyset$ and
$\bm{\epsilon}=\bm{\epsilon}^{\prime}$, in which case one can check that it is
not trivial. ∎

######  Theorem 10.2.3.

Let $\pi_{1}$ and $\pi_{2}$ be cuspidal automorphic representations of
$\operatorname{GL}_{2}(\mathbb{A})$ of weights $\bm{m}+2$ and $\bm{m}$,
respectively, where $\bm{m}=(m_{v})_{v|\infty}$ for positive integers $m_{v}$.
Assume that the central characters of $\pi_{1}$ and $\pi_{2}$ agree and have
infinity type $\omega_{\bm{m}}$. Let $\Pi_{S_{f}}$ be as in (5.3.1) for a set
$S_{f}$ of finite places of $F$ such that $|S_{f}|\geq 2$ is even, and choose
a coefficient field $E\supset\mathbb{Q}(\bm{m})$ over which $\pi_{i}$ and
$\Pi_{S_{f}}$ are defined. Then there exists a triple $(\pi,K,\Sigma)$ as in
Definition 10.1.3 and a Hodge class

$\xi\in\operatorname{Hdg}_{E}(\pi,K,\Sigma)\subset
H^{4d}_{!}(S(\operatorname{\mathbf{GSp}}_{4})\times
S(\operatorname{\mathbf{GL}}_{2}),\mathcal{W}_{\bm{m},E})(2d)\left[\Pi_{S_{f}}\boxtimes\pi_{2}^{\vee}\right]$

such that the induced map

$\xi_{\ast}:H_{!}^{3d}(S(\operatorname{\mathbf{GSp}}_{4}),\mathcal{V}_{\bm{m},E})(d)[\Pi_{S_{f}}]\to
H_{!}^{d}(S(\operatorname{\mathbf{GL}}_{2}),\mathcal{V}^{\prime}_{\bm{m},E})[\pi_{2,f}]$

is of the form

$\Pi_{S_{f}}^{E}\otimes
H_{!}^{3d}(S(\operatorname{\mathbf{GSp}}_{4}),\mathcal{V}_{\bm{m},E})_{\Pi_{S_{f}}}(d)\xrightarrow{\ell\otimes
s}\pi_{2,f}^{E}\otimes
H_{!}^{d}(S(\operatorname{\mathbf{GL}}_{2}),\mathcal{V}^{\prime}_{\bm{m},E})_{\pi_{2,f}},$

where $s$ is a surjection and $\ell$ is a nontrivial $E$-linear map. Moreover,
the image of $\xi$ in $\ell$-adic étale cohomology is
$\operatorname{Gal}(\overline{\mathbb{Q}}/F^{c})$-equivariant for all $\ell$.

###### Proof.

The same argument as for Theorem 7.2.5 implies $\xi_{\ast}$ is always a pure
tensor $\ell\otimes s$, and the map $s$ is always either trivial or a
surjection. Thus it suffices to show that there exists a _complex_ Hodge class
$\xi\in\operatorname{Hdg}_{\mathbb{C}}(\pi,K,\Sigma)$ with $\xi_{\ast}\neq 0.$
However, this is guaranteed by Lemma 9.3.2 and Proposition 10.2.2. ∎

## References

  * [1] Jeffrey Adams, Nicolás Arancibia, and Paul Mezo.  Equivalent definitions of arthur packets for real classical groups.  arXiv preprint arXiv:2108.05788, 2021.
  * [2] Jeffrey Adams and Joseph F Johnson.  Endoscopic groups and packets of non-tempered representations.  Compositio Mathematica, 64(3):271–309, 1987.
  * [3] Nicolás Arancibia.  Characteristic cycles, micro local packets and packets with cohomology.  arXiv preprint arXiv:1906.09214, 2019.
  * [4] James Arthur.  The Endoscopic classification of representations orthogonal and symplectic groups, volume 61.  American Mathematical Soc., 2013.
  * [5] Don Blasius and Jonathan Rogawski.  Galois representations for hilbert modular forms.  Bulletin (New Series) of the American Mathematical Society, 21(1):65–69, 1989.
  * [6] Gerd Faltings.  Endlichkeitssätze für abelsche varietäten über zahlkörpern.  Inventiones mathematicae, 73(3):349–366, 1983.
  * [7] Philipp Fleig, Henrik P. A. Gustafsson, Axel Kleinschmidt, and Daniel Persson.  Fourier Coefficients of Eisenstein Series on SL(2,A), page 155–169.  Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2018.
  * [8] Wee Teck Gan, Benedict H Gross, and Dipendra Prasad.  Branching laws for classical groups: the non-tempered case.  Compositio Mathematica, 156(11):2298–2367, 2020.
  * [9] Wee Teck Gan, Yannan Qiu, and Shuichiro Takeda.  The regularized siegel–weil formula (the second term identity) and the rallis inner product formula.  Inventiones mathematicae, 198(3):739–831, 2014.
  * [10] Günter Harder.  Cohomology of arithmetic groups.  preprint, 2019.
  * [11] Michael Harris and Stephen S Kudla.  Arithmetic automorphic forms for the nonholomorphic discrete series of gsp (2).  Duke Mathematical Journal, 66(1):59–121, 1992.
  * [12] Haruzo Hida.  Modules of congruence of hecke algebras and l-functions associated with cusp forms.  American Journal of Mathematics, 110(2):323–382, 1988.
  * [13] Roger Howe.  Transcending classical invariant theory.  Journal of the American Mathematical Society, 2(3):535–552, 1989\.
  * [14] Atsushi Ichino and Kartik Prasanna.  Hodge classes and the jacquet-langlands correspondence.  arXiv preprint arXiv:1806.10563, 2018.
  * [15] Herv´e Jacquet.  Automorphic forms on $GL(2)$: Part II.  Springer, 1972.
  * [16] Robert E Kottwitz.  Shimura varieties and $\lambda$-adic representations.  Automorphic forms, Shimura varieties, and L-functions, 1:161–210, 1990.
  * [17] Stephen S Kudla and Stephen Rallis.  Degenerate principal series and invariant distributions.  Israel Journal of Mathematics, 69(1):25–45, 1990.
  * [18] Stephen S Kudla and Stephen Rallis.  A regularized siegel-weil formula: the first term identity.  Annals of Mathematics, pages 1–80, 1994.
  * [19] Si Ying Lee.  Eichler-shimura relations for shimura varieties of hodge type.  arXiv preprint arXiv:2006.11745, 2020.
  * [20] Si Ying Lee.  Semisimplicity of$\backslash$’etale cohomology of certain shimura varieties.  arXiv preprint arXiv:2206.07283, 2022.
  * [21] Francesco Lemma.  Algebraic cycles and residues of degree 8 l-functions of gsp (4)$\times$ gl (2).  International Mathematics Research Notices, 2020(12):3614–3648, 2020.
  * [22] Eduard Looijenga.  $l^{2}$-cohomology of locally symmetric varieties.  Compositio Mathematica, 67(1):3–20, 1988.
  * [23] Colette Moeglin and Jean-Loup Waldspurger.  Spectral decomposition and Eisenstein series: a paraphrase of the scriptures.  Number 113. Cambridge University Press, 1995.
  * [24] Jan Nekovar and Anthony J Scholl.  Introduction to plectic cohomology.  Advances in the theory of automorphic forms and their L-functions, 664:321–337, 2016.
  * [25] Masami Ohta.  On the zeta function of an abelian scheme over the shimura curve.  Japanese journal of mathematics. New series, 9(1):1–25, 1983.
  * [26] Stephen Rallis.  Langlands’ functoriality and the weil representation.  American Journal of Mathematics, 104(3):469–515, 1982.
  * [27] Stephen Rallis.  On the howe duality conjecture.  Compositio Mathematica, 51(3):333–399, 1984.
  * [28] Brooks Roberts.  The theta correspondence for similitudes.  Israel Journal of Mathematics, 94(1):285–317, 1996.
  * [29] Brooks Roberts.  Global l-packets for gsp (2) and theta lifts.  Doc. Math, 6:247–314, 2001.
  * [30] Leslie Saper and Mark Stern.  L2-cohomology of arithmetic varieties.  Annals of Mathematics, pages 1–69, 1990.
  * [31] F Shahidi.  Amer. j. math.  On certain L-functions, 103:297–355, 1981.
  * [32] Richard Taylor.  On thel-adic cohomology of siegel threefolds.  Inventiones mathematicae, 114(1):289–310, 1993.
  * [33] David A Vogan and Gregg J Zuckerman.  Unitary representations with non-zero cohomology.  Compositio Mathematica, 53(1):51–90, 1984.
  * [34] J-L Waldspurger.  Quelques propriétés arithmétiques de certaines formes automorphes sur gl $(2)$.  Compositio Mathematica, 54(2):121–171, 1985.
  * [35] Rainer Weissauer.  Four dimensional galois representations.  Astérisque, 302:67–150, 2005.
  * [36] Rainer Weissauer.  Endoscopy for GSp (4) and the cohomology of Siegel modular threefolds.  Springer, 2009.
  * [37] Steven Zucker.  Locally homogeneous variations of hodge structure.  Enseign. Math.(2), 27(3-4):243–276, 1981.