c}(Y^{(2)}_{\mathcal{K}_{2}},{\mathcal{L}}^{(2)}(\boldsymbol{n}_{\boldsymbol{\lambda}};{\mathbf{C}}))[\pm,\pi^{(2)}_{\rm
fin}]\cap H^{1}_{\rm
c}(Y^{(2)}_{\mathcal{K}_{2}},{\mathcal{L}}^{(2)}(\boldsymbol{n}_{\boldsymbol{\lambda}};E_{\pi^{(2)}})).$

For each signature $\pm$, fix a generator $\eta^{\pm}_{\pi^{(2)}}$ of the
$1$-dimensional $E_{\pi^{(2)}}$-space $H^{1}_{\rm
c}(Y^{(2)}_{\mathcal{K}_{2}},{\mathcal{L}}^{(2)}(\boldsymbol{n}_{\boldsymbol{\lambda}};E_{\pi^{(2)}}))[\pm,\pi^{(2)}_{\rm
fin}]$. Then we define complex numbers
$\Omega^{\pm}_{\pi^{(2)}}\in{\mathbf{C}}^{\times}$ so that
$\delta^{\pm}_{\pi^{(2)}}=\Omega^{\pm}_{\pi^{(2)}}\eta^{\pm}_{\pi^{(2)}}\;(\text{double
sign in the same order})$. Note that each of $\Omega^{\pm}_{\pi^{(2)}}$ is
uniquely determined from $\pi^{(2)}$ up to multiplication of elements in
$E^{\times}_{\pi^{(2)}}$. We call $\Omega_{\pi^{(2)}}$ the periods of
$\pi^{(2)}$.

Next let us discuss the period of $\pi^{(3)}$. For each finite place $v$ of
${\mathbf{Q}}$, there exists a unique function
$W^{\mathrm{ess}}_{\pi^{(3)},v}$ in
${\mathcal{W}}(\pi^{(3)}_{v},\psi_{v})^{{\mathcal{K}}_{3}({\mathfrak{N}})_{v}}$
satisfying

$\displaystyle\int_{{\rm N}_{2}({\mathbf{Q}}_{v})\backslash{\rm
GL}_{2}({\mathbf{Q}}_{v})}W^{\mathrm{ess}}_{\pi^{(3)},v}\left(\iota(g)\right)W^{\circ}_{\pi^{(2)},v}(g)|\det
g|_{v}^{s-\frac{1}{2}}{\rm d}g=L(s,\pi^{(3)}_{v}\times\pi^{(2)}_{v}),$

where $\mathrm{N}_{2}$ denotes the uniponent radical of the upper triangular
Borel subgroup of $\mathrm{GL}_{2}$. The Whittaker function
$W^{\mathrm{ess}}_{\pi^{(3)},v}$ is called the essential vector of
$\pi^{(3)}_{v}$, whose existence and uniqueness is verified in [JPSS81, page
208, 211, Théor$\grave{\rm e}$m]. We also define archimedean Whittaker
functions
$W^{\alpha}_{\pi^{(3)},\infty}\in{\mathcal{W}}(\pi^{(3)}_{\infty},\psi_{\infty})$
for $-\lambda_{3}\leq\alpha\leq\lambda_{3}$ according to [HIM, Sections 4.6
and 9.4]. Recall that $\pi^{(3)}_{\infty}$ has the minimal
$\mathrm{O}_{3}(\mathbf{R})$-type $\tau^{(3)}_{\mathbf{\lambda}}$ for
$\boldsymbol{\lambda}^{(3)}=(\lambda_{3},\delta)=(l_{3}+1,\delta)$. Let
$\varphi^{(3)}_{+}\colon
V^{(3)}_{\boldsymbol{\lambda}}\to{\mathcal{W}}(\pi^{(3)}_{\infty},\psi_{\infty})$
be a $K_{3}$-homomorphism satisfying the following formula on the radial parts
for each $z_{\boldsymbol{a}}:=z^{a_{1}}_{1}z^{a_{2}}_{2}z^{a_{3}}_{3}\in
V^{(3)}_{\lambda}$ (see [HIM, Theorem 4.21 and Section 9.4]):

$\displaystyle\varphi^{(3)}_{+}$ $\displaystyle(z_{\boldsymbol{a}})({\rm
diag}(y_{1}y_{2}y_{3},y_{2}y_{3},y_{3}))$
$\displaystyle=(-1)^{a_{1}}\sqrt{-1}^{a_{2}}\frac{y_{1}y_{2}(y_{2}y_{3})^{3\nu_{3}}}{(4\pi\sqrt{-1})^{2}}$
$\displaystyle\qquad\times\int_{t_{2}}\int_{t_{1}}\frac{\Gamma_{\mathbf{C}}(t_{1}+\nu_{3}+\frac{\lambda_{3}-1}{2})\Gamma_{\mathbf{R}}(t_{2}+\nu_{3}+a_{1})}{\Gamma_{\mathbf{R}}(t_{1}+t_{2}+a_{1}+a_{3})}\Gamma_{\mathbf{C}}\left(t_{2}-\nu_{3}+\frac{\lambda_{3}-1}{2}\right)\Gamma_{\mathbf{R}}(t_{2}-\nu_{3}+a_{3})y^{-t_{1}}_{1}y^{-t_{2}}_{2}\,{\rm
d}t_{1}{\rm d}t_{2}.$

The path integrals $\int_{t_{2}}$ and $\int_{t_{1}}$ are defined in a similar
way to the case of ${\rm GL}_{2}$. Then define $W^{\alpha}_{\pi^{(3)},\infty}$
to be $\varphi^{(3)}_{+}(v^{\boldsymbol{\lambda}}_{\alpha})$ for each
$-\lambda_{3}\leq\alpha\leq\lambda_{3}$, and set
$W^{\alpha}_{\pi^{(3)}}=W^{\alpha}_{\pi^{(3)},\infty}\otimes\bigotimes_{v\text{:finite}}W^{\mathrm{ess}}_{\pi^{(3)},v}$.
We define $f^{\boldsymbol{\lambda}}_{\alpha}$ as the cuspidal automorphic form
on ${\rm GL}_{3}({\mathbf{Q}}_{\mathbf{A}})$ corresponding to
$W^{\alpha}_{\pi^{(3)}}$, that is, $f^{\boldsymbol{\lambda}}_{\alpha}$ is
defined to be the inverse Fourier transform of $W^{\alpha}_{\pi^{(3)}}$. Then
we set
$\boldsymbol{f}=\begin{pmatrix}f^{\boldsymbol{\lambda}}_{\lambda_{3}}&f^{\boldsymbol{\lambda}}_{\lambda_{3}-1}&\dotsc&f^{\boldsymbol{\lambda}}_{-\lambda_{3}}\end{pmatrix}$
and consider its image $\delta^{(3)}(\boldsymbol{f})$ under the
Eichler–Shimura map. Note that it is indeed an element of the $\pi^{(3)}_{\rm
fin}$-isotypic component $H^{2}_{\rm
cusp}(Y^{(3)}_{\mathcal{K}_{3}},{\mathcal{L}}^{(3)}(\boldsymbol{w}_{\boldsymbol{\lambda}};{\mathbf{C}}))[\pi^{(3)}_{\rm
fin}]$ of $H^{2}_{\rm
cusp}(Y^{(3)}_{\mathcal{K}_{3}},{\mathcal{L}}^{(3)}(\boldsymbol{w}_{\boldsymbol{\lambda}};{\mathbf{C}}))$,
which is of dimension 1 over ${\mathbf{C}}$. Similarly to the $n=2$ case, the
space $H^{2}_{\rm
cusp}(Y^{(3)}_{\mathcal{K}_{3}},{\mathcal{L}}^{(3)}(\boldsymbol{w}_{\boldsymbol{\lambda}};{\mathbf{C}}))[\pi^{(3)}_{\rm
fin}]$ is defined over the field of rationality $E_{\pi^{(3)}}$ of
$\pi^{(3)}$, and its $E_{\pi^{(3)}}$-structure is given as

$\displaystyle H^{2}_{\rm
cusp}(Y^{(3)}_{\mathcal{K}_{3}},{\mathcal{L}}^{(3)}(\boldsymbol{w}_{\boldsymbol{\lambda}};E_{\pi^{(3)}}))[\pi^{(3)}_{\rm
fin}]=H^{2}_{\rm
cusp}(Y^{(3)}_{\mathcal{K}_{3}},{\mathcal{L}}^{(3)}(\boldsymbol{w}_{\boldsymbol{\lambda}};{\mathbf{C}}))[\pi^{(3)}_{\rm
fin}]\cap
H^{2}(Y^{(3)}_{\mathcal{K}_{3}},{\mathcal{L}}^{(3)}(\boldsymbol{w}_{\boldsymbol{\lambda}};E_{\pi^{(3)}})).$

Hence we can choose a generator $\eta_{\pi^{(3)}}$ of the $1$-dimensional
$E_{\pi^{(3)}}$-space $H^{2}_{\rm
cusp}(Y^{(3)}_{\mathcal{K}_{3}},{\mathcal{L}}^{(3)}(\boldsymbol{w}_{\boldsymbol{\lambda}};E_{\pi^{(3)}}))[\pi^{(3)}_{\rm
fin}]$, and the universal coefficient theorem yields that there exists a
complex number $\Omega_{\pi^{(3)}}\in{\mathbf{C}}^{\times}$ satisfying
$\delta^{(3)}(\boldsymbol{f})=\Omega_{\pi^{(3)}}\eta_{\pi^{(3)}}$. From its
construction, $\Omega_{\pi^{(3)}}$ is determined up to multiplication of
elements of $E^{\times}_{\pi^{(3)}}$. We call $\Omega_{\pi^{(3)}}$ the period
for $\pi^{(3)}$.

######  Remark 7.1.

We impose that $\mathcal{K}_{2}$ coincides with
$\mathrm{GL}_{2}(\widehat{\mathbf{Z}})$ in the construction of the periods,
but this condition seems too restrictive. In fact, Iwahori fixed vectors play
important roles in the construction of the $p$-adic $L$-functions for ${\rm
GL}_{n+1}\times{\rm GL}_{n}$ (for example, see [Jan]). However, carefully
observing use of essential vectors in [JPSS81], we do not think that it is so
easy to be obvious to give an explicit relation between local zeta integrals
and the $L$-factors at ramified places when $\mathcal{K}_{2}$ does not
coincide with $\mathrm{GL}_{2}(\widehat{\mathbf{Z}})$. Since the main theme of
this article is to give a precise cohomological interpretation of archimedean
zeta integrals, we postpone the study of local zeta integrals at ramified
places to the future work.

###  7.2. Main result

Let $\nabla^{\boldsymbol{n}_{m}}\iota^{*}{\rm p}^{\ast}_{3}\eta_{\pi^{(3)},x}$
(resp. ${\rm p}^{\ast}_{2}\eta^{\pm}_{\pi^{(2)},x}$) be the restriction of
$\nabla^{\boldsymbol{n}_{m}}\iota^{*}{\rm p}^{\ast}_{3}\eta_{\pi^{(3)}}$
(resp. ${\rm p}^{\ast}_{2}\eta^{\pm}_{\pi^{(2)}}$) to
${\mathcal{Y}}^{(2)}_{{\mathcal{K}}_{2},x}$ for $x\in{\rm
Cl}^{+}_{\mathbf{Q}}({\mathcal{K}}_{2})$. The aim of this paper is to evaluate
the following summation of the cohomological cup products

$\displaystyle I(m,\pi^{(3)},\pi^{(2)},\varphi):=\sum_{x\in{\rm
Cl}^{+}_{\mathbf{Q}}({\mathcal{K}}_{2})}\varphi(x){\rm
Tw}_{[\boldsymbol{n}_{m}]}\left(\nabla^{\boldsymbol{n}_{m}}\iota^{*}{\rm
p}^{\ast}_{3}\eta_{\pi^{(3)},x}\cup{\rm
p}^{\ast}_{2}\eta^{\pm}_{\pi^{(2)},x}\right).$

As we have already explained in Remark 7.1, we suppose that
${\mathcal{K}}_{2}={\rm GL}_{2}(\widehat{\mathbf{Z}})$. Hence if the conductor
of $\varphi$ is not trivial, $I(m,\pi^{(3)},\pi^{(2)},\varphi)$ vanishes.
Therefore we may assume that $\varphi$ is the trivial character
${\mathbf{1}}$. We write $I(m,\pi^{(3)},\pi^{(2)})$ (resp.
${\mathcal{Z}}(s;\boldsymbol{f},h^{\pm})$) for the cup product
$I(m,\pi^{(3)},\pi^{(2)},{\mathbf{1}})$ (resp. for the zeta integral
${\mathcal{Z}}(s;\boldsymbol{f},h^{\pm},\mathbf{1})$) for the sake of
simplicity.

Set $(-1)^{m+\delta+w_{\boldsymbol{\lambda}}+1}=\pm 1$. Then the constructions
of $\eta_{\pi^{(3)}}$ and $\eta^{\pm}_{\pi^{(2)}}$ imply the following
identity (see Proposition 6.3 and Corollary 6.4):

(7.1) $\displaystyle\begin{aligned}
I(m,\pi^{(3)},\pi^{(2)})=\frac{1}{\Omega_{\pi^{(3)}}\Omega^{\pm}_{\pi^{(2)}}}\cdot\sqrt{-1}^{\frac{l_{3}}{2}+2l_{2}-m-1}C(\boldsymbol{w}_{\boldsymbol{\lambda}},\boldsymbol{n}_{m}){\mathcal{Z}}(s;\boldsymbol{f},h^{+})|_{s=\frac{1}{2}+[\boldsymbol{n}_{m}]}.\end{aligned}$

The main theorem in this article is the evaluation of the integral in the
right-hand side of (7.1). The result is as follows:

######  Theorem 7.2.

Suppose that $(-1)^{m+\delta+\frac{l_{3}}{2}}=\pm 1$. Then we have

$\displaystyle
I(m,\pi^{(3)},\pi^{(2)})=(-1)^{\delta}\sqrt{-1}^{\frac{l_{3}}{2}-m+1}\binom{\frac{l_{3}}{2}-1}{m-\frac{l_{3}}{2}-1}\binom{\frac{l_{3}}{2}-1}{\frac{l_{3}}{2}+l_{2}-m}\frac{L(m,{\mathcal{M}}(\pi^{(3)}\times\pi^{(2)}))}{\Omega_{\pi^{(3)}}\Omega^{\pm}_{\pi^{(2)}}}$

If $(-1)^{m+\delta+\frac{l_{3}}{2}}\neq\pm 1$, then
$I(m,\pi^{(3)},\pi^{(2)})=0$.

Note that a standard argument yields that the zeta integral appearing in (7.1)
is unfolded to

$\displaystyle{\mathcal{Z}}(s;\boldsymbol{f},h^{+})=\int_{{\rm
N}_{2}({\mathbf{Q}}_{\mathbf{A}})\backslash{\rm
GL}_{2}({\mathbf{Q}}_{\mathbf{A}})}W^{-\lambda_{2}}_{\pi^{(3)}}(\iota(g))W^{+}_{\pi^{(2)}}(g)|\det
g|^{s-\frac{1}{2}}_{\mathbf{A}}{\rm d}g,$

and hence the proof of Theorem 7.2 is reduced to evaluation of the local zeta
integral above.

###### Proof..

We firstly recall the notation and settings; we have
$\pi^{(3)}_{\infty}\cong\mathrm{Ind}^{\mathrm{GL}_{3}(\mathbf{R})}_{\mathrm{P}_{2,1}(\mathbf{R})}(D_{\nu_{3},l_{3}}\boxtimes\chi_{\nu_{3}}^{\delta})$
and $\pi^{(2)}_{\infty}\cong D_{\nu_{2},l_{2}}$ where

$\displaystyle\nu_{2}$ $\displaystyle=-\frac{l_{2}}{2}+\frac{1}{2},$
$\displaystyle\nu_{3}$ $\displaystyle=-\frac{l_{3}}{2}+1$
$\displaystyle\boldsymbol{\lambda}^{(3)}$
$\displaystyle=(\lambda_{3},\delta),\qquad\lambda_{3}=l_{3}+1,$
$\displaystyle\boldsymbol{\lambda}^{(2)}$
$\displaystyle=(\lambda_{2},0),\qquad\lambda_{2}=l_{2}+1,$
$\displaystyle\boldsymbol{n}_{\boldsymbol{\lambda}}$
$\displaystyle=(n_{\boldsymbol{\lambda}},0)=(l_{2}-1),$
$\displaystyle\boldsymbol{w}_{\boldsymbol{\lambda}}$
$\displaystyle=(w_{\boldsymbol{\lambda}},w_{\boldsymbol{\lambda}},w_{\boldsymbol{\lambda}})=\left(\dfrac{l_{3}}{2}-1,\dfrac{l_{3}}{2}-1,\dfrac{l_{3}}{2}-1\right),$
$\displaystyle\boldsymbol{n}_{m}$
$\displaystyle=(n_{m,1},n_{m,2})=(l_{2}-1,m-l_{2}-1),$
$\displaystyle[\boldsymbol{n}_{m}]$ $\displaystyle=m-2.$

See (4.1), (5.1), (5.2) and (6.5) for details. In particular, we see that
$(-1)^{m+\delta+w_{\boldsymbol{\lambda}}+1}=(-1)^{m+\delta+\frac{l_{3}}{2}}$
holds and obtain the identity of the $L$-functions
$L(s,\mathcal{M}(\pi^{(3)}\times\pi^{(3)}))=L(s-\frac{3}{2},\pi^{(3)}\times\pi^{(2)})$;
see (4.4). Since $\frac{1}{2}+[\boldsymbol{n}_{m}]=m-\frac{3}{2}$, the
identity yields that

$\displaystyle
L(s,\pi^{(3)}\times\pi^{(2)})|_{s=\frac{1}{2}+[\boldsymbol{n}_{m}]}=L(m,{\mathcal{M}}(\pi^{(3)}\times\pi^{(2)}))$

holds. For $s\in{\mathbf{C}}$ with ${\rm Re}(s)\gg 0$, the unfolding of the
zeta integral yields that

$\displaystyle{\mathcal{Z}}(s;\boldsymbol{f},h^{+})$ $\displaystyle=\int_{{\rm
N}_{2}({\mathbf{R}})\backslash{\rm
GL}_{2}({\mathbf{R}})}W^{-\lambda_{2}}_{\pi^{(3)},\infty}\left(\iota(g)\right)W^{+}_{\pi^{(2)},\infty}(g)|\det
g|^{s-\frac{1}{2}}_{\infty}{\rm d}g_{\infty}$
$\displaystyle\qquad\quad\times\prod_{v\text{:finite}}\int_{{\rm
N}_{2}({\mathbf{Q}}_{v})\backslash{\rm
GL}_{2}({\mathbf{Q}}_{v})}W_{\pi^{(3)},v}\left(\iota(g)\right)W_{\pi^{(2)},v}(g)|\det
g|^{s-\frac{1}{2}}_{v}{\rm d}g_{v}.$

By the property of the essential vector (see [JPSS81, page 208,
Théor$\grave{\rm e}$m (ii)] for example), we find that

$\displaystyle\int_{{\rm N}_{2}({\mathbf{Q}}_{v})\backslash{\rm
GL}_{2}({\mathbf{Q}}_{v})}W_{\pi^{(3)},v}\left(\iota(g)\right)W_{\pi^{(2)},v}(g)|\det
g|^{[\boldsymbol{n}_{m}]}{\rm
d}g=L(s,\pi^{(3)}_{v}\times\pi^{(2)}_{v})|_{s=\frac{1}{2}+[\boldsymbol{n}_{m}]}=L_{v}(m,{\mathcal{M}}(\pi^{(3)}\times\pi^{(2)})).$

We compute the archimedean local integral. Recall the $K_{2}$-homomorphism
$\varphi^{(2)}_{-}\colon
V^{(2)}_{\boldsymbol{\lambda}^{(2)}}\to{\mathcal{W}}(\pi^{(2)}_{\infty},\psi^{-1})$
(resp. the $K_{3}$-homomorphism $\varphi^{(3)}_{+}\colon
V^{(3)}_{\boldsymbol{\lambda}^{(3)}}\to{\mathcal{W}}(\pi^{(3)}_{\infty},\psi)$)
introduced in Section 7.1. Also recall the $K_{2}$-equivariant homomorphisms
$\iota^{\boldsymbol{\lambda}^{(3)}}_{\boldsymbol{\lambda}^{(2)}}\colon
V_{\boldsymbol{\lambda}^{(2)}}^{(2)}\hookrightarrow
V^{(3)}_{\boldsymbol{\lambda}^{(3)}}$ and
$\mathrm{P}^{\boldsymbol{\lambda}^{(3)}}_{\boldsymbol{\lambda}^{(2)}}\colon
V^{(3)}_{\boldsymbol{\lambda}^{(3)}}\twoheadrightarrow
V_{\boldsymbol{\lambda}^{(2)}}^{(2)}$ introduced in Section 3.6, which we
abbreviate as $\iota_{\boldsymbol{\lambda}}$ and
$\mathrm{P}_{\boldsymbol{\lambda}}$ to lighten the notation (see also [HIM,
Section 9.5]). Define a ${\mathbf{C}}$-bilinear pairing
$\langle\cdot,\cdot\rangle:V^{(2)}_{\boldsymbol{\lambda}^{(2)}}\otimes_{\mathbf{C}}V^{(2)}_{\boldsymbol{\lambda}^{(2)}}\to{\mathbf{C}}$
by

$\displaystyle\langle
v^{\boldsymbol{\lambda}^{(2)}}_{q},v^{\boldsymbol{\lambda}^{(2)}}_{q^{\prime}}\rangle=\delta_{0,q+q^{\prime}}$

for each $q,q^{\prime}\in\\{\pm\lambda_{2}\\}$; see [HIM, (9.2)] for details.
Here $\delta_{\ast,\ast}$ denotes Kronecker’s delta symbol. Then due to [HIM,
Corollary 9.7], we have

$\displaystyle\int_{{\rm N}_{2}({\mathbf{R}})\backslash{\rm
GL}_{2}({\mathbf{R}})}$ $\displaystyle
W^{-\lambda_{2}}_{\pi^{(3)},\infty}\left(\iota(g)\right)W^{+}_{\pi^{(2)},\infty}\left(g\right)\lvert\det
g\rvert^{s-\frac{1}{2}}_{\infty}{\rm d}g_{\infty}$ $\displaystyle=\int_{{\rm
N}_{2}({\mathbf{R}})\backslash{\rm
GL}_{2}({\mathbf{R}})}\varphi^{(3)}_{+}(v^{\boldsymbol{\lambda}^{(3)}}_{-\lambda_{3}})(\iota(g))\varphi^{(2)}_{-}(v^{\boldsymbol{\lambda}^{(2)}}_{\lambda_{2}})(g)|\det
g|^{s-\frac{1}{2}}_{\infty}{\rm d}g_{\infty}$
$\displaystyle=(-1)^{\lambda_{2}}\langle\mathrm{P}_{\boldsymbol{\lambda}}\left(v^{\boldsymbol{\lambda}^{(3)}}_{-\lambda_{2}}\right),v^{\boldsymbol{\lambda}^{(2)}}_{\lambda_{2}}\rangle\cdot
L(s,\pi^{(3)}_{\infty}\times\pi^{(2)}_{\infty}).$

Since
$\iota_{\boldsymbol{\lambda}}((-1)^{\delta}v^{\boldsymbol{\lambda}^{(2)}}_{-\lambda_{2}})=v^{\boldsymbol{\lambda}^{(3)}}_{-\lambda_{2}}$
holds, we have the identity

$\displaystyle\langle\mathrm{P}_{\boldsymbol{\lambda}}\left(v^{\boldsymbol{\lambda}^{(3)}}_{-\lambda_{2}}\right),v^{\boldsymbol{\lambda}^{(2)}}_{\lambda_{2}}\rangle=\langle(-1)^{\delta}v^{\boldsymbol{\lambda}^{(2)}}_{-\lambda_{2}},v^{\boldsymbol{\lambda}^{(2)}}_{\lambda_{2}}\rangle=(-1)^{\delta}.$

Hence, using this identity, we find that

$\displaystyle\sqrt{-1}^{n_{m,1}-n_{m,2}+w_{\boldsymbol{\lambda}}}C(\boldsymbol{w}_{\boldsymbol{\lambda}},\boldsymbol{n}_{m})$
$\displaystyle\int_{{\rm N}_{2}({\mathbf{R}})\backslash{\rm
GL}_{2}({\mathbf{R}})}W^{-n_{1}-2}_{\pi^{(3)},\infty}\left(\iota(g)\right)W^{+}_{\pi^{(2)},\infty}\left(g\right)|\det
g|^{s-\frac{1}{2}}_{\infty}{\rm d}g_{\infty}$
$\displaystyle=\sqrt{-1}^{n_{m,1}-n_{m,2}+w_{\boldsymbol{\lambda}}}C(\boldsymbol{w}_{\boldsymbol{\lambda}},\boldsymbol{n}_{m})\cdot(-1)^{\lambda_{2}+\delta}\cdot
L(s,\pi^{(3)}_{\infty}\times\pi^{(2)}_{\infty})$
$\displaystyle=(-1)^{\delta+l_{2}+1}\sqrt{-1}^{\frac{l_{3}}{2}+2l_{2}-m-1}\binom{\frac{l_{3}}{2}-1}{m-\frac{l_{3}}{2}-1}\binom{\frac{l_{3}}{2}-1}{\frac{l_{3}}{2}+l_{2}-m}L(s,\pi^{(3)}_{\infty}\times\pi^{(2)}_{\infty}).$

This proves the statement (note that we have
$(-1)^{l_{2}+1}\sqrt{-1}^{\frac{l_{3}}{2}+2l_{2}-m-1}=\sqrt{-1}^{\frac{l_{3}}{2}-m+1}$).
∎

######  Remark 7.3.

According to [Coa89, page 103 (4)], the modified Euler factor
${\mathcal{L}}^{\sqrt{-1}}_{\infty}({\mathcal{M}}(\pi^{(3)}\times\pi^{(2)})(m))$
at $\infty$ of the $p$-adic $L$-functions for
${\mathcal{M}}(\pi^{(3)}\times\pi^{(2)})$ is given by

$\displaystyle{\mathcal{L}}^{\sqrt{-1}}_{\infty}({\mathcal{M}}(\pi^{(3)}\times\pi^{(2)})(m))$
$\displaystyle=(\sqrt{-1})^{\frac{l_{3}}{2}+l_{2}-3m}L_{\infty}(m,{\mathcal{M}}(\pi^{(3)}\times\pi^{(2)}))$
$\displaystyle=(-1)^{\delta+m}\sqrt{-1}^{l_{2}-1}\cdot(-1)^{\delta}\sqrt{-1}^{\frac{l_{3}}{2}-m+1}L_{\infty}(m,{\mathcal{M}}(\pi^{(3)}\times\pi^{(2)})).$

Hence, if we normalize
$\nabla^{\boldsymbol{n}_{m}}=\nabla_{2w_{\boldsymbol{\lambda}}-n_{m,1}-n_{m,2},n_{m,2}}$
in (3.3) so that

$\displaystyle\widetilde{\nabla}^{\boldsymbol{n}_{m}}=(-1)^{\delta+m}\sqrt{-1}^{l_{2}-1}C(\boldsymbol{w}_{\boldsymbol{\lambda}},\boldsymbol{n}_{m})^{-1}\nabla^{\boldsymbol{n}_{m}},$

then $\widetilde{\nabla}^{\boldsymbol{n}_{m}}$ defines a map
$\iota^{*}{\mathcal{L}}^{(3)}(\boldsymbol{w}_{\boldsymbol{\lambda}};A)_{/{\mathcal{Y}}^{(2)}_{{\mathcal{K}}_{2}}}\longrightarrow{\mathcal{L}}^{(2)}(\boldsymbol{n}_{m};A)_{/{\mathcal{Y}}^{(2)}_{{\mathcal{K}}_{2}}}$
between local systems in the same way as (6.1). Then Theorem 7.2 implies that
the archimedean local integral appearing in the evaluation of ${\rm
Tw}_{[\boldsymbol{n}_{m}]}\left(\widetilde{\nabla}^{\boldsymbol{n}_{m}}\iota^{*}{\rm
p}^{\ast}_{3}\eta_{\pi^{(3)}}\cup{\rm
p}^{\ast}_{2}\eta^{\pm}_{\pi^{(2)}}\right)$ coincides with the modified local
$L$-factor
${\mathcal{L}}^{\sqrt{-1}}_{\infty}({\mathcal{M}}(\pi^{(3)}\times\pi^{(2)})(m))$
at $\infty$ for each $m\in\mathrm{Crit}(\boldsymbol{\lambda})$.

######  Remark 7.4.

We compare Theorem 7.2 with [Rag16, Theorem 2.48, Theorem 2.50] and [Jan19,
Corollary 4.12].

  1. (i)

We briefly recall a description of the critical values for ${\rm
GL}_{3}\times{\rm GL}_{2}$ presented by Raghuram in [Rag16]. Let
$\Pi,\Sigma,\epsilon,\eta$ be as defined in [Rag16], which we can interpret as
follows by using notations in this article:

$\displaystyle\Pi$ $\displaystyle=\pi^{(3)},$ $\displaystyle\Sigma$
$\displaystyle=\pi^{(2)},$ $\displaystyle\epsilon$
$\displaystyle=(-1)^{m+\delta+1+w_{\boldsymbol{\lambda}}}=(-1)^{m+\delta+\frac{l_{3}}{2}},$
$\displaystyle\eta$ $\displaystyle=\pm.$

Then Raghuram has defined a complex number
$\langle[\Pi_{\infty}]^{\epsilon},[\Sigma_{\infty}]^{\eta}\rangle$ in [Rag16,
(2.47)] by using a cup product of cohomology classes. The result in [Sun17]
yields that $\langle[\Pi_{\infty}]^{\epsilon},[\Sigma_{\infty}]^{\eta}\rangle$
does not equal zero (see [Rag16, Theorem 2.48]). Raghram also defines
$p^{\epsilon,\eta}_{\infty}(\boldsymbol{w}_{\boldsymbol{\lambda}},\boldsymbol{n}_{\boldsymbol{\lambda}})$
to be $\langle[\Pi_{\infty}]^{\epsilon},[\Sigma_{\infty}]^{\eta}\rangle^{-1}$,
and this
$p^{\epsilon,\eta}_{\infty}(\boldsymbol{w}_{\boldsymbol{\lambda}},\boldsymbol{n}_{\boldsymbol{\lambda}})$
shows up in his formula for the critical values in [Rag16, Theorem 2.50]. Note
that, in this article, we write the weight of $\Pi$ and $\Sigma$ (in the sense
of [Rag16]) as $\boldsymbol{w}_{\boldsymbol{\lambda}}$ and
$\boldsymbol{n}_{\boldsymbol{\lambda}}$ respectively. In [Rag16, Theorem
2.50], the complex number
$p^{\epsilon,\eta}_{\infty}(\boldsymbol{w}_{\boldsymbol{\lambda}},\boldsymbol{n}_{\boldsymbol{\lambda}})$
appears as an “unknown constant”, but due to the proof of Theorem 7.2, we find
the following explicit formula on it:

$\qquad\qquad\left(\Omega_{\pi^{(3)}}\Omega^{\pm}_{\pi^{(2)}}p^{\epsilon,\eta}_{\infty}(\boldsymbol{w}_{\boldsymbol{\lambda}},\boldsymbol{n}_{\boldsymbol{\lambda}})\right)^{-1}=\frac{\langle[\Pi_{\infty}]^{\epsilon},[\Sigma_{\infty}]^{\eta}\rangle}{\Omega_{\pi^{(3)}}\Omega^{\pm}_{\pi^{(2)}}}\\\
=(-1)^{\delta}\sqrt{-1}^{\frac{l_{3}}{2}-m+1}\binom{\frac{l_{3}}{2}-1}{m-\frac{l_{3}}{2}-1}\binom{\frac{l_{3}}{2}-1}{\frac{l_{3}}{2}+l_{2}-m}\frac{L_{\infty}(m,{\mathcal{M}}(\pi^{(3)}\times\pi^{(2)}))}{\Omega_{\pi^{(3)}}\Omega^{\pm}_{\pi^{(2)}}}.$

Hence Theorem 7.2 refines [Rag16, Theorem 2.50] and it is consistent with
[Coa89, page 107, Period Conjecture].

  2. (ii)

Let us recall one of the main results proposed in [Jan19]. According to
[Jan19, (25) and Section 4.3, p. 6569], define a pairing
$e_{\infty}(|\cdot|^{m}_{\infty},\cdot,\cdot)\colon{\mathcal{W}}(\pi^{(3)}_{\infty},\psi_{\infty})^{K_{3}}\otimes{\mathcal{W}}(\pi^{(2)}_{\infty},\psi^{-1}_{\infty})^{K_{2}}\to{\mathbf{C}}$
so that

$\displaystyle
e_{\infty}(|\cdot|^{m}_{\infty},{W}_{3},W_{2})=\frac{1}{L(m-\frac{3}{2},\pi^{(3)}_{\infty}\times\pi^{(2)}_{\infty})}\int_{{\rm
N}_{2}({\mathbf{R}})\backslash{\rm
GL}_{2}({\mathbf{R}})}{W}_{3}(\iota(g))W_{2}(g)|\det
g|^{m-\frac{3}{2}}_{\infty}{\rm d}g_{\infty}$

for ${W}_{3}\in{\mathcal{W}}(\pi^{(3)}_{\infty},\psi_{\infty})^{K_{3}}$ and
$W_{2}\in{\mathcal{W}}(\pi^{(2)}_{\infty},\psi^{-1}_{\infty})^{K_{2}}$. Let
$E/{\mathbf{Q}}$ be a sufficiently large finite extension of $\mathbf{Q}$
(depending on the weight of $\pi_{\infty}^{(3)}$). Take arbitrary
$m,m^{\prime}\in{\mathbf{Z}}$ so that that $m-m^{\prime}$ is even and both
$L(m-\frac{3}{2},\pi^{(3)}\times\pi^{(2)})$ and
$L(m^{\prime}-\frac{3}{2},\pi^{(3)}\times\pi^{(2)})$ are critical. Then
[Jan19, Corollary 4.12] yields that, for each
${W}_{3}\in{\mathcal{W}}(\pi^{(3)}_{\infty},\psi_{\infty})^{K_{3}}$ and
$W_{2}\in{\mathcal{W}}(\pi^{(2)}_{\infty},\psi^{-1}_{\infty})^{K_{2}}$, there
exists an element $c_{m,m^{\prime}}\in E$ (a priori depending on $m$ and
$m^{\prime}$) satisfying

$\displaystyle
e_{\infty}(|\cdot|^{m}_{\infty},{W}_{3},W_{2})=c_{m,m^{\prime}}e_{\infty}(|\cdot|^{m^{\prime}}_{\infty},{W}_{3},W_{2}).$

On the other hand, assuming $W_{3}=W^{-\lambda_{2}}_{\pi^{(3)},\infty}$ and
$W_{2}=W^{+}_{\pi^{(2)},\infty}$, we can verify by using [HIM, Corollary 9.7]
that

(7.2) $\displaystyle
e_{\infty}(|\cdot|^{m}_{\infty},{W}_{3},W_{2})=(-1)^{\delta+\lambda_{2}}\in E$

holds for each $m$, which is independent of $m$. Therefore we can conclude
that $c_{m,m^{\prime}}$ equals $1$ for each $m$ and $m^{\prime}$, and Theorem
7.2 thus refines the period relations in [Jan19, Theorem 4.13] in the
$\mathrm{GL}_{3}\times\mathrm{GL}_{2}$ case. For general $n$, [Jan19] obtains
similar result under some conditions; see [Jan19, Conjecture 4.3, Conjecture
4.6] for details. Meanwhile, Ishii and Miyazaki recently generalize the
results of [HIM] to the case ${\rm GL}_{n+1}\times{\rm GL}_{n}$ over
${\mathbf{C}}$ in [IM], which also refine the results in [Jan19]
unconditionally.

Since there had been no explicit formula for the archimedean zeta integral
$\langle[\Pi_{\infty}]^{\epsilon},[\Sigma_{\infty}]^{\eta}\rangle$ or
$e_{\infty}(|\cdot|^{m}_{\infty},{W}_{3},W_{2})$, the definition of periods
for ${\mathcal{M}}(\pi^{(3)}\times\pi^{(2)})$ in [Rag16] and [Jan] depends on
the critical points $m$; compare with [Jan, Theorem A]. From the view point of
Coates and Perrin-Riou’s conjecture (refer to [CP89], [Coa89, page 107, Period
Conjecture, page 111, Principal Conjecture]), the archimedean zeta integral
must be given as a product of the period
$\Omega_{\pi^{(3)}}\Omega^{\pm}_{\pi^{(2)}}$ and
${\mathcal{L}}^{\sqrt{-1}}_{\infty}({\mathcal{M}}(\pi^{(3)}\times\pi^{(2)})(m))$
in Remark 7.3. Januszewski’s result implies that his periods coincides with
products of $\Omega_{\pi^{(3)}}\Omega^{\pm}_{\pi^{(2)}}$,
${\mathcal{L}}^{\sqrt{-1}}_{\infty}({\mathcal{M}}(\pi^{(3)}\times\pi^{(2)})(m))$
and implicit algebraic numbers depending on $m$, but we think that it is
crucial to remove this ambiguous multiple. See also [Jan, Introduction] on
this point. We expect that the identity (7.2) should be useful for the further
refinement of the interpolation formula of the $p$-adic Rankin–Selberg
$L$-functions for ${\rm GL}_{3}\times{\rm GL}_{2}$ constructed in [Jan,
Theorem A].

###  7.3. Algebraicity

Since Theorem 7.2 refines the formula in [Rag16, Theorem 2.50], it immediately
deduces the following corollary:

######  Corollary 7.5.

(compare with [Rag16, Theorem 2.50]) Let $E(\pi^{(3)},\pi^{(2)})$ be a number
field containing $\sqrt{-1}$, $E_{\pi^{(2)}}$ and $E_{\pi^{(3)}}$. Then, if
$(-1)^{m+\delta_{3}+\frac{l_{3}}{2}}=\pm 1$, we have

$\displaystyle\frac{L(m,{\mathcal{M}}(\pi^{(3)}\times\pi^{(2)}))}{\Omega_{\pi^{(3)}}\Omega^{\pm}_{\pi^{(2)}}}\in
E(\pi^{(3)},\pi^{(2)}).$

Assuming Deligne’s conjecture on critical values [Del79, Conjecture 1.8]
holds, we can deduce from Corollary 7.5 an interesting period relation, which
implies that Raghuram and Shahidi’s period $\Omega_{\pi^{(3)}}$ for
$\pi^{(3)}$ has a motivic background as well as the canonical periods
$\Omega^{\pm}_{\pi^{(2)}}$. We summarize the argument in the following remark:

######  Remark 7.6.

Let us admit the existence of the motive ${\mathcal{M}}[\pi^{(3)}]$ and the
validity of Deligne’s conjecture [Del79, Conjecture 1.8]. In this remark, we
write $a\sim b$ for $a,b\in\mathbf{C}$ if $a=bc$ holds for some
$c\in\overline{\mathbf{Q}}^{\times}$. For a pure motive ${\mathcal{M}}$ over
${\mathbf{Q}}$, let $\delta({\mathcal{M}})$ and $c^{\pm}({\mathcal{M}})$ be
the periods for ${\mathcal{M}}$ defined in [Del79]. Besides these periods,
Yoshida defines period invariants $c_{p}({\mathcal{M}})$ of type
$\\{(\underbrace{2,\dotsc,2}_{p},\underbrace{1,\dotsc,1}_{t-2p},\underbrace{0,\dotsc,0}_{p});(1,1)\\}$
for integers $p$ satisfying a certain admissibility condition in [Yos01,
Section 2]; here $t$ denotes the number of jumps of the Hodge filtration on
$H_{B}(\mathcal{M})\otimes\mathbf{C}$. Then applying [Yos01, Proposition 12],
we find that222Here we temporarily denote
$\pi^{(3)}_{\infty}=\mathrm{Ind}^{\mathrm{GL}_{3}(\mathbf{R})}_{\mathrm{P}_{2,1}(\mathbf{R})}(D_{\nu_{3},l_{3}}\boxtimes\chi_{\nu_{3}}^{\delta_{3}})$
to avoid the notational confusion concerning $\delta$.

(7.3) $\displaystyle c^{\pm}({\mathcal{M}}(\pi^{(3)}\times\pi^{(2)}))\sim
c_{1}({\mathcal{M}}[\pi^{(3)}])\delta({\mathcal{M}}[\pi^{(2)}])c^{\pm(-1)^{\delta_{3}+\frac{l_{3}}{2}}}({\mathcal{M}}[\pi^{(2)}])$

where $c_{1}(\mathcal{M}[\pi^{(3)}])$ denotes the period invariant of
$\mathcal{M}[\pi^{(3)}]$ of type $\\{(2,1,0);(1,1)\\}$. Note that
$\delta({\mathcal{M}}[\pi^{(2)}])\sim(2\pi\sqrt{-1})^{-l_{2}}$ holds by
[Del79, (7.8.2)]. Moreover from [Del79, (5.1.8)] we also have the relation

$\displaystyle
c^{\pm}({\mathcal{M}}(\pi^{(3)}\times\pi^{(2)}))\sim(2\pi\sqrt{-1})^{-d^{\pm}({\mathcal{M}}(\pi^{(3)}\times\pi^{(2)}))m}c^{\pm(-1)^{m}}({\mathcal{M}}(\pi^{(3)}\times\pi^{(2)})(m)).$

Then Corollary 7.5 and Deligne’s conjecture yield that, if
$(-1)^{m+\delta_{3}+\frac{l_{3}}{2}}=\pm 1$ holds, both of the following
ratios are algebraic numbers:

$\displaystyle\frac{L(0,{\mathcal{M}}(\pi^{(3)}\times\pi^{(2)})(m))}{\Omega_{\pi^{(3)}}\Omega^{\pm}_{\pi^{(2)}}}$
$\displaystyle\sim\frac{L_{\rm
fin}(0,{\mathcal{M}}(\pi^{(3)}\times\pi^{(2)})(m))}{(2\pi\sqrt{-1})^{3m-l_{2}-\frac{l_{3}}{2}}\Omega_{\pi^{(3)}}\Omega^{\pm}_{\pi^{(2)}}}\sim\frac{L_{\rm
fin}(0,{\mathcal{M}}(\pi^{(3)}\times\pi^{(2)})(m))}{(2\pi\sqrt{-1})^{3m-\frac{l_{3}}{2}}\delta({\mathcal{M}}[\pi^{(2)}])\Omega_{\pi^{(3)}}\Omega^{\pm}_{\pi^{(2)}}},$
$\displaystyle\frac{L_{\rm
fin}(0,{\mathcal{M}}(\pi^{(3)}\times\pi^{(2)})(m))}{c^{+}({\mathcal{M}}(\pi^{(3)}\times\pi^{(2)})(m))}$
$\displaystyle\sim\frac{L_{\rm
fin}(0,{\mathcal{M}}(\pi^{(3)}\times\pi^{(2)})(m))}{(2\pi\sqrt{-1})^{3m}c^{(-1)^{m}}({\mathcal{M}}(\pi^{(3)}\times\pi^{(2)}))}.$

Hence (7.3) shows that

$\displaystyle(2\pi\sqrt{-1})^{-\frac{l_{3}}{2}}\delta({\mathcal{M}}[\pi^{(2)}])\Omega_{\pi^{(3)}}\Omega^{\pm}_{\pi^{(2)}}\sim
c^{(-1)^{m}}({\mathcal{M}}(\pi^{(3)}\times\pi^{(2)}))\sim
c_{1}({\mathcal{M}}[\pi^{(3)}])\delta({\mathcal{M}}[\pi^{(2)}])c^{(-1)^{\delta_{3}+\frac{l_{3}}{2}+m}}({\mathcal{M}}[\pi^{(2)}]).$

On the canonical periods $\Omega^{\pm}_{\pi^{(2)}}$, Hida shows in [Hid94,
Theorem 8.1] that

$\displaystyle 2\pi\sqrt{-1}\Omega^{\mp}_{\pi^{(2)}}\sim
c^{\pm}({\mathcal{M}}[\pi^{(2)}](1))\sim(2\pi\sqrt{-1})c^{\mp}({\mathcal{M}}[\pi^{(2)}])$

holds. Furthermore we also find that $c_{1}({\mathcal{M}}[\pi^{(3)}])\sim
c^{+}({\mathcal{M}}[\pi^{(3)}])c^{-}({\mathcal{M}}[\pi^{(3)}])$ holds due to
[Yos01, Theorem 3] and comparison of the types of both sides. Combining all of
them, we can derive the period relation

$\displaystyle\Omega_{\pi^{(3)}}\sim(2\pi\sqrt{-1})^{\frac{l_{3}}{2}}c_{1}({\mathcal{M}}[\pi^{(3)}])\sim(2\pi\sqrt{-1})^{\frac{l_{3}}{2}}c^{+}({\mathcal{M}}[\pi^{(3)}])c^{-}({\mathcal{M}}[\pi^{(3)}])$

under the condition $(-1)^{m+\delta_{3}+\frac{l_{3}}{2}}=\pm 1$. This explains
the motivic background of the Whittaker period $\Omega_{\pi^{(3)}}$.

##  A. Appendix: Construction of the Eichler–Shimura maps for ${\rm GL}_{3}$

We retain the settings and notation in Section 5; in particular, let
$\pi^{(3)}$ denote a cohomological irreducible cuspidal automorphic
representation of $\mathrm{GL}_{3}(\mathbf{Q}_{\mathbf{A}})$, which has the
minimal $\mathrm{O}_{3}(\mathbf{R})$-type $\tau^{(3)}_{\boldsymbol{\lambda}}$
with
$\boldsymbol{\lambda}=\boldsymbol{\lambda}^{(3)}=(\lambda_{3},\delta)\in\Lambda_{3}^{\mathrm{coh}}$
at the infinite place. In Appendix A we summarize the construction of the
Eichler–Shimura map for $\mathrm{GL}_{3}$; namely, we briefly explain how to
attach a specific class $\delta^{(3),i}(\boldsymbol{f})$ of the
$(\mathfrak{gl}_{3}(\mathbf{R}),K_{3})$-cohomology
$H^{i}(\mathfrak{gl}_{3}(\mathbf{R}),K_{3};H_{\pi^{(3)},K_{3}}\otimes
L^{(3)}(\boldsymbol{w}_{\boldsymbol{\lambda}};\mathbf{C}))$ to a cusp form
$\boldsymbol{f}=\begin{pmatrix}f^{\boldsymbol{\lambda}}_{\lambda_{3}}&f^{\boldsymbol{\lambda}}_{\lambda_{3}-1}&\dotsc&f^{\boldsymbol{\lambda}}_{-\lambda_{3}}\end{pmatrix}\in\mathcal{S}^{(3)}_{\boldsymbol{\lambda}}(\mathcal{K}_{3})$
for $i=2$ and $3$, where
$K_{3}=\mathbf{R}^{\times}_{>0}\mathrm{SO}_{3}(\mathbf{R})$ and
$\mathcal{K}_{3}$ is an open compact subgroup of
$\mathrm{GL}_{3}(\mathbf{Q}_{\mathbf{A},\mathrm{fin}})$ acting trivially on
$\pi^{(3)}$.

Recall that we have defined $\mathcal{P}_{3}$ as
$\mathfrak{gl}_{3}(\mathbf{R})/\mathrm{Lie}(K_{3})$ and
$\mathcal{P}_{3,\mathbf{C}}$ as its complexification. Set
$L^{(3)}(\boldsymbol{w}_{\boldsymbol{\lambda}};\mathbf{C})$ be the irreducible
representation of $\mathrm{GL}_{3}(\mathbf{R})$ defined as in Section 3.2 for
$\boldsymbol{w}_{\boldsymbol{\lambda}}=(w_{\boldsymbol{\lambda}},w_{\boldsymbol{\lambda}},w_{\boldsymbol{\lambda}})=\left(\frac{\lambda_{3}-3}{2},\frac{\lambda_{3}-3}{2},\frac{\lambda_{3}-3}{2}\right)$.
Then we have already seen in Section 5.3 that the cohomology group
$H^{i}(\mathfrak{gl}_{3}(\mathbf{R}),K_{3};H_{\pi^{(3)},K_{3}}\otimes
L^{(3)}(\boldsymbol{w}_{\boldsymbol{\lambda}};\mathbf{C}))$ is of dimension
$1$ and is isomorphic to

$\displaystyle\mathrm{Hom}_{\mathrm{SO}_{3}(\mathbf{R})}\left(\bigwedge^{i}\mathcal{P}_{3,\mathbf{C}},H_{\pi^{(3)},K_{3}}\otimes
L^{(3)}(\boldsymbol{w}_{\boldsymbol{\lambda}};\mathbf{C})\right)\cong\left(H_{\pi^{(3)},K_{3}}\otimes
L^{(3)}(\boldsymbol{w}_{\boldsymbol{\lambda}};\mathbf{C})\otimes\bigwedge^{i}\mathcal{P}_{3,\mathbf{C}}^{*}\right)^{\mathrm{SO}_{3}(\mathbf{R})}.$

Therefore, in order to define the Eichler–Shimura map $\delta^{(3),i}$, it
suffices to construct a non-trivial element of the rightmost group from a
given cusp form
$\boldsymbol{f}=\begin{pmatrix}f^{\boldsymbol{\lambda}}_{\lambda_{3}}&f^{\boldsymbol{\lambda}}_{\lambda_{3}-1}&\dotsc&f^{\boldsymbol{\lambda}}_{-\lambda_{3}}\end{pmatrix}$.
We first analyze the action of $\mathrm{SO}_{3}(\mathbf{R})$ on the wedge
product $\bigwedge^{i}\mathcal{P}_{3,\mathbf{C}}^{*}$, and then we construct a
cohomology class so that the actions of $\mathrm{SO}_{3}(\mathbf{R})$ on
$H_{\pi^{(3)},K_{3}}$,
$L^{(3)}(\boldsymbol{w}_{\boldsymbol{\lambda}};\mathbf{C})$ and
$\bigwedge^{i}\mathcal{P}^{*}_{3,\mathbf{C}}$ are canceled out.

###  A.1. Action on the wedge product of $\mathcal{P}_{3,\mathbf{C}}^{*}$

Recall the definition of the basis $\\{X_{2},X_{1},X_{0},X_{-1},X_{-2}\\}$ of
$\mathcal{P}_{3,\mathbf{C}}$ from Section 5.3 (see (5.4)). Lemma 5.2 implies
that this basis gives an isomorphism between $\mathcal{P}_{3,\mathbf{C}}$ and
$\tau^{(3)}_{(2,0)}$ as $\mathrm{SO}_{3}(\mathbf{R})$-modules. For $i=2$ and
$3$, the wedge product $\bigwedge^{i}\mathcal{P}_{3,\mathbf{C}}$ is of
dimension 10 and, due to weight calculation, we readily observe that its
$\mathbf{C}$-linear dual $\bigwedge^{i}\mathcal{P}_{3,\mathbf{C}}^{*}$ is
isomorphic to $\tau^{(3),\vee}_{(1,0)}\oplus\tau^{(3),\vee}_{(3,0)}$ as
$\mathrm{SO}_{3}(\mathbf{R})$-modules (here
$\tau^{(3),\vee}_{\boldsymbol{\lambda}}$ denotes the contragredient of
$\tau^{(3)}_{\boldsymbol{\lambda}}$). Now we determine an explicit basis of
the $\tau^{(3),\vee}_{(3,0)}$-isotypic component of
$\bigwedge^{i}\mathcal{P}_{3,\mathbf{C}}^{*}$.

Let $\\{v^{\boldsymbol{\lambda}}_{j}\\}_{-\lambda_{3}\leq j\leq\lambda_{3}}$
be the basis of $\tau^{(3)}_{\boldsymbol{\lambda}}$ defined in Section 3.5 and
$\\{v^{\boldsymbol{\lambda},*}_{j}\\}_{-\lambda_{3}\leq j\leq\lambda_{3}}$ the
dual basis of $\tau^{(3),\vee}_{\boldsymbol{\lambda}}$.

######  Lemma A.1.

Suppose that $i=2$ or $3$. Then there exist explicit elements
$\boldsymbol{\omega}^{i}_{j}$ $(-3\leq j\leq 3)$ such that the correspondence
$v^{(3,0),*}_{j}\mapsto\boldsymbol{\omega}^{i}_{j}$ gives rise to an
$\mathrm{SO}_{3}(\mathbf{R})$-equivariant injection of
$\tau^{(3),\vee}_{(3,0)}$ into $\bigwedge^{i}\mathcal{P}_{3,\mathbf{C}}^{*}$.
In particular, we have
$\wedge^{i}\mathrm{Ad}^{*}(u)\begin{pmatrix}\boldsymbol{\omega}_{3}^{i}&\dotsc&\boldsymbol{\omega}_{-3}^{i}\end{pmatrix}=\begin{pmatrix}\boldsymbol{\omega}_{3}^{i}&\dotsc&\boldsymbol{\omega}_{-3}^{i}\end{pmatrix}{}^{\mathrm{t}}\\!M_{(3,0)}(u)^{-1}$
for every $u\in\mathrm{SO}_{3}(\mathbf{R})$.

###### Proof..

For $2\geq j>j^{\prime}>j^{\prime\prime}\geq-2$, let us abbreviate the wedge
product $X_{j}\wedge X_{j^{\prime}}$ and $X_{j}\wedge X_{j^{\prime}}\wedge
X_{j^{\prime\prime}}$ as $X_{j,j^{\prime}}$ and
$X_{j,j^{\prime},j^{\prime\prime}}$ respectively. Note that
$\mathbf{B}^{2}=\\{X_{j,j^{\prime}}\\}_{2\geq j>j^{\prime}\geq-2}$ (resp.
$\mathbf{B}^{3}=\\{X_{j,j^{\prime},j^{\prime\prime}}\\}_{2\geq
j>j^{\prime}>j^{\prime\prime}\geq-2}$) forms a basis of
$\bigwedge^{2}\mathcal{P}_{3,\mathbf{C}}$ (resp.
$\bigwedge^{3}\mathcal{P}_{3,\mathbf{C}}$). Set

$\mathbf{X}^{i}=\\\
\begin{cases}\begin{pmatrix}X_{2,1}&X_{2,0}&X_{2,-1}&X_{2,-2}&X_{1,0}&X_{1,-1}&X_{1,-2}&X_{0,-1}&X_{0,-2}&X_{-1,-2}\end{pmatrix}&\text{for
$i=2$},\\\
\begin{pmatrix}X_{2,1,0}&X_{2,1,-1}&X_{2,1,-2}&X_{2,0,-1}&X_{2,0,-2}&X_{2,-1,-2}&X_{1,0,-1}&X_{1,0,-2}&X_{1,-1,-2}&X_{0,-1,-2}\end{pmatrix}&\text{for
$i=3$}.\end{cases}$

For each $u\in\mathrm{SO}_{3}(\mathbf{R})$, let $\widetilde{M}^{i}(u)$ denote
the matrix presentation of the action of $u$ on
$\bigwedge^{i}\mathcal{P}_{3,\mathbf{C}}$ with respect to the basis
$\mathbf{X}^{i}$; namely set
$\wedge^{i}\mathrm{Ad}(u)\mathbf{X}^{i}=\mathbf{X}^{i}\widetilde{M}^{i}(u)$.
Then, by direct calculation using Mathematica and Maxima, we observe that
$\widetilde{P}^{i,-1}\widetilde{M}^{i}(u)\widetilde{P}^{i}=\begin{pmatrix}M_{(1,0)}(u)&0\\\
0&M_{(3,0)}(u)\end{pmatrix}$ holds for

$\displaystyle\widetilde{P}^{2}$
$\displaystyle=\begin{pmatrix}0&0&0&1&0&0&0&0&0&0\\\
0&0&0&0&\frac{1}{2}&0&0&0&0&0\\\ -\frac{1}{5}&0&0&0&0&\frac{1}{5}&0&0&0&0\\\
0&-\frac{1}{10}&0&0&0&0&\frac{1}{20}&0&0&0\\\
\frac{3}{5}&0&0&0&0&\frac{2}{5}&0&0&0&0\\\
0&\frac{1}{5}&0&0&0&0&\frac{2}{5}&0&0&0\\\
0&0&-\frac{1}{5}&0&0&0&0&\frac{1}{5}&0&0\\\
0&0&\frac{3}{5}&0&0&0&0&\frac{2}{5}&0&0\\\ 0&0&0&0&0&0&0&0&\frac{1}{2}&0\\\
0&0&0&0&0&0&0&0&0&1\end{pmatrix},$ $\displaystyle\widetilde{P}^{3}$
$\displaystyle=\begin{pmatrix}0&0&0&3&0&0&0&0&0&0\\\ 0&0&0&0&1&0&0&0&0&0\\\
1&0&0&0&0&\frac{1}{5}&0&0&0&0\\\ -2&0&0&0&0&\frac{3}{5}&0&0&0&0\\\
0&\frac{1}{2}&0&0&0&0&\frac{3}{10}&0&0&0\\\ 0&0&1&0&0&0&0&\frac{1}{5}&0&0\\\
0&-4&0&0&0&0&\frac{3}{5}&0&0&0\\\ 0&0&-2&0&0&0&0&\frac{3}{5}&0&0\\\
0&0&0&0&0&0&0&0&1&0\\\ 0&0&0&0&0&0&0&0&0&3\end{pmatrix}.$

Let us define $\mathrm{d}\mathbf{X}^{i}$ similarly to $\mathbf{X}^{i}$
replacing $X_{j,j^{\prime}}$ and $X_{j,j^{\prime},j^{\prime\prime}}$ by
$\mathrm{d}X_{j,j^{\prime}}:=\mathrm{d}X_{j}\wedge\mathrm{d}X_{j^{\prime}}$
and
$\mathrm{d}X_{j,j^{\prime},j^{\prime\prime}}:=\mathrm{d}X_{j}\wedge\mathrm{d}X_{j^{\prime}}\wedge\mathrm{d}X_{j^{\prime\prime}}$
respectively (recall that $\\{\mathrm{d}X_{j}\\}_{-2\leq j\leq 2}$ denotes the
dual basis of $\mathcal{P}_{3,\mathbf{C}}$). Then it suffices to define
$\\{\boldsymbol{\omega}^{i}_{j}\\}_{-3\leq j\leq 3}$ as

$\displaystyle\begin{pmatrix}\boldsymbol{\omega}^{i}_{3}&\boldsymbol{\omega}^{i}_{2}&\boldsymbol{\omega}^{i}_{1}&\boldsymbol{\omega}^{i}_{0}&\boldsymbol{\omega}^{i}_{-1}&\boldsymbol{\omega}^{i}_{-2}&\boldsymbol{\omega}^{i}_{-3}\end{pmatrix}=\mathrm{d}\mathbf{X}^{i}P^{i}$

for

(A.1) $\displaystyle P^{2}$ $\displaystyle=\begin{pmatrix}1&0&0&0&0&0&0\\\
0&2&0&0&0&0&0\\\ 0&0&3&0&0&0&0\\\ 0&0&0&4&0&0&0\\\ 0&0&1&0&0&0&0\\\
0&0&0&2&0&0&0\\\ 0&0&0&0&3&0&0\\\ 0&0&0&0&1&0&0\\\ 0&0&0&0&0&2&0\\\
0&0&0&0&0&0&1\end{pmatrix},$ $\displaystyle P^{3}$
$\displaystyle=\begin{pmatrix}\frac{1}{3}&0&0&0&0&0&0\\\ 0&1&0&0&0&0&0\\\
0&0&2&0&0&0&0\\\ 0&0&1&0&0&0&0\\\ 0&0&0&\frac{8}{3}&0&0&0\\\ 0&0&0&0&2&0&0\\\
0&0&0&\frac{1}{3}&0&0&0\\\ 0&0&0&0&1&0&0\\\ 0&0&0&0&0&1&0\\\
0&0&0&0&0&0&\frac{1}{3}\end{pmatrix}.$

Note that $P^{i}\in\mathrm{M}_{10,7}(\mathbf{C})$ is a matrix obtained by
removing first three columns of ${}^{\mathrm{t}}\\!\widetilde{P}^{i,-1}$. ∎

###  A.2. Construction of the cohomology class

Recall from Section 3.2 that the representation space of the irreducible
representation $L^{(3)}(\boldsymbol{w}_{\boldsymbol{\lambda}};\mathbf{C})$ is
a subspace of
$\mathbf{C}[X,Y,Z;A,B,C]_{w_{\boldsymbol{\lambda}},w_{\boldsymbol{\lambda}}}$,
the space of polynomials in variables $X,Y,Z,A,B,C$ which are homogeneous of
degree $w_{\boldsymbol{\lambda}}$ both in $X,Y,Z$ and $A,B,C$. Recall also
that the irreducible representation $\tau^{(3)}_{\boldsymbol{\lambda}}$ of
$\mathrm{SO}_{3}(\mathbf{R})$ is realized as a certain quotient of the space
of homogeneous polynomials of degree $\lambda_{3}$ in variables
$z_{1},z_{2},z_{3}$. Now let us consider

$P(X,Y,Z,A,B,C,z_{1},z_{2},z_{3}):=\left(\begin{pmatrix}X&Y&Z\end{pmatrix}\begin{pmatrix}z_{1}\\\
z_{2}\\\
z_{3}\end{pmatrix}\right)^{w_{\boldsymbol{\lambda}}}\otimes\left(\begin{pmatrix}A&B&C\end{pmatrix}\begin{pmatrix}z_{1}\\\
z_{2}\\\
z_{3}\end{pmatrix}\right)^{w_{\boldsymbol{\lambda}}}\begin{pmatrix}v_{3}^{(3,\delta)}&\dotsc&v_{-3}^{(3,\delta)}\end{pmatrix}\\\
\in(\mathbf{C}[X,Y,Z;A,B,C]_{w_{\boldsymbol{\lambda}}}\otimes\mathbf{C}[z_{1},z_{2},z_{3}]_{\lambda_{3}}/\mathcal{V}_{\boldsymbol{\lambda}})^{\oplus
7}.$

######  Lemma A.2.

Let $P(X,Y,Z,A,B,C,z_{1},z_{2},z_{3})$ be as above.

  1. (i)

There exists a unique matrix

$\hskip
40.00006pt\mathcal{P}(X,Y,Z,A,B,C)=\begin{pmatrix}\mathcal{P}_{\lambda_{3},3}(X,Y,Z,A,B,C)&\dotsc&\mathcal{P}_{\lambda_{3},-3}(X,Y,Z,A,B,C)\\\
\vdots&\ddots&\vdots\\\
\mathcal{P}_{-\lambda_{3},3}(X,Y,Z,A,B,C)&\dotsc&\mathcal{P}_{-\lambda_{3},-3}(X,Y,Z,A,B,C)\end{pmatrix}\\\
\in\mathrm{M}_{2\lambda_{3}+1,7}(\mathbf{C}[X,Y,Z;A,B,C]_{w_{\boldsymbol{\lambda}},w_{\boldsymbol{\lambda}}})$

such that
$P(X,Y,Z,A,B,C,z_{1},z_{2},z_{3})=\begin{pmatrix}v^{\boldsymbol{\lambda}}_{\lambda_{3}}&\dotsc&v^{\boldsymbol{\lambda}}_{-\lambda_{3}}\end{pmatrix}\mathcal{P}(X,Y,Z,A,B,C)$
holds. Moreover every entry of $\mathcal{P}(X,Y,Z,A,B,C)$ is an element of
$L^{(3)}(\boldsymbol{w}_{\boldsymbol{\lambda}};\mathbf{C})$ all of whose
coefficients are contained in $\mathbf{Z}[2^{-1},\sqrt{-1}]$.

  2. (ii)

For each $u\in\mathrm{SO}_{3}(\mathbf{R})$, we have

$\displaystyle(\varrho_{\boldsymbol{\lambda}}^{(3)}(u)\mathcal{P})(X,Y,Z,A,B,C)=M_{\boldsymbol{\lambda}}^{-1}(u)\mathcal{P}(X,Y,Z,A,B,C)M_{(3,0)}(u).$

  3. (iii)

For a cusp form
$\boldsymbol{f}=\begin{pmatrix}f^{\boldsymbol{\lambda}}_{\lambda_{3}}&f^{\boldsymbol{\lambda}}_{\lambda_{3}-1}&\dotsc&f^{\boldsymbol{\lambda}}_{-\lambda_{3}}\end{pmatrix}\in\mathcal{S}^{(3)}_{\boldsymbol{\lambda}}(\mathcal{K}_{3})$,
the element $\delta^{(3),i}(\boldsymbol{f})$ defined as

$\displaystyle\delta^{(3),i}(\boldsymbol{f})=\begin{pmatrix}f^{\boldsymbol{\lambda}}_{\lambda_{3}}&f^{\boldsymbol{\lambda}}_{\lambda_{3}-1}&\dotsc&f^{\boldsymbol{\lambda}}_{-\lambda_{3}}\end{pmatrix}\mathcal{P}(X,Y,Z,A,B,C)\begin{pmatrix}\boldsymbol{\omega}^{i}_{3}\\\
\vdots\\\ \boldsymbol{\omega}^{i}_{-3}\end{pmatrix}$

gives rise to a class of
$H^{i}_{\mathrm{cusp}}(Y^{(3)}_{\mathcal{K}_{3}},\mathcal{L}^{(3)}(\boldsymbol{w}_{\boldsymbol{\lambda}};\mathbf{C}))$.

###### Proof..

  1. (i)

The first statement obviously follows from the identity

$\left(\begin{pmatrix}X&Y&Z\end{pmatrix}\begin{pmatrix}z_{1}\\\ z_{2}\\\
z_{3}\end{pmatrix}\right)^{w_{\boldsymbol{\lambda}}}\otimes\left(\begin{pmatrix}A&B&C\end{pmatrix}\begin{pmatrix}z_{1}\\\
z_{2}\\\
z_{3}\end{pmatrix}\right)^{w_{\boldsymbol{\lambda}}}=(Xz_{1}+Yz_{2}+Zz_{3})^{w_{\boldsymbol{\lambda}}}\otimes(Az_{1}+Bz_{2}+Cz_{3})^{w_{\boldsymbol{\lambda}}}\\\
=\left(\dfrac{X-\sqrt{-1}Y}{2}v^{(1,0)}_{1}-\dfrac{X+\sqrt{-1}Y}{2}v^{(1,0)}_{-1}+Zv^{(1,0)}_{0}\right)^{w_{\boldsymbol{\lambda}}}\otimes\left(\dfrac{A-\sqrt{-1}B}{2}v^{(1,0)}_{1}-\dfrac{A+\sqrt{-1}B}{2}v^{(1,0)}_{-1}+Cv^{(1,0)}_{0}\right)^{w_{\boldsymbol{\lambda}}}$

and the multiplicativity of $v^{\boldsymbol{\lambda}}_{\mu}$’s (see (3.5);
recall that we have $2w_{\boldsymbol{\lambda}}+3=\lambda_{3}$). We can also
verify the second statement by observing

$\displaystyle\iota_{w_{\boldsymbol{\lambda}},w_{\boldsymbol{\lambda}}}$
$\displaystyle\left((Xz_{1}+Yz_{2}+Zz_{3})^{w_{\boldsymbol{\lambda}}}\otimes(Az_{1}+Bz_{2}+Cz_{3})^{w_{\boldsymbol{\lambda}}}\right)$
$\displaystyle=\left(\dfrac{\partial^{2}}{\partial X\partial
A}+\dfrac{\partial^{2}}{\partial Y\partial B}+\dfrac{\partial^{2}}{\partial
Z\partial
Z}\right)(Xz_{1}+Yz_{2}+Zz_{3})^{w_{\boldsymbol{\lambda}}}\otimes(Az_{1}+Bz_{2}+Cz_{3})^{w_{\boldsymbol{\lambda}}}$
$\displaystyle=w^{2}_{\boldsymbol{\lambda}}(z_{1}^{2}+z_{2}^{2}+z_{3}^{2})(Xz_{1}+Yz_{2}+Zz_{3})^{w_{\boldsymbol{\lambda}}-1}\otimes(Az_{1}+Bz_{2}+Cz_{3})^{w_{\boldsymbol{\lambda}}-1}\equiv
0\mod{\mathcal{V}_{\boldsymbol{\lambda}}}.$

  2. (ii)

Let $u\in\mathrm{SO}_{3}(\mathbf{R})$. By definition, we have

$\displaystyle(\varrho_{\boldsymbol{w}_{\boldsymbol{\lambda}}}(u)P)$
$\displaystyle(X,Y,Z,A,B,C,z_{1},z_{2},z_{3})$
$\displaystyle=\left(\begin{pmatrix}X&Y&Z\end{pmatrix}u{}^{t}u\begin{pmatrix}z_{1}\\\
z_{2}\\\
z_{3}\end{pmatrix}\right)^{w_{\boldsymbol{\lambda}}}\otimes\left(\begin{pmatrix}A&B&C\end{pmatrix}{}^{t}u^{-1}{}^{t}u\begin{pmatrix}z_{1}\\\
z_{2}\\\
z_{3}\end{pmatrix}\right)^{w_{\boldsymbol{\lambda}}}\tau^{(3)}_{(3,0)}(u)\begin{pmatrix}v_{3}^{(3,0)}&\dotsc&v_{3}^{(3,0)}\end{pmatrix}$
$\displaystyle=P(X,Y,Z,A,B,C,z_{1},z_{2},z_{3})M_{(3,0)}(u)=\begin{pmatrix}v^{\boldsymbol{\lambda}}_{\lambda_{3}}&\dotsc&v^{\boldsymbol{\lambda}}_{-\lambda_{3}}\end{pmatrix}\mathcal{P}(X,Y,Z,A,B,C)M_{(3,0)}(u).$

On the other hand, we can calculate
$(\varrho_{\boldsymbol{w}_{\boldsymbol{\lambda}}}(u)P)(X,Y,Z,A,B,C,z_{1},z_{2},z_{3})$
by using (i) as

$\displaystyle(\varrho_{\boldsymbol{w}_{\boldsymbol{\lambda}}}(u)P)(X,Y,Z,A,B,C,z_{1},z_{2},z_{3})$
$\displaystyle=\tau^{(3)}_{\boldsymbol{\lambda}}(u)\begin{pmatrix}v^{\boldsymbol{\lambda}}_{\lambda_{3}}&\dotsc&v^{\boldsymbol{\lambda}}_{-\lambda_{3}}\end{pmatrix}(\varrho_{\boldsymbol{w}_{\boldsymbol{\lambda}}}(u)\mathcal{P})(X,Y,Z,A,B,C)$
$\displaystyle=\begin{pmatrix}v^{\boldsymbol{\lambda}}_{\lambda_{3}}&\dotsc&v^{\boldsymbol{\lambda}}_{-\lambda_{3}}\end{pmatrix}M_{\lambda}(u)(\varrho_{\boldsymbol{w}_{\boldsymbol{\lambda}}}(u)\mathcal{P})(X,Y,Z,A,B,C).$

Comparing these two equations, we obtain the desired result.

  3. (iii)

Since
$H^{i}_{\mathrm{cusp}}(Y^{(3)}_{\mathcal{K}_{3}},\mathcal{L}^{(3)}(\boldsymbol{w}_{\boldsymbol{\lambda}};\mathbf{C}))\cong(H_{\pi^{(3)},K_{3}}\otimes
L^{(3)}(\boldsymbol{w}_{\boldsymbol{\lambda}};\mathbf{C})\otimes\bigwedge^{i}\mathcal{P}_{3,\mathbf{C}}^{*})^{\mathrm{SO}_{3}(\mathbf{R})}\otimes(\pi^{(3)}_{\mathrm{fin}})^{\mathcal{K}_{3}}$,
it suffices to check that the archimedean component of
$\delta^{(3),i}(\boldsymbol{f})_{\infty}$ is fixed by the action of
$\mathrm{SO}_{3}(\mathbf{R})$, which obviously follows from (ii); for each
$u\in\mathrm{SO}_{3}(\mathbf{R})$ we have

$\displaystyle\qquad\quad(\varrho_{\boldsymbol{w}_{\boldsymbol{\lambda}}}$
$\displaystyle(u)\delta^{(3),i})(\boldsymbol{f})_{\infty}$
$\displaystyle=\tau^{(3)}_{\boldsymbol{\lambda}}(u)\begin{pmatrix}f^{\boldsymbol{\lambda}}_{\lambda_{3},\infty}&\dotsc&f^{\boldsymbol{\lambda}}_{-\lambda_{3},\infty}\end{pmatrix}(\varrho_{\boldsymbol{w}_{\boldsymbol{\lambda}}}(u)\mathcal{P}(X,Y,Z,A,B,C))\,\,{\vphantom{\wedge^{i}}}^{\mathrm{t}}\\!\\!\left(\wedge^{i}\mathrm{Ad}^{*}(u)\begin{pmatrix}\boldsymbol{\omega}^{i}_{3}&\dotsc&\boldsymbol{\omega}^{i}_{-3}\end{pmatrix}\right)$
$\displaystyle=\begin{pmatrix}f^{\boldsymbol{\lambda}}_{\lambda_{3},\infty}&\dotsc&f^{\boldsymbol{\lambda}}_{-\lambda_{3},\infty}\end{pmatrix}M_{\boldsymbol{\lambda}}(u)\bigl{\\{}M_{\boldsymbol{\lambda}}^{-1}(u)\mathcal{P}(X,Y,Z,A,B,C)M_{(3,0)}(u)\bigr{\\}}\,M_{(3,0)}(u)^{-1}\begin{pmatrix}\boldsymbol{\omega}^{i}_{3}\\\
\vdots\\\ \boldsymbol{\omega}^{i}_{-3}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}f^{\boldsymbol{\lambda}}_{\lambda_{3},\infty}&\dotsc&f^{\boldsymbol{\lambda}}_{-\lambda_{3},\infty}\end{pmatrix}\mathcal{P}(X,Y,Z,A,B,C)\begin{pmatrix}\boldsymbol{\omega}^{i}_{3}\\\
\vdots\\\
\boldsymbol{\omega}^{i}_{-3}\end{pmatrix}=\delta^{(3),i}(\boldsymbol{f})_{\infty}.$

∎

###  A.3. Change of the coordinates

We first introduce the coordinates $(y_{1},y_{2},x_{1},x_{2},x_{3})$ of (the
archimedean component of) the symmetric space
$Y^{(3)}_{\mathcal{K}_{3},\infty}=\mathrm{GL}_{3}(\mathbf{Q})\backslash\mathrm{GL}_{3}(\mathbf{R})/\mathbf{R}^{\times}_{>0}\mathrm{SO}_{3}(\mathbf{R})$,
and then calculate the presentations of the differential forms
$\\{\boldsymbol{\omega}^{i}_{j}\\}_{-3\leq j\leq 3}$ obtained in the previous
subsection with respect to this coordinates. All the results of this
subsection are obtained by longsome but straightforward computations
(especially concerning big matrices), and thus we omit their proofs333We have
checked out all the computations by using Mathematica and Maxima.. We first
introduce the explicit Iwasawa decomposition formula for
$\mathrm{GL}_{3}(\mathbf{R})$ as follows;

######  Lemma A.3.

For $g=\begin{pmatrix}a&b&c\\\ d&e&f\\\
g&h&i\end{pmatrix}\in\mathrm{GL}_{3}(\mathbf{R})$, define
$x_{1},x_{2},x_{3},y_{1}$ and $y_{2}$ as follows:

$\displaystyle x_{1}$
$\displaystyle=x_{1}(g)=\dfrac{dg+eh+fi}{g^{2}+h^{2}+i^{2}},$ $\displaystyle
x_{3}$ $\displaystyle=x_{3}(g)=\dfrac{ag+bh+ci}{g^{2}+h^{2}+i^{2}},$
$\displaystyle y_{1}$
$\displaystyle=y_{1}(g)=\sqrt{\dfrac{d^{2}+e^{2}+f^{2}}{g^{2}+h^{2}+i^{2}}-x^{2}_{1}},$
$\displaystyle x_{2}$
$\displaystyle=x_{2}(g)=\dfrac{1}{y_{1}^{2}}\left(\dfrac{ad+be+cf}{g^{2}+h^{2}+i^{2}}-x_{1}x_{3}\right),$
$\displaystyle y_{2}$
$\displaystyle=y_{2}(g)=\dfrac{1}{y_{1}}\sqrt{\dfrac{a^{2}+b^{2}+c^{2}}{g^{2}+h^{2}+i^{2}}-x_{3}^{2}-x_{2}^{2}y_{1}^{2}}.$

Then we have

$\displaystyle g=\begin{pmatrix}y_{1}y_{2}&y_{1}x_{2}&x_{3}\\\
0&y_{1}&x_{1}\\\ 0&0&1\end{pmatrix}\cdot\pm(g^{2}+h^{2}+i^{2})k\qquad\text{for
}k\in\mathrm{SO}_{3}(\mathbf{R}).$

Due to Lemma A.3, we can consider
$(y_{1}(g),y_{2}(g),x_{1}(g),x_{2}(g),x_{3}(g))$ as the coordinates of $g$ on
$Y^{(3)}_{\mathcal{K}^{(3)},\infty}$. Set
$F(g):=\begin{pmatrix}y_{1}(g)y_{2}(g)&y_{1}(g)x_{2}(g)&x_{3}(g)\\\
0&y_{1}(g)&x_{1}(g)\\\ 0&0&1\end{pmatrix}$ and let $\mathrm{d}F$ denote the
differential of $F$ at the identity matrix $1_{3}$.

######  Lemma A.4.

Let $\\{X_{j}\\}_{-2\leq j\leq 2}$ denote the basis of
$\mathcal{P}_{3,\mathbf{C}}$ introduced in Section 5.3. Then we have

$\mathrm{d}F\begin{pmatrix}X_{2}&X_{1}&X_{0}&X_{-1}&X_{-2}\end{pmatrix}\\\
=\begin{pmatrix}\left(\frac{\partial}{\partial
y_{1}}\right)_{1_{3}}&\left(\frac{\partial}{\partial
y_{2}}\right)_{1_{3}}&\left(\frac{\partial}{\partial
x_{1}}\right)_{1_{3}}&\left(\frac{\partial}{\partial
x_{2}}\right)_{1_{3}}&\left(\frac{\partial}{\partial
x_{3}}\right)_{1_{3}}\end{pmatrix}\begin{pmatrix}1&0&1&0&1\\\ -2&0&0&0&-2\\\
0&\sqrt{-1}&0&\sqrt{-1}&0\\\ -2\sqrt{-1}&0&0&0&2\sqrt{-1}\\\
0&1&0&-1&0\end{pmatrix}.$

Taking dual of Lemma A.4, we obtain

(A.6)
$(F^{*})^{-1}\begin{pmatrix}\mathrm{d}X_{2}&\mathrm{d}X_{1}&\mathrm{d}X_{0}&\mathrm{d}X_{-1}&\mathrm{d}X_{-2}\end{pmatrix}\\\
=\begin{pmatrix}(\mathrm{d}y_{1})_{1_{3}}&(\mathrm{d}y_{2})_{1_{3}}&(\mathrm{d}x_{1})_{1_{3}}&(\mathrm{d}x_{2})_{1_{3}}&(\mathrm{d}x_{3})_{1_{3}}\end{pmatrix}\,\,\raisebox{25.00003pt}{${}^{\mathrm{t}}$}\\!\\!\begin{pmatrix}1&0&1&0&1\\\
-2&0&0&0&-2\\\ 0&\sqrt{-1}&0&\sqrt{-1}&0\\\ -2\sqrt{-1}&0&0&0&2\sqrt{-1}\\\
0&1&0&-1&0\end{pmatrix}^{-1}.$

Next we extend (A.6) to the identity between global differential forms.

######  Lemma A.5.

Let $g\in\mathrm{GL}_{3}(\mathbf{R})$. Then we have

$\begin{pmatrix}(\mathrm{d}y_{1})_{g}&(\mathrm{d}y_{2})_{g}&(\mathrm{d}x_{1})_{g}&(\mathrm{d}x_{2})_{g}&(\mathrm{d}x_{3})_{g}\end{pmatrix}=\begin{pmatrix}(\mathrm{d}y_{1})_{1_{3}}&(\mathrm{d}y_{2})_{1_{3}}&(\mathrm{d}x_{1})_{1_{3}}&(\mathrm{d}x_{2})_{1_{3}}&(\mathrm{d}x_{3})_{1_{3}}\end{pmatrix}\circ
L_{g}^{-1}\\\
=\begin{pmatrix}(\mathrm{d}y_{1})_{1_{3}}&(\mathrm{d}y_{2})_{1_{3}}&(\mathrm{d}x_{1})_{1_{3}}&(\mathrm{d}x_{2})_{1_{3}}&(\mathrm{d}x_{3})_{1_{3}}\end{pmatrix}\dfrac{\partial(y_{1}(g^{-1}),y_{2}(g^{-1}),x_{1}(g^{-1}),x_{2}(g^{-1}),x_{3}(g^{-1}))}{\partial(y_{1},y_{2},x_{1},x_{2},x_{3})}\\\
=\begin{pmatrix}(\mathrm{d}y_{1})_{1_{3}}&(\mathrm{d}y_{2})_{1_{3}}&(\mathrm{d}x_{1})_{1_{3}}&(\mathrm{d}x_{2})_{1_{3}}&(\mathrm{d}x_{3})_{1_{3}}\end{pmatrix}\begin{pmatrix}\frac{1}{y_{1}}&0&0&0&0\\\
0&\frac{1}{y_{2}}&0&0&0\\\ 0&0&\frac{1}{y_{1}}&0&-\frac{x_{2}}{y_{1}y_{2}}\\\
0&0&0&\frac{1}{y_{2}}&0\\\ 0&0&0&0&\frac{1}{y_{1}y_{2}}\end{pmatrix}$

where $L_{g}$ denotes the left translation.

Combining (A.6) with Lemma A.5, we have

(A.7)
$(F^{*})^{-1}\left(\begin{pmatrix}\mathrm{d}X_{2}&\mathrm{d}X_{1}&\mathrm{d}X_{0}&\mathrm{d}X_{-1}&\mathrm{d}X_{-2}\end{pmatrix}\circ
L_{g}^{-1}\right)\\\
=\begin{pmatrix}\mathrm{d}y_{1}&\mathrm{d}y_{2}&\mathrm{d}x_{1}&\mathrm{d}x_{2}&\mathrm{d}x_{3}\end{pmatrix}\begin{pmatrix}0&0&\frac{1}{y_{1}}&0&0\\\
-\frac{1}{4y_{2}}&0&\frac{1}{2y_{2}}&0&-\frac{1}{4y_{2}}\\\
0&-\frac{x_{2}+\sqrt{-1}y_{2}}{2y_{1}y_{2}}&0&\frac{x_{2}-\sqrt{-1}y_{2}}{2y_{1}y_{2}}&0\\\
\frac{\sqrt{-1}}{4y_{2}}&0&0&0&-\frac{\sqrt{-1}}{4y_{2}}\\\
0&\frac{1}{2y_{1}y_{2}}&0&-\frac{1}{2y_{1}y_{2}}&0\end{pmatrix}.$

By taking wedge products of (A.7) and considering appropriate base change
given in Section A.1, we can explicitly describe the differential forms
$\\{\boldsymbol{\omega}^{i}_{j}\\}_{-3\leq j\leq 3}$ introduced in Section A.1
with respect to the coordinates $(y_{1},y_{2},x_{1},x_{2},x_{3})$ of
$Y^{(3)}_{\mathcal{K}_{3},\infty}$. We summarize the results in the following
proposition.

######  Proposition A.6.

Set $\varsigma_{1}=\mathrm{d}y_{1}$, $\varsigma_{2}=\mathrm{d}y_{2}$,
$\varsigma_{3}=\mathrm{d}x_{1}$, $\varsigma_{4}=\mathrm{d}x_{2}$ and
$\varsigma_{5}=\mathrm{d}x_{3}$. For $i=2$ and $3$, define
$\boldsymbol{\varsigma}^{i}$ as

$\displaystyle\boldsymbol{\varsigma}^{i}=\begin{cases}\begin{pmatrix}\varsigma_{2,1}&\varsigma_{2,0}&\varsigma_{2,-1}&\varsigma_{2,-2}&\varsigma_{1,0}&\varsigma_{1,-1}&\varsigma_{1,-2}&\varsigma_{0,-1}&\varsigma_{0,-2}&\varsigma_{-1,-2}\end{pmatrix}&\text{for
$i=2$},\\\
\begin{pmatrix}\varsigma_{2,1,0}&\varsigma_{2,1,-1}&\varsigma_{2,1,-2}&\varsigma_{2,0,-1}&\varsigma_{2,0,-2}&\varsigma_{2,-1,-2}&\varsigma_{1,0,-1}&\varsigma_{1,0,-2}&\varsigma_{1,-1,-2}&\varsigma_{0,-1,-2}\end{pmatrix}&\text{for
$i=3$}\end{cases}$

where $\varsigma_{j,j^{\prime}}$ and
$\varsigma_{j,j^{\prime},j^{\prime\prime}}$ abbreviate
$\varsigma_{j}\wedge\varsigma_{j^{\prime}}$ and
$\varsigma_{j}\wedge\varsigma_{j^{\prime}}\wedge\varsigma_{j^{\prime\prime}}$
respectively. Then the differential forms
$\begin{pmatrix}\boldsymbol{\omega}^{i}_{3}&\dotsc&\boldsymbol{\omega}_{-3}^{i}\end{pmatrix}$
is described on $Y^{(3)}_{\mathcal{K}_{3},\infty}$ as
$\boldsymbol{\varsigma}^{i}Q^{i}$ for

$\displaystyle Q^{2}$
$\displaystyle=\begin{pmatrix}0&\frac{1}{2y_{1}y_{2}}&0&0&0&-\frac{1}{2y_{1}y_{2}}&0\\\
0&0&\frac{x_{2}+\sqrt{-1}y_{2}}{2y_{1}^{2}y_{2}}&0&\frac{x_{2}-\sqrt{-1}y_{2}}{2y_{1}^{2}y_{2}}&0&0\\\
0&-\frac{\sqrt{-1}}{2y_{1}y_{2}}&0&0&0&-\frac{\sqrt{-1}}{2y_{1}y_{2}}&0\\\
0&0&-\frac{1}{2y_{1}^{2}y_{2}}&0&-\frac{1}{2y_{1}^{2}y_{2}}&0&0\\\
\frac{x_{2}+\sqrt{-1}y_{2}}{8y_{1}y_{2}^{2}}&0&-\frac{x_{2}-5\sqrt{-1}y_{2}}{8y_{1}y_{2}^{2}}&0&-\frac{x_{2}+5\sqrt{-1}y_{2}}{8y_{1}y_{2}^{2}}&0&\frac{x_{2}-\sqrt{-1}y_{2}}{8y_{1}y_{2}^{2}}\\\
0&-\frac{\sqrt{-1}}{4y_{2}^{2}}&0&\frac{\sqrt{-1}}{2y_{2}^{2}}&0&-\frac{\sqrt{-1}}{4y_{2}^{2}}&0\\\
-\frac{1}{8y_{1}y_{2}^{2}}&0&\frac{1}{8y_{1}y_{2}^{2}}&0&\frac{1}{8y_{1}y_{2}^{2}}&0&-\frac{1}{8y_{1}y_{2}^{2}}\\\
\frac{\sqrt{-1}x_{2}-y_{2}}{8y_{1}y_{2}^{2}}&0&-\frac{3(\sqrt{-1}x_{2}+y_{2})}{8y_{1}y_{2}^{2}}&0&\frac{3(\sqrt{-1}x_{2}-y_{2})}{8y_{1}y_{2}^{2}}&0&-\frac{\sqrt{-1}x_{2}+y_{2}}{8y_{1}y_{2}^{2}}\\\
0&0&0&\frac{\sqrt{-1}}{y_{1}^{2}y_{2}}&0&0&0\\\
\frac{\sqrt{-1}}{8y_{1}y_{2}^{2}}&0&-\frac{3\sqrt{-1}}{8y_{1}y_{2}^{2}}&0&\frac{3\sqrt{-1}}{8y_{1}y_{2}^{2}}&0&-\frac{\sqrt{-1}}{8y_{1}y_{2}^{2}}\end{pmatrix},$
$\displaystyle Q^{3}$
$\displaystyle=\begin{pmatrix}\frac{x_{2}+\sqrt{-1}y_{2}}{24y_{1}^{2}y_{2}^{2}}&0&\frac{x_{2}-\sqrt{-1}y_{2}}{8y_{1}^{2}y_{2}^{2}}&0&\frac{x_{2}+\sqrt{-1}y_{2}}{8y_{1}^{2}y_{2}^{2}}&0&\frac{x_{2}-\sqrt{-1}y_{2}}{24y_{1}^{2}y_{2}^{2}}\\\
0&0&0&-\frac{\sqrt{-1}}{3y_{1}y_{2}^{2}}&0&0&0\\\
-\frac{1}{24y_{1}^{2}y_{2}^{2}}&0&-\frac{1}{8y_{1}^{2}y_{2}^{2}}&0&-\frac{1}{8y_{1}^{2}y_{2}^{2}}&0&-\frac{1}{24y_{1}^{2}y_{2}^{2}}\\\
\frac{\sqrt{-1}x_{2}-y_{2}}{24y_{1}^{2}y_{2}^{2}}&0&\frac{\sqrt{-1}x_{2}+y_{2}}{8y_{1}^{2}y_{2}^{2}}&0&-\frac{\sqrt{-1}x_{2}-y_{2}}{8y_{1}^{2}y_{2}^{2}}&0&-\frac{\sqrt{-1}x_{2}+y_{2}}{24y_{1}^{2}y_{2}^{2}}\\\
0&0&0&-\frac{\sqrt{-1}}{6y_{1}^{3}y_{2}}&0&0&0\\\
\frac{\sqrt{-1}}{24y_{1}^{2}y_{2}^{2}}&0&\frac{\sqrt{-1}}{8y_{1}^{2}y_{2}^{2}}&0&-\frac{\sqrt{-1}}{8y_{1}^{2}y_{2}^{2}}&0&-\frac{\sqrt{-1}}{24y_{1}^{2}y_{2}^{2}}\\\
\frac{\sqrt{-1}x_{2}-y_{2}}{48y_{1}y_{2}^{3}}&0&-\frac{\sqrt{-1}x_{2}-3y_{2}}{16y_{1}y_{2}^{3}}&0&\frac{\sqrt{-1}x_{2}+3y_{2}}{16y_{1}y_{2}^{3}}&0&-\frac{\sqrt{-1}x_{2}+y_{2}}{48y_{1}y_{2}^{3}}\\\
0&-\frac{\sqrt{-1}}{8y_{1}^{2}y_{2}^{2}}&0&-\frac{\sqrt{-1}}{12y_{1}^{2}y_{2}^{2}}&0&-\frac{\sqrt{-1}}{8y_{1}^{2}y_{2}^{2}}&0\\\
\frac{\sqrt{-1}}{48y_{1}y_{2}^{3}}&0&-\frac{\sqrt{-1}}{16y_{1}y_{2}^{3}}&0&\frac{\sqrt{-1}}{16y_{1}y_{2}^{3}}&0&-\frac{\sqrt{-1}}{48y_{1}y_{2}^{3}}\\\
0&\frac{1}{8y_{1}^{2}y_{2}^{2}}&0&0&0&-\frac{1}{8y_{1}^{2}y_{2}^{2}}&0\end{pmatrix}.$

##  B. Appendix: Formulas for ${\mathcal{P}}(X,Y,Z,A,B,C)$

We retain the notations in Sections 3.3 and 5.3. In Appendix B, we summarize
several formulas concerning the matrix introduced in (5.6) or Lemma A.2 (i):

$\displaystyle{\mathcal{P}}(X,Y,Z,A,B,C)=\left({\mathcal{P}}_{\alpha,\beta}(X,Y,Z,A,B,C)\right)_{\begin{subarray}{c}\lambda_{3}\geq\alpha\geq-\lambda_{3}\\\
3\geq\beta\geq-3\end{subarray}}\in{\rm
M}_{2\lambda_{3}+1,7}({\mathbf{C}}[X,Y,Z;A,B,C]_{w_{\boldsymbol{\lambda}}}).$

######  Lemma B.1.

For each $-\lambda_{3}\leq\alpha\leq\lambda_{3}$ and $-3\leq\beta\leq 3$, we
have

$\displaystyle{\mathcal{P}}_{\alpha,\beta}(X,Y,Z,A,B,C)$
$\displaystyle=2^{\alpha-\beta}\sum_{\begin{subarray}{c}-\alpha-a+2j+k=w_{\boldsymbol{\lambda}},\\\
a+\beta+2b+c=w_{\boldsymbol{\lambda}}\end{subarray}}(-2^{-2})^{j+b}{}_{w_{\boldsymbol{\lambda}}}H_{j-a,j-\alpha,k}\cdot{}_{w_{\boldsymbol{\lambda}}}H_{b+a,b+\beta,c}$
$\displaystyle\qquad\times\left((-X+\sqrt{-1}Y)^{\alpha-a}(X^{2}+Y^{2})^{j-\alpha}Z^{k}\right)\otimes\left((-A+\sqrt{-1}B)^{a-\beta}(A^{2}+B^{2})^{b+\beta}C^{c}\right),$

where ${}_{e}H_{i,j,k}\in{\mathbf{Z}}$ is defined as in (6.7).

###### Proof..

Using the identities

$\displaystyle z_{1}X+z_{2}Y+z_{3}Z$
$\displaystyle=-\dfrac{1}{2}(z_{1}+\sqrt{-1}z_{2})(-X+\sqrt{-1}Y)-\dfrac{1}{2}(-z_{1}+\sqrt{-1}z_{2})(X+\sqrt{-1}Y)+z_{3}Z,$
$\displaystyle z_{1}A+z_{2}B+z_{3}C$
$\displaystyle=-\dfrac{1}{2}(z_{1}+\sqrt{-1}z_{2})(-A+\sqrt{-1}B)-\dfrac{1}{2}(-z_{1}+\sqrt{-1}z_{2})(A+\sqrt{-1}B)+z_{3}C,$

we expand
$(z_{1}X+z_{2}Y+z_{3}Z)^{w_{\boldsymbol{\lambda}}}\otimes(z_{1}A+z_{2}B+z_{3}C)^{w_{\boldsymbol{\lambda}}}$
as

$(z_{1}X+z_{2}Y+z_{3}Z)^{w_{\boldsymbol{\lambda}}}\otimes(z_{1}A+z_{2}B+z_{3}C)^{w_{\boldsymbol{\lambda}}}\\\
\begin{aligned}
\qquad=\sum_{\begin{subarray}{c}i+j+k=w_{\boldsymbol{\lambda}}\\\
a+b+c=w_{\boldsymbol{\lambda}}\end{subarray}}{}_{w_{\boldsymbol{\lambda}}}H_{i,j,k}\cdot{}_{w_{\boldsymbol{\lambda}}}H_{a,b,c}&\left((-\frac{1}{2})^{i+j}(z_{1}+\sqrt{-1}z_{2})^{i}(-X+\sqrt{-1}Y)^{i}(-z_{1}+\sqrt{-1}z_{2})^{j}(X+\sqrt{-1}Y)^{j}z^{k}_{3}Z^{k}\right)\\\
&\otimes\left((-\frac{1}{2})^{a+b}(z_{1}+\sqrt{-1}z_{2})^{a}(-A+\sqrt{-1}B)^{a}(-z_{1}+\sqrt{-1}z_{2})^{b}(A+\sqrt{-1}B)^{b}z^{c}_{3}C^{c}\right).\end{aligned}$

Changing the variables $i$ to $i+j$ and $a$ to $a+b$, the expanded polynomial
above is calculated as

$\begin{aligned} \sum_{\begin{subarray}{c}i+2j+k=w_{\boldsymbol{\lambda}}\\\
a+2b+c=w_{\boldsymbol{\lambda}}\end{subarray}}(-1)^{i+j+a+b}2^{-(i+2j+a+2b)}{}_{w_{\boldsymbol{\lambda}}}H_{i+j,j,k}\cdot{}_{w_{\boldsymbol{\lambda}}}H_{a+b,b,c}&\left((z_{1}+\sqrt{-1}z_{2})^{i}(-X+\sqrt{-1}Y)^{i}(X^{2}+Y^{2})^{j}z^{2j+k}_{3}Z^{k}\right)\\\
&\otimes\left((z_{1}+\sqrt{-1}z_{2})^{a}(-A+\sqrt{-1}B)^{a}(A^{2}+B^{2})^{b}z^{2b+c}_{3}C^{c}\right)\end{aligned}\\\
\begin{aligned} =\sum_{\begin{subarray}{c}i+2j+k=w_{\boldsymbol{\lambda}}\\\
a+2b+c=w_{\boldsymbol{\lambda}}\end{subarray}}(-1)^{i+j+a+b}2^{-(i+2j+a+2b)}&{}_{w_{\boldsymbol{\lambda}}}H_{i+j,j,k}\cdot{}_{w_{\boldsymbol{\lambda}}}H_{a+b,b,c}(z_{1}+\sqrt{-1}z_{2})^{i+a}z^{2w_{\boldsymbol{\lambda}}-i-a}_{3}\\\
&\times\left((-X+\sqrt{-1}Y)^{i}(X^{2}+Y^{2})^{j}Z^{k}\right)\otimes\left((-A+\sqrt{-1}B)^{a}(A^{2}+B^{2})^{b}C^{c}\right)\end{aligned}\\\
\begin{aligned} \begin{subarray}{c}i\mapsto i-a\\\
=\end{subarray}\sum_{\begin{subarray}{c}i-a+2j+k=w_{\boldsymbol{\lambda}}\\\
a+2b+c=w_{\boldsymbol{\lambda}}\end{subarray}}(-1)^{i+j+b}2^{-(i+2j+2b)}&{}_{w_{\boldsymbol{\lambda}}}H_{i+j-a,j,k}\cdot{}_{w_{\boldsymbol{\lambda}}}H_{a+b,b,c}v^{(2w_{\boldsymbol{\lambda}},0)}_{i}\\\
&\times\left((-X+\sqrt{-1}Y)^{i-a}(X^{2}+Y^{2})^{j}Z^{k}\right)\otimes\left((-A+\sqrt{-1}B)^{a}(A^{2}+B^{2})^{b}C^{c}\right).\end{aligned}$

Here we use the definition $v^{(\lambda,\delta)}_{\pm\mu}=(\pm
z_{1}+\sqrt{-1}z_{2})^{\mu}z^{\lambda-\mu}_{3}\in
V^{(3)}_{\boldsymbol{\lambda}}$ for $0\leq\mu\leq\lambda$ (see Section 3.5 for
details). Hence, for each $-3\leq\beta\leq 3$, we compute

$(z_{1}X+z_{2}Y+z_{3}Z)^{w_{\boldsymbol{\lambda}}}\otimes(z_{1}A+z_{2}B+z_{3}C)^{w_{\boldsymbol{\lambda}}}v^{(3,\delta)}_{\beta}\\\
\begin{aligned} =\sum_{\begin{subarray}{c}i-a+2j+k=w_{\boldsymbol{\lambda}}\\\
a+2b+c=w_{\boldsymbol{\lambda}}\end{subarray}}(-1)^{i+j+b}2^{-(i+2j+2b)}&{}_{w_{\boldsymbol{\lambda}}}H_{i+j-a,j,k}\cdot{}_{w_{\boldsymbol{\lambda}}}H_{a+b,b,c}v^{(2w_{\boldsymbol{\lambda}}+3,\delta)}_{i+\beta}\\\
&\times\left((-X+\sqrt{-1}Y)^{i-a}(X^{2}+Y^{2})^{j}Z^{k}\right)\otimes\left((-A+\sqrt{-1}B)^{a}(A^{2}+B^{2})^{b}C^{c}\right)\end{aligned}\\\
\begin{aligned}
\stackrel{{\scriptstyle\begin{subarray}{c}i\mapsto\alpha-\beta\\\ a\mapsto
a-\beta\end{subarray}}}{{=}}\sum_{\alpha}v^{\boldsymbol{\lambda}}_{\alpha}\sum_{\begin{subarray}{c}\alpha-a+2j+k=w_{\boldsymbol{\lambda}}\\\
a-\beta+2b+c=w_{\boldsymbol{\lambda}}\end{subarray}}(-1)^{\alpha-\beta+j+b}&2^{-(\alpha-\beta+2j+2b)}{}_{w_{\boldsymbol{\lambda}}}H_{\alpha+j-a,j,k}\cdot{}_{w_{\boldsymbol{\lambda}}}H_{a-\beta+b,b,c}\\\
&\times\left((-X+\sqrt{-1}Y)^{\alpha-a}(X^{2}+Y^{2})^{j}Z^{k}\right)\otimes\left((-A+\sqrt{-1}B)^{a-\beta}(A^{2}+B^{2})^{b}C^{c}\right)\end{aligned}\\\
\begin{aligned} \stackrel{{\scriptstyle\begin{subarray}{c}j\mapsto j-\alpha\\\
b\mapsto
b+\beta\end{subarray}}}{{=}}\sum_{\alpha}v^{\boldsymbol{\lambda}}_{\alpha}\sum_{\begin{subarray}{c}-\alpha-a+2j+k=w_{\boldsymbol{\lambda}}\\\
a+\beta+2b+c=w_{\boldsymbol{\lambda}}\end{subarray}}&(-1)^{j+b}2^{-(-\alpha+\beta+2j+2b)}{}_{w_{\boldsymbol{\lambda}}}H_{j-a,j-\alpha,k}\cdot{}_{w_{\boldsymbol{\lambda}}}H_{b+a,b+\beta,c}\\\
&\times\left((-X+\sqrt{-1}Y)^{\alpha-a}(X^{2}+Y^{2})^{j-\alpha}Z^{k}\right)\otimes\left((-A+\sqrt{-1}B)^{a-\beta}(A^{2}+B^{2})^{b+\beta}C^{c}\right)\end{aligned}$

by using multiplicativity (3.5). Comparing with the definition of
$\mathcal{P}(X,Y,Z,A,B,C)$, we find the statement. ∎

The image of ${\mathcal{P}}_{\alpha,\beta}$ under the differential operator
$\nabla^{\boldsymbol{n}}$ in (3.3) is computed as follows:

######  Lemma B.2.

Let $S=-X+\sqrt{-1}Y$ and $T=X+\sqrt{-1}Y$. Then we have

(B.1)
$\nabla^{\boldsymbol{n}}{\mathcal{P}}_{\alpha,\beta}(X,Y)=2^{-n_{1}}(-1)^{w_{\boldsymbol{\lambda}}+\alpha}\sqrt{-1}^{n_{2}+w_{\boldsymbol{\lambda}}}S^{\frac{n_{1}+\alpha-\beta}{2}}T^{\frac{n_{1}-\alpha+\beta}{2}}\\\
\times\sum_{b\in\mathbf{Z}}(-1)^{b}{}_{w_{\boldsymbol{\lambda}}}H_{b+\frac{n_{1}+\alpha+\beta}{2}+n_{2}-w_{\boldsymbol{\lambda}},-b+\frac{n_{1}-\alpha-\beta}{2},2w_{\boldsymbol{\lambda}}-n_{1}-n_{2}}\cdot{}_{w_{\boldsymbol{\lambda}}}H_{-(b+\beta)+w_{\boldsymbol{\lambda}}-n_{2},b+\beta,n_{2}}.$

###### Proof..

Let $\nabla_{k,l}$ be the differential operator which is introduced in (3.2).
By using Lemma B.1, we find that

$\displaystyle\left(2^{\alpha-\beta}\right)^{-1}$
$\displaystyle\nabla_{k,l}{\mathcal{P}}_{\alpha,\beta}(X,Y,Z,A,B,C)$
$\displaystyle=\left(2^{\alpha-\beta}\right)^{-1}\frac{1}{k!\,l!}\frac{\partial^{k}}{\partial
Z^{k}}\otimes\frac{\partial^{l}}{\partial
C^{l}}{\mathcal{P}}_{\alpha,\beta}(X,Y,Z,A,B,C)|_{A=-Y,B=X,Z=C=0}$
$\displaystyle=\sum_{\begin{subarray}{c}-\alpha-a+2j=w_{\boldsymbol{\lambda}}-k\\\
a+\beta+2b=w_{\boldsymbol{\lambda}}-l\end{subarray}}(-2^{-2})^{j+b}{}_{w_{\boldsymbol{\lambda}}}H_{j-a,j-\alpha,k}\cdot{}_{w_{\boldsymbol{\lambda}}}H_{b+a,b+\beta,l}$
$\displaystyle\hskip
60.00009pt\times\left((-X+\sqrt{-1}Y)^{\alpha-a}(X^{2}+Y^{2})^{j-\alpha}\right)\otimes\left((-A+\sqrt{-1}B)^{a-\beta}(A^{2}+B^{2})^{b+\beta}\right)|_{A=-Y,B=X}$
$\displaystyle=\sum_{\begin{subarray}{c}-\alpha-a+2j=w_{\boldsymbol{\lambda}}-k\\\
a+\beta+2b=w_{\boldsymbol{\lambda}}-l\end{subarray}}(-2^{-2})^{j+b}{}_{w_{\boldsymbol{\lambda}}}H_{j-a,j-\alpha,k}\cdot{}_{w_{\boldsymbol{\lambda}}}H_{b+a,b+\beta,l}$
$\displaystyle\hskip
150.00023pt\times(-X+\sqrt{-1}Y)^{\alpha-a}(X^{2}+Y^{2})^{j-\alpha}(Y+\sqrt{-1}X)^{a-\beta}(X^{2}+Y^{2})^{b+\beta}.$

Put $S=-X+\sqrt{-1}Y$ and $T=X+\sqrt{-1}Y$. Then the above equation is
calculated as

$\displaystyle\sum_{\begin{subarray}{c}-\alpha-a+2j=w_{\boldsymbol{\lambda}}-k\\\
a+\beta+2b=w_{\boldsymbol{\lambda}}-l\end{subarray}}(2^{-2})^{j+b}(-1)^{a-\alpha}\sqrt{-1}^{a-\beta}{}_{w_{\boldsymbol{\lambda}}}H_{j-a,j-\alpha,k}\cdot{}_{w_{\boldsymbol{\lambda}}}H_{b+a,b+\beta,l}S^{j+b}T^{j+b-\alpha+\beta}.$

Now substitute $(k,l)=(2w_{\boldsymbol{\lambda}}-n_{1}-n_{2},n_{2})$ as in
Section 3.3. Since

$\displaystyle
2j+2b-\alpha+\beta=2w_{\boldsymbol{\lambda}}-k-l=2w_{\boldsymbol{\lambda}}-(2w_{\boldsymbol{\lambda}}-n_{1}-n_{2})-n_{2}=n_{1},$

we see that $j+b=\frac{n_{1}+\alpha-\beta}{2}$ holds. Moreover we have

$\displaystyle\begin{cases}-\alpha-a+2j=w_{\boldsymbol{\lambda}}-k=n_{1}+n_{2}-w_{\boldsymbol{\lambda}},\\\
a+\beta+2b=w_{\boldsymbol{\lambda}}-l=w_{\boldsymbol{\lambda}}-n_{2},\end{cases}\Longleftrightarrow\quad\begin{cases}a=-2b-\beta-
n_{2}+w_{\boldsymbol{\lambda}}\\\
j=-b+\frac{n_{1}+\alpha-\beta}{2}.\end{cases}$

Substituing these, we obtain the statement. ∎

The following lemma is a special case that $\alpha-\beta=\pm n_{1}$ in Lemma
B.2.

######  Lemma B.3.

We find that

$\displaystyle\nabla^{\boldsymbol{n}}{\mathcal{P}}_{\alpha,\beta}(X,Y)$
$\displaystyle=(-2^{-1})^{n_{1}}\sqrt{-1}^{n_{2}+w_{\boldsymbol{\lambda}}}\binom{w_{\boldsymbol{\lambda}}}{n_{1}+n_{2}-w_{\boldsymbol{\lambda}}}\binom{w_{\boldsymbol{\lambda}}}{w_{\boldsymbol{\lambda}}-n_{2}}\times\begin{cases}(-1)^{n_{2}}T^{n_{1}}&\text{when
}\alpha-\beta=-n_{1},\\\ (-1)^{w_{\boldsymbol{\lambda}}}S^{n_{1}}&\text{when
}\alpha-\beta=n_{1}.\end{cases}$

###### Proof..

Suppose that $\alpha-\beta=-n_{1}$. Substituting this, we find that (B.1)
becomes

$\displaystyle 2^{n_{1}}$
$\displaystyle\sqrt{-1}^{-(n_{2}+w_{\boldsymbol{\lambda}})}\nabla^{\boldsymbol{n}}{\mathcal{P}}_{\alpha,\beta}(X,Y)$
$\displaystyle=(-1)^{w_{\boldsymbol{\lambda}}}T^{n_{1}}\sum_{b\in\mathbf{Z}}(-1)^{b+\alpha}{}_{w_{\boldsymbol{\lambda}}}H_{b+\frac{n_{1}+\alpha+\beta}{2}+n_{2}-w_{\boldsymbol{\lambda}},-b+\frac{n_{1}-\alpha-\beta}{2},2w_{\boldsymbol{\lambda}}-n_{1}-n_{2}}\cdot{}_{w_{\boldsymbol{\lambda}}}H_{-(b+\beta)+w_{\boldsymbol{\lambda}}-n_{2},b+\beta,n_{2}}$
$\displaystyle\stackrel{{\scriptstyle b\mapsto
b-\alpha}}{{=}}(-1)^{w_{\boldsymbol{\lambda}}}T^{n_{1}}\sum_{b\in\mathbf{Z}}(-1)^{b}{}_{w_{\boldsymbol{\lambda}}}H_{b+\frac{n_{1}-(\alpha-\beta)}{2}+n_{2}-w_{\boldsymbol{\lambda}},-b+\frac{n_{1}+(\alpha-\beta)}{2},2w_{\boldsymbol{\lambda}}-n_{1}-n_{2}}\cdot{}_{w_{\boldsymbol{\lambda}}}H_{-b+(\alpha-\beta)+w_{\boldsymbol{\lambda}}-n_{2},b-(\alpha-\beta),n_{2}}$
$\displaystyle=(-1)^{w_{\boldsymbol{\lambda}}}T^{n_{1}}\sum_{b\in\mathbf{Z}}(-1)^{b}{}_{w_{\boldsymbol{\lambda}}}H_{b+n_{1}+n_{2}-w_{\boldsymbol{\lambda}},-b,2w_{\boldsymbol{\lambda}}-n_{1}-n_{2}}\cdot{}_{w_{\boldsymbol{\lambda}}}H_{-(b+n_{1}+n_{2}-w_{\boldsymbol{\lambda}}),b+n_{1},n_{2}}.$

This summation does not vanish if and only if
$b=w_{\boldsymbol{\lambda}}-n_{1}-n_{2}$ holds. In this case the above
summation becomes

$\displaystyle(-1)^{w_{\boldsymbol{\lambda}}}$ $\displaystyle
T^{n_{1}}\sum_{b\in\mathbf{Z}}(-1)^{b}{}_{w_{\boldsymbol{\lambda}}}H_{b+n_{1}+n_{2}-w_{\boldsymbol{\lambda}},-b,2w_{\boldsymbol{\lambda}}-(n_{1}+n_{2})}\cdot{}_{w_{\boldsymbol{\lambda}}}H_{-(b+n_{1}+n_{2}-w_{\boldsymbol{\lambda}}),b+n_{1},n_{2}}$
$\displaystyle=(-1)^{w_{\boldsymbol{\lambda}}}T^{n_{1}}(-1)^{w_{\boldsymbol{\lambda}}-n_{1}-n_{2}}{}_{w_{\boldsymbol{\lambda}}}H_{0,n_{1}+n_{2}-w_{\boldsymbol{\lambda}},2w_{\boldsymbol{\lambda}}-n_{1}-n_{2}}\cdot{}_{w_{\boldsymbol{\lambda}}}H_{0,w_{\boldsymbol{\lambda}}-n_{2},n_{2}}$
$\displaystyle=(-1)^{n_{1}+n_{2}}T^{n_{1}}\binom{w_{\boldsymbol{\lambda}}}{n_{1}+n_{2}-w_{\boldsymbol{\lambda}}}\binom{w_{\boldsymbol{\lambda}}}{w_{\boldsymbol{\lambda}}-n_{2}}.$

This proves the statement in the case where $\alpha-\beta=-n_{1}$ holds. The
case where $\alpha-\beta=n_{1}$ holds is verified similarly, but in this case,
we should substitute $b=n_{1}$ in the last step, which causes the difference
of the signature. ∎

## Acknowledgements

The authors are grateful to Miki Hirano, Taku Ishii and Tadashi Miyazaki for
kindly sending the authors their preprints [HIM] and [IM], which motivate this
research project and play crucial roles in the present article. This work was
partially supported by the Research Institute for Mathematical Sciences, a
Joint Usage/Research Center located in Kyoto University (RIMS satellite
meeting “Special values of automorphic $L$-functions and their $p$-adic
properties” held at Miyama-cho, Kyoto in 2016). The first (resp. second)
author is supported by JSPS Grant-in-Aid for Young Scientists Grant Number
JP18K13395 (resp. for Young Scientists (B) Grant Number JP17K14174).

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where the remainder terms are specified as follows:

$\displaystyle\mathbb{E}\bigl{[}\bar{\mathscr{M}}^{\mathrm{T}\textrm{-}\mathrm{Mil},(\mu),j}_{t_{k+1},t_{k}}|\mathcal{F}_{t_{k}}\bigr{]}=0,\quad
0\leq k\leq 2^{\ell-1}-1,$ (81)

and for any $p\geq 2$ there exist constants $C_{1},C_{2}>0$ independent of
$\mu\in(0,1)$ such that

$\displaystyle\max_{0\leq k\leq
2^{\ell-1}-1}\mathbb{E}\bigl{[}|\bar{\mathscr{M}}^{\mathrm{T}\textrm{-}\mathrm{Mil},(\mu),j}_{t_{k+1},t_{k}}|^{p}\bigr{]}\leq
C_{1}\mu^{p}\Delta_{\ell-1}^{3p/2};$ $\displaystyle\max_{0\leq k\leq
2^{\ell-1}-1}\mathbb{E}\bigl{[}|\bar{\mathscr{N}}^{\mathrm{T}\textrm{-}\mathrm{Mil},(\mu),j}_{t_{k+1},t_{k}}|^{p}\bigr{]}\leq
C_{2}\mu^{2p}\Delta_{\ell-1}^{2p}.$

Similarly, it holds that

$\displaystyle\widetilde{X}^{\mathrm{T}\textrm{-}\mathrm{Mil},(\mu),[\ell],j}_{t_{k+1}}=\widetilde{X}^{\mathrm{T}\textrm{-}\mathrm{Mil},(\mu),[\ell],j}_{t_{k}}+\sum_{0\leq
m\leq
d}\mu^{\mathbf{1}_{m>0}}\sigma_{m}^{j}\bigl{(}\widetilde{X}^{\mathrm{T}\textrm{-}\mathrm{Mil},(\mu),[\ell]}_{t_{k}}\bigr{)}\Delta
B_{t_{k+1},t_{k}}^{m}$
$\displaystyle\qquad+\tfrac{1}{2}\mathscr{L}_{0}\sigma_{0}^{j}\bigl{(}\widetilde{X}^{\mathrm{T}\textrm{-}\mathrm{Mil},(\mu),[\ell]}_{t_{k}}\bigr{)}\Delta\eta_{t_{k+1},t_{k}}^{00}+\mu^{2}\sum_{1\leq
m_{1},m_{2}\leq
d}\mathscr{L}_{m_{1}}\sigma_{m_{2}}^{j}\bigl{(}\widetilde{X}^{\mathrm{T}\textrm{-}\mathrm{Mil},(\mu),[\ell]}_{t_{k}}\bigr{)}\Delta\eta_{t_{k+1},t_{k}}^{m_{1}m_{2}}$
$\displaystyle\qquad+\tfrac{\mu^{2}}{2}\sum_{1\leq m_{1},m_{2}\leq
d}\mathscr{L}_{m_{1}}\sigma_{m_{2}}^{j}\bigl{(}\widetilde{X}^{\mathrm{T}\textrm{-}\mathrm{Mil},(\mu),[\ell]}_{t_{k}}\bigr{)}\bigl{(}\Delta
B_{t_{k+1},t_{k+1/2}}^{m_{1}}\Delta B_{t_{k+1/2},t_{k}}^{m_{2}}-\Delta
B_{t_{k+1/2},t_{k}}^{m_{1}}\Delta B_{t_{k+1},t_{k+1/2}}^{m_{2}}\bigr{)}$
$\displaystyle\qquad+\widetilde{\mathscr{M}}^{~{}\mathrm{T}\textrm{-}\mathrm{Mil},(\mu),j}_{t_{k+1},t_{k}}+\widetilde{\mathscr{N}}^{~{}\mathrm{T}\textrm{-}\mathrm{Mil},(\mu),j}_{t_{k+1},t_{k}},$
(82)

where the remainder terms
$\widetilde{\mathscr{M}}_{t_{k+1},t_{k}}^{~{}\mathrm{T}\textrm{-}\mathrm{Mil},(\mu),j}$
and
$\widetilde{\mathscr{N}}_{t_{k+1},t_{k}}^{~{}\mathrm{T}\textrm{-}\mathrm{Mil},(\mu),j}$
satisfy the same property as the terms
$\bar{\mathscr{M}}_{t_{k+1},t_{k}}^{\mathrm{T}\textrm{-}\mathrm{Mil},(\mu),j}$
and
$\bar{\mathscr{N}}_{t_{k+1},t_{k}}^{\mathrm{T}\textrm{-}\mathrm{Mil},(\mu),j}$,
respectively.

###### Proof.

The proof follows the same argument of [10, Lemma 4.7] and Lemma B.1. We here
only mention that in the case of truncated Milstein scheme, the fine
discretization produces the deterministic $\mathcal{O}(\Delta_{n}^{2})$-term
as:

$\displaystyle\mathscr{L}_{0}\sigma_{0}(\bar{X}_{t_{k}}^{\textrm{T-Mil},(\mu),[\ell]})\Delta_{\ell}^{2}=\tfrac{1}{2}\mathscr{L}_{0}\sigma_{0}(\bar{X}_{t_{k}}^{\textrm{T-Mil},(\mu),[\ell]})\Delta\eta_{t_{k+1},t_{k}}^{00},$
(83)

whereas in the case of the weak second order scheme, we have:

$\displaystyle\mathscr{L}_{0}\sigma_{0}(\bar{X}_{t_{k}}^{f,(\mu),[\ell]})\bigl{\\{}\Delta_{\ell}^{2}+\Delta\eta_{t_{k+1/2},t_{k}}^{00}+\Delta\eta_{t_{k+1},t_{k+1/2}}^{00}\bigr{\\}}=\mathscr{L}_{0}\sigma_{0}(\bar{X}_{t_{k}}^{f,(\mu),[\ell]})\Delta\eta_{t_{k+1},t_{k}}^{00}.$
(84)

The rest of the proof relies on the same discussion in the proof of Lemma B.1,
and thus we omit the detailed proof. ∎

######  Lemma B.3.

Let $1\leq j\leq N$, $1\leq\ell\leq L$, $0\leq k\leq 2^{\ell-1}-1$ and
$t_{k}=k\Delta_{\ell-1}$. It holds that

$\displaystyle\begin{aligned}
\hat{X}^{f,(\mu),[\ell],j}_{t_{k+1}}&=\hat{X}^{f,(\mu),[\ell],j}_{t_{k}}+\sum_{0\leq
m\leq
d}\mu^{\mathbf{1}_{m>0}}\sigma_{m}^{j}(\hat{X}^{f,(\mu),[\ell]}_{t_{k}})\Delta
B_{t_{k+1},t_{k}}^{m}\\\
&\quad+\mathscr{L}_{0}\sigma_{0}^{j}(\hat{X}_{t_{k}}^{f,(\mu),[\ell]})\Delta\eta_{t_{k+1},t_{k}}^{00}+\mu^{2}\sum_{1\leq
m_{1},m_{2}\leq
d}\mathscr{L}_{m_{1}}\sigma_{m_{2}}^{j}(\hat{X}_{t_{k}}^{f,(\mu),[\ell]})\Delta\eta_{t_{k+1},t_{k}}^{m_{1}m_{2}}\\\
&\quad+\hat{\mathscr{M}}_{t_{k+1},t_{k}}^{f,(\mu),j}+\hat{\mathscr{N}}_{t_{k+1},t_{k}}^{f,(\mu),j},\end{aligned}$
(85)

where the remainder terms are specified as follows:

$\displaystyle\mathbb{E}\bigl{[}\hat{\mathscr{M}}_{t_{k+1},t_{k}}^{f,(\mu),j}|\mathcal{F}_{t_{k}}\bigr{]}=0,\quad
0\leq k\leq 2^{\ell-1}-1,$ (86)

and for any $p\geq 2$, there exist constants $C_{1},C_{2}>0$ independent of
$\mu\in(0,1)$ such that

$\displaystyle\max_{0\leq k\leq
2^{\ell-1}-1}\mathbb{E}\bigl{[}|\hat{\mathscr{M}}_{t_{k+1},t_{k}}^{f,(\mu),j}|^{p}\bigr{]}\leq
C_{1}\mu^{p}\Delta_{\ell-1}^{3p/2},\quad\max_{0\leq k\leq
2^{\ell-1}-1}\mathbb{E}\bigl{[}|\hat{\mathscr{N}}_{t_{k+1},t_{k}}^{f,(\mu),j}|^{p}\bigr{]}\leq
C_{2}\mu^{2p}\Delta_{\ell-1}^{2p}.$ (87)

###### Proof.

Using the argument in the proof of Lemma 2.1 in Appendix A.3 together with
Lemma B.1, we conclude. ∎

######  Lemma B.4.

Let $1\leq j\leq N$, $1\leq\ell\leq L$, $0\leq k\leq 2^{\ell-1}-1$ and
$t_{k}=k\Delta_{\ell-1}$. It holds that

$\displaystyle\begin{aligned}
\hat{X}_{t_{k+1}}^{c,(\mu),[\ell-1],j}&=\hat{X}_{t_{k}}^{c,(\mu),[\ell-1],j}+\sum_{0\leq
m\leq d}\sigma_{m}^{j}(\hat{X}_{t_{k}}^{c,(\mu),[\ell-1]})\Delta
B_{t_{k+1},t_{k}}^{m}\\\
&\quad+\mathscr{L}_{0}\sigma_{0}^{j}(\hat{X}_{t_{k}}^{c,(\mu),[\ell-1]})\Delta\eta^{00}_{t_{k+1},t_{k}}+\mu^{2}\sum_{1\leq
m_{1},m_{2}\leq
d}\mathscr{L}_{m_{1}}\sigma_{m_{2}}^{j}(\hat{X}_{t_{k}}^{c,(\mu),[\ell-1]})\Delta\eta^{m_{1}m_{2}}_{t_{k+1},t_{k}}\\\
&\quad+\hat{\mathscr{M}}_{t_{k+1},t_{k}}^{c,(\mu),j}+\hat{\mathscr{N}}_{t_{k+1},t_{k}}^{c,(\mu),j},\end{aligned}$

where the remainder terms are specified as follows:

$\displaystyle\mathbb{E}\bigl{[}\hat{\mathscr{M}}_{t_{k+1},t_{k}}^{c,(\mu),j}|\mathcal{F}_{t_{k}}\bigr{]}=0,\quad
0\leq k\leq 2^{\ell-1}-1,$ (88)

and for any $p\geq 2$ there exist constants $C_{1},C_{2}>0$ independent of
$\mu\in(0,1)$ such that

$\displaystyle\max_{0\leq k\leq
2^{\ell-1}-1}\mathbb{E}\bigl{[}|\hat{\mathscr{M}}_{t_{k+1},t_{k}}^{c,(\mu),j}|^{p}\bigr{]}\leq
C_{1}\mu^{p}\Delta_{\ell-1}^{3p/2},\quad\max_{0\leq k\leq
2^{\ell-1}-1}\mathbb{E}\bigl{[}|\hat{\mathscr{N}}_{t_{k+1},t_{k}}^{c,(\mu),j}|^{p}\bigr{]}\leq
C_{2}\mu^{2p}\Delta_{\ell-1}^{2p}.$ (89)

###### Proof.

From the definition of the standard/antithetic coarse discretizations (6)/(7)
in the main text with the dependence of the small parameter $\mu$, we deduce
the result by applying the argument used in the proof of Lemma A.3 in Appendix
A.4. ∎

######  Lemma B.5.

Let $1\leq j\leq N$, $1\leq\ell\leq L$, $0\leq k\leq 2^{\ell-1}-1$ and
$t_{k}=k\Delta_{\ell-1}$. It holds that

$\displaystyle\hat{X}^{\mathrm{T}\textrm{-}\mathrm{Mil},(\mu),[\ell],j}_{t_{k+1}}$
$\displaystyle=\hat{X}^{\mathrm{T}\textrm{-}\mathrm{Mil},(\mu),[\ell],j}_{t_{k}}+\sum_{0\leq
m\leq
d}\mu^{\mathbf{1}_{m>0}}\sigma_{m}^{j}(\hat{X}^{\mathrm{T}\textrm{-}\mathrm{Mil},(\mu),[\ell]}_{t_{k}})\Delta
B_{t_{k+1},t_{k}}^{m}$
$\displaystyle\quad+\tfrac{1}{2}\mathscr{L}_{0}\sigma_{0}^{j}(\hat{X}_{t_{k}}^{\mathrm{T}\textrm{-}\mathrm{Mil},(\mu),[\ell]})\Delta\eta_{t_{k+1},t_{k}}^{00}+\mu^{2}\sum_{1\leq
m_{1},m_{2}\leq
d}\mathscr{L}_{m_{1}}\sigma_{m_{2}}^{j}(\hat{X}_{t_{k}}^{\mathrm{T}\textrm{-}\mathrm{Mil},(\mu),[\ell]})\Delta\eta_{t_{k+1},t_{k}}^{m_{1}m_{2}}$
$\displaystyle\quad+\hat{\mathscr{M}}_{t_{k+1},t_{k}}^{\mathrm{T}\textrm{-}\mathrm{Mil},(\mu),j}+\hat{\mathscr{N}}_{t_{k+1},t_{k}}^{\mathrm{T}\textrm{-}\mathrm{Mil},(\mu),j},$

where the remainder terms are specified as follows:

$\displaystyle\mathbb{E}\bigl{[}\hat{\mathscr{M}}_{t_{k+1},t_{k}}^{\mathrm{T}\textrm{-}\mathrm{Mil},(\mu),j}|\mathcal{F}_{t_{k}}\bigr{]}=0,\quad
0\leq k\leq 2^{\ell-1}-1,$

and for any $p\geq 2$, there exist constants $C_{1},C_{2}>0$ independent of
$\mu\in(0,1)$ such that

$\displaystyle\max_{0\leq k\leq
2^{\ell-1}-1}\mathbb{E}\bigl{[}|\hat{\mathscr{M}}_{t_{k+1},t_{k}}^{\mathrm{T}\textrm{-}\mathrm{Mil},(\mu),j}|^{p}\bigr{]}\leq
C_{1}\mu^{p}\Delta_{\ell-1}^{3p/2},\quad\max_{0\leq k\leq
2^{\ell-1}-1}\mathbb{E}\bigl{[}|\hat{\mathscr{N}}_{t_{k+1},t_{k}}^{\mathrm{T}\textrm{-}\mathrm{Mil},(\mu),j}|^{p}\bigr{]}\leq
C_{2}\mu^{2p}\Delta_{\ell-1}^{2p}.$ (90)

###### Proof.

The result is deduced from Lemma B.2 and the argument used in the proof of
[10, Lemma 4.9], and we omit the detailed proof. ∎

###  B.4 Proof of Theorem 2.2

###### Proof.

Let $t_{k}=k\Delta_{\ell-1},\,0\leq k\leq 2^{\ell-1}$. We first show (23).
From Lemma B.3 and B.4, we obtain: for $1\leq k\leq 2^{\ell-1}$ and $1\leq
j\leq N$,

$\displaystyle\hat{X}^{f,(\mu),[\ell],j}_{t_{k}}-\hat{X}^{c,(\mu),[\ell-1],j}_{t_{k}}=\sum_{0\leq
i\leq k-1}\sum_{0\leq m\leq
d}\mu^{\mathbf{1}_{m>0}}\bigl{(}\sigma_{m}^{j}(\hat{X}^{f,(\mu),[\ell]}_{t_{i}})-\sigma_{m}^{j}(\hat{X}^{c,(\mu),[\ell-1]}_{t_{i}})\bigr{)}\Delta
B_{t_{i+1},t_{i}}^{m}$ $\displaystyle\qquad+\sum_{0\leq i\leq
k-1}\bigl{(}\mathscr{L}_{0}\sigma_{0}^{j}(\hat{X}^{f,(\mu),[\ell]}_{t_{i}})-\mathscr{L}_{0}\sigma_{0}^{j}(\hat{X}^{c,(\mu),[\ell-1]}_{t_{i}})\bigr{)}\Delta\eta_{t_{i+1},t_{i}}^{00}$
$\displaystyle\qquad+\mu^{2}\sum_{0\leq i\leq k-1}\sum_{1\leq m_{1},m_{2}\leq
d}\bigl{(}\mathscr{L}_{m_{1}}\sigma_{m_{2}}^{j}(\hat{X}^{f,(\mu),[\ell]}_{t_{i}})-\mathscr{L}_{m_{1}}\sigma_{m_{2}}^{j}(\hat{X}^{c,(\mu),[\ell-1]}_{t_{i}})\bigr{)}\Delta\eta_{t_{i+1},t_{i}}^{m_{1}m_{2}}$
$\displaystyle\qquad+\sum_{0\leq i\leq
k-1}\bigl{\\{}\hat{\mathscr{M}}_{t_{i+1},t_{i}}^{f,(\mu),j}+\hat{\mathscr{N}}_{t_{i+1},t_{i}}^{f,(\mu),j}+\hat{\mathscr{M}}_{t_{i+1},t_{i}}^{c,(\mu),j}+\hat{\mathscr{N}}_{t_{i+1},t_{i}}^{c,(\mu),j}\bigr{\\}},$
(91)

where the terms
$\hat{\mathscr{M}}_{t_{i+1},t_{i}}^{f,(\mu),j},\,\hat{\mathscr{N}}_{t_{i+1},t_{i}}^{f,(\mu),j},\,\hat{\mathscr{M}}_{t_{i+1},t_{i}}^{c,(\mu),j},\,\hat{\mathscr{N}}_{t_{i+1},t_{i}}^{c,(\mu),j}$
are defined in Lemma B.3 and B.4. Then, we introduce: for $0\leq n\leq
2^{\ell-1}$ and $p\geq 2$,

$\displaystyle\mathcal{S}_{n}^{\textrm{Weak-2},(\mu)}=\mathbb{E}\Bigl{[}\max_{0\leq
k\leq
n}\bigl{\|}\hat{X}_{t_{k}}^{f,(\mu),[\ell]}-\hat{X}_{t_{k}}^{c,(\mu),[\ell-1]}\bigr{\|}^{p}\Bigr{]}.$

We have from (B.4) that

$\displaystyle\mathcal{S}_{n}^{\textrm{Weak-2},(\mu)}$ $\displaystyle\leq
c_{1}\sum_{1\leq j\leq N}\mathbb{E}\Bigl{[}\max_{0\leq k\leq
n}\bigl{|}\hat{X}_{t_{k}}^{f,(\mu),[\ell],j}-\hat{X}_{t_{k}}^{c,(\mu),[\ell-1],j}\bigr{|}^{p}\Bigr{]}$
$\displaystyle\leq
c_{2}\Bigl{(}\mu^{p}\Delta_{\ell-1}^{p}+\Delta_{\ell-1}\sum_{0\leq k\leq
n-1}\mathcal{S}_{k}^{\textrm{Weak-2},(\mu)}\Bigr{)}$ (92)

for some constants $c_{1},c_{2}>0$ independent of $\mu\in(0,1)$, where we have
used the same argument in the proof of [10, Lemma 4.10] to get (92). Applying
the discrete Grönwall inequality to (92), we obtain (23).

Subsequently, we prove (24). It follows from Lemma B.5 that: for $1\leq k\leq
2^{\ell-1}$ and $1\leq j\leq N$,

$\displaystyle\hat{X}^{\textrm{T-Mil},(\mu),[\ell],j}_{t_{k}}-\bar{X}^{\textrm{T-Mil},(\mu),[\ell-1],j}_{t_{k}}$
$\displaystyle=\sum_{0\leq i\leq k-1}\sum_{0\leq m\leq
d}\mu^{\mathbf{1}_{m>0}}\bigl{(}\sigma_{m}^{j}(\hat{X}^{\textrm{T-Mil},(\mu),[\ell]}_{t_{i}})-\sigma_{m}^{j}(\bar{X}^{\textrm{T-Mil},(\mu),[\ell-1]}_{t_{i}})\bigr{)}\Delta
B_{t_{i+1},t_{i}}^{m}$ $\displaystyle\quad+\mu^{2}\sum_{0\leq i\leq
k-1}\sum_{1\leq m_{1},m_{2}\leq
d}\bigl{(}\mathscr{L}_{m_{1}}\sigma_{m_{2}}^{j}(\hat{X}^{\textrm{T-Mil},(\mu),[\ell]}_{t_{i}})-\mathscr{L}_{m_{1}}\sigma_{m_{2}}^{j}(\bar{X}^{\textrm{T-Mil},(\mu),[\ell-1]}_{t_{i}})\bigr{)}\Delta\eta_{t_{i+1},t_{i}}^{m_{1}m_{2}}$
$\displaystyle\quad+\sum_{0\leq i\leq
k-1}\tfrac{1}{2}\mathscr{L}_{0}\sigma_{0}^{j}(\hat{X}^{\textrm{T-Mil},(\mu),[\ell]}_{t_{i}})\Delta\eta_{t_{i+1},t_{i}}^{00}-\sum_{0\leq
i\leq
k-1}\mathscr{L}_{0}\sigma_{0}^{j}(\bar{X}^{\textrm{T-Mil},(\mu),[\ell-1]}_{t_{i}})\Delta\eta_{t_{i+1},t_{i}}^{00}$
$\displaystyle\quad+\sum_{0\leq i\leq
k-1}\bigl{\\{}\hat{\mathscr{M}}_{t_{i+1},t_{i}}^{\textrm{T-Mil},(\mu),j}+\hat{\mathscr{N}}_{t_{i+1},t_{i}}^{\textrm{T-Mil},(\mu),j}\bigr{\\}},$
(93)

where the terms
$\hat{\mathscr{M}}_{t_{i+1},t_{i}}^{\textrm{T-Mil},(\mu),j},\,\hat{\mathscr{N}}_{t_{i+1},t_{i}}^{\textrm{T-Mil},(\mu),j}$
are defined in Lemma B.5. We define: for $0\leq n\leq 2^{\ell-1}$ and $p\geq
2$,

$\displaystyle\mathcal{S}_{n}^{\textrm{T-Mil},(\mu)}=\mathbb{E}\Bigl{[}\max_{0\leq
k\leq
n}\bigl{\|}\hat{X}_{t_{k}}^{\textrm{T-Mil},(\mu),[\ell]}-\bar{X}_{t_{k}}^{\textrm{T-Mil},(\mu),[\ell-1]}\bigr{\|}^{p}\Bigr{]}.$

Then, it follows from (B.4) that

$\displaystyle\mathcal{S}_{n}^{\textrm{T-Mil},(\mu)}\leq
C\Bigl{(}\Delta_{\ell-1}^{p}+\Delta_{\ell-1}\sum_{0\leq k\leq
n-1}\mathcal{S}_{k}^{\textrm{T-Mil},(\mu)}\Bigr{)}$ (94)

for some $C>0$ independent of $\mu\in(0,1)$, where we have used again the
argument in the proof of [10, Lemma 4.10]. We here note that the first term of
the R.H.S. of (94) is now independent of $\mu\in(0,1)$ and comes from:

$\displaystyle\mathbb{E}\left[\max_{0\leq k\leq n}\Bigl{|}\sum_{0\leq i\leq
k-1}\tfrac{1}{2}\mathscr{L}_{0}\sigma_{0}^{j}(\hat{X}^{\textrm{T-Mil},(\mu),[\ell]}_{t_{i}})\Delta\eta_{t_{i+1},t_{i}}^{00}-\sum_{0\leq
i\leq
k-1}\mathscr{L}_{0}\sigma_{0}^{j}(\bar{X}^{\textrm{T-Mil},(\mu),[\ell-1]}_{t_{i}})\Delta\eta_{t_{i+1},t_{i}}^{00}\Bigr{|}^{p}\right]$
(95) $\displaystyle\leq c_{1}n^{p-1}\sum_{0\leq i\leq
n-1}|\Delta\eta_{t_{i+1},t_{i}}^{00}|^{p}\leq
c_{2}n^{p}\Delta_{\ell-1}^{2p}\leq c_{2}T^{p}\Delta_{\ell-1}^{p},$

for some $c_{1},c_{2}>0$ independent of $\mu\in(0,1)$. In particular, due to
‘$\tfrac{1}{2}$’, the term (95) is not bounded as the second term of the
R.H.S. of (94) in the case of the truncated Milstein scheme. Finally, applying
the discrete Grönwall inequality to (94), we conclude. ∎

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