# A cohomological interpretation of archimedean zeta integrals for ${\rm
GL}_{3}\times{\rm GL}_{2}$

Takashi Hara and Kenichi Namikawa  Department of Mathematics, College of
Liberal Arts, Tsuda University, 2-1-1 Tsuda-machi, Kodaira City, Tokyo
187-8577, Japan<EMAIL_ADDRESS>Faculty of Mathematics, Kyushu University,
744 Motooka, Nishi-Ku, Fukuoka, 819-0395, Japan<EMAIL_ADDRESS>
###### Abstract.

By studying an explicit form of the Eichler–Shimura map for ${\rm GL}_{3}$, we
describe a precise relation between critical values of the complete
$L$-function for the Rankin–Selberg convolution ${\rm GL}_{3}\times{\rm
GL}_{2}$ and the cohomological cup product of certain rational cohomology
classes which are uniquely determined up to rational scalar multiples from the
cuspidal automorphic representations under consideration. This refines
rationality results on critical values due to Raghuram et al.

###### 2010 Mathematics Subject Classification:

Primary 11F67, Secondary 11F75, 11F70

###### Contents

  1. 1 Introduction
  2. 2 Notation
  3. 3 Models of finite dimensional representations
  4. 4 Motives associated to the Rankin–Selberg convolution ${\rm GL}_{3}\times{\rm GL}_{2}$
  5. 5 Eichiler–Shimura maps for ${\rm GL}_{3}$
  6. 6 Cohomological cup products and global zeta integrals for ${\rm GL}_{3}\times{\rm GL_{2}}$
  7. 7 Evaluation of period integrals
  8. A Appendix: Construction of the Eichler–Shimura maps for ${\rm GL}_{3}$
  9. B Appendix: Formulas for ${\mathcal{P}}(X,Y,Z,A,B,C)$

##  1\. Introduction

The relation between certain cohomology classes and special values of
$L$-functions is a fascinating subject in number theory. Some of special
values of $L$-functions can be interpreted in terms of certain cohomology
classes, and rationality of the special values is often deduced from
rationality of those classes. In [Del79], Deligne introduces the notion of
periods associated to pure motives by using the comparison isomorphisms
between their Betti and de Rham realizations, and he makes a conjecture which
yields that critical values of the $L$-functions associated to them coincide
with his periods up to multiplication of algebraic numbers. Meanwhile, Clozel
conjectures the existence of motives associated to cohomological cuspical
automorphic representations of ${\rm GL}_{n}$ in [Clo90, Conjecture 4.5],
which provides us with motivic interpretation of automorphic $L$-functions
concerning the general linear groups. In the present article, we focus on the
critical values of the Rankin–Selberg $L$-functions associated to
cohomological cuspidal automorphic representations of ${\rm GL}_{3}\times{\rm
GL}_{2}$, and provide their precise cohomological interpretation.

Rationality of critical values of the Rankin–Selberg $L$-functions is firstly
studied by Manin [Man72] and Shimura [Shim76, Theorems 1 and 4] for ${\rm
GL}_{2}\times{\rm GL}_{1}$. Generalizing these classical works, Mahnkopf
considers in [Mah05] rationality of critical values of the Rankin–Selberg
$L$-functions of ${\rm GL}_{n+1}\times{\rm GL}_{n}$ for general $n$, based on
the generalized modular symbol method proposed by Kazhdan, Mazur and Schmidt
in [KMS00]. Meanwhile, Raghuram and Shahidi introduce the notion of Whittaker
periods in [RS08, Definition/Proposition 3.3] for irreducible cuspidal
automorphic representations $\pi$ of ${\rm GL}_{n}(F_{\mathbf{A}})$, where $F$
is a number field. They define their periods by using the comparison
isomorphisms between the Betti and de Rham cohomology groups of the symmetric
spaces attached to ${\rm GL}_{n}$, which are regarded as substitutes of
Clozel’s conjectural motives associated to $\pi$. Furthermore, Raghuram shows
that the critical values of the Rankin–Selberg $L$-functions for ${\rm
GL}_{n+1}\times{\rm GL}_{n}$ coincide with, up to multiplication of algebraic
numbers, the product of Raghuram–Shahidi’s Whittaker periods and the inverse
of certain archimedean zeta integrals (see [Rag16, Theorem 2.50] for details).
This implies that one should precisely study the archimedean zeta integrals
appearing as the equality [Rag16, (2.47)] in order to discuss rationality of
critical values of the Rankin–Selberg $L$-functions with respect to
Raghuram–Shahidi’s periods. Recently Sun has proved in [Sun17] that these
archimedean zeta integrals do not vanish, and Januszewski has studied in
[Jan19] their behaviors under Tate twists. However, as far as the authors
know, what is known on these archimedean zeta integrals is fairly little
except for Sun and Januszewski’s results. The explicit study of the
archimedean zeta integrals is essentially important even for the study of the
$p$-adic Rankin–Selberg $L$-functions for ${\rm GL}_{n+1}\times{\rm GL}_{n}$
made in [Jan], since it is necessary to establish certain Kummer-type
congruences (which are introduced as the Manin congruences in [Jan]). In this
article, based on the recent precise study of archimedean zeta integrals for
$\mathrm{GL}_{3}\times\mathrm{GL}_{2}$ made by Hirano, Ishii and Miyazaki in
[HIM], we show that the archimedean zeta integrals under consideration
coincide with the multiples of the desired $\Gamma$-factors with certain
subtle but explicit constants when $n$ equals $2$. Our explicit study of the
archimedean zeta integrals should be considered as a refinement of Raghuram,
Sun and Januszewski’s works, and we expect that it will be useful for the
study of the $p$-adic Rankin–Selberg $L$-functions for ${\rm GL}_{3}\times{\rm
GL}_{2}$.

In order to write down the precise statement of the main theorem in this
article, we first review the critical values of the Rankin–Selberg
$L$-functions for $\mathrm{GL}_{3}\times\mathrm{GL}_{2}$. Put
$\Sigma_{\mathbf{Q}}$ as the set of all places of ${\mathbf{Q}}$, and denote
by $\infty$ its unique infinite place. For $n=2$ or $3$, let
$\pi^{(n)}=\bigotimes^{\prime}_{v\in\Sigma_{\mathbf{Q}}}\pi^{(n)}_{v}$ denote
a cohomological irreducible cuspidal automorphic representation of ${\rm
GL}_{n}({\mathbf{Q}}_{\mathbf{A}})$. Since $\pi^{(2)}$ is cohomological, the
archimedean part $\pi^{(2)}_{\infty}$ of $\pi^{(2)}$ is a discrete series
representation $D_{\nu_{2},l_{2}}$ of ${\rm GL}_{2}({\mathbf{R}})$, and (the
archimedean part of) the central character $\omega_{\pi^{(2)}}$ of $\pi^{(2)}$
satisfies $\omega_{\pi^{(2)},\infty}(t)=t^{2\nu_{2}}$ for every
$t\in{\mathbf{R}}^{\times}$ with $t>0$. We further find that, since
$\pi^{(3)}$ is also cohomological, the archimedean part $\pi^{(3)}_{\infty}$
of $\pi^{(3)}$ is a generalized principal series representation of ${\rm
GL}_{3}({\mathbf{R}})$ parabolically induced from the direct product of a
discrete series representation $D_{\nu_{3},l_{3}}$ of ${\rm
GL}_{2}({\mathbf{R}})$ and a character $\chi_{\nu_{3}}^{\delta}$ of
${\mathbf{R}}^{\times}$. See Section 4.1 for details on the parameters of
these representations. Since our method used in this article is deeply based
upon the modular symbol method adopted in [Mah05], we always have to assume
that the inequality $(1\leq)\,l_{2}<l_{3}$ holds.

We prefer to use the language of the motive attached to $\pi^{(n)}$, since it
makes expositions concise. We let ${\mathcal{M}}[\pi^{(n)}]$ denote the
(conjectural) motive over ${\mathbf{Q}}$ attached to $\pi^{(n)}$ satisfying

$\displaystyle
L(s,{\mathcal{M}}[\pi^{(n)}])=L\left(s-\frac{n-1}{2},\pi^{(n)}\right),$

although it is not yet constructed except for the case $n=2$. The expected
properties of ${\mathcal{M}}[\pi^{(n)}]$ is precisely written down in [Clo90],
and we briefly summarize them in Section 4. Note that our main result (Theorem
1.1) does not depend on conjectural hypotheses such as the existence of
$\mathcal{M}[\pi^{(3)}]$. Define ${\mathcal{M}}(\pi^{(3)}\times\pi^{(2)})$ to
be the tensor product
${\mathcal{M}}[\pi^{(3)}]\otimes{\mathcal{M}}[\pi^{(2)}]$ of motives. Then the
complete Rankin–Selberg $L$-function $L(s,\pi^{(3)}\times\pi^{(2)})$ is
interpreted as the (complete) $L$-function associated to the motive
$\mathcal{M}(\pi^{(3)}\times\pi^{(2)})$; namely we obtain the identity of the
$L$-functions

$\displaystyle
L(s,{\mathcal{M}}(\pi^{(3)}\times\pi^{(2)}))=L\left(s-\frac{3}{2},\pi^{(3)}\times\pi^{(2)}\right).$

Let us normalize the parameter $\nu_{n}$ as
$\nu_{n}=-\frac{l_{n}}{2}+\frac{n-1}{2}$, so that
${\mathcal{M}}(\pi^{(3)}\times\pi^{(2)})$ is pure of weight $l_{2}+l_{3}$.
Then, by studying the Hodge decomposition of the motive
$\mathcal{M}(\pi^{(3)}\times\pi^{(2)})$, we find that $m\in\mathbf{Z}$ is a
critical point of $L(s,{\mathcal{M}}(\pi^{(3)}\times\pi^{(2)}))$ if and only
if it satisfies the following inequalities

$\displaystyle\begin{cases}\dfrac{l_{3}}{2}+1\leq
m\leq\dfrac{l_{3}}{2}+l_{2}&\text{when }l_{2}\leq\dfrac{l_{3}}{2}\text{
holds},\\\ l_{2}+1\leq m\leq l_{3},&\text{when
}\dfrac{l_{3}}{2}<l_{2}<l_{3}\text{ holds}.\end{cases}$

See also the argument at the end of Section 4.2. In the rest of Introduction,
we always suppose that $m$ satisfies the above condition.

Now let us briefly recall the strategy to study critical values
$L(m,{\mathcal{M}}(\pi^{(3)}\times\pi^{(2)}))$ via the (generalized) modular
symbol method utilized in [KMS00, Mah05, Rag16] and [Jan]. Let
${\mathcal{K}}_{n}$ be an open compact subgroup of ${\rm
GL}_{n}(\widehat{\mathbf{Z}})$ and $Y^{(n)}_{\mathcal{K}_{n}}$ the
corresponding symmetric space ${\rm GL}_{n}({\mathbf{Q}})\backslash{\rm
GL}_{n}({\mathbf{Q}}_{\mathbf{A}})/{\mathbf{R}}^{\times}_{>0}{\rm
SO}_{n}({\mathbf{R}}){\mathcal{K}}_{n}$. Suppose that ${\mathcal{K}}_{2}={\rm
GL}_{2}(\widehat{\mathbf{Z}})$ and that ${\mathcal{K}}_{3}$ is the mirabolic
subgroup of ${\rm GL}_{3}(\widehat{\mathbf{Z}})$ consisting of matrices whose
bottom rows are congruent to $(0,0,1)$ modulo an ideal ${\mathfrak{N}}$ of
$\widehat{\mathbf{Z}}$ (see also Remark 7.1 on the assumption
$\mathcal{K}_{2}=\mathrm{GL}_{2}(\widehat{\mathbf{Z}})$). We also assume that
$\pi^{(2)}$ has a ${\mathcal{K}}_{2}$-fixed vector and that $\mathfrak{N}$ is
minimal among ideals of $\widehat{\mathbf{Z}}$ such that $\pi^{(3)}$ has a
${\mathcal{K}}_{3}$-fixed vector. Since $\pi^{(n)}$ is cohomological, we can
define cohomology classes $\eta_{\pi^{(3)}}$ and $\eta^{\pm}_{\pi^{(2)}}$ in
certain cohomology groups of rational local systems on
$Y^{(3)}_{\mathcal{K}_{3}}$ and $Y^{(2)}_{\mathcal{K}_{2}}$ respectively (see
Section 7.1 for the definition of these classes). Raghuram and Shahidi define
their periods $\Omega_{\pi^{(3)}}$ and $\Omega^{\pm}_{\pi^{(2)}}$ attached to
$\pi^{(3)}$ and $\pi^{(2)}$ in [RS08] as the ratios of $\eta_{\pi^{(3)}}$ and
$\eta^{\pm}_{\pi^{(2)}}$ to the images of appropriate cusp forms of
$\pi^{(3)}$ and $\pi^{(2)}$ under the Eichler–Shimura maps $\delta^{(3)}$ and
$\delta^{(2)}$ respectively. One of the notable features in this article is
that we give an explicit construction of the Eichler–Shimura map
$\delta^{(3)}$ for $\mathrm{GL}_{3}$ (see Section 5.3 and Appendix A for
details), which enables us to proceed computation of the archimedean local
zeta integrals. Furthermore, let us consider the natural projection

$\displaystyle{\rm p}_{n}$
$\displaystyle:{\mathcal{Y}}^{(n)}_{\mathcal{K}_{n}}:={\rm
GL}_{n}({\mathbf{Q}})\backslash{\rm GL}_{n}({\mathbf{Q}}_{\mathbf{A}})/{\rm
SO}_{n}({\mathbf{R}}){\mathcal{K}}_{n}\longrightarrow
Y^{(n)}_{\mathcal{K}_{n}}.$

By considering ${\rm GL}_{2}$ as a subgroup of ${\rm GL}_{3}$ via the
embedding $g\mapsto\begin{pmatrix}g&\\\ &1\end{pmatrix}$, the branching rule
for irreducible algebraic representations of ${\rm GL}_{3}$ and ${\rm GL}_{2}$
induces maps from local systems on ${\mathcal{Y}}^{(3)}_{\mathcal{K}_{3}}$ to
those on ${\mathcal{Y}}^{(2)}_{\mathcal{K}_{2}}$; therefore, using the
branching rule, we can define cohomology classes
$\nabla^{\boldsymbol{n}_{m}}\iota^{*}{\rm p}^{\ast}_{3}\eta_{\pi^{(3)}}$ for
$\boldsymbol{n}_{m}=(l_{2}-1,m-l_{2}-1)$. See Section 3.3 for the definition
of the map $\nabla^{\boldsymbol{n}}$ for $\boldsymbol{n}=(n_{1},n_{2})$ coming
from the branching rule. Then the rationality of critical values
$L(m,{\mathcal{M}}(\pi^{(3)}\times\pi^{(2)}))$ is verified by a study of the
cup product of two cohomology classes
$\nabla^{\boldsymbol{n}_{m}}\iota^{*}{\rm p}^{\ast}_{3}\eta_{\pi^{(3)}}$ and
${\rm p}^{\ast}_{2}\eta_{\pi^{(2)}}$.

One of the difficulties concerning the modular symbol method is to study the
archimedean zeta integrals appearing in the unfolding of the cup product
$I(m,\pi^{(3)},\pi^{(2)})$ of $\nabla^{\boldsymbol{n}_{m}}\iota^{*}{\rm
p}^{\ast}_{3}\eta_{\pi^{(3)}}$ and ${\rm p}^{\ast}_{2}\eta^{\pm}_{\pi^{(2)}}$;
see the arguments in [Rag16, Section 2.5.3.6]. In this article, by giving an
effective form of the Eichler–Shimura maps for ${\rm GL}_{3}$ and using a
detailed study of archimedean Whittaker functions of $\pi^{(n)}$ made in
[HIM], we refine the definition of Raghuram and Shahidi’s periods
$\Omega_{\pi^{(3)}}$ and $\Omega^{\pm}_{\pi^{(2)}}$, and then give an explicit
formula for the archimedean zeta integral appearing in the computation of the
cup product pairing. The main result in this article is as follows:

######  Theorem 1.1.

(=Theorem 7.2) Suppose that $(-1)^{m+\delta+\frac{l_{3}}{2}}=\pm 1$ holds.
Then we have

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

where $\binom{a}{b}$ denotes the binomial coefficient $\frac{a!}{b!(a-b)!}$.
If $(-1)^{m+\delta+\frac{l_{3}}{2}}\neq\pm 1$ holds, we have
$I(m,\pi^{(3)},\pi^{(2)})=0$.

Theorem 1.1 is derived from explicit evaluations of archimedean zeta integrals
in [Rag16, (2.49)], and hence this refines a formula proposed in [Rag16,
Theorem 2.50]. In particular, Deligne’s conjecture implies that
$\Omega_{\pi^{(3)}}\Omega^{\pm}_{\pi^{(2)}}$ coincides with Deligne’s period
$c^{+}({\mathcal{M}}(\pi^{(3)}\times\pi^{(2)})(m))$ up to multiplication of an
algebraic number, and thus Theorem 1.1 clarifies a motivic background of
Raghuram–Shahidi’s Whittaker period
$\Omega_{\pi^{(3)}}\Omega^{\pm}_{\pi^{(2)}}$. See Remarks 7.3 and 7.4 for more
details on the result in [Rag16] and the compatibility with Deligne’s
conjecture. See also Remark 7.6 (and [Yos01]) on the motivic interpretation of
each of $\Omega_{\pi^{(3)}}$ and $\Omega^{\pm}_{\pi^{(2)}}$.

We would also like to explain a motivation of Theorem 1.1 from a viewpoint of
the study of the $p$-adic $L$-functions. In a series of his papers,
Januszewski constructs the $p$-adic Rankin–Selberg $L$-functions for ${\rm
GL}_{n+1}\times{\rm GL}_{n}$ (see [Jan] and references therein for details).
However, some of expected important properties proposed in Coates and Perrin-
Riou’s conjecture [CP89], [Coa89] have not been established yet. Among such
kinds of properties, we take up problems on effects of choices of periods on
the Manin conguruences. Indeed, the periods which Januszewski introduces in
[Jan, Theorem A] are defined critical pointwisely, and they a priori depend on
the choice of critical points $m$. Such dependence of periods on critical
points also appears in [Rag16, Theorem 1.1] for the completely same reason.
Meanwhile, it is important to determine an explicit relation between cup
product pairings and critical values including the signature, powers of
$\sqrt{-1}$ and so on, to establish congruences between the critical values at
different critical points. Theorem 1.1 exhibits this explicit relation and
hence we expect that the formula in Theorem 1.1 will be useful for the further
study of the $p$-adic Rankin–Selberg $L$-functions. We also mention that this
point is also a motivation of Januszewski’s study on period relations made in
[Jan19], and Theorem 1.1 refines the main result in [Jan19] in the case of
${\rm GL}_{3}\times{\rm GL}_{2}$. See Remark 7.4 for the relation between
Januszewski’s works and ours.

The organization of this article is as follows. Section 2 is devoted to a
preparation of necessary general notation. Section 3 is a summary on finite
dimensional representations which we use throughout this article. We remark
that here we propose not only general theory but also explicit models of
representations under consideration. Moreover, we give a branching rule for
the pair $({\rm GL}_{3},{\rm GL}_{2})$ in an explicit manner (see Section 3.3
for details), which is one of the keys in the present article. In Section 4,
we introduce our setting on automorphic representations for which we consider
the Rankin–Selberg $L$-function, and then survey the theory of motives
associated with $\mathrm{GL}_{3}\times\mathrm{GL}_{2}$ according to [Clo90].
The fundamental tool in this article is to give an appropriate form of the
Eichler–Shimura map for ${\rm GL}_{3}$, which is a map from the space of cusp
forms in $\pi^{(3)}$ to a cohomology group of $Y^{(3)}_{\mathcal{K}_{3}}$ of
degree 2 with certain local coefficients. In Section 5, we introduce notion of
cusp forms in this article and the Eichler–Shimura maps for ${\rm GL}_{2}$ and
${\rm GL}_{3}$. Although the Eichler–Shimura map for ${\rm GL}_{3}$ (resp. for
${\rm GL}_{2}$) to a cohomology group of degree $2$ (resp. of degree $1$) is
sufficient for the purpose of this article, we can also consider the
Eichler–Shimura map for ${\rm GL}_{3}$ to a cohomology group of degree $3$. We
shall sketch the construction of the Eichler–Shimura maps for
$\mathrm{GL}_{3}$ into not only the cohomology groups of degree $2$ but also
those of degree $3$ in Appendix A, because it has independent significance in
itself beyond the scope of this article, We then summarize explicit formulas
on differential forms and their connections to the global integrals in Section
6. Some of the formulas are deduced from longsome but straightforward
computations, which we postpone to Appendix B for the sake of readability of
this article. We finally propose the definition of periods
$\Omega_{\pi^{(3)}}$ and $\Omega_{\pi^{(2)}}^{\pm}$, and complete the proof of
Theorem 1.1 in Section 7. We also discuss a motivic interpretation of our
periods and compare our results with the previous researches.

We end the introduction with a remark on the base field. The method in this
article is given in the adèlic language, and hence the main theorem (Theorem
1.1) is generalized to the case where the base field is totally real in a
straightforward way. We leave it to the reader to make the notation and the
argument concise.

##  2\. Notation

Denote by ${\mathbf{Q}}$ (resp. ${\mathbf{Z}}$, ${\mathbf{R}}$,
${\mathbf{C}}$) the rational number field (resp. the ring of the rational
integers, the real number field, the complex number field). Let
${\mathbf{Q}}_{\mathbf{A}}$ be the ring of adèles of $\mathbf{Q}$ and
${\mathbf{Q}}_{\mathbf{A},{\rm fin}}$ the subring of finite adèles. Write the
set of all places of ${\mathbf{Q}}$ as $\Sigma_{\mathbf{Q}}$ and denote by
$\infty$ its unique infinite place. For each
$x=(x_{v})_{v\in\Sigma_{\mathbf{Q}}}\in{\mathbf{Q}}_{\mathbf{A}}$, we put
$x_{\rm
fin}=(x_{v})_{v\in\Sigma_{\mathbf{Q}}\setminus\\{\infty\\}}\in{\mathbf{Q}}_{\mathbf{A},{\rm
fin}}$. Let $G$ be an algebraic group $G$ over ${\mathbf{Q}}$. Then, for each
adèlic point $g\in G({\mathbf{Q}}_{\mathbf{A}})$, we write the infinite (resp.
finite) part of $g$ as $g_{\infty}$ (resp. $g_{\rm fin}$). We denote the
$v$-th component of $g$ by $g_{v}$ for each $v\in\Sigma_{\mathbf{Q}}$. We
always use similar notation for each adèlic object.

In this article, we use the terminology “$L$-function” to denote a complete
one. For instance, for each motive ${\mathcal{M}}$ over ${\mathbf{Q}}$, the
$L$-function $L(s,{\mathcal{M}})$ is defined to be the product of the
$\Gamma$-factor $L_{\infty}(s,{\mathcal{M}})$ and the finite part $L_{\rm
fin}(s,{\mathcal{M}})$.

##  3\. Models of finite dimensional representations

In this section, we summarize classifications and models of finite dimensional
representations appearing in this article. We also introduce isomorphisms
inducing so-called “the branching rules” in explicit forms. In particular, we
follow [HIM, Section 3.1, 4.1] on the descriptions of representation spaces
and the branching rule for the orthogonal groups introduced in Sections 3.4,
3.5 and 3.6.

###  3.1. Irreducible representations of ${\rm GL}_{2}$

Let $\mathcal{A}$ be a (commutative) integral domain of characteristic $0$.
For ${\boldsymbol{n}}=(n_{1},n_{2})$ with $n_{1}\in{\mathbf{Z}},n_{1}\geq 0$
and $n_{2}\in{\mathbf{Z}}$, let
$L^{(2)}({\boldsymbol{n}};{\mathcal{A}})={\mathcal{A}}[X,Y]_{n_{1}}$ be the
set of homogenous polynomials of degree $n_{1}$ in variables $X$ and $Y$ with
coefficients in $\mathcal{A}$. Define an action
$\varrho^{(2)}_{{\boldsymbol{n}}}$ of ${\rm GL}_{2}({\mathcal{A}})$ on
$L^{(2)}({\boldsymbol{n}};{\mathcal{A}})$ as follows:

$\displaystyle\varrho^{(2)}_{{\boldsymbol{n}}}(g)P(X,Y)$ $\displaystyle=(\det
g)^{n_{2}}P((X,Y)g)$ $\displaystyle\text{for }P\in
L^{(2)}({\boldsymbol{n}};{\mathcal{A}})\text{ and }g\in{\rm
GL}_{2}({\mathcal{A}}).$

When $\mathcal{A}$ is a field, it is well known that
$(\varrho^{(2)}_{{\boldsymbol{n}}},L^{(2)}({\boldsymbol{n}};{\mathcal{A}}))$
gives an irreducible algebraic representation of ${\rm GL}_{2}({\mathcal{A}})$
of highest weight $(n_{1}+n_{2},n_{2})$. We finally define the central
character
$\omega_{\boldsymbol{n}}\colon\mathcal{A}^{\times}\rightarrow\mathcal{A}^{\times}$
of $L^{(2)}(\boldsymbol{n};\mathcal{A})$ so that
$\varrho^{(2)}_{\boldsymbol{n}}(x1_{2})P(X,Y)=\omega_{\boldsymbol{n}}(x)P(X,Y)$
holds for each $x\in\mathcal{A}^{\times}$ and $P(X,Y)\in
L^{(2)}(\boldsymbol{n};\mathcal{A})$.

###  3.2. Irreducible representations of ${\rm GL}_{3}$

Following [FH13, Section 12], we can explicitly construct an irreducible
algebraic representation $L^{(3)}(\boldsymbol{w};{\mathbf{C}})$ of ${\rm
GL}_{3}({\mathbf{R}})$ for ${\boldsymbol{w}}=(w_{1}^{+},w_{1}^{-},w_{2})$,
with $w_{1}^{+},w_{1}^{-},w_{2}\in{\mathbf{Z}}$ and $w_{1}^{+},w_{1}^{-}\geq
0$. The highest weight of $L^{(3)}({\boldsymbol{w}};{\mathbf{C}})$ will be
given by $(w_{1}^{+}+w_{2},w_{2},-w_{1}^{-}+w_{2})$. Later we mainly use this
representation for the case where $w_{1}^{+}=w_{1}^{-}=w_{2}$ holds. Since the
construction is purely algebraic, we construct
$L^{(3)}({\boldsymbol{w}};\mathcal{A})$ for a general integral domain
$\mathcal{A}$ of characteristic $0$.

Let $\mathcal{A}$ be a (commutative) integral domain of characteristic $0$ and
set ${\boldsymbol{w}}=(w_{1}^{+},w_{1}^{-},w_{2})$ with
$w_{1}^{+},w_{1}^{-},w_{2}\in\mathbf{Z}$ and $w_{1}^{+},w_{1}^{-}\geq 0$. Let
${\rm Std}_{3}={\mathcal{A}}^{3}$ be the standard representarion of ${\rm
GL}_{3}({\mathcal{A}})$, ${\rm Symm}^{w_{1}^{+}}{\rm Std}_{3}$ the
$w_{1}^{+}$-th symmetric tensor product of ${\rm Std}_{3}$ and ${\rm
Symm}^{w_{1}^{-}}{\rm Std}^{\vee}_{3}$ the $w_{1}^{-}$-th symmetric tensor
product of the contragradient of $\mathrm{Std}_{3}$. We identify ${\rm
Symm}^{w_{1}^{+}}{\rm Std}_{3}\otimes_{\mathcal{A}}{\rm Symm}^{w_{1}^{-}}{\rm
Std}^{\vee}_{3}$ with the set
${\mathcal{A}}[X,Y,Z;A,B,C]_{w_{1}^{+},w_{1}^{-}}$ of homogenous polynomials
of degree $w_{1}^{+}$ in variables $X,Y,Z$ and of degree $w_{1}^{-}$ in
variables $A,B,C$. Define an action $\varrho^{(3)}_{{\boldsymbol{w}}}$ of
${\rm GL}_{3}({\mathcal{A}})$ on
${\mathcal{A}}[X,Y,Z;A,B,C]_{w_{1}^{+},w_{1}^{-}}$ as follows:

$\displaystyle\varrho^{(3)}_{{\boldsymbol{w}}}(g)P(X,Y,Z;A,B,C)$
$\displaystyle=(\det g)^{w_{2}}P((X,Y,Z)g;(A,B,C){}^{\rm t}\\!g^{-1})$
$\displaystyle\qquad\qquad\text{for
}P\in{\mathcal{A}}[X,Y,Z;A,B,C]_{w_{1}^{+},w_{1}^{-}}\text{ and }g\in{\rm
GL}_{3}({\mathcal{A}}).$

We also define a differential operator $\iota_{w_{1}^{+},w_{1}^{-}}$ to be

$\displaystyle\iota_{w_{1}^{+},w_{1}^{-}}=\frac{\partial^{2}}{\partial
X\partial A}+\frac{\partial^{2}}{\partial Y\partial
B}+\frac{\partial^{2}}{\partial Z\partial
C}:{\mathcal{A}}[X,Y,Z;A,B,C]_{w_{1}^{+},w_{1}^{-}}\to{\mathcal{A}}[X,Y,Z;A,B,C]_{w_{1}^{+}-1,w_{1}^{-}-1}.$

We find that $\iota_{w_{1}^{+},w_{1}^{-}}$ is equivariant with respect to the
actions $\varrho_{{\boldsymbol{w}}}^{(3)}$ and
$\varrho^{(3)}_{\boldsymbol{w}-(1,1,0)}$ of ${\rm GL}_{3}({\mathcal{A}})$.
Then define $L^{(3)}({\boldsymbol{w}};{\mathcal{A}})$ as the kernel of
$\iota_{w_{1}^{+},w_{1}^{-}}$. If $\mathcal{A}$ is a field, we may readily
verify that $\iota_{w_{1}^{+},w_{1}^{-}}$ is surjective, and thus the
dimension of $L^{(3)}({\boldsymbol{w}};\mathcal{A})$ is calculated as follows:

It is known that $L^{(3)}({\boldsymbol{w}};\mathcal{A})$ is an irreducible
algebraic representation of ${\rm GL}_{3}({\mathcal{A}})$ when $\mathcal{A}$
is a field (see [FH13, Section 13.2] for details). We finally define the
central character
$\omega_{\boldsymbol{w}}\colon\mathcal{A}^{\times}\rightarrow\mathcal{A}^{\times}$
of $L^{(3)}(\boldsymbol{w};\mathcal{A})$ so that

(3.1)
$\displaystyle\varrho^{(3)}_{\boldsymbol{w}}(x1_{3})P(X,Y,Z;A,B,C)=\omega_{\boldsymbol{w}}(x)P(X,Y,Z;A,B,C)$

holds for each $x\in\mathcal{A}^{\times}$ and $P(X,Y,Z;A,B,C)\in
L^{(3)}(\boldsymbol{w};\mathcal{A})$.

###  3.3. An explicit branching rule for $(\mathrm{GL}_{3},\mathrm{GL}_{2})$

Let us regard ${\rm GL}_{2}$ as a subgroup of ${\rm GL}_{3}$ via the fixed
embedding

$\displaystyle g\mapsto\iota(g):=\begin{pmatrix}g&\\\ &1\end{pmatrix}.$

Then, it is well known that, for $w_{1}^{+},w_{1}^{-}\in\mathbf{Z}$ with
$w_{1}^{+},w_{1}^{-}\geq 0$, an irreducible representation
$L^{(3)}(w_{1}^{+},w_{1}^{-},0\,;{\mathbf{C}})$ of
$\mathrm{GL}_{3}(\mathbf{R})$ of highest weight $(w_{1}^{+},0,-w_{1}^{-})$ is
decomposed as a $\mathrm{GL}_{2}({\mathbf{R}})$-representation as follows
(see, for example, [GW99, Theorem 8.1.1]):

$\displaystyle
L^{(3)}(w_{1}^{+},w_{1}^{-},0\,;\mathbf{C})|_{\mathrm{GL}_{2}(\mathbf{R})}\cong\bigoplus_{-w_{1}^{-}\leq-j\leq
0\leq i\leq w_{1}^{+}}L^{(2)}(i+j,-j;\mathbf{C}).$

Note that $L^{(2)}(i+j,-j;\mathbf{C})$ is an algebraic irreducible
representation of $\mathrm{GL}_{2}(\mathbf{R})$ of highest weight $(i,-j)$.
Such a decomposition is called the branching rule for the pair
$(\mathrm{GL}_{3},\mathrm{GL}_{2})$. In this subsection, we will construct an
explicit $\mathrm{GL}_{2}$-equivariant isomorphism as above in terms of the
models introduced in Sections 3.1 and 3.2, which would induce a decomposition
of local systems pulled back to the symmetric space
${\mathcal{Y}}^{(2)}_{\mathcal{K}_{2}}=\mathrm{GL}_{2}(\mathbf{Q})\backslash\mathrm{GL}_{2}(\mathbf{Q_{A}})/\mathrm{SO}_{2}(\mathbf{R})\mathcal{K}_{2}$;
see Section 6 for details.

Continuing from the previous subsection, let ${\mathcal{A}}$ be a
(commutative) integral domain of characteristic $0$, and assume that
$w_{1}^{+}!\,w_{1}^{-}!$ is invertible in ${\mathcal{A}}$. For each $0\leq
k\leq w_{1}^{+}$, $0\leq l\leq w_{1}^{-}$ and
$P\in{\mathcal{A}}[X,Y,Z;A,B,C]_{w_{1}^{+},w_{1}^{-}}$, set

(3.2)
$\displaystyle(\nabla_{k,l}P)(X,Y)=\frac{1}{k!\,l!}\frac{\partial^{k+l}P}{\partial
Z^{k}\partial C^{l}}(X,Y,0\,;-Y,X,0).$

Then one sees that $\nabla_{k,l}P$ is a polynomial in variables $X$ and $Y$,
which is homogenous of degree $w_{1}^{+}+w_{1}^{-}-k-l$. The following lemma
shows that $\nabla_{k,l}$ is a ${\rm GL}_{2}({\mathcal{A}})$-equivariant map:

######  Lemma 3.1.

For each $g\in{\rm GL}_{2}({\mathcal{A}})$, we have

$\displaystyle\left(\nabla_{k,l}\left((\varrho^{(3)}_{w_{1}^{+},w_{1}^{-},0}(\iota(g))P\right)\right)(X,Y)=(\det
g)^{-(w_{1}^{-}-l)}\left(\nabla_{k,l}P\right)((X,Y)g).$

In particular, $\nabla_{k,l}$ induces a ${\rm
GL}_{2}({\mathcal{A}})$-equivariant homomorphism

$\displaystyle\nabla_{k,l}\colon
L^{(3)}(w_{1}^{+},w_{1}^{-},0\,;\mathcal{A})|_{\rm
GL_{2}({\mathcal{A}})}\longrightarrow
L^{(2)}(w_{1}^{+}+w_{1}^{-}-k-l,-(w_{1}^{-}-l);\mathcal{A}).$

###### Proof..

For $g=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in{\rm GL}_{2}({\mathcal{A}})$,
we find that

$\displaystyle{}^{\rm t}\iota(g)^{-1}=(\det g)^{-1}\begin{pmatrix}d&-c&0\\\
-b&a&0\\\ 0&0&\det g\end{pmatrix}.$

Hence, by definition, we obtain the following equality:

$\displaystyle\left(\varrho^{(3)}_{w_{1}^{+},w_{1}^{-},0}(\iota(g))P\right)(X,Y,Z;A,B,C)$
$\displaystyle=P((X,Y,Z)\iota(g);(A,B,C){}^{\rm t}\iota(g)^{-1})$
$\displaystyle=(\det g)^{-w_{1}^{-}}P(aX+cY,bX+dY,Z;dA-bB,-cA+aB,(\det g)C).$

Using the chain rule, we can calculate the derivatives of
$\varrho^{(3)}_{w_{1}^{+},w_{1}^{-},0}(\iota(g))P$ as

$\frac{1}{k!\,l!}\frac{\partial^{k+l}}{\partial Z^{k}\partial
C^{l}}\left(\left(\varrho^{(3)}_{w_{1}^{+},w_{1}^{-},0}(\iota(g))P\right)(X,Y,Z;A,B,C)\right)\\\
=\frac{1}{k!\,l!}(\det g)^{-w_{1}^{-}+l}\frac{\partial^{k+l}P}{\partial
Z^{k}\partial C^{l}}(aX+cY,bX+dY,Z;dA-bB,-cA+aB,(\det g)C).$

Substituting $A=-Y,B=X$ and $C=Z=0$, we finally obtain the desired equality:

$\displaystyle\left(\nabla_{k,l}\left((\varrho^{(3)}_{w_{1}^{+},w_{1}^{-},0}(\iota(g))P\right)\right)(X,Y)=$
$\displaystyle\frac{1}{k!\,l!}(\det
g)^{-w_{1}^{-}+l}\frac{\partial^{k+l}P}{\partial Z^{k}\partial
C^{l}}(aX+cY,bX+dY,0;-(bX+dY),aX+cY,0)$ $\displaystyle=$ $\displaystyle(\det
g)^{-(w_{1}^{-}-l)}(\nabla_{k,l}P)(aX+cY,bX+dY)$ $\displaystyle=$
$\displaystyle(\det g)^{-(w_{1}^{-}-l)}(\nabla_{k,l}P)((X,Y)g).\qed$

By Lemma 3.1, we can define a ${\rm GL}_{2}({\mathcal{A}})$-equivariant
homomorphism

$\displaystyle{\boldsymbol{\nabla}}=(\nabla_{k,l})_{\begin{subarray}{c}0\leq
k\leq w_{1}^{+}\\\ 0\leq l\leq
w_{1}^{-}\end{subarray}}:L^{(3)}(w_{1}^{+},w_{1}^{-},0\,;\mathcal{A})|_{{\rm
GL}_{2}({\mathcal{A}})}\longrightarrow\bigoplus^{w_{1}^{+}}_{k=0}\bigoplus^{w_{1}^{-}}_{l=0}L^{(2)}(w_{1}^{+}+w_{1}^{-}-k-l,-(w_{1}^{-}-l);\mathcal{A}).$

######  Proposition 3.2.

Suppose that ${\mathcal{A}}$ is a field. Then the homomorphism
${\boldsymbol{\nabla}}$ is an isomorphism of
$\mathrm{GL}_{2}(\mathcal{A})$-modules.

###### Proof..

Since we have shown the ${\rm GL}_{2}({\mathcal{A}})$-equivariance of
${\boldsymbol{\nabla}}$ in Lemma 3.1, it suffices to show that the
${\mathcal{A}}$-dimension of the source of ${\boldsymbol{\nabla}}$ equals the
${\mathcal{A}}$-dimension of the target of ${\boldsymbol{\nabla}}$. The
${\mathcal{A}}$-dimensiton of the source has been already calculated in (3.2).
It is also easy to calculate the ${\mathcal{A}}$-dimension of the target as

$\sum^{w_{1}^{+}}_{k=0}\sum^{w_{1}^{-}}_{l=0}{\rm
dim}_{\mathcal{A}}L^{(3)}(w_{1}^{+}+w_{1}^{-}-k-l,-(w_{1}^{-}-l);{\mathcal{A}})\\\\[-5.0pt]
\begin{aligned}
&=\sum^{w_{1}^{+}}_{k=0}\sum^{w_{1}^{-}}_{l=0}(w_{1}^{+}+w_{1}^{-}-k-l+1)\\\
&=(w_{1}^{+}+w_{1}^{-}+1)(w_{1}^{+}+1)(w_{1}^{-}+1)-\frac{1}{2}w_{1}^{+}(w_{1}^{+}+1)(w_{1}^{-}+1)-(w_{1}^{+}+1)\cdot\frac{1}{2}w_{1}^{-}(w_{1}^{-}+1)\\\
&=\frac{1}{2}(w_{1}^{+}+1)(w_{1}^{-}+1)(w_{1}^{+}+w_{1}^{-}+2).\end{aligned}$

This completes the proof. ∎

Taking the ($w_{2}$-th) Tate twists for $w_{2}\in\mathbf{Z}$ into accounts, we
finally obtain the explicit decomposition

$\displaystyle{\boldsymbol{\nabla}}=(\nabla_{k,l})_{\begin{subarray}{c}0\leq
k\leq w_{1}^{+}\\\ 0\leq l\leq
w_{1}^{-}\end{subarray}}:L^{(3)}(w_{1}^{+},w_{1}^{-},w_{2};\mathcal{A})|_{{\rm
GL}_{2}({\mathcal{A}})}\longrightarrow\bigoplus^{w_{1}^{+}}_{k=0}\bigoplus^{w_{1}^{-}}_{l=0}L^{(2)}(w_{1}^{+}+w_{1}^{-}-k-l,-(w_{1}^{-}-l)+w_{2};\mathcal{A}).$

To lighten the notation, let us set
${\boldsymbol{n}}=(n_{1},n_{2})=(w_{1}^{+}+w_{1}^{-}-k-l,-(w_{1}^{-}-l)+w_{2})$
and

(3.3)
$\displaystyle\nabla^{\boldsymbol{n}}=\nabla^{n_{1},n_{2}}=\nabla_{w_{1}^{+}+w_{2}-n_{1}-n_{2},w_{1}^{-}-w_{2}+n_{2}}.$

Then the above decomposition is described as

(3.4)
$\displaystyle{\boldsymbol{\nabla}}=(\nabla^{{\boldsymbol{n}}})_{\boldsymbol{n}\in\Xi_{2}(\boldsymbol{w})}:L^{(3)}(\boldsymbol{w};\mathcal{A})|_{{\rm
GL}_{2}({\mathcal{A}})}\longrightarrow\bigoplus_{\boldsymbol{n}\in\Xi_{2}(\boldsymbol{w})}L^{(2)}(\boldsymbol{n};\mathcal{A}),$

where

$\displaystyle\Xi_{2}(\boldsymbol{w})=$
$\displaystyle\left\\{(n_{1},n_{2})\in{\mathbf{Z}}\mid(n_{1},n_{2})=(w_{1}^{+}+w_{1}^{-}-k-l,-(w_{1}^{-}-l)+w_{2})\text{\;
for }0\leq{}^{\exists}k\leq w_{1}^{+},\,0\leq{}^{\exists}l\leq
w_{1}^{-}\right\\}$ $\displaystyle=$
$\displaystyle\left\\{(n_{1},n_{2})\in{\mathbf{Z}}\mid w_{2}\leq
n_{1}+n_{2}\leq w_{1}^{+}+w_{2},\,w_{2}-w_{1}^{-}\leq n_{2}\leq
w_{2}\right\\}.$

###  3.4. Irreducible representation of ${\rm O}_{2}$

Set
$\Lambda_{2}=\\{(0,1)\\}\cup\\{(\lambda,0)\mid\lambda\in{\mathbf{Z}},\lambda\geq
0\\}$ and let $\boldsymbol{\lambda}=(\lambda,\delta)$ be an element of
$\Lambda_{2}$. We define an action $\tau^{(2)}_{\boldsymbol{\lambda}}$ of
$\mathrm{O}_{2}(\mathbf{R})$ on $\mathbf{C}[X,Y]_{\lambda}$ by
$\tau^{(2)}_{\boldsymbol{\lambda}}=\varrho^{(2)}_{\boldsymbol{\lambda}}|_{\mathrm{O}_{2}(\mathbf{R})}$,
and define
$v^{(2),\boldsymbol{\lambda}}_{\pm\lambda}\in\mathbf{C}[X,Y]_{\lambda}$
as111Note that both $v^{(2),(0,0)}_{0}$ and $v^{(2),(0,1)}_{0}$ equal $1$ as
an element of $V^{(2)}_{(0,0)}=V^{(2)}_{(0,1)}=\mathbf{C}$, but
${\mathrm{O}}_{2}(\mathbf{R})$ acts on them in different manners.

$\displaystyle v^{(2),\boldsymbol{\lambda}}_{\pm\lambda}$ $\displaystyle=(\pm
X+\sqrt{-1}Y)^{\lambda}$ (double sign in the same order).

Then the action of $\mathrm{O}_{2}(\mathbf{R})$ on
$v^{(2),\boldsymbol{\lambda}}_{\pm\lambda}$ is described as follows:

$\displaystyle\tau^{(2)}_{\boldsymbol{\lambda}}\left(\begin{pmatrix}\cos\theta&\sin\theta\\\
-\sin\theta&\cos\theta\end{pmatrix}\right)v^{(2),\boldsymbol{\lambda}}_{\pm\lambda}$
$\displaystyle=e^{\pm\sqrt{-1}\lambda\theta}v^{(2),\boldsymbol{\lambda}}_{\pm\lambda},$
$\displaystyle\tau^{(2)}_{\boldsymbol{\lambda}}\left(\begin{pmatrix}-1&0\\\
0&1\end{pmatrix}\right)v^{(2),\boldsymbol{\lambda}}_{\pm\lambda}$
$\displaystyle=(-1)^{\delta}v^{(2),\boldsymbol{\lambda}}_{\mp\lambda}.$

Hence the subspace $V^{(2)}_{\boldsymbol{\lambda}}:=\langle
v^{(2),\boldsymbol{\lambda}}_{\lambda},v^{(2),\boldsymbol{\lambda}}_{-\lambda}\rangle_{\mathbf{C}}$
of $\mathbf{C}[X,Y]_{\lambda}$ gives an irreducible representation of ${\rm
O}_{2}({\mathbf{R}})$, and hereafter we write
$\tau^{(2)}_{\boldsymbol{\lambda}}$ to denote the action of ${\rm
O}_{2}({\mathbf{R}})$ on this space. Then each irreducible representation of
${\rm O}_{2}({\mathbf{R}})$ is classified as
$\tau^{(2)}_{\boldsymbol{\lambda}}$ for some
$\boldsymbol{\lambda}\in\Lambda_{2}$.

###  3.5. Irreducible representation of ${\rm O}_{3}$

Set $\Lambda_{3}=\\{(\lambda,\delta)\mid\lambda\in{\mathbf{Z}},\,\lambda\geq
0,\,\delta\in\mathbf{Z}/2\mathbf{Z}\\}$ and let
$\boldsymbol{\lambda}=(\lambda,\delta)$ be an element of $\Lambda_{3}$. We
define an action $\tau^{(3)}_{\boldsymbol{\lambda}}$ of ${\rm
O}_{3}({\mathbf{R}})$ on ${\mathbf{C}}[z_{1},z_{2},z_{3}]_{\lambda}$ by

$\displaystyle\tau^{(3)}_{\boldsymbol{\lambda}}(u)P(z_{1},z_{2},z_{3})=(\det
u)^{\delta}P((z_{1},z_{2},z_{3})u)$

for each $u\in{\rm O}_{3}({\mathbf{R}})$ and
$P\in{\mathbf{C}}[z_{1},z_{2},z_{3}]_{\lambda}$. Then the subspace
${\mathcal{V}}_{\boldsymbol{\lambda}}$ defined as
${\mathcal{V}}_{\boldsymbol{\lambda}}=(z^{2}_{1}+z^{2}_{2}+z^{2}_{3}){\mathbf{C}}[z_{1},z_{2},z_{3}]_{\lambda-2}$
when $\lambda\geq 2$ and ${\mathcal{V}}_{\boldsymbol{\lambda}}=0$ otherwise is
stable under the action $\tau^{(3)}_{\boldsymbol{\lambda}}$ of ${\rm
O}_{3}({\mathbf{R}})$. One may verifiy that the quotient space
$V^{(3)}_{\boldsymbol{\lambda}}={\mathbf{C}}[z_{1},z_{2},z_{3}]_{\lambda}/{\mathcal{V}}_{\boldsymbol{\lambda}}$
gives rise to an irreducible representation of ${\rm O}_{3}({\mathbf{R}})$,
which we also denote by $\tau^{(3)}_{\boldsymbol{\lambda}}$ (see also [HIM,
Section 4.1]). Then each irreducible representation of ${\rm
O}_{3}({\mathbf{R}})$ is classified as $\tau^{(3)}_{\boldsymbol{\lambda}}$ for
some $\boldsymbol{\lambda}\in\Lambda_{3}$. It is easy to see that the
dimension of $V^{(3)}_{\boldsymbol{\lambda}}$ is equal to $2\lambda+1$.

Next we construct an explicit basis of the representation space
$V^{(3)}_{\boldsymbol{\lambda}}$. For every $\mu\in{\mathbf{Z}}$ with
$0\leq\mu\leq\lambda$, let $v^{(3),\boldsymbol{\lambda}}_{\pm\mu}$ be elements
of $V^{(3)}_{\boldsymbol{\lambda}}$ defined as

$\displaystyle v^{(3),\boldsymbol{\lambda}}_{\pm\mu}$ $\displaystyle=(\pm
z_{1}+\sqrt{-1}z_{2})^{\mu}z^{\lambda-\mu}_{3}\mod{\mathcal{V}}_{\boldsymbol{\lambda}}$
$\displaystyle\text{(double sign in the same order)}.$

We abbreviate $v^{(3),\boldsymbol{\lambda}}_{\pm\mu}$ to
$v^{\boldsymbol{\lambda}}_{\pm\mu}$ if no confusion likely occurs. Then
$\\{v^{(3),\boldsymbol{\lambda}}_{\pm\mu}\mid 0\leq\mu\leq\lambda\\}$ forms a
basis of $V^{(3)}_{\boldsymbol{\lambda}}$. The multiplicativity among these
elements

(3.5) $\displaystyle v^{(3),\boldsymbol{\lambda}}_{\pm\mu}\times
v^{(3),\boldsymbol{\lambda}^{\prime}}_{\pm\mu^{\prime}}$
$\displaystyle=v^{(3),\boldsymbol{\lambda}+\boldsymbol{\lambda}^{\prime}}_{\pm\mu\pm\mu^{\prime}}$
$\displaystyle\text{for
}\boldsymbol{\lambda}=(\lambda,\delta),\,\boldsymbol{\lambda}^{\prime}=(\lambda^{\prime},\delta^{\prime})\in\Lambda_{3}\text{
and }0\leq\mu\leq\lambda,\,0\leq\mu^{\prime}\leq\lambda^{\prime}$

is immediately deduced from their definition.

For later use, we here introduce the matrix representation
$M_{\boldsymbol{\lambda}}(u)\in{\rm GL}_{2\lambda+1}({\mathbf{C}})$ of the
action of $u\in{\rm O}_{3}({\mathbf{R}})$ on $V^{(3)}_{\boldsymbol{\lambda}}$
with respect to the above specific basis; namely we set

(3.6)
$\displaystyle\begin{pmatrix}\tau^{(3)}_{\boldsymbol{\lambda}}(u)v^{(3),\boldsymbol{\lambda}}_{\lambda}&\tau^{(3)}_{\boldsymbol{\lambda}}(u)v^{(3),\boldsymbol{\lambda}}_{\lambda-1}&\dotsc&\tau^{(3)}_{\boldsymbol{\lambda}}(u)v^{(3),\boldsymbol{\lambda}}_{-\lambda}\end{pmatrix}=\begin{pmatrix}v^{(3),\boldsymbol{\lambda}}_{\lambda}&v^{(3),\boldsymbol{\lambda}}_{\lambda-1}&\dotsc&v^{(3),\boldsymbol{\lambda}}_{-\lambda}\end{pmatrix}M_{\lambda}(u).$

###  3.6. An explicit branching rule for $({\mathrm{O}_{3},\mathrm{O}_{2}})$

We finally emphasize that, by using the specific bases introduced in the
previous subsections, we can explicitly describe the branching rule for
$({\mathrm{O}}_{3}(\mathbf{R}),{\mathrm{O}}_{2}(\mathbf{R}))$ with respect to
the fixed inclusion
${\mathrm{O}}_{2}(\mathbf{R})\hookrightarrow{\mathrm{O}}_{3}(\mathbf{R});u\mapsto\begin{pmatrix}u&\\\
&1\end{pmatrix}$ as follows. For each
$\boldsymbol{\lambda}=(\lambda,\delta)\in\Lambda_{3}$, define a subset
$\Sigma(\boldsymbol{\lambda})$ of $\Lambda_{2}$ as

$\displaystyle\Sigma(\boldsymbol{\lambda})=\left\\{(0,\delta)\right\\}\cup\left\\{(\mu,0)\mid\mu\in{\mathbf{Z}},1\leq\mu\leq\lambda\right\\}\quad\subseteq\Lambda_{2}.$

For each $\boldsymbol{\mu}=(\mu,\delta)\in\Sigma(\boldsymbol{\lambda})$, the
correspondence $v^{(2),\boldsymbol{\mu}}_{\pm\mu}\mapsto(\pm
1)^{\delta}v^{(3),\boldsymbol{\lambda}}_{\pm\mu}$ defines an
${\mathrm{O}}_{2}(\mathbf{R})$-equivariant homomorphism
$\iota^{\boldsymbol{\lambda}}_{\boldsymbol{\mu}}:V^{(2)}_{\boldsymbol{\mu}}\to
V^{(3)}_{\boldsymbol{\lambda}}$ (see also [HIM, Lemma 9.3]), which induces an
isomorphism of ${\mathrm{O}}_{2}(\mathbf{R})$-representations

$\displaystyle\bigoplus_{\boldsymbol{\mu}\in\Sigma(\boldsymbol{\lambda})}V^{(2)}_{\boldsymbol{\mu}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}V^{(3)}_{\boldsymbol{\lambda}}|_{{\mathrm{O}}_{2}(\mathbf{R})}.$

Obviously there also exists an ${\rm O}_{2}(\mathbf{R})$-homomorphism
${\mathrm{P}}^{\boldsymbol{\lambda}}_{\boldsymbol{\mu}}\colon
V^{(3)}_{\boldsymbol{\lambda}}|_{{\mathrm{O}}_{2}(\mathbf{R})}\to
V^{(2)}_{\boldsymbol{\mu}}$ satisfying ${\rm
P}^{\boldsymbol{\lambda}}_{\boldsymbol{\mu}}\circ\iota^{\boldsymbol{\lambda}}_{\boldsymbol{\mu}}=\mathrm{id}_{V_{\boldsymbol{\mu}}^{(2)}}$.

##  4\. Motives associated to the Rankin–Selberg convolution ${\rm
GL}_{3}\times{\rm GL}_{2}$

After briefly reviewing the notion of cohomological automorphic
representations of $\mathrm{GL}_{n}(\mathbf{Q}_{\mathbf{A}})$, we summarize in
this section the settings on the automorphic representations of ${\rm
GL}_{3}({\mathbf{Q}}_{\mathbf{A}})$, ${\rm GL}_{2}({\mathbf{Q}}_{\mathbf{A}})$
and their Rankin–Selberg convolution, which we consider throughout this
article. We also introduce the notion of their archimedean $L$-fuctors and
critical points, and discuss their relations to the (conjectural) associated
motives, according to [Del79] and [Clo90]. Some of materials are also dealt
with in [Mah05] and [Rag16]. Beyond these generalities, we would emphasize
here explicit models and parameters of the representations under
consideration, which are fundamental tools in this article. We always consider
that all the automorphic $L$-functions are normalized so that they satisfy the
functional equations between $L(s,-)$ and $L(1-s,-^{\vee})$ throughout this
article (where ∨ denotes the contragredient).

###  4.1. Generalities on automorphic representations of ${\rm GL}_{n}$

In this subsection, we briefly review general theory on cohomological
automorphic representations of ${\rm GL}_{n}$. Let $\pi^{(n)}$ be an
irreducible cuspidal automorphic representation of ${\rm
GL}_{n}({\mathbf{Q}}_{\mathbf{A}})$. We first recall the notion of the
Langlands parameter of $\pi^{(n)}$ according to [Kna94, Section 2] and
[Clo90]. For this purpose, let us introduce several notation on
representations of the Weil group $W_{\mathbf{R}}$ of the real number field
${\mathbf{R}}$.

Here we realize the Weil group $W_{\mathbf{R}}$ as the disjoint union
${\mathbf{C}}^{\times}\sqcup({\mathbf{C}}^{\times}j)$ where $j$ satisfies
$j^{2}=-1$ and $jzj=-z^{c}$ for arbitrary $z\in{\mathbf{C}}^{\times}$, where
$z^{c}$ denotes the complex conjugate of $z$. Then, for every
$\nu\in{\mathbf{C}}$, $\delta\in\\{0,1\\}$ and $l\in{\mathbf{Z}}$ with $l\geq
0$, define a $1$-dimensional representation
$\phi^{\delta}_{\nu}:W_{\mathbf{R}}\to{\mathbf{C}}^{\times}$ and a
$2$-dimensional representation $\phi_{\nu,l}:W_{\mathbf{R}}\to{\rm
GL}_{2}({\mathbf{C}})$ of $W_{\mathbf{R}}$ as follows:

$\displaystyle\phi^{\delta}_{\nu}(z)$
$\displaystyle=(zz^{c})^{\nu}\hphantom{\begin{pmatrix}(z^{c}/z)^{\frac{l}{2}}&0\\\
0&(z/z^{c})^{\frac{l}{2}}\end{pmatrix}}\;\quad\text{for
}z\in{\mathbf{C}}^{\times},$ $\displaystyle\phi^{\delta}_{\nu}(j)$
$\displaystyle=(-1)^{\delta},$ $\displaystyle\phi_{\nu,l}(z)$
$\displaystyle=(zz^{c})^{\nu}\begin{pmatrix}(z^{c}/z)^{\frac{l}{2}}&0\\\
0&(z/z^{c})^{\frac{l}{2}}\end{pmatrix}\quad\text{for
}z\in{\mathbf{C}}^{\times},$ $\displaystyle\phi_{\nu,l}(j)$
$\displaystyle=\begin{pmatrix}0&(-1)^{l}\\\ 1&0\end{pmatrix}.$

Let $\chi^{\delta}_{\nu}:{\mathbf{R}}^{\times}\to{\mathbf{C}}^{\times}$ be a
character of $\mathbf{R}^{\times}$ defined by $\chi^{\delta}_{\nu}(t)={\rm
sgn}(t)^{\delta}|t|^{\nu}_{\infty}$, and $D_{\nu,l}$ an irredicible
representation of $\mathrm{GL}_{2}(\mathbf{R})$ satisfying

$\displaystyle D_{\nu,l}(t1_{2})$ $\displaystyle=t^{2\nu}\quad\text{for
}t\in{\mathbf{R}}^{\times}\text{ with }t>0,$ $\displaystyle D_{\nu,l}|_{{\rm
SL}_{2}({\mathbf{R}})}$ $\displaystyle\cong D^{+}_{l}\oplus D^{-}_{l},$

where $D^{+}_{l}$ (resp. $D^{-}_{l}$) denotes the holomorphic (resp. anti-
holomorphic) discrete series representation of ${\rm SL}_{2}({\mathbf{R}})$ of
lowest weight $l+1$ (resp. of highest weight $-l-1$). Then
$\phi^{\delta}_{\nu}$ (resp. $\phi_{\nu,l}$) corresponds to
$\chi^{\delta}_{\nu}$ (resp. $D_{\nu,l}$) via the (archimedean) local
Langlands correspondence; see [Kna94, Theorem 2] for details.

It is well known that, when the automorphic representation $\pi^{(n)}$ under
consideration is cohomological (or regular algebraic in some literature),
strong constraints are imposed on the Langlands parameter of its archimedean
part. We do not review here the precise definition of cohomological
automorphic representations, and just quote the following classification
results which are necessary later. For details on cohomological automorphic
representations, see Clozel’s exposition [Clo90, Section 3.5] (refer also to
[Mah05, Section 3.1.1] and [GR14, Section 2.3] for brief summaries).

######  Proposition 4.1.

([Clo90, Lemme de pureté 4.9]) Let $\pi^{(n)}$ be a cohomological irreducible
cuspidal automorphic representation of
$\mathrm{GL}_{n}(\mathbf{Q}_{\mathbf{A}})$. Then the Langlands parameter of
the archimedean part $\pi^{(n)}_{\infty}$ of $\pi^{(n)}$ is described as
follows$:$

  1. (i)

When $n$ is even, set $n=2m$. Then the Langlands parameter of $\pi^{(n)}$ is
given by

$\displaystyle\bigoplus^{m}_{i=1}\phi_{\nu_{i},l_{i}}\quad\text{with
}l_{1}>l_{2}>\cdots>l_{m}$

such that $\nu_{i}\pm\dfrac{l_{i}}{2}-\dfrac{n-1}{2}$ are rational integers
for each $1\leq i\leq m$. Furthermore, the purity implies that

$\displaystyle
w:=2\nu_{i}-n+1\quad\left(=\left(\nu_{i}-\frac{l_{i}}{2}-\frac{n-1}{2}\right)+\left(\nu_{i}+\frac{l_{i}}{2}-\frac{n-1}{2}\right)\right)$

is independent of $i;$ namely, the values of $\nu_{1},\nu_{2},\dotsc,\nu_{m}$
coincide with one another.

  2. (ii)

When $n$ is odd, set $n=2m+1$. Then the Langlands parameter of $\pi^{(n)}$ is
given by

$\displaystyle\phi^{\delta}_{\nu}\oplus\bigoplus^{m}_{i=1}\phi_{\nu_{i},l_{i}}\quad\text{with
}l_{1}>l_{2}>\cdots>l_{m}$

such that both $\nu-\dfrac{n-1}{2}$ and
$\nu_{i}\pm\dfrac{l_{i}}{2}-\dfrac{n-1}{2}$ for each $1\leq i\leq m$ are
rational integers. Furthermore, the purity implies that

$\displaystyle w$ $\displaystyle:=2\nu-n+1=2\nu_{i}-n+1$
$\displaystyle\left(=2\left(\nu-\dfrac{n-1}{2}\right)=\left(\nu_{i}-\frac{l_{i}}{2}-\frac{n-1}{2}\right)+\left(\nu_{i}+\frac{l_{i}}{2}-\frac{n-1}{2}\right)\right)$

is independent of $i;$ namely, the values of
$\nu,\nu_{1},\nu_{2},\dotsc,\nu_{m}$ coincide with one another.

In both cases, we say that $\pi^{(n)}$ is pure of weight $w$.

######  Remark 4.2.

When $n$ is odd, Proposition 4.1 (ii) implies that $\nu$ is an integer, and we
readily see from purity that every $\nu_{i}$ is also an integer. Therefore,
due to the same statement, we observe that every $\dfrac{l_{i}}{2}$ is an
integer, or in other words, every weight parameter $l_{i}$ should be even when
$n$ is odd.

We end this subsection by introducing Clozel’s conjecture for the existence of
motives associated to cohomological irreducible cuspidal automorphic
representations of $\mathrm{GL}_{n}(\mathbf{Q}_{\mathbf{A}})$ according to
[Clo90, Section 4]. Let $L(s,\pi^{(n)})$ denote the (complete) automorphic
$L$-function associated to $\mathrm{GL}_{n}$, which is constructed by Godement
and Jacquet in [GJ72].

######  Conjecture 4.3.

([Clo90, Conjecture 4.5]) Let $\pi^{(n)}$ be a cohomological irreducible
cuspidal automorphic representation of
$\mathrm{GL}_{n}(\mathbf{Q}_{\mathbf{A}})$ which is pure of weight $w$ in the
sense of Proposition 4.1. Then there exists a pure motive
${\mathcal{M}}[\pi^{(n)}]$ of rank $n$ and of weight $-w$ which satisfies

$\displaystyle
L(s,{\mathcal{M}}[\pi^{(n)}])=L\left(s-\frac{n-1}{2},\pi^{(n)}\right).$

###  4.2. Critical values and motives for
$\mathrm{GL}_{3}\times\mathrm{GL}_{2}$

In this subsection, we focus on the cases where $n=2$ and $3$, and study the
Rankin–Selberg $L$-function $L(s,\pi^{(3)}\times\pi^{(2)})$ constructed by
Jacquet, Piatetski-Shapiro and Shalika in [JPSS83]. We especially discuss the
critical points of $L(s,\pi^{(3)}\times\pi^{(2)})$ and their motivic
interpretations.

We retain the notations in Section 4.1; in particular, let $\pi^{(3)}$ and
$\pi^{(2)}$ be cohomological irreducible cuspidal automorphic representations
of $\mathrm{GL}_{3}(\mathbf{Q}_{\mathbf{A}})$ and
$\mathrm{GL}_{2}(\mathbf{Q}_{\mathbf{A}})$ respectively. Then Proposition 4.1
and Remark 4.2 yield that the Langlands parameters of $\pi^{(2)}_{\infty}$ and
$\pi^{(3)}_{\infty}$ are described as $\phi_{\nu_{2},l_{2}}$ and
$\phi_{\nu_{3}}^{\delta}\oplus\phi_{\nu_{3},l_{3}}$ respectively, where
$\nu_{2}\in\frac{1}{2}\mathbf{Z}$, $\nu_{3}\in{\mathbf{Z}}$,
$\delta\in\mathbf{Z}/2\mathbf{Z}$, $l_{2}\in{\mathbf{Z}}$ with $l_{2}\geq 1$
and $l_{3}\in 2\mathbf{Z}$ with $l_{3}\geq 2$. Thus there exist isomorphisms
of $(\mathfrak{gl}_{n}(\mathbf{R}),{\mathrm{O}}_{n}(\mathbf{R}))$-modules

$\displaystyle\pi^{(2)}_{\infty}\cong
D_{\nu_{2},l_{2}},\quad\pi^{(3)}_{\infty}\cong{\rm Ind}^{\rm
GL_{3}({\mathbf{R}})}_{{\rm
P}_{2,1}({\mathbf{R}})}(D_{\nu_{3},l_{3}}\boxtimes\chi_{\nu_{3}}^{\delta})$

due to the Langlands classification (refer to [Kna94, Section 2]). Here ${\rm
Ind}^{\rm GL_{3}({\mathbf{R}})}_{{\rm
P}_{2,1}({\mathbf{R}})}(D_{\nu_{3},l_{3}}\boxtimes\chi_{\nu_{3}}^{\delta})$
denotes the generalized principal series representation parabolically induced
from $D_{\nu_{3},l_{3}}$ and $\chi_{\nu_{3}}^{\delta}$. From this observation,
we see that the minimal ${\rm O}_{2}({\mathbf{R}})$-type of
$\pi^{(2)}_{\infty}$ is given by $\tau^{(2)}_{(l_{2}+1,0)}$, and thus

$\displaystyle\Lambda^{\rm
coh}_{2}:=\left\\{(\lambda_{2},0)\mid\lambda_{2}\in{\mathbf{Z}},\,\lambda_{1}\geq
2\right\\}$

classifies the isomorphism classes of minimal ${\rm
O}_{2}({\mathbf{R}})$-types of (the archimedean parts of) cohomological
irreducible cuspidal automorphic representations of ${\rm
GL}_{2}({\mathbf{Q}}_{\mathbf{A}})$. Similarly, the minimal ${\rm
O}_{3}({\mathbf{R}})$-type of $\pi^{(3)}_{\infty}$ is given by
$\tau^{(3)}_{(l_{3}+1,\delta)}$, and thus

$\displaystyle\Lambda^{\rm
coh}_{3}:=\left\\{(\lambda_{3},\delta)\mid\lambda_{3}\in
1+2{\mathbf{Z}},\,\lambda_{3}\geq
3,\,\delta\in\mathbf{Z}/2\mathbf{Z}\right\\}$

classifies the isomorphism classes of minimal ${\rm
O}_{3}({\mathbf{R}})$-types of (the archimedean parts of) cohomological
irreducible cuspidal automorphic representations of ${\rm
GL}_{3}({\mathbf{Q}}_{\mathbf{A}})$.

From now on we normalize the action of the center of $\mathrm{GL}_{n}$ on
$\pi^{(n)}$ for $n=2$ and $3$ so that the parameters $\nu_{2}$ and $\nu_{3}$
satisfy the following equalities:

(4.1) $\displaystyle\nu_{2}$ $\displaystyle=-\frac{l_{2}}{2}+\frac{1}{2},$
$\displaystyle\nu_{3}$ $\displaystyle=-\frac{l_{3}}{2}+1.$

These normalizations simplify the description of the Hodge filtrations of the
motives $\mathcal{M}[\pi^{(2)}]$ and $\mathcal{M}[\pi^{(3)}]$; see (4.3)
below. Furthermore we always assume that the inequality

(4.2) $\displaystyle l_{2}<l_{3}$

holds, since we can study the special values of
$L(s,\pi^{(3)}\times\pi^{(2)})$ only under this condition by using the modular
symbol method. Now let us recall the definitions of the $L$-factor
$L_{\infty}(s,\pi^{(3)}\times\pi^{(2)})$ and the $\epsilon$-factor
$\epsilon_{\infty}(s,\pi^{(3)}\times\pi^{(2)},\psi_{\infty})$ at the infinite
place. Firstly, for $\pi^{(n)}$ with $n=2$ or $3$, the local $L$-factors and
the local $\epsilon$-factors are defined as

$\displaystyle
L_{\infty}(s,\pi^{(n)})=\begin{cases}\Gamma_{\mathbf{C}}\left(s+\nu_{2}+\dfrac{l_{2}}{2}\right)&\text{for
}n=2,\\\
\Gamma_{\mathbf{C}}\left(s+\nu_{3}+\dfrac{l_{3}}{2}\right)\Gamma_{\mathbf{R}}\left(s+\nu_{3}+\delta\right)&\text{for
}n=3,\end{cases}\quad\epsilon_{\infty}(s,\pi^{(n)},\psi_{\infty})=\begin{cases}\sqrt{-1}^{l_{2}+1}&\text{for
}n=2,\\\ \sqrt{-1}^{l_{3}+1+\delta}&\text{for }n=3,\end{cases}$

where $\Gamma_{\mathbf{R}}(s)$ and $\Gamma_{\mathbf{C}}(s)$ denote
$\pi^{-s/2}\Gamma(s/2)$ and $2(2\pi)^{-s}\Gamma(s)$ respectively, and
$\psi_{\infty}\colon\mathbf{R}\rightarrow\mathbf{C}^{\times}$ is the standard
additive character defined as $\psi_{\infty}(x)=\exp(2\pi\sqrt{-1}x)$. Since
the tensor product of the Langlands parameters associated to $\pi^{(3)}$ and
$\pi^{(2)}$ is easily calculated as

$\displaystyle\left(\phi_{\nu_{3},l_{3}}\oplus\phi^{\delta}_{\nu_{3}}\right)\otimes\phi_{\nu_{2},l_{2}}=\phi_{\nu_{3}+\nu_{2},l_{3}+l_{2}}\oplus\phi_{\nu_{3}+\nu_{2},|l_{3}-l_{2}|}\oplus\phi_{\nu_{3}+\nu_{2},l_{2}},$

we also find that the local $L$-factor and the local $\epsilon$-factor of the
Rankin–Selberg convolution $\pi^{(3)}\times\pi^{(2)}$ at the infinite place
are given as

$\displaystyle L_{\infty}(s,\pi^{(3)}\times\pi^{(2)})$
$\displaystyle=\Gamma_{\mathbf{C}}\left(s+\nu_{3}+\nu_{2}+\frac{l_{3}+l_{2}}{2}\right)\Gamma_{\mathbf{C}}\left(s+\nu_{3}+\nu_{2}+\frac{l_{3}-l_{2}}{2}\right)\Gamma_{\mathbf{C}}\left(s+\nu_{3}+\nu_{2}+\frac{l_{2}}{2}\right),$
$\displaystyle\epsilon_{\infty}(s,\pi^{(3)}\times\pi^{(2)},\psi_{\infty})$
$\displaystyle=\sqrt{-1}^{2l_{3}+l_{2}+3}$

(here we take the assumption (4.2) into accounts). Mimicking Deligne’s
definition [Del79, Definition 1.3], we call an integer $m$ a critical point of
$L(s,\pi^{(3)}\times\pi^{(2)})$ if neither
$L_{\infty}(s,\pi^{(3)}\times\pi^{(2)})$ nor
$L_{\infty}(1-s,(\pi^{(3)}\times\pi^{(2)})^{\vee})$ has a pole at
$s=m-\dfrac{3}{2}$. One can find an explanation for the shift $-\dfrac{3}{2}$
at (4.4) below.

Now let us determine the region of the critical points of
$L(s,\pi^{(3)}\times\pi^{(2)})$.

######  Proposition 4.4.

  1. (i)

Suppose that $l_{2}\leq\dfrac{l_{3}}{2}$ holds. Then $m$ is a critical point
of $L(s,\pi^{(3)}\times\pi^{(2)})$ if and only if the inequality
$\dfrac{l_{3}}{2}+1\leq m\leq\dfrac{l_{3}}{2}+l_{2}$ holds.

  2. (ii)

Suppose that $\dfrac{l_{3}}{2}<l_{2}<l_{3}$ holds. Then $m$ is a critical
point of $L(s,\pi^{(3)}\times\pi^{(2)})$ if and only if the inequality
$l_{2}+1\leq m\leq l_{3}$ holds.

###### Proof..

The statements directly follow from the arguments in [Rag16, Section 2.4.5.3].
However we briefly give a proof for the sake of the reader’s convenience and
the completeness. Due to the functoriality of the local Langlands
correspondence, the Langlands parameters of the contragredients
$\pi^{(2),\vee}_{\infty}$ and $\pi^{(3),\vee}_{\infty}$ are described as
follows:

$\displaystyle\phi_{\nu_{2},l_{2}}^{\vee}$
$\displaystyle\cong\phi_{-\nu_{2},l_{2}},$
$\displaystyle\phi^{\delta,\vee}_{\nu}\oplus\phi^{\vee}_{\nu_{3},l_{3}}$
$\displaystyle\cong\phi^{\delta}_{-\nu}\oplus\phi_{-\nu_{3},l_{3}}.$

Therefore the Langlands parameter of $(\pi^{(3)}\times\pi^{(2)})^{\vee}$ is
calculated as
$\phi_{-\nu_{3}-\nu_{2},l_{3}+l_{2}}\oplus\phi_{-\nu_{3}-\nu_{2},\lvert
l_{3}-l_{2}\rvert}\oplus\phi_{-\nu_{3}-\nu_{2},l_{2}}$, and by using (4.1) and
(4.2), we obtain

$\displaystyle L_{\infty}\left(s-\dfrac{3}{2},\pi^{(3)}\times\pi^{(2)}\right)$
$\displaystyle=\Gamma_{\mathbf{C}}(s)\Gamma_{\mathbf{C}}(s-l_{2})\Gamma_{\mathbf{C}}\left(s-\dfrac{l_{3}}{2}\right),$
$\displaystyle
L_{\infty}\left(1-\left(s-\dfrac{3}{2}\right),\left(\pi^{(3)}\times\pi^{(2)}\right)^{\vee}\right)$
$\displaystyle=\Gamma_{\mathbf{C}}(l_{3}+l_{2}+1-s)\Gamma_{\mathbf{C}}(l_{3}+1-s)\Gamma_{\mathbf{C}}\left(\dfrac{l_{3}}{2}+l_{2}+1-s\right).$

Therefore in order for $m\in\mathbf{Z}$ to be a critical point of
$L(s,\pi^{(3)}\times\pi^{(2)})$, it must satisfy both of the inequalities

$\displaystyle m$
$\displaystyle\geq\max\left\\{l_{2}+1,\dfrac{l_{3}}{2}+1\right\\}$ and
$\displaystyle m$
$\displaystyle\leq\min\left\\{l_{3},\dfrac{l_{3}}{2}+l_{2}\right\\},$

which verifies the statement. ∎

We finally discuss the motivic aspects of these critical points. For $n=2$ and
$3$, the Betti realization $H_{\rm B}({\mathcal{M}}[\pi^{(n)}])$ of
${\mathcal{M}}[\pi^{(n)}]$ has the following Hodge decomposition due to (4.1):

(4.3) $\displaystyle\begin{aligned} H_{\rm
B}({\mathcal{M}}[\pi^{(2)}])\otimes{\mathbf{C}}&=H^{l_{2},0}({\mathcal{M}}[\pi^{(2)}])\oplus
H^{0,l_{2}}({\mathcal{M}}[\pi^{(2)}]),\\\ H_{\rm
B}({\mathcal{M}}[\pi^{(3)}])\otimes{\mathbf{C}}&=H^{l_{3},0}({\mathcal{M}}[\pi^{(3)}])\oplus
H^{\frac{l_{3}}{2},\frac{l_{3}}{2}}({\mathcal{M}}[\pi^{(3)}])\oplus
H^{0,l_{3}}({\mathcal{M}}[\pi^{(3)}]).\end{aligned}$

Define ${\mathcal{M}}(\pi^{(3)}\times\pi^{(2)})$ to be the tensor product
${\mathcal{M}}[\pi^{(3)}]\otimes{\mathcal{M}}[\pi^{(2)}]$ of the motives
$\mathcal{M}[\pi^{(3)}]$ and $\mathcal{M}[\pi^{(2)}]$. Then (conjecturally) we
obtain the identity of the $L$-functions

(4.4) $\displaystyle
L(s,{\mathcal{M}}(\pi^{(3)}\times\pi^{(2)}))=L\left(s-\frac{3}{2},\pi^{(3)}\times\pi^{(2)}\right).$

The Hodge decomposition of $\mathcal{M}(\pi^{(3)}\times\pi^{(2)})(m)$ is
calculated as follows (here we abbreviate
$\mathcal{M}(\pi^{(3)}\times\pi^{(2)})(m)$ as $\mathcal{M}_{\pi,m}$):

$H_{B}(\mathcal{M}_{\pi,m})\otimes_{\mathbf{Q}}\mathbf{C}\cong
H^{l_{3}+l_{2}-m,-m}(\mathcal{M}_{\pi,m})\oplus
H^{l_{3}-m,l_{2}-m}(\mathcal{M}_{\pi,m})\oplus
H^{\frac{l_{3}}{2}+l_{2}-m,\frac{l_{3}}{2}-m}(\mathcal{M}_{\pi,m})\\\ \oplus
H^{\frac{l_{3}}{2}-m,\frac{l_{3}}{2}+l_{2}-m}(\mathcal{M}_{\pi,m})\oplus
H^{l_{2}-m,l_{3}-m}(\mathcal{M}_{\pi,m})\oplus
H^{-m,l_{3}+l_{2}-m}(\mathcal{M}_{\pi,m}).$

As is explained in [Del79, Section 1], the motive
$\mathcal{M}(\pi^{(3)}\times\pi^{(2)})(m)$ is critical if and only if the set

$\displaystyle\\{(p,q)\in\mathbf{Z}^{2}\mid
H^{p,q}(\mathcal{M}(\pi^{(3)}\times\pi^{(2)})(m))\neq 0\\}$

belongs to the union of $\\{(p,q)\mid p\leq-1,\,q\geq 0\\}$ and $\\{(p,q)\mid
p\geq 0,\,q\leq-1\\}$, since the motive
$\mathcal{M}(\pi^{(3)}\times\pi^{(2)})(m)$ does not contain diagonal
components. This observation also verifies the statement of Proposition 4.4,
if one admits the existence of the motive $\mathcal{M}[\pi^{(3)}]$.

##  5\. Eichiler–Shimura maps for ${\rm GL}_{3}$

Due to Clozel [Clo90, Section 3.5], the cuspidal cohomology groups of the
symmetric space associated to $\mathrm{GL}_{n}$ with appropriate local systems
are well understood, and the finite parts of cohomological irreducible
cuspidal automorphic representations of ${\rm GL}_{n}$ are realized in these
cohomology groups. It is necessary for us to specify certain cohomology
classes to clarify their connection to global zeta integrals and their
algebraic and $p$-adic properties. Such distinguished cohomology classes are
constructed from appropriate cusp forms on ${\rm GL}_{n}$, and hence we obtain
a map form the space of cusp forms on ${\rm GL}_{n}$ to the cuspidal
cohomology groups with certain local coefficients, which we call the
Eichler–Shimura map for ${\rm GL}_{n}$. The Eichler–Shimura map for ${\rm
GL}_{2}$ is a classical object, and its explicit description can be found in
standard references (often in terms of elliptic modular forms; see [Hid94,
Vat99] for instance). In this section, we give an explicit description of the
Eichler–Shimura map for ${\rm GL}_{3}$ analogously to the ${\rm GL}_{2}$ case,
which will give explicit evaluations of the zeta integrals and clarify their
connections to Whittaker periods.

Let $n$ denotes $2$ or $3$. In Section 5.1, we propose descriptions of cusp
forms on ${\rm GL}_{n}(\mathbf{Q}_{\mathbf{A}})$ in an appropriate way for our
purpose. Then we introduce several local systems on the symmetric space
$Y^{(n)}_{\mathcal{K}_{n}}$ in Section 5.2. The Eichler–Shimura map for ${\rm
GL}_{n}$ is introduced in Section 5.3. In this article, we only need the
Eichler–Shimura map taking values in the cohomology group of bottom degree.
However, the explicit description of the Eichler–Shimura map taking values in
the cohomology group of top degree will be also important for further studies
of zeta integrals, and hence we briefly summarize its construction in Appendix
A.

###  5.1. Cusp forms

Here we summarize some notation on automorphic representations $\pi^{(3)}$ and
$\pi^{(2)}$ in this article. For $n=2$ and $3$, denote by $\pi^{(n)}$ a
cohomological irreducible cuspidal automorphic representation of ${\rm
GL}_{n}({\mathbf{Q}}_{\mathbf{A}})$ of minimal ${\rm
O}_{n}({\mathbf{R}})$-type $\tau^{(n)}_{\boldsymbol{\lambda}^{(n)}}$ for
$\boldsymbol{\lambda}^{(n)}=(\lambda_{n},\delta_{n})\in\Lambda_{n}^{\mathrm{coh}}$
(recall the definition of $\Lambda^{\mathrm{coh}}_{n}$ from Section 4.2). We
often abbreviate $\tau^{(n)}_{\boldsymbol{\lambda}^{(n)}}$ as
$\tau^{(n)}_{\boldsymbol{\lambda}}$ if there is no risk of confusion.
Furthermore, since the second component of $\boldsymbol{\lambda}^{(2)}$ always
equals $0$, we remove the subindex $3$ from the second component of
$\boldsymbol{\lambda}^{(3)}$ and just write
$\boldsymbol{\lambda}^{(3)}=(\lambda_{3},\delta)$. Let $\phi_{\nu_{2},l_{2}}$
(resp. $\phi_{\nu_{3},l_{3}}\oplus\phi_{\nu_{3}}^{\delta}$) be the Langlands
parameter of $\pi^{(2)}_{\infty}$ (resp. $\pi^{(3)}_{\infty}$); then we have

(5.1) $\displaystyle\lambda_{3}$ $\displaystyle=l_{3}+1,$
$\displaystyle\lambda_{2}$ $\displaystyle=l_{2}+1.$

If we denote by $\omega_{\pi^{(n)}}$ the central character of $\pi^{(n)}$, the
evaluation $\omega_{\pi^{(n)},\infty}(-1)$ at $-1$ of its archimedean
component $\omega_{\pi^{(n)},\infty}$ is described by these parameters as
follows (note that $\lambda_{3}=l_{3}+1$ is odd since $l_{3}$ is even due to
Remark 4.2):

$\displaystyle\omega_{\pi^{(2)},\infty}(-1)$
$\displaystyle=(-1)^{\lambda_{2}},$
$\displaystyle\omega_{\pi^{(3)},\infty}(-1)$
$\displaystyle=(-1)^{\lambda_{3}+\delta}=-(-1)^{\delta}.$

For $n=2$ or $3$, let $\phi:{\rm GL}_{n}({\mathbf{Q}})\backslash{\rm
GL}_{n}({\mathbf{Q}}_{\mathbf{A}})\to{\mathbf{C}}$ be a cuspidal automorphic
form associated with $\pi^{(n)}$; namely, $\pi^{(n)}$ is isomorphic to the
infinite dimensional irreducible subquotient of the right regular
representation of ${\rm GL}_{n}({\mathbf{Q}}_{\mathbf{A}})$ generated by
$\phi$. Suppose that $\pi^{(n)}$ is fixed by the action of an open compact
subgroup ${\mathcal{K}}_{n}$ of ${\rm GL}_{n}({\mathbf{Q}}_{\mathbf{A},{\rm
fin}})$. Note that ${\rm
Hom}_{{\mathrm{O}}_{n}({\mathbf{R}})}(\tau^{(n)}_{\boldsymbol{\lambda}},\pi^{(n)}_{\infty})$
is of dimenstion $1$ by Schur’s lemma. Hence, for $n=2$, we find a pair
$(h^{+},h^{-})$ of cuspidal automorphic forms $h^{\pm}:{\rm
GL}_{2}({\mathbf{Q}})\backslash{\rm
GL}_{2}({\mathbf{Q}}_{\mathbf{A}})\to{\mathbf{C}}$ associated with $\pi^{(2)}$
satisfying

$\displaystyle\begin{pmatrix}h^{+}(gtu)&h^{-}(gtu)\end{pmatrix}$
$\displaystyle=t^{2\nu_{2}}\begin{pmatrix}h^{+}(g)e^{\sqrt{-1}\lambda_{2}\theta}&h^{-}(g)e^{-\sqrt{-1}\lambda_{2}\theta}\end{pmatrix}$
$\displaystyle\begin{pmatrix}h^{+}(g\epsilon)&h^{-}(g\epsilon)\end{pmatrix}$
$\displaystyle=\begin{pmatrix}h^{-}(g)&h^{+}(g)\end{pmatrix}$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:NormNu})}}{{=}}t^{(-l_{2}+1)}\begin{pmatrix}h^{+}(g)e^{\sqrt{-1}(l_{2}+1)\theta}&h^{-}(g)e^{-\sqrt{-1}(l_{2}+1)\theta}\end{pmatrix},$

for each $g\in{\rm GL}_{2}({\mathbf{Q}}_{\mathbf{A}})$,
$t\in{\mathbf{R}}^{\times}_{>0}$, $u=\begin{pmatrix}\cos\theta&\sin\theta\\\
-\sin\theta&\cos\theta\end{pmatrix}\in{\rm SO}_{2}({\mathbf{R}})$ and
$\epsilon=\begin{pmatrix}-1&0\\\ 0&1\end{pmatrix}$. Since the archimedean part
of $h^{+}$ (resp. $h^{-}$) belongs to $D^{+}_{l_{2}}$ (resp. $D^{-}_{l_{2}}$),
we call $h^{+}$ (resp. $h^{-}$) a holomorphic (resp. an anti-holomorphic)
cuspidal automorphic form associated with $\pi^{(2)}$, which is uniquely
determined up to scalar multiples. For each
$\boldsymbol{\lambda}^{(2)}\in\Lambda^{\mathrm{coh}}_{2}$, let
${\mathcal{S}}^{(2)}_{\boldsymbol{\lambda}^{(2)}}({\mathcal{K}}_{2})$ (often
abbreviated as $\mathcal{S}^{(2)}_{\boldsymbol{\lambda}}(\mathcal{K}_{2})$) be
the ${\mathbf{C}}$-vector space spanned by both of holomorphic and anti-
holomorphic cuspidal automorphic forms associated with certain cohomological
irreducible cuspidal automorphic representations of ${\rm
GL}_{2}({\mathbf{Q}}_{\mathbf{A}})$, which are invariant under the right
translation by ${\mathcal{K}}_{2}$ and whose archimedean parts have the
minimal ${\mathrm{O}}_{2}({\mathbf{R}})$-type
$\tau^{(2)}_{\boldsymbol{\lambda}}$.

In a similar way, for $n=3$, we find a tuple
$\begin{pmatrix}f^{\boldsymbol{\lambda}}_{\lambda_{3}}&f^{\boldsymbol{\lambda}}_{\lambda_{3}-1}&\dotsc&f^{\boldsymbol{\lambda}}_{-\lambda_{3}}\end{pmatrix}$
consisting of cuspidal automorphic forms
$f^{\boldsymbol{\lambda}}_{\alpha}\colon{\rm
GL}_{3}({\mathbf{Q}})\backslash{\rm
GL}_{3}({\mathbf{Q}}_{\mathbf{A}})\to{\mathbf{C}}$ for
$-\lambda_{3}\leq\alpha\leq\lambda_{3}$ associated with $\pi^{(3)}$ satisfying

$\displaystyle\begin{pmatrix}f^{\boldsymbol{\lambda}}_{\lambda_{3}}(gtu)&f^{\boldsymbol{\lambda}}_{\lambda_{3}-1}(gtu)&\dotsc&f^{\boldsymbol{\lambda}}_{-\lambda_{3}}(gtu)\end{pmatrix}$
$\displaystyle=\begin{pmatrix}f^{\boldsymbol{\lambda}}_{\lambda_{3}}(g)&f^{\boldsymbol{\lambda}}_{\lambda_{3}-1}(g)&\dotsc&f^{\boldsymbol{\lambda}}_{-\lambda_{3}}(g)\end{pmatrix}t^{3\nu_{3}}M_{\boldsymbol{\lambda}}(u)$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:NormNu})}}{{=}}\begin{pmatrix}f^{\boldsymbol{\lambda}}_{\lambda_{3}}(g)&f^{\boldsymbol{\lambda}}_{\lambda_{3}-1}(g)&\dotsc&f^{\boldsymbol{\lambda}}_{-\lambda_{3}}(g)\end{pmatrix}t^{-\frac{3}{2}l_{3}+3}M_{\boldsymbol{\lambda}}(u)$

for each $g\in{\rm GL}_{3}({\mathbf{Q}}_{\mathbf{A}})$,
$t\in{\mathbf{R}}^{\times}_{>0}$ and $u\in{\rm O}_{3}({\mathbf{R}})$; recall
the definition of the matrix $M_{\boldsymbol{\lambda}}(u)$ from (3.6). Here we
abbreviate $\boldsymbol{\lambda}^{(3)}$ as $\boldsymbol{\lambda}$ to lighten
the notation. We regard this tuple as a vector and denote by
${\boldsymbol{f}}$, which we call a cusp form associated with $\pi^{(3)}$. For
each $\boldsymbol{\lambda}^{(3)}\in\Lambda^{\mathrm{coh}}_{3}$, define
${\mathcal{S}}^{(3)}_{\boldsymbol{\lambda}^{(3)}}({\mathcal{K}}_{3})$ (often
abbreviated as $\mathcal{S}^{(3)}_{\boldsymbol{\lambda}}(\mathcal{K}_{3})$) to
be the ${\mathbf{C}}$-vector space spanned by cusp forms associated with
certain cohomological irreducible cuspidal automorphic representations of
${\rm GL}_{3}({\mathbf{Q}}_{\mathbf{A}})$, which are invariant under the right
translation by ${\mathcal{K}}_{3}$ and whose archimedean parts have the
minimal ${\mathrm{O}}_{3}({\mathbf{R}})$-type
$\tau^{(3)}_{\boldsymbol{\lambda}}$.

###  5.2. Local systems

Let $n$ denote $2$ or $3$. For each open compact subgroup
${\mathcal{K}}_{n}\subset{\rm GL}_{n}({\mathbf{Q}}_{{\mathbf{A}},{\rm fin}})$,
define the corresponding symmetric space as $Y^{(n)}_{{\mathcal{K}}_{n}}={\rm
GL}_{n}({\mathbf{Q}})\backslash{\rm
GL}_{n}({\mathbf{Q}}_{\mathbf{A}})/{\mathbf{R}}^{\times}_{>0}{\rm
SO}_{n}({\mathbf{R}}){\mathcal{K}}_{n}$. In this subsection, we introduce
local systems $\mathcal{L}^{(n)}(*;A)$ on $Y^{(n)}_{{\mathcal{K}}_{n}}$ coming
from finite dimensional irreducible representations $L^{(n)}(*;A)$ of ${\rm
GL}_{n}$ for a field $A$ of characteristic $0$. They will play crucial roles
to give a cohomological interpretation of cusp forms on ${\rm
GL}_{n}({\mathbf{Q}}_{\mathbf{A}})$ in the next subsection.

Let $\boldsymbol{n}=(n_{1},n_{2})\in\mathbf{Z}^{2}$ and
$\boldsymbol{w}=(w_{1}^{+},w_{1}^{-},w_{2})\in\mathbf{Z}^{3}$ be as in
Sections 3.1 and 3.2, where $n_{1},w_{1}^{+},w_{1}^{-}\geq 0$. Then we define
the local system ${\mathcal{L}}^{(n)}(*;A)$ to be the sheaf of locally
constant sections of the projection $\mathrm{pr}_{1}$ defined as

$\displaystyle{\rm GL}_{n}({\mathbf{Q}})\backslash\left({\rm
GL}_{n}({\mathbf{Q}}_{\mathbf{A}})/{\mathbf{R}}^{\times}_{>0}{\rm
SO}_{n}({\mathbf{R}}){\mathcal{K}}_{n}\times
L^{(n)}(*;A)\right)\stackrel{{\scriptstyle{\rm
pr}_{1}}}{{\longrightarrow}}Y^{(n)}_{{\mathcal{K}}_{n}},$

where $*$ denotes $\boldsymbol{n}$ for $n=2$ and $\boldsymbol{w}$ for $n=3$.
We are especially interested in the local system associated to a cohomological
irreducible cuspidal automorphic representation $\pi^{(n)}$ of
$\mathrm{GL}_{n}(\mathbf{Q}_{\mathbf{A}})$ whose archimedean component has the
minimal $\mathrm{O}_{3}(\mathbf{R})$-type $\tau^{(n)}_{\boldsymbol{\lambda}}$.
For
$\boldsymbol{\lambda}^{(n)}=(\lambda_{n},\delta_{n})\in\Lambda^{\mathrm{coh}}_{n}$,
set $n_{\boldsymbol{\lambda}}=\lambda_{2}-2\,(=l_{2}-1)$ and
$w_{\boldsymbol{\lambda}}=\dfrac{\lambda_{3}-3}{2}\,\left(=\dfrac{l_{3}}{2}-1\right)$.
Then define $\boldsymbol{n}_{\boldsymbol{\lambda}}\in\mathbf{Z}^{2}$ and
$\boldsymbol{w}_{\boldsymbol{\lambda}}\in\mathbf{Z}^{3}$ as

(5.2) $\displaystyle\boldsymbol{n}_{\boldsymbol{\lambda}}$
$\displaystyle=(n_{\boldsymbol{\lambda}},0)=(\lambda_{2}-2,0),$
$\displaystyle\boldsymbol{w}_{\boldsymbol{\lambda}}$
$\displaystyle=(w_{\boldsymbol{\lambda}},w_{\boldsymbol{\lambda}},w_{\boldsymbol{\lambda}})=\left(\dfrac{\lambda_{3}-3}{2},\dfrac{\lambda_{3}-3}{2},\dfrac{\lambda_{3}-3}{2}\right).$

Note that $w_{\boldsymbol{\lambda}}$ is indeed an integer since $\lambda_{3}$
is odd due to the definition of $\Lambda^{\mathrm{coh}}_{3}$. For later use,
we define the set $\Xi^{\rm coh}_{3}$ of cohomological weights of local
systems for ${\rm GL}_{3}$ to be $\Xi^{\rm
coh}_{3}=\left\\{\left(w,w,w\right)\mid w\in{\mathbf{Z}},w\geq 0\right\\}$.

###  5.3. Eichler–Shimura maps

Let $\boldsymbol{\lambda}^{(n)}$ be an element of
$\Lambda^{\mathrm{coh}}_{n}$. Due to Clozel [Clo90, Section 3.5], the
structure of the cuspidal cohomology group $H^{i}_{\rm
cusp}(Y^{(n)}_{{\mathcal{K}_{n}}},{\mathcal{L}}^{(n)}(*;{\mathbf{C}}))$ with
coefficients in appropriate local systems is well understood, and described by
various cuspidal automorphic representations of given minimal
$\mathrm{O}_{n}(\mathbf{R})$-type. In this subsection, we construct a specific
class in the cuspidal cohomology group from an appropriate cusp form on
$\mathrm{GL}_{n}(\mathbf{Q}_{\mathbf{A}})$. For later computations, we here
give an appropriate form of this construction, which we call the
Eichler–Shimura map.

For $n=2$ and $3$, let $\pi^{(n)}$ be a cohomological irreducible cuspidal
automorphic representation of $\mathrm{GL}_{n}(\mathbf{Q}_{\mathbf{A}})$ whose
archimedean part $\pi^{(n)}_{\infty}$ has the minimal
$\mathrm{O}_{n}(\mathbf{R})$-type $\tau^{(n)}_{\boldsymbol{\lambda}}$. Set
$K_{n}={\mathbf{R}}^{\times}_{>0}{\rm SO}_{n}({\mathbf{R}})$, and let
$H_{\pi^{(n)},K_{n}}$ denote the
$(\mathfrak{gl}_{n}({\mathbf{R}}),K_{n})$-module associated with
$\pi^{(n)}_{\infty}$. In (4.1), we have normalized the action of
${\mathbf{R}}^{\times}_{>0}$ on $H_{\pi^{(n)},K_{n}}$ so that

$\displaystyle\pi_{\infty}^{(n)}(t)(v)=t^{n\nu_{n}}v=\begin{cases}t^{-l_{2}+1}v=t^{-\lambda_{2}+2}v=t^{-n_{\boldsymbol{\lambda}}}v&\text{for
}n=2,\\\
t^{-\frac{3}{2}l_{3}+3}v=t^{-3\frac{\lambda_{3}-3}{2}}v=t^{-3w_{\boldsymbol{\lambda}}}v&\text{for
}n=3\end{cases}$

holds for each $t\in{\mathbf{R}}^{\times}_{>0}$ and $v\in
H_{\pi^{(n)},K_{n}}$.

Let $*_{\boldsymbol{\lambda}}$ denote $\boldsymbol{n}_{\boldsymbol{\lambda}}$
if $n$ equals $2$ and $\boldsymbol{w}_{\boldsymbol{\lambda}}$ if $n$ equals
$3$. Then it is known that the $\pi^{(n)}_{\rm fin}$-isotypic component
$H^{i}(Y^{(n)}_{\mathcal{K}_{n}},{\mathcal{L}}^{(n)}(*_{\boldsymbol{\lambda}};{\mathbf{C}}))[\pi^{(n)}_{\rm
fin}]$ of
$H^{i}(Y^{(n)}_{\mathcal{K}_{n}},{\mathcal{L}}^{(n)}(*_{\boldsymbol{\lambda}};{\mathbf{C}}))$
satisfies

$\displaystyle
H^{i}(Y^{(n)}_{\mathcal{K}_{n}},{\mathcal{L}}^{(n)}(*_{\boldsymbol{\lambda}};{\mathbf{C}}))[\pi^{(n)}_{\rm
fin}]=H^{i}(\mathfrak{gl}_{n}({\mathbf{R}}),K_{n};H_{\pi^{(n)},K_{n}}\otimes
L^{(n)}(*_{\boldsymbol{\lambda}};{\mathbf{C}}))\otimes_{\mathbf{C}}\pi^{(n),{\mathcal{K}}_{n}}_{\rm
fin}.$

In order to construct an distinguished cohomology class, it is necessary to
describe the $(\mathfrak{gl}_{n}(\mathbf{R}),K_{n})$-cohomology group
$H^{i}(\mathfrak{gl}_{n}({\mathbf{R}}),K_{n};H_{\pi^{(n)},K_{n}}\otimes
L^{(n)}(*_{\boldsymbol{\lambda}};{\mathbf{C}}))$ in an explicit manner. The
following well-known theorem is due to Clozel [Clo90, Lemme 3.14]. Here we
adopt the formalism presented by Mahnkopf in [Mah05].

######  Theorem 5.1.

(refer to [Mah05, Section 3.1.2] and [GR14, Section 5.5]) We have

$\displaystyle
H^{i}(\mathfrak{gl}_{2}({\mathbf{R}}),K_{2};H_{\pi^{(2)},K_{2}}\otimes
L^{(2)}(\boldsymbol{n}_{\boldsymbol{\lambda}};{\mathbf{C}}))\cong\begin{cases}{\mathbf{C}}^{2}&\text{for
}i=1,\\\ 0&\text{otherwise}.\end{cases}$

We also find that

$\displaystyle
H^{i}(\mathfrak{gl}_{3}({\mathbf{R}}),K_{3};H_{\pi^{(3)},K_{3}}\otimes
L^{(3)}(\boldsymbol{w}_{\boldsymbol{\lambda}};{\mathbf{C}}))\cong\begin{cases}{\mathbf{C}}&\text{for
}i=2\text{ and }3,\\\ 0&\text{otherwise}.\end{cases}$

In the present article, we are interested in the cohomology group
$H^{b_{n}}(\mathfrak{gl}_{n}({\mathbf{R}}),K_{n};H_{\pi^{(n)},K_{n}}\otimes
L^{(n)}(*_{\boldsymbol{\lambda}};{\mathbf{C}}))$ of bottom degree
$b_{n}:=[\frac{n^{2}}{4}]$. Note that the cohomology of top degree has also
independent interest beyond the scope of this article, hence we give an
explicit basis of the cohomology group
$H^{3}(\mathfrak{gl}_{3}({\mathbf{R}}),K_{3};H_{\pi^{(3)},K_{3}}\otimes
L^{(3)}(\boldsymbol{w}_{\boldsymbol{\lambda}};{\mathbf{C}}))$ in Appendix A.

Let ${\mathcal{P}}_{n}$ be the quotient Lie algebra
$\mathfrak{gl}_{n}({\mathbf{R}})/{\rm Lie}(K_{n})$ and
${\mathcal{P}}_{n,{\mathbf{C}}}={\mathcal{P}}_{n}\otimes_{\mathbf{R}}{\mathbf{C}}$
its complexification. By using [BW80, II, Proposition 3.1], we obtain a
natural isomorphism

(5.3) $\displaystyle
H^{i}(\mathfrak{gl}_{n}({\mathbf{R}}),K_{n};H_{\pi^{(n)},K_{n}}\otimes
L^{(n)}(*_{\boldsymbol{\lambda}};{\mathbf{C}}))\cong{\rm Hom}_{{\rm
SO}_{n}({\mathbf{R}})}\left(\bigwedge^{i}{\mathcal{P}}_{n,{\mathbf{C}}},H_{\pi^{(n)},K_{n}}\otimes
L^{(n)}(*_{\boldsymbol{\lambda}};{\mathbf{C}})\right).$

To study the module appearing in the right-hand side of (5.3), we first
analyze the adjoint action of $\mathrm{SO}_{n}(\mathbf{R})$ on
$\mathcal{P}_{n,\mathbf{C}}$ with respect to a certain specific basis. Let
$\\{H,E\\}$ be a basis of $\mathcal{P}_{2,\mathbf{C}}$ defined by

$\displaystyle H$ $\displaystyle=\frac{1}{2}\begin{pmatrix}1&0\\\
0&-1\end{pmatrix},$ $\displaystyle E$
$\displaystyle=\frac{1}{2}\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}.$

Similarly, let $\\{X_{2},X_{1},X_{0},X_{-1},X_{-2}\\}$ be a basis of
$\mathcal{P}_{3,\mathbf{C}}$ defined as follows:

(5.4) $\displaystyle\begin{aligned} A_{1}&=\begin{pmatrix}1&0&0\\\ 0&-1&0\\\
0&0&0\end{pmatrix},&A_{2}&=\begin{pmatrix}1&0&0\\\ 0&1&0\\\
0&0&-2\end{pmatrix},&N_{1}&=\begin{pmatrix}0&1&0\\\ 1&0&0\\\
0&0&0\end{pmatrix},&N_{2}&=\begin{pmatrix}0&0&1\\\ 0&0&0\\\
1&0&0\end{pmatrix},&N_{3}&=\begin{pmatrix}0&0&0\\\ 0&0&1\\\
0&1&0\end{pmatrix},\\\
X_{2}&=A_{1}+\sqrt{-1}N_{1},&X_{1}&=N_{2}+\sqrt{-1}N_{3},&X_{0}&=A_{2},&X_{-1}&=N_{2}-\sqrt{-1}N_{3},&X_{-2}&=A_{1}-\sqrt{-1}N_{1}.\end{aligned}$

The following lemma is a consequence of a direct calculation:

######  Lemma 5.2.

For each $u\in{\rm SO}_{2}({\mathbf{R}})$, we have ${\rm
Ad}(u)\begin{pmatrix}E&H\end{pmatrix}=\begin{pmatrix}E&H\end{pmatrix}u^{-2}$.
For each $u\in{\rm SO}_{3}({\mathbf{R}})$, we have

$\displaystyle{\rm
Ad}(u)\begin{pmatrix}X_{2}&X_{1}&X_{0}&X_{-1}&X_{-2}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}X_{2}&X_{1}&X_{0}&X_{-1}&X_{-2}\end{pmatrix}M_{(2,0)}(u),$

where the matrix $M_{(2,0)}(u)\in{\rm GL}_{5}({\mathbf{C}})$ is defined as in
(3.6).

First let us consider the $\mathrm{GL}_{2}$ case. Let
$\\{\mathrm{d}H,\mathrm{d}E\\}$ denote the dual basis of
$\mathcal{P}_{2,\mathbf{C}}^{*}$, where $\mathcal{P}_{2,\mathbf{C}}^{*}$
denotes the linear dual space of $\mathcal{P}_{2,\mathbf{C}}$. Let ${\rm
Ad}^{*}$ be the contragredient action of ${\rm Ad}$. Then the following
formula is easily deduced from the first satement of Lemma 5.2; for each
$u=\begin{pmatrix}\cos\theta&\sin\theta\\\
-\sin\theta&\cos\theta\end{pmatrix}\in{\rm SO}_{2}({\mathbf{R}})$, we have

$\displaystyle{\rm Ad}^{\ast}(u)\begin{pmatrix}{\rm dE}-\sqrt{-1}{\rm
d}H&-{\rm dE}-\sqrt{-1}{\rm d}H\end{pmatrix}=\begin{pmatrix}{\rm
dE}-\sqrt{-1}{\rm d}H&-{\rm dE}-\sqrt{-1}{\rm
d}H\end{pmatrix}\begin{pmatrix}e^{-2\sqrt{-1}\theta}&0\\\
0&e^{2\sqrt{-1}\theta}\end{pmatrix}.$

Hence, for each holomorphic (resp. anti-holomorphic) cusp form $h^{+}$ (resp.
$h^{-}$) in ${\mathcal{S}}^{(2)}_{\boldsymbol{\lambda}}({\mathcal{K}}_{2})$,
the element

$\displaystyle\delta^{(2)}(h^{\pm})=h^{\pm}(\mp
X+\sqrt{-1}Y)^{n_{\boldsymbol{\lambda}}}(\pm{\rm dE}-\sqrt{-1}{\rm
d}H)\quad\bigl{(}=h^{\pm}(\mp X+\sqrt{-1}Y)^{\lambda_{2}-2}(\pm{\rm
dE}-\sqrt{-1}\mathrm{d}H)\bigr{)}$

is indeed contained in ${\rm Hom}_{{\rm
SO}_{2}({\mathbf{R}})}\left({\mathcal{P}}_{2,{\mathbf{C}}},H_{\pi^{(2)},K_{2}}\otimes
L^{(2)}(\boldsymbol{n}_{\boldsymbol{\lambda}};{\mathbf{C}})\right)\otimes_{\mathbf{C}}(\pi^{(2)}_{\rm
fin})^{\mathcal{K}_{2}}$. We thus obtain $\delta^{(2)}$ a map

$\displaystyle\delta^{(2)}:{\mathcal{S}}^{(2)}_{\boldsymbol{\lambda}}({\mathcal{K}}_{2})\longrightarrow
H^{1}_{\rm
cusp}(Y^{(2)}_{{\mathcal{K}}_{2}},{\mathcal{L}}^{(2)}({\boldsymbol{n}}_{\boldsymbol{\lambda}};{\mathbf{C}}))\,;h^{\pm}\mapsto\delta^{(2)}(h^{\pm})$

which is called the Eichler–Shimula map for $\mathrm{GL}_{2}$. It is obvious
that $\delta^{(2)}$ is a Hecke equivariant homomorphism from its construction.

Now let us construct a similar map for the ${\rm GL}_{3}$ case; to be precise,
we will define a Hecke equivariant homomorphism

$\displaystyle\delta^{(3)}:{\mathcal{S}}^{(3)}_{\boldsymbol{\lambda}}({\mathcal{K}}_{3})\longrightarrow
H^{2}_{\rm
cusp}(Y^{(3)}_{{\mathcal{K}}_{3}},{\mathcal{L}}^{(3)}(\boldsymbol{w}_{\boldsymbol{\lambda}};{\mathbf{C}}))$

for $\boldsymbol{\lambda}^{(3)}=(\lambda_{3},\delta)\in\Lambda^{\rm coh}_{3}$.
For this purpose, we have to introduce some more notation. Let $P^{2}\in{\rm
M}_{10,7}({\mathbf{C}})$ be the matrix which is introduced in (A.1). Then we
define
${\boldsymbol{\omega}}_{i}\in\bigwedge^{2}{\mathcal{P}}^{\ast}_{3,{\mathbf{C}}}$
for $-3\leq i\leq 3$ to be

(5.5) $\displaystyle\begin{aligned}
\begin{pmatrix}{\boldsymbol{\omega}}_{3}&\boldsymbol{\omega}_{2}&\dotsc&{\boldsymbol{\omega}}_{-3}\end{pmatrix}=\begin{pmatrix}{\rm
d}X_{2}\wedge{\rm d}X_{1}&{\rm d}X_{2}\wedge{\rm d}X_{0}&\dotsc&{\rm
d}X_{-1}\wedge{\rm d}X_{-2}\end{pmatrix}P^{2}.\end{aligned}$

Here $\\{{\rm d}X_{2},{\rm d}X_{1},{\rm d}X_{0},{\rm d}X_{-1},{\rm
d}X_{-2}\\}$ is the dual basis of $\mathcal{P}^{*}_{3,\mathbf{C}}$ and
$\\{\mathrm{d}X_{i}\wedge\mathrm{d}X_{j}\\}_{2\geq i>j\geq-2}$ is ordered
lexicographically. Consider

$\displaystyle P(X,Y,Z,A,B,C,z_{1},z_{2},z_{3})$
$\displaystyle:=\left((X,Y,Z)\begin{pmatrix}z_{1}\\\ z_{2}\\\
z_{3}\end{pmatrix}\right)^{w_{\boldsymbol{\lambda}}}\otimes\left((A,B,C)\begin{pmatrix}z_{1}\\\
z_{2}\\\
z_{3}\end{pmatrix}\right)^{w_{\boldsymbol{\lambda}}}\begin{pmatrix}v^{(3,\delta)}_{3}&v^{(3,\delta)}_{2}&\dotsc&v^{(3,\delta)}_{-3}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}v^{\boldsymbol{\lambda}}_{\lambda_{3}}&v^{\boldsymbol{\lambda}}_{\lambda_{3}-1}&\dotsc&v^{\boldsymbol{\lambda}}_{-\lambda_{3}}\end{pmatrix}{\mathcal{P}}(X,Y,Z,A,B,C)$

(recall that $w_{\boldsymbol{\lambda}}=\dfrac{\lambda_{3}-3}{2}$). Then we can
write down ${\mathcal{P}}(X,Y,Z,A,B,C)$ as follows:

(5.6)
$\displaystyle{\mathcal{P}}(X,Y,Z,A,B,C)=\begin{pmatrix}{\mathcal{P}}_{\lambda_{3},3}(X,Y,Z,A,B,C)&\cdots&{\mathcal{P}}_{\lambda_{3},-3}(X,Y,Z,A,B,C)\\\
\vdots&\ddots&\vdots\\\
{\mathcal{P}}_{-\lambda_{3},3}(X,Y,Z,A,B,C)&\cdots&{\mathcal{P}}_{-\lambda_{3},-3}(X,Y,Z,A,B,C)\end{pmatrix},$

which is an element of ${\rm
M}_{2\lambda_{3}+1,7}({\mathbf{C}}[X,Y,Z;A,B,C]_{w_{\boldsymbol{\lambda}},w_{\boldsymbol{\lambda}}})$.

######  Lemma 5.3.

  1. (i)

The tuple $\\{{\boldsymbol{\omega}}_{3},\ldots,{\boldsymbol{\omega}}_{-3}\\}$
forms a basis of $\bigwedge^{2}{\mathcal{P}}^{\ast}_{3,{\mathbf{C}}}$.
Moreover, for each $u\in{\rm SO}_{3}({\mathbf{R}})$, we have

$\displaystyle\wedge^{2}{\rm
Ad}^{\ast}(u)\begin{pmatrix}{\boldsymbol{\omega}}_{3}&\boldsymbol{\omega}_{2}&\dotsc&{\boldsymbol{\omega}}_{-3}\end{pmatrix}=\begin{pmatrix}{\boldsymbol{\omega}}_{3}&\boldsymbol{\omega}_{2}&\dotsc&{\boldsymbol{\omega}}_{-3}\end{pmatrix}{}^{\rm
t}\\!M_{(3,0)}(u)^{-1}.$

  2. (ii)

Every entry ${\mathcal{P}}_{\alpha,\beta}(X,Y,Z,A,B,C)$ 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 $\mathbf{Z}[2^{-1},\sqrt{-1}]$.

  3. (iii)

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

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

###### Proof..

See Lemma A.1 for (i) and Lemma A.2 for (ii) and (iii) in Appendix A. ∎

Let
$\boldsymbol{f}=\begin{pmatrix}f^{\boldsymbol{\lambda}}_{\lambda_{3}}&f^{\boldsymbol{\lambda}}_{\lambda_{3}-1}&\dotsc&f^{\boldsymbol{\lambda}}_{-\lambda_{3}}\end{pmatrix}$
be an element in
${\mathcal{S}}^{(3)}_{\boldsymbol{\lambda}}({\mathcal{K}}_{3})$. Define

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

Since $\delta^{(3)}(\boldsymbol{f})$ gives an element in ${\rm Hom}_{{\rm
SO}_{3}({\mathbf{R}})}\left(\bigwedge^{2}{\mathcal{P}}_{3,{\mathbf{C}}},H_{\pi^{(3)},K_{3}}\otimes
L^{(3)}(\boldsymbol{w}_{\boldsymbol{\lambda}};{\mathbf{C}})\right)\otimes(\pi^{(3)}_{\rm
fin})^{\mathcal{K}_{3}}$ due to Lemma 5.3 (iii), we obtain a Hecke equivariant
map

$\displaystyle\delta^{(3)}\colon{\mathcal{S}}^{(3)}_{\boldsymbol{\lambda}}({\mathcal{K}}_{3})\longrightarrow
H^{2}_{\rm
cusp}(Y^{(3)}_{{\mathcal{K}}_{3}},{\mathcal{L}}^{(3)}(\boldsymbol{w}_{\boldsymbol{\lambda}};{\mathbf{C}}))\,;\boldsymbol{f}\longmapsto\delta^{(3)}(\boldsymbol{f}).$

##  6\. Cohomological cup products and global zeta integrals for ${\rm
GL}_{3}\times{\rm GL_{2}}$

The purpose of this section is to introduce the connection between the
cohomology classes constructed in the previous section and the global zeta
integrals for $\mathrm{GL}_{3}\times\mathrm{GL}_{2}$, which is given by
precise study of the cohomological cup product pairing based on an explicit
description of the branching rule for the pair (${\rm GL}_{3},{\rm GL}_{2}$)
provided in Section 3.3. After introducing the cup product pairing which we
shall use in Section 6.1, we consider in Section 6.2 the cup product of the
Eichler–Shimura classes constructed in the previous section. In Section 6.3,
we firstly give a cohomological interpretation of the partial zeta integrals
(see Proposition 6.3). This immediately deduces a cohomological interpretation
of the global zeta integrals, which we call Birch lemma, borrowing the name
customarily used in the research of $L$-functions (see Corollary 6.4).

###  6.1. Description of the cup product pairing

Let $\mathcal{Y}^{(n)}_{\mathcal{K}_{n}}$ be the symmetric space ${\rm
GL}_{n}({\mathbf{Q}})\backslash{\rm GL}_{n}({\mathbf{Q}}_{\mathbf{A}})/{\rm
SO}_{n}({\mathbf{R}}){\mathcal{K}}_{n}$, which is equipped with the natural
surjection ${\rm p}_{n}:{\mathcal{Y}}^{(n)}_{\mathcal{K}_{n}}\longrightarrow
Y^{(n)}_{\mathcal{K}_{n}}$. We denote by
$\iota:{\mathcal{Y}}^{(2)}_{\mathcal{K}_{2}}\longrightarrow{\mathcal{Y}}^{(3)}_{\mathcal{K}_{3}}$
the natural map induced by the inclusion $\iota:{\rm
GL}_{2}({\mathbf{Q}}_{\mathbf{A}})\to{\rm
GL}_{3}({\mathbf{Q}}_{\mathbf{A}})\,;g\mapsto\begin{pmatrix}g&\\\
&1\end{pmatrix}$. Note that
$\iota:{\mathcal{Y}}^{(2)}_{\mathcal{K}_{2}}\longrightarrow{\mathcal{Y}}^{(3)}_{\mathcal{K}_{3}}$
is a proper map. Hereafter, we always assume that
${\mathcal{K}}_{2}=\iota^{-1}({\mathcal{K}}_{3})$. In this subsection, we
consider a certain cup product on ${\mathcal{Y}}^{(2)}_{\mathcal{K}_{2}}$,
which will play a crucial role for a cohomological interpretation of critical
values of $L(s,{\mathcal{M}}(\pi^{(3)}\times\pi^{(2)}))$.

We shall use the same notation as the previous sections. In particular, let us
consider the weight parameters
$\boldsymbol{w}=(w,w,w)\in\Xi_{3}^{\mathrm{coh}}$ and
$\boldsymbol{n}=(n_{1},n_{2})\in\Xi_{2}(\boldsymbol{w})$ for the local systems
(see Sections 5.2 and 3.3 for the definitions of $\Xi^{\mathrm{coh}}_{3}$ and
$\Xi_{2}(\boldsymbol{w})$ respectively). Let $A$ be a subfield of
${\mathbf{C}}$. Recall from Section 3.3 that we have constructed the explicit
branching rule isomorphism (3.4):

$\displaystyle{\boldsymbol{\nabla}}=(\nabla^{{\boldsymbol{n}}})_{\boldsymbol{n}\in\Xi_{2}(\boldsymbol{w})}:L^{(3)}(\boldsymbol{w};A)|_{{\rm
GL}_{2}(A)}\longrightarrow\bigoplus_{\boldsymbol{n}\in\Xi_{2}(\boldsymbol{w})}L^{(2)}(\boldsymbol{n};A).$

For each $\boldsymbol{n}=(n_{1},n_{2})\in\Xi_{2}(\boldsymbol{w})$, define a
morphism between local systems on $\mathcal{Y}^{(2)}_{\mathcal{K}_{2}}$

(6.1)
$\displaystyle\iota^{*}\mathrm{p}_{3}^{*}{\mathcal{L}}^{(3)}(\boldsymbol{w};A)_{/{\mathcal{Y}}^{(2)}_{{\mathcal{K}}_{2}}}\longrightarrow\mathrm{p}_{2}^{*}{\mathcal{L}}^{(2)}(\boldsymbol{n};A)_{/{\mathcal{Y}}^{(2)}_{{\mathcal{K}}_{2}}};(g,P_{\iota(g)})\longmapsto(g,\nabla^{\boldsymbol{n}}P_{\iota(g)})$

induced by $\nabla^{\boldsymbol{n}}$, for which we use the same symbol
$\nabla^{\boldsymbol{n}}$.

Now consider the ${\rm GL}_{2}({\mathbf{Q}})$-equivariant non-degenerate
pairing induced by

$\displaystyle[\cdot,\cdot]_{\boldsymbol{n}}\colon
L^{(2)}(\boldsymbol{n},A)\otimes_{A}L^{(2)}(n_{1},0;A)\longrightarrow
L^{(2)}(0,[\boldsymbol{n}];A);\,X^{i}Y^{n_{1}-i}\otimes
X^{j}Y^{n_{1}-j}\longmapsto\begin{cases}(-1)^{i}\binom{n_{1}}{i}^{-1}&\text{for
}i+j=n_{1},\\\ 0&\text{otherwise}\end{cases}$

where we set $[\boldsymbol{n}]:=n_{1}+n_{2}$. Let
$\widetilde{A}_{[\boldsymbol{n}]}$ be the local system on
${\mathcal{Y}}^{(2)}_{{\mathcal{K}}_{2}}$, which is determined by
$L^{(2)}(0,[\boldsymbol{n}];A)$. For an open compact subgroup $U$ of
$\widehat{\mathbf{Z}}^{\times}$, we define the narrow ray class group ${\rm
Cl}^{+}_{\mathbf{Q}}(U)$ of $\mathbf{Q}$ modulo $U$ by ${\rm
Cl}^{+}_{\mathbf{Q}}(U)={\mathbf{Q}}^{\times}\backslash{\mathbf{Q}}^{\times}_{\mathbf{A}}/{\mathbf{R}}^{\times}_{>0}U$.
Then the determinant map defines a morphism

$\displaystyle{\rm
det}_{\mathcal{K}_{2}}\colon{\mathcal{Y}}^{(2)}_{\mathcal{K}_{2}}\longrightarrow{\rm
Cl}^{+}_{\mathbf{Q}}(\det{\mathcal{K}}_{2});\,[g]\longmapsto[\det g].$

In the following arguments, we write ${\rm
Cl}^{+}_{\mathbf{Q}}(\det{\mathcal{K}}_{2})$ as ${\rm
Cl}^{+}_{\mathbf{Q}}({\mathcal{K}}_{2})$ for the sake of simplicity. For each
$x\in{\rm Cl}^{+}_{\mathbf{Q}}({\mathcal{K}}_{2})$, let
${\mathcal{Y}}^{(2)}_{\mathcal{K}_{2},x}$ denote the inverse image of $x$
under $\det_{\mathcal{K}_{2}}$. We use the same symbol
$\widetilde{A}_{[\boldsymbol{n}]}$ for the pullback of
$\widetilde{A}_{[\boldsymbol{n}]}$ to
${\mathcal{Y}}^{(2)}_{\mathcal{K}_{2},x}$ under the natural inclusion
${\mathcal{Y}}^{(2)}_{{\mathcal{K}_{2}},x}\rightarrow{\mathcal{Y}}^{(2)}_{\mathcal{K}_{2}}$.
Then the paring $[\cdot,\cdot]_{\boldsymbol{n}}$ induces a cup product on
cohomology groups of $\mathcal{Y}^{(2)}_{\mathcal{K}_{2},x}$:

(6.2) $\displaystyle\cup\colon
H^{2}({\mathcal{Y}}^{(2)}_{\mathcal{K}_{2},x},\nabla^{\boldsymbol{n}}\iota^{\ast}{\rm
p}^{\ast}_{3}{\mathcal{L}}^{(3)}(\boldsymbol{w};A))\otimes_{A}H^{1}_{\rm
c}({\mathcal{Y}}^{(2)}_{\mathcal{K}_{2},x},{\rm
p}^{\ast}_{2}{\mathcal{L}}^{(2)}(n_{1},0;A))\longrightarrow H^{3}_{\rm
c}({\mathcal{Y}}^{(2)}_{\mathcal{K}_{2},x},\widetilde{A}_{[\boldsymbol{n}]}).$

Let $\widetilde{A}$ be the trivial local system on
${\mathcal{Y}}^{(2)}_{\mathcal{K}_{2},x}$, which is determined by $A$. Since a
character

$\displaystyle{\mathbf{Q}}^{\times}_{\mathbf{A}}\longrightarrow{\mathbf{C}}^{\times};x\longmapsto\frac{|x|_{\mathbf{A}}}{x_{\infty}}$

takes values in ${\mathbf{Q}}^{\times}$, the following morphism between local
systems on ${\mathcal{Y}}^{(2)}_{\mathcal{K}_{2},x}$ is well defined:

$\displaystyle{\rm
Tw}_{[\boldsymbol{n}]}:\widetilde{A}_{[\boldsymbol{n}]/{\mathcal{Y}}^{(2)}_{{\mathcal{K}}_{2},x}}\longrightarrow{\widetilde{A}}_{/{\mathcal{Y}}^{(2)}_{{\mathcal{K}}_{2},x}};(g,a_{g})\longmapsto(g,\frac{|\det
g|^{[\boldsymbol{n}]}_{\mathbf{A}}}{(\det
g_{\infty})^{[\boldsymbol{n}]}}a_{g}).$

Composing (6.2) with the map induced from ${\rm Tw}_{[\boldsymbol{n}]}$, the
pairing (6.2) is rewritten as

(6.3) $\displaystyle\cup\colon
H^{2}({\mathcal{Y}}^{(2)}_{\mathcal{K}_{2},x},\nabla^{\boldsymbol{n}}\iota^{\ast}{\rm
p}^{\ast}_{3}{\mathcal{L}}^{(3)}(\boldsymbol{w};A))\otimes_{A}H^{1}_{\rm
c}({\mathcal{Y}}^{(2)}_{\mathcal{K}_{2},x},{\rm
p}^{\ast}_{2}{\mathcal{L}}^{(2)}(n_{1},0;A))\longrightarrow H^{3}_{\rm
c}({\mathcal{Y}}^{(2)}_{\mathcal{K}_{2},x},\widetilde{A})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}A.$

Here the rightmost map in (6.3) is the same as the upper horizontal one in the
commutative diagram

$\displaystyle\begin{CD}H^{3}_{\rm
c}({\mathcal{Y}}^{(2)}_{\mathcal{K}_{2},x},\widetilde{A})@>{\sim}>{}>A\\\
@V{}V{}V@V{}V{}V\\\ H^{3}_{\rm
c}({\mathcal{Y}}^{(2)}_{\mathcal{K}_{2},x},\widetilde{\mathbf{C}})@>{\sim}>{\int}>{\mathbf{C}}\end{CD}$

where the vertical maps are induced by the natural inclusion
$A\to{\mathbf{C}}$ and the lower horizontal map is obtained by the integration
on ${\mathcal{Y}}^{(2)}_{\mathcal{K}_{2},x}$.

###  6.2. Cup product of the Eichler–Shimura classes

In this subsection, we return to the automorphic setting and consider the cup
product of the Eichler–Shimura classes $\delta^{(3)}(\boldsymbol{f})$ and
$\delta^{(2)}(h^{\pm})$ constructed in Section 5.3. For
$\boldsymbol{\lambda}^{(2)}=(\lambda_{2},0)\in\Lambda^{\mathrm{coh}}_{2}$ and
$\boldsymbol{\lambda}^{(3)}=(\lambda_{3},\delta)\in\Lambda^{\mathrm{coh}}_{3}$
satisfying the inequality $\lambda_{2}<\lambda_{3}$, we consider a
cohomological cuspidal automorphic representations $\pi^{(2)}$ and $\pi^{(3)}$
such that

$\displaystyle\pi^{(2)}_{\infty}\cong
D_{\nu_{2},l_{2}},\quad\pi^{(3)}_{\infty}\cong{\rm Ind}^{\rm
GL_{3}({\mathbf{R}})}_{{\rm
P}_{2,1}({\mathbf{R}})}(D_{\nu_{3},l_{3}}\boxtimes\chi_{\nu,\delta_{3}}),$
$\displaystyle\lambda_{2}=l_{2}+1,\quad\nu_{2}=-\frac{l_{2}}{2}+\frac{1}{2},\quad\quad\lambda_{3}=l_{3}+1,\quad\nu_{3}=\nu=-\frac{l_{3}}{2}+1.$

We specialize the construction of Section 6.1 to

$\displaystyle\boldsymbol{n}_{\boldsymbol{\lambda}}$
$\displaystyle=(n_{\boldsymbol{\lambda}},0)=(\lambda_{2}-2,0)=(l_{2}-1,0),$
$\displaystyle\boldsymbol{w}_{\boldsymbol{\lambda}}$
$\displaystyle=(w_{\boldsymbol{\lambda}},w_{\boldsymbol{\lambda}},w_{\boldsymbol{\lambda}})=\left(\dfrac{\lambda_{3}-3}{2},\dfrac{\lambda_{3}-3}{2},\dfrac{\lambda_{3}-3}{2}\right)=\left(\dfrac{l_{3}}{2}-1,\dfrac{l_{3}}{2}-1,\dfrac{l_{3}}{2}-1\right)$

as in (5.2). We can easily calculate $\Xi_{2}(\boldsymbol{w}_{\lambda})$ as

$\displaystyle\Xi_{2}(\boldsymbol{w}_{\boldsymbol{\lambda}})$
$\displaystyle=\left\\{(n_{1},n_{2})\in{\mathbf{Z}}\mid
w_{\boldsymbol{\lambda}}\leq n_{1}+n_{2}\leq 2w_{\boldsymbol{\lambda}},\,0\leq
n_{2}\leq w_{\boldsymbol{\lambda}}\right\\}$
$\displaystyle=\left\\{(n_{1},n_{2})\in{\mathbf{Z}}\ \middle|\ 0\leq
n_{1}\leq\dfrac{l_{3}}{2}-1,\,-n_{1}+\frac{l_{3}}{2}-1\leq
n_{2}\leq\frac{l_{3}}{2}-1\right\\}$
$\displaystyle\qquad\qquad\cup\left\\{(n_{1},n_{2})\in{\mathbf{Z}}\ \middle|\
\frac{l_{3}}{2}-1<n_{1}\leq l_{3}-2,\,0\leq n_{2}\leq-n_{1}+l_{3}-2\right\\}.$

In order to consider the cup product of the Eichler–Shimura classes, we should
focus on the branching rule map $\nabla^{\boldsymbol{n}}$ for
$\boldsymbol{n}=(n_{1},n_{2})\in\Xi_{2}(\boldsymbol{w}_{\lambda})$ with
$n_{1}=n_{\boldsymbol{\lambda}}=l_{2}-1$. The condition for
$(n_{\boldsymbol{\lambda}},n_{2})$ to be contained in
$\Xi_{2}(\boldsymbol{w}_{\boldsymbol{\lambda}})$ is described as follows:

(6.4) $\begin{cases}\dfrac{l_{3}}{2}-l_{2}\leq
n_{2}\leq\dfrac{l_{3}}{2}-1&\text{for }1\leq l_{2}\leq\dfrac{l_{3}}{2},\\\
0\leq n_{2}\leq l_{3}-l_{2}-1&\text{for }\dfrac{l_{3}}{2}<l_{2}\leq
l_{3}-1\end{cases}\Longleftrightarrow\begin{cases}\dfrac{l_{3}}{2}+1\leq
n_{2}+l_{2}+1\leq\dfrac{l_{3}}{2}+l_{2}&\text{for }0\leq
l_{2}\leq\dfrac{l_{3}}{2},\\\ l_{2}+1\leq n_{2}+l_{2}+1\leq l_{3}&\text{for
}\dfrac{l_{3}}{2}<l_{2}\leq l_{3}-1.\end{cases}$

If we set $m=n_{2}+l_{2}+1$, the region of $m$ in (6.4) completely coincides
with that in Proposition 4.4. We define $\mathrm{Crit}(\boldsymbol{\lambda})$
or $\mathrm{Crit}(\pi)$ as the set of integers $m$ satisfying the same
inequalities as those which $n_{2}+l_{2}+1$ satisfies in (6.4), and for each
$m\in\mathrm{Crit}(\boldsymbol{\lambda})$, set

(6.5)
$\displaystyle\boldsymbol{n}_{m}=(n_{m,1},n_{m,2})=(\lambda_{2}-2,m-\lambda_{2})=(l_{2}-1,m-l_{2}-1).$

Now let us consider the cup product of the Eichler–Shimura classes. For
$m\in\mathrm{Crit}(\boldsymbol{\lambda})$, let
$\nabla^{\boldsymbol{n}_{m}}\iota^{\ast}{\rm
p}^{\ast}_{3}\delta^{(3)}(\boldsymbol{f})_{x}$ (resp. ${\rm
p}^{\ast}_{2}\delta^{(2)}(h^{\pm})_{x}$) be the restriction of
$\nabla^{\boldsymbol{n}_{m}}\iota^{*}{\rm
p}^{\ast}_{3}\delta^{(3)}(\boldsymbol{f})$ (resp. ${\rm
p}^{\ast}_{2}\delta^{(2)}(h^{\pm})_{x}$) to
${\mathcal{Y}}^{(2)}_{{\mathcal{K}}_{2},x}$ for $x\in{\rm
Cl}^{+}_{\mathbf{Q}}({\mathcal{K}}_{2})$. Then the zeta integral for the
Rankin–Selberg convolution ${\rm GL}_{3}\times{\rm GL}_{2}$ is described as a
summation of the cup products in (6.3) of elements
$\nabla^{\boldsymbol{n}_{m}}\iota^{\ast}{\rm
p}^{\ast}_{3}\delta^{(3)}({\boldsymbol{f}})_{x}\in
H^{2}({\mathcal{Y}}^{(2)}_{\mathcal{K}_{2},x},\nabla^{\boldsymbol{n}_{m}}\iota^{\ast}{\rm
p}^{\ast}_{3}{\mathcal{L}}^{(3)}(\boldsymbol{w}_{\boldsymbol{\lambda}};A))$
and ${\rm p}^{\ast}_{2}\delta^{(2)}(h^{\pm})_{x}\in H^{1}_{\rm
c}({\mathcal{Y}}^{(2)}_{\mathcal{K}_{2},x},{\rm
p}^{\ast}_{2}{\mathcal{L}}^{(2)}(\boldsymbol{n}_{\boldsymbol{\lambda}};A))$.
To be more precise, the evaluation of the summation

(6.6) $\displaystyle\sum_{x\in{\rm
Cl}^{+}_{\mathbf{Q}}({\mathcal{K}}_{2})}\varphi(x){\rm
Tw}_{[\boldsymbol{n}_{m}]}\left(\nabla^{\boldsymbol{n}_{m}}\iota^{\ast}{\rm
p}^{\ast}_{3}\delta^{(3)}(\boldsymbol{f})_{x}\cup{\rm
p}^{\ast}_{2}\delta^{(2)}(h^{\pm})_{x}\right)$

coincides with the global zeta integral, where
$\varphi\colon{\mathbf{Q}}^{\times}\backslash{\mathbf{Q}}^{\times}_{\mathbf{A}}\to{\mathbf{C}}^{\times}$
is a Hecke character of finite order factoring through
$\mathrm{Cl}^{+}_{\mathbf{Q}}(\mathcal{K}_{2})$. To evaluate the cup product
in an explicit way, it is necessary to write down
$\nabla^{\boldsymbol{n}_{m}}\iota^{\ast}{\rm
p}^{\ast}_{3}\delta^{(3)}(\boldsymbol{f})$ and ${\rm
p}^{\ast}_{2}\delta^{(2)}(h^{\pm})$ as differential forms (with local
coefficients) on ${\mathcal{Y}}^{(2)}_{\mathcal{K}_{2}}$. We shall do it in
the next subsection.

For later use, we here summarize as the following Lemma the computation of the
image of the polynomial ${\mathcal{P}}_{\alpha,\beta}(X,Y,Z,A,B,C)\in
L^{(3)}(w_{\boldsymbol{\lambda}};\mathbf{C})$ introduced in Section 5.3 under
the map $\nabla^{\boldsymbol{n}}$. We postpone its proof to Appendix B because
it is based on a little complicated but straightforward calculations. For
$e,i,j,k\in{\mathbf{Z}}$ with $e\geq 0$, define an integer
${}_{e}H_{i,j,k}\in{\mathbf{Z}}$ by

(6.7) $\displaystyle(X+Y+Z)^{e}=\sum_{i,j,k}{}_{e}H_{i,j,k}X^{i}Y^{j}Z^{k}.$

######  Lemma 6.1.

Set $S=-X+\sqrt{-1}Y$ and $T=X+\sqrt{-1}Y$. Then, for every
$\boldsymbol{n}=(n_{1},n_{2})\in\Xi_{2}(\boldsymbol{w}_{\boldsymbol{\lambda}})$,
we have

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

In particular, if $\alpha-\beta=\pm n_{1}$ holds,
$\nabla^{\boldsymbol{n}}{\mathcal{P}}_{\alpha,\beta}(X,Y)$ is described as

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

###### Proof..

See Lemmas B.2 and B.3 in Appendix B. ∎

###  6.3. Birch lemma

To evaluate the cup product (6.6), it is necessary to write it down as an
integral on ${\mathcal{Y}}^{(2)}_{\mathcal{K}_{2}}$. We can do it by using the
explicit form of the Eichler–Shimura map introduced in Section 5.3. Some of
explicit formulas proposed here require longsome but straightforward
computations for their proofs, and hence we omit such calculations in this
subsection and summarize them in Appendix A.

First let us recall the explicit formula for ${\rm
p}^{\ast}_{2}\delta^{(2)}(h^{\pm})$, which is well known in terms of elliptic
modular forms.

######  Lemma 6.2.

For each $g\in{\mathcal{Y}}^{(2)}_{\mathcal{K}_{2}}$, we have

$\displaystyle{\rm
p}^{\ast}_{2}\delta^{(2)}(h^{\pm})_{g}=h^{\pm}(g)\varrho^{(2)}_{\boldsymbol{n}_{\boldsymbol{\lambda}}}(g_{\infty})(\mp
X+\sqrt{-1}Y)^{n_{\boldsymbol{\lambda}}}(\pm\frac{1}{2}{\rm
d}E-\frac{\sqrt{-1}}{2}{\rm d}H)\circ L^{-1}_{g_{\infty}},$

where $L_{g_{\infty}}$ denotes the left translation
$\left(T_{g_{\mathrm{fin}}}{\mathcal{Y}}^{(2)}_{\mathcal{K}_{2}}\right)_{\mathbf{C}}\cong\left(\mathfrak{gl}_{2}(\mathbf{R})/\mathfrak{so}_{2}(\mathbf{R})\right)_{\mathbf{C}}\to\left(T_{g}{\mathcal{Y}}^{(2)}_{\mathcal{K}_{2}}\right)_{\mathbf{C}}$
by $g_{\infty}$.

Due to the Iwasawa decomposition of $\mathrm{GL}_{2}(\mathbf{R})$, we define
the coordinates $(x_{2},y_{1},y_{2})$ of the archimedean component of
${\mathcal{Y}}^{(2)}_{\mathcal{K}_{2}}$ as

$\displaystyle{\rm
GL}_{2}({\mathbf{Q}})y_{1}\begin{pmatrix}y_{2}^{\frac{1}{2}}&y_{2}^{-\frac{1}{2}}x_{2}\\\
0&y_{2}^{-\frac{1}{2}}\end{pmatrix}g_{\rm fin}{\rm
SO}_{2}({\mathbf{R}}){\mathcal{K}}_{2},\quad g_{{\mathrm{fin}}}\in{\rm
GL}_{2}({\mathbf{Q}}_{\mathbf{A},\mathrm{fin}}).$

With respect to this coordinate, the differential form
${\boldsymbol{\xi}}_{\pm}:=(\pm\frac{1}{2}{\rm d}E-\frac{\sqrt{-1}}{2}{\rm
d}H)\circ L^{-1}_{g}$ is described as

$\displaystyle{\boldsymbol{\xi}}_{\pm}=e^{2\sqrt{-1}\theta}\frac{\pm{\rm
d}x_{2}+\sqrt{-1}{\rm d}y_{2}}{y_{2}}$

for each
$g=y_{1}\begin{pmatrix}y_{2}^{\frac{1}{2}}&y_{2}^{-\frac{1}{2}}x_{2}\\\
0&y_{2}^{-\frac{1}{2}}\end{pmatrix}\begin{pmatrix}\cos\theta&\sin\theta\\\
-\sin\theta&\cos\theta\end{pmatrix}$.

We next describe an explicit formula for
$\nabla^{\boldsymbol{n}_{m}}\iota^{*}{\rm
p}^{\ast}_{3}\delta^{(3)}(\boldsymbol{f})$. Using the explicit Iwasawa
decomposition of $\mathrm{GL}_{3}(\mathbf{R})$ proposed in Lemma A.3, we
define the coordinates $(t,x_{1},x_{2},x_{3},y_{1},y_{2})$ of the archimedean
component of ${\mathcal{Y}}^{(3)}_{\mathcal{K}_{3}}$ as

$\displaystyle{\rm
GL}_{3}({\mathbf{Q}})t\begin{pmatrix}y_{1}y_{2}&y_{1}x_{2}&x_{3}\\\
0&y_{1}&x_{1}\\\ 0&0&1\end{pmatrix}g_{\rm fin}{\rm
SO}_{3}({\mathbf{R}}){\mathcal{K}}_{3},\quad g_{{\mathrm{fin}}}\in{\rm
GL}_{3}({\mathbf{Q}}_{\mathbf{A},\mathrm{fin}}).$

Proposition A.6 describes how the basis
$\\{{\boldsymbol{\omega}}_{3},\ldots,{\boldsymbol{\omega}}_{-3}\\}$ of
$\bigwedge^{2}{\mathcal{P}}^{\ast}_{3,{\mathbf{C}}}$ introduced in (5.5) is
presented with respect to this coordinates; namely

(6.10)
$\displaystyle\begin{pmatrix}{\boldsymbol{\omega}}_{3}&\boldsymbol{\omega}_{2}&\dotsc&{\boldsymbol{\omega}}_{-3}\end{pmatrix}\circ
L^{-1}_{g_{\infty}}=\begin{pmatrix}{\rm d}y_{1}\wedge{\rm d}y_{2}&{\rm
d}y_{1}\wedge{\rm d}x_{1}&\dotsc&{\rm d}x_{2}\wedge{\rm
d}x_{3}\end{pmatrix}Q^{2}$

holds where $Q^{2}$ is the matrix introduced in Proposition A.6 (in the right-
hand side, the $2$-forms are ordered lexicographically with respect to the
order ${\rm d}y_{1}\prec{\rm d}y_{2}\prec{\rm d}x_{1}\prec{\rm
d}x_{2}\prec{\rm d}x_{3}$). Since $\iota$ induces
$(x_{2},y_{1},y_{2})\mapsto(0,0,x_{2},0,y_{1}y_{2}^{-\frac{1}{2}},y_{2})$, the
pullback of the differential forms on $\mathcal{Y}^{(3)}_{\mathcal{K}_{3}}$
along $\iota$ is described as

$\displaystyle\iota^{*}\begin{pmatrix}{\rm d}t&{\rm d}x_{1}&{\rm d}x_{2}&{\rm
d}x_{3}&{\rm d}y_{1}&{\rm d}y_{2}\end{pmatrix}=\begin{pmatrix}0&0&{\rm
d}x_{2}&0&y_{2}^{-\frac{1}{2}}{\rm
d}y_{1}-\dfrac{1}{2}y_{1}y_{2}^{-\frac{3}{2}}{\rm d}y_{2}&{\rm
d}y_{2}\end{pmatrix}.$

Taking this into accounts, we immediately obtain

$\displaystyle\iota^{\ast}{\rm p}^{\ast}_{3}{\boldsymbol{\omega}}_{\pm 3}$
$\displaystyle=\iota^{\ast}{\rm p}^{\ast}_{3}{\boldsymbol{\omega}}_{\pm
1}=0,\quad\iota^{\ast}{\rm
p}^{\ast}_{3}{\boldsymbol{\omega}}_{0}=\frac{\sqrt{-1}}{2y^{2}_{2}}{\rm
d}y_{2}\wedge{\rm d}x_{2},$ $\displaystyle\iota^{\ast}{\rm
p}^{\ast}_{3}{\boldsymbol{\omega}}_{\pm 2}$
$\displaystyle=\pm\frac{1}{2y_{1}y_{2}}{\rm d}y_{1}\wedge{\rm
d}y_{2}-\frac{\sqrt{-1}}{2y_{1}y_{2}}{\rm d}y_{1}\wedge{\rm
d}x_{2}-\frac{\sqrt{-1}}{4y^{2}_{2}}{\rm d}y_{2}\wedge{\rm d}x_{2}$

by computing the product of matrices in (6.10) explicitly. Hence the
differential forms $\iota^{\ast}{\rm
p}^{\ast}_{3}{\boldsymbol{\omega}}_{i}\wedge{\boldsymbol{\xi}}_{\mp}$ for
$i=3,\ldots,-3$ are computed as follows:

(6.11) $\displaystyle\begin{aligned} \iota^{\ast}{\rm
p}^{\ast}_{3}{\boldsymbol{\omega}}_{i}\wedge{\boldsymbol{\xi}}_{\mp}&=0,\quad\text{for
}i\neq\pm 2,\\\ \iota^{\ast}{\rm p}^{\ast}_{3}{\boldsymbol{\omega}}_{\pm
2}\wedge{\boldsymbol{\xi}}_{\mp}&=\left(\pm\frac{1}{2y_{1}y_{2}}{\rm
d}y_{1}\wedge{\rm d}y_{2}-\frac{\sqrt{-1}}{2y_{1}y_{2}}{\rm d}y_{1}\wedge{\rm
d}x_{2}-\frac{\sqrt{-1}}{4y^{2}_{2}}{\rm d}y_{2}\wedge{\rm
d}x_{2}\right)\wedge\frac{\mp{\rm d}x_{2}+\sqrt{-1}{\rm d}y_{2}}{y_{2}}\\\
&=\frac{1+1}{2}\times\frac{{\rm d}y_{1}\wedge{\rm d}x_{2}\wedge{\rm
d}y_{2}}{y_{1}y^{2}_{2}}=\frac{{\rm d}y_{1}\wedge{\rm d}x_{2}\wedge{\rm
d}y_{2}}{y_{1}y^{2}_{2}}\quad(\text{double sign in the same
order}).\end{aligned}$

Now define ${\rm d}g={\rm d}g_{\infty}{\rm d}g_{\mathrm{fin}}$ to be a Haar
measure on ${\rm GL}_{2}({\mathbf{Q}}_{\mathbf{A}})$ which satisfies

(6.12) $\displaystyle{\rm d}g_{\infty}=\frac{{\rm d}y_{1}\wedge{\rm
d}x_{2}\wedge{\rm d}y_{2}}{y_{1}y^{2}_{2}}{\rm d}u\qquad\text{for
}g_{\infty}=y_{1}\begin{pmatrix}y^{\frac{1}{2}}_{2}&y^{-\frac{1}{2}}_{2}x_{2}\\\
0&y^{-\frac{1}{2}}_{2}\end{pmatrix}u,\;u\in{\rm O}_{2}({\mathbf{R}}),\;{\rm
vol}({\rm d}u,{\rm O}_{2}({\mathbf{R}}))=1$

and ${\rm vol}({\rm d}g_{\rm fin},{\rm GL}_{2}(\widehat{\mathbf{Z}}))=1$. We
summarize the arguments so far in the following proposition:

######  Proposition 6.3.

For
$\boldsymbol{n}_{m}=(n_{m,1},n_{m,2})=(\lambda_{2}-2,m-\lambda_{2})=(l_{2}-1,m-l_{2}-1)$,
set

$\displaystyle\epsilon$
$\displaystyle=\bigotimes_{v\in\Sigma_{\mathbf{Q}}\setminus\\{\infty\\}}1_{2}\otimes\begin{pmatrix}-1&0\\\
0&1\end{pmatrix}\in{\rm GL}_{2}({\mathbf{Q}_{A}}),$ $\displaystyle\quad
C(\boldsymbol{w}_{\boldsymbol{\lambda}},\boldsymbol{n}_{m})$
$\displaystyle=\binom{w_{\boldsymbol{\lambda}}}{n_{m,1}+n_{m,2}-w_{\boldsymbol{\lambda}}}\binom{w_{\boldsymbol{\lambda}}}{w_{\boldsymbol{\lambda}}-n_{m,2}}=\binom{\frac{l_{3}}{2}-1}{m-\frac{l_{3}}{2}-1}\binom{\frac{l_{3}}{2}-1}{\frac{l_{3}}{2}+l_{2}-m}.$

Then we have

$\displaystyle{\rm Tw}_{[\boldsymbol{n}_{m}]}$
$\displaystyle\left(\nabla^{\boldsymbol{n}_{m}}\iota^{*}{\rm
p}^{\ast}_{3}\delta^{(3)}(\boldsymbol{f})_{x}\cup{\rm
p}^{\ast}_{2}\delta^{(2)}(h^{+})_{x}\right)$
$\displaystyle=\sqrt{-1}^{\frac{l_{3}}{2}+2l_{2}-m-1}C(\boldsymbol{w}_{\boldsymbol{\lambda}},\boldsymbol{n}_{m})\cdot
2[{\rm
GL}_{2}(\widehat{\mathbf{Z}}):{\mathcal{K}}_{2}]\int_{{\mathcal{Y}}^{(2)}_{\mathcal{K}_{2}},x}f^{\boldsymbol{\lambda}}_{-\lambda_{2}}\left(\iota(g)\right)h^{+}(g)\lvert\det
g\rvert_{\mathbf{A}}^{m-2}{\rm d}g.$

We also find that

$\displaystyle{\rm
Tw}_{[\boldsymbol{n}_{m}]}\left(\nabla^{\boldsymbol{n}_{m}}\iota^{*}{\rm
p}^{\ast}_{3}\delta^{(3)}(\boldsymbol{f})_{x\det\epsilon}\cup{\rm
p}^{\ast}_{2}\delta^{(2)}(h^{-})_{x\det\epsilon}\right)=(-1)^{m+\delta+1+w_{\boldsymbol{\lambda}}}{\rm
Tw}_{[\boldsymbol{n}_{m}]}\left(\nabla^{\boldsymbol{n}_{m}}\iota^{*}{\rm
p}^{\ast}_{3}\delta^{(3)}(\boldsymbol{f})_{x}\cup{\rm
p}^{\ast}_{2}\delta^{(2)}(h^{+})_{x}\right).$

###### Proof..

The identities (6.11) and the normalization (6.12) of the measure ${\rm d}g$
imply that

$\displaystyle{\rm
Tw}_{[\boldsymbol{n}_{m}]}\left(\nabla^{\boldsymbol{n}_{m}}\iota^{*}{\rm
p}^{\ast}_{3}\delta^{(3)}(\boldsymbol{f})_{x}\cup{\rm
p}^{\ast}_{2}\delta^{(2)}(h^{+})_{x}\right)$ $\displaystyle\begin{aligned}
&=\sum_{\alpha}2[{\rm
GL}_{2}(\widehat{\mathbf{Z}}):{\mathcal{K}}_{2}]\int_{{\mathcal{Y}}^{(2)}_{\mathcal{K}_{2}},x}f^{\boldsymbol{\lambda}}_{\alpha}(\iota(g))h^{+}(g)\left[\varrho^{(2)}_{\boldsymbol{n}_{m}}(g_{\infty})\nabla^{\boldsymbol{n}_{m}}{\mathcal{P}}_{\alpha,-2}(X,Y),\varrho^{(2)}_{\boldsymbol{n}_{\boldsymbol{\lambda}}}(g_{\infty})(-X+\sqrt{-1}Y)^{n_{\boldsymbol{\lambda}}}\right]_{\boldsymbol{n}_{m}}\frac{|\det
g|^{[\boldsymbol{n}_{m}]}_{\mathbf{A}}}{(\det
g_{\infty})^{[\boldsymbol{n}_{m}]}}{\rm d}g\\\ &=2[{\rm
GL}_{2}(\widehat{\mathbf{Z}}):{\mathcal{K}}_{2}]\sum_{\alpha}[\nabla^{\boldsymbol{n}_{m}}{\mathcal{P}}_{\alpha,-2}(X,Y),(-X+\sqrt{-1}Y)^{n_{\boldsymbol{\lambda}}}]_{\boldsymbol{n}_{m}}\int_{{\mathcal{Y}}^{(2)}_{\mathcal{K}_{2}},x}f^{\boldsymbol{\lambda}}_{\alpha}(\iota(g))h^{+}(g)\lvert\det
g\rvert_{\mathbf{A}}^{[\boldsymbol{n}_{m}]}{\rm d}g.\end{aligned}$

Set $S=-X+\sqrt{-1}Y$, $T=X+\sqrt{-1}Y$ and $g_{0}=\begin{pmatrix}-1&1\\\
\sqrt{-1}&\sqrt{-1}\end{pmatrix}$. Then, due to the equivariance of the
pairing $[\cdot,\cdot]_{\boldsymbol{n}_{m}}$, we find that

$\displaystyle[T^{n_{\boldsymbol{\lambda}}},S^{n_{\boldsymbol{\lambda}}}]_{\boldsymbol{n}_{m}}=[\det(g_{0})^{-n_{m,2}}\varrho^{(2)}_{\boldsymbol{n}_{m}}(g_{0})Y^{n_{\boldsymbol{\lambda}}},\varrho^{(2)}_{\boldsymbol{n}_{\boldsymbol{\lambda}}}(g_{0})X^{n_{\boldsymbol{\lambda}}}]_{\boldsymbol{n}_{m}}=(\det
g_{0})^{[\boldsymbol{n}_{m}]-n_{m,2}}[Y^{n_{\boldsymbol{\lambda}}},X^{n_{\boldsymbol{\lambda}}}]_{\boldsymbol{n}_{m}}=(-2\sqrt{-1})^{n_{m,1}}.$

Hence Lemma 6.1 yields that

Here $\delta_{\ast,\ast}$ denotes Kronecker’s delta. The formulas
$[\boldsymbol{n}_{m}]=m-2$,
$n_{m,1}-n_{m,2}+w_{\boldsymbol{\lambda}}=\dfrac{l_{3}}{2}+2l_{2}-m-1$ and the
first identity of (6.3) show the first statement.

We prove the second statement. A similar argument shows that

$\displaystyle{\rm Tw}_{[\boldsymbol{n}_{m}]}$
$\displaystyle\left(\nabla^{\boldsymbol{n}_{m}}\iota^{\ast}{\rm
p}^{\ast}_{3}\delta^{(3)}(\boldsymbol{f})_{x\det\epsilon}\cup{\rm
p}^{\ast}_{2}\delta^{(2)}(h^{-})_{x\det\epsilon}\right)$ $\displaystyle=2[{\rm
GL}_{2}(\widehat{\mathbf{Z}}):{\mathcal{K}}_{2}]\sum_{\alpha}\left[\nabla^{\boldsymbol{n}_{m}}{\mathcal{P}}_{\alpha,2}(X,Y),(X+\sqrt{-1}Y)^{n_{\boldsymbol{\lambda}}}\right]_{\boldsymbol{n}_{m}}\int_{{\mathcal{Y}}^{(2)}_{\mathcal{K}_{2}},x\det\epsilon}f^{\boldsymbol{\lambda}}_{\alpha}(\iota(g))h^{-}(g)\lvert\det
g\rvert_{\mathbf{A}}^{[\boldsymbol{n}_{m}]}{\rm d}g$ $\displaystyle=2[{\rm
GL}_{2}(\widehat{\mathbf{Z}}):{\mathcal{K}}_{2}]\sum_{\alpha}[\nabla^{\boldsymbol{n}_{m}}{\mathcal{P}}_{\alpha,2}(X,Y),(X+\sqrt{-1}Y)^{n_{\boldsymbol{\lambda}}}]_{\boldsymbol{n}_{m}}\int_{{\mathcal{Y}}^{(2)}_{\mathcal{K}_{2}},x}f^{\boldsymbol{\lambda}}_{\alpha}(\iota(g\epsilon))h^{-}(g\epsilon)\lvert\det
g\epsilon\rvert_{\mathbf{A}}^{[\boldsymbol{n}_{m}]}{\rm d}(g\epsilon)$
$\displaystyle=2[{\rm
GL}_{2}(\widehat{\mathbf{Z}}):{\mathcal{K}}_{2}]\sum_{\alpha}[\nabla^{\boldsymbol{n}_{m}}{\mathcal{P}}_{\alpha,2}(X,Y),(X+\sqrt{-1}Y)^{n_{\boldsymbol{\lambda}}}]_{\boldsymbol{n}_{m}}\int_{{\mathcal{Y}}^{(2)}_{\mathcal{K}_{2}},x}(-1)^{\delta}f^{\boldsymbol{\lambda}}_{-\alpha}(\iota(g))h^{+}(g)\lvert\det
g\rvert_{\mathbf{A}}^{[\boldsymbol{n}_{m}]}\cdot(-1){\rm d}g.$

This and the second identity of (6.3) show that

$\displaystyle{\rm Tw}_{[\boldsymbol{n}_{m}]}$
$\displaystyle\left(\nabla^{\boldsymbol{n}_{m}}\iota^{*}{\rm
p}^{\ast}_{3}\delta^{(3)}(\boldsymbol{f})_{x\det\epsilon}\cup{\rm
p}^{\ast}_{2}\delta^{(2)}(h^{-})_{x\det\epsilon}\right)$
$\displaystyle=\sqrt{-1}^{-n_{m,1}+n_{m,2}-w_{\boldsymbol{\lambda}}}C(\boldsymbol{w}_{\boldsymbol{\lambda}},\boldsymbol{n}_{m})\cdot(-1)^{\delta+1}2[{\rm
GL}_{2}(\widehat{\mathbf{Z}}):{\mathcal{K}}_{2}]\int_{{\mathcal{Y}}^{(2)}_{\mathcal{K}_{2}},x}f^{\boldsymbol{\lambda}}_{-\lambda_{2}}(\iota(g))h^{+}(g)\lvert\det
g|_{\mathbf{A}}^{[\boldsymbol{n}_{m}]}{\rm d}g$
$\displaystyle=(-1)^{[\boldsymbol{n}_{m}]+w_{\boldsymbol{\lambda}}+\delta+1}\nabla^{n}\iota^{*}{\rm
p}^{\ast}_{3}\delta^{(3)}(\boldsymbol{f})_{x}\cup{\rm
p}^{\ast}_{2}\delta^{(2)}(h^{+})_{x}.$

This proves the statement. ∎

######  Corollary 6.4.

(Birch Lemma) Let
$\varphi:{\mathbf{Q}}^{\times}\backslash{\mathbf{Q}}^{\times}_{\mathbf{A}}\to{\mathbf{C}}^{\times}$
be a Hecke character of finite order. Suppose that $\varphi$ is trivial on
$\det\mathcal{K}_{2}$. Then we have

$\displaystyle I(m,\boldsymbol{f},h^{\pm},\varphi)$
$\displaystyle:=\sum_{x\in{\rm
Cl}^{+}_{\mathbf{Q}}({\mathcal{K}}_{2})}\varphi(x){\rm
Tw}_{[\boldsymbol{n}_{m}]}\left(\nabla^{\boldsymbol{n}_{m}}\iota^{*}{\rm
p}^{\ast}_{3}\delta^{(3)}(\boldsymbol{f})_{x}\cup\left({\rm
p}^{\ast}_{2}\delta^{(2)}(h^{+})_{x}\pm{\rm
p}^{\ast}_{2}\delta^{(2)}(h^{-})_{x}\right)\right)$
$\displaystyle\begin{aligned}
\;\,=\sqrt{-1}^{\frac{l_{3}}{2}+2l_{2}-m-1}&C(\boldsymbol{w}_{\boldsymbol{\lambda}},\boldsymbol{n}_{m})\cdot
2[{\rm GL}_{2}(\widehat{\mathbf{Z}}):{\mathcal{K}}_{2}]\\\
&\times\left(1\pm(-1)^{m+\delta+1+w_{\boldsymbol{\lambda}}}\varphi_{\infty}(-1)\right)\int_{{\mathcal{Y}}^{(2)}_{\mathcal{K}_{2}}}f^{\boldsymbol{\lambda}}_{-\lambda_{2}}\left(\iota(g)\right)h^{+}(g)\varphi(\det
g)|\det g|^{m-2}_{\mathbf{A}}{\rm d}g.\end{aligned}$

###### Proof..

Proposition 6.3 shows that

$\displaystyle I(m,\boldsymbol{f},h^{\pm},\varphi)$
$\displaystyle=\sum_{x\in{\rm
Cl}^{+}_{\mathbf{Q}}({\mathcal{K}}_{2})}\varphi(x){\rm
Tw}_{[\boldsymbol{n}_{m}]}\left(\nabla^{\boldsymbol{n}_{m}}\iota^{*}{\rm
p}^{\ast}_{3}\delta^{(3)}(\boldsymbol{f})_{x}\cup{\rm
p}^{\ast}_{2}\delta^{(2)}(h^{+})_{x}\right)$
$\displaystyle\quad\quad\quad\quad\quad\pm\sum_{x\in{\rm
Cl}^{+}_{\mathbf{Q}}({\mathcal{K}}_{2})}\varphi(x\det\epsilon){\rm
Tw}_{[\boldsymbol{n}_{m}]}\left(\nabla^{\boldsymbol{n}_{m}}\iota^{*}{\rm
p}^{\ast}_{3}\delta^{(3)}(\boldsymbol{f})_{x\det\epsilon}\cup{\rm
p}^{\ast}_{2}\delta^{(2)}(h^{-})_{x\det\epsilon}\right)$
$\displaystyle=\sum_{x\in{\rm
Cl}^{+}_{\mathbf{Q}}({\mathcal{K}}_{2})}\varphi(x){\rm
Tw}_{[\boldsymbol{n}_{m}]}\left(\nabla^{\boldsymbol{n}_{m}}\iota^{*}{\rm
p}^{\ast}_{3}\delta^{(3)}(\boldsymbol{f})_{x}\cup{\rm
p}^{\ast}_{2}\delta^{(2)}(h^{+})_{x}\right)$
$\displaystyle\quad\quad\quad\quad\quad\pm(-1)^{m+\delta+1+w_{\boldsymbol{\lambda}}}\sum_{x\in{\rm
Cl}^{+}_{\mathbf{Q}}({\mathcal{K}}_{2})}\varphi_{\infty}(-1)\varphi(x){\rm
Tw}_{[\boldsymbol{n}_{m}]}\left(\nabla^{\boldsymbol{n}_{m}}\iota^{*}{\rm
p}^{\ast}_{3}\delta^{(3)}(\boldsymbol{f})_{x}\cup{\rm
p}^{\ast}_{2}\delta^{(2)}(h^{+})_{x}\right)$
$\displaystyle=\left(1\pm(-1)^{m+\delta+1+w_{\boldsymbol{\lambda}}}\varphi_{\infty}(-1)\right)\sum_{x\in{\rm
Cl}^{+}_{\mathbf{Q}}({\mathcal{K}}_{2})}\varphi(x){\rm
Tw}_{[\boldsymbol{n}_{m}]}\left(\nabla^{\boldsymbol{n}_{m}}\iota^{*}{\rm
p}^{\ast}_{3}\delta^{(3)}(\boldsymbol{f})_{x}\cup{\rm
p}^{\ast}_{2}\delta^{(2)}(h^{+})_{x}\right).$

This proves the statement. ∎

Now define the zeta integral ${\mathcal{Z}}(s;\boldsymbol{f},h^{+},\varphi)$
to be

$\displaystyle{\mathcal{Z}}(s;\boldsymbol{f},h^{+},\varphi)=\int_{{\mathcal{Y}}^{(2)}_{\mathcal{K}_{2}}}f^{\boldsymbol{\lambda}}_{-\lambda_{2}}\left(\iota(g)\right)h^{+}(g)\varphi(\det
g)|\det g|^{s-\frac{1}{2}}_{\mathbf{A}}{\rm d}g.$

In the next section, we will calculate explicit evaluations of the zeta
integrals ${\mathcal{Z}}(m-\frac{3}{2};\boldsymbol{f},h^{+},\mathbf{1})$ for
each $m\in\mathrm{Crit}(\boldsymbol{\lambda})$, and relate them with the
critical values $L(m,{\mathcal{M}}(\pi^{(3)}\times\pi^{(2)}))$ for certain
vectors $\boldsymbol{f}$ and $h^{\pm}$. We also clarify the relation between
these critical values and the periods of
${\mathcal{M}}(\pi^{(3)}\times\pi^{(2)})$. The choice of data, namely the
vectors $\boldsymbol{f}$, $h^{\pm}$ and the periods, will be also proposed
there.

######  Remark 6.5.

Corollary 6.4 implies that the condition on signature
$(-1)^{m+\delta+1+w_{\boldsymbol{\lambda}}}\neq\pm\varphi_{\infty}(-1)$ is
necessary for the zeta integral
$\mathcal{Z}(m-\frac{3}{2};\boldsymbol{f},h^{+},\varphi)$ (and thus the
critical value $L(m,\mathcal{M}(\pi^{(3)}\times\pi^{(2)})(\varphi))$) not to
vanish. Such a condition on signature also appears in [Rag16, Theorem 1.1];
see also [Rag16, Sections 2.5.3.5, 2.5.5.2]. We have derived the equality
$(-1)^{m+\delta+1+w_{\boldsymbol{\lambda}}}=\pm\varphi_{\infty}(-1)$ from the
explicit evaluation of cup products in the proof of Corollary 6.4. However,
one can also observe the necessity of the same condition formally by studying
the action of $\epsilon$ on the cup product paring as follows. First note that
the cup product $I(m,\boldsymbol{f},h^{\pm},\varphi)$ coincides with the
following summation:

$\displaystyle\sum_{x\in{\rm
Cl}^{+}_{\mathbf{Q}}({\mathcal{K}}_{2})}\varphi(x\det{\epsilon}){\rm
Tw}_{[\boldsymbol{n}_{m}]}\left(\nabla^{\boldsymbol{n}_{m}}\iota^{*}{\rm
p}^{\ast}_{3}\delta^{(3)}(\boldsymbol{f})_{x\det\epsilon}\cup\left({\rm
p}^{\ast}_{2}\delta^{(2)}(h^{+})_{x\det\epsilon}\pm{\rm
p}^{\ast}_{2}\delta^{(2)}(h^{-})_{x\det\epsilon}\right)\right).$

Since $\epsilon$ normalizes $\mathcal{K}_{2}K_{2}$, we see that $\epsilon$
acts as the right translation on the local systems appearing in Sections 5.2
and 6.1. We denote the action of $\epsilon$ on the local systems by
$\epsilon\,\bullet-$. Then we find that

$\displaystyle{\rm Tw}_{[\boldsymbol{n}_{m}]}$
$\displaystyle\left(\nabla^{\boldsymbol{n}_{m}}\iota^{*}{\rm
p}^{\ast}_{3}\delta^{(3)}(\boldsymbol{f})_{x\det\epsilon}\cup\left({\rm
p}^{\ast}_{2}\delta^{(2)}(h^{+})_{x\det\epsilon}\pm{\rm
p}^{\ast}_{2}\delta^{(2)}(h^{-})_{x\det\epsilon}\right)\right)$
$\displaystyle=\epsilon\bullet\left({\rm
Tw}_{[\boldsymbol{n}_{m}]}\left(\nabla^{\boldsymbol{n}_{m}}\iota^{*}{\rm
p}^{\ast}_{3}\delta^{(3)}(\boldsymbol{f})_{x}\cup\left({\rm
p}^{\ast}_{2}\delta^{(2)}(h^{+})_{x}\pm{\rm
p}^{\ast}_{2}\delta^{(2)}(h^{-})_{x}\right)\right)\right)$ $\displaystyle={\rm
Tw}_{[\boldsymbol{n}_{m}]}\left(\det(\epsilon)^{-[\boldsymbol{n}_{m}]}\epsilon\bullet\left(\nabla^{\boldsymbol{n}_{m}}\iota^{*}{\rm
p}^{\ast}_{3}\delta^{(3)}(\boldsymbol{f})_{x}\right)\cup\epsilon\bullet\left({\rm
p}^{\ast}_{2}\delta^{(2)}(h^{+})_{x}\pm{\rm
p}^{\ast}_{2}\delta^{(2)}(h^{-})_{x}\right)\right).$

Lemma 3.1 shows that

$\displaystyle\epsilon\bullet\left(\nabla^{\boldsymbol{n}_{m}}\iota^{*}{\rm
p}^{\ast}_{3}\delta^{(3)}(\boldsymbol{f})_{x}\right)=\nabla^{\boldsymbol{n}_{m}}\iota^{*}{\rm
p}^{\ast}_{3}\left(\iota(\epsilon)\bullet\delta^{(3)}(\boldsymbol{f})_{x}\right).$

The action of $\iota(\epsilon)$ on
$H^{2}(\mathfrak{gl}_{3}({\mathbf{R}}),K_{3};H_{\pi^{(3)},K_{3}}\otimes
L^{(3)}(\boldsymbol{w}_{\boldsymbol{\lambda}};{\mathbf{C}}))$ coincides with
the action of $-1_{3}\in{\rm GL}_{3}({\mathbf{R}})$, which is given by the
multiplication of
$\omega_{\pi^{(3)},\infty}(-1)\omega_{\boldsymbol{w}_{\boldsymbol{\lambda}}}(-1)$;
recall from Section 3.2 that $\omega_{\boldsymbol{w_{\lambda}}}$ denotes the
central character of
$L^{3}(\boldsymbol{w}_{\boldsymbol{\lambda}};\mathbf{C})$. Now we may readily
check from the definitions that

$\displaystyle\omega_{\pi^{(3)},\infty}(-1)$ $\displaystyle=(-1)^{\delta+1}$
and $\displaystyle\omega_{\boldsymbol{w}_{\boldsymbol{\lambda}}}(-1)$
$\displaystyle=(-1)^{\frac{l_{3}}{2}+1}.$

Meanwhile, the definition of $\delta^{(2)}(h^{+})\pm\delta^{(2)}(h^{-})$
immediately shows that

$\displaystyle\epsilon\bullet\left({\rm
p}^{\ast}_{2}\delta^{(2)}(h^{+})_{x}\pm{\rm
p}^{\ast}_{2}\delta^{(2)}(h^{-})_{x}\right)=(\pm 1)\left({\rm
p}^{\ast}_{2}\delta^{(2)}(h^{+})_{x}\pm{\rm
p}^{\ast}_{2}\delta^{(2)}(h^{-})_{x}\right).$

Summarizing the above argument, we deduce the identity

$\displaystyle
I(m,\boldsymbol{f},h^{\pm},\varphi)=\varphi_{\infty}(-1)(-1)^{[\boldsymbol{n}_{m}]}\omega_{\pi^{(3)},\infty}(-1)\omega_{\boldsymbol{w}_{\boldsymbol{\lambda}}}(-1)(\pm
1)I(m,\boldsymbol{f},h^{\pm},\varphi).$

Therefore it is necessary to assume the condition
$\pm\varphi_{\infty}(-1)=(-1)^{[\boldsymbol{n}_{m}]+\delta+\frac{l_{3}}{2}}=(-1)^{m+\delta+w_{\boldsymbol{\lambda}}+1}$
for the non-vanishing of the cup product
$I(m,\boldsymbol{f},h^{\pm},\varphi)$.

##  7\. Evaluation of period integrals

In this section, we prove the main theorem (Theorem 7.2) in this article. In
Section 7.1, we recall Raghuram and Shahidi’s definition of Whittaker periods
for $\pi^{(2)}$ and $\pi^{(3)}$ (introduced in [RS08]), and then refine them.
In Section 7.2, we propose a proof of Theorem 7.2. We also discuss the
algebraicity of critical values $L(m,{\mathcal{M}}(\pi^{(3)}\times\pi^{(2)}))$
in Section 7.3. Namely, we try to give a motivic interpretation of the periods
defined in Section 7.1 in terms of Yoshida’s period invariants (see Remark
7.6).

###  7.1. Periods

In this subsection, we introduce a specific vector
$\boldsymbol{f}\in{\mathcal{S}}^{(3)}_{\boldsymbol{\lambda}}({\mathcal{K}}_{3})$
(resp. specific vectors
$h^{\pm}\in{\mathcal{S}}^{(2)}_{\boldsymbol{\lambda}}({\mathcal{K}}_{2})$)
associated with $\pi^{(3)}$ (resp. $\pi^{(2)}$). According to the idea in
[RS08], we define the period $\Omega_{\pi^{(3)}}$ of $\pi^{(3)}$ (resp. the
periods $\Omega^{\pm}_{\pi^{(2)}}$ of $\pi^{(2)}$) depending on the choice of
the vector $\boldsymbol{f}$ (resp. the vectors $h^{\pm}$), based on the
comparison isomorphism between the Betti and the de Rham cohomology groups of
the symmetric space $Y^{(3)}_{{\mathcal{K}}_{3}}$ (resp.
$Y^{(2)}_{{\mathcal{K}}_{2}}$). Note that the periods
$\Omega^{\pm}_{\pi^{(2)}}$ of $\pi^{(2)}$ constructed in this subsection are
known as the canonical periods in literatures (see [Hid94] and [Vat99] for
instance), and they are determined uniquely up to multiplication of elements
in a field of rationality for $\pi^{(2)}$. We will define the period
$\Omega_{\pi^{(3)}}$ of $\pi^{(3)}$ in an analogous way to the construction of
the canonical periods of $\pi^{(2)}$, which is one of reasons why we introduce
an explicit form of the Eichler–Shimura map for ${\rm GL}_{3}$.

We here emphasize the difference of the definition of periods between the
present article and [RS08]. In [RS08], Raghuram and Shahidi define periods for
cohomological irreducible cuspidal automorphic representations $\pi^{(n)}$ of
${\rm GL}_{n}({\mathbf{Q}}_{\mathbf{A}})$ by fixing the following data:

  * —

signature $\varepsilon$ of the action of ${\mathbf{R}}^{\times}{\rm
O}_{n}({\mathbf{R}})/K_{n}$ on
$H^{b_{n}}(\mathfrak{gl}_{n}({\mathbf{R}}),K_{n};H_{\pi^{(n)},K_{n}}\otimes
L^{(n)}(*_{\boldsymbol{\lambda}};{\mathbf{C}}))$ where $K_{n}$ denotes
$\mathbf{R}^{\times}_{>0}\mathrm{SO}_{n}(\mathbf{R})$. We denote the eigen
space of its action by
$H^{b_{n}}(\mathfrak{gl}_{n}({\mathbf{R}}),K_{n};H_{\pi^{(n)},K_{n}}\otimes
L^{(n)}(*_{\boldsymbol{\lambda}};{\mathbf{C}}))[\varepsilon]$;

  * —

a new vector in $\pi^{(n)}_{\rm fin}$;

  * —

a vector ${\boldsymbol{w}}_{\infty}$ in
$H^{b_{n}}(\mathfrak{gl}_{n}({\mathbf{R}}),K_{n};H_{\pi^{(n)},K_{n}}\otimes
L^{(n)}(*_{\boldsymbol{\lambda}};{\mathbf{C}}))[\varepsilon]$.

Since Raghuram and Shahidi do not specify explicitly how to choose an
archimedean vector ${\boldsymbol{w}}_{\infty}$, a certain archimedean local
integral depending on the choice of $\boldsymbol{w}_{\infty}$ and the critical
points remains in their formulas of period integrals in [Rag16, Theorem 2.50];
see [Rag16, (2.49)] for the precise description of their archimedean local
integral. It is known that this archimedean local integral does not vanish by
Kasten and Schmidt [KS13] in the ${\rm GL}_{3}\times{\rm GL}_{2}$ case and by
Sun [Sun17] in the general ${\rm GL}_{n+1}\times{\rm GL}_{n}$ case. Moreover,
Januszewski has shown that the ratio of archimedean local integrals at each
critical points is given by a multiple of a power of $2\pi\sqrt{-1}$ with an
implicit rational number; for details see [Jan19, Corollary 4.12]. However, it
is important to determine this implicit rational number for the application to
the study of the Manin congruences among the critical values [Jan] and to the
construction of the $p$-adic $L$-functions for ${\rm GL}_{n+1}\times{\rm
GL}_{n}$. For this purpose, we introduce a distinguished vector
${\boldsymbol{w}}_{\infty}$ by fixing a normalization of the Whittaker
functions in this subsection. The algebraicity result on the critical values
proposed in the present article (Corollary 7.5) refines the one proposed in
[Rag16, Theorem 2.50] for the ${\rm GL}_{3}\times{\rm GL}_{2}$ case. See also
Remark 7.4 for the relation with the series of recent works of Januszewski
(for instance [Jan19] and [Jan]).

We specify a distinguished vector in ${\boldsymbol{w}}_{\infty}$ in
$H^{b_{n}}(\mathfrak{gl}_{n}({\mathbf{R}}),K_{n};H_{\pi^{(n)},K_{n}}\otimes
L^{(n)}(*_{\boldsymbol{\lambda}};{\mathbf{C}}))[\varepsilon]$ by using the
explicit description of the Eichler–Shimura maps, which has been already
studied in Section 5.3, and the explicit formulas of the radial parts of the
archimedean Whittaker functions. In this subsection, using the explicit and
thorough studies of the archimedean Whittaker functions for ${\rm GL}_{3}$
presented in the recent works by Hirano, Ishii and Miyazaki (see [HIM16] and
[HIM]), we refine the definition of the periods in [RS08,
Definition/Proposition 3.3].

Let
$\psi:{\mathbf{Q}}\backslash{\mathbf{Q}}_{\mathbf{A}}\to{\mathbf{C}}^{\times}$
be the additive character satisfying $\psi_{\infty}(x)={\rm
exp}(2\pi\sqrt{-1}x)$ for each $x\in{\mathbf{R}}$. Denote by
${\mathcal{W}}(\pi^{(3)},\psi)$ the $\psi$-Whittaker model of $\pi^{(3)}$ and
by ${\mathcal{W}}(\pi^{(2)},\psi^{-1})$ the $\psi^{-1}$-Whittaker model of
$\pi^{(2)}$. Each of ${\mathcal{W}}(\pi^{(3)},\psi)$ and
${\mathcal{W}}(\pi^{(2)},\psi^{-1})$ decomposes into the restricted tensor
product as follows:

$\displaystyle{\mathcal{W}}(\pi^{(3)},\psi)$
$\displaystyle={\bigotimes_{v}}^{\prime}{\mathcal{W}}(\pi^{(3)}_{v},\psi_{v}),$
$\displaystyle{\mathcal{W}}(\pi^{(2)},\psi^{-1})$
$\displaystyle={\bigotimes_{v}}^{\prime}{\mathcal{W}}(\pi^{(2)}_{v},\psi^{-1}_{v}).$

Now let us recall the notion of the field of rationality $E_{\pi^{(n)}}$ of
$\pi^{(n)}$ according to [Clo90, Section 3]; see also [RS08, Section 3.1].
Write $V$ as a representation space of $\pi^{(n)}_{\rm fin}$. For each
$\sigma\in{\rm Aut}({\mathbf{C}})$, define ${}^{\sigma}\pi^{(n)}_{\rm fin}$ to
be $V\otimes_{{\mathbf{C}},\sigma^{-1}}{\mathbf{C}}$, and set
$S(\pi^{(n)}_{\rm fin})=\\{\sigma\in{\rm
Aut}({\mathbf{C}})\mid{}^{\sigma}\pi^{(n)}_{\rm fin}=\pi^{(n)}_{\rm fin}\\}$.
Then ${}^{\sigma}\pi^{(n)}_{\rm fin}$ is indeed the finite part of a
cohomological cuspidal automorphic representation ${}^{\sigma}\pi$ of ${\rm
GL}_{n}({\mathbf{Q}}_{\mathbf{A}})$, and $E_{\pi^{(n)}}:=\\{x\in\mathbf{C}\mid
x^{\sigma}=x\text{ for }{}^{\forall}\sigma\in S(\pi^{(n)}_{\rm fin})\\}$ is a
finite extension of ${\mathbf{Q}}$ (see [Clo90, Théor$\grave{\rm e}$m 3.13]).
We call $E_{\pi^{(n)}}$ the field of rationality of $\pi^{(n)}$.

In the rest of this article, we impose several constraints on
$\mathcal{K}_{2}$ and $\mathcal{K}_{3}$. First we suppose that
$\pi^{(2)}_{\mathrm{fin}}$ has a nonzero vector fixed by the action of
$\mathrm{GL}_{2}(\widehat{\mathbf{Z}})$, and set
$\mathcal{K}_{2}=\mathrm{GL}_{2}(\widehat{\mathbf{Z}})$. Next, for an ideal
${\mathfrak{N}}$ of $\widehat{\mathbf{Z}}$, we introduce an open compact
subgroup ${\mathcal{K}}_{3}({\mathfrak{N}})$ of
$\mathrm{GL}_{3}(\widehat{\mathbf{Z}})$ defined as

$\displaystyle{\mathcal{K}}_{3}({\mathfrak{N}})=\left\\{g=\begin{pmatrix}g_{ij}\end{pmatrix}_{1\leq
i,j\leq 3}\in{\rm GL}_{3}(\widehat{\mathbf{Z}})\ \middle|\
g_{31},g_{32}\in{\mathfrak{N}},\ g_{33}\equiv 1\pmod{\mathfrak{N}}\right\\}.$

Suppose that there exists an ideal $\mathfrak{N}$ of $\widehat{\mathbf{Z}}$
such that $\pi^{(3)}_{\mathrm{fin}}$ has a nonzero vector fixed by the action
of $\mathcal{K}_{3}(\mathfrak{N})$. We choose a minimal ideal ${\mathfrak{N}}$
satisfying this property, and set
$\mathcal{K}_{3}=\mathcal{K}_{3}(\mathfrak{N})$. Refer to Remark 7.1 for these
assumptions.

First let us discuss the period of $\pi^{(2)}$. For each finite place $v$ of
$\mathbf{Q}$, the $v$-component $\pi^{(2)}_{v}$ of $\pi^{(2)}$ is unramified
by assumption, and thus there exists a unique function
$W^{\circ}_{\pi^{(2)},v}\in\mathcal{W}(\pi^{(2)}_{v},\psi_{v}^{-1})^{\mathrm{GL}(\mathbf{Z}_{v})}$
which is called by Shintani the class-$1$ Whittaker function on
$\mathrm{GL}_{2}(\mathbf{Q}_{v})$; see [Shin76] for details. We now define
archimedean Whittaker functions
$W^{\pm}_{\pi^{(2)},\infty}\in\mathcal{W}(\pi^{(2)}_{\infty},\psi^{-1}_{\infty})$
according to [HIM, Corollary 3.6, Section 9.3]. Recall from Section 4.2 that
$\pi^{(2)}_{\infty}$ has the minimal $\mathrm{O}_{2}(\mathbf{R})$-type
$\tau^{(2)}_{\boldsymbol{\lambda}}$ with
$\boldsymbol{\lambda}^{(2)}=(\lambda_{2},0)=(l_{2}+1,0)$. Let
$\varphi^{(2)}_{-}\colon
V^{(2)}_{\boldsymbol{\lambda}}\to{\mathcal{W}}(\pi^{(2)}_{\infty},\psi^{-1}_{\infty})$
be a $K_{2}$-homomorphism satisfying the following formulas on the radial
parts (see [HIM, Corollary 3.6, Section 9.3]):

$\displaystyle\varphi^{(2)}_{-}((-X+\sqrt{-1}Y)^{\lambda_{2}})({\rm
diag}(y_{1}y_{2},y_{2}))$
$\displaystyle=\displaystyle\frac{y^{\frac{1}{2}}_{1}y^{2\nu_{2}}_{2}}{2\pi\sqrt{-1}}\int_{t}\Gamma_{\mathbf{C}}\left(t+\nu_{2}+\frac{\lambda_{2}-1}{2}\right)y^{-t}_{1}{\rm
d}t,$ $\displaystyle\varphi^{(2)}_{-}((X+\sqrt{-1}Y)^{\lambda_{2}})({\rm
diag}(y_{1}y_{2},y_{2}))$ $\displaystyle=0.$

Here $\int_{t}$ denotes an integration on a vertical line from ${\rm
Re}(t)-\sqrt{-1}\infty$ to ${\rm Re}(t)+\sqrt{-1}\infty$ for an appropriate
complex number $t\in{\mathbf{C}}$ with the sufficiently large real part so
that all the poles of the integrand is in the left-side of the vertical line.
Then define $W^{\pm}_{\pi^{(2)},\infty}$ to be $\varphi^{(2)}_{-}((\pm
X+\sqrt{-1}Y)^{\lambda_{2}})$, and set
$W^{\pm}_{\pi^{(2)}}=W^{\pm}_{\pi^{(2)},\infty}\otimes\bigotimes_{v\colon\text{finite}}W^{\circ}_{\pi^{(2)},v}$.
We define $h^{\pm}$ as the cuspidal automorphic forms on
$\mathrm{GL}_{2}(\mathbf{Q}_{\mathbf{A}})$ corresponding to
$W^{\pm}_{\pi^{(2)}}$, that is, $h^{\pm}$ are defined to be the inverse
Fourier transform of $W^{\pm}_{\pi^{(2)}}$. Then define the cohomology classes
$\delta^{\pm}_{\pi^{(2)}}$ in
$H^{1}(Y^{(2)}_{\mathcal{K}_{2}},\mathcal{L}^{(2)}(\boldsymbol{n}_{\boldsymbol{\lambda}};\mathbf{C}))$
as $\delta^{\pm}_{\pi^{(2)}}=\delta^{(2)}(h^{+})\pm\delta^{(2)}(h^{-})$. Since
$\epsilon=\bigotimes_{v\colon\text{finite}}1_{2}\otimes\begin{pmatrix}-1&0\\\
0&1\end{pmatrix}$ normalizes $\mathcal{K}_{2}K_{2}$ where
$K_{2}=\mathbf{R}_{>0}^{\times}\mathrm{SO}_{2}(\mathbf{R})$, it acts on
$H^{\bullet}_{\ast}(Y^{(2)}_{\mathcal{K}_{2}},{\mathcal{L}}(\boldsymbol{n}_{\boldsymbol{\lambda}};{\mathbf{C}}))$;
hereafter $*$ denotes $\emptyset$ or ${\rm c}$. We denote by
$H^{\bullet}_{\ast}(Y^{(2)}_{\mathcal{K}_{2}},{\mathcal{L}}^{(2)}(\boldsymbol{n}_{\boldsymbol{\lambda}};{\mathbf{C}}))[\pm]$
its $(\pm 1)$-eigenspaces with respect to the action of $\epsilon$, and write
the $\pi^{(2)}_{\rm fin}$-isotypic components of
$H^{\bullet}_{\ast}(Y^{(2)}_{\mathcal{K}_{2}},{\mathcal{L}}^{(2)}(\boldsymbol{n}_{\boldsymbol{\lambda}};{\mathbf{C}}))[\pm]$
as
$H^{\bullet}_{\ast}(Y^{(2)}_{\mathcal{K}_{2}},{\mathcal{L}}^{(2)}(\boldsymbol{n}_{\boldsymbol{\lambda}};{\mathbf{C}}))[\pm,\pi_{\rm
fin}]$. Then each of $H^{1}_{\rm
c}(Y^{(2)}_{\mathcal{K}_{2}},{\mathcal{L}}^{(2)}(\boldsymbol{n}_{\boldsymbol{\lambda}};{\mathbf{C}}))[\pm,\pi_{\rm
fin}]$ is of dimension $1$ and it is spanned by $\delta^{\pm}_{\pi^{(2)}}$.
Furthermore the spaces $H^{1}_{\rm
c}(Y^{(2)}_{\mathcal{K}_{2}},{\mathcal{L}}^{(2)}(\boldsymbol{n}_{\boldsymbol{\lambda}};{\mathbf{C}}))[\pm,\pi_{\rm
fin}]$ are defined over the field of rationality $E_{\pi^{(2)}}$ of
$\pi^{(2)}$, and their $E_{\pi^{(2)}}$-structures are given as

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