# Coherent cohomology of Shimura varieties, motivic cohomology, and
archimedean $L$-packets

Gyujin Oh

###### Abstract.

We formulate an analogue of the archimedean motivic action conjecture of
Prasanna–Venkatesh for _irregular_ cohomological automorphic forms on Shimura
varieties, which appear on multiple degrees of coherent cohomology of Shimura
varieties. Such multiple appearances are due to many infinity types in a
single $L$-packet with equal minimal $K$-types. Accordingly, we formulate the
conjecture comparing periods of forms of _different automorphic
representations_. We provide evidences for the conjecture by showing its
compatibility with existing conjectures on periods of automorphic forms. The
conjectures suggest the existence of certain operations which move between
different infinity types in an $L$-packet.

###### Contents

  1. 1 Introduction
    1. 1.1 Archimedean $L$-packet and the matter of choosing automorphic realizations
    2. 1.2 Comparison of periods of different automorphic representations
    3. 1.3 Generalized complex conjugations
    4. 1.4 Summary
    5. 1.5 Problems and questions
    6. 1.6 Notation
  2. 2 Archimedean $L$-packets and the motivic action conjecture
    1. 2.1 $(\mathfrak{p},K)$-cohomology and automorphic forms
    2. 2.2 Metrics on cohomology
    3. 2.3 Archimedean motivic action conjecture for Shimura varieties
    4. 2.4 An approach towards the motivic action conjecture
  3. 3 Evidence I: The case of $\operatorname{Sp}_{4}$
    1. 3.1 Whittaker periods via cohomological period integrals
    2. 3.2 Hodge-linear algebra
    3. 3.3 Completion of the proof
  4. 4 Evidence II: The case of $\operatorname{SU}(2,1)$
    1. 4.1 Whittaker periods via cohomological period integrals
    2. 4.2 Hodge-linear algebra
    3. 4.3 Completion of the proof
  5. 5 Towards motivic action conjecture for rationality of classes
    1. 5.1 The case of Hilbert modular forms: comparison with [Ho]
    2. 5.2 Desiderata for generalized complex conjugations
  6. A Beilinson’s conjecture over a general number field
    1. A.1 Chow motives
    2. A.2 Beilinson’s conjecture for Chow and Grothendieck motives
  7. B Deligne cohomology and Lie algebra cohomology
    1. B.1 Nondegenerate limit of discrete series as constituents of reducible principal series
    2. B.2 Action of the $\operatorname{Ext}$-space
    3. B.3 Deligne cohomology as an $\operatorname{Ext}$-space

##  1\. Introduction

The motivic action conjecture of Venkatesh posits that, roughly speaking, for
a Hecke eigensystem $h$, there is a natural action of the motivic cohomology
of the adjoint motive associated to $h$ on the $h$-isotypic part of the
rational cohomology of locally symmetric spaces. This has many incarnations
which shed new light on various parts of the Langlands program. However, the
conjectures have been mostly restricted to the case of “$\delta\neq 0$,”
namely when the reductive group in concern $G$ has no compact Cartan subgroup.
This in particular excludes the case when the locally symmetric space is a
_Shimura variety_. As a locally symmetric space that is not a Shimura variety
so far has not been related to algebraic geometry (although see [GT]), the
motivic action conjectures seemed to be extremely difficult to approach.

On the other hand, there have been expectations that a similar conjecture
would exist for automorphic forms over Shimura varieties with _irregular
weight_. The easiest instance is the case of weight one modular forms; they
appear in both $H^{0}$ and $H^{1}$ of the modular curve of the same line
bundle. The main purpose of the paper is to give a formulation of such
conjecture for general Shimura varieties, and provide somewhat intricate
evidences using well-known results and conjectures regarding periods of
automorphic forms. The following is a generalization of the archimedean
motivic action conjecture to general Shimura varieties.

######  Conjecture 1 ((Archimedean motivic action conjecture for Shimura
varieties)).

Let $\lambda$ be a nondegenerate singular analytically integral character, and
let $\Pi$ satisfy Assumption 2.5, with
$\Pi_{\infty}\in\mathfrak{P}_{\lambda}$. Let
$\mathcal{M}=H_{M}^{1}((\operatorname{Ad}\Pi)_{\mathcal{O}_{E}},\overline{\mathbb{Q}}(1))$
and
$\mathcal{H}^{i}=H^{i}(X_{G}(\Gamma)_{\overline{\mathbb{Q}}},[V])[\Pi_{f}]$,
where both are regarded as $\overline{\mathbb{Q}}$-vector spaces equipped with
Hermitian bilinear forms, induced from a fixed admissible bilinear form on
$\mathfrak{g}_{\mathbb{C}}$ (see §2.2). Then, there is an isometry between
graded $\overline{\mathbb{Q}}$-vector spaces equipped with Hermitian metrics,

$\wedge^{*}\mathcal{M}^{*}\otimes\mathcal{H}^{i_{\min}}\cong\bigoplus_{i=i_{\min}}^{i_{\max}}\mathcal{H}^{i},$

where $V$ is the automorphic vector bundle coming from Levi (see Notation)
such that $\Pi_{f}$ appears in multiple degrees of its cohomology,
$i_{\min}=\min\\{i\mid H^{i}(X_{G}(\Gamma),[V])[\Pi_{f}]\neq 0\\}$, and
$i_{\max}$ is defined analogously.

This is the analogue of [PV, Conjecture 1.2.1], although new subtleties arise
in the irregular weight case as we will see. Under some standard conjectures
on periods, we show that the conjecture indeed holds in a few low-dimensional
cases.

######  Theorem 2.

Under certain mild conditions and several standard conjectures on periods (see
below), for $G=\operatorname{Sp}_{4}$ and $\operatorname{SU}(2,1)$, Conjecture
1 is true. In other words, the action of adjoint motivic cohomology group on
the coherent cohomology groups of Shimura variety respects
$\overline{\mathbb{Q}}^{\times}$-structure.

The “mild conditions” are that, informally speaking, the finite part is
globally generic, and that there are newforms; see Assumption 2.5. These
conditions exist to have a clean statement of the conjectures. More
importantly, the “standard conjectures on periods” are summarized in
Assumption 2.22, which includes the Lapid–Mao conjecture and the Beilinson
conjectures. The proof of Theorem 2.18 uses the same idea of [PV], but
requires more machinery, as the whole setup is about comparing periods of
_different automorphic representations_.

The cases where Theorem 2 is proved are when the Hecke eigensystem appears in
two consecutive degrees of cohomology. In this case, or more generally, when
we restrict the statement of Conjecture 1 into the relation between the top
and bottom degrees, the conjecture can be made into a statement that does not
refer to motivic cohomology.

More precisely, let $\Pi=\Pi_{f}\otimes\Pi_{\infty}$ be a cuspidal automorphic
representation satisfying Assumption 2.5, and let $\lambda$ be the
infinitesimal character of $\Pi_{\infty}$, which is singular and nondegenerate
(see Notation). Let $V$ be the automorphic vector bundle coming from the Levi
(see Notation), such that $H^{i}(\mathfrak{p},K;V\otimes\Pi_{\infty})\neq 0$
for some $i$, and $\Pi_{\min},\Pi_{\max}$ be the members of the archimedean
$L$-packet of $\Pi_{\infty}$ such that the degree that $\Pi_{\min}$
($\Pi_{\max}$, respectively) has nontrivial $(\mathfrak{p},K)$-cohomology with
coefficient in $V$ is the minimum (maximum, respectively) in the $L$-packet.
Let $i_{\min}$ ($i_{\max}$, respectively) be the degree, and let
$f_{\min}\in\Pi_{f}^{\operatorname{new}}\otimes\Pi_{\min}^{\operatorname{new}}$
and
$f_{\max}\in\Pi_{f}^{\operatorname{new}}\otimes\Pi_{\max}^{\operatorname{new}}$
(see Definition 2.6) such that $[f_{\min}],[f_{\max}]\in
H^{*}(X_{G}(\Gamma),V)$ (the harmonic Dolbeault forms corresponding to the
automorphic forms; see Definition 2.8 for a precise definition) are defined
over $\overline{\mathbb{Q}}$. Then, under the Beilinson conjectures, the
information on the top and bottom degrees in Conjecture 1 is precisely that

$\frac{\langle f_{\min},f_{\min}\rangle}{\langle
f_{\max},f_{\max}\rangle}\sim_{\overline{\mathbb{Q}}^{\times}\cap\mathbb{R}}\left|\pi^{i_{\min}-i_{\max}}\frac{L_{\infty}(1,\Pi,\operatorname{Ad})}{L_{\infty}(0,\Pi,\operatorname{Ad})}\cdot\frac{L(1,\Pi,\operatorname{Ad})}{\operatorname{vol}F^{1}H_{\operatorname{dR}}(\operatorname{Ad}\Pi)}\right|^{2},$

where $\langle\cdot,\cdot\rangle$ is the Petersson inner product, and the
volume is computed with respect to the metric induced by any weak polarization
(see [PV, §2.2.3])111The volume is independent of the choice of weak
polarization, see [PV, Lemma 2.2.2].. In particular, this statement is
_equivalent to Conjecture 1 if there are only two degrees that $\Pi_{f}$
appear in the cohomology_, assuming Beilinson conjectures.

At first sight, the information on top and bottom degrees might seem to less
interesting, as one might guess there is an extra duality that one can compare
the top and bottom degrees. Indeed, in [PV], the top and bottom degrees were
complementary, so that they are related via duality. On the other hand, in the
irregular setting, there are numerous cases where the top and bottom degrees
are not complementary. Rather, they are determined by the position of the
infinitesimal character inside the Weyl chambers. Indeed, we work with two
examples, $G=\operatorname{Sp}_{4}$ and $\operatorname{SU}(2,1)$, and both
cases we work with the choice of $\lambda$ where the minimum and maximum
degree of appearances are $H^{0}$ and $H^{1}$, respectively.

###  1.1. Archimedean $L$-packet and the matter of choosing automorphic
realizations

We now explain the new subtleties of the irregular weight case. In view of
automorphic cohomology, the phenomenon of a weight one modular form appearing
in $H^{0}$ and $H^{1}$ actually involves _two different representations_. If
we denote $\omega$ to be the weight one line bundle or the corresponding
representation of $\operatorname{SO}(2)$, then for a weight one modular
newform $f$,

$H^{0}(X,\omega)[f]=H^{0}(\mathfrak{p},\operatorname{SO}(2);D_{0}^{+}\otimes\omega),$
$H^{1}(X,\omega)[f]=H^{1}(\mathfrak{p},\operatorname{SO}(2);D_{0}^{-}\otimes\omega),$

where $\mathfrak{p}=\mathfrak{p}_{-}\oplus\mathfrak{s}\mathfrak{o}(2)$,
$\mathfrak{p}_{-}$ is the anti-holomorphic tangent space and $D_{0}^{+}$ and
$D_{0}^{-}$ are the holomorphic and the antiholomorphic discrete series,
respectively. Indeed, it is $\overline{f}d\overline{z}$ that appears in
$H^{1}(X,\omega)$ as a class in Dolbeult cohomology, and $\overline{f}$ is an
_antiholomorphic_ modular form of weight $-1$.

In general, if the same Hecke eigensystem appears in multiple degrees of
cohomology of the same automorphic vector bundle of a Shimura variety, then
each such instance is actually represented by the so-called
$(\mathfrak{p},K)$-cohomology of different automorphic representations. More
precisely, the finite part ($G(\mathbb{A}_{f})$-representation) remains the
same, while the infinity type varies inside an archimedean $L$-packet. Such
phenomenon happens precisely when the infinitesimal character of the infinity
type lies on the walls of Weyl chambers.

The action of $\wedge^{*}\mathfrak{a}_{G}^{*}$ in [PV] is most naturally
thought as the self-$\operatorname{Ext}$-algebra of a representation. On the
other hand, in our case, the representations corresponding to the target and
the source of the action is different, so the action cannot be thought as an
$\operatorname{Ext}$-_algebra_ , but merely as an $\operatorname{Ext}$-group.
Furthermore, the translation of the action into the automorphic cohomology
context also depends on the choice of _automorphic realization maps
$\Pi_{f}\otimes\Pi_{\infty}\hookrightarrow\mathcal{A}(G)$_ for each
$\Pi_{\infty}$; namely, each realization map can be always scaled by a scalar,
but we want to compare them as a whole. This problem did not arise in _op.
cit._ , as a choice of a single automorphic realization map would rigidify the
situation. In turn, we had no choice but to formulate a slightly weaker
Conjecture 1 that asserts the motivic action conjecture on the level of metric
spaces.

###  1.2. Comparison of periods of different automorphic representations

We briefly explain the strategy of the proof of Theorem 2. As does in [PV], we
most notably assume _Beilinson’s conjectures (for Chow motives)_. The main new
feature in this paper is that, because we need to compare periods of two
different representations, we need two different conjectures on periods, one
for each representation. In both evidences, we will compare periods of a
_holomorphic automorphic form_ and a _generic automorphic form_. For the
holomorphic form, we will need the _refined Gan–Gross–Prasad conjecture_
(referred as Ichino–Ikeda conjecture in _op. cit._), although in some
instances this requirement can be avoided by using the doubling construction
of standard $L$-functions. For the generic form, we will need the _Lapid–Mao
conjecture_ , which relates the value of a Whittaker function to a certain
$L$-value.

One also needs a way to detect rationality of classes for both types of forms.
For the holomorphic forms, we can use Fourier expansion, but for those
appearing in higher coherent cohomology, we need a new machinery. We will
develop the so-called _cohomological period integrals_ for higher coherent
cohomology, which realizes integral representations of $L$-functions as cup
product pairings in coherent cohomology. The cohomological interpretation of
such integral representations (or, at least, their appearance in the
literature) is relatively new ([LPSZ], [Oh]).

###  1.3. Generalized complex conjugations

The archimedean motivic action conjecture as stated in [PV] gives a recipe of
rational cohomology classes, whereas our Conjecture 1 is a statement on
metrics. To formulate a similar conjecture in the setting of coherent
cohomology of Shimura varieties, we need a way to rigidify between different
automorphic representations in an archimedean $L$-packet. Indeed, in
retrospect, even in the easy case of modular forms, one needs complex
conjugation to go between holomorphic and antiholomorphic limits of discrete
series. Unfortunately, beyond the case of modular forms, there is no known
general operation that can move between different infinity types. We will
tentatively name such an operation a  _generalized complex conjugation_ ,
which should send an automorphic form of a certain infinity type to an
automorphic form of another infinity type in the same $L$-packet. It seems
inevitable to come up with such an operation to formulate the full conjecture
on rational cohomology classes.

The generalized complex conjugations should be naturally understood in the
context of a “derived” local-global compatiblity in some sense, and their
existence is also suggested by the existence of similar operations in the
analogous settings over the $p$-adic fields (Kottwitz’s conjectures, e.g.
[FaMa]) and over the function fields (excursion operators, e.g. [Laf]).
Following the suggestions of Joseph Wolf, we will investigate the nature of
generalized complex conjugations using the theory of Penrose transforms.

On the other hand, if the associated Hermitian symmetric space is a product of
copies of the upper half planes, one can come up with an operation that
changes one infinity type to another by taking complex conjugation at certain
variables. This is a _partial complex conjugation_ , studied in [Ha2]. Using
partial complex conjugations, we can formulate a conjecture on rationality of
cohomology classes in the case of Hilbert modular forms of partial weight one.
There exists a prior work of [Ho] on the motivic action conjecture for Hilbert
modular forms of parallel weight one, which similarly uses partial complex
conjugations. We compare our conjecture in the Hilbert modular form case with
the conjecture of [Ho], and explain the evidences given in _op. cit._ are also
consistent with our conjecture.

###  1.4. Summary

In §2, we take an efficient route to the statement of the Archimedean motivic
action conjecture (Conjecture 2.13) and its more accessible variant, the
Period conjecture (Conjecture 2.15). The objective of the section is to set
the conjecture in a context. We in particular defer the abstract discussion of
how to derive the conjectures, parallel to those of [PV, §2-§5], to later
sections, as it requires more advanced theory on representation theory of real
groups.

In §3 and §4, we provide our main evidence for the Period conjecture
(Conjecture 2.15). Similarly to [PV, §7], we prove that, for
$G=\operatorname{Sp}_{4}$ and $\operatorname{SU}(2,1)$, the Period conjecture
is compatible with several well-accepted conjectures on periods of automorphic
forms, such as the Beilinson’s conjectures, the Lapid–Mao conjecture and the
refined Gan–Gross–Prasad conjectures. To use these, we review how certain
period integrals can be interpreted as cup product pairings of (higher)
coherent cohomology classes on Shimura varieties.

In §5, we discuss the issue on formulating a motivic action conjecture on
rationality of coherent cohomology classes. Most notably, we suggest the
notion of _generalized complex conjugations_ , which move between different
members of a single archimedean $L$-packet. In §5.1, we formulate a precise
conjecture in the case of Hilbert modular forms using partial complex
conjugations, and compare our conjecture with the conjecture of [Ho]. In §5.2,
we spell out conditions that the generalized complex conjugations should
satisfy, and formulate the full conjecture assuming their existence. In
Appendix A, we review the formulation of Beilinson’s conjecture for motives
over a general number field, as many references state the conjecture for only
$\mathbb{Q}$-motives. Finally, we develop a representation theory background
in Appendix B, parallel to [PV, §2-§4]. Although Appendix B is independent of
the development of the rest of the paper, the section is suggestive of a
correct foundation in which the motivic action conjectures need to be
developed.

###  1.5. Problems and questions

There are several interesting questions that arise in this work.

  1. (1)

Place the generalized complex conjugations in the context of some form of
“derived” local-global compatibility, motivated by the strong form of Arthur
conjectures as realized in the function field case via excursion operators as
in [Laf]. A correct formulation should be in accordance with the existing
statements of derived local-global compatibility as in [Fe] and [Zhu].

  2. (2)

The compatibility between the Period Conjecture, Conjecture 2.15, and the
existing period conjectures relies on the yet-to-be-calculated archimedean
zeta integrals. These can be conducted using explicit integral formulae of
(generalized) Whittaker functions, e.g. [KO], [Od].

  3. (3)

As we deal with motives over a more general number field and Shimura varieties
over a number field other than $\mathbb{Q}$, in every aspect of our
discussion, the choice of a complex embedding is always implicit. In
particular, there must be a relation between the conjectures in this paper for
the _conjugates of Shimura varieties_ (e.g. [Va]).

  4. (4)

It seems extremely hard to detect rationality of coherent cohomology classes
if the Hecke operators can only cut a space that is of dimension larger than
one. For example, if $X$ is a Hilbert modular surface, and if $\omega$ is the
parallel weight one line bundle, then it seems extremely difficult to
determine whether a class in $H^{1}(X,\omega)$ is defined over
$\overline{\mathbb{Q}}$, or even to produce a class in it.

  5. (5)

It is expected that the motivic action conjecture will involve $L$-packets
even in the case of $\delta>0$. It may be possible to formulate the conjecture
for the same eigensystem appearing in cohomology with _different
coefficients_.

###  1.6. Notation

Let $G$ be a connected reductive algebraic group over $\mathbb{Q}$. For
simplicity, let us assume that $G$ is quasisplit, $G(\mathbb{R})$ is
connected, and that the center of $G$ does not have a nontrivial
$\mathbb{R}$-split torus. Also, we assume that there exists a _twisting
element_ in the sense of [BG, Definition 5.2.1]222This is to avoid the
subtlety of difference between $C$-algebraicity and $L$-algebraicity.. Let
$\mathfrak{g}_{\mathbb{Q}}$ be the $\mathbb{Q}$-Lie algebra of $G$, and let
$\mathfrak{g}_{\mathbb{R}},\mathfrak{g}_{\mathbb{C}}$ be its base change to
$\mathbb{R}$ and $\mathbb{C}$, respectively. We occasionally drop the
subscript for $\mathbb{C}$. Let $W_{G}$ be the Weyl group of $G$. We endow an
invariant, $\theta$-invariant, $\mathbb{R}$-invariant bilinear form $B$ on
$\mathfrak{g}_{\mathbb{R}}$, such that $B(X,\theta(X))$ is negative definite,
where $\theta$ is the Cartan involution. We will use this to talk about inner
product on weight space, Riemannian metric on the Hermitian symmetric domain,
etc. In specific examples, we may and will choose $B$ to induce a preferred
Riemannian metric on the Hermitian symmetric space (for example, one may want
the Riemannian metric to be $dxdy$ on the upper half plane
$\mathbb{H}=\\{x+iy\mid y>0\\}$). We also use $B$ to any other bilinear form
induced from $B$.

Suppose further that $G$ gives rise to a _Shimura variety_ , which means that
there is a symmetric space $X$ for $G(\mathbb{R})$ which can be endowed with a
structure of Hermitian symmetric domain (which we will fix). Fix a point $h\in
X$, which gives rise to a _Hodge cocharacter_
$h:\mathbb{S}=\operatorname{Res}_{\mathbb{C}/\mathbb{R}}\mathbb{G}_{m,\mathbb{C}}\rightarrow
G_{\mathbb{R}}$ which in turn induces a real Hodge structure of weight $0$ on
$\mathfrak{g}_{\mathbb{R}}$,
$\mathfrak{g}=\mathfrak{g}^{-1,1}\oplus\mathfrak{g}^{0,0}\oplus\mathfrak{g}^{1,-1}$.
Given an open compact subgroup $\Gamma\subset G(\mathbb{A}_{f})$, there exists
a quasi-projective variety $Y_{G}(\Gamma)$, a Shimura variety, defined over a
number field $E$, whose complex points333Note that the choice of a point in a
Hermitian symmetric domain gives the reflex field as a subfield of
$\mathbb{C}$, so there is a _preferred complex embedding_ ; e.g. [Va, Notation
4.6]. In particular, one can expect that the statement of the conjecture
depends a priori on the choice of a Hermitian symmetric domain. It could be
interesting to check if our conjecture is consistent with conjugation of
Shimura varieties. have an analytification isomorphic to the double quotient
$G(\mathbb{Q})\backslash(X\times G(\mathbb{A}_{f})/\Gamma)$.

Let $K\subset G(\mathbb{R})$ be the stabilizer of $h$. Let $T$ be the Cartan
subgroup of $K$. Then, $X\cong G(\mathbb{R})/K$ and
$\mathfrak{g}^{0,0}=\mathfrak{k}:=\operatorname{Lie}(K)_{\mathbb{C}}$. We
denote $\mathfrak{p}_{+}=\mathfrak{g}^{-1,1}$,
$\mathfrak{p}_{-}=\mathfrak{g}^{1,-1}$ and
$\mathfrak{p}=\mathfrak{k}\oplus\mathfrak{p}_{-}$. Then, $\mathfrak{p}$ is a
parabolic subalgebra of $\mathfrak{g}$, giving rise to a parabolic subgroup
$P\subset G_{\mathbb{C}}$ with $\operatorname{Lie}P=\mathfrak{p}$. We also fix
once and for all a positive system of roots for $\mathfrak{k}$. The
holomorphic tangent space of $X$ at $h$ is identified with $\mathfrak{p}_{+}$,
so there is a $G(\mathbb{R})$-equivariant embedding of complex manifolds
$X\rightarrow\check{D}:=G(\mathbb{C})/P(\mathbb{C})$, sending $h\mapsto
P(\mathbb{C})$. In this regard, $K=G(\mathbb{R})\cap P(\mathbb{C})$, and
$K(\mathbb{C})$ is the Levi subgroup of $P(\mathbb{C})$. Also, any finite-
dimensional holomorphic representation $V$ of $P(\mathbb{C})$ gives rise to an
algebraic vector bundle over $Y_{G}(\Gamma)$, an _automorphic vector bundle_ ,
denoted $[V]$, which is an algebraization of the pullback of the holomorphic
vector bundle on $X$ which in turn is the restriction of the vector bundle
$G(\mathbb{C})\times^{P(\mathbb{C})}V\rightarrow\check{D}$ on $\check{D}$. If
$V$ factors through $K(\mathbb{C})$, namely if it is induced from a
representation of $K(\mathbb{C})$, we will call $[V]$ an automorphic vector
bundle _coming from the Levi_. If not, we will call $V$ _nearly_ , following
[LPSZ].

To save space, we may abbreviate some words with repeated appearances:
discrete series into DS, limit of discrete series into LDS, and nondegenerate
limit of discrete series (see the paragraph before Theorem 2.4 for its
definition) into NLDS. All real group representations are thought as
$(\mathfrak{g},K)$-modules.

For automorphic representations, their $L$-functions are normalized so that
$\frac{1}{2}$ is the center of symmetry. For pure motives, theire
$L$-functions are normalized so that, if $w$ is its weight, $\frac{w+1}{2}$ is
the center of symmetry. For both kinds of $L$-functions, $w$ is called the
_motivic weight_ of the $L$-function. For an automorphic representation
$\Pi=\Pi_{f}\otimes\Pi_{\infty}$ of $G(\mathbb{A})$, the field of rationality
$F_{\Pi}$ is the fixed field of the isomorphism class of $\Pi_{f}$ as a
$G(\mathbb{A}_{f})$-representation ([Cl, §3.1]). An inner product on the space
of automorphic forms, denoted $\mathcal{A}(G)$, can be given as either the
$L^{2}$-norm on $G(\mathbb{Q})/G(\mathbb{A})$ with respect to the Tamagawa
measure or the measure coming from the Riemannian metric of the symmetric
space, as the norms are all scaled by the same scalar factor. The second norm
is the same as the usual _Petersson norm_ , which we will denote as
$\langle,\rangle_{P}$.

For the integral representations, we fix a nontrivial additive character
$\psi=\prod_{p}\psi_{p}$ of $\mathbb{A}/\mathbb{Q}$. For a cuspidal
automorphic form $\varphi=\prod_{p}\varphi_{p}$ of cuspidal automorphic
representation $\pi=\bigotimes_{p}^{\prime}\pi_{v}$ of $G(\mathbb{A})$, its
Whittaker transform $W_{\varphi}(g)$ is defined as
$W_{\varphi}(g)=\int_{N_{G}(\mathbb{Q})\backslash
N_{G}(\mathbb{A})}\varphi(ng)\psi(n^{-1})dn$, for some choice of $N_{G}$ that
needs to be specified when talking about Whittaker model. Locally, a generic
$G(\mathbb{Q}_{p})$-representation $\pi_{p}$ is isomorphic to the space of
functions
$\mathcal{W}_{p}:=\\{W_{p}:G(\mathbb{Q}_{p})\rightarrow\mathbb{C}\mid
W_{p}(ng)=\psi_{p}(n)W_{p}(g)\text{ for }n\in N_{G}(\mathbb{Q}_{p})\\}$; one
can choose isomorphisms (a local _Whittaker model_)
$\pi_{p}\xrightarrow{\sim}\mathcal{W}_{p}$, $f_{p}\mapsto W_{f_{p}}$ that are
compatible with the global Whittaker model; namely,
$W_{\varphi}(g)=\prod_{p}W_{\varphi_{p}}(g_{p})$.

We use notational convention for motives as in [PV, §2], except that we will
deal with motives over a more general number field. In that case, we put the
complex embedding in the subscript, such as $M_{\sigma}$,
$\operatorname{comp}_{B,\operatorname{dR},\sigma}$, etc.

##  2\. Archimedean $L$-packets and the motivic action conjecture

In this section, we take the shortest path to the statement of the Archimedean
motivic action conjecture for Shimura varieties, Conjecture 2.13. More
abstract justification of the formulation of the Conjecture, including a
parallelism between [PV] and our conjecture, is discussed in Appendix B and
§5.2.

###  2.1. $(\mathfrak{p},K)$-cohomology and automorphic forms

Firstly, we quickly review how coherent cohomology of Shimura varieties is
related to automorphic forms via the theory of $(\mathfrak{p},K)$-cohomology.
Recall that, as the singular cohomology of locally symmetric spaces can be
calculated in terms of $(\mathfrak{g},K)$-cohomology, the coherent cohomology
of Shimura varieties can be calculated in terms of the so-called
$(\mathfrak{p},K)$-cohomology. By reinterpreting what the Dolbeault cohomology
calculates in the setting of Shimura varieties, one gets the following

######  Proposition 2.1 ((See [Su, (2.12)])).

We have

$H^{i}(Y_{G}(\Gamma),[V])\cong
H^{i}(\mathfrak{p},K;C^{\infty}(G(\mathbb{Q})\backslash
G(\mathbb{A})/\Gamma)^{K\mathrm{-finite}}\otimes V),$

where the left hand side is analytic cohomology, and $V$ is understood as a
$(\mathfrak{p},K)$-module with trivial $\mathfrak{p}$-action.

Furthermore, there is an analogue of Franke’s theorem for coherent cohomology
of Shimura varieties.

######  Theorem 2.2 ((Su, [Su, Theorem 6.7])).

For any sufficiently refined polyhedral cone decomposition $\Sigma$, there is
a natural Hecke-equivariant isomorphism

$H^{i}(X_{G}^{\Sigma}(\Gamma),[V]^{\operatorname{can}})\cong
H^{i}(\mathfrak{p},K;\mathcal{A}(G)^{\Gamma}\otimes V),$

where $X_{G}^{\Sigma}(\Gamma)$ is the corresponding toroidal compactification,
$[V]^{\operatorname{can}}$ is the canonical extension of $[V]$, and
$\mathcal{A}(G)$ is the space of automorphic forms, namely the space of right
$K$-finite, $Z(\mathfrak{g})$-finite smooth functions on
$G(\mathbb{Q})\backslash G(\mathbb{A})$ of moderate growth.

######  Remark 2.3.

We would be only interested in a part of coherent cohomology localized at a
cuspidal Hecke eigensystem, so it is unlikely that the full power of Su’s
theorem is required.

The calculation of coherent cohomology is therefore about
$(\mathfrak{p},K)$-cohomology of automorphic representations. As far as the
coherent cohomology is concerned, the choice of $\Sigma$ is ineffective, so we
may occasionally drop the superscript $\Sigma$ if there is no confusion.

We will be interested in the situation where a Hecke eigensystem appears in
multiple degrees of coherent cohomology. Namely, for an admissible
$G(\mathbb{A}_{f})$-representation $\Pi_{f}$ and an automorphic vector bundle
$\mathcal{E}$ of $Y_{G}(\Gamma)$, we are interested in

$H^{*}(X_{G}(\Gamma),\mathcal{E}^{\operatorname{can}})[\Pi_{f}].$

By Theorem 2.2, we have a canonical isomorphism

$H^{*}(X_{G}(\Gamma),\mathcal{E}^{\operatorname{can}})\cong\left(\bigoplus_{\Pi_{f}\otimes\Pi_{\infty}\subset\mathcal{A}(G)}H^{*}(\mathfrak{p},K;\Pi_{\infty}\otimes
E)\right)\otimes_{\mathbb{C}}\Pi_{f}^{\Gamma},$

where the sum runs over all automorphic representations with the finite part
being $\Pi_{f}$, and $E$ is the algebraic $P(\mathbb{C})$-representation such
that $[E]=\mathcal{E}$. Thus, it is possible that _several different
$\Pi_{\infty}$’s can appear in the decomposition_, if
$H^{*}(\mathfrak{p},K;\Pi_{\infty}\otimes E)\neq 0$ for several different
$\Pi_{\infty}$’s. Indeed, this can be the case if, for example, some of
$\Pi_{\infty}$ is a nondegenerate limit of discrete series (NLDS) and not a
discrete series (DS); recall that a _nondegenerate limit of discrete series_
is a limit of discrete series whose infinitesimal character is not orthogonal
to any compact root. Unlike the case of “$\delta>0$” as in [PV], the
appearance of single Hecke eigensystem in multiple cohomological degrees in
our setting necessarily implies that, by the following Theorem, there are
_many different archimedean representations involved_ :

######  Theorem-Definition 2.4 ((See [VZ], [Sc2])).

Let $\Pi_{\infty}$ be the $(\mathfrak{g},K)$-module associated to a DS or a
NLDS representation of $G(\mathbb{R})$. Then, there is a unique $0\leq
i\leq\dim X$ and a finite-dimensional irreducible $K$-representation $V$ such
that $H^{i}(\mathfrak{p},K;\Pi_{\infty}\otimes V)\neq 0$. Furthermore,
$\dim_{\mathbb{C}}H^{i}(\mathfrak{p},K;\Pi_{\infty}\otimes V)=1$. We will
denote $i_{\Pi_{\infty}}$ and $V_{\Pi_{\infty}}$ for the $i$ and $V$
corresponding to $\Pi_{\infty}$.

Indeed, the above Theorem says that a single archimedean representation can
only contribute to a single degree. We will see that the _raison d’être_ of
appearance of a Hecke eigensystem in multiple degrees is that the infinity
type $\Pi_{\infty}$ can change in an archimedean $L$-packet without changing
the finite part. We will see in detail in Appendix B how an archimedean
$L$-packet (rather than a single $G(\mathbb{R})$-representation) appears in
the context of motivic action. For now, we move on to the formulation of the
“metric” conjecture which does not involve abstract real group representation
theory nor Lie algebra cohomology. From now on, for the sake of simplicity, we
assume the following

######  Assumption 2.5.

Let $\Pi=\Pi_{f}\otimes\Pi_{\infty}$, with $\Pi_{f}=\prod_{p<\infty}\Pi_{p}$,
be a cuspidal automorphic automorphic representation with $\Pi_{\infty}$ an
NLDS (see Notation)444This would ensure that $F_{\Pi}$, the field of
definition, is a number field, by our assumption in Notation that there exists
a twisting element in the sense of [BG, Definition 5.2.1]. Note that the
twisting element indeed exists for $G=\operatorname{Sp}_{4}$ (as it is split
and has simply-connected derived subgroup) and $\operatorname{SU}(2,1)$ (as
the half-sum of positive roots is integral).. We hereafter assume the
following:

$\Pi_{f}$ is globally generic. For each $p<\infty$, there exists a compact
open subgroup $\Gamma_{p}\leq G(\mathbb{Z}_{p})$ such that
$\dim_{\mathbb{C}}\Pi_{p}^{\Gamma_{p}}=1$. If $G=\operatorname{Sp}_{4}$, a
holomorphic Siegel modular newform $f_{\Pi}$ in
$\Pi_{f}\otimes\Pi_{\infty}^{\operatorname{hol}}$, where
$\Pi_{\infty}^{\operatorname{hol}}$ is a holomorphic (limit of) discrete
series, has a nontrivial special Bessel period $B(f_{\Pi},F)\neq 0$ for some
imaginary quadratic field $F$. Here,
$B(f_{\Pi},F)=\sum_{A=\left(\begin{smallmatrix}a&b/2\\\
b/2&c\end{smallmatrix}\right),4ac-b^{2}=D_{F},A\sim M^{T}AM\text{ for
}M\in\operatorname{SL}_{2}(\mathbb{Z})}a_{A}$, where
$f_{\Pi}(Z)=\sum_{S}a_{S}e^{2\pi i\operatorname{tr}(SZ)}$ is the $q$-expansion
of $f_{\Pi}$, and $-D_{F}$ is the discriminant of $F$. If
$G=\operatorname{SU}(2,1)$, defined using an imaginary quadratic field $K$,
(1) there exists a split place $v$ such that $\Pi_{v}$ is supercuspidal, and
(2) if $p$ is not split in $K$ where $\Pi_{p}$ is also not unramified, either
$\Pi_{p}$ is supercuspidal or the stabilizer of an anisotropic vector
$\operatorname{SU}(1,1)(\mathbb{Q}_{p})\subset\operatorname{SU}(2,1)(\mathbb{Q}_{p})$
is compact.

The purpose of Assumption 2.5 is to isolate the effect of archimedean
$L$-packet phenomenon amongst others and to use existing results on the
refined Gan–Gross–Prasad conjectures. We believe that, for example, it would
be not difficult to formulate the Conjectures without (2.5).

######  Definition 2.6.

Under Assumption 2.5, let $\Gamma(\Pi_{f})=\Gamma:=\prod_{p}\Gamma_{p}$. Also,
when we say a vector $f\in\Pi$ is a _newform_ , it means
$f=\prod_{p}f_{p}\otimes f_{\infty}$ such that, not only
$f_{p}\in\Pi_{p}^{\Gamma_{p}}$, but also $f_{\infty}$ is a highest weight
vector of the minimal $K$-type of $\Pi_{\infty}$ (which exists as we will be
only concerned about either DS or LDS). We define
$\Pi_{f}^{\operatorname{new}}=\prod_{p}\Pi_{p}^{\Gamma_{p}}$, and also
$\Pi_{\infty}^{\operatorname{new}}$ to be the one-dimensional
$\mathbb{C}$-vector subspace of $\Pi_{\infty}$ generated by highest weight
vectors of the minimal $K$-type. In particular, $f$ being a newform means that
$f\in\Pi_{f}^{\operatorname{new}}\otimes\Pi_{\infty}^{\operatorname{new}}$.

In view of Theorem-Definition 2.4, the above “newform” appears in

$H^{i_{\Pi_{\infty}}}(X_{G}(\Gamma),[V_{\Pi_{\infty}}]^{\operatorname{can}})[\Pi_{f}],$

which we denote by $H^{i_{\Pi_{\infty}}}(X)[\Pi_{f}]$.

Note that a choice of a highest weight vector of $V_{\Pi_{\infty}}$ induces a
natural isomorphism

$H^{i_{\Pi_{\infty}}}(\mathfrak{p},K;\Pi_{\infty}\otimes
V_{\Pi_{\infty}})\cong\Pi_{\infty}^{\operatorname{new}},$

defined as follows. Given $v\in\Pi_{\infty}^{\operatorname{new}}$, define a
$K$-homomorphism

$f:\left(\wedge^{i_{\Pi_{\infty}}}(\mathfrak{p}/\mathfrak{k})\right)\otimes
V_{\Pi_{\infty}}^{*}\rightarrow\Pi_{\infty},$

by sending the highest weight vector (induced from the choice of a highest
weight vector of $V_{\Pi_{\infty}}$ and the roots of $\mathfrak{g}$) of the
highest $K$-type of the source to $v$ and sending all other $K$-types to zero.
This defines a class in
$\operatorname{Hom}_{K}(\wedge^{i_{\Pi_{\infty}}}(\mathfrak{p}/\mathfrak{k}),\Pi_{\infty}\otimes
V_{\Pi_{\infty}})$ which is closed in the corresponding Chevalley–Eilenberg
complex for the $(\mathfrak{p},K)$-cohomology, thus a class in
$H^{i_{\Pi_{\infty}}}(\mathfrak{p},K;\Pi_{\infty}\otimes V_{\Pi_{\infty}})$.

######  Remark 2.7.

In this paper, each conjecture will consider a fixed finite type $\Pi_{f}$ and
a coefficient vector bundle $V$, so in that context we firstly fix a choice of
a highest weight vector of $V$ before anything else.

######  Definition 2.8.

For
$f=\prod_{p\leq\infty}f_{p}\in\Pi_{f}^{\operatorname{new}}\otimes\Pi_{\infty}^{\operatorname{new}}$,
we define $[f]\in H^{i_{\Pi_{\infty}}}(X)[\Pi_{f}]$ to be the class
corresponding to

$[f_{\infty}]\otimes\prod_{p<\infty}f_{p}\in
H^{i_{\Pi_{\infty}}}(\mathfrak{p},K;\Pi_{\infty}\otimes
V_{\Pi_{\infty}})\otimes\Pi_{f}^{\operatorname{new}}\subset
H^{i_{\Pi_{\infty}}}(X)[\Pi_{f}],$

where $[f_{\infty}]\in H^{i_{\Pi_{\infty}}}(\mathfrak{p},K;\Pi_{\infty}\otimes
V_{\Pi_{\infty}})$ corresponds to
$f_{\infty}\in\Pi_{\infty}^{\operatorname{new}}$ via the above natural
isomorphism $H^{i_{\Pi_{\infty}}}(\mathfrak{p},K;\Pi_{\infty}\otimes
V_{\Pi_{\infty}})\cong\Pi_{\infty}^{\operatorname{new}}$.

###  2.2. Metrics on cohomology

To formulate the main Conjecture 2.13, we need to define metrics on the
coherent cohomology of Shimura varieties as well as motivic cohomology. To
define metrics on both motivic cohomology and Dolbeault cohomology, we fix an
_admissible bilinear form_ on $\mathfrak{g}_{\mathbb{C}}$ in the following
sense.

######  Definition 2.9 ((Admissible bilinear form)).

An _admissible bilinear form_ $B$ on $\mathfrak{g}_{\mathbb{C}}$ is an
invariant, $\theta$-invariant Hermitian bilinear form on
$\mathfrak{g}_{\mathbb{C}}$ such that the following conditions are satisfied.

  1. (1)

It is a natural extension of a $\mathbb{R}$-valued bilinear form on
$\mathfrak{g}_{\mathbb{R}}$ such that $B(X,\theta(X))$ is negative definite,
where $\theta$ is the Cartan involution that fixes
$\mathfrak{k}_{\mathbb{R}}$.

  2. (2)

It is $\overline{\mathbb{Q}}$-valued on
$\mathfrak{g}_{\overline{\mathbb{Q}}}$.

For example, if $G(\mathbb{R})$ is semisimple, the extention of the Killing
form as a Hermitian bilinear form on $\mathfrak{g}_{\mathbb{C}}$ is an
admissible bilinear form.

Firstly, the coherent cohomology of toroidal compactifications of Shimura
varieties

$H^{*}(X_{G}(\Gamma),[V]^{\operatorname{can}}),$

can be given a Hermitian metric555This coincides with the metric on the Lie
algebra cohomology (see e.g. [BW, §II.2]), and this will be reviewed later in
Appendix B., induced from a Hermitian metric on the Hermitian symmetric domain
and the automorphic vector bundle (which is always possible by the compactness
of $K$, see e.g. [BKK, §5]), which is in turn induced from our choice of
admissible bilinear form on $\mathfrak{g}_{\mathbb{C}}$. The choice of
Hermitian metric induces Hermitian metrics on the entries of Dolbeault complex
$\mathscr{A}^{0,*}([V]^{\operatorname{can}})$. By taking the formal adjoint
$\overline{\partial}^{*}$ of $\overline{\partial}$, one can define the
Laplacian
$\Delta=\overline{\partial}\overline{\partial}^{*}+\overline{\partial}^{*}\overline{\partial}$
on each entry of Dolbeault complex. By Hodge theory, the Dolbeault cohomology
$H^{i}([V]^{\operatorname{can}})$ is identified with the space of _harmonic
$(0,i)$-forms_ $\mathscr{H}^{i}([V]^{\operatorname{can}})$, which is just the
kernel of the Laplacian $\Delta$. The restriction of Hermitian metric on
$\mathscr{A}^{0,i}([V]^{\operatorname{can}})$ to the space of harmonic
$(0,i)$-forms gives rise to a Hermitian metric on the Dolbeault cohomology.

On the other hand, the motivic cohomology that would have to appear in the
motivic action conjectures is that of the adjoint motive. As in [PV, §4.2], we
assume a conjecture on the existence of adjoint motive.

######  Conjecture 2.10.

For $\Pi=\Pi_{f}\otimes\Pi_{\infty}$ satisfying Assumption 2.5, there exists
an adjoint motive $\operatorname{Ad}\Pi$, in the sense of [PV, Definition
4.2.1], over the reflex field $E$ with coefficients in $F_{\Pi}$.

To endow a Hermitian metric on the adjoint motivic cohomology
$H^{1}_{M}((\operatorname{Ad}\Pi)_{\mathcal{O}_{E}},\overline{\mathbb{Q}}(1))$,
consider the Beilinson regulator for $\operatorname{Ad}\Pi$. Recall that the
Beilinson regulator is a map from motivic cohomology to Deligne cohomology,

$H^{1}_{M}((\operatorname{Ad}\Pi)_{\mathcal{O}_{E}},\mathbb{Q}(1))\rightarrow
H_{\mathscr{D}}^{1}((\operatorname{Ad}\Pi)_{\mathbb{R}},\mathbb{R}(1)),$

where the Deligne cohomology group of a motive $M$ over a number field $k$ is
defined as (see [Ra, (6.1.22)])

$H_{\mathscr{D}}^{i}(M_{\mathbb{R}},A)=\prod_{w\text{ complex places of
$k$}}H_{\mathscr{D}}^{i}(M\times_{k,w}\mathbb{C},A)\times\prod_{w\text{ real
places of $k$}}H_{\mathscr{D}}^{i}(M\times_{k,w}\mathbb{R},A).$

Under the Beilinson’s conjectures, the Beilinson regulator gives rise to an
isomorphism

$H^{1}_{M}((\operatorname{Ad}\Pi)_{\mathcal{O}_{E}},\mathbb{Q}(1))\otimes_{\mathbb{Q}}\mathbb{R}\xrightarrow{\sim}H_{\mathscr{D}}^{1}((\operatorname{Ad}\Pi)_{\mathbb{R}},\mathbb{R}(1)).$

######  Remark 2.11.

Note that the Betti realization, on which the Beilinson regulator depends,
_depends on the complex embedding_ $E\hookrightarrow\mathbb{C}$. In our
discussion, we use the preferred embedding that came with the datum of reflex
field3. In the following discussions, we always use this complex embedding.

From §B.3, we know that the target of the Beilinson regulator is identified
with $\widehat{\mathfrak{g}}^{\varphi(W_{\mathbb{C}/\mathbb{R}})}$, where
$\varphi:W_{\mathbb{C}/\mathbb{R}}\rightarrow{}^{L}G$ is the corresponding
Lanvlands parameter, and that this is identified as a Lie subalgebra of
$\widehat{\mathfrak{t}}$. We define a Hermitian bilinear form on
$\widehat{\mathfrak{t}}$ as the dual Hermitian bilinear form of the one we
chose for $\mathfrak{t}$, and this restricts to a Hermitian bilinear form on
the Deligne cohomology.

######  Definition 2.12.

Let $X$ be a finite-dimensional $\overline{\mathbb{Q}}$-vector space, together
with an embedding $\iota:\overline{\mathbb{Q}}\hookrightarrow\mathbb{C}$. A
_Hermitian bilinear form_ on $X$ is a Hermitian metric on
$X\otimes_{\overline{\mathbb{Q}},\iota}\mathbb{C}$.

By [PV, Lemma 2.2.2], the volume of
$H_{M}^{1}((\operatorname{Ad}\Pi)_{\mathcal{O}_{E}},\overline{\mathbb{Q}}(1))$
is in fact independent of choice of the admissible Hermitian bilinear form.

###  2.3. Archimedean motivic action conjecture for Shimura varieties

Now, we are able to state the Archimedean motivic action conjecture for
Shimura varieties as follows.

######  Conjecture 2.13 ((Archimedean motivic action conjecture for Shimura
varieties, metric version)).

Let $\Pi=\Pi_{f}\otimes\Pi_{\infty}$ satisfy Assumption 2.5, with
$\Pi_{\infty}$ being a NLDS but not being a DS. Let
$\mathcal{M}=H_{M}^{1}((\operatorname{Ad}\Pi)_{\mathcal{O}_{E}},\overline{\mathbb{Q}}(1))$
and $\mathcal{H}^{i}=H^{i}(X)[\Pi_{f}]$, where both are regarded as
$\overline{\mathbb{Q}}$-vector spaces equipped with a Hermitian bilinear form
(see Definition 2.12), induced from a fixed admissible Hermitian bilinear form
on $\mathfrak{g}_{\mathbb{C}}$. Then, there is an isomorphism of graded
$\overline{\mathbb{Q}}$-vector spaces equipped with Hermitian metrics,

$\wedge^{*}\mathcal{M}^{*}\otimes\mathcal{H}^{i_{\min}}\cong\bigoplus_{i=i_{\min}}^{i_{\max}}\mathcal{H}^{i},$

where $i_{\min}$ and $i_{\max}$ are the bottom and top degrees, respectively,
of appearance of $\Pi_{f}$ in the cohomology $H^{*}(X)[\Pi_{f}]$.

######  Remark 2.14.

It seems that, to descend the coefficient field from $\overline{\mathbb{Q}}$
to a number field, one may have to take a field larger than $EF_{\Pi}$ (the
compositum of the reflex field and the field of definition of $\Pi_{f}$) even
in the case of $\operatorname{SL}_{2}(\mathbb{Q})$; see [Ho, Corollary 4.6].

The relationship between the above Conjecture and the philosophy of motivic
action conjectures will be fully discussed in Appendix B.

A special subset of the Main conjecture (Conjecture 2.13) concerning the norms
of _top and bottom degrees_ can be formulated without reference to motivic
cohomology, assuming Beilinson’s conjectures for Chow motives [PV, Conjecture
2.1.1].

######  Conjecture 2.15 ((Comparison of top and bottom in Conjecture 2.13)).

Let $\Pi$ be as in Conjceture 2.13. Let $V$ be the automorphic vector bundle
coming from the Levi, such that
$H^{i}(\mathfrak{p},K;V\otimes\Pi_{\infty})\neq 0$ for some $i$, and
$\Pi_{\min},\Pi_{\max}$ be the members of the archimedean $L$-packet of
$\Pi_{\infty}$ such that the degree that $\Pi_{\min}$ ($\Pi_{\max}$,
respectively) has nontrivial $(\mathfrak{p},K)$-cohomology with coefficient in
$V$ is the minimum (maximum, respectively) cohomological degree, denoted
$i_{\min}$ ($i_{\max}$, respectively) in the $L$-packet. Let
$f_{\min}\in\Pi_{f}^{\operatorname{new}}\otimes\Pi_{\min}^{\operatorname{new}}$
and
$f_{\max}\in\Pi_{f}^{\operatorname{new}}\otimes\Pi_{\max}^{\operatorname{new}}$
such that $[f_{\min}],[f_{\max}]\in H^{*}(X)[\Pi_{f}]$ are defined over
$\overline{\mathbb{Q}}$. Then,

$\frac{\langle f_{\min},f_{\min}\rangle_{P}}{\langle
f_{\max},f_{\max}\rangle_{P}}\sim_{\overline{\mathbb{Q}}^{\times}\cap\mathbb{R}}\left|\pi^{i_{\min}-i_{\max}}\frac{L_{\infty}(1,\Pi,\operatorname{Ad})}{L_{\infty}(0,\Pi,\operatorname{Ad})}\cdot\frac{L(1,\Pi,\operatorname{Ad})}{\operatorname{vol}F^{1}H_{\operatorname{dR}}(\operatorname{Ad}\Pi)}\right|^{2},$

where the volume is computed with respect to the metric induced by any weak
polarization (see [PV, §2.2.3])666The volume is independent of the choice of
weak polarization, see [PV, Lemma 2.2.2]..

######  Proposition 2.16.

Assuming the Beilinson conjecture, Conjecture 2.15 is equivalent to the
isometry statement in Conjecture 2.13 for top and bottom degrees,

$\wedge^{\mathrm{top}}\mathcal{M}^{*}\otimes\mathcal{H}^{i_{\min}}\cong\mathcal{H}^{i_{\max}}.$

In particular, if $i_{\max}=i_{\min}+1$, Conjecture 2.13 and Conjecture 2.15
are equivalent.

###### Proof.

The key is to compare the top and the bottom degrees of the graded vector
spaces and relate them with the volumes, which is independent of the choice of
metric. Namely, we need to prove the analogue of [PV, Lemma 2.2.2],

$\operatorname{vol}_{S}H^{1}_{M}((\operatorname{Ad}\Pi)_{\mathcal{O}_{E}},\overline{\mathbb{Q}}(1))\sim_{\overline{\mathbb{Q}}^{\times}}\frac{L^{*}(0,\operatorname{Ad}\Pi)}{\operatorname{vol}_{S}F^{1}H_{\operatorname{dR}}(\operatorname{Ad}\Pi)},$

for $S$ the Hermitian inner product induced by a weak polarization, and the
statement will follow after applying functional equation. This now follows
from the Beilinson’s conjecture over a general number fields, as in [Ra, §6].
Namely, there is a fundamental exact sequence, [Ra, (6.4.2)],

$0\rightarrow
F^{1}H_{\operatorname{dR}}((\operatorname{Ad}\Pi)_{\mathbb{C}})\rightarrow
H_{B}^{0}((\operatorname{Ad}\Pi)_{\mathbb{R}},\mathbb{C})\rightarrow
H_{\mathscr{D}}^{1}((\operatorname{Ad}\Pi)_{\mathbb{R}},\mathbb{C}(1))\rightarrow
0,$

where

$H_{B}^{0}((\operatorname{Ad}\Pi)_{\mathbb{R}},\mathbb{C})=\prod_{w\text{
complex places of
$E$}}H_{B}^{0}(M\times_{E,w}\mathbb{C},\mathbb{C})\times\prod_{w\text{ real
places of $E$}}H_{B}^{0}(M\times_{E,w}\mathbb{R},\mathbb{C}),$

is similarly defined as the “real Deligne cohomology”. The Beilinson’s
conjecture over a general number field says that the determinant of the
fundamental exact sequence (2.3) has incompatible
$\overline{\mathbb{Q}}$-rational structures, and are off precisely by
$L^{*}(0,\operatorname{Ad}\Pi)$ (see [Ra, §6.4]):

$\det(H_{B}^{0}((\operatorname{Ad}\Pi)_{\mathbb{R}},\overline{\mathbb{Q}}))L^{*}(0,\operatorname{Ad}\Pi)\sim_{\overline{\mathbb{Q}}^{\times}}\det
F^{1}H_{\operatorname{dR}}((\operatorname{Ad}\Pi)_{\overline{\mathbb{Q}}})\cdot\det(H_{M}^{1}((\operatorname{Ad}\Pi)_{\mathcal{O}_{E}},\overline{\mathbb{Q}}(1))).$

Regarding this as an equality inside the determinant of the fundamental exact
sequence (2.3), computing the volumes would give the desired statement. ∎

We will later prove Conjecture 2.13 in special cases where Conjecture 2.15 is
equivalent to Conjecture 2.13 (namely, the appearances of Hecke eigensystem
span over two degrees).

######  Example 2.17 ((Sanity check: $\operatorname{SL}_{2}$)).

The simplest example is the case of $G=\operatorname{SL}_{2,\mathbb{Q}}$,
where the conjecture is about _weight one elliptic modular forms_. Let $f\in
S_{1}(\Gamma)$ be a weight one cuspidal new eigenform with Fourier
coefficients in $\overline{\mathbb{Q}}$, generating an automorphic
representation $\Pi$. Then, the complex conjugate
$\overline{f}\in\overline{\Pi}$, which also satisfies
$\langle\overline{f},\overline{f}\rangle_{P}=\langle f,f\rangle_{P}$. Thus, we
are led to the following question: for what $c\in\mathbb{C}^{\times}$ does
$c\overline{f}$ define a $\overline{\mathbb{Q}}$-coherent cohomology class in
$H^{1}(X(\Gamma),\omega)$? After calculating the archimedean $L$-factors
(which is elementary), Conjecture 2.15 says that

$|c|^{-2}\sim_{\overline{\mathbb{Q}}^{\times}\cap\mathbb{R}}\left(\pi^{-2}\frac{L(1,\Pi,\operatorname{Ad})}{\operatorname{vol}F^{1}H_{\operatorname{dR}}(\operatorname{Ad}\Pi)}\right)^{2}.$

Note that $\Pi$ is a pure weight $0$ motive, and so is $\operatorname{Ad}\Pi$;
therefore, $F^{1}H_{\operatorname{dR}}(\operatorname{Ad}\Pi)=0$, which means
that

$|c|\sim_{\overline{\mathbb{Q}}^{\times}\cap\mathbb{R}}\frac{\pi^{2}}{L(1,\Pi,\operatorname{Ad})}.$

It is well-known that
$L(1,\Pi,\operatorname{Ad})\pi^{-2}\sim_{\overline{\mathbb{Q}}^{\times}}\langle
f,f\rangle_{P}$. Thus, the conjecture says

$|c|\sim_{\overline{\mathbb{Q}}^{\times}\cap\mathbb{R}}\frac{\pi}{\langle
f,f\rangle_{P}}.$

Let $c\overline{f}d\overline{z}$ define a Dolbeault cohomology class
$f^{\vee}\in H^{1}(X(\Gamma),\omega)$, which is, by definition, defined over
$\overline{\mathbb{Q}}$. By Serre duality, $\langle
f,f^{\vee}\rangle_{S}\in\overline{\mathbb{Q}}$, where $\langle-,-\rangle_{S}$
denotes the Serre duality.

On the other hand, the Serre duality in this case coincides with Petersson
inner product scaled by the factor of $\frac{1}{2\pi i}$ (e.g. [DMOS, p. 22]),
namely $2\pi i\langle f,f^{\vee}\rangle_{S}=\langle
f,\overline{c}f\rangle_{P}=\overline{c}\langle f,f\rangle_{P}$. Thus, means
that $c\sim_{\overline{\mathbb{Q}}^{\times}}\frac{1}{\langle
f,f\rangle_{P}}$,777Indeed, $\langle f,f\rangle_{P}$ is a real number. which
is consistent with our conjecture. That these facts can be realized as an
instance of motivic action conjecture was realized in [HV] and was explicitly
spelled out in [Ho].

###  2.4. An approach towards the motivic action conjecture

Unlike [PV] or the case of $\operatorname{SL}_{2}$, the top and bottom degrees
are most of the time not complementary (namely,
$i_{\Pi_{\min}}+i_{\Pi_{\max}}\neq\dim_{\mathbb{C}}X$), so the conjecture has
nothing to do with any form of duality. We will nevertheless prove a form of
the conjecture by relating this with _cohomological period integrals_. These
are integral representations of certain $L$-functions that

  * •

apply to automorphic forms appearing in higher coherent cohomology,

  * •

and admit an interpretation as cup product pairing in coherent cohomology.

This is useful in verifying our conjectures as the coherent cohomological cup
product can _detect rationality of higher coherent cohomology classes_.

We will show that a strategy similar to [PV, §7] can also show that our period
conjecture, Conjecture 2.15, is true in certain cases.

######  Theorem 2.18.

Let $G$ be either $\operatorname{Sp}_{4}$ or $\operatorname{SU}(2,1)$, and let
$\Pi$ be a globally generic cuspidal automorphic representation of
$G(\mathbb{A}_{\mathbb{Q}})$ satisfying Assumption 2.5. Assume the working
hypothesis on periods, Assumption 2.22. Then, Conjecture 2.15 is true, up to
the factor of an archimedean zeta integral (see Remarks 3.6, 4.5).

In the two cases, we will compare periods of _holomorphic LDS_ appearing in
$H^{0}$ and _generic LDS_ appearing $H^{1}$. The tools that we will use are
summarized in the following table.

LDS type | Detecting rationality | Relation with Petersson norm
---|---|---
Holomorphic ($H^{0}$) | Rational Fourier coefficient |  | Doubling method, Refined GGP conjectures
---
Generic ($H^{>0}$) | Cohomological period integrals | Lapid–Mao conjecture

A slightly more detailed outline is as follows. We would need to know how
$\overline{\mathbb{Q}}$-algebraicity of coherent cohomology classes in $H^{0}$
and $H^{1}$ is related to Petersson norms. For $H^{0}$, the classes are
represented by holomorphic automorphic forms, where their algebraicity is
detectable by Fourier coefficients. In the language of periods, these are
related to Bessel or Fourier–Jacobi periods, which are related to Petersson
norms via the refined Gan–Gross–Prasad conjectures. For $H^{1}$, the classes
are represented by generic automorphic forms888In both cases of our concern,
the infinity type corresponding to $H^{1}$ belongs to a generic (L)DS, which
is a numerical coincidence that only happens in certain special examples.. The
Petersson norms of corresponding automorphic forms are related to the
_Whittaker periods_ via the _Lapid–Mao conjecture_ [LM]. Due to its simple
statement, we recall the conjecture here:

######  Conjecture 2.19 (([LM])).

Let $\Pi$ be a globally generic representation, satisfying (2.5) of Assumption
2.5. Let $f=\otimes_{p}f_{p}\in\Pi$ be a newform. Then,

$\frac{\langle
f,f\rangle}{|W(1)|^{2}}\sim_{\mathbb{Q}^{\times}}\frac{L(1,\Pi,\operatorname{Ad})}{\Delta_{G}(1)|W_{\infty}(1)|^{2}},$

where $\langle,\rangle$ is the $L^{2}$-norm (as opposed to the Petersson
norm), $\Delta_{G}(s)$ is the $L$-function of the dual to the Artin motive
attached to $G$ as defined in [Gr], and $W$ ($W_{\infty}$, respectively) is
the Whittaker function of $f$ ($f_{\infty}$, respectively).

The Whittaker periods are then related to $\overline{\mathbb{Q}}$-algebraicity
of coherent cohomology classes via cohomological period integrals. On one
hand, the period integral has automorphic interpretation, which connects to
Whittaker periods. On the other hand, the period integral has cohomological
interpretation, so that in particular it descends to $\overline{\mathbb{Q}}$,
hence detects $\overline{\mathbb{Q}}$-algebraicity.

Finally, as the conjectures are formulated using invariants coming from the
motivic formalism, we would be working with the corresponding motives and
compute motivic invariants (dubbed “Hodge-linear algebra”). We will work with
the adjoint motive, as in Conjecture 2.10, as well as the motive of the given
automorphic representation999The construction of those motives is a subtle
matter, as we work with limits of discrete series. Indeed, the corresponding
Galois representations have been constructed, but only by using congruences.:

######  Conjecture 2.20.

For $\Pi$ as in Assumption 2.5, there exists a motive $M_{\Pi}$ that is
uniquely characterized by [Cl, §4.3.3].

######  Remark 2.21.

The adjoint motive of Conjecture 2.10 is the motive associated to
$L(s,\Pi,\operatorname{Ad})$ in the above sense.

Thus, the “working hypothesis” is as follows.

######  Assumption 2.22 ((Working hypothesis)).

We assume the following conjectures101010Even though the refined
Gan–Gross–Prasad conjectures are being mentioned throughout the paper, they
are not required as an assumption, because we have an unconditional
alternative result using the doubling method (and they are equivalent if one
assumes Beilinson’s conjectures anyways).. There are numerous instances of
these conjectures being verified, and we do not attempt to list them here.

  1. (1)

_Beilinson’s conjecture for Chow motives_ , Conjecture 2.1.1 of [PV].

  2. (2)

_Lapid–Mao conjecture_ , Conjecture 2.19.

  3. (3)

_Existence of motives_ , Conjecture 2.10 and Conjecture 2.20.

##  3\. Evidence I: The case of $\operatorname{Sp}_{4}$

We first provide the evidence for the motivic action conjecture (Conjecture
2.13) for the case of certain irregular automorphic forms on
$G(\mathbb{A})=\operatorname{Sp}_{4}(\mathbb{A})$. In this case, we will use
an integral reprsentation of the spinor $L$-function by Novodvorsky ([No]),
whose coherent cohomological interpretation was given by [LPSZ].

Let $\Pi$ be a globally generic cuspidal automorphic representation of
$G=\operatorname{GSp}_{4}(\mathbb{A}_{\mathbb{Q}})$, namely that the Whittaker
transform (see §1.6, Notation) defines a realization of $\Pi$ as a space of
functions on $G(\mathbb{Q})\backslash G(\mathbb{A})$ satisfying a
transformation property under the $N(\mathbb{A})$-action. We choose unramified
vectors $\varphi_{v}^{0}\in\Pi_{v}$ for $v$ finite with $\Pi_{v}$ unramified
such that, if $\psi_{v}$ is unramified, $W_{\varphi_{v}^{0}}(1)=1$. Suppose
that $\Pi$ satisfies Assumption 2.5.

######  Definition 3.1.

Let $M\supset F_{\Pi}$ be a number field. A $\psi$-Whittaker function $W$ on
$G$ is called to be _defined over $M$_ if it takes values in $M(\mu_{\infty})$
and satisfies

$\sigma(W(g))=W(w(\kappa(\sigma))g),$

for all $g\in G(\mathbb{A}_{f})$ and
$\sigma\in\operatorname{Gal}(\overline{\mathbb{Q}}/M)$, where
$w(x)=\operatorname{diag}(x^{3},x^{2},x,1)$ and
$\kappa:\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\rightarrow\widehat{\mathbb{Z}}^{\times}$
is the cyclotomic character.

######  Remark 3.2.

  1. (1)

The convention is made so that, if $\psi$ is unramified, the unramified
$\psi$-Whittaker function $W$ with $W(1)=1$ is defined over $F_{\Pi}$. As
$L(s,\Pi,\operatorname{Ad})$ is regular at $s=1$, $W(1)$ is nonzero by [LM,
§3.1]. Thus, $W$ is defined over $M$ if and only if $W(1)\in M^{\times}$.

  2. (2)

Indeed, our conditions on $G$ and $\Pi$ imply that $F_{\Pi}$ is a number field
(see Notation). Thus, the space of $\psi$-Whittaker functions, which naturally
has an action of $\operatorname{Aut}(\mathbb{C}/\mathbb{Q})$ as in [GHL,
§4.1], is fixed by $\operatorname{Aut}(\mathbb{C}/F_{\Pi})$. Thus, there
exists a nontrivial $\psi$-Whittaker function defined over $M$.

###  3.1. Whittaker periods via cohomological period integrals

The purpose of this subsection is to prove the following

######  Theorem 3.3.

Let
$\varphi\in\Pi^{\operatorname{new}}=\Pi_{f}^{\operatorname{new}}\otimes\Pi_{\infty}^{\operatorname{new}}$.
Let $F$ be an imaginary quadratic field such that the special Bessel period of
$\Pi$ for $F$ is not identically zero (see (2.5) of Assumption 2.5). If
$[\varphi]$ (see Definition 2.8) is defined over a number field
$F^{\prime}\supset FF_{\Pi}$, then
${\Lambda\left(1/2,\Pi\right)\Lambda\left(1/2,\Pi\otimes\chi_{F}\right)}W_{\varphi}$
is a nontrivial $\psi$-Whittaker function defined over $F^{\prime}$, where
$\chi_{F}$ is the quadratic character associated to $F$.

###### Proof.

Let $C\in\mathbb{C}^{\times}$ be such that $\frac{W_{\varphi}}{C}$ is defined
over $F^{\prime}$, which exists by Remark 3.2(2). We would like to show that

$C\Lambda(1/2,\Pi)\Lambda(1/2,\Pi\otimes\chi_{F})\in F^{\prime}{}^{\times}.$

We first prove that the above quantity is nonzero. This is because, by [FuMo,
(1.26)], the nonvanishing of $L(1/2,\Pi)L(1/2,\Pi\otimes\chi_{F})$ is
equivalent to nonvanishing of the special Bessel period $B(f_{\Pi},F)$ in
(2.5) of Assumption 2.5; that the special Bessel period defined in this paper
is the Bessel period (up to explicit nonzero scalar) in [FuMo] follows from
the calculation of [DPSS, Proposition 3.5].

We now consider Novodvorsky’s integral representation of spinor $L$-function.
This is, roughly speaking, the period integral of $\varphi$ times an
Eisenstein series over an embedded product of two modular curves. More
precisely, there is an embedding of
$H=\operatorname{GL}_{2}\times_{\operatorname{GL}_{1}}\operatorname{GL}_{2}$
into $G$, which is most naturally thought as
$\operatorname{SO}(2,2)\hookrightarrow\operatorname{SO}(3,2)$. Let $B\subset
H$ be the upper triangular Borel. For
$\Phi_{1},\Phi_{2}:\mathbb{A}^{2}\rightarrow\mathbb{C}$ and
$\chi_{1},\chi_{2}$ a unitary Grössencharcter, we define an Eisenstein series
on $H$ with respect to $B$, $E(h,\chi_{i},\Phi_{i},s_{i})$. The following are
well-known (e.g. [LPSZ, Proposition 7.3]).

######  Proposition 3.4.

Suppose we are taking the “weight $k$-section” for $\Phi_{i,\infty}$.

  1. (1)

For $-\frac{k}{2}+1\leq s_{1},s_{2}\leq\frac{k}{2}$ half-integral, equivalent
modulo $1$ to $\frac{k}{2}$, this defines a nearly holomorphic form on $H$,
which can be thought as an $H^{0}$ class of some automorphic vector bundle
over a Shimura variety for $H$.

  2. (2)

If $\chi_{i}$’s and $\Phi_{i}$’s are valued in a number field $F$, then the
$H^{0}$ class is defined over $F$.

Let $(\lambda_{1},\lambda_{2})$ be the Harish–Chandra character of the
(necessarily generic) (L)DS $\Pi_{\infty}$. Let $\Gamma_{H}=H\cap\Gamma$, and
let $i:X_{H}(\Gamma_{H})\hookrightarrow X_{G}(\Gamma)$ be the closed embedding
of certain toroidal compactifications of the corresponding Shimura
varieties111111That the closed immertion of open Shimura varieties extends to
a closed immersion of toroidal compactifications with respect to certain
refinements of polyhedral cone decomposition is achieved by [Lan2], but the
situation is simplier in this case, because $X_{H}$, being a productof modular
curves, is unique.. Then, Novodvorsky’s integral representation can be
understood via a cup product pairing

$H^{2}(X_{G},V)\otimes_{\mathbb{C}}H^{0}(X_{H},W)\xrightarrow{(\operatorname{id},i_{*})}H^{2}(X_{G},V)\otimes_{\mathbb{C}}H^{1}(X_{G},W^{\prime})\xrightarrow{\cup}H^{3}(X_{G},V\otimes
W^{\prime})\xrightarrow{S}H^{0}(X_{G},\mathcal{O})=\mathbb{C},$

where $\cup$ is the cohomological cup product and $S$ is the Serre duality
pairing, induced from a morphism of algebraic
$\overline{\mathbb{Q}}$-representations of $K_{\infty}$, $V\otimes
W^{\prime}\rightarrow\mathfrak{g}^{-1,1}$ (namely, the pairing is normalized
such that, on the level of representations of $K_{\infty}$, the
$\overline{\mathbb{Q}}$-structures are compatible). Here, $W$ and $W^{\prime}$
are certain automorphic vector bundles over $X_{H}$ and $X_{G}$, respectively,
corresponding to $[V_{H}(\lambda_{1}-\lambda_{2}-1,0)]\otimes\omega_{H}(1,1)$
and $[\widetilde{L}_{1}]$, if we use the notation of [LPSZ, §6]. The
reinterpretation, done in [LPSZ, §7.4], of Novodvosky’s integral asserts that,
given an imaginary quadratic field $F$, there is a coherent cohomology class
$[E]\in H^{0}(X_{H}(\Gamma_{H}),W)$ defined over $FF_{\Pi_{f}}$, which
corresponds to a nearly-holomorphic Eisenstein series $E$ under the Hodge
splitting of [LPSZ, §6.3], such that

$\langle[\varphi],[E]\rangle=C\Lambda\left(1/2,\Pi\right)\Lambda\left(1/2,\Pi\otimes\chi_{F}\right),$

where $\chi_{F}$ is the Hecke character corresponding to $F$. As
$\langle,\rangle$ is defined over $\mathbb{Q}$, the left hand side is in
$F^{\prime}$, which is what we are looking for. ∎

###  3.2. Hodge-linear algebra

We now put the relevant Hodge-linear algebra that proves the case of
$\operatorname{Sp}_{4}$. For the sake of simplicity, we assume we work with a
parallel weight $(2,2)$ Siegel modular form, or Harish–Chandra parameter
$(1,0)$, although the Hodge-linear algebra calculation stays the same for
general weights. Our goal is to convert, using elementary linear algebra,
$\operatorname{vol}F^{1}H_{\operatorname{dR}}(\operatorname{Ad}M)$ into an
expression that involves Deligne’s periods $c^{+}(M),c^{-}(M),\delta(M)$ whose
definitions will be recalled later. We will then be able to express
$\operatorname{vol}F^{1}H_{\operatorname{dR}}(\operatorname{Ad}M)$ with
$L$-values, using Deligne’s conjectures. We will prove

######  Proposition 3.5.

For a motive $M$ associated to $\Pi$ (in the sense of Conjecture 2.20), we
have

$\operatorname{vol}F^{1}H_{\operatorname{dR}}(\operatorname{Ad}M)\sim_{\overline{\mathbb{Q}}^{\times}}\frac{\sqrt{c^{+}(M)c^{-}(M)}^{3}}{\delta(M)^{3/2}}.$

###### Proof.

The motive $M$ of $\Pi$ should be of rank $4$ and weight $1$, with the Hodge
decomposition

$H_{B}(M)\otimes_{\mathbb{Q}}\mathbb{C}=H^{1,0}(M)\oplus
H^{0,1}(M),\quad\dim_{\mathbb{C}}H^{1,0}(M)=\dim_{\mathbb{C}}H^{0,1}(M)=2,$

which is of the type of the Hodge structure defined by the corresponding
archimedean Langlands parameter. In this case,

$\delta(M)\text{ }\left(\text{$c^{\pm}(M)$, resp.}\right)\text{
}\in\mathbb{C}^{\times}/\mathbb{Q}^{\times},$

is the determinant of the comparison map

$H_{B}(M)\otimes\mathbb{C}\xrightarrow{\sim}H_{\operatorname{dR}}(M)\otimes\mathbb{C}\text{
}\left(\text{$H_{B}(M)^{+}\otimes\mathbb{C}\rightarrow
H_{B}(M)\otimes\mathbb{C}\xrightarrow{\sim}H_{\operatorname{dR}}(M)\otimes\mathbb{C}\twoheadrightarrow(H_{\operatorname{dR}}(M)/F^{1}H_{\operatorname{dR}}(M))\otimes\mathbb{C}$,
resp.}\right),$

with respect to the bases coming from the underlying $\mathbb{Q}$-structures
on both sides.

Let $e_{1}^{+},e_{2}^{+}$ be a $\mathbb{Q}$-basis of $H_{B}(M)^{+}$ and
$e_{1}^{-},e_{2}^{-}$ be a $\mathbb{Q}$-basis of $H_{B}(M)^{-}$. Let
$f_{1},f_{2}\in F^{1}H_{\operatorname{dR}}(M)$ be a $\mathbb{Q}$-basis, and
$g_{1},g_{2}\in H_{\operatorname{dR}}(M)/F^{1}H_{\operatorname{dR}}(M)$ be a
$\mathbb{Q}$-basis, and $\widetilde{g_{1}},\widetilde{g_{2}}\in
H_{\operatorname{dR}}(M)$ be lifts of $g_{1},g_{2}$. Given two
$\mathbb{C}$-bases of $H_{B}(M)\otimes\mathbb{C}\cong
H_{\operatorname{dR}}(M)\otimes\mathbb{C}$, we can write an expression

$\begin{pmatrix}e_{1}^{+}&e_{2}^{+}&e_{1}^{-}&e_{2}^{-}\end{pmatrix}=\begin{pmatrix}f_{1}&f_{2}&\widetilde{g_{1}}&\widetilde{g_{2}}\end{pmatrix}\begin{pmatrix}A&B\\\
C&D\end{pmatrix},$

for $A,B,C,D\in M_{2}(\mathbb{C})$. Note that by definition
$\delta(M)=\det\left(\begin{smallmatrix}A&B\\\ C&D\end{smallmatrix}\right)$.

We have canonical isomorphisms

$H^{1,0}(M)\cong F^{1}H_{\operatorname{dR}}(M)\otimes\mathbb{C},\quad
H^{0,1}(M)\cong\frac{H_{\operatorname{dR}}(M)}{F^{1}H_{\operatorname{dR}}(M)}\otimes\mathbb{C}.$

Let $f_{1,B},f_{2,B}\in H^{1,0}(M)$ and $g_{1,B},g_{2,B}\in H^{0,1}(M)$ be the
images of $f_{1},f_{2},g_{1},g_{2}$ under the above canonical isomorphisms.
Then $f_{i,B}$ and $f_{i}$ coincide as elements of
$H_{B}(M)\otimes\mathbb{C}\cong H_{\operatorname{dR}}(M)\otimes\mathbb{C}$,
whereas $\widetilde{g_{i}}-g_{i,B}\in H^{1,0}(M)$. So,

$\begin{pmatrix}f_{1}&f_{2}&\widetilde{g_{1}}&\widetilde{g_{2}}\end{pmatrix}=\begin{pmatrix}f_{1,B}&f_{2,B}&g_{1,B}&g_{2,B}\end{pmatrix}\begin{pmatrix}1_{2}&M\\\
0_{2}&1_{2}\end{pmatrix}.$

In particular, if we write

$\begin{pmatrix}e_{1}^{+}&e_{2}^{+}&e_{1}^{-}&e_{2}^{-}\end{pmatrix}=\begin{pmatrix}f_{1,B}&f_{2,B}&g_{1,B}&g_{2,B}\end{pmatrix}\begin{pmatrix}A^{\prime}&B^{\prime}\\\
C^{\prime}&D^{\prime}\end{pmatrix},$

then $\det\left(\begin{smallmatrix}A&B\\\
C&D\end{smallmatrix}\right)=\det\left(\begin{smallmatrix}A^{\prime}&B^{\prime}\\\
C^{\prime}&D^{\prime}\end{smallmatrix}\right)$.

Also, there must be relations

$c_{B}(f_{1,B})=ag_{1,B}+bg_{2,B},$ $c_{B}(f_{2,B})=cg_{1,B}+dg_{2,B}.$

Using that $F_{\infty}e_{i}^{+}=e_{i}^{+}$, $F_{\infty}e_{i}^{-}=-e_{i}^{-}$
and that $F_{\infty}(f_{i,B})=c_{B}(f_{i,B})$ and
$F_{\infty}(g_{i,B})=c_{B}(g_{i,B})$ (as in [PV, Lemma, §8.2.1]), one has

$\displaystyle c_{B}(A^{\prime}_{1i}f_{1,B}+A^{\prime}_{2i}f_{2,B})$
$\displaystyle=$ $\displaystyle
C^{\prime}_{1i}g_{1,B}+C^{\prime}_{2i}g_{2,B},$ $\displaystyle
c_{B}(B^{\prime}_{1i}f_{1,B}+B^{\prime}_{2i}f_{2,B})$ $\displaystyle=$
$\displaystyle-(D^{\prime}_{1i}g_{1,B}+D^{\prime}_{2i}g_{2,B}),$

for $i=1,2$. This can be packaged into
$C^{\prime}=\left(\begin{smallmatrix}a&c\\\
b&d\end{smallmatrix}\right)A^{\prime}$,
$D^{\prime}=-\left(\begin{smallmatrix}a&c\\\
b&d\end{smallmatrix}\right)B^{\prime}$. So

$\delta(M)=\det\begin{pmatrix}A^{\prime}&B^{\prime}\\\
\left(\begin{smallmatrix}a&c\\\
b&d\end{smallmatrix}\right)A^{\prime}&-\left(\begin{smallmatrix}a&c\\\
b&d\end{smallmatrix}\right)B^{\prime}\end{pmatrix}=\det\begin{pmatrix}A^{\prime}&B^{\prime}\\\
0_{2}&-2\left(\begin{smallmatrix}a&c\\\
b&d\end{smallmatrix}\right)B^{\prime}\end{pmatrix}=4\det\begin{pmatrix}a&c\\\
b&d\end{pmatrix}\det A^{\prime}\det B^{\prime}.$

Note on the other hand that $c^{+}(M)=\det C$, $c^{-}(M)=\det D$. As
$C^{\prime}=C$, $D^{\prime}=D$, we see that

$c^{+}(M)c^{-}(M)\sim_{\overline{\mathbb{Q}}^{\times}}\delta(M)\det\begin{pmatrix}a&c\\\
b&d\end{pmatrix}.$

Note that, as $M$ is self-dual, $M^{\vee}\cong M(1)$, which implies
$\operatorname{Ad}M=(\operatorname{Ad}M)^{*}=(\operatorname{Sym}^{2}M)(1)$. We
are also led to calculate
$\operatorname{vol}F^{1}H_{\operatorname{dR}}(\operatorname{Ad}M)$ (we know it
does not depend on the choice of weak polarization up to
$\overline{\mathbb{Q}}^{\times}$-ambiguity). We know that
$(\operatorname{vol}F^{1}H_{\operatorname{dR}}(\operatorname{Ad}M))^{2}\sim_{\overline{\mathbb{Q}}^{\times}}\lambda$,
where $\varphi(c_{B}(v^{+}))=\lambda v^{-}$. Here, $v^{+},v^{-}$ are
$\mathbb{Q}$-bases vectors for $\det
F^{1}H_{\operatorname{dR}}(\operatorname{Ad}M)$ and
$\det\left(\frac{H_{\operatorname{dR}}(\operatorname{Ad}M)}{F^{0}H_{\operatorname{dR}}(\operatorname{Ad}M)}\right)$,
respectively, and $\varphi:\wedge^{\dim
F^{1}H_{\operatorname{dR}}(\operatorname{Ad}M)}(H_{\operatorname{dR}}(M)\otimes\mathbb{C})\rightarrow\det\left(\frac{H_{\operatorname{dR}}(\operatorname{Ad}M)}{F^{0}H_{\operatorname{dR}}(\operatorname{Ad}M)}\right)$
is the natural projection. As
$\operatorname{Ad}M=(\operatorname{Sym}^{2}M)(1)$, we can take
$f_{1}^{2},f_{1}f_{2},f_{2}^{2}$ and $g_{1}^{2},g_{1}g_{2},g_{2}^{2}$ as
$\mathbb{Q}$-bases of $F^{1}H_{\operatorname{dR}}(\operatorname{Ad}M)$ and
$\frac{H_{\operatorname{dR}}(\operatorname{Ad}M)}{F^{0}H_{\operatorname{dR}}(\operatorname{Ad}M)}$,
respectively. Now from the known relations,

$\displaystyle c_{B}(f_{1}^{2})$ $\displaystyle\equiv$ $\displaystyle
a^{2}g_{1}^{2}+2abg_{1}g_{2}+b^{2}g_{2}^{2},$ $\displaystyle
c_{B}(f_{1}f_{2})$ $\displaystyle\equiv$ $\displaystyle
acg_{1}^{2}+(ad+bc)g_{1}g_{2}+bdg_{2}^{2},$ $\displaystyle c_{B}(g_{1}g_{2})$
$\displaystyle\equiv$ $\displaystyle
c^{2}g_{1}^{2}+2cdg_{1}g_{2}+d^{2}g_{2}^{2},$

where $\equiv$ is mod
$F^{0}H_{\operatorname{dR}}(\operatorname{Ad}M)\otimes\mathbb{C}$. So

$\lambda=\det\begin{pmatrix}a^{2}&2ab&b^{2}\\\ ac&ad+bc&bd\\\
c^{2}&2cd&d^{2}\end{pmatrix}=(ad-bc)^{3}.$

Therefore, we obtain

$\operatorname{vol}F^{1}H_{\operatorname{dR}}(\operatorname{Ad}M)\sim_{\overline{\mathbb{Q}}^{\times}}\frac{\sqrt{c^{+}(M)c^{-}(M)}^{3}}{\delta(M)^{3/2}}.$

∎

###  3.3. Completion of the proof

###### Proof of Theorem 2.18 for $\operatorname{Sp}_{4}$.

Now we apply the relevant period conjectures for this case. Let
$f_{\operatorname{hol}}$, $f_{\operatorname{gen}}$ be newforms (see Assumption
2.5) in $\Pi_{f}\otimes\Pi_{\operatorname{hol}}$,
$\Pi_{f}\otimes\Pi_{\operatorname{gen}}$, respectively, where
$\Pi_{\operatorname{hol}}$ is the corresponding holomorphic NLDS in the
$L$-packet of $\Pi_{\operatorname{gen}}$. Let us assume that
$[f_{\operatorname{hol}}]$ and $[f_{\operatorname{gen}}]$ are defined over
$\overline{\mathbb{Q}}$. Let $F$ be the imaginary quadratic field as in
Theorem 3.3. Then, a theorem of Furusawa–Morimoto121212See [FuMo, Theorem 1]
for the case of discrete series; the case of limit of discrete series is also
an upcoming work of them. 131313This is a specific case of the refined
Gan–Gross–Prasad conjecture for Bessel periods, [Li, Conjecture 2.5]. This
special case is also sometimes called _Böcherer’s conjecture_. implies that

$\frac{\langle
f_{\operatorname{hol}},f_{\operatorname{hol}}\rangle}{|B(f_{\operatorname{hol}},F)|^{2}}\sim_{\overline{\mathbb{Q}}^{\times}}\pi^{-6}\frac{L(1,\Pi,\operatorname{Ad})}{L(1/2,\Pi)L(1/2,\Pi\otimes{\chi_{F}})}.$

Since $[f_{\operatorname{hol}}]$(see Definition 2.8) is defined over
$\overline{\mathbb{Q}}$, we know that $B(f_{\operatorname{hol}},F)$, a
$\mathbb{Q}$-linear combination of Fourier coefficients of
$f_{\operatorname{hol}}$, is in $\overline{\mathbb{Q}}$. Thus, in the setting
of Theorem 2.18, $\langle
f_{\operatorname{hol}},f_{\operatorname{hol}}\rangle\sim_{\overline{\mathbb{Q}}^{\times}}\pi^{-6}\frac{L(1,\Pi,\operatorname{Ad})}{L(1/2,\Pi)L(1/2,\Pi\otimes\chi_{F})}$.

We now relate the $\overline{\mathbb{Q}}$-rationality of
$f_{\operatorname{gen}}$ with $\langle
f_{\operatorname{gen}},f_{\operatorname{gen}}\rangle$. Note that Theorem 3.3
is about the relationship between rationality of Whittaker functions and that
of coherent cohomology classes, for the anti-generic LDS (namely, those
appearing in $H^{2}$ of coherent cohomology). Thus, as
$[f_{\operatorname{gen}}]$ is defined over $\overline{\mathbb{Q}}$, by Serre
duality,

$\langle f_{\operatorname{gen}},f_{\operatorname{gen}}\rangle_{P}=(2\pi
i)^{3}\langle[f_{\operatorname{gen}}],[\overline{f}_{\operatorname{gen}}]\rangle_{\operatorname{coh}},$

where $\langle-,-\rangle_{\operatorname{coh}}$ is the cohomological cup
product defined over $\overline{\mathbb{Q}}$ normalized so that it is induced
from the $\overline{\mathbb{Q}}$-morphism of algebraic
$K_{\infty}$-representations
$W\otimes\operatorname{Hom}(W,\mathfrak{g}^{-1,1})\rightarrow\mathfrak{g}^{-1,1}$.
If we let $C\in\mathbb{C}^{\times}$ be such that
$C[\overline{f}_{\operatorname{gen}}]$ is defined over
$\overline{\mathbb{Q}}$, we see that the RHS is
$\sim_{\overline{\mathbb{Q}}^{\times}}\pi^{3}C^{-1}$. On the other hand, by
Theorem 3.3,

$\Lambda(1/2,\Pi)\Lambda(1/2,\Pi\otimes\chi_{F})W_{C\overline{f}_{\operatorname{gen}}}=C\Lambda(1/2,\Pi)\Lambda(1/2,\Pi\otimes\chi_{F})W_{\overline{f}_{\operatorname{gen}}},$

is defined over $\overline{\mathbb{Q}}$. We now invoke the Lapid–Mao
conjecture, Conjecture 2.19:

$\frac{\langle
f_{\operatorname{gen}},f_{\operatorname{gen}}\rangle}{|W(1)|^{2}}\sim_{\overline{\mathbb{Q}}^{\times}}\pi^{9}\cdot\frac{L(1,\Pi,\operatorname{Ad})}{|W_{\infty}(1)|^{2}}.$

By Remark 3.2(1), $W_{\overline{f}_{\operatorname{gen}}}(1)\neq 0$, so that

$C\Lambda(1/2,\Pi)\Lambda(1/2,\Pi\otimes\chi_{F})W_{\overline{f}_{\operatorname{gen}}}(1)\in\overline{\mathbb{Q}}^{\times}.$

Since
$W_{\overline{f}_{\operatorname{gen}}}(1)=\overline{W_{f_{\operatorname{gen}}}(1)}$,
we have

$\pi^{9}\frac{L(1,\Pi,\operatorname{Ad})}{|W_{\infty}(1)|^{2}}|W_{f_{\operatorname{gen}}}(1)|^{2}\mathrel{\overset{\makebox{\text{\tiny
Lapid--Mao}}}{\sim_{\overline{\mathbb{Q}}^{\times}}}}\langle
f_{\operatorname{gen}},f_{\operatorname{gen}}\rangle\mathrel{\overset{\makebox{\text{\tiny(B)\quad}}}{\sim_{\overline{\mathbb{Q}}^{\times}}}}\pi^{3}C^{-1}\mathrel{\overset{\makebox{\text{\tiny(C)\quad}}}{\sim_{\overline{\mathbb{Q}}^{\times}}}}\pi^{3}\overline{W_{f_{\operatorname{gen}}}(1)}\Lambda(\frac{1}{2},\Pi)\Lambda(\frac{1}{2},\Pi\otimes\chi_{F}),$

or

$W_{f_{\operatorname{gen}}}(1)\sim_{\overline{\mathbb{Q}}^{\times}}\pi^{-6}\frac{|W_{\infty}(1)|^{2}\Lambda(\frac{1}{2},\Pi)\Lambda(\frac{1}{2},\Pi\otimes\chi_{F})}{L(1,\Pi,\operatorname{Ad})}\sim_{\overline{\mathbb{Q}}^{\times}}\pi^{-10}\frac{|W_{\infty}(1)|^{2}L(\frac{1}{2},\Pi)L(\frac{1}{2},\Pi\otimes\chi_{F})}{L(1,\Pi,\operatorname{Ad})}.$

We need to compute $\frac{\langle
f_{\operatorname{hol}},f_{\operatorname{hol}}\rangle_{P}}{\langle
f_{\operatorname{gen}},f_{\operatorname{gen}}\rangle_{P}}$, which can be now
seen as follows.

$\displaystyle\frac{\langle
f_{\operatorname{hol}},f_{\operatorname{hol}}\rangle_{P}}{\langle
f_{\operatorname{gen}},f_{\operatorname{gen}}\rangle_{P}}$
$\displaystyle\mathrel{\overset{\makebox{\text{\tiny Lapid--Mao +
(A)}}}{\sim_{\overline{\mathbb{Q}}^{\times}}}}\frac{\pi^{-6}\frac{L(1,\Pi,\operatorname{Ad})}{L(1/2,\Pi)L(1/2,\Pi\otimes\chi_{F})}}{\pi^{9}|W_{f_{\operatorname{gen}}}(1)|^{2}\frac{L(1,\Pi,\operatorname{Ad})}{|W_{\infty}(1)|^{2}}}$
$\displaystyle=\pi^{-15}\frac{|W_{\infty}(1)|^{2}}{|W_{f_{\operatorname{gen}}}(1)|^{2}L(1/2,\Pi)L(1/2,\Pi\otimes\chi_{F})}$
$\displaystyle\mathrel{\overset{\makebox{\text{\tiny(D)}}}{\sim_{\overline{\mathbb{Q}}^{\times}}}}\pi^{-15}\frac{|W_{\infty}(1)|^{2}}{\pi^{-20}\frac{|W_{\infty}(1)|^{4}L(1/2,\Pi)^{3}L(1/2,\Pi\otimes\chi_{F})^{3}}{L(1,\Pi,\operatorname{Ad})^{2}}}$
(E)
$\displaystyle\sim_{\overline{\mathbb{Q}}^{\times}}\frac{\pi^{9}}{|W_{\infty}(1)|^{2}}\left(\frac{L_{\infty}(1,\Pi,\operatorname{Ad})}{L_{\infty}(0,\Pi,\operatorname{Ad})}\cdot\frac{L(1,\Pi,\operatorname{Ad})}{\frac{\sqrt{L(1/2,\Pi)L(1/2,\Pi\otimes\chi_{\mathbb{Q}(i)})}^{3}}{\pi^{3}}}\right)^{2},$

which involves elementary calculation of archimedean $L$-factors.

On the other hand, by Deligne’s conjectures, Proposition 3.5 implies that

$\operatorname{vol}F^{1}H_{\operatorname{dR}}(\operatorname{Ad}M)\sim_{\overline{\mathbb{Q}}^{\times}}\frac{\sqrt{L(1/2,\Pi)L(1/2,\Pi\otimes\chi_{\mathbb{Q}(i)})}^{3}}{\pi^{3}}.$

Thus, the parenthized term in (E) is precisely the RHS of Conjecture 2.15,
which finishes the proof. ∎

######  Remark 3.6.

Unfortunately, for now, the author has been unable to calculate
$W_{\infty}(1)$. It is however very believable that the archimedean zeta
integral against a preferred, nice test vector is equal to an archimedean
local $L$-factor, which is $\sim_{\overline{\mathbb{Q}}^{\times}}$ half-
integral powers of $\pi$. In our case, there is even an explicit integral
expression of $W_{\infty}(1)$, as written in [CI]:

$W_{\infty}(1)=16e^{-2\pi}\pi^{\frac{7}{2}}\int_{c-i\infty}^{c+i\infty}\pi^{-2s}\Gamma(s+\frac{1}{2})^{2}U(s+\frac{1}{2},1,4\pi)\Gamma(s)\frac{ds}{2\pi
i},$

where
$U(a,b,z)=\frac{1}{\Gamma(a)}\int_{0}^{\infty}e^{-zt}t^{a-1}(1+t)^{b-a-1}dt$
is the confluent hypergeometric function of the second kind.

##  4\. Evidence II: The case of $\operatorname{SU}(2,1)$

In this section, we provide the evidence for the Period conjecture (Conjecture
2.15) for the case of certain irregular automorphic forms on
$G(\mathbb{A})=\operatorname{SU}(2,1)(\mathbb{A})$. In this case, we will use
an integral reprsentation of the base-change $L$-function by Gelbart and
Piatetski-Shapiro ([GPS], completed by [KO]), whose coherent cohomological
interpretation was given by [Oh].

Let $\Pi$ be a globally generic cuspidal automorphic representation of
$G=\operatorname{SU}(2,1)(\mathbb{A}_{\mathbb{Q}})$ as in the previous
subsection that also satisfies Assumption 2.5.

######  Definition 4.1.

Let $M\supset F_{\Pi}$ be a number field. A $\psi$-Whittaker function $W$ on
$G$ is called to be _defined over $M$_ if it takes values in $M(\mu_{\infty})$
and satisfies

$\sigma(W(g))=W(w(\kappa(\sigma))g),$

for all $g\in G(\mathbb{A}_{f})$ and
$\sigma\in\operatorname{Gal}(\overline{\mathbb{Q}}/M)$, where
$w(x)=\operatorname{diag}(x,1,x^{-1})$ and
$\kappa:\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\rightarrow\widehat{\mathbb{Z}}^{\times}$
is the cyclotomic character.

Again, by Remark 3.2(1), $W$ is defined over $M$ if and only if $W(1)\in
M^{\times}$.

###  4.1. Whittaker periods via cohomological period integrals

The purpose of this section is to prove the following

######  Theorem 4.2.

Let
$\varphi\in\Pi^{\operatorname{new}}=\Pi_{f}^{\operatorname{new}}\otimes\Pi_{\infty}^{\operatorname{new}}$.
If $[\varphi]$ (see Definition 2.8) is defined over a number field
$F^{\prime}\supset F_{\Pi}$, then $\Lambda\left(1/2,BC(\Pi)\right)W_{\varphi}$
is a nontrivial $\psi$-Whittaker function defined over $F^{\prime}$.

###### Proof.

Let $C\in\mathbb{C}^{\times}$ be such that $\frac{W_{\varphi}}{C}$ is defined
over $F^{\prime}$, which exists by Remark 3.2(2). We would like to show that

$C\Lambda(1/2,BC(\Pi))\in F^{\prime}{}^{\times}.$

First of all, this is indeed nonzero as $BC(\Pi)$ is cuspidal and tempered.

We consider Gelbart–Piatetski-Shapiro integral representation of base change
$L$-function. This is, roughly speaking, the period integral of $\varphi$
times an Eisenstein over an embedded modular curve. More precisely, there is
an embedding of $H=\operatorname{U}(1,1)$ into $G$. Let $B\subset H$ be the
upper-triangular Borel. For $\Phi:\mathbb{A}^{2}\rightarrow\mathbb{C}$ and
$\chi$ a unitary Grössencharacter of the imaginary quadratic field $F$ used
for the definition of unitary groups, we define an Eisenstein series on $H$
with respect to $B$, $E(h,\chi,\Phi,s)$. The same arithmeticity condition as
Proposition 3.4 applies, as the objects involved are elliptic modular forms.

Using the notation of Example B.3(3), let $(m,n)=(a-b,b-c)$ be the
Harish–Chandra character141414Note that $G$ is now $\operatorname{SU}(2,1)$,
so only the differences matter. of $\Pi_{\infty}$. Let
$\Gamma_{H}=H\cap\Gamma$, and let $i:X_{H}(\Gamma_{H})\hookrightarrow
X_{G}(\Gamma)$ be the closed embedding of closed Shimura varieties, as before.
Then, Gelbart–Piatetski-Shapiro’s integral representation can be understood
via a cup product pairing

$H^{1}(X_{G},V)\otimes_{\mathbb{C}}H^{0}(X_{H},W)\xrightarrow{(\operatorname{id},i_{*})}H^{1}(X_{G},V)\otimes_{\mathbb{C}}H^{1}(X_{G},W^{\prime})\xrightarrow{\cup}H^{2}(X_{G},V\otimes
W^{\prime})\xrightarrow{S}H^{0}(X_{G},\mathcal{O})=\mathbb{C},$

where $\cup$ is the cohomological cup product, $S$ is a Serre duality pairing,
normalized as in the proof of Theorem 3.3 (namely, the pairing induced from a
morphism of $K_{\infty}$-representations defined over
$\overline{\mathbb{Q}}$). Here, $W$ and $W^{\prime}$ are certain automorphic
vector bundles over $X_{H}$ and $X_{G}$, respectively, corresponding to
$[V_{H}(|m-n|-1)]\otimes\omega_{H}(1,0)$ and $[\widetilde{L}_{1}]$, if we use
the notation analogous to [LPSZ, §6]. The reinterpretation, done in [Oh], of
Gelbart–Piatetski-Shapiro’s integral asserts that there is a coherent
cohomology class $[E]\in H^{0}(X_{H}(\Gamma_{H}),W)$ defined over
$FF_{\Pi_{f}}$, which corresponds to a nearly-holomorphic Eisenstein series
$E$ under the Hodge splitting as in [LPSZ, §6.3], such that

$\langle[\varphi],[E]\rangle=C\Lambda\left(1/2,BC(\Pi)\right),$

where $BC$ means base-change. As $\langle,\rangle$ is defined over
$\mathbb{Q}$, the left hand side is in $F^{\prime}$. On the other hand, the
right hand side is nonzero as observed above. Thus, the desired statement
follows. ∎

###  4.2. Hodge-linear algebra

We now conduct relevant calculations in Hodge-linear algebra to prove Theorem
2.18 for $G=\operatorname{SU}(2,1)$. For the sake of simplicity, we assume
that we work with the case of Harish-Chandra character $(1,1,0)$; the same
calculation yields the proof for general weights. Our goal is, as in §3.2, to
convert $\operatorname{vol}F^{1}H_{\operatorname{dR}}(\operatorname{Ad}M)$
into an expression that involves Deligne’s periods. We will prove

######  Proposition 4.3.

For a motive $M$ associated with $\Pi$ in the sense of Conjecture 2.20, we
have

$\operatorname{vol}F^{1}H_{\operatorname{dR}}(\operatorname{Ad}M)\sim_{\overline{\mathbb{Q}}^{\times}}\frac{\pi^{3}c^{+}(BC(M))^{3/2}}{\delta(M_{\overline{\sigma}})^{1/2}}.$

###### Proof.

In this case, the motive $M$ is a motive over $F$ with coefficients in
$\mathbb{Q}$. If we denote $\sigma:F\rightarrow\mathbb{C}$ by the preferred
complex embedding, then as the minimal $K$-type is just $(1,1,1)$, by the
recipe in [HLS, 2.3],

$H_{B}(M_{\sigma})\otimes_{\mathbb{Q}}\mathbb{C}=\mathrel{\underset{\makebox{\text{\tiny$H^{1,0}$}}}{\underbrace{\mathbb{C}v_{1}\oplus\mathbb{C}v_{2}}}}\oplus\mathrel{\underset{\makebox{\text{\tiny$H^{0,1}$}}}{\underbrace{\mathbb{C}v_{3}}}},\quad
H_{B}(M_{\overline{\sigma}})\otimes_{\mathbb{Q}}\mathbb{C}=\mathrel{\underset{\makebox{\text{\tiny$H^{-1,0}$}}}{\underbrace{\mathbb{C}\overline{v}_{1}\oplus\mathbb{C}\overline{v}_{2}}}}\oplus\mathrel{\underset{\makebox{\text{\tiny$H^{0,-1}$}}}{\underbrace{\mathbb{C}\overline{v}_{3}}}}.$

We can choose $v_{i}$ and $\overline{v}_{i}$ so that
$F_{\infty}(v_{i})=\overline{v}_{i}$. Then,
$BC(M):=\operatorname{Res}_{F/\mathbb{Q}}M_{F}=M_{\sigma}\oplus
M_{\overline{\sigma}}(-1)$ and $\operatorname{Ad}M=M_{\sigma}\otimes
M_{\overline{\sigma}}$. The Deligne period $\delta$ of a motive, as before, is
the determinant of the Betti-to-de Rham comparison map with respect to natural
underlying $\mathbb{Q}$-structures. Also, $c^{+}(BC(M))$ in this case would be
the determinant of the map

$H_{B}(BC(M))^{+}\otimes\mathbb{C}\hookrightarrow
H_{B}(BC(M))\otimes\mathbb{C}\xrightarrow{\sim}H_{\operatorname{dR}}(BC(M))\otimes\mathbb{C}\twoheadrightarrow(H_{\operatorname{dR}}(BC(M))/F^{1}H_{\operatorname{dR}}(BC(M)))\otimes\mathbb{C},$

with respect to the natural underlying $\mathbb{Q}$-structures.

Let $e_{1}^{+},e_{2}^{+},e_{3}^{+}$ be a $\mathbb{Q}$-basis of
$H_{B}(BC(M))^{+}$, $e_{1}^{-},e_{2}^{-},e_{3}^{-}$ be a $\mathbb{Q}$-basis of
$H_{B}(BC(M))^{-}$, $f_{1}f_{2},f_{3}\in F^{1}H_{\operatorname{dR}}(BC(M))$ be
an $F$-basis, $g_{1},g_{2},g_{3}\in
H_{\operatorname{dR}}(BC(M))/F^{1}H_{\operatorname{dR}}(BC(M))$ be an
$F$-basis, and $\widetilde{g}_{1},\widetilde{g}_{2},\widetilde{g}_{3}\in
H_{\operatorname{dR}}(BC(M))$ be lifts of $g_{1},g_{2},g_{3}$. We can further
assume that $f_{1},f_{2}\in F^{1}H_{\operatorname{dR}}(M_{\sigma})$, $f_{3}\in
F^{1}H_{\operatorname{dR}}(M_{\overline{\sigma}})$,
$\widetilde{g}_{1},\widetilde{g}_{2}\in
H_{\operatorname{dR}}(M_{\overline{\sigma}})$, $\widetilde{g}_{3}\in
H_{\operatorname{dR}}(M_{\sigma})$. Given two $\mathbb{C}$-bases of
$H_{B}(M)\otimes\mathbb{C}\cong H_{\operatorname{dR}}(M)\otimes\mathbb{C}$, we
can express the map into a matrix,

$\begin{pmatrix}e_{1}^{+}&e_{2}^{+}&e_{3}^{+}&e_{1}^{-}&e_{2}^{-}&e_{3}^{-}\end{pmatrix}=\begin{pmatrix}f_{1}&f_{2}&f_{3}&\widetilde{g}_{1}&\widetilde{g}_{2}&\widetilde{g}_{3}\end{pmatrix}\begin{pmatrix}A&B\\\
C&D\end{pmatrix},$

for $A,B,C,D\in M_{3}(\mathbb{C})$.

Under the canonical isomorphisms

$H^{1,0}(BC(M))\cong F^{1}H_{\operatorname{dR}}(BC(M))\otimes\mathbb{C},\quad
H^{0,1}(BC(M))\cong\frac{H_{\operatorname{dR}}(BC(M))}{F^{1}H_{\operatorname{dR}}(BC(M))}\otimes\mathbb{C},$

let

$f_{1,B},f_{2,B},f_{3,B}\in H^{1,0}(BC(M)),\qquad g_{1,B},g_{2,B},g_{3,B}\in
H^{0,1}(BC(M)),$

be the images of $f_{1},f_{2},f_{3},g_{1},g_{2},g_{3}$ under the above
isomorphisms. Then, by the same argument as in §3.2,

$\begin{pmatrix}e_{1}^{+}&e_{2}^{+}&e_{3}^{+}&e_{1}^{-}&e_{2}^{-}&e_{3}^{-}\end{pmatrix}=\begin{pmatrix}f_{1,B}&f_{2,B}&f_{3,B}&{g}_{1,B}&{g}_{2,B}&{g}_{3,B}\end{pmatrix}\begin{pmatrix}A^{\prime}&B^{\prime}\\\
C&D\end{pmatrix}.$

As $c_{B}(f_{1,B}),c_{B}(f_{2,B}),c_{B}(f_{3,B})$ and
$g_{1,B},g_{2,B},g_{3,B}$ are two $\mathbb{C}$-bases of $H^{0,1}(BC(M))$,
there is a system of linear relations

$\begin{pmatrix}c_{B}(f_{1,B})&c_{B}(f_{2,B})&c_{B}(f_{3,B})\end{pmatrix}=\begin{pmatrix}g_{1,B}&g_{2,B}&g_{3,B}\end{pmatrix}X,$

for some $X\in\operatorname{GL}_{3}(\mathbb{C})$. Using the same technique as
[PV, Lemma, §8.2.1], we see that $C=XA^{\prime},D=XB^{\prime}$, which
similarly implies that

$c^{+}(BC(M))^{2}\sim_{\overline{\mathbb{Q}}^{\times}}\delta(BC(M))\det X,$

where we used $c^{+}(BC(M))\sim_{\overline{\mathbb{Q}}}c^{-}(BC(M))$ due to
the decomposition $BC(M)=M_{\sigma}\oplus M_{\overline{\sigma}}(-1)$ as in
[HL, (8)]. Furthermore, as $g_{1,B},g_{2,B}\in H^{0,1}(M_{\sigma})$ and
$g_{3,B}\in H^{0,1}(M_{\overline{\sigma}}(-1))$, it turns out that $X$ is of
form

$\begin{pmatrix}Y&\\\ &\mu\end{pmatrix},\quad
Y\in\operatorname{GL}_{2}(\mathbb{C}),\mu\neq 0.$

We know that
$(\operatorname{vol}F^{1}H_{\operatorname{dR}}(\operatorname{Ad}M))^{2}\sim_{\overline{\mathbb{Q}}^{\times}}\lambda$,
where $\varphi(c_{B}(v^{+}))=\lambda v^{-}$, $v^{+}=(f_{1}\otimes
f_{3})\wedge(f_{2}\otimes f_{3})$ and $v^{-}=(g_{1}\otimes
g_{3})\wedge(g_{2}\otimes g_{3})$. This implies that $\lambda=\mu^{2}\det
Y=\mu\det X$.

Now, as in [Ha4, §1.2], consider the determinant motive $\det(M)$. An
$F$-rational basis vector of $H_{\operatorname{dR}}(\det(M_{\sigma}))$ can be
taken as $v_{\sigma}:=f_{1}\wedge f_{2}\wedge\widetilde{g}_{3}$, and similarly
$v_{\overline{\sigma}}:=\widetilde{g}_{1}\wedge\widetilde{g}_{2}\wedge f_{3}$
can be taken as an $F$-rational basis vector of
$H_{\operatorname{dR}}(\det(M_{\overline{\sigma}}))$. On the other hand, if we
take $e_{\sigma}$ to be a $\mathbb{Q}$-rational basis vector of
$H_{B}(\det(M_{\sigma}))$, then
$F_{\infty}(e_{\sigma})=:e_{\overline{\sigma}}$ is a $\mathbb{Q}$-rational
basis vector of $H_{B}(\det(M_{\overline{\sigma}}))$. Then
$e_{\sigma}=\delta(M_{\sigma})v_{\sigma}$ and
$e_{\overline{\sigma}}=\delta(M_{\overline{\sigma}})v_{\overline{\sigma}}$, so

$c_{B}(v_{\sigma})=\delta(M_{\sigma})^{-1}e_{\overline{\sigma}}.$

On the other hand,

$c_{B}(v_{\sigma})=F_{\infty}(v_{\sigma})=(\det
Y\cdot\mu^{-1})v_{\overline{\sigma}},$

so

$\delta(M_{\overline{\sigma}})=\mu^{-1}\delta(M_{\sigma})\det Y.$

On the other hand, due to the polarization, we have [Ha4, (1.2.5)],

$\delta(M_{\overline{\sigma}})=\delta(M_{\sigma})^{-1}(2\pi i)^{-6}.$

Thus,

$\pi^{6}\delta(M_{\sigma})^{2}\det Y\sim_{\overline{\mathbb{Q}}^{\times}}\mu,$

so

$\mu^{2}\sim_{\overline{\mathbb{Q}}^{\times}}\pi^{6}\delta(M_{\sigma})^{2}\det
X\sim_{\overline{\mathbb{Q}}^{\times}}c^{+}(BC(M))^{2}\pi^{6}\frac{\delta(M_{\sigma})}{\delta(M_{\overline{\sigma}})}\sim_{\overline{\mathbb{Q}}^{\times}}c^{+}(BC(M))^{2}\pi^{12}\delta(M_{\sigma})^{2},$

or

$\mu\sim_{\overline{\mathbb{Q}}^{\times}}\pi^{6}\delta(M_{\sigma})c^{+}(BC(M)),$

which implies that

$(\operatorname{vol}F^{1}H_{\operatorname{dR}}(\operatorname{Ad}M))^{2}\sim_{\overline{\mathbb{Q}}^{\times}}\frac{\pi^{6}}{\delta(M_{\overline{\sigma}})}c^{+}(BC(M))^{3},$

as desired.∎

###  4.3. Completion of the proof

###### Proof of Theorem 2.18 for $\operatorname{SU}(2,1)$.

Let $f_{\operatorname{hol}},f_{\operatorname{gen}}$ be newforms (see
Assumption 2.5) in $\Pi_{f}\otimes\Pi_{\operatorname{hol}}$ and
$\Pi_{f}\otimes\Pi_{\operatorname{gen}}$, respectively, such that
$[f_{\operatorname{hol}}],[f_{\operatorname{gen}}]$ are defined over
$\overline{\mathbb{Q}}$. Then, under the assumptions of (2.5) of Assumption
2.5151515There is an extra condition on large residue characterstic in _loc.
cit._ , but this restriction is recently removed by [BP, Theorem 1]., [Zha,
Theorem 1.2]161616This is a special case of the refined Gan–Gross–Prasad
conjecture for Fourier–Jacobi periods (see e.g. [Xu, §1.1]), which is also
often referred as the Ichino–Ikeda conjecture. implies that

$\frac{\langle
f_{\operatorname{hol}},f_{\operatorname{hol}}\rangle_{P}}{|FJ(f_{\operatorname{hol}})|^{2}}\sim_{\overline{\mathbb{Q}}^{\times}}\pi^{-2}\frac{L(1,\Pi,\operatorname{Ad})}{L(1/2,BC(\Pi))},$

where $FJ(f_{\operatorname{hol}})$ is the special Fourier–Jacobi period of
$f_{\operatorname{hol}}$, defined by

$FJ(f_{\operatorname{hol}})=\int_{\operatorname{SU}(1,1)(\mathbb{Q})\backslash\operatorname{SU}(1,1)(\mathbb{A})}f_{\operatorname{hol}}(h)dh,$

integrated against the Tamagawa measure. This period is in turn expressed as
an inner product of (algebraic) theta functions,

$FJ(f_{\operatorname{hol}})\sim_{\overline{\mathbb{Q}}^{\times}}\langle
a(0,f_{\operatorname{hol}})(v),1\rangle,$

where $a(0,f_{\operatorname{hol}})$ is the zero-th Fourier–Jacobi coefficient
of $f_{\operatorname{hol}}$ and $1$ is the constant function (regarded as a
trivial theta function). Note that the nonvanishing of
$FJ(f_{\operatorname{hol}})$ (and thus $a(0,f_{\operatorname{hol}})$) is also
a part of the content of [Zha, §1.1].

By [Lan1], $a(0,f_{\operatorname{hol}})$ is identified with the algebraic
Fourier–Jacobi coefficient, and in particular is in
$\overline{\mathbb{Q}}^{\times}$, as $[f_{\operatorname{hol}}]$ is defined
over $\overline{\mathbb{Q}}$. Thus, we have $\langle
f_{\operatorname{hol}},f_{\operatorname{hol}}\rangle_{P}\sim_{\overline{\mathbb{Q}}^{\times}}\pi^{-2}\frac{L(1,\Pi,\operatorname{Ad})}{L(1/2,BC(\Pi))}$.

On the other hand, we exploit the fact that $H^{1}$ is the middle degree of
the Shimura variety. Note that the infinity type of
$\overline{f}_{\operatorname{gen}}$ is a generic LDS as well (with different
infinitesimal character from $f_{\operatorname{gen}}$). Suppose
$C\in\mathbb{C}^{\times}$ is a constant where
$C[\overline{f}_{\operatorname{gen}}]$ is defined over
$\overline{\mathbb{Q}}$. Then,

(G) $\displaystyle\langle
f_{\operatorname{gen}},f_{\operatorname{gen}}\rangle_{P}$ $\displaystyle=(2\pi
i)^{2}\langle[f_{\operatorname{gen}}],[\overline{f}_{\operatorname{gen}}]\rangle_{\operatorname{coh}}$
$\displaystyle\sim_{\overline{\mathbb{Q}}^{\times}}\pi^{2}C^{-1},$

where the cohomological cup product pairing
$\langle-,-\rangle_{\operatorname{coh}}$ is the Serre duality pairing induced
from a $\overline{\mathbb{Q}}$-morphism of algebraic
$K_{\infty}$-representations
$V\otimes\operatorname{Hom}(V,\mathfrak{g}^{-1,1})\rightarrow\mathfrak{g}^{-1,1}$.

By Theorem 4.2,

$\Lambda(1/2,BC(\Pi))W_{C\overline{f}_{\operatorname{gen}}}=C\Lambda(1/2,BC(\Pi))W_{\overline{f}_{\operatorname{gen}}},$

is defined over $\overline{\mathbb{Q}}$. We now invoke the Lapid–Mao
conjecture (Conjecture 2.19), which says

$\langle
f_{\operatorname{gen}},f_{\operatorname{gen}}\rangle_{P}\sim_{\mathbb{Q}^{\times}}|W_{f_{\operatorname{gen}}}(1)|^{2}\frac{L(1,\Pi,\operatorname{Ad})}{|W_{\infty}(1)|^{2}}.$

By Remark 3.2(1), $W_{\overline{f}_{\operatorname{gen}}}(1)\neq 0$, and
$C\Lambda(1/2,BC(\Pi))W_{\overline{f}_{\operatorname{gen}}}(1)\in\overline{\mathbb{Q}}^{\times}$
as observed in the beginning of the section. Thus, we have

$|W_{f_{\operatorname{gen}}}(1)|^{2}\frac{L(1,\Pi,\operatorname{Ad})}{|W_{\infty}(1)|^{2}}\mathrel{\overset{\makebox{\text{\tiny
Lapid--Mao}}}{\sim_{\overline{\mathbb{Q}}^{\times}}}}\langle
f_{\operatorname{gen}},f_{\operatorname{gen}}\rangle_{P}\mathrel{\overset{\makebox{\text{\tiny(F)\quad}}}{\sim_{\overline{\mathbb{Q}}^{\times}}}}\pi^{2}\Lambda(1/2,BC(\Pi))\overline{W_{f_{\operatorname{gen}}}(1)}.$

By using
$W_{\overline{f}_{\operatorname{gen}}}(1)=\overline{W_{f_{\operatorname{gen}}}(1)}$,
we have

$W_{f_{\operatorname{gen}}}(1)\sim_{\overline{\mathbb{Q}}^{\times}}\pi^{2}\frac{|W_{\infty}(1)|^{2}\Lambda(1/2,BC(\Pi))}{L(1,\Pi,\operatorname{Ad})}.$

On the other hand, Theorem 4.2 says

$W_{f_{\operatorname{gen}}}(1)\sim_{\overline{\mathbb{Q}}^{\times}}\frac{1}{\Lambda(1/2,BC(\Pi))},$

which gives an extra relationship that we can utilize; namely,

$\Lambda(1,\Pi,\operatorname{Ad})\sim_{\overline{\mathbb{Q}}^{\times}}\Lambda(1/2,BC(\Pi))^{2}|W_{\infty}(1)|^{2}\pi^{2}.$

For example, by combining (F) and (G), we have

$\langle
f_{\operatorname{gen}},f_{\operatorname{gen}}\rangle_{P}\sim_{\overline{\mathbb{Q}}^{\times}}\pi^{4}\frac{|W_{\infty}(1)|^{2}\Lambda(1/2,BC(\Pi))^{2}}{L(1,\Pi,\operatorname{Ad})}.$

######  Remark 4.4.

Note that this is already observed in [Ha4, Corollary 1.3.5], under the
assumption of Deligne’s conjectures. In particular, assuming Deligne’s
conjectures, we deduce that
$|W_{\infty}(1)|\sim_{\overline{\mathbb{Q}}^{\times}\pi^{\mathbb{Z}}}1$.

Combining these, we get

$\displaystyle\frac{\langle
f_{\operatorname{hol}},f_{\operatorname{hol}}\rangle_{P}}{\langle
f_{\operatorname{gen}},f_{\operatorname{gen}}\rangle_{P}}$
$\displaystyle\mathrel{\overset{\makebox{\text{\tiny Lapid--Mao +
(H)}}}{\sim_{\overline{\mathbb{Q}}^{\times}}}}\frac{\pi^{-2}\frac{L(1,\Pi,\operatorname{Ad})}{L(1/2,BC(\Pi))}}{\pi^{4}\frac{|W_{\infty}(1)|^{2}\Lambda(1/2,BC(\Pi))^{2}}{L(1,\Pi,\operatorname{Ad})}}$
$\displaystyle\sim_{\overline{\mathbb{Q}}^{\times}}\frac{1}{|W_{\infty}(1)|^{2}}\frac{L(1,\Pi,\operatorname{Ad})^{2}}{L(1/2,BC(\Pi))^{3}}$
$\displaystyle\sim_{\overline{\mathbb{Q}}^{\times}}\frac{\pi^{18}}{|W_{\infty}(1)|^{2}}\left(\frac{L_{\infty}(1,\Pi,\operatorname{Ad})}{L_{\infty}(0,\Pi,\operatorname{Ad})}\cdot\frac{L(1,\Pi,\operatorname{Ad})}{L(1/2,BC(\Pi))^{3/2}}\right)^{2},$
$\displaystyle\sim_{\overline{\mathbb{Q}}^{\times}}\frac{\pi^{21}}{|W_{\infty}(1)|^{2}}\left(\frac{L_{\infty}(1,\Pi,\operatorname{Ad})}{L_{\infty}(0,\Pi,\operatorname{Ad})}\cdot\frac{L(1,\Pi,\operatorname{Ad})}{\operatorname{vol}F^{1}H_{\operatorname{dR}}(\operatorname{Ad}M)}\right)^{2},$

by Deligne’s conjecture applied to Proposition 4.3. We are done, as the
paranthesized term is the RHS of Conjecture 2.15.∎

######  Remark 4.5.

Similarly to Remark 3.6, $W_{\infty}(1)$ is given by the inverse Mellin
transform of the formulae given in [KO, Theorem 5.5], which the author at the
moment is unable to compute.

##  5\. Towards motivic action conjecture for rationality of classes

The original motivic action conjecture of [PV] involves the rational structure
of singular cohomology. To derive a similar conjecture, we would have to come
up with a way to normalize all the (choice of newforms of) automorphic
representations at once. Recall that, in the case of modular forms, this is
done by using complex conjugation. Unfortunately, so far there is no general
construction of an operation that can move between different infinity types.
We will tentatively name such an operation a _generalized complex
conjugation_. Approaching the generalized complex conjugations using the
theory of cycle spaces and Penrose transform will be the subject of the
author’s forthcoming work. For now, we will have to content ourselves with a
preliminary analysis on what a generalized complex conjugation should be.

On the other hand, as the name suggests, the usual complex conjugation can be
used when the symmetric space is the upper half plane. More generally, if the
symmetric space is a product of upper half planes (e.g. in the case of Hilbert
modular varieties), then the complex conjugations with respect to each
variable would be a good candidate for generalized complex conjugation; these
are called _partial complex conjugations_ [Ha2] in the literature. In this
case, we can deduce a precise conjecture on the
$\overline{\mathbb{Q}}$-rational structure of coherent cohomology, from the
generalities of Appendix B. There is an existing work of Horawa exactly on
this problem [Ho], and we will compare our conjecture with that of _op. cit._
In particular, we observe that the numerical evidences given in _op. cit._ are
compatible with both conjectures.

###  5.1. The case of Hilbert modular forms: comparison with [Ho]

The work [Ho] states a similar conjecture, Conjecture 3.21 of _op. cit._ , on
what archimedean motivic action should be for Hilbert modular forms of partial
weight one. It uses the partial complex conjugation, which utilizes the fact
that every (limit of) discrete series for
$\operatorname{SL}_{2}(\mathbb{R})^{d}$ is holomorphic or antiholomorphic in
each variable.

######  Definition 5.1 ((Partial complex conjugations)).

Let $F$ be a totally real field of degree $d$ and $\varphi$ be a holomorphic
automorphic form for
$G=\operatorname{Res}_{F/\mathbb{Q}}\operatorname{GL}_{2,F}$, seen as a
holomorphic function on the symmetric space for $G(\mathbb{R})$,
$(\mathbb{C}-\mathbb{R})^{d}$, of weight $(k_{1},\cdots,k_{d};r)$ where
$k_{i}\geq 1$, $k_{i}\equiv r(\operatorname{mod}2)$ for $i=1,\cdots,d$. For
$I\subset\\{1,\cdots,d\\}$, $\varphi^{I}$ is the automorphic form for
$G=\operatorname{Res}_{F/\mathbb{Q}}\operatorname{SL}_{2,F}$, defined by
$\varphi^{I}(g)=\varphi(gJ^{I})$ for $J^{I}=(J_{1}^{I},\cdots,J_{d}^{I})$
given by $J_{j}^{I}=\left(\begin{smallmatrix}-1&0\\\
0&1\end{smallmatrix}\right)$ if $j\in I$ and $J_{j}^{I}=\operatorname{id}_{2}$
if $j\notin I$. This is called the _partial complex conjugation_.

The main conjecture of op. cit., [Ho, Conjecture 3.21], describes the
rationality of cohomology classes in terms of partial complex conjugations. In
particular, it implies that the decomposition
$H^{i}(X)[\Pi_{f}]=H^{i}(\mathfrak{p},K;\omega\otimes
I(\lambda))\otimes\Pi_{f}^{\Gamma}=\left(\bigoplus_{\pi\in\mathfrak{P}_{\lambda}}H^{i}(\mathfrak{p},K;\omega\otimes\pi)\right)\otimes\Pi_{f}^{\Gamma}$
descends to $\overline{\mathbb{Q}}$.

We have not been successful in approaching the conjecture. On the other hand,
based on the materials developed in Appendix B, we suggest a slightly
different conjecture. We use the language of [Ho, §3] belwo.

######  Conjecture 5.2 ((Motivic action conjecture for Hilbert modular
varieties; compare with [Ho, Conjecture 3.21])).

Let $f$ be a parallel weight one form. For each $u\in U_{f}^{\vee}$, let
$u_{i}\in(\operatorname{Ad}^{0}M\otimes_{\iota}\mathbb{C})^{\sigma_{i}c_{0}\sigma_{i}^{-1}}\cong\mathbb{C}$
be the $\sigma_{i}$-component of $U_{f}^{\vee}\otimes_{\iota}\mathbb{C}$ as in
[Ho, Proposition 3.2], where the isomorphism is given by the natural
$\overline{\mathbb{Q}}$-structure on $\mathfrak{sl}_{2}^{d}$. Then, for every
$u\in U_{f}^{\vee}$ not in the kernel of the pairing of [Ho, Lemma 3.1],

$2\pi
i\sum_{i=1}^{d}\frac{\omega_{f}^{\\{i\\}}}{\log(|\tau\otimes\iota(u)|)}\in
H^{1}(X(\Gamma),\omega),$

defines a cohomology class over $\overline{\mathbb{Q}}$.

Conjecture 5.2 suggests that the decomposition of coherent cohomology as a
$(\mathfrak{p},K)$-cohomology of archimedean representations is not in general
$\overline{\mathbb{Q}}$-rational. This is compatible with the corrected
version of Conjecture 3.21 of _op. cit._.

In [Ho, §5], a numerical evidence in favor of the conjecture of [Ho] is given
for base change forms in the case of Hilbert modular forms for real quadratic
fields. We claim that, for such Hilbert modular forms, the two conjectures
coincide:

######  Proposition 5.3.

Let $f$ be a Hilbert modular eigen-cuspform of parallel weight one for a real
quadratic field $F$. If $f$ is a base change form, then [Ho, Conjecture 3.21]
implies Conjecture 5.2.

###### Proof.

Let $\sigma_{1},\sigma_{2}$ be the two real embeddings of $F$, and suppose $f$
is a base change form of $f_{0}$. Indeed, it is shown in [Ho, Corollary 5.2]
that the space of $\overline{\mathbb{Q}}$-Stark units for $f$,
$U_{f}\otimes\overline{\mathbb{Q}}$, is naturally isomorphic to
$(U_{f_{0}}\otimes\overline{\mathbb{Q}})^{\oplus 2}$, and the decomposition is
compatible with the Beilinson regulator. In particular, the
$\overline{\mathbb{Q}}$-vector space spanned by the log of Stark units of
${f}$ is exactly the $\overline{\mathbb{Q}}$-vector space spanned by the log
of Stark units of $f_{0}$. In particular, both Conjecture 5.2 and [Ho,
Conjecture 3.21] are equivalent to the statement that

$\overline{\mathbb{Q}}\omega_{f}^{\\{1\\}}\oplus\overline{\mathbb{Q}}\omega_{f}^{\\{2\\}}=\frac{\log(u_{f_{0}})}{2\pi
i}H^{1}(X(\Gamma)_{\overline{\mathbb{Q}}},\omega),$

where $u_{f_{0}}\in U_{f_{0}}$. ∎

Indeed, a base change form satisfies an extra symmetry with respect to the
“change of two upper half planes”, namely

$\mathbb{H}^{2}\xrightarrow{(x,y)\mapsto(y,x)}\mathbb{H}^{2},$

and this extra symmetry guarantees that the $\overline{\mathbb{Q}}$-splitting
of our form is compatible with the $\overline{\mathbb{Q}}$-splitting of the
form in [Ho, Conjecture 3.21].

###  5.2. Desiderata for generalized complex conjugations

To have a normalized choice of newforms simultaneously, we would like a
certain way to relate different newforms. Example 2.17 suggests that the
complex conjugation should play a role in the tentative statement of the full
motivic action conjecture regarding rationality of cohomology classes. On the
other hand, the complex conjugation can go back and forth between only two
types of LDS’s. For example, it sends a vector in the “holomorphic LDS” to
that in the “anti-holomorphic LDS.” Since a general motivic action conjecture
involves many more LDS’s, we suggest that there are _generalized complex
conjugations_ that can go between any of $\pi\in\mathfrak{P}_{\lambda}$.

We denote a generalized complex conjugation, sending an automorphic form
$v\in\Pi\otimes A_{C}(\lambda)$ to another automorphic form in $\Pi\otimes
A_{C^{\prime}}(\lambda)$, by $c_{C,C^{\prime}}$. There are several desired
properties:

  * •

$c_{C,C^{\prime}}$ is $\mathbb{C}$-linear,

  * •

If $C_{\operatorname{hol}},C_{\mathrm{antihol}}$ are in
$\mathfrak{C}_{\lambda}$,
$c_{C_{\operatorname{hol}},C_{\mathrm{antihol}}}(f)=\overline{f}$,

  * •

$\langle f,f\rangle_{P}=\langle
c_{C,C^{\prime}}(f),c_{C,C^{\prime}}(f)\rangle_{P}$ (Condition (B)),

  * •

$c_{C^{\prime},C^{\prime\prime}}\circ c_{C,C^{\prime}}=c_{C,C^{\prime\prime}}$
and $c_{C,C}=\operatorname{id}$,

  * •

$c_{C_{1}^{\prime}\times C_{2}^{\prime},C_{1}\times
C_{2}}=(c_{C_{1}^{\prime},C_{1}},c_{C_{2}^{\prime},C_{2}})$, for
$G(\mathbb{R})=G_{1}(\mathbb{R})\times G_{2}(\mathbb{R})$.

It is still unclear how to formulate a set of conditions which will uniquely
characterize $c_{C,C^{\prime}}$’s. Although the nature of generalized complex
conjugations still remains mysterious, using the ideas of Penrose transform
and its related geometry, it could be possible to construct the purported
generalized complex conjugations, following the suggestion by Joseph Wolf.
This is the subject of the author’s forthcoming work.

######  Remark 5.4.

The last bullet point suggests that the _partial complex conjugation_ (e.g.
[Ha2], [Sh1]) serves the role of generalized complex conjugations in the case
of Hilbert modular forms . Unfortunately, for $C_{\operatorname{hol}}$ and
$C_{\mathrm{anithol}}$ to be both in $\mathfrak{C}_{\lambda}$, $\lambda$ has
to be orthogonal to all compact roots, and this is allowed only if there is no
compact root (as we exclude degenerate limit of discrete series from our
discussion). Thus, we cannot use the usual complex conjugation besides when
the associated symmetric space is a product of several upper half planes.

######  Remark 5.5.

It is a relatively well-accepted technique in the case of unitary groups to
use theta correspondence to move between different infinity types, as
suggested by the recipe of [Pr]. Indeed, for a unitary group, [HLS] proves
that the theta correspondences and character twists act transitively upon the
full Vogan $L$-packet (see [Vo2]). However, due to the idiosyncrasies of the
recipe for the theta correspondence, it is still unclear whether the theta
correspondence should be the generalized complex conjugation in this case.

######  Remark 5.6.

We also speculate that this is the archimedean version of _excursion
operators_ (first appeared in the work of [Laf] on the global Langlands
correspondence over function fields, and extended to the context of mixed
characteristic local Langlands via Kottwitz’s conjecture, e.g. [FaMa], [RV]).
Indeed, the isotypic decomposition with respect to the excursion algebra
canonically decomposes the automorphic spectrum into $L$-indistinguishable
pieces, which would mean that the excursion operators can go around different
members of an $L$-packet.

Under the hypothesis on existence of generalized complex conjugations
$c_{C,C^{\prime}}$, we can formulate the motivic action conjecture in the
Shimura variety context in its full form.

######  Conjecture 5.7.

Let $\lambda,\Pi$ as in Conjecture 2.13. Let
$f_{h}\in\Pi_{f}^{\operatorname{new}}\otimes\pi_{h}^{\operatorname{new}}$ be a
newform such that $[f_{h}]$ (see Definition 2.8) is defined over
$\overline{\mathbb{Q}}$. Let
$\mathcal{E}_{\lambda}\cong\operatorname{Ext}^{1}_{(\mathfrak{p},K)}(I(\lambda),\pi_{h})$
be defined such that
$\mathbb{C}\alpha_{i}\cong\operatorname{Ext}^{1}_{(\mathfrak{p},K)}(\pi_{\\{1,\cdots,n_{\lambda}\\}-\\{i\\}},\pi_{h})$
sends $1\alpha_{i}$ to the homomorphism
$c_{C_{h},C_{\\{1,\cdots,n_{\lambda}\\}-\\{i\\}}}(f_{h})\mapsto f_{h}$ (using
the identification from Proposition B.5). Then, for $v\in
H_{M}^{1}((\operatorname{Ad}^{*}\Pi)_{\mathcal{O}_{E}},\overline{\mathbb{Q}}(1))\subset
H_{\mathscr{D}}^{1}((\operatorname{Ad}^{*}\Pi)_{\mathbb{R}},\mathbb{C}(1))$,
$a(v)\cdot f_{l}$ defines a coherent cohomology class $[a(v)\cdot f_{l}]\in
H^{i_{l}+1}(X_{G}(\Gamma),[V_{A(\lambda)}])$ that is defined over
$\overline{\mathbb{Q}}$.

One can easily state a similar conjecture for the action of
$\bigwedge^{*}H_{M}^{1}((\operatorname{Ad}\Pi)_{\mathcal{O}_{E}},\overline{\mathbb{Q}}(1))$,
but it is no deeper than the conjecture stated above.

There is basically one known case of what generalized complex conjugation
should be, and it is the case of Hilbert modular forms.

##  Appendix A Beilinson’s conjecture over a general number field

In this section, we recall the statement of _Beilinson’s conjecture_ we will
need in the paper. A usual formulation of the conjecture involves motives over
$\mathbb{Q}$ with coefficients in $\mathbb{Q}$, but we would have to relax
both to be arbitrary number fields. A standard reference of this matter is
[Ra, §6].

###  A.1. Chow motives

We recall the definition of Chow motives over a number field $k$,
$\mathscr{M}_{k,\operatorname{rat}}$ in [PV, §2.1.1]. defined by cohomological
correspondences up to rational equivalence. If $k$ is a number field, then for
a Chow motive $M\in\mathscr{M}_{k,\operatorname{rat}}$, there are the
following cohomology theories, motivated by the cohomology theories of smooth
proper $k$-varieties.

  * •

For each prime $\ell$, there is $\ell$-adic cohomology
$H^{i}(M_{\overline{K}},\mathbb{Q}_{\ell}(r))$, which is a finite-dimensional
$\ell$-adic representation of $\operatorname{Gal}(\overline{K}/K)$.

  * •

For each embedding $\sigma:k\hookrightarrow\mathbb{C}$, there is Betti
cohomology $H^{i}_{B}(M_{\sigma},\mathbb{Q}(r))$, which is a pure
$\mathbb{Q}$-Hodge structure. If $\sigma$ is a real embedding, it is equipped
with the _infinite Frobenius_ $\operatorname{Fr}_{\sigma,\infty}$. On
$H^{i}_{B}(M_{\sigma},\mathbb{Q}(r))\otimes_{\mathbb{Q}}\mathbb{C}$, the
involution $\operatorname{Fr}_{\sigma,\infty}\otimes c_{B}$ preserves the
Hodge decomposition, where $c_{B}$ is the complex conjugation on the second
factor. If $\sigma$ is a complex embedding,
$\operatorname{Fr}_{\sigma,\infty}$ is rather an isomorphism of
$\mathbb{Q}$-vector spaces

$\operatorname{Fr}_{\sigma,\infty}:H_{B}^{i}(M_{\sigma},\mathbb{Q}(r))\xrightarrow{\sim}H^{i}_{B}(M_{\overline{\sigma}},\mathbb{Q}(r)).$

  * •

There is de Rham cohomology $H^{i}_{\operatorname{dR}}(M)(r)$, which is a
finite-dimensional $k$-vector space, equipped with a decreasing filtration
$F^{k}H_{\operatorname{dR}}^{i}(M)(r)$.

We will not care much about $\ell$-adic realization, as it plays no role in
the paper. For each embedding $\sigma:k\hookrightarrow\mathbb{C}$, there is a
comparison isomorphism

$\operatorname{comp}_{\sigma}:H_{B}^{i}(M_{\sigma},\mathbb{Q}(r))\otimes_{\mathbb{Q}}\mathbb{C}\cong
H_{\operatorname{dR}}^{i}(M)(j)\otimes_{k,\sigma}\mathbb{C},$

such that $\oplus_{p\geq k}H^{p,q}$ corresponds to
$(F^{k}H_{\operatorname{dR}})\otimes\mathbb{C}$. If $\sigma$ is a real
embedding, $\operatorname{Fr}_{\sigma,\infty}\otimes c_{B}$ corresponds to
$1\otimes c_{\operatorname{dR}}$, where $c_{\operatorname{dR}}$ is the complex
conjugation on the second factor of
$H_{\operatorname{dR}}^{i}(M)(j)\otimes_{k,\sigma}\mathbb{C}$. If $\sigma$ is
a complex embedding, $\operatorname{Fr}_{\sigma,\infty}\otimes c_{B}$
corresponds to $1\otimes c_{\operatorname{dR}}$ in the sense that there is a
commutative diagram

$\textstyle{H_{B}^{i}(M_{\sigma},\mathbb{Q}(r))\otimes_{\mathbb{Q}}\mathbb{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{comp}_{\sigma}}$$\scriptstyle{\operatorname{Fr}_{\sigma,\infty}\otimes
c_{B}}$$\textstyle{H_{\operatorname{dR}}^{i}(M)(j)\otimes_{k,\sigma}\mathbb{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1\otimes
c_{\operatorname{dR}}}$$\textstyle{H_{B}^{i}(M_{\overline{\sigma}},\mathbb{Q}(r))\otimes_{\mathbb{Q}}\mathbb{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{comp}_{\overline{\sigma}}}$$\textstyle{H_{\operatorname{dR}}^{i}(M)(j)\otimes_{k,\overline{\sigma}}\mathbb{C}}$

Another key player is the Deligne cohomology. For a complex smooth projective
variety $X$ and a subring $A\subset\mathbb{C}$ invariant under complex
conjugation, the Deligne cohomology $H_{\mathscr{D}}^{i}(X,A(r))$ is defined
as the hypercohomology of the complex

$A(r)_{\mathscr{D}}:A(r)\rightarrow\mathcal{O}_{X}\xrightarrow{d}\Omega_{X}^{1}\xrightarrow{d}\cdots\xrightarrow{d}\Omega_{X}^{r-1},$

regarded as a complex of analytic sheaves. This admits a complex conjugation
of coefficients, denoted $c_{\mathscr{D}}$, which is induced from the complex
conjugation on $A(r)_{\mathscr{D}}$. Similarly, there is infinite Frobenius
for Deligne cohomology.

In the formulation of Beilinson’s conjecture over $\mathbb{Q}$, a central role
is played by the cohomology of $M_{\mathbb{R}}$. It plays the same role in
Beilinson’s conjecture over general number fields, even though it may sound
peculiar to consider the base-change of $M$ to $\mathbb{R}$ even if $k$ is not
a real field.

######  Definition A.1.

Given a subfield $A\subset\mathbb{C}$ stable under complex conjugation, define

$H_{\operatorname{dR}}^{i}(M_{\mathbb{R}})(r)=\left(\bigoplus_{\sigma:k\hookrightarrow\mathbb{C}}H_{\operatorname{dR}}^{i}(M)(r)\otimes_{k,\sigma}\mathbb{C}\right)^{1\otimes
c_{\operatorname{dR}}}$
$H_{B}^{i}(M_{\mathbb{R}},A(r))=\left(\bigoplus_{\sigma:k\hookrightarrow\mathbb{C}}H_{B}^{i}(M_{\sigma},\mathbb{Q}(r))\otimes_{\mathbb{Q}}A\right)^{\oplus_{\sigma}\operatorname{Fr}_{\sigma,\infty}\otimes
c_{B}}$
$H_{\mathscr{D}}^{i}(M_{\mathbb{R}},A(r))=\left(\bigoplus_{\sigma:k\hookrightarrow\mathbb{C}}H_{\mathscr{D}}^{i}(M_{\sigma},A(r))\right)^{\oplus_{\sigma}\operatorname{Fr}_{\sigma,\infty}\otimes
c_{\mathscr{D}}}.$

Concretely, cohomology of $M_{\mathbb{R}}$ is the part fixed by
$\operatorname{Fr}_{\infty}\otimes c_{B}=1\otimes c_{\operatorname{dR}}$ via
Betti-de Rham comparison isomorphism. Furthermore, there is a Chern class map

$r_{\mathscr{D}}:H_{M}^{i}(M,\mathbb{Q}(r))\rightarrow
H_{\mathscr{D}}^{i}(M_{\mathbb{R}},\mathbb{R}(r)).$

The source of Chern class map is too large, and we choose a subspace
$H_{M}^{i}(M_{\mathcal{O}_{k}},\mathbb{Q}(r))\subset
H_{M}^{i}(M,\mathbb{Q}(r))$ consisting of classes that “extend to a good
proper model of $M$ over $\mathcal{O}_{k}$”. If $M=h(X)$ for a smooth proper
$k$-variety $X$, which has a regular proper model $\mathfrak{X}$ over
$\mathcal{O}_{k}$, then

$H_{M}^{i}(M_{\mathcal{O}_{k}},\mathbb{Q}(r))=H_{M}^{i}(M,\mathbb{Q}(r))\cap(\operatorname{im}(K_{2r-i}\mathfrak{X}\rightarrow
K_{2r-i}X)\otimes\mathbb{Q}),$

where the latter image of $K$-theory groups is the image via the Chern class
characters, and this definition is independent of choice of $\mathfrak{X}$. In
general, using alterations, Scholl defined this subspace in [Sch, Theorem
1.1.6] and showed that this is a unique way to assign subspaces satisfying
various natural properties. The restriction of Chern class charcater into the
integral subspace,

$r_{\mathscr{D}}:H^{i}_{M}(M_{\mathcal{O}_{k}},\mathbb{Q}(r))\rightarrow
H_{\mathscr{D}}^{i}(M_{\mathbb{R}},\mathbb{R}(r)),$

is called the Beilinson regulator.

###  A.2. Beilinson’s conjecture for Chow and Grothendieck motives

From now on, we assume that $r\geq\frac{i+1}{2}$171717This is equivalent to
that the weight of $M$, $i-2r$, is negative. This is not a restriction as one
can always reduce to this case possibly after using functional equation..
Beilinson’s conjecture is formulated using fundamental exact sequences, which
we review. From the definition of Deligne cohomology, for a complex smooth
projective variety $X$, there is a long exact sequence

$\cdots\rightarrow
H^{i-1}_{B}(X,A(r))\rightarrow\frac{H^{i-1}_{B}(X,\mathbb{C})}{F^{r}H_{\operatorname{dR}}^{i-1}(X)}\rightarrow
H_{\mathscr{D}}^{i}(X,A(r))\rightarrow H_{B}^{i}(X,A(r))\rightarrow\cdots.$

As this long exact sequence intertwines the involutions that define the
cohomology of $M_{\mathbb{R}}$, for a Chow motive $M$ defined over a number
field $k$,

$H_{\mathscr{D}}^{i}(M_{\mathbb{R}},\mathbb{R}(r))=\frac{H_{B}^{i-1}(M_{\mathbb{R}},\mathbb{C})}{F^{r}H_{\operatorname{dR}}^{i-1}(M_{\mathbb{R}})+H_{B}^{i-1}(M_{\mathbb{R}},\mathbb{R}(r))}=\frac{H_{B}^{i-1}(M_{\mathbb{R}},\mathbb{R}(r-1))}{F^{r}H_{\operatorname{dR}}^{i-1}(M_{\mathbb{R}})},$

which gives rise to two fundamental exact sequences,

$0\rightarrow F^{r}H_{\operatorname{dR}}^{i-1}(M_{\mathbb{R}})\rightarrow
H_{B}^{i-1}(M_{\mathbb{R}},\mathbb{R}(r-1))\rightarrow
H_{\mathscr{D}}^{i}(M_{\mathbb{R}},\mathbb{R}(r))\rightarrow 0,$ $0\rightarrow
H_{B}^{i-1}(M_{\mathbb{R}},\mathbb{R}(r))\rightarrow\frac{H_{\operatorname{dR}}^{i-1}(M_{\mathbb{R}})}{F^{r}H_{\operatorname{dR}}^{i-1}(M_{\mathbb{R}})}\rightarrow
H_{\mathscr{D}}^{i}(M_{\mathbb{R}},\mathbb{R}(r))\rightarrow 0.$

The first two entries of the two fundamental exact sequences as above have
natural $\mathbb{Q}$-structures, yielding the $\mathbb{Q}$-structure on $\det
H_{\mathscr{D}}^{i}(M_{\mathbb{R}},\mathbb{R}(r))$. Let $\mathscr{R}$ be the
one from the first sequence, and $\mathscr{D}\mathscr{R}$ be the one from the
second exact sequence.

######  Conjecture A.2 ((Beilinson’s conjecture)).

Suppose either $r>\frac{i}{2}+1$, or $r=\frac{i}{2}+1$ and there is no Tate
cycle, namely
$H_{\operatorname{\acute{e}t}}^{i}(M_{\overline{k}},\mathbb{Q}_{\ell}(i/2))^{\operatorname{Gal}(\overline{k}/k)}=0$.
Then, the following hold.

  1. (1)

The Beilinson regulator

$r_{\mathscr{D}}:H_{M}^{i+1}(M_{\mathcal{O}_{k}},\mathbb{Q}(r))\otimes\mathbb{R}\rightarrow
H_{\mathscr{D}}^{i+1}(M_{\mathbb{R}},\mathbb{R}(r)),$

is an isomorphism.

  2. (2)

Let $\mathscr{M}$ be the $\mathbb{Q}$-structure defined on $\det
H_{\mathscr{D}}^{i+1}(M_{\mathbb{R}},\mathbb{R}(r))$ via the Beilinson
regulator and the $\mathbb{Q}$-structure of the motivic cohomology. Then,

$\mathscr{M}=L(h^{-i}(M^{\vee}),1-r)^{*}\mathscr{R}=L(h^{i}(M),r)\mathscr{D}\mathscr{R}.$

######  Remark A.3.

The motives we will be working with come from automorphic representations in
the sense of [Cl, §4.3.3], and for that purpose, we would rather like to work
with _Grothendieck motives_ , where the equivalence relation used is numerical
equivalence, which is equivalent to homological equivalence under the Standard
Conjecture D, which we have to assume. Fortunately, [PV, §2.1.9] works with
arbitrary base field, so a similar set of assumptions would naturally lead to
Beilinson’s conjecture for Grothendieck motives.

##  Appendix B Deligne cohomology and Lie algebra cohomology

In this appendix, we develop a representation theoretic background parallel to
[PV, §2-§4]. This is to correctly guess the motivic action conjectures, namely
Conjecture 2.13 and Conjecture 5.7.

###  B.1. Nondegenerate limit of discrete series as constituents of reducible
principal series

We need to understand what kinds of infinity types can appear in the coherent
cohomology of Shimura varieties, given the finite part and the coefficient.
Fortunately, the $(\mathfrak{p},K)$-cohomology of unitary
$G(\mathbb{R})$-representations is computed by Vogan–Zuckerman. As far as
$G(\mathbb{R})$-representations are concerned, we will be only interested in
discrete series, or in general nondegenerate limits of discrete series.

For a (L)DS, its $L$-packet consists of all (L)DS with the same infinitesimal
character. Thus each such $L$-packet is consisted of $|W_{G}|/|W_{K}|$
elements, and in particular, upon choosing a system of positive roots
$\Delta_{K}^{+}$ for $K$, can be indexed by Weyl chambers which makes all
roots in $\Delta_{K}^{+}$ positive. Given an infinitesimal character $\lambda$
and a Weyl chamber $C$, let $A_{C}(\lambda)$ be the corresponding (L)DS. This
is in accordance with the notation of [VZ]. In particular,
$A_{C}(\lambda)=A_{\mathfrak{q}}(\lambda_{C})$ for
$\mathfrak{q}=\mathfrak{k}\oplus\mathfrak{p}_{C}$, where $\mathfrak{p}_{C}$ is
the subalgebra of noncompact roots in $\Delta_{C}^{+}$, the system of positive
roots determined by $C$, and $\lambda_{C}$ is the Weyl conjugate of $\lambda$
that is contained in $C$. By the relatively straightforward nature of
$K$-multiplicities of such representations, we have the following formulae.

######  Proposition B.1.

Given $\lambda$ and $C$ as above, we have the following,

$i_{A_{C}(\lambda)}=\dim_{\mathbb{C}}(\mathfrak{p}_{-}\cap\mathfrak{p}_{C}),$
$V_{A_{C}(\lambda)}=V(\lambda_{C}+2\rho(\mathfrak{p}_{+})),$

the highest weight representation of $K$ with highest weight
$\lambda_{C}+2\rho(\mathfrak{p}_{+})$, where $2\rho(\mathfrak{p}_{+})$ is the
sum of all roots in $\mathfrak{p}_{+}$.

Therefore, if $\lambda$ lies on some wall of the Weyl chambers, then it can
happen that $V_{A_{C}(\lambda)}=V_{A_{C^{\prime}}(\lambda)}$ for different
Weyl chambers $C,C^{\prime}$. This is the setting we will be interested in. In
such cases, we drop the subscript $C$ if there is no issue of confusion.

######  Theorem B.2.

Let $\lambda$ be a nondegenerate analytically integral character. Let
$\mathfrak{C}_{\lambda}=\\{C\text{ Weyl chamber}\mid\lambda\in C\\}$, and
$\mathfrak{P}_{\lambda}=\\{A_{C}(\lambda)\mid C\in\mathfrak{C}_{\lambda}\\}$.
Then, there is a parabolic subgroup $Q\subset G(\mathbb{R})$ and a discrete
series representation $\rho$ of the Levi $M_{Q}$ such that

$I(\lambda):=\operatorname{Ind}_{Q}^{G}\rho=\bigoplus_{\pi\in\mathfrak{P}_{\lambda}}\pi,$

where $\operatorname{Ind}_{Q}^{G}\rho$ is the normalized induction. These
satisfy the following properties.

  1. (1)

$Q$ is a parabolic subgroup which is minimal with respect to the property that
the Langlands parameter $\varphi:W_{\mathbb{C}/\mathbb{R}}\rightarrow{}^{L}G$
corresponding to the representations in $\mathfrak{P}_{\lambda}$ can be
arranged so that $\varphi(W_{\mathbb{C}/\mathbb{R}})\subset{}^{L}Q$.
Furthermore, if we denote $T_{M_{Q}}$ by a Cartan subgroup of $M_{Q}$, then
one can further assume that
$\varphi(\mathbb{C}^{\times})\subset{}^{L}T_{M_{Q}}$ and that
$\varphi(W_{\mathbb{C}/\mathbb{R}})$ normalizes ${}^{L}T_{M_{Q}}$.

  2. (2)

$|\mathfrak{C}_{\lambda}|$ is a power of $2$, and
$\lambda^{\perp}\subset\mathfrak{g}$ is spanned by a superorthogonal set of
real roots.

  3. (3)

The infinitesimal character of $\rho$ is the restriction of $\lambda$.

###### Proof.

We freely use the terminology of [Kn]. The property (3) is clear. The content
of [Kn, §14.15] implies that any NLDS appears as a direct summand of such
principal series with multiplicity one; this is via repeated application of
generalized Schmid identities [Kn, Theorem 14.68]. The relation between $Q$
and the Langlands parameter follows from the discussion before [La, Lemma 1].
Since the parabolic subgroup appears as Cayley transform of the minimal
parabolic in the sense of [KZ], and one chooses noncompact simple roots for
the Cayley transform, which in fact form a superorthogonal set of roots by
[Kn, Theorem 14.64], based only on the infinitesimal character and not the
chamber, it follows that

$\\{\operatorname{Ind}_{Q}^{G}\widetilde{\rho}\mid\widetilde{\rho}\text{ DS,
infinitesimal character $=$ the restriction of $\lambda$}\\},$

sees every NLDS with infinitesimal character $\lambda$ at least once as its
constituent. The number of constituents is exactly $|W_{G}|/|W_{K}|$, so each
such NLDS appears exactly once. Each $\operatorname{Ind}_{Q}^{G}\rho$ has
constituents $\pi=A_{C}(\lambda)$ such the representative of $\lambda$
(thought as a Weyl orbit) in $C$ is a fixed character (namely, $C$’s that
appear are all adjacent to a single representative of the Weyl orbit of
$\lambda$). This is an equivalence relation, so this implies $I(\lambda)$ is
precisely consisted of $A_{C}(\lambda)$ where $C$ contains $\lambda$ (regarded
as an actual character). As the $R$-group is a direct sum of bunch of
$\mathbb{Z}/2\mathbb{Z}$ by [Kn, §14.15], we get (2). ∎

######  Example B.3.

We explain how Theorem B.2 is realized in some examples. In the figures, red
arrows are the compact roots, so NLDS’s are those lying on a wall not
orthogonal to red arrows.

  1. (1)

$\operatorname{SL}_{2}(\mathbb{R})$. Let
$Q\subset\operatorname{SL}_{2}(\mathbb{R})$ be the upper triangular Borel, and
let $\rho=\det/|\det|:Q\rightarrow\\{\pm 1\\}$. Then
$I_{Q}^{\operatorname{SL}_{2}(\mathbb{R})}\rho=D_{0}^{+}\oplus D_{0}^{-}$, the
sum of the two NLDS, holomorphic (“weight $1$”) and anti-holomorphic (“weight
$-1$”).

  2. (2)

$\operatorname{Sp}_{4}(\mathbb{R})$ (e.g. [Mu]). There are four types of LDS,
two of them being holomorphic and anti-holomorphic, respectively, and the
other two being large (i.e., maximal Gelfand–Kirillov dimension). We call the
one adjacent to the holomorphic chamber _generic_ and the other one adjacent
to the anti-holomorphic chamber _anti-generic_. Using the notation of [Mu], a
singular infinitesimal character is of one of the forms $(p,0)$, $(0,-p)$ or
$(p,-p)$ with $p\in\mathbb{N}$. The Langlands parameter $\varphi_{\lambda}$
for infinitesimal character $\lambda=(a,b)$ is given by

$\varphi_{\lambda}(re^{i\theta})=\operatorname{diag}(e^{i(a+b)\theta},e^{i(a-b)\theta},e^{-i(a+b)\theta},e^{-i(a-b)\theta}),\quad\varphi_{\lambda}(j)=\begin{pmatrix}0&(-1)^{a+b}I_{2}\\\
I_{2}&0\end{pmatrix}.$

     * •

$\lambda=(p,0)$ lies on the wall between the holomorphic chamber and the
generic chamber. Then, $Q$ is the so-called _Klingen parabolic_ , whose Levi
is $\operatorname{GL}_{1}(\mathbb{R})\times\operatorname{SL}_{2}(\mathbb{R})$,
and $\rho$ is the (trivial extension of the) holomorphic DS $D_{p}^{+}$ of
$\operatorname{SL}_{2}(\mathbb{R})$ of infinitesimal character $p$ (or,
equivalently, weight $p+1$).

     * •

$\lambda=(p,-p)$ lies on the wall between the generic chamber and the anti-
generic chamber. Then, $Q$ is the so-called _Siegel parabolic_ , whose Levi is
$\operatorname{GL}_{2}(\mathbb{R})$, and $\rho$ is the DS $D_{2p}$ of
$\operatorname{GL}_{2}(\mathbb{R})$ with central charcater $2p$ (which is as
$\operatorname{SL}_{2}(\mathbb{R})$-representation the same as
$D_{2p}^{+}\oplus D_{2p}^{-}$).

     * •

$\lambda=(0,-p)$ lies on the wall between the anti-generic chamber and the
anti-holomorphic chamber. This situation is complex-conjugate to the situation
of $\lambda=(p,0)$. Thus, $Q$ is again the Klingen parabolic, but $\rho$ is
the anti-holomorphic DS of the same infinitesimal character.

  3. (3)

$\operatorname{U}(2,1)$ (e.g. [Wa], [Ro, §12]). There are three types of LDS,
_holomorphic_ , _generic_ and _anti-holomorphic_. There are two typse of
NLDS’s , those lying on the wall between the holomorphic chamber and the
generic chamber, and those lying on the wall betweent the generic chamber and
the anti-holomorphic chamber. The Langlands parameters for (L)DS are of the
form

$\varphi(z)=\operatorname{diag}((z/\overline{z})^{a},(z/\overline{z})^{b},(z/\overline{z})^{c}),\quad\varphi(j)=\begin{pmatrix}&&1\\\
&-1\\\ 1\end{pmatrix},$

for $a,b,c\in\mathbb{Z}$, and the parameter only depends on the unordered set
$\\{a,b,c\\}$. Ordering $a\geq b\geq c$, the two types of NLDS can occur for
$a=b>c$ and $a>b=c$.

     * •

$a=b>c$ lies on the wall between the holomorphic chamber and the generic
chamber. Then, $Q$ is the upper-triangular Borel (when $\operatorname{U}(2,1)$
is seen as the unitary group for the diagonal Hermitian matrix such as
$\operatorname{diag}(1,1,-1)$), whose Levi is $\mathbb{C}^{\times}\times
S^{1}$, and $\rho$ is the character
$\operatorname{diag}(\alpha,\beta,\overline{\alpha}^{-1})\mapsto\alpha^{a}\beta^{b}\overline{\alpha}^{-c}$
for $\alpha\in\mathbb{C}^{\times}$, $\beta\in S^{1}$.

     * •

$a>b=c$ lies on the wall between the generic chamber and the anti-holomorphic
chamber. The situation is completely symmetric to the previous case.

###  B.2. Action of the $\operatorname{Ext}$-space

We can now build an archimedean realization of motivic action from abstract
nonsense. Recall that for a $(\mathfrak{p},K)$-module $M$,
$H^{i}(\mathfrak{p},K;M)$ can be regarded as
$\operatorname{Ext}_{(\mathfrak{p},K)}^{i}(\mathbf{1},M)$, the $i$-th
$\operatorname{Ext}$ group in the category of $(\mathfrak{p},K)$-modules,
where $\mathbf{1}$ is the trivial module. Thus, there is a natural action

$\operatorname{Ext}_{(\mathfrak{p},K)}^{m}(M,N)\times\operatorname{Ext}_{(\mathfrak{p},K)}^{n}(\mathbf{1},M)\rightarrow\operatorname{Ext}^{m+n}_{(\mathfrak{p},K)}(\mathbf{1},N).$

In particular, if $\lambda$ is a nondegenerate singular character as above,
with $\pi_{1},\pi_{2}\in\mathfrak{P}_{\lambda}$ with
$i_{\pi_{1}}<i_{\pi_{2}}$, there is a natural action