The Residual Spectrum of $F_{4}$ Arising from Degenerate Eisenstein Series

Thesis Submitted in Partial Fulfillment
of the Requirements for the Degree of
“DOCTOR OF PHILOSOPHY”

By
Hezi Halawi

Submitted to the Senate of Ben-Gurion University of the Negev

April 2021

Beer Sheva

The Residual Spectrum of $F_{4}$ Arising From Degenerate Eisenstein Series

Thesis Submitted in Partial Fulfillment
of the Requirements for the Degree of
“DOCTOR OF PHILOSOPHY”

By
Hezi Halawi

Submitted to the Senate of
Ben-Gurion University of the Negev

Approved by the advisor:
Approved by the Dean of the Kreitman School of Advanced Graduate Studies:

April 2021

Beer Sheva

This work was carried out under the supervision of Prof. Nadya Gurevich
In the Department of Mathematics
Faculty of Natural Sciences

Research-Student’s Affidavit when Submitting the Doctoral Thesis for Judgment

I, Hezi Halawi, whose signature appears below, hereby declare that (Please
mark the appropriate statements):

X I have written this Thesis by myself, except for the help and guidance
offered by my Thesis Advisors.

X The scientific materials included in this Thesis are products of my own
research, culled from the period during which I was a research student.

This Thesis incorporates research materials produced in cooperation with
others, excluding the technical help commonly received during experimental
work. Therefore, I am attaching another affidavit stating the contributions
made by myself and the other participants in this research, which has been
approved by them and submitted with their approval.

Date:  07.04.2021  Student’s name: Hezi Halawi Signature:

## Acknowledgments

First I would like to thank my advisor Nadya Gurevich for endless hours of
fruitful discussions and for her consistent guidance. With her support,
personal help and exceptional patience, I made it through the thesis.

I would like to thank my dearest friend Nir Schwartz for tolerating all my
breaking points and for being there for me personally and professionally. He
helped me a lot with the brainstorming and the editorial discussions.

I would also like to thank my friend and colleague Avner Segal for our
mathematical discussions and for helping me taking my firsts steps in the
field.

In addition I would like to show my gratitude to my loving family for being
there and believing in me all those years. Without them, I couldn’t do
anything.

I also would like to thank the referees for reading through this thesis and
for making many helpful remarks that improved this thesis.

Last but not least I wish to thank from the bottom of my heart to my beloved
wife for giving me a supportive shoulder and helping me pushing through until
I finish this project.

###### Contents

  1. Abstract
  2. 1 Introduction
    1. 1.1 Decomposition of the square integrable spectrum
  3. 2 Preliminaries
  4. 3 Representation theory
    1. 3.1  Branching rule process
    2. 3.2 Aubert involution
    3. 3.3 Local intertwining operators
    4. 3.4 Iwahori–Hecke algebra
  5. 4 Local part - Results
    1. 4.1 Outline
    2. 4.2 $P=P_{4},z_{0}=\frac{5}{2}$
    3. 4.3 $P=P_{1},z_{0}=1$
    4. 4.4 $P=P_{3},z_{0}=\frac{1}{2}$
  6. 5 Global results
    1. 5.1 Notations - recollection
    2. 5.2 Main result
    3. 5.3 The image of $\Lambda_{P_{2}}(\frac{5}{2})$
    4. 5.4 The image of $\Lambda_{P}(z_{0})$
    5. 5.5 The proof of the second part
  7. A Zampera’s argument
  8. B Branching rule triples
    1. B.1 Different types of triples
    2. B.2 Triples of type $C_{n}$
    3. B.3 Triples of type $B_{3}$
  9. C Computational issues
    1. C.1 Preparations
    2. C.2 The proof of Proposition C.0.1
  10. D Local information
  11. E Global Information

## Abstract

Let $G$ be a split group of type $F_{4}$ defined over a number field. We study
the square-integrable automorphic representations of $G$ that can be realized
as leading terms of degenerate Eisenstein series associated to various maximal
parabolic subgroups. These representations appear in the residual spectrum.
The local representation theory over finite places plays a central role in our
work. Key words: Automorphic forms, Eisenstein series, Residual spectrum.

##  Chapter 1 Introduction

The degenerate Eisenstein series are an important class of automorphic forms.
Their residues and more general leading terms have been extensively studied
for two main reasons:

  * •

They are known to produce small automorphic representations, such as minimal
representations (see [GRS97]).

  * •

The degenerate Eisenstein series are used as an analytic ingredient in various
Rankin-Selberg integral representations. Hence their analytic behavior often
governs the analytic behavior of $L$-functions (see [GPSR87]).

In our M.Sc. thesis, we studied the poles of degenerate spherical Eisenstein
series for exceptional groups. In this thesis, which is a natural continuation
of the project, we study the leading terms of degenerate Eisenstein series of
the split group of exceptional type $F_{4}$. Our results go far beyond the
spherical case. To formulate our main theorem, we first introduce the general
setting.

####  1.0.1 General setting

Let $F$ be a number field. We denote by $\mathbb{A}$ the ring of adeles over
$F$. There is a decomposition $\mathbb{A}=\mathbb{A}_{f}\times F_{\infty}$,
where $\mathbb{A}_{f}$ is the ring of finite adeles and
$F_{\infty}=\Pi_{\nu|\infty}F_{\nu}$.

Let $G$ be a split, simply-connected group defined over a number field $F$,
let $G(\mathbb{A})=G(\mathbb{A}_{f})\times G_{\infty}$ be the group of points
over adeles and let $K=K_{f}\times K_{\infty}$ be a maximal compact subgroup
of $G(\mathbb{A})$.

For a maximal parabolic subgroup $P$ of $G$ and a complex number $z$,  we
denote by $\varpi_{P}$ the fundamental weight that corresponds to the unique
simple root in the Lie algebra of the radical of $P$. Consider a degenerate
principal series representation
$\Pi_{P}(z)=i^{G(\mathbb{A})}_{P(\mathbb{A})}\left(z\varpi_{P}\right)$ of the
adelic group $G(\mathbb{A})$. Denote by
$\Pi_{P,K_{\infty}}(z)\subset\Pi_{P}(z)$ the $G(\mathbb{A})$ representation
generated by the subspace of $K_{\infty}$-fixed standard sections.

For any standard $K_{\infty}$-fixed section $f$, the degenerate Eisenstein
series $E_{P}(f,z)$ converges for $Re(z)\gg 0$ and defines an automorphic
form. Moreover, it admits a meromorphic continuation to the complex plane. If
$E_{P}(f,z)$ has a pole at $z=z_{0}$, then the leading term of its Laurent
expansion around the point defines an automorphic form on $G(\mathbb{A})$. It
is expected that for a real non-negative number $z_{0}$, the order of the pole
of $E_{P}(f,z)$ at $z=z_{0}$ is bounded, as $f$ runs over the standard
sections. Then the leading term of $E_{P}(z)$ defines a
$G(\mathbb{A}_{f})$-equivariant operator from $\Pi^{K_{\infty}}_{P}(z_{0})$ to
the space of automorphic forms $\mathcal{A}(G)$. Let
$\mathcal{A}_{deg,K_{\infty}}(G)$ be the space of automorphic forms generated
by the leading terms of degenerate Eisenstein series $E_{P}(z)$ associated
with maximal parabolic subgroups at real positive points. It contains a
subspace $\mathcal{A}_{deg,K_{\infty}}^{2}(G)$ of square integrable functions.
The completion of the latter space with respect to the natural inner product,
denoted by $L^{2}_{deg}(G,K_{\infty}),$ is a natural representation of
$G(\mathbb{A})$. Note that $L^{2}_{deg}(G,K_{\infty})$ is a subspace of the
discrete residual spectrum $L^{2}_{res}(G(F)\backslash G(\mathbb{A}))$. Each
of its irreducible constituents, obtained as a leading term of $E_{P}(z)$
around $z=z_{0},$ is naturally a quotient of the degenerate principal series
$\Pi_{P,K_{\infty}}(z_{0}).$

####  1.0.2  The goal

The goal of the thesis is to describe completely the decomposition of the
space $L^{2}_{deg}(G,K_{\infty})$ for the split group $G$ of exceptional type
$F_{4}$. We enumerate the maximal parabolic subgroups of $G$ as
$P_{1},\ldots,P_{4}$. We focus on the data
$\mathcal{P}=\\{(P_{1},1),(P_{3},\frac{1}{2}),(P_{4},\frac{5}{2}))\\}$. For
$(P_{i},z_{i})\in\mathcal{P}$ and a finite place $\nu$, the local degenerate
principal series $\Pi_{P,\nu}(z_{0})$ admits a maximal semi-simple quotient
$\pi^{i}_{1,\nu}\oplus\pi^{i}_{2,\nu}$, where $\pi^{i}_{1,\nu}$ is the unique
spherical quotient. For an infinite place $\nu$, the representation
$\pi^{i}_{1,\nu}$ is the unique spherical quotient of $\Pi_{P,\nu}(z_{0})$.
Consequently, the maximal semi-simple quotient of
$\Pi_{P_{i},K_{\infty}}(z_{i})$ has the form

$\oplus_{|\mathcal{S}|<\infty}\pi^{\mathcal{S},i},\quad\pi^{\mathcal{S},i}=\left(\otimes_{\nu\in\mathcal{S}}\pi^{i}_{2,\nu}\right)\otimes\left(\otimes_{\nu\notin\mathcal{S}}\pi^{i}_{1,\nu}\right).$

Here, $\mathcal{S}$ runs over finite sets of finite places of $F$.

With these notations, we can formulate our main theorem.

######  Theorem 1.0.1.

The space $L^{2}_{deg}(G,K_{\infty})$ is multiplicity free. Any irreducible
constituent $\Pi\subset L^{2}_{deg}(G,K_{\infty})$ is realized as the leading
term of a degenerate Eisenstein series $E_{P_{i}}(z)$ at $z=z_{i}$ for
$(P_{i},z_{i})\in\\{(P_{1},1),(P_{3},\frac{1}{2}),(P_{4},\frac{5}{2}),(P_{2},\frac{5}{2})\\}$.

  1. 1.

The Eisenstein series $E_{P_{2}}(z)$ has a simple pole at $z=\frac{5}{2}$, and
the image of its leading term is the trivial representation.

  2. 2.

The Eisenstein series $E_{P_{1}}(z)$ has a simple pole at $z=1$, and the image
of its leading term is $\Sigma_{1}=\oplus_{|\mathcal{S}|\,{\rm
is\,even}}\pi^{\mathcal{S},1}$.

  3. 3.

The Eisenstein series $E_{P_{4}}(z)$ has a simple pole at $z=\frac{5}{2}$, and
the image of its leading term is $\Sigma_{4}=\oplus_{|\mathcal{S}|\,{\rm
is\,even}}\pi^{\mathcal{S},4}$.

  4. 4.

The Eisenstein series $E_{P_{3}}(z)$ has a double pole at $z=\frac{1}{2}$, and
the image of its leading term is $\Sigma_{3}=\oplus_{|\mathcal{S}|\neq
1}\pi^{\mathcal{S},3}$.

For every $i$, the representations $\pi^{\mathcal{S},i}$ are nearly equivalent
for all $\mathcal{S}$ and their local constituents at a place $\nu$ belonging
to the same local Arthur packet, see [AG91]. In the first three cases, the
local constituents are expected to constitute the entire local Arthur packet.
In the fourth case, the local constituents are not expected to exhaust it. Our
description agrees with Arthur’s multiplicity formula.

####  1.0.3 Main difficulties

During the work on the thesis, we had to resolve problems in the local
representation theory as well as problems of a global nature.

The local problems include:

  * •

Description of the constituents of various degenerate principal series and
their Jacquet modules. For this purpose we invoked the branching rule method.

  * •

Holomorphicity of local intertwining operators at certain points, often
restricted to lines, a description of their images and a description of the
action. For this purpose we used computations in the Iwahori-Hecke algebra.

The global problem includes the study of the poles of the constant term of
Eisenstein series. The constant term can be written as a sum over numerous
terms, with some terms having higher order poles than the sum. Using the local
theory, we prove all the cancellations.

In the next section we shall explain how our results fit into the framework of
the Langlands spectral decomposition and Arthur’s conjectures.

###  1.1 Decomposition of the square integrable spectrum

We begin with some notation. Let $F_{\nu}$ denote the completion of the number
field $F$ with respect to a valuation $\nu$. If $\nu$ is non-archimedean,
$\mathcal{O}_{\nu}$ stands for the ring of integers of $F_{\nu}$.

Let $\mathbf{G}$ be a semi-simple, simply connected split group with a trivial
center defined over $F$. Then $\mathbf{G}(F_{\nu})$ (resp.
$\mathbf{G}(\mathbb{A})$) stands for the group of its $F_{\nu}$ (resp.
$\mathbb{A}$) points. We fix a maximal split torus $\mathbf{T}$ of
$\mathbf{G}$ and a Borel subgroup $\mathbf{T}\subset\mathbf{B}$. We also
denote by $\mathit{W}_{\mathbf{G}}$ the Weyl group of $\mathbf{G}$.

####  1.1.1 The discrete spectrum $L^{2}_{disc}(\mathbf{G})$

One of the main tasks in the field of automorphic forms concerns the
decomposition of the space
$L^{2}(\mathbf{G}(F)\backslash\mathbf{G}(\mathbb{A}))$, which is a unitary
representation of $\mathbf{G}(\mathbb{A})$. To relax the notation, we denote
it by $L^{2}(\mathbf{G})$. Recall that
$L^{2}(\mathbf{G})=L^{2}_{cont}(\mathbf{G})\oplus L^{2}_{disc}(\mathbf{G}),$
where $L^{2}_{disc}(\mathbf{G})$ is the sum of all irreducible
subrepresentations and $L^{2}_{cont}(\mathbf{G})$ stands for its orthogonal
complement.

The space $L^{2}_{disc}(\mathbf{G})$ admits further decomposition into the
cuspidal spectrum $L^{2}_{cusp}(\mathbf{G})$ and its orthogonal complement
$L^{2}_{res}(\mathbf{G})$, which is called ”the residual spectrum.”

For any irreducible representation $\pi$, we denote by $m(\pi)$,
$m_{res}(\pi)$, $m_{cusp}(\pi)$ its multiplicity in
$L^{2}_{disc}(\mathbf{G})$,$L^{2}_{res}(\mathbf{G})$ and
$L^{2}_{cusp}(\mathbf{G})$ respectively. Set

$\displaystyle L^{2}_{res}(\mathbf{G})$
$\displaystyle=\widehat{\oplus}m_{res}(\pi)\pi,$ $\displaystyle
L^{2}_{cusp}(\mathbf{G})$ $\displaystyle=\widehat{\oplus}m_{cusp}(\pi)\pi.$
(1.1.1)

##### The Langlands decomposition

According to Langlands, [Lan76], the space $L^{2}_{disc}(\mathbf{G})$ admits
another decomposition with respect to cuspidal data; namely,

$L^{2}_{disc}(\mathbf{G})=\widehat{\oplus}_{[\mathbf{M},\sigma]}L^{2}_{[\mathbf{M},\sigma],disc},$

where the sum ranges over all the pairs of a standard Levi subgroup
$\mathbf{M}$ and a cuspidal representation $\sigma$ of
$\mathbf{M}(\mathbb{A})$, up to equivalence. Moreover, Langlands proved that
$L^{2}_{[\mathbf{M},\sigma],disc}(\mathbf{G})$ is generated by iterated
residues of Eisenstein series associated with $(\mathbf{M},\sigma)$. It is
worth noting that

$\displaystyle L^{2}_{cusp}(\mathbf{G})$
$\displaystyle=L^{2}_{[\mathbf{G},\sigma],disc}(\mathbf{G})$
$\displaystyle\text{ and }\quad
L^{2}_{res}(\mathbf{G})=\widehat{\oplus}_{[\mathbf{M},\sigma]}L^{2}_{[M,\sigma],disc}(\mathbf{G})\text{,
where }\mathbf{M}\subsetneq\mathbf{G}.$

Let us focus on the cuspidal data $[\mathbf{T},\operatorname{\mathbf{1}}]$.
The space $L^{2}_{[\mathbf{T},\operatorname{\mathbf{1}}],disc}(\mathbf{G})$ is
generated by iterated residues of $E_{B}(\lambda)$, where $\lambda$ is an
unramified character of $\mathbf{T}(F)\backslash\mathbf{T}(\mathbb{A})$. The
Eisenstein series $E_{B}(\lambda)$ is a meromorphic function of several
complex variables, and the computations of its iterated residues are very
involved as the $\operatorname{rank}(\mathbf{G})$ increases.

On the other hand, a degenerate Eisenstein series $E_{P}(z)$ corresponds to a
standard maximal parabolic subgroup $\mathbf{P}$, which is a function of one
complex variable $z$. These series appear naturally as a restriction of
$E_{B}(\lambda)$ to a specific line. Thus the space $L^{2}_{deg}(\mathbf{G})$
described earlier is a subspace of
$L^{2}_{[\mathbf{T},\operatorname{\mathbf{1}}],disc}(\mathbf{G}).$

##### Decomposition of the discrete spectrum a la Arthur

Arthur provided a conjectural decomposition of $L^{2}_{disc}(\mathbf{G})$ in
terms of Arthur parameters. Recall that an Arthur parameter is a morphism,
$\psi\>:\>\mathcal{L}_{F}\times\operatorname{SL}_{2}(\mathbb{C})\rightarrow{}^{L}G.$
Here, $\mathcal{L}_{F}$ is the conjectural Langlands group, and the properties
of the parameter are listed in [AG91, Part I. 3].

An Arthur parameter $\psi$ is called unipotent if its restriction to
$\mathcal{L}_{F}$ is trivial and the image of a non-trivial unipotent element
of $\operatorname{SL}_{2}(\mathbb{C})$ belongs to a distinguished unipotent
orbit of ${}^{L}G$. Thus, conjugacy classes of unipotent parameters stay in
one-to-one correspondence with distinguished unipotent orbits of ${}^{L}G$.

From now on, we shall assume that the Arthur parameter $\psi$ is unipotent.
Let $S_{\psi}$ be the finite group of connected components of
$\operatorname{Cent}_{{}^{L}G}(\operatorname{Im}\psi)$.

The parameter $\psi$ gives rise, by restriction, to a family
$\left\\{\psi_{\nu}\right\\}$ of local Arthur parameters
$\psi_{\nu}\>:\>\mathcal{L}_{F_{\nu}}\times\operatorname{SL}_{2}\left(\mathbb{C}\right)\>\>\rightarrow{}^{L}G$.
Since $\psi$ is unipotent,

$C_{\psi_{\nu}}=\operatorname{Cent}_{{}^{L}G}(\operatorname{Im}\psi_{\nu})=C_{\psi}$

and the group of components $S_{\psi_{\nu}}=S_{\psi}$ for all $\nu$.

The group $\mathbf{S_{\psi}}$ is a finite group scheme and
$\mathbf{S_{\psi}}(\mathbb{A})=\Pi^{\prime}_{\nu}\mathbf{S_{\psi}}(F_{\nu})=\Pi^{\prime}_{\nu}S_{\psi_{\nu}}$.
The group $S_{\psi}=\mathbf{S_{\psi}}(F)$ is embedded diagonally into
$\mathbf{S_{\psi}}(\mathbb{A})$.

For each irreducible representation $\eta=\otimes\eta_{\nu}$ of
$S_{\psi}(\mathbb{A})$, we define $m(\eta)$ to be the multiplicity of $\eta$
in the space $L^{2}(S_{\psi}(F)\backslash S_{\psi}(\mathbb{A}))$. Explicitly,

$m(\eta)=\frac{1}{|S_{\psi}|}\left(\sum_{s\in
S_{\psi}}\operatorname{tr}\eta(s)\right).$ (1.1.2)

We can formulate Arthur conjectures for the unipotent parameters.

  1. 1.

Existence of local A-packets ([Art89, Section 6]). For every $\nu$ there
exists a set $\Pi_{\psi,\nu}$, called a local A-packet, of admissible
unitarizable representations of $\mathbf{G}(F_{\nu})$, parameterized by the
set of the irreducible representations of $S_{\psi_{\nu}}:$

$\Pi_{\psi,\nu}=\left\\{\pi(\psi_{\nu},\eta_{\nu})\>:\>\eta_{\nu}\in\widehat{S_{\psi_{\nu}}}\right\\}.$

The representation $\pi(\psi_{\nu},Triv)$ is necessarily spherical with the
prescribed Satake parameter. In the case of real group this can be found in
[ABV92].

  2. 2.

For any irreducible representation $\eta=\otimes\eta_{\nu}$ of
$S_{\psi}(\mathbb{A})$, one can form a representation
$\pi(\eta)=\otimes_{\nu}\pi(\psi_{\nu},\eta_{\nu})$ of $G(\mathbb{A}).$ The
global Arthur packet $\Pi_{\psi}$ is defined to be

$\Pi_{\psi}=\\{\pi(\eta),\quad\eta\in\widehat{S_{\psi}(\mathbb{A})}\\}.$

######  Conjecture A.

(multiplicity formula) The multiplicity of $\pi(\eta)$ in
$L^{2}_{disc}(\mathbf{G})$ equals $m(\eta)$ defined in 1.1.2.

Assuming the conjecture, there exists a subspace $L^{2}_{\psi}(\mathbf{G})$ of
$L^{2}_{disc}(\mathbf{G})$ such that

$L^{2}_{\psi}(\mathbf{G})=\oplus_{\eta}m(\eta)\pi(\eta).$

##### The main theorem and Arthur’s conjectures

It is expected that
$L^{2}_{[\mathbf{T},\operatorname{\mathbf{1}}],disc}(\mathbf{G})$, and hence
the space of interest $L^{2}_{deg}(G,K_{\infty})$, is contained in
$\oplus_{\psi}L^{2}_{\psi}(\mathbf{G}),$ where the sum runs over the unipotent
Arthur parameters $\psi$.

We now can interpret 1.0.1 in terms of Arthur’s conjectures. The complex dual
group of $\mathbf{G}$ is ${}^{L}G=F_{4}(\mathbb{C})$. It admits $4$
distinguished unipotent orbits labeled as in [CM93].

  * •

The principal orbit $F_{4}$. In this case, $S_{\psi}=1$. The local and global
packets are singletons and consist of the trivial representation, which is
contained in
$L^{2}_{[\mathbf{T},\operatorname{\mathbf{1}}],disc}(\mathbf{G})$. The Satake
parameter of the trivial representation is $-\sum_{i=1}^{4}\varpi_{i}$ .

  * •

The sub-regular orbit $F_{4}(\alpha_{1})$. In this case, one has
$S_{\psi}\simeq\mathbb{Z}_{2}$. The local packet is expected to be
$\\{\pi_{1,\nu}^{4},\pi_{2,\nu}^{4}\\}$. Every $\pi^{\mathcal{S},4}$ where $S$
is a finite set of finite is expected to belong to the global packet. Arthur’s
conjecture predicts that $m(\pi^{\mathcal{S},4})=1$ for $|\mathcal{S}|$ even
and zero otherwise. This agrees with our description of $\Sigma_{4}$. The
Satake parameter of $\pi^{4}_{\nu}$ is $-\varpi_{1}-\varpi_{3}-\varpi_{4}$.

  * •

The orbit $F_{4}(\alpha_{2})$. In this case, one has
$S_{\psi}\simeq\mathbb{Z}_{2}$. The description is similar to the above and
agrees with our description of $\Sigma_{1}$. The Satake parameter of
$\pi^{1}_{\nu}$ is $-\varpi_{1}-\varpi_{3}$.

  * •

The orbit $F_{4}(\alpha_{3})$. In this case, one has $S_{\psi}\simeq S_{4}$.
There are five irreducible representations of $S_{4}$, among them the trivial
representation and the unique two-dimensional representation. Consequently,
the Arthur packet is expected to have $5$ elements, among them the spherical
representation $\pi^{3}_{1,\nu}$ and the representation $\pi^{3}_{2,\nu}.$
Hence, the global Arthur packet contains the representations
$\pi^{\mathcal{S},3}.$ According to Arthur’s multiplicity formula

$m(\pi^{\mathcal{S},3})=\frac{1}{24}\left(2^{|\mathcal{S}|}+3\cdot
2^{|\mathcal{S}|}+8(-1)^{|\mathcal{S}|}\right).$

This number is zero if $|S|=1,$ and a positive integer otherwise. This again
agrees with our description of $\Sigma_{3}$. Note that we only obtain the
realization of $\pi^{\mathcal{S},3}$ in the residual spectrum. The Satake
parameter of $\pi^{3}_{\nu}$ is $-\varpi_{3}$. .

#### Relation to the Springer correspondence

The Springer correspondence is an injective map from the set of irreducible
representations of $\mathit{W}_{\mathbf{G}}$, the Weyl group of $\mathbf{G}$,
into the set of pairs $(\mathcal{O},\eta)$, where $\mathcal{O}$ is a unipotent
orbit of ${}^{L}G$ and $\eta$ is an irreducible representation of the finite
group $S_{\mathcal{O}}$ (see [CM93, Chapter 10] and [ABV92]).

For a given $\mathcal{O}$, we denote by $Springer(\mathcal{O})$ the set of all
$\eta$ such that $(\mathcal{O},\eta)$ belongs to the image of the
correspondence.

Arthur’s conjecture does not indicate which representations in the packet
$\Pi_{\psi}$ occur in the cuspidal spectrum, which do so in the residual and
which occur in both.

Given a unipotent Arthur parameter $\psi$, for every $\nu$ we denote by
$\Pi_{\psi_{\nu},res}\subset\Pi_{\psi_{\nu}}$ the set of representations that
occur as a local component of a representation

$\pi\subset
L^{2}_{[\mathbf{T},\operatorname{\mathbf{1}}],disc}(\mathbf{G})\subset
L^{2}_{res}(\mathbf{G}).$

Let $\psi$ be a unipotent parameter corresponding to the unipotent orbit
$\mathcal{O}$. Note that $S_{\psi}=S_{\mathcal{O}}$. It is expected that the
representations of $\Pi_{\psi_{\nu},res}$ are parameterized by those
representations $\eta$ of $S_{\psi_{\nu}}=S_{\mathcal{O}}$ that belong to
$Springer(\mathcal{O})$.

This expectation holds in the following cases:

  * •

$\mathbf{G}$ is a classical split orthogonal or symplectic group (see
[Moe94]).

  * •

$\mathbf{G}$ is a split group of type $G_{2}$ (see [Kim96]).

This expectation holds also in the case where $\mathbf{G}$ is of type $F_{4}$
and $\psi$ corresponds to the orbits $F_{4}$ or $F_{4}(\alpha_{1})$ or
$F_{4}(\alpha_{2})$. By [Car93, p.432] one has
$|Springer(F_{4}\left(\alpha_{1}\right))|=|Springer\left(F_{4}(\alpha_{2})\right)|=2$.
In the case where $\psi$ corresponds to the orbit $F_{4}(\alpha_{3})$, the
expectation is that $|\Pi_{\psi_{\nu},res}|=4$. In this thesis, we only show
that
$\left\\{\pi_{1,\nu}^{3},\pi_{2,\nu}^{3}\right\\}\subset\Pi_{\psi_{\nu},res}$.
In such a case, $L^{2}_{deg}(\mathbf{G})$ is expected not to exhaust all of
$L^{2}_{[\mathbf{T},\operatorname{\mathbf{1}}],disc}(\mathbf{G})$.

#### The outline of the thesis

  * •

In Chapter 2 (Preliminaries) we set our notations. We define the local and
global degenerate principal series. We also review the degenerate Eisenstein
series $E_{P}(\cdot)$ and its constant term, $E_{P}^{0}(\cdot)$, along
$\mathbf{U}$, the unipotent radical of $\mathbf{B}$, and we introduce global
normalizing factors and normalized local intertwining operators.

  * •

In Chapter 3 we develop tools for studying degenerate principal series over
local non-Archimedean fields. Specifically, we review the necessary properties
of Iwahori-Hecke algebra and explain the concept of the branching rule method.

  * •

In Chapter 4 we show holomorphicity of certain local normalized intertwining
operators and compute their images. In addition, we completely describe the
structure of relevant degenerate principal series.

  * •

In Chapter 5 we prove our main result 5.2.1.

  * •

In Appendix B we list and prove all the branching rules needed for the
computations in Chapter 4.

  * •

Some of the proofs in the thesis rely on the data supplied in tables that
appear in Appendix D and Appendix E. These tables were obtained using Sagemath
[The17]. Most of them may also be obtained manually. In Appendix C, we
elaborate on the computational aspects of the thesis.

##  Chapter 2 Preliminaries

Let $F$ be a number field. For each place $\nu$ of $F$, let $F_{\nu}$ denote
the completion of $F$ with respect to $\nu$. The ring of adeles of $F$ is
denoted by $\mathbb{A}=\mathbb{A}_{F}$. If $\nu$ is a finite place, the ring
of integers of $F_{\nu}$ is denoted by $\mathcal{O}_{\nu}$. Also, $q_{\nu}$
denotes the cardinality of the residue field. We let $\varpi_{\nu}$ denote a
uniformizer of $F_{\nu}$.

###### G as an algebraic group.

Let $\mathbf{G}$ be a split, simply-connected, reductive group defined over
$F$ with a maximal split torus $\mathbf{T}$. Let $\mathfrak{g}$ denote the Lie
algebra of $\mathbf{G}$.

Let $\mathbf{X}(\mathbf{T})=\operatorname{Hom}(\mathbf{T},\mathbb{G}_{m})$ be
the character group of $\mathbf{T}$, and let
$\mathbf{Y}(\mathbf{T})=\operatorname{Hom}(\mathbb{G}_{m},\mathbf{T})$ be the
cocharacter group. Both $\mathbf{X}(\mathbf{T})$ and $\mathbf{Y}(\mathbf{T})$
are free abelian groups of rank $\dim\mathbf{T}$. Since
$\operatorname{Aut}(\mathbb{G}_{m})=\mathbb{Z}$, there is a canonical pairing

$\left\langle,\right\rangle\>:\>\mathbf{X}(\mathbf{T})\times\mathbf{Y}(\mathbf{T})\rightarrow\mathbb{Z}.$

The maximal torus $\mathbf{T}$ acts on the Lie algebra $\mathfrak{g}$ via the
adjoint action. Since $\mathbf{G}$ is reductive, the zero eigenspace of
$\mathfrak{g}$ with respect to $\mathbf{T}$ is exactly the Lie algebra
$\mathfrak{t}$ of $\mathbf{T}$ itself. Therefore we can decompose
$\mathfrak{g}=\mathfrak{t}\oplus_{\alpha\in\Phi_{\mathbf{G}}}\mathfrak{g}_{\alpha},$
where $\Phi_{\mathbf{G}}$ is the set of all
$0\neq\alpha\in\mathbf{X}(\mathbf{T})$ such that
$\mathfrak{g}_{\alpha}=\left\\{x\in\mathfrak{g}\>:\>t.x=\alpha(t)x\>\forall
t\in\mathbf{T}\right\\}\neq\left\\{0\right\\}$. The set $\Phi_{\mathbf{G}}$ is
called the set of roots of $\mathbf{G}$.

We choose a Borel subgroup $\mathbf{B}$ such that
$\mathbf{T}\subset\mathbf{B}$, and denote its Lie algebra by $\mathfrak{b}$.
There exists a closed, connected, normal subgroup of $\mathbf{B}$ denoted by
$\mathbf{U}$, such that $\mathbf{B}=\mathbf{T}\mathbf{U}$ is a semi–direct
product. Therefore, since $\mathbf{T}$ acts on $\mathbf{B}$, it also acts on
$\mathfrak{b}$. Hence, by choosing $\mathbf{B}$, we also obtain a choice of
positive roots $\Phi_{\mathbf{G}}^{+}$ defined by
$\alpha\in\Phi_{\mathbf{G}}^{+}$ iff
$\mathfrak{g}_{\alpha}\subset\mathfrak{b}$. We define the negative roots to be
$\Phi_{\mathbf{G}}^{-}=\Phi_{\mathbf{G}}\setminus\Phi_{\mathbf{G}}^{+}$. A
positive root $\alpha$ is called a simple root if it cannot be written as a
sum of two positive roots. The set of simple roots is denoted by
$\Delta_{\mathbf{G}}$.

For every root $\alpha\in\Phi_{\mathbf{G}}$, one can associate a homomorphism
$x_{\alpha}\>:\>\mathbb{G}_{a}\rightarrow\mathbf{G}$ such that for every
$c\in\mathbb{G}_{a}$ and $t\in\mathbf{T}$, it holds
$tx_{\alpha}(c)t^{-1}=x_{\alpha}(\alpha(t)c)$. Moreover, we can define a
morphism $\phi_{\alpha}\>:\>\operatorname{SL}_{2}\rightarrow\mathbf{G}$ such
that

$\displaystyle\phi_{\alpha}\left(\begin{array}[]{rr}1&c\\\
0&1\end{array}\right)$ $\displaystyle=x_{\alpha}(c)$
$\displaystyle\phi_{-\alpha}\left(\begin{array}[]{rr}1&0\\\
c&1\end{array}\right)$ $\displaystyle=x_{-\alpha}(c).$

Define $\alpha^{\vee}\>:\>\mathbb{G}_{m}\rightarrow\mathbf{T}$ by
$\alpha^{\vee}(c)=\phi_{\alpha}\left(\begin{array}[]{rr}c&0\\\
0&c^{-1}\end{array}\right)$. The element
$\alpha^{\vee}\in\mathbf{Y}(\mathbf{T})$, and it is called the coroot
associated with the root $\alpha$.

Let $\Omega=\left\\{\varpi_{\alpha}\right\\}_{\alpha\in\Delta_{\mathbf{G}}}$
be elements of $\mathfrak{t}^{\ast}$ such that
$\left\langle\varpi_{\alpha},\beta^{\vee}\right\rangle=\delta_{\alpha,\beta}$.
Here, $\delta_{\alpha,\beta}$ is the Kronecker delta. These elements are
called fundamental weights.

We set
$\mathfrak{a}_{\mathbf{T},\mathbb{C}}^{\ast}=\mathbf{X}(\mathbf{T})\otimes_{\mathbb{Z}}\mathbb{C}$.
We note that
$\mathfrak{a}^{\ast}_{\mathbf{T},\mathbb{C}}=\mathfrak{z}^{\ast}\oplus\operatorname{Span}_{\mathbb{C}}\left\\{\Omega\right\\}$,
where

$\mathfrak{z}^{\ast}=\left\\{x\in\mathfrak{a}^{\ast}_{\mathbf{T},\mathbb{C}}\>:\>\left\langle
x,\alpha^{\vee}\right\rangle=0\>\forall\alpha\in\Delta_{\mathbf{G}}\right\\}.$

Every root $\alpha\in\Phi_{\mathbf{G}}$ defines
$w_{\alpha}\in\operatorname{Aut}(\mathbf{X}(\mathbf{T}))$ and
$w_{\alpha^{\vee}}\in\operatorname{Aut}(\mathbf{Y}(\mathbf{T}))$ by the
formula

$\displaystyle w_{\alpha}(x)$ $\displaystyle=x-\left\langle
x,\alpha^{\vee}\right\rangle\alpha,$ $\displaystyle
w_{\alpha^{\vee}}(y)=y-\left\langle\alpha,y\right\rangle\alpha^{\vee}.$

We denote by $\operatorname{Norm}_{\mathbf{G}}(\mathbf{T})$ the normalizer of
$\mathbf{T}$ in $\mathbf{G}$. The Weyl group of $\mathbf{G}$ is defined as
$\mathit{W}_{\mathbf{G}}=:\operatorname{Norm}_{\mathbf{G}}(\mathbf{T})/\mathbf{T}$.
It is known that $\mathit{W}_{\mathbf{G}}=\left\langle
w_{\alpha}\>:\>\alpha\in\Delta_{\mathbf{G}}\right\rangle$ and it is a finite
group.

###### Parabolic subgroup.

A subgroup $\mathbf{P}$ is called a standard parabolic subgroup if
$\mathbf{B}\subseteq\mathbf{P}$. There exists a correspondence between
standard parabolic subgroups and subsets of $\Delta_{\mathbf{G}}$ via the
formula $\Theta\leftrightarrow\mathbf{P}_{\Theta}$, where
$\mathbf{P}_{\Theta}=\left\langle\mathbf{B},w_{\alpha}\>:\>\alpha\in\Theta\right\rangle.$
In addition, each standard parabolic subgroup $\mathbf{P}_{\Theta}$ admits a
Levi decomposition
$\mathbf{P}_{\Theta}=\mathbf{M}_{\Theta}\mathbf{N}_{\Theta}$, where
$\mathbf{M}_{\Theta}$ stands for the Levi subgroup and $\mathbf{N}_{\Theta}$
denotes the unipotent radical of $\mathbf{P}_{\Theta}$. Moreover, the set of
simple roots of $\mathbf{M}_{\Theta}$ is $\Theta$, and also
$\Phi_{\mathbf{M}_{\Theta}}=\Phi_{\mathbf{G}}\cap\operatorname{Span}_{\mathbb{Z}}\left\\{\Theta\right\\}$.
We let $\mathit{W}_{\mathbf{M}_{\Theta}}=\left\langle
w_{\alpha}\>:\>\alpha\in\Theta\right\rangle$. For a standard Levi subgroup
$\mathbf{M}$, we denote by $\mathbf{X}(\mathbf{M})$ (resp.
$\mathbf{Y}(\mathbf{M})$) the characters (resp. cocharacter) group of
$\mathbf{M}$. Set
$\mathfrak{a}_{\mathbf{M},\mathbb{C}}^{\ast}=\mathbf{X}(\mathbf{M})\otimes_{\mathbb{Z}}\mathbb{C}.$
As before,
$\mathfrak{a}^{\ast}_{\mathbf{M},\mathbb{C}}=\mathfrak{z}^{\ast}\oplus\operatorname{Span}_{\mathbb{C}}\left\\{\varpi_{\alpha}\>:\>\alpha\in\Delta_{\mathbf{G}}\setminus\Delta_{\mathbf{M}}\right\\}$.
We set

$\rho_{\mathbf{M}}=\frac{1}{2}\sum_{\alpha\in\Phi_{\mathbf{G}}^{+}\setminus\Phi^{+}_{\mathbf{M}}}\alpha.$
(2.0.1)

It is well known that
$\rho_{\mathbf{M}}\in\mathfrak{a}_{\mathbf{M},\mathbb{C}}^{\ast}$.

Among the standard parabolic subgroups, there are proper maximal parabolic
subgroups
$\mathbf{P}_{i}=\mathbf{P}_{\Delta_{G}\setminus\left\\{\alpha_{i}\right\\}}$.
We also denote by $\mathbf{M}_{i}$ the Levi part of $\mathbf{P}_{i}$.

######  The group of type $F_{4}$.

From now on we assume that $\mathbf{G}$ is a split group of type $F_{4}$. Let
us order the elements of $\Delta_{\mathbf{G}}$ with respect to the labeling of
the Dynkin diagram of $\mathbf{G}$, given by

$\alpha_{1}$$\alpha_{2}$$\alpha_{3}$$\alpha_{4}$

In addition, there are four maximal standard parabolic subgroups. According to
[Asg00, Section 4] one has

$\displaystyle\mathbf{M}_{1}$ $\displaystyle\simeq GSp_{6},$
$\displaystyle\mathbf{M}_{2}$
$\displaystyle\simeq\mathbf{M}_{3}\simeq\frac{\operatorname{GL}_{1}\times\operatorname{SL}_{2}\times\operatorname{SL}_{3}}{A},$
$\displaystyle\mathbf{M}_{4}$ $\displaystyle\simeq GSpin_{7},$ (2.0.2)

where
$A=\left\\{\left(\zeta,\zeta^{3}I_{2},\zeta^{2}I_{3}\right)\>:\>\zeta^{6}=1\right\\}\simeq\left\langle\zeta_{6}\right\rangle$.
Here $\zeta_{6}$ is a $6^{\textup{th}}$ primitive root of unity, and
$I_{2},I_{3}$ are the identity elements in $\operatorname{SL}_{2}$ and
$\operatorname{SL}_{3}$. For shorthand writing, we let $w_{i}$ stand for
$w_{\alpha_{i}}$ where $\alpha_{i}\in\Delta_{\mathbf{G}}$. The action of the
Weyl group is as follows:

$\displaystyle w_{1}\left(\varpi_{\alpha_{1}}\right)$
$\displaystyle=-\varpi_{\alpha_{1}}+\varpi_{\alpha_{2}}$ $\displaystyle
w_{j}\left(\varpi_{\alpha_{1}}\right)=\varpi_{\alpha_{1}}\quad\text{for}\quad
j\in\left\\{2,3,4\right\\}$ (Weyl Action) $\displaystyle
w_{2}\left(\varpi_{\alpha_{2}}\right)$
$\displaystyle=\varpi_{\alpha_{1}}-\varpi_{\alpha_{2}}+2\varpi_{\alpha_{3}}$
$\displaystyle
w_{j}\left(\varpi_{\alpha_{2}}\right)=\varpi_{\alpha_{2}}\quad\text{for}\quad
j\in\left\\{1,3,4\right\\}$ $\displaystyle
w_{3}\left(\varpi_{\alpha_{3}}\right)$
$\displaystyle=\varpi_{\alpha_{2}}-\varpi_{\alpha_{3}}+\varpi_{\alpha_{4}}$
$\displaystyle
w_{j}\left(\varpi_{\alpha_{3}}\right)=\varpi_{\alpha_{3}}\quad\text{for}\quad
j\in\left\\{1,2,4\right\\}$ $\displaystyle
w_{4}\left(\varpi_{\alpha_{4}}\right)$
$\displaystyle=\varpi_{\alpha_{3}}-\varpi_{\alpha_{4}}$ $\displaystyle
w_{j}\left(\varpi_{\alpha_{4}}\right)=\varpi_{\alpha_{4}}\quad\text{for}\quad
j\in\left\\{1,2,3\right\\}.$

Since $\mathbf{G}$ is semi-simple, it follows that
$\mathfrak{a}^{\ast}_{\mathbf{T},\mathbb{C}}=\operatorname{Span}_{\mathbb{C}}\left\\{\varpi_{\alpha}\>:\>\alpha\in\Delta_{\mathbf{G}}\right\\}$.

###### Representations.

We let $G(E)$ denote the group of $E$–points of $\mathbf{G}$ where
$E\in\left\\{F,F_{\nu},\mathcal{O}_{\nu},\mathbb{A}\right\\}$. Let
$K_{\nu}=\mathbf{G}(\mathcal{O}_{\nu})$ for a finite place $\nu$ and
$K_{\infty}$ be a maximal compact subgroup of $G(F_{\nu})$ in cases where
$\nu$ is infinite. We note that $K_{\nu}$ is a maximal compact subgroup of
$G(F_{\nu})$ for every $\nu$. Set $\mathbf{K}=\prod_{\nu}K_{\nu}$. Note that
$\mathbf{K}$ is a maximal compact subgroup of $G(\mathbb{A})$. By abuse of
notation, for every $E\in\left\\{F,F_{\nu},\mathbb{A}\right\\}$, the element
$\lambda\in\mathfrak{a}^{\ast}_{\mathbf{T},\mathbb{C}}$ can be realized as
$\lambda\>:\>T(E)\rightarrow\mathbb{C}$ via composition with the norm map.

Given a standard parabolic subgroup $\mathbf{P}=\mathbf{M}\mathbf{N}$ and
$\lambda\in\mathfrak{a}_{\mathbf{M},\mathbb{C}}^{\ast}$, we form a
representation of $P(\mathbb{A})$ by extending $\lambda$ trivially on
$N(\mathbb{A})$. We consider the normalized induced representation for
$\lambda\in\mathfrak{a}_{\mathbf{M},\mathbb{C}}^{\ast}$

$i_{M(\mathbb{A})}^{G(\mathbb{A})}(\lambda)=\operatorname{Ind}_{P(\mathbb{A})}^{G(\mathbb{A})}(\lambda).$

It is realized on the space of all $C^{\infty}$ and right $\mathbf{K}$–finite
functions on $G(\mathbb{A})$, which satisfy

$f(mng)=\rho_{\mathbf{M}}(m)\lambda(m)f(g),$

where $m\in M(\mathbb{A}),n\in N(\mathbb{A}),g\in F_{4}(\mathbb{A})$ and
$\rho_{\mathbf{M}}$ as in (2.0.1).

Note that
$\operatorname{Ind}_{P(\mathbb{A})}^{G(\mathbb{A})}(\lambda)=\otimes_{\nu}^{{}^{\prime}}\operatorname{Ind}_{P(F_{\nu})}^{G(F_{\nu})}(\lambda_{\nu})$.

###### Degenerate Eisenstein series.

We fix a maximal parabolic subgroup
$\mathbf{P}_{i}=\mathbf{M}_{i}\mathbf{N}_{i}$. Since
$\mathfrak{a}_{\mathbf{M}_{i},\mathbb{C}}^{\ast}=\operatorname{Span}\left\\{\varpi_{i}\right\\}$,
we may write $i_{M_{i}(\mathbb{A})}^{G(\mathbb{A})}(z\varpi_{{}_{i}})$. A
section $f_{z}\in i_{M_{i}(\mathbb{A})}^{G(\mathbb{A})}(z\varpi_{i})$ is
called:

  * •

Standard, if its restriction to $\mathbf{K}$ does not depend on $z$. Note that
in the literature it is also known as ”flat section”.

  * •

Spherical, if it is standard and $\mathbf{K}$–invariant. In addition, if
$f(1)=1$, we call $f$ a normalized section. Since $\dim
i_{M_{i}(\mathbb{A})}^{G(\mathbb{A})}(z\varpi_{i})^{\mathbf{K}}=1$, the
normalized spherical section is unique, in which case we denote it as $f^{0}$.

Given a standard section $f_{z}\in
i_{M_{i}(\mathbb{A})}^{G(\mathbb{A})}(z\varpi_{i})$ such that
$f=f_{z}\Big{|}_{\mathbf{K}}$, we define the Eisenstein series

$\displaystyle E_{P_{i}}(f,z,g)=\sum_{\gamma\in P_{i}(F)\backslash
G(\mathbb{A})}f_{z}(\gamma g).$ (2.0.3)

The Eisenstein series $E_{P_{i}}(f,z,g)$ converges for
$z\in\mathfrak{Z}^{+}_{i},$ where

$\displaystyle\mathfrak{Z}^{+}_{i}$
$\displaystyle=\left\\{z\in\mathbb{C}\>:\>\operatorname{Re}{\left\langle
z\varpi_{i}-\rho^{\mathbf{M}_{i}}_{\mathbf{T}},\alpha^{\vee}\right\rangle}>1\>:\forall\alpha\in\Phi_{\mathbf{G}}^{+}\setminus\Phi_{\mathbf{M}_{i}}^{+}\right\\},$
$\displaystyle\rho_{\mathbf{T}}^{\mathbf{M}_{i}}$
$\displaystyle=-\rho_{\mathbf{M}_{i}}+\rho_{\mathbf{T}}.$

The converges of the degenerate Eisenstein series follows from [Lan76].

######  Remark 2.0.1.

One has
$\mathfrak{Z}^{+}_{i}=\left\\{z\in\mathbb{C}\>:\>\operatorname{Re}{z}>\left\langle\rho_{\mathbf{M}_{i}},\alpha^{\vee}_{i}\right\rangle\right\\}$.

###### Proof.

For every $\beta\in\Phi_{\mathbf{G}}$, let us write
$\beta^{\vee}=\sum_{\alpha\in\Delta_{G}}n_{\alpha}\alpha^{\vee}$ and set
$h(\beta^{\vee})=\sum_{\alpha\in\Delta_{G}}n_{\alpha}$. By definition
$\left\langle\rho_{\mathbf{T}},\beta^{\vee}\right\rangle=h(\beta^{\vee})$.

Let $w_{\mathbf{M}_{i}}\in\mathit{W}_{\mathbf{M}_{i}}$ be the longest Weyl
element in $\mathit{W}_{\mathbf{M}_{i}}$. Note that

  * •

$w_{\mathbf{M}_{i}}(\beta)\in\Phi_{\mathbf{G}}^{+}\setminus\Phi_{\mathbf{M}_{i}}^{+}$
for every $\beta\in\Phi_{\mathbf{G}}^{+}\setminus\Phi_{\mathbf{M}_{i}}^{+}$.

  * •

$w_{\mathbf{M}_{i}}(\rho_{\mathbf{T}}^{\mathbf{M}_{i}})=-\rho_{\mathbf{T}}^{\mathbf{M}_{i}}$.

  * •

$w_{\mathbf{M}_{i}}(\varpi_{i})=\varpi_{i}$.

Given a $\beta\in\Phi_{\mathbf{G}}^{+}\setminus\Phi_{\mathbf{M}_{i}}^{+}$, we
write
$w_{\mathbf{M}_{i}}(\beta^{\vee})=\sum_{\alpha\in\Delta_{G}}m_{\alpha}\alpha^{\vee}$.
Since $m_{\alpha_{i}}\geq 1$, we deduce that

$\displaystyle\operatorname{Re}\left\langle
z\varpi_{i}-\rho_{\mathbf{T}}^{\mathbf{M}_{i}},\beta^{\vee}\right\rangle$
$\displaystyle=\operatorname{Re}\left\langle
w_{\mathbf{M}_{i}}\left(z\varpi_{i}-\rho_{\mathbf{T}}^{\mathbf{M}_{i}}\right),w_{\mathbf{M}_{i}}(\beta^{\vee})\right\rangle$
$\displaystyle=\operatorname{Re}\left\langle\left(z\varpi_{i}+\rho_{\mathbf{T}}^{\mathbf{M}_{i}}\right),w_{\mathbf{M}_{i}}(\beta^{\vee})\right\rangle$
$\displaystyle=\operatorname{Re}\left(m_{\alpha_{i}}\left(z-\left\langle\rho_{\mathbf{M}_{i}},\alpha^{\vee}_{i}\right\rangle\right)+h(w_{\mathbf{M}_{i}}(\beta^{\vee}))\right)$
$\displaystyle\geq\operatorname{Re}\left(m_{\alpha_{i}}\left(z-\left\langle\rho_{\mathbf{M}_{i}},\alpha^{\vee}_{i}\right\rangle\right)+1\right),$

showing that if
$\operatorname{Re}{z}>\left\langle\rho_{\mathbf{M}_{i}},\alpha^{\vee}_{i}\right\rangle$
then $z\in\mathfrak{Z}^{+}_{i}$.

In the opposite direction recall that

$\left\langle
z\varpi_{i}-\rho_{\mathbf{T}}^{\mathbf{M}_{i}},w_{\mathbf{M}_{i}}^{-1}(\alpha^{\vee}_{i})\right\rangle=\left\langle
z\varpi_{i}+\rho_{\mathbf{T}}^{\mathbf{M}_{i}},\alpha_{i}^{\vee}\right\rangle=z-\left\langle\rho_{\mathbf{M}_{i}},\alpha^{\vee}_{i}\right\rangle+1.$

Thus, if
$\operatorname{Re}z\leq\left\langle\rho_{\mathbf{M}_{i}},\alpha^{\vee}_{i}\right\rangle$
it follows that $z\not\in\mathfrak{Z}^{+}_{i}$. ∎

In addition, $E_{P_{i}}$ admits a meromorphic continuation to $\mathbb{C}$. At
points where $E_{P_{i}}(f,z,g)$ is holomorphic, it gives an automorphic form
of $G(\mathbb{A})$ as a function of $g$.

We say that the Eisenstein series has a pole of finite order $l\geq 0$ for
$z_{0}$, if for every $g\in G(\mathbb{A})$ and for every standard section
$f_{z}$ such that $f_{z}\Big{|}_{\mathbf{K}}=f$, one has

$\left(\Lambda_{P_{i}}^{l}(z_{0})\right)(f)(g)=(z-z_{0})^{l}E_{P_{i}}(f,z,g)\Big{|}_{z=z_{0}}$

that is a holomorphic function at $z=z_{0}$, and there is a standard section
such that $\Lambda_{P_{i}}^{l}(z_{0})(f)$ is not zero.

In this case, the mapping $f\mapsto\Lambda_{P_{i}}^{l}(z_{0})(f)$ is an
automorphic form of $G(\mathbb{A})$.

One way to study the poles of Eisenstein series is via $E_{P_{i}}^{0}(f,z,g)$
and its constant term along the Borel subgroup $\mathbf{B}$. Recall that

$E_{P_{i}}^{0}(f,z,g)=\int_{U(F)\backslash U(\mathbb{A})}E_{P_{i}}(f,z,ug)du.$
(2.0.4)

######  Theorem 2.0.2.

[MW95] The degenerate Eisenstein series $E_{P_{i}}(f,z,g)$ admits a pole of
order $n$ at $z=z_{0}$ if and only if $E_{P}^{0}(f,z,g)$ admits a pole of
order $n$ at $z=z_{0}$.

Next, we would like to rewrite (2.0.4). For this purpose, we recall the global
intertwining operators and their normalizing factors.

###### Global intertwining operator.

Given $w\in\mathit{W}_{\mathbf{G}}$ and
$\lambda\in\mathfrak{a}^{\ast}_{\mathbf{T},\mathbb{C}}$, we formally define a
global intertwining operator

$\mathcal{M}_{w}(\lambda)\>:\>\operatorname{Ind}_{B(\mathbb{A})}^{G(\mathbb{A})}(\lambda)\rightarrow\operatorname{Ind}_{B(\mathbb{A})}^{G(\mathbb{A})}(w\lambda)$

by the formula

$\left(\mathcal{M}_{w}(\lambda)(f)\right)(g)=\int_{U(\mathbb{A})\cap
wU(\mathbb{A})w^{-1}\backslash U(\mathbb{A})}f(w^{-1}ug)du.$ (2.0.5)

As a function of $\lambda$, $\mathcal{M}_{w}(\lambda)$ converges in some right
half plane and admits a meromorphic continuation to whole
$\mathfrak{a}^{\ast}_{\mathbf{T},\mathbb{C}}$. In addition, the intertwining
operator $\mathcal{M}_{w}(\lambda)$ is factorizable in the sense that if
$f=\otimes_{\nu}f_{\nu}$ then

$\mathcal{M}_{w}(\lambda)f=\otimes_{\nu}\mathcal{M}_{w,\nu}(\lambda_{\nu})f_{\nu}.$
(2.0.6)

Here

$\left(\mathcal{M}_{w,\nu}(\lambda_{\nu})(f)\right)(g)=\int_{U(F_{\nu})\cap
wU(F_{\nu})w^{-1}\backslash U(F_{\nu})}f(w^{-1}ug)du.$ (2.0.7)

More details on the local intertwining operator can be found in Section 3.3.
Additionally, $\mathcal{M}_{w}(\lambda)$ satisfies the following cocycle
relation [MW95, II.1.6]. Assume that $w=u_{1}u_{2}$, where
$l(w)=l(u_{1})+l(u_{2})$, then

$\mathcal{M}_{w}(\lambda)=\mathcal{M}_{u_{1}}(u_{2}\lambda)\circ\mathcal{M}_{u_{2}}(\lambda).$
(2.0.8)

In particular, writing $w=w_{\alpha_{i_{k}}}\cdot
w_{\alpha_{i_{k-1}}}\cdot\dots w_{\alpha_{i_{1}}}$, where $l(w)=k$ and
$w_{\alpha_{i_{j}}}$ are simple reflections, one has

$\displaystyle\mathcal{M}_{w}(\lambda)$
$\displaystyle=\mathcal{M}_{w_{\alpha_{i_{k}}}}(w^{{}^{\prime}}_{k-1}\lambda)\circ\mathcal{M}_{w_{\alpha_{i_{k-1}}}}(w^{{}^{\prime}}_{k-2}\lambda)\circ\dots\mathcal{M}_{w_{\alpha_{i_{1}}}}(\lambda),$
$\displaystyle w^{{}^{\prime}}_{j}=w_{\alpha_{i_{j}}}\cdot
w_{\alpha_{i_{j-1}}}\cdot\dots w_{\alpha_{i_{1}}}.$

We recall the properties of $\mathcal{M}_{w}(\lambda)$. For every
$\lambda\in\mathfrak{a}_{\mathbf{T},\mathbb{C}}^{\ast}$ and a place $\nu$, we
set
$f_{\lambda,\nu}^{0}\in\operatorname{Ind}_{B(F_{\nu})}^{G(F_{\nu})}(\lambda_{\nu})$
to be the normalized spherical vector. Then

$\mathcal{M}_{w,\nu}(\lambda_{\nu})f^{0}_{\lambda,\nu}=C_{w,\nu}(\lambda_{\nu})f_{w\lambda,\nu}^{0},$

where

$\displaystyle C_{w,\nu}(\lambda_{\nu})$ $\displaystyle=\prod_{\alpha\in
R(w)}\frac{\zeta_{\nu}(\left\langle\lambda_{\nu},\alpha^{\vee}\right\rangle)}{\zeta_{\nu}(\left\langle\lambda_{\nu},\alpha^{\vee}\right\rangle+1)}\quad\text{
here }\quad R(w)=\left\\{\alpha>0\>:\>w\alpha<0\right\\}.$ (2.0.9)

This is known as the Gindikin–Karpeleviˇc formula.

Returning back to our case, let $\mathbf{P}_{i}$ be a maximal parabolic
subgroup with a Levi decomposition
$\mathbf{P}_{i}=\mathbf{M}_{i}\mathbf{N}_{i}$. We introduce the character
$\chi_{P_{i},z}\in\mathfrak{a}^{\ast}_{T,\mathbb{C}}$, where

$\chi_{P_{i},z}=z\varpi_{\alpha_{i}}-\rho^{\mathbf{M}_{i}}_{\mathbf{T}}.$
(2.0.10)

We omit the subscript $P_{i}$ when there is no confusion. Extending
$\chi_{P_{i},z}$ trivially on $U(\mathbb{A})$, we obtain a character of
$B(\mathbb{A})$. Consider the representation
$\operatorname{Ind}_{B(\mathbb{A})}^{G(\mathbb{A})}(\chi_{P_{i},z})$. Note
that by induction in stages, as an abstract representation,
$\operatorname{Ind}_{P_{i}(\mathbb{A})}^{G(\mathbb{A})}(z\varpi_{i})$ is a
subrepresentation of
$\operatorname{Ind}_{B(\mathbb{A})}^{G(\mathbb{A})}(\chi_{P_{i},z})$. Thus,
the restriction of $\mathcal{M}_{w}(\chi_{P_{i},z})$ to
$\operatorname{Ind}_{P_{i}(\mathbb{A})}^{G(\mathbb{A})}(z\varpi_{i})$ defines
an intertwining operator

$\mathcal{M}_{w}(z)\>:\>\operatorname{Ind}_{P_{i}(\mathbb{A})}^{G(\mathbb{A})}(z\varpi_{i})\rightarrow\operatorname{Ind}_{B(\mathbb{A})}^{G(\mathbb{A})}(w\chi_{P_{i},z}).$

By a standard unfolding as in [GRS97] and [Lan76] (2.0.4) can be rewritten as

$E_{P_{i}}^{0}(f,g,z)=\sum_{w\in W(P_{i},G)}\mathcal{M}_{w}(z)f(g),$ (2.0.11)

where $W(P_{i},G)$ is the set of shortest representatives of
$\mathit{W}_{\mathbf{G}}/\mathit{W}_{\mathbf{M}_{i}}$.

By abuse of notation, for every place $\nu$ we let $f^{0}_{z,\nu}$ (resp.
$f^{0}_{w.z}$) be the normalized spherical vector of
$\operatorname{Ind}_{P(F_{\nu})}^{G(F_{\nu})}$ (resp.
$\operatorname{Ind}_{B(\mathbb{A})}^{G(\mathbb{A})}(w.\chi_{P_{i},z})$). Given
a factorizable standard section
$f_{z}\in\operatorname{Ind}_{P_{i}(\mathbb{A})}^{G(\mathbb{A})}(z\varpi_{i})$
such that $f=\otimes_{\nu}f_{\nu}$ for all $\nu\not\in\mathcal{S}$
($\mathcal{S}$ is a finite set of places, including the archimedean place)
$f_{z,\nu}=f_{z,\nu}^{0}$. Then one has

$\displaystyle\mathcal{M}_{w}(z)f$
$\displaystyle=\left(\otimes_{\nu\in\mathcal{S}}\mathcal{M}_{w,\nu}(z)f_{\nu}\right)\otimes_{\nu\not\in\mathcal{S}}C_{w,\nu}(z)f^{0}_{w.z}$
$\displaystyle=C_{w}(z)\left(\otimes_{\nu\in\mathcal{S}}\left(C_{w,\nu}(z)\right)^{-1}\mathcal{M}_{w,\nu}(z)f_{\nu}\right)\otimes\otimes_{\nu\not\in\mathfrak{S}}f_{w.z,\nu}^{0}$
$\displaystyle=C_{w}(z)\left(\otimes_{\nu\in\mathcal{S}}\mathcal{N}_{w,\nu}(z)f_{\nu}\right)\otimes\otimes_{\nu\not\in\mathfrak{S}}f^{0}_{w.z,\nu}.$
(2.0.12)

Here,

$\displaystyle C_{w,\nu}(z)$ $\displaystyle=C_{w}(\chi_{P_{i},z}),$
$\displaystyle C_{w}(z)$ $\displaystyle=\prod_{\nu}C_{w,\nu}(z),$
$\displaystyle\mathcal{N}_{w,\nu}(z)=\frac{1}{C_{w,\nu}(z)}\mathcal{M}_{w}(z).$

In other words,

$C_{w}(z)=\prod_{\alpha\in
R(w)}\frac{\zeta(\left\langle\chi_{P_{i},z},\alpha^{\vee}\right\rangle)}{\zeta(\left\langle\chi_{P_{i},z},\alpha^{\vee}\right\rangle+1)},$
(2.0.13)

where $\zeta(z)$ is the completed zeta function of $F$, and
$R(w)=\left\\{\alpha>0\>:\>w\alpha<0\right\\}$.

Recall that $\zeta(z)$ is a meromorphic function that satisfies the functional
equation $\zeta(z)=\zeta(1-z)$. Furthermore, $\zeta(z)$ admits simple poles at
$z=0,1$ and non zero for $\operatorname{Re}(z)>1$.

In conclusion, the analytic behavior of $\mathcal{M}_{w}(z)f_{z}$ depends on
the analytic behavior of $C_{w}(z)$ and $\mathcal{N}_{w,\nu}(z)$ for
$\nu\not\in\mathcal{S}$. In the next chapter, we recall the representation
theory over finite places. We also recall the properties of
$\mathcal{N}_{\nu,w}(z)$ over finite places in Section 3.3.

##  Chapter 3 Representation theory

Throughout the entire chapter, $F$ stands for a non-Archimedean local field of
characteristic zero. Let $\mathcal{O}$ denote its ring of integers and
$\mathcal{P}$ be the maximal ideal of $\mathcal{O}$. Let
$q=|\mathcal{O}/\mathcal{P}|$ and $\mathbb{F}_{q}$ be the residue field. We
let $\varpi$ denote a uniformizer of $F$.

Let $H$ be the group of $F$ points of a split, reductive, simply-connected
group $\mathbf{H}$ defined over $F$. Let $T$ be a maximal split torus of $H$,
and let $B=TU$ be a Borel subgroup of $H$, where $U$ is the unipotent radical
of $B$. Set $K=H(\mathcal{O})$, which is a maximal compact subgroup of $H$.

###### Characters.

Recall that for a Levi subgroup $M$ of $H$, we denote the group of rational
characters of $M$ as $\mathbf{X}(M)$. We also set
$\mathfrak{a}_{M,\mathbb{C}}^{\ast}=\mathbf{X}(M)\otimes_{Z}\mathbb{C}$. By
abuse of notation, we let $\lambda\in\mathfrak{a}_{M,\mathbb{C}}^{\ast}$
denote a complex character of $M$, by $\lambda(m)=|\lambda(m)|,$ where
$|\cdot|$ is an absolute value of $F$. A character
$\lambda\in\mathfrak{a}_{M,\mathbb{C}}^{\ast}$ is called unramified if
$\lambda\Big{|}_{M\cap K}=1$. An unramified character of $M$ has the form

$\lambda=\sum_{\alpha\in\Delta_{G}\setminus\Delta_{M}}z_{\alpha}\varpi_{\alpha}.$

Here, $(z_{\alpha})$ are complex numbers, and $z_{\alpha}\varpi_{\alpha}$ is
interpreted as $|\varpi_{\alpha}(\cdot)|^{z_{\alpha}}$.

###### Representations.

The category of complex, smooth, admissible representations of finite length
of $H$ is denoted by $\operatorname{Rep}(H)$. For a standard Levi subgroup $M$
of a parabolic subgroup $P$, we recall two functors associating
representations of $M$ and $H$:

  * •

For a representation $\Omega$ of $M$, we denote the normalized parabolic
induction of $\Omega$ to $H$ by $i_{M}^{H}(\Omega)$. In particular,
$i_{M}^{H}\>:\>\operatorname{Rep}(H)\rightarrow\operatorname{Rep}(M)$.

  * •

For a representation $\pi$ of $H$, we denote the normalized Jacquet functor of
$\pi$ along $P$ by $r^{H}_{M}\left(\pi\right)$. In particular,
$r_{M}^{H}:\>\>\operatorname{Rep}(H)\rightarrow\operatorname{Rep}(M)$.

These two functors are adjoint to each other; namely, they satisfy the
Frobenius reciprocity

$\operatorname{Hom}_{H}(\pi,i_{M}^{H}(\sigma))\simeq\operatorname{Hom}_{M}(r^{H}_{M}\left(\pi\right),\sigma).$
(3.0.1)

It is worth mentioning that for a $1$–dimensional representation $\Omega$ of
$M$, one has $r^{M}_{T}\left(\Omega\right)=\Omega\Big{|}_{T}-\rho_{T}^{M}$
where $\rho_{T}^{M}=\frac{1}{2}\times\sum_{\alpha\in\Phi^{+}_{M}}\alpha$.

###### Grothendieck ring.

In parts of this thesis, it is convenient to consider representations of a
group $H$ as elements in the Grothendieck ring $\mathcal{R}(H)$. To avoid
confusion, we denote by $\left[\pi\right]$ the class of a representation $\pi$
of $H$.

Given an irreducible representation $\sigma$ of $H$ and
$\pi\in\operatorname{Rep}(H)$, we denote by $mult\left(\sigma,\pi\right)$ the
multiplicity of $\sigma$ in the Jordan–Hölder series of $\pi$. We write that
$\sigma\leq\pi$ if $mult\left(\sigma,\pi\right)\geq 1$. Note that if
$\left[\pi\right]=\left[\pi_{1}\right]+\left[\pi_{2}\right]$, then for every
irreducible representation $\sigma$ of $H$, one has

$mult\left(\sigma,\pi\right)=mult\left(\sigma,\pi_{1}\right)+mult\left(\sigma,\pi_{2}\right).$

Furthermore, we write $\pi_{1}\leq\pi_{2}$ if for every irreducible
representation $\sigma$ of $H$,

$mult\left(\sigma,\pi_{1}\right)\leq mult\left(\sigma,\pi_{2}\right).$

For a standard Levi subgroup $M\subset H$, the functors $i_{M}^{H},r_{M}^{H}$
are exact. Hence they give rise to homomorphisms, which by abuse of notation
are denoted also by $i_{M}^{H},r_{M}^{H}$, between the rings $\mathcal{R}(M)$
and $\mathcal{R}(H)$.

We recall the following fundamental results (see [BZ77, Lemma. 2.12] , [Cas74,
Theorem 6.3.6]).

######  Lemma 3.0.1 (Geometric Lemma).

For the Levi subgroups $L,M$ of $H$, let $W^{M,L}$ be the set of shortest
representatives in $\mathit{W}_{H}$ of
$\mathit{W}_{L}\backslash\mathit{W}_{H}/\mathit{W}_{M}$. For a smooth
representation $\Omega$ of $M$, one has

$\left[r^{H}_{L}\left(i_{M}^{H}(\Omega)\right)\right]=\sum_{w\in
W^{M,L}}\left[i_{L^{{}^{\prime}}}^{L}\circ w\circ
r^{M}_{M^{\prime}}\left(\Omega\right)\right],$

where for $w\in W^{M,L}$

$\displaystyle M^{{}^{\prime}}$ $\displaystyle=M\cap w^{-1}(L),$
$\displaystyle L^{{}^{\prime}}$ $\displaystyle=w(M)\cap L.$

In particular, for an admissible representation $\Omega$ of $M$, the Jacquet
functor $r^{H}_{T}\left(i_{M}^{H}\left(\Omega\right)\right)$, as an element of
$\mathcal{R}(T)$, is a finite sum of one-dimensional representations of $T$.
Each such representation is called  an exponent of
$i_{M}^{H}\left(\Omega\right)$.

######  Remark 3.0.2.

For $\lambda\in\mathfrak{a}_{M,\mathbb{C}}^{\ast}$ the following holds

$\left[r^{H}_{T}\left(i_{M}^{H}(\lambda)\right)\right]=\sum_{w\in
W^{T,M}}w\cdot\lambda_{0},\quad\text{where}\quad\lambda_{0}=r^{M}_{T}\left(\lambda\right)=\lambda-\rho^{M}_{T}.$

######  Lemma 3.0.3.

Let $\pi$ be an irreducible representation of $H$ such that
$\pi\hookrightarrow i_{T}^{H}(\lambda_{0})$. Then $\lambda_{0}\leq
r^{H}_{T}\left(\pi\right)$.

###### Proof.

Follows immediately from the Frobenius reciprocity. ∎

The following Lemma plays a crucial role in our work. We refer to it as the
central character argument [HS20, Lemma 3.12].

######  Lemma 3.0.4.

Let $\pi$ be an irreducible constituent of a principal series. If $\lambda\leq
r^{H}_{T}\left(\pi\right)$, then there is an embedding of $\pi$ into
$i_{T}^{H}(\lambda)$.

###  3.1  Branching rule process

The main reference for this section is [HS21, Section 3]. For completeness of
presentation we recall it. The purpose of the branching rule process is to
provide partial knowledge of $r^{H}_{T}\left(\pi\right)$ for an irreducible
representation $\pi$ of $H$. Specifically, given an irreducible representation
$\pi$ of $H$ and an exponent $\lambda\leq r^{H}_{T}\left(\pi\right)$, the tool
produces a set $\mathcal{S}$ of exponents of $\pi$ and numbers
$\left\\{m_{\lambda}\right\\}_{\lambda\in\mathcal{S}}$ such that

$\displaystyle\sum_{\lambda\in\mathcal{S}}m_{\lambda}\lambda\leq
r^{H}_{T}\left(\pi\right).$ (3.1.1)

Note that this also provides a lower bound for $\dim
r^{H}_{T}\left(\pi\right)$.

Since the Jacquet functor is exact, if $\pi_{1}\leq\pi$ then
$r^{H}_{T}\left(\pi_{1}\right)\leq r^{H}_{T}\left(\pi\right)$. In addition,
for every representation $\pi\in\mathcal{R}(H)$ one has

$r^{H}_{T}\left(\pi\right)=r^{M}_{T}\left(r^{H}_{M}\left(\pi\right)\right).$

We say that the representations $\pi_{1}$ and $\pi_{2}$ of $H$ are Jacquet-
equivalent if
$\left[r^{H}_{T}\left(\pi_{1}\right)\right]=\left[r^{H}_{T}\left(\pi_{2}\right)\right].$
Given an irreducible representation $\sigma$ of $H$, we denote by
$\mathcal{S}_{\sigma,H}$ the set of all irreducible representations which are
Jacquet–equivalent to $\sigma$.

######  Definition 3.1.1.

Let $M$ be a standard Levi subgroup of $H$, $\sigma$ be an irreducible
representation of $M$ and $\lambda_{0}$ is a character of $T$. The triple
$(M,\sigma,\lambda_{0})$ is called a branching triple (also known as branching
rule) if $\sigma$ is the unique, irreducible representation of $M$, up to
Jacquet equivalence, such that $\lambda_{0}\leq r^{M}_{T}\left(\sigma\right)$.
The multiplicity of $\lambda_{0}$ in $r^{M}_{T}\left(\sigma\right)$ is denoted
by $n_{\sigma,M}(\lambda_{0})$.

######  Lemma 3.1.2.

Let $(M,\sigma,\lambda_{0})$ be a branching triple. Let $\pi$ be an
irreducible representation of $H$ such that
$mult(\lambda_{0},r^{H}_{T}\left(\pi\right))=k>0$. Then

$\left[r^{H}_{T}\left(\pi\right)\right]\geq\left\lceil\frac{k}{n_{\sigma,M}(\lambda_{0})}\right\rceil\times\left[r^{M}_{T}\left(\sigma\right)\right].$

###### Proof.

Recall that
$\left[r^{H}_{T}\left(\pi\right)\right]=\left[r^{M}_{T}\left(r^{H}_{M}\left(\pi\right)\right)\right]$.
Moreover, one has

$\left[r^{H}_{M}\left(\pi\right)\right]=\sum
m_{\tau}\times\left[\tau\right]\quad
m_{\tau}=mult\left(\tau,r^{H}_{M}\left(\pi\right)\right).$

Since $(M,\sigma,\lambda_{0})$ is a branching triple and
$k\times\lambda_{0}\leq r^{H}_{T}\left(\pi\right)$ for $k>0$, it follows that

$\sum_{\tau\in\mathcal{S}_{M,\sigma}}m_{\tau}\tau\leq
r^{H}_{M}\left(\pi\right).$

Thus, one has

$\left[r^{H}_{T}\left(\pi\right)\right]\geq
r^{M}_{T}\left(\sum_{\tau\in\mathcal{S}_{M,\sigma}}m_{\tau}\times\tau\right)=\left(\sum_{\tau\in\mathcal{S}_{M,\sigma}}{m_{\tau}}\right)\times\left[r^{M}_{T}\left(\sigma\right)\right].$

In particular,

$\left(\sum_{\tau\in\mathcal{S}_{M,\sigma}}m_{\tau}\right)\geq\left\lceil\frac{k}{n_{\sigma,M}(\lambda_{0})}\right\rceil.$

Thus, the claim follows. ∎

In Appendix B we list all the branching triples used in this thesis.

###  3.2 Aubert involution

The main reference for this section is [Ban02, Section 3]. In the Grothendieck
ring, there is an involutive action $D_{H}$ called the Aubert involution. The
Aubert involution is defined as follows

$D_{H}=\sum_{\Phi\subseteq\Delta_{H}}(-1)^{|\Phi|}i_{M_{\Phi}}^{H}\circ
r^{H}_{M_{\Phi}}.$ (3.2.1)

In particular,

$D_{T}(\lambda)=\lambda\quad\forall\lambda\in\mathfrak{a}_{T,\mathbb{C}}^{\ast}.$
(3.2.2)

The Aubert involution satisfies the following properties (all the equalities
are in the Grothendieck ring):

  1. $(Aub_{1})$

$D_{H}\circ i_{M}^{H}=i_{M}^{H}\circ D_{M}$.

  2. $(Aub_{2})$

For a standard Levi subgroup $M=M_{\Theta}$, let $w$ be the longest Weyl-
element of the set $\left\\{w\in\mathit{W}_{H}\ :w^{-1}(\Theta)>0\right\\}$.
Then

$r_{M}^{H}\circ D_{H}=w\circ D_{w^{-1}(M)}\circ r_{w^{-1}(M)}^{H}.$

Note that $w^{-1}(M)$ is also a standard Levi subgroup. In the special case
where $M=T$, one has

$r_{T}^{H}\circ D_{H}=w\circ r_{T}^{H},$

where $w\in\mathit{W}_{H}$ is the longest Weyl element.

  3. $(Aub_{3})$

$D_{H}$ takes any irreducible representation to an irreducible representation.

Combining $(Aub_{1})$ and (3.2.2) one has

$D_{H}(i_{T}^{H}(\lambda))=i_{T}^{H}\left(D_{T}(\lambda)\right)=i_{T}^{H}(\lambda).$
(3.2.3)

###  3.3 Local intertwining operators

Recall that $\mathit{W}_{H}\simeq N_{H}(T)/T$, where $N_{H}(T)$ is the
normalizer of $T$. Hence, it also acts on $\mathfrak{a}_{T,\mathbb{C}}^{\ast}$
as follows

$\left(w\lambda\right)(t)=\lambda\left(w^{-1}tw\right),\quad
w\in\mathit{W}_{G},\quad\lambda\in\mathfrak{a}_{T,\mathbb{C}}^{\ast},\quad
t\in T.$

Let $\lambda\in\mathfrak{a}_{T,\mathbb{C}}^{\ast}$ and $w\in\mathit{W}_{G}$.
We define, formally, the local intertwining operator
$\mathcal{M}_{w}(\lambda)\>:\>i_{T}^{H}(\lambda)\rightarrow
i_{T}^{H}(w\lambda)$ by the formula

$(\mathcal{M}_{w}(\lambda)f)(g)=\int_{\left(U\cap wUw^{-1}\right)\backslash
U}f(w^{-1}ug)du.$ (3.3.1)

As long the integral in (3.3.1) converges, it defines an intertwining operator
between $i_{T}^{H}(\lambda)$ and $i_{T}^{H}(w\lambda)$. The local intertwining
operator $\mathcal{M}_{w}(\lambda)$ satisfies the following properties:

  1. $(A_{1})$

$\mathcal{M}_{w}(\lambda)$ converges for $\lambda$ such that
$\operatorname{Re}\left(\left\langle\lambda,\alpha^{\vee}\right\rangle\right)>0$
for every $\alpha\in R(w)=\left\\{\alpha\>:\>w\alpha<0\right\\}$.

  2. $(A_{2})$

$\mathcal{M}_{w}(\lambda)$ admits a meromorphic continuation to all
$\mathfrak{a}_{T,\mathbb{C}}^{\ast}$.

  3. $(A_{3})$

If $w=u_{1}u_{2}$ and $l(w)=l(u_{1})+l(u_{2})$, then
$\mathcal{M}_{w}(\lambda)=\mathcal{M}_{u_{1}}(u_{2}\cdot\lambda)\circ\mathcal{M}_{u_{2}}(\lambda)$.

  4. $(A_{4})$

Suppose that $Re(\left\langle\lambda,\alpha^{\vee}\right\rangle)>0$ for some
$\alpha\in\Delta_{H}$. Then $\ker\mathcal{M}_{w_{\alpha}}(\lambda)\neq 0$ if
and only if $\left\langle\lambda,\alpha^{\vee}\right\rangle=1$.

  5. $(A_{5})$

Let $f^{0}_{\lambda}\in i_{T}^{G}(\lambda)$ denote the normalized spherical
vector. Then

$\left(\mathcal{M}_{w}(\lambda)f^{0}_{\lambda}\right)=C_{w}(\lambda)f^{0}_{w\cdot\lambda},$

where

$C_{w}(\lambda)=\prod_{\alpha\in
R(w)}\frac{\zeta(\left\langle\lambda,\alpha^{\vee}\right\rangle)}{\zeta(\left\langle\lambda,\alpha^{\vee}\right\rangle+1)}.$

Here, $\zeta(z)=\frac{1}{1-q^{-z}}$ for
$z\in\mathbb{C}\setminus\left\\{0\right\\}$. This property is known as the
Gindikin-Karpelevich formula.

For global reasons, it is customary to use the local normalized intertwining
operator

$\mathcal{N}_{w}(\lambda)=\prod_{\alpha\in
R(w)}\frac{\zeta\left(\left\langle\lambda,\alpha^{\vee}\right\rangle+1\right)}{\zeta\left(\left\langle\lambda,\alpha^{\vee}\right\rangle\right)}\mathcal{M}_{w}(\lambda).$

The normalized intertwining operator $\mathcal{N}_{w}(\lambda)$ also satisfies
$(A_{1})$–$(A_{4})$, while $(A_{3})$ holds in an even wider generality,
namely,

$\mathcal{N}_{u_{1}u_{2}}(\lambda)=\mathcal{N}_{u_{1}}(u_{2}\lambda)\circ\mathcal{N}_{u_{2}}(\lambda)\quad\text{
for any }\quad u_{1},u_{2}\in\mathit{W}_{H}.$

In addition, the analog of $(A_{5})$ is that

$\mathcal{N}_{w}(\lambda)f^{0}_{\lambda}=f^{0}_{w\lambda}.$

######  Lemma 3.3.1.

Let $w=w_{\alpha}$. Then $\mathcal{N}_{w}(\lambda)$ is holomorphic for
$\operatorname{Re}\left(\left\langle\lambda,\alpha^{\vee}\right\rangle\right)\neq-1$.

###### Proof.

Follows from [Win78]. ∎

Given $\lambda\in\mathfrak{a}_{M,\mathbb{C}}^{\ast}$, by induction in stages
one has

$\pi=i_{M}^{H}(\lambda)\hookrightarrow i_{T}^{H}(\lambda-\rho_{T}^{M}).$

Thus, the restrictions of $\mathcal{M}_{w}(\lambda-\rho^{M}_{T})$ and
$\mathcal{N}_{w}(\lambda-\rho^{M}_{T})$ to $\pi$ define the maps

$\mathcal{M}_{w},\mathcal{N}_{w}\>:\>\pi\rightarrow
i_{T}^{H}(w\cdot\left(\lambda-\rho^{M}_{T}\right)).$

Note that

$\displaystyle\pi$ $\displaystyle=i_{M}^{H}(\lambda)$ $\displaystyle
i_{T}^{G}(\lambda+\rho^{M}_{T})\overset{\mathcal{N}_{w_{M}^{0}}}{\twoheadrightarrow}\pi.$

Here, $w_{M}^{0}\in\mathit{W}_{M}$ is the longest Weyl element.

We end this section with a useful Lemma:

######  Lemma 3.3.2.

Let $\pi$ be an irreducible constituent of a principal series $\Pi$. Suppose
that $\lambda_{a.d}\leq r^{H}_{T}\left(\pi\right)$, where
$\lambda_{a.d}=\sum_{\alpha\in\Delta_{H}}c_{\alpha}\varpi_{\alpha}$ with
$c_{\alpha}\leq 0$; then $\pi$ is spherical.

###### Proof.

By the central character argument, one has $\pi\hookrightarrow
i_{T}^{H}(\lambda_{a.d})$. Let

$\Theta=\left\\{\alpha\in\Delta_{H}\>:\>\left\langle\lambda_{a.d},\alpha^{\vee}\right\rangle=0\right\\}=\left\\{\alpha\in\Delta_{H}\>:\>c_{\alpha}=0\right\\}.$

Set $W_{\Theta}=\left\langle w_{\alpha}\>:\>\alpha\in\Theta\right\rangle$.
Thus, (OR) asserts that

$\left|W_{M_{\Theta}}\right|\big{|}mult\left(\lambda_{a.d},r^{G}_{T}\left(\pi\right)\right)$

Geometric Lemma implies

$mult\left(\lambda_{a.d},r^{H}_{T}\left(i_{T}^{H}(\lambda_{a.d})\right)\right)=\operatorname{Stab}_{\mathit{W}_{H}}(\lambda_{a.d})=\left|W_{M_{\Theta}}\right|.$

The last equality holds since $\lambda_{a.d}$ is real, and hence its
stabilizer is generated by the generalized reflections $w_{\beta}$ for
$\beta\in\Phi$ being orthogonal to $\lambda_{a.d}$. As follows, if $\sigma$ is
an irreducible constituent of $i_{T}^{H}(\lambda_{a.d})$ such that
$\lambda_{a.d}\leq r^{H}_{T}\left(\sigma\right)$, then $\sigma=\pi$.

We continue by showing that $\pi$ is spherical. For this purpose, we proceed
as follows:

  * •

Let $w_{0}\in\mathit{W}_{H}$ be the longest Weyl element. Note that

$\lambda_{d}=w_{0}\lambda_{a.d}=\sum_{\alpha\in\Delta_{H}}d_{\alpha}\varpi_{\alpha}$

for some $d_{\alpha}\geq 0$.

  * •

Set

$\Theta^{\prime}=\left\\{\alpha\in\Delta_{H}\>:\>\left\langle\lambda_{d},\alpha^{\vee}\right\rangle=0\right\\}=\left\\{\alpha\in\Delta_{H}\>:\>d_{\alpha}=0\right\\}$

By induction in stages one has

$i_{T}^{H}(\lambda_{d})\simeq
i_{M_{\Theta^{\prime}}}^{H}\left(i_{T}^{M_{\Theta^{\prime}}}\left(0\right)\otimes\lambda_{d}\right).$

The representation $i_{T}^{M_{\Theta^{\prime}}}\left(0\right)$ is tempered and
irreducible. Furthermore, $\lambda_{d}\in\mathbf{X}(M_{\Theta^{\prime}})$ and
for every $\beta\in\Delta_{H}\setminus\Theta^{\prime}$ one has
$\left\langle\lambda_{d},\beta^{\vee}\right\rangle>0$. Thus,
$i_{T}^{H}(\lambda_{d})$ is a standard module. Therefore, according to
Langlands classification theory, it admits a unique irreducible quotient.

  * •

Write $w_{0}=w_{0,M^{\prime}}w_{M^{\prime}}$ where
$w_{0,M^{\prime}}\in\mathit{W}_{H}/\mathit{W}_{M_{\Theta^{\prime}}}$ is the
shortest representative of $w_{0}$ and
$w_{M^{\prime}}\in\mathit{W}_{M_{\Theta^{\prime}}}$ is the longest Weyl
element. Note that $w_{M^{\prime}}\lambda_{d}=\lambda_{d}$. Thus,
$w_{0,M^{\prime}}\lambda_{d}=\lambda_{a.d}$.

Moreover, for every $\beta\in\Phi^{+}_{H}\setminus\Phi_{M_{\Theta}^{\prime}}$
one has $\left\langle\lambda_{d},\beta^{\vee}\right\rangle>0$. Thus, the
intertwining operator $\mathcal{M}_{w_{0,M^{\prime}}}(\lambda)$ is holomorphic
at $\lambda=\lambda_{d}$. To summarize

$\mathcal{M}_{w_{0,M^{\prime}}}(\lambda_{d})\>:\>i_{T}^{H}(\lambda_{d})\to
i_{T}^{H}(\lambda_{a.d})$

In particular,
$\mathcal{M}_{w_{0,M^{\prime}}}(\lambda_{d})f^{0}_{\lambda_{d}}=C\times
f^{0}_{\lambda_{a.d}}$ with $C\neq 0$. Here $C$ is the local Gindikin-
Karpelevich factor.

  * •

On the other hand, [BW00, Chapter XI. Proposition 2.6] shows that the image of
$\mathcal{M}_{w_{0,M^{\prime}}}(\lambda_{d})$ is irreducible. Thus,
$J=\operatorname{Im}\mathcal{M}_{w_{0,M^{\prime}}}(\lambda_{d})$ is spherical.

  * •

Recall that $J\subset i_{T}^{H}(\lambda_{a.d})$; thus, one has
$\lambda_{a.d}\leq r^{H}_{T}\left(J\right)$, and therefore $\pi=J$.

∎

###  3.4 Iwahori–Hecke algebra

A key tool in studying the structure of unramified principal series
$\pi=i_{T}^{H}(\lambda)$, where $\lambda\in\mathfrak{a}_{T,\mathbb{C}}^{\ast}$
is the Iwahori–Hecke algebra. This section is dedicated to presenting the main
properties of this algebra. For more information about the structure of the
Iwahori–Hecke algebra and its modules, see [Pra05] and [HKP10].

Let $\mathbf{H}$ be a split, semi-simple, simply-connected group scheme such
that $H=\mathbf{H}(F)$, and let us assume that $\mathbf{H}$ is defined over
$\mathcal{O}$. Let $\mathbf{B},\mathbf{T}$ be a Borel subgroup and a maximal
split torus such that $B=\mathbf{B}(F)$ and $T=\mathbf{T}(F)$. We set
$K=\mathbf{H}(\mathcal{O})$. Let

$\Psi\>:\>K\>\rightarrow\mathbf{H}(\mathbb{F}_{q})$

denote the projection modulo $\mathcal{P}$.

Let $J=\Psi^{-1}(\mathbf{B}(\mathbb{F}_{q}))$ be an Iwahori subgroup of $H$.
The subgroup $J$ plays an important role in the study of unramified principal
series representations of $H$. By [Cas80, Proposition 2.7], if $\lambda$ is
unramified, then $\Pi=i_{T}^{H}(\lambda)$ is generated by its Iwahori fixed
vectors and so are all of its subquotients.

The Iwahori–Hecke algebra, $\mathcal{H}$, consists of all compactly supported
$J$ –bi-invariant complex functions on $H(F)$, namely,

$\mathcal{H}=\left\\{f\in C_{c}(H)\>:\>f(j_{1}hj_{2})=f(h)\quad\forall
j_{1},j_{2}\in J,\quad h\in H\right\\}.$

The multiplication in $\mathcal{H}$ is given by convolution, and the measure
of $J$ is set to be $1$. By [Bor76], there is an equivalence of categories
between the category of admissible representations of $H$ which are generated
by their $J$–fixed vectors and the category of finitely generated
$\mathcal{H}$–modules. This equivalence of categories sends an admissible
representation $\pi$ of $H$ to the $\mathcal{H}$–module $\pi^{J}$ of $J$–fixed
vectors in $\pi$. Thus, in order to study the structure of
$i_{T}^{H}(\lambda)$, it is sufficient to study the corresponding finite
dimensional $\mathcal{H}$–module.

####  3.4.1 The Bernstein presentation and unramified principal series

The Iwahori–Hecke algebra $\mathcal{H}$ can be described in terms of
generators and relations. One such presentation is known as the Bernstein
presentation. The algebra $\mathcal{H}$ admits two important subalgebras:

  * •

A finite dimensional algebra $\mathcal{H}_{0}=\left\langle
T_{w_{\alpha}}\>\>:\>\alpha\in\Delta_{H}\right\rangle=\operatorname{Span}_{\mathbb{C}}\left\\{T_{w}\>:\>w\in\mathit{W}_{H}\right\\}$
of dimension $|\mathit{W}_{H}|$.

  * •

An infinite dimensional commutative algebra

$\Theta=\left\langle\theta_{\alpha^{\vee}}\>:\>\alpha\in\Delta_{H}\right\rangle=\operatorname{Span}_{\mathbb{C}}\left\\{\theta_{x}\>:\>x\in\mathbb{Z}[\Delta^{\vee}_{H}]\right\\},$

where $\mathbb{Z}[\Delta^{\vee}_{H}]$ is the coroot lattice.

Together, these two subalgebras generate $\mathcal{H}$. The relations between
$T_{w_{\alpha}}$ and $\theta_{\alpha^{\vee}}$ are listed in [Pra05, Section
3]. As a vector space,
$\mathcal{H}\simeq\mathcal{H}_{0}\otimes_{\mathbb{C}}\Theta$.

Given an unramified principal series $\Pi=i_{T}^{H}(\lambda)$, we describe the
left $\mathcal{H}$–module, $\mathcal{H}(\lambda)$, corresponding to it by the
equivalence of categories of [Bor76]. The module $\mathcal{H}(\lambda)$ is
given as follows:

  * •

As a vector space, $\mathcal{H}(\lambda)=\mathcal{H}_{0}$.

  * •

The action of $\mathcal{H}_{0}\subset\mathcal{H}$ on $\mathcal{H}(\lambda)$ is
given by left multiplication.

  * •

By the Bernstein presentation, the action of $\Theta$ on
$\mathcal{H}(\lambda)$ is determined by the action of the generators
$\theta_{\alpha^{\vee}}\in\Theta$ on $T_{e}\in\mathcal{H}(\lambda)$. Let

$\theta_{\alpha^{\vee}}\cdot
T_{e}=q^{\left\langle\lambda,\varpi_{\alpha}^{\vee}\right\rangle}T_{e}.$

####  3.4.2 Intertwining operators

We recall the normalized intertwining operators $\mathcal{N}_{w}(\lambda)$,
which were introduced in Section 3.3 for
$\lambda\in\mathfrak{a}_{T,\mathbb{C}}^{\ast}$. Let $\Pi=i_{T}^{H}(\lambda)$;
these operators induce a map $\mathcal{N}_{w}(\lambda)\Big{|}_{\Pi^{J}}$ of
$\mathcal{H}$–modules. By [Pra05, Section 2], the action of
$\mathcal{N}_{w_{\alpha}}(\lambda)\Big{|}_{\Pi^{J}}$ for $\alpha\in\Delta_{H}$
is given by right-multiplication by the following element

$n_{w_{\alpha}}(\lambda)=\frac{q-1}{q^{z+1}-1}T_{e}+\frac{q^{z}-1}{q^{z+1}-1}T_{w_{\alpha}}\in\mathcal{H}_{0},\quad\text{where}\>\>z=\left\langle\lambda,\alpha^{\vee}\right\rangle.$

Suppose that $\Re z>-1$. Then, $\mathcal{N}_{w_{\alpha}}(\lambda)$ is
holomorphic there. Furthermore, considered as an element of
$\operatorname{End}(H_{0})$,
$\mathcal{N}_{w_{\alpha}}(\lambda)\Big{|}_{\Pi^{J}}$ is a diagonalizable
linear operator with two eigenvalues given by

$\displaystyle\lambda_{1}$
$\displaystyle=\frac{q-1}{q^{z+1}-1}+q\frac{q^{z}-1}{q^{z+1}-1}=1$ and
$\displaystyle\lambda_{2}$
$\displaystyle=\frac{q-1}{q^{z+1}-1}-\frac{q^{z}-1}{q^{z+1}-1}=\frac{q-q^{z}}{q^{z+1}-1},$

with the exception of $z=0$, where $n_{w_{\alpha}}(\lambda)=T_{e}$ is the
identity element and $\mathcal{N}_{w_{\alpha}}(\lambda)=\operatorname{Id}$.
Thus, $\mathcal{N}_{w_{\alpha}}(\lambda)\Big{|}_{\Pi^{J}}$ has a kernel if and
only if $\lambda_{2}=0$, which happens only if $z\in 1+\frac{2\pi
i}{\log(q)}\mathbb{Z}$.

It follows that for $z\in\mathbb{R}$, the injectivity of
$\mathcal{N}_{w_{\alpha}}(\lambda)\Big{|}_{\Pi^{J}}$ does not depend on the
value of $q$.

####  3.4.3 The submodule $\mathcal{H}_{P}(\lambda)$ of $\mathcal{H}(\lambda)$

Let $P$ be a parabolic subgroup of $H$ with Levi part $M$, and let $\lambda$
be an unramified character of $M$ with respect to $M\cap K$. We denote the
longest Weyl element of $\mathit{W}_{M}$ by $w^{0}_{M}$. Let
$\pi=i_{M}^{H}(\lambda)$ with $\lambda\in\mathfrak{a}_{M,\mathbb{C}}^{\ast}$
and $\mathcal{H}_{P}(\lambda)=\pi^{J}$.

We recall that

$\pi=i_{M}^{H}(\lambda)=\operatorname{Im}\mathcal{M}_{w_{M}^{0}}(\lambda+\rho^{M}_{T})=\operatorname{Im}\mathcal{N}_{w_{M}^{0}}(\lambda+\rho^{M}_{T}).$

It follows that $\mathcal{H}_{P}(\lambda)$ is the image of
$\mathcal{N}_{w_{M}^{0}}(\lambda+\rho^{M}_{T})$, and hence it has a basis
given by

$\left\\{T_{u}\cdot triv:u\in\mathit{W}_{H}/\mathit{W}_{M}\right\\}\quad\text{
where }\quad triv=\sum_{w\in\mathit{W}_{M}}T_{w}.$ (3.4.1)

Another way to derive (3.4.1) is by [Jan95, Lemma 2.1.4].

Let $\lambda_{0}=r^{M}_{T}\left(\lambda\right)=\lambda-\rho^{M}_{T}$ and
$\mathcal{H}_{P}(\lambda)=\pi^{J}$. By the equivalence of categories of
[Pra05], for any $w\in\mathit{W}_{H}$, the operator
$\mathcal{N}_{w}(\lambda_{0})\Big{|}_{\pi}$ has a non-trivial kernel if and
only if $\mathcal{N}_{w}(\lambda_{0})\Big{|}_{\pi^{J}}$ has a non-trivial
kernel. Here, the benefit of the finite dimension of $\mathcal{H}_{0}$ (resp.
$\mathcal{H}_{P}(\lambda)$) enters into consideration, namely,

$\ker\mathcal{N}_{w}(\lambda_{0})\Big{|}_{\pi}\neq
0\iff\operatorname{Rank}\left(\Lambda\right)<\dim\mathcal{H}_{P}(\lambda),$

where $\Lambda$ is the matrix representation of
$\mathcal{N}_{w}(\lambda_{0})\Big{|}_{\pi^{J}}$ with respect to the basis of
$\mathcal{H}_{P}(\lambda)$ given in (3.4.1) and the basis of
$\mathcal{H}(w\cdot\lambda_{0})$ given by
$\left\\{T_{w}:w\in\mathit{W}_{H}\right\\}$.

##  Chapter 4 Local part - Results

###  4.1 Outline

We fix the following notations for the entire chapter. Let $\nu$ be a finite
place of $F$. We let $G$ denote the group of $F_{\nu}$–points of a split,
simply-connected group of type $F_{4}$ defined over $F$. We fix a maximal
split torus $T$ of $G$. Given a maximal parabolic subgroup $P=MN$ associated
with $\Delta_{G}\setminus\left\\{\alpha_{i}\right\\}$, we denote by
$\pi_{P_{i},z}=i_{M_{i}}^{G}(z\varpi_{\alpha_{i}})$. We omit the subscript
$P_{i}$ when there is no place for confusion. Recall that

######  Theorem 4.1.1.

[CJ10, Theorem 6.1] Let
$(P,z_{0})\in\left\\{(P_{1},1),(P_{3},\frac{1}{2}),(P_{4},\frac{5}{2})\right\\}$.
The representation $\pi_{P,z_{0}}$ admits a unique irreducible
subrepresentation and a maximal semi-simple quotient of length two.

For $w\in\mathit{W}_{G}$ we set $\mathcal{N}_{w}^{L}(\lambda_{0})$ to be the
restriction of $\mathcal{N}_{w}(\lambda)$ to the line
$L\subset\mathfrak{a}_{T,\mathbb{C}}^{\ast}$ at the point $\lambda_{0}\in L$.
If $L=\chi_{P,z}$ and $w\in W(P,G)$, we simply write
$\mathcal{N}_{w}(z)=\mathcal{N}_{w}^{\chi_{P,z}}(\chi_{P,z})$.

We outline the results of this chapter and explain the ideas behind the
proofs.

######  Proposition 4.1.2.

Let
$(P,z_{0})\in\left\\{(P_{1},1),(P_{3},\frac{1}{2}),(P_{4},\frac{5}{2})\right\\}$.
Then for every $w\in W(P,G)$, the normalized intertwining operator
$\mathcal{N}_{w}(z)\Big{|}_{\pi_{z}}$ is holomorphic at $z=z_{0}$.

###### Proof.

The proof appears in 4.2.1, 4.3.1, 4.4.1. However, the proofs share a common
idea, which we introduce here.

Note that for a line $L\subset\mathfrak{a}_{T,\mathbb{C}}^{\ast}$ and
$w=u_{l}u_{l-1}\cdots u_{1}$, one has

$\mathcal{N}_{w}^{L}(\lambda)=\mathcal{N}_{u_{l}}^{u_{l-1}^{{}^{\prime}}L}(u_{l-1}^{{}^{\prime}}\lambda)\circ\mathcal{N}_{u_{l-1}}^{u_{l-2}^{{}^{\prime}}L}(u_{l-2}^{{}^{\prime}}\lambda)\circ\dots\circ\mathcal{N}_{u_{2}}^{u_{1}L}(u_{1}\lambda)\circ\mathcal{N}_{u_{1}}^{L}(\lambda),$

where $u_{i-1}^{{}^{\prime}}=u_{i-1}\cdot u_{i-2}\cdots u_{i}$. Thus, if for
every $i$ the operator
$\mathcal{N}_{u_{i}}^{u_{i-1}^{{}^{\prime}}L}(u_{i-1}^{{}^{\prime}}\lambda)$
is holomorphic at $\lambda=\lambda_{0}$, then $\mathcal{N}_{w}^{L}(\lambda)$
is also holomorphic at $\lambda=\lambda_{0}$ as a composition of holomorphic
operators.

In each case, we show that for each $w\in W(P,G)$, the operator
$\mathcal{N}_{w}(z)\Big{|}_{\pi_{z}}$ can be written as a composition of
holomorphic operators of one of the following types and is therefore
holomorphic:

  * •

We say that the pair $(w,\lambda_{0})$ defines an operator of $(H)$ type if
for every $\alpha\in R(w)$ one has
$\left\langle\lambda_{0},\alpha^{\vee}\right\rangle\neq-1$. In other words,
the operator $\mathcal{N}_{w}(\lambda)$ is holomorphic at
$\lambda=\lambda_{0}$. In particular, for each line
$L\subset\mathfrak{a}_{T,\mathbb{C}}^{\ast}$ such that $\lambda_{0}\in L$, the
operator $\mathcal{N}_{w}^{L}(\lambda)$ is holomorphic at
$\lambda=\lambda_{0}$.

  * •

We say that the triple $(u,\chi,L)$ defines an operator of $(Z)$ type if the
triple $(u,\chi,L)$ satisfies the conditions of A.0.1. In particular, by
A.0.1, the operator $\mathcal{N}_{w}^{L}(\lambda)$ is holomorphic at
$\lambda=\chi$.

  * •

Let $L\subset\mathfrak{a}_{T,\mathbb{C}}^{\ast}$ be a line and let
$\sigma_{\lambda}$ be a family of representations such that
$\sigma_{\lambda}\hookrightarrow i_{T}^{G}(\lambda)$ for every $\lambda\in L$.
We say that the quadruple $(w,L,\lambda_{0},\sigma_{\lambda_{0}})$ defines an
operator of $(S)$ type, if
$\mathcal{N}_{w}^{L}(\lambda)\Big{|}_{\sigma_{\lambda}}$ is holomorphic at
$\lambda=\lambda_{0}$.

∎

By 4.1.2, each normalized intertwining operator
$\mathcal{N}_{w}(z)\Big{|}_{\pi_{z}}$ is holomorphic at $z=z_{0}$. Hence, its
image is well defined. Let
$\Sigma_{w}=\operatorname{Im}\mathcal{N}_{w}(z_{0})\Big{|}_{\pi_{z_{0}}}$. A
central problem appearing in the local part is how to describe $\Sigma_{w}$.
We start by finding the structure of $\pi_{z_{0}}$, which is a main ingredient
in describing $\Sigma_{w}$. The structure of $\pi_{z_{0}}$ is summarized in
4.1.3. Henceforth we denote
$\Pi_{i}=i_{M_{i}}^{G}\left(z_{0}\cdot\varpi_{\alpha_{i}}\right)$.

######  Theorem 4.1.3.

  1.   2. 1.

If $i=1$ and $z_{0}=1$, one has
$\left(\Pi_{1}\right)_{s.s}=\tau^{1}+\pi_{1}^{1}+\pi_{2}^{1}$. Here,
$\tau^{1}$ is the unique irreducible subrepresentation, and
$\pi_{1}^{1}\oplus\pi_{2}^{1}$ is the maximal semi–simple quotient of
$\Pi_{1}$.

  3. 2.

If $i=3$ and $z_{0}=\frac{1}{2}$, one has
$\left(\Pi_{3}\right)_{s.s}=\tau^{3}+\sigma_{1}^{3}+\sigma_{2}^{3}+\pi_{1}^{3}+\pi_{2}^{3}$.
Here, $\tau^{3}$ is the unique irreducible subrepresentation, and
$\pi_{1}^{3}\oplus\pi_{2}^{3}$ is the maximal semi–simple quotient of
$\Pi_{3}$ and
$\sigma_{1}^{3}\oplus\sigma_{2}^{3}\hookrightarrow\Pi_{3}/\tau^{3}$.

  4. 3.

If $i=4$ and $z_{0}=\frac{5}{2}$, one has
$\left(\Pi_{4}\right)_{s.s}=\tau^{4}+\pi_{1}^{4}+\pi_{2}^{4}$. Here,
$\tau^{4}$ is the unique irreducible subrepresentation, and
$\pi_{1}^{4}\oplus\pi_{2}^{4}$ is the maximal semi–simple quotient of
$\Pi_{4}$.

Moreover, for each irreducible constituent $\sigma$ of
$\Pi_{1},\Pi_{3},\Pi_{4}$, we describe
$\left(r^{G}_{T}\left(\sigma\right)\right)_{s.s}$. In each case, the
multiplicity of $\sigma$ is at most one.

For global reasons, we have a great interest in the following case. Recall
that by 4.1.1, $\Pi_{i}$ admits a maximal semi-simple quotient of length two
$\pi_{1}^{i}\oplus\pi_{2}^{i}$. Assume that there are $u_{1},u_{2}\in W(P,G)$
with the following properties:

  * •

$\Sigma_{u_{1}}=\Sigma_{u_{2}}\simeq\pi_{1}^{i}\oplus\pi_{2}^{i}$.

  * •

Write $u_{2}=su_{1}$. Note that
$s\in\operatorname{Stab}_{\mathit{W}_{G}}(u_{1}\chi_{P,z_{0}})$.

Set

$E_{s}(z)=\mathcal{N}_{s}^{u_{1}\chi_{P,z}}(u_{1}\chi_{P,z})\Big{|}_{\Sigma_{u_{1},z}}.$

where
$\Sigma_{u_{1},z}=\operatorname{Im}\mathcal{N}_{u_{1}}(z)\Big{|}_{\pi_{z}}$.

In these cases, it is crucial to understand the properties of $E_{s}(z)$ at
$z=z_{0}$. These properties will play a major role in describing the residual
representation realized using the leading term of the Eisenstein series.

Their properties are summarized in 4.1.4.

######  Proposition 4.1.4.

With the above notations, one has $E_{s}(z)$, which is holomorphic at
$z=z_{0}$. In addition, there are $a_{1},a_{2}\in\mathbb{C}$ such that
$E_{s}(z_{0})\Big{|}_{\pi_{j}^{i}}=a_{j}\operatorname{Id}$.

###### Proof.

By 4.1.2, the operator $\mathcal{N}_{u_{i}}(z)\Big{|}_{\pi_{z}}$ is
holomorphic at $z=z_{0}$ for $i\in\left\\{1,2\right\\}$. Note that

$\mathcal{N}_{u_{2}}(z)\Big{|}_{\pi_{z}}=E_{s}(z)\circ\mathcal{N}_{u_{1}}(z)\Big{|}_{\pi_{z}}.$

Since $\mathcal{N}_{u_{2}}(z)\Big{|}_{\pi_{z}}$ is holomorphic at $z=z_{0}$,
it follows that $E_{s}(z)$ restricted to the image of
$\mathcal{N}_{u_{i}}(z)\Big{|}_{\pi_{z}}$ is holomorphic at $z=z_{0}$. Since
$\Sigma_{u_{1}}=\Sigma_{u_{2}}\simeq\pi_{1}^{i}\oplus\pi_{2}^{i}$, it follows
that $E_{s}(z_{0})\in\operatorname{End}_{G}\left(\Sigma_{u_{1}}\right)$. By
Schur’s lemma, there are $a_{1},a_{2}\in\mathbb{C}$ such that

$E_{s}(z_{0})\Big{|}_{\pi_{j}^{i}}=a_{j}\operatorname{Id}$

. ∎

The scalars $a_{i}$ are determined in Subsection 4.2.4, Subsection 4.3.4,
Subsection 4.4.4.

###  4.2 $P=P_{4},z_{0}=\frac{5}{2}$

####  4.2.1  Holomorphicity of normalized intertwining operators

######  Proposition 4.2.1.

For each $w\in W(P,G)$, the normalized intertwining operator
$\mathcal{N}_{w}(z)\Big{|}_{\pi_{z}}$ is holomorphic at $z=z_{0}$.

###### Proof.

It is sufficient to show that $\mathcal{N}_{w}(z)$ is holomorphic at
$z=z_{0}$. With the definitions of 4.1.2 , we show that each $w\in W(P,G)$ can
be written as a composition of holomorphic operators of one of two types:
$(H)$ and $(Z)$. For the reader’s convenience, LABEL:Table::_P4_::images
records the decomposition of each $w\in W(P,G)$. ∎

######  Remark 4.2.2.

Let $u\in W(P,G)$ such that
$\dim\left(\operatorname{Im}\mathcal{N}_{u}(z_{0})\right)^{J}=24$. Then
$\mathcal{N}_{u,\nu}(z)$ is holomorphic at $z=z_{0}$ for Archimedean places as
well. This can be proved along the lines of 4.2.1

####  4.2.2 The structure of $\Pi_{4}=i_{M_{4}}^{G}(\frac{5}{2})$

We specify the following exponents of $\Pi_{4}$.

$\displaystyle\lambda_{0}=$ $\displaystyle[-1,-1,-1,7],\text{ the initial
exponent of $\Pi_{4}$. }$ $\displaystyle\lambda_{a.d}=$
$\displaystyle[-1,0,-1,-1],\text{ the anti-dominant exponent in the Weyl orbit
of $\lambda_{0}$. }$ $\displaystyle\lambda_{1}=$ $\displaystyle
w_{\alpha_{3}}\lambda_{a.d}=[-1,-1,1,-2].$ $\displaystyle\lambda_{2}=$
$\displaystyle w_{\alpha_{4}}\lambda_{1}=[-1,-1,-1,2].$

Recall that by [CJ10, Theorem 6.1], $\Pi_{4}$ admits a unique irreducible
subrepresentation $\tau^{4}$ and a maximal semi–simple quotient of length two,
$\pi_{1}^{4}\oplus\pi_{2}^{4}$. In addition, with the notations of [CJ10,
Theorem 6.1], $\pi_{1}^{4}$ is an irreducible constituent of $\Pi_{4}$ such
that $\lambda_{a.d}\leq r^{G}_{T}\left(\pi_{1}^{4}\right)$.

######  Proposition 4.2.3.

The Jacquet module $\left[r^{G}_{T}\left(\sigma\right)\right]$ of
$\sigma\in\left\\{\pi_{1}^{4},\pi_{2}^{4},\tau^{4}\right\\}$ is described in
LABEL:Table::P4. Precisely,

$\left[r^{G}_{T}\left(\sigma\right)\right]=\sum n_{\lambda}\lambda,$

where $\lambda$ and $n_{\lambda}$ are given in LABEL:Table::P4.

###### Proof.

In order to simplify notations, all calculations are preformed in the relevant
Grothendieck ring. This allows us to suppress the notations
$\left[\cdot\right]$.

Note that

$r^{G}_{T}\left(\tau^{4}\right)+r^{G}_{T}\left(\pi_{1}^{4}\right)+r^{G}_{T}\left(\pi_{2}^{4}\right)\leq
r^{G}_{T}\left(\Pi_{4}\right)$

and $r^{G}_{T}\left(\Pi_{4}\right)$ can be easily computed by the Geometric
Lemma. This calculation is given in the second column of LABEL:Table::P4.

Recall that $\pi_{1}^{4}$ is an irreducible constituent having
$\lambda_{a.d}\leq r^{G}_{T}\left(\pi_{1}^{4}\right)$. An application of the
following sequence of branching rules yields:

  * •

Branching rule of type (OR) on $\lambda_{a.d}$ yields
$2\times\lambda_{a.d}\leq r^{G}_{T}\left(\pi_{1}^{4}\right)$.

  * •

Branching rule of type ($C_{2}(b)$) on $\lambda_{a.d}$ with respect to
$M_{\alpha_{2},\alpha_{3}}$ yields

$2\times\lambda_{a.d}+w_{\alpha_{3}}\lambda_{a.d}\leq
r^{G}_{T}\left(\pi_{1}^{4}\right).$

  * •

Branching rule of type ($A_{2}$) on $\lambda_{a.d}$ with respect to
$M_{\alpha_{1},\alpha_{2}}$ yields

$2\times\lambda_{a.d}+w_{\alpha_{1}}\lambda_{a.d}\leq
r^{G}_{T}\left(\pi_{1}^{4}\right).$

  * •

Branching rule of type ($A_{1}$) on $w_{\alpha_{3}}\lambda_{a.d}$ with respect
to $M_{\alpha_{4}}$ yields

$\underbrace{w_{\alpha_{3}}\lambda_{a.d}}_{\lambda_{1}}+\underbrace{w_{\alpha_{4}}w_{\alpha_{3}}\lambda_{a.d}}_{\lambda_{2}}\leq
r^{G}_{T}\left(\pi_{1}^{4}\right).$

To summarize, one has

$2\times\lambda_{a.d}+w_{\alpha_{1}}\lambda_{a.d}+\lambda_{1}+\lambda_{2}\leq
r^{G}_{T}\left(\pi_{1}^{4}\right).$ (4.2.1)

By our notation, $\tau^{4}$ is the unique irreducible subrepresentation of
$\Pi_{4}$. By induction in stages, one has $\Pi_{4}\hookrightarrow
i_{T}^{G}(\lambda_{0})$. In particular, $\tau^{4}$ is an irreducible
subrepresentation of $i_{T}^{G}(\lambda_{0})$. Hence, by Frobenius
reciprocity, $\lambda_{0}\leq r^{G}_{T}\left(\tau^{4}\right)$.

Since $mult\left(\lambda_{0},r^{G}_{T}\left(\Pi_{4}\right)\right)=1$, one has

$1\leq mult\left(\lambda_{0},r^{G}_{T}\left(\tau^{4}\right)\right)\leq
mult\left(\lambda_{0},r^{G}_{T}\left(\Pi_{4}\right)\right)=1.$

Thus $mult\left(\lambda_{0},r^{G}_{T}\left(\tau^{4}\right)\right)=1$. An
application of a sequence of branching rules of type ($A_{1}$) yields

$\displaystyle\lambda_{0}+w_{\alpha_{4}}\lambda_{0}+w_{\alpha_{3}}w_{\alpha_{4}}\lambda_{0}+w_{\alpha_{2}}w_{\alpha_{3}}w_{\alpha_{4}}\lambda_{0}$
$\displaystyle+w_{\alpha_{1}}w_{\alpha_{2}}w_{\alpha_{3}}w_{\alpha_{4}}\lambda_{0}+w_{\alpha_{3}}w_{\alpha_{2}}w_{\alpha_{3}}w_{\alpha_{4}}\lambda_{0}$
$\displaystyle+w_{\alpha_{4}}w_{\alpha_{3}}w_{\alpha_{2}}w_{\alpha_{3}}w_{\alpha_{4}}\lambda_{0}$
$\displaystyle+w_{\alpha_{2}}w_{\alpha_{3}}w_{\alpha_{2}}w_{\alpha_{3}}w_{\alpha_{4}}\lambda_{0}+w_{\alpha_{4}}w_{\alpha_{1}}w_{\alpha_{3}}w_{\alpha_{2}}w_{\alpha_{3}}w_{\alpha_{4}}\lambda_{0}$
$\displaystyle+w_{\alpha_{2}}w_{\alpha_{1}}w_{\alpha_{3}}w_{\alpha_{2}}w_{\alpha_{3}}w_{\alpha_{4}}\lambda_{0}w_{\alpha_{2}}w_{\alpha_{4}}w_{\alpha_{1}}w_{\alpha_{3}}w_{\alpha_{2}}w_{\alpha_{3}}w_{\alpha_{4}}\lambda_{0}$
$\displaystyle+w_{\alpha_{3}}w_{\alpha_{2}}w_{\alpha_{4}}w_{\alpha_{1}}w_{\alpha_{3}}w_{\alpha_{2}}w_{\alpha_{3}}w_{\alpha_{4}}\lambda_{0}$
$\displaystyle+w_{\alpha_{2}}w_{\alpha_{3}}w_{\alpha_{2}}w_{\alpha_{4}}w_{\alpha_{1}}w_{\alpha_{3}}w_{\alpha_{2}}w_{\alpha_{3}}w_{\alpha_{4}}\lambda_{0}$
$\displaystyle+w_{\alpha_{3}}w_{\alpha_{2}}w_{\alpha_{1}}w_{\alpha_{3}}w_{\alpha_{2}}w_{\alpha_{3}}w_{\alpha_{4}}\lambda_{0}$
$\displaystyle+w_{\alpha_{4}}w_{\alpha_{3}}w_{\alpha_{2}}w_{\alpha_{1}}w_{\alpha_{3}}w_{\alpha_{2}}w_{\alpha_{3}}w_{\alpha_{4}}\lambda_{0}$
$\displaystyle+w_{\alpha_{3}}w_{\alpha_{4}}w_{\alpha_{3}}w_{\alpha_{2}}w_{\alpha_{1}}w_{\alpha_{3}}w_{\alpha_{2}}w_{\alpha_{3}}w_{\alpha_{4}}\lambda_{0}$
$\displaystyle+w_{\alpha_{2}}w_{\alpha_{3}}w_{\alpha_{4}}w_{\alpha_{3}}w_{\alpha_{2}}w_{\alpha_{1}}w_{\alpha_{3}}w_{\alpha_{2}}w_{\alpha_{3}}w_{\alpha_{4}}\lambda_{0}$
$\displaystyle\leq r^{G}_{T}\left(\tau^{4}\right).$ (4.2.2)

On the other hand, we know that the length of $\Pi_{4}$ is at least three.
However, if $\lambda\not\in\left\\{\lambda_{1},\lambda_{2}\right\\}$ and
$\lambda\leq r^{G}_{T}\left(\Pi_{4}\right)$, it follows that

$mult\left(\lambda,r^{G}_{T}\left(\Pi_{4}\right)\right)=br\left(\lambda,r^{G}_{T}\left(\tau^{4}\right)\right)+br\left(\lambda,r^{G}_{T}\left(\pi_{1}^{4}\right)\right),$

where $br\left(\lambda,r^{G}_{T}\left(\sigma\right)\right)$ is a lower bound
of the multiplicity of $\lambda$ in $r^{G}_{T}\left(\sigma\right)$. In
addition, for $\lambda\in\left\\{\lambda_{1},\lambda_{2}\right\\}$, one has
$mult\left(\lambda,r^{G}_{T}\left(\Pi_{4}\right)\right)=2$.

This, combined with (4.2.1) and (4.2.2), implies that for the third
irreducible constituent $\pi_{2}^{4}$, one has

$r^{G}_{T}\left(\pi_{2}^{4}\right)\leq\lambda_{1}+\lambda_{2}.$

An application of ($A_{1}$) branching rule with respect to $M_{\alpha_{4}}$
yields

$\lambda_{1}\leq r^{G}_{T}\left(\sigma\right)\iff\lambda_{2}\leq
r^{G}_{T}\left(\sigma\right).$

Hence,

$r^{G}_{T}\left(\pi_{2}^{4}\right)=\lambda_{1}+\lambda_{2}.$

Let us summarize. For every $\lambda\leq r^{G}_{T}\left(\Pi_{4}\right)$, one
has

$br\left(\lambda,r^{G}_{T}\left(\tau^{4}\right)\right)+br\left(\lambda,r^{G}_{T}\left(\pi_{1}^{4}\right)\right)+br\left(\lambda,r^{G}_{T}\left(\pi_{2}^{4}\right)\right)=mult\left(\lambda,r^{G}_{T}\left(\Pi_{4}\right)\right).$

Thus, we conclude that
$mult\left(\lambda,r^{G}_{T}\left(\tau^{4}\right)\right)=br\left(\lambda,r^{G}_{T}\left(\tau^{4}\right)\right)$
and

$\displaystyle mult\left(\lambda,r^{G}_{T}\left(\pi_{1}^{4}\right)\right)$
$\displaystyle=br\left(\lambda,r^{G}_{T}\left(\pi_{1}^{4}\right)\right),$
$\displaystyle mult\left(\lambda,r^{G}_{T}\left(\pi_{2}^{4}\right)\right)$
$\displaystyle=br\left(\lambda,r^{G}_{T}\left(\pi_{2}^{4}\right)\right).$

∎

######  Corollary 4.2.4.

  1. 1.

$\Pi_{4}$ is of length three. In particular,
$\Pi_{4}/\tau^{4}\simeq\pi_{1}^{4}\oplus\pi_{2}^{4}$.

  2. 2.

One has

$\displaystyle\dim
r^{G}_{T}\left(\pi_{1}^{4}\right)=\dim\left(\pi_{1}^{4}\right)^{J}$
$\displaystyle=5,$ $\displaystyle\dim
r^{G}_{T}\left(\pi_{2}^{4}\right)=\dim\left(\pi_{2}^{4}\right)^{J}$
$\displaystyle=2,$ $\displaystyle\dim
r^{G}_{T}\left(\tau^{4}\right)=\dim\left(\tau^{4}\right)^{J}$
$\displaystyle=17.$

With the same notations as above one has

######  Proposition 4.2.5.

For every $\lambda\in\mathit{W}_{G}\cdot\lambda_{a.d}$, one has
$mult\left(\pi_{i}^{4},i_{T}^{G}(\lambda)\right)=1$ for $i=1,2$.

###### Proof.

Given $\lambda\in\mathit{W}_{G}\cdot\lambda_{a.d}$, an application of the
Geometric Lemma asserts that
$\left(i_{T}^{G}(\lambda)\right)_{s.s}=\left(i_{T}^{G}(\lambda_{a.d})\right)_{s.s}$.
Thus, it is enough to show that
$mult\left(\pi_{i}^{4},i_{T}^{G}(\lambda_{a.d})\right)=1$ for
$i\in\left\\{1,2\right\\}$.

Let $\lambda\in\mathit{W}_{G}\cdot\lambda_{a.d}$. Geometric lemma yields that

$mult\left(\lambda,i_{T}^{G}(\lambda_{a.d})\right)=|\operatorname{Stab}_{\mathit{W}_{G}}(\lambda)|.$

Since $|\operatorname{Stab}_{\mathit{W}_{G}}(\lambda)|$ does not depend on the
point of the orbit, it follows that

$mult\left(\lambda,i_{T}^{G}(\lambda_{a.d})\right)=|\operatorname{Stab}_{\mathit{W}_{G}}(\lambda_{a.d})|=2.$

By LABEL:Table::_P4_::images, one has

$\displaystyle
mult\left(\lambda_{a.d},r^{G}_{T}\left(\pi_{1}^{4}\right)\right)=$
$\displaystyle 2,$ $\displaystyle
mult\left(\lambda_{1},r^{G}_{T}\left(\pi_{1}^{4}\right)\right)=$
$\displaystyle 1,$ $\displaystyle
mult\left(\lambda_{1},r^{G}_{T}\left(\pi_{2}^{4}\right)\right)=$
$\displaystyle 1.$

Hence,

$mult\left(\lambda_{a.d},r^{G}_{T}\left(\pi_{1}^{4}\right)\right)=mult\left(\lambda_{a.d},r^{G}_{T}\left(i_{T}^{G}(\lambda_{a.d})\right)\right)=2.$

In conclusion, $mult\left(\pi_{1}^{4},i_{T}^{G}(\lambda_{a.d})\right)=1$. Note
that,

$\underbrace{mult\left(\lambda_{1},r^{G}_{T}\left(\pi_{1}^{4}\right)\right)}_{=1}+mult\left(\lambda_{1},r^{G}_{T}\left(\pi_{2}^{4}\right)\right)=mult\left(\lambda_{1},i_{T}^{G}(\lambda_{a.d})\right)=2.$

Thus, $mult\left(\pi_{2}^{4},i_{T}^{G}(\lambda_{a.d})\right)=1$. ∎

######  Proposition 4.2.6.

The representation $\pi_{1}^{4}$ is spherical.

###### Proof.

A direct consequence of 3.3.2. ∎

######  Proposition 4.2.7.

Let $\lambda\in\left\\{\lambda_{1},\lambda_{2}\right\\}$. Then,
$\pi_{1}^{4}\oplus\pi_{2}^{4}\hookrightarrow i_{T}^{G}(\lambda)$.

###### Proof.

By LABEL:Table::P4, one has

$\displaystyle mult\left(\lambda,r^{G}_{T}\left(\pi_{1}^{4}\right)\right)=$
$\displaystyle 1,$ $\displaystyle
mult\left(\lambda,r^{G}_{T}\left(\pi_{2}^{4}\right)\right)=$ $\displaystyle
1.$

Hence, the central character argument yields that both $\pi_{1}^{4}$ and
$\pi_{2}^{4}$ have an embedding into $i_{T}^{G}(\lambda_{1})$ and
$i_{T}^{G}(\lambda_{2})$. In particular,
$\pi_{1}^{4}\oplus\pi_{2}^{4}\hookrightarrow i_{T}^{G}(\lambda)$. ∎

####  4.2.3 Images of $\Sigma_{w}$

######  Lemma 4.2.8.

Let $w\in W(P,G)$.

  1. 1.

If $\dim\Sigma_{w}^{J}=5$, then $\Sigma_{w}\simeq\pi_{1}^{4}$.

  2. 2.

If $\dim\Sigma_{w}^{J}=24$, then $\Sigma_{w}\simeq\Pi_{4}$.

  3. 3.

If $\dim\Sigma_{w}^{J}=7$, then
$\Sigma_{w}\simeq\pi_{1}^{4}\oplus\pi_{2}^{4}$.

###### Proof.

Recall that $\Sigma_{w}$ is a component of $\Pi_{4}$. Thus, in the
Grothendieck ring, one has

$\Sigma_{w}=m_{\tau}\times\tau^{4}+m_{1}\times\pi_{1}^{4}+m_{2}\times\pi_{2}^{4}$

for suitable $m_{\tau},m_{1},m_{2}\in\left\\{0,1\right\\}$. In particular,

$\dim\Sigma_{w}^{J}=m_{\tau}\times\dim\left(\tau^{4}\right)^{J}+m_{1}\times\dim\left(\pi_{1}^{4}\right)^{J}+m_{2}\times\dim\left(\pi_{2}^{4}\right)^{J}.$

Thus, $\dim\Sigma_{w}^{J}=5$ (resp. $\dim\Sigma_{w}^{J}=24$) implies that
$\Sigma_{w}\simeq\pi_{1}^{4}$ (resp. $\Sigma_{w}\simeq\Pi_{4}$).

Additionally, if $\dim\Sigma_{w}^{J}=7$, then
$(\Sigma_{w})_{s.s}=\pi_{1}^{4}+\pi_{2}^{4}$. By LABEL:Table::_P4_::images,
$\dim\Sigma_{w}^{J}=7$ implies that $\Sigma_{w}\hookrightarrow
i_{T}^{G}(\lambda_{1})$ or $\Sigma_{w}\hookrightarrow i_{T}^{G}(\lambda_{2})$.

By 4.2.7 , it follows that there is an embedding of
$\pi_{1}^{4}\oplus\pi_{2}^{4}$ into $i_{T}^{G}(\lambda_{1})$ and
$i_{T}^{G}(\lambda_{2})$. By 4.2.5, the claim follows. ∎

####  4.2.4 Action of the stabilizer

In this subsection, we set

$\displaystyle u_{1}$
$\displaystyle=\left(w_{1}w_{2}w_{3}w_{4}w_{3}w_{2}w_{3}w_{1}w_{2}w_{3}w_{4}\right),$
$\displaystyle u_{1,1}$
$\displaystyle=\left(w_{3}w_{2}w_{3}\right)\left(w_{1}w_{2}w_{3}w_{4}w_{3}w_{2}w_{3}w_{1}w_{2}w_{3}w_{4}\right),$
$\displaystyle u_{2}$
$\displaystyle=w_{1}w_{2}w_{3}w_{4}w_{2}w_{3}w_{1}w_{2}w_{3}w_{4},$
$\displaystyle u_{2,1}$
$\displaystyle=\left(w_{4}\right)\cdot\left(w_{3}w_{2}w_{3}\right)\cdot\left(w_{1}w_{2}w_{3}w_{4}w_{3}w_{2}w_{3}w_{1}w_{2}w_{3}w_{4}\right).$

Note that $u_{i}\chi_{P,z_{0}}=u_{i,1}\chi_{P,z_{0}}$ for $i=1,2$. Moreover,
using braid relations, one has

$u_{2,1}=\left(w_{4}\right)\cdot\left(w_{3}w_{2}w_{3}\right)\cdot\left(w_{4}\right)\cdot\left(w_{1}w_{2}w_{3}w_{4}w_{2}w_{3}w_{1}w_{2}w_{3}w_{4}\right).$

######  Proposition 4.2.9.

One has $\Sigma_{u_{1}}=\Sigma_{u_{1,1}}$ and
$\Sigma_{u_{2}}=\Sigma_{u_{2,1}}$.

###### Proof.

By 4.2.8, for each $u\in\left\\{u_{1},u_{1,1},u_{2},u_{2,1}\right\\}$ one has
$\Sigma_{u}\simeq\pi_{1}^{4}\oplus\pi_{2}^{4}$. By 4.2.5, it follows that
$\Sigma_{u_{1}}=\Sigma_{u_{1,1}}$ and $\Sigma_{u_{2}}=\Sigma_{u_{2,1}}$. ∎

Observe that, $\lambda_{1}=u_{1}\chi_{P,z_{0}}$ and
$\lambda_{2}=u_{2}\chi_{P,z_{0}}$. Let $s=w_{3}w_{2}w_{3}$. Put

$E_{s}(z)=\mathcal{N}_{s}^{u_{1}\chi_{P,z}}(u_{1}\chi_{P,z}).$

######  Proposition 4.2.10.

One has $E_{s}(z)$ that is holomorphic at $z=z_{0}$. In addition,
$E_{s}(z_{0})$ is an isomorphism such that

$E_{s}(z_{0})\Big{|}_{\pi_{1}^{4}}=\operatorname{Id},\quad
E_{s}(z_{0})\Big{|}_{\pi_{2}^{4}}=-\operatorname{Id}.$

###### Proof.

An application of A.0.1, with the following arguments:

$\displaystyle u$ $\displaystyle=s,$ $\displaystyle\chi$
$\displaystyle=u_{1}\chi_{P,z_{0}},$ $\displaystyle L$
$\displaystyle=u_{1}\chi_{P,z_{0}}$

yields that $E_{s}(z)$ is holomorphic at $z=z_{0}$. Moreover, with the
notations of Appendix A, one has $a_{1}=1$. A.0.3 yields that $E_{s}(z_{0})$
is a diagonalizable operator with two eigenvalues $\pm 1$. Note that
$E_{s}(z_{0})\>:i_{T}^{G}(\lambda_{1})\rightarrow i_{T}^{G}(\lambda_{1})$.
Thus, by putting together 4.2.5, 4.2.7 and Schur’s Lemma, there are
$b_{1},b_{2}\in\mathbb{C}$ such that
$E_{s}(z_{0})\Big{|}_{\pi_{i}^{4}}=b_{i}\operatorname{Id}$. By
LABEL:Table::P4, one has

$\displaystyle
mult\left(\lambda_{a.d},r^{G}_{T}\left(\pi_{1}^{4}\right)\right)$
$\displaystyle=2,$ $\displaystyle
mult\left(\lambda_{a.d},r^{G}_{T}\left(\pi_{2}^{4}\right)\right)$
$\displaystyle=0.$ $\displaystyle
mult\left(\lambda_{1},r^{G}_{T}\left(\pi_{1}^{4}\right)\right)$
$\displaystyle=1,$ $\displaystyle
mult\left(\lambda_{1},r^{G}_{T}\left(\pi_{2}^{4}\right)\right)$
$\displaystyle=1.$

In particular, $\pi_{2}^{4}$ is not a subrepresentation of
$i_{T}^{G}(\lambda_{a.d})$, and hence,
$\pi_{2}^{4}\hookrightarrow\ker\mathcal{N}_{w_{3}}(u_{1}\chi_{P,z_{0}})$.
Thus, A.0.3 implies that
$E_{s}(z_{0})\Big{|}_{\pi_{2}^{4}}=-\operatorname{Id}$. On the other hand, by
4.2.6, $\pi_{1}^{4}$ is the unique irreducible spherical constituent of
$\Pi_{4}$. Hence

$E_{s}(z_{0})\Big{|}_{\pi_{1}^{4}}=\operatorname{Id}.$

∎

######  Proposition 4.2.11.

Let $u=w_{4}w_{3}w_{2}w_{3}w_{4}$. Let
$E^{1}_{u}(z)=\mathcal{N}_{s}^{u_{2}\chi_{P,z}}(u_{2}\chi_{P,z})$. Then
$E^{1}_{u}(z)$ is holomorphic at $z=z_{0}$. In addition, one has

$E_{u}^{1}(z_{0})\Big{|}_{\pi_{1}^{4}}=\operatorname{Id},\quad
E_{u}^{1}(z_{0})\Big{|}_{\pi_{2}^{4}}=-\operatorname{Id}.$

###### Proof.

Let

$\displaystyle F_{1}(z)$
$\displaystyle=\mathcal{N}_{w_{4}}^{u_{1,1}\chi_{P,z}}(u_{1,1}\chi_{P,z})\>:\>i_{T}^{G}(u_{1,1}\chi_{P,z})\rightarrow
i_{T}^{G}(u_{1,2}\chi_{P,z}),$ $\displaystyle F_{2}(z)$
$\displaystyle=\mathcal{N}_{w_{4}}^{u_{2}\chi_{P,z}}(u_{2}\chi_{P,z})\>:\>i_{T}^{G}(u_{2}\chi_{P,z})\rightarrow
i_{T}^{G}(u_{1}\chi_{P,z}).$

With the notations of 4.2.10, one has

$E^{1}_{u}(z)=F_{1}(z)\circ E_{s}(z)\circ F_{2}(z).$

Since $E_{s}(z)$ is holomorphic at $z=z_{0}$, it suffices to show that both
$F_{1}(z)$ and $F_{2}(z)$ are holomorphic at $z=z_{0}$. Note that

$\displaystyle\left\langle u_{2}\chi_{P,z_{0}},\alpha^{\vee}_{4}\right\rangle$
$\displaystyle=-2,$ $\displaystyle\left\langle
w_{3}w_{2}w_{3}u_{1}\chi_{P,z_{0}},\alpha^{\vee}_{4}\right\rangle$
$\displaystyle=2.$

Hence, both $F_{1}(z)$ and $F_{2}(z)$ are holomorphic at $z=z_{0}$. Moreover,
both $F_{1}(z_{0})$ and $F_{2}(z_{0})$ are isomorphisms. Note
$\left(w_{3}w_{2}w_{3}u_{1}\right)\cdot\chi_{P,z_{0}}=\left(w_{4}u_{2}\right)\cdot\chi_{P,z_{0}}$.
Hence,

$F=F_{1}(z_{0})\circ
F_{2}(z_{0})\in\operatorname{End}\left(i_{T}^{G}(u_{2}\chi_{P,z_{0}})\right).$

By the properties of the normalized intertwining operator, one has
$F=\operatorname{Id}$. Thus, the action of $E^{1}_{u}(z_{0})$ on each
irreducible subrepresentaion $\pi_{i}^{4}$ is determined by the action of
$E_{s}(z_{0})$. By 4.2.10, the claims follows. ∎

In LABEL:Table::_P4_::images we summarize all the information for
$i_{M_{4}}^{G}(z_{0})$ that is needed for the global part.

###  4.3 $P=P_{1},z_{0}=1$

####  4.3.1  Holomorphicity of normalized intertwining operators

######  Proposition 4.3.1.

For each $w\in W(P,G)$, the normalized intertwining operator
$\mathcal{N}_{w}(z)\Big{|}_{\pi_{z}}$ is holomorphic at $z=z_{0}$.

###### Proof.

With the definitions of 4.1.2 , we shall show that each $w\in W(P,G)$ can be
written as a composition of holomorphic operators of one of three types:
$(H)$, $(Z)$ and $(S)$. Let

$\displaystyle u_{1}$
$\displaystyle=w_{2}w_{3}w_{4}w_{2}w_{3}w_{1}w_{2}w_{3}w_{4}w_{1}w_{2}w_{3}w_{2}w_{1},$
$\displaystyle u_{2}$
$\displaystyle=w_{1}w_{2}w_{3}w_{4}w_{2}w_{3}w_{1}w_{2}w_{3}w_{4}w_{1}w_{2}w_{3}w_{2}w_{1}.$

For $w\in W(P,G)\setminus\left\\{u_{1},u_{2}\right\\}$, this decomposition is
recorded in LABEL:Table::_P1_::images. For $u_{1},u_{2}$, we shall use braid
relations in order to preform a decomposition. Note that

$\displaystyle u_{1}$
$\displaystyle=\underbrace{\left(w_{2}w_{3}w_{4}w_{2}w_{3}w_{2}\right)}_{u}\underbrace{\left(w_{1}w_{2}w_{3}w_{4}w_{2}w_{3}w_{2}w_{1}\right)}_{w^{\prime}},$
$\displaystyle u_{2}$
$\displaystyle=\left(w_{1}\right)\underbrace{\left(w_{2}w_{3}w_{4}w_{2}w_{3}w_{2}\right)}_{u}\underbrace{\left(w_{1}w_{2}w_{3}w_{4}w_{2}w_{3}w_{2}w_{1}\right)}_{w^{\prime}}.$

In 4.3.3 we show that
$(u,w^{\prime}\chi_{P,z},\operatorname{Im}\mathcal{N}_{w^{\prime}}(z)\Big{|}_{\pi_{z}},w^{\prime}\chi_{P,z_{0}})$
defines an operator of $S$ type. It is easy to verify that both
$(w^{\prime},\chi_{P,z_{0}})$ and $(w_{4},u_{1}\chi_{P,z_{0}})$ define an
operator of $H$ type. Thus, $u_{1}$ is of type $SH$ and $u_{2}$ is of type
$HSH$ ∎

######  Remark 4.3.2.

Let $u\in W(P,G)$ such that
$\dim\left(\operatorname{Im}\mathcal{N}_{u}(z_{0})\right)^{J}=24$. Then
$\mathcal{N}_{u,\nu}(z)$ is holomorphic at $z=z_{0}$ for Archimedean places as
well. This can be proved along the lines of 4.3.1

######  Proposition 4.3.3.

Let $u=w_{2}w_{3}w_{4}w_{2}w_{3}w_{2}$ and
$w^{\prime}=w_{1}w_{2}w_{3}w_{4}w_{2}w_{3}w_{2}w_{1}$. The quadruplet
$(u,w^{\prime}\chi_{P,z},\operatorname{Im}\mathcal{N}_{w^{\prime}}(z)\Big{|}_{\pi_{z}},w^{\prime}\chi_{P,z_{0}})$
defines an operator of $S$ type.

###### Proof.

By induction in stages, one has

$\displaystyle\pi_{z}\hookrightarrow\sigma_{z}$
$\displaystyle=i_{M_{1}}^{G}\left(i_{M_{\alpha_{3},\alpha_{4}}}^{M_{1}}(-\rho^{M_{1}}_{M_{\alpha_{3},\alpha_{4}}})\otimes
z\varpi_{\alpha_{1}}\right)$ $\displaystyle\simeq
i_{M_{\alpha_{3},\alpha_{4}}}^{G}(z\varpi_{1}+3\varpi_{1}-2\varpi_{2}),$

where $M_{\alpha_{3},\alpha_{4}}$ stands for the standard Levi subgroup $M$
such that $\Delta_{M}=\left\\{\alpha_{3},\alpha_{4}\right\\}$ and

$\rho^{M_{1}}_{M_{\alpha_{3},\alpha_{4}}}=\rho^{M_{1}}_{T}-\rho^{M_{\alpha_{3},\alpha_{4}}}_{T}-\rho^{M_{1}}_{T}=-3\varpi_{1}+2\varpi_{2}.$

Note that
$w^{\prime}\left\\{\alpha_{3},\alpha_{4}\right\\}=\left\\{\alpha_{3},\alpha_{4}\right\\}$.
Thus, as long as $\mathcal{N}_{w^{\prime}}(z)\Big{|}_{\pi_{z}}$ is
holomorphic, the image lies in

$i_{M_{\alpha_{3},\alpha_{4}}}^{G}\left(w^{\prime}\cdot(z\varpi_{1}+3\varpi_{1}-2\varpi_{2})\right)=\underbrace{i_{M_{\alpha_{3},\alpha_{4}}}^{G}(z(-2\varpi_{1}+\varpi_{2})-\varpi_{2})}_{\Sigma_{z}}.$

Thus,

$\mathcal{N}_{w^{\prime}}(z)\Big{|}_{\pi_{z}}\>:\>i_{M_{1}}^{G}(z\varpi_{1})\rightarrow
i_{M_{\alpha_{3},\alpha_{4}}}^{G}(z(-\varpi_{1}+\varpi_{2})-\varpi_{2}).$

Hence, it is enough to show that
$(u,w^{\prime}\chi_{P,z},\Sigma_{z},w^{\prime}\chi_{P,z_{0}})$ defines an
operator of $S$ type. We transfer the issue to the Iwahori–Hecke algebra.
Namely, let $n_{u}(z)$ be the operator associated with
$\mathcal{N}_{u}^{w^{\prime}\chi_{P,z}}(w^{\prime}\chi_{P,z})$ in the
Iwahori–Hecke algebra. We show that $n_{u}(z)\Big{|}_{\Sigma_{z}^{J}}$ is
holomorphic at $z=z_{0}$.

Recall that

$\Sigma_{z}^{J}=\operatorname{Span}\left\\{Triv\cdot
T_{u^{\prime}}\>:\>u^{\prime}\in W(M_{\alpha_{3},\alpha_{4}},G)\right\\}.$

Here
$Triv=1+T_{w_{4}w_{3}}+T_{w_{4}}+T_{w_{3}w_{4}w_{3}}+T_{w_{3}w_{4}}+T_{w_{3}}$
and $W(M_{\alpha_{3},\alpha_{4}},G)$ in the set of shortest representatives of
$\mathit{W}_{G}/\mathit{W}_{M_{\alpha_{3},\alpha_{4}}}$.

Since

$\operatorname{Im}n_{u}(z)\Big{|}_{\Sigma_{z}^{J}}=Span\left\\{T_{u^{\prime}}\cdot
Triv\cdot n_{u}(z)\>\>\>u^{\prime}\in
W(M_{\alpha_{3},\alpha_{4}},G)\right\\},$

we reduce the problem to show the holomorphicity of one element $Triv\cdot
n_{u}(z)$ at $z=1$.

Recall that for a simple reflection $w_{i}$, we let
$n_{w_{i}}(z)=\frac{q-1}{q^{z+1}-1}+\frac{q^{z}-1}{q^{z+1}-1}.$

Then,

$n_{u}(z)=n_{w_{2}}(z)\underbrace{\left(n_{w_{3}}(2z-1)n_{w_{2}}(z-1)n_{w_{4}}(2z-2)n_{w_{3}}(2z-3)\right)}_{e(z)}n_{w_{2}}(z-2).$

Direct computations show that

$\displaystyle\lim_{z\to z_{0}}e(z)$
$\displaystyle=\frac{1}{2(q+1)}\cdot\left(T_{w_{2}}+T_{w_{3}w_{2}}+2\cdot
T_{w_{3}w_{4}}+2\cdot T_{w_{4}}+4\cdot T_{w_{3}}\right)$
$\displaystyle-\frac{1}{2q(q+1)}\cdot\left(T_{w_{2}w_{3}}+T_{w_{3}w_{2}w_{3}}+2\cdot
T_{w_{4}w_{3}}+2\cdot T_{w_{3}w_{4}w_{3}}\right)$
$\displaystyle-\frac{(q-1)}{(q+1)}\cdot T_{1},$ $\displaystyle\lim_{z\to
z_{0}}Triv\cdot n_{w_{2}}(z)e(z)$
$\displaystyle=\frac{Triv\cdot\left(T_{w_{2}}+1\right)}{q+1}.$

Note that $\operatorname{ord}\limits_{z=z_{0}}n_{w_{2}}(z-2)=1$. Thus, in
order to show that $Triv\cdot n_{u}(z)$ is holomorphic at $z=z_{0}$, it
suffices to show that

$\frac{Triv(T_{w_{2}}+T_{e})}{q+1}\Lambda(n_{w_{2}}(z-2))=0.$

Here, $\Lambda(n_{w_{2}}(z-2))$ is the leading term of $n_{w_{2}}(z-2)$ around
$z=z_{0}$.

Note that

$\Lambda(n_{w_{2}}(z-2))=\frac{1}{\log
q}\left((q-1)+(q^{-1}-1)T_{w_{2}}\right).$

However,

$\left(1+T_{w_{2}}\right)\cdot\left((q-1)+(q^{-1}-1)T_{w_{2}}\right)=\left(q-1\right)\left(1+T_{w_{2}}\right)\left(q-T_{w_{2}}\right)=0.$

Hence, the claim follows. ∎

####  4.3.2  Structure of $\Pi_{1}=i_{M_{1}}^{G}(1)$

We specify the following exponents of $\Pi_{1}$:

$\displaystyle\lambda_{0}=$ $\displaystyle[4,-1,-1,-1],\text{ the initail
exponent of $\Pi_{1}$ }.$ $\displaystyle\lambda_{a.d}=$
$\displaystyle[-1,0,-1,0],\text{ the anti-dominant exponent in the Weyl orbit
of $\lambda_{0}$ }.$ $\displaystyle\lambda_{1}=$ $\displaystyle
w_{\alpha_{2}}w_{\alpha_{3}}\lambda_{a.d}=[-2,1,-1,-1].$
$\displaystyle\lambda_{2}=$ $\displaystyle
w_{\alpha_{1}}\lambda_{1}=[2,-1,-1,-1].$

Recall that by [CJ10, Theroem 6.1], $\Pi_{1}$ admits a unique irreducible
subrepresentation $\tau^{1}$ and a maximal semi–simple quotient of length two,
$\pi_{1}^{1}\oplus\pi_{2}^{1}$. In addition, with their notations,
$\pi_{1}^{1}$ is an irreducible constituent of $\Pi_{1}$ such that
$\lambda_{a.d}\leq r^{G}_{T}\left(\pi_{1}^{1}\right)$.

######  Proposition 4.3.4.

The Jacquet module $r^{G}_{T}\left(\sigma\right)_{s.s}$ of
$\sigma\in\left\\{\pi_{1}^{1},\pi_{2}^{1},\tau^{1}\right\\}$ is described in
LABEL:Table::P1. Specifically,

$r^{G}_{T}\left(\sigma\right)_{s.s}=\sum n_{\lambda}\lambda,$

where $\lambda$ and $n_{\lambda}$ are given in LABEL:Table::P1.

###### Proof.

In order to simplify the notations, all calculations are preformed in the
relevant Grothendieck ring. This allows us to suppress the notation
$\left[\cdot\right]$.

Note that

$r^{G}_{T}\left(\tau^{1}\right)+r^{G}_{T}\left(\pi_{1}^{1}\right)+r^{G}_{T}\left(\pi_{2}^{1}\right)\leq
r^{G}_{T}\left(\Pi_{1}\right)$

and $r^{G}_{T}\left(\Pi_{1}\right)$ can be easily computed by the Geometric
Lemma. This calculation is given in the second column of LABEL:Table::P1.

Recall that $\pi_{1}^{1}$ is an irreducible constituent having
$\lambda_{a.d}\leq r^{G}_{T}\left(\pi_{1}^{1}\right)$. An application of the
following sequence of branching rules yields

  * •

Branching rule of type (OR) on $\lambda_{a.d}$ yields
$4\times\lambda_{a.d}\leq r^{G}_{T}\left(\pi_{1}^{1}\right)$.

  * •

Branching rule of type ($C_{3}(a)$) on $\lambda_{a.d}$ with respect to
$M_{\alpha_{2},\alpha_{3},\alpha_{4}}$ yields

$4\times\lambda_{a.d}+2\times
w_{\alpha_{3}}\lambda_{a.d}+w_{\alpha_{2}}w_{\alpha_{3}}\lambda_{a.d}\leq
r^{G}_{T}\left(\pi_{1}^{1}\right).$

  * •

Branching rule of type ($A_{2}$) on $\lambda_{a.d}$ with respect to
$M_{\alpha_{1},\alpha_{2}}$ yields

$4\times\lambda_{a.d}+2\times w_{\alpha_{1}}\lambda_{a.d}\leq
r^{G}_{T}\left(\pi_{1}^{1}\right).$

  * •

Branching rule of type ($A_{2}$) on $w_{\alpha_{1}}\lambda_{a.d}$ with respect
to $M_{\alpha_{3},\alpha_{4}}$ yields

$2\times
w_{\alpha_{1}}\lambda_{a.d}+w_{\alpha_{3}}w_{\alpha_{1}}\lambda_{a.d}\leq
r^{G}_{T}\left(\pi^{1}_{1}\right).$

Continuing by an application of all possible ($A_{1}$) rules yields

$\displaystyle 4\times\lambda_{a.d}+2\times
w_{\alpha_{3}}\lambda_{a.d}+2\times w_{\alpha_{1}}\lambda_{a.d}$
$\displaystyle+w_{\alpha_{3}}w_{\alpha_{1}}\lambda_{a.d}+w_{\alpha_{2}}w_{\alpha_{3}}w_{\alpha_{1}}\lambda_{a.d}+\lambda_{1}+\lambda_{2}$
$\displaystyle+w_{\alpha_{3}}w_{\alpha_{2}}w_{\alpha_{3}}w_{\alpha_{1}}\lambda_{a.d}+w_{\alpha_{4}}w_{\alpha_{3}}w_{\alpha_{2}}w_{\alpha_{3}}w_{\alpha_{1}}\lambda_{a.d}$
$\displaystyle\leq r^{G}_{T}\left(\pi_{1}^{1}\right).$ (4.3.1)

By induction in stages, one has $\Pi_{1}\hookrightarrow
i_{T}^{G}(\lambda_{0})$. By our notations, $\tau^{1}$ is the unique
irreducible subrepresentation of $\Pi_{1}$. In particular,
$\tau^{1}\hookrightarrow i_{T}^{G}(\lambda_{0})$. Hence, by Frobenius
reciprocity, $\lambda_{0}\leq r^{G}_{T}\left(\tau^{1}\right)$.

Since $mult\left(\lambda_{0},r^{G}_{T}\left(\Pi_{!}\right)\right)=1$, one has

$1\leq mult\left(\lambda_{0},r^{G}_{T}\left(\tau^{1}\right)\right)\leq
mult\left(\lambda_{0},r^{G}_{T}\left(\Pi_{1}\right)\right)=1.$

Thus, $mult\left(\lambda_{0},r^{G}_{T}\left(\tau^{1}\right)\right)=1$. An
application of a sequence of branching rules of type ($A_{1}$) yields

$\displaystyle\lambda_{0}+w_{\alpha_{1}}\lambda_{0}+w_{\alpha_{2}}w_{\alpha_{1}}\lambda_{0}+w_{\alpha_{3}}w_{\alpha_{2}}w_{\alpha_{1}}\lambda_{0}+w_{\alpha_{2}}w_{\alpha_{3}}w_{\alpha_{2}}w_{\alpha_{1}}\lambda_{0}$
(4.3.2)
$\displaystyle+w_{\alpha_{4}}w_{\alpha_{3}}w_{\alpha_{2}}w_{\alpha_{1}}\lambda_{0}+w_{\alpha_{4}}w_{\alpha_{2}}w_{\alpha_{3}}w_{\alpha_{2}}w_{\alpha_{1}}\lambda_{0}w_{\alpha_{1}}w_{\alpha_{4}}w_{\alpha_{2}}w_{\alpha_{3}}w_{\alpha_{2}}w_{\alpha_{1}}\lambda_{0}$
$\displaystyle\leq r^{G}_{T}\left(\tau^{1}\right).$

On the other hand, we know that the length of $\Pi_{1}$ is at least three.
Noting that $\lambda\not\in\left\\{\lambda_{1},\lambda_{2}\right\\}$ and
$\lambda\leq r^{G}_{T}\left(\Pi_{1}\right)$, it follows that

$mult\left(\lambda,r^{G}_{T}\left(\Pi_{1}\right)\right)=br\left(\lambda,r^{G}_{T}\left(\tau^{1}\right)\right)+br\left(\lambda,r^{G}_{T}\left(\pi_{1}^{1}\right)\right),$

where $br\left(\lambda,r^{G}_{T}\left(\sigma\right)\right)$ is a lower bound
for the multiplicity of $\lambda$ in $r^{G}_{T}\left(\sigma\right)$.

In addition, for $\lambda\in\left\\{\lambda_{1},\lambda_{2}\right\\}$ one has
$mult\left(\lambda,r^{G}_{T}\left(\Pi_{1}\right)\right)=2$. This, combined
with (4.3.2) and 4.3.2, implies that the third irreducible constituent
$\pi_{2}^{1}$ has

$r^{G}_{T}\left(\pi^{1}_{2}\right)\leq\lambda_{1}+\lambda_{2}.$

An application of ($A_{1}$) branching rule with respect to $M_{\alpha_{1}}$
yields

$\lambda_{1}\leq r^{G}_{T}\left(\sigma\right)\iff\lambda_{2}\leq
r^{G}_{T}\left(\sigma\right).$

Hence,

$r^{G}_{T}\left(\pi_{2}^{1}\right)=\lambda_{1}+\lambda_{2}.$

Thus, we conclude that
$mult\left(\lambda,r^{G}_{T}\left(\tau^{1}\right)\right)=br\left(\lambda,r^{G}_{T}\left(\tau^{1}\right)\right)$
and

$\displaystyle mult\left(\lambda,r^{G}_{T}\left(\pi_{1}^{1}\right)\right)$
$\displaystyle=br\left(\lambda,r^{G}_{T}\left(\pi_{1}^{1}\right)\right),$
$\displaystyle mult\left(\lambda,r^{G}_{T}\left(\pi_{2}^{1}\right)\right)$
$\displaystyle=br\left(\lambda,r^{G}_{T}\left(\pi_{2}^{1}\right)\right).$

∎

######  Corollary 4.3.5.

  1. 1.

$\Pi_{1}$ is of length three. In particular
$\Pi_{1}/\tau^{1}\simeq\pi_{1}^{1}\oplus\pi_{2}^{1}$.

  2. 2.

One has

$\displaystyle\dim
r^{G}_{T}\left(\pi_{1}^{1}\right)=\dim\left(\pi_{1}\right)^{J}$
$\displaystyle=14,$ $\displaystyle\dim
r^{G}_{T}\left(\pi^{1}_{2}\right)=\dim\left(\pi_{2}^{1}\right)^{J}$
$\displaystyle=2,$ $\displaystyle\dim
r^{G}_{T}\left(\tau^{1}\right)=\dim\left(\tau^{1}\right)^{J}$
$\displaystyle=8.$

######  Proposition 4.3.6.

The representation $\pi_{1}^{1}$ is spherical.

###### Proof.

A direct consequence of 3.3.2. ∎

With the same notations as above, one has

######  Proposition 4.3.7.

For every $\lambda\in\mathit{W}_{G}\cdot\lambda_{a.d}$, one has
$mult\left(\pi_{i}^{1},i_{T}^{G}(\lambda)\right)=1$ for $i=1,2$.

###### Proof.

The proof of $mult\left(\pi_{1}^{1},i_{T}^{G}(\lambda_{a.d}\right)=1$ follows
through the same lines as in 4.2.5. In order to show that
$mult\left(\pi_{2}^{1},i_{T}^{G}(\lambda_{a.d})\right)=1$, we proceed as
follows:

Let $\lambda_{3}=w_{\alpha_{3}}\lambda_{1}=[-2,0,1,-2]$. Note that
$\lambda_{3}\not\leq
r^{G}_{T}\left(\pi_{1}^{1}\right),r^{G}_{T}\left(\pi_{2}^{1}\right)$. Thus,
there exists an irreducible constituent $\sigma$ of $i_{T}^{G}(\lambda_{a.d})$
such that $\lambda_{3}\leq r^{G}_{T}\left(\sigma\right)$.

Applying (OR) and ($C_{2}(b)$) implies that

$2\times\lambda_{3}+\lambda_{1}\leq r^{G}_{T}\left(\sigma\right).$

Note that:

  * •

If $mult\left(\lambda_{3},r^{G}_{T}\left(\sigma\right)\right)=2$, then there
is another irreducible constituent $\sigma_{1}$ with

$2\times\lambda_{3}+\lambda_{1}\leq r^{G}_{T}\left(\sigma_{1}\right).$

Hence,

$\displaystyle
mult\left(\lambda_{1},r^{G}_{T}\left(i_{T}^{G}(\lambda_{a.d})\right)\right)$
$\displaystyle\geq
mult\left(\lambda_{1},r^{G}_{T}\left(\pi_{1}^{1}\right)\right)+mult\left(\lambda_{1},r^{G}_{T}\left(\pi_{2}^{1}\right)\right)$
$\displaystyle+mult\left(\lambda_{1},r^{G}_{T}\left(\sigma\right)\right)+mult\left(\lambda_{1},r^{G}_{T}\left(\sigma^{1}\right)\right)$
$\displaystyle\geq
1+1+1+1=mult\left(\lambda_{1},r^{G}_{T}\left(i_{T}^{G}(\lambda_{a.d})\right)\right).$

Hence, $\pi^{1}_{2}$ appears with multiplicity one in
$i_{T}^{G}(\lambda_{a.d})$.

  * •

If $mult\left(\lambda_{3},r^{G}_{T}\left(\sigma\right)\right)=4$, then

$4\times\lambda_{3}+2\times\lambda_{1}\leq r^{G}_{T}\left(\sigma\right).$

As before, we show that $\pi^{1}_{2}$ appears with multiplicity one in
$i_{T}^{G}(\lambda_{a.d})$.

∎

######  Proposition 4.3.8.

Let $\lambda\in\left\\{\lambda_{1},\lambda_{2}\right\\}$. Then,
$\pi_{1}^{1}\oplus\pi_{2}^{1}\hookrightarrow i_{T}^{G}(\lambda)$.

###### Proof.

As in 4.2.7. ∎

####  4.3.3 Determine $\Sigma_{w}$

######  Lemma 4.3.9.

Let $w\in W(P,G)$.

  1. 1.

If $\dim\Sigma_{w}^{J}=14$, then $\Sigma_{w}\simeq\pi_{1}^{1}$.

  2. 2.

If $\dim\Sigma_{w}^{J}=24$, then $\Sigma_{w}\simeq\Pi_{1}$.

  3. 3.

If $\dim\Sigma_{w}^{J}=16$, then
$\Sigma_{w}\simeq\pi_{1}^{1}\oplus\pi_{2}^{1}$.

###### Proof.

The proof is the same as in 4.2.8. ∎

####  4.3.4 Action of the stabilizer

In this subsection, we set

$\displaystyle u_{1}$
$\displaystyle=\left(w_{1}w_{2}w_{3}w_{4}w_{2}w_{3}w_{2}w_{1}\right),$
$\displaystyle u_{1,1}$
$\displaystyle=\left(w_{2}w_{3}w_{4}w_{2}w_{3}w_{2}\right)\left(w_{1}w_{2}w_{3}w_{4}w_{2}w_{3}w_{2}w_{1}\right),$
$\displaystyle u_{2}$
$\displaystyle=\left(w_{2}w_{3}w_{4}w_{2}w_{3}w_{2}w_{1}\right),$
$\displaystyle u_{2,1}$
$\displaystyle=\left(w_{1}\right)\left(w_{2}w_{3}w_{4}w_{2}w_{3}w_{2}\right)\left(w_{1}\right)\left(w_{2}w_{3}w_{4}w_{2}w_{3}w_{2}w_{1}\right).$

Note that $\lambda_{1}=u_{1}\chi_{P,z_{0}}=u_{1,1}\chi_{P,z_{0}}$ and
$\lambda_{2}=u_{2}\chi_{P,z_{0}}=u_{2,1}\chi_{P,z_{0}}$.

######  Proposition 4.3.10.

One has $\Sigma_{u_{1}}=\Sigma_{u_{1,1}}$ and
$\Sigma_{u_{2}}=\Sigma_{u_{2,1}}$.

###### Proof.

Same as the proof of 4.2.9. ∎

Let $s=w_{2}w_{3}w_{4}w_{2}w_{3}w_{2}$. Set
$E_{s}(z)=\mathcal{N}_{s}^{u_{1}\chi_{P,z}}(u_{1}\chi_{P,z})\Big{|}_{\mathcal{N}_{u_{1}}(\pi_{z})}$.

######  Proposition 4.3.11.

Then $E_{s}(z)$ is holomorphic at $z=z_{0}$ and

$\displaystyle E_{s}\Big{|}_{\pi_{1}^{1}}$ $\displaystyle=\operatorname{Id},$
$\displaystyle E_{s}\Big{|}_{\pi_{2}^{1}}$ $\displaystyle=-\operatorname{Id}.$

###### Proof.

The holomorphicity in the content of 4.3.3. Note that
$E_{s}(z_{0})\in\operatorname{End}(\Sigma_{u_{1}})$. Hence by Schur’s Lemma,
it acts by a scalar on each summand. Since $\pi_{1}^{1}$ is spherical, it
follows that $E_{s}\Big{|}_{\pi_{1}^{1}}=\operatorname{Id}$.

We proceed as follows:

  * •

By the proof of [CJ10, Theorem 6.1] and 4.3.3, it follows that

$\operatorname{Im}\mathcal{N}_{u_{1}}(z_{0})\Big{|}_{\pi_{z_{0}}}\subset
i_{M_{\alpha_{3},\alpha_{4}}}^{G}(-2\varpi_{2})\simeq
i_{M_{1}}^{G}\left(i_{M_{\alpha_{3},\alpha_{4}}}^{M_{1}}(\tau)\right),$

where $\tau=1_{GL_{3}}\rtimes|\cdot|^{-2}$.

  * •

By [CJ10, Lemma 5.1(5)], one has
$i_{M_{\alpha_{3},\alpha_{4}}}^{M_{1}}(\tau)=\sigma_{1}\oplus\sigma_{2}$.

  * •

The proof of [CJ10, Theorem 6.1] yields $\pi_{i}^{1}\hookrightarrow
i_{M_{1}}^{G}\left(\sigma_{i}\right)$. Hence it is enough to understand
$E_{s}(z_{0})\Big{|}_{i_{M_{1}}^{G}(\sigma_{i})}$.

  * •

With the notations of B.2.3, one has
$\Sigma_{1}=\operatorname{Res}^{M_{1}}_{M_{1}^{der}}(i_{M_{\alpha_{3},\alpha_{4}}}^{M_{1}}(\tau))$.
In particular, it is of length two. Since
$i_{M_{\alpha_{3},\alpha_{4}}}^{M_{1}}(\tau)$ is of length two also, we deduce
that

$\Sigma_{1}=\operatorname{Res}^{M_{1}}_{M_{1}^{der}}(\sigma_{1})\oplus\operatorname{Res}^{M_{1}}_{M_{1}^{der}}(\sigma_{2}).$

  * •

Going through the proof of [BJ04, Proposition 4.1], it follows that the action
of $E_{s}(z_{0})$ is determined via its action on
$\operatorname{Res}^{M_{1}}_{M_{1}^{der}}(\sigma_{i})$. By [Han15], the
operator admits two eigenvalues $\pm 1$. On each summand it acts differently.
Hence,

$E_{s}\Big{|}_{\pi_{2}^{1}}=-\operatorname{Id}.$

∎

######  Proposition 4.3.12.

Let $u=w_{1}w_{2}w_{3}w_{4}w_{2}w_{3}w_{2}w_{1}$. We set

$E^{1}_{u}(z)=\mathcal{N}_{s}^{u_{2}\chi_{P,z}}(u_{2}\chi_{P,z})\Big{|}_{\operatorname{Im}\mathcal{N}_{u_{2}}(\pi_{z})}.$

Then $E^{1}_{u}(z)$ is holomorphic at $z=z_{0}$. In addition, one has

$E_{u}^{1}(z_{0})\Big{|}_{\pi_{1}^{1}}=\operatorname{Id},\quad
E_{u}^{1}(z_{0})\Big{|}_{\pi_{2}^{1}}=-\operatorname{Id}.$

###### Proof.

As in 4.2.11. ∎

In LABEL:Table::_P1_::images we summarize all the information for
$i_{M_{1}}^{G}(z_{0})$ needed for the global part.

###  4.4 $P=P_{3},z_{0}=\frac{1}{2}$

####  4.4.1  Holomorphicity of normalized intertwining operators

######  Proposition 4.4.1.

For any $w\in W(P,G)$, the operator $\mathcal{N}_{w}(z)$ is holomorphic at
$z=z_{0}$.

###### Proof.

It is sufficient to show that $\mathcal{N}_{w}(z)$ is holomorphic at
$z=z_{0}$. With the definitions of 4.1.2, we shall show that each $w\in
W(P,G)$ can be written as a composition of holomorphic operators of one of two
types: $(H)$ and $(Z)$. For the convenience of the reader, we record the
decomposition of each $w\in W(P,G)$ in LABEL:Table::_P3_::images.

Note that the above decomposition is not necessarily of a reduced form.
However, except for the word listed below, the reduced word admits such a
decomposition. The unique word for which such a decomposition does not exist
is

$w=w_{2}w_{3}w_{1}w_{2}w_{3}w_{4}w_{3}w_{2}w_{3}w_{1}w_{2}w_{3}w_{4}w_{2}w_{3}w_{1}w_{2}w_{3}.$

In this case we shall show that it is of the type $HZHZHZH$.

It is easy to verify that $w=w_{4}u,$ where

$u=\left(w_{2}w_{3}w_{1}w_{2}\right)\cdot\left(w_{3}w_{4}\right)\cdot\left(w_{3}w_{2}w_{3}\right)\cdot\left(w_{1}w_{2}w_{3}w_{4}\right)\cdot\left(w_{3}w_{2}w_{3}\right)\cdot\left(w_{1}w_{2}w_{3}.\right)$

Note that $u$ is of type $ZHZHZH$ and $(w_{4},u\lambda_{0})$ is of type $H$.
Thus the claim follows. ∎

We end this section with the following observation:

######  Remark 4.4.2.

Let $u\in W(P,G)$ such that
$\dim\left(\operatorname{Im}\mathcal{N}_{u}(z_{0})\right)^{J}>61$. Then
$\mathcal{N}_{u,\nu}(z)$ is holomorphic at $z=z_{0}$ for Archimedean places as
well. It can be proved along the lines of 4.4.1.

####  4.4.2  The structure of $\Pi_{3}=i_{M_{3}}^{G}(\frac{1}{2})$

For the remaining part, we shall specify the following exponents of
$\Pi_{3}=i_{M_{3}}^{G}(\frac{1}{2})$. Let

$\displaystyle\lambda_{a.d}$ $\displaystyle=[0,0,-1,0],$
$\displaystyle\lambda_{0}$ $\displaystyle=[-1,-1,3,-1],$
$\displaystyle\lambda_{1}$ $\displaystyle=[0,-1,1,-1],$
$\displaystyle\lambda_{2}$ $\displaystyle=[1,0,-1,-1].$

Note that $\lambda_{a.d}$ (resp. $\lambda_{0}$) is the anti–dominant (resp.
initial exponent) of $\Pi_{3}$. According to [CJ10, Thm 6.1 ], the
representation $\Pi_{3}$ admits a unique irreducible subrepresentation
$\tau^{3}$ and a maximal semi-simple quotient $\pi_{1}^{3}\oplus\pi_{2}^{3}$.
Moreover, with the notations of [CJ10, Thm 6.1 ], $\pi_{1}^{3}$ is the unique
irreducible constituent of $\Pi_{3}$, having $\lambda_{a.d}$ in its Jacquet
module. On the other hand, $\pi_{1}^{3},\pi_{2}^{3}$ are the only irreducible
constituents having $\lambda_{1}$ in their Jacquet module.

We start this subsection by giving a lower bound for $\dim
r^{G}_{T}\left(\pi_{1}^{3}\right)$ and $\dim
r^{G}_{T}\left(\pi_{2}^{3}\right)$. Then, using several computations in the
Iwahori–Hecke algebra, we shall show that the length of $\Pi_{3}$ is at least
five. We end this subsection by proving that its length is exactly $5$ and
determining its structure.

By 4.4.1, each normalized intertwining operator
$\mathcal{N}_{w}^{\chi_{P,z}}(\lambda)$ is holomorphic at
$\lambda=\lambda_{0}=\chi_{P,z_{0}}$. Thus, one can consider its restriction
to $\Pi_{3}$. For shorthand writing, for every $w\in W(P,G)$ we denote by
$\Sigma_{w}=\operatorname{Im}\left(\mathcal{N}_{w}^{\chi_{P,z}}(\lambda_{0})\right)\Big{|}_{\Pi_{3}}$.

All the calculations in this section are preformed in the relevant
Grothendieck ring. This allows us to suppress the notation
$\left[\cdot\right]$.

#####  Determine $r^{G}_{T}\left(\pi_{1}^{3}\right)$

In this section we give a lower bound for $\dim
r^{G}_{T}\left(\pi_{1}^{3}\right)$.

######  Lemma 4.4.3.

The representation $\pi_{1}^{3}$ is a common irreducible constituent of
$\Pi_{3}=i_{M_{3}}^{G}(\frac{1}{2})$ and
$\Pi^{\prime}=i_{M_{2}}^{G}(\frac{1}{2})$. Moreover,
$mult\left(\lambda_{2},r^{G}_{T}\left(\pi_{1}^{3}\right)\right)=2$ and

$mult\left(w_{\alpha_{1}}w_{\alpha_{3}}\lambda_{2},r^{G}_{T}\left(\pi_{1}^{3}\right)\right)\leq
2.$

###### Proof.

In order to show that $\pi_{1}^{3}$ is a common irreducible constituent of
$\Pi^{\prime},\Pi_{3}$, we apply Tadic’s criterion [Tad98, Lemma 3.1] with the
following arguments:
$i_{T}^{G}(\lambda_{a.d}),\Pi_{3},\Pi^{\prime},\lambda_{a.d}$. Indeed,

$mult\left(\lambda_{a.d},i_{T}^{G}(\lambda_{a.d}\right)=mult\left(\lambda_{a.d},\Pi^{\prime}\right)=mult\left(\lambda_{a.d},\Pi_{3}\right)=12.$

Hence, $\Pi^{\prime},\Pi_{3}$ share a common irreducible constituent such that
$\lambda_{a.d}$ appears in their Jacquet module. However, since $\pi_{1}^{3}$
is the unique irreducible constituent of $\Pi_{3}$ for which $\lambda_{a.d}$
appears in its Jacquet module, the first claim follows. In order to prove the
second part, we proceed as follows:

  * •

Recall that $\lambda_{a.d}\leq r^{G}_{T}\left(\pi_{1}^{3}\right)$. Therefore,
by the central character argument, $\pi_{1}^{3}\hookrightarrow
i_{T}^{G}(\lambda_{a.d})$. Since $i_{T}^{G}(\lambda_{a.d})$ is a standard
module, by Langlands, it admits a unique irreducible subrepresentation.
Namely, $\pi_{1}^{3}$ is the unique irreducible subrepresentation of
$i_{T}^{G}(\lambda_{a.d})$.

  * •

Note that $\lambda_{a.d}=u\cdot\lambda_{0}$. Where
$u=w_{3}w_{4}w_{2}w_{3}w_{1}w_{2}w_{3}w_{4}w_{3}w_{1}w_{2}w_{3}$, one has
$\Sigma_{u}\subset i_{T}^{G}(\lambda_{a.d})$. Since $\pi_{1}^{3}$ is the
unique irreducible subrepresentation of $i_{T}^{G}(\lambda_{a.d})$, it follows
that $\pi_{1}^{3}$ is also a subrepresentation of $\Sigma_{u}$.

  * •

Write $u=\left(w_{3}w_{2}w_{1}\right)\cdot u^{\prime}$ where
$u^{\prime}=w_{4}w_{3}w_{2}w_{3}w_{4}w_{3}w_{1}w_{2}w_{3}$.
LABEL:Table::_P3_::images implies that $\Sigma_{u^{\prime}}\simeq\Sigma_{u}$.
In particular, $\pi_{1}^{3}$ is a subrepresentation of
$\Sigma_{u^{\prime}}\subset i_{T}^{G}(u^{\prime}\lambda_{0})$. Note that
$u^{\prime}\lambda_{0}=\lambda_{2}$, and thus, $\pi_{1}^{3}\hookrightarrow
i_{T}^{G}(\lambda_{2})$. Therefore by Frobenius reciprocity, $\lambda_{2}\leq
r^{G}_{T}\left(\pi_{1}^{3}\right)$.

  * •

By (OR) it follows that
$2|mult\left(\lambda_{2},r^{G}_{T}\left(\pi_{1}^{3}\right)\right)$. However,
$mult\left(\lambda_{2},r^{G}_{T}\left(\Pi_{3}\right)\right)=2$. Thus the
second part follows.

In order to prove the third part we note that

$mult\left(w_{\alpha_{3}}w_{\alpha_{1}}\lambda_{2},r^{G}_{T}\left(\pi_{1}^{3}\right)\right)\leq
mult\left(w_{\alpha_{3}}w_{\alpha_{1}}\lambda_{2},r^{G}_{T}\left(\Pi^{\prime}\right)\right)=2.$

∎

######  Lemma 4.4.4.

$\dim r^{G}_{T}\left(\pi_{1}^{3}\right)\geq 42$.

###### Proof.

By 4.4.3, $\lambda_{a.d}+\lambda_{2}\leq r^{G}_{T}\left(\pi_{1}^{3}\right)$.
We proceed by applying the following sequence of branching rules:

  * •

Applying (OR) with respect to $\lambda_{a.d}$ implies that
$12\times\lambda_{a.d}\leq r^{G}_{T}\left(\pi_{1}^{3}\right)$.

  * •

Applying ($B_{3}(a)$) on $\lambda_{a.d}$ with respect to
$M_{\alpha_{1},\alpha_{2},\alpha_{3}}$ yields

$12\times\lambda_{a.d}+8\times w_{\alpha_{3}}\lambda_{a.d}+4\times
w_{\alpha_{2}}w_{\alpha_{3}}\lambda_{a.d}\leq
r^{G}_{T}\left(\pi_{1}^{3}\right).$

  * •

Applying ($C_{3}(a)$) on $\lambda_{a.d}$ with respect to
$M_{\alpha_{2},\alpha_{3},\alpha_{4}}$ gives

$12\times\lambda_{a.d}+6\times w_{\alpha_{3}}\lambda_{a.d}+3\times
w_{\alpha_{2}}w_{\alpha_{3}}\lambda_{a.d}\leq
r^{G}_{T}\left(\pi_{1}^{3}\right).$

However, we have shown that
$mult\left(w_{\alpha_{3}}\lambda_{a.d},r^{G}_{T}\left(\pi_{1}^{3}\right)\right)=8$.
LABEL:Table::C3::_a yields

$\operatorname{Res}^{GSp_{6}}_{Sp_{6}}\left(r^{G}_{M_{\alpha_{2},\alpha_{3},\alpha_{4}}}\left(\pi_{1}\right)\right)\geq
2\times D_{Sp_{6}}(\sigma_{0})+3\times\sigma_{0}.$

Thus,

$\begin{split}4\times w_{\alpha_{4}}w_{\alpha_{3}}\lambda_{a.d}+2\times
w_{\alpha_{3}}\lambda_{a.d}+2\times
w_{\alpha_{2}}w_{\alpha_{4}}w_{\alpha_{3}}\lambda_{a.d}&\\\ +2\times
w_{\alpha_{3}}w_{\alpha_{2}}w_{\alpha_{4}}w_{\alpha_{3}}\lambda_{a.d}+12\times\lambda_{a.d}+6\times
w_{\alpha_{3}}\lambda_{a.d}+3\times
w_{\alpha_{2}}w_{\alpha_{3}}\lambda_{a.d}&\leq
r^{G}_{T}\left(\pi_{1}^{3}\right).\end{split}$

In other words, we also invoke ($C_{3}(b)$) twice.

  * •

Applying (OR) and ($C_{3}(d)$) on $\lambda_{2}$ implies

$2\times\lambda_{2}+w_{\alpha_{3}}\lambda_{2}+w_{\alpha_{2}}w_{\alpha_{3}}\lambda_{2}\leq
r^{G}_{T}\left(\pi_{1}^{3}\right).$

  * •

By the Geometric Lemma, one has
$mult\left(w_{\alpha_{2}}w_{\alpha_{3}}\lambda_{2},r^{G}_{T}\left(\Pi_{3}\right)\right)=0$.
In particular,
$mult\left(w_{\alpha_{2}}w_{\alpha_{3}}\lambda_{2},r^{G}_{T}\left(\pi_{1}^{3}\right)\right)=0$.
Hence, by B.3.4, it follows that we can invoke ($B_{3}(c)$) once, namely,

$2\times\lambda_{2}+w_{\alpha_{1}}\lambda_{2}+w_{\alpha_{3}}\lambda_{2}+2\times
w_{\alpha_{1}}w_{\alpha_{3}}\lambda_{2}\leq
r^{G}_{T}\left(\pi_{1}^{3}\right).$

Note that
$w_{\alpha_{1}}\lambda_{2}=w_{\alpha_{2}}w_{\alpha_{3}}\lambda_{a.d}$. As a
result,
$mult\left(w_{\alpha_{2}}w_{\alpha_{3}}\lambda_{a.d},r^{G}_{T}\left(\pi_{1}^{3}\right)\right)\geq
5$ one occurrence from ($B_{3}(c)$) and four occurrences from ($B_{3}(a)$).

  * •

Note that $\lambda_{3}\leq r^{G}_{T}\left(\pi_{1}^{3}\right),$ where
$\lambda_{3}=w_{\alpha_{1}}w_{\alpha_{3}}\lambda_{2}=[-1,0,1,-2]$. Applying
($C_{3}(c)$) on $\lambda_{3}$ with respect to
$M_{\alpha_{2},\alpha_{3},\alpha_{4}}$ implies that

$2\times\lambda_{3}+2\times
w_{\alpha_{4}}\lambda_{3}+w_{\alpha_{3}}\lambda_{3}+w_{\alpha_{3}}w_{\alpha_{4}}\lambda_{3}\leq
r^{G}_{T}\left(\pi_{1}^{3}\right).$

To summarize, we applied the following rules of type $C_{3}$

Rule Type $(\sigma)$ | Multiplicity (m) | $m\times\dim r^{M_{\alpha_{2},\alpha_{3},\alpha_{4}}}_{T}\left(\sigma\right)$
---|---|---
($C_{3}(a)$) | 3 | 21
($C_{3}(b)$) | 2 | 10
($C_{3}(c)$) | 1 | 6
($C_{3}(d)$) | 1 | 4

Thus, we have shown that $\dim r^{G}_{T}\left(\pi_{1}^{3}\right)\geq 41$.
However, if $\dim r^{G}_{T}\left(\pi_{1}^{3}\right)=41$, the implication is
that
$mult\left(w_{\alpha_{2}}w_{\alpha_{3}}\lambda_{a.d},r^{G}_{T}\left(\pi_{1}^{3}\right)\right)=4$.
But we have shown that

$mult\left(w_{\alpha_{2}}w_{\alpha_{3}}\lambda_{a.d},r^{G}_{T}\left(\pi_{1}^{3}\right)\right)\geq
5.$

Thus the claim follows.

∎

##### Determine $r^{G}_{T}\left(\pi_{2}^{3}\right)$

The aim of this subsection is to demonstrate that
$r^{G}_{T}\left(\pi_{2}^{3}\right)\geq 19$. We fix the following notations.

$\displaystyle u_{1}$ $\displaystyle=w_{1}w_{2}w_{3}w_{4}w_{1}w_{2}w_{3},$
$\displaystyle u_{1}^{{}^{\prime}}$
$\displaystyle=\left(w_{2}w_{3}\right)u_{1},$ $\displaystyle u_{2}$
$\displaystyle=w_{3}w_{2}w_{3}w_{1}w_{2}w_{3}w_{4}w_{3}w_{2}w_{3}w_{1}w_{2}w_{3},$
$\displaystyle u_{2}^{{}^{\prime}}$
$\displaystyle=\left(w_{2}w_{3}w_{4}\right)u_{2},$ $\displaystyle\lambda_{s}$
$\displaystyle=[-1,-1,1,1],$ $\displaystyle\lambda_{s}^{{}^{\prime}}$
$\displaystyle=[0,-1,0,1],$ $\displaystyle u$
$\displaystyle=w_{1}w_{2}w_{3}w_{4}w_{3}w_{2}w_{3}w_{1}w_{2}w_{3},$
$\displaystyle s$ $\displaystyle=w_{3}w_{2}w_{3},$ $\displaystyle L$
$\displaystyle=u\chi_{P,z},$ $\displaystyle E(z)$
$\displaystyle=\mathcal{N}_{w_{3}w_{2}w_{3}}^{L}(u\chi_{P,z}).$ (4.4.1)

######  Proposition 4.4.5.

One has $\pi_{2}^{3}\oplus\pi_{2}^{3}\hookrightarrow i_{T}^{G}(\lambda_{s})$.
In particular

$mult\left(\lambda_{s},r^{G}_{T}\left(\pi_{2}^{3}\right)\right)\geq 2.$

###### Proof.

The proof goes as follows: In 4.4.6, we show that $E(z)$ is holomorphic at
$z=z_{0}$. In addition,
$E(z_{0})\in\operatorname{End}_{G}\left(i_{T}^{G}(\lambda_{s})\right)$. We
also show that

$i_{T}^{G}(\lambda_{s})=V_{1}\oplus V_{-1},$

where $V_{1},V_{-1}$ are eigenspaces of $E(z_{0})$.

Thus, it suffices to show that $\pi_{2}^{3}\hookrightarrow V_{1}$ and
$\pi_{2}^{3}\hookrightarrow V_{-1}$. In 4.4.8, we show that
$\pi_{1}^{3}\oplus\pi_{2}^{3}\hookrightarrow V_{1}$. Hence
$\pi_{2}^{3}\hookrightarrow V_{1}$. 4.4.9 implies that
$\pi_{2}^{3}\hookrightarrow V_{-1}$ . The second part follows immediately from
Frobenius reciprocity. ∎

######  Lemma 4.4.6.

One has $u\chi_{P,z_{0}}=su\chi_{P,z_{0}}=\lambda_{s}$. In addition, the
operator $E(z)$ is holomorphic at $z=z_{0}$. Furthermore, one has

$i_{T}^{G}(\lambda_{s})=V_{1}\oplus V_{-1},$

where $V_{i}$ are eigenspaces of $E(z_{0})$.

###### Proof.

A direct computation shows that $u\chi_{P,z_{0}}=su\chi_{P,z_{0}}$. The other
parts follow immediately from an application of A.0.3 with the following
parameters:

$\displaystyle\chi$ $\displaystyle=\lambda_{s},$ $\displaystyle\alpha$
$\displaystyle=\alpha_{3},$ $\displaystyle m$ $\displaystyle=u\varpi_{3},$
$\displaystyle s$ $\displaystyle=w_{3}w_{2}w_{3},$ $\displaystyle L=$
$\displaystyle u\chi_{P,z}=(z-z_{0})m+\chi.$

This shows that $i_{T}^{G}(\lambda_{s})=V_{1}\oplus V_{-1}$. Here,
$V_{1},V_{-1}$ are eigenspaces for the action of
$E(z_{0})=\mathcal{N}_{s}^{L}(u\chi_{P,z_{0}})$. ∎

Let $E=E(z_{0})$.

######  Lemma 4.4.7.

One has $\Sigma_{u_{1}^{{}^{\prime}}}=\Sigma_{u_{2}^{{}^{\prime}}}$ and
$\pi_{1}^{3}\oplus\pi_{2}^{3}\hookrightarrow\Sigma_{u_{1}^{{}^{\prime}}}$.

###### Proof.

The first item follows by a direct computation in the Iwahori-Hecke algebra.
Namely, we show that