# Toward the endoscopic classification of unipotent representations of
$p$-adic $G_{2}$

Clifton Cunningham  University of Calgary<EMAIL_ADDRESS>,  Andrew
Fiori  University of Lethbridge<EMAIL_ADDRESS>and  Qing Zhang  Korea
Advanced Institute of Science and Technology<EMAIL_ADDRESS>
###### Abstract.

We begin this paper by reviewing the Langlands correspondence for unipotent
representations of the exceptional group of type $G_{2}$ over a $p$-adic field
$F$ and present it in an explicit form. Then we compute all ABV-packets, as
defined in [CFMMX] following ideas from Vogan’s 1993 paper The local Langlands
Conjecture, and prove that these packets satisfy properties derived from the
expectation that they are generalized A-packets. We attach distributions to
ABV-packets for $G_{2}$ and its endoscopic groups and study a geometric
endoscopic transfer of these distributions. This paper builds on earlier work
by the same authors.

Clifton Cunningham gratefully acknowledges the support of NSERC Discovery
Grant RGPIN-2020-05220

Andrew Fiori thanks and acknowledges both the University of Lethbridge for
their financial support as well as the support of NSERC Discovery Grant
RGPIN-2020-05316.

Qing Zhang would like to thank the support NSFC grant 11801577, a postdoc
fellowship from the Pacific Institute for the Mathematical Sciences (PIMS)

###### Contents

  1. 0 Introduction
    1. 0.1 Background and motivation
    2. 0.2 Fundamental properties of ABV-packets and packet coefficients
    3. 0.3 Distributions attached to ABV-packets
    4. 0.4 Lifting stable distributions from endoscopic groups
    5. 0.5 Next steps
    6. 0.6 Geometry and arithmetic
    7. 0.7 Notation
    8. 0.8 Acknowledgements
  2. 1 Geometry of the moduli space of Langlands parameters for $G_{2}$
    1. 1.1 Roots for $G_{2}$ and ${\widehat{G_{2}}}$
    2. 1.2 Moduli spaces of Langlands parameters
    3. 1.3 Prehomogeneous vector spaces
    4. 1.4 The geometry of the moduli space of Langlands parameters
  3. 2 Unipotent representations for $G_{2}$
    1. 2.1 The Langlands correspondence
    2. 2.2 Vogan’s version of the local Langlands correspondence
    3. 2.3 Kazhdan-Lusztig conjecture
    4. 2.4 Aubert involution and the Fourier transform
    5. 2.5 A generalization of the component group of an Arthur parameter
    6. 2.6 Calculation of ABV-packet coefficients
    7. 2.7 Fundamental properties of ABV-packet coefficients for $G_{2}$
    8. 2.8 A geometric interpretation of Muic’s reducibility points
    9. 2.9 Unramified Arthur parameters for $G_{2}$
    10. 2.10 Distributions attached to ABV-packets
  4. 3 Unipotent representations of endoscopic groups for $G_{2}$
    1. 3.1 Endoscopic groups for $G_{2}$
    2. 3.2 Unipotent representations of $T=\operatorname{GL}_{1}^{2}$
    3. 3.3 Unipotent representations of $\operatorname{GL}_{2}$
    4. 3.4 Unipotent representations of $\operatorname{SO}_{4}$ and its pure inner forms
    5. 3.5 Unipotent representations of $\operatorname{PGL}_{3}$ and its pure inner forms
  5. 4 Geometric endoscopy
    1. 4.1 Lifting Langlands parameters
    2. 4.2 Regular conormal vectors and Arthur parameters
    3. 4.3 Statement of the fixed-point formula
    4. 4.4 Prehomogeneous vector subspaces
    5. 4.5 $V$-conormal
    6. 4.6 Restriction
    7. 4.7 Proof of Theorem 4.2
    8. 4.8 Geometric lifting of stable distributions

##  0\. Introduction

###  0.1. Background and motivation

Local Arthur packets are finite sets $\Pi_{\psi}(G(F))$ of irreducible
admissible representations of a $p$-adic $G(F)$, characterized by endoscopic
character identities for symplectic and quasi-split orthogonal groups $G$ (and
their inner twists over $F$) in [Arthur:book]. Arthur packets generalize
tempered $L$-packets and they remedy certain drawbacks of $L$-packets
associated with stable distributions. The description of Arthur packets in
[Arthur:book], relies heavily on the trace formula. These notions have been
extended to some other groups, including unitary groups in [Mok:Unitary], but
not yet to exceptional groups; however, some candidates for local Arthur
packets for $G_{2}(F)$ were proposed in [GGJ], [GG] and [Gan-Savin]. In this
paper we use the microlocal geometry of the moduli space of unramified
Langlands parameters to calculate ABV-packets, as defined in [CFMMX] building
on ideas from [Vogan:Langlands], for all unipotent representations of
$G_{2}(F)$ and show that these packets are compatible with endoscopy in a very
precise sense.

Vogan’s work [Vogan:Langlands] shows that there are advantages to studying the
representation theory of $G(F)$ together with that of its pure inner forms
$G^{\delta}(F)$ simultaneously; in this way one passes from $L$-packets
$\Pi_{\phi}(G(F))$ to "pure" $L$-packets $\Pi^{\text{pure}}_{\phi}(G)$ and
from Arthur packets $\Pi_{\psi}(G(F))$ to "pure" Arthur packets
$\Pi^{\text{pure}}_{\psi}(G)$. As explained in [CFMMX], when Arthur’s work is
viewed from this perspective it determines a function
$\Pi^{\text{pure}}_{\psi}(G)\to\mathsf{Rep}(A_{\psi})$ where $A_{\psi}$ is the
component group of $Z_{{\widehat{G}}}(\psi)$.

A geometric construction of Arthur packets for any $p$-adic group is proposed
in [CFMMX] building on ideas introduced by [Vogan:Langlands] and also by an
adaptation of ideas from [ABV]. In fact, this geometric construction is more
general: for any Langlands parameter $\phi$ for $G(F)$, [CFMMX] constructs a
group $A_{\phi}^{{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}}$,
and proposes a packet
$\Pi_{\phi}^{{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}}(G)$ of
irreducible representations of $G(F)$ and its pure inner forms, called the
ABV-packet of $\phi$, and a natural map
$\Pi_{\phi}^{{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}}(G)\to\mathsf{Rep}(A_{\phi}^{{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}})$.
If $\phi$ happens to be of Arthur type, then
$A_{\phi}^{{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}}=A_{\psi}$
for the associated Arthur parameter $\psi$. The main conjecture of [CFMMX]
says that when $\phi$ is of Arthur type then
$\Pi_{\phi}^{{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}}(G)=\Pi^{\text{pure}}_{\psi}(G)$
and the map
$\Pi_{\phi}^{{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}}(G)\to\mathsf{Rep}(A_{\phi}^{{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}})$
should be the same as $\Pi^{\text{pure}}_{\psi}(G)\to\mathsf{Rep}(A_{\psi})$
when $G$ is a classical group, where $\psi$ is the associated Arthur
parameter. The construction in [CFMMX] proposes a concrete way to compute
Arthur packets for classical groups, as illustrated by the numerous examples
in [CFMMX], and also provides a method to generalize Arthur packets to
exceptional groups and non-Arthur type local Langlands parameters.

###  0.2. Fundamental properties of ABV-packets and packet coefficients

To give some details of our results, we recall that, in
[Lusztig:Classification1] Lusztig provided a version of the local Langlands
correspondence for unipotent representations and showed that the unipotent
representations are classified by unramified parameters. Using that, in this
paper we compute the ABV-packets defined in [CFMMX], for each unramified local
Langlands parameter $\phi$ of $G_{2}(F)$, and thus we obtain a decomposition

$\Pi(G_{2}(F))_{\text{unip}}=\mathop{\bigcup}\limits_{\phi\in\Phi(G_{2}(F))_{\text{unr}}}\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}(G_{2}(F)),$

where $\Pi(G_{2}(F))_{\text{unip}}$ is the set of isomorphism classes of
irreducible unipotent representations of $G_{2}(F)$. For every ABV-packet
$\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}(G_{2}(F))$,
the general, canonical, geometric constructions in [CFMMX] also give a group
$A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}$ and a
function

$\begin{array}[]{rcl}\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}(G_{2}(F))&\to&\mathsf{Rep}(A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi})\\\
\pi&\mapsto&\langle\ ,\pi\rangle\end{array}$

and thus a function $\langle\ ,\
\rangle:A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}\times\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}(G_{2}(F))\to{\bar{\mathbb{Q}}}$;
we refer to this function as the ABV-packet coefficients for $\phi$.

In this paper we argue that ABV-packet coefficients provide a canonical way to
extend the notion of Arthur packets to all unipotent representations of the
exceptional $p$-adic group $G_{2}$, including those that are not of Arthur
type. For instance, in Theorem 2.2 we show that the ABV-packet coefficients
extend the local Langlands correspondence for unipotent representations of
$G_{2}(F)$: for all unramified Langlands parameters
$\phi:W^{\prime}_{F}\to{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G_{2}}}$, the
following diagram commutes

${\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}(G_{2}(F))}$${\widehat{A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}}}$${\Pi_{\phi}(G_{2}(F))}$${\widehat{A_{\phi}}}$$\scriptstyle{\mathrm{LLC}}$

where
$\widehat{A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}}$
(resp. $\widehat{A_{\phi}}$) denotes the set of characters of irreducible
representations of
$A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}$ (resp.
$A_{\phi}$). Here the local Langlands correspondence is normalized such that
if $\pi\in\Pi_{\phi}(G_{2}(F))$ is generic or spherical then the corresponding
representation of $A_{\phi}$ is trivial. We give other fundamental properties
of ABV-packet coefficients in Theorem 2.2. For instance, Part 5 of Theorem 2.2
shows that for every unramified Langlands parameter $\phi$ there is an
unramified Langlands parameter ${\hat{\phi}}$, with the same infinitesimal
parameter as ${\hat{\phi}}$, such that the Aubert involution defines a
bijection

$\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}(G_{2}(F))\to\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\hat{\phi}}(G_{2}(F)).$

###  0.3. Distributions attached to ABV-packets

We study distributions defined by ABV-packet coefficients. To do this we
introduce an algebraic group
$\mathcal{S}^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}$
contained in $Z_{{\widehat{G}}}(\phi)$ such that
$A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}=\pi_{0}(\mathcal{S}^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi})$
and
$\mathcal{S}^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}=Z_{{\widehat{G_{2}}}}(\psi)$
if $\phi$ is of Arthur type $\psi$. For
$s\in\mathcal{S}^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}$,
we set

$\Theta_{\phi,s}{\,:=\,}\sum_{\pi\in\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi_{\psi}}(G_{2}(F))}(-1)^{\dim(\phi)-\dim(\pi)}\langle
s,\pi\rangle\ \Theta_{\pi},$ (1)

as in Definition 4, where we identify $s$ with its image under
$\mathcal{S}^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}\to
A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}$ and where
$\Theta_{\pi}$ is the Harish-Chandra distribution character determined by the
admissible representation $\pi$. The terms $\dim(\phi)$ and $\dim(\pi)$ are
defined in Section 2.2. We show that if $\phi$ is of Arthur type $\psi$ then
$(-1)^{\dim(\phi)-\dim(\pi)}\langle s,\pi\rangle=\langle
s_{\psi}\,s,\pi\rangle,$ where $s_{\psi}$ is the image of $\psi(1,-1)\in
Z_{{\widehat{G}}_{2}}(\psi)$; in this case, Equation (1) takes the more
familiar form

$\Theta_{\psi,s}=\sum_{\pi\in\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi_{\psi}}(G_{2}(F))}\langle
s_{\psi}s,\pi\rangle\ \Theta_{\pi}.$

In Theorem 2.14 we prove that if $\pi$ is an irreducible unipotent
representation of $G_{2}(F)$ then $\Theta_{\pi}$ can be expressed in terms of
the distributions $\Theta_{\phi,s}$, letting $\phi$ range over unramified
$L$-parameters with the same infinitesimal parameter as $\pi$ and letting $s$
range over representatives for the components of
$\mathcal{S}^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}$.
This is an important part of the endoscopic classification promised in the
title of this paper. We also show that the span of the distributions
$\Theta_{\phi,s}$, as $s$ ranges over
$\mathcal{S}^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}$,
coincides with the span of the distributions $\Theta_{\pi}$, as $\pi$ ranges
over $\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}$,
if and only if
$\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}(G_{2}(F))\to\widehat{A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}}$
is bijective. In this theorem we also show that if
$\Theta_{\phi}{\,:=\,}\Theta_{\phi,1}$ is stable when $\phi$ is elliptic (see
Section 2.10), then $\Theta_{\phi}$ is stable for all $\phi$.

###  0.4. Lifting stable distributions from endoscopic groups

We show the ABV-packet coefficients $\langle\ ,\ \rangle$ are compatible with
the theory of endoscopy in the following sense. Let $(G,s,\xi)$ be an
endoscopic triple for $G_{2}$. Unlike the group $G_{2}$, the endoscopic groups
$G$ may admit pure inner forms. Recall that a pure inner form for $G$ is a
cocycle $\delta\in Z^{1}(F,G)$ and that $\delta$ determines an inner form
$G^{\delta}$ of $G$. We calculate the ABV-packets and ABV-packet coefficients
for the endoscopic group $G$. We generalize the definition of the
distributions above Equation (1) from $G_{2}$ to the pure inner forms of
endoscopic groups $G$ for $G_{2}$. In Definition 8 we introduce a "lifting" of
stable distributions attached to ABV-packets for $G(F)$ to invariant
distributions on $G_{2}(F)$ and prove Theorem 4.6: if
$\phi:W^{\prime}_{F}\to{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G}}$ is an unramified
Langlands parameter that is $\xi$-conormal, in the sense of Definition 6, then

$\operatorname{Lift}_{(G,s,\xi)}\Theta^{G}_{\phi}=\Theta_{\xi\circ\phi,s},$
(2)

for all
$s\in\mathcal{S}^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}$,
for
$\mathcal{S}^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}$
given in Definition 1. We show that if $\phi$ is of Arthur type then it is
always $\xi$-conormal. Since $G(F)$ is also an endoscopic group for the inner
form $G^{\delta}(F)$ we also define "lifting" from stable distribution
attached to ABV-packets for $G(F)$ to invariant distributions on
$G^{\delta}(F)$ and, in Theorem 4.6, we show that if
$\phi:W^{\prime}_{F}\to{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G}}$ is an unramified
Langlands parameter that is relevant to $G^{\delta}(F)$, in the sense of
[Borel:Automorphic], then the lift of $\Theta^{G}_{\phi}$ is
$e(\delta)\Theta^{G^{\delta}}_{\phi}$, where $e(\delta)$ is the Kottwitz sign.
Conjecture 1 predicts that $\Theta_{\xi\circ\phi,s}$ is the Langlands-Shelstad
transfer of the stable distribution $\Theta^{G}_{\phi}$.

All this comes together in Theorem 4.7, in which we show that ABV-packet
coefficients are uniquely characterized by the properties given in this paper.
This result also presents the endoscopic classification of unipotent
representations of $G_{2}(F)$ promised in the title of this paper, stated here
only for tempered representations to simplify the exposition. Let $\pi$ be a
tempered unipotent representation of $G_{2}(F)$ with Langlands parameter
$\phi$. Every $s\in\mathcal{S}_{\psi}$ determines an endoscopic triple
$(G,s,\xi)$ and a factorization $\phi=\xi\circ\phi^{s}$, for a Langlands
parameter $\phi^{s}$ for $G$. In this case, Theorem 4.7 gives

$\Theta_{\pi}=\sum_{(G,s,\xi)}(-1)^{\dim(\phi^{s})-\dim(\pi)}\frac{\overline{\langle
s,\pi\rangle}}{{|Z_{A_{\phi}}(s)|}}\
\operatorname{Lift}_{(G,s,\xi)}\Theta^{G}_{\phi^{s}},$ (3)

where the sum is taken over equivalence classes of endoscopic triples
$(G,s,\xi)$ with
$s\in\mathcal{S}^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}$
and where we identify $s$ with its image under
$\mathcal{S}^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}\to
A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}$ in the
calculation of $Z_{A_{\phi}}(s)$. This result is generalized from tempered
unipotent representations to a wider class of unipotent representations in
Theorem 4.7.

###  0.5. Next steps

Before we can declare that the ABV-packets of this paper are indeed
generalized Arthur packets for $G_{2}(F)$, the following two issues must be
addressed: (i) we must show that $\Theta^{G}_{\phi}$ is a stable distribution
for every unramified Langlands $\phi$ of $G(F)$ of Arthur type, where $G(F)$
is an endoscopic group for $G_{2}(F)$; and (ii) we must show that, if
$(G,s,\xi)$ is an endoscopic triple for $G_{2}(F)$, then

$\Theta^{G}_{\phi}(f^{s})=\Theta_{\xi\circ\phi,s}(f)$

where $f^{s}\in\mathcal{H}(G(F))$ is the Langlads-Shelstad transfer of
$f\in\mathcal{H}(G_{2}(F))$; in other words, we must show that
$\Theta_{\xi\circ\phi,s}$ is the endoscopic lift of $\Theta^{G}_{\phi}$. In
fact, it is a conjecture of Vogan that (i) is true for all Langlands
parameters, not just those of Arthur type. This is elementary for unramified
parameters except when $G=G_{2}$ and, as we show in Theorem 2.14, in that case
it is sufficient to show that $\Theta_{\phi}$ is stable for the four elliptic,
tempered, unipotent $L$-packets in Table 2.10.1; see also Remark 2.16. We are
working on a proof of this statement. We will also prove a generalization of
(ii) that asks only that $\phi$ is $s$-conormal, as in Definition 9, again
generalizing from the case of Langlands parameters of Arthur type.

###  0.6. Geometry and arithmetic

ABV-packet coefficients are defined by a microlocal analysis of the moduli
space of unramified Langlands parameters, following [CFMMX], which builds on
[Vogan:Langlands]. As in [CFMMX] and [CFZ:cubics], we define the
_infinitesimal parameter_ of a Langlands parameter
$\phi:W^{\prime}_{F}\to{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G_{2}}}$ is
$\lambda_{\phi}:W_{F}\to{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G_{2}}}$ to be
$\lambda_{\phi}(w)=\phi(w,\operatorname{diag}(|w|^{1/2},|w|^{-1/2}))$. There
is a natural moduli space for Langlands parameters with the same infinitesimal
parameter as $\phi$, introduced by Vogan and used extensively in [CFMMX] and
[CFZ:cubics]. This moduli space is denoted by $V_{\lambda_{\phi}}$ in this
paper and it naturally carries an action of the reductive group
$H_{\lambda_{\phi}}{\,:=\,}Z_{{\widehat{G_{2}}}}(\lambda_{\phi})$; it is, in
particular, a prehomogeneous vector space. In this paper we review the
bijection between simple objects in the category
$\mathsf{Per}_{H_{\lambda_{\phi}}}(V_{\lambda_{\phi}})$ equivariant perverse
sheaves on $V_{\lambda_{\phi}}$ and irreducible admissible representations of
$G_{2}(F)$ for which the Langlands parameter has infinitesimal parameter
$\lambda_{\phi}$.

We show that the Aubert involution of admissible representations appears as
the Fourier transform of equivariant perverse sheaves on the other side of
this bijection. We also show how to recognize when a unipotent representation
is tempered, Arthur type, unitary, generic, spherical or supercuspidal purely
in terms of the geometry of the corresponding simple equivariant perverse
sheaf. To explain this, we say that $\phi$ is _open_ (resp. _closed_) if the
$Z_{{\widehat{G_{2}}}}(\lambda_{\phi})$-orbit of $\phi$ is open (resp. closed)
in $V_{\lambda_{\phi}}$. If $\pi$ is tempered then its Langlands parameter is
open but the converse is not true; in this sense, the notion of open
parameters generalizes the notion of parameters bounded upon restriction to
$W_{F}$. We remark that a parameter
$\phi:W^{\prime}_{F}\to{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G_{2}}}$ is closed if
and only if it is trivial on the unipotent part of
$\operatorname{SL}_{2}({\mathbb{C}})$ appearing in
$W^{\prime}_{F}{\,:=\,}W_{F}\times\operatorname{SL}_{2}({\mathbb{C}})$.
Returning to the claim that ABV-packet coefficients provide an extension of
the Langlands correspondence and Arthur packets, we use these notions to prove
fundamental properties of ABV-packet coefficients in Theorem 2.2; for example,
in Part 2 of Theorem 2.2 we show that if
$\phi:W^{\prime}_{F}\to{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G_{2}}}$ is open or
closed then
$\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}(G_{2}(F))\to\widehat{A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}}$
is bijective. In [CFZ] we found examples when
$\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}(G_{2}(F))\to\widehat{A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}}$
is not bijective. In Part 4 of Theorem 2.2 we show that a representation
$\pi\in\Pi_{\phi}(G_{2}(F))$ is spherical if and only if $\phi$ is closed and
$\langle\ ,\pi\rangle$ is the trivial representation of
$A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}$, and in
this case, the only ABV-packet that contains $\pi$ is
$\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}(G_{2}(F))$.

###  0.7. Notation

Throughout this paper $F$ denotes a $F$-adic field, $\varphi$ a fixed
uniformizer in the ring $\mathcal{O}_{F}$ of integers of $F$ and $q$ denotes
the cardinality of the residue field of $F$; we place no lower bound on $q$.

We will use "polar notation" $\chi=\mu\nu^{a}$ for complex characters $\chi$
of $F^{\times}$, where $\nu:F^{\times}\to{\mathbb{C}}^{\times}$ is the
unramified character defined by $\nu(\varpi)=q$ and where
$\mu:F^{\times}\to{\mathbb{C}}^{\times}$ is unitary and $a\in{\mathbb{R}}$.
When subscripts are necessary, this notation becomes
$\chi_{i}=\mu_{i}\nu^{a_{i}}$. We will use the notation $\theta_{n}$ for a
fixed, unitary character of $F^{\times}$ of order $n$.

Similarly, we use the "$q$-polar notation" $z=uq^{a}$ for complex numbers
$z\in{\mathbb{C}}^{\times}$ where $u$ is unitary and $a\in{\mathbb{R}}$.

By abuse of notation, we also denote a fixed primitive complex $n$-th root-of-
unity by $\theta_{n}$ and write $\vartheta_{n}$ for the corresponding
character of
$\langle\theta_{n}\rangle{\,:=\,}\\{1,\theta_{n},\ldots,\theta_{n}^{n-1}\\}$,
so $\vartheta_{n}(\theta_{n})=\theta_{n}$. When the order $n$ is understood by
context, we will write $\vartheta$ for this character.

We write $\varepsilon$ for the sign character of $S_{3}$ and $\varrho$ for the
reflection representation of $S_{3}$.

For a split group $G$ over $F$ with a fixed Borel $B$ and torus $T$ over $F$,
we set $I^{G}(\sigma){\,:=\,}\text{Ind}_{B(F)}^{G(F)}(\sigma)$, where $\sigma$
is a representation of $T(F)$, inflated to $B(F)$; we write $J^{G}(\sigma)$
for the Langlands quotient $I^{G}(\sigma)$ when this induced representation is
a standard module. More generally, if $P$ is a parabolic subgroup of $G$ with
Levi subgroup $M$, we set
$I_{P}^{G}(\sigma){\,:=\,}\text{Ind}_{P(F)}^{G(F)}(\sigma)$; we write
$J_{P}^{G}(\sigma)$ for the Langlands quotient of $I_{P}^{G}(\sigma)$ when
this induced representation is a standard module.

For an algebraic group $G$ over $F$, denote by $1_{G}$ the trivial
representation of $G(F)$ and $\mathrm{St}_{G}$ the Steinberg representation of
$G(F)$.

For representations of $G_{2}(F)$, we follow the notations from [Muic] except
that the Steinberg representation of $\operatorname{GL}_{2}(F)$ is denoted by
$\mathrm{St}_{\operatorname{GL}_{2}}$ instead of $\delta(1)$ as in [Muic].

There are 4 cuspidal unipotent representations of $G_{2}(\mathbb{F}_{q})$:
they appear as $G_{2}[1]$, $G_{2}[-1]$, $G_{2}[\theta_{3}]$ and
$G_{2}[\theta^{2}_{3}]$ in [Carter]*p.460. For a cuspidal unipotent
representation $\sigma$ of $G_{2}(\mathbb{F}_{q})$, we use the shorthand
$I_{0}(\sigma)$ for
$\mathrm{cInd}_{G_{2}({\mathcal{O}}_{F})}^{G_{2}(F)}(\sigma)$. These
supercuspidal representations also appear in [CO], where they are denoted by
$v_{6}=I_{0}(G_{2}[1])$, $v_{7}=I_{0}(G_{2}[-1])$,
$v_{8}=I_{0}(G_{2}[\theta_{3}])$ and $v_{9}=I_{0}(G_{2}[\theta^{2}_{3}])$. The
last two of these also appear in [Savin]*Section 4, where they are denoted by
$\pi^{\prime}[\nu]$ and $\pi^{\prime}[\nu^{2}]$.

Throughout this paper, we use the notation ${\mathbbm{1}}_{X}$ for the
constant sheaf on $X$, or simply ${\mathbbm{1}}$ if the support $X$ is
understood.

###  0.8. Acknowledgements

The authors gratefully acknowledge and thank Goran Muić for explaining his
work on admissible representations of $G_{2}(F)$ and Maartin Solleveld for
helpful comments on the first draft this paper. The authors would also like to
thank all the members of the Voganish Project, especially Ahmed Moussaoui,
Jerrod Smith and Geoff Vooys.

##  1\. Geometry of the moduli space of Langlands parameters for $G_{2}$

###  1.1. Roots for $G_{2}$ and ${\widehat{G_{2}}}$

Let $T$ be a fixed maximal split torus of $G_{2}$; the corresponding set of
roots are denoted by $R{\,:=\,}R(G_{2},T)$. We write $W$ for the Weyl group
for this root system $R$. Let ${\gamma_{1}},{\gamma_{2}}:T\rightarrow
F^{\times}$ be a choice simple roots of $G_{2}$ with ${\gamma_{1}}$ short and
${\gamma_{2}}$ long so that the positive roots $R$ are

${\gamma_{1}},{\gamma_{2}},{\gamma_{1}}+{\gamma_{2}},2{\gamma_{1}}+{\gamma_{2}},3{\gamma_{1}}+{\gamma_{2}},3{\gamma_{1}}+2{\gamma_{2}}.$

The coroots are denoted $R^{\vee}{\,:=\,}R(G_{2},T)^{\vee}$, with
${\gamma_{1}^{\vee}},{\gamma_{2}^{\vee}}:F^{\times}\to T$ the coroots of
${\gamma_{1}}$ and ${\gamma_{2}}$. We fix the isomorphism $T\rightarrow
F^{\times}\times F^{\times}$ by

$t\mapsto((2{\gamma_{1}}+{\gamma_{2}})(t),({\gamma_{1}}+{\gamma_{2}})(t)).$

We use the notation $m:F^{\times}\times F^{\times}\to T$ for the inverse of
this isomorphism. We have

${\gamma_{1}^{\vee}}(a)=m(a,a^{-1}),\qquad\text{\ and\
}\qquad{\gamma_{2}^{\vee}}(a)=m(1,a).$

We denote by ${\widehat{G_{2}}}$ the dual group of $G_{2}$ over $\mathbb{C}$
and ${\widehat{T}}$ the dual torus and let
${\hat{R}}{\,:=\,}R({\widehat{G_{2}}},{\widehat{T}})$ be the roots of
${\widehat{G_{2}}}$ with the usual identification of coroots of $G_{2}$ with
roots of ${\widehat{G_{2}}}$. We again write $W$ for the Weyl group for the
root system ${\hat{R}}$. Denote by ${\hat{\gamma}}_{1}\in{\hat{R}}$ (resp.
${\hat{\gamma}}_{2}\in{\hat{R}}$) the image of ${\gamma_{1}^{\vee}}$ (resp.
${\gamma_{2}^{\vee}}$) under this identification. Then $\widehat{G}_{2}$ is a
complex reductive group of type $G_{2}$, with simple roots
${{\hat{\gamma}}_{1}},{{\hat{\gamma}}_{2}}$. where ${{\hat{\gamma}}_{1}}$ is a
long root of $\widehat{G}_{2}$ and ${{\hat{\gamma}}_{2}}$ is a short root of
$\widehat{G}_{2}$. As above, we fix an isomorphism
${\widehat{T}}\rightarrow\mathbb{C}^{\times}\times\mathbb{C}^{\times}$ by

$t\mapsto(({\hat{\gamma}_{1}}+2{\hat{\gamma}_{2}})(t),({\hat{\gamma}_{1}}+{\hat{\gamma}_{2}})(t))$

and we write
$\widehat{m}:\mathbb{C}^{\times}\times\mathbb{C}^{\times}\to{\widehat{T}}$ for
the inverse of this isomorphism. Note that we have

${{\hat{\gamma}}_{1}}(\widehat{m}(x,y))=x^{-1}y^{2},\qquad\mathrm{\ and\
}\qquad{{\hat{\gamma}}_{2}}(\widehat{m}(x,y))=xy^{-1}.$ (4)

We will denote by
${{\hat{\gamma}}_{1}}^{\vee},{{\hat{\gamma}}_{2}}^{\vee}:\mathbb{C}^{\times}\rightarrow{\widehat{T}}$
the coroots of ${{\hat{\gamma}}_{1}}$ and ${{\hat{\gamma}}_{2}}$. We have
again that

${{\hat{\gamma}}_{1}}^{\vee}(a)={\widehat{m}}(1,a),\quad{{\hat{\gamma}}_{2}}^{\vee}(a)={\widehat{m}}(a,a^{-1}).$

Observe that under the identification
$R(G_{2},T)=R({\widehat{G_{2}}},{\widehat{T}})^{\vee}$, we have
${\gamma_{1}}={{\hat{\gamma}}_{1}}^{\vee}$,
${\gamma_{2}}={{\hat{\gamma}}_{2}}^{\vee}$,
$({{\hat{\gamma}}_{1}}+{{\hat{\gamma}}_{2}})^{\vee}=3{{\hat{\gamma}}_{1}}^{\vee}+{{\hat{\gamma}}_{2}}^{\vee}$,
$({{\hat{\gamma}}_{1}}+2{{\hat{\gamma}}_{2}})^{\vee}=3{{\hat{\gamma}}_{1}}^{\vee}+2{{\hat{\gamma}}_{2}}^{\vee}$,
$({{\hat{\gamma}}_{1}}+3{{\hat{\gamma}}_{2}})^{\vee}={{\hat{\gamma}}_{1}}^{\vee}+{{\hat{\gamma}}_{2}}^{\vee}$
and
$(2{{\hat{\gamma}}_{1}}+3{{\hat{\gamma}}_{2}})^{\vee}=2{{\hat{\gamma}}_{1}}^{\vee}+{{\hat{\gamma}}_{2}}^{\vee}$.

###  1.2. Moduli spaces of Langlands parameters

Set $W^{\prime}_{F}{\,:=\,}W_{F}\times\operatorname{SL}_{2}({\mathbb{C}})$.
Given a Langlands parameter
$\phi:W_{F}^{\prime}\to{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G_{2}}}$, recall that
its infinitesimal parameter
$\lambda_{\phi}:W_{F}\to{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G_{2}}}$ is defined
by $\lambda_{\phi}(w)=\phi(w,\operatorname{diag}(|w|^{1/2},|w|^{-1/2}))$.

For a given infinitesimal parameter
$\lambda:W_{F}\to{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G_{2}}}$, there are only a
finite number of ${\widehat{G_{2}}}$-conjugacy classes of Langlands parameters
$\phi$ with $\lambda=\lambda_{\phi}$. As explained in [CFMMX] more generally,
each unramified infinitesimal parameter
$\lambda:W_{F}\to{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G_{2}}}$ determines a
prehomogeneous vector space

$V_{\lambda}=\\{x\in\operatorname{Lie}{\widehat{G_{2}}}{\ |\
}\operatorname{Ad}(\lambda(w))x=q^{|w|}x,\forall w\in W_{F}\\},$

equipped with an action of the group

$H_{\lambda}{\,:=\,}\\{g\in{\widehat{G_{2}}}{\ |\
}\lambda(w)g\lambda(w)^{-1}=g,\forall w\in W_{F}\\}.$

acting via the adjoint action of ${\widehat{G_{2}}}$ on
$\operatorname{Lie}{\widehat{G_{2}}}$. Using this action we can assume that
$\lambda({\operatorname{Fr}})\in{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{T}}$ and
define

${\hat{R}}_{\lambda}{\,:=\,}\\{{\hat{\gamma}}\in
R({\widehat{G}},{\widehat{T}}){\ |\
}\operatorname{Lie}U_{\hat{\gamma}}\subseteq V_{\lambda}\\},$

where $U_{\hat{\gamma}}\subset{\widehat{G_{2}}}$ is the root subgroup for
${\hat{\gamma}}$. Then $V_{\lambda}$ is a moduli space for all Langlands
parameters $\phi$ with $\lambda_{\phi}=\lambda$ and
$X_{\lambda}{\,:=\,}{\widehat{G_{2}}}\times_{H_{\lambda}}V_{\lambda}$ is a
moduli space for all Langlands parameters $\phi$ for which $\lambda_{\phi}$ is
${\widehat{G_{2}}}$-conjugate to $\lambda$. Note that ${\widehat{G_{2}}}$ acts
naturally on $X_{\lambda}$ while $H_{\lambda}$ acts on $V_{\lambda}$.

These notions lead to the following classification of unramified infinitesimal
parameters $\lambda$ for $G_{2}(F)$.

######  Proposition 1.1.

For every unramified
$\lambda:W_{F}\to{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G_{2}}}$, the set
${\hat{R}}_{\lambda}$ is $W$-conjugate to one of the following subsets of
${\hat{R}}$, also pictured in Table 1.2.1:

  1. (1)

${\hat{R}}_{\lambda}=\emptyset$;

  2. (2)

${\hat{R}}_{\lambda}=\\{\hat{\gamma}_{1}+2\hat{\gamma}_{2}\\}$;

  3. (3)

${\hat{R}}_{\lambda}=\\{2\hat{\gamma}_{1}+3\hat{\gamma}_{2}\\}$;

  4. (4)

${\hat{R}}_{\lambda}=\\{\hat{\gamma}_{1},2\hat{\gamma}_{1}+3\hat{\gamma}_{2}\\}$;

  5. (5)

${\hat{R}}_{\lambda}=\\{\hat{\gamma}_{1},\hat{\gamma}_{1}+2\hat{\gamma}_{2}\\}$;

  6. (6)

${\hat{R}}_{\lambda}=\\{\hat{\gamma}_{2},\hat{\gamma}_{1}+\hat{\gamma}_{2}\\}$;

  7. (7)

${\hat{R}}_{\lambda}=\\{\hat{\gamma}_{1},\hat{\gamma}_{1}+3\hat{\gamma}_{2}\\}$;

  8. (8)

${\hat{R}}_{\lambda}=\\{\hat{\gamma}_{1},\hat{\gamma}_{2}\\}$;

  9. (9)

${\hat{R}}_{\lambda}=\\{\hat{\gamma}_{1},\hat{\gamma}_{1}+\hat{\gamma}_{2},\hat{\gamma}_{1}+2\hat{\gamma}_{2},\hat{\gamma}_{1}+3\hat{\gamma}_{2}\\}$.

Each case above determines a moduli space $V_{\lambda}$ of unramified
Langlands parameters with infinitesimal parameter $\lambda$.

Table 1.2.1. The classification of unramified infinitesimal parameters
$\lambda$ for $G_{2}(F)$ in terms of the subsets ${\hat{R}}_{\lambda}$
appearing in Proposition 1.1. Black root vectors with a halo appear in
${\hat{R}}_{\lambda}$ and so the corresponding root spaces form $V_{\lambda}$.
Vectors in black without a halo determine root groups appearing in the
reductive group $H_{\lambda}$ acting on $V_{\lambda}$

###### Proof.

The possibilities for $\lambda$ are classified by the associated
prehomogeneous vector spaces $V_{\lambda}$ as they appear in
$\operatorname{Lie}{\widehat{G_{2}}}$ up to ${\widehat{G}}_{2}$-conjugacy.

Case 1 describes all $\lambda$ for which $V_{\lambda}=0$. These are the
$\lambda$ for which ${\hat{\gamma}}(\lambda({\operatorname{Fr}}))\neq q$ for
every root ${\hat{\gamma}}\in{\hat{R}}$. In this case the group $H_{\lambda}$
is ${\widehat{G_{2}}}$ or one of the subgroups
$\operatorname{SL}_{3}(\mathbb{C})$, $\operatorname{SO}_{4}(\mathbb{C})$,
$\operatorname{GL}_{2}^{{\hat{\gamma}}_{1}}({\mathbb{C}})$,
$\operatorname{GL}_{2}^{{\hat{\gamma}}_{2}}({\mathbb{C}})$ or ${\widehat{T}}$,
where $U_{\hat{\gamma}}\subset\operatorname{GL}_{2}^{\hat{\gamma}}$.

Cases 2 and 3 describe all $\lambda$ for which $\dim V_{\lambda}=1$; in these
cases, $V_{\lambda}=\mathrm{Span}\\{X_{\hat{\gamma}}\\}$ for some root
${\hat{\gamma}}$ of ${\widehat{G}}_{2}$ and $H_{\lambda}$ is either
${\widehat{T}}$ or
$\operatorname{GL}_{2}^{\hat{\gamma}^{\prime}}({\mathbb{C}})$ for a root
${\hat{\gamma}^{\prime}}$ perpendicular to ${\hat{\gamma}}$. Up to
${\widehat{G}}_{2}$-conjugacy, there are only two cases: either
${\hat{\gamma}}$ is short or long; in Case 2 we consider the short root
${\hat{\gamma}}={{\hat{\gamma}}_{1}}+2{{\hat{\gamma}}_{2}}$, to which
${{\hat{\gamma}}_{1}}$ is perpendicular, and in Case 3 we consider the long
root ${\hat{\gamma}}=2{{\hat{\gamma}}_{1}}+3{{\hat{\gamma}}_{2}}$, to which
${{\hat{\gamma}}_{2}}$ is perpendicular.

Cases 4, 5, 6, 7 and 8 describe all $\lambda$ for which $\dim V_{\lambda}=2$;
in these cases,
$V_{\lambda}=\mathrm{Span}\\{X_{\widehat{\gamma}},X_{\widehat{\gamma}^{\prime}}\\}$
for roots ${\widehat{\gamma}}$ and $X_{\hat{\gamma}^{\prime}}$ of
${\widehat{G}}_{2}$ and $H_{\lambda}$ is either ${\widehat{T}}$ or
$\operatorname{GL}_{2}^{\hat{\gamma}_{0}}$ for a root ${\hat{\gamma}_{0}}$
such that
${\widehat{\gamma}^{\prime}}={\widehat{\gamma}}+{\widehat{\gamma}_{0}}$.

If $\dim V_{\lambda}>2$ there is only one possibility for $\lambda$ and this
is Case 9, treated in [CFZ:cubics]. When $\dim V_{\lambda}>2$ this forces
$\dim V_{\lambda}=4$ and that

$V_{\lambda}=\mathrm{Span}\\{X_{\widehat{\gamma}},X_{{\widehat{\gamma}}+{\widehat{\gamma}_{0}}},X_{{\widehat{\gamma}}+2{\widehat{\gamma}_{0}}},X_{{\widehat{\gamma}}+3{\widehat{\gamma}_{0}}}\\}$

for roots $\hat{\gamma}$ and ${\widehat{\gamma}_{0}}$. The group that acts on
$V_{\lambda}$ in this case is
$H_{\lambda}=\operatorname{GL}_{2}^{\widehat{\gamma}_{0}}$. There are exactly
six such $V_{\lambda}$, all ${\widehat{G}}_{2}$-conjugate to
$\mathrm{Span}\\{X_{{{\hat{\gamma}}_{1}}},X_{{{\hat{\gamma}}_{1}}+{{\hat{\gamma}}_{2}}},X_{{{\hat{\gamma}}_{1}}+2{{\hat{\gamma}}_{2}}},X_{{{\hat{\gamma}}_{1}}+3{{\hat{\gamma}}_{2}}}\\}$,
in which case $H_{\lambda}=\operatorname{GL}_{2}^{{{\hat{\gamma}}_{2}}}$. ∎

###  1.3. Prehomogeneous vector spaces

This classification of infinitesimal parameters $\lambda$ for $G_{2}$ by the
associated prehomogeneous vector spaces $H_{\lambda}\times V_{\lambda}\to
V_{\lambda}$ given in Proposition 1.1 may be simplified further by examining
the categories $\mathsf{Per}_{H_{\lambda}}(V_{\lambda})$ that arise as
$\lambda$ goes through the cases appearing there.

######  Proposition 1.2.

If $\lambda:W_{F}\to{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G_{2}}}$ is an
unramified infinitesimal parameter for $G_{2}(F)$ then the category
$\mathsf{Per}_{H_{\lambda}}(V_{\lambda})$ is equivalent to
$\mathsf{Per}_{H}(V)$ where $H\times V\to V$ is one of the following five
prehomogeneous vector spaces.

  1. (1)

$V=0$ and $H$ is an algebraic group, not necessarily connected (trivial
action);

  2. (2)

$V=\mathbb{A}^{1}$ with $H=\operatorname{GL}_{1}$-action given by $t\cdot
x=tx$ (scalar multiplication);

  3. (3)

$V=\mathbb{A}^{2}$ with $H=\operatorname{GL}_{2}$-action $h\cdot
x=det(h)^{n}hx$, for non-negative integer $n$ (twisted matrix multiplication);

  4. (4)

$V=\mathbb{A}^{2}$ with $H=\operatorname{GL}_{1}^{2}$-action
$(t_{1},t_{2})\cdot(x_{1},x_{2})=(t_{1}x_{1},t_{1}t_{2}^{n}x_{2})$, for
positive integer $n$ (toric variety);

  5. (5)

$V=\mathbb{A}^{4}$ with $H=\operatorname{GL}_{2}$-action $h\cdot
x=(\det^{-1}\otimes\operatorname{Sym}^{3})(h)(x)=\det(h)^{-1}\operatorname{Sym}^{3}(h)x$
(normalized Symmetric cube).

###### Proof.

We begin the proof with a general observation. Suppose $H$ acts on $V$ and let
$u:H^{\prime}\to H$ be an epimorphism of algebraic groups with connected
kernel. Let $H^{\prime}$ act on $V$ through $u$. Then $\mathsf{Per}_{H}(V)$
and $\mathsf{Per}_{H^{\prime}}(V)$ are equivalent. To see this, recall that an
$H$-equivariant perverse sheaf on $V$ is a perverse sheaf $\mathcal{F}$
equipped with an isomorphism $\mu:m^{*}\mathcal{F}\to p^{*}\mathcal{F}$
satisfying a so-called cocycle condition. Consider the functor
$\mathsf{Per}_{H}(V)\to\mathsf{Per}_{H^{\prime}}(V)$ defined on objects by
$(\mathcal{F},\mu)\mapsto(\mathcal{F},(u\times\operatorname{id})^{*}\mu)$ and
on arrows by $\varphi\mapsto\varphi$. This functor is essentially surjective.
Since $u\times\operatorname{id}:H^{\prime}\times V\to H\times V$ is smooth
with connected fibres of equal dimension $\dim\ker u$ the functor
$(u\times\operatorname{id})^{*}$ is full and faithful by, for example,
[BBD]*Proposition 4.2.5. Thus, $\mathsf{Per}_{H}(V)$ and
$\mathsf{Per}_{H^{\prime}}(V)$ are equivalent. We refer to this equivalence as
an instance of base-change, below, and in this case we will say that the
prehomogeneous vector spaces $H\times V\to V$ and $H^{\prime}\times V\to V$
are equivalent.

The cases below all refer to Proposition 1.1.

  1. (Case 1)

The prehomogeneous vector space in Proposition 1.1, Case 1 is $V=0$ for a
connected group acting on $0$, which is an instance of the prehomogeneous
vector space 1. By the base-change argument above, this is equivalent to 1 for
trivial group $H$.

  2. (Case 2)

There are two group actions appearing in Proposition 1.1, Case 2, either
$H_{\lambda_{\ref{infcase:1-short}}}={\widehat{T}}$ or
$H_{\lambda_{\ref{infcase:1-short}}}=\operatorname{GL}_{2}$. In the former
case, observe that
${\hat{\gamma}_{1}}+2{\hat{\gamma}_{2}}:{\widehat{T}}\to{\operatorname{GL}_{1}}$
is an epimorphism of algebraic groups with connected fibre so, by the
paragraph above,
$\mathsf{Per}_{H_{\lambda_{\ref{infcase:1-short}}}}(V_{\lambda_{\ref{infcase:1-short}}})$
is equivalent to $\mathsf{Per}_{{\operatorname{GL}_{1}}}(\mathbb{A}^{1})$ for
the prehomogeneous vector space 2. In the latter case, observe that
$\det:{\widehat{T}}\to{\operatorname{GL}_{1}}$ is an epimorphism of algebraic
groups with connected fibre so, by the base-change argument above
$\mathsf{Per}_{H_{\lambda_{\ref{infcase:1-short}}}}(V_{\lambda_{\ref{infcase:1-short}}})$
is again equivalent to
$\mathsf{Per}_{{\operatorname{GL}_{1}}}(\mathbb{A}^{1})$ for the
prehomogeneous vector space 2.

  3. (Case 3)

In Proposition 1.1, Case 3 there are two group actions to consider: either
$H_{\lambda_{\ref{infcase:2-long}}}={\widehat{T}}$ or
$H_{\lambda_{\ref{infcase:2-long}}}=\operatorname{GL}_{2}$. Here,
$V_{\lambda_{\ref{infcase:2-long}}}=\mathbb{A}^{1}$. If
$H_{\lambda_{\ref{infcase:2-long}}}={\widehat{T}}$ then the action is
$h.x=\det(h)x$. Using the base-change argument above we have
$\mathsf{Per}_{H_{\lambda_{\ref{infcase:2-long}}}}(V_{\lambda_{\ref{infcase:2-long}}})=\mathsf{Per}_{{\operatorname{GL}_{1}}}(\mathbb{A}^{1})$
since
$2{\hat{\gamma}_{1}}+3{\hat{\gamma}_{2}}:{\widehat{T}}\to{\operatorname{GL}_{1}}$
is an epimorphism of algebraic groups with connected fibre. If
$H_{\lambda_{\ref{infcase:2-long}}}=\operatorname{GL}_{2}$ then the action is
$h\cdot x=\det(h)x$. Then, using the base-change argument above,
$\mathsf{Per}_{H_{\lambda_{\ref{infcase:2-long}}}}(V_{\lambda_{\ref{infcase:2-long}}})=\mathsf{Per}_{{\operatorname{GL}_{1}}}(\mathbb{A}^{1})$
since $\det:\operatorname{GL}_{2}\to{\operatorname{GL}_{1}}$ is an epimorphism
of algebraic groups with connected fibre. Thus,
$\mathsf{Per}_{H_{\lambda_{\ref{infcase:2-long}}}}(V_{\lambda_{\ref{infcase:2-long}}})$
is equivalent $\mathsf{Per}_{H}(V)$ for the prehomogeneous vector space 2.

  4. (Case 4)

The prehomogeneous vector space in Proposition 1.1, Case 4 is
$V_{\lambda_{\ref{infcase:3}}}=\mathbb{A}^{2}$ and
$H_{\lambda_{\ref{infcase:3}}}=\operatorname{GL}_{2}$; this is equivalent to
the prehomogeneous vector space 3 for $n=1$.

  5. (Case 5)

The prehomogeneous vector space in Proposition 1.1, Case 5 is
$V_{\lambda_{\ref{infcase:4-D2}}}=\mathbb{A}^{2}$ and
$H_{\lambda_{\ref{infcase:4-D2}}}={\widehat{T}}$; this is equivalent to the
prehomogeneous vector space 4 for $n=2$.

  6. (Case 6)

The prehomogeneous vector space in Proposition 1.1, Case 6 is
$V_{\lambda_{\ref{infcase:5}}}=\mathbb{A}^{2}$ and
$H_{\lambda_{\ref{infcase:5}}}=\operatorname{GL}_{2}$; this is equivalent to
the prehomogeneous vector space 3 for $n=0$.

  7. (Case 7)

The prehomogeneous vector space in Proposition 1.1, Case 7 is
$V_{\lambda_{\ref{infcase:6-A2}}}=\mathbb{A}^{2}$ and
$H_{\lambda_{\ref{infcase:6-A2}}}={\widehat{T}}$; this is equivalent to the
prehomogeneous vector space 4 for $n=3$.

  8. (Case 8)

The prehomogeneous vector space in Proposition 1.1, Case 8 is
$V_{\lambda_{\ref{infcase:7-reg}}}=\mathbb{A}^{2}$ and
$H_{\lambda_{\ref{infcase:7-reg}}}={\widehat{T}}$; this is equivalent to the
prehomogeneous vector space 4 for $n=1$.

  9. (Case 9)

The prehomogeneous vector space in Proposition 1.1, Case 9 is
$V_{\lambda_{\ref{infcase:8-sub}}}=\mathbb{A}^{4}$ and
$H_{\lambda_{\ref{infcase:8-sub}}}=\operatorname{GL}_{2}$; as explained in
[CFZ:cubics]*Proposition 3.1, this is equivalent to the prehomogeneous vector
space 5. Category
$\mathsf{Per}_{\operatorname{GL}_{2}}(\det^{-1}\otimes\operatorname{Sym}^{3})$
appeared in [CFZ:cubics].∎

###  1.4. The geometry of the moduli space of Langlands parameters

In this section we make a microlocal study of the five prehomogeneous vector
spaces appearing in Proposition 1.2. In Section 2.6 we use this to determine
the ABV-packets for $G_{2}(F)$.

Let $H\times V\to V$ be a prehomogeneous vector space. Let $V^{*}$ be the dual
vector space to $V$ and let $H\times V^{*}\to V^{*}$ be the adjoint action.
Then $H\times H$ acts on $T^{*}(V)=V\times V^{*}$ by
$(h,k)\cdot(x,y){\,:=\,}(h\cdot x,k\cdot y)$ and $T^{*}(V)$ is also a
prehomogeneous vector space. Now let $[\,\ ]:V\times
V^{*}\to\operatorname{Lie}H$ be the momentum map and consider the conormal
variety

$\Lambda{\,:=\,}\\{(x,y)\in T^{*}(V),\ [x,y]=0\\}.$

This Lagrangian subspace $\Lambda\subseteq T^{*}(V)$ carries the diagonal
$H$-action $h\cdot(x,y){\,:=\,}(h\cdot x,h\cdot y)$. This is a bundle of
vector spaces with group actions: for each $x\in V$, the fibre

$\Lambda_{x}=\\{y\in V^{*},\ [x,y]=0\\}$

caries a natural action of $Z_{H}(x)$. It is not always the case that
$\Lambda_{x}$ is a prehomogeneous vector space.

######  Proposition 1.3.

If $V$ is one of the 5 prehomogeneous vector spaces appearing in Proposition
1.2 then $\Lambda$ is a bundle of prehomogeneous vector spaces.

When the conormal variety $\Lambda$ to $V$ is a bundle of prehomogeneous
vector spaces, we write $\Lambda_{x}^{\text{sreg}}$ for the open dense
$Z_{H}(x)$-orbit in $\Lambda_{x}$ and $\Lambda^{\text{reg}}_{C}$ for the open
dense $H$-orbit in $\Lambda_{C}$, for each $H$-orbit $C\subseteq V$.

In this section we study the conormal varieties to the prehomogeneous vector
spaces appearing in Proposition 1.2. In each case we:

  * •

find all $H$-orbits $C\subseteq V$ and all dual $H$-orbits $C^{*}\subseteq
V^{*}$;

  * •

calculate each equivariant fundamental group $A_{C}{\,:=\,}\pi_{0}(Z_{H}(x))$,
for $x\in C$;

  * •

enumerate the simple objects in $\mathsf{Per}_{H}(V)$;

  * •

find the Fourier transform of each simple equivariant perverse sheaf on $V$;

  * •

show that the conormal variety $\Lambda$ is a bundle of prehomogeneous vector
spaces;

  * •

calculate the equivariant fundamental group
$A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{C}$ of
$\Lambda_{C}^{\text{sreg}}$, for each $H$-orbit $C$ in $V$;

  * •

calculate
$\operatorname{\mathsf{N}\hskip-1.0pt\mathsf{E}\hskip-1.0pt\mathsf{v}\hskip-1.0pt\mathsf{s}}_{C_{j}}{\mathcal{I\hskip-1.0ptC}}(\mathcal{L}_{C_{i}})\in\mathsf{Loc}_{H}(\Lambda_{C_{j}}^{\text{sreg}})$
for every simple ${\mathcal{I\hskip-1.0ptC}}(\mathcal{L}_{C_{i}})$ in
$\mathsf{Per}_{H}(V)$, where
$\operatorname{\mathsf{N}\hskip-1.0pt\mathsf{E}\hskip-1.0pt\mathsf{v}\hskip-1.0pt\mathsf{s}}$
is defined in [CFMMX]*Section 7.10.

These geometric calculations, and their applications to representation theory,
follow the ideas presented in the examples treated in [CFMMX]*Part II and in
[CFZ:cubics].

  1. (1)

$\mathsf{Per}_{H}(0)$ is the category of finite dimensional representations of
$\pi_{0}(H)$. Then $\Lambda=0$ and the functor
$\operatorname{\mathsf{N}\hskip-1.0pt\mathsf{E}\hskip-1.0pt\mathsf{v}\hskip-1.0pt\mathsf{s}}$
is the identity.

  2. (2)

For the action of $\operatorname{GL}_{1}$ on $\mathbb{A}^{1}$ given by scalar
multiplication there are two ${\operatorname{GL}_{1}}$-orbits: $C_{-}=\\{0\\}$
and its complement, $C_{1}$. Then
$\mathsf{Per}_{{\operatorname{GL}_{1}}}(\mathbb{A}^{1})$ has two simple
objects,
${\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{0}})={\mathbbm{1}}_{C_{0}}^{!}[0]$,
the extension-by-zero of ${\mathbbm{1}}_{C_{0}}$ and
${\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{1}})={\mathbbm{1}}_{\mathbb{A}^{1}}[1]$.
The conormal varieties are given as follows. First observe that
$\Lambda_{0}=T_{0}(\mathbb{A}^{1}){\ \cong\ }\mathbb{A}^{1}$ and
$\Lambda_{0}^{\text{sreg}}{\ \cong\ }\\{y\in\mathbb{A}^{1}{\ |\ }y\neq 0\\}$;
note that this is a single orbit under the $\operatorname{GL}_{1}$-action. If
$x\in C_{1}$ then $\Lambda_{x}=\\{(x,y){\ |\ }xy=0\\}=\\{(x,0)\\}$ which
itself is a $\operatorname{GL}_{1}$-orbit. In this way we see that
$T^{*}_{C_{0}}(\mathbb{A}^{1})^{\text{sreg}}$ and
$T^{*}_{C_{1}}(\mathbb{A}^{1})^{\text{sreg}}$ are both isomorphic, as
$\operatorname{GL}_{1}$-spaces, to $C_{1}$ and consequently, equivariant local
systems on these spaces are naturally identified with finite-dimensional
vector spaces. The functor
$\operatorname{\mathsf{N}\hskip-1.0pt\mathsf{E}\hskip-1.0pt\mathsf{v}\hskip-1.0pt\mathsf{s}}:\mathsf{Per}_{\operatorname{GL}_{1}}(\mathbb{A}^{1})\to\mathsf{Loc}_{\operatorname{GL}_{1}}(T^{*}_{\operatorname{GL}_{1}}(\mathbb{A}^{1})_{\text{sreg}})$
is given by the following table:

$\begin{array}[]{c| cc | c}\mathsf{Per}_{\operatorname{GL}_{1}}(\mathbb{A}^{1})&\mathsf{Loc}_{\operatorname{GL}_{1}}(\Lambda_{C_{0}}^{\text{sreg}})&\mathsf{Loc}_{\operatorname{GL}_{1}}(\Lambda_{C_{1}}^{\text{sreg}})&\text{Fourier}\\\ \hline\cr\hline\cr{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{0}})&{\mathbbm{1}}_{\Lambda_{C_{0}}^{\text{sreg}}}&0&{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{1}})\\\ {\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{1}})&0&{\mathbbm{1}}_{\Lambda_{C_{1}}^{\text{sreg}}}&{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{0}})\end{array}$

  3. (3)

For the twisted matrix multiplication action of $\operatorname{GL}_{2}$ on
$\mathbb{A}^{2}$, again there are only two $H$-orbits, $C_{0}=\\{(0,0)\\}$ and
the complement $C_{1}$ and only two simple objects in
$\mathsf{Per}_{\operatorname{GL}_{2}}(\mathbb{A}^{2})$ are
${\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{0}})={\mathbbm{1}}_{C_{0}}^{!}[0]$
and
${\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{1}})={\mathbbm{1}}_{\mathbb{A}^{2}}[2]$.
Again, $T^{*}_{C_{0}}(\mathbb{A}^{2})^{\text{sreg}}$ and
$T^{*}_{C_{1}}(\mathbb{A}^{2})^{\text{sreg}}$ are both isomorphic, as
$\operatorname{GL}_{2}$-spaces, to ${\bar{C}_{1}}$, which is a single
$\operatorname{GL}_{2}$-orbit and also has trivial equivariant fundamental
group. The functor
$\operatorname{\mathsf{N}\hskip-1.0pt\mathsf{E}\hskip-1.0pt\mathsf{v}\hskip-1.0pt\mathsf{s}}:\mathsf{Per}_{\operatorname{GL}_{2}}(\mathbb{A}^{2})\to\mathsf{Loc}_{\operatorname{GL}_{2}}(T^{*}_{\operatorname{GL}_{2}}(\mathbb{A}^{2})_{\text{sreg}})$
is given by

$\begin{array}[]{c| cc | c }\mathsf{Per}_{\operatorname{GL}_{2}}(\mathbb{A}^{2})&\mathsf{Loc}_{\operatorname{GL}_{2}}(\Lambda_{C_{0}}^{\text{sreg}})&\mathsf{Loc}_{\operatorname{GL}_{2}}(\Lambda_{C_{1}}^{\text{sreg}})&\text{Fourier}\\\ \hline\cr\hline\cr{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{0}})&{\mathbbm{1}}_{\Lambda_{C_{0}}^{\text{sreg}}}&0&{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{1}})\\\ {\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{1}})&0&{\mathbbm{1}}_{\Lambda_{C_{1}}^{\text{sreg}}}&{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{0}})\end{array}$

  4. (4)

Consider the action of $H=\operatorname{GL}_{1}^{2}$ on $V=\mathbb{A}^{2}$ by
$(t_{1},t_{2}).(x_{1},x_{2})=(t_{1}x_{1},t_{1}t_{2}^{n}x_{2})$ for positive
integer $n$. There are four $H$-orbits in this case: $C_{0}=\\{(0,0)\\}$,
$C_{1}=\\{(x_{1},0){\ |\ }x_{1}\neq 0\\}$, $C_{2}=\\{(0,x_{2}){\ |\ }x_{2}\neq
0\\}$ and the open orbit $C_{3}=\\{(x_{1},x_{2}){\ |\ }x_{1}x_{2}\neq 0\\}$.
The equivariant fundamental groups of these orbits are trivial with the
exception of $C_{3}$, for which $A_{C_{3}}$ is the group of $n$-th roots-of-
unity. There are $4+(n-1)$ simple objects in this category:
${\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{0}})={\mathbbm{1}}_{C_{0}}^{!}$,
${\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{1}})={\mathbbm{1}}_{\bar{C}_{1}}^{!}[1]$,
${\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{2}})={\mathbbm{1}}_{\bar{C}_{2}}^{!}[1]$,
${\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{3}})={\mathbbm{1}}_{\mathbb{A}^{2}}[2]$
and ${\mathcal{I\hskip-1.0ptC}}(\vartheta_{C_{3}})$, where $\vartheta$ is a
non-trivial character of the group of $n$th roots-of-unity. For $x\in C_{0}$
or $C_{3}$ we have $\Lambda_{x}{\ \cong\ }{\bar{C}_{3}}$ which has a unique
open $\operatorname{GL}_{1}^{2}$-orbit. For $x\in C_{1}$,
$\Lambda_{x}=\\{((x_{1},0),(y_{1},y_{2}))\in T^{*}(V){\ |\
}x_{1}y_{1}=0\\}=\\{((x_{1},0),(0,y_{2})){\ |\ }y_{2}\\}{\ \cong\
}{\bar{C}_{2}}$ which also has a unique $\operatorname{GL}_{1}^{2}$-orbit;
likewise for $x\in C_{2}$, $\Lambda_{x}{\ \cong\ }{\bar{C}_{1}}$. It follows
that $\Lambda_{C}$ has a unique open $H$-orbit, $\Lambda_{C}^{\text{sreg}}$,
for every $H$-orbit $C\subset V$. A simple calculation shows that the
equivariant fundamental group of $\Lambda_{C}^{\text{sreg}}$ is the group
$\langle\theta_{n}\rangle$ of $n$th roots-of-unity, for every $H$-orbit
$C\subset V$. The functor
$\operatorname{\mathsf{N}\hskip-1.0pt\mathsf{E}\hskip-1.0pt\mathsf{v}\hskip-1.0pt\mathsf{s}}$
is given by the following table, where the last row is removed if $n=1$ and
duplicated if $n>2$ for every non-trivial character $\vartheta$.

$\begin{array}[]{c | cccc | c}\mathsf{Per}_{\operatorname{GL}_{1}^{2}}(\mathbb{A}^{2})&\mathsf{Loc}_{\operatorname{GL}_{1}^{2}}(\Lambda_{C_{0}}^{\text{sreg}})&\mathsf{Loc}_{\operatorname{GL}_{1}^{2}}(\Lambda_{C_{1}}^{\text{sreg}})&\mathsf{Loc}_{\operatorname{GL}_{1}^{2}}(\Lambda_{C_{2}}^{\text{sreg}})&\mathsf{Loc}_{\operatorname{GL}_{1}^{2}}(\Lambda_{C_{3}}^{\text{sreg}})&\text{Fourier }\\\ \hline\cr\hline\cr{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{0}})&{\mathbbm{1}}_{\Lambda_{C_{0}}^{\text{sreg}}}&0&0&0&{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{3}})\\\ {\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{1}})&0&{\mathbbm{1}}_{\Lambda_{C_{1}}^{\text{sreg}}}&0&0&{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{2}})\\\ {\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{2}})&0&0&{\mathbbm{1}}_{\Lambda_{C_{2}}^{\text{sreg}}}&0&{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{1}})\\\ {\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{3}})&0&0&0&{\mathbbm{1}}_{\Lambda_{C_{3}}^{\text{sreg}}}&{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{0}})\\\ \hline\cr{\mathcal{I\hskip-1.0ptC}}(\vartheta_{C_{3}})&\vartheta_{\Lambda_{0}^{\text{sreg}}}&\vartheta_{\Lambda_{1}^{\text{sreg}}}&\vartheta_{\Lambda_{1}^{\text{sreg}}}&\vartheta_{\Lambda_{2}^{\text{sreg}}}&{\mathcal{I\hskip-1.0ptC}}(\vartheta_{C_{3}})\end{array}$

  5. (5)

The functor
$\operatorname{\mathsf{N}\hskip-1.0pt\mathsf{E}\hskip-1.0pt\mathsf{v}\hskip-1.0pt\mathsf{s}}$
for $H=\operatorname{GL}_{2}$ acting on $V=\mathbb{A}^{4}$ by
$h.x=\det^{-1}(h)\operatorname{Sym}^{3}(h)(x)$ (nomalized Symmetric cube) was
calculated in [CFZ:cubics]. We recall the result in the table below, in which
$\vartheta{\,:=\,}\vartheta_{2}$ is the sign character of
$\langle\theta_{2}\rangle$ and, as explained in Section 0.7, $\varepsilon$ is
the sign character of $S_{3}$ and $\varrho$ is the character of the reflection
representation of $S_{3}$.

$\begin{array}[]{ c | cccc | c }\mathsf{Per}_{\operatorname{GL}_{2}}(V)&\mathsf{Loc}_{\operatorname{GL}_{2}}(\Lambda_{C_{0}}^{\mathrm{sreg}})&\mathsf{Loc}_{\operatorname{GL}_{2}}(\Lambda_{C_{1}}^{\mathrm{sreg}})&\mathsf{Loc}_{\operatorname{GL}_{2}}(\Lambda_{C_{2}}^{\mathrm{sreg}})&\mathsf{Loc}_{\operatorname{GL}_{2}}(\Lambda_{C_{3}}^{\mathrm{sreg}})&\text{Fourier}\\\ \hline\cr\hline\cr{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{0}})&{\mathbbm{1}}_{\Lambda_{0}^{\mathrm{reg}}}&0&0&0&{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{3}})\\\ {\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{1}})&\varrho_{\Lambda_{0}^{\mathrm{reg}}}&{\mathbbm{1}}_{\Lambda_{1}^{\mathrm{reg}}}&0&0&{\mathcal{I\hskip-1.0ptC}}(\varrho_{C_{3}})\\\ {\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{2}})&0&\vartheta_{\Lambda_{1}^{\mathrm{reg}}}&{\mathbbm{1}}_{\Lambda_{2}^{\mathrm{reg}}}&0&{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{2}})\\\ {\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{3}})&0&0&0&{\mathbbm{1}}_{\Lambda_{3}^{\mathrm{reg}}}&{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{1}})\\\ {\mathcal{I\hskip-1.0ptC}}(\varrho_{C_{3}})&0&0&\vartheta_{\Lambda_{2}^{\mathrm{reg}}}&\varrho_{\Lambda_{3}^{\mathrm{reg}}}&{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{0}})\\\ {\mathcal{I\hskip-1.0ptC}}(\varepsilon_{C_{3}})&\varepsilon_{\Lambda_{0}^{\mathrm{reg}}}&{\mathbbm{1}}_{\Lambda_{1}^{\mathrm{reg}}}&\vartheta_{\Lambda_{2}^{\mathrm{reg}}}&\varepsilon_{\Lambda_{3}^{\mathrm{reg}}}&{\mathcal{I\hskip-1.0ptC}}(\varepsilon_{C_{3}})\end{array}$

##  2\. Unipotent representations for $G_{2}$

In this section we use the classification of infinitesimal parameters for
$G_{2}(F)$ in Proposition 1.1, together with the results of Section 1.4, to
enumerate all irreducible unipotent representations of $G_{2}(F)$ and to find
their Langlands parameters. In this way we give an explicit form of the
Langlands correspondence for unipotent representations of $G_{2}(F)$. At the
same time, we will find the ABV-packet coefficients and start assembling the
proof of Theorem 2.2.

###  2.1. The Langlands correspondence

The Langlands correspondence for unipotent representations of $G_{2}(F)$ is a
special case of [Lusztig:Classification1], though that paper uses a
normalization of the correspondence that is different from the one we give
here. Instead, here we use a normalization of the local Langlands
correspondence that satisfies the desiderata of [GGP]*§9; in particular, it
satisfies [Gross-Prasad]*Conjecture 2.6, which specifies the Langlands
parameter of generic representations for a particular infinitesimal parameter.
Moreover, for a choice of hyperspecial group, our normalization also specifies
the enhanced Langlands parameter of spherical representation for a particular
infinitesimal parameter, i.e., if $\pi\in\Pi_{\phi}(G_{2}(F))$ is spherical,
then $\phi$ is trivial on the $\operatorname{SL}_{2}(\mathbb{C})$ part and the
corresponding representation of $\widehat{A}_{\phi}$ is trivial. This choice
of normalization is compatible with the requirement in the work of Arthur
[Arthur:book] and agrees with the local Langlands correspondence appearing in
[Solleveld]. As in [Waldspurger:SOimpair]*p. 804, this normalization is
obtained by composing Lusztig’s correspondence with the Aubert involution.

The Langlands correspondence for unipotent representations of $G_{2}(F)$ is
given in Table 2.1.1, also in Table 2.7.1, making reference to the
classification of unramified infinitesimal parameters in Proposition 1.1 and
the enumeration of unramified Langlands parameters for $G_{2}(F)$ given in
Section 2.6. Both tables use notation from [Muic] for irreducible admissible
representations of $G_{2}(F)$, modified as explained in Section 0.7. In Table
2.1.1, characters of $A_{\phi}$ are listed in the column with $\phi$ at the
top and the $L$-packet $\Pi_{\phi}(G_{2}(F))$ is found by gathering together
the representations $\pi$ for which the character of $A_{\phi}$ is non-zero,
thus defining the bijection

$\Pi_{\phi}(G_{2}(F))\to\widehat{A_{\phi}},$

for all unramified Langlands parameters for $G_{2}(F)$. In this way, Table
2.1.1 defines $\pi(\phi,\rho)$ for each enhanced Langlands parameter
$(\phi,\rho)$, for the $\rho\in\widehat{A_{\phi}}$.

There are precisely three non-singleton $L$-packets of unipotent
representations of $G_{2}(F)$: $\Pi_{\phi_{\ref{infcase:4-D2}d}}(G_{2}(F))$,
$\Pi_{\phi_{\ref{infcase:6-A2}d}}(G_{2}(F))$ and
$\Pi_{\phi_{\ref{infcase:8-sub}d}}(G_{2}(F))$, of which the first has order
$2$ and the other two have order $3$. Note that the group $A_{\phi}$ is
trivial except when $\phi$ is $\phi_{\ref{infcase:4-D2}d}$,
$\phi_{\ref{infcase:6-A2}d}$ or $\phi_{\ref{infcase:8-sub}d}$, in which cases
$A_{\phi}$ is $\langle\theta_{2}\rangle$, $\langle\theta_{3}\rangle$ and
$S_{3}$, respectively. These three Langlands parameters are all of Arthur type
and are distinguished by the fact that they have elliptic endoscopy, as
explained in Section 4.

Table 2.1.1. The local Langlands correspondence for unipotent representations
of $G_{2}(F)$. See Section 2.1 for how to read this table.

$\begin{array}[]{r|c|cc|cc|cc|cccc|cc|cccc|cccc|cccc}&\lx@intercol\hfil\lambda_{\ref{infcase:0}}\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil\lambda_{\ref{infcase:1-short}}\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil\lambda_{\ref{infcase:2-long}}\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil\lambda_{\ref{infcase:3}}\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil\lambda_{\ref{infcase:4-D2}}\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil\lambda_{\ref{infcase:5}}\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil\lambda_{\ref{infcase:6-A2}}\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil\lambda_{\ref{infcase:7-reg}}\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil\lambda_{\ref{infcase:8-sub}}\hfil\lx@intercol\\\
&&&&&&&&&&&&&&&&&&&&&&&&&\operatorname{PGL}_{3}\\\
&T&T&\operatorname{GL}_{2}&T&\operatorname{GL}_{2}&T&\operatorname{GL}_{2}&T&\operatorname{GL}_{2}&\operatorname{GL}_{2}&\operatorname{SO}_{4}&T&\operatorname{GL}_{2}&T&\operatorname{GL}_{2}&\operatorname{GL}_{2}&\operatorname{PGL}_{3}&T&\operatorname{GL}_{2}&\operatorname{GL}_{2}&G_{2}&T&\operatorname{GL}_{2}&\operatorname{GL}_{2}&\operatorname{SO}_{4}\\\
&\phi_{\ref{infcase:0}}&\phi_{\ref{infcase:1-short}a}&\phi_{\ref{infcase:1-short}b}&\phi_{\ref{infcase:2-long}a}&\phi_{\ref{infcase:2-long}b}&\phi_{\ref{infcase:3}a}&\phi_{\ref{infcase:3}b}&\phi_{\ref{infcase:4-D2}a}&\phi_{\ref{infcase:4-D2}b}&\phi_{\ref{infcase:4-D2}c}&\phi_{\ref{infcase:4-D2}d}&\phi_{\ref{infcase:5}a}&\phi_{\ref{infcase:5}b}&\phi_{\ref{infcase:6-A2}a}&\phi_{\ref{infcase:6-A2}b}&\phi_{\ref{infcase:6-A2}c}&\phi_{\ref{infcase:6-A2}d}&\phi_{\ref{infcase:7-reg}a}&\phi_{\ref{infcase:7-reg}b}&\phi_{\ref{infcase:7-reg}c}&\phi_{\ref{infcase:7-reg}d}&\phi_{\ref{infcase:8-sub}a}&\phi_{\ref{infcase:8-sub}b}&\phi_{\ref{infcase:8-sub}c}&\phi_{\ref{infcase:8-sub}d}\\\
\hline\cr
I(\mu_{1}\nu^{a_{1}}\otimes\mu_{2}\nu^{a_{2}})&1&&&&&&&&&&&&&&&&&&&&&&&&\\\
\hline\cr I_{\gamma_{2}}(a-1/2,\mu\circ\det)&&1&0&&&&&&&&&&&&&&&&&&&&&&\\\
I_{\gamma_{2}}(a-1/2,\mu\otimes\mathrm{St}_{\operatorname{GL}_{2}})&&0&1&&&&&&&&&&&&&&&&&&&&&&\\\
\hline\cr I_{\gamma_{1}}(a-1/2,\mu\circ\det)&&&&1&0&&&&&&&&&&&&&&&&&&&&\\\
I_{\gamma_{1}}(a-1/2,\mu\otimes\mathrm{St}_{\operatorname{GL}_{2}})&&&&0&1&&&&&&&&&&&&&&&&&&&&\\\
\hline\cr
I_{{\gamma_{1}}}(1/6,\theta_{3}^{n}\circ\det)&&&&&&1&0&&&&&&&&&&&&&&&&&&\\\
I_{{\gamma_{1}}}(1/6,\theta_{3}^{n}\otimes\mathrm{St}_{\operatorname{GL}_{2}})&&&&&&0&1&&&&&&&&&&&&&&&&&&\\\
\hline\cr
J_{\gamma_{2}}(1,I^{\operatorname{GL}_{2}}(1\otimes\theta_{2}))&&&&&&&&1&0&0&0&&&&&&&&&&&&&&\\\
J_{{\gamma_{1}}}(1/2,\theta_{2}\otimes\mathrm{St}_{\operatorname{GL}_{2}})&&&&&&&&0&1&0&0&&&&&&&&&&&&&&\\\
J_{{\gamma_{2}}}(1/2,\theta_{2}\otimes\mathrm{St}_{\operatorname{GL}_{2}})&&&&&&&&0&0&1&0&&&&&&&&&&&&&&\\\
\pi(\theta_{2})&&&&&&&&0&0&0&1&&&&&&&&&&&&&&\\\
I_{0}(G_{2}[-1])&&&&&&&&0&0&0&\vartheta_{2}&&&&&&&&&&&&&&\\\ \hline\cr
I_{\gamma_{2}}(3/2,1_{\operatorname{GL}_{2}})&&&&&&&&&&&&1&0&&&&&&&&&&&&\\\
I_{\gamma_{2}}(3/2,\mathrm{St}_{\operatorname{GL}_{2}})&&&&&&&&&&&&0&1&&&&&&&&&&&&\\\
\hline\cr
J_{\gamma_{2}}(1,I^{\operatorname{GL}_{2}}(\theta_{3}\otimes\theta_{3}^{-1}))&&&&&&&&&&&&&&1&0&0&0&&&&&&&&\\\
J_{{\gamma_{1}}}(1/2,\theta_{3}^{2}\otimes\mathrm{St}_{\operatorname{GL}_{2}})&&&&&&&&&&&&&&0&1&0&0&&&&&&&&\\\
J_{{\gamma_{1}}}(1/2,\theta_{3}\otimes\mathrm{St}_{\operatorname{GL}_{2}})&&&&&&&&&&&&&&0&0&1&0&&&&&&&&\\\
\pi(\theta_{3})&&&&&&&&&&&&&&0&0&0&1&&&&&&&&\\\
I_{0}(G_{2}[\theta_{3}])&&&&&&&&&&&&&&0&0&0&\vartheta_{3}&&&&&&&&\\\
I_{0}(G_{2}[\theta_{3}^{2}])&&&&&&&&&&&&&&0&0&0&\vartheta_{3}^{2}&&&&&&&&\\\
\hline\cr 1_{G_{2}}&&&&&&&&&&&&&&&&&&1&0&0&0&&&&\\\
J_{\gamma_{1}}(3/2,\mathrm{St}_{\operatorname{GL}_{2}})&&&&&&&&&&&&&&&&&&0&1&0&0&&&&\\\
J_{{\gamma_{2}}}(5/2,\mathrm{St}_{\operatorname{GL}_{2}})&&&&&&&&&&&&&&&&&&0&0&1&0&&&&\\\
\mathrm{St}_{G_{2}}&&&&&&&&&&&&&&&&&&0&0&0&1&&&&\\\ \hline\cr
J_{{\gamma_{2}}}(1,I^{\operatorname{GL}_{2}}(1\otimes
1))&&&&&&&&&&&&&&&&&&&&&&1&0&0&0\\\
J_{\gamma_{1}}(1/2,\mathrm{St}_{\operatorname{GL}_{2}})&&&&&&&&&&&&&&&&&&&&&&0&1&0&0\\\
J_{{\gamma_{2}}}(1/2,\mathrm{St}_{\operatorname{GL}_{2}})&&&&&&&&&&&&&&&&&&&&&&0&0&1&0\\\
\pi(1)^{\prime}&&&&&&&&&&&&&&&&&&&&&&0&0&0&1\\\
\pi(1)&&&&&&&&&&&&&&&&&&&&&&0&0&0&\varrho\\\
I_{0}(G_{2}[1])&&&&&&&&&&&&&&&&&&&&&&0&0&0&\varepsilon\end{array}$

Recall that we have classified all unramified infinitesimal parameters for
$G_{2}(F)$ in Section 1.2. Working through the 9 cases in Proposition 1.1, in
Section 2.6 we find, in each case, all Langlands parameters $\phi$ with given
infinitesimal parameter $\lambda$. Using results from [CFMMX], we compute the
component group $A_{\phi}$. We find that these groups are trivial in all cases
except the unique tempered Langlands parameter with infinitesimal parameters
given by Cases 5, 7 and 9. In all other cases, except these three, since
$A_{\phi}$ is trivial, we use the Langlands classification to find the
corresponding admissible representation and note that
$\Pi_{\phi}(G_{2}(F))\to\mathsf{Rep}(A_{\phi})$ is trivial in these cases.
This is done case-by-case in Section 2.6. It then remains to consider only the
Langlands parameters with infinitesimal parameters given by Cases 5, 7 and 9,
specifically the three tempered (i.e., bounded upon restriction to $W_{F}$)
parameters denoted by $\phi_{\ref{infcase:4-D2}d}$,
$\phi_{\ref{infcase:6-A2}d}$ and $\phi_{\ref{infcase:8-sub}d}$. These three
are of Arthur type and are lifted from Arthur parameters of elliptic
endoscopic groups, as explained in Section 4. We found the local Langlands
correspondence for parameters with infinitesimal parameter
$\lambda_{\ref{infcase:8-sub}}$ in [CFZ:cubics]*Theorem 2.5. The local
Langlands correspondence for parameters with infinitesimal parameter
$\lambda_{\ref{infcase:4-D2}}$ and $\lambda_{\ref{infcase:6-A2}}$ is obtained
by similar arguments.

###  2.2. Vogan’s version of the local Langlands correspondence

Let $\lambda$ be an unramifed infinitesimal parameter of $G_{2}(F)$ and let
$\Pi_{\lambda}(G_{2}(F))$ be the set of (equivalence classes of) irreducible
smooth representations of $G_{2}(F)$ with infinitesimal parameter $\lambda$.
Associated with $\lambda$, we have the Vogan variety $V_{\lambda}$ and the
group $H_{\lambda}$. If $\phi$ is an unramified Langlands parameter with
infinitesimal parameter $\lambda$, let $C_{\pi}\subset V_{\lambda}$ be the
$H_{\lambda}$-orbit corresponding to $\phi$. For use below, we define
$\dim(\phi){\,:=\,}\dim C_{\phi}$ and $\dim(\pi){\,:=\,}\dim C_{\phi}$ when
$\phi$ is the Langlands parameter for $\pi$.

Let $\Pi_{\phi}(G_{2}(F))$ be the $L$-packet of $\phi$, whose existence was
proved in [Lusztig:Classification1]. Since $A_{\phi}$ is both the component
group of $\phi$ and the equivariant fundamental group of $C_{\phi}$, the
Langlands correspondence, as presented in Section 2.1, now determines a
bijection

$\mathcal{L}:\Pi_{\phi}(G_{2}(F))\to\mathsf{Loc}_{H_{\lambda}}(C_{\phi})_{/\mathrm{iso}}^{\mathrm{simple}}$

such that $\mathcal{L}(\pi)$ corresponds to the representation $\langle\
,\pi\rangle_{\phi}$ of $A_{\phi}$. Letting $\phi$ run over all $L$-parameters
with infinitesimal parameter $\lambda$, this defines a bijection

$\mathcal{P}:\Pi_{\lambda}(G_{2}(F))\to\mathsf{Per}_{H_{\lambda}}(V_{\lambda})_{/\mathrm{iso}}^{\mathrm{simple}}$

such that $\mathcal{P}(\pi)={\mathcal{I\hskip-1.0ptC}}(\mathcal{L}(\pi))$. We
make use of this bijection in Section 2.6. In Case 9, this bijection was made
explicit in [CFZ:cubics].

###  2.3. Kazhdan-Lusztig conjecture

Using the case-by-case calculations of Section 2.6, we have verified the
Kazhdan-Lusztig conjecture, as stated in [CFMMX]*Section 10.3.3, for
$G_{2}(F)$; this extends the result of [CFZ:cubics]*Section 2.10.

###  2.4. Aubert involution and the Fourier transform

We have also confirmed that the Aubert involution, as calculated in [Muic],
matches the Fourier transform of the corresponding simple perverse sheaves,
using the bijection of Section 2.2, for all unipotent representations of
$G_{2}(F)$, confirming the expectation of [CFMMX]*Section 10.3.4 and extending
the result of [CFZ:cubics]*Section 2.13.

###  2.5. A generalization of the component group of an Arthur parameter

We now introduce the algebraic group
$\mathcal{S}^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}$
promised in the introduction.

######  Corollary 2.1.

Let $\lambda:W_{F}\to{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G_{2}}}$ be an
unramified infinitesimal parameter for $G_{2}(F)$. Let $V_{\lambda}$ be the
moduli space of Langlands parameters with infinitesimal parameter $\lambda$.
The conormal variety $\Lambda_{\lambda}$ is a bundle of prehomogeneous vector
spaces over $V_{\lambda}$.

###### Proof.

Infinitesimal parameters $\lambda$ are classified by by the associated
prehomogeneous vector space $V_{\lambda}$ in Proposition 1.1 and these in turn
are expressed in terms of the five prehomogeneous vector spaces in Proposition
1.2. In Section 1.4 we saw that the conormal variety of each of these five
prehomogeneous vector spaces all have the property that they’re bundles of
prehomogeneous vector spaces. ∎

######  Definition 1.

Notation as above. Let $x$ be the point for $\phi$ in the moduli space $V$,
where $\lambda$ is the infinitesimal parameter of $\phi$. Let
$\Lambda_{\lambda}$ be the conormal variety and recall, by Corollary 2.1, that
this is a bundle of prehomogeneous vector spaces above $V_{\lambda}$. Set
$\mathcal{S}^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}{\,:=\,}Z_{{\widehat{G_{2}}}}(x,y)$,
where $x\in V$ is a point in the moduli space $V$ that maps to the Langlands
parameter $\phi$ and for $y\in\Lambda_{x}^{\text{sreg}}$. This variety is
independent of the choice of $y$. Also set
$A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}{\,:=\,}\pi_{0}(\mathcal{S}^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi})$,
equipped with the natural homomorphism
$\mathcal{S}^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}\to
A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}$. It
follows immediately from [CFMMX]*Prop. 6.1.1, $\phi$ is of Arthur type $\psi$,
then
$\mathcal{S}^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}=\mathcal{S}_{\psi}$.

Definition 1 is needed to find the ABV-packet coefficients $\langle\ ,\
\rangle:\mathcal{S}^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}\times\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}(G_{2}(F))\to{\bar{\mathbb{Q}}}$,
which we do in the next section.

###  2.6. Calculation of ABV-packet coefficients

In this section we calculate $\pi\mapsto\langle,\pi\rangle$ for every
unipotent representation $\pi$ of $G_{2}(F)$. The results are summarized in
Table 2.6.1, in which characters of
$A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}$ are
listed in the column with $\phi$ at the top, classified by infinitesimal
parameters. We list the endoscopic groups for $\phi$ above $\phi$. ABV-packets
$\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}(G_{2}(F))$
are found by gathering together the representations $\pi$ for which the
character of
$A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}$ is non-
zero.

Table 2.6.1. ABV-packets and ABV-packet coefficients for unipotent
representations of $G_{2}(F)$. See Section 2.6 for how to read this table.

$\begin{array}[]{r|c|cc|cc|cc|cccc|cc|cccc|cccc|cccc}&\lx@intercol\hfil\lambda_{\ref{infcase:0}}\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil\lambda_{\ref{infcase:1-short}}\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil\lambda_{\ref{infcase:2-long}}\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil\lambda_{\ref{infcase:3}}\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil\lambda_{\ref{infcase:4-D2}}\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil\lambda_{\ref{infcase:5}}\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil\lambda_{\ref{infcase:6-A2}}\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil\lambda_{\ref{infcase:7-reg}}\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil\lambda_{\ref{infcase:8-sub}}\hfil\lx@intercol\\\
&&&&&&&&&&&&&&&&&&&&&&&&&\operatorname{PGL}_{3}\\\
&T&T&\operatorname{GL}_{2}&T&\operatorname{GL}_{2}&T&\operatorname{GL}_{2}&T&\operatorname{GL}_{2}&\operatorname{GL}_{2}&\operatorname{SO}_{4}&T&\operatorname{GL}_{2}&T&\operatorname{GL}_{2}&\operatorname{GL}_{2}&\operatorname{PGL}_{3}&T&\operatorname{GL}_{2}&\operatorname{GL}_{2}&G_{2}&T&\operatorname{GL}_{2}&\operatorname{GL}_{2}&\operatorname{SO}_{4}\\\
&\phi_{\ref{infcase:0}}&\phi_{\ref{infcase:1-short}a}&\phi_{\ref{infcase:1-short}b}&\phi_{\ref{infcase:2-long}a}&\phi_{\ref{infcase:2-long}b}&\phi_{\ref{infcase:3}a}&\phi_{\ref{infcase:3}b}&\phi_{\ref{infcase:4-D2}a}&\phi_{\ref{infcase:4-D2}b}&\phi_{\ref{infcase:4-D2}c}&\phi_{\ref{infcase:4-D2}d}&\phi_{\ref{infcase:5}a}&\phi_{\ref{infcase:5}b}&\phi_{\ref{infcase:6-A2}a}&\phi_{\ref{infcase:6-A2}b}&\phi_{\ref{infcase:6-A2}c}&\phi_{\ref{infcase:6-A2}d}&\phi_{\ref{infcase:7-reg}a}&\phi_{\ref{infcase:7-reg}b}&\phi_{\ref{infcase:7-reg}c}&\phi_{\ref{infcase:7-reg}d}&\phi_{\ref{infcase:8-sub}a}&\phi_{\ref{infcase:8-sub}b}&\phi_{\ref{infcase:8-sub}c}&\phi_{\ref{infcase:8-sub}d}\\\
\hline\cr
I(\mu_{1}\nu^{a_{1}}\otimes\mu_{2}\nu^{a_{2}})&1&&&&&&&&&&&&&&&&&&&&&&&&\\\
\hline\cr I_{\gamma_{2}}(a-1/2,\mu\circ\det)&&1&0&&&&&&&&&&&&&&&&&&&&&&\\\
I_{\gamma_{2}}(a-1/2,\mu\otimes\mathrm{St}_{\operatorname{GL}_{2}})&&0&1&&&&&&&&&&&&&&&&&&&&&&\\\
\hline\cr I_{\gamma_{1}}(a-1/2,\mu\circ\det)&&&&1&0&&&&&&&&&&&&&&&&&&&&\\\
I_{\gamma_{1}}(a-1/2,\mu\otimes\mathrm{St}_{\operatorname{GL}_{2}})&&&&0&1&&&&&&&&&&&&&&&&&&&&\\\
\hline\cr
I_{{\gamma_{1}}}(1/6,\theta_{3}^{n}\circ\det)&&&&&&1&0&&&&&&&&&&&&&&&&&&\\\
I_{{\gamma_{1}}}(1/6,\theta_{3}^{n}\otimes\mathrm{St}_{\operatorname{GL}_{2}})&&&&&&0&1&&&&&&&&&&&&&&&&&&\\\
\hline\cr
J_{\gamma_{2}}(1,I^{\operatorname{GL}_{2}}(1\otimes\theta_{2}))&&&&&&&&1&0&0&0&&&&&&&&&&&&&&\\\
J_{{\gamma_{1}}}(1/2,\theta_{2}\otimes\mathrm{St}_{\operatorname{GL}_{2}})&&&&&&&&0&1&0&0&&&&&&&&&&&&&&\\\
J_{{\gamma_{2}}}(1/2,\theta_{2}\otimes\mathrm{St}_{\operatorname{GL}_{2}})&&&&&&&&0&0&1&0&&&&&&&&&&&&&&\\\
\pi(\theta_{2})&&&&&&&&0&0&0&1&&&&&&&&&&&&&&\\\
I_{0}(G_{2}[-1])&&&&&&&&\vartheta_{2}&\vartheta_{2}&\vartheta_{2}&\vartheta_{2}&&&&&&&&&&&&&&\\\
\hline\cr
I_{\gamma_{2}}(3/2,1_{\operatorname{GL}_{2}})&&&&&&&&&&&&1&0&&&&&&&&&&&&\\\
I_{\gamma_{2}}(3/2,\mathrm{St}_{\operatorname{GL}_{2}})&&&&&&&&&&&&0&1&&&&&&&&&&&&\\\
\hline\cr
J_{\gamma_{2}}(1,I^{\operatorname{GL}_{2}}(\theta_{3}\otimes\theta_{3}^{-1}))&&&&&&&&&&&&&&1&0&0&0&&&&&&&&\\\
J_{{\gamma_{1}}}(1/2,\theta_{3}^{2}\otimes\mathrm{St}_{\operatorname{GL}_{2}})&&&&&&&&&&&&&&0&1&0&0&&&&&&&&\\\
J_{{\gamma_{1}}}(1/2,\theta_{3}\otimes\mathrm{St}_{\operatorname{GL}_{2}})&&&&&&&&&&&&&&0&0&1&0&&&&&&&&\\\
\pi(\theta_{3})&&&&&&&&&&&&&&0&0&0&1&&&&&&&&\\\
I_{0}(G_{2}[\theta_{3}])&&&&&&&&&&&&&&\vartheta_{3}&\vartheta_{3}&\vartheta_{3}&\vartheta_{3}&&&&&&&&\\\
I_{0}(G_{2}[\theta_{3}^{2}])&&&&&&&&&&&&&&\vartheta_{3}^{2}&\vartheta_{3}^{2}&\vartheta_{3}^{2}&\vartheta_{3}^{2}&&&&&&&&\\\
\hline\cr 1_{G_{2}}&&&&&&&&&&&&&&&&&&1&0&0&0&&&&\\\
J_{\gamma_{1}}(3/2,\mathrm{St}_{\operatorname{GL}_{2}})&&&&&&&&&&&&&&&&&&0&1&0&0&&&&\\\
J_{{\gamma_{2}}}(5/2,\mathrm{St}_{\operatorname{GL}_{2}})&&&&&&&&&&&&&&&&&&0&0&1&0&&&&\\\
\mathrm{St}_{G_{2}}&&&&&&&&&&&&&&&&&&0&0&0&1&&&&\\\ \hline\cr
J_{{\gamma_{2}}}(1,I^{\operatorname{GL}_{2}}(1\otimes
1))&&&&&&&&&&&&&&&&&&&&&&1&0&0&0\\\
J_{\gamma_{1}}(1/2,\mathrm{St}_{\operatorname{GL}_{2}})&&&&&&&&&&&&&&&&&&&&&&\varrho&1&0&0\\\
J_{{\gamma_{2}}}(1/2,\mathrm{St}_{\operatorname{GL}_{2}})&&&&&&&&&&&&&&&&&&&&&&0&\vartheta_{2}&1&0\\\
\pi(1)^{\prime}&&&&&&&&&&&&&&&&&&&&&&0&0&0&1\\\
\pi(1)&&&&&&&&&&&&&&&&&&&&&&0&0&\vartheta_{2}&\varrho\\\
I_{0}(G_{2}[1])&&&&&&&&&&&&&&&&&&&&&&\varepsilon&1&\vartheta_{2}&\varepsilon\end{array}$

######  Definition 2.

Let $\phi$ be an unramified Langlands parameter for $G_{2}(F)$. Following
[CFMMX], we set

$\Pi_{\phi}^{{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}}(G_{2}(F)){\,:=\,}\\{\pi\in\Pi_{\lambda}(G_{2}/F)|~{}\operatorname{\mathsf{N}\hskip-1.0pt\mathsf{E}\hskip-1.0pt\mathsf{v}\hskip-1.0pt\mathsf{s}}_{C_{\phi}}(\mathcal{P}(\pi))\neq
0\\}.$

For
$s\in\mathcal{S}^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}$
and
$\pi\in\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}(G_{2}(F))$,
ABV-packet coefficients are given by

$\langle
s,\pi\rangle{\,:=\,}\operatorname{trace}_{s}\operatorname{\mathsf{N}\hskip-1.0pt\mathsf{E}\hskip-1.0pt\mathsf{v}\hskip-1.0pt\mathsf{s}}_{C_{\phi}}\mathcal{P}(\pi).$
(5)

In this definition we make implicit use of the homomorphism
$\mathcal{S}^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}\to
A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}$.

Let $\lambda:W_{F}\to\widehat{G_{2}}$ be an unramified infinitesimal parameter
for $G_{2}(F)$. Then $\lambda$ is determined by
$\lambda({\rm{Fr}})=\widehat{m}(u_{1}q^{a_{1}},u_{2}q^{a_{2}})$ where $u_{1}$
and $u_{2}$ are unitary complex numbers and $a_{1}$ and $a_{2}$ are real
numbers. Then

$\displaystyle\lambda(w)$
$\displaystyle=\widehat{m}(u_{1}^{\operatorname{ord}(w)},u_{2}^{\operatorname{ord}(w)})\cdot\widehat{m}(|w|^{a_{1}},|w|^{a_{2}})$
$\displaystyle=(2{{\hat{\gamma}}_{1}}^{\vee}+{{\hat{\gamma}}_{2}}^{\vee})(u_{2}^{\operatorname{ord}(w)}|w|^{a_{2}})\cdot({{\hat{\gamma}}_{1}}^{\vee}+{{\hat{\gamma}}_{2}}^{\vee})((u_{1}u_{2}^{-1})^{\operatorname{ord}(w)}|w|^{a_{2}-a_{1}}).$

Let $\chi=\chi_{1}\otimes\chi_{2}:T\to\mathbb{C}^{\times}$ be the dual
character of $\lambda:W_{F}\to{\widehat{T}}$. By the above form of $\lambda$,
we know that $\chi_{1}=\mu_{2}\nu^{a_{2}}$ and
$\chi_{2}=\mu_{1}\mu_{2}^{-1}\nu^{a_{1}-a_{2}}$, i.e., $\chi$ is given by

$\chi=\mu_{2}\nu^{a_{2}}\otimes\mu_{1}\mu_{2}^{-1}\nu^{a_{1}-a_{2}},$ (6)

where $\mu_{i}$ is the unitary unramified character of $F^{\times}$ defined by
$\mu_{i}(\varpi)=u_{i}$.

  1. (Case 1)

Suppose $\phi:W^{\prime}_{F}\to{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G_{2}}}$ is a
Langlands parameter with infinitesimal parameter
$\lambda_{\text{\ref{infcase:0}}}$ corresponding to Case 1 from Proposition
1.1. Then $\lambda_{\text{\ref{infcase:0}}}:W_{F}\to\widehat{G_{2}}$ is an
unramified infinitesimal parameter such that $\dim
V_{\lambda_{\text{\ref{infcase:0}}}}=0$, so
${\hat{\gamma}}(\lambda_{\text{\ref{infcase:0}}}({\operatorname{Fr}}))\neq q$
for every root ${\hat{\gamma}}\in{\hat{R}}$; this is Case 1 from Proposition
1.1. If we write
$\lambda_{\text{\ref{infcase:0}}}({\operatorname{Fr}})={\hat{m}}(u_{1}q^{a}_{1},u_{2}q^{a_{2}})$
then this case is equivalent to the condition

$\\{u_{2}^{2}u_{1}^{-1}q^{2a_{2}-a_{1}},u_{1}u_{2}^{-1}q^{a_{1}-a_{2}},u_{2}q^{a_{2}},u_{1}q^{a_{1}},u_{1}^{2}u_{2}^{-1}q^{2a_{1}-a_{2}},u_{1}u_{2}q^{a_{1}+a_{2}}\\}\cap\\{q,q^{-1}\\}=\emptyset.$

In this case $H_{\lambda_{\text{\ref{infcase:0}}}}$ is the dual of an
endoscopic group for $\widehat{G_{2}}$; so, besides $\widehat{G_{2}}$ itself,
$H_{\lambda_{\text{\ref{infcase:0}}}}$ is $\operatorname{SL}_{3}$ or
$\operatorname{SO}_{4}$ or one of the Levi subgroups
$\operatorname{GL}_{2}^{{{\hat{\gamma}}_{1}}}$,
$\operatorname{GL}_{2}^{{{\hat{\gamma}}_{2}}}$ or ${\widehat{T}}$. For each
such $\lambda_{\text{\ref{infcase:0}}}$, there is a unique $\phi$ with
$\lambda_{\phi}=\lambda_{\text{\ref{infcase:0}}}$, i.e.,
$\phi(w,x)=\lambda_{\text{\ref{infcase:0}}}(w)$. In this case, there is a
unique Langlands parameter with infinitesimal parameter
$\lambda_{\ref{infcase:0}}$ which is defined by
$\phi_{\ref{infcase:0}}(w,x)=\lambda_{\ref{infcase:0}}(w)$. If
$a_{1}=a_{2}=0$, then $\chi_{1}$ and $\chi_{2}$ are unitary. Notice that there
is a unique order 2 unramified character of $F^{\times}$ and thus
$\chi_{1},\chi_{2}$ cannot be two distinct order 2 unramified characters. In
this case, $I(\chi_{1}\otimes\chi_{2})$ is irreducible by [Keys]. If one of
$a_{1},a_{2}$ is nonzero, then at least one of $\chi_{1},\chi_{2}$ is non-
unitary. One easily translates the condition
$\hat{\gamma}(\lambda_{\ref{infcase:0}}({\rm{Fr}}))\neq q^{\pm
1},\forall\hat{\gamma}\in R(\widehat{G_{2}},{\widehat{T}})$ to the conditions
on $\chi_{1},\chi_{2}$, which are exactly the conditions of [Muic]*Proposition
3.1. Thus, the induced representation $I(\chi_{1}\otimes\chi_{2})$ is also
irreducible by [Muic]*Proposition 3.1. In both cases, the unramified local
Langlands correspondence says that
$\pi(\phi_{\ref{infcase:0}})=I(\chi_{1}\otimes\chi_{2})$ and the $L$-packet is
a singleton. Here, $\phi_{\ref{infcase:0}}(w,x)=\lambda_{\ref{infcase:0}}(w)$,
given above. There are no coronal representations to the admissible
representation $I(\chi_{1}\otimes\chi_{2})$; accordingly, this representation
is its own ABV-packet in this case. This is a direct consequence of the simple
observation that there is but one simple object in
$\mathsf{Per}_{H_{\lambda_{\ref{infcase:0}}}}(V_{\lambda_{\ref{infcase:0}}})$;
see also 1.

  2. (Case 2)

Suppose $\phi:W^{\prime}_{F}\to{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G_{2}}}$ is a
Langlands parameter with infinitesimal parameter
$\lambda_{\text{\ref{infcase:1-short}}}$ corresponding to Case 2 from
Proposition 1.1. Then
$V_{\lambda_{\ref{infcase:1-short}}}=\mathrm{Span}\\{X_{{{\hat{\gamma}}_{1}}+2{{\hat{\gamma}}_{2}}}\\}$
and
$({{\hat{\gamma}}_{1}}+2{{\hat{\gamma}}_{2}})(\lambda_{\ref{infcase:1-short}}({\operatorname{Fr}}))=q$
and ${\hat{\gamma}}(\lambda_{\ref{infcase:1-short}}({\operatorname{Fr}}))\neq
q$ for every
${\hat{\gamma}}\in{\hat{R}}-\\{\widehat{\gamma}_{1}+2\widehat{\gamma}_{2}\\}$;
in other words, if we write
$\lambda_{\ref{infcase:1-short}}({\operatorname{Fr}})=(u_{1}q^{a_{1}},u_{2}q^{a_{2}})$,
then $u_{1}u_{2}=1$, $a_{1}+a_{2}=1$ and

$\\{u^{-3}q^{2-3a},u^{2}q^{2a-1},u^{-1}q^{1-a},uq^{a},u^{3}q^{3a-1}\\}\cap\\{q,q^{-1}\\}=\emptyset,$

where $u{\,:=\,}u_{1}$ and $a{\,:=\,}a_{1}$ and
$H_{\lambda_{\ref{infcase:1-short}}}={\widehat{T}}$ if $u^{2}q^{2a-1}\neq 1$
and
$H_{\lambda_{\ref{infcase:1-short}}}=\operatorname{GL}_{2}^{\hat{\gamma}_{1}}$
if $u^{2}q^{2a-1}=1$.

The action of $H_{\lambda_{\ref{infcase:1-short}}}$ on
$V_{\lambda_{\ref{infcase:1-short}}}$ is given by

$\begin{array}[]{rcll}\widehat{m}(x,y).rX_{{{\hat{\gamma}}_{1}}+2{{\hat{\gamma}}_{2}}}&=&xrX_{{{\hat{\gamma}}_{1}}+2{{\hat{\gamma}}_{2}}},&\textrm{
if }u^{2}q^{2a-1}\neq 1,\\\
g.rX_{{{\hat{\gamma}}_{1}}+2{{\hat{\gamma}}_{2}}}&=&\det(g)rX_{{{\hat{\gamma}}_{1}}+2{{\hat{\gamma}}_{2}}},&\textrm{
if }u^{2}q^{2a-1}=1.\end{array}$

This action has two orbits: the closed (zero) orbit and the open orbit. The
corresponding Langlands parameters are explicitly given by

$\begin{array}[]{rcll}\phi_{\ref{infcase:1-short}a}(w,x)&=&\lambda_{\ref{infcase:1-short}}(w),&\text{closed};\\\
\phi_{\ref{infcase:1-short}b}(w,x)&=&\widehat{m}(1,u^{\operatorname{ord}(w)}|w|^{a-1/2})\iota_{{{\hat{\gamma}}_{1}}+2{{\hat{\gamma}}_{2}}}(x),&\text{open}.\end{array}$

The simple objects in $\mathsf{Per}_{H}(V)$ are described in 2 using the base-
change argument in the proof of Proposition 1.2.

By Equation (6), the dual character of $\lambda_{\ref{infcase:1-short}}$ is
$\chi=\mu\nu^{a}\otimes\mu^{-1}\nu^{1-a}$, where
$\mu:F^{\times}\rightarrow\mathbb{C}^{1}$ is the unramified unitary character
determined by $\mu(\varpi)=u$. We consider the representation
$I(\mu\nu^{a}\otimes\mu^{-1}\nu^{1-a})$. In the Grothendieck group, we have

$\displaystyle I(\mu\nu^{a}\otimes\mu^{-1}\nu^{1-a})$
$\displaystyle=I(\mu\nu^{a-1}\otimes\nu)$
$\displaystyle=I_{\gamma_{2}}(a-1/2,I^{\operatorname{GL}_{2}}(\mu\nu^{1/2}\otimes\mu\nu^{-1/2}))$
$\displaystyle=I_{\gamma_{2}}(a-1/2,\mu\otimes\mathrm{St}_{\operatorname{GL}_{2}})+I_{\gamma_{2}}(a-1/2,\mu\circ\det).$

By [Muic]*Theorem 3.1 (ii), the two representations,
$I_{\gamma_{2}}(a-1/2,\mu\otimes\mathrm{St}_{\operatorname{GL}_{2}})$ and
$I_{\gamma_{2}}(a-1/2,\mu\circ\det)$, are irreducible. For these
representations, the local Langlands correspondence is given by

$\begin{array}[]{rcl}\Pi_{\phi_{\ref{infcase:1-short}a}}(G_{2}(F))&=&\\{I_{\gamma_{2}}(a-1/2,\mu\circ\det)\\},\\\
\Pi_{\phi_{\ref{infcase:1-short}b}}(G_{2}(F))&=&\\{I_{\gamma_{2}}(a-1/2,\mu\otimes\mathrm{St}_{\operatorname{GL}_{2}})\\}.\end{array}$

It follows that the equivariant perverse sheaf corresponding to
$I_{\gamma_{2}}(a-1/2,\mu\circ\det)$ is
${\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{\ref{infcase:1-short}a}})$,
where ${\mathbbm{1}}_{C_{\ref{infcase:1-short}a}}$ is the constant local
system on $C_{\ref{infcase:1-short}a}=\\{0\\}$ in $V=\mathbb{A}^{1}$, and the
equivariant perverse sheaf corresponding to
$I_{\gamma_{2}}(a-1/2,\mu\otimes\mathrm{St}_{\operatorname{GL}_{2}})$ is
${\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{\ref{infcase:1-short}b}})$,
where ${\mathbbm{1}}_{C_{\ref{infcase:1-short}b}}$ is the constant local
system on the open orbit $C_{\ref{infcase:1-short}b}$. Taking the case $n=1$
of Section 2 and then using the base-change argument in the proof of
Proposition 1.2, it follows that the simple objects in
$\mathsf{Per}_{H_{\lambda}}(V_{\lambda})$ are
${\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{\ref{infcase:1-short}a}})$ and
${\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{\ref{infcase:1-short}b}})$ only,
and that
$\operatorname{\mathsf{N}\hskip-1.0pt\mathsf{E}\hskip-1.0pt\mathsf{v}\hskip-1.0pt\mathsf{s}}_{C_{\ref{infcase:1-short}a}}{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{\ref{infcase:1-short}a}})={\mathbbm{1}}_{\Lambda_{C_{\ref{infcase:1-short}a}}}$
and
$\operatorname{\mathsf{N}\hskip-1.0pt\mathsf{E}\hskip-1.0pt\mathsf{v}\hskip-1.0pt\mathsf{s}}_{C_{\ref{infcase:1-short}b}}{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{\ref{infcase:1-short}b}})={\mathbbm{1}}_{\Lambda_{C_{\ref{infcase:1-short}b}}}$
while
$\operatorname{\mathsf{N}\hskip-1.0pt\mathsf{E}\hskip-1.0pt\mathsf{v}\hskip-1.0pt\mathsf{s}}_{C_{\ref{infcase:1-short}a}}{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{\ref{infcase:1-short}b}})=0$
and
$\operatorname{\mathsf{N}\hskip-1.0pt\mathsf{E}\hskip-1.0pt\mathsf{v}\hskip-1.0pt\mathsf{s}}_{C_{\ref{infcase:1-short}b}}{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{\ref{infcase:1-short}a}})=0$.
Therefore, there are no coronal representations for the $L$-packets above, and
we have

$\begin{array}[]{rcl}\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi_{\ref{infcase:1-short}a}}(G_{2}(F))&=&\Pi_{\phi_{\ref{infcase:1-short}a}}(G_{2}(F)),\\\
\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi_{\ref{infcase:1-short}b}}(G_{2}(F))&=&\Pi_{\phi_{\ref{infcase:1-short}b}}(G_{2}(F)).\end{array}$

These calculations follow from the results of Section 2.

  3. (Case 3)

Suppose $\phi$ is a Langlands parameter with infinitesimal parameter
$\lambda_{\ref{infcase:2-long}}$ from Case 3 of Proposition 1.1. This case is
quite similar to the previous case. The infinitesimal parameter
$\lambda_{\ref{infcase:2-long}}$ can be represented by
$\lambda_{\ref{infcase:2-long}}({\operatorname{Fr}})=\widehat{m}(uq^{a},u^{-1}q^{1-a})$
with $u\in\mathbb{C}^{1},a\in\mathbb{R}$ satisfying
$\widehat{\gamma}(\lambda_{\ref{infcase:2-long}}({\operatorname{Fr}})=\widehat{m}(uq^{a},u^{-1}q^{1-a}))\neq
q$ for all
$\widehat{\gamma}\in\widehat{R}-\\{2\widehat{\gamma}_{1}+3\widehat{\gamma}_{2}\\}$,
which is equivalent to

$\\{u^{-3}q^{2-3a},u^{2}q^{2a-1},u^{-1}q^{1-a},uq^{a},u^{3}q^{3a-1}\\}\cap\\{q,q^{-1}\\}=\emptyset.$

We have
$V_{\lambda_{\ref{infcase:2-long}}}=\mathrm{Span}\\{X_{2{{\hat{\gamma}}_{1}}+3{{\hat{\gamma}}_{2}}}\\}$.
In this case, $H_{\lambda_{\ref{infcase:2-long}}}={\widehat{T}}$ if
$u^{2}q^{2a-1}\neq 1$, and
$H_{\lambda_{\ref{infcase:2-long}}}=\operatorname{GL}_{2,{{\hat{\gamma}}_{2}}}$
if $u^{2}q^{2a-1}=1$. The action of $H_{\lambda_{\ref{infcase:2-long}}}$ on
$V_{\lambda_{\ref{infcase:2-long}}}$ is given by

$\begin{array}[]{rcll}\widehat{m}(x,y).rX_{2{{\hat{\gamma}}_{1}}+3{{\hat{\gamma}}_{2}}}&=&xyrX_{2{{\hat{\gamma}}_{1}}+3{{\hat{\gamma}}_{2}}},&\textrm{
if }u^{2}q^{2a-1}\neq 1,\\\
g.rX_{2{{\hat{\gamma}}_{1}}+3{{\hat{\gamma}}_{2}}}&=&\det(g)rX_{2{{\hat{\gamma}}_{1}}+3{{\hat{\gamma}}_{2}}},&\textrm{
if }u^{2}q^{2a-1}=1.\end{array}$

This action has two orbits – the zero orbit and the open orbit – and each
Langlands parameter has a trivial component group. Representatives for the two
$L$-parameters are

$\begin{array}[]{rcl}\phi_{\ref{infcase:2-long}b}(w,x)&=&\lambda_{\ref{infcase:2-long}}(w)\\\
\phi_{\ref{infcase:2-long}b}(w,x)&=&\widehat{m}(u^{\operatorname{ord}(w)}|w|^{a-1/2},u^{-\operatorname{ord}(w)}|w|^{-a+1/2})\cdot\iota_{2{{\hat{\gamma}}_{1}}+3{{\hat{\gamma}}_{2}}}(x).\end{array}$

By Equation (6), the dual character of $\lambda_{\ref{infcase:2-long}}$ is
$\mu^{-1}\nu^{1-a}\otimes\mu^{2}\nu^{2a-1},$ where $\mu$ is the unitary
unramified character of $F^{\times}$ determined by $\mu(\varpi)=u$. In the
Grothendieck group, we have

$\displaystyle I(\mu^{-1}\nu^{1-a}\otimes\mu^{2}\nu^{2a-1})$
$\displaystyle=I(\mu\nu^{a}\otimes\mu\nu^{a-1})$
$\displaystyle=I_{\gamma_{1}}(a-1/2,I^{\operatorname{GL}_{2}}(\mu\nu^{1/2}\otimes\mu\nu^{-1/2}))$
$\displaystyle=I_{\gamma_{1}}(a-1/2,\mu\otimes\mathrm{St}_{\operatorname{GL}_{2}})+I_{\gamma_{1}}(a-1/2,\mu\circ\det).$

By [Muic]*Theorem 3.1 (i), the representations
$I_{\gamma_{1}}(a-1/2,\mu\otimes\mathrm{St}_{\operatorname{GL}_{2}})$ and
$I_{\gamma_{1}}(a-1/2,\mu\cdot\det)$ are irreducible. The Langlands
correspondence is given in given in Table 2.7.1. Arguing as in Case 2 or using
our microlocal analysis of the prehomogeneous vector space 2, the ABV-packet
coefficients are given in Table 2.6.1.

  4. (Case 4)

Suppose $\phi$ is an unramified Langlands parameter with infinitesimal
parameter $\lambda_{\ref{infcase:3}}$ from Case 4 of Proposition 1.1. We can
take $\lambda_{\ref{infcase:3}}({\operatorname{Fr}})=\widehat{m}(\theta
q^{1/3},\theta^{2}q^{2/3})$ with $\theta=\theta_{3}^{n}$ for $n=0,1,2$ and
$H_{\lambda_{\ref{infcase:3}}}=\operatorname{GL}_{2}^{{{\hat{\gamma}}_{1}}+3{{\hat{\gamma}}_{2}}}$.
To see this note that
${{\hat{\gamma}}_{1}}(\lambda_{\ref{infcase:3}}({\operatorname{Fr}}))=q$ and
$(2{{\hat{\gamma}}_{1}}+3{{\hat{\gamma}}_{2}})(\lambda({\operatorname{Fr}}))=q$
and ${\hat{\gamma}}(\lambda_{\ref{infcase:3}}({\operatorname{Fr}}))\neq q$ for
every other ${\hat{\gamma}}\in{\hat{R}}$, so if
$\lambda_{\ref{infcase:3}}({\operatorname{Fr}})={\widehat{m}}(u_{1}q^{a_{1}},u_{2}q^{a_{2}})$
then $u_{2}=u_{1}^{2}\in\\{1,\theta_{3},\theta_{3}^{2}\\}$ and
$a_{1}=\frac{1}{3}$, $a_{2}=\frac{2}{3}$. The action of
$H_{\lambda_{\ref{infcase:3}}}$ on $V_{\lambda_{\ref{infcase:3}}}$ is
$\det\otimes\operatorname{Sym}^{1}$, which is to say, $h\cdot x=\det(h)hx$
where $hx$ is matrix multiplication. This is case 3 for $n=1$. In particular,
there are two orbits of this action: the zero orbit and the open orbit. The
corresponding Langlands parameters are given by

$\begin{array}[]{rcl}\phi_{\ref{infcase:3}a}(w,x)&=&\lambda_{\ref{infcase:3}}(w),\\\
\phi_{\ref{infcase:3}b}(w,x)&=&\widehat{m}(\theta^{\operatorname{ord}(w)}|w|^{1/3},\theta^{2\operatorname{ord}(w)}|w|^{1/6})\cdot\iota_{{{\hat{\gamma}}_{1}}}(x).\end{array}$

By Equation (6), the dual character of $\lambda_{\ref{infcase:3}}$ is
$\chi=\vartheta\nu^{2/3}\otimes\vartheta^{2}\nu^{-1/3}$ where $\vartheta$ is
the unramified character of $F^{\times}$ dual to $\theta$. In the Grothendieck
group, we have

$\displaystyle I(\theta_{3}^{n}\nu^{2/3}\otimes\theta_{3}^{2n}\nu^{-1/3})$
$\displaystyle=I_{\gamma_{1}}(1/6,I^{\operatorname{GL}_{2}}(\theta_{3}^{n}\nu^{1/2}\otimes\theta_{3}^{-n}\nu^{-1/2}))$
$\displaystyle=I_{{\gamma_{1}}}(1/6,\theta_{3}^{n}\otimes\mathrm{St}_{\operatorname{GL}_{2}})+I_{\gamma_{1}}(1/6,\theta^{n}_{3}\circ\det).$

The Langlands correspondence is given in given in Table 2.7.1. Using our
microlocal analysis of the prehomogeneous vector space 3, for $n=1$, the ABV-
packet coefficients are given in Table 2.6.1.

  5. (Case 5)

Let $\lambda_{\ref{infcase:4-D2}}$ be the infinitesimal parameter appearing in
Case 5 of Proposition 1.1. Then
$\lambda_{\ref{infcase:4-D2}}({\operatorname{Fr}})=\widehat{m}(q,-q)$ and
$V_{\lambda_{\ref{infcase:4-D2}}}=\mathrm{Span}\\{X_{{{\hat{\gamma}}_{1}}},X_{{{\hat{\gamma}}_{1}}+2{{\hat{\gamma}}_{2}}}\\}$
and $H_{\lambda_{\ref{infcase:4-D2}}}={\widehat{T}}$. The action of
${\widehat{T}}$ on $V_{\lambda_{\ref{infcase:4-D2}}}$ is given by

$\widehat{m}(x,y).(r_{1}X_{{{\hat{\gamma}}_{1}}}+r_{2}X_{{{\hat{\gamma}}_{1}}+2{{\hat{\gamma}}_{2}}})=y^{2}x^{-1}r_{1}X_{{{\hat{\gamma}}_{1}}}+xr_{2}X_{{{\hat{\gamma}}_{1}}+2{{\hat{\gamma}}_{2}}}.$

There are 4 orbits: $C_{\ref{infcase:4-D2}a}=\\{0\\}$,
$C_{\ref{infcase:4-D2}b}=\\{r_{1}X_{{{\hat{\gamma}}_{1}}}{\ |\ }r_{1}\neq
0\\}$,
$C_{\ref{infcase:4-D2}c}=\\{r_{2}X_{{{\hat{\gamma}}_{1}}+2{{\hat{\gamma}}_{2}}}{\
|\ }r_{2}\neq 0\\}$ and
$C_{\ref{infcase:4-D2}d}=\\{r_{1}X_{{{\hat{\gamma}}_{1}}}+r_{2}X_{{{\hat{\gamma}}_{1}}+2{{\hat{\gamma}}_{2}}}{\
|\ }r_{1},r_{2}\neq 0\\}$. We found
$\mathsf{Per}_{H_{\lambda_{\ref{infcase:4-D2}}}}(V_{\lambda_{\ref{infcase:4-D2}}})$
in Proposition 1.2. In particular, there are four
$H_{\lambda_{\ref{infcase:4-D2}}}$-orbits in
$V_{\lambda_{\ref{infcase:4-D2}}}$: $C_{\text{\ref{infcase:4-D2}a}}=\\{0\\}$,
$C_{\text{\ref{infcase:4-D2}b}}=\\{r_{1}X_{{{\hat{\gamma}}_{1}}}{\ |\
}r_{1}\neq 0\\}$,
$C_{\text{\ref{infcase:4-D2}c}}=\\{r_{2}X_{{{\hat{\gamma}}_{1}}+2{{\hat{\gamma}}_{2}}}{\
|\ }r_{2}\neq 0\\}$ and
$C_{\text{\ref{infcase:4-D2}d}}=\buildrel{}_{\circ}\over{V}=\\{r_{1}X_{{{\hat{\gamma}}_{1}}}+r_{2}X_{{{\hat{\gamma}}_{1}}+2{{\hat{\gamma}}_{2}}}{\
|\ }r_{1},r_{2}\neq 0\\}$. Corresponding Langlands parameters are given by

$\begin{array}[]{rcl}\phi_{\text{\ref{infcase:4-D2}a}}(w,x)&=&\lambda(w),\\\
\phi_{\text{\ref{infcase:4-D2}b}}(w,x)&=&\widehat{m}(|w|,(-1)^{\operatorname{ord}(w)}|w|^{1/2})\cdot\iota_{{{\hat{\gamma}}_{1}}}(x),\\\
\phi_{\text{\ref{infcase:4-D2}c}}(w,x)&=&\widehat{m}(1,(-1)^{\operatorname{ord}(w)}|w|^{1/2})\cdot\iota_{{{\hat{\gamma}}_{1}}+2{{\hat{\gamma}}_{2}}}(x),\\\
\phi_{\text{\ref{infcase:4-D2}d}}(w,x)&=&\widehat{m}(1,(-1)^{\operatorname{ord}(w)})\iota_{{{\hat{\gamma}}_{1}}}(x)\iota_{{{\hat{\gamma}}_{1}}+2{{\hat{\gamma}}_{2}}}(x).\end{array}$

Note that we have
$A_{\phi_{\ref{infcase:4-D2}d}}=\langle{\widehat{m}}(1,-1)\rangle{\ \cong\
}\langle\theta_{2}\rangle$ and the component groups of the other 3 Langlands
parameters are trivial.

The dual character of $\lambda_{\ref{infcase:4-D2}}$ is
$\theta_{2}\nu\otimes\theta_{2}$ by Equation (6). Recall that $\theta_{2}$ is
the unique unramified order 2 character of $F^{\times}$. By [Muic]*Proposition
4.1, we have

$\displaystyle\begin{split}I(\theta_{2}\nu\otimes\theta_{2})&=I_{\gamma_{1}}(1/2,\theta_{2}\otimes\mathrm{St}_{\operatorname{GL}_{2}})+I_{\gamma_{1}}(1/2,\theta_{2}\circ\det)\\\
&=I_{\gamma_{2}}(1/2,\theta_{2}\otimes\mathrm{St}_{\operatorname{GL}_{2}})+I_{{\gamma_{2}}}(1/2,\theta_{2}\circ\det),\end{split}$
(7)

and

$\begin{split}\begin{array}[]{rcl}I_{\gamma_{1}}(1/2,\theta_{2}\otimes\mathrm{St}_{\operatorname{GL}_{2}})&=&\pi(\theta_{2})+J_{\gamma_{1}}(1/2,\theta_{2}\otimes\mathrm{St}_{\operatorname{GL}_{2}}),\\\
I_{{\gamma_{2}}}(1/2,\theta_{2}\otimes\mathrm{St}_{\operatorname{GL}_{2}})&=&\pi(\theta_{2})+J_{\gamma_{2}}(1/2,\theta_{2}\otimes\mathrm{St}_{\operatorname{GL}_{2}}),\\\
I_{\gamma_{1}}(1/2,\theta_{2}\circ\det)&=&J_{\gamma_{2}}(1,I^{\operatorname{GL}_{2}}(1\otimes\theta_{2}))+J_{\gamma_{2}}(1/2,\theta_{2}\otimes\mathrm{St}_{\operatorname{GL}_{2}}),\\\
I_{\gamma_{2}}(1/2,\theta_{2}\circ\det)&=&J_{\gamma_{2}}(1,I^{\operatorname{GL}_{2}}(1\otimes\theta_{2}))+J_{\gamma_{1}}(1/2,\theta_{2}\otimes\mathrm{St}_{\operatorname{GL}_{2}}),\end{array}\end{split}$
(8)

As in [Muic], we write $\pi(\theta_{2})$ for the unique irreducible sub-
representation of $I(\theta_{2}\nu\otimes\theta_{2})$; note that
$\pi(\theta_{2})$ is square integrable by [Muic]*Proposition 4.1. and all of
the above equations are in the Grothendieck group of representations of
$G_{2}(F)$.

The proof of the Langlands correspondence
$\Pi_{\phi}(G_{2}(F))\to\widehat{A_{\phi}}$ for all unramified Langlands
parameters for $G_{2}(F)$ with infinitesimal parameter
$\lambda_{\ref{infcase:4-D2}}$ is similar to that of [CFZ:cubics]*Theorem 2.5,
which follows from some well-known facts. The Langlands correspondence is
given in Table 2.1.1; see also Table 2.7.1. Here we only give some details of
the proof of
$\pi(\phi_{\text{\ref{infcase:4-D2}d}},\vartheta_{2})=I_{0}(G_{2}[-1]){\,:=\,}\mathrm{cInd}_{G_{2}({\mathcal{O}}_{F})}^{G_{2}(F)}(G_{2}[-1])$.
Using [Lusztig:Intersectioncohomology]*p.270, the local system
$(\phi_{\text{\ref{infcase:4-D2}d}},\vartheta_{2})$ is cuspidal and thus
$\pi(\phi_{\text{\ref{infcase:4-D2}d}},\vartheta_{2})$ is a depth zero
supercuspidal representation of the form
$I_{0}(\sigma){\,:=\,}\mathrm{cInd}_{G_{2}({\mathcal{O}}_{F})}^{G_{2}(F)}(\sigma),$
where $\sigma$ is a unipotent cuspidal representation of
$G_{2}(\mathbb{F}_{q})$. There are exactly four unipotent cuspidal
representations of $G_{2}(\mathbb{F}_{q})$; see [Carter]*p.460. To determine
which one the representation $\sigma$ is, we use the formal degree conjecture
of [HII], which is known in this case by [FOS]. A simple calculation shows
that the $L$-factor is given by

$L(s,\phi_{\text{\ref{infcase:4-D2}d}},\operatorname{Ad})=\frac{1}{(1+q^{-2-s})(1+a^{-1-s})(1-q^{-1-s})^{2}},$

and the epsilon
$\epsilon(s,\phi_{\text{\ref{infcase:4-D2}d}},\operatorname{Ad})=q^{10(1/2-s)}.$
Thus

$\gamma(0,\phi_{\text{\ref{infcase:4-D2}d}},\operatorname{Ad})=\frac{q^{9}}{(q+1)(q^{3}+1)}.$

Here the $\epsilon$-factor and the $\gamma$-factor are computed with respect
to a fixed additive character of $F$, which is omitted from the notation. The
formal degree conjecture asserts that

$\frac{\deg(\sigma)}{\mu(G_{2}({\mathcal{O}}_{F}))}=\frac{\dim(\vartheta_{2})}{|A_{\phi_{\text{\ref{infcase:4-D2}d}}}|}|\gamma(0,\phi_{\text{\ref{infcase:4-D2}d}},\operatorname{Ad})|,$

where $\mu(G_{2}({\mathcal{O}}_{F}))$ is the Haar measure of
$G_{2}({\mathcal{O}}_{F})$ normalized as in [HII], i.e.,

$\mu(G_{2}({\mathcal{O}}_{F}))=q^{-\dim(G_{2}(F))}\cdot|G_{2}(\mathbb{F}_{q})|=q^{-8}(q^{6}-1)(q^{2}-1).$

Thus we get

$\deg(\sigma)=\frac{q(q^{6}-1)(q^{2}-1)}{(q^{3}+1)(q+1)}.$

Comparing it with the 4 unipotent cupsidal representations of
$G_{2}(\mathbb{F}_{q})$, we get $\sigma=G_{2}[-1]$.

Using the microlocal analysis of the prehomogeneous vector space 4 for $n=2$
from Section 1.4, the ABV-packet coefficients are given by in Table 2.6.1.

  6. (Case 6)

We now find the ABV-packets and coefficients for Langlands parameters with
infinitesimal parameter $\lambda_{\ref{infcase:5}}$ from Case 6 of Proposition
1.1. The infinitesimal parameter $\lambda_{\ref{infcase:5}}$ is determined by
$\lambda_{\ref{infcase:5}}({\operatorname{Fr}})=\widehat{m}(q^{2},q)$. In this
case, we have
$V_{\lambda_{\ref{infcase:5}}}=\mathrm{Span}\\{X_{\hat{\gamma}_{2}},X_{\hat{\gamma}_{1}+\hat{\gamma}_{2}}\\}$
and $H_{\lambda_{\ref{infcase:5}}}=\operatorname{GL}_{2}^{\hat{\gamma}_{1}}$.
The action of $H_{\lambda_{\ref{infcase:5}}}$ on
$V_{\lambda_{\ref{infcase:5}}}$ is equivalent to matrix multiplication,
depending on an isomorphism of $\operatorname{GL}_{2}^{\hat{\gamma}_{1}}$ with
$\operatorname{GL}_{2}$. Category
$\mathsf{Per}_{H_{\lambda_{\ref{infcase:5}}}}(V_{\lambda_{\ref{infcase:5}}}){\
\cong\ }\mathsf{Per}_{\operatorname{GL}_{2}}(\mathbb{A}^{2})$, with reference
to the prehomogeneous vector space 3 for $n=0$. In particular, this action has
two orbits: the zero orbit $C_{\ref{infcase:5}a}$ and the open orbit
$C_{\ref{infcase:5}b}$. Corresponding Langlands parameters are given by

$\begin{array}[]{rcl}\phi_{\ref{infcase:5}a}(w,x)&=&\lambda_{\ref{infcase:5}}(w),\\\
\phi_{\ref{infcase:5}b}(w,x)&=&\widehat{m}(|w|^{3/2},|w|^{3/2})\cdot\iota_{\hat{\gamma}_{2}}(x).\end{array}$

By Equation (6), the dual character of $\lambda_{\ref{infcase:5}}$ is
$\nu\otimes\nu$. In the Grothendieck group, we have

$\displaystyle I(\nu\otimes\nu)$
$\displaystyle=I_{\gamma_{2}}(3/2,\pi(\nu^{1/2},\nu^{-1/2}))$
$\displaystyle=I_{\gamma_{2}}(3/2,\mathrm{St}_{\operatorname{GL}_{2}})+I_{\gamma_{2}}(3/2,1_{\operatorname{GL}_{2}}).$

The Langlands correspondence is given in given in Table 2.7.1. Using our
microlocal analysis of the prehomogeneous vector space 3, for $n=0$, the ABV-
packet coefficients are given in Table 2.6.1.

  7. (Case 7)

The ABV-packets and coefficients for Langlands parameters with infinitesimal
parameter $\lambda_{\ref{infcase:6-A2}}$ defined by
$\lambda_{\ref{infcase:6-A2}}({\operatorname{Fr}})=\widehat{m}(\theta_{3}q,\theta_{3}^{2}q)$
as in Case 7 of Proposition 1.1 are calculated as follows. In this case, we
have
$V_{\lambda_{\ref{infcase:6-A2}}}=\mathrm{Span}\\{X_{{{\hat{\gamma}}_{1}}},X_{{{\hat{\gamma}}_{1}}+3{{\hat{\gamma}}_{2}}}\\}$
and $H={\widehat{T}}$ and the action of ${\widehat{T}}$ on
$V_{\lambda_{\ref{infcase:6-A2}}}$ is given by

$\widehat{m}(x,y).(r_{1}X_{{{\hat{\gamma}}_{1}}}+r_{2}X_{{{\hat{\gamma}}_{1}}+3{{\hat{\gamma}}_{2}}})=y^{2}x^{-1}r_{1}X_{{{\hat{\gamma}}_{1}}}+x^{2}y^{-1}r_{2}X_{{{\hat{\gamma}}_{1}}+3{{\hat{\gamma}}_{2}}}.$

Using the base-change argument in the proof of Proposition 1.2 we find that
$\mathsf{Per}_{H_{\lambda_{\ref{infcase:6-A2}}}}(V_{\lambda_{\ref{infcase:6-A2}}})$
is equivalent to the category of equivariant perverse sheaves on the
prehomogeneous vector space 4 for $n=3$. There are 4 orbits of this action:
$C_{\ref{infcase:6-A2}a}=\\{0\\},C_{\ref{infcase:6-A2}b}=\\{r_{1}X_{{{\hat{\gamma}}_{1}}}{\
|\ }r_{1}\neq 0\\}$,
$C_{\ref{infcase:6-A2}c}=\\{r_{2}X_{{{\hat{\gamma}}_{1}}+3{{\hat{\gamma}}_{2}}},r_{2}\neq
0\\}$ and
$C_{\ref{infcase:6-A2}d}=\\{r_{1}X_{{{\hat{\gamma}}_{1}}}+r_{2}X_{{{\hat{\gamma}}_{1}}+3{{\hat{\gamma}}_{2}}},r_{1},r_{2}\neq
0\\}$. Corresponding Langlands parameters are given by

$\begin{array}[]{rcl}\phi_{\ref{infcase:6-A2}a}(w,x)&=&\lambda(w),\\\
\phi_{\ref{infcase:6-A2}b}(w,x)&=&\widehat{m}(\theta_{3}^{\operatorname{ord}(w)}|w|,\theta_{3}^{2\operatorname{ord}(w)}|w|^{1/2})\cdot\iota_{{\hat{\gamma}}_{1}}(x),\\\
\phi_{\ref{infcase:6-A2}c}(w,x)&=&\widehat{m}(\theta_{3}^{\operatorname{ord}(w)}|w|^{1/2},\theta_{3}^{2\operatorname{ord}(w)}|w|)\cdot\iota_{{{\hat{\gamma}}_{1}}+3{{\hat{\gamma}}_{2}}}(x),\\\
\phi_{\ref{infcase:6-A2}d}(w,x)&=&\widehat{m}(\theta_{3}^{\operatorname{ord}(w)},\theta_{3}^{2\operatorname{ord}(w)})\varphi_{X_{{{\hat{\gamma}}_{1}}}+X_{{{\hat{\gamma}}_{1}}+3{{\hat{\gamma}}_{2}}}}(x),\end{array}$

where
$\varphi_{X_{{{\hat{\gamma}}_{1}}}+X_{{{\hat{\gamma}}_{1}}+3{{\hat{\gamma}}_{2}}}}(x):\operatorname{SL}_{2}(\mathbb{C})\to\widehat{G_{2}}$
is a group homomorphism determined by the $\mathfrak{sl}_{2}$-triple $(e,f,h)$
with
$e=X_{{\hat{\gamma}}_{1}}+X_{{{\hat{\gamma}}_{1}}+3{{\hat{\gamma}}_{2}}}$,
$f=2X_{-{{\hat{\gamma}}_{1}}}+2X_{-({{\hat{\gamma}}_{1}}+3{{\hat{\gamma}}_{2}})}$
and
$h=2H_{{{\hat{\gamma}}_{1}}}+2H_{{{\hat{\gamma}}_{1}}+3{{\hat{\gamma}}_{2}}}$.
We find that $A_{\phi_{\ref{infcase:6-A2}d}}$ has order $3$ and the component
groups of the other 3 parameters are trivial.

The local Langlands correspondence in this case is obtained by arguments
similar to the proof in case 5 and is given in Table 2.7.1. Using the
microlocal analysis of the prehomogeneous vector space 4 for $n=3$ from
Section 1.4, the ABV-packet coefficients are given in Table 2.6.1.

  8. (Case 8)

Recall Case 8 from Proposition 1.1. The infinitesimal parameter
$\lambda_{\ref{infcase:7-reg}}$ is given by
$\lambda_{\ref{infcase:7-reg}}(w)=\widehat{m}(|w|^{3},|w|^{2})$. In this case,
we have $V=\\{r_{1}X_{\hat{\gamma}_{1}}+r_{2}X_{\hat{\gamma}_{2}}{\ |\
}r_{1},r_{2}\in\mathbb{C}\\}$ and
$H_{\lambda_{\ref{infcase:7-reg}}}={\widehat{T}}$. The action of
$H_{\lambda_{\ref{infcase:7-reg}}}$ on $V_{\lambda_{\ref{infcase:7-reg}}}$ is
given by

$\widehat{m}(x,y).(r_{1}X_{\hat{\gamma}_{1}}+r_{2}X_{\hat{\gamma}_{2}})=y^{2}x^{-1}r_{1}X_{\hat{\gamma}_{1}}+xy^{-1}r_{2}X_{\hat{\gamma}_{2}}.$

This action has 4 orbits $C_{\ref{infcase:7-reg}a}=\\{0\\}$,
$C_{\ref{infcase:7-reg}b}=\\{r_{1}X_{\hat{\gamma}_{1}},r_{1}\neq 0\\}$,
$C_{\ref{infcase:7-reg}c}=\\{r_{2}X_{\hat{\gamma}_{2}},r_{2}\neq 0\\}$ and
$C_{\ref{infcase:7-reg}d}=\\{r_{1}X_{\hat{\gamma}_{1}}+r_{2}X_{\hat{\gamma}_{2}}{\
|\ }r_{1},r_{2}\neq 0\\}.$ The 4 corresponding Langlands parameters are given
by

$\begin{array}[]{rcl}\phi_{\ref{infcase:7-reg}a}(w,x)&=&\lambda_{\ref{infcase:7-reg}}(w),\\\
\phi_{\ref{infcase:7-reg}b}(w,x)&=&\widehat{m}(|w|^{3},|w|^{3/2})\cdot\iota_{\hat{\gamma}_{1}}(x),\\\
\phi_{\ref{infcase:7-reg}c}(w,x)&=&\widehat{m}(|w|^{5/2},|w|^{5/2})\cdot\iota_{\hat{\gamma}_{2}}(x),\\\
\phi_{\ref{infcase:7-reg}d}(w,x)&=&\varphi_{\mathrm{reg}}(x),\end{array}$

where
$\varphi_{\mathrm{reg}}:\operatorname{SL}_{2}(\mathbb{C})\to\widehat{G_{2}}$
is the group homomorphism determined by the regular orbit, i.e.,
$\varphi_{\mathrm{reg}}$ is group homomorphism determined by an
$\mathfrak{sl}_{2}$-triple $(e,f,h)$ with
$e=X_{\hat{\gamma}_{1}}+X_{\hat{\gamma}_{2}}$. Each Langlands parameter has
trivial component group.

The dual character of $\lambda_{\ref{infcase:7-reg}}$ is $\nu^{2}\otimes\nu$.
All of the representations of with infinitesimal Langlands parameter
$\lambda_{\ref{infcase:7-reg}}$ are components of $I(\nu^{2}\otimes\nu)$. From
the decomposition of $I(\nu^{2}\otimes\nu)$ given in [Muic]*Proposition 4.4,
we easily find the local Langlands correspondence in this case which is given
in Table 2.7.1. Using 4 with $n=1$ we find the ABV-coefficients which are
given in Table 2.6.1

  9. (Case 9)

Case 9 from Proposition 1.1 was studied in [CFZ:cubics]. The Langlands
correspondence is given in Table 2.7.1 and the ABV-coefficient appear as the
last block of Table 2.6.1. We remark that
$A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi_{\ref{infcase:8-sub}a}}=A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi_{\ref{infcase:8-sub}d}}=S_{3}$
and
$A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi_{\ref{infcase:8-sub}b}}=A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi_{\ref{infcase:8-sub}c}}=\langle\theta_{2}\rangle$.

###  2.7. Fundamental properties of ABV-packet coefficients for $G_{2}$

Let $x_{\phi}$ be the point for $\phi$ in the moduli space
$V_{\lambda_{\phi}}$; let $C_{\phi}$ be the $H_{\lambda_{\phi}}$-orbit of
$x_{\phi}$ in $V_{\lambda_{\phi}}$. Pick $y\in V^{*}_{\lambda_{\phi}}$ such
that $(x_{\phi},y)\in T^{*}_{C_{\phi}}(V_{\lambda_{\phi}})_{\text{sreg}}$.
Then

$A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}=\pi_{0}(T^{*}_{C_{\phi}}(V_{\lambda_{\phi}}),(x_{\phi},y)).$

Let $p:T^{*}_{C_{\phi}}(V_{\lambda_{\phi}})\to C_{\phi}$ be projection. Then
$p$ induces the group homomorphism
$\pi_{1}(p,(x_{\phi},y)):\pi_{1}(T^{*}_{C_{\phi}}(V_{\lambda_{\phi}}),(x_{\phi},y))\to\pi_{1}(C_{\phi},x_{\phi})$
which in turn induces a group homomorphism between the equivariant quotients,
$A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}\to
A_{\phi}$. Pre-composition with this group homomorphism defines the map
$\mathsf{Rep}(A_{\phi})\to\mathsf{Rep}(A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi})$
appearing in Theorem 2.2.

Table 2.7.1. $L$-packets and ABV-packets for all unipotent representations of
$G_{2}(F)$, arranged by the cases appearing Proposition 1.1. Each row gives an
ABV-packet, formed as the union of an $L$-packet and its coronal
representations. Notation and proofs are presented in Section 2.6 drawing on
results from Proposition 1.2.

$\begin{array}[]{| c | c | c |}\hline\cr\text{$L$-parameter}&\text{$L$-packet}&\text{Coronal}\\\ \phi&\Pi_{\phi}(G_{2}(F))&\text{representations}\\\ \hline\cr\hline\cr\phi_{\ref{infcase:0}}&I(\mu_{1}\nu^{a_{1}}\otimes\mu_{2}\nu^{a_{2}})&\\\ \hline\cr\hline\cr\phi_{\ref{infcase:1-short}a}&I_{\gamma_{2}}(a-1/2,\mu\circ\det)&\\\ \phi_{\ref{infcase:1-short}b}&I_{\gamma_{2}}(a-1/2,\mu\otimes\mathrm{St}_{\operatorname{GL}_{2}})&\\\ \hline\cr\phi_{\ref{infcase:2-long}a}&I_{\gamma_{1}}(a-1/2,\mu\circ\det)&\\\ \phi_{\ref{infcase:2-long}b}&I_{\gamma_{1}}(a-1/2,\mu\otimes\mathrm{St}_{\operatorname{GL}_{2}})&\\\ \hline\cr\hline\cr\phi_{\ref{infcase:3}a}&I_{{\gamma_{1}}}(1/6,\theta_{3}^{n}\circ\det)&\\\ \phi_{\ref{infcase:3}b}&I_{{\gamma_{1}}}(1/6,\theta_{3}^{n}\otimes\mathrm{St}_{\operatorname{GL}_{2}})&\\\ \hline\cr\phi_{\ref{infcase:4-D2}a}&J_{\gamma_{2}}(1,I^{\operatorname{GL}_{2}}(1\otimes\theta_{2}))&I_{0}(G_{2}[-1])\\\ \phi_{\ref{infcase:4-D2}b}&J_{{\gamma_{1}}}(1/2,\theta_{2}\otimes\mathrm{St}_{\operatorname{GL}_{2}})&I_{0}(G_{2}[-1])\\\ \phi_{\ref{infcase:4-D2}c}&J_{{\gamma_{2}}}(1/2,\theta_{2}\otimes\mathrm{St}_{\operatorname{GL}_{2}})&I_{0}(G_{2}[-1])\\\ \phi_{\ref{infcase:4-D2}d}&\pi(\theta_{2}),I_{0}(G_{2}[-1])&\\\ \hline\cr\phi_{\ref{infcase:5}a}&I_{\gamma_{2}}(3/2,1_{\operatorname{GL}_{2}})&\\\ \phi_{\ref{infcase:5}b}&I_{{\gamma_{2}}}(3/2,\mathrm{St}_{\operatorname{GL}_{2}})&\\\ \hline\cr\phi_{\ref{infcase:6-A2}a}&J_{\gamma_{2}}(1,I^{\operatorname{GL}_{2}}(\theta_{3}\otimes\theta_{3}^{-1}))&I_{0}(G_{2}[\theta_{3}]),I_{0}(G_{2}[\theta_{3}^{2}])\\\ \phi_{\ref{infcase:6-A2}b}&J_{{\gamma_{1}}}(1/2,\theta_{3}^{2}\otimes\mathrm{St}_{\operatorname{GL}_{2}})&I_{0}(G_{2}[\theta_{3}]),I_{0}(G_{2}[\theta_{3}^{2}])\\\ \phi_{\ref{infcase:6-A2}c}&J_{{\gamma_{1}}}(1/2,\theta_{3}\otimes\mathrm{St}_{\operatorname{GL}_{2}})&I_{0}(G_{2}[\theta_{3}]),I_{0}(G_{2}[\theta_{3}^{2}])\\\ \phi_{\ref{infcase:6-A2}d}&\pi(\theta_{3}),I_{0}(G_{2}[\theta_{3}]),I_{0}(G_{2}[\theta_{3}^{2}])&\\\ \hline\cr\phi_{\ref{infcase:7-reg}a}&1_{G_{2}}&\\\ \phi_{\ref{infcase:7-reg}b}&J_{\gamma_{1}}(3/2,\mathrm{St}_{\operatorname{GL}_{2}})&\\\ \phi_{\ref{infcase:7-reg}c}&J_{{\gamma_{2}}}(5/2,\mathrm{St}_{\operatorname{GL}_{2}})&\\\ \phi_{\ref{infcase:7-reg}d}&\mathrm{St}_{G_{2}}&\\\ \hline\cr\hline\cr\phi_{\ref{infcase:8-sub}a}&J_{{\gamma_{2}}}(1,I^{\operatorname{GL}_{2}}(1\otimes 1))&J_{\gamma_{1}}(1/2,\mathrm{St}_{\operatorname{GL}_{2}}),I_{0}(G_{2}[1])\\\ \phi_{\ref{infcase:8-sub}b}&J_{\gamma_{1}}(1/2,\mathrm{St}_{\operatorname{GL}_{2}})&J_{{\gamma_{2}}}(1/2,\mathrm{St}_{\operatorname{GL}_{2}}),I_{0}(G_{2}[1])\\\ \phi_{\ref{infcase:8-sub}c}&J_{{\gamma_{2}}}(1/2,\mathrm{St}_{\operatorname{GL}_{2}})&\pi(1),I_{0}(G_{2}[1])\\\ \phi_{\ref{infcase:8-sub}d}&\pi(1)^{\prime},\pi(1),I_{0}(G_{2}[1])&\\\ \hline\cr\end{array}$

######  Theorem 2.2.

Let $\phi:W^{\prime}_{F}\to{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G_{2}}}$ be an
unramified Langlands parameter.

  1. (1)

The function $\pi\mapsto\langle\ ,\pi\rangle$ extends the local Langlands
correspondence: for all unramified Langlands parameters
$\phi:W^{\prime}_{F}\to{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G_{2}}}$, the diagram

${\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}(G_{2}(F))}$${\widehat{A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}}}$${\Pi_{\phi}(G_{2}(F))}$${\widehat{A_{\phi}}}$$\scriptstyle{\pi\mapsto\langle\
,\pi\rangle}$$\scriptstyle{\pi\mapsto\langle\
,\pi\rangle_{\phi}}$$\scriptstyle{\mathrm{LLC}}$ (9)

commutes, where
$\widehat{A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}}$
(resp. $\widehat{A_{\phi}}$) denotes the set of characters of irreducible
representations of
$A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}$ (resp.
$A_{\phi}$).

  2. (2)

If $\phi:W^{\prime}_{F}\to{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G_{2}}}$ is open
or closed then $\pi\mapsto\langle\ ,\pi\rangle$ is a bijection
$\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}(G_{2}(F))\to\widehat{A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}}$.

  3. (3)

If $\phi$ is bounded upon restriction to $W_{F}$ then all the representations
in
$\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}(G_{2}(F))$
are tempered. If $\phi$ is not bounded upon restriction to $W_{F}$ then
$\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}(G_{2}(F))$
may contain non-tempered representations and $\langle\ ,\ \rangle$ may not be
bijective.

  4. (4)

A representation $\pi\in\Pi_{\phi}(G_{2}(F))$ is spherical if and only if
$\phi$ is closed and $\langle\ ,\pi\rangle$ is the trivial representation of
$A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}$. In this
case, the only ABV-packet that contains $\pi$ is
$\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}(G_{2}(F))$.
A representation $\pi\in\Pi_{\phi}(G_{2}(F))$ is generic if and only if $\phi$
is open and $\langle\ ,\pi\rangle$ is the trivial representation of
$A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}$.

  5. (5)

For every unramified Langlands parameter $\phi$ there is an unramified
Langlands parameter ${\hat{\phi}}$, with the same infinitesimal parameter as
$\phi$, such that the Aubert involution defines a bijection

$\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}(G_{2}(F))\to\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\hat{\phi}}(G_{2}(F)).$

The $Z_{{\widehat{G_{2}}}}(\lambda_{\phi})$-orbit of $\phi$ in
$V_{\lambda_{\phi}}$ is dual to the transpose of the
$Z_{{\widehat{G_{2}}}}(\lambda_{\phi})$-orbit of ${\hat{\phi}}$ in
$V_{\lambda_{\phi}}$.

###### Proof.

All these results follow from a study of Table 2.6.1 together with results
from [Muic].

  1. 1

Table 2.6.1 shows that
$r:\mathsf{Rep}(A_{\phi})\to\mathsf{Rep}(A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi})$
preserves irreducibility and is injective. It follows from [CFMMX, Theorem
7.10.1 (d)] that if $\pi\in\Pi_{\phi}$ then $\langle\
,\pi\rangle=r(\mathop{LLC}(\pi))$ where
$\mathop{LLC}(\pi)\in\widehat{A_{\phi}}$ is the representation given by the
local Langlands correspondence applied to $\pi$. Alternately, this can be seen
by comparing Tables 2.1.1 and 2.6.1.

  2. 2

In Cases 1, 2, 3, 4, 6 and 8,
$\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}(G_{2}(F))\to\widehat{A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}}$
is a bijection since
$\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}(G_{2}(F))=\Pi_{\phi}(G_{2}(F))$.
Observe also that every Langlands parameter is either open or closed in cases
Cases 1, 2, 3, 4 and 6. Inspect Table 2.6.1 for Cases 5, 7 and 9.

  3. 3

Note that when $\phi|_{W_{F}}$ is bounded, then $\phi$ is open and
$\Pi_{\phi}^{{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}}(G_{2}(F))=\Pi_{\phi}(G_{2}(F))$.
It follows from the previous explicit local Langlands correspondence that each
$\Pi_{\phi}(G_{2}(F))$ consists of tempered representations when
$\phi|_{W_{F}}$ is bounded.

  4. 4

Recall the general fact that if $P=MN$ is a parabolic subgroup of a reductive
group $G$ over $F$ and if $\sigma$ is an irreducible representation of $M$,
then the parabolic induction $\text{Ind}_{P}^{G}(\sigma)$ has a unique
spherical component if $\sigma$ is spherical and has a unique generic
component if $\sigma$ is generic. The assertion follows from this general
fact. For example, in the Case 5, from Equation (7) and Equation (8), we see
that $\pi(\mu)$ is generic and $J_{\gamma_{2}}(1,\pi(1,\mu))$ is spherical.
All of the other cases are similar.

  5. 5

The Aubert involution is computed in [Muic]. The involution
$\phi\mapsto{\hat{\phi}}$ is given by:
${\hat{\phi}}_{\ref{infcase:0}}=\phi_{\ref{infcase:0}}$;
${\hat{\phi}}_{\ref{infcase:1-short}a}=\phi_{\ref{infcase:1-short}b}$;
${\hat{\phi}}_{\ref{infcase:2-long}a}=\phi_{\ref{infcase:2-long}b}$;
${\hat{\phi}}_{\ref{infcase:3}a}=\phi_{\ref{infcase:3}b}$;
${\hat{\phi}}_{\ref{infcase:4-D2}a}=\phi_{\ref{infcase:4-D2}d}$,
${\hat{\phi}}_{\ref{infcase:4-D2}b}=\phi_{\ref{infcase:4-D2}c}$;
${\hat{\phi}}_{\ref{infcase:5}a}=\phi_{\ref{infcase:5}b}$;
${\hat{\phi}}_{\ref{infcase:6-A2}a}=\phi_{\ref{infcase:6-A2}d}$,
${\hat{\phi}}_{\ref{infcase:6-A2}b}=\phi_{\ref{infcase:6-A2}c}$;
${\hat{\phi}}_{\ref{infcase:7-reg}a}=\phi_{\ref{infcase:7-reg}d}$,
${\hat{\phi}}_{\ref{infcase:7-reg}b}=\phi_{\ref{infcase:7-reg}c}$ ;
${\hat{\phi}}_{\ref{infcase:8-sub}a}=\phi_{\ref{infcase:8-sub}d}$,
${\hat{\phi}}_{\ref{infcase:8-sub}b}=\phi_{\ref{infcase:8-sub}c}$. The result
now follows from Table 2.6.1.∎

######  Remark 2.3.

With reference to 2, all unramified Langlands parameters of $G_{2}(F)$ are
either open or closed with the exception of $\phi_{\ref{infcase:4-D2}b}$,
$\phi_{\ref{infcase:4-D2}c}$, $\phi_{\ref{infcase:6-A2}b}$,
$\phi_{\ref{infcase:6-A2}c}$, $\phi_{\ref{infcase:8-sub}b}$ and
$\phi_{\ref{infcase:8-sub}c}$. For all but the last two, $\pi\mapsto\langle\
,\pi\rangle$ defines a bijection
$\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}(G_{2}(F))\to\widehat{A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}}$
and in these two cases, the map is surjective.

###  2.8. A geometric interpretation of Muic’s reducibility points

Many results of [Muic] on reducibility points of induced representations of
$G_{2}$ can be explained by the geometry of the action of $H_{\lambda}$ on
$V_{\lambda}$. The following corollary is a collection of previous
discussions, which explains the reducibility results of [Muic]*Proposition 3.1
and Theorem 3.1 using geometry.

######  Proposition 2.4.

Let $\lambda:W_{F}\to{\widehat{T}}\to{\widehat{G_{2}}}$ be an unramified
infinitesimal parameter with Vogan variety $V_{\lambda}$ and the group
$H_{\lambda}$ acting on it. Let
$\chi=\chi_{1}\otimes\chi_{2}:T(F)\rightarrow\mathbb{C}^{\times}$ be the
character dual to $\lambda$, which is determined by Equation (6).

  1. (1)

The representation $I(\chi_{1}\otimes\chi_{2})$ is irreducible if and only if
$V_{\lambda}=\\{0\\};$

  2. (2)

Suppose that $V_{\lambda}\neq\\{0\\}$. Then we can write
$I(\chi_{1}\otimes\chi_{2})=I_{\gamma}(s,\mu\otimes\mathrm{St}_{\operatorname{GL}_{2}})+I_{\gamma}(s,\mu\circ\det)$
for $\gamma\in\\{{\gamma_{1}},{\gamma_{2}}\\}$, $s\in\mathbb{C}$ and a unitary
character $\mu$. Moreover, the two representations,
$I_{\gamma}(s,\mu\otimes\mathrm{St}_{\operatorname{GL}_{2}})$ and
$I_{\gamma}(s,\mu\circ\det)$, are irreducible if and only if the action of
$H_{\lambda}$ on $V_{\lambda}$ has exactly two orbits.

###### Proof.

Item 1 is a corollary of the discussion in Case 1, Section 2.6. If
$V_{\lambda}\neq 0$, from the above case by case study, we see that we can
write $I(\chi_{1}\otimes\chi_{2})$ as a sum,
$I_{\gamma}(s,\mu\otimes\mathrm{St}_{\operatorname{GL}_{2}})+I_{\gamma}(s,\mu\circ\det)$.
Note that in the cases in 2, 3, 6 and 4, the action of $H_{\lambda}$ on
$V_{\lambda}$ has exactly two orbits, and the corresponding
$I(s,\mu\otimes\mathrm{St}_{\operatorname{GL}_{2}})$ and $I(s,\mu\circ\det)$
are irreducible. While in the cases in subsections 5, 7, 8 and 9, the action
of $H_{\lambda}$ on $V_{\lambda}$ has more than two orbits and the
corresponding $I(s,\mu\otimes\mathrm{St}_{\operatorname{GL}_{2}})$ and
$I(s,\mu\circ\det)$ are reducible. ∎

This corollary shows that how the geometry of the action of $H_{\lambda}$ on
$V_{\lambda}$ can determine the representations, which is exactly the spirit
of [CFMMX].

###  2.9. Unramified Arthur parameters for $G_{2}$

######  Definition 3.

Let us say that a Langlands parameter
$\phi:W_{F}\times\operatorname{SL}_{2}(\mathbb{C})\to{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G_{2}}}$
is of Arthur type $\psi$ if there exists an Arthur parameter
$\psi:W_{F}\times\operatorname{SL}_{2}(\mathbb{C})\times\operatorname{SL}_{2}(\mathbb{C})\to{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G_{2}}}$,
such that $\phi=\phi_{\psi}$, where
$\phi_{\psi}(w,x)=\psi(w,x,\operatorname{diag}(|w|^{1/2},|w|^{-1/2}))$. We say
that an irreducible admissible representation $\pi$ of $G_{2}(F)$ is of Arthur
type if there is a Langlands parameter $\phi$ of Arthur type such that
$\pi\in\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}(G_{2}(F))$;
if such a $\phi$ exists, it is generally not unique.

######  Remark 2.5.

If $\phi$ is of Arthur type then the Arthur parameter $\psi$, above, is
unique. If $\pi$ is of Arthur type then the Langlands parameter $\phi$, above,
is generally not unique.

######  Remark 2.6.

Definition 3 opens the door to a dangerous misapprehension. Every Arthur
parameter $\psi$ determines a Langlands parameter $\phi_{\psi}$; this function
from Arthur parameters to Langlands parameters is injective. We say that a
Langlands parameter is of Arthur type if it lies in the image of this
function. On the other hand, an irreducible admissible representation is said
to be of Arthur type if it lies in an Arthur packet - typically more than one.
It can happen that a representation of Arthur type may have a Langlands
parameters that is not of Arthur type. An example of this phenomenon appears
in [CFMMX]*Section 16. Specifically, the Arthur packet
$\Pi_{\psi_{2}}(\operatorname{SO}_{7}(F))$ contains two representations,
$\pi_{2}^{+}$ and $\pi_{3}^{-}$; the Langlands parameter for $\pi_{2}^{+}$ is
of Arthur type $\psi_{2}$, but the Langlands parameter for $\pi_{3}^{-}$ is
not of Arthur type; see [CFMMX]*Table 16.5.1. Thus $\pi_{3}^{-}$ is an Arthur
type representations whose Langlands parameter is not of Arthur type. On the
other hand, this phenomenon does not happen for any unipotent representations
of $G_{2}(F)$ \- see Corollary 2.9.

We now determine which unramified Langlands parameters for $G_{2}(F)$ are of
Arthur type.

######  Proposition 2.7.

Table 4.1.1 presents necessary and sufficient conditions for an unramified
Langlands parameter $\phi$ of $G_{2}(F)$ to be of Arthur type.

###### Proof.

Note that $\psi|_{W_{F}\times\\{1\\}\times\\{1\\}}$ is bounded by definition
of Arthur parameter. Moreover, we can write
$\psi(w,x,y)=\psi(w,1,1)\psi(1,x,1)\psi(1,1,y)$. The image of $\psi(1,x,1)$
and $\psi(1,1,y)$ is either 1 or a copy of $\operatorname{SL}_{2}(\mathbb{C})$
in ${\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G_{2}}}$. Observe that the images of
$\psi(w,1,1)$, $\psi(1,x,1)$ and $\psi(1,1,y)$ are commutative and both
$d\psi\left(1,\left(\begin{smallmatrix}0&1\\\
0&0\end{smallmatrix}\right),1\right)$ and
$d\psi\left(1,1,\left(\begin{smallmatrix}0&1\\\
0&0\end{smallmatrix}\right)\right)$ are in $V_{\lambda}$ with
$\lambda=\lambda_{\psi}$, where $d\psi$ is the differential of $\psi$.

  1. (Case 1)

In this case, we have
$\phi_{\ref{infcase:0}}(w,x)=\widehat{m}(u_{1}^{\operatorname{ord}(w)},u_{2}^{\operatorname{ord}(w)})\cdot\widehat{m}(|w|^{a_{1}},|w|^{a_{2}})$
with $u_{1},u_{2}\in\mathbb{C}^{1},a_{1},a_{2}\in\mathbb{R}$ and the Vogan
variety is $\left\\{{0}\right\\}$. From the above observation,
$\psi(1,x,y)=1$. Thus in this case, $\phi_{\ref{infcase:0}}$ is of Arthur type
if and only if $\psi_{\ref{infcase:0}}(w,x,y)=\phi_{\ref{infcase:0}}(w,x)$ is
an Arthur parameter, which is equivalent to $a_{1}=a_{2}=0$.

  2. (Case 2)

In this case, $V_{\lambda_{\ref{infcase:1-short}}}$ has dimension one. Let
$\psi$ be an Arthur parameter with
$\lambda_{\psi}=\lambda_{\ref{infcase:1-short}}$. From the above observation,
we get at least one of $\psi(1,x,1)$ and $\psi(1,1,y)$ is trivial. Suppose
that $\psi_{\ref{infcase:1-short}b}$ is an Arthur parameter with
$\phi_{\ref{infcase:1-short}b}=\phi_{\psi_{\ref{infcase:1-short}b}}$. Then
$\psi_{\ref{infcase:1-short}b}(1,x,1)=\iota_{\hat{\gamma}_{1}+2\hat{\gamma}_{2}}(x)$
and thus $\psi_{\ref{infcase:1-short}b}(1,1,y)=1$. Now the condition
$\phi_{\ref{infcase:1-short}b}=\phi_{\psi_{\ref{infcase:1-short}b}}$ implies
that $\psi(w,1,1)=\widehat{m}(1,u^{\operatorname{ord}(w)}|w|^{a-1/2})$ with
$u\in\mathbb{C}^{1},a\in\mathbb{R}$, which is supposed to be bounded. Thus we
get $a=1/2$. Note that, when $a=1/2$, the corresponding Arthur parameter
$\psi_{\ref{infcase:1-short}b}(w,x,y)=\phi_{\ref{infcase:1-short}}(w,x)$. It
is easy to see that $\phi_{\ref{infcase:1-short}a}$ is of Arthur type if and
only if $a=1/2$ and in this case the corresponding Arthur parameter is given
by
$\psi_{\ref{infcase:1-short}a}(w,x,y)=\psi_{\ref{infcase:1-short}b}(w,y,x)$.

  3. (Case 3)

This case is similar to Case 2 and thus omitted.

  4. (Case 4)

Recall that in this case, $V_{\lambda}$ is spanned by
$X_{\widehat{\gamma}_{1}}$ and
$X_{2\widehat{\gamma}_{1}+3\widehat{\gamma}_{2}}$. Notice that these two roots
are not orthogonal to each other. Suppose that $\phi_{\ref{infcase:3}b}$ is of
Arthur type with Arthur parameter $\psi_{\ref{infcase:3}b}$. We have
$\psi_{\ref{infcase:3}b}(1,x,1)=\phi_{\ref{infcase:3}b}(1,x)=\iota_{\widehat{\gamma}_{1}}(x)$.
Since $\psi_{\ref{infcase:3}b}(1,1,y)$ is commutative with
$\iota_{\widehat{\gamma}_{1}}$ and
$(d\psi_{\ref{infcase:3}b})\left(1,1,\left(\begin{smallmatrix}1&1\\\
0&1\end{smallmatrix}\right)\right)\in V_{\lambda}$, we must have
$\psi_{\ref{infcase:3}b}(1,1,y)=1$. This implies that
$\psi_{\ref{infcase:3}b}(w,1,1)=\phi_{\ref{infcase:3}b}(w,1)$ which is
unbounded. Thus there is no such Arthur parameter and
$\phi_{\ref{infcase:3}b}$ is not of Arthur type. Similarly,
$\phi_{\ref{infcase:3}a}$ is also not of Arthur type.

  5. (Case 5)

Every Langlands parameter in this case is of Arthur type and the corresponding
Arthur parameter can be given explicitly,

$\displaystyle\psi_{\ref{infcase:4-D2}a}(w,x,y)$
$\displaystyle=\widehat{m}(1,(-1)^{\operatorname{ord}(w)})\iota_{\widehat{\gamma}_{1}}(y)\iota_{\widehat{\gamma}_{1}+2\widehat{\gamma}_{2}}(y),$
$\displaystyle\psi_{\ref{infcase:4-D2}b}(w,x,y)$
$\displaystyle=\widehat{m}(1,(-1)^{\operatorname{ord}(w)})\iota_{\widehat{\gamma}_{1}}(x)\iota_{\widehat{\gamma}_{1}+2\widehat{\gamma}_{2}}(y),$
$\displaystyle\psi_{\ref{infcase:4-D2}c}(w,x,y)$
$\displaystyle=\widehat{m}(1,(-1)^{\operatorname{ord}(w)})\iota_{\widehat{\gamma}_{1}}(y)\iota_{\widehat{\gamma}_{1}+2\widehat{\gamma}_{2}}(x),$
$\displaystyle\psi_{\ref{infcase:4-D2}d}(w,x,y)$
$\displaystyle=\widehat{m}(1,(-1)^{\operatorname{ord}(w)})\iota_{\widehat{\gamma}_{1}}(x)\iota_{\widehat{\gamma}_{1}+2\widehat{\gamma}_{2}}(x).$

  6. (Case 6)

In this case, both parameters are not of Arthur type and the reason is the
same as in the case 4.

  7. (Case 7)

The Langlands parameters $\phi_{\ref{infcase:6-A2}a}$ and
$\phi_{\ref{infcase:6-A2}d}$ are of Arthur type and the corresponding Arthur
parameters are given by

$\psi_{\ref{infcase:6-A2}d}(w,x,y)=\phi_{\ref{infcase:6-A2}d}(w,x),\quad\psi_{\ref{infcase:6-A2}a}(w,x,y)=\psi_{\ref{infcase:6-A2}d}(w,y,x).$

The parameters $\phi_{\ref{infcase:6-A2}b}$ and $\phi_{\ref{infcase:6-A2}c}$
are not of Arthur type for the following reason. Suppose that, for example,
$\phi_{\ref{infcase:6-A2}c}$ is of Arthur type and the corresponding Arthur
parameter is $\psi_{\ref{infcase:6-A2}c}$. Similar reason as above,
$\psi_{\ref{infcase:6-A2}c}(1,1,y)$ should be commutative with
$\iota_{\widehat{\gamma}_{1}+3\widehat{\gamma}_{2}}(x)$ and also
$(d\psi_{\ref{infcase:6-A2}c})\left(1,1,\left(\begin{smallmatrix}1&1\\\
0&1\end{smallmatrix}\right)\right)$ should be in
$V_{\lambda_{\ref{infcase:6-A2}}}$. One can check that such
$\psi_{\ref{infcase:6-A2}c}(1,1,y)$ does not exist.

  8. (Case 8)

The situation is similar to the above case. The parameters
$\phi_{\ref{infcase:7-reg}a}$ and $\phi_{\ref{infcase:7-reg}d}$ are of Arthur
type and the corresponding Arthur parameters are given by

$\psi_{\ref{infcase:7-reg}d}(w,x,y)=\phi_{\ref{infcase:7-reg}d}(w,x),\quad\psi_{\ref{infcase:7-reg}a}(w,x,y)=\psi_{\ref{infcase:7-reg}d}(w,y,x).$

The parameters $\phi_{\ref{infcase:7-reg}b}$ and $\phi_{\ref{infcase:7-reg}c}$
are not of Arthur type for a similar reason as in the last case.

  9. (Case 9)

All of the 4 Langlands parameters are of Arthur type as explained in
[CFZ:cubics].∎

######  Remark 2.8.

For a unipotent conjugacy class of ${\widehat{G_{2}}}$, there is a
corresponding conjugacy class of group homomorphisms,
$\varphi:\operatorname{SL}_{2}(\mathbb{C})\to{\widehat{G_{2}}}$, which
determine Arthur parameters $\psi$ by the formula $\psi(w,x,y)=\varphi(y)$.
There are a total of 5 unipotent conjugacy classes of ${\widehat{G_{2}}}$, see
[GG]*2.7, for example. The Arthur parameters of these 5 unipotent conjugacy
classes are: $\psi_{\ref{infcase:0}}$ when $u_{1}=u_{2}=1$ and $a_{1}=a_{2}=0$
(corresponding to the trivial conjugacy class),
$\psi_{\ref{infcase:1-short}a}$ when $u=1$ (corresponding to the unipotent
conjugacy class of the short root, i.e., $\mathcal{O}_{short}$ in the notation
of [GG]*2.7), $\psi_{\ref{infcase:2-long}a}$ when $u=1$ (corresponding to
$\mathcal{O}_{long}$), $\psi_{\ref{infcase:7-reg}a}$ (corresponding to
$\mathcal{O}_{reg}$), and $\psi_{\ref{infcase:8-sub}a}$ (corresponding to
$\mathcal{O}_{subreg}$).

######  Corollary 2.9.

A unipotent representation $\pi$ of $G_{2}(F)$ is of Arthur type if and only
if its Langlands parameter is of Arthur type.

###### Proof.

We have found all unramified Arthur parameters in Proposition 2.7. Using Table
2.6.1, we find all unipotent representations of Arthur type. Using Table 2.1.1
we find the Langlands parameters of these representations. Using Proposition
2.7 again, we find that these Langlands parameters are all of Arthur type. ∎

Arthur has conjectured [Arthur:Conjectures] that an irreducible admissible
representation is unitary if and only if it is of Arthur type. In Corollary
2.10 we confirm this conjecture for unipotent representations of $G_{2}(F)$.

######  Corollary 2.10.

A unipotent representation of $G_{2}(F)$ is unitary if and only if is of
Arthur type.

###### Proof.

We use [Muic] to find which unipotent representations are unitary. We use
Proposition 2.7 and Table 2.6.1 to find which unipotent representations are or
Arthur type. Direct inspection reveals that these two sets are the same. ∎

Let $\phi:W^{\prime}_{F}\to{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G_{2}}}$ be a
Langlands parameter for $G_{2}$ over $F$. We set $\dim(\phi){\,:=\,}\dim
C_{\phi}$. For $\pi\in\Pi(G_{2}(F))$, set $\dim(\pi){\,:=\,}\dim(\phi_{\pi})$
where $\phi_{\pi}$ is the Langlands parameter for $\pi$.

######  Proposition 2.11.

If $\phi$ is an unramified Langlands parameter of Arthur type $\psi$ then
$\mathcal{S}^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}=\pi_{0}(Z_{{\widehat{G}}}(\psi))$
and