$s_{\phi}{\,:=\,}s_{\psi}{\,:=\,}\psi(1,1,-1)\in\mathcal{S}^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}$
has the property

$\langle s_{\phi},\pi\rangle=(-1)^{\dim(\phi)-\dim(\pi)}\langle 1,\pi\rangle,$

for every
$\pi\in\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}(G_{2}(F))$.

###### Proof.

According to Definition 1, the value of $\langle s,\pi\rangle$, for any
$s\in\mathcal{S}^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}$,
depends only on the image of $s$ under the map
$\mathcal{S}^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}\to
A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}$, so we
identify $s$ with its image in the following argument. The proof is based on a
case by case consideration. If
$A_{\phi}^{{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}}=1$, this
follows from that $\langle~{},\pi\rangle\neq 0$ if and only if
$\Pi_{\phi}=\\{\pi\\}$. The case 9 is considered in [CFZ:cubics]. Thus it
suffices to consider the case (5), (7a) and (7d). We only consider the case
(7a) and omit the details for the other cases. Notice that
$s_{\phi_{\ref{infcase:6-A2}a}}^{2}=1$ while
$A_{\phi_{\ref{infcase:6-A2}a}}^{{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}}$
has order 3 and thus $\langle
s_{\ref{infcase:6-A2}a},\pi\rangle_{\phi_{\ref{infcase:6-A2}a}}=\langle
1,\pi\rangle_{\phi_{\ref{infcase:6-A2}a}}$ for all
$\pi\in\Pi^{{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}}_{\phi_{\ref{infcase:6-A2}a}}$.
Thus it suffices to check that
$(-1)^{\dim(\phi_{\ref{infcase:6-A2}a})-\dim(\pi)}=1$ for all
$\pi\in\Pi_{\phi_{\ref{infcase:6-A2}a}}^{{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}}(G_{2}(F))$.
Note that $\dim({\phi_{\ref{infcase:6-A2}a}})=0$ and $\dim(\pi)=0$ or $2$ by
the description of
$\Pi_{\phi_{\ref{infcase:6-A2}a}}^{{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}}.$
The result follows. ∎

######  Remark 2.12.

Clearly, Proposition 2.11 is true for many Langlands parameters that are not
of Arthur type. For example, when
$A_{\phi}^{{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}}=1$,
Proposition 2.11 is true for $s=1$, regardless of whether $\phi$ is of Arthur
type or not. However, there indeed exist $\phi$ for which
$(-1)^{\dim(\phi)-\dim(\pi)}\langle 1,\pi\rangle$ does not take the form
$\langle s,\pi\rangle$ for any
$s\in\mathcal{S}_{\phi}^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}$.
Here is one example. Let $\phi=\phi_{\text{\ref{infcase:6-A2}b}}$ and
$\pi=\mathrm{cInd}_{G_{2}({\mathcal{O}}_{F})}^{G_{2}(F)}(G_{2}[\theta_{3}])$.
Then $\langle~{},\pi\rangle=\vartheta_{3}$ according Table 2.6.1. Note that
$\dim(\phi)=1$ and $\dim(\pi)=2$ and thus $(-1)^{\dim(\phi)-\dim(\pi)}\langle
1,\pi\rangle=-1$. On the other hand, we have $\langle
s,\pi\rangle=\vartheta_{3}(s)\in\left\\{1,\theta_{3},\theta_{3}^{2}\right\\}.$
Thus $(-1)^{\dim(\phi)-\dim(\pi)}\langle 1,\pi\rangle$ does not take the form
$\langle s,\pi\rangle$ for any
$s\in\mathcal{S}_{\phi}^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}$,
in this case. The same is true for $\phi=\phi_{\text{\ref{infcase:6-A2}c}}$ by
a similar argument.

###  2.10. Distributions attached to ABV-packets

######  Definition 4.

For any Langlands parameter
$\phi:W_{F}^{\prime}\to{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G_{2}}}$ and any
$s\in\mathcal{S}^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}$
we define

$\Theta_{\phi,s}{\,:=\,}\sum_{\pi\in\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi_{\psi}}(G_{2}(F))}\operatorname{trace}_{s}\left(\operatorname{\mathsf{N}\hskip-1.0pt\mathsf{E}\hskip-1.0pt\mathsf{v}\hskip-1.0pt\mathsf{s}}_{C_{\phi}}[\dim(\phi)]\mathcal{P}(\pi)[-\dim(\pi)]\right)\Theta_{\pi}.$
(10)

Using Definition 1 this takes the equivalent form

$\Theta_{\phi,s}=\sum_{\pi\in\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}(G_{2}(F))}(-1)^{\dim(\phi)-\dim(\pi)}\
\langle s,\pi\rangle\ \Theta_{\pi}.$ (11)

We will also use the notation
$\Theta^{G_{2}}_{\phi}{\,:=\,}\Theta_{\phi,1}^{G_{2}}$.

######  Proposition 2.13.

If $\phi$ is of Arthur type $\psi$ then

$\Theta_{\phi,s}=\mathop{\sum}\limits_{\pi\in\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}(G_{2}(F))}\langle
s_{\psi}\,s,\pi\rangle\ \Theta_{\pi}.$ (12)

###### Proof.

Using Definitions 1, 4 and Proposition 2.11, we have

$\begin{array}[b]{rcl}\Theta_{\phi,s}&=&\mathop{\sum}\limits_{\pi\in\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}(G_{2}(F))}\operatorname{trace}_{s}\left(\operatorname{\mathsf{N}\hskip-1.0pt\mathsf{E}\hskip-1.0pt\mathsf{v}\hskip-1.0pt\mathsf{s}}_{C_{\phi}}[\dim(\phi)]\mathcal{P}(\pi)[-\dim(\pi)]\right)\
\Theta_{\pi}\\\
&=&\mathop{\sum}\limits_{\pi\in\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}(G_{2}(F))}\operatorname{trace}_{s}\left(\operatorname{\mathsf{N}\hskip-1.0pt\mathsf{E}\hskip-1.0pt\mathsf{v}\hskip-1.0pt\mathsf{s}}_{C_{\phi}}[\dim(\phi)]\mathcal{P}(\pi)[-\dim(\pi)]\right)\
\Theta_{\pi}\\\
&=&\mathop{\sum}\limits_{\pi\in\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}(G_{2}(F))}(-1)^{\dim(\phi)-\dim(\pi)}\operatorname{trace}_{s}\left(\operatorname{\mathsf{N}\hskip-1.0pt\mathsf{E}\hskip-1.0pt\mathsf{v}\hskip-1.0pt\mathsf{s}}_{C_{\phi}}\mathcal{P}(\pi)\right)\
\Theta_{\pi}\\\
&=&\mathop{\sum}\limits_{\pi\in\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}(G_{2}(F))}(-1)^{\dim(\phi)-\dim(\pi)}\langle
s,\pi\rangle\ \Theta_{\pi}\\\
&=&\mathop{\sum}\limits_{\pi\in\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}(G_{2}(F))}\langle
s_{\psi}\,s,\pi\rangle\ \Theta_{\pi}.\end{array}\qed$

The right hand side of Equation (12) recalls the distributions used by Arthur
for classical groups in [Arthur:book] and parameters of Arthur type. As we
noticed in Remark 2.12, there are Langlands parameters $\phi$ for which the
conclusion of Proposition 2.11 does not hold. Thus Definition 4 provides a
generalization of the distributions used by Arthur. In the the next result we
will try to justify that it is indeed the correct generalization.

In order to state the next result, let us say that an Langlands parameter for
$G_{2}(F)$ is elliptic if it does not factor through the $L$-group of Levi
subgroup of $G_{2}(F)$. We will discuss endoscopy for $G_{2}(F)$ in Sections 3
and 4. We simply enumerate the elliptic unramified Langlands parameters for
$G_{2}(F)$ in Table 2.10.1 .

Table 2.10.1. Elliptic unramified Langlands parameters for $G_{2}(F)$ and
their $L$-packets. The representations appearing in these $L$-packets form the
complete list of elliptic unipotent representations of $G_{2}(F)$; see [CO],
where they are denoted by $v_{1}=\mathrm{St}_{G_{2}}$,
$v_{2}=\pi(1)^{\prime}$, $v_{3}=\pi(1)$, $v_{4}=\pi(\theta_{3})$,
$v_{5}=\pi(\theta_{2})$, $v_{6}=I_{0}(G_{2}[1])$, $v_{7}=I_{0}(G_{2}[-1])$,
$v_{8}=I_{0}(G_{2}[\theta_{3}])$ and $v_{9}=I_{0}(G_{2}[\theta^{2}_{3}])$.

$\begin{array}[]{| c | l| c | c |}\hline\cr\text{$L$-parameter}&\text{$L$-packet = ABV-packet}&\text{Endoscopic}&\text{Component groups}\\\ \phi&\Pi_{\phi}(G_{2}(F))=\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}(G_{2}(F))&\text{groups}&A_{\phi}=A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}\\\ \hline\cr\hline\cr\phi_{\ref{infcase:4-D2}d}&\pi(\theta_{2}),I_{0}(G_{2}[-1])&\operatorname{SO}_{4}&\langle\theta_{2}\rangle\\\ \phi_{\ref{infcase:6-A2}d}&\pi(\theta_{3}),I_{0}(G_{2}[\theta_{3}]),I_{0}(G_{2}[\theta^{2}_{3}])&\operatorname{PGL}_{3}&\langle\theta_{3}\rangle\\\ \phi_{\ref{infcase:7-reg}d}&\mathrm{St}_{G_{2}}&G_{2}&1\\\ \phi_{\ref{infcase:8-sub}d}&\pi(1)^{\prime},\pi(1),I_{0}(G_{2}[1])&\operatorname{SO}_{4},\operatorname{PGL}_{3}&S_{3}\\\ \hline\cr\end{array}$

######  Theorem 2.14.

Let $\phi$ be an unramified Langlands parameter for $G_{2}(F)$; recall
$\mathcal{S}^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}$
from Definition 1 and $\Theta_{\phi,s}$ from Definition 4, for
$s\in\mathcal{S}^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}$.

  1. (1)

The span of the distributions

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

is equal to the span of the distributions $\Theta_{\pi}$, as $\pi$ ranges over
$\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}(G_{2}(F))$
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 a bijection. In this case, the inverse of the linear system of equations
above is

$\Theta_{\pi}=\sum_{s\in\mathcal{A}^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}}(-1)^{\dim(C_{\phi}^{s})-\dim(\pi)}\frac{\overline{\langle
s,\pi\rangle}}{{|Z_{A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}}(s)|}}\
\Theta_{\phi,s},\qquad\qquad\pi\in\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}(G_{2}(F)),$

where $\phi$ is a Langlands parameter for $\pi$, the taken over
representatives $s$ of the fibres of
$\mathcal{S}^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}\to
A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}$; here
$\dim(C_{\phi}^{s})$ is the dimension of the $s$-fixed points of the orbit
$C_{\phi}\subseteq V_{\lambda_{\phi}}$. More generally, if $\pi$ is a
unipotent representation of $G_{2}(F)$ then its distribution character
$\Theta_{\pi}$ can be written as a linear combination of the distributions
$\Theta_{\phi,s}$, by letting $\phi$ range over unramified $L$-parameters for
$G_{2}(F)$ with the same infinitesimal parameter as $\pi$, and letting $s$
range over
$\mathcal{S}^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}$.

  2. (2)

Suppose $\Theta_{\phi}$ is stable when $\phi$ is elliptic and open. Then
$\Theta_{\phi}$ is stable for all unramified $\phi$. Moreover, these
distributions form a basis for the space of stable unipotent distributions,
letting $\phi$ range over unramified $L$-parameters for $G_{2}(F)$.

###### Proof.

Let $\phi$ be an unramified Langlands parameter for $G_{2}(F)$.

  1. 1

$\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}(G_{2}(F))\to\widehat{A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}}$
is a bijection in all cases except when $\phi=\phi_{\ref{infcase:8-sub}b}$ and
$\phi=\phi_{\ref{infcase:8-sub}c}$. Moreover,
$A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}$ is
trivial unless the infinitesimal parameter for $\phi$ is given by Cases 5, 7
or 9. Accordingly, there are precisely 10 non-trivial cases of the first
claim: $\phi_{\ref{infcase:4-D2}a}$, $\phi_{\ref{infcase:4-D2}b}$,
$\phi_{\ref{infcase:4-D2}c}$, $\phi_{\ref{infcase:4-D2}d}$,
$\phi_{\ref{infcase:6-A2}a}$, $\phi_{\ref{infcase:6-A2}b}$,
$\phi_{\ref{infcase:6-A2}c}$, $\phi_{\ref{infcase:6-A2}d}$,
$\phi_{\ref{infcase:8-sub}a}$ and $\phi_{\ref{infcase:8-sub}d}$. Consider the
first of these cases, namely $\phi=\phi_{\ref{infcase:6-A2}a}$. Recall that
$A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi_{\ref{infcase:6-A2}a}}=\langle\theta_{3}\rangle$
and let
$s_{3}\in\mathcal{S}^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi_{\ref{infcase:6-A2}a}}$
be a representative for $\theta_{3}$ above
$\mathcal{S}^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi_{\ref{infcase:6-A2}a}}\to
A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi_{\ref{infcase:6-A2}a}}$.
Then

$\begin{array}[]{rcl}\Theta_{\phi_{\ref{infcase:6-A2}a}}&=&\Theta_{\pi(\phi_{\ref{infcase:6-A2}a})}-\Theta_{\pi(\phi_{\text{\ref{infcase:6-A2}d}},\vartheta_{3})}-\Theta_{\pi(\phi_{\text{\ref{infcase:6-A2}d}},\vartheta_{3}^{2})}\\\
\Theta_{\phi_{\ref{infcase:6-A2}a},s_{3}}&=&\Theta_{\pi(\phi_{\ref{infcase:6-A2}a})}-\theta_{3}\Theta_{\pi(\phi_{\text{\ref{infcase:6-A2}d}},\vartheta_{3})}-\theta_{3}^{2}\Theta_{\pi(\phi_{\text{\ref{infcase:6-A2}d}},\vartheta_{3}^{2})}\\\
\Theta_{\phi_{\ref{infcase:6-A2}a},s_{3}^{2}}&=&\Theta_{\pi(\phi_{\ref{infcase:6-A2}a})}-\theta_{3}^{2}\Theta_{\pi(\phi_{\text{\ref{infcase:6-A2}d}},\vartheta_{3})}-\theta_{3}\Theta_{\pi(\phi_{\text{\ref{infcase:6-A2}d}},\vartheta_{3}^{2})},\end{array}$

where
$\pi(\phi_{\ref{infcase:6-A2}a})=J_{\gamma_{2}}(1,I^{\operatorname{GL}_{2}}(\theta_{3}\otimes\theta_{3}^{-1}))$,
$\pi(\phi_{\ref{infcase:6-A2}d},\vartheta_{3})=I_{0}(G_{2}[\theta_{3}])$ and
$\pi(\phi_{\ref{infcase:6-A2}d},\vartheta_{3}^{2})=I_{0}(G_{2}[\theta_{3}^{2}])$.
The transition matrix for this system of equations is the product of the
matrix for $\langle s,\pi\rangle$, which is the character table for
$A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi_{\ref{infcase:6-A2}a}}=\langle\theta_{3}\rangle$,
and the diagonal, self-inverse matrix of signs $(-1)^{\dim(\phi)-\dim(\pi)}$:

$\begin{pmatrix}1&-1&-1\\\ 1&-\theta_{3}&-\theta^{2}_{3}\\\
1&-\theta_{3}^{2}&-\theta_{3}\end{pmatrix}=\begin{pmatrix}1&1&1\\\
1&\theta_{3}&\theta^{2}_{3}\\\
1&\theta_{3}^{2}&\theta_{3}\end{pmatrix}\begin{pmatrix}1&0&0\\\ 0&-1&0\\\
0&0&1\end{pmatrix}.$

So,

$\begin{pmatrix}1&-1&-1\\\ 1&-\theta_{3}&-\theta^{2}_{3}\\\
1&-\theta_{3}^{2}&-\theta_{3}\end{pmatrix}^{-1}=\begin{pmatrix}1&0&0\\\
0&-1&0\\\ 0&0&1\end{pmatrix}\begin{pmatrix}1/3&1/3&1/3\\\
1/3&{\bar{\theta}}_{3}/3&{\bar{\theta}}^{2}_{3}/3\\\
1/3&{\bar{\theta}}^{2}_{3}/3&{\bar{\theta}}_{3}/3\end{pmatrix}$

To see the formula for the inverse given in the Theorem is correct, it only
remains to verify that the diagonal entries in the first matrix are given by
$(-1)^{\dim(C_{\phi}^{s})-\dim(C_{\pi})}$, which is done case-by-case. All the
other case follow by the same argument. We illustrate a non-Abelian case,
$\phi=\phi_{\ref{infcase:8-sub}d}$, for which
$A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi_{\ref{infcase:8-sub}d}}=S_{3}$.
Let $s_{2}$ and $s_{3}$ be representatives for the two non-trivial conjugacy
classes in $S_{3}$, of orders $2$ and $3$, respectively. Then

$\begin{array}[]{rcl}\Theta_{\phi_{\ref{infcase:8-sub}d}}&=&\Theta_{\pi(\phi_{\ref{infcase:8-sub}d},1)}+2\Theta_{\pi(\phi_{\ref{infcase:8-sub}d},\varrho)}+\Theta_{\pi(\phi_{\ref{infcase:8-sub}d},\varepsilon)}\\\
\Theta_{\phi_{\ref{infcase:8-sub}d},s_{2}}&=&\Theta_{\pi(\phi_{\ref{infcase:8-sub}d},1)}-\Theta_{\pi(\phi_{\ref{infcase:8-sub}d},\varepsilon)}\\\
\Theta_{\phi_{\ref{infcase:8-sub}d},s_{3}}&=&\Theta_{\pi(\phi_{\ref{infcase:8-sub}d},1)}-\Theta_{\pi(\phi_{\ref{infcase:8-sub}d},\varrho)}+\Theta_{\pi(\phi_{\ref{infcase:8-sub}d},\varepsilon)},\end{array}$

where $\pi(\phi_{\ref{infcase:8-sub}d},1)=\pi(1)^{\prime}$,
$\pi(\phi_{\ref{infcase:8-sub}d},\varrho)=\pi(1)$ and
$\pi(\phi_{\ref{infcase:8-sub}d},\varepsilon)=I_{0}(G_{2}[1])$. The matrix for
this system of equations is the character table for $S_{3}$. The two examples
above illustrate that both the complex conjugation and the denominator
${|Z_{A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}}(s)|}$
are needed for $G_{2}(F)$.

Note from Table 2.6.1 that
$\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}(G_{2}(F))\to\widehat{A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}}$
is a bijection for all unramified Langlands parameters for $G_{2}(F)$ except
$\phi=\phi_{\ref{infcase:8-sub}b}$ and $\phi=\phi_{\ref{infcase:8-sub}c}$. For
$\phi=\phi_{\ref{infcase:8-sub}b}$ we have
$A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi_{\ref{infcase:8-sub}b}}=\langle\theta_{2}\rangle$
and, after picking representative for $\theta_{2}$ above
$\mathcal{S}^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi_{\ref{infcase:8-sub}b}}\to
A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi_{\ref{infcase:8-sub}b}}$,
we find

$\begin{array}[]{rcl}\Theta_{\phi_{\ref{infcase:8-sub}b}}&=&\Theta_{\pi(\phi_{\ref{infcase:8-sub}b})}-\Theta_{\pi(\phi_{\ref{infcase:8-sub}c})}+\Theta_{\pi(\phi_{\ref{infcase:8-sub}d},\varepsilon)}\\\
\Theta_{\phi_{\ref{infcase:8-sub}b},s_{2}}&=&\Theta_{\pi(\phi_{\ref{infcase:8-sub}b})}-\theta_{2}\Theta_{\pi(\phi_{\ref{infcase:8-sub}c})}+\Theta_{\pi(\phi_{\ref{infcase:8-sub}d},\varepsilon)},\end{array}$

where
$\pi(\phi_{\ref{infcase:8-sub}b})=J_{\gamma_{1}}(1/2,\mathrm{St}_{\operatorname{GL}_{2}})$,
$\pi(\phi_{\ref{infcase:8-sub}c})=J_{{\gamma_{2}}}(1/2,\mathrm{St}_{\operatorname{GL}_{2}})$
and, as above, $\pi(\phi_{\ref{infcase:8-sub}d},\varepsilon)=I_{0}(G_{2}[1])$.
While it is not possible to express
$\Theta_{\pi(\phi_{\ref{infcase:8-sub}b})}$,
$\Theta_{\pi(\phi_{\ref{infcase:8-sub}b})}$ and
$\Theta_{\pi(\phi_{\ref{infcase:8-sub}b})}$ as a linear combination of
$\Theta_{\phi_{\ref{infcase:8-sub}b}}$ and
$\Theta_{\phi_{\ref{infcase:8-sub}b},s_{2}}$, we can write
$\Theta_{\pi(\phi_{\ref{infcase:8-sub}d},\varepsilon)}$ in terms of
$\Theta_{\phi_{\ref{infcase:8-sub}d}}$ ,
$\Theta_{\phi_{\ref{infcase:8-sub}d},s_{2}}$ and
$\Theta_{\phi_{\ref{infcase:8-sub}d},s_{3}}$, as above, and then express
$\Theta_{\pi(\phi_{\ref{infcase:8-sub}b})}$,
$\Theta_{\pi(\phi_{\ref{infcase:8-sub}b})}$ and
$\Theta_{\pi(\phi_{\ref{infcase:8-sub}b})}$ in terms of
$\Theta_{\phi_{\ref{infcase:8-sub}b}}$,
$\Theta_{\phi_{\ref{infcase:8-sub}b},s_{2}}$,
$\Theta_{\phi_{\ref{infcase:8-sub}d}}$ ,
$\Theta_{\phi_{\ref{infcase:8-sub}d},s_{2}}$ and
$\Theta_{\phi_{\ref{infcase:8-sub}d},s_{3}}$.

The case $\phi=\phi_{\ref{infcase:8-sub}c}$ is similar in that there are more
representations in the packet
$\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi_{\ref{infcase:8-sub}c}}(G_{2}(F))$
than there are conjugacy classes in
$A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi_{\ref{infcase:8-sub}c}}$,
but again, we can write all the distribution characters for representations in
$\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi_{\ref{infcase:8-sub}c}}(G_{2}(F))$
using the distributions $\Theta_{\phi_{\ref{infcase:8-sub}c}}$ and
$\Theta_{\phi_{\ref{infcase:8-sub}c},s_{2}}$ together with the distributions
$\Theta_{\phi_{\ref{infcase:8-sub}d}}$,
$\Theta_{\phi_{\ref{infcase:8-sub}d},s_{2}}$ and
$\Theta_{\phi_{\ref{infcase:8-sub}d},3}$, as above.

  2. 2

This follows from calculations that are made by working through the
classification of unipotent representations of $G_{2}(F)$ appearing in Section
1.2. In Case 9 this is [CFZ:cubics]*Theorem 2.16. We give the details here of
the proof in case 7. By [Muic]*Proposition 4.2, in the Grothendieck group of
representations of $G_{2}(F)$, we have

$\begin{array}[]{rcl}M(\phi_{\ref{infcase:6-A2}a})&=&I(\nu\theta_{3}\otimes\theta_{3})=\pi(\phi_{\text{\ref{infcase:6-A2}d}})+\pi(\phi_{\text{\ref{infcase:6-A2}b}})+\pi(\phi_{\text{\ref{infcase:6-A2}c}})+\pi(\phi_{\ref{infcase:6-A2}a}),\\\
M(\phi_{\text{\ref{infcase:6-A2}b}})&=&I_{\gamma_{1}}(1/2,\theta_{3}^{-1}\otimes\mathrm{St}_{\operatorname{GL}_{2}})=\pi(\phi_{\text{\ref{infcase:6-A2}d}})+\pi(\phi_{\text{\ref{infcase:6-A2}b}}),\\\
M(\phi_{\text{\ref{infcase:6-A2}c}})&=&I_{\gamma_{1}}(1/2,\theta_{3}\otimes\mathrm{St}_{\operatorname{GL}_{2}})=\pi(\phi_{\text{\ref{infcase:6-A2}d}})+\pi(\phi_{\text{\ref{infcase:6-A2}c}}),\end{array}$

where $M(\phi)$ means the standard module of $\pi(\phi)$ for a Langlands
parameter $\phi$. From the expression of $\Theta_{\phi}$ for each $\phi$
appeared in the proof of last part, we easily see that

$\begin{pmatrix}\Theta_{\phi_{\text{\ref{infcase:6-A2}a}}}\\\
\Theta_{\phi_{\text{\ref{infcase:6-A2}b}}}\\\
\Theta_{\phi_{\text{\ref{infcase:6-A2}c}}}\\\
\Theta_{\phi_{\text{\ref{infcase:6-A2}d}}}\end{pmatrix}=\begin{pmatrix}1&-1&-1&1\\\
0&1&0&-1\\\ 0&0&1&-1\\\
0&0&0&1\end{pmatrix}\begin{pmatrix}\Theta_{M(\phi_{\text{\ref{infcase:6-A2}a}})}\\\
\Theta_{M(\phi_{\text{\ref{infcase:6-A2}b}})}\\\
\Theta_{M(\phi_{\text{\ref{infcase:6-A2}c}})}\\\
\Theta_{\phi_{\text{\ref{infcase:6-A2}d}}}\end{pmatrix}.$

Note that each standard module is stable since it is a representation either
induced from $\operatorname{GL}_{2}(F)$ or
$T=\operatorname{GL}_{1}(F)\times\operatorname{GL}_{1}(F)$. This shows that if
$\Theta_{\phi_{\text{\ref{infcase:6-A2}d}}}$ is stable, then each
$\Theta_{\phi}$ is stable for every $\phi$ in Case 7. For the rest cases, the
proof is similar and thus omitted.

∎

######  Remark 2.15.

The appearance of $\dim(C_{\phi}^{s})$ in Theorem 4.7 is a foreshadowing of
the geometric constructions in Section 4 on geometric endoscopy.

######  Remark 2.16.

It is widely expected that $\Theta_{\phi}$ is stable when $\phi$ is elliptic,
but it seems that this has not been proved; see [CFZ:cubics]*Remark 2.17. Note
that, if confirmed 2, could then be strengthened to: $\Theta_{\phi}$ is stable
for all unramified $\phi$.

##  3\. Unipotent representations of endoscopic groups for $G_{2}$

###  3.1. Endoscopic groups for $G_{2}$

Let $(G,s,\xi)$ be an endoscopic triple for $G_{2}$. Since
$s\in{\widehat{G_{2}}}$ is semisimple we choose $s\in{\widehat{T}}$. Since
$Z_{{\widehat{G_{2}}}}(s)$ is connected, $G$ is split over $F$ and
$\xi:{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G}}\to{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G_{2}}}$
is given by
${\widehat{G}}{\,:=\,}Z_{{\widehat{G_{2}}}}(s)\hookrightarrow{\widehat{G_{2}}}$;
see [Kottwitz-Shelstad]*(2.1.4a) and (2.1.4b). The endoscopic triples for
$G_{2}$ are naturally arranged into the following six families, of which the
last three are elliptic.

  1. (1)

$G=T=\operatorname{GL}_{1}^{2}$, $s={\hat{m}}(x,y)$ regular, $x^{-1}y^{2}\neq
1$, $xy^{-1}\neq 1$, $x\neq 1$, $y\neq 1$, $x^{2}y^{-1}\neq 1$, $xy\neq 1$;

  2. (2)

$G=\operatorname{GL}_{2}^{\hat{\gamma}_{2}}$, $s={\hat{m}}(x,x)$, $x^{2}\neq
1$;

  3. (3)

$G=\operatorname{GL}_{2}^{\hat{\gamma}_{1}}$, $s={\hat{m}}(x^{2},x)$,
$x^{2}\neq 1$, $x^{3}\neq 1$;

  4. (4)

$G=\operatorname{SO}_{4}$ split over $F$, $s={\hat{m}}(1,-1)$;

  5. (5)

$G=\operatorname{PGL}_{3}$, $s={\hat{m}}(\theta_{3}^{2},\theta_{3})$;

  6. (6)

$G=G_{2}$ over $F$ (necessarily split), $s=1$.

In Sections 3.3, 3.4 and 3.5 we study the unipotent representations of
$\operatorname{GL}_{2}$, $\operatorname{SO}_{4}$ and $\operatorname{PGL}_{3}$
and their inner forms, respectively, including the Langlands correspondence
and ABV-packets for these representations.

We will also study distributions attached to ABV-packets, for which we require
the following definition, related closely to [CFMMX]*Definition 3.

Table 3.1.1. Endoscopic relations for $G_{2}$. The dotted arrows
$G\dashrightarrow G^{\prime}$ indicate that $G$ is an endoscopic group for
$G^{\prime}$ and therefore the existence of an injective admissible
homomorphism
$\xi:{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G}}\to{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G^{\prime}}}$.
There $T^{\prime}$ is the form of the torus $T=\operatorname{GL}_{1}^{2}$ that
splits over the quadratic unramified extension of $F$.

${G_{2}(F)}$${\operatorname{PGL}_{3}^{\delta}(F)}$${\operatorname{PGL}_{3}(F)}$${\operatorname{SO}_{4}(F)}$${\operatorname{SO}_{4}^{\delta}(F)}$${\operatorname{PGL}_{3}^{\delta^{\prime}}(F)}$${\operatorname{GL}_{2}^{\hat{\gamma}_{1}}(F)}$${\operatorname{GL}_{2}^{\hat{\gamma}_{2}}(F)}$${T(F)}$${T^{\prime}(F)}$

Let $(G,s,\xi)$ be an endoscopic triple for $G_{2}$. Recall from
[CFMMX]*Section 8.1, Definition 1 that
$\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}(G/F)$ is
a subset of $\Pi^{\mathrm{pure}}(G/F)$, thus contained in the set of
equivalence classes of irreducible admissible representations of $G(F)$ and
all its pure inner forms. Let $\delta\in Z^{1}(F,G)$ be a pure inner form of
$G$; recall that $G^{\delta}$ denotes form of $G$ determined by the image of
$\delta$ under $Z^{1}(F,G)\to Z^{1}(F,\text{Aut}(G))$. Now let
$\phi:W^{\prime}_{F}\to{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G}}$ be an unramified
Langlands parameter. It is possible that the $L$-packet
$\Phi_{\phi}(G^{\delta}(F))$ is empty, since we have not demanded that the
Langlands parameter
$\phi:W^{\prime}_{F}\to{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G}}$ is relevant to
$G(F)$ in the sense of [Borel:Automorphic]*Section 8.3 (ii). If $\phi$ is
relevant to $G^{\delta}(F)$, then $\Phi_{\phi}(G^{\delta}(F))$ is non-empty.

We write
$\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}(G^{\delta}(F))$
for the subset of
$\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}(G/F)$
that contains only representations of $G^{\delta}(F)$. Again, it is possible
that
$\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}(G^{\delta}(F))$
is empty when $\phi$ is not relevant to $G^{\delta}(F)$.

######  Definition 5.

Let $\phi:W^{\prime}_{F}\to{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G}}$ be an
unramified Langlands parameter; let $s$ be an element of
$\mathcal{S}^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\xi\circ\phi}$.
Set

$\Theta_{\phi,s}^{G^{\delta}}{\,:=\,}e(\delta)\sum_{\pi\in\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}(G^{\delta}(F))}\hskip
0.0pt\operatorname{trace}_{s}\left(\operatorname{\mathsf{N}\hskip-1.0pt\mathsf{E}\hskip-1.0pt\mathsf{v}\hskip-1.0pt\mathsf{s}}_{\phi}[\dim(\phi)]\mathcal{P}(\pi)[-\dim(\pi)]\right)\
\Theta_{\pi}.$

When written using ABV-packet coefficients for $G^{\delta}$ using Definition
1, the definition of $\Theta_{\phi,s}^{G^{\delta}}$, above, takes the form:

$\Theta_{\phi,s}^{G^{\delta}}=e(\delta)\sum_{\pi\in\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}(G^{\delta}(F))}\hskip
0.0pt(-1)^{\dim(\phi)-\dim(\pi)}\ \langle s,\pi\rangle\ \Theta_{\pi}.$

We also set $\Theta_{\phi}^{G^{\delta}}{\,:=\,}\Theta_{\phi,1}^{G^{\delta}}$.

###  3.2. Unipotent representations of $T=\operatorname{GL}_{1}^{2}$

Unipotent representations of $T(F)$ are simply unramified characters
$\chi_{1}\otimes\chi_{2}:T(F)\to{\mathbb{C}}^{\times}$. The Langlands
parameter for this character is $W^{\prime}_{F}\to{\widehat{T}}$ defined by
$(w,x)\mapsto\varphi_{1}(w)\times\varphi_{2}(w)$ where $\varphi_{i}$
corresponds to $\chi_{i}$ under class field theory.

###  3.3. Unipotent representations of $\operatorname{GL}_{2}$

The category of unipotent representations of the group
$\operatorname{GL}_{2}(F)$ is precisely the category of unramified principal
series representations of $\operatorname{GL}_{2}(F)$, corresponding therefore
to the subcategory of smooth representations of $\operatorname{GL}_{2}(F)$
appearing in the Bernstein decomposition indexed by the inertial class of the
trivial representation of $T(F)$. In particular, there are no supercuspidal
unipotent representations of $\operatorname{GL}_{2}(F)$. In this section we
partition the set $\Pi(\operatorname{GL}_{2}(F))_{\text{unip}}$ of unipotent
representations of $\operatorname{GL}_{2}(F)$ into $L$-packets and also
describe the corresponding $L$-parameters, thus giving the Langlands
correspondence for these representations. Since this case is very well known,
we omit all proofs here.

Table 3.3.1. $L$-packets for unipotent representations of
$\operatorname{GL}_{2}(F)$: each row in an $L$-packet. The notation and the
corresponding $L$-parameters are explained below.

$\begin{array}[]{| c | c | l | }\hline\cr\text{$L$-parameter}&\text{$L$-packet}&\text{Arthur}\\\ \phi&\Pi_{\phi}(\operatorname{GL}_{2}(F))&\text{type?}\\\ \hline\cr\hline\cr\phi_{\ref{ssec:GL2}.0}&\text{Ind}_{B_{2}(F)}^{\operatorname{GL}_{2}(F)}(\chi_{1}\otimes\chi_{2})&\text{unitary $\chi_{1}$, $\chi_{2}$}\\\ \hline\cr\hline\cr\phi_{\ref{ssec:GL2}.1a}&(\chi\circ\det)\otimes{\mathbbm{1}}_{\operatorname{GL}_{2}}&\text{unitary $\chi$ }\\\ \phi_{\ref{ssec:GL2}.1b}&(\chi\circ\det)\otimes\mathrm{St}_{\operatorname{GL}_{2}}&\text{unitary $\chi$}\\\ \hline\cr\end{array}$

We fix the standard maximal split torus $T(F)$ of $\operatorname{GL}_{2}(F)$.
Let $B_{2}(F)$ be the standard Borel subgroup of $\operatorname{GL}_{2}(F)$.

  1. 3.3.0

Let $\chi_{1}$ and $\chi_{2}$ be unramified complex characters of
$\operatorname{GL}_{1}(F)=F^{\times}$ and let $\chi_{1}\otimes\chi_{2}$ be the
associated character of $T(F)$. Then the parabolically induced representation

$I^{\operatorname{GL}_{2}}(\chi_{1}\otimes\chi_{2}){\,:=\,}\text{Ind}_{B_{2}(F)}^{\operatorname{GL}_{2}(F)}(\chi_{1}\otimes\chi_{2})$

is irreducible if and only if $\chi_{1}\chi_{2}^{-1}\neq\nu^{\pm 1}$, where
$\nu$ is the unramified character of $\operatorname{GL}_{1}(F)$ defined by
$\nu(\varpi)=q$. In this case,
$\text{Ind}_{B_{2}(F)}^{\operatorname{GL}_{2}(F)}(\chi_{1}\otimes\chi_{2})$ is
its own $L$-packet. The $L$-parameter
$\phi:W^{\prime}_{F}\to{\widehat{\operatorname{GL}_{2}}}$ for this irreducible
representation is

$\phi_{\ref{ssec:GL2}.0}(w,x)=\begin{pmatrix}\varphi_{1}(w)&0\\\
0&\varphi_{2}(w)\end{pmatrix}$

where $\varphi_{i}:W_{F}\to{\widehat{\operatorname{GL}_{1}}}$ is the character
corresponding to $\chi_{i}:\operatorname{GL}_{1}(F)\to{\mathbb{C}}^{\times}$
under local class field theory. This $L$-parameter is of Arthur type if and
only if $\chi_{1}$ and $\chi_{2}$ are unitary.

  2. 3.3.1

Now suppose $\chi_{1}$ and $\chi_{2}$ are unramified and
$\chi_{1}\chi_{2}^{-1}=\nu$. Then we may write $\chi_{1}=\chi\nu^{1/2}$ and
$\chi_{2}=\chi\nu^{-1/2}$, in which case

$\text{Ind}_{B_{2}(F)}^{\operatorname{GL}_{2}(F)}(\chi\nu^{1/2}\otimes\chi\nu^{-1/2})=(\chi\circ\det)\otimes\text{Ind}_{B_{2}(F)}^{\operatorname{GL}_{2}(F)}(\nu^{1/2}\otimes\nu^{-1/2}).$

The unique irreducible sub-representation of
$\text{Ind}_{B_{2}(F)}^{\operatorname{GL}_{2}(F)}(\nu^{1/2}\otimes\nu^{-1/2})$
is the Steinberg representation, $\mathrm{St}_{\operatorname{GL}_{2}}$,
whereas the trivial representation,
${\mathbbm{1}}_{\operatorname{GL}_{2}(F)}$, is the unique irreducible
quotient. The $L$-parameter for
$(\chi\circ\det)\otimes{\mathbbm{1}}_{\operatorname{GL}_{2}(F)}$ is

$\phi_{\ref{ssec:GL2}.1a}(w,x)=\varphi(w)\begin{pmatrix}{|w|}^{1/2}&0\\\
0&{|w|}^{-1/2}\end{pmatrix},$

where $\varphi:W_{F}\to\mathbb{C}^{\times}$ is the character corresponding to
$\chi:F^{\times}\to{\mathbb{C}}^{\times}$ under class field theory. The
$L$-parameter for $(\chi\circ\det)\otimes\mathrm{St}_{\operatorname{GL}_{2}}$
is

$\phi_{\ref{ssec:GL2}.1b}(w,x)=\varphi(w)x.$

Both $L$-parameters appearing in this case are of Arthur type if and only if
$\chi$ is unitary. We remark that
$(\chi\circ\det)\otimes{\mathbbm{1}}_{\operatorname{GL}_{2}(F)}$ and
$(\chi\circ\det)\otimes\mathrm{St}_{\operatorname{GL}_{2}}$ are interchanged
by Aubert-Zelevinski duality.

This completes the description of the Langlands correspondence for unipotent
representations of $\operatorname{GL}_{2}(F)$. Each $L$-packet for
$\operatorname{GL}_{2}(F)$ is its own ABV-packet.

Theorems 2.2 and 2.14 are largely trivial in the case of
$\operatorname{GL}_{2}(F)$ so we refrain from stating them here except to say
that [CFMMX]*Conjecture 1 is true for unramified representations of
$\operatorname{GL}_{2}(F)$.

######  Remark 3.1.

The category of unipotent representations of $\operatorname{GL}_{2}(F)$ is
equivalent to the category of modules over the affine Hecke algebra for
$\operatorname{GL}_{2}(F)$. As such, Table 3.3.1 lists the simple objects in
this module category.

###  3.4. Unipotent representations of $\operatorname{SO}_{4}$ and its pure
inner forms

In this section we find the ABV-packets and ABV-packet coefficients for all
unipotent representations of $\operatorname{SO}_{4}$ and its pure inner forms.
The $p$-adic group $\operatorname{SO}_{4}$ has two pure inner forms, since
$H^{1}(F,\operatorname{SO}_{4})$ has order $2$, and
$H^{1}(F,\operatorname{SO}_{4})\to H^{1}(F,\text{Aut}(\operatorname{SO}_{4}))$
is injective. Let $\delta\in Z^{1}(F,\operatorname{SO}_{4})$ be a
representative for the non-trivial class in $H^{1}(F,\operatorname{SO}_{4})$.
Let $\operatorname{SO}_{4}^{\delta}$ be the inner form of
$\operatorname{SO}_{4}$ attached to this cocycle. We partition the set
$\Pi(\operatorname{SO}_{4}(F))_{\text{unip}}$ of unipotent representations of
the split orthogonal group $\operatorname{SO}_{4}(F)$ into $L$-packets and
also describe the corresponding $L$-parameters. We do the same for the non-
split inner form $\operatorname{SO}_{4}^{\delta}$ of $\operatorname{SO}_{4}$.
The group $\operatorname{SO}_{4}^{\delta}$ is the orthogonal group for the
quadratic space coming from the norm on the unique quaternion division algebra
over $F$.

Table 3.4.1. ABV-packets for all unipotent representations of
$\operatorname{SO}_{4}(F)$ and its inner form
$\operatorname{SO}_{4}^{\delta}(F)$

$\begin{array}[]{| c | c | c | l | }\hline\cr\text{$L$-parameter}&\text{Pure $L$-packet}&\text{Coronal}&\text{Parameter of}\\\ \phi&\Pi^{\text{pure}}_{\phi}(\operatorname{SO}_{4}/F)&\text{representations}&\text{Arthur type?}\\\ \hline\cr\hline\cr\phi_{\ref{ssec:SO4}.0}&I^{\operatorname{SO}_{4}}(\chi_{1}\otimes\chi_{2})&&\text{unitary $\chi_{1}$, $\chi_{2}$}\\\ \hline\cr\hline\cr\phi_{\ref{ssec:SO4}.0^{\prime}}&\pi_{4},\pi_{4}^{\prime}&&\text{yes}\\\ \hline\cr\hline\cr\phi_{\ref{ssec:SO4}.1a}&I_{\beta_{1}}(\chi\circ\det)&&\text{unitary $\chi$}\\\ \phi_{\ref{ssec:SO4}.1b}&I_{\beta_{1}}(\chi\otimes\mathrm{St}_{\operatorname{GL}_{2}})&&\text{unitary $\chi$}\\\ \hline\cr\hline\cr\phi_{\ref{ssec:SO4}.2a}&I_{\beta_{2}}(\chi\circ\det)&&\text{unitary $\chi$}\\\ \phi_{\ref{ssec:SO4}.2b}&I_{\beta_{2}}(\chi\otimes\mathrm{St}_{\operatorname{GL}_{2}})&&\text{unitary $\chi$}\\\ \hline\cr\hline\cr\phi_{\ref{ssec:SO4}.3a}&\chi_{\operatorname{SO}_{4}(F)}&\chi_{\operatorname{SO}_{4}^{\delta}(F)}&\text{yes}\\\ \phi_{\ref{ssec:SO4}.3b}&J_{\beta_{1}}(1/2,\chi\otimes\mathrm{St}_{\operatorname{GL}_{2}})&\chi_{\operatorname{SO}_{4}^{\delta}(F)}&\text{yes}\\\ \phi_{\ref{ssec:SO4}.3c}&J_{\beta_{2}}(1/2,\chi\otimes\mathrm{St}_{\operatorname{GL}_{2}})&\chi_{\operatorname{SO}_{4}^{\delta}(F)}&\text{yes}\\\ \phi_{\ref{ssec:SO4}.3d}&\chi_{\operatorname{SO}_{4}(F)}\otimes\mathrm{St}_{\operatorname{SO}_{4}},\chi_{\operatorname{SO}_{4}^{\delta}(F)}&&\text{yes}\\\ \hline\cr\end{array}$

#### Langlands correspondence

The local Langlands correspondence for $\operatorname{SO}_{4}(F)$ may be
assembled as follows. We start with the observation that
$\operatorname{GSO}_{4}(F)\cong(\operatorname{GL}_{2}(F)\times\operatorname{GL}_{2}(F))/\Delta(F^{\times})$,
see [Gross-Prasad]*(15.1) for example, and therefore the local Langlands
correspondence for $\operatorname{GSO}_{4}(F)$ follows from that of
$\operatorname{GL}_{2}(F)$. By the general theory of [Gelbart-Knapp] and [Gan-
Takeda], the local Langlands correspondence for $\operatorname{SO}_{4}(F)$ can
be obtained in the following way. Notice that
$\widehat{\operatorname{GSO}_{4}(F)}\cong\operatorname{GSpin}_{4}(\mathbb{C})$
and
$\widehat{\operatorname{SO}_{4}(F)}\cong\operatorname{SO}_{4}(\mathbb{C})$.
Let
$\mathrm{std}:\operatorname{GSpin}_{4}(\mathbb{C})\to\operatorname{SO}_{4}(\mathbb{C})$
be the standard homomorphism. Let
$\phi:W_{F}\times\operatorname{SL}_{2}(\mathbb{C})\to\operatorname{SO}_{4}(\mathbb{C})$
be a local Langlands parameter for $\operatorname{SO}_{4}(F)$ and let
$\widetilde{\phi}$ be a local Langlands parameter for
$\operatorname{GSO}_{4}(F)$ such that $\mathrm{std}\circ\widetilde{\phi}=\phi$
and let $\Pi_{\widetilde{\phi}}(\operatorname{GSO}_{4}(F))$ be the
corresponding local $L$-packet for $\operatorname{GSO}_{4}(F)$, which is a
singleton. Assume that
$\Pi_{\widetilde{\phi}}(\operatorname{GSO}_{4}(F))=\\{\widetilde{\pi}\\}$.
Then $\widetilde{\pi}|_{\operatorname{SO}_{4}}$ is multiplicity free and
$\Pi_{\phi}(\operatorname{SO}_{4}(F))=\mathrm{JH}(\widetilde{\pi})$, where
$\mathrm{JH}(\widetilde{\pi})=\\{\mathrm{constituents~{}of~{}}\widetilde{\pi}|_{\operatorname{SO}_{4}(F)}\\}$.
Consider the subgroup
$F^{\times}\cdot\operatorname{SO}_{4}(F)\subset\operatorname{GSO}_{4}(F)$, we
have $\operatorname{GSO}_{4}(F)/(F^{\times}\cdot\operatorname{SO}_{4}(F))\cong
F^{\times}/F^{\times,2}$. By [Gelbart-Knapp]*Lemma 2.1, we have

$\big{|}\Pi_{\phi}(\operatorname{SO}_{4}(F))\big{|}=\big{|}\\{\chi:F^{\times}/F^{\times,2}\rightarrow{\mathbb{C}}^{\times}:\widetilde{\pi}\otimes\chi=\widetilde{\pi}\\}\big{|}.$

This allows us to find the local Langlands correspondence for unipotent
representations of $\operatorname{SO}_{4}(F)$ explicitly, as follows.

We realize $\operatorname{GSO}_{4}(F)$ and $\operatorname{SO}_{4}(F)$ as
$\operatorname{GSO}_{4}(F)=\\{g\in\operatorname{GL}_{4}(F):g^{t}Jg=\lambda(g)J,\lambda(g)\in
F^{\times}\\},$ and
$\operatorname{SO}_{4}(F)=\\{g\in\operatorname{GSO}_{4}(F):\lambda(g)=1\\}$,
where

$J=\begin{pmatrix}0&0&0&1\\\ 0&0&1&0\\\ 0&1&0&0\\\ 1&0&0&0\end{pmatrix}.$

We denote the two simple roots of $\operatorname{SO}_{4}$ (and
$\operatorname{GSO}_{4}$) by $\beta_{1}$ and $\beta_{2}$ and let $r\mapsto
x_{\beta_{i}}(r)$ be the one parameter subgroup associated with $\beta_{i}$,
which can be realized as

$x_{\beta_{1}}(r)=\begin{pmatrix}1&r&0&0\\\ 0&1&0&0\\\ 0&0&1&-r\\\
0&0&0&1\end{pmatrix},\quad x_{\beta_{2}}(r)=\begin{pmatrix}1&0&r&0\\\
0&1&0&-r\\\ 0&0&1&0\\\ 0&0&0&1\end{pmatrix}.$

For $i=1,2$, the simple root $\beta_{i}$ define an embedding
$\iota_{\delta_{i}}:\operatorname{GL}_{2}(F)\to\operatorname{GSO}_{4}(F)$
which can be explicitly described as

$\iota_{\beta_{1}}\left(\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\right)=\begin{pmatrix}a&b&0&0\\\ c&d&0&0\\\ 0&0&a&-b\\\
0&0&-c&d\end{pmatrix},\quad x_{\beta_{2}}\left(\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\right)=\begin{pmatrix}a&0&b&0\\\ 0&a&0&-b\\\ c&0&d&0\\\
0&-c&0&d\end{pmatrix}.$

Note that $\lambda(\iota_{\beta_{i}}(g))=\det(g)$ for
$g\in\operatorname{GL}_{2}(F)$. In particular, $\iota_{\beta_{i}}$ defines an
embedding
$\iota_{\beta_{i}}:\operatorname{SL}_{2}(F)\to\operatorname{SO}_{4}(F)$.
Moreover, the map $\iota_{\beta_{i}}$ define an isomorphism
$\operatorname{GSO}_{4}(F)\cong(\operatorname{GL}_{2}(F)\times\operatorname{GL}_{2}(F))/(\Delta(F^{\times}))$,
where $\Delta(F^{\times})=\\{(a,a):a\in
F^{\times}\\}\subset\operatorname{GL}_{2}(F)\times\operatorname{GL}_{2}(F)$.
Thus an irreducible representation of $\operatorname{GSO}_{4}(F)$ is of the
form $\pi_{1}\boxtimes\pi_{2}$ with $\omega_{\pi_{1}}\cdot\omega_{\pi_{2}}=1$,
where $\pi_{i}$ is an irreducible representation of $\operatorname{GL}_{2}(F)$
and $\omega_{\pi_{i}}$ is the central character of $\pi_{i}$. Let $\chi$ be a
character of $F^{\times}$, we denote
$I^{\operatorname{GL}_{2}}(\chi)=\text{Ind}_{B_{\operatorname{GL}_{2}}}^{\operatorname{GL}_{2}(F)}(\chi\otimes\chi^{-1})$.

Let $T_{\operatorname{SO}_{4}}$ be the maximal torus of
$\operatorname{SO}_{4}(F)$ which consists of elements of the form
$t(a,b):=\operatorname{diag}(a,b,b^{-1},a^{-1})$ for $a,b\in F^{\times}$. Let
$B_{\operatorname{SO}_{4}}$ be the upper triangular Borel subgroup of
$\operatorname{SO}_{4}(F)$. Let $\chi_{1},\chi_{2}$ be a pair of characters of
$F^{\times}$. We denote $\chi_{1}\otimes\chi_{2}$ the character of
$T_{\operatorname{SO}_{4}}$ defined by
$\chi_{1}\otimes\chi_{2}(t(a,b))=\chi_{1}(a)\chi_{2}(b)$. Denote
$I^{\operatorname{SO}_{4}}(\chi_{1}\otimes\chi_{2}){\,:=\,}\text{Ind}_{B_{\operatorname{SO}_{4}}}^{\operatorname{SO}_{4}(F)}(\chi_{1}\otimes\chi_{2})$.
Note that $I^{\operatorname{SO}_{4}}(\chi_{1}\otimes\chi_{2})\cong
I^{\operatorname{SO}_{4}}(\chi_{2}\otimes\chi_{1})$ in the Grothendieck group
of representations of $\operatorname{SO}_{4}(F)$.

Up to conjugacy, the group $\operatorname{SO}_{4}$ has 2 maximal parabolics
$P_{i}=M_{i}N_{i}$ with $x_{\beta_{i}}\in M_{i}$ for $i=1,2$, where $M_{i}$ is
the Levi subgroup of $P_{i}$. Note that $M_{i}\cong\operatorname{GL}_{2}(F)$.
Let $\sigma$ be an irreducible smooth representation of
$\operatorname{GL}_{2}(F)\cong M_{i}$. Denote
$I_{P_{i}}(\sigma):=\text{Ind}_{P_{i}}^{\operatorname{SO}_{4}(F)}(\sigma)$.

The dual group $\operatorname{SO}_{4}(\mathbb{C})$ is realized in a similar
way as $\operatorname{SO}_{4}(F)$ described above. In particular, if we call
$\widehat{\beta}_{i}$ the root of $\operatorname{SO}_{4}(\mathbb{C})$ which is
dual to $\beta_{i}$, then the root space $x_{\hat{\beta}_{i}}$ is described
using the same matrix as $x_{\beta_{i}}$. Similarly,
$\iota_{\widehat{\beta}_{i}}$ defines an embedding
$\operatorname{SL}_{2}(\mathbb{C})\to\operatorname{SO}_{4}(\mathbb{C})$.
Denote
${\hat{t}}(a,b)=\operatorname{diag}(a,b,b^{-1},a^{-1})\in\operatorname{SO}_{4}(\mathbb{C})$.

  1. 3.4.0

Let $\chi_{1}$ and $\chi_{2}$ be unramified characters of $F^{\times}$ with
$\chi_{1}\chi_{2},\chi_{1}\chi_{2}^{-1}\neq\nu^{\pm 1}$, we consider the
representation $I^{\operatorname{SO}_{4}}(\chi_{1}\otimes\chi_{2})$. Then

$I^{\operatorname{SO}_{4}}(\chi_{1}\otimes\chi_{2})=(I^{\operatorname{GL}_{2}}(\chi^{\prime}_{1})\boxtimes
I^{\operatorname{GL}_{2}}(\chi^{\prime}_{2}))|_{\operatorname{SO}_{4}(F)},$

where $\chi^{\prime}_{1},\chi^{\prime}_{2}$ are unramified characters of
$F^{\times}$ such that
$\chi_{1}=\chi^{\prime}_{1}\chi^{\prime}_{2},\chi_{2}=\chi^{\prime}_{2}/\chi^{\prime}_{1}$.
We remark that $(\chi^{\prime}_{1},\chi^{\prime}_{2})$ are uniquely determined
by $(\chi_{1},\chi_{2})$ up to a twist by $\theta_{2}$, where $\theta_{2}$ is
the unique unramified quadratic character of $F^{\times}$. There are two cases
to consider.

    1. (i)

Suppose $(\chi_{1},\chi_{2})\neq(\theta_{2},1)$ and
$(\chi_{1},\chi_{2})\neq(1,\theta_{2})$. Then the representation
$I^{\operatorname{SO}_{4}}(\chi_{1}\otimes\chi_{2})$ is irreducible. The
Langlands parameter is given by

$\phi_{\ref{ssec:SO4}.0}(w,x)={\hat{t}}(\varphi_{1}(w),\varphi_{2}(w)),$

where $\varphi_{i}:W_{F}\to\mathbb{C}^{\times}$ is the character dual to
$\chi_{i}$, and

$\Pi_{\phi_{\ref{ssec:SO4}.0}}(\operatorname{SO}_{4}(F))=\\{I^{\operatorname{SO}_{4}}(\chi_{1}\otimes\chi_{2})\\}.$

The moduli space of Langlands parameters with the infinitesimal parameter
$\lambda_{\ref{ssec:SO4}.0}$ for $\phi_{\ref{ssec:SO4}.0}$ is 1:
$V_{\lambda_{\ref{ssec:SO4}.0}}=\\{0\\}$ and where
$H_{\lambda_{\ref{ssec:SO4}.0}}$ is equal to: $\operatorname{GL}_{2}$, when
exactly one of $\varphi_{1}\varphi_{2},\varphi_{1}\varphi_{2}^{-1}$ is 1;
$\operatorname{SO}_{4}$, when
$\varphi_{1}\varphi_{2}=\varphi_{1}\varphi_{2}^{-1}=1$;
${\widehat{T}}=\operatorname{GL}_{1}^{2}$ in all other cases where
$(\varphi_{1},\varphi_{2})\neq(\theta_{2},1)$ and
$(\varphi_{1},\varphi_{2})\neq(1,\theta_{2})$.

    2. (ii)

Suppose $(\chi_{1},\chi_{2})=(\theta_{2},1)$ or
$(\chi_{1},\chi_{2})=(1,\theta_{2})$. Then

$I^{\operatorname{SO}_{4}}(\theta_{2}\otimes
1)=I^{\operatorname{SO}_{4}}(1\otimes\theta_{2})$

and this representation is a direct sum of two irreducible tempered
representations of $\operatorname{SO}_{4}(F)$. We denote these two
representations by $\pi_{4}$ and $\pi_{4}^{\prime}$. Only one of
$\pi_{4},\pi_{4}^{\prime}$ is
$\operatorname{SO}_{4}({\mathcal{O}}_{F})$-spherical, i.e., has a nonzero
vector fixed by the hyperspecial group
$\operatorname{SO}_{4}({\mathcal{O}}_{F})$, where ${\mathcal{O}}_{F}$ is the
ring of integers of $F$, and we choose notation so that $\pi_{4}$ is
$\operatorname{SO}_{4}({\mathcal{O}}_{F})$-spherical. The Langlands parameter
for $\pi_{4}$ and $\pi^{\prime}_{4}$ is

$\phi_{\ref{ssec:SO4}.0^{\prime}}(w,x)={\hat{t}}(\varphi_{1}(w),\varphi_{2}(w)),$

where $\varphi_{i}$ corresponds to $\chi_{i}$ under class field theory. and

$\Pi_{\phi_{\ref{ssec:SO4}.0^{\prime}}}(\operatorname{SO}_{4}(F))=\\{\pi_{4},\pi^{\prime}_{4}\\}.$

The moduli space of Langlands parameters with the infinitesimal parameter
$\lambda_{\ref{ssec:SO4}.0^{\prime}}$ for $\phi_{\ref{ssec:SO4}.0^{\prime}}$
is 1: $V_{\lambda_{\ref{ssec:SO4}.0^{\prime}}}=\\{0\\}$ where
$H_{\lambda_{\ref{ssec:SO4}.0}}$ is the disconnected group ${\rm S}({\rm
O}_{2}\times{\rm O}_{2})$, namely, the group generated by ${\widehat{T}}$ and
$J$.

  2. 3.4.1

Let $\chi$ be an unramified character of $F^{\times}$ with
$\chi^{2}\neq\nu^{\pm 1}$. The irreducible components of
$I(\chi\nu^{1/2}\otimes\chi\nu^{-1/2})$ are
$I_{\beta_{1}}(\chi\otimes\mathrm{St}_{\operatorname{GL}_{2}})$ and
$I_{\beta_{1}}(\chi\circ\det)$, where
$I_{\beta_{1}}(\chi\otimes\mathrm{St}_{\operatorname{GL}_{2}})=(\mathrm{St}_{\operatorname{GL}_{2}}\boxtimes
I^{\operatorname{GL}_{2}}(\chi))|_{\operatorname{SO}_{4}(F)}$ and
$I_{\beta_{1}}(\chi\circ\det)=(1_{\operatorname{GL}_{2}}\boxtimes
I^{\operatorname{GL}_{2}}(\chi))|_{\operatorname{SO}_{4}(F)}$. The local
Langlands parameter of $I_{\beta_{1}}(\chi\circ\det)$ is given by

$\phi_{\ref{ssec:SO4}.1a}(w,x)={\hat{t}}(\varphi(w)|w|^{1/2},\varphi(w)|w|^{-1/2}),$

and the Langlands parameter of
$I_{\beta_{1}}(\chi\otimes\mathrm{St}_{\operatorname{GL}_{2}})$ is given by

$\phi_{\ref{ssec:SO4}.1b}(w,x)\to{\hat{t}}(\varphi(w),\varphi(w))\iota_{\widehat{\beta}_{1}}(x),$

where $\varphi:W_{F}\to\mathbb{C}^{\times}$ is the dual character of
$\chi:F^{\times}\to\mathbb{C}^{\times}$. The Vogan variety
$V_{\lambda_{\ref{ssec:SO4}.1}}$ is the root space
$\mathfrak{u}_{\hat{\beta}_{1}}$ with
$H_{\lambda_{\ref{ssec:SO4}.1}}={\widehat{T}}$ action given by $t\cdot
x=\beta_{1}(t)x$. This is equivalent to 2, and the two orbits in this
prehomogenous vector space, $C_{0}$ and $C_{1}$, match
$\phi_{\ref{ssec:SO4}.1a}$ and $\phi_{\ref{ssec:SO4}.1b}$, respectively.

  3. 3.4.2

Let $\chi$ be an unramified character with $\chi^{2}\neq\nu^{\pm 1}$. The
irreducible components of $I(\chi\nu^{1/2}\otimes\chi^{-1}\nu^{1/2})$ are
$I_{\beta_{2}}(\chi\otimes\mathrm{St}_{\operatorname{GL}_{2}})$ and
$I_{\beta_{2}}(\chi\circ\det)$ where
$I_{\beta_{2}}(\chi\otimes\mathrm{St}_{\operatorname{GL}_{2}})=(I^{\operatorname{GL}_{2}}(\chi)\boxtimes\mathrm{St}_{\operatorname{GL}_{2}})|_{\operatorname{SO}_{4}(F)}$
and $I_{\beta_{2}}(\chi\circ\det)=(I^{\operatorname{GL}_{2}}(\chi)\boxtimes
1_{\operatorname{GL}_{2}})|_{\operatorname{SO}_{4}(F)}$. The local Langlands
parameter for $I_{\beta_{2}}(\chi\circ\det)$ is

$\phi_{\ref{ssec:SO4}.2a}\left(w,x\right)={\hat{t}}(\varphi(w)|w|^{1/2},\varphi(w)^{-1}|w|^{1/2}),$

and the Langlands parameter for
$I_{\beta_{2}}(\chi\otimes\mathrm{St}_{\operatorname{GL}_{2}})$ is

$\phi_{\ref{ssec:SO4}.2b}\left(w,\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\right)={\hat{t}}(\varphi(w),\varphi(w)^{-1})\iota_{\widehat{\beta}_{2}}(x),$

where $\varphi:W_{F}\to\mathbb{C}^{\times}$ is the dual character of $\chi$.
The Vogan variety $V_{\lambda_{\ref{ssec:SO4}.2}}$ is the root space
$\mathfrak{u}_{\hat{\beta}_{2}}$ with
$H_{\lambda_{\ref{ssec:SO4}.2}}={\widehat{T}}$ action given by $t\cdot
x=\widehat{\beta}_{2}(t)x$. This is equivalent to 2, and the two orbits in
this prehomogenous vector space, $C_{0}$ and $C_{1}$, match
$\phi_{\ref{ssec:SO4}.2a}$ and $\phi_{\ref{ssec:SO4}.2b}$, respectively.

  4. 3.4.3

Let $\chi$ be an unramified character of $F^{\times}$ such that $\chi^{2}=1$.
Thus $\chi$ is either trivial or $\theta_{2}$. The representation
$I(\chi\nu\otimes\chi)$ has 4 irreducible components:

$\chi_{\operatorname{SO}_{4}(F)},\
J_{\beta_{1}}(1/2,\chi\otimes\mathrm{St}_{\operatorname{GL}_{2}}),\
J_{\beta_{2}}(1/2,\chi\otimes\mathrm{St}_{\operatorname{GL}_{2}}),\
\chi_{\operatorname{SO}_{4}(F)}\otimes\mathrm{St}_{\operatorname{SO}_{4}},$

where $\chi_{\operatorname{SO}_{4}}$ is the character $\chi$ composed with the
spinor norm of $\operatorname{SO}_{4}(F)$,
$\mathrm{St}_{\operatorname{SO}_{4}}$ is the Steinberg representation of
$\operatorname{SO}_{4}(F)$, which is the unique subrepresentation of
$I^{\operatorname{SO}_{4}}(\nu\otimes 1)$, and
$J_{\beta_{i}}(s,\chi\circ\mathrm{St}_{\operatorname{GL}_{2}})$ is the unique
quotient of
$I_{\beta_{i}}(\nu^{s}\chi\otimes\mathrm{St}_{\operatorname{GL}_{2}})$ for
$s>0$. These 4 representations can also expressed as

$\begin{array}[]{ll}\chi_{\operatorname{SO}_{4}}=(\chi\circ\det\boxtimes
1_{\operatorname{GL}_{2}})|_{\operatorname{SO}_{4}},&J_{\beta_{1}}(1/2,\chi\otimes\mathrm{St}_{\operatorname{GL}_{2}})=(\mathrm{St}_{\operatorname{GL}_{2}}\boxtimes\chi\circ\det)|_{\operatorname{SO}_{4}(F)},\\\
\mathrm{St}_{\operatorname{SO}_{4}}=(\mathrm{St}_{\operatorname{GL}_{2}}\boxtimes\mathrm{St}_{\operatorname{GL}_{2}})|_{\operatorname{SO}_{4}(F)},&J_{\beta_{2}}(1/2,\chi\otimes\mathrm{St}_{\operatorname{GL}_{2}})=(\chi\circ\det\boxtimes\mathrm{St}_{\operatorname{GL}_{2}})|_{\operatorname{SO}_{4}}.\end{array}$

The local Langlands parameters of these 4 representations are given, in order,
by

$\displaystyle\phi_{\ref{ssec:SO4}.3a}(w,x)$
$\displaystyle={\hat{t}}(\varphi(w)|w|,\varphi(w)),$
$\displaystyle\phi_{\ref{ssec:SO4}.3b}(w,x)$
$\displaystyle={\hat{t}}(\varphi(w)|w|^{1/2},\varphi(w)|w|^{1/2})\iota_{\widehat{\beta}_{1}}(x),$
$\displaystyle\phi_{\ref{ssec:SO4}.3c}(w,x)$
$\displaystyle={\hat{t}}(\varphi(w)|w|^{1/2},\varphi(w)|w|^{-1/2})\iota_{\widehat{\beta}_{2}}(x),$
$\displaystyle\phi_{\ref{ssec:SO4}.3d}(w,x)$
$\displaystyle={\hat{t}}(\varphi(w),\varphi(w))\iota_{\widehat{\beta}_{1}}(x)\iota_{\widehat{\beta}_{2}}(x).$

Here $\varphi:W_{F}\to\mathbb{C}^{\times}$ is the dual character of the
quadratic character $\chi:F^{\times}\to\mathbb{C}^{\times}$. Let
$\lambda_{\ref{ssec:SO4}.3}:W_{F}\to{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{\operatorname{SO}_{4}}}$
be the infinitesimal parameter for any of these Langlands parameters; thus,

$\lambda_{\ref{ssec:SO4}.3}(w)={\hat{t}}(\varphi(w)|w|,\varphi(w)).$

The moduli space of Langlands parameters with infinitesimal parameter
$\lambda_{\ref{ssec:SO4}.3}$ is the direct sum
$\mathfrak{u}_{\hat{\beta}_{1}}\oplus\mathfrak{u}_{\hat{\beta}_{2}}$ of the
root space for the simple roots $\hat{\beta}_{1}$ and $\hat{\beta}_{2}$
equipped with the group action of ${\widehat{T}}$ given by
$t\cdot(x_{1},x_{2}){\,:=\,}(\hat{\beta}_{1}(t)x_{1},\hat{\beta}_{2}(t)x_{2})$.
As a prehomogeneous vector space,
$V_{\lambda_{\ref{ssec:SO4}.3}}=\mathbb{A}^{2}$ with this
$H_{\lambda_{\ref{ssec:SO4}.3}}=\operatorname{GL}_{1}^{2}$ action is
equivalent to 4 for $n=2$.

#### ABV-packets

Table 3.4.1 presents the ABV-packets for all unipotent representations of
$\operatorname{SO}_{4}$ and its pure inner form
$\operatorname{SO}_{4}^{\delta}$, each as a union of a pure $L$-packet and its
coronal representations.

We now sketch how the calculations in Table 3.4.2 are made; the arguments are
very similar to those in Section 3.5 so we will be brief here. The bijection
between
$\Pi^{\mathrm{pure}}_{\lambda_{\ref{ssec:SO4}.3}}(\operatorname{SO}_{4}/F)$
and the six simple perverse sheaves in
$\mathsf{Per}_{{\widehat{T}}}(\mathfrak{u}_{\hat{\beta}_{1}}\oplus\mathfrak{u}_{\hat{\beta}_{2}})$
is given by the following table.

$\begin{array}[]{r|cccc}&C_{\ref{ssec:SO4}.3a}&C_{\ref{ssec:SO4}.3b}&C_{\ref{ssec:SO4}.3c}&C_{\ref{ssec:SO4}.3d}\\\
\hline\cr\chi_{\operatorname{SO}_{4}(F)}&{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{\ref{ssec:SO4}.3a}})&&&\\\
J_{\beta_{1}}(1/2,\chi\otimes\mathrm{St}_{\operatorname{GL}_{2}})&&{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{\ref{ssec:SO4}.3b}})&&\\\
J_{\beta_{2}}(1/2,\chi\otimes\mathrm{St}_{\operatorname{GL}_{2}})&&&{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{\ref{ssec:SO4}.3c}})&\\\
\chi_{\operatorname{SO}_{4}(F)}\otimes\mathrm{St}_{\operatorname{SO}_{4}}&&&&{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{\ref{ssec:SO4}.3d}})\\\
\chi_{\operatorname{SO}_{4}^{\delta}(F)}&&&&{\mathcal{I\hskip-1.0ptC}}(\mathcal{D}_{C_{\ref{ssec:SO4}.3d}})\end{array}$

where $\mathcal{D}$ is the character of $A_{\phi_{\ref{ssec:SO4}.3}}=\\{\pm
1\\}$ corresponding to the pure inner form $\delta\in
Z^{1}(F,\operatorname{SO}_{4})$.

To illustrate the calculations that this table summarizes, consider the case
$\phi_{\ref{ssec:SO4}.3}$. The component group of $\phi_{\ref{ssec:SO4}.3}$ is
$\\{\pm 1\\}$ and thus the pure $L$-packet of $\phi_{\ref{ssec:SO4}.3}$ has 2
elements:

$\Pi_{\phi_{\ref{ssec:SO4}.3}}^{\mathrm{pure}}(\operatorname{SO}_{4})=\\{\mathrm{St}_{\operatorname{SO}_{4}}(\chi),\chi_{\operatorname{SO}_{4}^{\delta}(F)}\\}.$

where $\chi_{\operatorname{SO}_{4}^{\delta}(F)}=1$ if $\chi=1$ and
$\chi_{\operatorname{SO}_{4}^{\delta}(F)}$ is the unique unramified quadratic
character of ${\operatorname{SO}_{4}^{\delta}(F)}$ if $\chi\neq 1$. Table
3.4.2 presents the ABV-packet coefficients promised in the Introduction as it
pertains to the Langlands parameter $\phi_{\ref{ssec:SO4}.3}$. From Table
3.4.2 we read off the ABV-packets in Case 3.4.3; the result appears in Table
3.4.1.

Table 3.4.2.  ABV-packet coefficients $\langle\ ,\ \rangle$ for unipotent
representations of $\operatorname{SO}_{4}$ with infinitesimal parameter
$\lambda_{\ref{ssec:SO4}.3}$. The first four rows refer to representations of
$\operatorname{SO}_{4}(F)$ while the fifth row refers to a representation of
the inner form $\operatorname{SO}_{4}^{\delta}(F)$.

$\begin{array}[]{|c||c|c|c|c|}\hline\cr\Pi^{\mathrm{pure}}_{\lambda_{\ref{ssec:SO4}.3}}(\operatorname{SO}_{4}/F)&\widehat{A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi_{\ref{ssec:SO4}.3a}}}&\widehat{A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi_{\ref{ssec:SO4}.3b}}}&\widehat{A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi_{\ref{ssec:SO4}.3c}}}&\widehat{A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi_{\ref{ssec:SO4}.3d}}}\\\
\hline\cr\hline\cr\chi_{\operatorname{SO}_{4}(F)}&{\mathbbm{1}}&0&0&0\\\
J_{\beta_{1}}(1/2,\chi\otimes\mathrm{St}_{\operatorname{GL}_{2}})&0&{\mathbbm{1}}&0&0\\\
J_{\beta_{2}}(1/2,\chi\otimes\mathrm{St}_{\operatorname{GL}_{2}})&0&0&{\mathbbm{1}}&0\\\
\mathrm{St}_{\operatorname{SO}_{4}}(\chi)&0&0&0&{\mathbbm{1}}\\\
\chi_{\operatorname{SO}_{4}^{\delta}(F)}&\delta&\delta&\delta&\delta\\\
\hline\cr\end{array}$

Properties 1, 2, 3 and 4 are elementary for $\operatorname{SO}_{4}(F)$ and its
pure inner forms. Proposition 2 is interesting for $\operatorname{SO}_{4}(F)$
and its pure inner forms.

######  Proposition 3.2.

The distributions $\Theta^{\operatorname{SO}_{4}}_{\phi}$ (resp.
$\Theta^{\operatorname{SO}_{4}^{\delta}}_{\phi}$) form a basis for the space
of invariant distributions spanned by characters of unipotent irreducible
admissible representations $\pi$ of $\operatorname{SO}_{4}(F)$ (resp.
$\operatorname{SO}_{4}^{\delta}(F)$). The distributions
$\Theta^{\operatorname{SO}_{4}}_{\phi}$ and
$\Theta^{\operatorname{SO}_{4}^{\delta}}_{\phi}$ are stable.

###### Proof.

In all cases except $\phi=\phi_{\ref{ssec:SO4}.0^{\prime}}$,
$\Pi_{\phi}(\operatorname{SO}_{4}(F))$ and
$\Pi_{\phi}(\operatorname{SO}^{\delta}_{4}(F))$ are singletons (or empty) and
$\Theta^{\operatorname{SO}_{4}}_{\phi}=\Theta_{\pi}$. To see that these are
all stable distributions, we may argue as follows.

In Case 3.4.0, the representations
$I^{\operatorname{SO}_{4}}(\chi_{1}\otimes\chi_{2})$ are stable because they
are standard modules, stably induced from representations of $T(F)$.

In Case 3.4.1 we have
$\displaystyle\Theta_{I^{\operatorname{SO}_{4}}(\chi\nu^{1/2}\otimes\chi\nu^{-1/2})}=\Theta_{I_{P_{1}}(\chi\otimes\mathrm{St}_{\operatorname{GL}_{2}})}+\Theta_{I_{P_{1}}(\chi\circ\det)}.$
Since the distribution
$\Theta_{I^{\operatorname{SO}_{4}}(\chi\nu^{1/2}\otimes\chi\nu^{-1/2})}$ is
stable (by an argument as above) and $\Theta_{I_{P_{1}}(\chi\circ\det)}$ is
stable by hand,
$\Theta_{I_{P_{1}}(\chi\circ\mathrm{St}_{\operatorname{GL}_{2}})}$ is also
stable. For Case 3.4.2 argue as in Case 3.4.1. In Case 3.4.3 we have

$\Theta_{I^{\operatorname{SO}_{4}}(\chi\nu\otimes\chi)}=\Theta_{J_{\emptyset}(\chi\nu\otimes\chi)}+\Theta_{J_{P_{1}}(1/2,\chi\otimes\mathrm{St}_{\operatorname{GL}_{2}})}+\Theta_{J_{P_{2}}(1/2,\chi\otimes\mathrm{St}_{\operatorname{GL}_{2}})}+\Theta_{\mathrm{St}_{\operatorname{SO}_{4}}(\chi)}.$

Since $I^{\operatorname{SO}_{4}}(\chi\nu\otimes\chi)$ is stable (argue as in
Case 3.4.0) and $J_{\emptyset}(\chi\nu\otimes\chi)$,
$J_{P_{1}}(1/2,\chi\otimes\mathrm{St}_{\operatorname{GL}_{2}})$, and
$J_{P_{2}}(1/2,\chi\otimes\mathrm{St}_{\operatorname{GL}_{2}})$ are stable
(argue as in Case 3.4.1) it follows that
$\Theta_{\mathrm{St}_{\operatorname{SO}_{4}}(\chi)}$ is stable. Finally,
$\chi_{\operatorname{SO}_{4}^{\delta}(F)}$ is stable because
${\mathbbm{1}}_{\operatorname{SO}_{4}^{\delta}(F)}$ is stable. Case
$\phi=\phi_{\ref{ssec:SO4}.0^{\prime}}$ is a well known:
$\Pi_{\phi_{\ref{ssec:SO4}.0^{\prime}}}(\operatorname{SO}_{4}(F))=\\{\pi_{4},\pi^{\prime}_{4}\\}$
where $\pi_{4}$ and $\pi_{4}^{\prime}$ are the two irreducible representations
appearing in the induced representation
$I^{\operatorname{SO}_{4}}(\chi_{2}\otimes 1)$ and the stable distribution
attached to this $L$-packet is
$\Theta^{\operatorname{SO}_{4}}_{\phi_{\ref{ssec:SO4}.0^{\prime}}}=\Theta_{\pi_{4}}+\Theta_{\pi^{\prime}_{4}}$.
∎

###  3.5. Unipotent representations of $\operatorname{PGL}_{3}$ and its pure
inner forms

In this section we find the ABV-packets and ABV-packet coefficients for all
unipotent representations of $\operatorname{PGL}_{3}$ and its pure inner
forms.

As in Section 3.3, the category of unipotent representations of
$\operatorname{PGL}_{3}(F)$ is precisely the category of unramified principal
series representations of $\operatorname{PGL}_{3}(F)$. In this section we
partition the set $\Pi(\operatorname{PGL}_{3}(F))_{\text{unip}}$ of unipotent
representations of $\operatorname{PGL}_{3}(F)$ into $L$-packets and also
describe the corresponding $L$-parameters. We do the same for the non-split
inner form $D^{\times}/F^{\times}$, where $D$ is a central division algebra of
degree $3$ over $F$. The $p$-adic group $\operatorname{PGL}_{3}$ has three
pure inner forms but only two inner forms: $H^{1}(F,\operatorname{PGL}_{3})$
has cardinality while its image in $H^{1}(F,\operatorname{PGL}_{3})\to
H^{1}(F,\text{Aut}(\operatorname{PGL}_{3}))$ has cardinality $2$. We remark
that this function of pointed sets is not surjective. Let
$\delta,\delta^{\prime}\in Z^{1}(F,\operatorname{PGL}_{3})$ be representatives
for the two non-trivial classes in $H^{1}(F,\operatorname{PGL}_{3})$. Let
$\operatorname{PGL}_{3}^{\delta},\operatorname{PGL}_{3}^{\delta^{\prime}}$ be
the inner forms of $\operatorname{PGL}_{3}$ attached to these cocycles; then
$\operatorname{PGL}_{3}^{\delta}{\ \cong\
}\operatorname{PGL}_{3}^{\delta^{\prime}}$ as algebraic groups over $F$,
although $[\delta]\neq[\delta^{\prime}]$ in $H^{1}(F,\operatorname{PGL}_{3})$.
We can take $\operatorname{PGL}_{3}^{\delta}(F)=D^{\times}/F^{\times}$ and
$\operatorname{PGL}_{3}^{\delta^{\prime}}(F)=(D^{\prime})^{\times}/F^{\times}$,
where $D^{\prime}$ is the opposite algebra of $D$.

#### Langlands correspondence

Table 3.5.1. Pure Arthur packets for unipotent representations of
$\operatorname{PGL}_{3}(F)$ and its pure inner forms
$\operatorname{PGL}_{3}^{\delta}$ and
$\operatorname{PGL}_{3}^{\delta^{\prime}}$: each row gives a pure $L$-packet
for an $L$-parameter and its coronal representations and therefore determines
a pure Arthur packet. Notation is explained below. Cases, indicated in the
left-hand column, gather representations together by infinitesimal parameter.

$\begin{array}[]{|c|c|c| l |}\hline\cr\text{$L$-parameter}&\text{Pure
$L$-packet}&\text{Coronal}&\text{Parameter of }\\\
\phi&\Pi^{\text{pure}}_{\phi}(\operatorname{PGL}_{3}/F)&\text{representations}&\text{Arthur
type?}\\\
\hline\cr\hline\cr\phi_{\ref{ssec:PGL3}.0}&\text{Ind}_{B_{3}(F)}^{\operatorname{GL}_{3}(F)}(\chi_{1}\otimes\chi_{2}\otimes\chi_{1}^{-1}\chi_{2}^{-1})&&\text{unitary
$\chi_{1}$, $\chi_{2}$}\\\
\hline\cr\hline\cr\phi_{\ref{ssec:PGL3}.1a}&I_{\alpha_{1}}((\chi\circ\det)\otimes\chi^{-2})&&\text{unitary
$\chi$}\\\
\phi_{\ref{ssec:PGL3}.1b}&I_{\alpha_{1}}(\chi\mathrm{St}_{\operatorname{GL}_{2}}\otimes\chi^{-2})&&\text{unitary
$\chi$}\\\
\hline\cr\hline\cr\phi_{\ref{ssec:PGL3}.2a}&I_{\alpha_{1}}((\chi\nu^{1/6}\circ\det)\otimes\chi\nu^{-1/3})&&\text{no}\\\
\phi_{\ref{ssec:PGL3}.2b}&I_{\alpha_{1}}(\chi\nu^{1/6}\mathrm{St}_{\operatorname{GL}_{2}}\otimes\chi\nu^{-1/3})&&\text{no}\\\
\hline\cr\phi_{\ref{ssec:PGL3}.3a}&\chi_{\operatorname{PGL}_{3}}&\chi_{\operatorname{PGL}_{3}^{\delta}(F)},\
\chi_{\operatorname{PGL}_{3}^{\delta^{\prime}}(F)}&\text{yes}\\\
\phi_{\ref{ssec:PGL3}.3b}&\chi\otimes
J_{\alpha_{1}}(\nu\otimes\nu^{-1/2}\mathrm{St}_{\operatorname{GL}_{2}})&\chi_{\operatorname{PGL}_{3}^{\delta}(F)},\
\chi_{\operatorname{PGL}_{3}^{\delta^{\prime}}(F)}&\text{no}\\\
\phi_{\ref{ssec:PGL3}.3c}&\chi\otimes
J_{\alpha_{1}}(\nu^{1/2}\mathrm{St}_{\operatorname{GL}_{2}}\otimes\nu^{-1})&\chi_{\operatorname{PGL}_{3}^{\delta}(F)},\
\chi_{\operatorname{PGL}_{3}^{\delta^{\prime}}(F)}&\text{no}\\\
\phi_{\ref{ssec:PGL3}.3d}&\chi\otimes\mathrm{St}_{\operatorname{PGL}_{3}},\
\chi_{\operatorname{PGL}_{3}^{\delta}(F)},\
\chi_{\operatorname{PGL}_{3}^{\delta^{\prime}}(F)}&&\text{yes}\\\
\hline\cr\end{array}$

Let $T$ be the standard maximal torus in $\operatorname{PGL}_{3}$ and let
$B_{3}$ be the standard Borel subgroup of $\operatorname{GL}_{3}$. We denote
the matching positive roots of $\operatorname{GL}_{3}$ by $\alpha_{1}$ and
$\alpha_{2}$. Let $P_{\alpha_{i}}$ be the standard maximal parabolic subgroup
of $\operatorname{GL}_{3}$ such that the root space $\alpha_{i}$ is in the
Levi of $P_{\alpha_{i}}$. Since
$\operatorname{PGL}_{3}(F)=\operatorname{GL}_{3}(F)/F^{\times}$, a
representation of $\operatorname{PGL}_{3}(F)$ is just a representation of
$\operatorname{GL}_{3}(F)$ with trivial central character. The cases treated
below group representations together by infinitesimal parameter.

  1. 3.5.0

Let $\chi_{1},\chi_{2}$ are characters of $F^{\times}$ such that
$\chi_{1}\chi_{2}^{-1}$, $\chi_{1}^{2}\chi_{2}$ and $\chi_{1}\chi_{2}^{2}$ are
not equal to $\nu$ and not equal to $\nu^{-1}$. Then the induced
representation

$\text{Ind}_{B_{3}(F)}^{\operatorname{GL}_{3}(F)}(\chi_{1}\otimes\chi_{2}\otimes\chi_{1}^{-1}\chi_{2}^{-1})$

is irreducible. Note that this representation has trivial central character
and thus can be viewed as a representation of $\operatorname{PGL}_{3}(F)$. We
denote this representation of $\operatorname{PGL}_{3}(F)$ by
$I^{\operatorname{PGL}_{3}}(\chi_{1}\otimes\chi_{2})$. The corresponding
Langlands parameter for $I^{\operatorname{PGL}_{3}}(\chi_{1}\otimes\chi_{2})$
is given by

$\phi_{\ref{ssec:PGL3}.0}(w,x)=\begin{pmatrix}\varphi_{1}(w)&0&0\\\
0&\varphi_{2}(w)&0\\\
0&0&\varphi_{1}(w)^{-1}\varphi_{2}(w)^{-1}\end{pmatrix}\in\operatorname{SL}_{3}(\mathbb{C}),$

where $\varphi_{i}:W_{F}\to\mathbb{C}^{\times}$ corresponds to
$\chi_{i}:F^{\times}\to\mathbb{C}^{\times}$ under class field theory.

  2. 3.5.1

Let $\chi$ be a character of $F^{\times}$ with $\chi^{3}\neq\nu^{\pm
1/2},\nu^{\pm 3/2}$. We consider the representation

$\text{Ind}_{B_{3}(F)}^{\operatorname{GL}_{3}(F)}(\chi\nu^{1/2}\otimes\chi\nu^{-1/2}\otimes\chi^{-2})$

of $\operatorname{GL}_{3}(F)$, which is also viewed as a representation of
$\operatorname{PGL}_{3}(F)$. This representation has length 2, and its two
irreducible components are
$I_{\alpha_{1}}(\chi\mathrm{St}_{\operatorname{GL}_{2}}\otimes\chi^{-2})$ and
$I_{\alpha_{1}}(\chi\circ\det\otimes\chi^{-2})$, which, adapting the notation
above, are written as

$I^{\operatorname{PGL}_{3}}_{\alpha_{1}}(\chi\mathrm{St}_{\operatorname{GL}_{2}}),\qquad\text{and}\qquad
I^{\operatorname{PGL}_{3}}_{\alpha_{1}}(\chi\circ\det).$

Here $I^{\operatorname{PGL}_{3}}_{\alpha_{1}}$ stands for
$\text{Ind}_{P_{\alpha_{1}}}^{\operatorname{GL}_{3}}$, and a representation of
$\operatorname{GL}_{3}(F)$ is viewed as a representation of
$\operatorname{PGL}_{3}(F)$ if it has trivial central character. The Langlands
parameter of $I_{\alpha_{1}}(\chi\circ\det\otimes\chi^{-2})$ is given by

$\phi_{\ref{ssec:PGL3}.1.a}(w,x)=\begin{pmatrix}\varphi(w)|w|^{1/2}&0&0\\\
0&\varphi(w)|w|^{-1/2}&0\\\
0&0&\varphi(w)^{-2}\end{pmatrix}\in\operatorname{SL}_{3}(\mathbb{C}),$

where $\varphi:W_{F}\rightarrow\mathbb{C}^{\times}$ is the dual character of
$\chi$. The Langlands parameter for
$I_{\alpha_{1}}(\chi\mathrm{St}_{\operatorname{GL}_{2}}\otimes\chi^{-2})$ is
given by

$\phi_{\ref{ssec:PGL3}.1b}\left(w,\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\right)=\begin{pmatrix}\varphi(w)a&\varphi(w)b&0\\\
\varphi(w)c&\varphi(w)d&0\\\
0&0&\varphi(w)^{-2}\end{pmatrix}=\begin{pmatrix}\varphi(w)x&\\\
&\varphi(w)^{-2}\end{pmatrix}\in\operatorname{SL}_{3}(\mathbb{C}).$

Note that the infinitesimal parameter for these representations is

$\lambda_{\ref{ssec:PGL3}.1a}(w)=\begin{pmatrix}\varphi(w)|w|^{1/2}&0&0\\\
0&\varphi(w)|w|^{-1/2}&0\\\ 0&0&\varphi(w)^{-2}\end{pmatrix}.$

  3. 3.5.2

Let $\chi$ be a character of $F^{\times}$ with $\chi^{3}=1$. Consider the
representation

$\text{Ind}_{B_{3}(F)}^{\operatorname{GL}_{3}(F)}(\chi\nu^{2/3}\otimes\chi\nu^{-1/3}\otimes\chi\nu^{-1/3}),$

which is also viewed as a representation of $\operatorname{PGL}_{3}(F)$. This
representation has length 2 and its two components are

$I_{\alpha_{1}}(\chi\nu^{1/6}\mathrm{St}_{\operatorname{GL}_{2}}\otimes\chi\nu^{-1/3})\quad\text{and}\quad
I_{\alpha_{1}}(\chi\nu^{1/6}\circ\det\otimes\chi\nu^{-1/3}).$

The Langlands parameter of
$I_{\alpha_{1}}(\chi\nu^{1/6}\circ\det\otimes\chi\nu^{-1/3})$ is given by

$\phi_{\ref{ssec:PGL3}.2a}(w,x)=\begin{pmatrix}\varphi(w)|w|^{2/3}&0&0\\\
0&\varphi(w)|w|^{-1/3}&0\\\
0&0&\varphi(w)|w|^{-1/3}\end{pmatrix}\in\operatorname{SL}_{3}(\mathbb{C}),$

and the Langlands parameter of
$I_{\alpha_{1}}(\chi\nu^{1/6}\mathrm{St}_{\operatorname{GL}_{2}}\otimes\chi\nu^{-1/3})$
is given by

$\phi_{\ref{ssec:PGL3}.2b}(w,x)=\begin{pmatrix}\varphi(w)|w|^{1/6}x&\\\
&\varphi(w)|w|^{-1/3}\end{pmatrix}\in\operatorname{SL}_{3}(\mathbb{C}).$

  4. 3.5.3

Let $\chi$ be a character of $F^{\times}$ with $\chi^{3}=1$. Consider the
induced representation

$\text{Ind}_{B_{3}(F)}^{\operatorname{GL}_{3}(F)}(\chi\nu\otimes\chi\otimes\chi\nu^{-1})$

of $\operatorname{GL}_{3}(F)$ (and also of $\operatorname{PGL}_{3}(F)$). This
representation has length 4 and its components are given by

$\begin{array}[]{cc}\chi_{\operatorname{PGL}_{3}(F)},&\chi\otimes
J_{\alpha_{2}}(\nu\otimes\nu^{-1/2}\mathrm{St}_{\operatorname{GL}_{2}}),\\\
\chi\otimes
J_{\alpha_{1}}(\nu^{1/2}\mathrm{St}_{\operatorname{GL}_{2}}\otimes\nu^{-1}),&\chi\otimes\mathrm{St}_{\operatorname{PGL}_{3}},\end{array}$

where $\chi_{\operatorname{PGL}_{3}(F)}=\chi\circ\det$,
$J_{\alpha_{1}}(\nu^{1/2}\mathrm{St}_{\operatorname{GL}_{2}}\otimes\nu^{-1})$
is the unique quotient of
$I_{\alpha_{1}}(\nu^{1/2}\mathrm{St}_{\operatorname{GL}_{2}}\otimes\nu^{-1})$
and $J_{\alpha_{2}}(\nu\otimes\nu^{-1/2}\mathrm{St}_{\operatorname{GL}_{2}}$
is defined similarly. The local Langlands parameters of these 4
representations are given, in order, by

$\displaystyle\phi_{\ref{ssec:PGL3}.3a}(w,x)$
$\displaystyle=\varphi(w)\begin{pmatrix}|w|&&\\\ &1&\\\
&&|w|^{-1}\end{pmatrix}\in\operatorname{SL}_{3}(\mathbb{C}),$
$\displaystyle\phi_{\ref{ssec:PGL3}.3b}(w,x)$
$\displaystyle=\varphi(w)\begin{pmatrix}|w|&\\\
&|w|^{-1/2}x\end{pmatrix}\in\operatorname{SL}_{3}(\mathbb{C}),$
$\displaystyle\phi_{\ref{ssec:PGL3}.3c}(w,x)$
$\displaystyle=\varphi(w)\begin{pmatrix}|w|^{1/2}x&\\\
&|w|^{-1}\end{pmatrix}\in\operatorname{SL}_{3}(\mathbb{C}),$
$\displaystyle\phi_{\ref{ssec:PGL3}.3d}(w,x)$
$\displaystyle=\varphi(w)\operatorname{Sym}^{2}(x)\in\operatorname{SL}_{3}(\mathbb{C}),$

where $\varphi:W_{F}\to C^{\times}$ is Langlands parameter for the quadratic
character $\chi$ and $\operatorname{Sym}^{2}$ is the symmetric square
representation of $\operatorname{SL}_{2}(\mathbb{C})$.

This completes the description of the Langlands correspondence for unipotent
representations of $\operatorname{PGL}_{3}(F)$.

#### ABV-packets

Table 3.5.1 presents the $L$-packets for all unipotent representations of
$\operatorname{PGL}_{3}(F)$. Returning to pure $L$-packets, we only explain
the case 3.5.3 since the others proceed by similar, simpler arguments. Denote
the last Langlands parameter in Case 3.5.3 by ${\phi}_{\ref{ssec:PGL3}.3d}$.
This Langlands parameter has component group $\mu_{3}$ so its pure $L$-packet
contains 3 admissible irreducible representations:

$\Pi_{{\phi}_{\ref{ssec:PGL3}.3d}}^{\mathrm{pure}}(\operatorname{PGL}_{3}/F)=\\{\chi\otimes\mathrm{St}_{\operatorname{PGL}_{3}},\chi_{\operatorname{PGL}_{3}^{\delta}(F)},\chi_{\operatorname{PGL}_{3}^{\delta^{\prime}}(F)}\\},$

where $\chi_{\operatorname{PGL}_{3}^{\delta}(F)}$ (resp.
$\chi_{\operatorname{PGL}_{3}^{\delta^{\prime}}(F)}$) is the character of
$\operatorname{PGL}_{3}^{\delta}(F)$ (resp.
$\operatorname{PGL}_{3}^{\delta^{\prime}}(F)$) obtained by composing $\chi$
with the reduced norm.

Table 3.5.2 presents the ABV-packet coefficients for the Langlands parameter
$\phi_{\ref{ssec:PGL3}.3}$. From Table 3.5.2 we read off the ABV-packets in
Case 3.5.3; the result appears in Table 3.5.1, where $\mathcal{D}$ (resp.
$\mathcal{D}^{\prime}$) is the character of $A_{\phi_{\ref{ssec:PGL3}.3d}}$
corresponding to the pure inner form $\delta$ (resp. $\delta^{\prime}$) in
$Z^{1}(F,\operatorname{PGL}_{3})$.

Table 3.5.2. ABV-packet coefficients $\langle\ ,\ \rangle$ for unipotent
representations of $\operatorname{PGL}_{3}$ and its pure inner forms with
infinitesimal parameter $\lambda_{\ref{ssec:PGL3}.3}$. The first four rows
refer to representations of $\operatorname{PGL}_{3}(F)$, while the fifth and
sixth rows are refer to the pure inner forms $\delta$ and $\delta^{\prime}$,
respectively. Below, we identify $\delta$ (resp. $\delta^{\prime}$) with the
character of $A_{\phi_{\ref{ssec:PGL3}.3d}}$ corresponding to the pure inner
form $\delta$ (resp. $\delta^{\prime}$) in $Z^{1}(F,\operatorname{PGL}_{3})$.
All microlocal fundamental groups are order $3$

$\begin{array}[]{|c||cccc|}\hline\cr\text{Irreps with inf parameter
$\lambda_{\ref{ssec:PGL3}.3}$}&\widehat{A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi_{\ref{ssec:PGL3}.3a}}}&\widehat{A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi_{\ref{ssec:PGL3}.3b}}}&\widehat{A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi_{\ref{ssec:PGL3}.3c}}}&\widehat{A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi_{\ref{ssec:PGL3}.3d}}}\\\
\hline\cr\hline\cr\chi_{\operatorname{PGL}_{3}(F)}&{\mathbbm{1}}&0&0&0\\\
\chi\otimes
J_{\alpha_{1}}(\nu\otimes\nu^{-1/2}\mathrm{St}_{\operatorname{GL}_{2}})&0&{\mathbbm{1}}&0&0\\\
\chi\otimes
J_{\alpha_{1}}(\nu^{1/2}\mathrm{St}_{\operatorname{GL}_{2}}\otimes\nu^{-1})&0&0&{\mathbbm{1}}&0\\\
\chi\otimes\mathrm{St}_{\operatorname{PGL}_{3}}&0&0&0&{\mathbbm{1}}\\\
\chi_{\operatorname{PGL}_{3}^{\delta}(F)}&\delta&\delta&\delta&\delta\\\
\chi_{\operatorname{PGL}_{3}^{\delta^{\prime}}(F)}&\delta^{\prime}&\delta^{\prime}&\delta^{\prime}&\delta^{\prime}\\\
\hline\cr\end{array}$

We now explain the calculations behind Table 3.5.2, as a special case of the
techniques of [CFMMX]; see also [CFZ:cubics]. Let
$\lambda_{\ref{ssec:PGL3}.3}:W_{F}\to\operatorname{SL}_{3}(\mathbb{C})$ be the
infinitesimal parameter of $\phi_{\ref{ssec:PGL3}.3d}$:

$\lambda_{\ref{ssec:PGL3}.3}(w){\,:=\,}\phi_{\ref{ssec:PGL3}.3d}\left(w,\operatorname{diag}({|w|}^{1/2},{|w|}^{-1/2})\right)=\operatorname{diag}({|w|},1,{|w|}^{-1}).$

The moduli space of Langlands parameters with infinitesimal parameter
$\lambda_{\ref{ssec:PGL3}.3}$ is the direct sum
$\mathfrak{u}_{\hat{\alpha}_{1}}\oplus\mathfrak{u}_{\hat{\alpha}_{2}}$ of the
root space for the simple roots $\hat{\alpha}_{1}$ and $\hat{\alpha}_{2}$,
equipped with the group action of ${\widehat{T}}$ given by
$t\cdot(x_{1},x_{2}){\,:=\,}(\hat{\alpha}_{1}(t)x_{1},\hat{\alpha}_{2}(t)x_{2})$.

Vogan’s version of the Langlands correspondence establishes a bijection
between simple objects in the category
$\mathsf{Per}_{{\widehat{T}}}(\mathfrak{u}_{\hat{\alpha}_{1}}\oplus\mathfrak{u}_{\hat{\alpha}_{2}})$
of ${\widehat{T}}$-equivariant perverse sheaves on
$\mathfrak{u}_{\hat{\alpha}_{1}}\oplus\mathfrak{u}_{\hat{\alpha}_{2}}$ and
irreducible admissible representations of $\operatorname{PGL}_{3}$ and its
pure inner forms with infinitesimal parameter $\lambda_{\ref{ssec:PGL3}.3}$.
We have already seen there are six of the latter – the four irreducible
subquotients of
$\text{Ind}_{B_{3}(F)}^{\operatorname{GL}_{3}(F)}(\chi\nu\otimes\chi\otimes\mu\nu^{-1})$
together with the characters $\chi_{\operatorname{PGL}_{3}^{\delta}(F)}$ and
$\chi_{\operatorname{PGL}_{3}^{\delta^{\prime}}(F)}$. Let us enumerate the
corresponding simple objects in
$\mathsf{Per}_{{\widehat{T}}}(\mathfrak{u}_{\hat{\alpha}_{1}}\oplus\mathfrak{u}_{\hat{\alpha}_{2}})$.
Let $C_{\ref{ssec:PGL3}.3a}$ be the trivial orbit in
$V_{\lambda_{\ref{ssec:PGL3}.3}}$, let $C_{\ref{ssec:PGL3}.3b}$ be the
${\widehat{T}}$-orbit of $X_{\hat{\alpha}_{1}}$, which is the point in the
moduli space for the Langlands parameter $\phi_{1}$ for $\chi\otimes
J_{\alpha_{1}}(\nu\otimes\nu^{-1/2}\mathrm{St}_{\operatorname{GL}_{2}})$; let
$C_{\ref{ssec:PGL3}.3c}$ be the ${\widehat{T}}$-orbit of
$X_{\hat{\alpha}_{2}}$, which is the point in the moduli space for the
Langlands parameter $\phi_{2}$ for $\chi\otimes
J_{\alpha_{1}}(\nu^{1/2}\mathrm{St}_{\operatorname{GL}_{2}}\otimes\nu^{-1})$.
Finally, let $C_{\ref{ssec:PGL3}.3d}$ be the ${\widehat{T}}$-orbit of
$X_{\hat{\alpha}_{1}}+X_{\hat{\alpha}_{2}}$; this orbit of the Langlands
parameter $\phi_{\ref{ssec:PGL3}.3d}$ for
$\mu\otimes\mathrm{St}_{\operatorname{PGL}_{3}}$. The equivariant fundamental
groups of these orbits are trivial with the exception of the open orbit
$C_{\ref{ssec:PGL3}.3d}$, which has equivariant fundamental group

$A_{\phi_{\ref{ssec:PGL3}.3d}}{\,:=\,}\pi_{1}^{{\widehat{T}}}(C_{\ref{ssec:PGL3}.3},X_{\hat{\alpha}_{1}}+X_{\hat{\alpha}_{2}})=\pi_{0}(Z_{{\widehat{T}}}(X_{\hat{\alpha}_{1}}+X_{\hat{\alpha}_{2}}))=\\{1,\theta_{3},\theta_{3}^{2}\\}.$

The bijection between
$\Pi^{\mathrm{pure}}_{\lambda_{\ref{ssec:PGL3}.3}}(\operatorname{PGL}_{3}/F)$
and the six simple perverse sheaves in
$\mathsf{Per}_{{\widehat{T}}}(\mathfrak{u}_{\hat{\alpha}_{1}}\oplus\mathfrak{u}_{\hat{\alpha}_{2}})$
are given by the following table.

$\begin{array}[]{r|cccc}&C_{\ref{ssec:PGL3}.3a}&C_{\ref{ssec:PGL3}.3b}&C_{\ref{ssec:PGL3}.3c}&C_{\ref{ssec:PGL3}.3d}\\\
\hline\cr\chi_{\operatorname{PGL}_{3}(F)}&{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{\ref{ssec:PGL3}.3a}})&&&\\\
\chi\otimes
J_{\alpha_{1}}(\nu\otimes\nu^{-1/2}\mathrm{St}_{\operatorname{GL}_{2}})&&{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{\ref{ssec:PGL3}.3b}})&&\\\
\chi\otimes
J_{\alpha_{1}}(\nu^{1/2}\mathrm{St}_{\operatorname{GL}_{2}}\otimes\nu^{-1})&&&{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{\ref{ssec:PGL3}.3c}})&\\\
\chi\otimes\mathrm{St}_{\operatorname{PGL}_{3}}&&&&{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{\ref{ssec:PGL3}.3d}})\\\
\chi_{\operatorname{PGL}_{3}^{\delta}(F)}&&&&{\mathcal{I\hskip-1.0ptC}}(\mathcal{D}_{C_{\ref{ssec:PGL3}.3d}})\\\
\chi_{\operatorname{PGL}_{3}^{\delta^{\prime}}(F)}&&&&{\mathcal{I\hskip-1.0ptC}}(\mathcal{D}^{\prime}_{C_{\ref{ssec:PGL3}.3d}}),\end{array}$

where $\mathcal{D}$ (resp. $\mathcal{D}^{\prime}$) is the local system
corresponding to the choice of primitive cube root of unity $\delta$ (resp.
$\delta^{\prime}$) appearing in the pure inner form $\delta\in
Z^{1}(F,\operatorname{PGL}_{3})$.

Now, for each irreducible admissible $\pi$ above, and for each Langlands
parameter $\phi$ with infinitesimal parameter $\lambda_{\ref{ssec:PGL3}.3}$,
we compute
$\operatorname{\mathsf{N}\hskip-1.0pt\mathsf{E}\hskip-1.0pt\mathsf{v}\hskip-1.0pt\mathsf{s}}_{C_{\phi}}\mathcal{P}(\pi)$,
where $\mathcal{P}(\pi)$ is the corresponding simple object in
$\mathsf{Per}_{{\widehat{T}}}(\mathfrak{u}_{\hat{\alpha}_{1}}\oplus\mathfrak{u}_{\hat{\alpha}_{2}})$.
The result is a local system in
$\Lambda_{C_{\phi}}^{\text{sreg}}{\,:=\,}T^{*}_{C_{\phi}}(\mathfrak{u}_{\hat{\alpha}_{1}}\oplus\mathfrak{u}_{\hat{\alpha}_{2}})_{\text{sreg}}$
and this may be viewed as a representation of the equivariant fundamental
group of $\Lambda_{C_{\phi}}^{\text{sreg}}$, which is
$A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}$. The
results are summarized in Table 3.5.2.

The properties in Theorems 2.2 and 2.14 are elementary for
$\operatorname{PGL}_{3}(F)$ and its pure inner forms as is [CFMMX]*Conjecture
1.

##  4\. Geometric endoscopy

###  4.1. Lifting Langlands parameters

Let $(G,s,\xi)$ be an endoscopic triple for $G_{2}$ and let
$\phi:W^{\prime}_{F}\to{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G}}$ be an unramified
Langlands parameter such that
$s\in\mathcal{S}^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\xi\circ\phi}$.
The Langlands parameter $\phi$ determines, or lifts to, a Langlands parameter
$\xi\circ\phi:W^{\prime}_{F}\to{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G_{2}}}$.
Matching the cases that appear in our classification of Langlands parameters
for $G$ and $G_{2}$ is a little complicated, because we have grouped
parameters into families based on the geometry created by the corresponding
infinitesimal parameters. Consequently, it happens that different members of
these families may lift to different families for $G_{2}$. To illustrate this
phenomenon, here we give the details for one of the six endoscopic groups for
$G_{2}$. Together with all other cases, this information is summarized in
Table 4.1.2.

  1. 5

$G=\operatorname{PGL}_{3}$, $s={\hat{m}}(\theta_{3}^{2},\theta_{3})$.

    1. (0)

The Langlands parameter $\phi_{\ref{ssec:PGL3}.0}$ lifts to
$\phi_{\ref{infcase:0}}$ if and only if
$\varphi_{1}({\operatorname{Fr}}),\varphi_{2}({\rm{Fr}}),\varphi_{1}({\rm{Fr}})\varphi_{2}({\rm{Fr}})\notin\\{q^{\pm
1}\\}$; lifts to $\phi_{\ref{infcase:1-short}a}$ if and only if one of
$\varphi_{1}({\operatorname{Fr}}),\varphi_{2}({\rm{Fr}}),\varphi_{1}({\rm{Fr}})\varphi_{2}({\rm{Fr}})$
is in $\\{q^{\pm 1}\\}$; and lifts to $\phi_{\ref{infcase:5}a}$ if and only if
two of
$\varphi_{1}({\operatorname{Fr}}),\varphi_{2}({\rm{Fr}}),\varphi_{1}({\rm{Fr}})\varphi_{2}({\rm{Fr}})$
are in $\\{q^{\pm 1}\\}$.

    2. (1)

The Langlands parameter $\phi_{\ref{ssec:PGL3}.1a}$ (resp.
$\phi_{\ref{ssec:PGL3}.1a}$) lifts to $\phi_{\ref{infcase:2-long}a}$ (resp.
$\phi_{\ref{infcase:2-long}b}$) if and only if
$\varphi({\rm{Fr}})\notin\\{-q^{\pm 1/2},q^{\pm 3/2}\\}$ ;
$\phi_{\ref{infcase:4-D2}a}$ (resp. $\phi_{\ref{infcase:4-D2}b}$) if and only
if $\varphi({\rm{Fr}})\in\\{-q^{\pm 1/2}\\}$; $\phi_{\ref{infcase:7-reg}a}$
(resp. $\phi_{\ref{infcase:7-reg}b}$) if and only if
$\varphi({\rm{Fr}})\in\\{q^{\pm 3/2}\\}$.

    3. (2)

The Langlands parameter $\phi_{\ref{ssec:PGL3}.2a}$ (resp.
$\phi_{\ref{ssec:PGL3}.2b}$) lifts to $\phi_{\ref{infcase:3}a}$ (resp.
$\phi_{\ref{infcase:3}b}$).

    4. (3)

The Langlands parameter $\phi_{\ref{ssec:PGL3}.3a}$ (resp.
$\phi_{\ref{ssec:PGL3}.3b}$, $\phi_{\ref{ssec:PGL3}.3c}$,
$\phi_{\ref{ssec:PGL3}.3d}$) lifts to $\phi_{\ref{infcase:6-A2}a}$ (resp.
$\phi_{\ref{infcase:6-A2}b}$, $\phi_{\ref{infcase:6-A2}b}$,
$\phi_{\ref{infcase:6-A2}b}$) if and only if $\varphi({\rm{Fr}})\neq 1$; and
lifts to $\phi_{\ref{infcase:8-sub}a}$ (resp. $\phi_{\ref{infcase:8-sub}b}$,
$\phi_{\ref{infcase:8-sub}c}$, $\phi_{\ref{infcase:8-sub}d}$) if and only if
$\varphi({\rm{Fr}})=1$.

If we turn the problem of lifting parameters on its head, our classification
of scheme for Langlands parameters is more illuminating: instead of lifting
parameters from endoscopic groups, begin with a Langlands parameter for
$G_{2}$ and find all endoscopic triples $(G,s,\xi)$ such that the Langlands
parameter factors through
$\xi:{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G}}\to{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G_{2}}}$.
This information appears in Table 4.1.1.

Table 4.1.1. Langlands parameters and their associated minimal endoscopic
groups. In the column labeled "minimal endoscopic group(s)" we indicate, for
each Langlands parameter $\phi$ of $G_{2}$, the minimal endoscopic groups $G$
such that the parameter factors $\phi$ through
$\xi:{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G}}\to{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G_{2}}}$.
The column labeled "Arthur type" gives necessary and sufficient conditions for
$\phi$ to be of Arthur type.

$\begin{array}[t]{| c | c | c | c c | c |}\hline\cr\text{$L$-parameter}&\text{Minimal}&\text{Arthur}&\lx@intercol\hfil\text{Component}\hfil\lx@intercol\vrule\lx@intercol&\\\ &\text{endoscopic}&\text{type}&\lx@intercol\hfil\text{groups}\hfil\lx@intercol\vrule\lx@intercol&\\\ \phi&\text{group(s)}&\text{parameter}&A_{\phi}&A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}&\dim(\phi)\\\ \hline\cr\hline\cr\phi_{\ref{infcase:0}}&T&\text{$a_{1}=a_{2}=0$}&1&1&0\\\ \hline\cr\hline\cr\phi_{\ref{infcase:1-short}a}&T&\text{$a=1/2$}&1&1&0\\\ \phi_{\ref{infcase:1-short}b}&\operatorname{GL}_{2}^{\gamma_{2}}&\text{$a=1/2$}&1&1&1\\\ \hline\cr\phi_{\ref{infcase:2-long}a}&T&\text{$a=1/2$}&1&1&0\\\ \phi_{\ref{infcase:2-long}b}&\operatorname{GL}_{2}^{\gamma_{1}}&\text{$a=1/2$}&1&1&1\\\ \hline\cr\hline\cr\phi_{\ref{infcase:3}.a}&T&\text{no}&1&1&0\\\ \phi_{\ref{infcase:3}.b}&\operatorname{GL}_{2}^{\gamma_{1}}&\text{no}&1&1&2\\\ \hline\cr\phi_{\ref{infcase:4-D2}a}&T&\text{yes}&1&\langle\theta_{2}\rangle&0\\\ \phi_{\ref{infcase:4-D2}b}&\operatorname{GL}_{2}^{\gamma_{1}}&\text{yes}&1&\langle\theta_{2}\rangle&1\\\ \phi_{\ref{infcase:4-D2}c}&\operatorname{GL}_{2}^{\gamma_{2}}&\text{yes}&1&\langle\theta_{2}\rangle&1\\\ \phi_{\ref{infcase:4-D2}d}&\operatorname{SO}_{4}&\text{yes}&\langle\theta_{2}\rangle&\langle\theta_{2}\rangle&2\\\ \hline\cr\phi_{\ref{infcase:5}a}&T&\text{no}&1&1&0\\\ \phi_{\ref{infcase:5}b}&\operatorname{GL}_{2}^{\gamma_{2}}&\text{no}&1&1&2\\\ \hline\cr\hline\cr\phi_{\ref{infcase:6-A2}a}&T&\text{yes}&1&\langle\theta_{3}\rangle&0\\\ \phi_{\ref{infcase:6-A2}b}&\operatorname{GL}_{2}^{\gamma_{1}}&\text{no}&1&\langle\theta_{3}\rangle&1\\\ \phi_{\ref{infcase:6-A2}c}&\operatorname{GL}_{2}^{\gamma_{1}}&\text{no}&1&\langle\theta_{3}\rangle&1\\\ \phi_{\ref{infcase:6-A2}d}&\operatorname{PGL}_{3}&\text{yes}&\langle\theta_{3}\rangle&\langle\theta_{3}\rangle&2\\\ \hline\cr\phi_{\ref{infcase:7-reg}a}&T&\text{yes}&1&1&0\\\ \phi_{\ref{infcase:7-reg}b}&\operatorname{GL}_{2}^{\gamma_{1}}&\text{no}&1&1&1\\\ \phi_{\ref{infcase:7-reg}c}&\operatorname{GL}_{2}^{\gamma_{2}}&\text{no}&1&1&1\\\ \phi_{\ref{infcase:7-reg}d}&G_{2}&\text{yes}&1&1&2\\\ \hline\cr\hline\cr\phi_{\ref{infcase:8-sub}a}&T&\text{yes}&1&S_{3}&0\\\ \phi_{\ref{infcase:8-sub}b}&\operatorname{GL}_{2}^{\gamma_{1}}&\text{yes}&1&\langle\theta_{2}\rangle&2\\\ \phi_{\ref{infcase:8-sub}c}&\operatorname{GL}_{2}^{\gamma_{2}}&\text{yes}&1&\langle\theta_{2}\rangle&3\\\ \phi_{\ref{infcase:8-sub}d}&\operatorname{SO}_{4},\operatorname{PGL}_{3}&\text{yes}&S_{3}&S_{3}&4\\\ \hline\cr\end{array}$

Table 4.1.2. Summary of possible lifts for members of each family of Langlands
parameters. The leftmost column gives endoscopic groups $G$ for $G_{2}$. The
column "Endoscopic Langlands parameters" refers to Sections 3.2 through 3.4.
The column "Arthur-type" indicates those $\xi$-conormal parameters that, when
unitary, are parameters of Arthur type; see Definition 6 for this notion. The
lifts appearing in the column "other regular" indicates all other
$\xi$-conormal parameters. In these last two cases, the information in these
two columns refers to the classification of unramified Langlands parameters
for $G_{2}$ appearing in Section 1.2. Finally, the column "irregular lifts"
indicates all remaining lifts.

$\begin{array}[]{ l l | l r | l}\text{Endoscopic}&\text{Endoscopic}&\lx@intercol\hfil\text{$s$-regular lifts}\hfil\lx@intercol\vrule\lx@intercol&\\\ \text{group}&\text{$L$-parameter}&\text{Arthur}&\text{Other}&\\\ &&\text{type}&\text{regular}&\text{Irregular lifts}\\\ \hline\cr\ref{endcase:torus}&\phi_{\ref{ssec:T}.0}&\phi_{\ref{infcase:0}}&&\phi_{\ref{infcase:1-short}a},\phi_{\ref{infcase:2-long}a},\phi_{\ref{infcase:3}a},\phi_{\ref{infcase:4-D2}a},\phi_{\ref{infcase:5}a},\phi_{\ref{infcase:6-A2}a},\phi_{\ref{infcase:7-reg}a},\phi_{\ref{infcase:8-sub}a}\\\ \hline\cr\ref{endcase:short}&\phi_{\ref{ssec:GL2}.0}&\phi_{\ref{infcase:0}}&&\phi_{\ref{infcase:1-short}a},\phi_{\ref{infcase:2-long}a},\phi_{\ref{infcase:3}a},\phi_{\ref{infcase:4-D2}a},\phi_{\ref{infcase:5}a},\phi_{\ref{infcase:6-A2}a},\phi_{\ref{infcase:7-reg}a},\phi_{\ref{infcase:8-sub}a}\\\ &\phi_{\ref{ssec:GL2}.1a}&\phi_{\ref{infcase:1-short}a}&\phi_{\ref{infcase:5}a}&\phi_{\ref{infcase:4-D2}a},\phi_{\ref{infcase:7-reg}a},\phi_{\ref{infcase:8-sub}a}\\\ &\phi_{\ref{ssec:GL2}.1b}&\phi_{\ref{infcase:1-short}b}&\phi_{\ref{infcase:5}b}&\phi_{\ref{infcase:4-D2}c},\phi_{\ref{infcase:7-reg}c},\phi_{\ref{infcase:8-sub}c}\\\ \hline\cr\ref{endcase:long}&\phi_{\ref{ssec:GL2}.0}&\phi_{\ref{infcase:0}}&&\phi_{\ref{infcase:1-short}a},\phi_{\ref{infcase:2-long}a},\phi_{\ref{infcase:3}a},\phi_{\ref{infcase:4-D2}a},\phi_{\ref{infcase:5}a},\phi_{\ref{infcase:6-A2}a},\phi_{\ref{infcase:7-reg}a},\phi_{\ref{infcase:8-sub}a}\\\ &\phi_{\ref{ssec:GL2}.1a}&\phi_{\ref{infcase:2-long}a}&\phi_{\ref{infcase:3}a}&\phi_{\ref{infcase:4-D2}a},\phi_{\ref{infcase:6-A2}a},\phi_{\ref{infcase:7-reg}a},\phi_{\ref{infcase:8-sub}a}\\\ &\phi_{\ref{ssec:GL2}.1b}&\phi_{\ref{infcase:2-long}b}&\phi_{\ref{infcase:3}b}&\phi_{\ref{infcase:4-D2}b},\phi_{\ref{infcase:6-A2}b},\phi_{\ref{infcase:7-reg}b},\phi_{\ref{infcase:8-sub}b}\\\ \hline\cr\ref{endcase:D2}&\phi_{\ref{ssec:SO4}.0}&\phi_{\ref{infcase:0}}&&\phi_{\ref{infcase:1-short}a},\phi_{\ref{infcase:2-long}a},\phi_{\ref{infcase:3}a},\phi_{\ref{infcase:4-D2}a},\phi_{\ref{infcase:5}a},\phi_{\ref{infcase:6-A2}a},\phi_{\ref{infcase:7-reg}a},\phi_{\ref{infcase:8-sub}a}\\\ &\phi_{\ref{ssec:SO4}.1a}&\phi_{\ref{infcase:1-short}a}&\phi_{\ref{infcase:5}a}&\phi_{\ref{infcase:7-reg}a}\\\ &\phi_{\ref{ssec:SO4}.1b}&\phi_{\ref{infcase:1-short}b}&\phi_{\ref{infcase:5}b}&\phi_{\ref{infcase:7-reg}c}\\\ &\phi_{\ref{ssec:SO4}.2a}&\phi_{\ref{infcase:2-long}a}&\phi_{\ref{infcase:3}a}&\phi_{\ref{infcase:6-A2}a},\phi_{\ref{infcase:7-reg}a}\\\ &\phi_{\ref{ssec:SO4}.2b}&\phi_{\ref{infcase:2-long}b}&\phi_{\ref{infcase:3}b}&\phi_{\ref{infcase:6-A2}b},\phi_{\ref{infcase:7-reg}b}\\\ &\phi_{\ref{ssec:SO4}.3a}&\phi_{\ref{infcase:4-D2}a},\phi_{\ref{infcase:8-sub}a}&&\\\ &\phi_{\ref{ssec:SO4}.3b}&\phi_{\ref{infcase:4-D2}b},\phi_{\ref{infcase:8-sub}b}&&\\\ &\phi_{\ref{ssec:SO4}.3c}&\phi_{\ref{infcase:4-D2}c},\phi_{\ref{infcase:8-sub}c}&&\\\ &\phi_{\ref{ssec:SO4}.3d}&\phi_{\ref{infcase:4-D2}d},\phi_{\ref{infcase:8-sub}d}&&\\\ \hline\cr\ref{endcase:A2}&\phi_{\ref{ssec:PGL3}.0}&\phi_{\ref{infcase:0}}&&\phi_{\ref{infcase:1-short}a},\phi_{\ref{infcase:5}a}\\\ &\phi_{\ref{ssec:PGL3}.1a}&\phi_{\ref{infcase:2-long}a}&&\phi_{\ref{infcase:4-D2}a},\phi_{\ref{infcase:7-reg}a}\\\ &\phi_{\ref{ssec:PGL3}.1b}&\phi_{\ref{infcase:2-long}b}&&\phi_{\ref{infcase:4-D2}b},\phi_{\ref{infcase:7-reg}b}\\\ &\phi_{\ref{ssec:PGL3}.2a}&&\phi_{\ref{infcase:3}a}&\\\ &\phi_{\ref{ssec:PGL3}.2b}&&\phi_{\ref{infcase:3}b}&\\\ &\phi_{\ref{ssec:PGL3}.3a}&\phi_{\ref{infcase:6-A2}a},\phi_{\ref{infcase:8-sub}a}&&\\\ &\phi_{\ref{ssec:PGL3}.3b}&&\phi_{\ref{infcase:6-A2}b}&\phi_{\ref{infcase:8-sub}b}\\\ &\phi_{\ref{ssec:PGL3}.3c}&&\phi_{\ref{infcase:6-A2}c}&\phi_{\ref{infcase:8-sub}b}\\\ &\phi_{\ref{ssec:PGL3}.3d}&\phi_{\ref{infcase:6-A2}d},\phi_{\ref{infcase:8-sub}d}&&\\\ \end{array}$

###  4.2. Regular conormal vectors and Arthur parameters

Let $(G,s,\xi)$ be an endoscopic triple for $G_{2}$ and let
$\phi:W^{\prime}_{F}\to{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G}}$ be an unramified
Langlands parameter. Let $\lambda$ be the infinitesimal parameter for $\phi$;
then $\xi\circ\lambda$ is the infinitesimal parameter for $\xi\circ\phi$. Now
consider the map
$\xi:{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G}}\to{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G_{2}}}$
on dual groups and the induced homomorphism
${\widehat{G}}\to{\widehat{G_{2}}}$, also denoted by $\xi$ below. Pass to
tangent spaces $d\xi:{\widehat{\mathfrak{g}}}\to{\widehat{\mathfrak{g}_{2}}}$
and restricted to Vogan varieties to define the embedding
$d\xi|_{V_{\lambda}}:V_{\lambda}\to V_{\xi\circ\lambda}$ such that the
subvariety $V_{\lambda}$ is the fixed-point set of the
$(V_{\xi\circ\lambda})^{s}=V_{\lambda}$. This induces
$d\xi|_{T^{*}(V_{\lambda})}:T^{*}(V_{\lambda})\to T^{*}(V_{\xi\circ\lambda})$
such that $\left(T^{*}(V_{\xi\circ\lambda})\right)^{s}=T^{*}(V_{\lambda})$;
recall that we may view $T^{*}(V_{\lambda})$ as a subvariety of
${\widehat{\mathfrak{g}}}$. Likewise, this induces a map of conormal bundles
$d\xi|_{\Lambda_{\lambda}}:\Lambda_{\lambda}\to\Lambda_{\xi\circ\lambda}$ such
that $\Lambda_{\lambda}=\left(\Lambda_{\xi\circ\lambda}\right)^{s}$; again, we
may view $\Lambda_{\lambda}$ as a subvariety of ${\widehat{\mathfrak{g}}}$.
Now let $x_{\phi}$ (resp. $x_{\xi\circ\phi}$) be the point in the moduli space
$V_{\lambda}$ (resp. $V_{\xi\circ\lambda}$) for
$\phi:W^{\prime}_{F}\to{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G}}$ (resp. for
$\xi\circ\phi:W^{\prime}_{F}\to{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G_{2}}}$) and
let $C_{\phi}$ (resp. $C_{\xi\circ\phi}$) be its $H_{\lambda}$-orbit (resp.
$H_{\xi\circ\lambda}$-orbit). Now $d\xi|_{\Lambda_{\lambda}}$ restricts to an
immersion
$d\xi|_{\Lambda_{\lambda,C_{\phi}}}:\Lambda_{\lambda,C_{\phi}}\to\Lambda_{\xi\circ\lambda,C_{\xi\circ\phi}}.$Now
consider the restriction

${\Lambda_{\lambda,C_{\phi}}}$${\Lambda_{\xi\circ\lambda,C_{\xi\circ\phi}}}$${\Lambda_{\lambda,C_{\phi}}^{\text{sreg}}}$$\scriptstyle{d\xi|_{\Lambda_{\lambda,C_{\phi}}}}$$\scriptstyle{d\xi|_{\Lambda_{\lambda,C_{\phi}}^{\text{sreg}}}}$

######  Definition 6.

Let $(G,s,\xi)$ be an endoscopic triple for $G_{2}$ and let
$\phi:W^{\prime}_{F}\to{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G}}$ be an unramified
Langlands parameter. Let us say that a Langlands parameter
$\phi:W^{\prime}_{F}\to{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G}}$ is
$\xi$-conormal, if the image of
$d\xi|_{\Lambda_{\lambda,C_{\phi}}^{\text{sreg}}}$ is contained in
$\Lambda_{\xi\circ\lambda,C_{\xi\circ\phi}}^{\text{sreg}}$. If $\phi$ is
$\xi$-conormal the map

$d\xi|_{\Lambda_{\lambda,C_{\phi}}^{\text{sreg}}}:{\Lambda_{\lambda,C_{\phi}}^{\text{sreg}}}\to\Lambda_{\xi\circ\lambda,C_{\xi\circ\phi}}^{\text{sreg}}$

induces a group homomorphism of equivariant fundamental groups, denoted by

$d\xi_{\phi}:A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}\to
A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\xi\circ\phi}.$

######  Proposition 4.1.

Let $(G,s,\xi)$ be an endoscopic triple for $G_{2}$ and let
$\phi:W^{\prime}_{F}\to{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G}}$ be an unramified
Langlands parameter. If $\phi$ is of Arthur type then $\phi$ is
$\xi$-conormal.

###### Proof.

Let $\psi:W^{\prime\prime}_{F}\to{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G}}$ be the
Arthur parameter such that $\phi(w,x)=\psi(w,x,d_{w})$. Set
$x{\,:=\,}d\psi(1,e,1)$ and $y{\,:=\,}d\psi(1,1,f)$. Then by [CFMMX]*Prop.
6.1.1 we have $(x,y)$ is a strongly regular conormal vector for $\phi$.
Consider
$\xi\circ\psi:W^{\prime\prime}_{F}\to{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G_{2}}}$.
Then again by [CFMMX]*Prop. 6.1.1 we have
$d\xi(x,y)=(d(\xi\circ\psi)(1,e,1),d(\xi\circ\psi)(1,1,f))$ is a strongly
regular conormal vector for $\xi\circ\phi$ . ∎

In Table 4.1.2 we present a complete list of all endoscopic triples
$(G,s,\xi)$ and all unramified Langlands parameters
$\phi:W^{\prime}_{F}\to{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G_{2}}}$ for which
$\phi$ is $\xi$-conormal. We now explain this table. The left-hand column of
Table 4.1.2 lists the endoscopic groups $G$ for $G_{2}$ as explained in
Section 3.1. Then, for each such $G$, we list the families of endoscopic
Langlands parameters
$\phi:W^{\prime}_{F}\to{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G}}$ as they appeared
in Sections 3.2 through 3.4. In Section 4.1 we saw that for each $\phi$ in
these families, there are multiple Langlands parameters for $G_{2}(F)$ to
which $\phi$ may lift, depending on the properties of $\phi$. Each row of
Table 4.1.2 lists the lifts $\xi\circ\phi$ that arise as $\phi$ ranges through
each family, partially ordered left to right along rows by the relative
dimension $\dim(\xi\circ\phi)-\dim(\phi)$; note that if this relative
dimension is $0$ then $\phi$ is trivially $\xi$-conormal.

###  4.3. Statement of the fixed-point formula

######  Theorem 4.2.

Let $(G,s,\xi)$ be an endoscopic triple for $G_{2}$ and let
$\phi:W^{\prime}_{F}\to{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G}}$ be an unramified
Langlands parameter for $G$. If $\phi$ is $\xi$-conormal then

$\operatorname{trace}_{s}\left(\operatorname{\mathsf{N}\hskip-1.0pt\mathsf{E}\hskip-1.0pt\mathsf{v}\hskip-1.0pt\mathsf{s}}_{\xi\circ\phi}[\dim(\xi\circ\phi)]\mathcal{P}\right)=\operatorname{trace}_{s}\left(\operatorname{\mathsf{N}\hskip-1.0pt\mathsf{E}\hskip-1.0pt\mathsf{v}\hskip-1.0pt\mathsf{s}}_{\phi}[\dim(\phi)]\
{\operatorname{res}}\mathcal{P}\right)$

for every
$\mathcal{P}\in\mathsf{Per}_{H_{\xi\circ\lambda}}(V_{\xi\circ\lambda})$.

The proof of Theorem 4.2 will occupy the rest of this Section, so we explain
the strategy here. We work through the list of endoscopic triples $(G,s,\xi)$
for $G_{2}$ and, in each case, we recall that classification of Langlands
parameters $\phi:W^{\prime}_{F}\to{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G}}$ in
terms of infinitesimal parameters in Sections 3.2 through 3.4. We identify
those Langlands parameters $\phi$ that are $\xi$-conormal, expanding on the
information presented in Table 4.1.2. We then recall the classification of
Vogan varieties $H_{\xi\circ\lambda}\times V_{\xi\circ\lambda}\to
V_{\xi\circ\lambda}$ in terms of prehomogeneous vector spaces $H\times V\to V$
from Section 1.2 and we observe that same list of five prehomogeneous vector
spaces capture all instances of the Vogan varieties $H_{\lambda}\times
V_{\lambda}\to V_{\lambda}$. We further show that the embeddings
$d\xi|_{V_{\lambda}}:V_{\lambda}\to V_{\xi\circ\lambda},$ explained at the
beginning of Section 4.2, all take the form $V^{s}\subseteq V$ where $V$ is
one of these five prehomogeneous vector spaces and $s\in H$ has finite order
(hence semisimple). We are then able to match $L$-parameters $\phi$ with
$H^{s}$-orbits $C^{\prime}\subseteq V^{s}$ and identify those that are
$\xi$-conormal with the corresponding property of the $H^{s}$-orbit
$C^{\prime}$; we refer to these orbits as $V$-conormal. This is the genesis of
the notion of $\xi$-conormal, in fact. We then prove the theorem for these
prehomogeneous vector spaces in the form of Proposition 4.4, for all
$V$-conormal orbits $C^{\prime}\subseteq V^{s}$.

###  4.4. Prehomogeneous vector subspaces

######  Lemma 4.3.

Let $H\times V\to V$ be one of the five prehomogenous vector spaces appearing
in Proposition 1.2; observe that $H$ is reductive in these cases. For
semisimple $s\in H$, set $H^{s}{\,:=\,}Z_{H}(s)$ and $V^{s}{\,:=\,}\\{x\in V{\
|\ }s(x)=x\\}$. Then $H^{s}\times V^{s}\to V^{s}$ be is also a prehomogeneous
vector space and again one of the five prehomogenous vector spaces appearing
in Proposition 1.2.

###### Proof.

When $V=V^{s}$ and $H=H^{s}$ this is a tautology, whereas $V=V^{s}$ and $H\neq
H^{s}$ only occurs for 1.

Up to the natural notion of equivalence, the possibilities for proper fix-
point prehomogenous vector subspaces are the following.

  1. 2

The prehomogeneous vector space $V=\mathbb{A}^{1}$ with
$H=\operatorname{GL}_{1}$-action (scalar multiplication) has only one proper
fixed-point prehomogeneous vector subspace:

    1. (i)

$V^{s}=\\{0\\}$ with $H^{s}=\operatorname{GL}_{1}$ (this is 1);

  2. 3

The prehomogeneous vector space $V=\mathbb{A}^{2}$ with
$H=\operatorname{GL}_{2}$-action given by twisted matrix multiplication
$h.x=\det(h)^{n}\,hx$ also has the following proper fixed-point prehomogeneous
vector subspaces:

    1. (i)

$V^{s}=\\{0\\}$ with $H^{s}=\operatorname{GL}_{2}$ or
$\operatorname{GL}_{1}^{2}$ (this is 1);

    2. (ii)

$V^{s}=\\{(x_{1},0){\ |\ }x_{1}\\}$ and $H^{s}=\operatorname{GL}_{1}^{2}$
(this is equivalent to 2);

  3. 4

The prehomogeneous vector space $V=\mathbb{A}^{2}$ with
$H=\operatorname{GL}_{1}^{2}$-action given by
$(t_{1},t_{2}).(x_{1},x_{2})=(t_{1}x_{1},t_{1}t_{2}^{n}x_{2})$, for positive
integer $n$, the following proper fixed-point prehomogeneous vector subspaces
are possible:

    1. (i)

$V^{s}=\\{(0,0)\\}$ and $H^{s}=H$ (this is 1);

    2. (ii)

$V^{s}=\\{(x_{1},0){\ |\ }x_{1}\\}$ and $H^{s}=H$ (this is 2);

    3. (iii)

$V^{s}=\\{(0,x_{2}){\ |\ }x_{2}\\}$ and $H^{s}=H$ (this is 2);

  4. 5

The prehomogeneous vector space $V=\mathbb{A}^{4}$ with
$H=\operatorname{GL}_{2}$ and action $\det^{-1}\otimes\operatorname{Sym}^{3}$
has the following proper fixed-point prehomogeneous vector subspaces, up to
isomorphism:

    1. (i)

$V^{s}=\\{(0,0,0,0)\\}$ and $H^{s}=\operatorname{GL}_{2}$ or
$\operatorname{GL}_{1}^{2}$ (this is 1);

    2. (ii)

$V^{s}=\\{(x_{1},0,0,0){\ |\ }x_{1}\\}$ and $H^{s}=\operatorname{GL}_{1}^{2}$
(this is equivalent to 2);

    3. (iii)

$V^{s}=\\{(0,x_{2},0,0){\ |\ }x_{2}\\}$ and $H^{s}=\operatorname{GL}_{1}^{2}$
(this is equivalent to 2);

    4. (iv)

$V^{s}=\\{(x_{1},0,x_{3},0){\ |\ }x_{1},x_{3}\\}$ and
$H^{s}=\operatorname{GL}_{1}^{2}$ (this is 4 for $n=2$);

    5. (v)

$V^{s}=\\{(x_{1},0,0,x_{4}){\ |\ }x_{1},x_{4}\\}$ and
$H^{s}=\operatorname{GL}_{1}^{2}$ (this is 4 for $n=3$). ∎

###  4.5. $V$-conormal

Let $H\times V\to V$ be one of the prehomogeneous vector spaces appearing in
Proposition 1.2 and let $H^{s}\times V^{s}\to V^{s}$ be one of the subspaces
appearing in Lemma 4.3, for semisimple $s\in H$. Then
$T^{*}(V)^{s}=T^{*}(V^{s})$, where $H$ acts on $T^{*}(V)$ by
$s\cdot(x,y){\,:=\,}(s\cdot x,s\cdot y)$. Recall that $\Lambda=\\{(x,y)\in
T^{*}(V){\ |\ }[x,y]=0\\}$; then $\Lambda^{s}=\\{(x^{\prime},y^{\prime})\in
T^{*}(V^{s}){\ |\ }[x^{\prime},y^{\prime}]=0\\}$.

Now let $C^{\prime}\subseteq V^{s}$ be an $H^{s}$-orbit. The $H$-orbit of
$C^{\prime}$, denoted by $H\cdot C^{\prime}\subseteq V$ and sometimes called
the saturation of $C^{\prime}$, is a single $H$-orbit, denoted below by $C$.
The intersection $C^{s}$ of $C$ with $V^{s}$ contains $C^{\prime}$ but may
contain other $H^{s}$-orbits: we write $C^{s}=\cup_{i}C^{\prime}_{i}$. Then,
$(\Lambda_{C})^{s}=\cup_{i}(\Lambda^{s})_{C^{\prime}_{i}}$ Passing to the
regular part of the conormal varieties we find that it can happen that
$(\Lambda^{\text{sreg}}_{C})^{s}$ and
$\cup_{i}(\Lambda^{s}_{C_{i}^{\prime}})^{\text{sreg}}$ are not equal. Of
course, this is essentially the same phenomenon discussed in Section 4.2.
Definition 6 now takes this form.

######  Definition 7.

With notation as above, let us say that an $H^{s}$-orbit $C^{\prime}\subseteq
V^{s}$ is $V$-conormal if
$(\Lambda^{s}_{C^{\prime}})^{\text{sreg}}\subseteq(\Lambda^{\text{sreg}}_{C})^{s}$,
where the $H$-orbit $C\subseteq V$ is the saturation of $C^{\prime}$. In this
case, this inclusion induces a map of equivariant fundamental groups
$A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{C^{\prime}}\to
A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{C}$.

Whenever $V=V^{s}$ and $H=H^{s}$ then we trivially have that every orbit is
$V$-conormal. For those cases under consideration in this paper where
$V=V^{s}$ and $H\neq H^{s}$ the same remains true. We note that this only
occurs for 1.

From the cases of prehomogenous vector spaces $V$ and proper subspaces $V^{s}$
appearing in Lemma 4.3, only the following $H^{s}$-orbits $C^{\prime}\subset
V^{s}$ are $V$-conormal:

  1. (1)

Case 3(ii). The closed orbit $C^{\prime}_{0}$ and the open orbit
$C^{\prime}_{1}$ in $V^{s}$ are $V$-conormal. We note that the saturation of
the closed $H^{s}$-orbit $C^{\prime}_{0}$ in $V^{s}$ is the closed $H$-orbit
$C_{0}$ in $V$; the saturation of the open $H^{s}$-orbit $C^{\prime}_{1}$ in
$V^{s}$ is the $H$-orbit $C_{1}$ in $V$, which is open.

  2. (2)

Case 5(iv). Every $H^{s}$-orbit $C^{\prime}$ in $V^{s}$ is $V$-conormal. These
orbits are denoted by $C^{\prime}_{0}$, $C^{\prime}_{1}$, $C^{\prime}_{2}$ and
$C^{\prime}_{3}$, where $C^{\prime}_{0}$ is closed and $C^{\prime}_{3}$ is
open and where $C^{\prime}_{1}$ is the $H^{s}$-orbit of $(1,0,0,0)$ and
$C^{\prime}_{2}$ is the $H^{s}$-orbit of $(0,0,1,0)$. The saturation of the
closed $H^{s}$-orbit $C^{\prime}_{0}$ is the closed $H$-orbit $C_{0}$, the
saturation of $C^{\prime}_{1}$ is $C_{1}$, the saturation of $C^{\prime}_{2}$
is $C_{2}$ and the saturation of the open $H^{s}$-orbit $C^{\prime}_{3}$ is
the open $H$-orbit $C_{3}$.

  3. (3)

Case 5(v). Only two of the four $H^{s}$-orbits $C^{\prime}$ in $V^{s}$ are
$V$-conormal. Let the $H^{s}$-orbits in $V^{s}$ be denoted by
$C^{\prime}_{0}$, $C^{\prime}_{1}$, $C^{\prime}_{2}$ and $C^{\prime}_{3}$,
where $C^{\prime}_{0}$ is closed and $C^{\prime}_{3}$ is open and where
$C^{\prime}_{1}$ is the $H^{s}$-orbit of $(1,0,0,0)$ while $C^{\prime}_{2}$ is
the $H^{s}$-orbit of $(0,0,0,1)$. Then the saturation of the closed
$H^{s}$-orbit $C^{\prime}_{0}$ is the closed $H$-orbit $C_{0}$, the saturation
of $C^{\prime}_{1}$ is $C_{1}$, the saturation of $C^{\prime}_{2}$ is $C_{1}$
again, and the saturation of the open $H^{s}$-orbit $C^{\prime}_{3}$ is the
open $H$-orbit $C_{3}$. Only $C^{\prime}_{0}$ and $C^{\prime}_{3}$ are
$V$-conormal.

###  4.6. Restriction

Let $(G,s,\xi)$ be an endscopic triple for $G_{2}$ over $F$. Let $\lambda$ be
an infinitesimal parameter for $G$. To prove Theorem 4.2 we will calculate the
equivariant restriction functor

${\operatorname{res}}:D_{H_{\lambda}}(V_{\xi\circ\lambda})\to
D_{H_{\lambda}}(V_{\lambda})$

on simple objects in the abelian category
$\mathsf{Per}_{H}(V_{\xi\circ\lambda})$. As we will see, the restriction
${\operatorname{res}}\,\mathcal{P}$ of an equivariant perverse sheaf is not
necessarily perverse. To address this issue, we replace
$\mathsf{Per}_{H}(V_{\lambda})$ with the full subcategory
$\mathsf{Per}_{H}^{\bullet}(V_{\lambda})$ of $D_{H}(V_{\lambda})$ generated by
shifts $\mathcal{P}[n]$ of equivariant perverse sheaves. In this section we
will see that ${\operatorname{res}}$ defines

${\operatorname{res}}:\mathsf{Per}^{\bullet}_{H_{\xi\circ\lambda}}(V_{\xi\circ\lambda})\to\mathsf{Per}^{\bullet}_{H_{\lambda}}(V_{\lambda})$

We use this fact to replace restriction along $d\xi:V_{\lambda}\to
V_{\xi\circ\lambda}$ with restriction along $V^{s}\hookrightarrow V$, where
$H\times V\to V$ is one of the five prehomogeneous vector spaces from
Proposition 1.2 and $s\in H$ is semisimple. The functor

${\operatorname{res}}:\mathsf{Per}^{\bullet}_{H}(V)\to\mathsf{Per}^{\bullet}_{H^{s}}(V^{s})$

on simple objects is given in Table 4.6.1 for the three prehomogeneous vector
spaces appearing semisimple $s\in H$ appearing in Section 4.5.

Table 4.6.1. Restrictions of Standard and Simple Perverse Sheaves

$\begin{array}[]{l l l || l l c
}\text{Case}&\text{Standard}&\text{Restriction}&\text{Perverse}&\text{Restriction}&\\\
&\text{sheaf}&&\text{sheaf}&&\\\
\hline\cr\hline\cr\ref{geocase-2}(ii)&{\mathbbm{1}}_{C_{0}}&{\mathbbm{1}}_{C_{0}^{\prime}}&{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{0}})&{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C^{\prime}_{0}})\\\
&{\mathbbm{1}}_{C_{1}}&{\mathbbm{1}}_{C_{1}^{\prime}}&{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{1}})&{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{1}^{\prime}})[1]\\\
\hline\cr\ref{geocase-
sub}(iv)&{\mathbbm{1}}_{C_{0}}&{\mathbbm{1}}_{C_{0}^{\prime}}&{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{0}})&{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C^{\prime}_{0}})&\\\
&{\mathbbm{1}}_{C_{1}}&{\mathbbm{1}}_{C_{1}^{\prime}}&{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{1}})&{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C^{\prime}_{1}})[1]&\\\
&{\mathbbm{1}}_{C_{2}}&{\mathbbm{1}}_{C_{2}^{\prime}}&{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{2}})&\mathcal{F}_{2}[2]\oplus{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C^{\prime}_{0}})[1]&\\\
&{\mathbbm{1}}_{C_{3}}&{\mathbbm{1}}_{C_{3}^{\prime}}&{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{3}})&{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{3}^{\prime}})[2]&\\\
&\varrho_{C_{3}}&{\mathbbm{1}}_{C_{3}^{\prime}}\oplus\rho_{C_{3}^{\prime}}&{\mathcal{I\hskip-1.0ptC}}(\varrho_{C_{3}})&\mathcal{F}_{3}[2]\oplus{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C^{\prime}_{0}})[2]\oplus{\mathcal{I\hskip-1.0ptC}}(\rho_{C^{\prime}_{3}})[2]&\\\
&\varepsilon_{C_{3}}&\rho_{C_{3}^{\prime}}&{\mathcal{I\hskip-1.0ptC}}(\varepsilon_{C_{3}})&{\mathcal{I\hskip-1.0ptC}}(\rho_{C^{\prime}_{3}})[2]&\\\
\hline\cr\ref{geocase-
sub}(v)&{\mathbbm{1}}_{C_{0}}&{\mathbbm{1}}_{C_{0}^{\prime}}&{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{0}})&{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C^{\prime}_{0}})&\\\
&{\mathbbm{1}}_{C_{1}}&{\mathbbm{1}}_{C_{1}^{\prime}}\oplus{\mathbbm{1}}_{C_{2}^{\prime}}&{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{1}})&\mathcal{F}_{4}[1]&\\\
&{\mathbbm{1}}_{C_{2}}&0&{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{2}})&\mathcal{F}_{4}[2]\oplus{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C^{\prime}_{0}})[1]&\\\
&{\mathbbm{1}}_{C_{3}}&{\mathbbm{1}}_{C_{3}^{\prime}}&{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{3}})&{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{3}^{\prime}})[2]&\\\
&\varrho_{C_{3}}&\rho_{C^{\prime}_{3}}\oplus\rho^{2}_{C^{\prime}_{3}}&{\mathcal{I\hskip-1.0ptC}}(\varrho_{C_{3}})&{\mathcal{I\hskip-1.0ptC}}(\rho_{C^{\prime}_{3}})[2]\oplus{\mathcal{I\hskip-1.0ptC}}(\rho^{2}_{C^{\prime}_{3}})[2]\oplus{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C^{\prime}_{0}})[2]&\\\
&\varepsilon_{C_{3}}&{\mathbbm{1}}_{C_{3}^{\prime}}&{\mathcal{I\hskip-1.0ptC}}(\varepsilon_{C_{3}})&\mathcal{F}_{5}[2]&\\\
\hline\cr\end{array}$

In Table 4.6.1 we find four indecomposable perverse sheaves, given here.

  * •

On 4 (for $n=2$) we define $\mathcal{F}_{2}={\mathbbm{1}}_{C_{2}\cup C_{1}\cup
C_{0}}[1]$ which is cohomologically equivalent to a complex

$\cdots\rightarrow
0\rightarrow({\mathbbm{1}}_{\overline{C_{2}}}\oplus{\mathbbm{1}}_{\overline{C_{1}}})\rightarrow{\mathbbm{1}}_{C_{0}}$

and hence sits in the exact sequence

$0\rightarrow{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{0}})\rightarrow\mathcal{F}_{2}\rightarrow{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{1}})\oplus{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{2}})\rightarrow
0,$

which we use to compute the functor
$\operatorname{\mathsf{N}\hskip-1.0pt\mathsf{E}\hskip-1.0pt\mathsf{v}\hskip-1.0pt\mathsf{s}}$
on these perverse sheaves in Table 4.7.2.

  * •

On 4 (for $n=2$) we define $\mathcal{F}_{3}={\mathbbm{1}}_{C_{3}\cup
C_{2}}[2]$ which is cohomologically equivalent to a complex

$\cdots\rightarrow 0\rightarrow{\mathbbm{1}}_{\overline{C_{3}}}\rightarrow
1_{\overline{C_{1}}}\rightarrow 0$

and hence sits in the exact sequence

$0\rightarrow{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{1}})\rightarrow\mathcal{F}_{3}\rightarrow{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{3}})\rightarrow
0,$

which we use to compute the functor
$\operatorname{\mathsf{N}\hskip-1.0pt\mathsf{E}\hskip-1.0pt\mathsf{v}\hskip-1.0pt\mathsf{s}}$
on these perverse sheaves in Table 4.7.2.

  * •

On 4 (for $n=2$) we define $\mathcal{F}_{4}={\mathbbm{1}}_{C_{2}\cup C_{1}\cup
C_{0}}[1]$ which is cohomologically equivalent to a complex

$\cdots\rightarrow
0\rightarrow({\mathbbm{1}}_{\overline{C_{2}}}\oplus{\mathbbm{1}}_{\overline{C_{1}}})\rightarrow{\mathbbm{1}}_{C_{0}}$

and hence sits in the exact sequence

$0\rightarrow{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{0}})\rightarrow\mathcal{F}_{4}\rightarrow{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{1}})\oplus{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{2}})\rightarrow
0,$

which we use to compute the functor
$\operatorname{\mathsf{N}\hskip-1.0pt\mathsf{E}\hskip-1.0pt\mathsf{v}\hskip-1.0pt\mathsf{s}}$
on these perverse sheaves in Table 4.7.3.

  * •

On 4 (for $n=2$) we define $\mathcal{F}_{5}={\mathbbm{1}}_{C_{3}}$ which is
cohomologically equivalent to a complex

${\mathbbm{1}}_{\overline{C_{3}}}[2]\rightarrow({\mathbbm{1}}_{\overline{C_{2}}}\oplus{\mathbbm{1}}_{\overline{C_{1}}})[1]\rightarrow{\mathbbm{1}}_{C_{0}}[0]$

and hence sits in the exact sequence

$0\rightarrow\mathcal{F}_{4}\rightarrow\mathcal{F}_{5}\rightarrow{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{3}})\rightarrow
0,$

which we use to compute the functor
$\operatorname{\mathsf{N}\hskip-1.0pt\mathsf{E}\hskip-1.0pt\mathsf{v}\hskip-1.0pt\mathsf{s}}$
on these perverse sheaves in Table 4.7.3.

###  4.7. Proof of Theorem 4.2

Let $H\times V\to V$ be one of the prehomogeneous vector spaces appearing in
Proposition 1.2. The result is immediate if $V=V^{s}$ and $H=H^{s}$ or in the
case 1 so we let $H^{s}\times V^{s}\to V^{s}$ be one of the proper
prehomogeneous vector spaces appearing in Lemma 4.3. Let $C^{\prime}$ be an
$H^{s}$-orbit in $V^{s}$ and let $C\subseteq V$ be the saturation of
$C^{\prime}$. In Proposition 4.4 we show that if $C^{\prime}$ is $V$-conormal
then the outside of the following diagram commutes.

${D_{H}(V)}$${D_{H}(\Lambda^{\text{sreg}}_{C})}$${D_{H}((\Lambda^{\text{sreg}}_{C})^{s})}$${{\mathbb{C}}}$${D_{H^{s}}(V^{s})}$${D_{H^{s}}((\Lambda^{s}_{C^{\prime}})^{\text{sreg}})}$$\scriptstyle{res}$$\scriptstyle{\operatorname{\mathsf{N}\hskip-1.0pt\mathsf{E}\hskip-1.0pt\mathsf{v}\hskip-1.0pt\mathsf{s}}_{C}[\dim
C]}$$\scriptstyle{{\operatorname{res}}}$$\scriptstyle{\operatorname{trace}_{s}}$$\scriptstyle{{\operatorname{res}}}$$\scriptstyle{\operatorname{\mathsf{N}\hskip-1.0pt\mathsf{E}\hskip-1.0pt\mathsf{v}\hskip-1.0pt\mathsf{s}}_{C^{\prime}}[\dim
C^{\prime}]}$$\scriptstyle{\operatorname{trace}_{s}}$

It is important to note that the square in this diagram does not commute.

######  Proposition 4.4.

Let $H\times V\to V$ be one of the prehomogeneous vector spaces appearing in
Proposition 1.2 and let $H^{s}\times V^{s}\to V^{s}$ be one of the
prehomogeneous vector spaces appearing in Lemma 4.3. Let $C^{\prime}$ be an
$H^{s}$-orbit in $V^{s}$. If $C^{\prime}$ is $V$-conormal then

$\operatorname{trace}_{s}\left(\operatorname{\mathsf{N}\hskip-1.0pt\mathsf{E}\hskip-1.0pt\mathsf{v}\hskip-1.0pt\mathsf{s}}_{C}[\dim
C]\
\mathcal{P}\right)=\operatorname{trace}_{s}\left(\operatorname{\mathsf{N}\hskip-1.0pt\mathsf{E}\hskip-1.0pt\mathsf{v}\hskip-1.0pt\mathsf{s}}_{C^{\prime}}[\dim
C^{\prime}]{\operatorname{res}}\,\mathcal{P}|_{V^{s}}\right),$ (13)

for all equivariant perverse sheaves $\mathcal{P}$ on $V$, where $C\subseteq
V$ is the saturation of $C^{\prime}$.

###### Proof.

We prove this by verifying Equation (13) by calculating the left- and right-
hand sides of Equation (13) independently, in each of the three cases
appearing in Section 4.5.

  1. (1)

Case 3(ii). The main calculations are summarized by the Table 4.7.1. The
calculations are completed by comparing $\operatorname{trace}_{s}$ of the
sheaves in the final two columns. In this case these are always $1$. We note
that for each $C_{i}^{\prime}$ the saturation is $C_{i}$.

Table 4.7.1. Summary of Calculations for Case 3(ii) $\mathcal{P}$ | ${\operatorname{res}}\ \mathcal{P}$ | $C^{\prime}$ | $\operatorname{\mathsf{N}\hskip-1.0pt\mathsf{E}\hskip-1.0pt\mathsf{v}\hskip-1.0pt\mathsf{s}}_{C}[\dim{C}]\ \mathcal{P}$ | $\operatorname{\mathsf{N}\hskip-1.0pt\mathsf{E}\hskip-1.0pt\mathsf{v}\hskip-1.0pt\mathsf{s}}_{C^{\prime}}[\dim{C^{\prime}}]\ {\operatorname{res}}\ \mathcal{P}$
---|---|---|---|---
${\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{0}})$ | ${\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{0}^{\prime}})$ | $C_{0}^{\prime}$ | ${\mathbbm{1}}_{\Lambda^{\text{sreg}}_{C_{0}}}[0]$ | ${\mathbbm{1}}_{\Lambda^{\text{sreg}}_{C^{\prime}_{0}}}[0]$
|  | $C_{1}^{\prime}$ | $0_{\Lambda^{\text{sreg}}_{C_{1}}}[0]$ | $0_{\Lambda^{\text{sreg}}_{C^{\prime}_{1}}}[0]$
${\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{1}})$ | ${\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{1}^{\prime}})[1]$ | $C_{0}^{\prime}$ | ${\mathbbm{1}}_{\Lambda^{\text{sreg}}_{C_{0}}}[0]$ | ${\mathbbm{1}}_{\Lambda^{\text{sreg}}_{C^{\prime}_{0}}}[0]$
|  | $C_{1}^{\prime}$ | ${\mathbbm{1}}_{\Lambda^{\text{sreg}}_{C_{1}}}[2]$ | ${\mathbbm{1}}_{\Lambda^{\text{sreg}}_{C^{\prime}_{1}}}[2]$
  2. (2)

Case 5(iv). This is the fixed-point prehomogeneous vector subspace
$H^{s}\times V^{s}\to V^{s}$ for
$s=\operatorname{diag}(-1,1)\in\operatorname{GL}_{2}$ and
$H=\operatorname{GL}_{2}$ acting on $V=\mathbb{A}^{4}$ by
$\det^{-1}\otimes\operatorname{Sym}^{3}$; then
$H^{s}=\operatorname{GL}_{1}^{2}$ and $V^{s}=\mathbb{A}^{2}$ with action
equivalent to 4 in the case $n=2$. Recall that we write $C^{\prime}_{0}$ for
the trivial $H^{s}$-orbit, $C^{\prime}_{1}$ for the $H^{s}$-orbit of
$(1,0,0,0)\in V^{s}$, $C^{\prime}_{2}$ for the $H^{s}$-orbit of $(0,0,1,0)\in
V^{s}$ and $C^{\prime}_{3}$ for the open $H^{s}$-orbit in $V^{s}$. The
equivariant fundamental group of each $H^{s}$-orbit is trivial except
$A_{C^{\prime}_{3}}=S_{2}$; let $\rho$ be the non-trivial quadratic character
of this group and let $\rho_{C^{\prime}_{3}}$ be the corresponding local
system on $C^{\prime}_{3}$.

The main calculations are summarized by the Table 4.7.2. The calculations are
completed by comparing $\operatorname{trace}_{s}$ of the sheaves in the final
two columns and accounting for shifts. Because the image of $s$ in
$A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{C}$ is order $2$
we have $\operatorname{trace}_{s}\varrho=0$,
$\operatorname{trace}_{s}\varepsilon=-1$, $\operatorname{trace}_{s}\rho=-1$
and $\operatorname{trace}_{s}\vartheta_{2}=-1$. We note that for each
$C_{i}^{\prime}$ the saturation is $C_{i}$.

Table 4.7.2. Summary of Calculations for Case 5(iv). In this table we write
$\vartheta:\\{1,\theta_{2}\\}\to{\mathbb{C}}^{\times}$ for the character
generally denoted in this paper by $\vartheta_{2}$, defined by
$\vartheta(\theta_{2})=\theta_{2}$.

$\begin{array}[]{cccccc}\mathcal{P}&{\operatorname{res}}\
\mathcal{P}&C^{\prime}&\operatorname{\mathsf{N}\hskip-1.0pt\mathsf{E}\hskip-1.0pt\mathsf{v}\hskip-1.0pt\mathsf{s}}_{C}[\dim{C}]\
\mathcal{P}&\operatorname{\mathsf{N}\hskip-1.0pt\mathsf{E}\hskip-1.0pt\mathsf{v}\hskip-1.0pt\mathsf{s}}_{C^{\prime}}[\dim{C^{\prime}}]\
{\operatorname{res}}\ \mathcal{P}\\\
\hline\cr\hline\cr{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{0}})&{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{0}^{\prime}})&C_{0}^{\prime}&{\mathbbm{1}}_{\Lambda^{\text{sreg}}_{C_{0}}}[0]&{\mathbbm{1}}_{\Lambda^{\text{sreg}}_{C^{\prime}_{0}}}[0]\\\
&&C_{1}^{\prime}&0&0\\\ &&C_{2}^{\prime}&0&0\\\ &&C_{3}^{\prime}&0&0\\\
\hline\cr{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{1}})&{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{1}^{\prime}})[1]&C_{0}^{\prime}&\varrho_{\Lambda^{\text{sreg}}_{C_{0}}}[0]&0\\\
&&C_{1}^{\prime}&{\mathbbm{1}}_{\Lambda^{\text{sreg}}_{C_{1}}}[2]&{\mathbbm{1}}_{\Lambda^{\text{sreg}}_{C^{\prime}_{1}}}[2]\\\
&&C_{2}^{\prime}&0&0\\\ &&C_{3}^{\prime}&0&0\\\
\hline\cr{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{2}})&\mathcal{F}_{2}[2]\oplus{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{0}^{\prime}})[1]&C_{0}^{\prime}&0&{\mathbbm{1}}_{\Lambda^{\text{sreg}}_{C^{\prime}_{0}}}[2]\oplus{\mathbbm{1}}_{\Lambda^{\text{sreg}}_{C^{\prime}_{0}}}[1]\\\
&&C_{1}^{\prime}&\vartheta_{\Lambda^{\text{sreg}}_{C_{1}}}[2]&{\mathbbm{1}}_{\Lambda^{\text{sreg}}_{C^{\prime}_{1}}}[3]\\\
&&C_{2}^{\prime}&{\mathbbm{1}}_{\Lambda^{\text{sreg}}_{C_{2}}}[3]&{\mathbbm{1}}_{\Lambda^{\text{sreg}}_{C^{\prime}_{2}}}[3]\\\
&&C_{3}^{\prime}&0&0\\\
\hline\cr{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{3}})&{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{3}^{\prime}})[2]&C_{0}^{\prime}&0&0\\\
&&C_{1}^{\prime}&0&0\\\ &&C_{2}^{\prime}&0&0\\\
&&C_{3}^{\prime}&{\mathbbm{1}}_{\Lambda^{\text{sreg}}_{C_{3}}}[4]&{\mathbbm{1}}_{\Lambda^{\text{sreg}}_{C^{\prime}_{3}}}[4]\\\
\hline\cr{\mathcal{I\hskip-1.0ptC}}(\varrho_{C_{3}})&(\mathcal{F}_{3}\oplus{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{0}^{\prime}})\oplus{\mathcal{I\hskip-1.0ptC}}(\rho_{C_{3}^{\prime}}))[2]&C_{0}^{\prime}&0&{\mathbbm{1}}_{\Lambda^{\text{sreg}}_{C^{\prime}_{0}}}[2]\oplus\vartheta_{\Lambda^{\text{sreg}}_{C^{\prime}_{0}}}[2]\\\
&&C_{1}^{\prime}&0&{\mathbbm{1}}_{\Lambda^{\text{sreg}}_{C^{\prime}_{1}}}[3]\oplus\vartheta_{\Lambda^{\text{sreg}}_{C^{\prime}_{1}}}[3]\\\
&&C_{2}^{\prime}&\vartheta_{\Lambda^{\text{sreg}}_{C_{2}}}[3]&\vartheta_{\Lambda^{\text{sreg}}_{C^{\prime}_{2}}}[3]\\\
&&C_{3}^{\prime}&\varrho_{\Lambda^{\text{sreg}}_{C_{3}}}[4]&{\mathbbm{1}}_{\Lambda^{\text{sreg}}_{C^{\prime}_{3}}}[4]\oplus\vartheta_{\Lambda^{\text{sreg}}_{C^{\prime}_{3}}}[4]\\\
\hline\cr{\mathcal{I\hskip-1.0ptC}}(\varepsilon_{C_{3}})&{\mathcal{I\hskip-1.0ptC}}(\rho_{C^{\prime}_{3}})[2]&C_{0}^{\prime}&\varepsilon_{\Lambda^{\text{sreg}}_{C_{0}}}[0]&\vartheta_{\Lambda^{\text{sreg}}_{C^{\prime}_{0}}}[2]\\\
&&C_{1}^{\prime}&{\mathbbm{1}}_{\Lambda^{\text{sreg}}_{C_{1}}}[2]&\vartheta_{\Lambda^{\text{sreg}}_{C^{\prime}_{1}}}[3]\\\
&&C_{2}^{\prime}&\vartheta_{\Lambda^{\text{sreg}}_{C_{2}}}[3]&\vartheta_{\Lambda^{\text{sreg}}_{C^{\prime}_{2}}}[3]\\\
&&C_{3}^{\prime}&\varepsilon_{\Lambda^{\text{sreg}}_{C_{3}}}[4]&\vartheta_{\Lambda^{\text{sreg}}_{C^{\prime}_{3}}}[4]\\\
\end{array}$

  3. (3)

Case 5(v). The fixed-point prehomogenous space $H^{s}\times V^{s}\to V^{s}$ is
equivalent to 4 in the case $n=3$. Without loss of generality, we take
$s=\operatorname{diag}(\theta_{3}^{2},\theta_{3})$. Recall that in Case 5(v)
there are only two $H^{s}$-orbits $C^{\prime}\subset V^{s}$ that are
$V$-conormal: the open orbit $C^{\prime}_{0}$ and the closed orbit
$C^{\prime}_{3}$.

The main calculations are summarized by Table 4.7.3. The calculations are
completed by comparing $\operatorname{trace}_{s}$ of the sheaves in the final
two columns and accounting for shifts. Because the image of $s$ in
$A^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{C}$ has order
$3$ (or order $1$) we have $\operatorname{trace}_{s}\vartheta_{2}=1$,
$\operatorname{trace}_{s}\varepsilon=1$,
$\operatorname{trace}_{s}\varrho=\theta_{3}+\theta_{3}^{2}=-1$ and
$\operatorname{trace}_{s}\rho\oplus\rho^{2}=\theta_{3}+\theta_{3}^{2}=-1$.
Note that saturation of $C_{1}^{\prime}$ and $C_{2}^{\prime}$ are both
$C_{1}$, these do not satisfy the hypotheses, and that these cases do not
always satisfy (13).∎

Table 4.7.3. Summary of Calculations for Case 5(v). In this table we write
$\vartheta:\\{1,\theta_{3},\theta_{3}^{2}\\}\to{\mathbb{C}}^{\times}$ for the
character generally denoted in this paper by $\vartheta_{3}$, defined by
$\vartheta(\theta_{3})=\theta_{3}$.

$\begin{array}[]{cccccc}\mathcal{P}&{\operatorname{res}}\
\mathcal{P}&C^{\prime}&\operatorname{\mathsf{N}\hskip-1.0pt\mathsf{E}\hskip-1.0pt\mathsf{v}\hskip-1.0pt\mathsf{s}}_{C}[\dim{C}]\
\mathcal{P}&\operatorname{\mathsf{N}\hskip-1.0pt\mathsf{E}\hskip-1.0pt\mathsf{v}\hskip-1.0pt\mathsf{s}}_{C^{\prime}}[\dim{C^{\prime}}]\
{\operatorname{res}}\ \mathcal{P}\\\
\hline\cr\hline\cr{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{0}})&{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{0}^{\prime}})&C_{0}^{\prime}&{\mathbbm{1}}_{\Lambda^{\text{sreg}}_{C_{0}}}[0]&{\mathbbm{1}}_{\Lambda^{\text{sreg}}_{C^{\prime}_{0}}}[0]\\\
&&C_{1}^{\prime}&0&0\\\ &&C_{2}^{\prime}&0&0\\\ &&C_{3}^{\prime}&0&0\\\
\hline\cr{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{1}})&\mathcal{F}_{4}[1]&C_{0}^{\prime}&\varrho_{\Lambda^{\text{sreg}}_{C_{0}}}[0]&{\mathbbm{1}}_{\Lambda^{\text{sreg}}_{C^{\prime}_{0}}}[1]\\\
&&C_{1}^{\prime}&{\mathbbm{1}}_{\Lambda^{\text{sreg}}_{C_{1}}}[2]&{\mathbbm{1}}_{\Lambda^{\text{sreg}}_{C^{\prime}_{1}}}[2]\\\
&&C_{2}^{\prime}&{\mathbbm{1}}_{\Lambda^{\text{sreg}}_{C_{1}}}[2]&{\mathbbm{1}}_{\Lambda^{\text{sreg}}_{C^{\prime}_{2}}}[2]\\\
&&C_{3}^{\prime}&0&0\\\
\hline\cr{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{2}})&\mathcal{F}_{4}[2]\oplus{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C^{\prime}_{0}})[1]&C_{0}^{\prime}&0&{\mathbbm{1}}_{\Lambda^{\text{sreg}}_{C^{\prime}_{0}}}[2]\oplus{\mathbbm{1}}_{\Lambda^{\text{sreg}}_{C^{\prime}_{0}}}[1]\\\
&&C_{1}^{\prime}&\vartheta_{\Lambda^{\text{sreg}}_{C_{1}}}[2]&{\mathbbm{1}}_{\Lambda^{\text{sreg}}_{C^{\prime}_{1}}}[3]\\\
&&C_{2}^{\prime}&\vartheta_{\Lambda^{\text{sreg}}_{C_{1}}}[2]&{\mathbbm{1}}_{\Lambda^{\text{sreg}}_{C^{\prime}_{1}}}[3]\\\
&&C_{3}^{\prime}&0&0\\\
\hline\cr{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{3}})&{\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C_{3}^{\prime}})[2]&C_{0}^{\prime}&0&0\\\
&&C_{1}^{\prime}&0&0\\\ &&C_{2}^{\prime}&0&0\\\
&&C_{3}^{\prime}&{\mathbbm{1}}_{\Lambda^{\text{sreg}}_{C_{3}}}[4]&{\mathbbm{1}}_{\Lambda^{\text{sreg}}_{C^{\prime}_{3}}}[4]\\\
\hline\cr{\mathcal{I\hskip-1.0ptC}}(\varrho_{C_{3}})&({\mathcal{I\hskip-1.0ptC}}({\mathbbm{1}}_{C^{\prime}_{0}})\oplus{\mathcal{I\hskip-1.0ptC}}(\vartheta_{C^{\prime}_{3}})\oplus{\mathcal{I\hskip-1.0ptC}}(\vartheta^{2}_{C^{\prime}_{3}}))[2]&C_{0}^{\prime}&0&{\mathbbm{1}}_{\Lambda^{\text{sreg}}_{C^{\prime}_{0}}}[2]\oplus\vartheta_{\Lambda^{\text{sreg}}_{C^{\prime}_{3}}}[2]\oplus\vartheta^{2}_{\Lambda^{\text{sreg}}_{C^{\prime}_{3}}}[2]\\\
&&C_{1}^{\prime}&0&\vartheta_{\Lambda^{\text{sreg}}_{C^{\prime}_{3}}}[3]\oplus\vartheta^{2}_{\Lambda^{\text{sreg}}_{C^{\prime}_{3}}}[3]\\\
&&C_{2}^{\prime}&0&\vartheta_{\Lambda^{\text{sreg}}_{C^{\prime}_{3}}}[3]\oplus\vartheta^{2}_{\Lambda^{\text{sreg}}_{C^{\prime}_{3}}}[3]\\\
&&C_{3}^{\prime}&\varrho_{\Lambda^{\text{sreg}}_{C_{3}}}[4]&\vartheta_{\Lambda^{\text{sreg}}_{C^{\prime}_{3}}}[4]\oplus\vartheta^{2}_{\Lambda^{\text{sreg}}_{C^{\prime}_{3}}}[4]\\\
\hline\cr{\mathcal{I\hskip-1.0ptC}}(\varepsilon_{C_{3}})&\mathcal{F}_{5}[2]&C_{0}^{\prime}&\varepsilon_{\Lambda^{\text{sreg}}_{C_{0}}}[0]&{\mathbbm{1}}_{\Lambda^{\text{sreg}}_{C^{\prime}_{0}}}[2]\\\
&&C_{1}^{\prime}&{\mathbbm{1}}_{\Lambda^{\text{sreg}}_{C_{1}}}[2]&{\mathbbm{1}}_{\Lambda^{\text{sreg}}_{C^{\prime}_{1}}}[3]\\\
&&C_{2}^{\prime}&{\mathbbm{1}}_{\Lambda^{\text{sreg}}_{C_{1}}}[2]&{\mathbbm{1}}_{\Lambda^{\text{sreg}}_{C^{\prime}_{2}}}[3]\\\
&&C_{3}^{\prime}&\varepsilon_{\Lambda^{\text{sreg}}_{C_{3}}}[4]&{\mathbbm{1}}_{\Lambda^{\text{sreg}}_{C^{\prime}_{3}}}[4]\\\
\end{array}$

###  4.8. Geometric lifting of stable distributions

From Section 3, recall Definition 5:

$\Theta_{\phi,s}^{G^{\delta}}{\,:=\,}e(\delta)\sum_{\pi\in\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\phi}(G^{\delta}(F))}\hskip
0.0pt\operatorname{trace}_{s}\left(\operatorname{\mathsf{N}\hskip-1.0pt\mathsf{E}\hskip-1.0pt\mathsf{v}\hskip-1.0pt\mathsf{s}}_{\phi}[\dim(\phi)]\mathcal{P}(\pi)[-\dim(\pi)]\right)\
\Theta_{\pi},$

where $(G,s,\xi)$ is an endoscopic triple for $G_{2}$ and $\delta\in
Z^{1}(G,F)$ is a pure inner form of $G$. Recall also that this distribution
can be written in the form

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

######  Remark 4.5.

If the Langlands parameter
$\phi:W^{\prime}_{F}\to{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G}}$ is not relevant
to $G(F)$, in the sense of [Borel:Automorphic], then
$\Theta_{\phi,s}^{G^{\delta}}$ may be trivial; for example,
$\Theta_{\phi_{\ref{ssec:PGL3}.0}}^{\operatorname{PGL}_{3}^{\delta}}=0$ for
non-trivial $\delta$. On the other hand, $\Theta_{\phi,s}^{G^{\delta}}$ can be
non-trivial even when
$\phi:W^{\prime}_{F}\to{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G}}$ is not relevant
to $G(F)$; for example, although $\phi_{\ref{ssec:PGL3}.3a}$ is not relevant
to ${\operatorname{PGL}_{3}^{\delta}}$ for non-trivial $\delta$,
$\Theta_{\phi_{\ref{ssec:PGL3}.3a},s}^{\operatorname{PGL}_{3}^{\delta}}=e(\delta)\Theta_{\chi_{\operatorname{PGL}_{3}^{\delta}(F)}},$
which is non-trivial.

######  Definition 8.

Let $(G,s,\xi)$ be an endoscopic triple for $G_{2}$. Let
$\phi:W^{\prime}_{F}\to{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G}}$ be an unramified
Langlands parameter. We define

$\operatorname{Lift}_{(G,s,\xi)}\Theta_{\phi}^{G}{\,:=\,}\sum_{\pi\in\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\xi\circ\phi}(G_{2}(F))}\hskip-20.0pt\operatorname{trace}_{s}\left(\operatorname{\mathsf{N}\hskip-1.0pt\mathsf{E}\hskip-1.0pt\mathsf{v}\hskip-1.0pt\mathsf{s}}_{\phi}[\dim(\phi)]\
{\operatorname{res}}\,\mathcal{P}(\pi)[-\dim(\pi)]\right)\ \Theta_{\pi},$

where
${\operatorname{res}}:D_{Z_{\widehat{G_{2}}}(\xi\circ\phi)}(V_{\xi\circ\lambda})\to
D_{Z_{\widehat{G}}(\phi)}(V_{\lambda})$ is the equivariant functor defined by
${\operatorname{res}}\,\mathcal{F}=\mathcal{F}|_{V_{\lambda}}$. If the context
make it clear what the endoscopic triple is, we will sometimes use the
abbreviated notation $\operatorname{Lift}_{G}^{G_{2}}$ for
$\operatorname{Lift}_{(G,s,\xi)}$. Let $\delta\in Z^{1}(F,G)$ be a pure inner
form of $G$. Then $(G,1,\operatorname{id})$ is an endoscopic triple for
$G^{\delta}$ over $F$ and, extending the definition above,

$\operatorname{Lift}_{G}^{G^{\delta}}\Theta_{\phi}^{G}{\,:=\,}e(\delta)\mathop{\sum}\limits_{\pi\in\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\xi\circ\phi}(G^{\delta}(F))}\operatorname{trace}_{1}\left(\operatorname{\mathsf{N}\hskip-1.0pt\mathsf{E}\hskip-1.0pt\mathsf{v}\hskip-1.0pt\mathsf{s}}_{\phi}[\dim(\phi)]\
{\operatorname{res}}\,\mathcal{P}(\pi)[-\dim(\pi)]\right)\ \Theta_{\pi}.$

######  Conjecture 1.

Let $(G,s,\xi)$ be an endoscopic triple for $G_{2}$ and let
$\phi:W^{\prime}_{F}\to{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G}}$ be a unramified
Langlands parameter that is $\xi$-conormal. The Langlands-Shelstad transfer of
$\Theta^{G}_{\phi}$ from $G(F)$ to $G_{2}(F)$ is
$\Theta_{\xi\circ\phi,s}^{G_{2}}$. Now let $\delta\in Z^{1}(F,G)$ be a pure
inner form for $G$ over $F$ and suppose
$\phi:W^{\prime}_{F}\to{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G}}$ is also relevant
to $G^{\delta}(F)$. Then the Jacquet-Langlands transfer of $\Theta^{G}_{\phi}$
from $G(F)$ to $G^{\delta}(F)$ is $e(\delta)\Theta_{\phi}^{G^{\delta}}$

######  Theorem 4.6.

Let $(G,s,\xi)$ be an endoscopic triple for $G_{2}$. Let
$\phi:W^{\prime}_{F}\to{\hskip-2.0pt\,{}^{L}\hskip-1.0pt{G}}$ be a unramified
Langlands parameter that is $\xi$-conormal. Then

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

Let $\delta\in Z^{1}(F,G)$ be a pure inner form of $G$ and suppose also that
$\phi$ is relevant to $G^{\delta}$. Then

$\operatorname{Lift}_{G}^{G^{\delta}}\Theta_{\phi}^{G}=e(\delta)\Theta_{\phi}^{G^{\delta}}.$

###### Proof.

Theorem 4.2, we have

$\operatorname{trace}_{s}\left(\operatorname{\mathsf{N}\hskip-1.0pt\mathsf{E}\hskip-1.0pt\mathsf{v}\hskip-1.0pt\mathsf{s}}_{\phi}[\dim(\phi)]\
\mathcal{P}\right)=\operatorname{trace}_{s}\left(\operatorname{\mathsf{N}\hskip-1.0pt\mathsf{E}\hskip-1.0pt\mathsf{v}\hskip-1.0pt\mathsf{s}}_{\xi\circ\phi}[\dim(\xi\circ\phi)]{\operatorname{res}}\mathcal{P}\right),$
(14)

for every simple object $\mathcal{P}$ in
$\mathsf{Per}_{Z_{\widehat{G_{2}}}(\xi\circ\lambda)}(V_{\xi\circ\lambda})$.
Thus,

$\begin{array}[]{rcl
r}\operatorname{Lift}_{(G,s,\xi)}\Theta^{G}_{\phi}&:=&\mathop{\sum}\limits_{\pi\in\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\xi\circ\phi}(G_{2}(F))}\hskip
0.0pt\operatorname{trace}_{s}\left(\operatorname{\mathsf{N}\hskip-1.0pt\mathsf{E}\hskip-1.0pt\mathsf{v}\hskip-1.0pt\mathsf{s}}_{\phi}[\dim(\phi)]{\operatorname{res}}\,\mathcal{P}(\pi)[-\dim(\pi)]\right)\
\Theta_{\pi}&\\\
&=&\mathop{\sum}\limits_{\pi\in\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\xi\circ\phi}(G_{2}(F))}\hskip
0.0pt\operatorname{trace}_{s}\left(\operatorname{\mathsf{N}\hskip-1.0pt\mathsf{E}\hskip-1.0pt\mathsf{v}\hskip-1.0pt\mathsf{s}}_{\xi\circ\phi}[\dim(\xi\circ\phi)]\mathcal{P}(\pi)[-\dim(\pi)]\right)\
\Theta_{\pi}&\\\ &=:&\Theta_{\xi\circ\phi,s}.&\end{array}$

Directly from the definitions we see

$\begin{array}[b]{rcl}\operatorname{Lift}_{G}^{G^{\delta}}\Theta_{\phi}^{G}&{\,:=}&e(\delta)\sum_{\pi\in\Pi^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}_{\xi\circ\phi}(G^{\delta}(F))}\operatorname{trace}_{1}\left(\operatorname{\mathsf{N}\hskip-1.0pt\mathsf{E}\hskip-1.0pt\mathsf{v}\hskip-1.0pt\mathsf{s}}_{\phi}[\dim(\phi)]\
{\operatorname{res}}\,\mathcal{P}(\pi)[-\dim(\pi)]\right)\ \Theta_{\pi}\\\