# Howe duality and Dichotomy
for exceptional theta correspondences

Wee Teck Gan and Gordan Savin  W.T.G.: Department of Mathematics, National
University of Singapore, 10 Lower Kent Ridge Road Singapore 119076
<EMAIL_ADDRESS>G. S.: Department of Mathematics, University of Utah, Salt
Lake City, UT<EMAIL_ADDRESS>
###### Abstract.

We study three exceptional theta correspondences for $p$-adic groups, where
one member of the dual pair is the exceptional group $G_{2}$. We prove the
Howe duality conjecture for these dual pairs and a dichotomy theorem, and
determine explicitly the theta lifts of all non-cuspidal representations.

###### 2000 Mathematics Subject Classification:

11F27, 11F70, 22E50

##  1\. Introduction

Let $F$ be a non-archimedean local field of characteristic $0$ and residue
characteristic $p$. In this paper, we study the local theta correspondence
furnished by the following diagram of dual pairs:

$\textstyle{{\rm
PGSp}_{6}}$$\textstyle{G_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\rm
PD^{\times}}$$\textstyle{{\rm PGL_{3}\rtimes{\mathbb{Z}}/2{\mathbb{Z}}}}$

where $D$ denotes a cubic division $F$-algebra, so that $PD^{\times}$ is the
unique inner form of ${\rm PGL}_{3}$. More precisely, one has the dual pairs

$\begin{cases}({\rm PGL}_{3}\rtimes{\mathbb{Z}}/2{\mathbb{Z}})\times
G_{2}\subset E_{6}\rtimes{\mathbb{Z}}/2{\mathbb{Z}}\\\ PD^{\times}\times
G_{2}\subset E_{6}^{D}\\\ G_{2}\times{\rm PGSp}_{6}\subset E_{7}\end{cases}$

where the exceptional groups of type $E$ are all of adjoint type. In each of
the three cases, the centralizer of $G_{2}$ is a group $H_{J}={\rm Aut}(J)$,
where $J$ is a Freudenthal-Jordan algebra of rank 3. One can thus consider the
restriction of the minimal representation $\Pi$ [GS05] of $E$ to the relevant
dual pair and obtain a local theta correspondence.

More precisely, if $\pi\in{\rm Irr}(G_{2})$ is an irreducible smooth
representation of $G_{2}$, then the maximal $\pi$-isotypic quotient of $\Pi$

$\Pi/\cap_{\phi\in{\rm Hom}_{G_{2}}(\Pi,\pi)}{\rm ker}(\phi).$

can be expressed as $\pi\otimes\Theta(\pi)$ for some smooth representation
$\Theta(\pi)$ of $H_{J}$ [MVW, Lemme III.4]. The representation $\Theta(\pi)$
is called the big theta lift of $\pi$, and its maximal semi-simple quotient
(cosocle) is denoted $\theta(\pi)$. We say that $\pi$ has nonzero theta lift
to $H_{J}$ if $\Theta(\pi)\neq 0$, or equivalently ${\rm
Hom}_{G_{2}}(\Pi,\pi)\neq 0$. Similarly, one can consider the theta lift from
$H_{J}$ to $G_{2}$ and have the analogous notions.

The first main result of this paper is the following dichotomy theorem:

######  Theorem 1.1.

Let $\pi\in{\rm Irr}(G_{2}(F))$. Then $\pi$ has nonzero theta lift to exactly
one of $PD^{\times}$ or ${\rm PGSp}_{6}(F)$.

The group ${\rm PGL}_{3}\rtimes{\mathbb{Z}}/2{\mathbb{Z}}$ is not featured in
the dichotomy theorem, but it is needed for some finer aspects of the theta
correspondences. For example, every irreducible discrete series representation
of $G_{2}$ lifts to a discrete series representation of precisely one of the
three groups. After the above dichotomy theorem, we consider the problem of
understanding these theta correspondences more precisely. These local theta
correspondences have all been studied to some extent by Maggard-Savin [MS],
Gross-Savin [GrS], Gan [G99], Savin [Sa], Gan-Savin [GS99, GS04] and Savin-
Weissman [SWe]. Though various neat results were obtained in the various
cases, they fall short of determining the theta correspondences completely.
One of the main results of this paper is the completion of the analysis begun
in these papers.

The main difficulty in studying these exceptional theta correspondences is
that, unlike the classical theta correspondence, one does not know a priori
the analog of the Howe duality conjecture. Namely, one does not know that
$\Theta(\pi)$ has finite length with unique irreducible quotient (that is,
$\theta(\pi)$ is irreducible if $\Theta(\pi)$ is nonzero). In this paper, we
show that the analog of the Howe duality conjecture holds for these dual
pairs. To summarize, we have:

######  Theorem 1.2.

The Howe duality conjecture holds for the three dual pairs considered here.
Namely, for $\pi_{1},\pi_{2}\in{\rm Irr}(G_{2}(F))$, $\Theta(\pi_{i})$ has
finite length and

$\dim{\rm Hom}_{H_{J}}(\theta(\pi_{1}),\theta(\pi_{2}))\leq\dim{\rm
Hom}_{G_{2}}(\pi_{1},\pi_{2}).$

Likewise, for $\tau\in{\rm Irr}(H_{J})$, $\Theta(\tau)$ has finite length with
unique irreducible quotient (if nonzero).

More precisely, we have:

(i) The theta correspondence for $PD^{\times}\times G_{2}$ defines an
injective map

$\theta_{D}:{\rm Irr}^{\heartsuit}(PD^{\times})\hookrightarrow{\rm
Irr}(G_{2}(F)),$

where ${\rm Irr}^{\heartsuit}(PD^{\times})\subset{\rm Irr}(PD^{\times})$ is
the subset of representations which have nonzero theta lift to $G_{2}$. If
$p\neq 3$, then ${\rm Irr}^{\heartsuit}(PD^{\times})={\rm Irr}(PD^{\times})$,
so that one has an injective map:

$\theta_{D}:{\rm Irr}(PD^{\times})\hookrightarrow{\rm Irr}(G_{2}(F))$

(ii) The theta correspondence for $({\rm
PGL}_{3}(F)\rtimes{\mathbb{Z}}/2{\mathbb{Z}})\times G_{2}$ defines an
injective map

$\theta_{B}:{\rm Irr}^{\heartsuit}({\rm
PGL}_{3}(F)\rtimes{\mathbb{Z}}/2{\mathbb{Z}})\hookrightarrow{\rm
Irr}(G_{2}(F)),$

where ${\rm Irr}^{\heartsuit}({\rm
PGL}_{3}(F)\rtimes{\mathbb{Z}}/2{\mathbb{Z}})\subset{\rm Irr}({\rm
PGL}_{3}(F)\rtimes{\mathbb{Z}}/2{\mathbb{Z}})$ is the subset of
representations which have nonzero theta lift to $G_{2}$. Moreover, one can
determine the subset ${\rm Irr}^{\heartsuit}({\rm
PGL}_{3}(F)\rtimes{\mathbb{Z}}/2{\mathbb{Z}})$ explicitly, and the image of
$\theta_{B}$ is disjoint from that of $\theta_{D}$ by the dichotomy theorem.

(iii) The theta correspondence for $G_{2}\times{\rm PGSp}_{6}$ defines an
injection

$\theta:{\rm Irr}(G_{2}(F))\smallsetminus{\rm
Im}(\theta_{D})\hookrightarrow{\rm Irr}({\rm PGSp}_{6}(F)).$

For an irreducible representation $\tau$ of $PD^{\times}$, the non-vanishing
of $\theta_{D}(\tau)$ is equivalent to existence of non-zero vectors in $\tau$
fixed by a maximal torus in $PD^{\times}$. The existence of such vectors has
been checked by Lonka and Tandon [LT] in the tame case, where $p\neq 3$. Thus,
if $p=3$, we do not know that all irreducible representations of $PD^{\times}$
lifts to $G_{2}$, but the lift is still one-to-one on the subset of those
representations that have nonzero lift.

In fact, for the three dual pairs, we determine the theta lift of all non-
supercuspidal representations of $G_{2}$, and the lift of supercuspidal
representations whose lift is not supercuspidal. The detailed statements are
in the main text, and we simply state the following qualitative result here:

######  Theorem 1.3.

The three theta correspondences satisfy the following properties: (i) The
correspondences preserve tempered representations.

(ii) Any discrete series representation of $G_{2}$ lifts to a discrete series
representation of precisely one of the three groups.

(iii) The correspondences are functorial for non-tempered representations.

As an illustration of our results, let $\pi_{\rm gen}$ be a Whittaker generic
(henceforth simply generic) supercuspidal representation of $G_{2}$. Then we
have the following description of $\Theta(\pi_{\rm gen})$, the big theta lift
to ${\rm PGSp}_{6}$, completing the results of [SWe]:

(a) If $\pi_{\rm gen}$ is not a theta lift from ${\rm
PGL}_{3}(F)\rtimes{\mathbb{Z}}/2{\mathbb{Z}}$, then $\Theta(\pi_{\rm gen})$ is
an irreducible generic supercuspidal representation of ${\rm PGSp}_{6}$.

(b) Otherwise, $\pi_{\rm gen}$ is a lift of a supercuspidal representation of
${\rm PGL}_{3}(F)\rtimes{\mathbb{Z}}/2{\mathbb{Z}}$. Now the irreducible
representations of ${\rm PGL}_{3}(F)\rtimes{\mathbb{Z}}/2{\mathbb{Z}}$ can be
described in terms of those of ${\rm PGL}_{3}$ as follows:

  * •

${\rm Ind}_{{\rm PGL}_{3}}^{{\rm
PGL}_{3}\rtimes{\mathbb{Z}}/2{\mathbb{Z}}}(\tau)\cong{\rm Ind}_{{\rm
PGL}_{3}}^{{\rm PGL}_{3}\rtimes{\mathbb{Z}}/2{\mathbb{Z}}}(\tau^{\vee})$ where
$\tau\ncong\tau^{\vee}\in{\rm Irr}({\rm PGL}_{3})$;

  * •

$\tau^{+}\ncong\tau^{-}$, two extensions of a self dual
($\tau\cong\tau^{\vee}$) representation of ${\rm PGL}_{3}$.

Accordingly, we have the two following two cases:

(b1) If $\theta_{B}({\rm Ind}_{{\rm PGL}_{3}}^{{\rm
PGL}_{3}\rtimes{\mathbb{Z}}/2{\mathbb{Z}}}(\tau))=\pi_{\rm gen}$, then

$\Theta(\pi_{\rm gen})\cong{\rm Ind}_{P_{3}}^{{\rm PGSp}_{6}}(\tau)\cong{\rm
Ind}_{P_{3}}^{{\rm PGSp}_{6}}(\tau^{\vee})$

where $P_{3}$ is a maximal parabolic of ${\rm PGSp}_{6}$ with Levi factor
${\rm GL}_{3}$.

(b2) If $\theta_{B}(\tau^{+})=\pi_{\rm gen}$, then
$\theta_{B}(\tau^{-})=\pi_{\rm deg}$ is an irreducible degenerate (i.e.
nongeneric) supercuspidal representation of $G_{2}$. Since $\tau$ is self-
dual, the induced representation ${\rm Ind}_{P_{3}}^{{\rm PGSp}_{6}}(\tau)$ is
reducible. It is a direct sum

${\rm Ind}_{P_{3}}^{{\rm PGSp}_{6}}(\tau)={\rm Ind}_{P_{3}}^{{\rm
PGSp}_{6}}(\tau)_{\rm gen}\oplus{\rm Ind}_{P_{3}}^{{\rm PGSp}_{6}}(\tau)_{\rm
deg}$

of two summands, with the first summand generic and the second degenerate. We
have:

$\Theta(\pi_{\rm gen})\cong{\rm Ind}_{P_{3}}^{{\rm PGSp}_{6}}(\tau)_{\rm
gen}\text{ and }\Theta(\pi_{\rm deg})\cong{\rm Ind}_{P_{3}}^{{\rm
PGSp}_{6}}(\tau)_{\rm deg}.$

To conclude the introduction, we would like to explain the general idea and
strategy for proving the Howe duality theorem. We begin with a discussion of
the statement:

  * (a)

$\Theta(\pi)$ has finite length.

This finiteness result is fundamental and it was shown by Kudla [K] for the
classical theta correspondence. The main tools used are his computation of the
Jacquet modules of the Weil representation (relative to maximal parabolic
subgroups of the two members of the dual pair) and his exploitation of the
doubling see-saw identity. One key consequence of the finite length of
$\Theta(\pi)$ is

  * (b)

If $\Theta(\pi)\neq 0$, then it has an irreducible quotient.

For the dual pairs considered in this paper, we will in fact first prove
statement (b) and then use it with other inputs to show (a).

Let us elaborate on this slightly subtle point and our strategy of proof. By
Bernstein’s decomposition, we may decompose

$\Theta(\pi)=\Theta(\pi)_{c}\oplus\Theta(\pi)_{nc}$

as the sum of its cuspidal part and non-cuspidal part. If $\Theta(\pi)_{c}$ is
nonzero, then it certainly has an irreducible quotient, since it is
semisimple. On the other hand, we shall show using Jacquet module computations
that

  * (c)

$\Theta(\pi)_{nc}$ has finite length and hence has an irreducible quotient if
it is nonzero.

The necessary Jacquet module computations are already available in the
literature [MS, Sa] when $H_{J}=PD^{\times}$ or ${\rm PGL}_{3}$ and are
partially available [MS, GrS] for $H_{J}={\rm PGSp}_{6}$. In §13, we complete
the remaining Jacquet module computations. We stress that the material in §13
is independent of the rest of the paper and could have been discussed earlier
in the paper; we have refrained from doing so, as the computations are rather
technical. Consequences of the results of §13 are then discussed in §14.

In any case, we show statement (b) by showing (c) via Jacquet modules; the
proof is written in §14.2. The statement (b) is used in the proof of the
dichotomy theorem (i.e. Theorem 1.1) for tempered representations in §6. For
nontempered representations, the Jacquet module computations (of [MS, GrS] and
§13-14) will tell us everything about their theta lifts.

For the statement (a) (finite length of $\Theta(\pi)$), it remains to show
that $\Theta(\pi)_{c}$ is of finite length. We shall show this together with
the Howe duality conjecture, by showing that $\Theta(\pi)_{c}$ is either
irreducible or $0$. This part of the argument may be considered the analog of
the doubling see-saw argument, though one would legitimately question what
that means in the setting of exceptional dual pairs.

It will be instructive to first recall the argument for a classical dual pair
${\rm Sp}(W)\times{\rm O}(V)$, where $W$ is a symplectic space and $V$ a
quadratic space. The Howe duality theorem was shown by examining the so-called
doubling see-saw diagram:

$\textstyle{{\rm
O}(V^{\square})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Sp}(W)\times{\rm
Sp}(W)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
O}(V)\times{\rm
O}(V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Sp}(W)^{\Delta}}$

where $V^{\square}=V+V^{-}$ is the doubled quadratic space. Starting from
$\pi,\pi^{\prime}\in{\rm Irr}({\rm O}(V))$, the resulting see-saw identity
gives

$\dim{\rm Hom}_{{\rm Sp}(W)}(\theta(\pi^{\prime}),\theta(\pi))\leq\dim{\rm
Hom}_{{\rm O}(V)\times{\rm O}(V)}(\Theta(1),\pi^{\prime}\otimes\pi^{\vee}).$

where $\Theta(1)$ is the big theta lift of the trivial representation of ${\rm
Sp}(W)^{\Delta}$ to ${\rm O}(V^{\square})$. By the local analog of the Siegel-
Weil formula, one identifies $\Theta(1)$ with a submodule of a certain
degenerate principal series representation $I$ on ${\rm O}(V^{\square})$. This
implies that, for $\pi$ outside a small family of representations,

$\dim{\rm Hom}_{{\rm O}(V)\times{\rm
O}(V)}(\Theta(1),\pi^{\prime}\otimes\pi^{\vee})\subset\dim{\rm Hom}_{{\rm
O}(V)\times{\rm O}(V)}(I,\pi^{\prime}\otimes\pi^{\vee}).$

Using Mackey theory, one can analyze the latter space and show that, for $\pi$
outside another small family of representations,

$\dim{\rm Hom}_{{\rm O}(V)\times{\rm
O}(V)}(I,\pi^{\prime}\otimes\pi^{\vee})\leq\dim{\rm Hom}_{{\rm
O}(V)}(\pi^{\prime},\pi).$

Taken together, one obtains the desired inequality

$\dim{\rm Hom}_{{\rm Sp}(W)}(\theta(\pi^{\prime}),\theta(\pi))\leq\dim{\rm
Hom}_{{\rm O}(V)}(\pi^{\prime},\pi)$

for $\pi$ outside a small family of representations. For this small family of
representations, one needs to do a separate argument.

Now for the exceptional dual pairs $G\times H$ studied in this paper, there is
no analog of the doubling see-saw; this is ultimately tied to the sporadic
nature of the geometry underlying exceptional groups. There is thus no direct
analog of the above argument. However, the above argument is a particular
manifestation of a general principle:

Theta correspondence typically relates or transfers a period on $G$ to a
period on $H$.

More precisely, given a subgroup $H_{1}\subset H$ and $\tau\in{\rm Irr}(H)$,
we may consider the space ${\rm Hom}_{H_{1}}(\tau,\chi)$ for some one-
dimensional character $\chi$ of $H_{1}$, typically trivial; let us call this
Hom space the $H_{1}$-period for $\tau$. Now, given

  * -

$\pi\in{\rm Irr}(G)$, with big theta lift $\Theta(\pi)$ on $H$,

  * -

$\tau\in{\rm Irr}(H)$ an irreducible quotient of $\Theta(\pi)$,

one typically obtains a statement of the form

$\text{$H_{1}$-period of $\tau$}\subset\text{$H_{1}$-period of
$\Theta(\pi)$}\cong\text{$G_{1}$-period of $\pi$}$

for some subgroup $G_{1}$ of $G$.

Now one can turn the table around. For an irreducible quotient $\tau$ of
$\Theta(\pi)$, one can consider the $G_{1}$-period of $\Theta(\tau)$, which
has $\pi$ as an irreducible quotient. One typically gets a statement

$\text{$G_{1}$-period of $\pi$}\subset\text{$G_{1}$-period of
$\Theta(\tau)$}\cong\text{$H_{2}$-period of $\tau$}$

for some subgroup $H_{2}$ on $H$.

Iterating this process, one obtains a family of periods relative to subgroups
$G_{i}$ on $G$ and $H_{i}$ on $H$ such that

$\text{$H_{i}$-period of $\tau$ }\subset\text{$H_{i}$-period of $\Theta(\pi)$
}\cong\text{$G_{i}$-period of $\pi$}$

and

$\text{$G_{i}$-period of $\pi$ }\subset\text{$G_{i}$-period of $\Theta(\tau)$
}\cong\text{$H_{i+1}$-period of $\tau$},$

leading to a chain of containment of periods of $\pi$ and $\tau$. One may call
this a game of ping-pong with periods. Now an (empirical) observation is that
the subgroups $G_{i}$ and $H_{i}$ become more and more reductive (as $i$
increases) and one ultimately obtains a reductive period. When that happens,
the next iteration will result in a seesaw diagram analogous to that in the
classical case above and the consideration of an appropriate degenerate
principal series representation.

Now the miracle is that a Mackey theory argument with this degenerate
principal series representation then returns us the initial $H_{1}$-period! In
other words, for some $i>1$, one has $H_{i}=H_{1}$, and this allows one to
complete the chain of containment of periods into a cycle. In particular, if
one of these period spaces is finite-dimensional, then this cycle of
containment is a cycle of equalities. This is the key step in our proof of the
Howe duality theorem for the dual pairs treated here. We shall play this game
of ping-pong with periods on two occasions, in §6 and §12. This seems to us to
be a rather robust method for proving the Howe duality conjecture and should
be applicable to other exceptional dual pairs, though the precise details will
undoubtedly be different in each case.

To be honest, just as in the classical case, this last step will hold for
representations of $G$ outside a small family, which needs to be treated
separately. One can characterize this exceptional family precisely, but we
prefer not to do it, and simply observe that tempered irreducible
representations of $G$ do not lie in this exceptional family. As mentioned
earlier, the theta lifts of nontempered representations can be explicitly
determined using the Jacquet module computations of [MS, GrS] and §13-14.

The main motivation for showing the results of this paper is the application
to the local Langlands correspondence for the exceptional group $G_{2}$. We
will treat this application in a sequel to this paper.

##  2\. The Group $G_{2}$

We begin by introducing the algebraic group $G_{2}$ over $F$.

###  2.1. Octonion algebra.

Let $\mathbb{O}$ be the split octonion algebra over $F$. Thus, $\mathbb{O}$ is
an $8$-dimensional non-associative and non-commutative $F$-algebra. It comes
equipped with a conjugation map $x\mapsto\overline{x}$ with associated norm
$N(x)=x\cdot\overline{x}=\overline{x}\cdot x$ and trace
$Tr(x)=x+\overline{x}$. Moreover, $N:\mathbb{O}\rightarrow F$ is a
nondegenerate quadratic form.

Every element $x$ of $\mathbb{O}$ satisfies its characteristic polynomial
$t^{2}-Tr(x)t+N(x)$. A nonzero element $x\in\mathbb{O}$ is said to be of rank
$1$ if $N(x)=0$. Otherwise it is of rank $2$, in which case the subalgebra
$F[x]$ of $\mathbb{O}$ generated by $x$ over $F$ is isomorphic to the
separable quadratic $F$-algebra $F[t]/(t^{2}-Tr(x)t+N(x))$. We denote by
$\mathbb{O}_{0}$ the 7-dimensional subspace of trace $0$ elements in
$\mathbb{O}$.

###  2.2. Automorphism group.

The group $G_{2}$ is the automorphism group of the $F$-algebra $\mathbb{O}$.
It is a split simple linear algebraic group of rank $2$ which is both simply
connected and adjoint. If we fix a maximal torus $T$ contained in a Borel
subgroup $B$, then we obtain a system of simple roots $\\{\alpha,\beta\\}$ of
$G_{2}$ relative to $(T,B)$, with $\alpha$ short and $\beta$ long. The
resulting root system is given by the following diagram.

$\alpha$$\beta$

The highest root is $\beta_{0}=3\alpha+2\beta$.

###  2.3. Maximal torus.

Following Muić, we will fix the isomorphism $T\cong\mathbb{G}_{m}^{2}$ by

$t\mapsto((2\alpha+\beta)(t),(\alpha+\beta)(t)).$

Any pair of characters $(\chi_{1},\chi_{2})$ of $F^{\times}$ thus define a
character $\chi_{1}\times\chi_{2}$ of $T$ by composition with the above
isomorphism.

###  2.4. Parabolic subgroups.

Up to conjugation, $G_{2}$ has 2 maximal parabolic subgroups which may be
described as follows. Let $V_{1}\subset V_{2}\subset\mathbb{O}_{0}$ be
subspaces of dimension $1$ and $2$ respectively on which the octonion
multiplication is identically zero. Let $P$ and $Q$ be the stabiliser of
$V_{2}$ and $V_{1}$ respectively. Then $P=MN$ and $Q=LU$ are the two maximal
parabolic subgroups of $G_{2}$. Moreover, their intersection $B=P\cap Q$ is a
Borel subgroup of $G_{2}$.

The Levi factor $M$ of $P$ is given by

$M\cong{\rm GL}(V_{2})\cong{\rm GL}_{2}.$

The isomorphism of $M$ with ${\rm GL}_{2}$ can be fixed so that the modulus
character of $M$ is $\delta_{P}=\det^{3}$. Its unipotent radical $N$ is a
5-dimensional Heisenberg group with 1-dimensional center $Z=U_{\beta_{0}}$.
The action of $M$ on $N/Z$ is isomorphic to $Sym^{3}(F^{2})\otimes\det^{-1}$.
Moreover, the generic $M(F)$-orbits on $N(F)/Z(F)$ is naturally parametrized
by the set of isomorphism classes of separable cubic $F$-algebras.

The Levi factor $L$ of $Q$ is given by

$L\cong{\rm GL}(V_{3}/V_{1})\cong{\rm GL}_{2}$

where

$V_{3}=\\{x\in\mathbb{O}_{0}:x\cdot y=0\,\,\text{for all $y\in V_{1}$}\\}.$

The isomorphism of $L$ with ${\rm GL}_{2}$ can be fixed so that the modulus
character of $L$ is $\delta_{Q}=\det^{5}$. The unipotent radical $U$ is a
5-dimensional 3-step nilpotent group:

$U=U_{0}\supset U_{1}\supset U_{2}\supset U_{3}=\\{1\\},$

such that

$U_{0}/U_{1}=U_{\alpha}\times U_{\alpha+\beta},\quad U_{1}/U_{2}\cong
U_{2\alpha+\beta}\quad U_{2}/U_{3}=U_{\beta_{0}}\times U_{\beta_{0}-\beta}.$

As representation of $L$, one has

$U_{0}/U_{1}\cong F^{2},\quad U_{1}/U_{2}\cong\det\quad U_{2}/U_{3}\cong
F^{2}\otimes\det.$

###  2.5. The subgroup ${\rm SL}_{3}$.

The subgroup of $G_{2}$ generated by the long root subgroups is isomorphic to
${\rm SL}_{3}$. The normaliser of ${\rm SL}_{3}$ in $G_{2}$ is a semidirect
product ${\rm SL}_{3}\rtimes{\mathbb{Z}}/2{\mathbb{Z}}$, with the nontrivial
element of ${\mathbb{Z}}/2{\mathbb{Z}}$ acting on ${\rm SL}_{3}$ as a pinned
outer automorphism. The subgroup ${\rm SL}_{3}$ is the pointwise stabilizer of
a quadratic subalgebra of $\mathbb{O}$ which is isomorphic to $F\times F$,
whereas the setwise stabilizer of such a subalgebra is ${\rm
SL}_{3}\rtimes{\mathbb{Z}}/2{\mathbb{Z}}$.

More generally, given a subalgebra of $\mathbb{O}$ which is isomorphic to a
quadratic field extension of $F$, the pointwise stabilizer of this subalgebra
is isomorphic to the quasi-split special unitary group ${\rm SU}_{3}^{K}$; the
setwise stabilizer of this subalgebra is ${\rm
SU}_{3}^{K}\rtimes{\mathbb{Z}}/2{\mathbb{Z}}$.

###  2.6. The dual group.

The Langlands dual group of $G_{2}$ is the complex Lie group
$G_{2}(\mathbb{C})$. In particular, one has the subgroups

${\rm SO}_{3}(\mathbb{C})\subset{\rm SL}_{3}(\mathbb{C})\subset{\rm
SL}_{3}(\mathbb{C})\rtimes{\mathbb{Z}}/2{\mathbb{Z}}\subset
G_{2}(\mathbb{C}).$

The centralizer of ${\rm SL}_{3}(\mathbb{C})$ in $G_{2}(\mathbb{C})$ is
$\mu_{3}$, and the centralizer of ${\rm SO}_{3}(\mathbb{C})$ in
$G_{2}(\mathbb{C})$ is $S_{3}=\mu_{3}\rtimes{\mathbb{Z}}/2{\mathbb{Z}}$. Let
${\rm SL}_{2,l}(\mathbb{C})$ be a long root ${\rm SL}_{2}$. Then the
centralizer of ${\rm SL}_{2,l}(\mathbb{C})$ in $G_{2}(\mathbb{C})$ is ${\rm
SL}_{2,s}(\mathbb{C})$, a short root ${\rm SL}_{2}$, and vice versa. Thus we
also have the subgroup

${\rm SL}_{2,l}(\mathbb{C})\times_{\mu_{2}}{\rm SL}_{2,r}(\mathbb{C})\cong{\rm
SO}_{4}(\mathbb{C})\subset G_{2}(\mathbb{C}).$

##  3\. Representations of $G_{2}$

In this section, we introduce some notations for the representations of
$G_{2}(F)$. In particular, we shall describe all non-supercuspidal
representations. The results in this section are almost entirely due to Muić
[Mu].

###  3.1. Principal series representations for $P$.

We first consider the principal series representations for the Heisenberg
parabolic subgroup $P=MN$, where $M\cong{\rm GL}_{2}$. Let $\tau$ be an
irreducible representation of $M$ with central character $\omega_{\tau}$ and
set

$I_{P}(\tau)={\rm Ind}_{P}^{G_{2}}\tau\text{ and }I_{P}(s,\tau)={\rm
Ind}_{P}^{G_{2}}(|\det|^{s}\cdot\tau)$

if we need to consider a family of induced representations. If $I_{P}(s,\tau)$
is a standard module, we will denote its unique Langlands quotient by
$J_{P}(s,\tau)$. Now we have:

######  Proposition 3.1.

(i) If $\tau$ is a unitary supercuspidal representation, then $I_{P}(s,\tau)$
is reducible if and only if $\tau^{\vee}\cong\tau$ (so $\omega_{\tau}^{2}=1$)
and one of the following holds:

  * •

$\omega_{\tau}=1$ and $s=1/2$, in which case there is a non-split short exact
sequence of length $2$,

$\begin{CD}0@>{}>{}>\delta_{P}(\tau)@>{}>{}>I_{P}(1/2,\tau)@>{}>{}>J_{P}(1/2,\tau)@>{}>{}>0,\end{CD}$

where $\delta_{P}(\tau)$ is a generic discrete series representation.

  * •

$\omega_{\tau}\neq 1$ and $s=0$, in which case

$I_{P}(0,\tau)=I_{P}(\tau)_{\mathrm{gen}}\oplus I_{P}(\tau)_{\mathrm{deg}}$

where $I_{P}(\tau)_{\mathrm{gen}}$ is generic.

(ii) If $\tau={\rm st}_{\chi}$ is a twisted Steinberg representation, then
$I_{P}(s,\tau)$ is irreducible except for the following cases:

  * •

$\chi=1$ and $s=\pm 3/2$ or $\pm 1/2$, in which case one has:

$\begin{CD}0@>{}>{}>{\rm St}_{G_{2}}@>{}>{}>I_{P}(3/2,{\rm
st})@>{}>{}>J_{P}(3/2,{\rm st})@>{}>{}>0,\end{CD}$

with ${\rm St}_{G_{2}}$ the Steinberg representation. On the other hand,
$I_{P}(1/2,{\rm st})$ has length $3$, with a unique irreducible submodule
$\pi_{\mathrm{gen}}[1]$ which is a generic discrete series representation, a
unique irreducible Langlands quotient $J_{P}(1/2,{\rm st})$ and a subquotient
$J_{Q}(1/2,{\rm st})$.

  * •

$\chi^{2}=1$ but $\chi\neq 1$ and $s=\pm 1/2$, in which case one has:

$\begin{CD}0@>{}>{}>\pi_{\mathrm{gen}}[\chi]@>{}>{}>I_{P}(1/2,{\rm
st}_{\chi})@>{}>{}>J_{P}(1/2,{\rm st}_{\chi})@>{}>{}>0\end{CD}$

where $\pi_{\mathrm{gen}}[\chi]$ is a generic discrete series representation.

  * •

$\chi^{3}=1$ but $\chi\neq 1$ and $s=\pm 1/2$, in which case one has:

$\begin{CD}0@>{}>{}>\pi_{\mathrm{gen}}[\chi]@>{}>{}>I_{P}(1/2,{\rm
st}_{\chi})@>{}>{}>J_{P}(1/2,{\rm<EMAIL_ADDRESS>
where $\pi_{\mathrm{gen}}[\chi]\cong\pi_{\mathrm{gen}}[\chi^{-1}]$ is a
generic discrete series representation.

(iii) If $\tau=\chi$ is 1-dimensional unitary, then $I_{P}(s,\tau)$ is
irreducible except in the following cases:

  * •

$\chi=1$ and $s=\pm 1/2$ or $\pm 3/2$, in which case one has:

$\begin{CD}0@>{}>{}>J_{Q}(5/2,{\rm
st})@>{}>{}>I_{P}(3/2,1)@>{}>{}>1_{G_{2}}@>{}>{}>0,\end{CD}$

whereas $I_{P}(1/2,1)$ is of length $3$, with a unique irreducible submodule
$\pi_{\mathrm{deg}}[1]$ which is a nongeneric discrete series representation,
a unique irreducible quotient $J_{Q}(1,\pi(1,1))$ and a subquotient
$J_{Q}(1/2,{\rm st})$.

  * •

$\chi^{2}=1$ but $\chi\neq 1$ and $s=\pm 1/2$, in which case one has:

$\begin{CD}0@>{}>{}>J_{Q}(1/2,{\rm
st}_{\chi})@>{}>{}>I_{P}(1/2,\chi<EMAIL_ADDRESS>
  * •

$\chi^{3}=1$ but $\chi\neq 1$ and $s=\pm 1/2$, in which case one has:

$\begin{CD}0@>{}>{}>J_{P}(1/2,{\rm st}_{\chi^{-1}})@>{}>{}>I_{P}(1/2,{\rm
st}_{\chi}<EMAIL_ADDRESS>
###  3.2. Principal series representations for $Q$.

Now we consider the principal series representations for the 3-step parabolic
subgroup $Q=LU$, where $L\cong{\rm GL}_{2}$. Let $\tau$ be an irreducible
unitary representation of $L$ with L-parameter $\phi_{\tau}$ and set

$I_{Q}(\tau)={\rm Ind}_{Q}^{G_{2}}\tau\text{ and }I_{Q}(s,\tau)={\rm
Ind}_{Q}^{G_{2}}|\det|^{s}\cdot\tau$

if we need to consider a family of induced representations. As before, we let
$J_{Q}(s,\tau)$ denote the unique Langlands quotient of $I_{Q}(s,\tau)$ if the
latter is a standard module. Then we have:

######  Proposition 3.2.

(i) If $\tau$ is unitary supercuspidal, then $I_{Q}(s,\tau)$ is reducible if
and only if $\tau^{\vee}\cong\tau$ (so $\omega_{\tau}^{2}=1$) and one of the
following holds:

  * •

$\omega_{\tau}=1$ and $s=\pm 1/2$, in which case one has:

$\begin{CD}0@>{}>{}>\delta_{Q}(\tau)@>{}>{}>I_{Q}(1/2,\tau)@>{}>{}>J_{Q}(1/2,\tau)@>{}>{}>0,\end{CD}$

where $\delta_{Q}(\tau)$ is a generic discrete series representation.

  * •

$\omega_{\tau}\neq 1$ (so $\tau$ is dihedral), ${\rm Im}(\phi_{\tau})=S_{3}$
and $s=\pm 1$,in which case one has:

$\begin{CD}0@>{}>{}>\pi_{\mathrm{gen}}[\tau]@>{}>{}>I_{Q}(1,\tau)@>{}>{}>J_{Q}(1,\tau)@>{}>{}>0,\end{CD}$

where $\pi_{\mathrm{gen}}[\tau]$ is a generic discrete series representation.

  * •

$\omega_{\tau}\neq 1$, ${\rm Im}(\phi_{\tau})\neq S_{3}$ and $s=0$, in which
case one has:

$I_{Q}(0,\tau)=I_{Q}(\tau)_{\mathrm{gen}}\oplus I_{Q}(\tau)_{\mathrm{deg}}$

where $I_{Q}(\tau)_{\mathrm{gen}}$ is generic.

(ii) If $\tau={\rm st}_{\chi}$ is a twisted Steinberg representation, the
$I_{Q}(s,\tau)$ is irreducible except for the following cases:

  * •

$\chi=1$ and $s=\pm 5/2$ or $\pm 1/2$, in which case one has

$\begin{CD}0@>{}>{}>{\rm St}_{G_{2}}@>{}>{}>I_{Q}(5/2,{\rm
st})@>{}>{}>J_{Q}(5/2,{\rm st})@>{}>{}>0,\end{CD}$

and

$\begin{CD}0@>{}>{}>\pi_{\mathrm{gen}}[1]\oplus\pi_{\mathrm{deg}}[1]@>{}>{}>I_{Q}(1/2,{\rm
st})@>{}>{}>J_{Q}(1/2,{\rm<EMAIL_ADDRESS>
Here $\pi_{\mathrm{gen}}[1]$ is the generic discrete series representation
already defined in Proposition 3.1(ii) (first bullet point) and
$\pi_{\mathrm{deg}}[1]$ is the nongeneric discrete series representation
already defined in Proposition 3.1(iii) (first bullet point).

  * •

$\chi^{2}=1$ but $\chi\neq 1$ and $s=\pm 1/2$, in which case one has:

$\begin{CD}0@>{}>{}>\pi_{\mathrm{gen}}[\chi]@>{}>{}>I_{Q}(1/2,{\rm
st}_{\chi})@>{}>{}>J_{Q}(1/2,{\rm<EMAIL_ADDRESS>
Here, $\pi_{\mathrm{gen}}[\chi]$ is the generic discrete series representation
defined in Proposition 3.1(ii) (second bullet point).

(iii) If $\tau=\chi$ is 1-dimensional unitary, then $I_{Q}(s,\tau)$ is
irreducible except in the following cases:

  * •

$\chi=1$ and $s=\pm 1/2$ or $\pm 5/2$, in which case one has:

$\begin{CD}0@>{}>{}>J_{P}(3/2,{\rm
st})@>{}>{}>I_{Q}(5/2,1)@>{}>{}>{1}_{G_{2}}@>{}>{}>0,\end{CD}$

whereas $I_{Q}(1/2,1)$ is of length $3$, with unique irreducible submodule
$J_{Q}(1/2,{\rm st})$, a unique irreducible quotient $J_{Q}(1,\pi(1,1))$ and
subquotient $J_{P}(1/2,{\rm st})$.

  * •

$\chi^{2}=1$ but $\chi\neq 1$ and $s=\pm 1/2$, in which case one has:

$\begin{CD}0@>{}>{}>J_{P}(1/2,{\rm
st}_{\chi})@>{}>{}>I_{Q}(1/2,\chi<EMAIL_ADDRESS>
###  3.3. Principal series representations for $B$.

We now consider the principal series representations induced from the Borel
subgroup $B$. More precisely, suppose that $\pi$ is a Langlands quotient of a
standard module

$I(s_{1},s_{2},\chi_{1},\chi_{2})\twoheadrightarrow\pi$

with

$s_{1}\geq s_{2}\geq 0$

and $\chi_{i}$ unitary characters of $F^{\times}$. Here, recall the convention
about characters of $T$ which we have fixed in §2.3. Then

$\pi\hookrightarrow
I(-s_{1},-s_{2},\chi_{1}^{-1},\chi_{2}^{-1})=I_{P}(\pi(\chi_{1}^{-1}|-|^{-s_{1}},\chi_{2}^{-1}|-|^{-s_{2}})).$

Now the representation
$\pi(\chi_{1}^{-1}|-|^{-s_{1}},\chi_{2}^{-1}|-|^{-s_{2}})$ of $M\cong{\rm
GL}_{2}$ is reducible if and only if

$\chi_{2}/\chi_{1}\cdot|-|^{s_{2}-s_{1}}=|-|^{-1},\quad\text{i.e.
$\chi_{1}=\chi_{2}$ and $s_{1}=s_{2}+1\geq 1$,}$

in which case one has

$\pi\hookrightarrow I_{P}(-s+\frac{1}{2},\chi_{1}^{-1}),\quad\text{with $s\geq
1$}.$

There is another, convenient, way to bookkeep the principal series ${\rm
Ind}_{B}^{G_{2}}(\chi)$. Let $\beta_{1},\beta_{2},\beta_{3}$ be three long
roots such that $\beta_{1}+\beta_{2}+\beta_{3}=0$. This triple is unique up
the action of the Weyl group of $G_{2}$. Then the corresponding co-roots
$\beta_{i}^{\vee}:F^{\times}\rightarrow T$ generate $T$, in particular, the
character $\chi$ defines three characters of $F^{\times}$ by
$\chi_{i}=\chi\circ\beta_{i}^{\vee}$ (and is determined by them). Clearly,
these characters satisfy $\chi_{1}\cdot\chi_{2}\cdot\chi_{3}=1$.

######  Proposition 3.3.

The induced representation ${\rm Ind}_{B}^{G_{2}}(\chi)$ is irreducible unless
one of the following two conditions hold:

  * •

$\chi_{i}=|\cdot|^{\pm 1}$ for some $i$ or $\chi_{i}/\chi_{j}=|\cdot|^{\pm 1}$
for a pair $i\neq j$.

  * •

The three characters $\chi_{i}$ are quadratic, non-trivial and pairwise
different. Then

${\rm Ind}_{B}^{G_{2}}(\chi)={\rm Ind}_{B}^{G_{2}}(\chi)_{\rm gen}\oplus{\rm
Ind}_{B}^{G_{2}}(\chi)_{\rm deg}$

where ${\rm Ind}_{B}^{G_{2}}(\chi)_{\rm gen}$ is generic.

###  3.4. Conjectural L-packets of $G_{2}$.

The above results allow one to give an enumeration of the non-cuspidal
representations of $G_{2}$. Using the desiderata of the conjectural local
Langlands correspondence (LLC) for $G_{2}$, we explain how one can assign
L-parameters to the noncuspidal representations of $G_{2}$, and hence
partition them into L-packets. Recall that an $L$-parameter of $G_{2}$ is an
admissible homomorphism

$\varphi:WD_{F}\longrightarrow G_{2}^{\vee}=G_{2}(\mathbb{C})$

of the Weil-Deligne group $WD=W_{F}\times{\rm SL}_{2}(\mathbb{C})$ to the dual
group $G_{2}(\mathbb{C})$, taken up to conjugacy by $G_{2}(\mathbb{C})$. Let

$A_{\varphi}=\pi_{0}(Z_{G_{2}}(\varphi))$

be the associated component group of $\varphi$. Then one expects that there
should be an L-packet

$\Pi_{\varphi}=\\{\pi(\rho):\rho\in\hat{A}_{\varphi}\\}\subset{\rm
Irr}(G_{2})$

associated to each $\varphi$, whose members are indexed by the characters of
$A_{\varphi}$, such that

${\rm Irr}(G_{2})=\bigcup_{\varphi}\Pi_{\varphi}.$

The non-tempered irreducible representations of $G_{2}$ are uniquely realized
as Langlands quotients of standard modules, so have the form $J_{P}(\tau)$,
$J_{Q}(\tau)$ or $J_{B}(\chi)$. The Levi factors of the parabolic subgroups
$P$, $Q$ aand $B$ are isomorphic to ${\rm GL}_{2}$ and ${\rm GL}_{1}\times{\rm
GL}_{1}$. Since the LLC for these groups are known, one can assign
L-parameters to the nontempered representations. For example, if
$\pi=J_{P}(\tau)$, and $\varphi_{\tau}:WD_{F}\longrightarrow M^{\vee}={\rm
GL}_{2}(\mathbb{C})$ is the L-parameter of $\tau$, then the L-parametrer of
$\pi=J_{P}(\tau)$ is the composite

$\varphi_{\pi}:WD_{F}\longrightarrow M^{\vee}\hookrightarrow
G_{2}^{\vee}=G_{2}(\mathbb{C}).$

Since the L-packets on the Levi subgroups are singletons, we see also that the
nontempered L-packets of $G_{2}$ are singletons, and $A_{\varphi_{\pi}}$ is
correspondingly trivial.

In other words, the non-tempered irreducible representations of $G_{2}$ are
naturally parametrized by the nontempered L-parameters of $G_{2}$; these are
the L-parameters $\varphi$ such that $\varphi(W_{F})$ is unbounded. In the
following, we will use this partial LLC to describe the effect of the various
theta correspondences on nontempered representations.

By the same token, since irreducible tempered representations which are not
square-integrable are uniquely realized as summands of principal series
representations induced from unitary square-integrable representations of Levi
factors, one can attach L-parameters to these tempered (but not square-
integrable) representations of $G_{2}$. The resulting L-parameters $\varphi$
have the property that $\varphi(W_{F})$ is bounded but $\varphi(WD_{F})$ is
contained in a proper Levi subgroup. The size of such a tempered L-packet now
depends on the number of irreducible summands in the corresponding
parabolically induced representations. From the results recalled in this
section, one sees that the size of a tempered L-packet $\Pi_{\varphi}$ is $1$
or $2$. One can verify that this is the same as the size of $A_{\varphi}$.
Moreover, in each tempered L-packet, there is a unique generic representation,
and this is assigned to the trivial character of $A_{\varphi}$. Thus, the LLC
for tempered non-discrete series representations of $G_{2}$ is also known, and
we may refer to this partial LLC for describing these representations.

Hence, the main issue with the LLC for $G_{2}$ comes down to the
classification of the square-integrable or discrete series representations by
discrete series L-parameters; these are the L-parameters $\varphi$ which do
not factor through any proper Levi subgroup, or equivalently whose centralizer
$C_{\varphi}=Z_{G_{2}}(\varphi)$ is finite. Guided by the desiderata of the
LLC, we can now describe the various families of discrete series L-parameters,
according to $\varphi({\rm SL}_{2})$, and list all non-supercuspidal members.

  1. (1)

$\varphi({\rm SL}_{2})$ is the principal ${\rm SL}_{2}$. Then $A_{\varphi}=1$
and the packet consists of the Steinberg representation:

$\Pi_{\varphi}=\\{{\rm St}_{G_{2}}\\}$

  2. (2)

$\varphi({\rm SL}_{2})={\rm SO}_{3}\subseteq{\rm SL}_{3}\subseteq G_{2}$; this
is the subregular ${\rm SL}_{2}$. The centralizer of ${\rm SO}_{3}$ in $G_{2}$
is the finite symmetric group $S_{3}$, so that $\varphi$ gives by restriction
a map $\phi:W_{F}\rightarrow S_{3}$. There are four cases to discuss:

     * •

$\phi(W_{F})=1$. Then $A_{\varphi}=S_{3}$. Let $1,r,\epsilon$ be the three
irreducible representations of $S_{3}$: the trivial, 2-dimensional and the
sign character respectively. Then

$\Pi_{\varphi}=\\{\pi(1)=\pi_{\rm gen}[1],\,\pi(r)=\pi_{\rm
deg}[1],\,\pi(\epsilon)=\pi_{\mathrm{sc}}[1]\\}$

where $\pi_{\rm gen}[1]$ is defined in Proposition 3.1(ii) (first bullet
point) and $\pi_{\rm deg}[1]$ is given in Proposition 3.1(iii) (first b.p.).
The representation $\pi(\epsilon)$ is a depth 0 supercuspidal representation
induced from a cuspidal unipotent representation of $G_{2}(\mathbb{F}_{q})$,
inflated to a hyperspecial maximal compact group. The cuspidal unipotent
representation is denoted in the literature by $G_{2}[1]$ and hence our
notation $\pi_{\mathrm{sc}}[1]$.

     * •

$\phi(W_{F})=\mu_{2}$. Then, by the local class field theory, $\phi$ defines a
quadratic character $\chi$ of $F^{\times}$. Let $1$ and $-1$ denote the
trivial and non-trivial characters of $A_{\varphi}=\mu_{2}$. Then

$\Pi_{\varphi}=\\{\pi(1)=\pi_{\rm gen}[\chi],\,\pi(-1)\\},$

where $\pi_{\rm gen}[\chi]$ is as defined in Proposition 3.1(ii) (second
b.p.). If the character $\chi$ is unramified, then $\pi(-1)=\pi_{\rm sc}[-1]$
is a depth 0 supercuspidal representation. It is induced from a cuspidal
unipotent representation of $G_{2}(\mathbb{F}_{q})$, denoted by $G_{2}[-1]$,
inflated to a hyperspecial maximal compact group.

     * •

$\phi(W_{F})=\mu_{3}$. Then, by local class field theory, $\phi$ defines a
cubic character $\chi$ of $F^{\times}$. Let $1$, $\omega$ and $\omega^{2}$
denote the characters of $A_{\varphi}=\mu_{3}$. Then

$\Pi_{\varphi}=\\{\pi(1)=\pi_{\rm
gen}[\chi],\,\pi(\omega),\,\pi(\omega^{2})\\},$

where $\pi_{\rm gen}[\chi]$ is as defined in Proposition 3.1(ii) (third b.p.).
If the character $\chi$ is unramified, then $\pi(\omega)=\pi_{\rm sc}[\omega]$
and $\pi(\omega^{2})=\pi_{\rm sc}[\omega^{2}]$ are induced from a cuspidal
unipotent representations of $G_{2}(\mathbb{F}_{q})$, denoted by
$G_{2}[\omega]$ and $G_{2}[\omega^{2}]$, inflated to a hyperspecial maximal
compact group.

     * •

$\phi(W_{F})=S_{3}$. Then $r\circ\phi$ corresponds to a supercuspidal
representation $\tau$ of ${\rm GL}_{2}$ (where we recall that $r$ denotes the
two-dimensional irreducible representation of $S_{3}$). In this case
$A_{\varphi}$ is trivial and

$\Pi_{\varphi}=\\{\pi_{\rm gen}[\tau]\\},$

where $\pi_{\rm gen}[\tau]$ is as defined in Proposition 3.2(i) (second b.p.).

  3. (3)

$\varphi({\rm SL}_{2})={\rm SL}_{2,s}$, a short root ${\rm SL}_{2}$. The
centralizer of ${\rm SL}_{2,s}$ in $G_{2}$ is ${\rm SL}_{2,l}$, a long root
${\rm SL}_{2}$. Then $\varphi$ gives, by restriction, a map form the Weil
group $\phi:W_{F}\rightarrow{\rm SL}_{2,l}$, that corresponds to supercuspidal
representation $\tau$ of ${\rm GL}_{2}$ with the trivial central character
(and hence $\tau\cong\tau^{\vee}$). In this case $A_{\varphi}=\mu_{2}$, and

$\Pi_{\varphi}=\\{\pi(1)=\delta_{P}(\tau),\,\pi(-1)\\},$

where $\delta_{P}(\tau)$ is as defined in Proposition 3.1(i) (first b.p.) and
$\pi(-1)$ is supercuspidal.

  4. (4)

$\varphi({\rm SL}_{2})={\rm SL}_{2,l}$, a long root ${\rm SL}_{2}$. The
centralizer of ${\rm SL}_{2,l}$ in $G_{2}$ is ${\rm SL}_{2,s}$, a long root
${\rm SL}_{2}$. Then $\varphi$ gives, by restriction, a map form the Weil
group $\phi:W_{F}\rightarrow{\rm SL}_{2,s}$, that corresponds to supercuspidal
representation $\tau$ of ${\rm GL}_{2}$ with the trivial central character
(and hence $\tau\cong\tau^{\vee}$). In this case $A_{\varphi}=\mu_{2}$, and

$\Pi_{\varphi}=\\{\pi(1)=\delta_{Q}(\tau),\,\pi(-1)\\},$

where $\delta_{Q}(\tau)$ is as defined in Proposition 3.2(i) (first b.p.) and
$\pi(-1)$ is supercuspidal.

  5. (5)

$\varphi(SL_{2})=1$. Then $\varphi:W_{F}\rightarrow G_{2}(\mathbb{C})$ gives
rise to an L-packet consisting entirely of supercuspidal representations of
$G_{2}$.

There has been some work towards the above conjectural LLC for $G_{2}$, most
notably [SWe] and [HKT]. At the moment, we simply wish to point out that all
the noncuspidal discrete series representations are fully accounted for by the
above classification scheme.

###  3.5. Local Fourier coefficients.

It will be useful to consider the twisted Jacquet modules of a representation
$\pi$ of $G_{2}$ along the unipotent radical $N$ of $P$. The $M$-orbits of
1-dimensional characters of $N$ are naturally indexed by cubic $F$-algebras,
with the generic orbits corresponding to étale cubic $F$-algebras. For any
such cubic $F$-algebra $E$, we shall write $\psi_{E}$ for a character of $N$
in the corresponding $M$-orbit. Then one may consider $\pi_{N,\psi_{E}}$. In
particular, we note:

######  Lemma 3.4.

For any irreducible, nongeneric, infinite dimensional representation $\pi$ of
$G_{2}$, there exists an étale cubic $F$-algebra $E$ such that
$\pi_{N,\psi_{E}}\neq 0$. Moreover, $\pi_{N,\psi_{E}}$ is finite-dimensional
for any étale $E$.

###### Proof.

Since $\pi$ is not Whittaker generic, and not the trivial representation, its
wave-front set is supported on subregular nilpotent orbits, which are
paramterized by cubic étale algebras, see [HMS] and [LS]. The non-vanishing of
$\pi_{N,\psi_{E}}$ for some $E$ and the finite-dimensionality of these spaces
is the main result of [MW] and [Va]. ∎

##  4\. Exceptional Dual Pairs

In this section, we briefly describe the dual pairs which intervene in this
paper and some structural results which will be important in the study of the
associated theta correspondences.

###  4.1. The group $M_{J}$.

Let $J$ be a Freudenthal-Jordan $F$-algebra [KMRT]. The algebra $J$ comes
equipped with a cubic norm form $N_{J}$, and we let

$M_{J}=\\{g\in{\rm GL}(J):N_{J}\circ g=N_{J}\\}.$

It contains the automorphism group ${\rm Aut}(J)$ as a subgroup. Now we
consider the $F$-vector space

$\mathfrak{g}_{J}=\mathfrak{sl}_{3}\oplus Lie(M_{J})\oplus(F^{3}\otimes
J)\oplus(F^{3}\otimes J)^{*}$

Then $\mathfrak{g}_{J}$ can be given the structure of a simple exceptional Lie
algebra (see, for example, [GS05]). We have the following cases of interest:

$\dim J$ | $1$ | $3$ | $9$ | $15$
---|---|---|---|---
$\mathfrak{g}_{J}$ | $G_{2}$ | $D_{4}$ | $E_{6}$ | $E_{7}$

We observe:

  * •

If $\dim J=3$, then $J$ is a cubic étale $F$-algebra $E$.

  * •

If $\dim J=9$ ,then $J$ corresponds to a pair $(B_{K},\tau)$ where $B_{K}$ is
a central simple algebra over an étale quadratic $F$-algebra $K$ and $\iota$
is an involution of the second kind. Thus, $J=B_{K}^{\iota}$ is the subspace
of $\iota$-symmetric elements. The split version of this algebra is
$M_{3}(F^{2})=M_{3}(F)\oplus M_{3}(F)$ where $\iota$ permutes two factors.
Thus $J=M_{3}(F)$, and ${\rm Aut}(M_{3})={\rm
PGL}_{3}\rtimes{\mathbb{Z}}/2{\mathbb{Z}}$.

  * •

If $\dim J=15$, then $J$ is $H_{3}(B_{F})$ is the space of all $3\times 3$
hermitian-symmetric matrices, where $B_{F}$ is a quaternion algebra over $F$.
The split version is when $B_{F}=M_{2}(F)$.

Let $G_{J}$ be the identity component of ${\rm Aut}(\mathfrak{g}_{J})$. If
$\dim J=9$ then

$\begin{CD}1@>{}>{}>G_{J}@>{}>{}>{\rm
Aut}(\mathfrak{g}_{J}<EMAIL_ADDRESS>
This short exact sequence need not split in general.

###  4.2. Dual pair $G_{2}\times{\rm Aut}(J)$.

We can now describe some dual pairs in $G_{J}$ or rather in ${\rm
Aut}(\mathfrak{g}_{J})$. It will be easier to do this on the level of Lie
algebras.

The centralizer of ${\rm Aut}(J)$ in $\mathfrak{g}_{J}$ is

$\mathfrak{sl}_{3}\oplus F^{3}\otimes 1_{J}\oplus F^{3}\otimes 1_{J}$

which one recognizes to be $\mathfrak{g}_{F}$ (i.e. taking $J=F$). Thus this
is a Lie subalgebra of type $G_{2}$, and we have a dual pair

$G_{2}\times{\rm Aut}(J)\subset{\rm Aut}(G_{J}).$

If $\dim J=9$, we recall that ${\rm Aut}(J)$ sits in a short exact sequence

$\begin{CD}1@>{}>{}>{\rm Aut}(J)^{0}@>{}>{}>{\rm
Aut}(J<EMAIL_ADDRESS>
If $J$ is associated to a pair $(B_{K},\tau)$, then ${\rm
Aut}(J)^{0}=PGU(B_{K},\tau)$ is an adjoint group of type $A_{2}$.

###  4.3. Dual pair ${\rm Aut}(i:E\rightarrow J)\times G_{E}$.

Now we fix an embedding $i:E\longrightarrow J$, where $E$ is a cubic étale
$F$-algebra. We have the subgroup

${\rm Aut}(i:E\rightarrow J)\subset{\rm Aut}(J).$

If $\dim J=9$ a detailed description of this group is in [GS14]. Its identity
component is a 2-dimensional torus. The centralizer of ${\rm
Aut}(i:E\rightarrow J)$ in $\mathfrak{g}_{J}$ contains

$\mathfrak{g}_{E}=\mathfrak{sl}_{3}\oplus\mathfrak{t}_{E}\oplus F^{3}\otimes
E\oplus(F^{3}\otimes E)^{*}$

where $E\hookrightarrow J$ via $i$ and $\mathfrak{t}_{E}\cong E^{0}$ is the
toral Lie subalgebra of trace 0 elements in $E$. This Lie algebra is
isomorphic to $Lie(G_{E})$ (where $G_{E}$ is the simply connected quasi-split
group $Spin_{8}^{E}$), and we have the dual pair

${\rm Aut}(i:E\rightarrow J)\times G_{E}\longrightarrow{\rm Aut}(G_{J}).$

Note that this map is not injective.

###  4.4. A see-saw diagram

The two dual pairs we described above fit together into a see-saw diagram:

(4.1) $\textstyle{G_{E}:={\rm
Spin}_{4}^{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Aut}(J)=:H_{J}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Aut}(i:E\rightarrow J)=:H_{J,E}}$

in ${\rm Aut}(G_{J})$. The various $J$’s of interest in this paper, and the
corresponding groups $H_{J}={\rm Aut}(J)$ and $H_{J,E}={\rm
Aut}(i:E\rightarrow J)$ are given in the table below.

$J$ | $D^{+}$ | $M_{3}(F)^{+}$ | $H_{3}(M_{2}(F))$
---|---|---|---
$H_{J}$ | $PD^{\times}$ | ${\rm PGL}_{3}\rtimes{\mathbb{Z}}/2{\mathbb{Z}}$ | ${\rm PGSp}_{6}$
$H_{J,E}$ | $PE^{\times}$ | $PE^{\times}\rtimes{\mathbb{Z}}/2{\mathbb{Z}}$ | ${\rm SL}_{2}(E)/\mu_{2}$

Here, note that $D^{+}$ denotes the Jordan algebra associated to a cubic
division $F$-algebra $D$, in which case $E$ is necessarily a field.

##  5\. The See-Saw Argument

In this section, we shall consider the see-saw identity arising from the
seesaw diagram (4.1) and pursue some of its consequences.

###  5.1. See-saw identity.

Suppose that $\pi\in{\rm Irr}(G_{2})$. Then we have the see-saw identity
associated with the seesaw (4.1):

(5.1) ${\rm Hom}_{H_{J,E}}(\Theta(\pi),\mathbb{C})\cong{\rm
Hom}_{G_{2}}(R_{J}(E),\pi)$

where

$R_{J}(E):=\Theta(1)$

is the big theta lift of the trivial representation of $H_{J,E}$. To make use
of this see-saw identity, we need to understand the representation $R_{J}(E)$
of ${\rm Spin}_{8}^{E}$. This has been studied in [GS15] and we recall the
relevant results there.

###  5.2. Degenerate principal series of ${\rm Spin}_{8}^{E}$.

Let $P_{E}=M_{E}\cdot N_{E}\subset{\rm Spin}_{8}^{E}$ be the Heisenberg
parabolic subgroup, so that its Levi factor is

$M_{E}\cong{\rm GL}_{2}(E)^{0}.$

Then the determinant map defines an algebraic character
$M_{E}\rightarrow\mathbb{G}_{m}$ which is a basis element of ${\rm
Hom}(M_{E},\mathbb{G}_{m})$. We may consider the degenerate principal series
representation

$I_{E}(s)={\rm Ind}_{P_{E}}^{{\rm Spin}_{8}^{E}}|\det|^{s}.$

In [GS15, S], the module structure of this family of degenerate principal
series representation has been determined. In particular, we have:

######  Proposition 5.2.

$R_{J}(E)\hookrightarrow I_{E}(s_{J})$

where

$s_{J}=\begin{cases}-1/2,\text{ if $J=D^{+}$ or $M_{3}(F)^{+}$;}\\\ 1/2,\text{
if $J=H_{3}(M_{2}(F))$.}\end{cases}$

The representation $I_{E}(1/2)$ has length $3$ when $E$ is a field and has
length $2$ otherwise. More precisely, it has a unique irreducible submodule
$V$ with quotient isomorphic to $R_{M_{3}(F)}(E)\oplus R_{D}(E)$ (where
$R_{D}(E)$ is interpreted to be $0$ when $E$ is not a field). Indeed, one has
the short exact sequence:

$\begin{CD}0@>{}>{}>R_{H_{3}(M_{2}(F))}(E)@>{}>{}>I_{E}(1/2<EMAIL_ADDRESS>
and

$\begin{CD}0@>{}>{}>V@>{}>{}>R_{H_{3}(M_{2}(F))}(E<EMAIL_ADDRESS>
In particular, when $E$ is not a field, $I_{E}(1/2)=R_{H_{3}(M_{2}(F))}(E)$.

As a consequence of the above discussion, we see that it is useful to
understand the Hom space

${\rm Hom}_{G_{2}}(I_{E}(s),\pi)\quad\text{ for $\pi\in{\rm Irr}(G_{2})$.}$

We shall study this in two ways.

###  5.3. Vanishing of an ${\rm Ext}^{1}$.

In view of the proposition, we see that there is an exact sequence

$\begin{CD}0@>{}>{}>{\rm Hom}_{G_{2}}(R_{D}(E),\pi)@>{}>{}>{\rm
Hom}_{G_{2}}(I_{E}(1/2),\pi)@>{}>{}>{\rm
Hom}_{G_{2}}(R_{H_{3}(M_{2}(F))}(E),\pi)\\\ @V{}V{}V\\\ {\rm
Ext}^{1}_{G_{2}}(R_{D}(E),\pi).\end{CD}$

Now we have the following useful fact:

######  Proposition 5.3.

Assume that $E$ is a field. If $\pi\in{\rm Irr}(G_{2})$ is tempered or has
cuspidal support different from $\pi_{\rm deg}[1]$, then

${\rm Ext}^{1}_{G_{2}}(R_{D}(E),\pi)=0,$

so that one has a short exact sequence

$\begin{CD}0\rightarrow{\rm Hom}_{G_{2}}(R_{D}(E),\pi)@>{}>{}>{\rm
Hom}_{G_{2}}(I_{E}(1/2),\pi)@>{}>{}>{\rm
Hom}_{G_{2}}(R_{H_{3}(M_{2}(F))}(E),\pi)\rightarrow 0.\end{CD}$

###### Proof.

One needs to understand $R_{D}(E)$ as a representation of $G_{2}$, and this is
essentially done in [Sa], where the dual pair correspondence for
$PD^{\times}\times G_{2}$ was studied. We shall recall the results of [Sa] in
greater detail later on. At this point, we simply note that as a
representation of $G_{2}$, $R_{D}(E)$ is the direct sum of a supercuspidal
representation (of infinite length) and the irreducible discrete series
representation $\pi_{\rm deg}[1]$, which is a constituent of $I_{Q}(1/2,{\rm
st})$. From this, the vanishing of ${\rm Ext}^{1}_{G_{2}}(R_{D}(E),\pi)$ for
those $\pi$ with different cuspidal support from $\pi_{\rm deg}[1]$ follows
immediately. On the other hand, if $\pi$ is tempered, then one also has ${\rm
Ext}^{1}(\pi_{\rm deg}[1],\pi)=0$ since discrete series representations are
projective in the category of tempered representations. ∎

###  5.4. $I_{E}(s)$ as $G_{2}$-module.

On the other hand, we may understand the restriction of $I_{E}(s)$ to $G_{2}$
using Mackey theory. The following is a key technical result:

######  Proposition 5.4.

As a representation of $G_{2}$, $I_{E}(s)$ admits an equivariant filtration

$0\subset I_{0}\subset I_{1}\subset I_{2}\subset I_{3}\subset I_{4}$

with successive quotients described as follows:

  * •

$I_{0}\cong{\rm ind}_{N}^{G_{2}}\bar{\psi}_{E}$;

  * •

$J_{1}:=I_{1}/I_{0}\cong I_{P}(\frac{s}{2}+\frac{1}{4},C^{\infty}_{c}({\rm
PGL}_{2}))$.

  * •

$J_{2}:=I_{2}/I_{1}\cong m_{E}\cdot I_{P}(\frac{s}{2}+\frac{1}{4},{\rm
ind}_{N}^{{\rm PGL}_{2}}\psi)$

  * •

$J_{3}:=I_{3}/I_{2}\cong m_{E}\cdot I_{Q}(s+1)$.

  * •

$J_{4}:=I_{4}/I_{3}\cong I_{P}(s+1)$.

Here,

$m_{E}=\begin{cases}3,\text{ if $E=F^{3}$;}\\\ 1,\text{ if $E=F\times K$;}\\\
0,\text{ if $E$ is a field.}\end{cases}$

The proposition implies that one has a short exact sequence

$\begin{CD}0@>{}>{}>{\rm
ind}_{N}^{G_{2}}\bar{\psi}_{E}@>{}>{}>I_{E}(s)@>{}>{}>\Sigma_{E}(s)@>{}>{}>0.\end{CD},$

from which one deduces an exact sequence:

$\begin{CD}0\rightarrow{\rm Hom}_{G_{2}}(\Sigma_{E}(s),\pi)\rightarrow{\rm
Hom}_{G_{2}}(I_{E}(s),\pi)\rightarrow{\rm
Hom}_{N}(\pi^{\vee},\psi_{E})\rightarrow{\rm
Ext}^{1}_{G_{2}}(\Sigma_{E}(s),\pi).\end{CD}$

We now specialize to $s=1/2$, where we need to be more precise.

######  Proposition 5.5.

Suppose that $\pi\in{\rm Irr}(G_{2})$ is tempered or has cuspidal support
along $Q$. Then

${\rm Hom}_{G_{2}}(I_{E}(1/2),\pi)\cong{\rm Hom}_{N}(\pi^{\vee},\psi_{E}).$

###### Proof.

We need to prove

${\rm Hom}_{G_{2}}(\Sigma_{E}(1/2),\pi)=0={\rm
Ext}^{1}_{G_{2}}(\Sigma_{E}(1/2),\pi).$

To that end, it suffices to prove the following lemma:

######  Lemma 5.6.

For all $i$ and $j$, ${\rm Ext}^{i}_{G_{2}}(J_{j},\pi)=0$, with $\pi$ as in
Proposition 5.5.

###### Proof.

Consider $J_{1}$ firstly. By the Frobenius reciprocity,

${\rm Ext}^{i}_{G_{2}}(J_{1},\pi)={\rm Ext}^{i}_{M}(|\det|^{1/2}\cdot
C^{\infty}_{c}({\rm PGL}_{2}),r_{\overline{P}}(\pi)).$

Since $\pi$ is tempered, the center of $M={\rm GL}_{2}$ acts on
$R_{\overline{P}}(\pi))$ by characters $\chi$ such that $|\chi(z)|=|z|^{t}$
where $t\leq 0$. On the other hand, the center of $M$ acts on
$|\det|^{1/2}\cdot C^{\infty}_{c}({\rm PGL}_{2})$ by $|z|$. Thus the right
hand side is 0. The other cases are dealt with in the same way.

∎

This completes the proof of Proposition 5.5. ∎

##  6\. Dichotomy

The goal of this section is to prove the following theorem:

######  Theorem 6.1.

For any representation $\pi\in{\rm Irr}(G_{2})$, $\pi$ has nonzero theta lift
to exactly one of $PD^{\times}$ or ${\rm PGSp}_{6}$.

To prove this dichotomy theorem, we need some preliminary results which are
consequences of the computation of the Jacquet modules of the minimal
representation $\Pi_{J}$ with respect to the various maximal parabolic
subgroups of $G_{2}$ and $H_{J}$. The required Jacquet module computations
were carried out in [Sa] when $H_{J}=PD^{\times}$ and in [MS, GS04] when
$H_{J}={\rm PGL}_{3}$. For $H_{J}={\rm PGSp}_{6}$, the Jacquet module
computations for some parabolic subgroups were carried out in [MS]. The
remaining ones will be done in §13 and some implications of these computations
are discussed in §14. We note that §13 is a self-contained section independent
of the rest of this paper. Hence, we first record some results from §13-14 and
the earlier references [Sa, MS, GS04] that we will use.

###  6.1. Consequences of Jacquet module computations.

We first note:

######  Lemma 6.2.

Consider the theta correspondence for $G_{2}\times H_{J}$ for the 3 cases of
$J$.

(i) Let $\pi\in{\rm Irr}(G_{2})$ and write

$\Theta_{J}(\pi)=\Theta_{J}(\pi)_{c}\oplus\Theta_{J}(\pi)_{nc}$

as a sum of its cuspidal and noncuspidal components. Then
$\Theta_{J}(\pi)_{nc}$ has finite length. In particular, if
$\Theta_{J}(\pi)\neq 0$, then it has an irreducible quotient.

(ii) Likewise, let $\tau\in{\rm Irr}(H_{J})$ and write

$\Theta_{J}(\tau)=\Theta_{J}(\tau)_{c}\oplus\Theta_{J}(\tau)_{nc}.$

Then $\Theta_{J}(\tau)_{nc}$ has finite length. In particular, if
$\Theta_{J}(\tau)\neq 0$, then it has an irreducible quotient.

###### Proof.

The case of $J=D^{+}$ follows from results of [Sa], whereas that for
$J=M_{3}(F)^{+}$ follows from [GS04]. The case of $J=H_{3}(M_{2}(F))$ is shown
in §14, based on the results of §13. ∎

In fact, the Jacquet module computations allow one to determine the theta lift
of non-tempered representations explicitly (see Theorem 14.1). We simply note
the following here:

######  Lemma 6.3.

(i) Let $\pi\in{\rm Irr}(G_{2})$ and $\tau\in{\rm Irr}(H_{J})$ be such that
$\pi\otimes\tau$ is a quotient of $\Pi_{J}$. Then

$\text{$\pi$ is tempered}\Longleftrightarrow\text{$\tau$ is tempered}.$

(ii) Let $\pi\in{\rm Irr}(G_{2})$ be non-tempered. Then $\pi$ has nonzero
theta lifting to ${\rm PGSp}_{6}$.

###### Proof.

For $H_{J}=PD^{\times}$ or ${\rm PGL}_{3}\rtimes{\mathbb{Z}}/2{\mathbb{Z}}$,
the desired results have been verified in [Sa] and [GS04] respectively. For
the case when $H_{J}={\rm PGSp}_{6}$, this is shown in Theorem 14.1 in §14. ∎

###  6.2. Reduction to non-generic tempered case.

With the above inputs in place, we can now begin the proof of the dichotomy
theorem. We note:

  * •

The dichotmy theorem holds for nontempered $\pi$. Indeed, if $\pi$ is non-
tempered, then Lemma 6.3(ii) says that $\pi$ has nonzero theta lift to ${\rm
PGSp}_{6}$, whereas [Sa] shows that $\pi$ has zero theta lift to
$PD^{\times}$.

  * •

The dichotmy theorem holds for generic $\pi$. Indeed, it was shown in [GS04,
Cor. 20] that a generic $\pi$ has nonzero theta lift to ${\rm PGSp}_{6}$ (see
also Cor. 11.2 below) and it was shown in [Sa] that $\pi$ has zero theta lift
to $PD^{\times}$.

Thus, to prove the dichotomy theorem, it remains to deal with non-generic
tempered $\pi$.

###  6.3. Weak dichotomy.

We first prove that a non-generic tempered $\pi$ has nonzero theta lift to one
of $PD^{\times}$ or ${\rm PGSp}_{6}$. Since $\pi$ is non-generic and infinite-
dimensional, there exists an étale cubic $F$-algebra $E$ such that ${\rm
Hom}_{N}(\pi^{\vee},\psi_{E})\neq 0$. By Proposition 5.5, we have an
isomorphism

${\rm Hom}_{G_{2}}(I_{E}(1/2),\pi)\twoheadrightarrow{\rm
Hom}_{N}(\pi^{\vee},\psi_{E})\ \neq 0.$

This implies, by Proposition 5.3,

${\rm Hom}_{G_{2}}(R_{D}(E),\pi)\neq 0\quad\text{or}\quad{\rm
Hom}_{G_{2}}(R_{H_{3}(M_{2}(F)}(E),\pi)\neq 0.$

By the see-saw identity (5.1), we deduce that

${\rm Hom}_{H_{E}}(\Theta_{J}(\pi),\mathbb{C})\neq 0$

for $J=D^{+}$ or $H_{3}(M_{2}(F))$. In particular, $\Theta_{J}(\pi)\neq 0$ for
$J=D^{+}$ or $H_{3}(M_{2}(F))$. We have thus verified that $\pi$ has nonzero
theta lift to at least one of $PD^{\times}$ or ${\rm PGSp}_{6}$.

###  6.4. Curious chain of containments.

It remains to show that a nongeneric tempered $\pi$ cannot lift to both
$PD^{\times}$ and ${\rm PGSp}_{6}$. Let
$\bar{\pi}\cong\pi\otimes_{\mathbb{C}}\mathbb{C}$ be the complex conjugate of
$\pi$. If $\pi$ is unitarizable (e.g. if $\pi$ is tempered), then
$\bar{\pi}\cong\pi^{\vee}$. Thus

$\pi_{\bar{\psi}_{E}}\cong\bar{\pi}_{\psi_{E}}\cong(\pi^{\vee})_{\psi_{E}}$

where, in the second isomorphism, we assume that $\pi$ is unitarizable. Since
the minimal representation $\Pi_{J}$ used in this paper is defined over
$\mathbb{R}$, we have a canonical isomorphism $\bar{\Pi}_{J}\cong\Pi_{J}$. It
follows that $\Theta_{J}(\bar{\pi})$ is the complex conjugate of
$\Theta_{J}(\pi)$.

We shall make use of the curious chain of containment given in the following
lemma; this is the first instance of the game of ping-pong with periods
discussed at the end of the introduction.

######  Lemma 6.4.

Let $\pi\in{\rm Irr}(G_{2})$ be tempered. For $J=D^{+}$, $M_{3}(F)$ or
$H_{3}(M_{2}(F))$, let $\tau\in{\rm Irr}(H_{J})$ be tempered and such that

${\rm Hom}_{G_{2}\times H_{J}}(\Pi_{J},\pi\boxtimes\tau)\neq 0.$

Then we have the following natural inclusions

$\displaystyle{\rm Hom}_{N}(\pi,\psi_{E})\subseteq{\rm
Hom}_{N}(\Theta(\tau),\psi_{E})\cong{\rm
Hom}_{H_{J,E}}(\tau^{\vee},\mathbb{C})\subseteq{\rm
Hom}_{H_{J,E}}(\Theta(\pi^{\vee}),\mathbb{C})\cong{\rm
Hom}_{G_{2}}(R_{J}(E),\pi^{\vee}).$

If one of these spaces is finite-dimensional, then all inclusions are
isomorphisms.

###### Proof.

The first inclusion arises from $\Theta(\tau)\twoheadrightarrow\pi$. The
second follows from

${\rm Hom}_{N}(\Theta(\tau),\psi_{E})\cong{\rm
Hom}_{H_{J}}((\Pi_{J})_{N,\psi_{E}},\tau)$

combined with (see Lemma 2.9 on page 213 in [GrS])

$(\Pi_{J})_{N,\psi_{E}}\cong\mathrm{ind}_{H_{J,E}}^{H_{J}}(1)$

and the Frobenius reciprocity. For the third, observe that
$\Theta_{J}(\bar{\pi})$ is the complex conjugate of $\Theta_{J}(\pi)$. Since
$\bar{\pi}\cong\pi^{\vee}$ and $\bar{\tau}\cong\tau^{\vee}$ and we have
$\Theta(\pi^{\vee})\twoheadrightarrow\tau^{\vee}$. The fourth is the see-saw
identity (5.1).

If any of the spaces is finite-dimensional, then ${\rm Hom}_{N}(\pi,\psi_{E})$
is finite-dimensional. If this space is finite dimensional then, since $\pi$
is tempered, by Propositions 5.3 and 5.5, one has

(6.5) $\dim{\rm Hom}_{G_{2}}(R_{J}(E),\pi^{\vee})\leq\dim{\rm
Hom}_{G_{2}}(I_{E}(1/2),\pi^{\vee})=\dim{\rm Hom}_{N}(\pi,\psi_{E}).$

It follows that all spaces have the same dimension and the lemma is proved. ∎

###  6.5. Conclusion of proof.

Using the lemma, we can now conclude the proof of Theorem 6.1.

Assume $\pi$ is tempered nongeneric and has nonzero theta lift to
$PD^{\times}$. Since $PD^{\times}$ is compact, one can find $\tau\in{\rm
Irr}(PD^{\times})$ such that $\tau$ is an irreducible quotient of
$\Theta_{D}(\pi)$. Moreover $\tau$ is tempered. Choose $E$ so that ${\rm
Hom}_{N}(\pi,\psi_{E})\neq 0$. We may now apply Lemma 6.4 with the chosen
$\tau$ and $E$ to deduce that

$d:=\dim{\rm Hom}_{G_{2}}(I_{E}(1/2),\pi)=\dim{\rm
Hom}_{G_{2}}(R_{D}(E),\pi)=\dim{\rm Hom}_{N}(\pi,\psi_{E})\neq 0$

Similarly, if $\pi$ has nonzero theta lift to ${\rm PGSp}_{6}$, then we may
find a tempered $\tau\in{\rm Irr}({\rm PGSp}_{6})$ such that $\tau$ is an
irreducible quotient of $\Theta(\pi)$ (by Lemma 6.2 and Lemma 6.3(i)). With
$E$ as above, an application of Lemma 6.4 shows that

$d=\dim{\rm Hom}_{G_{2}}(I_{E}(1/2),\pi)=\dim{\rm
Hom}_{G_{2}}(R_{H_{3}(M_{2}(F))}(E),\pi)=\dim{\rm Hom}_{N}(\pi,\psi_{E})\neq
0$

Moreover, since all these dimensions are finite, one deduces by Proposition
5.3 that

$d=\dim{\rm Hom}_{G_{2}}(I_{E}(1/2),\pi)=\dim{\rm
Hom}_{G_{2}}(R_{D}(E),\pi)+\dim{\rm
Hom}_{G_{2}}(R_{H_{3}(M_{2}(F))}(E),\pi)=2d.$

This gives the desired contradiction and completes the proof of Theorem 6.1.

###  6.6. Uniqueness results

As further applications of Lemma 6.4, we may now derive two multiplicity one
statements which will play a key role in the reminder of the paper. These
statements are the first steps towards the proof of the Howe duality theorem.

######  Proposition 6.6.

Let $\tau\in{\rm Irr}(H_{J})$ be tempered. Let $\pi\in{\rm Irr}(G_{2})$ be a
tempered, non-generic quotient of $\Theta_{J}(\tau)$. Then $\pi$ is the unique
irreducible tempered quotient of $\Theta_{J}(\tau)$.

###### Proof.

Since $\pi$ is non-generic, there exists $E$ such that $d=\dim{\rm
Hom}_{N}(\pi,\psi_{E})$ is finite and non-zero. By Lemma 6.4, $\dim{\rm
Hom}_{N}(\Theta(\tau),\psi_{E})=d$. Let $\pi^{\prime}$ be another tempered
irreducible quotient of $\Theta_{J}(\tau)$. We apply Lemma 6.4 to
$\pi^{\prime}$ and $\tau$ to deduce that $\dim{\rm
Hom}_{N}(\pi^{\prime},\psi_{E})=d$. Since $\pi\oplus\pi^{\prime}$ is a
quotient of $\Theta_{J}(\tau)$,

$d=\dim{\rm Hom}_{N}(\Theta_{J}(\tau),\psi_{E})\geq\dim{\rm
Hom}_{N}(\pi,\psi_{E})+\dim{\rm Hom}_{N}(\pi^{\prime},\psi_{E})=2d,$

a contradiction. ∎

######  Proposition 6.7.

Let $\pi\in{\rm Irr}(G_{2})$ be tempered and non-generic. Then
$\Theta_{J}(\pi)$ cannot have two tempered irreducible quotients. In
particular, the cuspidal representation $\Theta_{J}(\pi)_{c}$ is irreducible
or $0$.

###### Proof.

Let $\tau_{1},\tau_{2}\in{\rm Irr}(H_{J})$, irreducible tempered, such that
$\Theta_{J}(\pi)\twoheadrightarrow\tau_{1}\oplus\tau_{2}$. Since $\pi$ is non-
generic there exists $E$ such that $d=\dim{\rm Hom}_{N}(\pi^{\vee},\psi_{E})$
is finite and non-zero. By Lemma 6.4, applied to $\pi^{\vee}$,
$\tau_{1}^{\vee}$ and then to $\pi^{\vee}$, $\tau_{2}^{\vee}$,

$d=\dim{\rm Hom}_{H_{J,E}}(\tau_{1},\mathbb{C})=\dim{\rm
Hom}_{H_{J,E}}(\Theta_{J}(\pi),\mathbb{C})=\dim{\rm
Hom}_{H_{J,E}}(\tau_{2},\mathbb{C}).$

Since $\tau_{1}\oplus\tau_{2}$ is a quotient of $\Theta(\pi)$,

$d=\dim{\rm Hom}_{H_{J,E}}(\Theta_{J}(\pi),\mathbb{C})\geq\dim{\rm
Hom}_{H_{J,E}}(\tau_{1},\mathbb{C})+\dim{\rm
Hom}_{H_{J,E}}(\tau_{2},\mathbb{C})=2d,$

a contradiction. ∎

Combining Proposition 6.6 and 6.7 with Lemmas 6.2(i) and 6.3(i), we deduce the
following corollary which may be considered as a first step towards the Howe
duality theorem.

######  Corollary 6.8.

Let $\pi\in{\rm Irr}(G_{2})$ be tempered and non-generic. Then
$\Theta_{J}(\pi)$ has finite length. If $\Theta_{J}(\pi)\neq 0$, then it has a
unique irreducible quotient $\theta(\pi)$ and $\theta(\pi)$ is tempered.
Moreover, for $\pi_{1},\pi_{2}\in{\rm Irr}(G_{2})$ tempered and non-generic,

$\theta(\pi_{1})\cong\theta(\pi_{2})\Longrightarrow\pi_{1}\cong\pi_{2}.$

###### Proof.

Writing $\Theta_{J}(\pi)=\Theta_{J}(\pi)_{c}\oplus\Theta_{J}(\pi)_{nc}$,
Proposition 6.7 tells us that $\Theta_{J}(\pi)_{c}$ is irreducible or $0$,
whereas Lemma 6.2(i) tells us that $\Theta_{J}(\pi)_{nc}$ has finite length.
Hence $\Theta_{J}(\pi)$ has finite length, so that its cosocle
$\theta_{J}(\pi)$ is a finite sum of irreducible representations. Moreover,
Lemma 6.3(i) says that $\theta_{J}(\pi)$ is tempered, and Proposition 6.7 then
shows the irreducibility of $\theta_{J}(\pi)$ if it is nonzero. The final
implication now follows by Proposition 6.6. ∎

In the rest of the paper, we shall examine each of the 3 dual pairs
$G_{2}\times H_{J}$ in turn and complete the proof of the Howe duality
conjecture.

##  7\. Theta Correspondence for $PD^{\times}\times G_{2}$

In this section, we discuss the theta correspondence for the dual pair
$PD^{\times}\times G_{2}$. A preliminary study of this dual pair
correspondence has been carried out by the second author in [Sa]. We first
recall the results established there.

Let $\Pi_{D}$ be the minimal representation of $PD^{\times}\times G_{2}$. Then
we have

$\Pi_{D}=\bigoplus_{\tau\in{\rm Irr}(PD^{\times})}\tau\boxtimes\Theta(\tau).$

The following was shown in [Sa]:

######  Proposition 7.1.

(i) If $\tau=1$ is the trivial representation of $PD^{\times}$, then

$\Theta(1)=\pi_{\rm deg}[1],$

the unipotent discrete series representation introduced in Proposition
3.1(iii) (first bullet point).

(ii) If $\tau$ is not the trivial representation, then $\Theta(\tau)$ is
nongeneric supercuspidal of finite length (possibly zero).

(iii) If $\tau=\chi$ is a nontrivial unramified cubic character, then

$\Theta(\chi)=\pi_{\rm sc}[\chi]\text{ and }\Theta(\chi^{2})=\pi_{\rm
sc}[\chi^{2}]$

the two depth 0 supercuspidal representations introduced in §3.4 (2) (third
bullet point).

(iv) For each cubic field extension $E/F$,

${\rm Hom}_{N}(\Theta(\tau),\psi_{E})\cong{\rm
Hom}_{PE^{\times}}(\tau,\mathbb{C}).$

We can now easily extend the above results. More precisely,

######  Theorem 7.2.

(i) For any $\tau\in{\rm Irr}(PD^{\times})$, $\Theta(\tau)$ is an irreducible
representation of $G_{2}$ if it is nonzero.

(ii) If $\tau_{1},\tau_{2}\in{\rm Irr}(PD^{\times})$ are such that
$\Theta(\tau_{1})\cong\Theta(\tau_{2})\neq 0$, then $\tau_{1}\cong\tau_{2}$.

(ii) If $p\neq 3$, then the map $\tau\mapsto\Theta(\tau)$ defines an injection

${\rm Irr}(PD^{\times})\hookrightarrow{\rm Irr}(G_{2}).$

Hence, the Howe duality theorem holds for $PD^{\times}\times G_{2}$, so that

$\dim{\rm Hom}_{G_{2}}(\theta(\tau_{1}),\theta(\tau_{2}))\leq\dim{\rm
Hom}_{PD^{\times}}(\tau_{1},\tau_{2})$

for any $\tau_{1},\tau_{2}\in{\rm Irr}(PD^{\times})$. In particular, for any
$\pi\in{\rm Irr}(G_{2})$, the representation $\Theta(\pi)$ of $PD^{\times}$ is
irreducible or zero.

###### Proof.

The first two parts follow from Propositions 6.6 and 6.7. As for (iii) we use

${\rm Hom}_{N}(\Theta(\tau),\psi_{E})\cong{\rm
Hom}_{PE^{\times}}(\tau,\mathbb{C}),$

so it suffices to show that there exists a field $E$ such that ${\rm
Hom}_{PE^{\times}}(\tau,\mathbb{C})\neq 0$. If $p\neq 3$, this was proved for
all irreducible $\tau$ by [LT], Theorem 2.4.

∎

##  8\. Theta Correspondence for $({\rm
PGL}_{3}\rtimes{\mathbb{Z}}/2{\mathbb{Z}})\times G_{2}$

In this section, we consider the theta correspondence for dual pair $({\rm
PGL}_{3}\rtimes{\mathbb{Z}}/2{\mathbb{Z}})\times G_{2}$ and prove various
results analogous to those in the last section. In fact, the theta
correspondence for ${\rm PGL}_{3}\times G_{2}$ was almost completely studied
in [GS04]. But the treatment there ignores the outer automorphism group of
${\rm PGL}_{3}$; this is akin to working with special orthogonal groups
instead of orthogonal groups in classical theta correspondence and is of
course undesirable. Thus, we shall complete the results of [GS04] in their
natural setting here. We extend the minimal representation of $E_{6}$ to
$E_{6}\rtimes{\mathbb{Z}}/2{\mathbb{Z}}$ so that ${\mathbb{Z}}/2{\mathbb{Z}}$
fixes the spherical vector.

###  8.1. Representations of $H={\rm
PGL}_{3}\rtimes{\mathbb{Z}}/2{\mathbb{Z}}$.

We realize ${\mathbb{Z}}/2{\mathbb{Z}}$, acting on ${\rm GL}_{3}$ as a pinned
automorphism preserving the standard pinning, i.e. acting via

$A\mapsto w_{0}\cdot{{}^{t}}A^{-1}\cdot
w_{0}^{-1}\quad\text{with}\quad\left(\begin{array}[]{ccc}&&1\\\ &-1&\\\
1&&\end{array}\right)$

Let $U\subset{\rm GL}_{3}$ be the maximal unipotent subgroup of upper
triangular matrices and let $\psi$ be a ${\mathbb{Z}}/2{\mathbb{Z}}$-invariant
Whittaker character of $U$. Then $\psi$ extends to two characters of
$U\rtimes{\mathbb{Z}}/2{\mathbb{Z}}$. Let $\psi\otimes 1$ be the extension
such that ${\mathbb{Z}}/2{\mathbb{Z}}$ acts trivially, and let
$\psi\otimes{\rm sign}$ be the other extension. If $\tau\in{\rm Irr}({\rm
PGL}_{3})$, then there are two possibilities:

  * •

if $\tau^{\vee}\ncong\tau$, then

$\tau^{+}:={\rm Ind}_{{\rm PGL}_{3}}^{H}\tau\cong{\rm Ind}_{{\rm
PGL}_{3}}^{H}\tau^{\vee}$

is irreducible. If $\tau$ is generic then

$\dim{\rm Hom}_{U\rtimes{\mathbb{Z}}/2{\mathbb{Z}}}(\tau^{+},\psi\otimes
1)=\dim{\rm Hom}_{U\rtimes{\mathbb{Z}}/2{\mathbb{Z}}}(\tau^{+},\psi\otimes{\rm
sign})=1.$

  * •

if $\tau\cong\tau^{\vee}$, then $\tau$ has two extensions to $H$, which differ
from each other by twisting with the unique quadratic character ${\rm
sign}:H\longrightarrow\langle\pm 1\rangle$ of $H$. When $\tau$ is generic (for
example when $\tau$ is tempered), we let $\tau^{+}$ denote the unique
extension of $\tau$ such that

$\dim{\rm Hom}_{U\rtimes{\mathbb{Z}}/2{\mathbb{Z}}}(\tau^{+},\psi\otimes
1)=1,$

and let $\tau^{-}$ denote the other extension. The only nongeneric and self-
dual representations of ${\rm PGL}_{3}$ are the Langlands quotients
$J_{B}(\mu)$, where $B=TU$ is the normalizer of $U$ and

$\mu=\chi|-|^{1/2}\otimes 1\otimes\chi|-|^{-1/2}$

is a character of $T$ such that $\chi^{2}=1$. In this case, $\theta(\tau)$ is
irreducible by [GS04], and we define $\tau^{+}$ by setting
$\theta(\tau^{+})=\theta(\tau)$ and $\theta(\tau^{-})=0$.

It follows from the above discussion that any irreducible representation of
$H$ is self-dual.

###  8.2. Whittaker models.

The following lemma summarizes some basic computations.

######  Lemma 8.1.

Let $\Pi$ be the minimal representation of split
$E_{6}\rtimes{\mathbb{Z}}/2{\mathbb{Z}}$. (i) Let
$\psi_{V}:V\rightarrow\mathbb{C}^{\times}$ be a Whitaker character for $G_{2}$
(so $V$ is a maximal unipotent subgroup of $G_{2}$). Then

$\Pi_{V,\psi_{V}}\cong{\rm
ind}_{U\rtimes{\mathbb{Z}}/2{\mathbb{Z}}}^{H}\psi\otimes 1.$

In particular, for any $\tau^{\epsilon}\in{\rm Irr}(H)$,

${\rm Hom}_{V}(\Theta(\tau^{\epsilon}),\psi_{V})\cong{\rm
Hom}_{U\rtimes{\mathbb{Z}}/2{\mathbb{Z}}}(\tau^{\epsilon},\psi\otimes 1)$

(ii) For any étale cubic $F$-algebra, we have:

${\rm Hom}_{N}(\Theta(\tau^{\epsilon}),\psi_{E})\cong{\rm
Hom}_{PE^{\times}\rtimes{\mathbb{Z}}/2{\mathbb{Z}}}(\tau^{\epsilon},\mathbb{C}).$

###  8.3. Our earlier results.

The following is a simple combination of results [GS04] and the previous
discussion:

######  Theorem 8.2.

For $\tau\in{\rm Irr}({\rm PGL}_{3})$, let
$\phi_{\tau}:WD_{F}\longrightarrow{\rm SL}_{3}(\mathbb{C})$ denote the
L-parameter of $\tau$. If $\tau$ is non-supercuspidal, then $\Theta(\tau)$ has
finite length. If $\tau\ncong\tau^{\vee}$ is supercuspidal, then
$\Theta(\tau)$ is irreducible supercuspidal. (This covers all $\tau\in{\rm
Irr}({\rm PGL}_{3})$ if $p\neq 2$). In these cases, set
$\theta(\tau^{\epsilon})$ to be the maximal semisimple quotient of
$\Theta(\tau^{\epsilon})$ for $\epsilon=\pm$.

More precisely, we have:

(i) If $\tau\neq\tau^{\vee}$, then $\theta(\tau^{+})$ is irreducible and
nonzero. If $\tau$ is generic, or supercuspidal, or a discrete series
representation, or tempered, so is $\theta(\tau^{+})$. When $\tau$ is not
supercuspidal, then $\theta(\tau^{+})$ is not supercuspidal and its
L-parameter is obtained by composing $\phi_{\tau}$ with the inclusion ${\rm
SL}_{3}(\mathbb{C})\subset G_{2}(\mathbb{C})$.

(ii) If $\tau=\tau^{\vee}$ and the parameter $\phi_{\tau}$ contains the
trivial representation, then $\theta(\tau^{-})=0$ and $\theta(\tau^{+})$ is
nonzero irreducible, non-discrete-series and its L-parameter is obtained by
composing $\phi_{\tau}$ with the inclusion ${\rm SL}_{3}(\mathbb{C})\subset
G_{2}(\mathbb{C})$.

(iii) If $\tau=\tau^{\vee}$ and the parameter $\phi_{\tau}$ does not contain
the trivial representation, then we have the following cases:

  * •

$\tau={\rm St}$, the Steinberg representation. Then

$\theta({\rm St}^{+})\oplus\theta({\rm St}^{-})=\pi_{\rm
gen}[1]\oplus\pi_{\mathrm{s}c}[1]$

where $\pi_{gen}[1]$ is the generic discrete series representation introduced
in Proposition 3.1(ii) and $\pi_{sc}[1]$ is the depth 0 supercuspidal
representation introduced in §3.4 (2).

  * •

$\tau$ is a tempered representation induced from a supercuspidal
representation $\sigma\cong\sigma^{\vee}$ of ${\rm GL}_{2}$ with a non-trivial
central character. Then

$\theta(\tau^{+})\oplus\theta(\tau^{-})={\rm Ind}_{P}^{G_{2}}(\sigma)={\rm
Ind}_{P}^{G_{2}}(\sigma)_{\rm gen}\oplus{\rm Ind}_{P}^{G_{2}}(\sigma)_{\rm
deg}$

  * •

$\tau$ is a tempered principal series induced from a triple of non-trivial
quadratic characters $(\chi_{1},\chi_{2},\chi_{3})$ such that
$\chi_{1}\cdot\chi_{2}\cdot\chi_{3}=1$ then

$\theta(\tau^{+})\oplus\theta(\tau^{-})={\rm Ind}_{B}^{G_{2}}(\chi)={\rm
Ind}_{B}^{G_{2}}(\chi)_{\rm gen}\oplus{\rm Ind}_{B}^{G_{2}}(\chi)_{\rm deg}$

where $\chi$ is the quadratic character of $T$ determined by
$(\chi_{1},\chi_{2},\chi_{3})$ as in §3.3.

  * •

$\tau$ is a self-dual supercuspidal representation (so $p=2$). Then
$\Theta(\tau^{\epsilon})$ is supercuspidal and

$\Theta(\tau^{+})\oplus\Theta(\tau^{-})=\pi_{\rm gen}\oplus\pi_{\rm deg}$

where $\pi_{\rm gen}$ is a generic irreducible supercuspidal representation,
while $\pi_{\rm deg}$ is a non-generic supercuspidal representation of unknown
length.

Observe that the only case for which we do not know that
$\Theta(\tau^{\epsilon})$ has finite length (and hence
$\theta(\tau^{\epsilon})$ is defined) is when $\tau$ is a self-dual
supercuspidal representation (so $p=2$). In this case, however, the last
bullet point states that $\Theta(\tau^{\epsilon})$ is supercuspidal and hence
semisimple. Hence, even in this exceptional case, we may set
$\theta(\tau^{\epsilon})=\Theta(\tau^{\epsilon})$. Moreover, observe that if
$\tau^{\epsilon}$ is nontempered, then $\theta(\tau^{\epsilon})$ is
irreducible nontempered and is completely determined by Theorem 8.2. On the
other hand, when $\tau^{\epsilon}$ is tempered, then so is every irreducible
summand of $\theta(\tau^{\epsilon})$. In particular, the results highlighted
in Lemma 6.2 and 6.3 hold in this case.

In the rest of the section, we shall complete the results above by completing
the unresolved parts of the theorem.

###  8.4. A miracle of Oberwolfach

Let $\tau\in{\rm Irr}({\rm PGL}_{3})$ be a self-dual supercuspidal
representation. The goal here is to show that $\Theta(\tau^{-})\neq 0$. Let
$Q=LU$ be the 3-step maximal parabolic subgroup of $G_{2}$. Recall that the
group $U$ has the 3-step filtration

$U\supset[U,U]\supset Z_{U}$

where $Z_{U}$ is the 2-dimensional center of $U$ and $U/Z_{U}$ is a
3-dimensional Heisenberg group with the center $[U,U]/Z_{U}$. Let $\psi$ be a
non-trivial character of $[U,U]$, trivial on $Z_{U}$. Then $\Pi_{[U,U],\psi}$
is naturally a $({\rm PGL}_{3}\rtimes{\mathbb{Z}}/2{\mathbb{Z}})\times{\rm
SL}_{2}$-module, where ${\rm SL}_{2}=[L,L]$. In order to describe
$\Pi_{[U,U],\psi}$, we need some additional notation.

Consider the action of the group ${\rm
GL}_{2}\rtimes{\mathbb{Z}}/2{\mathbb{Z}}$ on $M_{2}(F)$, with elements in
${\rm GL}_{2}(F)$ acting by conjugation and the nontrivial element of
${\mathbb{Z}}/2{\mathbb{Z}}$ acting via:

$X\mapsto\left(\begin{array}[]{cc}0&1\\\ 1&0\end{array}\right)\cdot
X^{t}\cdot\left(\begin{array}[]{cc}0&1\\\ 1&0\end{array}\right).$

This action preserves the determinant (quadratic) form on $M_{2}(F)$ and
descends to the quotient group

${\rm PGL}_{2}(F)\times{\mathbb{Z}}/2{\mathbb{Z}}\cong\\{\pm
1\\}\times\mathrm{SO}_{3}=\mathrm{O}_{3}.$

On the space $C_{c}(M_{2}(F))$, we have a Weil representation of
$\mathrm{O}_{3}\times{{\rm SL}}_{2}$, which we may regard as a representation
of ${\rm GL}_{2}(F)\rtimes{\mathbb{Z}}/2{\mathbb{Z}}$. Then the following
lemma follows by a standard computation:

######  Lemma 8.3.

We have an isomorphism of $({\rm
PGL}_{3}\rtimes{\mathbb{Z}}/2{\mathbb{Z}})\times{\rm SL}_{2}$-modules:

$\Pi_{[U,U],\psi}\cong\mathrm{ind}_{{\rm
GL}_{2}\rtimes{\mathbb{Z}}/2{\mathbb{Z}}}^{{\rm
PGL}_{3}\rtimes{\mathbb{Z}}/2{\mathbb{Z}}}(C_{c}(M_{2}(F))$

where ${\rm GL}_{2}$ is embedded in ${\rm PGL}_{3}$ via

$\left(\begin{array}[]{cc}a&b\\\
c&d\end{array}\right)\mapsto\left(\begin{array}[]{ccc}a&&b\\\ &1&\\\
c&&d\end{array}\right).$

Using the lemma, we can now prove:

######  Proposition 8.4.

Let $\tau\in{\rm Irr}({\rm PGL}_{3})$ be a self-dual supercuspidal
representation. Then $\Theta(\tau^{-})\neq 0$.

###### Proof.

It suffices to show that $\tau^{-}$ is a quotient of $\Pi_{[U,U],\psi}$, in
fact we shall show that $\tau^{-}$ is a quotient of ${\rm
SL}_{2}$-coinvaraints of $\Pi_{[U,U],\psi}$. Decompose
$M_{2}(F)=M^{\circ}(F)\oplus F$, where $M_{2}(F)^{\circ}$ is the subspace of
trace 0 elements and $F$ is the center. Accordingly, the Weil representation
of $\mathrm{O}_{3}\times\widetilde{{\rm SL}}_{2}$ on $C_{c}(M_{2}(F))$
decomposes as a tensor product

$C_{c}(M_{2}(F))=C_{c}(M^{\circ}_{2}(F))\otimes C_{c}(F)$

where $\mathrm{O}_{3}$ acts trivially on $C_{c}(F)$ and $\tilde{{\rm SL}}_{2}$
acts by the Weil representations $\rho_{\psi}$. Recall that as an
$\widetilde{{\rm SL}}_{2}$-module, $\rho_{\psi}$ decomposes as a sum

$\rho_{\psi}=\rho_{\psi}^{+}\oplus\rho_{\psi}^{-}$

of even and odd Weil representations. Let $\Theta(\rho_{\bar{\psi}}^{+})$ and
$\Theta(\rho_{\bar{\psi}}^{-})$ be the theta lifts of their contragredients to
$\mathrm{O}_{3}$, via the Weil representation on $C_{c}(M^{\circ}_{2}(F))$
with respect to $\psi$. Thus the ${\rm SL}_{2}$-coinvariant of
$\Pi_{[U,U],\psi}$ is given by

$\mathrm{ind}_{{\rm GL}_{2}\rtimes{\mathbb{Z}}/2{\mathbb{Z}}}^{{\rm
PGL}_{3}\rtimes{\mathbb{Z}}/2{\mathbb{Z}}}(\Theta(\rho_{\bar{\psi}}^{+}))\oplus\mathrm{ind}_{{\rm
GL}_{2}\rtimes{\mathbb{Z}}/2{\mathbb{Z}}}^{{\rm
PGL}_{3}\rtimes{\mathbb{Z}}/2{\mathbb{Z}}}(\Theta(\rho_{\bar{\psi}}^{-})).$

Let $\mathrm{st}$ be the Steinberg representation of ${\rm SO}(3)\cong{\rm
PGL}_{2}$. We extend $\mathrm{st}$ to two representations $\mathrm{st}^{+}$
and $\mathrm{st}^{-}$ of $\mathrm{O}(3)$ by letting $-1\in\mathrm{O}(3)$ act
by $1$ and $-1$ respectively. Then
$\Theta(\rho_{\psi}^{-})\cong\mathrm{st}^{-}$ while $\Theta(\rho_{\psi}^{+})$
is the principal series representation with the trivial representation as a
quotient and $\mathrm{st}^{+}$ as a submodule. Since $\tau^{-}$ is a
supercuspidal representation, it suffices to show that it is a quotient of

$\mathrm{ind}_{{\rm GL}_{2}\rtimes{\mathbb{Z}}/2{\mathbb{Z}}}^{{\rm
PGL}_{3}\rtimes{\mathbb{Z}}/2{\mathbb{Z}}}(\mathrm{st}^{+})\oplus\mathrm{ind}_{{\rm
GL}_{2}\rtimes{\mathbb{Z}}/2{\mathbb{Z}}}^{{\rm
PGL}_{3}\rtimes{\mathbb{Z}}/2{\mathbb{Z}}}(\mathrm{st}^{-}).$

It is known that any generic representation of ${\rm GL}_{2}$, in particular
$\mathrm{st}$, is a quotient of $\tau$. Hence either $\mathrm{st}^{+}$ or
$\mathrm{st}^{-}$ is a quotient of $\tau^{-}$. Now the proposition follows
from Frobenius reciprocity.

∎

###  8.5. Main result.

We shall now strengthen Theorem 8.2.

######  Theorem 8.5.

For any $\tau\in{\rm Irr}({\rm PGL}_{3})$, let
$\phi_{\tau}:WD_{F}\longrightarrow{\rm SL}_{3}(\mathbb{C})$ denote the
L-parameter of $\tau$.

(i) The representation $\Theta(\tau^{\epsilon})$ is zero if and only if
$\phi_{\tau}$ contains the trivial representation (so $\tau\cong\tau^{\vee}$)
and $\epsilon=-$.

(ii) For any $\epsilon=\pm$, $\Theta(\tau^{\epsilon})$ has finite length with
unique irreducible quotient $\theta(\tau^{\epsilon})$ (if it is nonzero).

(iii) $\theta(\tau^{\epsilon})$ is generic if and only if $\tau$ is generic
and $\epsilon=+$.

(iv) Suppose that $\theta(\tau^{\epsilon})\neq 0$. If $\tau$ is a discrete
series (resp. tempered) representation, so is $\theta(\tau^{\epsilon})$.
Moreover, $\theta(\tau^{\epsilon})$ is supercuspidal if and only if $\tau$ is
supercuspidal or $\tau^{\epsilon}={\rm St}^{-}$.

(v) If
$\theta(\tau_{1}^{\epsilon_{1}})\cong\theta(\tau_{2}^{\epsilon_{2}})\neq 0$,
then $\tau_{1}^{\epsilon_{1}}=\tau_{2}^{\epsilon_{2}}$.

In particular, the Howe duality theorem holds for the dual pair $({\rm
PGL}_{3}\rtimes{\mathbb{Z}}/2{\mathbb{Z}})\times G_{2}$:

$\dim{\rm
Hom}_{G_{2}}(\theta(\tau_{1}^{\epsilon_{1}}),\theta(\tau_{2}^{\epsilon_{2}}))\leq\dim{\rm
Hom}_{{\rm
PGL}_{3}\rtimes{\mathbb{Z}}/2{\mathbb{Z}}}(\tau_{1}^{\epsilon_{1}},\tau_{2}^{\epsilon_{2}})$

for any $\tau_{1}^{\epsilon_{1}},\tau_{2}^{\epsilon_{2}}\in{\rm Irr}({\rm
PGL}_{3}\rtimes{\mathbb{Z}}/2{\mathbb{Z}})$. Moreover, for $\pi\in{\rm
Irr}(G_{2})$, $\Theta(\pi)$ is a finite length representation of ${\rm
PGL}_{3}$ with a unique irreducible quotient (if nonzero).

###### Proof.

(i) From Theorem 8.2, it remains to show that $\Theta(\tau^{\epsilon})$ is
nonzero for those representations $\tau$ as in Theorem 8.2(iii) and any
$\epsilon=\pm$. Consider first the Steinberg representation. Recall that
$\pi_{\rm gen}[1]$ is generic while $\pi_{\rm sc}[1]$ is not. It follows, from
Lemma 8.1 part (ii), that $\pi_{\rm gen}[1]$ is a summand of $\theta(\rm
St^{+})$. Furthermore, by Proposition 6.6, $\pi_{\rm sc}[1]$ cannot be a
summand of $\theta(\rm St^{+})$. Hence

$\theta(\rm St^{+})=\pi_{\rm gen}[1]\text{ and }\theta(\rm St^{-})=\pi_{\rm
sc}[1].$

The same argument works in the other three cases to show that
$\theta(\tau^{+})$ is the generic $G_{2}$ summand and $\theta(\tau^{-})$ is
the degenerate summand. Moreover, in the last case of Theorem 8.2, where
$\tau$ is a self-dual supercuspidal representation (so $p=2$), we deduce by
Proposition 6.6 again that $\pi_{\rm deg}$ is nonzero irreducible.

(ii) This follows from Theorem 8.2 and the irreducibility of $\pi_{\rm deg}$
in the proof of (i) above.

(iii) and (iv): These summarize what we already know from Theorem 8.2.

(v) Suppose that

$\pi:=\theta(\tau_{1}^{\epsilon_{1}})\cong\theta(\tau_{2}^{\epsilon_{2}})\neq
0.$

If $\pi$ is non-supercuspidal, then $\tau_{1}$ and $\tau_{2}$ are both non-
supercuspidal. The desired equality
$\tau_{1}^{\epsilon_{1}}\cong\tau_{2}^{\epsilon_{2}}$ follows from the results
of [GS04] and our new understanding in (i) (which determines
$\theta(\tau^{\epsilon})$ for those $\tau$ in Theorem 8.2(iii))

Suppose that $\pi$ is supercuspidal. Then $\tau_{i}^{\epsilon_{i}}$ is either
supercuspidal or $St^{-}$, in which case both $\tau_{1}$ and $\tau_{2}$ are
generic discrete series representations. By (iii), we deduce that
$\epsilon_{1}=\epsilon_{2}$. Hence, it remains to show that
$\tau_{1}=\tau_{2}$ or $\tau_{2}^{\vee}$. We now consider the following two
cases:

  * (a)

Suppose $\tau_{1}\ncong\tau_{1}^{\vee}$ and $\tau_{2}\ncong\tau_{2}^{\vee}$.
Then $\epsilon_{1}=\epsilon_{2}=+$ and $\pi$ is an irreducible generic
supercuspidal representation.

Consider, for $i=1$ or $2$, the induced representation ${\rm
Ind}_{P_{3}}^{{\rm PGSp}_{6}}(\tau_{i})$, where $P_{3}$ is the Siegel
parabolic subgroup. Its normalized Jacquet functor with respect to $P_{3}$ is
$\tau_{i}\oplus\tau_{i}^{\vee}$. Since $\tau_{i}\neq\tau_{i}^{\vee}$, it
follows that ${\rm Ind}_{P_{3}}^{{\rm PGSp}_{6}}(\tau_{i})$ is an irreducible
generic tempered representation.

By the computation of the Jacquet module of the minimal representation
$\Pi^{\prime}$ of $G_{2}\times{\rm PGSp}_{6}$ along $P_{3}$ given in [MS], we
deduce that $\pi\otimes{\rm Ind}_{P_{3}}^{{\rm PGSp}_{6}}(\tau_{i})$ is an
irreducible quotient of $\Pi^{\prime}$. By [SWe], a generic representation
cannot lift to two different generic representaitons of ${\rm PGSp}_{6}$.
Hence, we must have

${\rm Ind}_{P_{3}}^{{\rm PGSp}_{6}}(\tau_{1})\cong{\rm Ind}_{P_{3}}^{{\rm
PGSp}_{6}}(\tau_{2}).$

By consideration of the Jacquet modules with respect to $P_{3}$, we see that
$\tau_{2}\cong\tau_{1}$ or $\tau_{1}^{\vee}$, as desired.

  * (b)

Assume now that $\tau_{1}=\tau_{1}^{\vee}$. In this case, we know that

$\theta(\tau_{1}^{+})=\pi_{gen}\quad\text{and}\quad\theta(\tau_{1}^{-})=\pi_{deg}.$

Moreover, the tempered representation ${\rm Ind}_{P_{3}}^{{\rm
PGSp}_{6}}(\tau_{1})$ is the sum of two representations, one of which is
generic and the other degenerate (see Proposition 10.3(i)). By the Jacquet
module of $\Pi^{\prime}$ again, we see that both $\pi_{gen}$ and $\pi_{deg}$
lifts to irreducible summands of ${\rm Ind}_{P_{3}}^{{\rm
PGSp}_{6}}(\tau_{1})$. Moreover, $\pi_{deg}$ cannot lift to a generic
representation of ${\rm PGSp}_{6}$ and hence must lift to the nondegenerate
summand [SWe]. By Proposition 6.6, it follows that $\pi_{gen}$ cannot lift to
the degenerate summand and thus must lift to the generic summand.

Now suppose that $\epsilon_{1}=\epsilon_{2}=+$, so that
$\pi=\theta(\tau_{1}^{+})=\theta(\tau_{2}^{+})$ is generic. Then as before, we
see that $\pi$ lifts to the generic summand of ${\rm Ind}_{P_{3}}^{{\rm
PGSp}_{6}}(\tau_{i})$ (regardless of whether $\tau_{2}$ is self-dual or not).
By Jacquet module consideration, we see that $\tau_{1}\cong\tau_{2}$. On the
other hand, if $\epsilon_{1}=\epsilon_{2}=-$, so that
$\pi=\theta(\tau_{1}^{-})=\theta(\tau_{2}^{-})$ is nongeneric, then
Proposition 6.7 implies that the nongeneric summand of ${\rm
Ind}_{P_{3}}^{{\rm PGSp}_{6}}(\tau_{1})$ is contained in ${\rm
Ind}_{P_{3}}^{{\rm PGSp}_{6}}(\tau_{2})$. Again, Jacquet module considerations
shows that $\tau_{1}\cong\tau_{2}$.

The inequality at the end of the theorem is simply a restatement of (v).
Finally, given $\pi\in{\rm Irr}(G_{2})$, we write

$\Theta(\pi)=\Theta(\pi)_{c}\oplus\Theta(\pi)_{nc}$

as a sum of its cuspidal and noncuspidal component. As we noted in Lemma 6.2,
the results of [GS04] imply that $\Theta(\pi)_{nc}$ has finite length. The
result in (v) shows that $\Theta(\pi)$ has a unique irreducible quotient if it
is nonzero, implying in particular that $\Theta(\pi)_{c}$ is either $0$ or
irreducible, and hence $\Theta(\pi)$ has finite length. ∎

##  9\. The group ${\rm PGSp}_{6}$

Before discussing the last dual pair $G_{2}\times{\rm PGSp}_{6}$, we need to
devote the next few sections to a discussion of the structure and
representations of ${\rm PGSp}_{6}$, as well as certain particular periods on
$G_{2}$ and ${\rm PGSp}_{6}$.

Let $e_{1},\ldots,e_{6}$ be the standard basis of $F^{6}$, where we have a
symplectic form defined by

$\omega(e_{1},e_{6})=\omega(e_{2},e_{5})=\omega(e_{3},e_{4})=1$

and all other $\omega(e_{i},e_{j})=0$ with $i<j$ . Let ${\rm GSp}_{6}$ be the
group of linear transformations $g$ of $F^{6}$, such that for some $\nu(g)\in
F^{\times}$

$\omega(gv,gw)=\nu(g)\cdot\omega(v,w)$

for all $v,w\in F^{6}$. Then $\nu:{\rm GSp}_{6}\rightarrow F^{\times}$ is the
similitude character.

Let $\tilde{P}_{1}$, $\tilde{P}_{2}$ and $\tilde{P}_{3}$ be maximal parabolic
subgroups of ${\rm GSp}_{6}$ defined as the stabilizers of subspaces

$\langle e_{1}\rangle\subset\langle e_{1},e_{2}\rangle\subset\langle
e_{1},e_{2},e_{3}\rangle$

respectively. For $i=1,2,3$, let $P_{i}\subseteq{\rm PGSp}_{6}$ be the
quotient of $\tilde{P}_{i}$ by the center of ${\rm GSp}_{6}$. The group ${\rm
PGSp}_{6}$ acts faithfully on $J=\wedge^{2}F^{6}\otimes i^{-1}$, and we shall
(partially) describe how the parabolic subgroups act on this module.

The group ${\rm PGSp}_{6}$ can be explicitly described in terms of its action
on $J$ as follows. Let $x_{ij}=e_{i}\wedge e_{j}\in J$ for $i\neq j$. On $J$,
we have a natural trilinear form $(x,y,z)$

$\wedge^{2}F^{6}\times\wedge^{2}F^{6}\times\wedge^{2}F^{6}\rightarrow\wedge^{6}F^{6}\cong
F.$

The group of linear transformations of $J$ preserving this form is ${\rm
SL}_{6}/\mu_{2}$ and ${\rm PGSp}_{6}={\rm Sp}_{6}/\mu_{2}$ is the subgroup
fixing

$e=x_{16}+x_{25}+x_{34}.$

The Levi factor $M_{3}$ of $P_{3}$, as an algebraic group, is isomorphic to
${\rm GL}_{3}/\mu_{2}$. Observe that group acts faithfully on
$\wedge^{2}F^{3}$, and since the latter is a three dimensional vector space,
this action gives an isomorphism ${\rm GL}_{3}/\mu_{2}\cong{\rm GL}_{3}$. Thus
we have an identification

$M_{3}={\rm GL}(\langle x_{12},x_{13},x_{23}\rangle).$

Under this identification, the maximal torus is given by diagonal matrices
$(t_{1},t_{2},t_{3})$. The three simple co-roots of ${\rm PGSp}_{6}$ are,
respectively,

$\alpha_{1}^{\vee}(t)=(1,t,t^{-1}),\hskip
5.0pt\alpha_{2}^{\vee}(t)=(t,t^{-1},1),\hskip
5.0pt\alpha_{3}^{\vee}(t)=(1,t,t).$

An unramified character $\chi$ of the maximal torus is given by a triple of
complex numbers $(s_{1},s_{2},s_{3})$

$\chi(t_{1},t_{2},t_{3})=|t_{1}|^{s_{1}}|t_{2}|^{s_{2}}|t_{3}|^{s_{3}}.$

The Weyl group action on the characters is somewhat different in this picture.
The simple reflections corresponding to the first two roots $\alpha_{1}$ and
$\alpha_{2}$ are the usual permutations of entries of $(s_{1},s_{2},s_{3})$,
however, the simple reflection corresponding to the third simple root
$\alpha_{3}$ is given by

$(s_{1},s_{2},s_{3})\mapsto(s_{1}+s_{2}+s_{3},-s_{3},-s_{2}).$

Thus the root hyperplanes are $s_{i}-s_{j}=0$ and $s_{i}+s_{j}=0$ for short
and long roots, respectively. This looks like a $D_{3}$ root system; however,
the Weyl-invariant quadratic form in this case is

$q(s_{1},s_{2},s_{3})=s_{1}^{2}+s_{2}^{2}+s_{3}^{3}-\frac{1}{4}(s_{1}+s_{2}+s_{3})^{2}$

rather than the usual dot product, and with this form, we have a realization
of the $C_{3}$ root system with simple roots

$\alpha_{1}=(0,1,-1),\hskip 5.0pt\alpha_{2}=(1,-1,0),\hskip
5.0pt\alpha_{3}=(0,2,2).$

This somewhat unconventional description of the $C_{3}$ root system is a
source of potential confusion, as one has the tendency to confound it with the
more familiar description of the root system of ${\rm Sp}_{6}$, but what we
have done here is definitely the natural way to set things up for ${\rm
PGSp}_{6}$.

The character $\chi$ is in the positive chamber if for every positive root
$\alpha$, $\chi(\alpha^{\vee}(t))=|t|^{s}$ for some $s\in\mathbb{C}$ such that
$\Re(s)>0$ (the real part). One checks that $\chi$ is positive if

$\Re(s_{1})>\Re(s_{2})>|\Re(s_{3})|.$

The modulus character of $M_{3}\cong{\rm GL}_{3}$ is

$\delta_{P_{3}}(m)=|\det(m)|^{2}.$

It follows that the Levi factor $M_{13}$ of $P_{13}=P_{1}\cap P_{3}$ is

$M_{13}={\rm GL}(\langle x_{12},x_{13}\rangle)\times{\rm GL}(\langle
x_{23}\rangle).$

The group $P_{13}$ is the stabilizer of the amber space $V_{2}=\langle
x_{12},x_{13}\rangle$.

Consider now the group $P_{2}$ and its Levi factor $M_{2}$. The standard Levi
factor of $\tilde{P}_{2}$ is ${\rm GL}_{2}\times{\rm GL}_{2}$ where the first
${\rm GL}_{2}$ acts on $\langle e_{1},e_{2}\rangle$ in the standard way, fixes
$\langle e_{3},e_{4}\rangle$ and acts by transpose-inverse on $\langle
e_{5},e_{6}\rangle$. The second ${\rm GL}_{2}$ acts on $\langle
e_{3},e_{4}\rangle$ in the standard way, by $\det$ on $\langle
e_{1},e_{2}\rangle$ and fixes $\langle e_{5},e_{6}\rangle$. The group $P_{2}$
is the stabilizer of the singular line $V_{1}=\langle x_{12}\rangle$, and the
Levi factor $M_{2}$ acts faithfully on the 4-dimensional subspace

$V_{4}=\langle x_{13},x_{23},x_{14},x_{24}\rangle$

preserving the quadratic form $x\mapsto(x,x,x_{56})$. If we identify
$x=ax_{14}+bx_{13}+cx_{24}+d_{23}$ with the matrix

$\left(\begin{array}[]{cc}a&b\\\ c&d\end{array}\right)$

then $(x,x,x_{56})=2\det(x)$. Thus, with $V_{4}$ identified with the set of
$2\times 2$ matrices, we have

$M_{2}\cong{\rm GL}_{2}\times{\rm GL}_{2}/{\rm
GL}_{1}^{\nabla}\quad\text{where ${\rm GL}_{1}^{\nabla}=\\{(t,t^{-1}):t\in{\rm
GL}_{1}\\}$},$

so that $(\alpha,\beta)\in M_{2}$ acts on $x\in V_{4}$ by $x\mapsto\alpha
x\bar{\beta}$ where $\bar{\beta}$ is the transpose of $\beta$. The element
$(\alpha,\beta)$ acts on the line $\langle x_{12}\rangle$ by
$\det(\alpha\beta)$. The modulus character is

$\delta_{P_{2}}((\alpha,\beta))=|\det(\alpha\beta)|^{5}.$

This sets up the necessary notation to discuss the representations of ${\rm
PGSp}_{6}$.

##  10\. Representations of $\rm PGSp_{6}$

In this section we list some irreducible non-supercuspidal representations of
${\rm PGSp}_{6}(F)$, relevant to this work. Observe that Langlands
parameterization of irreducible representations is known for all of Levi
factors of parabolic subgroups of ${\rm PGSp}_{6}(F)$ (by Gan and Takeda [GT]
for $M_{1}\cong{\rm GSp}_{4}$). Thus, following Shahidi [Sh], reducibility
points of generalized principal series can be computed using $L$-functions of
Langlands parameters.

###  10.1. Principal series representations for $P_{2}$.

We first consider certain principal series representations for the parabolic
subgroup $P_{2}=M_{2}N_{2}$, where $M_{2}\cong{\rm GL}_{2}\times{\rm
GL}_{2}/{\rm GL}_{1}^{\nabla}$. Let $\tau$ be an irreducible representation of
${\rm GL}_{2}$ with L-parameter $\phi_{\tau}$ and the central character
$\omega_{\tau}$. Set

$I_{2}(\tau\otimes\tau)={\rm Ind}_{P_{2}}^{{\rm
PGSp}_{6}}\tau\otimes\tau\text{ and }I_{2}(s,\tau\otimes\tau)={\rm
Ind}_{P_{2}}^{G_{2}}(|\det|^{s}\tau)\otimes(|\det|^{s}\tau)$

if we need to consider a family of induced representations. Then we have:

######  Proposition 10.1.

(i) If $\tau$ is unitary supercuspidal, then $I_{2}(s,\tau\otimes\tau)$ is
reducible if and only if $\tau^{\vee}\cong\tau$ (so $\omega_{\tau}^{2}=1$) and
one of the following holds:

  * •

$\omega_{\tau}=1$ and $s=\pm 1/2$, in which case one has:

$\begin{CD}0@>{}>{}>\delta_{2}(\tau)@>{}>{}>I_{2}(1/2,\tau\otimes\tau)@>{}>{}>J_{2}(1/2,\tau\otimes\tau)@>{}>{}>0,\end{CD}$

where $\delta_{2}(\tau)$ is a generic discrete series representation.

  * •

$\omega_{\tau}\neq 1$ (so $\tau$ is dihedral), ${\rm Im}(\phi_{\tau})=S_{3}$
and $s=\pm 1$,in which case one has:

$\begin{CD}0@>{}>{}>\sigma_{\mathrm{gen}}[\tau]@>{}>{}>I_{2}(1,\tau\otimes\tau)@>{}>{}>J_{2}(1,\tau\otimes\tau)@>{}>{}>0,\end{CD}$

where $\sigma_{\mathrm{gen}}[\tau]$ is a generic discrete series
representation.

  * •

$\omega_{\tau}\neq 1$, ${\rm Im}(\phi_{\tau})\neq S_{3}$ and $s=0$, in which
case one has:

$I_{2}(0,\tau\otimes\tau)=I_{2}(\tau\otimes\tau)_{\mathrm{gen}}\oplus
I_{2}(\tau\otimes\tau)_{\mathrm{deg}}$

where $I_{2}(\tau\otimes\tau)_{\mathrm{gen}}$ is generic.

(ii) If $\tau={\rm st}_{\chi}$ is a twisted Steinberg representation, then
$I_{2}(s,\tau\otimes\tau)$ is irreducible except for the following cases:

  * •

$\chi=1$ and $s=\pm 5/2$ or $\pm 1/2$, in which case one has

$\begin{CD}0@>{}>{}>{\rm St}_{{\rm PGSp}_{6}}@>{}>{}>I_{2}(5/2,{\rm
st}\otimes{\rm st})@>{}>{}>J_{2}(5/2,{\rm st}\otimes{\rm
st})@>{}>{}>0,\end{CD}$

and

$\begin{CD}0@>{}>{}>{\rm Ind}_{P_{3}}^{{\rm PGSp}_{6}}({\rm St})_{\rm
gen}@>{}>{}>I_{2}(1/2,{\rm st}\otimes{\rm st})@>{}>{}>J_{2}(1/2,{\rm
st}\otimes{\rm<EMAIL_ADDRESS>
  * •

$\chi^{2}=1$ but $\chi\neq 1$ and $s=\pm 1/2$, in which case one has:

$\begin{CD}0@>{}>{}>\sigma_{\mathrm{gen}}[\chi]@>{}>{}>I_{2}(1/2,{\rm
st}_{\chi}\otimes{\rm st}_{\chi})@>{}>{}>J_{2}(1/2,{\rm st}_{\chi}\otimes{\rm
st}_{\chi})@>{}>{}>0,\end{CD}$

where $\sigma_{\mathrm{gen}}[\chi]$ is a generic discrete series
representation.

###  10.2. Principal series representations for $P_{13}$.

Now we consider certain principal series representations for the parabolic
subgroup $P_{13}=M_{13}N_{13}$, where $M_{13}\cong{\rm GL}_{2}\times{{\rm
GL}_{1}}$. Let $\tau$ be an irreducible representation of ${\rm GL}_{2}$ with
the central character $\omega_{\tau}$. Set

$I_{13}(\tau\otimes 1)={\rm Ind}_{P_{13}}^{{\rm PGSp}_{6}}\tau\otimes 1\text{
and }I_{13}(s,\tau\otimes 1)={\rm Ind}_{P_{13}}^{G_{2}}(|\det|^{s}\tau)\otimes
1$

if we need to consider a family of induced representations. In the more
familiar language of representations of ${\rm Sp}_{6}$, the restriction of
$I_{13}(s,\tau\otimes 1)$ to ${\rm Sp}_{6}$ is a principal series induced from
$|\cdot|^{2s}\omega_{\tau}\otimes|\det|^{s}\tau$. In particular, if $\tau$ is
unitary tempered and $s>0$, this is a standard module. We have:

######  Proposition 10.2.

If $\tau$ is unitary supercuspidal, then $I_{13}(s,\tau\otimes 1)$ is
reducible if and only if $\tau^{\vee}\cong\tau$ (so $\omega_{\tau}^{2}=1$) and
one of the following holds:

  * •

$\omega_{\tau}=1$ and $s=\pm 1/2$, in which case $I_{13}(1/2,\tau\otimes 1)$
has length 4 and has a unique irreducible submodule $\delta_{13}(\tau)$, which
is a generic discrete series representation.

  * •

$\omega_{\tau}\neq 1$, ${\rm Im}(\phi_{\tau})\neq S_{3}$ and $s=0$, in which
case one has:

$I_{13}(0,\tau\otimes 1)=I_{13}(\tau\otimes 1)_{\mathrm{gen}}\oplus
I_{13}(\tau\otimes 1)_{\mathrm{deg}}$

where $I_{13}(\tau\otimes 1)_{\mathrm{gen}}$ is generic.

###  10.3. Principal series representations for $P_{3}$.

Now we consider certain principal series representations for the parabolic
subgroup $P_{3}=M_{3}N_{3}$, where $M_{3}\cong{\rm GL}_{3}$. Let $\tau$ be an
irreducible representation of ${\rm GL}_{3}$. We set

$I_{3}(\tau)={\rm Ind}_{P_{3}}^{{\rm PGSp}_{6}}\tau.$

Then we have [Ta, Example 7.7 and Theorem 7.9:]:

######  Proposition 10.3.

(i) Assume that $\tau$ is discrete series representation with trivial central
character. Then we have two cases:

  * •

If $\tau\neq\tau^{\vee}$ then

$I_{3}(\tau)\cong I_{3}(\tau^{\vee})$

is irreducible.

  * •

If $\tau\cong\tau^{\vee}$ then

$I_{3}(\tau)=I_{3}(\tau)_{\rm gen}\oplus I_{3}(\tau)_{\rm deg}$

where $I_{3}(\tau)_{\rm gen}$ is generic.

(ii) Let $\chi_{1},\chi_{2},\chi_{3}$ be three characters of $F^{\times}$ such
that $\chi_{1}\cdot\chi_{2}\cdot\chi_{3}=1$, and let
$\tau=\tau(\chi_{1},\chi_{2},\chi_{3})$ be the associated principal series
representation of ${\rm GL}_{3}(F)$ (which is possibly reducible). Then the
induced representation $I_{3}(\tau)$ is irreducible unless one of the
following two conditions hold:

  * •

$\chi_{i}=|\cdot|^{\pm 1}$ for some $i$ or $\chi_{i}/\chi_{j}=|\cdot|^{\pm 1}$
for a pair $i\neq j$.

  * •

The three characters $\chi_{i}$ are quadratic, non-trivial and pairwise
different. Then

$I_{3}(\tau)=I_{3}(\tau)_{\rm gen}\oplus I_{3}(\tau)_{\rm deg}$

where $I_{3}(\tau)_{\rm gen}$ is generic.

###  10.4. Principal series representations for $P_{1}$.

Now we consider certain principal series representations for the parabolic
subgroup $P_{1}=M_{1}N_{1}$, where $M_{1}\cong{\rm GSp}_{4}$. Let $\tau$ be an
irreducible representation of ${\rm GSp}_{4}$. We set

$I_{1}(\tau)={\rm Ind}_{P_{1}}^{{\rm PGSp}_{6}}\tau\text{ and
}I_{1}(s,\tau)={\rm Ind}_{P_{1}}^{{\rm PGSp}_{6}}|\nu|^{s}\tau$

where $\nu$ is the similitude character of ${\rm GSp}_{4}$. Let $\tau$ be an
irreducible supercuspidal representation of ${\rm GSp}_{4}(F)$ with trivial
central character. Let $\varphi_{\tau}:WD_{F}\rightarrow{\rm
Spin}_{5}\cong{\rm Sp}(4)$ be its Langlands parameter [GT].

######  Proposition 10.4.

Assume that $\tau$ is a supercuspidal representation of ${\rm GSp}_{4}(F)$
with trivial central character such that the parameter ${\rm
std}\circ\varphi_{\tau}$ contains the trivial representation, where ${\rm
std}$ denotes the 5-dimensional standard representation of ${\rm Spin}_{5}$.
Then $I_{1}(s,\tau)$ is reducible if and only if $s=\pm 1/2$, in which case
one has:

$\begin{CD}0@>{}>{}>\delta_{1}(\tau)@>{}>{}>I_{1}(1/2,\tau)@>{}>{}>J_{1}(1/2,\tau)@>{}>{}>0,\end{CD}$

where $\delta_{1}(\tau)$ is a discrete series representation.

##  11\. Fourier-Jacobi and Shalika periods

In this section, we introduce and study a Fourier-Jacobi-type model for the
group $G_{2}$ and a Shalika period for ${\rm PGSp}_{6}$. These are some of the
periods that will appear when we consider a game of ping-pong with periods for
the dual pair $G_{2}\times{\rm PGSp}_{6}$, as discussed at the end of the
introduction.

###  11.1. Whittaker periods

We begin by recalling the following results about Whittaker periods from
[GS04, Prop. 19 and Cor. 20], see also the appendix of [HKT].

######  Proposition 11.1.

Let $\Pi$ be the minimal representation of $E_{7}$ and let
$(V^{\prime},\psi_{V^{\prime}})$ be a Whittaker datum for ${\rm PGSp}_{6}$ (so
$V^{\prime}$ is a maximal unipotent subgroup and $\psi_{V^{\prime}}$ a generic
character of $V^{\prime}$). Then we have an isomorphism of $G_{2}$-modules:

$\Pi_{V^{\prime},\psi^{\prime}}\cong{\rm ind}_{V}^{G_{2}}\psi_{V}$

where $(V,\psi_{V})$ is a Whittaker datum for $G_{2}$.

######  Corollary 11.2.

(i) If $\pi\in{\rm Irr}(G_{2})$ is generic and $\Theta(\pi)$ is its big theta
lift to ${\rm PGSp}_{6}$, then

$\dim{\rm Hom}_{V^{\prime}}(\Theta(\pi),\psi)=1$

so that $\Theta(\pi)$ contains a unique irreducible generic subquotient and
thus is nonzero.

(ii) If $\pi\in{\rm Irr}(G_{2})$ is non-generic and $\tau\in{\rm Irr}({\rm
PGSp}_{6})$ is generic, then

${\rm Hom}_{G_{2}\times{\rm PGSp}_{6}}(\Pi,\pi\otimes\tau)=0.$

###  11.2. Fourier-Jacobi period of $G_{2}$

Let $Q=LU$ be the 3-step maximal parabolic subgroup of $G_{2}$. Recall that
$[L,L]\cong{\rm SL}_{2}$ corresponds to the long simple root $\beta$. Thus
$V=U_{\beta}U$ is the unipotent radical of the standard Borel subgroup of
$G_{2}$. If we set $J=[L,L]U$, then the quotient of $J$ by the center of $U$
is the Jacobi group. Let $\rho_{\psi}$ be the unique irreducible
representation of the 2-fold cover $\tilde{J}$, such that the center of
$U/[U,U]\cong U_{2\alpha+\beta}$ acts by $\psi$. If $\sigma$ is a genuine
representation of $\widetilde{{\rm SL}}_{2}$, we have a representation of $J$
on $\sigma\otimes\rho_{\psi}$. For $\pi\in{\rm Irr}(G_{2})$ and genuine
$\sigma\in{\rm Irr}(\tilde{{\rm SL}_{2}})$, the Fourier-Jacobi period of $\pi$
with respect to $\sigma$ is the space

${\rm Hom}_{J}(\pi,\sigma\otimes\rho_{\psi})\cong{\rm Hom}_{G_{2}}(\pi,{\rm
Ind}_{J}^{G_{2}}\sigma\otimes\rho_{\psi}).$

Recall that $\widetilde{{\rm SL}}_{2}$ has a family of principal series
representations $I_{\psi}(s)$ such that when $s=1/2$, one has a short exact
sequence:

$0\rightarrow\mathrm{St}_{\psi}\rightarrow
I_{\psi}(1/2)\rightarrow\rho^{+}_{\psi}\rightarrow 0.$

Moreover, the contragredient of $I_{\psi}(1/2)$ is $I_{\bar{\psi}}(-1/2)$.

We shall use the following enhanced uniqueness of the Fourier-Jacobi model (we
say enhanced since the principal series is not assumed irreducible):

######  Lemma 11.3.

If $\pi\in{\rm Irr}(G_{2})$ is generic and tempered (or has cuspidal support
on $Q$), then

${\rm Hom}_{G_{2}}({\rm
ind}^{G_{2}}_{J}(I_{\psi}(-s)\otimes\rho_{\bar{\psi}}),\pi^{\vee})\cong{\rm
Hom}_{G_{2}}(\pi,{\rm
Ind}^{G_{2}}_{J}(I_{\bar{\psi}}(s)\otimes\rho_{\psi}))\cong\mathbb{C}.$

for all $s<1/2$.

###### Proof.

Let $B=T_{\beta}U_{\beta}\subset[L,L]\cong{\rm SL}_{2}$ be the Borel subgroup,
where $T_{\beta}$ is the one-dimensional torus generated by the image of the
simple coroot $\beta^{\vee}:{\rm GL}_{1}\longrightarrow T$. Observe that for
any genuine character $\chi$ of $\widetilde{B}$, we have an isomorphism of
$J$-modules

${\rm Ind}_{\widetilde{B}}^{\widetilde{\rm
SL}_{2}}(\chi)\otimes\rho_{\psi}\cong{\rm Ind}_{BU}^{J}(\chi\cdot\rho_{\psi})$

where $f\otimes v$ is mapped to a function on ${\rm SL}_{2}$ given by
$g\mapsto f(g)\cdot\rho_{\psi}(g)(v)$. Since $\rho_{\psi}$ restricted to
$\widetilde{B}U$ is induced, using the transitivity of induction, it is easy
to check that

$I_{\bar{\psi}}(s)\otimes\rho_{\psi}\cong{\rm
ind}_{T_{\beta}N}^{J}(|\cdot|^{s+3/2}\cdot\psi_{2\alpha+\beta}),$

where $N$ is the unipotent radical of the maximal parabolic $P$ (so
$V=U_{\alpha}N$) and $\psi_{2\alpha+\beta}$ is a character of $N$ nontrivial
only on the root space $U_{2\alpha+\beta}\subset N$ as the subscript
indicates. Here the induction is not normalized.

Thus, it follows by Frobenius reciprocity that

${\rm Hom}_{G_{2}}(\pi,{\rm
Ind}^{G_{2}}_{J}(I_{\bar{\psi}}(s)\otimes\rho_{\psi}))\cong{\rm
Hom}_{T_{\beta}N}(\pi,|\cdot|^{s+3/2}\cdot\psi_{2\alpha+\beta})$

and to prove the lemma, it suffices to show that

$\dim{\rm
Hom}_{T_{\beta}N}(\pi,|\cdot|^{s+3/2}\cdot\psi_{2\alpha+\beta})=1\quad\text{
if $s<1/2$.}$

We shall prove this in a somewhat roundabout fashion.

Let $\psi_{\alpha}$ be the character of $U$ obtained by restricting a
Whittaker character of $V$. As the notation indicates, $\psi_{\alpha}$ is only
supported on the root space $U_{\alpha}\subset U$. Let $\gamma=3\alpha+2\beta$
be the highest root. Observe that $\gamma$ is perpendicular to $\alpha$ and
therefore, $\pi_{U,\psi_{\alpha}}$ is naturally a module for
$T_{\gamma}U_{\beta}$, a mirabolic subgroup of $L\cong{\rm GL}_{2}$. Recall
that the mirabolic subgroup has a unique irreducible infinite-dimensional
(generic) representation, and all others are one-dimensional with the trivial
action of $U_{\beta}$. The infinite-dimensional representation is realized on
$C_{c}(T_{\gamma})$, so it is a regular representation of $T_{\gamma}\cong{\rm
GL}_{1}$. Since $\pi$ is Whittaker generic, $C_{c}(T_{\gamma})$ has to appear
with multiplicity one in $\pi_{U,\psi_{\alpha}}$. Thus we have an exact
sequence of $T_{\gamma}U_{\beta}$-modules

$0\rightarrow
C_{c}(T_{\gamma})\rightarrow\pi_{U,\psi_{\alpha}}\rightarrow\pi_{V,\psi_{\alpha}}\rightarrow
0$

where the third term is the $U_{\beta}$-coinvariants. Next, observe that
$\pi_{V,\psi_{\alpha}}$ is a quotient of $\pi_{N}$. Since we have assumed that
$\pi$ is tempered, the group $T_{\gamma}$ acts on $\pi_{V,\psi_{\alpha}}$ by
characters which are, after taking the absolute value, equal to
$|\cdot|^{t+3}$ where $t\geq 0$. It follows that

$\dim{\rm Hom}_{T_{\gamma}U}(\pi,|\cdot|^{s+3}\cdot\psi_{\alpha})=1\quad\text{
if $s<0$.}$

However, we are still not done since the group $T_{\gamma}U$ is not even
conjugate to $T_{\beta}N$. The last step requires the technique of root
exchange.

Let $N^{\prime}$ be obtained by adding $U_{-\beta}$ to $U$ and removing
$U_{\alpha+\beta}$, so that $N^{\prime}$ is conjugate to $N$ (by the simple
Weyl reflection $w_{\beta}$). Now we claim that there is an isomorphism

${\rm Hom}_{T_{\gamma}U}(\pi,|-|^{s+3}\cdot\psi_{\alpha})\cong{\rm
Hom}_{T_{\gamma}N^{\prime}}(\pi,|-|^{s+2}\cdot\psi_{\alpha}).$

which sends $\ell$ on the LHS to an element $\ell^{\prime}$ on the RHS defined
by the convergent integral

$\ell^{\prime}(v)=\int_{U_{-\beta}}\ell(\pi(u)\cdot v)~{}du$

Conversely, we can recover $\ell$ from $\ell^{\prime}$ by integrating over
$U_{\alpha+\beta}$. Assuming the claim, we complete the proof of the lemma by
observing that the pair $(T_{\beta}N,|-|^{s+2}\cdot\psi_{2\alpha+\beta})$ is
conjugate to the pair $(T_{\gamma}N^{\prime},|-|^{s+2}\cdot\psi_{\alpha})$ (by
the Weyl element $w_{\alpha}\cdot w_{\beta}$).

To justify the root exchange argument in the claim, we observe that
$U_{-\beta}$ and $U_{\alpha+\beta}$ generate a Heisenberg group with center
$U_{\alpha}$, modulo higher order commutators. More precisely, consider the
group

$V^{\prime}=U\cdot U_{\beta}=N^{\prime}\cdot U_{\alpha+\beta},$

which is a maximal unipotent subgroup of $G_{2}$ and hence conjugate to $V$
(by the simple reflection $w_{\beta}$). If we consider the lower central
series of the unipotent group $V^{\prime}$:

$V^{\prime}\supset[V^{\prime},V^{\prime}]=V^{\prime}_{1}\supset
V^{\prime}_{2}=[V^{\prime},V^{\prime}_{1}]\supset
V^{\prime}_{3}\supset\\{1\\},$

then $V/V^{\prime}_{2}$ is the Heisenberg group in question with center
$V^{\prime}_{1}/V^{\prime}_{2}=U_{\alpha}$. Note moreover that the elements
$\ell$ and $\ell^{\prime}$ in the two Hom spaces in the claim both factors
through $\pi_{V^{\prime}_{2}}$ (which is a module for the Heisenberg group
$V^{\prime}/V^{\prime}_{2}$). With this observation, the justification of the
claim is given by the following lemma, included as a convenience to the
reader. ∎

######  Lemma 11.4.

Let $H$ be a Heisenberg group. Let $\pi$ be a smooth $H$-module. Let $X$ and
$Y$ be two maximal abelian subgroups of $H$. Let $\psi_{X}$ and $\psi_{Y}$ be
characters of $X$ and $Y$, agreeing on the intersection $X\cap Y$, and non-
trivial on the center of $H$. Then we have an isomorphism ${\rm
Hom}_{X}(\pi,\psi_{X})\cong{\rm Hom}_{Y}(\pi,\psi_{Y})$,
$\ell\mapsto\ell^{\prime}$, defined by

$\ell^{\prime}(v)=\int_{Y/X\cap Y}\ell(\pi(y)v)~{}dy.$

###### Proof.

By the Frobenius reciprocity, we have

${\rm Hom}_{X}(\pi,\psi_{X})\cong{\rm Hom}_{H}(\pi,{\rm
Ind}_{X}^{H}\psi_{X})\text{ and }{\rm Hom}_{Y}(\pi,\psi_{Y})\cong{\rm
Hom}_{H}(\pi,{\rm Ind}_{Y}^{H}\psi_{Y}).$

We also have an isomorphism ${\rm Ind}_{X}^{H}\psi_{X}\cong{\rm
Ind}_{Y}^{H}\psi_{Y}$, where every $f\in{\rm Ind}_{X}^{H}\psi_{X}$ goes to
$f^{\prime}\in{\rm Ind}_{Y}^{H}\psi_{Y}$ defined by

$f^{\prime}(h)=\int_{Y/X\cap Y}f(yh)\bar{\psi}_{Y}(y)~{}dy.$

The lemma follows by combining this isomorphism with the two Frobenius
reciprocity isomorphisms. ∎

###  11.3. Shalika period on ${\rm PGSp}_{6}$.

We shall now discuss a Shalika period on ${\rm PGSp}_{6}$.

Recall the maximal parabolic subgroup $P_{2}=M_{2}N_{2}$ of ${\rm PGSp}_{6}$
with Levi factor $M_{2}\cong({\rm GL}_{2}\times{\rm GL}_{2})/{\rm
GL}_{1}^{\nabla}$ and unipotent radical $N_{2}$. Let $\psi_{2}$ be a generic
character of $N_{2}(F)$. The stabilizer of $\psi_{2}$ in the Levi group
$M_{2}$ is the diagonally embedded ${\rm PGL}_{2}^{\Delta}$. The Shalika
subgroup of ${\rm PGSp}_{6}$ is the semi-direct product

$S=PGL_{2}^{\Delta}\ltimes N_{2}$

and the Shalika character $\psi_{S}$ is the character $\psi_{2}$ extended to
$S(F)$ (trivially on ${\rm PGL}_{2}(F)$). For any smooth representation $\pi$
of ${\rm PGSp}_{6}(F)$, the Shalika period of $\pi$ is the coinvariant space
$\pi_{S,\psi_{S}}$.

This Shalikla period has already been exploited in [SWe]. Indeed, the
following was shown in [SWe, Lemma 4.5]:

######  Proposition 11.5.

Let $\Pi$ be the minimal representation of $E_{7}$ and $(V,\psi_{V})$ a
Whittaker datum for $G_{2}$. Then

$\Pi_{V,\psi_{V}}\cong{\rm ind}_{S}^{{\rm PGSp}_{6}}\psi_{S}$

as ${\rm PGSp}_{6}$-modules.

###  11.4. Shalika period of $\Pi$.

We now consider the minimal representation $\Pi$ of the dual pair
$G_{2}\times{\rm PGSp}_{6}$ and determine its Shalika period
$\Pi_{S,\psi_{S}}$ as a representation of $G_{2}(F)$. To describe the answer,
we need to introduce some more notations.

The group ${\rm PGL}_{2}$ acts by conjugation on the space $M_{2}(F)$ of
$2\times 2$ matrices, preserving the determinant (quadratic) form. As we saw
in §8.4, there is a Weil representation of ${\rm PGL}_{2}\times{\rm SL}_{2}$
on $C_{c}(M_{2}(F))$ which decomposes as a tensor product

$C_{c}(M_{2}(F))=C_{c}(M^{\circ}_{2}(F))\otimes\rho_{\psi}$

where $M_{2}^{\circ}(F)$ is the space of trace zero matrices. We view
$C_{c}(M^{\circ}_{2}(F))\otimes\rho_{\psi}$ as a representation of the group
$J=[L,L]U\subset G_{2}$ introduced in §11.2, where the first factor is a
representation of $\widetilde{\rm SL}_{2}$ and $\rho_{\psi}$ is the
irreducible representation of $\tilde{J}$ introduced in §11.2. With the group
${\rm PGL}_{2}$ acting trivially on $\rho_{\psi}$, we see that
$C_{c}(M_{2}(F))$ becomes a representation of ${\rm PGL}_{2}\times J$.

We are now ready to compute $\Pi_{S,\psi_{S}}$. Firstly, we have

$\Pi_{N_{2},\psi_{2}}\cong{\rm
ind}_{J}^{G_{2}}(C_{c}(M^{\circ}_{2}(F))\otimes\rho_{\psi})$

as $G_{2}\times{\rm PGL}_{2}$-modules. It remains to compute the ${\rm
PGL}_{2}$-coinvariant of the RHS. We need the following:

######  Lemma 11.6.

Let $H\subset G$ and $L$ be three $p$-adic groups. Let $W$ be a smooth
$H\times L$-module, and $\tau$ an irreducible representation of $L$. Let
$\Theta(\tau)\otimes\tau$ be the maximal $\tau$-isotypic quotient of $W$. If
${\rm Ext}^{1}_{L}(\tau,\tau)=0$ then

${\rm ind}_{H}^{G}\Theta(\tau)\otimes\tau$

is the maximal $\tau$-isotypic quotient of ${\rm ind}_{H}^{G}W$. Here ${\rm
ind}$ stands for induction with compact support.

###### Proof.

Since ${\rm Ext}^{1}_{L}(\tau,\tau)=0$, the kernel of the projection of $W$ on
$\Theta(\tau)\otimes\tau$ does not have $\tau$ as a quotient. Thus, it
suffices to prove that if ${\rm Hom}_{L}(W,\tau)=0$, then ${\rm Hom}_{L}({\rm
ind}_{H}^{G}W,\tau)=0$. We shall prove that

${\rm Hom}_{L}(({\rm ind}_{H}^{G}W)^{K},\tau)=0$

for any open compact subgroup $K$ of $G$. Write $G=\cup_{i\in I}Hg_{i}K$ where
$I$ is an index set, and set $K_{i}=H\cap g_{i}Kg_{i}^{-1}$ for every $i\in
I$. Then, as an $L$-module, $({\rm ind}_{H}^{G}W)^{K}$ is a direct sum of
$W^{K_{i}}$. Since $W^{K_{i}}$ is a direct summand of the $L$-module $W$, it
follows that ${\rm Hom}_{L}(W^{K_{i}},\tau)=0$, and this proves the lemma. ∎

We apply Lemma 11.6 taking $H\subset G$ to be $J\subset G_{2}$ and $L={\rm
PGL}_{2}$. Since ${\rm Ext}^{1}_{{\rm PGL}_{2}}(1,1)=0$, the lemma implies
that computing ${\rm PGL}_{2}$-coinvariants of $\Pi_{N_{2},\psi_{2}}$ boils
down to computing the ${\rm PGL}_{2}$-coinvariant of
$C_{c}(M^{\circ}_{2}(F))$, where it is the full degenerate principal series
$I_{\bar{\psi}}(1/2)$. We have shown:

######  Proposition 11.7.

As a representation of $G_{2}(F)$, one has:

$\Pi_{S,\psi_{S}}\cong(\Pi_{N_{2},\psi_{2}})_{{\rm PGL}_{2}}\cong{\rm
ind}_{J}^{G_{2}}(I_{\bar{\psi}}(1/2)\otimes\rho_{\psi}).$

##  12\. Howe Duality for $G_{2}\times{\rm PGSp}_{6}$: Tempered Case

After the preparation of the previous 3 sections, we are now in a position to
begin our study of the theta correspondence for the dual pair $G_{2}\times{\rm
PGSp}_{6}$. In this section, we shall show the Howe duality theorem for
tempered representations of $G_{2}$. The key is to show the analog of
Propositions 6.6 and 6.7 for generic representations of $G_{2}$. This will
rely on another curious chain of containment given in the following lemma,
which comes from a consideration of a game of ping-pong with periods.

######  Lemma 12.1.

Let $\Pi$ be the minimal representation of $E_{7}$. Let $\pi\in{\rm
Irr}(G_{2})$ be tempered and let $\psi_{V}:V\rightarrow\mathbb{C}^{\times}$ be
a Whittaker character for $G_{2}$. Let $H={\rm PGSp}_{6}$ and $\tau\in{\rm
Irr}(H)$ be tempered such that

${\rm Hom}_{G_{2}\times H}(\Pi,\pi\boxtimes\tau)\neq 0.$

Then we have the following natural inclusions

$\displaystyle{\rm Hom}_{V}(\pi,\psi_{V})\subseteq{\rm
Hom}_{V}(\Theta(\tau),\psi_{V})\cong{\rm
Hom}_{S}(\tau^{\vee},\bar{\psi}_{S})\subseteq{\rm
Hom}_{S}(\Theta(\pi^{\vee}),\bar{\psi}_{S})\cong{\rm Hom}_{G_{2}}({\rm
ind}^{G_{2}}_{J}I_{\psi}(1/2)\otimes\rho_{\bar{\psi}},\pi^{\vee}).$

If $\pi$ is generic, then all of these spaces are one-dimensional.

###### Proof.

We examine each containment in turn:

  * •

The first inclusion arises from the surjection
$\Theta(\tau)\twoheadrightarrow\pi$.

  * •

The second follows from the identity

${\rm Hom}_{V}(\Theta(\tau),\psi_{V})\cong{\rm Hom}_{V\times
H}(\Pi,\psi_{V}\boxtimes\tau)\cong{\rm Hom}_{H}(\Pi_{V,\psi_{V}},\tau)$

combined with Proposition 11.5 (i.e. [SWe, Lemma 4,.5]):

$\Pi_{V,\psi_{V}}\cong\mathrm{ind}_{S}^{H}\psi_{S}$

and the Frobenius reciprocity.

  * •

For the third, observe that $\Theta(\bar{\pi})$ is the complex conjugate of
$\Theta(\pi)$. Since $\bar{\pi}\cong\pi^{\vee}$ and
$\bar{\tau}\cong\tau^{\vee}$ and we have
$\Theta(\pi^{\vee})\twoheadrightarrow\tau^{\vee}$.

  * •

The fourth follows from the identity,

${\rm Hom}_{S}(\Theta(\pi^{\vee}),\bar{\psi}_{S})\cong{\rm Hom}_{S\times
G_{2}}(\Pi,\bar{\psi}_{S}\boxtimes\pi^{\vee})\cong{\rm
Hom}_{G_{2}}(\Pi_{S,\bar{\psi}_{S}},\pi^{\vee})$

combined with Proposition 11.7:

$\Pi_{S,\bar{\psi}_{S}}\cong{\rm
ind}_{J}^{G_{2}}I_{\psi}(1/2)\otimes\rho_{\bar{\psi}}$

and Frobenius reciprocity.

If $\pi$ is generic, then the first and the last spaces are one-dimensional,
with the latter by Lemma 11.3 applied to $s=-1/2<0$. Hence, all spaces in the
chain are one-dimensional. ∎

We can now obtain the following two propositions as consequences of Lemma
12.1.

######  Proposition 12.2.

Let $\tau\in{\rm Irr}({\rm PGSp}_{6})$ be tempered. Then $\Theta(\tau)$ cannot
have two irreducible tempered and generic quotients.

###### Proof.

Let $\pi_{1},\pi_{2}\in{\rm Irr}(G_{2})$ be tempered and generic such that
$\Theta(\tau)\twoheadrightarrow\pi_{1}\oplus\pi_{2}$. Then

$\dim\Theta(\tau)_{V,\psi_{V}}\geq 2.$

On the other hand, $\dim\Theta(\tau)_{V,\psi_{V}}=1$ by Lemma 12.1, which is a
contradiction. ∎

Remark: This proposition is proved in [SWe] using the uniqueness of Shalika
functional, however, the proof of this uniqueness is difficult. The proof here
is based on Lemma 11.3 whose proof is very elementary, based on the Frobenius
reciprocity and the root exchange technique.

######  Proposition 12.3.

Let $\pi\in{\rm Irr}(G_{2})$ be tempered and generic. Then $\Theta(\pi)$
cannot have two tempered irreducible quotients. In particular, its cuspidal
component $\Theta(\pi)_{c}$ is irreducible or $0$.

###### Proof.

Let $\tau_{1},\tau_{2}\in{\rm Irr}({\rm PGSp}_{6})$ be irreducible tempered
and such that $\Theta(\pi)\twoheadrightarrow\tau_{1}\oplus\tau_{2}$. By Lemma
12.1, applied to $\pi^{\vee}$, $\tau_{1}^{\vee}$ and then to $\pi^{\vee}$,
$\tau_{2}^{\vee}$, one has:

$1=\dim{\rm Hom}_{S}(\tau_{1},\psi_{S})=\dim{\rm
Hom}_{S}(\Theta(\pi),\psi_{S})=\dim{\rm Hom}_{S}(\tau_{2},\psi_{S}).$

Since $\tau_{1}\oplus\tau_{2}$ is a quotient of $\Theta(\pi)$,

$1=\dim{\rm Hom}_{S}(\Theta(\pi),\psi_{S})\geq\dim{\rm
Hom}_{S}(\tau_{1},\psi_{S})+\dim{\rm Hom}_{S}(\tau_{2},\psi_{S})=2,$

which is a contradiction. ∎

Combining Propositions 12.2 and 12.3 with the results of §6, we can now show
the Howe duality theorem for tempered representations of $G_{2}$:

######  Theorem 12.4.

Let $\pi\in{\rm Irr}(G_{2})$ be tempered and consider its big theta lift
$\Theta(\pi)$ on ${\rm PGSp}_{6}$. Then

(i) $\Theta(\pi)$ has finite length and a unique irreducible quotient
$\theta(\pi)$ (if nonzero), which is tempered.

(ii) Moereover, for tempered $\pi_{1},\pi_{2}\in{\rm Irr}(G_{2})$,

$0\neq\theta(\pi_{1})\cong\theta(\pi_{2})\Longrightarrow\pi_{1}\cong\pi_{2}.$

###### Proof.

(i) We have seen (i) for non-generic $\pi$ in Corollary 6.8. The proof for
generic $\pi$ is the same, using Lemmas 6.2(i) and 6.3(ii), as well as
Proposition 12.3.

(ii) If one of $\pi_{1}$ or $\pi_{2}$ is nongeneric, then the desired result
follows by Proposition 6.6. If $\pi_{1}$ and $\pi_{2}$ are both generic, then
the desired result follows by Proposition 12.2.

∎

We also point out the following corollary:

######  Corollary 12.5.

Let $\pi\in{\rm Irr}(G_{2})$ be generic, supercuspidal and not a theta lift
from ${\rm PGL}_{3}$. Then $\Theta(\pi)$ is generic, supercuspidal and
irreducible.

###### Proof.

By [SWe], we have known that $\Theta(\pi)$ is generic and supercuspidal (hence
tempered and semisimple), but now we know by Proposition 12.3 that it is also
irreducible. ∎

##  13\. Jacquet Modules

The purpose of this section is to compute the various Jacquet modules of the
minimal representation of $E_{7}$ with respect to the maximal parabolic
subgroups of $G_{2}$ and ${\rm PGSp}_{6}$. We note that the results of this
section are entirely self-contained, and do not depend on any prior results in
this paper. As consequences of the results here, we deduce Lemmas 6.2 and 6.3
for the dual pair $G_{2}\times{\rm PGSp}_{6}$. Indeed, we shall determine in
Theorem 14.1 the theta lifts of all non-tempered representations of $G_{2}$
and ${\rm PGSp}_{6}$ precisely.

###  13.1. Jacquet functors for $G_{2}$

Recall that $P=MN$ and $Q=LU$ are the two maximal parabolic subgroups of
$G_{2}$ as before, in standard position relative to a maximal split torus $T$
in $G_{2}$ and a choice of positive roots, so that $P\cap Q$ is a Borel
subgroup. In particular, their Lie algebras arise from $\mathbb{Z}$-gradings
given by two fundamental co-characters. Since $G_{2}$ is contained in $E_{7}$,
as a memeber of the dual pair, the two co-characters give
$\mathbb{Z}$-gradings of the Lie algebra of $E_{7}$, defining parabolic
subgroups $\mathcal{P}=\mathcal{M}\mathcal{N}$ and
$\mathcal{Q}=\mathcal{L}\mathcal{U}$ of $E_{7}$, whose intersections with
$G_{2}$ are $P$ and $Q$, respectively. The Lie types of the Levi factors
$\mathcal{M}$ and $\mathcal{L}$ are $D_{6}$ and $A_{1}\times A_{5}$, as
explained in [GS99]. In the rest of the paper, we shall fix the following:

  * •

The group $P$ acts on $C_{c}({\rm GL}_{2})$ and $C_{c}({\rm GL}_{1})$ by left
translation by $g$ and $\det(g)$, respectively, via the identification
$M\cong{\rm GL}_{2}$ .

  * •

The group $Q$ acts on $C_{c}({\rm GL}_{2})$ and $C_{c}({\rm GL}_{1})$ by left
translation by $g$ and $\det(g)$, respectively, via the identification
$L\cong{\rm GL}_{2}$.

  * •

Let $\bar{B}$ be the group of lower-triangular matrices in ${\rm GL}_{2}$.
Then $\bar{B}$ acts on $C_{c}({\rm GL}_{1})$ by right translation by the
$(1,1)$ matrix entry of $g\in\bar{B}$.

We identify $M\cong{\rm GL}_{2}$ such that the action of $M$ on $N/[N,N]$ is
the symmetric cube of the standard representation of ${\rm GL}_{2}$ twisted by
determinant inverse. In particular, a scalar matrix $(z,z)$ in ${\rm GL}_{2}$
acts by $z$. We have [MS, Theorem 6.1],

######  Proposition 13.1.

Let $H={\rm PGSp}_{6}$. As a ${\rm GL}_{2}\times H$-module, $r_{P}(\Pi)$ (the
normalized Jacquet functor) has a filtration with three successive sub
quotients (top to bottom):

  1. (1)

$\delta_{P}^{-1/2}\cdot\Pi_{\mathcal{N}}=\Pi_{D_{6}}\cdot|\det|^{1/2}\oplus\Pi_{\emptyset}\cdot|\det|^{3/2}$.

  2. (2)

${\rm Ind}_{\bar{B}\times P_{2}}^{{\rm GL}_{2}\times H}(\delta\cdot C_{c}({\rm
GL}_{1}))$.

  3. (3)

${\rm Ind}_{P_{13}}^{H}C_{c}({\rm GL}_{2})$.

Here, note that:

  * -

In (1), the center of $M\cong{\rm GL}_{2}$ acts trivially on both
$\Pi_{D_{6}}$ and $\Pi_{\emptyset}$, the minimal and the trivial
representation of the Levi $\mathcal{M}$.

  * -

In (2), $\delta=|\cdot|^{-1/2}\times|\cdot|$ is a character of the group
$\bar{B}$ of lower triangular matrices in ${\rm GL}_{2}$.

Let $W$ be the Weil representation for the similitude dual pair ${\rm
GL}_{2}\times\mathrm{GSO}_{4}$; see [Ro] where theta correspondences for
similitude groups are treated in detail. Observe that
$\mathrm{GSO}_{4}\cong({\rm GL}_{2}\times{\rm GL}_{2})/{\rm GL}_{1}^{\nabla}$,
with the isomorphism realized by latter acting on the space
$\mathbb{M}_{2}(F)$ of $2\times 2$ matrices by left and right multiplication
and the quadratic form given by the determinant. We identify the first factor
${\rm GL}_{2}$ with $L$ so that the action of $L$ on $U/[U,U]$ is the standard
representation of ${\rm GL}_{2}$. The irreducible quotients of $W$ are
$\pi^{\vee}\otimes\pi\otimes\pi$, where $\pi$ is an irreducible representation
of ${\rm GL}_{2}$. We need a slight refinement of this to the big theta lifts.

######  Lemma 13.2.

Consider the similitude theta correspondence for the dual pair ${\rm
GL}_{2}\times\mathrm{GSO}_{4}$ on $W$. Let $\pi$ be an irreducible generic
representation of ${\rm GL}_{2}$. Then $\Theta(\pi^{\vee})=\pi\otimes\pi$ and
$\Theta(\pi\otimes\pi)=\pi^{\vee}$.

###### Proof.

Let $V\cong F$ be a maximal unipotent subgroup of ${\rm GL}_{2}$ and $\psi$ a
non-trivial character of $V$. We shall use that $W_{V,\psi}=C_{c}({\rm
GL}_{2})$, the regular representation of ${\rm GL}_{2}$, where $V$ is in any
of the three ${\rm GL}_{2}$.

We have $\Theta(\pi\otimes\pi)\otimes(\pi\otimes\pi)$ as a quotient of $W$.
Apply the functor of $(V,\psi)$-coinvariants, with $V$ sitting in one of ${\rm
GL}_{2}$ factors of $\mathrm{GSO}_{4}$, to conclude that
$\Theta(\pi\otimes\pi)\otimes\pi$ is a quotient of the regular representation
of ${\rm GL}_{2}$. This implies that $\Theta(\pi\otimes\pi)\cong\pi^{\vee}$,
as desired. In the other direction, similar arguing shows that
$\Theta(\pi^{\vee})$ cannot be an extension of $\pi\otimes\pi$ by
$\pi\otimes\pi$. Thus the lemma holds except perhaps when $\pi$ is a character
twist of the Steinberg representation ${\rm st}$. For example, $\Theta({\rm
st})$ could be a non trivial extension of ${\rm st}\otimes{\rm st}$ by
$1\otimes 1$. But

${\rm Ext}^{1}_{{\rm SO}_{4}}(1\otimes 1,{\rm st}\otimes{\rm st})=0$

thus one can have a non-trivial extension of these two representations of
$\mathrm{GSO}_{4}$ only if the center of $\mathrm{GSO}_{4}$ does not act by
scalars on $\Theta({\rm st})$. But it does, since the centers of the three
${\rm GL}_{2}$ are identified, the action on $\Theta({\rm st})$ is equal to
the action on ${\rm st}$ where it is scalar action.

∎

######  Proposition 13.3.

Let $H={\rm PGSp}_{6}$. As a ${\rm GL}_{2}\times H$-module, $r_{Q}(\Pi)$ (the
normalized Jacquet functor) has a filtration with three successive sub
quotients (top to bottom):

  1. (1)

$\delta_{Q}^{-1/2}\cdot\Pi_{\mathcal{U}}=\Pi_{A_{5}}\cdot|\det|^{3/2}\oplus\Pi_{A_{1}}\cdot|\det|^{2}$.

  2. (2)

${\rm Ind}_{\bar{B}\times P_{2}}^{{\rm GL}_{2}\times H}(\delta\cdot C_{c}({\rm
GL}_{1}))$.

  3. (3)

${\rm Ind}_{P_{2}}^{H}W$.

  * -

In (1), the center of $L\cong{\rm GL}_{2}$ acts trivially on both
$\Pi_{A_{5}}$ and $\Pi_{A_{1}}$, the minimal and a principal series
representation of the two factors of $\mathcal{L}$.

  * -

In (2) $\delta=|\cdot|^{1/2}\times|\cdot|$ is a character of the group
$\bar{B}$ of lower triangular matrices in ${\rm GL}_{2}$.

###### Proof.

This proposition is entirely similar to Proposition 6.8 in [GS99], which
treated the case of non-split form of $H$, except the character $\delta$ was
not determined there. This is done as follows. For a generic character $\chi$
of ${\rm GL}_{2}$, representations $I_{Q}(\chi)$ and $I_{2}(\chi\otimes\chi)$
are both irreducible and $I_{Q}(\chi)\otimes I_{2}(\chi\otimes\chi)$ is a
quotient of $\Pi$, this follows from the bottom factor (3) of the filtration.
Hence $r_{Q}(I_{Q}(\chi))\otimes I_{2}(\chi\otimes\chi)$ is a quotient of
$r_{P_{2}}(\Pi)$. Now determining $\delta$ is an easy exercise using
$r_{Q}(I_{Q}(\chi))$. ∎

###  13.2. Non-tempered representations

We enumerate the nontempered irreducible representations of $G_{2}$ using the
discussion from Section 3. Let $P=MN$ and $Q=LU$ be the two maximal parabolic
subgroups in $G_{2}$ as before. Their Levi groups are isomorphic to ${\rm
GL}_{2}$. Let $\tau$ be a representation of ${\rm GL}_{2}$, and let
$I_{P}(\tau)$ and $I_{Q}(\tau)$ be the corresponding normalized induced
representations of $G_{2}$. Irreducible, non-tempered representations of
$G_{2}$ are described as follows, where $\tau$ is irreducible, and
$\omega_{\tau}$ is the central character of $\tau$.

  * (a)

Unique irreducible quotient of $I_{Q}(\tau)$ where $\tau$ is an unramified
twist of a tempered representation such that $|\omega_{\tau}|=|\cdot|^{s}$ for
some $s>0$.

  * (b)

Unique irreducible quotient of $I_{P}(\tau)$ where $\tau$ is an unramified
twist of a tempered representation such that $|\omega_{\tau}|=|\cdot|^{s}$ for
some $s>0$.

  * (c)

Unique quotient of $I_{P}(\tau)$ where $\tau$ is the unique quotient of a
representation induced from an ordered pair of characters $\chi_{1},\chi_{2}$
such that $|\chi_{1}|=|\cdot|^{s_{1}}$, $|\chi_{2}|=|\cdot|^{s_{2}}$ where
$s_{1}>s_{2}>0$.

In (a) and (b), $I_{Q}(\tau)$ and $I_{P}(\tau)$ are standard modules, while in
(c), $I_{P}(\tau)$ is a quotient of a standard module associated to the
minimal parabolic $P\cap Q$. In any case, each of these induced
representations has a unique irreducible quotient which we denote by
$J_{Q}(\tau)$ in (a) and by $J_{P}(\tau)$ in (b) and (c). These
representations $J_{Q}(\tau)$ and $J_{P}(\tau)$ exhaust the irreducible
nontempered representations of $G_{2}$.

We also enumerate some relevant nontempered representations of ${\rm
PGSp}_{6}$. Let $P_{i}=M_{i}N_{i}$, $i=1,2,3$ be three maximal parabolic