2.1 Theta correspondences

In [Reference Gan and SavinGS23], we studied the local theta correspondence for the following dual pairs:

where D denotes a cubic division F-algebra. More precisely, one has the dual pairs

$$\begin{align*}\begin{cases} (\mathrm{PGL}_3 \rtimes {\mathbb Z}/2{\mathbb Z}) \times G_2 \subset E_6 \rtimes {\mathbb Z}/2{\mathbb Z} \\ \mathrm{PD}^{\times} \times G_2 \subset E_6^{D} \\ G_2 \times \mathrm{PGSp_6} \subset E_7 \end{cases} \end{align*}$$

where the exceptional groups of type E are of adjoint type. One can thus consider the restriction of the minimal representation of E to the relevant dual pair and obtain a local theta correspondence. In particular, for a representation $\pi $ of one member of a dual pair, one has a big theta lift $\Theta (\pi )$ on the other member of the dual pair, and its maximal semisimple quotient $\theta (\pi )$ . In [Reference Gan and SavinGS23], the following theorem was shown:

Theorem 2.1. (i) (Howe duality) The Howe duality theorem holds for the above dual pairs, i.e. $\Theta (\pi )$ has finite length and its maximal semisimple quotient $\theta (\pi )$ is irreducible or zero.

(ii) (Theta dichotomy) Let $\pi \in \mathrm {Irr}(G_2)$ . Then $\pi $ has nonzero theta lift to exactly one of $\mathrm {PD}^{\times }$ or $\mathrm {PGSp}_6$ . In particular, one has a decomposition:

$$\begin{align*}\mathrm{Irr}(G_2) = \mathrm{Irr}^{\heartsuit}(G_2) \sqcup \mathrm{Irr}^{\spadesuit}(G_2) \end{align*}$$

where $\mathrm {Irr}^{\heartsuit }(G_2)$ consists of those irreducible representations which participate in theta correspondence with $\mathrm {PGSp_6}$ and $\mathrm {Irr}^{\spadesuit }(G_2)$ consists of those which participate in theta correspondence with $\mathrm {PD}^{\times }$ .

(iii) More precisely, one has:

1. (a) the theta correspondence for $\mathrm {PD}^{\times } \times G_2$ defines an injective map

   $$\begin{align*}\theta_{D} : \mathrm{Irr}^{\spadesuit}(G_2) \hookrightarrow \mathrm{Irr}(\mathrm{PD}^{\times}), \end{align*}$$
   which is bijective if $p\ne 3$ . Moreover, $\mathrm {Irr}^{\spadesuit }(G_2)$ is contained in the subset $\mathrm {Irr}_{ds}(G_2)$ of discrete series representations.

2. (b) the theta correspondence for $G_2 \times \mathrm {PGSp_6}$ defines an injection

   $$\begin{align*}\theta: \mathrm{Irr}^{\heartsuit}(G_2) \hookrightarrow \mathrm{Irr} (\mathrm{PGSp_6}). \end{align*}$$
   The map $\theta $ carries tempered representations to tempered representations.

3. (c) the theta correspondence for $(\mathrm {PGL}_3\rtimes {\mathbb Z}/2{\mathbb Z}) \times G_2$ defines an injective map

   $$\begin{align*}\theta_{M_3} : \mathrm{Irr}^{\clubsuit}(G_2) \hookrightarrow \mathrm{Irr} (\mathrm{PGL}_3 \rtimes {\mathbb Z}/2{\mathbb Z}), \end{align*}$$
   where $\mathrm {Irr}^{\clubsuit }(G_2) \subset \mathrm {Irr}^{\heartsuit }(G_2)$ is the subset of representations which participates in theta correspondence with $\mathrm {PGL}_3 \rtimes {\mathbb Z}/2{\mathbb Z}$ . The map $\theta _{M_3}$ respects tempered (resp. discrete series) representations. Moreover, its image has been completely determined.

By Theorem 2.1(iii), as a refinement of the decomposition of $\mathrm {Irr}(G_2)$ in Theorem 2.1(i), one has a further decomposition

(2.2) $$ \begin{align} \mathrm{Irr}^{\heartsuit}(G_2) = \mathrm{Irr}^{\diamondsuit}(G_2) \sqcup \mathrm{Irr}^{\clubsuit}(G_2), \end{align} $$

where $\mathrm {Irr}^{\clubsuit }(G_2)$ consists of those representations which participate in theta correspondence with $\mathrm {PGL}_3\rtimes {\mathbb Z}/2{\mathbb Z}$ and $ \mathrm {Irr}^{\diamondsuit }(G_2)$ consists of those which participate exclusively in the theta correspondence with $\mathrm {PGSp_6}$ . Further, we have a trichotomy result for discrete series representations:

Proposition 2.3. Each irreducible discrete series representation of $G_2$ has a nonzero discrete series theta lift to exactly one of $\mathrm {PD}^{\times }$ , $\mathrm {PGL}_3 \rtimes {\mathbb Z}/2{\mathbb Z}$ or $\mathrm {PGSp_6}$ . Setting $\mathrm {Irr}^{\bullet }_{ds}(G_2) = \mathrm {Irr}_{ds}(G_2) \cap \mathrm {Irr}^{\bullet }(G_2)$ , this trichotomy is given by the disjoint union

$$\begin{align*}\mathrm{Irr}_{ds}(G_2) = \mathrm{Irr}^{\spadesuit}_{ds}(G_2) \sqcup \mathrm{Irr}^{\clubsuit}_{ds}(G_2) \sqcup \mathrm{Irr}_{ds}^{\diamondsuit}(G_2). \end{align*}$$

In fact, the results of [Reference Gan and SavinGS23] give a precise determination of the maps $\theta _{M_3}$ and $\theta $ on nonsupercuspidal (in particular nontempered) representations. These precise results are too intricate to recall here; we refer the reader to [Reference Gan and SavinGS23].

2.2 Maps of L-parameters

The above dichotomy results on the representation theory side is reflected to some extent by analogous results on the side of L-parameters. Recall the natural morphisms of Langlands dual groups

which induce natural maps

Moreover, the outer automorphism of $\mathrm {SL}_3({\mathbb C})$ induces an action of ${\mathbb Z}/2{\mathbb Z}$ on $\Phi (\mathrm {PGL}_3)$ . We note:

Lemma 2.4. (i) The map $\iota ^{\prime }_*$ gives an injection

$$\begin{align*}\Phi(\mathrm{PGL}_3)/ _{{\mathbb Z}/2{\mathbb Z}} \hookrightarrow \Phi(G_2), \end{align*}$$

whose image we denote by $\Phi ^{\spadesuit \clubsuit }(G_2)$ . It restricts to an injection

$$\begin{align*}\Phi_{ds}(\mathrm{PGL}_3)/_{{\mathbb Z}/2{\mathbb Z}} \hookrightarrow \Phi_{ds}(G_2), \end{align*}$$

and the image of $\Phi _{ds}(\mathrm {PGL}_3)$ is characterized as those $\phi \in \Phi _{ds}(G_2)$ such that the local L-factor $L(s, \mathrm {std} \circ \iota \circ \phi )$ has a pole at $s=0$ .

Moreover, for $\phi \in \Phi _{ds}(\mathrm {PGL}_3)$ with component group $S_{\phi } = \mu _3$ , one has

$$\begin{align*}S_{\iota^{\prime}_*(\phi)} = \begin{cases} \mu_3 \text{ if } \phi\ \text{is not self-dual;} \\ S_3 \text{ if } \phi\ \text{is self-dual.} \end{cases} \end{align*}$$

(ii) The map $\iota _*$ gives an injection

$$\begin{align*}\iota_*: \Phi(G_2) \longrightarrow \Phi(\mathrm{PGSp_6}) \end{align*}$$

which restricts to an injection

$$\begin{align*}\Phi^{\diamondsuit}_{ds}(G_2) := \Phi_{ds}(G_2) \smallsetminus \Phi^{\spadesuit \clubsuit}(G_2) \longrightarrow \Phi_{ds}(\mathrm{PGSp_6}). \end{align*}$$

The image of $\Phi ^{\diamondsuit }_{ds}(G_2)$ is characterized as those $\phi ' \in \Phi _{ds}(\mathrm {PGSp_6})$ such that the local L-factor $L(s, {\mathrm {spin}} \circ \phi ')$ has a pole at $s=0$ . Moreover, for $\phi \in \Phi ^{\diamondsuit }_{ds}( G_2)$ with component group $S_{\phi }$ , one has

$$\begin{align*}\iota_*: S_{\phi} \cong S_{\iota_*(\phi)}/ Z(\mathrm{Spin}_7), \end{align*}$$

where $Z(\mathrm {Spin}_7) \cong \mu _2$ is the center of $\mathrm {Spin}_7({\mathbb C})$ .

(iii) The map

$$\begin{align*}\mathrm{std}_* \circ \iota_*: \Phi(G_2) \longrightarrow \Phi(\mathrm{Sp_6}) \end{align*}$$

is injective and restricts to an injection

$$\begin{align*}\Phi^{\diamondsuit}_{ds}(G_2) \hookrightarrow \Phi_{ds}(\mathrm{Sp_6}). \end{align*}$$

Proof. (i) If $\phi _1, \phi _2 \in \Phi (\mathrm {PGL}_3)$ are conjugate in $G_2(\mathbb C)$ , then the following semi-simple representations are isomorphic:

$$\begin{align*}\phi _1+ \phi_1^{\vee} +1 \cong \phi _2+ \phi_2^{\vee} +1. \end{align*}$$

We need to show that this isomorphism implies $\phi _1\cong \phi _2$ or $\phi _1\cong \phi _2^{\vee }$ . This is obvious if $\phi _1$ is irreducible. If $\phi _1$ contains an irreducible two dimensional summand, then that summand must be contained in $\phi _2$ or $\phi _2^{\vee }$ . Assume that it is contained in $\phi _2$ . The one-dimensional summand of $\phi _1$ is the determinant inverse of the two-dimensional summand, hence it is also contained in $\phi _2$ and thus $\phi _1\cong \phi _2$ . We leave the case of three summands as an exercise. Assume $\phi \in \Phi _{ds}(\mathrm {PGL}_3)$ . Then the centralizer of $\phi $ in $\mathrm {SL}_3(\mathbb C)$ is $\mu _3$ . The stabilizer in $G_2(\mathbb C)$ of $1\subset 1+ \phi + \phi ^{\vee }$ is $\mathrm {SL}_3(\mathbb C) \rtimes {\mathbb Z}/2{\mathbb Z}$ . It follows that the centralizer of $\phi $ in $G_2(\mathbb C)$ is either $\mu _3$ or $S_3$ , depending on whether $\phi \not \cong \phi ^{\vee }$ or $\phi \cong \phi ^{\vee }$ respectively.

For (ii) and (iii), we refer the reader to [Reference Gross and SavinGrS2, Chapter 4, Section 1].

3.1 First definition of $\mathcal {L}$

We shall give two slightly different constructions of the map $\mathcal {L}$ .

For the first construction (which was sketched in the introduction), we make use of the decomposition

$$\begin{align*}\mathrm{Irr}(G_2) = \mathrm{Irr}^{\spadesuit}(G_2) \sqcup \mathrm{Irr}^{\heartsuit}(G_2) \end{align*}$$

introduced in Theorem 2.1(i). First consider $\pi \in \mathrm {Irr}^{\spadesuit }(G_2) \subset \mathrm {Irr}_{ds}(G_2)$ , and set $\tau = \theta _D(\pi )$ for a unique $\tau \in \mathrm {Irr}(\mathrm {PD}^{\times })$ . By the Jacquet-Langlands correspondence and the LLC for $\mathrm {PGL}_3$ , one has a discrete L-parameter

$$\begin{align*}\phi_{\tau} : WD_F \longrightarrow \mathrm{SL}_3({\mathbb C}). \end{align*}$$

Composing this with the natural inclusion

$$\begin{align*}\iota' : \mathrm{SL}_3({\mathbb C}) \longrightarrow G_2({\mathbb C}), \end{align*}$$

we set

$$\begin{align*}\mathcal{L}(\pi) = \iota' \circ \phi_{\tau}: WD_F \longrightarrow G_2({\mathbb C}), \end{align*}$$

which is an element of $\Phi _{ds}(G_2)$ by Lemma 2.4(i).

For $\pi \in \mathrm {Irr}^{\heartsuit }(G_2)$ , let $\sigma = \theta (\pi ) \in \mathrm {Irr}(\mathrm {PGSp_6})$ and consider its restriction

$$\begin{align*}\mathrm{rest}(\sigma) \in \mathrm{Irr}(\mathrm{Sp_6})/_{\mathrm{PGSp_6}}. \end{align*}$$

By the LLC for $\mathrm {Sp_6}$ , $\mathrm {rest}(\sigma )$ gives rise to an L-parameter

$$\begin{align*}\phi_{\mathrm{rest}(\sigma)} \in \Phi(\mathrm{Sp_6}). \end{align*}$$

In view of Lemma 2.4, it remains to verify:

Proposition 3.1. For $\pi \in \mathrm {Irr}^{\heartsuit }(G_2)$ , with $\sigma = \theta (\pi ) \in \mathrm {Irr}(\mathrm {PGSp_6})$ ,

$$\begin{align*}\phi_{\mathrm{rest}(\sigma)} \in \mathrm{Im}(\mathrm{std}_* \circ \iota_*), \end{align*}$$

so that it gives rise to a uniquely determined element $\mathcal {L}(\pi ) \in \Phi (G_2)$ by Lemma 2.4(iii).

We shall verify this in §3.3 by global means.

3.2 Second definition of $\mathcal {L}$

Our second construction is closely related to the first and is based on the more refined decomposition

$$\begin{align*}\mathrm{Irr}(G_2) = \mathrm{Irr}^{\spadesuit}(G_2) \sqcup \mathrm{Irr}^{\clubsuit}(G_2) \sqcup \mathrm{Irr}^{\diamondsuit}(G_2) \end{align*}$$

in (2.2).

For $\pi \in \mathrm {Irr}^{\spadesuit }(G_2)$ or $\mathrm {Irr}^{\diamondsuit }(G_2) \subset \mathrm {Irr}^{\heartsuit }(G_2)$ , the definition of $\mathcal {L}(\pi )$ is as in the first construction above. Thus, the difference only arises for those $\pi \in \mathrm {Irr}^{\clubsuit }(G_2) \subset \mathrm {Irr}^{\heartsuit }(G_2)$ . Such a $\pi $ has nonzero theta lift to a unique $\tilde {\tau } \in \mathrm {Irr}(\mathrm {PGL}_3 \rtimes {\mathbb Z}/2{\mathbb Z})$ . Restricting $\tilde {\tau }$ to $\mathrm {PGL}_3$ gives a well-defined element

$$\begin{align*}\{ \tau, \tau^{\vee} \} \in \mathrm{Irr}(\mathrm{PGL}_3) / _{{\mathbb Z}/2{\mathbb Z}}, \end{align*}$$

and thus an L-parameter

$$\begin{align*}\phi_{\tilde{\tau}}: WD_F \longrightarrow \mathrm{SL}_3({\mathbb C}) \end{align*}$$

which is well-defined up to the outer automorphism action. By Lemma 2.4, we may set

$$\begin{align*}\mathcal{L}(\pi) = \iota' \circ \phi_{\tilde{\tau}} \in \Phi(G_2).\end{align*}$$

There is of course a need to reconcile the two definitions for $\pi \in \mathrm {Irr}^{\clubsuit }(G_2)$ . This follows readily from the explicit theta correspondences computed in [Reference Gan and SavinGS23, Thm. 8.2, Thm. 8.5, Thm. 15.1 and Thm. 15.2].

3.3 Proof of Proposition 3.1

To complete the construction of $\mathcal {L}$ , it remains to verify Proposition 3.1. For a non-supercuspidal representation $\pi \in \mathrm {Irr}^{\heartsuit }(G_2)$ , we have determined in [Reference Gan and SavinGS23, Thm. 15.1, Thm 15.2 and Thm. 15.3] the representation $\theta (\pi ) \in \mathrm {Irr}(\mathrm {PGSp_6})$ explicitly. From this, Proposition 3.1 follows readily. Indeed, the results of [Reference Gan and SavinGS23] shows that $\mathcal {L}(\pi ) \in \Phi (G_2)$ is given precisely by the discussion in [Reference Gan and SavinGS23, §3.5].

It remains to treat supercuspidal $\pi \in \mathrm {Irr}^{\heartsuit }(G_2)$ . We shall use a global argument.

Let k be a totally real number field with a place w such that $k_w \cong F$ . Let $\mathbb {O}$ be a totally definite octonion algebra over F with automorphism group $G = \mathrm {Aut}(\mathbb {O})$ . Then $G_v$ is anisotropic at all archimedean places v and $G_v$ is the split $G_2$ at all finite places v. By Proposition 10.5 and Corollary 10.6 in Appendix A, we can find a cuspidal automorphic representation $\Pi = \otimes _v \Pi _v$ of G so that

• • $\Pi _w \cong \pi $ ;

• • $\Pi _{\infty }$ has sufficiently regular infinitesimal character at each archimedean place $\infty $ ;

• • $\Pi _u \cong $ the Steinberg representation $\mathrm {St}$ at some finite place $u \ne w$ .

• • $\Pi $ has nonzero cuspidal global theta lift $\Sigma := \theta (\Pi )$ to the split group $\mathrm {PGSp_6}$ over k.

Here, in the last bullet point, we are considering the dual pair $G \times \mathrm {PGSp_6}$ in the k-rank 3 form of $E_7$ (commonly denoted by $E_{7,3}$ ); this global theta correspondence was studied in [Reference Gross and SavinGrS2].

Now the nonzero cuspidal representation $\theta (\Pi ) =: \Sigma = \otimes _v \Sigma _v$ of $\mathrm {PGSp_6}$ satisfies:

• • $\Sigma _w$ is the representation $\theta (\pi )$ of $\mathrm {PGSp_6}(k_w)$ .

• • $\Sigma _{\infty }$ is a holomorphic discrete series with sufficiently regular infinitesimal character for every archimedean place $\infty $ .

• • $\Sigma _u$ is the Steinberg representation of $\mathrm {PGSp_6}(k_u)$ .

By Arthur [Reference ArthurA], the restriction of $\Sigma $ to $\mathrm {Sp_6}$ has a cuspidal transfer $\mathcal {A}(\mathrm {rest}(\Sigma ))$ to $\mathrm {GL}_7$ (because of the Steinberg local component at u) which is regular algebraic at all real places. Now at each finite place v, it follows by local-global compatibility of the transfer from classical groups to $\mathrm {GL}_N$ that the local L-parameter of $\mathcal {A}(\mathrm { rest}(\Sigma ))_v$ is the local L-parameter of $\mathrm {rest}(\Sigma _v)$ . Moreover, by a result of Chenevier [Reference ChenevierC, Thm. E], at all finite places v, the local L-parameter of $\mathcal {A}(\mathrm {rest}(\Sigma ))_v$ factors through $G_2({\mathbb C})$ , uniquely up to $G_2$ -conjugacy. Hence, the local L-parameter of $\mathrm {rest}(\Sigma _w) = \mathrm {rest}(\theta (\pi ))$ takes value in $G_2({\mathbb C}) \subset \mathrm {SO}_7({\mathbb C})$ , as desired.

We have thus completed the proof of Proposition 3.1.

3.4 Initial properties

The definition of $\mathcal {L}$ given above and the results of [Reference Gan and SavinGS23] allow us to read off some initial properties of $\mathcal {L}$ readily. We highlight some of these here:

• • (Nontempered representations) As mentioned above, the results of [Reference Gan and SavinGS23, Thm. 15.1] show that our definition of $\mathcal {L}$ gives the Langlands parametrization of nontempered representations discussed in [Reference Gan and SavinGS23, §3.5] (based on the consideration of usual desiderata). In particular, $\mathcal {L}$ gives a bijection

  $$\begin{align*}\mathcal{L}: \mathrm{Irr}_{nt}(G_2) \longleftrightarrow \Phi_{nt}(G_2) \end{align*}$$
  where the subscript “nt” stands for “nontempered”. In particular, nontempered L-packets of $G_2$ are singletons.

• • (Tempered representations) The results of [Reference Gan and SavinGS23, Thm 15.2, 15.3] imply that one has a commutative diagram

  

• • (Tempered but non-discrete series) One has a surjection

  $$\begin{align*}\mathcal{L}: \mathrm{Irr}_t(G_2) \smallsetminus \mathrm{Irr}_{ds}(G_2) \twoheadrightarrow \Phi_t(G_2) \smallsetminus \Phi_{ds}(G_2) \end{align*}$$
  such that the fiber over $\phi $ has size $1$ or $2$ , according to whether $|S_{\phi }| =1$ or $2$ . The fiber $\mathcal {L}^{-1}(\phi )$ has a unique generic representation, and we attach it to the trivial character of $S_{\phi }$ .

• • (Summary) In summary, we have established items (i), (ii) and (iii) of the Main Theorem. Moreover, the commutativity of the two diagrams in (iv) of the Main Theorem follows by the construction of $\mathcal {L}$ , as does the characterization of $\mathcal {L}$ given in (v) of the Main Theorem. Furthermore, we have also demonstrated (vii) of the Main Theorem for non-discrete-series parameters $\phi $ (for all p).

• • (Compatibility with global Langlands)

  We can also show (viii) of the Main Theorem. In the context of (viii) in the Main Theorem, the global theta lift $\Sigma $ of $\Pi $ to $\mathrm {PGSp_6}$ is globally generic, regular algebraic and cuspidal with a Steinberg local component. Moreover, the transfer of $\mathrm {rest}(\Sigma )$ to $\mathrm {GL}_7$ is a regular algebraic cuspidal automorphic representation $\mathcal {A}(\mathrm {rest}(\Sigma ))$ (with cuspidality a consequence of the Steinberg local component). Hence, the Galois representation $\rho _{\Pi }$ is associated with $\mathcal {A}(\mathrm {rest}(\Sigma ))$ . By [Reference ChenevierC, Thm. 6.4], $\rho _{\Pi }$ takes value in $G_2(\overline {{\mathbb Q}}_l)$ (after conjugation). By local-global compatibility [Reference Taylor and YoshidaTY, Reference CaraianiCa], the local Galois representation $\rho _{\Pi , v}$ at each finite place v corresponds to the local L-parameter of $\mathrm {rest}(\Sigma _v)$ . But by the construction of $\mathcal {L}$ , the local L-parameter of $\mathrm {rest}(\Sigma _v)$ is $\mathrm {std} \circ \iota \circ \mathcal {L}(\Pi _v)$ . This establishes (viii).

• • (Preservation of $\gamma $ -factors) We now show (ix) of the Main Theorem. Let us first briefly recall the definition of the relevant $\gamma $ -factors given in [Reference Gan and SavinGS24]. The results of [Reference Gan and SavinGS23] can be summarized as the construction of a map

  $$\begin{align*}\mathrm{Lif} : \mathrm{Irr}(G_2) \longrightarrow \mathrm{Irr}(\mathrm{GL}_7) \end{align*}$$
  which is given in the following commutative diagram:
  
  Here,

  1. – the two $\theta $ ’s refer to the local theta correspondence;

  2. – JL refers to the Jacquet-Langlands transfer;

  3. – rest refers to the restriction of representations;

  4. – Art refers to the functorial transfer of irreducible representations from $\mathrm {Sp_6}$ to $\mathrm {GL}_7$ established by Arthur;

  5. – $\boxplus : \mathrm {Irr}_{ds}(\mathrm {PGL}_3) \longrightarrow \mathrm {Irr}(\mathrm {GL}_7)$ is the map

     $$\begin{align*}\tau \mapsto \tau \times 1 \times \tau^{\vee}. \end{align*}$$

  With the above preparation, we can now define the local $\gamma $ -factors. Fix a nontrivial additive character $\psi : F \rightarrow {\mathbb C}^{\times }$ . For $\pi \in \mathrm {Irr}(G_2)$ and $\tau \in \mathrm {Irr}(\mathrm {GL}_r)$ , set

  $$\begin{align*}\gamma(s, \pi \times \tau,\psi) := \gamma(s, \mathrm{Lif}(\pi) \times \tau, \psi). \end{align*}$$
  where the $\gamma $ -factor on the RHS is the Rankin-Selberg $\gamma $ -factor defined by Jacquet-Piatetski-Shapiro-Shalika and Shahidi [Reference Gelbart and ShahidiGeSh].

  Let $\phi _{\pi }: WD_F \longrightarrow G_2({\mathbb C})$ and $\phi _{\tau }: WD_F \longrightarrow \mathrm {GL}_r({\mathbb C})$ be the L-parameters of $\pi $ and $\tau $ respectively and recall that $\mathrm {std}: G_2({\mathbb C}) \longrightarrow \mathrm {GL}_7({\mathbb C})$ is the standard degree 7 irreducible representation of $G_2({\mathbb C})$ . Then it follows from the construction of the LLC for $G_2$ and the definition of the relevant $\gamma $ -factors that one has:

  $$\begin{align*}\gamma(s, \pi \times \tau, \psi) = \gamma(s, (\mathrm{std} \circ \phi_{\pi}) \otimes \phi_{\tau}, \psi), \end{align*}$$
  using the fact that the LLC for $\mathrm {GL}_r$ respects the Rankin-Selberg $\gamma $ -factors. In other words, (ix) follows.

  The reader will no doubt complain that our definition of the $\gamma $ -factor makes the statement (ix) almost a formality, which would not be inaccurate description. However, the main result of [Reference Gan and SavinGS24] states that the family of invariants $\gamma (s, \pi \times \tau ,\psi )$ can be characterized by a (usual) list of properties which is independent of the LLC, including multiplicativity and the global functional equations (for certain cuspidal representations). This characterization of the $\gamma $ -factors shows that the definition given in [Reference Gan and SavinGS24] (and recalled above) is the only possible candidate, and it is this which gives non-trivial content to the statement (ix).

  Finally, the characterization of $\mathcal {L}$ by (ix) asserted in (x) follows from the standard argument that the preservation of $\gamma $ -factors of pairs allows one to determine the 7-dimensional representation $\mathrm { std} \circ \phi _{\pi }$ uniquely (for tempered $\pi $ ). Since $\mathrm {GL}_7({\mathbb C})$ -conjugacy of $G_2({\mathbb C})$ -valued L-parameters implies $G_2({\mathbb C})$ -conjugacy (as we have exploited in the construction of $\mathcal {L}$ ), it follows that $\phi _{\pi }$ is uniquely characterized up to $G_2({\mathbb C})$ -conjugacy,

3.5 The packets for $\Phi _{ds}^{\spadesuit \clubsuit }(G_2)$

By the above discussion, it remains to understand the map

$$\begin{align*}\mathcal{L}: \mathrm{Irr}_{ds}(G_2) \longrightarrow \Phi_{ds}(G_2) \end{align*}$$

on discrete series representations. In fact, by construction, $\mathcal {L}$ restricts to give:

$$\begin{align*}\mathcal{L}^{\spadesuit \clubsuit}: \mathrm{Irr}_{ds}^{\spadesuit}(G_2) \sqcup \mathrm{Irr}_{ds}^{\clubsuit}(G_2) \longrightarrow \Phi_{ds}^{\spadesuit \clubsuit}(G_2). \end{align*}$$

and

$$\begin{align*}\mathcal{L}^{\diamondsuit}: \mathrm{Irr}_{ds}^{\diamondsuit}(G_2) \longrightarrow \Phi_{ds}^{\diamondsuit}(G_2). \end{align*}$$

The results of [Reference Gan and SavinGS23, §7 and §8] allow us to understand the fibers of the map $\mathcal {L}^{\spadesuit \clubsuit }$ quite precisely.

Proposition 3.2. The map $\mathcal {L}^{\spadesuit \clubsuit }$ is surjective. Moreover, for $\phi \in \Phi ^{\spadesuit \clubsuit }_{ds}(G_2)$ , the fiber $\mathcal {L}^{-1}(\phi )$ contains a unique generic element (belonging to $\mathrm {Irr}_{ds}^{\clubsuit }(G_2)$ ). Further, for $p \ne 3$ , one has a natural bijection

$$\begin{align*}\mathcal{L}^{-1}(\phi) \longleftrightarrow \mathrm{Irr} (S_{\phi}) \end{align*}$$

such that the unique generic element corresponds to the trivial character of $S_{\phi }$ . For $p= 3$ , we only know that there is an injection $ \mathcal {L}^{-1}(\phi ) \hookrightarrow \mathrm {Irr} (S_{\phi })$ in general, though this injection is bijective when $\phi |_{\mathrm {SL}_2}$ is the subregular $\mathrm {SL}_2$ in $G_2({\mathbb C})$ .

Proof. As almost all statements of the proposition follow from [Reference Gan and SavinGS23, §7 and §8], we will focus on the main new information here: the natural bijection

$$\begin{align*}\mathcal{L}^{-1}(\phi) \longleftrightarrow \mathrm{Irr} (S_{\phi}) \end{align*}$$

when $p \ne 3$ . To obtain this, we need to set things up rather carefully.

Let $M_3$ be the F-algebra of $3 \times 3$ -matrices, with associated Jordan algebra $M_3^+$ . The automorphism group $\mathrm {Aut}(M_3^+)$ of $M_3^+$ is a disconnected group whose identity component is $H = \mathrm {PGL}_3$ , with Langlands dual group $H^{\vee }= \mathrm {SL}_3({\mathbb C})$ . Note that $H^1(F, H) = H^1(F, \mathrm {PGL}_3)$ classifies the isomorphism classes of central simple F-algebras of degree $3$ , with the distinguished point corresponding to $M_3$ . Moreover, the invariant map gives a bijection:

$$\begin{align*}\mathrm{inv}: H^1(F, H) = H^1(F, \mathrm{PGL}_3) \longrightarrow {\mathbb Z}/3{\mathbb Z}. \end{align*}$$

For each nontrivial $j \in {\mathbb Z}/3{\mathbb Z}$ , the associated central simple algebra $D_j$ is a cubic division algebra. Moreover, $D_1$ and $D_2$ are opposite of each other, so that the inverse map $x \mapsto x^{-1}$ gives a canonical isomorphism of multiplicative groups $i: D_1^{\times } \cong D_2^{\times }$ .

Suppose that $\tau $ is a discrete series representation of $\mathrm {PGL}_3$ . Then for $j \in {\mathbb Z}/3{\mathbb Z}$ , one has a Jacquet-Langlands transfer $JL_{D_j}(\tau )$ of $\tau $ to $D_j^{\times }$ . Via the isomorphism $i: D_1^{\times } \cong D_2^{\times }$ , we may compare $JL_{D_1}(\tau )$ and $JL_{D_2}(\tau )$ . One has

$$\begin{align*}JL_{D_1}(\tau)^{\vee} \cong i^*(JL_{D_2}(\tau)). \end{align*}$$

Consider now $[\phi ] \in \Phi _{ds}^{\spadesuit \clubsuit }(G_2)$ , which is a $G_2$ -conjugacy class of homomorphisms $\phi : WD_F \longrightarrow G_2({\mathbb C})$ . One has a projective system (relative to $G_2$ -conjugacy) of centralizer subgroups $Z_{G_2}(\phi )$ which are finite of order $3$ , and thus identified with the component groups $S_{\phi }$ . We denote this projective system of local component groups by $S_{[\phi ]}$ and set $C_{[\phi ]} = Z_{G_2}(S_{[\phi ]})$ : this is a projective system of subgroups of $G_2({\mathbb C})$ which are simply-connected of type $A_2$ . We fix a projective system (relative to $G_2$ -conjugacy) of isomorphisms

$$\begin{align*}i_{[\phi]} : H^{\vee} = \mathrm{SL}_3({\mathbb C}) \longrightarrow C_{[\phi]}. \end{align*}$$

Note that there are basically two inner automorphism classes of such projective systems of isomorphisms, exchanged by an outer automorphism of $\mathrm {SL}_3({\mathbb C})$ .

By virtue of $i_{[\phi ]}$ , one has:

1. (a) $[\phi ]$ gives rise to a well-defined $\mathrm {SL}_3({\mathbb C})$ -conjugacy classes of maps

   $$\begin{align*}[\rho]: WD_F \longrightarrow \mathrm{SL}_3({\mathbb C}), \end{align*}$$
   i.e. a discrete series L-parameter of $H = \mathrm {PGL}_3$ , so that
   $$\begin{align*}[\phi] = i_{[\phi]} \circ [\rho]. \end{align*}$$
   Let $\tau _{\rho }$ be the discrete series representation of $\mathrm {PGL}_3$ with L-parameter $\rho $ .

2. (b) One also has a projective system of isomorphism

   $$\begin{align*}i_{[\phi]} : Z(H^{\vee}) = Z(\mathrm{SL}_3({\mathbb C})) \longrightarrow S_{[\phi]} = Z(C_{[\phi]}) \end{align*}$$
   inducing a projection system of bijections:
   
   Hence, an element $\eta \in \mathrm {Irr}(S_{[\phi ]})$ gives rise to a central simple algebra $D_{j(\eta )}$ of degree $3$ , determined by its invariant $j(\eta ) \in {\mathbb Z}/3{\mathbb Z}$ .

3. (c) By combining (a) and (b), one sees that $\eta \in \mathrm {Irr}(S_{[\phi ]})$ gives rise to a representation

   $$\begin{align*}\tau_{\phi, \eta} = JL_{D_{j(\eta)}}(\tau_{\rho}) \in \mathrm{Irr}(PD_{j(\eta)}^{\times}). \end{align*}$$

Now the central simple algebra $D_{j(\eta )}$ , or rather its associated Jordan algebra $D_{j(\eta )}^+$ gives rise to a local theta correspondence for the dual pair

$$\begin{align*}\mathrm{Aut}(D_{j(\eta)}^+) \times G_2 \subset E_6^{D_{j(\eta)}^+}. \end{align*}$$

This is the local theta correspondence used in the definition of $\theta _{D}$ and $\theta _{M_3}$ in the Main Theorem. Observe that $D_{j(\eta )}^+ \cong D_{-j(\eta )}^+$ and we have canonical (up to inner automorphisms) isomorphisms

$$\begin{align*}PD_{j(\eta)}^{\times} = \mathrm{Aut}(D_{j(\eta)}^+)^0 \cong \mathrm{Aut}(D_{-j(\eta)}^+)^0 = PD_{-j(\eta)}^{\times}, \end{align*}$$

where the composite $i : PD_{j(\eta )}^{\times } \rightarrow PD_{-j(\eta )}^{\times }$ is given by $i(x)=x^{-1}$ (up to inner automorphisms). Hence one can define a representation

$$\begin{align*}\pi_{\eta} := \theta( \tau_{\phi, \eta}) = \theta( JL_{D_{j(\eta)}(\tau_{\rho}}) ) \cong \theta( JL_{D_{-j(\eta)}}(\tau_{\rho}^{\vee}) ), \end{align*}$$

where for the last isomorphism we used the fact that $i^* ( JL_{D_{j(\eta )}}(\tau _{\rho }))= JL_{D_{-j(\eta )}}(\tau _{\rho }^{\vee })$ as observed earlier. When $p \ne 3$ , we have seen in [Reference Gan and SavinGS23] that $\pi _{\eta }$ is nonzero irreducible. When $p =3$ , we only know this nonvanishing when $\eta $ is trivial or if $\phi $ gives the subregular $\mathrm {SL}_2$ when restricted to the Deligne $\mathrm {SL}_2$ .

It is clear from the construction of $\mathcal {L}$ that one has

$$\begin{align*}\mathcal{L}^{-1}([\phi]) = \{ \pi_{\eta}: \eta \in \mathrm{Irr} (S_{[\phi]}) \} \quad \text{for each}\ [\phi] \in \Phi^{\spadesuit\clubsuit}_{ds}(G_2). \end{align*}$$

This gives the natural parametrization of the fibers of $\mathcal {L}^{\spadesuit \clubsuit }$ in the proposition.

Note that this parametrization does not depend on the choice of the projective system of isomorphism $i_{[\phi ]}$ . Indeed, there are 2 such choices as noted above, but changing the choice replaces $\rho $ by $\rho ^{\vee }$ in (a) above (and hence $\tau _{\rho }$ by $\tau _{\rho }^{\vee }$ ) and j by $-j$ in (b). Hence, under this new regime, $\eta \in \mathrm {Irr}(S_{[\phi ]})$ is associated by the same process to

$$\begin{align*}\theta(JL_{D_{-j(\eta)}}(\tau_{\rho}^{\vee})) \cong \theta( JL_{D_{j(\eta)}}(\tau_{\rho}) ) = \pi_{\eta}, \end{align*}$$

as asserted.

After the above proposition, we see that to prove the Main Theorem, it remains to analyze the map

$$\begin{align*}\mathcal{L}^{\diamondsuit}: \mathrm{Irr}_{ds}^{\diamondsuit}(G_2) \longrightarrow \Phi_{ds}^{\diamondsuit}(G_2). \end{align*}$$

For example, a first key question to address is whether this map is surjective. Before addressing such questions, let us describe in the following two sections some ingredients we shall use.

4.1 Triality and Spin representations

We begin with a brief discussion of the phenomenon of triality. Recall that there are 3 different maps over F:

$$\begin{align*}f_1, f_2, f_3: \mathrm{SO}_8 \longrightarrow \mathrm{PGSO}_8 \end{align*}$$

where the groups $\mathrm {SO}_8$ and $\mathrm {PGSO}_8$ are split. These three morphisms are non-conjugate under $\mathrm {PGSO}_8$ but are cyclically permuted by an order 3 outer automorphism of $\mathrm {PGSO}_8$ (the triality automorphism). Moreover, they induce corresponding morphisms

$$\begin{align*}f_1^{\vee}, f_2^{\vee}, f_3^{\vee} : \mathrm{Spin}_8({\mathbb C}) \longrightarrow \mathrm{SO}_8({\mathbb C}) \end{align*}$$

on the dual side. Without loss of generality, let us fix

$$\begin{align*}f_1 : \mathrm{SO}_8 \hookrightarrow \mathrm{GSO}_8 \longrightarrow \mathrm{PGSO}_8 \end{align*}$$

and call it the “standard $\mathrm {SO}_8$ ”. Then $f_1^{\vee } = \mathrm {std}: \mathrm {Spin}_8({\mathbb C}) \longrightarrow \mathrm {SO}_8({\mathbb C})$ will be called the standard representation of $\mathrm {Spin}_8({\mathbb C})$ . On the other hand, $f_2^{\vee }$ and $f_3^{\vee }$ will be called the half-spin representations of $\mathrm {Spin}_8({\mathbb C})$ .

Consider now the standard embedding

Then as representations of $\mathrm {Spin}_7({\mathbb C})$ , one has:

$$\begin{align*}f_1^{\vee} \circ i = \mathrm{std} \oplus 1 \quad \text{and} \quad f_2^{\vee} \circ i = f_3^{\vee} \circ i = {\mathrm{spin}}. \end{align*}$$

Thus, one has a very useful and convenient description of the Spin representation of $\mathrm {Spin}_7({\mathbb C})$ .

4.2 Implication

We observe some implications in representation theory.

For an irreducible generic representation $\sigma $ of $\mathrm {PGSp_6}$ , and $\sigma ^{\flat } \in \mathrm {Irr}(\mathrm {Sp_6})$ contained in the restriction $\mathrm {rest}(\sigma )$ , one may consider the usual (isometry) theta lift of $\sigma ^{\flat }$ from $\mathrm {Sp_6}$ to $\mathrm {SO}_8$ , as well as the similitude theta lift of $\sigma $ to $\mathrm {PGSO}_8$ . Both these theta lifts are nonzero and one has the following compatibility:

$$\begin{align*}\theta(\sigma^{\flat}) \subset \theta(\sigma)|_{\mathrm{SO}_8} = f_1^*(\theta(\sigma)). \end{align*}$$

If the L-parameter of $\sigma ^{\flat }$ is $\phi ^{\flat }$ , then the L-parameter of any irreducible constituent of $f_1^*(\theta (\sigma ))$ is $\phi ^{\flat } \oplus 1$ .

On the other hand, if one considers $f_2^*(\theta (\sigma ))$ instead, one potentially gets a very different representation of $\mathrm {SO}_8$ with a very different L-parameter. Indeed, consider the special case when $\sigma $ is unramified with Satake parameter $s \in \mathrm {Spin}_7({\mathbb C})$ . Then one can show that $\theta (\sigma )$ is unramified with Satake parameter $i(s) \in \mathrm {Spin}_8({\mathbb C})$ . Thus, the Satake parameter for $f_1^*(\theta (\sigma ))$ is $f_1^{\vee }(i(s)) = 1 \oplus s \in \mathrm {SO}_8({\mathbb C})$ (as we noted above), whereas that of $f_2^*(\theta (\sigma ))$ is

$$\begin{align*}f_2^{\vee}(i(s)) = {\mathrm{spin}} (s). \end{align*}$$

This suggests that the map on generic representations of $\mathrm {PGSp_6}$ given by

$$\begin{align*}\sigma\mapsto \theta(\sigma) \mapsto f_2^*(\theta(\sigma)) \in \mathrm{Irr}(\mathrm{SO}_8)/ _{\mathrm{PGSO}_8} \end{align*}$$

is nothing but the Langlands functorial lifting corresponding to the spin representation

$$\begin{align*}{\mathrm{spin}}: \mathrm{Spin}_7({\mathbb C}) \longrightarrow \mathrm{SO}_8({\mathbb C}) \longrightarrow \mathrm{GL}_8({\mathbb C}), \end{align*}$$

after one composes the above with the functorial lifting

$$\begin{align*}\mathcal{A}: \mathrm{Irr}(\mathrm{SO}_8) \longrightarrow \mathrm{Irr}(\mathrm{GL}_8)\end{align*}$$

provided by Cogdell-Kim-Piatetski-Shapiro-Shahidi [Reference Cogdell, Kim, Piatetski-Shapiro and ShahidiCKPSS]) or Arthur [Reference ArthurA].

4.3 Spin lifting

Motivated by the above, we can now define the Spin lifting

$$\begin{align*}{\mathrm{spin}}_*: \mathrm{Irr}_{gen}(\mathrm{PGSp_6}) \longrightarrow \mathrm{Irr}(\mathrm{GL}_8)\end{align*}$$

by

$$\begin{align*}{\mathrm{spin}}_*(\sigma) = \mathcal{A} \left( f_2^*(\theta(\sigma)) \right) \in \mathrm{Irr}(\mathrm{GL}_8). \end{align*}$$

Observe that the above construction can be carried out globally as well. Namely, if $\Sigma $ is a globally generic cuspidal automorphic representation of $\mathrm {PGSp_6}$ over a number field k, then the global theta lift $\theta (\Sigma )$ on $\mathrm {PGSO}_8$ is globally generic and thus nonzero. Suppose further that $\theta (\Sigma )$ is cuspidal. Then $f_2^*(\theta (\Sigma ))$ gives rise to a submodule of globally generic cusp forms on $\mathrm {SO}_8$ , all of whose summands are weak spin lifting of $\Sigma $ . Its transfer $\mathcal {A} \left ( f_2^*(\theta (\Sigma )) \right )$ to $\mathrm {GL}_8$ (à la Cogdell-Kim-Piatetski-Shapiro-Shahidi [Reference Cogdell, Kim, Piatetski-Shapiro and ShahidiCKPSS]) is then an isobaric automorphic representation of $\mathrm {GL}_8$ (of orthogonal type): this is the global analog of the local construction above. Moreover, for each place v, one has:

$$\begin{align*}{\mathrm{spin}}_*(\Sigma)_v \cong {\mathrm{spin}}_*(\Sigma_v) \in \mathrm{Irr}(\mathrm{GL}_8(k_v)). \end{align*}$$

4.4 A key property

We shall now show a key property of our local spin lifting ${\mathrm {spin}}_*$ .

Proof. We shall prove this via a classical global to local argument. Choose a totally complex number field k with some finite set $T \sqcup \{ u,w\}$ of finite places (with T nonempty) such that $k_u\cong k_w \cong F \cong k_v$ for all $v\in T$ . By the globalization result in Proposition 10.7 of Appendix A, there exists a globally generic cuspidal automorphic representation $\Sigma $ such that

• • $\Sigma _v\cong \sigma $ for all $v \in T$ ;

• • $\Sigma _u$ is the Steinberg representation $\mathrm {St}$ ;

• • $\Sigma _{w}$ is a generic supercuspidal representation $\delta $ ;

• • $\Sigma _v$ is a spherical representation at all other finite places v.

Fix a global additive character $\Psi $ of $k \backslash \mathbb {A}_k$ with $\Psi _u=\Psi _w = \psi =\Psi _v$ for all $v \in T$ . For a sufficiently large set S of places, containing $T \sqcup \{u,w\}$ and all archimedean places, we have a global functional equation

$$\begin{align*}L^S(1-s, \Sigma^{\vee}, \mathrm{spin}) = \prod_{v\in S} \gamma(s, \Sigma_v, \mathrm{spin}, \Psi_v)\cdot L^S(s, \Sigma ,\mathrm{spin}) \end{align*}$$

where $L^S$ denotes the partial L-function. Since $\Sigma $ is generic, its theta lift $\theta (\Sigma )$ to $\mathrm {PGSO}_8$ is non-zero and generic. Moreover, since $\Sigma _u$ is the Steinberg representation, its theta lift to $\mathrm {PGSO}_6$ (the smaller group in the Rallis-Witt tower) is 0. Hence $\theta (\Sigma )$ is cuspidal.

By the global analog of the Spin lifting discussed in the previous subsection, we have an isobaric generic automorphic representation

$$\begin{align*}{\mathrm{spin}}_*(\Sigma) = \mathcal{A} \left( f_2^*(\theta(\Sigma)) \right) \end{align*}$$

on $\mathrm {GL}_8$ . Likewise, we have the global functional equation

$$\begin{align*}L^S(1-s, {\mathrm{spin}}_*(\Sigma)^{\vee}, \mathrm{std}) = \prod_{v\in S} \gamma(s, {\mathrm{spin}}_*(\Sigma)_v, \mathrm{std}, \Psi_v)\cdot L^S(s, {\mathrm{spin}}_*( \Sigma), \mathrm{std}). \end{align*}$$

Next, by the theta correspondence of unramified representations, observe that

$$\begin{align*}L^S(s, \Sigma, \mathrm{spin}) =L^S(s, {\mathrm{spin}}_*(\Sigma), \mathrm{std}). \end{align*}$$

Moreover, one can readily check that

$$\begin{align*}\gamma(s, \Sigma_v, \mathrm{spin}, \Psi_v) = \gamma(s, {\mathrm{spin}}_*(\Sigma)_v, \mathrm{std}, \Psi_v) \quad \text{for all}\ v \notin T \sqcup \{u, w \}. \end{align*}$$

Indeed, the local components at finite places outside of $T \sqcup \{ u,w\}$ are generic principal series representations and hence their spin lifts and $\gamma $ -factors are easily computable. For complex groups, the theta correspondence is explicitly known, [Reference Adams and BarbaschAB] (observe here that $\mathrm {PGSp_6}(\mathbb C)=\mathrm {Sp_6}(\mathbb C)/\mu _2$ so the usual theta correspondence is already a correspondence for similitude groups), so that one can check the identities at complex places. Hence, we deduce that

$$\begin{align*}\gamma(s, \Sigma_u, \mathrm{spin}, \Psi_u) \cdot \gamma(s, \Sigma_{w}, \mathrm{spin}, \Psi_{w}) \cdot \prod_{v \in T} \gamma(s, \Sigma_v, \mathrm{spin}, \Psi_v) \end{align*}$$

$$\begin{align*}= \gamma(s, {\mathrm{spin}}_*(\Sigma)_u, \mathrm{std}, \Psi_u) \cdot \gamma(s, {\mathrm{spin}}_*(\Sigma)_{w}, \mathrm{std}, \Psi_{w}) \cdot \prod_{v \in T} \gamma(s, {\mathrm{spin}}_*(\Sigma)_v, \mathrm{std}, \Psi_v). \end{align*}$$

This gives the identity for representations of $G_2(F)$ :

$$\begin{align*}\gamma(s, \mathrm{St}, \mathrm{spin}, \psi) \cdot \gamma(s, \delta, \mathrm{spin}, \psi) \cdot \gamma(s, \sigma, \mathrm{spin}, \psi)^{|T|} \end{align*}$$

$$\begin{align*}= \gamma(s, {\mathrm{spin}}_*(\mathrm{St}), \mathrm{std}, \psi) \cdot \gamma(s, {\mathrm{spin}}_*(\delta), \mathrm{std}, \psi) \cdot \gamma(s, {\mathrm{spin}}_*(\sigma), \mathrm{std}, \psi)^{|T|}. \end{align*}$$

Since we can vary the size of T, by considering the quotient of two such identities (say with $|T| = 1$ and $2$ respectively), we deduce that

$$\begin{align*}\gamma(s, \sigma, \mathrm{spin}, \psi) = \gamma(s, {\mathrm{spin}}_*(\sigma), \mathrm{std}, \psi) \end{align*}$$

as desired. This proves (i).

For (ii), globalize $\chi $ to a quadratic Hecke character $\mathcal {\chi }$ of $\mathbb {A}_k^{\times }$ . By examining places outside S, one checks readily that ${\mathrm {spin}}_*(\Sigma \otimes \mathcal {\chi })$ and ${\mathrm {spin}}_*(\Sigma ) \otimes \mathcal {\chi }$ are nearly equivalent. By the strong multiplicity one theorem for $\mathrm {GL}_8$ , it follows that ${\mathrm {spin}}_*(\sigma \otimes \chi ) \cong {\mathrm {spin}}_*(\sigma )\otimes \chi $ , as desired.

5.1 Results of Kret-Shin

Assume that $\Sigma $ is a cuspidal automorphic representation of $\mathrm {PGSp_6}$ over a totally real number field k such that

• • $\Sigma _{\infty }$ is a sufficiently regular generic discrete series representation at each archimedean place $\infty $ ;

• • $\Sigma _u$ is the Steinberg representation for some finite place u of k.

Then Kret-Shin [Reference Kret and ShinKS, Thm. Reference ArthurA] constructed a global Galois representation

satisfying:

• • for almost all places v, $\rho _{\Sigma }(\mathrm {Frob}_v)_{ss}$ is conjugate to the Satake parameter $s_{\Sigma _v}$ in $\mathrm {Spin}_7(\overline {{\mathbb Q}}_l)$ ;

• • ${\mathrm {std}}\circ \rho _{\Sigma }$ is the Galois representation associated to the restriction $\mathrm {rest}(\Sigma )$ of $\Sigma $ to $\mathrm {Sp_6}$ .

5.2 Spin lifting and Kret-Shin parameters

We shall apply the results of Kret-Shin to the following problem. Suppose that $\sigma $ is an irreducible generic discrete series representation of $\mathrm {PGSp_6}$ . Then $\sigma $ determines a pair

$$\begin{align*}(\phi^{\flat}, \phi^{\#}) \in \Phi(\mathrm{Sp_6}) \times \Phi(\mathrm{GL}_8) \end{align*}$$

where $\phi ^{\flat }$ is the L-parameter of $\mathrm {rest}(\sigma )$ and $\phi ^{\#}$ is the L-parameter of the Spin lifting ${\mathrm {spin}}_*(\sigma )$ . On the other hand, one has the natural map

$$\begin{align*}\mathrm{std}_* \times {\mathrm{spin}}_*: \Phi(\mathrm{PGSp_6}) \longrightarrow \Phi(\mathrm{Sp_6}) \times \Phi(\mathrm{GL}_8) \end{align*}$$

and it is natural to ask if there exists $\phi \in \Phi (\mathrm {PGSp_6})$ such that

$$\begin{align*}\mathrm{std}_*(\phi) = \phi^{\flat} \quad \text{and} \quad {\mathrm{spin}}_*(\phi) = \phi^{\#}. \end{align*}$$

Such a $\phi $ would be a natural candidate for an L-parameter of $\sigma $ . The results of Kret-Shin allow us to show the existence of such a $\phi $ .

For a given $\sigma \in \mathrm {Irr}_{gen,ds}(\mathrm {PGSp_6})$ , we will call a $\phi $ which satisfies the conditions of Proposition 5.1 a Kret-Shin parameter of $\sigma $ . Unfortunately, it is not true that $\sigma $ necessarily has a unique Kret-Shin parameter, because $\mathrm {Spin}_7({\mathbb C})$ is not acceptable [Reference Chenevier and GanCG1]. However, any two Kret-Shin parameters are related to each other via twisting by a quadratic character. The following says that in some cases, the Kret-Shin parameter is unique.

Proof. Let k be a number field as in the proof of Proposition 5.1 and consider the split group $G_2$ over k. By Proposition 10.7 and Lemma 10.8 in Appendix A, we can find a globally generic cuspidal representation $\Pi $ such that $\Pi _w = \pi $ and $\Pi _u$ is the Steinberg representation for some finite place $u \ne w$ . The global theta lift of $\Pi $ to $\mathrm {PGSp_6}$ is then a nonzero globally generic cuspidal representation $\Sigma $ of $\mathrm {PGSp_6}$ . As in the proof of Proposition 5.1, by considering the global Spin lifting of $\Sigma $ to $\mathrm {GL}_8$ and looking at unramified places, we deduce by the strong multiplicity one theorem that

$$\begin{align*}{\mathrm{spin}}_*(\Sigma) = 1 \boxplus \mathrm{std}_*(\Sigma). \end{align*}$$

This implies that the L-parameter of ${\mathrm {spin}}_*(\sigma )$ is $1 + \phi ^{\flat }$ . On the other hand, it is also the case that

$$\begin{align*}{\mathrm{spin}} \circ \iota \circ \mathcal{L}(\pi) = 1 + \phi^{\flat}. \end{align*}$$

Hence, $\iota \circ \mathcal {L}(\pi )$ is a Kret-Shin parameter for $\sigma $ .

Suppose $\phi $ is another Kret-Shin parameter for $\sigma $ . Then $\phi $ is a quadratic twist of $\iota \circ \mathcal {L}(\pi )$ and ${\mathrm {spin}} \circ \phi = 1 \oplus \phi ^{\flat }$ . It follows by [Reference Harris, Khare and ThorneHKT, Lemma 4.6] (which is a group theoretic result based on [Reference GriessGr95]) that $\iota \circ \mathcal {L}(\pi )$ and $\phi $ are in fact conjugate in $\mathrm {Spin}_7({\mathbb C})$ , i.e

$$\begin{align*}\phi = \iota \circ \mathcal{L}(\pi) \in \Phi(\mathrm{PGSp_6}).\end{align*}$$

This completes the proof of the proposition.

8.1 Uniqueness of generic member

We first show:

Proof. Since the existence part has been shown in Proposition 6.1, the main issue here is the uniqueness. Suppose, for the sake of contradiction, that $\pi $ and $\pi '$ are two distinct generic representations of $G_2$ such that $\mathcal {L}(\pi ) = \mathcal {L}(\pi ') = \phi $ . Then $\pi $ and $\pi '$ belongs to $\mathrm {Irr}_{ds}^{\diamondsuit }(G_2)$ and we set

$$\begin{align*}\sigma = \theta(\pi) \quad \text{and} \quad \sigma' = \theta(\pi'). \end{align*}$$

Both $\sigma $ and $\sigma '$ are distinct generic discrete series representations of $\mathrm {PGSp_6}$ (by Howe duality) belonging to $\tilde {\Pi }_{\phi ^{\flat }}$ (see (7.1)). Hence, there is a quadratic character $\chi $ such that $\sigma ' \cong \sigma \otimes \chi \ncong \sigma $ .

Now by Proposition 5.2, $\sigma $ has (unique) Kret-Shin parameter $\iota \circ \phi $ which is an element of $\tilde {\Phi }_{\phi ^{\flat }}$ (see (7.2). On the other hand,

• • by Proposition 4.1(ii), since $\sigma ' \cong \sigma \otimes \chi $ , $\sigma '$ has Kret-Shin parameter $(\iota \circ \phi ) \otimes \chi $ ;

• • by Proposition 5.2, $\sigma ' = \theta (\pi ')$ has Kret-Shin parameter $\iota \circ \phi $ .

By the uniqueness part of Proposition 5.2, we have

$$\begin{align*}\iota \circ \phi = (\iota \circ \phi) \otimes \chi \in \tilde{\Phi}_{\phi^{\flat}}. \end{align*}$$

Hence the quadratic character $\chi $ stabilizes $\iota \circ \phi \in \tilde {\Phi }_{\phi ^{\flat }}$ but does not stabilize $\sigma \in \tilde {\Pi }_{\phi ^{\flat }}$ . This contradicts statement (e) of §7.

8.2 A distinguished Xu’s packet

In view of Proposition 8.1, we can now pick out a distinguished Xu’s packet contained in $\tilde {\Pi }_{\phi ^{\flat }}$ . Namely, we set

$$\begin{align*}\tilde{\Pi}^{X\ast}_{\phi^{\flat}} := \text{the Xu's packet containing the unique generic representation}\ \theta(\pi)\ \text{with}\ \mathcal{L}(\pi) = \phi. \end{align*}$$

Another way to say this is:

$$\begin{align*}\tilde{\Pi}^{X\ast}_{\phi^{\flat}} := \text{the Xu's packet containing the unique generic}\ \sigma \in \tilde{\Pi}_{\phi^{\flat}}\ \text{with nonzero theta lift to}\ G_2. \end{align*}$$

We shall show:

Proposition 8.2. Let $\phi \in \Phi _{ds}^{\diamondsuit }(G_2)$ . The local theta correspondence defines a bijection

$$\begin{align*}\mathcal{L}^{-1}(\phi) \longleftrightarrow \tilde{\Pi}^{X\ast}_{\phi^{\flat}}. \end{align*}$$

8.3 Parametrization of fiber of $\mathcal {L}$

Let us assume this proposition for a moment and explain how it leads to a proof of (vii) of the Main Theorem. It was shown in [Reference Gross and SavinGrS2, Section 4, Prop. 1.10] that the inclusion $\iota $ gives rise to an isomorphism

$$\begin{align*}S_{\phi} \cong S_{\iota\circ \phi}/ Z(\mathrm{Spin}_7), \end{align*}$$

and thus a bijection

$$\begin{align*}\mathrm{Irr} (S_{\iota\circ \phi}/ Z(\mathrm{Spin}_7)) \longleftrightarrow \mathrm{Irr} (S_{\phi}). \end{align*}$$

On the other hand, statement (c) of §7 gives a bijection

$$\begin{align*}\tilde{\Pi}^{X\ast}_{\phi^{\flat}} \longleftrightarrow \mathrm{Irr} (S_{\iota\circ \phi}/ Z(\mathrm{Spin}_7)). \end{align*}$$

Combining this with Proposition 8.2, we obtain a bijection

$$\begin{align*}\mathcal{L}^{-1}(\phi) \longleftrightarrow \tilde{\Pi}^{X\ast}_{\phi^{\flat}} \longleftrightarrow \mathrm{Irr} (S_{\iota\circ \phi}/ Z(\mathrm{Spin}_7)) \longleftrightarrow \mathrm{Irr} (S_{\phi}). \end{align*}$$

This finishes the proof of (vii) of the Main Theorem.

8.4 One in, all in

It remains to prove Proposition 8.2. The following is the key lemma:

Proof. It suffices to show that if $\sigma \in \tilde {\Pi }^X_{\phi ^{\flat }}$ has nonzero theta lift to $G_2$ , then so does any other $\sigma ' \in \tilde {\Pi }^X_{\phi ^{\flat }}$ . This will proceed by a global argument.

Suppose that $\sigma = \theta (\pi )$ for $\pi \in \mathrm {Irr}^{\diamondsuit }_{ds}(G_2)$ . Note that all non-supercuspidal representations in $\mathrm {Irr}^{\diamondsuit }_{ds}(G_2)$ are generic (see [Reference Gan and SavinGS23, Thm. 15.3]). We shall globalize $\pi $ to a cuspidal representation $\Pi $ in one of the following two ways, depending on the type of $\pi $ .

1. (i) Suppose that $\pi $ is generic (this includes all non-supercuspidal $\pi $ ). Let k be a totally real number field with two places $w_1$ and $w_2$ such that $k_{w_1}\cong k_{w_2}\cong F$ . Let $\mathbb O$ be the split octonion algebra over k, and let $G=\mathrm {Aut}(\mathbb O)$ . Fix an additional place w and a generic supercuspidal representation $\delta $ of $G(k_w)$ such that $\theta (\delta )$ is a generic supercuspidal representation of $H(k_w)$ . (Using [Reference GanG] there are such representations $\delta $ of depth 0.) By Proposition 10.7 and Lemma 10.8, one can find a globally generic cuspidal automorphic representation $\Pi =\otimes _v \Pi _v$ of G such that

   • – $\Pi _{\infty }$ is a discrete series representation, with the infinitesimal character sufficiently away from walls, for every archimedean place $\infty $ ;

   • – $\Pi _{w_1}\cong \Pi _{w_2}\cong \pi $ ;

   • – $\Pi _w = \delta $ .

   Then the global theta lift of $\Pi $ to $\mathrm {PGSp_6}$ is a globally generic cuspidal representation $\Sigma $ satisfying

   • – $\Sigma _{\infty }$ is a discrete series representation for every archimedean place $\infty $ ;

   • – $\Sigma _{w_1}\cong \Sigma _{w_2}\cong \sigma = \theta (\pi )$ ;

   • – the restriction of $\Sigma $ to $\mathrm {Sp_6}$ is attached to a generic A-parameter.

   Here, the first item holds by the matching of infinitesimal characters [Reference Huang, Pandžić and SavinHPS] and the fact that, at infinitesimal characters of discrete series sufficiently away from walls, only discrete series representations are unitary. The third item is assured by the first.

2. (ii) Suppose that $\pi $ is supercuspidal. Let k be a totally real number field as in (i). Let $\mathbb O$ be the totally definite octonion algebra over k, and let $G=\mathrm {Aut}(\mathbb O)$ . By Proposition 10.5 and Corollary 10.6, one can find a cuspidal automorphic representation $\Pi $ such that:

   • – $\Pi _{w_1} \cong \Pi _{w_2} \cong \pi $ ;

   • – $\Pi _u$ is the Steinberg representation for some other finite place u;

   • – $\Pi $ has nonzero cuspidal global theta lift $\Sigma $ on $\mathrm {PGSp_6}$ .

   Thus, $\Sigma $ is a nonzero cuspidal automorphic representation of $\mathrm {PGSp_6}$ satisfying:

   • – $\Sigma _{w_1}\cong \Sigma _{w_2}\cong \sigma = \theta (\pi )$ ;

   • – $\Sigma _u$ is the Steinberg representation;

   • – the restriction of $\Sigma $ to $\mathrm {Sp_6}$ is attached to a generic A-parameter.

Here the last item is a consequence of the second.

Now let $\sigma '$ be a representation in $\tilde {\Pi }^X_{\phi ^{\flat }}$ . Then by the Arthur multiplicity formula for $\mathrm {PGSp_6}$ established by Xu, there exists a cuspidal automorphic representation $\Sigma '$ satisfying:

• • $\Sigma ^{\prime }_v \cong \Sigma _v$ for all $v \ne w_1$ or $w_2$ ;

• • $\Sigma ^{\prime }_{w_1} \cong \Sigma ^{\prime }_{w_2} \cong \sigma '$ .

In addition, one has an equality of partial Spin L-functions:

$$\begin{align*}L^S(s, \Sigma', {\mathrm{spin}}) = L^S(s, \Sigma, {\mathrm{spin}}) = \zeta^S(s) \cdot L^S(s, \Sigma, \mathrm{std}). \end{align*}$$

In particular, since $L^S(s, \Sigma , \mathrm {std})$ is nonzero at $s=1$ (as $\mathrm {rest}(\Sigma )$ is associated with a generic A-parameter), we see that $L^S(s, \Sigma ', {\mathrm {spin}})$ has a pole at $s=1$ . By [Reference Gan and SavinGS20, Thm. 1.1], $\Sigma '$ has nonzero global theta lift to a form of $G_2$ over k. Specializing to the place $w_1$ , we see that $\sigma '$ has nonzero local theta lift to the split $G_2$ , as desired. This concludes the proof.

8.5 Proof of Proposition 8.2

Using Lemma 8.3, we can now give the proof of Proposition 8.2. Lemma 8.3 implies that all members of the distinguished Xu’s packet $\tilde {\Pi }^{X \ast }_{\phi ^{\flat }}$ has nonzero theta lift to $G_2$ (since the generic member does). On the other hand, for any other Xu’s packet contained in $\tilde {\Pi }_{\phi ^{\flat }}$ , the generic member does not participate in theta correspondence with $G_2$ by Proposition 8.1. Hence, Lemma 8.3 implies that none of the members does. This proves Proposition 8.2.

10.1 A local non-vanishing result

Let F be a p-adic field and ${\mathbb O}$ be an octonion algebra over F, and let $G=\mathrm {Aut}({\mathbb O})$ . Let D be a quaternion algebra over F and fix an embedding $D \hookrightarrow \mathbb {O}$ , unique up to conjugation by G. Let $G_D\subset G$ be the pointwise stabilizer of $D\subset {\mathbb O}$ . Then $G_D$ is isomorphic to the group of norm one elements in D.

Let $H= \mathrm {PGSp_6}(F)$ , and $P=MN$ the Siegel maximal parabolic subgroup. Let $\hat N$ be the unitary dual of the unipotent radical N. The group M acts on $\hat N$ with open orbits parameterized by quaternion algebras D: the stabilizer in M of an element in the orbit parameterized by is isomorphic to $\mathrm {Aut}(D)$ .

Recall that we have the dual pair $G \times H \subset E_7$ and the minimal representation $\Pi $ of $E_7$ induces a local theta correspondence $\theta : \mathrm {Irr}^{\heartsuit }(G) \longrightarrow \mathrm {Irr}(H)$ . Now we note:

Proof. We claim that there exists $\psi _N$ , in one of the open orbits, such that $\theta (\pi )_{N,\psi _N} \neq 0$ . If not then any summand of $\theta (\pi )$ restricted to $\mathrm {Sp_6}(F)$ is of Howe’s N-rank 2, and thus a classical theta lift of a unitary representation of an orthogonal group $\mathrm {O}(2)$ [Reference LiLi]. But these theta lifts are not tempered, hence the claim

Let $\Pi $ be the minimal representation of the group $E_7$ , so that $\pi \otimes \theta (\pi )$ is its quotient. From [Reference Gross and SavinGrS2, Section 5, Lemma 3.4] (case (4) in the proof) we have

$$\begin{align*}\Pi_{N, \psi_N}= \mathrm{ind}^G_{G_D}(1) \end{align*}$$

where D is such that $\psi _N$ belongs to the open orbit parameterized by D. The lemma follows.

Proof. Recall that $\theta (\mathrm {St})$ is the Steinberg representation of H. Let U be a maximal unipotent subgroup of H containing N. By [Reference Chan and SavinCS, Prop. 5], the restriction of the Steinberg representation of H to U is the regular representation of U. It follows at once that $\theta (\mathrm {St})_{N,\psi _N} \neq 0$ for any character $\psi _N$ of N. Hence, arguing as in Lemma 10.1, $\mathrm {St}$ is a quotient of $\mathrm {ind}^G_{G_D}(1)$ with D non-split. Since

$$\begin{align*}0\neq \mathrm{Hom}_G( \mathrm{ind}^G_{G_D}(1), \mathrm{St}) \cong \mathrm{Hom}_G(\mathrm{St}, \mathrm{Ind}^G_{G_D}(1)) \cong \mathrm{Hom}_{G_D}( \mathrm{St}, \mathbb C) \end{align*}$$

and $G_D$ is anisotropic, there exists a non-zero vector in $\mathrm {St}$ fixed by $G_D$ . The corresponding matrix coefficient gives us

$$\begin{align*}\mathrm{St} \subset L^2(G_D \backslash G)\subset L^2(G).\\[-36pt]\end{align*}$$

We have a complementary result for the compact Lie group $G^c_2(\mathbb {R})$ which is the automorphism group of the octonion division $\mathbb {R}$ -algebra.

Lemma 10.4. Let $\mathbb H$ be the quaternion division $\mathbb {R}$ -algebra. For any irreducible (finite-dimensional) representation $\pi $ of $G^c_2(\mathbb {R})$ , one has $\pi ^{G_{\mathbb H}(\mathbb {R})} \ne 0$ , so that

$$\begin{align*}\pi \subset L^2(G_{\mathbb H}(\mathbb{R}) \backslash G^c_2(\mathbb{R})). \end{align*}$$

10.2 First globalization result

Let k be a totally real number field and $\mathbb O$ be a totally definite octonion algebra over k, so that $\mathbb O_{\infty }$ is non-split for all archimedean places $\infty $ of k, and set $G=\mathrm {Aut}({\mathbb O})$ . For any place v of k, write $G_v=G(k_v)$ for simplicity. Then $G_v$ is anisotropic for archimedean places v and split otherwise.

Let $D\subseteq \mathbb O$ be a quaternion subalgebra. Define

$$\begin{align*}L^2_{D}(G(k)\backslash G(\mathbb A)) \subset L^2(G(k)\backslash G(\mathbb A)) \end{align*}$$

be the Hilbert subspace which is orthogonal to the span of all automorphic representations $\Pi =\otimes _v \Pi _v$ such that

$$\begin{align*}\int_{G_D(k)\backslash G_D(\mathbb A)} f(g) ~dg = 0 \end{align*}$$

for all $f\in \Pi $ . Hence, for any $\Pi \subset L^2_{D}(G(k)\backslash G(\mathbb A))$ , the global period integral over $G_D$ is nonzero on $\Pi $ .

Proof. Pick D such that for all $i=1, \ldots , n$ ,

$$\begin{align*}\pi_i \subset L^2(G_{D,v_i} \backslash G_{v_i} ). \end{align*}$$

This is possible by Corollary 10.2, Lemma 10.3 and Lemma 10.4. We can now proceed following [Reference Sakellaridis and VenkateshSV, Theorem 16.3.2 and Remark 16.4.1]. Using Poincaré series arising from compactly supported functions on $G_D\backslash G$ , one shows that

$$\begin{align*}\bigotimes_{i=1}^n L^2(G_{D, v_i} \backslash G_{v_i}) \quad \text{is weakly contained in} \quad L^2_{D} (G(k)\backslash G(\mathbb A)) \end{align*}$$

in the sense of Definition 11.5. Hence, $\otimes _i \pi _i$ is weakly contained there as well. The proposition now follows from Corollary 11.7, since supercuspidal representations and $\mathrm {St}$ are isolated in the unitary dual of $G_{v_i}$ by Proposition 11.3 (for the Steinberg representation).

We consider the global theta correspondence for the dual pair $G \times H$ in the adjoint group of absolute type $E_7$ (and k-rank $3$ ), that corresponds to the Albert algebra $J_3(\mathbb O)$ via the Koecher-Tits construction, with respect to the minimal representation constructed in [Reference Hanzer and SavinHS, Theorem 6.4].

10.3 Second globalization result

Assume now that k is an arbitrary number field and G is a simple split group over k. Let U be the unipotent radical of a Borel subgroup of G, and fix a Whittaker character $\psi = \prod _v \psi _v: U(\mathbb A)/U(k) \rightarrow \mathbb C^{\times }$ . In particular, we have a notion of local and automorphic Whittaker functional (relative to $\psi $ ) and the notion of (Whittaker) generic representations, both locally and globally.

Fix a place w of k and a $\psi _w$ -generic supercuspidal representation $\sigma $ of $G(k_w)$ . Let

$$\begin{align*}L^2_{\psi-\mathrm{gen}, \sigma} (G(k)\backslash G(\mathbb A)) \subset L_{cusp}^2(G(k)\backslash G(\mathbb A)) \end{align*}$$

be the Hilbert subspace of the automorphic discrete spectrum of G which is orthogonal to the span of all square-integrable automorphic representations $\Pi =\otimes _v \Pi _v$ such that either the automorphic $\psi $ -Whittaker functional vanishes on $\Pi $ or $\Pi _w\ncong \sigma $ . Then any $\Pi \subset L^2_{\psi -\mathrm {gen}, \sigma } (G(k)\backslash G(\mathbb A))$ is globally (Whittaker) generic and $\Pi _w \cong \sigma $ .

For any place v, let $L^2_{\psi _v}(U(k_v)\backslash G(k_v))$ be the space of square integrable Whittaker functions. Recall that all generic square integrable representations of $G(k_v)$ are direct summands of $L^2_{\psi _v}(U(k_v)\backslash G(k_v))$ [Reference Sakellaridis and VenkateshSV].

Proof. Again, using Poincaré series [Reference Sakellaridis and VenkateshSV, Theorem 16.3.2 and Remark 16.4.1], one proves that

$$\begin{align*}L^2_{\psi_{v_1}}(U(k_{v_1})\backslash G(k_{v_1}))\otimes \cdots \otimes L^2_{\psi_{v_n}}(U(k_{v_n})\backslash G(k_{v_n})) \end{align*}$$

is weakly contained in $L^2_{\psi -\mathrm {gen}, \sigma } (G(k)\backslash G(\mathbb A))$ . In particular, $\delta _1\otimes \cdots \otimes \delta _n$ is weakly contained there as well. The existence of the globalization $\Pi $ now follows from Proposition 11.6 and the hypothesis of isolation of the $\delta _i$ ’s. The extra control at the places different from w and $v_i$ is achieved by using appropriate test functions at these other places as the input for the Poincaré series, see [Reference Gan and IchinoGI, proof of Prop. Reference ArthurA.2].

In applications of Proposition 10.7, one would need to verify the hypothesis of isolation in the proposition. For classical groups and their associated similitude groups, the isolation for all generic discrete series representations is a consequence of estimates towards the Ramanujan conjecture for globally generic cuspidal representations; see [Reference Ichino, Lapid and MaoILM, Prop. A.5 and Lemma A.2]. The following verifies it for the group $G_2$ to the extent we shall need it in this paper.

Proof. We argue by way of contradiction. Assume that there exists a sequence $\pi _n$ of local components of automorphic representations in $L^2_{\mathrm {gen}, \sigma } (G(k)\backslash G(\mathbb A))$ such that $\delta $ is a limit point of the sequence in Fell topology. Since discrete series representations are isolated in the tempered dual, the $\pi _n$ ’s may be assumed to be nontempered for any n. Further, since any irreducible $\Pi \subset L^2_{\mathrm {gen}, \sigma } (G(k)\backslash G(\mathbb A))$ has a nonzero globally generic global theta lift $\Sigma \subset L^2_{\mathrm {gen}, \theta (\sigma )} (H(k)\backslash H(\mathbb A))$ (see [Reference Harris, Khare and ThorneHKT, appendix]), the local theta lifts $\theta (\pi _n)$ are generic and unitarizable. We shall show that $\theta (\pi _i)$ converges to $\theta (\delta )$ and this will give the desired contradiction, since as we remarked before the lemma, $\theta (\delta )$ is known to be isolated in $L^2_{\mathrm {gen}, \theta (\sigma )} (H(k)\backslash H(\mathbb A))$ .

By [Reference TadićTa1], the supercuspidal support of $\delta $ is the limit of the sequence of supercuspidal supports of $\pi _n$ . For generic representations, the theta correspondence is continuous for supercupidal supports. Thus the supercuspidal support of $\theta (\delta )$ is the limit of the sequence of supercuspidal supports of $\theta (\pi _n)$ . Now this should imply that $\theta (\delta )$ is a limit point of a subsequence of $\theta (\pi _n)$ , since any sequence of generic unitary representations should have a generic representation as a cluster point. For example, this will be true if genericity can be detected by a certain K-type. Since we could not find a reference for this result, we proceed as follows, using Proposition 11.2 shown below.

By Proposition 11.2, on passing to a subsequence of the given $\pi _n$ ’s, we may realize $\delta $ as a limit point of a sequence of non-tempered representations $\pi _n=\mathrm {Ind}_P^G(\tau _n)$ such that $\delta $ is a subquotient of $\mathrm {Ind}_P^G(\tau )$ , where $\tau $ is an irreducible generic representation which is a limit point of the sequence $\tau _n$ . Now the local theta correspondence for non-tempered representation is known by [Reference Gan and SavinGS23]. In particular, there exists a parabolic subgroup $Q=LU \subset H$ , depending on P, such that

$$\begin{align*}\theta(\pi_n) =\mathrm{Ind}_Q^H(\nu_n), \end{align*}$$

where $\nu _n$ is irreducible and explicitly constructed from $\tau _n$ .

Let $\nu $ be the irreducible representation of L explicitly constructed from $\tau $ in the same way as $\nu _n$ is constructed from $\tau _n$ . Then it follows (by the continuity of induction) that the irreducible subquotients of $\mathrm {Ind}_Q^H(\nu )$ are limit points of the sequence $\mathrm {Ind}_Q^H(\nu _n) = \theta (\pi _n)$ . Now the irreducible representation $\nu $ is generic (since $\tau $ is generic), so that $\mathrm {Ind}_Q^H(\nu )$ has a unique irreducible generic subquotient. Since the supercuspidal support of $\mathrm {Ind}_Q^H(\nu )$ is the same as the superccuspidal support of $\theta (\delta )$ , we see that $\theta (\delta )$ is the unique irreducible generic subquotient of $\mathrm {Ind}_Q^H(\nu )$ . In particular, $\theta (\delta )$ is a limit point of $\theta (\pi _n)$ . Thus we have proved that $\theta (\delta )$ is not isolated in $L^2_{\mathrm {gen}, \theta (\sigma )} (H(k)\backslash H(\mathbb A))$ , giving the desired contradiction.

11.1 Steinberg is isolated

Using an argument due to Miličić, we prove that the Steinberg representation of G is isolated, with respect of the Fell topology, in the unitary dual of G. (We refer the reader to [Reference Bekka, de la Harpe and ValetteBHV, Appendix F], for a nice summary of Fell topology, definitions and basic properties). Here we work with smooth representations.

We first recall some basic notions:

• • An irreducible unitarizable representation $\pi $ of G is a limit point of a sequence $\pi _n$ of irreducible unitarizable representations $\pi _n$ of G if for some (or equivalently for any) matrix coefficient $\langle \pi (g) v, v\rangle $ of $\pi $ , there is a sequence of matrix coefficients $\langle \pi _n(g) v_n, v_n\rangle $ of $\pi _n$ converging uniformly on all compact sets in G to $\langle \pi (g) v, v\rangle $ .

• • An irreducible unitarizable representation $\pi $ of G is a cluster point of the sequence $\pi _n$ if it is a limit point of a subsequence of $\pi _n$ .

Let $L[\pi _n]$ and $C[\pi _n]$ denote the sets of limit and cluster points of $\pi _n$ respectively, so that clearly,

$$\begin{align*}L[\pi_n] \subseteq C[\pi_n]. \end{align*}$$

However, since the unitary dual is not a Hausdorff space, a nonempty $L[\pi _n]$ does not have to be a singleton and the inclusion $L[\pi _n] \subseteq C[\pi _n]$ can be strict. However, as explained by Miličić, one can have the equality $L[\pi _n] = C[\pi _n]$ under some conditions.

More precisely, let $\chi _{\pi _n}$ be the character distribution of $\pi _n$ . We say that a sequence $\chi _{\pi _n}$ is weakly converging if, for every $f\in C_c(G)$ , the sequence of complex numbers $\chi _{\pi _n}(f)$ is convergent. We have the following result [Reference MiličićMi, Thm. 5]; see also [Reference TadićTa1].

Theorem 11.1. Let $\pi _n$ be a sequence of irreducible unitarizable representations of G. Assume that the corresponding sequence of characters is weakly converging. Then there exists a finite set S of unitary representations such that

$$\begin{align*}\lim_{n\rightarrow \infty} \chi_{\pi_n}(f) = \sum_{\sigma \in S} n_{\sigma} \chi_{\sigma}(f) \end{align*}$$

for all $f\in C_c(G)$ , where $n_{\sigma }>0$ are natural numbers. Moreover, one has

$$\begin{align*}S=L[\pi_n]=C[\pi_n]. \end{align*}$$

We shall need the following proposition which, for real groups, was established by Vogan [Reference VoganVo] in a greater generality.