Theta correspondence and Arthur packets

We continue to let $F$ be a number field. Let $(G,H)$ be a reductive dual pair over $F$ in stable range and $H$ is the larger group, that is, the split rank of $H$ is greater than the dimension of the standard representation of $G$ (we shall make it more precisely later). In some early works of Howe [How82] and J-S. Li [Li97], they defined the so-called “low rank representations” of classical groups, and showed that theses representations can be characterized by the theta lift from some smaller groups. These works suggest the possibility to study the automorphic spectrum of $G$ in terms of $H$, using the theta correspondence between $(G,H)$. This idea was first exploited by Gan-Ichino [GI18]. In their work, they put $G=Mp(2n)$, and $H=SO(2r+1)$, such that $r>2n$. By computing some unramified theta lifts and partial $L$-functions, they attached to each near equivalence class of $G$ an elliptic $A$-parameter (i.e. showed the existence of the weak lift to $GL(2n)$ through the standard representation of the dual group); they also observed that, for those automorphic representations $\pi$ of $G$ with tempered elliptic $A$-parameters, any automorphic realization of $\pi$ must be cuspidal, which implies that

where $m_{disc}$ means the multiplicity in the automorphic discrete spectrum, and $\theta^{abs}(\pi)$ means the abstract theta lift of $\pi$ to $H$ (i.e. restricted tensor product of the local theta lift at each local place). Combining this with some knowledge on the local theta lift, they proved the Arthur’s multiplicity formula for the tempered part of the automorphic discrete spectrum of $Mp(2n)$.

In our previous paper [CZ21], we put $G$ to be an even orthogonal or unitary group (not necessarily quasi-split), and $H$ be a symplectic or quasi-split unitary group according to $G$. We established the same results as in [GI18]. Besides, we also observed that the multiplicity preservation (i.e. equality (1.1)) also holds if the Witt index of $G$ is less than or equal to one. Hence we obtained a description for the full automorphic discrete spectra of those even orthogonal or unitary groups, by “pulling back” the Arthur’s multiplicity formula for $H$ to $G$ through the theta lift. For each local place of $F$, we defined the so-called “$\theta$-packets” of $G$ to be the theta lift of certain $A$-packets of $H$ (see Section 2.5), and “glue” them together to get some global packets; we showed that the automorphic discrete spectrum of $G$ can be decomposed according to these global packets (see Theorem 8.13). We should emphasize here that, essentially our results in this paper are independent of the results in [CZ21]; we use [CZ21] only as one of the motivations for the results shown in this paper.

Notice that at each place of $F$, the definition of the “$\theta$-packets” is purely local; also, although we assumed that the Witt index of $G$ is less or equal to one, the localization of $G$ at local places could be quite general. It makes sense to ask: are these “$\theta$-packets” dependent on the choice of $H$? What are these “$\theta$-packets”? It turns out that, at least at non-Archimedean places, $\theta$-packets are independent of the choice of $H$; moreover, at those places where our $G$ is quasi-split, we show that the $\theta$-packets are the same as the $A$-packets defined by Arthur and Mok. These are the main results in this paper. Indeed, these questions are already asked in the study of local theta correspondence, known as the “Adams conjecture”.

Now let $F$ be a local field of characteristic $0$. In his paper [Ada89] Section 4, Adams proposed the following conjecture, which describes the local theta lift in terms of $A$-parameters:

Our results in this paper can be more or less regarded as a refined version of Conjecture 1.1 (B) here: we not only prove the bijection between packets as (multi) sets, we also show the consistency of “labelings”, i.e. the characters of component groups attached to the representations inside the packets. However, in our results, we use the terminology “$\theta$-packets” rather than “$A$-packets”, due to two reasons:

• •

  for quasi-split groups, our results imply the two terminologies are the same, so there is no harm; but for non quasi-split groups, at present $A$-packets (under the framework of Arthur, both locally and gloablly, cf. [Art13] Chapter 9, or [KMSW14] Chapter 1.6 and 1.7) are not avaliable, so we use $\theta$-packets for substitutions;

• •

  as explained in the last subsection, a motivation of studying this question is to obtain some results for non quasi-split groups from quasi-split groups via theta lifts; motivated by this purpose and taking the results of [CZ21] into account, it is natural to use the $\theta$-packets.

We should also mention to readers that Mœglin has done many wonderful works on Conjeture 1.1 in [Mœg11c], based on her explicit construction of local $A$-packets for classical groups. Next we briefly recall her results.

For a classical group $G$ over some $p$-adic field and a local $A$-parameter $\psi$ of $G$, Mœglin has construct a packet $\Pi_{\psi}^{M}(G)$ consisting of irreducible unitary representations using the techniques of Jacquet modules. To be more precise, she defined a collection of unitary representations

where the parameter $(\underline{t},\underline{\eta})$ runs over some set $\Sigma_{\psi}(G)$ (which can be writen down explicitly according to $\psi$). These representations are either zero or irreducible, and the packet $\Pi_{\psi}^{M}(G)$ is simply the collection of all non-zero guys. When $G$ is a symplectic group or quasi-split orthogonal group, it was proved in [Xu17a] that $\Pi_{\psi}^{M}(G)=\Pi_{\psi}^{A}(G)$. To distinguish various notions, we shall call the packet $\Pi_{\psi}^{M}(G)$ “$M$-packet”.

Now, let $(G,H)$ be an orthogonal-symplectic dual pair such that the dimension of the standard representation of $\prescript{L}{}{G}$ is not greater than that of $\prescript{L}{}{H}$, and $\psi$ be a local $A$-parameter for the group $G$. Then, for each $(\underline{t},\underline{\eta})\in\Sigma_{\psi}(G)$, she constructed a pair $(\widetilde{\underline{t}},\widetilde{\underline{\eta}})\in\Sigma_{\theta(\psi)}(H)$, where $\theta(\psi)$ is the local $A$-parameter for $H$ predicted by Conjecture 1.1. Under some technical assumptions on the local $A$-parameter $\psi$, she asserted that

if the LHS and the RHS are both non-zero (see [Mœg11c] 5.2 Théorème). She also showed some non-vanishing criterion for the representation $\pi\left(\psi,\underline{t},\underline{\eta}\right)$ along the proof. Hence in some sense, she has proved Conjecture 1.1 (A) under her assumptions. Her method is purely local. To prove these results, she mastered the Kudla’s filtration and Jacquet modules very carefully. Moreover, she pointed out that Conjecture 1.1 (A) is generally not true by giving some counter-examples.

If we suppose further that $(G,H)$ is in the stable range, then Mœglin’s technical assumptions are automatically satisfied. From her results, one can easily check the following:

• •

  the assignment $(\underline{t},\underline{\eta})\mapsto(\widetilde{\underline{t}},\widetilde{\underline{\eta}})$ induces a bijection

  $$\theta:\bigsqcup\Sigma_{\psi}(G^{\prime})\longrightarrow\Sigma_{\theta(\psi)}(H),$$

  where $G^{\prime}$ runs over all pure inner forms of $G$;

• •

  the representation $\pi\left(\psi,\underline{t},\underline{\eta}\right)$ is non-zero if and only if $\pi\left(\theta(\psi),\widetilde{\underline{t}},\widetilde{\underline{\eta}}\right)$ is non-zero.

It follows that Conjecture 1.1 (B) holds (with “$A$-packets” replaced by “$M$-packets”).

However, we still want to look for an independent proof of Conjecture 1.1 (B), due to the following reasons:

• •

  As we have explained, one of the motivations of studying this question is to obtain some results for non quasi-split groups from quasi-split groups via theta lifts. Hence we want to look for an approach which is free of using results from non quasi-split groups.

• •

  Except for for the bijectivity, for our purpose, we also need to show the consistency of the “labelings”. To pass from Mœglin’s parametrization to that of Arthur, one still needs to do some computions following [Xu17a].

In the next subsection, we briefly describe the idea of our approach.

Our idea of the proof is very simple: we use global methods as much as possible. For quasi-split classical groups, the Arthur’s multiplicity formula implies that any localization of an irreducible unitary representation occuring in the automorphic discrete spectrum lies in a local $A$-packet. One can image that, if the Arthur’s multiplicity formula has been established for all classical groups, then Conjecture 1.1 (B) should simply follows from the combination of some easy computations at unramified places and the Arthur’s multiplicity formula. From this point of view, many of our lemmas/ propositions in this paper indeed reduce to appropriately globalize a (local) representation. However, since the Arthur’s multiplicity formula for general non quasi-split groups has not been established yet, we still need to appeal to Mœglin’s explicit constructions of $p$-adic local $A$-packets to deal with some cases. But we should emphasize that our approach only rely on her results for quasi-split groups. As for the “labelings”, we shall use the intertwining relation to interpret the “labelings” of a local $A$-packet as some representation-theoretical quantities. Then we can compute the “labelings” using the same techniques as in [GI16].

For Archimedean places, we also expect that Conjecture 1.1 (B) holds. Indeed, combining results in [Cos09] and [MR19], one can conclude that Conjecture 1.1 (B) holds for unitary dual-pairs when the $A$-parameter $\psi$ is Adams-Johnson. We will not consider Archimedean places in this paper.

Now we give a summary of the layout of this paper. We formulate the main theorems in Section 2, taking the chance to recall some preliminaries. After doing some preparation work in Section 3 and recalling some results from other papers that we will use in Section 4, we prove our first result (independence of $\theta$-packets as sets on the choice of some data used in their construction) in Section 5, and we also prove some complementary results in Section 6. Then in Section 7 we recall the local intertwining relation by Arthur, and state an alternative version of it. Finally in Section 8, we prove the local intertwining relation for non quasi-split groups using some techniques developed by Gan-Ichino, and finish the proof of our main results; after that, we briefly summarize some expected and known properties of $\theta$-packets.

Assume $F$ is local for a moment. With respect to the non-trivial additive character $\psi_{F}$ of $F$ and the auxliary data $(\chi_{V},\chi_{W})$, one can define the Weil representation $\omega$ of $G\times H$. For any irreducible representation $\pi$ of $G$, the maximal $\pi$-isotypic quotient of $\omega$ is of the form

for some smooth representation $\Theta(\pi)$ of $H$ of finite length. Then by the Howe duality [How89], [Wal90], [GT16a], [GT16b], the maximal semi-simple quotient $\theta(\pi)$ of $\Theta(\pi)$ is either zero or irreducible. Similarly, for any irreducible representation $\sigma$ of $H$, we can define $\Theta(\sigma)$ and $\theta(\sigma)$.

Suppose next that $F$ is a number field. Fix a non-trivial additive character $\psi_{F}$ of $F\backslash\mathbb{A}$, and also characters $(\chi_{V},\chi_{W})$. Let $\pi=\otimes_{v}\pi_{v}$ be an abstract irreducible representation of $G(\mathbb{A})$ (i.e. a collection of local irreducible representations $\pi_{v}$ of $G(F_{v})$ for all places $v$ of $F$, such that $\pi_{v}$ is unramified for almost all $v$). At each place $v$ of $F$, we can form the local theta lift $\theta(\pi_{v})$ with respect to $(\psi_{F,v},\chi_{V,v},\chi_{W,v})$. Assume that they are all non-vanishing. Then $\theta(\pi_{v})$ is irreducible for all $v$ and is unramified for almost all $v$. Hence we may define an abstract irreducible representation

of $H(\mathbb{A})$. We call $\theta^{abs}(\pi)$ the abstract theta lift of $\pi$ to $H(\mathbb{A})$.

The notion of rank for unitary representations was first introduced by Howe [How82] in the case of symplectic groups and was extended to the case of classical groups by J-S. Li [Li89a]. Following [Li89a], we say that an irreducible unitary representation of $H=H\left(W_{(r)}\right)$ is of low rank if its rank is less than $r$. Such representations are obtained by theta lifts as follows.

Let $F$ be a local field. Assume $\dim V<r$. In particular, the reductive dual pair $(G,H)$ is in the stable range. Then for any irreducible representation $\pi$ of $G$, its theta lift $\theta(\pi)$ to $H$ is non-vanishing. Moreover, if $\pi$ is unitary, then by [Li89b], so is $\theta(\pi)$. In [Li89a], J-S. Li showed that:

This result has a global analog. Let $F$ be a number field and $\sigma=\otimes_{v}\sigma_{v}$ an irreducible unitary representation of $H(\mathbb{A})$ which occurs as a subrepresentation of $\mathcal{A}(H)$, where $\mathcal{A}(H)$ is the space of automorphic forms of $H$. Then, by [Li89b] and [Li97], we have:

Finally, we recall another result of J-S. Li, which allows us to lift square-integrable automorphic representations of $G(\mathbb{A})$ to $H(\mathbb{A})$. For any irreducible representation $\pi$ of $G(\mathbb{A})$, we define its multiplicities $m(\pi)$ and $m_{disc}(\pi)$ by

where $\mathcal{A}^{2}(G)=\mathcal{A}(G)\cap L^{2}_{disc}(G)$. Obviously, $m_{disc}(\pi)\leq m(\pi)$. Likewise, if $\sigma$ is an irreducible representation of $H(\mathbb{A})$, we have its multiplicities $m(\sigma)$ and $m_{disc}(\sigma)$. By [Li97], we have

We briefly recall some terminologies and results from [Art13] (also [AG17]), [Mok15], and some other papers.

First let $F$ be a local field of characteristic $0$. A local $A$-parameter for the group $G$ is a homomorphism

where $L_{F}$ is the Weil-Deligne group of $F$. If there is no further explanations, we will assume that the image of the Weil group under a local $A$-parameter is bounded by default. By composing this homomorphism with the standard representation of $\prescript{L}{}{G}$, we can regard a local $A$-parameter as a (conjugate) self-dual representation of $L_{E}\times SL_{2}(\mathbb{C})$ with certain parity. We denote by $\Psi(G)$ the set of local $A$-parameters for $G$. Following Arthur, we define

where $z_{\psi}$ is the image of $-I\in\widehat{G}$ in $\mathcal{S}_{\psi}$. We shall call $\mathcal{S}_{\psi}$ or $\overline{\mathcal{S}_{\psi}}$ the component group associated to the local $A$-parameter $\psi$. If we write $\psi$ as

where each $\psi_{i}=\phi_{i}\boxtimes S_{d_{i}}$ is an irreducible representation of $L_{E}\times SL_{2}(\mathbb{C})$, and $I_{\psi}$ is the index set of this summation, then as explicated in [GGP12] Section 8, $\mathcal{S}_{\psi}$ has an explicit description in the form

where on the RHS, the product runs over the subset $I^{\prime}_{\psi}$ of $I_{\psi}$ containing all $i\in I_{\psi}$ such that $\psi_{i}$ is of the same parity as $\psi$; each element in the canonical basis $\{a_{i}\}$ of $\mathcal{S}_{\psi}$ corresponds to such a $\psi_{i}$. Under this identification, we have

and $\overline{\mathcal{S}_{\psi}}=\mathcal{S}_{\psi}/\langle z_{\psi}\rangle$; where again, the summation on the RHS runs over all $i$ such that $\psi_{i}$ is of the same parity as $\psi$.

When $V=V^{+}$, i.e. $G=G^{*}$ is quasi-split, thanks to Arthur and Mok, we can talk about the local $A$-packet $\Pi_{\psi}^{A}(G^{*})$ associated to the $A$-parameter $\psi$: this is a finite (multi) set of irreducible unitary representations of $G^{*}$, together with a map to the Pontryagin dual of the component group

This map depends on the choice of a Whittaker datum $\mathscr{W}$ of $G^{*}$. The local $A$-packet $\Pi_{\psi}^{A}(G^{*})$ can be also regarded as a representation of $\overline{\mathcal{S}_{\psi}}\times G^{*}$ by setting

where the summation on the RHS runs over all irreducible unitary representations of $G^{*}$ in $\Pi_{\psi}^{A}(G^{*})$. Sometimes we shall adopt this point of view without any further explanation.

Similarly, one can define the local $A$-parameter for the group $H$, and for a local $A$-parameter $\psi_{H}$ of $H$, one can define and describe the component group $\overline{\mathcal{S}_{\psi_{H}}}$ in the same manner. Again, according to Arthur and Mok’s works, we can talk about the local $A$-packet $\Pi_{\psi_{H}}^{A}(H)$ associated to the $A$-parameter $\psi_{H}$, which can be regarded as a representation of $\overline{\mathcal{S}_{\psi_{H}}}\times H$ by setting

where $\mathscr{W}^{\prime}$ is a Whittaker datum of $H$, and the summation on the RHS runs over all irreducible unitary representations of $H$ in $\Pi_{\psi_{H}}^{A}(H)$.

Now we turn to the global classifications. Let $F$ be a number field. Two irreducible representations $\pi=\otimes_{v}\pi_{v}$ and $\pi^{\prime}=\otimes_{v}\pi^{\prime}_{v}$ of $G(\mathbb{A})$ are said to be nearly equivalent if $\pi_{v}$ and $\pi^{\prime}_{v}$ are equivalent for almost all places $v$ of $F$. The decomposition of automorphic discrete spectrum of $G$ into near equivalence classes will be expressed in terms of elliptic $A$-parameters. Recall that an elliptic $A$-parameter for $G$ is nothing but a formal finite sum

where

• •

  $\rho_{i}$ is an irreducible (conjugate) self-dual cuspidal automorphic representation of $GL_{n_{i}}(\mathbb{A}_{E})$;

• •

  $S_{d_{i}}$ is the $d_{i}$-dimensional irreducible representation of $SL_{2}(\mathbb{C})$;

• •

  $\sum_{i}n_{i}d_{i}=\dim V$;

• •

  If $d_{i}$ is odd, then $\rho_{i}$ is

  $$\begin{cases}\textit{orthogonal}\quad&\textit{Case $O$};\\ \textit{conjugate symplectic }\quad&\textit{Case $U_{0}$};\\ \textit{conjugate orthogonal }\quad&\textit{Case $U_{1}$}.\end{cases}$$

• •

  If $d_{i}$ is even, then $\rho_{i}$ is

  $$\begin{cases}\textit{symplectic}\quad&\textit{Case $O$};\\ \textit{conjugate orthogonal }\quad&\textit{Case $U_{0}$};\\ \textit{conjugate symplectic }\quad&\textit{Case $U_{1}$}.\end{cases}$$

• •

  If $(\rho_{i},d_{i})=(\rho_{j},d_{j})$, then $i=j$;

• •

  In Case $O$, if we denote the central character of $\rho_{i}$ by $\omega_{i}$, then

  $$\prod_{i}\omega_{i}^{d_{i}}=\chi_{V}.$$

If further $d_{i}=1$ for all $i$, then we say that $\psi$ is generic. We denote the set of all elliptic $A$-parameters by $\Psi_{ell}(G)$. For each place $v$ of $F$, let

be the localization of $\psi$ at $v$. Here

and we use the local Langlands correspondence for the general linear groups. We associate to it an $L$-parameter $\phi_{\psi_{v}}:L_{F_{v}}\rightarrow\prescript{L}{}{G_{v}}$ by

We have

When $G=G^{*}$ is quasi-split, a further decomposition of each near equivalence class is avaliable, known as the Arthur’s multiplicity formula for $G^{*}$. Fix a global Whittaker datum $\mathscr{W}$ of $G^{*}$. Given an elliptic $A$-parameter $\psi$, we define the global packet $\Pi_{\psi}^{A}(G^{*})$ associated to $\psi$ as the restricted tensor product of the local $A$-packets

We then have a map

where $x\in\mathcal{S}_{\psi}$ and $x_{v}$ is the localization of $x$ at $v$. We can also define the so-called canonical sign character $\epsilon_{\psi}\in\widehat{\mathcal{S}_{\psi}}$ following [Art13] page 47, or [Mok15] page 29. We put

Then the main global Theorems in [Art13] and [Mok15] assert that

Similarly, one can define the global $A$-parameters for the group $H$, and for a global elliptic $A$-parameter $\psi_{H}$ of $H$, one can define the associated global $A$-packets by gluing local $A$-packets in the same manner. Again, according to Arthur and Mok’s works, Theorem 2.5 and Theorem 2.6 also holds for $H$.

Since the local or global classification of both $G^{*}$ and $H$ depend on the choices of Whittaker data on $G^{*}$ and $H$, and the theta lift also depends on the choice of an additive character, we need to choose these data in a compatible way. We now briefly describe the way we choose these data.

Let $F$ be a local or global field. Firstly we fix a non-trivial additive character $\psi_{F}$ of $F$ (or $F\backslash\mathbb{A}$ if $F$ is global). Next we fix an Whittaker datum $\mathscr{W}$ of $G^{*}$ as follows.

In Case $O$, $G^{*}=G(V^{+})$ is an even orthogonal groups. As explicated at the beginning of this section, we fix an isometry

for some $d,c\in F$, where $\mathcal{H}$ is the (orthogonal) hyperbolic plane. The images of $1,X\in F[X]$ in $V_{(d,c)}$ are denoted by $e,e^{\prime}$, respectively. For $1\leq k\leq n-1$, we write the $k$-th hyperbolic plane $\mathcal{H}=Fv_{k}+Fv_{k}^{*}$ with

and we set

We denote by $B=TU$ the $F$-rational Borel subgroup of $G^{*}$ stabilizing the complete flag

where $T$ is the $F$-rational torus stabilizing the lines $Fv_{i}$ for $1\leq k\leq n-1$. We define a generic character $\mu_{c}$ of $U$ by

Let $\mathscr{W}=(U,\mu_{c})$. Note that in fact $\mathscr{W}$ does not depend on the choice of the additive character $\psi_{F}$, but only depends on the constant $c$ we have picked up.

In Case $U_{0}$, $G^{*}=G(V^{+})$ is an even unitary groups. Recall that we have fixed a trace zero element $\delta\in E^{\times}$. Let $V^{+}$ be the $2n$-dimensional Hermitian space over $E$ such that $G^{*}=U(V^{+})$. Since $G^{*}$ quasi-split, the Witt index of $V^{+}$ is $n$. We choose a basis $\{v_{i},v_{i}^{*}~{}|~{}i=1,\cdots,n\}$ of $V^{+}$ such that

for $1\leq i,j\leq n$. We set

for $1\leq i,j\leq n$. We denote by $B=TU$ the $F$-rational Borel subgroup of $G^{*}$ stabilizing the complete flag

where $T$ is the $F$-rational torus stabilizing the lines $Ev_{i}$ for $1\leq k\leq n$. We define a generic character $\mu$ of $U$ by

Let $\mathscr{W}=(U,\mu)$.

In Case $U_{1}$, there is an unique Whittaker datum $\mathscr{W}$ of $G^{*}$.

Finally we fix a Whittaker datum $\mathscr{W}^{\prime}$ of $H$ as follows.

In Case $O$, $W$ is the $2r$-dimensional symplectic space. We choose a basis $\{w_{i},w_{i}^{*}~{}|~{}i=1,\cdots,r\}$ of $W$ such that

for $1\leq i,j\leq r$. We set

for $1\leq i,j\leq r$. We denote by $B^{\prime}=T^{\prime}U^{\prime}$ the $F$-rational Borel subgroup of $H$ stabilizing the complete flag

where $T^{\prime}$ is the $F$-rational torus stabilizing the lines $Fw_{i}$ for $1\leq k\leq r$. We define a generic character $\mu^{\prime}_{c}$ of $U^{\prime}$ by

where the constant $c$ is the one appearing in the isometry $V^{+}\simeq V_{(d,c)}+\mathcal{H}^{n-1}$ we have fixed. Let $\mathscr{W}^{\prime}=(U^{\prime},\mu^{\prime}_{c})$. In this case we also define another generic character $\mu^{\prime}_{1}$ of $U^{\prime}$ by