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FOURIER-JACOBI MODELS FOR REAL
SYMPLECTIC-METAPLECTIC GROUPS: THE
BASIC CASE

Cheng Chen, Rui Chen, Jialiang Zou

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Cheng Chen, Rui Chen, Jialiang Zou. FOURIER-JACOBI MODELS FOR REAL SYMPLECTIC-METAPLECTIC GROUPS: THE BASIC CASE. 2026. hal-05499207

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Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
FOURIER–JACOBI MODELS FOR REAL SYMPLECTIC-METAPLECTIC GROUPS: THE BASIC CASE

CHENG CHEN, RUI CHEN, AND JIALIANG ZOU

ABSTRACT. In this paper, we generalize the method in [GI16] [Ato18] to the field of real numbers and prove the basic tempered case of the local Gan-Gross-Prasad conjecture for Fourier-Jacobi models of symplectic-metaplectic groups, based on the tempered case of the conjecture for Bessel models proved in [CL22].

CONTENTS

<table><tr><td>1. Introduction</td><td>1</td></tr><tr><td>Notation</td><td>4</td></tr><tr><td>Acknowledgement</td><td>5</td></tr><tr><td>2. Local Langlands correspondence</td><td>5</td></tr><tr><td>2.1. Weil groups and L-parameters</td><td>5</td></tr><tr><td>2.2. Local Langlands correspondence for classical groups</td><td>6</td></tr><tr><td>2.3. Local Langlands correspondence for metaplectic groups</td><td>9</td></tr><tr><td>3. Local Gan-Gross-Prasad Conjecture</td><td>9</td></tr><tr><td>3.1. Symplectic-metaplectic case</td><td>9</td></tr><tr><td>3.2. Special orthogonal case</td><td>10</td></tr><tr><td>4. Preliminaries and Toolkits</td><td>11</td></tr><tr><td>4.1. Schwartz induction and Borel's lemma</td><td>11</td></tr><tr><td>4.2. Representation theory of Jacobi groups</td><td>12</td></tr><tr><td>4.3. MVW-involution</td><td>17</td></tr><tr><td>4.4. Local theta lifts</td><td>17</td></tr><tr><td>4.5. A vanishing result for Hom spaces</td><td>20</td></tr><tr><td>5. Uniqueness and epsilon dichotomy of the Fourier-Jacobi model</td><td>20</td></tr><tr><td>5.1. Desideratum I: uniqueness and epsilon dichotomy</td><td>20</td></tr><tr><td>5.2. Stable range seesaw I: some non-tempered Bessel models</td><td>22</td></tr><tr><td>5.3. Peeling off the non-tempered part I: Bessel model</td><td>22</td></tr><tr><td>6. Existence of the Fourier-Jacobi model</td><td>25</td></tr><tr><td>6.1. Desideratum II: Existence</td><td>25</td></tr><tr><td>6.2. Stable range seesaw II: some non-tempered Fourier-Jacobi models</td><td>26</td></tr><tr><td>6.3. Peeling off the non-tempered part II: Fourier-Jacobi model</td><td>26</td></tr><tr><td>References</td><td>30</td></tr></table>

1. INTRODUCTION

The aim of this paper is to prove the basic tempered case of the local Gan-Gross-Prasad conjecture [GGP12] for symplectic-metaplectic Fourier-Jacobi models over the field $\mathbb{R}$.

<footnote n="2020">2020 Mathematics Subject Classification. Primary 22E50 22E45; Secondary 20G20.</footnote>

<footnote>Key words and phrases. Gan-Gross-Prasad conjecture, Schwartz homology.</footnote>
Let $W$ be a symplectic vector space over $\mathbb{R}$. Let $\widetilde{\text{Sp}}(W)$ be the associated symplectic group and $\widetilde{\text{Sp}}(W)$ the metaplectic group, i.e., the unique non-split double cover of $\text{Sp}(W)$. For a non-trivial additive character $\psi$ of $F$, let $\omega_\psi$ denote the Weil representation of $\widetilde{\text{Sp}}(W)$ associated with $\psi$ (see Section 4.2). We set

$$G = \text{Sp}(W) \times \widetilde{\text{Sp}}(W) \quad \text{and} \quad H = \Delta \widetilde{\text{Sp}}(W),$$

where $\Delta \widetilde{\text{Sp}}(W)$ denotes the diagonal embedding of $\widetilde{\text{Sp}}(W)$ into $G$. We denote by $\text{Irr}(\widetilde{\text{Sp}}(W))$ the set of irreducible admissible Casselman-Wallach representations of $\widetilde{\text{Sp}}(W)$ and $\text{Irr}_{\text{gen}}(\widetilde{\text{Sp}}(W))$ the set of genuine irreducible admissible Casselman-Wallach representations of $\widetilde{\text{Sp}}(W)$. For $\pi \in \text{Irr}(\widetilde{\text{Sp}}(W))$ and $\tilde{\pi} \in \text{Irr}_{\text{gen}}(\widetilde{\text{Sp}}(W))$, we define the multiplicity

$$(1.0.1) \qquad m(\pi \boxtimes \tilde{\pi}) = \dim \text{Hom}_H(\pi \boxtimes \tilde{\pi}, \omega_\psi).$$

The well-known multiplicity-one theorem asserts that

$$m(\pi \boxtimes \tilde{\pi}) \le 1.$$

This theorem was proved in [SZ12, LS13]. The local Gan-Gross-Prasad conjecture provides a refinement of this multiplicity-one theorem, by precisely describing the behavior of $m(\pi \boxtimes \tilde{\pi})$ for $\pi, \tilde{\pi}$ belonging to generic L-packets.

To state the precise conjecture, we first recall the local Langlands correspondence (“LLC” for short). We denote by $W_\mathbb{R}$ the Weil group of $\mathbb{R}$. For a connected reductive group $G$ over $\mathbb{R}$ and a local L-parameter

$$\phi : W_\mathbb{R} \to \mathbb{L}G,$$

there is a finite subset $\Pi_\phi(G)$ of $\text{Irr}(G(\mathbb{R}))$, called the L-packet of $G$ associated to $\phi$. Following the work of Vogan [Vog93], the Vogan L-packet attached to $\phi$ is defined as

$$\Pi_{\phi} = \bigsqcup_{G'} \Pi_{\phi}(G'),$$

where $G'$ runs over all pure inner forms of $G$. The advantage of working with $\Pi_\phi$ is that, after fixing a Whittaker datum $w$ for a quasi-split pure inner form $G^*$ of $G$, there exists a bijection

$$\mathcal{J}_w : \Pi_\phi \leftrightarrow \widetilde{S}_\phi, \quad \pi \leftrightarrow \eta_\pi,$$

where $\widetilde{S}_\phi$ denotes the set of characters of the component group

$$S_\phi := \pi_0(\text{Cent}_{\hat{G}}(\text{Im}(\phi))).$$

In [AB98], Adams and Barbasch also extended the LLC to the metaplectic group $\widetilde{\text{Sp}}(W)$, by using the local Shimura-Waldspurger correspondence. We briefly recall their construction. Assume that $\dim W = 2n$, and let $V^+ = \text{SO}(V^+)$. Given an L-parameter $\phi$ of $\widetilde{\text{Sp}}(W)$, one can also regard $\phi$ as an L-parameter of $\text{SO}(V^+)$. We then define the (Vogan) L-packet to be

$$\theta_{V,W,\psi} : \text{Irr}_{\text{gen}}(\widetilde{\text{Sp}}(W)) \leftrightarrow \prod_{\substack{\dim V = 2n+1 \\ \text{disc}(V) = 1}} \text{Irr}(\text{SO}(V)).$$

Note that these quadratic spaces $V$ with $\dim V = 2n+1$ and $\text{disc}(V) = 1$ exactly parametrize all pure inner forms of $\text{SO}(V^+)$. Therefore, one can define the L-group of $\widetilde{\text{Sp}}(W)$ to be the same as that of $\text{SO}(V^+)$. Given an L-parameter $\phi$ of $\widetilde{\text{Sp}}(W)$, one can also regard $\phi$ as an L-parameter of $\text{SO}(V^+)$. We then define the (Vogan) L-packet to be

$$\Pi_{\phi}(\widetilde{\text{Sp}}(W)) = \{\pi \in \text{Irr}_{\text{gen}}(\widetilde{\text{Sp}}(W)) \mid \theta_{V,W,\psi}(\pi) \in \Pi_{\phi}\}.$$

For $\pi \in \Pi_\phi(\widetilde{\text{Sp}}(W))$, we set

$$\eta_{\pi} := \mathcal{J}_{w}(\theta_{V,W,\psi}(\pi)) \in \widetilde{S}_{\phi}.$$
This provides a natural bijection

$$ \pi \in \Pi_{\phi}(\widetilde{\mathrm{Sp}}(W)) \quad \longleftrightarrow \quad \eta_{\pi} \in \widehat{S}_{\phi}. $$

The main result of this paper is the so-called “basic tempered case” of the local Gan–Gross–Prasad conjecture [GGP12, Conjecture 17.1, 17.3] for symplectic–metaplectic Fourier–Jacobi models.

**Theorem 1.0.1.** The following statements hold:

(1) **Multiplicity-one:** For every tempered L-parameter $\phi$ of $\mathrm{Sp}(W)$ and every tempered L-parameter $\tilde{\phi}$ of $\widetilde{\mathrm{Sp}}(W)$, we have

$$ \sum_{\pi \in \Pi_{\phi}} \sum_{\tilde{\pi} \in \Pi_{\tilde{\phi}}} m(\pi \boxtimes \tilde{\pi}) = 1. $$

(2) **Epsilon-dichotomy:** The unique pair $(\pi, \tilde{\pi}) \in \Pi_{\phi} \times \Pi_{\tilde{\phi}}$ such that

$$ m(\pi \boxtimes \tilde{\pi}) = 1 $$

corresponds to the distinguished characters of component groups

$$ \eta_{\pi} \times \eta_{\tilde{\pi}} = \eta_{\tilde{\phi}} \times \eta_{\phi}, $$

where $\eta_{\tilde{\phi}} \times \eta_{\phi}$ is defined in terms of local root numbers as in (3.1.1).

We should remark that the local Gan–Gross–Prasad conjecture is originally formulated in a much more general context. There are several different settings, which can be broadly divided into *Bessel models* and *Fourier-Jacobi models*.

* The **Bessel models** concern special orthogonal groups and unitary groups. For special orthogonal groups, these are often referred to as the *local Gross–Prasad conjecture*, originally introduced in [GP92, GP94].

* The **Fourier–Jacobi models** concern symplectic–metaplectic groups and unitary groups. In the symplectic–metaplectic case, one can consider groups of the form $\mathrm{Sp}(W) \times \widetilde{\mathrm{Sp}}(W')$, where $W$ and $W'$ are not necessarily of the same dimension; also, one can consider generic L-parameters instead of tempered L-parameters. The case we are considering in this paper, namely, when $W = W'$ and L-parameters are tempered, is called the basic tempered case (of Fourier-Jacobi models).

The general form of this conjecture can be reduced to the basic tempered cases: when the base field is non-Archimedean, this was done in [GGP12, Theorem 15.1, 16.1]; when the base field is Archimedean, this was done in the work of the first author [Che23].

Over the non-Archimedean local field, the local Gan-Gross-Prasad conjecture is known in all cases, by the works of Waldspurger [Wal10, Wal12a, Wal12b, Wal12c, Wal12d], Mœglin–Waldspurger [MW12], Beuzart–Plessis [BP14, BP16], Gan–Ichino [GI16] and Atobe [Ato18].

Over the complex number field $\mathbb{C}$, since the (Vogan) L-packet contains only one element, the multiplicity formula in [Che23, Theorem A] implies the local Gan-Gross-Prasad conjecture.

Over the real number field $\mathbb{R}$, the current status of the conjecture is as follows:

* For Bessel models of unitary groups, Beuzart–Plessis proved the multiplicity one part of the conjecture for tempered L-parameters in [BP14, BP16]. H. He proved the full conjecture for discrete L-parameters in [He17] and Xue proved the full conjecture for tempered L-parameters in [Xue23]. Both [He17] and [Xue23] use theta correspondence as the main tool. These results are extended to generic L-parameters by Xue in [Xue20] using the Schwartz homology.
* The local Gross–Prasad conjecture for tempered L-parameters was proved in [Luo20, CL22], following Waldspurger’s approach. These results are extended to generic L-parameters by the first author in [Che23].

* For Fourier–Jacobi models of unitary groups, Xue proved the conjecture in [Xue24]. He first used (almost) equal rank theta correspondence to reduce the basic tempered case to the Bessel models for unitary group, and then extended to generic L-parameters using a similar strategy as in [Xue20].

The only remaining case is the Fourier–Jacobi model for symplectic–metaplectic groups. In this paper, we establish the conjecture for the basic tempered case. Together with the reduction results of [Che23], this settles the last outstanding case of the local Gan–Gross–Prasad conjecture, thereby bringing this long program to completion.

Next, we outline our strategy for proving Theorem 1.0.1. The method of [GI16, Ato18] (over non-archimedean local fields) allows one to reduce the Fourier–Jacobi models to the Bessel models via the (almost) equal rank theta correspondence. Two key ingredients are needed in this approach:

* the irreducibility of the “big theta lifts” of tempered representations; and

* the description of theta lifts in terms of L-parameters (the so-called Prasad’s conjecture).

Over $\mathbb{R}$, however, the irreducibility of the “big theta lifts” in the (almost) equal rank case is not known in general. To circumvent this difficulty, we instead employ the stable range theta lift, for which the irreducibility of the “big theta lifts” has been established in [LM15]. Using the stable range theta lift, we are led naturally to certain non-tempered Gan–Gross–Prasad problems (see [GGP20]), both in the Bessel and Fourier–Jacobi cases. We then relate these non-tempered cases to the tempered ones by the mean of non-tempered reductions, which generalizes the arguments of [MW12, §1.4] from the non-Archimedean setting to the real number field $\mathbb{R}$. We hope that our approach here can shed some light for non-tempered Gan–Gross–Prasad problems. We should mention that Zhe Li and Shanwen Wang are also considering the basic tempered case of Fourier–Jacobi models, but with a different approach.

This paper is organized as follows. In Section 2, we review the local Langlands correspondence for the groups that appear in this work. In Section 3 we recall the statements of the local Gross–Prasad conjecture and the local Gan–Gross–Prasad conjecture for the symplectic–metaplectic Fourier–Jacobi models. In Section 4 we introduce some basic notions of Schwartz analysis and the properties of the theta correspondence that we will use. In Section 5, we prove the uniqueness part of the conjecture, namely: if there exists a pair of representations $(\pi, \tilde{\pi})$ in a Vogan L-packet that admit a nonzero Fourier–Jacobi model, then they must be the pair of representations determined by the distinguished characters defined in (3.1.1). Finally, in Section 6 we establish the existence of the Fourier–Jacobi model, i.e., there does exist a pair $(\pi, \tilde{\pi})$ in each Vogan L-packet with a nonzero Fourier–Jacobi model. Combining the results of Sections 5 and 6, we complete the proof of Theorem 1.0.1.

**Notation.** We end this introduction by setting some notations and conventions. Throughout the paper, we fix a non-trivial additive character $\psi$ of $\mathbb{R}$. For any $a \in \mathbb{R}^\times$, we define $\psi_a(x) = \psi(ax)$ for $x \in \mathbb{R}$. For a Nash group $G$, its closed Nash subgroup $H$, and a representation $\sigma$ of $H$, we denote by $\mathrm{Ind}_H^G(\sigma)$ and $\mathrm{ind}_H^G(\sigma)$ the usual normalized smooth induction and the normalized Schwartz induction from $H$ to $G$, respectively. In our proof of the main theorem, we will also make use of unnormalized inductions, which we shall denote by $\mathfrak{Ind}$ and $\mathfrak{ind}$ respectively. Since we will work with classical groups, we shall employ the standard notations

$$ \pi_1 \times \pi_2 \quad \text{and} \quad \tau \times \pi $$
to denote normalized parabolic inductions of general linear groups and classical groups. All tensor products in this paper should be understood as completed tensor products.

**Acknowledgement.** The first author is supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 101034255. The third author is supported by the Simons Foundation. The authors thank Zhibin Geng, Zhe Li, Shanwen Wang, Hang Xue and Hao Ying for helpful discussions.

## 2. LOCAL LANGLANDS CORRESPONDENCE

In this section, we recall the notion of Vogan L-packets for special orthogonal groups, symplectic groups, and metaplectic groups over $\mathbb{R}$ following [GGP12] and [Vog93].

### 2.1. Weil groups and L-parameters.
Recall that the Weil group $W_{\mathbb{R}}$ of $\mathbb{R}$ is defined to be

$$W_{\mathbb{R}} := \mathbb{C}^{\times} \sqcup \mathbb{C}^{\times} j,$$

with $j^2 = -1$ and $j \cdot z \cdot j^{-1} = \bar{z}$ for $z \in \mathbb{C}^{\times}$. There is an isomorphism $W_{\mathbb{R}}^{ab} \cong \mathbb{R}^{\times}$, where the superscript “ab” means the abelianization. Denote by $1$ and sgn the trivial and sign characters of $\mathbb{R}^{\times}$ and view them as characters of $W_{\mathbb{R}}$ via the isomorphism above. For $x \in \frac{1}{2}\mathbb{Z}$, we define a two-dimensional induced representation

$$D_x = \mathrm{Ind}_{\mathbb{C}^{\times}}^{W_{\mathbb{R}}} \left( \frac{z^{2x}}{|z|^x} \right).$$

Then $D_x \cong D_{-x}$ and they are irreducible unless $x = 0$, in which case $D_0 \cong 1 \oplus \mathrm{sgn}$. Moreover, $D_x$ is orthogonal if and only if $x \in \mathbb{Z}$, and $D_x$ is symplectic if and only if $x \in \frac{1}{2} + \mathbb{Z}$. Any irreducible self-dual representation of $W_{\mathbb{R}}$ is either

* 1-dimensional, which factors through $W_{\mathbb{R}}^{ab} \cong \mathbb{R}^*$ and is therefore equal to the trivial character $1$ or the sign character sgn; or

* 2-dimensional, which is isomorphic to $D_x$ for some $x \in \frac{1}{2}\mathbb{Z} \setminus \{0\}$.

For a finite-dimensional representation $M$ of $W_{\mathbb{R}}$, we set $\epsilon(M, \psi)$ to be the local root number defined in [Jac09, §16]. These local root numbers are determined by the following rules:

(1) For two representations $M_1$ and $M_2$ of $W_{\mathbb{R}}$, one has

$$(2.1.1) \qquad \epsilon(M_1 \oplus M_2, \psi) = \epsilon(M_1, \psi) \cdot \epsilon(M_2, \psi);$$

(2) Let $\psi_a(x) = \psi(ax)$, then we have

$$(2.1.2) \qquad \epsilon(M, \psi_a) = \det M(a) \cdot \epsilon(M, \psi);$$

(3) Set $\psi_0(x) = e^{2\pi i x}$, then the local root numbers associated with irreducible representations of $W_{\mathbb{R}}$ are given by

$$(2.1.3) \qquad \epsilon(M, \psi_0) = \begin{cases} (-i)^{\epsilon} & \text{if } M = \mathrm{sgn}^{\epsilon} | \cdot |^s, \\ (-i)^{2|x|+1} & \text{if } M = D_x | \cdot |^s, \end{cases}$$

where $\epsilon \in \{0, 1\}$, $x \in \frac{1}{2}\mathbb{Z}$, and $s \in \mathbb{C}$.

Let $G$ be a connected reductive algebraic group over $\mathbb{R}$. We denote by $\hat{G}$ the Langlands dual group of $G$, and by $^LG := \hat{G} \times \mathrm{Gal}(\mathbb{C}/\mathbb{R})$ the L-group of $G$. We say that a continuous homomorphism

$$\phi : W_{\mathbb{R}} \to {}^LG$$

is an L-parameter of $G$ if:

* elements in the image $\mathrm{Im}(\phi) \subset {}^LG$ are semi-simple;
• the composition of $\phi$ with the projection map $^LG \to \text{Gal}(\mathbb{C}/\mathbb{R})$ is the natural projection map $W_{\mathbb{R}} \to \text{Gal}(\mathbb{C}/\mathbb{R})$.

Let $\Phi(G)$ be the set of the $\hat{G}$-conjugacy classes of L-parameters for $G$. An L-parameter $\phi$ is called **tempered** if $\text{Im}(\phi)$ is bounded. We denote by $\Phi_{\text{temp}}(G)$ the subset of the conjugacy class of tempered L-parameters of $G$. For an L-parameter $\phi$, we denote by $S_{\phi} = \text{Cent}_{\hat{G}}(\text{Im}(\phi))$ the centralizer of the image of $\phi$ in $\hat{G}$, and set $S_{\phi} := \pi_0(S_{\phi})$.

2.2. **Local Langlands correspondence for classical groups.** Let $G$ be a connected reductive group over $\mathbb{R}$, the celebrated local Langlands correspondence, established by R. Langlands in [Lan73] and Shelstad in [She08a, She08b, She10], asserts that:

(1) There is a finite to one surjective map

$$ \mathcal{L}: \bigsqcup_{G'} \text{Irr}(G') \to \Phi(G), $$

where $G'$ runs over the isomorphism classes of pure inner forms of $G$, and $\text{Irr}(G')$ is the set of irreducible Casselman-Wallach representations of $G'(\mathbb{R})$ (see [Cas89], [Wal94]). For each $\phi \in \Phi(G)$, we denote by $\Pi_{\phi} = \mathcal{L}^{-1}(\phi)$ the Vogan L-packet associated to $\phi$ and $\Pi_{\phi}(G') = \Pi_{\phi} \cap \text{Irr}(G')$ the local L-packet of $G'$ associated to $\phi$. The map $\mathcal{L}$ preserves temperedness, i.e., $\pi \in \text{Irr}(G')$ is tempered if and only if $\mathcal{L}(\pi)$ is a tempered L-parameter.

(2) Let $G^*$ be a quasi-split pure inner form of $G$. A **Whittaker datum** of $G^*$ is a conjugacy class of pairs $\mathfrak{w} = (B, \mu)$, where $B$ is an $\mathbb{R}$-rational Borel subgroup of $G^*$ with unipotent radical $U$, and $\mu$ is a generic character of $U(\mathbb{R})$. Fix a Whittaker datum $\mathfrak{w}$ of $G^*$, then there exists a bijection

$$ \mathcal{J}_{\mathfrak{w}}: \Pi_{\phi} \to \text{Irr}(S_{\phi}), $$

such that the unique $\mathfrak{w}$-generic representation inside $\Pi_{\phi}$ corresponds to the trivial character of $S_{\phi}$.

In the case that $G$ is a classical group, given an L-parameter $\phi \in \Phi(G)$, by composing $\phi$ with the standard representation of $^LG$, we can often regard $\phi$ as a semi-simple representation of $W_{\mathbb{R}}$ with some additional self-duality condition. Now we give a more concrete description of L-parameters, component groups, pure inner forms and Whittaker data of special orthogonal and symplectic groups following [GGP12, §10, 11].

**The symplectic group.** Let $(W, \langle \cdot, \cdot \rangle)$ be a $2n$-dimensional symplectic space and $G = \text{Sp}(W)$. Then $\hat{G} = \text{SO}_{2n+1}(\mathbb{C})$ and $^LG = \text{SO}_{2n+1}(\mathbb{C}) \times \text{Gal}(\mathbb{C}/\mathbb{R})$.

• An L-parameter $\phi$ of $G$ gives rise to an orthogonal representation $M$ of $W_{\mathbb{R}}$ with dim $M = 2n+1$ and $\det(M) = 1$. Let $\Phi(G)$ be the set of isomorphism classes of $(2n+1)$-dimensional orthogonal representations of $W_{\mathbb{R}}$ with determinant 1. Then the map $\phi \mapsto M$ gives a bijection between $\Phi(G)$ and $\Phi(G)$. In later sections, we often refer to $M$ as an L-parameter of $G$ via the above bijection and put $\Pi_M(G') = \Pi_{\phi}(G')$.

• We denote by $C_M$ the centralizer of the representation $M$ and $A_M = \pi_0(C_M)$ be the component group of $C_M$. We also let

$$ C_M^+ = \{a \in C_M | \det(a) = 1\} $$

and $A_M^+$ be the image of $C_M^+$ in $A_M$. Since $M$ is orthogonal, it admits a decomposition

$$ M = P + \sum_{i} m_{i} M_{i} + P^{\vee}, $$
where $M_i$ are pairwise distinct irreducible orthogonal subrepresentations of $M$, $m_i$ is the multiplicity of $M_i$ in $M$, and any subrepresentation of $P$ is not orthogonal. Then

$$A_M \cong \prod_i (\mathbb{Z}/2\mathbb{Z}) a_i$$

with a canonical basis $\{a_i\}$ corresponding to orthogonal subrepresentations $\{M_i\}$, and $A_M^+$ is the kernel of the map

$$A_M \to \mathbb{Z}/2\mathbb{Z}, \quad a_i \mapsto (-1)^{\dim M_i}.$$

In particular, $A_M^+$ is an index 2 subgroup of $A_M$ since there is some $M_i$ that is 1-dimensional. According to [GGP12, Theorem 8.1], we have $S_\phi \cong A_M^+$. For each $c \in \mathbb{R}^\times$, we define a character $\delta_{M,c}$ of $C_M^+$ by setting

$$(2.2.1) \qquad \delta_{M,c}(a) = \det (M^a)(c),$$

where $a \in C_M^+$, and $M^a$ is the $(-1)$-eigenspace of $a$ in $M$. This character $\delta_{M,c}$ factors through $A_M^+$, and we regard it as a character of $A_M^+$.

* The only pure inner form of $G$ is itself, so we have

$$\Pi_{\phi} = \Pi_{\phi}(G).$$

* By [GGP12, §12], the set of Whittaker data for $G = \text{Sp}(W)$ is in bijection with the set of $\mathbb{R}^{\times 2}$-orbits of non-trivial characters $\psi$ of $\mathbb{R}$. In this paper, we fix a non-trivial character $\psi$ and denote by $\mathfrak{w}$ the Whittaker datum of $G = \text{Sp}(W)$ corresponding to $\psi$.

* Let $\pi \in \Pi_M$. It follows from [Kal13, Theorem 4.9]) that $\pi^\vee \in \Pi_M$, and

$$(2.2.2) \qquad \mathcal{J}_{\mathfrak{w}}(\pi^{\vee}) = \mathcal{J}_{\mathfrak{w}}(\pi) \otimes \delta_{M,-1}.$$

**The odd special orthogonal group.** Let $(V, q)$ be a $(2n+1)$-dimensional quadratic space and $G = \text{SO}(V)$. Recall that the discriminant of $V$ is defined by the formula

$$(2.2.3) \qquad \text{disc}(V) = (-1)^n \det (\langle e_i, e_j \rangle)_{i,j} \in \mathbb{R}^\times / \mathbb{R}^{\times 2},$$

where $\{e_i\}$ is an orthogonal basis of $V$, and

$$(2.2.4) \qquad \langle x, y \rangle = \frac{1}{2} (q(x+y) - q(x) - q(y)), \quad \text{for } x, y \in V$$

is the symmetric bilinear form associated to $q$. In this case, we have $\hat{G} = \text{Sp}_{2n}(\mathbb{C})$ and $^LG = \text{Sp}_{2n}(\mathbb{C}) \times \text{Gal}(\mathbb{C}/\mathbb{R})$.

* An L-parameter $\phi$ of $G$ gives rise to a symplectic representation $M$ of $W_\mathbb{R}$ with $\dim M = 2n$. Let $\Phi(G)$ be the set of isomorphism classes of $2n$-dimensional symplectic representations of $W_\mathbb{R}$. Then the map $\phi \mapsto M$ gives a bijection between $\Phi(G)$ and $\Phi(G)$. In later sections we often refer to $M$ as an L-parameter of $G$ via the above bijection and put $\Pi_M(G) = \Pi_\phi(G)$.

* We define the groups $C_M, C_M^+, A_M, A_M^+$ in the same way as in the case of symplectic groups Section 2.2. Since $M$ is symplectic, it admits a decomposition

$$M = P + \sum_i m_i M_i + P^\vee,$$

where $M_i$ are pairwise distinct irreducible symplectic subrepresentations of $M$, $m_i$ is the multiplicity of $M_i$ in $M$, and any subrepresentation of $P$ is not symplectic. Then

$$A_M \cong \prod_i (\mathbb{Z}/2\mathbb{Z}) a_i,$$
with a canonical basis {$a_i$} corresponding to the symplectic subrepresentations {$M_i$}, and $A_M^+ = A_M$ in this case since every $M_i$ is 2-dimensional. According to [GGP12, Theorem 8.1] we have $S_\phi \cong A_M^+$.

* The pure inner forms of $G$ are of the form $G' = \text{SO}(V')$, where $V'$ is a quadratic space of the same dimension and discriminant as $V$. If $G = \text{SO}(p, q)$ with $p+q = 2n+1$, then these $V'$ are parametrized by indices $(p', q')$ such that

$$p' + q' = 2n + 1, \quad \text{and} \quad p' \equiv p \quad \text{mod } 2.$$

We have

$$\Pi_{\phi} = \bigsqcup_{V'} \Pi_{\phi}(\text{SO}(V')).$$

* By [GGP12, §12], there is a unique choice of Whittaker datum and we denote it by $\mathfrak{w}'$.

**The even special orthogonal group.** Let $(V, q)$ be a $2n$-dimensional quadratic space and $G = \text{SO}(V)$. The discriminant $\text{disc}(V)$ is defined as

$$(2.2.5) \qquad \text{disc}(V) = (-1)^n \det (\langle e_i, e_j \rangle)_{i,j} \in \mathbb{R}^\times / \mathbb{R}^{\times 2},$$

where {$e_i$} an orthogonal basis of $V$ and $\langle \cdot, \cdot \rangle$ is the symmetric bilinear form associated to $q$ defined in Equation (2.2.4). The discriminant character of $V$ is

$$\chi_V = \begin{cases} 1 & \text{if } \text{disc}(V) \in \mathbb{R}^{\times 2}, \\ \text{sgn} & \text{otherwise.} \end{cases}$$

Then $\hat{G} = \text{SO}_{2n}(\mathbb{C})$ and $^L G = \text{SO}_{2n}(\mathbb{C}) \times \text{Gal}(\mathbb{C}/\mathbb{R})$. The action of $\text{Gal}(\mathbb{C}/\mathbb{R})$ on $\text{SO}_{2n}(\mathbb{C})$ is trivial if $\text{disc}(V) \in \mathbb{R}^{\times 2}$ and is non-trivial otherwise.

* An L-parameter $\phi$ of $G$ gives rise to an orthogonal representation $M$ of $W_\mathbb{R}$ with $\dim M = 2n$ and $\det(M) = \chi_V$. Let $\Phi(G)$ be the set of isomorphism classes of $2n$-dimensional orthogonal representations of $W_\mathbb{R}$ with determinant $\chi_V$, and $\Phi^\epsilon(G)$ the subset of $\Phi(G)$ consisting of those $M$ containing an 1-dimensional orthogonal subrepresentation. The map $\phi \mapsto M$ is surjective from $\Phi(G)$ to $\Phi(G)$. The fiber of an orthogonal representation $M \in \Phi(G)$ contains one element if $M \in \Phi^\epsilon(G)$ and two elements otherwise. In this paper we only need to consider L-parameters $\phi$ whose associated representation $M \in \Phi^\epsilon(G)$, so in later sections we can still refer to $M$ as an L-parameter of $G$ via the above map, and put $\Pi_M(G) = \Pi_\phi(G)$ without confusion.

* We define the groups $C_M, C_M^+, A_M, A_M^+$ in the same way as in the case of symplectic groups Section 2.2. Since $M$ is orthogonal, it admits a decomposition

$$M = P + \sum_{i} m_i M_i + P^{\vee},$$

where $M_i$ are pairwise distinct irreducible orthogonal subrepresentations of $M$, $m_i$ is the multiplicity of $M_i$ in $M$, and any subrepresentation of $P$ is not orthogonal. Then

$$A_M \cong \prod_{i} (\mathbb{Z}/2\mathbb{Z}) a_i,$$

with a canonical basis {$a_i$} corresponding to orthogonal subrepresentations {$M_i$}. Depending on whether $M \in \Phi^\epsilon(G)$ or not, $A_M^+$ is either an index 2 subgroup of $A_M$ or $A_M$ itself. According to [GGP12, Theorem 8.1] we have $S_\phi \cong A_M^+$. For each $c \in \mathbb{R}^\times$, we define a character $\delta_{M,c}$ of $A_M^+$ in the same way as in (2.2.1).
* The pure inner forms of $G$ are of the form $G' = \text{SO}(V')$, where $V'$ is a quadratic space of the same dimension and discriminant as $V$. If $G = \text{SO}(p, q)$ with $p+q = 2n$, then these $V'$ are parametrized by indices $(p', q')$ such that

$$p' + q' = 2n, \quad \text{and} \quad p' \equiv p \quad \text{mod } 2.$$

We have

$$\Pi_{\phi} = \bigsqcup_{V'} \Pi_{\phi}(\text{SO}(V')).$$

* Let $G^* = \text{SO}(V^+)$ be a quasi-split pure inner form of $G$. By [GGP12, §12], the set of Whittaker data for $G^*$ is in bijection with the set of $\text{SO}(V^+)$-orbits of non-isotropic lines $L \subset V^+$ such that $L^{\perp}$ is split. One can check that two such non-isotropic lines $L$ and $L'$ lie in the same orbit if and only if $\text{disc}(L) = \text{disc}(L')$. If $\text{disc}(L) = c$, we denote by $w_c$ the Whittaker datum corresponding to $L$. Note that $w_c$ does not depend on the choice of the additive character $\psi$.

* For two Whittaker data $w_c$ and $w_{c'}$, where $c, c' \in \mathbb{R}^\times / (\mathbb{R}^\times)^2$, the discrepancy of the choices of Whittaker data is governed by the following formula (see [Kal13, Theorem 3.3]):

$$(2.2.6) \qquad \mathcal{J}_{w_{c'}}(\pi) = \mathcal{J}_{w_c}(\pi) \otimes \delta_{M,c'/c}, \quad \text{for } \pi \in \Pi_{\phi}.$$

**2.3. Local Langlands correspondence for metaplectic groups.** Let $W$ be a $2n$-dimensional symplectic space and $G = \widetilde{\text{Sp}}(W)$. Following [GGP12, §11], we define $\tilde{G} = \text{Sp}_{2n}(\mathbb{C})$ and $^LG = \text{Sp}_{2n}(\mathbb{C}) \times \text{Gal}(\mathbb{C}/\mathbb{R})$. The concrete descriptions of L-parameters and component groups of metaplectic groups align with those of special odd orthogonal groups. To distinguish them, we shall typically denote an L-parameter for $G = \widetilde{\text{Sp}}(W)$ by the symbol $\tilde{\phi}$. Following [GGP12, §11], we define the LLC for $G = \widetilde{\text{Sp}}(W)$ by using the Shimura-Waldspurger correspondence. Recall that the Shimura-Waldspurger correspondence [AB98] gives a bijection

$$(2.3.1) \qquad \theta_{V,W,\psi} : \text{Irr}_{\text{gen}}(\widetilde{\text{Sp}}(W)) \to \prod_{\substack{V=2n+1 \\ \text{disc}(V)=1}}^{\dim V} \text{Irr}(\text{SO}(V)).$$

Combining this bijection with the LLC for special odd orthogonal groups, we obtain a finite to one surjective map

$$(2.3.2) \qquad \mathcal{L}_{\psi} : \text{Irr}_{\text{gen}}(\widetilde{\text{Sp}}(W)) \to \Phi(G) \\ \qquad \qquad \qquad \qquad \pi \mapsto \mathcal{L}(\theta_{V,W,\psi}(\pi)).$$

Moreover, as the only “pure inner form” of the group $G = \widetilde{\text{Sp}}(W)$ is itself, we set $\Pi_{\tilde{\phi}} = \Pi_{\tilde{\phi}}(G) = \mathcal{L}_{\psi}^{-1}(\tilde{\phi})$. We also obtain a parametrization of each packet by setting

$$(2.3.3) \qquad \mathcal{J}_{\psi} : \Pi_{\tilde{\phi}} \to \widetilde{S}_{\tilde{\phi}} \\ \qquad \qquad \qquad \qquad \pi \mapsto \mathcal{J}_{\mathfrak{w}}(\theta_{V,W,\psi}(\pi)).$$

### 3. LOCAL GAN-GROSS-PRASAD CONJECTURE

In this section we recall the statement of the local Gan-Gross-Prasad conjecture for symplectic-metaplectic Fourier-Jacobi models and special orthogonal Bessel models following [GGP12]. We retain the notation in Section 2 in this section.

**3.1. Symplectic-metaplectic case.** In this subsection, we state the conjecture for the symplectic-metaplectic case. Let $W$ be a symplectic space over $\mathbb{R}$.
**Distinguished character.** We first recall the distinguished characters defined in [GGP12, §17].
Let $\tilde{\phi} : W_{\mathbb{R}} \to \text{Sp}(M)$ (resp. $\phi : W_{\mathbb{R}} \to \text{SO}(N)$) be a tempered L-parameter of $\widetilde{\text{Sp}}(W)$ (resp. $\text{Sp}(W)$), where $M$ (resp. $N$) is the symplectic (resp. orthogonal) representation of $W_{\mathbb{R}}$ associated to $\tilde{\phi}$ (resp. $\phi$). Put $N_1 = N \oplus 1$. We define a character $\eta_{N_1}$ of $C_M$ and a character $\eta_M$ of $C_{N_1}^+$ as follows: for $a \in C_M$ and $b \in C_{N_1}^+$, we set

$$ (3.1.1) \qquad \left\{ \begin{aligned} \eta_{N_1}(a) &= \epsilon (M^a \otimes N_1, \psi) \cdot \det(M^a) (-1)^{\frac{\dim N_1}{2}} \cdot \det(N_1) (-1)^{\frac{\dim M^a}{2}}; \\ \eta_M(b) &= \epsilon (M \otimes N_1^b, \psi) \cdot \det(M) (-1)^{\frac{\dim N_1^b}{2}} \cdot \det(N_1^b) (-1)^{\frac{\dim M}{2}}, \end{aligned} \right. $$

where $M^a$ and $N_1^b$ are the $(-1)$-eigenspaces for $a$ in $M$ and $b$ in $N_1$, respectively, and $\epsilon(\cdots)$ is the local root number defined by (2.1.1) and (2.1.3). The characters $\eta_{N_1}$ and $\eta_M$ factor through $A_M$ and $A_{N_1}^+$. Note that $S_{\tilde{\phi}} \cong A_M$, and $S_{\phi} \cong A_N^+$ is a subgroup of $A_{N_1}^+$. We shall regard $\eta_M$ as a character of $S_{\phi} \cong A_N^+$ by restriction.

**Statement of the conjecture.** The projection map and the identity map define a diagonal embedding

$$ \Delta : \widetilde{\text{Sp}}(W) \to \text{Sp}(W) \times \widetilde{\text{Sp}}(W). $$

Recall that we have fixed a non-trivial additive character $\psi$. This determines a Weil representation $\omega_{\psi}$ of $\widetilde{\text{Sp}}(W)$, a Whittaker datum $\mathfrak{w}$ of $\text{Sp}(W)$ (see Section 2.2), as well as the map $\mathcal{J}_{\psi}$ in the LLC for the metaplectic group $\widetilde{\text{Sp}}(W)$ (see (2.3.3)). The following theorem, known as the symplectic-metaplectic case of the local Gan-Gross-Prasad conjecture, is the main result of this paper.

**Theorem 3.1.1.** Let $\tilde{\phi} : W_{\mathbb{R}} \to \text{Sp}(M)$ be a tempered L-parameter of $\widetilde{\text{Sp}}(W)$, and $\phi : W_{\mathbb{R}} \to \text{SO}(N)$ a tempered L-parameter of $\text{Sp}(W)$. Let $\tilde{\pi} \in \Pi_{\tilde{\phi}}$ and $\pi \in \Pi_{\phi}$. Then

$$ \operatorname{Hom}_{\widetilde{\text{Sp}}(W)} (\pi \boxtimes \tilde{\pi}, \omega_{\psi}) \neq 0 \iff \mathcal{J}_{\psi}(\tilde{\pi}) \times \mathcal{J}_{\mathfrak{w}}(\pi) = \eta_{N_1} \times \eta_M, $$

where $\eta_{N_1}$ and $\eta_M$ are the characters defined in (3.1.1).

## 3.2. Special orthogonal case.
In this subsection, we recall the local Gross-Prasad conjecture (i.e. local Gan-Gross-Prasad conjecture for special orthogonal groups) over $\mathbb{R}$ proved in [CL22, Theorem 2.3.2]. Let $V$ be a quadratic space over $\mathbb{R}$ and $V'$ be a non-degenerate subspace of $V$ of codimension 1. Assume that both $\text{SO}(V)$ and $\text{SO}(V')$ are quasi-split. Let $V^{\text{even}}$ (resp. $V^{\text{odd}}$) be the unique even (resp. odd) dimensional space in $\{V, V'\}$.

**Distinguished character.** We first recall the distinguished characters defined in [GGP12, §17].
Let $\phi : W_{\mathbb{R}} \to \text{O}(M)$ (resp. $\phi' : W_{\mathbb{R}} \to \text{Sp}(N)$) be a tempered L-parameter of $\text{SO}(V^{\text{even}})$ (resp. $\text{SO}(V^{\text{odd}})$), where $M$ (resp. $N$) is the orthogonal (resp. symplectic) representation of $W_{\mathbb{R}}$ associated to $\phi$ (resp. $\phi'$). We define a character $\eta_N$ of $C_M^+$ and a character $\eta_M$ of $C_N$ as follows: for each $a \in C_M^+$ and $b \in C_N$, we set

$$ (3.2.1) \qquad \left\{ \begin{aligned} \eta_N(a) &= \epsilon (M^a \otimes N, \psi) \cdot \det(M^a) (-1)^{\frac{\dim N}{2}} \cdot \det(N) (-1)^{\frac{\dim M^a}{2}}; \\ \eta_M(b) &= \epsilon (M \otimes N^b, \psi) \cdot \det(M) (-1)^{\frac{\dim N^b}{2}} \cdot \det(N^b) (-1)^{\frac{\dim M}{2}}, \end{aligned} \right. $$

where $M^a$ and $N^b$ are the $(-1)$-eigenspaces for $a$ in $M$ and $b$ in $N$, respectively, and $\epsilon(\cdots)$ is the local root number defined in Equation (2.1.1) and Equation (2.1.3). The characters $\eta_N$ and $\eta_M$ factor through $A_M^+ \cong S_{\phi}$ and $A_N \cong S_{\phi'}$.
**Statement of Conjecture.** We say a pure inner form $G_\alpha = \text{SO}(V_\alpha) \times \text{SO}(V'_\alpha)$ of $G = \text{SO}(V) \times \text{SO}(V')$ is relevant, if there is an embedding $V'_\alpha \hookrightarrow V_\alpha$ of quadratic spaces, such that $V_\alpha/V'_\alpha \cong V/V'$ as quadratic spaces. If this is the case, then there is a natural embedding $\text{SO}(V'_\alpha) \hookrightarrow \text{SO}(V_\alpha)$. Hence, we have a diagonal embedding

$$ \Delta : \text{SO}(V'_\alpha) \to \text{SO}(V_\alpha) \times \text{SO}(V'_\alpha). $$

Recall that there is a unique choice of Whittaker datum for $\text{SO}(V^{\text{odd}})$, which we denote by $\mathfrak{w}$. Let $c = \text{disc}(V^{\text{odd}})$. Following [GGP12, §17], we shall use the Whittaker datum $\mathfrak{w}_c$ of $\text{SO}(V^{\text{even}})$ (see Section 2.2) to fix the internal parametrization of L-packets in the LLC. The following theorem, known as the local Gan-Gross-Prasad conjecture for special orthogonal groups, has been proved in [CL22, Thm. 2.3.2].

**Theorem 3.2.1.** Let $\phi : W_\mathbb{R} \to O(M)$ be a tempered L-parameter of $\text{SO}(V^{\text{even}})$, and $\phi' : W_\mathbb{R} \to \text{Sp}(N)$ a tempered L-parameter of $\text{SO}(V^{\text{odd}})$. Let $\sigma^{\text{even}} \boxtimes \sigma^{\text{odd}}$ be an irreducible tempered representation of a relevant pure inner form $G_\alpha = \text{SO}(V_\alpha^{\text{even}}) \times \text{SO}(V_\alpha^{\text{odd}})$, such that $\sigma \in \Pi_\phi$ and $\sigma' \in \Pi_{\phi'}$. Then one has

$$ (3.2.2) \qquad \operatorname{Hom}_{\operatorname{SO}(V'_\alpha)} (\sigma^{\text{even}} \boxtimes \sigma^{\text{odd}}, \mathbb{C}) \neq 0 \iff \mathcal{J}_{\mathfrak{w}_c}(\sigma^{\text{even}}) \times \mathcal{J}_{\mathfrak{w}'}(\sigma^{\text{odd}}) = \eta_N \times \eta_M, $$

where $\eta_N$ and $\eta_M$ are the characters defined in (3.2.1), and $c = \text{disc}(V^{\text{odd}})$.

# 4. PRELIMINARIES AND TOOLKITS

In this section, we recall some preliminaries and toolkits for later proofs.

**4.1. Schwartz induction and Borel's lemma.** We recall some basic notions of Schwartz analysis in this subsection. We shall work in the setting of the so-called almost linear Nash groups, i.e., a Nash group admitting a homomorphism to some general linear group with finite kernel. All groups we are working with in this paper (orthogonal, symplectic, metaplectic and Jacobi groups, as well as their closed subgroups) belong to this class of groups. Let $G$ be an almost linear Nash group, we denote by $\mathcal{S}\text{mod}(G)$ the category of Fréchet representations of moderate growth of $G$.

Let $H$ be a Nash subgroup of $G$, and $(\pi_H, V) \in \mathcal{S}\text{mod}(H)$. Then there is a continuous map

$$ I : \mathcal{S}(G, V) \to C^\infty(G, V), \quad f \mapsto \left( g \mapsto \int_H \pi_H(h) f (h^{-1}g) dh \right) $$

from the space of Schwartz functions $\mathcal{S}(G, V)$ to the space of smooth functions $C^\infty(G, V)$, with the image landing in the usual un-normalized induction $\mathfrak{In}\mathfrak{d}_H^G V$. The image of $I$, equipped with the quotient topology of $\mathcal{S}(G, V)$, is called the Schwartz induction of $\pi_H$ and will be denoted by $\mathfrak{in}\mathfrak{d}_H^G V$. We also define the *normalized Schwartz induction* of $V$ from $H$ to $G$ to be

$$ (4.1.1) \qquad \mathrm{ind}_H^G V := \left( \mathfrak{in}\mathfrak{d}_H^G \left( V \cdot \delta_H^{1/2} \right) \right) \cdot \delta_G^{-1/2}. $$

Here $\delta_H$ and $\delta_G$ are the modulus characters of $H$ and $G$. In particular, if $G$ is a reductive group and $P$ is a parabolic subgroup of $G$, then $\mathrm{ind}_P^G V = \mathfrak{in}\mathfrak{d}_P^G (V \cdot \delta_P^{1/2})$ coincides with the usual (normalized) parabolic induction.

For a representation $(\pi, V) \in \mathcal{S}\text{mod}(G)$, we put

$$ V_G = V / \langle \pi(g)v - v \mid g \in G, v \in V \rangle, $$

equipped with the quotient topology. It might not be Hausdorff. By definition, we have

$$ V_G^* \cong \operatorname{Hom}_G (V, \mathbb{1}_G), $$
where $V_G^*$ is the continuous linear dual of $V_G$, and $1_G$ is the trivial representation of $G$. The next proposition is a version of the Frobenius reciprocity in the Schwartz analysis setting. Readers may consult [CS21, Proposition 6.6, Theorem 6.8] for more details.

**Proposition 4.1.1.** Let $H$ be a Nash subgroup of $G$ and $(\pi, V) \in \mathcal{S}^{\text{mod}}(H)$. Then

$$ ((\text{in}\delta_H^G (V \otimes \delta_H)) \otimes \delta_G^{-1})_G \simeq V_H. $$

**Corollary 4.1.2.** Let $G$ be a reductive Nash group, and $H$ a reductive Nash subgroup of $G$. Let $(\pi, V)$ be a Casselman-Wallach representation of $H$, and $(\sigma, W)$ a Casselman-Wallach representation of $G$. Then we have

$$ \operatorname{Hom}_G (\text{in}\delta_H^G (V \otimes \delta_H), W \otimes \delta_G) \cong \operatorname{Hom}_H (W^\vee|_H, V^\vee), $$
where $W^\vee$ stands for the contragredient representation of $V$, and likewise for $W^\vee$.

*Proof.* Since $(\pi, V)$ and $(\sigma, W)$ are Casselman-Wallach representations, it follows from [Che23, Lemma 2.2.22] that we have

$$ \begin{aligned} \operatorname{Hom}_G (\text{in}\delta_H^G (V \otimes \delta_H), W \otimes \delta_G) &\cong \operatorname{Hom}_G (\text{in}\delta_H^G (V \otimes \delta_H) \otimes W^\vee \otimes \delta_G^{-1}, 1_G) \\ &\cong \operatorname{Hom}_G (\text{in}\delta_H^G ((V \otimes W^\vee)|_H) \otimes \delta_H) \otimes \delta_G^{-1}, 1_G) \end{aligned} $$

Then by Theorem 4.1.1, we have

$$ \begin{aligned} \operatorname{Hom}_G (\text{in}\delta_H^G ((V \otimes W^\vee)|_H) \otimes \delta_H) \otimes \delta_G^{-1}, 1_G) &\cong \operatorname{Hom}_H (V \otimes W^\vee|_H, 1_H) \\ &\cong \operatorname{Hom}_H (W^\vee|_H, V^\vee). \end{aligned} $$

□

In later proofs we also need to do some orbit analysis. We now introduce a version of Borel's lemma, following [CS21, Proposition 8.2, 8.3] and [Xue20, Proposition 2.5]. Let $\mathcal{X}$ be a Nash manifold and $\mathcal{Z}$ a closed Nash submanifold. We put $\mathcal{U} = \mathcal{X} \setminus \mathcal{Z}$. Let $\mathcal{E}$ be a tempered vector bundle over $\mathcal{X}$, then the "extension by zero" of Schwartz sections over $\mathcal{U}$ gives us the following exact sequence:

$$ 0 \to \Gamma^S (\mathcal{U}, \mathcal{E}) \to \Gamma^S (\mathcal{X}, \mathcal{E}) \to \Gamma^S_{\mathcal{Z}} (\mathcal{X}, \mathcal{E}) \to 0, $$

where the symbol $\Gamma^S$ stands for taking Schwartz sections, and

$$ \Gamma_{\mathcal{Z}}^S (\mathcal{X}, \mathcal{E}) := \Gamma^S (\mathcal{X}, \mathcal{E}) / \Gamma^S (\mathcal{U}, \mathcal{E}). $$

Let $N_{\mathcal{Z}|\mathcal{X}}^*$ be the conormal bundle of $\mathcal{Z}$ and $N_{\mathcal{Z}|\mathcal{X}}^*$ the complexification of $N_{\mathcal{Z}|\mathcal{X}}^*$.

**Proposition 4.1.3.** There is a descending filtration on $\Gamma_{\mathcal{Z}}^S (\mathcal{X}, \mathcal{E})$ indexed by $k \in \mathbb{N}$

$$ \Gamma_{\mathcal{Z}}^S (\mathcal{X}, \mathcal{E}) = \Gamma_{\mathcal{Z}}^S (\mathcal{X}, \mathcal{E})_0 \supset \Gamma_{\mathcal{Z}}^S (\mathcal{X}, \mathcal{E})_1 \supset \Gamma_{\mathcal{Z}}^S (\mathcal{X}, \mathcal{E})_2 \supset \dots, $$

such that

$$ \Gamma_{\mathcal{Z}}^S (\mathcal{X}, \mathcal{E}) = \lim_{\leftarrow \leftarrow \leftarrow} \Gamma_{\mathcal{Z}}^S (\mathcal{X}, \mathcal{E}) / \Gamma_{\mathcal{Z}}^S (\mathcal{X}, \mathcal{E})_k, $$

with each graded pieces are isomorphic to

$$ \Gamma^S (\mathcal{Z}, \text{Sym}^k \mathcal{N}_{\mathcal{Z}|\mathcal{X}}^* \otimes \mathcal{E}|_{\mathcal{Z}}), \quad k \in \mathbb{N}. $$

Moreover, if $\mathcal{X}$ is a $G$-Nash manifold, $\mathcal{Z}$ is stable under the $G$-action, and $\mathcal{E}$ is a tempered $G$-bundle, then this filtration is stable under the $G$-action.

4.2. **Representation theory of Jacobi groups.** In this subsection, we recall some preliminaries on Weil representations and the representation theory of Jacobi groups.
**The Weil representation.** Let $W$ be a symplectic vector space over $\mathbb{R}$. We denote by

$$ \mathrm{Hei}(W) := \mathbb{G}_a \oplus W $$

the Heisenberg group associated with $W$, with multiplication law

$$ (z_1, w_1) + (z_2, w_2) = \left( z_1 + z_2 + \frac{1}{2} \langle w_1, w_2 \rangle, w_1 + w_2 \right). $$

The symplectic group $\mathrm{Sp}(W)$ acts naturally on $\mathrm{Hei}(W)$ by

$$ (4.2.1) \qquad g \cdot (z, w) = (z, g \cdot w), \quad \text{for } z \in \mathbb{G}_a, w \in W, g \in \mathrm{Sp}(W). $$

We define the *metaplectic Jacobi group* to be

$$ \tilde{\mathcal{J}}(W) := \widetilde{\mathrm{Sp}}(W) \ltimes \mathrm{Hei}(W), $$

where the action of $\widetilde{\mathrm{Sp}}(W)$ on $\mathrm{Hei}(W)$ factors through the natural action of $\mathrm{Sp}(W)$ on $\mathrm{Hei}(W)$ given by Equation (4.2.1). For a non-trivial additive character $\psi$ of $\mathbb{R}$, by the well-known Stone-von Neumann theorem, there is a unique irreducible representation $\omega_\psi$ of $\mathrm{Hei}(W)$ with central character $\psi$, called the *Weil representation*. This representation $\omega_\psi$ uniquely extends to $\tilde{\mathcal{J}}(W)$. By a slight abuse of notation, we use the same symbol to denote the restriction of $\omega_\psi$ to $\widetilde{\mathrm{Sp}}(W)$, which is the one appearing in the main theorem.

**Fourier-Jacobi functors.** In this section, we review the representation theory of Jacobi groups and reinterpret the Hom space in (1.0.1) in terms of it. The point is that, in terms of Jacobi groups, this Hom space becomes completely parallel to the Bessel case recalled in Section 3.2.

We retain the notation in Section 4.2. The *Jacobi group* associated with $W$ is defined as

$$ \mathcal{J}(W) := \mathrm{Sp}(W) \ltimes \mathrm{Hei}(W), $$

where the action of $\mathrm{Sp}(W)$ on $\mathrm{Hei}(W)$ is given by Equation (4.2.1). We denote by $\mathcal{S}\mathrm{mod}_\psi(J(W))$ the category of smooth Fréchet representations of moderate growth of $\mathcal{J}(W)$ with central character $\psi$, and by $\mathcal{S}\mathrm{mod}_{\mathrm{gen}}(\widetilde{\mathrm{Sp}}(W))$ the category of genuine smooth Fréchet representations of moderate growth of $\widetilde{\mathrm{Sp}}(W)$. There are two functors:

$$ (4.2.2) \qquad \begin{gathered} \mathcal{F}_\psi : \mathcal{S}\mathrm{mod}_{\mathrm{gen}}(\widetilde{\mathrm{Sp}}(W)) \to \mathcal{S}\mathrm{mod}_\psi(J(W)) \\ \pi \mapsto \pi \otimes \omega_\psi \end{gathered} $$

and

$$ (4.2.3) \qquad \begin{gathered} \mathcal{G}_\psi : \mathcal{S}\mathrm{mod}_\psi(J(W)) \to \mathcal{S}\mathrm{mod}_{\mathrm{gen}}(\widetilde{\mathrm{Sp}}(W)) \\ \sigma \mapsto \mathrm{Hom}_{\mathrm{Hei}(W)}(\omega_\psi, \sigma). \end{gathered} $$

These functors are called *Fourier-Jacobi functors*. By [Sun12, Proposition 4.2], we have:

**Proposition 4.2.1.** *These two functors $\mathcal{F}_\psi$ and $\mathcal{G}_\psi$ are mutually inverse equivalences of categories between $\mathcal{S}\mathrm{mod}_{\mathrm{gen}}(\widetilde{\mathrm{Sp}}(W))$ and $\mathcal{S}\mathrm{mod}_\psi(J(W))$.*

Recall that in the Fourier-Jacobi model (see (1.0.1)), one studies the Hom space

$$ (4.2.4) \qquad \mathrm{Hom}_H(\pi \boxtimes \tilde{\pi}, \omega_\psi) \cong \mathrm{Hom}_H(\pi \boxtimes (\tilde{\pi} \otimes \bar{\omega}_\psi), \mathbb{1}_H), $$

where $H = \Delta\widetilde{\mathrm{Sp}}(W)$ is the diagonal embedding of $\widetilde{\mathrm{Sp}}(W)$ into $G = \mathrm{Sp}(W) \times \widetilde{\mathrm{Sp}}(W)$, $\pi \in \mathrm{Irr}(\mathrm{Sp}(W))$ and $\tilde{\pi} \in \mathrm{Irr}_{\mathrm{gen}}(\widetilde{\mathrm{Sp}}(W))$. Note that by Theorem 4.2.1, the tensor product $\tilde{\pi} \otimes \bar{\omega}_\psi \cong \mathcal{F}_\psi(\tilde{\pi})$ can be regarded as an irreducible representation of $\mathcal{J}(W)$. Therefore, if we set

$$ G' = \mathrm{Sp}(W) \times \mathcal{J}(W) \quad \text{and} \quad H' = \Delta\mathrm{Sp}(W), $$
where $\Delta\mathrm{Sp}(W)$ denotes the diagonal embedding of $\mathrm{Sp}(W)$ into $G'$, then the right hand side of Equation (4.2.4) can be rewritten as

$$ (4.2.5) \qquad \mathrm{Hom}_{H'}(\pi \boxtimes \mathcal{F}_{\bar{\psi}}(\tilde{\pi}), 1_H). $$

Comparing the Hom space in Equation (4.2.5) with the Bessel model (3.2.2), one can heuristically regard the Jacobi group $J(W)$ as a substitution of $SO(V)$ in the Fourier-Jacobi model. Alternatively, one can also consider corank two Fourier-Jacobi models as follows. Let $W^+ = W \oplus \mathcal{H}'$, where $\mathcal{H}'$ is the (symplectic) hyperbolic plane with a standard basis $\{e, f\}$. We write elements in $\mathrm{Sp}(W^+)$ as block matrices using the decomposition $W^+ = \langle e \rangle \oplus W \oplus \langle f \rangle$. Then we can embed $J(W)$ into $\mathrm{Sp}(W^+)$ by setting

$$ g \mapsto \begin{pmatrix} 1 & & \\ & g & \\ & & 1 \end{pmatrix} \quad \text{and} \quad (z, w) \mapsto \begin{pmatrix} 1 & w_e^* & z \\ & 1 & w \\ & & 1 \end{pmatrix}, \quad \text{for } g \in \mathrm{Sp}(W) \text{ and } (z, w) \in \mathrm{Hei}(W). $$

Here $w_e^* \in \mathrm{Hom}(W, \langle e \rangle)$ is defined by $w_e^*(v) = \langle w, v \rangle \cdot e$ for $v \in W$. One can easily check that

$$ (4.2.6) \qquad J(W) \cong \mathrm{Stab}_{\mathrm{Sp}(W^+)}(e) $$

via this embedding. Set

$$ G'' = J(W) \times \mathrm{Sp}(W^+) \quad \text{and} \quad H'' = \Delta J(W), $$

where $\Delta J(W)$ denotes the diagonal embedding of $J(W)$ into $G''$. Let $\tilde{\pi} \in \mathrm{Irr}_{\mathrm{gen}}(\widetilde{\mathrm{Sp}}(W))$ and $\pi^+ \in \mathrm{Irr}(\mathrm{Sp}(W^+))$. Then, the corank two Fourier-Jacobi model defined in [GGP12, §12] can be written as

$$ (4.2.7) \qquad \mathrm{Hom}_{H''}(\mathcal{F}_{\bar{\psi}}(\tilde{\pi}) \boxtimes \pi^+, 1_{H''}). $$

This is again an analogy of the Bessel model; in this case the Jacobi group $J(W)$ plays the role of $SO(V')$. We remark that Theorem 1.0.1 for above Hom space can be reduced to the case of (4.2.5). We refer the readers to [Che23] for this reduction step.

Next we define the notion of “parabolic induction” for Jacobi groups, and show that the functors defined in (4.2.2) and (4.2.3) are compatible with “parabolic inductions” on both sides. We should remark here that “parabolic subgroups” for Jacobi groups defined below are not the usual notion of parabolic subgroups.

We first define the notion of “parabolic subgroups”. Consider the case of “maximal parabolics”. Recall that for classical groups, maximal parabolic subgroups are stabilizers of points in Grassmannians; thus, we need to find suitable substitutions for Grassmannians. For each $0 \le k \le \frac{1}{2} \dim W$, let $\mathcal{F}_{W,k}$ be the flag variety of $\mathrm{Sp}(W)$ consisting of the $k$-dimensional isotropic subspaces of $W$. Consider the dual tautological bundle

$$ (4.2.8) \qquad \mathcal{F}_{W,k}^J := \{ (X, v^*) \mid X \in \mathcal{F}_{W,k}, v^* \in X^* \} $$

of $\mathcal{F}_{W,k}$. There is a natural transitive action of $J(W)$ on $\mathcal{F}_{W,k}^J$ given by

$$ g \cdot (X, v^*) = (g \cdot X, g \cdot v^*), \qquad \text{for } g \in \mathrm{Sp}(W); $$

$$ (z, w) \cdot (X, v^*) = (X, v^* + \iota_X(w)), \qquad \text{for } (z, w) \in \mathrm{Hei}(W). $$

Here $X^*$ is the linear dual of $X$; $g \cdot v^*$ is a linear functional on $g \cdot X$ defined by the formula $(g \cdot v^*)(v) = v^* (g^{-1} \cdot v)$ for $v \in g \cdot X$; and $\iota_X : W \to W/X^\perp \cong X^*$ is the natural projection map. For each point $(X, v^*) \in \mathcal{F}_{W,k}^J$, we define a corresponding “maximal parabolic subgroup”

$$ P_{X,v^*} := \mathrm{Stab}_{J(W)}(X, v^*). $$
All such subgroups $P_{X,v^*}$ are conjugate as $(X, v^*)$ varies in $\mathcal{F}_{W,k}^J$. To describe them explicitly, we choose a point $(X, 0) \in \mathcal{F}_{W,k}^J$. Let $W_0 = X^\perp/X$, equipped with the symplectic form induced from $W$. Then

$$P_{X,0} \cong Q_X \rtimes \mathrm{Hei}(X^\perp),$$

where $Q_X$ is the maximal parabolic subgroup of $\mathrm{Sp}(W)$ stabilizing the isotropic subspace $X$, and for each subspace $Y \subset W$, possibly degenerate, we set $\mathrm{Hei}(Y) = \mathbb{G}_a \oplus Y$, viewed as a subgroup of $\mathrm{Hei}(W)$. There is a natural projection

$$P_{X,0} \to M_{X,0} := \mathrm{GL}(X) \times \mathrm{J}(W_0),$$

and we refer to $M_{X,0}$ as the “Levi quotient” of $P_{X,0}$. To simplify notations, when there is no confusion, we shall suppress “0” from the subscript and write

$$P_{X,0} = P_X \quad \text{and} \quad M_{X,0} = M_X.$$

For each $(X, v^*) \in \mathcal{F}_{W,k}^J$, put

$$X_{v^*} := \{x + v^*(x)e \mid x \in X\} \subseteq W^+.$$

One checks that $X_{v^*}$ is an isotropic subspace of $W^+$ with $\dim X_{v^*} = k$. This gives a $J(W)$-equivariant embedding

$$i_k : \mathcal{F}_{W,k}^J \to \mathcal{F}_{W^+,k}, \\ (X, v^*) \mapsto X_{v^*},$$

where $J(W)$ acts on $\mathcal{F}_{W^+,k}$ through the isomorphism (4.2.6). The image of $i_k$ can be described as

$$i_k (\mathcal{F}_{W,k}^J) = \{ Y \in \mathcal{F}_{W^+,k} | Y \subseteq \langle e \rangle^\perp \text{ and } \langle e \rangle \not\subseteq Y \}.$$

There is also a natural $J(W)$-equivariant surjection

$$p_k : \mathcal{F}_{W,k}^J \to \mathcal{F}_{W,k}, \\ (X, v^*) \mapsto X,$$

where the action of $J(W)$ on $\mathcal{F}_{W,k}$ factors through the quotient map $p : J(W) \to \mathrm{Sp}(W)$. A point $(X, v^*) \in \mathcal{F}_{W,k}^J$ specifies maximal parabolic subgroups $Q_{X_{v^*}}$ and $Q_X$ of $\mathrm{Sp}(W^+)$ and $\mathrm{Sp}(W)$, stabilizing isotropic subspaces $X_{v^*}$ and $X$ respectively. By using maps $i_k$ and $p_k$ defined above, one deduces that

$$(4.2.9) \qquad P_{X,v^*} = Q_{X_{v^*}} \cap \mathrm{Sp}(W^+) \quad \text{and} \quad p(P_{X,v^*}) = Q_X.$$

These discussions describe “maximal parabolic subgroups” of Jacobi groups. We shall also call “maximal parabolic subgroups” corank-1 “parabolic subgroups”.

General “parabolic subgroups” of Jacobi groups can be defined similarly: consider a flag $\underline{X}$ of isotropic subspaces

$$X_1 \subseteq X_2 \subseteq \cdots \subseteq X_r \subseteq W,$$

and a vector $v^* \in X_r^*$. Let $P_{\underline{X},v^*}$ be the “parabolic subgroup” of $J(W)$ stabilizing this flag $\underline{X}$ and $v^*$. The group $J(W)$ acts transitively on such pairs of flags and dual vectors, hence all stabilizers are conjugate and isomorphic. We fix a representative $(\underline{X}, 0)$ and set $P_{\underline{X}} = P_{\underline{X},0}$ to simplify the notation. Then we have

$$P_{\underline{X}} \cong Q_{\underline{X}} \rtimes \mathrm{Hei}(X_r^\perp),$$

where $Q_{\underline{X}}$ is the parabolic subgroup of $\mathrm{Sp}(W)$ stabilizing the above flag $\underline{X}$, and the “Levi quotient” of $P_{\underline{X}}$ is defined to be

$$M_{\underline{X}} \cong \mathrm{GL}(X_1) \times \mathrm{GL}(X_2/X_1) \times \cdots \times \mathrm{GL}(X_r/X_{r-1}) \times \mathrm{J}(W_0),$$

with $W_0 \cong X_r^\perp/X_r$.
Next we define the notion of “parabolic induction” for Jacobi groups. In the case of “maximal parabolics”, let $X \in \mathcal{F}_{W,k}$ be an isotropic subspace of $W$, and $P_X = P_{X,0}$ the “maximal parabolic subgroup” of $J(W)$ associated with $(X,0) \in \mathcal{F}_{W,k}^J$. For any $\tau \in \mathcal{S}mod(\mathrm{GL}(X))$ and $\pi_0^J \in \mathcal{S}mod_\psi(J(W_0))$, where $W_0 = X^\perp/X$, we define the “normalized parabolic induction”

$$ (4.2.10) \qquad \tau \rtimes \pi_0^J := \mathrm{ind}_{P_X}^{J(W)} (\tau \boxtimes \pi_0^J) \\ = \mathrm{ind}_{P_X}^{J(W)} \left( (\tau \boxtimes \pi_0^J) \otimes \delta_{P_X}^{\frac{1}{2}} \right). $$

Here $\tau \boxtimes \pi_0^J$ is a representation of the “Levi quotient” $M_X = \mathrm{GL}(X) \times \mathrm{J}(W_0)$, and we also regard it as a representation of $P_X$ by inflation. The modulus character $\delta_{P_X}$ factors through the projection to $\mathrm{GL}(X)$ and is given by

$$ (4.2.11) \qquad \delta_{P_X}(a) = |\det a|^{\dim W_0 + k + 2}, \quad \text{for } a \in \mathrm{GL}(X). $$

Similarly, for a general “parabolic subgroup” $P_X$ of $J(W)$, where $X$ is a flag of isotropic subspaces of $W$ as before, and for any $\tau_i \in \mathcal{S}mod(\mathrm{GL}(X_i/X_{i-1}))$ and $\pi_0^J \in \mathcal{S}mod_\psi(J(W_0))$, we define the normalized parabolic induction to be

$$ \tau_1 \times \dots \times \tau_r \rtimes \pi_0^J := \mathrm{ind}_{P_X}^{J(W)} (\tau_1 \boxtimes \dots \boxtimes \tau_r \boxtimes \pi_0^J). $$

On the other hand, for metaplectic groups we already have the notions of parabolic subgroups and inductions. Parabolic subgroups of $\widetilde{\mathrm{Sp}}(W)$ are precisely preimages of those of $\mathrm{Sp}(W)$. Let $X$ be an isotropic subspace of $W$, $Q_X$ the parabolic subgroup of $\mathrm{Sp}(W)$ stabilizing $X$, and $\tilde{Q}_X$ the preimage of $Q_X$ in $\widetilde{\mathrm{Sp}}(W)$. We let $U_X$ be the unipotent radical of $\tilde{Q}_X$ and $\tilde{M}_X$ the Levi quotient. Then we have

$$ \tilde{M}_X \cong \widetilde{\mathrm{GL}}(X) \times_{\mu_2} \widetilde{\mathrm{Sp}}(W_0), $$

where $\widetilde{\mathrm{GL}}(X)$ is a double cover of $\mathrm{GL}(X)$, with the underlying set

$$ \widetilde{\mathrm{GL}}(X) = \mathrm{GL}(X) \times \{\pm 1\} $$

and the multiplication law

$$ (g_1, \epsilon_1) \cdot (g_2, \epsilon_2) = (g_1 g_2, \epsilon_1 \epsilon_2 \cdot (\det g_1, \det g_2)_{\mathbb{R}}). $$

Here $(\cdot, \cdot)_{\mathbb{R}}$ denotes the Hilbert symbol. According to [AB98, §1], there is a genuine character $\chi_\psi$ of $\widetilde{\mathrm{GL}}(X)$. By tensoring with this genuine character $\chi_\psi$, one obtains a bijection between the set of equivalence classes of representations of $\mathrm{GL}(X)$ and the set of equivalence classes of genuine representations of $\widetilde{\mathrm{GL}}(X)$. Let $\tau \in \mathcal{S}mod(\mathrm{GL}(X))$ and $\tilde{\pi}_0 \in \mathcal{S}mod_{\mathrm{gen}}(\widetilde{\mathrm{Sp}}(W_0))$. We can form the normalized parabolic induction

$$ \tau \rtimes \tilde{\pi}_0 := \mathrm{ind}_{\tilde{Q}_X}^{\widetilde{\mathrm{Sp}}(W)} (\tau \chi_\psi \boxtimes \tilde{\pi}_0) \\ = \mathrm{ind}_{\tilde{Q}_X}^{\widetilde{\mathrm{Sp}}(W)} \left( (\tau \chi_\psi \boxtimes \tilde{\pi}_0) \otimes \delta_{\tilde{Q}_X}^{\frac{1}{2}} \right). $$

Here the modulus character $\delta_{\tilde{Q}_X}$ factors through the projection to $\mathrm{GL}(X)$, and is given by

$$ (4.2.12) \qquad \delta_{\tilde{Q}_X}(a) = |\det a|^{\dim W_0 + k + 1}, \quad \text{for } a \in \mathrm{GL}(X). $$

Similarly, for a general parabolic subgroup $\tilde{Q}_X$ of $\widetilde{\mathrm{Sp}}(W)$, where $X$ is a flag of isotropic subspaces of $W$ as before, and for any $\tau_i \in \mathcal{S}mod(\mathrm{GL}(X_i/X_{i-1}))$ and $\tilde{\pi}_0 \in \mathcal{S}mod_{\mathrm{gen}}(\widetilde{\mathrm{Sp}}(W_0))$, we define the normalized parabolic induction to be

$$ \tau_1 \times \dots \times \tau_r \rtimes \tilde{\pi}_0 := \mathrm{ind}_{\tilde{Q}_X}^{\widetilde{\mathrm{Sp}}(W)} (\tau_1 \chi_\psi \boxtimes \dots \boxtimes \tau_r \chi_\psi \boxtimes \tilde{\pi}_0). $$
**Proposition 4.2.2.** The functors $\mathcal{F}_{\psi}$ and $\mathcal{G}_{\psi}$ defined in Equation (4.2.2) and Equation (4.2.3) are compatible with “parabolic induction functors” on both sides. i.e., we have

$$ (4.2.13) \qquad \mathcal{F}_{\psi}(\tau_1 \times \cdots \times \tau_r \times \tilde{\pi}_0) \cong \tau_1 \times \cdots \times \tau_r \times \mathcal{F}_{\psi}(\tilde{\pi}_0) $$

and likewise

$$ (4.2.14) \qquad \mathcal{G}_{\psi}(\tau_1 \times \cdots \times \tau_r \times \pi_0^J) \cong \tau_1 \times \cdots \times \tau_r \times \mathcal{G}_{\psi}(\pi_0^J). $$

*Proof.* In the case of “maximal parabolic subgroups”, the desired conclusion is shown during the proof of [LS13, Proposition 4.3]. For general “parabolic subgroups”, the desired conclusion follows from induction in stages.

□

As a direct consequence of Theorem 4.2.1 and Theorem 4.2.2, we obtain a Langlands classification of irreducible representations of Jacobi groups. Let $\pi^J \in \mathcal{S}\mathrm{mod}_{\psi}(J(W))$ be an irreducible representation. Then there exists a “parabolic induction”

$$ \tau_1 | \det |^{a_1} \times \cdots \times \tau_r | \det |^{a_r} \times \pi_0^J, $$

such that $\pi^J$ is the unique irreducible quotient of it. Here for each $1 \le i \le r$, $\tau_i$ is either a unitary character of $\mathrm{GL}_1(\mathbb{R})$ or a discrete series of $\mathrm{GL}_2(\mathbb{R})$, $a_i$ is a real number, such that $a_1 \ge \cdots \ge a_r > 0$; $\pi_0^J$ is an irreducible representation of some appropriate Jacobi group, such that $\mathcal{G}_{\psi}(\pi_0^J)$ is tempered. We denote by

$$ \pi = LQ (\tau_1 | \det |^{a_1} \times \cdots \times \tau_r | \det |^{a_r} \times \pi_0^J), $$

and call $\pi$ the “Langlands quotient” of $\tau_1 | \det |^{a_1} \times \cdots \times \tau_r | \det |^{a_r} \times \pi_0^J$. In the case that $\mathcal{F}_{\psi}(\pi^J)$ is tempered, we say that $\pi^J$ is tempered.

**4.3. MVW-involution.** Let $V$ be a non-degenerate quadratic or symplectic space over $\mathbb{R}$, and let $G = G(V)$ be the isometry group associated with $V$ (or the metaplectic group if $V$ is symplectic). There exists a remarkable covariant exact functor on $\mathcal{S}\mathrm{mod}(G)$ (or $\mathcal{S}\mathrm{mod}_{\mathrm{gen}}(G)$ if $G$ is a metaplectic group), called the *MVW-involution*:

**Theorem 4.3.1.** There is a covariant exact functor

$\mathrm{MVW}: \mathcal{S}\mathrm{mod}(G) \to \mathcal{S}\mathrm{mod}(G) \quad (\text{or } \mathcal{S}\mathrm{mod}_{\mathrm{gen}}(G) \to \mathcal{S}\mathrm{mod}_{\mathrm{gen}}(G) \text{ if } G \text{ is metaplectic}),$

sending each (genuine) representation $\pi$ to another $\pi^{\mathrm{MVW}}$, satisfies the following properties:

(1) MVW is involutive, i.e., $(\pi^{\mathrm{MVW}})^{\mathrm{MVW}} \cong \pi$ for all $\pi \in \mathcal{S}\mathrm{mod}(G)$ (or $\mathcal{S}\mathrm{mod}_{\mathrm{gen}}(G)$ if $G$ is a metaplectic group);

(2) if $\pi \in \mathrm{Irr}(G)$ (or $\mathrm{Irr}_{\mathrm{gen}}(G)$ if $G$ is a metaplectic group), then $\pi^{\mathrm{MVW}} \cong \pi^{\vee}$;

(3) for a parabolic induced representation $\tau \times \pi_0$, one has

$$ (\tau \times \pi_0)^{\mathrm{MVW}} \cong \tau \times \pi_0^{\mathrm{MVW}}. $$

*Proof.* See [MVW87, Chapter 4.II.1].

□

**Remark 4.3.2.** By Theorem 4.2.1, we may transport the MVW-functor to $\mathcal{S}\mathrm{mod}_{\psi}(J(W))$ via the category equivalence with $\mathcal{S}\mathrm{mod}_{\mathrm{gen}}(\widetilde{\mathrm{Sp}}(W))$. In particular, the resulting functor on $\mathcal{S}\mathrm{mod}_{\psi}(J(W))$ satisfies the same properties as in Theorem 4.3.1.

**4.4. Local theta lifts.** In this subsection, we recall some facts about theta lifts for symplectic-orthogonal reductive dual pairs.
**Theta lifting and Howe duality.** We fix a non-trivial additive character $\psi$ of $\mathbb{R}$. Let $V$ and $W$ be a quadratic and a symplectic space over $\mathbb{R}$, respectively. Then as explicated in [Kud94], one can associate a Weil representation $\omega = \omega_{V,W,\psi}$ to the group

$$ \begin{cases} \widetilde{\text{Sp}}(W) \times \text{O}(V), & \text{if dim } V \text{ is odd;} \\ \text{Sp}(W) \times \text{O}(V), & \text{if dim } V \text{ is even.} \end{cases} $$

Given $\pi \in \text{Irr}_{\text{gen}}(\widetilde{\text{Sp}}(W))$ or $\pi \in \text{Irr}(\text{Sp}(W))$, one can consider the maximal $\pi$-isotypic quotient

$$ \pi \boxtimes \Theta_{V,W,\psi}(\pi) $$

of $\omega$. Here $\Theta_{V,W,\psi}(\pi)$ is the multiplicity space, which is indeed a finite length representation of $\text{O}(V)$. The celebrated Howe duality conjecture, which was proved by Waldspurger [Wal90] and Gan-Takeda [GT16], asserts that:

* the maximal semi-simple quotient $\theta_{V,W,\psi}(\pi)$ of $\Theta_{V,W,\psi}(\pi)$ is either zero or irreducible;

* if $\pi_1 \not\cong \pi_2$ are two non-isomorphic irreducible smooth representations such that both $\theta_{V,W,\psi}(\pi_1)$ and $\theta_{V,W,\psi}(\pi_2)$ are nonzero, then $\theta_{V,W,\psi}(\pi_1) \not\cong \theta_{V,W,\psi}(\pi_2)$.

Similarly, given $\sigma \in \text{Irr}(\text{O}(V))$, we obtain smooth finite length representations $\Theta_{W,V,\psi}(\sigma)$ and $\theta_{W,V,\psi}(\sigma)$ of $\text{Sp}(W)$ or $\widetilde{\text{Sp}}(W)$.

**(Almost) equal rank theta lifts and Prasad's conjecture.** To prove our main result using a seesaw argument, we need to describe the local theta lift for even orthogonal-symplectic dual pairs in terms of the LLC. In the case that

$$ |\dim V - \dim W| \le 2, $$

such descriptions has been provided by the so-called Prasad's conjecture (now a theorem due to [AB98] [Ada95] [Pau05]).

**Theorem 4.4.1.** Suppose that

$$ \dim V = \dim W + 2 $$

is even. Let $\phi: W_\mathbb{R} \to \text{SO}(M)$ be a tempered L-parameter of $\text{Sp}(W)$, where $M$ is the orthogonal representation of $W_\mathbb{R}$ associated to $\phi$. Let $\pi \in \Pi_\phi(\text{Sp}(W))$. If its small theta lift $\sigma := \theta_{V,W,\psi}(\pi)$ is non-zero, then:

(1) the restriction of $\sigma$ to $\text{SO}(V)$ is irreducible, which we shall still denote by $\sigma$;

(2) as a representation of $\text{SO}(V)$, we have $\sigma \in \Pi_{\theta(\phi)}(\text{SO}(V))$, where $\theta(\phi)$ is a tempered L-parameter of $\text{SO}(V)$ corresponding to the orthogonal representation

$$ \theta(M) = M \oplus 1 $$

of $W_\mathbb{R}$; moreover, if we let $c = \text{disc}(V)$, then under the natural embedding $A_M^+ \hookrightarrow A_{\theta(M)}^+$, we have

$$ \mathcal{J}_{w-c}(\sigma)|_{A_M^+} = \mathcal{J}_w(\pi). $$

Moreover, for each $\sigma \in \Pi_{\theta(\phi)}(\text{SO}(V))$, there exists a $\pi \in \Pi_\phi(\text{Sp}(W))$ such that $\sigma \cong \theta_{V,W,\psi}(\pi)|_{\text{SO}(V)}$.

**Remark 4.4.2.** This theorem is essentially proved by A. Paul in [Pau05]. But we should remind readers that Paul's paper is not written in the fashion of the LLC as we stated. For the translation of Paul's result, readers can consult to [CZ24, Theorem 6.11] and [Kak24, Theorem 8.1].
**Stable range theta lifts.** Besides the (almost) equal rank case, there is another extreme case, namely the stable range case. In this section, we shall recall some results on the stable range theta lifts. Let $r_V$ be the Witt index of $V$, and similarly $r_W$ the Witt index of $W$. The following result of Loke-Ma [LM15, Theorem A] is a key ingredient in the proof of our main result.

**Theorem 4.4.3.**

(1) Suppose that $r_V \ge \dim W$. Let $\pi$ be an irreducible unitary (genuine) representation of $\widetilde{\mathrm{Sp}}(W)$ or $\mathrm{Sp}(W)$, depending on $\dim V$. Then its big theta lift $\Theta_{V,W,\psi}(\pi)$ is irreducible.

(2) Similarly, suppose that $r_W \ge \dim V$. Let $\sigma$ be an irreducible unitary representation of $\mathrm{O}(V)$. Then its big theta lift $\Theta_{W,V,\psi}(\sigma)$ is irreducible.

Combining this irreducibility result with the induction principle, we can describe the stable range theta lifts under some technical conditions.

**Corollary 4.4.4.**

(1) Fix an orthogonal space $V$ over $F$ with $\mathrm{disc}(V) = 1$. Let $\pi$ be an irreducible unitary (genuine) representation of $\widetilde{\mathrm{Sp}}(W)$ or $\mathrm{Sp}(W)$ depending on the parity of $\dim V$. Suppose that

$$s = \frac{\dim V - \dim W - 1}{2} \ge 0,$$

and $\theta_{V,W,\psi}(\pi)$ is non-zero. For any $k \in \mathbb{N}$, let $V_k = V \oplus \mathcal{H}^k$, where $\mathcal{H}$ is the (orthogonal) hyperbolic plane. Then for any $k \ge \dim W$ there is a surjection

$$| \cdot |^{k+s-\frac{1}{2}} \times \dots \times | \cdot |^{s+\frac{1}{2}} \times \theta_{V,W,\psi}(\pi) \to \Theta_{V_k,W,\psi}(\pi).$$

(2) Similarly, let $\sigma$ be an irreducible unitary representation of $\mathrm{O}(V)$ with $\mathrm{disc}(V) = 1$. Suppose that

$$s' = \frac{\dim W - \dim V + 1}{2} \ge 0,$$

and $\theta_{W,V,\psi}(\sigma)$ is non-zero. For any $k \in \mathbb{N}$, let $W_k = W \oplus \mathcal{H}'^k$, where $\mathcal{H}'$ is the (symplectic) hyperbolic plane. Then for any $k \ge \dim V$ there is a surjection

$$| \cdot |^{k+s'-\frac{1}{2}} \times \dots \times | \cdot |^{s'+\frac{1}{2}} \times \theta_{W,V,\psi}(\sigma) \to \Theta_{W_k,V,\psi}(\sigma).$$

*Proof.* We only prove (1) here, the proof of (2) is similar. By the induction principle [Pau05, §5.2], we know that there is a non-zero map

$$\Theta_{V_k,W,\psi}(\pi) \to \mathrm{ind}_{Q_k}^{\mathrm{O}(V_k)} (|\det|^{-\frac{s-k}{2}} \boxtimes \theta_{V,W,\psi}(\pi)),$$

where $Q_k$ is the standard parabolic subgroup of $\mathrm{O}(V_k)$ with Levi component $\mathrm{GL}_k \times \mathrm{O}(V)$. Since $k \ge \dim W$, the dual-pair $\widetilde{\mathrm{Sp}}(W) \times \mathrm{O}(V_k)$ (or $\mathrm{Sp}(W) \times \mathrm{O}(V_k)$) is in the stable range. It follows from Theorem 4.4.3 that $\Theta_{V_k,W,\psi}(\pi)$ is irreducible. Therefore the above map is injective. Applying both the MVW-involution functor (see Theorem 4.3.1) and the contragredient functor to the above map, we deduce that $\Theta_{V_k,W,\psi}(\pi)$ is a quotient of $\mathrm{Ind}_{Q_k}^{\mathrm{O}(V_k)} (|\det|^{s+\frac{k}{2}} \boxtimes \theta_{W,V,\psi}(\sigma))$, which is also a quotient of

$$| \cdot |^{k+s-\frac{1}{2}} \times \dots \times | \cdot |^{s+\frac{1}{2}} \times \theta_{V,W,\psi}(\pi)$$

as desired.

□
4.5. **A vanishing result for Hom spaces.** In this subsection we record a vanishing result for Hom spaces between certain parabolic induced representations and Langlands quotients of standard modules; this will be used in Section 5.3 and Section 6.3.

**Proposition 4.5.1.** Let $V$ be either an orthogonal or symplectic space, $G = G(V)$ the isometry group (or the metaplectic/ Jacobi group when $V$ is symplectic), and

$$ \pi = LQ (\tau_1 | \det |^{a_1} \times \cdots \times \tau_r | \det |^{a_r} \times \pi_0) $$

be an irreducible representation of $G$. Here

$$ a_1 \ge \cdots \ge a_r > 0 $$

are real numbers, $\tau_i$'s are either unitary characters of $GL_1(\mathbb{R})$ or discrete series of $GL_2(\mathbb{R})$, and $\pi_0$ is a tempered representation of $G_0 = G(V_0)$ for some non-degenerate subspace $V_0 \subset V$ of the proper dimension. Then, for any:

* $\tau$ unitary character of $GL_1$ or discrete series of $GL_2$;

* real number $a > a_1$;

* $\sigma$ a representation (not necessarily of finite length) of $G_1 = G(V_1)$ for some non-degenerate subspace $V_1 \subset V$ of the proper dimension,

we have

$$ \mathrm{Hom}_G (\tau | \det |^a \times \sigma, \pi) = 0. $$

*Proof.* In the case that $G$ is a classical group or metaplectic group, this has been proved in [Che23, Lemma 4.1.4] and [Che23, Theorem A.0.1]. In the case that $G$ is a Jacobi group, this follows from the case of metaplectic group, combined with Theorem 4.2.1 and Theorem 4.2.2. $\square$

<figure><img src="image_3.png" /></figure>

## 5. UNIQUENESS AND EPSILON DICHOTOMY OF THE FOURIER-JACOBI MODEL

We shall prove the uniqueness and epsilon dichotomy for the Fourier-Jacobi model in a Vogan L-packet in this section, and leave the proof of existence in the next section.

**5.1. Desideratum I: uniqueness and epsilon dichotomy.** We retain the notation in Section 3.1. Our desideratum in this section is the following.

**Theorem 5.1.1.** Let $\tilde{\phi}: W_{\mathbb{R}} \to \mathrm{Sp}(M)$ be a tempered L-parameter of $\widetilde{\mathrm{Sp}}(W)$, and $\phi: W_{\mathbb{R}} \to \mathrm{SO}(N)$ a tempered L-parameter of $\mathrm{Sp}(W)$, where $M$ is the symplectic representation of $W_{\mathbb{R}}$ associated to $\tilde{\phi}$, and $N$ is the orthogonal representation associated to $\phi$. If there exists some $\tilde{\pi} \in \Pi_{\tilde{\phi}}(\widetilde{\mathrm{Sp}}(W))$ and $\pi \in \Pi_{\phi}(\mathrm{Sp}(W))$ such that

$$ (5.1.1) \qquad \mathrm{Hom}_{\widetilde{\mathrm{Sp}}(W)} (\tilde{\pi} \boxtimes \pi, \omega_{\psi}) \neq 0, $$

then we must have

$$ \mathcal{J}_{\psi}(\tilde{\pi}) \times \mathcal{J}_{\omega}(\pi) = \eta_{N_1} \times \eta_{M} $$

as in Theorem 3.1.1.

So now suppose that

$$ \mathrm{Hom}_{\widetilde{\mathrm{Sp}}(W)} (\tilde{\pi} \boxtimes \pi, \omega_{\psi}) \cong \mathrm{Hom}_{\widetilde{\mathrm{Sp}}(W)} ((\tilde{\pi} \boxtimes \pi) \otimes \bar{\omega}_{\psi}, \mathbb{C}) \neq 0. $$

This assumption is equivalent to

$$ \mathrm{Hom}_{\mathrm{Sp}(W)} (\tilde{\pi} \boxtimes \bar{\omega}_{\psi}, \pi^{\vee}) \neq 0. $$
Here $\mathrm{Sp}(W) \hookrightarrow \widetilde{\mathrm{Sp}}(W) \times_{\{\pm 1\}} \widetilde{\mathrm{Sp}}(W)$ is the diagonal embedding. By (2.3.1), there exists a unique $2n + 1$-dimensional orthogonal space $V'$ with $\mathrm{disc}(V') = 1$, such that $\tilde{\pi} \cong \theta_{W,V',\psi}(\sigma')$ for an irreducible tempered representation $\sigma'$ of $\mathrm{O}(V')$. Let $L$ be an $1$-dimensional quadratic space with $\mathrm{disc}(L) = -1$, $e \in L$ a generator of $L$, and

$$ (5.1.2) \qquad V = V' \oplus L. $$

We have $\mathrm{disc}(V) = 1$ by (2.2.3) and (2.2.5). Consider the seesaw diagram:

<figure><img src="image_4.png" /></figure>

Then the associated seesaw identity reads:

$$ (\times.0) \qquad \mathrm{Hom}_{\mathrm{Sp}(W)} (\Theta_{W,V',\psi}(\sigma') \boxtimes \omega_{\bar{\psi}}, \pi^{\vee}) = \mathrm{Hom}_{\mathrm{O}(V')} (\Theta_{V,W,\psi}(\pi^{\vee}), \sigma'). $$

Note that $\omega_{\bar{\psi}} \simeq \bar{\omega}_{\psi}$, and $\tilde{\pi} = \theta_{W,V',\psi}(\sigma')$ is a quotient of $\Theta_{W,V',\psi}(\sigma')$. We deduce from our assumption that

$$ (5.1.3) \qquad \mathrm{Hom}_{\mathrm{O}(V')} (\Theta_{V,W,\psi}(\pi^{\vee}), \sigma') \neq 0, $$

and in particular $\sigma := \theta_{V,W,\psi}(\pi^{\vee})$ is non-zero. The key point of the proof of Theorem 5.1.1 is to show the following.

**Proposition 5.1.2.** In the context of above, we have

$$ \mathrm{Hom}_{\mathrm{O}(V')} (\sigma, \sigma') \neq 0. $$

Note that if one can show that $\Theta_{V,W,\psi}(\pi^{\vee})$ is irreducible, then this proposition would simply follow from (5.1.3). When the base field is non-Archimedean, one can show the irreducibility of $\Theta_{V,W,\psi}(\pi^{\vee})$ using Kudla's filtration (see [GI14, Proposition C.4]). However in the Archimedean case, the lack of Kudla's filtration makes it difficult to show the irreducibility of $\Theta_{V,W,\psi}(\pi^{\vee})$ directly.

Assuming this proposition, we now end up this subsection with the proof of Theorem 5.1.1. The proof of Proposition 5.1.2 will be left to the rest subsections.

*Proof of Theorem 5.1.1.* Let $\theta(\tilde{\phi})$ and $\theta(\phi)$ be the tempered L-parameters of $\mathrm{SO}(V')$ and $\mathrm{SO}(V)$ with associated orthogonal representations $M$ and $\theta(N) = N \oplus 1$ of $W_{\mathbb{R}}$, respectively. By using the construction of the LLC for metaplectic groups (see (2.3.2) and (2.3.3)), we know that the restriction of $\sigma'$ to $\mathrm{SO}(V')$ is irreducible and lies in the L-packet $\Pi_{\theta(\tilde{\phi})}(\mathrm{SO}(V'))$, and satisfies

$$ (5.1.4) \qquad \mathcal{J}_{\mathfrak{w}'}(\sigma') = \mathcal{J}_{\psi}(\tilde{\pi}). $$

Similarly, applying Theorem 4.4.1, we know that the restriction of $\sigma$ to $\mathrm{SO}(V)$ is irreducible and lies in the L-packet $\Pi_{\theta(\phi)}(\mathrm{SO}(V))$, and satisfies

$$ (5.1.5) \qquad \mathcal{J}_{\mathfrak{w}_{-1}}(\sigma)|_{A_N^+} = \mathcal{J}_{\mathfrak{w}}(\pi^{\vee}). $$

Indeed, one can express $\mathcal{J}_{\mathfrak{w}}(\pi)$ in terms of $\mathcal{J}_{\mathfrak{w}_1}(\sigma)$:

$$ \begin{align} \mathcal{J}_{\mathfrak{w}}(\pi) &= \mathcal{J}_{\mathfrak{w}}(\pi^{\vee}) \otimes \delta_{N,-1} \tag{2.2.2} \\ &= \mathcal{J}_{\mathfrak{w}_{-1}}(\sigma)|_{A_N^+} \otimes \delta_{N,-1} \tag{5.1.5} \\ &= (\mathcal{J}_{\mathfrak{w}_1}(\sigma) \otimes \delta_{N_1,-1})|_{A_N^+} \otimes \delta_{N,-1} \tag{2.2.6} \\ &= \mathcal{J}_{\mathfrak{w}_1}(\sigma)|_{A_N^+}, \notag \end{align} $$
where in the last step we have make use of the obvious equality $\delta_{N_1,-1}|_{A_N^+} = \delta_{N,-1}$. On the other hand, by Theorem 5.1.2 we have

$$ \mathrm{Hom}_{\mathrm{SO}(V')} (\sigma, \sigma') \neq 0. $$

Hence, by Theorem 3.2.1,

$$ (5.1.7) \quad \left\{ \begin{array}{ll} \mathcal{J}_{\mathfrak{W}_1}(\sigma)(a) = \epsilon(N_1^a \otimes M, \psi) \cdot \det(N_1^a)(-1)^{\frac{\dim M}{2}} \cdot \det(M)(-1)^{\frac{\dim N_1^a}{2}}, & \text{for } a \in A_{N_1}^+; \\ \mathcal{J}_{\mathfrak{W}}(\sigma')(b) = \epsilon(N_1 \otimes M^b, \psi) \cdot \det(M^b)(-1)^{\frac{\dim N_1}{2}} \cdot \det(N_1)(-1)^{\frac{\dim M^b}{2}}, & \text{for } b \in A_M. \end{array} \right. $$

Combining (5.1.4), (5.1.6) and (5.1.7), we obtain the desired equality in Theorem 5.1.1. This completes the proof. $\square$

<figure><img src="image_5.png" /></figure>

**5.2. Stable range seesaw I: some non-tempered Bessel models.** To prove Proposition 5.1.2, we shall make use of some stable range seesaw diagram. For any $k \in \mathbb{N}$, set

$$ V_k^\dagger = V^\dagger + \mathcal{H}^k, $$

where $\mathcal{H}$ is the (orthogonal) hyperbolic plane, and $\dagger \in \{\emptyset, t\}$. Note that both $\sigma$ and $\sigma'$ are tempered, we consider the Langlands quotient of the standard modules

$$ (5.2.1) \qquad \sigma_k = LQ \left( |\cdot|^k \times \dots \times |\cdot|^1 \rtimes \sigma \right), $$

and also

$$ (5.2.2) \qquad \sigma'_k = LQ \left( |\cdot|^{k-\frac{1}{2}} \times \dots \times |\cdot|^{\frac{1}{2}} \rtimes \sigma' \right). $$

According to Corollary 4.4.4, we know that if $k \ge \dim W$, then $\sigma_k \cong \Theta_{V_k,W,\psi}(\pi^\vee)$ and $\sigma'_k \cong \Theta_{V'_k,W,\psi}(\tilde{\pi})$. Thus we fix a positive integer $t \ge \dim W$. Consider the following seesaw diagram:

<figure><img src="image_6.png" /></figure>

Then the associated seesaw identity reads:

$$ (\blacksquare.1) \qquad \mathrm{Hom}_{\mathrm{Sp}(W)} \left( \Theta_{W,V'_t,\psi} (\sigma'_t) \boxtimes \omega_{\bar{\psi}}^{\vee}, \pi^{\vee} \right) = \mathrm{Hom}_{\mathrm{O}(V'_t)} \left( \Theta_{V_t,W,\psi} (\pi^{\vee}), \sigma'_t \right). $$

Since $\tilde{\pi} \cong \theta_{W,V'_t,\psi}(\sigma'_t)$ is a quotient of $\Theta_{W,V'_t,\psi}(\sigma'_t)$, and $\sigma_t \cong \Theta_{V_t,W,\psi}(\pi^\vee)$, the Equation (5.1.1) and our assumption (5.1.1) imply that

$$ (5.2.3) \qquad \mathrm{Hom}_{\mathrm{O}(V'_t)} (\sigma_t, \sigma'_t) \neq 0. $$

This Hom space is a non-tempered Bessel model. In the next section, we shall “peel off” the non-tempered part of (5.2.3) to deduce that $\mathrm{Hom}_{\mathrm{O}(V')} (\sigma, \sigma') \neq 0$.

**5.3. Peeling off the non-tempered part I: Bessel model.** This section is devoted to proving the following lemma.

**Lemma 5.3.1.** In the context of the previous subsection, suppose that

$$ (5.3.1) \qquad \mathrm{Hom}_{\mathrm{O}(V'_{k+1})} (\sigma_{k+1}, \sigma'_{k+1}) \neq 0 $$

for some non-negative integer $k$. Then

$$ \mathrm{Hom}_{\mathrm{O}(V'_k)} (\sigma_k, \sigma'_k) \neq 0. $$
*Proof.* By (5.2.1), we have a natural surjection

$$| \cdot |^{k+1} \rtimes \sigma_k \to \sigma_{k+1}.$$

Then assumption (5.3.1) implies that

$$(5.3.2) \qquad \mathrm{Hom}_{\mathcal{O}(V'_{k+1})} (|\cdot|^{k+1} \rtimes \sigma_k, \sigma'_{k+1}) \neq 0.$$

We fix two vectors $e_{k+1}, f_{k+1} \in V'_{k+1}$, such that they form a (orthogonal) hyperbolic plane, i.e.,

$$\langle e_{k+1}, e_{k+1} \rangle = \langle f_{k+1}, f_{k+1} \rangle = 0 \quad \text{and} \quad \langle e_{k+1}, f_{k+1} \rangle = 1,$$

and we have decompositions

$$V'_{k+1} = \langle e_{k+1} \rangle \oplus V'_{k} \oplus \langle f_{k+1} \rangle \quad \text{and} \quad V_{k+1} = \langle e_{k+1} \rangle \oplus V_{k} \oplus \langle f_{k+1} \rangle.$$

Let $Q_X$ be the parabolic subgroup of $O(V_{k+1})$ stabilizing the isotropic line $X = \langle e_{k+1} \rangle$, with Levi component $M_X \cong \mathrm{GL}_1(\mathbb{R}) \times O(V_k)$. Let $\mathcal{X} = O(V_{k+1})/Q_X$ be the flag variety of $O(V_{k+1})$ consisting of all isotropic lines in $V_{k+1}$, and $\mathcal{E}$ the $O(V_{k+1})$-equivariant tempered vector bundle on $\mathcal{X}$ whose fiber representation at $X \in \mathcal{X}$ is

$$(| \cdot |^{k+1} \boxtimes \sigma_k) \otimes \delta_{Q_X}^{\frac{1}{2}}.$$

Here, the modulus character $\delta_{Q_X}$ factors through the projection to $\mathrm{GL}_1(\mathbb{R})$, and

$$(5.3.3) \qquad \delta_{Q_X}(a) = |a|^{\dim V_k} \quad \text{for } a \in \mathrm{GL}_1(\mathbb{R}).$$

Then we have an isomorphism of $O(V_{k+1})$-representations

$$(5.3.4) \qquad \Gamma^S(\mathcal{X}, \mathcal{E}) \cong |\cdot|^{k+1} \rtimes \sigma_k.$$

Now we can appeal to the Mackey theory (or more precisely, Borel's lemma Theorem 4.1.3). There are two $O(V'_{k+1})$-orbits on $\mathcal{X}$:

* the closed orbit $\mathcal{Z}$, consisting of isotropic lines in $V_{k+1}$ that are contained in $V'_{k+1}$;

* the open orbit $\mathcal{U}$, consisting of isotropic lines in $V_{k+1}$ that are not contained in $V'_{k+1}$.

We have a short exact sequence of $O(V'_{k+1})$-representations

$$0 \to \Gamma^S(\mathcal{U}, \mathcal{E}) \to \Gamma^S(\mathcal{X}, \mathcal{E}) \to \Gamma^S_{\mathcal{Z}}(\mathcal{X}, \mathcal{E}) \to 0,$$

where $\Gamma^S_{\mathcal{Z}}(\mathcal{X}, \mathcal{E}) := \Gamma^S(\mathcal{X}, \mathcal{E})/\Gamma^S(\mathcal{U}, \mathcal{E})$. This induces an exact sequence of Hom spaces

$$\mathrm{Hom}_{\mathcal{O}(V'_{k+1})} (\Gamma^S_{\mathcal{Z}}(\mathcal{X}, \mathcal{E}), \sigma'_{k+1}) \to \mathrm{Hom}_{\mathcal{O}(V'_{k+1})} (\Gamma^S(\mathcal{X}, \mathcal{E}), \sigma'_{k+1}) \to \mathrm{Hom}_{\mathcal{O}(V'_{k+1})} (\Gamma^S(\mathcal{U}, \mathcal{E}), \sigma'_{k+1}).$$

We claim that the open orbit has non-zero contribution, namely

$$\mathrm{Hom}_{\mathcal{O}(V'_{k+1})} (\Gamma^S(\mathcal{U}, \mathcal{E}), \sigma'_{k+1}) \neq 0.$$

Note that by (5.3.2), (5.3.4) and the exact sequence above, to show our claim, it suffices to show that

$$\mathrm{Hom}_{\mathcal{O}(V'_{k+1})} (\Gamma^S_{\mathcal{Z}}(\mathcal{X}, \mathcal{E}), \sigma'_{k+1}) = 0.$$

We analyze $\Gamma^S_{\mathcal{Z}}(\mathcal{X}, \mathcal{E})$ using Theorem 4.1.3. Choose the representative $X = \langle e_{k+1} \rangle \in \mathcal{Z}$. Then its stabilizer $\mathrm{Stab}_{\mathcal{O}(V'_{k+1})}(X) = Q'_X$ is a parabolic subgroup of $O(V'_{k+1})$, with Levi component $M'_X \cong \mathrm{GL}_1(\mathbb{R}) \times O(V'_k)$. Hence $\mathcal{E}|_{\mathcal{Z}}$ is isomorphic to a $O(V'_{k+1})$-equivariant bundle over $\mathcal{Z} \cong \mathrm{O}(V'_{k+1})/Q'_X$ whose fiber representation at $X \in \mathcal{Z}$ is

$$\left( \left( | \cdot |^{k+1} \boxtimes \sigma_k \right) \otimes \delta_{Q_X}^{\frac{1}{2}} \right) \Big|_{Q'_X} \cong | \cdot |^{k+1+\frac{\dim V_k}{2}} \boxtimes \left( \sigma_k |_{\mathcal{O}(V'_k)} \right) \quad (\text{by (5.3.3)}).$$
Note that the modulus character $\delta_{Q'_X}$ factors through the projection to $\text{GL}_1(\mathbb{R})$, and we have
$\delta_{Q'_X}(a) = |a|^{\dim V'_k}$ for $a \in \text{GL}_1(\mathbb{R})$. It follows that

$$
\begin{align*}
\Gamma^S (\mathcal{Z}, \mathcal{E}|_{\mathcal{Z}}) &\cong \mathrm{in}\partial_{Q'_X}^{\mathrm{O}(V'_{k+1})} \left( |\cdot|^{k+1+\frac{\dim V_k}{2}} \boxtimes (\sigma_k|_{\mathrm{O}(V'_k)}) \right) \\
&\cong \mathrm{in}\partial_{Q'_X}^{\mathrm{O}(V'_{k+1})} \left( |\cdot|^{k+\frac{3}{2}} \boxtimes (\sigma_k|_{\mathrm{O}(V'_k)}) \right) \\
&= |\cdot|^{k+\frac{3}{2}} \times (\sigma_k|_{\mathrm{O}(V'_k)})
\end{align*}
$$

as $O(V'_{k+1})$-representations. On the other hand, the complexified conormal bundle $N^*_{\mathcal{Z}|\mathcal{X}}$ is a
$O(V'_{k+1})$-equivariant bundle over $\mathcal{Z}$ whose fiber representation at $X \in \mathcal{Z}$ is sgn$|\cdot| \boxtimes 1_{O(V'_{k})}$. By
Theorem 4.1.3, there is an $O(V'_{k+1})$-equivariant filtration on $\Gamma^S_{\mathcal{Z}}(\mathcal{X}, \mathcal{E})$, with graded pieces

$$
\Gamma^S (\mathcal{Z}, \mathrm{Sym}^j \mathcal{N}_{\mathcal{Z}|\mathcal{X}}^* \otimes \mathcal{E}|_{\mathcal{Z}}) \cong (\mathrm{sgn}^j | \cdot |^{k+\frac{3}{2}+j}) \times (\sigma_k|_{\mathrm{O}(V'_k)}) \quad \text{for } j \in \mathbb{N}.
$$

Recall that in the construction of $\sigma'_{k+1}$ (see (5.2.2)), all “exponents” in the standard module of $\sigma'_{k+1}$
are bounded above by $k + \frac{1}{2}$, hence we are in a situation that Theorem 4.5.1 is applicable. It follows
that

$$
\operatorname{Hom}_{\mathcal{O}(V'_{k+1})} \left( \left( \operatorname{sgn}^j \cdot |^{k+\frac{3}{2}+j} \right) \times \left( \sigma_k |_{\mathcal{O}(V'_{k})} \right), \sigma'_{k+1} \right) = 0
$$

for all $j \in \mathbb{N}$. We conclude that $\operatorname{Hom}_{\mathcal{O}(V'_{k+1})}(\Gamma_{\mathcal{Z}}^{\mathcal{S}}(\mathcal{X}, \mathcal{E}), \sigma'_{k+1}) = 0$ and our claim holds.

Next we compute the contribution of the open orbit. Recall that $V = V' \oplus \langle e \rangle$ with $\langle e, e \rangle_V = -1$ (see (5.1.2)). Choose a representative

$$
Y = \left\langle e + \frac{e_{k+1} + f_{k+1}}{\sqrt{2}} \right\rangle \in \mathcal{U}.
$$

Then $\text{Stab}_{\text{O}(V'_{k+1})}(Y) \cong \text{O}(V_k)$, and $\mathcal{E}|_{\mathcal{U}}$ is isomorphic to a $\text{O}(V'_{k+1})$-equivariant bundle over $\mathcal{U}$ whose fiber representation at $Y$ is $\sigma_k$. Therefore,

$$
\Gamma^S(\mathcal{U}, \mathcal{E}) \cong \mathrm{in}\partial_{\mathrm{O}(V_k)}^{\mathrm{O}(V'_{k+1})} (\sigma_k).
$$

By the Frobenius reciprocity in Theorem 4.1.2, we obtain that

$$
\begin{align*}
\operatorname{Hom}_{\mathcal{O}(V'_{k+1})} (\Gamma^S(\mathcal{U}, \mathcal{E}), \sigma'_{k+1}) &\cong \operatorname{Hom}_{\mathcal{O}(V'_{k+1})} \left( \operatorname{in}\partial_{\mathcal{O}(V_k)}^{\mathcal{O}(V'_{k+1})} (\sigma_k), \sigma'_{k+1} \right) \\
&\cong \operatorname{Hom}_{\mathcal{O}(V_k)} ((\sigma'_{k+1})^{\vee}, \sigma_k^{\vee}) \\
&\cong \operatorname{Hom}_{\mathcal{O}(V_k)} (\sigma'_{k+1}, \sigma_k).
\end{align*}
$$

Here in the last isomorphism, we also make use of the fact that irreducible representations of
orthogonal groups are self-dual. Our claim in the previous paragraph then implies that

$$
\operatorname{Hom}_{\mathcal{O}(V_k)} (\sigma'_{k+1}, \sigma_k) \neq 0.
$$

Note that by our construction (5.2.2), there is also a natural surjection

$$
| \cdot |^{k+\frac{1}{2}} \times \sigma'_{k} \to \sigma'_{k+1}.
$$

Applying the same argument above using this surjection, we obtain that

$$
\operatorname{Hom}_{\mathcal{O}(V'_k)} (\sigma_k, \sigma'_k) \neq 0
$$

as desired.

<figure><img src="image_7.png" /></figure>

Using Theorem 5.3.1 iteratively, we conclude from (5.2.3) that $\operatorname{Hom}_{\mathcal{O}(V')}(\sigma, \sigma') \neq 0$. This completes the proof of Theorem 5.1.2.
## 6. EXISTENCE OF THE FOURIER–JACOBI MODEL

We shall prove the existence of the Fourier–Jacobi model in a Vogan L-packet in this section.

**6.1. Desideratum II: Existence.** We retain the notations in Section 5.1. Our desideratum in this section is the following.

**Theorem 6.1.1.** Let $\tilde{\phi}: W_R \to \text{Sp}(M)$ be a tempered L-parameter of $\widetilde{\text{Sp}}(W)$, and $\phi: W_R \to \text{SO}(N)$ a tempered L-parameter of $\text{Sp}(W)$, where $M$ is the symplectic representation of $W_R$ associated to $\tilde{\phi}$, and $N$ is the orthogonal representation associated to $\phi$. Then there exists an $\tilde{\pi} \in \Pi_{\tilde{\phi}}(\widetilde{\text{Sp}}(W))$ and $\pi \in \Pi_{\phi}(\text{Sp}(W))$, such that the Hom space

$$ \operatorname{Hom}_{\widetilde{\text{Sp}}(W)} (\tilde{\pi} \boxtimes \pi, \omega_{\psi}) \neq 0. $$

Similar to the proof of Theorem 5.1.1, we first write

$$ \operatorname{Hom}_{\widetilde{\text{Sp}}(W)} (\tilde{\pi} \boxtimes \pi, \omega_{\psi}) \cong \operatorname{Hom}_{\text{Sp}(W)} (\tilde{\pi} \boxtimes \bar{\omega}_{\psi}, \pi^{\vee}). $$

We will use Bessel models and a seesaw identity argument to produce a non-zero element in the right hand side Hom space of the above equation.

We fix a $2n + 1$-dimensional quadratic space $V'$ of discriminant $+1$, and set $V := V' \oplus L$, where $L$ is an $1$-dimensional quadratic space with $\text{disc}(L) = -1$. Let $\theta(\tilde{\phi})$ and $\theta(\phi)$ be the tempered L-parameters of $\text{SO}(V')$ and $\text{SO}(V)$ with associated orthogonal representations $M$ and $\theta(N) = N \oplus 1$ of $W_R$, respectively. Then by Theorem 3.2.1, there exists a relevant pure inner form of $\text{SO}(V') \times \text{SO}(V)$, corresponding to a pair of quadratic spaces $V'_{\alpha} \subset V_{\alpha}$, together with irreducible representations $\sigma'_{\alpha} \in \Pi_{\theta(\tilde{\phi})}(\text{SO}(V'_{\alpha}))$ and $\sigma_{\alpha} \in \Pi_{\theta(\phi)}(\text{SO}(V_{\alpha}))$, such that

$$ \operatorname{Hom}_{\text{SO}(V'_{\alpha})} (\sigma_{\alpha}, \sigma'_{\alpha}) \neq 0. $$

By Theorem 4.4.1, we know that there exists an irreducible representation $\pi$ of $\text{Sp}(W)$, such that $\pi^{\vee} \in \Pi_{\phi}(\text{Sp}(W))$, and

$$ \sigma_{\alpha} = \theta_{V,W,\psi} (\pi^{\vee}) |_{\text{SO}(V_{\alpha})}. $$

It follows from [Kal13, Theorem 4.9]) that one also has $\pi \in \Pi_{\phi}(\text{Sp}(W))$. Let $\sigma = \theta_{V,W,\psi}(\pi^{\vee})$ be an irreducible representation of $\text{O}(V_{\alpha})$. Then we have

$$ \operatorname{Hom}_{\text{SO}(V'_{\alpha})} (\sigma_{\alpha}, \sigma'_{\alpha}) \cong \operatorname{Hom}_{\text{O}(V'_{\alpha})} (\sigma, \operatorname{Ind}_{\text{SO}(V'_{\alpha})}^{\text{O}(V'_{\alpha})} (\sigma'_{\alpha})). $$

This implies that there exists an irreducible summand $\sigma'$ of $\operatorname{Ind}_{\text{SO}(V'_{\alpha})}^{\text{O}(V'_{\alpha})} (\sigma'_{\alpha})$ such that

$$ (6.1.1) \qquad \operatorname{Hom}_{\text{O}(V'_{\alpha})} (\sigma, \sigma') \neq 0. $$

Now consider the seesaw diagram as in (6.1.0). The associated seesaw identity reads

$$ \operatorname{Hom}_{\text{Sp}(W)} (\Theta_{W,V'_{\alpha},\psi}(\sigma') \boxtimes \omega_{\bar{\psi}}, \pi^{\vee}) = \operatorname{Hom}_{\text{O}(V'_{\alpha})} (\Theta_{V_{\alpha},W,\psi}(\pi^{\vee}), \sigma'). $$

Note that $\omega_{\bar{\psi}} \cong \bar{\omega}_{\psi}$, and $\sigma$ is a quotient of $\Theta_{V_{\alpha},W,\psi}(\pi^{\vee})$. We conclude from (6.1.1) and above seesaw identity that

$$ \operatorname{Hom}_{\text{Sp}(W)} (\Theta_{W,V'_{\alpha},\psi}(\sigma') \boxtimes \omega_{\bar{\psi}}, \pi^{\vee}) \neq 0. $$

In particular, we know that $\Theta_{W,V'_{\alpha},\psi}(\sigma') \neq 0$. Let $\tilde{\pi} = \theta_{W,V'_{\alpha},\psi}(\sigma')$. It follows from (2.3.2) that $\tilde{\pi} \in \Pi_{\tilde{\phi}}(\widetilde{\text{Sp}}(W))$. Again, due to the issue of big theta lift, we are not able to conclude

$$ \operatorname{Hom}_{\text{Sp}(W)} (\tilde{\pi} \boxtimes \bar{\omega}_{\psi}, \pi^{\vee}) \neq 0. $$

from the non-vanishing of the Hom space on the left hand side of above seesaw identity.
## 6.2. Stable range seesaw II: some non-tempered Fourier-Jacobi models.
As in Section 5, we circumvent the issue of the big theta lift by working with some stable range theta lifts. For any non-negative integer $k$, we put

$$W_k = W \oplus \mathcal{H}'^k,$$

where $\mathcal{H}'$ is the (symplectic) hyperbolic plane. Note that both $\tilde{\pi}$ and $\pi$ are tempered, we consider the Langlands quotient of the standard modules

$$(6.2.1) \qquad \tilde{\pi}_k = LQ \left( |\cdot|^{k-\frac{1}{2}} \times \dots \times |\cdot|^{\frac{1}{2}} \times \tilde{\pi} \right),$$

and also

$$(6.2.2) \qquad \pi_k^{\vee} = LQ \left( |\cdot|^{k-1} \times \dots \times |\cdot|^{1} \times \pi_1^{\vee} \right),$$

where $\pi_1^{\vee} := \theta_{W_1, V_{\alpha}, \psi}(\sigma)$ is tempered. Since $\sigma$ has non-zero theta lift to $\text{Sp}(W)$, by [GI14, Proposition C.1] one knows that $\pi_1^{\vee}$ is a direct summand of $|\cdot|^0 \times \pi^{\vee}$. According to Corollary 4.4.4, we know that if $k \ge \text{dim } V$, then $\pi_k^{\vee} \cong \Theta_{W_k, V_{\alpha}, \psi}(\sigma)$ and $\tilde{\pi}_k \cong \Theta_{W_k, V'_{\alpha}, \psi}(\sigma')$. We fix a positive integer $t > \text{dim } V$. Consider the following seesaw diagram:

<figure><img src="image_8.png" /></figure>

Then the associated seesaw identity reads:

$$(\star.2) \qquad \text{Hom}_{\text{Sp}(W_t)} (\Theta_{W_t, V'_\alpha, \psi}(\sigma') \boxtimes \bar{\omega}_\psi, \pi_t^\vee) = \text{Hom}_{\text{O}(V'_\alpha)} (\Theta_{V_\alpha, W_t, \psi}(\pi_t^\vee), \sigma').$$

Note that $\Theta_{W_t, V'_\alpha, \psi}(\sigma') \cong \tilde{\pi}_t$, and $\sigma$ is a quotient of $\Theta_{V_\alpha, W_t, \psi}(\pi_t^\vee)$. Therefore, we deduce from (6.1.1) and (★.2) that

$$\text{Hom}_{\text{Sp}(W_t)} (\tilde{\pi}_t \boxtimes \bar{\omega}_\psi, \pi_t^\vee) \neq 0.$$

In the next subsection, we shall use an analog of Lemma 5.3.1 to “peel off” the non-tempered part of the Hom space above.

## 6.3. Peeling off the non-tempered part II: Fourier-Jacobi model.
This section is devoted to proving the following lemma.

**Lemma 6.3.1.** *Suppose that*

$$(6.3.1) \qquad \text{Hom}_{\text{Sp}(W_{k+1})} (\tilde{\pi}_{k+1} \boxtimes \bar{\omega}_{\psi}, \pi_{k+1}^{\vee}) \neq 0$$

for some non-negative integer $k$. Then

$$\text{Hom}_{\text{Sp}(W_k)} (\tilde{\pi}_k \boxtimes \bar{\omega}_\psi, \pi_k^{\vee}) \neq 0.$$

*Proof.* The proof of this lemma is parallel to that of Theorem 5.3.1; we therefore only sketch it. By Theorem 4.2.1 we know that when restricted to $\text{Sp}(W_{k+1})$, one has

$$\tilde{\pi}_{k+1} \boxtimes \bar{\omega}_{\psi} \cong \mathcal{F}_{\tilde{\psi}}(\tilde{\pi}_{k+1}).$$

As explained in (4.2.5), we can rephrase (6.3.1) as a branching problem from the Jacobi group $J(W_{k+1})$ to the symplectic group $\text{Sp}(W_{k+1})$ through the natural embedding $\text{Sp}(W_{k+1}) \hookrightarrow J(W_{k+1})$. By Theorem 4.2.2 and the construction of $\tilde{\pi}_{k+1}$ (see (6.2.1)), there is a surjection

$$|\cdot|^{k+\frac{1}{2}} \times \mathcal{F}_{\tilde{\psi}}(\tilde{\pi}_k) \to \mathcal{F}_{\tilde{\psi}}(\tilde{\pi}_{k+1}),$$
where we recall that $|\cdot|^{k+\frac{1}{2}} \rtimes \mathcal{F}_{\bar{\psi}}(\tilde{\pi}_k)$ on the left hand side is the normalized “parabolic induction” of $J(W_{k+1})$ (see (4.2.10) and (4.2.11)). Our assumption (6.3.1) then implies that

$$ (6.3.2) \qquad \mathrm{Hom}_{\mathrm{Sp}(W_{k+1})} \left( |\cdot|^{k+\frac{1}{2}} \rtimes \mathcal{F}_{\bar{\psi}}(\tilde{\pi}_k), \pi_{k+1}^{\vee} \right) \neq 0. $$

We fix two vectors $e_{k+1}, f_{k+1} \in W_{k+1}$, such that they form a (symplectic) hyperbolic plane, and we have a decomposition

$$ W_{k+1} = \langle e_{k+1} \rangle \oplus W_k \oplus \langle f_{k+1} \rangle. $$

Let $\mathcal{X} = \mathcal{F}_{W_{k+1},1}^J$ be the “flag variety” of $J(W_{k+1})$ defined in (4.2.8); it consists of pairs $(X, v^*)$ where $X$ is an isotropic line in $W_{k+1}$ and $v^* \in X^*$. We fix a base point $(Y, 0) \in \mathcal{F}_{W_{k+1},1}^J$, where $Y = \langle e_{k+1} \rangle$, and let $P_Y = \mathrm{Stab}_{J(W_{k+1})}(Y, 0)$ be a “parabolic subgroup” of $J(W_{k+1})$, with “Levi quotient” $M_Y \cong \mathrm{GL}_1(\mathbb{R}) \times J(W_k)$. Let $\mathcal{E}$ be the $J(W_{k+1})$-equivariant tempered vector bundle on $\mathcal{X}$ whose fiber representation at $(Y, 0) \in \mathcal{X}$ is

$$ (|\cdot|^{k+\frac{1}{2}} \rtimes \mathcal{F}_{\bar{\psi}}(\tilde{\pi}_k)) \otimes \delta_{P_Y}^{\frac{1}{2}}. $$

Then we have

$$ \Gamma^S(\mathcal{X}, \mathcal{E}) \cong |\cdot|^{k+\frac{1}{2}} \rtimes \mathcal{F}_{\bar{\psi}}(\tilde{\pi}_k) $$

as $J(W_{k+1})$-representations. There are two $\mathrm{Sp}(W_{k+1})$-orbits on $\mathcal{X}$:

* the closed orbit $\mathcal{Z}$, consisting of pairs $(X, 0)$;

* the open orbit $\mathcal{U}$, consisting of pairs $(X, v^*)$ with $v^* \neq 0$.

This yields an $\mathrm{Sp}(W_{k+1})$-equivariant short exact sequence

$$ 0 \to \Gamma^S(\mathcal{U}, \mathcal{E}) \to \Gamma^S(\mathcal{X}, \mathcal{E}) \to \Gamma^S_{\mathcal{Z}}(\mathcal{X}, \mathcal{E}) \to 0, $$

where $\Gamma^S_{\mathcal{Z}}(\mathcal{X}, \mathcal{E}) := \Gamma^S(\mathcal{X}, \mathcal{E})/\Gamma^S(\mathcal{U}, \mathcal{E})$. It induces an exact sequence of Hom spaces

$$ \mathrm{Hom}_{\mathrm{Sp}(W_{k+1})} (\Gamma_{\mathcal{Z}}^{S}(\mathcal{X}, \mathcal{E}), \pi_{k+1}^{\vee}) \to \mathrm{Hom}_{\mathrm{Sp}(W_{k+1})} (\Gamma^{S}(\mathcal{X}, \mathcal{E}), \pi_{k+1}^{\vee}) \to \mathrm{Hom}_{\mathrm{Sp}(W_{k+1})} (\Gamma^{S}(\mathcal{U}, \mathcal{E}), \pi_{k+1}^{\vee}). $$

We claim that the open orbit has non-zero contribution, namely

$$ \mathrm{Hom}_{\mathrm{Sp}(W_{k+1})} (\Gamma^S(\mathcal{U}, \mathcal{E}), \pi_{k+1}^{\vee}) \neq 0. $$

Note that by (6.3.2) and the exact sequence above, it suffices to show that

$$ \mathrm{Hom}_{\mathrm{Sp}(W_{k+1})} (\Gamma_{\mathcal{Z}}^{S}(\mathcal{X}, \mathcal{E}), \pi_{k+1}^{\vee}) = 0. $$

We analyze $\Gamma_{\mathcal{Z}}^S(\mathcal{X}, \mathcal{E})$ using Theorem 4.1.3. Choose the representative $(Y, 0) \in \mathcal{Z}$. Then

$$ Q_Y := \mathrm{Stab}_{\mathrm{Sp}(W_{k+1})} (Y, 0) $$

is the parabolic subgroup of $\mathrm{Sp}(W_{k+1})$ stabilizing $Y$, with Levi component $\mathrm{GL}_1(\mathbb{R}) \times \mathrm{Sp}(W_k)$. The modulus character of $Q_Y$ factor through $\mathrm{GL}_1(\mathbb{R})$, and $\delta_{Q_Y}(a) = |a|^{\dim W_{k+2}}$ for $a \in \mathrm{GL}_1(\mathbb{R})$. It follows that

$$ \begin{align*} \Gamma^S(\mathcal{Z}, \mathcal{E}|_{\mathcal{Z}}) &\cong \mathrm{in}\delta_{Q_Y}^{\mathrm{Sp}(W_{k+1})} \left( \left( |\cdot|^{k+\frac{1}{2}} \rtimes \mathcal{F}_{\bar{\psi}}(\tilde{\pi}_k) \right) \otimes \delta_{P_Y}^{\frac{1}{2}} \right|_{Q_Y} \\ &\cong \mathrm{in}\delta_{Q_Y}^{\mathrm{Sp}(W_{k+1})} \left( |\cdot|^{k+2+\frac{\dim W_k}{2}} \rtimes \left( \mathcal{F}_{\bar{\psi}}(\tilde{\pi}_k) \right)_{|\mathrm{Sp}(W_k)} \right) \\ &\cong |\cdot|^{k+1} \rtimes \left( \mathcal{F}_{\bar{\psi}}(\tilde{\pi}_k) \right)_{|\mathrm{Sp}(W_k)} \end{align*} $$

as $\mathrm{Sp}(W_{k+1})$-representations. On the other hand, the complexified conormal bundle $\mathcal{N}_{\mathcal{Z}|\mathcal{X}}^*$ is a $\mathrm{Sp}(W_{k+1})$-equivariant bundle over $\mathcal{Z}$ whose fiber representation at $(Y, 0) \in \mathcal{Z}$ is $\mathrm{sgn}|\cdot| \rtimes \mathbb{1}_{\mathrm{Sp}(W_k)}$. By Theorem 4.1.3, there is an $\mathrm{Sp}(W_{k+1})$-equivariant filtration on $\Gamma_{\mathcal{Z}}^S(\mathcal{X}, \mathcal{E})$, with graded pieces

$$ \Gamma^S(\mathcal{Z}, \mathrm{Sym}^j \mathcal{N}_{\mathcal{Z}|\mathcal{X}}^* \otimes \mathcal{E}|_{\mathcal{Z}}) \cong (\mathrm{sgn}^j|\cdot|^{k+j+1}) \rtimes (\mathcal{F}_{\bar{\psi}}(\tilde{\pi}_k)|_{\mathrm{Sp}(W_k)}). $$
Recall that in the construction of $\pi_{k+1}^{\vee}$ (see (6.2.2)), all “exponents” in the standard module of $\pi_{k+1}^{\vee}$ are bounded above by $k$, hence we are in a situation that Theorem 4.5.1 is applicable. It follows that

$$ \mathrm{Hom}_{\mathrm{Sp}(W_{k+1})} \left( \left( \left( \mathrm{sgn}^j \cdot |^{k+j+1} \right) \times \left( \mathcal{F}_{\tilde{\psi}}(\tilde{\pi}_k) \middle|_{\mathrm{Sp}(W_k)} \right), \pi_{k+1}^{\vee} \right) = 0 $$

for all $j \in \mathbb{N}$. We conclude that $\mathrm{Hom}_{\mathrm{Sp}(W_{k+1})} (\Gamma_{\mathcal{Z}}^{\mathcal{S}}(\mathcal{X}, \mathcal{E}), \pi_{k+1}^{\vee}) = 0$ and our claim holds.

Now, to analyze the contribution of the open orbit, we choose the representative $(Y, f_{k+1}) \in \mathcal{U}$, where $f_{k+1}$ is regarded as a linear functional on $Y$ through the pairing on $W_{k+1}$. By (4.2.6) we have

$$ \mathrm{Stab}_{\mathrm{Sp}(W_{k+1})} (Y, f_{k+1}) \cong \mathcal{J}(W_k). $$

Hence $\mathcal{E}|_{\mathcal{U}}$ is isomorphic to a $\mathrm{Sp}(W_{k+1})$-equivariant bundle over $\mathcal{U}$ whose fiber representation at $(Y, f_{k+1})$ is

$$ (| \cdot |^{k+\frac{1}{2}} \boxtimes \mathcal{F}_{\tilde{\psi}}(\tilde{\pi}_k)) \otimes \delta_{\tilde{P}_Y}^{\frac{1}{2}}|_{J(W_k)} \cong \mathcal{F}_{\tilde{\psi}}(\tilde{\pi}_k). $$

Therefore

$$ \Gamma^{\mathcal{S}}(\mathcal{U}, \mathcal{E}) \cong \mathrm{in}\partial_{J(W_k)}^{\mathrm{Sp}(W_{k+1})} \mathcal{F}_{\tilde{\psi}}(\tilde{\pi}_k). $$

By Theorem 4.1.1, one has

$$ \begin{align*} \mathrm{Hom}_{\mathrm{Sp}(W_{k+1})} (\Gamma^{\mathcal{S}}(\mathcal{U}, \mathcal{E}), \pi_{k+1}^{\vee}) &\cong \mathrm{Hom}_{\mathrm{Sp}(W_{k+1})} \left( \mathrm{in}\partial_{J(W_k)}^{\mathrm{Sp}(W_{k+1})} \mathcal{F}_{\tilde{\psi}}(\tilde{\pi}_k), \pi_{k+1}^{\vee} \right) \\ &\cong \mathrm{Hom}_{\mathrm{Sp}(W_{k+1})} \left( \pi_{k+1} \otimes \mathrm{in}\partial_{J(W_k)}^{\mathrm{Sp}(W_{k+1})} \mathcal{F}_{\tilde{\psi}}(\tilde{\pi}_k), \mathbb{1}_{\mathrm{Sp}(W_{k+1})} \right) \\ &\cong \mathrm{Hom}_{J(W_k)} \left( \pi_{k+1}|_{J(W_k)} \otimes \mathcal{F}_{\tilde{\psi}}(\tilde{\pi}_k), \mathbb{1}_{J(W_k)} \right). \end{align*} $$

Note that the last $\mathrm{Hom}$ space can be rewritten as

$$ \begin{align*} \mathrm{Hom}_{J(W_k)} \left( \pi_{k+1}|_{J(W_k)} \otimes \mathcal{F}_{\tilde{\psi}}(\tilde{\pi}_k), \mathbb{1}_{J(W_k)} \right) &\cong \mathrm{Hom}_{J(W_k)} \left( \left( \pi_{k+1}|_{J(W_k)} \right)_{Z, \psi} \otimes (\bar{\omega}_{\psi} \otimes \tilde{\pi}_k), \mathbb{1}_{J(W_k)} \right) \\ &\cong \mathrm{Hom}_{\widetilde{\mathrm{Sp}}(W_k)} \left( \left( \left( \pi_{k+1}|_{J(W_k)} \right)_{Z, \psi} \otimes \bar{\omega}_{\psi} \right)_{\mathrm{Hei}(W_k)} \otimes \tilde{\pi}_k, \mathbb{1}_{\widetilde{\mathrm{Sp}}(W_k)} \right) \\ &\cong \mathrm{Hom}_{\widetilde{\mathrm{Sp}}(W_k)} \left( \mathcal{G}_{\psi} \left( \left( \pi_{k+1}|_{J(W_k)} \right)_{Z, \psi} \right) \otimes (\omega_{\psi} \otimes \bar{\omega}_{\psi})_{\mathrm{Hei}(W_k)} \otimes \tilde{\pi}_k, \mathbb{1}_{\widetilde{\mathrm{Sp}}(W_k)} \right). \end{align*} $$

Here, the subscript $Z, \psi$ means taking the maximal quotient such that the center $Z$ of the Heisenberg group $\mathrm{Hei}(W_k)$ acts by $\psi$, and in the last isomorphism, we make use of [Sun12, Proposition 4.2]. An easy computation shows that $(\omega_{\psi} \otimes \bar{\omega}_{\psi})_{\mathrm{Hei}(W_k)} \cong \mathbb{C}$. Therefore the last $\mathrm{Hom}$ space above is isomorphic to

$$ \begin{align*} \mathrm{Hom}_{\widetilde{\mathrm{Sp}}(W_k)} \left( \mathcal{G}_{\psi} \left( \left( \pi_{k+1}|_{J(W_k)} \right)_{Z, \psi} \right) \otimes \tilde{\pi}_k, \mathbb{1}_{\widetilde{\mathrm{Sp}}(W_k)} \right) &\cong \mathrm{Hom}_{\widetilde{\mathrm{Sp}}(W_k)} \left( \mathcal{G}_{\psi} \left( \left( \pi_{k+1}|_{J(W_k)} \right)_{Z, \psi} \right), \tilde{\pi}_k^{\vee} \right) \\ &\cong \mathrm{Hom}_{J(W_k)} \left( \pi_{k+1}|_{J(W_k)}, \mathcal{F}_{\psi}(\tilde{\pi}_k^{\vee}) \right). \end{align*} $$

Our claim in the previous paragraph then implies that

$$ (6.3.3) \qquad \mathrm{Hom}_{J(W_k)} (\pi_{k+1}, \mathcal{F}_{\psi}(\tilde{\pi}_k^{\vee})) \neq 0. $$

As explicated in (4.2.7), this $\mathrm{Hom}$ space can be rephrased as a corank two Fourier-Jacobi model.

Next we reduce the corank two model above back to the basic case by appealing to the Mackey theory again. Applying the MVW-involution in Theorem 4.3.1 to $\pi_{k+1}^{\vee}$ constructed in (6.2.2), we deduce that

$$ \pi_{k+1} \cong \mathrm{LQ} (| \cdot |^k \times \dots \times | \cdot |^1 \rtimes \pi_1). $$
In particular, there is a surjection

$$| \cdot |^k \rtimes \pi_k \to \pi_{k+1}.$$

Substituting above surjection into (6.3.3), we deduce that

$$(6.3.4) \qquad \mathrm{Hom}_{J(W_k)} (|\cdot|^k \rtimes \pi_k, \mathcal{F}_\psi(\tilde{\pi}_k^\vee)) \neq 0.$$

Let $\mathcal{X}' = \mathrm{Sp}(W_{k+1})/Q_Y$ be the flag variety of $\mathrm{Sp}(W_{k+1})$ consisting of all isotropic lines in $W_{k+1}$, and let $\mathcal{E}'$ be the $\mathrm{Sp}(W_{k+1})$-equivariant tempered vector bundle whose fiber representation at $Y \in \mathcal{X}'$ is

$$(|\cdot|^k \boxtimes \pi_k) \otimes \delta_{Q_Y}^{\frac{1}{2}}.$$

Then one has

$$\Gamma^S(\mathcal{X}', \mathcal{E}') \cong |\cdot|^k \rtimes \pi_k$$

as $\mathrm{Sp}(W_{k+1})$-representations. There are three $J(W_k)$-orbits on $\mathcal{X}'$:

* the closed orbit $\mathcal{Z}'$, consisting of the line $Y$, which is a single point;

* the middle orbit $\mathcal{M}'$, consisting of isotropic lines in $Y^\perp \setminus Y$;

* the open orbit $\mathcal{U}'$, consisting of isotropic lines in $W_{k+1} \setminus Y^\perp$.

Set $\mathcal{V}' = \mathcal{U}' \cup \mathcal{M}'$. Then there is a $J(W_k)$-equivariant short exact sequence

$$0 \to \Gamma^S(\mathcal{V}', \mathcal{E}') \to \Gamma^S(\mathcal{X}', \mathcal{E}') \to \Gamma_{\mathcal{Z}'}^S(\mathcal{X}', \mathcal{E}') \to 0,$$

where $\Gamma_{\mathcal{Z}'}^S(\mathcal{X}', \mathcal{E}') \cong \Gamma^S(\mathcal{X}', \mathcal{E}')/\Gamma^S(\mathcal{V}', \mathcal{E}')$. This induces an exact sequence of Hom spaces

$$\mathrm{Hom}_{J(W_k)} (\Gamma_{\mathcal{Z}'}^S(\mathcal{X}', \mathcal{E}'), \mathcal{F}_\psi(\tilde{\pi}_k^\vee)) \to \mathrm{Hom}_{J(W_k)} (\Gamma^S(\mathcal{X}', \mathcal{E}'), \mathcal{F}_\psi(\tilde{\pi}_k^\vee)) \to \mathrm{Hom}_{J(W_k)} (\Gamma^S(\mathcal{V}', \mathcal{E}'), \mathcal{F}_\psi(\tilde{\pi}_k^\vee)).$$

By Theorem 4.1.3, there is a filtration on $\Gamma_{\mathcal{Z}'}^S(\mathcal{X}', \mathcal{E}')$. One checks easily that the center $Z(J(W_k))$ acts trivially on each graded piece

$$\Gamma^S(\mathcal{Z}', \mathrm{Sym}^j \mathcal{N}_{\mathcal{Z}'|\mathcal{X}'}^* \otimes \mathcal{E}'|_{\mathcal{Z}'}),$$

for all $j \in \mathbb{N}$. On the other hand, the center $Z(J(W_k))$ acts on $\mathcal{F}_\psi(\tilde{\pi}_k^\vee)$ by the non-trivial character $\psi$. By comparing central characters, we obtain that

$$\mathrm{Hom}_{J(W_k)} (\Gamma_{\mathcal{Z}'}^S(\mathcal{X}', \mathcal{E}'), \mathcal{F}_\psi(\tilde{\pi}_k^\vee)) = 0.$$

Together with (6.3.4) and above exact sequence of Hom spaces, this implies

$$(6.3.5) \qquad \mathrm{Hom}_{J(W_k)} (\Gamma^S(\mathcal{V}', \mathcal{E}'), \mathcal{F}_\psi(\tilde{\pi}_k^\vee)) \neq 0.$$

The decomposition $\mathcal{V}' = \mathcal{M}' \cup \mathcal{U}'$ also yields a $J(W_k)$-equivariant short exact sequence

$$0 \to \Gamma^S(\mathcal{U}', \mathcal{E}') \to \Gamma^S(\mathcal{V}', \mathcal{E}') \to \Gamma_{\mathcal{M}'}^S(\mathcal{V}', \mathcal{E}') \to 0,$$

where $\Gamma_{\mathcal{M}'}^S(\mathcal{V}', \mathcal{E}') \cong \Gamma^S(\mathcal{V}', \mathcal{E}')/\Gamma^S(\mathcal{U}', \mathcal{E}')$, and hence induce another exact sequence of Hom spaces

$$\mathrm{Hom}_{J(W_k)} (\Gamma_{\mathcal{M}'}^S(\mathcal{V}', \mathcal{E}'), \mathcal{F}_\psi(\tilde{\pi}_k^\vee)) \to \mathrm{Hom}_{J(W_k)} (\Gamma^S(\mathcal{V}', \mathcal{E}'), \mathcal{F}_\psi(\tilde{\pi}_k^\vee)) \to \mathrm{Hom}_{J(W_k)} (\Gamma^S(\mathcal{U}', \mathcal{E}'), \mathcal{F}_\psi(\tilde{\pi}_k^\vee)).$$

Once again we claim that

$$(6.3.6) \qquad \mathrm{Hom}_{J(W_k)} (\Gamma^S(\mathcal{U}', \mathcal{E}'), \mathcal{F}_\psi(\tilde{\pi}_k^\vee)) \neq 0.$$

Note that by (6.3.5) and above exact sequence, to show our claim, it suffices to show that

$$\mathrm{Hom}_{J(W_k)} (\Gamma_{\mathcal{M}'}^S(\mathcal{V}', \mathcal{E}'), \mathcal{F}_\psi(\tilde{\pi}_k^\vee)) = 0.$$

We now analyze $\Gamma_{\mathcal{M}'}^S(\mathcal{V}', \mathcal{E}')$ using Borel's lemma Theorem 4.1.3. We fix two vectors $e_k, f_k \in W_k$, such that they form a (symplectic) hyperbolic plane, and we have a decomposition

$$W_k = \langle e_k \rangle \oplus W_{k-1} \oplus \langle f_k \rangle.$$
Choose the representative $Y' = \langle e_k \rangle \in \mathcal{M}'$. By (4.2.9), we know that $P_{Y'} := \text{Stab}_{J(W_k)}(Y')$ is a "parabolic subgroup" of $J(W_k)$ with "Levi quotient" $M_{Y'} \cong \text{GL}_1(\mathbb{R}) \times J(W_{k-1})$. By computations similar to those in the previous case, one obtains

$$ \Gamma^S (\mathcal{M}', \mathcal{E}'|_{\mathcal{M}'}) \cong | \cdot |^{k+\frac{1}{2}} \times (\pi_k|_{J(W_{k-1})}). $$

On the other hand, the complexified conormal bundle $\mathcal{N}_{\mathcal{M}'|V'}^*$ is a $J(W_k)$-equivariant bundle over $\mathcal{M}'$ whose fiber representation at $Y'$ is $\text{sgn}|\cdot|\boxtimes 1_{J(W_{k-1})}$. By Theorem 4.1.3, there is a $J(W_k)$-equivariant filtration on $\Gamma_{\mathcal{M}'(V', \mathcal{E}')}^S$, with graded pieces

$$ (6.3.7) \qquad \Gamma^S (\mathcal{M}', \text{Sym}^j \mathcal{N}_{\mathcal{M}'|V'}^* \otimes \mathcal{E}'|_{\mathcal{M}'}) \cong (\text{sgn}^j|\cdot|^{k+j+\frac{1}{2}}) \boxtimes (\pi_k|_{J(W_{k-1})}) \quad \text{for } j \in \mathbb{N}. $$

Using Theorem 4.5.1 together with (6.2.1), we deduce that

$$ \operatorname{Hom}_{J(W_k)} \left( \left( \operatorname{sgn}^j | \cdot |^{k+j+\frac{1}{2}} \right) \times (\pi_k|_{J(W_{k-1})}), \mathcal{F}_{\psi}(\tilde{\pi}_k^{\vee}) \right) = 0 \quad \text{for all } j \in \mathbb{N}, $$

and therefore $\operatorname{Hom}_{J(W_k)}(\Gamma_{\mathcal{M}'(V', \mathcal{E}')}^S, \mathcal{F}_{\psi}(\tilde{\pi}_k^{\vee})) = 0$ as desired.

Finally, choose the representative $\langle f_{k+1} \rangle \in \mathcal{U}'$. Then $\text{Stab}_{J(W_k)}(\langle f_{k+1} \rangle) \cong \text{Sp}(W_k)$ and

$$ \Gamma^S (\mathcal{U}', \mathcal{E}') \cong \text{in}\partial_{\text{Sp}(W_k)}^{J(W_k)} \left( \left( (|\cdot|^k \boxtimes \pi_k) \otimes \delta_{Q_{Y'}}^{\frac{1}{2}} \right) \bigg|_{\text{Sp}(W_k)} \right) \cong \text{in}\partial_{\text{Sp}(W_k)}^{J(W_k)} \pi_k. $$

Note that there is a natural non-degenerate $J(W_k)$-invariant pairing

$$ \mathcal{F}_{\psi}(\tilde{\pi}_k^{\vee}) \times \mathcal{F}_{\bar{\psi}}(\tilde{\pi}_k) \to \mathbb{C}. $$

By using this pairing and Theorem 4.1.1, we have

$$ \begin{align} \operatorname{Hom}_{J(W_k)} (\Gamma^S (\mathcal{U}', \mathcal{E}'), \mathcal{F}_{\psi}(\tilde{\pi}_k^{\vee})) &\cong \operatorname{Hom}_{J(W_k)} (\text{in}\partial_{\text{Sp}(W_k)}^{J(W_k)} \pi_k, \mathcal{F}_{\psi}(\tilde{\pi}_k^{\vee})) \tag{6.3.8} \\ &\hookrightarrow \operatorname{Hom}_{J(W_k)} (\mathcal{F}_{\bar{\psi}}(\tilde{\pi}_k) \otimes \text{in}\partial_{\text{Sp}(W_k)}^{J(W_k)} \pi_k, 1_{J(W_k)}) \\ &\cong \operatorname{Hom}_{\text{Sp}(W_k)} (\mathcal{F}_{\bar{\psi}}(\tilde{\pi}_k) |_{\text{Sp}(W_k)} \otimes \pi_k, 1_{\text{Sp}(W_k)}) \end{align} $$

Together with (6.3.6), this shows that

$$ \operatorname{Hom}_{\text{Sp}(W_k)} (\tilde{\pi}_k \otimes \bar{\omega}_{\psi}, \pi_k^{\vee}) \neq 0, $$

which completes the proof. $\square$

Applying Theorem 6.3.1 iteratively, we conclude that $\operatorname{Hom}_{\text{Sp}(W)}(\tilde{\pi} \boxtimes \bar{\omega}_{\psi}, \pi^{\vee}) \neq 0$. This completes the proof of Theorem 6.1.1.

## REFERENCES

[AB98] Jeffrey Adams and Dan Barbasch. Genuine representations of the metaplectic group. *Compositio Mathematica*, 113(1):23–66, 1998.

[Ada95] Jeffrey Adams. Genuine representations of the metaplectic group and epsilon factors. In *Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994)*, pages 721–731. Birkhäuser, Basel, 1995.

[Ato18] Hiraku Atobe. The local theta correspondence and the local Gan–Gross–Prasad conjecture for the symplectic-metaplectic case. *Mathematische Annalen*, 371(1):225–295, 2018.

[BP14] Raphaël Beuzart-Plessis. Expression d'un facteur epsilon de paire par une formule intégrale. *Canadian Journal of Mathematics*, 66(5):993–1049, 2014.

[BP16] Raphaël Beuzart-Plessis. La conjecture locale de Gross-Prasad pour les représentations tempérées des groupes unitaires, 2016.

[Cas89] William Casselman. Canonical extensions of Harish-Chandra modules to representations of G. *Canadian Journal of Mathematics*, 41(3):385–438, 1989.
[Che23] Cheng Chen. Multiplicity formula for induced representations: Bessel and Fourier-Jacobi models over Archimedean local fields. *arXiv preprint arXiv:2308.02912*, 2023.

[CL22] Cheng Chen and Zhilin Luo. The local Gross-Prasad conjecture over $\mathbb{R}$: Epsilon dichotomy. *arXiv preprint arXiv:2204.01212*, 2022.

[CS21] Yangyang Chen and Binyong Sun. Schwartz homologies of representations of almost linear Nash groups. *J. Funct. Anal.*, 280(7):Paper No. 108817, 50, 2021.

[CZ24] Rui Chen and Jialiang Zou. Arthur's multiplicity formula for even orthogonal and unitary groups. *Journal of the European Mathematical Society*, 2024.

[GGP12] Wee Teck Gan, Benedict H. Gross, and Dipendra Prasad. Symplectic local root numbers, central critical $L$ values, and restriction problems in the representation theory of classical groups. Number 346, pages 1–109. 2012. Sur les conjectures de Gross et Prasad. I.

[GGP20] Wee Teck Gan, Benedict H. Gross, and Dipendra Prasad. Branching laws for classical groups: the non-tempered case. *Compos. Math.*, 156(11):2298–2367, 2020.

[GI14] Wee Teck Gan and Atsushi Ichino. Formal degrees and local theta correspondence. *Invent. Math.*, 195(3):509–672, 2014.

[GI16] Wee Teck Gan and Atsushi Ichino. The Gross–Prasad conjecture and local theta correspondence. *Inventiones mathematicae*, 206(3):705–799, 2016.

[GP92] Benedict H Gross and Dipendra Prasad. On the decomposition of a representation of $SO_n$ when restricted to $SO_{n-1}$. *Canadian Journal of Mathematics*, 44(5):974–1002, 1992.

[GP94] Benedict H Gross and Dipendra Prasad. On irreducible representations of $SO(2n+1) \times SO(2m)$. *Canadian Journal of Mathematics*, 46(5):930–950, 1994.

[GT16] Wee Teck Gan and Shuichiro Takeda. A proof of the Howe duality conjecture. *J. Amer. Math. Soc.*, 29(2):473–493, 2016.

[He17] Hongyu He. On the Gan-Gross-Prasad conjecture for $U(p,q)$. *Invent. Math.*, 209(3):837–884, 2017.

[Jac09] Hervé Jacquet. Archimedean Rankin-Selberg integrals. *Contemporary Mathematics*, 14:57, 2009.

[Kak24] Hirotaka Kakuhama. Local theta correspondences and Langlands parameters for rigid inner twists. *arXiv preprint arXiv:2409.00805*, 2024.

[Kal13] Tasho Kaletha. Genericity and contragredience in the local Langlands correspondence. *Algebra Number Theory*, 7(10):2447–2474, 2013.

[Kud94] Stephen S. Kudla. Splitting metaplectic covers of dual reductive pairs. *Israel J. Math.*, 87(1-3):361–401, 1994.

[Lan73] RP Langlands. On the classification of irreducible representations of real algebraic groups. preprint (sic), IAS, Princeton, 1973.

[LM15] Hung Yean Loke and Jiajun Ma. Invariants and $K$-spectrums of local theta lifts. *Compos. Math.*, 151(1):179–206, 2015.

[LS13] Yifeng Liu and Binyong Sun. Uniqueness of Fourier-Jacobi models: the Archimedean case. *Journal of Functional Analysis*, 265(12):3325–3344, 2013.

[Luo20] Zhilin Luo. A local trace formula for the local Gan-Gross-Prasad conjecture for special orthogonal groups. *arXiv preprint arXiv:2009.13947*, 2020.

[MVW87] Colette Mœglin, Marie-France Vignéras, and Jean-Loup Waldspurger. *Correspondances de Howe sur un corps p-adique*, volume 1291 of *Lecture Notes in Mathematics*. Springer-Verlag, Berlin, 1987.

[MW12] Colette Mœglin and Jean-Loup Waldspurger. La conjecture locale de Gross-Prasad pour les groupes spéciaux orthogonaux: le cas général. *Astérisque*, 347:167–216, 2012.

[Pau05] Annegret Paul. On the Howe correspondence for symplectic-orthogonal dual pairs. *J. Funct. Anal.*, 228(2):270–310, 2005.

[She08a] D. Shelstad. Tempered endoscopy for real groups. I. Geometric transfer with canonical factors. In *Representation theory of real reductive Lie groups*, volume 472 of *Contemp. Math.*, pages 215–246. Amer. Math. Soc., Providence, RI, 2008.

[She08b] D. Shelstad. Tempered endoscopy for real groups. III. Inversion of transfer and $L$-packet structure. *Represent. Theory*, 12:369–402, 2008.

[She10] D. Shelstad. Tempered endoscopy for real groups. II. Spectral transfer factors. In *Automorphic forms and the Langlands program*, volume 9 of *Adv. Lect. Math. (ALM)*, pages 236–276. Int. Press, Somerville, MA, 2010.

[Sun12] BinYong Sun. On representations of real Jacobi groups. *Sci. China Math.*, 55(3):541–555, 2012.

[SZ12] Binyong Sun and Chen-Bo Zhu. Multiplicity one theorems: the Archimedean case. *Annals of Mathematics*, pages 23–44, 2012.

[Vog93] David A Vogan. The local Langlands conjecture. *Contemporary Mathematics*, 145:305–305, 1993.
[Wal90] J.-L. Waldspurger. Démonstration d'une conjecture de dualité de Howe dans le cas *p*-adique, *p* ≠ 2. In *Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Part I (Ramat Aviv, 1989)*, volume 2 of *Israel Math. Conf. Proc.*, pages 267–324. Weizmann, Jerusalem, 1990.

[Wal94] Nolan R Wallach. Real reductive groups II. *Bull. Amer. Math. Soc*, 30:157–158, 1994.

[Wal10] J-L Waldspurger. Une formule intégrale reliée à la conjecture locale de Gross-Prasad. *Compositio Mathematica*, 146(5):1180–1290, 2010.

[Wal12a] Jean-Loup Waldspurger. Calcul d'une valeur d'un facteur $\varepsilon$ par une formule intégrale par. *Astérisque*, 347:1–102, 2012.

[Wal12b] Jean-Loup Waldspurger. La conjecture locale de Gross-Prasad pour les représentations tempérées des groupes spéciaux orthogonaux. *Astérisque*, No. 347:103–165, 2012.

[Wal12c] Jean-Loup Waldspurger. Une formule intégrale reliée à la conjecture locale de Gross-Prasad, 2e partie: extension aux représentations tempérées. *Astérisque*, 346:171–312, 2012.

[Wal12d] Jean-Loup Waldspurger. Une variante d'un résultat de Aizenbud, Gourevitch, Rallis et Schiffmann. *Astérisque*, no. 346:313–318, 2012.

[Xue20] Hang Xue. Bessel models for unitary groups and Schwartz homology. preprint, 2, 2020.

[Xue23] Hang Xue. Bessel models for real unitary groups: the tempered case. *Duke Math. J.*, 172(5):995–1031, 2023.

[Xue24] Hang Xue. Fourier-Jacobi models for real unitary groups. *J. Funct. Anal.*, 287(12):Paper No. 110645, 41, 2024.

UNIVERSITÉ PARIS CITÉ, SORBONNE UNIVERSITÉ, CNRS, IMJ-PRG, BÂTIMENT SOPHIE GERMAIN, 8 PLACE AURÉLIE NEMOURS, F-75013 PARIS, FRANCE

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