1.1. Biquadratic extension

Let F be a non-Archimedean local field of characteristic $0$ and $E \neq K$ two distinct quadratic field extensions of F. Let $L=E \otimes _F K$ so that L is a biquadratic extension of F. We thus have the picture:

In particular, we have set:

$$\begin{align*}\operatorname{Gal}(E/F) \simeq \operatorname{Gal}(L/K) \simeq \langle\sigma\rangle, \quad \textit{and} \quad \operatorname{Gal}(K/F) \simeq \operatorname{Gal}(L/E) \simeq \langle\tau\rangle. \end{align*}$$

We also fix a nontrivial additive character $\psi _F$ of F and set $\psi _K=\psi _F\circ \operatorname {Tr}_{K/F}$ . In this paper, when we talk about Weil representations or theta correspondence, we always use the additive character $\psi _F$ or $\psi _K$ (see Section 2.1).

1.2. Skew-Hermitian spaces

Let V be an n-dimensional skew-Hermitian space over E. There are exactly two such spaces, which are distinguished by their sign

$$\begin{align*}\epsilon(V)=\omega_{E/F}(\delta^{-n}\cdot\mathrm{disc} V), \end{align*}$$

where $\mathrm {disc} V=(-1)^{n(n-1)/2}\cdot \det V$ , and $\delta $ is a fixed trace zero element in $E^\times $ . As observed in [Reference Gan, Gross and PrasadGGP23, Lem. 8.1], the scalar extension $V_K = V \otimes _F K$ is a distinguished split skew-Hermitian space over L whose isomorphism class is independent of the choice of V. In particular, if we continue to use the trace zero element $\delta \in L^\times $ to define the sign of $V_K$ , then we always have $\epsilon (V_K)=+1$ .

1.3. Twisted GGP problem

We come now to the restriction problem to be studied. For the skew-Hermitian space V over E, we have the Weil representation $\omega _{V,\mu }$ , where $\mu $ is a conjugate-symplectic character of $E^\times $ . Then we are interested in determining

$$\begin{align*}m_{V}(\pi,\mu) = \dim \operatorname{Hom}_{\operatorname{U}(V)}(\pi, \omega_{V,\mu}) \quad \textit{for } \pi\in \operatorname{Irr}\left(\operatorname{U}(V_K)\right). \end{align*}$$

Here is the main local conjecture for the twisted GGP problem:

We remark that [Reference Gan, Gross and PrasadGGP23] also formulated a conjecture in the case $E = K$ and showed that, in this case, the conjecture can be reduced to the case of discrete series representations of $\operatorname {U}(V_K) \simeq \operatorname {GL}(V)$ . However, we do not deal with the case $E = K$ in this paper.

In [Reference Gan, Gross and PrasadGGP23, Sect. 9 & 10], the three authors have proved that:

1.4. Main result

Our main result is the following theorem.

Theorem 1.3. Let M be a tempered L-parameter for $\operatorname {U}(V_K)$ of the form

$$\begin{align*}M = M_1+\cdots +M_n \end{align*}$$

with each $M_i$ one-dimensional and conjugate self-dual of parity $(-1)^{n-1}$ . Then Conjecture 1.1 holds for M.

The parity condition on each $M_i$ is equivalent to requiring that the L-parameter M is of good parity. Note that, though these tempered L-parameters M are maximally reducible and hence not the most general in the p-adic case, they are the ones whose L-packets are of maximal size. Hence, in some sense, they provide the most stringent test for Conjecture 1.1.

Recall that by properties of the LLC, tempered L-packets can be constructed using irreducible parabolic induction from good parity L-packets. An immediate corollary of our result (combining with [Reference Gan, Gross and PrasadGGP23, Thm. 10.1]) is that we may complete Theorem 1.2(2) above:

1.5. Idea of proof

The main tool for the proof of Theorem 1.3 is the theta correspondence. Using theta correspondence, we shall effectively show that Theorem 1.3 for the case $\dim V = n+1$ can be reduced to the case for $\dim V = n$ . In this way, for the type of tempered L-parameters M considered in Theorem 1.3, we may use theta correspondence to successively strip off the irreducible summands $M_i$ one at a time and reduce the conjecture for such M’s to the case when $\dim V = 1$ . In fact, since the conjecture has been shown for $\dim V \leq 2$ , we could have formulated a slightly more general main result. We content ourselves with just the following corollary:

This is because all endoscopic tempered L-packets of $\operatorname {U}_3$ can be constructed by theta lifting from tempered L-packets of $\operatorname {U}_2$ .

The rest of the paper is devoted to the proof of Theorem 1.3. In §2, we study a local theta lift of a Weil representation of a unitary group to the edge of the stable range. The main point here is to show that the resulting big theta lift is irreducible. Then in §3, we show how the conjecture in dimension $n+1$ can be reduced to that in dimension n by invoking two seesaw arguments. In the proofs, we have made use of the so-called Adams’ conjecture, which describes the theta correspondence in terms of (conjectural) A-packets. But our result is not conditional on the construction of A-packets; we refer the readers to Remark 2.2 for details.

2.1. Local theta correspondence

We first recall the basic setup of the local theta correspondence. Let $F\subset E$ be a quadratic extension of non-Archimedean local fields, V an skew-Hermitian space of dimension n and W an Hermitian space of dimension m. We shall use the symbol $\mathcal {H}$ (resp. $\mathcal {H}'$ ) to denote the skew-Hermitian (resp. Hermitian) hyperbolic plane.

To consider the theta correspondence for the reductive dual pair $\operatorname {U}(V)\times \operatorname {U}(W)$ , one requires some additional data:

• • a nontrivial additive character $\psi _F$ of F;

• • a pair of characters $\chi _V$ and $\chi _W$ of $E^\times $ such that

  $$ \begin{align*} \chi_V\big|_{F^\times}=\omega_{E/F}^{\dim V}\quad\textit{and}\quad\chi_W\big|_{F^\times}=\omega_{E/F}^{\dim W}. \end{align*} $$

To elaborate, the tensor product $V\otimes W$ has a natural symplectic form, which induces a natural map

$$ \begin{align*} \operatorname{U}(V)\times \operatorname{U}(W)\longrightarrow \operatorname{Sp}(V\otimes W). \end{align*} $$

One has the metaplectic $S^1$ -cover $\operatorname {Mp}(V\otimes W)$ of $\operatorname {Sp}(V\otimes W)$ , and the character $\psi _F$ determines a Weil representation $\omega _{\psi _F}$ of $\operatorname {Mp}(V\otimes W)$ . The datum $(\psi _F,\chi _V,\chi _W)$ then allows one to specify a splitting of the metaplectic cover over $\operatorname {U}(V)\times \operatorname {U}(W)$ following [Reference KudlaKud94]. Hence, we have a Weil representation $\omega =\omega _{V,W}$ of $\operatorname {U}(V)\times \operatorname {U}(W)$ .

As explicated in [Reference KudlaKud94] and [Reference Harris, Kudla and SweetHKS96], the splitting over $\operatorname {U}(V)$ is determined by $(\psi _F,\chi _W)$ , whereas that of $\operatorname {U}(W)$ by $(\psi _F,\chi _V)$ . In particular, taking W such that $\dim W=1$ and $\chi _W=\mu $ a conjugate symplectic character of $E^\times $ , one gets a splitting over $\operatorname {U}(V)$ associated to $(\psi _F,\mu )$ , and thus a Weil representation $\omega _{V,\mu }$ of $\operatorname {U}(V)$ , which is the one appearing in the main conjecture.

Given an irreducible representation $\pi $ of $\operatorname {U}(V)$ , the maximal $\pi $ -isotypic quotient of $\omega $ is of the form

$$ \begin{align*} \Theta(\pi)\boxtimes\pi \end{align*} $$

for some smooth representation $\Theta (\pi )$ of $\operatorname {U}(W)$ of finite length. By the Howe duality [Reference WaldspurgerWal90] [Reference Gan and TakedaGT16a] [Reference Gan and TakedaGT16b], we have:

• • The maximal semisimple quotient $\theta (\pi )$ of $\Theta (\pi )$ is irreducible if $\Theta (\pi )$ is nonzero;

• • If $\pi _1\not \simeq \pi _2$ are two nonisomorphic irreducible smooth representations of $\operatorname {U}(V)$ such that both $\theta (\pi _1)$ and $\theta (\pi _2)$ are nonzero, then $\theta (\pi _1)\not \simeq \theta (\pi _2)$ .

2.2. A refinement of Adams’ conjecture

Next, we give a description of the theta correspondence in terms of A-parameters. We fix a nontrivial additive character $\psi _F$ once and for all. Assume that

$$\begin{align*}m=\dim W \geq n=\dim V \geq 1. \end{align*}$$

Fix a pair of splitting characters $(\chi _V,\chi _W)$ , and consider the theta correspondence between $\operatorname {U}(V) \times \operatorname {U}(W)$ with respect to it.

Let $\Psi $ be a local A-parameter of $\operatorname {U}(V)$ . If we write it as a sum of irreducible subrepresentations

$$\begin{align*}\Psi=\sum_i \rho_i S_{a_i} \boxtimes S_{b_i}, \end{align*}$$

we say that $\Psi $ is of good parity if $\rho _i S_{a_i} \boxtimes S_{b_i}$ is conjugate self-dual of parity $(-1)^{n-1}$ for all i. Here, we are following Atobe–Gan’s notation [Reference Atobe and GanAG17] on irreducible representations of the Weil–Deligne group $WD_E=W_E\times \operatorname {SL}_2(\mathbb C)$ ; we omit the tensor symbol between $\rho _i$ and $S_{a_i}$ to distinguish finite-dimensional representations of the Weil–Deligne $\operatorname {SL}_2(\mathbb C)$ and the Arthur $\operatorname {SL}_2(\mathbb C)$ .

Recall that for each local A-parameter $\Psi $ , the local A-packet $\Pi _\Psi (\operatorname {U}(V))$ is also equipped with a map (depending on the choice of the additive character $\psi _F$ )

$$\begin{align*}\mathcal{J}:\Pi_\Psi(\operatorname{U}(V)) \longrightarrow \operatorname{Irr} A_\Psi, \end{align*}$$

where $A_\Psi $ is the component group associated to $\Psi $ . For example, if $\Psi $ is a local A-parameter of good parity as above, then

$$\begin{align*}A_\Psi = \sum_j \mathbb{Z}/2\mathbb{Z} \, a_j \end{align*}$$

is a free $\mathbb {Z}/2\mathbb {Z}$ -module with a canonical basis $\{a_j\}_j$ , where j runs over a representative set of inequivalent subrepresentations of $\Psi $ .

2.3. A result of Atobe

The following lemma, which is essentially due to Atobe, is useful to us in the later proofs. Let $\nu $ be the normalized absolute value of $E^\times $ .

Lemma 2.4. Let $G_0=\operatorname {U}(V_0)$ be the unitary group associated to some Hermitian (or skew-Hermitian) space $V_0$ , and $\Psi _0$ an A-parameter of $G_0$ . Suppose that $\Psi _0$ is of good parity, multiplicity free and trivial on the Weil–Deligne $\operatorname {SL}_2(\mathbb C)$ . Let $\rho $ be an irreducible representation of $W_E$ and $x\in \frac {1}{2}\mathbb {Z}$ positive such that

$$\begin{align*}\rho\boxtimes S_{2x-1}\subset\Psi_0\quad \textit{and} \quad \rho\boxtimes S_{2x+1}\not\subset\Psi_0. \end{align*}$$

Then for any $\pi _0\in \Pi _{\Psi _0}(G_0)$ , we have a nonsplit exact sequence:

$$\begin{align*}0 \longrightarrow \pi \longrightarrow \rho\nu^{-x}\rtimes \pi_0 \longrightarrow \pi' \longrightarrow 0, \end{align*}$$

where $\pi $ is the unique irreducible subrepresentation and $\pi '$ is the unique irreducible quotient of $\rho \nu ^{-x}\rtimes \pi _0$ . In particular, the length of the induced representation $\rho \nu ^{-x}\rtimes \pi _0$  is  $2$ .

2.4. Some local A-packets

Now, we use the Adams’ conjecture to describe Weil representations. Let $E^1$ be the subgroup of $E^\times $ consists of norm $1$ elements. Let $\chi _0$ be a character of $E^1$ and $\chi $ the character of $E^\times $ obtained from $\chi _0$ by base change; we may regard $\chi $ as the L-parameter of the unitary group $E^1$ corresponding to $\chi _0$ . We denote by $\omega _{V,\mu }[\chi ]$ the maximal subrepresentation of $\omega _{V,\mu }$ such that the center of $\operatorname {U}(V)$ acts by $\chi _0$ . When $n=1$ , the representation $\omega _{V,\mu }[\chi ]$ has been studied by [Reference MoenMoe87] and [Reference RogawskiRog92]. So we shall concentrate on the case $n \geq 2$ .

Proof. Let $L_1$ be the one-dimensional Hermitian space associated to $1\in F^\times $ . Let $\chi _V$ be a character of $E^\times $ such that $\chi _V~\big |_{F^\times }=\omega _{E/F}^n$ and $\Omega _{L_1,V}$ the Weil representation associated to $\operatorname {U}(L_1)\times \operatorname {U}(V)$ with respect to the splitting character $(\mu ,\chi _V)$ . Then we have

$$\begin{align*}\Omega_{L_1,V}~\Big|_{\operatorname{U}(V)}=\omega_{V,\mu}. \end{align*}$$

Hence, $\omega _{V,\mu }[\chi ]$ can be regarded as the theta lift of the character $\chi \mu ^{-n}\chi _V$ . Thus, our first assertion follows from Theorem 2.1, and the second follows from Theorem 2.3.

2.5. Irreducibility of big theta lifts

Finally, we investigate the irreducibility of the big theta lift of $\omega _{V,\mu }[\chi ]$ . We shall work in a slightly more general setting.

We retain the notations of Section 2.2. From now on, we assume that m is even and $m\geq \max \{2n-2,n\}$ . Let

$$\begin{align*}\Psi=\delta+\mu\boxtimes S_{n-1} \end{align*}$$

be a local A-parameter of $\operatorname {U}(V)$ , where $\delta $ and $\mu $ are conjugate self-dual characters of parity $(-1)^{n-1}$ and $-1$ , respectively. Our goal is to show the following.

Theorem 2.7. For any $\pi \in \Pi _\Psi (\operatorname {U}(V))$ , the big theta lift $\Theta (\pi )$ to $\operatorname {U}(W)$ is irreducible if it is nonzero. Moreover, we have

$$\begin{align*}\operatorname{Ext}_{\operatorname{U}(V)}^i(\Omega,\pi)_{sm}=0 \end{align*}$$

for all $i>0$ . Here, $\Omega $ is the Weil representation associated to $\operatorname {U}(V)\times \operatorname {U}(W)$ , and the subscript ‘ $sm$ ’ stands for taking the $\operatorname {U}(W)$ -smooth vectors.

We shall prove this theorem by induction on the dimension of V. Let $x_n=-n/2+1$ . According to Mœglin [Reference MœglinMœg06, Sect. 2.4], we know that:

Lemma 2.9. Assume that $n\geq 3$ , and let $\pi \in \Pi _\Psi (\operatorname {U}(V))$ . If $\pi $ is not supercuspidal, then there exists a unique $\pi _0\in \Pi _{\Psi _0}(\operatorname {U}(V_0))$ such that

$$\begin{align*}\pi\hookrightarrow\mu\nu^{x_n}\rtimes\pi_0. \end{align*}$$

Here,

$$\begin{align*}\Psi_0=\delta+\mu\boxtimes S_{n-3}, \end{align*}$$

and $V_0$ is a subspace of V such that $V\simeq V_0 \oplus \mathcal {H}$ .

Using this fact, we now do the induction step.

Proposition 2.10. In the context of Lemma 2.9, assume that

$$\begin{align*}\pi\hookrightarrow\mu\nu^{x_n}\rtimes\pi_0. \end{align*}$$

Let $W_0$ be a subspace of W such that $W\simeq W_0 \oplus \mathcal {H}'$ . Consider the theta correspondence of $\operatorname {U}(V_0)\times \operatorname {U}(W_0)$ (with respect to the same splitting characters). Then if Theorem 2.7 holds for $\pi _0$ , it also holds for $\pi $ .

Proof of Proposition 2.10

Let P be the standard parabolic subgroup of $\operatorname {U}(V)$ with Levi component $\operatorname {GL}_1\times \operatorname {U}(V_0)$ . For $i\geq 0$ , by the (derived version of) Frobenius reciprocity, we have

$$ \begin{align*} \operatorname{Ext}^i_{\operatorname{U}(V)}(\Omega,\mu\nu^{x_n}\rtimes\pi_0) = \operatorname{Ext}^i_{\operatorname{GL}_1\times \operatorname{U}(V_0)}(R_P\Omega,\mu\nu^{x_n}\boxtimes\pi_0), \end{align*} $$

where $R_P$ is the normalized Jacquet module along P. To compute the right-hand side of above, one can appeal to the Kudla’s filtration. There is a two-step filtration on $R_P\Omega $ :

$$\begin{align*}R_P\Omega=R^0 \supset R^1 \supset R^2=0, \end{align*}$$

whose successive quotient $J^a=R^a/R^{a+1}$ can be described as follows:

$$\begin{align*}J^0=\chi_W\nu^{\frac{m-n+1}{2}}\boxtimes \Omega_0, \end{align*}$$

and

$$\begin{align*}J^1=\operatorname{Ind}_{\operatorname{GL}_1\times\operatorname{U}(V_0)\times Q}^{\operatorname{GL}_1\times\operatorname{U}(V_0)\times \operatorname{U}(W)}\left(\mathcal{S}(E^\times)\boxtimes \Omega_{00}\right). \end{align*}$$

Here:

• • $\Omega _0$ is the Weil representation associated to $\operatorname {U}(V_0)\times \operatorname {U}(W)$ ;

• • Q is a maximal parabolic subgroup of $\operatorname {U}(W)$ stabilizing an isotropic line of W; the Levi subgroup of Q is isomorphic to $\operatorname {GL}_1\times \operatorname {U}(W_0)$ ;

• • $\mathcal {S}(E^\times )$ is the space of Schwartz functions on $E^\times $ , equipped with the natural action of two copies of $\operatorname {GL}_1$ (twisted by the splitting characters);

• • $\Omega _{00}$ is the Weil representation associated to $\operatorname {U}(V_0)\times \operatorname {U}(W_0)$ .

Applying the functor $\operatorname {Hom}_{\operatorname {GL}_1\times \operatorname {U}(V_0)}(\cdot ,\mu \nu ^{x_n}\boxtimes \pi _0)$ to the short exact sequence

$$\begin{align*}0\longrightarrow J^1 \longrightarrow R_P\Omega \longrightarrow J^0 \longrightarrow 0, \end{align*}$$

we get a long exact sequence

$$ \begin{align*} \cdots\longrightarrow \operatorname{Ext}^i(J^0,\mu\nu^{x_n}\boxtimes\pi_0)_{sm} \longrightarrow \operatorname{Ext}^i(R_P\Omega &,\mu\nu^{x_n}\boxtimes\pi_0)_{sm}\\ \longrightarrow \operatorname{Ext}^i(J^1 &,\mu\nu^{x_n}\boxtimes\pi_0)_{sm} \longrightarrow \operatorname{Ext}^{i+1}(J^0,\mu\nu^{x_n}\boxtimes\pi_0)_{sm} \longrightarrow\cdots \end{align*} $$

Here, for the exactness of taking $\operatorname {U}(W)$ -smooth vectors, one may refer to [Reference Adams, Prasad and SavinAPS17, Lem. 5.14, Lem. 7.4]. Since by our assumptions ${x_n} \ne \frac {m-n+1}{2}$ , we know that

$$\begin{align*}\operatorname{Ext}^i(J^0,\mu\nu^{x_n}\boxtimes\pi_0)=0 \end{align*}$$

for all $i\geq 0$ . This implies that

$$ \begin{align*} \operatorname{Ext}^i(R_P\Omega,\mu\nu^{x_n}\boxtimes\pi_0)_{sm} \simeq & \operatorname{Ext}^i(J^1,\mu\nu^{x_n}\boxtimes\pi_0)_{sm}\\ =&\left(\mu'\right)^c\nu^{x_n}\rtimes \operatorname{Ext}^i(\Omega_{00},\pi_0)_{sm}, \end{align*} $$

where $\mu '=\mu \chi _W^{-1}\chi _V$ . Here, in the last equality, we have made use of (an $\operatorname {Ext}$ -version of) [Reference Adams, Prasad and SavinAPS17, Lem. 2.6] and the Künneth formula [Reference Adams, Prasad and SavinAPS17, Lem. 3.3]. In particular, we get

$$\begin{align*}\operatorname{Hom}_{\operatorname{U}(V)}(\Omega,\mu\nu^{x_n}\rtimes\pi_0)_{sm}\simeq\left(\mu'\right)^c\nu^{x_n}\rtimes \Theta(\pi_0)^\vee \end{align*}$$

and

(♠) $$ \begin{align} \operatorname{Ext}^i_{\operatorname{U}(V)}(\Omega,\mu\nu^{x_n}\rtimes\pi_0)_{sm} = 0 \quad \textit{for }i>0 \end{align} $$

by our induction hypothesis. Similarly, we also have $-{x_n} \ne \frac {m-n+1}{2}$ . The same argument gives that

(♣) $$ \begin{align} \operatorname{Ext}^i_{\operatorname{U}(V)}(\Omega,\mu\nu^{-x_n}\rtimes\pi_0)_{sm} = 0 \quad \textit{for }i>0. \end{align} $$

Now, note that Lemma 2.4 asserts that $\mu \nu ^{x_n}\rtimes \pi _0$ is of length $2$ . Let $\pi '$ be the unique irreducible quotient of $\mu \nu ^{x_n}\rtimes \pi _0$ , the sequence

$$\begin{align*}0\longrightarrow \pi \longrightarrow \mu\nu^{x_n}\rtimes\pi_0 \longrightarrow \pi' \longrightarrow 0 \end{align*}$$

is exact. Applying the functor $\operatorname {Hom}_{\operatorname {U}(V)}(\Omega ,\cdot )$ to this short exact sequence and taking $\operatorname {U}(W)$ -smooth vectors, we get

(♡) $$ \begin{align} 0\rightarrow \operatorname{Hom}(\Omega,\pi)_{sm}\rightarrow \operatorname{Hom}(\Omega,\mu\nu^{x_n}\rtimes\pi_0)_{sm}&\rightarrow \operatorname{Hom}(\Omega,\pi')_{sm}\rightarrow \operatorname{Ext}^1(\Omega,\pi)_{sm} \rightarrow \nonumber\\ \cdots\rightarrow\operatorname{Ext}^i(\Omega,\mu\nu^{x_n}\rtimes\pi_0)_{sm}\rightarrow\operatorname{Ext}^i(\Omega,\pi')_{sm}&\rightarrow\operatorname{Ext}^{i+1}(\Omega,\pi)_{sm}\rightarrow\operatorname{Ext}^{i+1}(\Omega,\mu\nu^{x_n}\rtimes\pi_0)_{sm}\rightarrow\cdots \end{align} $$

It follows from Equation (♠) that

$$\begin{align*}\operatorname{Ext}^i(\Omega,\pi')_{sm}\simeq\operatorname{Ext}^{i+1}(\Omega,\pi)_{sm} \quad \textit{for }i>0. \end{align*}$$

Dually, apply both the contragredient and MVW-involution (see [Reference Mœglin, Vignéras and WaldspurgerMVW, Chap. 4.II.1]) to ${\mu \nu ^{x_n}\rtimes \pi _0}$ , we get a dualized short exact sequence

$$\begin{align*}0\longrightarrow \pi' \longrightarrow \mu\nu^{-x_n}\rtimes\pi_0 \longrightarrow \pi \longrightarrow 0. \end{align*}$$

Similar to the argument above, this short exact sequence leads to a long exact sequence

(♢) $$ \begin{align} 0\rightarrow \operatorname{Hom}(\Omega,\pi')_{sm}\rightarrow \operatorname{Hom}(\Omega,\mu\nu^{-x_n}\rtimes\pi_0)_{sm}&\rightarrow \operatorname{Hom}(\Omega,\pi)_{sm}\rightarrow \operatorname{Ext}^1(\Omega,\pi')_{sm} \rightarrow \nonumber\\ \cdots\rightarrow\operatorname{Ext}^i(\Omega,\mu\nu^{-x_n}\rtimes\pi_0)_{sm}\rightarrow\operatorname{Ext}^i(\Omega,\pi)_{sm}&\rightarrow\operatorname{Ext}^{i+1}(\Omega,\pi')_{sm}\rightarrow\operatorname{Ext}^{i+1}(\Omega,\mu\nu^{-x_n}\rtimes\pi_0)_{sm}\rightarrow\cdots \end{align} $$

which combining with Equation (♣) similarly implies that

$$\begin{align*}\operatorname{Ext}^i(\Omega,\pi)_{sm}\simeq\operatorname{Ext}^{i+1}(\Omega,\pi')_{sm} \quad \textit{for }i>0. \end{align*}$$

Playing ‘Ping-Pong’, one can see that $\operatorname {Ext}^i(\Omega ,\pi )$ is periodic:

$$\begin{align*}\operatorname{Ext}^i(\Omega,\pi)_{sm} \simeq \operatorname{Ext}^{i+2}(\Omega,\pi)_{sm} \quad \textit{for }i>0. \end{align*}$$

Since the higher extensions vanish when the degree is sufficiently large [Reference BernsteinBer92, Pg. 98, Sect. 4.2], these groups $\operatorname {Ext}^i(\Omega ,\pi )_{sm}$ must vanish for all $i>0$ with no other choice. The same reason also gives the vanishing of higher extensions of $\pi '$ .

Suppose that $\Theta (\pi )\ne 0$ . It remains to show that $\Theta (\pi )$ is irreducible. Thanks to the vanishing of higher extensions, we deduce from the long exact sequence (♡) that

$$\begin{align*}0\longrightarrow \Theta(\pi)^\vee \longrightarrow \left(\mu'\right)^c\nu^{x_n}\rtimes \Theta(\pi_0)^\vee \longrightarrow \Theta(\pi')^\vee \longrightarrow 0 \end{align*}$$

is exact. In particular, $\Theta (\pi _0)$ must be nonzero, hence irreducible by our induction hypothesis. It then follows from Theorem 2.1 that $\Theta (\pi _0)$ lies in $\Pi _{\theta (\Psi _0)}(\operatorname {U}(W_0))$ , where

$$\begin{align*}\theta(\Psi_0)=\delta\chi_W^{-1}\chi_V+\mu\chi_W^{-1}\chi_V\boxtimes S_{n-3}+\chi_V\boxtimes S_{m-n}. \end{align*}$$

Now, we claim that the induced representation $\left (\mu '\right )^c\nu ^{x_n}\rtimes \Theta (\pi _0)^\vee $ is of length $2$ , and the two subquotients are nonisomorphic. Indeed, if $m-n=1$ and $\delta =\chi _W$ , one can easily check this by hand. Otherwise, note that:

• • $\theta (\Psi _0)$ is of good parity, multiplicity free and trivial on the Weil–Deligne $\operatorname {SL}_2(\mathbb C)$ ;

• • $\mu '\boxtimes S_{-2x_n-1}\subset \theta (\Psi _0)$ but $\mu '\boxtimes S_{-2x_n+1}\not \subset \theta (\Psi _0)$ .

In short, we are again in a situation such that we can appeal to Lemma 2.4, from which our claim follows. Therefore, it suffices to check that $\Theta (\pi ') \ne 0$ . We shall argue by contradiction to show this. Suppose on the contrary that $\Theta (\pi ') = 0$ . Then on the one hand, we have

$$\begin{align*}\Theta(\pi)^\vee \simeq \left(\mu'\right)^c\nu^{x_n}\rtimes \Theta(\pi_0)^\vee, \end{align*}$$

which implies that $\left (\mu '\right )^c\nu ^{x_n}\rtimes \Theta (\pi _0)^\vee $ has socle $\theta (\pi )^\vee $ . On the other hand, we also deduce from the long exact sequence (♢) that

$$\begin{align*}0\longrightarrow \Theta(\pi')^\vee \longrightarrow \left(\mu'\right)^c\nu^{-x_n}\rtimes \Theta(\pi_0)^\vee \longrightarrow \Theta(\pi)^\vee \longrightarrow 0 \end{align*}$$

is exact. Since we had assumed that $\Theta (\pi ') = 0$ , this exact sequence implies that

$$\begin{align*}\left(\mu'\right)^c\nu^{-x_n}\rtimes \Theta(\pi_0)^\vee \simeq \Theta(\pi)^\vee. \end{align*}$$

Applying both the contragredient and the MVW-involution, we get

$$\begin{align*}\Theta(\pi)^{MVW} \simeq \left(\mu'\right)^c\nu^{x_n}\rtimes \Theta(\pi_0)^\vee, \end{align*}$$

which implies that $\left (\mu '\right )^c\nu ^{x_n}\rtimes \Theta (\pi _0)^\vee $ also has cosocle $\theta (\pi )^\vee $ . This contradicts our claim. Thus, $\Theta (\pi ') \ne 0$ as desired.

Now, we can prove our goal.

3.1. Two seesaw diagrams: uniqueness

If there is an irreducible tempered representation $\Pi $ in the L-packet $\Pi _M$ corresponding to $\eta \in \operatorname {Irr} A_M$ such that

$$\begin{align*}m_V(\Pi,\mu)\ne 0, \end{align*}$$

we would like to lift $\Pi $ to some unitary group of n-variables to obtain some information. Let $\{a_i\}_{i=1}^r$ be a canonical basis of $A_M$ , where each $a_i$ corresponds to some irreducible subrepresentation $M_i$ of M (so $a_1$ corresponds to $M_1$ ). We set $\epsilon =\eta (a_1)$ and W the unique n-dimensional Hermitian space over L of sign $\epsilon $ . Let $\left (\mathcal {X}_V,\mathcal {X}_W\right )$ be a pair of characters of $L^\times $ such that

$$\begin{align*}\mathcal{X}_V~\big|_{K^\times}=\omega_{L/K}^{n+1} \quad \textit{and} \quad \mathcal{X}_W=M_{1}. \end{align*}$$

Then one can consider the theta correspondence between $\operatorname {U}(V_K)\times \operatorname {U}(W)$ with respect to the splitting character $\left (\mathcal {X}_V,\mathcal {X}_W\right )$ . By [Reference Gan and IchinoGI16, Sect. 4.6(P2)], one knows that there is an irreducible tempered representation $\Sigma $ of $\operatorname {U}(W)$ such that

$$\begin{align*}\Pi=\Theta(\Sigma) \end{align*}$$

is the big theta lift of $\Sigma $ . Indeed, one knows that $\Sigma $ has the L-parameter

$$\begin{align*}\theta(M)=M_0\cdot\mathcal{X}_W^{-1}\mathcal{X}_V \end{align*}$$

and corresponds to the character $\theta (\eta ) = \eta ~\big |_{A_{\theta (M)}}$ . Consider the following seesaw diagram:

(♮.1)

where:

• • $\mathcal {R} W$ is the restriction of scalar of W to E;

• • the theta correspondence between $\operatorname {U}(V_K)\times \operatorname {U}(W)$ is with respect to some splitting characters $(\mathcal {X}_V,\mathcal {X}_W)$ ;

• • the theta correspondence between $\operatorname {U}(V)\times \operatorname {U}\left (\mathcal {R} W\right )$ is with respect to some splitting characters $(\chi _V,\chi _W)$ ;

• • to make use of this seesaw diagram, we choose these splitting characters so that:

  $$\begin{align*}\mathcal{X}_V=\chi_V\circ\operatorname{Nm}_{L/E} \quad \textit{and} \quad \chi_W=\mathcal{X}_W~\big|_{E^\times}; \end{align*}$$

• • $\chi $ is the L-parameter of the central character of the restriction of $\Pi $ to $\operatorname {U}(V)$ , that is,

  $$\begin{align*}\chi=\det(M)~\big|_{E^\times}. \end{align*}$$

Then by the seesaw identity, we get

(✠.1) $$ \begin{align} m_V\left(\Pi,\mu\right) = \dim \operatorname{Hom}_{\operatorname{U}(W)} \left(\Lambda, \Sigma\right). \end{align} $$

In particular, $\Lambda $ is nonzero. By Lemma 2.6, Theorem 2.1 and Theorem 2.7, we know that:

• • $\omega _{V,\mu }[\chi ]$ lies in the A-packet $\Pi _{\Psi _{M,\mu }}(\operatorname {U}(V))$ , where

  $$\begin{align*}\Psi_{M,\mu}=\chi\cdot\mu^{-n}+\mu\boxtimes S_n; \end{align*}$$

• • $\Lambda $ is an irreducible unitary representation lies in the A-packet $\Pi _{\theta (\Psi _{M,\mu })}\left (\operatorname {U}\left (\mathcal {R} W\right )\right )$ , where

  $$ \begin{align*} \theta\left(\Psi_{M,\mu}\right) &= \Psi_{M,\mu}\cdot\chi_W^{-1}\chi_V+\chi_V\boxtimes S_{n-1} \\ &= \chi\cdot\mu^{-n}\cdot\chi_W^{-1}\chi_V + \chi_V\boxtimes S_{n-1} + \mu\cdot\chi_W^{-1}\chi_V\boxtimes S_n. \end{align*} $$

To compute the right-hand side of equality (✠.1), we shall use another seesaw diagram:

(♮.2)

where:

• • $V^\flat $ is an n-dimensional skew-Hermitian space over E which will be suitably chosen later, and $V^\flat _K$ is its scalar extension to L;

• • the theta correspondence between $\operatorname {U}(V^\flat )\times \operatorname {U}\left (\mathcal {R} W\right )$ is with respect to some splitting characters $(\chi _{V^\flat },\chi ^{\prime }_W)$ ;

• • the theta correspondence between $\operatorname {U}(V^\flat _K)\times \operatorname {U}(W)$ is with respect to some splitting characters $(\mathcal {X}_{V^\flat },\mathcal {X}^{\prime }_W)$ ;

• • to make use of this seesaw diagram, we choose these splitting characters so that:

  $$\begin{align*}\mathcal{X}_{V^\flat}=\chi_{V^\flat}\circ\operatorname{Nm}_{L/E} \quad \textit{and} \quad \chi^{\prime}_W=\mathcal{X}^{\prime}_W~\big|_{E^\times}; \end{align*}$$

• • $\omega $ is some irreducible unitary representation of $\operatorname {U}(V^\flat )$ which will also be suitably chosen later.

We would like to choose these data appropriately such that $\omega $ is the theta lift of certain character of $\operatorname {U}_1$ , and $\Lambda =\Theta (\omega )$ . To make this possible, we need to pick up these splitting characters very carefully. Let

$$\begin{align*}\chi_{V^\flat}=\mu\cdot\chi_W^{-1}\chi_V \quad \textit{and} \quad \mathcal{X}^{\prime}_W= M_1^{-1} \cdot \Upsilon, \end{align*}$$

where $\Upsilon $ is a conjugate orthogonal character of $L^\times $ so that

$$\begin{align*}\Upsilon ~\Big|_{E^\times} = \mu^2. \end{align*}$$

It is not hard to see that such $\Upsilon $ exists. Then again by Theorem 2.1 one can see that $\omega $ (if exists) lies in the A-packet $\Pi _{\Psi ^\flat }(\operatorname {U}(V^\flat ))$ , where

$$\begin{align*}\Psi^\flat= \chi^\flat\cdot\mu^{-n+1} + \mu\boxtimes S_{n-1}, \quad \textit{with } \chi^\flat=\det(M/M_1)~\big|_{E^\times}. \end{align*}$$

Indeed, we have:

Proposition 3.2. Let $V^\flat $ be the n-dimensional skew-Hermitian space of sign

(†) $$ \begin{align} \epsilon\left(V^\flat\right)=\begin{cases} +1 \quad & \textit{if }n=2 \textit{ and }\chi^\flat=\mu^2, \\ \epsilon(V)\cdot\epsilon\left(\mathcal{R} W\right)\cdot\epsilon\left(\frac{1}{2},\operatorname{As}^+_{L/E}(M_{1})\cdot\mu^{-1}, \psi_{E,\delta}\right)\cdot\omega_{E/F}(-1)^n \quad & \textit{otherwise}, \end{cases} \end{align} $$

and

$$\begin{align*}\omega=\omega_{V^\flat,\mu}[\chi^\flat]. \end{align*}$$

Then $\Lambda $ is the (big) theta lift of $\omega $ to $\operatorname {U}(\mathcal {R} W)$ , that is, $\Lambda =\Theta (\omega )$ .

Proof. We first check the special case that $n=2$ and $\chi ^\flat =\mu ^2$ . So

$$\begin{align*}\theta\left(\Psi_{M,\mu}\right)=\chi_V+\mu\cdot\chi_W^{-1}\chi_V\boxtimes S_2+\left(\chi_V^c\right)^\vee \end{align*}$$

and

$$\begin{align*}\Lambda\subset \chi_V \rtimes (\chi_0\circ\det), \end{align*}$$

where $\chi _0$ is the character of $E^1$ whose base change to $E^\times $ is $\mu \cdot \chi _W^{-1}\chi _V$ . By the induction principle, one knows that the theta correspondence between $\operatorname {U}(V^\flat )\times \operatorname {U}(\mathcal {R} W)$ defines a bijection

$$\begin{align*}\theta:\Pi_\phi(\operatorname{U}(V^\flat))\longrightarrow\Pi_{\theta\left(\Psi_{M,\mu}\right)}(\operatorname{U}(\mathcal{R} W)), \end{align*}$$

where $\phi =\mu +\mu $ is an L-parameter of $\operatorname {U}(V^\flat )$ . Hence, $\Lambda $ is the (big) theta lift of some $\omega \in \Pi _\phi (\operatorname {U}(V^\flat ))$ . To check that $\omega =\omega _{V^\flat ,\mu }[\chi ^\flat ]$ , one can compute the character $\eta ^\flat \in \operatorname {Irr} A_\phi $ associated to $\omega $ . Recall that $\Lambda $ is also the theta lift of $\omega _{V,\mu }[\chi ]$ . If we denote by $\eta \in \operatorname {Irr} A_{\Psi _{M,\mu }}$ and $\theta (\eta )\in \operatorname {Irr} A_{\theta (\Psi _{M,\mu })}$ the character associated to $\omega _{V,\mu }[\chi ]$ and $\Lambda $ , respectively, then by Lemma 2.6 and Theorem 2.3, we have

$$ \begin{align*} \theta(\eta)(a)&=\eta(a)=\epsilon\left(\frac{1}{2},\chi_W\mu^{-1},\psi_{E,\delta}\right). \end{align*} $$

Here, $a\in A_{\theta (\Psi _{M,\mu })}$ is the basis element corresponding to $\chi _V$ , and we regard $A_\phi $ and $A_{\Psi _{M,\mu }}$ as subgroups of $A_{\theta (\Psi _{M,\mu })}$ . Apply Theorem 2.3 again, we get

$$ \begin{align*} \eta^\flat(a)&=\theta(\eta)(a)\cdot\epsilon\left(\frac{1}{2},\chi_V\cdot\chi_{V^\flat}^{-1},\psi_{E,\delta}\right)=1. \end{align*} $$

This implies that $\omega =\omega _{V^\flat ,\mu }[\chi ^\flat ]$ .

Now, excluding the special case above, we prove the general case. It would be convenient to consider the cases of odd and even n separately. In the following, we check the case of odd n in full details.

Let $e_1$ , $e_{n-1}$ and $e_{n}$ be the basis elements of $A_{\theta \left (\Psi _{M,\mu }\right )}$ corresponding to $\chi \cdot \mu ^{-n}\cdot \chi _W^{-1}\chi _V$ , $\chi _V\boxtimes S_{n-1}$ and $\mu \cdot \chi _W^{-1}\chi _V\boxtimes S_n$ , respectively. Then:

• • $A_{\Psi _{M,\mu }}$ can be regarded as the subgroup of $A_{\theta \left (\Psi _{M,\mu }\right )}$ generated by $e_1$ and $e_n$ ;

• • $A_{\Psi ^\flat }$ can be regarded as the subgroup of $A_{\theta \left (\Psi _{M,\mu }\right )}$ generated by $e_1$ and $e_{n-1}$ .

Recall that $\omega _{V,\mu }[\chi ]\in \Pi _{\Psi _{M,\mu }}(\operatorname {U}(V))$ corresponds to the character $\nu _{n+1}$ of $A_{\Psi _{M,\mu }}$ such that

$$\begin{align*}\nu_{n+1}: (e_1,e_n) \longmapsto \left(1,\epsilon(V)\right). \end{align*}$$

Then by Theorem 2.3, $\Lambda =\Theta \left (\omega _{V,\mu }[\chi ]\right )$ corresponds to the character $\nu $ of $A_{\theta \left (\Psi _{M,\mu }\right )}$ such that

$$\begin{align*}\nu: (e_1,e_n)\longmapsto \left(\epsilon\left(\frac{1}{2},\chi\cdot\mu^{-n}\cdot\chi_W^{-1},\psi_{E,\delta}\right),\, \epsilon(V)\cdot\epsilon\left(\frac{1}{2},\mu\cdot\chi_W^{-1}\boxtimes S_n, \psi_{E,\delta}\right)\right). \end{align*}$$

The evaluation of $\nu $ at $e_{n-1}$ can be determined by its evaluation at $(e_1,e_n)$ and the sign of $\mathcal {R} W$ . To be more precise, $\nu $ takes $e_{n-1}$ to

$$ \begin{align*} &\epsilon(V)\cdot\epsilon\left(\mathcal{R} W\right)\cdot\epsilon\left(\frac{1}{2},\chi\cdot\mu^{-n}\cdot\chi_W^{-1},\psi_{E,\delta}\right)\cdot\epsilon\left(\frac{1}{2},\mu\cdot\chi_W^{-1}\boxtimes S_n, \psi_{E,\delta}\right) \\ =\ &\epsilon(V)\cdot\epsilon\left(\mathcal{R} W\right)\cdot\epsilon\left(\frac{1}{2},\chi^\flat\cdot\mu^{-n},\psi_{E,\delta}\right)\cdot\epsilon\left(\frac{1}{2},\operatorname{As}^+_{L/E}(M_{1})\cdot\mu^{-1}, \psi_{E,\delta}\right)\cdot\omega_{E/F}(-1). \end{align*} $$

Hence, if we let $V^\flat $ be the n-dimensional skew-Hermitian space as in Equation (†), then again by Theorem 2.3, one can check that:

• • $\omega _{V^\flat ,\mu }[\chi ^\flat ]\in \Pi _{\Psi ^\flat }(\operatorname {U}(V^\flat ))$ corresponding to the character $\nu _{n}$ of $A_{\Psi ^\flat }$ such that

  $$\begin{align*}\nu_{n}: (e_1, e_{n-1})\longmapsto \left(\epsilon\left(\frac{1}{2},\chi^\flat\cdot\mu^{-n},\psi_{E,\delta}\right),\, \epsilon\left(V^\flat\right)\cdot\epsilon\left(\frac{1}{2},\chi^\flat\cdot\mu^{-n},\psi_{E,\delta}\right)\right); \end{align*}$$

• • the theta lift of $\omega _{V^\flat ,\mu }[\chi ^\flat ]$ to $\operatorname {U}(\mathcal {R} W)$ is nonzero and exactly equal to $\Lambda $ .

These complete the proof of the case when n is odd.

Similarly, when n is even, $\omega _{V,\mu }[\chi ]\in \Pi _{\Psi _{M,\mu }}(\operatorname {U}(V))$ corresponds to

$$\begin{align*}\nu_{n+1}: (e_1,e_n) \longmapsto \left(\epsilon\left(\frac{1}{2},\chi\mu^{-n-1},\psi_{E,\delta}\right),\epsilon(V)\epsilon\left(\frac{1}{2},\chi\mu^{-n-1},\psi_{E,\delta}\right)\right). \end{align*}$$

By Theorem 2.3, $\Lambda $ corresponds to $\nu \in \operatorname {Irr} A_{\theta \left (\Psi _{M,\mu }\right )}$ such that $\nu ~\big |_{A_{\Psi _{M,\mu }}}=\nu _{n+1}$ , so

$$\begin{align*}\nu(e_{n-1})= \epsilon(V)\cdot\epsilon\left(\mathcal{R} W\right). \end{align*}$$

Then again one can appeal to Theorem 2.3 to show that the theta lift of $\omega _{V^\flat ,\mu }[\chi ^\flat ]$ is exactly $\Lambda $ .

With this proposition in hand, we get

(✠.2) $$ \begin{align} m_V(\Pi,\mu) = \dim\operatorname{Hom}_{\operatorname{U}(W)}(\Lambda,\Sigma) = m_{V^\flat}(\Pi^\flat,\mu) \end{align} $$

is nonzero. In particular, $\Pi ^\flat $ is nonzero. By [Reference Gan and IchinoGI16, Sect. 4.4(P1)], we know that:

• • The sign of the Hermitian space W is given by

  $$\begin{align*}\epsilon(W)=\epsilon\left(\frac{1}{2}, M_0\cdot {^{\tau}M_{1}}\cdot\mu^{-1}\circ\operatorname{Nm}_{L/E}, \,\psi_{L,\delta}\right), \end{align*}$$
  where $\psi _{L,\delta }=\psi _F\left (\operatorname {Tr}_{L/F}(\delta \cdot ~)\right )$ .

• • $\Pi ^\flat $ is an irreducible tempered representation has L-parameter $M^\flat =M^{\prime }_0$ and corresponds to $\eta ^\flat $ , where

  $$\begin{align*}M^{\prime}_0=M_0\cdot {^{\tau}M_{1}}\cdot M_{1}^{-1}\cdot\Upsilon\cdot\mu^{-1}\circ\operatorname{Nm}_{L/E}, \end{align*}$$
  and
  $$\begin{align*}\eta^\flat(a_i)/\eta(a_i) = \epsilon\left(\frac{1}{2}, M_i\cdot{^{\tau}M_{1}}\cdot\mu^{-1}\circ\operatorname{Nm}_{L/E}, \,\psi_{L,\delta}\right) \end{align*}$$
  for all $i\geq 2$ .

Also note that

$$\begin{align*}\epsilon\left(\mathcal{R} W\right)=\epsilon(W)\cdot\omega_{K/F}\left(\delta^2\right)^n\cdot\omega_{E/F}(-1)^n. \end{align*}$$

Substitute these into Equation (†), we get

(††) $$ \begin{align}\epsilon\left(V^\flat\right)=\epsilon(V)&\cdot\epsilon \left(\frac{1}{2}, \operatorname{Ind}_L^E\left(^{\tau}M_{1} \otimes (M/M_{1})\right)\cdot\mu^{-1}, \psi_{E,\delta}\right)\nonumber\\&\cdot\epsilon\left(\frac{1}{2},\operatorname{As}^+_{L/E}(M_{1})\cdot\mu^{-1}, \psi_{E,\delta}\right)\cdot\omega_{K/F}\left(\delta^2\right)^n. \end{align} $$

Now, if we assume that Conjecture 1.1 holds for the L-parameter $M^\flat $ , then it follows that:

1. (1) The multiplicity $m_{V}(\Pi ,\mu )=1$ .

2. (2) V is the unique $(n+1)$ -dimensional Hermitian space over E predicted by the formula in Conjecture 1.1(3). Indeed, note that for any semisimple representation N and any character $\mathcal {X}$ of $WD_L$ , we have

   $$\begin{align*}\operatorname{As}^+(N\cdot\mathcal{X})=\operatorname{As}^+(N)\cdot\left(\mathcal{X}~\Big|_{E^\times}\right). \end{align*}$$
   Combining this with Conjecture 1.1(3) for $M^\flat $ , we know that
   $$ \begin{align*} \epsilon\left(V^\flat\right)&=\epsilon\left(\frac{1}{2}, \operatorname{As}^+_{L/E}\left(M^\flat\right)\otimes\mu^{-1}, \psi_{E,\delta}\right)\cdot\omega_{K/F}\left(\delta^2\right)^{n(n-1)/2}\\ &=\epsilon\left(\frac{1}{2}, \operatorname{As}^+_{L/E}\left(M_0\right)\otimes\mu^{-1}, \psi_{E,\delta}\right)\cdot\omega_{K/F}\left(\delta^2\right)^{n(n-1)/2}. \end{align*} $$
   Then applying the equality (††), we get
   $$ \begin{align*}\epsilon(V)&=\epsilon\left(V^\flat\right)\cdot\epsilon \left(\frac{1}{2}, \operatorname{Ind}_L^E\left(^{\tau}M_{1} \otimes (M/M_{1})\right)\cdot\mu^{-1}, \psi_{E,\delta}\right)\\&\qquad\qquad\cdot\epsilon\left(\frac{1}{2},\operatorname{As}^+_{L/E}(M_{1})\cdot\mu^{-1}, \psi_{E,\delta}\right)\cdot\omega_{K/F}\left(\delta^2\right)^n\\ &=\epsilon\left(\frac{1}{2}, \operatorname{As}^+_{L/E}(M)\otimes\mu^{-1}, \psi_{E,\delta}\right)\cdot\omega_{K/F}\left(\delta^2\right)^{n(n+1)/2}. \end{align*} $$

3. (3) $\Pi $ is the unique member in $\Pi _{M}$ predicted by the formula in Conjecture 1.1(4). Similar to (2), it follows from Conjecture 1.1(4) that

   $$ \begin{align*}\eta^\flat(a_i) &= \epsilon \left(\frac{1}{2}, \operatorname{Ind}_L^E\left(^{\tau}M_i \otimes (M_0/M_i)\right)\cdot\mu^{-1}, \psi_{E,\delta}\right)\\ &= \epsilon\left(\frac{1}{2}, {^{\tau}M_i} \otimes (M_0/M_i)\cdot\mu^{-1}\circ\operatorname{Nm}_{L/E}, \psi_{L,\delta}\right) \end{align*} $$
   for all $i\geq 2$ . Hence,
   $$ \begin{align*} \eta(a_i)&=\eta^\flat(a_i)\cdot\epsilon\left(\frac{1}{2}, M_i\cdot{^{\tau}M_{1}}\cdot\mu^{-1}\circ\operatorname{Nm}_{L/E}, \,\psi_{L,\delta}\right)\\ &=\epsilon\left(\frac{1}{2}, {^{\tau}M_i} \otimes (M/M_i)\cdot\mu^{-1}\circ\operatorname{Nm}_{L/E}, \,\psi_{L,\delta}\right)\\ &=\epsilon \left(\frac{1}{2}, \operatorname{Ind}_L^E\left(^{\tau}M_i \otimes (M/M_i)\right)\cdot\mu^{-1}, \psi_{E,\delta}\right) \end{align*} $$
   for all $i\geq 2$ . On the other hand, recall that $\eta (a_1)=\epsilon (W)$ . This implies the desired equality
   $$ \begin{align*}\eta(a_1)&=\epsilon\left(\frac{1}{2}, M_0\cdot {^{\tau}M_{1}}\cdot\mu^{-1}\circ\operatorname{Nm}_{L/E}, \,\psi_{L,\delta}\right) \\ &=\epsilon \left(\frac{1}{2}, \operatorname{Ind}_L^E\left(^{\tau}M_1 \otimes (M/M_1)\right)\cdot\mu^{-1}, \psi_{E,\delta}\right). \end{align*} $$

The computation above shows that there is at most one $\Pi $ in the L-packet $\Pi _M$ such that $m_V(\Pi ,\mu )\ne 0$ .

3.2. Reversing two seesaw diagrams: existence

Conversely, still under the assumption that Conjecture 1.1 holds for the L-parameter $M^\flat $ , we can produce an irreducible tempered representation $\Pi '\in \Pi _M$ such that

$$\begin{align*}m_{V'}(\Pi',\mu)\ne 0 \end{align*}$$

for some $(n+1)$ -dimensional skew-Hermitian space $V'$ , from the unique irreducible tempered representation $\Pi ^\flat \in \Pi _{M^\flat }$ such that

$$\begin{align*}m_{V^\flat}(\Pi^\flat,\mu) \ne 0. \end{align*}$$

We do it by applying the two seesaw diagrams reversely as follows. First, consider an analog of the seesaw diagram (♮.2) (using the same splitting characters):

where $W'$ is the unique n-dimensional Hermitian space over L chosen by the theta dichotomy [Reference Gan and IchinoGI16, Sect. 4.4(P1)]; that is, the theta lift $\Sigma '$ of $\Pi ^\flat $ to $\operatorname {U}(W')$ is nonzero. Symmetrically, $\Pi ^\flat =\Theta \left (\Sigma '\right )$ is the big theta lift of $\Sigma '$ . By the seesaw identity, we have

(✠.3) $$ \begin{align} m_{V^\flat}\left(\Pi^\flat,\mu\right)=\dim\operatorname{Hom}_{\operatorname{U}(W')}\left(\Lambda',\Sigma'\right). \end{align} $$

In particular, $\Lambda '$ is nonzero. It then follows from Theorem 2.1 and Theorem 2.3 that $\Lambda '$ is an irreducible unitary representation lies in the A-packet $\Pi _{\theta (\Psi _{M,\mu })}\left (\operatorname {U}\left (\mathcal {R} W'\right )\right )$ , where

$$ \begin{align*} \theta\left(\Psi_{M,\mu}\right) &= \chi\cdot\mu^{-n}\cdot\chi_W^{-1}\chi_V + \chi_V\boxtimes S_{n-1} + \mu\cdot\chi_W^{-1}\chi_V\boxtimes S_n. \end{align*} $$

Next, we shall use an analog of the seesaw diagram (♮.1). The following is an analog of the key Proposition 3.2.

Proposition 3.3. Let $V'$ be the $(n+1)$ -dimensional skew-Hermitian space of sign

(‡) $$ \begin{align} \epsilon\left(V^\prime\right)= \epsilon\left(V^\flat\right)\cdot\epsilon\left(\mathcal{R} W'\right)\cdot\epsilon\left(\frac{1}{2},\operatorname{As}^+_{L/E}(M_{1})\cdot\mu^{-1}, \psi_{E,\delta}\right)\cdot\omega_{E/F}(-1)^n \end{align} $$

and

$$\begin{align*}\omega'=\omega_{V^\prime,\mu}[\chi]. \end{align*}$$

Then $\Lambda '$ is the (big) theta lift of $\omega '$ to $\operatorname {U}(\mathcal {R} W')$ , that is, $\Lambda '=\Theta (\omega ')$ . Here, we are using the same splitting characters as described in Equation (♮.1).

Proof. We first check the special case that $n=2$ and $\chi ^\flat =\mu ^2$ . So

$$\begin{align*}\Psi_{M,\mu}=\chi_W+\mu\boxtimes S_2 \end{align*}$$

and

$$\begin{align*}\theta\left(\Psi_{M,\mu}\right)=\chi_V+\mu\cdot\chi_W^{-1}\chi_V\boxtimes S_2+\left(\chi_V^c\right)^\vee. \end{align*}$$

Let $V"$ and $\mathcal {R} W"$ be the companion spaces of $V'$ and $\mathcal {R} W'$ respectively. Consider the following map given by the theta correspondence:

$$ \begin{align*} \theta_{4}:\operatorname{Irr} \operatorname{U}(V')\sqcup \operatorname{Irr} \operatorname{U}(V")\longrightarrow \operatorname{Irr} \operatorname{U}(\mathcal{R} W')\sqcup \operatorname{Irr} \operatorname{U}(\mathcal{R} W"), \end{align*} $$

where

$$ \begin{align*} \pi\mapsto\begin{cases} \theta_{\mathcal{R} W'}(\pi)\in\operatorname{Irr}\left(\operatorname{U}(\mathcal{R} W')\right) \quad &\textit{if } \theta_{\mathcal{R} W'}(\pi)\ne 0;\\ \theta_{\mathcal{R} W"}(\pi)\in\operatorname{Irr}\left(\operatorname{U}(\mathcal{R} W")\right) \quad &\textit{otherwise}. \end{cases} \end{align*} $$

This map is well defined by the theta dichotomy. Using Theorem 2.1, One can easily check by hand that this map restricts to a bijection

$$\begin{align*}\theta_4: \Pi_{\Psi_{M,\mu}} (\operatorname{U}(V'))\sqcup \Pi_{\Psi_{M,\mu}} (\operatorname{U}(V"))\longrightarrow \Pi_{\theta\left(\Psi_{M,\mu}\right)} (\operatorname{U}(\mathcal{R} W'))\sqcup \Pi_{\theta\left(\Psi_{M,\mu}\right)} (\operatorname{U}(\mathcal{R} W")). \end{align*}$$

Hence, $\Lambda '$ is the (big) theta lift of some

$$\begin{align*}\omega'\in \Pi_{\Psi_{M,\mu}} (\operatorname{U}(V'))\sqcup \Pi_{\Psi_{M,\mu}} (\operatorname{U}(V")). \end{align*}$$

To check that $\omega '=\omega _{V',\mu }[\chi ]$ , one can use Theorem 2.3 and Lemma 2.6 to compute the character $\eta '\in \operatorname {Irr} A_{\Psi _{M,\mu }}$ associated to $\omega '$ . We omit the details.

Excluding the special case above, the theta correspondence between $\operatorname {U}(V')\times \operatorname {U}(\mathcal {R} W')$ is in the situation of Theorem 2.1(2). It follows that the theta lift $\Theta (\omega _{V',\mu }[\chi ])$ to $\operatorname {U}(\mathcal {R} W')$ is nonvanishing. So the proof of the general case comes down to a computation of the labellings similar to the proof of Proposition 3.2. We shall not repeat the tedious computation here.

Now, we can consider the following seesaw diagram, with respect to the same splitting characters as described in Equation (♮.1):

Again by the seesaw identity, we have

(✠.4) $$ \begin{align} m_{V'}(\Pi',\mu) = \dim\operatorname{Hom}_{\operatorname{U}(W)}(\Lambda',\Sigma') = m_{V^\flat}(\Pi^\flat,\mu) \end{align} $$

is nonzero. In particular, $\Pi '$ is nonzero. By [Reference Gan and IchinoGI16, Sect. 4.6(P2)], we know that $\Pi '$ is an irreducible tempered representation of $\operatorname {U}(V^{\prime }_K)$ lies in the L-packet $\Pi _M$ . The construction above shows the existence of $\Pi '\in \Pi _M$ such that $m_{V'}(\Pi ',\mu )\ne 0$ .

3.3. Conclusion

In summary, we have shown that:

Proposition 3.4. Let $V_0$ be an n-dimensional Hermitian space over E and $M^{\prime }_0$ a tempered L-parameter for the unitary group $\operatorname {U}(V_{0,K})$ . Assume that Conjecture 1.1 holds for the L-parameter $M^{\prime }_0$ . Then it also holds for the L-parameter of the form

$$\begin{align*}M = M^{\prime}_0 \cdot\mathcal{X} + M_1, \end{align*}$$

where $\mathcal {X}$ is any conjugate symplectic character of $L^\times $ , and $M_1$ is any conjugate self-dual character of $L^\times $ of parity $(-1)^n$ .

Proof. As we have explicated above, given such an L-parameter M, one can construct an L-parameter $M^\flat $ of $\operatorname {U}(V_{0,K})$ . As long as Conjecture 1.1 holds for the L-parameter $M^\flat $ , it also holds for M. On the other hand, from the construction of $M^\flat $ , one can see that

$$\begin{align*}M^\flat = M^{\prime}_0\cdot\mathcal{Y} \end{align*}$$

for some conjugate orthogonal character $\mathcal {Y}$ of $L^\times $ . Thus, by Lemma 3.1, Conjecture 1.1 holds for $M^\flat $ .

Now, we can prove the main result of this paper.

The reader may notice the similarity of our setup with the paper [Reference XueXue23] of Hang Xue, in which he showed the Bessel case of the local GGP conjecture for unitary groups over $\mathbb {R}$ . There, he worked also with L-parameters M of the same form as those in Theorem 1.3. Indeed, we are partly inspired by his results to consider these M’s. However, the inductive argument in our proof is different from that in [Reference XueXue23] (not to mention that the setting of our result is different).

We end up this paper with a remark on the global conjecture [Reference Gan, Gross and PrasadGGP23, Conj. 11.1]. One can expect to prove the global conjecture for the near equivalence class

$$\begin{align*}M=M_1+\cdots+M_n \end{align*}$$

with each $M_i$ conjugate self-dual automorphic character of $\operatorname {GL}_1$ of parity $(-1)^{n-1}$ by using the same argument. Instead of the Adams’ conjecture used in this paper, one will need to show an analog of the Siegel–Weil formula in the global case so that one can compare the theta integrals of $\omega _{V,\mu }$ and $\omega _{V^\flat ,\mu }$ . More precisely, let $\Omega _{V}$ and $\Omega _{V^\flat }$ be the Weil representation associated to $\operatorname {U}(V)\times \operatorname {U}(\mathcal {R} W)$ and $\operatorname {U}(V^\flat )\times \operatorname {U}(\mathcal {R} W)$ , respectively, one needs to compare

$$\begin{align*}\int_{[\operatorname{U}(V)]}\theta_\varphi(g,h)f(g)\,dg \quad \textit{for }\varphi\in\Omega_V,\ f\in\omega_{V,\mu},\ g\in\operatorname{U}(V),\ h\in\operatorname{U}(\mathcal{R} W) \end{align*}$$

and

$$\begin{align*}\int_{[\operatorname{U}(V^\flat)]}\theta_{\varphi'}(g',h)f'(g')\,dg' \quad \textit{for }\varphi'\in\Omega_{V^\flat},\ f'\in\omega_{V^\flat,\mu},\ g'\in\operatorname{U}(V^\flat),\ h\in\operatorname{U}(\mathcal{R} W). \end{align*}$$

Unfortunately, these theta integrals diverge in general. So one has to properly regularize these theta integrals first. Once a global analog of Proposition 3.2 has been established, the remaining parts should go over smoothly.