Local theta correspondence of tempered representations and Langlands parameters

Abstract

In this paper, we give an explicit determination of the theta lifting for symplectic-orthogonal and unitary dual pairs over a nonarchimedean field F of characteristic 0. We determine when theta lifts of tempered representations are nonzero, and determine the theta lifts in terms of the local Langlands correspondence.

Appendices

Appendix A: Preparations for the local Langlands correspondence

In this appendix, we recall some basic results on standard gamma factors, Plancherel measures, and local factors associated to representations of Weil–Deligne groups.

1.1 A.1 Standard gamma factors

Fix a non-trivial additive character $\psi$ of F. For $\pi \in \mathrm {Irr}(G(W_n))$ and a character $\chi$ of $E^\times$, let $\gamma (s,\pi ,\chi ,\psi )$ be the standard $\gamma$-factor defined by Lapid–Rallis [27] using the doubling method. For its properties, see [9, 27] and [11, §10, §11]. The property which we need is as follows:

Proposition A.1

([11, Theorem 11.2]) Let $\pi \in \mathrm {Irr}_\mathrm {temp}(G(W_{n}))$. Assume that $\Theta _{V_m,W_n}(\pi )\not =0$ and $l=n-m+\epsilon _0>0$. Then $\gamma (s,\pi ,\chi _V^{-1},\psi )$ has a pole at $s=\frac{l+1}{2}$.

1.2 A.2 Plancherel measures

Let G be a reductive group over F and $P=MU$ be a parabolic subgroup of G. For $\pi \in \mathrm {Irr}(M)$, consider the normalized induced representation

$$\begin{aligned} I_P^G(\pi ):=\mathrm {Ind}_P^G(\pi ). \end{aligned}$$

We define an intertwining operator

$$\begin{aligned} J_{\overline{P}|P}(\pi ):I_P^G(\pi ) \rightarrow I_{\overline{P}}^G(\pi ) \end{aligned}$$

by the integral

$$\begin{aligned} J_{\overline{P}|P}(\pi )f(g)=\int _{\overline{U}} f(\overline{u}g)d\overline{u} \quad \text {for } f \in I_P^G(\pi ), \end{aligned}$$

where $\overline{P}=M\overline{U}$ is the parabolic subgroup of G opposite to P. More precisely, the above integral converges if $\pi$ belongs to a certain cone in its Bernstein component (which is a complex manifold), and admits a meromorphic continuation to the whole Bernstein component, being given by a rational function in $\pi$ (see [46, Théorème IV.1.1]). Then there exists a rational function $\mu$ of $\pi$ such that

$$\begin{aligned} J_{P|\overline{P}}(\pi ) \circ J_{\overline{P}|P}(\pi ) = \mu (\pi )^{-1}. \end{aligned}$$

The rational function $\mu$ is called the Plancherel measure associated to $I_P^G(\pi )$ (though some reader might find it more appropriate to use the term Plancherel measure or density for the product of the function $\mu$ with the formal degree of $\pi$ if $\pi$ is essentially square-integrable). The function $\mu$ is only well-defined up to a scalar since it depends on the choice of Haar measures on U and $\overline{U}$. We choose Haar measures as in [11, §B.2], which are determined by $\psi$. We denote the corresponding Plancherel measure by $\mu _\psi$.

Let $(V_m,W_n)$ be as in Sect. 2.2, and put $W_{n_1}=W_{n}+\mathbb {H}^k$ and $V_{m_1}=V_{m}+\mathbb {H}^k$ with $n_1=n+2k$ and $m_1=m+2k$. We consider the maximal parabolic subgroups $P=M_PU_P$ and $Q=M_QU_Q$ of $G(W_{n_1})$ and $H(V_{m_1})$ with Levi components

$$\begin{aligned} M_P=\mathrm {GL}_k(E)\times G(W_{n}) \quad \text {and}\quad M_Q=\mathrm {GL}_k(E)\times H(V_{m}), \end{aligned}$$

respectively.

Theorem A.2

([11, Theorem 12.1]) Let $\pi \in \mathrm {Irr}(G(W_{n}))$ and put $\sigma =\theta _{V_{m},W_{n}}(\pi )$. Assume that $\sigma \not =0$, so that $\sigma \in \mathrm {Irr}(H(V_{m}))$. For $\tau \in \mathrm {Irr}(\mathrm {GL}_k(E))$ and $s\in \mathbb {C}$, we put $\tau _s=\tau |\det |_E^s$. Then we have

$$\begin{aligned} \frac{\mu _\psi \left( \tau _s\chi _V \otimes \pi \right) }{\mu _\psi \left( \tau _s\chi _W\otimes \sigma \right) }&=\gamma \left( s-\frac{l-1}{2},\tau ,\psi _E\right) \cdot \gamma \left( -s-\frac{l-1}{2},\tau ^\vee ,\psi _E^{-1}\right) . \end{aligned}$$

For metaplectic groups, we have to replace $\mathrm {GL}_k(E)$ with its double cover $\widetilde{\mathrm {GL}}_k(E)$. More precisely, see [13, § 2.2–§ 2.5] and [11, §2.5 and §2.6].

1.3 A.3 Representations of Weil–Deligne groups

We denote by $W_E$ and $WD _E=W_E \times \mathrm {SL}_2(\mathbb {C})$ the Weil group and Weil–Deligne group of E, respectively. Let $I_E$ be the inertia subgroup of $W_E$. We fix a geometric Frobenius element $\mathrm {Frob}_E$ of $W_E$.

If $E\not =F$, we regard $W_E$ as a subgroup $W_F$ such that $W_F/W_E \cong \mathrm {Gal}(E/F)$ and fix $s\in W_F \setminus W_E$. If $E=F$, we put $s=1$.

Let M be a finite dimensional vector space over $\mathbb {C}$. We say that a homomorphism $\phi : WD _E \rightarrow \mathrm {GL}(M)$ is a representation of $WD _E$ if

• $\phi (\mathrm {Frob}_E)$ is semi-simple;

• the restriction of $\phi$ to $W_E$ is smooth;

• the restriction of $\phi$ to $\mathrm {SL}_2(\mathbb {C})$ is algebraic.

We call $\phi$ tempered if the image of $W_E$ is bounded. Let $\phi ^\vee$ be the contragredient representation of $\phi$ defined by $\phi ^\vee (w)={}^t \phi (w)^{-1}$. We define a representation ${}^c\phi$ of $WD _E$ by ${}^c\phi (w)=\phi (sws^{-1})$. Then the equivalence class of ${}^c\phi$ is independent of the choice of s.

Fix $b\in \{\pm 1\}$. We say that $\phi$ is conjugate self-dual of sign b if there exists a non-degenerate bilinear form $B:M \times M \rightarrow \mathbb {C}$ such that

$$\begin{aligned} \left\{ \begin{aligned}&B\left( \phi (w)x,\phi (sws^{-1})y\right) = B(x,y),\\&B(y,x) = b \cdot B(x,\phi (s^2)y) \end{aligned} \right. \end{aligned}$$

for $x,y\in M$ and $w\in WD _E$. In this case, $\phi$ is equivalent to ${}^c\phi ^\vee$. If $E=F$, then $s=1$ and ${}^c\phi =\phi$. In this case, we call $\phi$ self-dual of sign b. We also say that $\phi$ is

$$\begin{aligned} \left\{ \begin{aligned}&\text {orthogonal }&\quad&\text {if }\phi \text { is self-dual of sign } +1,\\&\text {symplectic }&\quad&\text {if }\phi \text { is self-dual of sign } -1,\\&\text {conjugate-orthogonal }&\quad&\text {if }\phi \text { is conjugate self-dual of sign } +1,\\&\text {conjugate-symplectic }&\quad&\text {if }\phi \text { is conjugate self-dual of sign } -1. \end{aligned} \right. \end{aligned}$$

More precisely, see [10, §3].

For each positive integer k, there exists a unique irreducible algebraic representation $S_k$ of $\mathrm {SL}_2(\mathbb {C})$ of dimension k. It is easy to see that $S_k$ is (conjugate) self-dual of sign $(-1)^{k-1}$. Moreover we have

$$\begin{aligned} S_a \otimes S_b \cong \bigoplus _{k=1}^{\min \left\{ a,b\right\} }S_{a+b+1-2k} =S_{a+b-1} \oplus S_{a+b-3} \oplus \cdots \oplus S_{\left| a-b\right| +1} \end{aligned}$$

for positive integers a and b. We can prove this isomorphism by computing the character of $S_a \otimes S_b$ using the highest weight theory for $\mathrm {SL}_2(\mathbb {C})$.

1.4 A.4 Local factors

We define local factors associated to representations of $WD _E$. Fix a non-trivial additive character $\psi '$ of E. A representation $\phi$ of $WD _E$ is written by

$$\begin{aligned} \phi =\bigoplus _{n\ge 1}\phi _n\boxtimes S_n, \end{aligned}$$

where $(\phi _n,M_n)$ is a representation of $W_E$. Let $M_n^{I_E}$ be the subspace of $M_n$ consisting of $I_E$-fixed vectors. Note that $M_n^{I_E}$ is a subrepresentation of $M_n$ and $\phi _n(\mathrm {Frob}_E) \in \mathrm {GL}(M_n^{I_E})$ is independent of the choice of $\mathrm {Frob}_E$. We define the local factors associated $\phi$ by

$$\begin{aligned} L(s,\phi )&=\prod _{n\ge 1} \det \left( {\mathbf{1}}-q_{E}^{-\left( s+\frac{n-1}{2}\right) }\phi _n(\mathrm {Frob}_E)\big |M_n^{I_E}\right) ^{-1}\\&=\prod _{n\ge 1}L\left( s+\frac{n-1}{2},\phi _n\right) ,\\ \varepsilon \left( s,\phi ,\psi '\right)&=\prod _{n\ge 1}\varepsilon \left( s,\phi _n,\psi '\right) ^n \det \left( -q_{E}^{\frac{1}{2}-s}\phi _n(\mathrm {Frob}_E)\big |M_n^{I_E}\right) ^{n-1},\\ \gamma \left( s,\phi ,\psi '\right)&=\varepsilon \left( s,\phi ,\psi '\right) \frac{L\left( 1-s,\phi ^\vee \right) }{L(s,\phi )}. \end{aligned}$$

For the definition of $\varepsilon (s,\phi _n,\psi ')$, see [44, §3]. For $c \in E^\times$, we define the non-trivial additive character $\psi '_c$ of E by $\psi '_c(x)=\psi '(cx)$. It is known that

$$\begin{aligned} \varepsilon \left( s, \phi , \psi '_c\right) =\det (\phi )(c) \cdot \left| c\right| _E^{\dim (\phi )\left( s - \frac{1}{2}\right) } \cdot \varepsilon (s,\phi ,\psi '). \end{aligned}$$

The local functional equation asserts that

$$\begin{aligned}&\gamma (s,\phi ,\psi ') \cdot \gamma \left( 1-s, \phi ^\vee , \psi '^{-1}\right) =1 \quad \text {or}\quad \\&\quad \varepsilon \left( s,\phi ,\psi '\right) \cdot \varepsilon \left( 1-s, \phi ^\vee , \psi '\right) =\det (\phi )(-1). \end{aligned}$$

In particular, if $\phi$ is self-dual with $\det (\phi )={\mathbf{1}}$, then $\varepsilon (1/2,\phi ,\psi ')$ is in $\{\pm 1\}$ and independent of $\psi '$. In this case, we write $\varepsilon (\phi ):=\varepsilon (1/2,\phi ,\psi ')$. For $a\not \equiv b\bmod 2$, we have

$$\begin{aligned} \varepsilon (S_a \otimes S_b)=(-1)^{\min \left\{ a,b\right\} }. \end{aligned}$$

If $E\not =F$ and $\phi$ is conjugate self-dual, then we write $\varepsilon (\phi ,\psi '):=\varepsilon (1/2,\phi ,\psi ')$. By [10, Propostition 5.1], if $E\not =F$ and ${}^c\psi '=\psi '^{-1}$, then $\varepsilon (\phi ,\psi ')\in \{\pm 1\}$. Here, ${}^c\psi '(x) = \psi '({}^c x)$ for $x \in E$, where ${}^c x$ is the conjugate of x.

We need some lemmas for local factors.

Lemma A.3

Let $\phi$ be an irreducible representation of $W_E$ and l be a positive integer. Then we have

$$\begin{aligned}&\varepsilon (s,\phi ,\psi ')^l\varepsilon \left( -s,\phi ^\vee ,\psi '^{-1}\right) ^l\\&\quad =\varepsilon \left( s-\frac{l-1}{2},\phi ,\psi '\right) \varepsilon \left( -s-\frac{l-1}{2},\phi ^\vee ,\psi '^{-1}\right) , \end{aligned}$$

and

$$\begin{aligned}&\gamma \left( s,\phi \otimes S_l,\psi '\right) \gamma \left( -s,\phi ^\vee \otimes S_l,\psi '^{-1}\right) \\&\quad =\gamma \left( s-\frac{l-1}{2},\phi ,\psi '\right) \gamma \left( -s-\frac{l-1}{2},\phi ^\vee ,\psi '^{-1}\right) . \end{aligned}$$

Proof

Straightforward. $\square$

Lemma A.4

Let $\psi '$ be a non-trivial additive character of E, $\phi$ be a representation of $WD _E$, and l be a positive integer. Assume that

• $\psi '|F={\mathbf{1}}$, i.e., ${}^c\psi '=\psi '^{-1}$ if $E\not =F$;

• $\phi$ is conjugate self-dual of sign $(-1)^{l-1}$ if $E\not =F$;

• $\phi$ is self-dual of sign $(-1)^{l-1}$ if $E=F$.

We define $\alpha _l(\phi )\in \{\pm 1\}$ by

$$\begin{aligned} \alpha _l(\phi )=\frac{\varepsilon \left( \phi \otimes S_{l+1},\psi '\right) }{\varepsilon \left( \phi \otimes S_{l-1},\psi '\right) } \times \left\{ \begin{aligned}&\det (\phi )(-1)&\quad&\text {if }E=F,\\&1&\quad&\text {if }E\not =F. \end{aligned} \right. \end{aligned}$$

Here, if $l=1$, then we interpret $\varepsilon (\phi \otimes S_{l-1},\psi ') :=1$.

• (1) Suppose that $\phi$ is irreducible. Then $\alpha _l(\phi )=-1$ if and only if $\phi =S_l$.

• (2) If $\phi =\phi _0\oplus {}^c\phi _0^\vee$, then $\alpha _l(\phi )=1$.

• (3) In general, $\alpha _l(\phi )=(-1)^{m_\phi (S_l)}$, where $m_\phi (S_l)$ is the multiplicity of $S_l$ in $\phi$.

Proof

Straightforward. $\square$

For a character $\chi$ of $E^\times$, we put

$$\begin{aligned} \delta (\chi = {\mathbf{1}}) = \left\{ \begin{aligned}&1&\quad&\text {if }\chi = {\mathbf{1}}, \\&-1&\quad&\text {if }\chi \not = {\mathbf{1}}. \end{aligned} \right. \end{aligned}$$

Lemma A.5

Let $\chi$ be a quadratic character of $E^\times$, and k be a positive integer. Then $\chi \otimes S_{2k}$ is a symplectic representation of $WD _E$, and satisfies

$$\begin{aligned} \varepsilon (\chi \otimes S_{2k}) = - \delta (\chi ={\mathbf{1}}) \cdot \chi (-1)^k. \end{aligned}$$

Proof

Since $\chi$ and $S_{2k}$ is self-dual representations of sign $+1$ and $-1$, respectively, we see that $\chi \otimes S_{2k}$ has sign $-1$. By the definition of the $\varepsilon$-factor, we have

$$\begin{aligned}&\varepsilon \left( \chi \otimes S_{2k}\right) = \varepsilon (\chi ,\psi )^{2k} \cdot \det \left( -\chi (\mathrm {Frob}_E) \big | \mathbb {C}(\chi )^{I_E}\right) ^{2k-1}\\&\quad =\chi (-1)^k \cdot \det \left( -\chi (\mathrm {Frob}_E) \big | \mathbb {C}(\chi )^{I_E}\right) ^{2k-1}, \end{aligned}$$

where $\mathbb {C}(\chi )$ denotes the space of $\chi$. If $\chi$ is ramified, then $\mathbb {C}(\chi )^{I_E}=0$ so that $\det (-\chi (\mathrm {Frob}_E) | \mathbb {C}(\chi )^{I_E})=1$. If $\chi$ is unramified, then we have

$$\begin{aligned}&\det \left( -\chi (\mathrm {Frob}_E) \big | \mathbb {C}(\chi )^{I_E}\right) \\&\quad = \left\{ \begin{aligned}&-1&\quad&\text {if }\chi ={\mathbf{1}},\\&1&\quad&\text {if }\chi \text { is the unique non-trivial unramified quadratic character}. \end{aligned} \right. \end{aligned}$$

Hence for any quadratic character $\chi$, we have $\det (-\chi (\mathrm {Frob}_E) | \mathbb {C}(\chi )^{I_E})=-\delta (\chi ={\mathbf{1}})$. $\square$

The following lemma is [13, Lemma 12.3] and [12, Lemma A.6].

Lemma A.6

Let $\phi _1$, $\phi _2$ be a tempered representations of $WD _E$ of the same dimension n. Assume that

$$\begin{aligned}&\gamma \left( s,\phi _1^\vee \otimes \phi _\rho , \psi '\right) \cdot \gamma \left( -s, \phi _1 \otimes \phi _\rho ^\vee , \psi '^{-1}\right) \\&\quad =\gamma \left( s,\phi _2^\vee \otimes \phi _\rho , \psi '\right) \cdot \gamma \left( -s, \phi _2 \otimes \phi _\rho ^\vee , \psi '^{-1}\right) \end{aligned}$$

for every irreducible representation $\phi _\rho$ of $W_E$. Then

$$\begin{aligned} \phi _1\cong \phi _2 \end{aligned}$$

as representations of $WD _E$.

Appendix B: Local Langlands correspondence

In this paper, we assume the local Langlands correspondence for classical groups, which parametrizes irreducible representations. For general linear groups, it was established by Harris–Taylor [18], Henniart [19], and Scholze [41]. For other classical groups, it is known by Arthur [1], Mok [35], and Kaletha–Mínguez–Shin–White [23], under some assumption on the stabilization of twisted trace formulas. For this assumption, see also the two books [34] of Mœglin–Waldspurger, and papers of Chaudouard–Laumon [7, 8]. For metaplectic groups, it was established by the second author and Savin [13]. In this appendix, we summarize some of its properties which are used in this paper.

1.1 B.1 Parameters and its component groups

In this subsection, we define parameters and its component groups for (possibly disconnected) classical groups. More precisely, see [1] and [10].

Fix $\epsilon \in \{\pm 1\}$. Let $V_m$ be an $\epsilon$-Hermitian space of dimension m and $G=H(V_m)$ be the isometry group of $V_m$. Let $\Phi (H(V_m))$ be the set of equivalence classes of representations $\phi$ of $WD _E$ of dimension m if $V_m$ is an even-dimensional orthogonal space, or of dimension $m-\epsilon _{0}$ otherwise, which are

$$\begin{aligned} \left\{ \begin{aligned}&\text {conjugate self-dual of sign } (-1)^{m-1},&\quad&\text {if }E\not =F,\\&\text {self-dual of sign } +1 \text { such that } \det (\phi )=\chi _V,&\quad&\text {if }E=F,\ \epsilon =+1\text { and }m\in 2\mathbb {Z},\\&\text {self-dual of sign } -\epsilon \text { such that } \det (\phi )={\mathbf{1}},&\quad&\text {otherwise}. \end{aligned} \right. \end{aligned}$$

In particular, if $E=F$, $\epsilon =+1$ and $m=1$, then $\Phi (H(V_1))=\{\text { the zero representation of } WD _E\}$. We call an element in $\Phi (H(V_m))$ a parameter for $H(V_m)$. We denote by $\Phi _\mathrm {temp}(H(V_{m}))$ the subset of equivalence classes of tempered parameters, i.e., the subset of $\phi \in \Phi (H(V_m))$ such that $\phi (W_E)$ is bounded.

If $E=F$ and $G=H(V_m)$, we denote by $\widehat{G}$ the Langlands dual group of G. It is given by

$$\begin{aligned} \widehat{G}= \left\{ \begin{aligned}&\mathrm {Sp}_{m-1}(\mathbb {C})&\quad&\text {if }E=F, \epsilon =+1 \text { and } m \text { is odd},\\&\mathrm {SO}_{m+1}(\mathbb {C})&\quad&\text {if }E=F, \epsilon =-1,\\&\mathrm {SO}_m(\mathbb {C})&\quad&\text {if }E=F, \epsilon =+1 \text { and } m \text { is even}. \end{aligned} \right. \end{aligned}$$

Let $\phi \in \Phi (H(V_m))$. We denote the space of $\phi$ by M and the $WD _E$-invariant bilinear form on M by B. Let

$$\begin{aligned} C_\phi= & {} \left\{ g \in \mathrm {GL}(M)\ \big |\ B(gx,gy)=B(x,y) \text { for any } x, y\in M, \right. \\&\left. \text { and } g \phi (w) = \phi (w) g \text { for any } w \in WD _E \right\} \end{aligned}$$

be the centralizer of $\mathrm {Im}(\phi )$ in $\mathrm {Aut}(M,B)$. Also we put

$$\begin{aligned} C_\phi ^+ = \left\{ \begin{aligned}&C_\phi \cap \mathrm {SL}(M)&\quad&\text {if }E=F \text { and } m \text { is even}, \\&C_\phi&\quad&\text {otherwise}. \end{aligned} \right. \end{aligned}$$

Finally, we define the component groups $A_\phi$ and $A_\phi ^+$ of $\phi$ by

$$\begin{aligned} A_\phi = \pi _0(C_\phi ) \quad \text {and}\quad A_\phi ^+ = \pi _0(C_\phi ^+), \end{aligned}$$

respectively.

Let $\phi \in \Phi (H(V_m))$. For an irreducible representation $\phi _0$ of $WD _E$, we denote the multiplicity of $\phi _0$ in $\phi$ by $m_\phi (\phi _0)$. We can decompose

$$\begin{aligned} \phi =m_1\phi _1+\cdots +m_r\phi _r+\phi '+{}^c\phi '^\vee , \end{aligned}$$

where $\phi _1,\dots , \phi _r$ are distinct irreducible (conjugate) self-dual representations of $WD _E$ of the same type as $\phi$, $m_i=m_\phi (\phi _i)$, and $\phi '$ is a sum of irreducible representations of $WD _E$ which are not (conjugate) self-dual of the same type as $\phi$. Then by [10, § 4], $A_\phi$ is described as follows:

$$\begin{aligned} A_\phi =\bigoplus _{i=1}^{r}(\mathbb {Z}/2\mathbb {Z}) a_i \cong (\mathbb {Z}/2\mathbb {Z})^r. \end{aligned}$$

Namely, $A_\phi$ is a free $\mathbb {Z}/2\mathbb {Z}$-module of rank r with a canonical basis $\{a_i\}$ indexed by the summands $\phi _i$ of $\phi$. For $a=a_{i_1}+\cdots +a_{i_k} \in A_\phi$ with $1\le i_1< \cdots < i_k \le r$, we put

$$\begin{aligned} \phi ^{a}=\phi _{i_1} \oplus \cdots \oplus \phi _{i_k}. \end{aligned}$$

Also, we denote

$$\begin{aligned} z_\phi :=\sum _{i=1}^r m_\phi (\phi _i)\cdot a_i =\sum _{i=1}^s m_i \cdot a_i \in A_\phi . \end{aligned}$$

This is the image of $-{\mathbf{1}}$ in $A_\phi$. We call $z_\phi$ the central element in $A_\phi$. The determinant map $\det :\mathrm {GL}(M) \rightarrow \mathbb {C}^\times$ gives a homomorphism

$$\begin{aligned} \det :A_\phi \rightarrow \mathbb {Z}/2\mathbb {Z}, \quad \sum _{i=1}^r \varepsilon _i a_i \mapsto \sum _{i=1}^r \varepsilon _i \cdot \dim (\phi _i), \end{aligned}$$

where $\varepsilon _i\in \{0,1\} = \mathbb {Z}/2\mathbb {Z}$. Then the group $A_\phi ^+$ can be described as follows ([10, Theorem 8.1]):

$$\begin{aligned} A_\phi ^+= \left\{ \begin{aligned}&\ker (\det )&\quad&\text {if }E=F \text { and } m \text { is even},\\&A_\phi&\quad&\text {otherwise}. \end{aligned} \right. \end{aligned}$$

We say that a parameter $\phi$ is discrete if $m_i=1$ for any $i=1,\dots , r$ and $\phi '=0$, i.e., $\phi$ is a multiplicity-free sum of irreducible (conjugate) self-dual representations of $WD _E$ of the same type as $\phi$. We denote by $\Phi _\mathrm {disc}(H(V_{m}))$ the subset of equivalence classes of discrete parameters. Then we have a sequence

$$\begin{aligned} \Phi _\mathrm {disc}(H(V_{m})) \subset \Phi _\mathrm {temp}(H(V_{m})) \subset \Phi (H(V_{m})). \end{aligned}$$

1.2 B.2 Local Langlands correspondence for connected classical groups

In this subsection, we introduce $\Pi (H(V_m))$ and state some properties of the local Langlands correspondence which we need.

First, we consider orthogonal groups. So we assume that $E=F$ and $\epsilon =+1$, and we write $H(V_m)=\mathrm {O}(V_m)$. We define equivalence relations $\sim _{\det }$ on $\mathrm {Irr}(\mathrm {O}(V_m))$ and $\sim _\varepsilon$ on $\mathrm {Irr}(\mathrm {SO}(V_m))$ by

$$\begin{aligned} \sigma \sim _{\det } \sigma \otimes \det \quad \text {and}\quad \sigma _0 \sim _{\varepsilon } \sigma _0^\varepsilon \end{aligned}$$

for $\sigma \in \mathrm {Irr}(\mathrm {O}(V_m))$ and $\sigma _0 \in \mathrm {Irr}(\mathrm {SO}(V_m))$. Here, we fix an element $\varepsilon \in \mathrm {O}(V_m) \setminus \mathrm {SO}(V_m)$ and define $\sigma _0^\varepsilon$ by $\sigma _0^\varepsilon (h)=\sigma _0(\varepsilon ^{-1}h\varepsilon )$ for $\sigma _0 \in \mathrm {Irr}(\mathrm {SO}(V_m))$ and $h \in \mathrm {SO}(V_m)$. Note that $\sigma |\mathrm {SO}(V_m) \cong (\sigma \otimes \det )|\mathrm {SO}(V_m)$ for $\sigma \in \mathrm {Irr}(\mathrm {O}(V_m))$, and $\mathrm {Ind}_{\mathrm {SO}(V_m)}^{\mathrm {O}(V_m)}(\sigma _0) \cong \mathrm {Ind}_{\mathrm {SO}(V_m)}^{\mathrm {O}(V_m)}(\sigma _0^\varepsilon )$ for $\sigma _0 \in \mathrm {Irr}(\mathrm {SO}(V_m))$. The restriction and the induction give a canonical bijection

$$\begin{aligned} \mathrm {Irr}(\mathrm {O}(V_m))/\sim _{\det } \longleftrightarrow \mathrm {Irr}(\mathrm {SO}(V_m))/\sim _\varepsilon . \end{aligned}$$

In [1], Arthur has parametrized not $\mathrm {Irr}(\mathrm {SO}(V_m))$ but $\mathrm {Irr}(\mathrm {SO}(V_m))/\sim _\varepsilon$. Via the above bijection, we translate the parametrization for $\mathrm {Irr}(\mathrm {O}(V_m))/\sim _{\det }$.

We return the general setting. Let E be either F or a quadratic extension of F, $V_m$ be an $\epsilon$-Hermitian space of dimension m for fixed $\epsilon \in \{\pm 1\}$, and $H(V_m)$ be the isometry group of $V_m$. We define $\Pi (H(V_m))$ by

$$\begin{aligned} \Pi (H(V_m))= \left\{ \begin{aligned}&\mathrm {Irr}(H(V_m))/\sim _{\det }&\quad&\text {if }E=F \text { and } \epsilon =+1,\\&\mathrm {Irr}(H(V_m))&\quad&\text {otherwise}. \end{aligned} \right. \end{aligned}$$

For $\pi \in \mathrm {Irr}(H(V_m))$, we denote the image of $\pi$ under the canonical map $\mathrm {Irr}(H(V_m)) \rightarrow \Pi (H(V_m))$ by $[\pi ]$. Also, we denote the image of $\mathrm {Irr}_*(H(V_m))$ in $\Pi (H(V_m))$ by $\Pi _*(H(V_m))$ for $*=\mathrm {disc}$ or $\mathrm {temp}$.

If $E\not =F$ or $\epsilon =+1$, then there exist exactly two Witt towers $\mathcal {V}$ and $\mathcal {V}'$ such that $V_m \in \mathcal {V}$ and

$$\begin{aligned} \left\{ \begin{aligned}&\dim (V_m) \equiv \dim (V'_{m'}) \bmod 2&\quad&\text {if }E\not =F, \\&\mathrm {disc}(V_{m}) = \mathrm {disc}(V'_{m'})&\quad&\text {if }E=F \text { and } \epsilon =+1 \end{aligned} \right. \end{aligned}$$

for $V'_{m'} \in \mathcal {V}'$. Let $\mathcal {V}^+$ be the Witt tower whose anisotropic space is

$$\begin{aligned} \left\{ \begin{aligned}&0&\quad&\text {if }E\not =F \text { and } m \text { is even},\\&(E,1)&\quad&\text {if }E\not =F, m \text { is odd and } \epsilon =+1,\\&(E,\delta )&\quad&\text {if }E\not =F, m \text { is odd and } \epsilon =-1,\\&0&\quad&\text {if }E=F, m \text { is even and } \mathrm {disc}(V_m)=1,\\&(F(\sqrt{d}), \mathrm {tr}_{F(\sqrt{d})/F})&\quad&\text {if }E=F, m \text { is even and } d :=\mathrm {disc}(V_m)\not =1 \text { in } F^\times /F^{\times 2},\\&(F, 2\mathrm {disc}(V_m))&\quad&\text {if }E=F \text { and } m \text { is odd}. \end{aligned} \right. \end{aligned}$$

We denote the other Witt tower by $\mathcal {V}^-$. A pure inner form of $H(V_m)$ is $H(V^+_m)$ or $H(V^-_m)$, where $V^\pm _m\in \mathcal {V}^\pm$. If $E=F$ and $\epsilon =-1$, a pure inner form of $H(V_m)$ is $H(V_m)$ itself only.

Now we are ready to describe the desiderata for the Langlands correspondence.

Desideratum B.1

• (1) There exists a canonical surjection

  $$\begin{aligned} \bigsqcup _{V_m^\bullet } \Pi (H(V_m^\bullet ))\rightarrow \Phi (H(V_m)), \end{aligned}$$

  where $V_m^\bullet$ runs over the spaces such that $H(V_m^\bullet )$ is a pure inner form of $H(V_m)$. For $\phi \in \Phi (H(V_m))$, we denote by $\Pi _\phi ^0$ the inverse image of $\phi$ under this map, and call $\Pi _\phi ^0$ the L-packet of $\phi$.

• (2) There exists a bijection

  $$\begin{aligned} \iota :\Pi _\phi ^0 \rightarrow \widehat{A_\phi ^+}, \end{aligned}$$

  which satisfies certain character identities. Here, we denote by $\widehat{A_\phi ^+}$ the Pontryagin dual of $A_\phi ^+$.

• (3) Let $[\pi ] \in \Pi _\phi ^0$ with $\iota ([\pi ])=\eta$. Then $[\pi ] \in \Pi (H(V^-_m))$ if and only if $z_\phi \in A_\phi ^+$ and $\eta (z_\phi )=-1$.

• (4) We have

  $$\begin{aligned} \bigsqcup _{V_m^\bullet } \Pi _*\left( H(V^\bullet _m)\right) = \bigsqcup _{\phi \in \Phi _*(H(V_m))}\Pi _\phi ^0 \end{aligned}$$

  for $* \in \{\mathrm {disc}, \mathrm {temp}\}$.

• (5) Assume that $\phi =\phi _\tau +\phi _0+{}^c\phi _\tau ^\vee$, where $\phi _0$ is an element in $\Phi _\mathrm {temp}(H(V_{m_0}))$ and $\phi _\tau$ is an irreducible tempered representation of $WD _E$ which corresponds to $\tau \in \mathrm {Irr}_\mathrm {temp}(\mathrm {GL}_k(E))$. Then the induced representation

  $$\begin{aligned} \mathrm {Ind}_Q^{H(V_m)}\left( \tau \otimes \pi _0\right) \end{aligned}$$

  is a direct sum of tempered representations of $H(V_m)$, where Q is a parabolic subgroup of $H(V_m)$ with Levi subgroup $L_Q=\mathrm {GL}_k(E) \times H(V_{m_0})$ and $\pi _0$ is (a representative of) an element in $\Pi _{\phi _0}^0$. The L-packet $\Pi _\phi ^0$ is given by

  $$\begin{aligned} \Pi _\phi ^0 = \left\{ [\pi ]\ \big |\ \pi \subset \mathrm {Ind}_Q^{H(V_m)}\left( \tau \otimes \pi _0\right) , [\pi _0] \in \Pi _{\phi _0}^0\right\} . \end{aligned}$$

  Moreover if $\pi \subset \mathrm {Ind}_Q^{H(V_m)}(\tau \otimes \pi _0)$, then $\iota ([\pi ])|A_{\phi _0}^+ = \iota ([\pi _0])$.

• (6) Assume that

  $$\begin{aligned} \phi =\phi _{\tau _1}\left| \cdot \right| ^{s_1} + \cdots +\phi _{\tau _r}\left| \cdot \right| ^{s_r} + \phi _0 +{}^c\left( \phi _{\tau _1}\left| \cdot \right| ^{s_1} + \cdots +\phi _{\tau _r}\left| \cdot \right| ^{s_r}\right) ^\vee , \end{aligned}$$

  where $\phi _0$ is an element in $\Phi _\mathrm {temp}(H(V_{m_0}))$, $\phi _{\tau _i}$ is an irreducible tempered representation of $WD _E$ which corresponds to $\tau _i \in \mathrm {Irr}_\mathrm {temp}(\mathrm {GL}_{k_i}(E))$, and $s_1 \ge \cdots \ge s_r>0$. Then the L-packet $\Pi _\phi ^0$ consists of (the equivalence classes of) the unique irreducible quotients $\pi$ of the standard modules

  $$\begin{aligned} \tau _1\left| \det \right| _F^{s_1} \times \cdots \times \tau _r\left| \det \right| _F^{s_r} \rtimes \pi _0, \end{aligned}$$

  where $\pi _0$ runs over (representatives of) elements of $\Pi _{\phi _0}^0$. Moreover if $\pi$ is the unique irreducible quotient of $\tau _1|\det |_F^{s_1} \times \cdots \times \tau _r|\det |_F^{s_r} \rtimes \pi _0$, then $\iota ([\pi ])|A_{\phi _0}^+ = \iota ([\pi _0])$.

• (7) The local Langlands correspondence respects the standard $\gamma$-factor. Namely, we have

  $$\begin{aligned} \gamma (s,\pi ,\chi ,\psi ) = \gamma \left( s, \phi \otimes \chi , \psi _E\right) \end{aligned}$$

  for $\pi \in \mathrm {Irr}(H(V_m))$ whose parameter is $\phi$, and any character $\chi$ of $E^\times$. Here, we put $\psi _E=\psi \circ \mathrm {tr}_{E/F}$.

• (8) The Plancherel measures are invariants of an L-packet. Namely, if $\pi _1,\pi _2$ have the same parameter $\phi$, then we have

  $$\begin{aligned} \mu _\psi \left( \tau _s\otimes \pi _1\right) =\mu _\psi \left( \tau _s\otimes \pi _2\right) \end{aligned}$$

  for any $\tau \in \mathrm {Irr}(\mathrm {GL}_k(E))$. In particular, by a result of Shahidi [42], we have

  $$\begin{aligned} \mu _\psi (\tau _s\otimes \pi )= & {} \gamma \left( s,\phi _\tau \otimes \phi ^\vee ,\psi _E\right) \cdot \gamma \left( -s,\phi _\tau ^\vee \otimes \phi ,\psi _E^{-1}\right) \\&\cdot \gamma \left( 2s,R\circ \phi _\tau ,\psi \right) \cdot \gamma \left( -2s,R\circ \phi _\tau ^\vee ,\psi ^{-1}\right) \end{aligned}$$

  for any $\pi$ whose parameter is $\phi \in \Phi (H(V_m))$, where

  $$\begin{aligned} R=\left\{ \begin{aligned}&\mathrm {Asai}^+&\quad&\text {if }E\not =F \text { and } m \text { is even},\\&\mathrm {Asai}^-&\quad&\text {if }E\not =F \text { and } m \text { is odd},\\&\mathrm {Sym}^2&\quad&\text {if }E=F, \epsilon =+1 \text { and } m \text { is odd},\\&\wedge ^2&\quad&\text {otherwise}. \end{aligned} \right. \end{aligned}$$

The desiderata B.1 (7) and (8), at least for quasi-split classical groups, should follow from [1] and [35], supplemented by some results of many others. For non-quasi-split unitary groups, see also [23] and [32, § 1.4, Theorem 1.4.1].

Remark B.2

The bijection $\iota :\Pi _\phi ^0 \rightarrow \widehat{A_\phi ^+}$ may not be canonical. It is determined by a choice of a Whittaker datum of a quasi-split pure inner form $H(V_m^\bullet )$. If m is odd, then $H(V_m^\bullet )$ has a unique Whittaker datum, so that $\iota$ is canonical. Otherwise, we choose the Whittaker datum such that

$$\begin{aligned} \iota =\left\{ \begin{aligned}&J_{\psi ^E}&\text { in } [12]&\quad&\text {if }E\not =F \text { and }\epsilon = +1,\\&J_\psi&\text { in } [12]&\quad&\text {if }E\not =F \text { and } \epsilon = -1,\\&\iota _{\mathfrak {w}_1}&\text { in } [2]&\quad&\text {if }E=F \text { and }\epsilon = +1,\\&\iota _{\mathfrak {w}'_1}&\text { in } [2]&\quad&\text {if }E=F \text { and }\epsilon = -1. \end{aligned} \right. \end{aligned}$$

Here, in the first case, we fix a nonzero element $\delta \in E$ such that $\mathrm {tr}_{E/F}(\delta )=0$ and put $\psi ^E(x) = \psi (\frac{1}{2}\mathrm {tr}_{E/F}(\delta x))$ for $x \in E$.

Remark B.3

If $H(V_m) = \mathrm {Sp}(V_m)$ is a symplectic group, then $z_\phi \not \in A_\phi ^+$ so that

$$\begin{aligned} A_\phi = A_\phi ^+ \oplus (\mathbb {Z}/2\mathbb {Z})z_\phi \end{aligned}$$

for each $\phi \in \Phi (\mathrm {Sp}(V_m))$. Hence we may identify $\widehat{A_\phi ^+}$ with

$$\begin{aligned} \left\{ \eta \in \widehat{A_\phi }\ \big |\ \eta (z_\phi ) = 1\right\} \subset \widehat{A_\phi }. \end{aligned}$$

If $H(V_m)$ is not an orthogonal group, we have $\Pi (H(V_m)) = \mathrm {Irr}(H(V_m))$. In this case, we set $\Pi _\phi = \Pi _\phi ^0$ for $\phi \in \Phi (H(V_m))$. Using Remark B.3, unless $H(V_m)$ is an orthogonal group, we may regard $\iota$ as an injection

$$\begin{aligned} \iota :\Pi _\phi \hookrightarrow \widehat{A_\phi }. \end{aligned}$$

If $\pi \in \Pi _{\phi }$ and $\iota (\pi )=\eta \in \widehat{A_\phi }$, we call $(\phi ,\eta )$ the L-parameter for $\pi$.

The L-parameter for the contragredient representation $\pi ^\vee$ of $\pi$ is described by Kaletha [22].

Proposition B.4

([22, Theorem 4.9]) Let $\pi \in \mathrm {Irr}(H(V_m))$ with L-parameter $(\phi _\pi ,\eta _\pi )$. We denote the L-parameter for $\pi ^\vee$ by $(\phi _{\pi ^\vee },\eta _{\pi ^\vee })$. Then we have $\phi _{\pi ^\vee } = \phi _\pi ^\vee$. In particular, the component groups $A_{\phi _\pi }$ and $A_{\phi _{\pi }^\vee }$ are canonically identified. Moreover, we have $\eta _{\pi ^\vee }=\eta _\pi \cdot \eta _0$, where $\eta _0$ is given by

$$\begin{aligned} \eta _0(a)= \left\{ \begin{aligned}&\omega _{E/F}(-1)^{\dim (\phi _\pi ^{a})}&\quad&\text {if }E\not =F \text { and } m \text { is even},\\&\det (\phi _\pi ^{a})(-1)&\quad&\text {if }E=F \text { and } \epsilon =-1,\\&1&\quad&\text {otherwise}\end{aligned} \right. \end{aligned}$$

for $a \in A_{\phi _\pi }$.

1.3 B.3 Local Langlands correspondence for full orthogonal groups

In this subsection, we explain the parametrization of $\mathrm {Irr}(\mathrm {O}(V_m))$. Through this subsection, we assume $E=F$ and $\epsilon =+1$, so that $H(V_{m})=\mathrm {O}(V_{m})$. For $\phi \in \Phi (\mathrm {O}(V_m))$, we define the L-packet $\Pi _\phi$ of $\mathrm {O}(V_m)$, which is a subset of $\sqcup _{V_m^\bullet }\mathrm {Irr}(\mathrm {O}(V_m^\bullet ))$ by the inverse image of $\Pi _\phi ^0$ under the canonical map

$$\begin{aligned} \bigsqcup _{V_m^\bullet }\mathrm {Irr}(\mathrm {O}(V_m^\bullet )) \rightarrow \bigsqcup _{V_m^\bullet }\Pi (\mathrm {O}(V_m^\bullet )) = \bigsqcup _{V_m^\bullet }\mathrm {Irr}(\mathrm {O}(V_m^\bullet ))/\sim _{\det }. \end{aligned}$$

In the rest of this subsection, we parametrize $\Pi _\phi$.

First, we assume that m is odd. Then $\mathrm {O}(V_{m}) = \mathrm {SO}(V_m) \times \{\pm {\mathbf{1}}_{V_m}\}$. Any representation $\pi \in \mathrm {Irr}(\mathrm {O}(V_m))$ is determined by its image $[\pi ]$ in $\mathrm {Irr}(\mathrm {O}(V_{m}))/\sim _{\det }$ and its central character $\omega _\pi :\{\pm {\mathbf{1}}_{V_m}\} \rightarrow \mathbb {C}^\times$. Hence we have a bijection

$$\begin{aligned} \Pi _\phi \rightarrow \widehat{A_\phi } \times \left\{ \pm 1\right\} ,\ \pi \mapsto \left( \iota ([\pi ]), \omega _\pi (-{\mathbf{1}}_{V_m^\bullet })\right) . \end{aligned}$$

If $\pi \in \Pi _\phi$ corresponds to $(\eta , \nu ) \in \widehat{A_\phi } \times \{\pm 1\}$, we call the triple $(\phi , \eta , \nu )$ the L-parameter for $\pi$.

Next, we assume that m is even. For $\phi \in \Phi (\mathrm {O}(V_m))$, we have an inclusion $A_\phi ^+ \subset A_\phi$, so that we obtain a canonical surjection

$$\begin{aligned} \widehat{A_\phi } \twoheadrightarrow \widehat{A_\phi ^+}. \end{aligned}$$

Proposition B.5

For $\phi \in \Pi _\phi$, we have $\#\Pi _\phi = \# \widehat{A_\phi }$. Moreover, the following are equivalent:

• $[A_\phi : A_\phi ^+]=2$;

• $\pi \otimes \det \not \cong \pi$ for some $\pi \in \Pi _\phi$;

• $\pi \otimes \det \not \cong \pi$ for any $\pi \in \Pi _\phi$.

Proof

This follows from [2, Proposition 3.2]. $\square$

We fix $\epsilon \in \mathrm {O}(V_{m}) \setminus \mathrm {SO}(V_m)$ as in [3], which depends on the choice of Whittaker datum. Then [1, Theorem 2.2.4] gives a bijection

$$\begin{aligned} \iota :\Pi _\phi \rightarrow \widehat{A_\phi } \end{aligned}$$

which satisfies a similar condition of Desiderata B.1 (2) – (8), and such that the diagram

is commutative. More precisely, see [3]. If $\pi \in \Pi _\phi$ and $\iota (\pi )=\eta$, we call $(\phi , \eta )$ the L-parameter for $\pi$.

1.4 B.4 Local Langlands correspondence for metaplectic groups

In this subsection, we explain the parametrization of $\mathrm {Irr}(\mathrm {Mp}(W_{2n}))$. Let $(W_{2n},V_m)$ be as in Sect. 2.2. Through this subsection, we assume $E=F$, $\epsilon =+1$ and $m=2n+1$, so that $G(W_{2n})=\mathrm {Mp}(W_{2n})$ and $H(V_{m})=\mathrm {O}(V_{2n+1})$.

First, we recall a result of Gan–Savin.

Theorem B.6

([13, Theorem 1.1 and Corollary 1.2]) Let $c \in F^\times / F^{\times 2}$. The theta correspondence gives a natural bijection (depending on the choice of $\psi$)

$$\begin{aligned} \mathrm {Irr}(\mathrm {Mp}(W_{2n})) \rightarrow \bigsqcup _{V_{2n+1}^\bullet }\mathrm {Irr}\left( \mathrm {O}(V_{2n+1}^\bullet )\right) /\sim _{\det } =\bigsqcup _{V_{2n+1}^\bullet }\Pi \left( \mathrm {O}(V_{2n+1}^\bullet )\right) , \end{aligned}$$

where the union is taken over all the isomorphism classes of orthogonal spaces $V_{2n+1}^\bullet$ over F with $\dim (V_{2n+1}^\bullet )=2n+1$ and $\mathrm {disc}(V_{2n+1}^\bullet )=c$.

We describe this map more precisely. Since we are in the p-adic setting, there are exactly two isomorphism classes $V_{2n+1}$ and $V_{2n+1}'$ such that $\dim (V_{2n+1}) = \dim (V_{2n+1}') = 2n+1$ and $\mathrm {disc}(V_{2n+1}) = \mathrm {disc}(V_{2n+1}')=c$. For $\pi \in \mathrm {Irr}(\mathrm {Mp}(W_{2n}))$, exactly one of two theta lifts $\Theta _{V_{2n+1},W_{2n}}(\pi )$ and $\Theta _{V_{2n+1}',W_{2n}}(\pi )$ is nonzero. If $\Theta _{V_{2n+1}^\bullet ,W_{2n}}(\pi )$ is nonzero, then the image of $\pi$ under this map is $[\theta _{V_{2n+1}^\bullet ,W_{2n}}(\pi )]$. Also, the inverse image can be described as follows: For $\sigma \in \mathrm {Irr}(\mathrm {O}(V_{2n+1}^\bullet ))$, exactly one of two theta lifts $\Theta _{W_{2n},V_{2n+1}^\bullet }(\sigma )$ and $\Theta _{W_{2n},V_{2n+1}^\bullet }(\sigma \otimes \det )$ is nonzero, and the image of $[\sigma ] \in \Pi (\mathrm {O}(V_{2n+1}^\bullet ))$ under the inverse map is the nonzero small theta lift $\theta _{W_{2n}, V_{2n+1}^\bullet }(\sigma )$ or $\theta _{W_{2n}, V_{2n+1}^\bullet }(\sigma \otimes \det )$.

Corollary B.7

The theta correspondence for $(\mathrm {Mp}(W_{2n}),\mathrm {O}(V_{2n+1}^\bullet ))$ with $\mathrm {disc}(V_{2n+1}^\bullet )=1$ and the local Langlands correspondence for $\mathrm {O}(V_{2n+1}^\bullet )$ gives a surjection (depending on $\psi$)

$$\begin{aligned} \mathrm {Irr}(\mathrm {Mp}(W_{2n})) \rightarrow \Phi (\mathrm {O}(V_{2n+1})). \end{aligned}$$

For $\phi \in \Phi (\mathrm {O}(V_{2n+1}))$, we denote by $\Pi _\phi$ the inverse image of $\phi$ under this map, and call $\Pi _\phi$ the L-packet of $\phi$. Moreover, the composition of $\iota$ for $\mathrm {O}(V_{2n+1})$ and the relevant theta lift gives a bijection (depending on $\psi$)

$$\begin{aligned} \iota :\Pi _\phi \rightarrow \widehat{A_\phi }. \end{aligned}$$

We define $\Phi (\mathrm {Mp}(W_{2n})) :=\Phi (\mathrm {O}(V_{2n+1}))$. For $*=\mathrm {disc}$ or $\mathrm {temp}$, we put $\Phi _*(\mathrm {Mp}(W_{2n})) :=\Phi _*(\mathrm {O}(V_{2n+1}))$. Then by [13, Theorem 1.3], we see that Desideratum B.1 (1), (2), (4), (5), (6), (7) and (8) for $R=\mathrm {Sym}^2$ are satisfied.

We also need to know the effect of theta correspondence on L-parameters for the pair for $(\mathrm {Mp}(W_{2n}), \mathrm {O}(V_{2n+1}))$ with $\mathrm {disc}(V_{2n+1})=c$. Then $\chi _{V}=\chi _c$, where $\chi _c$ is the quadratic character of $F^\times$ associated to $c \in F^\times / F^{\times 2}$.

Theorem B.8

We write $c = \mathrm {disc}(V_{2n+1})$. Let $\pi \in \mathrm {Irr}(\mathrm {Mp}(W_{2n}))$ and $\sigma \in \mathrm {Irr}(\mathrm {O}(V_{2n+1}))$ with L-parameters $(\phi _\pi ,\eta _\pi )$ and $(\phi _\sigma , \eta _\sigma )$, respectively. Assume that $\sigma =\theta _{V_{2n+1},W_{2n}}(\pi )$. Then we have the following:

• (1) We have

  $$\begin{aligned} \phi _\sigma = \phi _\pi \otimes \chi _c. \end{aligned}$$

  In particular, we have a canonical identification $A_{\phi _\pi } = A_{\phi _\sigma }$.

• (2) The characters $\eta _\pi$ and $\eta _{\sigma }$ are related by

  $$\begin{aligned} \eta _\sigma (a) / \eta _\pi (a) = \varepsilon (\phi _\pi ^{a}) \cdot \varepsilon \left( \phi _\pi ^{a} \otimes \chi _c\right) \cdot \chi _c(-1)^{\frac{1}{2}\dim (\phi _\pi ^{a})} \in \{\pm 1\} \end{aligned}$$

  for any $a \in A_{\phi _\pi } = A_{\phi _\sigma }$.

• (3) Let $(\phi _{\pi ^\vee },\eta _{\pi ^\vee })$ be the L-parameter for $\pi ^\vee \in \mathrm {Irr}(\mathrm {Mp}(W_{2n}))$. Then we have

  $$\begin{aligned} \phi _{\pi ^\vee }= & {} \phi _\pi \otimes \chi _{-1} \quad \text {and }\\ \eta _{\pi ^\vee }(a) / \eta _\pi (a)= & {} \varepsilon \left( \phi _\pi ^{a}\right) \cdot \varepsilon \left( \phi _\pi ^{a} \otimes \chi _{-1}\right) \cdot \chi _{-1}(-1)^{\frac{1}{2}\dim (\phi _\pi ^{a})} \in \left\{ \pm 1\right\} \end{aligned}$$

  for any $a \in A_{\phi _\pi } = A_{\phi _{\pi ^\vee }}$.

Proof

This follows from [13, Theorem 1.5]. See also [2, § 3.6]. $\square$

Appendix C: Gross–Prasad conjecture

To prove main theorems, we used two highly non-trivial results. One is the Gross–Prasad conjecture, which gives an answer for restriction problems. The other is Prasad’s conjectures, which describe the local theta correspondence for (almost) equal rank cases. In this appendix, we state the Gross–Prasad conjecture (GP).

The Gross–Prasad conjecture consists of four cases; orthogonal, hermitian, symplectic-metaplectic, and skew-hermitian cases. For each case, the statements are slightly different. So we state each case separately. We refer the reader to [10, §6 and §18] for a discussion of the various subtleties in the definition of the characters which appear in the statements of conjecture.

First, we state the GP conjecture for the orthogonal cases.

Theorem C.1

(GP conjecture for the orthogonal cases) For an orthogonal space $V_m^\bullet$, we put $V_{m+1}^\bullet =V_m^\bullet \oplus L_{(-1)^{m+1}}$, where $L_{(-1)^{m+1}}$ is the orthogonal space of dimension 1 and discriminant $(-1)^{m+1}$. We set $V_{\mathrm {even}}$ and $V_{\mathrm {odd}}$ so that

$$\begin{aligned} \left\{ V_{\mathrm {even}},\ V_{\mathrm {odd}}\right\} = \left\{ V_m,\ V_{m+1}\right\} \quad \text {and}\quad \dim (V_{\mathrm {even}}) \in 2\mathbb {Z}. \end{aligned}$$

For $\phi \in \Phi _\mathrm {temp}(\mathrm {O}(V_{\mathrm {even}}))$, $\phi ' \in \Phi _\mathrm {temp}(\mathrm {O}(V_{\mathrm {odd}}))$ and $\nu \in \{\pm 1\}$, there exists a unique pair $(\sigma ,\sigma ') \in \Pi _\phi \times \Pi _{\phi '}$ such that

• $\sigma \otimes \sigma '$ is a representation of $\mathrm {O}(V_m^\bullet ) \times \mathrm {O}(V_{m+1}^\bullet )$ for some $V_m^\bullet$;

• the central character of $\sigma '$ corresponds to $\nu$;

• $\mathrm {Hom}_{\mathrm {O}(V_m^\bullet )}(\sigma \otimes \sigma ', \mathbb {C})\not =0$.

Moreover, $\iota (\sigma )$ and $\iota (\sigma ')$ are given by

$$\begin{aligned}&\left\{ \begin{aligned} \iota (\sigma )(a)&= \varepsilon \left( \phi ^{a} \otimes \phi '\right) \cdot \det (\phi ^{a})(-1)^{\frac{1}{2}\dim (\phi ')} \cdot \nu ^{\dim (\phi ^{a})},\\ \iota (\sigma ')(a')&= \varepsilon \left( \phi \otimes \phi '^{a'}\right) \cdot \det (\phi )(-1)^{\frac{1}{2}\dim (\phi '^{a'})} \end{aligned} \right. \end{aligned}$$

for $a \in A_\phi$ and $a' \in A_{\phi '}$.

The GP conjecture for the special orthogonal cases was proven in [47,48,49,50]. In [3], the authors extended this result to the full orthogonal cases under an assumption on LLC for $\mathrm {O}(V_{2n})$.

Secondly, we state the GP conjecture for the hermitian cases.

Theorem C.2

(GP conjecture for the hermitian cases) Suppose that $E \not =F$. For a hermitian space $V_m^\bullet$, we put $V_{m+1}^\bullet =V_m^\bullet \oplus L_{(-1)^{m}}$, where $L_{(-1)^{m}}$ is the hermitian space of dimension 1 and discriminant $(-1)^{m}$. For $\phi \in \Phi _\mathrm {temp}(\mathrm {U}(V_{m}))$ and $\phi ' \in \Phi _\mathrm {temp}(\mathrm {U}(V_{m+1}))$, there exists a unique pair $(\sigma ,\sigma ') \in \Pi _\phi \times \Pi _{\phi '}$ such that $\sigma \otimes \sigma '$ is a representation of $\mathrm {U}(V_m^\bullet ) \times \mathrm {U}(V_{m+1}^\bullet )$ for some $V_m^\bullet$, and

$$\begin{aligned} \mathrm {Hom}_{\mathrm {U}(V_m^\bullet )}\left( \sigma \otimes \sigma ', \mathbb {C}\right) \not =0. \end{aligned}$$

Moreover, $\iota (\sigma )$ and $\iota (\sigma ')$ are given by

$$\begin{aligned}&\left\{ \begin{aligned} \iota (\sigma )(a)&= \omega _{E/F}(-1)^{(m+1)\dim (\phi ^a)} \cdot \varepsilon \left( \phi ^{a} \otimes \phi ', \psi ^E_{2}\right) ,\\ \iota (\sigma ')(a')&= \omega _{E/F}(-1)^{m\dim (\phi '^{a'})} \cdot \varepsilon \left( \phi \otimes \phi '^{a'}, \psi ^E_{2}\right) \end{aligned} \right. \end{aligned}$$

for $a \in A_\phi$ and $a' \in A_{\phi '}$.

The GP conjecture for the hermitian cases was proven in [4,5,6].

Thirdly, we state the GP conjecture for the symplectic-metaplectic cases.

Theorem C.3

(GP conjecture for the symplectic-metaplectic cases) Let $W_n$ be a symplectic space. For $c\in F^\times$, we denote by $\omega _{\psi _c}$ be the Weil representation of $\mathrm {Mp}(W_n \otimes L_1)$ associated to the additive character $\psi _c(x):=\psi (cx)$ of F, where $L_1$ is the orthogonal space of dimension 1 and discriminant 1. For $\phi \in \Phi _\mathrm {temp}(\mathrm {Sp}(W_n))$ and $\phi ' \in \Phi _\mathrm {temp}(\mathrm {Mp}(W_{n}))$, there exists a unique pair $(\pi ,\pi ') \in \Pi _\phi \times \Pi _{\phi '}$ such that

$$\begin{aligned} \mathrm {Hom}_{\mathrm {Mp}(W_n)}(\pi \otimes \pi ', \omega _{\psi _c})\not =0. \end{aligned}$$

Moreover, $\iota (\pi )$ and $\iota (\pi ')$ are given by

$$\begin{aligned}&\left\{ \begin{aligned} \iota (\pi )(a)&= \varepsilon \left( \phi ^{a}\chi _c \otimes \phi '\right) \cdot \varepsilon \left( \phi \chi _c \otimes \phi '\right) ^{\det (a)}\\&\qquad \cdot \det (\phi ^{a})(-1)^{\frac{1}{2}\dim (\phi ')} \cdot \det (\phi ^a)(c),\\ \iota (\pi ')(a')&= \varepsilon \left( \phi \chi _c \otimes \phi '^{a'}\right) \cdot \varepsilon (\phi '^{a'}) \cdot \chi _c(-1)^{\frac{1}{2}\dim (\phi '^{a'})} \end{aligned} \right. \end{aligned}$$

for $a \in A_\phi$ and $a' \in A_{\phi '}$.

The GP conjecture for the symplectic-metaplectic cases was proven by [2] when $c=1$. For general c, it follows from [10, Proposition 18.1] and the case when $c=1$.

Finally, we state the GP conjecture for the skew-hermitian cases.

Theorem C.4

(GP conjecture for the skew-hermitian cases) Suppose that $E \not =F$. Let $W_n$ be a skew-hermitian space. For a character $\chi$ of $E^\times$ such that $\chi |F^\times =\omega _{E/F}$, we denote by $\omega _{\psi ,\chi }$ the Weil representation of $\mathrm {U}(W_n^\bullet )$ associated to $\psi$ and $\chi$. For $\phi , \phi ' \in \Phi _\mathrm {temp}(\mathrm {U}(W_n))$, there exists a unique pair $(\pi ,\pi ') \in \Pi _\phi \times \Pi _{\phi '}$ such that $\pi$ and $\pi '$ are representations of the same group $\mathrm {U}(W_n^\bullet )$ and

$$\begin{aligned} \mathrm {Hom}_{\mathrm {U}(W_n^\bullet )}\left( \pi \otimes \pi ', \omega _{\psi ,\chi }\right) \not =0. \end{aligned}$$

Moreover, $\iota (\pi )$ and $\iota (\pi ')$ are given by

$$\begin{aligned}&\left\{ \begin{aligned} \iota (\pi )(a)&=\varepsilon \left( \phi ^{a} \otimes \phi ' \otimes \chi ^{-1}, \psi ^E_{2}\right) ,\\ \iota (\pi ')(a')&=\varepsilon \left( \phi \otimes \phi '^{a'} \otimes \chi ^{-1}, \psi ^E_{2}\right) \end{aligned} \right. \end{aligned}$$

for $a \in A_\phi$ and $a' \in A_{\phi '}$.

The GP conjecture for the skew-hermitian cases was proven by [12]. We also use the following form.

Corollary C.5

Let the notation be as above. For $\phi , \phi ' \in \Phi _\mathrm {temp}(\mathrm {U}(W_n))$, there exists a unique pair $(\pi ,\pi ') \in \Pi _\phi \times \Pi _{\phi '}$ such that $\pi$ and $\pi '$ are representations of the same group $\mathrm {U}(W_n^\bullet )$ and

$$\begin{aligned} \mathrm {Hom}_{\mathrm {U}(W_n^\bullet )}\left( \pi \otimes \pi ', \overline{\omega _{\psi ,\chi }}\right) \not =0. \end{aligned}$$

Moreover, $\iota (\pi )$ and $\iota (\pi ')$ are given as follows:

$$\begin{aligned} \left\{ \begin{aligned} \iota (\pi )(a)&= \omega _{E/F}(-1)^{\dim (\phi ^a)} \cdot \varepsilon \left( \phi ^{a} \otimes \phi ' \otimes \chi , \psi ^E_{2}\right) ,\\ \iota (\pi ')(a')&= \omega _{E/F}(-1)^{\dim (\phi '^{a'})} \cdot \varepsilon \left( \phi \otimes \phi '^{a'} \otimes \chi , \psi ^E_{2}\right) \end{aligned} \right. \end{aligned}$$

for $a \in A_\phi$ and $a' \in A_{\phi '}$.

Proof

Since $\pi$ and $\pi '$ are tempered, we have $\pi ^\vee =\overline{\pi }$ and $\pi '^\vee =\overline{\pi '}$. The assertion follows from Theorem C.4 and Proposition B.4. $\square$

We also need the following lemma.

Lemma C.6

Let $V_m$ be a Hermitian space of dimension m and $W_n$ be a skew-Hermitian space of dimension n. Put $V_{m+1}=V_m \oplus L$ for some line L. If $E=F$, we set $G(W_n)$ and $G'(W_n)$ to be $\mathrm {Sp}(W_n)$ or $\mathrm {Mp}(W_n')$ such that $\{G(W_n), G(W_n')\}=\{\mathrm {Sp}(W_n), \mathrm {Mp}(W_n)\}$. Let $\omega =\omega _{\psi _c}$ or $\omega _{\psi ,\chi }$.

• (1) For $\sigma \in \mathrm {Irr}_\mathrm {temp}(H(V_{m+1}))$, there exists $\sigma ' \in \mathrm {Irr}_\mathrm {temp}(H(V_m))$ such that $\mathrm {Hom}_{H(V_m)}(\sigma \otimes \sigma ', \mathbb {C})\not =0$.

• (2) For $\pi \in \mathrm {Irr}_\mathrm {temp}(G(W_n))$, there exists $\pi ' \in \mathrm {Irr}_\mathrm {temp}(G'(W_n))$ such that $\mathrm {Hom}_{G(W_n)}(\pi \otimes \pi ', \omega )\not =0$.

Proof

The proof is similar to that of Lemma 12.5 in [13]. The absolutely convergence of double integrals which we need are proven in [21] for orthogonal cases, [16] for hermitian cases, [52] for symplectic-metaplectic cases, and [51] for skew-hermitian cases. $\square$

Appendix D: Prasad’s conjectures

In this appendix, we state Prasad’s conjectures [40], which are the other highly non-trivial results.

Let $(V_m,W_n)$ be as in Sect. 2.2. We have fixed a non-trivial additive character $\psi$ of F, and $\delta \in E^\times$ such that $\mathrm {tr}_{E/F}(\delta )=0$ if $E\not =F$. Recall that we put

$$\begin{aligned} \psi ^E_c(x)=\psi \left( \frac{c}{2}\mathrm {tr}_{E/F}(\delta x)\right) \end{aligned}$$

for $x\in E$ and $c \in F^\times$. If $c=1$, we simply write $\psi ^E=\psi ^E_1$. For a representation $\phi$ of $WD _E$, we write $\varepsilon (\phi ,\psi _c^E)=\varepsilon (1/2, \phi ,\psi _c^E)$.

First, we state Prasad’s conjecture for the equal rank case:

Theorem D.1

(Prasad’s conjecture for the equal rank case) Assume that $E\not =F$ and $m=n$. Hence $G(W_n)=\mathrm {U}(W_n)$ and $H(V_m^\pm )=\mathrm {U}(V_n^\pm )$. Let $\pi \in \mathrm {Irr}(\mathrm {U}(W_n))$ with L-parameter $(\phi ,\eta )$. Then we have the following:

• (1) There is a unique pure inner form $\mathrm {U}(V_n^\bullet )$ of $\mathrm {U}(V_n)$ such that $\Theta _{V_n^\bullet ,W_n}(\pi )$ is nonzero.

• (2) For given $\mathrm {U}(V_n^\bullet )$, the theta lift $\Theta _{V_n^\bullet ,W_n}(\pi )$ is nonzero if and only if

  $$\begin{aligned} \varepsilon (\phi \otimes \chi _V^{-1}, \psi ^E_2) = \omega _{E/F}\left( \delta ^{-n} \cdot \mathrm {disc}(V_n^\bullet ) \cdot \mathrm {disc}(W_n)\right) . \end{aligned}$$

• (3) Suppose $\Theta _{V_n^\bullet ,W_n}(\pi )$ is nonzero. Let $(\theta (\phi ),\theta (\eta ))$ be the L-parameter of $\theta _{V_n^\bullet ,W_n}(\pi )$. Then $\theta (\phi )=\phi \otimes \chi _V^{-1}\chi _W$. In particular, we have a canonical identification $A_\phi = A_{\theta (\phi )}$. Moreover, we have

  $$\begin{aligned} \theta (\eta )(a)/\eta (a)=\varepsilon \left( \phi ^{a}\otimes \chi _V^{-1},\psi ^E_2\right) \end{aligned}$$

  for $a \in A_\phi =A_{\theta (\phi )}$.

Next, we state Prasad’s conjecture for the almost equal rank case. If $E=F$ and $\epsilon =-1$, then $G(W_n)=\mathrm {O}(W_n)$ and $H(V_m)=\mathrm {Sp}(V_m)$. Recall that for $\pi \in \mathrm {Irr}(\mathrm {O}(W_n))$, we may consider the two theta lifts $\Theta _{V_m,W_n}(\pi )$ and $\Theta _{V_m,W_n}(\pi \otimes \det )$.

Theorem D.2

(Prasad’s conjecture for the almost equal rank case) Assume that $l=n-m+\epsilon _0=-1$. Let $\pi \in \mathrm {Irr}(G(W_n))$ with L-parameter $(\phi ,\eta )$. Then we have the following:

• (i) Suppose that $\phi$ does not contain $\chi _V$.

  • (a) For any pure inner form $H(V_m^\bullet )$ of $H(V_m)$, the theta lift $\Theta _{V_m^\bullet ,W_n}(\pi )$ is nonzero.

  • (b) Let $(\theta (\phi ),\theta (\eta ))$ be the L-parameter of $\theta _{V_m^\bullet ,W_n}(\pi )$. Then we have $\theta (\phi ) = (\phi \otimes \chi _V^{-1}\chi _W) \oplus \chi _W$. Hence there is a canonical injection $A_\phi \hookrightarrow A_{\theta (\phi )}$.

  • (c) We have $[A_{\theta (\phi )}:A_\phi ]=2$.

  • (d) The character $\theta (\eta )$ of $A_{\theta (\phi )}$ satisfies

    $$\begin{aligned} \theta (\eta )|A_\phi = \eta . \end{aligned}$$

• (ii) Suppose that $\phi$ contains $\chi _V$.

  • (a) Exactly one of two theta lifts $\Theta _{V_m,W_n}(\pi )$ and $\Theta _{V'_m,W_n}(\pi )$ (or $\Theta _{V_m,W_n}(\pi )$ and $\Theta _{V_m,W_n}(\pi \otimes \det )$) is nonzero.

  • (b) $\Theta _{V_m^\bullet ,W_n}(\pi )$ is nonzero if and only if

    $$\begin{aligned} \left\{ \begin{aligned}&\eta (z_\phi + e_1) = 1&\quad&\text {if }G(W_n)=\mathrm {O}(W_n) \text { and } H(V_m)=\mathrm {Sp}(V_m), \\&V_m^\bullet \in \mathcal {V}^{\eta (z_\phi )}&\quad&\text {otherwise}. \end{aligned} \right. \end{aligned}$$

    Here, $e_1$ is the element in $A_\phi$ corresponding to $\chi _V$.

  • (c) Suppose that $\Theta _{V_m^\bullet ,W_n}(\pi )$ is nonzero. Let $(\theta (\phi ),\theta (\eta ))$ be the L-parameter of $\theta _{V_m^\bullet ,W_n}(\pi )$. Then $\theta (\phi ) = (\phi \otimes \chi _V^{-1}\chi _W) \oplus \chi _W$. Hence there is a canonical injection $A_\phi \hookrightarrow A_{\theta (\phi )}$.

  • (d) We have $[A_{\theta (\phi )}:A_\phi ]=1$.

  • (e) The character $\theta (\eta )$ of $A_{\theta (\phi )}$ satisfies

    $$\begin{aligned} \theta (\eta )|A_\phi = \eta . \end{aligned}$$

Prasad’s conjectures (Theorems D.1 and D.2) are established by [12] when $E\not =F$. When $E=F$, Theorem D.2 is proven by [2] and [3].

By the conservation relation (Proposition 2.5), for any $\pi \in \mathrm {Irr}(G(W_n))$, we have

$$\begin{aligned} m^\mathrm {down}(\pi ) \le n+\epsilon _0+1. \end{aligned}$$

If $m^\mathrm {down}(\pi )= n+ \epsilon _0 +1$, then $m^\mathrm {up}(\pi )=m^\mathrm {down}(\pi )=n+ \epsilon _0 +1$. Namely, both of two theta lifts $\Theta _{V_m^\bullet ,W_n}(\pi )$ with $m=n+ \epsilon _0 +1$ are nonzero. In this case, $\phi$ does not contain $\chi _V$ by Theorem D.2.