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.

1 Introduction

The theory of local theta correspondence was initiated by Roger Howe almost 40 years ago and has since been a major theme in representation theory and the theory of automorphic forms. In this paper, we shall address some basic questions concerning the local theta correspondence. Let us briefly recall the setup in broad strokes, leaving the precise exposition to the main body of the paper.

Let F be a nonarchimedean local field of characteristic 0 and let E be F itself or a quadratic field extension of F. Fix $\epsilon = \pm 1$ and set $\epsilon _0 = \epsilon$ if $E = F$ and $\epsilon _0 = 0$ if E is a quadratic field. Consider a $-\epsilon$-Hermitian space $W_n$ over E of dimension n with associated isometry group $\mathrm {U}(W_n)$. Likewise, let $V_m$ be an $\epsilon$-Hermitian space over E of dimension m with associated isometry group $\mathrm {U}(V_m)$. Then

$$\begin{aligned} \mathrm {U}(W_n) \times \mathrm {U}(V_m) \longrightarrow \mathrm {Sp}( \mathrm {Res}_{E/F}(W_n \otimes _E V_m)) \end{aligned}$$

forms a reductive dual pair in the above symplectic group.

After fixing some extra data, the dual pair $\mathrm {U}(W_n) \times \mathrm {U}(V_m)$ has a Weil representation $\omega _{W_n,V_m}$. For an irreducible representation $\pi$ of $\mathrm {U}(W_n)$, the maximal $\pi$-isotypic quotient of $\omega _{W_n, V_m}$ has the form

$$\begin{aligned} \pi \boxtimes \Theta _{W_n, V_m}(\pi ) \end{aligned}$$

for some smooth representation $\Theta _{W_n, V_m}(\pi )$ of $\mathrm {U}(V_m)$ (known as the big theta lift of $\pi$). It was shown by Kudla that $\Theta _{W_n, V_m}(\pi )$ has finite length (possibly zero). The following basic result is known as the Howe duality conjecture (see [14, 15, 20, 45]):

Theorem 1.1

If $\Theta _{W_n, V_m}(\pi )$ is nonzero, then it has a unique irreducible quotient $\theta _{W_n, V_m}(\pi )$.

We call $\theta _{W_n, V_m}(\pi )$ the small theta lift of $\pi$ to $\mathrm {U}(V_m)$ and shall interpret it to be 0 if $\Theta _{W_n, V_m}(\pi )$ is zero. After the above theorem, it is natural to consider the following two basic problems:

Problem A

Determine precisely when $\theta _{W_n, V_m}(\pi )$ is nonzero.

Problem B

Determine $\theta _{W_n, V_m}(\pi )$ precisely when it is nonzero.

In this paper, we shall address these two problems for tempered representations $\pi$.

To formulate answers to these two problems, especially Problem B, it is necessary to have some sort of classification of irreducible representations of the groups $\mathrm {U}(W_n)$ and $\mathrm {U}(V_m)$. Such a classification is provided by the local Langlands correspondence (LLC). The recent results of Arthur [1], Mok [35], Kaletha–Mínguez–Shin–White [23] and Gan–Savin [13] meant that the LLC is almost completely known for the groups considered in this paper.

The LLC classifies the irreducible representations $\pi$ of $\mathrm {U}(W_n)$ by their L-parameters $(\phi , \eta )$, where

$$\begin{aligned} \phi : WD _E \rightarrow \mathrm {GL}_N(\mathbb {C}) \end{aligned}$$

is a conjugate self-dual representation of the Weil–Deligne group $WD _E = W_E \times \mathrm {SL}_2(\mathbb {C})$ of a certain sign, and

$$\begin{aligned} \eta \in \mathrm {Irr}(A_{\phi }) \end{aligned}$$

is an irreducible character of the component group $A_{\phi }$ associated to $\phi$. We may think of $\phi$ as the last name of the representation $\pi$ and $\eta$ its first name. Thus we shall address Problems A and B in terms of the last names and first names of tempered representations.

Before going on, let us give a reformulation of Problem A. Let $\mathcal {V}= (V_m)$ be a Witt tower of $\epsilon$-Hermitian spaces over E so that $V_{m+2} = V_m + \mathbb {H}$, where $\mathbb {H}$ is the hyperbolic plane. In particular, $m = \dim _E(V_m)$ is of a fixed parity. Then one has a Witt tower of local theta correspondence associated to the dual pair $\mathrm {U}(W_n) \times \mathrm {U}(V_m)$. It is known by Kudla that the number

$$\begin{aligned} m_{\mathcal {V}}(\pi ) = \min \left\{ m\ \big |\ \Theta _{V_m,W_n}(\pi ) \not = 0 \right\} \end{aligned}$$

is finite. Moreover, $\Theta _{V_m,W_n}(\pi ) \not = 0$ for all $m \ge m_{\mathcal {V}}(\pi )$. The number $m_{\mathcal {V}}(\pi )$ is called the first occurrence index of $\pi$ in the Witt tower $\mathcal {V}$. Addressing Problem A for $\pi$ is equivalent to determining the first occurrence index $m_{\mathcal {V}}(\pi )$ of $\pi$ in every Witt tower $\mathcal {V}$.

For this purpose, the so-called conservation relation reduces our workload by half. More precisely, given any Witt tower $\mathcal {V}$, there is a companion Witt tower $\mathcal {V}' = (V'_m)$. We shall denote the two Witt towers by $(V_m^+)$ and $(V_m^-)$ and denote the first occurrence indices of $\pi$ by $m^{\pm }(\pi )$ accordingly. The conservation relation, shown by Kudla–Rallis [26] and Sun–Zhu [43], says that

$$\begin{aligned} m^+(\pi ) + m^-(\pi ) = 2 \cdot ( n+ \epsilon _0 +1). \end{aligned}$$

This shows that

$$\begin{aligned} m^{\mathrm {down}}(\pi ) = \min \left\{ m^+(\pi ), m^-(\pi )\right\} \le n + \epsilon _0 +1 \end{aligned}$$

and

$$\begin{aligned} m^{\mathrm {up}}(\pi ) = \max \left\{ m^+(\pi ), m^-(\pi )\right\} \ge n + \epsilon _0 +1. \end{aligned}$$

To address Problems A and B, we need to determine:

• the value of $m^{\mathrm {down}}(\pi )$ and which of $m^{\pm }(\pi )$ it is equal to;

• the L-parameter $(\theta ^{\pm }_m(\phi ), \theta ^{\pm }_m(\eta ))$ of $\theta _{V^{\pm }_m,W_n}(\pi )$ if it is nonzero;

in terms of the L-parameter $(\phi , \eta )$ of $\pi$.

Let us describe our results in the special case of discrete series representations when $\mathrm {U}(W) \times \mathrm {U}(V) = \mathrm {Mp}_{2n} \times \mathrm {O}_{2m+1}$. More precisely, let $W_{2n}$ be the 2n-dimensional symplectic space and $V^{\pm }_{2m+1}$ be the two $(2m+1)$-dimensional quadratic spaces of discriminant 1, with $V_{2m+1}^+$ the split quadratic space. Let $\pi$ be an irreducible (genuine) discrete series representation of $\mathrm {Mp}(W_{2n})$, with L-parameter $(\phi , \eta )$. Thus

$$\begin{aligned} \phi = \bigoplus _{i=1}^r \phi _i \end{aligned}$$

is a direct sum of distinct irreducible symplectic representations of the Weil–Deligne group $WD _F = W_F \times \mathrm {SL}_2(\mathbb {C})$ of F and $\eta$ is a character of the component group

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

which is a $\mathbb {Z}/2\mathbb {Z}$-vector space with a canonical basis $\{a_i \}$ indexed by the summands $\phi _i$ of $\phi$. Let $z_{\phi }$ denote the element $\sum _{i=1}^r a_i \in A_{\phi }$. On the other hand, since $\mathrm {O}(V_{2m+1}^{\pm }) \cong \mathrm {SO}(V_{2m+1}^{\pm }) \times \mathbb {Z}/2\mathbb {Z}$, an irreducible representation of $\mathrm {O}(V_{2m+1}^{\pm })$ is parametrized by $(\phi ', \eta ', \nu ')$ where

• $\phi '$ is a symplectic representation of $WD _F$;

• $\eta '$ is an irreducible character of the component group $A_{\phi '}$;

• $\nu ' = \pm 1$ is a sign, with $\nu '=1$ corresponding to the trivial character of $\mathbb {Z}/2\mathbb {Z}$.

Now we consider the theta liftings of $\pi$ to the two Witt towers $\mathcal {V}^{\pm }$. The conservation relation says that

$$\begin{aligned} m^{\mathrm {down}}(\pi ) + m^{\mathrm {up}}(\pi ) = 4n+4, \end{aligned}$$

so that

$$\begin{aligned} m^{\mathrm {down}}(\pi ) \le 2n+1 \quad \text {and} \quad m^{\mathrm {up}}(\pi ) \ge 2n+3. \end{aligned}$$

Our main results in this case are summarized in the following three theorems:

Theorem 1.2

• (1) $m^{\mathrm {down}}(\pi ) = m^{\epsilon }(\pi )$ if and only if $\epsilon = \eta ( z_{\phi })$. We call $\mathcal {V}^{ \eta ( z_{\phi })}$ the going-down tower, and $\mathcal {V}^{ -\eta ( z_{\phi })}$ the going-up tower.

• (2) Consider the set $\mathcal {T}$ containing 0 and all even integers $l >0$ satisfying the following conditions:

  • (chain condition) $\phi$ contains $S_2 + S_4 +\cdots +S_l$, where $S_k$ denotes the (unique) k-dimensional irreducible representation of $\mathrm {SL}_2(\mathbb {C})$;

  • (initial condition) if $e_k$ denotes the basis element of $A_{\phi }$ associated to $S_k$, then $\eta (e_2) = 1$;

  • (alternating condition) $\eta (e_i) = - \eta (e_{i+2})$ for even $2 \le i \le l-2$.

Let

$$\begin{aligned} l(\pi ) = \max \, \mathcal {T}. \end{aligned}$$

Then

$$\begin{aligned} m^{\mathrm {down}}(\pi ) = 2n+1 - l(\pi ) \quad \text {and} \quad m^{\mathrm {up}}(\pi ) = 2n+3 + l(\pi ). \end{aligned}$$

In particular, the above theorem addresses Problem A.

Theorem 1.3

Consider the going-down tower $\mathcal {V}^{\eta (z_{\phi })}$. For each $V_{2m+1}$ in this Witt tower, with $2m+1 \ge m^{\mathrm {down}}(\pi ) = 2n+1- l(\pi )$, consider the theta lift $\theta _{W_{2n}, V_{2m+1}}(\pi )$ and let its L-parameter be given by $(\theta _{2m+1}(\phi ), \theta _{2m+1}(\eta ), \nu _{2m+1}(\phi , \eta ))$.

• (1) One has:

  $$\begin{aligned} \nu _{2m+1}(\phi , \eta ) = \eta (z_{\phi }) \cdot \epsilon (1/2, \phi ). \end{aligned}$$

• (2) If $m < n$, then

  $$\begin{aligned} \theta _{2m+1}(\phi ) = \phi - S_{2n-2m}. \end{aligned}$$

  Hence $\theta _{2m+1}(\phi )$ is a discrete series parameter and there is a natural injection $A_{\theta _{2m+1}(\phi )} \hookrightarrow A_{\phi }$. For the basis element $a_i$ of $A_{\theta _{2m+1}(\phi )}$ associated to an irreducible summand $\phi _i$, one has

  $$\begin{aligned} \theta _{2m+1}(\eta ) (a_i) / \eta (a_i)&= \epsilon \left( 1/2, \phi _i \otimes S_{2(n-m)-1}\right) \cdot \epsilon (1/2, \phi _i)\\&= \left\{ \begin{aligned}&-1&\quad&\text {if }\phi _i = S_{2k} \text { for some } 1 \le k \le n-m{-}1, \\&1&\quad&\text {otherwise}. \end{aligned}\right. \end{aligned}$$

• (3) If $m = n$, then

  $$\begin{aligned} \theta _{2m+1}(\phi ) = \phi \quad \text {and}\quad \theta _{2m+1}(\eta ) = \eta . \end{aligned}$$

  Hence $\theta _{2m+1}(\phi )$ is a discrete series parameter.

• (4) If $m > n$, then $\theta _{2m+1}(\pi )$ is non-tempered and is the unique Langlands quotient of the standard module

  $$\begin{aligned} \times _{i=1}^{m-n} \left| \cdot \right| ^{m-n+ \frac{1}{2}-i} \rtimes \theta _{2n+1}(\pi ). \end{aligned}$$

  In particular,

  $$\begin{aligned} \theta _{2m+1}(\phi ) = \phi \oplus \left( \bigoplus _{i=1}^{m-n} \left| \cdot \right| ^{m-n +\frac{1}{2} - i} \oplus \left| \cdot \right| ^{-\left( m-n + \frac{1}{2} -i\right) } \right) , \end{aligned}$$

  so that there is a natural identification $A_{\theta _{2m+1}(\phi )} \cong A_{\theta _{2n+1}(\phi )}$, and

  $$\begin{aligned} \theta _{2m+1}(\eta ) = \theta _{2n+1}(\eta ). \end{aligned}$$

Theorem 1.4

Consider the going-up tower $\mathcal {V}^{-\eta (z_{\phi })}$. For each $V_{2m+1}$ in this Witt tower, consider the theta lift $\theta _{W_{2n}, V_{2m+1}}(\pi )$ and let its L-parameter be given by $(\theta _{2m+1}(\phi ), \theta _{2m+1}(\eta ), \nu _{2m+1}(\phi , \eta ))$.

• (1) One has:

  $$\begin{aligned} \nu _{2m+1}(\phi , \eta ) = \eta (z_{\phi }) \cdot \epsilon (1/2, \phi ). \end{aligned}$$

• (2) If $2m+1 = m^{\mathrm {up}}(\pi )$, then $\theta _{2m+1}(\pi )$ is a tempered representation with

  $$\begin{aligned} \theta _{2m+1}(\phi ) = \phi + S_{l(\pi ) +2}, \end{aligned}$$

  so that there is a natural inclusion

  $$\begin{aligned} A_{\phi } \hookrightarrow A_{\theta _{2m+1}(\phi )}. \end{aligned}$$

  For the basis element $a_i$ of $A_{\theta _{2m+1}(\phi )}$ associated to an irreducible summand $\phi _i$, one has

  $$\begin{aligned} \theta _{2m+1}(\eta )(a_i) / \eta (a_i)&= \epsilon (1/2, \phi _i \otimes S_{l(\pi )+1}) \cdot \epsilon (1/2, \phi _i)\\&= \left\{ \begin{aligned}&-1&\quad&\text {if }\phi _i = S_{2k} \text { for some } 1 \le k \le l(\pi )/2,\\&1&\quad&\text {otherwise}. \end{aligned}\right. \end{aligned}$$

• (3) If $2m+1 > m^{\mathrm {up}}(\pi )$ (so that $m-n-1 - l(\pi ) > 0$), then $\theta _{2m+1}(\pi )$ is non-tempered and is the unique Langlands quotient of the standard module

  $$\begin{aligned} \times _{i=1}^{m-n-1-l(\pi )/2} \left| \cdot \right| ^{m-n+ \frac{1}{2} - i} \rtimes \theta _{m^{\mathrm {up}}(\pi )}(\pi ). \end{aligned}$$

  In particular,

  $$\begin{aligned} \theta _{2m+1}(\phi ) = \phi \oplus S_{l(\pi )+2} \oplus \left( \bigoplus _{i=1}^{m-n-1 - l(\pi )/2} |\cdot |^{m-n +\frac{1}{2} - i} \oplus \left| \cdot \right| ^{-(m-n +\frac{1}{2} -i)} \right) , \end{aligned}$$

  so that there is a natural identification $A_{\theta _{2m+1}(\phi )} \cong A_{\theta _{m^{\mathrm {up}}(\pi )}(\phi )}$ and

  $$\begin{aligned} \theta _{2m+1}(\eta ) = \theta _{m^{\mathrm {up}}(\pi )}(\eta ). \end{aligned}$$

Taken together, the above two theorems give precise determination of the theta lifts of any discrete series representation $\pi$ of $\mathrm {Mp}(W_{2n})$. In the case of tempered $\pi$, the results are in the same spirit, though slightly more involved to state.

We note that Problems A and B have been extensively studied by Muić [36,37,38,39], Mœglin [31, 32] and Matić [28,29,30], at least for the symplectic-orthogonal dual pairs and the metaplectic-orthogonal dual pairs. Their work uses the Mœglin–Tadić classification of discrete series representations of classical groups in terms of supercuspidal representations. At that point, the Mœglin–Tadić classification was conditional, and it may be viewed as a preliminary form of the LLC. As such, the formulation of the answers to Problems A and B in the various papers of Muić may seem somewhat complicated, as are the proofs. The formulation of our main results and their proofs are neater and more transparent. There are several reasons for this:

• the LLC affords a more efficient language to describe the answers;

• the theory of local theta correspondence is in a more mature state today than at the time of Muić’s work. For example, the conservation relation is now known and we do exploit it to simplify life;

• we make use of a wider spectrum of tools than Muić. For example, we use results of Gan–Ichino [11] on the behaviour of the standard gamma factors and Plancherel measures in the local theta correspondence, as well as results of Gan–Takeda [14] and Gan–Savin [13]. In the proofs of some of these results, the doubling see-saw diagram plays a crucial role. In addition, Problems A and B in the almost equal rank case were resolved in [12] for the unitary case and [2] for symplectic-orthogonal case by the local intertwining relation given by Arthur [1]. Muić, on the other hand, mainly made use of the computation of Jacquet modules and Kudla’s filtration.

However, the main innovation of this paper is the exploitation of the local Gross–Prasad conjecture (GP), which is now established, in addressing Problems A and B. Recall that the GP conjecture comes in two flavours: the Bessel case and the Fourier–Jacobi case. For tempered representations, the Bessel case was proved by Waldpsurger [47,48,49,50] for special orthogonal groups, and Beuzart-Plessis [4,5,6] for unitary groups. In [12] and [2], the Fourier–Jacobi case (for tempered representations) was deduced from the Bessel case by using the theta correspondence in the almost equal rank case. In particular, in the almost equal rank case, Problems A and B were fully addressed in [12] for unitary dual pairs, [2] and [3] for symplectic-orthogonal dual pairs, and [13] for metaplectic-orthogonal dual pairs, and these allow one to deduce the Fourier–Jacobi case of the GP conjecture from the Bessel case. In this paper, with the GP conjecture in hand, we turn the table around and use it to understand the theta correspondence for general dual pairs.

Let us give a brief summary of the contents of this paper. After describing some background material on theta correspondence and the LLC in Sects. 2 and 3, our main results are given in Sect. 4. In order not to overburden the reader with too much background material, we have placed the more precise description of LLC in Appendices A and B. The local Gross–Prasad conjecture and Prasad’s conjectures (which resolve Problems A and B for almost equal rank dual pairs) are placed in Appendices C and D, respectively. Note that in a prequel to this paper [3], we have discussed the LLC for full orthogonal groups and established the GP conjecture for full orthogonal groups. Finally the proofs of the main results are given in Sects. 5 and 6.

2 Local theta correspondence

In this section, we fix some notations.

2.1 Fields

Let F be a nonarchimedean local field of characteristic 0 and residue characteristic p. Let $\mathfrak {o}_F$ be the ring of integers of F, $\mathfrak {p}_F$ be the maximal ideal of $\mathfrak {o}_F$, $\varpi _F$ be a uniformizer of $\mathfrak {o}_F$, and $q_F$ be the cardinality of $\mathfrak {o}_F/\mathfrak {p}_F$. The absolute value $|\cdot |_F$ on F is normalized by $|\varpi _F|_F=q_F^{-1}$. We fix a non-trivial additive character $\psi$ of F.

Let E be either F itself or a quadratic extension of F, and $\omega _{E/F}$ be the quadratic character of $F^\times$ corresponding to E via the local class field theory. We denote the generator of $\mathrm {Gal}(E/F)$ by c. We define a non-trivial additive character $\psi _E$ of E by $\psi _E=\psi \circ \mathrm {tr}_{E/F}$. If $E\not =F$, we fix an element $\delta \in E^\times$ such that $\mathrm {tr}_{E/F}(\delta )=0$, and set

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

for $x\in E$ and $a \in F^\times$. If $a=1$, we simply write $\psi ^E=\psi ^E_1$. One should not confuse $\psi _E$ with $\psi ^E$. If $E=F$, we set

$$\begin{aligned} \psi _a(x)=\psi (ax) \end{aligned}$$

for $x\in F$ and $a \in F^\times$.

2.2 Spaces

Fix $\epsilon =\pm 1$ in $E^\times$. Let

$$\begin{aligned} W_n&=\text {a }-\epsilon \text {-Hermitian space over } E \text { of dimension } n \text { over } E,\\ V_m&=\text {an } \epsilon \text {-Hermitian space over } E \text { of dimension } m \text { over } E. \end{aligned}$$

We set

$$\begin{aligned} l=n-m+\epsilon _0 \quad \text {with}\quad \epsilon _0=\left\{ \begin{aligned}&\epsilon&\quad&\text {if }E=F,\\&0&\quad&\text {if }E\not = F, \end{aligned} \right. \end{aligned}$$

and

$$\begin{aligned} \kappa =\left\{ \begin{aligned}&1&\quad&\text {if }l \text { is odd }, \\&2&\quad&\text {if }l \text { is even }. \end{aligned} \right. \end{aligned}$$

We define the discriminant $\mathrm {disc}(V_m)$ and $\mathrm {disc}(W_n)$ as in [11, §2.2]. Note that

$$\begin{aligned} \mathrm {disc}(V_m) \in \left\{ \begin{aligned}&F^\times /F^{\times 2}&\quad&\text {if }E=F,\\&F^\times /N_{E/F}(E^\times )&\quad&\text {if }E\not =F \text { and } \epsilon =+1,\\&\delta ^m \cdot F^\times /N_{E/F}(E^\times )&\quad&\text {if }E\not =F \text { and } \epsilon =-1. \end{aligned} \right. \end{aligned}$$

2.3 Groups

We will consider the isometry groups associated to the pair $(V_m,W_n)$ of $\pm \epsilon$-Hermitian spaces. More precisely, we set:

$$\begin{aligned} G(W_n)=\left\{ \begin{aligned}&\text {the metaplectic group } \mathrm {Mp}(W_n),&\quad&\text {if }E=F, \epsilon =+1 \text { and } m \text { is odd },\\&\text {the isometry group of } W_n,&\quad&\text {otherwise}. \end{aligned} \right. \end{aligned}$$

We define $H(V_m)$ similarly by switching the roles of $W_n$ and $V_m$.

For a vector space X over E, we denote the general linear group of X by $\mathrm {GL}(X)$. Let $\det _X=\det _{\mathrm {GL}(X)}$ be the determinant on $\mathrm {GL}(X)$.

2.4 Representations

Let G be a p-adic group. We denote the category of smooth representations of G by $\mathrm {Rep}(G)$. Let $\mathrm {Irr}(G)$ be the set of equivalence classes of irreducible smooth (genuine) representations of G. We also denote by $\mathrm {Irr}_\mathrm {temp}(G)$ (resp. $\mathrm {Irr}_\mathrm {disc}(G)$) the subset of $\mathrm {Irr}(G)$ of classes of irreducible tempered representations (resp. discrete series representations).

For a parabolic subgroup $P=MN$ of G, let $\delta _P$ be the modulus character of P. For $(\pi _0,\mathcal {V}_0) \in \mathrm {Rep}(M)$, we define the normalized induction $\mathrm {Ind}_P^G(\pi _0)$ by the space of smooth functions $f :G \rightarrow \mathcal {V}_0$ such that

$$\begin{aligned} f(mng)=\delta _P(m)^{\frac{1}{2}} \cdot \pi _0(m)f(g) \quad \text {for } m \in M, n\in N \text { and } g \in G. \end{aligned}$$

The group G acts on $\mathrm {Ind}_P^G(\pi _0)$ by right translation. For $(\pi ,\mathcal {V}) \in \mathrm {Rep}(G)$, we define the normalized Jacquet module $R_P(\pi )$ by $R_P(\pi )=\mathcal {V}/\mathcal {V}(N)$, where $\mathcal {V}(N)$ is the subspace generated by $\pi (n)v-v$ for $n\in N$ and $v\in \mathcal {V}$. Note that $\mathcal {V}(N)$ is an M-subrepresentation of $\mathcal {V}$. The group M acts on $R_P(\pi )$ by

$$\begin{aligned} m\cdot \left( v \bmod \mathcal {V}(N)\right) = \delta _P(m)^{-\frac{1}{2}} \cdot \pi (m)v \bmod \mathcal {V}(N) \end{aligned}$$

for $m \in M$ and $v \in \mathcal {V}$.

We have the normalized induction functor

$$\begin{aligned} \mathrm {Ind}_P^G :\mathrm {Rep}(M) \rightarrow \mathrm {Rep}(G) \end{aligned}$$

and the normalized Jacquet functor

$$\begin{aligned} R_P :\mathrm {Rep}(G) \rightarrow \mathrm {Rep}(M). \end{aligned}$$

Let $\overline{P}= M\overline{N}$ be the opposite parabolic subgroup to P. Then there exist two Frobenius reciprocity formulas:

$$\begin{aligned} \mathrm {Hom}_G\left( \pi ,\mathrm {Ind}_P^G(\pi _0)\right) \cong \mathrm {Hom}_M(R_P(\pi ),\pi _0) \quad \text {(standard Frobenius reciprocity)} \end{aligned}$$

and

$$\begin{aligned} \mathrm {Hom}_G\left( \mathrm {Ind}_P^G(\pi _0),\pi \right) \cong \mathrm {Hom}_M\left( \pi _0, R_{\overline{P}}(\pi )\right) \quad \text {(Bernstein's Frobenius reciprocity)}. \end{aligned}$$

2.5 Parabolic inductions

We shall use Tadić’s notation for induced representations. Let $W_n$ be a $-\epsilon$-Hermitian space, and $G(W_n)$ as in Sect. 2.3. If $X_t$ is a t-dimensional isotropic subspace of $W_n$, we decompose

$$\begin{aligned} W = X_t \oplus W_{n-2t} \oplus X_t^*, \end{aligned}$$

where $X_t^*$ is a t-dimensional isotropic subspace of $W_n$ such that $X_t \oplus X_t^*$ is non-degenerate, and $W_{n-2t}$ is the orthogonal complement of $X_t \oplus X_t^*$ in $W_n$. We denote by $P(X_t) = L(X_t) \cdot U(X_t)$ the maximal parabolic subgroup stabilizing $X_t$, where $L(X_t) = \mathrm {GL}(X_t) \times G(W_{n-2t})$ is the Levi subgroup of $P(X_t)$ stabilizing $X_t^*$. If $\tau \in \mathrm {Irr}( \mathrm {GL}(X_t))$ and $\pi _0 \in \mathrm {Irr}(G(W_{n-2t}))$, we write

$$\begin{aligned} \tau \rtimes \pi _0 :=\mathrm {Ind}_{P(X_t)}^{G(W_n)}\left( \tau \otimes \pi _0\right) . \end{aligned}$$

More generally, a standard parabolic subgroup P of G(W) has the Levi factor of the form $\mathrm {GL}_{n_1}(E) \times \cdots \times \mathrm {GL}_{n_r}(E) \times G(W_{n_0})$, and we set

$$\begin{aligned} \tau _1 \times \cdots \times \tau _r \rtimes \pi _0 :=\mathrm {Ind}_P^{G(W_n)}\left( \tau _1 \otimes \cdots \otimes \tau _r \otimes \pi _0\right) , \end{aligned}$$

where $\tau _i$ is a representation of $\mathrm {GL}_{n_i}(E)$ and $\pi _0$ is a representation of $G(W_{n_0})$. When $G(W_n) = \mathrm {Mp}(W_n)$ is a metaplectic group, we will follow the convention of [13, § 2.2–2.5] for the normalized parabolic induction.

2.6 Galois conjugate

Recall that c denotes the generator of $\mathrm {Gal}(E/F)$. Let X be a vector space over E of dimension t. Choose a basis $\{x_j\}$ of X, and we set

$$\begin{aligned} i :\mathrm {GL}_t(E) \rightarrow \mathrm {GL}(X),\quad g\mapsto \left[ (x_1,\dots ,x_t)\mapsto (x_1,\dots ,x_t)g\right] . \end{aligned}$$

For a representation $\tau$ of $\mathrm {GL}(X)$, we define the c-conjugate ${}^c\tau$ of $\tau$ by

$$\begin{aligned} {}^c\tau (h):=\tau \left( i \circ c \circ i^{-1}(h)\right) \end{aligned}$$

for $h \in \mathrm {GL}(X)$. Let $\{x'_j\}$ be another basis of X and we denote by $i' :\mathrm {GL}_t(E) \rightarrow \mathrm {GL}(X)$ the corresponding map. If $A \in \mathrm {GL}_t(E)$ satisfies

$$\begin{aligned} (x'_1,\dots ,x'_t)=(x_1,\dots ,x_t) \cdot A, \end{aligned}$$

then we have $i'(g)=i(A \cdot g \cdot A^{-1})$, and so

$$\begin{aligned} i' \circ c \circ i'^{-1}(h)=i\left( A \cdot {}^c A^{-1}\right) \cdot i \circ c \circ i^{-1}(h) \cdot i\left( A \cdot {}^c A^{-1}\right) ^{-1} \end{aligned}$$

for $h \in \mathrm {GL}(X)$. This shows that the equivalence class of ${}^c\tau$ is independent of the choice of a basis of X.

2.7 $\mathrm {MVW}$ functor

Let $\delta$ be an F-linear automorphism on $W_n$ such that $\delta G(W_n) \delta ^{-1} = G(W_n)$. For a representation $\pi$ of $G(W_n)$, we denote by $\pi ^\delta$ the representation of $G(W_n)$ defined by conjugation, i.e., $\pi ^\delta (g) = \pi ( \delta g \delta ^{-1})$. The following proposition is in Chapter 4.$\mathrm {II}$.1 in [33].

Proposition 2.1

Let $\pi$ be an irreducible admissible representation of $G(W_n)$ and $\pi ^\vee$ be the contragredient of $\pi$. Let $\delta$ be an E-conjugate linear automorphism on $W_n$ such that

$$\begin{aligned} \langle \delta x, \delta y \rangle = \langle y,x \rangle \end{aligned}$$

for $x, y \in W_n$. Here, $\langle -,- \rangle$ denotes the Hermitian pairing of $W_n$. Then, $\pi ^\delta \cong \pi ^\vee$.

Fix $\delta$ as in Proposition 2.1. We define a functor

$$\begin{aligned} \mathrm {MVW}:\mathrm {Rep}(G(W_n)) \rightarrow \mathrm {Rep}(G(W_n)) \end{aligned}$$

by $\pi ^\mathrm {MVW}= \pi ^\delta$. Note that $\mathrm {MVW}$ is independent of the choice of $\delta$. By the definition and Proposition 2.1, we see that

• $\mathrm {MVW}$ is an involution, i.e., $(\pi ^\mathrm {MVW})^\mathrm {MVW}\cong \pi$;

• $\mathrm {MVW}$ is a covariant functor;

• $\mathrm {Ind}_{P(X_t)}^{G(W_n)}(\tau \otimes \pi _0)^\mathrm {MVW}\cong \mathrm {Ind}_{P(X_t)}^{G(W_n)}({}^c\tau \otimes \pi _0^\mathrm {MVW})$ for $\tau \in \mathrm {Irr}(\mathrm {GL}(X_t))$ and $\pi _0 \in \mathrm {Rep}(G(W_{n-2t}))$;

• if $\pi$ is irreducible, then $\pi ^\mathrm {MVW}\cong \pi ^\vee$.

We will use $\mathrm {MVW}$ in the following form.

Lemma 2.2

Let P be a standard parabolic subgroup of $G(W_n)$ with the Levi factor of the form $\mathrm {GL}_{n_1}(E) \times \cdots \times \mathrm {GL}_{n_r}(E) \times G(W_{n_0})$. Then for $\tau _i \in \mathrm {Irr}(\mathrm {GL}_{n_i}(E))$, $\pi _0 \in \mathrm {Irr}(G(W_{n_0}))$ and $\pi \in \mathrm {Irr}(G(W_{n}))$, the following are equivalent:

• (1) $\pi$ is a subrepresentation of $\tau _1\times \cdots \times \tau _r \rtimes \pi _0$;

• (2) $\pi$ is a quotient of ${}^c\tau _1^\vee \times \cdots \times {}^c\tau _r^\vee \rtimes \pi _0$.

Proof

Use both the contragredient functor and the $\mathrm {MVW}$ functor. $\square$

2.8 Weil representations

Let $(V,W)=(V_m,W_n)$ be as in Sect. 2.2. We consider the Weil representation of the pair $G(W) \times H(V)$. We fix a pair of characters ${\varvec{\chi }}=(\chi _{V_m},\chi _{W_n})$ of $E^\times$ as in [11, §3.2]. When there is no fear of confusion, $\chi _{V_m}$ and $\chi _{W_n}$ are simply denoted by $\chi _V$ and $\chi _W$, respectively. Note that ${}^c \chi _V^{-1}=\chi _V$ and ${}^c \chi _W ^{-1} =\chi _W$. Moreover, if $V_m$ (resp. $W_n$) is a symplectic space, then $\chi _V={\mathbf{1}}$ (resp. $\chi _W={\mathbf{1}}$). These data and $\psi$ give a splitting $G(W) \times H(V) \rightarrow \mathrm {Mp}(W \otimes V)$ of the dual pair. More precisely, see [17, 25] and [11, § 3.3]. Pulling back the Weil representation of $\mathrm {Mp}(W \otimes V)$ to $G(W) \times H(V)$ via this splitting, we obtain the associated Weil representation $\omega _{V,W,{\varvec{\chi }},\psi }$ of $G(W) \times H(V)$. We simply write $\omega _{V,W}$ for the Weil representation.

2.9 Theta correspondence

Let $\omega _{V,W}$ be the Weil representation of $G(W) \times H(V)$. For $\pi \in \mathrm {Irr}(G(W))$, the maximal $\pi$-isotypic quotient of $\omega _{V,W}$ is of the form

$$\begin{aligned} \pi \boxtimes \Theta _{V,W}(\pi ), \end{aligned}$$

where $\Theta _{V,W}(\pi )$ is a smooth representation of H(V). We emphasize that $\Theta _{V,W}(\pi )$ depends on ${\varvec{\chi }}$ and $\psi$ also. It was shown by Kudla [24] that $\Theta _{V,W}(\pi )$ is either zero or of finite length.

The following result is proven by Waldspurger [45] when $p\not =2$ and by [14, 15] in general.

Theorem 2.3

(Howe duality conjecture) If $\Theta _{V,W}(\pi )$ is nonzero, then $\Theta _{V,W}(\pi )$ has a unique irreducible quotient $\theta _{V,W}(\pi )$.

2.10 First occurrence and tower property

Fix $\epsilon =\pm 1$. Let $W_n$ be a $-\epsilon$-Hermitian space as in Sect. 2.2. For an anisotropic $\epsilon$-Hermitian space $V_{m_0}$ and $r\ge 0$, we put

$$\begin{aligned} V_{m_0+2r}=V_{m_0} \oplus \mathbb {H}^r, \end{aligned}$$

where $\mathbb {H}$ is the hyperbolic plane. The collection

$$\begin{aligned} \mathcal {V}=\left\{ V_{m_0+2r}\ \big |\ r\ge 0\right\} \end{aligned}$$

is called a Witt tower of spaces. Note that $\mathrm {disc}(V_m)$ and the parity of $\dim (V_m)$ depend only on the Witt tower $\mathcal {V}$ to which $V_m$ belongs. One can consider a tower of the theta correspondence associated to reductive dual pairs $\{(G(W_n),H(V_{m}))\ | \ V_m \in \mathcal {V}\}$. For $\pi \in \mathrm {Irr}(G(W_n))$, we have the representation $\Theta _{V_{m},W_n}(\pi )$ of $H(V_{m})$. The number

$$\begin{aligned} m_\mathcal {V}(\pi ) = \min \left\{ m\ \big |\ \Theta _{V_{m},W_n}(\pi )\not =0\right\} \end{aligned}$$

is finite and is called the first occurrence index of $\pi$ for the Witt tower $\mathcal {V}$, and the representation $\theta _{V_{m_\mathcal {V}(\pi )},W_n}(\pi )$ is called the first occurrence of $\pi$ for this Witt tower.

The following proposition is often referred to as the tower property of theta correspondence (see [24]).

Proposition 2.4

Let $m_\mathcal {V}(\pi )$ be the first occurrence index of $\pi$ for the Witt tower $\mathcal {V}=\{V_{m}\}$. Then we have $\Theta _{V_{m},W_n}(\pi )\not =0$ for any $m \ge m_\mathcal {V}(\pi )$.

If $E\not =F$ or $\epsilon =+1$, for a given Witt tower $\mathcal {V}=\{V_m\}$, there exists another Witt tower $\mathcal {V}'=\{V'_{m'}\}$ such that

• $V_m \ncong V'_{m'}$;

• $\dim (V_m)\equiv \dim (V'_{m'})\bmod 2$;

• $\mathrm {disc}(V_{m})=\mathrm {disc}(V'_{m'})$ if $E=F$ and $\epsilon =+1$.

We call $\mathcal {V}'$ the companion Witt tower of $\mathcal {V}$. Also, by a companion space of $V_m$, we mean $V_m$ or $V_m'$. For each $\pi \in \mathrm {Irr}(G(W_n))$, we may consider two first occurrence indices $m_\mathcal {V}(\pi )$ and $m_{\mathcal {V}'}(\pi )$. Let $\mathcal {V}^+=\{V^+_{m}\}$ be the Witt tower whose anisotropic space is

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

Here, we consider $V_m$ as a vector space equipped with a suitable Hermitian pairing. For example, by $(F, 2\mathrm {disc}(V_m))$, we mean the one dimensional space equipped with the bilinear form

$$\begin{aligned} (x,y) \mapsto 2d xy, \end{aligned}$$

where $d \in F^\times$ satisfies $d \bmod F^{\times 2} = \mathrm {disc}(V_m)$ in $F^\times / F^{\times 2}$. Note that this space has discriminant $\mathrm {disc}(V_m)$. We denote the other Witt tower by $\{V^-_{m}\}$. Then for each $\pi \in \mathrm {Irr}(G(W_n))$, we have two first occurrence indices $m^\pm (\pi ) :=m_{\mathcal {V}^\pm }(\pi )$.

On the other hand, if $E=F$ and $\epsilon =-1$, then there is only a single tower of symplectic spaces $\mathcal {V}=\{V_{m}\}$. In this case, a companion space of $V_m$ is just $V_m$. However, since $\pi$ is a representation of the orthogonal group $G(W_n)=\mathrm {O}(W_n)$, we may consider its twist $\pi \otimes \det$. Thus we have the two towers of theta lifts

$$\begin{aligned} \Theta _{V_{m},W_{n}}(\pi ) \quad \text {and}\quad \Theta _{V_{m},W_{n}}(\pi \otimes \det ). \end{aligned}$$

Hence we may define two first occurrence indices for each $\pi \in \mathrm {Irr}(G(W_n))$. When n is odd, we define $m^\pm (\pi )$ by

$$\begin{aligned} m^\pm (\pi ):= & {} \min \left\{ m\ \big |\ \Theta _{V_m,W_n}(\pi ') \not =0 \text { with } \pi ' \in \left\{ \pi , \pi \otimes \det \right\} \right. \\&\left. \text { such that } \pi '(-{\mathbf{1}}_{W_n})=\pm \mathrm {id}\right\} . \end{aligned}$$

When n is even, we define $m^\pm (\pi )$ by

$$\begin{aligned} m^+(\pi ) :=\min \Big \{ \min \big \{m\ \big |\ \Theta _{V_m,W_n}(\pi ) \not =0\big \},\ \min \big \{m\ \big |\ \Theta _{V_m,W_n}(\pi \otimes \det ) \not =0\big \}\Big \},\\ m^-(\pi ) :=\max \Big \{ \min \big \{m\ \big |\ \Theta _{V_m,W_n}(\pi ) \not =0\big \},\ \min \big \{m\ \big |\ \Theta _{V_m,W_n}(\pi \otimes \det ) \not =0\big \} \Big \}. \end{aligned}$$

Hence, in this case, $m^+(\pi )\le m^-(\pi )$ by convention.

In any case, for each $\pi \in \mathrm {Irr}(G(W_n))$, we have two first occurrence indices $m^\pm (\pi )$. We put

$$\begin{aligned} m^{\mathrm {up}}(\pi )=\max \left\{ m^+(\pi ), m^-(\pi )\right\} \quad \text {and}\quad m^{\mathrm {down}}(\pi )=\min \left\{ m^+(\pi ), m^-(\pi )\right\} . \end{aligned}$$

The following proposition is often referred to as the conservation relation (see [43]).

Proposition 2.5

For any $\pi \in \mathrm {Irr}(G(W_{n}))$, we have

$$\begin{aligned} m^\mathrm {up}(\pi )+m^\mathrm {down}(\pi )=2n+2+2\epsilon _0. \end{aligned}$$

This relation shows that

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

If we put

$$\begin{aligned} l=n-m^\mathrm {down}(\pi )+\epsilon _0, \end{aligned}$$

then we have $l \ge -1$.

3 Parametrization of irreducible representations

In this section, we explain the local Langlands correspondence (LLC) quickly. More precisely, see Appendix B.

Let $WD _E=W_E \times \mathrm {SL}_2(\mathbb {C})$ be the Weil–Deligne group of E. We define $\Phi (H(V_m))$, which is a set of equivalence classes of representations of $WD _E$, in the various cases as follows:

$$\begin{aligned} \left\{ \begin{aligned} \Phi (\mathrm {O}(V_m))&=\left\{ \phi : WD _F \rightarrow \mathrm {Sp}(m-1,\mathbb {C})\right\} /\cong ,&\quad&\text {if }m \text { is odd},\\ \Phi (\mathrm {Sp}(V_m))&=\left\{ \phi : WD _F \rightarrow \mathrm {SO}(m+1,\mathbb {C})\right\} /\cong ,\\ \Phi (\mathrm {O}(V_m))&=\left\{ \phi : WD _F \rightarrow \mathrm {O}(m,\mathbb {C})\ \big |\ \det (\phi )=\chi _{V}\right\} /\cong ,&\quad&\text {if }m \text { is even},\\ \Phi (\mathrm {Mp}(V_m))&=\left\{ \phi : WD _F \rightarrow \mathrm {Sp}(m,\mathbb {C})\right\} /\cong . \end{aligned} \right. \end{aligned}$$

For the unitary group $\mathrm {U}(m)$, we define $\Phi (\mathrm {U}(m))$ to be the set of equivalence classes of conjugate self-dual representations of $WD _E$ of sign $(-1)^{m-1}$. For the notion of conjugate self-dual representations, see Appendix A.3.

We say that $\phi \in \Phi (H(V_m))$ is tempered if $\phi (W_E)$ is bounded. We denote by $\Phi _\mathrm {temp}(H(V_m))$ the subset of equivalence classes of tempered $\phi$. For $\phi \in \Phi (H(V_m))$, we denote by $L(s,\phi )$, $\varepsilon (s,\phi ,\psi ')$, and $\gamma (s,\phi ,\psi ')$ the L-, $\varepsilon$-, and $\gamma$-factors associated to $\phi$, respectively. Here, $\psi '$ is a non-trivial additive character of E. The root number $\varepsilon (1/2,\phi ,\psi ')$ is also denoted by $\varepsilon (\phi )$ or $\varepsilon (\phi ,\psi ')$.

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 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 of the same type as $\phi$. We define the component group $A_\phi$ by

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

Namely, $A_\phi$ is a free $\mathbb {Z}/2\mathbb {Z}$-module of rank r and $\{a_1, \dots , a_r\}$ is a basis of $A_\phi$ with $a_i$ associated to $\phi _i$. 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 set

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

We call $z_\phi$ the central element in $A_\phi$. There is a homomorphism

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

where $\varepsilon _i\in \{0,1\} = \mathbb {Z}/2\mathbb {Z}$.

The LLC classifies $\mathrm {Irr}(H(V_m))$ as follows:

Desideratum 3.1

• (1) There exists a partition

  $$\begin{aligned} \bigsqcup _{V_m^\bullet }\mathrm {Irr}(H(V_m^\bullet )) = \bigsqcup _{\phi \in \Phi (H(V_m))}\Pi _\phi , \end{aligned}$$

  where $V_m^\bullet$ runs over all companion spaces of $V_m$. We call $\Pi _\phi$ the L-packet of $\phi$.

• (2) $\pi \in \mathrm {Irr}(H(V_m^\bullet ))$ is tempered if and only if $\pi$ belongs to $\Pi _\phi$ for tempered $\phi$.

• (3) There exists a map

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

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

• (4) The map $\iota$ is surjective unless $H(V_m) = \mathrm {Sp}(V_m)$ is a symplectic group. In this case, the image of $\iota$ is given by

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

• (5) The map $\iota$ is injective unless $H(V_m)=\mathrm {O}(V_m)$ is an odd orthogonal group (i.e., m is odd). In this case, each fiber of this map is of the form

  $$\begin{aligned} \left\{ \pi ,\ \pi \otimes \det \right\} . \end{aligned}$$

  Hence the map

  $$\begin{aligned} \Pi _\phi \rightarrow \widehat{A_\phi } \times \left\{ \pm 1\right\} ,\ \pi \mapsto \left( \iota (\pi ), \omega _\pi (-1)\right) \end{aligned}$$

  is bijective, where $\omega _\pi$ is the central character of $\pi$.

• (6) Suppose that $V_m^-$ exists. Then $\pi \in \Pi _\phi$ is a representation of $H(V_m^-)$ if and only if $\iota (\pi )(z_\phi )=-1$.

Therefore, unless $H(V_m)=\mathrm {O}(V_m)$ is an odd orthogonal group, $\pi \in \mathrm {Irr}(H(V_m))$ is parametrized by $(\phi ,\eta )$, where $\phi \in \Phi (H(V_m))$ and $\eta \in \widehat{A_\phi }$. If $H(V_m)=\mathrm {O}(V_m)$ is an odd orthogonal group, $\pi \in \mathrm {Irr}(H(V_m))$ is parametrized by the triple $(\phi , \eta , \nu )$, where $\phi \in \Phi (H(V_m))$, $\eta \in \widehat{A_\phi }$ and $\nu \in \{\pm 1\}$. The pair $(\phi ,\eta )$ is called the L-parameter for $\pi$. We also call $\phi$ and $\eta$ the last name and the first name of $\pi$, respectively.

Remark 3.2

The map $\iota :\Pi _\phi \rightarrow \widehat{A_\phi }$ may not be canonical. To specify $\iota$, we need to choose a Whittaker datum for $H(V_m)$. More precisely, see Remark B.2 below.

Suppose that $H(V_m)=\mathrm {O}(V_m)$ is an even orthogonal group (i.e., m is even). Then the following are equivalent:

• $\phi \in \Phi (\mathrm {O}(V_m))$ contains an irreducible orthogonal representation of $WD _F$ of odd dimension;

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

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

4 Main results

The purpose of this paper is to describe theta lifts of tempered representations in terms of the local Langlands correspondence. In this section, we state the main results over 3 theorems. Though we formulate the main results as 3 theorems, these are proven together (in Sect. 6).

We denote by $S_r$ the unique irreducible algebraic representation of $\mathrm {SL}_2(\mathbb {C})$ of dimension r. When $\chi$ is a character of $E^\times$, we regard $\chi$ as a character of $W_E$ via the local class field theory $W_E^{\mathrm {ab}} \cong E^\times$, so that $\chi S_r = \chi \boxtimes S_r$ is a representation of $WD _E = W_E \times \mathrm {SL}_2(\mathbb {C})$. The first main theorem gives an answer to Problem A in Sect. 1 for tempered representations.

Theorem 4.1

Let $(V_m,W_n)$ and $\kappa \in \{1,2\}$ be as in Sect. 2.2, and $\pi \in \mathrm {Irr}_\mathrm {temp}(G(W_n))$ with L-parameter $(\phi ,\eta )$.

• (1) Consider the set $\mathcal {T}$ containing $\kappa -2$ and all integers $l >0$ with $l \equiv \kappa \bmod 2$ satisfying the following conditions:

  • (chain condition) $\phi$ contains $\chi _V S_{r}$ for $r=\kappa , \kappa +2, \dots , l$;

  • (odd-ness condition) the multiplicity $m_\phi (\chi _V S_{r})$ is odd for $r=\kappa , \kappa +2, \dots , l-2$;

  • (initial condition) if $\kappa =2$, then

    $$\begin{aligned} \eta (e_2)= \left\{ \begin{aligned}&\epsilon \cdot \delta (\chi _V={\mathbf{1}})&\quad&\text {if }E=F \text { and } m \not \equiv n \bmod 2,\\&-1&\quad&\text {if }E\not =F \text { and } m \equiv n \bmod 2; \end{aligned} \right. \end{aligned}$$

  • (alternating condition) $\eta (e_{r})=-\eta (e_{r+2})$ for $r=\kappa , \kappa +2, \dots , l-2$.

  Here, $e_r$ is the element in $A_\phi$ corresponding to $\chi _V S_r$, and for a character $\chi$, we put

  $$\begin{aligned} \delta (\chi ={\mathbf{1}})=\left\{ \begin{aligned}&+1&\quad&\text {if }\chi ={\mathbf{1}},\\&-1&\quad&\text {otherwise}. \end{aligned} \right. \end{aligned}$$

  Let

  $$\begin{aligned} l(\pi ) = \max \, \mathcal {T}. \end{aligned}$$

  Then

  $$\begin{aligned} m^{\mathrm {down}}(\pi ) = n + \epsilon _0 - l(\pi ) \quad \text {and} \quad m^{\mathrm {up}}(\pi ) = n+2+\epsilon _0 + l(\pi ). \end{aligned}$$

• (2) If $l(\pi )=-1$, then $m^\mathrm {up}(\pi )=m^\mathrm {down}(\pi )$. Suppose that $l(\pi )\ge 0$. Then $\phi$ contains $\chi _V$ if $\kappa =1$. Moreover, $m^\mathrm {down}(\pi )=m^\alpha (\pi )$ if and only if

  $$\begin{aligned} \alpha =\left\{ \begin{aligned}&\eta (z_\phi +e_1)&\quad&\text {if }\kappa =1,\\&\eta (z_\phi )\cdot \varepsilon (\phi )\cdot \varepsilon \left( \phi \otimes \chi _V\right) \cdot \chi _V(-1)^{\frac{n}{2}}&\quad&\text {if }E=F, m\not \equiv n \bmod 2 \text { and } \epsilon =+1,\\&\eta (z_\phi )\cdot \varepsilon (\phi )&\quad&\text {if }E=F, m\not \equiv n \bmod 2 \text { and } \epsilon =-1,\\&\eta (z_\phi ) \cdot \varepsilon \left( \phi \otimes \chi _V^{-1}, \psi _2^E\right)&\quad&\text {if }E\not =F \text { and } m\equiv n \bmod 2. \end{aligned} \right. \end{aligned}$$

We call $\mathcal {V}^\mathrm {down}:=\mathcal {V}^{\alpha }$ (resp. $\mathcal {V}^\mathrm {up}:=\mathcal {V}^{-\alpha }$) the going-down tower (resp. the going-up tower) with respect to $\pi$.

Remark 4.2

Recall that when $(G(W_n),H(V_m)) = (\mathrm {O}(W_n), \mathrm {Sp}(V_{m}))$ with even n, by the definition, $m^\mathrm {down}(\pi ) = m^+(\pi )$ for each $\pi \in \mathrm {Irr}(\mathrm {O}(W_n))$ (see Sect. 2.10). In this case, (2) asserts that if $\pi \in \mathrm {Irr}(\mathrm {O}(W_n))$ satisfies that $\Theta _{V_m,W_n}(\pi ) \not =0$ and $\Theta _{V_m,W_n}(\pi \otimes \det ) =0$ for some $m \le n$, then the L-parameter $(\phi ,\eta )$ of $\pi$ satisfies that $\phi \supset {\mathbf{1}}$ and $\eta (z_\phi + e_1) =1$. This follows from Prasad’s conjecture (Theorem D.2 below).

The proof of Theorem 4.1 is given in Sect. 6. We give an indication for the relevant results. To prove (1), it is enough to show the following two statements:

• If $\Theta _{V_m^\bullet ,W_n}(\pi ) \not =0$, then $l:=n-m+\epsilon _0 \in \mathcal {T}$.

• $n-m^\mathrm {down}(\pi )+\epsilon _0+2 \not \in \mathcal {T}$.

For the first assertion, (chain condition) and (odd-ness condition) follow from Corollary 6.2, and (initial condition) and (alternating condition) follow from Proposition 6.8. The second assertion follows from Corollary 6.13, Proposition 6.10 and Prasad’s conjecture (Theorem D.2). The assertion (2) follows from Prasad’s conjecture (Theorem D.2) together with a comparison of central elements $z_\phi$ (Proposition 6.7) unless $E=F$, $m\not \equiv n \bmod 2$ and $\epsilon =-1$. In this case, we compare the central character of $\pi \in \mathrm {Irr}(\mathrm {O}(W_n))$ with the central element $z_\phi$ (Proposition 6.20).

The second and third main theorems describe the L-parameter for $\theta _{V_m,W_n}(\pi )$.

Theorem 4.3

Let $(V_m,W_n)$ and $\kappa \in \{1,2\}$ be as in Sect. 2.2, and $\pi \in \mathrm {Irr}_\mathrm {temp}(G(W_n))$ with L-parameter $(\phi ,\eta )$. Assume that $V_m$ belongs to the going-down tower $\mathcal {V}^\mathrm {down}$, $m\ge m^\mathrm {down}(\pi )$ and $m\equiv m^\mathrm {down}(\pi ) \bmod 2$. Put $m_1=n+\epsilon _0+2-\kappa$. Let $(\theta _m(\phi ),\theta _m(\eta ))$ be the L-parameter for $\theta _{V_m,W_n}(\pi )$.

• (1) If $m^\mathrm {down}(\pi ) \le m < m_1$, then

  $$\begin{aligned} \theta _m(\phi )=(\phi \otimes \chi _V^{-1}\chi _W)-\chi _W S_l, \end{aligned}$$

  where $l=n-m+\epsilon _0>0$. In particular, there is a canonical injection $A_{\theta _m(\phi )} \hookrightarrow A_{\phi }$. If $l=1$, then we have $\eta | A_{\theta _{m}(\phi )} = \theta _{m}(\eta )$. If $l>1$, then $\theta _m(\eta )(a)/\eta (a)$ is equal to

  $$\begin{aligned} \left\{ \begin{aligned}&\varepsilon (\phi ^a\chi _{V}^{-1} \otimes S_{l-1}) \cdot \varepsilon (\phi ^a) \cdot \chi _{V}(-1)^{\frac{1}{2}\dim (\phi ^a)}&\quad&\text {if }E=F, \epsilon = +1 \text { and } m \text { is odd},\\&\varepsilon \left( \phi ^a\chi _{V}^{-1} \otimes S_{l-1}\right) \cdot \varepsilon \left( \phi ^a\chi _{W}\right) \cdot \chi _{W}(-1)^{\frac{1}{2}\dim (\phi ^a)}&\quad&\text {if }E=F, \epsilon = -1 \text { and } n \text { is odd},\\&\varepsilon \left( \phi ^a\chi _{V}^{-1} \otimes S_{l-1}\right) \cdot \det \left( \phi ^a \chi _V^{-1}\right) (-1)^{\frac{l-1}{2}}&\quad&\text {if }E=F \text { and } m, n \text { are even},\\&\varepsilon \left( \phi ^a\chi _{V}^{-1} \otimes S_{l-1},\psi ^E_{2}\right)&\quad&\text {if }E\not =F,\\ \end{aligned} \right. \end{aligned}$$

  for any $a \in A_{\theta _m(\phi )} \subset A_{\phi }$.

• (2) If $m=m_1$ and $\kappa =1$, then

  $$\begin{aligned} \theta _{m_1}(\phi )=\left( \phi \otimes \chi _V^{-1}\chi _W\right) \oplus \chi _W. \end{aligned}$$

  In particular, there is a canonical injection $A_{\phi } \hookrightarrow A_{\theta _{m_1}(\phi )}$. Moreover, we have $\theta _{m_1}(\eta ) | A_{\phi } =\eta$.

• (3) If $m=m_1$ and $\kappa =2$, then

  $$\begin{aligned} \theta _{m_1}(\phi )=\phi \otimes \chi _V^{-1}\chi _W. \end{aligned}$$

  In particular, there is a canonical identification $A_{\phi } = A_{\theta _{m_1}(\phi )}$. Moreover, $\theta _m(\eta )(a)/\eta (a)$ is equal to

  $$\begin{aligned} \left\{ \begin{aligned}&\varepsilon (\phi ^a) \cdot \varepsilon \left( \phi ^a \otimes \chi _V^{-1}\chi _W\right) \cdot \left( \chi _V^{-1}\chi _W\right) (-1)^{\frac{1}{2}\dim (\phi ^a)}&\quad&\text {if }E=F,\\&\varepsilon \left( \phi ^a \otimes \chi _V^{-1}, \psi ^E_2\right)&\quad&\text {if }E\not =F \end{aligned} \right. \end{aligned}$$

  for any $a \in A_{\theta _{m_1}(\phi )} = A_{\phi }$.

• (4) If $m>m_1$, then $\theta _m(\phi )$ is equal to

  $$\begin{aligned} \theta _{m_1}(\phi ) \oplus \left( \bigoplus _{i=1}^{(m-m_1)/2} \left( \chi _W\left| \cdot \right| _E^{\frac{m-n-\epsilon _0+1}{2}-i} \oplus \chi _W\left| \cdot \right| _E^{-\frac{m-n-\epsilon _0+1}{2}+i}\right) \right) . \end{aligned}$$

  In particular, there is a canonical identification $A_{\theta _m(\phi )} = A_{\theta _{m_1}(\phi )}$. Moreover, we have $\theta _{m}(\eta ) | A_{\theta _{m_1}(\phi )} =\theta _{m_1}(\eta )$.

• (5) If $(G(W_n),H(V_m))=(\mathrm {Mp}(W_n),\mathrm {O}(V_m))$ with odd m, then $\theta _{V_m,W_n}(\pi )$ is parametrized by $(\theta _m(\phi ),\theta _m(\eta ),\nu _m(\phi ,\eta ))$ with

  $$\begin{aligned} \nu _m(\phi ,\eta )= \eta (z_{\phi }) \cdot \varepsilon (\phi ) \cdot \chi _V(-1)^{\frac{n}{2}}. \end{aligned}$$

Remark 4.4

In Theorem 4.1 (2), we note that

$$\begin{aligned} \left[ A_{\theta _{m_1}(\phi )} : A_\phi \right] = \left\{ \begin{aligned}&1&\quad&\text {if }\phi \text { contains } \chi _V,\\&2&\quad&\text {if }\phi \text { does not contain } \chi _V. \end{aligned} \right. \end{aligned}$$

If $\phi$ does not contain $\chi _V$, then $m^+(\pi ) = m^-(\pi ) = m_1$ for any $\pi \in \Pi _\phi$ by Theorem 4.1 (1), and $z_{\theta _{m_1}(\phi )}$ is not contained in $A_\phi$. The value $\theta _{m_1}(\eta )(z_{\theta _{m_1}(\phi )})$ is determined by Desideratum 3.1 (4) or (6).

The assertion (1) will be shown in Sect. 6.2. The assertions (2) and (3) are the (almost) equal rank cases (Theorems B.8, D.1 and D.2). The assertion (4) follows from [14, Proposition 3.2] (see Proposition 5.6 below). The assertion (5) is Proposition 6.19.

Theorem 4.5

Let $(V_m,W_n)$ and $\kappa \in \{1,2\}$ be as in Sect. 2.2, and $\pi \in \mathrm {Irr}_\mathrm {temp}(G(W_n))$ with L-parameter $(\phi ,\eta )$. Assume that $V_m$ belongs to the going-up tower $\mathcal {V}^\mathrm {up}$ and

$$\begin{aligned} m \ge m^\mathrm {up}(\pi )\ge n+\epsilon _0+2 \quad \text {i.e.,}\quad l(\pi )\ge 0. \end{aligned}$$

Let $(\theta _m(\phi ),\theta _m(\eta ))$ be the L-parameter for $\theta _{V_m,W_n}(\pi )$. We put $l=m-n-\epsilon _0-2 \ge 0$.

• (1) Suppose that $m=m^\mathrm {up}(\pi )$ so that $l=l(\pi )$. If $l=0$ or $m_\phi (\chi _V S_l)$ is odd, then

  $$\begin{aligned} \theta _{m}(\phi )=\left( \phi \otimes \chi _V^{-1}\chi _W\right) \oplus \chi _W S_{l+2}, \end{aligned}$$

  so that $\theta _{V_m,W_n}(\pi )$ is tempered. In particular, there is a canonical injection $A_{\phi } \hookrightarrow A_{\theta _{m}(\phi )}$. Moreover, $\theta _{m}(\eta )(a)/\eta (a)$ is equal to

  $$\begin{aligned} \left\{ \begin{aligned}&\varepsilon \left( \phi ^a\chi _{V}^{-1} \otimes S_{l+1}\right) \cdot \varepsilon (\phi ^a) \cdot \chi _{V}(-1)^{\frac{1}{2}\dim (\phi ^a)}&\quad&\text {if }E=F, \epsilon = +1 \text { and } m \text { is odd},\\&\varepsilon \left( \phi ^a\chi _{V}^{-1} \otimes S_{l+1}\right) \cdot \varepsilon \left( \phi ^a\chi _{W}\right) \cdot \chi _{W}(-1)^{\frac{1}{2}\dim (\phi ^a)}&\quad&\text {if }E=F, \epsilon = -1 \text { and } n \text { is odd},\\&\varepsilon \left( \phi ^a\chi _{V}^{-1} \otimes S_{l+1}\right) \cdot \det \left( \phi ^a\chi _V^{-1}\right) (-1)^{\frac{l+1}{2}}&\quad&\text {if }E=F \text { and } m, n \text { are even},\\&\varepsilon \left( \phi ^a\chi _{V}^{-1} \otimes S_{l+1},\psi ^E_{2}\right)&\quad&\text {if }E\not =F,\\ \end{aligned}\right. \end{aligned}$$

  for $a \in A_\phi \subset A_{\theta _{m}(\phi )}$.

• (2) Suppose that $m=m^\mathrm {up}(\pi )$ so that $l=l(\pi )$. If $l>0$ and $m_\phi (\chi _V S_l)=2h>0$, then

  $$\begin{aligned} \theta _{m^\mathrm {up}}(\phi )= \Big (\big (\phi \otimes \chi _V^{-1}\chi _W\big )-\chi _WS_l \Big ) \oplus \Big ( \chi _W S_{l+1} \otimes \big (\big |\cdot \big |_E^{\frac{1}{2}}+\big |\cdot \big |_E^{-\frac{1}{2}}\big ) \Big ), \end{aligned}$$

  so that $\theta _{V_m,W_n}(\pi )$ is not tempered. In this case,

  $$\begin{aligned} \pi \subset \chi _V \mathrm {St}_l \times \cdots \times \chi _V \mathrm {St}_l \rtimes \pi _0, \end{aligned}$$

  where $\pi _0 \in \mathrm {Irr}_\mathrm {temp}(G(W_{n-2lh}))$ has the L-parameter $(\phi _0,\eta _0)$ given by $\phi _0=\phi -(\chi _V \mathrm {St}_l)^{\oplus 2h}$ and $\eta _0=\eta |A_{\phi _0}$. Then

  $$\begin{aligned} m_0:= & {} m^\mathrm {up}(\pi _0)=m-2lh-2 \quad \text {and}\quad \\ \theta _{m_0}(\phi _0)= & {} \left( \phi \otimes \chi _V^{-1}\chi _W\right) -(\chi _WS_l)^{\oplus (2h-1)}. \end{aligned}$$

  In particular, there is a canonical identification $A_{\theta _{m_0}(\phi _0)} = A_{\theta _{m}(\phi )}$. Moreover, we have $\theta _{m}(\eta ) | A_{\theta _{m_0}(\phi _0)} =\theta _{m_0}(\eta _0)$.

• (3) Suppose that $m>m_1:=m^\mathrm {up}(\pi )$. Then $\theta _m(\phi )$ is equal to

  $$\begin{aligned} \theta _{m_1}(\phi )\oplus \left( \bigoplus _{i=1}^{(m-m_1)/2} \left( \chi _W\left| \cdot \right| _E^{\frac{m-n-\epsilon _0+1}{2}-i} \oplus \chi _W\left| \cdot \right| _E^{-\frac{m-n-\epsilon _0+1}{2}+i}\right) \right) . \end{aligned}$$

  In particular, there is a canonical identification $A_{\theta _m(\phi )} = A_{\theta _{m_1}(\phi )}$. Moreover, we have $\theta _{m}(\eta ) | A_{\theta _{m_1}(\phi )} =\theta _{m_1}(\eta )$.

• (4) If $(G(W_n),H(V_m))=(\mathrm {Mp}(W_n),\mathrm {O}(V_m))$ with odd m, then $\theta _{V_m,W_n}(\pi )$ is parametrized by $(\theta _m(\phi ),\theta _m(\eta ),\nu _m(\phi ,\eta ))$ with

  $$\begin{aligned} \nu _m(\phi ,\eta )= \eta (z_{\phi }) \cdot \varepsilon (\phi ) \cdot \chi _V(-1)^{\frac{n}{2}}. \end{aligned}$$

Remark 4.6

Note that in (1), $A_{\phi }$ can have index 2 in $A_{\theta _{m}(\phi )}$. In this case, we see that

$$\begin{aligned} A_{\theta _{m}(\phi )} =A_\phi \oplus (\mathbb {Z}/2\mathbb {Z})e_{l(\pi )+2}. \end{aligned}$$

By Theorem 4.1 (1), we have $\theta _{m}(\eta )(e_{l(\pi )+2}) = - \theta _{m}(\eta )(e_{l(\pi )})$. Together with this equation, we see that (1) describes $\theta _{m}(\eta )$ completely.

Under the assumption of Theorem 4.5 (1), we will show that $\theta _{V_m,W_n}(\pi )$ is tempered (Corollary 6.13). If we knew the temperedness of $\theta _{V_m,W_n}(\pi )$, we obtain Theorem 4.5 (1) by applying Theorem 4.3 (1) to $\theta _{V_m,W_n}(\pi )$. The assertions (2) and (3) will be shown in Sects. 6.6 and 6.7, respectively. The assertion (4) is Propositions 6.19.

The twisted epsilon factors appearing in Theorems 4.3 and 4.5 can be computed by using the following lemma.

Lemma 4.7

Let $l \ge 3$ be an integer, $\chi _V$ be a character of $E^\times$ and $\phi$ be a representation of $WD _E$ such that $\phi \chi _V^{-1}$ is (conjugate) self-dual of sign $(-1)^{l-1}$.

• (1) If $E=F$ and l is even, then

  $$\begin{aligned} \varepsilon \left( \phi \chi _V^{-1} \otimes S_{l-1}\right) = (-1)^{m_\phi \left( \chi _V S_{l-2}\right) + \cdots + m_\phi (\chi _V S_{2})} \cdot \varepsilon \left( \phi \chi _V^{-1}\right) . \end{aligned}$$

• (2) If $E=F$ and l is odd, then

  $$\begin{aligned} \varepsilon \left( \phi \chi _V^{-1} \otimes S_{l-1}\right) \cdot \det \left( \phi \chi _V^{-1}\right) (-1)^{\frac{l-1}{2}} = (-1)^{m_\phi \left( \chi _V S_{l-2}\right) + \cdots + m_\phi (\chi _V S_{1})}. \end{aligned}$$

• (3) If $E\not =F$ and l is even, then

  $$\begin{aligned} \varepsilon \left( \phi \chi _V^{-1} \otimes S_{l-1}, \psi _{2}^E\right) = (-1)^{m_\phi (\chi _V S_{l-2}) + \cdots + m_\phi (\chi _V S_{2})} \cdot \varepsilon \left( \phi \chi _V^{-1}, \psi _{2}^E\right) . \end{aligned}$$

• (4) If $E\not =F$ and l is odd, then

  $$\begin{aligned} \varepsilon (\phi \chi _V^{-1} \otimes S_{l-1}, \psi _{2}^E) = (-1)^{m_\phi (\chi _V S_{l-2}) + \cdots + m_\phi (\chi _V S_{1})}. \end{aligned}$$

Proof

This follows from Lemma A.4. $\square$

5 Irreducibility and temperedness of theta lifts

In this section, we recall Kudla’s filtration of the normalized Jacquet module of Weil representations, and prove the irreducibility and temperedness of theta lifts.

5.1 Kudla’s filtration and irreducibility of big theta lifts

Let $(V_m,W_n)$ be a pair of spaces as in Sect. 2.2. We denote the anisotropic space in the Witt tower $\mathcal {V}=\{V_{m}\}$ by $V_{\mathrm {an}}$. Decompose

$$\begin{aligned} W_n=X_k + W_{n-2k} + X_k^* \quad \text {and}\quad V_m=Y_a + V_{m-2a} + Y_a^*, \end{aligned}$$

where $X_k, X_k^*$ (resp. $Y_a, Y_a^*$) are k-dimensional (resp. a-dimensional) isotropic subspaces of $W_n$ (resp. $V_m$) such that $X_k + X_k^*$ (resp. $Y_a + Y_a^*$) is non-degenerate, and $W_{n-2k}$ (resp. $V_{m-2a}$) is the orthogonal complement of $X_k + X_k^*$ (resp. $Y_a + Y_a^*$) in $W_n$ (resp. $V_m$). Let $P(X_k)$ (resp. $Q(Y_a)$) be the maximal parabolic subgroup of $G(W_n)$ (resp. $H(V_m)$) stabilizing $X_k$ (resp. $Y_a$). We denote the normalized Jacquet functor with respect to a parabolic subgroup P by $R_P$.

The following lemma is called Kudla’s filtration.

Lemma 5.1

([24]) The normalized Jacquet module $R_{P(X_k)}(\omega _{V_m,W_n})$ has an equivariant filtration

$$\begin{aligned} R_{P(X_k)}(\omega _{V_m,W_n})=R^0 \supset R^1 \supset \cdots \supset R^k \supset R^{k+1}=0, \end{aligned}$$

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

$$\begin{aligned} J^a=\mathrm {Ind}_{P(X_{k-a},X_k) \times G(W_{n-2k}) \times Q(Y_a)} ^{\mathrm {GL}(X_k) \times G(W_{n-2k}) \times H(V_{m})} \left( \chi _V\left| {\det }_{X_{k-a}}\right| _E^{\lambda _{k-a}} \otimes \mathcal {S}(\mathrm {Isom}(Y_a,X'_a)) \otimes \omega _{V_{m-2a},W_{n-2k}}\right) , \end{aligned}$$

where

• $\lambda _{k-a}=(m-n+k-a-\epsilon _0)/2$;

• $X_k=X_{k-a}+X'_a$ with $\dim (X_{k-a})=k-a$ and $\dim (X'_a)=a$, and $P(X_{k-a},X_k)$ is the maximal parabolic subgroup of $\mathrm {GL}(X_k)$ stabilizing $X_{k-a}$;

• $\mathrm {Isom}(Y_a,X'_a)$ is the set of invertible E-conjugate linear maps from $Y_a$ to $X'_a$ and $\mathcal {S}(\mathrm {Isom}(Y_a,X'_a))$ is the space of locally constant compactly supported functions on $\mathrm {Isom}(Y_a,X'_a)$;

• $\mathrm {GL}(X'_a) \times \mathrm {GL}(Y_a)$ acts on $\mathcal {S}(\mathrm {Isom}(Y_a,X'_a))$ as

  $$\begin{aligned} ((g,h) \cdot f)(x)=\chi _V(\det (g))\chi _W(\det (h))f(g^{-1}\cdot x \cdot h) \end{aligned}$$

  for $(g,h) \in \mathrm {GL}(X'_a) \times \mathrm {GL}(Y_a)$, $f \in \mathcal {S}(\mathrm {Isom}(Y_a,X'_a))$ and $x \in \mathrm {Isom}(Y_a,X'_a)$.

If $m-2a < \dim (V_{\mathrm {an}})$, we interpret $R^a$ and $J^a$ to be 0.

For a representation $\mathcal {U}$ of a totally disconnected locally compact group G, we denote by $\mathcal {U}_\infty$ the smooth part of $\mathcal {U}$, i.e., the G-submodule of smooth vectors in $\mathcal {U}$. Note that for $\pi \in \mathrm {Irr}(G(W_n))$, we have an isomorphism

$$\begin{aligned} \mathrm {Hom}_{G(W_{n})}\left( \omega _{V_{m},W_{n}}, \pi \right) _\infty \cong \Theta _{V_m,W_n}(\pi )^\vee \end{aligned}$$

as representations of $H(V_m)$. In the following proposition, in order to simplify notation, we will also use $\Theta _{V_m, W_n}(\pi )^\vee$ to denote the left hand side of the above equation when $\pi$ is admissible but possibly reducible.

The following proposition is useful.

Proposition 5.2

Assume that $l=n-m+\epsilon _0>0$ and $k>0$. Let $\pi _0$ be an admissible representation of $G(W_{n-2k})$, and $\tau$ be an irreducible essentially discrete series representation of $\mathrm {GL}(X_k)$. Then we have: the space $\mathrm {Hom}_{\mathrm {GL}(X_k) \times G(W_{n-2k})}(J^a, \chi _V {}^c\tau ^\vee \otimes \pi _0)_\infty$ is isomorphic to

$$\begin{aligned} \left\{ \begin{aligned}&\mathrm {Ind}_{Q(Y_k)}^{H(V_m)}\left( \chi _W^{-1} \tau ^\vee \otimes \Theta _{V_{m-2k},W_{n-2k}} (\pi _0)^\vee \right)&\quad&\text {if }a=k,\\&\mathrm {Ind}_{Q(Y_{k-1})}^{H(V_m)}\left( \chi _W^{-1} \mathrm {St}_{k-1}\left| {\det }_{Y_{k-1}}\right| _E^{\frac{k-l+1}{2}} \otimes \Theta _{V_{m-2k+2},W_{n-2k}}(\pi _0)^\vee \right)&\quad&\text {if }a=k{-}1 \text { and } \tau =\mathrm {St}_{k} \left| {\det }_{X_k}\right| _E^{\frac{l-k}{2}},\\&0&\quad&\text {otherwise}\end{aligned} \right. \end{aligned}$$

as representations of $H(V_m)$.

Proof

We put $\tau '={}^c\tau ^\vee$. For $a=k$, it is easy to see that

$$\begin{aligned}&\mathrm {Hom}_{\mathrm {GL}(X_k) \times G(W_{n-2k})}\left( J^k, \chi _V \tau ' \otimes \pi _0\right) _\infty \\&\quad \cong \mathrm {Ind}_{Q(Y_k)}^{H(V_m)}\left( \chi _W^{-1} \tau ^\vee \otimes \mathrm {Hom}_{G(W_{n-2k})}\left( \omega _{V_{m-2k},W_{n-2k}}, \pi _0\right) _\infty \right) \end{aligned}$$

(c.f., [13, p. 1674–1676]).

Next, we assume that $a<k$. By Bernstein’s Frobenius reciprocity, we have

$$\begin{aligned}&\mathrm {Hom}_{\mathrm {GL}(X_k) \times G(W_{n-2k})}\left( J^a, \chi _V \tau ' \otimes \pi _0\right) \\&\quad \cong \mathrm {Hom}_{\mathrm {GL}(X_{k-a}) \times \mathrm {GL}(X'_a) \times G(W_{n-2k})} \left( \chi _V\left| {\det }_{X_{k-a}}\right| _E^{\lambda _{k-a}} \otimes \mathcal {S}\left( \mathrm {Isom}(Y_a,X'_a)\right) \right. \\&\quad \left. \otimes \omega _{V_{m-2a},W_{n-2k}}, R_{\overline{P(X_{k-a},X_k)}}(\chi _V\tau ') \otimes \pi _0\right) , \end{aligned}$$

where $\overline{P(X_{k-a},X_k)}$ is the parabolic subgroup of $\mathrm {GL}(X_k)$ opposite to $P(X_{k-a},X_k)$. By [53, Proposition 9.5], the normalized Jacquet module $R_{\overline{P(X_{k-a},X_k)}}(\chi _V\tau ')$ is given by

$$\begin{aligned} R_{\overline{P(X_{k-a},X_k)}}\left( \chi _V\tau '\right) \cong \chi _V \tau _1 \left| {\det }_{X_{k-a}}\right| _E^{e_1} \otimes \chi _V \tau _2 \left| {\det }_{X'_{a}}\right| _E^{e_2}, \end{aligned}$$

where $\tau _1$ (resp. $\tau _2$) is an irreducible (unitary) discrete series representation of $\mathrm {GL}(X_{k-a})$ (resp. $\mathrm {GL}(X'_a)$), and $e_1, e_2 \in \mathbb {R}$ such that

$$\begin{aligned} e_1 < e_2 \quad \text {and}\quad e_1 \cdot (k-a) + e_2 \cdot a=0. \end{aligned}$$

Since $\mathrm {GL}(X_{k-a})$ acts on $\chi _V|{\det }_{X_{k-a}}|_E^{\lambda _{k-a}} \otimes \mathcal {S}(\mathrm {Isom}(Y_a,X'_a)) \otimes \omega _{V_{m-2a},W_{n-2k}}$ by the character $\chi _V|{\det }_{X_{k-a}}|_E^{\lambda _{k-a}}$, if $\mathrm {Hom}_{\mathrm {GL}(X_k) \times G(W_{n-2k})}(J^a, \chi _V \tau ' \otimes \pi _0)\not =0$, then we must have $k-a=1$. Moreover, by results of Zelevinsky (see [38, p. 105]), we must have $\tau '=\mathrm {St}_k|{\det }_{X_k}|_E^e$ for some $e \in \mathbb {R}$. Then we have

$$\begin{aligned} e_1=e-\frac{k-1}{2} \quad \text {and}\quad e_2=e+\frac{1}{2}. \end{aligned}$$

We must have $e_1=\lambda _1$ so that $e=(k-l)/2$. In this case, we have

$$\begin{aligned}&\mathrm {Hom}_{\mathrm {GL}(X_k) \times G(W_{n-2k})}\left( J^{k-1}, \chi _V \tau ' \otimes \pi _0\right) _\infty \\&\quad \cong \mathrm {Hom}_{\mathrm {GL}(X'_{k-1}) \times G(W_{n-2k})} \left( \mathcal {S}\left( \mathrm {Isom}\left( Y_{k-1},X'_{k-1}\right) \right) \otimes \omega _{V_{m-2k+2},W_{n-2k}}, \chi _V\mathrm {St}_{k-1}\left| {\det }_{X'_{k-1}}\right| _E^{\frac{k-l+1}{2}} \otimes \pi _0\right) _\infty ,\\&\quad \cong \mathrm {Ind}_{Q(Y_{k-1})}^{H(V_m)}\left( \chi _W^{-1} \mathrm {St}_{k-1}\left| {\det }_{X'_{k-1}}\right| _E^{\frac{k-l+1}{2}} \otimes \mathrm {Hom}_{G(W_{n-2k})}\left( \omega _{V_{m-2k+2},W_{n-2k}}, \pi _0\right) _\infty \right) \end{aligned}$$

(c.f., [13, p. 1674–1676]). Hence the proposition follows. $\square$

Corollary 5.3

We put $n_0=n-2k$ and $m_0=m-2k$. Let $\pi \in \mathrm {Irr}(G(W_n))$, $\pi _0\in \mathrm {Irr}(G(W_{n_0}))$ and $\tau$ be an irreducible essentially discrete series representation of $\mathrm {GL}(X_k)$. Assume that

• $l=n-m+\epsilon _0>0$;

• $\tau \not \cong \mathrm {St}_k|{\det _{X_k}}|_E^{\frac{l-k}{2}}$;

• $\mathrm {Ind}_{P(X_k)}^{G(W_n)}(\chi _V \tau \otimes \pi _0) \twoheadrightarrow \pi$.

Then we have

$$\begin{aligned} \mathrm {Ind}_{Q(Y_k)}^{H(V_m)}\left( \chi _W \tau \otimes \Theta _{V_{m_0},W_{n_0}}(\pi _0)\right) \twoheadrightarrow \Theta _{V_m,W_n}(\pi ). \end{aligned}$$

Proof

By Lemma 2.2, we have $\pi \hookrightarrow \mathrm {Ind}_{P(X_k)}^{G(W_n)}(\chi _V {}^c\tau ^\vee \otimes \pi _0)$. Hence we have

$$\begin{aligned} \Theta _{V_m,W_n}(\pi )^\vee&\cong \mathrm {Hom}_{G(W_n)}\left( \omega _{V_m,W_n}, \pi \right) _\infty \\&\hookrightarrow \mathrm {Hom}_{G(W_n)}\left( \omega _{V_m,W_n}, \mathrm {Ind}_{P(X_k)}^{G(W_n)} \left( \chi _V {}^c\tau ^\vee \otimes \pi _0\right) \right) _\infty \\&\cong \mathrm {Hom}_{\mathrm {GL}(X_k) \times G(W_{n_0})}\left( R_{P(X_k)}\left( \omega _{V_m,W_n}\right) , \chi _V {}^c\tau ^\vee \otimes \pi _0\right) _\infty . \end{aligned}$$

Since $\tau \not \cong \mathrm {St}_k|{\det }_{X_k}|_E^{\frac{l-k}{2}}$, by Proposition 5.2, we have

$$\begin{aligned}&\mathrm {Hom}_{\mathrm {GL}(X_k) \times G(W_{n_0})}\left( R_{P(X_k)}\left( \omega _{V_m,W_n}\right) , \chi _V {}^c\tau ^\vee \otimes \pi _0\right) _\infty \\&\quad \hookrightarrow \mathrm {Hom}_{\mathrm {GL}(X_k) \times G(W_{n_0})}\left( J^k, \chi _V {}^c\tau ^\vee \otimes \pi _0\right) _\infty \\&\quad \cong \left( \mathrm {Ind}_{Q(Y_k)}^{H(V_m)} \left( \chi _W \tau \otimes \Theta _{V_{m_0},W_{n_0}}(\pi _0)\right) \right) ^\vee . \end{aligned}$$

Taking the contragredient functor, we get the corollary. $\square$

Corollary 5.3 implies an irreducibility condition of big theta lifts.

Proposition 5.4

Let $\pi \in \mathrm {Irr}(G(W_{n}))$ whose last name is $\phi \in \Phi (G(W_n))$. Assume that

• $\pi$ is tempered;

• $\Theta _{V_m,W_n}(\pi )\not =0$ for $l=n-m+\epsilon _0>0$;

• $\phi$ contains $\chi _V S_l$ with multiplicity one.

Then $\Theta _{V_m,W_n}(\pi )$ is irreducible and tempered.

Proof

We prove this corollary by induction on n. If $\pi$ is a discrete series representation, then by a similar argument as that for [11, Proposition C.1], we see that all irreducible subquotients of $\Theta _{V_m,W_n}(\pi )$ are discrete series representations. Hence $\Theta _{V_m,W_n}(\pi )$ is a direct sum of irreducible discrete series representations, and so $\Theta _{V_m,W_n}(\pi )$ is irreducible by the Howe duality conjecture (Theorem 2.3).

Suppose that $\pi$ is not a discrete series representation. Then there exist $\tau \in \mathrm {Irr}_\mathrm {disc}(\mathrm {GL}(X_k))$ and $\pi _0 \in \mathrm {Irr}_\mathrm {temp}(G(W_{n_0}))$ with $n_0=n-2k$ such that $\mathrm {Ind}_{P(X_k)}^{G(W_n)}(\chi _V \tau \otimes \pi _0) \twoheadrightarrow \pi$. By our assumption, $\tau \not \cong \mathrm {St}_l$. Also, $\tau \not \cong \mathrm {St}_{k}|{\det }_{X_k}|_E^{\frac{l-k}{2}}$ since $\tau$ is discrete series. Hence we can apply Corollary 5.3 to $\pi$. We have

$$\begin{aligned} \mathrm {Ind}_{Q(Y_k)}^{H(V_m)}\left( \chi _W \tau \otimes \Theta _{V_{m_0},W_{n_0}}(\pi _0)\right) \twoheadrightarrow \Theta _{V_m,W_n}(\pi ). \end{aligned}$$

By the induction hypothesis, we see that $\Theta _{V_{m_0},W_{n_0}}(\pi _0)$ is irreducible and tempered. Hence so is $\Theta _{V_{m},W_{n}}(\pi )$, by the Howe duality conjecture (Theorem 2.3) and the fact that the induced representation is semisimple and tempered. $\square$

5.2 Temperedness of theta lifts 1

First, we prove the following proposition.

Proposition 5.5

Let $\pi \in \mathrm {Irr}(G(W_{n}))$ be such that $\Theta _{V_m,W_n}(\pi )\not =0$. Assume one of the following:

• (1) $\pi$ is tempered and $m \le n+1+\epsilon _0$;

• (2) $\pi$ is a discrete series representation and $\Theta _{V_m,W_n}(\pi )$ is the first lift to the going-up tower $\mathcal {V}^\mathrm {up}$ so that $m=m^\mathrm {up}(\pi )$.

Then all irreducible subquotients of $\Theta _{V_{m},W_{n}}(\pi )$ are tempered.

Proof

The first case is similar to [11, Proposition C.1]. Hence we consider the second case. So we assume that $\pi$ is a discrete series representation and $m=m^\mathrm {up}(\pi )$.

Fix an $H(V_{m})$-invariant filtration of $\Theta _{V_{m},W_{n}}(\pi )$:

$$\begin{aligned} \Theta _{V_{m},W_{n}}(\pi ) =\Sigma _0 \supset \Sigma _1 \supset \cdots \supset \Sigma _c \supset \Sigma _{c+1}=0 \end{aligned}$$

such that

$$\begin{aligned} \Pi _i:=\Sigma _i/\Sigma _{i+1} \end{aligned}$$

is irreducible for any i. Suppose that $\Pi _k$ is non-tempered. We may assume that $\Pi _i$ is tempered for $i=0,\dots ,k-1$. Then there exists a maximal parabolic subgroup Q of $H(V_{m})$ with Levi component $L_Q=\mathrm {GL}_t(E)\times H(V_{m_0})$ such that

$$\begin{aligned} \Pi _k \hookrightarrow \mathrm {Ind}_Q^{H(V_{m})}(\tau |\det |_E^{-s_0} \otimes \sigma _0), \end{aligned}$$

where $\tau \in \mathrm {Irr}_\mathrm {disc}(\mathrm {GL}_t(E))$, $s_0>0$ and $\sigma _0\in \mathrm {Irr}(H(V_{m_0}))$. By a similar argument as that for [11, Proposition C.1], we have a nonzero $H(V_{m})$-map

$$\begin{aligned} \Theta _{V_{m},W_{n}}(\pi ) \rightarrow \mathrm {Ind}_Q^{H(V_{m})}\left( \tau \left| \det \right| _E^{-s_0} \otimes \sigma _0\right) . \end{aligned}$$

Hence we have

$$\begin{aligned} \pi ^\vee&\hookrightarrow \mathrm {Hom}_{H(V_{m})}\left( \omega _{V_{m},W_{n}}, \mathrm {Ind}_Q^{H(V_{m})}\left( \tau \left| \det \right| _E^{-s_0} \otimes \sigma _0\right) \right) \\&\cong \mathrm {Hom}_{\mathrm {GL}_t(E)\times H(V_{m_0})} \left( R_Q\left( \omega _{V_{m},W_{n}}\right) , \tau \left| \det \right| _E^{-s_0} \otimes \sigma _0\right) , \end{aligned}$$

where $R_Q$ denotes the normalized Jacquet functor with respect to Q. The last $\mathrm {Hom}$ space has been studied precisely in the proof of [14, Proposition 3.1]. According to (the proof of) this proposition, one of the following must occur:

• (a) $\Theta _{V_{m-2},W_{n}}(\pi )\not =0$;

• (b) $\mathrm {Ind}_{P(X_a)}^{G(W_{n})}(\chi _V\mathrm {St}_a \otimes \pi _0)\twoheadrightarrow \pi$ for some a and $\pi _0\in \mathrm {Irr}_\mathrm {temp}(G(W_{n_0}))$.

However, (a) can not occur since $\Theta _{V_{m},W_{n}}(\pi )$ is the first occurrence. Also, (b) contradicts that $\pi$ is a discrete series representation. This completes the proof. $\square$

We also need the following proposition in [14]:

Proposition 5.6

([14, Proposition 3.2]) Let $\pi \in \mathrm {Irr}(G(W_n))$. Assume that $l=n-m+\epsilon _0 \le 0$ and $\theta _{V_m,W_n}(\pi )$ is nonzero and tempered. We put $V_{m+2r}=V_m\oplus \mathbb {H}^r$ for $r\ge 0$. Then $\theta _{V_{m+2r},W_n}(\pi )$ is the unique irreducible quotient of the standard module

$$\begin{aligned} \chi _W\left| \cdot \right| _E^{\frac{2r-1-l}{2}} \times \chi _W\left| \cdot \right| _E^{\frac{2r-3-l}{2}} \times \cdots \times \chi _W\left| \cdot \right| _E^{\frac{1-l}{2}} \rtimes \theta _{V_m,W_n}(\pi ). \end{aligned}$$

This proposition implies Theorem 4.3 (4). In fact, [14, Proposition 3.2] can be applied to a more general situation as we shall show in Proposition 6.18 below. Theorem 4.5 (3) is proven by showing that we can apply [14, Proposition 3.2] to $\theta _{V_{m^\mathrm {up}(\pi )},W_n}(\pi )$, which may be non-tempered, for $\pi \in \mathrm {Irr}_\mathrm {temp}(G(W_n))$.

Also, Proposition 5.6 implies the following.

Corollary 5.7

Let $\pi \in \mathrm {Irr}(G(W_n))$. Assume that $l=n-m+\epsilon _0 < -1$ and $\theta _{V_m,W_n}(\pi )$ is nonzero and tempered. Let $V_{m_0}$ be the space which belongs to the same Witt tower as $V_m$, and $l_0=n-m_0+\epsilon _0=0$ or $-1$. Then $\Theta _{V_{m_0},W_n}(\pi )=0$.

Proof

If $\theta _{V_{m_0},W_n}(\pi )$ were nonzero, it must be tempered by Proposition 5.5, so that $\theta _{V_{m},W_n}(\pi )$ is non-tempered by Proposition 5.6. This contradicts the temperedness of $\theta _{V_m,W_n}(\pi )$. $\square$

6 Proof of main theorems

In this section, we prove Theorems 4.1, 4.3 and 4.5.

6.1 Correspondence of last names

First, we study the correspondence of last names.

Proposition 6.1

Let $\pi \in \mathrm {Irr}_\mathrm {temp}(G(W_{n}))$ with L-parameter $(\phi ,\eta )$. Assume that $\Theta _{V_m,W_n}(\pi )\not =0$ with $l=n-m+\epsilon _0>0$. Then $\phi$ contains $\chi _V S_{l}$.

Proof

Consider the standard gamma factors (see Appendix A.1). By Proposition A.1 and Desideratum B.1 (7), the gamma factor

$$\begin{aligned} \gamma \left( s,\phi \otimes \chi _V^{-1},\psi _E\right) =\varepsilon \left( s,\phi \otimes \chi _V^{-1},\psi _E\right) \frac{L\left( 1-s,\phi ^\vee \otimes \chi _V^{-1}\right) }{L\left( s,\phi \otimes \chi _V^{-1}\right) } \end{aligned}$$

has a pole at $s=\frac{l+1}{2}$. This implies that $L(1-s,\phi ^\vee \otimes \chi _V^{-1})$ has a pole at $s=\frac{l+1}{2}$. We decompose

$$\begin{aligned} \phi =\bigoplus _{i\ge 1}\phi _i \otimes S_i, \end{aligned}$$

where $\phi _i$ is a tempered representation of $W_E$. Then we have

$$\begin{aligned} L\left( 1-s,\phi ^\vee \otimes \chi _V^{-1}\right) =\prod _{i\ge 1}L\left( 1-s+\frac{i-1}{2},\phi _i^\vee \otimes \chi _V^{-1}\right) . \end{aligned}$$

Since $\phi _i$ is tempered, only $L(1-s+\frac{l-1}{2},\phi _{l}\otimes \chi _V^{-1})$ can have a pole at $s=\frac{l+1}{2}$. Moreover, if it has a pole, then $\phi _{l}\otimes \chi _V^{-1}$ must contain the trivial representation. Hence the proposition follows. $\square$

Corollary 6.2

Let $\pi \in \mathrm {Irr}_\mathrm {temp}(G(W_{n}))$ with L-parameter $(\phi ,\eta )$. Assume that $\Theta _{V_m,W_n}(\pi )\not =0$ with $l=n-m+\epsilon _0>0$. Define $\kappa \in \{1,2\}$ by $\kappa \equiv l \bmod 2$. Then $\phi$ contains $\chi _V S_{r}$ for $r=\kappa , \kappa +2, \dots , l$. Moreover, the multiplicity $m_\phi (\chi _V S_{r})$ is odd for $r=\kappa , \kappa +2, \dots , l-2$.

Proof

By Propositions 6.1 and 2.4, we see that $\phi$ contains $\chi _V S_{r}$ for $r=\kappa , \kappa +2, \dots , l$.

By an induction on n, we prove that $m_\phi (\chi _V S_r)$ is odd for any $r=\kappa +2i$ with $0 \le i < (l-\kappa )/2$. We may assume that $m_\phi (\chi _V S_r) \ge 2$. Then we can write

$$\begin{aligned} \phi =\chi _V S_r \oplus \phi _0 \oplus \chi _V S_r \end{aligned}$$

for some $\phi _0 \in \Phi _\mathrm {temp}(G(W_{n_0}))$ with $n_0=n-2r$. We can find $\pi _0 \in \mathrm {Irr}_\mathrm {temp}(G(W_{n_0}))$ such that there is a surjection $\mathrm {Ind}_{P(X_r)}^{G(W_n)}(\chi _V \mathrm {St}_{r} \otimes \pi _0) \twoheadrightarrow \pi$. Then the L-parameter of $\pi _0$ is given by $(\phi _0,\eta |A_{\phi _0})$. Since $r < l$, by Corollary 5.3, we have a surjection $\mathrm {Ind}_{Q(Y_r)}^{H(V_m)}(\chi _W \mathrm {St}_{r} \otimes \Theta _{V_{m_0},W_{n_0}}(\pi _0)) \twoheadrightarrow \Theta _{V_m,W_n}(\pi )$ with $m_0=m-2r$. In particular, $\Theta _{V_{m_0},W_{n_0}}(\pi _0)$ is nonzero. Since $n_0-m_0+\epsilon _0=l$, by the induction hypothesis, we see that $m_{\phi _0}(\chi _V S_{r})$ is odd. Therefore $m_\phi (\chi _V S_{r})=m_{\phi _0}(\chi _V S_{r})+2$ is also odd. $\square$

Corollary 6.2 gives the (chain condition) and the (odd-ness condition) in Theorem 4.1 (1). Note that it is possible that $m_\phi (\chi _V S_l)$ is even as we shall see later. The parity of $m_\phi (\chi _V S_l)$ determines the temperedness of the first occurrence $\theta _{V'_{m^\mathrm {up}(\pi )},W_{n}}(\pi )$ to the going-up tower (Corollary 6.13).

Next, we determine the last name of theta lifts in a special case for the going-down tower.

Theorem 6.3

Let $\pi \in \mathrm {Irr}_\mathrm {temp}(G(W_n))$ whose last name is $\phi \in \Phi _\mathrm {temp}(G(W_n))$. Assume that $\Theta _{V_m,W_n}(\pi )\not =0$ with $l=n-m+\epsilon _0>0$. Put

$$\begin{aligned} \theta _{V_m,W_n}(\phi )=\left( \phi \otimes \chi _V^{-1}\chi _W\right) -\chi _W S_{l}. \end{aligned}$$

Then $\theta _{V_m,W_n}(\phi )\in \Phi (H(V_{m}))$ and it is the last name of $\theta _{V_m,W_n}(\pi )$.

Proof

Let $\phi _{\theta (\pi )}$ be the last name of $\theta _{V_m,W_n}(\pi )$. Consider the Plancherel measure (see Appendix A.2). By Theorem A.2, we have

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

for any supercuspidal representation $\tau$ of $\mathrm {GL}_k(E)$. Using Desideratum B.1 (8) and Lemma A.3, for any irreducible representation $\phi _\tau$ of $W_E$, we have

$$\begin{aligned}&\gamma \left( s,\phi _\tau \chi _W\otimes \phi _{\theta (\pi )}^\vee ,\psi _E\right) \gamma \left( -s,\left( \phi _\tau \chi _W\right) ^\vee \otimes \phi _{\theta (\pi )},\psi _E^{-1}\right) \\&\quad =\frac{\gamma \left( s,\phi _\tau \chi _V\otimes \phi ^\vee ,\psi _E\right) \gamma \left( -s,\left( \phi _\tau \chi _V\right) ^\vee \otimes \phi ,\psi _E^{-1}\right) }{\gamma \left( s-\frac{l-1}{2},\phi _\tau ,\psi _E\right) \gamma \left( -s-\frac{l-1}{2},\phi _\tau ^\vee ,\psi _E^{-1}\right) }\\&\quad =\gamma \left( s,\phi _\tau \chi _W \otimes \theta _{V_m,W_n}(\phi )^\vee ,\psi _E\right) \gamma \left( -s,\left( \phi _\tau \chi _W\right) ^\vee \otimes \theta _{V_m,W_n}(\phi ),\psi _E^{-1}\right) . \end{aligned}$$

By Proposition 5.5 (1), we see that $\phi _{\theta (\pi )}$ is tempered. Hence by Lemma A.6, we have

$$\begin{aligned} \phi _{\theta (\pi )}=\theta _{V_m,W_n}(\phi ), \end{aligned}$$

as desired. In particular, we have $\theta _{V_m,W_n}(\phi )\in \Phi (H(V_{m}))$. $\square$

6.2 Correspondence of first names

In this subsection, we compare the first name of $\theta _{V_m,W_n}(\pi )$ with the one of $\pi$. To do this, we need the following lemma.

Lemma 6.4

Let $\pi \in \mathrm {Irr}(G(W_n))$. Assume that $\Theta _{V_m,W_n}(\pi )\not =0$ and all irreducible subquotients of $\Theta _{V_m,W_n}(\pi )$ are tempered. Then all irreducible subquotients of $\Theta _{V_m,W_n}(\pi )$ belong to the same L-packet.

Proof

This follows from [12, Lemma A.1], [11, Lemma B.2, Proposition B.3] and [12, Lemma A.6]. $\square$

In the following theorem, to avoid a confusion, we denote the characters associated to $V_m$ and $W_n$ by $\chi _{V_m}$ and $\chi _{W_n}$, respectively.

Theorem 6.5

Let $\pi \in \mathrm {Irr}_\mathrm {temp}(G(W_n))$ with L-parameter $(\phi , \eta )$. Assume that $\Theta _{V_m,W_n}(\pi ) \not =0$ with $l = n-m+\epsilon _0 > 1$. Let $(\theta (\phi ), \theta (\eta ))$ be the L-parameter for $\theta _{V_m,W_n}(\pi ) \in \mathrm {Irr}(H(V_m))$. Then we have

$$\begin{aligned}&\theta (\eta )(a)/\eta (a) =\\&\left\{ \begin{aligned}&\varepsilon \left( \phi ^a\chi _{V_m}^{-1} \otimes S_{l-1}\right) \cdot \varepsilon (\phi ^a) \cdot \chi _{V_m}(-1)^{\frac{1}{2}\dim (\phi ^a)}&\quad&\text {if }E=F, \epsilon = +1 \text { and } m \text { is odd},\\&\varepsilon \left( \phi ^a\chi _{V_m}^{-1} \otimes S_{l-1}\right) \cdot \varepsilon \left( \phi ^a\chi _{W_n}\right) \cdot \chi _{W_n}(-1)^{\frac{1}{2}\dim (\phi ^a)}&\quad&\text {if }E=F, \epsilon = -1 \text { and } n \text { is odd},\\&\varepsilon \left( \phi ^a\chi _{V_m}^{-1} \otimes S_{l-1}\right) \cdot \det \left( \phi ^a\chi _{V_m}^{-1}\right) (-1)^{\frac{l-1}{2}} \cdot \nu ^{\det (a)}&\quad&\text {if }E=F \text { and } m, n \text { are even},\\&\varepsilon \left( \phi ^a\chi _{V_m}^{-1} \otimes S_{l-1},\psi ^E_{2}\right)&\quad&\text {if }E\not =F,\\ \end{aligned} \right. \end{aligned}$$

where the constant $\nu \in \{\pm 1\}$ is given by

$$\begin{aligned} \nu = (-1)^{\frac{l-1}{2}} \cdot \eta (e_1+e_l). \end{aligned}$$

Proof

If $E\not =F$, we choose a character $\chi$ of $E^\times$ such that $\chi | F^\times =\omega _{E/F}$. We shall treat the cases $\epsilon = +1$ and $\epsilon = -1$ separately.

Suppose that $\epsilon = +1$. Put

$$\begin{aligned} \omega =\left\{ \begin{aligned}&\omega _{\psi }&\quad&\text {if }E=F,\\&\omega _{\psi ,\chi }&\quad&\text {if }E\not =F. \end{aligned} \right. \end{aligned}$$

Let L be the Hermitian space of dimension 1 such that

$$\begin{aligned} \mathrm {disc}(L)= \left\{ \begin{aligned}&(-1)^{m+1}&\quad&\text {if }E=F,\\&(-1)^{m}&\quad&\text {if }E\not =F. \end{aligned} \right. \end{aligned}$$

Put $V_{m+1}=V_m \oplus L$. If $E\not =F$, we set $\chi _L = \chi ^{(-1)^{m}}$ and $\chi _{V_{m+1}}=\chi _{V_m}\chi _L$. We denote by $(G'(W_n),H(V_{m+1}))$ the pair of groups associated to $(V_{m+1},W_n)$ defined in Sect. 2.3. By Lemma C.6, we can find $\pi ' \in \mathrm {Irr}_\mathrm {temp}(G'(W_{n}))$ such that

$$\begin{aligned} \left\{ \begin{aligned}&\mathrm {Hom}_{G'(W_{n})}\left( \pi \otimes \pi ', \omega \right) \not =0&\quad&\text {if }E{=}F \text { and } m\equiv 0 \bmod 2, \text { or } E\not =F \text { and } m\equiv 1 \bmod 2,\\&\mathrm {Hom}_{G'(W_{n})}\left( \pi \otimes \pi ', \overline{\omega }\right) \not =0&\quad&\text {otherwise}\end{aligned} \right. \end{aligned}$$

so that

$$\begin{aligned} \left\{ \begin{aligned}&\mathrm {Hom}_{G'(W_{n})}\left( \pi \otimes \overline{\omega }, \pi '^\vee \right) \not =0&\quad&\text {if }E=F \text { and } m\equiv 0 \bmod 2, \text { or } E\not =F \text { and } m\equiv 1 \bmod 2,\\&\mathrm {Hom}_{G'(W_{n})}\left( \pi \otimes \omega , \pi '^\vee \right) \not =0&\quad&\text {otherwise}. \end{aligned} \right. \end{aligned}$$

We put $\sigma =\theta _{V_m,W_n}(\pi ) \in \mathrm {Irr}_\mathrm {temp}(H(V_m))$. Since $\pi \cong \theta _{W_n,V_m}(\sigma )$, we have

$$\begin{aligned} \mathrm {Hom}_{G'(W_{n})}\left( \Theta _{W_n,V_m}(\sigma ) \otimes \overline{\omega }, \pi '^\vee \right) \supset \mathrm {Hom}_{G'(W_{n})}\left( \pi \otimes \overline{\omega }, \pi '^\vee \right) \not =0 \end{aligned}$$

or

$$\begin{aligned} \mathrm {Hom}_{G'(W_{n})}\left( \Theta _{W_n,V_m}(\sigma ) \otimes \omega , \pi '^\vee \right) \supset \mathrm {Hom}_{G'(W_{n})}\left( \pi \otimes \omega , \pi '^\vee \right) \not =0. \end{aligned}$$

The see-saw diagram

implies that

$$\begin{aligned} \mathrm {Hom}_{H(V_m)}\left( \Theta _{V_{m+1},W_n}(\pi '^\vee ), \sigma \right) \not =0. \end{aligned}$$

Hence $\Theta _{V_{m+1},W_{n}}(\pi '^\vee )$ has an irreducible subquotient $\sigma '$ such that

$$\begin{aligned} \mathrm {Hom}_{H(V_m)}(\sigma ', \sigma )\not =0 \quad \text {so that}\quad \mathrm {Hom}_{H(V_m)}\left( \sigma ^\vee \otimes \sigma ',\mathbb {C}\right) \not =0. \end{aligned}$$

Since $\sigma ^\vee$ and $\sigma '$ are tempered, they are unitary, so that $\overline{\sigma ^\vee } \cong \sigma$ and $\overline{\sigma '} \cong \sigma '^\vee$. Hence we have

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

By the GP conjectures (Theorems C.1 – C.4 and Corollary C.5), we have

$$\begin{aligned} \eta (a)&= \left\{ \begin{aligned}&\varepsilon \left( \phi ^a \otimes \phi _{\pi '} \chi _{-1}\right) \cdot \varepsilon (\phi ^a) \cdot \chi _{-1}(-1)^{\frac{1}{2}\dim (\phi ^a)}&\quad&\text {if }E=F \text { and } m \text { is odd},\\&\varepsilon \left( \phi ^a \otimes \phi _{\pi '}\right) \cdot \varepsilon \left( \phi \otimes \phi _{\pi '}\right) ^{\det (a)} \cdot \det (\phi ^a)(-1)^{\frac{1}{2}\dim (\phi _{\pi '})}&\quad&\text {if }E=F \text { and } m \text { is even},\\&\varepsilon \left( \phi ^a \otimes \phi _{\pi '} \otimes \chi ^{-1}, \psi ^E_{2}\right)&\quad&\text {if }E\not =F \text { and } m \text { is odd},\\&\omega _{E/F}(-1)^{\dim (\phi ^a)} \cdot \varepsilon \left( \phi ^a \otimes \phi _{\pi '} \otimes \chi , \psi ^E_{2}\right)&\quad&\text {if }E\not =F \text { and } m \text { is even},\\ \end{aligned} \right. \\ \theta (\eta )(a)&= \left\{ \begin{aligned}&\varepsilon \left( \theta (\phi )^a \otimes \phi _{\sigma '^\vee }\right) \cdot \det \left( \phi _{\sigma '^\vee }\right) (-1)^{\frac{1}{2}\dim \left( \theta (\phi )^a\right) }&\quad&\text {if }E=F \text { and } m \text { is odd},\\&\varepsilon \left( \theta (\phi )^a \otimes \phi _{\sigma '^\vee }\right) \cdot \det (\theta (\phi )^a)(-1)^{\frac{1}{2}\dim \left( \phi _{\sigma '^\vee }\right) } \cdot \nu \left( \sigma '^\vee \right) ^{\det (a)}&\quad&\text {if }E=F \text { and } m \text { is even},\\&\omega _{E/F}(-1)^{(m+1)\dim (\theta (\phi )^a)} \cdot \varepsilon \left( \theta (\phi )^a \otimes \phi _{\sigma '^\vee }, \psi ^E_{2}\right)&\quad&\text {if }E\not =F \end{aligned} \right. \end{aligned}$$

for $a \in A_{\theta (\phi )} \subset A_{\phi }$. Here,

• $\phi _{\pi '}$ and $\phi _{\sigma '^\vee }$ are the last names of $\pi '$ and $\sigma '^\vee$, respectively;

• $\nu (\sigma '^\vee ) \in \{\pm 1\}$ is the central value of $\sigma '^\vee$, i.e., $\sigma '^\vee (-{\mathbf{1}}_{V_{m+1}}) = \nu (\sigma '^\vee ) \cdot \mathrm {id}$.

By Theorem 6.3, Lemma 6.4, Proposition B.4 and Theorem B.8, we have

$$\begin{aligned} \theta (\phi )=\left( \phi \otimes \chi _{V_m}^{-1} - S_l\right) \otimes \chi _{W_n} \quad \text {so that}\quad \theta (\phi )^a = \phi ^a \otimes \chi _{V_m}^{-1}\chi _{W_n} \end{aligned}$$

and

$$\begin{aligned} \phi _{\sigma '^\vee }= \left\{ \begin{aligned}&\left( \phi _{\pi '} \otimes \chi _{V_{m+1}} - S_{l-1}\right) \otimes \chi _{W_{n}}^{-1}&\quad&\text {if }E\not =F \text { or } m \text { is odd},\\&\left( \phi _{\pi '} \otimes \chi _{V_{m+1}}\chi _{-1} - S_{l-1}\right) \otimes \chi _{W_{n}}^{-1}&\quad&\text {if }E=F \text { and } m \text { is even}. \end{aligned} \right. \end{aligned}$$

Therefore we have

$$\begin{aligned}&\theta (\eta )(a)/\eta (a) =\\&\quad \left\{ \begin{aligned}&\varepsilon \left( \phi ^a\chi _{V_m}^{-1} \otimes S_{l-1}\right) \cdot \varepsilon (\phi ^a) \cdot \chi _{V_m}(-1)^{\frac{1}{2}\dim (\phi ^a)}&\quad&\text {if }E=F \text { and } m \text { is odd},\\&\varepsilon \left( \phi ^a\chi _{V_m}^{-1} \otimes S_{l-1}\right) \cdot \det \left( \phi ^a\chi _{V_m}^{-1}\right) (-1)^{\frac{l-1}{2}} \cdot \nu ^{\det (a)}&\quad&\text {if }E=F \text { and } m \text { is even},\\&\varepsilon \left( \phi ^a\chi _{V_m}^{-1} \otimes S_{l-1},\psi ^E_{2}\right)&\quad&\text {if }E\not =F,\\ \end{aligned} \right. \end{aligned}$$

for some constant $\nu \in \{\pm 1\}$.

We shall determine this constant $\nu \in \{\pm 1\}$. So we assume that $E=F$ and m is even, hence $G(W_n) = \mathrm {Sp}(W_n)$ and $H(V_m) = \mathrm {O}(V_m)$. Since $\sigma =\theta _{V_m,W_n}(\pi ) \in \mathrm {Irr}_\mathrm {temp}(\mathrm {O}(V_m))$ satisfies that $\pi \cong \theta _{W_n,V_m}(\sigma )$ is nonzero and tempered, by Corollary 5.7, we have $\Theta _{W_m,V_m}(\sigma ) = 0$. By Prasad conjecture (Theorem D.2), we have $\theta (\eta )(z_{\theta (\phi )} + e_1) = -1$. Since $z_{\theta (\phi )} + e_1 = z_\phi + e_1 + e_l$ in $A_\phi$, we have $\eta (z_{\theta (\phi )} + e_1) = \eta (e_1+e_l)$. On the other hand, if $a = z_\phi + e_1 + e_l$, we have

$$\begin{aligned}&\varepsilon \left( \phi ^a\chi _{V_m}^{-1} \otimes S_{l-1}\right) \cdot \det \left( \phi ^a\chi _{V_m}^{-1}\right) (-1)^{\frac{l-1}{2}} \cdot \nu ^{\det (a)} \\&\quad = \varepsilon \left( \phi \chi _{V_m}^{-1} \otimes S_{l-1}\right) \cdot \varepsilon \left( (S_1 \oplus S_l) \otimes S_{l-1}\right) \cdot \chi _{V_{m}}(-1)^{\frac{l-1}{2}}\cdot \nu . \end{aligned}$$

We have $\varepsilon ((S_1 \oplus S_l) \otimes S_{l-1}) = -(-1)^{l-1} = -1$. Also, since $\det (\phi \chi _{V_m}^{-1}) = \chi _{V_m}$, by Lemma A.4 and the (odd-ness condition) proved in Corollary 6.2, we know

$$\begin{aligned} \varepsilon \left( \phi \chi _{V_m}^{-1} \otimes S_{l-1}\right) \cdot \chi _{V_m}(-1)^{\frac{l-1}{2}}= & {} (-1)^{m_\phi \left( \chi _{V_m}S_{l-2}\right) + m_{\phi }\left( \chi _{V_m}S_{l-4}\right) + \cdots + m_{\phi }\left( \chi _{V_m}S_1\right) }\\= & {} (-1)^{\frac{l-1}{2}}. \end{aligned}$$

Hence we have $\nu = (-1)^{\frac{l-1}{2}} \cdot \eta (e_1+e_l)$, as desired.

Now suppose that $\epsilon = -1$. Then $n\ge m-\epsilon _0+2$. If $n \le 2-\epsilon _0$, then $n=2-\epsilon _0$ and $m=0$. In this case, the only representation of $G(W_n)$ which participates in the theta correspondence with $H(V_0)$ is the trivial representation, so that we have nothing to prove. In the other cases, there is a line L in $W_{n}$ such that

$$\begin{aligned} \mathrm {disc}(L) = \left\{ \begin{aligned}&(-1)^{n}&\quad&\text {if }E=F,\\&(-1)^{n-1}&\quad&\text {if }E \not =F. \end{aligned} \right. \end{aligned}$$

Let $W_{n-1}$ be the orthogonal complement of L in $W_n$. If $E\not =F$, we set $\chi _L=\chi ^{(-1)^{n-1}}$ and $\chi _{W_{n-1}}=\chi _{W_n}\chi ^{(-1)^n}$. By Lemma C.6, we can find $\pi ' \in \mathrm {Irr}_\mathrm {temp}(G(W_{n-1}))$ such that

$$\begin{aligned} \mathrm {Hom}_{G(W_{n-1})}(\pi \otimes \pi ', \mathbb {C})\not =0 \quad \text {so that}\quad \mathrm {Hom}_{G(W_{n-1})}(\pi , \pi '^\vee )\not =0. \end{aligned}$$

We put $\sigma =\theta _{V_m,W_n}(\pi ) \in \mathrm {Irr}_\mathrm {temp}(H(V_m))$. Since $\pi \cong \theta _{W_n,V_m}(\sigma )$, we have

$$\begin{aligned} \mathrm {Hom}_{G(W_{n-1})}\left( \Theta _{W_n,V_m}(\sigma ), \pi '^\vee \right) \supset \mathrm {Hom}_{G(W_{n-1})}(\pi , \pi '^\vee )\not =0. \end{aligned}$$

The see-saw diagram

implies that

$$\begin{aligned} \left\{ \begin{array}{l@{\qquad }l} \mathrm {Hom}_{H(V_m)}\left( \Theta _{V_m,W_{n-1}}(\pi '^\vee ) \otimes \omega , \sigma \right) \not =0&{} \text {if } E=F \text { and } n \equiv 0 \bmod 2,\\ &{}\text {or } E\not =F \text { and } n \equiv 1 \bmod 2,\\ \mathrm {Hom}_{H(V_m)}\left( \Theta _{V_m,W_{n-1}}(\pi '^\vee ) \otimes \overline{\omega }, \sigma \right) \not =0 &{} \text {otherwise}, \end{array} \right. \end{aligned}$$

where we put

$$\begin{aligned} \omega =\left\{ \begin{aligned}&\omega _{\psi }&\quad&\text {if }E=F,\\&\omega _{\psi ,\chi }&\quad&\text {if }E\not =F. \end{aligned} \right. \end{aligned}$$

Hence $\Theta _{V_m,W_{n-1}}(\pi '^\vee )$ has an irreducible subquotient $\sigma '$ such that

$$\begin{aligned} \left\{ \begin{array}{l@{\qquad }l} \mathrm {Hom}_{H(V_m)}\left( \sigma ' \otimes \omega , \sigma \right) \not =0 &{} \text {if } E=F \text { and } n \equiv 0 \bmod 2,\\ &{}\text {or } E\not =F \text { and } n \equiv 1 \bmod 2,\\ \mathrm {Hom}_{H(V_m)}\left( \sigma ' \otimes \overline{\omega }, \sigma \right) \not =0 &{}\text {otherwise}, \end{array} \right. \end{aligned}$$

so that

$$\begin{aligned} \left\{ \begin{array}{l@{\qquad }l} \mathrm {Hom}_{H(V_m)}\left( \sigma \otimes \sigma '^\vee , \omega \right) \not =0 &{}\text {if } E=F \text { and } n \equiv 0 \bmod 2, \\ &{}\text {or } E\not =F \text { and } n \equiv 1 \bmod 2,\\ \mathrm {Hom}_{H(V_m)}\left( \sigma \otimes \sigma '^\vee , \overline{\omega }\right) \not =0 &{} \text {otherwise}. \end{array} \right. \end{aligned}$$

Here we use the fact that $\sigma$, $\sigma '$ and $\omega$ are unitary. By the GP conjectures (Theorems C.1–C.4 and Corollary C.5), we have

$$\begin{aligned} \eta (a)&= \left\{ \begin{aligned}&\varepsilon (\phi ^a \otimes \phi _{\pi '}) \cdot \det (\phi _{\pi '})(-1)^{\frac{1}{2}\dim (\phi ^a)}&\quad&\text {if }E=F \text { and } n \text { is odd},\\&\varepsilon (\phi ^a \otimes \phi _{\pi '}) \cdot \det (\phi ^a)(-1)^{\frac{1}{2}\dim (\phi _{\pi '})} \cdot \nu (\pi ')^{\det (a)}&\quad&\text {if }E=F \text { and } n \text { is even},\\&\omega _{E/F}(-1)^{(n-1)\dim (\phi ^a)} \cdot \varepsilon \left( \phi ^a \otimes \phi _{\pi '}, \psi ^E_{2}\right)&\quad&\text {if }E\not =F, \end{aligned} \right. \\ \theta (\eta )(a)&= \left\{ \begin{aligned}&\varepsilon \left( \theta (\phi )^a \otimes \phi _{\sigma '^\vee } \chi _{-1}\right) \cdot \varepsilon \left( \theta (\phi )^a\right) \cdot \chi _{-1}(-1)^{\frac{1}{2}\dim (\theta (\phi )^a)}&\quad&\text {if }E=F \text { and } n \text { is odd},\\&\varepsilon \left( \theta (\phi )^a \otimes \phi _{\sigma '^\vee }\right) \cdot \varepsilon \left( \theta (\phi ) \otimes \phi _{\sigma '^\vee }\right) ^{\det (a)}\\&\quad \cdot \det \left( \theta (\phi )^a\right) (-1)^{\frac{1}{2} \dim \left( \phi _{\sigma '^\vee }\right) }&\quad&\text {if }E=F \text { and } n \text { is even},\\&\varepsilon \left( \theta (\phi )^a \otimes \phi _{\sigma '^\vee } \otimes \chi ^{-1}, \psi ^E_{2}\right)&\quad&\text {if }E\not =F \text { and } n \text { is odd},\\&\omega _{E/F}(-1)^{\dim (\theta (\phi )^a)} \cdot \varepsilon \left( \theta (\phi )^a \otimes \phi _{\sigma '^\vee } \otimes \chi , \psi ^E_{2}\right)&\quad&\text {if }E\not =F \text { and } n \text { is even} \end{aligned} \right. \end{aligned}$$

for $a \in A_{\theta (\phi )} \subset A_{\phi }$. Here,

• $\phi _{\pi '}$ and $\phi _{\sigma '^\vee }$ are the last names for $\pi '$ and $\sigma '^\vee$, respectively;

• $\nu (\pi ') \in \{\pm 1\}$ is the central value of $\pi '$, i.e., $\pi '(-{\mathbf{1}}_{V_{m+1}}) = \nu (\pi ') \cdot \mathrm {id}$.

By Theorem 6.3, Lemma 6.4, Proposition B.4 and Theorem B.8, we have

$$\begin{aligned} \theta (\phi )=\left( \phi \otimes \chi _{V_m}^{-1} - S_l\right) \otimes \chi _{W_n} \quad \text {so that}\quad \theta (\phi )^a = \phi ^a \otimes \chi _{V_m}^{-1}\chi _{W_n} \end{aligned}$$

and

$$\begin{aligned} \phi _{\sigma '^\vee }= \left\{ \begin{aligned}&\left( \phi _{\pi '} \otimes \chi _{V_m} - S_{l-1}\right) \otimes \chi _{W_{n-1}}^{-1}&\quad&\text {if }E\not =F \text { or } n \text { is odd},\\&\left( \phi _{\pi '} \otimes \chi _{V_m}- S_{l-1}\right) \otimes \chi _{W_{n-1}}^{-1}\chi _{-1}&\quad&\text {if }E=F \text { and } n \text { is even}. \end{aligned} \right. \end{aligned}$$

Therefore we have

$$\begin{aligned} \theta (\eta )(a)/\eta (a) =\left\{ \begin{aligned}&\varepsilon \left( \phi ^a\chi _{V_m}^{-1} \otimes S_{l-1}\right) \cdot \varepsilon \left( \phi ^a\chi _{V_m}^{-1}\chi _{W_n}\right) \\&\quad \cdot \chi _{W_n}(-1)^{\frac{1}{2}\dim (\phi ^a)}&\quad&\text {if }E=F \text { and } n \text { is odd},\\&\varepsilon \left( \phi ^a\chi _{V_m}^{-1} \otimes S_{l-1}\right) \cdot \det \left( \phi ^a\chi _{V_m}^{-1}\right) \\ {}&\quad (-1)^{\frac{l-1}{2}} \cdot \nu ^{\det (a)}&\quad&\text {if }E=F \text { and } n \text { is even},\\&\varepsilon \left( \phi ^a\chi _{V_m}^{-1} \otimes S_{l-1},\psi ^E_{2}\right)&\quad&\text {if }E\not =F,\\ \end{aligned} \right. \end{aligned}$$

for some constant $\nu \in \{\pm 1\}$.

We shall determine this constant $\nu \in \{\pm 1\}$. So we assume that $E=F$ and n is even, hence $G(W_n) = \mathrm {O}(W_n)$ and $H(V_m) = \mathrm {Sp}(V_m)$. Note that $\theta (\eta )(z_{\theta (\phi )}) = 1$. Also, by Prasad conjecture (Theorem D.2), we have $\eta (z_{\phi } + e_1) = 1$. Since $z_{\theta (\phi )} = z_\phi + e_l$ in $A_\phi$, we have $\eta (z_{\theta (\phi )}) = \eta (e_1+e_l)$. On the other hand, if $a = z_\phi + e_l$, we have

$$\begin{aligned}&\varepsilon \left( \phi ^a\chi _{V_m}^{-1} \otimes S_{l-1}\right) \cdot \det \left( \phi ^a\chi _{V_m}^{-1}\right) (-1)^{\frac{l-1}{2}} \cdot \nu ^{\det (a)} \\&\quad = \varepsilon \left( \phi \chi _{V_m}^{-1} \otimes S_{l-1}\right) \cdot \varepsilon \left( S_l \otimes S_{l-1}\right) \cdot \chi _{W_n}(-1)^{\frac{l-1}{2}} \cdot \nu . \end{aligned}$$

We have $\varepsilon (S_l \otimes S_{l-1}) = (-1)^{l-1}=1$. Also, by Lemma A.4 and the (odd-ness condition) proved in Corollary 6.2, we have

$$\begin{aligned} \varepsilon \left( \phi \chi _{V_m}^{-1} \otimes S_{l-1}\right) \cdot \chi _{W_n}(-1)^{\frac{l-1}{2}}= & {} (-1)^{m_\phi \left( \chi _{V_m}S_{l-2}\right) + m_{\phi }\left( \chi _{V_m}S_{l-4}\right) + \cdots + m_{\phi }(\chi _{V_m}S_1)}\\= & {} (-1)^{\frac{l-1}{2}}. \end{aligned}$$

Hence we have $\nu = (-1)^{\frac{l-1}{2}} \cdot \eta (e_1+e_l)$, as desired. This completes the proof. $\square$

Remark 6.6

Suppose that $E=F$ and m, n are even. After Proposition 6.8, which shows the (alternating condition), we will obtain $\eta (e_1+e_l) = (-1)^{\frac{l-1}{2}}$ so that $\nu =1$. By using Theorems 4.3 (5) and 4.5 (4), which are proven in Proposition 6.19, we can obtain $\nu =1$ directly.

6.3 Comparison of central elements

Let $(V_m,W_n)$ and $l=n-m+\epsilon _0$ be as in Sect. 2.2. Let $\pi \in \mathrm {Irr}_\mathrm {temp}(G(W_n))$. Assume that $l \ge 2$ and $\sigma =\theta _{V_m,W_n}(\pi )\not =0$ so that $\sigma \in \mathrm {Irr}(H(V_m))$. We denote the L-parameters for $\pi$ and $\sigma$ by $(\phi _\pi ,\eta _\pi )$ and $(\phi _\sigma ,\eta _\sigma )$, respectively. In this subsection, we compare “$\eta _\pi (z_{\phi _\pi })$” with “$\eta _\sigma (z_{\phi _\sigma })$”.

Let $\phi \in \Phi (G(W_n))$ (resp. $\phi ' \in \Phi (H(V_m))$). If $l=n-m+\epsilon _0$ is odd and $\phi$ contains $\chi _V$ (resp. $\phi '$ contains $\chi _W$), then we denote by $e_1$ the element in $A_\phi$ (resp. $A_{\phi '}$) corresponding to $\chi _V$ (resp. $\chi _W$), i.e., $\phi ^{e_1}=\chi _V$ (resp. $\phi '^{e_1} = \chi _W$) (for the definition of $\phi ^a$, see Sect. 3).

Proposition 6.7

Let $\pi \in \mathrm {Irr}_\mathrm {temp}(G(W_n))$ such that $\sigma =\theta _{V_m,W_n}(\pi ) \in \mathrm {Irr}(H(V_m))$ is nonzero. Assume that $l=n-m+\epsilon _0 \ge 2$. We denote the L-parameters for $\pi$ and $\sigma$ by $(\phi _\pi ,\eta _\pi )$ and $(\phi _\sigma ,\eta _\sigma )$, respectively. Then we have the following:

• (1) If l is odd, then $\phi _\pi \supset \chi _V$ and $\phi _\sigma \supset \chi _W$.

• (2) If $E=F$, $m \not \equiv n \bmod 2$ and $\epsilon =+1$, then

  $$\begin{aligned} \eta _\pi (z_{\phi _\pi })=\eta _\sigma (z_{\phi _\sigma }) \cdot \varepsilon (\phi _\pi ) \cdot \varepsilon \left( \phi _\pi \otimes \chi _V\right) \cdot \chi _V(-1)^{\frac{n}{2}}. \end{aligned}$$

• (3) If $E=F$, $m \not \equiv n \bmod 2$ and $\epsilon =-1$, then

  $$\begin{aligned} \eta _\pi (z_{\phi _\pi })=-\eta _\sigma (z_{\phi _\sigma }) \cdot \delta (\chi _W={\mathbf{1}}) \cdot \varepsilon (\phi _\pi ) \cdot \varepsilon (\phi _\pi \otimes \chi _W) \cdot \chi _W(-1)^{\frac{n-1}{2}}. \end{aligned}$$

• (4) If $E=F$, $m \equiv n \bmod 2$ and $\epsilon = +1$, then $\eta _\pi (z_{\phi _\pi }+e_1) = \eta _\sigma (z_{\phi _\sigma })$.

• (5) If $E=F$, $m \equiv n \bmod 2$ and $\epsilon = -1$, then $\eta _\pi (z_{\phi _\pi }) = -\eta _\sigma (z_{\phi _\sigma } + e_1)$.

• (6) If $E\not =F$ and l is even, then

  $$\begin{aligned} \eta _\sigma (z_{\phi _\sigma }) = \varepsilon \left( \phi _\pi \otimes \chi _V^{-1}, \psi ^E_2\right) \cdot \eta _\pi (z_{\phi _\pi }). \end{aligned}$$

• (7) If $E\not =F$ and l is odd, then $\eta _\pi (e_1)=-\eta _\sigma (e_1)$ and $\eta _\pi (z_{\phi _\pi }+e_1)=\eta _\sigma (z_{\phi _\sigma })$.

Proof

(1) follows from Corollary 6.2 and Theorem 6.3.

The proofs of (2)–(7) are similar. So we prove (3) only. By the assumption, $G(W_n)=\mathrm {O}(W_n)$ is an odd orthogonal group and $H(V_m)=\mathrm {Mp}(V_m)$ is a metaplectic group. By Theorem B.6, there is unique $W_{m+1}^\bullet$ such that $\pi '=\theta _{W^\bullet _{m+1},V_m}(\sigma )$ is nonzero. Let $(\phi _{\pi '},\eta _{\pi '})$ be the L-parameter for $\pi '$. Note that $\theta _{W_n,V_m}(\sigma )=\pi$ is tempered and $m-n-\epsilon _0<-1$. By applying Corollary 5.7 to $\sigma \in \mathrm {Irr}(\mathrm {Mp}(V_m))$ and $\mathrm {O}(W_n)$, we have $\Theta _{W_{m+1},V_m}(\sigma )=0$, where $W_{m+1}$ is the space which belongs to the same Witt tower as $W_n$. This implies that $W_{m+1}^\bullet \not = W_{m+1}$. Since

$$\begin{aligned} \eta _\pi (z_{\phi _\pi })&= \left\{ \begin{aligned}&+1&\quad&\text {if }\mathrm {O}(W_n) \text { is split}, \\&-1&\quad&\text {if }\mathrm {O}(W_n) \text { is not split}, \end{aligned}\right. \\ \eta _{\pi '}\left( z_{\phi _{\pi '}}\right)&= \left\{ \begin{aligned}&+1&\quad&\text {if }\mathrm {O}\left( W_{m+1}^\bullet \right) \text { is split}, \\&-1&\quad&\text {if }\mathrm {O}\left( W_{m+1}^\bullet \right) \text { is not split}, \end{aligned} \right. \end{aligned}$$

we have

$$\begin{aligned} \eta _\pi (z_{\phi _\pi }) = -\eta _{\pi '}(z_{\phi _{\pi '}}). \end{aligned}$$

On the other hand, by Theorem B.8, we have

$$\begin{aligned} \eta _{\pi '}\left( z_{\phi _{\pi '}}\right) = \eta _\sigma (z_{\phi _\sigma }) \cdot \varepsilon (\phi _\sigma ) \cdot \varepsilon \left( \phi _\sigma \otimes \chi _W\right) \cdot \chi _W(-1)^{\frac{m}{2}}. \end{aligned}$$

Since $\phi _\sigma = (\phi _\pi \otimes \chi _V^{-1} \chi _W) \oplus \chi _W S_l$ by Theorem 6.3, using Lemma A.5, we have

$$\begin{aligned} \eta _{\pi '}(z_{\phi _{\pi '}})&=\eta _\sigma (z_{\phi _\sigma }) \cdot \delta (\chi _W={\mathbf{1}}) \cdot \varepsilon (\phi _\pi ) \cdot \varepsilon \left( \phi _\pi \otimes \chi _W\right) \cdot \chi _W(-1)^{\frac{n-1}{2}}. \end{aligned}$$

Hence we obtain (3).

Using Theorems B.8, D.1, D.2, Proposition 2.4, and Corollary 5.7, the proofs of the other cases are similar to that of the above case. $\square$

If $l\in \{-1,0,1\}$, then we see that a similar assertion holds by using Theorem B.8 and Prasad’s conjectures (Theorems D.1 and D.2). This implies Theorem 4.1 (2) unless $E=F$, $m \not \equiv n \bmod 2$ and $\epsilon =-1$. In this case, $G(W_n) = \mathrm {O}(W_n)$ is an odd orthogonal group, and the first occurrence indices $m^\pm (\pi )$ can be determined by the central character of $\pi \in \mathrm {Irr}(\mathrm {O}(W_n))$. Hence the remaining issue of Theorem 4.1 (2) is a relation between the central character of $\pi$ and theta lifts $\Theta _{V_m,W_n}(\pi )$. It will be treated in Sect. 6.8 (Proposition 6.20).

6.4 Character conditions

In this subsection, we derive the (initial condition) and the (alternating condition) in Theorem 4.1 (1).

Proposition 6.8

Let $\pi \in \mathrm {Irr}_\mathrm {temp}(G(W_n))$ with L-parameter $(\phi ,\eta )$. Assume that $\Theta _{V_m,W_n}(\pi )\not =0$ and $l=n-m+\epsilon _0 \ge 2$. Define $\kappa \in \{1,2\}$ by $\kappa \equiv l \bmod 2$. Let $e_{i}$ be the element in $A_\phi$ corresponding to $\chi _V S_{i} \subset \phi$. Then we have

$$\begin{aligned} \eta (e_{\kappa +2i+2})=-\eta (e_{\kappa +2i}) \end{aligned}$$

for $0 \le i < (l-\kappa )/2$. Moreover, if $\kappa =2$, then

$$\begin{aligned} \eta (e_2)= \left\{ \begin{aligned}&\epsilon \cdot \delta (\chi _V={\mathbf{1}})&\quad&\text {if }E=F \text { and } m \not \equiv n \bmod 2,\\&-1&\quad&\text {if }E\not =F \text { and } m \equiv n \bmod 2. \end{aligned} \right. \end{aligned}$$

Proof

Let $(\theta (\phi ),\theta (\eta ))$ be the L-parameter for $\theta _{V_m,W_n}(\pi )$. Note that

$$\begin{aligned} z_\phi = z_{\theta (\phi )}+e_{l}. \end{aligned}$$

By applying Proposition 6.7 and Theorem 6.5 to $a=z_{\theta (\phi )} \in A_{\theta (\phi )} \subset A_\phi$, we have

$$\begin{aligned}&\eta (e_l)=\frac{\eta (z_\phi )}{\theta (\eta )(z_{\theta (\phi )})} \cdot \frac{\theta (\eta )\left( z_{\theta (\phi )}\right) }{\eta \left( z_{\theta (\phi )}\right) }\\&\quad =\left\{ \begin{aligned}&\delta (\chi _{V}={\mathbf{1}})\cdot \varepsilon (\phi \chi _{V}^{-1} \otimes S_{l-1}) \cdot \varepsilon \left( \phi \chi _V^{-1}\right)&\quad&\text {if }E=F, \epsilon = +1 \text { and } m \text { is odd},\\&-\varepsilon \left( \phi \otimes S_{l-1}\right) \cdot \varepsilon (\phi )&\quad&\text {if }E=F, \epsilon = -1 \text { and } n \text { is odd},\\&\eta (e_1) \cdot \varepsilon \left( \phi \chi _{V}^{-1} \otimes S_{l-1}\right) \cdot \chi _{V}(-1)^{\frac{l-1}{2}}&\quad&\text {if }E=F, \epsilon =+1 \text { and } m\equiv n \equiv 0\bmod 2,\\&-\varepsilon \left( \phi \chi _{V}^{-1},\psi ^E_2\right) \cdot \varepsilon (\phi \chi _{V}^{-1} \otimes S_{l-1},\psi ^E_{2})&\quad&\text {if }E\not =F \text { and } m \equiv n \bmod 2,\\&\eta (e_1) \cdot \varepsilon \left( \phi \chi _{V}^{-1} \otimes S_{l-1},\psi ^E_{2}\right)&\quad&\text {if }E\not =F \text { and } m \not \equiv n \bmod 2. \end{aligned} \right. \end{aligned}$$

If $E=F$, $\epsilon =-1$ and $m\equiv n \equiv 0\bmod 2$, applying Proposition 6.7 and Theorem 6.5 to $a=z_{\theta (\phi )}+e_1 = z_\phi + e_1 + e_l \in A_{\theta (\phi )} \subset A_\phi$. we obtain

$$\begin{aligned} \eta (e_l)&=\frac{\eta (z_\phi +e_1)}{\theta (\eta ) \left( z_{\theta (\phi )}+e_1\right) } \cdot \frac{\theta (\eta )\left( z_{\theta (\phi )} +e_1\right) }{\eta \left( z_{\theta (\phi )}+e_1\right) }\\&=\eta (e_1) \cdot \epsilon \left( \phi \chi _V^{-1} \otimes S_{l-1}\right) \cdot \chi _W(-1)^{\frac{l-1}{2}}. \end{aligned}$$

By the tower property (Proposition 2.4), a similar equation for $\eta (e_{l-2i})$ holds for $i=0,1,\dots ,(l-\kappa )/2$. In particular, if $\kappa =2$, then we have

$$\begin{aligned} \eta (e_2)= \left\{ \begin{aligned}&\epsilon \cdot \delta (\chi _V={\mathbf{1}})&\quad&\text {if }E=F \text { and } m \not \equiv n \bmod 2,\\&-1&\quad&\text {if }E\not =F \text { and } m \equiv n \bmod 2. \end{aligned} \right. \end{aligned}$$

Moreover, we have

$$\begin{aligned}&\frac{\eta \left( e_{\kappa +2i+2}\right) }{\eta \left( e_{\kappa +2i}\right) } =\frac{\varepsilon \left( \phi \chi _{V}^{-1} \otimes S_{\kappa +2i+1},\psi ^E_{2}\right) }{\varepsilon \left( \phi \chi _{V}^{-1} \otimes S_{\kappa +2i-1},\psi ^E_{2}\right) } \times \left\{ \begin{aligned}&\det \left( \phi \chi _{V}^{-1}\right) (-1)&\quad&\text {if }E=F,\\&1&\quad&\text {if }E\not =F \end{aligned} \right. \end{aligned}$$

for $0 \le i <(l-\kappa )/2$. By Lemma A.4 and (odd-ness condition) in Theorem 4.1 (1), this is equal to $(-1)^{m_\phi (\chi _{V}S_{\kappa +2i})}=-1$. $\square$

This is the (initial condition) and the (alternating condition) in Theorem 4.1 (1). In particular, we have

$$\begin{aligned} n-m^\mathrm {down}(\pi )+\epsilon _0 \in \mathcal {T}\end{aligned}$$

for any $\pi \in \mathrm {Irr}_\mathrm {temp}(G(W_n))$.

Also, when $E=F$ and m, n are both even (so that l is odd), we have

$$\begin{aligned} \eta (e_1+e_l) = (-1)^{\frac{l-1}{2}}. \end{aligned}$$

Theorem 6.5 together with this equation implies Theorem 4.3 (1).

Remark 6.9

We may apply the result shown above (i.e., Theorem 4.3 (1)) to the going-up tower sometimes. Under the notation and assumption of Theorem 4.5 (1), we will show that $\theta _{V_m,W_n}(\pi )$ is tempered (Corollary 6.13). If we knew the temperedness of $\theta _{V_m,W_n}(\pi )$, Theorem 4.3 (1) implies Theorem 4.5 (1).

The following proposition says that $l(\pi )=\max \,\mathcal {T}=n-m^\mathrm {down}(\pi )+\epsilon _0$ in a special case.

Proposition 6.10

Let $\pi \in \mathrm {Irr}_\mathrm {temp}(G(W_n))$ with L-parameter $(\phi ,\eta )$. Assume that

• $l=n-m^\mathrm {down}(\pi )+\epsilon _0 \ge 0$;

• $\phi$ contains $\chi _{V}S_{l-2i}$ for $i=-1,0,\dots , (l-\kappa )/2$;

• the first occurrence $\sigma ^\mathrm {up}=\theta _{V'_{m^\mathrm {up}(\pi )},W_n}(\pi )$ to the going-up tower $\mathcal {V}^\mathrm {up}$ is tempered.

Then we have:

• (1) If $l=0$, then the (initial condition) in Theorem 4.1 (1) does not hold. Namely,

  $$\begin{aligned} \eta (e_2)= \left\{ \begin{aligned}&-\epsilon \cdot \delta (\chi _V={\mathbf{1}})&\quad&\text {if }E=F \text { and } m \not \equiv n \bmod 2,\\&+1&\quad&\text {if }E\not =F \text { and } m \equiv n \bmod 2. \end{aligned}\right. \end{aligned}$$

• (2) If $l>0$, then $m_\phi (\chi _{V}S_l)$ is odd, and the (alternating condition) in Theorem 4.1 (1) does not hold. Namely,

  $$\begin{aligned} \eta (e_{l+2}+e_{l})=-(-1)^{m_\phi (\chi _{V}S_l)}=+1. \end{aligned}$$

Proof

First, we prove (2). Let $(\phi _\sigma ,\eta _\sigma )$ be the L-parameter for $\sigma ^\mathrm {up}$. Note that $\sigma ^\mathrm {up}$ is tempered by the assumption, and $m^\mathrm {up}(\pi ) - n - \epsilon _0 = l+2 \ge 2$ by the conservation relation (Proposition 2.5). By applying Theorem 6.3, Corollary 6.2 and Proposition 6.8 to $\sigma ^\mathrm {up}$, we have

$$\begin{aligned} \phi _\sigma =\left( \phi \otimes \chi _{V}^{-1}\chi _{W}\right) \oplus \chi _{W}S_{l+2}, \end{aligned}$$

and we see that $m_{\phi _\sigma }(\chi _{W}S_l)=m_\phi (\chi _{V}S_l)$ is odd, and $\eta _\sigma (e_{l+2}+e_{l})=-1$. Therefore it is enough to show $\eta (e_{l+2}+e_{l})/\eta _\sigma (e_{l+2}+e_{l})=-1$. It follows from Theorem 4.3 (1). Hence we have (2). The proof of (1) is similar. $\square$

By Proposition 5.5, if $\pi$ is a discrete series representation, then the first occurrence $\sigma ^\mathrm {up}=\theta _{V'_{m^\mathrm {up}(\pi )},W_n}(\pi )$ is tempered. Hence by Proposition 6.10, we see that

$$\begin{aligned} n-m^\mathrm {down}(\pi )+\epsilon _0+2=l+2 \not \in \mathcal {T}\end{aligned}$$

if $\pi$ is a discrete series representation. This completes the proof of Theorem 4.1 (1) for discrete series representations.

6.5 Temperedness of theta lifts 2

In this subsection, we discuss whether the first occurrence $\sigma =\theta _{V'_{m^\mathrm {up}(\pi )},W_n}(\pi )$ for the going-up tower $\mathcal {V}^\mathrm {up}$ is tempered or not.

Let $\pi \in \mathrm {Irr}_\mathrm {temp}(G(W_n))$ with L-parameter $(\phi ,\eta )$. Assume that $l=n-m^\mathrm {down}(\pi )+\epsilon _0 \ge 0$. Define $\kappa \in \{1,2\}$ by $\kappa \equiv l \bmod 2$. Then by Corollary 6.2, we know that $\phi$ contains $\chi _V S_{\kappa +2i}$ for $0 \le i \le (l-\kappa )/2$, and $m_\phi (\chi _V S_{\kappa +2i})$ is odd for $0 \le i < (l-\kappa )/2$. Note that $m^\mathrm {up}(\pi ) - n - \epsilon _0 = l+2 \ge 2$.

Decompose $\phi =\phi ' \oplus \phi _0 \oplus {}^c\phi '^\vee$ with $\phi _0 \in \Phi _\mathrm {disc}(G(W_{n_0}))$. Assume that

$$\begin{aligned} \chi _V\tau _1 \times \cdots \times \chi _V \tau _r \rtimes \pi _0 \twoheadrightarrow \pi \end{aligned}$$

for some $\tau _i \in \mathrm {Irr}_\mathrm {disc}(\mathrm {GL}_{k_i}(E))$ and $\pi _0 \in \mathrm {Irr}_\mathrm {disc}(G(W_{n_0}))$ with $n_0=n-2\sum _{i=1}^r k_i$, so that the L-parameter of $\pi _0$ is given by $(\phi _0,\eta |A_{\phi _0})$. If $m \ge n+\epsilon _0$, then by a similar argument as that for [11, Proposition C.4], we have

$$\begin{aligned} \chi _W\tau _1 \times \cdots \times \chi _W \tau _r \rtimes \Theta _{V_{m_0}, W_{n_0}}(\pi _0) \twoheadrightarrow \Theta _{V_m,W_n}(\pi ), \end{aligned}$$

where $m_0=m-2\sum _{i=1}^r k_i$. In particular, if $\Theta _{V_m,W_n}(\pi )$ is nonzero, then $\Theta _{V_{m_0}, W_{n_0}}(\pi _0)$ is also nonzero.

Lemma 6.11

Suppose that $m^\mathrm {down}(\pi )<m^\mathrm {up}(\pi )$. Then the going-down tower $\mathcal {V}^\mathrm {down}$ with respect to $\pi$ is also the going-down tower $\mathcal {V}^\mathrm {down}$ with respect to $\pi _0$.

Proof

Set $m=n+\epsilon _0+2-\kappa$. Then $l=n-m+\epsilon _0=\kappa -2 \in \{0,-1\}$. A tower $\mathcal {V}$ is the going-down tower with respect to $\pi$ if and only if $\Theta _{V_m,W_n}(\pi )$ is nonzero for $V_m \in \mathcal {V}$. In this case, $\Theta _{V_{m_0}, W_{n_0}}(\pi _0)$ is also nonzero for $V_{m_0} \in \mathcal {V}$. This shows that $\mathcal {V}$ is also the going-down tower with respect to $\pi _0$. $\square$

We determine the first occurrence index of $\pi _0$ in terms of the one of $\pi$.

Proposition 6.12

Let notation be as above. If $m^\mathrm {down}(\pi )=n+\epsilon _0-l$ with $l>0$, then

$$\begin{aligned} m^\mathrm {down}(\pi _0)=\left\{ \begin{aligned}&n_0+\epsilon _0-l&\quad&\text {if }m_\phi (\chi _V S_l) \text { is odd},\\&n_0+\epsilon _0-l+2&\quad&\text {if }m_\phi (\chi _V S_l) \text { is even}. \end{aligned} \right. \end{aligned}$$

Proof

Note that we have proven Theorem 4.1 (1) for the discrete series representation $\pi _0$. By Corollary 6.2, we see that $m_\phi (\chi _V S_{\kappa +2i})$ is odd for $0 \le i < (l-\kappa )/2$, where we define $\kappa \in \{1,2\}$ by $\kappa \equiv l \bmod 2$. If $m_\phi (\chi _V S_l)$ is even, then by applying Theorem 4.1 (1) to $\pi _0$, we have $m^\mathrm {down}(\pi _0)=n_0+\epsilon _0-l+2$.

Suppose that $m_\phi (\chi _V S_l)$ is odd. Note that $m^\mathrm {up}(\pi )=n+\epsilon _0+l+2$. By Lemma 6.11 and a remark before this lemma, we have $m^\mathrm {up}(\pi _0) \le n_0+\epsilon _0+l+2$. Hence $m^\mathrm {down}(\pi _0) \ge n_0+\epsilon _0-l$. On the other hand, by applying Theorem 4.1 (1) to $\pi _0$, we have $m^\mathrm {down}(\pi _0) \le n_0+\epsilon _0-l$. Therefore we have $m^\mathrm {down}(\pi _0) = n_0+\epsilon _0-l$. $\square$

Corollary 6.13

Let $\pi \in \mathrm {Irr}_\mathrm {temp}(G(W_n))$ with L-parameter $(\phi ,\eta )$. Assume that $m^\mathrm {down}(\pi )=n+\epsilon _0-l$ with $l\ge 0$, so that $m^\mathrm {up}(\pi )=n+\epsilon _0+l+2$. Let $\sigma =\theta _{V'_{m^\mathrm {up}(\pi )},W_n}(\pi )$ be the first occurrence for the going-up tower $\mathcal {V}^\mathrm {up}$.

• (1) If $l=0$, then $\sigma$ is tempered.

• (2) Suppose that $l>0$. Then $\sigma$ is tempered if and only if $m_\phi (\chi _V S_l)$ is odd.

Proof

We prove (2). The proof of (1) is similar. So we assume that $l>0$.

If $\sigma$ is tempered, then we have proven that $m_\phi (\chi _V S_l)$ is odd in Proposition 6.10.

Conversely, suppose that $m_\phi (\chi _V S_l)$ is odd. We may assume that

$$\begin{aligned} \chi _V\tau _1 \times \cdots \times \chi _V \tau _r \rtimes \pi _0 \twoheadrightarrow \pi \end{aligned}$$

for some $\tau _i \in \mathrm {Irr}_\mathrm {disc}(\mathrm {GL}_{k_i}(E))$ and $\pi _0 \in \mathrm {Irr}_\mathrm {disc}(G(W_{n_0}))$ with $n_0=n-2\sum _{i=1}^r k_i$. As we have seen before Lemma 6.11, we have

$$\begin{aligned} \chi _W\tau _1 \times \cdots \times \chi _W \tau _r \rtimes \Theta _{V'_{m_0}, W_{n_0}}(\pi _0) \twoheadrightarrow \Theta _{V'_m,W_n}(\pi ), \end{aligned}$$

where $m_0=m-2\sum _{i=1}^r k_i$ and $m = m^{\mathrm {up}}(\pi )$. Hence there exists an irreducible subquotient $\sigma _0$ of $\Theta _{V'_{m_0}, W_{n_0}}(\pi _0)$ such that

$$\begin{aligned} \chi _W\tau _1 \times \cdots \times \chi _W \tau _r \rtimes \sigma _0 \twoheadrightarrow \sigma . \end{aligned}$$

Since $m_\phi (\chi _V S_l)$ is odd, by Proposition 6.12 together with the conservation relation (Proposition 2.5), we see that $\Theta _{V'_{m_0}, W_{n_0}}(\pi _0)$ is the first lift of a discrete series representation $\pi _0$ to the going-up tower $\mathcal {V}^\mathrm {up}$. By Proposition 5.5 (2), the irreducible subquotient $\sigma _0$ of $\Theta _{V'_{m_0}, W_{n_0}}(\pi _0)$ is tempered. Therefore, $\sigma$ is also tempered. $\square$

Corollary 6.13 and Proposition 6.12 imply that

$$\begin{aligned} n-m^\mathrm {down}(\pi )+\epsilon _0+2=l+2 \not \in \mathcal {T}\end{aligned}$$

for any tempered representation $\pi$. Hence we have $l(\pi )=\max \,\mathcal {T}=n-m^\mathrm {down}(\pi )+\epsilon _0$. This completes the proof of Theorem 4.1 (1). Also, using Corollary 6.13, we obtain Theorem 4.5 (1) from Theorem 4.3 (1), as we noted in Remark 6.9.

6.6 Non-tempered first lifts

In this subsection, we prove Theorem 4.5 (2).

Let $\pi \in \mathrm {Irr}_\mathrm {temp}(G(W_n))$ with L-parameter $(\phi ,\eta )$. Assume that $l = l(\pi ) = n - m^\mathrm {down}(\pi ) + \epsilon _0 > 0$. Theorem 4.1 (1) implies that

• $\phi$ contains $\chi _V S_l, \chi _V S_{l-2}, \dots , \chi _V S_\kappa$, where $\kappa \in \{1,2\}$ is defined by $\kappa \equiv l \bmod 2$;

• $m_\phi (\chi _V S_{\kappa +2i})$ is odd for $0 \le i < (l - \kappa )/2$.

We put $m=m^\mathrm {up}(\pi )$. Note that $m-n-\epsilon _0=l+2$. Let $\sigma = \theta _{V_{m},W_{n}}(\pi )$ be the first occurrence of $\pi$ to the going-up tower $\mathcal {V}^\mathrm {up}$. By Corollary 6.13, we see that $\sigma$ is non-tempered if and only if $m_\phi (\chi _V S_l)$ is even. In this subsection, we assume these conditions.

Suppose that $\sigma$ is the Langlands quotient of the standard module

$$\begin{aligned} \tau _1 \left| \cdot \right| _E^{s_1} \times \cdots \times \tau _r \left| \cdot \right| _E^{s_r} \rtimes \sigma _0, \end{aligned}$$

where $\tau _i \in \mathrm {Irr}_\mathrm {disc}(\mathrm {GL}_{k_i}(E))$, $\sigma _0 \in \mathrm {Irr}_\mathrm {temp}(H(V_{m_0}))$, $2k_1+\cdots +2k_r+m_0=m$, and $s_1 \ge \cdots \ge s_r >0$.

First, we have the following:

Proposition 6.14

For any $i=1,\dots ,r$, the exponent $s_i$ is in $(1/2)\mathbb {Z}$.

Proof

Consider the Plancherel measure (see Appendix A.2). By Theorem A.2, we have

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

for any $\tau \in \mathrm {Irr}( \mathrm {GL}_k(E))$. In particular, by Desideratum B.1 (8), we have

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

Let $\mathcal {A}$ be the set of $s_0 \in \mathbb {C}$ such that the left hand side of the above equation has a pole at $s=s_0$ for some unitary supercuspidal representation $\tau$ of $\mathrm {GL}_k(E)$. Looking at the right hand side, we see that

$$\begin{aligned} \left\{ \mathrm {Re}(s_0)\ \big |\ s_0 \in \mathcal {A}\right\} \subset \frac{1}{2}\mathbb {Z}. \end{aligned}$$

Let $\phi _{\tau _i}$ be the irreducible representation of $WD _E$ corresponding to $\tau _i$. We may decompose $\phi _{\tau _i} \cong \phi _i \boxtimes S_{d_i}$, where $\phi _i$ is an irreducible representation of $W_E$ and $d_i$ is a positive integer. Since

$$\begin{aligned} \phi _\sigma = \phi _{\tau _1}\left| \cdot \right| _E^{s_1} \oplus \cdots \oplus \phi _{\tau _r}\left| \cdot \right| _E^{s_r} \oplus \phi _{\sigma _0} \oplus {}^c\phi _{\tau _r}^\vee \left| \cdot \right| _E^{-s_r} \oplus \cdots \oplus {}^c\phi _{\tau _1}^\vee \left| \cdot \right| _E^{-s_1}, \end{aligned}$$

we have

$$\begin{aligned}&\gamma \left( s,\chi _W\phi _\tau \otimes \phi _\sigma ^\vee , \psi _E\right) \cdot \gamma \left( -s,\chi _W^{-1}\phi _\tau ^\vee \otimes \phi _\sigma , \psi _E^{-1}\right) \\&\quad =\Big [\prod _{i=1}^r \gamma \big (s-s_i, \chi _W\phi _\tau \otimes \phi _{\tau _i}^\vee , \psi _E\big ) \gamma \big (s+s_i, \chi _W\phi _\tau \otimes {}^c\phi _{\tau _i}, \psi _E\big )\\&\quad \times \gamma \big (-s-s_i, \chi _W^{-1}\phi _\tau ^\vee \otimes \phi _{\tau _i}^\vee , \psi _E^{-1}\big ) \gamma \big (-s+s_i, \chi _W^{-1}\phi _\tau ^\vee \otimes {}^c\phi _{\tau _i}, \psi _E^{-1}\big )\Big ]\\&\quad \times \gamma \big (s,\chi _W\phi _\tau \otimes \phi _{\sigma _0}^\vee , \psi _E\big ) \cdot \gamma \big (-s,\chi _W^{-1}\phi _\tau ^\vee \otimes \phi _{\sigma _0}, \psi _E^{-1}\big ). \end{aligned}$$

Now suppose that some $s_j$ is not in $(1/2)\mathbb {Z}$. We may assume that $s_i \not \in (1/2)\mathbb {Z}$ and $s_i$ satisfies that

$$\begin{aligned} \max \left\{ s_j+\frac{d_j-1}{2}\ \big |\ s_j \not \in \frac{1}{2}\mathbb {Z}\right\} = s_i + \frac{d_i-1}{2}. \end{aligned}$$

Taking $\phi _\tau = \chi _W^{-1} \phi _i$, in the above equation, we see that $\gamma (s,\chi _W\phi _\tau \otimes \phi _\sigma ^\vee , \psi _E) \cdot \gamma (-s,\chi _W^{-1}\phi _\tau ^\vee \otimes \phi _\sigma , \psi _E^{-1})$ has a pole at $s= 1+ s_i + (d_i-1)/2$ since the local gamma factor $\gamma (s-s_i, \chi _W\phi _\tau \otimes \phi _{\tau _i}^\vee , \psi _E)$ has a pole at this point. Hence $1+ s_i + (d_i-1)/2 \in \mathcal {A}$ but $1+ s_i + (d_i-1)/2 \not \in (1/2)\mathbb {Z}$. This is a contradiction. $\square$

Corollary 6.15

We have $s_i=1/2$ and $\tau _i=\chi _W \mathrm {St}_{l+1}$ for any $i=1,\dots ,r$.

Proof

By [14, Proposition 3.1], we know that $s_1=1/2$ and $\tau _1=\chi _W \mathrm {St}_{l+1}$. Hence we have $s_i=1/2$ for any $i=1,\dots ,r$. Since each $\tau _i$ is a discrete series representation of a general linear group, we can interchange $\tau _i$ with $\tau _1$ (see e.g., [53]). Hence we have $\tau _i=\chi _W \mathrm {St}_{l+1}$ for any $i=1,\dots ,r$. $\square$

The following is the key result.

Proposition 6.16

We have $r=1$.

Proof

By (the proof of) Proposition 3.1 in [14], we can find an irreducible representation $\sigma _1$ of $H(V_{m_1})$ such that

$$\begin{aligned} \mathrm {Ind}_{Q(Y_{l+1})}^{H(V_m)}\left( \chi _W \mathrm {St}_{l+1}\left| \cdot \right| _E^{1/2} \otimes \sigma _1\right) \twoheadrightarrow \sigma , \end{aligned}$$

and

$$\begin{aligned} \mathrm {Ind}_{P(X_l)}^{G(W_n)}\left( \chi _V \mathrm {St}_{l} \otimes \Theta _{W_{n_1},V_{m_1}}(\sigma _1)\right) \twoheadrightarrow \pi , \end{aligned}$$

where we put $m_1=m-2(l+1)$ and $n_1=n-2l$. We have to show that $\sigma _1$ is tempered. Suppose for the sake of contradiction that $\sigma _1$ is not tempered. Then by Corollary 6.15, there exists $\sigma _2 \in \mathrm {Irr}(H(V_{m_2}))$ such that

$$\begin{aligned} \mathrm {Ind}_{Q(Y'_{l+1})}^{H(V_{m_1})}\left( \chi _W \mathrm {St}_{l+1}\left| \cdot \right| _E^{1/2} \otimes \sigma _2\right) \twoheadrightarrow \sigma _1, \end{aligned}$$

where $m_2 = m_1 - 2(l+1)$ and $V_{m_1}= Y'_{l+1} \oplus V_{m_2} \oplus (Y'_{l+1})^*$. Since $m_1-n_1-\epsilon _0=l$, by Corollary 5.3, we have

$$\begin{aligned} \mathrm {Ind}_{P(X'_{l+1})}^{G(W_{n_{1}})} \left( \chi _V \mathrm {St}_{l+1}\left| \cdot \right| _E^{1/2} \otimes \Theta _{W_{n_2},V_{m_2}}(\sigma _2)\right) \twoheadrightarrow \Theta _{W_{n_{1}},V_{m_{1}}}(\sigma _{1}), \end{aligned}$$

where $n_2=n_1-2(l+1)$ and $W_{n_1}= X'_{l+1} \oplus W_{n_2} \oplus (X'_{l+1})^*$. Combining these maps, we have

$$\begin{aligned} \chi _V \mathrm {St}_l \times \chi _V \mathrm {St}_{l+1}\left| \cdot \right| _E^{1/2} \rtimes \Theta _{W_{n_2},V_{m_2}}(\sigma _2) \twoheadrightarrow \pi . \end{aligned}$$

This contradicts the hypothesis that $\pi$ is tempered by Casselman’s criterion. $\square$

Now we are ready to prove Theorem 4.5 (2). More precisely, we prove the following theorem:

Theorem 6.17

Let $\pi \in \mathrm {Irr}_\mathrm {temp}(G(W_n))$ with L-parameter $(\phi ,\eta )$. Assume that

• $l = l(\pi ) = n - m^\mathrm {down}(\pi ) + \epsilon _0 > 0$;

• $m_\phi (\chi _V S_l)=2h$ for some $h>0$.

We write $\phi = \phi _0 \oplus (\chi _V S_l)^{\oplus 2h}$. Put $n_0=n-2hl$ and $m_0=m-2hl-2$. Let $\pi _0 \in \mathrm {Irr}_\mathrm {temp}(G(W_{n_0}))$ such that

$$\begin{aligned} \chi _V\mathrm {St}_l \times \dots \chi _V \mathrm {St}_l \rtimes \pi _0 \twoheadrightarrow \pi , \end{aligned}$$

so that the L-parameter of $\pi _0$ is $(\phi _0,\eta |A_{\phi _0})$. Here, $\chi _V \mathrm {St}_l$ appears h-times. We set $m=m^\mathrm {up}(\pi )$ and let $\sigma =\theta _{V_m,W_n}(\pi )$ be the first occurrence of $\pi$ to the going-up tower $\mathcal {V}^\mathrm {up}$. Then we have

$$\begin{aligned} \chi _W \mathrm {St}_{l+1}\left| \cdot \right| _E^{1/2} \times \chi _W\mathrm {St}_l \times \cdots \times \chi _W\mathrm {St}_l \rtimes \sigma _0 \twoheadrightarrow \sigma , \end{aligned}$$

where $\sigma _0=\theta _{V_{m_0},W_{n_0}}(\pi _0)$, and $\chi _W\mathrm {St}_l$ appears $(h - 1)$-times. In particular, if we denote the L-parameter for $\sigma$ (resp. $\sigma _0$) by $(\phi _\sigma , \eta _\sigma )$ (resp. $(\phi _{\sigma _0}, \eta _{\sigma _0})$), then we have

$$\begin{aligned} \phi _{\sigma _0}= & {} \phi \chi _V^{-1}\chi _W - (\chi _W S_l)^{\oplus (2h-1)} \quad \text {and} \quad \\ \phi _\sigma= & {} \phi _{\sigma _0} +\left( \chi _W S_l\right) ^{\oplus 2(h-1)} + \chi _W S_{l+1} \otimes \left( \left| \cdot \right| _E^{1/2}+\left| \cdot \right| _E^{-1/2}\right) . \end{aligned}$$

Moreover the canonical injection $A_{\phi _{\sigma _0}} \hookrightarrow A_{\phi _\sigma }$ is in fact bijective, and we have $\eta _\sigma |A_{\phi _{\sigma _0}}=\eta _{\sigma _0}$.

Proof

By [14, Proposition 3.1], we can find $\sigma _1 \in \mathrm {Irr}(H(V_{m_1}))$ and $\pi _1 \in \mathrm {Irr}(G(W_{n_1}))$ with $m_1=m-2(l+1)$ and $n_1=n-2l$ such that

$$\begin{aligned} \mathrm {Ind}_{Q(Y_{l+1})}^{H(V_m)}\left( \chi _W \mathrm {St}_{l+1} \left| \cdot \right| _E^{1/2} \otimes \sigma _1\right) \twoheadrightarrow \sigma , \quad \mathrm {Ind}_{P(X_l)}^{G(W_n)}\left( \chi _V \mathrm {St}_l \otimes \pi _1\right) \twoheadrightarrow \pi \end{aligned}$$

and $\pi _1$ is a subquotient of $\Theta _{W_{n_1},V_{m_1}}(\sigma _1)$. Proposition 6.16 says that $\sigma _1$ is tempered. Hence $\theta _{W_{n_1},V_{m_1}}(\sigma _1)$ belongs to the same L-packet as $\pi _1$ by Proposition 5.5 and Lemma 6.4. Therefore we have

$$\begin{aligned} \phi _\sigma&=\phi _{\sigma _1} + \chi _W S_{l+1} \otimes \left( \left| \cdot \right| _E^{1/2}+\left| \cdot \right| _E^{-1/2}\right) \\&=\left( \phi _{\pi _1}\chi _V^{-1}\chi _W + \chi _W S_l\right) +\chi _W S_{l+1} \otimes \left( \left| \cdot \right| _E^{1/2}+\left| \cdot \right| _E^{-1/2}\right) \\&=\phi \chi _V^{-1}\chi _W - \chi _W S_l +\chi _W S_{l+1} \otimes \left( \left| \cdot \right| _E^{1/2}+\left| \cdot \right| _E^{-1/2}\right) , \end{aligned}$$

where we denote by $\phi _{\sigma _1}$ and $\phi _{\pi _1}$ the last names for $\sigma _1$ and $\pi _1$, respectively.

In particular, there exists $\sigma _0 \in \mathrm {Irr}_\mathrm {temp}(H(V_{m_0}))$ whose L-parameter is $(\phi _{\sigma _0},\eta _{\sigma _0})$ with

$$\begin{aligned} \phi _{\sigma _0}=\phi \chi _V^{-1}\chi _W - (\chi _W S_l)^{\oplus (2h-1)}, \quad \eta _{\sigma _0}=\eta _\sigma |A_{\phi _{\sigma _0}} \end{aligned}$$

such that

$$\begin{aligned} \chi _W \mathrm {St}_{l+1}\left| \cdot \right| _E^{1/2} \times \chi _W\mathrm {St}_l \times \cdots \times \chi _W\mathrm {St}_l \rtimes \sigma _0 \twoheadrightarrow \sigma , \end{aligned}$$

with $\chi _W\mathrm {St}_l$ occuring $h-1$ times. Note that $\chi _W\mathrm {St}_l \times \cdots \times \chi _W\mathrm {St}_l \rtimes \sigma _0$ is irreducible since $\phi _{\sigma _0}$ contains $\chi _W S_l$, so that

$$\begin{aligned} \chi _W \mathrm {St}_{l+1}\left| \cdot \right| _E^{1/2} \times \chi _W\mathrm {St}_l \times \cdots \times \chi _W\mathrm {St}_l \rtimes \sigma _0 \end{aligned}$$

is a standard module, which has a unique Langlands quotient. We have to show that $\sigma _0=\theta _{V_{m_0}, W_{n_0}}(\pi _0)$. Since $\chi _W \mathrm {St}_{l+1}|\cdot |_E^{1/2}$ and $\chi _W\mathrm {St}_l$ are not linked, we have

$$\begin{aligned}&\chi _W \mathrm {St}_{l+1}\left| \cdot \right| _E^{1/2} \times \chi _W\mathrm {St}_l \times \cdots \times \chi _W\mathrm {St}_l \rtimes \sigma _0 \\&\quad \cong \chi _W\mathrm {St}_l \times \cdots \times \chi _W\mathrm {St}_l \times \chi _W \mathrm {St}_{l+1}\left| \cdot \right| _E^{1/2} \rtimes \sigma _0. \end{aligned}$$

For the linked-ness and its properties, see [53] (in particular, see [53, Theorem 9.7]). By Lemma 2.2, we have

$$\begin{aligned} \sigma \hookrightarrow \chi _W\mathrm {St}_l \times \cdots \times \chi _W\mathrm {St}_l \times \chi _W \mathrm {St}_{l+1}\left| \cdot \right| _E^{-1/2} \rtimes \sigma _0. \end{aligned}$$

Since $m-n-\epsilon _0=l+2$, by applying Corollary 5.3 to $\chi _W\mathrm {St}_l \times \cdots \times \chi _W\mathrm {St}_l \times \chi _W \mathrm {St}_{l+1}|\cdot |_E^{-1/2} \rtimes \sigma _0$, we have

$$\begin{aligned}&\pi ^\vee \hookrightarrow \Theta _{W_{n},V_m}(\sigma )^\vee \cong \mathrm {Hom}_{H(V_m)}\left( \omega _{V_m,W_n}, \sigma \right) _\infty \\&\quad \hookrightarrow \mathrm {Hom}_{H(V_m)}\left( \omega _{V_m,W_n}, \chi _W\mathrm {St}_l \times \cdots \times \chi _W\mathrm {St}_l \times \chi _W \mathrm {St}_{l+1}\left| \cdot \right| _E^{-1/2} \rtimes \sigma _0\right) _\infty \\&\quad \cong \chi _V\mathrm {St}_l \times \cdots \times \chi _V\mathrm {St}_l \\&\qquad \rtimes \mathrm {Hom}_{H(V_{m-2(h-1)l})}\left( \omega _{V_{m-2(h-1)l},W_{n-2(h-1)l}}, \mathrm {Ind}_{Q(Y_{l+1})}^{H(V_{m-2(h-1)l})}\left( \chi _W \mathrm {St}_{l+1}\left| \cdot \right| _E^{-1/2} \otimes \sigma _0\right) \right) _\infty . \end{aligned}$$

We cannot apply Corollary 5.3 to

$$\begin{aligned} \mathrm {Hom}_{H(V_{m-2(h-1)l})}(\omega _{V_{m-2(h-1)l},W_{n-2(h-1)l}}, \mathrm {Ind}_{Q(Y_{l+1})}^{H(V_{m-2(h-1)l})}( \chi _W \mathrm {St}_{l+1}|\cdot |_E^{-1/2} \otimes \sigma _0))_\infty . \end{aligned}$$

According to Proposition 5.2, $J^{l}$ and $J^{l+1}$ can contribute. However, since

$$\begin{aligned}&\mathrm {Hom}_{\mathrm {GL}(Y_{l+1}) \times H(V_{m_0})}\big (J^{l+1}, \chi _W \mathrm {St}_{l+1}\big |\cdot \big |_E^{-1/2} \otimes \sigma _0\big )_\infty \\&\quad \cong \big ( \mathrm {Ind}_{P(X_{l+1})}^{G(W_{n_0+2l})}\big (\chi _V \mathrm {St}_{l+1}\big |\cdot \big |_E^{1/2} \otimes \Theta _{W_{n_0-2},V_{m_0}}(\sigma _0)\big ) \big )^\vee , \end{aligned}$$

we have

$$\begin{aligned}&\mathrm {Hom}_{G(W_n)}\\&\quad \left( \pi ^\vee , \chi _V\mathrm {St}_l \times \cdots \times \chi _V\mathrm {St}_l \rtimes \mathrm {Hom}_{\mathrm {GL}(Y_{l+1}) \times H(V_{m_0})}(J^{l+1}, \chi _W \mathrm {St}_{l+1}\left| \cdot \right| _E^{-1/2} \otimes \sigma _0)_\infty \right) =0 \end{aligned}$$

by Casselman’s temperedness criterion. Hence we have

$$\begin{aligned} \pi ^\vee&\hookrightarrow \chi _V\mathrm {St}_l \times \cdots \times \chi _V\mathrm {St}_l \rtimes \mathrm {Hom}_{\mathrm {GL}(Y_{l+1}) \times H(V_{m_0})}\left( J^{l}, \chi _W \mathrm {St}_{l+1}|\cdot |_E^{-1/2} \otimes \sigma _0\right) _\infty \\&\cong \chi _V\mathrm {St}_l \times \cdots \times \chi _V\mathrm {St}_l \times \left( \chi _V\mathrm {St}_l \rtimes \Theta _{W_{n-2hl},V_{m-2hl-2}}(\sigma _0)^\vee \right) \end{aligned}$$

by Proposition 5.2. In particular, there exists an irreducible subquotient $\pi _0'$ of $\Theta _{W_{n_0},V_{m_0}}(\sigma _0)$ such that

$$\begin{aligned} \chi _V\mathrm {St}_l \times \cdots \times \chi _V\mathrm {St}_l \rtimes \pi '_0 \twoheadrightarrow \pi , \end{aligned}$$

where $\chi _V\mathrm {St}_l$ appears h-times. This implies that the L-parameter for $\pi _0'$ is given by $(\phi _0, \eta |A_{\phi _0})$, which is the same as the one for $\pi _0$. Also, if $G(W_n)$ is an odd orthogonal group, the central character of $\pi _0'$ coincides with the one of $\pi _0$. Hence we have $\pi _0' \cong \pi _0$. Since $\phi _{\sigma _0}$ contains $\chi _W S_l$ with multiplicity one, by Proposition 5.4, we see that $\Theta _{W_{n_0},V_{m_0}}(\sigma _0)$ is irreducible, and so $\Theta _{W_{n_0},V_{m_0}}(\sigma _0)=\theta _{W_{n_0},V_{m_0}}(\sigma _0)=\pi _0$. In other words, we have $\sigma _0=\theta _{V_{m_0}, W_{n_0}}(\pi _0)$. This completes the proof. $\square$

6.7 Higher lifts

In this subsection, we prove Theorem 4.5 (3).

Let $\pi \in \mathrm {Irr}_\mathrm {temp}(G(W_n))$ with L-parameter $(\phi ,\eta )$, and $\sigma =\theta _{V_m,W_n}(\pi ) \in \mathrm {Irr}(H(V_m))$ be the first occurrence to the going-up tower i.e., $m=m^\mathrm {up}(\pi )$. Assume that $\sigma$ is non-tempered. Then $l(\pi )+2=m-n-\epsilon _0>2$. Let $\sigma ' = \theta _{V_{m'},W_n}(\pi )$ be a higher lift, i.e., $m'>m$. The assertion of Theorem 4.5 (3) follows from [14, Proposition 3.2] if we knew that this proposition can be applied to $\sigma$ and $\sigma '$. So what we have to show is as follows:

Proposition 6.18

We can apply [14, Proposition 3.2] to $\sigma$ and $\sigma '$. Namely, the same assertion as Proposition 5.6 is true for $\sigma =\theta _{V_m,W_n}(\pi )$ and $\sigma ' = \theta _{V_{m'},W_n}(\pi )$.

Proof

We freely use the notation of [14]. According to the proof of Proposition 3.2 in [14], it suffices show that only the 0-th piece $R_0$ of the filtration of Lemma 2.2 in [14] can contribute in the proof of Proposition 3.2 in [14] for $\sigma$ and $\sigma '$.

Suppose that $R_t$ contributes for some $t>0$. Then we have a nonzero $\mathrm {GL}(Y_t)$-homomorphism

$$\begin{aligned} \chi _W\left| {\det }_{Y_t}\right| ^{s} \rightarrow R_{\overline{Q(Y_t)}}(\sigma ), \end{aligned}$$

where

• $V_m=Y_t+V_{m_0}+Y_t^*$ with $m_0=m-2t$;

• $s=(m+r-n-\epsilon _0)/2+t/2>0$ for some $r\ge 0$.

See also the argument after Lemma 2.2 in [14].

Put

$$\begin{aligned} \Sigma = \sum _{f\in \mathrm {Hom}_{\mathrm {GL}(Y_t)}\left( \chi _W\left| {\det }_{Y_t}\right| ^s, R_{\overline{Q(Y_t)}}(\sigma )\right) } \mathrm {Im}(f). \end{aligned}$$

This is a $\mathrm {GL}(Y_t)\times H(V_{m_0})$-subrepresentation of $R_{\overline{Q(Y_t)}}(\sigma )$ of the form

$$\begin{aligned} \Sigma = \chi _W\left| {\det }_{Y_t}\right| ^s \boxtimes \Sigma _0, \end{aligned}$$

where $\Sigma _0$ is a nonzero smooth representation of $H(V_{m_0})$. Since $R_{\overline{Q(Y_t)}}(\sigma )$ is finite length, so is $\Sigma _0$. Hence we can find an irreducible subrepresentation $\sigma _0$ of $\Sigma _0$. We obtain a nonzero $\mathrm {GL}(Y_t)\times H(V_{m_0})$-homomorphism

$$\begin{aligned} \chi _W\left| {\det }_{Y_t}\right| ^s \boxtimes \sigma _0 \rightarrow R_{\overline{Q(Y_t)}}(\sigma ). \end{aligned}$$

By Bernstein’s Frobenius reciprocity, we have a surjection

$$\begin{aligned} \mathrm {Ind}_{Q(Y_t)}^{H(V_m)}\left( \chi _W\left| {\det }_{Y_t}\right| ^s \boxtimes \sigma _0\right) \twoheadrightarrow \sigma . \end{aligned}$$

By Lemma 2.2, this surjection gives an injection

$$\begin{aligned} \sigma \hookrightarrow \mathrm {Ind}_{Q(Y_t)}^{H(V_m)}\left( \chi _W\left| {\det }_{Y_t}\right| ^{-s} \boxtimes \sigma _0\right) . \end{aligned}$$

Hence we have

$$\begin{aligned} \pi ^*&\hookrightarrow \mathrm {Hom}_{H(V_m)}\left( \omega _{V_m,W_n}, \sigma \right) \\&\hookrightarrow \mathrm {Hom}_{H(V_m)}\left( \omega _{V_m,W_n}, \mathrm {Ind}_{Q(Y_t)}^{H(V_m)}\left( \chi _W\left| {\det }_{Y_t}\right| ^{-s} \boxtimes \sigma _0\right) \right) \\&\cong \mathrm {Hom}_{\mathrm {GL}(Y_t) \times H(V_{m_0})}\left( R_{Q(Y_t)}\left( \omega _{V_m,W_n}\right) , \chi _W\left| {\det }_{Y_t}\right| ^{-s} \boxtimes \sigma _0\right) . \end{aligned}$$

By Kudla’s filtration (Lemma 5.1), we see that there is a nonzero homomorphism

$$\begin{aligned} \pi ^\vee \rightarrow \mathrm {Hom}_{\mathrm {GL}(Y_t) \times H(V_{m_0})}\left( J^a, \chi _W\left| {\det }_{Y_t}\right| ^{-s} \boxtimes \sigma _0\right) _\infty \end{aligned}$$

for some $0\le a \le t$.

First, consider the case when $0 \le a < t$. By the definition of the normalized Jacquet module, we have

$$\begin{aligned} R_{\overline{Q(Y_{t-a},Y_t)}}\left( \chi _W\left| {\det }_{Y_t}\right| ^{-s}\right) = \chi _W \left| {\det }_{Y_{t-a}}\right| ^{-s+a/2} \boxtimes \chi _W \left| {\det }_{Y'_a}\right| ^{-s-(t-a)/2}. \end{aligned}$$

Note that $\mathrm {GL}(Y_{t-a})$ acts on $J^a$ by the character

$$\begin{aligned} \chi _W \left| {\det }_{Y_{t-a}}\right| ^{\left( n-m+\epsilon _0+t-a\right) /2}. \end{aligned}$$

Since $t-a>0 \ge -r/2$, we have

$$\begin{aligned} \left( n-m+\epsilon _0+t-a\right) /2 \not = -(m+r-n-\epsilon _0)/2-t/2+a/2. \end{aligned}$$

Hence we have

$$\begin{aligned} \mathrm {Hom}_{G(W_n)}\left( \pi ^\vee , \mathrm {Hom}_{\mathrm {GL}(Y_t) \times H(V_{m_0})}\left( J^a, \chi _W\left| {\det }_{Y_t}\right| ^{-s} \boxtimes \sigma _0\right) _\infty \right) =0. \end{aligned}$$

We conclude that there must be an injection

$$\begin{aligned} \pi ^\vee \hookrightarrow \mathrm {Hom}_{\mathrm {GL}(Y_t) \times H(V_{m_0})}\left( J^t, \chi _W\left| {\det }_{Y_t}\right| ^{-s} \boxtimes \sigma _0\right) _\infty . \end{aligned}$$

However,

$$\begin{aligned} \mathrm {Hom}_{\mathrm {GL}(Y_t) \times H(V_{m_0})}\big (J^t, \chi _W\big |{\det }_{Y_t}\big |^{-s} \boxtimes \sigma _0\big )_\infty \cong \big (\mathrm {Ind}_{P_t}^{G(W_{n})} \big (\chi _V\big |{\det }_{X_t}\big |^s \boxtimes \Theta _{W_{n_0},V_{m_0}}(\sigma _0)\big )\big )^\vee . \end{aligned}$$

Since $s>0$, it has no irreducible tempered subrepresentations by Casselman’s criterion.

We obtain a contradiction, so that $R_t$ cannot contribute for any $t>0$. $\square$

6.8 Central characters of representations of odd orthogonal groups

Recall that for an odd orthogonal group $\mathrm {O}(V_m)$, our local Langlands correspondence described in Sect. 3 or Appendix B parametrizes $\mathrm {Irr}(\mathrm {O}(V_m))$ by the triples $(\phi ,\eta ,\nu )$. More precisely, a pair $(\phi ,\eta )$ corresponds to the set

$$\begin{aligned} \left\{ \sigma ,\ \sigma \otimes \det \right\} \end{aligned}$$

for some $\sigma \in \mathrm {Irr}(\mathrm {O}(V_m))$, and

$$\begin{aligned} \nu :\mathrm {Irr}(\mathrm {O}(V_m)) \rightarrow \left\{ \pm 1\right\} \end{aligned}$$

is given by the central character, i.e., $\sigma (-{\mathbf{1}}_{V_m})=\nu (\sigma ) \cdot \mathrm {id}$ for $\sigma \in \mathrm {Irr}(\mathrm {O}(V_m))$.

In this subsection, we consider the theta correspondence for $(\mathrm {Mp}(W_n),\mathrm {O}(V_m))$, i.e., $E=F$, $\epsilon =+1$, m is odd and n is even. We prove Theorems 4.3 (5), 4.5 (4) and complete the proof of Theorem 4.1 (2). Namely, we treat the following two issues:

• (1) For $\pi \in \mathrm {Irr}_\mathrm {temp}(\mathrm {Mp}(W_n))$ with $\theta _{V_m,W_n}(\pi )\not =0$, determine $\nu (\theta _{V_m,W_n}(\pi ))$.

• (2) For $\sigma \in \mathrm {Irr}_\mathrm {temp}(\mathrm {O}(V_m))$, determine which tower $\{\Theta _{W_n, V_m}(\sigma )\}_n$ or $\{\Theta _{W_n, V_m}(\sigma \otimes \det )\}_n$ is the going-down tower.

First, we consider (1). Let $\pi \in \mathrm {Irr}(\mathrm {Mp}(W_n))$ and assume that $\sigma =\theta _{V_m,W_n}(\pi )$ is nonzero so that $\sigma \in \mathrm {Irr}(\mathrm {O}(V_m))$. We define $\epsilon (V)\in \{\pm 1\}$ by

$$\begin{aligned} \epsilon (V)=\left\{ \begin{aligned}&+1&\quad&\text {if }\mathrm {O}(V_m) \text { is split},\\&-1&\quad&\text {if }\mathrm {O}(V_m) \text { is non-split}. \end{aligned} \right. \end{aligned}$$

Note that $\epsilon (V)=\eta _\sigma (z_{\phi _\sigma })$ by Desideratum B.1 (3), where $(\phi _\sigma ,\eta _\sigma )$ is the L-parameter for $\sigma$. The following proposition is Theorem 4.3 (5) and Theorem 4.5 (4).

Proposition 6.19

Let $\pi \in \mathrm {Irr}_\mathrm {temp}(\mathrm {Mp}(W_n))$ with L-parameter $(\phi _\pi ,\eta _\pi )$. Assume that $\sigma =\theta _{V_m,W_n}(\pi )$ is nonzero. Then we have

$$\begin{aligned} \nu (\sigma )=\eta _\pi (z_{\phi _\pi }) \cdot \varepsilon (\phi _\pi ) \cdot \chi _V(-1)^{\frac{n}{2}}. \end{aligned}$$

Proof

The Schrodinger model of the Weil representation allows one to relate the central characters of $\pi$ and $\sigma$. In particular, if $z(\pi )$ denotes the central sign of $\pi$ as defined in [13, Pg. 1658], we have:

$$\begin{aligned} z(\pi )=\nu (\sigma )\cdot \chi _V(-1)^{\frac{n}{2}}. \end{aligned}$$

On the other hand, by the properties of the local Shimura correspondence [13, Theorem 1.4] and the definition of LLC for $\mathrm{Mp}(W_n)$, we see that

$$\begin{aligned} z(\pi )=\varepsilon (\phi _{\pi })\cdot \eta _{\pi }(z_{\phi _{\pi }}). \end{aligned}$$

Combining the two equations gives the desired result. $\square$

Next, we consider (2). Let $\sigma \in \mathrm {Irr}(\mathrm {O}(V_m))$. We compare the two towers $\{\Theta _{W_n, V_m}(\sigma )\}_n$ and $\{\Theta _{W_n, V_m}(\sigma \otimes \det )\}_n$.

Proposition 6.20

Let $\sigma \in \mathrm {Irr}_\mathrm {temp}(\mathrm {O}(V_m))$ with L-parameter $(\phi _\sigma , \eta _\sigma , \nu _\sigma )$. Then $\{\Theta _{W_n, V_m}(\sigma )\}_n$ is the going-down tower with respect to $\sigma$, i.e.,

$$\begin{aligned} \min \left\{ n\ \big |\ \Theta _{W_n, V_m}(\sigma ) \not =0\right\} \le \min \left\{ n\ \big |\ \Theta _{W_n, V_m}(\sigma \otimes \det ) \not =0\right\} \end{aligned}$$

if and only if

$$\begin{aligned} \nu _\sigma = \eta _\sigma (z_{\phi _\sigma }) \cdot \varepsilon (\phi _\sigma ). \end{aligned}$$

Proof

Note that $\{\Theta _{W_n, V_m}(\sigma )\}_n$ is the going-down tower if and only if $\Theta _{W_{m-1},V_m}(\sigma )$ is nonzero. This is equivalent to $\nu _\sigma = \epsilon (V) \cdot \varepsilon (\phi _\sigma ) = \eta _\sigma (z_{\phi _\sigma }) \cdot \varepsilon (\phi _\sigma )$ by [11, Theorem 11.1]. $\square$

Together with Proposition 6.7, this completes the proof of Theorem 4.1 (2).

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.