Quaternion distinguished generic representations of $\mathrm{GL}_{2n}$

TL;DR

This paper classifies generic representations of isplaystyle GL_{2m}(E) distinguished by GL_m(D) over a quadratic extension, linking representation theory with base change lifts and quaternion algebra structures.

Contribution

It provides a classification of GL_m(D)-distinguished generic representations of GL_{2m}(E) using Zelevinsky classification and relates it to base change lifts from unitary groups.

Findings

Classification of distinguished representations via Zelevinsky parameters.

A sufficient condition for base change lift representations to be distinguished.

Connection between quaternion algebra embeddings and representation distinction.

Abstract

Let $E/F$ be a quadratic extension of non-Archimedean local fields of characteristic 0. Let $D$ be the unique quaternion division algebra over $F$ and fix an embedding of $E$ to $D$. Then, $\mathrm{GL}_m(D)$ can be regarded as a subgroup of $\mathrm{GL}_{2m}(E)$. Using the method of Matringe, we classify irreducible generic $\mathrm{GL}_m(D)$-distinguished representations of $\mathrm{GL}_{2m}(E)$ in terms of Zelevinsky classification. Rewriting the classification in terms of corresponding representations of the Weil-Deligne group of $E$, we prove a sufficient condition for a generic representation in the image of the unstable base change lift from the unitary group $\mathrm{U}_{2m}$ to be $\mathrm{GL}_m(D)$-distinguished.

Equations

$${\int }_{Z({\mathbb{A}}_F){\mathrm{GL}}_{2m}(F)\backslash {\mathrm{GL}}_{2m}({\mathbb{A}}_F)}f(h)dh\ne 0.$$

$${\psi }_E(x)=\psi \left(\frac{1}{2}{\mathrm{Tr}}_{E/F}(x)\right),x\in E.$$

$$\left\{\begin{array}[]{l}B(\tau m,s\tau s^{-1}n)=B(m,n)\\ B(n,m)=b\cdot B(m,s^{2}n)\end{array}\right.\hskip 10.0pt\tau\in WD_{E},\ m,n\in M$$

$$M=\underset{i\in I_{+}}{⨁}{V}_i\otimes M_i+\underset{i\in I_{-}}{⨁}{V}_i\otimes M_i+\underset{i\in I_0}{⨁}{V}_i\otimes (M_i+(M_i^c{)}^{\vee }).$$

$$C\cong \prod _{i\in I_{+}}\mathrm{O}(V_i)\times \prod _{i\in I_{-}}\mathrm{Sp}(V_i)\times \prod _{i\in I_0}\mathrm{GL}(V_i).$$

$$A=A_M\cong (\mathbb{Z}/2\mathbb{Z}{)}^k,$$

$$M^a=\{m\in M;am=-m\}.$$

$$\eta (a)=(-1{)}^{\dim M^a},a\in A.$$

$$d(h{h}^{\prime })={\delta }_H(h^{\prime }{)}^{-1}dh$$

$$f(hg)={\delta }_H^{1/2}(h)\sigma (h)f(g)$$

$$f(hg)=\sigma (h)f(g)$$

$$t\cdot \theta (n)=\theta (t^{-1}nt),n\in N.$$

$$\left(\begin{array}[]{ccc}g_{1}&\ast&\ast\\ &\ddots&\ast\\ &&g_{r}\end{array}\right)\in G,\hskip 10.0ptg_{i}\in G_{n_{i}}.$$

$${\sigma }_{P^{\prime }}(m)j_{P^{\prime }}(v)={\delta }_{P^{\prime }}^{-1/2}(m^{\prime })j_{P^{\prime }}(\sigma (m^{\prime })v),v\in V,m^{\prime }\in M^{\prime }.$$

$$\theta (u)={\psi }_E\left(\sum _{i=1}^{n-1}{u}_{i,i+1}\right),u=(u_{i,i+1})\in N.$$

$$\begin{array}[]{l}\tilde{\phi}(x)=(\phi_{1}(x),\phi_{1}(sxs^{-1}))\rtimes\phi_{2}(x),\hskip 10.0ptx\in WD_{E}\\ \tilde{\phi}((s,1))=(1,\phi_{1}(s^{2}))\rtimes s.\end{array}$$

$$\begin{array}[]{l}\operatorname{As}^{+}((g,h)\rtimes x)(a\otimes b)=(ga)\otimes(hb),\hskip 10.0pt(g,h)\in\mathrm{GL}_{n}(\mathbb{C})\times\mathrm{GL}_{n}(\mathbb{C}),\ x\in W_{E}\\ \operatorname{As}^{+}((1,1)\rtimes s)(a\otimes b)=b\otimes a.\end{array}$$

$$\begin{array}[]{l}\operatorname{As}^{-}((g,h)\rtimes x)(a\otimes b)=(ga)\otimes(hb),\hskip 10.0pt(g,h)\in\mathrm{GL}_{n}(\mathbb{C})\times\mathrm{GL}_{n}(\mathbb{C}),\ x\in W_{E}\\ \operatorname{As}^{-}((1,1)\rtimes s)(a\otimes b)=-b\otimes a.\end{array}$$

$$G^{\prime }=\{g\in G;{}^t{g}^{c}Jg=J\}.$$

$$J=J_{n}=\left(\begin{array}[]{cc}&w\\ -w&\\ \end{array}\right),\hskip 10.0ptw=w_{m}=\left(\begin{array}[]{ccc}&&1\\ &\reflectbox{$\ddots$}&\\ 1&&\\ \end{array}\right)\in\mathrm{GL}_{m}.$$

$${\theta }_{\kappa }^{\prime }(u)={\psi }_E\left(\sum _{i=1}^{m-1}{u}_{i,i+1}+\kappa u_{m,m+1}\right),u=(u_{i,j})\in N^{\prime }.$$

$$sg{s}^{-1}=J{}^t{g}^{-1}{J}^{-1},g\in {\mathrm{GL}}_n(\mathbb{C}).$$

$$\chi (sx{s}^{-1})=\chi (x{)}^{-1}$$

$$\chi (s^2)=-1.$$

$$b{c}_{\chi }:{}^L{G}^{\prime }\to {}^L{G}_{/F}$$

$$\begin{array}[]{ll}bc_{\chi}(g\rtimes 1)=(g,\ J\,\,{}^{t}\!g^{-1}J^{-1})\rtimes 1,&g\in\mathrm{GL}_{n}(\mathbb{C});\\ bc_{\chi}(\mathbf{1}_{n}\rtimes x)=(\chi(x)\cdot\mathbf{1}_{n},\ \chi(x)^{-1}\cdot\mathbf{1}_{n})\rtimes x,&x\in W_{E};\\ bc_{\chi}(\mathbf{1}_{n}\rtimes s)=(-\mathbf{1}_{n},\mathbf{1}_{n})\rtimes s.&\end{array}$$

$$\begin{array}[]{ll}bc_{\mathbf{1}}(g\rtimes 1)=(g,\ J\,\,{}^{t}\!g^{-1}J^{-1})\rtimes 1,&g\in\mathrm{GL}_{n}(\mathbb{C});\\ bc_{\mathbf{1}}(\mathbf{1}_{n}\rtimes x)=(\mathbf{1}_{n},\ \mathbf{1}_{n})\rtimes x,&x\in W_{E};\\ bc_{\mathbf{1}}(\mathbf{1}_{n}\rtimes s)=(\mathbf{1}_{n},\mathbf{1}_{n})\rtimes s.&\end{array}$$

$$J({\theta }^{\prime }):{\Pi }_{{\varphi }^{\prime }}\to \mathrm{Irr}(A_M).$$

$$(x_1+x_2\sqrt{\tau })(y_1+y_2\sqrt{\tau })=(x_1{y}_1+x_2{y}_2^c\tau )+(x_1{y}_2+x_2{y}_1^c)\sqrt{\tau }$$

$$\varepsilon_{1}=\left(\begin{array}[]{cc}&\tau\\ 1&\end{array}\right),\hskip 10.0pt\varepsilon_{m}=\left(\begin{array}[]{cc}\varepsilon_{1}&\\ &\varepsilon_{m-1}\end{array}\right).$$

Full text

Introduction

Let $F$ be a non-Archimedean local field of characteristic 0. Let $G$, $H$ be algebraic groups over $F$ and suppose $H$ is a closed subgroup of $G$. A smooth representation $\pi$ of $G(F)$ is said to be $H(F)$-distinguished if it has a nonzero $H(F)$-invariant linear form. This paper concentrate on the case of $(G,H)=(\operatorname{Res}_{E/F}\mathrm{GL}_{2m},\mathrm{GL}_{m}(D))$, where $E$ is a quadratic extension of $F$, $\operatorname{Res}_{E/F}$ denotes the restriction of scalars and $D$ is the unique quaternion division algebra over $F$. Given an embedding of $E$ to $D$, $H$ can be regarded as a subgroup of $G$. We classify irreducible smooth generic $H(F)$-distinguished representations of $G(F)$ and relate it to a functorial lift from the quasi-split unitary group $\mathrm{U}_{2m}$.

In order to explain the background, we consider the global setting. Let $E/F$ be a quadratic extension of number fields and $\mathbb{A}_{E}$, $\mathbb{A}_{F}$ the rings of adeles of $E$ and $F$, respectively. Let $\pi$ be an irreducible cuspidal automorphic representation of $\mathrm{GL}_{2m}(\mathbb{A}_{E})$. Then, $\pi$ is said to be $\mathrm{GL}_{2m}$-distinguished if there is a cusp form $f$ in the space of $\pi$ which has a nonzero period integral over $\mathrm{GL}_{2m}$:

[TABLE]

Here, $Z$ denotes the center of $\mathrm{GL}_{2m}$. Flicker and Rallis (see [F]) conjectured that $\pi$ is $\mathrm{GL}_{2m}$-distinguished if and only if it is an unstable base change lift of a generic cuspidal automorphic representation of the quasi-split unitary group $\mathrm{U}_{2m}(\mathbb{A}_{F})$.

Let $D$ be a quaternion algebra over $F$ which $E$ embedds. Then, distinguished cuspidal representations of $\mathrm{GL}_{2m}(\mathbb{A}_{E})$ with respect to $\mathrm{GL}_{m}(D)$ is defined similarly as above. In [S], the author defined a non-degenerate character $\theta^{\prime}_{\tau}$ associated to $D$ for the quasi-split unitary group and conjectured that $\pi$ is $\mathrm{GL}_{m}(D)$-distinguished if and only if it is an unstable base change lift of a $\theta^{\prime}_{1}$-generic and $\theta^{\prime}_{\tau}$-generic cuspidal representation of $\mathrm{U}_{2m}(\mathbb{A}_{F})$. This conjecture slightly generalizes that of Flicker and Rallis.

Now, we go back to the local setting. Recall that $G=\operatorname{Res}_{E/F}\mathrm{GL}_{2m}$. Let $WD_{F}$ be the Weil-Deligne group of $F$ and ${}^{L}G$, ${}^{L}\mathrm{U}_{2m}$ be the $L$-groups of $G$ and $\mathrm{U}_{2m}$. For an irreducible smooth representation $\pi$ of $G(F)=\mathrm{GL}_{2m}(E)$, there corresponds an $L$-parameter $\tilde{\phi}\,\colon WD_{F}\rightarrow{}^{L}G$ by local Langlands correspondence for $G$ which was established by Harris, Taylor [HT] and Henniart [H]. See section 1 for more details.

According to the local Langlands correspondence in a form proposed by Vogan, for an $L$-parameter $\phi^{\prime}\,\colon WD_{F}\rightarrow{}^{L}\mathrm{U}_{2m}$ of $\mathrm{U}_{2m}$, we should obtain a finite set $\Pi_{\phi^{\prime}}$ of irreducible smooth representations of pure inner forms of $\mathrm{U}_{2m}$ called the Vogan $L$-packet of $\phi^{\prime}$. In section 2, we will review the definition of the unstable base change map $bc_{\chi}$ from ${}^{L}\mathrm{U}_{2m}$ to ${}^{L}G$. For an $L$-parameter $\phi^{\prime}$ of $\mathrm{U}_{2m}$, the composition $bc_{\chi}\circ\phi^{\prime}$ is an $L$-parameter of $G$. Hence, if we assume the local Langlands correspondence for $\mathrm{U}_{2m}$, we get a map from the set of irreducible smooth representations of (pure inner forms of) $\mathrm{U}_{2m}(F)$ to that of $G(F)$, sending the representations in $\Pi_{\phi^{\prime}}$ to the representation corresponding to $bc_{\chi}\circ\phi^{\prime}$. We call this conjectural map the unstable base change lift.

Many researchers studied the following local analogue of the conjecture of Flicker and Rallis:

Statement. Let $\pi$ be an irreducible representation of $G(F)=\mathrm{GL}_{2m}(E)$. Let $\tilde{\phi}$ be the corresponding $L$-parameter. Then, $\pi$ is $\mathrm{GL}_{2m}(F)$-distinguished if and only if there exists an $L$-parameter $\phi^{\prime}\,\colon WD_{F}\rightarrow{}^{L}\mathrm{U}_{2m}$ of $\mathrm{U}_{2m}(F)$ so that $\tilde{\phi}=bc_{\chi}\circ\phi^{\prime}$.

The reader may see, for example [AR]. Though this naive analogue is not valid in general, it is still expected to be true for a wide class of representations. For example, this statement is true for generic representations (see [GGM, Theorem 4.9] and [M10]).

In this paper, we consider the local analogue of [S, Conjecture 2] and characterize $\mathrm{GL}_{m}(D)$-distinguished generic representations of $\mathrm{GL}_{2m}(E)$ in terms of an unstable base change lift of a generic representation of $\mathrm{U}_{2m}(F)$. The main result is as follows:

Theorem 0.1**.**

Let $\pi$ be an irreducible generic representation of $\mathrm{GL}_{2m}(E)$. Let $\tilde{\phi}$ be the corresponding $L$-parameter. If there exists an $L$-parameter $\phi^{\prime}$ of $\mathrm{U}_{2m}(F)$ which is generic with respect to any non-degenerate characters and satisfies $\tilde{\phi}=bc_{\chi}\circ\phi^{\prime}$, then $\pi$ is $\mathrm{GL}_{m}(D)$-distinguished.

Unlike the global conjecture, the converse implication does not hold for local representations. In order to get more precise description of $H(F)$-distinguished representations, we need to generalize the conjecture of Prasad. For a connected reductive quasi-split group $G$ over $F$, Prasad [P] introduced a quasi-split group $H^{\mathrm{op}}$ over $F$ which is isomorphic to $H$ over $E$. He formulated a conjectural formula [P, Conjecture 2] for the dimension of the space of $H(F)$-invariant linear forms on irreducible smooth representations of $H(E)$ in terms of the degree of the base change map from $H^{\mathrm{op}}(F)$ to $H(E)$. In particular, he conjectured that an $H(F)$-distinguished representation of $H(E)$ arises from base change from $H^{\mathrm{op}}$. For this conjecture, several low dimensional examples are known; see [L18] and [L19]. A recent remarkable work of Beuzart-Plessis [BP] established an integral formula computing the dimension of the $H(F)$-invariant linear forms for square-integrable representations. As a corollary, he proved that this dimension is preserved under the Jacquet-Langlands correspondence. This result verifies a consequence of the conjecture of Prasad and plays an important role in the proof of our main theorem.

In our setting, $H=\mathrm{GL}_{m}(D)$ is a non quasi-split inner form of $H^{\prime}=\mathrm{GL}_{2m}$ and they are isomorphic over $E$. It is natural to ask whether the dimension of the space of $H(F)$-invariant linear forms on representations of $H(E)$ can be described in terms of the base change from $H^{\prime\mathrm{op}}$. We hope that the result of this paper will be reinterpreted in relative Langlands frameworks in the sense of Prasad, generalized to non quasi-split groups.

We briefly explain the contents of this paper. In section 1, we fix necessary notations and recall some facts about representations of Weil-Deligne groups. In section 2, we review the conjectural local Langlands correspondence and known results. In section 3, we quote some results of Matringe about double coset decomposition of $G$. In section 4, we summarize the Zelevinsky classification of representations of $G(F)$. In section 5, using the method of Matringe [M10], we classify $H$-distinguished generic representations of $G(F)$ in terms of Zelevinsky classification. Rewriting that classification in terms of representations of $WD_{E}$, we obtain the main theorem in the last section.

1. Preliminaries

Let $E/F$ be a quadratic extension of non-Archimedean local fields of characteristic 0. Fix a nontrivial additive character $\psi$ of $F$. Define the character $\psi_{E}$ of $E$ by

[TABLE]

Denote the action of the nontrivial element of the Galois group $\mathrm{Gal}(E/F)$ by $x\mapsto x^{c}$. We consider various objects on which $\mathrm{Gal}(E/F)$ acts. By abuse of notation, we write all of these actions by the same symbol ${\cdot}^{c}$. Denote by $\mathrm{N}_{E/F}$ (resp. $\mathrm{Tr}_{E/F}$) the norm map (resp. trace map) from $E$ to $F$. Let $\omega=\omega_{E/F}$ be the unique nontrivial character of $F^{\times}/\mathrm{N}_{E/F}(E^{\times})\cong\mathrm{Gal}(E/F)$. Sometimes we regard it as a quadratic character of $F^{\times}$.

Let $W_{F}$ be the Weil group of $F$. Recall that this is a subgroup of the absolute Galois group of $F$ whose Abelianization is isomorphic to $F^{\times}$. Hence, $\omega$ can be regarded as a character of $W_{F}$. Define the Weil-Deligne group $WD_{F}$ of $F$ by $WD_{F}=W_{F}\times\mathrm{SL}_{2}(\mathbb{C})$. Similarly, we denote by $W_{E}$ and $WD_{E}$ the Weil group and Weil-Deligne group of $E$. Note that $W_{E}$ is a subgroup of $W_{F}$ of index $2$ and the quotient $W_{F}/W_{E}$ is naturally identified with $\mathrm{Gal}(E/F)$. We fix an element $s$ of $W_{F}$ outside $W_{E}$ once for all. The image of $s$ in $W_{F}/W_{E}$ is the nontrivial element of $\mathrm{Gal}(E/F)$.

1.1. Representations of $WD_{E}$

Let $M$ be a finite dimensional complex vector space. A homomorphism $\underline{\phi}\,\colon WD_{E}\rightarrow\mathrm{GL}(M)$ is called a representation of $WD_{E}$ if

• •

the image of a geometric Frobenius element of $W_{E}$ is semisimple;

• •

the restriction of $\underline{\phi}$ to $W_{E}$ is smooth;

• •

the restriction of $\underline{\phi}$ to $\mathrm{SL}_{2}(\mathbb{C})$ is algebraic.

For an $L$-parameter $\phi\,\colon WD_{E}\rightarrow\mathrm{GL}_{n}(\mathbb{C})\times W_{E}$, we write by $\underline{\phi}\,\colon WD_{E}\rightarrow\mathrm{GL}(M)$ the associated representation of $WD_{E}$. Here, $M$ is a complex vector space of dimension $n$. We recall some facts about conjugate self-dual representations of $WD_{E}$ from [GGP, Section 4]. We say that the representation $M$ of $WD_{E}$ is conjugate self-dual if there exists a non-degenerate bilinear form $B\,\colon M\times M\rightarrow\mathbb{C}$ which satisfies

[TABLE]

with some $b\in\{\pm 1\}$. We call $b$ the sign of $B$. If $b=+1$, $M$ is called conjugate-orthogonal and if $b=-1$, $M$ is called conjugate-symplectic. For a conjugate self-dual representation $(\underline{\phi},M)$ of $WD_{E}$, we have a decomposition

[TABLE]

Here, each $M_{i}$ is an irreducible representation of $WD_{E}$ and $V_{i}$ is a space of multiplicity satisfying

• •

for $i\in I_{+}$, $M_{i}$ is conjugate self-dual of sign $b$;

• •

for $i\in I_{-}$, $M_{i}$ is conjugate self-dual of sign$-b$;

• •

for $i\in I_{0}$, $M_{i}$ is not conjugate self-dual.

The restriction of the form $B$ to each summand of the above decomposition induces a non-degenerate bilinear form on each $V_{i}$. These pairings have sign $+1$ for $i\in I_{+}$ and $-1$ for $i\in I_{-}$. Let $C=C(M,B)$ be the subgroup of $\mathrm{Aut}(M,B)\subset\mathrm{GL}(M)$ which centralizes the image of $WD_{E}$. Due to the above decomposition, we have

[TABLE]

Thus, the component group of $C$ is

[TABLE]

where $k$ is the order of $I_{+}$.

For a semisimple element $a$ in $C$, set

[TABLE]

Note that the parity of the dimension of $M^{a}$ only depends on the image of $a$ in $A$. We define the quadratic character $\eta$ of $A$ by

[TABLE]

In particular, $\eta$ is the trivial character if and only if $\dim\,M_{i}$ is even for all $i\in I_{+}$.

1.2. Groups and representations

Throughout this paper, for an algebraic group $G$ over $F$, we write the group of $F$-rational points by $G$. Denote by $Z_{G}$ or simply by $Z$ the center of $G$.

A representation of $G$ always means an admissible representation of $G$. For a representation $(\pi,V)$ of $G$, denote its contragredient i.e. the smooth dual by $(\pi^{\vee},V^{\vee})$. For a closed subgroup $H$ of $G$ and its left Haar measure $dh$, there exists a continuous character $\delta_{H}\,\colon H\rightarrow\mathbb{R}_{>0}$ which satisfies

[TABLE]

for all $h^{\prime}\in H$. We call this the modulus character of $H$. For a representation $(\sigma,W)$ of $H$, denote by $I_{H}^{G}W$ the space of locally constant functions $f\,\colon G\rightarrow W$ satisfying

[TABLE]

for all $h\in H$ and $g\in G$. The representation of $G$ on this space given by right translation is called the normalized induced representation from $(\sigma,W)$ and denoted by $(I_{H}^{G}\sigma,I_{H}^{G}W)$.

Similarly, let $\operatorname{ind}_{H}^{G}W$ be the space of locally constant functions $f\,\colon G\rightarrow W$ satisfying

[TABLE]

for all $h\in H$ and $g\in G$ and its support is compact modulo $H$. The representation of $G$ on this space given by right translation is called (unnormalized) compactly induced representation from $(\sigma,W)$ and denoted by $(\operatorname{ind}_{H}^{G}\sigma,\operatorname{ind}_{H}^{G}W)$.

Write the trivial representation of $H$ by $\mathbf{1}_{H}$ or simply by $\mathbf{1}$. For a character $\chi$ of $H$, we say that a representation $\pi$ of $G$ is $(H,\chi)$-distinguished if the space $\operatorname{Hom}_{H}(\pi,\chi)$ has a nontrivial element. If $\chi$ is the trivial character, $(H,\chi)$-distinguished representations are simply called $H$-distinguished.

Suppose that $G$ is a quasi-split reductive group. Take a Borel subgroup $B$ of $G$ defined over $F$. Let $T$ be a maximal $F$-subtorus of $G$ contained in $B$ and $N$ the unipotent radical of $B$ so that we have $B=T\ltimes N$. For a character $\theta$ of $N$ and an element $t$ of $T$, we define a character $t\cdot\theta$ of $N$ by

[TABLE]

This defines an action of $T$ on the set of characters of $N$. A character $\theta$ of $N$ is called non-degenerate or generic if its stabilizer in $T$ is $Z_{G}$. The set of non-degenerate characters of $N$ is stable under the action of $T$. Suppose that $\theta$ is a non-degenerate character of $N$. A representation $\pi$ of $G$ is called $\theta$-generic if it is $(N,\theta)$-distinguished. This only depends on the $T$-orbit of $\theta$.

2. Local Langlands correspondence

2.1. General linear groups and Asai representations

Set $G=G_{n}=\mathrm{GL}_{n}(E)$. This can be viewed as a group of $F$-rational points of $\operatorname{Res}_{E/F}\mathrm{GL}_{n}$. Here, $\operatorname{Res}_{E/F}$ stands for the restriction of scalars. The finite symmetric group $\mathfrak{S}_{n}$ is identified with the subgroup of $G$ consisting of permutation matrices.

For a partition $\lambda=(n_{1},\ldots,n_{r})$ of $n$, let $P_{\lambda}$ be the parabolic subgroup of $G$ consisting of matrices of the form

[TABLE]

A subgroup of this form is called a standard parabolic subgroup. Let $M_{\lambda}$ be the Levi subgroup of $P_{\lambda}$ consisting of matrices of the form $\mathrm{diag}(g_{1},\ldots,g_{r})$ with $g_{i}\in G_{n_{i}}$. A subgroup of this form is called a standard Levi subgroup. We write the unipotent radical of $P_{\lambda}$ by $N_{\lambda}$. A parabolic subgroup of $M_{\lambda}$ is called standard if it is a product of standard parabolic subgroups of each component $G_{n_{i}}$.

The standard parabolic subgroup corresponding to the partition $(1,\ldots,1)$ is a Borel subgroup of $G$ and denoted by $B$. The standard Levi subgroup of $B$ is a maximal torus of $G$ and denoted by $T$. Denote by $N$ the unipotent radical of $B$.

Let $\lambda=(n_{1},\ldots,n_{r})$ be a partition of $n$ and $(\sigma_{i},V_{i})$ a representation of $G_{n_{i}}$ for each $i$. Set $(\sigma,V)=(\sigma_{1}\boxtimes\cdots\boxtimes\sigma_{r},V_{1}\otimes\cdots\otimes V_{r})$. This is a representation of $M_{\lambda}$. We extend $\sigma$ to a representation of $P_{\lambda}$ so that $N_{\lambda}$ acts trivially on $V$. We write the normalized induced representation $I_{P_{\lambda}}^{G}\sigma$ by $\sigma_{1}\times\cdots\times\sigma_{r}$.

Let $P^{\prime}=M^{\prime}N^{\prime}$ be a standard parabolic subgroup of $M_{\lambda}$. We denote by $V(P^{\prime})$ the subspace of $V$ spanned by elements of the form $\sigma(n^{\prime})v-v$ with $v\in V$ and $n^{\prime}\in N^{\prime}$. Set $V_{P^{\prime}}=V/V(P^{\prime})$ and we call this the Jacquet module of $V$ with respect to $P^{\prime}$. We write the natural projection $V\rightarrow V_{P^{\prime}}$ by $j_{P^{\prime}}$ and define the representation $\sigma_{P^{\prime}}$ of $M^{\prime}$ on $V_{P^{\prime}}$ by

[TABLE]

Let $\theta$ be the non-degenerate character of $N$ given by

[TABLE]

There is only one $T$-orbit of non-degenerate character of $N$. Thus, we simply say a representation of $G$ is generic if it is $\theta$-generic.

If we regard $G$ as a group over $E$, its $L$-group is given by ${}^{L}G_{/E}=\mathrm{GL}_{n}(\mathbb{C})\times W_{E}$. On the other hand, if we regard it as a group over $F$, its $L$-group is given by ${}^{L}G_{/F}=(\mathrm{GL}_{n}(\mathbb{C})\times\mathrm{GL}_{n}(\mathbb{C}))\rtimes W_{F}$. Here, the action of $W_{F}$ on $\mathrm{GL}_{n}(\mathbb{C})\times\mathrm{GL}_{n}(\mathbb{C})$ factors through $W_{F}/W_{E}\cong\mathrm{Gal}(E/F)$ and the action of $s$ is the permutation of the first and second components. For an $L$-parameter $\phi\,\colon WD_{E}\rightarrow{}^{L}G_{/E}$, we define the corresponding $L$-parameter $\tilde{\phi}\,\colon WD_{F}\rightarrow{}^{L}G_{/F}$ by

[TABLE]

Here, each $\phi_{i}$ is the composition of $\phi$ with the $i$-th projection of ${}^{L}G_{/E}$. The equivalence class of $\tilde{\phi}$ does not depend on the choice of $s$ and this construction provides a bijection between two sets of equivalence classes of $L$-parameters. If we simply say an $L$-parameter of $G$, it means an $L$-parameter which takes its values in ${}^{L}G_{/E}$.

By Harris, Taylor [HT] and Henniart [H], the local Langlands conjecture for $G$ was established. Thus there is a bijection between the set of equivalence classes of $L$-parameters of $G$ and the set of irreducible representations of $G(F)$.

We define a complex representation $\operatorname{As}^{+}$ of ${}^{L}G_{/F}$ on $\mathbb{C}^{n}\otimes\mathbb{C}^{n}$ as follows: for $a\otimes b\in\mathbb{C}^{n}\otimes\mathbb{C}^{n}$,

[TABLE]

Similarly, we define a representation $\operatorname{As}^{-}$ of ${}^{L}G_{/F}$ on the same space as follows: for $a\otimes b\in\mathbb{C}^{n}\otimes\mathbb{C}^{n}$,

[TABLE]

In hornor of T. Asai, we call these representations Asai representations.

According to [GGP, Proposition 7.4], if we identify $\mathbb{C}^{n}\otimes\mathbb{C}^{n}$ with the space $\mathrm{M}_{n}(\mathbb{C})$ of $n$ by $n$ matrices, the stabilizer in $\operatorname{As}^{(-1)^{n-1}}$ of a vector corresponding to an invertible matrix is isomorphic to the $L$-group of the unitary group $\mathrm{U}_{n}$. We say that an $L$-parameter $\tilde{\phi}\,\colon WD_{F}\rightarrow{}^{L}G_{/F}$ fixes a non-degenerate element of $\operatorname{As}^{\pm}$ if the fixed point subspace of the image of $\operatorname{As}^{\pm}\circ\,\tilde{\phi}$ contains an invertible matrix.

2.2. Even unitary groups and unstable base change lift

From now on, we assume that $n=2m$ is even. Let $G^{\prime}=G^{\prime}_{m}$ be the quasi-split even unitary group of rank $m$ defined by

[TABLE]

Here,

[TABLE]

Let $B^{\prime}$ denote the Borel subgroup of $G^{\prime}$ consisting of upper triangular matrices. Denote by $T^{\prime}$ the maximal torus in $B^{\prime}$ consisting of diagonal matrices and by $N^{\prime}$ the unipotent radical of $B^{\prime}$.

There are two $T^{\prime}$-orbits of non-degenerate characters of $N^{\prime}$. We take representatives $\theta^{\prime}_{\kappa}$ of these orbits given by

[TABLE]

Here, $\kappa$ runs over a set of representatives of $F^{\times}/\mathrm{N}_{E/F}(E^{\times})$ (cf. [GGP, Proposition 12.1]). We denote simply by $\theta^{\prime}$ the character corresponding to the element of $F^{\times}/\mathrm{N}_{E/F}(E^{\times})$.

The $L$-group of $G^{\prime}$ is given by ${}^{L}G^{\prime}=\mathrm{GL}_{n}(\mathbb{C})\rtimes W_{F}$. Here, the action of $W_{F}$ factors through $W_{F}/W_{E}\cong\mathrm{Gal}(E/F)$ and the action of $s$ is given by

[TABLE]

Take a character $\chi$ of $E^{\times}$ whose restriction to $F^{\times}$ coincides with $\omega$ and regard it as a character of $W_{E}$ as before. Note that we have

[TABLE]

for $x\in W_{E}$ and

[TABLE]

We define the embedding of $L$-groups

[TABLE]

as follows:

[TABLE]

For an irreducible representation $\pi$ of $G(F)$, let $\tilde{\phi}$ be the corresponding $L$-parameter. We say that $\pi$ arises from unstable base change lift of an $L$-parameter of $G^{\prime}$ if there exists an $L$-parameter $\phi^{\prime}$ of $G^{\prime}$ which satisfies $\tilde{\phi}=bc_{\chi}\circ\phi^{\prime}$.

One can define another embedding $bc_{\mathbf{1}}$ by

[TABLE]

The next result is well-known.

Lemma 2.1**.**

For an $L$-parameter $\phi$ of $G$, the following conditions are equivalent:

• (1)

$\tilde{\phi}$* fixes a non-degenerate element of $\operatorname{As}^{+}$;*

• (2)

there is an $L$-parameter $\phi^{\prime}$ of $G^{\prime}$ which satisfies $\tilde{\phi}=bc_{\chi}\circ\phi^{\prime}$;

• (3)

$\underline{\phi}$* is conjugate-orthogonal.*

Moreover, if $\phi$ satisfies these conditions, we have $L(s,\operatorname{Ad}\circ\,\phi^{\prime})=L(s,\operatorname{As}^{+}\circ\,\tilde{\phi})$ with $\phi^{\prime}$ as in (2).

Remark 2.1**.**

Here, $L(s,\operatorname{Ad}\circ\,\phi^{\prime})$ and $L(s,\operatorname{As}^{+}\circ\,\tilde{\phi})$ are $L$-functions associated with representations of Weil-Deligne groups. For their definition, see [GR, Section 2].

Proof.

Equivalence of (1) and (2) is [GGP, Corollary 8.2]. Equivalence of (1) and (3) is [GGP, Proposition 7.5]. See also [Mok, Lemma 2.2.1]. The last assertion is [GGP, Proposition 7.4].

Note that we use the embedding $bc_{\chi}$ to associate a representation of $WD_{E}$ with an $L$-parameter of $G^{\prime}$ instead of $bc_{\mathbf{1}}$. This is why the conditions above are different from those in [GGP]. ∎

2.3. $L$-packet

In this subsection, we describe the desiderata for the local Langlands correspondence for $G^{\prime}$. For known results and precise conjectures, see [Mok] and [GGP].

For an $L$-parameter $\phi^{\prime}$ of $G^{\prime}$, set $\tilde{\phi}=bc_{\chi}\,\circ\phi^{\prime}$. Let $\phi$ be the $L$-parameter of $G$ corresponding to $\tilde{\phi}$ and $\underline{\phi}\,\colon WD_{E}\rightarrow\mathrm{GL}(M)$ be the representation of $WD_{E}$ attached to $\phi$.

For each $L$-parameter $\phi^{\prime}$, it is conjectured that there exists a finite set $\Pi_{\phi^{\prime}}$ of equivalence classes of irreducible representations of pure inner forms of $G^{\prime}$ which satisfies following conditions:

• (1)

The order of $\Pi_{\phi^{\prime}}$ is equal to the number of irreducible representations of the finite group $A_{M}$, hence the order of $A_{M}$. Moreover, a generic character $\theta^{\prime}$ of $N^{\prime}$ determines a bijection

[TABLE]

Here, $\operatorname{Irr}(A_{M})$ is the set of irreducible representations of $A_{M}$.

• (2)

The number of $\theta^{\prime}$-generic representations contained in $\Pi_{\phi^{\prime}}$ is at most 1. The finite set $\Pi_{\phi^{\prime}}$ contains a $\theta^{\prime}$-generic representation if and only if $L(s,\operatorname{Ad}\circ\,\phi^{\prime})$ is holomorphic at $s=1$. This part is called the conjecture of Gross-Prasad and Rallis. If this is the case, we say that $\phi^{\prime}$ or $\Pi_{\phi^{\prime}}$ is generic. If $\Pi_{\phi^{\prime}}$ is generic, the $\theta^{\prime}$-generic representation in $\Pi_{\phi^{\prime}}$ corresponds to the trivial representation of $A_{M}$ via the map $J(\theta^{\prime})$. In addition, the $\theta^{\prime}_{\tau}$-generic representation in $\Pi_{\phi^{\prime}}$ corresponds to the quadratic character $\eta$ defined in section 1.1. Here, $\tau$ denotes a representative of the nontrivial element of $F^{\times}/\mathrm{N}_{E/F}(E^{\times})$. We say that $\phi^{\prime}$ is generic with respect to any non-degenerate characters if $\phi^{\prime}$ is generic and $\eta$ is trivial.

• (3)

An element of $\Pi_{\phi^{\prime}}$ corresponding to a character $\mu$ of $A_{M}$ via the map $J(\theta^{\prime})$ is a representation of $G^{\prime}$ if and only if $\mu(-1)=+1$.

The finite set $\Pi_{\phi^{\prime}}$ is called the $L$-packet corresponding to $\phi^{\prime}$.

3. Quaternion algebra and symmetric subgroup

3.1. Definitions

Denote the unique quaternion division algebra over $F$ by $D$ and we fix an embedding of $E$ to $D$. Fix a representative $\tau$ of the nontrivial element of $F^{\times}/\mathrm{N}_{E/F}(E^{\times})$. Then there is an element $\sqrt{\tau}$ in $D^{\times}$ which satisfies

• •

as a vector space over $E$, $D=E\oplus E\sqrt{\tau}$;

• •

we have

[TABLE]

for any $x_{1},x_{2},y_{1},y_{2}\in E$.

Define inductively an element $\varepsilon_{m}$ of $G_{m}$ by

[TABLE]

Let $\sigma$ be the involution of $G$ given by

[TABLE]

The subgroup $H=H_{m}$ of fixed points of $\sigma$ is isomoprhic to $\mathrm{GL}_{m}(D)$. For $m=1$ case, $H_{1}$ consists of matrices of the form

[TABLE]

with $a,b\in E$ satisfying $a^{2}-\tau b^{2}\neq 0$. Thus, an explicit isomorphism $H_{1}\cong D^{\times}$ is given by

[TABLE]

In general, write an element $h$ of $H_{m}$ in the form of a 2 by 2 block matrix $(h_{i,j})_{1\leq i,j\leq m}$. Each entry $h_{i,j}$ is of the form (2) and it corresponds to an element of $D$ similarly as above. This observation provides an isomorphism $H_{m}\cong\mathrm{GL}_{m}(D)$.

Let $\lambda=(n_{1},\ldots,n_{r})$ be a partition of $n$. Suppose each $n_{i}$ is even and write $n_{i}=2m_{i}$. Then, $P_{\lambda}\cap H$ is a parabolic subgroup of $H$ which is, under the above isomorphism, sent to the parabolic subgroup of $\mathrm{GL}_{m}(D)$ consisting of matrices of the form

[TABLE]

We call these parabolic subgroups standard. Let $K_{H}$ be a maximal compact subgroup of $H$ which satisfies $H=(P_{\lambda}\cap H)K_{H}$ for any standard parabolic subgroups $P_{\lambda}\cap H$.

Let $x=x_{m}$ be the element of $\mathfrak{S}_{n}$ defined by

[TABLE]

Set $H^{\prime}=H^{\prime}_{m}=x^{-1}Hx$. This is the subgroup of fixed points of the involution $\sigma^{\prime}\,\colon g\mapsto\varepsilon^{\prime}g^{c}{\varepsilon^{\prime}}^{-1}$, where

[TABLE]

Hence, $H^{\prime}$ consists of matrices of the form

[TABLE]

The map \left(\begin{array}[]{cc}A&\tau B\\ w_{m}B^{c}w_{m}&w_{m}A^{c}w_{m}\end{array}\right)\mapsto A+\sqrt{\tau}B\in\mathrm{GL}_{m}(D) defines an isomorphism from $H^{\prime}$ to $\mathrm{GL}_{m}(D)$.

3.2. Double coset decomposition

Let $\lambda=(n_{1},\ldots,n_{r})$ be a partition of $n$. In this section, we recall some results on the double coset decomposition $H^{\prime}\backslash G/P_{\lambda}$ from [M17].

Let $\mathcal{S}(\lambda)$ be the set of $r$ by $r$ matrices $S=(s_{i,j})$ which satisfies following conditions:

• •

all entries are non-negative integers;

• •

for any $1\leq i,j\leq r$, $s_{i,j}=s_{j,i}$, i.e. $S=\,{}^{t}\!S$;

• •

for any $1\leq i\leq r$, $s_{i,i}\in 2\mathbb{Z}_{\geq 0}$;

• •

for any $1\leq i\leq r$, $\sum_{j=1}^{r}s_{i,j}=n_{i}$.

Take an element $S$ of $\mathcal{S}(\lambda)$. For each $1\leq i\leq r$, set $t_{i}=s_{i,i}/2$ and $d_{i}=\sum_{j=i}^{r}s_{i,j}-t_{i}$. Then, we obtain a partition $(d_{1},\ldots,d_{r})$ of $m$ and a partition $(t_{i},s_{i,i+1},s_{i,i+2},\ldots,s_{i,r})$ of $d_{i}$ for each $1\leq i\leq r$. Define an element $w_{S}$ as follows: for any $1\leq l\leq r$,

• (w1)

if there are integers $i$ and $k$ satisfying $1\leq i\leq r$ and $1\leq k\leq t_{i}$ so that $l$ can be expressed as

[TABLE]

we set

[TABLE]

• (w2)

if there are integers $i$ and $k$ satisfying $1\leq i<j\leq r$ and $1\leq k\leq s_{i,j}$ so that $l$ can be expressed as

[TABLE]

we set

[TABLE]

• (w3)

if there are integers $i$ and $k$ satisfying $1\leq i\leq r$ and $1\leq k\leq t_{i}$ so that $l$ can be expressed as

[TABLE]

we set

[TABLE]

• (w4)

if there are integers $i$ and $k$ satisfying $1\leq j<i\leq r$ and $1\leq k\leq s_{i,j}$ so that $l$ can be expressed as

[TABLE]

we set

[TABLE]

Proposition 3.1** ([M17], Proposition 3.1).**

For a partition $\lambda$ of $n$, we have

[TABLE]

An element $S=(s_{i,j})$ of $\mathcal{S}(\lambda)$ can be identified with a subpartition

[TABLE]

of $\lambda$. Here, 0-entries are ignored. Set $\varepsilon^{\prime}_{S}=w_{S}\varepsilon^{\prime}w_{S}^{-1}$. We define an involution $\sigma^{\prime}_{S}$ of $G$ by $\sigma^{\prime}_{S}\,\colon g\mapsto\varepsilon^{\prime}_{S}g^{c}{\varepsilon^{\prime}_{S}}^{-1}$. Note that $P_{S}$ and $M_{S}$ is stable under $\sigma^{\prime}_{S}$. An element $m$ of $M_{S}$ is of the form $m=\mathrm{diag}(m_{1,1},m_{1,2},\ldots,m_{r,r-1},m_{r,r})$ with $m_{i,j}\in G_{s_{i,j}}$. Of course components where $s_{i,j}=0$ are eliminated. Then, $m$ is contained in $M_{S}^{\sigma^{\prime}_{S}}$ if and only if the following conditions are satisfied:

• •

for any $1\leq i\leq r$, $m_{i,i}=\varepsilon^{\prime}_{t_{i}}m_{i,i}^{c}{\varepsilon^{\prime}_{t_{i}}}^{-1}$ i.e. $m_{i,i}\in H^{\prime}_{t_{i}}$;

• •

for any $1\leq i\neq j\leq r$, $m_{i,j}=w_{s_{i,j}}m_{j,i}^{c}w_{s_{i,j}}^{-1}$.

Hence, we get

[TABLE]

Set $P^{\prime}_{S}=M\cap P_{S}$. This is a parabolic subgroup of $M$.

Proposition 3.2** ([M17], Proposition 3.3).**

We have $\delta_{P_{S}^{\sigma^{\prime}_{S}}}^{2}|_{M_{S}^{\sigma^{\prime}_{S}}}=\delta_{P^{\prime}_{S}}\delta_{P}|_{M_{S}^{\sigma^{\prime}_{S}}}=\delta_{P_{S}}|_{M_{S}^{\sigma^{\prime}_{S}}}$.

3.3. Some results on $H^{\prime}$-distinguished representations

We record some results about $H^{\prime}$-distinguished representations of $G$. One can prove these results by almost the same arguments as in [F] with minor modification. So, we do not give a proof or only give a sketch.

Proposition 3.3** (See [F], Proposition 10).**

For any $g\in G$, there are $x,y\in H^{\prime}$ with $g^{-1}=x\sigma^{\prime}(g)y$.

Proposition 3.4** (See [F], Proposition 11).**

For an irreducible representation $\pi$ of $G$, the dimension of $\operatorname{Hom}_{H^{\prime}}(\pi,\mathbb{C})$ is at most 1.

Proof.

We sketch the proof. For details, see [F, Proposition 11].

Applying the theorem of Gel’fand and Kazhdan, and due to the above proposition, we are reduced to show the following claim:

For an irreducible representation $\pi$ of $G$, if $\operatorname{Hom}_{H^{\prime}}(\pi,\mathbb{C})$ is nonzero, then $\operatorname{Hom}_{H^{\prime}}(\pi^{\vee},\mathbb{C})$ is nonzero.

Let $l$ be an element of $\operatorname{Hom}_{H^{\prime}}(\pi,\mathbb{C})$. We define a representation $\pi^{\prime}$ of $G$ over the space of $\pi$ by

[TABLE]

Then, $\pi^{\vee}$ is isomorphic to $\pi^{\prime}$. Since $H^{\prime}$ is stable under the involution $g\mapsto w_{n}\,{}^{t}\!g^{-1}w_{n}^{-1}$, $l$ is an element of $\operatorname{Hom}_{H^{\prime}}(\pi^{\prime},\mathbb{C})$. This proves the above claim. ∎

Proposition 3.5** (See [F], Proposition 12).**

If an irreducible representation $\pi$ of $G$ is $H$-distinguished, then $\pi$ is conjugate self-dual, i.e. $\pi^{c}\cong\pi^{\vee}$.

4. Zelevinsky classification

In this section, we summarize some results on classification of irreducible representations of $G$ by Zelevinsky [Z].

First, we recall the classification of irreducible essentially square-integrable representations.

Proposition 4.1**.**

Let $l$ be a divisor of $n$ and set $\lambda=(n/l,\ldots,n/l)$ be a partition of $n$. For an irreducible supercuspidal representation $\rho$ of $G_{n/l}$, the representation $\rho|\mathrm{det}|^{(1-l)/2}\times\cdots\times\rho|\mathrm{det}|^{(l-1)/2}$ of $G$ has a unique irreducible subrepresentation $\operatorname{St}_{l}(\rho)$. This is essentially square-integrable. Conversely, any irreducible essentially square-integrable representations of $G$ can be written in this form uniquely.

With the notation of the above proposition, let $\underline{\phi}$ be the representation of $WD_{E}$ corresponding to $\rho$. Since $\rho$ is supercuspidal, the restriction of $\underline{\phi}$ to $\mathrm{SL}_{2}(\mathbb{C})$ is the trivial representation. We denote the restriction of $\underline{\phi}$ to $W_{E}$ again by $\underline{\phi}$. Let $\operatorname{Sp}(l)$ be the unique irreducible algebraic $l$-dimensional representation of $\mathrm{SL}_{2}(\mathbb{C})$. Then, the representation of $WD_{E}$ corresponding to $\operatorname{St}_{l}(\rho)$ is $\underline{\phi}\boxtimes\operatorname{Sp}(l)$.

A Jacquet module of $\operatorname{St}_{l}(\rho)$ is given as follows:

Lemma 4.2**.**

Let $l$ be a divisor of $n$ and $\rho$ be an irreducible supercuspidal representation of $G_{n/l}$. Set $\pi=\operatorname{St}_{l}(\rho)$ and denote by $V$ the space of $\pi$. Let $\lambda=(n_{1},\ldots,n_{r})$ be a partition of $n$ and set $P=P_{\lambda}$. Then, $V_{P}\neq 0$ ifa nd only if there is a non-negative integer $k_{i}$ for each $1\leq i\leq r$ satisfying $n_{i}=k_{i}\cdot n/l$. If this is the case, we have

[TABLE]

By Langlands classification, all irreducible representations of reductive groups appear in the composition factors of parabolically induced representations from essentially square-integrable representations. In the case at hand, more precise classification is known.

Theorem 4.3**.**

Let $\lambda=(n_{1},\ldots,n_{r})$ be a partition of $n$ and $\Delta_{i}$ be an irreducible essentially square-integrable representation of $G_{n_{i}}$. We write the absolute value of the central character of each $\Delta_{i}$ by $|\mathrm{det}|^{t_{i}}$ with real number $t_{i}$. Reordering $\Delta_{i}$’s so that we have

[TABLE]

the representation $\Delta_{1}\times\cdots\times\Delta_{r}$ has a unique irreducible quotient that we denote as $\Delta_{1}\boxplus\cdots\boxplus\Delta_{r}$. This representation is independent of the reordering which satisfies (4) and any irreducible representations of $G$ can be expressed in this form. Moreover, this expression is unique up to permutation.

Generic representations are classified in terms of the above classification as follows:

Proposition 4.4**.**

Let $\pi=\operatorname{St}_{l_{1}}(\rho_{1})\boxplus\cdots\boxplus\operatorname{St}_{l_{r}}(\rho_{r})$ be an irreducible representation of $G$. Here,

• •

$\lambda=(n_{1},\ldots,n_{r})$* is a partition of $n$;*

• •

$l_{i}$* is a divisor of $n_{i}$;*

• •

$\rho_{i}$* is an irreducible supercuspidal representation of $G_{n_{i}/l_{i}}$.*

Then, $\pi$ is generic if and only if there are no $i$, $j$ with $1\leq i\neq j\leq r$ and a positive integer $d$ satisfying the two conditions ($\heartsuit$1)–($\heartsuit$2):

• ($\heartsuit$1)

$\max\{1,l_{i}-l_{j}+1\}\leq d\leq l_{i}$* and $n_{i}/l_{i}=n_{j}/l_{j}$;*

• ($\heartsuit$2)

$\rho_{i}\cong\rho_{j}\otimes|\mathrm{det}|^{d+(l_{j}-l_{i})/2}$.

If this is the case, $\pi\cong\operatorname{St}_{l_{w(1)}}(\rho_{w(1)})\times\cdots\times\operatorname{St}_{l_{w(r)}}(\rho_{w(r)})$ for any permutation $w\in\mathfrak{S}_{r}$.

Remark 4.1**.**

Let $\pi$ be an irreducible representation of $G$ and $\underline{\phi}$ be the corresponding representation of $WD_{E}$. The above characterization of generic representations can be restated as follows: $\pi$ is generic if and only if $L(s,\operatorname{Ad}\circ\,\underline{\phi})$ is holomorphic at $s=1$. Suppose that $\pi$ is generic and there is an $L$-parameter $\phi^{\prime}$ of $G^{\prime}$ such that $\tilde{\phi}=bc_{\chi}\circ\,\phi^{\prime}$. Combining the above characterization of generic representations with its analog for $G^{\prime}$ (§2.3 (2)), one can see that $\Pi_{\phi^{\prime}}$ is generic. Another way to prove this is to use the equation ([M10, Theorem 5.3])

[TABLE]

and the fact that $L(s,\pi,\operatorname{As}^{+})$ is holomorphic at $s=1$ ([AM, Proposition 7.1]).

5. Classification of distinguished representations

First, we recall the classification of $\mathrm{GL}_{n}(F)$-distinguished generic representations of $G$ by Matringe. Note that this result also holds for odd $n$.

Theorem 5.1** ([M10], Theorem 5.2).**

Let $\pi$ be an irreducible generic representation of $G$ and write $\pi=\Delta_{1}\boxplus\cdots\boxplus\Delta_{r}$ as in Theorem 4.3. Here, $\lambda=(n_{1},\ldots,n_{r})$ is a partition of $n$ and each $\Delta_{i}$ is an irreducible essentially square-integrable representation of $G_{n_{i}}$. Then, $\pi$ is $\mathrm{GL}_{n}(F)$-distinguished if and only if there are an integer $1\leq k\leq r/2$ and a reordering of $\Delta_{i}$’s satisfying

• (1)

for all $1\leq i\leq k$, $\Delta_{2i}^{c}\cong\Delta_{2i-1}^{\vee}$;

• (2)

for all $2k+1\leq i\leq r$, $\Delta_{i}$ is $\mathrm{GL}_{n_{i}}(F)$-distinguished.

We have a similar classification for $H$-distinguished generic representations.

Theorem 5.2**.**

Let $\pi$ be an irreducible generic representation of $G$ and write $\pi=\Delta_{1}\boxplus\cdots\boxplus\Delta_{r}$ as in Theorem 4.3. Here, $\lambda=(n_{1},\ldots,n_{r})$ is a partition of $n$ and each $\Delta_{i}$ is an irreducible essentially square-integrable representation of $G_{n_{i}}$. Then, $\pi$ is $H$-distinguished if and only if there are an integer $1\leq k\leq r/2$ and a reordering of $\Delta_{i}$’s satisfying

• (1)

for all $1\leq i\leq k$, $\Delta_{2i}^{c}\cong\Delta_{2i-1}^{\vee}$;

• (2)

for all $2k+1\leq i\leq r$, $n_{i}=2m_{i}$ is even and $\Delta_{i}$ is $H_{m_{i}}$-distinguished.

Comparing these two classification results, we obtain the next corollary.

Corollary 5.3**.**

An irreducible generic representation of $G$ is $\mathrm{GL}_{n}(F)$-distinguished if it is $H$-distinguished.

The next lemma is the key to the proof of Theorem 5.2.

Lemma 5.4**.**

Let $\lambda=(n_{1},\ldots,n_{r})$ be a partition of $n$ and $\Delta_{i}$ an irreducible essentially square-integrable representation of $G_{n_{i}}$ for each $1\leq i\leq r$. Set $\Delta=\Delta_{1}\boxtimes\cdots\boxtimes\Delta_{r}$ and regard is as a representation of $M_{\lambda}$. Suppose that the Jacquet module $\Delta_{P^{\prime}_{S}}$ is $M_{S}^{\sigma^{\prime}_{S}}$-distinguished for some $S=(s_{i,j})\in\mathcal{S}(\lambda)$. Then, there are an integer $1\leq k\leq r/2$ and a reordering of $\Delta_{i}$’s satisfying (1) and (2) in Theorem 5.2.

Proof.

The proof is the same as [M10, Lemma 5.5]. We write each $\Delta_{i}$ in the form of $\operatorname{St}_{l_{i}}(\rho_{i})$. Here, $l_{i}$ is a divisor of $n_{i}$ and $\rho_{i}$ is an irreducible supercuspidal representation of $G_{n_{i}/l_{i}}$. We may assume

[TABLE]

Since we have $\Delta_{P^{\prime}_{S}}\neq\{0\}$, there exists a non-negative integer $k_{i,j}$ satisfying $s_{i,j}=k_{i,j}\cdot n_{i}/l_{i}$ for each $1\leq i,j\leq r$. Hence we have

[TABLE]

and

• •

for any $1\leq i\leq r$, $\Delta_{i,i}$ is $H^{\prime}_{t_{i}}$-distinguished;

• •

for any $1\leq i,j\leq r$, $\Delta_{j,i}^{c}\cong\Delta_{i,j}^{\vee}$.

By induction on $r$, we show that there are an integer $1\leq k\leq r/2$ and reordering of $\Delta_{i}$’s which satisfy the conditions (1) and (2) in Theorem 5.2. Set $i_{0}=\min\{1\leq i\leq r\,;\,s_{1,i}\neq 0\}$.

Case 1.  If $i_{0}=1$, $\Delta_{1,1}$ is conjugate self-dual since it is $H^{\prime}_{t_{1}}$-distinguished. Thus we get $\rho_{1}^{\vee}\cong\rho_{1}^{c}|\mathrm{det}|^{l_{1}-k_{1,1}}$. Therefore, $\Delta_{1}^{\vee}$ is isomorphic to $\operatorname{St}_{l_{1}}(\rho_{1}^{c}|\mathrm{det}|^{l_{1}-k_{1,1}})$. Take $1\leq j\leq r$ with $\Delta_{1}^{\vee}\cong\Delta_{j}^{c}$. Then, $n_{1}=n_{j}$, $l_{1}=l_{j}$ and $\rho_{1}$ is isomorphic to $\rho_{j}|\mathrm{det}|^{k_{1,1}-l_{1}}$. From ($\heartsuit$1) and($\heartsuit$2), one can see that $k_{1,1}=l_{1}$, i.e. $s_{1,1}=n_{1}$ and $s_{1,j}=0$ for any $1<j\leq r$. Hence we have $\Delta_{1}=\Delta_{1,1}$ and this is $H^{\prime}_{m_{1}}$-distinguished. By the induction hypothesis, there is a reordering of $\Delta_{2},\ldots,\Delta_{r}$ which satisfies two conditions of Theorem 5.2 for suitable $1\leq k\leq(r-1)/2$.

Case 2.  If $i_{0}>1$, note that we have $k_{1,j}=0$ for any $1\leq j\leq i_{0}$. Since $\Delta_{1,i_{0}}^{c}\cong\operatorname{St}_{k_{1,i_{0}}}(\rho_{1}^{c}|\mathrm{det}|^{(l_{1}-k_{1,i_{0}})/2})$ is isomorphic to $\Delta_{i_{0},1}^{\vee}\cong\operatorname{St}_{k_{i_{0},1}}(\rho_{i_{0}}^{\vee}|\mathrm{det}|^{(k_{i_{0},1}-l_{i_{0}})/2})$, we see that $n_{1}/l_{1}=n_{i_{0}}/l_{i_{0}}$ and $\rho_{i_{0}}^{\vee}\cong\rho_{1}^{c}|\mathrm{det}|^{(l_{1}+l_{i_{0}})/2-k_{1,i_{0}}}$. Therefore, we obtain

[TABLE]

Take $1\leq j\leq r$ so that $\Delta_{i_{0}}^{\vee}\cong\Delta_{j}^{c}$. Then we have $n_{j}/l_{j}=n_{i_{0}}/l_{i_{0}}=n_{1}/l_{1}$ and $\rho_{1}\cong\rho_{j}|\mathrm{det}|^{k_{1,i_{0}}-(l_{1}+l_{i_{0}})/2}$. Moreover, $n_{j}=n_{i_{0}}$ implies $l_{j}=l_{i_{0}}$. Since we have $l_{j}-k_{1,i_{0}}<l_{j}$, the conditions ($\heartsuit$1), ($\heartsuit$2) and the assumption (5) imply

[TABLE]

Since $l_{1}\geq k_{1,i_{0}}$, we obtain $l_{1}=k_{1,i_{0}}$, i.e. $s_{1,i_{0}}=n_{1}$ and for any $1\leq j\neq i_{0}\leq r$, we see that $s_{1,j}=0$. Hence $\Delta_{1}=\Delta_{1,i_{0}}$.

We have two cases: $\Delta_{1}$ is conjugate self-dual (Case 2-a) or $\Delta_{1}$ is not conjugate self-dual (Case 2-b).

Case 2-a.  If $\Delta_{1}=\Delta_{1,i_{0}}$ is conjugate self-dual, $\Delta_{1,i_{0}}^{\vee}\cong\Delta_{i_{0},1}^{c}$ is conjugate self-dual. Hence we get $\rho_{i_{0}}^{\vee}\cong\rho_{i_{0}}^{c}|\mathrm{det}|^{l_{i_{0}}-l_{1}}$. Thus, $\Delta_{i_{0}}^{\vee}$ is isomorphic to $\mathrm{St}_{l_{i_{0}}}(\rho_{i_{0}}^{c}|\mathrm{det}|^{l_{i_{0}}-l_{1}})$. Take $1\leq j\leq r$ so that $\Delta_{i_{0}}^{\vee}$ is isomorphic to $\Delta_{j}^{c}$. Then we have $n_{i_{0}}/l_{i_{0}}=n_{j}/l_{j}$ and $\rho_{j}\cong\rho_{i_{0}}|\mathrm{det}|^{l_{i_{0}}-l_{1}}=\rho_{i_{0}}|\mathrm{det}|^{l_{j}-l_{1}}$. The conditions ($\heartsuit$1) and ($\heartsuit$2) imply $l_{j}=l_{1}$. Hence we have $l_{i_{0}}=l_{1}=k_{1,i_{0}}=k_{i_{0},1}$, i.e. $n_{i_{0}}=s_{i_{0},1}$ and $s_{i_{0},j}=0$ for any $1<j\leq r$. Therefore, one can see that $\Delta_{i_{0}}=\Delta_{i_{0},1}\cong\Delta_{1,i_{0}}^{\vee}\cong\Delta_{1}$. By the induction hypothesis, there is a reordering of $\Delta_{2},\ldots,\Delta_{i_{0}-1},\Delta_{i_{0}+1},\ldots,\Delta_{r}$ which satisfies two conditions of Theorem 5.2 for some $1\leq k\leq r/2-1$.

Case 2-b.  If $\Delta_{1}=\Delta_{1,i_{0}}$ is not conjugate self-dual, $\Delta_{2}^{c}\cong\Delta_{1}^{\vee}$. Note that $n_{1}=n_{2}$ and $l_{1}=l_{2}$. Set $j_{0}=\min\{1\leq j\leq r\,;\,s_{2,j}\neq 0\}$. If $j_{0}=2$, one can obtain $\Delta_{2}=\Delta_{2,2}$ by the similar arguments as in the Case 1. Since $\Delta_{2}$ is $H^{\prime}_{m_{2}}$-distinguished, by the induction hypothesis, there is a reordering of $\Delta_{1},\Delta_{3},\ldots,\Delta_{r}$ which satisfies two conditions of Theorem 5.2 for some $1\leq k\leq(r-1)/2$. Hereafter, we assume that $j_{0}\neq 2$.

Suppose that $j_{0}>2$. Since $\Delta_{2,j_{0}}^{c}=\mathrm{St}_{k_{2,j_{0}}}(\rho_{2}^{c}|\mathrm{det}|^{(l_{2}-k_{2,j_{0}})/2})$ is isomorphic to $\Delta_{j_{0},2}^{\vee}\cong\mathrm{St}_{k_{2,j_{0}}}(\rho_{j_{0}}^{\vee}|\mathrm{det}|^{k_{j_{0},1}+(k_{j_{0},2}-l_{j_{0}})/2})$, we obtain

[TABLE]

If $j_{0}=i_{0}$, we have $\rho_{i_{0}}^{\vee}\cong\rho_{1}^{c}|\mathrm{det}|^{(l_{i_{0}}-l_{1})/2}$ and $\rho_{i_{0}}^{\vee}\cong\rho_{2}^{c}|\mathrm{det}|^{-k_{j_{0},2}+(l_{i_{0}}-l_{1})/2}$ as we see above. Then we get $\rho_{2}\cong\rho_{1}|\mathrm{det}|^{k_{j_{0},2}}$. From ($\heartsuit$1) and ($\heartsuit$2), we see that $k_{j_{0},2}=0$, which contradicts the choice of $j_{0}$. Thus, $j_{0}\neq i_{0}$.

Take $1\leq j\leq r$ so that $\Delta_{j_{0}}^{\vee}\cong\Delta_{j}^{c}$. Since we have

[TABLE]

we see that $n_{j}/l_{j}=n_{j_{0}}/l_{j_{0}}=n_{2}/l_{2}$ and $\rho_{j}=\rho_{2}|\mathrm{det}|^{-k_{j_{0},2}+(l_{j}+l_{2})/2}$. From the conditions ($\heartsuit$1), ($\heartsuit$2) and the assumption (5), one can see that $l_{2}=k_{j_{0},2}$ as $l_{j}\geq l_{j}-k_{j_{0},2}\geq l_{j}-l_{2}\geq 0$. Hence $s_{2,j_{0}}=n_{2}$ and $s_{2,j}=0$ for any $1\leq j\neq j_{0}\leq r$. Therefore we obtain $\Delta_{2}=\Delta_{2,j_{0}}$.

If $j_{0}=1$, we have $s_{1,2}=s_{2,1}\neq 0$ and hence $i_{0}=2$. As we have seen, $\Delta_{1}=\Delta_{1,2}$. This implies $\Delta_{2}^{c}\cong\Delta_{1,2}^{\vee}\cong\Delta_{2,1}^{c}$. Hence we get $\Delta_{2}=\Delta_{2,1}$.

In any case, we see that $s_{2,j}=0$ for all $1\leq j\neq j_{0}\leq r$, i.e. $k_{2,j_{0}}=l_{2}=l_{1}$, $n_{2}=s_{2,j_{0}}$. Since $\Delta_{1}^{c}\cong\Delta_{i_{0},1}^{\vee}\cong\mathrm{St}_{l_{1}}(\rho_{i_{0}}^{\vee}|\mathrm{det}|^{(l_{1}-l_{i_{0}})/2})$ is isomorphic to $\Delta_{2}^{\vee}\cong\Delta_{j_{0},2}^{c}\cong\mathrm{St}_{l_{2}}(\rho_{j_{0}}^{c}|\mathrm{det}|^{(l_{j_{0}}-l_{2})/2})$, we get $\rho_{i_{0}}^{\vee}\cong\rho_{j_{0}}^{c}|\mathrm{det}|^{(l_{j_{0}}+l_{i_{0}})/2-l_{1}}$. Take $1\leq j^{\prime}\leq r$ so that $\Delta_{i_{0}}^{\vee}\cong\Delta_{j^{\prime}}^{c}$. Then we have $n_{i_{0}}/l_{i_{0}}=n_{j^{\prime}}/l_{j^{\prime}}$ and $\rho_{j^{\prime}}\cong\rho_{j_{0}}|\mathrm{det}|^{(l_{j_{0}}+l_{j^{\prime}})/2-l_{1}}$. If $l_{1}<l_{j_{0}}$, ($\heartsuit$1) and ($\heartsuit$2) imply $l_{j^{\prime}}=l_{1}$. If this is the case,

[TABLE]

Hence we get $\Delta_{i_{0}}\cong\Delta_{2}$. By the induction hypothesis, there is a reordering of $\Delta_{2},\ldots,\Delta_{i_{0}-1},\Delta_{i_{0}+1},\ldots,\Delta_{r}$ which satisfies two conditions of Theorem 5.2 for some $1\leq k\leq(r-1)/2$. If $l_{1}=l_{j_{0}}$, $\Delta_{j_{0}}\cong\Delta_{1}$ since we have $\rho_{j_{0}}^{c}\cong\rho_{i_{0}}^{\vee}|\mathrm{det}|^{(l_{1}-l_{i_{0}})/2}\cong\rho_{1}^{c}$. By the induction hypothesis, there is a reordering of $\Delta_{1},\Delta_{3},\ldots,\Delta_{j_{0}-1},\Delta_{j_{0}+1},\ldots,\Delta_{r}$ which satisfies two conditions of Theorem 5.2 for some $1\leq k\leq(r-1)/2$.

This completes the proof of the lemma. ∎

Proof of Theorem 5.2.

Suppose that $\pi$ is $H^{\prime}$-distinguished. We simply write $P_{\lambda}$ by $P$. There is a total order $\geq$ on the set $W_{\lambda}=\{w_{S}\,;\,S\in\mathcal{S}(\lambda)\}$ which satisfies following condition:

For any $w\in W_{\lambda}$,

[TABLE]

is open in $G$ and $PwH^{\prime}$ is closed in $G_{\geq w}$.

Let $W_{i}$ be the space of $\Delta_{i}$ for each $1\leq i\leq r$ and set $(\Delta,W)=(\Delta_{1}\boxtimes\cdots\boxtimes\Delta_{r},W_{1}\otimes\cdots\otimes W_{r})$. For each $w\in W_{\lambda}$, let $\mathcal{F}_{w}$ be the $H^{\prime}$-submodule of $I_{P}^{G}W$ consisting of elements whose support is contained in $G_{\geq w}$. Set $\mathcal{F}_{w^{+}}=\sum_{w^{\prime}>w}\mathcal{F}_{w}$. The map $\mathcal{F}_{w}\rightarrow\operatorname{ind}_{w^{-1}Pw\cap H^{\prime}}^{H^{\prime}}(w^{-1}(\delta_{P}^{-1/2}\Delta))$ given by sending $f\in\mathcal{F}_{w}$ to the function $h\mapsto f(wh)$ on $H^{\prime}$ induces an isomorphism of $H^{\prime}$-modules

[TABLE]

Take a divisor $l_{i}$ of $n_{i}$ and an irreducible supercuspidal representation $\rho_{i}$ of $G_{n_{i}/l_{i}}$ so that we have $\Delta_{i}\cong\operatorname{St}_{l_{i}}(\rho_{i})$. Since $\pi$ is $H^{\prime}$-distinguished, there is a $w\in W_{\lambda}$ such that $\operatorname{ind}_{w^{-1}Pw\cap H^{\prime}}^{H^{\prime}}(w^{-1}(\delta_{P}^{1/2}\Delta))$ has a nonzero $H^{\prime}$-invariant linear form. Let $S$ be the element of $\mathcal{S}(\lambda)$ with $w=w_{S}$. Then, we have $P\cap wH^{\prime}w^{-1}=P_{S}^{\sigma^{\prime}_{S}}$. By Frobenius reciprocity, we obtain

[TABLE]

By Proposition 3.2, we see that the restrictions of $\delta_{P_{S}^{\sigma^{\prime}_{S}}}^{-1}\delta_{P}^{1/2}$ and $\delta_{P^{\prime}_{S}}^{-1/2}$ to $P_{S}^{\sigma^{\prime}_{S}}$ are equal. Hence we get $\operatorname{Hom}_{P_{S}^{\sigma^{\prime}_{S}}}(\delta_{P^{\prime}_{S}}^{-1/2}\Delta,\mathbf{1})\neq\{0\}$. Since we have $M\cap N_{S}=N^{\prime}_{S}\subset N_{S}^{\sigma^{\prime}_{S}}N$, the Jacquet module $\Delta_{P^{\prime}_{S}}$ is $M_{S}^{\sigma^{\prime}_{S}}$-distinguished. By the above lemma, there are an integer $1\leq k\leq r/2$ and a reordering of $\Delta_{i}$’s which satisfy the two conditions in the theorem.

Conversely, suppose that there are an integer $1\leq k\leq r/2$ and a reordering of $\Delta_{i}$’s which satisfy the two conditions in the theorem. For each $2k+1\leq i\leq r$, take a nonzero element $\xi_{i}$ of $\operatorname{Hom}_{H_{m_{i}}}(\Delta_{i},\mathbf{1})$.

For $1\leq i\leq k$, let $Q_{i}$ be the standard parabolic subgroup of $G_{2n_{2i-1}}$ corresponding to the partition $(n_{2i-1},n_{2i-1})$. Let $\tau_{i,s}=\Delta_{2i-1}|\mathrm{det}|^{s}\times\Delta_{2i}|\mathrm{det}|^{-s}$ be a representation of $G_{2n_{2i-1}}$ with a parameter $s\in\mathbb{C}$. Since $\Delta_{2i}^{c}$ is isomorphic to $\Delta_{2i-1}^{\vee}$, there is a nonzero $G_{n_{2i-1}}$-invariant linear form $\gamma_{i}$ on $\Delta_{2i-1}\otimes\Delta_{2i}^{c}$. Take a flat section $v_{i,s}$ of $\tau_{i,s}$ and set $v_{i}=v_{i,0}$. Then, by [BD, Theorem 2.8, Theorem 2.26], the integral

[TABLE]

converges absolutely for $\mathrm{Re}(s)\gg 0$ and has meromorphic continuation to whole $s$-plane. Here,

[TABLE]

Hence, this integral defines a nonzero element of $\mathrm{Hom}_{H_{n_{2i-1}}}(\tau_{i,s},\mathbf{1})$. Let $\xi_{i}(v)$ be the leading coefficient of its Laurent expansion at $s=0$, the map $v\mapsto\xi_{i}(v)$ provides a nonzero element of $\operatorname{Hom}_{H_{n_{2i-1}}}(\tau_{i},\mathbf{1})$.

Let $\tilde{P}=P_{\tilde{\lambda}}$ be the standard parabolic subgroup of $G$ corresponding to the partition $\tilde{\lambda}=(2n_{1},2n_{3},\ldots,2n_{2k-1},n_{2k+1},\ldots,n_{r})$ of $n$. We define the representation $(\tilde{\Delta},\tilde{W})$ of $\tilde{M}=M_{\tilde{\lambda}}$ by

[TABLE]

so that we have $\pi=\mathrm{Ind}_{\tilde{P}}^{G}(\tilde{\Delta})$. Let $\tilde{\xi}$ be the element of $\operatorname{Hom}_{H\cap\tilde{M}}(\tilde{\Delta},\mathbf{1})$ given by

[TABLE]

Recall that we have $(\tilde{P}\cap H)K_{H}=H$ since $\tilde{P}\cap H$ is a standard parabolic subgroup of $H$. For an element $\phi$ of $\pi=\mathrm{Ind}_{\tilde{P}}^{G}(\tilde{\Delta})$, set

[TABLE]

Then, $\xi$ defines a nonzero element of $\operatorname{Hom}_{H}(\pi,\mathbf{1})$. Therefore, $\pi$ is $H$-distinguished. ∎

6. Main theorem

Lemma 6.1**.**

Let $\Delta$ be an irreducible essentially square-integrable representation of $G$. Then, following conditions are equivalent:

• (1)

$\Delta$* is $\mathrm{GL}_{m}(D)$-distinguished;*

• (2)

$\Delta$* is $\mathrm{GL}_{n}(F)$-distinguished;*

• (3)

$\Delta$* arises from unstable base change lift;*

• (4)

$\Delta$* arises from unstable base change lift of an $L$-parameter of $G^{\prime}$ which is generic with respect to any non-degenerate characters.*

Proof.

Equivalence of (1) and (2) is [BP, Theorem 1]. Let us prove equivalence of (2) and (3). By [M09, Theorem 3.7], $\Delta$ is $\mathrm{GL}_{n}(F)$-distinguished if and only if $L(s,\Delta,\operatorname{As}^{+})$ has a pole at $s=0$. Take a divisor $l$ of $n$ and an irreducible supercuspidal representation $\rho$ of $G_{n/l}$ so that $\Delta=\operatorname{St}_{l}(\rho)$. Let $\underline{\phi_{\rho}}$ (resp. $\underline{\phi}$) be the representation of $WD_{E}$ associated with $\rho$ (resp. $\Delta$). We write the restriction of $\underline{\phi_{\rho}}$ to $W_{E}$ by the same symbol. Then, we have $\underline{\phi}=\underline{\phi_{\rho}}\boxtimes\operatorname{Sp}(l)$. By [M09, Proposition 4.1], we have

[TABLE]

Since $L(s,\omega\otimes\rho,\operatorname{As}^{+})=L(s,\rho,\operatorname{As}^{-})$, $L(s,\Delta,\operatorname{As}^{+})$ has a pole at$s=0$ if and only if one of the following conditions holds:

• •

$l$ is odd and $\widetilde{\phi_{\rho}}$ fixes a non-degenerate vector of $\operatorname{As}^{+}$

• •

$l$ is even and $\widetilde{\phi_{\rho}}$ fixes a non-degenerate vector of $\operatorname{As}^{-}$.

When $l$ is odd, $\operatorname{Sp}(l)$ is orthogonal and when $l$ is even, $\operatorname{Sp}(l)$ is symplectic. Besides, $\widetilde{\phi_{\rho}}$ fixes a non-degenerate vector of $\operatorname{As}^{+}$ (resp. $\operatorname{As}^{-}$) if and only if $\underline{\phi_{\rho}}$ is conjugate-orthogonal (resp. conjugate-symplectic), see [GGP, Proposition 7.5]. Hence, $\Delta$ is $\mathrm{GL}_{n}(F)$-distinguished if and only if $\underline{\phi}$ is conjugate-orthogonal. By Lemma 2.1, we obtain equivalence of (2) and (3).

Finally, we show that (3) implies (4). Take an $L$-parameter $\phi^{\prime}$ of $G^{\prime}$ with $\tilde{\phi}=bc_{\chi}\circ\,\phi^{\prime}$. By Remark 4.1, $\Pi_{\phi^{\prime}}$ is generic. Since $n$ is even, the quadratic character $\eta$ of $A_{\phi}$ is trivial. Hence $\phi^{\prime}$ is generic with respect to any non-degenerate characters. ∎

Remark 6.1**.**

By a similar argument as the proof of the above lemma, one can see that following conditions for an irreducible essentially square-integrable representation $\Delta$ of $G$ are equivalent:

• •

$\Delta$ is $\mathrm{GL}_{n}(F)$-distinguished (resp. $(\mathrm{GL}_{n}(F)),\omega$-distinguished);

• •

the representation of $WD_{E}$ corresponding to $\underline{\phi}$ is conjugate-orthogonal (resp. conjugate-symplectic).

These equivalence hold even when $n$ is odd.

Theorem 6.2**.**

Let $\pi$ be an irreducible generic representation of $G$. Consider the following statements:

• (A)

$\pi$* arises from unstable base change lift of an $L$-parameter of $G^{\prime}$ which is generic with respect to any non-degenerate characters;*

• (B)

$\pi$* is $H$-distinguished.*

Then, (A) implies (B).

Proof.

Suppose that (A) holds. Let $\lambda=(n_{1},\ldots,n_{r})$ be a partition of $n$ and $\Delta_{i}$ an irreducible essentially square-integrable representation of $G_{n_{i}}$ with $\pi=\Delta_{1}\times\cdots\times\Delta_{r}$. Denote by $\underline{\phi}$ the representation of $WD_{E}$ corresponding to $\pi$. By Lemma 2.1, $\underline{\phi}$ is conjugate-orthogonal. Hence, there is an integer $1\leq k^{\prime}\leq k\leq r/2$ and a reordering of $\Delta_{i}$’s satisfying

• (1)

for all $1\leq i\leq k^{\prime}$, $\Delta_{2i-1}$ is not conjugate self-dual and $\Delta_{2i}^{c}\cong\Delta_{2i-1}^{\vee}$;

• (2)

for all $k^{\prime}+1\leq i\leq k$, $\Delta_{2i-1}$ is $(\mathrm{GL}_{n_{2i-1}}(F),\omega)$-distinguished and $\Delta_{2i-1}\cong\Delta_{2i}$;

• (3)

for all $2k+1\leq i\leq r$, $\Delta_{i}$ is $\mathrm{GL}_{n_{i}}(F)$-distinguished.

Note that $\Delta_{i}$’s appearing in (1) (resp. (2), (3)) correspond to the irreducible factors of $\underline{\phi}$ which is not conjugate self-dual (resp. conjugate-symplectic, conjugate-orthogonal), cf. Remark 6.1. Since the quadratic character $\eta$ of $A_{\phi}$ is trivial, $n_{i}=2m_{i}$ is even for all $2k+1\leq i\leq r$. Hence by [BP, Theorem 1], each $\Delta_{i}$ is $H_{m_{i}}$-distinguished for $2k+1\leq i\leq r$. By Theorem 5.2, $\pi$ is $H$-distinguished, i.e. (B) holds. ∎

Remark 6.2**.**

The converse direction (B)$\Rightarrow$(A) does not hold in general. In fact, suppose $m$ is odd and $\Delta$ is a $\mathrm{GL}_{m}(F)$-distinguished irreducible essentially square-integrable representation of $G_{m}$. Then, a representation $\pi=\Delta\times\Delta$ of $G_{2m}$ is $H_{m}$-distinguished and does not satisfy the condition (A).

Acknowledgement. I would like to thank Tamotsu Ikeda, Hiraku Atobe and Masao Oi for giving me many helpful comments. I am also grateful for the anonymous referees for pointing out some inaccuracies in an earlier draft, especially the equation (3) in Lemma 4.2 and the numerous comments which greatly improve the manuscript. This research does not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.