Functoriality for the exterior square of $\operatorname {GL}_{4}$ and the symmetric fourth of $\operatorname {GL}_{2}$
By Henry H. Kim, with an appendix by Dinakar Ramakrishnan, with an appendix co-authored by Peter Sarnak
Abstract
In this paper we prove the functoriality of the exterior square of cusp forms on $\operatorname {GL}_{4}$ as automorphic forms on $\operatorname {GL}_{6}$ and the symmetric fourth of cusp forms on $\operatorname {GL}_{2}$ as automorphic forms on $\operatorname {GL}_{5}$. We prove these by applying a converse theorem of Cogdell and Piatetski-Shapiro to analytic properties of certain $L$-functions obtained by the Langlands-Shahidi method. We give several applications: First, we prove the weak Ramanujan property of cuspidal representations of $\operatorname {GL}_{4}$ and the absolute convergence of the exterior square $L$-functions of $\operatorname {GL}_{4}$. Second, we prove that the fourth symmetric power $L$-functions of cuspidal representations of $\operatorname {GL}_{2}$ are entire, except for those of dihedral and tetrahedral type. Third, we prove the bound $\frac {3}{26}$ for Hecke eigenvalues of Maass forms over any number field.
1. Introduction
Let $\wedge ^{2}: \operatorname {GL}_{n}(\mathbb{C})\longrightarrow \operatorname {GL}_{N}(\mathbb{C})$, where $N=\frac {n(n-1)}{2}$, be the map given by the exterior square. Then Langlands’ functoriality predicts that there is a map from cuspidal representations of $\operatorname {GL}_{n}$ to automorphic representations of $\operatorname {GL}_{N}$, which satisfies certain canonical properties. To explain, let $F$ be a number field, and let $\mathbb{A}$ be its ring of adeles. Let $\pi =\bigotimes _{v} \pi _{v}$ be a cuspidal (automorphic) representation of $\operatorname {GL}_{n}(\mathbb{A})$. In what follows, a cuspidal representation always means a unitary one. Now by the local Langlands correspondence, $\wedge ^{2}\pi _{v}$ is well defined as an irreducible admissible representation of $\operatorname {GL}_{N}(F_{v})$ for all $v$ (the work of Harris-Taylor ReferenceH-T and Henniart ReferenceHe2 on $p$-adic places and of Langlands ReferenceLa4 on archimedean places). Let $\wedge ^{2}\pi =\bigotimes _{v} {\wedge ^{2}\pi _{v}}$. It is an irreducible admissible representation of $\operatorname {GL}_{N}(\mathbb{A})$. Then Langlands’ functoriality in this case is equivalent to the fact that $\wedge ^{2}\pi$ is automorphic.
Note that $\wedge ^{2}(\operatorname {GL}_{2}(\mathbb{C}))\simeq \operatorname {GL}_{1}(\mathbb{C})$ and in fact for a cuspidal representation $\pi$ of $\operatorname {GL}_{2}(\mathbb{A})$,$\wedge ^{2}\pi =\omega _{\pi }$, the central character of $\pi$. Furthermore, $\wedge ^{2}(\operatorname {GL}_{3}(\mathbb{C}))\simeq \operatorname {GL}_{3}(\mathbb{C})$. In this case, given a cuspidal representation $\pi$ of $\operatorname {GL}_{3}(\mathbb{A})$,$\wedge ^{2}\pi =\tilde{\pi }\otimes \omega _{\pi }$, where $\tilde{\pi }$ is the contragredient of $\pi$.
In this paper, we look at the case $n=4$. Let $\pi =\bigotimes _{v} \pi _{v}$ be a cuspidal representation of $\operatorname {GL}_{4}(\mathbb{A})$. What we prove is weaker than the automorphy of $\wedge ^{2}\pi$. We prove (Theorem 5.3.1)
Theorem A
Let $T$ be the set of places where $v|2, 3$ and $\pi _{v}$ is a supercuspidal representation. Then there exists an automorphic representation $\Pi$ of $\operatorname {GL}_{6}(\mathbb{A})$ such that $\Pi _{v}\simeq \wedge ^{2}\pi _{v}$ if $v\notin T$. Moreover, $\Pi$ is of the form $\operatorname {Ind}\tau _{1}\otimes \cdots \otimes \tau _{k}$, where the $\tau _{i}$’s are all cuspidal representations of $\operatorname {GL}_{n_{i}}(\mathbb{A})$.
The reason why we have the exceptional places $T$, especially for $v|2$, is due to the fact that supercuspidal representations of $\operatorname {GL}_{4}(F_{v})$ are very complicated when $v|2$. We use the Langlands-Shahidi method and a converse theorem of Cogdell-Piatetski-Shapiro to prove the above theorem (cf. ReferenceCo-PS1, ReferenceKi-Sh2). We expect many applications of this result. Among them, we mention two: First, we prove the weak Ramanujan property of cuspidal representations of $\operatorname {GL}_{4}(\mathbb{A})$ (Proposition 6.3; see Definition 3.6 for the notation).
Second, we prove the existence of the symmetric fourth lift of a cuspidal representation of $\operatorname {GL}_{2}(\mathbb{A})$ as an automorphic representation of $\operatorname {GL}_{5}(\mathbb{A})$. More precisely, let $\operatorname {GL}_{2}(\mathbb{C})\longrightarrow \operatorname {GL}_{m+1}(\mathbb{C})$ be the symmetric $m$th power (the $m+1$-dimensional irreducible representation of $\operatorname {GL}_{2}(\mathbb{C})$ on symmetric tensors of rank $m$). Let $\pi =\bigotimes _{v} \pi _{v}$ be a cuspidal representation of $\operatorname {GL}_{2}(\mathbb{A})$ with central character $\omega _{\pi }$. By the local Langlands correspondence, $\operatorname {Sym}^{m}(\pi _{v})$ is well defined for all $v$. Hence Langlands’ functoriality predicts that $\operatorname {Sym}^{m}(\pi )=\bigotimes _{v} \operatorname {Sym}^{m}(\pi _{v})$ is an automorphic representation of $\operatorname {GL}_{m+1}(\mathbb{A})$. Gelbart and Jacquet ReferenceGe-J proved that $\operatorname {Sym}^{2}(\pi )$ is an automorphic representation of $\operatorname {GL}_{3}(\mathbb{A})$. We proved in ReferenceKi-Sh2 that $\operatorname {Sym}^{3}(\pi )$ is an automorphic representation of $\operatorname {GL}_{4}(\mathbb{A})$ as a consequence of the functorial product $\operatorname {GL}_{2}\times \operatorname {GL}_{3}\longrightarrow \operatorname {GL}_{6}$, corresponding to the tensor product map $\operatorname {GL}_{2}(\mathbb{C})\times \operatorname {GL}_{3}(\mathbb{C})\longrightarrow \operatorname {GL}_{6}(\mathbb{C})$.
$\operatorname {Sym}^{4}(\pi )$ is an automorphic representation of $\operatorname {GL}_{5}(\mathbb{A})$. If $\operatorname {Sym}^{3}(\pi )$ is cuspidal, $\operatorname {Sym}^{4}(\pi )$ is either cuspidal or induced from cuspidal representations of $\operatorname {GL}_{2}(\mathbb{A})$ and $\operatorname {GL}_{3}(\mathbb{A})$.
Here we stress that there is no restriction on the places as opposed to the case of the exterior square lift.
Theorem B is obtained by applying Theorem A to $\operatorname {Sym}^{3}(\pi )\otimes \omega _{\pi }^{-1}$. For simplicity, we write $A^{m}(\pi )=\operatorname {Sym}^{m}(\pi )\otimes \omega _{\pi }^{-1}$. We prove that
This implies that $A^{4}(\pi )$ is an automorphic representation of $\operatorname {GL}_{5}(\mathbb{A})$, and so is $\operatorname {Sym}^{4}(\pi )$.
An immediate corollary is that we have a new estimate for Ramanujan and Selberg’s conjectures using ReferenceLu-R-Sa. Namely, let $\pi$ be a cuspidal representation of $\operatorname {GL}_{2}(\mathbb{A})$. Let $\pi _{v}$ be a local (finite or infinite) spherical component, given by $\pi _{v}=\operatorname {Ind}(|\ |_{v}^{s_{1v}},|\ |_{v}^{s_{2v}})$. Then $|\operatorname {Re}(s_{iv})|\leq \frac {3}{26}.$ If $F=\mathbb{Q}$ and $v=\infty$, this condition implies that $\lambda _{1}\geq \frac {40}{169}\approx 0.237$, where $\lambda _{1}$ is the first positive eigenvalue for the Laplace operator on the corresponding hyperbolic space.
In a joint work with Sarnak in Appendix 2ReferenceKi-Sa, by considering the twisted symmetric square $L$-functions of the symmetric fourth (cf. ReferenceBDHI), we improve the bound further, at least over $\mathbb{Q}$, namely, $\operatorname {Re}(s_{ip})\leq \frac {7}{64}.$ As for the first positive eigenvalue for the Laplacian, we have $\lambda _{1}\geq \frac {975}{4096}\approx 0.238.$
In ReferenceKi-Sh3, we determine exactly when $A^{4}(\pi )$ is cuspidal. We show that $A^{4}(\pi )$ is not cuspidal and $A^{3}(\pi )$ is cuspidal if and only if there exists a non-trivial quadratic character $\eta$ such that $A^{3}(\pi )\simeq A^{3}(\pi )\otimes \eta$, or equivalently, there exists a non-trivial grössencharacter $\chi$ of $E$ such that $(\operatorname {Ad}(\pi ))_{E}\simeq (\operatorname {Ad}(\pi ))_{E}\otimes \chi$, where $E/F$ is the quadratic extension, determined by $\eta$. We refer to that paper for many applications of symmetric cube and symmetric fourth: The analytic continuation and functional equations are proved for the 5th, 6th, 7th, 8th and 9th symmetric power $L$-functions of cuspidal representations of $\operatorname {GL}_{2}$. It has immediate application for Ramanujan and Selberg’s bounds and the Sato-Tate conjecture: Let $\pi _{v}$ be an unramified local component of a cuspidal representation $\pi =\bigotimes _{v} \pi _{v}$. Then it is shown that $q_{v}^{-\frac {1}{9}}<|\alpha _{v}|, |\beta _{v}|<q_{v}^{\frac {1}{9}}$, where the Hecke conjugacy class of $\pi _{v}$ is given by $diag(\alpha _{v}, \beta _{v})$. Furthermore, if $a_{v}=\alpha _{v}+\beta _{v}$, then for every $\epsilon >0$, there are sets $T^{+}$ and $T^{-}$ of positive lower (Dirichlet) density such that $a_{v}>1.68...-\epsilon$ for all $v\in T^{+}$ and $a_{v}<-1.68...+\epsilon$ for all $v\in T^{-}$.
In ReferenceKi5, we give an example of automorphic induction for a non-normal quintic extension whose Galois closure is not solvable. In fact, the Galois group is $A_{5}$, the alternating group on five letters. The key observation, due to Ramakrishnan is that the symmetric fourth of the 2-dimensional icosahedral representation is equivalent to the 5-dimensional monomial representation of $A_{5}$ (see ReferenceBu). It should be noted that the only complete result for non-normal automorphic induction before this is for non-normal cubic extension due to ReferenceJ-PS-S2 as a consequence of the converse theorem for $\operatorname {GL}_{3}$.
We now explain the content of this paper. In Section 2, we recall a converse theorem of Cogdell and Piatetski-Shapiro and the definition of weak lift and strong lift. In Section 3, we study the analytic properties of the automorphic $L$-functions which we need for the converse theorem, namely, $L(s,\sigma \otimes \pi ,\rho _{m}\otimes \wedge ^{2}\rho _{4})$, where $\sigma$ is a cuspidal representation of $\operatorname {GL}_{m}(\mathbb{A})$,$m=1,2,3,4$, and $\pi$ is a cuspidal representation of $\operatorname {GL}_{4}(\mathbb{A})$. The automorphic $L$-functions appear in the constant term of the Eisenstein series coming from the split spin group $\operatorname {Spin}(2n)$ (the $D_{n}-3$ case in ReferenceSh3). Hence we can apply the Langlands-Shahidi method ReferenceKi1, ReferenceKi2, ReferenceKi-Sh2, ReferenceSh1–ReferenceSh3.
In Section 4, we first obtain a weak exterior square lift by applying the converse theorem to $\wedge ^{2}\pi =\bigotimes _{v} \wedge ^{2}\pi _{v}$, with $S$ being a finite set of finite places, where $\pi _{v}$ is unramified for $v<\infty$ and $v\notin S$. In this case, the situation is simpler because if $\sigma \in \mathcal{T}^{S}(m)$ as in the statement of the converse theorem, one of $\sigma _{v}$ or $\pi _{v}$ is in the principal series for $v<\infty$. Here one has to note the following: In the converse theorem, the $L$-function$L(s,\sigma _{v}\times \Pi _{v})$ is the Rankin-Selberg $L$-function defined by either integral representations ReferenceJ-PS-S or the Langlands-Shahidi method. They are the same, and they are an Artin $L$-function due to the local Langlands correspondence. However, the $L$-function$L(s,\sigma _{v}\otimes \pi _{v},\rho _{m}\otimes \wedge ^{2}\rho _{4})$ is defined by the Langlands-Shahidi method ReferenceSh1 as a normalizing factor of intertwining operators which appear in the constant term of the Eisenstein series. The equality of two $L$-functions which are defined by completely different methods is not obvious at all. The same is true for $\epsilon$-factors. Indeed, a priori we do not know the equality when $\pi _{v}$ is a supercuspidal representation, even if $\sigma _{v}$ is a character of $F_{v}^{\times }$. Hence we need to proceed in two steps as in ReferenceRa1, namely, first, we do the good case when none of $\pi _{v}$ is supercuspidal, and then we do the general case, following Ramakrishnan’s idea of descent ReferenceRa1. It is based on the observation of Henniart ReferenceHe1 that a supercuspidal representation of $\operatorname {GL}_{n}(F_{v})$ becomes a principal series after a solvable base change. Here one needs an extension of Proposition 3.6.1 of ReferenceRa1 to isobaric automorphic representations (from cuspidal automorphic representations). Appendix 1 provides the extension. We may avoid using the descent method, hence Appendix 1 altogether, by using the stability of $\gamma$-factors as in ReferenceCKPSS (see Remark 4.1 for more detail). We hope to pursue this in the future. Indeed, for the special case of the functoriality of $\wedge ^{2}(A^{3}(\pi ))$, hence the symmetric fourth of $\operatorname {GL}_{2}$, we do not need it. (See Remark 7.2.)
The converse theorem only provides a weak lift $\Pi$ which is equivalent to a subquotient of $\operatorname {Ind}|\det |^{r_{1}}\tau _{1}\otimes \cdots \otimes |\det |^{r_{k}}\tau _{k}$, where the $\tau _{i}$’s are (unitary) cuspidal representations of $\operatorname {GL}_{n_{i}}$ and $r_{i}\in \mathbb{R}$. If $\pi$ satisfies the weak Ramanujan property, it immediately implies $r_{1}=\cdots =r_{k}=0$. In general, we show that $r_{1}=\cdots =r_{k}=0$ by comparing the Hecke conjugacy classes of $\wedge ^{2}\pi$ and $\Pi$.
In Section 5.1, we give a new proof of the existence of the functorial product corresponding to the tensor product map $\operatorname {GL}_{2}(\mathbb{C})\times \operatorname {GL}_{2}(\mathbb{C})\longrightarrow \operatorname {GL}_{4}(\mathbb{C})$. It is originally due to Ramakrishnan ReferenceRa1. However, we give a proof, based entirely on the Langlands-Shahidi method. As a corollary, we obtain the Gelbart-Jacquet lift $\operatorname {Ad}(\pi )$ReferenceGe-J as an automorphic representation of $\operatorname {GL}_{3}(\mathbb{A})$ for a cuspidal representation $\pi$ of $\operatorname {GL}_{2}(\mathbb{A})$ by showing that $\pi \boxtimes \tilde{\pi }=\operatorname {Ad}(\pi )\boxplus 1$.
In Section 5.2, we construct all local lifts $\Pi _{v}$ in the sense of Definition 2.2 and show that unless $v|2,3$ and $\pi _{v}$ is a supercuspidal representation, $\Pi _{v}$ is in fact $\wedge ^{2}\pi _{v}$, the one given by the local Langlands correspondence ReferenceH-T, ReferenceHe2. Here is how it is done: Note that if $v\nmid 2$, any supercuspidal representation of $\operatorname {GL}_{4}(F_{v})$ is induced, i.e., corresponds to $\operatorname {Ind}(W_{F_{v}},W_{K},\mu )$, where $K/F_{v}$ is an extension of degree 4 (not necessarily Galois) and $\mu$ is a character of $K^{\times }$. (This is the so-called tame case. See, for example, ReferenceH, p. 179 for references.) Also thanks to Harris’ work ReferenceH, we have automorphic induction for non-Galois extensions. Namely, there exists a cuspidal representation $\pi$ which corresponds to $\operatorname {Ind}(W_{F},W_{E},\chi )$, where $E_{w}=K$,$w|v$, and $\chi$ is a grössencharacter of $E$ such that $\chi _{w}=\mu$. Likewise, if $v\nmid 2,3$, any supercuspidal representation $\sigma _{v}$ of $\operatorname {GL}_{m}(F_{v})$,$m=1,2,3,4$, is induced. We embed $\sigma _{v}$ as a local component of a cuspidal representation using automorphic induction. We can compare the functional equations of $L(s,\sigma \otimes \pi ,\rho _{m}\otimes \wedge ^{2}\rho _{4})$ and the corresponding Artin $L$-function and obtain our assertion that the local lift we constructed is equivalent to the one given by the local Langlands correspondence. (If $v|3$, we need to twist by supercuspidal representations of $\operatorname {GL}_{3}(F_{v})$, where there can be supercuspidal representations which are not induced. The global Langlands correspondence is not available for them.)
In Section 5.3, by applying the converse theorem twice to $\Pi =\bigotimes _{v} \Pi _{v}$ with $S_{1}=\{v_{1}\}$,$S_{2}=\{v_{2}\}$, where $v_{1},v_{2}$ are any finite places, we prove that $\Pi$ is an automorphic representation of $\operatorname {GL}_{6}(\mathbb{A})$.
In Section 7, we prove that if $\pi$ is a cuspidal representation of $\operatorname {GL}_{2}(\mathbb{A})$, then $A^{4}(\pi )$ is an automorphic representation of $\operatorname {GL}_{5}(\mathbb{A})$. Here we need to be careful because of the exceptional places $T$ in the discussion of the exterior square lift. We first prove that there exists an automorphic representation $\Pi$ of $\operatorname {GL}_{5}(\mathbb{A})$ such that $\Pi _{v}\simeq A^{4}(\pi _{v})$ if $v\notin T$. Next we show that this is true for $v\in T$. If $v|3$, any supercuspidal representation of $\operatorname {GL}_{2}(F_{v})$ is monomial, and hence it can be embedded into a monomial cuspidal representation of $\operatorname {GL}_{2}(\mathbb{A})$. If $v|2$, any extraordinary supercuspidal representation of $\operatorname {GL}_{2}(F_{v})$ is of tetrahedral type or octahedral type (see ReferenceG-L, p. 121). Hence in this case, the global Langlands correspondence is available ReferenceLa3, ReferenceTu. We can compare the functional equations of $L(s,\sigma \times A^{4}(\pi ))$ and the corresponding Artin $L$-function and obtain our assertion.
Finally, we emphasize that for the functoriality of $A^{4}(\pi )$, we do not need the full functoriality of the exterior square of $\operatorname {GL}_{4}$; first of all, one does not need the comparison of Hecke conjugacy classes in Section 4.1, since $A^{3}(\pi )$ satisfies the weak Ramanujan property. Secondly, one does not need the method of base change and Ramakrishnan’s descent argument (hence Appendix 1), because we can prove the equality of $\gamma$-factors for supercuspidal representations directly (see Remark 7.2 for the details).
2. Converse theorem
Throughout this paper, let $F$ be a number field, and let $\mathbb{A}=\mathbb{A}_{F}$ be the ring of adeles. We fix an additive character $\psi =\bigotimes _{v} \psi _{v}$ of $\mathbb{A}/F$. Let $\rho _{m}$ be the standard representation of $\operatorname {GL}_{m}(\mathbb{C})$.
First recall a converse theorem from ReferenceCo-PS1.
Suppose $\Pi =\bigotimes _{v} \Pi _{v}$ is an irreducible admissible representation of $\operatorname {GL}_{n}(\mathbb{A})$ such that $\omega _{\Pi }=\bigotimes _{v} \omega _{\Pi _{v}}$ is a grössencharacter of $F$. Let $S$ be a finite set of finite places, and let $\mathcal{T}^{S}(m)$ be a set of cuspidal representations of $\operatorname {GL}_{m}(\mathbb{A})$ that are unramified at all places $v\in S$. Suppose $L(s,\sigma \times \Pi )$ is nice (i.e., entire, bounded in vertical strips and satisfies a functional equation) for all cuspidal representations $\sigma \in \mathcal{T}^{S}(m)$,$m<n-1$. Then there exists an automorphic representation $\Pi '$ of $\operatorname {GL}_{n}(\mathbb{A})$ such that $\Pi _{v}\simeq \Pi _{v}'$ for all $v\notin S$.
Let $\pi =\bigotimes _{v} \pi _{v}$ be a cuspidal representation of $\operatorname {GL}_{4}(\mathbb{A})$. In order to apply the converse theorem, we need to do the following:
(1)
For all $v$, find an irreducible representation $\Pi _{v}$ of $\operatorname {GL}_{6}(F_{v})$ such that$$\begin{gather*} \gamma (s,\sigma _{v}\otimes \pi _{v},\rho _{m}\otimes \wedge ^{2}\rho _{4},\psi _{v})= \gamma (s,\sigma _{v}\times \Pi _{v},\psi _{v}),\\ L(s,\sigma _{v}\otimes \pi _{v},\rho _{m}\otimes \wedge ^{2}\rho _{4})= L(s,\sigma _{v}\times \Pi _{v}), \end{gather*}$$
for all $\sigma _{v}$, where $\sigma =\bigotimes _{v} \sigma _{v}\in \mathcal{T}^{S}(m)$,$m=1,2,3,4$.
(2)
Prove the analytic continuation and functional equation of the $L$-functions$L(s,\sigma \otimes \pi , \rho _{m}\otimes \wedge ^{2}\rho _{4})$.
(3)
Prove that $L(s,\sigma \otimes \pi , \rho _{m}\otimes \wedge ^{2}\rho _{4})$ is entire for $\sigma \in \mathcal{T}^{S}(m)$,$m=1,2,3,4$.
(4)
Prove that $L(s,\sigma \otimes \pi , \rho _{m}\otimes \wedge ^{2}\rho _{4})$ is bounded in vertical strips for $\sigma \in \mathcal{T}^{S}(m)$,$m=1,2,3,4$.
Hence the equalities of $\gamma$ and $L$-factors imply the equality of $\epsilon$-factors.
The $L$-function$L(s,\sigma \otimes \pi , \rho _{m}\otimes \wedge ^{2}\rho _{4})$ and the $\gamma$-factor$\gamma (s,\sigma _{v}\otimes \pi _{v},\rho _{m}\otimes \wedge ^{2}\rho _{4},\psi _{v})$ are available from the Langlands-Shahidi method, by considering the split spin group $\operatorname {Spin}(2n)$ with the maximal Levi subgroup $\mathbf{M}$ whose derived group is $\operatorname {SL}_{n-3}\times \operatorname {SL}_{4}$. We will study the analytic properties of the $L$-functions in the next section; (2) is well known by Shahidi’s work ReferenceSh3; (4) is the result of ReferenceGe-Sh. We will especially study (3); in general, the $L$-functions$L(s,\sigma \otimes \pi , \rho _{m}\otimes \wedge ^{2}\rho _{4})$ may not be entire. Our key idea is to apply the converse theorem to the twisting set $\mathcal{T}^{S}(m)\otimes \chi$, where $\chi _{v}$ is highly ramified for $v\in S$. Then for $\sigma \in \mathcal{T}^{S}(m)\otimes \chi$, the $L$-function$L(s,\sigma \otimes \pi , \rho _{m}\otimes \wedge ^{2}\rho _{4})$ is entire. Observe that $L(s,(\sigma \otimes \chi )\times \Pi )=L(s,\sigma \times (\Pi \otimes \chi ))$. Hence applying the converse theorem with the twisting set $\mathcal{T}^{S}(m)\otimes \chi$ is equivalent to applying the converse theorem for $\Pi \otimes \chi$ with the twisting set $\mathcal{T}^{S}(m)$ (see ReferenceCo-PS2).
We will address problem (1) in Section 4. We have a natural candidate for $\Pi _{v}$, namely, $\wedge ^{2}\pi _{v}$, the one given by the local Langlands correspondence (see Section 4 for the detail). However, proving the equalities in (1) is not so obvious due to the fact that two $L$-functions on the left and on the right are defined in completely different manners. The right-hand side is the Rankin-Selberg $L$-functionReferenceJ-PS-S defined by either integral representations or the Langlands-Shahidi method, which in turn is an Artin $L$-function due to the local Langlands correspondence. We note that if $\Pi _{v}$ is not generic, then we write $\Pi _{v}$ as a Langlands quotient of an induced representation $\Xi _{v}$, which is generic, and we define the $\gamma$- and $L$-factors$\gamma (s,\sigma _{v}\times \Pi _{v},\psi _{v})=\gamma (s,\sigma _{v}\times \Xi _{v},\psi _{v})$ and $L(s,\sigma _{v}\times \Pi _{v})=L(s,\sigma _{v}\times \Xi _{v})$.
The left-hand side is defined in the Langlands-Shahidi method ReferenceSh1 as a normalizing factor of intertwining operators which appear in the constant term of the Eisenstein series. Proving (1) is equivalent to the fact that Shahidi’s $L$-functions and $\gamma$-factors on the left are those of Artin factors. It is clearly true if $\sigma _{v}\otimes \pi _{v}$ is unramified. Shahidi has shown that (1) is true when $v=\infty$ReferenceSh7.
Remark 2.1
Eventually we are going to prove in Section 5 that $\Pi _{v}$ on the right side of (1) is generic in our case. However, $\Pi _{v}$ is not generic in general. For example, if $\pi _{v}$ is given by the principal series $\operatorname {Ind}_{B}^{\operatorname {GL}_{4}}\, |\ |^{\frac {1}{4}}\otimes |\ |^{\frac {1}{4}}\otimes |\ |^{-\frac {1}{4}}\otimes |\ |^{-\frac {1}{4}}$, then $\Pi _{v}=\wedge ^{2}\pi _{v}$ is the unique quotient of $\operatorname {Ind}_{B}^{\operatorname {GL}_{6}}\, |\ |^{\frac {1}{2}}\otimes |\ |^{-\frac {1}{2}}\otimes 1\otimes 1\otimes 1\otimes 1$, namely, $\operatorname {Ind}_{\operatorname {GL}_{2}\times \operatorname {GL}_{1}\times \operatorname {GL}_{1}\times \operatorname {GL}_{1}\times \operatorname {GL}_{1}}^{\operatorname {GL}_{6}} |\det |\otimes 1\otimes 1\otimes 1\otimes 1$. Hence in the course of applying the converse theorem, we need to deal with such non-generic representations on the right side of (1). However, in the definition of Shahidi’s $\gamma$- and $L$-factors on the left side of (1), we only deal with generic representations, since any local components of a cuspidal representation of $\operatorname {GL}_{n}(\mathbb{A})$ are generic. By a well-known result, any generic representation of $\operatorname {GL}_{n}(F_{v})$ is always a full induced representation.
We were not able to prove (1) for $\Pi _{v}=\wedge ^{2}\pi _{v}$ when $v|2,3$ and $\pi _{v}$ is a supercuspidal representation of $\operatorname {GL}_{4}(F_{v})$. Hence we make the following definition.
Definition 2.2
Let $\pi =\bigotimes _{v} \pi _{v}$ be a cuspidal representation of $\operatorname {GL}_{4}(\mathbb{A})$. We say that an automorphic representation $\Pi$ of $\operatorname {GL}_{6}(\mathbb{A})$ is a strong exterior square lift of $\pi$ if for every $v$,$\Pi _{v}$ is a local lift of $\pi _{v}$ in the sense that
for all generic irreducible representations $\sigma _{v}$ of $\operatorname {GL}_{m}(F_{v})$,$1\leq m\leq 4$.
If the above equality holds for almost all $v$, then $\Pi$ is called weak lift of $\pi$.
In Section 4, we apply the converse theorem with $S$ being a finite set of finite places such that $\pi _{v}$ is unramified for $v\notin S$,$v<\infty$. Then if $\pi _{v}$ is ramified, the local components of the twisting representations at $S$ are unramified and hence the equalities in (1) become simpler. In this way, we first find a weak lift in Section 4 and use it to define all local lifts in Section 5 and to obtain the strong lift.
We record the following proposition which is very useful in proving (1).
Let $\sigma _{1v}$ ($\sigma _{2v}$, resp.) be an irreducible generic admissible representation of $\operatorname {GL}_{k}(F_{v})$ ($\operatorname {GL}_{l}(F_{v})$, resp.) with parametrization $\phi _{i}: W_{F_{v}}\times \operatorname {SL}_{2}(\mathbb{C})\longrightarrow \operatorname {GL}_{k}(\mathbb{C})$ ($\operatorname {GL}_{l}(\mathbb{C})$, resp.) by the local Langlands correspondence ReferenceH-T, ReferenceHe2. Let $L(s,\phi _{1}\otimes \phi _{2})$ be the Artin $L$-function; let $L_{1}(s,\sigma _{1v}\times \sigma _{2v})$ be the Rankin-Selberg $L$-function defined by integral representation ReferenceJ-PS-S; and let $L_{2}(s,\sigma _{1v}\times \sigma _{2v})$ be the Langlands-Shahidi $L$-function defined as a normalizing factor for intertwining operators ReferenceSh1. Then we have the equality
We have similar equalities for $\gamma$- and $\epsilon$-factors.
Proof.
The equality $L(s,\phi _{1}\otimes \phi _{2})=L_{1}(s,\sigma _{1v}\times \sigma _{2v})$ is the local Langlands correspondence (the work of Harris-Taylor ReferenceH-T and Henniart ReferenceHe2 on $p$-adic places and of Langlands ReferenceLa4 on archimedean places). Similar equalities hold for $\gamma$- and $\epsilon$-factors.
The equality $L_{1}(s,\sigma _{1v}\times \sigma _{2v})=L_{2}(s,\sigma _{1v}\times \sigma _{2v})$ is due to Shahidi (ReferenceSh7 for archimedean places and ReferenceSh4, Theorem 5.1 for $p$-adic places; see ReferenceSh6, p 282 for the explanation of why the constant $\omega _{2}^{m}(-1)$ disappears). Similar equalities hold for $\gamma$- and $\epsilon$-factors.
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For the sake of completeness, we recall how $L$- and $\epsilon$-factors are defined from the Langlands-Shahidi method ReferenceSh1, Section 7. Let $\mathbf{G}$ be a quasi-split reductive group defined over a number field $F$. Let $\mathbf{M}$ be a maximal Levi subgroup. Let $\pi$ be a generic cuspidal representation of $\mathbf{M}(\mathbb{A})$. From the theory of local coefficients, which come from intertwining operators, a $\gamma$-factor$\gamma (s,\pi _{v},r_{i},\psi _{v})$ is defined for every generic irreducible admissible representation $\pi _{v}$ and certain finite-dimensional representation $r_{i}$’s. If $\pi _{v}$ is tempered, $L(s,\pi _{v},r_{i})$ is defined to be
where $P_{\pi _{v},i}$ is the unique polynomial satisfying $P_{\pi _{v},i}(0)=1$ such that $P_{\pi _{v},i}(q_{v}^{-s})$ is the numerator of $\gamma (s,\pi _{v},r_{i},\psi _{v})$. We define the $\epsilon$-factor using the identity $\gamma (s,\pi _{v},r_{i},\psi _{v})=\epsilon (s,\pi _{v},r_{i},\psi _{v}) \frac {L(1-s,\tilde{\pi }_{v},r_{i})}{L(s,\pi _{v},r_{i})}.$ Hence if $\pi _{v}$ is tempered, then the $\gamma$-factor canonically defines both the $L$-factor and the $\epsilon$-factor. If $\pi _{v}$ is non-tempered, write it as a Langlands quotient of an induced representation and we can write the corresponding intertwining operator as a product of rank-one operators. For these rank-one operators, there correspond $\gamma$- and $L$-factors and we define $\gamma (s,\pi _{v},\psi _{v})$ and $L(s,\pi _{v},r_{i})$ to be the product of these rank-one $\gamma$- and $L$-factors. We then define $\epsilon$-factor to satisfy the above relation.
Recall the multiplicativity of $\gamma$-factors (cf. ReferenceSh7). We suppress the subscript $v$ until the end of Section 2. Let $\pi$ be an irreducible generic admissible representation of $M=\mathbf{M}(F)$. Suppose $\pi \subset \operatorname {Ind}_{M_{\theta } N_{\theta }}^{M}\, \sigma \otimes 1$, where $M_{\theta }N_{\theta }$,$\theta \subset \Delta$, is a parabolic subgroup of $M$ and $\sigma$ is an irreducible generic admissible representation of $M_{\theta }$. Let $\theta '=w(\theta )\subset \Delta$ and fix a reduced decomposition $w=w_{n-1}\cdots w_{1}$ of $w$ as in ReferenceSh2, Lemma 2.1.1. Then for each $j$, there exists a unique root $\alpha _{j}\in \Delta$ such that $w_{j}(\alpha _{j})<0$. For each $j$,$2\leq j\leq n-1$, let $\bar{w}_{j}=w_{j-1}\cdots w_{1}$. Set $\bar{w}_{1}=1$. Also let $\Omega _{j}=\theta _{j}\cup \{\alpha _{j}\}$, where $\theta _{1}=\theta$,$\theta _{n}=\theta '$, and $\theta _{j+1}=w_{j}(\theta _{j})$,$1\leq j\leq n-1$. Then the group $M_{\Omega _{j}}$ contains $M_{\theta _{j}}N_{\theta _{j}}$ as a maximal parabolic subgroup and $w_{j}(\sigma )$ is a representation of $M_{\theta _{j}}$. The $L$-group${}^{L} M_{\theta }$ acts on $V_{i}$. Given an irreducible constituent of this action, there exists a unique $j$,$1\leq j\leq n-1$, which under $w_{j}$ is equivalent to an irreducible constituent of the action of ${}^{L} M_{\theta _{j}}$ on the Lie algebra of ${}^{L} N_{\theta _{j}}$. We denote by $i(j)$ the index of this subspace of the Lie algebra of ${}^{L} N_{\theta _{j}}$. Finally, let $S_{i}$ denote the set of all such $j$’s where $S_{i}$, in general, is a proper subset of $1\leq j\leq n-1$.
Proposition 2.4 (ReferenceSh1, (3.13) (multiplicativity of $\gamma$-factors)).
For each $j\in S_{i}$ let $\gamma (s,w_{j}(\sigma ),r_{i(j)},\psi )$ denote the corresponding factor. Then
We follow the exposition in ReferenceSh6, p. 280. Let $\phi : W_{F}\times \operatorname {SL}_{2}(\mathbb{C})\longrightarrow {}^{L} M$ be the parametrization of $\pi$. Then $\phi$ factors through ${}^{L} M_{\theta }$, i.e., there exists $\phi ': W_{F}\times \operatorname {SL}_{2}(\mathbb{C}) \longrightarrow {}^{L} M_{\theta }$ such that $\phi =i\circ \phi '$, where $i: {}^{L} M_{\theta }\hookrightarrow {}^{L} M$. Let $r_{i}'=r_{i}|_{{}^{L} M_{\theta }}$. Then $r_{i}'=\bigoplus _{j} r_{i(j)}$, and
Given an irreducible component of $r_{i}|_{{}^{L} M_{\theta }}$, there exists a unique $j$, which under $w_{j}$ makes this component equivalent to an irreducible constituent of the action of ${}^{L} M_{\theta _{j}}$ on the Lie algebra of ${}^{L} N_{\theta _{j}}$. Hence we have
Proposition 2.5
Let $\pi ,\sigma$ be as in Proposition 2.4. Suppose $\pi$ is tempered and $\gamma (s,w_{j}(\sigma ),r_{i(j)},\psi )$ is an Artin factors for each $j\in S_{i}$, namely, $\gamma (s,w_{j}(\sigma ),r_{i(j)},\psi ) =\gamma (s,\phi ',r_{i(j)},\psi )$ for each $j$. Then $\gamma (s,\pi ,r_{i},\psi )$ and $L(s,\pi ,r_{i})$ are also Artin factors.
Proof.
Clear from the multiplicativity formulas. Since $\pi$ is tempered, $\gamma$-factors determine the $L$-factors uniquely.
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Because of Proposition 2.5, we are reduced to the supercuspidal case when verifying that Shahidi’s $\gamma$- and $L$-factors are Artin factors. Later on, in many situations, all the rank-one factors in Proposition 2.5 are the Rankin-Selberg $\gamma$- and $L$-factors for $\operatorname {GL}_{n}\times \operatorname {GL}_{m}$, and by Proposition 2.3, they are Artin factors.
Proposition 2.6 (multiplicativity of $L$-factors).
Suppose $\pi ,\sigma$ to be as in Proposition 2.4. Suppose $\pi$ is tempered and $\sigma$ is a discrete series. Suppose Conjecture 7.1 of ReferenceSh1 is valid for every $L(s,w_{j}(\sigma ),r_{i(j)})$,$j\in S_{i}$. Then
Now let $\pi$ be a non-tempered irreducible generic admissible representation of $M=M(F_{v})$. Then $\pi$ is the unique quotient of an induced representation $\operatorname {Ind}_{M_{\theta }N_{\theta }}^{M}\, \sigma \otimes 1$, where $M_{\theta }N_{\theta }$,$\theta \subset \Delta$, is a parabolic subgroup of $M$ and $\sigma$ is an irreducible generic quasi-tempered representation of $M_{\theta }$. (In many cases when the standard module conjecture is known, $\pi =\operatorname {Ind}_{M_{\theta }N_{\theta }}^{M}\, (\sigma \otimes 1$).) Then by the definition of $L$-factors,
In the multiplicativity of $\gamma$-factors (Proposition 2.4), we realized $\pi$ as a subrepresentation of an induced representation. On the other hand, in the above, $\pi$ is realized as a quotient. However, this does not matter, since local coefficients of two equivalent representations are the same.
Remark 2.3
Even though it is not necessary, we remark that we can define $L(s,\pi ,r_{i})$, even when $\pi$ is non-generic as long as it has generic supercuspidal support. Write $\pi$ as the Langlands quotient of $\Xi =\operatorname {Ind}_{M_{\theta }N_{\theta }}^{M}\, \sigma \otimes 1$. Just define $\gamma (s,\pi ,r_{i},\psi )=\gamma (s,\Xi ,r_{i},\psi )$ using the formula in Proposition 2.4, and define $L(s,\pi ,r_{i})$ using the formula in Proposition 2.5. These definitions agree with those of the Rankin-Selberg $\gamma$- and $L$-factors in the sense of ReferenceJ-PS-S (see the paragraph before Remark 2.1), and hence Proposition 2.3 holds without the genericity condition.
For example, let $\pi _{v}=\mu \circ \det$ be a character of $\operatorname {GL}_{2}(F_{v})$, which is the Langlands quotient of $\operatorname {Ind}\mu |\ |^{\frac {1}{2}}\otimes \mu |\ |^{-\frac {1}{2}}$. Then the standard $L$-function$L(s,\pi _{v})$ is obtained by considering the induced representation $\operatorname {Ind}_{\operatorname {GL}_{2}\times \operatorname {GL}_{1}}^{\operatorname {GL}_{3}}\, \pi _{v}|\det |^{\frac {s}{2}}\otimes |\ |^{-\frac {s}{2}}$, which is a quotient of $\operatorname {Ind}_{B}^{\operatorname {GL}_{3}}\, \mu |\ |^{\frac {1}{2}+\frac {s}{2}}\otimes \mu |\ |^{-\frac {1}{2}+\frac {s}{2}}\otimes |\ |^{-\frac {s}{2}}$. Hence $\gamma (s,\pi _{v},\psi _{v})= \gamma (s+\frac {1}{2},\mu , \psi _{v})\gamma (s-\frac {1}{2},\mu ,\psi _{v})$, and $L(s,\pi _{v})= L(s+\frac {1}{2},\mu )L(s-\frac {1}{2},\mu )$ if $\mu$ is unramified. On the other hand, if $\sigma _{v}$ is the Steinberg representation, which is the subrepresentation of $\operatorname {Ind}\mu |\ |^{\frac {1}{2}} \otimes \mu |\ |^{-\frac {1}{2}}$, then $\gamma (s,\sigma _{v},\psi _{v})=\gamma (s,\pi _{v},\psi _{v})$. However, by the definition of the $L$-factor, there is a cancellation, and $L(s,\sigma _{v})=L(s+\frac {1}{2},\mu )$.
3. Analytic properties of the $L$-functions
Consider the $D_{n}-3$ case in ReferenceSh3, $n=4,5,6,7$: Let $\mathbf{G}=\operatorname {Spin}(2n)$ be the split spin group. It is, up to isomorphism, the unique simply-connected group of type $D_{n}$. We can think of it as a two-fold covering group of $SO(2n)$, namely, there is a 2 to 1 map $\phi : \operatorname {Spin}(2n)\longrightarrow SO(2n)$. Let $\mathbf{T}$ be a maximal torus of $\mathbf{G}$.
Let $\theta =\{\alpha _{1}=e_{1}-e_{2},...,\alpha _{n-4}=e_{n-4}-e_{n-3}, \alpha _{n-2}=e_{n_{2}}-e_{n-1}, \alpha _{n-1}=e_{n-1}-e_{n}, \alpha _{n}=e_{n-1}+e_{n}\}=\Delta -\{\alpha _{n-3}\}$. Let $\mathbf{T}\subset \mathbf{M}_{\theta }=\mathbf{M}$ be the Levi subgroup of $\mathbf{G}$ generated by $\theta$, and let $\mathbf{P}=\mathbf{M}\mathbf{N}$ be the corresponding standard parabolic subgroup of $\mathbf{G}$. Let $\mathbf{A}$ be the connected component of the center of $\mathbf{M}$:
Since $\mathbf{G}$ is simply connected, the derived group $\mathbf{M}_{D}$ of $\mathbf{M}$ is simply connected, and hence $\mathbf{M}_{D}\simeq \operatorname {SL}_{n-3}\times \operatorname {SL}_{4}$. Then
Under the identification $\mathbf{M}_{D}\simeq \operatorname {SL}_{n-3}\times \operatorname {SL}_{4}$,$H_{\alpha _{1}}(t)H_{\alpha _{2}}(t^{2})\cdots H_{\alpha _{n-4}}(t^{n-4})$ is an element in $\operatorname {SL}_{n-3}$, and $H_{\alpha _{n-1}}(t)H_{\alpha _{n-2}}(t^{2})H_{\alpha _{n}}(t)$ is an element in $\operatorname {SL}_{4}$. Using this, it is easy to see that
We note that it is independent of the choices of the roots of unity which show up.
Let $\sigma ,\pi$ be cuspidal representations of $\operatorname {GL}_{n-3}(\mathbb{A}), \operatorname {GL}_{4}(\mathbb{A})$ with central characters $\omega _{1}, \omega _{2}$, resp. Let $\Sigma$ be a cuspidal representation of $\mathbf{M}(\mathbb{A})$, induced by $f$ and $\sigma ,\pi$. (More precisely,Footnote^{1} Thanks are due to Prof. Shahidi who pointed this out.^{✖} we need to proceed in the following way: $\mathbf{M}(\mathbb{A}) \mathbb{A}^{*}$ is co-compact in $\operatorname {GL}_{n-3}(\mathbb{A}) \times \operatorname {GL}_{4}(\mathbb{A})$, where $\mathbb{A}^{*}$ is embedded as the center of, say, the first factor. Consequently $\sigma \otimes \pi |_{f(M)},\ M=\mathbf{M}(\mathbb{A})$, decomposes to a direct sum of irreducible cuspidal representations of $M$. Let $\Sigma$ be any irreducible constituent of this direct sum. As we shall see, its choice is irrelevant.)
Let $\Sigma _{v}$ be the unramified representation of $\mathbf{M}(F_{v})$, given by $\sigma _{v}, \pi _{v}$’s. Then $\Sigma _{v}$ is induced from the character $\chi$ of the torus. We have
For ramified places, let $L(s,\Sigma _{v},r_{1})$ and $L(s,\Sigma _{v},r_{2})$ be the ones defined in ReferenceSh1, Section 7. Observe that in particular, if $v=\infty$, then $L(s,\pi _{v},r_{i})$ is the corresponding Artin $L$-function (cf. ReferenceSh7) in each case.
Let $I(s,\Sigma _{v})$ be the induced representation, and let $N(s,\Sigma _{v},w_{0})$ be the normalized local intertwining operator ReferenceKi1, (2.1):
where $A(s,\Sigma _{v},w_{0})$ is the unnormalized intertwining operator. In ReferenceKi4, we showed that $N(s,\Sigma _{v},w_{0})$ is holomorphic and non-zero for $\operatorname {Re}(s)\geq \frac {1}{2}$ for all $v$. For the sake of completeness, we give a proof.
Proposition 3.1
The normalized local intertwining operators $N(s,\Sigma _{v},w_{0})$ are holomorphic and non-zero for $\operatorname {Re}(s)\geq \frac {1}{2}$ for all $v$.
Proof.
We proceed as in ReferenceKi2, Proposition 3.4. If $\Sigma _{v}$ is tempered, then the unnormalized operators are holomorphic and non-zero for $\operatorname {Re}(s)>0$. We only need to verify Conjecture 7.1 of ReferenceSh1, namely, $L(s,\Sigma _{v},r_{i})$ is holomorphic for $\operatorname {Re}(s)>0$: for archimedean places, $L(s,\Sigma _{v},r_{i})$ is an Artin $L$-function, and hence our assertion follows. For $p$-adic places, by the multiplicativity of $L$-factors (Proposition 2.6), $L(s,\Sigma _{v},r_{i})$ is a product of rank-one $L$-functions for discrete series. The rank-one factors are Rankin-Selberg $L$-functions for $\operatorname {GL}_{k}\times \operatorname {GL}_{l}$, and the cases $D_{n}-2$ and $D_{n}-3$. The first two cases are well known (ReferenceSh1, Proposition 7.2). The $D_{n}-3$ case is a result of ReferenceAs.
If $\Sigma _{v}$ is non-tempered, we write $I(s,\Sigma _{v})$ as in ReferenceKi1, p. 841,
where $\pi _{0}$ is a tempered representation of $\mathbf{M}_{0}(F_{v})$ and $\mathbf{P}_{0}=\mathbf{M}_{0}\mathbf{N}_{0}$ is another parabolic subgroup of $\mathbf{G}$. We can identify the normalized operator $N(s,\Sigma _{v},w_{0})$ with the normalized operator $N(s\tilde{\alpha }+\Lambda _{0},\pi _{0},w_{0})$, which is a product of rank-one operators attached to tempered representations (cf. ReferenceZh, Proposition 1).
Here $\tilde{\alpha }=e_{1}+\cdots +e_{n-3}$;$\Lambda _{0}=r_{1}e_{1}+r_{2}e_{2}+\cdots + (-r_{2})e_{n-4}+(-r_{1})e_{n-3}+ (r_{1}'+r_{2}')e_{n-2}+(r_{1}'-r_{2}')e_{n-1}$, where $\frac {1}{2}>r_{1}\geq \cdots \geq r_{[\frac {n-3}{2}]}\geq 0$,$\frac {1}{2}> r_{1}'\geq r_{2}'\geq 0$. Here $r_{i}=0$ if $\pi _{1v}$ is tempered. The same is true for $\pi _{2v}$. Hence
All the rank-one operators are operators attached to tempered representations of a parabolic subgroup whose Levi subgroup has a derived group isomorphic to $\operatorname {SL}_{k}\times \operatorname {SL}_{l}$ inside a group whose derived group is $\operatorname {SL}_{k+l}$, unless $r_{1}'=r_{2}'\ne 0$, in which case the rank-one operator is for $D_{k}-2$. It is the case when $\pi _{2}'=\operatorname {Ind}|\det |^{r'}\rho \otimes |\det |^{-r'}\rho$, where $\rho$ is a tempered representation of $\operatorname {GL}_{2}$.
In the first case, the operators are restrictions to $\operatorname {SL}_{k+l}$ of corresponding standard operators for $\operatorname {GL}_{k+l}$. By ReferenceM-W2, Proposition I.10 one knows that these rank-one operators are holomorphic for $\operatorname {Re}(s)>-1$. Hence by identifying roots of $G$ with respect to a parabolic subgroup with those of $G$ with respect to the maximal torus, it is enough to check $\operatorname {Re}(\langle s\tilde{\alpha }+\Lambda _{0},\beta ^{\vee }\rangle )>-1$ for all positive roots $\beta$ if $\operatorname {Re}(s)\geq \frac {1}{2}$. We observed that the least value of $\operatorname {Re}(\langle s\tilde{\alpha }+ \Lambda _{0},\beta ^{\vee }\rangle )$ is $\operatorname {Re}(s)-r_{1}-(r_{1}'+r_{2}')$ which is larger than $-1$, if $\operatorname {Re}(s)\geq \frac {1}{2}$.
Now suppose we are in the exceptional case, namely, $\pi _{2}'=\operatorname {Ind}|\det |^{r'}\rho \otimes |\det |^{-r'}\rho$, where $\rho$ is a tempered representation of $\operatorname {GL}_{2}$. Then by direct computation, we see that $N(s\tilde{\alpha }+\Lambda _{0},\pi _{0},\tilde{w})$ is a product of the following three operators:
where $s\tilde{\alpha }'+\Lambda _{0}'=(s+r_{1})e_{1}+\cdots + (s-r_{1})e_{n-3}$ and $\omega _{\rho }$ is the central character of $\rho$. The first operator is the operator for $D_{k}-2$ and it is in the corresponding positive Weyl chamber and is holomorphic for $\operatorname {Re}(s)\geq \frac {1}{2}$ (ReferenceKi1, Lemma 2.4). The last two operators are the operators for $\operatorname {GL}_{k}\times \operatorname {GL}_{1}$. Since $\operatorname {Re}(s-2r'-r_{1})>-1$ if $\operatorname {Re}(s)\geq \frac {1}{2}$, they are holomorphic. Consequently, $N(s\tilde{\alpha }+\Lambda _{0},\pi _{0},\tilde{w}_{0})$ is holomorphic for $\operatorname {Re}(s)\geq \frac {1}{2}$. By Zhang’s lemma (cf. ReferenceKi2, Lemma 1.7, ReferenceZh), it is non-zero as well.
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We recall some general results in the next two propositions. Let $\mathbf{G}$ be a quasi-split group defined over a number field $F$, and let $\mathbf{P}=\mathbf{M}\mathbf{N}$ be a maximal parabolic subgroup over $F$. Let $\Sigma$ be a cuspidal representation of $\mathbf{M}(\mathbb{A})$.
If $w_{0}\Sigma \ncong \Sigma$,$\prod _{i=1}^{m} L_{S}(1+is,\Sigma ,r_{i})$ has no zeros for $\operatorname {Re}(s)>0$.
Remark 3.1
Since the Eisenstein series $E(s,f,g,P)$ is holomorphic for $\operatorname {Re}(s)=0$, we see that $\prod _{i=1}^{m} L_{S}(1+ is,\Sigma ,r_{i})$ has no zeros for $\operatorname {Re}(s)=0$ either. Since the local $L$-functions$L(s,\Sigma _{v},r_{i})$ have no zeros, the completed $L$-function$\prod _{i=1}^{m} L(1+ is,\Sigma ,r_{i})$ has no zeros for $\operatorname {Re}(s)\geq 0$.
Let $S$ be a finite set of finite places where $\pi _{v}$ is unramified if $v<\infty$ and $v\notin S$. Fix $\chi$ be a grössencharacter of $F$ such that $\chi _{v}$ is highly ramified for at least one $v\in S$. Let $\Sigma _{\chi }$ be the cuspidal representation of $\mathbf{M}(\mathbb{A})$, induced by the map $f: \mathbf{M}\longrightarrow \operatorname {GL}_{n-3}\times \operatorname {GL}_{4}$ and $\sigma \otimes \chi ,\pi$. Then the central character of $\Sigma _{\chi }$ is
Note that $w_{0}(\omega _{\Sigma _{\chi }})=\omega _{\Sigma _{\chi }}^{-1}$. Hence if $\chi _{v}$ is highly ramified (say, $\chi _{v}^{24}$ is ramified), then