American Mathematical Society

Functoriality for the exterior square of upper G upper L 4 and the symmetric fourth of upper G upper L 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 upper G upper L 4 as automorphic forms on upper G upper L 6 and the symmetric fourth of cusp forms on upper G upper L 2 as automorphic forms on upper G upper L 5 . We prove these by applying a converse theorem of Cogdell and Piatetski-Shapiro to analytic properties of certain upper L -functions obtained by the Langlands-Shahidi method. We give several applications: First, we prove the weak Ramanujan property of cuspidal representations of upper G upper L 4 and the absolute convergence of the exterior square upper L -functions of upper G upper L 4 . Second, we prove that the fourth symmetric power upper L -functions of cuspidal representations of upper G upper L 2 are entire, except for those of dihedral and tetrahedral type. Third, we prove the bound three-twenty-sixths for Hecke eigenvalues of Maass forms over any number field.

1. Introduction

Let logical-and Superscript 2 Baseline colon upper G upper L Subscript n Baseline left-parenthesis double-struck upper C right-parenthesis long right-arrow upper G upper L Subscript upper N Baseline left-parenthesis double-struck upper C right-parenthesis , where upper N equals StartFraction n left-parenthesis n minus 1 right-parenthesis Over 2 EndFraction , be the map given by the exterior square. Then Langlands’ functoriality predicts that there is a map from cuspidal representations of upper G upper L Subscript n to automorphic representations of upper G upper L Subscript upper N , which satisfies certain canonical properties. To explain, let upper F be a number field, and let double-struck upper A be its ring of adeles. Let pi equals circled-times Underscript v Endscripts pi Subscript v be a cuspidal (automorphic) representation of upper G upper L Subscript n Baseline left-parenthesis double-struck upper A right-parenthesis . In what follows, a cuspidal representation always means a unitary one. Now by the local Langlands correspondence, logical-and pi Subscript v is well defined as an irreducible admissible representation of upper G upper L Subscript upper N Baseline left-parenthesis upper F Subscript v Baseline right-parenthesis 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 logical-and pi equals circled-times Underscript v Endscripts logical-and pi Subscript v . It is an irreducible admissible representation of upper G upper L Subscript upper N Baseline left-parenthesis double-struck upper A right-parenthesis . Then Langlands’ functoriality in this case is equivalent to the fact that logical-and pi is automorphic.

Note that logical-and left-parenthesis upper G upper L 2 left-parenthesis double-struck upper C right-parenthesis right-parenthesis asymptotically-equals upper G upper L 1 left-parenthesis double-struck upper C right-parenthesis and in fact for a cuspidal representation pi of upper G upper L 2 left-parenthesis double-struck upper A right-parenthesis , logical-and pi equals omega Subscript pi , the central character of pi . Furthermore, logical-and left-parenthesis upper G upper L 3 left-parenthesis double-struck upper C right-parenthesis right-parenthesis asymptotically-equals upper G upper L 3 left-parenthesis double-struck upper C right-parenthesis . In this case, given a cuspidal representation pi of upper G upper L 3 left-parenthesis double-struck upper A right-parenthesis , logical-and pi equals pi overTilde circled-times omega Subscript pi , where pi overTilde is the contragredient of pi .

In this paper, we look at the case n equals 4 . Let pi equals circled-times Underscript v Endscripts pi Subscript v be a cuspidal representation of upper G upper L 4 left-parenthesis double-struck upper A right-parenthesis . What we prove is weaker than the automorphy of logical-and pi . We prove (Theorem 5.3.1)

Theorem A

Let upper T be the set of places where v vertical-bar 2 comma 3 and pi Subscript v is a supercuspidal representation. Then there exists an automorphic representation normal upper Pi of upper G upper L 6 left-parenthesis double-struck upper A right-parenthesis such that normal upper Pi Subscript v Baseline asymptotically-equals logical-and pi Subscript v if v not-an-element-of upper T . Moreover, normal upper Pi is of the form upper I n d tau 1 circled-times ellipsis circled-times tau Subscript k , where the tau Subscript i ’s are all cuspidal representations of upper G upper L Subscript n Sub Subscript i Baseline left-parenthesis double-struck upper A right-parenthesis .

The reason why we have the exceptional places upper T , especially for v vertical-bar 2 , is due to the fact that supercuspidal representations of upper G upper L 4 left-parenthesis upper F Subscript v Baseline right-parenthesis are very complicated when v vertical-bar 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 upper G upper L 4 left-parenthesis double-struck upper A right-parenthesis (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 upper G upper L 2 left-parenthesis double-struck upper A right-parenthesis as an automorphic representation of upper G upper L 5 left-parenthesis double-struck upper A right-parenthesis . More precisely, let upper G upper L 2 left-parenthesis double-struck upper C right-parenthesis long right-arrow upper G upper L Subscript m plus 1 Baseline left-parenthesis double-struck upper C right-parenthesis be the symmetric m th power (the m plus 1 -dimensional irreducible representation of upper G upper L 2 left-parenthesis double-struck upper C right-parenthesis on symmetric tensors of rank m ). Let pi equals circled-times Underscript v Endscripts pi Subscript v be a cuspidal representation of upper G upper L 2 left-parenthesis double-struck upper A right-parenthesis with central character omega Subscript pi . By the local Langlands correspondence, upper S y m Superscript m Baseline left-parenthesis pi Subscript v Baseline right-parenthesis is well defined for all v . Hence Langlands’ functoriality predicts that upper S y m Superscript m Baseline left-parenthesis pi right-parenthesis equals circled-times Underscript v Endscripts upper S y m Superscript m Baseline left-parenthesis pi Subscript v Baseline right-parenthesis is an automorphic representation of upper G upper L Subscript m plus 1 Baseline left-parenthesis double-struck upper A right-parenthesis . Gelbart and Jacquet ReferenceGe-J proved that upper S y m squared left-parenthesis pi right-parenthesis is an automorphic representation of upper G upper L 3 left-parenthesis double-struck upper A right-parenthesis . We proved in ReferenceKi-Sh2 that upper S y m cubed left-parenthesis pi right-parenthesis is an automorphic representation of upper G upper L 4 left-parenthesis double-struck upper A right-parenthesis as a consequence of the functorial product upper G upper L 2 times upper G upper L 3 long right-arrow upper G upper L 6 , corresponding to the tensor product map upper G upper L 2 left-parenthesis double-struck upper C right-parenthesis times upper G upper L 3 left-parenthesis double-struck upper C right-parenthesis long right-arrow upper G upper L 6 left-parenthesis double-struck upper C right-parenthesis .

We prove (Theorem 7.3.2)

Theorem B

upper S y m Superscript 4 Baseline left-parenthesis pi right-parenthesis is an automorphic representation of upper G upper L 5 left-parenthesis double-struck upper A right-parenthesis . If upper S y m cubed left-parenthesis pi right-parenthesis is cuspidal, upper S y m Superscript 4 Baseline left-parenthesis pi right-parenthesis is either cuspidal or induced from cuspidal representations of upper G upper L 2 left-parenthesis double-struck upper A right-parenthesis and upper G upper L 3 left-parenthesis double-struck upper A right-parenthesis .

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 upper S y m cubed left-parenthesis pi right-parenthesis circled-times omega Subscript pi Superscript negative 1 . For simplicity, we write upper A Superscript m Baseline left-parenthesis pi right-parenthesis equals upper S y m Superscript m Baseline left-parenthesis pi right-parenthesis circled-times omega Subscript pi Superscript negative 1 . We prove that

logical-and left-parenthesis upper A cubed left-parenthesis pi right-parenthesis right-parenthesis equals upper A Superscript 4 Baseline left-parenthesis pi right-parenthesis squared-plus omega Subscript pi Baseline period

This implies that upper A Superscript 4 Baseline left-parenthesis pi right-parenthesis is an automorphic representation of upper G upper L 5 left-parenthesis double-struck upper A right-parenthesis , and so is upper S y m Superscript 4 Baseline left-parenthesis pi right-parenthesis .

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 upper G upper L 2 left-parenthesis double-struck upper A right-parenthesis . Let pi Subscript v be a local (finite or infinite) spherical component, given by pi Subscript v Baseline equals upper I n d left-parenthesis StartAbsoluteValue EndAbsoluteValue Subscript v Superscript s Super Subscript 1 v Superscript Baseline comma StartAbsoluteValue EndAbsoluteValue Subscript v Superscript s Super Subscript 2 v Superscript Baseline right-parenthesis . Then StartAbsoluteValue upper R e left-parenthesis s Subscript i v Baseline right-parenthesis EndAbsoluteValue less-than-or-equal-to three-twenty-sixths period If upper F equals double-struck upper Q and v equals normal infinity , this condition implies that lamda 1 greater-than-or-equal-to StartFraction 40 Over 169 EndFraction almost-equals 0.237 , where lamda 1 is the first positive eigenvalue for the Laplace operator on the corresponding hyperbolic space.

In a joint work with Sarnak in Appendix 2 ReferenceKi-Sa, by considering the twisted symmetric square upper L -functions of the symmetric fourth (cf. ReferenceBDHI), we improve the bound further, at least over double-struck upper Q , namely, upper R e left-parenthesis s Subscript i p Baseline right-parenthesis less-than-or-equal-to seven-sixty-fourths period As for the first positive eigenvalue for the Laplacian, we have lamda 1 greater-than-or-equal-to StartFraction 975 Over 4096 EndFraction almost-equals 0.238 period

In ReferenceKi-Sh3, we determine exactly when upper A Superscript 4 Baseline left-parenthesis pi right-parenthesis is cuspidal. We show that upper A Superscript 4 Baseline left-parenthesis pi right-parenthesis is not cuspidal and upper A cubed left-parenthesis pi right-parenthesis is cuspidal if and only if there exists a non-trivial quadratic character eta such that upper A cubed left-parenthesis pi right-parenthesis asymptotically-equals upper A cubed left-parenthesis pi right-parenthesis circled-times eta , or equivalently, there exists a non-trivial grössencharacter chi of upper E such that left-parenthesis upper A d left-parenthesis pi right-parenthesis right-parenthesis Subscript upper E Baseline asymptotically-equals left-parenthesis upper A d left-parenthesis pi right-parenthesis right-parenthesis Subscript upper E Baseline circled-times chi , where upper E slash upper 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 upper L -functions of cuspidal representations of upper G upper L 2 . It has immediate application for Ramanujan and Selberg’s bounds and the Sato-Tate conjecture: Let pi Subscript v be an unramified local component of a cuspidal representation pi equals circled-times Underscript v Endscripts pi Subscript v . Then it is shown that q Subscript v Superscript negative one-ninth Baseline less-than StartAbsoluteValue alpha Subscript v Baseline EndAbsoluteValue comma StartAbsoluteValue beta Subscript v Baseline EndAbsoluteValue less-than q Subscript v Superscript one-ninth , where the Hecke conjugacy class of pi Subscript v is given by d i a g left-parenthesis alpha Subscript v Baseline comma beta Subscript v Baseline right-parenthesis . Furthermore, if a Subscript v Baseline equals alpha Subscript v Baseline plus beta Subscript v , then for every epsilon greater-than 0 , there are sets upper T Superscript plus and upper T Superscript minus of positive lower (Dirichlet) density such that a Subscript v Baseline greater-than 1.68 period period period minus epsilon for all v element-of upper T Superscript plus and a Subscript v Baseline less-than negative 1.68 period period period plus epsilon for all v element-of upper T Superscript minus .

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 upper 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 upper 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 upper G upper L 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 upper L -functions which we need for the converse theorem, namely, upper L left-parenthesis s comma sigma circled-times pi comma rho Subscript m Baseline circled-times logical-and rho 4 right-parenthesis , where sigma is a cuspidal representation of upper G upper L Subscript m Baseline left-parenthesis double-struck upper A right-parenthesis , m equals 1 comma 2 comma 3 comma 4 , and pi is a cuspidal representation of upper G upper L 4 left-parenthesis double-struck upper A right-parenthesis . The automorphic upper L -functions appear in the constant term of the Eisenstein series coming from the split spin group upper S p i n left-parenthesis 2 n right-parenthesis (the upper D Subscript n Baseline minus 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 logical-and pi equals circled-times Underscript v Endscripts logical-and Superscript 2 Baseline pi Subscript v , with upper S being a finite set of finite places, where pi Subscript v is unramified for v less-than normal infinity and v not-an-element-of upper S . In this case, the situation is simpler because if sigma element-of script upper T Superscript upper S Baseline left-parenthesis m right-parenthesis as in the statement of the converse theorem, one of sigma Subscript v or pi Subscript v is in the principal series for v less-than normal infinity . Here one has to note the following: In the converse theorem, the upper L -function upper L left-parenthesis s comma sigma Subscript v Baseline times normal upper Pi Subscript v Baseline right-parenthesis is the Rankin-Selberg upper L -function defined by either integral representations ReferenceJ-PS-S or the Langlands-Shahidi method. They are the same, and they are an Artin upper L -function due to the local Langlands correspondence. However, the upper L -function upper L left-parenthesis s comma sigma Subscript v Baseline circled-times pi Subscript v Baseline comma rho Subscript m Baseline circled-times logical-and rho 4 right-parenthesis 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 upper 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 Subscript v is a supercuspidal representation, even if sigma Subscript v is a character of upper F Subscript v Superscript times . Hence we need to proceed in two steps as in ReferenceRa1, namely, first, we do the good case when none of pi Subscript 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 upper G upper L Subscript n Baseline left-parenthesis upper F Subscript v Baseline right-parenthesis 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 logical-and left-parenthesis upper A cubed left-parenthesis pi right-parenthesis right-parenthesis , hence the symmetric fourth of upper G upper L 2 , we do not need it. (See Remark 7.2.)

The converse theorem only provides a weak lift normal upper Pi which is equivalent to a subquotient of upper I n d StartAbsoluteValue det EndAbsoluteValue Superscript r 1 Baseline tau 1 circled-times ellipsis circled-times StartAbsoluteValue det EndAbsoluteValue Superscript r Super Subscript k Baseline tau Subscript k , where the tau Subscript i ’s are (unitary) cuspidal representations of upper G upper L Subscript n Sub Subscript i and r Subscript i Baseline element-of double-struck upper R . If pi satisfies the weak Ramanujan property, it immediately implies r 1 equals ellipsis equals r Subscript k Baseline equals 0 . In general, we show that r 1 equals ellipsis equals r Subscript k Baseline equals 0 by comparing the Hecke conjugacy classes of logical-and pi and normal upper Pi .

In Section 5.1, we give a new proof of the existence of the functorial product corresponding to the tensor product map upper G upper L 2 left-parenthesis double-struck upper C right-parenthesis times upper G upper L 2 left-parenthesis double-struck upper C right-parenthesis long right-arrow upper G upper L 4 left-parenthesis double-struck upper C right-parenthesis . 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 upper A d left-parenthesis pi right-parenthesis ReferenceGe-J as an automorphic representation of upper G upper L 3 left-parenthesis double-struck upper A right-parenthesis for a cuspidal representation pi of upper G upper L 2 left-parenthesis double-struck upper A right-parenthesis by showing that pi squared-times pi overTilde equals upper A d left-parenthesis pi right-parenthesis squared-plus 1 .

In Section 5.2, we construct all local lifts normal upper Pi Subscript v in the sense of Definition 2.2 and show that unless v vertical-bar 2 comma 3 and pi Subscript v is a supercuspidal representation, normal upper Pi Subscript v is in fact logical-and pi Subscript v , the one given by the local Langlands correspondence ReferenceH-T, ReferenceHe2. Here is how it is done: Note that if v does-not-divide 2 , any supercuspidal representation of upper G upper L 4 left-parenthesis upper F Subscript v Baseline right-parenthesis is induced, i.e., corresponds to upper I n d left-parenthesis upper W Subscript upper F Sub Subscript v Subscript Baseline comma upper W Subscript upper K Baseline comma mu right-parenthesis , where upper K slash upper F Subscript v is an extension of degree 4 (not necessarily Galois) and mu is a character of upper K Superscript 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 upper I n d left-parenthesis upper W Subscript upper F Baseline comma upper W Subscript upper E Baseline comma chi right-parenthesis , where upper E Subscript w Baseline equals upper K , w vertical-bar v , and chi is a grössencharacter of upper E such that chi Subscript w Baseline equals mu . Likewise, if v does-not-divide 2 comma 3 , any supercuspidal representation sigma Subscript v of upper G upper L Subscript m Baseline left-parenthesis upper F Subscript v Baseline right-parenthesis , m equals 1 comma 2 comma 3 comma 4 , is induced. We embed sigma Subscript v as a local component of a cuspidal representation using automorphic induction. We can compare the functional equations of upper L left-parenthesis s comma sigma circled-times pi comma rho Subscript m Baseline circled-times logical-and rho 4 right-parenthesis and the corresponding Artin upper 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 vertical-bar 3 , we need to twist by supercuspidal representations of upper G upper L 3 left-parenthesis upper F Subscript v Baseline right-parenthesis , 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 normal upper Pi equals circled-times Underscript v Endscripts normal upper Pi Subscript v with upper S 1 equals StartSet v 1 EndSet , upper S 2 equals StartSet v 2 EndSet , where v 1 comma v 2 are any finite places, we prove that normal upper Pi is an automorphic representation of upper G upper L 6 left-parenthesis double-struck upper A right-parenthesis .

In Section 7, we prove that if pi is a cuspidal representation of upper G upper L 2 left-parenthesis double-struck upper A right-parenthesis , then upper A Superscript 4 Baseline left-parenthesis pi right-parenthesis is an automorphic representation of upper G upper L 5 left-parenthesis double-struck upper A right-parenthesis . Here we need to be careful because of the exceptional places upper T in the discussion of the exterior square lift. We first prove that there exists an automorphic representation normal upper Pi of upper G upper L 5 left-parenthesis double-struck upper A right-parenthesis such that normal upper Pi Subscript v Baseline asymptotically-equals upper A Superscript 4 Baseline left-parenthesis pi Subscript v Baseline right-parenthesis if v not-an-element-of upper T . Next we show that this is true for v element-of upper T . If v vertical-bar 3 , any supercuspidal representation of upper G upper L 2 left-parenthesis upper F Subscript v Baseline right-parenthesis is monomial, and hence it can be embedded into a monomial cuspidal representation of upper G upper L 2 left-parenthesis double-struck upper A right-parenthesis . If v vertical-bar 2 , any extraordinary supercuspidal representation of upper G upper L 2 left-parenthesis upper F Subscript v Baseline right-parenthesis 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 upper L left-parenthesis s comma sigma times upper A Superscript 4 Baseline left-parenthesis pi right-parenthesis right-parenthesis and the corresponding Artin upper L -function and obtain our assertion.

Finally, we emphasize that for the functoriality of upper A Superscript 4 Baseline left-parenthesis pi right-parenthesis , we do not need the full functoriality of the exterior square of upper G upper L 4 ; first of all, one does not need the comparison of Hecke conjugacy classes in Section 4.1, since upper A cubed left-parenthesis pi right-parenthesis 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 upper F be a number field, and let double-struck upper A equals double-struck upper A Subscript upper F be the ring of adeles. We fix an additive character psi equals circled-times Underscript v Endscripts psi Subscript v of double-struck upper A slash upper F . Let rho Subscript m be the standard representation of upper G upper L Subscript m Baseline left-parenthesis double-struck upper C right-parenthesis .

First recall a converse theorem from ReferenceCo-PS1.

Theorem 2.1 (ReferenceCo-PS1).

Suppose normal upper Pi equals circled-times Underscript v Endscripts normal upper Pi Subscript v is an irreducible admissible representation of upper G upper L Subscript n Baseline left-parenthesis double-struck upper A right-parenthesis such that omega Subscript normal upper Pi Baseline equals circled-times Underscript v Endscripts omega Subscript normal upper Pi Sub Subscript v is a grössencharacter of upper F . Let upper S be a finite set of finite places, and let script upper T Superscript upper S Baseline left-parenthesis m right-parenthesis be a set of cuspidal representations of upper G upper L Subscript m Baseline left-parenthesis double-struck upper A right-parenthesis that are unramified at all places v element-of upper S . Suppose upper L left-parenthesis s comma sigma times normal upper Pi right-parenthesis is nice (i.e., entire, bounded in vertical strips and satisfies a functional equation) for all cuspidal representations sigma element-of script upper T Superscript upper S Baseline left-parenthesis m right-parenthesis , m less-than n minus 1 . Then there exists an automorphic representation normal upper Pi prime of upper G upper L Subscript n Baseline left-parenthesis double-struck upper A right-parenthesis such that normal upper Pi Subscript v Baseline asymptotically-equals normal upper Pi prime Subscript v for all v not-an-element-of upper S .

Let pi equals circled-times Underscript v Endscripts pi Subscript v be a cuspidal representation of upper G upper L 4 left-parenthesis double-struck upper A right-parenthesis . In order to apply the converse theorem, we need to do the following:

(1)

For all v , find an irreducible representation normal upper Pi Subscript v of upper G upper L 6 left-parenthesis upper F Subscript v Baseline right-parenthesis such that StartLayout 1st Row gamma left-parenthesis s comma sigma Subscript v Baseline circled-times pi Subscript v Baseline comma rho Subscript m Baseline circled-times logical-and rho 4 comma psi Subscript v Baseline right-parenthesis equals gamma left-parenthesis s comma sigma Subscript v Baseline times normal upper Pi Subscript v Baseline comma psi Subscript v Baseline right-parenthesis comma 2nd Row upper L left-parenthesis s comma sigma Subscript v Baseline circled-times pi Subscript v Baseline comma rho Subscript m Baseline circled-times logical-and rho 4 right-parenthesis equals upper L left-parenthesis s comma sigma Subscript v Baseline times normal upper Pi Subscript v Baseline right-parenthesis comma EndLayout

for all sigma Subscript v , where sigma equals circled-times Underscript v Endscripts sigma Subscript v Baseline element-of script upper T Superscript upper S Baseline left-parenthesis m right-parenthesis , m equals 1 comma 2 comma 3 comma 4 .

(2)

Prove the analytic continuation and functional equation of the upper L -functions upper L left-parenthesis s comma sigma circled-times pi comma rho Subscript m Baseline circled-times logical-and rho 4 right-parenthesis .

(3)

Prove that upper L left-parenthesis s comma sigma circled-times pi comma rho Subscript m Baseline circled-times logical-and rho 4 right-parenthesis is entire for sigma element-of script upper T Superscript upper S Baseline left-parenthesis m right-parenthesis , m equals 1 comma 2 comma 3 comma 4 .

(4)

Prove that upper L left-parenthesis s comma sigma circled-times pi comma rho Subscript m Baseline circled-times logical-and rho 4 right-parenthesis is bounded in vertical strips for sigma element-of script upper T Superscript upper S Baseline left-parenthesis m right-parenthesis , m equals 1 comma 2 comma 3 comma 4 .

Recall the equalities:

StartLayout 1st Row StartLayout 1st Row 1st Column Blank 2nd Column gamma left-parenthesis s comma sigma Subscript v Baseline circled-times pi Subscript v Baseline comma rho Subscript m Baseline circled-times logical-and rho 4 comma psi Subscript v Baseline right-parenthesis 2nd Row 1st Column Blank 2nd Column equals epsilon left-parenthesis s comma sigma Subscript v Baseline circled-times pi Subscript v Baseline comma rho Subscript m Baseline circled-times logical-and rho 4 comma psi Subscript v Baseline right-parenthesis StartFraction upper L left-parenthesis 1 minus s comma sigma overTilde Subscript v Baseline circled-times pi overTilde Subscript v Baseline comma rho Subscript m Baseline circled-times logical-and rho 4 right-parenthesis Over upper L left-parenthesis s comma sigma Subscript v Baseline circled-times pi Subscript v Baseline comma rho Subscript m Baseline circled-times logical-and rho 4 right-parenthesis EndFraction comma EndLayout 2nd Row gamma left-parenthesis s comma sigma Subscript v Baseline times normal upper Pi Subscript v Baseline comma psi Subscript v Baseline right-parenthesis equals epsilon left-parenthesis s comma sigma Subscript v Baseline times normal upper Pi Subscript v Baseline comma psi Subscript v Baseline right-parenthesis StartFraction upper L left-parenthesis 1 minus s comma sigma overTilde Subscript v Baseline times normal upper Pi overTilde Subscript v Baseline right-parenthesis Over upper L left-parenthesis s comma sigma Subscript v Baseline times normal upper Pi Subscript v Baseline right-parenthesis EndFraction period EndLayout

Hence the equalities of gamma and upper L -factors imply the equality of epsilon -factors.

The upper L -function upper L left-parenthesis s comma sigma circled-times pi comma rho Subscript m Baseline circled-times logical-and rho 4 right-parenthesis and the gamma -factor gamma left-parenthesis s comma sigma Subscript v Baseline circled-times pi Subscript v Baseline comma rho Subscript m Baseline circled-times logical-and rho 4 comma psi Subscript v Baseline right-parenthesis are available from the Langlands-Shahidi method, by considering the split spin group upper S p i n left-parenthesis 2 n right-parenthesis with the maximal Levi subgroup bold upper M whose derived group is upper S upper L Subscript n minus 3 Baseline times upper S upper L 4 . We will study the analytic properties of the upper 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 upper L -functions upper L left-parenthesis s comma sigma circled-times pi comma rho Subscript m Baseline circled-times logical-and rho 4 right-parenthesis may not be entire. Our key idea is to apply the converse theorem to the twisting set script upper T Superscript upper S Baseline left-parenthesis m right-parenthesis circled-times chi , where chi Subscript v is highly ramified for v element-of upper S . Then for sigma element-of script upper T Superscript upper S Baseline left-parenthesis m right-parenthesis circled-times chi , the upper L -function upper L left-parenthesis s comma sigma circled-times pi comma rho Subscript m Baseline circled-times logical-and rho 4 right-parenthesis is entire. Observe that upper L left-parenthesis s comma left-parenthesis sigma circled-times chi right-parenthesis times normal upper Pi right-parenthesis equals upper L left-parenthesis s comma sigma times left-parenthesis normal upper Pi circled-times chi right-parenthesis right-parenthesis . Hence applying the converse theorem with the twisting set script upper T Superscript upper S Baseline left-parenthesis m right-parenthesis circled-times chi is equivalent to applying the converse theorem for normal upper Pi circled-times chi with the twisting set script upper T Superscript upper S Baseline left-parenthesis m right-parenthesis (see ReferenceCo-PS2).

We will address problem (1) in Section 4. We have a natural candidate for normal upper Pi Subscript v , namely, logical-and pi Subscript 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 upper L -functions on the left and on the right are defined in completely different manners. The right-hand side is the Rankin-Selberg upper L -function ReferenceJ-PS-S defined by either integral representations or the Langlands-Shahidi method, which in turn is an Artin upper L -function due to the local Langlands correspondence. We note that if normal upper Pi Subscript v is not generic, then we write normal upper Pi Subscript v as a Langlands quotient of an induced representation normal upper Xi Subscript v , which is generic, and we define the gamma - and upper L -factors gamma left-parenthesis s comma sigma Subscript v Baseline times normal upper Pi Subscript v Baseline comma psi Subscript v Baseline right-parenthesis equals gamma left-parenthesis s comma sigma Subscript v Baseline times normal upper Xi Subscript v Baseline comma psi Subscript v Baseline right-parenthesis and upper L left-parenthesis s comma sigma Subscript v Baseline times normal upper Pi Subscript v Baseline right-parenthesis equals upper L left-parenthesis s comma sigma Subscript v Baseline times normal upper Xi Subscript v Baseline right-parenthesis .

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 upper L -functions and gamma -factors on the left are those of Artin factors. It is clearly true if sigma Subscript v Baseline circled-times pi Subscript v is unramified. Shahidi has shown that (1) is true when v equals normal infinity ReferenceSh7.

Remark 2.1

Eventually we are going to prove in Section 5 that normal upper Pi Subscript v on the right side of (1) is generic in our case. However, normal upper Pi Subscript v is not generic in general. For example, if pi Subscript v is given by the principal series upper I n d Subscript upper B Superscript upper G upper L 4 Baseline StartAbsoluteValue EndAbsoluteValue Superscript one-fourth circled-times StartAbsoluteValue EndAbsoluteValue Superscript one-fourth Baseline circled-times StartAbsoluteValue EndAbsoluteValue Superscript negative one-fourth Baseline circled-times StartAbsoluteValue EndAbsoluteValue Superscript negative one-fourth , then normal upper Pi Subscript v Baseline equals logical-and pi Subscript v is the unique quotient of upper I n d Subscript upper B Superscript upper G upper L 6 Baseline StartAbsoluteValue EndAbsoluteValue Superscript one-half circled-times StartAbsoluteValue EndAbsoluteValue Superscript negative one-half Baseline circled-times 1 circled-times 1 circled-times 1 circled-times 1 , namely, upper I n d Subscript upper G upper L 2 times upper G upper L 1 times upper G upper L 1 times upper G upper L 1 times upper G upper L 1 Superscript upper G upper L 6 Baseline StartAbsoluteValue det EndAbsoluteValue circled-times 1 circled-times 1 circled-times 1 circled-times 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 upper L -factors on the left side of (1), we only deal with generic representations, since any local components of a cuspidal representation of upper G upper L Subscript n Baseline left-parenthesis double-struck upper A right-parenthesis are generic. By a well-known result, any generic representation of upper G upper L Subscript n Baseline left-parenthesis upper F Subscript v Baseline right-parenthesis is always a full induced representation.

We were not able to prove (1) for normal upper Pi Subscript v Baseline equals logical-and pi Subscript v when v vertical-bar 2 comma 3 and pi Subscript v is a supercuspidal representation of upper G upper L 4 left-parenthesis upper F Subscript v Baseline right-parenthesis . Hence we make the following definition.

Definition 2.2

Let pi equals circled-times Underscript v Endscripts pi Subscript v be a cuspidal representation of upper G upper L 4 left-parenthesis double-struck upper A right-parenthesis . We say that an automorphic representation normal upper Pi of upper G upper L 6 left-parenthesis double-struck upper A right-parenthesis is a strong exterior square lift of pi if for every v , normal upper Pi Subscript v is a local lift of pi Subscript v in the sense that

StartLayout 1st Row gamma left-parenthesis s comma sigma Subscript v Baseline circled-times pi Subscript v Baseline comma rho Subscript m Baseline circled-times logical-and rho 4 comma psi Subscript v Baseline right-parenthesis equals gamma left-parenthesis s comma sigma Subscript v Baseline times normal upper Pi Subscript v Baseline comma psi Subscript v Baseline right-parenthesis comma 2nd Row upper L left-parenthesis s comma sigma Subscript v Baseline circled-times pi Subscript v Baseline comma rho Subscript m Baseline circled-times logical-and rho 4 right-parenthesis equals upper L left-parenthesis s comma sigma Subscript v Baseline times normal upper Pi Subscript v Baseline right-parenthesis comma EndLayout

for all generic irreducible representations sigma Subscript v of upper G upper L Subscript m Baseline left-parenthesis upper F Subscript v Baseline right-parenthesis , 1 less-than-or-equal-to m less-than-or-equal-to 4 .

If the above equality holds for almost all v , then normal upper Pi is called weak lift of pi .

In Section 4, we apply the converse theorem with upper S being a finite set of finite places such that pi Subscript v is unramified for v not-an-element-of upper S , v less-than normal infinity . Then if pi Subscript v is ramified, the local components of the twisting representations at upper 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).

Proposition 2.3 (ReferenceSh4).

Let sigma Subscript 1 v ( sigma Subscript 2 v , resp.) be an irreducible generic admissible representation of upper G upper L Subscript k Baseline left-parenthesis upper F Subscript v Baseline right-parenthesis ( upper G upper L Subscript l Baseline left-parenthesis upper F Subscript v Baseline right-parenthesis , resp.) with parametrization phi Subscript i Baseline colon upper W Subscript upper F Sub Subscript v Subscript Baseline times upper S upper L 2 left-parenthesis double-struck upper C right-parenthesis long right-arrow upper G upper L Subscript k Baseline left-parenthesis double-struck upper C right-parenthesis ( upper G upper L Subscript l Baseline left-parenthesis double-struck upper C right-parenthesis , resp.) by the local Langlands correspondence ReferenceH-T, ReferenceHe2. Let upper L left-parenthesis s comma phi 1 circled-times phi 2 right-parenthesis be the Artin upper L -function; let upper L 1 left-parenthesis s comma sigma Subscript 1 v Baseline times sigma Subscript 2 v Baseline right-parenthesis be the Rankin-Selberg upper L -function defined by integral representation ReferenceJ-PS-S; and let upper L 2 left-parenthesis s comma sigma Subscript 1 v Baseline times sigma Subscript 2 v Baseline right-parenthesis be the Langlands-Shahidi upper L -function defined as a normalizing factor for intertwining operators ReferenceSh1. Then we have the equality

upper L left-parenthesis s comma phi 1 circled-times phi 2 right-parenthesis equals upper L 1 left-parenthesis s comma sigma Subscript 1 v Baseline times sigma Subscript 2 v Baseline right-parenthesis equals upper L 2 left-parenthesis s comma sigma Subscript 1 v Baseline times sigma Subscript 2 v Baseline right-parenthesis period

We have similar equalities for gamma - and epsilon -factors.

Proof.

The equality upper L left-parenthesis s comma phi 1 circled-times phi 2 right-parenthesis equals upper L 1 left-parenthesis s comma sigma Subscript 1 v Baseline times sigma Subscript 2 v Baseline right-parenthesis 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 upper L 1 left-parenthesis s comma sigma Subscript 1 v Baseline times sigma Subscript 2 v Baseline right-parenthesis equals upper L 2 left-parenthesis s comma sigma Subscript 1 v Baseline times sigma Subscript 2 v Baseline right-parenthesis 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 Superscript m Baseline left-parenthesis negative 1 right-parenthesis disappears). Similar equalities hold for gamma - and epsilon -factors.

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For the sake of completeness, we recall how upper L - and epsilon -factors are defined from the Langlands-Shahidi method ReferenceSh1, Section 7. Let bold upper G be a quasi-split reductive group defined over a number field upper F . Let bold upper M be a maximal Levi subgroup. Let pi be a generic cuspidal representation of bold upper M left-parenthesis double-struck upper A right-parenthesis . From the theory of local coefficients, which come from intertwining operators, a gamma -factor gamma left-parenthesis s comma pi Subscript v Baseline comma r Subscript i Baseline comma psi Subscript v Baseline right-parenthesis is defined for every generic irreducible admissible representation pi Subscript v and certain finite-dimensional representation r Subscript i ’s. If pi Subscript v is tempered, upper L left-parenthesis s comma pi Subscript v Baseline comma r Subscript i Baseline right-parenthesis is defined to be

upper L left-parenthesis s comma pi Subscript v Baseline comma r Subscript i Baseline right-parenthesis equals upper P Subscript pi Sub Subscript v Subscript comma i Baseline left-parenthesis q Subscript v Superscript negative s Baseline right-parenthesis Superscript negative 1 Baseline comma

where upper P Subscript pi Sub Subscript v Subscript comma i is the unique polynomial satisfying upper P Subscript pi Sub Subscript v Subscript comma i Baseline left-parenthesis 0 right-parenthesis equals 1 such that upper P Subscript pi Sub Subscript v Subscript comma i Baseline left-parenthesis q Subscript v Superscript negative s Baseline right-parenthesis is the numerator of gamma left-parenthesis s comma pi Subscript v Baseline comma r Subscript i Baseline comma psi Subscript v Baseline right-parenthesis . We define the epsilon -factor using the identity gamma left-parenthesis s comma pi Subscript v Baseline comma r Subscript i Baseline comma psi Subscript v Baseline right-parenthesis equals epsilon left-parenthesis s comma pi Subscript v Baseline comma r Subscript i Baseline comma psi Subscript v Baseline right-parenthesis StartFraction upper L left-parenthesis 1 minus s comma pi overTilde Subscript v Baseline comma r Subscript i Baseline right-parenthesis Over upper L left-parenthesis s comma pi Subscript v Baseline comma r Subscript i Baseline right-parenthesis EndFraction period Hence if pi Subscript v is tempered, then the gamma -factor canonically defines both the upper L -factor and the epsilon -factor. If pi Subscript 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 upper L -factors and we define gamma left-parenthesis s comma pi Subscript v Baseline comma psi Subscript v Baseline right-parenthesis and upper L left-parenthesis s comma pi Subscript v Baseline comma r Subscript i Baseline right-parenthesis to be the product of these rank-one gamma - and upper 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 upper M equals bold upper M left-parenthesis upper F right-parenthesis . Suppose pi subset-of upper I n d Subscript upper M Sub Subscript theta Subscript upper N Sub Subscript theta Superscript upper M Baseline sigma circled-times 1 , where upper M Subscript theta Baseline upper N Subscript theta , theta subset-of normal upper Delta , is a parabolic subgroup of upper M and sigma is an irreducible generic admissible representation of upper M Subscript theta . Let theta prime equals w left-parenthesis theta right-parenthesis subset-of normal upper Delta and fix a reduced decomposition w equals w Subscript n minus 1 Baseline ellipsis w 1 of w as in ReferenceSh2, Lemma 2.1.1. Then for each j , there exists a unique root alpha Subscript j Baseline element-of normal upper Delta such that w Subscript j Baseline left-parenthesis alpha Subscript j Baseline right-parenthesis less-than 0 . For each j , 2 less-than-or-equal-to j less-than-or-equal-to n minus 1 , let w overbar Subscript j Baseline equals w Subscript j minus 1 Baseline ellipsis w 1 . Set w overbar Subscript 1 Baseline equals 1 . Also let normal upper Omega Subscript j Baseline equals theta Subscript j Baseline union StartSet alpha Subscript j Baseline EndSet , where theta 1 equals theta , theta Subscript n Baseline equals theta prime , and theta Subscript j plus 1 Baseline equals w Subscript j Baseline left-parenthesis theta Subscript j Baseline right-parenthesis , 1 less-than-or-equal-to j less-than-or-equal-to n minus 1 . Then the group upper M Subscript normal upper Omega Sub Subscript j contains upper M Subscript theta Sub Subscript j Baseline upper N Subscript theta Sub Subscript j as a maximal parabolic subgroup and w Subscript j Baseline left-parenthesis sigma right-parenthesis is a representation of upper M Subscript theta Sub Subscript j . The upper L -group Superscript upper L Baseline upper M Subscript theta acts on upper V Subscript i . Given an irreducible constituent of this action, there exists a unique j , 1 less-than-or-equal-to j less-than-or-equal-to n minus 1 , which under w Subscript j is equivalent to an irreducible constituent of the action of Superscript upper L Baseline upper M Subscript theta Sub Subscript j on the Lie algebra of Superscript upper L Baseline upper N Subscript theta Sub Subscript j . We denote by i left-parenthesis j right-parenthesis the index of this subspace of the Lie algebra of Superscript upper L Baseline upper N Subscript theta Sub Subscript j . Finally, let upper S Subscript i denote the set of all such j ’s where upper S Subscript i , in general, is a proper subset of 1 less-than-or-equal-to j less-than-or-equal-to n minus 1 .

Proposition 2.4 (ReferenceSh1, (3.13) (multiplicativity of gamma -factors)).

For each j element-of upper S Subscript i let gamma left-parenthesis s comma w Subscript j Baseline left-parenthesis sigma right-parenthesis comma r Subscript i left-parenthesis j right-parenthesis Baseline comma psi right-parenthesis denote the corresponding factor. Then

gamma left-parenthesis s comma pi comma r Subscript i Baseline comma psi right-parenthesis equals product Underscript j element-of upper S Subscript i Baseline Endscripts gamma left-parenthesis s comma w Subscript j Baseline left-parenthesis sigma right-parenthesis comma r Subscript i left-parenthesis j right-parenthesis Baseline comma psi right-parenthesis period

We follow the exposition in ReferenceSh6, p. 280. Let phi colon upper W Subscript upper F Baseline times upper S upper L 2 left-parenthesis double-struck upper C right-parenthesis long right-arrow Superscript upper L Baseline upper M be the parametrization of pi . Then phi factors through Superscript upper L Baseline upper M Subscript theta , i.e., there exists phi prime colon upper W Subscript upper F Baseline times upper S upper L 2 left-parenthesis double-struck upper C right-parenthesis long right-arrow Superscript upper L Baseline upper M Subscript theta Baseline such that phi equals i ring phi prime , where i colon Superscript upper L Baseline upper M Subscript theta Baseline right-arrow with hook Superscript upper L Baseline upper M . Let r prime Subscript i Baseline equals r Subscript i Baseline vertical-bar Sub Superscript upper L Subscript upper M Sub Subscript theta Subscript . Then r prime Subscript i Baseline equals circled-plus Underscript j Endscripts r Subscript i left-parenthesis j right-parenthesis , and

gamma left-parenthesis s comma phi comma r Subscript i Baseline comma psi right-parenthesis equals product Underscript j Endscripts gamma left-parenthesis s comma phi prime comma r Subscript i left-parenthesis j right-parenthesis Baseline comma psi right-parenthesis period

Given an irreducible component of r Subscript i Baseline vertical-bar Sub Superscript upper L Subscript upper M Sub Subscript theta Subscript , there exists a unique j , which under w Subscript j makes this component equivalent to an irreducible constituent of the action of Superscript upper L Baseline upper M Subscript theta Sub Subscript j on the Lie algebra of Superscript upper L Baseline upper N Subscript theta Sub Subscript j . Hence we have

Proposition 2.5

Let pi comma sigma be as in Proposition 2.4. Suppose pi is tempered and gamma left-parenthesis s comma w Subscript j Baseline left-parenthesis sigma right-parenthesis comma r Subscript i left-parenthesis j right-parenthesis Baseline comma psi right-parenthesis is an Artin factors for each j element-of upper S Subscript i , namely, gamma left-parenthesis s comma w Subscript j Baseline left-parenthesis sigma right-parenthesis comma r Subscript i left-parenthesis j right-parenthesis Baseline comma psi right-parenthesis equals gamma left-parenthesis s comma phi prime comma r Subscript i left-parenthesis j right-parenthesis Baseline comma psi right-parenthesis for each j . Then gamma left-parenthesis s comma pi comma r Subscript i Baseline comma psi right-parenthesis and upper L left-parenthesis s comma pi comma r Subscript i Baseline right-parenthesis are also Artin factors.

Proof.

Clear from the multiplicativity formulas. Since pi is tempered, gamma -factors determine the upper 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 upper 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 upper L -factors for upper G upper L Subscript n Baseline times upper G upper L Subscript m , and by Proposition 2.3, they are Artin factors.

Next we have ReferenceSh6, Theorem 5.2

Proposition 2.6 (multiplicativity of upper L -factors).

Suppose pi comma 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 upper L left-parenthesis s comma w Subscript j Baseline left-parenthesis sigma right-parenthesis comma r Subscript i left-parenthesis j right-parenthesis Baseline right-parenthesis , j element-of upper S Subscript i . Then

upper L left-parenthesis s comma pi comma r Subscript i Baseline right-parenthesis equals product Underscript j element-of upper S Subscript i Baseline Endscripts upper L left-parenthesis s comma w Subscript j Baseline left-parenthesis sigma right-parenthesis comma r Subscript i left-parenthesis j right-parenthesis Baseline right-parenthesis period

Now let pi be a non-tempered irreducible generic admissible representation of upper M equals upper M left-parenthesis upper F Subscript v Baseline right-parenthesis . Then pi is the unique quotient of an induced representation upper I n d Subscript upper M Sub Subscript theta Subscript upper N Sub Subscript theta Superscript upper M Baseline sigma circled-times 1 , where upper M Subscript theta Baseline upper N Subscript theta , theta subset-of normal upper Delta , is a parabolic subgroup of upper M and sigma is an irreducible generic quasi-tempered representation of upper M Subscript theta . (In many cases when the standard module conjecture is known, pi equals upper I n d Subscript upper M Sub Subscript theta Subscript upper N Sub Subscript theta Subscript Superscript upper M Baseline left-parenthesis sigma circled-times 1 ).) Then by the definition of upper L -factors,

Proposition 2.7

Let pi comma sigma be as above. Then

upper L left-parenthesis s comma pi comma r Subscript i Baseline right-parenthesis equals product Underscript j element-of upper S Subscript i Baseline Endscripts upper L left-parenthesis s comma w Subscript j Baseline left-parenthesis sigma right-parenthesis comma r Subscript i left-parenthesis j right-parenthesis Baseline right-parenthesis comma gamma left-parenthesis s comma pi comma r Subscript i Baseline comma psi right-parenthesis equals product Underscript j element-of upper S Subscript i Baseline Endscripts gamma left-parenthesis s comma w Subscript j Baseline left-parenthesis sigma right-parenthesis comma r Subscript i left-parenthesis j right-parenthesis Baseline comma psi right-parenthesis period

Remark 2.2

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 upper L left-parenthesis s comma pi comma r Subscript i Baseline right-parenthesis , even when pi is non-generic as long as it has generic supercuspidal support. Write pi as the Langlands quotient of normal upper Xi equals upper I n d Subscript upper M Sub Subscript theta Subscript upper N Sub Subscript theta Superscript upper M Baseline sigma circled-times 1 . Just define gamma left-parenthesis s comma pi comma r Subscript i Baseline comma psi right-parenthesis equals gamma left-parenthesis s comma normal upper Xi comma r Subscript i Baseline comma psi right-parenthesis using the formula in Proposition 2.4, and define upper L left-parenthesis s comma pi comma r Subscript i Baseline right-parenthesis using the formula in Proposition 2.5. These definitions agree with those of the Rankin-Selberg gamma - and upper 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 Subscript v Baseline equals mu ring det be a character of upper G upper L 2 left-parenthesis upper F Subscript v Baseline right-parenthesis , which is the Langlands quotient of upper I n d mu StartAbsoluteValue EndAbsoluteValue Superscript one-half circled-times mu StartAbsoluteValue EndAbsoluteValue Superscript negative one-half . Then the standard upper L -function upper L left-parenthesis s comma pi Subscript v Baseline right-parenthesis is obtained by considering the induced representation upper I n d Subscript upper G upper L 2 times upper G upper L 1 Superscript upper G upper L 3 Baseline pi Subscript v Baseline StartAbsoluteValue det EndAbsoluteValue Superscript StartFraction s Over 2 EndFraction circled-times StartAbsoluteValue EndAbsoluteValue Superscript minus StartFraction s Over 2 EndFraction , which is a quotient of upper I n d Subscript upper B Superscript upper G upper L 3 Baseline mu StartAbsoluteValue EndAbsoluteValue Superscript one-half plus StartFraction s Over 2 EndFraction circled-times mu StartAbsoluteValue EndAbsoluteValue Superscript negative one-half plus StartFraction s Over 2 EndFraction circled-times StartAbsoluteValue EndAbsoluteValue Superscript minus StartFraction s Over 2 EndFraction . Hence gamma left-parenthesis s comma pi Subscript v Baseline comma psi Subscript v Baseline right-parenthesis equals gamma left-parenthesis s plus one-half comma mu comma psi Subscript v Baseline right-parenthesis gamma left-parenthesis s minus one-half comma mu comma psi Subscript v Baseline right-parenthesis , and upper L left-parenthesis s comma pi Subscript v Baseline right-parenthesis equals upper L left-parenthesis s plus one-half comma mu right-parenthesis upper L left-parenthesis s minus one-half comma mu right-parenthesis if mu is unramified. On the other hand, if sigma Subscript v is the Steinberg representation, which is the subrepresentation of upper I n d mu StartAbsoluteValue EndAbsoluteValue Superscript one-half circled-times mu StartAbsoluteValue EndAbsoluteValue Superscript negative one-half , then gamma left-parenthesis s comma sigma Subscript v Baseline comma psi Subscript v Baseline right-parenthesis equals gamma left-parenthesis s comma pi Subscript v Baseline comma psi Subscript v Baseline right-parenthesis . However, by the definition of the upper L -factor, there is a cancellation, and upper L left-parenthesis s comma sigma Subscript v Baseline right-parenthesis equals upper L left-parenthesis s plus one-half comma mu right-parenthesis .

3. Analytic properties of the upper L -functions

Consider the upper D Subscript n Baseline minus 3 case in ReferenceSh3, n equals 4 comma 5 comma 6 comma 7 : Let bold upper G equals upper S p i n left-parenthesis 2 n right-parenthesis be the split spin group. It is, up to isomorphism, the unique simply-connected group of type upper D Subscript n . We can think of it as a two-fold covering group of upper S upper O left-parenthesis 2 n right-parenthesis , namely, there is a 2 to 1 map phi colon upper S p i n left-parenthesis 2 n right-parenthesis long right-arrow upper S upper O left-parenthesis 2 n right-parenthesis . Let bold upper T be a maximal torus of bold upper G .

Let theta equals StartSet alpha 1 equals e 1 minus e 2 comma period period period comma alpha Subscript n minus 4 Baseline equals e Subscript n minus 4 Baseline minus e Subscript n minus 3 Baseline comma alpha Subscript n minus 2 Baseline equals e Subscript n 2 Baseline minus e Subscript n minus 1 Baseline comma alpha Subscript n minus 1 Baseline equals e Subscript n minus 1 Baseline minus e Subscript n Baseline comma alpha Subscript n Baseline equals e Subscript n minus 1 Baseline plus e Subscript n Baseline EndSet equals normal upper Delta minus StartSet alpha Subscript n minus 3 Baseline EndSet . Let bold upper T subset-of bold upper M Subscript theta Baseline equals bold upper M be the Levi subgroup of bold upper G generated by theta , and let bold upper P equals bold upper M bold upper N be the corresponding standard parabolic subgroup of bold upper G . Let bold upper A be the connected component of the center of bold upper M :

StartLayout 1st Row 1st Column Blank 2nd Column bold upper A equals left-parenthesis intersection Underscript alpha element-of theta Endscripts k e r alpha right-parenthesis Superscript 0 Baseline 2nd Row 1st Column Blank 2nd Column equals StartLayout Enlarged left-brace 1st Row StartSet upper H Subscript alpha 1 Baseline left-parenthesis t right-parenthesis upper H Subscript alpha 2 Baseline left-parenthesis t squared right-parenthesis ellipsis upper H Subscript alpha Sub Subscript n minus 3 Subscript Baseline left-parenthesis t Superscript n minus 3 Baseline right-parenthesis upper H Subscript alpha Sub Subscript n minus 2 Subscript Baseline left-parenthesis t Superscript n minus 3 Baseline right-parenthesis upper H Subscript alpha Sub Subscript n minus 1 Subscript Baseline left-parenthesis t Superscript StartFraction n minus 3 Over 2 EndFraction Baseline right-parenthesis upper H Subscript alpha Sub Subscript n Subscript Baseline left-parenthesis t Superscript StartFraction n minus 3 Over 2 EndFraction Baseline right-parenthesis colon t element-of upper F overbar Superscript asterisk Baseline EndSet comma 2nd Row for n odd comma 3rd Row StartSet upper H Subscript alpha 1 Baseline left-parenthesis t squared right-parenthesis upper H Subscript alpha 2 Baseline left-parenthesis t Superscript 4 Baseline right-parenthesis ellipsis upper H Subscript alpha Sub Subscript n minus 3 Subscript Baseline left-parenthesis t Superscript 2 left-parenthesis n minus 3 right-parenthesis Baseline right-parenthesis upper H Subscript alpha Sub Subscript n minus 2 Subscript Baseline left-parenthesis t Superscript 2 left-parenthesis n minus 3 right-parenthesis Baseline right-parenthesis upper H Subscript alpha Sub Subscript n minus 1 Subscript Baseline left-parenthesis t Superscript n minus 3 Baseline right-parenthesis upper H Subscript alpha Sub Subscript n Subscript Baseline left-parenthesis t Superscript n minus 3 Baseline right-parenthesis colon t element-of upper F overbar Superscript asterisk Baseline EndSet comma 4th Row for n even period EndLayout EndLayout

Since bold upper G is simply connected, the derived group bold upper M Subscript upper D of bold upper M is simply connected, and hence bold upper M Subscript upper D Baseline asymptotically-equals upper S upper L Subscript n minus 3 Baseline times upper S upper L 4 . Then

bold upper A intersection bold upper M Subscript upper D Baseline equals StartLayout Enlarged left-brace 1st Row StartSet upper H Subscript alpha 1 Baseline left-parenthesis t right-parenthesis upper H Subscript alpha 2 Baseline left-parenthesis t squared right-parenthesis ellipsis upper H Subscript alpha Sub Subscript n minus 4 Subscript Baseline left-parenthesis t Superscript n minus 4 Baseline right-parenthesis upper H Subscript alpha Sub Subscript n minus 1 Subscript Baseline left-parenthesis t Superscript StartFraction n minus 3 Over 2 EndFraction Baseline right-parenthesis upper H Subscript alpha Sub Subscript n Subscript Baseline left-parenthesis t Superscript StartFraction n minus 3 Over 2 EndFraction Baseline right-parenthesis colon t Superscript n minus 3 Baseline equals 1 EndSet comma 2nd Row for n odd comma 3rd Row StartSet upper H Subscript alpha 1 Baseline left-parenthesis t squared right-parenthesis upper H Subscript alpha 2 Baseline left-parenthesis t Superscript 4 Baseline right-parenthesis ellipsis upper H Subscript alpha Sub Subscript n minus 4 Subscript Baseline left-parenthesis t Superscript 2 left-parenthesis n minus 4 right-parenthesis Baseline right-parenthesis upper H Subscript alpha Sub Subscript n minus 1 Subscript Baseline left-parenthesis t Superscript n minus 3 Baseline right-parenthesis upper H Subscript alpha Sub Subscript n Subscript Baseline left-parenthesis t Superscript n minus 3 Baseline right-parenthesis colon t Superscript 2 left-parenthesis n minus 3 right-parenthesis Baseline equals 1 EndSet comma 4th Row for n even period EndLayout

We identify bold upper A with upper G upper L 1 . Then

bold upper M asymptotically-equals left-parenthesis upper G upper L 1 times upper S upper L Subscript n minus 3 Baseline times upper S upper L 4 right-parenthesis slash left-parenthesis bold upper A intersection bold upper M Subscript upper D Baseline right-parenthesis period

We define a map f overbar colon bold upper A times bold upper M Subscript upper D Baseline long right-arrow upper G upper L 1 times upper G upper L 1 times upper S upper L Subscript n minus 3 Baseline times upper S upper L 4 by

f overbar colon left-parenthesis a left-parenthesis t right-parenthesis comma x comma y right-parenthesis long right-arrow from bar StartLayout Enlarged left-brace 1st Row 1st Column left-parenthesis t comma t Superscript StartFraction n minus 3 Over 2 EndFraction Baseline comma x comma y right-parenthesis comma 2nd Column for n odd comma 2nd Row 1st Column left-parenthesis t squared comma t Superscript n minus 3 Baseline comma x comma y right-parenthesis comma 2nd Column for n even comma EndLayout

which induces a map

f colon bold upper M long right-arrow upper G upper L Subscript n minus 3 Baseline times upper G upper L 4 period

Under the identification bold upper M Subscript upper D Baseline asymptotically-equals upper S upper L Subscript n minus 3 Baseline times upper S upper L 4 , upper H Subscript alpha 1 Baseline left-parenthesis t right-parenthesis upper H Subscript alpha 2 Baseline left-parenthesis t squared right-parenthesis ellipsis upper H Subscript alpha Sub Subscript n minus 4 Baseline left-parenthesis t Superscript n minus 4 Baseline right-parenthesis is an element in upper S upper L Subscript n minus 3 , and upper H Subscript alpha Sub Subscript n minus 1 Baseline left-parenthesis t right-parenthesis upper H Subscript alpha Sub Subscript n minus 2 Baseline left-parenthesis t squared right-parenthesis upper H Subscript alpha Sub Subscript n Baseline left-parenthesis t right-parenthesis is an element in upper S upper L 4 . Using this, it is easy to see that

f left-parenthesis upper H Subscript alpha Sub Subscript n minus 3 Subscript Baseline left-parenthesis t right-parenthesis right-parenthesis equals left-parenthesis d i a g left-parenthesis 1 comma period period period comma 1 comma t right-parenthesis comma d i a g left-parenthesis 1 comma 1 comma t comma t right-parenthesis right-parenthesis period

We note that it is independent of the choices of the roots of unity which show up.

Let sigma comma pi be cuspidal representations of upper G upper L Subscript n minus 3 Baseline left-parenthesis double-struck upper A right-parenthesis comma upper G upper L 4 left-parenthesis double-struck upper A right-parenthesis with central characters omega 1 comma omega 2 , resp. Let normal upper Sigma be a cuspidal representation of bold upper M left-parenthesis double-struck upper A right-parenthesis , induced by f and sigma comma pi . (More precisely,Footnote1 Thanks are due to Prof. Shahidi who pointed this out.✖ we need to proceed in the following way: bold upper M left-parenthesis double-struck upper A right-parenthesis double-struck upper A Superscript asterisk is co-compact in upper G upper L Subscript n minus 3 Baseline left-parenthesis double-struck upper A right-parenthesis times upper G upper L 4 left-parenthesis double-struck upper A right-parenthesis , where double-struck upper A Superscript asterisk is embedded as the center of, say, the first factor. Consequently sigma circled-times pi vertical-bar Subscript f left-parenthesis upper M right-parenthesis Baseline comma upper M equals bold upper M left-parenthesis double-struck upper A right-parenthesis , decomposes to a direct sum of irreducible cuspidal representations of upper M . Let normal upper Sigma be any irreducible constituent of this direct sum. As we shall see, its choice is irrelevant.)

The central character of normal upper Sigma is

omega Subscript normal upper Sigma Baseline equals StartLayout Enlarged left-brace 1st Row 1st Column omega 1 omega 2 Superscript StartFraction n minus 3 Over 2 EndFraction Baseline comma 2nd Column for n odd comma 2nd Row 1st Column omega 1 squared omega 2 Superscript n minus 3 Baseline comma 2nd Column for n even period EndLayout

Now suppose sigma Subscript v Baseline comma pi Subscript v Baseline are unramified representations, given by

sigma Subscript v Baseline equals pi left-parenthesis mu 1 comma period period period comma mu Subscript n minus 3 Baseline right-parenthesis comma pi Subscript v Baseline equals pi left-parenthesis nu 1 comma nu 2 comma nu 3 comma nu 4 right-parenthesis period

Let normal upper Sigma Subscript v be the unramified representation of bold upper M left-parenthesis upper F Subscript v Baseline right-parenthesis , given by sigma Subscript v Baseline comma pi Subscript v Baseline ’s. Then normal upper Sigma Subscript v is induced from the character chi of the torus. We have

StartLayout 1st Row chi ring upper H Subscript alpha 1 Baseline left-parenthesis t right-parenthesis equals mu 1 mu 2 Superscript negative 1 Baseline left-parenthesis t right-parenthesis comma ellipsis comma chi ring upper H Subscript alpha Sub Subscript n minus 4 Subscript Baseline left-parenthesis t right-parenthesis equals mu Subscript n minus 4 Baseline mu Subscript n minus 3 Superscript negative 1 Baseline left-parenthesis t right-parenthesis comma 2nd Row chi ring upper H Subscript alpha Sub Subscript n minus 1 Subscript Baseline left-parenthesis t right-parenthesis equals nu 1 nu 2 Superscript negative 1 Baseline left-parenthesis t right-parenthesis comma chi ring upper H Subscript alpha Sub Subscript n minus 2 Subscript Baseline left-parenthesis t right-parenthesis equals nu 2 nu 3 Superscript negative 1 Baseline left-parenthesis t right-parenthesis comma chi ring upper H Subscript alpha Sub Subscript n Subscript Baseline left-parenthesis t right-parenthesis equals nu 3 nu 4 Superscript negative 1 Baseline left-parenthesis t right-parenthesis comma 3rd Row chi left-parenthesis a left-parenthesis t right-parenthesis right-parenthesis equals omega Subscript normal upper Sigma Sub Subscript v Subscript Baseline left-parenthesis t right-parenthesis period EndLayout

Since f left-parenthesis upper H Subscript alpha Sub Subscript n minus 3 Subscript Baseline left-parenthesis t right-parenthesis right-parenthesis equals left-parenthesis d i a g left-parenthesis 1 comma period period period comma 1 comma t right-parenthesis comma d i a g left-parenthesis 1 comma 1 comma t comma t right-parenthesis right-parenthesis , we have

chi ring upper H Subscript alpha Sub Subscript n minus 3 Subscript Baseline left-parenthesis t right-parenthesis equals mu Subscript n minus 3 Baseline nu 3 nu 4 period

Hence, we see that, for almost all v ,

StartLayout 1st Row 1st Column upper L left-parenthesis s comma normal upper Sigma Subscript v Baseline comma r 1 right-parenthesis 2nd Column equals upper L left-parenthesis s comma sigma Subscript v Baseline circled-times pi Subscript v Baseline comma rho Subscript n minus 3 Baseline circled-times logical-and rho 4 right-parenthesis comma 2nd Row 1st Column upper L left-parenthesis s comma normal upper Sigma Subscript v Baseline comma r 2 right-parenthesis 2nd Column equals upper L left-parenthesis s comma sigma Subscript v Baseline comma logical-and circled-times omega Subscript 2 v Baseline right-parenthesis period EndLayout

For ramified places, let upper L left-parenthesis s comma normal upper Sigma Subscript v Baseline comma r 1 right-parenthesis and upper L left-parenthesis s comma normal upper Sigma Subscript v Baseline comma r 2 right-parenthesis be the ones defined in ReferenceSh1, Section 7. Observe that in particular, if v equals normal infinity , then upper L left-parenthesis s comma pi Subscript v Baseline comma r Subscript i Baseline right-parenthesis is the corresponding Artin upper L -function (cf. ReferenceSh7) in each case.

Let upper I left-parenthesis s comma normal upper Sigma Subscript v Baseline right-parenthesis be the induced representation, and let upper N left-parenthesis s comma normal upper Sigma Subscript v Baseline comma w 0 right-parenthesis be the normalized local intertwining operator ReferenceKi1, (2.1):

upper A left-parenthesis s comma normal upper Sigma Subscript v Baseline comma w 0 right-parenthesis equals StartFraction upper L left-parenthesis s comma normal upper Sigma Subscript v Baseline comma r 1 right-parenthesis upper L left-parenthesis 2 s comma normal upper Sigma Subscript v Baseline comma r 2 right-parenthesis Over upper L left-parenthesis 1 plus s comma normal upper Sigma Subscript v Baseline comma r 1 right-parenthesis upper L left-parenthesis 1 plus 2 s comma normal upper Sigma Subscript v Baseline comma r 2 right-parenthesis EndFraction StartFraction upper N left-parenthesis s comma normal upper Sigma Subscript v Baseline comma w 0 right-parenthesis Over epsilon left-parenthesis s comma normal upper Sigma Subscript v Baseline comma r 1 comma psi Subscript v Baseline right-parenthesis epsilon left-parenthesis 2 s comma normal upper Sigma Subscript v Baseline comma r 2 comma psi Subscript v Baseline right-parenthesis EndFraction comma

where upper A left-parenthesis s comma normal upper Sigma Subscript v Baseline comma w 0 right-parenthesis is the unnormalized intertwining operator. In ReferenceKi4, we showed that upper N left-parenthesis s comma normal upper Sigma Subscript v Baseline comma w 0 right-parenthesis is holomorphic and non-zero for upper R e left-parenthesis s right-parenthesis greater-than-or-equal-to one-half for all v . For the sake of completeness, we give a proof.

Proposition 3.1

The normalized local intertwining operators upper N left-parenthesis s comma normal upper Sigma Subscript v Baseline comma w 0 right-parenthesis are holomorphic and non-zero for upper R e left-parenthesis s right-parenthesis greater-than-or-equal-to one-half for all v .

Proof.

We proceed as in ReferenceKi2, Proposition 3.4. If normal upper Sigma Subscript v is tempered, then the unnormalized operators are holomorphic and non-zero for upper R e left-parenthesis s right-parenthesis greater-than 0 . We only need to verify Conjecture 7.1 of ReferenceSh1, namely, upper L left-parenthesis s comma normal upper Sigma Subscript v Baseline comma r Subscript i Baseline right-parenthesis is holomorphic for upper R e left-parenthesis s right-parenthesis greater-than 0 : for archimedean places, upper L left-parenthesis s comma normal upper Sigma Subscript v Baseline comma r Subscript i Baseline right-parenthesis is an Artin upper L -function, and hence our assertion follows. For p -adic places, by the multiplicativity of upper L -factors (Proposition 2.6), upper L left-parenthesis s comma normal upper Sigma Subscript v Baseline comma r Subscript i Baseline right-parenthesis is a product of rank-one upper L -functions for discrete series. The rank-one factors are Rankin-Selberg upper L -functions for upper G upper L Subscript k Baseline times upper G upper L Subscript l , and the cases upper D Subscript n Baseline minus 2 and upper D Subscript n Baseline minus 3 . The first two cases are well known (ReferenceSh1, Proposition 7.2). The upper D Subscript n Baseline minus 3 case is a result of ReferenceAs.

If normal upper Sigma Subscript v is non-tempered, we write upper I left-parenthesis s comma normal upper Sigma Subscript v Baseline right-parenthesis as in ReferenceKi1, p. 841,

upper I left-parenthesis s comma normal upper Sigma Subscript v Baseline right-parenthesis equals upper I left-parenthesis s alpha overTilde plus normal upper Lamda 0 comma pi 0 right-parenthesis equals upper I n d Subscript bold upper M 0 left-parenthesis upper F Sub Subscript v Subscript right-parenthesis bold upper N 0 left-parenthesis upper F Sub Subscript v Subscript right-parenthesis Superscript bold upper G left-parenthesis upper F Super Subscript v Superscript right-parenthesis Baseline pi 0 circled-times q Superscript mathematical left-angle s alpha overTilde plus normal upper Lamda 0 comma upper H Super Subscript upper P 0 Superscript left-parenthesis right-parenthesis mathematical right-angle Baseline comma

where pi 0 is a tempered representation of bold upper M 0 left-parenthesis upper F Subscript v Baseline right-parenthesis and bold upper P 0 equals bold upper M 0 bold upper N 0 is another parabolic subgroup of bold upper G . We can identify the normalized operator upper N left-parenthesis s comma normal upper Sigma Subscript v Baseline comma w 0 right-parenthesis with the normalized operator upper N left-parenthesis s alpha overTilde plus normal upper Lamda 0 comma pi 0 comma w 0 right-parenthesis , which is a product of rank-one operators attached to tempered representations (cf. ReferenceZh, Proposition 1).

Here alpha overTilde equals e 1 plus ellipsis plus e Subscript n minus 3 ; normal upper Lamda 0 equals r 1 e 1 plus r 2 e 2 plus ellipsis plus left-parenthesis minus r 2 right-parenthesis e Subscript n minus 4 Baseline plus left-parenthesis minus r 1 right-parenthesis e Subscript n minus 3 Baseline plus left-parenthesis r prime 1 plus r prime 2 right-parenthesis e Subscript n minus 2 Baseline plus left-parenthesis r prime 1 minus r prime 2 right-parenthesis e Subscript n minus 1 , where one-half greater-than r 1 greater-than-or-equal-to ellipsis greater-than-or-equal-to r Subscript left-bracket StartFraction n minus 3 Over 2 EndFraction right-bracket Baseline greater-than-or-equal-to 0 , one-half greater-than r prime 1 greater-than-or-equal-to r prime 2 greater-than-or-equal-to 0 . Here r Subscript i Baseline equals 0 if pi Subscript 1 v is tempered. The same is true for pi Subscript 2 v . Hence

s alpha overTilde plus normal upper Lamda 0 equals left-parenthesis s plus r 1 right-parenthesis e 1 plus ellipsis plus left-parenthesis s minus r 1 right-parenthesis e Subscript n minus 3 Baseline plus left-parenthesis r prime 1 plus r prime 2 right-parenthesis e Subscript n minus 2 Baseline plus left-parenthesis r prime 1 minus r prime 2 right-parenthesis e Subscript n minus 1 Baseline period

All the rank-one operators are operators attached to tempered representations of a parabolic subgroup whose Levi subgroup has a derived group isomorphic to upper S upper L Subscript k Baseline times upper S upper L Subscript l inside a group whose derived group is upper S upper L Subscript k plus l , unless r prime 1 equals r prime 2 not-equals 0 , in which case the rank-one operator is for upper D Subscript k Baseline minus 2 . It is the case when pi prime 2 equals upper I n d StartAbsoluteValue det EndAbsoluteValue Superscript r prime Baseline rho circled-times StartAbsoluteValue det EndAbsoluteValue Superscript minus r prime Baseline rho , where rho is a tempered representation of upper G upper L 2 .

In the first case, the operators are restrictions to upper S upper L Subscript k plus l of corresponding standard operators for upper G upper L Subscript k plus l . By ReferenceM-W2, Proposition I.10 one knows that these rank-one operators are holomorphic for upper R e left-parenthesis s right-parenthesis greater-than negative 1 . Hence by identifying roots of upper G with respect to a parabolic subgroup with those of upper G with respect to the maximal torus, it is enough to check upper R e left-parenthesis mathematical left-angle s alpha overTilde plus normal upper Lamda 0 comma beta Superscript logical-or Baseline mathematical right-angle right-parenthesis greater-than negative 1 for all positive roots beta if upper R e left-parenthesis s right-parenthesis greater-than-or-equal-to one-half . We observed that the least value of upper R e left-parenthesis mathematical left-angle s alpha overTilde plus normal upper Lamda 0 comma beta Superscript logical-or Baseline mathematical right-angle right-parenthesis is upper R e left-parenthesis s right-parenthesis minus r 1 minus left-parenthesis r prime 1 plus r prime 2 right-parenthesis which is larger than negative 1 , if upper R e left-parenthesis s right-parenthesis greater-than-or-equal-to one-half .

Now suppose we are in the exceptional case, namely, pi prime 2 equals upper I n d StartAbsoluteValue det EndAbsoluteValue Superscript r prime Baseline rho circled-times StartAbsoluteValue det EndAbsoluteValue Superscript minus r prime Baseline rho , where rho is a tempered representation of upper G upper L 2 . Then by direct computation, we see that upper N left-parenthesis s alpha overTilde plus normal upper Lamda 0 comma pi 0 comma w overTilde right-parenthesis is a product of the following three operators:

upper N left-parenthesis s alpha overTilde Superscript prime Baseline plus normal upper Lamda prime 0 comma pi Subscript 1 v Baseline circled-times rho circled-times rho comma w prime 0 right-parenthesis comma

upper N left-parenthesis left-parenthesis s minus 2 r Superscript prime Baseline right-parenthesis alpha overTilde Superscript prime Baseline plus normal upper Lamda prime 0 comma pi Subscript 1 v Baseline circled-times omega Subscript rho Baseline comma w prime 0 right-parenthesis , and

upper N left-parenthesis left-parenthesis s plus 2 r Superscript prime Baseline right-parenthesis alpha overTilde Superscript prime Baseline plus normal upper Lamda prime 0 comma pi Subscript 1 v Baseline circled-times omega Subscript rho Baseline comma w prime 0 right-parenthesis ,

where s alpha overTilde Superscript prime Baseline plus normal upper Lamda prime 0 equals left-parenthesis s plus r 1 right-parenthesis e 1 plus ellipsis plus left-parenthesis s minus r 1 right-parenthesis e Subscript n minus 3 and omega Subscript rho is the central character of rho . The first operator is the operator for upper D Subscript k Baseline minus 2 and it is in the corresponding positive Weyl chamber and is holomorphic for upper R e left-parenthesis s right-parenthesis greater-than-or-equal-to one-half (ReferenceKi1, Lemma 2.4). The last two operators are the operators for upper G upper L Subscript k Baseline times upper G upper L 1 . Since upper R e left-parenthesis s minus 2 r prime minus r 1 right-parenthesis greater-than negative 1 if upper R e left-parenthesis s right-parenthesis greater-than-or-equal-to one-half , they are holomorphic. Consequently, upper N left-parenthesis s alpha overTilde plus normal upper Lamda 0 comma pi 0 comma w overTilde Subscript 0 Baseline right-parenthesis is holomorphic for upper R e left-parenthesis s right-parenthesis greater-than-or-equal-to one-half . By Zhang’s lemma (cf. ReferenceKi2, Lemma 1.7, ReferenceZh), it is non-zero as well.

■

We recall some general results in the next two propositions. Let bold upper G be a quasi-split group defined over a number field upper F , and let bold upper P equals bold upper M bold upper N be a maximal parabolic subgroup over upper F . Let normal upper Sigma be a cuspidal representation of bold upper M left-parenthesis double-struck upper A right-parenthesis .

Proposition 3.2 (Langlands ReferenceLa2, Lemma 7.5 or ReferenceKi1, Proposition 2.1).

Unless w 0 normal upper Sigma asymptotically-equals normal upper Sigma , the global intertwining operator upper M left-parenthesis s comma normal upper Sigma comma w 0 right-parenthesis is holomorphic for upper R e left-parenthesis s right-parenthesis greater-than-or-equal-to 0 .

Proposition 3.3 (ReferenceKi1, Lemma 2.3).

If w 0 normal upper Sigma approximately-but-not-actually-equals normal upper Sigma , product Underscript i equals 1 Overscript m Endscripts upper L Subscript upper S Baseline left-parenthesis 1 plus i s comma normal upper Sigma comma r Subscript i Baseline right-parenthesis has no zeros for upper R e left-parenthesis s right-parenthesis greater-than 0 .

Remark 3.1

Since the Eisenstein series upper E left-parenthesis s comma f comma g comma upper P right-parenthesis is holomorphic for upper R e left-parenthesis s right-parenthesis equals 0 , we see that product Underscript i equals 1 Overscript m Endscripts upper L Subscript upper S Baseline left-parenthesis 1 plus i s comma normal upper Sigma comma r Subscript i Baseline right-parenthesis has no zeros for upper R e left-parenthesis s right-parenthesis equals 0 either. Since the local upper L -functions upper L left-parenthesis s comma normal upper Sigma Subscript v Baseline comma r Subscript i Baseline right-parenthesis have no zeros, the completed upper L -function product Underscript i equals 1 Overscript m Endscripts upper L left-parenthesis 1 plus i s comma normal upper Sigma comma r Subscript i Baseline right-parenthesis has no zeros for upper R e left-parenthesis s right-parenthesis greater-than-or-equal-to 0 .

Let upper S be a finite set of finite places where pi Subscript v is unramified if v less-than normal infinity and v not-an-element-of upper S . Fix chi be a grössencharacter of upper F such that chi Subscript v is highly ramified for at least one v element-of upper S . Let normal upper Sigma Subscript chi be the cuspidal representation of bold upper M left-parenthesis double-struck upper A right-parenthesis , induced by the map f colon bold upper M long right-arrow upper G upper L Subscript n minus 3 Baseline times upper G upper L 4 and sigma circled-times chi comma pi . Then the central character of normal upper Sigma Subscript chi is

omega Subscript normal upper Sigma Sub Subscript chi Baseline equals StartLayout Enlarged left-brace 1st Row 1st Column omega 1 chi Superscript m Baseline omega 2 Superscript StartFraction n minus 3 Over 2 EndFraction Baseline comma 2nd Column for n odd comma 2nd Row 1st Column omega 1 squared chi Superscript 2 m Baseline omega 2 Superscript n minus 3 Baseline comma 2nd Column for n even period EndLayout

Note that w 0 left-parenthesis omega Subscript normal upper Sigma Sub Subscript chi Subscript Baseline right-parenthesis equals omega Subscript normal upper Sigma Sub Subscript chi Superscript negative 1 . Hence if chi Subscript v is highly ramified (say, chi Subscript v Superscript 24 is ramified), then

w 0 left-parenthesis omega Subscript normal upper Sigma Sub Subscript chi Subscript Baseline right-parenthesis not-equals omega Subscript normal upper Sigma Sub Subscript chi Subscript Baseline comma

for m equals 1 comma 2 comma 3 comma 4 . Therefore,

w 0 left-parenthesis normal upper Sigma Subscript chi Baseline right-parenthesis not-asymptotically-equals normal upper Sigma Subscript chi Baseline comma

for m equals 1 comma 2 comma 3 comma 4 . Hence by Propositions 3.1 and 3.2,

Proposition 3.4

Let chi be as above. Then for all cuspidal representations sigma element-of script upper T Superscript upper S Baseline left-parenthesis m right-parenthesis , m equals 1 comma 2 comma 3 comma 4 , upper L left-parenthesis s comma left-parenthesis sigma circled-times chi right-parenthesis circled-times pi comma rho Subscript m Baseline circled-times logical-and rho 4 right-parenthesis